Phonon Distribution Research Articles

A deterministic solver for the transport of the AlGaN/GaN 2D electron gas including hot-phonon and degeneracy effects

The transport of the two-dimensional electron gas formed at an AlGaN/GaN heterostructure in the presence of strain polarization fields is investigated. For this purpose, we develop a deterministic multigroup model to the Boltzmann transport equations. The envelope wave functions for the confined electrons are calculated using a self-consistent Poisson–Schrödinger solver. The electron gas degeneracy and hot phonons are included in our transport equations. Numerical results are given for the dependence of macroscopic quantities on the electric field strength and on time and for the electron and phonon distribution functions. We compare our results to those of Monte Carlo simulations and with experiments.

Phonon dynamics of theSn∕Ge(111)surface at room temperature and low temperature determined by helium-atom scattering

We report on a high-resolution helium atom scattering study of the surface lattice dynamics, for both the room temperature $(\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3})\mathrm{R}30\ifmmode^\circ\else\textdegree\fi{}$ and the low-temperature $(3\ifmmode\times\else\texttimes\fi{}3)$ structures formed by $1∕3$ of a monolayer of Sn on Ge(111). At room temperature a pronounced lowering of the energy of the transversally polarized phonons along the $\overline{\ensuremath{\Gamma}\mathrm{M}}$ direction is observed for wave vectors in the vicinity of $Q=0.6\phantom{\rule{0.3em}{0ex}}{\mathrm{\AA{}}}^{\ensuremath{-}1}$, the $\overline{\mathrm{K}}$ point of the room temperature $(\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3})\mathrm{R}30\ifmmode^\circ\else\textdegree\fi{}$ structure. The temperature dependence of this mode is analyzed between 140 and 300 K. We conclude that the softening of this mode is related with the $(\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3})\mathrm{R}30\ifmmode^\circ\else\textdegree\fi{}\ensuremath{\leftrightarrow}(3\ifmmode\times\else\texttimes\fi{}3)$ phase transition. Phonon distributions curves are determined for both phases and the two high symmetry directions. The experimental data are explained from the backfolding of the slightly modified $\mathrm{Ge}(111)\ensuremath{-}(1\ifmmode\times\else\texttimes\fi{}1)$ Rayleigh wave and from the modes related to the Sn induced superstructures, in good agreement with first-principles calculations.

Lattice-dynamical calculation of phonon scattering at ideal Si–Ge interfaces

Detailed phonon scattering at an ideal Si–Ge interface is studied with a linear lattice dynamics model. Frequency dependent transmission coefficients indicate the significance of acoustic-optical phonon mode conversion at the interface. Applied to multiple interfaces, the method shows how the overall thermal resistance approaches a finite (Bloch mode) limit with the increasing number of interfaces in absence of other scattering mechanisms. The dependence of thermal resistance on the superlattice layer thickness is not significant even in the interface-scattering-only limit we study. We also assess errors incurred by the finite domain size and classical statistics in molecular dynamics simulations of interface thermal resistance. Results suggest that using 6×6 unit cells in the transverse directions, a tractable size for such simulations, will incur only a 5% error in the predicted thermal resistance. Similarly, the error due to the classical (Boltzmann) phonon distribution in molecular dynamics simulations is predicted to be less than 10% for temperatures above 300 K.

Negative refraction of phonons and acoustic lensing effect of a crystalline slab

We study how good a flat slab of a bulk crystalline solid with a large elastic anisotropy exhibits a lensing action for phonons or sound waves. The slowness and group-velocity surfaces of an ideal elastic solid for a flat phonon lens are analyzed in the geometrical acoustic approximation. These surfaces are compared with the corresponding surfaces of an existing bulk crystal (a zinc crystal) with hexagonal symmetry. To demonstrate the lensing effect we calculate the intensity distribution of phonons emitted from a point source in an isotropic medium (on one side of the lens), propagating through the slab lens and then transmitted into the isotropic medium in the other side. A similar calculation for sound waves with a finite-difference-time-domain method is performed to see the effects neglected in the geometrical acoustic approximation, that is, the effects of finite wavelength, mode conversion, and finite transmission at the interfaces.

Acoustic lensing effects of anisotropic bulk slabs exhibiting negative refraction

We study lensing effects of crystalline solids for phonons. The slowness surface of an ideal elastic solid used for a flat phonon lens is analyzed and compared with the corresponding surface of an existing bulk crystal (a zinc crystal) with hexagonal symmetry. The lensing effects are demonstrated by calculating the intensity distribution of phonons in the system consisting of a flat lens sandwiched in between isotropic media served as a phonon source and a focal region. A finite-difference-time-domain calculation is also performed to see the effects for ultrasound waves. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Theoretical assessment of electronic transport in InN

Among the group-III nitrides, InN displays markedly unusual electronic transport characteristics due to its smaller effective mass, high peak velocity and high background electron concentration. First, a non-local empirical pseudopotential band structure of InN is obtained in the light of recent experimental and first-principles results. This is utilized within an ensemble Monte Carlo framework to illuminate the interesting transport properties. It is observed that InN has a peak velocity which is about 75% higher than that of GaN while at higher fields its saturation velocity is lower than that of GaN. Because of the strongly degenerate regime brought about by the high background electron concentration, the electron–electron interaction is also investigated, but its effect on the steady-state and transient velocity–field characteristics is shown to be negligible. Finally, hot phonon generation due to excessive polar optical phonon production in the electron scattering and relaxation processes is accounted for. The main findings are the appreciable reduction in the saturation drift velocity and the slower recovery from the velocity overshoot regime. The time evolution of the hot phonon distribution is analysed in detail and it is observed to be extremely anisotropic, predominantly along the electric force direction.

Phonon effects in molecular transistors: Quantal and classical treatment

We present a comprehensive theoretical treatment of the effect of electron-phonon interactions in molecular transistors, including both quantal and classical limits and we study both equilibrated and out of equilibrium phonons. We present detailed results for conductance, noise and phonon distribution in two regimes. One involves temperatures large as compared to the rate of electronic transitions on and off the dot; in this limit our approach yields classical rate equations, which are solved numerically for a wide range of parameters. The other regime is that of low temperatures and weak electron-phonon coupling where a perturbative approximation in the Keldysh formulation can be applied. The interplay between the phonon-induced renormalization of the density of states on the quantum dot and the phonon-induced renormalization of the dot-lead coupling is found to be important. Whether or not the phonons are able to equilibrate in a time rapid compared to the transit time of an electron through the dot is found to affect the conductance. Observable signatures of phonon equilibration are presented. We also discuss the nature of the low-T to high-T crossover.

Kinetic flux-vector splitting schemes for the hyperbolic heat conduction

A kinetic solver is developed for the initial and boundary value problems of the symmetric hyperbolic moment system. This nonlinear system of equations is related to the heat conduction in solids at low temperatures. The system consists of a conservation equation for the energy density e and a balance equation for the heat flux Q i , where e and Q i are the four basic fields of the theory. We use kinetic flux vector splitting (KFVS) scheme to solve these equations in one and two space dimensions. The flux vectors of the equations are splitted on the basis of the local equilibrium distribution of phonons. The resulting computational procedure is efficient and straightforward to implement. The second-order accuracy of the scheme is achieved by using MUSCL-type reconstruction and min-mod nonlinear limiters. The solutions exhibit second-order accuracy and satisfactory resolution of gradients with no spurious oscillations. The scheme is extended to the two-dimensional case in a usual dimensionally split manner. In order to prescribe the boundary data, we need the knowledge of the e and Q i . However, in experiments only one of the quantities can be controlled at the boundary. This problem is removed by using a continuity condition. It turned out that after some short time energy and heat flux are related to each other according to Rankine–Hugoniot jump relations. To illustrate the performance of the KFVS scheme, we perform several one- and two-dimensional test computations. For the comparison of our results, we use high order central schemes. The present study demonstrates that this kinetic method is effective in handling such problems.

Lattice dynamics and the high-pressure equation of state of Au

Elastic constants and zone-boundary phonon frequencies of gold are calculated by total energy electronic structure methods to twofold compression. A generalized force constant model is used to interpolate throughout the Brillouin zone and evaluate moments of the phonon distribution. The moments are used to calculate the volume dependence of the Gruneisen parameter in the fcc solid. Using these results with ultrasonic and shock data, we formulate the complete free energy for solid Au. This free energy is given as a set of closed form expressions, which are valid to compressions of at least V/V_0 = 0.65 and temperatures up to melting. Beyond this density, the Hugoniot enters the solid-liquid mixed phase region. Effects of shock melting on the Hugoniot are discussed within an approximate model. We compare with proposed standards for the equation of state to pressures of ~200 GPa. Our result for the room temperature isotherm is in very good agreement with an earlier standard of Heinz and Jeanloz.

Electron and phonon distribution relaxation in metal films under a femtosecond laser pulse

The simultaneous time evolution of electron and phonon distributions in a gold film submitted to a femtosecond laser pulse has been theoretically investigated. A system of two coupled time-dependent Boltzmann equations describing electron and phonon dynamics has been solved by considering electron–electron (e–e) and electron–phonon (e–p) collisions. Phonon–phonon (p–p) collisions have been neglected. The collision probability of each process is derived by quantum-mechanical first order perturbation theory. The injected laser energy (source term) is directly absorbed by the electron gas through electron–photon collisions and a perturbation term is added to the Boltzmann equation for electrons. However, phonons do not directly absorb energy from the laser but they are affected by the irradiation due to their collisional interaction with electrons. The results emphasize the transient non-equilibrium behavior of the electron distribution function and details of electron and phonon relaxation dynamics.

A deterministic solution method for the coupled system of transport equations for the electrons and phonons in polar semiconductors

We present a multigroup model of the Boltzmann equations governing the transient transport regime in polar semiconductors. Efforts have been made to give an accurate description of the coupled hot-electron hot-phonon system, which allows us to study the modifications of the carrier and longitudinal optical phonon distribution functions in comparison to the usual equilibrium phonon calculations. Computations are performed for InP, taking into account all the relevant scattering mechanisms. We investigate the response of the coupled electron?phonon system to a step-like high dc electric field pulse. Moreover, we discuss the relation between our model and a matrix method.

Angular distribution of a pulse of low-energy phonons in liquid4He

We have made time-resolved measurements of the angular distribution of low-energy phonons (l-phonons) that are created by a heater in liquid helium, at several powers and pulse lengths, in order to investigate the temporal and spatial development of such l-phonon pulses. The angular distributions show a mesa shape with a flat top in the direction normal to the heater. The width of the flat top increases with power and pulse length. We argue that the flat top is due to the creation of high-energy phonons (h-phonons). These are created until the temperature of the central region of the l-phonon sheet reaches 0.7 K, when the creation of h-phonons essentially stops. Outside the flat top, the angular distribution is due to the lateral spreading of the l-phonon sheet. A model for this is described in the preceding paper [I.N. Adamenko, K. E. Nemchenko, V.A. Slipko, and A.F.G. Wyatt, preceding paper, Phys. Rev. B 68, 134507 (2003)] which gives a framework for considering these measurements.

Low-temperature saturation of variable-range hopping and delocalization of electron states by entanglement with phonons in two dimensions

We investigate low-temperature transport properties in a two-dimensional electronic system coupled with phonons. It is found that some of the localized states will be delocalized by the entanglement with phonons even when the temperature approaches the absolute zero. The distribution of phonons near the zero temperature is departed from the Boson distribution because the entangled states are preferred from the energy point of view. This leads to the low-temperature saturation of the variable-range hopping of electrons and the existence of a metal-like behavior in two dimensions.

Analytic Study on Some Properties of the Holstein Model**The project supported by National Natural Science Foundation of China under Grant No. 19875048 and the Foundations of the Chinese Academy of Sciences and Education Ministry of China

A new ansatz is presented for analytic study of the properties of the Holstein model. New analytic results for the polaronic band structure, ground state energy, phonon distribution, hopping probability of electron, and effective masses, of the Holstein molecular crystal model, are given in one dimension. All the analytic results obtained are in accord with the numerical results completed recently and valid both in the adiabatic and nonadiabatic regimes, and accurate enough in the region of validity.

On the suprathermal distribution in an anisotropic phonon system in He II

The equation that describes the suprathermal distribution of high-energy phonons (h phonons) created in anisotropic phonon systems in superfluid helium is obtained. The solution of this equation enables the derivation of the value of suprathermal ratio S as the ratio of the actual distribution to the Bose–Einstein one, its dependences on the momentum of the h phonons, the anisotropy parameters, and the temperature of the low-energy phonons from which the h phonons are created. We analyze this equation to obtain an estimate of the value of the ratio between the h-phonon number density in anisotropic and isotropic phonon systems and draw conclusions about the dependence of S on the relevant parameters.

A Coupled Cellular Automata Representation of Nanoscale Transport Across Semiconductor Interfaces

ABSTRACTThin-film devices pose unique challenges to device simulators. Relaxation approximations to the Boltzmann Transport Equation may be vitiated as the width and length of the thin-film components approach the mean free path lengths of electrons and phonons. Concomitant with these reduced lengths, surface and other interface scattering mechanisms exert greater influences on the transport processes. Consequently, it may not be effective to use transport models that depend upon analytic or other pre-ordained representations of electron, phonon, and photon distribution functions. Finally, thin-film simulations must account for tunneling processes, local defects, nanoscale dopant gradients, nanoscale roughness, and nanoscale variations in local geometries. Consequently, it is desirable to develop a nanoscale simulation technique that (1) can arbitrarily vary material properties in real space while (2) tracking reciprocal space scattering for arbitrary and volatile distributions of electrons, phonons, and photons. The Discrete State Simulation (DSS) forms the backbone of such a simulation. Written with the run-time flexibility of object-oriented code, the Discrete State Simulation (DSS) is a coupled cellular automata (CA) simulator that builds upon the objects and rules of quantum mechanics. The DSS represents global non-equilibrium processes as patterns that emerge through an ensemble of scattering events that are localized at vibronic nodes. The run-time flexibility of the DSS enables dynamic rule construction - computational algorithms that evolve in response to the conditions that are being simulated. The reported DSS effort simulated nanoscale transport processes at dopant-defined junctions by coupling electron-phonon and electron-photon scattering mechanisms. Using electronic band structures, phonon band structures, and deformation potentials analogous to silicon <100> material properties, the DSS generated data depicted the dynamic band bending induced by femtosecond variations in applied electric fields.

Effects of Screening and Finite Phonon Energy on the Transport in Quantized Layers in Compound Semiconductors

Analytical expressions for the momentum relaxation times of the conduction electrons in a non-degenerate two dimensional electron gas in the surface of a compound semiconductor have been obtained for interactions with the piezoelectric and deformation potential acoustic phonons taking due account of the screening of the perturbing potential under the the condition of low lattice temperature when the phonon energy cannot be neglected in comparison to the average thermal energy of the electrons and for that matter the equipartition approximation for the phonon distribution is hardly valid. The relaxation times calculated for inversion layers in GaAs and ZnO are found to depend upon the carrier energy, the lattice temperature and the impurity concentration in rather complex manners which are significantly different from what follows from the traditional approach of either neglecting the phonon energy or disregarding the process of screening. It is seen how the finite value of the phonon energy and the screening of the perturbing potential change the mobility characteristics significantly at the low lattice temperatures. The temperature dependence of the zero field mobility that one obtains using the relaxation times calculated here is quite different from the traditional laws.

Behaviour of Polariton in a Simple Model

Behaviour of a simple model polariton system is restudied. Thedistribution of phonons in polariton never is sub-Poisson given bycomputation of the thermal counterpart of the Q parameter and the matrixm(3). The polariton complex as a whole shows non-classical behaviourbelow a threshold temperature, which depends on the photon-phonon couplingstrength, when the distribution of phonons is always classical at anytemperature.

Piezoelectric interaction in degenerate semiconductors at low lattice temperatures

The theory of quasi-elastic intravalley scattering of free electrons in a degenerate semiconductor material due to piezoelectric interaction has been developed under the condition of low temperature when the well-known approximations of the traditional theory are hardly valid. The theory reveals complex and altogether different dependence of the scattering rates upon the carrier energy and the lattice temperature. The temperature dependence of the mobility as obtained from the scattering rates calculated here is also quite different from what follows from the traditional approximations. The numerical results for CdS and Si reveal how significantly does the degeneracy level or the true phonon distribution or the quasi-elasticity of the collisions affect the scattering rates and the mobility characteristics at low temperatures.

LATTICE INSTABILITY OF 2H-TaSe2

The charge density wave (CDW) in the layered compound 2H-TaSe 2 at low temperatures has a commensurate phase, which causes super lattice points to appear in the Brillöuin zone of the undistorted phase. A Born-von Karman formalism has been employed for the calculation of phonon frequency distribution curves of 2H-TaSe 2 both in the normal and in the commensurate charge density wave (CCDW) phases. A folding technique has been adopted for the calculation in the CCDW phase. The phonon distribution for both the phases have been reported. With these distributions the thermal properties such as specific heat capacity, Debye Waller factor W(k) and thermal conductivity have been worked out, and compared with the available experimental results.