Semiclassical Boltzmann Transport Equation Research Articles

Unusual lattice thermal conductivity in the simple crystalline compounds TlXTe2(X=Ga,In)

It is generally believed that the lattice thermal conductivity in crystalline solids is mainly carried by phonons, whereas diffusons are regarded as the most important vibrational entity in amorphous solids. Contrary to this belief, here we show that the behavior of lattice thermal conductivity (LTC) of simple crystalline $\mathrm{Tl}X\mathrm{T}{\mathrm{e}}_{2}(X=\mathrm{Ga},\mathrm{In})$ compounds cannot be characterized either by phonons or diffusons alone. The calculated LTC based on first-principles density functional theory and the semiclassical Boltzmann transport equation (BTE) is only half of the experimental value. We show that the relatively weak bonding between Tl atoms and rigid $X\mathrm{T}{\mathrm{e}}_{2}$ chains gives rise to strong vibrational anharmonicity, which suppresses the mean free path of a large portion of phonon modes to even below the Ioffe-Regel limit and thus invalidates the precondition of the semiclassical BTE. We further show that the ultralow LTC of $\mathrm{Tl}X\mathrm{T}{\mathrm{e}}_{2}(X=\mathrm{Ga},\mathrm{In})$ can be rationalized by using the two-channel transport model. This finding would enhance the understanding of the underlying heat transport behavior in such materials.

Thermoelectric performance of monolayer Bi2Te2Se of ultra low lattice thermal conductivity

The electronic structure and thermoelectric properties of monolayer Bi2Te2Se were studied by density functional theory and semi-classical Boltzmann transport equation. The band gap with TB-mBJ can be improved for monolayer Bi2Te2Se. Monolayer Bi2Te2Se have ultra-low thermal conductivity comparing with other well-known two-dimensional materials. The monolayer Bi2Te2Se can improve electrical conductivities. ZT increases with increasing temperature for monolayer Bi2Te2Se. Comparing to GGA, TB-mBJ has larger ZT value in p-type doping. Monolayer Bi2Te2Se have larger ZT comparing with other well-known two-dimensional materials. Our calculated results show that our calculation greatly underestimates ZT value, therefore, monolayer Bi2Te2Se should have a higher ZT value.

Vertex corrections to the dc conductivity in anisotropic multiband systems

For an isotropic single-band system, it is well known that the semiclassical Boltzmann transport theory within the relaxation time approximation and the Kubo formula with the vertex corrections provide the same result with the $(1-\cos\theta)$ factor in the inverse transport relaxation time. In anisotropic multiband systems, the semiclassical Boltzmann transport equation is generalized to coupled integral equations relating transport relaxation times at different angles in different bands. Using the Kubo formula, we study the vertex corrections to the dc conductivity in anisotropic multiband systems and derive the relation satisfied by the transport relaxation time for both elastic and inelastic scatterings, verifying that the result is consistent with the semiclassical approach.

Quantification of uncertainties in thermoelectric properties of materials from a first-principles prediction method: An approach based on Gaussian process regression

We present the electron-phonon averaged via Gaussian process regression (EPA-GPR) method, in which the electron-phonon coupling matrix is represented as a function of two energies and is in turn modeled as a Gaussian process. The EPA-GPR method can be used as an efficient method to estimate thermoelectric properties of materials for fast-screening applications, comparable to the original electron-phonon averaged (EPA) method and the electron-phonon averaged via moving-least-squares (EPA-MLS) method. The EPA-GPR method does not require specification of any open parameter, unlike the other EPA-related methods, since all the hyperparameters in the model can be unambiguously estimated within the type II maximum likelihood (ML-II) approximation. Thus, the EPA-GPR method is a parameter-free estimation method. Additionally, the concept of Gaussian processes in the EPA-GPR method allows us to quantify the uncertainty in estimated properties of thermoelectric materials. One can randomly realize the electron-phonon coupling coefficients from the identified Gaussian process, and those realized samples can be further analyzed in the solution process of the semiclassical Boltzmann transport equation for charge carriers. The results of the semiclassical Boltzmann transport equation provide the statistical properties of the thermoelectric properties of interest. The means, standard deviations, histograms, and confidence intervals of the Seebeck coefficient, the electrical conductivity, and the power factor can be constructed and analyzed. The proposed EPA-GPR method is applied to a $p$-type half-Heusler compound, i.e., HfCoSb, as a case example, the results of which clearly present the advantages of the method.

H-CaS and h-CaSe nanosheets in CaX (X = O, S, Se and Te) series: promising thermoelectric materials under DFT investigation

The present work reports systematic study of a series of calcium chalcogenides CaX (X = O, S, Se and Te) in the architecture of rock-salt and hexagonal monolayer phases. Using first principle investigation within density functional theory (DFT) framework, we have computed the equilibrium structure and phonon dispersion curves for the dynamic stability, which follows the calculation of electronic properties like electronic band structure and projected density of state for the considered chalcogenide series. Furthermore, the thermoelectric properties such as thermal and electrical conductivities, Seebeck coefficient (S) and figure of merit (ZT) of the considered compounds are computed using the semi-classical Boltzmann transport equations (BTE). The present work reports the monolayer calcium chalcogenides as potential candidate for thermoelectric applications.

Structural, electronic and transport properties of silicene on graphene substrate

In this paper, we investigate the structural, electronic and transport properties of graphene/silicene substrate system, Si2C6 in top stacking configuration using density functional theory and semi-classical Boltzmann transport equation. From electronic band structure, the Si2C6 substrate system exhibits semi-metallic properties and charge transfer due to multiple band crossings at K-point, H-point, along H–A-line, A–G-line and G–K-line. Dirac cone is preserved in the band structure of Si2C6 due to the weak band interaction between graphene substrate and silicene layer. Using BoltzTraP, the study on electronic transport properties revealed electrical conductivity and Seebeck coefficient of Si2C6 have a higher thermoelectric response in the p-type doping concentration. Major peaks are found in p-type region of Seebeck coefficient, suggesting that major charge carrier in Si2C6 is hole.

Simulation of the Impact of Ionized Impurity Scattering on the Total Mobility in Si Nanowire Transistors.

Nanowire transistors (NWTs) are being considered as possible candidates for replacing FinFETs, especially for CMOS scaling beyond the 5-nm node, due to their better electrostatic integrity. Hence, there is an urgent need to develop reliable simulation methods to provide deeper insight into NWTs’ physics and operation, and unlock the devices’ technological potential. One simulation approach that delivers reliable mobility values at low-field near-equilibrium conditions is the combination of the quantum confinement effects with the semi-classical Boltzmann transport equation, solved within the relaxation time approximation adopting the Kubo–Greenwood (KG) formalism, as implemented in this work. We consider the most relevant scattering mechanisms governing intraband and multi-subband transitions in NWTs, including phonon, surface roughness and ionized impurity scattering, whose rates have been calculated directly from the Fermi’s Golden rule. In this paper, we couple multi-slice Poisson–Schrödinger solutions to the KG method to analyze the impact of various scattering mechanisms on the mobility of small diameter nanowire transistors. As demonstrated here, phonon and surface roughness scattering are strong mobility-limiting mechanisms in NWTs. However, scattering from ionized impurities has proved to be another important mobility-limiting mechanism, being mandatory for inclusion when simulating realistic and doped nanostructures, due to the short range Coulomb interaction with the carriers. We also illustrate the impact of the nanowire geometry, highlighting the advantage of using circular over square cross section shapes.

Phonon transport and thermoelectric properties of semiconducting Bi2Te2X (X = S, Se, Te) monolayers.

Confinement or dimensionality reduction is a novel strategy to reduce the lattice thermal conductivity and, consequently, to improve the thermoelectric conversion performance. Bismuth and tellurium based low-dimensional materials have great potential in this regard. The phonon transport and thermoelectric properties of Bi2Te2X (X = S, Se, Te) monolayers are systematically investigated by employing density functional theory and the Boltzmann transport equation. The calculated lattice thermal conductivity of these 2D systems ranges from ∼1.3 W m-1 K-1 (Bi2Te2Se) to ∼1.5 W m-1 K-1 (Bi2Te3) for a 10 μm system size at room temperature and considering spin-orbit coupling in harmonic force constants. This remarkably low lattice thermal conductivity is attributed to small group velocities and enhanced anharmonic phonon scattering rates. A detailed analysis is presented in terms of mode-level phonon group velocities, anharmonic scattering rates and phonon mean free paths. Our results reveal that the thermal transport in these 2D systems is dominated by in-plane transverse acoustic modes. Additionally, the thermal conductivity can be further reduced by decreasing the sample size due to phonon-boundary scattering. The thermoelectric properties including the Seebeck coefficient, power factor and electrical conductivity are calculated using the semi-classical Boltzmann transport equation within the rigid band approximation. The low thermal conductivities coupled with their high carrier mobilities lead to good thermoelectric power factors. With optimal carrier doping, a figure of merit ∼0.6 can be achieved at room temperature, which increases to ∼0.8 at 700 K, thus making them promising candidates for thermoelectric applications.

Hole transport in selenium semiconductors using density functional theory and bulk Monte Carlo

Amorphous selenium is emerging as a viable large-area imaging detector with avalanche multiplication gain for low-light and low-dose radiation detection applications. A key feature of its avalanche process is that only holes become “hot” carriers and undergo impact ionization. Thus, understanding the transport of non-equilibrium hot holes in extended states is pivotal to all the device applications. One of the interesting aspects of elemental selenium is the similar general feature of the electronic structure for various phase modifications. This stems from the strikingly similar short-range order between the crystalline and amorphous phases of selenium. At high electric fields, hole mobility in amorphous selenium loses its activated behavior and saturates with transport shifted entirely from localized to extended states. Thus, we expect the general details of the extended-state hole-phonon interaction in the amorphous phase to be described by the band-transport lattice theory of its crystalline counterparts, namely, monoclinic and trigonal selenium. To that effect and due to the intrinsic meta-stability of the monoclinic phase and high trap density in prepared specimens, we study hole transport in crystalline trigonal selenium semiconductors using a bulk Monte Carlo technique to solve the semi-classical Boltzmann transport equation. We validated our transport model by showing the excellent match between experimentally calculated hole drift mobilities with that calculated using the bulk Monte Carlo technique. Furthermore, calculations of the field-dependent carrier energy showed that holes in selenium can break the thermal equilibrium and get hot at which point the rate of energy gain from the applied electric field exceeds that of energy loss from the lattice.

Thermoelectric properties of monolayer Sb2Te3

The successful demonstration of monolayer films as promising thermoelectric materials highlights alternative strategies to nanostructuring for achieving high thermoelectric efficiency. Due to this reason, the electronic structure and thermoelectric properties of the monolayer Sb2Te3 are studied by using the density functional theory and the semiclassical Boltzmann transport equation. The dynamical stability of the monolayer Sb2Te3 can be guaranteed by the absence of imaginary frequencies in the phonon band structure. The monolayer Sb2Te3 can reduce the lattice thermal conductivity. The Seebeck coefficient S of the p-type monolayer Sb2Te3 is almost three times as high as those of the n-type monolayer Sb2Te3. The power factor for p-type doping is significantly larger than that for the n-type doping. Our calculated ZT values for the monolayer Sb2Te3 are far higher than those of nanomaterials Sb2Te3, bulk Sb2Te3, and the eutectic PbTe-Sb2Te3 composites, indicating that the thermoelectric performance of low-dimensional structure is indeed superior.

A theoretical investigation of structural, mechanical, electronic and thermoelectric properties of orthorhombic CH3NH3PbI3

The structural, mechanical, electronic and thermoelectric properties of the low temperature orthorhombic perovskite phase of CH3NH3PbI3 have been investigated using density functional theory (DFT). Elastic parameters bulk modulus B, Young’s modulus E, shear modulus G, Poisson’s ratio ν and anisotropy value A have been calculated by the Voigt–Reuss–Hill averaging scheme. Phonon dispersions of the structure were investigated using a finite displacement method. The relaxed system is dynamically stable, and the equilibrium elastic constants satisfy all the mechanical stability criteria for orthorhombic crystals, showing stability against the influence of external forces. The lattice thermal conductivity was calculated within the single-mode relaxation-time approximation of the Boltzmann equation from first-principles anharmonic lattice dynamics calculations. Our results show that lattice thermal conductivity is anisotropic, and the corresponding lattice thermal conductivity at 150 K was found to be 0.189, 0.138, and 0.530 Wm−1K−1 in the a, b, and c directions. Electronic structure calculations demonstrate that this compound has a DFT direct band gap at the gamma point of about 1.57 eV. The electronic transport properties have been calculated by solving the semiclassical Boltzmann transport equation on top of DFT calculations, within the constant relaxation time approximation. The Seebeck coefficient S is almost constant from 50 to 150 K. At temperatures 100 and 150 K, the maximal figure of merit is found to be 0.06 and 0.122 in the direction of the c-axis, respectively.

Theoretical and experimental analyses to determine the effects of crystal orientation and grain size on the thermoelectric properties of oblique deposited bismuth telluride thin films

The effects of crystal orientation and grain size on the thermoelectric properties of Bi2Te3 thin films were investigated by conducting experimental and theoretical analyses. To vary the crystal orientation and grain size, we performed oblique deposition, followed by thermal annealing treatment. The crystal orientation decreased as the oblique angle was increased, while the grain size was not changed significantly. The thermoelectric properties were measured at room temperature. A theoretical analysis was performed using a first principles method based on density functional theory. Then the semi-classical Boltzmann transport equation was used in the relaxation time approximation, with the effect of grain size included. Furthermore, the effect of crystal orientation was included in the calculation based on a simple semi-experimental model. A maximum power factor of 11.6 µW/(cm·K2) was obtained at an oblique angle of 40°. The calculated thermoelectric properties were in very good agreement with the experimentally measured values.

Monolayer PdSe2: A promising two-dimensional thermoelectric material

Motivated by the recent experimental synthesis of two-dimensional semiconducting film PdSe2, we investigate the electronic and thermal transport properties of PdSe2 monolayer by using the density functional theory and semiclassical Boltzmann transport equation. The calculated results reveal anisotropic transport properties. Low lattice thermal conductivity about 3 Wm−1 K −1 (300K) along the x direction is obtained, and the dimensionless thermoelectric figure of merit can reach 1.1 along the x direction for p-type doping at room temperature, indicating the promising thermoelectric performance of monolayer PdSe2.

Spin currents of charged Dirac particles in rotating coordinates

The semiclassical Boltzmann transport equation of charged, massive fermions in a rotating frame of reference, in the presence of external electromagnetic fields is solved in the relaxation time approach to establish the distribution function up to linear order in the electric field in rotating coordinates, centrifugal force and the derivatives. The spin and spin current densities are calculated by means of this distribution function at zero temperature up to the first order. It is shown that the nonequilibrium part of the distribution function yields the spin Hall effect for fermions constrained to move in a plane perpendicular to the angular velocity and magnetic field. Moreover it yields an analogue of Ohm’s law for spin currents whose resistivity depends on the external magnetic field and the angular velocity of the rotating frame. Spin current densities in three-dimensional systems are also established.

Thermoelectric properties of nanocrystalline Sb2Te3 thin films: experimental evaluation and first-principles calculation, addressing effect of crystal grain size

The effect of crystal grain size on the thermoelectric properties of nanocrystalline antimony telluride (Sb2Te3) thin films was investigated by experiments and first-principles studies using a developed relaxation time approximation. The Sb2Te3 thin films were deposited on glass substrates using radio-frequency magnetron sputtering. To change the crystal grain size of the Sb2Te3 thin films, thermal annealing was performed at different temperatures. The crystal grain size, lattice parameter, and crystal orientation of the thin films were estimated using XRD patterns. The carrier concentration and in-plane thermoelectric properties of the thin films were measured at room temperature. A theoretical analysis was performed using a first-principles study based on density functional theory. The electronic band structures of Sb2Te3 were calculated using different lattice parameters, and the thermoelectric properties were predicted based on the semi-classical Boltzmann transport equation in the relaxation time approximation. In particular, we introduced the effect of carrier scattering at the grain boundaries into the relaxation time approximation by estimating the group velocities from the electronic band structures. Finally, the experimentally measured thermoelectric properties were compared with those obtained by calculation. As a result, the calculated thermoelectric properties were found to be in good agreement with the experimental results. Therefore, we can conclude that introducing the effect of carrier scattering at the grain boundaries into the relaxation time approximation contributes to enhance the accuracy of a first-principles calculation relating to nanocrystalline materials.

Hybrid functional calculations of electronic and thermoelectric properties of GaS, GaSe, and GaTe monolayers.

Motivated by the recent success in the experimental synthesis of two-dimensional group-III monochalcogenides, we investigate the structural, electronic and thermoelectric properties of GaS, GaSe and GaTe monolayers based on the first-principles approach by using density functional theory and the semi-classical Boltzmann transport equation. The electronic band structures demonstrate that the GaS, GaSe, and GaTe monolayers are wide bandgap semiconductors having an indirect bandgap of 3.73 eV, 3.27 eV, and 2.46 eV respectively. We have evaluated the thermoelectric properties comprising the Seebeck coefficient, electrical conductivity per relaxation time, electronic thermal conductivity per relaxation time, electronic specific heat, thermoelectric figure of merit and power factor at various temperatures from 50 K to 800 K. It is found that GaS, GaSe and GaTe monolayers could be suitable candidates for low temperature efficient thermoelectric materials because of the high value of the thermoelectric figure of merit below room temperature.

Thermoelectric properties of 3D topological insulator: Direct observation of topological surface and its gap opened states

We report thermoelectric (TE) properties of topological surface Dirac states (TSDS) in three-dimensional topological insulators (3D-TIs) purely isolated from the bulk by employing single crystal Bi$_{2-x}$Sb$_x$Te$_{3-y}$Se$_y$ films epitaxially grown in the ultrathin limit. Two intrinsic nontrivial topological surface states, a metallic TSDS (m-TSDS) and a gap-opened semiconducting topological state (g-TSDS), are successfully observed by electrical transport, and important TE parameters (electrical conductivity (${\sigma}$), thermal conductivity (${\kappa}$), and thermopower ($S$)) are accurately determined. Pure m-TSDS gives $S$=-44 {\mu}VK$^{-1}$, which is an order of magnitude higher than those of the conventional metals and the value is enhanced to -212 {\mu}VK$^{-1}$ for g-TSDS. It is clearly shown that the semi-classical Boltzmann transport equation (SBTE) in the framework of constant relaxation time (${\tau}$) most frequently used for conventional analysis cannot be valid in 3D-TIs and strong energy dependent relaxation time ${\tau}(E)$ beyond the Born approximation is essential for making intrinsic interpretations. Although ${\sigma}$ is protected on the m-TSDS, ${\kappa}$ is greatly influenced by the disorder on the topological surface, giving a dissimilar effect between topologically protected electronic conduction and phonon transport.

Structural, electronic, mechanical, and thermoelectric properties of a novel half Heusler compound HfPtPb

We explore the structural, electronic, mechanical, and thermoelectric properties of a new half Heusler compound HfPtPb, an all metallic heavy element, recently proposed to be stable [Gautier et al., Nat. Chem. 7, 308 (2015)]. In this work, we employ density functional theory and semi-classical Boltzmann transport equations with constant relaxation time approximation. The mechanical properties, such as shear modulus, Young's modulus, elastic constants, Poisson's ratio, and shear anisotropy factor, have been investigated. The elastic and phonon properties reveal that this compound is mechanically and dynamically stable. Pugh's ratio and Frantsevich's ratio demonstrate its ductile behavior, and the shear anisotropic factor reveals the anisotropic nature of HfPtPb. The band structure predicts this compound to be a semiconductor with a band gap of 0.86 eV. The thermoelectric transport parameters, such as Seebeck coefficient, electrical conductivity, electronic thermal conductivity, and lattice thermal conductivity, have been calculated as a function of temperature. The highest value of Seebeck coefficient is obtained for n-type doping at an optimal carrier concentration of 1.0 × 1020 e/cm3. We predict the maximum value of figure of merit (0.25) at 1000 K. Our investigation suggests that this material is an n-type semiconductor.

Magnetothermoelectric effects in graphene and their dependence on scatterer concentration, magnetic field, and band gap

Using a semiclassical Boltzmann transport equation approach, we derive analytical expressions for electric and thermoelectric transport coefficients of graphene in the presence and absence of a magnetic field. Scattering due to acoustic phonons, charged impurities, and vacancies is considered in the model. Seebeck (Sxx) and Nernst (N) coefficients are evaluated as functions of carrier density, temperature, scatterer concentration, magnetic field, and induced band gap, and the results are compared to experimental data. Sxx is an odd function of Fermi energy, while N is an even function, as observed in experiments. The peak values of both coefficients are found to increase with the decreasing scatterer concentration and increasing temperature. Furthermore, opening a band gap decreases N but increases Sxx. Applying a magnetic field introduces an asymmetry in the variation of Sxx with Fermi energy across the Dirac point. The formalism is more accurate and computationally efficient than the conventional Green's function approach used to model transport coefficients and can be used to explore transport properties of other materials with Dirac cones such as Weyl semimetals.

Intraband memory function and memory-function conductivity formula in doped graphene

The generalized self-consistent field method is used to describe intraband relaxation processes in a general multiband electronic system with presumably weak residual electron-electron interactions. The resulting memory-function conductivity formula is shown to have the same structure as the result of a more accurate approach based on the quantum kinetic equation. The results are applied to heavily doped and lightly doped graphene. It is shown that the scattering of conduction electron by phonons leads to the redistribution of the intraband conductivity spectral weight over a wide frequency range, however, in a way consistent with the partial transverse conductivity sum rule. The present form of the intraband memory function is found to describe correctly the scattering by quantum fluctuations of the lattice, at variance with the semiclassical Boltzmann transport equations, where this scattering channel is absent. This is shown to be of fundamental importance in quantitative understanding of the reflectivity data measured in lightly doped graphene as well as in different low-dimensional strongly correlated electronic systems, such as the cuprate superconductors.