Thermoelectric Properties of HfSe2 Monolayer

In this work detailed investigation of thermoelectric properties of HfSe2 monolayer is investigated employing Boltzmann transport formalism. We present numerical calculations of thermoelectric properties namely, electrical conductivity, σ, electronic thermal conductivity, Ke , diffusion thermopower, Sd and thermoelectric figure of merit, ZT , considering the scattering of electrons from acoustic phonons and charged impurities. We find that HfSe2 monolayer possess extremely low electronic thermal conductivity and relatively high thermopower (~ 100µV/K). Our calculations show that thermoelectric figure of merit is more than one at higher temperatures thus making HfSe2 monolayer suitable candidate for thermoelectric applications.


INTRODUCTION
Transition metal dichalcogenides (TMDCs) have received great scientific interest due to their intriguing properties in recent times.Already it is well established that these two-dimensional TMDC materials being good thermoelectric(TE) materials beside possessing suitable band gap tunabilty, low dimensionality, low thermal conductivity and ultra high electrical conductivity.Hf-based TMDC compounds have comparatively high thermoelectric efficiencies due to their very low thermal conductivity.HfSe2 monolayer is under intense investigation due to its better thermoelectric properties [1][2][3][4].HfSe2 is n-type semiconductor with narrow band gap , high dielectric constant and low thermal conductivity makes it suitable for electronic and thermoelectric applications.Further, it resembles silicon in many aspects [5,6].Hence it becomes inevitable to investigate the thermoelectric properties of HfSe2 layers.
The conversion of heat to electricity efficiency of a TE material is measured by the dimensionless TE figure of merit (ZT).The search for superior TE materials requires materials with condition such that the dimensionless figure of merit is as large as possible; , where S,  and  are, respectively, thermopower (TP), electrical conductivity(EC) and thermal conductivity (TC) of the material, T being absolute temperature.There are experimental and theoretical reports on thermoelectric (TE) properties of MX2 monolayer's largely on MoS2 and WS2 layers [7][8][9][10][11].Özbal et al [8] using first principle calculations investigated the thermoelectric properties of few TMDCs and oxides.ab initio based calculations on density functional formalism have predicted HfSe2 based nanostructures possess higher ZT [9].Khan et al have shown that ZrS2-HfSe2 heterostructures at 1220 K exhibits larger power factor [10].Ding et al have investigated the TE properties of HfSe2 and ZrSe2 monolayers and found that the TE properties of HfSe2 is superior to ZrSe2 at a higher temperatures [11].In the present work, we have made the detailed theoretical investigation of TE properties of HfSe2 monolayer using Boltzmann formalism.

THEORY
The transport equations relating transport coefficients in the presence of an electric field E and temperature gradient, T  may be written as [12]; ( ) Here, J and U are respectively electric current density and the heat current density, We consider a perfect two dimensional monolayer with thickness, t and employing the Boltzmann transport formalism, with T in the plane of the monolayer , we obtain the expression for transport coefficients Kn as Here, vk is velocity of carriers in state k with relaxation time τ(Ek).In the presence of E with 0 = T , we obtain the expression for electrical conductivity as Under open circuit condition, we obtain the expression for diffusion thermopower, Sd and electronic contribution to thermal conductivity, e, respectively, as follows The main scattering mechanisms operative in 2D TMDC systems are electron-acoustical phonon and electron-charged impurity scattering.To assess contribution from acoustic phonons and charged impurity we have made a study of thermoelectric properties of HfSe2 single layer.The expressions for the electron relaxation times, which can be obtained from the Fermi's Golden rule, are taken from literature for two-dimensional systems [13][14].In next section we present the numerical calculations.
Figure 1 represents electronic thermal conductivity, κe for the monolayer HfSe2 over wide temperature range(4<T<300K).Curves a and b depict acoustic phonons and charged impurities limited κe, respectively.Curve 1 represents overall κe.It is seen from calculations that for T<10K , the contribution from the charged impurities is significant for the parameters considered .With the increasing the temperature, we notice that the acoustic phonon scattering contribution become increasingly significant in limiting κe throughout the temperature range.Further, our calculations show that transverse acoustic phonons are more dominant than that of longitudinal phonons.Figure .2 variation of electrical conductivity in HfSe2 monolayers with temperature.Curves a and b show the contributions to σ, when the electrons are scattered from acoustic phonons and impurities, respectively.Curve 1 shows the variation of the total σ, calculated with total relaxation time given by Mattheissen's rule.It is seen that throughout the temperature range acoustic phonon scattering is dominant and only at very low temperatures (T < 10 K), there is an influence of impurities.
We have also calculated the diffusion thermopower, Sd.We find that at low temperature Sd increases linearly before saturating at higher temperature.The Sd is found to be ~ 200 µ V/K at room temperature.It is seen that the ZT increases almost linearly before saturating.At room temperature is seen that HfSe2 monolayers possesses ZT more than one which is very significant from application point of view.The detailed calculation considering contribution from lattice to TC and all relevant scattering mechanisms will be vital to gauge the thermoelectric performance of HfSe2 monolayers.

T(K)
B

Figure 1 .
Figure 1.Variation of electronic thermal conductivity, κe in HfSe2 monolayer with temperature.Curves a and b respectively, represent contribution from acoustic phonons and charged impurities.Curve 1 is total κe calculated using Matthessien rule.

Figure 2 :
Figure 2: Variation of, σ in HfSe2 with temperature .Curve a and b respectively, represents contribution from acoustic phonons and charged impurities.Curve 1 depicts is total σ.

Figure 3
Figure 3  depicts the numerical results of thermoelectric figure of merit, ZT.It is seen that the ZT increases almost linearly before saturating.At room temperature is seen that HfSe2 monolayers possesses ZT more than one which is very significant from application point of view.The detailed calculation considering contribution from lattice to TC and all relevant scattering mechanisms will be vital to gauge the thermoelectric performance of HfSe2 monolayers.