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Optical phonons and elasticity of diamond at megabar stresses: ab-initio calculations
Physica 139 & 140B (1986) 202-205 North-Holland, Amsterdam
OPTICAL PHONONS AND ELASTICITY OF DIAMOND AT MEGABAR STRESSES: AB-INITIO CALCULATIONS O.H. NIELSEN NORD1TA, Blegdamsve] 17, DK-2100 Copenhagen, Denmark and Laboratoire de Physique des Solides associd au CNRS, Universite P. et M. Curie, Tour 13, 4, PI. Jussieu, F-75230 Paris Cedex 05, France
The optical F-phonon and the stress-strain relations in diamond are investigated for general uniaxial and hydrostatic stresses up to several megabar. Theoretical calculations are carried out using ab-initio pseudopotentials within local-densityfunctional theory. All second-order and third-order elastic properties, including the internal-strain effect, are calculated. The splittings and shifts of the F-phonon are derived both in the linear and nonlinear regimes. A number of quantities are predicted, and where experiments are available good agreement is found.
I. Introduction
The general equation-of-state as well as the vibrational properties of diamond at high pressures are relatively little known despite the long period of application of diamonds in highpressure experiments such as the diamond anvil cell technique. The purpose of the present work is to study the equation-of-state and the optical F-phonon of diamond subjected to hydrostatic as well as general uniaxial stresses well into the megabar regime. The diamond equation-of-state specifies the macroscopic stress as well as microscopic atomic displacements within the unit cell (internal strain), in response to an external uniform strain, i.e. hydrostatic as well as uniaxial compressions and expansions. The internal strains must be treated in any microscopic theory, but are usually difficult to determine experimentally [1]. By Taylor-expansion of the total energy the equation-of-state can be represented in terms of nth order (n = 2,3 . . . . ) elastic constants and internal strain constants. The optical F-phonon is normally threefold degenerate with a frequency of 1332.5 cm -1 (see e.g. [2]). This phonon mode has been reproduced to better than 1% in recent ab-initio calculations using the "frozen-phonon" technique [3]. The behaviour of this phonon mode under external strain is of particular interest. Hydrostatic corn-
pression will increase the phonon frequency, whereas symmetry-lowering uniaxial strains lift the degeneracy partially or fully. Very recently the hydrostatic shift (the Gr/ineisen parameter) has been determined by position-sensitive Raman scattering on diamond samples situated in a diamond anvil cell at pressures up to 300kbar [4, 5]. The present work investigates not only the hydrostatic shift, but in addition the general behaviour of the F-phonon under uniaxial pressures. Our calculations employ plane wave basis sets together with ab-initio pseudopotentials [6], and treat the electron exchange and correlation within local-density-functional theory [7]. Further details have been presented elsewhere [8, 9].
2. Linear and nonlinear elastic properties
The lattice constant, a, and the bulk modulus, B, were calculated, and the results presented in table I agree well with other ab-initio calculations [6, 10] and with experimental values. The linear and nonlinear elastic properties of diamond were investigated using the calculated macroscopic stress together with atomic forces for solving the internal-strain problem. This method has given accurate results for Si, Ge and GaAs [9]. The elastic properties are conventionally described in terms of Lagrangian strain 77and stress t
0378-4363/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
O.H. Nielsen I Optical phonons and elasticity of diamond
vector of all relative displacements of atomic pairs (relative to the perfectly strained lattice), which give the new equilibrium atomic positions. The Aj and Ajk are second-order and third-order internal strain tensors, respectively. The A 4 is essentially the Kleinman [13] internal-strain parameter ~':
Table I Equilibrium lattice constant a, and second-order and thirdorder elastic properties of diamond. ~" is the equilibrium internal-strain parameter, and Ajk/a are the third-order internal-strain parameters
a B cn
c~2 c44 r cm cn2 cl23 cl, 4 c 166 c456
A 14/a A 161a A,/a
Calculation
Experiment
Unit
3.55 4.4(1) 10.6(2) 1.25(3) 5.57(2) O. 108(1) -63(3) -8(1) 0(4) 0(3) - 26( 1 ) - 13(I) 0.39(4) 0.31(2)
3.567 4.42(5) 10.79(5) 1.24(5) 5.78(2) -
,~ Mbar Mbar Mbar Mbar
= -4AJa
-
-
0.55(1)
-
[11]. The general elastic relations are [12]: ti = E Cij~ j + 1 E Cijk~j~ k + ' ' "
(1)
U = 2 A j r l j + 1 E njkrljrh: + " " j jk
(2)
j
jk
(3)
.
For diamond (and zincblende) structure crystals the symmetry permits derivation of all properties using a few selected strains along the (100)-, (110)-, and (lll)-directions, and a change of volume [14]. The calculations are carried out for strains in the range - 0 . 2 0 ~< r / ~ +0.05. For both (110)- and ( 111 )-strains the internal displacement u is determined. The fitted cij and the ci~k are shown in table I. The case of (111)-strain is shown in fig. 1. The c~j agree with experimental values to about 5%. Only incomplete experimental data exist for comparison with the c~jk [15], and the measured combinations of c~jk (see [11]) agree with our results to within 1-2 experimental error bars. Besides, an anharmonic valence-force-field model has been fitted to various experimental data [2] and gives c~jk in good agreement with our values. The value for ~" is the first prediction for diamond by an ab-initio method, and in addition
Mbar Mbar Mbar Mbar Mbar Mbar
-
203
where c~j and c~jk are the second-order and thirdorder elastic constants, respectively, u is the 2500 \
200o
~
o.4
~
0.3
1500
0.2
100o
0.1
©
tl
o.o
500
\
-0.2
0
--500
I
-0.2
1
I
I
-0.1
(lll)-strain
I
0.0
I
~-~.
0.1
r]4
Fig. 1. Diagonal (ti) and off-diagonal (t4) stresses for diamond with strain ~74along (111). (Diagonal strain ~1 = 0.) Corresponding internal strain ~ is shown on the righthand scale. Points denote calculations, solid curves are given by eqs. (1) a n d (2) and the parameters in table I, and dotted curves are the full fourth-order fits.
O.H. Nielsen / optical phonons and elasticity of diamond
204
the third-order internal-strain parameters have been predicted for the first time.
Ajk/a
1800
1600
7
55¸ ©
3. Optical F-phonon at large stresses
~
1400 ©
In the present work the optical F-phonon is described by a 3 by 3 "phonon-frequency tensor" ~. For cubic diamond it is the phonon frequency w0 times the unit tensor. The variation of g2 up to second order in strain follows closely the analysis of stress in section 2:
1 ,~, j
]
(4)
Jk
where/2~j and g2~j~are dimensionless second-order and third-order phonon-strain tensors, and 6~ is the unit tensor. There are three and six independent elements Y20and g2~jk , respectively, which are indexed as the elastic constants. Calculations are carried out for the strains used in section 2 for the elastic properties, and the phonon frequencies are fitted to extract the g20 and O/jk (table II). The case of (lll)-strain is shown in fig. 2. The g2~jhave been measured by Grimsditch et al. [2]. Accurate determinations of the hydrostatic shift Awo/P,which to first order is O)0(~'~11 + 29212)/3B , have been made recently by Boppart et al. [5] and Hanfland et al. [4]. These data agree well with the values presented in table II. The change Awo/P has very recently been Table II Phonon frequency w0 (in cm -j) at equilibrium, its hydrostatic pressure-induced shift A%/p (in cm-~/kbar), and secondorder and third-order phonon-strain constants Calculation
Experiment
f2 u 12,2 12,,
1298. 0.282(2) - 1.33(1) -0.76(1) - 1.06(1)
1332.5 0.287 - 1.45(3) -0.76(3) -0.95(9)
0.,
0.0(1)
-
1.5(3) 2.2(9) 1.8(4) -0.6(2) -1.3(1)
-
wo
A~oo/P
12,12 J~,23
~144 'f~166 0456
0 e~ I L-
1200
55
1000 / 800
d
I
--0L )
J --0.1
I
I 0.00
I
I 0.1
(111)-st r a i n ?-]4 Fig. 2. Optical F-phonon at the strains in fig. 1. Polarization parallel and perpendicular to strain as well as average (dashed curve) are plotted. Points denote calculations, solid curves are given by eq. (4) and the parameters in table II, and dotted curves are the full fourth-order fits.
calculated by Fahy et al. [4], and the value in table II agrees well with this work. However, at large hydrostatic pressures we find a weaker sublinear behaviour. If we fit our frequencypressure results with the Birch-type equation of state [4], we find the values a = 4 4 9 G P a and b = 0.455.
Acknowledgements Discussions with Richard M. Martin, Karel Kunc and Sverre Froyen are gratefully acknowledged. Computational facilities were provided by the Scientific Committee of the CCVR (Centre de Calcul Vectoriel pour la Recherche), France. This work was supported in part by a NATO grant, and by a grant from the Danish Natural Science Research Council.
References [1] H. D'Amour, W. Denner, H. Schulz and M. Cardona, J. Appl. Crystallogr. 15 (1982) 148. [2] M.H. Grimsditch, E. Anastassakis and M. Cardona, Phys. Rev. B18 (1978) 901. [3] D. Vanderbilt, S.G. Louie and M.L. Cohen, Phys. Rev. Lett. 53 (1984) 1497. [4] M. Hanfland, K. Syassen, S. Fahy, S.G. Louie and M.L. Cohen, Phys. Rev. B31 (1985) 6896.
O.H. Nielsen / Optical phonons and elasticity of diamond [5] H. Boppart, J. van Straaten and I.F. Silvera, Phys. Rev. B32 (1985) 1423. [6] G.B. Bachelet, H.S. Greenside, G.A. Baraff and M. Schliiter, Phys. Rev. B24 (1981) 4745. [7] W. Kohn and L.J. Sham, Phys. Rev. 140 (1965) Al133; 145 (1966) B561. [8] O.H. Nielsen and R.M. Martin, Phys. Rev. Lett. 50 (1983) 697. [9] O.H. Nielsen and R.M. Martin, Phys. Rev. B, 32 (1985) 3792. in: Electronic Structure, Dynamics and Quantum Structural Properties of Condensed Matter, ed. J.T. Devreese (Plenum, New York, 1985). [10] Ref. [6]; M.T. Yin and M.L. Cohen, Phys. Rev. B24 (1981) 6121;
[11]
[12] [13] [14] [15]
205
J.R. Chelikowsky and S.G. Louie, Phys. Rev, B29 (1984) 3470; R. Biswas, R.M. Martin, R.J. Needs and O.H. Nielsen, Phys. Rev. B30 (1984) 3210. R.N. Thurston and K. Brugger, Phys. Rev. 133 (1964) A1604; K. Brugger, Phys. Rev. 133 (1964) A1611. C.S.G. Cousins, J. Phys. CI1 (1978) 4867; 15 (1982) 1857. L. Kleinman, Phys. Rev. 128 (1962) 2614. O.H. Nielsen, Phys. Rev. B, in print. H.J. McSkimin and P. Andreatch, J. Appl. Phys. 43 (1972) 2944.
OPTICAL PHONONS AND ELASTICITY OF DIAMOND AT MEGABAR STRESSES: AB-INITIO CALCULATIONS O.H. NIELSEN NORD1TA, Blegdamsve] 17, DK-2100 Copenhagen, Denmark and Laboratoire de Physique des Solides associd au CNRS, Universite P. et M. Curie, Tour 13, 4, PI. Jussieu, F-75230 Paris Cedex 05, France
The optical F-phonon and the stress-strain relations in diamond are investigated for general uniaxial and hydrostatic stresses up to several megabar. Theoretical calculations are carried out using ab-initio pseudopotentials within local-densityfunctional theory. All second-order and third-order elastic properties, including the internal-strain effect, are calculated. The splittings and shifts of the F-phonon are derived both in the linear and nonlinear regimes. A number of quantities are predicted, and where experiments are available good agreement is found.
I. Introduction
The general equation-of-state as well as the vibrational properties of diamond at high pressures are relatively little known despite the long period of application of diamonds in highpressure experiments such as the diamond anvil cell technique. The purpose of the present work is to study the equation-of-state and the optical F-phonon of diamond subjected to hydrostatic as well as general uniaxial stresses well into the megabar regime. The diamond equation-of-state specifies the macroscopic stress as well as microscopic atomic displacements within the unit cell (internal strain), in response to an external uniform strain, i.e. hydrostatic as well as uniaxial compressions and expansions. The internal strains must be treated in any microscopic theory, but are usually difficult to determine experimentally [1]. By Taylor-expansion of the total energy the equation-of-state can be represented in terms of nth order (n = 2,3 . . . . ) elastic constants and internal strain constants. The optical F-phonon is normally threefold degenerate with a frequency of 1332.5 cm -1 (see e.g. [2]). This phonon mode has been reproduced to better than 1% in recent ab-initio calculations using the "frozen-phonon" technique [3]. The behaviour of this phonon mode under external strain is of particular interest. Hydrostatic corn-
pression will increase the phonon frequency, whereas symmetry-lowering uniaxial strains lift the degeneracy partially or fully. Very recently the hydrostatic shift (the Gr/ineisen parameter) has been determined by position-sensitive Raman scattering on diamond samples situated in a diamond anvil cell at pressures up to 300kbar [4, 5]. The present work investigates not only the hydrostatic shift, but in addition the general behaviour of the F-phonon under uniaxial pressures. Our calculations employ plane wave basis sets together with ab-initio pseudopotentials [6], and treat the electron exchange and correlation within local-density-functional theory [7]. Further details have been presented elsewhere [8, 9].
2. Linear and nonlinear elastic properties
The lattice constant, a, and the bulk modulus, B, were calculated, and the results presented in table I agree well with other ab-initio calculations [6, 10] and with experimental values. The linear and nonlinear elastic properties of diamond were investigated using the calculated macroscopic stress together with atomic forces for solving the internal-strain problem. This method has given accurate results for Si, Ge and GaAs [9]. The elastic properties are conventionally described in terms of Lagrangian strain 77and stress t
0378-4363/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
O.H. Nielsen I Optical phonons and elasticity of diamond
vector of all relative displacements of atomic pairs (relative to the perfectly strained lattice), which give the new equilibrium atomic positions. The Aj and Ajk are second-order and third-order internal strain tensors, respectively. The A 4 is essentially the Kleinman [13] internal-strain parameter ~':
Table I Equilibrium lattice constant a, and second-order and thirdorder elastic properties of diamond. ~" is the equilibrium internal-strain parameter, and Ajk/a are the third-order internal-strain parameters
a B cn
c~2 c44 r cm cn2 cl23 cl, 4 c 166 c456
A 14/a A 161a A,/a
Calculation
Experiment
Unit
3.55 4.4(1) 10.6(2) 1.25(3) 5.57(2) O. 108(1) -63(3) -8(1) 0(4) 0(3) - 26( 1 ) - 13(I) 0.39(4) 0.31(2)
3.567 4.42(5) 10.79(5) 1.24(5) 5.78(2) -
,~ Mbar Mbar Mbar Mbar
= -4AJa
-
-
0.55(1)
-
[11]. The general elastic relations are [12]: ti = E Cij~ j + 1 E Cijk~j~ k + ' ' "
(1)
U = 2 A j r l j + 1 E njkrljrh: + " " j jk
(2)
j
jk
(3)
.
For diamond (and zincblende) structure crystals the symmetry permits derivation of all properties using a few selected strains along the (100)-, (110)-, and (lll)-directions, and a change of volume [14]. The calculations are carried out for strains in the range - 0 . 2 0 ~< r / ~ +0.05. For both (110)- and ( 111 )-strains the internal displacement u is determined. The fitted cij and the ci~k are shown in table I. The case of (111)-strain is shown in fig. 1. The c~j agree with experimental values to about 5%. Only incomplete experimental data exist for comparison with the c~jk [15], and the measured combinations of c~jk (see [11]) agree with our results to within 1-2 experimental error bars. Besides, an anharmonic valence-force-field model has been fitted to various experimental data [2] and gives c~jk in good agreement with our values. The value for ~" is the first prediction for diamond by an ab-initio method, and in addition
Mbar Mbar Mbar Mbar Mbar Mbar
-
203
where c~j and c~jk are the second-order and thirdorder elastic constants, respectively, u is the 2500 \
200o
~
o.4
~
0.3
1500
0.2
100o
0.1
©
tl
o.o
500
\
-0.2
0
--500
I
-0.2
1
I
I
-0.1
(lll)-strain
I
0.0
I
~-~.
0.1
r]4
Fig. 1. Diagonal (ti) and off-diagonal (t4) stresses for diamond with strain ~74along (111). (Diagonal strain ~1 = 0.) Corresponding internal strain ~ is shown on the righthand scale. Points denote calculations, solid curves are given by eqs. (1) a n d (2) and the parameters in table I, and dotted curves are the full fourth-order fits.
O.H. Nielsen / optical phonons and elasticity of diamond
204
the third-order internal-strain parameters have been predicted for the first time.
Ajk/a
1800
1600
7
55¸ ©
3. Optical F-phonon at large stresses
~
1400 ©
In the present work the optical F-phonon is described by a 3 by 3 "phonon-frequency tensor" ~. For cubic diamond it is the phonon frequency w0 times the unit tensor. The variation of g2 up to second order in strain follows closely the analysis of stress in section 2:
1 ,~, j
]
(4)
Jk
where/2~j and g2~j~are dimensionless second-order and third-order phonon-strain tensors, and 6~ is the unit tensor. There are three and six independent elements Y20and g2~jk , respectively, which are indexed as the elastic constants. Calculations are carried out for the strains used in section 2 for the elastic properties, and the phonon frequencies are fitted to extract the g20 and O/jk (table II). The case of (lll)-strain is shown in fig. 2. The g2~jhave been measured by Grimsditch et al. [2]. Accurate determinations of the hydrostatic shift Awo/P,which to first order is O)0(~'~11 + 29212)/3B , have been made recently by Boppart et al. [5] and Hanfland et al. [4]. These data agree well with the values presented in table II. The change Awo/P has very recently been Table II Phonon frequency w0 (in cm -j) at equilibrium, its hydrostatic pressure-induced shift A%/p (in cm-~/kbar), and secondorder and third-order phonon-strain constants Calculation
Experiment
f2 u 12,2 12,,
1298. 0.282(2) - 1.33(1) -0.76(1) - 1.06(1)
1332.5 0.287 - 1.45(3) -0.76(3) -0.95(9)
0.,
0.0(1)
-
1.5(3) 2.2(9) 1.8(4) -0.6(2) -1.3(1)
-
wo
A~oo/P
12,12 J~,23
~144 'f~166 0456
0 e~ I L-
1200
55
1000 / 800
d
I
--0L )
J --0.1
I
I 0.00
I
I 0.1
(111)-st r a i n ?-]4 Fig. 2. Optical F-phonon at the strains in fig. 1. Polarization parallel and perpendicular to strain as well as average (dashed curve) are plotted. Points denote calculations, solid curves are given by eq. (4) and the parameters in table II, and dotted curves are the full fourth-order fits.
calculated by Fahy et al. [4], and the value in table II agrees well with this work. However, at large hydrostatic pressures we find a weaker sublinear behaviour. If we fit our frequencypressure results with the Birch-type equation of state [4], we find the values a = 4 4 9 G P a and b = 0.455.
Acknowledgements Discussions with Richard M. Martin, Karel Kunc and Sverre Froyen are gratefully acknowledged. Computational facilities were provided by the Scientific Committee of the CCVR (Centre de Calcul Vectoriel pour la Recherche), France. This work was supported in part by a NATO grant, and by a grant from the Danish Natural Science Research Council.
References [1] H. D'Amour, W. Denner, H. Schulz and M. Cardona, J. Appl. Crystallogr. 15 (1982) 148. [2] M.H. Grimsditch, E. Anastassakis and M. Cardona, Phys. Rev. B18 (1978) 901. [3] D. Vanderbilt, S.G. Louie and M.L. Cohen, Phys. Rev. Lett. 53 (1984) 1497. [4] M. Hanfland, K. Syassen, S. Fahy, S.G. Louie and M.L. Cohen, Phys. Rev. B31 (1985) 6896.
O.H. Nielsen / Optical phonons and elasticity of diamond [5] H. Boppart, J. van Straaten and I.F. Silvera, Phys. Rev. B32 (1985) 1423. [6] G.B. Bachelet, H.S. Greenside, G.A. Baraff and M. Schliiter, Phys. Rev. B24 (1981) 4745. [7] W. Kohn and L.J. Sham, Phys. Rev. 140 (1965) Al133; 145 (1966) B561. [8] O.H. Nielsen and R.M. Martin, Phys. Rev. Lett. 50 (1983) 697. [9] O.H. Nielsen and R.M. Martin, Phys. Rev. B, 32 (1985) 3792. in: Electronic Structure, Dynamics and Quantum Structural Properties of Condensed Matter, ed. J.T. Devreese (Plenum, New York, 1985). [10] Ref. [6]; M.T. Yin and M.L. Cohen, Phys. Rev. B24 (1981) 6121;
[11]
[12] [13] [14] [15]
205
J.R. Chelikowsky and S.G. Louie, Phys. Rev, B29 (1984) 3470; R. Biswas, R.M. Martin, R.J. Needs and O.H. Nielsen, Phys. Rev. B30 (1984) 3210. R.N. Thurston and K. Brugger, Phys. Rev. 133 (1964) A1604; K. Brugger, Phys. Rev. 133 (1964) A1611. C.S.G. Cousins, J. Phys. CI1 (1978) 4867; 15 (1982) 1857. L. Kleinman, Phys. Rev. 128 (1962) 2614. O.H. Nielsen, Phys. Rev. B, in print. H.J. McSkimin and P. Andreatch, J. Appl. Phys. 43 (1972) 2944.