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A calculation of the isotope frequency shifts of local mode caused by interstitial oxygen in single crystal silicon
Solid State Communications, Vol. 65, No. 10, pp. 1247-1251, 1988. Printed in Great Britain.
0038-1098/88 $3.00 + .00 Pergamon Press plc
A C A L C U L A T I O N OF T H E ISOTOPE F R E Q U E N C Y SHIFTS OF LOCAL M O D E CAUSED BY I N T E R S T I T I A L O X Y G E N IN S I N G L E CRYSTAL SILICON Xu Zhengyi and Ge Peiwen Institute of Physics, Academia Sinica, P.O. Box 603, Beijing, P.R. China and Gu Benyuan* Dipartimento di Fisica, Universit/t "La Sapienza", 00185 Roma, Italy (Received 28 September 1987; in revised f o r m 20 November 1987 by M. Balkanski)
In the present paper, a calculation of the isotope frequency shifts of local mode caused by interstitial oxygen in silicon is presented by using a 9-atom cluster model based on the force constant approach. We find that the calculated result is not better than that of a single Si20 molecular model. Even if the effects of non-central force and local distortion of lattice are taken into account, the result is still not expected well. In order to obtain the best fit with the experimental isotopic shifts, it is necessary to introduce the variation of the S i 4 ) force constant with the different isotopes. It turns out that the Si-O force constant increases with increasing the mass of silicon atom but decreases with increasing the mass of oxygen atom. 1. I N T R O D U C T I O N M A N Y PHYSICAL P R O P E R T I E S of silicon devices made of crystal silicon grown by Czochraski method depend sensitivity on impurity oxygens. Therefore, the studies of local modes caused by the impurity oxygens in silicon have attracted the extensive attention in the past years. In 1956 Kaiser et al. [1] have at first reported the infrared absorption of the impurity oxygen in silicon. Since then many further investigations were reported in the literatures [2-4]. However, to our knowledge, some problems still remain unclear. Experiments show that the impurity oxygen atoms situate at the interstitial sites of the silicon lattice, and the corresponding infrared absorption lines appear at 1203, 1136, 517 and 29.3cm l, in which the last line is only observed at low temeprature [3]. At the earliest the first three lines were designated to v l(A1), v3(BI) and v2(A1) of SizO molecular, respectively. Later on Bosomworth et al. [3] assigned the lines 517 and 29.3 cm-~ to the modes v~ and v2, respectively, however, the line 1203cm -~ was designed to v3 + 2v2. Bosomworth et al. [3] and Pajot et al. [4] also investigated the isotope shifts of the stretched mode v3 * On leave from the Institute of Physics, Academia Sinica, P.O. Box 603, Beijing, P.R. China. To whom all correspondence should be sent.
in detail. Although these isotope frequency shifts can be explained well based on a Si20 molecular model, the calculated result can be improved further. Pajot et al. [4] found that if the masses of all the silicon atoms and their isotopes in the Si20 molecular artificially increase 3, then the calculated isotope shifts can fit with the experimental data better than before. They interpreted the reason taking the above mentioned treatment is that the molecular Si20 in the silicon can not be regarded as a completely isolated group, this molecular is bonded with the rest of silicon atoms. They introduce simply a phenomenological interaction mass to reflect this effect. This influence of the interaction between the Si20 molecular and the rest of atoms in the silicon lattice on the infrared absorption spectrum can be rigorously considered. In the present paper we present a calculation of the isotope shifts of local mode caused by interstitial oxygen in c-silicon based on a 9-atom cluster model. We find that this more rigorous treatment can not bring result better than that of the single Si20 molecular model. We also consider the influence of non-central force and the local distortion of lattice on the infrared absorption spectra. However, the result still is not expected well. Only under the consideration of the variation of the Si-O force constant with the different isotopes, the calculated result shows a good agreement with the experimental data.
1247
1248
O X Y G E N IN S I N G L E CRYSTAL SILICON
/I
IX,.~-/~ . . . . .
C) sl
Fig. 1. The positions of Og, its nearest and the second neighbour Si atoms as well as the vibration pattern of the atoms for the stretched mode. [The length of arrow is proportional to the atomic vibration amplitude. The length of the arrows attached to the second neighbour atoms is magnified by one hundred times. The relative vibration amplitudes of the atoms are as follows: u(O~) =
(0.38, 0.38, 0.38),
u(0) =
(-0.12,-0.15,-0.18)
u(l) =
(-0.18,-0.15,-0.12),
u(2) =
( - 0 . 5 6 , 0.81, 0.84) x 10 -2
u(3) =
(0.64, - 0 . 3 8 , 0.70) x 10 -2,
u(4) =
(0.50, 0.53, - 0 . 2 0 ) x 10 -2
u(5) =
(0.76, 0.73, - 0 . 4 4 ) × 10 -2 ,
u(6) =
(0.63, - 0 . 2 4 , 0.57) × 10 -2
u(7) =
( - 0 . 0 5 , 0.46, 0.44) × 10 2.]
|s2/3_ \$2/3
2/3 C2/2
V = ~all(r?~ + r~),
(1)
where r0, r~ express the lengths for the bonds Si(0)4)~ and Si(1)-Og, respectively. In the local coordinate system X'Y'Z', the force constants are defined by. c~2 V
Ou~(i)Ou~(j)'
*
p
cr,'c = x , y , z ' ,
(2)
where u~(i) is the a-th component of displacement vector of the i-th atom. Thus, the force constant matrix between Si(0) and Oi is
f sin20 O'(O, Oi) = -alll~nOcosO
sin0cos0 cos20
, (3)
0
An interstitial oxygen atom (thereafter referred to O~) in silicon breaks a Si-Si bond and forms new bonds Si-O-Si. For each O~ there are six equivalent positions available around a line joining the two silicon atoms bonded with it. Let this line be along [1 1 1] direction. Therefore, the O~ may situate at a position in the equal-dividing plane between the plane (0 1 1)
C2/2 - w/~SC $2/3 - ~ S C
We now consider a 9-atom cluster composed of O~ atom and its eight neighbour Si atoms (see Fig. 1). For diamond-type crystal the force constants up to sixth neighbour have been given by Hermen under the form of matrix [5]. In this paper we adopt the same symbols, including the serial number of Si atoms in Fig. 1, as those of Hermen's paper [5]. We denote the j-th Si atom as Si(j). Since the oxygen atom squeezes into the interstitial site of the silicon lattice, it causes a local distortion of the silicon lattice, so that the distance between Si(0) and Si(1) increases from 2.34 to 3.20A [6]. Therefore, the interaction between Si(0) and Si(1) can be neglected. Assuming the interaction between Si and Oz atoms is central force, then the interaction potential can be written as [7]
O'~(i,j)
2. LOCAL MODES CAUSED BY INTERSTITIAL OXYGEN
{$2/3 +
Vol. 65, No. 10
where 20 is the bond angle of Si(0)-Oi-Si(1). Let the x'- axis be parallel to [1 1 1] direction, and y' 11 [1 01] direction. Through some standard algebraic operations, it is readily obtained an expression of the force constant matrix under the crystallographic coordinate system XYZ of silicon, that is
0(0, Oi) = -all,
$2/3 _ C2/2
\
s2/3+
)
(4)
$2/3 + x/~66SC $2/3 + C2/2 + x//~SC
and (T 1 0) or at one of the other five equivalent positions, which can be easily found by rotating the original position to an angle nn/3 (n = 1, 2, 3.. 5) round the [1 1 1] axis.
where S = sin 0 and C = cos 0. We can also find the following relation: • (1, Oi) =
(I)(0, Oi)10~ 0-
(5)
OXYGEN IN SINGLE CRYSTAL SILICON
Vol. 65, No. 10
With the consideration of the relationship to be satisfied by the force constants [8], Z+O(i,j)
ZjO(i,j)
=
=
0,
(6)
we obtain
(2x3+
@(Oi, Oi)
=
2S2/3 2S2 /3 -
/
at1
C2
2S2/3
2S2/3
2S2 /3
2S2 /3 + C 2
=
A
1
(a) B a s e d on the m o d e l o f a 9-atom cluster with central f o r c e constants .
(7)
,
0
for
pattern of the atoms is described by the corresponding eigenvector. Since the 9-atom cluster given in Fig. 1 has 6'2symmetry, therefore, all the vibrational modes are infrared-active. 3. C A L C U L A T E D R E S U L T S
However, Oi,(j,j) ( j = 2, 3 . . . 7) have the same forms as those of the perfect silicon crystal. That is
¢b(j,j)
1249
j = 2, 3, 4 . . . 7.
The non-zero force constants of Si atom up to the fifth neighbour are as follows [10]: =
0.53141 x 105dyncm -~,
fl =
0.37573 × 105 d y n c m -~,
p =
0.03632 x 105 d y n c m -~,
2 =
- 0 . 0 7 2 4 3 x 105dyncm-~,
v =
0.03095 x 105 d y n c m -1,
6 =
0.03151 x 105dyncm ~,
(8)
p'"
=
0.00487 x 105 d y n c m ~,
The relationship between the force constants given by H e r m a n [5] and the valence force constants presented by Solbrig [9] has been established by Gu et al. [10]. One has
2"
=
0.01948 x 105 d y n c m 1,
v'" =
0.00487 × 105 d y n c m - I ,
6"
0.00974 × 105dyncm ~.
A =
Let 0 = 82 °. Substituting above mentioned force constants into the dynamical matrix and considering at l as an adjustable parameter, for the configuration 28Si160-28Si, we solve the eigenvalue problem of a 27 x 27 matrix and find that if al~ = 4.7502 x 105 dyn cm -1, then one of the vibrational frequencies of the local modes approaches to 1136.0cm -I . The corresponding vibrational pattern of the atoms is shown in Fig. 1. Although the interaction among the Si(0), Si(1) and the other silicon atoms is taken into account, the vibration of Si(0)~O:-Si(1) still remains in the plane to be comprised by the equilibrium positions of these three atoms, this vibration mode is similar with the v3 mode under the Si20 molecular model. The vibration amplitude of the other Si atoms is two orders of magnitude less than that of Oi. Therefore this mode can be referred to a stretched mode. While considering the isotope effect, it requires only to change the mass of silicon and oxygen atoms in the dynamical matrix. The calculated result is shown in the second colume of Table 1. It is seen that this result is worse than that of the single Si20 molecular model (see the fourth and fifth columns of Table 1).
4~ + 8p + 42 + 8#'" + 42'".
(9)
For ~(0, 0) and ~(1, 1), since the bond Si(0)-Si(l) is broken, and both Si(0) and Si(1) are bonded with O~ forming two new bonds Si-O~, consequently, the form of the corresponding force constant matrix should be modified as
O(j,j)
=
A
1
--
o -
-
(I)(j, Oi) , for
j = 0, 1.
(10)
For the Og atom we consider only the interaction between the O+ and its nearest neighbour silicon atoms, consequently, one has • (j,O,)
=
0, for
j = 2, 3, 4 . . . 7.
(11)
All the rest of force constant matrices interest can be easily derived referring to H e r m a n ' s paper [5]. The dynamical matrix is D~(i,j)
=
O ~ ( i , j ) / ( M ~ M j ) ~/2,
(12)
where Mi is the mass of the i-th Si atom. The frequency of the vibration mode is determined by the eigenvalue of the dynamical matrix D,~ (i, j ) , and the vibrational
=
(b) N o n - c e n t r a l f o r c e case In the general case, the interaction potential a m o n g Oi, Si(0) and Si(1) can be written as follows:
1250
OXYGEN IN SINGLE CRYSTAL SILICON
Vol. 65, No. 10
Table 1. Experimental results and the calculated isotope frequency shifts for the stretched mode of interstitial oxygen in silicon crystal Configulation 28Si-160-28Si 28Sij60-29Si
285i-160-3°5i 295i-160298i 29Si-160--3°Si 3°5i-160-3°5i 28Si-:O-28Si 28Si-~80-28Si 28Si-180-29Si 28Si-180-3°Si
Ava
Avb
Ag (cm t) Avc [4]
Avd [4]
0 2.37 4.55 4.77 6.99 9.23 25.55 48.72 51.21 53.51
0 1.91 3.57 3.85 5.54 7.24 26.90 51.34 53.41 55.26
0 2.22 4.29 4.43 6.51 8.58 26.18 49.99 52.31 54.48
0 1.86 3.60 3.71 5.46 7.20 26.77 51.13 53.07 54.90
Exp. [4] 0 1.90 + 0.08 3.64 _____0.08 5.54 7.20 26.9 51.36 53.41 55.26
___ 0.09 _+ 0.08 + 0.20 ___ 0.06 ___ 0.09 __+ 0.15
The adjustable parameter an = 4.7502 x 105 d y n c m -~, the force constants are independent of isotopes. b The Si-O force constant is isotope-dependent. The force constant is taken from Table 2. c The calculated result based on the single Si20 molecular model. d The calculated result based on the single Si20 molecular model with a phenomenological interaction mass modification. a
V = ½ [a,,(r~ + ~) + a33~, + 2at2ror, + 2al3(rolro + rolrl)],
(13)
where r0~ expresses the Si(0)-Si(i) bond length. Analogously to the case of the single Si20 molecular, when (all - a 1 2 ) . . . . . . . tral = (all)central, for a 9-atom cluster model with non-central force, we can derive the same frequency of the stretched mode as that with central force. Therefore this new consideration can not bring a good result either. The other frequency of the mode depends on a~2, al3 and a33. Since the number of the spectral lines available is very a few, therefore, there is no way to reasonably determine these force constants. This calculation scheme is not useful. (c) The effect of the local distortion of lattice If we assume that the equilibrium position of only the nearest neighbour Si atoms of O; occurs a change since the oxygen squeezes into the interstitial site of the silicon lattice, thus the distance between the first and the second neighbour Si atoms of O; decreases 18%. Assume that the force constant is inversely varied as the cube of the bond length, therefore, the relevant force constants might increase 70%. While increasing the ct and fl values by 1.7 times larger than the usual values, the calculated result exhibits only an improvement a little. However, it is not reason to imagine that the ~ and fl have so much change. (d) The variation of the Si-O force constant with the
different isotopes All the calculation schemes as mentioned above can not bring the isotope shifts of the stretched mode
of the interstitial oxygen in silicon better than those in agreement with experiment of the single molecular model. Therefore, it is necessary to introduce the variation of the Si-O force constant with the different isotopes. This variation may originate from the anharmonic term in the atomic interaction potential [11]. For example, the mass of deuterium atom is twice times as heavy as that of hydrogen atom, the H - H and H - D bond lengths could be different, in fact, this difference has been observed in the experiment [12]. The dependence of the force constants on the isotopes has also been shown by Bai et al. [13]. Through adjusting the force constants all (285i-160), a~ (29Si-160) and a~ (3°5i-160), we can calculate the vibration frequency of the following configurations: 28Si ~60-28Si, 285i160 295i, 285i-160-3°Si, 29Si-160-3°5i, and 3°Si 1603°Si, and make the result fit with the experimental data in a very good agreement (see the third column in Table 1). The variation of the Si-O force constant a~ with the different isotopes is shown in Table 2.
Table 2. The variation of the Si-O force constant a~ with the different isotopes Bond
al, ( x 105 dyn cm)
.28Si 160 295i-160 3°5i-160 285i-170 28Si IsO
4.7502 4.7580 4.7672 4.7385 4.7270 4.7346 4.7426
295i 180 3°5i-180
Vol. 65, No. 10
O X Y G E N IN S I N G L E C R Y S T A L SILICON
1251
It can be seen from Table 2 that the force constant a~ increases with increasing the mass of silicon atom but decreases with increasing the mass of the oxygen atom.
Acknowledgements - - One of authors (B.Y. Gu) has carried out this work with the support of the ICTP program for Training and Research in Italian Laboratories, Trieste, Italy.
4. DISCUSSIONS
REFERENCES
It is seen from Fig. 1 that the vibration mode corresponding to the frequency 1136 cm- ~shows very strong localizability. The vibration amplitude of the second neighbour silicon atoms of Oi is two orders of magnitude less than that of the oxygen atom. T h e vibration amplitude for the third neighbour Si atoms of Oi is so much less that it can be completely neglected. Since the force constant of Si-O is about one order of magnitude larger than that of Si-Si, and the mass of the oxygen atom is four-seventh of the silicon atom, therefore, in the dynamical matrix, only the submatrices D(Si, Oi) and D(Oi, Oi) present a large contribution. Consequently, the size of a 9-atom cluster used for our calculations is larger enough to reflect the influence of the Si lattice on the interstitial oxygen atom. The consideration of the second neighbour silicon atoms of O~ is necessary for the calculation of the isotope frequency shifts. In particular, for the low frequency mode their influence can not be ignored. In the above calculations, for the second neighbour silicon atoms of Oi, an average mass is used. However, the effect of their isotope mass on the calculated result is also examined. Taking the isotope mass modification, the variance of frequency for the stretched mode is less than 0.01 cm ~. So this effect can be ignored.
1. 2. 3. 4.
5. 6. 7. 8. 9. 10. 11. 12. 13.
W. Kaiser, P.H. Keck & C.F. Lange, Phys. Rev. 101, 1264 (1956). H.J. Hrostowski & R.H. Kaiser, Phys. Rev. 107, 966 (1957). D.R. Bosomworth, W. Hayes, A.R.L. Spray & G.D. Watkins, Proc. Roy. Soc. (London) A317, 133 (1970). B. Pajot & B. Cales, Infrared Spectroscopy of Interstitial Oxygen in Silicon, in Oxygen, Car-
bon, Hydrogen and Nitrogen in Crystalline Silicon, p. 39 (Edited by J.C. Mikkelsen, Jr., S.J. Pearton, J.W. Corbett and S.J. Pennycook), MRS, Pittsburgh, (1985). F. Herman, J. Phys. Chem. Solids 8, 405 (1959). B. Pajot, H.J. Stein, B. Cales & C. Nand, J. Electron Soc., 132, 3034 (1985). G. Hertzberg, Infrared and Raman Spectra of Polyatomic molecules, Van Nostrand, New York, (1945). M. Lax, Symmetry Principles in Solid State and Molecular Physics, Wiley, London, (1974). A.W. Solbrig, Jr. J. Phys. Chem. Solid 8, 1761 (1971). Gu Benyuan, Xu Zhengyi & Ge Peiwen, Scientia Sinica (ser. A.) 28, 740 (1985). L.M. Xie, Private Communication. W.B. Olson, et al., J. Mol. Spectroscopy, 66, 314 (1977). G.R. Bai, et al., to be published in Scientia Sinica (ser. A.).
0038-1098/88 $3.00 + .00 Pergamon Press plc
A C A L C U L A T I O N OF T H E ISOTOPE F R E Q U E N C Y SHIFTS OF LOCAL M O D E CAUSED BY I N T E R S T I T I A L O X Y G E N IN S I N G L E CRYSTAL SILICON Xu Zhengyi and Ge Peiwen Institute of Physics, Academia Sinica, P.O. Box 603, Beijing, P.R. China and Gu Benyuan* Dipartimento di Fisica, Universit/t "La Sapienza", 00185 Roma, Italy (Received 28 September 1987; in revised f o r m 20 November 1987 by M. Balkanski)
In the present paper, a calculation of the isotope frequency shifts of local mode caused by interstitial oxygen in silicon is presented by using a 9-atom cluster model based on the force constant approach. We find that the calculated result is not better than that of a single Si20 molecular model. Even if the effects of non-central force and local distortion of lattice are taken into account, the result is still not expected well. In order to obtain the best fit with the experimental isotopic shifts, it is necessary to introduce the variation of the S i 4 ) force constant with the different isotopes. It turns out that the Si-O force constant increases with increasing the mass of silicon atom but decreases with increasing the mass of oxygen atom. 1. I N T R O D U C T I O N M A N Y PHYSICAL P R O P E R T I E S of silicon devices made of crystal silicon grown by Czochraski method depend sensitivity on impurity oxygens. Therefore, the studies of local modes caused by the impurity oxygens in silicon have attracted the extensive attention in the past years. In 1956 Kaiser et al. [1] have at first reported the infrared absorption of the impurity oxygen in silicon. Since then many further investigations were reported in the literatures [2-4]. However, to our knowledge, some problems still remain unclear. Experiments show that the impurity oxygen atoms situate at the interstitial sites of the silicon lattice, and the corresponding infrared absorption lines appear at 1203, 1136, 517 and 29.3cm l, in which the last line is only observed at low temeprature [3]. At the earliest the first three lines were designated to v l(A1), v3(BI) and v2(A1) of SizO molecular, respectively. Later on Bosomworth et al. [3] assigned the lines 517 and 29.3 cm-~ to the modes v~ and v2, respectively, however, the line 1203cm -~ was designed to v3 + 2v2. Bosomworth et al. [3] and Pajot et al. [4] also investigated the isotope shifts of the stretched mode v3 * On leave from the Institute of Physics, Academia Sinica, P.O. Box 603, Beijing, P.R. China. To whom all correspondence should be sent.
in detail. Although these isotope frequency shifts can be explained well based on a Si20 molecular model, the calculated result can be improved further. Pajot et al. [4] found that if the masses of all the silicon atoms and their isotopes in the Si20 molecular artificially increase 3, then the calculated isotope shifts can fit with the experimental data better than before. They interpreted the reason taking the above mentioned treatment is that the molecular Si20 in the silicon can not be regarded as a completely isolated group, this molecular is bonded with the rest of silicon atoms. They introduce simply a phenomenological interaction mass to reflect this effect. This influence of the interaction between the Si20 molecular and the rest of atoms in the silicon lattice on the infrared absorption spectrum can be rigorously considered. In the present paper we present a calculation of the isotope shifts of local mode caused by interstitial oxygen in c-silicon based on a 9-atom cluster model. We find that this more rigorous treatment can not bring result better than that of the single Si20 molecular model. We also consider the influence of non-central force and the local distortion of lattice on the infrared absorption spectra. However, the result still is not expected well. Only under the consideration of the variation of the Si-O force constant with the different isotopes, the calculated result shows a good agreement with the experimental data.
1247
1248
O X Y G E N IN S I N G L E CRYSTAL SILICON
/I
IX,.~-/~ . . . . .
C) sl
Fig. 1. The positions of Og, its nearest and the second neighbour Si atoms as well as the vibration pattern of the atoms for the stretched mode. [The length of arrow is proportional to the atomic vibration amplitude. The length of the arrows attached to the second neighbour atoms is magnified by one hundred times. The relative vibration amplitudes of the atoms are as follows: u(O~) =
(0.38, 0.38, 0.38),
u(0) =
(-0.12,-0.15,-0.18)
u(l) =
(-0.18,-0.15,-0.12),
u(2) =
( - 0 . 5 6 , 0.81, 0.84) x 10 -2
u(3) =
(0.64, - 0 . 3 8 , 0.70) x 10 -2,
u(4) =
(0.50, 0.53, - 0 . 2 0 ) x 10 -2
u(5) =
(0.76, 0.73, - 0 . 4 4 ) × 10 -2 ,
u(6) =
(0.63, - 0 . 2 4 , 0.57) × 10 -2
u(7) =
( - 0 . 0 5 , 0.46, 0.44) × 10 2.]
|s2/3_ \$2/3
2/3 C2/2
V = ~all(r?~ + r~),
(1)
where r0, r~ express the lengths for the bonds Si(0)4)~ and Si(1)-Og, respectively. In the local coordinate system X'Y'Z', the force constants are defined by. c~2 V
Ou~(i)Ou~(j)'
*
p
cr,'c = x , y , z ' ,
(2)
where u~(i) is the a-th component of displacement vector of the i-th atom. Thus, the force constant matrix between Si(0) and Oi is
f sin20 O'(O, Oi) = -alll~nOcosO
sin0cos0 cos20
, (3)
0
An interstitial oxygen atom (thereafter referred to O~) in silicon breaks a Si-Si bond and forms new bonds Si-O-Si. For each O~ there are six equivalent positions available around a line joining the two silicon atoms bonded with it. Let this line be along [1 1 1] direction. Therefore, the O~ may situate at a position in the equal-dividing plane between the plane (0 1 1)
C2/2 - w/~SC $2/3 - ~ S C
We now consider a 9-atom cluster composed of O~ atom and its eight neighbour Si atoms (see Fig. 1). For diamond-type crystal the force constants up to sixth neighbour have been given by Hermen under the form of matrix [5]. In this paper we adopt the same symbols, including the serial number of Si atoms in Fig. 1, as those of Hermen's paper [5]. We denote the j-th Si atom as Si(j). Since the oxygen atom squeezes into the interstitial site of the silicon lattice, it causes a local distortion of the silicon lattice, so that the distance between Si(0) and Si(1) increases from 2.34 to 3.20A [6]. Therefore, the interaction between Si(0) and Si(1) can be neglected. Assuming the interaction between Si and Oz atoms is central force, then the interaction potential can be written as [7]
O'~(i,j)
2. LOCAL MODES CAUSED BY INTERSTITIAL OXYGEN
{$2/3 +
Vol. 65, No. 10
where 20 is the bond angle of Si(0)-Oi-Si(1). Let the x'- axis be parallel to [1 1 1] direction, and y' 11 [1 01] direction. Through some standard algebraic operations, it is readily obtained an expression of the force constant matrix under the crystallographic coordinate system XYZ of silicon, that is
0(0, Oi) = -all,
$2/3 _ C2/2
\
s2/3+
)
(4)
$2/3 + x/~66SC $2/3 + C2/2 + x//~SC
and (T 1 0) or at one of the other five equivalent positions, which can be easily found by rotating the original position to an angle nn/3 (n = 1, 2, 3.. 5) round the [1 1 1] axis.
where S = sin 0 and C = cos 0. We can also find the following relation: • (1, Oi) =
(I)(0, Oi)10~ 0-
(5)
OXYGEN IN SINGLE CRYSTAL SILICON
Vol. 65, No. 10
With the consideration of the relationship to be satisfied by the force constants [8], Z+O(i,j)
ZjO(i,j)
=
=
0,
(6)
we obtain
(2x3+
@(Oi, Oi)
=
2S2/3 2S2 /3 -
/
at1
C2
2S2/3
2S2/3
2S2 /3
2S2 /3 + C 2
=
A
1
(a) B a s e d on the m o d e l o f a 9-atom cluster with central f o r c e constants .
(7)
,
0
for
pattern of the atoms is described by the corresponding eigenvector. Since the 9-atom cluster given in Fig. 1 has 6'2symmetry, therefore, all the vibrational modes are infrared-active. 3. C A L C U L A T E D R E S U L T S
However, Oi,(j,j) ( j = 2, 3 . . . 7) have the same forms as those of the perfect silicon crystal. That is
¢b(j,j)
1249
j = 2, 3, 4 . . . 7.
The non-zero force constants of Si atom up to the fifth neighbour are as follows [10]: =
0.53141 x 105dyncm -~,
fl =
0.37573 × 105 d y n c m -~,
p =
0.03632 x 105 d y n c m -~,
2 =
- 0 . 0 7 2 4 3 x 105dyncm-~,
v =
0.03095 x 105 d y n c m -1,
6 =
0.03151 x 105dyncm ~,
(8)
p'"
=
0.00487 x 105 d y n c m ~,
The relationship between the force constants given by H e r m a n [5] and the valence force constants presented by Solbrig [9] has been established by Gu et al. [10]. One has
2"
=
0.01948 x 105 d y n c m 1,
v'" =
0.00487 × 105 d y n c m - I ,
6"
0.00974 × 105dyncm ~.
A =
Let 0 = 82 °. Substituting above mentioned force constants into the dynamical matrix and considering at l as an adjustable parameter, for the configuration 28Si160-28Si, we solve the eigenvalue problem of a 27 x 27 matrix and find that if al~ = 4.7502 x 105 dyn cm -1, then one of the vibrational frequencies of the local modes approaches to 1136.0cm -I . The corresponding vibrational pattern of the atoms is shown in Fig. 1. Although the interaction among the Si(0), Si(1) and the other silicon atoms is taken into account, the vibration of Si(0)~O:-Si(1) still remains in the plane to be comprised by the equilibrium positions of these three atoms, this vibration mode is similar with the v3 mode under the Si20 molecular model. The vibration amplitude of the other Si atoms is two orders of magnitude less than that of Oi. Therefore this mode can be referred to a stretched mode. While considering the isotope effect, it requires only to change the mass of silicon and oxygen atoms in the dynamical matrix. The calculated result is shown in the second colume of Table 1. It is seen that this result is worse than that of the single Si20 molecular model (see the fourth and fifth columns of Table 1).
4~ + 8p + 42 + 8#'" + 42'".
(9)
For ~(0, 0) and ~(1, 1), since the bond Si(0)-Si(l) is broken, and both Si(0) and Si(1) are bonded with O~ forming two new bonds Si-O~, consequently, the form of the corresponding force constant matrix should be modified as
O(j,j)
=
A
1
--
o -
-
(I)(j, Oi) , for
j = 0, 1.
(10)
For the Og atom we consider only the interaction between the O+ and its nearest neighbour silicon atoms, consequently, one has • (j,O,)
=
0, for
j = 2, 3, 4 . . . 7.
(11)
All the rest of force constant matrices interest can be easily derived referring to H e r m a n ' s paper [5]. The dynamical matrix is D~(i,j)
=
O ~ ( i , j ) / ( M ~ M j ) ~/2,
(12)
where Mi is the mass of the i-th Si atom. The frequency of the vibration mode is determined by the eigenvalue of the dynamical matrix D,~ (i, j ) , and the vibrational
=
(b) N o n - c e n t r a l f o r c e case In the general case, the interaction potential a m o n g Oi, Si(0) and Si(1) can be written as follows:
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OXYGEN IN SINGLE CRYSTAL SILICON
Vol. 65, No. 10
Table 1. Experimental results and the calculated isotope frequency shifts for the stretched mode of interstitial oxygen in silicon crystal Configulation 28Si-160-28Si 28Sij60-29Si
285i-160-3°5i 295i-160298i 29Si-160--3°Si 3°5i-160-3°5i 28Si-:O-28Si 28Si-~80-28Si 28Si-180-29Si 28Si-180-3°Si
Ava
Avb
Ag (cm t) Avc [4]
Avd [4]
0 2.37 4.55 4.77 6.99 9.23 25.55 48.72 51.21 53.51
0 1.91 3.57 3.85 5.54 7.24 26.90 51.34 53.41 55.26
0 2.22 4.29 4.43 6.51 8.58 26.18 49.99 52.31 54.48
0 1.86 3.60 3.71 5.46 7.20 26.77 51.13 53.07 54.90
Exp. [4] 0 1.90 + 0.08 3.64 _____0.08 5.54 7.20 26.9 51.36 53.41 55.26
___ 0.09 _+ 0.08 + 0.20 ___ 0.06 ___ 0.09 __+ 0.15
The adjustable parameter an = 4.7502 x 105 d y n c m -~, the force constants are independent of isotopes. b The Si-O force constant is isotope-dependent. The force constant is taken from Table 2. c The calculated result based on the single Si20 molecular model. d The calculated result based on the single Si20 molecular model with a phenomenological interaction mass modification. a
V = ½ [a,,(r~ + ~) + a33~, + 2at2ror, + 2al3(rolro + rolrl)],
(13)
where r0~ expresses the Si(0)-Si(i) bond length. Analogously to the case of the single Si20 molecular, when (all - a 1 2 ) . . . . . . . tral = (all)central, for a 9-atom cluster model with non-central force, we can derive the same frequency of the stretched mode as that with central force. Therefore this new consideration can not bring a good result either. The other frequency of the mode depends on a~2, al3 and a33. Since the number of the spectral lines available is very a few, therefore, there is no way to reasonably determine these force constants. This calculation scheme is not useful. (c) The effect of the local distortion of lattice If we assume that the equilibrium position of only the nearest neighbour Si atoms of O; occurs a change since the oxygen squeezes into the interstitial site of the silicon lattice, thus the distance between the first and the second neighbour Si atoms of O; decreases 18%. Assume that the force constant is inversely varied as the cube of the bond length, therefore, the relevant force constants might increase 70%. While increasing the ct and fl values by 1.7 times larger than the usual values, the calculated result exhibits only an improvement a little. However, it is not reason to imagine that the ~ and fl have so much change. (d) The variation of the Si-O force constant with the
different isotopes All the calculation schemes as mentioned above can not bring the isotope shifts of the stretched mode
of the interstitial oxygen in silicon better than those in agreement with experiment of the single molecular model. Therefore, it is necessary to introduce the variation of the Si-O force constant with the different isotopes. This variation may originate from the anharmonic term in the atomic interaction potential [11]. For example, the mass of deuterium atom is twice times as heavy as that of hydrogen atom, the H - H and H - D bond lengths could be different, in fact, this difference has been observed in the experiment [12]. The dependence of the force constants on the isotopes has also been shown by Bai et al. [13]. Through adjusting the force constants all (285i-160), a~ (29Si-160) and a~ (3°5i-160), we can calculate the vibration frequency of the following configurations: 28Si ~60-28Si, 285i160 295i, 285i-160-3°Si, 29Si-160-3°5i, and 3°Si 1603°Si, and make the result fit with the experimental data in a very good agreement (see the third column in Table 1). The variation of the Si-O force constant a~ with the different isotopes is shown in Table 2.
Table 2. The variation of the Si-O force constant a~ with the different isotopes Bond
al, ( x 105 dyn cm)
.28Si 160 295i-160 3°5i-160 285i-170 28Si IsO
4.7502 4.7580 4.7672 4.7385 4.7270 4.7346 4.7426
295i 180 3°5i-180
Vol. 65, No. 10
O X Y G E N IN S I N G L E C R Y S T A L SILICON
1251
It can be seen from Table 2 that the force constant a~ increases with increasing the mass of silicon atom but decreases with increasing the mass of the oxygen atom.
Acknowledgements - - One of authors (B.Y. Gu) has carried out this work with the support of the ICTP program for Training and Research in Italian Laboratories, Trieste, Italy.
4. DISCUSSIONS
REFERENCES
It is seen from Fig. 1 that the vibration mode corresponding to the frequency 1136 cm- ~shows very strong localizability. The vibration amplitude of the second neighbour silicon atoms of Oi is two orders of magnitude less than that of the oxygen atom. T h e vibration amplitude for the third neighbour Si atoms of Oi is so much less that it can be completely neglected. Since the force constant of Si-O is about one order of magnitude larger than that of Si-Si, and the mass of the oxygen atom is four-seventh of the silicon atom, therefore, in the dynamical matrix, only the submatrices D(Si, Oi) and D(Oi, Oi) present a large contribution. Consequently, the size of a 9-atom cluster used for our calculations is larger enough to reflect the influence of the Si lattice on the interstitial oxygen atom. The consideration of the second neighbour silicon atoms of O~ is necessary for the calculation of the isotope frequency shifts. In particular, for the low frequency mode their influence can not be ignored. In the above calculations, for the second neighbour silicon atoms of Oi, an average mass is used. However, the effect of their isotope mass on the calculated result is also examined. Taking the isotope mass modification, the variance of frequency for the stretched mode is less than 0.01 cm ~. So this effect can be ignored.
1. 2. 3. 4.
5. 6. 7. 8. 9. 10. 11. 12. 13.
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