Time-dependent Boltzmann Transport Equation Research Articles

Analytical Solution of the Elastic Boltzmann Transport Equation in an Infinite Uniform Medium Using Cumulant Expansion

We study the analytical solution of the time-dependent elastic Boltzmann transport equation in an infinite uniform isotropic medium with an arbitrary phase function. We calculate (1) the exact distribution in angle, (2) the spatial cumulants at any angle, exact up to an arbitrary high order n. At the second order, n = 2, an analytical, hence extremely useful combined distribution in position and angle, is obtained as a function of time. This distribution is Gaussian in position, but not in angle. The average center and spread of the half-width are exact. By the central limit theorem the complete distribution approaches this Gaussian distribution as the number of collisions (or time) increases. The center of this distribution advances in time, and an ellipsoidal contour that grows and changes shape provides a clear picture of the time evolution of the particle migration from near ballistic, through snake-like, and into the final diffusive regime. This second-order cumulant approximation also provides the c...

A transient solution of the Boltzmann equation exposes energy overshoot in semiconductor devices

A method is developed to analyze the transient response of semiconductor devices in phase space. This is achieved by solving the space and time dependent electron Boltzmann transport equation self-consistently with the Poisson and transient hole-current-continuity equation. The result gives the details of the time evolution of the distribution function. The method is applied to analyze a bipolar junction transistor. The model predicts the limits in which the steady-state response approximation can be applied. The model exposes a transient overshoot in the high energy tail of the distribution function.

Boundary-condition-induced current overshoot in short GaAs samples

The effect of boundary conditions on current overshoot observed in transient nonstationary simulations of electron transport in submicrometer bars of GaAs is described. Noting that the one-valley model for GaAs does not give rise to observable overshoot when the spatially independent Boltzmann transport equation (BTE) is solved, significant current overshoot is reported when the same model is used in the solution of the spatially dependent BTE with replacement boundary conditions. The spatially unidimensional, time-dependent BTE is solved for undoped GaAs samples with lengths varying between 0.2 and 1 mu m. The amount of current overshoot varies depending on the length of the sample and is seen to be as high as 10%. The phenomenon is explained using the moments of the Boltzmann equation.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Picosecond time evolution of photoexcited hot-electron mobility in GaAs and the speed of photoresponse

The transient mobility of hot electrons photoexcited in undoped GaAs by subpicosecond laser pulses is calculated. For this, we solve the time-dependent Boltzmann transport equation in the presence of a low-frequency, weak electric field. The attention is focused mainly on the role of intracentral Γ valley scattering in determining the delay in the mobility rise on the picosecond time scale, and the hot-electron energies are assumed to be below the thresholds for possible side-valley transfers (Γ→L,X). We consider the mobility response under two separate conditions of excited carrier density, namely, (1) low-density excitations for which the electron–longitudinal phonon (LO) Fröhlich interaction initially dominates in the carrier relaxation and (2) high-density excitations for which the electron-electron interaction is faster than all other collisions. The mobility of hot electrons is very small (&amp;lt;1000 cm2/V s) just after photogeneration. It rises to its maximum value with a time constant decided by the various scattering processes which are influenced by the values of the carrier density and the lattice temperature. We find that the mobility rise can be quite slow up to the scale of several picoseconds, even when the possible delay due to side-valley scattering is absent.

The calculation of electron transient response in semiconductors by the Monte Carlo technique

It is shown that the multiparticle Monte Carlo technique allows one to find the solution of the time-dependent Boltzmann transport equation. This solution is independent of the electron self-scattering rate.

New Approach to the Solution of the Time-Dependent Boltzmann Transport Equation

We present a new method of solving analytically the time-dependent Boltzmann transport equation. Two examples are given to demonstrate the use of this method to predict ac and dc conductivities in nondegenerate single-valley simiconductors.

Quantum Theory of Electrical Transport in a Magnetic Field. I

The relationship between the quantum theory of electrical transport in the presence of a magnetic field and the corresponding Boltzmann transport equation is established for a simple system. The model consists of noninteracting free electrons being elastically scattered by an arbitrary potential in the presence of uniform electric and magnetic fields. Without the aid of a representation, the exact gauge-dependent Liouville equation for the density operator of this system is transformed into a completely gauge-independent equation satisfied by a new density operator. The new density operator is shown to give the correct current density, using the ordinary gauge-independent free-particle velocity operator. No approximations are made in performing the transformations, and the physical content of the new gauge-independent transport formalism is identical in all respects with that contained in the initial gauge-dependent equations. A new density matrix is defined which is essentially the Fourier sum of the matrix of the gauge-independent density operator. By considering the scattering potential resulting from a set of fixed impurity centers, it is shown that the diagonal elements of this new density matrix satisfy the ordinary time-dependent Boltzmann transport equation in which the spatial gradient term appears explicitly. The possible existence of a spatial variation in the average density of scatterers is also taken into account. The final equation for the quantum-mechanical distribution function represents the result obtained by treating the electric field and the effective scattering potential to the lowest possible order in which they contribute.

Preliminary Calculation for Experiments Using Osaka University HITS, (I)

In order to discuss the ambiguity inherent in the diffusion approximation applied to a small-sized medium including a pulsed neutron source, the time variation and the spatial distribution of thermal neutron angular flux are calculated in P 3-approximation of the time-dependent Boltzmann transport equation. With the discussion on the existence of a set of real and complex eigenvalues, the real eigenvalues for the lowest and some higher modes are computed numerically and the corresponding eigenfunctions are derived under Marshak's boundary condition. The results are compared with those obtained in the diffusion approximation and the physical interpretations of resultant asymptotic and non-asymptotic thermal neutron flux are given.

均質核反応系の&lt;i&gt;n&lt;/i&gt;群Boltzmann方程式の解法

核反応で発生した高速中性子が熱中性子に減速される過程を取り扱う方法は多くの著者により考察されてきたが,ここでは減速過程にさらに分裂により高速に戻ってくる中性子の効果を取り入れたn群のBoltzmann方程式を水素を減速材とする均質無限系について正確に解くことを試みた。分裂スペクトルは単色としたが,単色でないときはBoltzmann方程式の線型性から単色のときの解を重畳することによって求めればよい。上述の核反応系での中性子数が時間的に定常の状態に漸近するための条件として,臨界条件が求められ,またそのときの中性子スペクトルが求められる。多群近似を扱うと各グループにてその速度則が v Σa-Σ2/2 則であることが示される。