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Accepted Manuscript
Electronic properties and the phonon band structure of PbTe
J.O. Akinlami, G.A. Adebayo, M.O. Omeike, J.A. Akindiilete, L.O. Abdulfatai
PII:
S2352-2143(17)30129-6
DOI:
10.1016/j.cocom.2017.10.005
Reference:
COCOM 106
To appear in:
Computational Condensed Matter
Received Date: 26 June 2017
Revised Date:
17 October 2017
Accepted Date: 18 October 2017
Please cite this article as: J.O. Akinlami, G.A. Adebayo, M.O. Omeike, J.A. Akindiilete, L.O. Abdulfatai,
Electronic properties and the phonon band structure of PbTe, Computational Condensed Matter (2017),
doi: 10.1016/j.cocom.2017.10.005.
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to
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ELECTRONIC PROPERTIES AND THE PHONON BAND STRUCTURE OF PbTe
J. O. Akinlami1*, G. A. Adebayo1, M. O. Omeike2 J. A. Akindiilete1 and L. O. Abdulfatai1,
1. Department of Physics, Federal University of Agriculture, Abeokuta, Ogun State, Nigeria.
2. Department of Mathematics, Federal University of Agriculture, Abeokuta, Ogun State, Nigeria
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ABSTRACT
Electronic properties of PbTe in its Rock-Salt (B1) phase have been investigated using density functional
theory (DFT) as well as the Phonon Band Structure of PbTe using density functional perturbation theory
(DFPT). The lattice parameter, bulk modulus and pressure derivative are calculated as 6.608 ?, 36.00
GPa and 3.39 respectively; The density of states (DOS) has a maximum peak of 3.2 states/eV at -3.00 eV,
the spherical charge distributions between the Pb+2 and Te-2 ions depict that the system is an ionic
compound; From the band structure, a direct band gap of 0.7968 eV was obtained which agreed well with
other calculations. We observed a large splitting value of 2.16 THZ between the longitudinal optic (LO)
and the transverse optic (TO) phonon branches along the ?- point. The vibrational density of states
(VDOS) has a maximum peak of 0.185 states/eV at 1.53 THZ. The values obtained for electronic
properties and the phonon band structure PbTe are essentially important for PbTe energy applications
such as thermoelectric, solar cells, biological imaging electroluminescent, infrared photodetectors and
optoelectronic devices.
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1. INTRODUCTION
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Lead telluride is a compound of lead and tellurium with chemical symbol PbTe. Lead Telluride is a
narrow gap semiconductor that occurs naturally as the mineral altaite. The colour of PbTe is gray and it
forms in a rock-salt face-centered cubic crystal structure (FCC) with ionic bonds. PbTe is a IV-VI
narrow-gap (Eg = 0.32 eV at 300 K) semiconductor material in the bulk [1]. This compound has some
interesting characteristics, such as low resistivity, small energy gaps, high melting points, high dielectric
constants (~1000) and large carrier mobilities [2]. PbTe is an important thermoelectric material due to its
good low vapour pressure, high melting point (924oC), and chemical stability. The best candidate for
understanding the thermoelectric behavior of PbTe under quantum confined conditions is the large
exciton Bohr radius (?B = 152 nm) which permits for a strong confinement of quantum within a large
size range [3]. So far considerable efforts have been made on enhancing the thermoelectric potential of
several existing material alloys, including silicides [4], half-Hauslers [5,6] and tellurides (e.g Bi [7], Pb
[8,9] and Ge [10,11]).
The previous theoretical studies of the phonon band structure, structural and electronic properties of
this Lead Telluride solid were made by many researchers [28 ? 32], using different methods. A direct
band gap was identified at the L-point of the high symmetry for PbTe in these theoretical calculations.
The overestimation and underestimation of the calculated lattice parameters by GGA and LDA were also
acknowledged. Likewise, the phonon dispersions identified that the Acoustic modes converge to zero
around ? point and has maximum frequency along the Optic mode with obvious LO-TO splitting.
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*Corresponding Author Email: [email protected]
In this study, we are investigating the electronic properties and phonon band structure such as lattice
parameter, bulk modulus, pressure derivatives, electronic density of states (DOS), charged density, band
structure, vibrational density of states (VDOS) and phonon dispersions of PbTe using Projected
Augumented Wave (PAW) of the Perdew-Burke-Ernerhof (PBE) exchange correlation for the
Generalized Gradient Approximation (GGA) executed in the Quantum Espresso package [12-16]. The
remaining part of the study is organized as follows. Method of calculation of electronic properties and
phonon band structure, results and discussions are presented in sections 2 and 3, while section 4 gives the
conclusions drawn from this work.
2. METHOD OF CALCULATION
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Ab-initio density functional theory (DFT) [12] and density functional perturbation (DFPT) [13-16]
within Projected Augumented Wave (PAW) [17] of the Perdew-Burke-Ernerhof (PBE) [18] exchange
correlation for the generalized gradient approximation (GGA) [19] implemented in the Quantum [20-22]
programme was used in the determination of electronic properties and phonon band structure of PbTe in
its Rock-Salt (B1) phase.
The Brillouin zone was performed automatically with 10�� k-point mesh according to the
Monkhorst-pack scheme [23].
Equation of states (consists of mathematical relationship between two or more state functions such as
pressure and volume) was used to determine the Pressure-Volume relationship which allows us to obtain
the equilibrium lattice parameter. We adopted the method proposed by Birch-Murnaghan [24-25] to fit the
data generated from energy-volume calculations:
(1)
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We later obtained the volume equation from the above pressure as follows:
(2)
Where P, V, , ,
are the Pressure, Volume, equilibrium volume, bulk modulus and bulk modulus
pressure derivative respectively.
A 10�� k-point grid and 4 phonon q-point grid with 0.06 Ryd marzari-vanderbilt smearing
were used for the self-consistent and the dynamic matrices calculations respectively. Phonon dispersions
[26] and vibrational density of states (VDOS) were later obtained from the interatomic force constants
derived by the Fourier interpolation of the dynamical matrices.
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3. RESULTS AND DISCUSSION
A. Results
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The Rock salt phase for the primitive cell of PbTe exhibits Face-center cubic (FCC) crystal structure.
The structural optimization, Electronic properties and the Phonon band structures are calculated within
the DFT and DFPT using the PAW exchange correlation functional of the GGA Approximation are
presented in this work and compared with experimental and some previous research works.
In order to determine the stable state of PbTe, the cutoff for wavefunction of our initial structure was
optimized as 60 Ryd (Fig. 1), the Brillion zone of the face centered crystal structure that is enough to
produce accurate result at minimum energy for PbTe was optimized as 10 x 10 x 10 Monkhorst park grid
(Fig. 2) and the lattice optimization was carried out to determine the stable state of the crystal with the
aid of Birch-Murnaghan [24-25] equation of states. The lattice parameter of the Rock-salt (B1) phase of
PbTe was calculated at minimum energy as 6.608 ? (Fig. 3), the pressure derivative and Bulk modulus
were also calculated as 36.000 GPa and 3.390 respectively.
The electronic band structure of PbTe in Rock-Salt (B1) Phase displays the energy range between 9.833 eV and 21.743 eV versus the wave-vectors ?, X, W, K, ?, L and U as presented in Figure 5. The
band path in Figure 4 was achieved by generating manual set of k-point with 55 k-points along the
selected points (?, X, W, K, ?, L, U) with the Xcrysden [27] package within QE. This plot mainly
displays the conduction and valence bands with Fermi level Ef at 7.4963 eV and band gap energy Eg =
0.7968 eV along L-path. The calculated density of states (DOS) shown in Figure 6 has maximum peak of
3.2 states/eV at a band energy of -3.00 eV and minimum peak of 1.37 states/eV at a band energy of 13.03
eV, several other peaks were observed at band energies of-2.82, 5.85, 6.57, 8.69, 10.76, 12.21, 15.98,
17.22, 17.93, and 19.39 eV. The result for electronic charge density at (1 1 0) plane with minimum,
maximum and number of levels are respectively 0, 0.09 and 6 is presented in Figure 7. Phonon band
structure in Figure 8(a) is plotted along the high symmetry directions of q-points
(??X?W?K???L?U) (Fig. 4). The calculated peak frequency optical mode with zero and nonzero phonon wavevectors are located at ?o = 3.35 THZ and ? = 3.49 THZ. The Brillouin zone for the FCC
lattice corresponding to 2 atom primitive cell has its vibrational modes around the ? point (q = 0). Figure
8(b) shows the vibrational density of states (VDOS) curve for PbTe. It has its first and second peaks at
phonon frequencies of 1.03 THZ and 1.40 THZ and several other peaks at phonon frequencies of 1.53,
1.58, 1.77, 1.98, 2.25, 2.40, 2.50, and 2.82 THZ. The Longitudinal Optic (LO) frequency and Transverse
Optic (TO) frequency of this system at ? point are 3.35 THZ and 1.19 THZ as displayed in Figure 9. The
longitudinal acoustic and transverse acoustic frequencies also observed at X point are 1.97 THZ and 1.61
THZ. The implications and the analysis of these are discussed in the next section.
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Figure 1: A plot of Total Energy versus ECUT curve for PbTe yielded a monotonic minimum.
Figure 2: A plot of Total Energy versus KPOINT curve for PbTe yielded a monotonic minimum.
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Figure 3: A plot of Total energy versus lattice parameter for PbTe yielded a Parabola with minimum at
6.608 ?.
Figure 4: The Band path selection of PbTe in Rock-Salt (B1) Phase along ??X?W?K???L?U
directions.
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Figure 5: The Electronic Band structure along the high symmetry points of PbTe in rock-salt (B1) Phase.
Figure 6: Density of States curve of PbTe in rock-salt (B1) phase.
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Figure 7: The Electronic charge density of PbTe in Rock-Salt (B1) Phase.
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(a)
(b)
Figure 8: (a) Phonon dispersions curve and (b) Vibrational density of states of PbTe in Rock-Salt (B1)
Phase.
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Figure 9: Phonon Dispersions curve displaying LO/TO splitting of PbTe in Rock- Salt (B1) Phase.
B. Discussion
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The calculated lattice parameter 6.608 ? agrees well with the GGA studies of Lach-hab et al. [28], John
et al. [29] and Zhang et al. [30] but overestimate the experimental lattice parameter by 2.26% which is
usual of it.
The calculated pressure derivative, 3.390 and bulk modulus, 36.00 GPa are lower than that of the other
studies reviewed but the pressure derivative agrees with the study of Lach-hab et al. [28]. The result for
the bulk modulus agrees well with the experimental value of Madelung [34]. The GGA functional was
used in both calculations and this validates the accuracy of the PAW GGA.
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Table 1: The comparison of the calculated values of equilibrium structural parameters of Lead Telluride
with previous works.
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PRESENT Theo[28]
WORK
Theo[29]
Theo[30]
Theo[31]
Exp[34]
Exp[33]
6.608
6.565
6.573
6.556
6.439
6.462
6.462
36.00
41.40
39.10
40.40
51.70
38.39
39.80
3.390
3.352
4.000
-
4.520
-
-
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The band structure of PbTe which crystallizes in the Rock-Salt structure shows the valence band,
conduction band, the position of the Fermi Level at energy scale of 7.49630 eV with the symmetry
position along the vertical lines. The valence band maximum (7.00389 eV) and the conduction band
minimum (7.80069 eV) (band edges) are located at the same point (L) in the Brillouin zone implying that
the material is a direct band-gap semiconductor. This is in agreement with previously reported results
from experimental and theoretical works shown in Table 2, except the little differences in our band-gap
value which may resulted from the different pseudopotentials used. Although, the band-gap value
predicted by Zhang et al. [30] is the closest to our calculated band-gap and the experimental value at 4.2
K also confirm the PAW GGA accuracy.
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Band-gap Energy
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Table 2: The Calculated Band-Gap Energy along L-Path compared with Theoretical and Experimental
works
0.7968
PRESENT WORK
Theo[28]
0.6450
Theo[30]
0.8060
Theo[32]
0.7300
Experimental (4.2K)[25]
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0.1900
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The electronic density of States (DOS) which represent the number of available states per unit volume
per unit energy [35] is shows the same trend of narrow band-gap semiconductor (between conduction
band minimum, 7.80069 eV and valence band maximum, 7.00389 eV) as we have in band structure.
Moreso, the structure of DOS has its peaks agreeing with Lach-hab et al. [28] and Sarankumar [36].
The dynamic charge density of the Rock-Salt is needed in order to predict the type of bond existing
between the atoms of the compounds which can be learnt from the charge density calculations.
The total electronic charge density in plane (110) was presented in order to confirm whether the
bonding type in PbTe is ionic or covalent. This figure shows that the Pb2+ and Te2- ions charge
distributions are spherical and thus supports the ionic bonding which can be depicted from the strong
sharing of electron. However, it is worthy of note that the ionic bond is very strong due to the strong
sharing between the atoms as shown above.
The phonon dispersions of the PbTe system calculated along the high symmetry
??X?W?K???L?U as shown in Figure 8(a) above reveals that the Acoustic modes converge to
zero around ? point and has maximum frequency along the Optic mode. The Vibrational Density of
States (VDOS) for PbTe is plotted on Figure 8(b) above. This VDOS has highest peak of 0.185 states/eV
and lowest peak of 0.04 states/eV at phonon frequencies of 1.53 THZ and 2.82 THZ but the majority of
the system states dominate high frequency levels which indicate that the system is stable. This is in
agreement with Zhang et al. [30] and Asaf [37].
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Figure 9 depicts that there is a large splitting (of value 2.16 THZ) between the Longitudinal Optic (LO)
branch and Transverse Optic (TO) branch at the ? point, due to the ionic charge effect of the PbTe
system, indicating that it?s a strong polar solid.
The Transverse Optic (TO) branches are very soft and there are anharmonic behaviours of the
Longitudinal acoustic (LA) branch and Transverse Optic (TO) branches couplings along the ??X and
??U directions in agreement with Zhang et al. [30] and Takuma et al. [38]. The calculated phonon
frequencies at high-symmetry points are shown in Table 3 and this is in support of Zhang et al. [30]
results and the theoretical work. The calculated longitudinal optic (LO) and transverse optic (TO)
frequency values are very close to the previously reported theoretical and experimental values. The
calculated Longitudinal Acoustic (LA) frequency and Transverse Acoustic (TA) frequency along the ?point have the same values but the values vary slightly along the L-point and X-point which could be as a
result of different pseudopotentials used [39].
LO
TO
LA
TA
PRESENT WORK
3.35
1.19
0
0
Theo[30]
3.42
1.04
0
0
3.42
0.95
0
0
PRESENT WORK
3.25
2.74
3.07
2.25
Theo[30]
3.24
2.80
2.79
1.66
Experimental[40]
2.86
2.72
2.74
1.63
PRESENT WORK
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2.39
2.17
1.97
1.61
Theo[30]
2.21
1.93
0.99
0.73
Experimental[40]
2.35
2.08
0.98
0.72
High Symmetry
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Results
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Experimental[40]
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Table 3: Calculated Phonon frequencies in THZ for PbTe at high symmetry compared with theoretical
and Experimental results.
4. CONCLUSIONS
We have investigated the Electronic properties and the phonon band structure of structurally optimized
Lead Telluride using the first principles density functional theory (DFT) and density functional
perturbation theory (DFPT) within the projected augumented wave (PAW) GGA exchange correlation
coefficient.
The equilibrium lattice parameter, bulk modulus, pressure derivative, band gap, density of states,
charge density, vibrational density of states and phonon dispersion of PbTe have been determined in its
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Rock-Salt (B1) Phase. The results obtained agree well with the existing experimental and theoretical data
but overestimated the experimental lattice parameter which confirms that GGA approximation predict a
larger crystal.
In this study, we found a direct band gap at 0.7968 eV which shows that Rock-Salt Lead Telluride is a
narrow-gap semiconductor. The calculated density of states was found to have maximum of 3.2 states/eV
at band energy of -3.00 eV.
The bonding type existing in the molecule of Lead Telluride has been studied and the spherical charge
distribution of the system has shown that it is an ionic compound.
The phonon dispersion of the system gives a very large splitting between the Transverse Optic (TO)
branch and Longitudinal Optic (LO) branch at the ?-point, 2.16 THZ, due to the ionic charge effect of the
PbTe system, indicating that it?s a strong polar solid. The Transverse Optic (TO) branches are very soft
and there are anharmonic behaviours of the Transverse Optic (TO) branch and Longitudinal Acoustic
(LA) branches couplings along the ??X and ??U directions which showed that the system is stable.
The vibrational density of states was found to have maximum value of 0.185 states/eV at a phonon
frequency of 1.53 THZ. PbTe is thus a structurally stable, narrow gap, ionic and strong polar
semiconductor material.
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