稀有金属(英文版) 2019,38(11),1015-1023

Electronical and thermoelectric properties of half-Heusler ZrNiPb under pressure in bulk and nanosheet structures for energy conversion

Hossein Asghar Rahnamaye Aliabad Zahra Nodehi Behrooz Maleki Azam Abareshi

Department of Physics,Hakim Sabzevari University

Department of Chemistry,Hakim Sabzevari University

作者简介：*Hossein Asghar Rahnamaye Aliabad,e-mail:Rahnama@hsu.ac.ir;Rahnamaye@gmail.com;

收稿日期：25 May 2018

Electronical and thermoelectric properties of half-Heusler ZrNiPb under pressure in bulk and nanosheet structures for energy conversion

Hossein Asghar Rahnamaye Aliabad Zahra Nodehi Behrooz Maleki Azam Abareshi

Department of Physics,Hakim Sabzevari University

Department of Chemistry,Hakim Sabzevari University

Abstract：

The pressure dependence of structural,electronic and thermoelectric properties of half-Heusler ZrNiPb was investigated in the bulk and nanosheet structures.In order to obtain the accurate results,the full-potential(linearized) augmented plane-wave(FP(L)APW)calculations were performed with the Perdew-BurkeErnzerhof generalized gradient approximation(PBE-GGA)and modified Becke-Johnson(mBJ) plus spin-orbit coupling(SOC).Obtained band gap values are in close agreement with the experimental results(<0.5 eV).The variations of the thermoelectric properties of the ZrNiPb were studied under different temperatures,carrier concentrations and the hydrostatic pressures.The results show that the hydrostatic pressure decreases the lattice constant value.The band structure calculations display that the band gap increases with pressure for the bulk state and it is 0 for the nanosheet of ZrNiPb [010].The highest value of figure of merit(ZT)=0.95 is found at 9.378 GPa at a carrier concentration of n=1 × 10¹⁸ cm^-3 at 250 K for p-type of ZrNiPb.

Keyword：

Electronic and thermoelectric properties; ZrNiPb; Bulk and nanosheet structures; The pressure effects;

Received： 25 May 2018

1 Introduction

Thermoelectric materials with high efficiency are used for energy conversion in the various sources such as heat,cooler and solar sources [ 1, 2, 3, 4, 5] .Half-Heusler thermoelectric compounds have been extensively studied for generation of renewable energies and converting waste heat into electricity due to their electronic structure and narrow band gap [ 6] .In the past few decades,thermoelectric properties of half-Heusler compounds have been investigated experimentally as well as theoretically.The efficiency of thermoelectric materials is defined by the figure of merit,ZT=S²σT/κ,where S,σ,T andκare the Seebeck coefficient,electrical conductivity,absolute temperature and the thermal conductivity,respectively [ 7] .

Recent theoretical studies show that the ZT values for half-Heusler compounds are 1.38,2.39 and 3.54 for NaMgX (X=Pb,Sb,As) [ 8] ,2.2 for LaPtSb [ 9] ,1.5 for FeNbSb at 1200 K [ 10] ,1.1 for FeNb_1-xTi_xSb with0.04≤x≤0.24 at 1100 K [ 11] ,1.1 for ZrNiSn at 1100 K [ 12] ,0.75 for ZrNiPb_0.38Sn_0.6Bi_0.02 [ 13] ,0.55 for ZrNiPb_0.98Bi_0.02 [ 13] and 0.3 for ZrNiPb [ 14] .Doped ZrNiPb with Hf improves the thermoelectric power factor [ 6] .The electronic structure and thermoelectric properties of ABPb (A=Hf,Zr;B=Ni,Pd) half-Heusler compounds have been investigated in Ref. [ 15] .They found that the band gap of half-Heusler compounds is narrow and the calculated power factor first increases and then decreases with the increase in carrier concentration.The Ni substitution by Pd decreases the lattice thermal conductivity of ZrPdPb compound for good thermoelectric applications [ 16] .

On the other hand,the experimental results demonstrate that the Seebeck coefficient and power factor of ZrNiPb compound are-153μV·K^-1 and 5.2μW·cm^-1·K^-2 at room temperature [ 17] .Also,the reported experimental ZT values are 0.50 at 1000 K for half-Heusler Zr_0.5Hf_0.5CoSb_0.8Sn_0.2 [ 18] ,0.35 for BiTeBr at 560 K [ 19] ,0.84 for Cu_1.99Li_0.01S at 900 K [ 20] ,0.55 for Cu_2.995Sm_0.005SbSe_3.95S_0.05 at 648 K [ 21] ,1.20 for Sedoped Bi_0.5Sb_1.5Te₃ alloys at 350 K [ 22] ,0.17-0.34 for Co_0.95Zn_0.05SbS_0.85Se_0.15 at 875 K [ 23] ,0.74 for SnSe+Ag2Se composite at 773 K [ 24] ,1.0 for (PbTe)_0.5(SnTe)_0.5at 873 K [ 25] and 0.83 for n-type Bi₂(Te,Se)3 at 373 K [ 26] .

The structural,optical and thermoelectric properties are sensitive to the applied pressure,and it was shown that physical properties of PbTe [ 27] ,p-Bi_2-xSb_xTe₃ (x=0,1.4,1.5 and 1.6) [ 28] ,CdGa₂S₄ [ 29] ,Ca₅Al₂Sb₆ [ 30] ,Sr₅Sn₂As₆ [ 31] and CoSb₃ [ 32] were significantly improved by the pressure.Therefore,in this study we calculated the thermoelectric properties of ZrNiPb halfHeusler compound under pressures from bulk to nanosheet structures.The crystal structure of half-Heusler ZrNiPb is face-centered cubic with F-43 m (No.216) space group.From the experimental results,the lattice constants and atomic positions of half-Heusler ZrNiPb were calculated under pressure [ 17] .Then by optimizing the total energy of ZrNiPb to the volume,the theoretical lattice constants were obtained at different pressures to 9.378 GPa.The nanosheet supercells of half-Heusler ZrNiPb compound were constructed based on the periodic boundary conditions with a suitable vacuum thickness along y direction [ 10] ,as shown in Fig.1.The vacuum thickness was selected about1.27 nm (it is twice the lattice parameter along the y direction) for all nanosheet supercells cloud not interact together.

2 Method of calculations

The electronic structure and thermoelectric properties were calculated by the density functional theory (DFT) approach and the Boltzmann transport theory as implemented in the Wien2k and BoltzTrap codes [ 33, 34] .The full-potential(linearized) augmented plane-wave (FP(L)APW) calculations were performed with the Perdew-Burke-Ernzerhof generalized gradient approximation (PBE-GGA) and the modified Tran-Blaha-modified Becke-Johnson (TB-mBJ)plus spin-orbit coupling (SOC) [ 35, 36] .In the present calculation,the separation energy between the core and valence states is-6.0 Ry.For calculation of the transport properties,a high-density k-point mesh with 24,000 kpoints was used in the first Brillouin zone.The iteration was halted when the energy difference was less than 0.0001Ry between steps.

Fig.1 Two crystal structures of ZrNiPb in a bulk and b nanosheet at [010] direction

The GGA method often underestimates the band gap and SOC modifies the band structures;therefore,for calculation of the thermoelectric properties we employed the TB-mBJ+SOC.A basic assumption of the rigid band approximation for doping of the compound is a shift of the Fermi level.At 0 K limit,the Fermi level (E_F) is definedas: where n and m are carrier con-centration and the band effective mass,respectively.We studied the thermoelectric properties varying with the carrier concentrations.All transport results were obtained for carrier concentrations of n=1×10¹⁸,1×10¹⁹,1×10²⁰,1×10²¹ and 1×10²² cm^-3 at 50 to 800 K.The thermoelectric results are very extended;^therefore,we just presented the Seebeck coefficient and the dimensionless ZT.The Seebeck coefficient (S) is dependent on the effective mass as follows:

where k_B,e,h,N_v,T,n,E(k), and are the Boltzmann constant,the electronic charge,the Planck constant,the band degeneracy,the temperature,the carrier concentration,the band energy,the mass components along three perpendicular directions (i=x,y,z) and band mass,respectively [ 37] .In these calculations,the relaxation time is treated as a constant and we have used the following relations:

whereαandβare tensor indices,σ_αβis the electrical conductivity tensor,N is the number ofk-points sampled,v_αβis the group velocity tensor,μis chemical potential,Ωis a unit cell volume,S is the Seebeck coefficient, is the thermal conductivity,and f_μis the Fermi-Dirac distribution function [ 38, 39] .

3 Results and discussion

3.1 Structural and electronic properties

In order to evaluate the optimized lattice constants under different pressures,we used the third-order Birch-Murnaghan equation of state as following [ 40] :

where P is the pressure,V₀,V,B₀ and B'₀ represent the reference unit cell volume,deformed unit cell volume,bulk modulus and derivative of the bulk modulus with respect to pressure,respectively.Obtained values for the bulk modulus and ground state energy are 104.65 GPa and-52,097.4587 Ry.The total energy was calculated as a function of the volume,as shown in Fig.2a.For calculation of the pressure values,we fitted the results with the Birch-Murnaghan equation of state.

The optimized lattice constants under pressure are shown in Fig.2b,which decreases with the increase in pressure.The calculated data for the changes of lattice constant were fitted with a(P)=6.254-0.018P equation.It is seen that the trend of lattice constant with pressure is linear.In order to understand the trend of energy band gap with different approximations at 0 GPa,we plotted the band structure of ZrNiPb by GGA,mBJ and GGA/mBJ+SOC in Fig.3.The first band gap (E_g) is the energy difference between the top of the valence band and the bottom of the conduction band.It can be seen that the nature of the band gap is indirect at a high symmetryΓ→X direction.Obtained band gap values are in close agreement with the band gaps reported by others [ 15] .The experimental results show that the fundamental band gap of ZrNiPb is smaller than 0.5 eV [ 17] .The other band gaps E_g1,E_g2 and E_g3 are the energy difference between the three lowest minimum energies at the K,Λand W directions.Figures 4 and 5 present the calculated band structures by mBJ+SOC and band gap values of E_g,E_g1,E_g2 and E_g3 for ZrNiPb at different hydrostatic pressures by GGA,mBJ and GGA/mBJ+SOC.It can be seen that all band gaps increase with pressure,but the increasing trend of E_g with pressure is slower than those of E_g1,E_g2 and E_g3.Figure 6 shows the effect of hydrostatic pressure on the band structure of nanosheet of ZrNiPb [ 10] .It is shown that the band gap value of the nanosheet of ZrNiPb [ 10] under the hydrostatic pressure is 0 and we can observe the metallic nature for this compound at the nanosheet structure.

The calculated density of states (DOS) is displayed in Fig.7 for the bulk and nanosheet of ZrNiPb.From Fig.7a,the bottom of the conduction band is mainly composed of the d states of Ni and Zr,while the top of the valence bands is made up the hybridized d state of Zr with the p state of Pb.There is a strong hybridization between the d states of Ni and Zr atoms.The localized Pb-d states are also observed in the energy range of-15 and-18 eV.The localized states in core regions play a key role in the optoelectronic transitions and electrical conductivity in the thermoelectric properties [ 31, 39] .Therefore,the appearance of the Pb-d state in the middle of the valence band may have an effective influence on the optical and thermoelectric properties of ZrNiPb.The calculated DOS of the nanosheet of ZrNiPb is shown in Fig.7b with a very strong hybridization between the d states of Ni and Zr atoms.It can be seen that the band gap value is 0 which confirms the previous band structure results.

Figure 8 shows the calculated total electronic density of states (DOS) under pressure for the bulk structure.The height of the peak closest to the Fermi level decreases by increasing the pressure.When the pressure is increased (by contraction of volume),the broadening of bands around the Fermi level expands,which results in the decrease of the density of states.On the other hand,the height of the peaks in the middle of the valence band (-15 to-20 eV,d state of Pb) is increased by increasing pressure.

Fig.2 a Unit cell energy as a function of unit cell volume and b pressure dependence of lattice parameter for ZrNiPb (bulk state)

Fig.3 Calculated band structures of ZrNiPb (bulk state) at zero pressure by a GGA,b GGA+SOC,c mBJ and d mBJ+SOC

Fig.4 Calculated band structures of ZrNiPb (bulk state) under different pressures by mBJ+SOC:a 0 GPa,b 5.536 GPa and c 9.378 GPa

Fig.5 Variation of band gap values of ZrNiPb (bulk state) with pressure by a GGA,b GGA+SOC,c mBJ and d mBJ+SOC

Fig.6 Calculated band structures of ZrNiPb (nanosheet at [010] direction) under different pressures by GGA

3.2 Seebeck coefficient

Figures 9 and 10 show the Seebeck coefficients as functions of temperature and carrier concentration under different pressures.Schmitt et al. [ 41] investigated thermoelectric properties of XNiSn (X=Ti,Zr,Hf) halfHeusler compounds.They showed that the GoldsmidSharp band gap,E_g=2e|S|_maxT_max where|S|_max and T_max are the maximum value of Seebeck coefficient and temperature,is a tool widely used to estimate the band gap from temperature-dependent Seebeck coefficient measurements.It was found that in the nanosheet of ZrNiPb [ 10] under the hydrostatic pressure with 0 band gaps,the maximum value of Seebeck coefficient (|S|_max) is～0;therefore,we investigated the thermoelectric properties of ZrNiPb in the bulk state.The Seebeck coefficient is positive for p-type ZrNiPb,while it is negative for n-type one.It can be seen that the Seebeck coefficient increases by increasing temperature for both n and p-types of the ZrNiPb,then decreases by increasing temperature.These behaviors are observed for n=1×10¹⁸ to 1×10²⁰.For higher carrier concentrations,n=1×10²⁰ to 1×10²²,the Seebeck coefficient increases linearly by increasing temperature for both n-and p-types of the ZrNiPb.These behaviors are due to the metallic properties of the ZrNiPb,and it is observed for other compounds such as Cd₃As₂ by Zhou et al. [ 42] .

Fig.7 Density of states (DOS) of ZrNiPb at 0 Pa for a bulk state and b nanosheet at [010] direction

Fig.8 Calculated DOS of ZrNiPb (bulk state) under different pressures by mBJ+SOC:a 0 GPa,b 5.536 GPa and c 9.378 GPa

Fig.9 Variation of Seebeck coefficient (S) for n-type of ZrNiPb (bulk state) with pressure,temperature and carrier concentration:a 0 GPa,b 5.536 GPa,and c 9.378 GPa

It can be seen that with the increase in carrier concentrations,the maximum peak of the Seebeck coefficient shifts toward higher temperature.The compressive pressures slowly increase the band gap value and partially enhance the Seebeck coefficient value.Our results show that the applied pressure does not change the band shapes in the first Brillouin zone at the bottom of the Fermi level.Therefore,the Seebeck coefficient (SC) values have little change in the p-type ZrNiPb in comparison with the n-type one.The maximal absolute value of 605μV·K^-1 is obtained for n=1×10¹⁸ cm^-3 at 300 K for p-type ZrNiPb by 9.378 GPa.Also,the maximal absolute value of-585μV·K^-1 is obtained for n=1×10¹⁸ cm^-3 at 300 K for n-type ZrNiPb by 0 GPa.

Fig.10 Variation of Seebeck coefficient (S) for p-type of ZrNiPb (bulk state) with pressure,temperature and carrier concentration:a 0 GPa,b 5.536 GPa,and c 9.378 GPa

Fig.11 Variation of figure of merit (ZT) for n-type of ZrNiPb (bulk state) with pressure,temperature and carrier concentration:a 0 GPa,b 5.536 GPa and c 9.378 GPa

Fig.12 Variation of figure of merit (ZT) for p-type of ZrNiPb (bulk state) with pressure,temperature and carrier concentration:a 0 GPa,b 5.536 GPa and c 9.378 GPa

3.3 Figure of merit

The calculated figure of merit (ZT) under pressure as a function of carrier concentration and temperature are shown in Figs.11 and 12 for n-and p-types of ZrNiPb,respectively.At lower carrier concentrations,n=1×10¹⁸and 1×10¹⁹ cm^-3,ZT first is constant,and then decreases with the increase in temperature.At n=1×10²⁰ cm^-3,the ZT increases with the increase in temperature;then,it reaches a maximum value and then decreases.It is also seen that at higher carrier concentration of n=1×10²¹and 1×10²² cm^-3,the ZT increases with the increase in temperature,but the slope of curve for n=1×10²¹ cm^-3is larger than n=1×10²² cm^-3.The highest value of ZT=0.95 at all pressures is found at 9.378 GPa at a carrier concentration ofn=1×10¹⁸ cm^-3 at 250 K for p-type of ZrNiPb.The maximum ZT value has also been observed to be 0.94 at 9.378 GPa at a carrier concentration of n=1×10¹⁸ cm^-3 at 300 K for n-type one.Thus,by increasing the pressure from 0 to 9.378 GPa at 250 and300 K,the ZT increases for p-and n-type one,respectively.

Fig.13 Variation of figure of merit (ZT) for a n-type and b p-type of ZrNiPb (bulk state) with carrier concentration at 300 K

To visualize the effect of the carrier concentration on the thermoelectric properties of the ZrNiPb,we plotted the ZT at 300 K versus the carrier concentration in Fig.13.It is achieved that the highest value for the figure of merit occurs at n=1×10¹⁸ cm^-3 and P=9.378 GPa,which yields the maximum ZT=0.948 and 0.944 for n-and p-types,respectively.

4 Conclusion

In this research,the DFT calculations were performed to investigate the structural,electronic and thermoelectric properties of ZrNiPb under pressure in the bulk and nanosheet structures.The results show that,when the pressure was applied to the bulk and nanosheet structures of the ZrNiPb,the lattice constants decrease.The nature of the band gap for the bulk state of the ZrNiPb is indirect at a high symmetryΓ→X direction,while it is 0 for the nanosheet of the ZrNiPb [ 10] with the metallic nature.Pressure has clear effects on the conduction bands,and it has little influences on the valence bands.Therefore,the applied pressure has non-negligible effects on the thermoelectric properties for the n-type and small effects in the p-type doping.Meanwhile,at 9.378 GPa,a value of ZT=0.95 is obtained for the p-type of ZrNiPb with carrier concentration of n=1×10¹⁸ cm^-3 at 250 K.Our DFT calculations suggest that the p-type of the ZrNiPb under high pressures has a better efficiency at low temperatures for thermoelectric applications.

Acknowledgements

We thank Prof.Blaha and Prof.Madsen of Vienna University of Technology,Austria,for their help in using of Wien2k and BoltzTrap packages.

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