碲化汞晶格热导反常压应变效应的第一性原理研究

姜恩来， 欧阳滔

(湘潭大学 物理与光电工程学院，湖南 湘潭 411105)

Effectively engineering the lattice thermal conductivity (hereafter denotedasκp) of materials is the key in current thermal science community, such as thermoelectric application[1-4]. Among the numerous methods, applying pressure[5-20]is regarded as one of the most worthwhile processes to modify the thermal transport property of materials, due to its robust tunability and flexibility to realize. Previous experiments reported thatκpof most bulk structures and weak interaction systems increases under pressure[5-8].Using density functional theory (DFT) calculations or Green-Kubo molecular dynamics (GK-MD) method, the similar trend has been demonstrated theoretically in iron[9], solid argon[10], diamond[11], silicon[12], cubic boron nitride[13], and also for some nanostructures[14,15]. A universal conclusion seems to be drawn that, the pressure always has a positive effect on theκpespecially for the bulk systems, owning to the enhancement of phonon group velocity (stiffness) and weakening of phonon-phonon scattering. In this paper, however, based on first-principles calculations,we observe a surprising decrease in theκpwith pressure for bulk II-VI mercury telluride (HgTe) system, which is a very promising thermoelectric material[16]. The abnormal phenomenon originates from the unexpected increase in anharmonic phonon scattering upon compression. The finding overthrows our common understanding thatκpincreases with pressure and extends the capability of conventional treatments on advancing the energy conversion performance of thermoelectric materials.

We first validated our calculation by comparing the phonon dispersion curves of HgTe under different pressures, which is done by solving the eigenvalues of the dynamical matrix constructed from the harmonic force constants. The results are shown in Fig.1. The phonon dispersion of HgTe under zero-pressure is in good agreement with previous theoretical studies and experimental measurements[23-24]. For example, the phonon frequency of the transverse optic (TO) mode atΓpoint is 3.47 THz and the experimental value is 3.54 THz[24], confirming the accuracy of the calculated force constants. It should be noted that the real space supercell method[25]cannot give the LO-TO splitting directly. Here the Born effective charges and dielectric constants are taken into account in the calculations. For better description of the thermal transport property of HgTe, this modification is used in the later on discussions. In order to check the validity of local density approximation, we run additional calculations using the Perdew, Burke, Ernzerhof generalized gradient approximation (PBE-GGA)[26]. We found that the phonon spectrum based on the PBE-GGA cannot describe the phonon spectrum very well (The results is not shown here). Therefore, we used the LDA throughout this work, which is also preferred in previous study[23]. Previous work reports that the spin-orbit interaction has weak influence on the phonon spectrum of HgTe[23]. However, we did not find noticeable change in the phonon dispersion curves when the spin-orbit interaction was switched on. Therefore, the spin-orbit interaction is not taken into consideration in this work. Upon compression, it can be seen that the low frequency transverse acoustic (TA) phonons change barely and even become soften aroundXandKpoints. As pressure increases, the longitudinal acoustic (LA) phonons and the optical phonons gradually shift to higher frequencies, similar to that observed in most of bulk materials, such as solid argon and diamond[10-11]. From phonon dispersion curves, one might intuitively expect thatκpof HgTe will also increase with pressure increasing. However, as we will see shortly, this is not the case.

We further validated our calculation by comparingκpof HgTe at zero pressure with experiments. In this paper, diamond is taken as an example for comparison because it possesses similar crystal structure with zinc blende HgTe. Moreover, diamond is a representative of many other materials with the same positive pressure-κprelationship[11]. Therefore, we can also compare our results with these data and check the reliability of our work. Theκpof diamond at room temperature is calculated to be 2 033 W/mK, which agrees reasonably with previous theoretical and experimental results[11]. For HgTe, at room temperature our calculation result is 10.46 W/mK, while the experimental value is only about 2.14 W/mK[27]. It looks like our ab initio results overestimates theκpof HgTe, but this is understandable considering that, the impurity scattering in the sample plays critical role as compared with the intrinsic phonon-phonon scattering. As described in previous study[27], the experimental samples contain hole impurities. Then, we considered the impurity scattering by incorporating a Rayleigh-type term[28]to the intrinsic phonon-phonon relaxation time by using the Matthiessen rule[29]. By taking an average hole concentration of 1.0 × 1017cm-3, the calculated thermal conductivity of HgTe matches very well with the experimental data.

Theκpof HgTe as a function of pressure is plotted in Fig.2. For comparison we also show the results of bulk diamond. We first notice that, theκpof diamond increases remarkably with pressure. These results are in good agreement with previous PBTE simulations[11]. At 60 GPa, the thermal conductivity of diamond increases by about 48%, very close to the value of 52% reported in Ref. 11. Theκpis doubled when the pressure is increased to 120 GPa. The enhancement inκpof diamond under pressure is attributed to the reduction of phonon-phonon scattering rates and the stiffening of phonon modes, due to the shift of phonon branches to high frequency under pressure[11].

A striking result in Fig.2 is the anomalous response ofκpof HgTe to pressure, which is sharply opposed to the case of diamond, despite the similarity of their cubic crystal structures. Theκpof HgTe decreases dramatically with pressure increasing. Up to 1.5 GPa, which is the phase transition point of HgTe (beyond that the zinc blende phase transits into cinnabar phase), the room temperatureκpis reduced by 67%. This trend is alien to that found previously for other bulk materials[11-13], where theκpincreases largely with pressure increasing. To the best of our knowledge, the abnormal trend for HgTe has not been reported for bulk materials so far.

We also calculated the Young’s modulus (EY) and shear modulus (G) of diamond and HgTe and compared the two cases in the inset of Fig.2. The calculatedEYandGfor diamond at zero pressure are 1 287 GPa and 577 GPa, respectively, while these values are 58.3 GPa and 22.3 GPa for HgTe, which agree very well with previous experimental data[30]. By applying pressure, bothEYandGof diamond increases in a conformable manner. However, for HgTe the trend of pressure dependentEYandGis bifurcate:EYincreases as pressure increases, whileGdecreases with pressure. Considering the anomalous pressure-dependentκp, we speculate that there might exist some relationship betweenκpandGof HgTe. This will be explained later.

Phonon scattering in a solid material is determined by its intrinsical harmonicity, whose magnitude can be qualitatively characterized by the Grüneisen parameters (γλ)[11,22]:

To correlate the anomalous phenomena with the atomic structure of HgTe, in Fig.5 we show the electron localization function (ELF)[32]of the ground state HgTe. We also calculated ELF for diamond for comparison. From the distribution of ELF, it is clearly seen that the electrons in diamond are mainly localized between two carbon atoms due to the nature of covalent bonding [Fig.5(a)]. In contrast, the electrons in HgTe present a fully different distribution: the electrons are mainly distributed around the Te atoms, instead of localizing between Hg and Te atoms [see Fig.5(b)]. It is because of this different distribution of ELF, the atoms in HgTe can slide with each other more easily than that in diamond, where the covalent bond has strong directivity and thus atom slide hardly occurs. This scenario is also supported by their opposing pressure dependentGas shown in the inset of Fig.2. On the one hand, the slide motion between Hg and Te atoms renders the lowGin HgTe; on the other hand, it hinders the phonon propagation and induces strong anharmonic scattering especially for the low energy acoustic phonons, which is the main reason for the lowκpof HgTe. TheGof HgTe decreases as the pressure increases, while it increases considerably with pressure for diamond. That is to say, the slide motion in HgTe becomes more intense and easier at high pressures. In this case, the anharmonic effect is enhanced in HgTe upon compression. As a result, bothGandκpof HgTe decreases with pressure increasing. It should be noted that the enhanced anharmonicity in HgTe under pressure is also supported by the previous experimental study[31]that thermal expansion of HgTe has positive to negative transition above room temperature when the pressure rises, meaning that the mode dependent Grüneisen parameters are mostly negative under compression. Based on our current study, we speculate that the negative pressure dependentκpis usually accompanied by lowG.

Before closing, it is worth pointing out that a similar observation of negative pressure dependence of thermal conductivity has been published very recently[33]. The authors attribute the anomalous thermal phenomenon to the different intrinsic scattering processes due to the large different mass ratio. However, in our work the mass difference between Hg and Te is very small and thus the negative pressure dependence of thermal conductivity in the HgTe cannot be explained by the mechanism presented in Ref. 33. Therefore, our results provide another new perspective to understand this anomalous thermal phenomenon and further clarify the relationship between negative Grüneisen parameters and lattice thermal conductivity[34].

In summary, we have presented the pressure effect on the phonon transport of bulk HgTe form Boltzmann transport equation based on first-principles calculations. In contrast to the increase ofκpobserved in many bulk materials, we uncover an anomalous decrease ofκpof HgTe with pressure. We identify the primary mechanism as the enhancement of negative Grüneisen parameters of TA modes, which results in the negative thermal expansion of HgTe at low temperatures and plays critical role in the lowκpdue to the phonon anharmonicity. Further, the unexpected phenomenon is associated with the unique electron distribution in HgTe inducing low shear modulus, which is directly associated with the aforementioned TA modes and gives rise to strong phonon-phonon scattering. Under pressure, this intrinsic effect further weakens the shear modulus and enhances the anharmonic phonon scattering in HgTe, and thus drivesκpto even lower values. This study provides new physical insights into the effect of pressure on the phonon transport of bulk materials and offers an effective way to decouple the electrical and phononic transport in thermoelectrics in terms of improving their energy conversion performance. Considering the small pressure (below 2 GPa) studied here, this anomalous pressure dependent thermal conductivity of HgTe could be demonstrated with current experimental techniques.

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