Semi-metals as potential thermoelectric materials

The best thermoelectric materials are believed to be heavily doped semiconductors. The presence of a band gap is assumed to be essential to achieve large thermoelectric power factor and figure of merit. In this work, we propose semi-metals with large asymmetry between conduction and valence bands as an alternative class of thermoelectric materials. To illustrate the idea, we study semi-metallic HgTe in details experimentally and theoretically. We employ ab initio calculations with hybrid exchange-correlation functional to accurately describe the electronic band structure in conjunction with the Boltzmann Transport theory to investigate the electronic transport properties. We calculate the lattice thermal conductivity using first principles calculations and evaluate the overall figure of merit. To validate our theoretical approach, we prepare semi-metallic HgTe samples and characterize their transport properties. Our first-principles calculations agree well with the experimental data. We show that intrinsic HgTe, a semimetal with large disparity in its electron and hole masses, has a high thermoelectric power factor that is comparable to the best known thermoelectric materials. Finally, we propose other possible materials with similar band structures as potential candidates for thermoelectric applications.

In this supplementary material, we provide the supporting information about the Hall coefficient measurements, the electrical conductivity fitting, the lattice thermal conductivity calculations and measurements.

S.1 Hall coefficient measurements.
In Fig. S1, we show the experimental data for the Hall effect resistance R xy as a function of an applied external magnetic field B in the temperature range 200 K ≤ T ≤ 400 K. The Hall coefficient R H can be extracted from the slope of the R xy (B) curve as where l is the sample thickness. The net carrier concentration and electron mobilities can be found from the Hall coefficient data as and are shown in Fig. S2 and Fig. S3 respectively. Here e is an elementary charge, R is the resistance of the sample without magnetic field.

S.2 Electrical conductivity fitting.
The electrical conductivity can be found using the following expression where V cell is a unit cell volume, E is energy, µ is the chemical potential, f µ is the Fermi-Dirac distribution function and σ(E) is the differential conductivity where g(E) is the density of states, v g (E) is a group velocity and τ (E) = 1/Γ(E) is the total relaxation time that is inversely proportional to the total scattering rate Γ(E). In our calculations we use g(E) and v g obtained with the HSE06 exchange-correlation functional. In the constant relaxation time approximation (CRTA), one assumes that τ (E) is constant and energy independent.
In this work, we consider the energy dependent scattering rates. We consider 3 types of carrier scattering including acoustic deformation potential scattering Γ ac (E), ionized impurity scattering Γ imp (E) and polar optical scattering Γ pop (E) [3]. As follows from the Matthiessen's rule, the total scattering rate Γ(E) is a sum of all three contributions. Overall, we have 4 fitting parameters A 1 , A 2 , A 3 and phonon energyhω.
Acoustic deformation potential scattering rate is Ionized impurity scattering rate is where n C is the net carrier concentration obtained from the Hall coefficient measurements (see Fig. S2) where T d = 450.9 K and n 0 = 16.01 · 10 17 cm −3 . Polar optical scattering rate is where n BE is the Bose-Einstein distribution function, v g is the group velocity. The first two terms represent the polar-optical absorption while the last two terms describe the emission.
The energy dependent scattering rates obtained from the fitting to experimental electrical conductivity for the samples before and after annealing are shown in Fig. S4.

S.3 Thermal conductivity measurements.
To obtain the thermal conductivity κ, we use the following formula where ρ is the measured density of a sample, D(T ) is the measured thermal diffusivity and c p (T ) is the theoretical specific heat capacity. The measured thermal diffusivity for the original ingot sample and the sample after the SPS is shown in Fig. S5. The ingot sample has an excess of Te atoms, and a lower density, ρ =7.82 ± 0.04 g/cm 3 , comparing to ρ = 7.98 ± 0.17 g/cm 3 after the SPS. The thermal diffusivity is higher for the ingot samples (black circles) than in the SPS samples (blue triangles), but does not change after the annealing of the SPS sample. The theoretical heat capacity is where α is the coefficient of thermal expansion, β T is the isothermal diffusivity, c V can be found from the Debye model where T D = 140 K is the Debye temperature. For the second term in Eq. S.11, we use the experimental values from Ref.
[2] and get the following expression for the specific heat c p (T ) = c V (T ) + 1.01 · 10 −2 T (S.13) The obtained heat capacity c p (T ) linearly changes from 0.158 JK −1 g −1 at T = 250 K to 0.171 JK −1 g −1 at T = 700 K S.4 Isotopic scattering for phonons.
In this work, we perform ab initio calculations solving the Boltzmann Transport Equation (BTE). The algorithm we use is described in details in Ref.
[1]. Apart from the intrinsic three-phonon scattering processes, we include the isotopic disorder scattering processes with rates given by (S.14) where q -phonon wave vector, j -phonon branch index, ω qj -frequency of phonon (q, j), n qj -Bose-Einstein distribution function, α -Cartesian coordinate, s -atom type, z sα qj -phonon eigenmode, g s 2 -isotopic fluctuation parameter We use the natural isotopic composition of Hg and Te as summarized in Table 1.
The lattice thermal conductivity can be written as where V cell is the unit cell volume, ν = {q, j}, c ν is the group velocity, F ν is the linear deviation of the out-of-equilibrium phonon distribution n out ν from its equilibrium value n ν n out It can be found from the solution of the Boltzmann Transport Equation.
In the relaxation time approximation (RTA) F RT A ν = Λ RT A ν = τ ν c ν . In the exact solution it plays a role of a vectorial mean free-path displacement. To find a scalar mean-free path Λ exact ν , one needs to project it onto velocity direction The lattice thermal conductivity can be rewritten as a function of one single variable Λ as where the accumulated thermal conductivity is defined as In Fig. S6 we show the difference in the accumulated thermal conductivities in the two approaches discussed above. As one can see, the mean free path distribution in the exact approach is shifted toward the longer values.

S.6 Summary of experimental results
In Fig. S7, we summarize the experimental data on electrical conductivity (

S.7 Seebeck coefficient convergence
In Fig. S8, we show the theoretical Seebeck coefficient of HgTe as a function of carrier concentration for n− and p−type samples of HgTe calculated at different initial DFT k-point grids in the CRTA. We use the GGA exchangecorrelation functional and the interpolated grid 20 times denser than DFT grid for this demonstration. While the full convergence is achieved at very dense grid 40x40x40 (black solid line), the coarser grid 20x20x20 has a maximum error of about ±12µV/K at concentrations corresponding to the maximum of the Seebeck coefficient. This coarser grid also reproduces well the positions of the Seebeck coefficient maxima as well as the overall Seebeck coefficient profile as a function of a carrier concentration. We note that our converged GGA results are consistent with the ones from the literature Ref. 15 where WIEN2K + GGA functional + BoltzTraP were used to calculate the transport properties of HgTe in the CRTA. The use of a coarser grid 20x20x20 for the calculations with HSE06 functional is enforced by its high computational cost. Figure S1: The Hall effect resistance R xy measured as a function of magnetic field B at different temperatures. Figure S2: The net carrier concentration obtained from the Hall coefficient measurements as a function of temperature for the samples before (blue diamonds) and after (red diamonds) annealing. The samples are found to be n-type. Figure S3: The experimental carrier mobilities as a function of temperature for the samples before (blue diamonds) and after (red diamonds) annealing. The mobilities are improved after annealing. In both samples, the mobilities decrease with temperature. Figure S4: Acoustic deformation potential (yellow curves), polar optical (maroon curves) and charged impurity (blue curves) scattering rates obtained from the fitting of experimental electrical conductivities in the samples after annealing. Figure S5: The temperature-dependent thermal diffusivity of HgTe ingot and SPS samples. The thermal diffusivity decreases after the SPS process, and both of ingot and SPS samples' thermal diffusivity reduce with increased temperature.   Different curves represent different initial DFT k-point grids: red curve -8x8x8, blue curve -20x20x20, green curve -30x30x30, black curve -40x40x40. The interpolated k-point grid is set to be 20 times more dense than the initial grid.