Boltzmann Transport Formalism Research Articles

Diffusive nature of thermal transport in stanene

Using the phonon Boltzmann transport formalism and density functional theory based calculations, we show that stanene has a low thermal conductivity. For a sample size of 1 × 1 μm(2) (L × W), the lattice thermal conductivities along the zigzag and armchair directions are 10.83 W m(-1) K(-1) and 9.2 W m(-1) K(-1) respectively, at room temperature, indicating anisotropy in thermal transport. The low values of thermal conductivity are due to large anharmonicity in the crystal resulting in high Grüneisen parameters, and low group velocities. The room temperature effective phonon mean free path is found to be around 17 nm indicating that the thermal transport in stanene is completely diffusive in nature. Furthermore, our study reveals the relative importance of the contributing phonon branches and that, at very low temperatures, the contribution to lattice thermal conductivity comes from the flexural acoustic (ZA) branch and at higher temperatures it is dominated by the longitudinal acoustic (LA) branch. We also show that the lattice thermal conductivity of stanene can further be reduced by tuning the sample size and creating rough surfaces at the edges. Such tunability of lattice thermal conductivity in stanene suggests its applications in thermoelectric devices.

Diameter dependence of thermoelectric power of semiconducting carbon nanotubes

We calculate the thermoelectric power (or thermopower) of many semiconducting single wall carbon nanotubes (s-SWNTs) within a diameter range $0.5--1.5\phantom{\rule{0.16em}{0ex}}\mathrm{nm}$ by using the Boltzmann transport formalism combined with an extended tight-binding model. We find that the thermopower of s-SWNTs increases as the tube diameter decreases. For some s-SWNTs with diameters less than $0.6\phantom{\rule{0.16em}{0ex}}\mathrm{nm}$, the thermopower can reach a value larger than $2000\phantom{\rule{0.16em}{0ex}}\ensuremath{\mu}\mathrm{V}/\mathrm{K}$ at room temperature, which is about 6 to 10 times larger than that found in commonly used thermoelectric materials. The large thermopower values may be attributed to the one dimensionality of the nanotubes and to the presence of large band gaps of the small-diameter s-SWNTs. We derive an analytical formula to reproduce the numerical calculation of the thermopower and we find that the thermopower of a given s-SWNT is directly related with its band gap. The formula also explains the shape of the thermopower as a function of tube diameter, which looks similar to the shape of the so-called Kataura plot of the band gap dependence on tube diameter.

Electronic thermal conduction in suspended graphene

The electronic thermal conductivity (ETC), κe, of suspended graphene (SG) is studied for 15<T<400K, following the Boltzmann transport formalism. The electrons are considered to be scattered from defects along with the intrinsic in-plane acoustic phonons, out-of-plane flexural phonons (FPs) and optical phonons. The ETC is evaluated by computing the first-order perturbation distribution function by directly solving the linearized Boltzmann equation by an iterative method. Numerical calculations of the temperature and concentration dependences of κe show the dominance of charged impurity scattering at lower temperatures (T<75K) and of FPs at higher temperatures. The results are compared with the commonly used low-temperature and high-energy relaxation time approximations. Our calculations are in good agreement with recent κe data extracted for high-mobility SG samples. The validity of Wiedemann–Franz law is also discussed.

One-dimensional behavior and high thermoelectric power factor in thin indium arsenide nanowires

Electrical conductivity and Seebeck coefficient of quasi-one-dimensional indium arsenide (InAs) nanowires with 20 nm diameter are investigated. The carrier concentration of the passivated nanowires was modulated by a gate electrode. A thermoelectric power factor of 1.7 × 10−3 W/m K2 was measured at room temperature. This value is at least as high as in bulk-InAs and exceeds by far typical values of thicker InAs nanowires with three-dimensional properties. The interpretation of the experimental results in terms of power-factor enhancement by one-dimensionality is supported by model calculations using the Boltzmann transport formalism.

Parametric modeling of energy filtering by energy barriers in thermoelectric nanocomposites

We present a parametric modeling of the thermoelectric transport coefficients based on a model previously used to interpret experimental measurements on the conductivity, σ, and Seebeck coefficient, S, in highly Boron-doped polycrystalline Si, where a very significant thermoelectric power factor (TPF) enhancement was observed. We have derived analytical formalism for the transport coefficients in the presence of an energy barrier assuming thermionic emission over the barrier for (i) non-degenerate and (ii) degenerate one-band semiconductor. Simple generic parametric equations are found that are in agreement with the exact Boltzmann transport formalism in a wide range of parameters. Moreover, we explore the effect of energy barriers in 1-d composite semiconductors in the presence of two phases: (a) the bulk-like phase and (b) the barrier phase. It is pointed out that significant TPF enhancement can be achieved in the composite structure of two phases with different thermal conductivities. The TPF enhancement is estimated as a function of temperature, the Fermi energy position, the type of scattering, and the barrier height. The derived modeling provides guidance for experiments and device design.

Temperature dependent screened electronic transport in gapped graphene

We report our theoretical calculations on the temperature and energy dependent electrical conductivity of gapped graphene within the framework of Boltzmann transport formalism. Since screening effects have known to be of vital importance in explaining the conductivity of gapless graphene therefore we first worked out the behavior of the temperature dependent polarization function for gapped graphene as a function of wave vector and band gap, respectively. Polarization of gapped graphene has been compared with that of gapless graphene, bilayer graphene, and 2DEG to see the effects of gap. It is found that the gapped graphene polarization function exhibits a strong dependence on temperature, wave vector and band gap and the effect translates to the conductivity of gapped graphene. The nature of conductivity in gapped graphene is observed to be non-monotonic ranging from good to poor to semiconducting. We also find that the conductivity computed as a function of temperature by averaging over quasi-particle energy significantly differs from that computed at Fermi energy, suggesting that a notable contribution to temperature dependent conductivity is made by electrons close to the Fermi level.

Phonon-limited electron mobility in III-nitride heterojunctions

Low-field electron mobility limited by phonons in GaN-, InN- and AlN-based heterojunctions (HJs) for temperatures T<300K is studied within the framework of Boltzmann transport formalism by an iterative method. Electrons are assumed to occupy the lowest subband and scattered by the inelastic polar LO phonons and by quasi-elastic acoustic phonons through the deformation potential and piezoelectric couplings. Numerical calculations of the energy dependence of the first-order perturbation distribution function ϕ(E) for the polar LO phonons and of the acoustic phonon relaxation rates τ−1(E) bring out the characteristic features of phonon scattering in nitride HJs. The temperature and concentration dependences of phonon-limited mobility are compared with the low-temperature and high-energy relaxation time approximations, commonly used for ϕ. Calculations, taking into account other relevant scattering mechanisms, obtain good agreement with available Hall mobility data for GaN/AlGaN HJ.

Electronic transport properties of pnictogen-substituted skutterudites with alkaline-earth fillers

ABSTRACTThe materials class of skutterudites is one of the promising thermoelectric materials due to its decent electronic properties and cage-like structural feature that can be filled with guest atoms. First principles calculations have been performed in order to investigate electronic band structures and related transport properties of pnictogen-substitued skutterudites filled with alkaline-earth elements (MxCo4A6Te6 where M=Ca, Sr, or Ba, A=Ge or Sn, and x=0.5 or 1). The Seebeck coefficient and the power factor, which are electronic transport properties related to thermoelectricity, are computed by using the Boltzmann transport formalism within the constant-relaxation-time-approximation. The results are compared against the corresponding properties of the unfilled pnictogen-substitued ternary skutterudites (CoA1.5Te1.5) to identify the effects of filling, based on which the potential of filled pnictogen-substituted skutterudites for thermoelectric applications is evaluated. The possible changes in the ionic character of the interatomic bonding, which was suspected to be an important scattering source, are probed by analyzing the projected density of states.

Nonequilibrium conductivity at quantum critical points

Quantum criticality provides an important route to revealing universal non-equilibrium behaviour. A canonical example of a quantum critical point is the Bose-Hubbard model, which we study under the application of an electric field. A Boltzmann transport formalism and $\epsilon$-expansion are used to obtain the non-equilibrium conductivity and current noise. This approach allows us to explicitly identify how a universal non-equilibrium steady state is maintained, by identifying the rate-limiting step in balancing Joule heating and dissipation to a heat bath. It also reveals that the non-equilibrium distribution function is very far from a thermal distribution.

Use of Atomistic Phonon Dispersion and Boltzmann Transport Formalism to Study the Thermal Conductivity of Narrow Si Nanowires

We study the thermal properties of ultra-narrow silicon nanowires (NW) with diameters from 3 nm to 12 nm. We use the modified valence-force-field method for computation of phononic dispersion and the Boltzmann transport equation for calculation of phonon transport. Phonon dispersion in ultra-narrow 1D structures differs from dispersion in the bulk and dispersion in thicker NWs, which leads to different thermal properties. We show that as the diameter of the NW is reduced the density of long-wavelength phonons per cross section area increases, which increases their relative importance in carrying heat compared with the rest of the phonon spectrum. This effect, together with the fact that low-frequency, low-wavevector phonons are affected less by scattering and have longer mean-free-paths than phonons in the rest of the spectrum, leads to a counter-intuitive increase in thermal conductivity as the diameter is reduced to the sub-ten-nanometers range. This behavior is retained in the presence of moderate boundary scattering.

Atomistic simulation of phonon and alloy limited hole mobility in Si1–xGex nanowires

Abstractmagnified imageThe role of alloy and phonon scattering is theoretically explored in 5 nm diameter SiGe nanowires at room temperature. Low‐field mobility calculations are performed by utilizing sp3d5ds*‐spin–orbit‐coupled tight binding model for electronic structure and Boltzmann transport formalism. Three different transport orientations 〈100〉, 〈110〉 and 〈111〉 are considered. Alloy scattering is found to play an important role in these Si1–xGex nanowires, leading to a characteristic ‘U’ shaped mobility curve as a function of alloy composition. It is concluded that to extract any advantage of higher Ge hole mobility by alloying, Ge% &gt; 70% is needed. Furthermore, the 〈111〉 channel orientation exhibits the highest hole mobility while 〈100〉 has the lowest hole mobility for any given alloy composition. (© 2013 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Electronic, vibrational, and transport properties of pnictogen-substituted ternary skutterudites

First principles calculations are used to investigate electronic band structure and vibrational spectra of pnictogen-substituted ternary skutterudites. We compare the results with the prototypical binary composition CoSb${}_{3}$ to identify the effects of substitutions on the Sb site, and evaluate the potential of ternary skutterudites for thermoelectric applications. Electronic transport coefficients are computed within the Boltzmann transport formalism assuming a constant relaxation time, using a methodology based on maximally localized Wannier function interpolation. Our results point to a large sensitivity of the electronic transport coefficients to carrier concentration and to scattering mechanisms associated with the enhanced polarity. The ionic character of the bonds is used to explain the detrimental effect on the thermoelectric properties.

Atomistic simulations of low-field mobility in Si nanowires: Influence of confinement and orientation

A simulation framework that couples atomistic electronic structures to Boltzmann transport formalism is developed and applied to calculate the transport characteristics of thin silicon nanowires (NWs) up to 12 nm in diameter. The sp${}^{3}$d${}^{5}$s${}^{*}$-spin-orbit-coupled atomistic tight-binding model is used for the electronic structure calculation. Linearized Boltzmann transport theory is applied, including carrier scattering by phonons, surface roughness (SRS), and impurities. We present a comprehensive investigation of the low-field mobility in silicon NWs considering i) $n$- and $p$-type NWs, ii) [100], [110], and [111] transport orientations, and iii) diameters from $D$ = 12 nm (electronically almost bulk-like) down to $D$ = 3 nm (ultra-scaled). The simulation results display strong variations in the characteristics of the different NW types. For $n$-type NWs, phonon scattering and SRS become stronger as the diameter is reduced and drastically degrade the mobility by up to an order of magnitude depending on the orientation. For the [111] and [110] $p$-type NWs, on the other hand, large mobility enhancements (on the order of \ensuremath{\sim}4\ifmmode\times\else\texttimes\fi{}) can be achieved as the diameter scales down to $D$ = 3 nm. This enhancement originates from the increase in the subband curvatures as the diameter is scaled. It overcompensates for the mobility reduction caused by SRS in narrow NWs and offers an advantage with diameter scaling. Our results may provide understanding of recent experimental measurements, as well as guidance in the design of NW channel devices with improved transport properties.

Numerical Analysis of Electronic Transport Characteristics in Dielectrics Irradiated by Ultrashort Pulsed Laser Using the Nonlocal Fokker-Planck Equation

ABSTRACT The ultimate goal of this article lies in investigating theoretically the electronic transport characteristics in fused silica (SiO2) irradiated by ultrashort pulsed lasers. The nonlocal type of multivariate Fokker-Planck equation is modeled on the basis of the Boltzmann transport formalism to describe the ultrashort pulsed laser-induced damage phenomena in the energy-position space, together with avalanche ionization, three-body recombination, and multiphoton ionization. From the results, it is observed that the recombination becomes prominent and contributes to reduce substantially the rate of increase in electron number density when the electron density exceeds a certain threshold. With very intense laser irradiation, a strong absorption of laser energy takes place and an initially transparent solid is converted to a metallic state, a phenomenon well known as laser-induced breakdown. It is also found that full ionization is provided at intensities above threshold, and all further laser energy is deposited within a thin skin depth. The absorption length is of the order of a wavelength at very high laser fluence, and it becomes thinner as laser fluence is larger. This is because the absorbed energy is no longer consumed for multiphoton ionization, but rather leads to a drastic increase in the absorption coefficient because of Joule heating.

Universal relationship between giant magnetoresistance and anisotropic magnetoresistance in spin valve multilayers

We measure the giant magnetoresistance (GMR) with the current both parallel and perpendicular to the direction of the magnetization in the ferromagnetic (FM) layers and thus probe the anisotropy of the effective mean free paths for the spin-up and spin-down electrons, seen in the anisotropic magnetoresistance. We find that the difference of the GMR in the two configurations, when expressed in terms of the sheet conductance, displays a nearly universal behavior as a function of GMR. On interpreting the results within the Boltzmann transport formalism we demonstrate the importance of bulk scattering for GMR.

Parallel transport in a quasi-two-dimensional electron gas subjected to an in-plane magnetic field

The transport coefficients Rxx and Sxx are calculated for electrons confined to a parabolic potential well Vc(z)=a2z2 and subjected to an in-plane magnetic field B=(B,0,0). A Boltzmann transport formalism is used to develop general formulae for all the transport coefficients in the presence of delta -function scatterers. The anisotropic scattering lifetime when B not=0 is evaluated. It allows an immediate calculation of Rxx and Sxx, which show the structure expected with increasing B as successive sub-band minima pass through the Fermi level. In particular Sxx exhibits a sign change and the behaviour of Rxx is consistent with recent experimental data. The formulae for Ryy and Syy requires the numerical iterative calculation of a renormalised velocity in the y direction for their evaluation.