Ashok T. Ramu

Rigorous calculation of the Seebeck coefficient and mobility of thermoelectric materials

The Seebeck coefficient of a typical thermoelectric material is calculated without recourse to the relaxation time approximation (RTA). To that end, the Boltzmann transport equation is solved in one spatial and two k-space coordinates by a generalization of the iterative technique first described by Rode. Successive guesses for the chemical potential profile are generated until current continuity and charge-neutrality in the bulk of the device are simultaneously satisfied. Both the mobility and Seebeck coefficient are calculated as functions of the temperature and the agreement to experimentally obtained values is found to be satisfactory. Comparison is made with the less accurate RTA result, which has the sole advantage of giving closed form expressions for the transport coefficients.

GaN-Based Integrated Lateral Thermoelectric Device for Micro-Power Generation

Lateral thermoelectric devices were fabricated using c-plane GaN thin films grown on sapphire by MOCVD. The device design is appropriate for on-chip integration for power generation in the 1 V and tens of µA range. The fabricated devices were measured to have a maximum open circuit voltage of 0.3 V with a maximum output power of 2.1 µW (=0.15 V×14 µA) at a relatively small temperature difference (ΔT) of 30 K and an average temperature (Tavg) of 508 K. In addition, the suitability of GaN for high temperature thermoelectric applications was confirmed by measurements at 825 K.

An enhanced Fourier law derivable from the Boltzmann transport equation and a sample application in determining the mean-free path of nondiffusive phonon modes

An enhanced Fourier law that we term the unified nondiffusive-diffusive (UND) phonon transport model is proposed in order to account for the effect of low-frequency phonon modes of long mean-free path that propagate concomitantly to the dominant high-frequency modes. The theory is based on spherical harmonic expansions of the phonon distribution functions, wherein the high-frequency mode distribution function is truncated at the first order in the expansion, while the low-frequency mode distribution function, which is farther out of thermal equilibrium, is truncated at the second order. As an illustrative application, the predictions of the proposed model are compared with data from a recent experiment that utilized the transient gratings method to investigate the deviation of thermal transport in a silicon membrane from the predictions of the Fourier law. The good fit of the experimental effective thermal conductivity (ETC) with the analytical solution derived in this work yields quantitative information...

Thermoelectric transport in the coupled valence-band model

The Boltzmann transport equation (BTE) is applied to the problem of thermoelectric transport in p-type semiconductors whose valence band-structure is describable in terms of two bands degenerate at the Γ point. The Seebeck coefficient and mobility are calculated from the solution to two coupled BTEs, one for each band, with interband scattering and scattering by inelastic mechanisms treated exactly by the application of an algorithm developed by the authors in an earlier work. Most treatments of this problem decouple the two bands by neglecting certain terms in the BTE, greatly simplifying the mathematics: the error in the Seebeck coefficient and mobility introduced by this approximation is quantified by comparing with the exact solution. Degenerate statistics has been assumed throughout, and the resulting formalism is therefore valid at high hole concentrations. Material parameters are used that have been deduced from optical, strain and other experiments often not directly related to hole transport. The...

A compact heat transfer model based on an enhanced Fourier law for analysis of frequency-domain thermoreflectance experiments

A recently developed enhanced Fourier law is applied to the problem of extracting thermal properties of materials from frequency-domain thermoreflectance (FDTR) experiments. The heat transfer model comprises contributions from two phonon channels: one a high-heat-capacity diffuse channel consisting of phonons of mean free path (MFP) less than a threshold value, and the other a low-heat-capacity channel consisting of phonons with MFP higher than this value that travel quasi-ballistically over length scales of interest. The diffuse channel is treated using the Fourier law, while the quasi-ballistic channel is analyzed using a second-order spherical harmonic expansion of the phonon distribution function. A recent analysis of FDTR experimental data suggested the use of FDTR in deriving large portions of the MFP accumulation function; however, it is shown here that the data can adequately be explained using our minimum-parameter model, thus highlighting an important limitation of FDTR experiments in exploring the accumulation function of bulk matter.

Analysis of the "3-Omega" method for substrates and thick films of anisotropic thermal conductivity

The heat equation in an anisotropic material under sinusoidal excitation is solved in cylindrical-polar coordinates for the geometry conventionally used in the measurement of thermal conductivity by the “3-Omega” method. Although the solution can be carried out in Cartesian coordinates using Fourier transforms, the inverse transforms have to be evaluated numerically. The current method allows us to make contact with, and extend, the analytical results of Cahill and Pohl for isotropic substrates. We find that the 3-Omega method experiment, if performed conventionally, measures the average thermal conductivity in the in-plane and cross-plane directions, for modest anisotropies. An alternate geometry with separate heater and thermometer lines is explored, where the temperature of the thermometer is expected on physical grounds to be selectively sensitive to the in-plane thermal conductivity. The selectivity is quantified using the analytic forms derived for the temperature profile, and the results are verifi...

On the solenoidal heat-flux in quasi-ballistic thermal conduction

The Boltzmann transport equation for phonons is recast directly in terms of the heat-flux by means of iteration followed by truncation at the second order in the spherical harmonic expansion of the distribution function. This procedure displays the heat-flux in an explicitly coordinate-invariant form, and leads to a natural decomposition into two components, namely, the solenoidal component in addition to the usual irrotational component. The solenoidal heat-flux is explicitly shown to arise by applying the heat-flux equation to a right-circular cylinder. These findings are important in the context of phonon resonators that utilize the strong quasi-ballistic thermal transport reported recently in silicon membranes at room temperature.

Measurement of the high-temperature Seebeck coefficient of thin films by means of an epitaxially regrown thermometric reference material

The Seebeck coefficient of a typical thermoelectric material, silicon-doped InGaAs lattice-matched to InP, is measured over a temperature range from 300 K to 550 K. By depositing and patterning a thermometric reference bar of silicon-doped InP adjacent to a bar of the material under test, temperature differences are measured directly. This is in contrast to conventional two-thermocouple techniques that subtract two large temperatures to yield a small temperature difference, a procedure prone to errors. The proposed technique retains the simple instrumentation of two-thermocouple techniques while eliminating the critical dependence of the latter on good thermal contact. The repeatability of the proposed technique is demonstrated to be ±2.6% over three temperature sweeps, while the repeatability of two-thermocouple measurements is about ±5%. The improved repeatability is significant for reliable reporting of the ZT figure of merit, which is proportional to the square of the Seebeck coefficient. The accuracy of the proposed technique depends on the accuracy with which the high-temperature Seebeck coefficient of the reference material may be computed or measured. In this work, the Seebeck coefficient of the reference material, n+ InP, is computed by rigorous solution of the Boltzmann transport equation. The accuracy and repeatability of the proposed technique can be systematically improved by scaling, and the method is easily extensible to other material systems currently being investigated for high thermoelectric energy conversion efficiency.

The impact of commonly used approximations on the computation of the Seebeck coefficient and mobility of polar semiconductors

Seebeck coefficient modeling and measurement has important applications in direct thermal to electrical energy conversion and solid-state physics. The computations of the Seebeck coefficient and mobility of polar semiconductors in the literature often employ certain approximations, notably the relaxation time approximation (RTA) and the truncation of the Boltzmann transport equation. We study the accuracy of these approximations as a function of the effective mass, temperature, and carrier concentration using a recently developed technique for rigorous solution of the Boltzmann transport equation. We find that the approximations give rise to considerable error in the computed Seebeck coefficients of heavily doped semiconductors with a low effective mass, and that the RTA is entirely inapplicable for the accurate computation of the mobility of several important materials.

Validation of a Unified Nondiffusive-Diffusive Phonon Transport Model for Nanoscale Heat Transfer Simulations

Heat transfer in the vicinity of nanoscale hot-spots is qualitatively different from that in the macroscale, which effect stems from the breakdown of Fourier law due to phonon nondiffusive transport. In this work, we validate a recently proposed alternative, high-fidelity phonon transport model, the unified nondiffusive-diffusive (UND) model, which takes into account the mixed ballistic-diffusive nature of heat transport, as well as reduces to the Fourier law as a limiting case. In the UND model, the nondiffusive phonons are treated using the Boltzmann transport equation, while the diffusive phonon gas is treated by the Fourier law. The numerical results of Maznev et al. for the geometry and spatial dependence of variables corresponding to the transient gratings experiments of Johnson et al. are used for validation of the model.© 2014 ASME