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Simulation of thermal diffuse scattering including a detailed phonon dispersion curve
Ultramicroscopy 86 (2001) 371–380
Simulation of thermal diffuse scattering including a detailed phonon dispersion curve David A. Mullera,*, Byard Edwardsb, Earl J. Kirklandc, John Silcoxc a
Bell Laboratories, Lucent Technologies, Room IE 356, 700 Mountain Avenue, Murray Hill, NJ 07974, USA b Physics Department, Cornell University, Ithaca, NY 14853, USA c School of Applied and Engineering Physics, Cornell University, Ithaca, NY 14853, USA Received 19 June 2000; received in revised form 27 September 2000
Abstract Thermal vibration of the atoms in a crystal give rise to a diffuse background in the diffraction pattern (in between the normal allowed Bragg reflections). The Einstein model for phonon vibrations in a crystal leads to Gaussian statistics for the phonons. However, the Einstein model ignores the possibility of correlation between the atoms. An accurate model of the phonon dispersion curves for silicon is used to generate a set of more accurate random atomic displacements. These displacements are used in a multislice-style simulation to gauge the validity of the Einstein approximation. The phonon dispersion curve yields a small additional oscillatory structure in the thermal diffuse scattering (TDS) pattern. This does not produce significant changes in the annular dark field scanning transmission electron microscope (ADF-STEM) image signal, but could have a large impact on convergent beam measurements of bond charges. # 2001 Published by Elsevier Science B.V. PACS: 61.14.Lj; 61.16.Bg; 87.64.Bx Keywords: Thermal diffuse scattering; CBED; Multislice simulation; ADF-STEM
1. Introduction At room temperature the atoms in the specimen vibrate. These vibrations produce effects in both diffraction patterns and electron microscope images. If the atoms in the specimen were perfectly stationary only a discrete set of diffraction angles are allowed and the diffraction pattern is a sequence of discrete spots. Random thermal vibrations of the atoms in the specimen produce *Corresponding author. Tel.: +1-908-582-8237; fax: +1908-582-3260. E-mail address: [email protected] (D.A. Muller).
a low-intensity diffuse background in between the normal diffraction peaks. This background intensity will be referred to as the thermal diffuse scattering or simply TDS. The majority of the TDS intensity can be accounted for by a simple Einstein model for thermal vibrations [1–3]. The Einstein model assumes that each atom in the crystal vibrates independent of all other atoms in the crystal. A real crystal, however, will have one or more preferred directions (i.e. the crystal axis), and neighboring atoms in the crystal may be strongly coupled to one another. Vibration along the preferred crystal axis may not be independent of
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vibrations in other directions and the vibration of neighboring atoms may not be independent. Correlations between atoms in the specimen are ignored in the simple Einstein model. A comprehensive discussion of the effect of thermal scattering on diffraction patterns is given by Warren [4], where it is shown that the Einstein approximation is exact if the atomic displacements are Gaussian, and more generally that it is the leading term in a power series expansion in the thermal displacements. The higher-order corrections due to correlations are given in Chapter 11 of Ref. [4] and depend on the specific model chosen to describe the phonon vibrations. The Debye model is often chosen for its simplicity, but it is only accurate at long wavelengths. Correlation functions for the Debye model [5] have been used in Bloch-wave simulations of annular dark-field (ADF) images [6]. As a result, Jesson and Pennycook [6] find that the Einstein model is not adequate in a Bloch-wave simulation where only the lowest order beams are retained. TDS from lattice vibrations can play an important role in the formation of annular darkfield images [7,6]. Further, charge density maps extracted from convergent beam diffraction patterns are exquisitely sensitive to the fitting and removal of the TDS background [8]. The chief attraction of the Einstein model for TDS is its simple analytic description of multiphonon excitations as a Gaussian envelope in real space [1,9], which is a consequence of the model assuming each atom acts as an independent oscillator. The Gaussian probability distribution of lattice vibrations is easy to include in the propagators of multislice simulations [10] and is implicit in the use of the Debye–Waller measure of thermal vibrations [4]. The success of the Einstein model probably arises from the result that a Gaussian is a good zeroth-order description of any random distribution that tends to a central limit. Warren [4] has shown that in a perturbation expansion of TDS, the leading term is always Gaussian, although the Debye–Waller factor may be different from that predicted by the Einstein model. Here, we explore the effect of multi-atom correlations and examine the validity of the Einstein model’s ability to reproduce convergent beam diffraction (CBED) patterns. We use a
detailed phonon model [11] that has been fitted to neutron diffraction data, including up to sixth nearest-neighbor interactions. The phonon modes are populated using Bose–Einstein statistics (so this is a true multi-phonon method). The only approximation made so far is the frozen phonon approximation [1,9] which allows us to use an elastic multislice simulation. It should be noted that the multislice method is itself an approximation. Charge redistributions due to bonding in the solid are also neglected – for silicon this will result in an error of a few percent or less in the lowerorder beams [12,13]). Detailed phonon models have been considered previously – usually as snapshots of molecular dynamics simulations [14,15]. These simulations, while providing realistic descriptions of the interactions between atoms, solve classical equations of motions for the ions. Consequently, while the correct phonon spectrum is generated, the population of phonon modes follows classical Boltzman statistics rather than the quantum Bose–Einstein statistics. In particular, the non-trivial zero-point motion is neglected. For silicon, the zero-point motion contributes almost half of the average vibration amplitude at room temperature [14]. (The effect should be less significant at the elevated temperatures used in Ref. [15].) We compare our detailed simulations with the measured, energy-filtered CBED patterns recorded by Xu, Loane and Silcox (XLS) [2] (Fig. 1). Previous work on modeling the TDS intensity in this experiment involved the frozen phonon method and the Einstein model of lattice vibration [1]. Although agreement with experiment was good, the average vibration amplitude is treated as a free parameter in the Einstein model and fitted to reproduce the ratio of TDS to HOLZ scattering. There are no such free parameters in the detailed phonon model (used here), so it provides a much stronger test of the frozen phonon method. It is also possible to turn off the interatomic correlations by randomizing the phonon phases (we did so accidentally in a previous report [16]) and so explore the effects of correlations in thermal vibrations between different atoms on the convergent beam pattern. We find the correlations are needed to reproduce the banding perpendicular to
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Fig. 1. Energy-filtered convergent beam electron diffraction patterns of silicon [1 0 0] measured by XLS [2]. The multislice simulation using an Einstein model of thermal vibrations (Fig. 1d) reproduces all the main features of the CBED pattern except the banding perpendicular to the Kikuchi lines.
the Kikuchi lines that is seen experimentally. Correlations are not needed however to simulate the annular dark field signal (or the Kikuchi lines themselves).
2. Phonon dispersion curves The positions of the atoms were calculated according to quantum-mechanical harmonic
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phonon theory [17] in which the phonon frequencies o and polarization vectors e are determined by diagonalizing the eigenvalue equation n X
0
0
Dss ðkÞ es ðk; lÞ ¼ o2l ðkÞes ðk; lÞ:
ð1Þ
s0 ¼1 0
Here Dss ðkÞ is the dynamical matrix, n is the number of atoms per unit cell, and k is the wavevector. s; s0 range from 1 to n, and l varies from 1 to 3n, denoting the mode. For silicon, with two atoms per primitive unit cell, n ¼ 2, and there are 6 bands at a given value of k. The dynamical matrix elements were taken from Jian, Kaiming, and Xide (JKX) [11]. There were a number of sign errors in the typesetting of this paper, which should be obvious from the symmetry of the equations. After correcting these errors, we checked the phonon dispersion curves along all major symmetry cuts to ensure they reproduced the neutron data to which the JKX matrix elements were fitted. Fig. 2 shows both the full phonon dispersion curve, as well as the Debye equation, along the G–X direction. The Debye model is clearly inadequate for all but the longest wavelengths and lowest temperatures. Given the phonon dispersion curves, the atomic displacements u can be calculated in terms of
Fig. 2. Phonon dispersion curves for the full parametrization of JKX [11] and the Debye model more commonly used to estimate thermal scattering.
normal coordinates q1 and q2 [17] N=2 1 X X 1 s s pffiffiffife ðk; lÞexp½2pik RðlÞ u ðlÞ ¼ pffiffiffiffiffiffiffiffiffi NM k 2 l þ es * ðk; lÞexp½ 2pik RðlÞ gq1 ðk; lÞ i þ pffiffiffifes ðk; lÞexp½2pik RðlÞ 2 es * ðk; lÞexp½ 2pik RðlÞ gq2 ðk; lÞ ;
ð2Þ
where RðlÞ is the lth lattice site, N is the number of unit cells (periodic boundary conditions are imposed), and M is the atomic mass. The sum is over N=2 wave vectors, as it includes only one of k and k. Thus, to determine the instantaneous atomic positions resulting from a measurement we need the values of the normal coordinates fqi ðk; lÞg. In the space of the normal coordinates, the phonon modes have been decoupled into independent harmonic oscillators. The probability distribution for the normal coordinates is then determined by the harmonic oscillator wave functions. At finite temperature, modes above the ground state are occupied according to Bose–Einstein statistics. The sum over all the excited states has a closed form in the normal coordinates, and the resulting probability density for qi ðk; lÞ is rigorously Gaussian at T > 0 [9], with sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi oM ho tanh Pðq; TÞ ¼ ph 2kB T oM 2 ho q tanh
exp ð3Þ h 2kB T and the values of fqi ðk; lÞg are determined randomly, using the probability distribution in Eq. (3). We calculated the instantaneous atomic coordinates for a system of 6 6 55 cubic (nonprimitive) eight-atom unit cells of silicon, with periodic boundary conditions. To maintain consistency with the boundary conditions, we calculated the phonons on a uniformly spaced 6 6 55 grid in the Brillouin zone of the cubic unit cell. The random numbers fqi ðk; lÞg were determined according to Eq. (3) using the random number generator ran2 and the subroutine gasdev
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[18]. The positions of all 15 840 atoms were then calculated using Eq. (2). Finally, it is worth noting that as the normal modes are populated according to Bose–Einstein statistics, this is a true multi-phonon method that includes the zero-point motion. There are no free parameters and the only unknown in the simulation is the temperature of the crystal. As we will be comparing the simulation to the XLS experiment [2], the temperature is constrained to lie between 293 and 299 K.
3. Multislice simulation of CBED and ADF-STEM images The multislice method of image simulation is a means of calculating the propagation of an electron wave function through a thick specimen including the effects of the geometrical thickness of the specimen and dynamical scattering in the specimen [19–21]. The multislice method was originally proposed for the simulation of CTEM images in which the incident wave function is a simple plane wave. It can also be extended to the simulation of convergent beam electron diffraction patterns [22] and STEM images [23] in which the incident wave function is a convergent focused probe. 3.1. Simulation of disordered solids One further modification to the standard multislice code is needed to efficiently simulate an arbitrary ensemble of atoms: A simple crystal with stationary atoms (and no thermal vibrations) can usually be decomposed into a small number of identical slices that are repeated in some pattern as the electron wave function progresses through the specimen. For example the [1 0 0] projection of silicon contains four different layers that are repeated in the sequence abcdabcdab. . . : The traditional form of a multislice simulation would calculate the transmission function for each of the four layers and store them in memory to be used over and over again. However, when the atoms in the specimen are allowed to vibrate the specimen no longer has a simple repeating structure. Each
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layer or slice is different from every other. There is no longer any reason to precompute the slices and save them. A multislice simulation of a disordered specimen becomes dominated by the calculation of the transmission functions. This calculation must be repeated for each layer because every layer is different. There is some incentive to optimize the speed of calculation of the transmission function because there will be many of them to calculate in a disordered specimen. It is actually quicker to calculate the potential of the specimen in real space by using a look-up table for the real space potential for each single atom [24]. The total specimen potential is a simple linear summation of atomic potentials ignoring electronic rearrangement due to bonding in the solid. The traditional method of calculating of the atomic potential is to form the reciprocal space structure factor in the first Born approximation and inverse Fourier transform. The potential of each atom in the specimen has a short range in real space of no more than about 3 A˚ but extends over large angles in reciprocal space. Therefore, the real space potential only has to be calculated at a few points surrounding each atom site in the transmission function rather than filling in all of reciprocal space out to high angles as in the structure factor calculation. The speed increase scales as the ratio of the area of the layer over the area of a single atom – roughly a factor of 100 improvement for the present simulation. The atomic potentials listed by Kirkland [25] were used for the simulations in this paper. These were derived from the multiconfigurational Dirac–Fock program of Grant et al. [26]. 3.2. Simulated convergent beam electron diffraction patterns A version of the standard multislice calculation was written to accept the ðx; y; zÞ coordinates for an amorphous structure and propagate the electron wave function through a non-periodic specimen. The source code and executables for this program (autoslice) are given in Ref. [25]. A CBED pattern was simulated by starting the multislice calculation from an incident focused probe. CBED
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patterns were averaged over 16 different thermal vibrations using the frozen phonon method. The slice sizes were 512 512 pixels which sets the maximum scattering angle at 190 mrad (beyond the second HOLZ line) and ensures that the HOLZ lines are sampled on a sufficiently fine grid. The slice size was increased to 1024 1024 pixels for 1 run to check convergence, and no significant differences were noted. The slice thickness was 1:4 A˚ to ensure that all vertical periodicities were maintained. The incident probe modeled the experimental conditions of XLS for a 100 keV VG-HB501 STEM with an analytic polepiece ðCs ¼ 3:3 mm; 1100 A˚ defocus; 7:5 mrad objective aperture) [2]. The simulation size (6 6 55 cubic unit cells of silicon, which is treated as a 15 840 atom amorphous material) was chosen to reproduce the 29.9 nm thick [1 0 0] oriented sample used by XLS. A multislice simulation of 1 phonon configuration took 21 min on a 180 MHz Pentium Pro. 44 phonon configurations were used to generate each image.
4. Results 4.1. The effect of correlations The inclusion of phonons in the simulation reduces the HOLZ line scattering and creates a background of thermal diffuse scattering (Fig. 3). The thermal diffuse background (first order TDS) can be reproduced with independent atomic vibrations – these were produced by randomizing the phases in the detailed phonon model to produce uncorrelated phonon vibrations. The bands perpendicular to the Kikuchi lines (second-order TDS) requires correlations between atoms and are only reproduced by the detailed phonon model. A discussion of the various orders of TDS scattering is given in Ref. [4, p. 205]. The first-order TDS describes the root mean square (RMS) displacement of atoms from their crystal sites and is the result of incoherent thermal scattering [4,27]. ffi For silicon, the RMS displacepffiffiffiffiffiffiffiffi ˚ The higher-order TDS ment is hu2 i ¼ 0:076 A.
Fig. 3. Simulated CBED patterns of 299 A˚ of (1 0 0) silicon. The left pattern is calculated without phonons. The upper right pattern from 44 configurations of the detailed phonon model. The lower right pattern is from 44 configurations of the detailed phonon model at 297 K. where the phonon phases have been scrambled to remove correlations between atoms. Notice that the bands perpendicular to the Kikuchi lines require the correlated phonons, but the Kikuchi lines themselves do not.
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scattering describes the degree of coherence of the thermal scattering between different atomic sites. The pair correlation function for two atoms, m and n, a distance rmn apart, and for a phonon of wavelength k is [4,5,27] Siðk:rmn Þ hðum un Þ2 i 2hu2 i 1 ; ð4Þ k:rmn where SiðxÞ ¼
Z
x
du 0
sin u ; u
ð5Þ
when k:rmn 41, then hðum un Þ2 i 2hu2 i. This means the peak width in the pairffi correlation pffiffiffiffiffiffiffiffiffiffi 2 i ¼ 0:1074 A ˚ function should tend to 2hup ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (which is the uncorrelated width hu2m i þ hu2n i) at large distances. The pair correlation function for silicon at 297 K (Fig. 4) shows that this is the case. In fact, ˚ the large distances appear to be about 4 A, second-neighbor distance. Only the first peak in the pair correlation function is narrowed by the introduction of correlations between atoms. This demonstrates that when averaged over all wavelengths, only the nearest-neighbor correlations survive at room temperature even though sixth-
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neighbor effects were included in the phonon model. The nearest-neighbor correlations, as suggested by Eq. (4), lead to an oscillatory pattern imposed on the first-order TDS, with a period characteristic of the inverse of an interatomic spacing, i.e. a Bragg spacing. This means the TDS will be peaked at Bragg peaks and reduced between. The result is that a Debye–Waller factor fitted only to Bragg reflections (especially low-order reflections) will appear smaller than one fitted to all the thermal scattering – or to the ratio of HOLZ=TDS scattering. This may have an effect on the extraction of bond charges from CBED measurements if standard Debye–Waller factors are used, or flat backgrounds across the CBED disks are assumed. For instance, Nuchter et al. noted that the best fitting results were obtained when the background fitting parameter was allowed to vary from disk to disk [28]. The variations in background intensity from disk to disk, and across a single disk are roughly 10% and 1%, respectively, depending on the size of the disk, and the order of reflection – the effect is more noticeable for the (4 4 0) than the (2 2 0) disk. 4.2. Comparison to experiment
Fig. 4. The pair correlation function for silicon at 297 K, calculated using the detailed phonon model, and with the phases randomized to remove correlations between atoms. Only the first peak in the pair correlation function is narrowed by the introduction of correlations between atoms. This demonstrates that when averaged over all wavelengths, only the nearestneighbor correlations survive – even though sixth-neighbor effects were included in the phonon model.
The experimental CBED pattern for (1 0 0) Silicon from XLS [2] is shown as an inset in the CBED pattern calculated using the detailed phonon simulation in Fig. 5. Agreement in intensity and location of the diffuse scattering, including the banding perpendicular to the Kikuchi lines is excellent. This should be contrasted with the Einstein model shown in Fig. 1d. The azimuthally integrated intensity from the simulated CBED pattern is compared to the energy-filtered measurements and Einstein simulation of XLS [2] (Fig. 6). The Einstein model is fitted to match the HOLZ=TDS scattering ratio and so is in good agreement at 1 k > 2:5 A˚ . The Einstein model is missing the 1 TDS oscillations from 1 to 2:5 A˚ . The detailedphonon model reproduces the HOLZ=TDS ratio at 297 K. It also reproduces the oscillatory TDS 1 scattering at k > 1:5 A˚ . There is also a 10% 1 disagreement at k51 A˚ , in part due to scanning
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Fig. 5. Comparison of the energy-filtered CBED pattern measured by XLS [2] with the detailed phonon simulation.
Fig. 6. Scattering intensities determined by azimuthally integrating the CBED pattern of Fig. 3. Both the detailed phonon simulation and the Einstein model accurately reproduce XLSs [2] energy-filtered measurements at large scattering angles. The Einstein model does 1 not reproduce the oscillations at 1–2 A˚ , while the detailed model does. The discrepancy at the first Bragg peak is most likely due to the use of free atom charge densities in the simulation, which differs most noticeably from the charge density of crystalline silicon at small scattering angles.
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Fig. 7. Annular dark field signal as a fraction of the incident beam for different collector inner angles. The Einstein model underestimates the ratio of HOLZ=TDS scattering by 10%, though the match could be improved by adjusting the temperature. The detailed phonon simulation is within 1% of XLSs [2] energy-filtered measurements.
distortions in the experiment. Another discrepancy is at small scattering angles where the charge density in crystalline silicon departs from the free-atom potential assumed in both the simulations. Annular dark field (ADF) images are formed by collecting the electrons scattered to large angles from a scanned, focused probe. Fig. 7 shows the ADF signal that is expected as the inner angles of the ADF detector is varied. The Einstein model underestimates the HOLZ=TDS ratio by 10%. This could be improved by better fitting. The detailed phonon simulation is within 1% of XLS’s energy filtered measurements. The same is true of the uncorrelated phonon simulation. We conclude that correlations between thermal vibrations on neighboring atoms are not needed to reproduce the ADF signal, and that a properly fitted Einstein model is sufficient. A similar result was found for conventional TEM images of sub-10 nm NiAl films [15].
5. Summary The discrepancies between the simulation using a detailed phonon model and the experimental
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data are smaller than the experimental error. The only significant approximation made in the simulation is the frozen phonon approximation (the multislice method itself is an approximation, as is neglecting charge redistribution due to bonding in the solid). For scattering angles beyond 30 mrad where the errors in the experimental data are on the order of 1% or so, the frozen phonon approximation is in excellent agreement with the measured, energy filtered CBED pattern. At smaller scattering angles, the distortions in the experimental image lead to errors of about 10%. Better experimental data is needed to see just where the frozen phonon approximation breaks down. Correlations between neighboring atoms are not needed to reproduce the ADF signal, but they are needed for the CBED pattern, where they introduce an oscillatory banding in the TDS scattering. As the bands are always peaked on Bragg disks, any Debye–Waller factor that is fitted only to the Bragg Disks will be overestimated. Higher-order correlations (beyond the nearest neighbor) do not seem to be needed to model the TDS background in CBED patterns, although they can be important in selected area diffraction patterns. The Einstein model is thus sufficiently accurate to be used in the simulation of ADF images. This is a very encouraging result as it has a simple analytic expression that can be included in both Bloch wave and multislice codes. Finally, the Einstein model does not reproduce the oscillatory TDS structure (at 20–40 mrad) which may affect the background models used in CBED measurements of bond charges. We make no claims for the validity of the method when insufficient beams are included in a simulation or for bright-field images which (unlike ADF) could be very sensitive to inelastic scattering. Finally, these results are important in setting confidence limits on the underlying multislice simulations. These play an important role in understanding the changes in the electron beam profile as it propagates through a crystal. The evolution of the beam profile is in turn needed to develop a full understanding of the origins of the EELS spectra in the sample, and the resolution limits in thick samples.
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Acknowledgements This work was supported by DOE grant DEFG02-87ER45322.
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[15] G. Mo¨bus, T. Gemming, P. Gumbsch, Acta Crystallogr. A 54 (1998) 83. [16] D.A. Muller, B. Edwards, E.J. Kirkland, J. Silcox, in: G.W. Bailey (Ed.), Microscopy and Microanalysis, Springer, New York, 1997, p. 1153. [17] M. Born, K. Huang, Dynamical Theory of Crystal Lattices, Oxford University Press, London, 1954. [18] W.H. Press, B.P. Flannery, S.A. Teukolsky, W.T. Vetterling, Numerical Recipes, 1st Edition, Cambridge University Press, Cambridge, 1988. [19] J.M. Cowley, A.F. Moodie, Acta Crystallogr. 10 (1957) 609. [20] P. Goodman, A.F. Moodie, Acta Crystallogr. A 30 (1974) 280. [21] D. van Dyck, in: P.W. Hawkes (Ed.), Advances in Electronics and Electron Physics, North-Holland, Amsterdam, 1985, Vol. 65, pp. 295–355. [22] J.M. Cowley, J.C.H. Spence, Ultramicroscopy 3 (1979) 433. [23] E. Kirkland, R.F. Loane, J. Silcox, Ultramicroscopy 23 (1987) 77. [24] M. Pan, P. Rez, J.M. Cowley, Proceedings of the 47th Annual Meeting of the Microscopy Society of America, Vol. 47, 1989, p. 478. [25] E.J. Kirkland, Advanced Computing in Electron Microscopy, Plenum, New York, 1998. [26] I.P. Grant, B.J. McKenzie, P.H. Norrington, Comp. Phys. Commun. 21 (1980) 207. [27] Z.L. Wang, Reflection Electron Microscopy and Spectroscopy for Surface Analysis, Cambridge University Press, Cambridge, 1996, pp. 254–257. [28] W. Nu¨chter, A.L. Weickenmeir, J. Mayer, Acta Crystallogr. A 54 (1998) 147.
Simulation of thermal diffuse scattering including a detailed phonon dispersion curve David A. Mullera,*, Byard Edwardsb, Earl J. Kirklandc, John Silcoxc a
Bell Laboratories, Lucent Technologies, Room IE 356, 700 Mountain Avenue, Murray Hill, NJ 07974, USA b Physics Department, Cornell University, Ithaca, NY 14853, USA c School of Applied and Engineering Physics, Cornell University, Ithaca, NY 14853, USA Received 19 June 2000; received in revised form 27 September 2000
Abstract Thermal vibration of the atoms in a crystal give rise to a diffuse background in the diffraction pattern (in between the normal allowed Bragg reflections). The Einstein model for phonon vibrations in a crystal leads to Gaussian statistics for the phonons. However, the Einstein model ignores the possibility of correlation between the atoms. An accurate model of the phonon dispersion curves for silicon is used to generate a set of more accurate random atomic displacements. These displacements are used in a multislice-style simulation to gauge the validity of the Einstein approximation. The phonon dispersion curve yields a small additional oscillatory structure in the thermal diffuse scattering (TDS) pattern. This does not produce significant changes in the annular dark field scanning transmission electron microscope (ADF-STEM) image signal, but could have a large impact on convergent beam measurements of bond charges. # 2001 Published by Elsevier Science B.V. PACS: 61.14.Lj; 61.16.Bg; 87.64.Bx Keywords: Thermal diffuse scattering; CBED; Multislice simulation; ADF-STEM
1. Introduction At room temperature the atoms in the specimen vibrate. These vibrations produce effects in both diffraction patterns and electron microscope images. If the atoms in the specimen were perfectly stationary only a discrete set of diffraction angles are allowed and the diffraction pattern is a sequence of discrete spots. Random thermal vibrations of the atoms in the specimen produce *Corresponding author. Tel.: +1-908-582-8237; fax: +1908-582-3260. E-mail address: [email protected] (D.A. Muller).
a low-intensity diffuse background in between the normal diffraction peaks. This background intensity will be referred to as the thermal diffuse scattering or simply TDS. The majority of the TDS intensity can be accounted for by a simple Einstein model for thermal vibrations [1–3]. The Einstein model assumes that each atom in the crystal vibrates independent of all other atoms in the crystal. A real crystal, however, will have one or more preferred directions (i.e. the crystal axis), and neighboring atoms in the crystal may be strongly coupled to one another. Vibration along the preferred crystal axis may not be independent of
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vibrations in other directions and the vibration of neighboring atoms may not be independent. Correlations between atoms in the specimen are ignored in the simple Einstein model. A comprehensive discussion of the effect of thermal scattering on diffraction patterns is given by Warren [4], where it is shown that the Einstein approximation is exact if the atomic displacements are Gaussian, and more generally that it is the leading term in a power series expansion in the thermal displacements. The higher-order corrections due to correlations are given in Chapter 11 of Ref. [4] and depend on the specific model chosen to describe the phonon vibrations. The Debye model is often chosen for its simplicity, but it is only accurate at long wavelengths. Correlation functions for the Debye model [5] have been used in Bloch-wave simulations of annular dark-field (ADF) images [6]. As a result, Jesson and Pennycook [6] find that the Einstein model is not adequate in a Bloch-wave simulation where only the lowest order beams are retained. TDS from lattice vibrations can play an important role in the formation of annular darkfield images [7,6]. Further, charge density maps extracted from convergent beam diffraction patterns are exquisitely sensitive to the fitting and removal of the TDS background [8]. The chief attraction of the Einstein model for TDS is its simple analytic description of multiphonon excitations as a Gaussian envelope in real space [1,9], which is a consequence of the model assuming each atom acts as an independent oscillator. The Gaussian probability distribution of lattice vibrations is easy to include in the propagators of multislice simulations [10] and is implicit in the use of the Debye–Waller measure of thermal vibrations [4]. The success of the Einstein model probably arises from the result that a Gaussian is a good zeroth-order description of any random distribution that tends to a central limit. Warren [4] has shown that in a perturbation expansion of TDS, the leading term is always Gaussian, although the Debye–Waller factor may be different from that predicted by the Einstein model. Here, we explore the effect of multi-atom correlations and examine the validity of the Einstein model’s ability to reproduce convergent beam diffraction (CBED) patterns. We use a
detailed phonon model [11] that has been fitted to neutron diffraction data, including up to sixth nearest-neighbor interactions. The phonon modes are populated using Bose–Einstein statistics (so this is a true multi-phonon method). The only approximation made so far is the frozen phonon approximation [1,9] which allows us to use an elastic multislice simulation. It should be noted that the multislice method is itself an approximation. Charge redistributions due to bonding in the solid are also neglected – for silicon this will result in an error of a few percent or less in the lowerorder beams [12,13]). Detailed phonon models have been considered previously – usually as snapshots of molecular dynamics simulations [14,15]. These simulations, while providing realistic descriptions of the interactions between atoms, solve classical equations of motions for the ions. Consequently, while the correct phonon spectrum is generated, the population of phonon modes follows classical Boltzman statistics rather than the quantum Bose–Einstein statistics. In particular, the non-trivial zero-point motion is neglected. For silicon, the zero-point motion contributes almost half of the average vibration amplitude at room temperature [14]. (The effect should be less significant at the elevated temperatures used in Ref. [15].) We compare our detailed simulations with the measured, energy-filtered CBED patterns recorded by Xu, Loane and Silcox (XLS) [2] (Fig. 1). Previous work on modeling the TDS intensity in this experiment involved the frozen phonon method and the Einstein model of lattice vibration [1]. Although agreement with experiment was good, the average vibration amplitude is treated as a free parameter in the Einstein model and fitted to reproduce the ratio of TDS to HOLZ scattering. There are no such free parameters in the detailed phonon model (used here), so it provides a much stronger test of the frozen phonon method. It is also possible to turn off the interatomic correlations by randomizing the phonon phases (we did so accidentally in a previous report [16]) and so explore the effects of correlations in thermal vibrations between different atoms on the convergent beam pattern. We find the correlations are needed to reproduce the banding perpendicular to
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Fig. 1. Energy-filtered convergent beam electron diffraction patterns of silicon [1 0 0] measured by XLS [2]. The multislice simulation using an Einstein model of thermal vibrations (Fig. 1d) reproduces all the main features of the CBED pattern except the banding perpendicular to the Kikuchi lines.
the Kikuchi lines that is seen experimentally. Correlations are not needed however to simulate the annular dark field signal (or the Kikuchi lines themselves).
2. Phonon dispersion curves The positions of the atoms were calculated according to quantum-mechanical harmonic
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phonon theory [17] in which the phonon frequencies o and polarization vectors e are determined by diagonalizing the eigenvalue equation n X
0
0
Dss ðkÞ es ðk; lÞ ¼ o2l ðkÞes ðk; lÞ:
ð1Þ
s0 ¼1 0
Here Dss ðkÞ is the dynamical matrix, n is the number of atoms per unit cell, and k is the wavevector. s; s0 range from 1 to n, and l varies from 1 to 3n, denoting the mode. For silicon, with two atoms per primitive unit cell, n ¼ 2, and there are 6 bands at a given value of k. The dynamical matrix elements were taken from Jian, Kaiming, and Xide (JKX) [11]. There were a number of sign errors in the typesetting of this paper, which should be obvious from the symmetry of the equations. After correcting these errors, we checked the phonon dispersion curves along all major symmetry cuts to ensure they reproduced the neutron data to which the JKX matrix elements were fitted. Fig. 2 shows both the full phonon dispersion curve, as well as the Debye equation, along the G–X direction. The Debye model is clearly inadequate for all but the longest wavelengths and lowest temperatures. Given the phonon dispersion curves, the atomic displacements u can be calculated in terms of
Fig. 2. Phonon dispersion curves for the full parametrization of JKX [11] and the Debye model more commonly used to estimate thermal scattering.
normal coordinates q1 and q2 [17] N=2 1 X X 1 s s pffiffiffife ðk; lÞexp½2pik RðlÞ u ðlÞ ¼ pffiffiffiffiffiffiffiffiffi NM k 2 l þ es * ðk; lÞexp½ 2pik RðlÞ gq1 ðk; lÞ i þ pffiffiffifes ðk; lÞexp½2pik RðlÞ 2 es * ðk; lÞexp½ 2pik RðlÞ gq2 ðk; lÞ ;
ð2Þ
where RðlÞ is the lth lattice site, N is the number of unit cells (periodic boundary conditions are imposed), and M is the atomic mass. The sum is over N=2 wave vectors, as it includes only one of k and k. Thus, to determine the instantaneous atomic positions resulting from a measurement we need the values of the normal coordinates fqi ðk; lÞg. In the space of the normal coordinates, the phonon modes have been decoupled into independent harmonic oscillators. The probability distribution for the normal coordinates is then determined by the harmonic oscillator wave functions. At finite temperature, modes above the ground state are occupied according to Bose–Einstein statistics. The sum over all the excited states has a closed form in the normal coordinates, and the resulting probability density for qi ðk; lÞ is rigorously Gaussian at T > 0 [9], with sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi oM ho tanh Pðq; TÞ ¼ ph 2kB T oM 2 ho q tanh
exp ð3Þ h 2kB T and the values of fqi ðk; lÞg are determined randomly, using the probability distribution in Eq. (3). We calculated the instantaneous atomic coordinates for a system of 6 6 55 cubic (nonprimitive) eight-atom unit cells of silicon, with periodic boundary conditions. To maintain consistency with the boundary conditions, we calculated the phonons on a uniformly spaced 6 6 55 grid in the Brillouin zone of the cubic unit cell. The random numbers fqi ðk; lÞg were determined according to Eq. (3) using the random number generator ran2 and the subroutine gasdev
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[18]. The positions of all 15 840 atoms were then calculated using Eq. (2). Finally, it is worth noting that as the normal modes are populated according to Bose–Einstein statistics, this is a true multi-phonon method that includes the zero-point motion. There are no free parameters and the only unknown in the simulation is the temperature of the crystal. As we will be comparing the simulation to the XLS experiment [2], the temperature is constrained to lie between 293 and 299 K.
3. Multislice simulation of CBED and ADF-STEM images The multislice method of image simulation is a means of calculating the propagation of an electron wave function through a thick specimen including the effects of the geometrical thickness of the specimen and dynamical scattering in the specimen [19–21]. The multislice method was originally proposed for the simulation of CTEM images in which the incident wave function is a simple plane wave. It can also be extended to the simulation of convergent beam electron diffraction patterns [22] and STEM images [23] in which the incident wave function is a convergent focused probe. 3.1. Simulation of disordered solids One further modification to the standard multislice code is needed to efficiently simulate an arbitrary ensemble of atoms: A simple crystal with stationary atoms (and no thermal vibrations) can usually be decomposed into a small number of identical slices that are repeated in some pattern as the electron wave function progresses through the specimen. For example the [1 0 0] projection of silicon contains four different layers that are repeated in the sequence abcdabcdab. . . : The traditional form of a multislice simulation would calculate the transmission function for each of the four layers and store them in memory to be used over and over again. However, when the atoms in the specimen are allowed to vibrate the specimen no longer has a simple repeating structure. Each
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layer or slice is different from every other. There is no longer any reason to precompute the slices and save them. A multislice simulation of a disordered specimen becomes dominated by the calculation of the transmission functions. This calculation must be repeated for each layer because every layer is different. There is some incentive to optimize the speed of calculation of the transmission function because there will be many of them to calculate in a disordered specimen. It is actually quicker to calculate the potential of the specimen in real space by using a look-up table for the real space potential for each single atom [24]. The total specimen potential is a simple linear summation of atomic potentials ignoring electronic rearrangement due to bonding in the solid. The traditional method of calculating of the atomic potential is to form the reciprocal space structure factor in the first Born approximation and inverse Fourier transform. The potential of each atom in the specimen has a short range in real space of no more than about 3 A˚ but extends over large angles in reciprocal space. Therefore, the real space potential only has to be calculated at a few points surrounding each atom site in the transmission function rather than filling in all of reciprocal space out to high angles as in the structure factor calculation. The speed increase scales as the ratio of the area of the layer over the area of a single atom – roughly a factor of 100 improvement for the present simulation. The atomic potentials listed by Kirkland [25] were used for the simulations in this paper. These were derived from the multiconfigurational Dirac–Fock program of Grant et al. [26]. 3.2. Simulated convergent beam electron diffraction patterns A version of the standard multislice calculation was written to accept the ðx; y; zÞ coordinates for an amorphous structure and propagate the electron wave function through a non-periodic specimen. The source code and executables for this program (autoslice) are given in Ref. [25]. A CBED pattern was simulated by starting the multislice calculation from an incident focused probe. CBED
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patterns were averaged over 16 different thermal vibrations using the frozen phonon method. The slice sizes were 512 512 pixels which sets the maximum scattering angle at 190 mrad (beyond the second HOLZ line) and ensures that the HOLZ lines are sampled on a sufficiently fine grid. The slice size was increased to 1024 1024 pixels for 1 run to check convergence, and no significant differences were noted. The slice thickness was 1:4 A˚ to ensure that all vertical periodicities were maintained. The incident probe modeled the experimental conditions of XLS for a 100 keV VG-HB501 STEM with an analytic polepiece ðCs ¼ 3:3 mm; 1100 A˚ defocus; 7:5 mrad objective aperture) [2]. The simulation size (6 6 55 cubic unit cells of silicon, which is treated as a 15 840 atom amorphous material) was chosen to reproduce the 29.9 nm thick [1 0 0] oriented sample used by XLS. A multislice simulation of 1 phonon configuration took 21 min on a 180 MHz Pentium Pro. 44 phonon configurations were used to generate each image.
4. Results 4.1. The effect of correlations The inclusion of phonons in the simulation reduces the HOLZ line scattering and creates a background of thermal diffuse scattering (Fig. 3). The thermal diffuse background (first order TDS) can be reproduced with independent atomic vibrations – these were produced by randomizing the phases in the detailed phonon model to produce uncorrelated phonon vibrations. The bands perpendicular to the Kikuchi lines (second-order TDS) requires correlations between atoms and are only reproduced by the detailed phonon model. A discussion of the various orders of TDS scattering is given in Ref. [4, p. 205]. The first-order TDS describes the root mean square (RMS) displacement of atoms from their crystal sites and is the result of incoherent thermal scattering [4,27]. ffi For silicon, the RMS displacepffiffiffiffiffiffiffiffi ˚ The higher-order TDS ment is hu2 i ¼ 0:076 A.
Fig. 3. Simulated CBED patterns of 299 A˚ of (1 0 0) silicon. The left pattern is calculated without phonons. The upper right pattern from 44 configurations of the detailed phonon model. The lower right pattern is from 44 configurations of the detailed phonon model at 297 K. where the phonon phases have been scrambled to remove correlations between atoms. Notice that the bands perpendicular to the Kikuchi lines require the correlated phonons, but the Kikuchi lines themselves do not.
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scattering describes the degree of coherence of the thermal scattering between different atomic sites. The pair correlation function for two atoms, m and n, a distance rmn apart, and for a phonon of wavelength k is [4,5,27] Siðk:rmn Þ hðum un Þ2 i 2hu2 i 1 ; ð4Þ k:rmn where SiðxÞ ¼
Z
x
du 0
sin u ; u
ð5Þ
when k:rmn 41, then hðum un Þ2 i 2hu2 i. This means the peak width in the pairffi correlation pffiffiffiffiffiffiffiffiffiffi 2 i ¼ 0:1074 A ˚ function should tend to 2hup ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (which is the uncorrelated width hu2m i þ hu2n i) at large distances. The pair correlation function for silicon at 297 K (Fig. 4) shows that this is the case. In fact, ˚ the large distances appear to be about 4 A, second-neighbor distance. Only the first peak in the pair correlation function is narrowed by the introduction of correlations between atoms. This demonstrates that when averaged over all wavelengths, only the nearest-neighbor correlations survive at room temperature even though sixth-
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neighbor effects were included in the phonon model. The nearest-neighbor correlations, as suggested by Eq. (4), lead to an oscillatory pattern imposed on the first-order TDS, with a period characteristic of the inverse of an interatomic spacing, i.e. a Bragg spacing. This means the TDS will be peaked at Bragg peaks and reduced between. The result is that a Debye–Waller factor fitted only to Bragg reflections (especially low-order reflections) will appear smaller than one fitted to all the thermal scattering – or to the ratio of HOLZ=TDS scattering. This may have an effect on the extraction of bond charges from CBED measurements if standard Debye–Waller factors are used, or flat backgrounds across the CBED disks are assumed. For instance, Nuchter et al. noted that the best fitting results were obtained when the background fitting parameter was allowed to vary from disk to disk [28]. The variations in background intensity from disk to disk, and across a single disk are roughly 10% and 1%, respectively, depending on the size of the disk, and the order of reflection – the effect is more noticeable for the (4 4 0) than the (2 2 0) disk. 4.2. Comparison to experiment
Fig. 4. The pair correlation function for silicon at 297 K, calculated using the detailed phonon model, and with the phases randomized to remove correlations between atoms. Only the first peak in the pair correlation function is narrowed by the introduction of correlations between atoms. This demonstrates that when averaged over all wavelengths, only the nearestneighbor correlations survive – even though sixth-neighbor effects were included in the phonon model.
The experimental CBED pattern for (1 0 0) Silicon from XLS [2] is shown as an inset in the CBED pattern calculated using the detailed phonon simulation in Fig. 5. Agreement in intensity and location of the diffuse scattering, including the banding perpendicular to the Kikuchi lines is excellent. This should be contrasted with the Einstein model shown in Fig. 1d. The azimuthally integrated intensity from the simulated CBED pattern is compared to the energy-filtered measurements and Einstein simulation of XLS [2] (Fig. 6). The Einstein model is fitted to match the HOLZ=TDS scattering ratio and so is in good agreement at 1 k > 2:5 A˚ . The Einstein model is missing the 1 TDS oscillations from 1 to 2:5 A˚ . The detailedphonon model reproduces the HOLZ=TDS ratio at 297 K. It also reproduces the oscillatory TDS 1 scattering at k > 1:5 A˚ . There is also a 10% 1 disagreement at k51 A˚ , in part due to scanning
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Fig. 5. Comparison of the energy-filtered CBED pattern measured by XLS [2] with the detailed phonon simulation.
Fig. 6. Scattering intensities determined by azimuthally integrating the CBED pattern of Fig. 3. Both the detailed phonon simulation and the Einstein model accurately reproduce XLSs [2] energy-filtered measurements at large scattering angles. The Einstein model does 1 not reproduce the oscillations at 1–2 A˚ , while the detailed model does. The discrepancy at the first Bragg peak is most likely due to the use of free atom charge densities in the simulation, which differs most noticeably from the charge density of crystalline silicon at small scattering angles.
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Fig. 7. Annular dark field signal as a fraction of the incident beam for different collector inner angles. The Einstein model underestimates the ratio of HOLZ=TDS scattering by 10%, though the match could be improved by adjusting the temperature. The detailed phonon simulation is within 1% of XLSs [2] energy-filtered measurements.
distortions in the experiment. Another discrepancy is at small scattering angles where the charge density in crystalline silicon departs from the free-atom potential assumed in both the simulations. Annular dark field (ADF) images are formed by collecting the electrons scattered to large angles from a scanned, focused probe. Fig. 7 shows the ADF signal that is expected as the inner angles of the ADF detector is varied. The Einstein model underestimates the HOLZ=TDS ratio by 10%. This could be improved by better fitting. The detailed phonon simulation is within 1% of XLS’s energy filtered measurements. The same is true of the uncorrelated phonon simulation. We conclude that correlations between thermal vibrations on neighboring atoms are not needed to reproduce the ADF signal, and that a properly fitted Einstein model is sufficient. A similar result was found for conventional TEM images of sub-10 nm NiAl films [15].
5. Summary The discrepancies between the simulation using a detailed phonon model and the experimental
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data are smaller than the experimental error. The only significant approximation made in the simulation is the frozen phonon approximation (the multislice method itself is an approximation, as is neglecting charge redistribution due to bonding in the solid). For scattering angles beyond 30 mrad where the errors in the experimental data are on the order of 1% or so, the frozen phonon approximation is in excellent agreement with the measured, energy filtered CBED pattern. At smaller scattering angles, the distortions in the experimental image lead to errors of about 10%. Better experimental data is needed to see just where the frozen phonon approximation breaks down. Correlations between neighboring atoms are not needed to reproduce the ADF signal, but they are needed for the CBED pattern, where they introduce an oscillatory banding in the TDS scattering. As the bands are always peaked on Bragg disks, any Debye–Waller factor that is fitted only to the Bragg Disks will be overestimated. Higher-order correlations (beyond the nearest neighbor) do not seem to be needed to model the TDS background in CBED patterns, although they can be important in selected area diffraction patterns. The Einstein model is thus sufficiently accurate to be used in the simulation of ADF images. This is a very encouraging result as it has a simple analytic expression that can be included in both Bloch wave and multislice codes. Finally, the Einstein model does not reproduce the oscillatory TDS structure (at 20–40 mrad) which may affect the background models used in CBED measurements of bond charges. We make no claims for the validity of the method when insufficient beams are included in a simulation or for bright-field images which (unlike ADF) could be very sensitive to inelastic scattering. Finally, these results are important in setting confidence limits on the underlying multislice simulations. These play an important role in understanding the changes in the electron beam profile as it propagates through a crystal. The evolution of the beam profile is in turn needed to develop a full understanding of the origins of the EELS spectra in the sample, and the resolution limits in thick samples.
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Acknowledgements This work was supported by DOE grant DEFG02-87ER45322.
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