J . Phys. Chem. 1990, 94, 1087-1090
example, we have not determined yet whether in the process of observing phonon-assisted laser operation macroscopic occupation of a phonon mode occurs and whether we are operating a phonon oscillator. High-pressure measurements, among others, may help resolve these questions. In previous high-pressure measurements on QWHs we (N.H.) with H. Drickamer and our students have been able to probe, because of easily resolved confined-particle exciton states, the “inside” of an energy band and not just the band edges.21 High-Q heat sinking of a QWH and photopumping present an opportunity to observe confined-particle transitions (recombination radiation) and phonon sidebands (not just bulk(21) Kirchoefer, S. W.; Holonyak, Jr., N.; Hess, K.; Gulino, D. A.; Drickamer, H. G.; Coleman, J. J.; Dapkus, P. D. Appl. Phys. Lett 1982, 40, 821.
1087
crystal band-edge processes) and see how they “tune” with pressure. These are experiments worth pursuing. We wish many more vigorous years for Harry Drickamer and the opportunity to do more high-pressure studies with him, including high-pressure versions of the experiments described here. Acknowledgment. We are grateful to John Bardeen for helpful discussions (N.H.) on macroscopic phonon occupation in photopumped QWHs. For technical assistance we thank B. L. Marshall, R. T. Gladin, and B. L. Payne. This work has been supported by the Army Research Office, Contract DAAL-03-89-K-0008, and by the National Science Foundation, Grants DMR 86-12860 and CDR 85-22666. Registry No. GaAs, 1303-00-0; AIGaAs, 37382- 15-3; Alo,6Gao,4As, 106804-30-2; In, 7440-74-6; sapphire, 1317-82-4.
X-ray Absorption Fine Structure Study of Amorphous Germanium under High Pressure J. Freund,* R. Ingalls, Department of Physics, University of Washington, Seattle, Washington 98195
and E. D. Crozier Physics Department, Simon Fraser University, Burnaby, B.C., Canada V5A 1S6 (Received: May 5, 1989) The X-ray absorption fine structure (XAFS) of amorphous germanium under pressures ranging from 0.0 to 8.9 GPa is investigated. Contrary to some reports from the literature we cannot find a transition to crystalline germanium at about 6 GPa. Analysis of the XAFS phases of amorphous germanium and copper, which is used as a pressure marker, allows calculation of the bulk modulus at zero pressure, Bo. of the bonds (as opposed to the bulk modulus of the bulk material, containing voids) and its pressure derivative, B,,’. They are 97 f 8 GPa and 6 f 2, respectively. Bo is 30% larger than in crystalline germanium. Fair agreement is obtained with the two- and three-body potential energy formalism of Stillinger and Weber (1985).
Introduction The high-pressure behavior of amorphous germanium (a-Ge) has been of considerable interest since the 1970s. In a series of papers the group of Minomura, Shimomura, Tamura, and collaboratorsI4 observed that a-Ge, which is semiconducting at zero pressure, becomes metallic at 6 GPa but probably remains amorphous. At 10 GPa the resistivity drops further and the structure probably changes to a white& (SSn) structure. Some years later, Minomuras-* reported that at 6-7 GPa a-Ge becomes metallic and transforms to a (distorted) white& structure or remains semiconducting with a phase change to a body-centered cubic (BC-8) or diamond (FC-2) structure (over an amorphous background). Above 10 GPa it becomes metallic and assumes the white& structure. In this paper we present strong evidence that our a-Ge sample remains completely amorphous up to 8.9 GPa. Our evidence includes both the Fourier transform of the extended X-ray absorption fine structure (EXAFS) and the shape of the X-ray absorption near-edge structure (XANES) and its first-energy derivative. Pressures higher than about 9 GPa cannot yet be ( I ) Minomura, S., et al. Tetrahedrally Bonded Amorphous Semiconductors; AIP Conf. Roc. No. 20; American Institute of Physics: New York, 1974; p 234. (2) Shimomura, O., et al. Philos. Mag. 1974, 29, 547. (3) Tamura, K., et al. The Properties of Liquid Metals, Taylor and Francis: London, 1973; p 295. (4) Tamura, K.; Fukushima, J.; Endo, H.; Asaumi, K. J. Phys. Soc. Jpn. 1914, 36, 5 5 8 . (5) Minomura, S., et al. Proceedings of the 7th International Conference on Amorphous and Liquid Semiconductors; University of Edinburgh: Edinburgh, 1977; p 53. ( 6 ) Minomura, S . High-pressure and Low-Temperature Physics; Plenum: New York, 1978; p 483. (7) Minomura, S. J . Phys. Colloq. 1981, 42, C4, Suppl. 10, 181. (8) Minomura, S.Amorphous Semiconductor Technologies and Devices, 1982; North Holland: Amsterdam, 1981; p 245.
0022-3654/90/2094-1087$02.50/0
routinely obtained with our pressure cell due to the difficulties with X-ray absorption spectroscopy, as described in our paper on copper under high p r e s ~ u r e . ~ In ref 9 we have shown that copper is an excellent pressure marker in XAFS experiments with errors of not more than 0.5 GPa. When the nearest-neighbor distance of a-Ge is extracted from the phase of the EXAFS and plotted versus pressure one is in a position to calculate the bulk modulus at zero pressure, Bo, and its first pressure derivative, B,,’, with any two-parameter isothermal equation of state. The bulk modulus we refer to here is the bulk modulus of the single Ge-Ge bonds, not the bulk modulus of the bulk material. Whereas in most materials both bulk moduli are the same, they are vastly different in amorphous materials with their variable amounts of voids. The voids in a-Ge, for example, collapse rapidly below 2 GPa, as was shown by Wu and Luo.Io Therefore, the two bulk moduli are very different in the low-pressure region. The bulk modulus of the bulk material can be calculated from measurements of the film thickness. The bond bulk modulus, on the other hand, can only be calculated from a knowledge of the compression on an atomic scale. Since EXAFS probes the short-range order it is particularly useful for this purpose. We find a bond bulk modulus that is about 30% larger than for crystalline germanium (c-Ge); i.e., a-Ge is harder than its crystalline counterpart once the voids have been expelled. Atomic potential models can be used to calculate the bond bulk modulus since it is proportional to the second spatial derivative of the potential. With the two- and three-body potential formalism of Stillinger and Weber” and the numerical values for Ge calculated by Ding and Andersen12 we find indeed that a-Ge must have a ~
~~~
(9) Freund, J.; Ingalls, R. Phys. Rev. B 1989, 39, 12537. (10) Wu, C. T.; Luo,H. L. J . Non-Cryst. Solids 1975, 18, 21. (11) Stillinger, F. H.; Weber, T. A. Phys. Rev. B 1985, 31, 5262.
0 1 9 9 0 American Chemical Society
1088 The Journal of Physical Chemistry, Vol. 94, No. 3, 1990
Freund et al.
I " " I " " I " " I " " r
0.15
I '
t
0
2
4
6
10
r (A) Figure 1 . The magnitude of the Fourier transform of k x ( k ) of amorphous germanium at 0.0 GPa (solid) and 8.9 GPa (dashed), and of crystalline germanium at 0.0 GPa (dotted) for comparison. The transform was taken over the range 4-10 A-' by using a modified 10% Gaussian window. The EXAFS phase shift has not been removed.
0.5
"
"
'
I
"
"
20
"
"
40
'
60
Figure 2. X A N E S of the K-edge of amorphous germanium at 0.0 GPa (solid) and 8.9 GPa (dashed), and of crystalline germanium at 0.0 GPa (dotted) for comparison. The step height was normalized to unity. The zero of the energy scale is the peak position. 0.3
(12) Ding, K.; Andersen, H. C. Phys. Rev. B 1986, 34, 6987. (13) Ingalls, R., et al. J . Appl. Phys. 1980, 51, 3158. (14) Bouldin, C. E. Ph.D. Thesis, University of Washington, Seattle, 1984. ( I 5) Bouldin, C.E.:Stern, E. A.; von Roedern, B.: Azoulay, J. Phys. Rev. B 1984, 30, 4462. (16) Rabe, P.; Tolkiehn, G ; Werner, A. J . Phys. C: Solid Stare Phys. 1979, 12, L545. (17) Crozier, E. D.; Seary, A. J. Can. J . Phys. 1981, 59, 876. (18) Bouldin, C. E.; Stern, E. A . EXAFS and Near Edge Structure III; Springer: Berlin, 1984; p 278. (19) Bouldin, C. E.; Stern, E. A.; von Roedern, B.; Azoulay, J. J . NonCryst. Solids 1984, 66, 105. (20) Stegemann, G.; Lengeier, B. J . Phys. Colloq. 1986, 47, C8, Suppl. 12, 407. (21) Wakagi, M.; Chigasaki, M.; Nomura, M. J. Phys. SOC.Jpn. 1987, 56,1765
"
0
Relative P h o t o n Energy (eV)
I
:
larger bond bulk modulus than c-Ge. Experimentation and Data Analysis The geometry of the high-pressure cell used in the experiments has been shown previ0us1y.l~ The a-Ge sample was obtained from B ~ u l d i n . ' ~ It J ~was prepared by sputtering Ge onto a 25 pm thick kapton substrate which was held at a constant temperature of 265 O C . The sputtering was done in an argon-filled chamber with a pressure of 7 X lo-* Pa. The deposition rate was 0.6 pm/h. Since the a-Ge film thus produced is only about 1 pm thick several layers have to be stacked in order to obtain a sufficient EXAFS signal. A copper foil of 5 pm thickness is put on top and the whole sample is embedded in soft epoxy which acts as a pressure medium. The sample of diameter 0.8 mm is centered in an inconel gasket of about 0.4-0.6 mm thickness. Three compression experiments were performed at wiggler beamline 4-1 of the Stanford Synchrotron Radiation Laboratory (SSRL) between May 1986 and November 1987. All experiments were made in transmission mode. Data analysis proceeds in the usual fashion, as described in ref 9: After background removal with a cubic spline and division by the step height the data are weighted with k1 and subjected to a Fourier transform. A Gaussian window is picked with a 10% height at k = 4.0 A-I, full height from 6.0 to 8.0 A-l, and a 10% height at 10.0 A-'. This produces one large peak centered in the vicinity of the nearest-neighbor distance of Ge which is well-defined in a-Ge and is the same as in c-Ge.1621 Then a square window is taken from about r = 1.63 to 2.52 A (depending on pressure) and the data are subjected to a Fourier back-transform. The ratio method is then applied between the zero-pressure set and a high-pressure set to extract AI?, the pressure-induced change in nearest-neighbor distance. Pressure-dependent energy shifts and third cumulants are found to be negligible, analogous to copper. The analysis of the copper EXAFS is described in ref
"
-20
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-20
I
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,
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,
,
,
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20
40
60
Relative P h o t o n Energy (eV) Figure 3. First energy derivative of the X A N E S of amorphous germanium at 0.0 GPa (solid) and 8.9 GPa (dashed), and of crystalline germanium at 0.0 GPa (dotted) for comparison.
9 and yields the pressure scale that we refer to in the following. The highest pressures reached were 7.7,8.9, and 7.1 GPa. The results presented hereinafter are representative of all three runs. In particular, we do not observe a crystallization through ageing of our sample which was already 3 years old at the time of our first experiment. Structural Investigation The Fourier transform of the EXAFS spectrum of a crystalline material exhibits a series of peaks, each peak representing one, sometimes several, neighboring shells. Amorphous materials, on the other hand, exhibit only one, at best two, peaks because in amorphous materials there is little correlation between atoms further apart. Since the phase of the EXAFS contribution from each single shell contains a k-dependent phase shift in addition to the distance term, 2kr, the peaks are not centered exactly at the distances of the respective shells, but typically at smaller values of r . Figure 1 shows the Fourier transforms of the EXAFS of a-Ge at 0.0 and 8.9 GPa and, for comparison, of c-Ge at 0.0 GPa. For both zero pressure and high pressure the Fourier transforms are those of a highly amorphous material because not even traces a r e evident of a second shell. The two little peaks accompanying the main peak are side lobes resulting from the truncation of the absorption spectrum through the kTspace window, as explained in the last section. The small bumps at higher r are noise. Figure 2 shows the XANES of the K-edge of a-Ge at 0.0 and 8.9 GPa, normalized to a step height of unity. The zero of the
The Journal of Physical Chemistry, Vol. 94, No. 3, 1990 1089
XAFS Study of Amorphous Germanium # I
0.98
l
l
l
l
l
l
,
l
l
,
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r
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from 2.77 to 5.03 g/cm3 can be explained with two parameters: pressure in the growth chamber and deposition rate. Viscor and Allan31 report that density also depends on the type of substrate. Clark et aL3, find that the formation of a quenchable high-pressure phase depends on the density of the noncrystalline Ge prior to the run. It is therefore quite reasonable to assume that highpressure phases, too, depend on the zero-pressure density. Also, it seems that any degree of crystallinity at zero pressure can be obtained by varying the temperature of the substrate at the time of deposition. I4,20.28,29,33,34
,
1 0
2
4
6
8
10
P (GPa) Figure 4. Relative volume, V/Vo, of amorphous germanium versus pressure. The solid lines are the data fits with the isothermal equations of state described in the text. The dotted line is the compression curve of crystalline germanium with Bo = 73.5 GPa and B,' = 4.4.
energy scale is the peak position. Figure 3 shows the first energy derivative of Figure 2. For comparison, both figures also show the behavior of c-Ge. The interpretation of the near-edge structure is more difficult since it defies a simple analytical treatment. A recent review on XANES was written by Durham:22 Qualitatively speaking, the XANES is sensitive to (1) the low-lying extended states which determine the electronic properties of a system, and (2) the spatial arrangement of the atoms, Le., distances and angles. It is multiple scattering that contributes to (2). EXAFS, on the other hand, is primarily due to single scattering and therefore easier to interpret. Some examples from the literature may be instructive to see how the XANES depends on the two factors cited: Rohler et show how the LIII-edgeschange when materials, such as SmS, EuPd2Si2,CeAI,, Ce, and EuO, undergo pressure-induced valence changes. Also, the pressure-induced structural phase transitions of RbCl and CuBr leave signatures on the K-edges, as shown by our g r o ~ p . ~ ~Finally, , ~ ' Sayers28and Proietti et al.29 show series of LIII-edgesand K-edges, respectively, of germanium samples at zero pressure that range from completely amorphous to completely crystalline: With increasing crystallinity the K-edge develops an increasingly complicated structure. We observe only minor changes of the a-Ge K-edge and its first pressure derivative. These changes occur rather randomly and must be attributed to variations of the absorption background rather than to real physical processes. In short, we do not observe the amorphous-to-crystalline phase transition in our sample of a-Ge. Also, judging from the fact that valence changes alter the XANES of many materials, we have reason to believe that our sample does not undergo the semiconductor-to-metal phase transition either. We believe that we get results different from Minomura's (op. cit.) probably because we used a different type of a-Ge. It is known that physical properties of a-Ge films depend on growth conditions. Tatsumi et aL30show that a startling range of densities
Bulk Modulus and Its Pressure Derivative When the changes of nearest-neighbor distance are converted to relative volume, V / Vo,and plotted versus pressure (Figure 4) an isothermal equation of state (EOS)can be fitted through the data points (solid lines). The fits are made with EOS, V
-= VO
[l - a In (1
+ bp)13
and its inverse
+
+
with a = 1/(3B,,' 1 ) and b = (3Bd 1)/3&. The advantage of working with an invertible EOS is that two values, for both Bo and B,,', are obtained which can be averaged. The differences are estimates of the errors incurred by fitting the set of data to that particular EOS. The results are Bo = 97 f 8 GPa and B,,' = 6 f 2. Similar results are obtained when other invertible two-parameter EOS are used. Literature values of Bo for c-Ge range from 72 to 75 GPa.3638 Values of B,,' for c-Ge are typically in the vicinity of 4.4,3639but a value as high as 6.3 is reported.40 Since c-Ge has no voids, these values refer to the bulk modulus of both the bonds and the bulk material. Figure 4 also shows the compression curve of c-Ge (dotted), calculated from eq l a with Bo = 73.5 GPa and B,,' =
4.4. No experimental results are available for a-Ge, neither for the bond bulk modulus nor for the bulk modulus of the bulk material. Theoretical continuous random network (CRN) model calculations are reported from Steinhardt et al.41 and G ~ t t m a n . ~C ~R N models do not contain voids. Both studies make use of the Keating potential energy formulation that consists of a two-body (bondstretching) and a three-body (mixed bond-stretching and anglebending) term. The root mean square (rms) angular distortions are about 7O and 1 l o , and the rms bond length distortions are about 1% and not specified, respectively. The studies show that the bond bulk modulus for a-Ge should be about 3 4 % lower than for c-Ge. But our Bo is about 30% higher. In order to investigate this discrepancy we make a calculation with the more recent potential energy formalism proposed by Stillinger and Weber1' and Ding and Andersen.12 If we note that each atom has four neighbors and each two-body potential energy is split among two atoms, the pair potential energy (Helmholtz (31) Viscor, P.; Allan, D. Thin Solid Films 1979, 62, 259. (32) Clark, J. B.; Dachille, F.; Shimada, M. J . Non-Cryst. Solids 1977,
(22) Durham, P. J. X-Ray Absorption, Wiley: New York, 1988; p 53. (23) Rohler, J.; Krill, G.; Kappler, J. P.; Ravet, M. F. EXAFS and Near Edge Structure; Springer: Berlin, 1983; p 213. (24) Rohler, J. EXAFS and Near Edge Structure III; Springer: Berlin, 1984; p 379. (25) Rohler, J., et al. EXAFS and Near Edge Structure III; Springer: Berlin, 1984; p 385. (26) Tranquada, J. M.; Ingalls, R.; Crozier, E. D. EXAFS and Near Edge Structure III; Springer: Berlin, 1984; p 374. (27) Tranquada, J. M.; Ingalls, R. Phys. Rev. E 1986, 34, 4267. (28) Sayers, D. E., et al. J . Non-Cryst. Solids 1985, 77,78, 237. (29) Proietti, M. G.; Mobilio, S.;Gargano, A. EXAFS and Near Edge Structure III; Springer: Berlin, 1984; p 26. (30) Tatsumi, Y . ; Honda, H.; Ikegami, K.; Naito, S.J . Phys. SOC.Jpn. 1987, 56, 2977.
-3,
1.1
LJ, IJ.
(33) Bouldin, C. E.; Stern, E. A. EXAFS and Near Edge Structure III; Springer: Berlin, 1984; p 273. (34) Johnson, G. W.; Brodie, D. E.; Crozier, E. D. Can. J . Phys., in press. (35) Freund, J.; Ingalls, R. J . Phys. Chem. Solids 1989, 50, 263. (36) Anderson, 0. L.J . Phys. Chem. Solids 1966, 27, 547. (37) Goncharova, V. A.; Chernysheva, E. V.; Voronov, F. F. Sou. Phys. Solid Stare 1983, 25, 21 18. (38) Menoni, C. S.;Hu, J. Z.; Spain, I. L. High Pressure in Science and Technology, North Holland: Amsterdam, 1984; Vol. 111, p 121. (39) Soma, T. Phys. Status Solidi ( E ) 1981, 104, 293. (40) Philip, J.; Breazeale, M. A. J . Appl. Phys. 1983, 54, 752. (41) Steinhardt, P.; Alben, R.; Weaire, D. J. Non-Cryst. Solids 1974,15, 199. (42) Guttman, L.Solid State Commun. 1977, 24, 21 1 .
Freund et al.
1090 The Journal of Physical Chemistry, Vol. 94, No. 3, I990
free energy at zero temperature) for one atom is given by
where r is in 8, and = 2.175 X lo-'* J. The superscript, (2), indicates two-body potential. From the definition of the bulk modulus, B = -V(ap/dVj, the definition of pressure, p = -(aF/aVj, and the relation dV = 47rrs2 ar,, with rs being the Wigner-Seitz radius, one obtains
Expressing r, in terms of the nearest-neighbor distance, r, of a diamond structure one obtains for the bulk modulus at zero pressure of the two-body potential energy
to three atoms, eq 6b must be multiplied by a factor of 2 again. In c-Ge the second-nearest-neighbor distance is 4.001 8, and the cosine of the angle between the two nearest neighbors is -1/3, so eq 6 is exactly zero. In a-Ge the radii and bond angles must be replaced by (Gaussian) distributions. If independence is assumed for the three variables, xi, xj, and tk, the three-body potential energy per atom becomes
4)
exp( - 2a,, exp(
-4) dx,
2ax2
The numerical value is 79.4 GPa, in good agreement with experiment. In a similar way one calculates the pressure derivative of BJ2) a
(Bd)(2) =
13 ( 1
3
~
- r?)
(4)
-
The numerical value is 2.26. The agreement with experiment is bad because the accuracy of the potential energy formulation decreases with each derivative taken. In a-Ge the radius, r, must be replaced by a (Gaussian) radius distribution. The two-body potential energy then becomes F 3.926 A, respectively. d3) = 9.564 X J per triangle. Each atom participates in six such triangles, and since each triangle belongs (43)Wooten, F.;Winer, K.; Weaire, D. Phys. Reu. Left. 1985, 54, 1392.
exp(
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-";)
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--I
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-
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2,3,1)
cyclic term (2,3,1
-
+
3,1,2) (7)
For the nearest-neighbor distances, rl and r2, the integration boundaries can be replaced by infinities and the first exponentials can be expanded so the integrals can be solved analytically. This procedure does not apply to the second-nearest-neighbor distance, r3. Since the second derivative with respect to r3is required, even a numerical solution of the x3integral is cumbersome. Therefore, we chose to solve eq 7 with a Monte Carlo integration producing f i 3 )with an accuracy of three significant figures after lo6 repetitions. This procedure was applied to nearest-neighbor distances of 2.40, 2.41, ..., 2.50 8, and the second derivative was obtained by fitting a parabola to the function f 1 3 ) (r). Estimates for uf, used by Wooten et al.43are 10.9' (Keating) and 11.4' (Weber). This gives +8.2 and +10.3 GPa, respectively, for Bd3). Application of the Stillinger-Weber potential can therefore account for an increase of the bulk modulus by 9.3-10.8 GPa when c-Ge is replaced by a-Ge. This is about one half of the observed difference. Two explanations are possible for the remaining half Firstly, the three-body potential used above may be inaccurate. Since the second spatial derivative is required, small errors in the three-body potential would be magnified in the calculation of the bulk modulus. We have already seen that the accuracy of the two-body potential is limited to the extent that only first and second spatial derivatives are reliable. Secondly, the theoretical rms distortions for bond length and bond angle used above may be too small. While EXAFS meas u r e m e n t ~ of ~ ~uX12 - ~ ~indicate that 2-395 of the bond length is a reasonable input, gXAFS is not able to estimate uf3correctly. For uat> loo, the second peak in the Fourier transform disappears. Therefore the only information EXAFS can yield is that 10' is a minimum. Acknowledgment. We thank N. Alberding, A. J. Seary, and K. R. Bauchspiess, our collaborators from Simon Fraser University, and J. E. Whitmore and B. Houser, our University of Washington group, for having done the experiments with us at SSRL. This work was supported by the Department of Energy Grants No. DE-AT06-83ER45038 and DE-FG06-84ER45 163, and by the Natural Science and Engineering Research Council of Canada. SSRL is supported by the U S . Department of Energy (Office of Basic Energy Sciences) and the National Institutes of Health (Biotechnology Research Program, Division of Research Resources). Registry No. Ge, 7440-56-4.
