Band structure and transport studies of copper selenide: An efficient thermoelectricmaterialKriti Tyagi, Bhasker Gahtori, Sivaiah Bathula, S. Auluck, and Ajay Dhar Citation: Applied Physics Letters 105, 173905 (2014); doi: 10.1063/1.4900927 View online: http://dx.doi.org/10.1063/1.4900927 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/105/17?ver=pdfcov Published by the AIP Publishing Articles you may be interested in High pressure effect on structure, electronic structure, and thermoelectric properties of MoS 2 J. Appl. Phys. 113, 013709 (2013); 10.1063/1.4772616 Microstructure, optical property, and electronic band structure of cuprous oxide thin films J. Appl. Phys. 110, 103503 (2011); 10.1063/1.3660782 On Landauer versus Boltzmann and full band versus effective mass evaluation of thermoelectric transportcoefficients J. Appl. Phys. 107, 023707 (2010); 10.1063/1.3291120 Erratum: “Band structures and thermoelectric properties of the clathrates Ba 8 Ga 16 Ge 30 , Sr 8 Ga 16 Ge 30 ,Ba 8 Ga 16 Si 30 , and Ba 8 In 16 Sn 30 ” [J. Chem. Phys. 115, 8060 (2001)] J. Chem. Phys. 116, 9545 (2002); 10.1063/1.1473816 Band structures and thermoelectric properties of the clathrates Ba 8 Ga 16 Ge 30 , Sr 8 Ga 16 Ge 30 , Ba 8 Ga16 Si 30 , and Ba 8 In 16 Sn 30 J. Chem. Phys. 115, 8060 (2001); 10.1063/1.1397324

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Band structure and transport studies of copper selenide: An efficientthermoelectric material

Kriti Tyagi, Bhasker Gahtori, Sivaiah Bathula, S. Auluck, and Ajay Dhara)

CSIR-Network of Institutes for Solar Energy, Materials Physics and Engineering Division,CSIR-National Physical Laboratory, Dr. K. S. Krishnan Road, New Delhi 110012, India

(Received 27 August 2014; accepted 17 October 2014; published online 30 October 2014)

We report the band structure calculations for high temperature cubic phase of copper selenide

(Cu2Se) employing Hartree-Fock approximation using density functional theory within the general-

ized gradient approximation. These calculations were further extended to theoretically estimate the

electrical transport coefficients of Cu2Se employing Boltzmann transport theory, which show a rea-

sonable agreement with the corresponding experimentally measured values. The calculated trans-

port coefficients are discussed in terms of the thermoelectric (TE) performance of this material,

which suggests that Cu2Se can be a potential p-type TE material with an optimum TE performance

at a carrier concentration of �4� 6� 1021cm�3. VC 2014 AIP Publishing LLC.

[http://dx.doi.org/10.1063/1.4900927]

Copper selenide (Cu2Se) is a binary compound, which

due to its unique properties is being extensively explored as

an “green energy material” for generation of renewable

green energy employing efficient photovoltaic devices1 and

more lately as a potential material for efficient thermoelec-

tric (TE) device applications.2 Cu2Se is typically a p-type

semiconductor with an indirect band gap of 1.23 eV, which

lies close to the optimum value for solar cell applications

and thus it has shown remarkable properties suitable for

application as an energy conversion material in photovoltaic

devices.3

The superionic behaviour of Cu2Se coupled with its

liquid-like nature has led to a thermoelectric figure-of-merit

(ZT)4 of 1.5 at 1000 K. The crystal structure of Cu2Se system

is complex and it exists in two phases, a low-temperature

monoclinic phase (LT) and a high temperature cubic phase

(HT)5 with a structural phase transition at �400 K. The HT

cubic phase of Cu2Se, with Copper (Cu) atoms occupying

random positions, is mainly responsible for its high ZT.

However, the crystal structure of LT phase of Cu2Se is still

not fully understood with different researchers reporting var-

ied structures, although the crystal structure of HT phase has

been determined to be cubic6 (space group Fm�3m). The

enhanced ZT of Cu2Se is primarily governed by its electrical

transport properties, which in turn originate from the elec-

tronic band structure.

In the recent past, a few attempts have been made to

determine the electronic band structure of Cu2Se. Rasander

et al.,7 have theoretically evaluated the electronic band struc-

ture of the cubic phase of Cu2Se using density functional

theory (DFT) calculations within the local approximations

including LDA, PBE, and AM05 as well as the non-local

hybrid PBE and HSE approximations.8 These authors have

determined a band gap of 0.47 eV and 0.30 eV using hybrid

PBE functional and LDA þ U approximations, respectively.

However, these authors observed that the band gap opens up

with an unphysically large value of interaction term U

(10 eV).7 More recently, Zhang et al.9 have predicted Cu2Se

to be semi-metallic using mBJ approximation, while on

application of an onsite Columbic interaction mBJ þ U

shows a finite band gap. Despite these few recent reported

studies7,9 on the band structure determination of Cu2Se

employing different approximations, there has been no

attempt to extend these first-principle calculations to deter-

mine the electronic transport properties and compare them

with the measured experimental data. However, in order to

understand the enhanced TE performance of the cubic phase

of Cu2Se, the theoretically calculated electronic transport

properties of this material need to be evaluated from the

band structure calculations and compared with the experi-

mental values to test the validity of these theoretical models

within approximation used for these calculations.

Emboldened by this premise, we report the band struc-

ture of Cu2Se employing Hartree-Fock (HF) approximation

and further extended the DFT calculations to evaluate the

electronic transport coefficients using BoltzTraP code.10 To

corroborate our theoretical calculations and to get a clear

understanding of the transport properties of Cu2Se, we have

carried out the electronic transport measurements of as-

synthesized Cu2Se (supplementary material23) and compared

the experimental data with theoretically calculated results.

We performed the theoretical band structure calculations

employing the full potential linearized augmented plane

wave10 (FP-LAPW) method in a scalar relativistic version,

based on DFT11 as implemented in the WIEN2K code.

Cu2Se crystallizes in a cubic structure with space group

Fm-3m.12 Electronic band structure and density of states

(DOS) calculations of Cu2Se were performed using Perdew-

Burke-Ernzerhof13 (PBE) and screened PBE-GGA hybrid

functional exchange-correlation potential. GGA functional is

the preferred choice in the present study as it is simple to

evaluate and gives near-chemical accuracy. Since this poten-

tial underestimates the band gap,17 therefore hybrid PBE

functional has been employed18 in the present study. The

electronic band structures were calculated along the symme-

try directions, using the calculated lattice constants. The

Kohn-Sham equations were solved using a basis of linearized

a)Author to whom correspondence should be addressed. Electronic mail:

adhar@nplindia.org. Tel.: þ91 11 4560 9456. Fax: þ91 11 4560 9310.

0003-6951/2014/105(17)/173905/5/$30.00 VC 2014 AIP Publishing LLC105, 173905-1

APPLIED PHYSICS LETTERS 105, 173905 (2014)

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augmented plane wave, wherein we have used

RmtKmax¼ 7.0. The potential and charge density in the

muffin-tin (MT) spheres were expanded in spherical and

non-spherical harmonics with lmax¼ 10 and 6, respectively;

however, in the interstitial region, the potential and the

charge density were represented by Fourier series. It was

found that using PBE-GGA self-consistency is obtained

using 560 k points in the irreducible Brillouin zone (IBZ),

while, for hybrid PBE-GGA functional, self-consistency is

obtained using 729 k and 35 k points. We extended our DFT

studies to calculate the transport properties using 6900 k-

points in IBZ, which were calculated employing Boltzmann

theory with constant scattering time approximation (CSTA)

as implemented in the BoltzTraP code.14–16 The CSTA

directly calculates the Seebeck coefficient as a function of

doping level and temperature, with no adjustable parameters.

The convergence of the total energy in the self-consistent

calculations is taken with respect to the total charge of the

system with a tolerance 0.0001 electron charges.

The resulting band structures within a few eV from band

gap are shown in Fig. 1. We observe that the hybrid PBE

potential gives roughly the same band structure as the PBE

potential except that in the former there is a energy shift and

the band gap opens up, as opposed to PBE potential, where

the top of valence band and bottom of conduction band touch

at the C-point. This is in agreement with the previous calcula-

tions.7 More recently, Zhang et al.9 employed mBJ method

and reported a bandgap of �0.4 eV using mBJ þ U approach.

In Fig. 1(a), flat conduction band along C to L is observed for

PBE, where the bottom of conduction band is only �0.1 eV

above the Fermi level. Corresponding to these functional, the

lowest conduction bands are degenerate with two valence

bands at C. Depending on the functional used, the band next

to top valence band at C-point behaves differently. For PBE

band structure, band next to top valence band is found at

1.19 eV below the Fermi level, while in case of hybrid PBE

functional this band reaches the Fermi level rendering a band

gap of �0.5 eV, which is slightly higher than bandgap

previously reported.7,9 It can be noticed in Fig. 1 that the

upper valence band has broadened while the conduction

bands are shifted towards higher energies. Most of the other

qualitative properties of the band in the two cases remain

unchanged. The striking common feature found in the two

band structure, shown in Fig. 1, is that both the highest va-

lence band and the lowest conduction band are found along

K-line (C to L) irrespective of the approximation considered.

Figures 2(a) and 2(b) show the total and partial electron

DOS, respectively, as evaluated using the PBE and hybrid

PBE functional which yield similar results. The calculations

done using PBE functional do not yield a gap between

FIG. 1. Electronic band structure for

HT cubic phase of Cu2Se calculated

using (a) PBE and (b) hybrid PBE

functionals. Fermi level is indicated by

EF.

FIG. 2. Total density of states (DOS) and partial DOS using (a) PBE approx-

imation and (b) hybrid PBE approximation.

173905-2 Tyagi et al. Appl. Phys. Lett. 105, 173905 (2014)

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occupied valence bands and the unoccupied conduction

bands, which may be due to the fact that the optical band gap

of Cu2Se is estimated to be around 1.23 eV, which is rather

small. DFT is known to underestimate the band gaps of

semiconductors and insulators when using local and semi-

local approximations.19 In order to investigate the band gap

of Cu2Se more accurately, further DOS calculations were

carried out using more complex and accurate hybrid PBE

functional. On comparing the DOS obtained using PBE and

hybrid PBE, it is observed that the overall shape of both is

quite similar. It is clear from the results in Fig. 2 that the va-

lence band for hybrid PBE consists of two regions: middle

region in between 5 and 7.5 eV below the Fermi level, which

is a mixture of states derived from both Cu and Se, and an

upper region from the Fermi level down to 3.5 eV, which

mostly consists of Cu 3d-states with a mixing of Se 4p states.

The obvious trend in hybrid PBE potential is the shift of

states towards lower energies as compared to PBE calcula-

tion. For hybrid PBE functional, the valence region in

between �8 eV and �5 eV is shifted downwards by �1 eV,

similar is the case for upper valence region. The Cu-s states

are also present in both the regions. It can be noted that there

is a Cu-d peak in the DOS at about 3 eV below the Fermi

level, which is positioned in between two regions of Cu-d

and Se-p hybridized regions, the lower region between about

�8 and �5 eV and the upper region from about �2 eV to the

Fermi level.

Figure 3(a) shows a direct comparison between the tem-

perature dependence of theoretically calculated and experi-

mentally determined Seebeck coefficient of Cu2Se at

different doping levels. It is clear from Fig. 3(a) that experi-

mental Seebeck coefficient values show a reasonably good

agreement with those determined theoretically at a carrier

concentration of �1021 cm�3, although some deviation is

observed at extreme low and high temperatures. A slight

change of slope observed for experimental Seebeck coeffi-

cient corresponds to the phase transition at �400 K, which is

well supported by the experimental differential scanning cal-

orimetry (DSC) curve (Fig. 3(b)). It should be noted that ex-

perimental Seebeck coefficient increases with temperature

well in agreement with theoretically determined value up to

�800 K after which it shows a more rapid increase with tem-

perature, as opposed to that based on DFT calculations. The

low temperature deviation of experimental Seebeck coeffi-

cient from theoretical one could be attributed to a to b phase

transition in Cu2Se. However, the observed deviation at high

temperature can be attributed to the fast diffusion rate of Cu

on Se atoms.4 Fig. 3(c) shows that the temperature depend-

ence of the electrical conductivity, derived theoretically at

different carrier concentrations and the inset shows the corre-

sponding experimental data (inset of Fig. 3(c)) for compari-

son. The deviation in experimental electrical conductivity

from the theoretical is clearly visible in the entire tempera-

ture range. This deviation at low temperature can be attrib-

uted to the a to b-phase transition in Cu2Se, whereas the

high temperature deviation between the two could be a con-

sequence of the limitation in our theoretical model used in

our calculations. For evaluating the electrical conductivity

using DFT, a constant value of s (10�14 s) has been used for

entire temperature range; however, in practice s varies with

temperature,20 which may be responsible for the continuous

increasing deviation between the theoretical and experimen-

tal electrical conductivity values, especially at higher tem-

perature. Despite these limitations, the order of both the

calculated and experimental electrical conductivity was

found to be in the same range. However, the overall agree-

ment between experiment and theory is quite reasonable, in

view of the various approximations employed in the theoreti-

cal calculations. The ZT of a TE material is determined by

its ZT¼ S2rT/j, where S is seebeck coefficient, r is the

electrical conductivity, T is the absolute temperature, and jis the thermal conductivity which is a contribution of elec-

tronic (je) and lattice (je) components. In the case of Cu2Se,

the major contribution to total thermal conductivity comes

from lattice.4 It has been already reported21 that in the re-

gime where lattice thermal conductivity dominates, the func-

tion S2r is an indicator of the performance of ZT.

FIG. 3. (a) Comparison of the temperature dependence of experimentally

measured and theoretically determined Seebeck coefficient at different car-

rier concentrations. (b) Typical DSC curve of as-synthesized Cu2Se showing

phase transition (c) the variation of electrical conductivity calculated at dif-

ferent carrier concentrations with temperature and inset of (c) shows the var-

iation of experimental electrical conductivity with temperature.

173905-3 Tyagi et al. Appl. Phys. Lett. 105, 173905 (2014)

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Calculated transport coefficients S and S2r/s for Cu2Se

as a function of temperature for different doping concentra-

tions for p and n-type Cu2Se are shown in Figs. 4 and 5,

respectively. r/s term can be calculated directly from first-

principle electronic structure as a function of T and doping

level, where s is inverse scattering rate. It is clear from

Figs. 4 and 5 that for both p and n-type doping, the lowest

value of doping level gives highest S at all the temperature

and the value of S decreases as the doping concentration is

increased. For an individual concentration, the value of S as

well as S2r/s increases with concentration for both p-type as

well as n-type doping. For n-type doping, the value of

Seebeck coefficient is less as compared to p-type doping in

the entire temperature range for different doping concentra-

tions. However, the value of S2r/s for n-type doping at

higher temperature is slightly more than p-type doping. This

increase in S2r/s value for n-type doping can be attributed to

the enhancement in electrical conductivity for n-type doping.

By Wiedmann-Franz law, increase in r is synonymous to

increase in j. The combined effect of interdependent param-

eters, i.e., S, r, and j, indicates low value of ZT for n-type

doping. Therefore, a p-type doping is suggested for optimum

TE performance in Cu2Se.

Boltzmann transport calculations for S and TE power

factor with respect to relaxation time were carried out

employing first-principle electronic structure of HT cubic

phase of Cu2Se. Figures 6(a) and 6(b) show the behaviour of

S and S2r/s, respectively, as a function of carrier concentra-

tion at different temperatures. Further, Fig. 6(a) shows that

although S increases with increasing temperature, but it

decreases with increasing carrier concentration, in accordance

with the “Pisarenko relation.”22 Further, Fig. 6(b) reveals that

peak value of S2r/s occurs at �4� 6� 1021cm�3, depending

on the temperature, although at high temperatures the effect

of bipolar conduction is clearly evident.21 Thus, a hole

FIG. 4. Temperature dependent Seebeck coefficient for both hole and elec-

tron doping for Cu2Se system calculated using DFT.

FIG. 5. Temperature dependent S2r/s for both hole and electron doping for

Cu2Se system calculated using DFT.

FIG. 6. Theoretically calculated electrical properties of Cu2Se as a function of

carrier concentration (a) Seebeck Coefficient and (b) power factor with respect

to relaxation time (S2r/s) for hole doping. Inset of Fig. 6(b) shows magnified

view of Fig. 6 for a carrier concentration range of 1� 1020–1� 1021 at differ-

ent temperatures.

173905-4 Tyagi et al. Appl. Phys. Lett. 105, 173905 (2014)

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doping concentration of �4� 6� 1021cm�3 appears to be

optimum condition for ideal TE performance of Cu2Se. The

bipolar effect is less prominent in the curve for 300 K indicat-

ing that improved TE performance at low temperature can be

gained by lowering the carrier concentration. Further, the

magnitude of thermopower in Cu2Se can be increased by

lowering the carrier concentration until the point where bipo-

lar conduction sets in.

In summary, we have reported the electrical transport

properties for high temperature cubic phase of Cu2Se using

local PBE and non-local hybrid PBE functional of DFT.

Employing PBE functional in our theoretical calculations,

Cu2Se shows a semi-metallic behavior, while a band gap of

�0.5 eV opens up using PBE hybrid functional. However, a

three-fold degeneracy remains intact irrespective of the func-

tional used. The experimentally determined and theoretically

calculated values of Seebeck coefficient and electrical con-

ductivity show a reasonable agreement, although they exhibit

deviation at both extreme low and high temperatures.

BoltzTrap code, which was used to calculate electronic

transport properties, suggests that the optimum TE perform-

ance in Cu2Se is expected to occur at a hole carrier concen-

tration of �4� 6� 1021cm�3.

This work was supported by CSIR-TAPSUN (CSIR-

NWP 54) programme entitled “Novel approaches for solarenergy conversion under technologies and products for solarenergy utilization through networking.” The authors are

grateful to the Director, Professor R. C. Budhani, for his

constant mentoring and support for this project. The

technical support rendered by Mr. Radhey Shyam and Mr.

Naval Kishor Upadhyay is also gratefully acknowledged.

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