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Structural and electronic properties of BCC tellurium under high pressure
ARTICLE IN PRESS
Physica B 363 (2005) 82–87 www.elsevier.com/locate/physb
Structural and electronic properties of BCC tellurium under high pressure F. El Haj Hassan, A. Hijazi, M. Zoaeter, F. Bahsoun Faculte´ des Sciences (I), Laboratoire de Physisque de Mate´riaux (LPM), Universite´ Libanaise, EL-Hadath, Beyrouth, Liban Received 21 October 2003; received in revised form 1 March 2005; accepted 2 March 2005
Abstract Using the full potential linearized augmented plane wave (FP-LAPW) method we present the structural and electronic properties of Tellurium in the BCC phase at high pressure. Apart from the electronic band structure in the BCC phase, the density of states (DOS) and the Fermi energies (EF) at various pressures are calculated. The equilibrium lattice constant, the phase transition pressure, the bulk modulus and its pressure derivative were found to be in good agreement with the experiment. Further, we have also calculated the electronic specific heat coefficient, which decreases with an increase in pressure. r 2005 Elsevier B.V. All rights reserved. PACS: 64.60.i; 71.15.Mb; 71.15.m; 71.15.Nc; 71.20.b Keywords: Tellurium; FP-LAPW; High pressure; Structural phase transition; b-Po type rhombohedral phase; BCC phase; Electronic band structure and DOS
1. Introduction Recently, the study of the pressure-induced structural phase transitions of the group-VIb elements O, S, Se, and Te have been progressed with the development of the high-pressure X-ray diffraction experimental techniques. The group-VIb elements, selenium and tellurium, are semiconductors at ambient pressure; but transform to metallic phase Corresponding author. Tel.: +961 5 460494;
fax: +961 5 461496. E-mail address: [email protected] (F. El Haj Hassan).
and exhibit superconductivity under high pressure. The stable form of these elements is hexagonal and consists of spiral chains parallel to the c-axis [1,2]. Previous high-pressure X-ray diffraction studies have shown that hexagonal Te undergoes four structural phase transitions with increasing pressure from its most stable hexagonal phase to a monoclinic phase at 4.5 GPa [2], to an orthorhombic phase at 6 GPa [2], to a b-Po-type structural phase at 11 GPa [3] and finally to the higher symmetry BCC structure at 27 GPa [4]. Theoretical investigations of tellurium have so far been limited to the band structure calculation
0921-4526/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2005.03.004
ARTICLE IN PRESS F. El Haj Hassan et al. / Physica B 363 (2005) 82–87
[5–8], and most of them have been performed for the trigonal phase (Te-I) [7,8]. Some early studies included computations for the rhombohedral (TeIV), the BCC (Te-V) phase [9] and the hypothetical simple cubic structure [10,11]. The self-consistent calculation for a high-pressure structure was done for the monoclinic phase (Te-II), by using an orthorhombic approximation to the monoclinic unit cell, and a local pseudopotential [12]. The band structure and superconductivity of BCC phase have also been calculated [13] using the linear muffin-tin orbital (LMTO) method within atomic sphere approximation (ASA). The aim of this paper is to apply the full potential linearized augmented plane wave (FPLAPW) method to study electronic and structural properties of the Te in the BCC phase, and also to obtain the high pressure phase transition from the b-Po type rhombohedral to BCC phase. After a brief description of the calculation method, we present a calculation of high-pressure structural phase transition and the structural parameters, then we give the obtained band structure and the density of states (DOS) for Te in the BCC phase. Finally we present the results of calculated electronic specific heat coefficient. Concluding remarks are presented at the end of the paper.
2. Method of calculation A full-potential linearized augmented plane wave method was used to calculate the structural and electronic properties of the BCC phase of Tellurium. The calculation was carried out with the WIEN97 code [14]. The self-consistent potentials and charge densities were treated essentially with no shape approximation such as a muffin-tin potential [15]. The calculations were performed by the density functional theory (DFT) [16]. The exchange-correlation potential was calculated by the generalized gradient approximation (GGA) using the scheme of Predew et al. [17]. We considered electrons in [Kr](3d)10 states as core electrons; which are treated as relaxed. We have used 104 k-points (grid of 14 14 14) for the irreducible zone integration for the total energy and a plane wave cut-off of 18 Ry. The muffin-tin
83
radius of 2.1 a.u. is used in the present calculation. Both the muffin-tin radius and the number of kpoints were varied to ensure convergence.
3. Results and discussion 3.1. Total energy calculation and phase transition The b-Po type structure has a trigonal Bravais lattice and is characterized by an edge distance of trigonal cell ar and angle ar formed by two trigonal axes. Alternatively the structure is characterized by a set of the hexagonal lattice constants, namely, ah and ch. The relation between two representations is expressed by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ah ch ar ¼ þ 3, (1) 3 ah 0sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi11 2 3@ ch þ 3A . sin ðar =2Þ ¼ 2 ah
(2)
On the other hand the BCC structure is characterized by only one lattice parameter ab. The BCC lattice is a special case ofpthe ffiffiffi b-Po type lattice when ar ¼ 109:471; a ¼ ð 3=2Þab ; ah ¼ rffi pffiffiffiffiffiffiffi 2ar sin ðar =2Þ and ch =ah ¼ 3=8 ¼ 0:612: Fig. 1 shows the total energy in the b-Po type rhombohedral structure plotted as a function of the rhombohedral angle ar for the volume V r ¼ ( 3 : From this figure it can be seen that each 23:2 A SC structure has the highest energy. The energy in the SC structure forms an energy barrier between the FCC and the BCC structures. Since the induced-pressure of b-Po type structure obtained by our calculations is characterized by ar ¼ 1041; we can naturally understand that the b-Po type rhombohedral can be transformed into the BCC structure at high pressures. Fig. 2 shows the total energy of the b-Po type rhombohedral and the BCC structures as a function of volume. The curves were obtained by calculating the total energy ET at many different volumes around equilibrium and by fitting the calculated values to the Murnaghan’s equation of
ARTICLE IN PRESS F. El Haj Hassan et al. / Physica B 363 (2005) 82–87
84
Total Energy (eV)
-184800.0 -184800.5 -184801.0 sc
fcc
-184801.5 -184802.0 50
60
bcc
70 80 90 100 Angle αρ (degree)
110
120
Fig. 1. Total energy as a function of the rhombohedral angle ar : The energy curve correspond to atomic volume V r ¼ ( 3: 23:2 A
-184799.0 Total energy (eV)
-184799.5 -184800.0 -184800.5 -184801.0 -184801.5
bcc (Te-V)
-184802.0
β-Po rhombohedral (Te-IV)
-184802.5
6 8 10 12 14 16 18 20 22 24 26 28 30 32 Volume (Å3/atom) Fig. 2. Total energy as a function of volume for the b-Po type rhombohedral (the broken line) and the BCC (the solid line) structures.
state (EOS) [18] " # 0 BV ðV 0 =V ÞB V 0B þ 1 þ E0 0 , E T ðV Þ ¼ 0 B B 1 B0 1 B PðV Þ ¼ 0 B
"
d2 E T B¼V , dV 2
V0 V
B0
Table 1 Calculated and experimental volumes par atom (V r ), the corresponding pressure (P), bulk modulus (B) and pressure derivatives of bulk modulus (B0 ) of rhombohedral phase, Te-IV Te-IV
P (GPa)
Vr (A˚3)
B (GPa)
B0
Present Experiment [4] Other calculation [9]
19.5 17.5 17.5
22.5 23.1 23.6
114 115 113.7
2.5 2 4.1
(3)
# 1 ,
the bulk modulus and its pressure derivative at V ¼ V 0 ; respectively. The pressure corresponding to a volume for both b-Po type rhombohedral and BCC structure has been calculated by using Eq. (4). We have used the value of V 0 from Ref. [9]. The calculated and experimental volumes per atom of the b-Po type rhombohedral structure characterized by the experimental angle [4] ar ¼ 1041 are reported in Table 1. The value of the volume obtained from the present work is 22.5 A˚3 corresponding to a pressure of 19.5 GPa, whereas the value obtained from X-ray diffraction work of Parthasarathy et al. [4] at a pressure 17.5 GPa is 23.1 A˚3. The tendency of the calculated volume variation is in a good agreement with that of the experiment. In Table 2 we reported the calculated volumes per atom, bulk modulus and its pressure for the BCC phase. It is clearly seen, that our calculated values are in better agreement with experiment than the other calculations. The experimental volume is smaller than the GGA computed volumes. The structural phase transition is determined by calculating the Gibbs’ free energy (G) [19] for two
(4)
(5)
where V and V 0 represent the atomic volume and its value at zero pressure, respectively, B and B0 are
Table 2 Calculated volume (Vb) and the corresponding pressure (P), bulk modulus (B) and pressure derivatives of bulk modulus (B0 ) of Te in the BCC phase (Te-V) compared to experiment and other theoretical works Te-V
P (GPa)
Vb (A˚3)
B (GPa)
B0
Present Experiment [4] Other calculations
29 33 33 [9] 23 [13]
21.12 20.6 20.5 [9] 23.04 [13]
403 425 216 [9]
2.66 5.0 4.5 [9]
ARTICLE IN PRESS F. El Haj Hassan et al. / Physica B 363 (2005) 82–87
phases, which is given by G ¼ E T þ PV þ TS: Since the theoretical calculations are performed at T ¼ 0 K; Gibbs’ free energy becomes equal to the enthalpy, H ¼ E T þ PV : For a given pressure, a stable structure is one for which enthalpy has its lowest value. For this work as shown in Fig. 3, before transition pressure (Pt ) the b-Po type rhombohedral phase has lower enthalpy and hence a stable structure; but after the transition pressure, the enthalpy of the BCC phase becomes lower and hence BCC becomes the stable phase. Variation of
-20 bcc
-40 -60
β-Po rhombohedral
-80 Pt
-100
10
15
20
25 30 35 Pressure (GPa)
40
45
50
3.2. Band structure and DOS The calculated band structure of BCC Te phase at equilibrium volume is presented in Fig. 4. The overall band profile exhibits characteristic features similar to other BCC sp elements. The bands coming in the lowest energy region are caused by 5 s atomic orbitals in which they are all occupied states. The next higher energy states are mainly contributed by 5p electrons. The upper bands (conduction bands) are mainly because of antibonding 5p and 5d states. Further, the overall profile of total DOS (Fig. 4) histogram agrees with earlier pseudopotential work [12]. Fig. 5 shows the partial densities of states (DOSs), the peaks present in the lower energy region are mainly because of the 5s electron whereas upper region mainly consists of 5p and 5d orbitals.
EF
Fig. 3. The variation of enthalpies with pressure in b-Po type rhombohedral and BCC structures for Te.
total energy with volume (Fig. 2) also confirms the phase stability of the two phases. At transition pressure the enthalpies for the two structures are equal. The transition pressure Pt found by our calculation is 26 GPa. The comparisons of this value with that of experimental value (27 GPa) shows a good agreement. To our knowledge there has been so far only one X-ray diffraction study of the Te-IV2Te-V transition [4].
10
5
5
Energy (eV)
10
EF
0
0
Energy (eV)
Enthalpy (E-13500 Ry)
0
85
-5
-5
-10
-10
-15
Γ
∆
H
G
Ν
Σ
Γ
Λ P 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 DOS (states/eV.cell)
-15
Fig. 4. Band structure along the principal high-symmetry points and the total DOS of Te in the BCC phase.
ARTICLE IN PRESS F. El Haj Hassan et al. / Physica B 363 (2005) 82–87
86
d-like
0.14 0.12
Table 3 Variation of DOS at Fermi energy NðE F Þ; electronic specificheat coefficient and valence band width as a function of V=V 0 for Te in the BCC phase
0.10 V/V0
N(EF) (states/ Ry. cell)
g (mJ/k2 mol)
Band width (eV)
1.00 0.90 0.80 0.70 0.60
10.03 8.83 7.80 7.09 6.41
1.74 1.53 1.35 1.23 1.11
14.46 15.45 16.70 18.33 20.51
0.08 0.06 0.04 0.02 0.00 0.7
p-like
DOS (states/eV.cell)
0.6
3.3. Electronic specific-heat coefficient
0.5
Pressure dependence of the electronic specificheat coefficient (g) that is a function of density of states is calculated using the expression
0.4 0.3 0.2 0.1 0.0 0.6
s-like
0.5 0.4
1 (6) g ¼ p2 NðE F Þk2B N A , 3 where, NðE F Þ is the density of states at the Fermi energy, kB is the Boltzmann’s constant and N A is the Avogadro’s number. The calculated specificheat coefficients for different values of V =V 0 are given in Table 3, which shows that it decreases with increase in pressure.
0.3 0.2
4. Conclusion 0.1 0.0 -15
-10
-5
0
5
10
Fig. 5. Calculated partial DOS per formula unit of Te in the BCC phase.
Further, it is seen that the Fermi level is shifting gradually to higher energies with increase in pressure. It may be because of the increase in electron concentration under pressure. DOS at the Fermi level decreases with increase in pressure. The conduction bandwidth (which is the difference in energy between Fermi level and lowest eigen value corresponding to G-point) becomes broader with increase in pressure. The conduction bandwidth and DOS at Fermi energy are given in Table 3 for various pressures.
We have presented a theoretical analysis of the structural and electronic properties of tellurium in the BCC phase high pressures. The result regarding the high-pressure structural phase transition agrees with the experimental data. We understood that with increasing pressure the next structure after the b-Po type is the BCC, which is consistent with the experiment. We have calculated the total energies as a function of volumes, fitted them with the Murnaghan equation of state, and estimated the transition pressure. The calculated equilibrium lattice parameters, bulk modulus and its pressure derivative are in agreement with the experimental values. The contribution of every atomic orbital to the electronic structure was detailed of Te in the BCC phase. The electronic specific heat coefficient has also obtained.
ARTICLE IN PRESS F. El Haj Hassan et al. / Physica B 363 (2005) 82–87
Acknowledgements We wish to thank P. Blaha, K. Schwarz and J. Luitz for providing their WIEN97 code and their help in using it. References [1] R. Keller, W.B. Holzapfel, H. Schulz, Phys. Rev. B 15 (1977) 4404. [2] K. Aoki, O. Shimomura, S. Minomura, J. Phys. Soc. Japan 48 (1980) 551. [3] C. Jamieson, D.B. McWhan, J. Chem. Phys. 43 (1965) 1149. [4] G. Parthasarathy, W.B. Holzapfel, Phys. Rev. B 37 (1988) 8499. [5] J.D. Johannopoulos, M. Schluter, M.L. Cohen, Phys. Rev. B 11 (1975) 2186. [6] G. Czack, D. Koschel, H.H. Kugier (Eds.), Gmelin Handbook of Inorganic Chemistry, Tellurium, vol. A2 (Supp.), Springer, Berlin, 1983, pp. 107–110. [7] T. Starkloff, J.D. Johannopoulos, J. Chem. Phys. 68 (1978) 579.
87
[8] H. Isomaki, J. Von Boehm, P. Krusius, Phys. Rev. B 22 (1980) 2945. [9] F. Kirchhoff, N. Binggeli, G. Galli, Phys. Rev. B 50 (1994) 9063. [10] L.R. Newkirk, C.C. Tsuei, Phys. Rev. B 4 (1971) 2321. [11] C. Weigel, R.P. Messmer, J.W. Corbett, Phys. Status Solidi B 57 (1973) 455. [12] G. Doerre, J.D. Joannopoulos, Phys. Rev. Lett. 43 (1979) 1040. [13] G. Kalpana, B. Palanivel, B. Kousaya, M. Rajagopalan, Physica B 191 (1993) 287. [14] P. Blaha, K. Schwarz, J. Luitz, Wien97, Viena University of Technology 1997. [Improved and updated Unix version of the original copyrighted WIEN-code, which was published by in: P. Blaha, K. Schwarz, P. Sorantin, S.B. Trickey (Eds.), Commun., vol. 59, 1990, p. 399]. [15] M. Weinert, J. Math. Phys. 22 (1981) 2433. [16] W. Kohn, L.J. Sham, Phys. Rev. B 140 (1965) 1133. [17] J.P. Perdew, S. Burke, M. Erzerhof, Phys. Rev. Lett. 77 (1964) 3865. [18] F.D. Murnaghan, Proc. Natl. Acad. Sci. USA 30 (1944) 5390. [19] M. Born, K. Huang, Dynamical Theory of Crystal Lattices, Clarendon, Oxford, 1954.
Physica B 363 (2005) 82–87 www.elsevier.com/locate/physb
Structural and electronic properties of BCC tellurium under high pressure F. El Haj Hassan, A. Hijazi, M. Zoaeter, F. Bahsoun Faculte´ des Sciences (I), Laboratoire de Physisque de Mate´riaux (LPM), Universite´ Libanaise, EL-Hadath, Beyrouth, Liban Received 21 October 2003; received in revised form 1 March 2005; accepted 2 March 2005
Abstract Using the full potential linearized augmented plane wave (FP-LAPW) method we present the structural and electronic properties of Tellurium in the BCC phase at high pressure. Apart from the electronic band structure in the BCC phase, the density of states (DOS) and the Fermi energies (EF) at various pressures are calculated. The equilibrium lattice constant, the phase transition pressure, the bulk modulus and its pressure derivative were found to be in good agreement with the experiment. Further, we have also calculated the electronic specific heat coefficient, which decreases with an increase in pressure. r 2005 Elsevier B.V. All rights reserved. PACS: 64.60.i; 71.15.Mb; 71.15.m; 71.15.Nc; 71.20.b Keywords: Tellurium; FP-LAPW; High pressure; Structural phase transition; b-Po type rhombohedral phase; BCC phase; Electronic band structure and DOS
1. Introduction Recently, the study of the pressure-induced structural phase transitions of the group-VIb elements O, S, Se, and Te have been progressed with the development of the high-pressure X-ray diffraction experimental techniques. The group-VIb elements, selenium and tellurium, are semiconductors at ambient pressure; but transform to metallic phase Corresponding author. Tel.: +961 5 460494;
fax: +961 5 461496. E-mail address: [email protected] (F. El Haj Hassan).
and exhibit superconductivity under high pressure. The stable form of these elements is hexagonal and consists of spiral chains parallel to the c-axis [1,2]. Previous high-pressure X-ray diffraction studies have shown that hexagonal Te undergoes four structural phase transitions with increasing pressure from its most stable hexagonal phase to a monoclinic phase at 4.5 GPa [2], to an orthorhombic phase at 6 GPa [2], to a b-Po-type structural phase at 11 GPa [3] and finally to the higher symmetry BCC structure at 27 GPa [4]. Theoretical investigations of tellurium have so far been limited to the band structure calculation
0921-4526/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2005.03.004
ARTICLE IN PRESS F. El Haj Hassan et al. / Physica B 363 (2005) 82–87
[5–8], and most of them have been performed for the trigonal phase (Te-I) [7,8]. Some early studies included computations for the rhombohedral (TeIV), the BCC (Te-V) phase [9] and the hypothetical simple cubic structure [10,11]. The self-consistent calculation for a high-pressure structure was done for the monoclinic phase (Te-II), by using an orthorhombic approximation to the monoclinic unit cell, and a local pseudopotential [12]. The band structure and superconductivity of BCC phase have also been calculated [13] using the linear muffin-tin orbital (LMTO) method within atomic sphere approximation (ASA). The aim of this paper is to apply the full potential linearized augmented plane wave (FPLAPW) method to study electronic and structural properties of the Te in the BCC phase, and also to obtain the high pressure phase transition from the b-Po type rhombohedral to BCC phase. After a brief description of the calculation method, we present a calculation of high-pressure structural phase transition and the structural parameters, then we give the obtained band structure and the density of states (DOS) for Te in the BCC phase. Finally we present the results of calculated electronic specific heat coefficient. Concluding remarks are presented at the end of the paper.
2. Method of calculation A full-potential linearized augmented plane wave method was used to calculate the structural and electronic properties of the BCC phase of Tellurium. The calculation was carried out with the WIEN97 code [14]. The self-consistent potentials and charge densities were treated essentially with no shape approximation such as a muffin-tin potential [15]. The calculations were performed by the density functional theory (DFT) [16]. The exchange-correlation potential was calculated by the generalized gradient approximation (GGA) using the scheme of Predew et al. [17]. We considered electrons in [Kr](3d)10 states as core electrons; which are treated as relaxed. We have used 104 k-points (grid of 14 14 14) for the irreducible zone integration for the total energy and a plane wave cut-off of 18 Ry. The muffin-tin
83
radius of 2.1 a.u. is used in the present calculation. Both the muffin-tin radius and the number of kpoints were varied to ensure convergence.
3. Results and discussion 3.1. Total energy calculation and phase transition The b-Po type structure has a trigonal Bravais lattice and is characterized by an edge distance of trigonal cell ar and angle ar formed by two trigonal axes. Alternatively the structure is characterized by a set of the hexagonal lattice constants, namely, ah and ch. The relation between two representations is expressed by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ah ch ar ¼ þ 3, (1) 3 ah 0sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi11 2 3@ ch þ 3A . sin ðar =2Þ ¼ 2 ah
(2)
On the other hand the BCC structure is characterized by only one lattice parameter ab. The BCC lattice is a special case ofpthe ffiffiffi b-Po type lattice when ar ¼ 109:471; a ¼ ð 3=2Þab ; ah ¼ rffi pffiffiffiffiffiffiffi 2ar sin ðar =2Þ and ch =ah ¼ 3=8 ¼ 0:612: Fig. 1 shows the total energy in the b-Po type rhombohedral structure plotted as a function of the rhombohedral angle ar for the volume V r ¼ ( 3 : From this figure it can be seen that each 23:2 A SC structure has the highest energy. The energy in the SC structure forms an energy barrier between the FCC and the BCC structures. Since the induced-pressure of b-Po type structure obtained by our calculations is characterized by ar ¼ 1041; we can naturally understand that the b-Po type rhombohedral can be transformed into the BCC structure at high pressures. Fig. 2 shows the total energy of the b-Po type rhombohedral and the BCC structures as a function of volume. The curves were obtained by calculating the total energy ET at many different volumes around equilibrium and by fitting the calculated values to the Murnaghan’s equation of
ARTICLE IN PRESS F. El Haj Hassan et al. / Physica B 363 (2005) 82–87
84
Total Energy (eV)
-184800.0 -184800.5 -184801.0 sc
fcc
-184801.5 -184802.0 50
60
bcc
70 80 90 100 Angle αρ (degree)
110
120
Fig. 1. Total energy as a function of the rhombohedral angle ar : The energy curve correspond to atomic volume V r ¼ ( 3: 23:2 A
-184799.0 Total energy (eV)
-184799.5 -184800.0 -184800.5 -184801.0 -184801.5
bcc (Te-V)
-184802.0
β-Po rhombohedral (Te-IV)
-184802.5
6 8 10 12 14 16 18 20 22 24 26 28 30 32 Volume (Å3/atom) Fig. 2. Total energy as a function of volume for the b-Po type rhombohedral (the broken line) and the BCC (the solid line) structures.
state (EOS) [18] " # 0 BV ðV 0 =V ÞB V 0B þ 1 þ E0 0 , E T ðV Þ ¼ 0 B B 1 B0 1 B PðV Þ ¼ 0 B
"
d2 E T B¼V , dV 2
V0 V
B0
Table 1 Calculated and experimental volumes par atom (V r ), the corresponding pressure (P), bulk modulus (B) and pressure derivatives of bulk modulus (B0 ) of rhombohedral phase, Te-IV Te-IV
P (GPa)
Vr (A˚3)
B (GPa)
B0
Present Experiment [4] Other calculation [9]
19.5 17.5 17.5
22.5 23.1 23.6
114 115 113.7
2.5 2 4.1
(3)
# 1 ,
the bulk modulus and its pressure derivative at V ¼ V 0 ; respectively. The pressure corresponding to a volume for both b-Po type rhombohedral and BCC structure has been calculated by using Eq. (4). We have used the value of V 0 from Ref. [9]. The calculated and experimental volumes per atom of the b-Po type rhombohedral structure characterized by the experimental angle [4] ar ¼ 1041 are reported in Table 1. The value of the volume obtained from the present work is 22.5 A˚3 corresponding to a pressure of 19.5 GPa, whereas the value obtained from X-ray diffraction work of Parthasarathy et al. [4] at a pressure 17.5 GPa is 23.1 A˚3. The tendency of the calculated volume variation is in a good agreement with that of the experiment. In Table 2 we reported the calculated volumes per atom, bulk modulus and its pressure for the BCC phase. It is clearly seen, that our calculated values are in better agreement with experiment than the other calculations. The experimental volume is smaller than the GGA computed volumes. The structural phase transition is determined by calculating the Gibbs’ free energy (G) [19] for two
(4)
(5)
where V and V 0 represent the atomic volume and its value at zero pressure, respectively, B and B0 are
Table 2 Calculated volume (Vb) and the corresponding pressure (P), bulk modulus (B) and pressure derivatives of bulk modulus (B0 ) of Te in the BCC phase (Te-V) compared to experiment and other theoretical works Te-V
P (GPa)
Vb (A˚3)
B (GPa)
B0
Present Experiment [4] Other calculations
29 33 33 [9] 23 [13]
21.12 20.6 20.5 [9] 23.04 [13]
403 425 216 [9]
2.66 5.0 4.5 [9]
ARTICLE IN PRESS F. El Haj Hassan et al. / Physica B 363 (2005) 82–87
phases, which is given by G ¼ E T þ PV þ TS: Since the theoretical calculations are performed at T ¼ 0 K; Gibbs’ free energy becomes equal to the enthalpy, H ¼ E T þ PV : For a given pressure, a stable structure is one for which enthalpy has its lowest value. For this work as shown in Fig. 3, before transition pressure (Pt ) the b-Po type rhombohedral phase has lower enthalpy and hence a stable structure; but after the transition pressure, the enthalpy of the BCC phase becomes lower and hence BCC becomes the stable phase. Variation of
-20 bcc
-40 -60
β-Po rhombohedral
-80 Pt
-100
10
15
20
25 30 35 Pressure (GPa)
40
45
50
3.2. Band structure and DOS The calculated band structure of BCC Te phase at equilibrium volume is presented in Fig. 4. The overall band profile exhibits characteristic features similar to other BCC sp elements. The bands coming in the lowest energy region are caused by 5 s atomic orbitals in which they are all occupied states. The next higher energy states are mainly contributed by 5p electrons. The upper bands (conduction bands) are mainly because of antibonding 5p and 5d states. Further, the overall profile of total DOS (Fig. 4) histogram agrees with earlier pseudopotential work [12]. Fig. 5 shows the partial densities of states (DOSs), the peaks present in the lower energy region are mainly because of the 5s electron whereas upper region mainly consists of 5p and 5d orbitals.
EF
Fig. 3. The variation of enthalpies with pressure in b-Po type rhombohedral and BCC structures for Te.
total energy with volume (Fig. 2) also confirms the phase stability of the two phases. At transition pressure the enthalpies for the two structures are equal. The transition pressure Pt found by our calculation is 26 GPa. The comparisons of this value with that of experimental value (27 GPa) shows a good agreement. To our knowledge there has been so far only one X-ray diffraction study of the Te-IV2Te-V transition [4].
10
5
5
Energy (eV)
10
EF
0
0
Energy (eV)
Enthalpy (E-13500 Ry)
0
85
-5
-5
-10
-10
-15
Γ
∆
H
G
Ν
Σ
Γ
Λ P 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 DOS (states/eV.cell)
-15
Fig. 4. Band structure along the principal high-symmetry points and the total DOS of Te in the BCC phase.
ARTICLE IN PRESS F. El Haj Hassan et al. / Physica B 363 (2005) 82–87
86
d-like
0.14 0.12
Table 3 Variation of DOS at Fermi energy NðE F Þ; electronic specificheat coefficient and valence band width as a function of V=V 0 for Te in the BCC phase
0.10 V/V0
N(EF) (states/ Ry. cell)
g (mJ/k2 mol)
Band width (eV)
1.00 0.90 0.80 0.70 0.60
10.03 8.83 7.80 7.09 6.41
1.74 1.53 1.35 1.23 1.11
14.46 15.45 16.70 18.33 20.51
0.08 0.06 0.04 0.02 0.00 0.7
p-like
DOS (states/eV.cell)
0.6
3.3. Electronic specific-heat coefficient
0.5
Pressure dependence of the electronic specificheat coefficient (g) that is a function of density of states is calculated using the expression
0.4 0.3 0.2 0.1 0.0 0.6
s-like
0.5 0.4
1 (6) g ¼ p2 NðE F Þk2B N A , 3 where, NðE F Þ is the density of states at the Fermi energy, kB is the Boltzmann’s constant and N A is the Avogadro’s number. The calculated specificheat coefficients for different values of V =V 0 are given in Table 3, which shows that it decreases with increase in pressure.
0.3 0.2
4. Conclusion 0.1 0.0 -15
-10
-5
0
5
10
Fig. 5. Calculated partial DOS per formula unit of Te in the BCC phase.
Further, it is seen that the Fermi level is shifting gradually to higher energies with increase in pressure. It may be because of the increase in electron concentration under pressure. DOS at the Fermi level decreases with increase in pressure. The conduction bandwidth (which is the difference in energy between Fermi level and lowest eigen value corresponding to G-point) becomes broader with increase in pressure. The conduction bandwidth and DOS at Fermi energy are given in Table 3 for various pressures.
We have presented a theoretical analysis of the structural and electronic properties of tellurium in the BCC phase high pressures. The result regarding the high-pressure structural phase transition agrees with the experimental data. We understood that with increasing pressure the next structure after the b-Po type is the BCC, which is consistent with the experiment. We have calculated the total energies as a function of volumes, fitted them with the Murnaghan equation of state, and estimated the transition pressure. The calculated equilibrium lattice parameters, bulk modulus and its pressure derivative are in agreement with the experimental values. The contribution of every atomic orbital to the electronic structure was detailed of Te in the BCC phase. The electronic specific heat coefficient has also obtained.
ARTICLE IN PRESS F. El Haj Hassan et al. / Physica B 363 (2005) 82–87
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