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Modelling of local modes in SixGe1−x and CxSiyGe1−x−y alloys to explore the local clustering of the species
Physica B 273}274 (1999) 616}619
Modelling of local modes in Si Ge and C Si Ge x 1~x x y 1~x~y alloys to explore the local clustering of the species Simon Scarle, Alison Mainwood* Physics Department, King's College London, Strand, London, WC2R 2LS, UK
Abstract The frequencies of the local vibrational modes due to defects within group IV alloys are a!ected by the local environment of the defects; therefore by investigating known modes we get an insight into alloy #uctuations near the defects. The local modes are modelled around defects in a range of local alloy arrangements to investigate the e!ects of clustering alloy species. An interatomic potential has been "tted to phonon frequencies for the pure materials at symmetry points in k-space, to local vibrational modes of lighter elements in otherwise pure local environments and to atomic positions predicted by local density calculations for certain alloy structures. Using this potential, the frequencies of local modes of substitutional carbon are found in Si Ge with various values of x. From a stochastic analysis of the x 1~x local environs to the carbon, the modes are weighted to produce the expected local vibrational spectrum of a Si Ge x 1~x alloy. The experimental local mode line shapes in the alloys deviate su$ciently from those predicted to show that the carbon substitutes preferentially in regions of high silicon content. ( 1999 Elsevier Science B.V. All rights reserved. Keywords: SiGe; CSiGe; Local vibrational modes; Defects
1. Introduction Carbon is added in small amounts to SiGe alloys as a way of altering the average lattice constant, and thus reduce the strain in layered structures built on silicon [1]. It is also possible to grow ternary group IV alloy Si Ge C , although since the solubility of carbon in x y 1~x~y bulk silicon or germanium is very low, only highly dilute carbon alloys are in equilibrium. The solubility of carbon can be orders of magnitude higher [2] near the surface. Since the bonding of the group IV elements has the same sp3 electronic structure, but the three elements have very di!erent band gaps, once the problems of solubility and clustering of the alloy species have been overcome, band gap engineered, strain-free quantum wells and superlattices could be produced. In this paper, we show how the local segregation of species in a SiGe or SiGeC alloy can be monitored by an
* Corresponding author. Tel.: #44-(0)-171-848-2044; fax: #44-(0)-171-848-2420. E-mail address: [email protected] (A. Mainwood)
examination of the local vibrational modes (LVMs) of defects in the alloy. A comparison of the observed LVM spectrum with the prediction produced by a stochastic arrangement of the species will allow one to investigate the local segregation of the alloy species.
2. Calculating the local mode spectrum An LVM is present at a defect when the local bonds are stronger or when the defect atoms are lighter than in the perfect lattice. Therefore, substitutional carbon in otherwise pure silicon or germanium has a clear single LVM. For 12C this LVM has a frequency of 605 cm~1 in silicon [3], and 531 cm~1 in germanium [4], whilst for 13C it is at 590 cm~1 in silicon [4], and 512 cm~1 in germanium [4]. In the alloys the carbon has multiple LVMs, the frequencies of which depend on the environment about the carbon. This can be seen as a two-level e!ect. The "rst level is from the direct nearest neighbours of the carbon. We are assuming here, that due to the low solubility, no carbon has another carbon as a nearest neighbour. This leaves
0921-4526/99/$ - see front matter ( 1999 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 9 ) 0 0 5 8 7 - 6
S. Scarle, A. Mainwood / Physica B 273}274 (1999) 616}619 Table 1 The local modes produced by each local environment [4]. In local environ, Si Ge : C means a substitutional C with nearest n m neighbours of n Si's and m Ge's Local environ
Number of modes
Symmetry in T point group $
Si : C 4 Si Ge : C 3 Si Ge : C 2 2 SiGe : C 3 Ge : C 4
1 2 3 2 1
T 2 A E 1 A B B 1 2 1 A E 1 T 2
"ve possible structures for the defect, as shown in Table 1. If all the next neighbours of the carbon are of the same type denoted by Si : C or Ge : C, the carbon is in a near 4 4 tetrahedral environment and the mode is approximately triply degenerate with T symmetry * approximately 2 because the further neighbours to the carbon may break this symmetry and split the T state slightly. For mix2 tures of species neighbouring the carbon, the symmetry is lower and there will be two or three LVMs. The second level of this e!ect depends on the overall ratio of SiGe, this a!ects the above splitting only very slightly, but the masses of the further neighbours shifts the frequencies of the LVMs between the germanium and silicon extremes. These e!ects were demonstrated by Ho!man et al. [4], who observed the carbon LVM in Si Ge alloys with x 1~x x varying from 0.5 to 1, and also used an ab initio model to "nd the LVM frequencies. They were able to show the splitting due to the di!erent species as nearest neighbours, but could not demonstrate the e!ect of the more distant neighbours because of the long computer time required to "nd the frequencies for each of a large number of alloy arrangements.
3. The model A valence force potential with a large parameter set has been used. i.
Parameters for interactions between atoms of the same species have been "tted to phonon frequencies for the pure materials at symmetry points in k-space. ii. The main interaction between an atom and a neighbour of a di!erent species has been derived from LVMs of lighter atoms in otherwise pure local environments [3}5]. iii. The positions of the atoms in the alloys depend on the interatomic potential, so the positions calculated by local density approximation ab initio calculations, using a super-cell of 64 atoms of given SiGe
617
and SiC structures, were used to optimise the remaining interaction parameters [6]. To calculate the LVM expected of carbon in a particular Si Ge alloy, a super-cell of 64 atoms was used x 1~x with the arrangement of species selected randomly in the correct ratio. This was repeated for each of the possible arrangements of nearest neighbours to the carbon and for 10 di!erent random arrangements of further neighbours, leading to 50 runs for each value of x, giving 150 LVM frequencies, some of them degenerate. We assumed an intrinsic line width of 8 cm~1 taken from the experimental value for the LVM in germanium [4] and a Gaussian line shape. Thus we constructed a spectrum corresponding to each ratio. If the species were arranged randomly, the probability of the neighbours to the carbon being Si Ge in an n 4~n alloy of Si Ge would be x 1~x 4! xn (1!x)4~n. (1) n!(4!n)! We can, therefore, construct the expected spectra for the carbon LVM in Si Ge for various values of x. x 1~x 4. Results and discussion The expected LVM spectra for four silicon-rich alloys (Si Ge , x"0.95, 0.85, 0.75, 0.65) are shown in x 1~x Fig. 1. The modes due to the arrangement of species among the nearest neighbours can be distinguished clearly. A comparison with the LVM spectra reported by Ho!man et al. [4] shows a rather di!erent distribution of modes. Note that their SiGe samples were implanted with carbon, so we can assume that the site the carbon arrives at is determined by the implantation, carbon migration and substitutional processes, not by the growth of the SiGeC alloy. Ho!man's spectra shows that the peak corresponding to four silicon neighbours around the carbon is the strongest feature for 0.5(x(1, whereas we would expect from our spectra that other modes would dominate as x decreases below 0.75. This suggests that either: 1. There are regions of the alloy where the silicon and germanium are slightly segregated, or 2. Carbon substitutes preferentially into sites surrounded with silicon. The second of these suggestions is supported by Kelires [7] and Berdin et al. [8], who have shown that Ge}C bonds are less strong than Si}C or Si}Ge bonds. In addition, Yang et al. [5] show that the Ge}Ge vibrational mode in the ternary SiGeC alloy is not a!ected by the concentration of carbon, whilst the Ge}Si mode is.
618
S. Scarle, A. Mainwood / Physica B 273}274 (1999) 616}619
Fig. 2. Density of phonon states in arbitary units for a stochastic arrangement of C Si Ge with the main modes iden0.01 0.80 0.19 ti"ed.
second case, only regions with x greater than the average would be observed by monitoring the LVM spectrum of carbon. We have analysed Ho!man's spectra qualitatively in the light of our calculated spectra, near the carbon atom, their spectra show a concentration of germanium which is consistently 5}10% lower than the x values implies. There is no evidence for a matching increase in x elsewhere in the crystal, so these results support the preferential substitution of carbon into regions of high silicon content. It was not possible to determine whether these silicon-rich regions were more abundant than the stochastic analysis would predict.
5. The ternary alloy
Fig. 1. Calculated LVM spectra for 13C in Si Ge alloys, x 1~x assuming stochastic arrangements of species, with the modes identi"ed by the number of neighbours to the carbon. The values of x are the same as those used by Ho!man et al. [4].
In the "rst case, we would expect the experimental spectra to show a wider spread of x values than the stochastic predictions * that is regions with x signi"cantly both greater and smaller than average. In the
A similar investigation of the ternary alloy can be made by looking at the density of phonon states and identifying the modes associated with particular combinations of species. Fig. 2 shows an example of the density of phonon states that we predict for a stochastic ternary alloy (C Si Ge ). The Si}C LVMs can clearly be 0.01 0.80 0.19 seen, whilst the Ge}C modes are starting to be lost into the Si}Si peak. An alternative approach is to look at the LVM spectra associated with an impurity. Boron is common in silicon and its alloys. It is a shallow acceptor, which means that the excess hole has a very large orbit, and the local elastic behaviour of the boron will be very similar to that of a carbon atom with the mass of boron. Therefore, apart from adjusting the parameters for B}C and B}Si to give the LVM frequencies observed [9,10] and scaling the B}Ge parameter by the same factor as B}Si, the parameters in the potential for boron and boron}carbon are the same as those for carbon. With these assumptions, the LVM spectra for boron and carbon in stochastic arrangements of C Si 0.02 0.05
S. Scarle, A. Mainwood / Physica B 273}274 (1999) 616}619
619
overall alloy ratios. Comparisons of these spectra with those observed experimentally will allow the segregation of the alloy to be quanti"ed.
6. Conclusions We have demonstrated a method by which the local segregation of the species in group IV alloys can be investigated by means of the LVM spectra of carbon or impurities in the alloy.
Acknowledgements We thank J.E. Lowther for the results quoted as Ref. [6]. SS thanks the Engineering and Physical Sciences Research Council of the U.K. for a studentship.
References
Fig. 3. Predicted LVM spectra for 10B in C Si Ge assumx y 1~x~y ing a stochastic arrangement of species.
Ge and C Si Ge are shown in Fig. 3. The 0.93 0.02 0.15 0.83 inclusion of boron has clearly made the Ge : C peak 4 wider than in the pure alloy, and most of the intermediate modes are now hidden under the Ge : B peak. The 4 Si : C peak appears stronger due to the intermediate 4 Si Ge : B modes having similar frequencies. Although n m not shown on these graphs for reasons of clarity, modes for C}B nearest neighbours were found at around 1000 and 720 cm~1. Again, all of these features shift with
[1] R.A. Soref, Proc. IEEE 81 (1993) 1687. [2] J. Terso!, Phys. Rev. Lett. 74 (1995) 5080. [3] R.C. Newman, J.B. Willis, J. Phys. Chem. Solids 26 (1965) 373. [4] L. Ho!man, J.C. Bath, J. Lundsgaard Hansen, A. Nylandsted Larsen, B. Bech Nielsen, P. Leary, R. Jones, S. OG berg, Mater. Sci. Forum 258}263 (1998) 97. [5] B.-K. Yang, M. Krishnamurthy, W.H. Weber, J. Appl. Phys. 84 (1998) 2011. [6] J.E. Lowther, private communication. [7] P.C. Kelires, Phys. Rev. Lett. 75 (1995) 1114. [8] M.A. Berding, A. Sher, M. van Schilfgaarde, Phys. Rev. B 56 (1997) 3885. [9] S.J. Breuer, P.R. Briddon, Phys. Rev. B 49 (1994) 10332. [10] C.P. Herrero, M. Stutzman, Phys. Rev. B 38 (1998) 12688.
Modelling of local modes in Si Ge and C Si Ge x 1~x x y 1~x~y alloys to explore the local clustering of the species Simon Scarle, Alison Mainwood* Physics Department, King's College London, Strand, London, WC2R 2LS, UK
Abstract The frequencies of the local vibrational modes due to defects within group IV alloys are a!ected by the local environment of the defects; therefore by investigating known modes we get an insight into alloy #uctuations near the defects. The local modes are modelled around defects in a range of local alloy arrangements to investigate the e!ects of clustering alloy species. An interatomic potential has been "tted to phonon frequencies for the pure materials at symmetry points in k-space, to local vibrational modes of lighter elements in otherwise pure local environments and to atomic positions predicted by local density calculations for certain alloy structures. Using this potential, the frequencies of local modes of substitutional carbon are found in Si Ge with various values of x. From a stochastic analysis of the x 1~x local environs to the carbon, the modes are weighted to produce the expected local vibrational spectrum of a Si Ge x 1~x alloy. The experimental local mode line shapes in the alloys deviate su$ciently from those predicted to show that the carbon substitutes preferentially in regions of high silicon content. ( 1999 Elsevier Science B.V. All rights reserved. Keywords: SiGe; CSiGe; Local vibrational modes; Defects
1. Introduction Carbon is added in small amounts to SiGe alloys as a way of altering the average lattice constant, and thus reduce the strain in layered structures built on silicon [1]. It is also possible to grow ternary group IV alloy Si Ge C , although since the solubility of carbon in x y 1~x~y bulk silicon or germanium is very low, only highly dilute carbon alloys are in equilibrium. The solubility of carbon can be orders of magnitude higher [2] near the surface. Since the bonding of the group IV elements has the same sp3 electronic structure, but the three elements have very di!erent band gaps, once the problems of solubility and clustering of the alloy species have been overcome, band gap engineered, strain-free quantum wells and superlattices could be produced. In this paper, we show how the local segregation of species in a SiGe or SiGeC alloy can be monitored by an
* Corresponding author. Tel.: #44-(0)-171-848-2044; fax: #44-(0)-171-848-2420. E-mail address: [email protected] (A. Mainwood)
examination of the local vibrational modes (LVMs) of defects in the alloy. A comparison of the observed LVM spectrum with the prediction produced by a stochastic arrangement of the species will allow one to investigate the local segregation of the alloy species.
2. Calculating the local mode spectrum An LVM is present at a defect when the local bonds are stronger or when the defect atoms are lighter than in the perfect lattice. Therefore, substitutional carbon in otherwise pure silicon or germanium has a clear single LVM. For 12C this LVM has a frequency of 605 cm~1 in silicon [3], and 531 cm~1 in germanium [4], whilst for 13C it is at 590 cm~1 in silicon [4], and 512 cm~1 in germanium [4]. In the alloys the carbon has multiple LVMs, the frequencies of which depend on the environment about the carbon. This can be seen as a two-level e!ect. The "rst level is from the direct nearest neighbours of the carbon. We are assuming here, that due to the low solubility, no carbon has another carbon as a nearest neighbour. This leaves
0921-4526/99/$ - see front matter ( 1999 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 9 ) 0 0 5 8 7 - 6
S. Scarle, A. Mainwood / Physica B 273}274 (1999) 616}619 Table 1 The local modes produced by each local environment [4]. In local environ, Si Ge : C means a substitutional C with nearest n m neighbours of n Si's and m Ge's Local environ
Number of modes
Symmetry in T point group $
Si : C 4 Si Ge : C 3 Si Ge : C 2 2 SiGe : C 3 Ge : C 4
1 2 3 2 1
T 2 A E 1 A B B 1 2 1 A E 1 T 2
"ve possible structures for the defect, as shown in Table 1. If all the next neighbours of the carbon are of the same type denoted by Si : C or Ge : C, the carbon is in a near 4 4 tetrahedral environment and the mode is approximately triply degenerate with T symmetry * approximately 2 because the further neighbours to the carbon may break this symmetry and split the T state slightly. For mix2 tures of species neighbouring the carbon, the symmetry is lower and there will be two or three LVMs. The second level of this e!ect depends on the overall ratio of SiGe, this a!ects the above splitting only very slightly, but the masses of the further neighbours shifts the frequencies of the LVMs between the germanium and silicon extremes. These e!ects were demonstrated by Ho!man et al. [4], who observed the carbon LVM in Si Ge alloys with x 1~x x varying from 0.5 to 1, and also used an ab initio model to "nd the LVM frequencies. They were able to show the splitting due to the di!erent species as nearest neighbours, but could not demonstrate the e!ect of the more distant neighbours because of the long computer time required to "nd the frequencies for each of a large number of alloy arrangements.
3. The model A valence force potential with a large parameter set has been used. i.
Parameters for interactions between atoms of the same species have been "tted to phonon frequencies for the pure materials at symmetry points in k-space. ii. The main interaction between an atom and a neighbour of a di!erent species has been derived from LVMs of lighter atoms in otherwise pure local environments [3}5]. iii. The positions of the atoms in the alloys depend on the interatomic potential, so the positions calculated by local density approximation ab initio calculations, using a super-cell of 64 atoms of given SiGe
617
and SiC structures, were used to optimise the remaining interaction parameters [6]. To calculate the LVM expected of carbon in a particular Si Ge alloy, a super-cell of 64 atoms was used x 1~x with the arrangement of species selected randomly in the correct ratio. This was repeated for each of the possible arrangements of nearest neighbours to the carbon and for 10 di!erent random arrangements of further neighbours, leading to 50 runs for each value of x, giving 150 LVM frequencies, some of them degenerate. We assumed an intrinsic line width of 8 cm~1 taken from the experimental value for the LVM in germanium [4] and a Gaussian line shape. Thus we constructed a spectrum corresponding to each ratio. If the species were arranged randomly, the probability of the neighbours to the carbon being Si Ge in an n 4~n alloy of Si Ge would be x 1~x 4! xn (1!x)4~n. (1) n!(4!n)! We can, therefore, construct the expected spectra for the carbon LVM in Si Ge for various values of x. x 1~x 4. Results and discussion The expected LVM spectra for four silicon-rich alloys (Si Ge , x"0.95, 0.85, 0.75, 0.65) are shown in x 1~x Fig. 1. The modes due to the arrangement of species among the nearest neighbours can be distinguished clearly. A comparison with the LVM spectra reported by Ho!man et al. [4] shows a rather di!erent distribution of modes. Note that their SiGe samples were implanted with carbon, so we can assume that the site the carbon arrives at is determined by the implantation, carbon migration and substitutional processes, not by the growth of the SiGeC alloy. Ho!man's spectra shows that the peak corresponding to four silicon neighbours around the carbon is the strongest feature for 0.5(x(1, whereas we would expect from our spectra that other modes would dominate as x decreases below 0.75. This suggests that either: 1. There are regions of the alloy where the silicon and germanium are slightly segregated, or 2. Carbon substitutes preferentially into sites surrounded with silicon. The second of these suggestions is supported by Kelires [7] and Berdin et al. [8], who have shown that Ge}C bonds are less strong than Si}C or Si}Ge bonds. In addition, Yang et al. [5] show that the Ge}Ge vibrational mode in the ternary SiGeC alloy is not a!ected by the concentration of carbon, whilst the Ge}Si mode is.
618
S. Scarle, A. Mainwood / Physica B 273}274 (1999) 616}619
Fig. 2. Density of phonon states in arbitary units for a stochastic arrangement of C Si Ge with the main modes iden0.01 0.80 0.19 ti"ed.
second case, only regions with x greater than the average would be observed by monitoring the LVM spectrum of carbon. We have analysed Ho!man's spectra qualitatively in the light of our calculated spectra, near the carbon atom, their spectra show a concentration of germanium which is consistently 5}10% lower than the x values implies. There is no evidence for a matching increase in x elsewhere in the crystal, so these results support the preferential substitution of carbon into regions of high silicon content. It was not possible to determine whether these silicon-rich regions were more abundant than the stochastic analysis would predict.
5. The ternary alloy
Fig. 1. Calculated LVM spectra for 13C in Si Ge alloys, x 1~x assuming stochastic arrangements of species, with the modes identi"ed by the number of neighbours to the carbon. The values of x are the same as those used by Ho!man et al. [4].
In the "rst case, we would expect the experimental spectra to show a wider spread of x values than the stochastic predictions * that is regions with x signi"cantly both greater and smaller than average. In the
A similar investigation of the ternary alloy can be made by looking at the density of phonon states and identifying the modes associated with particular combinations of species. Fig. 2 shows an example of the density of phonon states that we predict for a stochastic ternary alloy (C Si Ge ). The Si}C LVMs can clearly be 0.01 0.80 0.19 seen, whilst the Ge}C modes are starting to be lost into the Si}Si peak. An alternative approach is to look at the LVM spectra associated with an impurity. Boron is common in silicon and its alloys. It is a shallow acceptor, which means that the excess hole has a very large orbit, and the local elastic behaviour of the boron will be very similar to that of a carbon atom with the mass of boron. Therefore, apart from adjusting the parameters for B}C and B}Si to give the LVM frequencies observed [9,10] and scaling the B}Ge parameter by the same factor as B}Si, the parameters in the potential for boron and boron}carbon are the same as those for carbon. With these assumptions, the LVM spectra for boron and carbon in stochastic arrangements of C Si 0.02 0.05
S. Scarle, A. Mainwood / Physica B 273}274 (1999) 616}619
619
overall alloy ratios. Comparisons of these spectra with those observed experimentally will allow the segregation of the alloy to be quanti"ed.
6. Conclusions We have demonstrated a method by which the local segregation of the species in group IV alloys can be investigated by means of the LVM spectra of carbon or impurities in the alloy.
Acknowledgements We thank J.E. Lowther for the results quoted as Ref. [6]. SS thanks the Engineering and Physical Sciences Research Council of the U.K. for a studentship.
References
Fig. 3. Predicted LVM spectra for 10B in C Si Ge assumx y 1~x~y ing a stochastic arrangement of species.
Ge and C Si Ge are shown in Fig. 3. The 0.93 0.02 0.15 0.83 inclusion of boron has clearly made the Ge : C peak 4 wider than in the pure alloy, and most of the intermediate modes are now hidden under the Ge : B peak. The 4 Si : C peak appears stronger due to the intermediate 4 Si Ge : B modes having similar frequencies. Although n m not shown on these graphs for reasons of clarity, modes for C}B nearest neighbours were found at around 1000 and 720 cm~1. Again, all of these features shift with
[1] R.A. Soref, Proc. IEEE 81 (1993) 1687. [2] J. Terso!, Phys. Rev. Lett. 74 (1995) 5080. [3] R.C. Newman, J.B. Willis, J. Phys. Chem. Solids 26 (1965) 373. [4] L. Ho!man, J.C. Bath, J. Lundsgaard Hansen, A. Nylandsted Larsen, B. Bech Nielsen, P. Leary, R. Jones, S. OG berg, Mater. Sci. Forum 258}263 (1998) 97. [5] B.-K. Yang, M. Krishnamurthy, W.H. Weber, J. Appl. Phys. 84 (1998) 2011. [6] J.E. Lowther, private communication. [7] P.C. Kelires, Phys. Rev. Lett. 75 (1995) 1114. [8] M.A. Berding, A. Sher, M. van Schilfgaarde, Phys. Rev. B 56 (1997) 3885. [9] S.J. Breuer, P.R. Briddon, Phys. Rev. B 49 (1994) 10332. [10] C.P. Herrero, M. Stutzman, Phys. Rev. B 38 (1998) 12688.