DFT Study on the Carrier Concentration and Temperature-Dependent Thermoelectric Properties of Antimony Selenide

Wepresent the thermoelectric properties ofAntimony Selenide (Sb 2 Se 3 ) obtained using first principles calculations.We investigated the electronic band structure using the FP-LAPW method within the sphere of the density functional theory. Thermoelectric propertieswere calculated usingBoltzTrap code using the constant relaxation time (τ) approximation at three different temperatures 300K, 600K, and 800K. Seebeck coefficient (S)was found to decreasewith increasing temperature, electrical conductivity (σ/τ)was almost constant in the entire temperature range, and electronic thermal conductivity (κ/τ) increased with increasing temperature. With increase in temperature S decreased from 1870 μV/K (at 300K) to 719μV/K (at 800K), electronic thermal conductivity increased from 1.56 × 10W/mK s (at 300K) to 3.92 × 10W/mK s (at 800K), and electrical conductivity decreased from 22 × 10/Ωms (at 300K) to 20 × 10/Ωms (at 800K). The thermoelectric properties were also calculated for different hole concentrations and the optimum concentration for a good thermoelectric performance over a large range of temperatures (from 300K to 1000K) was found for hole concentration around 10 cm.


Introduction
Thermoelectric (TE) materials are a wonderful class of materials that enable us to have direct conversion from heat to electricity and vice versa.The efficiency of a thermoelectric material is characterized by a dimensionless parameter called the figure of merit: where  is the Seebeck coefficient,  is temperature,  is electrical conductivity, and  el and   are the electron and phonon contributions to total thermal conductivity.
To have the maximum figure of merit, one needs a material with large Seebeck coefficient, large electrical conductivity, and low thermal conductivity.
The A 2 B 3 (A = Sb and Bi; B = S, Se, and Te) series consists of layered chalcogenide semiconductors that have attracted considerable interest, mainly due to their exceptional thermoelectric properties.More recently, three members of this series, namely, the Bi 2 Te 3 , Sb 2 Te 3 , and Bi 2 Se 3 compounds, were shown to exhibit topological insulating properties [1,2].This discovery has revitalized the scientific interest in these materials.At ambient conditions, the A 2 B 3 (A = Sb and Bi; B = S, Se, and Te) family is divided into two structural classes: the heavier Bi 2 Se 3 , Bi 2 Te 3 , and Sb 2 Te 3 compounds adopt a rhombohedral structure (space group R-3m), which is composed of layers of AB 6 octahedra stacked perpendicular to the long -axis; the lighter Bi 2 S 3 , Sb 2 S 3 , and Sb 2 Se 3 materials on the other hand, crystallize in an orthorhombic phase (space group Pnma, U 2 S 3 -type), made up of AB 7 and AB 7+1 polyhedra.
Sb 2 Se 3 is a promising TE material with a high Seebeck coefficient ∼1800 V/K, but its figure of merit is limited by its low electrical conductivity which is ∼10 −6 to 10 −2 Ω −1 m −1 for bulk Sb 2 Se 3 at 300 K [3][4][5].In order to improve its TE performance, various nanostructures of the material have At ambient conditions, Sb 2 Se 3 adopts an orthorhombic structure (U 2 S 3 -type, space group: 62.Pnma).There are two nonequivalent Sb cation sites in this phase.One of the Sb cations is coordinated by seven Se anions and the coordination around the second Sb site can be described as (7+1) with respect to the Se ions [9].
The fractional coordinates of atoms in a unit cell of Sb 2 Se 3 [10] are given in Table 1.

Computational Details
Sb 2 Se 3 has an orthorhombic structure belonging to 62.Pnma space group.Experimental lattice parameters were used for computation [10,11].The band structure of Sb 2 Se 3 was calculated under the purview of density functional theory using the WIEN2k [12] package which uses FP-LAPW (Full Potential Linear Augmented Plane Wave) method with local orbitals.In this method, the wave functions are expanded in spherical harmonics inside nonoverlapping atomic spheres of radius RMT and in plane waves in the interstitial region.The PBE (Perdew, Burke, and Ernzerhof) and mBJ (modified Becke-Johnson) exchange correlation potentials were used for calculations.The RK max (the product of RMT min and K max ) value was chosen to be 8.The maximum value of  ( max ) for the expansion of spherical harmonics in the atomic spheres is taken as 10.A k mesh of 3000 k points was used in the first Brillouin zone.
The transport properties such as Seebeck coefficient, electrical conductivity, and electronic thermal conductivity were calculated using the BoltzTraP code [13] that solves the semiclassical Boltzmann transport equation under the constant relaxation time () approximation.A dense k mesh of 50,000 k points in the first Brillouin zone was used for transport properties calculation.Lpfac (lattice points factor) value of 5 was used in these calculations.In the constant relaxation time approximation  is independent of  while , , and power factor  2  can only be obtained in units of .To obtain their actual values, one may either use experimental values of  if available or use available models for energy and temperature-dependent relaxation time in simpler systems like PbTe [14,15].Since there is no experimental data available on the relaxation time in this material, we only give the electrical conductivity, electronic thermal conductivity, and thermoelectric power factor scaled in terms of .

Results and Discussion
3.1.Band Structure and Density of States.The band structure of Sb 2 Se 3 was calculated along the high symmetry points of the Brillouin zone.The band structures were calculated using PBE and mBJ exchange correlation potential (Figures 1(a) and 1(b)).The band gap of Sb 2 Se 3 according to the PBE was found to be ∼0.9 eV.The experimental indirect band gap observed was around 1.2 eV [16] which is obtained from allowed indirect Tauc plot fitting of optical absorption spectrum [17].The Tauc equation used for determining the indirect band gap from optical absorption spectrum is given by where ℎ is the photon energy and   is the indirect energy band gap.So using the PBE exchange correlation potential, the band gap is underestimated.From the analysis of band structure using mBJ potential, we observe that it is an indirect band gap semiconductor with a band gap of 1.21 eV which is closer to the experimental value [18].
The total density of states was also calculated and is shown in Figure 2. From the total density of states plot, it is evident that the valence and conduction band are equally dense near the band edge.Also a band gap of 1.20 eV can be observed between the band edges.

Transport Properties as a Function of Chemical Potential.
The thermoelectric properties were calculated using the BoltzTraP code in the constant relaxation time approximation.The calculated properties were plotted as a function of chemical potential at 300 K, 600 K, and 800 K.
Before presenting the calculated results we discuss briefly what is meant by chemical potential and what factors affect its value.
The chemical potential determines the center of the Fermi-Dirac distribution function.In case of electrons, the probability of occupancy of a quantum state with energy  is given by the Fermi-Dirac distribution function () = 1/(1+exp((−)/)), where  is the chemical potential.The total number of electrons per unit volume in a crystal is given by  = ∫ ∞ 0 (, , )(), where () is the density of states and (, , ) is the Fermi-Dirac distribution function.This equation determines the chemical potential as a function of temperature for a given carrier concentration .
The position of chemical potential () plays an important role with regard to the transport properties.Position of  in the band structure determines which electrons in the valence or conduction band take part in the electronic transport and thus influences both the conductivity and the Seebeck coefficient.Doping or substitution is used to manipulate the chemical potential by varying the number of valence electrons in compounds and alloys.By definition,  = 0 corresponds to the top of the valence band in semiconductors [20].Chemical potential is determined by the temperature and by the total number of carriers.The behaviour of the chemical potential with varying temperature and doping is interesting with regard to thermoelectric properties.Figure 9 shows the calculated variation of chemical potential with respect to temperature in the case of Sb 2 Se 3 .It has a slight variation with respect to temperature.
From the Seebeck plot (Figure 3) it is clear that at  = 0, the value of Seebeck coefficient is 222 V/K (at  = 300 K).Hence, it is a -type semiconductor (since  > 0 at  = 0).Also, maximum value of  (1870 V/K) at 300 K is close to the value of 1800 V/K reported experimentally [8].Seebeck coefficient () was found to decrease with increasing temperature: with increase in temperature  decreased from 1870 V/K (at 300 K) to 719 V/K (at 800 K).
The electronic thermal conductivity (/, Figure 5) is found to increase with temperature.Electronic thermal conductivity increased from 1.56 × 10 15 W/m K s (at 300 K) to 3.92 × 10 15 W/m K s (at 800 K).
From the plot of thermoelectric power factor ((S 2 /), Figure 6), it is evident that it is highest for -type Sb (∼67 × 10 10 W m −1 K −2 s −1 ) at  = 0.12 Ry at 800 K.For -type Sb 2 Se 3, the maximum value is exhibited at  = −0.025Ry at 800 K. From Figure 7(b), we observe that till a certain temperature, the conductivity is independent of temperature (e.g., for  = 10 16 cm −3 , till 600 K the conductivity is constant).But as the temperature is increased further, bipolar transport sets up, leading to exponential increase in the conductivity [21].

Variation of Transport Properties with
The same is the case for the electronic thermal conductivity.As the concentration is increased, the recombination rate increases.Thus, the bipolar transport is set up at a considerably higher temperature.As the concentration is increased from 10 16 cm −3 to 10 18 cm −3 , the temperature at which the bipolar transport is set up increases from ∼600 K to ∼850 K.
The Seebeck coefficient (Figure 7(a)) is almost constant till a particular temperature.But as the bipolar transport is set up, the Seebeck coefficient decreases.This is because of the opposing nature of the currents setup by electrons and holes due to thermal gradient.Also, we can observe that at high temperatures, the Seebeck coefficient is negative.This is attributed to the fact that the thermally excited electrons start to dominate transport and the semiconductor changes from -type to type.
From the power factor graph (Figure 7(d)), we see a dip at 740 K for a concentration of 10 16 cm −3 .This dip is because of change in the sign of Seebeck coefficient at this temperature.Beyond this dip, the value of Seebeck coefficient is negative.
At lower temperatures, the Seebeck coefficient monotonically decreases with increasing hole concentration (Figure 8).At high temperatures, however, there is an optimum concentration of holes at which the Seebeck coefficient is highest.It corresponds to ∼5 × 10 17 cm −3 holes (at 800 K).

Conclusion
The study of Sb 2 Se 3 using the density functional theory proved insightful.It has been observed that Sb 2 Se 3 shows optimal thermoelectric performance at high temperatures (∼ 800 K) for a doping concentration of ∼10 19 cm −3 holes.

Figure 5 :Figure 6 :
Figure 5: Variation of electronic thermal conductivity (/) with respect to chemical potential.

Figure 7 :
Figure 7: (a) Seebeck coefficient, (b) electrical conductivity, (c) electronic thermal conductivity, and (d) power factor with respect to temperature at various hole doping concentrations.

Figure 8 :Figure 9 :
Figure 8: Plot of Seebeck coefficient versus hole concentration at different temperatures.