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Band structure and structural stability of the high-pressure phases of the group VIb elements
J. Phys. Chem. Solids Vol. 56, No. 314, pp. 551-554. 1995 Copyright Q 1995 Ekvier Science Ltd Printed in Great Britain. All rights reserved 0022.3697/95 $9.50 + 0.00
Pergamon
BAND STRUCTURE AND STRUCTURAL STABILITY OF THE HIGH-PRESSURE PHASES OF THE GROUP VIb ELEMENTS ATSUSHI NISHIKAWA,~
KOMAJIRO NIIZEKI,$ KAORU OHNO
KOICHI SHINDO$ and
tChiba-Keizai College, 4-3-30 Todoroki-cho, Inage-ku, Chiba 263, Japan ZDepartment of Physics, Faculty of Science, Tohoku University, Sendai 980, Japan $Coilege of Humanities and Social Science, lwate University, Morioka 020, Japan qInstitute for Materials Research, Tohoku University, Sendai 980, Japan Abstract-The group VIb elements, such as sulfur, selenium and tellurium, exhibit similar successive structural phase transitions under pressure. The electronic band-structure of the high-pressure phases of these elements is investigated on the basis of first principles calculation. In particular, the structural stability of the P-PO type rhombohedral phase of selenium has been studied, and the result suggests that the rhombohedrai structure becomes unstable owing to monoclinic distortion. Keywords: A. selenium, C. ah initio calculations, D. electronic structure, D. phase transitions.
1. INTRODUCTION It is interesting to study the electronic structure of the group VIb elements, such as sulfur, selenium and tellurium under high pressure, because they undergo several pressure-induced structural phase transitions. [i-l 1] They also show the semiconductor-metal transition under pressure 112, 131.The high-pressure metallic phases of Se and Te are superconducting at sufficiently low temperatures [ 121. Recent experimental studies show a close similarity of the structural sequence between selenium and telIurium under pressure: from a hexagonal phase to a base-centered monoclinic phase, to a base-centered orthorhombic phase (bco), to a ~-PO type rhombohedrai phase, and finally to the body-centered-cubic phase, that is, they undergo systematic transitions from a spiral chain structure under normal conditions to a layer structure during the intermediate states, and finally to a higher symmetry closed packed structure under high pressure. Moreover, the structures of the high-pressure phases of sulfur have been found to be isomorphous to those of selenium and tellurium. However, the theoretical studies so far are limited only to low pressures due to the lack of structural data. We have performed extensive calculations on the band-structure of the high-pressure phases of selenium and tellurium based on first principles [ 141.
The purpose of the present paper is to investigate the structural stability of the @-PO type rhombohedral phase of the group VIb elements, which appears at 162 GPa for S, at 60 GPa for Se, and at 11 GPa for Te, respectively, and we wish to focus our attention on the transition to the rhombohedrai phase; which structure becomes stable under lower pressure before the transition? A proposed structural model was base-centered orthorhombic for S [lo] and Se [7,9], and simple orthorhombic for Te [3]. Each of them consists of the same lattice with a puckered layer. The lattice belongs to the space group D,,, while the rhombohedral lattice corresponds to D,,,. Thus if the transition from orthorhombic to rhombohedral structure occurs as in Refs [4,9, 1I], it has to be a first order one from the consideration of the symmetry. However, the observation of the diffraction lines and the small volume discontinuity of this phase transition suggest that this transition seems to be of second order. This discrepancy attracts our strong attention. 2. CALCULATION
We have investigated the band structures and the total energies of high pressure phases of the group VIb elements on the basis of the local density-functional formalism. We employ the Ceperiey-Alder form of the exchange-correlation potential [IS, 161. In the calculation of the total energies we have nsed 551
552
A. NISHIKAWA et al.
the norm-conserving pseudopotential developed by Bachelet, Hamman and Schhiter [17]. The one electron wave functions are expanded into plane waves: the cut-off energy of the plane waves is taken to be 16 Ry. In this paper we will focus on the rhombohedral phase of selenium among the group VIb elements. Five structural phase transitions are observed in selenium under pressure, and we will deal with the Se-IV phase and the Se-V phase (the ~-PO type rhombohedral structure). We have, first of all, made calculations on the band structures of the rhombohedral structure and bco structure. We have calculated the total energy as a function of the atomic volume for the rhombohedral structure, and at each volume we have optimized the ratio c/a to find the minimum energy. On the other hand, the bco structure with a puckered layer structure has four atoms in a unit cell and the two atoms at (&-f, f ~0) are associated with each lattice point. We have optimized one internal parameter v, though we have used the experimental values of the lattice constants [9]. The effect of the change of the lattice constants at fixed volume on the total energy is less than 1 mRy in the relevant region of the lattice constants, and the change of the total energy differences between the two structures at fixed volume is also less than 1 mRy with the increase of the cut-off energy of the plane waves. Then we have also considered other crystal structures, which can be derived from the rhombohedral structure. 3. RESULTS
The calculated total energies of Se are shown as a function of the atomic volume in the volume range for the Se-IV and Se-V phase in Fig. 1. The total energies of the bco structure are higher than those of
!_,..,IpQ_j z t,,,i,,,i8j 13
14 Volume
15 (A3/atom)
Fig. 1. The calculated total energy for the rhombohedral structure (solid line) and the bco structure (solid circle) of Se. The experimental values of the lattice constants are used for the bco structure.
lo : 05
m (4
4
0
Energy (Ryd.) Fig. 2. The calculated DOS for the rhombohedral structure (a) and bco structure (b) of Se at the atomic volume 15.76 A’. The vertical line denotes the Fermi-level. The lattice constants are a = 4.086 A and c = 3.269A in the hexagonal coordinates for the rhombohedral structure and a = 4.019 A, b = 6.062 8, and c = 2.587 A, and the internal parameter v = 0.125 for the bco structure.
the rhombohedral structure and thus the bco structure is not stable compared with the rhombohedral structure, though the bco structure is suggested as a structural model for the Se-IV phase. The calculated density of states (DOS) for the rhombohedral and bco structures is shown in Fig. 2. Both the rhombohedral and bco phase are metallic, and the lower part of the DOS mainly consists of 4s bands and the next part is mainly derived from 4p bands. Then the Fermi-level stands in the middle of the 4p bands. The Fermi-level in the bco structure is located near the peak of the DOS, whereas that in the rhombohedral structure is located between the peaks. Therefore the band structure energy of the bco structure is not expected to be lower than that of the rhombohedral structure. Then we consider othLr crystal structures which are derived from the /?-PO type rhombohedral structure. From the symmetry consideration we have chosen a base-centered monoclinic structure with six atoms in a unit cell as shown in Fig. 3. The monoclinic distortion from the rhombohedral structure is described by two operations. One of these is a change of the angle /I from 90”, and the other is a displacement of the positions of atoms in a unit cell from the symmetrical position. We have calculated the band structures and the densities of states for the various monoclinic structures described above. As a result we have found that the displacement of the atomic positions along the u(x)-axis yields a significant effect on the band structure. Thus we present a typical example of the calculated results in Fig. 4, in which we have only considered the displacement of the atomic positions
Band structure and structural stability of the high-pressure phases of the group Vlb elements
553
a Fig. 3. The rhombohedral structure and the base-centered monoclinic structure. In the latter lattice, the basis consists of three atoms at (O,O,O),(213 - ZA,0, l/3 -I-v), (i/3 + U, 0, 213- t:), and they are shown by black, dark gray, and light gray circles, respectively. The atomic sites with same color lie in a same lattice plane perpendicular to the c-axis. The rhombohedral structure corresponds to the case of a/b = J3, u = v = 0. and p = 90”.
along the u-axis. The result of the rhombohedral structure is also shown. This monoclinic phase is also metallic. The lower three bands mainly consist of 4s-orbitais, and the upper bands are mainly derived from clp-orbitals. The distortion of the atomic position along the u-axis brings about a split of the band along the A and C-axes and yiefds a dip near the Fermi-level of the density of states. This suggests the possibility that the rhombohedral structure becomes unstable owing to the displacement of the atomic positions in a unit cell.
4. DISCUSSION
We have found by the total energy calculation that the proposed bco structure with a puckered layer structure is not stable as the Iower pressure phase of the rhombohedral phase. Our band structure calculation suggests that the rhombohedral structure becomes unstable owing to the monoclinic distortion. Recent diffraction experiments also suggest the appearance of a monoclinic structure before the transition to the rhombohedral phase [18]. A detailed
(a) .
(b)
dt
I- ~
-i.o*
r
Or
T
d
x,r,z
DOS Fig. 4. (a) A ty ical example of the band structure and the DOS for the monoclinic structure at the atomic volume 15.76 11 The horizontal lines denote the Fermi-level. The lattice constants are a = 7.077 8, b = 4.086 a and c = 3.269 A, the internal parameters u = 0.03 and 1’= 0.00, and the angle B = 90’. The A and the C-axis lie in the q-plane, whereas the A-axis lies along the z-axis perpendicular to the sy-plane. (b) The band structure for the rhombohedral structure at the same atomic volume reduced to the Brillouin zone of the monoclinic lattice is also shown. See also the DOS for the rhombohedral structure of Fig. 2.
554
A. NISHIKAWA
total energy calculation for the monoclinic structure of the group VIb elements is now in progress. Acknowledgement-This work is supported by a Grant-inAid for Scientific Research from the ministry of Education, Science and Culture.
REFERENCES I. Jamieson J. C. and McWhan D. B. J. Chem. Phys. 43, 1149 (1965). 2. Mao H. K., Zou G. and Bell P. M. Carnegie Inst. Washington, Yearbook 80, 283 (1980). 3. Aoki K., Shimomura 0. and Minomura S. J. Phys. Sac. Jpn 48, 551 (1980). 4. Parthasarathy G. and Holzapfel W. B. Phys. Rev. B37, 8499 (1988). 5. Parthasarathy G. and Holzapfel W. B. Phys. Rev. B38, 10105 (1988). 6. Akahama Y., Kobayashi M. and Kawamura H. Solid State Commun. 83, 273 (1992).
et al.
7. Akahama Y., Kobayashi M. and Kawamura H. So/id Slate Commun. 83, 269 (1992). 8. Kruger T. and Holzapfel W. B. Phys. Rev. Let{. 69, 305 (1992). 9. Akahama Y., Kobayashi M. and Kawamura H. Phys. Rev. B47, 20 (1993). 10. Akahama Y., Kobayashi M. and Kawamura H. Phy.~. Rev. EM, 6862 (1993). 11. Luo H.. Greene R. Cl. and Ruoff A. L. Phvs. Rev. LeIt. 71, 2943 (1993). 12. Akahama Y., Kobayashi M. and Kawamura H. Solid State Commun. 84, 803 (1992). 13. Luo H., Desgreniers S., Vohra Y. K. and Ruoff A. L. Phys. Rev. Lett. 67, 2998 (1991). 14. Nishikawa A., Niizeki K. and Shindo K. Jpn. J. Appl. Phvs. Suool. 32-l. 48 (1993). 15. Ceperley’i). M. and Alder B. J. Phys. Rev. Lert. 45, 566 (1980). 16. Perdew J. and Zunger A. Phys. Rev. 823, 5048 (1981). 17. Bachelet G. B., Hamann D. R. and Schliiter M. PhJs.7. Rev. B26, 4199 (1982). 18. Akahama Y. private communication.
Pergamon
BAND STRUCTURE AND STRUCTURAL STABILITY OF THE HIGH-PRESSURE PHASES OF THE GROUP VIb ELEMENTS ATSUSHI NISHIKAWA,~
KOMAJIRO NIIZEKI,$ KAORU OHNO
KOICHI SHINDO$ and
tChiba-Keizai College, 4-3-30 Todoroki-cho, Inage-ku, Chiba 263, Japan ZDepartment of Physics, Faculty of Science, Tohoku University, Sendai 980, Japan $Coilege of Humanities and Social Science, lwate University, Morioka 020, Japan qInstitute for Materials Research, Tohoku University, Sendai 980, Japan Abstract-The group VIb elements, such as sulfur, selenium and tellurium, exhibit similar successive structural phase transitions under pressure. The electronic band-structure of the high-pressure phases of these elements is investigated on the basis of first principles calculation. In particular, the structural stability of the P-PO type rhombohedral phase of selenium has been studied, and the result suggests that the rhombohedrai structure becomes unstable owing to monoclinic distortion. Keywords: A. selenium, C. ah initio calculations, D. electronic structure, D. phase transitions.
1. INTRODUCTION It is interesting to study the electronic structure of the group VIb elements, such as sulfur, selenium and tellurium under high pressure, because they undergo several pressure-induced structural phase transitions. [i-l 1] They also show the semiconductor-metal transition under pressure 112, 131.The high-pressure metallic phases of Se and Te are superconducting at sufficiently low temperatures [ 121. Recent experimental studies show a close similarity of the structural sequence between selenium and telIurium under pressure: from a hexagonal phase to a base-centered monoclinic phase, to a base-centered orthorhombic phase (bco), to a ~-PO type rhombohedrai phase, and finally to the body-centered-cubic phase, that is, they undergo systematic transitions from a spiral chain structure under normal conditions to a layer structure during the intermediate states, and finally to a higher symmetry closed packed structure under high pressure. Moreover, the structures of the high-pressure phases of sulfur have been found to be isomorphous to those of selenium and tellurium. However, the theoretical studies so far are limited only to low pressures due to the lack of structural data. We have performed extensive calculations on the band-structure of the high-pressure phases of selenium and tellurium based on first principles [ 141.
The purpose of the present paper is to investigate the structural stability of the @-PO type rhombohedral phase of the group VIb elements, which appears at 162 GPa for S, at 60 GPa for Se, and at 11 GPa for Te, respectively, and we wish to focus our attention on the transition to the rhombohedrai phase; which structure becomes stable under lower pressure before the transition? A proposed structural model was base-centered orthorhombic for S [lo] and Se [7,9], and simple orthorhombic for Te [3]. Each of them consists of the same lattice with a puckered layer. The lattice belongs to the space group D,,, while the rhombohedral lattice corresponds to D,,,. Thus if the transition from orthorhombic to rhombohedral structure occurs as in Refs [4,9, 1I], it has to be a first order one from the consideration of the symmetry. However, the observation of the diffraction lines and the small volume discontinuity of this phase transition suggest that this transition seems to be of second order. This discrepancy attracts our strong attention. 2. CALCULATION
We have investigated the band structures and the total energies of high pressure phases of the group VIb elements on the basis of the local density-functional formalism. We employ the Ceperiey-Alder form of the exchange-correlation potential [IS, 161. In the calculation of the total energies we have nsed 551
552
A. NISHIKAWA et al.
the norm-conserving pseudopotential developed by Bachelet, Hamman and Schhiter [17]. The one electron wave functions are expanded into plane waves: the cut-off energy of the plane waves is taken to be 16 Ry. In this paper we will focus on the rhombohedral phase of selenium among the group VIb elements. Five structural phase transitions are observed in selenium under pressure, and we will deal with the Se-IV phase and the Se-V phase (the ~-PO type rhombohedral structure). We have, first of all, made calculations on the band structures of the rhombohedral structure and bco structure. We have calculated the total energy as a function of the atomic volume for the rhombohedral structure, and at each volume we have optimized the ratio c/a to find the minimum energy. On the other hand, the bco structure with a puckered layer structure has four atoms in a unit cell and the two atoms at (&-f, f ~0) are associated with each lattice point. We have optimized one internal parameter v, though we have used the experimental values of the lattice constants [9]. The effect of the change of the lattice constants at fixed volume on the total energy is less than 1 mRy in the relevant region of the lattice constants, and the change of the total energy differences between the two structures at fixed volume is also less than 1 mRy with the increase of the cut-off energy of the plane waves. Then we have also considered other crystal structures, which can be derived from the rhombohedral structure. 3. RESULTS
The calculated total energies of Se are shown as a function of the atomic volume in the volume range for the Se-IV and Se-V phase in Fig. 1. The total energies of the bco structure are higher than those of
!_,..,IpQ_j z t,,,i,,,i8j 13
14 Volume
15 (A3/atom)
Fig. 1. The calculated total energy for the rhombohedral structure (solid line) and the bco structure (solid circle) of Se. The experimental values of the lattice constants are used for the bco structure.
lo : 05
m (4
4
0
Energy (Ryd.) Fig. 2. The calculated DOS for the rhombohedral structure (a) and bco structure (b) of Se at the atomic volume 15.76 A’. The vertical line denotes the Fermi-level. The lattice constants are a = 4.086 A and c = 3.269A in the hexagonal coordinates for the rhombohedral structure and a = 4.019 A, b = 6.062 8, and c = 2.587 A, and the internal parameter v = 0.125 for the bco structure.
the rhombohedral structure and thus the bco structure is not stable compared with the rhombohedral structure, though the bco structure is suggested as a structural model for the Se-IV phase. The calculated density of states (DOS) for the rhombohedral and bco structures is shown in Fig. 2. Both the rhombohedral and bco phase are metallic, and the lower part of the DOS mainly consists of 4s bands and the next part is mainly derived from 4p bands. Then the Fermi-level stands in the middle of the 4p bands. The Fermi-level in the bco structure is located near the peak of the DOS, whereas that in the rhombohedral structure is located between the peaks. Therefore the band structure energy of the bco structure is not expected to be lower than that of the rhombohedral structure. Then we consider othLr crystal structures which are derived from the /?-PO type rhombohedral structure. From the symmetry consideration we have chosen a base-centered monoclinic structure with six atoms in a unit cell as shown in Fig. 3. The monoclinic distortion from the rhombohedral structure is described by two operations. One of these is a change of the angle /I from 90”, and the other is a displacement of the positions of atoms in a unit cell from the symmetrical position. We have calculated the band structures and the densities of states for the various monoclinic structures described above. As a result we have found that the displacement of the atomic positions along the u(x)-axis yields a significant effect on the band structure. Thus we present a typical example of the calculated results in Fig. 4, in which we have only considered the displacement of the atomic positions
Band structure and structural stability of the high-pressure phases of the group Vlb elements
553
a Fig. 3. The rhombohedral structure and the base-centered monoclinic structure. In the latter lattice, the basis consists of three atoms at (O,O,O),(213 - ZA,0, l/3 -I-v), (i/3 + U, 0, 213- t:), and they are shown by black, dark gray, and light gray circles, respectively. The atomic sites with same color lie in a same lattice plane perpendicular to the c-axis. The rhombohedral structure corresponds to the case of a/b = J3, u = v = 0. and p = 90”.
along the u-axis. The result of the rhombohedral structure is also shown. This monoclinic phase is also metallic. The lower three bands mainly consist of 4s-orbitais, and the upper bands are mainly derived from clp-orbitals. The distortion of the atomic position along the u-axis brings about a split of the band along the A and C-axes and yiefds a dip near the Fermi-level of the density of states. This suggests the possibility that the rhombohedral structure becomes unstable owing to the displacement of the atomic positions in a unit cell.
4. DISCUSSION
We have found by the total energy calculation that the proposed bco structure with a puckered layer structure is not stable as the Iower pressure phase of the rhombohedral phase. Our band structure calculation suggests that the rhombohedral structure becomes unstable owing to the monoclinic distortion. Recent diffraction experiments also suggest the appearance of a monoclinic structure before the transition to the rhombohedral phase [18]. A detailed
(a) .
(b)
dt
I- ~
-i.o*
r
Or
T
d
x,r,z
DOS Fig. 4. (a) A ty ical example of the band structure and the DOS for the monoclinic structure at the atomic volume 15.76 11 The horizontal lines denote the Fermi-level. The lattice constants are a = 7.077 8, b = 4.086 a and c = 3.269 A, the internal parameters u = 0.03 and 1’= 0.00, and the angle B = 90’. The A and the C-axis lie in the q-plane, whereas the A-axis lies along the z-axis perpendicular to the sy-plane. (b) The band structure for the rhombohedral structure at the same atomic volume reduced to the Brillouin zone of the monoclinic lattice is also shown. See also the DOS for the rhombohedral structure of Fig. 2.
554
A. NISHIKAWA
total energy calculation for the monoclinic structure of the group VIb elements is now in progress. Acknowledgement-This work is supported by a Grant-inAid for Scientific Research from the ministry of Education, Science and Culture.
REFERENCES I. Jamieson J. C. and McWhan D. B. J. Chem. Phys. 43, 1149 (1965). 2. Mao H. K., Zou G. and Bell P. M. Carnegie Inst. Washington, Yearbook 80, 283 (1980). 3. Aoki K., Shimomura 0. and Minomura S. J. Phys. Sac. Jpn 48, 551 (1980). 4. Parthasarathy G. and Holzapfel W. B. Phys. Rev. B37, 8499 (1988). 5. Parthasarathy G. and Holzapfel W. B. Phys. Rev. B38, 10105 (1988). 6. Akahama Y., Kobayashi M. and Kawamura H. Solid State Commun. 83, 273 (1992).
et al.
7. Akahama Y., Kobayashi M. and Kawamura H. So/id Slate Commun. 83, 269 (1992). 8. Kruger T. and Holzapfel W. B. Phys. Rev. Let{. 69, 305 (1992). 9. Akahama Y., Kobayashi M. and Kawamura H. Phys. Rev. B47, 20 (1993). 10. Akahama Y., Kobayashi M. and Kawamura H. Phy.~. Rev. EM, 6862 (1993). 11. Luo H.. Greene R. Cl. and Ruoff A. L. Phvs. Rev. LeIt. 71, 2943 (1993). 12. Akahama Y., Kobayashi M. and Kawamura H. Solid State Commun. 84, 803 (1992). 13. Luo H., Desgreniers S., Vohra Y. K. and Ruoff A. L. Phys. Rev. Lett. 67, 2998 (1991). 14. Nishikawa A., Niizeki K. and Shindo K. Jpn. J. Appl. Phvs. Suool. 32-l. 48 (1993). 15. Ceperley’i). M. and Alder B. J. Phys. Rev. Lert. 45, 566 (1980). 16. Perdew J. and Zunger A. Phys. Rev. 823, 5048 (1981). 17. Bachelet G. B., Hamann D. R. and Schliiter M. PhJs.7. Rev. B26, 4199 (1982). 18. Akahama Y. private communication.