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Lattice relaxation around impurity atoms in semiconductors – arsenic in silicon – a comparison between experiment and theory
Nuclear Instruments and Methods in Physics Research B 200 (2003) 60–65 www.elsevier.com/locate/nimb
Lattice relaxation around impurity atoms in semiconductors – arsenic in silicon – a comparison between experiment and theory V. Koteski, N. Ivanovic 1, H. Haas, E. Holub-Krappe, H.-E. Mahnke
*
Hahn-Meitner-Institut Berlin GmbH, Bereich Strukturforschung, Glienicker Strasse 100, D-14109 Berlin, Germany
Abstract We have measured the lattice relaxation around As in Si at a homogeneous As concentration of 4 1018 cm3 by EXAFS spectroscopy. From the absorption spectra, distances up to the 4th shell could be extracted. A sizeable misfit due to an increased distance is only observed for the 1st shell. Complementing our experimental work we have performed ab initio calculations based on the density functional theory with the WIEN97 package which uses the linearised augmented plane wave method and with the FHI96md program which uses first-principles pseudo-potentials and a plane wave basis set to investigate the size dependence of the super-cells constructed around one substitutional As atom. The calculations yielded good agreement with our EXAFS experiment so that the determined relaxations can be used as a solid basis for further interpretations of derived parameters such as hyperfine interaction parameters in defect complexes. Ó 2002 Elsevier Science B.V. All rights reserved. PACS: 61.72.Tt; 61.72.)y; 61.10.Ht; 71.15.Ap; 71.15.Mb; 71.55.Cn Keywords: Lattice relaxation; Dopants in silicon; Fluorescence detected X-ray absorption; Calculations with DFT theories with LAPW and pseudo-potential methods
1. Introduction The incorporation of impurity atoms for doping of semiconductors is widely applied in the fabrication of semiconductor devices. However, in many cases, there is still a large gap between their engineered use in highly sophisticated applications * Corresponding author. Tel.: +49-30-8062-2715; fax: +4930-8062-2293. E-mail address: [email protected] (H.-E. Mahnke). 1 A Institute of Nuclear Science, P.O. On leave from VINC Box 522, 11001, Belgrade, Yugoslavia.
and the understanding of underlying processes involved in the incorporation and leading to the function as dopant. Many properties and features of incorporated impurity atoms are influenced by the lattice distortions around the impurities and lattice distortions accompanying the formation of complexes in general. Modern theoretical studies on doping problems of compound semiconductors and on defect centers therefore have to include lattice relaxations in their calculations. Experimental information on the relaxation of neighbouring atoms is contained in the the analysis of hyperfine interaction (HFI) parameters
0168-583X/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 5 8 3 X ( 0 2 ) 0 1 6 7 5 - 0
V. Koteski et al. / Nucl. Instr. and Meth. in Phys. Res. B 200 (2003) 60–65
measured in electron paramagnetic resonance or perturbed angular correlation (PAC) experiments, although in a very indirect way. The interpretation of such data depends on reliable calculations of the HFI parameters for which the precise knowledge of structural relaxation is crucial. E.g. a change of 1% in the next neighbour distance can result in a 10% different electric field gradient, the typical HFI parameter in PAC spectroscopy. A direct experimental access to lattice distortions is provided by the X-ray absorption spectroscopy (EXAFS). With todayÕs improved techniques, this method allows to measure distances of the absorbing impurity atom to the neighbouring atomic shells up to 4th order. Lattice relaxations around impurities in semiconductors have been measured up to now only in a few experiments. Calculations, with density functional approaches, typically show satisfactory agreement. The small number of experiments certainly reflects the mismatch of concentrations: the higher the impurity concentrations the easier is the detection of the absorption, in contrast to the requirements for the doping of semiconductors. In the case of Si doped with arsenic, lattice relaxation has been determined by EXAFS on bulk material in the high-concentration regime [1] and on doped epilayers [2]. A few calculations were performed following different approaches with some of them being in fair agreement with the earlier EXAFS data [3–5]. At high impurity concentrations an influence of As–As pairs or even larger clusters (e.g. As4 ) has been discussed and investigated [6]. Additionally, since As is a shallow donor, the far reaching electron wave function may influence the atomic position and depend on the concentration due to overlapping densities. It is therefore desirable to extend earlier investigations to much lower concentrations. Similarly, the results of theoretical calculations with the super-cell approach have to be checked for possible cell size dependence. In the following contribution we present new experimental results on the relaxation around As in Si with samples with a homogeneous As concentration down to 4 1018 cm3 , together with theoretical investigations with the linearised augmented plane wave (LAPW) method as implemented in the WIEN97 program package as well as with the density functional theory (DFT) code
61
with pseudo-potentials FHI96md to follow the super-cell size dependence. 2. Experiment We have measured the absorption fine structure at the K-edge of arsenic in the fluorescence yield mode at the X1 beamline of HASYLAB at DESY. The fluorescence was detected with a 5-segment Ge-detector at 90°, in line with the polarisation vector of the incoming synchrotron radiation. To reduce the influence from elastic and inelastic scattering of the incoming radiation, critical absorption was employed using a foil made out of Ge powder mixed with polyethylene. The crystals were either Czochralski-grown (CZ) Si homogeneously doped with As with a conductance of 3:5–5:5 103 X cm (corresponding to a concentration of 1:2–2 1019 cm3 ) [7] or float-zone (FZ) material with an As concentration of 4 1018 cm3 [8]. For the measurements, the samples were mounted either on a He flow or a liquid nitrogen cryostat, and the temperature was kept at approximately 17, 78 K or at room temperature. In order to check for the possible influence of Bragg reflections in the elastic scattering background, one sample was crushed to prepare a powdered sample. From the experimental absorption spectra the pure oscillatory part was obtained by subtraction of the smoothly varying atomic-like absorption and normalisation by the net increase at the edge using the AUTOBK program [9]. The resulting vðEÞ spectra, transformed into k-space, were then analysed, and Fourier transforms were formed after applying a linear weighting with k following the standard FEFF procedure [10]. An example for the original energy spectrum is given in Fig. 1. Fig. 2 illustrates the radial distribution function (magnitude of the Fourier transform) for the low-concentration As in Si sample at different temperatures. In the fit program theoretical standards are calculated for the backscattering amplitudes and phase shifts according to the given atomic arrangements. Thus, structural parameters such as inter-atomic distances, mean-square displacements, and, if not known, co-ordination numbers can thereby be extracted from fits to the experimental data.
62
V. Koteski et al. / Nucl. Instr. and Meth. in Phys. Res. B 200 (2003) 60–65
Fig. 1. X-ray absorption spectrum around the As K-edge in arsenic doped Si with a concentration of 2 1019 cm3 , taken at a temperature of approximately 18 K (original energy spectrum together with the k-transformed spectrum in the insert).
Fig. 2. Radial distribution function for the low concentration (4 1018 cm3 ) of As in Si at different temperatures as Fourier transforms of the k-spectra in the k-range between 3.8 and 11 1 . The data are shown as full lines and the best fits as dotted A . lines. The fit was performed within the range of 1.5–4.5 A
3. Theory Calculations of the structural relaxation around the As atom in silicon were performed by employing a super-cell approach within the frame-
work of the DFT. The LAPW method was used to calculate the relaxation around one substitutional As atom in a 32-atom super-cell of Si atoms. In addition, in order to study the convergence of the calculations in a more systematic way, ab initio pseudo-potential calculations were carried out with super-cell sizes of 16, 32 and 64 atoms. The diamond lattice of Si was constructed by use of a 2 2 2 super-cell in fcc, bcc, or sc structure, respectively. The experimental value of the lattice ) was used. All calcuconstant of bulk Si (5.43 A lations were performed both for the neutral and the singly positive charge state of the defect. In the LAPW method, as implemented in the WIEN97 program [11], the unit cell is divided into atom-centred muffin-tin (MT) spheres (with radii RMT ) and interstitial regions. In the interstitial region, a basis set of plane waves is used, augmented to spherical harmonics within the MT spheres. In the present calculations we have used RMT of 1.22 for As and 1.06 A for Si. The plane wave cut-off A parameter RMT Kmax was set to 7.5. The exchange and correlation effects were treated with the generalised gradient approximation [12]. The As 4s, 4p and 3d, and Si 3s and 3p states were considered as valence states. The Brillouin zone (BZ) was sampled with 64 points, reduced to 8 k points in the irreducible part of the BZ. The atoms were moved according to the Hellman–Feynman forces, with . the force convergence criterion set to 0.025 eV/A We have tested this kind of calculation previously for the isovalent impurity system of a single Zn atom replacing a Cd atom in the zinc-blende type CdTe semiconductor. Taking a super-cell of 32 atoms, i.e. Cd15 ZnTe16 , we have found perfect agreement for the nearest and next nearest shell distances around Zn between our WIEN97 calculation and our EXAFS measurement on Zn in Cd1x Znx Te with x ¼ 0:05 [13]. The pseudo-potential calculations (PP) were performed with the FHI96md package [14]. The Hamann norm-conserving pseudo-potentials [15] were used along with an energy cut-off of 18 Ry for the plane wave expansions. In contrast to the LAPW calculations, the As 3d-electrons were included in the pseudo-core. The BZ integration was carried out using the 4 4 4 Monkhorst–Pack scheme [16]. The structural relaxation was done
V. Koteski et al. / Nucl. Instr. and Meth. in Phys. Res. B 200 (2003) 60–65
Fig. 3. Relaxation of the nearest-neighbour (NN) Si atoms around the As impurity from the ideal lattice sites as function of the calculation steps during the motion towards the equilibrium position (upper part) along with the magnitude of the Hellman– Feynman forces that act upon the NN atoms (lower part). The negative values of the forces mean that they are directed toward the As atom. The results are given for the positively charged 64atom super-cell.
according to the damped Newton scheme. Within this scheme the damping and the mass parameters of the ionic motion are chosen in such a way that the ions move towards their equilibrium distances in an oscillatory-like motion, which can be seen in Fig. 3. In this calculation, the force convergence . criterion was set to 0.005 eV/A The positions of all atoms were allowed to relax, keeping the symmetry around the As impurity atom (Td point group) fixed. This assumption was tested on a super-cell of 32 atoms, allowing relaxation without symmetry constrains. Starting with atomic positions slightly randomised to avoid any predetermination of symmetry, the total energy minimisation with respect to the atomic positions restored the Td symmetry around the As atom.
of free parameters in the fit. As variables we have used: the energy shift parameter E0As , the passive 2 electron amplitude reduction factor S0As , and the 2 mean-square displacements rAsSi for the various shells. The path distances RAsSi up to the 4th shell (the 4th shell distance was set fixed to the pure Si value) were chosen as common to the different temperature sets, since the lattice-constant variation of Si between room temperature and He temperature is negligible within our accuracy. The co-ordination numbers for each shell were set according to the Si diamond structure. The three largest contributions from multiple scattering paths were included with fixed values for the paths lengths. Concentrating on the geometry, the results for the distances from the central As atom to the 1st, 2nd and 3rd Si shell are summarised in Table 1. The errors given are slightly increased over the actual errors from the fits to allow for variations e.g. in the limits of the ranges in k- and r-space. As is easily realised when the distances are compared to the pure values of Si, there is a sizeable misfit for the 1st shell with distance increase of more than 3%, while the higher shells are not affected. We have not found a significant difference in the different concentrations, and also not for the different temperatures at which the samples were measured. Since our concentrations are lower by more than one order of magnitude compared to the ones used in [1], we can safely state to be far away from possible influences due to pair or cluster formation of arsenic.
Table 1 Nearest-neighbour (RNN ), next nearest-neighbour (RNNN ) and second next nearest neighbour (RNNNN ) distances in pure Si (2nd row), around As in Si as determined from our EXAFS experiment (3rd row) and, as representative for the calculations (see Table 2), values obtained with the pseudo-potential code with a 64-atom positively charged super-cell around As (right row)
4. Results and discussion Measurements of absorption spectra at different temperatures enabled us to perform simultaneous fits to the data which helped to reduce the number
63
Si 2.351 3.840 4.502 . All values are given in A RNN RNNN RNNNN
Si:As
PP64þ
2.43(1) 3.87(1) 4.53(2)
2.42 3.85 4.51
64
V. Koteski et al. / Nucl. Instr. and Meth. in Phys. Res. B 200 (2003) 60–65
Table 2 Nearest-neighbour and next nearest-neighbour As–Si distances (RNN , RNNN ) and the relaxation energy Erel (the total energy difference between the relaxed and unrelaxed configuration) as calculated for the different super-cell sizes and charge states Supercell size
16-fcc 32-bcc 64-sc
RAs–As ) (A
7.68 9.40 10.86
PP ) R0NN (A
Rþ NN (A)
R0NNN ) (A
Rþ NNN ) (A
2.429 2.424 2.421
2.419 2.412 2.416
3.840 3.848 3.851
3.840 3.848 3.852
LAPW ) R 0 (A
Rþ NN (A)
R0NNN ) (A
Rþ NNN ) (A
2.427
2.421
3.849
3.851
NN
PP 0 (eV) Erel
þ Erel (eV)
0.20 0.12 0.14
0.19 0.11 0.13
The distance between two impurity atoms RAs–As is also given as a function of the super-cell size.
Matters of concern in modeling the relaxation using super-cell methods are the convergence in terms of force limits and the convergence with respect to cell size. Charging the super-cell and compensating the extra charge in the background charge also allows to perform the calculation for the positively charged As atom and compare it with the result for the neutral defect. Since the WIEN program puts more emphasis on the calculation of derived quantities such as electric field gradients, it is based on the highly accurate allelectron LAPW method, quite demanding from the computational point of view. The convergence towards zero force was therefore tested within the FHI96md procedure, as is illustrated in Fig. 3, and the test on the cell size dependence was performed with the much faster pseudo-potentials approach as well. The results are summarised in Table 2. There is obviously full agreement for the two calculations with the same super-cell size. The major difference in treating the atomic orbitals at the As atom, namely the treatment of the 3d electrons either as core states or within the valence band, has no influence on the atomic positions. Somewhat surprising is the small difference for a positively charged or a neutral As atom. This could explain why we did not find a significant change when going from room temperature down to 18 K. In summary, we have determined the lattice relaxation around the As dopant atom in Si at a concentration level where influences due to pairing or clustering can be safely neglected. Calculations with super-cell methods are in good agreement with our experimental results. One can thus use with confidence such calculations in the interpretation of measured properties for more complicated defect complexes.
Acknowledgements This work was partly supported by the Bundesminister f€ ur Bildung und Forschung under the grant 03MK4HMI8. The authors are grateful to the HASYLAB staff at DESY, in particular to N. Haack and E. Welter. We thank H. Rossner for his advice on analysis and fitting of the data. We further thank J. Weber from the TU Dresden for providing us with samples and for fruitful discussions on the problem of lattice relaxation around dopants in Si, and P. Becker from the PTB Braunschweig for letting us use one of his samples matching our concentration needs. Finally, we gratefully acknowledge the help of G. Schwarz from the FHI Berlin in setting up the input parameters for the FHI96md computer code.
References [1] A. Erbil, W. Weber, G.S. Cargill, R.F. Boehme, Phys. Rev. B 34 (1986) R1392. [2] S. Wei, H. Oyanagi, H. Kawanami, K. Sakamoto, T. Sakamoto, K. Tamura, N. Saini, K. Uosaku, J. Appl. Phys. 82 (1997) 4810. [3] M. Scheffler, Physica 146B (1987) 176. [4] F. Bechstedt, W. Harrison, Phys. Rev. B 39 (1989) 5041. [5] A. Settels, Diss. RWTH Aachen 2000, Berichte des FZ J€ ulich, J€ ul-3731. [6] K.C. Pandey, A. Erbil, G.S. Cargill, R.F. Boehme, D. Vanderbilt, Phys. Rev. Lett. 61 (1988) 1282. [7] P. Becker, Physikalisch-Technische Baudesanstalt, D38116 Braunschweig, private communication. [8] J. Bollmann, J. Weber, TU Dresden, Institut f€ ur Tuftemperaturphysik, D-01062 Dresden, private communication. [9] M. Newville, P. Livins, Y. Yacoby, J.J. Rehr, E.A. Stern, Phys. Rev. B 47 (1993) 14126.
V. Koteski et al. / Nucl. Instr. and Meth. in Phys. Res. B 200 (2003) 60–65 [10] J.J. Rehr, J. Mustre de Leon, S.I. Zabinsky, R.C. Albers, J. Am. Chem. Soc. 113 (1991) 5135. [11] P. Blaha, K. Schwarz, J. Luitz, WIEN97, Karlheinz Schwarz, Techn. Univ. Wien, Austria, 1999, ISBN 39501031-0-4. [12] J.P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 77 (1996) 3865.
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[13] V. Koteski, B. Reinhold, H. Haas, E. Holub-Krappe, H.-E. Mahnke, D. Wruck, Hyperfine Interactions 136/137 (2001) 681. [14] M. Bockstedte, A. Kley, J. Neugebauer, M. Scheffler, Comp. Phys. Comm. 107 (1997) 187. [15] D.R. Hamann, Phys. Rev. B 40 (1989) 2980. [16] H.J. Monkhorst, J.D. Pack, Phys. Rev. B 13 (1976) 5188.
Lattice relaxation around impurity atoms in semiconductors – arsenic in silicon – a comparison between experiment and theory V. Koteski, N. Ivanovic 1, H. Haas, E. Holub-Krappe, H.-E. Mahnke
*
Hahn-Meitner-Institut Berlin GmbH, Bereich Strukturforschung, Glienicker Strasse 100, D-14109 Berlin, Germany
Abstract We have measured the lattice relaxation around As in Si at a homogeneous As concentration of 4 1018 cm3 by EXAFS spectroscopy. From the absorption spectra, distances up to the 4th shell could be extracted. A sizeable misfit due to an increased distance is only observed for the 1st shell. Complementing our experimental work we have performed ab initio calculations based on the density functional theory with the WIEN97 package which uses the linearised augmented plane wave method and with the FHI96md program which uses first-principles pseudo-potentials and a plane wave basis set to investigate the size dependence of the super-cells constructed around one substitutional As atom. The calculations yielded good agreement with our EXAFS experiment so that the determined relaxations can be used as a solid basis for further interpretations of derived parameters such as hyperfine interaction parameters in defect complexes. Ó 2002 Elsevier Science B.V. All rights reserved. PACS: 61.72.Tt; 61.72.)y; 61.10.Ht; 71.15.Ap; 71.15.Mb; 71.55.Cn Keywords: Lattice relaxation; Dopants in silicon; Fluorescence detected X-ray absorption; Calculations with DFT theories with LAPW and pseudo-potential methods
1. Introduction The incorporation of impurity atoms for doping of semiconductors is widely applied in the fabrication of semiconductor devices. However, in many cases, there is still a large gap between their engineered use in highly sophisticated applications * Corresponding author. Tel.: +49-30-8062-2715; fax: +4930-8062-2293. E-mail address: [email protected] (H.-E. Mahnke). 1 A Institute of Nuclear Science, P.O. On leave from VINC Box 522, 11001, Belgrade, Yugoslavia.
and the understanding of underlying processes involved in the incorporation and leading to the function as dopant. Many properties and features of incorporated impurity atoms are influenced by the lattice distortions around the impurities and lattice distortions accompanying the formation of complexes in general. Modern theoretical studies on doping problems of compound semiconductors and on defect centers therefore have to include lattice relaxations in their calculations. Experimental information on the relaxation of neighbouring atoms is contained in the the analysis of hyperfine interaction (HFI) parameters
0168-583X/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 5 8 3 X ( 0 2 ) 0 1 6 7 5 - 0
V. Koteski et al. / Nucl. Instr. and Meth. in Phys. Res. B 200 (2003) 60–65
measured in electron paramagnetic resonance or perturbed angular correlation (PAC) experiments, although in a very indirect way. The interpretation of such data depends on reliable calculations of the HFI parameters for which the precise knowledge of structural relaxation is crucial. E.g. a change of 1% in the next neighbour distance can result in a 10% different electric field gradient, the typical HFI parameter in PAC spectroscopy. A direct experimental access to lattice distortions is provided by the X-ray absorption spectroscopy (EXAFS). With todayÕs improved techniques, this method allows to measure distances of the absorbing impurity atom to the neighbouring atomic shells up to 4th order. Lattice relaxations around impurities in semiconductors have been measured up to now only in a few experiments. Calculations, with density functional approaches, typically show satisfactory agreement. The small number of experiments certainly reflects the mismatch of concentrations: the higher the impurity concentrations the easier is the detection of the absorption, in contrast to the requirements for the doping of semiconductors. In the case of Si doped with arsenic, lattice relaxation has been determined by EXAFS on bulk material in the high-concentration regime [1] and on doped epilayers [2]. A few calculations were performed following different approaches with some of them being in fair agreement with the earlier EXAFS data [3–5]. At high impurity concentrations an influence of As–As pairs or even larger clusters (e.g. As4 ) has been discussed and investigated [6]. Additionally, since As is a shallow donor, the far reaching electron wave function may influence the atomic position and depend on the concentration due to overlapping densities. It is therefore desirable to extend earlier investigations to much lower concentrations. Similarly, the results of theoretical calculations with the super-cell approach have to be checked for possible cell size dependence. In the following contribution we present new experimental results on the relaxation around As in Si with samples with a homogeneous As concentration down to 4 1018 cm3 , together with theoretical investigations with the linearised augmented plane wave (LAPW) method as implemented in the WIEN97 program package as well as with the density functional theory (DFT) code
61
with pseudo-potentials FHI96md to follow the super-cell size dependence. 2. Experiment We have measured the absorption fine structure at the K-edge of arsenic in the fluorescence yield mode at the X1 beamline of HASYLAB at DESY. The fluorescence was detected with a 5-segment Ge-detector at 90°, in line with the polarisation vector of the incoming synchrotron radiation. To reduce the influence from elastic and inelastic scattering of the incoming radiation, critical absorption was employed using a foil made out of Ge powder mixed with polyethylene. The crystals were either Czochralski-grown (CZ) Si homogeneously doped with As with a conductance of 3:5–5:5 103 X cm (corresponding to a concentration of 1:2–2 1019 cm3 ) [7] or float-zone (FZ) material with an As concentration of 4 1018 cm3 [8]. For the measurements, the samples were mounted either on a He flow or a liquid nitrogen cryostat, and the temperature was kept at approximately 17, 78 K or at room temperature. In order to check for the possible influence of Bragg reflections in the elastic scattering background, one sample was crushed to prepare a powdered sample. From the experimental absorption spectra the pure oscillatory part was obtained by subtraction of the smoothly varying atomic-like absorption and normalisation by the net increase at the edge using the AUTOBK program [9]. The resulting vðEÞ spectra, transformed into k-space, were then analysed, and Fourier transforms were formed after applying a linear weighting with k following the standard FEFF procedure [10]. An example for the original energy spectrum is given in Fig. 1. Fig. 2 illustrates the radial distribution function (magnitude of the Fourier transform) for the low-concentration As in Si sample at different temperatures. In the fit program theoretical standards are calculated for the backscattering amplitudes and phase shifts according to the given atomic arrangements. Thus, structural parameters such as inter-atomic distances, mean-square displacements, and, if not known, co-ordination numbers can thereby be extracted from fits to the experimental data.
62
V. Koteski et al. / Nucl. Instr. and Meth. in Phys. Res. B 200 (2003) 60–65
Fig. 1. X-ray absorption spectrum around the As K-edge in arsenic doped Si with a concentration of 2 1019 cm3 , taken at a temperature of approximately 18 K (original energy spectrum together with the k-transformed spectrum in the insert).
Fig. 2. Radial distribution function for the low concentration (4 1018 cm3 ) of As in Si at different temperatures as Fourier transforms of the k-spectra in the k-range between 3.8 and 11 1 . The data are shown as full lines and the best fits as dotted A . lines. The fit was performed within the range of 1.5–4.5 A
3. Theory Calculations of the structural relaxation around the As atom in silicon were performed by employing a super-cell approach within the frame-
work of the DFT. The LAPW method was used to calculate the relaxation around one substitutional As atom in a 32-atom super-cell of Si atoms. In addition, in order to study the convergence of the calculations in a more systematic way, ab initio pseudo-potential calculations were carried out with super-cell sizes of 16, 32 and 64 atoms. The diamond lattice of Si was constructed by use of a 2 2 2 super-cell in fcc, bcc, or sc structure, respectively. The experimental value of the lattice ) was used. All calcuconstant of bulk Si (5.43 A lations were performed both for the neutral and the singly positive charge state of the defect. In the LAPW method, as implemented in the WIEN97 program [11], the unit cell is divided into atom-centred muffin-tin (MT) spheres (with radii RMT ) and interstitial regions. In the interstitial region, a basis set of plane waves is used, augmented to spherical harmonics within the MT spheres. In the present calculations we have used RMT of 1.22 for As and 1.06 A for Si. The plane wave cut-off A parameter RMT Kmax was set to 7.5. The exchange and correlation effects were treated with the generalised gradient approximation [12]. The As 4s, 4p and 3d, and Si 3s and 3p states were considered as valence states. The Brillouin zone (BZ) was sampled with 64 points, reduced to 8 k points in the irreducible part of the BZ. The atoms were moved according to the Hellman–Feynman forces, with . the force convergence criterion set to 0.025 eV/A We have tested this kind of calculation previously for the isovalent impurity system of a single Zn atom replacing a Cd atom in the zinc-blende type CdTe semiconductor. Taking a super-cell of 32 atoms, i.e. Cd15 ZnTe16 , we have found perfect agreement for the nearest and next nearest shell distances around Zn between our WIEN97 calculation and our EXAFS measurement on Zn in Cd1x Znx Te with x ¼ 0:05 [13]. The pseudo-potential calculations (PP) were performed with the FHI96md package [14]. The Hamann norm-conserving pseudo-potentials [15] were used along with an energy cut-off of 18 Ry for the plane wave expansions. In contrast to the LAPW calculations, the As 3d-electrons were included in the pseudo-core. The BZ integration was carried out using the 4 4 4 Monkhorst–Pack scheme [16]. The structural relaxation was done
V. Koteski et al. / Nucl. Instr. and Meth. in Phys. Res. B 200 (2003) 60–65
Fig. 3. Relaxation of the nearest-neighbour (NN) Si atoms around the As impurity from the ideal lattice sites as function of the calculation steps during the motion towards the equilibrium position (upper part) along with the magnitude of the Hellman– Feynman forces that act upon the NN atoms (lower part). The negative values of the forces mean that they are directed toward the As atom. The results are given for the positively charged 64atom super-cell.
according to the damped Newton scheme. Within this scheme the damping and the mass parameters of the ionic motion are chosen in such a way that the ions move towards their equilibrium distances in an oscillatory-like motion, which can be seen in Fig. 3. In this calculation, the force convergence . criterion was set to 0.005 eV/A The positions of all atoms were allowed to relax, keeping the symmetry around the As impurity atom (Td point group) fixed. This assumption was tested on a super-cell of 32 atoms, allowing relaxation without symmetry constrains. Starting with atomic positions slightly randomised to avoid any predetermination of symmetry, the total energy minimisation with respect to the atomic positions restored the Td symmetry around the As atom.
of free parameters in the fit. As variables we have used: the energy shift parameter E0As , the passive 2 electron amplitude reduction factor S0As , and the 2 mean-square displacements rAsSi for the various shells. The path distances RAsSi up to the 4th shell (the 4th shell distance was set fixed to the pure Si value) were chosen as common to the different temperature sets, since the lattice-constant variation of Si between room temperature and He temperature is negligible within our accuracy. The co-ordination numbers for each shell were set according to the Si diamond structure. The three largest contributions from multiple scattering paths were included with fixed values for the paths lengths. Concentrating on the geometry, the results for the distances from the central As atom to the 1st, 2nd and 3rd Si shell are summarised in Table 1. The errors given are slightly increased over the actual errors from the fits to allow for variations e.g. in the limits of the ranges in k- and r-space. As is easily realised when the distances are compared to the pure values of Si, there is a sizeable misfit for the 1st shell with distance increase of more than 3%, while the higher shells are not affected. We have not found a significant difference in the different concentrations, and also not for the different temperatures at which the samples were measured. Since our concentrations are lower by more than one order of magnitude compared to the ones used in [1], we can safely state to be far away from possible influences due to pair or cluster formation of arsenic.
Table 1 Nearest-neighbour (RNN ), next nearest-neighbour (RNNN ) and second next nearest neighbour (RNNNN ) distances in pure Si (2nd row), around As in Si as determined from our EXAFS experiment (3rd row) and, as representative for the calculations (see Table 2), values obtained with the pseudo-potential code with a 64-atom positively charged super-cell around As (right row)
4. Results and discussion Measurements of absorption spectra at different temperatures enabled us to perform simultaneous fits to the data which helped to reduce the number
63
Si 2.351 3.840 4.502 . All values are given in A RNN RNNN RNNNN
Si:As
PP64þ
2.43(1) 3.87(1) 4.53(2)
2.42 3.85 4.51
64
V. Koteski et al. / Nucl. Instr. and Meth. in Phys. Res. B 200 (2003) 60–65
Table 2 Nearest-neighbour and next nearest-neighbour As–Si distances (RNN , RNNN ) and the relaxation energy Erel (the total energy difference between the relaxed and unrelaxed configuration) as calculated for the different super-cell sizes and charge states Supercell size
16-fcc 32-bcc 64-sc
RAs–As ) (A
7.68 9.40 10.86
PP ) R0NN (A
Rþ NN (A)
R0NNN ) (A
Rþ NNN ) (A
2.429 2.424 2.421
2.419 2.412 2.416
3.840 3.848 3.851
3.840 3.848 3.852
LAPW ) R 0 (A
Rþ NN (A)
R0NNN ) (A
Rþ NNN ) (A
2.427
2.421
3.849
3.851
NN
PP 0 (eV) Erel
þ Erel (eV)
0.20 0.12 0.14
0.19 0.11 0.13
The distance between two impurity atoms RAs–As is also given as a function of the super-cell size.
Matters of concern in modeling the relaxation using super-cell methods are the convergence in terms of force limits and the convergence with respect to cell size. Charging the super-cell and compensating the extra charge in the background charge also allows to perform the calculation for the positively charged As atom and compare it with the result for the neutral defect. Since the WIEN program puts more emphasis on the calculation of derived quantities such as electric field gradients, it is based on the highly accurate allelectron LAPW method, quite demanding from the computational point of view. The convergence towards zero force was therefore tested within the FHI96md procedure, as is illustrated in Fig. 3, and the test on the cell size dependence was performed with the much faster pseudo-potentials approach as well. The results are summarised in Table 2. There is obviously full agreement for the two calculations with the same super-cell size. The major difference in treating the atomic orbitals at the As atom, namely the treatment of the 3d electrons either as core states or within the valence band, has no influence on the atomic positions. Somewhat surprising is the small difference for a positively charged or a neutral As atom. This could explain why we did not find a significant change when going from room temperature down to 18 K. In summary, we have determined the lattice relaxation around the As dopant atom in Si at a concentration level where influences due to pairing or clustering can be safely neglected. Calculations with super-cell methods are in good agreement with our experimental results. One can thus use with confidence such calculations in the interpretation of measured properties for more complicated defect complexes.
Acknowledgements This work was partly supported by the Bundesminister f€ ur Bildung und Forschung under the grant 03MK4HMI8. The authors are grateful to the HASYLAB staff at DESY, in particular to N. Haack and E. Welter. We thank H. Rossner for his advice on analysis and fitting of the data. We further thank J. Weber from the TU Dresden for providing us with samples and for fruitful discussions on the problem of lattice relaxation around dopants in Si, and P. Becker from the PTB Braunschweig for letting us use one of his samples matching our concentration needs. Finally, we gratefully acknowledge the help of G. Schwarz from the FHI Berlin in setting up the input parameters for the FHI96md computer code.
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