Fundamentals and Special Applications of Non-Contact Scanning Force Microscopy

ADVANCES IN ELECTRONICS AND ELECTRON PHYSICS. VOL. 81

Fundamentals and Special Applications of Non-contact Scanning Force Microscopy U . HARTMANN Institute of Thin Film and Ion Technology. KFA-Julich. Federal Republic of Germany

I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . I1. Probe-Sample Interactions in Non-contact Scanning Force Microscopy . . . . A . Methodical Outline. . . . . . . . . . . . . . . . . . . . . . . B. Van der Waals Forces . . . . . . . . . . . . . . . . . . . . . . C . Ionic Forces . . . . . . . . . . . . . . . . . . . . . . . . . D . Squeezing of Individual Molecules: Solvation Forces . . . . . . . . . . E . Capillary Forces . . . . . . . . . . . . . . . . . . . . . . . . F . Patch Charge Forces . . . . . . . . . . . . . . . . . . . . . . 111. Electric Force Microscopy Used as a Servo Technique . . . . . . . . . . . A . Fundamentals of Electrostatic Probe-Sample Interactions . . . . . . . . B. Operational Conditions . . . . . . . . . . . . . . . . . . . . . IV . Theory of Magnetic Force Microscopy . . . . . . . . . . . . . . . . . A . Basics of Contrast Formation . . . . . . . . . . . . . . . . . . . B . Properties of Ferromagnetic Microprobes . . . . . . . . . . . . . . C . Contrast Modeling . . . . . . . . . . . . . . . . . . . . . . . D . Sensitivity, Lateral Resolution, and Probe Optimization Concepts . . . . . E . Scanning Susceptibility Microscopy . . . . . . . . . . . . . . . . . F . Applications of Magnetic Force Microscopy . . . . . . . . . . . . . V . Aspects of Instrumentation . . . . . . . . . . . . . . . . . . . . . VI . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . .

49 51 51

53 102 112 119 127 129 129 131 133 133 138 157

182 183 189 191 195 197 197

I . INTRODUCTION In 1986 Gerd Binnig and Heinrich Rohrer shared the Nobel Prize in Physics for inventing the scanning tunneling microscope (STM) and discovering that it can image the surface of a conducting sample with unprecedented resolution (Binnig and Rohrer. 1982). The instrument utilizes an atomically sharp tip which is placed sufficiently close to the sample so that tunneling of electrons between the two is possible . The tunneling current as a function of position of the tip across the sample provides an image that reflects the local density of electronic states at the Fermi level of the uppermost atoms at the surface of the sample. On the other hand. the close proximity of probe and 49

Copyright 0 1994 by Academic Press. Inc. All rights of reproduction in any form reserved. ISBN 0-12-014729-7

50

U.HARTMANN

FIGURE1. Schematic of the solid-vacuum transition. N denotes the position of the outermost atomic nuclei serving as reference plane. I is the extent of the inner electrons, which is typically 10-30 picometers. The probability density of valence/conduction band electrons usually drops with a decay length V / L between 0.1 and 1 nm. The extent of electromagnetic surface modes, which are responsible for the van der Waals (VDW) interaction, is about 100 nm. Static fields resulting from electric and magnetic charge distributions within the solid may have various extents E / M ranging from a few nanometers up to a macroscopic dimension. [The illustration is based on a presentation previously given by Pohl (1991).]

sample results in a mutual force which is of the same order of magnitude as that of interatomic forces in a solid. This latter phenomenon gave rise to a novel development, the atomic force microscope (AFM), which was presented by Gerd Binnig, Calvin Quate, and Christoph Gerber in 1986. Here, the probing tip is part of a tiny cantilever beam. Probe-sample forces F are detected according to Hooke’s Law, F = -k.s, from microscopic deflections s of a cantilever with spring constant k . Unlike the tunneling microscope, the force microscope is by no means restricted to conducting probes and samples and it is not restricted to probe-sample separations in the angstrom regime. Thus, by modifying the working distance, probe-sample interactions of varying decay lengths become accessible, as shown in Fig. 1. Tip-sample interactions at atomically close separations predominantly result from the overlap of tip and sample electronic wavefunctions. Thus, the “contact” mode of operation of the force microscope is dominated by short-range interatomic forces. Conceptually, the contact mode of imaging is like using a stylus profilometer to measure the topography of surface atoms. The AFM achieves sub-nanometer to atomic resolution by using a very small loading force - typically to lo-” N - which makes the area of contact between tip and sample exceedingly small.

FUNDAMENTALS OF NON-CONTACT FORCE MICROSCOPY

51

With a sufficient increase in the probe-sample separation, long-range electromagnetic interactions dominate, as shown in Fig. I . In the “noncontact” mode of force microscope operation, both the force acting on the probing tip and the spatial resolution obtained upon detecting a certain type of interaction critically depend on the probe-sample separation and on the mesoscopic to macroscopic geometry of the microprobe’s apex region. The present work is devoted to a review of basic fundamentals and of some important applications of non-contact-mode force microscopy. In Section I1 a detailed discussion of the various surface forces that may occur between the probe and the sample of a force microscope is presented. Section 111 gives a brief introduction to electric force microscopy, which is realized by externally applying an electrostatic potential difference between probe and sample. Section IV is devoted to the basics of magnetic force microscopy, which currently appears to be the most important application of the force microscope in the non-contact mode of operation. Finally, some general principles of instrumentation are discussed in Section V. Concerning the terminology used throughout the present work, “scanning force microscopy” and “scanning force microscope” (both abbreviated by SFM) denote the technique and the instrument in the most general sense (contact or non-contact mode of operation), in contrast to “atomic force microscopy” and “atomic force microscope” (both abbreviated by AFM), which always refer to the contact mode of operation. Unfortunately, this terminology was not used in a consistent way throughout the earlier literature. Since the present work can only cover part of the many facets of the still rapidly growing field of non-contact SFM, the reader is referred to some previously presented excellent general introductions and reviews, among which are the recent articles by Wickramasinghe (1990) and by Rugar and Hansma (l990), as well as the book by Sarid (1991) and the book chapters by Meyer and Heinzelmann (1992), by Wickramasinghe (1992), and by Burnham and Colton (1992).

11. PROBE-SAMPLE INTERACTIONS I N NON-CONTACT SCANNING FORCE MICROSCOPY A . Methodical Outline A general theory concerning the long-range probe-sample interactions effective in non-contact scanning force microscopy (SFM), i.e., at probesample separations well beyond the regime of overlap of the electron wave

52

U. HARTMANN

equation nonequil. thermdyn

generalized Derjaguin approximation

forces

1

t

.

General theorv of noncontactiru? SFM

*

FIGURE 2. Schematic of the approach toward a general theory of non-contact scanning force microscopy.

functions, is a rather ambitious project. Even in the absence of externally applied electro- and magnetostatic interactions, the approach has to account for various intermolecular and surface forces which are, however, ultimately all of electromagnetic origin. Figure 2 gives a survey of the different components which generally contribute to the total probe-sample interaction. In the absence of any contamination on probe and sample surface, i.e., under clean UHV conditions, an ever-present long-range interaction is provided by van der Waals forces. In this area theory starts with some well-known results from quantum electrodynamics. In order to account for the typical geometry involved in an SFM, i.e., a sharp probe opposite to a flat or curved sample surface, the Derjaguin geometrical approximation is used, which essentially reduces the inherent many-body problem to a twobody approach. Under ambient conditions surface contaminants, e g , water films, are generally present on probe and sample. Liquid films on solids often give rise to a surface charge, and thus to an electrostatic interaction between probe and sample. The effect of these ionic forces is treated by classical Poisson-Boltzmann statistics, where the particular probe-sample geometry is again accounted for by employing the Derjaguin approximation. If the probe-sample separation is reduced to a few molecular diameters liquids can no longer be treated by a pure continuum approach. The discrete molecular structure gives rise to solvation forces which are due to the long-range ordering of liquid molecules in the gap between probe and sample. Finally, capillary condensation is a common phenomenon in SFM under ambient conditions. In this area the well-known Laplace equation provides an appropriate starting basis. Capillary action is then treated in terms of two extreme approaches: While the first is for liquid films strictly obeying a

FUNDAMENTALS O F NON-CONTACT FORCE MICROSCOPY

53

thermodynamic equilibrium behavior represented by the Kelvin equation, the second approach is for liquids which are actually not in thermodynamic equilibrium. I t must be emphasized that the general situation in non-contact SFM is governed by a complex interplay of all the aforementioned contributions. The situation is further complicated by the fact that not all of these contributions are simply additive. The following detailed discussion relies on a macroscopic point of view. All material properties involved are treated in terms of isotropic bulk considerations, and even properties attributed to individual molecules are consequently deduced from the overall macroscopic behavior of the solids or liquids composed by these molecules. The considerations concerning the presence of liquids in SFM are of course not restricted to “parasitic” effects due to contaminating films, but in particular also apply to the situation where the SFM is completely operated in a liquid immersion medium or where just the properties of a liquid film, e.g., of a polymeric layer on top of a substrate, are of interest. B. Van der W a d s Forces 1. Generul Description of the Phenomenon

Macroscopic van der Waals (VDW) forces arise from the interplay of electromagnetic field fluctuations with boundary conditions on ponderable bodies. These field fluctuations result from zero-point quantum vibrations as well as from thermal agitation of permanent electronic multipoles and extend well beyond the surface of any absorbing medium - partly as traveling waves, partly as exponentially damped “evanescent” waves. According to this particular picture Lifshitz calculated the mutual attraction of two semi-infinite dielectric slabs separated by an intervening vacuum gap (Lifshitz, 1955/56). Since the Lifshitz “random field approach” involves a solution of the full Maxwell equations rather than of the simpler Laplace tin

substrate

substrate

FIGURF 3. Distribution of virtual photons associated with probe and sample. At close proximity an exchange of virtual photons takes place, giving rise to VDW interactions.

54

U. HARTMANN

equation, retardation effects are accounted for in a natural way. The well-known fundamental results of the London (Eisenschitz and London, 1930) and Casimir (Casimir and Polder, 1948; Casimir, 1948) theories are obtained as specific cases of this general approach. Since the VDW interaction between any two bodies occurs through the fluctuating electromagnetic field, it stands to reason that the following alternative viewpoint could be developed: As schematically shown in Fig. 3 for the typical probe-sample arrangement involved in SFM, the fluctuating electromagnetic field can be considered in terms of a distribution of virtual photons associated with probe and sample. Now, if both come into close proximity, an exchange of these virtual photons occurs, giving rise to a macroscopic force between probe and sample. This alternative viewpoint is actually the basis for a treatment of the problem by methods of quantum field theory. Using the formidable apparatus of the Matsubara-Fradkin-Green function technique of quantum statistical mechanics, Dzyaloshinskii, Lifshitz, and Pitaevskii ( 1961) rederived the Lifshitz two-slab result and extended the approach to the presence of any intervening medium filling the gap between the dielectric slabs. Subsequently, several other approaches to the general problem of electromagnetic interaction between macroscopic bodies, all more or less equivalent, have been developed by various authors (see, for example, Mahanty and Ninham, 1976). In the present context the most important aspect common to all this work is the following: On a microscopic level, the origin of the dispersion forces between two molecules is linked to a process which can be described by the induction of polarization on one due to the instantaneous polarization field of the other. However, this process is seriously affected by a third molecule placed near the two. The macroscopic consequence is that VDW forces are in general highly nonadditive. For example, if two perfectly conducting bodies (a perfect conductor may be considered as the limit of a London superconductor, as the penetration depth approaches zero) mutually interact via VDW forces, only bounding surface layers will contribute to the interaction, while the interiors of the bodies are completely screened. Thus, the interaction can certainly not be characterized by straightforward pairwise summation of isotropic intermolecular contributions, at least not in this somewhat fictitious case. However, it is precisely the explicit assumption of the additivity of two-body intermolecular pair potentials which is the basis of the classical Hamaker approach (1937). Granted additivity, the interaction between any two macroscopic bodies which have well-defined geometric shapes and uniform molecular densities, can be calculated by a simple double-volume integration. In spite of its apparent limitations, the Hamaker approach not

FUNDAMENTALS OF NON-CONTACT FORCE MICROSCOPY

55

only has the virtue of ease in comprehension, but works over a wider range than would at first be thought possible. The most conspicuous result is that the additivity approach yields the correct overall power law dependence of the VDW interaction between two arbitrarily shaped macroscopic bodies on the separation between them. Although not rigorously proved, this appears to hold for the London limit, i.e., for a completely nonretarded interaction, as well as for the Casimir limit, i.e., complete radiation field retardation (Hartmann, 1990a). From the field point of view, the geometrical boundary conditions associated with the SFM’s probe-sample arrangement lead to tremendous mathematical difficulties in a rigorous calculation of VDW interactions, especially if retardation is included. Actually, several rather involved mathematical detours by various authors have shown that the key problem of a precise calculation of the magnitude of VDW forces as a function of separation of interacting bodies which exhibit curved surfaces can be solved fairly unambiguously only in some elementary cases involving spherical configurations (see, for example, Mahanty and Ninham, 1976). Because of all the aforementioned difficulties, it appears quite clear why a rigorous treatment of VDW interactions in SFM has not yet been presented. On the one hand, field theories are extremely complicated and tend to obscure the physical processes giving rise to the probe-sample forces. On the other hand, although two-body forces generally provide the dominant contribution, the explicit assumption of pairwise molecular additivity of VDW interactions of the many-particle system simply does not hold. The corrections due to many-body effects are generally essential in order to estimate whether the VDW interaction of a given tip-sample arrangement is within or well beyond the experimentally accessible regime. In what follows, a treatment of VDW interactions in non-contact SFM is proposed, which is based on elements of both the quantum field DLP theory and the Hamaker additivity approach. While some basic results from field theory provide an appropriate starting point, a characterization of material dielectric contributions, and a final analysis of the limitations of the developed framework, the additivity approach allows to account in a practical way, in terms of reasonable approximations, for the particular geometrical boundary conditions involved. In this sense the resulting model can best be referred to as a “renormalized Hamaker approach.” 2. The Two-Slab Problem: Separation of Geometrical and Material Properties The DLP theory (Dzyaloshinskii et al., 1961) gives the exact result for the electromagnetic interaction of two dielectric slabs separated by a third

56

U. HARTMANN

dielectric material of arbitrary thickness:

with j = I 2, and 7j(%

iVm1

P) = 47%

&ZiJPZ/C.

(2e)

In this somewhat complex expression, f ( z ) is the “VDW pressure,” i.e., the force per unit surface area exerted on the two slabs as a function of their separation z . kT is the thermal agitation energy, c the speed of light, and h Planck’s constant. p is simply an integration constant, and a , P, y, and 7 j are functions of p and the characteristic frequencies vm.The three media involved are completely characterized by their dielectric permittivities E ~ with j = 1,2,3, where “3” corresponds to the intervening medium. The summation in Eq. ( 1 ) entails calculating the functions tJ at discrete imaginmeans that only the first term of the sum has ary frequencies iv,, where to be multiplied by The dielectric permittivities at imaginary frequency are related to the imaginary parts of the dielectric permittivities taken at real frequency by the well-known Kramers-Kronig relation,

i.

(i)o

The imaginary parts of the complex permittivities €,([) = t;(E) + i c y ( < ) entering Eq. ( 3 ) are always positive and determine the dissipation of energy as a function of field frequency. The values of E, at purely imaginary arguments which enter Eqs. (1) and (2) are thus real quantities which decrease monotonically from their electrostatic limits E , ~to 1 for vrn-+ 00. Separation

,

FUNDAMENTALS OF NON-CONTACT FORCE MICROSCOPY

57

of the entropic and quantum mechanical contributions involved in Eq. (1) can now simply be performed by considering the zero-frequency (rn = 0) and the nonzero-frequency contributions separately. In order to ensure convergence of &heintegral, it is wise to follow the transformation procedure originally given by Lifshitz (1956). With y = mp, one obtains

where

For m = 0, one thus has a. = 0 and PO = A130A230, where the latter quantity is determined by the electrostatic limit of A,,(iv) =

t,

(iv)- c3 (iv) iv) €3 (iv)’

€/(

+

given for v = 0. Using the definite integral

one finally obtains

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U. HARTMANN

where 3 “ Am A m 130 m 3 230 e- 4- k TmC =l

incorporates all material properties in terms of the three static dielectric constants cia. A(.) characterizes the purely entropic contribution to the total VDW pressure given by Eq. (1) and involves a simple inverse power law dependence on the separation z of the two slabs. The zero frequency force is due to the thermal agitation of permanent electric dipoles present in the three media and includes Debye and Keesom contributions. For reasons of consistency with the following treatment of the quantum mechanical dispersion contribution the material properties are all incorporated into the socalled “entropic Hamaker constant” given by Eq. (8b). It should be noted that the latter quantity cannot exceed a value of [3<\3)/4]kT (C denotes Riemann’s zeta function), which is about 3.6 x J or 22.5meV for T = 300K. The maximum value is obtained for c10,c20 + 00 and €30 = 1, i.e., for two perfect conductors interacting across vacuum. According to Eq. (8b), the entropic Hamaker constant becomes negative if the static dielectric constant of the intervening medium is just in between those of the two dielectric slabs. This then leads via Eq. (8a) to a repulsion of the slabs. To discuss the dispersion force contribution resulting from zero-point quantum fluctuations, the nonzero frequency terms (m> 0) in Eq. (1) have to be evaluated. According to Eq. (2a), the discrete frequencies are given by 4 . 3 ~x IOl3Hz at room temperature. Since this is clearly beyond typical rotational relaxation frequencies of a molecule, the effective dielectric contributions according to Eq. (3) are solely determined by electronic polarizabilities. Absorption frequencies related to the latter are usually located somewhere in the UV region. However, with respect to this regime the urns are very close together. Thus, since one has from Eq. (2a) dm = (h/27rkT)dv, one applies the transformation

m=l

(9)

to Eq. (1) and obtains

where a(iv,p), P ( i v , p ) , and ~ ( u , i v , pare ) given by Eqs. (2b-e) - now, however, for a continuous electromagnetic spectrum. Since v l according

59

FUNDAMENTALS OF NON-CONTACT FORCE MICROSCOPY

to Eq. (2a) is much smaller than the prominent electronic absorption frequencies, the spectral integration in Eq. (10) can be performed from zero to infinity. Following the DLP approach (Dzyaloshinskii et al., 1961), the asymptotic VDW pressuref(z + 0) is given from Eq. (10) by

with

Using the identity given in Eq. (7), one can rewrite this as

In almost all cases of practical interest, where some experimental results have to be compared with theory, restriction to the first term of the preceding sum should be sufficient, where corrections due to higher-order terms are always less than 1 - 1/[(3) = 16.7% of the m = 1 term. Equation (1 1) characterizes the dispersion contribution to the total VDW pressure acting on the two slabs in the London limit, i.e., in the absence of radiation field retardation at small separation z . The inverse power law dependence is exactly the same as in Eq. (8a). From the Hamaker point of view this is not so surprising, since intermolecular Debye, Keesom, and London forces all exhibit the same dependence on the separation of two molecules, l / r 7 (see, for example, Israelachvili, 1985). However, contrarily to the entropic Hamaker constant given by Eq. (8b) the "nonretarded Hamaker constant" according to Eq. (12b) now involves the detailed dielectric behavior of the three media through the complete electromagnetic spectrum. Since Hn is thus related to dynamic electronic polarizabilities, while He is related to zero frequency orientational processes, there is generally no close relation between both quantities. In the opposite limit of large separation between the two dielectric slabls, the asymptotic VDW pressure f ( z 4 m) obtained from Eq. (10) is given according to the DLP result (Dzyaloshinskii et al., 1961) by

-

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U . HARTMANN

cr(0,p) and p(0,p) are again given by Eqs. (2b-d), but now in the static limit of the eIectronic polarizability. Using Eq.(7) one can rewrite the preceding as

Equation (13) characterizes the VDW pressure due to zero-point quantum fluctuations in the Casimir limit, i.e., for total radiation field retardation. A glance at Eq. ( 1 1) shows that, as in the case of two interacting molecules (Casimir and Polder, 1948; Casimir, 1948), retardation leads to an increase of the power law index by unity. However, the material properties now enter through Eq. (l4b) in terms of dielectric permittivities E,(O), j = 1,2,3, depending on the electronic polarizabilities in the electrostatic limit. Thus, ~ ~ (must 0 ) not be confused with orientational contributions E , ~determining the entropic Hamaker constant in Eq. (8b). H,[cl (0),~ 2 ( 0 ) c3(0)] , is called the “retarded Hamaker constant.” In spite of having already performed a tour de force of rather lengthy calculations, one is still at a point where one only has the VDW pressure acting upon two semi-infinite dielectric slabs separated by a third dielectric medium of arbitrary thickness. However, this is actually still the only geometrical arrangement for which a rigorous solution of equations of the form of Eq. (1) has been presented, which is equally valid at all separations and for any material combination. Without fail this means that the adaption of the preceding results to the SFM configuration must involve several serious manipulations of the basic results obtained from field theory. A certain problem in handling the formulae results from the convolution of material and geometrical properties present in the integrand of the complete dispersion force solution in Eq. (10). A separation of both, as in the case of the entropic component given by Eq. (sa), is only obtained for the London and Casimir limits characterized by Eqs. (1 1) and ( I 3), respectively. However, a straightforward interpolation between both asymptotic regimes is given by Hn tanh (x132/z> f(2)=- 67r 23

3

where (16) is a characteristic wavelength which indicates the onset of retardation. X132is determined by the electronic contributions to the dielectric permittivities via the quotient of the nonretarded Hamaker constant, according to Eq. (12b), and the retarded constant, according to Eq. (14b). This approximation is based on the assumption that Hn and H, have the same sign. It turns out x132

= 6.1rHr/Hn

FUNDAMENTALS OF NON-CONTACT FORCE MICROSCOPY

61

that this assumption does not hold for any material combination of the two slabs and the intervening medium (see concluding remarks in Section II.B.5). It is fairly obvious that Eqs. (15) and (16) combined immediately yield the London and Casimir limit. This simple analytical approximation of the complex exact result, Eq. (lo), provides an accuracy which is more than sufficient for SFM applications. If entropic contributions are included, the total VDW pressure then is given by He + H , tanh -

This latter result shows that, while retardation causes a transition from an initial 1 / z 3 to a 1/z4 distance dependence of the dispersion contribution, the interaction is dominated by entropic contributions at very large separations, giving again a l/z3 inverse power law (Hartmann, 1991a). However, as will be shown later, this phenomenon is well beyond the regime which is accessible to SFM. 3 . Transition to Renormalized Molecular Interactions

The macroscopic DLP theory (Dzyaloshinskii et al., 1961) can be used to derive the effective interaction of any two individual molecules within two dielectric slabs exhibiting a macroscopic VDW interaction. Accounting for an intervening dielectric medium of permittivity e3 (iv),the intermolecular force is given in the nonretarded limit by F,(z) = - A / z I,

(18a)

where z is the intermolecular distance and

a;(&)are the dynamic electronic “excess polarizabilities” of the two interacting molecules in the immersion medium. For c3 = 1, i.e., interaction in vacuum, the 0;(iv)’s become the ordinary polarizabilities ctj(iv) of isolated molecules, and Eqs. ( 1 8) are identical with the well-known London formula (Eisenschitz and London, 1930). On the other hand, the retarded limit gives (Dzyaloshinskii et al., 1961; Israelachvili, 1972a) FJZ) =

with

-s/z 8 .

(19a)

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U . HARTMANN

where the electronic contributions now have to be considered in their electrostatic limits. For c3 = 1 and a; = a,(O), the preceding result coincides with the classical Casimir-Polder result (Casimir and Polder, 1948; Casimir, 1948). Since these results have been derived from the macroscopic DLP theory (Dzyaloshinskii et al., 1961), the excess electronic polarizabilities reflect molecular properties that are generally not directly related to the behavior of the isolated molecule, but rather to its behavior in an environment composed by all molecules of the macroscopic arrangement under consideration, e.g., of the two-slab arrangement. The molecular constants A and B thus involve an implicit renormalization with respect to the dielectric and geometrical properties of the complete macroscopic environment. This means in particular that a;(iv) is not solely determined by the overall dielectric permittivities of all three media involved, but varies if for a given material combination only the geometry of the system is modified. Consequently, if a;(iv)is considered in this way, it involves corrections for many-body effects. Using the intermolecular interactions given in Eqs. (18) and (19) within the Hamaker approach (Hamaker, 1937), which involves volume integration of these pairwise interactions to obtain the macroscopic VDW force, yields the correct result if A and B are renormalized in an appropriate way. If, for example, the excess dielectric polarizability a;(iv) of a sphere of radius R and permittivity c,(iv), a; (iu) = 47rc0c3( i ~ ) 2 ~ ,(iv) 3 R

(20a)

with

is introduced into Eqs. (18) and (19) for two spherical particles separated by a distance d, one obtains the accurate result for the macroscopic dispersion interaction of the particles in the London and Casimir limits, respectively: H , R:RI F”(d)= - - _ _ 67r d’ ’

where the nonretarded Hamaker constant is given by

FUNDAMENTALS OF NON-CONTACT FORCE MICROSCOPY

63

and

with retarded Hamaker constant 161hc

Z A I 3 (0)2A23(0)

For an arbitrary geometrical configuration consisting of two macroscopic bodies with volumes VI and V 2 , the Hamaker approach is given by the sixfold integral

where Fn,rdenotes the nonretarded or retarded macroscopic dispersion force and fn,, is the renormalized two-body intermolecular contribution according to Eqs. (18) and (19). Equation (23) applied to the two-slab arrangement yields

F,(d) = - TPl P2A 36d3 ~

and F,(d) = -

X P I P2B ~

70d4

’

where pI and p2 are the molecular densities. Comparison of Eqs. (24a) and ( 1 1 ) as well as of Eqs. (24b) and ( 1 3) yields the effective molecular constants A and B in terms of their “two-slab renormalization:”

and Using Eqs. (12b) and (28b), one obtains from Eq. (25a) with reasonable accuracy pc$(iv) = 2 ~ ~ ~ ~ ( i v ) A ~ ~ ( i v )

(26)

for the effective excess dynamic polarizability of an individual molecule ‘7,’’ where A,,(iv) is defined in Eq. (6). Employing this result in a threefold Hamaker integration and using Eqs. (20), the nonretarded interaction between a small particle or a molecule “2” and a semi-infinite dielectric slab

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U. HARTMANN

“1” is given by

~1

Hn -, F n ( d )= - 67r d 4

with

The corresponding result for the retarded interaction can easily be derived from the original DLP work (Dzyaloshinskii et al., 1961):

with

The result holds for arbitrary dielectric constants q ( 0 ) .If especially ~ ~ (is0 ) sufficiently small (I5), the preceding result simplifies to H,

=

23hc 40.1r2m

If one has a metallic half space, is simply given by

(0) + 00, the retarded Hamaker constant

3hc - 47r2JE30

H -

(o)2A23(o)

2A23

(O).

While Eqs. (25) are ultimately the basis for the renormalized Hamaker approach used in the following, Eqs. (27) and (28) play a role in modeling processes of molecular-scale surface manipulation involving physisorption of large nonpolar molecules (see Section II.B.9). Equations (22) and (28) are finally used to check the limits of the presented theory as provided by size effects (see Section II.B.8). In order to analyze the behavior of a large molecule near a substrate surface, it is convenient to extend the somewhat empirical interpolation

FUNDAMENTALS OF NON-CONTACT FORCE MICROSCOPY

65

given by Eqs. ( 1 5) and (16) to the particleesubstrate dispersion interaction. Equations (27) and (28) can then be combined as Hn F(d) = - R23

tanh(X132/d)

67r

d4

1

with the retardation wavelength given by A132 =

Hr 6~ -. Hn

This approach is valid for d >> R. 4. The Efleect qf Probe Geometry

In order to model the probe-sample interaction in SFM, the general expression for the VDW pressure previously obtained now has to be adapted to the particular geometrical boundary conditions involved. Since the actual mesoscopic geometry of the employed sharp probes, i.e., the shape at nanometer scale near the apex region, is generally not known in detail, it is convenient to analyze the effect of probe geometry by considering some basic tip shapes exhibiting a cylindrical symmetry. Additionally accounting for a certain curvature of the sample surface, one obtains the geometrical arrangement shown in Fig. 4. The force between the two curved bodies can be obtained in a straightforward way by integrating the interaction between the circular regions of infinitesimal area 27rxdx on one surface and the opposite surface, which is assumed to be locally flat and a distance C = d + zI z2 away. The error involved in this approximation is thus due to the assumption of local flatness of one surface usually of the sample surface, since the probing tip should be much sharper. However, since the VDW interaction according to Eq. (17) exhibits an overall l / z 3 distance dependence at small separations, those contributions of the force field in Fig. 4 involving increasing distances to the probe’s volume element under consideration exhibit a rapid damping with respect to near-field contributions. This effect is further enhanced by

+

force field

sample FlCiuRE

4. Basic geometry in the Derjaguin approximation

66

U. HARTMANN

radiation field retardation gradually leading to a 1 / z 4 inverse power law for large distances as given by the z + 00 limit of Eq. (17). According to Fig. 4, the VDW force between probe and sample is given by

where f ( 5 ) is simply the previously obtained VDW pressure between two slabs separated by an arbitrary medium of local thickness <, with <(x) = z I(x) z 2 ( x )+ d, and d is the distance between the apices of probe and sample. The relation between the cross-sectional radius x and the vertical coordinate z is given by

+

x = z tan 4,

for a cone with half-cone angle

(31a)

4,

for a paraboloid with semiaxes R , and R,,

for an ellipsoid with the preceding semiaxes. To unify calculations, it is convenient to define an effective measure of curvature by R

=

{

tan4 R:/2R,

(cone) (paraboloid).

R:/R,

(ellipsoid)

Combining Eqs. (31) and (32), one immediately obtains xdx =

(cone) (paraboloid, ellipsoid)

(33a)

with

EX = R * R 2 / ( R 1 +R2),

(33b) where R I and R2 characterize the curvature of probe and sample, respectively. Inserting these substitutions for xdx into Eq. (30) yields

for two opposite conical surfaces and F(d) = 2xRu(d)

FUNDAMENTALS OF NON-CONTACT FORCE MICROSCOPY

67

for two opposite paraboloidal or ellipsoidal surfaces, respectively.

44 =

1;

dCf (0

(35)

is the total VDW energy per unit area of two flat surfaces separated by the same distance d as the apices of probe and local sample protrusion (see Fig. 4). For the special case of two interacting spheres with radii R l , R2 >> d, the preceding treatment is known as the Derjaguin approximation (Derjaguin, 1934). It should be emphasized that it is not necessary to explicitly specify the type of interaction f ( C ) which enters Eq. (30). The Derjaguin formulae (34) are thus valid for any type of interaction law, whether attractive, repulsive, or oscillating. In order to check the effect of probe geometry in detail, the dispersion pressure given by Eqs. (15) and (16) is inserted into Eqs. (34) and (35). One thus obtains for the conical arrangement

dx In cosh x.

-

In the nonretarded and retarded limits, wheref(C) in Eqs. (34) and (35) follows a simple inverse power law I/
used in Eq. (30) yields the particularly simple results and F,(d) = -7rH,R2/3d2 (38b) as nonretarded and retarded limits of Eq. (36). According to Eq. (34b), the dispersion force for two opposing paraboloidal or ellipsoidal surfaces is given by

F(d)= - -

-

~

'132

-1 1

x132

Xi321d

0

Expansion of the ln(cosh) terms for large and small arguments leads to the nonretarded and retarded limits given by F,(d)

=

-H,,R/6d2

(40a)

and

F,(d) = -27rH,R/3d3.

(40b)

68

U . HARTMANN

If the sample locally exhibits an atomically flat surface, Eq. (33b) simplifies to lim W = R , RZ+X

where, according to Eq. (32), R is the effective radius of apex curvature for a paraboloidal or ellipsoidal probe and R = tan 4 for a conical probe. Apart from describing the probe-sample dispersion interation, Eq. (39) also characterizes the adsorption of a large nonpolar molecule or small particle on an atomically flat substrate surface. Directly at the surface, one has R >> d, and the nonretarded dispersion interaction is given by Eq. (40a). On the other hand, if the particle is initially far away from the substrate, R << d, the interaction is given by Eqs. (29). This involves a nonretarded transition of the dispersion force from a l / d 2 dependence at small distances to l i d 4 dependence at large distances, and finally a transition to l i d 5 at very large distances, which is due to retardation. If the retardation wavelength A132 of the particleesubstrate arrangement is assumed to be independent of the particle's distance from the surface, i.e., if A132 is the same in eqs. ( I S ) and (29a), then the particle-substrate dispersion interaction is modeled by

(9) 2

F ( d ) = 27rRw(d) tanh

,

where R characterizes the dimension of the particle according to Eq. (32), and w ( d ) is the specific energy obtained for the two-slab system as given in Eq. (35). The nonretarded l / d 2 to l i d 4 transition is determined by the transition length 1 H , from Eq. (27b) H , from Eq. (12b)'

K

which is more or less close to R. It can easily be verified that Eqs. (42) satisfy the limiting results given in Eq. (40a) for d << R and in Eq. (29) for d >> R. Equations (42) allow the modeling of particle or molecule physisorption processes if the involved dispersion interactions are governed by bulk dielectric properties. Figure 5(a) shows the decrease of the dispersion force for increasing working distance for a conical probe according to Eq. (36) and for a paraboloidal or ellipsoidal probe according to Eq. (39). The curve for the interaction of two slabs is given by Eq. (15), and the physisorption curve for a small particle or large molecule with a flat surface is obtained from Eqs. (42), where xf32/x132 = 0.1 was used as a somewhat typical nonretarded transition length. For reference, the interaction between small particles with

FUNDAMENTALS OF NON-CONTACT FORCE MICROSCOPY

69

F I G U R 5E. Dispersion interaction for some rundamental arrangements. [a) shows the normalized force as a function o f separation, where denotes the material-dependent retardation wavelength. (b) shows the corresponding differential power law index to be used in an inverse power law ansatz for the distance dependence of the dispersion force.

70

U.HARTMANN

radii R , and R2 - according to Eqs. (21a) and (22a) modeled by F ( d ) = -(Hn/6r)R:R2 tanh(XI3,/d)/d7, with Xi32 given by Eq. (16) - is also indicated. If one performs the transformation (Hn/6r)R:R: A and B / A , with the molecular constants A and B given by Eqs. (l8b) and ( 19b), the latter curve corresponds to the intermolecular dispersion force and includes retardation effects. Even if all these formulae for the dispersion force involve the same material-dependent retardation wavelength XI 32, the gradual onset of retardation effects is clearly determined by the mesoscopic geometry of the interacting bodies (Hartmann, 1991b). This phenomenon is clarified by considering the differential power law index, -+

-+

k(d) =

-

da dd

- InF(d),

(43)

which has to be applied if the VDW force F ( d ) is approximated for a given distance d by an inverse power law of type F ( d ) N 1/dk(&). Application of Eq. (43) yields for the two-slab arrangement the simple result

wheref(d) is the VDW pressure according to Eq. (15). For a paraboloidal or ellipsoidal SFM probe, one obtains from Eq. (34b) k ( 4 = df(d)/w(dh

(44b) wheref(d) and w ( d ) are the VDW pressure according to Eq. ( I S ) and the VDW energy per unit surface area according to Eq. (35) obtained from the two-slab arrangement. The result for the conical probe is obtained from Eq. (34a):

The preceding results describe in detail the geometry-dependent transition to retardation for an extremely blunt probe (cylindrical, i.e., two-slab arrangement), for a realistic probe type (paraboloidal or ellipsoidal), and for the limit of an atomically sharp probe (conical). The distance dependence of the differential power law index according to Eqs. (44) is shown in Fig. 5(b). Additionally, the physisorption behavior of a small particle or large molecule onto a flat surface, obtained by applying Eq. (43) to Eq. (42a), is indicated. In the present context, the most important result from Fig. 5 is that VDW forces drop with an l/d2 inverse power law for the most realistic probe geometries, i.e., for paraboloidal or ellipsoidal apices, in the nonretarded

FUNDAMENTALS OF NON-CONTACT FORCE MICROSCOPY

71

limit. This clearly indicates a long-range probe-sample interaction in comparison with the exponential decrease of a tunneling current or the short-range manifestation of interatomic repulsive forces resulting from electronic orbital overlap. Consequently, the spatial resolution which can be obtained in VDW microscopy should be determined by the mesoscopic probe dimensions at nanometer scale near the probe's apex, and of course by the probe-sample separation (Hartmann, 1991b). An estimation of the lateral resolution is obtained by determining the probe diameter A ( d ) corresponding to the center of interaction. The latter is determined by the maximum contribution of the integrand in Eq. (30), i.e., by

Accounting for the l/Ck dependence off,((), this can be replaced by the more convenient form k

1--x--0. ax

<

According to Eq. (1 1) the nonretarded VDW pressure involves k = 3. Using the geometrical relations given in Fig. 4, one obtains the remarkably simple results

A , ( d ) = 24-d

(47a)

for the resolution of a paraboloidal or ellipsoidal probe, and

A , ( d ) = Rd

(47b)

for a conical probe. R is related by Eqs. (32) and (33b) to the effective dimensions of probe apex and local sample protrusion. For an atomically flat sample, R + R simply characterizes the sharpness of the probe. While the minimum resolvable lateral dimension for a conical probe is proportional to the working distance d, paraboloidal or ellipsoidal probes exhibit a square root dependence on the working distance. While Eqs. (47) give the lateral resolution in VDW microscopy in terms of simple analytical results, more elaborate solutions can only be obtained numerically (Moiseev et al., 1988). Apart from quantifying the microscope's resolution, Eqs. (47) additionally tell us that the resolution is independent of material properties in the considered nonretarded limit. This is, however, not valid if retardation becomes effective. In this case the power law index k in Eq. (46) becomes distance- and material-dependent. Finally, it should be emphasized that the solutions obtained largely analytically for forces, power law indices, and lateral resolutions by using the approximate result for the dispersion pressure in Eq. (15) can be

72

U. HARTMANN

numerically obtained in an exact way by using the DLP result from Eq. ( l o ) , whenever the dispersion pressure f ( d ) for the two-slab arrangement is needed. Consequently, the corresponding specific dispersion energy in Eq. (35) has to be calculated by directly integrating Eq. (10). The result is then

+ In [ I

-

P(iv,p)exp

(-du,iv,P))l),

where a, p, and q are defined as in Eq. (10) for z = d. The entropic component always additionally present is rigorously given by Eq. (8). Integration of the latter equation immediately yields the entropic VDW energy w e ( d ) per unit surface area. Equations (8), (lo), and (48) then provide the general framework for a rigorous numerical calculation of probe-sample forces via Eqs. (34), of power law indices via Eqs. (44), and of lateral resolutions at any working distance via Eq. (45). However, the major advantage of the analytical treatment just presented is that it emphasizes the physical processes giving rise to VDW interaction in SFM, while the rigorous numerical treatment ultimately based on Eq. ( 1 ) tends to obscure the basic physical aspects because of considerable mathematical complexities.

5. Dielectric Contributions: The Hamaker Constants Apart from the probe and sample geometry considered earlier, the magnitude of VDW forces in SFM is determined by the detailed dielectric properties of probe, sample, and an immersion medium which may be present in the intervening gap. The real dielectric permittivities taken at imaginary frequencies, cj(iu), enter the two-slab dispersion pressure in Eq. (15) via the nonretarded Hamaker constant H , and via the retardation wavelength XI3*. The latter quantity is determined according to Eq. (16) by the ratio of H , to the retarded Hamaker constant H,. For a given probe-sample geometry, H , and H , thus completely determine the magnitude of the resulting force as well as the onset of retardation effects. The following discussion is devoted to a calculation of the Hamaker constants in terms of only two characteristic material properties: the optical refractive index and the effective electronic absorption wavelength. The energy absorption spectrum of any medium for frequencies from zero through to the ultraviolet (UV) regime is characterized by jjm 7

(49)

73

F U N D A M E N T A L S OF NON-CONTACT F O R C E MICROSCOPY

where the first non-unitary term describes the effect of possible Debye rotational relaxation processes, and the second models absorption using a Lorentz harmonic oscillator model of the dielectric (see, for example, Mahanty and Ninham, 1976). The characteristic constants cjl, vjl, vim are given in pertinent tables of dielectric data. The damping coefficients rj, associated with the Lorentz oscillations are rather difficult to determine and are not known in most cases. However, since for dielectrics the widths of the absorption spectra are always small compared with the absorption frequencies, i.e., y,,,<< v,,, the term rmv/v$can be dropped to a satisfactory approximation in Eq. (49). If the static dielectric constant is denoted by f j 0 , and if one only has one prominent rotational absorption peak for v = vj,,ot and one prominent electronic absorption peak for v = vj,,, the preceding may be written as

,fi,,,,

where n is the optical refractive index. While vrot is typically given by microwave or lower frequencies, v, is located in the UV regime, and for most materials of practical interest one has v, FZ 3 x lOI5Hz. If there are m individual electronic absorption frequencies, Eq. (50) has to be replaced by

where n,,.] + , = 1 . In the far-UV and soft x-ray regime, all matter responds like a free electron gas (see, for example, Landau and Lifshitz, 1960), and the response function changes to €,(ZV) =

1

+

(52)

where v, is now the free electron gas plasma frequency. This latter expression also characterizes approximately the dielectric permittivity of a metal from zero through the visible to the soft x-ray regime. In the intermediate regime between far UV and soft x-ray, there is little knowledge of ~(iv). However, some reasonable interpolation schemes may be constructed (Mahanty and Ninham, 1976). According to the preceding result, matter may roughly be subdivided into three classes of dielectric behavior, as shown in Fig. 6. For water, the simplest Debye rotational relaxation and some closely spaced infrared (IR) bands lead to variations of ~ ( bbelow ) the UV regime. Thus, ~ ( b ) has to be evaluated according to Eq. (51), conveniently using effective

74

U. HARTMANN 1

log v/Hz FIGURE 6 . Dielectric permittivity on the imaginary frequency axis as a function of real frequency for water, typical hydrocarbons, and typical metals. v, = 3 x lo” Hz is taken as the prominent electronic absorption frequency. The visible regime is indicated for reference.

values for refractive indices and absorption frequencies in the IR and UV regime (see, for example, Mahanty and Ninham, 1976), respectively. On the other hand, typical hydrocarbons (liquid or crystallized) exhibit a constant E(1’v)from zero frequency through the optical regime. The complex absorption spectrum in the near-UV regime is conveniently summarized by taking mean values corresponding to the first ionization potential (Mahanty and Ninham, 1976). In this case, cj(iv) is simply approximated by Eq. (50), where only Lorentz harmonic contributions have to be considered. The third class of dielectric behavior belongs to metals. In this case cj(iv) is simply given by Eq. (52), where typical plasma frequencies are 3-5 x 1015Hz (see, for example, Israelachvili, 1985), and cjo 4 00. According to Eqs. (6) and (12b), the nonretarded Hamaker constant H , is determined by the dielectric response functions of probe, sample, and intervening medium given according to Eqs. (50)-(52). Figure 7 shows the spectral contributions to the VDW interaction for some material combinations of practical interest. The dispersion force in the nonretarded regime is directly proportional to the area under a curve. The Hamaker constant is obviously most sensitive to spectral features between about 1 and 10-20 eV. This involves, for example, the widths of typical band gaps in semiconductors. The maximum H,, is found for two typical metals interacting across vacuum. If one metal is replaced by mica, a representative dielectric, H , becomes

FUNDAMENTALS O F NON-CONTACT FORCE MICROSCOPY

3.0

I

*

-ll''oll

l'--rl'l

across vocuurn across water

""""'

"

75

m ~

hv (eV) FIGURE 7. Spectral contributions to the nonretarded Hamaker constant H , for some material combinations of practical interest. H , is directly proportional to the area under a curve.

considerably smaller, and the maximum of the corresponding curve in Fig. 7 is slightly shifted to higher spectral energies. If both interacting materials are mica, H , further decreases, and the maximum is further shifted to the right. Two water films interacting across saturated air exhibit a still smaller H,, where the maximum is now slightly shifted to lower energies with respect to the mica/mica interaction. The long low-energy tail of the water/water curve is due to the IR spectral contributions, as discussed earlier. Filling the intervening gap between the two interacting media with water considerably reduces the H , values, respectively. This is not due to a Debye orientational process of the highly polar water molecules, but results from the dynamic electronic contribution e 3 ( i v ) > 1 to the response functions A13(iv)and A23(iv)defined in Eq. (6). The maxima of the curves in Fig. 7 are slightly shifted to lower energies with respect to the vacuum values. If each of the three media involved is characterized with respect to its dielectric permittivity by Eq. (50),and if all media exhibit approximately the same electronic absorption frequency v,, the nonretarded Hamaker constant H , according to Eq. (12b) can be evaluated analytically (Israelachvili, 198S), where sufficient accuracy is obtained if only the first term of the sum is considered. Since possible low-frequency rotational processes corresponding to the first term in Eq. (SO) are represented by the entropic constant He given according to Eq. (Sb), H , is simply given in terms of

76

U.HARTMANN 0.50.

1

I

I

0.40.

s" L

,"

\c I

0.30

0.20 0.10 ,

0.00 --0 1

0.25

I

1.oo

I

2.00

I

3.00

I0

FIGURE 8. Nonretarded Hamaker constant for dielectric systems as a function of optical refractive indices of probe, sample, and intervening medium ( n 3 ) . ve denotes the prominent electronic absorption frequency.

the three optical refractive indices rtj ( j = I , 2 , 3 ) for probe, sample, and immersion medium, and the absorption frequency v,:

The detailed behavior of H , as a function of the refractive indices is shown in Fig. 8. Let n2 be the smaller index with respect to the probe-sample ensemble. A reasonable range covering almost all dielectrics is given by 1/4 5 n 2 / n 3 5 4, where n 3 is the index of the intervening medium. If n 2 / n 3> 1, H , is always positive and is usually given by a point in between the curves for nl = n 2 and nl = 4. Most dielectric materials, however, lie in between the curves nl = n 2 5 2 and NI = 2. If the optical refractive index of the immersion medium matches that of either probe or sample, H , becomes zero and the nonretarded force vanishes. For n 2 / n 3 < 1, one has to distinguish between two regimes: If nl < n 3 , Hn is again positive and is located somewhere in between the curve n l = n 2 and the abscissa which corresponds to n 3 = n I . However, if nl > n 3 , H , becomes negative and, according to Eq. (1 I ) , the nonretarded dispersion pressure becomes repulsive. Almost all dielectrics in this regime lie in between the abscissa and the curve n1 = 4n2, while most of them are limited by the curve n, = 2n2.

FUNDAMENTALS O F NON-CONTACT FORCE MICROSCOPY

77

The main conclusions from the preceding analysis of H , are: (i) If probe and sample materials are interchanged, the nonretarded dispersion force remains the same. (ii) For interactions in vacuum or for identical probe and sample materials, the nonretarded force is always attractive. (iii) If probe and sample are made from dielectrics with different refractive indices n l and n 2 , and if there is an intervening medium with n3 > 1, the nonretarded force may be attractive (n3 < n l , n 2 or n 3 > n l , n 2 ) ,vanishing ( n 3 matches either nl or n 2 or both of them), or repulsive (nl < n 3 < n 2 or nl > n3 > n 2 ) . (iv) According to Fig. 8, most of the dielectric material combinations produce a Hamaker constant H , 5 hv,/lO. With a typical value of ve : 3x Hz, one has H , 5 2 x l o p i 9J (1.2eV). H , has to be added to the entropic Hamaker constant He defined in Eq. (8b) to obtain the total nonretarded VDW force given for the z << XI32 limit of Eq. (17). Depending on the static values E , ~( j = I , 2 , 3 ) , the entropic force component may also be attractive, vanishing, or repulsive, where signs and magnitudes of H , and He are generally not correlated. However, the maximum value found for He (see Section II.B.2) is only 1.5% of the limiting value given earlier for H,. This clearly implies that entropic VDW forces play only a minor role in SFM applications (Hartmann, 1991~). If either the probe or the sample is made from a typical metal, characterized by Eq. ( 5 2 ) , the nonretarded Hamaker constant is given by

where ve is the prominent absorption frequency of the system, n 2 the optical refractive index of the remaining dielectric, and n3 that of the intervening medium. Equation (54) permits an estimate of the maximum repulsive dispersion force that can be obtained: n 2 << n 3 yields H , = -(3/8fi)n;hve/(n,' n 3 ) , which gives for large 113 and ve = 3 x l o i 5Hz the upper limit lHnl 5 5 x J (3.1 eV). For two dielectrics with different absorption frequencies veI and ve2,interacting across vacuum, one obtains

+

which was already presented by Israelachvili (1985). If either probe or sample is a metal, one finds

H --h

"-8fi

3

78

U .HARTMANN

and, if both probe and sample are metallic, H - - t i 3- ,

" - 8Jz

veive2 vei ve2

(57)

+

which reduces to the particularly simple result H , = (3/16&)hv, (Israelachvili, 1985) if both metals have the same electronic absorption frequency. Assuming a free-electron gas plasma frequency of 5 x l O I 5 Hz, one obtains H , 5 9 x loi9J (5.4eV) as a realistic upper limit for metallic probeesample arrangements. The dependence of the nonretarded Hamaker constants on the electronic absorption frequencies of probe and sample is shown in Fig. 9 in detail. For given values of vel and ve2,the resulting H, is always highest if both probe and sample are metals. The metal/dielectric arrangement yields lower values depending on the optical refractive index n of the dielectric (either probe or sample). The lowest values of H , are obtained if probe and sample are dielectric, where the magnitude of the nonretarded dispersion force now depends on nl and n2. Anyway, an increase of the absorption frequencies veland ve2always leads to an increase of H,.

- dielectric/dielectric --

rnetal/rnetal or rnetal/dielectn'c

n 1 =4.0

0.3

u,/u, (metal/metal), (n+ ;

else

1)"*v2/v,

FIGURE 9. Nonretarded Hamaker constant H , as a function of the prominent absorption frequencies v, and v2 of the probe-sample combination. n, and n2 denote the optical refractive indices if dielectric materials are involved.

79

FUNDAMENTALS OF NON-CONTACT FORCE MICROSCOPY

1.o

I

I

I

I

(a>

0.8-

-

0.6-

-

0.4-

-

0.2-

-

n

r3

c

0

'= . 6

Ng

5 I

0.0 0.0

I

-.6 0.0

I

I

0.2

0.4

I

0.4

0.2

I

I

I

1 .o

I

1 .o

0.6

0.8

I

0.8

0.6

"2ln3 FIGURE 10. Retarded Hamaker constant H , as a function of effective refractive indices (infrared and visible). (a) shows the positive values of H , , where both probe and sample have larger indices than the intervening medium (n3).(b) shows the situation if the indices of probe and sample are smaller than that of the immersion medium. The dotted lines indicate results from the low-permittivity analytical approximation. Both numerical and analytical results correspond to the first term of the infinite series involved.

80

U. HARTMANN

The preceding results obtained for H , are only part of the whole story. The total VDW pressure according to Eq. (1 7) is completely characterized if, apart from He and H , , the retarded Hamaker constant H , according to Eq. (14b) is also calculated. H , depends on the static electronic limits ~ ~ (of0 ) the dielectric response functions of probe, sample, and intervening medium. Since the relative magnitudes of ~ ~ ( (0j = ) 1’2’3) are in general related neither to the overall behavior of the functions t,(iv) [Eq. (12b) via (6)] over the complete electromagnetic spectrum, nor to the quasistatic orientational contributions ejo [Eq. (8b) via (6)], H , is apriori not closely related to H , and He with respect to sign and magnitude. Apart from the magnitude of the dispersion pressure in the retarded limit given by Eq. (13), H , determines together with H , via the retardation wavelength [Eq. (16)] the onset of retardation effects. The electrostatic limits of the electronic permittivity components are given from Eq. (51) by cj(0)total

2

2

- (cjo - nil) = njl

+

ej(0)electronic.

(58)

As for most hydrocarbons (see Fig. 6) the njI’soften equal the usual optical refractive indices nj. However, as in the case of water, which is of particular practical importance for many SFM experiments, nil is sometimes determined by lower-frequency (IR) absorption bands. However, introduction of generalized refractive indices ni ranging from unity to infinity in Eq. (l4b) permits a unified analysis of H , for all material combinations, i.e., metals and dielectrics. The resulting values of H,, as depending on the individual refractive indices nil = n,, are shown in Fig. 10. Let n 2 be the smaller index for the probe--sample system under consideration. n 3 is the index of the intervening immersion medium. If n 2 > n 3 (Fig. IOa), H , is always positive, and its magnitude is given by a point in between the curves for n l = n 2 and nl + 00. For n l , n 2+ 00 (two interacting metal slabs), one obtains from Eq. (14b) H

n hc - 480

n3’

(59)

which gives, according to Eq. (13), a retarded dispersion pressure which is completely independent of the nature of the employed metals - a property that does not hold for small distances, where the dispersion force according to Eqs. (11) and (12b) depends on higher-frequency contributions to the dielectric response functions which are generally different for different metals. For n 3 = 1, Eq. (59) coincides with the well-known Casimir result (Casimir, 1948). If only typical dielectric materials are involved, Eq. (14b) may be evaluated analytically (Israelachvili, 1972a) by expanding cr(0,p)

FUNDAMENTALS OF NON-CONTACT FORCE MICROSCOPY

81

and P ( 0 , p ) for small ~,(o)/c~(O),j = I , 2. The approximate result is

with A,, according to Eq. (6). The validity of this approximation, depending on the magnitudes of n l / n 3 and n2/n3, can be obtained from Fig. 10. For two interacting metals, i.e., for A,,(O), A23(0) + 1, the low-permittivity approximation still predicts the correct order of magnitude for H I . More precisely, Eq. (60) yields 69C(4)/(27r4) M 38% of the correct value given by Eq. (59). If one has nJ/n3= 1 at least for one of the quotients ( j = I , 2), H , becomes zero and the retarded dispersion force vanishes. If n 2 < n 3 (Fig. lob), one has to distinguish between two regimes: If also nl < n3, HI is again positive and is located in between the abscissa, corresponding to n l = n 3 , and the curve nl = n 2 . The maximum value of this latter curve is obtained for n , /n3, n 2 / n 3 -.+ 0. The low-permittivity approximation, which underestimates the exact value, yields in this case again 38% of the value given by Eq. (59). On the other hand, if n l > n3, H , becomes negative and is given by some point in between the abscissa (nl = n 3 ) and the curve for nl + 00. The approximation for the minimum of this latter curve, obtained for n 2 / n 3 i 0, i.e., A13, A13 --t 1 in Eq. (60), gives a magnitude of 38% of the value in Eq. (59), which is, according to Fig. 10, an underestimate of the exact value. The maximum repulsive dispersion force that can be obtained for any material combination is obtained from the condition SH1/Sn3 = 0, where nl + CG is an obvious boundary condition to achieve high HI values. The use of Eq. (60) yields n 2 = nj(\/5 - 2)'12 and a maximum repulsive retarded dispersion force with a magnitude of about 22% of the value in Eq. (59). This is again slightly underestimated with respect to the exact value numerically obtained from Eq. (14b). It should be emphasized that the entropic Hamaker constant He scales with kT; the nonretarded constant with hv,; and the retarded constant with hc. The absolute maximum obtained for H , is for two metals interacting across vacuum and amounts according to Eq. (59) to H I = 1.2 x Jm (7.4eVnm). Comparison of Eqs. (60) and (53) confirms that the previous statements (i)--(iii) characterizing the behavior of the nonretarded force can be directly extended to the retarded force, however, where one now has to consider the low-frequency indices nil (Eq. (51)) instead of the ordinary optical indices n,. If there is no absorption in the IR regime, the situation is simple, and = n, as in the case of hydrocarbons. However, strong IR absorption, as in the case of water, considerably complicates the situation: The relative weight of different frequency regimes (IR, visible, and UV) becomes a sensitive function of separation between probe and sample. At

82

U. HARTMANN

small distances (nonretarded regime) the interaction is dominated by UV fluctuations. With increasing distance these contributions are progressively damped, leading to a dominance of visible and then IR contributions. For very large separations the interaction would finally be dominated by Debye rotational relaxation processes. This complicated behavior may in principle be characterized by treating the different spectral components according to Eq. (50) additively in terms of separate Hamaker constants and retardation wavelengths. In the present context, the major point is that, because of a missing correlation between the magnitudes of nj and nj, H , and H , may have differrent signs, i.e., the VDW force may be attractive at small probesample separation and exhibit a retardation-induced transition to repulsion at larger separations, or vice versa. In this case the simple analytical approximation of the DLP theory given in Eqs. (17) breaks down. However, even in this case it is possible to keep the concept of separating geometrical and dielectric contributions. The DLP result from Eq. (1) may now be modeled by

where the definitions of He, H,, H,, and X132 remain totally unchanged. This

--

repulsive

-

Y

0

-

-3-

II

-

-6-

0.1

0.5

1.0

5.0

10.0

z / b FIGURE 1 I. Dispersion pressure for the two-slab configuration as a function of separation. If the system exhibits strong infrared absorption, a retardation-induced transition from attraction to repulsion (or vice versa) may occur. An overall attractive (or repulsive) interaction occurs if nonretarded and retarded Hamaker constants have the same sign.

83

FUNDAMENTALS OF NON-CONTACT FORCE MICROSCOPY

formally implies the occurrence of negative retardation wavelengths which are obtained according to Eq. (16) if H , and H , have different signs. Equation (61) exhibits the same behavior as Eq. (17) for the zs)(A132 limits, but additionally demonstrates a retardation-induced transition between the attractive and repulsive regimes at z = A132, as can be seen in Fig. 1 1 . A completely blunt (cylindrical) SFM probe would detect a force exactly corresponding to the curves obtained for the VDW pressure. However, according to Eq. (34b), more realistic probe models (paraboloidal, ellipsoidal) predict a measurement of forces being proportional to the specific VDW energy given via an integration of Eq. (61). This implies that the transition distance measured with a paraboloidal or ellipsoidal probe is somewhat smaller than that measured with a cylindrical probe (Al32). The smallest transition separation is, according to Eq. (34a), obtained for a conical probe. The intriguing conclusion is that for a probe-sample interaction which does not involve a monotonic distance dependence, the force measured at a given probe-sample separation may be attractive for one probe and repulsive for another with different apex geometry. defined in Eq. (16) depends on the The retardation wavelength dielectric response functions of probe, sample, and intervening medium. Retardation effects of the radiation field between probe and sample become noticeable if the probe-sample separation is comparable with AIj2. The retardation wavelength is thus closely related to the prominent absorption wavelength A, = c / u , of the material combination, which is usually about IOOnm, i.e., within the UV regime. The actual onset of retardation effects, manifest in a gradual increase of the differential power law index k according to Eq. (43), is then for a given material combination determined by the probe geometry (see Section II.B.4). In the following, some simple analytical results for X132 are presented which allow a straightforward verification of the relevance of retardation effects for most material combinations of practical importance to SFM. Combining Eqs. (53) and (60), one obtains the retardation wavelength for a solely dielectric material combination. First-order approximation yields A132

=2 23& 1

207T

%I

(A n + v) ~ 3 ’(~ ,/-+{z,,. n,2 rill +n,, n,?

I=,

1

-

3

ye

)1. (62)

If the system does not exhibit effective IR absorption, i.e., n,l = nj ( j = 1,2,3), the product in parentheses reduces to unity and A132 is solely determined by the ordinary optical refractive indices and the prominent electronic UV absorption frequency. If the probe or the sample is

84

U. HARTMANN

metallic, Eqs. (54) and (60) yield the approximate result

where the product in parentheses again becomes unity in the absence of IR absorption. If dielectric probe and sample have different absorption frequencies and if they interact across vacuum, Eqs. (55) and (60) approximately give

which again simplifies for rill = nl. If either the probe or the sample is metallic, one obtains from Eqs. (56) and (60)

1.00 1

I

I

I

_ -

0.80-

<

N

2

0.700.600.50-

4

metal/dielectrk

0.10-

0.00

----------__________

-

nnl=1.5 l=1.0 n1=2.0 n1=4.0

_

dielectric/dielectriC

1 1

_

7

I

2

I

3

I

4

I

5

"2 FIGURE 12. Retardation wavelength X13z as a function of the optical refractive indices of probe and/or sample interacting across vacuum. u, is the prominent electronic absorption frequency of the system of which the absence of infrared absorption bands is assumed. The upper limit provided by the metal-metal arrangement is indicated for reference.

FUNDAMENTALS OF NON-CONTACT FORCE MICROSCOPY

85

with the aforementioned simplification for n Z 1= n2. If a purely metallic probe-sample system interacts across vacuum, Eqs. (57) and (59) yield the exact result

which only involves the free-electron gas plasma frequencies as characteristics of the metals. If one in particular has clue, = c/ueZ= A,, amounts to 93% of A,. A glance at Eqs. (62)-(65) shows that this can be considered as an upper limit for any material combination with n,, = n, ( j = 1,2,3), i.e., for arrangements where IR absorption only plays a minor role. On the other hand, large values of A132 are obtained according to Eq. (16) if H , is nearly vanishing and H , is determined by IR absorption. Figure 12 shows typical values of A132 obtained in an accurate way by numerically solving Eqs. (12b) and (14b). The maximum value for A132 in a solely dielectric probe-sample arrangement is about 31% of A,. For a metal/dielectric combination this value amounts to 37%. Both values are considerably lower than the aforementioned value, which may be obtained for a metal/metal combination of probe and sample. Typical values of A132 are 20-35% of A, if one does not have a purely metallic arrangement.

v e d v e1 FIGURE13. Retardation wavelength as a function of the prominent ultraviolet absorption frequencies vel and ve2 of probe and sample. n l and n2 denote the ordinary optical refractive indices if dielectric materials are involved. The curves are valid for systems without effective absorption bands in the infrared regime.

86

U. HARTMANN

The rigorous solution for as a function of the prominent UV absorption frequencies involved is shown in Fig. 13. The minimum value of X132 for a metal/metal arrangement is about 46% of A,, = c/vel if ve2 + vel. For a metal/dielectric or dielectric/dielectric combination, XIj2 can be much smaller depending on the optical refractive indices involved. -1

1

N

E C

'=.

5I C

0

0 -

, ' ' I

-

I: metal-air

-2n

1

-

-3-4-

-

-5-

-

-6-

-

-7-

I -

-8-9

II 111 IV V

-

-10,

1

I

5

8

" ' I

10

I

50

'

'

8

1

100

z (nm>

FIGURE 14. (a) shows the two-slab VDW pressure as a function of separation for some representative material combinations. (b) shows the corresponding retardation-induced increase of the differential power law indices.

FUNDAMENTALS OF NON-CONTACT FORCE MICROSCOPY

87

Systems with extremely low retardation wavelength may be constructed according to Eq. (62). Suitable material combinations consist of a dielectric probe, sample, and immersion medium with an appropriate choice of refractive indices, minimising X132. Most effective would be a match of the IR indices, n31 of the immersion medium with n i l and/or n21 of the probe-sample combination. Unfortunately, reliable IR data are not available for most materials. For material combinations which do not exhibit pronounced 1R absorption, the ordinary optical indices nj ( j = 1,2,3) should all be as large as possible, where a highly refractive immersion medium ( n 3 ) is especially effective. In this way, retardation wavelengths smaller than 10 nm are generated, which opens the way for an experimental confirmation of radiation field retardation effects by SFM (see Section II.B.6).

6. On the Observability of van der Waals Forces The framework for calculating VDW forces for any material combination and any probe geometry as a function of probe-sample separation is now complete. The material properties of a certain system are characterized by the three Hamaker constants He,H , , and Hr according to Eqs. (8b), (12b), and (14b). This includes the determination of the retardation wavelength via Eq. (16).The total VDW pressure f ( z ) for the two-slab arrangement is then given by Eqs. (17) or (61) in terms of a reasonable approximation. For relevant probe geometries, the VDW interaction is characterized by Eq. (34b), which involves the probe’s effective radius of curvature. An estimate of the resulting lateral resolution is obtained from Eq. (47a). Figure 14 shows the typical order of magnitude of the two-slab VDW pressure as well as the material-dependent onset of retardation effects for some representative material combinations. The dielectric data used for these model calculations are given in Table I. In the regime from 1 to TABLE 1 DIELECTRIC DATAUSEDFOR THE CALCULATIONS’

Metdl/air/rnetal Micaiairirnica H20pdir/H20 Hydrocarbon/air/hydrocarbon Mica/HzO/mica

40 10

3.7 7.1 2.0

0.30 0.17 0.29 0.04 0.2 1

130 9.3 4.5 8.7 2.0

61 20 23 23 17

’ For reasons of comparison, the present data are deduced from the basic data given by lsraelachvili ( I972 b, 1985). For water, infrared absorption contributions have been neglected.

88

U . HARTMANN

1-

I

1

“ “ I

probe radius: 1bOnrn

U

-5-

I I

I

-6-

-7

(a> 1

I

5

‘

~~~1

10

--__

i-. I

I

I I

I

I I

50

’

r

-

3

1 I0

d (nm>

A

n

E

FIGURE15. VDW interaction of a IOOnm metal probe with a metal and a mica substrate under clean vacuum conditions, respectively. The retardation wavelengths X for the metal/metal and metal/mica configurations are indicated. The cntropic limit determines the absolute roomtemperature maximum for thermally agitated interaction contributions. Deviations from a linear decrease of the curves with increasing probe-sample separation reflect the gradual onset of retardation effects. The indicated experimental limits are accessible by state-of-the-art instruments. (a) shows the forces measured upon static operation of the force microscope and (b) the vertical force derivative, detected in the dynamic mode.

FUNDAMENTALS O F NON-CONTACT FORCE MICROSCOPY

89

100 nm separation, the pressure drops by six to seven orders of magnitude in ambient air (or vacuum). As mentioned before, two typical metals yield the strongest possible interaction. Mica, representing a typical dielectric material, yields a pressure of about 25% of the metal value in the nonretarded regime and of about 7% in the retarded limit, respectively. Crystallized hydrocarbons and water are most frequently the sources of surface contaminations. These media only exhibit a small VDW pressure with respect to the metal limit. Consequently, if initially clean metal surfaces become contaminated by films of hydrocarbons or water, the VDW interaction may decrease by 80-90% or more for a given width of the intervening air gap. If the complete intervening gap between two mica surfaces is filled with water, the VDW pressure drops with respect to the air (or vacuum) value by about 80%. The onset of retardation effects also critically depends on the system composition. The metal system yields the highest The hydrocarbon and water values are about the same. The retardation wavelength for two mica slabs in air (vacuum) is reduced by about 17% if the intervening gap is filled with water. As an example of direct practical relevance, Fig. 15 shows the VDW interaction between a realistic metal probe (paraboloidal or ellipsoidal)

mica fused quartz

I

I

.

I

"'I"

z 5

I

0.05

I

->

v

polystyrene

o

hydrocarb.

attractive

LL

o.oot

probe-sample distance: 2nm

1.2

1.3

1.4

1.5

1.6

1.7

n FIGURE 16. V D W force between a metal probe operated in a benzene immersion and with various dielectric substrates at a fixed probeesample distance. n denotes the sample's ordinary optical refractive index. For purposes of comparison refractive indices and absorption indices have been choscn according to Isrdelachvih (1985).

90

U. HARTMANN

with a mesoscopic radius of apex curvature of l00nm and two different atomically flat substrates; a typical metal and mica, representing a typical dielectric. Assuming an experimental sensitivity of 10 pN, which is not unrealistic for present-day UHV-SFM systems, forces should be detectable up to about 20nm for the metal sample and up to about 10nm for mica. Radiation field retardation becomes effective just near these probesample separations. The entropic limit, according to Eq. (8a) with He = 3.6 x J, indicates that thermally agitated VDW forces could only be measured at working distances 5 1 nm. In the dynamic or “ac” mode of SFM, the vertical force derivative F ’ ( d ) = 6F/Sd is detected. An accessible experimental sensitivity may be given by 10 pN/m. This extends the measurable regime up to about 70nm for the metal sample and up to about 50 nm for mica. According to Fig. I3b, this clearly involves the onset of retardation effects. Performance of SFM in an immersion medium generally offers the possibility to choose material combinations yielding attractive, repulsive, or just vanishing VDW interactions between probe and substrate. Assuming a metal probe operated in a benzene immersion at a fixed probe-sample separation, Fig. 16 gives the resulting VDW forces for various dielectric substrates as a function of the ordinary optical refractive index n of the sample according to Eq. (53). While polytetrafluoroethylene (PTFE), CaF2, and fused quartz with n < 1.5 produce repulsive nonretarded VDW forces, polyvinylchloride (PVC), polystyrene, and mica with n > 1.5 yield attractive forces. Crystallized hydrocarbons just match the index of benzene, n = 1.5, and the VDW force reduces to the small entropic contribution. 7. The Effect of Adsorbed Surface Layers The analytical solutions for the VDW pressure of the two-slab configuration given in Eqs. (17) and (61) allow straightforward extension to multilayer configurations. Figure 17 shows the basic geometry for two slabs “1” and

FIGURE 17. Basic geometry of the four-slab arrangement used to analyze the interaction of two bulk media which have surfaces covered with adsorbed layers.

FUNDAMENTALS OF NON-CONTACT FORCE MICROSCOPY

91

“2,” both with an adsorbed surface film “4.” An arbitrary intervening substance is denoted by “3.” If we extend previous results found for the nonretarded interaction (Mahanty and Ninham, 1976; Israelachvili, 1972a) to an analysis for arbitrary separations d and film thicknesses t41 and t42, the VDW pressure is given by .fad(Z) =.h34(z) -h41(z

+ t41)

-f342(z+

t42) + f i 4 2 ( z +

t41 f t42).

(67)

The subscripts of type kIm denote the material combination which actually has to be considered to calculate an individual term of this expression. k and m denote the opposite slabs, respectively, and 1 the intervening medium. The solution of this four-slab problem is thus reduced to the calculation of four “partial” VDW pressures involving four sets of Hamaker constants. Since these partial pressures have different entropic, nonretarded, and retarded magnitudes and varying retardation wavelengths, the distance dependence of .fad is generally much more complex than that for the two-slab arrangement, in particular, if there are fk,,-terms showing a retardation-induced 5.0

-

I ‘ ~ ~ PTFE/adsorbate/immersion 1

4.0

‘

~ ’‘ ’l ~ “ ‘ ‘ ‘ 1 ‘ , ‘ medium/adsorbate/mica 1

. T

3,O hydrocarbon adsorbate, H20 immersion

LL

0

2.0

LL

5 0 u-

hydrocarbon adsorbate, vacuum

---__ --

1.o

.-

H20 adsorbate,.,.’ vacuum

---

I

1

-“-\ __

_ _ C C

I

1.o

r

’

“““I

10.0

‘

‘

‘

-

~

curved surf

- slabs ‘

~

100.0

~r I

‘

‘

I

.

1( 0.0

FIGURE18. Model calculation showing the effect of adsorbed hydrocarbon (liquid or crystallized) or water layers on the interaction between polytetrafluoroethylene (PTFE) and a mica surface. The quotients f,d/f and F a d / F denote the force ratios obtained for adsorbatecovered surfaces with respect to clean surfaces, for planar and paraboloidally or ellipsoidally curved surfaces, respectively. The adsorbate thickness t is assumed to be the same on PTFE and mica. d denotes the width of the intervening gap, either for vacuum or water immersion. The curved- and planar-surface curves for hydrocarbon adsorbate in vacuum cannot be distinguished within the accuracy of the plot. The dashed curves would be detected with a typical probe in dc-mode force microscopy, while the solid lines reflect ac data.

92

U . HARTMANN

changeover between attractive and repulsive regimes according to Eq. (61). However, it follows immediately from Eq. (67) that f a d ( z )+ f 4 3 4 ( z ) for t 4 , / z lt 4 2 / ~+ 00; for large thicknesses of the adsorbed surface layers the VDW pressure is solely determined by the interaction of the layers ~ 0, one across the intervening medium. On the other hand, if t 4 1 / ~t, 4 2 / + immediately finds,fad(z) + f i 3 * ( z ) = f ( z ) , which is simply the solution of the two-slab problem according to Eq. (17) or (61). In the latter case the interaction is dominated by the interaction of the two bulk media across the intervening medium. Figure 18 exemplarily shows the considerable differences of the VDW interactions which occur if initially clean polytetrafluoroethylene (PTFE) and mica surfaces adsorb typical hydrocarbons (liquid or crystallized) or water. The adsorption of hydrocarbons slightly increases the vacuum forces. However, if the intervening gap is filled with water, the magnitude of the VDW forces increases by about a factor of four with respect to the interaction of clean surfaces across water. Water adsorption in air considerably reduces the forces with respect to clean surfaces. In all cases involving adsorbed surface layers, the bulk interaction value is not approached before the intervening gap exceeds the layer thickness by two to three orders of magnitude. This clearly emphasizes the fact that VDW interactions are highly surface-sensitive: Even a monolayer adsorbed on a substrate considerably modifies the probe-sample interaction with respect to the clean substrate up to separations of several nanometers. The situation is additionally complicated by the fact that the difference in VDW force measured between clean and coated substrate surfaces also depends on the probe geometry (see Fig. 18). This intriguing phenomenon is due to the integral equations (34) determining the probe-sample force from the two-slab pressure.

8. Size, Shape, and Surface EfSects: Limitations of the Theory The rigorously macroscopic analysis of VDW interactions in SFM implicitly exhibits some apparent shortcomings which are ultimately due to the particular mesoscopic, i.e., nanometer-scale, physical properties of sharp probes and corrugated sample surfaces exhibiting deviations from ordinary bulk physics. To obtain an upper quantitative estimate for those errors resulting from size and shape effects, it is convenient to apply the present formalism to some particular worst-case configurations for which exact results from quantum field theory are available for comparison. Two such arrangements which have been subject to rigorous treatments are two interacting spheres and a sphere interacting with a semi-infinite slab. These configurations do reflect worst-case situations insofar as the sphere of finite

FUNDAMENTALS OF NON-CONTACT FORCE MICROSCOPY

93

size emphasizes geometrical errors involved in the Derjaguin approximation as well as shape-induced deviations from a simple bulk dielectric behavior. Size and shape effects occur when the probe-sample separation becomes comparable with the effective mesoscopic probe radius as defined in Eqs. (32). Since realistic probe radii are generally of the order of the retardation wavelengths defined by Eq. (16), the following analysis is restricted to the retarded limit of probe-sample dispersion interaction to obtain an upper boundary for the involved errors. However, extension of the treatment to arbitrary probe-sample separations is straightforward. According to the basic Hamaker approach given in Eq. (23), the dispersion interaction between a sphere of radius R1 and a single molecule at a distance z from the sphere’s center is simply given by 2 2

r’

where p 1 is the molecular density within the sphere and B the molecular interaction constant given by Eq. (19b). The interaction between two spheres separated by a distance d is then given by

where R2 and p 2 are radius and molecular density of the second sphere and is taken from Eq. (68). The interaction between a sphere and a semiinfinite slab is obtained without problems by analytically evaluating the preceding integrals and letting one of the radii go to infinity. However, from reasons clarified later, the limiting behavior for d >> R is more interesting in the present context. For two identical spheres ( R 1= R2 = R, pI = p2 = p ) , one obtains

f,(z)

16 F ( d ) = - -T 9

R6 p B s , d

and for the sphere-slab configuration ( R I = R , R2 F(d)= -

8 105

-T

2 2

+

m, p1 = p2 = p ) ,

R’

p B7 d

Both results have already been derived in Eqs. (22a) and (28a). If one now assumes that the screening of the radiation field by the near-surface molecules is the same as for the two-slab configuration, the microscopic quantity p 2 B is related to the macroscopic Hamaker constant by Eq. (25b). Especially for ideal metals, which may be considered as the limit of a London superconductor, as the penetration depth approaches zero, one

94

U. HARTMANN

obtains via eq. (59) for an interaction in vacuum 7n2 R 6 F ( d ) = - -hc 7 27 d

(72)

from Eq. (70), and r2 R3 F ( d ) = - - hc 3 (73) 90 from Eq. (71). However, these results are not completely correct, since the surface screening of the radiation field is affected by the actual curvature of the interacting surfaces. The correct result for the two-sphere configuration is obtained by using the Hamaker constant given in Eq. (22b). For perfectly conducting spheres, as considered in the present case, one has, apart from the electric polarizability, to account for the magnetic polarizability, which provides an additional contribution of 50% of the electric component to the total polarizability (see, for example, Jackson, 1975). Appropriate combination of electric and magnetic dipole photon contributions yields 2A13(0)2A23(O) = A2(0) = Ak(0) + AL(0) + (14/23)AEM(0)(Fienberg and Sucher, 1970; Feinberg, 1974), where A,(O) = 1 and A,(O) = are the pure electric and magnetic contributions, respectively, and AEM(0) = is due to an interference of electric and magnetic dipole photons. Inserting A2(0) = 143/92 into Eq. (22b) then ultimately leads to

1

4

which has been previously derived by a more involved treatment (Feinberg and Sucher, 1970; Feinberg, 1974). Comparison with Eq. (72) shows that the two-slab renormalization underestimates the sphere-sphere VDW force by about 19%, which is due to the reduced screening of the curved surfaces. For the sphere-slab arrangement, the rigorous result is obtained by using the Hamaker constant given in Eq. (28d) for a perfectly conducting metal sphere. Using 2A23(0)= this yields

5,

9 R3 F ( d ) = - -hc -, (75) 87r2 d S which is in agreement with a previous result (Datta and Ford, 1981) obtained by different methods of theory. A comparison with Eq. (73) yields a slight underestimate of about 4% due to the two-slab renormalization. At very small separations, d << R, the Derjaguin approach according to Eq. (40b) yields the correct results 7r 2 R F ( d ) = - -hc 7 1440 d

95

FUNDAMENTALS OF NON-CONTACT FORCE MICROSCOPY

0.01

1 .oo

0.10

10.00

100.00

d/R

-15

(b)

I

'

~

~

.

'

~

-

asymptotic limit .

'~ - I~ ' ' ~ - I

-

~

'

~

~

~

T

-

r

FIGURE 19. Retarded vacuum dispersion force between two spheres (a) and between a sphere and a semi-infinite slab (b). R denotes the sphere's radius, F the magnitude of the attractive force, and d the surface-to-surface distance. The upper curves correspond to perfectly conducting constituents, while the lower ones characterize identical dielectrics. The dashed lines indicate results from short- and long-distance approximations.

96

U. HARTMANN

for the sphere-sphere interaction, and 7r 2 R F(d) = - - hc -5 (77) 720 d for the sphere-slab interaction, where in both cases the Hamaker constant according to Eq. (59) has to be used; radiation field screening is about the same for planar and very smoothly curved surfaces. Comparison of Eqs. (74) and (75) with (76) and (77) shows that the dispersion force changes from l i d 3 to a l / d 8 dependence for the two spheres, and from a l/d' to a I /d5 dependence for the sphere-slab arrangement. Both cases are correctly modeled by the Hamaker approach according to Eq. (69), as shown in Fig. 19. For d >, R/10, the Derjaguin approximations exhibit increasing deviations from the Hamaker curves. Deviations in radiation field screening with respect to the two-slab configuration gradually occur and reach the aforementioned asymptotic values when the Hamaker curves approach the asymptotic limit. Figure 19 additionally includes results of the preceding comparative study for interacting dielectrics. In this case, surface screening is much less pronounced, as for perfectly conducting bodies. Thus, the Hamaker approach with two-slab renormalization yields almost accurate results at any interaction distance and for arbitrarily curved surfaces. The major conclusion that can be drawn from this worst-case scenario is that the maximum error due to surface screening of a probe with unknown electric and magnetic form factors amounts, at large distances, to 10% for an arbitrarily corrugated sample surface and to 4% for an atomically flat substrate. At ordinary working distances, d << R , and for dielectrics, geometry-modified screening is completely negligible (see also Mostepanenko and Sokolov, 1988). Another shortcoming of the present theory is that it implicitly neglects multipole contributions beyond exchange of dipole photons. In general, for probe-sample separations greater than about one nanometer, the exchange of dipole photons generally overshadows that due to dipole-quadrupole and higher multipole exchange processes. However, for smaller separations, as present in contact-mode SFM, and for some particular material combinations also at larger separations, multipole interactions assume increasing importance. For the retarded interaction of perfectly conducting spheres, the total force including the interference between electric and magnetic quadrupole photons (Feinberg and Sucher, 1970; Feinberg, 1974) is shown in Fig. 19 in addition to the pure electric dipole contribution. However, in most cases these corrections are of little relevance in the present context. A much more serious obstacle for a rigorous characterization of VDW interactions in SFM results from the explicit assumption of isotropic

FUNDAMENTALS O F NON-CONTACT FORCE MICROSCOPY

97

bulk dielectric permittivities of probe, sample, and immersion medium. Especially if probe and sample are in close proximity, this assumption is subject to dispute, in view of the microstructure of solids (Zarenba and Kohn, 1976) and liquids (see Section 1I.D) and the existence of particular surface states. This may require specific correlations to account for the particular molecularscale surface dielectric permittivities. However, unfortunately little reliable information on specific surface dielectric properties has become available so far. Further progress is also needed in appropriately treating the behavior of real metal tips and substrates. Especially for sharp probes and substrate protrusions, the delocalized electrons, moving under the influence of the radiation field fluctuations, require a specific nonlocal microscopic treatment (Girad, 1991). 9. Applicution of van der Wuuls Forces f o r Molecular-Scale Analysis and Surface Munipulution Manipulation of substrate surfaces by using scanned probe devices assumes an increasing importance. In particular, the deposition of individual molecules or small particles provides an approach to study microscopic electronic or mechanic properties and to achieve positional control of interaction processes. First experimental results (Eigler and Schweizer, 1990) imply that VDW forces may play an important role within this field. Figure 20 shows some proposals for the analysis and manipulation of small particles or molecules by a systematic employment of VDW interactions. A small particle or molecule physisorbed on a flat substrate may be moved in close contact to the substrate by a “sliding process,” as shown in the upper left image of Fig. 20. The VDW bonds between particle and substrate and between particle and tip have to ensure on the one hand the fixing of the particle between tip and substrate during sliding, and on the other hand the anchoring to the substrate during final withdrawal of the tip. A liquid environment permits the variation of the nonretarded Hamaker constant for the tip-substrate interaction over a wide range, preferably according to Eqs. (53) or (54). PTFE may be considered as a promising universal substrate material, since its optical refractive index ( n = 1.359) is lower than that of several liquids yielding repulsive interactions with respect to most tip materials. Especially for water immersion, n = 1.333, the interaction between PTFE and any tip material should almost vanish. PTFE can be easily modified to render its surface hydrophilic. Another surface manipulation process (Fig. 20, upper right) involves the elevation of the particle if the tip-particle VDW bond is stronger than that between particle and substrate. The particle can thus be transported over larger distances and obstacles. Deposition is performed at a place where the

c

FIGURE 20. Manipulation (upper row) and analysis (lower row) of small particles and molecules by systematically employing VDW interactions.

FUNDAMENTALS OF NON-CONTACT FORCE MICROSCOPY

99

substrate-particle interaction is stronger than the tip-particle interaction. For this purpose the tip should be dielectric, and the place of position of the PTFE substrate should have a higher optical refractive index than the tip. As was shown in Section II.B.7, even a monolayer coverage on top of the PTFE substrate can considerably increase the particle-substrate Hamaker constant and can raise the interaction above that between tip and particle. A liquid immersion again allows control of the perturbative tip-substrate interaction. The lack of really reproducible, well-characterized, and mesoscopically (nanometer scale) sharp probes in standard SFM systems is the most apparent obstacle for high resolution VDW measurements. The “molecular tip array” (MTA) philosophy developed by Drexler (1991) could alleviate this problem. The proposed geometry is shown in the lower left of Fig. 20. The sample, shaped as a small bead, is attached to the microscope’s cantilever by adhesion. Small particles or macromolecules are adsorbed on the flat substrate surface. Since the bead’s radius R’ is assumed to be very large compared to the effective molecular radius, which is according to Eq. (32) given by R = 2R:/R, for an ellipsoidal molecule with semiaxes R , and R,, an individial molecule underneath the bead images the gently curved surface upon raster scanning the substrate with respect to the bead. The lateral resolution follows directly from Eq. (47a) and amounts to A,(d)

=2

R . d m .

While the interaction between bead and molecule at arbitrary separations may be obtained from Eqs. (42), the total nonretarded VDW force for separations being large compared with the effective molecular radius, i.e., d > R:/R,, is, according to Eqs. (27), H n ~ 6 , F,(d) = - 67r R:d4‘

(79)

For A << R : / R , Eq. (40a) yields F,(d)

1 H,

--

1

6

~

R: R,d2‘

The substantial bead radius raises the issue of unwanted surface forces. The bead-substrate interaction is, according to Eq. (40a), 1 R* Fi(d)= - - H i 6 (2R, + d ) ”

with a Hamaker constant preferably according to Eqs. (53) or (54). The

100

U. HARTMANN

ratio of this “parasitic” force to the imaging force is, for d >> R $ / R , ,

For a close bead-molecule separation, d << R $ / R , , one obtains

The preceding quotients may be considered as “noise-to-signal ratios” and should be much smaller than unity. The considerable potential of MTA imaging is emphasized if one somewhat quantifies the preceding design analysis. For simplicity, arbitrary spherical macromolecules with R , = R , = 1 nm are assumed. A moleculebead separation of d = 1 nm separates the VDW interaction from shortrange forces due to orbital overlap. Under these conditions, Eq. (78) yields a lateral resolution of A, = 1.3 nm for the VDW imaging of the bead’s surface. Using a somewhat typical Hamaker constant of 1.5 x J (see Section II.B.5), the force according to Eq. (80) amounts to F, = 25pN, which is within reach of present technology. Suppression of parasitic forces F,’ requires, according to Eq. (83), a Hamaker constant H,* which is less than 4% of H , . This may easily be achieved by using PTFE substrates in combination with an aqueous immersion. Potential tip structures may predominantly include single-chain proteins, proteins with bound partially exposed ligands, or nanometer-scale crystalline particles (Drexler, 199I). The considerable capabilities of modern organic synthesis and biotechnology offer broad freedom in molecular tip design. MTA technology would permit the quasi-simultaneous use of a broad varity of tips scattered across the substrate. This may include tips of different composition, electric charge, magnetization, and orientation. A tip density of more than 1,000/pm2 has been considered as reasonable (Drexler, 1991). First results in obtaining suitable metallic bead-cantilever systems have been reported by Lemke et al. (1990). Apart from VDW imaging of the surface of a spherical sample at ultrahigh spatial resolution, the MTA technology may be well suited to obtaining a deeper insight into molecular electronics and mechanics (Fig. 20, lower right). Using an SFM with a conducting tip-cantilever system, simultaneous tunneling and force measurements may be performed on a single molecule. This may help to clarify the process of tunneling through localized electronic states in organic molecules by detecting the tip-moleculesubstrate tunneling current Zas a function of the tunneling voltage V and the force exerted on the molecule.

FUNDAMENTALS OF NON-CONTACT FORCE MICROSCOPY

101

10. Some Concluding Remarks The present analysis emphasizes that VDW forces play an important role in SFM. At probe-sample separations less than a few nanometers, the force drops according to F,(d) = - ( H , / 6 ) R / d 2 , where R is the probe’s effective radius of curvature, H , the nonretarded Hamaker constant, and d the probe-sample separation. A somewhat representative value for the force at r l = 1 nm is IF,I/R = 10mN/m. While the interaction is always attractive in dry air or vacuum, it may be attractive, repulsive, or even vanishing if the gap between probe and sample is filled with a liquid medium. Thermally activated processes generally play a minor role, and it is in almost all cases sufficient to analyze the dispersion part of the forces. Non-contact VDW microscopy is capable of providing information on surface dielectric permittivities at sub- 100 nm resolution. The technique is sensitive even to monolayer coverages of a substrate. Important future fields of application are the investigation of liquid/air (vapour) interfaces (Mate et al., 1989) and the imaging of soft (biological) samples including individual macromolecules. As in contact-mode SFM ( i e , AFM), where VDW forces have a substantial influence on the net force balance, and thus on the probeesample contact radius, the long-range interactions may play a role in other noncontact modes of operation, i t . , in electric and magnetic force microscopy, if these are performed at low working distances ( z 1 nm). However, in this latter context VDW forces may be reduced in a welldefined way by covering the sample surface with a suitable dielectric and/ or using an adapted liquid immersion medium. Finally, some open questions with respect to the general subject of VDW interactions in SFM should be listed: (i) In what way may the effective dielectric permittivities deviate from the assumed anisotropic bulk dielectric properties, especially for sharp metal tips? (ii) Is the present rigorously macroscopic treatment satisfactory down to probeesample separations which involve electron-orbital overlap, or is a special nonlocal microscopic treatment needed? (iii) May VDW forces be externally stimulated in a measurable way by electromagnetic irradiation, preferably at wavelengths between IR and UV? Such an excitation, beyond zero-point fluctuations, would permit the performance of “scanning force spectroscopy” as a technique to sense the spectral variation of surface dielectric permit tivities. (iv) Do excited surface states, i.e., surface plasmons (see, for example, Rather, 1988, as well as several articles on plasmon observation by

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U . HARTMANN

STM) have a measurable effect on the probe-sample VDW interaction? These questions are considered as some major future challenges for elaborate SFM experiments on the VDW forces. Additional questions are concerned with the delicate interplay of VDW forces with other interactions to be discussed in the following. C . Ionic Forces 1. Probe-Sample Charging in Ambient Liquids

Situations in which VDW forces solely determine the probe-sample interaction in SFM are in general restricted to an operation under clean vacuum conditions. Under ambient conditions, which are often present in SFM experiments, long-range electrostatic forces are frequently additionally involved, and the interplay of these latter and VDW forces has important consequences. If wetting films are present on probe and sample or if the intervening gap is filled with a liquid, surface charging may come about essentially in two ways (see, for example, Israelachvili, 1985); (i) by ionization or dissociation of ionizable surface groups, and (ii) by adsorption of ions onto initially uncharged surfaces. Whatever the actual mechanism, the equilibrium final surface charge is balanced by a diffuse atmosphere of counterions close to the surfaces, resulting in the so-called “double layer,” (see Fig. 21). The electrostatic interaction between probe and sample is closely related to the counterion concentration profile.

FIGURE 21. Diffuse counterion atmosphere near the surfaces of two slabs which exhibit a certain surface charge density u. The intervening gap of thickness d contains a solution with a static dielectric constant c. Vo and X denote voltage and separation between fictitious centric planes of the near-surface counterion profiles.

FUNDAMENTALS OF NON-CONTACT FORCE MICROSCOPY

103

Since the intervening homogeneously dielectric gap between two semiinfinite equally charged slabs is field-free, the counterions do not experience an attractive electrostatic force toward the surfaces. The ionic concentration profile is solely determined by interionic electrostatic repulsion and the entropy of mixing, while the amount of surface charge only controls the total number of counterions. For simplicity, an identical charge density 0 is assumed on the opposite surfaces, and one also assumes electroneutrality of the complete two-slab arrangement. The resulting nonlinear second-order Poisson-Boltzmann differential equation (see, for example, Israelachvili, 1985) leads to a general form of the so-called contact value theorem, The ionic excess osmotic pressure is, for a given thermal activation energy, simply proportional to the excess counterion density, present directly in front of the surfaces of the charged slabs. p ( m ) is the ionic surface density for an isolated charged surface. Since the intervening liquid does not contain a bulk electrolytic reservoir, one has

where t is the static dielectric constant of the immersion fluid. The two-slab counterion surface concentration amounts to

where n denotes the ionic valency. The characteristic length X depends, for a given valency and for given values of and E , on the separation d of the slabs and has to fulfill the following condition: 2kT neX

d 2X

-tan -

+

u

-=

tc0

0.

(87)

Insertion of Eqs. (85) and (86) into Eq. (84) shows t h a t f ( d ) in Eq. (84) may be associated with the pressure in the interspace of a simple parallelplate capacitor with C / s = cco/z. If one applies a certain voltage, Vo, this pressure amounts to f ( z ) = -€toVt/2z2.The analogy holds if the virtual separation of the capacitor plates is given by z = InlX(d), while one has to apply an imaginary voltage Vo = i2kT/e,

(88)

which results in a repulsive interaction between the plates. It should be noted that this elementary voltage is completely independent of the type of counterions. The magnitude amounts to 52.5 mV at room temperature.

104

U. HARTMANN

The two-slab ionic pressure is now simply given by

lnlX thus obviously represents the separation of the fictitious capacitor plates which may be associated with the maxima of the near-surface counterion concentration profiles (see Fig. 21). Figure 22 shows the dependence of X for monovalent ions in water on the separation d between two slabs for three different surface charge densities, computed according to Eq. (87). X decreases with decreasing d and increasing 0.For high surface charge densities and at large separations, X becomes proportional to d. In this limit the pressure becomes

which is known as the Langmuir relation. This relation may be used to calculate the equilibrium thickness and disjoining pressure of wetting films on probe and sample in SFM systems. In the oppposite limit d + 0, Eq. (87) yields X + d N ( q k T / n e o ) d , and the pressure according to Eq. (89) is

0.5 0.0

1

I

5

8

’

, ‘ I

10

I

50

I

,

‘

.

100

d (nm> FIGURE22. Characteristic separation length X for monovalent counterions in water as a function of separation of two surfaces exhibiting an equal charge density 0.The latter quantity is given in electrons per surface area, where I e-/0.8nm2 = 0.2C/m2 represents a typical value for a fully ionized surface.

FUNDAMENTALS OF NON-CONTACT FORCE MICROSCOPY

105

given by f ( d ) = iVoa/nd,

which describes a real and repulsive pressure because V , is imaginary and a and n have opposite signs. This simple equation is considered to be of particular importance for SFM experiments involving deionized immersion liquids and a moderate charging of probe and sample surfaces. However, the total interaction between probe and sample in a liquid environment must of course also include the V D W force. Unlike the ionic force, VDW interactions are largely insensitive to variations of the counterion concentration, while they are highly sensitive to those surface reactions ultimately leading to the ionic forces, i.e., dissociation or adsorption processes (see Section II.B.7). Thus, for any given probe-sample-immersion configuration, the total interaction is obtained by simple linear superposition of V D W and ionic contributions. The comparison of Eqs. (11) and (91) shows that the VDW force generally exceeds the ionic force at small separations of the interacting surfaces, while, according to Eq. (90), the ionic force is dominant at large separations. If the VDW force is attractive, this results in a transition from repulsive to attractive interactions if the -5

I

1

-12

I

1

l

1

I

l

-

l

1

repulsive

van der Waals‘.,

1

l

I

I

2

j

i 4 Q

I

I

1

l

l

7 8 9 1 0

FIGURF 23. Interplay of ionic and VDW pressure as a function of separation between two planar surfaces interacting in pure water. Surface charging is assumed t o result from a monovalent ionization process. The long-dashed lines correspond t o the pure repulsive ionic J yields an attractive VDW interaction force. A typical nonretarded Hamaker constant of following the short-dashed straight line. The resulting total pressure is given by the solid lines which show a zero-axis crossing for the two lower charge densities.

106

U. HARTMANN

probe approaches the sample, as shown in Fig. 23. Even for highly charged surfaces, the VDW force causes deviations from the simple ionic double layer behavior up to surface separations of more than a nanometer. For low surface charge densities, both contributions may interplay throughout the whole regime that is interesting for SFM experiments. If the VDW force is attractive, the total pressure generally changes from repulsion to attraction below 10 nm separation of the surfaces. If two slabs are finally forced into molecular contact, the pressure pushing the trapped counterions toward the surfaces dramatically increases according to Eq. (91). The high ionic pressure may initiate “charge regulation processes,” e.g., readsorption of counterions onto original surface sites. As a result the surface charge density exhibits a reduction with decreasing distance between the slabs. The ionic force thus falls below the value predicted by Eq. (91). However, charge regulation is expected to be of little importance in noncontact SFM, since probe-sample separations are generally well above the molecular diameter. Moreover, for a sharp tip close to a flat substrate, charge regulation would be restricted to the tip’s very apex, while the major part of the interaction comes about from longer-range contributions. Thus, Eq. (89) should be a good basis to calculate the actual ionic probe-sample interaction via the framework developed in Section II.B.4. 2. The Efect of an Electrolyte Solution

The treatment in Section II.C.1 was based on the assumption that the immersion medium is a pure liquid, i.e., that it only contains a certain counterion concentration just compensating the total surface charge of probe and sample. This assumption is generally not strictly valid for SFM systems involving wetting films on probe and sample or liquid immersion: Pure water at pH 7 contains M (1 M = 1 mol/dm3 corresponds to a number density of 6 x 1026/m3)of H 3 0 f and OH- ions. Many biological samples exhibit ion concentrations about 0.2 M resulting from dissociated inorganic salts. A bulk reservoir of electrolyte ions has a profound effect on the ionic probe-sample interaction. For an isolated surface, covered with a charge density 0 and immersed in a monovalent electrolyte solution of bulk concentration Pb, the surface electrostatic potential is given by $ o ( ~ pb) , = -iVo arsinh

D

J8ebPb ’

which is a convenient form of the Grahame relation (see, for example, Hiemenz, 1977). The imaginary potential difference Vo is defined in Eq.

FUNDAMENTALS OF NON-CONTACT FORCE MICROSCOPY

I07

n

-0

al

1

n

0

>

v

W

v)

-0

E

n

E C

v

0 4

FIGURE24. Debye length and surface potential of an isolated charged surface as a function of bulk electrolytic concentration.

(88), and

denotes the Debye length. The dependence of qb0 and AD on the bulk electrolytic concentration is shown in Fig. 24. If X,,U/EE << -iV0/2, then Eq. (92) yields (94) which represents a parallel-plate capacitor, the plates of which exhibit a surface charge density u and are separated by AD. A glance at Fig. 24 shows that this proportionality between surface potential and charge density holds up to surface potentials of about -iV0/2. The Debye length characterizes the separation of the effective centric plane of the counterion profile from the charged surface, as shown in Fig. 25. According to the Gouy-Chapman approach (see, for example, Hiemenz, 1977), the electrostatic potential at any separation z from the isolated surface is given by $O(a, Pb) = ITAD(Pb)/EEOj

$(z)

=

-i2 Voartanh [exp (-./AD)

tanh (iqbO/2V O ) ] ,

(953)

which reduces to

$(z)

=

-i2V0 tanh ( i y ! ~ ~ / 2exp V ~ (-./AD) )

(933)

108

U. HARTMANN

FIGURE 25. Diffuse counterion atmosphere near probe and substrate which both exhibit a surface charge density 6.The intervening gap of width d contains an electrolytic solution with a static dielectric constant t . The Debye length AD characterizes the separation of the centric ,I and @,, planes of the counterion clouds from the surfaces of probe and sample, respectively. $ denote the surface and midplane potentials.

for z >> AD and/or do << -i2V0. Especially for low surface potentials, Q0 << -iVo, this latter relation may be represented by the Debye-Hiickel approximation $ ( z ) = $Oexp (-z/AD).

(95c) The ionic pressure between two equally charged surfaces may now be calculated according to Eq. (84). However, it is more convenient to use the contact value theorem in the alternative form (see, for example, Israelachvili, 1985)

f(.)= kT[prn(z)- P b l i

(96) which is the excess osmotic pressure of the ions in the midplane over the bulk pressure. Since the bulk ionic concentration pb is known, the problem is reduced to the calculation of the midplane ionic concentration pm, which is related to the midplane potential $,(z) by the Boltzmann ansatz. This leads to f ( z ) = 4kTpbsinh2[i$,,(z)/VO].

(97a)

In the “weak overlap approximation,” &,(z) is found by a linear superposition of the potentials of the isolated surfaces produced at 212, i.e., $m

(z)= 2 $ ( ~ / 2 )

(97b)

where $(z/2) is given by Eq. (95). The resulting ionic pressure according to the full weak overlap approximation, i.e., Eqs. (97) combined with (95a), is shown in Fig. 26 for various surface charge densities and two electrolyte concentrations. At any surface charge density, more dilute electrolytes with long Debye screening lengths (see Eq. (93)) lead to a stronger repulsion

F U N D A M E N T A L S OF N O N - C O N T A C T FORCE MICROSCOPY

109

F I G U R E 26. Repulsive ionic pressure between two slabs with equal surface charge density n in an aqueous monovalent electrolyte of bulk concentration p a s a function of surface-to-surface separation.

between the slabs than concentrated electrolytes. The difference in magnitude and decay of the electrolytic ionic pressure with respect to the pure liquid results shown in Fig. 23 is striking. While dynamic-mode SFM essentially detects the pressure according to Eq. (97), the actual ionic force exerted on a probe of a given radius R, according to Eq. (32), has to be obtained via the Derjaguin formulae according to Eqs. (34) and (35). Calculations considerably simplify if a small midplane potential can be assumed, i.e., q ! << ~ ~-EVo. This latter condition is satisfied if d > 2xD and/or $yJ << -iVo, where d is the probe-sample separation and the potential of an isolated surface which is assumed to approximately represent the real surface potential (see Fig. 25). Expansion of the sinh term in Eq. (97a) and insertion of Eq. (95b) via (97b) yields the particularly simple result

F ( d ) = 1287rR(e 2 P 2~ 3X ~ / C tanh2(i$,,/2V0) E ~ ) exp (-d/XD).

(98)

Especially if one has low Surface potentials, $ 5 - iV0/2, application of Eq. (94) together with the Debye- Hiickel approximation, Eq. (95c), yields via second-order expansion of Eq. (97a)

These results show that the ionic double layer forces in an electrolytic environment drop exponentially with probe-sample separation. The decay

110

U . HARTMANN

’\,0.5

-0.7 ‘,O.d\\ \

\

\

\

-0.8 , \ \ \ \ \ \ 0.05\ ,

’

\

’\

\

\

\

\

--

u=0.01C/mj u=O.O5C/m‘

‘, \

\

\

FIGURE 27. Total probe-sample force as a function of working distance if the force microscope is operated under electrolytic immersion. The curves correspond to two different surface charge densities 0 and to various bulk electrolytic concentrations p.

length is given by the Debye screening length, which only depends on the bulk electrolytic concentration. The behavior is in strong contrast to ionic forces in non-electrolytic immersions exhibiting a logarithmic to 1/ d force law as discussed in the previous section. However, as for pure liquids, the total probe-sample force also has to include the VDW component. The interference of ionic and VDW forces is well known from the classical Derjaguin-Landau-Verwey-Overbeck (DLVO) theory of lyophobic colloid stability (see, for example, Hiemenz, 1977). Figure 27 shows representative curves of the total probe-sample interaction that may occur if SFM is performed under electrolytic immersion. The results, shown for two different surface charges and various bulk electrolytic concentrations, may be generalized as follows: For highly charged surfaces and dilute electrolytes, there is strong repulsion more or less throughout the whole regime relevant to non-contact SFM. For electrolytes of higher concentration, the total probe-sample interaction exhibits a minimum (attractive), preferably at probe-sample separations of a few nanometers. If surface charging is relatively low, the force exhibits a broad maximum (repulsive) some nanometers away from the substrate surface, and approaches the pure VDW curve for increasing electrolyte concentration. The preceding analysis has important bearings on VDW and magnetic force microscopy performed under liquid immersion. Unwanted ionic forces due to surface dissociation processes of probe and sample can largely be

FUNDAMENTALS OF NON-CONTACT FORCE MICROSCOPY

111

suppressed by adding an appropriate amount of inorganic salts to the immersion fluid. For magnetic measurements performed at ultralow working distances, the total nonmagnetic force may be reduced by approximately compensating repulsive ionic and attractive VDW forces for a given average probe-sampling spacing. Equation (99) can immediately be extended to the situation of electrolytic immersions containing divalent ions or a mixture of ions of arbitrary valency nj and bulk particle densities p b j . This only requires the use of a generalized Debye screening length (see, for example, Israelachvili, 1985), now given by

However, it must be emphasized that the weak overlap approximation used to derive the probe-sample ionic interaction is implicitly based on the assumption that the working distance is well beyond AD. For smaller separations one must resort to numerical solutions of the Poisson-Boltzmann equation (see, for example, Hiemenz, 1977). For example, pure water at pH 7 exhibits a room temperature value of AD M 950nm, while AD N 0.8 nm is found for ocean water, and AD M 0.7 nm for many biological samples (see Israelachvili, 1985). However, for almost all SFM applications it is satisfactory to use the simple analytical results obtained in Section II.C.1 if AD k 1 nm, while the results just given are preferred for smaller Debye lengths. Finally, it may be instructive to address at least one of the open problems in the field of collective VDW-ionic interaction in SFM. The preceding treatment was consequently based on the assumption that both contributions can be evaluated separately, where the total force exerted on the probe is then given by a linear superposition. However, fundamental statistical mechanics does not provide any firm basis for this treatment (Mahanty and Ninham, 1976). A rigorous ab initio ansatz would involve a Poisson-Boltzmann equation,

V2($

+ 4) =

-

(101a)

which contains the sum of the equilibrium ionic potential $ and the fluctuating VDW potential 4. Linearization yields

For a sharp probe in close proximity to a substrate, $ may be a complicated function of position and may show deviations from the simple behavior

112

U. HARTMANN

assumed in the preceding treatment; Eq. (101b) in general is extremely difficult to solve in a self-consistent way. However, it confirms that, at least for high ionic mobility, VDW and ionic forces are not completely independent. Detailed SFM experiments on ionic forces may help in future to further clarify this point. If the detailed nature of the interaction is understood, SFM of ionic forces would be particularly valuable to measure surface charge densities at high spatial resolution. This may also include externally superimposed electrostatic potential differences between probe and sample. D . Squeezing of Individual Molecules: Solvation Forces

The theories of VDW and ionic probe-sample interactions discussed so far are pure continuum theories in which immersion liquids present in the intervening gap between probe and sample are solely treated in terms of bulk properties, such as dielectric permittivity and average ionic concentration. This treatment breaks down when the probe-sample separation is decreased to some molecular diameters. In this regime the discrete molecular nature of immersion media can no longer be ignored, since the effective intermolecular pair potentials in the liquids become a sensitive, anisotropic function of the distance between probe and sample. This phenomenon may cause quite long-range ordering effects of the liquid molecules (Nicholson and Personage, 1982; Rickayzen and Richmond, 1985), as shown in Fig. 28. Attractive interaction between the trapped molecules and the surfaces of probe and sample together with the geometric constraining effect give rise to density oscillations which may extend over several molecular diameters. Forces related to these ordering phenomena are known as “solvation forces.” An excess near-surface molecular density is, according to the contact value theorem, Eq. (84),related to a repulsive pressure between two slabs in close proximity. Modeling of probe-sample solvation forces thus consists

FIGURE 28. Squeezing of individual liquid molecules between probe apex and substrate leads to long-range ordering phenomena. The resulting molecular density oscillations exhibit a periodicity roughly equal to the molecular diameter 6.

FUNDAMENTALS OF NON-CONTACT FORCE MICROSCOPY

113

in calculating the excess surface molecular density for the two slabs separated by a certain distance with respect to that of a free surface. Over the past years there has been much study of the liquid structure near constraining walls (Chui, 1991). Different theoretical approaches to the problem include linear theories, nonlinear density-functional theories, and Monte Carlo simulations. However, the somewhat controversial results indicate that the field is yet not fully exploited. Thus, the present treatment is devoted to a derivation of a representative order of magnitude of the effects and to an analysis of the physics behind solvation interactions in SFM. As intuitively expected from the simple-minded model in Fig. 28, all theoretical work (Nicholson and Personage, 1982; Rickayzen and Richmond, 1985; Chui, 1991) as well as some experimental observations (Israelachvili, 1985) have invariably confirmed an oscillating near-surface excess molecular density which may roughly be modeled by P(Z)

- p ( m ) = pocos

(2xz/6)exp ( - z l d ) ,

(102)

where z denotes the separation of the two slabs and 6 the effective molecular diameter of the intervening fluid. The empirical ansatz describes an exponentially damped oscillatory variation of the surface excess molecular density. po determines the excess density if the gap between the slabs just equals one molecular diameter. The solvation force acting on a typical SFM probe is then given, according to Eqs. (34b) and (84), by integrating Eq. (102): F ( d ) = F(S)[cos(27rd/6) - 27rsin (27rd/6)] exp (1

-

d/6),

(1 03a)

where one approximately has

F ( 6 ) = k7'po6R/(2~exp( I ) ) ,

(l03b)

with an effective probe radius according to Eqs. (32). The problem of estimating a somewhat realistic order of magnitude of the force is now reduced to an estimation of po, i.e., of the molecular density if the probesample spacing is just one molecular diameter 6. This problem is of course hard to solve in general, since po is expected to be sensitive to the geometry of the opposing surfaces (Chui, 1991). However, a rough estimate may be obtained by considering an upper limit of the total order-disorder difference of an ideal hard sphere liquid. In the total ordering limit, i.e., solidification of the hard sphere molecules between probe and sample in a close-packed lattice, the maximum number density would be p ( S ) = d / S 3 . If it is further assumed that the excess near-surface molecular density of a free surface is almost negligible with respect to this value, i.e., p(00) << p ( 6 ) , Eq. (102)

114

U. HARTMANN

yields po = a e x p ( l)/S3. Thus, one obtains from Eq. (103b) F ( 6 ) / R = (l/d%)kT/6*.

(104)

This particularly simple relationship represents an upper limit of the solvation force per unit probe radius measured at a probe-sample separation of one molecular diameter for an ideal hard sphere VDW immersion liquid. For 6 = 1 nm, one obtains a value of about 1 mN/m. This is 10 times smaller than the typical VDW magnitude mentioned in Section 1I.B. 10. The oscillating solvation force according to Eq. (103a) is shown in Fig. 29 in comparison with a small attractive VDW interaction. While the empirical two-slab pressure according to Eq. (102) exhibits a maximum when the gap width d corresponds to multiples of the molecular diameter 6, the force measured with a paraboloidal or ellipsoidal probe exhibits, according to Eq. (103a), a shift of the molecular peaks by about 25% of the molecular diameter toward lower gap values. The amplitude of the force oscillations increases with the square of the reciprocal of the molecular diameter, while the latter also determines the characteristic decay length with increasing probe-sample separation. The total probe-sample interaction at molecular probe-sample separations is of course composed of both solvation and VDW interactions. The solvation forces result from density fluctuations of the molecules trapped between

/

/

/

/

/

H ’,

=

J

_-

6= 1 .Onm

0 FIGURE 29. Oscillatory solvation force per unit probe radius as a function of probe-sample separation for two different molecular diameters. A weak attractive VDW force is shown for reference.

FUNDAMENTALS OF NON-CONTACT FORCE MICROSCOPY

115

probe and substrate which are due to long-range molecular ordering processes. The VDW force between probe and sample is sensitive to the dielectric permittivity of the intervening gap, which in turn depends on the actual molecular density of the immersion fluid. Thus, it is clear that solvation and VDW forces cannot be treated separately to obtain the total interaction by linear superposition. Both components act collectively rather than simply additively. Since the VDW theory developed in Section 1I.B is a pure continuum theory, it is convenient to treat the molecular ordering processes in a quasi-macroscopic way. This can be done via the ClausiusMossotti equation, t o [ c ( i v ) - I]/[c(iv)

+ 2]p,

(105) which relates the effective molecular polarizability of hard sphere molecules in the gas phase to the dielectric permittivity and average molecular density which would be measured for a macroscopic ensemble of the molecules. Since the effective polarizability of non-interacting hard sphere molecules is invariant to density fluctuations, a certain average molecular density p , deviating from the bulk liquid density p b , transforms into a modification of the dielectric permittivity through C Y ( ~ Y= )

where c b = f ( P b ) corresponds to the permittivity of the bulk liquid. Under the assumed boundary conditions, Eq. (106) holds for any spectral contribution, i.e., for static orient as well as for higher-frequency electronic permittivities. The consequence is, that, according to Eqs. (8b) and (12b), the entropic and nonretarded Hamaker constants exhibit a density-induced modulation. If one assumes that the average molecular density p(d) within the volume between probe and sample is approximately equal to the surface molecular density p ( d ) , Eq. (102) yields ( 107a) p(d)/Pb = p(O0)/pb[l + cos (2Td/6)exp (17 - d/6)1, with (1 07b) 17 = 1 + In [ P ( 6 ) / P ( W ) - 11. At this point some heuristic assumptions have to be made concerning the ratio of the free surface density p ( m ) to the bulk density of the liquid pb, as well as the ratio of the gap's excess surface density p ( 6 ) to p ( m ) . A reasonable assumption is that the permittivity of the small gap between probe and sample approaches its vacuum value somewhere between d = 36/4 and d = 612, when there is no space left to trap any liquid molecules (see Fig. 28). According to Eq. (106), c = 1 requires' p = 0, and

116

U. HARTMANN

i,

thus, according to Eq. (107a), > 2 which in turn yields for the gap-induced increase of molecular ordering 2.6 > p ( 6 ) / p ( c o ) 2 1.6. In other words, the packing fraction of molecules on probe and sample surface increases by a factor of 1.6 to 2.6 when the probe approaches the sample surface and finally reaches a separation corresponding to only one molecular diameter. The second free parameter left in Eq. (107a) is the effective excess molecular packing fraction p ( o o ) / p b , which is simply not known for a system consisting of a sharp tip opposite to an arbitrarily shaped sample surface. Information on this quantity can only be obtained by performing Monte Carlo simulations under realistic boundary conditions. However, first results obtained for the structure of hard spheres near flat or spherical walls (Chui, 1991) suggest that the packing fraction is a complicated function of molecular diameter and constraining wall geometry and may by far exceed unity. Because of these uncertainties, it is convenient to choose a somewhat pragmatic way. Equation (107a) is strictly valid only for probe-sample separations of a few molecular diameters since it was assumed p ( d ) = p ( d ) . However, to ensure bulk convergence for large probe-sample separation, one has to fulfill p(m)= P b , This in turn formally requires p(oo) = P b . This pragmatic approach permits at least an order of magnitude estimate of oscillatory VDW forces without two many ambiguous parameters. A typical result for a metal-dielectric combination of probe and sample immersed in an ideal hard sphere liquid with p ( 6 ) / 6 ( 0 0 ) = 2.1 (which can be considered as a somewhat typical value according to the preceding analysis) is shown in Fig. 30. The oscillating Hamaker constant has been obtained according to and e 3 [ p ( d ) ]according to Eqs. (106) and Eq. (54) with n 3 ( d )= (107). The oscillating refractive index n3 of the immersion liquid transforms into a huge “overshoot” of the nonretarded Hamaker constant with respect to its bulk value. If the bulk index of the immersion fluid is close to that of the dielectric (sample), the originally purely attractive interaction may become repulsive for certain probe-sample separations, while it is solely attractive but oscillating if probe and sample are made from the same material. Refractive index and Hamaker constant both exhibit the exponential damping ultimately resulting from the decrease of the molecular excess osmotic pressure according to Eq. (102). They are completely out of phase, but both show the molecular periodicity.

m,

’

The upper limit Is additionally constrained by the fact that c ( 6 ) must of course be finite. Convergence of Eq. (106) requires p(6) < pb(tb 2 ) / ( f h - I). However, this criterion only becomes relevant if the excess surface density for the gap between probe and sample is almost the same as for the free surfaces, and if this free surface molecular density is much higher than the bulk liquid density. For p(00) = P b , used in the following, p ( 6 ) / p ( m ) < 2.6 can be considered as the relevant criterion for all immersion liquids (with c b < 2.9).

+

FUNDAMENTALS OF NON-CONTACT FORCE MICROSCOPY

117

d (nm) FICXJRE 30. Periodic molecular ordering of molecules trapped between probe and sample causes oscillations of the effective optical refractive index n3 of the immersion fluid, the bulk index of which is given by n j m . These oscillations transform into a huge periodic variation of the nonretarded Hamaker constant H , with respect to its bulk value Hnm.6 denotes the effective molecular diameter of the immersion medium and nz the refractive index of the dielectric sample. The probing tip is metallic.

The total VDW solvation force exerted on the probe is now obtained by a linear superposition of the osmotic contribution according to Eq. (103a) and the VDW contribution according to Eq. (40a) using a densitymodulated Hamaker constant according to Eqs. (6), (12b), (106), and (107). A typical result, again for a metal-dielectric combination of probe and sample, is shown in Fig. 31. The total interaction still shows the molecular periodicity 6. However, since osmotic and VDW contributions are mutually phase-shifted in a complicated way, the oscillating curve does generally not peak when the probe-sample separation exactly equals a multiple of half the molecular diameter. The damping at small probesample distances is stronger than that of the excess osmotic pressure in Eq. ( 1 02) and approaches the latter a few molecular diameters away from the sample surface. At very small probe-sample separations, i.e., just before interatomic repulsion occurs, the total interaction approaches the VDW continuum expected for a vacuum interaction between probe and sample. However, if probe and substrate are separated by more than about one molecular diameter, the giant oscillations of the VDW solvation force exceed by far the continuum of VDW forces.

1 I8

U. HARTMANN

-a

I

0.1

I

0.5

" , ' I

1 .o

I

5.0

d (nm> FIGURE 31. Force per unit probe radius as a function of probe-sample separation for a metal/dielectric (optical refractive index n2) configuration of probe and sample, immersed in a hard-sphere liquid with an effective molecular diameter 6 and a bulk optical refractive index n3 m. Superposition of the oscillatory VDW and osmotic contributions yields the total force exerted on the probing tip. For reference, the VDW curve resulting from the pure continuum theory is also shown.

Finally, it should be emphasized that the field of solvation force phenomena in SFM is completely open and, to the author's knowledge, no detailed observation of an oscillating attractive/repulsive interaction at molecular working distances has ever been reported up to the present time. However, the present theoretical analysis confirms that, at least for some model configurations, oscillatory solvation forces should be detectable. Quite promising inert immersion liquids, which contain fairly rigid spherical or quasi-spherical molecules, are, for example, octamethylcyclotetrasiloxane (OMCTS, nonpolar, 6 M 0.9 nm), carbon tetrachloride (nonpolar, 6 M 0.28 nm), cyclohexane (nonpolar, 6 M 0.29 nm, and propylenecarbonate (highly polar hydrogen-bonding, 6 M 0.5 nm) (Israelachvili, 1985). SFM measurements on these and other immersion liquids could help provide a deeper insight into molecular ordering processes near surfaces and in small cavities. As already emphasized with respect to VDW and ionic interactions, solvation forces certainly have to be accounted for as unwanted contributions, if electric or magnetic force microscopy is performed at ultralow working distances and under liquid immersion. In general, the situation is complicated by the fact that VDW, ionic, and solvation forces may contribute to the total probesample interaction in a non-additive way. Unfortunately, this is only part of

FUNDAMENTALS OF NON-CONTACT FORCE MICROSCOPY

119

the whole story. If SFM experiments are performed under aqueous immersion, or if only trace amounts of water are present - and this is the case for almost all experiments under ambient conditions - hydrophilic and hydrophobic interactions must often additionally be taken into account (Israelachvili, 1985). The phenomena are mainly of entropic origin and result from the rearrangement of water molecules if probe and sample come into close contact. In this sense hydrophilic and hydrophobic forces clearly belong to the general field of solvation forces; however, macroscopic experiments (Israelachvili, 1985) confirm that they are generally not well characterized by the simple theory presented here. Hydration forces result whenever water molecules strongly bind to hydrophilic surface groups of probe and sample. A strong repulsion results, which exhibits an exponential decay over a few molecular diameters (Israelachvili, 1985). In the opposite situation, for hydrophobic probe and sample, the rearrangement of water molecules in the overlapping solvation zones results in a strong attractive interaction. These phenomena once again show that water is one of the most complicated liquids that we know. However, its importance in SFM experiments under ambient conditions must not be emphasized, and more detailed information on its microscopic behavior is of great importance.

E. Capillary Forces Under humid conditions, a liquid bridge between probe and sample can be formed in two different ways: by spontaneous capillary condensation of

FIGURE32. Capillary interaction between the probe and a substrate which has a surface covered with a liquid adsorbate. When the probe is dipped into the adsorbate the liquid surface exhibits curvature near the probe’s surface (left side). Withdrawal of the probe or spontaneous capillary condensation before the probe contacts the liquid surface results in an elongated liquid bridge (right side).

120

U. HARTMANN

vapours, and by direct dipping of the tip into a wetting film which is present on top of the substrate surface. Capillary condensation is a first-order phase transition whereby the undersaturated vapour condenses in the small cavity between probe apex and sample surface. Because of surface tension, a liquid bridge between probe and sample results in a mutual attraction. At thermodynamic equilibrium, the meniscus radii according to Fig. 32 are related to the relative vapour pressure by the well-known Kelvin equation (see, for example, Adamson, 1976),

where C denotes the universal gas constant and p, M , y are the mass density, the molar mass, and the specific surface free energy or surface tension of the liquid forming the capillary. Since p < p s , the Kelvin mean radius, l r K l = r 1 r 2 / ( r 1 r 2 ) , for a concave meniscus as in Fig. 32 is negative. Figure 33 shows the equilibrium Kelvin radius for a water capillary between probe and sample as a function of relative humidity of the experimental environment. For r K -+ -00, i.e., for a relative humidity approaching loo%, the swelling capillary degenerates to a wetting film. In the opposite extreme, at a relative humidity of a few percent, no capillary is formed, or a preexisting capillary evaporates, since the Kelvin radius approaches molecular dimensions.

+

FIGURE 33. Equilibrium dimension of the Kelvin radius for a water capillary between probe and sample.

FUNDAMENTALS OF NON-CONTACT FORCE MICROSCOPY

121

The mutual attractioil of probe and sample results from the Laplace pressure,

( 109a) within the liquid bridge. The total capillary force exerted on the probe is thus given by

F(4

= .X2(4r/rk,

(109b)

where, as shown in Fig. 32, x is the radius of the area where the meniscus is in contact with the probe’s surface. The problem is thus to determine this radius as a function of the probe-substrate separation, since the Kelvin radius is known at thermodynamic equilibrium from Eq. (108). One first considers the situation sketched in the left part of Fig. 32, i.e., the probesubstrate separation d is less than or equal to the adsorbate thickness t . For simplicity, an ideally wetting liquid with vanishing contact angle at the probe is considered. From Eqs. (31) one obtains the relation x 2 = 2Rz, where the effective probe radius is determined by Eq. (32). From geometrical considerations one then immediately obtains

z

M

t

-

d + rI[l

+ R / ( R+ rl)],

(1 10)

which is valid for thin adsorbate films with t << R . Since r l << r2, one has a good approximation r l M -rk. The force according to Eq. (109b) is thus given by F ( d ) = - 2 ~ R y [ l+ R / ( R - rk) - ( t - d ) / r k ] .

(111) Force-versus-distance curves according to this relation contain complete information about an adsorbate layer. At low partial vapour pressure leading to -rk << R, the force measured for a virgin probe-adsorbate contact, FIR = - 4 ~ 7 , permits a measurement of the adsorbate’s surface tension. The Kelvin radius is directly obtained from the slope dF/?ld = - 2 n y R / r k . Finally, the adsorbate thickness may be obtained by a simple dipping experiment, whereby F ( d ) is detected for 0 5 d 5 f. The maximum capillary force is obtained just before the tip touches the substrate: F ( 0 ) = -27ry(2rk - t ) R / r K for -rk << R and F ( 0 ) = 2.rrytR/rk for - r h << t . For a water film the specific surface free energy is 73mJ/m2 (see, for example, Israelachvili, 1985). The capillary force acting on a probe which dips into a water film on top of the sample is thus IFI/R > 0.9 N/m, which is about 90 times the typical VDW magnitude mentioned in Section 1I.B. 10. When the probe is withdrawn after it has contact with the liquid adsorbate, an elongated capillary is formed as shown in the right part of Fig. 32. Since for d 2 t both meniscus radii r I and r2 now vary over a considerable

122

U . HARTMANN

range, the calculation of the probe-sample capillary force is slightly more complicated than in the previous situation. Simple geometrical arguments lead to

x = R(r2

+ r , ) / ( R+ r l )

(1 12a)

and

x = RJ1

-

[R

+d -

+

( 1 12b)

t -r,]/[R rl],

where, at thermodynamic equilibrium, rl and r2 are additionally related to each other by Eq. (108). After a little algebra, the radius of the probecapillary contact area is determined by the solution of the following cubic equation:

x 3 - (d - t ) x 2 + 2R(2rk + d - t)x - R ( d - t ) 2 = 0,

(113)

which is valid for d - t << R . The result can of course be obtained analytically, but is then a little bit unwieldy. For d - t >> -rk, rl M ( d - t)/2 and r2 M -rk << R lead, according to Eqs. (1 12a) and (109b), to the asymptotic force

+

F ( d ) = r y R 2 ( d - Z)2/rk(2R d - t ) 2 ,

(1 14)

from which an upper limit of F ( m ) = r y R 2 / r k can be deduced for the capillary force. For example, this upper limit amounts for a lOOnm

- - - - _ _----_ _

-250

I

0

!

lb

I

20

I

30

1

40

I

50

I

60

I

70

--

90

x

50 X

1

80

I

90

1 10

FIGURE34. Capillary force as a function of probe-substrate separation. The substrate is covered with a 5 nm water film of fixed thickness, while the probe-sample interaction is shown for three different values of the ambient relative humidity.

FUNDAMENTALS OF NON-CONTACT FORCE MICROSCOPY

123

probe, interacting with the substrate via a water meniscus to 1.5pN at an ambient humidity of 50-60%. This order of magnitude shows that capillary forces in SFM are generally much stronger than any aforementioned interaction. Figure 34 shows some force-versus-distance curves obtained according to Eqs. (1 1 I), (109b), and (1 13). When the approaching probe first touches the water film of 5nm thickness, it experiences a sudden attractive force of magnitude 47ryR, which is, for a 100nm probe, about 90nN. The probe then penetrates the adsorbate layer and exhibits a linear increase in attractive force according to Eq. ( 1 11). The maximum value achieved just before touching the substrate depends on the ambient humidity, where the highest values are obtained in a relatively dry atmosphere. If the probe is withdrawn before making contact with the substrate, the attractive force decreases reversibly until the adsorbate-air interface is reached. From then on an elongated capillary is formed, leading to a pronounced hystereris effect. The curves are now described by Eqs. (109b) and (113). Upon further withdrawal of the probe, the force first exhibits a further decrease until some minimum value close to zero is reached. From this point it increases again, approaching the asymptotic behavior according to Eq. (1 14) (not shown). If thermodynamic equilibrium conditions would be present throughout the complete measurement, the capillary between probe and sample would assume an arbitrary length, while the smaller meniscus radius r2 becomes equal to the Kelvin radius - r K , i.e., the circumference of the meniscus becomes stable. However, since the adsorbate film has a finite thickness, material transport into the growing capillary is disrupted at some time, leading to an irreversibility whereby the force suddenly vanishes. The same occurs if the capillary is destroyed by external perturbations (vibration, air currents). These results are only precise for nearly spherical probes and vanishing contact angle. However, the analysis can immediately be extended to paraboloidal or ellipsoidal probes, even of low aspect ratios R,/R, (see Eq. (32)) and to arbitrary contact angles. The main results predicted by the preceding treatment remain unchanged. A particularly interesting feature is related to Eq. (113). A careful analytical examination shows that, for a certain regime of the substrate-sample separation, the cubic equation involves three real roots, all leading to stable solutions for the force according to Eq. (109b). This implies the possibility of discontinuous transitions between different force curves upon variation of the probe-sample separation. It should further be noted that the Kelvin equation (108) as well as the Laplace relation ( 109a) are strictly macroscopic equations, i.e., to ensure validity, the system has to be in thermodynamic equilibrium and the Kelvin radius must well exceed the molecular diameter. For very small Kelvin radii,

124

U . HARTMANN

the liquid's surface tension is no longer a constant. While for simple Lennard-Jones liquids such as cyclohexane or benzene a macroscopic behavior is already manifest at molecular Kelvin radii, for water a value of about 5 nm is assumed (Israelachvili, 1985), which corresponds, according to Fig. 33, to a relative humidity of 80%. The other important question is whether an ideal thermodynamic equilibrium can be assumed for an arbitrary adsorbate. In the extreme situation, where an adsorbate exhibits a nearly vanishing evaporation rate since it forms a stable film on top of the substrate, the simple free-liquid equilibrium conditions considered earlier are no longer valid. In this case, the Kelvin equation (log), which controls the interplay of the meniscus radii r1 and r2, has to be replaced by a relation representing the condition of zero material transport into the capillary. According to Fig. 32 the meniscus volume for d 2 t is given by I

"

(115a) with

zl=R+d-t-rlz2 = rl

"-}.

R2-x

(115b)

Zero material transport is then ensured by the constraint a V /a d = 0. This latter condition then relates r 2 ( d ) to r l ( d ) . The additional use of Eqs. ( 1 12) then leads to a first-order nonlinear differential equation for the meniscus-probe contact radius x . The numerical solution permits, together with Eq. (109b), a calculation of the capillary force for any probe-sample separation d. However, in the most interesting regime t 5 d << R , the differential equation can be considerably simplified which leads to an analytical solution for x . z 1 x z2 = rl and rl = ( x 2 2R[d - t ] ) / 4 R inserted into Eq. (1 15a) yields the simple equilibrium condition

+

(

2 [ d - t]

:)

+-

-

+ x = 0,

(116a)

with the solution x =

4Rri d - tNrK'

(116b)

FUNDAMENTALS OF NON-CONTACT FORCE MICROSCOPY

125

The capillary force according to Eq. (109b) is thus

F(d)=

TK

d-t-TK’

which matches the result of Eq. (1 11) for d = t. On the other hand, if d - t >> -rK, r l zz ( d - t ) / 2 - rK directly inserted into Eq. (1 12b) yields via Eq. (l09b) the asymptotic behavior

This latter relation clearly shows that the meniscus between probe and substrate may extend over probe-sample separations several times exceeding the adsorbate thickness, and even exceeding the probe radius. The capillary instability point can be estimated by considering the decrease of the meniscus radius r2 (see Fig. 32) with increasing probe-sample separation d. Using r l = ( d - t ) / 2 , the combination of Eqs. (112) yields r2 = 4 2 R 2 + R ( d - t ) - ( d - t)/2,

(119a)

where the capillary becomes unstable if r2 = -rK. This gives a critical probe-substrate separation of d=2(1+&)R+t~t+5.5R.

(119b)

Thus, an ideal, externally unperturbed capillary may extend over more than a hundred nanometers. The force obtained according to the constant-volume equilibrium via Eq. (117) exhibits a behavior completely different from that shown in Fig. 34. The result for a perfluoropolyether (PFPE) polymer liquid film adsorbed on a substrate is shown in Fig. 35. The surface tension and Kelvin radius values were taken from Mate et a/. (1989). Upon approach to the sample, a sudden attractive force is exerted on the probe, when it first touches the adsorbate film. The linear behavior upon dipping the probe into the adsorbate film is again described by Eq. ( 1 1 1 ) . Upon withdrawal, again a considerable hysteresis occurs, since an elongated liquid bridge is now formed between probe and adsorbate surface. This leads to a monotonic decrease of the force with increasing probe-sample separation - initially according to Eq. ( 1 17), and then, in the asymptotic regime, according to Eq. ( 118). The theoretical result shown in Fig. 35 is in good quantitative agreement with experimental results on PFPE polymer liquid films presented by Mate et al. (1989). The existence of long-range capillary forces has been demonstrated by several experimental results. Detailed measurements for water were presented by Wciscnhorn et al. (1989). However, a detection of the pure

126

U. HARTMANN 2

n

z 5

I

l

l

l

l

l

l

l

l

l

r

l

,

,

l

,

,

,

,

l

,

,

,

,

l

l

l

l

l

-2-

--._______-_--______-----------

-6-

:approach

-

-10-

-

-14-

-

FIGURE 35. Capillary force as a function of probe-substrate separation. The adsorbate thickness is assumed to be 30nm. The model calculation actually applies to an adsorbed perfluoropolyether polymer liquid film.

capillary forces appears to be difficult in some cases because the intermediate contact of the probe with the substrate yields additional adhesion forces which may considerably modify the curves shown in Figs. 34 and 35. Yet, not enough experimental data are available to strictly decide whether liquid adsorbates in general exhibit capillary forces according to Fig. 34 or according to Fig. 35, or if they in general exhibit a more complex intermediate behavior. However, especially for very thin films showing clear capillary formation at highly undersaturated vapour pressure, it is likely that Eqs. (1 17) and (1 18) are valid. It is interesting that the force according to these relations does not involve the multistabilities which occur if the free-liquid thermodynamic equilibrium equation ( 1 13) is used to derive the probemeniscus contact radius. In spite of these uncertainties, the preceding analysis has an important implication for force measurements in the presence of liquids: If the SFM probe is dipped into a liquid adsorbate of a new nanometers thickness, the force always exhibits the linear dependence on the probe-sample separation d given by Eq. (1 11). Since all forces dealt with before involve a nonlinear dependence on the probe-sample separation, they can be measured in the presence of capillary forces. A complete immersion of the whole SFM can thus be reduced to dipping only the tip into an immersion film of sufficient thickness, while ultimate corrections of the measured force-versus-distance curves are simply linear. Additionally, capillary forces may be used to lock

FUNDAMENTALS OF NON-CONTACT FORCE MICROSCOPY

127

in the non-contacting probe at a certain separation from the substrate surface, instead of using electrostatic servo forces. However, in general, much more experimental information on capillary phenomena is needed to further elucidate this complex but interesting area (see Evens et al., 1986). A more detailed examination should also include the effect of electrostatic fields on capillary equilibrium, since voltages between probe and substrate are used quite often in scanned probe microscopy. F. Patch Charge Forces

While the present work was in preparation, an additional force not included in Fig. 2 was shown to be highly relevant to explain some unusual effects in previous SFM data. Burnham et al. (1992) suggested that very long-ranged, mostly attractive but sometimes repulsive probe-sample forces may be attributed to work function anisotropies and their associated patch charges. The work function is very sensitive to perturbations at a material’s surface. Even if the surface is ideally clean and free of defects, different crystallographic orientations are associated with differences in the work function. If one takes out an electron or one region of a material with work function Q l and puts it back into another region with work function Q2 (a, # Q 2 ) , the energy - (P2) is not conserved. Energy conservation requires that the two regions are at different electrostatic potentials V1 and V2 such that V2 - V1 = - a2 (Ashcroft and Mermin, 1976). Thus, the surface charge density must vary across the sample. Burnham et ul. (1992) were the first to discuss the interaction of the resulting patch charges between two distinct bodies. As an exemplary electrostatic interaction, patch charge forces scale with respect to their range with the characteristic sizes of the patches involved. Some concrete examples were already provided by Harper (1 967). A rigorous calculation would necessarily include all of the patches on the tip, sample, and nearby instrumentation, since charge neutrality must be maintained: a hopelessly complicated task. Burnham et al. (1992) applied in a first ansatz a simple discrete-charge model using the method of images (see, for example, Jackson, 1975). They found the following important implications.

(i) The patch charge effect is not limited to conducting probes and samples. For insulators, however, the work function has to be adequately defined since no electrons are present at the Fermi level (Harper, 1967). (ii) Patch charge forces are generally much longer-ranged than VDW forces. The simple point-charge model yielded a linear dependence of the force on the probe-sample separation at separations that were

128

U. HARTMANN

small compared with the probe’s radius. A more rigorous treatment, however, will of course yield a certain dependence of the forceversus-distance curve on the probe geometry, as for the VDW interaction discussed in Section 1I.B. (iii) If the two patch charges on probe and sample which provide the dominant contribution to the interaction have the same sign, the force can be repulsive at large probe-sample separations. It is attractive at sufficiently small separations independently of being attractive or repulsive at large separations. A transition from repulsion to attraction has been experimentally observed and was shown to be in good agreement with the patch charge model (Burnham et al., 1992). Such a transition in the force-versus-distance curve under well-defined experimental conditions (clean surfaces, no intervening medium, no remanent discrete charges involved) can be considered as the most characteristic signature of patch charge forces, since the initial repulsion is hardly explainable by the exclusive occurrence of the previously discussed surface forces. (iv) Since the electrostatic is caused by local variations in the work function, patch charge forces should critically depend on the surface crystallographic perfection and on the amount of adsorbate covering. This is clearly a good point of application for experimental work in which patch charge forces could be varied in situ by suitable surface treatments. (v) The magnitude of the patch charge interaction is related to the dielectric constants in much the same way as the retarded Hamaker constants discussed in the context of VDW forces. However, because patch charge forces are determined by atomically thin surface layers, the involved dielectric constants are real surface quantities, while retarded Hamaker constants involve a finite electromagnetic penetration depth (see Section 1I.B). As Burnham et al. (1992) have emphasized, recognizing the existence of patch charge forces has some important consequences for SFM. First of all, it strongly suggests that previous interpretations of force curves are incomplete. Patch charge forces are very likely to significantly contribute to the “jump-to-contact’’ phenomenon (Landman et al., 1990). Furthermore, local work function variations should be detectable by force microscopy and thus by a method that is completely independent of the well-established techniques including STM. A significant advantage of this is that work function measurements by SFM are not restricted to conductors. In order to further clarify the role of patch charge forces in SFM, experiments have to be performed on samples with well-known local variations in

FUNDAMENTALS OF NON-CONTACT FORCE MICROSCOPY

129

work function. That should in particular yield more information on the typical range of patch charge interactions. A completely unknown aspect is the patch distribution on typical microtips (metallic or nonmetallic) as used for SFM. Improved equipotential calculations should yield information on the influence of probe geometry and of patch size and shape, as well as on fine structure details of force-versus-distance curves. 111. ELECTRIC FORCEMICROSCOPY USEDAS

A

SERVO TECHNIQUE

A . Fundamentals of Electrostatic Probe-Sample Interactions

To ensure proper operation of the force microscope’s feedback loop upon scanning in a mode of constant probe-sample interaction, the interaction must be a monotonically varying function of the probe-sample separation. In general, both surface forces as well as magnetostatic forces may locally be attractive, repulsive, or vanishing. Thus, an additional “servo force” is required to control the gap between probe and sample in a well-defined way. Scanning force microscopy (SFM) has proven capable of sensing Coulomb forces resulting from a charging of probe and sample at high spatial resolution (Stern et al., 1988). In principle, the sensitivity of SFM is high enough to detect single free electrons (Rugar and Hansma, 1990). It is thus convenient to use the electrostatic interaction between probe and sample for servo purposes. In order to model the electrostatic interaction between the SFM probe and a flat sample surface, the probe is often approximated by a sphere. However, this is of course a poor approximation for a sharp SFM probe. Force-versus-distance curves calculated according to this oversimplified model only permit an order-of-magnitude estimate of the electrostatic interaction. As already emphasized in the context of Section 11, approximation of the probe by a paraboloid of rotational symmetry is a first step towards a better characterization of probeesample interactions. The parabolic coordinates are given by (see, for example, Moon and Spencer, 1961) x = ptvcos 0,

( 120a)

y = pusin 0,

( 1 20b)

z

= (p2- u2)/2

( 120c)

The probe’s surface is then determined by v = vo, 0 5 p < 03, and 0 5 0 5 27r. The quantity uo (0 5 uo 5 03) determines the sharpness of the

130

U. HARTMANN

probe. Using this paraboloidal approximation, the problem of calculating the electrostatic interaction between probe and sample is reduced to a wellknown boundary value problem of classical potential theory. The solution of Laplace’s equation for the electrostatic potential may conveniently be expressed in terms of 3essel functions of order zero for p and L/ combined with the usual trigonometric functions for -9 (see, for example, Moon and Spencer, 196 1) . If the probe carries a homogeneous surface charge density, rather than exhibiting a constant electric surface potential with respect to the sample, the electrostatic potential is of the form

4 = (Y + @Inpl

(121)

where Q and @ are determined by the sharpness of the probe, by its surface charge density, and by the dielectric constant of the intervening medium between probe and sample. Once the potential has been determined, the corresponding electrostatic force between probe and sample is obtained by the standard methods of potential theory (Morse and Feshbach, 1953). At small probe-sample separations, z << R , surface curvature may become substantial. In this limiting regime, the Derjaguin approximation (Derjaguin, 1934), which was already discussed in Section 11.3.4, allows one to a certain degree to account for a smooth surface curvature. The approximation yields for the electrostatic force between probe and sample

F = - 7 r q V 2 (R , R , / [ R ,

+ R,])/d,

( 122)

where 6 is the static dielectric constant of the intervening medium between probe and sample and V is the applied voltage. R , and R , are the effective radii of curvature of probe and sample surface, respectively, and d is the probe-sample separation. If the sample is locally flat, the geometric factor in Eq. (122) reduces to the probe’s radius of curvature R , . If the probe consists of a polarizable material rather than of a conducting material, an electric field manifest between the electrodes of probe and sample causes the occurrence of free electric charges on the probe’s surface and in its interior. The resulting charge distribution may conveniently be modeled by approximating the probe’s apex region by a homogeneously polarized prolate spheroid, as shown in Fig. 36. The prolate spheroidal coordinates are given by (see, for example, Moon and Spencer, 1961) x = a sinh 7 sin -9 cos

c1

(123a)

y = a sinh q sin 6 sin El

(123b)

z = acoshqcos-9,

( 123c)

where a determines the focal position of the probe. The probe’s surface is

FUNDAMENTALS OF NON-CONTACT FORCE MICROSCOPY

131

FIGURE36. Model used for calculating the electrostatic interaction between a dielectric probe and a dielectric or metallic sample.

then given by q = qo (1 5 qo 5 oo), which determines the sharpness, and 0 5 0 5 7r, 0 5 5 27r. The problem of calculating the electrostatic potential for the probe-sample arrangement is thus again reduced to a standard problem of potential theory. The elementary solutions of Laplace’s equation are of the form (see Moon and Spencer, 1961)

<

q5 = Pn(coshq)Pn(cosO), Q,(coshq)P,(cosO),etc.,

(1 24)

where P, and Qn are Legendre functions of the first and second kind of order n. The complete solution for the potential is then obtained by a linear combination of the elementary solutions. As pointed out in Section TI, the use of dielectric probes is of particular importance with respect to the analysis of long-range surface forces. The Laplace solution for dielectric probes, Eq. (124), is directly transferable to magnetic microsensors used in magnetic force microscopy. B . Operational Conditions

In order to maintain a constant force F o r compliance F’ between probe and sample the microscope’s servo increases the probe-sample spacing over regions of strong attractive interaction and reduces the spacing over regions of weak attraction. The resulting contours of constant F o r F ’ , z = z ( x , y ) , are determined by where FS” is the surface force contribution, F i ) the electrostatic servo contribution, FA) the magnetic contribution, and FJ’) the constant force or compliance maintained by the servo system. For purposes of contrast modeling, Eq. (125) is solved by standard iterative methods, where the results presented in Section I1 may be employed to account for Fi’). Suitable results for F i ) are presented in Section IV. It has been demonstrated

132

U. HARTMANN

that the electrostatic contribution in F f ) in Eq. (125) may be used to separate topographic and magnetic information, when an additional sinusoidal voltage between probe and sample is superimposed (Schonenberger and Alvarado, 1990a). If the working distance is sufficiently large, the surface force contribution FJ’) in Eq. (125) may often be neglected. The magnetic response of the microscope may be linearized by choosing F f ) >> F$. Thus, the local variation of the probe-sample spacing is given by

Az(X, y ) = F$ (X, y ) / (aF$’ / a z )d, where d = z(F$)) is the average spacing. If magnetic forces are absent, Eq. (125) provides an elegant method of obtaining information about the effective probe radius and about the surface forces. If the decay rate of the relevant surface force is given by a C R , / d ” inverse power law, and if the probe-sample separation is sufficiently small so that eq. (122) can be applied for the electrostatic interaction, then Eq. (126) yields T ~ E O (V

f &)*/d+ C / d ” = Fo/R,,

( 1 27)

where Fo is the maintained force and R, the probe’s effective radius, while the sample is considered to be perfectly flat. Additionally to the applied voltage, the contact potential q$ between probe and sample is included. If the relevant surface force is the nonretarded VDW force, then C = Hn/6 and n = 2, where H , is the nonretarded Hamaker constant (see Section II.B.4). Equation (127) thus yields for a constant-compliance measurement V

+ & ) * / d 2+ H n / 3 d 3= F d / R , .

TTTETTE~(

(128)

If V is ramped from negative to positive values while maintaining Fd, the d( V ) curves approach straight lines for voltages I V I 2 1 V, since the VDW interaction can be ignored in this regime. The slope is given by d d / d V = ,,/mcoR,/F,’

(129)

and allows the determination of the effective probe radius R , . The asymptotes of d( V ) intersect at V = -I&, which allows a measurement of contact potentials. Finally, a determination of the minimum probe-sample separation, do = d( V = -$,.), allows via

H,

= 3Fddi/R,

( 130)

a measurement of the nonretarded Hamaker constant. For the retarded

FUNDAMENTALS OF NON-CONTACT FORCE MICROSCOPY

133

VDW interaction one obtains (see Section II.B.4)

H,

= Fdd;/2nRp.

(131)

First direct measurements of Hamaker constants according to this method have recently been presented by den Boef (1991), where some reasonable values for the constants were obtained. Complete VDW images of a sample may be obtained by recording d( V ) curves at any probe position. In this context it seems worthwhile to emphasize the relationship between lateral and vertical resolution inherent to all force microscopes (Hartmann et al., 1991):

F(’)(x+ AX,^

+ AY)- F ( ’ ) ( x , ~=) min ( F ( ’ ) ) .

(132)

Two points of the sample surface, the lateral positions of which are mutually shifted by Ax and Ay, are distinguishable if they produce a variation in force F or compliance F’ which is equal to the force sensitivity min ( F ) or compliance sensitivity min ( F ’ ) . For optimized instruments, min ( F ( ’ ) )is mainly limited by the thermal vibration of the cantilever (Sarid, 1991). The thermal vibration amplitude is determined by the equipartition theorem and is given by

~o((Ad)~= ) / k2T / 2 ,

(133)

where co is the spring constant of the free cantilever. For example, for a co = 1 N/m cantilever at room temperature, the rms thermal vibration amplitude amounts to 0.06 nm. Under typical operation conditions, thermal noise limits the detectable compliance to min ( F ’ ) M Njm.

IV. THEORY OF MAGNETIC FORCE MICROSCOPY A . Basics of Contrast Formation

If the probe and sample of a scanning force microscope (SFM) exhibit a magnetostatic coupling, the magnetic force microscope (MFM) is realized. The manifestation of magnetostatic interactions is obvious if a sharp ferromagnetic tip is brought into close proximity to the surface of a ferromagnetic sample. Raster-scanning of the tip across the surface then allows the detection of spatial variations of the probe-sample magnetic interaction. However, contrary to electromagnetic surface forces and to externally applied electrostatic tip-sample interactions, the long-range magnetostatic coupling is not directly determined by the mesoscopic probe geometry, but rather by the internal magnetic structure of the ferromagnetic probe. As

134

U. HARTMANN

shown in the following, this complicates matters extremely and requires a detailed discussion of contrast formation. A sharp ferromagnetic needle exhibits in a natural way a considerable magnetic shape anisotropy which forces the magnetization vector field near the probe's apex to predominantly align with the axis of symmetry of the probe. On the other hand, sufficiently far away from the apex region, where the probe's cross-sectional area is almost constant, the more or less complex natural domain structure is established. This domain structure depends on the detailed material properties represented by the exchange, magnetocrystalline anisotropy, and magnetostriction energies. Lattice defects, stresses, and the surface topology exhibit an additional influence on the domain structure (see, for example, Chikazumi, 1964). Because of this complicated situation, it is necessary to develop reasonable magnetic models to describe the experimentally observed features of magnetostatic probe-sample interaction as accurately as possible. Since it is generally hopelessly complicated to derive the actual magnetization vector field of a probe from first principles (Brown, 1963), it is reasonable to apply the model shown in Fig. 37. The unknown magnetization vector field near the probe's apex, with all its surface and volume charges, is modeled by a homogeneously magnetized prolate spheroid of suitable dimension, while the magnetic response of the probe outside this fictitious domain is completely neglected. The second assumption is that the dimensions and the magnitude of the homogeneous magnetization of the detector domain are both completely rigid, i.e., independent of external stray fields produced by the sample (Hartmann, 1988). In this way the micromagnetic problem is simplified to a magnetostatic problem. It is shown in what follows that this model allows a simulation of almost all experimental results obtained so far. Moreover, the concept of assuming a single prolate spheroidal domain which is magnetically effective

FIGURE 37. Effective-domain model used for contrast analysis in magnetic force microscopy.

FUNDAMENTALS OF NON-CONTACT FORCE MICROSCOPY

135

approaches reality for bulk ferromagnetic probes surprisingly well, as shown in Fig. 38. Using this “effective-domain model,” the problem is now to determine the probe’s magnetic properties and the probe-sample magnetostatic interaction for a variety of experimentally relevant cases. The magnetostatic potential created by any ferromagnetic sample is given by

where Ms(r’) is the sample magnetization vector field and s’ an outward normal vector from the sample surface. The first two-dimensional integral covers all surface charges created by magnetization components perpendicular to the bounding surface, while the latter three-dimensional integral contains the volume magnetic charges resulting from interior divergences of the magnetization vector field. The stray field is then given by

FIGURE 38. Transmission electron microscope image of an electrolytically prepared nickel tip taken in the Foucault mode of Lorentz microscopy (see, for example, Reimer, 1984). Details of probe fabrication and characterization were given by Lemke et al. (1989) and McVitie and Hartmann (1991). The dark pattern outside the tip reflects the stray field component oriented along the probe’s axis and allows a determination of the relevant apex domain dimensions. Note that apart from the apex domain, no sources of flux escape are observable. The irregularities on the probe’s surface are nonmagnetic contaminations probably resulting from the fabrication process.

136

U. HARTMANN

H,(r) = -v$$(r). The magnetostatic free energy of a microprobe exposed to this stray field is

where $ , ( T I ) is given by Eq. (134) and Mp(r’) is the magnetization vector field of the probe. The resulting force is then given by F(r) = -v$(r). This ansatz is rigorously valid for any probe involving an arbitrary magnetization field Mp(r).The first integral, taken over the complete surface of the probe, covers the interaction of the stray field with free surface charges, while the latter volume integral involves the probe’s dipole moment as well as possible volume divergences. According to the effective-domain model (see Fig. 37), Mp(r) is divergence-free, and the latter integral in Eq. (135) reduces to the dipole response exhibited by the probe. In many cases of contrast interpretation, it turns out that even further simplification of the probe’s magnetic behavior yields satisfactory results (Hartmann, 1989a, 1990b). The effective monopole and dipole moments of the probe, resulting from a multipole expansion of Eq. (135), are projected into a fictitious probe of infinitesimal size which is located an appropriate distance away from the sample surface. The a priori unknown magnetic moments as well as the effective probe-sample separation are treated as free parameters to be fitted to the experimental data. This is known as the “point-probe approximation” (Hartmann, 1989a). The force acting on the probe which is immersed into the near-surface sample

FIGURE39. Four basic geometrical arrangements often met in magnetic force microscopy. Only the force component along the vector n is usually detected. In the most simple situation, the probe’s magnetic moment m and the average normal vector from the sample surface are on the axis (a). In (b) n and z are on the axis, while m is arbitrarily tilted. (c) involves a situation in which m is perpendicular to the sample surface, while an off-axis force component is detected. (d) reflects the general situation, where all involved vectors are mutually tilted.

FUNDAMENTALS OF NON-CONTACT FORCE MICROSCOPY

microfield is then given by F

= po(q

137

+ m.v)H,

which implicitly involves the condition v x H = 0. q and m are the probe’s effective monopole and dipole moments. However, as shown in Fig. 39, this force is generally not directly detected by MFM. Usually the instrument detects the vertical component of the cantilever deflection. (Some instruments, however, allow a simultaneous detection of lateral components.) The detected force component is thus rather given by Fd = n-F, where n is the outward unit normal from the cantilever surface. Well-defined different orientations of the probe with respect to the sample, as shown in Fig. 39, then allow the successive detection of lateral as well as vertical field components. Putting Eq. (136) into component form, one gets the more illustrative result

which is the basis of contrast modeling if the MFM is operated in the static mode. However, most instruments are operated in the dynamic mode, where the probe-sample separation is periodically modulated with an oscillation amplitude which is small compared with the average probe-sample distance. In this case the compliance component Fd(r) = (n.v)(n.F(r)), with F(r) according to Eq. (136), is detected. Contrast modeling is then based upon

which involves, apart from monopole moment and dipole components, “pseudo-potentials” $p = aq/dxi and “pseudo-charges’’ q p k i = dmk/axi.2 These pseudo-contributions result from the fact that the actual magnetic response of a real probe of finite size clearly depends on its position with respect to the sample surface. This aspect, which has been completely neglected in previous models, is further clarified in Section 1V.B. In the present context the most important consequence is that in ac mode v’q = I could of course also be associated with a “pseudo-current” and v.m = Vv.M with a “pseudo-divergence” of the probe magnetization within the volume V . However, in the context of Eq. (138), the component form is emphasized and the denotations “pseudo-potential” and “pseudo-charge” are thus preferred.

138

U. HARTMANN

MFM, it is not only the second derivatives of the field components that contribute to the ultimately observed contrast but, according to Eq. (138), also the first derivatives, as well as the field components themselves. The number of field derivatives entering Eqs. (137) and (138) is reduced by v x H(r) = 0, leading to

The most serious limitation of the point-probe approximation is of course that low-pass filtering of the sample’s stray field configuration due to the finite probe size is completely neglected (Schonenberger and Alvarado, 1990a). This latter effect can be accounted for by applying a low-pass filter of type

where r = ( p ,d ) determines that geometrical center of the probe which is at a height d above the sample surface. p‘ is a cross-sectional radial vector whose range is determined by a certain effective probe diameter A. B. Properties of Ferromagnetic Microprobes

1. Bulk Probes Most force sensors in MFM have been fabricated from fine electrochemically etched ferromagnetic wires (Lemke et al., 1990), predominantly made out of nickel and iron. For these soft magnetic materials the apex domain is consistently found to exhibit a major axis length (see Fig. 37), between a few hundred nanometers and about one micrometer (Schonenberger and Alvarado,l990a; Goddenhenrich et al., 1990a), while for some tips, lengths of more than 10 micrometers have been found (Rugar et al., 1990). The actual extent of the apex domain is closely related to the sharpness of the tip. According to the effective-domain model introduced in Section IV.A, the shape-anisotropy field is related to the on-axis demagnetization coefficient by (141a) H , = M ( I - 3N>)/2, where M denotes the probe’s magnetization. N> depends on the aspect ratio a (minor to major semiaxis) of the apex domain (Bozorth, 1951), which is determined by the near-apex geometry of the probe. Figure 40 shows H , as a function of a. For purposes of comparison, the magnetocrystalline

FUNDAMENTALS OF NON-CONTACT FORCE MICROSCOPY

0.01

0.1

0.4

0.2

0.6

0.8

139

1.0

a FIGURE40. Shape-anisotropy field normalized to the tip magnetization as a function of aspect ratio. For purposes of comparison, the magnetocrystalline anisotropy fields for the most frequently used tip materials are indicated.

anisotropy fields HK

= 21Kl/pOM,

(141b)

with the room-temperature anisotropy constants IKyI = 4.1 x lo5 J/m3, 4.8 x 104J/m3, and 4.5 x lo3J/m3 for cobalt, iron, and nickel are indicated, respectively. The on-axis orientation of the near-apex domain becomes unstable if H s ( a ) 5 H K . Thus, for iron and nickel even relatively blunt probes still exhibit on-axis polarization, while for cobalt at least a sharpness of ( x 5 0.3 is required. Once the actual dimensions of the effective domain are determined, e.g., by direct observation as in Fig. 38, contrast modeling is based either directly on Eq. (135) or is performed in the point-probe approximation represented by Eqs. (137) and (138). Suitable algorithms for calculating the sample stray field are discussed in Section 1V.C. However, an essential question in the present context of probe characterization is that of stray fields, which are produced by the probes themselves, as can clearly be seen in Fig. 38. At sufficiently small probe-sample separations, this highly focused microfield may perturb the near-surface magnetization of soft magnetic samples (Hartmann, 1988; Goddenhenrich rt al., 1988; Mamin et al., 1989; Scheinfein et al., 1990). However, apart from destructiveness, new concepts for MFM may arise from the availability of highly focused magnetic stray field sources, as provided by sharp tips. This latter aspect is discussed in

140

U. HARTMANN

detail in Section 1V.E. In any case, the effective-domain model permits a fairly realistic analysis of the stray field configuration produced by typical MFM probes. It is convenient to start the analysis with a paramagnetic, prolate spheroidal particle exposed to an external, homogeneous magnetic field. The situation is thus in complete analogy to the case of a dielectric tip exposed to an electric field as discussed in Section 111. Boundary conditions for the present magnetic Dirichlet problem with axial symmetry are

where, as before, prolate spheroidal coordinates ( q , 8 ,E ) have been applied. Thus, a cos qo equals the major semi-axis R , of the spheroidal particle under consideration. The externally applied field Ho is parallel to R,, and p denotes the relative permeability of the particle. 4i and $e are the interior and exterior magnetostatic potentials, and 4o is a less important gauge constant. The particular solutions of Laplace's equation are

4e(v16 ) = do + Hoa(cosh 71 + [ ( P - 1 )/c,~J cash vosinh 7 o Q i (coshqo)) cos 6, ( 143a)

and 4 i ( q , S ) = 40+ H o ~ / c , , , ( Q ~ ( c o s h ~ o ) c o-sQ ~ ~i ( c o s h ~ ~ ) s i n ~ ~ ) (143b) cosh q cos 8, with c ~ , ,= , ~Q;(cosh 770) cosh 70- pQi(cosh 70)sinhqo.

( 143c)

Q, (x) is the Legendre function of second kind and order one, with Q ( ( x )= dQ/dx. The corresponding fields are then easily derived by applying

where u, and He(qi 0) =

Ug

are unit vectors. Thus, one obtains at any exterior point

Ho Jsin2 q sin2 e

+

{

-

[sinhq coo 'I

+

-

cosh qo sinh qoQ,'(cosh q)

cosh qo sinh qOQi(cosh

FUNDAMENTALS OF NON-CONTACT FORCE MICROSCOPY

141

An experimentally important situation (see Section 1V.E) is given by replacing the paramagnetic probe by an ideal, soft ferromagnetic one of high permeability, >> 1. Then Eq. (145) becomes

which allows a calculation of the field for an MFM probe which is polarized by an exciter coil as used in an early experiment (Martin and Wickramasinghe, 1987). The maximum field is produced at the probe’s apex. 6 = 0 and 7 4 q0 yields

This equation quantifies the well-known result that a sufficiently sharp, soft magnetic tip produces an apex stray field which by far exceeds the driving field produced by the exciter coil. The lower field limit is of course given if the tip degenerates to a sphere, qo -+ 00, which yields He,min = 3Ho. Returning to the paramagnetic case, Eq. (143b) yields the interior field

HI

=

- ( ~ o H ~ / ~ , ~ ~ ) r Q ; ( c o s h ~ o) cQo is(hc ~o soh ~ o ) s i n h ~ o ] ~(148) z,

with z = acosh vcos 0. The homogeneous demagnetizing field in the interior of a prolate spheroidal particle is usually characterized in terms of the principal demagnetization coefficient N , (Bozorth, 1951):

H,

Ho - N>M,

( 149a)

where M is the induced magnetization. Considering the magnetic induction (1 49b) one immediately obtains (1 50a) and thus (1 50b)

142

U. HARTMANN

Comparison with Eq. (148) yields N>(VO)=

Qi(cosh710)

QI(cash 70)- Ql(cosh V O ) coth 710

(151)

for the relevant geometrical demagnetization coefficient. Now, it is straightforward to deal with a usual MFM probe exhibiting the spontaneous magnetization M . The interior magnetostatic potential corresponding to the demagnetizing field is then given by

d,(q,8)= MaN>(Vo) coshVcoshf?,

( 152a)

while at any exterior point

4 e (v,e ) = Ma[l - N > ( V O 11[cash v o / Q I (cash v o 1I Q 1 (cash 7) cos 6, ( 152b) which is related to the stray field. Figure 41 shows the modified equipotentials about a typical MFM probe for (interior) N , / ( 1 - N > ) 4 (exterior)’

(153)

The vertical stray field component,

x/a FIGURE 41. Equipotentials about a typical magnetic force probe, a denotes the focal distance to the center of the apex domain.

FUNDAMENTALS OF NON-CONTACT FORCE MICROSCOPY

143

is of particular importance. Via Eq. (144) one obtains H , ( q , 8 ) = - M[1 - N,q0)](~~~hq0/Q1(~~~h(r10)](sinh2 qQ~(coshq)cos20

+ cosh qQ,(cosh q ) sin28]/[sin2q + sin281.

(1 5 5 )

Along the probe’s axis of symmetry, 8 = 0, this reduces to Hz ( z ) = - W1 - N> (11011[cash QO / Q 1 (cash T O)I Q I’(cash 77).

(156)

Figure 42 shows the decay of the axial stray field component with increasing distance to the apex for various values of the aspect ratio Q = R,/R, inherent to the effective domain. Q determines the maximum stray field directly at the apex as well as the decay rate. A sphere, a = 1, which is used by many authors for contrast analysis in MFM, yields the minimum apex field strength H , = 2M/3 and a maximum decay length. A sphere is certainly a rather poor approximation for a sharp MFM probe with Q << 1 . With decreasing aspect ratio the apex field strength increases, while the characteristic range of the stray field decreases. Thus, if a minimum or maximum stray field is required for an experiment, the appropriate aspect ratio of the probe is determined by the desired probe-sample separation, e.g., tunneling distances or a working distance of more than 100nm.

i

0 . 9 b

0.34

0.001

\

0.01 0

0.l o o

1 .do0

FIGURE 42. Vertical stray field component produced by typical magnetic force probes as a function of separation from the probe’s apex for various values of the aspect ratio LY = R , / R , . R < ,,denote the minor and major semiaxes of the erective apex domain. M is the spontaneous magnetization of the probe.

144

U. HARTMANN

1

30

a FIGURE43. Vertical stray field component directly at the apex of magnetic force sensors as a function of aspect ratio. The approximation is obtained by first-order expansion of the accurate result.

For tunneling experiments involving ferromagnetic microprobes, the vertical field strength directly at the apex of the tip is of great importance. This quantity is obtained with 77 = qo in Eq. ( 1 56) and is shown in Fig. 43 as a function of the aspect ratio. Probes with (Y 5 0.03 exhibit a stray field almost equal to the spontaneous magnetization, while the minimum value of H , = 2M/3 is obtained when the tip degenerates to a sphere. This result shows that MFM tips can produce high magnetic fields in extremely small areas (10 nm scale) at the sample surface. In order to quickly estimate the actual field magnitude for a sharp tip, first-order expansion of Eq. (156) leads to the convenient form H,(o)

=

M[I

+ ( a 2 / 2In) ( a 2 / 2 ) ] .

( 1 57)

Results obtained by this approximation are also indicated in Fig. 43. This analysis shows that contrast formation in MFM is governed by a kind of uncertainty principle. Sharp probes which would yield a high resolution from the geometrical point of view are likely to perturb the magnetic object via their high stray fields, while dull probes produce less perturbing stray field but also exhibit a reduced lateral resolution. The radial stray field component produced by the microprobe plays a role, for example, if the probe is raster-scanned across a magnetic object which is sensitive to in-plane pinning forces. Such a probe-induced pinning has been observed for highly mobile interdomain boundaries in iron

FUNDAMENTALS OF NON-CONTACT FORCE MICROSCOPY

145

0.30 0.25

0.20 0.15 0.10

I 0.05 0.00 I -.05 -.lo -.15

-.20 -.25

-.30

in combination with Eq. (144) yields H , ( 7 / ,19)=

M [1

-

N,(~o)][coshqO/Ql (cosh qo)][sinhvsin0cos8/ sin2 v

+ sin2 O]lQl(cosh v) - cosh ~)Q{(coshq)].

(159)

The behavior of the radial stray field component at the sample surface is shown in Fig. 44 for probe-sample contact. The maximum magnitude is almost independent of the aspect ratio (Y and is about 30% of the saturation magnetization. However, the peak's sharpness and its distance to the probe's axis of symmetry are sensitive functions of a. At large probesample separations (see Fig. 45), the initially pronounced field peaks of sharp probes are smeared out and gradually diminish, while a spherical probe still produces a more or less pronounced peak. The philosophy inherent to the effective-domain model is to approach the net magnetic response exhibited by an MFM probe which involves a more or less complex internal micromagnetic configuration by considering the purely magnetostatic behavior of an idealized apex domain (see Fig. 37). The total magnetic dipole moment of the probe according to this model is

146

U. HARTMANN

-.040

,

-10

. ,

-0

.

,

-6

14

12

0

2

4

6

0

10

dR< FIGURE 45. Same as in Fig. 44 (note the difference in scaling of the abscissa), but for a probe--sample separation that equals the minor semiaxis of the apex domain.

then simply given by m = VM, where V is the volume and M the spontaneous magnetization of the effective apex domain. However, according to Eq. (135), the magnetostatic free energy of the probe, which is exposed to the near-surface microfield of the sample, involves an integration of the spatially varying magnetostatic potential over the probe’s bounding surface and of its gradient, i.e., of the stray field, over the complete volume of the probe. This implies that for sample stray fields whose characteristic vertical decay lengths X are much smaller than the major axis 2 R , of the probe’s apex domain, only a small part of this domain is really relevant for contrast formation. The probe’s effective volume element obviously involves a “magnetic monopole moment,” since the net surface charge of the tip no longer vanishes. This latter aspect was completely neglected in almost all previous discussions of contrast formation. If only a small part of the probe’s apex domain contributes to contrast formation, the tiny magnetic object obviously images the dull probe. This latter phenomenon is of course the reason for the fairly high lateral resolutions which have been obtained in some experiments (Grutter et af., 1990b; Hobbs et al., 1989; Moreland and Rice, 1990), where magnetic objects much smaller than the actual probe diameter, 2 R , , have been imaged. To obtain a complete deconvolution of probe and sample properties, as desired in the point-probe approximation (see Section IV.A), effective magnetic monopole and dipole moments have to be attributed to the probe.

147

FUNDAMENTALS OF NON-CONTACT FORCE MICROSCOPY

These have to be compatible with the average range of the sample microfield under investigation. If X is the characteristic vertical decay length, and if the probe-sample separation is given by d, then the actual range of interaction of the microfield with the probe’s apex domain is given by 6 = X - d. The net magnetic charge and dipole moment, carried by the probe, are obtained from simple geometrical arguments: ( 160a)

= - x ( ~ 6 ( 2 R ,- 5 ) M

and

m

2 .2

= T (Y

5 (R,

-

( 160b)

6/3)M,

where M = 3= JMlu,is assumed in accordance to the effective-domain model shown in Fig. 37. The apex domain is completely determined by the aspect ratio (Y = R J R , , the major semiaxis R,, and the spontaneous magnetization M , as before. The effective interaction range 6 in turn determines the relevant probe diameter

A = 2ad-6, (161) which is, according to Eq. (140), an important quantity to model size

-

-.2-

dp-..

-.4-

-

-.6-

-2.-1

.o

a = 0.1 I

I

I

I

I

I

1

I

FIGURE46. Constitutional parameters of the advanced point-probe approximation which allow a fairly realistic estimate of the effective monopole and dipole moments, q and m,as well as the effective probe diameter A. q%pand yp are the pseudo-potential and the pseudo-charge effective in the dynamic mode of operation. The total magnetic charge of the probe is denoted by Q.S is the effective interaction range which is normalized with respect to the probe’s major semiaxis. The assumed aspect ratio may be considered as typical.

148

U. HARTMANN

effects in the advanced point-probe approximation. The effective, moments implicitly depend on the working distance via S = X - d. Thus, if d is periodically modulated upon dynamic operation of the MFM, the magnetic nonlinearity of the sensor leads to the "pseudo-contributions" (162a) 4p = -27ra(R, - 6 ) M , which reflects a potential, and

q p = w,

( 162b)

which gives an additional magnetic charge. These pseudo-contributions have already been taken into account in Eq. (138). Figure 46 shows the dependence of all quantities required to characterize an MFM probe in terms of the advanced point-probe model on the effective interaction range. Up to now it was implicitly assumed that the local microfield produced by the sample exhibits a characteristic decay length X in the vertical direction, while the lateral variation is long-range with respect to the effective probe diameter A. However, this situation is of course somewhat pathological. Generally, the radial range of interaction also involves a certain decay length p , In this case, it is convenient to define a modified vertical range of interaction by

6'

= R, -

4R;

- ( p / c ~ ) (with ~ p

5 aR,).

(163)

Straightforward geometrical arguments then lead, with respect to Eqs. (160)-(162), to the transformations

46)

+

( 164a)

(1(6*),

m ( s ) + 7rp2(6 - S' ) M + m ( S * ) ,

( 164b)

4pm

dJ,(S*)l

( 164c)

T P 2 M + qp@* ).

( 164d)

qp(S)

+

+

The nominal probe diameter is directly determined by the radial range of interaction:

A

-+

2p.

( 164e)

Equations (164) are the basis of contrast modeling for a bulk ferromagnetic probe interacting with a microfield which locally involves a vertical decay length X and a radial in-plane decay length p . 2. Thin-Film Probes While, up to the present time, most MFM experiments have been performed using bulk ferromagnetic tip-cantilever systems, some results have also been

FUNDAMENTALS OF NON-CONTACT FORCE MICROSCOPY

149

obtained by coating nonmagnetic wire tips with ferromagnetic thin films (Rugar et ul., 1990; Mamin et al., 1990; den Boef, 1990; Sneoka et al., 1991). The coating was performed either by sputter or electrolytic deposition of the ferromagnetic material. However, today, the most advanced cantilevers are microfabricated from silicon, silicon oxide ( S O 2 , Si203), or silicon nitride (Si3N4) by photolithographic techniques (see, for example, Albrecht and Quate, 1988). The integrated extremely sharp tips (Wolter et al., 1991) with apex radii down to about 5nm can easily be coated with soft or hard magnetic materials (Griitter et al., 1990a, 1991). The obvious advantages of microfabricated MFM sensors are the excellent mechanical properties, the extremely sharp integrated tips, and of course the possibility of batch fabrication for a variety of magnetic coatings, which relieves the tedium of etching each individual sensor. However, from the magnetic point of view it is a priori not so clear whether there are considerable advantages over the conventional bulk probes, since no detailed analysis of the magnetic properties of thin-film tips has yet been presented. The effective-domain model presented in Section IV.B.1 for bulk ferromagnetic probes permits a straightforward extension to thin film probes. Instead of directly considering the isolated ferromagnetic thin film deposited on a nonmagnetic probe, it is more convenient to apply the model shown in Fig. 47. The thin layer is modeled with respect to its magnetic properties by considering two fictitious ellipsoidal bulk probes, the foci (or apices) of which are shifted with respect to each other by the film thickness, and the magnetization vectors of which are antiparallel. Both probes are assumed to exhibit the bulk magnetic properties discussed in Section 1V.B.I. This two-probe model implicitly assumes that the thin film has a thickness which is sufficient to ensure bulk ferromagnetic properties. Now the magnetic analysis is straightforward, and the complete framework developed for bulk probes can be used. The aforementioned geometrical boundary condition, which concerns the relative apex positions of the fictitious bulk probes, allows the modeling of

FIGURE 47. Two-probe model applied to the analysis of thin-film magnetic force probes.

150

U . HARTMANN

two different types of thin-film tips. For the type-I probe it is assumed that the semiaxes of the inner probe are given by

where R>,< are the corresponding axes of the outer probe and t is the film thickness. Transfer of these conditions into prolate spheroidal coordinates yields a* =

ad1

-

( 166a)

2t/a(cosh 770 - sinh ~0

for the focal position, and CoshV; = (a/a*)[coshVO- t / a ]

(1 66b)

for the sharpness of the inner probe. Additionally, one obtains cash v* = (u/u*)cash 77,

( 166c)

which determines, together with 8' = 0 and <* = <, the location of any exterior point with respect to the inner probe. For the type-I1 thin-film probe, the geometrical boundary conditions are chosen as

0.0

-.5

5 -1.0

-1.5

,

I

--

_. Type I I

FIGURE48. The two basic types of thin-film magnetic sensors. f denotes the film thickness and n the focal length of the outer spheroid.

151

FUNDAMENTALS OF NON-CONTACT FORCE MICROSCOPY

This yields a* = a( 1

(1 68a) for the focal position of the inner probe, which exhibits the same aspect ratio as the outer one, i.e., 77; = 770. (1 68b) -

t/R,)

Hence, as before, (1 68c)

cash Q* = (u /L z *cash ) 7,

as well as 0' = 0 and E' = [. The geometrical differences between the two probe types are clarified in Fig. 48. Both types are fairly realistic and could be fabricated by suitably adjusting the parameters for thin-film deposition. With respect to the differences in magnetic behavior, it is interesting to compute the magnetostatic potential according to Eq. (153). According to the two-probe model, the total potential of a thin-film probe is given by 4,(77,0> = 4(% 0 ) - 4*(77,0 ) )

(169) t ) = 4a.,T,o. (q*,0 ) . The potentials are calculated according to where 4*(~, Eqs. ( 1 52) for any interior (71 5 7 ; ) and exterior (77 2 v0) point. Within the thin film (77; 5 77 5 q o ) , one has to use the interior potential for 4 and the exterior one for #*. The result of such a calculation is shown in Fig. 49 for a type-I probe of exactly the same outer geometry as used in Fig. 41. The comparison confirms that the stray field of the thin-film probe exhibits a 0.0 -.5

>

-1

.o

0

-1.5

-2.0 t/a

-2.5

5

= 0.1 1

-1.0

I

-.5

I

0.0

I

0.5

I

1 .o

5

x/a FIGUH 49~ ~Equipotentials for a thin-film magnetic force sensor of type 1.

152

U. HARTMANN

0.0

1

-.5

R

-1.0

u #

-1.5

I I

+ 0.005

-2.0 t/a

-2.5

= 0.005 aM

aM

= 0.1

-i.o

j

-15

I

I

0.5

0.0

I

1.o

5

x/a FIGURE 50. Same as in Fig. 49, but for a type-I1 force sensor.

t

0.001

Y-V.

0.01

8

I

I

0.010

0.100

1.ooo

d/R> FIGURE 51. Axial vertical stray field component produced by a thin-film probe with respect to that of a bulk probe of same outer geometry. ddenotes the distance to the probe's apex. R , is the major semiaxis, a the aspect ratio, and t the film thickness.

FUNDAMENTALS OF NON-CONTACT FORCE MICROSCOPY

153

somewhat smaller range than that of the bulk probe. Because of the geometrical boundary conditions, some of the equipotentials close through the interior of the probe, thus yielding a homogeneous “demagnetizing” field. On the other hand, near the apex the equipotentials exclusively close through the thin film (not shown for reasons of clarity). The corresponding result for a type-I1 probe is shown in Fig. 50. Because of only slight modifications of the geometrical boundary conditions, this probe type shows a substantial reduction in stray field range with respect to type I. The interior is now completely field-free and all equipotentials close through the magnetic layer. The preceding analysis of the two basic types of thin film sensors clearly shows that, if the major goal is a reduction of the stray field range, a thinfilm tip should be of type 11. However, the dispute concerning thin-film versus bulk probes is of course also related to the question of absolute field magnitude (Griitter et al., 1991). The two-probe model permits an easy calculation of the stray field produced by a thin film probe, by applying Eqs. ( 1 55)-( 159) to the two fictitious bulk probes and by deriving the net field according to Eq. (169) by a subtraction of the field produced by the inner probe from that produced by the outer probe. The result for the axial field component along the probe’s axis of symmetry is shown in Fig. 51 for different film thicknesses and aspect ratios of a type-I1 thin-film probe. With respect to a bulk probe which exactly fits the outer geometry of the thin-film

2-

0.40 _____________

0

N

I

.Y-

O

N

I

0.01

0.05

0.10

0.50

1

a F I C ~ J R52E Apex field strength of thin-film probes with respect to those of geometrically equivalent bulk probes as a function of aspect ratio for three different coating thicknesses.

154

U. HARTMANN

probe, the stray field of the latter shows a more rapid decrease with increasing distance from the apex. This behavior is most pronounced for sharp probes (small aspect ratios a ) and small film thicknesses. Dull probes show a stronger field reduction than sharp probes right at the apex, but a relatively small decay rate. If the experimental boundary conditions are given in terms of a certain desired probe-sample separation, Fig. 51 permits a determination of that specific film thickness, which yields the minimum perturbing stray field for a tip of given sharpness. For tunneling experiments involving ferromagnetic probes, the field strength directly at the apex of the probe is of predominant importance. Figure 52 shows the apex field strengths produced by type-I1 thn-film probes with respect to those exhibited by equivalent bulk probes. For very small aspect ratios, a 5 0.01, there is no reduction in field strength at all. However, with increasing a, the field reduction exhibited by the thin-film probes becomes more and more pronounced, where thinner films produce less stray field than thicker ones. As a concrete example, a type-I1 thin-film probe with a major semiaxis of 500 nm, a film thckness of 25 nm, and an aspect ratio of 0.5 produces only 40% of the stray field which would be produced by an equivalent bulk probe (same outer geometry, same ferromagnetic material). The radial stray field magnitude in the sample plane at apex-sample contact, d = 0, is shown in Fig. 53. Again the reduction in stray field 1

I

I

I

FIGURE 53. Radial stray field component produced by thin-film probes with respect to that produced by equivalent bulk probes for vanishing probe-sample separation as a function of the radial distance to the probe's apex. For a given aspect ratio a , the upper, middle, and lower curves correspond to thick, medium, and thin magnetic coatings.

FUNDAMENTALS OF NON-CONTACT FORCE MICROSCOPY

155

increases with increasing distance from the apex. As for the vertical field component, the effect is most pronounced for dull probes and small coating thicknesses. Note that, according to Fig. 44, the radial field components of both thin film and bulk probes vanish at the apex, r = 0. However, the limiting stray field ratio remains finite. With respect to the dispute concerning thin-film versus bulk probes, the following conclusion can be drawn. An advantage of thin-film probes, apart from their advanced mechanical properties, is the reduced stray field to which the probe is exposed under typical MFM operation conditions, i.e., a probe-sample separation of at least 10 nm. At smaller separations, thin-film probes exhibit almost the same magnitude of stray field as bulk probes. This latter aspect, however, may also be considered as an advantage for applications where just a highly focused stray field is desired, as discussed in Section 1V.E. The major advantage of thin-film probes is of course that they are in general mesoscopically much sharper than bulk probes. Up to now, thin-film probes have only been considered with respect to the stray field that they produce. Intuitively, it is obvious from the two-probe model shown in Fig. 47 that both monopole and dipole moments of thinfilm probes are greatly reduced with respect to a geometrically equivalent bulk probe. However, due to the close vicinity of opposite free surface magnetic charges, the monopole moment appears to be much more greatly reduced than the dipole moment. The consequence is thus that a thin-film probe produces a magnetic image of a given sample which is generally different from an image produced by a geometrically equivalent bulk probe. The differences with respect to the sensor behavior are manifested in modifications of the magnetic moments, which are given in Eqs. (160) and (162) for bulk probes. The two-probe model provides a simple transformation procedure solely based on geometrical arguments:

( 170a) for the monopole moment, m ( 6 )+ m ( 6 ) - ( 1

-

t/R,)3m(6

-

t)

(170b)

for the dipole moment,

( 170c) for the pseudo-potential, and

( 170d)

156

U. HARTMANN

for the pseudo-charge. 6 is the effective range of interaction, as before. The nominal probe diameter A(6) remains the same, since the outer geometry of the probe does not change. The preceding magnetic moments with respect to those obtained for bulk probes are shown in Fig. 54 as a function of interaction range. The obtained results clearly emphasize the fact that the reduction in stray field has to be paid for by a reduction in magnetic sensitivity. As a concrete example, one obtains from Fig. 51 for a thin-film probe with an aspect ratio a = 0.3, a major semiaxis R , = 800 nm and a coating thickness of t = 50 nm, a reduction of the axial vertical stray field component H , by about 40% with respect to a comparable bulk probe at a working distance d = 50nm. If the sample stray field exhibits a vertical decay length of X = d 6 = 250nm, Fig. 54 then shows that the corresponding reduction of the monopole moment is about 70% and that of the dipole moment about 50% with respect to the bulk probe. If the local radial stray field range is governed by a decay length p, a modified vertical interaction range 6* is defined, as in Eq. (163). If 6 is smaller than the film thickness t , the magnetic moments of the thin-film probe are the same as for the bulk probe, i.e., they are given by

+

1 .oo

0.80 0.60

0.10

0.08 0.06

0.02

0.01

0.0

'zero:

0.1

0.2 0.3

0.4 0.5 0.6 0.7

m,

0.8 0.9

1.0

FIGURE 54. Magnetic moments of thin-film probes with respect to those of equivalent bulk probes as a function of characteristic interaction range. q denotes the monopole moment, m the dipole moment, dp the pseudo-potential, and qp the pseudo-charge. Q is the total magnetic charge of the probe.

FUNDAMENTALS OF NON-CONTACT FORCE MICROSCOPY

157

Eqs. (164). On the other hand, for 6*2 t the moments are obtained via Eqs. (170) by replacing 6 by 6*. C. Contrast Modeling 1. Treatment of’ Periodic Charge Distributions Once the magnetic properties of the microprobe are determined, contrast modeling requires an appropriate solution for the microfield profile which is produced by the magnetic sample under investigation. Thus, it is not so surprising that MFM has led to a renascence of the development of advanced algorithms for stray field calculation. Such calculations received much interest almost 30 years ago, during a period when contrast formation underlying the Bitter colloid technique was extensively investigated. The first attempts to detect near-surface microfield profiles by scanned solid-state probes, e.g., Hall and Permalloy induction probes, were even performed at that time (Carey and Isaac, 1966). However, today advanced observation techniques and theoretical methods provide a much deeper insight into the near-surface magnetization of ferromagnetic samples. Microfield calculation is then based, for a given configuration of the sample’s magnetization vector field, on the integral equation (134). Fortunately, the solution can be obtained for almost all samples which have been studied by MFM in an analytical way. The effective-domain model then offers two alternatives: either a combined surface/volume integration over the idealized ellipsoidal domain of the probe, or a direct employment of the stray field components and their spatial derivatives in terms of the advanced point-probe approximation represented by Eqs. (137)-( 140). While the first way always involves some computational effort, the point-probe ansatz can be performed in a straightforward analytical way. Especially if, in terms of Eq. (l40), the finite probe size is accounted for, the data obtained from the advanced point-probe approximation turn out to be consistent with almost all experimental data which have been presented so far (Hartmann et al., 1991). In the following, contrast modeling is performed for some cases of particular experimental relevance. It turns out that classical potential theory combined with the introduction of free magnetic charges, as already used in Section IV.B, is a convenient concept to understand the contrast produced by an MFM. If one has an arbitrary two-dimensional periodic magnetic charge distribution at the sample surface, Fourier expansion of the charge density is given by (171a)

158

U. HARTMANN

with umn

=

1 J’” 4 4n2 0

1;

dV(J, 7)exp (-i

[M+ w1)

1

(171b)

where [ = x/Lx, Q = y/L,,, and where L,, Ly define the unit cell of area 4n2L,Ly. The Laplace equation, 0’4 = 0, valid exterior to the sample, together with the condition of continuity of B = po(H M), yields via H = -V+ the stray field produced by the periodically charged sample. Directly at the surface, z = 0, one thus obtains for the vertical stray field component

+

Y , 0) = 4x7 Y , )/2.

(172)

The exterior solution for the Fourier coefficients of the magnetic potential are thus dmn = -(nmn/2vmn)

~ X (P- v m n z ) i

(173a)

with the “spatial frequencies” (1 73b)

The complete exterior Laplace solution is thus given by

An important area of application to MFM is the analysis of thin-film structures, e.g., of recording media. If the probe-sample separation becomes comparable with or even exceeds the film thickness, the stray fields of both the top and the bottom sample surface contribute to the contrast. Thus, if t is the film thickness,

4(r) = -

C 1

m=-mn=-m

( g m n / v m n ) sinh tvmnf/2>~

X ( P i + WI~

- ~mnz)t

(175) where z is the vertical distance measured from the center of the film. This kind of treatment of periodic magnetic charge distributions was originally used in some classical work devoted to an analysis of the magnetostatic stability exhibited by certain periodic domain arrangements (Kittel, 1956). The applicability to highly symmetric problems in MFM is of course fairly obvious (Mansuripur, 1989; Schonenberger and Alvarado, 1990a). The form of Eq. ( 175) is particularly suitable for numerical computation involving standard two-dimensional FFT algorithms.

FUNDAMENTALS OF NON-CONTACT FORCE MICROSCOPY

FIGURE

159

55. Schematic of a thin-film longitudinal recording medium.

2. Longitudinal Recording Media In longitudinal magnetic recording, the recording head is flown over the medium with a spacing of a few hundred nanometers or less. Upon writing, oppositely magnetized regions with head-to-head and tail-to-tail transitions are created, as shown in Fig. 55. Since the transitions of width S involve free magnetic charges, a stray field is generated which transmits the stored bit configuration to the recording head. MFM is thus a particularly useful method of analysis, since it detects the generated stray field profile, which is detected by the recording head upon reading operation. Since the stray field is produced right at the transitions between the antiparallel magnetic regions, the detailed internal structure of the transition regimes is of great importance. The latter is determined by demagnetizing effects in the medium. The line charge approximation used in earlier approaches to MFM contrast formation (Mamin et al., 1988; Hartmann, 1989b; Wadas et al., 1990) may thus be an inexpedient approximation. An approximation commonly used in recording physics is (Rugar et al., 1990) M,(x)

=

-(2M/7r) arctan (x/S),

(176)

where M , denotes the in-plane magnetization component near the transition which is centered at x = 0, as shown in Fig. 55. 6 denotes the characteristic transition width and M the spontaneous magnetization in the uniformly magnetized regions. An estimate of 6 may be obtained, for example, from the Williams-Comstock model (Williams and Comstock, 1972). Substitution of Eq. (176) into Eq. (134) yields the stray field for an isolated transition (Potter, 1970): x(t

+z)

-

7r

for the in-plane component, and H,(x,z)

M xft

= -

2

x2

arctan

+s +z y

+ (z+

x2

+ x62z + 6z]

(177a)

(1 77b)

160

U. HARTMANN

1

0.15

0.10-

1

-

8

1

1

d/w=O.l, t)w=0.05

~

"

'

I

"

'

z/w=O.l

-

-.lo-.15

'

(b) I

I

I

I

1

.

8

9

I

a

1

I

'

_I

Hx

H,

"

B

'

FIGURE 56. Contributions to the magnetic contrast produced by a longitudinal recording medium. The field components are considered with respect to the in-plane spontaneous magnetization and are plotted as a function of lateral position. w denotes the spacing between the individual transitions. 6 denotes the effective transition width for which a representative value has been chosen. (a) shows the contrast contributions directly at the surface of the medium, together with the magnetization divergence. In (b) the working distance has been increased and is now equal to one-tenth of the transition spacing.

FUNDAMENTALS OF NON-CONTACT FORCE MICROSCOPY

161

for the vertical component. The total field is obtained by a linear superposition procedure,

l)”H(xwhere w is the spacing between the transitions. The stray field components together with the quantity d M , / a x , derived from Eq. (176), are shown in Fig. 56. According to Eq. ( 1 37), these stray field components predominantly produce the MFM contrast, if the probe mainly exhibits a monopole moment. While in Fig. 56a the contributions to the M F M contrast at vanishing working distance are shown, Fig. 56b shows the effect of increasing probe-sample separation. Apart from a decrease in magnitude, fine details of the field get lost. It should be noted that the MFM tip is still considered as a point probe of infinite lateral resolution. According to Eq. (175) the loss in information is due to the predominant damping of higher Fourier components at increasing working distance. This behavior involves a certain similarity to the “point-spread phenomenon” dealt with in common optics. Apart from the two field components, their first derivatives with respect to x and z also contribute, according to Eq. ( 1 37), to the contrast if the M F M is operated in the static mode. To obtain a better understanding of the relationship between the various contributions, Fig. 57 shows these field quantities, to which the dipole moment of the probe is sensitive. Because of the constraint given in Eq. (139), only three out of four derivatives are required to model the MFM contrast. Again the loss in information with increasing working distance is fairly obvious. Finally, the second derivatives, which according to Eq. (137) are relevant in dynamic-mode MFM, are shown in Fig. 58. Apart from the constraint given in Eq. (139), the symmetry of the arrangement yields d2H,/az2 = -d2H2/dx2. Thus, one has to calculate three out of six possible second derivatives. In general, all components shown in Figs. 56 and 57 contribute to the MFM contrast in the dc mode of operation, while in the ac mode the components shown in Fig. 58 provide additional contributions. For an arbitrary probeesample arrangement, as schematically shown in Fig. 39d, the ultimate contrast is obtained by a linear combination of the various field quantities, where, according to Eqs. (137) and (138), these are weighted by the corresponding magnetic moments of the probe and by the actual orientation of the cantilever with respect to the sample surface. The finite probe size is additionally accounted for by low-pass filtering according to Eq. (140).The probe’s effective magnetic moments and its effective diameter may either be treated as free parameters fitted to the experimental data, or

162

U. HARTMANN

may be estimated from the characteristic lateral and vertical decay rates of the stray field components. However, it must be emphasized that these quantities, which characterize the probe's response, are strictly dependent on the microfield profile under investigation. Calibration of the probe thus always refers to the particular sample used for calibration, rather than to the

1.2-

I

1

I

I

a/w=o.1,

,

-

8

'

1

I

"

"

I

"

"

t/w=0.05

-

0.8- z/w=O.l

3

v

-.4-

-

-.8-

-1.2

--

(b) I

I

I

I

'

'

'

'

I

'

'

'

'

_I

-

8H2/8z

8HU/8z 8H,/Ou

'

'

'

'

FUNDAMENTALS OF NON-CONTACT FORCE MICROSCOPY

163

given probe operated on any arbitrary magnetic sample. This latter fact has not been recognized in earlier experimental approaches to probe calibration (Goddenhenrich et af., 1990b). On the one hand, the interplay of five basic contrast contributions in staticmode MFM and eight occurring in the dynamic mode often makes the

b / w = O . l , t/w=0.05 15

Y X

2rc)

\ I cb

>3 U

-10 -15

FIGURE58. Same as in Fig. 56, but for the second derivatives of the stray field components. These quantities become relevant if the force microscope is operated in the dynamic mode.

164

U. HARTMANN

interpretation of experimental data from longitudinal recording media difficult. On the other hand, the well-defined relative orientations of probe and sample, as shown in Fig. 39, may greatly reduce the number of individual contrast contributions. The minimum number is two, e.g., H, and d H Z / d z , for dc microscopy, and three, e.g., H,, dH,/dz, and d2H,/dz2, for ac operation. By successively modifying the relative orientation of probe and sample in order to catch both the in-plane and the vertical field quantities (see Fig. 39), a complete characterization of the recording medium may be achieved (Schonenberger and Alvarado, 1990a; Rugar et al., 1990).

3. Vertical Recording Media The basic geometry underlying the two-dimensional problem is shown in Fig. 59. A uniaxial magnetic anisotropy forces the magnetization to assume an orientation perpendicular to the sample surface. Contrarily to the longitudinal media discussed in the previous section, the magnetic charge density is established along the magnetized regions. The detailed internal structure of the transition zones is thus less important in this case, and an abrupt transition approximation may be used for simplicity. It is convenient to employ a one-dimensional form of the Fourier ansatz given in Eq. (175):

Hence, the stray field components are

H Z ( x , z )=

-'4M"E2n + 1 7r

n=O

W

W

(1 80a)

FIGURE 59. Schematic of a thin-film vertical recording medium magnetized in a squarewave pattern.

165

FUNDAMENTALS OF NON-CONTACT FORCE MICROSCOPY

and 2nf 1

W

W

( 180b) These contributions to the M F M contrast are shown in Fig. 60. Since the assumed dimensions for the relative film thickness and working distance, t / w and z / w , are the same as for the longitudinal medium discussed in the previous section, the results shown in Fig. 60 are directly comparable with those shown in Fig. 56b. The maxima of H , and the zero-axis crossings of H , are located for the vertical medium at lateral positions given by odd multiples of w / 2 . The in-plane component, H,, exhibits exactly the same oscillation amplitude as the vertical component H,. For the longitudinal medium (see Fig. 56b), the maxima of H , and the zero-axis crossings of H , are located at the magnetization transitions, i.e., they are phase-shifted by w/2 with respect to the vertical medium. The maximum magnitude of H , is only about 60% of that exhibited by H,. From the technological point of view it is important that the oscillation amplitudes of H , differ only by a factor of 1.9 for the two media, in spite of the fact that the magnetically

*

0.15

0.10-

8

8

8

9

1

1

8

t/w=0.05

8

I

1

'

"

'

I

'

"

'

z/w=O.l

.I, I,

!

1 I

0.052 \

I

'\,

0.00-

-

-.05'

I

-

-.lo-.15 -1

.o

I

1

I

,

I

-.5

,

~

'

1

0.0

"

"

"

"

'

-

0.5

H,

1.o

x/w FIGURE 60. Contributions to the magnetic contrast for a vertical recording medium which shows a square-wave magnetization pattern. As before, I denotes the film thickness, w the spacing between the transitions, z the working distance, and M the spontaneous magnetization.

166

U. HARTMANN

charged area corresponding to a single stored bit is so much more extended for the vertical medium. However, directly at the surface H , exhibits, according to Eq. (172), a value which equals half the spontaneous magnetization of the vertical medium, while the longitudinal medium reaches only 28% of the inherent magnetization.

25 20

0

FIGURE 61. Same as in Fig. 60, but for the relevant first (a) and second (b) field derivatives.

FUNDAMENTALS OF NON-CONTACT F O R C E MICROSCOPY

167

Apart from the constraining condition, Eq. (139), the following identities, which are directly obtained from Eqs. (180), immediately yield the remaining contrast contributions:

aH,

-

x

dz (i

+

1)

(181a)

and thus

(181b) The relevant field quantities shown in Fig. 61 directly correspond to those shown in Figs. 57b and 58b for the longitudinal recording medium. 4. Magneto-optic Recording Media A very promising alternative to longitudinal recording is magneto-optic recording. The concept has received much attention mainly due to the high areal storage density which could be achieved (Rugar et af., 1987). Magneto-optic recording materials exhibit a uniaxial magnetic anisotropy which forces the magnetization to an orientation perpendicular to the film plane. The complete recording process consists of a magneto-thermal writing process and a magneto-optic reading process. Marks are written by locally heating the medium with a focused laser beam above the Curie temperature while an external bias field is present, the orientation of which is antiparallel to the local magnetization vector. After cooling below the Curie temperature, a reverse magnetic domain is formed, which is schematically shown in Fig. 62. The information is read back via Faraday rotation of a polarized laser beam reflected off the written domains. Since the cylindrical domains may be written in arbitrary patterns, it is convenient to treat the problem of MFM contrast formation first for an isolated mark. The stray field produced by an ensemble of domains is then

FIGURE 62. Schematic of a circular domain written into a magneto-optic recording medium.

168

U. HARTMANN

obtained by a linear superposition of the individual domain contributions. Thus, a Fourier ansatz as in Eq. (175) is not convenient in this case. According to Fig. 62, the boundary value problem is three-dimensional, however, involving symmetry of rotation about the vertical axis. Upon evaluating the magnetic potential according to Eq. (134), the volume integral can be dropped because the magnetization is homogeneous throughout the film thickness. Insertion of the magnetic charge density profile for the top surface in Fig. 62 yields the potential for a medium of infinite thickness:

where polar coordinates (r', 0 ) are applied, and where r is the radial distance to the center of the domain. Expansion of the integrands into power series and use of the indentity

then yields

This form of the potential, which is an alternative form to the commonly used expansion in terms of zonal harmonics (Morse and Feshbach, 1953), is particularly suitable for a quick numerical evaluation of the M F M contrast contributions. The finite thickness t of the medium is accounted for by the transformation

d(r,2 )

+

d ( r ,z ) - d ( r ,z

+t).

(185)

The resulting field components are shown in Fig. 63. Two features are particularly important. (i) The maximum radial stray field component exceeds the maximum vertical component. (ii) Far away from the domain,

FUNDAMENTALS OF NON-CONTACT FORCE MICROSCOPY 0.15 0.10-

L 0.1

z/w t/w

I

I

1

I

I

I

I1 I I

= 0.05

I I

-

1 1

-

0.055

\

I

--- - - - - _ _ _ - _

0.00-:------

-.05-

I

I

I

1

-

I 1 I I I 1 I I I I

-.lo-

-

_-

I 1

-.15

-2.0

169

-i.5

-i.o

I

-.5

I

0.0

I

0.5

Hr

- H, I I 1.0 1.5

2.0

the vertical field component is solely determined by the spontaneous magnetization and the film thickness. However, approaching the transition zone, H2 first shows a negative “overshoot,” then a positive one, and finally reaches a local minimum at the center of the domain. Thus, the presence of the domain locally raises the vertical stray field far above the magnitude obtained for the uniformly magnetized medium. The remaining contrast contributions, numerically evaluated according to Eq. (184), are shown in Fig. 64. It is interesting that all these contributions exhibit their peak values close to the magnetization transition zone. Apart from the constraint given in Eq. (139), the symmetry of the arrangement yields a 2 H z l a r 2= - a 2 H z / d z 2 .

5 . Type-II Supuconductors It is of current importance to estimate the stray field produced by a hexagonal vortex lattice manifest in a type-I1 superconductor which is exposed to an external magnetic field oriented perpendicular to the superconductor’s surface. Several groups are presently working on an experimental detection of the Abrikosov vortex lattice by means of MFM. A detailed discussion of contrast formation thus seems worthwhile. From the symmetry point of view, the magnetostatic boundary value problem exhibits a certain similarity to a hexagonal arrangement of

170

U. HARTMANN

uniformly magnetized cylindrical domains within a nonmagnetic environment. Because of the hexagonal symmetry of the lattice, it is convenient to slightly modify the Fourier ansatz given in Eq. (171): ( 186a)

Hz(r,z = 0) = x f i ( G ) e x p ( i G - r ) , c

z/w = 0.1

I!It

0.8 x-

(0

\ I rn n

I

\ 3 W

\I

-.a-

\ I

-'*O-

!

-1.2

(a) I

I

I

I

I

- - 8Hr/%r -- %Hz/%r

I

8H7/8Z

-

-

I

25 20

r/w FIGURE 64. Same as in Fig. 63, but for the relevant first (a) and second (b) field derivatives.

FUNDAMENTALS OF NON-CONTACT FORCE MICROSCOPY

171

with

( 186b) This Fourier ansatz refers to the vertical stray field component at the surface of the superconductor, and the expansion is performed with respect to the reciprocal lattice represented by

which is an arbitrary reciprocal lattice vector. a is the real-space lattice constant of the flux lattice, and f,, denotes the area of the corresponding unit cell (see Fig. 76). h, k are arbitrarily chosen natural numbers. M denotes a fictitious magnetization: The vortices have simply been modeled by cylindrical domains of a certain radius A. Each of these domains is assumed to carry the homogeneous vertical “magnetization” M = 2@0/(p 0 r X 2 ) . $0 denotes the elementary flux quantum. One thus obtains, for example, for niobium diselenide (NbSe2, A = 69nm), a value of p o M = 0.27T. The material is of particular importance for the experimental investigations, since samples usually show a high degree of crysallographic perfection. According to Eq. (172), the fictitious charge density produced at the surface of the superconductor is just twice the magnetization M . The flux produced per unit cell, p0Mf,,/2, then equals a0. Using IrdOexp (-iGr’cosO) = 27rJo(Gr’)

( 188a)

and (188b) where Jo and J I are Bessel functions of order zero and one, one obtains for the Fourier coefficients in Eq. (186)

H ( G ) = TMAJI (AG)/f,,G.

(189)

Commonly, these Fourier coefficients are replaced by “form factors” (see, for example, Hiibener, 1979):

172

U. HARTMANN

where hz(r') is the vertical stray field component which would be produced by an isolated vortex directly at the surface of the superconductor. According to the present approximation, one thus obtains F(G) = 2J,(XG)/AG.

(191)

Hence, the surface vertical field component produced by the complete vortex lattice is, according to Eq. (186a), given by

where H ( G ) = @ o F ( G ) / p 0 f uwas c used. Equation (192) allows, according to Eq. (173), the reconstruction of the complete exterior magnetic potential:

The obvious shortcoming of the preceding approach is that the detailed interior magnetic structure of the vortices has been neglected (see, for example, Hubener, 1975). However, all information about the interior structure of a vortex is represented by the form factors F ( G ) weighting the individual Fourier components in Eq. ( 1 93). Thus, an advanced approach must be based on the employment of more realistic form factors. It has been shown that these quantities can be experimentally obtained from neuron diffraction measurements (Schelten, 1974). The alternative possibility is to apply a more appropriate model for the magnetic behavior in the vicinity of a vortex core. The approach presented by Clem (1 974a, 1975b) appears to be particularly suitable for purposes of MFM contrast modeling. In treating the core of an isolated vortex, Clem assumes a normalized order parameter $(r)

= I$(r)Iexp

(-4,

( 194a)

where r denotes the radial coordinate, 77 the phase, and l$(r)l = r / d r 2

+ 6:.

( 194b)

tVis

a variational core-radius parameter. Substitution into the second Ginzburg-Landau equations yields (Clem, 1975a, 1975b) h,(r)

=

(J 7 Z / A L ) ]

(@0/2"POXLJV) [KO

/[KI (Jv/XL)I

(195)

for the surface stray field produced by an individual vortex. KO and K 1 denote the McDonald functions of order zero and one (see, for example, Abramowitz and Stegun, 1964). Substitution into Eq. (190) yields for the

FUNDAMENTALS OF NON-CONTACT FORCE MICROSCOPY

173

form factors

where XL denotes the London penetration depth of the superconductor at a given temperature. The variational core-radius parameter tvis determined by minimizing the energy per unit length of a vortex. This leads to the constraint (Clem, 1975a, 1975b) 6

= J 2 [ 1 - ~02(€v/XL)/K:(~v/XL)IXL/€V,

( 197)

where K denotes the Ginzburg-Landau parameter of the superconductor. Using K = 9, one obtains for niobium diselenide a normalized core radius of ["/XL = 0.15 at OK. Niobium with K = 1.4 yields <,,/AL = 0.84. Clem (1975a, 197%) has further shown that linear superposition according to Eq. (193) can be used to obtain the stray field at arbitrary K values and vortex spacings, provided that the correct spatially dependent magnitude of the order parameter, given in Eq. (194), is used. Overlapping of the vortices can be approximately accounted for by replacing XL by a fielddependent penetration depth Xeff

=~

L / ~ ( I W O ) l * ) >

(198)

where (I$(Ho)12)is the spatial average of the order parameter in Eq. (194b), which is now depending on the externally applied field Ho. Hence, for overlapping vortices the form factors in Eq. (196) exhibit a field dependence not only via the lattice constant u, but also via the modified penetration depth XL -+ X,n. The unit cell's area is related to the externally applied field by Jfo = @ o / P o f u c .

(1 99)

Thus, one obtains from Eq. (193) Hz(r,z)

=

Ho

F(G)exp (iG.r - Gz)

(200a)

G

for the vertical stray field component, and Hr(r,z ) = -iHo

F(G exp (zG-r - Gz)uc

(200b)

G

for the in-plane field component, where uG = G / G is a unit vector. A closer investigation of these equations shows that higher Fourier components are rapidly damped, since F ( G ) is, according to Eq. (196), monotonically

174

U. HARTMANN

P

FIGURE65. Contours of constant field magnitudes for an Abrikosov vortex lattice with a lattice constant a, which is twice the London penetration depth AL. On the left, the normalized vertical field oscillations Hz/(2FjHo) - 1 are shown. On the right, the corresponding in-plane field oscillations Hr/4F, Ho are shown. The maximum and minimum values obtained directly at the surface of the superconductor are indicated. Since no special assumptions on the interior vortex structure enter the calculations, the shown flux distribution applies to any type-I1 superconducting material.

FUNDAMENTALS O F NON-CONTACT FORCE MICROSCOPY

175

decreasing with increasing G. This behavior is enhanced by the increasing exponential damping for an increasing distance z to the sample surface. The first Fourier component obtained for G = 0 in Eq. (200a) yields the external field Ho, while this component vanishes for the in-plane stray field H,. Since F ( G ) = F(G) only depends on the magnitude of the reciprocal lattice vector, Eq. (192) shows that all reciprocal lattice vectors with dh2 hk k 2 = n, where n is a natural number, yield the same form factor. Figure 65 shows stray field profiles according to Eqs. (200). For the vertical field component, only the oscillations superimposed on the external bias field have been considered. The calculations were performed in a first-order approximation involving only the six reciprocal lattice vectors with Ghk = 27ra/fu,. The sum of all corresponding form factors is denoted by F, . The maximum vertical field component is then max (H,) = (1 + 6 F I ) H o , while min ( H , ) = ( 1 - 3F,)Ho. For the in-plane component, one obtains max ( H , ) = 4 d 3 F 1 H oand min (H,) = 0. Locations of these peaks are marked in Fig. 65. Equations (200) are the basis for MFM contrast modeling. The various field derivatives which are required according to Eqs. (137) and (138) all show the same symmetry as either H , or H,. From the experimental point of view, the most important question concerns the maximum variation in force or compliance obtained upon raster-scanning a typical ferromagnetic probe across the vortex lattice. In order to deduce a representative value, a bulk iron probe with a saturation flux density p o M = 2.1 T, aspect ratio CY = 0.5,

+ +

FIGURE 66. Possible deformation of the vortex lattice due to the highly focused microfield produced by a ferromagnetic probe. Within the indicated circular area underneath the probe, the lattice constant changes to u'.f,, denotes the undistorted unit cell. In the lower parts of the images, cross-sections are shown, which are taken along the indicated line scans. (a) shows a situation in which the probe's stray field is parallel to the external bias field, which leads to (I* < u. In (b) both fields are antiparallel, which results in a* > a .

176

U. HARTMANN

and semiaxis domain length R , = 500nm is assumed (see Section 1V.B.1 for a description of the parameters). The sample is niobium diselenide, where the material parameters &/XL = 0.15 and K = 9 are used. The externally applied flux density is taken as poHo = 120mT in order to avoid strong vortex overlap. Since both lateral and vertical stray field components involve a characteristic decay range, effective lateral and vertical ranges of probe-sample interaction, p and S', have to be determined (see Section 1V.B.I). For p the half-width-half-maximum of the stray field taken right above the center of a vortex seems reasonable. The modified vertical range 6' is then determined by Eq. (163). For a working distance of 5nm, the maximum force variation amounts to 319 pN, where the finite probe size has been accounted for in terms of Eq. (140). The corresponding maximum compliance detected in the dynamic mode of operation is 89mN/m. While the first value may just be in reach of present technology, the second should be clearly detectable. However, because of an effective probe diameter A = 56 nm, the expected lateral resolution is rather poor. The lateral forces exerted on the vortex ensemble exhibit a maximum value of 330 pN. However, this is only part of the whole story. Up to now, the stray field produced by the ferromagnetic probe itself has completely been neglected. The highly focused microfield superimposed on the externally applied field, in principle, (i) may nucleate vortices underneath the probe; (ii) may lead to a strong repulsive force between probe and sample, which is due to the local flux expulsion; (iii) may cause a deformation of the vortex lattice as schematically shown in Fig. 66. Issues (i) and (ii) are discussed in more detail in Section 1V.E. 6. Interdomain Boundaries in Ferromugnets

Interdomain boundaries are the natural magnetization transition zones between adjacent domains of different magnetic polarization. Within these transition zones the magnetization vector rotates continuously, thus forming a domain wall of finite thickness. In general, the wall thickness is determined by the exchange, magnetocrystalline anisotropy, magnetostriction, and magnetostatic energies inherent to the ferromagnet (see, for example, Chikazumi, 1964). Typical transition widths range from the nanometer scale for hard magnetic materials up to more than a micron for very soft materials. At the intersection with the crystal surface, complex two- or three-dimensional flux closure configurations generally occur in soft magnetic materials (Hubert, 1975). This near-surface modification of the wall structure is due to a natural energy minimization behavior inherent to the wall: Extended free surface magnetic charges are avoided by a suitable rotation of the wall's magnetization vector field.

FUNDAMENTALS OF NON-CONTACT FORCE MICROSCOPY

177

FIGURE67. Schematic of a 180" Bloch wall (upper image). The type-I approximation models the wall by a plane of infinitesimal thickness, which carries the specific dipole moment 40. The type-11 approximation is based on a finite wall thickness 6 and a homogeneous magnetization M . The near-surface profile of the wall is modeled by an ellipsoidal cylinder. The numerical approach accounts for the complex (asymmetric) near-surface profile of the wall's magnetization vector field M(r). the lower image shows a 180" Nee1 wall.

Nucleation, free motion, and annihilation of domain boundaries, as well as their interactions with the crystal lattice, determine the magnetization process of a ferromagnet, the technical relevance of which need not be emphasized. A study of interdomain boundaries in ferromagnets by means of MFM is thus of importance with respect to both basic as well as applied research. The upper part of Fig. 67 schematically shows a 180" Bloch wall as it occurs, e.g., in iron. Within the wall the magnetization vector rotates between the two antiparallel, adjacent domains and exhibits a component perpendicular to the sample surface. The stray field calculation can be based on three different models of varying complexity. In the type-I approximation, the wall is modeled by a plane of infinitesimal thickness which carries a homogeneous dipole moment &, per unit area. Near-surface fine structures of the wall are neglected. The finite width of a symmetric wall can be accounted for in a first-order approximation by modeling the wall, close to the sample surface, by a cylinder with an ellipsoidal cross-section. This is denoted as the type-I1 approximation in Fig. 67. Finally, the accurate approach consists of a numerical calculation of the internal wall structure by means of energy minimization procedures (see Hubert, 1975). Advanced calculations have recently been performed for iron and Permalloy (Scheinfein rt a/., 1991; Aharoni and Jakubovics, 1991). Previous approaches to MFM contrast formation (Hartmann and Heiden, 1988; Hartmann, 1989c; Hartmann et al., 1991) have shown that

178

U. HARTMANN

the experimental data obtained from domain walls can generally be modeled by use of the type-I approximation in Fig. 67. The solution of the twodimensional problem (infinite extent of the wall along the y axis in Fig. 67) is obtained from Eq. (134):

H(r) = (40/2.)r/r2, (201) where H and r are vectors within the x-y plane in Fig. 67. For a straight wall of constant thickness p , the specific dipole moment is given by 4o = Mb, where M is the wall’s magnetization, which is assumed to be uniform and perpendicular to the sample surface. Assuming wall widths of S = 10-100 nm, one obtains for iron ( p o M = 2.1 T) specific dipole moments of po40= lop8Wb/m. Formally, q50 could also be associated with a magnetic potential or an electric current. In the latter case one would obtain values of I = 18180mA for iron. The basic contributions to the MFM contrast, calculated according to Eq. (201), are shown in Fig. 68 for a working distance which is equal to the wall width d. Already at this distance the stray field profile is much wider than the wall. This result can again be attributed to the general phenomenon of loss in higher Fourier components, as discussed in Section IV.C.1. The fact that the stray field profiles calculated according to the different approaches shown in Fig. 67 are almost the same a few hundred nanometers above the sample surface is the reason why contrast modeling according to the simple type-I approximation yields a satisfactory agreement

-5

I

-4

I

-3

I

-2

I

-1

I

0

x/A

I

1

I

2

-

H;

I

3

I

4

5

FIGURE 68. Field components contributing to the M F M contrast of a 180” Bloch wall. Mis the spontaneous magnetization and 6 the characteristic wall width. z denotes the working distance.

FUNDAMENTALS O F NON-CONTACT FORCE MICROSCOPY

179

with experimental data (Hartmann and Heiden, 1988; Hartmann, 1989~; Hartmann et al., 1991). The additional field quantities required for MFM contrast modeling are shown in Fig. 69. A second basic wall type occurs in very thin ferromagnetic films. Within these Nee1 walls (see lower part of Fig. 67), the magnetization rotation is 0.20

I

I

I

1

I

z/a= 1

I

/'?,

0.10 CL]

b-0

n

0.00

I-\

I

I

I

'

I

,/ ;

x-

> I

I

I

'

\

',

\I'

',

____------------_______

5

2 W

-.lo

-.20

(a)

-.30(b) -.35

-14

-13

-12

I

I

0

-'1

,

1

,I

, I '

-

I

I

I

I

I

',I

I

'

1

I

_ - BH,/Bx I

2

I

3

-_-

ffH,/Bz' B'H,/Bx'

1

I

4

-

1

FIGURE69. Same as in Fig. 68, but referring to the first (a) and second (b) field derivatives of the Bloch wall.

180

U . HARTMANN

perpendicular to that of a Bloch wall (see Chikazumi, 1964; Hubert 1975). This mode of rotation leads to a reduction in magnetostatic energy. Stray field calculation can be performed using the same basic approaches as for the Bloch wall. With respect to the type-I approximation for the Bloch wall, the dipole plane has to be rotated by 90" to obtain the corresponding approximation for the Nee1 wall. The stray field components are thus obtained by the following transformation procedure:

The stray field of interdomain boundaries in materials of finite thickness t is then obtained by the transformation H&:

z ) -i ffx,,(x, z ) - f L , , ( X , Z

+ 4.

(203)

7 . The Detection of Electrical Currents A filament which carries an electrical current I exhibits the magnetic field

H

=

(1/27r)1 x r / r 2 ,

(204) where r is the radial vector with respect to the center of the filament. The field shows the same decay rates as derived for interdomain boundaries in the previous section. Equation (204) permits an estimate of the sensitivity of MFM with respect to the detection of electrical currents. For a bulk iron probe with a semiaxis domain length of R , = 500nm and an aspect ratio of Q = 0.5 (see Section 1V.B.I for a description of the parameters) which is raster-scanned at a height of 10 nm across the filament, the minimum detectable current is 350pA in the dynamic mode of operation if a compliance sensitivity of N/m is assumed. The effective probe diameter derived from the interaction decay ranges (see Section 1V.B.1) amounts to A = 20 nm, which is a first-order approximation for the obtained lateral resolution. If instead of the filament a conductor of rectangular cross-section is considered, the vertical field component is given by

H,

=

z:

-(-1. 27rwt

+ xf

2

XL2+ZT2 x,2+zy

z; - - In

(arctan 7 z: - arctan x,

2

x;2+z;2 x,2+z;2

-

arctan

which is a standard solution of Laplace's equation for the magnetic

181

FUNDAMENTALS OF NON-CONTACT FORCE MICROSCOPY 0.16

e

I

I

8

1

0

I

I

I

I

I

I

I

9

-

11

z/w = 0.1

0.140.1 20.10-

1

8

I1

I1 I1 I 1 I1 I I I 1 I 1

t / w = 0.1

0.08-

I # I Ill I Ill

0.06-

-

I Ill

0.04-

-

0.02-

0.00

-

-.02-

-.04-.06-

I 1 \ I I1

-5

__

I'

-.08-

-.lo

-

I

14

-I3

I

-2

~

-1

1

0

~

1

1

filament

- rectangular 2

~

3

1

4

~

5

1

'

FIGURF 70 Magnetic field components produced by a conductor of rectangular crosssection (thickness 1, width WJ) and by a filament of infinitesimal cross-section. x and z denote the lateral and vertical coordinates, and I denotes the applied current.

vector potential (Morse and Feshbach, 1953). Using the abbreviations z: = z f t / 2 and x$ = x 5 w / 2 , the in-plane field component Ci, is also obtained from Eq. (205), provided that i$ + x: and x i + z:. Both coordinates x and z are measured from the center of the conductor. The basic contributions to the MFM contrast for both the filament and the rectangular conductor as shown in Fig. 70. The smaller decay rate of the field produced by a conductor of finite size increases the effective interaction ranges experienced by the probe, and thus leads to an enhanced current sensitivity of the MFM at reduced lateral resolution with respect to the filament. A preliminary experiment (Goddenhenrich et a/., 1990b) has demonstrated the potential of MFM to detect electrical currents in microfabricated planar devices. The local imaging of inhomogeneous current distributions in materials and lithographically prepared devices is considered an especially promising new field of application to MFM. D . Sensitivity, Lateral Resolution, and Probe Optimization Concepts

The main strength of MFM, compared to the various other magnetic imaging techniques, is its capability to achieve high spatial resolution (typically better than a hundred nanometers) on technically relevant samples with little or no

1

~

1

182

U. HARTMANN

sample preparation. The ability to handle real-world samples, complete with overcoats and substrates, greatly simplifies the imaging process with respect to, for example, electron microscopic techniques. Major improvements concerning the lateral resolution mainly rely on new concepts of probe fabrication. If the dipole-dipole interaction between two ferromagnetic spheres at a center-to-center separation of 10 nm is considered, 4,300 Bohr magnetons per sphere yield a force which is just in reach of present technology for a microscope which is operated in the static mode. If iron spheres are considered, the corresponding radius of a sphere would be 3.8nm. In the dynamic mode of operation, 2,150 Bohr magnetons would be detectable, corresponding to a radius of 2 nm for the iron spheres. If the monopole interaction between two magnetically charged disks is considered, the minimum radius would be 4.3 nm in the static mode and 1 . 1 nm in the dynamic mode. These considerations are of course somewhat simple-minded. However, the derived quantities may well be considered as some ultimate limits of MFM with respect to sensitivity and lateral resolution. Figure 71 illustrates the basic design concepts for optimized magnetic dipole and monopole probes. Another promising probe type is the superparamagnetic probe shown in Fig. 72. Because of weak or even missing shape and crystalline anisotropies, the magnetization within the probe's effective domain exhibits field-induced free Nee1 rotation. Using such a probe, the detected force component is Fd = poVMn-vH,and the detected compliance is Fi = poVM(n.v)(n.vH),where V is the domain volume, M the spontaneous magnetization, H the stray field magnitude, and n the cantilever's unit normal vector. The main difference in contrast formation with respect to ferromagnetic probes is that the interaction is always attractive. A first step toward the fabrication of superparamagnetic probes has been presented by Lemke el al. (1990). While the aforementioned

optimized ferromagnetic probes FIGURE 71. Design for optimized ferromagnetic force sensors

FUNDAMENTALS O F NON-CONTACT FORCE MICROSCOPY

183

FIGURE 72. Schematic of a superparamagnetic probe.

optimization concepts are concerned with advanced probe geometries, it seems also promising to look for other probe materials. Antiferromagnetic, ferrimagnetic, and metamagnetic materials (see, for example, Chikazumi, 1964) may be promising alternatives if it is possible to restrict their net magnetization to the near-apex regime of the probe. However, little information is available concerning the size-affected magnetic behavior of these materials close to the apex of sharp tips.

E. Scanning Susceptibility Microscopy Scanning susceptibility microscopy (SSM) is proposed as a new technique which is closely related to MFM (Hartmann et al., 1991). The highly focused microfield of ferromagnetic probes is used to induce a magnetic response of the sample. If the sample is nonmagnetic but conducting, the probe which is vibrating close to the sample surface generates eddy currents in the near-surface regime of the sample. This leads to repulsive forces between probe and sample, which depend on the electric conductivity of the sample at a local scale. If the sample is a soft ferromagnet, SSM is capable of detecting the static and dynamic susceptibility of the sample perpendicular to its surface. In this case the attractive magnetostatic component interplays with the repulsive eddy current component. Since the magnetic susceptibility is a complicated function of field frequency and

FIOURE

73. Schematic of a sensor suitable for scanning susceptibility microscopy.

184

U. HARTMANN

magnitude, it is desirable to equip the SSM with a soft magnetic tip which is polarized by an exciter coil, as shown in Fig. 73. An interresting application to SSM is the investigation of superconductors. A first step toward a calculation of the forces arising when a ferromagnetic microprobe is approached to a superconductor was recently presented by Hug et al. (1991). The probe was modeled by a magnetic point charge, and the sample was considered as an ideal London superconductor, where full account has been taken for the finite penetration depth XL. Certain limitations of the model result from the fact that the probe is assumed to be a magnetic monopole of fixed moment. The detailed analysis of the magnetic behavior of real MFM probes, presented in Section IV.B, however, has shown that the stray field does not simply exhibit a monopole character, but also contains considerable dipole components, especially when the probe-sample separation becomes comparable with the dimensions of the effective apex domain. In the following a model is presented that accounts on the one hand for the finite probe size and on the other hand for the presence of vortices in the superconductor. With respect to the rigorous London model (Hug et al., 1991), the real situation is simplified by assuming complete flux penetration into the superconductor up to a depth equal to XL and complete flux expulsion beyond XL. The magnitude of the probe-sample interaction derived in the following is thus a lower limit of the accurate value and approaches the latter for increasing probe-sample separation (Hug et al., 1991). The boundary condition H L ( z = -AL) = 0, corresponding to a complete Meissner effect, is met in the usual way by considering an image probe identical to the real probe and equidistant below the plane z = -AL. According to the effective-domain model presented in Section IV.B, the microprobe is represented by its monopole moment q, and its dipole moment m, and by an effective probe diameter A. The total repulsive force between the probe and its magnetostatic image is thus composed by a monopole-monopole component FMM= (1/4TPo)q2/4(d+ XL)',

(206a)

by a monopole-dipole component FMD= (1/4TPo)qm/4(d+ X L ) ~ ,

(206b)

and by a dipole-dipole component

FDD= ( 1 / 4 ~ ~ & 1 ~ / 8X (L )d~+,

(206c)

where d is the distance of the probe's apex to the sample surface. According to Eqs. (160) q and m are sensitive functions of the effective magnetostatic interaction range 5. Since the real probe interacts with its image, the

FUNDAMENTALS OF NON-CONTACT FORCE MICROSCOPY

185

interaction range is thus equal to the distance between the probe and its image, 6 = 2 ( d + XL). Substitution of Eqs. (160) into Eqs. (170) then yields the force between the superconductor, which is characterized by its London penetration depth XL, and the finite probe, which is characterized by its semiaxis domain length R,, its aspect ratio a , and its saturation magnetization M . It is convenient to have some upper estimates for the forces at hand, which are obtained when the probing tip touches the surface of the superconductor. Assuming XL << R,, one gets FOMM = 7rtr2R:pOM2,

(207a)

FOMD = “FOMM

(207b)

FODD

(3cr2/2)FOMM.

(207c)

These characteristic forces are completely independent of the superconducting material, as long as XL << R , is provided. Corresponding compliance values are defined as Fd = -FO/XL. Figure 74 shows force-versus-distance and compliance-versus-distance curves for the individual contributions according to Eqs. (206). These curves clearly show that it is not justifiable to neglect the monopole-dipole and dipole-dipole contributions to the total

0.400.200.107

0.08: 0.04-

0.020.01

I

0.1

I

0.5

’

i i

# ‘ I

d/hL

1.0

5.0 I m

t

r

10.0

FIGURE74. Normalized force ( F )- and compliance ( F ‘ )-versus-distance curves obtained for the interaction of a ferromagnetic probe of finite size with a superconductor from the flux expulsion model. XL is the London penetration depth. MM, MD, and DD denote the monopole-monopole. monopole-dipole, and dipole-dipole contributions. The normalization constants F, and F,; are solely determined by the probe’s magnetic properties.

186

U. HARTMANN

n

E

'1

NbSep

100.0

W

iL n

Z

C

v

LL

__ -

T=O

0.1

I

I

F' F

FIGURE 75. Force ( F ) - and compliance ( F ' )-versus-distance curves expected for niobium and niobium diselenide if the measurement is performed with a bulk ion probe as commonly used.

force. Both of these contributions exceed the monopole-monopole contribution for a probe of finite size. Force- and compliance-versus-distance curves in absolute units are shown in Fig. 75 for niobium and niobium diselenide at zero temperature. The curves refer to a realistic bulk iron probe with R, = 500nm, (Y = 0.5, and ,uoM = 2.1 T (see Section IV.B.1 for a description of these parameters). Both force and compliance exhibit magnitudes which should easily be detectable, at least from the point of instrumental sensitivity. The complete flux expulsion model a priori excludes the phenomenon of probe-induced vortex nucleation in a type-I1 superconductor. On the

FIGURE 76. FIGURE 76. Schematic Schematic of of aa probe-induced probe-induced vortex vortex nucleation nucleation process. process

FUNDAMENTALS OF NON-CONTACT FORCE MICROSCOPY

187

other hand, it has been shown in Section 1V.B that sharp ferromagnetic probes may produce quite strong near-apex stray fields which may by far exceed the lower critical field of several type-I1 superconductors. The result could be a partial flux expulsion combined with a partial flux penetration through vortices, as shown in Fig. 76. Some evidence for this behavior has been found in experiments concerned with the levitation of a macroscopic magnet over a type-I1 superconductor (Hellmann et al., 1988). Probeinduced vortex nucleation processes should be detectable in terms of sudden peaks in the force and compliance versus distance curves shown in Fig. 75. Returning to the complete flux expulsion model, a preexisting vortex lattice should be detectable in terms of periodic force or compliance modulations obtained upon raster-scanning the probe across the superconductor. It is convenient to introduce a spatially dependent penetration depth X(v) = A L / l N r ) l l (208) where I@(r)l is the magnitude of the Clem order parameter introduced in Eq. (194b). Figure 77 shows the spatial dependence of X for niobium and niobium diselenide obtained for a center-to-center spacing of the vortices which equals twice the London penetration depth XL. While for

0

FIGURE 77. Variation of the spatially dependent penetration depth A. AL denotes the London penetration depth and tl,the variational core-radius parameter. The solid curve refers to niobium diselenide and the dashed curve to niobium.

188

U. HARTMANN 1

FIGURE 78. Relative force variations for niobium and niobium diselenide obtained upon scanning the probe across an isolated vortex. The broad profiles are produced by a conventional probe, as presently used, while the sharp profiles refer to the optimized probes shown in Fig. 71.

niobium diselenide no substantial overlap of the vortices is obtained, niobium already shows strong overlap. Now, if the probe is raster-scanned across the superconductor, the presence of the vortices affects the manifestation of a complete magnetic image of the probe within the superconductor. Within the present linear approximation, this effect can be accounted for by replacing XL in Eqs. (206) by X(r) from Eq. (208). The resulting variation in force obtained upon raster-scanning the probe across an isolated vortex is shown in Fig. 78. Because of the finite probe size, which has been fully accounted for, the profile of the vortex is considerably broadened. However, the relative variation in force is about 10% for niobium diselenide and about 35% for niobium considering a realistic bulk iron probe. These values would also be obtained if vortex nucleation takes place during recording of a force-versus-distance curve as shown in Fig. 75. Vortex nucleation should thus clearly be detectable. The sharp profiles additionally indicated in Fig. 78 model the contrast which would be obtained by using the optimized probes shown in Fig. 71. F. Applications of Magnetic Force Microscopy

The very first magnetic force microscopy (MFM) images were obtained by Martin and Wickramasinghe in 1987. In a poineering experiment they

FUNDAMENTALS OF NON-CONTACT FORCE MICROSCOPY

189

imaged the field profile of a thin film magnetic recording head which was driven by an oscillating head current. In this way they obtained images which were free from nonmagnetic artifacts. Using a slightly modified separation technique, Schonenberger et al. (1 990) simultaneously measured the magnetic and topographic structure of a recording head. Another issue of particular importance in magnetic recording technology is the performance of the recording medium. Thus, it is not surprising that the high-resolution analysis of recording media has received considerable interest in the MFM literature (see Sarid, 1991). An excellent review and extension to the field of longitudinal media has been given by Rugar el a1 (1990). Imaging of stray fields produced by recording media is of great technical importance since contrast formation is closely related to the readback operation of the recording head. The parasitic “media noise,” the erasure, and overwrite behavior have been studied in great detail for various media (Rugar r t al., 1990; Sarid, 1991). Media exhibiting perpendicular anisotropy are of particular importance to magneto-optic recording. In fact, magneto-optic materials were among the first samples examined by MFM (Martin et al., 1988). One of the most recent investigations in this field is the MFM analysis of multilayer structures (den Boef, 1990), which are quite promising candidates for erasable magneto-optic storage schemes. Natural domain structures generally have a much more complex topology than artificially created magnetization patterns in storage media. The analysis is important with respect to both basic research and technical application (Hartmann rr al., 1991). A variety of material has been investigated (see Sarid, 1991). Apart from the aspect of material research the study of naturally established magnetic fine structure provides information on the ultimate capabilities of MFM with respect to sensitivity and resolution. Griitter et at. ( 1990b) observed magnetic features on rapidly quenched FeNdB and claimed a lateral resolution of IOnm. Hobbs et al. (1989) achieved about 25 nm resolution on FbFeCo thin film. A particular challenge of the field of natural domain configurations is the imaging of interdomain boundaries. The very first clear image of an individual domain boundary was obtained by Goddenhenrich et al. (1988), who simultaneously applied MFM and Kerr microscopy to study 180” Bloch walls in iron whiskers. The major problems in imaging interdomain boundaries result on the one hand from the relatively small microfields (Hartmann and Heiden, 1988; Hartmann, 1989c, 1990b, 1990c) produced by the walls, and on the other hand from the perturbative influence of the probe (Goddenhenrich et al., 1988; Hartmann, 1988; Mamin et al., 1989; Griitter et ul., 1990a). Nonetheless, even fine structures within domain boundaries have been resolved (Mamin et al., 1989; Goddenhenrich et al., 1990a).

190

U. HARTMANN

As of this writing, the author counts at least 60 publications on MFM and more than 10 groups working in this field. Reviews on various aspects of MFM have been given by Martin et al. (1989), Hartmann et al. (1990), Hartmann (1990), Sarid (1991), Hartmann et al. (1991), and Griitter et ul. (1992).

V. ASPECTS OF INSTRUMENTATION Considering the beautiful high-resolution topographic SMF data, the ultrasensitive measurements of electromagnetic surface forces, the detection of individual magnetic interdomain boundaries, and the verification of the Coulomb field associated with the minute charge corresponding to only one electron, the most obvious question is: How does an instrument look which is capable of providing us with such striking data? The central part of any SFM is the microprobe which interacts more or less locally with the surface of the sample. The microprobe is in most cases a ~~

deflection sensor-

m s i c r o p r o b e flexible cantilever sample

1

I

FIGURE 79. Cantilever deflection sensor schemes used in force microscopy. Basically, probe-sample interactions are converted into a physical quantity Q . With electron tunneling, Q corresponds to the current J between cantilever and tip counterelectrode which are at angstrom separation. With optical interferometry, Q is determined by the intensity I of interfering light beams. The beam deflection technique relates Q to the position-dependent intensity I measured with a two-element photodetector for a light beam reflected o n top of the cantilever. The capacitance detector measures deflection-induced changes in capacitance between cantilever and reference electrode, yielding Q in terms of C. The bimorph piezosensor directly converts deflections into a voltage induced between its electrodes, thus relating Q to V .

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sharp tip, while for some special applications smoothly curved probes have been used (Hartmann et al., 1992). The force-sensing probe must have, apart from its well-defined geometry, certain material properties, such as conductivity, dielectric permittivity, permanent magnetization, etc., which determine the type and strength of the interaction with the sample. The microprobe is attached to - or part of - a soft cantilever spring whose geometrical and material properties are chosen such that the respective interaction between probe and sample is detected via measuring the cantilever deflection. Although such a tip-cantilever ensemble is part of all force microscopes, the details of implementation vary. The original AFM (Binnig et al., 1986), for example, used a handmade cantilever spring formed from a piece of gold foil approximately 1 mm long. A small diamond stylus glued to the foil served as tip. Some of the best sensors for electric and magnetic force microscopy have been fabricated from fine, electrochemically etched wires. Today, the most advanced SFM cantilevers are microfabricated from silicon, silicon oxide ( S O z and Si203), or silicon nitride using lithography and etching techniques well established in micromechanics. Typical lateral cantilever dimensions are of the order of l00pm with thicknesses of the order of I pm. Typical spring constants are in the range of 0.1 to I N/m, and resonant frequencies are 10 to 100 kHz. The other critical component of the SFM is the sensor that detects the cantilever’s deflection. Ideally, the sensor should have subangstrom sensitivity and should exert a negligible force on the cantilever. According to Fig. 79, the deflection schemes which convert the cantilever’s mechanical status into some nonmechanical quantity, are divided into two basic types: electronic and optical systems. Electron tunneling, which was the method originally applied to Binnig el al. (1986), has the virtue of being extremely sensitive: The tunneling current between two conducting surfaces changes exponentially with distance, typically by a factor of 10 per angstrom of displacement. The resolution could thus be as high as A.Although excellent results have been achieved by many groups, tunneiing generally has the disadvantage that its performance can be degraded if the tunneling surfaces become contaminated. Another electrical method consists of the capacitance detection system. The basic philosophy is to monitor the cantilever deflections by measuring the varying capacitance between the free end of the lever and a fixed reference electrode. The virtue of this method is that it is relatively simple to implement even under UHV and low-temperature conditions. Goddenhenrich et al. (l990a) demonstrated an instrument optimized for magnetic imaging, while Neubauer et al. (1990) presented a dual capacitance sensing system for simultaneous measurements of vertical and lateral force

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components. An advantage of the capacitance detection method is that it is by far not as sensitive as electron tunneling to local surface imperfections of cantilever and reference electrodes. However, the large effective electrode area also causes the main drawback of the capacitance detection system. The interelectrode spacing is limited by the “snapping distance” at which the derivative of electrostatic and surface forces is equal to the lever’s spring constant. Decreasing the interelectrode spacing below this distance causes a jump to contact of the lever. The performance of the capacitance detection SFM is thus mainly limited by appreciable electrostatic and surface interelectrode forces. The third electrical cantilever-deflection sensor scheme is the bimorph piezosensor (Anders and Heiden, 1988), which directly converts, via the piezoelectric effect, deflections into a voltage between the electrodes of the piezoelement. This elegant method has suffered so far from the problem of tailoring bimorph piezoelements to cantilevers of suitable spring constant. The tendency, however, to microfabricate complete scanning probe microscopes on a chip (see, for example, Quate, 1990) will most likely increase the importance of the bimorph piezocantilevers, which are well suited for integrative microfabrication techniques. Optical deflection-sensing schemes are subdivided into two basic classes (see Fig. 79): beam deflection and interferometry. Optical methods average the rough surface of a cantilever and exert an almost negligible force which generally makes them the better alternative with respect to the electrical detection schemes. In a beam deflection system (Meyer and Amer, 1988; Alexander el al., 1989), a collimated laser beam is focused on the lever and is reflected into a two-segment photodetector: The photocurrents of both elements of the detector are fed into a differential amplifier whose output signal is proportional to the cantilever’s deflection. All optical elements are at large distances from the force sensing lever. That offers advantages for some implementations (e.g., UHV) of the instrument, but also raises the sensitivity to externally caused directional fluctuations in the optical paths. The beam deflection has been successfully used in many experiments and was employed in the first commercially available SFMs. Interferometer-based systems have taken various different forms. In a homodyne detection system, the flexible cantilever beam and a fixed optical flat form a Fabry-Perot-type interferometer (McClelland, 1987). The incident laser beam first passes through a beam splitter and is then incident on the Fabry-Perot. The beam reflected back from the latter is incident on the same beam splitter, by which part of it is deflected into a photodetector. In a differential homodyne system, a fraction of the laser power, serving as a reference beam, is diverted by a beam splitter to a first photodetector. The

193

FUNDAMENTALS OF NON-CONTACT FORCE MICROSCOPY

light passing through the beam splitter, serving as a signal beam, is incident on the Fabry-Perot, reflected back, and deflected into a second photodiode. The differential signal of the two photodetectors is used to image the force acting on the probe. Stability problems due to the large physical path difference between the reference beam and the light reflected back from the cantilever have been surmounted by using a fiber-optic technique (Rugar, 1989) that places a reference reflector within microns of the cantilever, or they have been overcome by using a polarization interferometer of Nomarski type (Schonenberger and Alvarado, 1989), splitting the incident light beam into two orthogonally polarized fractions reflected off the free and supported ends of the cantilever. Heterodyne detection systems (Martin and Wickramasinghe, 1987) completely eliminate the effects of the drift in the optical path length. This is achieved by introducing a frequency shift between reference and signal beams employing an acousto-optical modulator. On the one hand, such a system exhibits an unprecedented sensitivity and stability, and on the other hand, it consists of a relatively complicated and expensive setup. A special kind of an interferometric deflection sensing scheme is the laserdiode feedback system (Sarid et al., 1988). Lasers are known to be extremely sensitive to optical feedback. Even a minute amount of light emitted by the laser that is fed back into its cavity can affect its operation drastically. In the laser-diode feedback system, the cantilever is positioned a few microns from the front facet of a laser diode. The lever and front facet act as a lossy Fabry-Perot, whose reflectivity determines the effective reflectivity of the front facet of the laser. A deflection-dependent laser power results, which is then measured with a photodetector which is integrated into the rear facet of the laser diode. The laser-diode feedback system has the virtue of being simple to assemble and align and involves only a few components. The theory of operation, however, is more complicated than for the other systems because of the highly nonlinear behavior of the laser. Today, most implementations of non-contact SFM are based on a dynamic mode of probe-sample interaction sensing rather than on a static mode as employed in the original AFM and most of today’s contact-mode applications of SFM. In the dynamic mode the cantilever is driven e.g., by a small piezoelectric element to vibrate near its resonant frequency. The presence of a nonuniform force F acts to modify the cantilever’s effective spring constant according to k = ko d F / d z , where ko is the spring constant of the isolated cantilever and d F / a z is the force gradient’s component along the cantilever’s normal vector. If one exerts, for example, an attractive force on the cantilever, the spring will effectively soften. As a result the resonant frequency will decrease, and this decrease is detected by measuring the amplitude, phase, or frequency shift of the vibration. -

~

+

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There are mainly three advantages of the dynamic mode of SFM operation: (i) The operation is removed from the regime where llfnoise has a significant contribution; (ii) the use of a phase-sensitive detection method increases the signal-to-noise ratio; and (iii) it is possible to use cantilever resonance enhancement to greatly increase the sensitivity. Martin et al. (1987) demonstrated a sensitivity to forces of 3 x N, where forces were obtained by numerically integrating the force derivatives obtained from force-versus-distance curves. Such a sensitivity would hardly be obtainable by a static measurement. Albrecht et al. (1991) demonstrated a sensitivity to force derivatives of 9 x 10-'N - a value very close to the ultimate thermal limit - using a feedback-driven lever. Schonenberger et al. (1990) demonstrated the advantage of simultaneously having information about both the static and dynamic cantilever status. That allows one, for example, to completely separate electro- and magnetostatic components of the probe-sample interaction. Apart from the tip-cantilever ensemble and the deflection-sensing scheme, three additional components are required for an SFM: (i) a feedback system to monitor and control changes in the cantilever status and, hence, in the force or the force derivative; (ii) a mechanical scanning system - usually piezoelectric - to move the sample with respect to the tip vertically and in a lateral raster pattern; and (iii) a display system to convert the measured data into an image. The scanning, feedback, and display systems are essentially the same as for an STM. The schematic diagram in Fig. 80 shows the basic setup of an SFM as operated in the constant interaction mode, which is preferred in most applications. With respect to the mechanical setup of an SFM system, care must be taken that its performance is not limited by external vibrations, such as

&

Error I

V(

Feedback z)=const

X, y ,

I

1

>

I

z-Piezo

Position

Probe-Sample Arrangement

f

FIGURE 80. Typical setup of a scanning force microscope operated in the constant-force or constant-compliance mode.

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from the laboratory building. The effect of external vibration is to cause unwanted motion of the tip with respect to the sample and deflection sensor. The ultimate immunity of the SFM to external vibration is determined by the ratio of the excitation frequency v to the lowest resonant frequency vo of the SFM. Hereby, the mechanical system consists of both the lever and the rest of the instrument. The amplitude of unwanted probe motion is attenuated by a factor of (v/v0)*for v << vo. Hence, for an external perturbation of frequency v = 10 Hz and amplitude 1 pm - which may be considered as a typical building vibration - an SFM of lowest resonant frequency vo = 10 kHz would exhibit a probe motion of 0.01 A. This level is well below the thermal vibration rms amplitude of 0.6A obtained for a 1 N/m cantilever at room temperature. Because cantilevers can be readily made with high resonant frequencies (some tens of kilohertz), the art of building good SFMs consists of making the mechanical components rigid and compact, especially in the path from cantilever to sample. An extensive discussion of details of operation and noise sources is found in the book by Sarid (1991).

VI. CONCLUSIONS Because of the rapid growth of the field of SFM in general, it does not appear to be very simple to predict the major future perspectives in this field and in particular for non-contact SFM. On the other hand, reflecting on the past accomplishments and critically analyzing the underlying theories for the various SFM applications presented in this work allows one to draw some conclusions concerning dominant future applications in science and technology, further instrumental improvements, and its ultimate capabilities. One of the definite scientific goals in SFM is to completely understand force-versus-distance curves. Starting at large probe-sample separation, i.e., in the pre-contact regime, then proceeding through the contact regime involving elastic and eventually plastic deformation of probe and sample, and subsequently going back to large probe-sample separation involves interactions caused by all of the electromagnetic contours of the surface of a solid indicated in Fig. 1. Even the pre-contact regime has not yet been completely understood, as the recent paper by Burnham et af. (1992) emphasizes. Major applications of non-contact SFM will clearly be provided by our efforts to completely understand the interaction between two solids at arbitrary separation and with an arbitrary intervening medium. Other more specific scientific applications of non-contact SFM will

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predominantly include the investigation of electric and magnetic microfields resulting from highly localized charge and spin arrangements. Because the major strength of non-contact SFM is the detection of electric and magnetic microfields and their separation from - or relation to topographic peculiarities, the major technological applications will consequently also be in that area. Engineers can make quantitative measurements on magnetic storage components and can obtain electric potential maps from integrated circuits. The trend is already toward developing instruments capable of handling complete magnetic disks or silicon wafers. The rate of instrumental improvement now, about seven years after the demonstration of the first force microscope (Binnig et al., 1986), exhibits a certain decrease. The first excellent commercial general-purpose instruments are available. These instruments are easy to use, and hence open SFM to a broad community of potential users. Nevertheless, considerable technical challenges do still exist. Among these is certainly the implementation of SFM under UHV or/and low-temperature conditions (see, for example, GieBibl et al., 1991). In particular, for non-contact SFM, the routine low-temperature implementation is of great importance because that would make the interesting electromagnetic properties of superconductors accessible to local investigations. A lot of engineering effort will also concentrate on fabricating tailor-made probes for certain applications. That will involve both the search for new probe materials and the improvement of microfabrication techniques. Ultimate capabilities of non-contact SFM follow directly from the presented theory, even if not all potential experimental capabilities have yet been achieved. These ultimate capabilities are all related to sensitivity and spatial resolution. The basic limit is provided by thermal noise. It is fairly obvious that for a perfectly designed instrument, thermal noise limits the sensitivity to forces or force gradients. Thermal noise, however, also ultimately limits the spatial resolution. For any given kind of interaction between probe and sample, two neighboring points of the sample surface are distinguishable only if they produce distinguishable signals. If future technologies allow SFM probes to be fabricated with arbitrary sharpness while keeping satisfactory mechanical properties, the required probe radius is solely determined by the requirement that the probe has to be large enough so that the interaction variation across the sample is well above the thermal noise limit. Sensitivity to forces or force gradients and spatial resolution are thus unequivocally related in non-contact SFM. From this universal relation it then follows that it would never be possible to image individual spins by magnetic force microscopy, but it is, for example, indeed possible to image the equivalent of only one electron charge smeared out over a certain area (Schonenberger and Alvarado, 1990b).

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ACKNOWLEDGMENTS Thanks are due to C . Heiden, KFA-Jiilich/University of GieBen, for his continuous interest in the present work and many helpful discussions. Part of the theory involved was developed during a seven-month stay at the Institute of Physics of the University of Basel, Switzerland. The author would like to thank H.-J. Giintherodt and his group, especially R. Wiesendanger, for the inspiring atmosphere. The author is indebted to N. A. Burnham, KFA-Jiilich, for her permanent scientific contribution and for the careful proofread of the manuscript. Thanks are due to E. Brauweiler for her technical assistance in the preparation of the manuscript.

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