- Email: [email protected]
Scanning soft X-ray imaging at 10 nm resolution
Ultramicroscopy 69 (1997)259-278
ELSEWIER
Scanning soft X-ray imaging at 10 nm resolution R.E. Burgeavb,*, X.-C. Yuana*b, J.N. Knauerb, M.T. Browneb, P. Charalambousb a Cavendish Laboratov, University of Cambridge, Madingley Road, Cambridge CB3 OHE, UK b Physics Department, King’s College London, Strand, London WC2R 2L.Y. UK
Received 18 February 1997;received in revised form 10 June 1997
Abstract The potential imaging and selected-area near-edge X-ray absorption fine structure spectroscopic (NEXAFS) performance of a new scanning probe X-ray microscope (SPXM) for soft X-rays from synchrotron radiation is examined by computer modelling; an optical design for the microscope is also given. Acceptable dwell times per pixel in image collection are predicted. The microscope is expected to have a spatial resolution for “water-window” X-ray wavelengths (2.3-4.4 nm) of about 10 nm for specimens up to 200 nm thick. The central component of the optical system is a tube collimator forming a probe a few wavelengths in diameter above a scanned specimen. The collimator is illuminated at near-normal incidence by a Fresnel zoneplate condenser, and its exit aperture is positioned a few wavelengths above the specimen. Physical understanding is gained by a two-dimensional (2D) waveguide approach, and by calculations using the full-vector theory of Maxwell’s equations. The calculations, because of computational limitations, are carried out mainly in 2D. The vector results agree well with a multislice scalar calculation in 2D which is then applied to 3D to describe the 3D probe imaging and to estimate the energy throughput. PACS: 07.85. - m; 07.60.Pb; 61.10.Xh Keywords:
X-ray microscopy; Scanning; Maxwell’s equations
1. Introduction
A new type of scanning X-ray microscope
probe transmission soft (SPXM) is introduced which op-
erates with synchrotron radiation. The potential performance of the SPXM is predicted by
*Corresponding author. Tel./fax: + 44 1223329130;e-mail: [email protected].
computational modelling and related experimental aspects are also considered. The microscope is predicted to have a point-to-point resolution in the water-window range of soft X-ray wavelengths (between the C and 0 K-absorption edges, 2.3-4.4 nm) of about 10 nm. The concept is to collimate narrow wavelength band soft X-rays, and produce a nanometre dimension scanning probe, using a long metallised tube a few X-ray wavelengths in diameter, to maintain the specimen within a few nanometres, typically
0304-3991/97/%17.000 1997 Elsevier Science B.V. All rights reserved PII SO304-3991(97)00050-8
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10-20 nm, of the probe aperture, and to detect the transmitted radiation in far-field. Imaging through a tube of nanometre width will provide sufficient X-ray throughput for data collection in a reasonable time through the use of an undulator beamline on a high brightness third generation synchrotron such as the ESRF facility at Grenoble and the ELETTRA synchrotron at Trieste. The SPXM has some analogies with the scanning near-field optical microscope (SNOM), but for the latter, both the scanning probe dimension and the probe-to-specimen separation are subwavelength [l, 23. The target resolution proposed here for soft X-rays is 2-3 wavelengths. A factor of importance to justify the potential of using X-rays, at similar resolutions to those achieved with SNOM, is the specimen thickness up to which the transmitted probe retains the dimension of the probe aperture. In the visible-optical case with a subwavelength dimension probe [3] this distance is close to one-half the aperture dimension, which is restrictive for 10-20 nm resolution in transmission. For soft X-rays the practical probe aperture dimension is several wavelengths and the thickness restriction found using SNOM for high resolution is significantly relaxed. The SPXM is introduced as a way, possibly an interim way, to increase the resolution of current X-ray microscopes operating with synchrotron radiation (see Ref. [4] for review and biological applications). The latter employ Fresnel zone plates (FZP) as objective lenses made by electron beam lithography [5-71 and the FZP have been subject to a process of slow and laborious improvement. The theoretical Rayleigh resolution of an ideal FZP that can be fabricated today, in the first diffraction order, is about 20 nm. In practice, the best current practical values for the resolution lie between 30-40 nm [8,9], though there is potential for further improvement which might overtake the SPXM and might include the use of higher FZP diffraction orders (and correspondingly, reduced radiation bandwidth). The SPXM, besides its projected improvement in image resolution, provides potential advantages for the measurement of near-edge X-ray absorption fine structure spectroscopy (NEXAFS) because the area (diameter 0.1-0.5 brn in harmony with zone plate resolution), selected for
absorption studies [lo-l21 will be reduced by a minimum factor of about 100 times. To predict the imaging performance of the SPXM, we have made calculations, variously at wavelengths of 2.4 and 3 nm within the water window, for slits (2D) and tubes (3D) with widths/ diameters between 5-20 nm. The consideration of the choice of optical theory for such dimensions has been central to this study and has benefitted from parallel research on proximity printing of submicron elements in integrated circuit replication by X-ray lithography [13, 143 and transmission imaging in SNOM [15, 161.
2. Experimental
and computational
2. I. Scattering parameters
overview
and computer models
We first consider experimental parameters guide the modelling based on the schematic rangement in Fig. 1.
to ar-
zone PI&
Order Sorting
Aperture
SL or Cylindrical
Fig. 1. Schematic diagram of scanning soft X-ray microscope (SPXM). The collimating aperture represents a slit in 2D imaging and a hollow cylinder for 3D. The condenser zone plate is apodised and the order-sorting aperture is matched to it.
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puter resources and software development. The magnitude of calculations for SNOM imaging [15-173 has dictated that published work so far has been for 2D, with the assumption, backed up by special cases, that the results in 2D are not misleading when extrapolated to 3D. The need for the rigorous theories of electromagnetic diffraction by subwavelength structures in near-field involving solutions to the full-vector Maxwell equations is generally accepted for SNOM imaging [2]. For proximity imaging with soft X-rays in VLSI [13, 141 using wavelengths near 1 nm and feature sizes of 100 nm, the features on X-ray masks resemble lossy dielectrics from tens to hundreds of wavelengths thick, as they do in X-ray microscopy, and the validity of Kirchoff boundary conditions which ignore fringing fields at boundaries has been questioned. In these circumstances, we decided to use fullvector Maxwell’s equations whenever possible, which means applied to 2D situations, and use a Fourier-transform-based scalar approach applicable in both 2D and 3D, validating its use in 3D by the measure of the agreement in a comparison between the vector and scalar sets of results for 2D. The slit edges and tube walls are assumed to be smooth on the scale of the soft X-ray wavelengths. At soft X-ray wavelengths the refractive index of materials has the form n = 1 - 6 + ifi. For illustration the material for which images will be calculated will be taken to be polymethyl methacrylate (PMMA), the refractive index of which is reasonably representative of biological material; PMMA is also an important material in lithographic masks. Values of n are taken from the Henke data [ 18, 193 and are given for gold and PMMA, and also calcium (Fig. 10) in Table 1. As the real part of the refractive index is less than
Synchrotron radiation is incident from a monochromator (bandwidth considered later) on to a FZP collimator. The FZP is of moderate resolution, say about 0.2 urn, with a diameter of 100 urn. It has a focal length of around 6 mm for wavelengths in the water window to allow space to insert the collimator tube and its adjustments. The FZP is apodised and an order-sorting aperture matched to the diameter of the apodising spot is inserted downstream to isolate the first-order focus and to prevent the transmission of the zero-order radiation to the specimen. The FZP illuminates a collimating slit for 2D geometry (or tube in 3D) within 1” of normal incidence. The specimen moves beneath the collimating aperture. For calculations in 2D using vector electromagnetic theory, the slit in Fig. 1, fabricated in gold, has parallel smooth sides with separation d in the xdirection, is infinite in y, with depth z. The depth of the slit is governed by the requirement that the X-ray intensity transmitted through the edges of the slit must be small. A representative value for a slit made of gold, in the water window, is 200 nm for a transmission of less than 1% of the incident intensity. Considering the electric vector of the incident X-rays, TE polarisation has the electric vector parallel to the length of the slit, i.e. parallel to y, while for TM the electric vector is parallel to x. For 3D scalar calculations the collimator is a smooth tube with diameter d but otherwise the apparatus is the same as shown in Fig. 1. The source to specimen separation is set by an accessory atomic force microscope (AFM); in principle, a second signal describing the surface topography could be recorded using the AFM tip at the same time as the X-ray transmission signal. The modelling of the tip-specimen interaction must be seen within the context of available comTable 1 Refractive Material
Gold Calcium PMMA
index data [19,20]
for gold, calcium,
and PMMA
for i = 2.4 and 3 nm. n = 1 - 6 + i/I
i. = 2.4 nm
1, = 3.0 nm
6
B
6
B
0.4274E - 2 0.9396E - 3 0.8731E - 3
0.4395E - 2 0.6747E - 3 0.1826E - 3
0.5386E - 2 0.9290E - 3 0.1389E - 2
0.5927E - 2 0.1238E - 2 0.3983E - 3
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unity, at a vacuum (air) to material boundary, with the X-rays incident from vacuum, the X-rays are totally externally reflected for grazing angles of incidence less than the critical angle (either sharply defined for p 6 6 or, as for a gold-walled collimating slit/tube where the critical angle is about 6”, gradually, for /I - 6). We have chosen to work with two quite different approaches to the solution of Maxwell’s equations in 2D. This has the advantage of comparison of results and the choice where a given method is more easily applied. The first is the coupled wave method [20,21] developed for the analysis of planar gratings, surface relief dielectric gratings, and metallic gratings. The required structural unit is repeated on a lattice and the incident electromagnetic wave, expanded in spatial harmonics, is propagated with boundary matching through each of a series of thin layers dividing the specimen thickness. The number of spatial harmonics is increased until convergence. There are no approximations and results are obtainable to arbitrary accuracy. The diffraction efficiencies of all orders of both transmitted and reflected waves are calculated. The second approach, also used for SNOM calculations (see Refs. [ 15, 17]), is the finite-difference time-domain method (FDTD). In the FDTD method the scattering volume is divided up into cells by a rectangular mesh. Each cell is much smaller than the wavelength. Maxwell’s equations are solved on the discretised grid without further assumptions giving the time-averaged electric field and also the Poynting vector, + E xH within each cell. For the scalar theory we require first validation by comparing the 2D results with the corresponding vector-coupled wave and FDTD results and second we need it to consider the specification of the transmitted probe and the throughput of transmitted energy in 3D. The first point to consider concerns wave polarisation, specifically excluded from scalar calculations. However, as will be shown in the calculations below, as in Ref. [14], the differences are small for soft X-rays between the diffraction of TE- and TM-polarised waves. The need for validation of scalar calculations in 2D is parallel to the situation that Kupka et al. [14] found themselves in for proximity printing with soft X-rays,
69 (1997) 259-278
and which they resolved satisfactorily using the Fourier-transform-based beam propagation method, as against the FDTD (also method of moments) vector calculations of Schattenburg et al. [13]. An extended argument is given by Kupka et al. in their justification. The Fourier-based method we have adopted is similar to the beam propagation method and is a modification [22] for soft X-rays of the multislice method developed for calculating electron images [23]. It is relevant to the argument that will link satisfactorily vector and scalar calculations to draw attention to the good agreement [22] between scalar calculations for the images of 0.1 urn diameter dielectric spheres for soft X-ray wavelengths and results calculated from the Mie theory [24] for spheres developed from the full Maxwell equations. For soft X-rays and the given scattering parameters the results of Mie theory are very insensitive to the plane of polarisation of the incident wave. 2.2. Computational
details
In the coupled wave method (2D) a 10 nm slit width, for example, is represented for calculation by a lattice of gold blocks with a 100 nm period and block separation 10 nm. The longer the grating period the more accurate is the representation of the slit diffraction. Limitations of computer memory restricted the total number of spatial harmonics included in the calculations to 131, including propagating and evanescent orders. The coupledwave method can deal in single calculations with the propagation of a plane wave through the collimating slit into air to define the electric field energy density of the X-ray probe incident on a specimen, and also describe the emergent probe on propagation onwards into and out of a transmission specimen. The FDTD calculation (2D) available to us was restricted to a maximum of 80 000 cells accounting for the plane of symmetry about the propagation direction of the X-rays passing through the collimating slit. The method was applied to the determination of the electric field energy density and the Poynting vector, or flux density, within the interior of gold slits with widths from 5 to 20 nm, 200 nm long in the beam direction, with 8 cells per
R.E. Burge et al. / Ultramicroscopy
wavelength which gave adequate convergence. Absorbing boundaries are introduced to prevent unwanted reflections from all edges of the mesh. Maxwell’s equations are solved on the mesh and the resultant E and H vectors and Poynting vector are calculated within each cell at a series of time steps. The maximum time step on the Courant criterion is 8.8E-19 s, the actual timestep was lE-19 s, and the time average was taken between 15 000 and 15 499 timesteps. The multislice (2D and 3D) scalar method was used to calculate the probe emergent from both the 200 nm long slit aperture and the 200 nm long cylindrical aperture, both fashioned in gold. For both 2D and 3D calculations the sampling [22] in x and y was 1.5 nm and adequate convergence was found after 64 slices. Results of the calculations were expressed as electric field energy density. Both the 2D and 3D waves have to be propagated through vacuum (air), through medium and from near-to-far field for detection. We have adopted the propagation method of Harvey [25] which avoids the explicit assumptions of the Rayleigh-Sommerfeld theory [26] that the diffracting aperture is large compared with the incident wavelength and the diffracted field is observed far from the aperture. Although the method is strictly a scalar theory, when the effect of wave polarisation is small the method can with confidence be applied throughout the entire space where diffraction occurs. Harvey gives two expressions for what he terms the general Rayleigh-Sommerfeld diffraction formula, one based on Fourier transform theory, the other a double integral equation. To have confidence in our handling of the theory we have used both expressions with effectively identical results. To calculate the distributions of intensity for images in far field, the complex amplitude distribution corresponding to the probe transmitted through an object pixel was propagated a distance of 3 mm downstream to a large aperture detector where the distribution was Fourier transformed to form the image. The convergence of the integration of the image signal was checked by varying the size of the detector. Absorption images, corresponding to an infinitely small probe, were derived from the Henke data [18,19] and scaled to the coupledwave results.
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The probe at the specimen is spatially coherent. However, because of the large detector the imaging process is incoherent [S] and consideration can be given to improving the image signal-to-noise ratio by deconvolution of the image intensity given the probe intensity distribution. 2.3. X-ray probe propagation:
Wave guiding
The collimating aperture, regarded as a slit in 2D or a cylindrical collimator in 3D, may be regarded as a waveguide. The essential physics of wave propagation is expressed in the 2D case which is now considered. A waveguide transmits a variety of modes [27], for example, guided modes, radiation modes, evanescent modes, according to the length of the guide, whether the guide is open or terminated and other factors. We consider for soft X-ray transmission, the dependence of the mode transmission on the width and length of a 2D guide in gold. The width is to be between 5 and 20 nm, and the length 200 nm. The input plane wave suffers diffraction at the ingoing aperture and suffers external reflection at the slit edges according to the grazing angle. A consideration has been made [14] of the propagation of guided soft X-ray modes through 2D slits. For a symmetrical 2D (or 3D) guide, the propagating guided modes are TE,, and TM,, (or HE1 1 in 3D), which never reach cut-off for a finite guide length, and other even-order modes TE2, TE4, and TM2, TM,, etc., according to the slit width; the odd guided modes are not supported in a symmetrical guide illuminated symmetrically. The field distribution in a lossy waveguide in the steady state can be described by a discrete set of guided modes and a continuum of radiation modes. The radiation modes redistribute the energy carried by the guided modes from the waveguide core into its cladding. Leaky modes correspond to series expansions in terms of radiation modes and may be interpreted as guided modes beyond the cutoff point. The following is a brief explanation of guided and leaky modes [28]. At any given slit width d, the waveguide can support a finite number of guided modes. Considering a ray picture, if the angle of incidence at the cladding reaches the critical angle, total reflection is lost and energy leaks into the
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characteristic length and the leaky modes have decay lengths less than 43 nm. Fig. 2 shows that the leaky TE2 mode has a decay length of 7 nm compared to the guided TEo mode of approximately 210 nm. To examine mode transmission for soft X-rays, calculations were made using the coupled-wave method for wave transmission down slits of fixed width e.g. 10 nm, and a range of slit lengths z, to determine the guide length at which the distribution of electric field energy was closely similar to the distributions corresponding to the propagation of the TE, and TM, modes only. For d = 10 nm and 3 nm wavelength, the transmitted probe shape is equivalent to that for the TEo mode for slit lengths greater than about 500 nm. This characteristic length increases rapidly as d increases. Thus, for the collimating 2D apertures relevant to the SPXM, with smooth parallel sides and widths from 5 to 20 nm, and of length 200 nm, the
cladding; the guided mode is said to have reached its cutoff. Leaky modes occur beyond cut-off, they decay quickly and they are important to the transition region between the initial excitation of the waveguide and the steady-state solution further down the guide. The l/e decay lengths of the guided and leaky modes as a function of the slit width are shown in Fig. 2 for normal incidence. At d = 10 nm, the waveguide supports only one guided mode and an infinite but discrete set of leaky modes. Guided modes are cut-off when the mode absorption equals the bulk absorption, which occurs at the characteristic depth of 43 nm for gold at II = 2.4 nm as shown by the dotted line in Fig. 2. Similar calculations to those in Fig. 2 for even-order TM modes produce propagation characteristics closely similar to those for the corresponding TE modes. The guided modes have l/e decay length > to the
20
30 width
40
50
d (nm)
Fig. 2. Absorption of even-order TE guided modes above dotted line (equivalent for TM modes) and leaky modes, below dotted line, for 2.4 nm wavelength X-rays propagated through a gold slit aperture of width d -C 50 nm. The dotted line represents the decay length for bulk absorption at which value all guided modes cut off except the zero order.
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dominant transmitted guided mode is TEo (TMJ and the differences in propagation of the planepolarised components are small. For all slit widths, the minimum slit length needed to define the collimator aperture is shorter than the transient length and the properties of the probe emerging from the slit will both differ from the fundamental TEo mode because of additional leaky modes of finite energy and change in details as the slit length changes.
3. Calculations of soft X-ray transmission and image resolution 3.1. Electric field energy and Poynting vector distributions for propagation through a 200 nm long gold 20 slit into vacuum For TE propagation
account must be taken of
E, (tangential), H, and H, field components and for
TM the appropriate components are H,, E, (tangential) and E, (normal). Results will be discussed either as electric field intensity IEyj2, lEx12, lEz12, or the Poynting vector jE,H,* for the total flux density. FDTD calculations in 2D for TE polarisation are shown in Fig. 3 of IE,12 for a number of z coordinates within the slit as far as the emergent slit face as an incident plane wave traverses the slit. Near the slit input, at z = 7.5 nm, the TE2 leaky mode is clearly visible but it attenuates rapidly with distance down the slit. There is a maximum in the transmitted intensity due to constructive interference between z = 50 nm and z = 100 nm. The loss of energy to the gold edges is clear, and for the constant width d, the full-width at half-maximum (FWHM) of the wave within the slit remains constant as the wave proceeds. Complete 2D contour plots are shown in Fig. 4, calculated by FDTD, of the electric field energy
0.10 r z=lOOnm
-10
0 x across aperture [nm]
10
Fig. 3. Snapshots of a 2D probe as it traverses a 200 nm long gold slit, 10 nm wide. Distances z value of plane wave entry into the slit. Calculations by FDTD method. IL= 3 nm.
-.-.-.____
20
are measured
30
downstream
from the
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69 (1997) 259-278
130
z
in
aperture
Fig. 4. Contour plot of the intensity distribution inside the slit aperture, calculation). The grey levels are linearly spaced.
density for the traverse of a 3 nm normal incidence plane wave down a 10 nm wide 200 nm long gold slit; the contours of magnetic field intensity are very similar. These results are for TE polarisation. The TE2 leaky wave component is visible and the maximum of transmitted intensity occurs about 60nm down the tube. As has been considered above the differences between the results for TE and TM are too small to be revealed at the given scale. The contours give a quantitative picture of the energy transfer into the gold cladding. The results for the Poynting vector show a smooth flow of flux at the centre parallel to the slit axis, and at the edges the flow is inclined towards the cladding consistent with the loss of energy as the wave propagates. The details are in harmony with the intensity contours in Fig. 4 and are not shown. The lowest curve in Fig. 3, which gives JE,l’ for a tangential component of E as the wave is on the
[nm]
TE tangential
component,
d = 10 nm, /z = 3 nm (FDTD
vector
point of emerging from the slit, describes the X-ray probe to be propagated downstream through the air gap, assumed to be 10 nm and on to the specimen. This calculation, by the FDTD method, gives results that are effectively identical with the corresponding calculation made by the coupled-wave method. and the agreement in this example was found to extend to others so as to allow the two methods to be used interchangeably. When the probe is propagated downstream using the general Rayleigh-Sommerfeld approach for the tangential components of both TE and TM, as before, the differences are very small. Calculations for the TM normal component IE,1’ show this to be of low but finite intensity as compared with the tangential component. Plots of the probe intensity distributions for a 10 nm slit width and 2.4 nm wavelength are given in Fig. 5a and Fig. 5b in the form of snapshots of the probe in vacuum for the tangential and normal components, respectively, of the
R.E. Burge et al. / Ultramicroscopy 69 (1997) 259-278
electric field energy density at different distances up to 400 nm downstream from the slit surface. Fig. 5a demonstrates a probe broadening out from an initial FWHM comparable with the slit width; the probe retains its integrity as a single Gaussianshaped peak until a distance z of about 200 nm. On the other hand, the normal component shows a double-peak structure which persists but broadens and weakens with the increase of distance downstream. The double peaks arise from the accumulation of electric charge at the slit edges: in the corresponding 3D case cylindrical waves arise from the edges of the circular aperture which have weak intensity and widen and blur out with increasing downstream distance and are of negligible intensity compared with the tangential components. The same intensity scales are used for Fig. 5a and Fig. 5b which show that the relative contribution of the normal field component to the total electric field energy density as compared with the tangential field is negligible. Because of the effective equivalence of the distributions of intensity in the X-ray probe for both polarisations, only the TE component is considered in 2D calculations below. Fig. 5c shows, for the same slit and wavelength parameters as for Fig. 5a and Fig. 5b, for comparison with Fig. 5a, calculations for the probe as propagated with the scalar multislice theory as applied in 2D. The probe is shown at the same positions downstream as before. The comparison shows that the general features of the vector and scalar calculations in 2D are very closely similar up to z values of about 200 nm and quite similar for larger distances downstream with similar changes in intensity levels with increasing z. The separations of the two peaks into which the intensity patterns divide at the larger z values is somewhat smaller in the scalar calculation relative to the vector results. Fig. 6 shows a wider range of results for slit widths of 5, 10, 15, 20 nm, for a gold slit 200 nm long and 2.4 nm X-rays, in the form of the variation of FWHM values for the probe widths as propagated through air for z values up to 400 nm. Each curve is superimposed on a curve which corresponds to the probe width if it were comprised entirely of TEO (TMO)-guided waves. The differences between the curves for the different slit widths correspond to the effects of the so-called leaky
261
modes for slit depths less than the transient distance. The curves in Fig. 6, especially for z less than 200 nm, for the four apertures are very similar despite the four fold difference in slit width. This similarity for small z is in harmony with the result that for all these slit widths the principal guided mode will be the fundamental mode which allows experimentally the use of a larger aperture with a significant gain in photon flux at each image point with small change in resolution. The curve for 5 nm slit width is closest to the TEO curve as the smaller the slit width the greater the discrimination against all other excited modes. In contrast, for propagation distances of greater than about 500 nm the transmitted probe approximates to the TEO mode because of the absorption of the leaky modes. To show that the nature of the probe depends on the resultant of the guided and leaky modes, the curve for the z-dependence of the FWHM for a 225 nm long and 10 nm wide slit is also shown on the plot for d = 10 nm. The curve for the longer slit shows a smaller half-width at given z than for the 200 nm long slit. Finally, concerning probe propagation through a specimen for d = 10 nm, using coupledwave theory, the variation was examined when, after a 1Onm air gap, the probe is propagated through a continuous medium with the properties of PMMA. The curve for PMMA was found to “shadow” that for the slit length of 225 nm, showing a small narrowing in probe half-width due to the real part of the refractive index. Considering further the relationship between the scanning probe description in 2D and 3D, in Fig. 7 are shown three sets of results expressing the probe intensity distribution in terms of FWHM as it propagates in air. The results are all for d = 10 nm, slit length 200nm in gold, wavelength 2.4 nm; the corresponding results for a wavelength of 3 nm are very similar. One set is the same as in Fig. 6 under 2D vector conditions. The other two sets are derived from the scalar multislice calculations as applied in 2D and in 3D as propagated by either a 2D or a 3D propagator. The FWHM in 3D is the half-width across a central slice. There is good agreement between all three data sets up to almost z = 200 nm. The perspective plots in Fig. 8 show the probe intensity distribution in 3D at four locations downstream up to 200 nm and correspond to
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h=2.4nm.
p=lOnm.
69 (1997) 259-278
Z=Onm
3.0~
1
TE. k2.4nm.
#=lOnm.
Z=60nm
O.BF
FE. h=2.4”m,
#= tO”m.
Z=BOnm
4
gyy---j
TE. h=2.4”m,
$=lO”m.
2=250”m
;_~
-40
-20 I
TE, h=2,4”m,
$=,O”m,
2=3OO”m
0.20~
TE, A=2.4”m,
Q=lO”m,
TM. k2.4M”.
p1onm.
2=350”m
O.ZLl~~j
1
TE,
h=2.4”m,
&)
#=,O”m,
40
20
2=4OOnm
f$-jyyJ
2=2Onm
i_ii
-40
a.ao*o~-q
-20
0 x (nmf
20
rcl
TM, k2.4”m,
$=IO”m.
Z=LLOnm
TM, X=2.4”m.
#=lO”m,
Z=2OO”m
_, ~~~~~~~ili~~~ -40
-20
TM, h=2.4”m,
-”
-”
I (in, =’
”
h=2.4”m,
$=lO”m,
Z=,OO”m
_,
TM.
h=2.4”m,
#=lO”m,
2=250”m
_,
TM,
h=2.4”m.
$t,=1onm.
z=400nm
~~~~
t_,L--Jjq
(b)
TM,
-40
-*o
II x h)
#=lO”m.
0
x[W
20
40
2=350nm
20
40
-40
-20
0
x(4
20
10
Fig. 5. (a) TE tangential component of probe propagation following incidence of a plane wave on a gold slit with d = 10 nm. 2D coupled-wave vector calculation through vacuum (air) for selected distances downstream from the slit exit. Wavelength 2.4 nm. (b) As 5(a) for TM normal component. (c) As .5(a) but 2D scalar multislice calculation.
R.E. Burge et al. / Ultramicroscopy A-2.4nm,
$=lOnm,
Z=Onm
A=2.4nm,
$=lOnm,
269
69 (1997) 259-278 Z=ZOnm
0.5ok
A=2.4nm,
)=lOnm,
Z-40nm
~~~
Fig. 5. (Continued)
a cylindrical collimating tube. The throughput values, i.e. the ratio of the probe intensity distribution integrated across the measurement plane and the intensity incident, as defined by the aperture of the collimator tube, as determined from the 3D scalar case, evaluated from the electric field intensity for circular apertures with diameters 5, 10, 15, 20nm, are, respectively, 6.2%, 31.7%, 78.2%, and 92.8%. These throughput values are considered to be closely representative of the values that would have been found in a full-3D vector calculation. All the above calculations have assumed normal incidence. When using a FZP, specified as in Section 2.1, as a collimator the maximum illumination angle to the normal is one degree. The effect of non-normal incidence on the probe emerging into air from a 10 nm slit was explored for a rather wider range of possible conditions using coupledwave theory and is shown in Fig. 9 for a range of angles to the normal up to the critical angle for gold
at a wavelength of 2.4 nm. As the incident angle increases the transmitted peak height falls, the peak suffers displacement, and the probe widens. However, for the small convergence angle of the chosen FZP collimator the effect on the transmitted probe is very small. 3.2. Intensity distributions for X-rays transmitted through dielectric materials: Imaging and spatial resolution To take advantage of the range of z where the FWHM of the probe is effectively constant, the probe-to-specimen distance should be minimised and a distance of 10 nm is assumed here. The variation of image resolution with specimen thickness is illustrated in Fig. 10 where using coupled-wave (2D vector) theory a comparison is made of calculated scanned images, sampled at 2 nm intervals, of a small particle of calcium placed first on top of
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X=2.4nm,
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h=2.4nm,
$=5nm,
aoF.“““.”
$= 1 Onm,
...’ .““‘..‘.“..““..‘i
““7 60
ot.. 0
100
Ol 200
Distance
h=2.4nm,
0
100
300
400
0
(nm)
$=15nm,
200
Distance
306
100
200 300 Distance (nm)
h=2.4nm,
400
(nm)
0
100
400
$=20nm,
200 300 Distance (nm)
400
Fig. 6. 2D coupled-wave calculations (full lines) for FWHM height on propagation of a plane wave by a 200 nm long gold slit into air. Slit widths 5 nm, 10 nm, 15 nm, 20 nm, as marked. Wavelength 2.4 nm. The FWHM values for a single zero order guided mode are shown by dashed lines for each d value. For d = 10 nm an additional line is shown (chain) representing propagation into air but for a slit 225 nm long.
a 400 nm thick PMMA supporting film and then underneath it. The probe-to-specimen (particle or uniform film) distance was 10 nm. The particle was 40 nm thick and 50 nm wide. For comparison the ideal kinematic absorption images are also shown calculated from the Henke refractive index data for a point probe. The calculations show that the levels of image intensity as the probe is scanned across abrupt edges are modified from the ideal Henke values but asymptote to these values some distance from the edges if the separation of specimen features is large enough to allow it. The edges themselves are displayed as edge spread functions from which the spatial resolution can be derived. The particle on the top is imaged with a resolution of about 10 nm and with a 60 nm resolution on the bottom. Further, vector coupled-wave calculations for scanned images are presented in Fig. 11 for a series of perpendicular steps in uniform PMMA films, for d = 10 nm and a wavelength of 2.4 nm. These calculations extend to a maximum thickness of
200 nm, which is the extent of the range of specimen thickness up to which the probe dimensions (see Fig. 7) are expected to remain closely constant. The film thicknesses selected were 50, 100, 200 nm, giving edge spread functions between 0 to 50 nm, 0 to 200 nm, 50 to 100 nm, 100 to 200 nm. Resolution values derived from edge spread functions adopt the width between the 20% and 80% intensity levels (60% criterion) or the lO-90% levels (80% criterion). Results for these two widths give a measure of the resolution, lying between 9 and 11 nm, according to the chosen criterion, for the thinner specimens, A more physically satisfactory assessment of image resolution may be made from the modulation transfer function (MTF) which can be calculated directly from the edge spread function by first differentiating to derive the line spread function and then inverse Fourier transforming to give the MTF. The MTF allows the setting of the minimum contrast level of image features said to be resolved.
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Coupled-Wave
0
20
Calculation (solid); Multislice Calculation 2-D (circle), 3-D (star)
40
60
80
100 z(nm)
120
140
160
160
200
Fig. 7. Comparison of values of full-width at half-height for 2D (vector, full line), 2D (scalar, circles), and 3D (scalar, stars) probes propagated downstream and measured from the slit exit surface. Plane wave incident, wavelength 2.4 nm, 200 nm long gold slit/cylinder.
MTF curves calculated from the four-edge spread functions of Fig. 1I are shown in Fig. 12 covering a range of image spatial frequencies and image contrast. Taking a (stringent) MTF value of 0.1 as in Ref. [S] the resolution varies from 7.4 nm for 50 nm thickness to 10.4 nm at 200 nm thickness; for a more realistic MTF value of 0.2, the resolution values are 8.6 nm (50 nm thick), 10.0 nm (100 nm thick), 12.2 nm (200 nm thick). There is a decrease in resolution with increasing thickness due to probe broadening, but the results are consistent with the expectation that the resolution, for the thinner specimens, will be better than 10 nm. Further calculations for d = 10 nm, but for slits longer than 200 nm, show a further narrowing of the probe, with correspondingly improved spatial resolution. The question of image deconvolution, given the specification of the scanning probe or an estimation of it, may be considered using, for example,
Wiener-Helstrom filter [29]. The level of success can be monitored by the improvement in signalto-noise ratio [30]. We draw attention to the extended examples of signal-to-noise improvement following deblurring given in Ref. [31] including the effects of noise and of incorrectly chosen probes.
4. Experimental aspects On the basis of measurement at the undulator beamline at the ELETTRA synchrotron and informed prediction at the ESRF, a photon flux is expected of between 5E9 and lEl0 ‘photons s-l into a 0.2 urn spot of an FZP used as a condenser (see Fig. 1) to illuminate the collimator tube at a chromatic resolving power n/An of some hundreds. The chromatic properties of the tube collimator, like the wavelength dependence of the
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Fig. 8. Three-dimensional plots of scalar calculations for the 3D probe intensity distributions at the given distances downstream from the exit slit surface. Plane wave incident, wavelength 2.4 nm, 200 nm long cylindrical gold collimator.
critical angle, have been shown above not to be critical by the interchangeable use for calculations of wavelengths of 2.4 and 3 nm. The FZP must be set accounting for the focus for the required wavelength. The allowable bandwidth of the incident radiation will be governed by the number of photons falling, out of focus, around the collimator tube and their contribution to the background at each pixel by transmission through the aperture edges. If the L/AA value, with the FZP focused at wavelength I, is set when the geometrical size of the probe is the same (0.2 pm) as the diffraction iimited
FZP focus, then n/An = 500 for the given dimensions. In experimental conditions, it may be necessary to increase the length of the collimator tube from 200 nm to perhaps 250 nm to reduce to neghgible levels the transmission through the tube edges. The order selecting aperture which isolates the first-order focus is not critical for this purpose for diffraction orders of significant relative intensity (perhaps for practical FZP up to the third-order), because of the angular selecting properties of the collimating slit/tube shown in Fig. 9, but is
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B=l
.O deg,
8=2.0
z=lOnm(air)
deg.
z=lOnm(oir)
0.6 0.6 .g $ 0.4 C
$ 2
0.4
0.2
-40
-20
0 Xh) 8=3.0
-40
deg,
8=5.0
deg,
40
-40
-20
20
40
deg,
-20 deg,
40
z= 1 Onm(oir)
0 Xbm) 8=6.0
z=lOnm(oir)
20
X&, 8=4.0
z=lOnm(oir)
0 Xh)
-20
20
20
z=lOnm(oir)
0.25
-40
-20
0 Xbm)
20
40
-40
-20
0 Xbm)
20
40
Fig. 9. Change of probe intensity at slit exit surface for the given angles of grazing incidence to the slit walls. Coupled-wave calculation in 2D. Plane wave, 200 nm long gold slit, wavelength 2.4 nm.
essential for screening the area around the collimating tube entrance and the specimen area from the zero-order radiation which has the full aperture of the FZP. A further feature of imaging with a collimating tube, as against focussing with a zone plate as in the usual scanning X-ray microscope where the depth of focus decreases with increasing resolution (and with higher-order focus), is the absence of a requirement for specimen focussing. The imaging beam is a projection of the transmitted X-ray probe, with, for specimens up to about 200 nm thick, an approximately constant probe width. A diagram of the proposed optical arrangement for the SPXM is shown in Fig. 13. It has a photon
detector and a specimen-observing optical microscope in far field. The inset diagram on the righthand side shows one possible arrangement for the collimating tube in the form of a commercially available AFM tip, pierced with a 10-20 nm hole, which not only passes X-rays but also maintains the probe-aperture-to-specimen distance at 10 nm by a standard cantilever/optical arrangement with a split detector to monitor accurately the position of the AFM tip in its proximity to the specimen. A pierced tip has been proposed previously for SNOM [32]. For X-rays in the water window the tube must be given a depth of 200-250nm by evaporation of a heavy metal; gold has been used in the above calculations. Experiments are in hand
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of scanning images calculated by coupled-wave method for wavelength 2.4 nm, of 50 nm wide, 40 nm thick particle Fig. 10. Comparison of calcium on a 460 nm thick supporting film of PMMA. Solid curve, particle on top of film nearest the probe; dashed curve particle on bottom of film furthest from the probe. Dotted line gives the Henke data.
with a 5 nm diameter 30 KeV energy gallium ion beam to drill the AFM tip, which is fabricated in silicon nitride, and very early results have produced apertures 30 nm in diameter. Silicon nitride is a good conductor of heat and order of magnitude calculations show that for all conceivable levels of photon flux the temperature rise of the AFM tip at steady state will not be greater than 1”. Using the microscope with a wet specimen will inevitably draw water into a collimating tube. Remembering that the wavelengths in mind are in the water window, calculations show that the probe emerging from a wet tube is not significantly changed relative to a dry tube. In complementary possibilities for probe fabrication, solid probes are now available commercially with tips having 10 nm radius from materials like silicon and silicon nitride which have high X-ray transmission and can be coated with heavy metal, possibly after ion beam
shaping. We note the development of carbon nanotubes [33] which can be 5-20 nm diameter and up to 1 urn long, and are now being used as nanoprobes in scanning probe microscopy, the developments in glass nanochannel arrays [34] fabricated so far with minimum diameters of 33 nm, and the cylindrical straight holes 10 nm in diameter and up to 6 urn long, in polycarbonate now commercially available. The last have been fabricated by exposure to collimated charged particles in a nuclear reactor. The carbon nanoprobes in particular, have mechanical properties which make them resistive to catastrophic failure on contacting the specimen. In practice, the walls of the collimating tubes may not be smooth at the scale of an X-ray wavelength. For a rough surface a broad diffusely scattered component occurs taking energy from the specular component. Fortunately in the present
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Normalised Absorption
0.16-
I 1
0
20
,
40
I
I
I
100 60 80 Specimen Co-ordinate (nm)
I
120
I I
140
160
Fig. 11. Normalised X-ray absorption and calculated edge spread functions by coupled-wave theory for density steps in PMMA up to 200 nm thick. Dotted line gives the Henke data. Wavelength 2.4 nm, d = 10 nm.
application, due to self-shadowing of the smaller features of surface roughness by the larger for incidence at small grazing angles, the diffuse component is minimised [35,36]. Further analysis awaits the characteristics of particular collimating tubes. Finally, it is important to estimate the dwell time of each pixel measurement for reasonable counting statistics. Taking the figure of lEl0 photons s-l in a spot of diameter 0.2 urn for X-rays with small bandwidth, and d = 10 nm, the dwell time to detect 1000 photons in ideal imaging is 40 us. This figure should be compared with the approximately l-5 ms dwell time per pixel at the highly successful “conventional” scanning X-ray microscope established at the National Synchrotron Light Source, Brookhaven, USA [4], not from the point of view that the predicted dwell time is some orders of magnitude less, rather to point out that, for a real experimental situation, there is room for the practi-
cal selection of parameters for the near-field instrument. The expected spatial resolution and the dwell time interact with the design of the scanning stage, which needs a minimum scanning step length of 5 nm and a maximum scanning frequency in the range l-10 kHz.
5. Conclusions It has been shown by computer modelling that a new scanning X-ray microscope, utilising a zone plate condenser and tube-aperture, with a pointto-point resolution of about 10 nm is possible for thin specimens for wavelengths in the water window. The practical resolution of the optical SNOM is similar to that projected for SPXM, but the specimen thickness possible for 10 nm resolution studies with soft X-rays is greatly increased to 200 nm as against 5-10 nm for the SNOM. Steps
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0.15 Frequency(l/nm)
0.2
0.25
0.3
Fig. 12. Modulation transfer functions calculated from the theoretical noise free edge spread functions in Fig. 1I. Circles denote a common line for the O-50 nm and 50-100 nm steps. The other lines represent the 100-200 nm step (dot-dash-dot) and O-200 nm step (dotted).
are in hand to implement the new microscope at the ESRF. Similar high resolution, in the sense of the size of specimen area selected for spectroscopic measurements is available for NEXAFS. In the latter application, similar to conventional zone plate scanning X-ray microscopy, the condenser zone plate must be focussed on the aperture for each X-ray wavelength. Discussion has taken place above, implicitly by the choice of a polymer specimen, mainly on the potential of the microscope for imaging and spectroscopy of biological or polymer specimens. In practice, due to the increased radiation damage for such specimens for high-resolution imaging at reasonable signal-to-noise ratios, it may be that studies of wet inorganic specimens as in corrosion and catalysis may be more appropriate, especially if the suggestion by Cazaux [37], that X-ray radiation damage occurs only in insulating materials, turns out to be correct. Alternatively, the possibility of cryomicroscopy [38] to minimise the effects of radi-
ation on the X-ray image can be considered by which the potentially diffusing molecular fragments are frozen in situ. The thickness limit of 200 nm at the highest resolution is less of a restriction in practice than it may appear. The reason is that for thicker specimens the resolution is limited by the overlapping of structural features in depth. For the same reason, a search through the literature for high-resolution work under current conditions with both X-ray microscopy and spectroscopy with the traditional zone plate instrument shows that specimens are frequently said to be about 0.1 pm thick. In the proposed instrument, removing the collimating tube and the AFM tip reveals a “standard” zone plate scanning X-ray microscope which can readily be used in a higher order FZP focus given reduced source bandwidth. Retaining tube and tip, and changing the source and detector for longer wavelength radiation, provides a SNOM.
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incident Beam /
I
CoBimator Tube and AFM Tip
zonePlate
Order Sorting Aperture
1 Specimen
Gas FiUed
Proportional counter
Schematic Diagram of Scanning Probe X-ray Microscope /
\
Fig. 13. Optical layout for proposed scanning near-field X-ray microscope. The inset diagram coated with evaporated gold to make a cylindrical collimator 20&250 nm deep, of diameter
Acknowledgements
Financial and material support was provided by the Leverhulme Trust, the Royal Society, Defence Research Agency, Malvern, Oxford Instruments PLC and Leo Electron Microscopy, Cambridge. We are grateful to Professor A. Howie for the provision of facilities at the Cavendish Laboratory, to FE1 Europe Ltd for assistance with ion beam milling, and to Dr. A. Hare for discussion of the multislice method.
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M.A. Paesler, York, 1996.
P.J. Moyer,
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Wiley, New
shows a pierced silicon nitride AFM tip l&20 nm.
[Z] D.W. Pohl, D. Courjon (Eds.), Near Field Optics, Kluwer, Netherlands, 1992. [3] A. Roberts, J. Appl. Phys. 65 (1989) 2896. [4] J. Kirz, C. Jacobsen, M. Howells, Quart. Rev. Biophys. 28 (1995) 33. [S] E. Anderson, D. Kern, Springer Ser. Opt. Sci. 67 (1992) 75. [6] P. Charalambous, P. Anastasi, R.E. Burge, K. Popova, Proc. SPIE vol. 2516 1994 p. 2. [7] C. David, B. Kaulich, R. Medenwaldt, M. Hettwer, N. Fay, M. Diehl. J. Thieme, G. Schmahl, J. Vat. Sci. B 13 (1995) 2762. [S] C. Jacobsen, S. Williams, E. Anderson. M.T. Browne, C.J. Buckley, D. Kern, J. Kirz, M. Rivers. X. Zhang, Opt. Commun. 86 (1991) 351. [9] G. Schmahl, P. Guttmann, G. Schneider, B. Niemann, C. David, T. Wilhein, J. Thieme, D. Rudolph, in: V.V. Aristov. A.I. Erko (Eds.), X-ray Microscopy IV, Bogorodski Pechatnik, Chernogovka. Moscow Region, 1994, pp. 196-206.
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[23] [24] [25] 1261 [27] [28] [29] [30] [31]
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J.M. Cowley, A.F. Moodie, Acta Crystallogr. 10 (1957) 609. G. Mie, Ann Phys. (ser. 4) 25 (1908) 377. J.E. Harvey, Amer. J. Phys. 47 (1979) 974. A. Sommerfeld, Optics, Lectures on Theoretical Physics, vol. IV, Academic Press, New York, 1954. A.W. Snyder, Optical Waveguide Theory, Chapman & Hall, London, 1983. D. Marcuse. Theory of Dielectric Optical Waveguides, 2nd ed.. Chapter 1, Academic Press, London Edition, 1991. C.W. Helstrom, J. Opt. Sot. Amer. 57 (1967) 3. L.D. Marks, Ultramicroscopy 62 (1996) 43. R.H.T. Bates, M.J. McDonnell. Image Restoration and Reconstruction. Clarendon Press, Oxford, 1986, Example I pp. 1417.; Example III pp. 92-97. F. Baida, D. Courjon, G. Tribillon, in: Near Field Optics, as Ref. [3]. pp. 71-78. H. Dai, J.H. Hafnar, A.G. Rinsler, D.J. Colbert. R.E. Smalley. Nature 384 (1996) 147. R.J. Tonucci, B.L. Justus, A.J. Campillo. C.E. Ford, Science 258 (1992)783. S.K. Sinha, E.B. Sirota, S. Garoff, H.B. Stanley, Phys. Rev. B 38 (1988) 2297. H.N. Chapman, K.A. Nugent, SW. Wilkins, Appl. Opt. 32 (1993) 6316, 6333. J. Cazauz, Ultamicroscopy 60 (1995) 411. G. Schneider, as Ref. [lo] pp. 181-195.
ELSEWIER
Scanning soft X-ray imaging at 10 nm resolution R.E. Burgeavb,*, X.-C. Yuana*b, J.N. Knauerb, M.T. Browneb, P. Charalambousb a Cavendish Laboratov, University of Cambridge, Madingley Road, Cambridge CB3 OHE, UK b Physics Department, King’s College London, Strand, London WC2R 2L.Y. UK
Received 18 February 1997;received in revised form 10 June 1997
Abstract The potential imaging and selected-area near-edge X-ray absorption fine structure spectroscopic (NEXAFS) performance of a new scanning probe X-ray microscope (SPXM) for soft X-rays from synchrotron radiation is examined by computer modelling; an optical design for the microscope is also given. Acceptable dwell times per pixel in image collection are predicted. The microscope is expected to have a spatial resolution for “water-window” X-ray wavelengths (2.3-4.4 nm) of about 10 nm for specimens up to 200 nm thick. The central component of the optical system is a tube collimator forming a probe a few wavelengths in diameter above a scanned specimen. The collimator is illuminated at near-normal incidence by a Fresnel zoneplate condenser, and its exit aperture is positioned a few wavelengths above the specimen. Physical understanding is gained by a two-dimensional (2D) waveguide approach, and by calculations using the full-vector theory of Maxwell’s equations. The calculations, because of computational limitations, are carried out mainly in 2D. The vector results agree well with a multislice scalar calculation in 2D which is then applied to 3D to describe the 3D probe imaging and to estimate the energy throughput. PACS: 07.85. - m; 07.60.Pb; 61.10.Xh Keywords:
X-ray microscopy; Scanning; Maxwell’s equations
1. Introduction
A new type of scanning X-ray microscope
probe transmission soft (SPXM) is introduced which op-
erates with synchrotron radiation. The potential performance of the SPXM is predicted by
*Corresponding author. Tel./fax: + 44 1223329130;e-mail: [email protected].
computational modelling and related experimental aspects are also considered. The microscope is predicted to have a point-to-point resolution in the water-window range of soft X-ray wavelengths (between the C and 0 K-absorption edges, 2.3-4.4 nm) of about 10 nm. The concept is to collimate narrow wavelength band soft X-rays, and produce a nanometre dimension scanning probe, using a long metallised tube a few X-ray wavelengths in diameter, to maintain the specimen within a few nanometres, typically
0304-3991/97/%17.000 1997 Elsevier Science B.V. All rights reserved PII SO304-3991(97)00050-8
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10-20 nm, of the probe aperture, and to detect the transmitted radiation in far-field. Imaging through a tube of nanometre width will provide sufficient X-ray throughput for data collection in a reasonable time through the use of an undulator beamline on a high brightness third generation synchrotron such as the ESRF facility at Grenoble and the ELETTRA synchrotron at Trieste. The SPXM has some analogies with the scanning near-field optical microscope (SNOM), but for the latter, both the scanning probe dimension and the probe-to-specimen separation are subwavelength [l, 23. The target resolution proposed here for soft X-rays is 2-3 wavelengths. A factor of importance to justify the potential of using X-rays, at similar resolutions to those achieved with SNOM, is the specimen thickness up to which the transmitted probe retains the dimension of the probe aperture. In the visible-optical case with a subwavelength dimension probe [3] this distance is close to one-half the aperture dimension, which is restrictive for 10-20 nm resolution in transmission. For soft X-rays the practical probe aperture dimension is several wavelengths and the thickness restriction found using SNOM for high resolution is significantly relaxed. The SPXM is introduced as a way, possibly an interim way, to increase the resolution of current X-ray microscopes operating with synchrotron radiation (see Ref. [4] for review and biological applications). The latter employ Fresnel zone plates (FZP) as objective lenses made by electron beam lithography [5-71 and the FZP have been subject to a process of slow and laborious improvement. The theoretical Rayleigh resolution of an ideal FZP that can be fabricated today, in the first diffraction order, is about 20 nm. In practice, the best current practical values for the resolution lie between 30-40 nm [8,9], though there is potential for further improvement which might overtake the SPXM and might include the use of higher FZP diffraction orders (and correspondingly, reduced radiation bandwidth). The SPXM, besides its projected improvement in image resolution, provides potential advantages for the measurement of near-edge X-ray absorption fine structure spectroscopy (NEXAFS) because the area (diameter 0.1-0.5 brn in harmony with zone plate resolution), selected for
absorption studies [lo-l21 will be reduced by a minimum factor of about 100 times. To predict the imaging performance of the SPXM, we have made calculations, variously at wavelengths of 2.4 and 3 nm within the water window, for slits (2D) and tubes (3D) with widths/ diameters between 5-20 nm. The consideration of the choice of optical theory for such dimensions has been central to this study and has benefitted from parallel research on proximity printing of submicron elements in integrated circuit replication by X-ray lithography [13, 143 and transmission imaging in SNOM [15, 161.
2. Experimental
and computational
2. I. Scattering parameters
overview
and computer models
We first consider experimental parameters guide the modelling based on the schematic rangement in Fig. 1.
to ar-
zone PI&
Order Sorting
Aperture
SL or Cylindrical
Fig. 1. Schematic diagram of scanning soft X-ray microscope (SPXM). The collimating aperture represents a slit in 2D imaging and a hollow cylinder for 3D. The condenser zone plate is apodised and the order-sorting aperture is matched to it.
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puter resources and software development. The magnitude of calculations for SNOM imaging [15-173 has dictated that published work so far has been for 2D, with the assumption, backed up by special cases, that the results in 2D are not misleading when extrapolated to 3D. The need for the rigorous theories of electromagnetic diffraction by subwavelength structures in near-field involving solutions to the full-vector Maxwell equations is generally accepted for SNOM imaging [2]. For proximity imaging with soft X-rays in VLSI [13, 141 using wavelengths near 1 nm and feature sizes of 100 nm, the features on X-ray masks resemble lossy dielectrics from tens to hundreds of wavelengths thick, as they do in X-ray microscopy, and the validity of Kirchoff boundary conditions which ignore fringing fields at boundaries has been questioned. In these circumstances, we decided to use fullvector Maxwell’s equations whenever possible, which means applied to 2D situations, and use a Fourier-transform-based scalar approach applicable in both 2D and 3D, validating its use in 3D by the measure of the agreement in a comparison between the vector and scalar sets of results for 2D. The slit edges and tube walls are assumed to be smooth on the scale of the soft X-ray wavelengths. At soft X-ray wavelengths the refractive index of materials has the form n = 1 - 6 + ifi. For illustration the material for which images will be calculated will be taken to be polymethyl methacrylate (PMMA), the refractive index of which is reasonably representative of biological material; PMMA is also an important material in lithographic masks. Values of n are taken from the Henke data [ 18, 193 and are given for gold and PMMA, and also calcium (Fig. 10) in Table 1. As the real part of the refractive index is less than
Synchrotron radiation is incident from a monochromator (bandwidth considered later) on to a FZP collimator. The FZP is of moderate resolution, say about 0.2 urn, with a diameter of 100 urn. It has a focal length of around 6 mm for wavelengths in the water window to allow space to insert the collimator tube and its adjustments. The FZP is apodised and an order-sorting aperture matched to the diameter of the apodising spot is inserted downstream to isolate the first-order focus and to prevent the transmission of the zero-order radiation to the specimen. The FZP illuminates a collimating slit for 2D geometry (or tube in 3D) within 1” of normal incidence. The specimen moves beneath the collimating aperture. For calculations in 2D using vector electromagnetic theory, the slit in Fig. 1, fabricated in gold, has parallel smooth sides with separation d in the xdirection, is infinite in y, with depth z. The depth of the slit is governed by the requirement that the X-ray intensity transmitted through the edges of the slit must be small. A representative value for a slit made of gold, in the water window, is 200 nm for a transmission of less than 1% of the incident intensity. Considering the electric vector of the incident X-rays, TE polarisation has the electric vector parallel to the length of the slit, i.e. parallel to y, while for TM the electric vector is parallel to x. For 3D scalar calculations the collimator is a smooth tube with diameter d but otherwise the apparatus is the same as shown in Fig. 1. The source to specimen separation is set by an accessory atomic force microscope (AFM); in principle, a second signal describing the surface topography could be recorded using the AFM tip at the same time as the X-ray transmission signal. The modelling of the tip-specimen interaction must be seen within the context of available comTable 1 Refractive Material
Gold Calcium PMMA
index data [19,20]
for gold, calcium,
and PMMA
for i = 2.4 and 3 nm. n = 1 - 6 + i/I
i. = 2.4 nm
1, = 3.0 nm
6
B
6
B
0.4274E - 2 0.9396E - 3 0.8731E - 3
0.4395E - 2 0.6747E - 3 0.1826E - 3
0.5386E - 2 0.9290E - 3 0.1389E - 2
0.5927E - 2 0.1238E - 2 0.3983E - 3
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unity, at a vacuum (air) to material boundary, with the X-rays incident from vacuum, the X-rays are totally externally reflected for grazing angles of incidence less than the critical angle (either sharply defined for p 6 6 or, as for a gold-walled collimating slit/tube where the critical angle is about 6”, gradually, for /I - 6). We have chosen to work with two quite different approaches to the solution of Maxwell’s equations in 2D. This has the advantage of comparison of results and the choice where a given method is more easily applied. The first is the coupled wave method [20,21] developed for the analysis of planar gratings, surface relief dielectric gratings, and metallic gratings. The required structural unit is repeated on a lattice and the incident electromagnetic wave, expanded in spatial harmonics, is propagated with boundary matching through each of a series of thin layers dividing the specimen thickness. The number of spatial harmonics is increased until convergence. There are no approximations and results are obtainable to arbitrary accuracy. The diffraction efficiencies of all orders of both transmitted and reflected waves are calculated. The second approach, also used for SNOM calculations (see Refs. [ 15, 17]), is the finite-difference time-domain method (FDTD). In the FDTD method the scattering volume is divided up into cells by a rectangular mesh. Each cell is much smaller than the wavelength. Maxwell’s equations are solved on the discretised grid without further assumptions giving the time-averaged electric field and also the Poynting vector, + E xH within each cell. For the scalar theory we require first validation by comparing the 2D results with the corresponding vector-coupled wave and FDTD results and second we need it to consider the specification of the transmitted probe and the throughput of transmitted energy in 3D. The first point to consider concerns wave polarisation, specifically excluded from scalar calculations. However, as will be shown in the calculations below, as in Ref. [14], the differences are small for soft X-rays between the diffraction of TE- and TM-polarised waves. The need for validation of scalar calculations in 2D is parallel to the situation that Kupka et al. [14] found themselves in for proximity printing with soft X-rays,
69 (1997) 259-278
and which they resolved satisfactorily using the Fourier-transform-based beam propagation method, as against the FDTD (also method of moments) vector calculations of Schattenburg et al. [13]. An extended argument is given by Kupka et al. in their justification. The Fourier-based method we have adopted is similar to the beam propagation method and is a modification [22] for soft X-rays of the multislice method developed for calculating electron images [23]. It is relevant to the argument that will link satisfactorily vector and scalar calculations to draw attention to the good agreement [22] between scalar calculations for the images of 0.1 urn diameter dielectric spheres for soft X-ray wavelengths and results calculated from the Mie theory [24] for spheres developed from the full Maxwell equations. For soft X-rays and the given scattering parameters the results of Mie theory are very insensitive to the plane of polarisation of the incident wave. 2.2. Computational
details
In the coupled wave method (2D) a 10 nm slit width, for example, is represented for calculation by a lattice of gold blocks with a 100 nm period and block separation 10 nm. The longer the grating period the more accurate is the representation of the slit diffraction. Limitations of computer memory restricted the total number of spatial harmonics included in the calculations to 131, including propagating and evanescent orders. The coupledwave method can deal in single calculations with the propagation of a plane wave through the collimating slit into air to define the electric field energy density of the X-ray probe incident on a specimen, and also describe the emergent probe on propagation onwards into and out of a transmission specimen. The FDTD calculation (2D) available to us was restricted to a maximum of 80 000 cells accounting for the plane of symmetry about the propagation direction of the X-rays passing through the collimating slit. The method was applied to the determination of the electric field energy density and the Poynting vector, or flux density, within the interior of gold slits with widths from 5 to 20 nm, 200 nm long in the beam direction, with 8 cells per
R.E. Burge et al. / Ultramicroscopy
wavelength which gave adequate convergence. Absorbing boundaries are introduced to prevent unwanted reflections from all edges of the mesh. Maxwell’s equations are solved on the mesh and the resultant E and H vectors and Poynting vector are calculated within each cell at a series of time steps. The maximum time step on the Courant criterion is 8.8E-19 s, the actual timestep was lE-19 s, and the time average was taken between 15 000 and 15 499 timesteps. The multislice (2D and 3D) scalar method was used to calculate the probe emergent from both the 200 nm long slit aperture and the 200 nm long cylindrical aperture, both fashioned in gold. For both 2D and 3D calculations the sampling [22] in x and y was 1.5 nm and adequate convergence was found after 64 slices. Results of the calculations were expressed as electric field energy density. Both the 2D and 3D waves have to be propagated through vacuum (air), through medium and from near-to-far field for detection. We have adopted the propagation method of Harvey [25] which avoids the explicit assumptions of the Rayleigh-Sommerfeld theory [26] that the diffracting aperture is large compared with the incident wavelength and the diffracted field is observed far from the aperture. Although the method is strictly a scalar theory, when the effect of wave polarisation is small the method can with confidence be applied throughout the entire space where diffraction occurs. Harvey gives two expressions for what he terms the general Rayleigh-Sommerfeld diffraction formula, one based on Fourier transform theory, the other a double integral equation. To have confidence in our handling of the theory we have used both expressions with effectively identical results. To calculate the distributions of intensity for images in far field, the complex amplitude distribution corresponding to the probe transmitted through an object pixel was propagated a distance of 3 mm downstream to a large aperture detector where the distribution was Fourier transformed to form the image. The convergence of the integration of the image signal was checked by varying the size of the detector. Absorption images, corresponding to an infinitely small probe, were derived from the Henke data [18,19] and scaled to the coupledwave results.
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The probe at the specimen is spatially coherent. However, because of the large detector the imaging process is incoherent [S] and consideration can be given to improving the image signal-to-noise ratio by deconvolution of the image intensity given the probe intensity distribution. 2.3. X-ray probe propagation:
Wave guiding
The collimating aperture, regarded as a slit in 2D or a cylindrical collimator in 3D, may be regarded as a waveguide. The essential physics of wave propagation is expressed in the 2D case which is now considered. A waveguide transmits a variety of modes [27], for example, guided modes, radiation modes, evanescent modes, according to the length of the guide, whether the guide is open or terminated and other factors. We consider for soft X-ray transmission, the dependence of the mode transmission on the width and length of a 2D guide in gold. The width is to be between 5 and 20 nm, and the length 200 nm. The input plane wave suffers diffraction at the ingoing aperture and suffers external reflection at the slit edges according to the grazing angle. A consideration has been made [14] of the propagation of guided soft X-ray modes through 2D slits. For a symmetrical 2D (or 3D) guide, the propagating guided modes are TE,, and TM,, (or HE1 1 in 3D), which never reach cut-off for a finite guide length, and other even-order modes TE2, TE4, and TM2, TM,, etc., according to the slit width; the odd guided modes are not supported in a symmetrical guide illuminated symmetrically. The field distribution in a lossy waveguide in the steady state can be described by a discrete set of guided modes and a continuum of radiation modes. The radiation modes redistribute the energy carried by the guided modes from the waveguide core into its cladding. Leaky modes correspond to series expansions in terms of radiation modes and may be interpreted as guided modes beyond the cutoff point. The following is a brief explanation of guided and leaky modes [28]. At any given slit width d, the waveguide can support a finite number of guided modes. Considering a ray picture, if the angle of incidence at the cladding reaches the critical angle, total reflection is lost and energy leaks into the
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characteristic length and the leaky modes have decay lengths less than 43 nm. Fig. 2 shows that the leaky TE2 mode has a decay length of 7 nm compared to the guided TEo mode of approximately 210 nm. To examine mode transmission for soft X-rays, calculations were made using the coupled-wave method for wave transmission down slits of fixed width e.g. 10 nm, and a range of slit lengths z, to determine the guide length at which the distribution of electric field energy was closely similar to the distributions corresponding to the propagation of the TE, and TM, modes only. For d = 10 nm and 3 nm wavelength, the transmitted probe shape is equivalent to that for the TEo mode for slit lengths greater than about 500 nm. This characteristic length increases rapidly as d increases. Thus, for the collimating 2D apertures relevant to the SPXM, with smooth parallel sides and widths from 5 to 20 nm, and of length 200 nm, the
cladding; the guided mode is said to have reached its cutoff. Leaky modes occur beyond cut-off, they decay quickly and they are important to the transition region between the initial excitation of the waveguide and the steady-state solution further down the guide. The l/e decay lengths of the guided and leaky modes as a function of the slit width are shown in Fig. 2 for normal incidence. At d = 10 nm, the waveguide supports only one guided mode and an infinite but discrete set of leaky modes. Guided modes are cut-off when the mode absorption equals the bulk absorption, which occurs at the characteristic depth of 43 nm for gold at II = 2.4 nm as shown by the dotted line in Fig. 2. Similar calculations to those in Fig. 2 for even-order TM modes produce propagation characteristics closely similar to those for the corresponding TE modes. The guided modes have l/e decay length > to the
20
30 width
40
50
d (nm)
Fig. 2. Absorption of even-order TE guided modes above dotted line (equivalent for TM modes) and leaky modes, below dotted line, for 2.4 nm wavelength X-rays propagated through a gold slit aperture of width d -C 50 nm. The dotted line represents the decay length for bulk absorption at which value all guided modes cut off except the zero order.
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dominant transmitted guided mode is TEo (TMJ and the differences in propagation of the planepolarised components are small. For all slit widths, the minimum slit length needed to define the collimator aperture is shorter than the transient length and the properties of the probe emerging from the slit will both differ from the fundamental TEo mode because of additional leaky modes of finite energy and change in details as the slit length changes.
3. Calculations of soft X-ray transmission and image resolution 3.1. Electric field energy and Poynting vector distributions for propagation through a 200 nm long gold 20 slit into vacuum For TE propagation
account must be taken of
E, (tangential), H, and H, field components and for
TM the appropriate components are H,, E, (tangential) and E, (normal). Results will be discussed either as electric field intensity IEyj2, lEx12, lEz12, or the Poynting vector jE,H,* for the total flux density. FDTD calculations in 2D for TE polarisation are shown in Fig. 3 of IE,12 for a number of z coordinates within the slit as far as the emergent slit face as an incident plane wave traverses the slit. Near the slit input, at z = 7.5 nm, the TE2 leaky mode is clearly visible but it attenuates rapidly with distance down the slit. There is a maximum in the transmitted intensity due to constructive interference between z = 50 nm and z = 100 nm. The loss of energy to the gold edges is clear, and for the constant width d, the full-width at half-maximum (FWHM) of the wave within the slit remains constant as the wave proceeds. Complete 2D contour plots are shown in Fig. 4, calculated by FDTD, of the electric field energy
0.10 r z=lOOnm
-10
0 x across aperture [nm]
10
Fig. 3. Snapshots of a 2D probe as it traverses a 200 nm long gold slit, 10 nm wide. Distances z value of plane wave entry into the slit. Calculations by FDTD method. IL= 3 nm.
-.-.-.____
20
are measured
30
downstream
from the
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69 (1997) 259-278
130
z
in
aperture
Fig. 4. Contour plot of the intensity distribution inside the slit aperture, calculation). The grey levels are linearly spaced.
density for the traverse of a 3 nm normal incidence plane wave down a 10 nm wide 200 nm long gold slit; the contours of magnetic field intensity are very similar. These results are for TE polarisation. The TE2 leaky wave component is visible and the maximum of transmitted intensity occurs about 60nm down the tube. As has been considered above the differences between the results for TE and TM are too small to be revealed at the given scale. The contours give a quantitative picture of the energy transfer into the gold cladding. The results for the Poynting vector show a smooth flow of flux at the centre parallel to the slit axis, and at the edges the flow is inclined towards the cladding consistent with the loss of energy as the wave propagates. The details are in harmony with the intensity contours in Fig. 4 and are not shown. The lowest curve in Fig. 3, which gives JE,l’ for a tangential component of E as the wave is on the
[nm]
TE tangential
component,
d = 10 nm, /z = 3 nm (FDTD
vector
point of emerging from the slit, describes the X-ray probe to be propagated downstream through the air gap, assumed to be 10 nm and on to the specimen. This calculation, by the FDTD method, gives results that are effectively identical with the corresponding calculation made by the coupled-wave method. and the agreement in this example was found to extend to others so as to allow the two methods to be used interchangeably. When the probe is propagated downstream using the general Rayleigh-Sommerfeld approach for the tangential components of both TE and TM, as before, the differences are very small. Calculations for the TM normal component IE,1’ show this to be of low but finite intensity as compared with the tangential component. Plots of the probe intensity distributions for a 10 nm slit width and 2.4 nm wavelength are given in Fig. 5a and Fig. 5b in the form of snapshots of the probe in vacuum for the tangential and normal components, respectively, of the
R.E. Burge et al. / Ultramicroscopy 69 (1997) 259-278
electric field energy density at different distances up to 400 nm downstream from the slit surface. Fig. 5a demonstrates a probe broadening out from an initial FWHM comparable with the slit width; the probe retains its integrity as a single Gaussianshaped peak until a distance z of about 200 nm. On the other hand, the normal component shows a double-peak structure which persists but broadens and weakens with the increase of distance downstream. The double peaks arise from the accumulation of electric charge at the slit edges: in the corresponding 3D case cylindrical waves arise from the edges of the circular aperture which have weak intensity and widen and blur out with increasing downstream distance and are of negligible intensity compared with the tangential components. The same intensity scales are used for Fig. 5a and Fig. 5b which show that the relative contribution of the normal field component to the total electric field energy density as compared with the tangential field is negligible. Because of the effective equivalence of the distributions of intensity in the X-ray probe for both polarisations, only the TE component is considered in 2D calculations below. Fig. 5c shows, for the same slit and wavelength parameters as for Fig. 5a and Fig. 5b, for comparison with Fig. 5a, calculations for the probe as propagated with the scalar multislice theory as applied in 2D. The probe is shown at the same positions downstream as before. The comparison shows that the general features of the vector and scalar calculations in 2D are very closely similar up to z values of about 200 nm and quite similar for larger distances downstream with similar changes in intensity levels with increasing z. The separations of the two peaks into which the intensity patterns divide at the larger z values is somewhat smaller in the scalar calculation relative to the vector results. Fig. 6 shows a wider range of results for slit widths of 5, 10, 15, 20 nm, for a gold slit 200 nm long and 2.4 nm X-rays, in the form of the variation of FWHM values for the probe widths as propagated through air for z values up to 400 nm. Each curve is superimposed on a curve which corresponds to the probe width if it were comprised entirely of TEO (TMO)-guided waves. The differences between the curves for the different slit widths correspond to the effects of the so-called leaky
261
modes for slit depths less than the transient distance. The curves in Fig. 6, especially for z less than 200 nm, for the four apertures are very similar despite the four fold difference in slit width. This similarity for small z is in harmony with the result that for all these slit widths the principal guided mode will be the fundamental mode which allows experimentally the use of a larger aperture with a significant gain in photon flux at each image point with small change in resolution. The curve for 5 nm slit width is closest to the TEO curve as the smaller the slit width the greater the discrimination against all other excited modes. In contrast, for propagation distances of greater than about 500 nm the transmitted probe approximates to the TEO mode because of the absorption of the leaky modes. To show that the nature of the probe depends on the resultant of the guided and leaky modes, the curve for the z-dependence of the FWHM for a 225 nm long and 10 nm wide slit is also shown on the plot for d = 10 nm. The curve for the longer slit shows a smaller half-width at given z than for the 200 nm long slit. Finally, concerning probe propagation through a specimen for d = 10 nm, using coupledwave theory, the variation was examined when, after a 1Onm air gap, the probe is propagated through a continuous medium with the properties of PMMA. The curve for PMMA was found to “shadow” that for the slit length of 225 nm, showing a small narrowing in probe half-width due to the real part of the refractive index. Considering further the relationship between the scanning probe description in 2D and 3D, in Fig. 7 are shown three sets of results expressing the probe intensity distribution in terms of FWHM as it propagates in air. The results are all for d = 10 nm, slit length 200nm in gold, wavelength 2.4 nm; the corresponding results for a wavelength of 3 nm are very similar. One set is the same as in Fig. 6 under 2D vector conditions. The other two sets are derived from the scalar multislice calculations as applied in 2D and in 3D as propagated by either a 2D or a 3D propagator. The FWHM in 3D is the half-width across a central slice. There is good agreement between all three data sets up to almost z = 200 nm. The perspective plots in Fig. 8 show the probe intensity distribution in 3D at four locations downstream up to 200 nm and correspond to
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h=2.4nm.
p=lOnm.
69 (1997) 259-278
Z=Onm
3.0~
1
TE. k2.4nm.
#=lOnm.
Z=60nm
O.BF
FE. h=2.4”m,
#= tO”m.
Z=BOnm
4
gyy---j
TE. h=2.4”m,
$=lO”m.
2=250”m
;_~
-40
-20 I
TE, h=2,4”m,
$=,O”m,
2=3OO”m
0.20~
TE, A=2.4”m,
Q=lO”m,
TM. k2.4M”.
p1onm.
2=350”m
O.ZLl~~j
1
TE,
h=2.4”m,
&)
#=,O”m,
40
20
2=4OOnm
f$-jyyJ
2=2Onm
i_ii
-40
a.ao*o~-q
-20
0 x (nmf
20
rcl
TM, k2.4”m,
$=IO”m.
Z=LLOnm
TM, X=2.4”m.
#=lO”m,
Z=2OO”m
_, ~~~~~~~ili~~~ -40
-20
TM, h=2.4”m,
-”
-”
I (in, =’
”
h=2.4”m,
$=lO”m,
Z=,OO”m
_,
TM.
h=2.4”m,
#=lO”m,
2=250”m
_,
TM,
h=2.4”m.
$t,=1onm.
z=400nm
~~~~
t_,L--Jjq
(b)
TM,
-40
-*o
II x h)
#=lO”m.
0
x[W
20
40
2=350nm
20
40
-40
-20
0
x(4
20
10
Fig. 5. (a) TE tangential component of probe propagation following incidence of a plane wave on a gold slit with d = 10 nm. 2D coupled-wave vector calculation through vacuum (air) for selected distances downstream from the slit exit. Wavelength 2.4 nm. (b) As 5(a) for TM normal component. (c) As .5(a) but 2D scalar multislice calculation.
R.E. Burge et al. / Ultramicroscopy A-2.4nm,
$=lOnm,
Z=Onm
A=2.4nm,
$=lOnm,
269
69 (1997) 259-278 Z=ZOnm
0.5ok
A=2.4nm,
)=lOnm,
Z-40nm
~~~
Fig. 5. (Continued)
a cylindrical collimating tube. The throughput values, i.e. the ratio of the probe intensity distribution integrated across the measurement plane and the intensity incident, as defined by the aperture of the collimator tube, as determined from the 3D scalar case, evaluated from the electric field intensity for circular apertures with diameters 5, 10, 15, 20nm, are, respectively, 6.2%, 31.7%, 78.2%, and 92.8%. These throughput values are considered to be closely representative of the values that would have been found in a full-3D vector calculation. All the above calculations have assumed normal incidence. When using a FZP, specified as in Section 2.1, as a collimator the maximum illumination angle to the normal is one degree. The effect of non-normal incidence on the probe emerging into air from a 10 nm slit was explored for a rather wider range of possible conditions using coupledwave theory and is shown in Fig. 9 for a range of angles to the normal up to the critical angle for gold
at a wavelength of 2.4 nm. As the incident angle increases the transmitted peak height falls, the peak suffers displacement, and the probe widens. However, for the small convergence angle of the chosen FZP collimator the effect on the transmitted probe is very small. 3.2. Intensity distributions for X-rays transmitted through dielectric materials: Imaging and spatial resolution To take advantage of the range of z where the FWHM of the probe is effectively constant, the probe-to-specimen distance should be minimised and a distance of 10 nm is assumed here. The variation of image resolution with specimen thickness is illustrated in Fig. 10 where using coupled-wave (2D vector) theory a comparison is made of calculated scanned images, sampled at 2 nm intervals, of a small particle of calcium placed first on top of
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R.E. Burge et al. / Ultramicroscopy
X=2.4nm,
69 (1997)259-278
h=2.4nm,
$=5nm,
aoF.“““.”
$= 1 Onm,
...’ .““‘..‘.“..““..‘i
““7 60
ot.. 0
100
Ol 200
Distance
h=2.4nm,
0
100
300
400
0
(nm)
$=15nm,
200
Distance
306
100
200 300 Distance (nm)
h=2.4nm,
400
(nm)
0
100
400
$=20nm,
200 300 Distance (nm)
400
Fig. 6. 2D coupled-wave calculations (full lines) for FWHM height on propagation of a plane wave by a 200 nm long gold slit into air. Slit widths 5 nm, 10 nm, 15 nm, 20 nm, as marked. Wavelength 2.4 nm. The FWHM values for a single zero order guided mode are shown by dashed lines for each d value. For d = 10 nm an additional line is shown (chain) representing propagation into air but for a slit 225 nm long.
a 400 nm thick PMMA supporting film and then underneath it. The probe-to-specimen (particle or uniform film) distance was 10 nm. The particle was 40 nm thick and 50 nm wide. For comparison the ideal kinematic absorption images are also shown calculated from the Henke refractive index data for a point probe. The calculations show that the levels of image intensity as the probe is scanned across abrupt edges are modified from the ideal Henke values but asymptote to these values some distance from the edges if the separation of specimen features is large enough to allow it. The edges themselves are displayed as edge spread functions from which the spatial resolution can be derived. The particle on the top is imaged with a resolution of about 10 nm and with a 60 nm resolution on the bottom. Further, vector coupled-wave calculations for scanned images are presented in Fig. 11 for a series of perpendicular steps in uniform PMMA films, for d = 10 nm and a wavelength of 2.4 nm. These calculations extend to a maximum thickness of
200 nm, which is the extent of the range of specimen thickness up to which the probe dimensions (see Fig. 7) are expected to remain closely constant. The film thicknesses selected were 50, 100, 200 nm, giving edge spread functions between 0 to 50 nm, 0 to 200 nm, 50 to 100 nm, 100 to 200 nm. Resolution values derived from edge spread functions adopt the width between the 20% and 80% intensity levels (60% criterion) or the lO-90% levels (80% criterion). Results for these two widths give a measure of the resolution, lying between 9 and 11 nm, according to the chosen criterion, for the thinner specimens, A more physically satisfactory assessment of image resolution may be made from the modulation transfer function (MTF) which can be calculated directly from the edge spread function by first differentiating to derive the line spread function and then inverse Fourier transforming to give the MTF. The MTF allows the setting of the minimum contrast level of image features said to be resolved.
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R.E. Burge et al. 1 Ultramicroscopy 69 (1997) 259-278
Coupled-Wave
0
20
Calculation (solid); Multislice Calculation 2-D (circle), 3-D (star)
40
60
80
100 z(nm)
120
140
160
160
200
Fig. 7. Comparison of values of full-width at half-height for 2D (vector, full line), 2D (scalar, circles), and 3D (scalar, stars) probes propagated downstream and measured from the slit exit surface. Plane wave incident, wavelength 2.4 nm, 200 nm long gold slit/cylinder.
MTF curves calculated from the four-edge spread functions of Fig. 1I are shown in Fig. 12 covering a range of image spatial frequencies and image contrast. Taking a (stringent) MTF value of 0.1 as in Ref. [S] the resolution varies from 7.4 nm for 50 nm thickness to 10.4 nm at 200 nm thickness; for a more realistic MTF value of 0.2, the resolution values are 8.6 nm (50 nm thick), 10.0 nm (100 nm thick), 12.2 nm (200 nm thick). There is a decrease in resolution with increasing thickness due to probe broadening, but the results are consistent with the expectation that the resolution, for the thinner specimens, will be better than 10 nm. Further calculations for d = 10 nm, but for slits longer than 200 nm, show a further narrowing of the probe, with correspondingly improved spatial resolution. The question of image deconvolution, given the specification of the scanning probe or an estimation of it, may be considered using, for example,
Wiener-Helstrom filter [29]. The level of success can be monitored by the improvement in signalto-noise ratio [30]. We draw attention to the extended examples of signal-to-noise improvement following deblurring given in Ref. [31] including the effects of noise and of incorrectly chosen probes.
4. Experimental aspects On the basis of measurement at the undulator beamline at the ELETTRA synchrotron and informed prediction at the ESRF, a photon flux is expected of between 5E9 and lEl0 ‘photons s-l into a 0.2 urn spot of an FZP used as a condenser (see Fig. 1) to illuminate the collimator tube at a chromatic resolving power n/An of some hundreds. The chromatic properties of the tube collimator, like the wavelength dependence of the
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Fig. 8. Three-dimensional plots of scalar calculations for the 3D probe intensity distributions at the given distances downstream from the exit slit surface. Plane wave incident, wavelength 2.4 nm, 200 nm long cylindrical gold collimator.
critical angle, have been shown above not to be critical by the interchangeable use for calculations of wavelengths of 2.4 and 3 nm. The FZP must be set accounting for the focus for the required wavelength. The allowable bandwidth of the incident radiation will be governed by the number of photons falling, out of focus, around the collimator tube and their contribution to the background at each pixel by transmission through the aperture edges. If the L/AA value, with the FZP focused at wavelength I, is set when the geometrical size of the probe is the same (0.2 pm) as the diffraction iimited
FZP focus, then n/An = 500 for the given dimensions. In experimental conditions, it may be necessary to increase the length of the collimator tube from 200 nm to perhaps 250 nm to reduce to neghgible levels the transmission through the tube edges. The order selecting aperture which isolates the first-order focus is not critical for this purpose for diffraction orders of significant relative intensity (perhaps for practical FZP up to the third-order), because of the angular selecting properties of the collimating slit/tube shown in Fig. 9, but is
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B=l
.O deg,
8=2.0
z=lOnm(air)
deg.
z=lOnm(oir)
0.6 0.6 .g $ 0.4 C
$ 2
0.4
0.2
-40
-20
0 Xh) 8=3.0
-40
deg,
8=5.0
deg,
40
-40
-20
20
40
deg,
-20 deg,
40
z= 1 Onm(oir)
0 Xbm) 8=6.0
z=lOnm(oir)
20
X&, 8=4.0
z=lOnm(oir)
0 Xh)
-20
20
20
z=lOnm(oir)
0.25
-40
-20
0 Xbm)
20
40
-40
-20
0 Xbm)
20
40
Fig. 9. Change of probe intensity at slit exit surface for the given angles of grazing incidence to the slit walls. Coupled-wave calculation in 2D. Plane wave, 200 nm long gold slit, wavelength 2.4 nm.
essential for screening the area around the collimating tube entrance and the specimen area from the zero-order radiation which has the full aperture of the FZP. A further feature of imaging with a collimating tube, as against focussing with a zone plate as in the usual scanning X-ray microscope where the depth of focus decreases with increasing resolution (and with higher-order focus), is the absence of a requirement for specimen focussing. The imaging beam is a projection of the transmitted X-ray probe, with, for specimens up to about 200 nm thick, an approximately constant probe width. A diagram of the proposed optical arrangement for the SPXM is shown in Fig. 13. It has a photon
detector and a specimen-observing optical microscope in far field. The inset diagram on the righthand side shows one possible arrangement for the collimating tube in the form of a commercially available AFM tip, pierced with a 10-20 nm hole, which not only passes X-rays but also maintains the probe-aperture-to-specimen distance at 10 nm by a standard cantilever/optical arrangement with a split detector to monitor accurately the position of the AFM tip in its proximity to the specimen. A pierced tip has been proposed previously for SNOM [32]. For X-rays in the water window the tube must be given a depth of 200-250nm by evaporation of a heavy metal; gold has been used in the above calculations. Experiments are in hand
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of scanning images calculated by coupled-wave method for wavelength 2.4 nm, of 50 nm wide, 40 nm thick particle Fig. 10. Comparison of calcium on a 460 nm thick supporting film of PMMA. Solid curve, particle on top of film nearest the probe; dashed curve particle on bottom of film furthest from the probe. Dotted line gives the Henke data.
with a 5 nm diameter 30 KeV energy gallium ion beam to drill the AFM tip, which is fabricated in silicon nitride, and very early results have produced apertures 30 nm in diameter. Silicon nitride is a good conductor of heat and order of magnitude calculations show that for all conceivable levels of photon flux the temperature rise of the AFM tip at steady state will not be greater than 1”. Using the microscope with a wet specimen will inevitably draw water into a collimating tube. Remembering that the wavelengths in mind are in the water window, calculations show that the probe emerging from a wet tube is not significantly changed relative to a dry tube. In complementary possibilities for probe fabrication, solid probes are now available commercially with tips having 10 nm radius from materials like silicon and silicon nitride which have high X-ray transmission and can be coated with heavy metal, possibly after ion beam
shaping. We note the development of carbon nanotubes [33] which can be 5-20 nm diameter and up to 1 urn long, and are now being used as nanoprobes in scanning probe microscopy, the developments in glass nanochannel arrays [34] fabricated so far with minimum diameters of 33 nm, and the cylindrical straight holes 10 nm in diameter and up to 6 urn long, in polycarbonate now commercially available. The last have been fabricated by exposure to collimated charged particles in a nuclear reactor. The carbon nanoprobes in particular, have mechanical properties which make them resistive to catastrophic failure on contacting the specimen. In practice, the walls of the collimating tubes may not be smooth at the scale of an X-ray wavelength. For a rough surface a broad diffusely scattered component occurs taking energy from the specular component. Fortunately in the present
21.5
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Normalised Absorption
0.16-
I 1
0
20
,
40
I
I
I
100 60 80 Specimen Co-ordinate (nm)
I
120
I I
140
160
Fig. 11. Normalised X-ray absorption and calculated edge spread functions by coupled-wave theory for density steps in PMMA up to 200 nm thick. Dotted line gives the Henke data. Wavelength 2.4 nm, d = 10 nm.
application, due to self-shadowing of the smaller features of surface roughness by the larger for incidence at small grazing angles, the diffuse component is minimised [35,36]. Further analysis awaits the characteristics of particular collimating tubes. Finally, it is important to estimate the dwell time of each pixel measurement for reasonable counting statistics. Taking the figure of lEl0 photons s-l in a spot of diameter 0.2 urn for X-rays with small bandwidth, and d = 10 nm, the dwell time to detect 1000 photons in ideal imaging is 40 us. This figure should be compared with the approximately l-5 ms dwell time per pixel at the highly successful “conventional” scanning X-ray microscope established at the National Synchrotron Light Source, Brookhaven, USA [4], not from the point of view that the predicted dwell time is some orders of magnitude less, rather to point out that, for a real experimental situation, there is room for the practi-
cal selection of parameters for the near-field instrument. The expected spatial resolution and the dwell time interact with the design of the scanning stage, which needs a minimum scanning step length of 5 nm and a maximum scanning frequency in the range l-10 kHz.
5. Conclusions It has been shown by computer modelling that a new scanning X-ray microscope, utilising a zone plate condenser and tube-aperture, with a pointto-point resolution of about 10 nm is possible for thin specimens for wavelengths in the water window. The practical resolution of the optical SNOM is similar to that projected for SPXM, but the specimen thickness possible for 10 nm resolution studies with soft X-rays is greatly increased to 200 nm as against 5-10 nm for the SNOM. Steps
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0.15 Frequency(l/nm)
0.2
0.25
0.3
Fig. 12. Modulation transfer functions calculated from the theoretical noise free edge spread functions in Fig. 1I. Circles denote a common line for the O-50 nm and 50-100 nm steps. The other lines represent the 100-200 nm step (dot-dash-dot) and O-200 nm step (dotted).
are in hand to implement the new microscope at the ESRF. Similar high resolution, in the sense of the size of specimen area selected for spectroscopic measurements is available for NEXAFS. In the latter application, similar to conventional zone plate scanning X-ray microscopy, the condenser zone plate must be focussed on the aperture for each X-ray wavelength. Discussion has taken place above, implicitly by the choice of a polymer specimen, mainly on the potential of the microscope for imaging and spectroscopy of biological or polymer specimens. In practice, due to the increased radiation damage for such specimens for high-resolution imaging at reasonable signal-to-noise ratios, it may be that studies of wet inorganic specimens as in corrosion and catalysis may be more appropriate, especially if the suggestion by Cazaux [37], that X-ray radiation damage occurs only in insulating materials, turns out to be correct. Alternatively, the possibility of cryomicroscopy [38] to minimise the effects of radi-
ation on the X-ray image can be considered by which the potentially diffusing molecular fragments are frozen in situ. The thickness limit of 200 nm at the highest resolution is less of a restriction in practice than it may appear. The reason is that for thicker specimens the resolution is limited by the overlapping of structural features in depth. For the same reason, a search through the literature for high-resolution work under current conditions with both X-ray microscopy and spectroscopy with the traditional zone plate instrument shows that specimens are frequently said to be about 0.1 pm thick. In the proposed instrument, removing the collimating tube and the AFM tip reveals a “standard” zone plate scanning X-ray microscope which can readily be used in a higher order FZP focus given reduced source bandwidth. Retaining tube and tip, and changing the source and detector for longer wavelength radiation, provides a SNOM.
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incident Beam /
I
CoBimator Tube and AFM Tip
zonePlate
Order Sorting Aperture
1 Specimen
Gas FiUed
Proportional counter
Schematic Diagram of Scanning Probe X-ray Microscope /
\
Fig. 13. Optical layout for proposed scanning near-field X-ray microscope. The inset diagram coated with evaporated gold to make a cylindrical collimator 20&250 nm deep, of diameter
Acknowledgements
Financial and material support was provided by the Leverhulme Trust, the Royal Society, Defence Research Agency, Malvern, Oxford Instruments PLC and Leo Electron Microscopy, Cambridge. We are grateful to Professor A. Howie for the provision of facilities at the Cavendish Laboratory, to FE1 Europe Ltd for assistance with ion beam milling, and to Dr. A. Hare for discussion of the multislice method.
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