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Self-consistent field electronic structure calculations for compressed magnesium oxide
1. Phys. Chm.
Solids. 1978. Vol. 39. pp. 255-251.
Pergamon Press.
Printed in Great Britain
SELF-CONSISTENT FIELD ELECTRONIC STRUCTURE CALCULATIONS FOR COMPRESSED MAGNESIUM OXIDEt DAVIDA. LIBERMAN Theoretical Division, Los Alamos Scientific Laboratory, Los Alamos, NM 87545,U.S.A. (Received
30 June 1977; accepted
30 August
1977)
Abstract-Recently there have been reports that magnesium oxide and other ionic solids have been converted into metallic conductors when subjected to pressures of about 1 Mbar. Electronic structure calculations for compressed magnesium oxide indicate that the gap between valence and conduction bands does not close until much higher pressures are reached-perhaps around 50 Mbar.
In the last few years there have been several reports[l, 21 of insulating materials being converted into electrical conductors by pressures which are believed to be about a million atmospheres. Some of these materials have been ionic crystals such as magnesium oxide and sodium chloride. In these substances the energy gap between the completely occupied valence band and the empty conduction band is likely to be 8 or 9 eV which is about l/3 of the Work performed under the auspices of the Energy Research and Development Administration.
atomic unit of energy. A naive guess of the pressure needed to close the band gap would be l/3 of the atomic unit of pressure which is about 100Mbar (e2/a04= 294 Mbar). Can this simple view of the matter be supported by more detailed considerations? A series of self-consistent field electronic structure calculations have been done for magnesium oxide over a wide range of densities for both the rock salt (Bl) structure, which is observed at low pressures, and the cesium chloride (82) structure, which is likely to occur at high pressures. The calculations yield the pressure at a
Pressure
7.0
-
6.0
-
- 1.0
I
8.0
1Mbar)
I
I
7.0
6.0
Lattice
Constant
(Bohr
5.0
radii1
Fig. I. Calculated energy gaps for magnesium oxide with rock salt (Bf) and cesium chloride (82) structures. The lattice parameters and pressures are for the rock salt structure. The cesium chloride energy gap is given for a lattice parameter corresponding to the same density as the rock salt energy gap. See Table 1for the lattice parameters and pressures for the cesium chloride structure. 255
2%
D. A. LIBERMAN
Table 1. Summary of calculated pressures and energy gaps and also pressures based on experimental StrUCtUIX! TYPO
Lattice Constant (Bohr radii
Density (g/cc)
Calculated Pressure (Mbar)
Measured Pressure [al (Mbar)
data
Calculated Energy Gap (eV)
Bl 82
8.500 5.355
2.942
-0.177 -0.159
31
7.958 5.013
3.585
0.033 0.036
0.000
B2
4.64 (T,i") 2.26 (M,l')
Bl B2
7.500 4.725
0.440 0.398
0.404
4.283
6.42 fT,T) 4.11 (M,T)
Bl 82
7.250 4.567
0.835 0.741
0.810
4.741
7.67 (T,x) 4.90 (M,M)
Bl
7.081
5.088
1.197 [bl
1.200
Bl 82
7.000 4.410
5.268
1.411 1.238
7.53 U,X) 5.25 (M,M)
Bl B2
6.500 4.095
6.579
3.47 3.02
6.98 U,X) 5.82 (M,M)
I31
B2
6.000 3.780
8.365
7.86 6.80
5.91 U,X) 6.43 (M,M)
Bl BZ
5.500 3.465
10.86
16.7 IL.8
4.01 (r,x) 5.64 (&f,r)
Bl B2
5.000 3.150
14.45
37.3 33.9
0.96 (T,X) 3.26 (M,r)
Bl B2
4.500 2.835
19.83
83.5 74.7
-2.11 (r,A) -0.61 (M,f?
Bl B2
4.000 2.520
28.23
192. 173.
Bl
3.500
42.14
461.
2.93 (r,T) 0.74 (M,T)
[a] The "Measured pressures" are from the 293'K isotherm obtained by B reduction of shock wave data (Carter> _ et -__, al Ref. [31, Table 3). [bl An interpolated value of the calculated pressure to compare with the highest experinental value.
given density and a description of the band structure. There is good agreement between the calculated pressures and those based on shock wave data[3] in the pressure range covered by the experiments. The calculations are believed to be still better in the very high pressure region beyond the reach of experiments. According to the calculations the merging of the valence and conduction bands does occur at pressures comparable to the 100Mbar given by the dimensional analysis argument. A full description of the self-consistent field method would be out of place here. Briefly, what is done is: (i) solve the one-electron SchrGdinger equation for a periodic potential by the Korringa-Kohn-Rostoker method[4]; (ii) from the wave functions (both core and valence) form an electron charge density: (iii) construct from this charge density an electrostatic potential and a local density exchange potential (using the Kohn-Sham prescription tAt high densities the valence electrons become more like a degenerate electron gas. ‘The local density form of exchange becomes more accurate and correlation effects become less important. The electron density in the outer part of an atomic cell becomes more uniform and, in consequence, the “muon-tin” approximation used in the construction of the potential function should cause smaller errors.
IS]): and then repeat this whole process until the changes from one iteration to the next become small. The self-consistent field method can be derived variationally from an approximate expression for the total energy which is closely related to the exact one. Because of this, changes in the calculated total energy caused by changes in a parameter such as the lattice constant should be quite good. This is the experience in previous calcuIations done with suficient care[6-8], and it is true here as well. The comparison of caIculated and measured pressures in Table 1 bears this out. In the very high compression region the calculated pressures should be even better than in the region below 1 Mbart where comparison with experiment already shows good agreement. Other results taken from seIf-consistent field calculations for both atoms and solids are not as good as the total energies and pressures. Energy levels of atoms obtained in this manner using the local density exchange approximation are rather poor, because Koopmans’ theorem does not bold. In solids Koopmans’ theorem is formally true, but the energy levels are still poor. For magnesium oxide in the rock salt structure the calculated energy gap between the valence and conduction bands is
Self-consistent field electronic structure calculations for compressed magnesium oxide
4.6 eV while measurement gives 7.8 eV[9]. The calculations underestimate the energy gap in uncompressed magnesium oxide, and we hope they will continue to give a lower bound to the energy gap at high pressures also. The calculated energy gaps are shown in Fig. 1 and are listed in Table 1, For both structures, the closing of the band gap and the insulator to metal transition requires pressures of around 50 Mbar. WhPe this prediction could easily be off by a factor of two, we think a transition at a very much lower pressure would require some unexpected mechanism. Ack~owf~dgemen~s-The author has had helpful conversations with B. T. Matthias, M. L. Cohen, and A. R. Williams. He is particularly grateful to Williams,J. F. Janak, and V. L. Moruzzi for providing the computer program with which the calculations were done. REFERENCES 1.
Kawai N. and NishiyamaA., Proc. Japan Acad. 50, 634, 72 (1974).
257
2. Vereshchagin L. F., Yakovlev E. N., Vinogradov B. V. and Sakun V. P., JETP Letrers 20, 246 (1974); YakovlevE. N., Vereshchagin L. F., Vinogradov B. V. and Timofeev Yu. A., Sov. Tech. Phys. Letters 2, 223 (1976). 3. Carter W. J., Marsh S. P., Fritz J. N. and McQueen R. G., In Accurate Characterization of the High Pressure Environment, Gajthersbu~, (Md., U.S.A.. M-18 Oct., 1968 (Edited by E. C.
Lloyd), p. 147-58. Natl. Bur. Stand. (U.S.), Spec. Publ. 326 (1971). 4. Korringa J., Physica 13, 392 (1946); Kohn W. and Rostoker N., Phys. Rev. 94, 1I I I (1954). 5. Kohn W. and Sham L. J., Phys. Rev. 140,All33 (1%5). 6. Liberman D. A., In Les Propri&s Physiques des Solides sow Pression, Colioques latemat~oaa~ du Centre National de la Reche~che Scjentj~que, G~eaobte. 1969. pp. 35-41. Editions du
CNRS, Paris (1970). 7. Janak J. F., Moruzzi V. L. and Williams A. R., Phys. Rev. B 12, 1257(1975);Phys. Rev. 815, 2854(1977). 8. Ross M. and Johnson K. W., Phys. Rev. B2,4709 (1970). 9. Whited R. C. and Walker W. C., Phys. Rev. Letters 22, 1428 (1969).
Solids. 1978. Vol. 39. pp. 255-251.
Pergamon Press.
Printed in Great Britain
SELF-CONSISTENT FIELD ELECTRONIC STRUCTURE CALCULATIONS FOR COMPRESSED MAGNESIUM OXIDEt DAVIDA. LIBERMAN Theoretical Division, Los Alamos Scientific Laboratory, Los Alamos, NM 87545,U.S.A. (Received
30 June 1977; accepted
30 August
1977)
Abstract-Recently there have been reports that magnesium oxide and other ionic solids have been converted into metallic conductors when subjected to pressures of about 1 Mbar. Electronic structure calculations for compressed magnesium oxide indicate that the gap between valence and conduction bands does not close until much higher pressures are reached-perhaps around 50 Mbar.
In the last few years there have been several reports[l, 21 of insulating materials being converted into electrical conductors by pressures which are believed to be about a million atmospheres. Some of these materials have been ionic crystals such as magnesium oxide and sodium chloride. In these substances the energy gap between the completely occupied valence band and the empty conduction band is likely to be 8 or 9 eV which is about l/3 of the Work performed under the auspices of the Energy Research and Development Administration.
atomic unit of energy. A naive guess of the pressure needed to close the band gap would be l/3 of the atomic unit of pressure which is about 100Mbar (e2/a04= 294 Mbar). Can this simple view of the matter be supported by more detailed considerations? A series of self-consistent field electronic structure calculations have been done for magnesium oxide over a wide range of densities for both the rock salt (Bl) structure, which is observed at low pressures, and the cesium chloride (82) structure, which is likely to occur at high pressures. The calculations yield the pressure at a
Pressure
7.0
-
6.0
-
- 1.0
I
8.0
1Mbar)
I
I
7.0
6.0
Lattice
Constant
(Bohr
5.0
radii1
Fig. I. Calculated energy gaps for magnesium oxide with rock salt (Bf) and cesium chloride (82) structures. The lattice parameters and pressures are for the rock salt structure. The cesium chloride energy gap is given for a lattice parameter corresponding to the same density as the rock salt energy gap. See Table 1for the lattice parameters and pressures for the cesium chloride structure. 255
2%
D. A. LIBERMAN
Table 1. Summary of calculated pressures and energy gaps and also pressures based on experimental StrUCtUIX! TYPO
Lattice Constant (Bohr radii
Density (g/cc)
Calculated Pressure (Mbar)
Measured Pressure [al (Mbar)
data
Calculated Energy Gap (eV)
Bl 82
8.500 5.355
2.942
-0.177 -0.159
31
7.958 5.013
3.585
0.033 0.036
0.000
B2
4.64 (T,i") 2.26 (M,l')
Bl B2
7.500 4.725
0.440 0.398
0.404
4.283
6.42 fT,T) 4.11 (M,T)
Bl 82
7.250 4.567
0.835 0.741
0.810
4.741
7.67 (T,x) 4.90 (M,M)
Bl
7.081
5.088
1.197 [bl
1.200
Bl 82
7.000 4.410
5.268
1.411 1.238
7.53 U,X) 5.25 (M,M)
Bl B2
6.500 4.095
6.579
3.47 3.02
6.98 U,X) 5.82 (M,M)
I31
B2
6.000 3.780
8.365
7.86 6.80
5.91 U,X) 6.43 (M,M)
Bl BZ
5.500 3.465
10.86
16.7 IL.8
4.01 (r,x) 5.64 (&f,r)
Bl B2
5.000 3.150
14.45
37.3 33.9
0.96 (T,X) 3.26 (M,r)
Bl B2
4.500 2.835
19.83
83.5 74.7
-2.11 (r,A) -0.61 (M,f?
Bl B2
4.000 2.520
28.23
192. 173.
Bl
3.500
42.14
461.
2.93 (r,T) 0.74 (M,T)
[a] The "Measured pressures" are from the 293'K isotherm obtained by B reduction of shock wave data (Carter> _ et -__, al Ref. [31, Table 3). [bl An interpolated value of the calculated pressure to compare with the highest experinental value.
given density and a description of the band structure. There is good agreement between the calculated pressures and those based on shock wave data[3] in the pressure range covered by the experiments. The calculations are believed to be still better in the very high pressure region beyond the reach of experiments. According to the calculations the merging of the valence and conduction bands does occur at pressures comparable to the 100Mbar given by the dimensional analysis argument. A full description of the self-consistent field method would be out of place here. Briefly, what is done is: (i) solve the one-electron SchrGdinger equation for a periodic potential by the Korringa-Kohn-Rostoker method[4]; (ii) from the wave functions (both core and valence) form an electron charge density: (iii) construct from this charge density an electrostatic potential and a local density exchange potential (using the Kohn-Sham prescription tAt high densities the valence electrons become more like a degenerate electron gas. ‘The local density form of exchange becomes more accurate and correlation effects become less important. The electron density in the outer part of an atomic cell becomes more uniform and, in consequence, the “muon-tin” approximation used in the construction of the potential function should cause smaller errors.
IS]): and then repeat this whole process until the changes from one iteration to the next become small. The self-consistent field method can be derived variationally from an approximate expression for the total energy which is closely related to the exact one. Because of this, changes in the calculated total energy caused by changes in a parameter such as the lattice constant should be quite good. This is the experience in previous calcuIations done with suficient care[6-8], and it is true here as well. The comparison of caIculated and measured pressures in Table 1 bears this out. In the very high compression region the calculated pressures should be even better than in the region below 1 Mbart where comparison with experiment already shows good agreement. Other results taken from seIf-consistent field calculations for both atoms and solids are not as good as the total energies and pressures. Energy levels of atoms obtained in this manner using the local density exchange approximation are rather poor, because Koopmans’ theorem does not bold. In solids Koopmans’ theorem is formally true, but the energy levels are still poor. For magnesium oxide in the rock salt structure the calculated energy gap between the valence and conduction bands is
Self-consistent field electronic structure calculations for compressed magnesium oxide
4.6 eV while measurement gives 7.8 eV[9]. The calculations underestimate the energy gap in uncompressed magnesium oxide, and we hope they will continue to give a lower bound to the energy gap at high pressures also. The calculated energy gaps are shown in Fig. 1 and are listed in Table 1, For both structures, the closing of the band gap and the insulator to metal transition requires pressures of around 50 Mbar. WhPe this prediction could easily be off by a factor of two, we think a transition at a very much lower pressure would require some unexpected mechanism. Ack~owf~dgemen~s-The author has had helpful conversations with B. T. Matthias, M. L. Cohen, and A. R. Williams. He is particularly grateful to Williams,J. F. Janak, and V. L. Moruzzi for providing the computer program with which the calculations were done. REFERENCES 1.
Kawai N. and NishiyamaA., Proc. Japan Acad. 50, 634, 72 (1974).
257
2. Vereshchagin L. F., Yakovlev E. N., Vinogradov B. V. and Sakun V. P., JETP Letrers 20, 246 (1974); YakovlevE. N., Vereshchagin L. F., Vinogradov B. V. and Timofeev Yu. A., Sov. Tech. Phys. Letters 2, 223 (1976). 3. Carter W. J., Marsh S. P., Fritz J. N. and McQueen R. G., In Accurate Characterization of the High Pressure Environment, Gajthersbu~, (Md., U.S.A.. M-18 Oct., 1968 (Edited by E. C.
Lloyd), p. 147-58. Natl. Bur. Stand. (U.S.), Spec. Publ. 326 (1971). 4. Korringa J., Physica 13, 392 (1946); Kohn W. and Rostoker N., Phys. Rev. 94, 1I I I (1954). 5. Kohn W. and Sham L. J., Phys. Rev. 140,All33 (1%5). 6. Liberman D. A., In Les Propri&s Physiques des Solides sow Pression, Colioques latemat~oaa~ du Centre National de la Reche~che Scjentj~que, G~eaobte. 1969. pp. 35-41. Editions du
CNRS, Paris (1970). 7. Janak J. F., Moruzzi V. L. and Williams A. R., Phys. Rev. B 12, 1257(1975);Phys. Rev. 815, 2854(1977). 8. Ross M. and Johnson K. W., Phys. Rev. B2,4709 (1970). 9. Whited R. C. and Walker W. C., Phys. Rev. Letters 22, 1428 (1969).