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Theory of Image Formation by Inelastically Scattered Electrons in the Electron Microscope
ADVANCES IN ELECTRONICS A N D ELECTRON PHYSICS. VOL. 65
Theory of Image Formation by Inelastically Scattered Electrons in the Electron Microscope H. KOHL AND H. ROSE lnstitut fur Anyewandte Physik Darmstadt. Federal Republic of Germany
I. Introduction . . . . . . . . . . . . . 11. The Mixed Dynamic Form Factor . . . . . . A. The Phase Problem in Electron Scattering . . B. Properties of the Mixed Dynamic Form Factor C. The Generalized Dielectric Function . . . . 111. Theory of Image Formation . . . . . . . . A. Image Formation in STEM . . , . . . . B. Image Formation in FBEM . . . . . . . C . The Reciprocity Theorem. . . . . . . . D. Phasecontrast . . . . , , . , . . . IV. Numerical Results. . . . . . . . , . . . A. Image of a single Atom in an FBEM. . . . B. Image of a Single Atom in STEM , . . . . C. Images of Assemblies of Atoms . , . . . . D. Image of a Surface Plasmon . . . . . . . V. Conclusion . . . . . . , . , , . . . . Appendix. . . . . . . . . . . . . . . References . . . . . . . . . . . . . .
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180 183 185 185 189 191 193 195 195 200 205 21 1 213 214 224
I. INTRODUCTION The main objective of electron microscopy is to obtain high-resolution images of thin specimens. Modern instruments yield point-to-point resolutions of about 3 A. Thus, it is possible to elucidate the properties of the object on an ultramicroscopic scale (Bethge and Heydenreich, 1982; Reimer, 1984). In such an instrument, the image is formed by electrons, which are scattered in various directions within the objective aperture. In electron energy loss spectroscopy (EELS), we measure the intensity of inelastically scattered electrons as a function of energy loss and scattering I73 Copyright 8 1985 by Academic Press, Inc. All rights of reproduction in any form reserved. ISBN 0-12-014665-7
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H. KOHL AND H. ROSE
angle. This technique is widely used in solid-state physics to obtain information about the dynamic properties of the specimen as a whole. For recent reviews see Raether (1980), Schnatterly (1979), Silcox (1979), and Colliex (1984). Recently, both techniques were combined under the name of analytical electron microscopy. Thus, it is now possible to obtain energy-loss information from a small volume of a few cubic nanometers. By registering only those electrons that have suffered a characteristic energy loss (say a K-loss corresponding to a particular element), we can quantitatively determine the elemental distribution in an object (Isaacson and Johnson, 1975; Egerton, 1979, 1984; Joy, 1979; Maher, 1979; Colliex, 1984; Colliex and Mory, 1984). An extensive bibliography has been given by Egerton and Egerton (1983). Experimental (Isaacson et al., 1974) and theoretical (Rose, 1976a,b) investigations show that images taken with inelastically scattered electrons are blurry as compared to an elastic image of the same object. This “delocalization of the interaction” makes the interpretation of inelastic images more difficult. Despite the striking progress in experimental abilities, a detailed theory of the process of image formation is still lacking. Most reviews on image formation treat the contribution of the inelastically scattered electrons as a deleterious side effect (Misell, 1973; Humphreys, 1979). In view of the recent progress in analytical electron microscopy, it seems worthwhile to examine image formation by inelastically scattered electrons in more detail-first, the scattering properties of the object (Section 11) followed by electron-optical considerations (Section 111) and image calculations (Section IV). 11. THE MIXEDDYNAMIC FORMFACTOR
A . The Phase Problem in Electron Scattering
The standard geometry for a scattering experiment is shown schematically in Fig. 1. The object is illuminated by an incident plane wave with wave vector k,. A far-field detector registers all electrons that have been scattered in the direction k, and determines their energy and angle of scatter. For simplicity, we shall restrict our discussion to the case of purely elastic scattering. The differential cross section dg/dQ gives the probability that an electron is scattered into the solid angle dQ. It is given by where f ( 0 , 4 ) denotes the scattering amplitude for the direction determined by the polar angle 0 and azimuth 4. The differential cross section is equal to
THEORY OF IMAGE FORMATION
i
175
.c ko
object
detector FIG.1. Conventional arrangement for diffraction measurements; K = k, - k, is the scattering vector, and k, and k, are the wave vectors of the incident and the emergent electron, respectively.
the square of the modulus of the scattering amplitude. For fast incident electrons with wave vector k,, scattered by a thin object, we can use the firstorder Born approximation to calculate the scattering amplitude. For scattering in the direction k,, we then find f(K)
=
-
m &
1
V(r) exp(iKr) d3r
where K = k, - k, denotes the scattering vector, V(r) the object potential, m, the electron mass, and 12 Planck’s constant. Thus, to first order, the scattering amplitude is proportional to the Fourier transform of the object potential. As can be seen from Eq. (l), we can extract only the modulus, not the phase, of the scattering amplitude from a diffraction experiment. Thus it is impossible to determine unambiguously the object potential from experimental diffraction data. This so-called phase problem has grave consequences for diffraction experiments, because there are an infinite number of functions with a prescribed modulus of the Fourier transform. (For example, a translation leads to a shift in phase of the scattering amplitude.) Even with additional information, it is often impossible to determine the object potential unambiguously. For example, it is impossible to distinguish between the potentials V(r) and P(r) = V ( - r), because T(K) = f’*(K), and
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H. KOHL AND H. ROSE
thus lY(K)I2 = If(K)12. This fact has serious consequences for crystallography, where a noncentrosymmetric structure cannot be distinguished from its inverse, even though the types of atoms in the unit cell and their scattering properties may be known. Thus, even with considerable amount of a priori information, we cannot determine the object structure by a scattering experiment as shown in Fig. 1. If we consider inelastic scattering, the situation is even more complicated. It is not possible to determine the spatial structure of the transition density of the excitation by examining the scattering data. All we can do is compare the measured data with theoretical calculations. However, as we have seen in the case of elastic scattering, agreement between theory and experiment does not necessarily mean that the theoretical model is correct. Thus, a method to measure the phase of the scattering amplitude is of great experimental interest. Let us now discuss a setup that allows us to determine the phase of the scattering amplitude (Fig. 2). A biprism splits the incident wave into two
K' K
FIG.2. Arrangement for determining phase effects in elastic and inelastic scattering. The object is illuminated by a coherent superposition of two plane waves with wave vectors k and k . The detector registers the intensity in the direction of k,. [From Kohl (1983a). Copyright 1983 North-Holland Publ. Co., Amsterdam.]
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177
FIG.3. Intensity distribution in the specimen plane when two inclined plane waves with relative phases 4 = 0 or 4 = n are impinging on the object. Each dot represents an atom, and a is the lattice constant. The origin is located on an atom. The directions of incidence of the two waves enclose an angle i j a . [From Kohl (1983a). Copyright 1983 North-Holland Publ. Co., Amsterdam.]
coherent plane waves, which form interference fringes in the object plane (Mollenstedt and Diiker, 1956). The initial state of the electron is then given by (1/42)(exp(ikr)
+ exp(i4) exp(ik’r))
(3)
where k and k‘ are the wave vectors of the two incident plane waves impinging on the object. The factor ei4 determines the phase relation between the two waves. To understand its meaning a little better, let us first calculate the intensity distribution in the object plane. It is given by I(r)
=
1 + cos[(k - k ) r - 41
(4)
The vector r denotes the position on the object. Suppose the object is periodic with a lattice vector ae,. We adjust k and k’ so that k - k = (2n/a)e, is a reciprocal lattice vector.’ Then the interference fringes have the same periodicity as the object. The phase 4 determines the position of the fringes on the lattice (Fig. 3). Depending on 4, either the atoms will be illuminated predominantly (4 = 0) or else the space in between ( 4 = n). The probability of registering an electron in the detector will depend on the position of the interference fringes on the object. The count rate of the detector is proportional to
1.m)+ exP(i4>.f(K’)l2
(5)
where K = k - k, and K’ = k’ - k, denote the two scattering vectors. By measuring the intensity as a function of K and K‘, we can obtain the phase of the scattering amplitude (Gabor, 1957; Hoppe, 1969a,b; Hoppe and Strube,
’ We use the convention a;aj = 2nsij, where a; is a reciprocal lattice vector and aj a lattice vector in real space. This definition, which is common in solid state physics, differs from the one used in crystallography (a;aj = S i j ) by a factor of 2n.
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H. KOHL AND H. ROSE
1969; Berndt and Doll, 1976, 1978, 1983). Formula (5) shows that an experiment as depicted in Fig. 2 cannot be completely described by calculating cross sections. The phases play an important role in this type of experiment. Let us now consider inelastic scattering events. The detector is assumed to register the electrons as a function of energy and angle of scatter. T o describe the scattering process, we must use the product states of the incident electrons and the object state. (The influence of exchange is neglected here.) The initial state is given by
+ exp(i4) exp(ik’r)] Im)
(l/J2)[exp(ikr)
(6)
where Im) denotes the initial state of the object. Here the object is assumed to be initially in a pure state. The final state is equal to eikf‘In). For fast electrons and thin objects, we can use the first-order Born approximation to determine the count rate. Since the detector registers only the direction and energy of the scattered electron but not the final state of the object, we must sum over all possible final object states with a given energy. Using Dirac notation
I kfn)
= eikf’In>
we obtain for the transition rate from the initial to the final state
+ exp(i4)( km I V I kf n ) (kfn I V I k’m) + exp( i&)(kml Vlkfn)(kfnl Vlkm))6(wm -
-
+
o, o)
(7)
Here V denotes the interaction potential between the incident electron and the object, while hw, and hw, are the energies of the initial and final state of the specimen, respectively. The first two terms stem from the scattering of the partial wave from k (k’) to k,. They appear likewise in a calculation of a cross section. The following two interference terms contain all spatial information about the object. For scattering experiments as shown in Fig. 1, the object properties can be described conveniently by means of a density-density correlation function (Van Hove, 1954; Platzman and Wolff, 1973). We shall generalize Van Hove’s result to the case of two (or more) interfering incident waves. With V
=
C V(r i
-
Ti)
THEORY OF IMAGE FORMATION
179
where V(r - ri) describes the interaction of the incident electron with the ith particle, we obtain
where V(K) =
s
V(r) exp( - iKr) d3r
is the Fourier transform of the interaction potential V(r). For Coulomb interactions, we obtain V(K)
= e;/EoK2
where - e , is the charge of the electron and c0 the dielectric constant of the vacuum. In Eq. (8), S(K, K’, 0)=
J 2n
m
-m
(pK(t)P-K,(o))T
exp(iot) d t
(9)
is the mixed dynamic form factor (Rose, 1976a,b), where PK(~) =
C~xPC-
iKrj(t)I
j
is the Fourier-transformed density operator in the Heisenberg representation and ( )T denotes the thermal average. In Eq. (8), p m gives the probability that the object was in the state Im) before the scattering took place. By inserting Eq. (8) into Eq. (7), we obtain
+ I V(K’)I2S(K‘,K’, o) + exp(i+)V(K)V*(K’)S(K, K’, o)+ exp( -$)V*(K)V(K’)S(K’,
w = (n/h’)[I V(K)I2S(K,K, o)
K, o)]
(10) The calculations are outlined in Appendix A. Here we have included a thermal average with respect to the initial states of the object. The transition rate w is given by a sum of four terms, each of which is a product of the Fouriertransformed density-density correlation function and two conjugate Fourier transforms of the interaction potential. Thus, Eq. (10) can be viewed as a generalization of the relation given by Van Hove (1954). The last two terms of Eq. (10) contain the function S(K, K’, w ) with K # K’, whereas in a conventional scattering experiment only terms with S(K, o)= S(K, K, o)occur. Our apparatus in Fig. 2, thus, enables us to obtain additional information about the object.
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H. KOHL AND H. ROSE
The first two terms in Eq. (10) can be obtained from a conventional scattering experiment. If we now measure the intensity with k and k’ fixed for two different phases 4, we can solve the two resulting equations for the two unknown values S(K, K’, w ) and S(K’, K, o). This procedure can be repeated for all values of K and K by varying the direction of the incident electrons and the detector. Thus, the function S(K, K’, o)is a quantity that can be measured experimentally (at least in principle). It contains information about the spatial distribution of the excitation within the object. (Subsections I1.B and 1I.C will deal with the properties of the mixed dynamic form factor. The more experimentally oriented reader can bypass them without loss of comprehension.) B. Properties of the Mixed Dynamic Form Factor
In Subsection II.A, we found that the mixed dynamic form factor S(K, K’, o)can be viewed as a generalization of the conventional form factor S(K, o).We therefore ask Which properties of S(K, o) are also true for S(K, K’, w)? First of all, let us consider two sum rules. The integral
j
m
-m
S(K, w> do
=
(PK(o)P-K(o))T
= S(K)
(1 1)
yields the static form factor S(K) (Kittel, 1964). By using Eq. (9) and interchanging the order of integration, we find
j
m
=
-m
=
(12)
Accordingly the right-hand side is called the mixed static form factor. Using the operator identity where K = I K 1, H is the Hamilton operator, and = po the number operator, we obtain the generalized Thomas-Reiche-Kuhn sum rule Jm -00
w{S(K, w )
+ S(-K,
o)}dw
=
hKZ ~
m0
N
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THEORY OF IMAGE FORMATION
Here N denotes the number of particles in the object (Kittel, 1964). By employing the double commutator [ [ H , &], P-K’], we obtain [[& PKI? P-K’I = -(hZKK’/mO)pK-K’
(15)
Taking the thermal average, we find
=
1 -
Z
1 exp(-pEm)(Em
-
m,n
En)((mlPKIn)(nlp-K’lm)
+ (ml P -K’I n>( n I PK Im)) cu
=
-A]
w{S(K, K’, 0) -m
+ S( - K’,
-
K, w ) } dw
where p m = e-DEm/Z denotes the occupation probability, Z the partition sum,
p = l/k, T, k , Boltzmann’s constant, and T the temperature. Thus,
The magnitude of the mixed dynamic form factor is related to the expectation value of the Fourier transform of the electron density for K - K’. The larger the density variation in the object, the more pronounced are the terms for K # K’ in the form factor. In contrast to the conventional dynamic form factor S(K, w), the mixed dynamic form factor is, in general, not a real quantity. Therefore, we cannot simply carry over the proofs for a certain property from S ( K , o ) to S(K, K’, w). Rather, we must check very carefully, if the proof for S(K, w ) makes use of the fact that the form factor is a real quantity. Symmetries of the object lead to further restrictions on the form factor. Suppose the specimen has a center of inversion at the origin. Then,
(dr,t)P(r’))T
=
(p(-r? L)d-r’))T
where p(r, t ) =
1 i
-
ri(t))
is the density operator in the Heisenberg representation and p(r) = p(r, 0). By using pK(t) = p(r, t)exp( -iKr) d3r, we find
S
S(K, K’, W ) = S( - K, - K’, W )
(17)
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H. KOHL AND H. ROSE
For a periodic object with lattice vectors a,, a 2 , a3, the relation ( ~ ( r+ ai, L > P ( ~+’ ai))T
=
S(K, K’, w ) = exp[i(K
K)a,]S(K, K‘, w )
( d r , t)dr’))T
leads to for i
=
-
(18)
1,2, 3. This means that either S(K, K’, w ) = 0, or else exp[i(K
-
K)ai]
= 1
The latter condition is equivalent to the condition K
-
K’ = g, where
3
g = Ch,a,
hiEH
i=l
is an arbitrary reciprocal lattice vector. In short, the mixed dynamic form factor is nonvanishing only if K - K’ = g. This can be understood by looking at Fig. 3. If K - K = g, the fringes have the same periodicity as the object. In that case, we illuminate predominantly one part of the unit cell, e.g., the atoms ( 4 = 0) or the space in between ( 4 = n). Classically speaking, the scattering is then characteristic for that part of the unit cell. If the difference K - K’ is unequal to any reciprocal lattice vector, the fringes on the object have a periodicity different from the lattice. The scattering is then given by an average over the elementary cell. By using a setup as shown in Fig. 2, we can determine the spatial structure of an excitation. If the object is invariant with respect to arbitrary translations, Eq. (18) is valid for any vector a. Then S(K, K’, w ) is nonzero only if the two scattering vectors are equal (K = K ) . In that case the scattered intensity no longer depends on the position of the fringes on the object. For such a free electron gas, the conventional form factor S(K, w ) gives a complete description of the scattering properties. Let us compare the conditions K = g and K‘ = g’ for elastic scattering with the corresponding condition K - K’ = g for inelastic scattering. Electrons scattered elastically can be found only if both scattering vectors coincide with reciprocal lattice vectors (K = g and K = g’). For inelastic scattering, the weaker rule K - K’ = g applies, because for such processes a quasi-momentum hq can be transferred to the crystal. Since we must sum over all possible final states with a given energy, many excitations contribute to S(K, K’, w ) for fixed o. Finally, we shall consider the influence of time-reversal symmetry on the mixed dynamic form factor. The relation S(K, K’, 0) = S( - K’,
-
K, 0)
(19)
THEORY OF IMAGE FORMATION
183
is a generalization of the well-known formula S(K, w ) = S( - K, o)(Kittel, 1964). To apply the time-reversal symmetry, all magnetic fields must be reversed (Landau and Liftshitz, 1965). If such magnetic fields are negligible, Eqs. ( 1 6) and (1 9) yield
For objects that are invariant under time reversal and inversion, the combination of Eqs. (1 7) and (19) yields
S(K, K‘, w ) = S(K’, K, W )
(21)
Since S(K, K’, w ) = S*(K’, K, w ) , this means that in these cases the mixed dynamic form factor is real. We could also exploit the implications of point symmetries of the object. A specific example (complete rotational symmetry) will be given in Appendix C. C. The Generalized Dielectric Function
The dissipation-fluctuation theorem yields a relation between the dissipative part of a linear response function and the fluctuations in thermodynamic equilibrium. The latter are described by correlation functions. Applying this theorem to electronic excitations, we find for the conventional form factor (Pines, 1962, 1964; Kittel, 1964; Geiger, 1968; Platzman and Wolff, 1973; Schnatterly, 1979; Raether, 1980)
S(K, to)
=
-{EOhK2V/nei[1 - exp(-phw)]} Im E-’(K, w )
(22)
where E denotes the dielectric function of the medium, V its volume, and Im the imaginary part. Hence, the dynamic form factor is proportional to the imaginary part of the dielectric response function E - which describes absorption effects in the object. Reading Eq. (22) from right to left, we find that we can calculate Im E - ‘(K, w ) from measured values of S(K, w). The real part can be obtained by use of the appropriate Kramers-Kronig relation. We now try to generalize Eq. (22) to include the mixed dynamic form factor S(K, K’, w). For the derivation, we must keep in mind that S(K, K’, w ) is a complex quantity due to the spatial Fourier transforms. Thus, appropriate care must be taken when generalizing properties from the conventional dynamic form factor to the mixed dynamic form factor. The corresponding linear response function is the generalized inverse dielectric function E&(o), which was introduced into solid-state physics by Adler (1962) and Wiser (1963). This function includes the so-called local field effects, which stem from the atomic structure of the specimen.
’,
184
H. KOHL AND H. ROSE
FIG.^. Spring model to visualize the polarization of a diatomic chain induced by a homogeneous external electric field.
The dielectric function relates the induced polarization to the applied external field. To understand its meaning, we consider a one-dimensional diatomic lattice (Fig. 4). This spring model describes the effect of the different polarizabilities of the two types of atoms. Suppose we apply a homogeneous external field to the object. The polarization itself will not be homogeneous; rather it will be the sum of Fourier components that have the periodicity of the lattice. It is convenient to use the external potential Oext,where E = grad Oexc and the induced charge eOpind = div P for the calculations. For an external potential Oext,the induced charge is given by (Adler, 1962)
+
Here g and g‘ are reciprocal lattice vectors; the indexes q g and q + g’ denote the respective Fourier components. The vector q is restricted to the first Brillouin zone. Calculation of such a dielectric function is so onerous a task that only few numerical results have been reported (e.g., Van Vechten and Martin, 1972; Hanke and Sham, 1975). Nevertheless, this dielectric function is of great interest in solid-state physics. Sturm (1982) has used the nearly free electron approximation to calculate it in the region of the plasmon frequency wp. He then discusses the influence of the band structure on plasmon dispersion and linewidth. For real solids, the charge density is nonuniformly distributed within the unit cell. Consequently the plasmon excitation is not a simple plane wave. Applying Bloch’s theorem, we find that it will be modulated by a lattice periodic function. Thus, there might even be plasmon bands. The conditions under which these could be detected are discussed in Sturm’s review (1982).
THEORY OF IMAGE FORMATION
185
The excited states of an insulator can also be calculated via the dielectric formulation. The lowest excitation energy is not given by the band gap; the electron-hole correlations must also be taken into account. Such excitonic effects have been discussed by Hanke (1978). For nonperiodic objects, the sum in Eq. (23) must be extended over all vectors K’. All other properties remain unaffected. The relationship between the mixed dynamic form factor and the dielectric function is given by S(K, K’, w ) = isOhV/2ne;[1
-
exp( -fiho)](K2~&w) - K’2~,&’(w)} (24)
The derivation is outlined in Appendix B. Since S(K, K’, w ) is a complex function, Eq. (24) is more complicated than the corresponding relation [Eq. (22)]. Also, the dissipative part of the response function is no longer given by its imaginary part (Johnson, 1974). The generalized dielectric response function can be obtained from the mixed dynamic form factor by the relation s&(w)
=
hKK,+ -in
a, ~
0 - 0’
4
s,hK2V ~
[1 -- exp( - Bhw‘)]S(K, K’, 0’) dw’
[1 - exp( - phw)]S(K, K’, w )
Here # denotes the principal value; the integral describes the dispersion, and the following term is the absorption in the object. 111. THEORY OF IMAGE FORMATION
A . Image Formation in S T E M To calculate the intensity distribution in the image, we must first determine the scattering properties of the object. We restrict our investigations here to objects thinner than the depth of field. The first-order Born and the small-angle approximation can then be used to describe the elastically scattered electrons (Cowley, 1975; Colliex and Mory, 1984). To explain image formation, a knowledge of the differential cross section is not sufficient, because it does not contain spatial information. In a semiclassical picture, we can consider the electron as a classical particle with impact parameter b and calculate the probability for the excitation of a particular object state (Fliigge, 1971). This procedure is equivalent to assuming infinite resolution (Ritchie, 1981).
186
H. KOHL AND H. ROSE
In the more general case both the incident electron and the object are treated quantum mechanically (Rose 1976a,b). The degree of coherence between incident plane waves coming from different directions must then be taken into account. Thus, image formation in a scanning transmission electron microscope (STEM) (Fig. 5) can be interpreted as a more complicated version of the biprism experiment depicted in Fig. 2. To calculate the incident wave packet at the object plane, we assume a plane wave in front of the objective lens. The object is located in its back-focal plane. There the wave function of the incident electron with wavenumber k , is given by
J ~ ( eexp~-iy,(e)i ) exp[-ik,(p n
exmoz0)
- po)ei d2e
(26)
where A ( @ )denotes the aperture function. For a circular aperture subtending an angle €lo, we find A ( @ )=
1 0
for 101 SOo otherwise
The phase shift Yo(8) = k0{(C3/4)8~- (Af/2>02)
(28) introduced by the lens depends on both the coefficient of the spherical aberration C3 and the defocus Af. The vectors po and p denote the position of an
object
aperture spectrometer FIG.5 . Schematic setup in STEM for obtaining images with inelastically scattered electrons. Here 0, is the limiting objective aperture angle, and 0,defines the spectrometer acceptance angle. [From Kohl (1983a). Copyright 1983 North-Holland Publ. Co., Amsterdam.]
THEORY OF IMAGE FORMATION
187
object point and the position of the center of the spot, respectively. The latter can be influenced by the deflection coils of the instrument. We now generalize Eq. (10) to the case where the incident electron wave packet is given by Eq. (26). The measured quantity is the current per unit energy dIldE, which is given by (Rose, 1976a,b) dI I , kk:E, d E n30;h E,
where D(O) =
1 0
for 10150, otherwise
is the detector function, I, the incident beam current, 0,the spectrometer acceptance angle (at the specimen), E , = 13.6eV is Rydberg's energy, and E , the energy of the incident electrons. To derive Eq. (29), we have employed the small-angle approximation. For small energy losses AE 4 E,, we then find the following expressions for the scattering vectors:
+ (0 - a)] K' = k0[0,e, + (0' O)] K
=
k,[O,e,
-
where 0, = AE/2E, is the characteristic scattering angle for the energy loss AE. The two integrations over 0 and 8' reflect the coherent summation of the plane wave components of the incident wave packet, which enters quadratically in the current. The integration over the angle O must be performed over the entire solid angle accepted by the spectrometer. The factor S(K, K', C O ) / K ~ K describes '~ the scattering properties of the object in the firstorder Born approximation. [For simplicity we have restricted our discussion to electronic excitations. Phonon excitations are discussed by Kohl (1983b).] Although a certain relationship exists between our theoretical approach of inelastic image formation and the light-optical concept of object transparency, the latter is in general not sufficient to describe correctly the coherence properties of inelastically scattered radiation. Hawkes (1978) tried to describe image formation by the inelastically scattered electrons by means of an inelastic specimen transparency function. Following a recent paper by Rose (1984), let us reformulate Eq. (29) in such a way that it can be compared more
188
H. KOHL AND H. ROSE
directly with Hawkes’ results. To do so we Fourier-transform all relevant quantities exp[i(Kp - K’p’)] d28 d2W and
s
a(p) = A(8) exp[ - iy,(8)] exp[ - ik08p] d28 Conversely,
j
s ( ~ ~ = :w(p, ~p’, )w ) exp[ - i(Kp - K’p’)] d2p d2p‘ and A ( @ )exp[ -iyo(8)]
=
(qj
~ ( p exp(ik,Op) ) d2p
-
(32)
(33)
(34)
(35)
The function a( - p) is the amplitude of the wave function in the object planeup to the phase exp(ikz)-where the Gaussian focus is centered at the origin. Inserting these expressions into Eq. (29), we obtain dl dE
-
I , kk:E, 7c3Bgh E, x exp[ik,@(p’
-
p”)]D(@) d2p’d2p”d 2 0
(36)
The quantity w(p, p’, w ) has been called the “cross-spectral object transparency” by Rose (1984). For purely elastic scattering by a static potential in the dark-field mode, we obtain lo dE-rc30ih
E,
6(w)
jI
ja(p
-
p‘)T(p‘) exp( - ik,@p’) d2p‘
- lo k:EH 6(w) j u * ( p - p’)u(p - p”)T*(p’)T(p”) 7c38gh E,
x exp[ik,O(p’ - p”)]D(@) d2p‘ d2p“d 2 0
(37)
To include inelastic scattering, Hawkes (1978) has generalized Eq. (37) by replacing the object transparency T(p) by a phenomenologic complex transparency function T(p, w), where w determines the energy loss. Comparing the resulting expression with Eq. (36), we find that w(p’, p”, w ) should have the form NP’, P”, 0)= T*(P’, w)T(p”, w )
(38)
THEORY OF IMAGE FORMATION
189
However, w(p’, p”, (0)factors in this way only in the special case of transitions between pure nondegenerate states. In practice dipole excitations play a dominant role. For free atoms, then, at least one of the two object states must be degenerate.Due to theorthogonalityoftheobject states((n1m) = Oforn # m), the partial waves of the scattered electrons leading to different object states must be added incoherently, even though they might have suffered the same energy loss. The factorization, however, implies that for coherent illumination, those partial waves that have suffered the same energy loss remain coherent. Therefore, the object is not described accurately by an inelastic object transparency function. Rose’s (1984) cross-spectral object transparency must be used for a correct description of the inelastic image. Equation (29) is based on an assumption of fully coherent illumination. The importance of partial coherent illumination has already been emphasized by Hawkes (1978), who also worked out the implications of the inelastic transparency of the restricted form [Eq. (38)]. Rose (1984) has given a unified treatment of elastic and inelastic scattering, including partial coherence, by use of a modified Glauber approximation.
B. Image Formation in FBEM Let us consider a fixed-beam electron microscope (FBEM) as shown schematically in Fig. 6. The object is illuminated by a plane wave with wave
Kohler illumination
object
4%
imaging energy filter image
FIG.6. Image formation in an FBEM equipped with an imaging energy filter. The angles 0, and 8, are the illumination cone angle and the objective aperture angle, respectively. [From Kohl (1983a). Copyright 1983 North-Holland Publ. Co., Amsterdam.]
190
H. KOHL AND H. ROSE
vector k,. An imaging energy filter is incorporated in the imaging system below the objective lens. The latter can be either a Castaing filter (Castaing and Henry, 1962; Ottensmeyer and Andrew, 1980) or an omega filter (Rose and Plies, 1974; Krahl 1982). Since its detailed properties are of no concern for our considerations, we have drawn it as a black box. We assume that it removes all electrons, except those whose energy lies within a prespecified range, without introducing an aberration into the image. After the scattering process, the state of the system composed of the object and the incident electron is given by
The expansion coefficients A,(r) denote the electron wave function if the object has been excited from Im) to In). We shall restrict our discussion to thin objects so that the first-order Born approximation can be used. If we had no objective lens, the coefficients A,@) would be given by A,(r)
= exp(ik0r) 6,,
+ f,(K)Cexp(ik,r>/rl
(40)
for r + co,wheref,(K) is the scattering amplitude for the excitation of the nth state of the object and
k,
= ,/kg
- 2m0AElh2
(41)
is the wave number after the scattering has taken place. In an electron microscope, however, the electrons are registered in the image plane below the objective lens. Thus, the influence of the lens on the partial waves of the scattered electron must be taken into account. Next, let us calculate the value of A,@) in the image plane. In free space every single function A,(r) obeys the wave equation AA,(r)
+ k;A,(r)
=0
(42)
Also, the value of A,(r) just below the object plane is known. Therefore, we can use the Kirchhoff integral to calculate A,@) in any other plane, just as is done for elastically scattered electrons (Lenz, 1965, 1971; Hanszen, 1971; Hawkes, 1973,1984; Howie, 1984). For each A,(r), we must use the appropriate wave number k,. Propagation of electron waves in electric and magnetic fields (electron lenses) has been thoroughly discussed by Glaser (1943, 1952, 1956) and Glaser and Schiske (1953a,b), who have shown that the effect of the lens can be treated as a phase shift. The total current density per unit energy dj/dE is given by a sum over the partial current densities obtained from the amplitudes A,. Thus, for coherent illumination, contributions from partial waves resulting in the same final state of the object must be added coherently,
THEORY OF IMAGE FORMATION
191
whereas contributions leading to different final states must be added incoherently. In this way we find for axial illumination
x exp{ - i[y(0, AE) - y(W, AE)]} exp[ik(0
- el)p] d20 d2el (43)
where
y(0, AE)
=
k[(C3/4)e4 - (Af/2)e2
+ C,(AE/2E,)e2]
(44)
denotes the phase shift due to the defocus Af, the spherical aberration C,, and the chromatic aberration C,. The two vectors K and K’ are given by
K = k,[B,eZ
+ 01,
K’ = k,[O,e,
+ el]
(45)
and j , is the current density in the image plane if no object is present. The vector p denotes the position in the image plane but is referred back to the object. Equation (43) can be reformulated into a real-space description, similar to Eq. (36). The corresponding expression has been discussed by Rose (1984). Quite often the object in FBEM is illuminated incoherently from all directions, forming an angle less than 0,with respect to the optic axis (Kohler illumination). We must then average the current density per unit energy over all directions of the incident electrons, thus yielding
x exp{ - i[y(0, AE) - ?(el, AE)]} expCikp(0 - el)] d20 d20‘ d2@# (46)
The similarity between this formula and the corresponding formula for the STEM [Eq. (29)] will be discussed in the following subsection.
C. The Reciprocity Theorem In light optics the reciprocity theorem states that if the positions of the source and detector are interchanged, the light amplitude at the detector remains the same (Sommerfeld, 1978; Born and Wolf, 1965). Suppose the amplitude is u(S) at the source point S and u(P) at the image point P . If we put a point source with strength u(S) at P, the reciprocity theorem tells us that the amplitude at S will be u(P). Applying this theorem to a setup as shown in Fig. 7, we find that the amplitude at the detector is the same, regardless of whether we have the
192
H. KOHL AND H. ROSE
bS
source
I
xp Pi
objective lens
I
FIG.7. Simple arrangement of an imaging system that illustrates the reciprocity theorem.
source u(S) at S and measure at various image points P and P' or whether we move the source u(S) over the image plane and fix the detector at S . The theorem of reciprocity also holds for electron scattering off a static potential (Landau and Lifshitz, 1965). Thus, the image in a STEM with an axial detector and in an FBEM with axial illumination are the same for equivalent conditions (Cowley, 1969; Zeitler and Thomson, 1970a,b). The movement of the (virtual) source in a STEM is generated by a double deflection element. Before discussing inelastic scattering processes, we must define the meaning of reciprocity in these more general cases. For elastic scattering, reciprocity is due to time-reversal symmetry (Landau and Lifshitz, 1965) and leads to an important property of the elastic scattering amplitude for scattering of a wave ki to k,: f ( k i , k,) = f ( - k , , -ki) (47) Generalizing this result to inelastic processes, we find
f A k i k,) = f * m 4 - k,, - ki) The indexes rn and n denote the initial and final object states Irn) and In), while Im*) and In*) are the time-reversed states. The reciprocity theorem states that the amplitude for a transition from Im) to In) is equal to the timereversed transition amplitude from In*) to I m*). If inelastic scattering is taken into account, the reciprocity theorem cannot be used to prove the equivalence of images taken in FBEM and STEM, because the state reciprocal to the final state in FBEM is given by 3
THEORY OF IMAGE FORMATION
193
It consists of a coherent superposition of excited object states, which is clearly unequal to the initial state in a STEM, where the object is in its ground state. Nevertheless, Eqs. (29) and (46) look very similar. We shall, therefore, compare the images in STEM and FBEM. Let us assume that the instruments operate under equivalent conditions, namely, that the defocus, aberration coefficients, and objective aperture angles are equal. Let us further assume that the illumination angle in FBEM is equal to the spectrometer acceptance angle in STEM. If we then consider electrons that have suffered one particular energy loss AE, we find that, apart from the different forefactors, two further differences occur. First, in STEM the incident plane wave k , is focused, whereas in an FBEM the objective lens operates on the electrons k,, which have lost an energy A E in the object. Thus, in FBEM the chromatic aberration leads to an additional term in the phase shift y, which can be compensated by refocusing properly for the particular energy E , - AE. Second, the factor exp[ik,p(O - e’)] in STEM corresponds to exp[ikp(O - e’)] in FBEM. Since the energy loss is generally very small, this difference is usually too small to be detectable. Pogany and Turner (1968) used a different definition of reciprocity. In their famous paper, they name the approximate equivalence of the intensities an approximate “reciprocity of intensities.” We do not follow their definition, because in optics and quantum mechanics the term reciprocity is used to denote an exact property of a quantity (an amplitude) that obeys a particular differential equation (Sommerfeld, 1978). The approximate equivalence in intensities has a different origin and should be named differently.
D. Phase Contrast Thin phase objects are visualized in light microscopy by the phasecontrast method (Zernike, 1935). By introducing a proper phase plate in the back-focal plane of the objective lens, we obtain an intensity distribution in the image that, in the case of weak-phase objects, is proportional to the local phase shift (Goodman, 1968). Scherzer (1949) showed that in electron microscopy the phase plate can be obtained approximately by proper choice of defocus (Scherzer focus). This phase-contrast method can be used for thin objects and simplifies image interpretation considerably. For describing the phase contrast, it is sufficient to assume that the electrons are scattered by a static potential. This leads to an interesting question. Suppose the electron is scattered off a single atom; the transfer of momentum to the atom then results in a change of energy of the incident electron. Why is it then possible to see phase contrast or any interference between the unscattered and scattered electron waves? (We are grateful to several colleagues for raising this question.)
194
H. KOHL AND H. ROSE
Obviously the point is rather subtle. Phase-contrast images stem from the interference between the unscattered and the elastically scattered waves. In this case the object remains in its initial state. We shall show that there is always a finite probability amplitude that the internal state of the object does not change. For an FBEM with axial illumination, we obtain (Eusemann and Rose, 1982) A(8) exp[-iy(O)] exp(ik,pe)(f*(K)),
1
d28
(50)
where (f(K))T denotes the expectation value (thermal average) of the elastic scattering amplitude and all other quantities are as defined in Subsection 1II.B. The first term in brackets gives the constant current density if no object is present; the integral denotes the current density variations, which depend linearly on the scattering amplitude of the object. Evaluation of (f(K))T yields (Kohl, 1983b)
where VJK) is the Fourier transform of the interaction potential between the incident electron and the pth atom, and (plr,JT is the expectation value of the (Fourier-transformed) particle density of the pth atom. For a thin crystal plate, we find (f(K))T
= -
2c
exp(iKR,) exPC- ~ ( K ) I F ( K ) V 3 K )
(52)
Ir
where R, is the position of the pth atom within the elementary cell,
is the Debye-Waller exponent, and F(K)
=
N 0
for K equal to a reciprocal lattice vector otherwise
(54)
where N is the number of elementary cells in the object, M , the mass of a p atom, and s the number of atoms in the unit cell. The indexes q and o indicate the wave vector and polarization of a phonon, respectively, and fiqa
= (exp(Phmqa) -
1I-l
(55)
is the mean occupation number of the phonon mode qo. The calculation is given by Kohl (1983b) and need not be repeated here. The contrast decreases
THEORY OF IMAGE FORMATION
195
by a factor exp( - W ) . [The factor is not exp( - 2 W ) because the phase contrast depends on the amplitude, not the intensity, of the scattered electron.] Thus, the reason for the occurrence of phase contrast in spite of phonon vibrations is the same as the reason for the occurrence of distinct Bragg reflections off a crystal.
IV. NUMERICAL RESULTS A . Image of a Single Atom in an FBEM
To obtain quantitative results, we shall proceed in two steps. First, we shall calculate the mixed dynamic form factor S(K, K’, 0). For an arbitrary object, this is a rather hopeless task. Therefore, we must settle for simple examples. In this section we consider imaging of a single atom in an FBEM (Fig. 6) with axial illumination and an imaging energy filter. This filter removes all electrons unless they have suffered a distinct energy loss AE corresponding to a particular (dipole-allowed) transition. The objective aperture angle is assumed to be sufficiently small so that we can use the dipole approximation. Figuratively speaking, this means that the resolution limit of the microscope is larger than the diameter of the atom. In this case the mixed dynamic form factor for this particular transition is given by S(K, K’, 0) =
1
25 ~
+1 (nJMI exp(-iKrj)In’J’M‘)
x
M ,M ’ j, k
x (n’J’M’I exp(iK’rk)InJM)G(o - Am)
j, k
=
CKK’G(o - Am)
The only assumption used in the last step is rotational symmetry of the object. The proof is outlined in Appendix C . Since the quantum states with various M for a given 5 are degenerate, we must average over the 25 + 1 initial states ( n J M ) and sum over all 25’ + 1 final states In’J’M’). Note carefully that in this approximation the mixed dynamic form factor S(K, K’, o)factors into a strength factor CG(m - Am) and a shape factor KK’. Thus, the shape of the intensity distribution is the same for all atoms and depends only on 8, and on the microscope setting. The particular transition enters only via the strength factor CG(o - Am).
196
H. KOHL AND H. ROSE
Inserting Eq. (56) into Eq. (43), we obtain
with
Here J,(x) and J , ( x ) denote Bessel functions of the first kind. The total current density is given by a sum of two contributions. The first part, l A l i12, stems from the excitation of a dipole parallel to the incident electron, while IA,I2 describes the excitations perpendicular to the optic axis. For our numerical calculations, we assume that the image is recorded in the Gaussian image plane. Furthermore, we neglect spherical and chromatical aberrations. The results obtained with these assumptions are displayed in Figs. 8-1 1. The intensity distribution in the image is shown in Fig. 8 as a function of the (normalized) distance kp8, = 3.83p/d from the atom; d = 0.612/8, is the instrumental resolution limit (for an ideal lens). The dashed curve denotes the contribution of the parallel excitations IA II 1'; the solid curve shows the total intensity. The ratio 8J8, of the characteristic scattering angle eE = AE/2E0 to the objective aperture angle B0 is a measure of the energy loss. For large energy losses the aperture is almost uniformly illuminated. The intensity distribution in the image is then almost proportional to the Airy distribution [J,(kp8,)/(kp8,)]2. At large losses the perpendicular excitations contribute very little to the total intensity. With decreasing energy loss, the image spot starts to broaden as a result of the delocalization of the inelastic interaction. Its influence on the spot shape begins to show up for 8J8, = 0.5. For lower losses ( O E / 8 0 = 0.1 and 0.02), we find a ring of high intensity around the atom. This ring is due to the perpendicular excitations, which dominate low-loss images. The spot size increases even further. These effects of decreasing energy loss can be seen even more clearly in Fig. 9. Here we have drawn the image intensity within a thin ring p dp around the atom. The solid curve shows the total intensity kp8,( [ A l2 1 A , 12) ; the dashed curve represents the contribution of the parallel excitations kp8,I A II 1.' For relatively large energy losses ( 8 J 8 , = 2.0 or OS), the intensity decreases rapidly within a short distance from the object. For lower-energy losses the image spot is much broader. Comparing the curves for 8J80 = 0.1 and those for 0.02, we find that the height of the first maximum hardly increases. Thus, the increase of the cross section is due to an increase of the
+
197
THEORY OF IMAGE FORMATION (b)
-
003
001
0
0 20 018
0.18
0 16
0.16
0 14
0.14
-6
-
012
a?
-2 010
-2
w
012
010
U
008
008
0 06
006
0 04
004
002 0
2
4
6
8
1
0
002 0
FIG.8. Radial intensity distribution in the FBEM image of an atom formed by inelastically scattered electrons with different relative energy losses OJB,. (a) @JOo = 2.0, (b) @,/O, = 0.5, (c) OJO, = 0.1, and (d) t),/O, = 0.02. The normalized radius kpB, = 3.83p/d is inversely proportional to the radius d of the Airy disk. The dashed curve represents the contribution of the parallel excitations ; the solid curve describes the total intensity.
intensity at a far distance from the center of the atom. The images of neighboring atoms will then overlap considerably, thus creating an undesirably high background intensity. Comparing the images at a given energy loss for various objective apertures, we must keep in mind that the scale at the axis changes also. To be specific, let us assume an energy loss AE = 120 eV and an incident energy of 60 keV. Then 0, = 1 x and the four energy-loss parameters correspond
198
H. KOHL AND H. ROSE 0301
lb)
025-
-z'
020015
I
I
I
m" 0 10
~
1 a
005-
07 0
5
10
15 k Pea
20
25
30
kpe,
FIG.9. Intensity contained within a ring of radius p and width dp in the inelastic atom images shown in Fig. 8 as a function of the normalized distance from the atom center. (a) OE/OO = 2.0, (b) O J O , = 0.5, (c) HE/@, = 0.1, and (d) OE/Oo = 0.02.
2 x lop3, 1 x and 5 x to objective aperture angles 8, = 5 x 10- '. The corresponding instrumental resolution limits are then given by d = 60,15,3, and 0.6 A, respectively. Thus kp8, = 1 corresponds to p % 15,4, 0.75, and 0.15 A in these images. We find that for increasing objective aperture angle the spot size decreases considerably. The instrumental resolution limit, however, can only be reached for a large energy-loss parameter (e,/e, = 2.0). We are thus led to the problem of determining the specimen resolution obtainable with inelastically scattered electrons. Since most objects consist of many atoms, rather than just two, use of the Rayleigh criterion would be inappropriate. For such problems other definitions, which relate the resolution to the radius of a circle containing a given percentage of the total intensity, have proven fruitful. The two most common definitions for the effective specimen resolution limit deff use either the radius of a circle that contains 84 % of all scattered electrons or else twice the 59 % radius. Both definitions are chosen so as to yield the same result as the Rayleigh criterion for a point scatterer. The fraction P(kp8,) of the total intensity falling into a disk of radius p is shown in Fig. 10. We find that the effective resolution limit increases with
$IL
199
THEORY OF IMAGE FORMATION
a
IO-
04
02 00
5
10
15 kPeo
(b)
(a)
20
25
30
0
5
10
15 kpe,
20
25
30
10,
(C)
25
30
kpe,
kpe,
FIG.10. Fraction P ( k p 0 , ) of the total intensity contained within a disk of radius p in the images shown in Fig. 8. (a) OJO, = 2.0, (b) O,/O, = 0.5, (c) @,/On = 0.1, and (d) OE/OO = 0.02.
decreasing energy loss. The two definitions for deffyield different numerical values, but the general trend remains unaffected. Some authors have suggested use of the equation as a rule of thumb for the effective resolution limit (Howie, 1981). This formula (or similar ones) can be inferred either from the uncertainty relation or else by replacing the aperture angle B0 by the characteristic scattering angle 0, in the Abbe formula
d
= O.6A/tI0
(60)
To justify this procedure, we presuppose that most of the electrons are scattered within a cone of OE around the optic axis. This implies that replacing the angular shape of the scattered intensity l/(0; + d2) with 1 for 0 S 8, and with zero otherwise will not affect the result too much. Looking at Fig. 10, we find that this crude rule of thumb does not describe reality very well. In particular, for low energy losses (0,/0, e l.O), the effective specimen resolution limit deffis much smaller than Eq. (59) indicates. The most important reason is the fact that the total intensity is the sum of two distinct parts. For low energy losses, the perpendicular excitations lead to the
200
H. KOHL AND H. ROSE
characteristic donut-shaped structure, which means that a large fraction of all electrons in the image are close to the atomic nucleus. These results are in qualitative agreement with recent experimental findings. Colliex et al. (1981), and Colliex (1982) have imaged small uranium clusters and found that the effective resolution limit, although worse than the point-to-point resolution of the instrument, is definitely better than we would expect from the rule of thumb [Eq. (59)]. This fact will be seen more clearly in the discussion of the image of a surface plasmon (Subsection 1V.D). Due to the poor signal-to-noise ratio in inelastic images, it is extremely difficult to measure intensity profiles. Thus, unfortunately, a detailed quantitative comparison is not yet possible. Another means for describing the properties of optical systems is to consider the diffractograms H(h/B,). They are the Fourier transforms of the image intensity and determine how well a periodicity in the object is transferred to the image. Some mathematical details are given in Appendix D. In Fig. 11 we have drawn the diffractograms for the images shown in Fig. 8. With decreasing energy loss, the decrease of H(6/8,) becomes steeper. The transfer of high-spatial frequencies in the object is then increasingly suppressed. For OE/O, 5 0.5, we find that the diffractograms even have a zero, meaning that this spatial frequency will not be found in the image. For 6 larger than this value, H(h/8,) is negative, thus resulting in a reversal of contrast. All results given in this chapter are also valid for a STEM with a small axial detector (0,4 6,) as can be seen from our discussion in Subsection 1II.C. The influence of the detector angle will be discussed in the following subsection. B. Image of a Single Atom in S T E M
To calculate the intensity distribution in a STEM image, we insert Eq. (56) into Eq. (29). We assume zero defocus (Af = 0) and neglect spherical aberration (C, = 0). In this case four of the six integrations can be performed analytically (Kohl, 1983b). The remaining integrals were evaluated numerically; some results are shown in Figs. 12-14. We have drawn the intensities for the energy loss parameters 8J8, = 2.0,0.5,0.1, and 0.02 for a fixed detector angle 0,.The dashed curve denotes the contribution of the excitations parallel to the optic axis. We see in all three figures that the spot size increases with decreasing energy loss. Figure 12 shows the intensity distribution for a comparatively = 0.28,). Here we note again the donut-shaped structure small detector (0, for low energy losses (OE/6, = 0.1 or 0.02). This is comparable to Fig. 8, but the effect is not as pronounced. Increasing the spectrometer acceptance angle
20 1
THEORY OF IMAGE FORMATION
"1
601
- 1 04 0
(b)
05
1
15
2
1.5
1 2
6/80
16 0 14 0
12 0
-? 9 -
r
10 0
80 60 40 20
00 -2 0
05
1
a / 00
15
2
-
0
5 05
1
0
618,
FIG.11. Diffractogram H(6/Bo) of the inelastic FBEM image of a single atom as a function of the normalized spatial frequency 6/B, in the case of axial illumination for four values of the characteristic energy loss parameter BJB,; (a) BJB0 = 2.0, (b) B J B , = 0.5, (c) BJB0 = 0.1, and (d) Os/Oo = 0.02. The dashed curve represents the contribution of the parallel excitations.
(0,= e,), we find only a very weak donut structure for 8J0, = 0.02. For an even larger detector angle (0,= XI,), the spot looks more and more like an Airy disk. At first sight it seems surprising that the spot shape is so strongly dependent on the detector angle. In imaging with elastically scattered electrons this dependence is very weak, so, in general, it can be neglected. Why is this different for inelastic scattering?
202
H. KOHL AND H. ROSE 0.25
(a)
1.0
0.20
d -
0.15
Y
Y
0.10
0.05
C
FIG.12. Radial intensity distribution of inelastic images obtained with a spectrometer acceptance angle (at the specimen) 0,= 0.28, for four values of the energy-loss parameter 8J8,; (a) OJ8, = 2.0, (b) OJO, = 0.5, (c) OE/Bo = 0.1, and (d) B J B , = 0.02. The dashed curve represents the contribution of the parallel dipole excitations.
Due to the small extension of the atomic charge distribution (RA < 0.5 A), an atom can be considered as a point scatterer for elastically scattered electrons at present resolution limits (d 2 2 8).Therefore, the electrons are scattered almost uniformly in all directions accepted by the detector. The detector angle, which determines the total current measured, then has no significant influence on the spot shape. With respect to inelastic scattering the atom cannot be considered a point scatterer, as is apparent from the spot shapes in Figs. 8 and 12-14. The strong
203
THEORY OF IMAGE FORMATION (b)
2c 18
\
16
-
z
-
14 12
10
b-.
8
E 4
\
' 0
FIG.13. Radial intensity distribution in the inelastic image of an atom where the spectrometer acceptance angle @, is equal to the objective aperture angle Oo. (a) OJOo = 2.0, (b) O,/H, = 0.5, (c) O,/O, = 0.1, and (d) OJO0 = 0.02.
dependence of the spot shape can be explained by the fact that electrons passing close to the nucleus have a large excitation probability, but due to their large deflection they contribute to the image only if the detector angle is sufficiently large. Therefore, the spot size decreases with increasing spectrometer acceptance angle. The perpendicular excitations play an important role for this effect. In their channeling experiments, Tafto and Krivanek (1982) observed the strong dependence of localization on scattering angle. They find that as the angle of scatter increases, the excitation is more localized.
204
H. KOHL AND H. ROSE
120-
(d)
100 -
100-
-
$-
80-
60-
3
80 60-
40-
40-
20-
20.
07
0
2
4
6 kpe,
8
1
0
0 ....... 0 2
4
6 kPe,
8
1
0
FIG.14. Radial intensity distribution in the inelastic STEM image of an atom obtained with a large detector 0,= 50, for four values of the energy-loss parameter BE/@,; (a) 8,/0, = 2.0, (b) Oe/Bo = 0.5, (c) 0,/0, = 0.1, and (d) 0,/0, = 0.02.
Maslen and Rossouw (1984) and Rossouw and Maslen (1984) evaluated the differential cross section for inelastic scattering in crystals. They used a hydrogenic model to calculate the mixed dynamic form factor for K-excitations (Maslen, 1983). Their results confirm the increase of localization for increasing scattering angle. Knowledge of the spot shape may be of special interest for high-resolution microanalysis. The current approach is to deduce the elemental concentration in a pixel from the number of counts for a characteristic energy loss (after subtraction of the background) by using a differential cross section (Isaacson
THEORY OF IMAGE FORMATION
205
and Johnson, 1975; Egerton, 1981a,b). However, this is true only if the spot is smaller than a pixel. The differential cross section only gives the total number of scattered electrons, regardless of where they appear in the image. If the resolution is so high that the image spot extends over several pixels, the spot shape must be considered in analyzing the data. As can be seen from Figs. 8 and 12- 14, this will be particularly important for high-resolution low-loss images (small 8,/0,). The Z-contrast method (Crewe et al., 1975; Carlemalm and Kellenberger, 1982; Colliex et al., 1984) is frequently used for obtaining high-contrast images of biological specimens. This method makes use of the fact that the ratio of the elastic to the total inelastic cross section is proportional to the atomic number Z (Lenz, 1954). Dividing the elastic by the inelastic image, we thus obtain an image whose intensity is proportional to the mean atomic number of the corresponding object point. A detailed theory of contrast, however, must take the different localizations of the elastic and inelastic images into account. As we divide the dark-field signal by the inelastic signal [lDF(p)/linc,(p)],the delocalization of the inelastic image is expected to lead to an increased contrast as compared to the ratio of the respective cross sections. Calculations of total inelastic images (summed over all possible energy losses) have been performed by Rose (1976a,b). Since the details of his work are too lengthy to be repeated here, we refer the interested reader to his original papers. C . Images of Assemblies of Atoms
So far we have been dealing with images of a single atom. In practice, however, any specimen consists of a large number of different atoms. Unfortunately, it is not possible to treat such an object in the same general manner as the case of a single atom. We shall discuss one approximation that gives some insight into the image formation process. The excitation spectrum can be divided into several parts. Quasi-elastic processes lead to energy losses smaller than the energy resolution of about 0.5-2 eV. Such losses are due to the excitation or annihilation of phonons. The low-loss region extends out to energy losses of about 50 eV. The dominant processes in this regime are plasmon excitations and transitions of valence electrons. The plasmon energies are not very sensitive to elemental composition, so their use for microanalysis is very difficult (Williams and Edington, 1976). The energies of valence excitations are strongly dependent on the chemical state of an element, so they can be used to identify molecular species (Isaacson, 1972; Johnson, 1979). We shall focus our attention on inner-shell losses whose energies are larger than 50 eV. The energies of the edges are characteristic for a particular
206
H. KOHL AND H. ROSE
element. The fine structure of the energy-loss spectrum yields valuable information on their local environment [extended energy-loss fine structure (EXELFS)] and on their chemical state [energy-loss near edge structure (ELNES)] (Stern, 1982). By registering the edge as such, we can obtain purely elemental information. In this latter case the natural approximation is to consider the atoms as single, independent entities. We thereby neglect the influence of the potential of the neighboring atoms on the secondary electron of an ionized atom. Since this EXELFS effect produces only small intensity oscillations in the energy-loss spectrum (typically less than 5 %), this approximation will be satisfactory. [Very close to the edge the error will be larger, as can be seen from ELNES and x-ray absorption near-edge structure (XANES) experiments (Pendry, 1983)l. By assuming the atoms to be independent from one another and using the form factor [Eq. (56)] for each, we obtain for an assembly of N atoms of one element in the object plane the total form factor N
S(K, K’, w ) = Ch(w - Aw)KK’
1exp[ - i(K - K’)pj]
j=1
(61)
where pj denotes the position of the jth atom. The excited states of the N atoms are degenerate, so we sum over all of them. Inserting this expression into Eq. (29) or Eq. (43), we find that the total intensity I , in the image is just the sum of the intensities due to the excitation of a single atom I , N
This situation differs sharply from that of an elastic dark-field image. There we must consider the interference terms between different atoms, because there is only one final state (which is equal to the initial state). Here the different final states can be classified by the number j of the atom that has been excited. All the contributions of the different atoms must be added incoherently, because the final object states are distinct. One possible way to determine the delocalization of the inelastic interaction experimentally is to measure the intensity distribution in the image of a straight edge of a thin foil (Isaacson et al., 1974). Suppose we take an amorphous object, whose density (per unit area) is given by
If the scatterers are independent, we obtain the intensity distribution by a convolution of the intensity distribution of a single atom with the particle
207
THEORY OF IMAGE FORMATION
density distribution. This is equivalent to multiplying the Fourier transform of the density times H ( 6 ) and using the inverse Fourier transform (Rose, 1976a,b). Thus
In Fig. 15 we have drawn the (normalized) intensity distribution in the inelastic image of the edge of a thin foil. We can see again that the image becomes blurrier as the energy loss decreases.
-100
-50
kxe,
kx0,
(a1
(b)
,
-50
kx0,
00 kxe,
(Ci
(d I
00
50
100
- 100
50
FIG.15. Normalized intensity distribution in the inelastic FBEM image of a straight edge for the case of axial illumination as a function of the distance x from the physical edge for four values of the energy-loss parameter 0JBo; (a) BJV, = 2.0, (b) B J B o = 0.5, (c) OE/BO = 0.1, and (d) f~,/O, = 0.02.
100
208
H. KOHL AND H. ROSE
Similarly we can calculate the intensity I, in the image of an amorphous sphere as
Some results for OJO, = 0.1 are shown in Fig. 16. To obtain dimensionless quantities, we have displayed 1J(8n2R,”).For resolution limits larger than the radius of the sphere ( d = 2R,), we find a donut shape again. For smaller resolution limits (d = R,), this structure disappears. The intensity distribution is then more similar to the projected density; still, there is some delocalization left. Images of spheres have been taken by Colliex et al. (1981) and Colliex (1982). These authors found that the spot is broader than the sphere when inelastically scattered electrons are used to form the image. Let us now consider the inelastic image of a thin crystal. We assume the crystal to be so thin that the first-order Born approximation still holds. In Subsection 1I.B we found that for periodic objects S(K, K’, w ) is nonvanishing only if K - K’ = g. We use only this property, which holds for the two dimensions of the object plane, and the small-angle approximation [Eq. (3 l)]. Then g is a two-dimensional, reciprocal lattice vector. We make no assumption about the particular excitation; i.e., the atoms may be strongly interacting. When we take a lattice image in STEM, we are free to choose the position of our detector with respect to the optic axis. The importance of the detector position for elastic lattice images has been thoroughly discussed by Cowley and Spence (1981) and Spence and Cowley (1978). To establish the main point, let us consider the biprism experiment as shown in Fig. 2. We have already found (Subsection 1I.B) that to see the crystal structure we must choose k and k’ so that their difference is equal to a reciprocal lattice vector g. The lattice fringes then have the same periodicity as the crystal. To detect any 018.
-a“ -
I L q 010-
008006-
o i
-* a
(bl
010
006
004-
002-
018
(a)
004 ....,. .-.. ,
\,
0
PlR,
-..
002
0
’... .
1
2
3
4
p/R,
FIG.16. Radial intensity distribution in the inelastic image of a sphere for two values of the ratio d / R , between the resolution limit d and the radius of the sphere R,; (a) d / R , = 2 and (b) d/R, = 1. A STEM with a point detector or an FBEM with axial illumination has been assumed.
THEORY OF IMAGE FORMATION
209
elastic signal, we must place our detector in a Bragg position with respect to the incident beams. In STEM, the object is illuminated by a coherent superposition of all plane waves within the illumination cone, which is limited by the objective aperture. This wave packet is diffracted and the crystal is resolved only if the detector is located in an area where (at least) two diffracted disks overlap (Fig. 17). If we want to register inelastically scattered electrons, we must still fix k and k so that k - k = g, but now we detect electrons in all directions. Any excitation in a crystal can be classified by its wave vector q (or its crystal
FIG. 17. Scheme demonstrating the formation of an elastic lattice image in STEM. The incident cone is Bragg reflected. The lattice fringes appear in the image only if the detector is placed at a position where two cones overlap (hatched area).
210
H. KOHL AND H. ROSE
momentum hq), which lies in the first Brillouin zone. The energy-loss spectrum in a given direction k, reflects only those excitations whose wave vector q differs from the scattering vectors K and K' by arbitrary reciprocal lattice vectors. If, on the other hand, we look at a specific energy loss AE, we register electrons only in those directions corresponding to an excitation with energy hw, = AE. Since in any real experiment we have a finite energy resolution, the directions will not be so well defined. To detect a lattice image with energy loss AE in a STEM, we must fulfill a condition for the detector similar to the one for the elastic case. The disk representing the incident wave packet must be tilted by q (with hw, = A E ) and then Bragg-reflected (Fig. 18). The detector must be placed so that at least two disks overlap. If the spectrometer resolution is worse than the bandwidth of the excitation, the position of spectrometer is irrelevant. Note carefully that for thin crystals such images are due to single-scattering inelastic processes only. The image then reflects only the spatial structure of the excitation. It should not be confused with inelastic images of thicker crystals, in which multiple elastic and inelastic scattering processes are dominant (Fujimoto and Kainuma, 1963; Howie, 1963, 1984). The latter images look similar to elastic images; the contrast, however, is reduced. (a)
crystal
ax
Fia. 18. Comparison of the formation of elastic and inelastic lattice images in STEM. In (a) and (b), the incident and Bragg reflected cones are represented by their axes; (c) and (d) show the cross section through the cones at the detector plane. Lattice fringes can only be observed if the detector is placed in one of the hatched areas. The deflection 0, = q/k, resulting from the excitation of the crystal must be considered when describing inelastic image formation.
THEORY OF IMAGE FORMATION
21 1
In an FBEM with an imaging energy filter the image is formed with all electrons of a given energy that pass through the objective aperture. To form an elastic lattice image in an FBEM, at least two diffracted beams or one diffracted beam together with the unscattered beam must pass through the objective aperture. For inelastic lattice images, we must take the additional tilt by the angle O4 = q / k into account. Then if two diffracted and tilted beams pass through the objective aperture, we can see lattice fringes. If the energy filter transmits a whole energy interval simultaneously, the total image is given by an incoherent superposition of all inelastic lattice images resulting from excitations whose energies are in the chosen interval. In practice it is very difficult to prepare a crystal thin enough so that the first-order Born approximation holds. The argument can be carried over to thicker crystals, if we take hq to be the difference of the crystal momenta between the final and initial states of the object. Inelastic lattice images of crystals have been obtained by Craven and Colliex (1977). A quantitative description of such images is rather complicated. Thus, it is not yet possible to compare their results with theoretical calculations. In the case of thin crystals and independent atoms, the Fourier transform of an image is related to the diffractrogram of a single atom H ( 6 ) via r
r
x exp[ - i k ~ ( p- pj)] d2(p - p j ) exp( - ik6pj)
For a periodic object, the last sum is nonvanishing only if k6 is equal to a reciprocal lattice vector. If the atomic positions are known, it is then possible to measure H ( 6 ) at points that correspond to reciprocal lattice vectors. D. Image of a Surface Plasmon
As a further example of an inelastic image, let us consider the image of a plasmon on the surface of a metallic sphere or, similarly, on the surface of a spherical void in a metal. The properties of such objects have gained increasing attention over recent years (Fujimoto and Kumaki, 1968; Natta, 1969; Ritchie, 1981; Schmeits, 1981; Marks, 1982; Manzke et al. 1983; Penn and Apell, 1983; Lundqvist, 1983; Ekardt, 1984). The procedure to determine the intensity distribution is similar to the one described in Subsection 1V.A. To calculate the mixed dynamic form factor, we use a plasmon model given by Ashley et al. (1974, 1976). We consider only dipole plasmons that have an
212
H. KOHL AND H. ROSE
excitation energy of AE = EP/$ for metallic spheres and AE = E p Z for voids, where E , is the energy of a bulk plasmon. For the calculations (Kohl, 1983a,b), we assume an FBEM with axial illumination or a STEM with a small axial detector and neglect all aberrations (y = 0). For finite resolution, we use d / R , as a parameter. The resolution limit d and the radius from the center of the sphere p are both normalized to the radius of the sphere R , . Figure 19 shows the intensity distribution for d / R , = 0.2. The total intensity (solid curves) is again a sum of two parts. One can be viewed as resulting from the parallel excitations (dashed curves), the other as due to the perpendicular excitations. For a rather large energy loss (O,/O, = O.l), we obtain a full spot, whereas for lower energy losses the image is donutshaped. This ring of high intensity at the surface of the sphere is due to the perpendicular excitations. Images of surface plasmons were already reported 15 years ago by Henoc and Henry (1970). They used a neutron-irradiated foil that they imaged in an FBEM with a Castaing-Henry filter (1962) by selecting the plasmon loss electrons. In this case the voids appear as bright spots on a dark background, whereas they can hardly be recognized in the elastic image. Some experimental findings have been obtained that confirm our theoretical predictions. During the past years, Batson (1982a,b, 1985) and Batson and Treacy (1980, 1982) imaged small aluminium spheres in a STEM by using only those electrons that had excited a surface plasmon. In the course of specimen preparation, their spheres had been covered with a thin oxide layer. In their experiments both the instrumental resolution limit ( d / R , 4 1 ) and the energy loss (OJO, 4 1) were very small. More recently Ach6che et al. (1985) experimented with metallic spheres of tin, gallium, and uranium. In these images they find bright rings at the surfaces. Unfortunately, we cannot yet compare their experimental results quantitatively with our predictions from theory, because they used all surface plasmons in their experiments so far, whereas we have considered only dipole plasmons. Furthermore, the finite spectrometer acceptance angle influences the spot shape as well.
"6
05
PI R,
1
15
FIG.19. Radial intensity distribution in the image of a sphere, obtained by using only those electrons, which have excited a dipole surface plasmon, (a) B,/O, = 0.1 and (b) &/B0 = 0.005.
THEORY OF IMAGE FORMATION
213
Nevertheless the qualitative agreement between theoretical and experimental results is encouraging. Batson and Treacy (1982) and Colliex and Mory (1984) state that the excitation process is amazingly localized, in contrast to expectations from the rule of thumb [Eq. (59)]. The intensity distribution in their images indicates a specimen resolution limit of less than 40 8, whereas Eq. (59) would indicate approximately 800 A. These experiments show that, in spite of the delocalization of the interaction, we can obtain images with fairly high resolution even at rather low energy losses (AE z 5-15 eV). V. CONCLUSION
We have outlined a quantum-mechanical theory of image formation in the electron microscope, which considers both elastically and inelastically scattered electrons. The calculation of the inelastic image necessitates knowledge of the local scattering and excitation properties of the object. This information is contained entirely in the mixed dynamic form factor S(K, K’, w), which allows us to determine the shape of the inelastic image of thin objects within the first-order Born approximation. The mixed dynamic form factor can be viewed as a generalization of the ordinary dynamic form factor S(K,w). We have investigated the properties of the mixed dynamic form factor.and its interrelation with the generalized dielectric function including local field effects. The function S(K, K’, w ) can be determined experimentally by using an interference experiment as shown in Fig. 2. Recently, Schulke (1982) has shown that the mixed dynamic form factor of crystalline objects can also be measured by Compton scattering of x rays if the crystal is in Bragg position. The theory yields information about the coherence properties of the scattered radiation. In the case of coherent illumination, all partial waves leading to the same final object state are coherent with each other. Partial waves, however, that result in different final object states are incoherent with each other. By considering these coherence properties we can explain the occurrence of phase contrast in spite of the fact that the object atoms can absorb recoil energy, which is connected with an energy loss of the incident electrons. In Section IV we have calculated the intensity distribution in the image of two model objects. The image of an atom formed by the inelastically scattered electrons (in dipole approximation) is much more blurred than that formed by the elastically scattered electrons. This delocalization, however, is not as pronounced as has been assumed previously by Howie (1981). For small energy losses we found a donut-shaped intensity distribution. Contrary to elastic images, inelastic images in STEM exhibit a pronounced dependence
214
H. KOHL AND H. ROSE
on the spectrometer acceptance angle. The spot size decreases with increasing detector angle. Our calculations of images formed by dipole-surface-plasmon-loss electrons show that for low energy losses and high resolution the image of a sphere consists of a ring of high intensity about the projected surface. The delocalization is again less pronounced than has been assumed previously by Howie (1981). Our predictions agree reasonably well with recent experimental findings (Batson, 1982a; Colliex, 1982). Summarizing, we can state that the large progress in analytical electron microscopy has enabled the experimentalists to obtain inelastic images with high spatial resolution. Although the interpretation of the inelastic images is more difficult than the interpretation of elastic images, they contain information which is not accessible otherwise. For example, inelastic images can serve as a means for elucidating the local electronic structure at boundaries and surfaces. Therefore, these techniques will undoubtedly gain increasing importance for solid-state research.
APPENDIX A . Calculation of the Transition Rate
The interaction potential of the incident electron with the object is given by the operator
b' =
1 V(r - rj) j
where r and rj denote the positions of the incident and thejth object electron, respectively. The matrix element can be written as (kml Vlkfn)
=
=
Similarly, we find
1 (ml .i
7
(ml
s
exp[i(kf
J exp[i(kf
-
k)r] V(r - rj) d3rln)
I-
- k)(r
-
rj)] v(r
-
rj) d3(r - rj)
THEORY OF IMAGE FORMATION
215
The two scattering vectors K and K’ are defined as
K=k-kJ,
K’=k’-k
f
The operator
is the Fourier transform of the particle density operator. With the procedure introduced by Van Hove (1954), the product of matrix elements appearing in Eq. (8) can be transformed into a density-density correlation function. Taking into account the probability p m that the system is initially in state Im), we obtain for the sum
where ( )T denotes the thermal average, H , is the Hamilton operator of the object, and pK(t)is the density operator in the Heisenberg representation. By introducing the mixed dynamic form factor (Rose, 1976a,b) as
we eventually arrive at Eq. (8). The use of correlation functions such as (A7) is often advantageous, because it is then not necessary to calculate the corresponding wave functions explicitly. For collective excitations, it is in many cases possible to determine equations of motion for the density operator, which in turn lead directly to the correlation function (Pines, 1962, 1964).
216
H. KOHL AND H. ROSE
B. Interrelation of the Mixed Dynamic Form Factor and the Generalized Dielectric Function
To calculate the reaction of a solid when an external electric potential cDext(r,t ) is applied, we generalize the treatment given by Platzman and Wolff (1973). The Hamilton operator H can be written as the sum of an unperturbed operator H , and a perturbation H,(t) = -e,
s
p(r)cDexc(r,t) d3r
(B1)
where -eo denotes the charge of the electron and p(r) the density operator. By utilizing the Fourier expansions
mext(r,t) =
1 7 1cDf'(t) exp(iKr) K
we obtain
where V is the volume of the solid. By employing methods based on the linear response theory (Platzman and Wolff, 1973), we derive the following expression for the expectation value of the Fourier-transformed density operator
By inserting the Fourier transforms with respect to time,
into Eq. (B5), we obtain
217
THEORY O F IMAGE FORMATION
where the function
is the response function and 6 > 0 is a small quantity tending to zero. The response function describes the reaction of pKwhen a periodic potential @Ft of frequency OI is applied to the object. We now show the relationship existing between this function and the mixed dynamic form factor. To avoid unnecessary terms, we consider only the case o # 0. (For o = 0, the term ( p K ) T d ( o ) and similar ones occur in the formulas.) By using I
Zlr,
r
U,
J-u.
(p-K,pK(t))T
exp(iwt) dt
= exp( -Bhw)
S(K, K’, w )
(B10)
we obtain at the limit 6 .+ 0
+
x exp[i(o - o’ i6)tl O0
a,
do’d t
S(K, K’, o’)[ 1 - exp( - Pho’)] do’ w - w’ + i6 S(K, K’, 0’) [1 - exp( - Phw’)] d o ’
ine; hV
-~ [1 - exp( - Phw)]S(K, K’, w )
where # denotes the principal value. It should be noted that taking the imaginary part of (B11) yields a relationship between the linear response function F(K, K‘, w ) and the mixed dynamic form factor only if S(K, K’, w ) is real (e.g., for K = K’ or when the object is centrosymmetric and invariant with respect to time reversal). This function, however, is generally complex and fulfills the relation S*(K, K’, W ) = S(K’, K, O )
(B12)
To obtain a relationship between F and S, we exchange the variables K and K’ in (B 1 I), which yields an additional complementary equation. By adding
218
H. KOHL AND H. ROSE
these equations, using relation (B 12), and taking the imaginary part, we obtain the real part of S(K, K’, w ) : Im i{F(K, K’, w ) + F(K, K, w ) } = (-ne#l/)[l
- exp(-flhw)]
Re S(K, K’, w )
(B13)
On the other hand, by subtracting the two equations and using Eq. (B12), we obtain the imaginary part of S(K, K , w ) as 2 1 - ne0 Im - {F(K, K’, w ) - F(K’, K, a)}= [l hV 2i ~
-
exp( -flhw)] Im S(K, K , w ) (B14)
If we multiply Eq. (B14) by i and add Eq. (B13), we arrive at the following relation between the response function and the mixed dynamic form factor: S(K, K’, w ) = {ihV/2nei[1 - exp( -flhw)]}(F(K, K‘, w ) - F*(K’, K, w ) } (B 15)
In solid-state physics, we often use a dielectric function, which describes the response of the system as a function of the total potential @Iot(r, t ) = aext(r, t ) mind(r,t), where Oindis the induced potential. This dielectric functions plays the central role in the self-consistent field approximation. Here we , was intromust use the generalized inverse dielectric function ~ & , ( w ) which duced by Adler (1962) and Wiser (1963) to describe local field effects. In accordance with Adler (1962), we write
+
This relationship can be expressed as
By using the definition
THEORY OF IMAGE FORMATION
219
of the inverse dielectric function as an inverse matrix with respect to EKK,(w), we find
From the Poisson equation, we obtain e,p$d(co) =
K‘
{E&,(w) - dKK,}@Zt(u)
(B20)
Comparison of Eq. (B20) with Eq. (B8) yields F ( K , K’, W )
= E ~ K ~ { C ; ; , ’ ,( OdKK,} >)
(B2 1)
By inserting this relationship into Eq. (B15), we find
This formula enables us to determine the mixed dynamic form factor from the inverse generalized dielectric function. C . The Mixed Dynamic Form Factor in Dipole Approximation
In the following derivation, we shall use the notation employed by Messiah (1964). As a special example we shall consider an arbitrary, rotationally symmetric system. Then an operator T can be expanded into tensor operators TF) of kth order:
We start by proving the indentity
1 (nJMITf’In’J’M’)((nJMIT$’)ln’J’M‘))*
M.M‘
where InJM) denotes a state with principal quantum number n, angular momentum J , and projection M . The term ( dIT(’)IIn’J’) is the reduced matrix element. By using the Wigner-Eckart theorem (Messiah, 1964)
220
H. KOHL AND H. ROSE
and the orthogonality of the 3j symbols
for -k 5 q 5 kand ) J - J‘I 2 k 5 J
+ J‘, we find
1 ( n J M I Tf)I n’J’M’)(( nJM IT$’)In’J‘M’))*
M.M ’
=
(nJI
In’J’)((nJI 1T”’)lIn’J’))* ( J M
M,M,
k J’)(.J q M’ M
k‘ q’
J’) M‘
This is precisely Eq. (C2). We now apply this identity to the sum ( n J M I Kr I n’J’M’)(n’J’M’ I K’r I n J M )
M ,M ’
(C5)
The operator r = x k r k is the sum of the position operators of the individual electrons. The z axis is assumed to point in the K direction, so that Kr = Kz. By using the rotation operator R, we can write K’r as K‘RzR-’. Since z = TL’) is the 0-th row of a dipole operator (k = l), we find 1
where R$,) denotes a row of the rotation matrix. Insertion into Eq. (C5) yields
1 ( n J M IK r 1 n’J’M’)((nJM 1 K’r In’J’M’))*
M ,M ’
K K’
x
M ,M ’ ,q
( n J M I Tb’)I n’J’M’)( ( n J M I T:) I n’J’M’))* R$‘
where we have used RSb = P,(cos 9), and 9 denotes the angle between the two vectors K and K’. This relationship was employed to derive Eq. (56).
221
THEORY OF IMAGE FORMATION
D. Calculation of the Diffractogram The normalized inelastic intensity distribution per unit frequency w = E / h in the image plane of an FBEM for axial illumination can be written as
x exp{ - i[y(O, A E ) - y(@, A E ) ] } expCikp(0 - el)] d20 d2el
(D1) where
K
= k,(O,e,
and
S(K, K', W ) = So
+ 0),
K'
= k,(O,e,
+ el)
r+dm'2 S(K, K', o)dw
o-d0/2
is the mixed dynamic form factor averaged over a small frequency interval SW.
In the following, we consider w to be fixed and write I(p), thus omitting the parameter o.For w # 0, this intensity is proportional to the current density distribution in FBEM, [Eq. ( 4 3 ) ] .The diffractogram H ( 6 ) is the Fourier transform of the intensity.
H(S) =
~
~
(2.)2 k2
s
I(p) exp( -ik6p) d2p
x exp[ikp(B
S(K, K', W ) -
0' - S ) ]
x exp{ - i[y(O, A E ) - y(@, =
ki
s s
A(e)A(el)
S(K, K', w ) K
~
x exp{ -icy(& A E ) - y(@, =
k;
A E ) ] } d20 d2el d2p
A(0)A(0 - 6)
S(K, K
K
~
~
A E ) ] } S(0 - el - 6) d20 d20'
k,6, o) K2(K - k 0 6 T
x exp{ -i[y(d, A E ) - y(l0
-
-
61, A E ) ] } d20
222
H. KOHL AND H. ROSE
The first equality in Eq. (D2) shows that for 6 = 0 we obtain the integrated intensity in the image. Thus, we obtain a quantity that is proportional to the effective cross section 4k E , d28 = -- H ( 0 ) hko Eo The total number of electrons in the image (given by the right-hand side) is equal to the number that pass through the hole in the aperture screen (lefthand side). To calculate the diffractogram of the image of a single atom in FBEM, we use the dipole approximation as outlined in Subsection 1V.A and Appendix C . Setting the “strength factor” C = Sw, we find Neglecting aberrations (y excitations
(D4) H ( 4 = H 11 (6) + HL(4 obtain for the contribution of the parallel
= 0), we
and for the perpendicular excitations
The integrations over the angle /3 enclosed by the two vectors 8 and 6 yield
H 11 (4
140:
I
0
I
-
for 8, 5 6 < 20, otherwise
(D7)
223
THEORY OF IMAGE FORMATION
0
(D8)
otherwise
where 8, is the aperture angle; 8, = AE/2Eo is the characteristic scattering angle, which is defined as the ratio of the energy loss AE and twice the energy of the incident electrons E,. The remaining integrations were performed numerically and are displayed in Fig. l l . For a STEM with an extended detector, the image intensity is
x exp{ - i[yo(e) - yo(@)]} exp[ikop(8 -
e’)] d28d2W d2@ (D9)
with the scattering vectors
K
= k,[dEe,
+ (8 a)], -
s
K‘ = k0[8,e,
+ (0’ - a)]
(D10)
Thus, we find for the diffractogram of the STEM image HsTEM(8) = k;
S(K,K - ko6, W) A(e)A(e - 6 ) ~ ( @ ) K2(K - ko6)2
x exp{i[yo(8) - yo(18 - SI)]} d28d2@
This result is a generalization of Eq. (D2). Correspondingly, we obtain
and the effective cross section. between HSTEM(0)
(D11)
224
H. KOHL AND H. ROSE
Comparing Eqs. (D12) and (Dll), we find that an image yields more information than the differential cross section. The latter is related to H ( 6 = 0), whereas in an image all spatial frequencies 6 between 0 and 20, occur.
ACKNOWLEDGMENTS We would like to thank Drs. M. Achtche (Laboratoire de Physique des Solides, Orsay, France), P. E. Batson (IBM, Yorktown Heights, New York), and D. Krahl (MPG-Fritz Haber Institut, Berlin, Germany) for sending us their experimental results prior to publication and Profs. R. F. Egerton (University of Alberta, Edmonton, Canada) and J. Kiibler (Technische Hochschule Darmstadt, Germany) for fruitful discussions. In particular, we acknowledge the stimulating cooperation with Dr. C. Colliex and the members of his group from the Laboratoire de Physique des Solides in Orsay, France. Thanks are also due to Dr. A. Levensohn (Albany, New York) for his assistance in editing the manuscript. The valuable comments of Drs. R. Eusemann (Technische Hochschule Darmstadt, Germany) and M. Haider (EMBL, Heidelberg, Germany) are gratefully acknowledged.
REFERENCES Acheche, M., Colliex, C., Kohl, H., and Trebbia, P. (1985). To be published. Adler, S. (1962). Phys. Rev. 126,413. Ashley, J. C., and Ferrell, T. L. (1976). Phys. Rev. B 14, 3277. Ashley, J. C., Ferrell, T. L., and Ritchie, R. H. (1974). Phys. Rev. B 10, 554. Batson, P. E. (1982a). UItramicroscopy 9,277. Batson, P. E. (1982b). Phys. Reu. Lett. 49,936. Batson, P. E. (1985). To be published. Batson, P. E., and Treacy, M. M. J. (1980). Proc. 38th Annu. E M S A Meet. San Francisco, California, p. 126. Batson, P. E., and Treacy, M. M. J. (1982). Unpublished report. Berndt, H., and Doll, R. (1976). Optik 46, 309. Berndt, H., and Doll, R. (1978). Optik 51,93. Berndt, H., and Doll, R. (1983). Optik 64, 349. Bethge, H., and Heydenreich, J. (1982). “Elektronenmikroskopie in der Festkorperphysik.” Springer-Verlag, Berlin and New York. Born, M., and Wolf, E. (1965). “Principles of Optics,” Chap. 8.3.2. Pergamon, Oxford. Carlemalm, E., and Kellenberger, E. (1982). EMBO J . 1, 63. Castaing, R., and Henry, L. (1962). C . R. Hebd. Seances Acad. Sci. 255,76. Colliex, C. (1982). “Electron Microscopy,” 10th Int. Congr. Hamburg, Vol. I, p. 159. Colliex, C. (1984). In “Advances in Optical and Electron Microscopy,” Vol. 9, (R. Barer and V. E. Cosslett, eds.), p. 65. Colliex, C., and Mory, C. (1984). In “Quantitative Electron Microscopy,” Proc. Scott. Uniu. Summer Sch. Phys. 25fh, GIusgow (J. Chapman and A. J. Craven, eds.), p. 149. Colliex, C., Krivanek, 0. L., and Trebbia, P. (1981). Inst. Phys. ConJ Ser. 61, 183.
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Colliex, C., Jeanguillaume, C., and Mory, C. (1984). J . Ultrastruct. Res. 88, 177. Cowley, J. M. (1969). Appl. Phys. Lett. 15,58. Cowley, J. M. (1975). “Diffraction Physics.” North-Holland Publ., Amsterdam. Cowley, J. M., and Spence, J. C. H. (1981). Ultramicroscopy 6, 359. Craven, A. J., and Colliex, C. (1977). J . Microsc. Spectrosc. Electr. 2,511. Crewe, A. V., Langmore, J. P., and Isaacson, M. S. (1975). In “Physical Aspects of Electron Microscopy and Microbeam Analysis,” (B. M. Siege1 and D. R. Beaman, eds.), p. 47. Wiley, New York. Egerton, R. F. (1979). Ultramicroscopy 4, 169. Egerton, R. F. (1981a). Proc. 39th Annu. E M S A Meer. Atlanta, Georgia, p. 198. Egerton, R. F. (1981b). J . Microsc. 123, 333. Egerton, R. F. (1984). In “Quantitative Electron Microscopy,” Proc. Scott. Univ. Summer Sch. Phys., 25th, G h s g o w (J. Chapman and A. J. Craven, eds.), p. 273. Egerton, R. F., and Egerton, M. (1983). In “Scanning Electron Microscopy,” Vol. I, (0.Johari, ed.) p. 119. SEM, OHare. Ekardt, W. (1984). Phys. Rev. Lett. 52,1925. Eusemann, R., and Rose, H. (1982). Ultramicroscopy 9,85. Fliigge, S . (1971). “Practical Quantum Mechanics 11,” p. 150. Springer-Verlag, Berlin and New York. Fujimoto, F., and Kainuma, Y. (1963). J . Phys. Soc. Jpn. 18, 1972. Fujimoto, F., and Kumaki, K. (1968). J . Phys. Soc. Jpn. 25, 1679. Gabor, D. (1957). Rev. M o d . Phys. 28,260. Geiger, J. (1968). “Elektronen und Festkorper.” Vieweg, Braunschweig. Glaser, W. (1943). Z . Phys. 121,647. Glaser, W. (1952). “Grundlagen der Elektronenoptik.” Springer, Wien. Glaser, W. (1956). In “Encyclopedia of Physics,” Vol. 33, (S. Flugge, ed.), p. 123. Springer-Verlag, Berlin and New York. Gla’ser, W., and Schiske, P. (1953a). Ann. Phys. (Leipzig) 12, 241. Glaser, W., and Schiske, P. (1953b). Ann. Phys. (Leipzig) 12, 267. Goodmann, J. W. (1968). “Introduction into Fourier Optics.” McGraw-Hill, New York. Hanke, W. (1978). Adu. Phys. 27,287. Hanke, W., and Sham, L. W. (1975). Phys. Reo. B 12,4501. Hanszen, K. J. (1971). In “Advances in Optical Electron Microscopy,” Vol. 4, (R. Barer and V. E. Cosslett, eds.), p. 1. Academic, New York. Hawkes, P. W. (1973). In “Image Processing and Computer-Aided Design in Electron Optics,” (P. W. Hawkes, ed.) p. 1 . Academic, London. Hawkes, P. W. (1978). Optik 50, 353. Hawkes, P. W. (1984). In “Quantitative Electron Microscopy,” Proc. Scott. Unio. Summer Sch. Phys., 25th. Glusgow (J. Chapman and A. J. Craven, eds.), p. 351. Henoc, P., and Henry, L. (1970). J . Phys. (Orsay, Fr.) 31 Suppl. 4C-1, 55. Hoppe, W. (1969a). Acta Cryst. A 25,495. Hoppe, W. (1969b). Acta Cryst. A 25, 508. Hoppe, W., and Strube, G. (1969). Acta Cryst. A 25,502. Howie, A. (1963). Proc. R . Soc. London Ser A 271,268. Howie, A. (1981). J . Microsc. 117, 11. Howie, A. (1984). In “Quantitative Electron Microscopy,” Proc. Scott. Uniu. Summer Sch. Phys., 25th, Glasgow (J. Chapman and A. J. Craven, eds.), p. 1. Humphreys, C. J. (1979). Rep. Progr. Phys. 42, 1825. Isaacson, M. S. (1972). J . Chem. Phys. 56, 1813. Isaacson, M. S., and Johnson, D. (1975). Ultramicroscopy 1, 33.
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Isaacson, M. S., Langmore, J., and Rose, H. (1974). Optik 41,92. Johnson, D. L. (1974). Phys. Rev. B 9,4475. Johnson, D. (1979). In “Introduction to Analytical Electron Microscopy,” (J. Hren, J. Goldstein, and D. Joy, eds.), p. 245. Plenum, New York. Joy, D. C. (1979). In “Introduction to Analytical Electron Microscopy,” (J. Hren, J. Goldstein, and D. Joy, eds.), p. 223. Plenum, New York. Kittel, C. (1964). “Quantum Theory of Solids.” Wiley, New York. Kohl, H. (1983a). Ultramicroscopy 11,53. Kohl, H. (1983b). Ph.D. thesis, Darmstadt. Unpublished. Krahl, D. (1982). “Electron Microscopy,” 10th Int. Congr. Hamburg, Vol. I, p. 173. Landau, L. D., and Liftshitz, E. M. (1965). “Quantum Mechanics.” Pergamon, Oxford. Lenz, F. (1954). 2. Naturforsch. 9a, 185. Lenz, F. (1965). Lab. Invest. 14, 808; see also “Quantitative Electron Microscopy,” (G. F. Bahr and E. H. Zeitler, eds.). Williams & Wilkens, Baltimore. Lenz, F. (1971). In “Electron Microscopy in Material Science,” (U. Valdrt, ed.), p. 540. Academic, New York. Lundqvist, S. (1983). In “Electron Correlations in Solids, Molecules and Atoms,’’ (J. T. Devreese and F. Brosens, eds.), p. 301. Plenum, New York. Maher, D. M. (1979). In “Introduction to Analytical Electron Microscopy,” (J. Hren, J. Goldstein, and D. Joy, eds.), p. 259. Plenum, New York. Manzke, R., Crecelius, and Fink, J. (1983). Phys. Rev. Lett. 51, 1095. Marks, L. D. (1982). Solid State Commun. 43, 727. Maslen, V. (1983). Journ. Phys. B 16,2065. Maslen, V., and Rossouw, C. J. (1984). Philos. Mag. A 49,735. Messiah, A. (1964). “Quantum Mechanics,” Vol. 11, Appendix C. North-Holland Pub]., Amsterdam. Misell, D. L. (1973). Adv. Electr. Electron Phys. 32,64. Mollenstedt, G.,and Diiker, H. (1956). Z. Phys. 145,377. Natta, M. (1969). Solid State Commun. 7,823. Ottensmeyer, F. P., and Andrew, J. W. (1980). J. Ultrastruct. Res. 72, 336. Pendry, J. B. (1983). Comments Solid State Phys. B10,219. Penn, D. R., and Apell, P. (1983). J. Phys. C 16,5729. Pines, D. (1962). “The Many Body Problem.” Benjamin, New York. Pines, D. (1964). “Elementary Excitations in Solids.” Benjamin, New York. Pogany, A. P., and Turner, P. S. (1968). Acta Crystallogr. Sec. A 24, 103. Platzman, P. M., and Wolff, P. A. (1973). “Solid State Physics, Suppl. 13,”chapter 2. Academic, New York. Raether, H. (1980). “Springer Tracts in Modern Physics,” Vol. 88, p. 1. Springer-Verlag, Berlin and New York. Reimer, L. (1984). “Springer Series in Optical Sciences,” Vol. 36, p. 1. Springer-Verlag, Berlin and New York. Ritchie, R. H. (1981). Philos. Mag. A44, 931. Rose, H. (1976a). Optik 45, 139. Rose, H. (1976b). Optik 45, 187. Rose, H. (1984). Ultramicroscopy 15, 173. Rose, H., and Plies, E. (1974). Optik 40, 336. Rossouw, C. J., and Maslen, V. W. (1984). Philos. Mag. A 49,743. Scherzer, 0.(1949). J . Appl. Phys. 20,20. Schmeits, M. (1981). J . Phys. C 14, 1203. Schnatterly, S. E. (1979). “Solid State Physics,” Vol. 34, p. 275. Academic, New York.
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Schiilke, W. (1982). Solid State Commun. 43,863. Silcox, J. (1979). In “Introduction to Analytical Electron Microscopy,” (J. Hren, J. Goldstein, and D. Joy, eds.), p. 295. Plenum, New York. Sommerfeld, A. (1978). “Vorlesungen uber Theoretische Physik,” Vol. VI, chapter 10F. Deutsch, Thun. Spence, J. C. H., and Cowley, J. M. (1978). Optik 50,129. Stern, E. A. (1982). Optik 61, 45. Sturm, K . (1982). Adu. Phys. 31, I . Tafta, J., and Krivanek, 0. L. (1982). Nucl. Instrum. Methods 194, 153. Van Hove, L. (1954). Phys. Rev. 95, 249. Van Vechten, J. A., and Martin, R. M. (1972). Phys. Rev. Lett. 28,446. Williams, D. B., and Edington, J. W. ( I 976). J . Microsc. 108, 1 13. Wiser, N. (1963). Phq’s. Rev. 129,62. Zeitler, E., and Thomson, M. G. R. (1970a). Optik 31,258. Zeitler, E., and Thomson, M. G. R. (1970b). Optik 31, 359. Zernike, F. (1935). Z.Techn. Phys. 16,454.
Theory of Image Formation by Inelastically Scattered Electrons in the Electron Microscope H. KOHL AND H. ROSE lnstitut fur Anyewandte Physik Darmstadt. Federal Republic of Germany
I. Introduction . . . . . . . . . . . . . 11. The Mixed Dynamic Form Factor . . . . . . A. The Phase Problem in Electron Scattering . . B. Properties of the Mixed Dynamic Form Factor C. The Generalized Dielectric Function . . . . 111. Theory of Image Formation . . . . . . . . A. Image Formation in STEM . . , . . . . B. Image Formation in FBEM . . . . . . . C . The Reciprocity Theorem. . . . . . . . D. Phasecontrast . . . . , , . , . . . IV. Numerical Results. . . . . . . . , . . . A. Image of a single Atom in an FBEM. . . . B. Image of a Single Atom in STEM , . . . . C. Images of Assemblies of Atoms . , . . . . D. Image of a Surface Plasmon . . . . . . . V. Conclusion . . . . . . , . , , . . . . Appendix. . . . . . . . . . . . . . . References . . . . . . . . . . . . . .
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. 173 . 174 . 174 .
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180 183 185 185 189 191 193 195 195 200 205 21 1 213 214 224
I. INTRODUCTION The main objective of electron microscopy is to obtain high-resolution images of thin specimens. Modern instruments yield point-to-point resolutions of about 3 A. Thus, it is possible to elucidate the properties of the object on an ultramicroscopic scale (Bethge and Heydenreich, 1982; Reimer, 1984). In such an instrument, the image is formed by electrons, which are scattered in various directions within the objective aperture. In electron energy loss spectroscopy (EELS), we measure the intensity of inelastically scattered electrons as a function of energy loss and scattering I73 Copyright 8 1985 by Academic Press, Inc. All rights of reproduction in any form reserved. ISBN 0-12-014665-7
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angle. This technique is widely used in solid-state physics to obtain information about the dynamic properties of the specimen as a whole. For recent reviews see Raether (1980), Schnatterly (1979), Silcox (1979), and Colliex (1984). Recently, both techniques were combined under the name of analytical electron microscopy. Thus, it is now possible to obtain energy-loss information from a small volume of a few cubic nanometers. By registering only those electrons that have suffered a characteristic energy loss (say a K-loss corresponding to a particular element), we can quantitatively determine the elemental distribution in an object (Isaacson and Johnson, 1975; Egerton, 1979, 1984; Joy, 1979; Maher, 1979; Colliex, 1984; Colliex and Mory, 1984). An extensive bibliography has been given by Egerton and Egerton (1983). Experimental (Isaacson et al., 1974) and theoretical (Rose, 1976a,b) investigations show that images taken with inelastically scattered electrons are blurry as compared to an elastic image of the same object. This “delocalization of the interaction” makes the interpretation of inelastic images more difficult. Despite the striking progress in experimental abilities, a detailed theory of the process of image formation is still lacking. Most reviews on image formation treat the contribution of the inelastically scattered electrons as a deleterious side effect (Misell, 1973; Humphreys, 1979). In view of the recent progress in analytical electron microscopy, it seems worthwhile to examine image formation by inelastically scattered electrons in more detail-first, the scattering properties of the object (Section 11) followed by electron-optical considerations (Section 111) and image calculations (Section IV). 11. THE MIXEDDYNAMIC FORMFACTOR
A . The Phase Problem in Electron Scattering
The standard geometry for a scattering experiment is shown schematically in Fig. 1. The object is illuminated by an incident plane wave with wave vector k,. A far-field detector registers all electrons that have been scattered in the direction k, and determines their energy and angle of scatter. For simplicity, we shall restrict our discussion to the case of purely elastic scattering. The differential cross section dg/dQ gives the probability that an electron is scattered into the solid angle dQ. It is given by where f ( 0 , 4 ) denotes the scattering amplitude for the direction determined by the polar angle 0 and azimuth 4. The differential cross section is equal to
THEORY OF IMAGE FORMATION
i
175
.c ko
object
detector FIG.1. Conventional arrangement for diffraction measurements; K = k, - k, is the scattering vector, and k, and k, are the wave vectors of the incident and the emergent electron, respectively.
the square of the modulus of the scattering amplitude. For fast incident electrons with wave vector k,, scattered by a thin object, we can use the firstorder Born approximation to calculate the scattering amplitude. For scattering in the direction k,, we then find f(K)
=
-
m &
1
V(r) exp(iKr) d3r
where K = k, - k, denotes the scattering vector, V(r) the object potential, m, the electron mass, and 12 Planck’s constant. Thus, to first order, the scattering amplitude is proportional to the Fourier transform of the object potential. As can be seen from Eq. (l), we can extract only the modulus, not the phase, of the scattering amplitude from a diffraction experiment. Thus it is impossible to determine unambiguously the object potential from experimental diffraction data. This so-called phase problem has grave consequences for diffraction experiments, because there are an infinite number of functions with a prescribed modulus of the Fourier transform. (For example, a translation leads to a shift in phase of the scattering amplitude.) Even with additional information, it is often impossible to determine the object potential unambiguously. For example, it is impossible to distinguish between the potentials V(r) and P(r) = V ( - r), because T(K) = f’*(K), and
176
H. KOHL AND H. ROSE
thus lY(K)I2 = If(K)12. This fact has serious consequences for crystallography, where a noncentrosymmetric structure cannot be distinguished from its inverse, even though the types of atoms in the unit cell and their scattering properties may be known. Thus, even with considerable amount of a priori information, we cannot determine the object structure by a scattering experiment as shown in Fig. 1. If we consider inelastic scattering, the situation is even more complicated. It is not possible to determine the spatial structure of the transition density of the excitation by examining the scattering data. All we can do is compare the measured data with theoretical calculations. However, as we have seen in the case of elastic scattering, agreement between theory and experiment does not necessarily mean that the theoretical model is correct. Thus, a method to measure the phase of the scattering amplitude is of great experimental interest. Let us now discuss a setup that allows us to determine the phase of the scattering amplitude (Fig. 2). A biprism splits the incident wave into two
K' K
FIG.2. Arrangement for determining phase effects in elastic and inelastic scattering. The object is illuminated by a coherent superposition of two plane waves with wave vectors k and k . The detector registers the intensity in the direction of k,. [From Kohl (1983a). Copyright 1983 North-Holland Publ. Co., Amsterdam.]
THEORY OF IMAGE FORMATION
177
FIG.3. Intensity distribution in the specimen plane when two inclined plane waves with relative phases 4 = 0 or 4 = n are impinging on the object. Each dot represents an atom, and a is the lattice constant. The origin is located on an atom. The directions of incidence of the two waves enclose an angle i j a . [From Kohl (1983a). Copyright 1983 North-Holland Publ. Co., Amsterdam.]
coherent plane waves, which form interference fringes in the object plane (Mollenstedt and Diiker, 1956). The initial state of the electron is then given by (1/42)(exp(ikr)
+ exp(i4) exp(ik’r))
(3)
where k and k‘ are the wave vectors of the two incident plane waves impinging on the object. The factor ei4 determines the phase relation between the two waves. To understand its meaning a little better, let us first calculate the intensity distribution in the object plane. It is given by I(r)
=
1 + cos[(k - k ) r - 41
(4)
The vector r denotes the position on the object. Suppose the object is periodic with a lattice vector ae,. We adjust k and k’ so that k - k = (2n/a)e, is a reciprocal lattice vector.’ Then the interference fringes have the same periodicity as the object. The phase 4 determines the position of the fringes on the lattice (Fig. 3). Depending on 4, either the atoms will be illuminated predominantly (4 = 0) or else the space in between ( 4 = n). The probability of registering an electron in the detector will depend on the position of the interference fringes on the object. The count rate of the detector is proportional to
1.m)+ exP(i4>.f(K’)l2
(5)
where K = k - k, and K’ = k’ - k, denote the two scattering vectors. By measuring the intensity as a function of K and K‘, we can obtain the phase of the scattering amplitude (Gabor, 1957; Hoppe, 1969a,b; Hoppe and Strube,
’ We use the convention a;aj = 2nsij, where a; is a reciprocal lattice vector and aj a lattice vector in real space. This definition, which is common in solid state physics, differs from the one used in crystallography (a;aj = S i j ) by a factor of 2n.
178
H. KOHL AND H. ROSE
1969; Berndt and Doll, 1976, 1978, 1983). Formula (5) shows that an experiment as depicted in Fig. 2 cannot be completely described by calculating cross sections. The phases play an important role in this type of experiment. Let us now consider inelastic scattering events. The detector is assumed to register the electrons as a function of energy and angle of scatter. T o describe the scattering process, we must use the product states of the incident electrons and the object state. (The influence of exchange is neglected here.) The initial state is given by
+ exp(i4) exp(ik’r)] Im)
(l/J2)[exp(ikr)
(6)
where Im) denotes the initial state of the object. Here the object is assumed to be initially in a pure state. The final state is equal to eikf‘In). For fast electrons and thin objects, we can use the first-order Born approximation to determine the count rate. Since the detector registers only the direction and energy of the scattered electron but not the final state of the object, we must sum over all possible final object states with a given energy. Using Dirac notation
I kfn)
= eikf’In>
we obtain for the transition rate from the initial to the final state
+ exp(i4)( km I V I kf n ) (kfn I V I k’m) + exp( i&)(kml Vlkfn)(kfnl Vlkm))6(wm -
-
+
o, o)
(7)
Here V denotes the interaction potential between the incident electron and the object, while hw, and hw, are the energies of the initial and final state of the specimen, respectively. The first two terms stem from the scattering of the partial wave from k (k’) to k,. They appear likewise in a calculation of a cross section. The following two interference terms contain all spatial information about the object. For scattering experiments as shown in Fig. 1, the object properties can be described conveniently by means of a density-density correlation function (Van Hove, 1954; Platzman and Wolff, 1973). We shall generalize Van Hove’s result to the case of two (or more) interfering incident waves. With V
=
C V(r i
-
Ti)
THEORY OF IMAGE FORMATION
179
where V(r - ri) describes the interaction of the incident electron with the ith particle, we obtain
where V(K) =
s
V(r) exp( - iKr) d3r
is the Fourier transform of the interaction potential V(r). For Coulomb interactions, we obtain V(K)
= e;/EoK2
where - e , is the charge of the electron and c0 the dielectric constant of the vacuum. In Eq. (8), S(K, K’, 0)=
J 2n
m
-m
(pK(t)P-K,(o))T
exp(iot) d t
(9)
is the mixed dynamic form factor (Rose, 1976a,b), where PK(~) =
C~xPC-
iKrj(t)I
j
is the Fourier-transformed density operator in the Heisenberg representation and ( )T denotes the thermal average. In Eq. (8), p m gives the probability that the object was in the state Im) before the scattering took place. By inserting Eq. (8) into Eq. (7), we obtain
+ I V(K’)I2S(K‘,K’, o) + exp(i+)V(K)V*(K’)S(K, K’, o)+ exp( -$)V*(K)V(K’)S(K’,
w = (n/h’)[I V(K)I2S(K,K, o)
K, o)]
(10) The calculations are outlined in Appendix A. Here we have included a thermal average with respect to the initial states of the object. The transition rate w is given by a sum of four terms, each of which is a product of the Fouriertransformed density-density correlation function and two conjugate Fourier transforms of the interaction potential. Thus, Eq. (10) can be viewed as a generalization of the relation given by Van Hove (1954). The last two terms of Eq. (10) contain the function S(K, K’, w ) with K # K’, whereas in a conventional scattering experiment only terms with S(K, o)= S(K, K, o)occur. Our apparatus in Fig. 2, thus, enables us to obtain additional information about the object.
180
H. KOHL AND H. ROSE
The first two terms in Eq. (10) can be obtained from a conventional scattering experiment. If we now measure the intensity with k and k’ fixed for two different phases 4, we can solve the two resulting equations for the two unknown values S(K, K’, w ) and S(K’, K, o). This procedure can be repeated for all values of K and K by varying the direction of the incident electrons and the detector. Thus, the function S(K, K’, o)is a quantity that can be measured experimentally (at least in principle). It contains information about the spatial distribution of the excitation within the object. (Subsections I1.B and 1I.C will deal with the properties of the mixed dynamic form factor. The more experimentally oriented reader can bypass them without loss of comprehension.) B. Properties of the Mixed Dynamic Form Factor
In Subsection II.A, we found that the mixed dynamic form factor S(K, K’, o)can be viewed as a generalization of the conventional form factor S(K, o).We therefore ask Which properties of S(K, o) are also true for S(K, K’, w)? First of all, let us consider two sum rules. The integral
j
m
-m
S(K, w> do
=
(PK(o)P-K(o))T
= S(K)
(1 1)
yields the static form factor S(K) (Kittel, 1964). By using Eq. (9) and interchanging the order of integration, we find
j
m
=
-m
=
(12)
Accordingly the right-hand side is called the mixed static form factor. Using the operator identity where K = I K 1, H is the Hamilton operator, and = po the number operator, we obtain the generalized Thomas-Reiche-Kuhn sum rule Jm -00
w{S(K, w )
+ S(-K,
o)}dw
=
hKZ ~
m0
N
181
THEORY OF IMAGE FORMATION
Here N denotes the number of particles in the object (Kittel, 1964). By employing the double commutator [ [ H , &], P-K’], we obtain [[& PKI? P-K’I = -(hZKK’/mO)pK-K’
(15)
Taking the thermal average, we find
=
1 -
Z
1 exp(-pEm)(Em
-
m,n
En)((mlPKIn)(nlp-K’lm)
+ (ml P -K’I n>( n I PK Im)) cu
=
-A]
w{S(K, K’, 0) -m
+ S( - K’,
-
K, w ) } dw
where p m = e-DEm/Z denotes the occupation probability, Z the partition sum,
p = l/k, T, k , Boltzmann’s constant, and T the temperature. Thus,
The magnitude of the mixed dynamic form factor is related to the expectation value of the Fourier transform of the electron density for K - K’. The larger the density variation in the object, the more pronounced are the terms for K # K’ in the form factor. In contrast to the conventional dynamic form factor S(K, w), the mixed dynamic form factor is, in general, not a real quantity. Therefore, we cannot simply carry over the proofs for a certain property from S ( K , o ) to S(K, K’, w). Rather, we must check very carefully, if the proof for S(K, w ) makes use of the fact that the form factor is a real quantity. Symmetries of the object lead to further restrictions on the form factor. Suppose the specimen has a center of inversion at the origin. Then,
(dr,t)P(r’))T
=
(p(-r? L)d-r’))T
where p(r, t ) =
1 i
-
ri(t))
is the density operator in the Heisenberg representation and p(r) = p(r, 0). By using pK(t) = p(r, t)exp( -iKr) d3r, we find
S
S(K, K’, W ) = S( - K, - K’, W )
(17)
182
H. KOHL AND H. ROSE
For a periodic object with lattice vectors a,, a 2 , a3, the relation ( ~ ( r+ ai, L > P ( ~+’ ai))T
=
S(K, K’, w ) = exp[i(K
K)a,]S(K, K‘, w )
( d r , t)dr’))T
leads to for i
=
-
(18)
1,2, 3. This means that either S(K, K’, w ) = 0, or else exp[i(K
-
K)ai]
= 1
The latter condition is equivalent to the condition K
-
K’ = g, where
3
g = Ch,a,
hiEH
i=l
is an arbitrary reciprocal lattice vector. In short, the mixed dynamic form factor is nonvanishing only if K - K’ = g. This can be understood by looking at Fig. 3. If K - K = g, the fringes have the same periodicity as the object. In that case, we illuminate predominantly one part of the unit cell, e.g., the atoms ( 4 = 0) or the space in between ( 4 = n). Classically speaking, the scattering is then characteristic for that part of the unit cell. If the difference K - K’ is unequal to any reciprocal lattice vector, the fringes on the object have a periodicity different from the lattice. The scattering is then given by an average over the elementary cell. By using a setup as shown in Fig. 2, we can determine the spatial structure of an excitation. If the object is invariant with respect to arbitrary translations, Eq. (18) is valid for any vector a. Then S(K, K’, w ) is nonzero only if the two scattering vectors are equal (K = K ) . In that case the scattered intensity no longer depends on the position of the fringes on the object. For such a free electron gas, the conventional form factor S(K, w ) gives a complete description of the scattering properties. Let us compare the conditions K = g and K‘ = g’ for elastic scattering with the corresponding condition K - K’ = g for inelastic scattering. Electrons scattered elastically can be found only if both scattering vectors coincide with reciprocal lattice vectors (K = g and K = g’). For inelastic scattering, the weaker rule K - K’ = g applies, because for such processes a quasi-momentum hq can be transferred to the crystal. Since we must sum over all possible final states with a given energy, many excitations contribute to S(K, K’, w ) for fixed o. Finally, we shall consider the influence of time-reversal symmetry on the mixed dynamic form factor. The relation S(K, K’, 0) = S( - K’,
-
K, 0)
(19)
THEORY OF IMAGE FORMATION
183
is a generalization of the well-known formula S(K, w ) = S( - K, o)(Kittel, 1964). To apply the time-reversal symmetry, all magnetic fields must be reversed (Landau and Liftshitz, 1965). If such magnetic fields are negligible, Eqs. ( 1 6) and (1 9) yield
For objects that are invariant under time reversal and inversion, the combination of Eqs. (1 7) and (19) yields
S(K, K‘, w ) = S(K’, K, W )
(21)
Since S(K, K’, w ) = S*(K’, K, w ) , this means that in these cases the mixed dynamic form factor is real. We could also exploit the implications of point symmetries of the object. A specific example (complete rotational symmetry) will be given in Appendix C. C. The Generalized Dielectric Function
The dissipation-fluctuation theorem yields a relation between the dissipative part of a linear response function and the fluctuations in thermodynamic equilibrium. The latter are described by correlation functions. Applying this theorem to electronic excitations, we find for the conventional form factor (Pines, 1962, 1964; Kittel, 1964; Geiger, 1968; Platzman and Wolff, 1973; Schnatterly, 1979; Raether, 1980)
S(K, to)
=
-{EOhK2V/nei[1 - exp(-phw)]} Im E-’(K, w )
(22)
where E denotes the dielectric function of the medium, V its volume, and Im the imaginary part. Hence, the dynamic form factor is proportional to the imaginary part of the dielectric response function E - which describes absorption effects in the object. Reading Eq. (22) from right to left, we find that we can calculate Im E - ‘(K, w ) from measured values of S(K, w). The real part can be obtained by use of the appropriate Kramers-Kronig relation. We now try to generalize Eq. (22) to include the mixed dynamic form factor S(K, K’, w). For the derivation, we must keep in mind that S(K, K’, w ) is a complex quantity due to the spatial Fourier transforms. Thus, appropriate care must be taken when generalizing properties from the conventional dynamic form factor to the mixed dynamic form factor. The corresponding linear response function is the generalized inverse dielectric function E&(o), which was introduced into solid-state physics by Adler (1962) and Wiser (1963). This function includes the so-called local field effects, which stem from the atomic structure of the specimen.
’,
184
H. KOHL AND H. ROSE
FIG.^. Spring model to visualize the polarization of a diatomic chain induced by a homogeneous external electric field.
The dielectric function relates the induced polarization to the applied external field. To understand its meaning, we consider a one-dimensional diatomic lattice (Fig. 4). This spring model describes the effect of the different polarizabilities of the two types of atoms. Suppose we apply a homogeneous external field to the object. The polarization itself will not be homogeneous; rather it will be the sum of Fourier components that have the periodicity of the lattice. It is convenient to use the external potential Oext,where E = grad Oexc and the induced charge eOpind = div P for the calculations. For an external potential Oext,the induced charge is given by (Adler, 1962)
+
Here g and g‘ are reciprocal lattice vectors; the indexes q g and q + g’ denote the respective Fourier components. The vector q is restricted to the first Brillouin zone. Calculation of such a dielectric function is so onerous a task that only few numerical results have been reported (e.g., Van Vechten and Martin, 1972; Hanke and Sham, 1975). Nevertheless, this dielectric function is of great interest in solid-state physics. Sturm (1982) has used the nearly free electron approximation to calculate it in the region of the plasmon frequency wp. He then discusses the influence of the band structure on plasmon dispersion and linewidth. For real solids, the charge density is nonuniformly distributed within the unit cell. Consequently the plasmon excitation is not a simple plane wave. Applying Bloch’s theorem, we find that it will be modulated by a lattice periodic function. Thus, there might even be plasmon bands. The conditions under which these could be detected are discussed in Sturm’s review (1982).
THEORY OF IMAGE FORMATION
185
The excited states of an insulator can also be calculated via the dielectric formulation. The lowest excitation energy is not given by the band gap; the electron-hole correlations must also be taken into account. Such excitonic effects have been discussed by Hanke (1978). For nonperiodic objects, the sum in Eq. (23) must be extended over all vectors K’. All other properties remain unaffected. The relationship between the mixed dynamic form factor and the dielectric function is given by S(K, K’, w ) = isOhV/2ne;[1
-
exp( -fiho)](K2~&w) - K’2~,&’(w)} (24)
The derivation is outlined in Appendix B. Since S(K, K’, w ) is a complex function, Eq. (24) is more complicated than the corresponding relation [Eq. (22)]. Also, the dissipative part of the response function is no longer given by its imaginary part (Johnson, 1974). The generalized dielectric response function can be obtained from the mixed dynamic form factor by the relation s&(w)
=
hKK,+ -in
a, ~
0 - 0’
4
s,hK2V ~
[1 -- exp( - Bhw‘)]S(K, K’, 0’) dw’
[1 - exp( - phw)]S(K, K’, w )
Here # denotes the principal value; the integral describes the dispersion, and the following term is the absorption in the object. 111. THEORY OF IMAGE FORMATION
A . Image Formation in S T E M To calculate the intensity distribution in the image, we must first determine the scattering properties of the object. We restrict our investigations here to objects thinner than the depth of field. The first-order Born and the small-angle approximation can then be used to describe the elastically scattered electrons (Cowley, 1975; Colliex and Mory, 1984). To explain image formation, a knowledge of the differential cross section is not sufficient, because it does not contain spatial information. In a semiclassical picture, we can consider the electron as a classical particle with impact parameter b and calculate the probability for the excitation of a particular object state (Fliigge, 1971). This procedure is equivalent to assuming infinite resolution (Ritchie, 1981).
186
H. KOHL AND H. ROSE
In the more general case both the incident electron and the object are treated quantum mechanically (Rose 1976a,b). The degree of coherence between incident plane waves coming from different directions must then be taken into account. Thus, image formation in a scanning transmission electron microscope (STEM) (Fig. 5) can be interpreted as a more complicated version of the biprism experiment depicted in Fig. 2. To calculate the incident wave packet at the object plane, we assume a plane wave in front of the objective lens. The object is located in its back-focal plane. There the wave function of the incident electron with wavenumber k , is given by
J ~ ( eexp~-iy,(e)i ) exp[-ik,(p n
exmoz0)
- po)ei d2e
(26)
where A ( @ )denotes the aperture function. For a circular aperture subtending an angle €lo, we find A ( @ )=
1 0
for 101 SOo otherwise
The phase shift Yo(8) = k0{(C3/4)8~- (Af/2>02)
(28) introduced by the lens depends on both the coefficient of the spherical aberration C3 and the defocus Af. The vectors po and p denote the position of an
object
aperture spectrometer FIG.5 . Schematic setup in STEM for obtaining images with inelastically scattered electrons. Here 0, is the limiting objective aperture angle, and 0,defines the spectrometer acceptance angle. [From Kohl (1983a). Copyright 1983 North-Holland Publ. Co., Amsterdam.]
THEORY OF IMAGE FORMATION
187
object point and the position of the center of the spot, respectively. The latter can be influenced by the deflection coils of the instrument. We now generalize Eq. (10) to the case where the incident electron wave packet is given by Eq. (26). The measured quantity is the current per unit energy dIldE, which is given by (Rose, 1976a,b) dI I , kk:E, d E n30;h E,
where D(O) =
1 0
for 10150, otherwise
is the detector function, I, the incident beam current, 0,the spectrometer acceptance angle (at the specimen), E , = 13.6eV is Rydberg's energy, and E , the energy of the incident electrons. To derive Eq. (29), we have employed the small-angle approximation. For small energy losses AE 4 E,, we then find the following expressions for the scattering vectors:
+ (0 - a)] K' = k0[0,e, + (0' O)] K
=
k,[O,e,
-
where 0, = AE/2E, is the characteristic scattering angle for the energy loss AE. The two integrations over 0 and 8' reflect the coherent summation of the plane wave components of the incident wave packet, which enters quadratically in the current. The integration over the angle O must be performed over the entire solid angle accepted by the spectrometer. The factor S(K, K', C O ) / K ~ K describes '~ the scattering properties of the object in the firstorder Born approximation. [For simplicity we have restricted our discussion to electronic excitations. Phonon excitations are discussed by Kohl (1983b).] Although a certain relationship exists between our theoretical approach of inelastic image formation and the light-optical concept of object transparency, the latter is in general not sufficient to describe correctly the coherence properties of inelastically scattered radiation. Hawkes (1978) tried to describe image formation by the inelastically scattered electrons by means of an inelastic specimen transparency function. Following a recent paper by Rose (1984), let us reformulate Eq. (29) in such a way that it can be compared more
188
H. KOHL AND H. ROSE
directly with Hawkes’ results. To do so we Fourier-transform all relevant quantities exp[i(Kp - K’p’)] d28 d2W and
s
a(p) = A(8) exp[ - iy,(8)] exp[ - ik08p] d28 Conversely,
j
s ( ~ ~ = :w(p, ~p’, )w ) exp[ - i(Kp - K’p’)] d2p d2p‘ and A ( @ )exp[ -iyo(8)]
=
(qj
~ ( p exp(ik,Op) ) d2p
-
(32)
(33)
(34)
(35)
The function a( - p) is the amplitude of the wave function in the object planeup to the phase exp(ikz)-where the Gaussian focus is centered at the origin. Inserting these expressions into Eq. (29), we obtain dl dE
-
I , kk:E, 7c3Bgh E, x exp[ik,@(p’
-
p”)]D(@) d2p’d2p”d 2 0
(36)
The quantity w(p, p’, w ) has been called the “cross-spectral object transparency” by Rose (1984). For purely elastic scattering by a static potential in the dark-field mode, we obtain lo dE-rc30ih
E,
6(w)
jI
ja(p
-
p‘)T(p‘) exp( - ik,@p’) d2p‘
- lo k:EH 6(w) j u * ( p - p’)u(p - p”)T*(p’)T(p”) 7c38gh E,
x exp[ik,O(p’ - p”)]D(@) d2p‘ d2p“d 2 0
(37)
To include inelastic scattering, Hawkes (1978) has generalized Eq. (37) by replacing the object transparency T(p) by a phenomenologic complex transparency function T(p, w), where w determines the energy loss. Comparing the resulting expression with Eq. (36), we find that w(p’, p”, w ) should have the form NP’, P”, 0)= T*(P’, w)T(p”, w )
(38)
THEORY OF IMAGE FORMATION
189
However, w(p’, p”, (0)factors in this way only in the special case of transitions between pure nondegenerate states. In practice dipole excitations play a dominant role. For free atoms, then, at least one of the two object states must be degenerate.Due to theorthogonalityoftheobject states((n1m) = Oforn # m), the partial waves of the scattered electrons leading to different object states must be added incoherently, even though they might have suffered the same energy loss. The factorization, however, implies that for coherent illumination, those partial waves that have suffered the same energy loss remain coherent. Therefore, the object is not described accurately by an inelastic object transparency function. Rose’s (1984) cross-spectral object transparency must be used for a correct description of the inelastic image. Equation (29) is based on an assumption of fully coherent illumination. The importance of partial coherent illumination has already been emphasized by Hawkes (1978), who also worked out the implications of the inelastic transparency of the restricted form [Eq. (38)]. Rose (1984) has given a unified treatment of elastic and inelastic scattering, including partial coherence, by use of a modified Glauber approximation.
B. Image Formation in FBEM Let us consider a fixed-beam electron microscope (FBEM) as shown schematically in Fig. 6. The object is illuminated by a plane wave with wave
Kohler illumination
object
4%
imaging energy filter image
FIG.6. Image formation in an FBEM equipped with an imaging energy filter. The angles 0, and 8, are the illumination cone angle and the objective aperture angle, respectively. [From Kohl (1983a). Copyright 1983 North-Holland Publ. Co., Amsterdam.]
190
H. KOHL AND H. ROSE
vector k,. An imaging energy filter is incorporated in the imaging system below the objective lens. The latter can be either a Castaing filter (Castaing and Henry, 1962; Ottensmeyer and Andrew, 1980) or an omega filter (Rose and Plies, 1974; Krahl 1982). Since its detailed properties are of no concern for our considerations, we have drawn it as a black box. We assume that it removes all electrons, except those whose energy lies within a prespecified range, without introducing an aberration into the image. After the scattering process, the state of the system composed of the object and the incident electron is given by
The expansion coefficients A,(r) denote the electron wave function if the object has been excited from Im) to In). We shall restrict our discussion to thin objects so that the first-order Born approximation can be used. If we had no objective lens, the coefficients A,@) would be given by A,(r)
= exp(ik0r) 6,,
+ f,(K)Cexp(ik,r>/rl
(40)
for r + co,wheref,(K) is the scattering amplitude for the excitation of the nth state of the object and
k,
= ,/kg
- 2m0AElh2
(41)
is the wave number after the scattering has taken place. In an electron microscope, however, the electrons are registered in the image plane below the objective lens. Thus, the influence of the lens on the partial waves of the scattered electron must be taken into account. Next, let us calculate the value of A,@) in the image plane. In free space every single function A,(r) obeys the wave equation AA,(r)
+ k;A,(r)
=0
(42)
Also, the value of A,(r) just below the object plane is known. Therefore, we can use the Kirchhoff integral to calculate A,@) in any other plane, just as is done for elastically scattered electrons (Lenz, 1965, 1971; Hanszen, 1971; Hawkes, 1973,1984; Howie, 1984). For each A,(r), we must use the appropriate wave number k,. Propagation of electron waves in electric and magnetic fields (electron lenses) has been thoroughly discussed by Glaser (1943, 1952, 1956) and Glaser and Schiske (1953a,b), who have shown that the effect of the lens can be treated as a phase shift. The total current density per unit energy dj/dE is given by a sum over the partial current densities obtained from the amplitudes A,. Thus, for coherent illumination, contributions from partial waves resulting in the same final state of the object must be added coherently,
THEORY OF IMAGE FORMATION
191
whereas contributions leading to different final states must be added incoherently. In this way we find for axial illumination
x exp{ - i[y(0, AE) - y(W, AE)]} exp[ik(0
- el)p] d20 d2el (43)
where
y(0, AE)
=
k[(C3/4)e4 - (Af/2)e2
+ C,(AE/2E,)e2]
(44)
denotes the phase shift due to the defocus Af, the spherical aberration C,, and the chromatic aberration C,. The two vectors K and K’ are given by
K = k,[B,eZ
+ 01,
K’ = k,[O,e,
+ el]
(45)
and j , is the current density in the image plane if no object is present. The vector p denotes the position in the image plane but is referred back to the object. Equation (43) can be reformulated into a real-space description, similar to Eq. (36). The corresponding expression has been discussed by Rose (1984). Quite often the object in FBEM is illuminated incoherently from all directions, forming an angle less than 0,with respect to the optic axis (Kohler illumination). We must then average the current density per unit energy over all directions of the incident electrons, thus yielding
x exp{ - i[y(0, AE) - ?(el, AE)]} expCikp(0 - el)] d20 d20‘ d2@# (46)
The similarity between this formula and the corresponding formula for the STEM [Eq. (29)] will be discussed in the following subsection.
C. The Reciprocity Theorem In light optics the reciprocity theorem states that if the positions of the source and detector are interchanged, the light amplitude at the detector remains the same (Sommerfeld, 1978; Born and Wolf, 1965). Suppose the amplitude is u(S) at the source point S and u(P) at the image point P . If we put a point source with strength u(S) at P, the reciprocity theorem tells us that the amplitude at S will be u(P). Applying this theorem to a setup as shown in Fig. 7, we find that the amplitude at the detector is the same, regardless of whether we have the
192
H. KOHL AND H. ROSE
bS
source
I
xp Pi
objective lens
I
FIG.7. Simple arrangement of an imaging system that illustrates the reciprocity theorem.
source u(S) at S and measure at various image points P and P' or whether we move the source u(S) over the image plane and fix the detector at S . The theorem of reciprocity also holds for electron scattering off a static potential (Landau and Lifshitz, 1965). Thus, the image in a STEM with an axial detector and in an FBEM with axial illumination are the same for equivalent conditions (Cowley, 1969; Zeitler and Thomson, 1970a,b). The movement of the (virtual) source in a STEM is generated by a double deflection element. Before discussing inelastic scattering processes, we must define the meaning of reciprocity in these more general cases. For elastic scattering, reciprocity is due to time-reversal symmetry (Landau and Lifshitz, 1965) and leads to an important property of the elastic scattering amplitude for scattering of a wave ki to k,: f ( k i , k,) = f ( - k , , -ki) (47) Generalizing this result to inelastic processes, we find
f A k i k,) = f * m 4 - k,, - ki) The indexes rn and n denote the initial and final object states Irn) and In), while Im*) and In*) are the time-reversed states. The reciprocity theorem states that the amplitude for a transition from Im) to In) is equal to the timereversed transition amplitude from In*) to I m*). If inelastic scattering is taken into account, the reciprocity theorem cannot be used to prove the equivalence of images taken in FBEM and STEM, because the state reciprocal to the final state in FBEM is given by 3
THEORY OF IMAGE FORMATION
193
It consists of a coherent superposition of excited object states, which is clearly unequal to the initial state in a STEM, where the object is in its ground state. Nevertheless, Eqs. (29) and (46) look very similar. We shall, therefore, compare the images in STEM and FBEM. Let us assume that the instruments operate under equivalent conditions, namely, that the defocus, aberration coefficients, and objective aperture angles are equal. Let us further assume that the illumination angle in FBEM is equal to the spectrometer acceptance angle in STEM. If we then consider electrons that have suffered one particular energy loss AE, we find that, apart from the different forefactors, two further differences occur. First, in STEM the incident plane wave k , is focused, whereas in an FBEM the objective lens operates on the electrons k,, which have lost an energy A E in the object. Thus, in FBEM the chromatic aberration leads to an additional term in the phase shift y, which can be compensated by refocusing properly for the particular energy E , - AE. Second, the factor exp[ik,p(O - e’)] in STEM corresponds to exp[ikp(O - e’)] in FBEM. Since the energy loss is generally very small, this difference is usually too small to be detectable. Pogany and Turner (1968) used a different definition of reciprocity. In their famous paper, they name the approximate equivalence of the intensities an approximate “reciprocity of intensities.” We do not follow their definition, because in optics and quantum mechanics the term reciprocity is used to denote an exact property of a quantity (an amplitude) that obeys a particular differential equation (Sommerfeld, 1978). The approximate equivalence in intensities has a different origin and should be named differently.
D. Phase Contrast Thin phase objects are visualized in light microscopy by the phasecontrast method (Zernike, 1935). By introducing a proper phase plate in the back-focal plane of the objective lens, we obtain an intensity distribution in the image that, in the case of weak-phase objects, is proportional to the local phase shift (Goodman, 1968). Scherzer (1949) showed that in electron microscopy the phase plate can be obtained approximately by proper choice of defocus (Scherzer focus). This phase-contrast method can be used for thin objects and simplifies image interpretation considerably. For describing the phase contrast, it is sufficient to assume that the electrons are scattered by a static potential. This leads to an interesting question. Suppose the electron is scattered off a single atom; the transfer of momentum to the atom then results in a change of energy of the incident electron. Why is it then possible to see phase contrast or any interference between the unscattered and scattered electron waves? (We are grateful to several colleagues for raising this question.)
194
H. KOHL AND H. ROSE
Obviously the point is rather subtle. Phase-contrast images stem from the interference between the unscattered and the elastically scattered waves. In this case the object remains in its initial state. We shall show that there is always a finite probability amplitude that the internal state of the object does not change. For an FBEM with axial illumination, we obtain (Eusemann and Rose, 1982) A(8) exp[-iy(O)] exp(ik,pe)(f*(K)),
1
d28
(50)
where (f(K))T denotes the expectation value (thermal average) of the elastic scattering amplitude and all other quantities are as defined in Subsection 1II.B. The first term in brackets gives the constant current density if no object is present; the integral denotes the current density variations, which depend linearly on the scattering amplitude of the object. Evaluation of (f(K))T yields (Kohl, 1983b)
where VJK) is the Fourier transform of the interaction potential between the incident electron and the pth atom, and (plr,JT is the expectation value of the (Fourier-transformed) particle density of the pth atom. For a thin crystal plate, we find (f(K))T
= -
2c
exp(iKR,) exPC- ~ ( K ) I F ( K ) V 3 K )
(52)
Ir
where R, is the position of the pth atom within the elementary cell,
is the Debye-Waller exponent, and F(K)
=
N 0
for K equal to a reciprocal lattice vector otherwise
(54)
where N is the number of elementary cells in the object, M , the mass of a p atom, and s the number of atoms in the unit cell. The indexes q and o indicate the wave vector and polarization of a phonon, respectively, and fiqa
= (exp(Phmqa) -
1I-l
(55)
is the mean occupation number of the phonon mode qo. The calculation is given by Kohl (1983b) and need not be repeated here. The contrast decreases
THEORY OF IMAGE FORMATION
195
by a factor exp( - W ) . [The factor is not exp( - 2 W ) because the phase contrast depends on the amplitude, not the intensity, of the scattered electron.] Thus, the reason for the occurrence of phase contrast in spite of phonon vibrations is the same as the reason for the occurrence of distinct Bragg reflections off a crystal.
IV. NUMERICAL RESULTS A . Image of a Single Atom in an FBEM
To obtain quantitative results, we shall proceed in two steps. First, we shall calculate the mixed dynamic form factor S(K, K’, 0). For an arbitrary object, this is a rather hopeless task. Therefore, we must settle for simple examples. In this section we consider imaging of a single atom in an FBEM (Fig. 6) with axial illumination and an imaging energy filter. This filter removes all electrons unless they have suffered a distinct energy loss AE corresponding to a particular (dipole-allowed) transition. The objective aperture angle is assumed to be sufficiently small so that we can use the dipole approximation. Figuratively speaking, this means that the resolution limit of the microscope is larger than the diameter of the atom. In this case the mixed dynamic form factor for this particular transition is given by S(K, K’, 0) =
1
25 ~
+1 (nJMI exp(-iKrj)In’J’M‘)
x
M ,M ’ j, k
x (n’J’M’I exp(iK’rk)InJM)G(o - Am)
j, k
=
CKK’G(o - Am)
The only assumption used in the last step is rotational symmetry of the object. The proof is outlined in Appendix C . Since the quantum states with various M for a given 5 are degenerate, we must average over the 25 + 1 initial states ( n J M ) and sum over all 25’ + 1 final states In’J’M’). Note carefully that in this approximation the mixed dynamic form factor S(K, K’, o)factors into a strength factor CG(m - Am) and a shape factor KK’. Thus, the shape of the intensity distribution is the same for all atoms and depends only on 8, and on the microscope setting. The particular transition enters only via the strength factor CG(o - Am).
196
H. KOHL AND H. ROSE
Inserting Eq. (56) into Eq. (43), we obtain
with
Here J,(x) and J , ( x ) denote Bessel functions of the first kind. The total current density is given by a sum of two contributions. The first part, l A l i12, stems from the excitation of a dipole parallel to the incident electron, while IA,I2 describes the excitations perpendicular to the optic axis. For our numerical calculations, we assume that the image is recorded in the Gaussian image plane. Furthermore, we neglect spherical and chromatical aberrations. The results obtained with these assumptions are displayed in Figs. 8-1 1. The intensity distribution in the image is shown in Fig. 8 as a function of the (normalized) distance kp8, = 3.83p/d from the atom; d = 0.612/8, is the instrumental resolution limit (for an ideal lens). The dashed curve denotes the contribution of the parallel excitations IA II 1'; the solid curve shows the total intensity. The ratio 8J8, of the characteristic scattering angle eE = AE/2E0 to the objective aperture angle B0 is a measure of the energy loss. For large energy losses the aperture is almost uniformly illuminated. The intensity distribution in the image is then almost proportional to the Airy distribution [J,(kp8,)/(kp8,)]2. At large losses the perpendicular excitations contribute very little to the total intensity. With decreasing energy loss, the image spot starts to broaden as a result of the delocalization of the inelastic interaction. Its influence on the spot shape begins to show up for 8J8, = 0.5. For lower losses ( O E / 8 0 = 0.1 and 0.02), we find a ring of high intensity around the atom. This ring is due to the perpendicular excitations, which dominate low-loss images. The spot size increases even further. These effects of decreasing energy loss can be seen even more clearly in Fig. 9. Here we have drawn the image intensity within a thin ring p dp around the atom. The solid curve shows the total intensity kp8,( [ A l2 1 A , 12) ; the dashed curve represents the contribution of the parallel excitations kp8,I A II 1.' For relatively large energy losses ( 8 J 8 , = 2.0 or OS), the intensity decreases rapidly within a short distance from the object. For lower-energy losses the image spot is much broader. Comparing the curves for 8J80 = 0.1 and those for 0.02, we find that the height of the first maximum hardly increases. Thus, the increase of the cross section is due to an increase of the
+
197
THEORY OF IMAGE FORMATION (b)
-
003
001
0
0 20 018
0.18
0 16
0.16
0 14
0.14
-6
-
012
a?
-2 010
-2
w
012
010
U
008
008
0 06
006
0 04
004
002 0
2
4
6
8
1
0
002 0
FIG.8. Radial intensity distribution in the FBEM image of an atom formed by inelastically scattered electrons with different relative energy losses OJB,. (a) @JOo = 2.0, (b) @,/O, = 0.5, (c) OJO, = 0.1, and (d) t),/O, = 0.02. The normalized radius kpB, = 3.83p/d is inversely proportional to the radius d of the Airy disk. The dashed curve represents the contribution of the parallel excitations ; the solid curve describes the total intensity.
intensity at a far distance from the center of the atom. The images of neighboring atoms will then overlap considerably, thus creating an undesirably high background intensity. Comparing the images at a given energy loss for various objective apertures, we must keep in mind that the scale at the axis changes also. To be specific, let us assume an energy loss AE = 120 eV and an incident energy of 60 keV. Then 0, = 1 x and the four energy-loss parameters correspond
198
H. KOHL AND H. ROSE 0301
lb)
025-
-z'
020015
I
I
I
m" 0 10
~
1 a
005-
07 0
5
10
15 k Pea
20
25
30
kpe,
FIG.9. Intensity contained within a ring of radius p and width dp in the inelastic atom images shown in Fig. 8 as a function of the normalized distance from the atom center. (a) OE/OO = 2.0, (b) O J O , = 0.5, (c) HE/@, = 0.1, and (d) OE/Oo = 0.02.
2 x lop3, 1 x and 5 x to objective aperture angles 8, = 5 x 10- '. The corresponding instrumental resolution limits are then given by d = 60,15,3, and 0.6 A, respectively. Thus kp8, = 1 corresponds to p % 15,4, 0.75, and 0.15 A in these images. We find that for increasing objective aperture angle the spot size decreases considerably. The instrumental resolution limit, however, can only be reached for a large energy-loss parameter (e,/e, = 2.0). We are thus led to the problem of determining the specimen resolution obtainable with inelastically scattered electrons. Since most objects consist of many atoms, rather than just two, use of the Rayleigh criterion would be inappropriate. For such problems other definitions, which relate the resolution to the radius of a circle containing a given percentage of the total intensity, have proven fruitful. The two most common definitions for the effective specimen resolution limit deff use either the radius of a circle that contains 84 % of all scattered electrons or else twice the 59 % radius. Both definitions are chosen so as to yield the same result as the Rayleigh criterion for a point scatterer. The fraction P(kp8,) of the total intensity falling into a disk of radius p is shown in Fig. 10. We find that the effective resolution limit increases with
$IL
199
THEORY OF IMAGE FORMATION
a
IO-
04
02 00
5
10
15 kPeo
(b)
(a)
20
25
30
0
5
10
15 kpe,
20
25
30
10,
(C)
25
30
kpe,
kpe,
FIG.10. Fraction P ( k p 0 , ) of the total intensity contained within a disk of radius p in the images shown in Fig. 8. (a) OJO, = 2.0, (b) O,/O, = 0.5, (c) @,/On = 0.1, and (d) OE/OO = 0.02.
decreasing energy loss. The two definitions for deffyield different numerical values, but the general trend remains unaffected. Some authors have suggested use of the equation as a rule of thumb for the effective resolution limit (Howie, 1981). This formula (or similar ones) can be inferred either from the uncertainty relation or else by replacing the aperture angle B0 by the characteristic scattering angle 0, in the Abbe formula
d
= O.6A/tI0
(60)
To justify this procedure, we presuppose that most of the electrons are scattered within a cone of OE around the optic axis. This implies that replacing the angular shape of the scattered intensity l/(0; + d2) with 1 for 0 S 8, and with zero otherwise will not affect the result too much. Looking at Fig. 10, we find that this crude rule of thumb does not describe reality very well. In particular, for low energy losses (0,/0, e l.O), the effective specimen resolution limit deffis much smaller than Eq. (59) indicates. The most important reason is the fact that the total intensity is the sum of two distinct parts. For low energy losses, the perpendicular excitations lead to the
200
H. KOHL AND H. ROSE
characteristic donut-shaped structure, which means that a large fraction of all electrons in the image are close to the atomic nucleus. These results are in qualitative agreement with recent experimental findings. Colliex et al. (1981), and Colliex (1982) have imaged small uranium clusters and found that the effective resolution limit, although worse than the point-to-point resolution of the instrument, is definitely better than we would expect from the rule of thumb [Eq. (59)]. This fact will be seen more clearly in the discussion of the image of a surface plasmon (Subsection 1V.D). Due to the poor signal-to-noise ratio in inelastic images, it is extremely difficult to measure intensity profiles. Thus, unfortunately, a detailed quantitative comparison is not yet possible. Another means for describing the properties of optical systems is to consider the diffractograms H(h/B,). They are the Fourier transforms of the image intensity and determine how well a periodicity in the object is transferred to the image. Some mathematical details are given in Appendix D. In Fig. 11 we have drawn the diffractograms for the images shown in Fig. 8. With decreasing energy loss, the decrease of H(6/8,) becomes steeper. The transfer of high-spatial frequencies in the object is then increasingly suppressed. For OE/O, 5 0.5, we find that the diffractograms even have a zero, meaning that this spatial frequency will not be found in the image. For 6 larger than this value, H(h/8,) is negative, thus resulting in a reversal of contrast. All results given in this chapter are also valid for a STEM with a small axial detector (0,4 6,) as can be seen from our discussion in Subsection 1II.C. The influence of the detector angle will be discussed in the following subsection. B. Image of a Single Atom in S T E M
To calculate the intensity distribution in a STEM image, we insert Eq. (56) into Eq. (29). We assume zero defocus (Af = 0) and neglect spherical aberration (C, = 0). In this case four of the six integrations can be performed analytically (Kohl, 1983b). The remaining integrals were evaluated numerically; some results are shown in Figs. 12-14. We have drawn the intensities for the energy loss parameters 8J8, = 2.0,0.5,0.1, and 0.02 for a fixed detector angle 0,.The dashed curve denotes the contribution of the excitations parallel to the optic axis. We see in all three figures that the spot size increases with decreasing energy loss. Figure 12 shows the intensity distribution for a comparatively = 0.28,). Here we note again the donut-shaped structure small detector (0, for low energy losses (OE/6, = 0.1 or 0.02). This is comparable to Fig. 8, but the effect is not as pronounced. Increasing the spectrometer acceptance angle
20 1
THEORY OF IMAGE FORMATION
"1
601
- 1 04 0
(b)
05
1
15
2
1.5
1 2
6/80
16 0 14 0
12 0
-? 9 -
r
10 0
80 60 40 20
00 -2 0
05
1
a / 00
15
2
-
0
5 05
1
0
618,
FIG.11. Diffractogram H(6/Bo) of the inelastic FBEM image of a single atom as a function of the normalized spatial frequency 6/B, in the case of axial illumination for four values of the characteristic energy loss parameter BJB,; (a) BJB0 = 2.0, (b) B J B , = 0.5, (c) BJB0 = 0.1, and (d) Os/Oo = 0.02. The dashed curve represents the contribution of the parallel excitations.
(0,= e,), we find only a very weak donut structure for 8J0, = 0.02. For an even larger detector angle (0,= XI,), the spot looks more and more like an Airy disk. At first sight it seems surprising that the spot shape is so strongly dependent on the detector angle. In imaging with elastically scattered electrons this dependence is very weak, so, in general, it can be neglected. Why is this different for inelastic scattering?
202
H. KOHL AND H. ROSE 0.25
(a)
1.0
0.20
d -
0.15
Y
Y
0.10
0.05
C
FIG.12. Radial intensity distribution of inelastic images obtained with a spectrometer acceptance angle (at the specimen) 0,= 0.28, for four values of the energy-loss parameter 8J8,; (a) OJ8, = 2.0, (b) OJO, = 0.5, (c) OE/Bo = 0.1, and (d) B J B , = 0.02. The dashed curve represents the contribution of the parallel dipole excitations.
Due to the small extension of the atomic charge distribution (RA < 0.5 A), an atom can be considered as a point scatterer for elastically scattered electrons at present resolution limits (d 2 2 8).Therefore, the electrons are scattered almost uniformly in all directions accepted by the detector. The detector angle, which determines the total current measured, then has no significant influence on the spot shape. With respect to inelastic scattering the atom cannot be considered a point scatterer, as is apparent from the spot shapes in Figs. 8 and 12-14. The strong
203
THEORY OF IMAGE FORMATION (b)
2c 18
\
16
-
z
-
14 12
10
b-.
8
E 4
\
' 0
FIG.13. Radial intensity distribution in the inelastic image of an atom where the spectrometer acceptance angle @, is equal to the objective aperture angle Oo. (a) OJOo = 2.0, (b) O,/H, = 0.5, (c) O,/O, = 0.1, and (d) OJO0 = 0.02.
dependence of the spot shape can be explained by the fact that electrons passing close to the nucleus have a large excitation probability, but due to their large deflection they contribute to the image only if the detector angle is sufficiently large. Therefore, the spot size decreases with increasing spectrometer acceptance angle. The perpendicular excitations play an important role for this effect. In their channeling experiments, Tafto and Krivanek (1982) observed the strong dependence of localization on scattering angle. They find that as the angle of scatter increases, the excitation is more localized.
204
H. KOHL AND H. ROSE
120-
(d)
100 -
100-
-
$-
80-
60-
3
80 60-
40-
40-
20-
20.
07
0
2
4
6 kpe,
8
1
0
0 ....... 0 2
4
6 kPe,
8
1
0
FIG.14. Radial intensity distribution in the inelastic STEM image of an atom obtained with a large detector 0,= 50, for four values of the energy-loss parameter BE/@,; (a) 8,/0, = 2.0, (b) Oe/Bo = 0.5, (c) 0,/0, = 0.1, and (d) 0,/0, = 0.02.
Maslen and Rossouw (1984) and Rossouw and Maslen (1984) evaluated the differential cross section for inelastic scattering in crystals. They used a hydrogenic model to calculate the mixed dynamic form factor for K-excitations (Maslen, 1983). Their results confirm the increase of localization for increasing scattering angle. Knowledge of the spot shape may be of special interest for high-resolution microanalysis. The current approach is to deduce the elemental concentration in a pixel from the number of counts for a characteristic energy loss (after subtraction of the background) by using a differential cross section (Isaacson
THEORY OF IMAGE FORMATION
205
and Johnson, 1975; Egerton, 1981a,b). However, this is true only if the spot is smaller than a pixel. The differential cross section only gives the total number of scattered electrons, regardless of where they appear in the image. If the resolution is so high that the image spot extends over several pixels, the spot shape must be considered in analyzing the data. As can be seen from Figs. 8 and 12- 14, this will be particularly important for high-resolution low-loss images (small 8,/0,). The Z-contrast method (Crewe et al., 1975; Carlemalm and Kellenberger, 1982; Colliex et al., 1984) is frequently used for obtaining high-contrast images of biological specimens. This method makes use of the fact that the ratio of the elastic to the total inelastic cross section is proportional to the atomic number Z (Lenz, 1954). Dividing the elastic by the inelastic image, we thus obtain an image whose intensity is proportional to the mean atomic number of the corresponding object point. A detailed theory of contrast, however, must take the different localizations of the elastic and inelastic images into account. As we divide the dark-field signal by the inelastic signal [lDF(p)/linc,(p)],the delocalization of the inelastic image is expected to lead to an increased contrast as compared to the ratio of the respective cross sections. Calculations of total inelastic images (summed over all possible energy losses) have been performed by Rose (1976a,b). Since the details of his work are too lengthy to be repeated here, we refer the interested reader to his original papers. C . Images of Assemblies of Atoms
So far we have been dealing with images of a single atom. In practice, however, any specimen consists of a large number of different atoms. Unfortunately, it is not possible to treat such an object in the same general manner as the case of a single atom. We shall discuss one approximation that gives some insight into the image formation process. The excitation spectrum can be divided into several parts. Quasi-elastic processes lead to energy losses smaller than the energy resolution of about 0.5-2 eV. Such losses are due to the excitation or annihilation of phonons. The low-loss region extends out to energy losses of about 50 eV. The dominant processes in this regime are plasmon excitations and transitions of valence electrons. The plasmon energies are not very sensitive to elemental composition, so their use for microanalysis is very difficult (Williams and Edington, 1976). The energies of valence excitations are strongly dependent on the chemical state of an element, so they can be used to identify molecular species (Isaacson, 1972; Johnson, 1979). We shall focus our attention on inner-shell losses whose energies are larger than 50 eV. The energies of the edges are characteristic for a particular
206
H. KOHL AND H. ROSE
element. The fine structure of the energy-loss spectrum yields valuable information on their local environment [extended energy-loss fine structure (EXELFS)] and on their chemical state [energy-loss near edge structure (ELNES)] (Stern, 1982). By registering the edge as such, we can obtain purely elemental information. In this latter case the natural approximation is to consider the atoms as single, independent entities. We thereby neglect the influence of the potential of the neighboring atoms on the secondary electron of an ionized atom. Since this EXELFS effect produces only small intensity oscillations in the energy-loss spectrum (typically less than 5 %), this approximation will be satisfactory. [Very close to the edge the error will be larger, as can be seen from ELNES and x-ray absorption near-edge structure (XANES) experiments (Pendry, 1983)l. By assuming the atoms to be independent from one another and using the form factor [Eq. (56)] for each, we obtain for an assembly of N atoms of one element in the object plane the total form factor N
S(K, K’, w ) = Ch(w - Aw)KK’
1exp[ - i(K - K’)pj]
j=1
(61)
where pj denotes the position of the jth atom. The excited states of the N atoms are degenerate, so we sum over all of them. Inserting this expression into Eq. (29) or Eq. (43), we find that the total intensity I , in the image is just the sum of the intensities due to the excitation of a single atom I , N
This situation differs sharply from that of an elastic dark-field image. There we must consider the interference terms between different atoms, because there is only one final state (which is equal to the initial state). Here the different final states can be classified by the number j of the atom that has been excited. All the contributions of the different atoms must be added incoherently, because the final object states are distinct. One possible way to determine the delocalization of the inelastic interaction experimentally is to measure the intensity distribution in the image of a straight edge of a thin foil (Isaacson et al., 1974). Suppose we take an amorphous object, whose density (per unit area) is given by
If the scatterers are independent, we obtain the intensity distribution by a convolution of the intensity distribution of a single atom with the particle
207
THEORY OF IMAGE FORMATION
density distribution. This is equivalent to multiplying the Fourier transform of the density times H ( 6 ) and using the inverse Fourier transform (Rose, 1976a,b). Thus
In Fig. 15 we have drawn the (normalized) intensity distribution in the inelastic image of the edge of a thin foil. We can see again that the image becomes blurrier as the energy loss decreases.
-100
-50
kxe,
kx0,
(a1
(b)
,
-50
kx0,
00 kxe,
(Ci
(d I
00
50
100
- 100
50
FIG.15. Normalized intensity distribution in the inelastic FBEM image of a straight edge for the case of axial illumination as a function of the distance x from the physical edge for four values of the energy-loss parameter 0JBo; (a) BJV, = 2.0, (b) B J B o = 0.5, (c) OE/BO = 0.1, and (d) f~,/O, = 0.02.
100
208
H. KOHL AND H. ROSE
Similarly we can calculate the intensity I, in the image of an amorphous sphere as
Some results for OJO, = 0.1 are shown in Fig. 16. To obtain dimensionless quantities, we have displayed 1J(8n2R,”).For resolution limits larger than the radius of the sphere ( d = 2R,), we find a donut shape again. For smaller resolution limits (d = R,), this structure disappears. The intensity distribution is then more similar to the projected density; still, there is some delocalization left. Images of spheres have been taken by Colliex et al. (1981) and Colliex (1982). These authors found that the spot is broader than the sphere when inelastically scattered electrons are used to form the image. Let us now consider the inelastic image of a thin crystal. We assume the crystal to be so thin that the first-order Born approximation still holds. In Subsection 1I.B we found that for periodic objects S(K, K’, w ) is nonvanishing only if K - K’ = g. We use only this property, which holds for the two dimensions of the object plane, and the small-angle approximation [Eq. (3 l)]. Then g is a two-dimensional, reciprocal lattice vector. We make no assumption about the particular excitation; i.e., the atoms may be strongly interacting. When we take a lattice image in STEM, we are free to choose the position of our detector with respect to the optic axis. The importance of the detector position for elastic lattice images has been thoroughly discussed by Cowley and Spence (1981) and Spence and Cowley (1978). To establish the main point, let us consider the biprism experiment as shown in Fig. 2. We have already found (Subsection 1I.B) that to see the crystal structure we must choose k and k’ so that their difference is equal to a reciprocal lattice vector g. The lattice fringes then have the same periodicity as the crystal. To detect any 018.
-a“ -
I L q 010-
008006-
o i
-* a
(bl
010
006
004-
002-
018
(a)
004 ....,. .-.. ,
\,
0
PlR,
-..
002
0
’... .
1
2
3
4
p/R,
FIG.16. Radial intensity distribution in the inelastic image of a sphere for two values of the ratio d / R , between the resolution limit d and the radius of the sphere R,; (a) d / R , = 2 and (b) d/R, = 1. A STEM with a point detector or an FBEM with axial illumination has been assumed.
THEORY OF IMAGE FORMATION
209
elastic signal, we must place our detector in a Bragg position with respect to the incident beams. In STEM, the object is illuminated by a coherent superposition of all plane waves within the illumination cone, which is limited by the objective aperture. This wave packet is diffracted and the crystal is resolved only if the detector is located in an area where (at least) two diffracted disks overlap (Fig. 17). If we want to register inelastically scattered electrons, we must still fix k and k so that k - k = g, but now we detect electrons in all directions. Any excitation in a crystal can be classified by its wave vector q (or its crystal
FIG. 17. Scheme demonstrating the formation of an elastic lattice image in STEM. The incident cone is Bragg reflected. The lattice fringes appear in the image only if the detector is placed at a position where two cones overlap (hatched area).
210
H. KOHL AND H. ROSE
momentum hq), which lies in the first Brillouin zone. The energy-loss spectrum in a given direction k, reflects only those excitations whose wave vector q differs from the scattering vectors K and K' by arbitrary reciprocal lattice vectors. If, on the other hand, we look at a specific energy loss AE, we register electrons only in those directions corresponding to an excitation with energy hw, = AE. Since in any real experiment we have a finite energy resolution, the directions will not be so well defined. To detect a lattice image with energy loss AE in a STEM, we must fulfill a condition for the detector similar to the one for the elastic case. The disk representing the incident wave packet must be tilted by q (with hw, = A E ) and then Bragg-reflected (Fig. 18). The detector must be placed so that at least two disks overlap. If the spectrometer resolution is worse than the bandwidth of the excitation, the position of spectrometer is irrelevant. Note carefully that for thin crystals such images are due to single-scattering inelastic processes only. The image then reflects only the spatial structure of the excitation. It should not be confused with inelastic images of thicker crystals, in which multiple elastic and inelastic scattering processes are dominant (Fujimoto and Kainuma, 1963; Howie, 1963, 1984). The latter images look similar to elastic images; the contrast, however, is reduced. (a)
crystal
ax
Fia. 18. Comparison of the formation of elastic and inelastic lattice images in STEM. In (a) and (b), the incident and Bragg reflected cones are represented by their axes; (c) and (d) show the cross section through the cones at the detector plane. Lattice fringes can only be observed if the detector is placed in one of the hatched areas. The deflection 0, = q/k, resulting from the excitation of the crystal must be considered when describing inelastic image formation.
THEORY OF IMAGE FORMATION
21 1
In an FBEM with an imaging energy filter the image is formed with all electrons of a given energy that pass through the objective aperture. To form an elastic lattice image in an FBEM, at least two diffracted beams or one diffracted beam together with the unscattered beam must pass through the objective aperture. For inelastic lattice images, we must take the additional tilt by the angle O4 = q / k into account. Then if two diffracted and tilted beams pass through the objective aperture, we can see lattice fringes. If the energy filter transmits a whole energy interval simultaneously, the total image is given by an incoherent superposition of all inelastic lattice images resulting from excitations whose energies are in the chosen interval. In practice it is very difficult to prepare a crystal thin enough so that the first-order Born approximation holds. The argument can be carried over to thicker crystals, if we take hq to be the difference of the crystal momenta between the final and initial states of the object. Inelastic lattice images of crystals have been obtained by Craven and Colliex (1977). A quantitative description of such images is rather complicated. Thus, it is not yet possible to compare their results with theoretical calculations. In the case of thin crystals and independent atoms, the Fourier transform of an image is related to the diffractrogram of a single atom H ( 6 ) via r
r
x exp[ - i k ~ ( p- pj)] d2(p - p j ) exp( - ik6pj)
For a periodic object, the last sum is nonvanishing only if k6 is equal to a reciprocal lattice vector. If the atomic positions are known, it is then possible to measure H ( 6 ) at points that correspond to reciprocal lattice vectors. D. Image of a Surface Plasmon
As a further example of an inelastic image, let us consider the image of a plasmon on the surface of a metallic sphere or, similarly, on the surface of a spherical void in a metal. The properties of such objects have gained increasing attention over recent years (Fujimoto and Kumaki, 1968; Natta, 1969; Ritchie, 1981; Schmeits, 1981; Marks, 1982; Manzke et al. 1983; Penn and Apell, 1983; Lundqvist, 1983; Ekardt, 1984). The procedure to determine the intensity distribution is similar to the one described in Subsection 1V.A. To calculate the mixed dynamic form factor, we use a plasmon model given by Ashley et al. (1974, 1976). We consider only dipole plasmons that have an
212
H. KOHL AND H. ROSE
excitation energy of AE = EP/$ for metallic spheres and AE = E p Z for voids, where E , is the energy of a bulk plasmon. For the calculations (Kohl, 1983a,b), we assume an FBEM with axial illumination or a STEM with a small axial detector and neglect all aberrations (y = 0). For finite resolution, we use d / R , as a parameter. The resolution limit d and the radius from the center of the sphere p are both normalized to the radius of the sphere R , . Figure 19 shows the intensity distribution for d / R , = 0.2. The total intensity (solid curves) is again a sum of two parts. One can be viewed as resulting from the parallel excitations (dashed curves), the other as due to the perpendicular excitations. For a rather large energy loss (O,/O, = O.l), we obtain a full spot, whereas for lower energy losses the image is donutshaped. This ring of high intensity at the surface of the sphere is due to the perpendicular excitations. Images of surface plasmons were already reported 15 years ago by Henoc and Henry (1970). They used a neutron-irradiated foil that they imaged in an FBEM with a Castaing-Henry filter (1962) by selecting the plasmon loss electrons. In this case the voids appear as bright spots on a dark background, whereas they can hardly be recognized in the elastic image. Some experimental findings have been obtained that confirm our theoretical predictions. During the past years, Batson (1982a,b, 1985) and Batson and Treacy (1980, 1982) imaged small aluminium spheres in a STEM by using only those electrons that had excited a surface plasmon. In the course of specimen preparation, their spheres had been covered with a thin oxide layer. In their experiments both the instrumental resolution limit ( d / R , 4 1 ) and the energy loss (OJO, 4 1) were very small. More recently Ach6che et al. (1985) experimented with metallic spheres of tin, gallium, and uranium. In these images they find bright rings at the surfaces. Unfortunately, we cannot yet compare their experimental results quantitatively with our predictions from theory, because they used all surface plasmons in their experiments so far, whereas we have considered only dipole plasmons. Furthermore, the finite spectrometer acceptance angle influences the spot shape as well.
"6
05
PI R,
1
15
FIG.19. Radial intensity distribution in the image of a sphere, obtained by using only those electrons, which have excited a dipole surface plasmon, (a) B,/O, = 0.1 and (b) &/B0 = 0.005.
THEORY OF IMAGE FORMATION
213
Nevertheless the qualitative agreement between theoretical and experimental results is encouraging. Batson and Treacy (1982) and Colliex and Mory (1984) state that the excitation process is amazingly localized, in contrast to expectations from the rule of thumb [Eq. (59)]. The intensity distribution in their images indicates a specimen resolution limit of less than 40 8, whereas Eq. (59) would indicate approximately 800 A. These experiments show that, in spite of the delocalization of the interaction, we can obtain images with fairly high resolution even at rather low energy losses (AE z 5-15 eV). V. CONCLUSION
We have outlined a quantum-mechanical theory of image formation in the electron microscope, which considers both elastically and inelastically scattered electrons. The calculation of the inelastic image necessitates knowledge of the local scattering and excitation properties of the object. This information is contained entirely in the mixed dynamic form factor S(K, K’, w), which allows us to determine the shape of the inelastic image of thin objects within the first-order Born approximation. The mixed dynamic form factor can be viewed as a generalization of the ordinary dynamic form factor S(K,w). We have investigated the properties of the mixed dynamic form factor.and its interrelation with the generalized dielectric function including local field effects. The function S(K, K’, w ) can be determined experimentally by using an interference experiment as shown in Fig. 2. Recently, Schulke (1982) has shown that the mixed dynamic form factor of crystalline objects can also be measured by Compton scattering of x rays if the crystal is in Bragg position. The theory yields information about the coherence properties of the scattered radiation. In the case of coherent illumination, all partial waves leading to the same final object state are coherent with each other. Partial waves, however, that result in different final object states are incoherent with each other. By considering these coherence properties we can explain the occurrence of phase contrast in spite of the fact that the object atoms can absorb recoil energy, which is connected with an energy loss of the incident electrons. In Section IV we have calculated the intensity distribution in the image of two model objects. The image of an atom formed by the inelastically scattered electrons (in dipole approximation) is much more blurred than that formed by the elastically scattered electrons. This delocalization, however, is not as pronounced as has been assumed previously by Howie (1981). For small energy losses we found a donut-shaped intensity distribution. Contrary to elastic images, inelastic images in STEM exhibit a pronounced dependence
214
H. KOHL AND H. ROSE
on the spectrometer acceptance angle. The spot size decreases with increasing detector angle. Our calculations of images formed by dipole-surface-plasmon-loss electrons show that for low energy losses and high resolution the image of a sphere consists of a ring of high intensity about the projected surface. The delocalization is again less pronounced than has been assumed previously by Howie (1981). Our predictions agree reasonably well with recent experimental findings (Batson, 1982a; Colliex, 1982). Summarizing, we can state that the large progress in analytical electron microscopy has enabled the experimentalists to obtain inelastic images with high spatial resolution. Although the interpretation of the inelastic images is more difficult than the interpretation of elastic images, they contain information which is not accessible otherwise. For example, inelastic images can serve as a means for elucidating the local electronic structure at boundaries and surfaces. Therefore, these techniques will undoubtedly gain increasing importance for solid-state research.
APPENDIX A . Calculation of the Transition Rate
The interaction potential of the incident electron with the object is given by the operator
b' =
1 V(r - rj) j
where r and rj denote the positions of the incident and thejth object electron, respectively. The matrix element can be written as (kml Vlkfn)
=
=
Similarly, we find
1 (ml .i
7
(ml
s
exp[i(kf
J exp[i(kf
-
k)r] V(r - rj) d3rln)
I-
- k)(r
-
rj)] v(r
-
rj) d3(r - rj)
THEORY OF IMAGE FORMATION
215
The two scattering vectors K and K’ are defined as
K=k-kJ,
K’=k’-k
f
The operator
is the Fourier transform of the particle density operator. With the procedure introduced by Van Hove (1954), the product of matrix elements appearing in Eq. (8) can be transformed into a density-density correlation function. Taking into account the probability p m that the system is initially in state Im), we obtain for the sum
where ( )T denotes the thermal average, H , is the Hamilton operator of the object, and pK(t)is the density operator in the Heisenberg representation. By introducing the mixed dynamic form factor (Rose, 1976a,b) as
we eventually arrive at Eq. (8). The use of correlation functions such as (A7) is often advantageous, because it is then not necessary to calculate the corresponding wave functions explicitly. For collective excitations, it is in many cases possible to determine equations of motion for the density operator, which in turn lead directly to the correlation function (Pines, 1962, 1964).
216
H. KOHL AND H. ROSE
B. Interrelation of the Mixed Dynamic Form Factor and the Generalized Dielectric Function
To calculate the reaction of a solid when an external electric potential cDext(r,t ) is applied, we generalize the treatment given by Platzman and Wolff (1973). The Hamilton operator H can be written as the sum of an unperturbed operator H , and a perturbation H,(t) = -e,
s
p(r)cDexc(r,t) d3r
(B1)
where -eo denotes the charge of the electron and p(r) the density operator. By utilizing the Fourier expansions
mext(r,t) =
1 7 1cDf'(t) exp(iKr) K
we obtain
where V is the volume of the solid. By employing methods based on the linear response theory (Platzman and Wolff, 1973), we derive the following expression for the expectation value of the Fourier-transformed density operator
By inserting the Fourier transforms with respect to time,
into Eq. (B5), we obtain
217
THEORY O F IMAGE FORMATION
where the function
is the response function and 6 > 0 is a small quantity tending to zero. The response function describes the reaction of pKwhen a periodic potential @Ft of frequency OI is applied to the object. We now show the relationship existing between this function and the mixed dynamic form factor. To avoid unnecessary terms, we consider only the case o # 0. (For o = 0, the term ( p K ) T d ( o ) and similar ones occur in the formulas.) By using I
Zlr,
r
U,
J-u.
(p-K,pK(t))T
exp(iwt) dt
= exp( -Bhw)
S(K, K’, w )
(B10)
we obtain at the limit 6 .+ 0
+
x exp[i(o - o’ i6)tl O0
a,
do’d t
S(K, K’, o’)[ 1 - exp( - Pho’)] do’ w - w’ + i6 S(K, K’, 0’) [1 - exp( - Phw’)] d o ’
ine; hV
-~ [1 - exp( - Phw)]S(K, K’, w )
where # denotes the principal value. It should be noted that taking the imaginary part of (B11) yields a relationship between the linear response function F(K, K‘, w ) and the mixed dynamic form factor only if S(K, K’, w ) is real (e.g., for K = K’ or when the object is centrosymmetric and invariant with respect to time reversal). This function, however, is generally complex and fulfills the relation S*(K, K’, W ) = S(K’, K, O )
(B12)
To obtain a relationship between F and S, we exchange the variables K and K’ in (B 1 I), which yields an additional complementary equation. By adding
218
H. KOHL AND H. ROSE
these equations, using relation (B 12), and taking the imaginary part, we obtain the real part of S(K, K’, w ) : Im i{F(K, K’, w ) + F(K, K, w ) } = (-ne#l/)[l
- exp(-flhw)]
Re S(K, K’, w )
(B13)
On the other hand, by subtracting the two equations and using Eq. (B12), we obtain the imaginary part of S(K, K , w ) as 2 1 - ne0 Im - {F(K, K’, w ) - F(K’, K, a)}= [l hV 2i ~
-
exp( -flhw)] Im S(K, K , w ) (B14)
If we multiply Eq. (B14) by i and add Eq. (B13), we arrive at the following relation between the response function and the mixed dynamic form factor: S(K, K’, w ) = {ihV/2nei[1 - exp( -flhw)]}(F(K, K‘, w ) - F*(K’, K, w ) } (B 15)
In solid-state physics, we often use a dielectric function, which describes the response of the system as a function of the total potential @Iot(r, t ) = aext(r, t ) mind(r,t), where Oindis the induced potential. This dielectric functions plays the central role in the self-consistent field approximation. Here we , was intromust use the generalized inverse dielectric function ~ & , ( w ) which duced by Adler (1962) and Wiser (1963) to describe local field effects. In accordance with Adler (1962), we write
+
This relationship can be expressed as
By using the definition
THEORY OF IMAGE FORMATION
219
of the inverse dielectric function as an inverse matrix with respect to EKK,(w), we find
From the Poisson equation, we obtain e,p$d(co) =
K‘
{E&,(w) - dKK,}@Zt(u)
(B20)
Comparison of Eq. (B20) with Eq. (B8) yields F ( K , K’, W )
= E ~ K ~ { C ; ; , ’ ,( OdKK,} >)
(B2 1)
By inserting this relationship into Eq. (B15), we find
This formula enables us to determine the mixed dynamic form factor from the inverse generalized dielectric function. C . The Mixed Dynamic Form Factor in Dipole Approximation
In the following derivation, we shall use the notation employed by Messiah (1964). As a special example we shall consider an arbitrary, rotationally symmetric system. Then an operator T can be expanded into tensor operators TF) of kth order:
We start by proving the indentity
1 (nJMITf’In’J’M’)((nJMIT$’)ln’J’M‘))*
M.M‘
where InJM) denotes a state with principal quantum number n, angular momentum J , and projection M . The term ( dIT(’)IIn’J’) is the reduced matrix element. By using the Wigner-Eckart theorem (Messiah, 1964)
220
H. KOHL AND H. ROSE
and the orthogonality of the 3j symbols
for -k 5 q 5 kand ) J - J‘I 2 k 5 J
+ J‘, we find
1 ( n J M I Tf)I n’J’M’)(( nJM IT$’)In’J‘M’))*
M.M ’
=
(nJI
In’J’)((nJI 1T”’)lIn’J’))* ( J M
M,M,
k J’)(.J q M’ M
k‘ q’
J’) M‘
This is precisely Eq. (C2). We now apply this identity to the sum ( n J M I Kr I n’J’M’)(n’J’M’ I K’r I n J M )
M ,M ’
(C5)
The operator r = x k r k is the sum of the position operators of the individual electrons. The z axis is assumed to point in the K direction, so that Kr = Kz. By using the rotation operator R, we can write K’r as K‘RzR-’. Since z = TL’) is the 0-th row of a dipole operator (k = l), we find 1
where R$,) denotes a row of the rotation matrix. Insertion into Eq. (C5) yields
1 ( n J M IK r 1 n’J’M’)((nJM 1 K’r In’J’M’))*
M ,M ’
K K’
x
M ,M ’ ,q
( n J M I Tb’)I n’J’M’)( ( n J M I T:) I n’J’M’))* R$‘
where we have used RSb = P,(cos 9), and 9 denotes the angle between the two vectors K and K’. This relationship was employed to derive Eq. (56).
221
THEORY OF IMAGE FORMATION
D. Calculation of the Diffractogram The normalized inelastic intensity distribution per unit frequency w = E / h in the image plane of an FBEM for axial illumination can be written as
x exp{ - i[y(O, A E ) - y(@, A E ) ] } expCikp(0 - el)] d20 d2el
(D1) where
K
= k,(O,e,
and
S(K, K', W ) = So
+ 0),
K'
= k,(O,e,
+ el)
r+dm'2 S(K, K', o)dw
o-d0/2
is the mixed dynamic form factor averaged over a small frequency interval SW.
In the following, we consider w to be fixed and write I(p), thus omitting the parameter o.For w # 0, this intensity is proportional to the current density distribution in FBEM, [Eq. ( 4 3 ) ] .The diffractogram H ( 6 ) is the Fourier transform of the intensity.
H(S) =
~
~
(2.)2 k2
s
I(p) exp( -ik6p) d2p
x exp[ikp(B
S(K, K', W ) -
0' - S ) ]
x exp{ - i[y(O, A E ) - y(@, =
ki
s s
A(e)A(el)
S(K, K', w ) K
~
x exp{ -icy(& A E ) - y(@, =
k;
A E ) ] } d20 d2el d2p
A(0)A(0 - 6)
S(K, K
K
~
~
A E ) ] } S(0 - el - 6) d20 d20'
k,6, o) K2(K - k 0 6 T
x exp{ -i[y(d, A E ) - y(l0
-
-
61, A E ) ] } d20
222
H. KOHL AND H. ROSE
The first equality in Eq. (D2) shows that for 6 = 0 we obtain the integrated intensity in the image. Thus, we obtain a quantity that is proportional to the effective cross section 4k E , d28 = -- H ( 0 ) hko Eo The total number of electrons in the image (given by the right-hand side) is equal to the number that pass through the hole in the aperture screen (lefthand side). To calculate the diffractogram of the image of a single atom in FBEM, we use the dipole approximation as outlined in Subsection 1V.A and Appendix C . Setting the “strength factor” C = Sw, we find Neglecting aberrations (y excitations
(D4) H ( 4 = H 11 (6) + HL(4 obtain for the contribution of the parallel
= 0), we
and for the perpendicular excitations
The integrations over the angle /3 enclosed by the two vectors 8 and 6 yield
H 11 (4
140:
I
0
I
-
for 8, 5 6 < 20, otherwise
(D7)
223
THEORY OF IMAGE FORMATION
0
(D8)
otherwise
where 8, is the aperture angle; 8, = AE/2Eo is the characteristic scattering angle, which is defined as the ratio of the energy loss AE and twice the energy of the incident electrons E,. The remaining integrations were performed numerically and are displayed in Fig. l l . For a STEM with an extended detector, the image intensity is
x exp{ - i[yo(e) - yo(@)]} exp[ikop(8 -
e’)] d28d2W d2@ (D9)
with the scattering vectors
K
= k,[dEe,
+ (8 a)], -
s
K‘ = k0[8,e,
+ (0’ - a)]
(D10)
Thus, we find for the diffractogram of the STEM image HsTEM(8) = k;
S(K,K - ko6, W) A(e)A(e - 6 ) ~ ( @ ) K2(K - ko6)2
x exp{i[yo(8) - yo(18 - SI)]} d28d2@
This result is a generalization of Eq. (D2). Correspondingly, we obtain
and the effective cross section. between HSTEM(0)
(D11)
224
H. KOHL AND H. ROSE
Comparing Eqs. (D12) and (Dll), we find that an image yields more information than the differential cross section. The latter is related to H ( 6 = 0), whereas in an image all spatial frequencies 6 between 0 and 20, occur.
ACKNOWLEDGMENTS We would like to thank Drs. M. Achtche (Laboratoire de Physique des Solides, Orsay, France), P. E. Batson (IBM, Yorktown Heights, New York), and D. Krahl (MPG-Fritz Haber Institut, Berlin, Germany) for sending us their experimental results prior to publication and Profs. R. F. Egerton (University of Alberta, Edmonton, Canada) and J. Kiibler (Technische Hochschule Darmstadt, Germany) for fruitful discussions. In particular, we acknowledge the stimulating cooperation with Dr. C. Colliex and the members of his group from the Laboratoire de Physique des Solides in Orsay, France. Thanks are also due to Dr. A. Levensohn (Albany, New York) for his assistance in editing the manuscript. The valuable comments of Drs. R. Eusemann (Technische Hochschule Darmstadt, Germany) and M. Haider (EMBL, Heidelberg, Germany) are gratefully acknowledged.
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