Classification of current instabilities during low-temperature breakdown in germanium

We present experimental investigations on the spatio-temporal nonlinear current flow in the post-breakdown regime of p-germanium at liquid-helium temperatures. The basic nonlinear effects are characterized in terms of the underlying semiconductor physics, taking into account the influence of different experimental parameters.     

Experimental progress in the nonlinear behavior of semiconductors

During the past five years spontaneous oscillations and chaotic behavior have been observed in the electronic transport of many semiconductors. Whereas the temperature range of some experiments extends up to room temperature, the majority of the measurements have been performed at liquid helium temperatures. We summarize these experimental developments and discuss their impact on the field of nonlinear dynamics which is rapidly progressing at present.     

Noise current spectrum in submicrometer samples

The noise current spectral density in submicrometer samples is computed using the Monte Carlo method. The normalized spectral density is found to decrease with sample length and increase with the field. The high-field noise is like shot noise and increases with current in agreement with experimental results.     

Noise and correlation functions of hot carriers in semiconductors

We present a unifying theory of electronic noise appropriate to semiconductor materials in the presence of electric fields of arbitrary strength. In addition to thermal noise, a classification scheme for excess noise indicating different microscopic sources of fluctuations responsible for number and mobility fluctuations is provided. On the basis of simple two-level models, numerical calculations using a Monte Carlo technique are performed for the case of p-type Si at 77 K. The primary quantity which is evaluated by the theory is the auto-correlation function of current fluctuations which, subsequently, is analyzed in terms of correlation functions of the relevant physical variables. Accordingly, the corresponding current spectral-densities are determined and then compared with direct experimental results and/or analytical expressions. Important subjects which have been investigated are: (i) the effect of field assisted ionization on generation-recombination noise from shallow impurity levels; (ii) the contribution to the total noise spectrum of cross-correlation terms coupling fluctuations in velocity with those in energy and number; (iii) the current random telegraph signal and the corresponding spectral density associated with a mobility fluctuator. In all cases the numerical calculations are found to be in satisfactory agreement with experiments and/or analytical expressions thus fully supporting the physical reliability of the theoretical approach here proposed.List of the Symbols Used e Absolute value of the electron charge - f Frequency - f Distribution function - g ₁ Scattering strength with the scatter in state 1 - g ₂ Scattering strength with the scatter in state 2 - Reduced Planck constant - j Total current density - j _c Conduction current density - j _d Displacement current density - j _x Component along the x direction of the total current density - k Carrier wavevector - m Carrier effective mass - m ₀ Free electron mass - r Position vector - s Average sound velocity - t Time - u Fraction of ionized carriers - u _i Random telegraph signal related to carrier state - u _m Random telegraph signal related to scatterer state - v _d Ensemble average of the free carrier drift-velocity - v _i Carrier group velocity - v _t Ensemble average of the carrier velocity in the direction transverse to the applied field - v _ix Component along the x direction of the carrier group velocity - v _d ^r Ensemble average of the reduced drift-velocity - v ^r _i Reduced velocity component in the field direction of the i-th particle - v _ix ^j Reduced velocity component along the x axis of the i-th particle in band j - v ^r _ix Reduced velocity component along the x axis of the i-th particle - x _d Ensemble average of the carrier displacement along the x direction from the initial position - x _i Displacement along the x direction of the i-th carrier from the initial position - y _i i-th stochastic parameter - A Cross-sectional area of a homogeneous sample - C _I Auto-correlation function of the total current fluctuations - Auto-correlation function of the total current fluctuations due to mobility fluctuations - D Diffusion coefficient - D _t K Optical deformation potential - E Electrical field strength - E Electric field - E _x Component of the electric field along the x direction - E ₁ ⁰ Acoustic deformation potential - G Conductance - I Total current - I ₀ Total current in the voltage noise operation - I _m Total current associated with mobility fluctuations - I ^V Total current in the current noise operation - K _B Boltzmann constant - L Length of a homogeneous sample - N Number of free carriers which are instantaneously present in the device - N _A Acceptor concentration - N _I Total number of carriers inside the device participating in the transport (here assumed to be constant in time) - N _T Total number of carriers which are instantaneously present in the device - S _I Spectral density of current fluctuations - S _V Spectral density of voltage fluctuations - Spectral density of current fluctuations associated with the mobility fluctuations - Spectral density of current fluctuations due to correlations between fluctuations in number and velocity - Spectral density of current fluctuations due to generation-recombination processes - Spectral density of current fluctuations due to free carrier drift-velocity fluctuations - S _I ^l Longitudinal component with respect to the applied field of the current spectral-density - S _I ^t Transverse component with respect to the applied field of the current spectral-density - T Absolute temperature - T _e Electron temperature - V Electrical potential - V ^I Electrical potential in the voltage noise operation - W Collision rate - Z Small signal impedance - Poole-Frenkel factor - Equilibrium generation rate - _E Field dependent generation rate - Typical energy for thermally escaping from the impurity level - v _d ⁽⁰⁾ Fluctuation of the ensemble average of the driftvelocity associated with Brownian-like motion - v _d ^r(0) Fluctuation of the ensemble average of the reduced drift-velocity associated with Brownian-like motion - Carrier energy - ₀ Vacuum permittivity - _a Energy of the acceptor level - _r Relative static dielectric constant - Angle between initial and final k states - _op Optical phonon equivalent temperature - Mobility - ₀ Chemical potential - ₁ Mobility with the fluctuating scatterer in state 1 - ₂ Mobility with the fluctuating scatterer in state 2 - ₀ Crystal density - _E Field dependent volume recombination rate - _eq Equilibrium volume recombination rate - Conductivity - _g Cross-section for impact ionization - _c Average scattering time - _g Generation time - _l Carrier lifetime - _m Scatterer lifetime - _m1 Mean value of the time spent by the fluctuating scatterer in state 1 - _m2 Mean value of the time spent by the fluctuating scatterer in state 2 - _r Average recombination time - _T Transit time - Scattering rate - _AB Correlation function of the two variables A and B     

Theoretical approaches to nonlinear and chaotic dynamics of generation-recombination processes in semiconductors

We present an overview of recent progress in the theoretical modelling of nonlinear and chaotic dynamics induced by generation and recombination processes of charge carriers. Impact ionisation of impurities is the autocatalytic, i.e. destabilizing, step of three different physical mechanisms for spontaneous, self-sustained oscillations of the carrier density. The restoring force is furnished by one of the following three processes: (i) dielectric relaxation of the internal electric field, (ii) energy relaxation of the hot carriers, and (iii) trapping at impurities, where the discrete nature of the individual generationrecombination processes is taken into account.     

Contact versus bulk noise in resistive layers

General formulae are derived to calculate the noise generated by a resistor of arbitrary shape. Contact and bulk noise have been taken into account. By calculating a particular example it will be shown that the influence of the contact and bulk noise can be changed by varying the resistor's shape.     

Hopping and band transport in amorphized semiconductors

Crystalline samples of Si, GaAs, GaP, InP, and CdTe have been rendered amorphous by bombardment with rare gas ions. DC conductivity and thermopower have been measured as a function of temperature in the interval between 15–500 K. In all cases, electron transport at low temperatures is characterized by non-simply activated processes of the hopping type transport, whereas band transport is observed at higher temperatures. The common and individual features of the different amorphous systems are discussed within the framework of existing transport theories.     

Magnetoplasma waves in anisotropic semiconductors and semimetals

The propagation of electromagnetic waves in solids with electron and hole energy surfaces of second order placed in a magnetic field is investigated. The deduced dispersion relation is compared with measurements of the Alfvén wave velocity in bismuth. The theoretical and experimental results are in good agreement. In particular, it is clearly shown that for special orientations of the magnetic field switching from one mode to the other occurs. Parameters of the energy surfaces of bismuth are given by fitting the theoretical curves to the measured phase velocity diagrams.     

Measurement of the hot-electron conductivity in semiconductors using ultrafast electric pulses

Semiconductor response to ultrafast electric pulses was investigated both theoretically and experimentally. The possibilities for hot-electron drift velocity estimation from a pulsed electric conductivity measurement were analysed. An optoelectronic arrangement with time resolution of 20 ps was used to perform such measurements on then-InSb andn-InAs single crystals. Negative differential mobility (n.d.m.) was observed in both semiconductors at high electric fields.     

A simple theory of the laser-induced damage in semiconductors

A simple theory is presented for the laser-induced damage in semiconductors in the range of low irradiation intensities. This theory avoids the solution of coupled differential equations, and takes into account characteristic physical properties of semiconductors—i.e. energy gap, carrier lifetime, and surface recombination velocity. The deduced equations permit us to estimate the damage threshold, or the minimum irradiation time required for damage to occur. Comparison is made with some experimental results reported in the literature and a reasonable agreement is found.     

Nonlinear response and chaos in semiconductors induced by impact ionization

A review is given of the nonlinear response and chaos induced by impact ionization of neutral shallow donors, observed in n-GaAs. Two kinds of the observation are described; (i) firing wave instability, and (ii) periodically driven current filament. For the firing wave instability, several important aspects are discussed including the selective excitation of the current filaments and the deterministic nature of the firing density wave. The nonlinear response of a periodically driven current filament has been investigated by applying a dc+ac bias of the form ofV _dc+V _ac sin(2f ₀ t), wheref ₀1 MHz. The carrier dynamics and the bifurcation routes to chaos are discussed in terms of the observed phase diagram and the bifurcation map. The deterministic nature of the strange attractors are described in detail in terms of the correlation dimension and the Kolmogorov entropy.     

On the origin of the hot-electron instabilities inn-InSb at crossed electric and magnetic fields

Coherent current oscillations with frequencies of several hundreds of megahertz and microwave generation at frequencies up to 40 GHz were investigated in then-InSb samples subjected to crossed electric and magnetic fields. They are shown as being caused by electric field redistribution and negative differential mobility in high-field regions.     

Low-temperature magnetoresistance due to weak localization in lightly doped semiconductors

The first observation of low-temperature magnetoresistance (MR) of interference nature in the case of a light doping is reported. The MR occurs in n- and p-type Ge samples at a frequency of 10 GHz at temperatures below 30 K in weak magnetic fields on the background of the classical MR effect associated with electrons in different valleys (n-Ge) and with heavy and light holes (p-Ge).     

A negative differential conductivity due to recombination and impact ionization in semiconductors at low temperatures

The probability of impact ionization and the recombination time are known to increase monotonically with the electric fieldE. I show that at low temperatures both functions achieve a maximum and decrease in the electric field range where the emission of optical phonons with subsequent impurity scattering dominate. This nonmonotonicity results in three different types of N-shaped negative differential conductivity (n-ndc). The carrier concentration and the current decrease whenE increases due to decreasing of the impact ionization probability for weakly compensated samples and of the recombination time for highly compensated samples. At the antithreshold electric-field impact ionization dies out, which results in a dramatic decrease of the current for intermediately compensated samples. This huge n-ndc could be used in a novel type of the Gunn diode. The essential increase of threshold electric field of impact ionization is also predicted, and the effect could enhance the efficiency of photodetectors.The research was supported by the Alexandervon-Humboldt-Foundation     

Nonlinear oscillations and chaotic behavior due to impact lonization of shallow donors in InSb

Both nonlinear oscillations and chaotic behavior in n-InSb are experimentally investigated for the case of impact ionization of shallow donors at low temperatures. Complex behavior including a simple periodic oscillation, a period-doubling route to chaos, and quasiperiodic behavior are observed with increasing electric field as the parameter. For the first time, a type of pitchfork bifurcation (period halving) is seen.     

Effect of temperature on the validity of the einstein relation in heavily doped semiconductors

An analytical expression of the modified form of the Einstein relation in heavily doped semiconductors in which Gaussian band tails are formed near the lower limit of heavy doping is derived for studying the temperature dependence of the diffusivity-mobility ratio of the carriers in such semiconductors. It is found that, with increasing temperature from relatively low values, the ratio first increases in a nonlinear manner and then decreases till, at high temperatures, it approaches its value corresponding to the non-degenerate condition resulting in a peak over a narrow range of temperatures.     

On the possibility of wake-field effects in the conductivity of doped semiconductors

The formation of a wake field around a swift ion passing through an electron gas and the resulting contribution to the stopping power acting on the ion is an intensively studied phenomenon in metals and semiconductors. The present investigation serves to clarify whether an analogous effect, namely the formation of wake fields and a corresponding contribution to the resistivity, might occur in the Galilei-transformed case of electronic transport in doped semiconductors where the gas of drifting charge carriers passes through an array of fixed impurity ions. By use of an appropriate dynamical screening theory we show that indeed a local plateau in the current density versus field characteristic has to be expected whenever the mean drift energy per carrier exceeds the sum of the mean thermal carrier energy and the zero-point energy of the longitudinal plasma mode of the carrier gas. However, our survey of the published literature suggests that this condition might be too stringent, at least for bulk materials and standard experimental situations, where the strong carrier heating in the high-field regime of relevance in combination with other drift-limiting mechanisms or interband electron-hole avalanching would always precede and prevent the formation of the wake.Dedicated to Prof. H.-J. Queisser on the occasion of his 60th birthday     

Oscillatory effective mass in degenerate narrow-gap semiconductors in a quantizing magnetic field

An attempt is made to study the effect of a quantizing magnetic field on the effective electron mass in degeneraten-type narrow-gap semiconductors at low temperatures. It is found, takingn-Hg_1−x Cd _x Te as an example, that the effective electron mass shows an oscillatory magnetic-field dependence as is expected because of the dependence of the effective mass in degenerate non-parabolic bands on Fermi energy which oscillates with changing magnetic field. The amplitude of oscillations is, however, found to be significantly influenced by the alloy composition whereas the period is found to be independent of the band non-parabolicity, i.e. of the compositional parameter in ternary semiconductors.     

Determination of the impurity concentration in heavily doped inhomogeneous semiconductors from the measurement of the low-temperature conductivity

A method has been developed for determining the effective concentration of shallow impuritiesN ^* reponsible for the low-temperature conductivity in the vicinity of the metal-insulator transition for inhomogeneous samples of n-Ge:As. The method is based on the measurement of two ratios of sample resistance -R(4.2 K)/R(300 K), =R(2.0 K)/R(4.2 K) and the conductivity (4.2 K). The next step consists of plotting and (4.2) vs . Assuming that is the most reliable parameter, one can calculate, after an averaging procedare, the corrected values of ^*, ^*(4.2) and the resistivity at room temperature ^*(300)=[ ^{* *}(4.2)]^–1. Finally, using the known dependence (300)=f(N) for homogeneous samples, one can obtain the values ofN ^*. The dependences ofN ^* on and on are plotted. The scaling behavior of the conductivity of the Ge:As samples with corrected values of ^*(4.2) andN ^* has been observed down toT=100 mK.     

Theory of hot-electron quantum diffusion in semiconductors

In the linear response regime close to equilibrium, the fluctuation-dissipation theorem relates linear transport coefficients via the well-known Green–Kubo or Einstein relation. The latter embodies a deep connection between fluctuations causing diffusion and dissipation, which are responsible for a finite mobility. Far from equilibrium, however, the Einstein relation is no longer valid so that both the mobility and diffusivity gain their own physical integrity. Consequently, beyond a linear response, both quantities have to be described by different approaches. Unfortunately, there is a strong imbalance of research activities devoted to the study of both transport mechanisms in semiconductors. On one hand, the rich physics of high-field quantum drift in semiconducting structures has a long history and has reached a high level of sophistication. On the other hand, there are only comparatively few and unsystematic studies that cover quantum diffusion of carriers under high-field conditions. This review aims at reducing this gap by presenting a unified approach to quantum drift and quantum diffusion. Starting from a semi-phenomenological basis, a quantum theory of transport coefficients is developed for one- as well as multi-band models. Physical implications are illustrated by selected applications whereby the quantum character of the approach is emphasized. Furthermore, the basic unified treatment of transport coefficients is extended by accounting for the two-time dependence of one-particle correlation functions in quantum statistics. As an application, a phononless transport mechanism is identified, which solely originates from the double-time nature of the evolution. Finally, additional examples are presented that illustrate the important role played by quantum diffusion in semiconductor physics.