- Email: [email protected]
Calculation of phonon density of states for amorphous Si
Solid State Communications, Vol. 50, No. 4, pp. 367-370, 1984. Printed in Great Britain.
0038-1098/84 $3.00 + .00 Pergamon Press Ltd.
CALCULATION OF PHONON DENSITY OF STATES FOR AMORPHOUS Si N. Ishli Department of Electrical Engineering and Electronic Engineering, Fukui Institute of Technology, Fukui 910, Japan and M. Kumeda and T. Shimizu Department of Electronics, Faculty of Technology, Kanazawa University, Kanazawa 920, Japan
(Received 21 January 1984 by H. Kawamura) The phonon density of states is calculated for amorphous $1 clusters with various bond angle and bond length fluctuations. From these results we conclude that the broadening of the width of the TO-like peak is due to the bond angle fluctuation, whereas the bond length fluctuation has a rather small effect on the phonon density of states. 1. INTRODUCTION v(r,e)
RAMAN SCATTERING has been used to investigate the structural disorder in amorphous Si (a-S0 [1-3] and revealed to be a useful tool for obtaining an extent of the structural disorder in amorphous network [3, 4]. There is still confusions, however, about what kind of disorder, such as the bond angle fluctuation, the bond length fluctuation or the topological change, affects most importantly the shape of the Raman scattering spectra in amorphous materials. Since the k-selection rule breaks down for amorphous materials, the Raman scattering spectra roughly correspond to the phonon density of states (PDOS). So calculations of the PDOS for various structures are useful to answer the above question. Meek concluded from theoretical calculations for amorphous Ge that the shape of the TO-like peak of the PDOS depends on the topological change and the bond angle fluctuation [5]. However, it seems that the influence of the bond length fluctuation on the shape of the TO-like peak can not be excluded from his results. In the present work we calculate the PDOS for Si clusters with various bond angle and bond length fluctuations and examine the relation between the PDOS and the kind of disorder in amorphous network. 2. CALCULATION METHOD We constructed an a-Si cluster with balls and sticks and read the three-dimensional coordinates of the centers of the balls. In order to relax the hand-made a-Si clusters, we minimized the Keating potential V(KP) [61, 367
=
y_
3 a
R
-
d2) 2
I,A
+~_ ~ 3 /~ (RI/," + ~ d 2 ) 2, I (zx,A') 8 ~ RIA'
(1)
where cxand 13are the force constants, d is the bond length of c-S1 (2.35 A). RI,~ is the position vector from an atom I to the atom IA bonding with the atom I. The properties of the model clusters used for calculations are listed in Table 1. Cluster A is a crystalline one. Clusters B and C were relaxed from the same handmade cluster by using/3/t~ = 0.15 and 0.01, respectively. Cluster D was relaxed from another hand-made cluster by using B/a = 0.01. We solved the following secular equation in order to obtain the phonon frequency, ~ , IMco21--DI = 0,
(2)
where M is the mass of a Si atom, 1 is a unit matrix, and D is a dynamical matrix, which is obtained from an exact dynamic matrix, Dkt:Kp ~ a 2 V(KP) u~ ) - 0q~0q~
allq=0,
(3)
by omitting the terms proportional to (Rye. R m -- d 2) and (RIA" RIA' + d2/3) [7], where q~ is the k component of the displacement of the atom I. The reason why we omitted these terms is as follows: When we diagonalized the D(KP) for the clusters obtained by minimizing V(KP), the 6-fold degenerate solution of co -- 0 which should correspond to the translational and rotational freedoms of the model cluster could not be obtained in some cases. It seems to be due to
368
CALCULATION OF PHONON DENSITY OF STATES FOR AMORPHOUS Si
Table 1. Model clusters and their properties used for calculations The number of total Si atoms
Cluster
A
123 110 110 120
B
C D
6~
i
!
5
"
,
Bond length fluctuation (percent)
Bond angle fluctuation (percent)
0 1.4 0.2 0.4
0 6.3 5.8 10.0
!
i
c-Si CALC. - EXR TA
"
'
Uj"
'=1
,
200
300
400
500
600
Fig. 2. The PDOS's calculated for a crystalline Si cluster with 123 atoms. The solid curve is one calculated by using equation (5) and the dot-and-dash curve is one without the correction of the surface effect. The PDOS calculated for a perfect Si crystal as shown in Fig. 1 is drawn again for the sake of comparison by the dotted curve. These three PDOS's, p (~o)'s, are normalized for f~ p (co) do: to be equal.
TO
A
500
neighbor atoms (atoms without dangling bonds) in the cluster. The sum Zj is over all atoms in the cluster. The surface effect on the calculated results is considerably excluded by using this equation as shown below.
i
600
(c° --} a6°')2 2 ,
Y
TA
FREQUENCY (cm-I)
(4)
where a = 8 cm -1 and Ni is a weight of the ith state. All Ni's are usually set equal to 1. However, in order to exclude the surface effect on the calculated results, we used Ni calculated by equation (5),
J
TO
05 0 (ZI ¢1
incomplete relaxation of the cluster. The complete relaxation requires long CPU time. The difference between the PDOS's obtained by using D(KP) and D is small. Therefore, we used D instead of D(KP). In order to obtain the PDOS, p (co), from the calculated phonon frequency, ~i, we used the following equation;
' Y ' us
PERFECT CRYSTAL
o
Fig. 1. The PDOS for crystalline Si. The solid curve is the calculated PDOS by choosing the force constants as ot = 4.85 x 104 dyne cm -1 [6] and/$ = O.15oL. The dotted curve zs one obtained experimentally [8].
-=
SURFACE CORRECTION 4
£
:/ "/ i
p(co) = ~{ Ni. exp
I
. . . . . NO SURFACE CORRECTION
1 1
200 300 400 FREQUENCY (cm-1 )
I
c-Si
I00
H
LO
I
C
I
LA
I
g
'
100
I
Vol. 50, No. 4
(5)
J
where u~ is the displacement of atom J in ith vibrational mode and the sum Z~ is over atoms with four nearest
3. RESULTS A N D DISCUSSION We first calculated the PDOS for crystalline Si. We used a = 4.85 x 104 dyne cm -1 [6] and ~/ot = 0.15 and calculated the phonon spectra for about 1000 unequal points in the first Brillouin zone. Since the calculated PDOS fairly well reproduces the experimental one [8] as shown in Fig. 1, these values of ct and/~ are used in the later calculations. Since the surface of a model amorphous cluster may affect the calculated PDOS when the cluster size is small, a question arises how large cluster size is necessary to obtain the reliable result. In order to check tins surface effect, we calculated the PDOS for a crystalline Si cluster with 123 atoms (cluster A in Table 1) by two different methods. One is carried out by using equation (5) and the other by setting all Ni's in equation (4) equal. The PDOS calculated by these two methods are shown in Fig. 2 together with one for a perfect Si crystal calculated above. It is found from Fig. 2 that the PDOS for the crystalline cluster with 123 atoms reproduces four peaks in the PDOS for the perfect crystal and that the use of equation (5) for calculating the PDOS considerably improves the surface effect (the shift of the
Vol. 50, No. 4
CALCULATION OF PHONON DENSITY OF STATES FOR AMORPHOUS Si 3
6
I
I
I
369
I
I
a-Si
=¢2
--
CLUSTER B
•
-- CLUSTER C
TO o
~/~ f
o
I
CLUSTER A (c-S=)
4
ft.
CLUSTER B (a-Sl)
~3 3
100
O 1,0 0 O_
200 300 400 FREQUENCY (cm -I)
500
600
TO .
I00
TA
Fig. 4. Comparison of the PDOS's for a-Si clusters with different amounts of the bond length fluctuation. The solid and dotted curves are for clusters B and C in Table 1, respectively. The normalization of the PDOS's is the same as for Fig. 2.
LO
200 300 400 FREQUENCY (cm-I)
500
600 I
Fig. 3. Comparison of the PDOS's for a-S1 cluster (cluster B in Table 1) and a crystalline Si cluster (cluster A in Table 1). The solid curve is for cluster B. The dotted curve is for cluster A (already shown by the solid curve in Fig. 2). The normalization of the PDOS's is the same as for Fig. 2.
I
CLUSTER B
"E
I
I
a-Si
I
TO
!
TA-like peak to low frequency region and the increase of the TA-like peak intensity comparing with the TOlike peak intensity). Accordingly, about 100 atoms are considered to be sufficient for the cluster to avoid the surface effect if we make use of equation (5). We used equation (5) in the following calculations for a-Si clusters with about 100 atoms. The PDOS for cluster B whose structural fluctuations are of typical values for a-Si obtained experimentally is shown in Fig. 3 together with one for cluster A. A remarkable difference between them exists on the shape of the TO-like peaks, although the positions of these peaks do not change. What is the origin of this difference? The bond length fluctuation hardly affects the PDOS of a-Si, because the difference between the PDOS for cluster B and that for cluster C is rather small as shown m Fig. 4, where clusters B and C have the same topology but their bond length fluctuations are very different as shown in Table 1. We next compare cluster B with cluster D m Table 1. The bond angle fluctuation of cluster D is about 1.6 times larger than that of cluster B and the topology of cluster D is not equal to that of cluster B. The PDOS's for clusters B and D are shown in Fig. 5. The width of the TO-like peak for cluster D broadens on the high frequency side in comparison with that for cluster B. Accordingly, we conclude that the shape of the TO-like peak depends on the bond angle fluctuation and/or the topology of the Si network as suggested by Meek [5].
100
200 300 400 FREQUENCY (cm -t)
500
,
600
Fig. 5. Comparison o f the PDOS's for a-Si clusters w i t h
different amounts of the bond angle fluctuation. The solid and dotted curves are for clusters B and D in Table 1, respectively. The normalization of the PDOS's is the same as for Fig. 2. Since we could not construct a-Si clusters with very different amounts of the bond angle fluctuation without changing topology, the effect of the bond angle fluctuation on the change of the TO-like peak could not be picked up separately from the effect of topological change. However, the former is considered to be dominant because the phonon frequency may be determined mostly by the short range structure. The topological change may only affect the PDOS through the change of the bond angle. Above calculations were carried out by assuming that ot and/3 do not depend on the structures. However, the possibility that ~, and/~ depend on the structures can not be excluded. In order to examine where and how the change of the value of ot and ~ affects the PDOS, the calculations were carried out for cluster B in two different cases: (1) the increase of t~ by 10% and /~ unchanged (2) ot unchanged and the increase of/~ by 30%. The results are as follows. The increase of ot makes the TO- and LO-like peaks shift toward the high frequency side by about 25 cm -1 but scarcely changes the shape of these peaks. On the other hand, the increase
370
CALCULATION OF PHONON DENSITY OF STATES FOR AMORPHOUS Si
of/3 makes the TA-like peak shift toward the high frequency side by about 20 cm -a but scarcely changes the shape of this peak [7]. In conclusion, it is found from the present calculation that the width of the TO-like peak broadens with an increase in the bond angle fluctuation, whereas the bond length fluctuation has a rather small effect. Acknowledgements - This work was partly supported by the Sunshine Project of the Ministry of International Trade and Industry of Japan. The authors are grateful to Mr M. Toyokura for executing a part of the computer calculations. Calculations were carried out with a FACOM M-170F computer at Kanazawa University Data Processing Center.
Vol. 50, No. 4
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8.
T. Ishidate, K. Inoue, K. Tsuji & S. Minomura, Solid State Commun. 42, 197 (1982). J.S. Lannin, L.J. Ptlione, S.T. Kshirsagar, R. Messier & R.C. Ross, Phys. Rev. B26, 3506 (1982). R. Tsu, J. Gonzalez-Hernandez, J. Doehler & S.R. Ovshinsky, Solid State Commun. 46, 79 (1983). A. Morimoto, S. Oozora, M. Kumeda & T. Shimizu, Solid State Commun. 47, 773 (1983). P.E. Meek, Phil. Mag. 33, 897 (1976). P.N. Keating, Phys. Rev. 145,637 (1966). R. Alben, D. Weaire, J.E. Smith, Jr. & M.H. Brodsky, Phys. Rev. BI1, 2271 (1975). G. Dolling & R.A. Cowley, Proc. Phys. Soc. London 88, 463 (1966).
0038-1098/84 $3.00 + .00 Pergamon Press Ltd.
CALCULATION OF PHONON DENSITY OF STATES FOR AMORPHOUS Si N. Ishli Department of Electrical Engineering and Electronic Engineering, Fukui Institute of Technology, Fukui 910, Japan and M. Kumeda and T. Shimizu Department of Electronics, Faculty of Technology, Kanazawa University, Kanazawa 920, Japan
(Received 21 January 1984 by H. Kawamura) The phonon density of states is calculated for amorphous $1 clusters with various bond angle and bond length fluctuations. From these results we conclude that the broadening of the width of the TO-like peak is due to the bond angle fluctuation, whereas the bond length fluctuation has a rather small effect on the phonon density of states. 1. INTRODUCTION v(r,e)
RAMAN SCATTERING has been used to investigate the structural disorder in amorphous Si (a-S0 [1-3] and revealed to be a useful tool for obtaining an extent of the structural disorder in amorphous network [3, 4]. There is still confusions, however, about what kind of disorder, such as the bond angle fluctuation, the bond length fluctuation or the topological change, affects most importantly the shape of the Raman scattering spectra in amorphous materials. Since the k-selection rule breaks down for amorphous materials, the Raman scattering spectra roughly correspond to the phonon density of states (PDOS). So calculations of the PDOS for various structures are useful to answer the above question. Meek concluded from theoretical calculations for amorphous Ge that the shape of the TO-like peak of the PDOS depends on the topological change and the bond angle fluctuation [5]. However, it seems that the influence of the bond length fluctuation on the shape of the TO-like peak can not be excluded from his results. In the present work we calculate the PDOS for Si clusters with various bond angle and bond length fluctuations and examine the relation between the PDOS and the kind of disorder in amorphous network. 2. CALCULATION METHOD We constructed an a-Si cluster with balls and sticks and read the three-dimensional coordinates of the centers of the balls. In order to relax the hand-made a-Si clusters, we minimized the Keating potential V(KP) [61, 367
=
y_
3 a
R
-
d2) 2
I,A
+~_ ~ 3 /~ (RI/," + ~ d 2 ) 2, I (zx,A') 8 ~ RIA'
(1)
where cxand 13are the force constants, d is the bond length of c-S1 (2.35 A). RI,~ is the position vector from an atom I to the atom IA bonding with the atom I. The properties of the model clusters used for calculations are listed in Table 1. Cluster A is a crystalline one. Clusters B and C were relaxed from the same handmade cluster by using/3/t~ = 0.15 and 0.01, respectively. Cluster D was relaxed from another hand-made cluster by using B/a = 0.01. We solved the following secular equation in order to obtain the phonon frequency, ~ , IMco21--DI = 0,
(2)
where M is the mass of a Si atom, 1 is a unit matrix, and D is a dynamical matrix, which is obtained from an exact dynamic matrix, Dkt:Kp ~ a 2 V(KP) u~ ) - 0q~0q~
allq=0,
(3)
by omitting the terms proportional to (Rye. R m -- d 2) and (RIA" RIA' + d2/3) [7], where q~ is the k component of the displacement of the atom I. The reason why we omitted these terms is as follows: When we diagonalized the D(KP) for the clusters obtained by minimizing V(KP), the 6-fold degenerate solution of co -- 0 which should correspond to the translational and rotational freedoms of the model cluster could not be obtained in some cases. It seems to be due to
368
CALCULATION OF PHONON DENSITY OF STATES FOR AMORPHOUS Si
Table 1. Model clusters and their properties used for calculations The number of total Si atoms
Cluster
A
123 110 110 120
B
C D
6~
i
!
5
"
,
Bond length fluctuation (percent)
Bond angle fluctuation (percent)
0 1.4 0.2 0.4
0 6.3 5.8 10.0
!
i
c-Si CALC. - EXR TA
"
'
Uj"
'=1
,
200
300
400
500
600
Fig. 2. The PDOS's calculated for a crystalline Si cluster with 123 atoms. The solid curve is one calculated by using equation (5) and the dot-and-dash curve is one without the correction of the surface effect. The PDOS calculated for a perfect Si crystal as shown in Fig. 1 is drawn again for the sake of comparison by the dotted curve. These three PDOS's, p (~o)'s, are normalized for f~ p (co) do: to be equal.
TO
A
500
neighbor atoms (atoms without dangling bonds) in the cluster. The sum Zj is over all atoms in the cluster. The surface effect on the calculated results is considerably excluded by using this equation as shown below.
i
600
(c° --} a6°')2 2 ,
Y
TA
FREQUENCY (cm-I)
(4)
where a = 8 cm -1 and Ni is a weight of the ith state. All Ni's are usually set equal to 1. However, in order to exclude the surface effect on the calculated results, we used Ni calculated by equation (5),
J
TO
05 0 (ZI ¢1
incomplete relaxation of the cluster. The complete relaxation requires long CPU time. The difference between the PDOS's obtained by using D(KP) and D is small. Therefore, we used D instead of D(KP). In order to obtain the PDOS, p (co), from the calculated phonon frequency, ~i, we used the following equation;
' Y ' us
PERFECT CRYSTAL
o
Fig. 1. The PDOS for crystalline Si. The solid curve is the calculated PDOS by choosing the force constants as ot = 4.85 x 104 dyne cm -1 [6] and/$ = O.15oL. The dotted curve zs one obtained experimentally [8].
-=
SURFACE CORRECTION 4
£
:/ "/ i
p(co) = ~{ Ni. exp
I
. . . . . NO SURFACE CORRECTION
1 1
200 300 400 FREQUENCY (cm-1 )
I
c-Si
I00
H
LO
I
C
I
LA
I
g
'
100
I
Vol. 50, No. 4
(5)
J
where u~ is the displacement of atom J in ith vibrational mode and the sum Z~ is over atoms with four nearest
3. RESULTS A N D DISCUSSION We first calculated the PDOS for crystalline Si. We used a = 4.85 x 104 dyne cm -1 [6] and ~/ot = 0.15 and calculated the phonon spectra for about 1000 unequal points in the first Brillouin zone. Since the calculated PDOS fairly well reproduces the experimental one [8] as shown in Fig. 1, these values of ct and/~ are used in the later calculations. Since the surface of a model amorphous cluster may affect the calculated PDOS when the cluster size is small, a question arises how large cluster size is necessary to obtain the reliable result. In order to check tins surface effect, we calculated the PDOS for a crystalline Si cluster with 123 atoms (cluster A in Table 1) by two different methods. One is carried out by using equation (5) and the other by setting all Ni's in equation (4) equal. The PDOS calculated by these two methods are shown in Fig. 2 together with one for a perfect Si crystal calculated above. It is found from Fig. 2 that the PDOS for the crystalline cluster with 123 atoms reproduces four peaks in the PDOS for the perfect crystal and that the use of equation (5) for calculating the PDOS considerably improves the surface effect (the shift of the
Vol. 50, No. 4
CALCULATION OF PHONON DENSITY OF STATES FOR AMORPHOUS Si 3
6
I
I
I
369
I
I
a-Si
=¢2
--
CLUSTER B
•
-- CLUSTER C
TO o
~/~ f
o
I
CLUSTER A (c-S=)
4
ft.
CLUSTER B (a-Sl)
~3 3
100
O 1,0 0 O_
200 300 400 FREQUENCY (cm -I)
500
600
TO .
I00
TA
Fig. 4. Comparison of the PDOS's for a-Si clusters with different amounts of the bond length fluctuation. The solid and dotted curves are for clusters B and C in Table 1, respectively. The normalization of the PDOS's is the same as for Fig. 2.
LO
200 300 400 FREQUENCY (cm-I)
500
600 I
Fig. 3. Comparison of the PDOS's for a-S1 cluster (cluster B in Table 1) and a crystalline Si cluster (cluster A in Table 1). The solid curve is for cluster B. The dotted curve is for cluster A (already shown by the solid curve in Fig. 2). The normalization of the PDOS's is the same as for Fig. 2.
I
CLUSTER B
"E
I
I
a-Si
I
TO
!
TA-like peak to low frequency region and the increase of the TA-like peak intensity comparing with the TOlike peak intensity). Accordingly, about 100 atoms are considered to be sufficient for the cluster to avoid the surface effect if we make use of equation (5). We used equation (5) in the following calculations for a-Si clusters with about 100 atoms. The PDOS for cluster B whose structural fluctuations are of typical values for a-Si obtained experimentally is shown in Fig. 3 together with one for cluster A. A remarkable difference between them exists on the shape of the TO-like peaks, although the positions of these peaks do not change. What is the origin of this difference? The bond length fluctuation hardly affects the PDOS of a-Si, because the difference between the PDOS for cluster B and that for cluster C is rather small as shown m Fig. 4, where clusters B and C have the same topology but their bond length fluctuations are very different as shown in Table 1. We next compare cluster B with cluster D m Table 1. The bond angle fluctuation of cluster D is about 1.6 times larger than that of cluster B and the topology of cluster D is not equal to that of cluster B. The PDOS's for clusters B and D are shown in Fig. 5. The width of the TO-like peak for cluster D broadens on the high frequency side in comparison with that for cluster B. Accordingly, we conclude that the shape of the TO-like peak depends on the bond angle fluctuation and/or the topology of the Si network as suggested by Meek [5].
100
200 300 400 FREQUENCY (cm -t)
500
,
600
Fig. 5. Comparison o f the PDOS's for a-Si clusters w i t h
different amounts of the bond angle fluctuation. The solid and dotted curves are for clusters B and D in Table 1, respectively. The normalization of the PDOS's is the same as for Fig. 2. Since we could not construct a-Si clusters with very different amounts of the bond angle fluctuation without changing topology, the effect of the bond angle fluctuation on the change of the TO-like peak could not be picked up separately from the effect of topological change. However, the former is considered to be dominant because the phonon frequency may be determined mostly by the short range structure. The topological change may only affect the PDOS through the change of the bond angle. Above calculations were carried out by assuming that ot and/3 do not depend on the structures. However, the possibility that ~, and/~ depend on the structures can not be excluded. In order to examine where and how the change of the value of ot and ~ affects the PDOS, the calculations were carried out for cluster B in two different cases: (1) the increase of t~ by 10% and /~ unchanged (2) ot unchanged and the increase of/~ by 30%. The results are as follows. The increase of ot makes the TO- and LO-like peaks shift toward the high frequency side by about 25 cm -1 but scarcely changes the shape of these peaks. On the other hand, the increase
370
CALCULATION OF PHONON DENSITY OF STATES FOR AMORPHOUS Si
of/3 makes the TA-like peak shift toward the high frequency side by about 20 cm -a but scarcely changes the shape of this peak [7]. In conclusion, it is found from the present calculation that the width of the TO-like peak broadens with an increase in the bond angle fluctuation, whereas the bond length fluctuation has a rather small effect. Acknowledgements - This work was partly supported by the Sunshine Project of the Ministry of International Trade and Industry of Japan. The authors are grateful to Mr M. Toyokura for executing a part of the computer calculations. Calculations were carried out with a FACOM M-170F computer at Kanazawa University Data Processing Center.
Vol. 50, No. 4
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8.
T. Ishidate, K. Inoue, K. Tsuji & S. Minomura, Solid State Commun. 42, 197 (1982). J.S. Lannin, L.J. Ptlione, S.T. Kshirsagar, R. Messier & R.C. Ross, Phys. Rev. B26, 3506 (1982). R. Tsu, J. Gonzalez-Hernandez, J. Doehler & S.R. Ovshinsky, Solid State Commun. 46, 79 (1983). A. Morimoto, S. Oozora, M. Kumeda & T. Shimizu, Solid State Commun. 47, 773 (1983). P.E. Meek, Phil. Mag. 33, 897 (1976). P.N. Keating, Phys. Rev. 145,637 (1966). R. Alben, D. Weaire, J.E. Smith, Jr. & M.H. Brodsky, Phys. Rev. BI1, 2271 (1975). G. Dolling & R.A. Cowley, Proc. Phys. Soc. London 88, 463 (1966).