Carrier Distribution Function Research Articles

Investigation of the influence of low- and high energetic optical phonons onto electronic transport parameters of n-Si

Novel general expressions for the energy and momentum loss rates of hot electrons due to their ν-th order interaction with non-polar optical phonons in covalent semiconductors are derived for the first time. In the calculation the distribution function of the charge carriers is approximated by a hot drifted Fermi-Dirac (FD-) function. Both loss rates are applied to determine selfconsistently for a 190 K (680 K) optical phonon the temperature, the drift velocity and the average energy loss of hot electrons in n-Si as a function of the lattice temperature T, the electron density n and the electron heating field strength E assumed to be parallel to the 〈1 1 1〉-orientation. The theoretical results obtained with the hot drifted FD-function are compared with those derived in the diffusion approximation. In the low field range (E < 0.1 kV cm−1) and in the low temperature range (T < 200 K) this approximation is found to be completely inadequate for 680 K phonon which couples in zero-order to the electrons.

Hopping relaxation and photoconductivity of disordered semiconductors

AbstractA method of evaluating the distribution function of hot localized carriers excited by an external electromagnetic wave is presented. The time‐dependent distribution function is used to find the hopping current relaxation for a quasi‐continuous distribution of states and the de photoconductivity at low temperatures.

Transient absorption spectra of a modulation-doped Ga0.47In0.53As/Al0.48In0.52As multiple quantum well structure measured by picosecond infrared pulses

Intersubband and intrasubband scattering of hot electron-hole pairs is studied with picosecond infrared pulses. The carrier distribution function is derived directly from the transient absorption spectra after selective excitation of carriers to the n=1 and 2 subbands. Band filling of the n=1 conduction band results in a strong bleaching of the corresponding interband absorption and in large changes of the refractive index. An absorption change due to the population of the n=2 subbands is not detected, indicating their fast relaxation within less than 3 ps. The hot plasma cools by LO phonon emission on a time scale of several tens of picoseconds.

Monte Carlo simulation of intersubband relaxation in semiconductor quantum wells

We have used an ensemble Monte Carlo simulation of semiconductor quantum wells to simulate recent time resolved optical experiments of intersubband relaxation. In the present work we model a single quantum well electron system with the inclusion of intra- and intersubband carrier-carrier scattering and nonequilibrium polar optical phonons. Simulated results with and without these basic mechanisms show that the experimentally measured long time constant pile-up of carriers in upper subbands in wide quantum wells appears to be associated with the thermal tail of the total carrier distribution function which remains hot for long times after the pump pulse due to hot phonon effects.

Time evolution of the phonon and ionized impurity limited 2D hot carrier distribution function in a n-GaAs square quantum well (QW)

The time evolution of the distribution function of hot electrons interacting with ionized impurities, acoustic and polar optical phonons is studied in terms of the Boltzmann integral equation. The calculated distribution functions are used to determine the charge carrier drift velocity, the average energy as well as the average momentum and energy loss rates as a function of time at constant values of the lattice temperature and charge carrier density. The stationary non-equilibrium values of these quantities are found to be established after 3–4 ps.

Analysis of carrier distribution function through Smith-Purcell effect in GaAs/GaAlAs heterostructures

We have observed for the first time the Smith-Purcell effect in GaAs-GaAlAs-heterostructures. A grating structure induces far infrared (FIR) emission from drifted carrier distributions, which is angle dependent and allows the determination of intrinsic carrier properties.

Band-tail relaxation in disordered systems at T=0

Expressions for the evolution of the energy-dependent carrier density and distribution function for hopping down the band tail at T=0 are derived which are valid for arbitrary initial conditions and forms of density-of-state tails. Carrier relaxation is calculated explicitly for hopping down an exponential band tail and different initial conditions. The corresponding time dependences of the peak shift, width and moments are calculated and the general asymptotic behaviour is discussed. The modifications due to recombination are shown to result in a crossover of the peak shift from the double- to a single-logarithmic time dependence; the accompanying change in the width displays a transition from a constant to a single-logarithmic dependence.

Principles of Population Inversion and Realization of Stimulated Far-IR Radiation Emission

We present new physical methods to obtain population inversion and electromagnetic wave generation in semiconductors based on deformation of the carrier distribution function in momentum space, and also laser action on optical transitions between light and heavy hole subbands, and NEMAG on a heavy hole cyclotron resonance, the inversion of the light hole population of Landau levels.

Equivalence of Lei and Ting balance equations for nonlinear electronic transport and a Boltzmann approach.

We discuss the connection between the Green's-function approach to nonlinear electronic transport in semiconductors proposed recently by Lei and Ting [Phys. Rev. B 32, 1112 (1985)] and the description of high-field transport based on the familiar nonlinear Boltzmann equation. We show that their balance equations are identical to those one obtains from the Boltzmann equation by assuming that the carrier distribution function is a displaced Fermi-Dirac distribution at the electron temperature ${T}_{e}$.

Balance-equation approach to high-field electronic transport: A linear-response theory in the scattering interactions.

The nonlinear balance equations for hot-electron transport previously suggested by us [Phys. Rev. B 32, 1112 (1985)] are obtained by following the standard prescription of conventional linear-response theory in calculating the density matrix contribution to first order in the impurity and phonon scattering interactions, after separation of the center-of-mass degrees of freedom from the relative motion of the electrons. Having the nature of a linear-response theory, this approach contains no assumption relating to the form of the carrier distribution function, which is certainly not a displaced equilibrium type (Fermi-Dirac or Maxwell) distribution function. It is a non-Boltzmann method in the sense that these balance equations cannot be obtained within a Boltzmann theory without employing an inconsistent ansatz.

A comprehensive transport model for semiconductor device simulation

In this paper a comprehensive carrier dynamical transport model for semiconductor device simulation is presented. The model consists of carrier, carrier momentum and carrier energy conservation relations derived using a perturbation solution for the carrier distribution function. Carrier degeneracy, multiple conduction sub-bands and ellipsoidal constant energy surfaces are accounted for, and the effective masses and band edges are assumed to be spatially inhomogeneous. The new formulation overcomes modelling inaccuracies of previous energy transport models based on a drifted Maxwellian distribution function, and for spatially homogeneous, non-degenerate semiconductors offers several computational advantages.

Alternative Quasi‐Classical Transport Equations. II. Boltzmann's Equation and the Electron Distribution Function for MIS Systems

AbstractIn Part I of this paper a new quasi‐classical formulation of transport equations in the framework of the local‐density approximation is reported including wave function effects due to the potential barrier near an interface. For practical applications to semiconductor devices the non‐equilibrium distribution function of the charge carriers is to be determined. In Part II this problem will be solved for a MIS field effect transistor as a typical example. For this system Boltzmann's equation is formulated and solved in a more general manner than before. The scattering processes in the bulk are considered together with partly diffuse interface scattering which is introduced by means of special boundary conditions. The transport properties of the MIS field effect transistor will be calculated later in Part III of this paper.

Subpicosecond dynamics of electron injection into GaAs/AlGaAs quantum wells

We model the subpicosecond evolution of a nonthermal electron distribution injected into a GaAs/AlGaAs quantum well using an ensemble Monte Carlo simulation which includes electron-electron scattering. The calculated results are in good agreement with the experimental time dependence of the carrier distribution function from recent bleaching experiments with carrier-carrier scattering the dominant mechanism contributing to band filling.

Transient process in an injection laser with a non-quasi-Fermi electron distribution function

A calculation is made of the reaction of a semiconductor laser to rapid changes in the injection current. An allowance is made for deviations of the nonequilibrium carrier distribution function from the quasi-Fermi form when these deviations are characterized by an effective relaxation time τeff governed by the rates of intraband relaxation and polarization decay. It is shown that for typical parameters of a heterojunction laser the nature of its transient characteristic changes radically (from deep relaxation pulsations to a practically instantaneous repetition of the changes in the injection current) when τeff increases from 10−13 to 10−12 sec.

Solution of a new nonlinear equation for the distribution of charge carriers in a semiconductor.

The solution of a recently obtained nonlinear differential equation for the distribution function of charge carriers in a semiconductor in an electric field is derived. It is given by ${\mathit{f}}_{\mathrm{SL}}$(x)={1+B[s/(x+s${)]}^{\mathrm{s}}$${\mathrm{e}}^{\mathrm{x}}$ ${\mathit{}}}^{\mathrm{\ensuremath{-}}1}$. This solution represents the symmetric part of the total distribution function. The nondimensional energy and applied electric field are x and \ensuremath{\surd}s , respectively, and B is a constant determined by normalization. The total distribution is given by the above and its derivative and is found to be rotationally symmetric about the electric field. This distribution reduces to the shifted Fermi-Dirac distribution for small s and to the Druyvesteyn distribution in the classical limit. An analytic expression for electrical conductivity is derived together with an approximate expression for the chemical potential in the small-electric-field limit. A generalized criterion for the classical versus quantum domains is discussed relevant to the present study. It is found that otherwise quantum domains become classical for sufficiently large applied electric fields.

Hot carriers in quasi-2-D polar semiconductors

This paper reviews hot carrier effects in quasi-2-D polar semiconductors (quantum wells and heterostructures), with special emphasis on the GaAs/AlGaAs system. After briefly introducing the basic concepts in hot carrier physics, we discuss theoretical calculations of carrier-phonon interactions and hot carrier energy loss rates to the lattice in quasi-2-D systems. We then discuss how these quantities are affected by degeneracy, plasma effects, and hot phonons. The bulk of the paper is devoted to a discussion of experimental results and their analysis. Three kinds of experiments are discussed: I-V and related transport measurements, direct time-of-flight measurements of velocity-field characteristics, and measurements which use optical spectroscopy to provide direct information about the carrier distribution function in the presence of external perturbations. The optical studies have given valuable new insight into the behavior of hot carrier relaxation processes in quasi-2-D systems from femtosecond to steady-state conditions.

Hall Factor in p‐Ge with Anisotropic Energy Spectrum

AbstractBy solving the kinetic Boltzmann equation, a quasi‐momentum distribution function for charge carriers in p‐Ge is found taking into account the non‐sphericity of the constant energy surfaces of light and heavy hole bands. Account is also taken of the anisotropic carrier scattering by ionized impurities and acoustic lattice vibrations. The magnetic field and temperature dependences of the Hall‐factor rH are calculated for fields 0 &lt; B ≦ 5 T and temperatures 77 K ≦ T ≦ = 260 K. The results calculated are compared with measured τH values obtained on oriented p‐Ge samples with free carrier concentration p ≈ 1.7 × 1014cm−3.

Nonlinear field dependence of carrier mobilities and irreversible thermodynamics in semiconductors

Irreversible thermodynamics and transport processes in semiconductors are studied by means of the Boltzmann equation for carrier distribution functions. By solving the equation by the modified moment method, we derive the extended Gibbs relation for entropy change and the evolution equations for fluxes. The steady-state solutions of the evolution equations yield diffusion fluxes and drift velocities which are nonlinear with respect to the field strength. Depending on the wave-number dependences of the scattering cross sections and the affinity of electrons in different valleys in the conduction band, the drift velocity and the mobility can exhibit a plateau or a negative differential mobility in the high-field regime. Calculated carrier drift velocities are compared with experimental data on the electron drift velocities in Si and $n$-Ge. The comparisons are in good agreement and show that the theory generally predicts qualitatively correct field dependences of electron drift velocities.

High-field electroluminescence

The theoretical efficiency for the direct conversion of electrical energy into luminescent emission, that is, electroluminescence (EL), is first evaluated from thermodynamics. High-field, collisional excitation EL is then introduced and compared with low-field, minority carrier injection EL and with cathodoluminescence. Some general characteristics of thin-film high-field EL are discussed. The basic mechanism of high-field EL is reviewed. It consists of three steps occuring in sequence: generation of charge carriers, acceleration of these carriers to optical energies, and their inelastic collisions to excite luminescent centers. Ballistic, streaming and Maxwellian hot carriers are distinguished, and their distribution functions discussed. The incompatibility of large cross sections for collisional excitation and stability of the excited luminescent centers in high fields is considered, taking account of the energies of the electronic states of dopants with respect to band structure. The present understanding of the structure and electronic states of transition metal and rare earth (RE) luminescent centers in II–VI and II–VII 2 semiconductors is evaluated. Particular attention is given to probable structures of molecular dopants such as REF 3 in ZnS. Recent advances in thin-film, high-field EL are summarized. The emphasis has been on ZnS:Mn and ZnSe:Mn films sandwiched between insulating films such as Y 2O 3 and operated with a.c. voltages. All three steps of the high-field EL mechanism occur in the II–VI layer, with some carrier generation at the II–VI/insulator junction. The spatial separation of the sequential steps in contiguous layers in composite EL cells is proposed, and evidence for separation of the acceleration and collision excitation in SiO/ZnF 2:Mn/SiO cells is presented. Some quantum optical effects in thin-film EL cells are discussed. Finally, some basic problems of high-field EL are reviewed. These include: the origin of charge carriers, the distribution function of the energetic carriers, structure and energy levels of complex dopants, and sources of inefficiencies and of saturation. Possible methods for achieving higher efficiencies than the current (1%) are proposed.

Hot electron phenomena and electroluminescence

Hot electron concepts borrowed from studies of semiconductors and insulators are critically evaluated within the context of high-field thin-film electroluminescence. Particular emphasis is placed on the models required to evaluate the carrier distribution functions and the description of strongly accelerated carriers in insulators. Streaming electrons are distinguished from true ballistic electrons and are shown to involve large energy losses despite their ability to accelerate to high energies. Recent models for impact ionisation in SiO 2 are compared with data on electroluminescent media.