Bifurcation phenomena in a periodically driven current filament and a conjecture on the turbulent patterns by computer simulations

Bifurcation routes to chaos in a periodically driven current filament have been studied by computer simulations. By an impact ionization model, theS-shaped currentvoltage curve is perturbed by the dc+ac bias ofE ₀+E _acsin(27f ₀t). The bifurcation maps are described as a function ofE ₀. In the prebreakdown region, the fractal basin boundary, the crisis and the intermittency are discussed, based on the general considerations of the carrier dynamics on the catastrophe manifold. The intermittent burst of the current filament is explained by the destabilization of the weak turbulence generated in the lower branch. In the diffusion-reaction model, the spatio-temporal mode patterns of the transverse carrier profile have revealed the competitive evolution of the hyper-freezing and the firing.     

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.     

Complex Ginzburg-Landau equation for nonlinear travelling waves in extrinsic semiconductors

We study the nonlinear state of a travelling-wave instability occurring close to the onset of impact ionization in extrinsic semiconductors. Our investigations are based on a complex Ginzburg-Landau equation (CGLE). For a simple generation-recombination function including impact ionization and thermal recombination of the charge carriers, we find a supercritical bifurcation of stable travelling waves for most parameter values. The results are compared with a numerical solution of the basic equations of motion. Furthermore, we expect that weak turbulence phenomena should be observed in semiconductors if their specific generation-recombination kinetics leads to a CGLE with appropriate coefficients.     

Quasiperiodicity and chaos in the Gunn effect

A model is proposed describing the Gunn effect in GaAs. Numerical results of the model exhibit frequency locking and the transition from quasiperiodicity to chaos under an externally applied periodic electrical field.     

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     

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.     

Logarithmic frequency scaling of semiconductor oscillations caused by a modified saddle-node bifurcation on a limit cycle

The oscillatory behavior of low-temperature impact ionization breakdown inp-type germanium is investigated experimentally. We explain the anomalous scaling behavior of a saddle-node bifurcation on a limit cycle in terms of a simple model approach. It represents the low-dimensional analog to a new type of intermittency proposed recently.     

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.     

Spiking in an activator-inhibitor model for elements with S-shaped negative differential conductivity

Self-organized spatio-temporal dynamics of electrical transport is described by a simple noncubic activator-inhibitor system derived for layered semiconductor structures. The model exhibits a novel mode of self-sustained oscillations due to current filaments switching on and off (spiking) which may be periodic or chaotic. Additionally, we obtain complex multifilamentary spatio-temporal patterns. As such phenomena have been observed in various devices exhibiting S-shaped negative differential conductivity, our model is suggested to describe a generic mechanism.     

Statistical appearance of nonuniform fluctuations in a wide-gap n-n-n-n device

A cross-correlation matrix applied for restoring the doping profile in an n⁺-n⁻-n-n⁺ device was reported recently [Y.-H. Shiau, Solid-State Electron. 50 (2006) 191]. In this paper we will show that this statistical method is very useful for detecting the dynamical processes embedded in semiconductor devices. In addition, extraction of nonuniform fluctuations hidden in this wide-gap semiconductor device could be helpful for clarifying the previous studies on several competing instabilities in InSb at 77 K [A. ?enys, G. Lasiene, K. Pyragas, Solid-State Electron. 35 (1992) 975; H. Ito, Y. Ueda, Phys. Lett. A 280 (2001) 312]. A general discussion about the application of the cross-correlation matrix to other pattern-forming systems is also given in the present study.     

Nonlinear electro-optic characteristic in a photoexcited semiconductor

We study the transport properties of a GaAs-based Gunn device under local optical excitation via direct numerical simulation. The simulation results show that the hysteretic transition in between quenched and transit modes. The key mechanism for this kind of transition is related to the formation of a stationary and nonuniform hole profile around the notch regime. Therefore, the development of optical control of the microwave output is reported. In addition, the influence of impact ionization on this nonlinear semiconductor is also discussed in the present study.     

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     

Saturation of ionization edge absorption by donors in germanium

A study of saturation of the absorption and photoconductivity of Sb and P donors in Ge for radiation of 90 m wavelength, i.e., of energy very closely above their ionization edges is presented at T=9.3 K. Under these conditions negligible heating by the excess radiation energy is expected, which provides a convenient opportunity to study the kinetics of photoionization and recombination. From these measurements we have determined the donor capture cross section of electrons at 9.3 K to be _c=(1.2±0.7)×10^–12cm^–2, and the relaxation time from the 2s to the ground state as ₂₁=(5.8±1.0)×10^–10s. The saturation intensity of the absorption coefficient is around three orders of magnitude higher than the saturation intensity of the photoconductivity. We explain the nonlinear photoconductivity by the Debye-Conwell dependence of the mobility on the number of photoionized donors and compensating acceptors.     

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.     

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     

Balance-equation approach to terahertz-field-driven magnetotransport in semiconductors

We investigate the magnetotransport in semiconductors under the influence of a dc or slowly-varying electric field, an intense polarized radiation field of terahertz frequency, and a uniform magnetic field, being in arbitrary directions and having arbitrary strengths. Effective force- and energy-balance equations are derived by using a gauge that the magnetic field and the high-frequency radiation field are described by a vector potential and the dc or slowly-varying field by a scalar potential, and by distinguishing the slowly-varying velocity from the rapidly-oscillating velocity related to the high-frequency field. These equations, which include the elastic photon process and all orders of multiphoton absorption and emission processes, are applied to the examination of the effect of a terahertz radiation on the magnetophonon resonance of the longitudinal resistivity in the transverse configuration in nonpolar and polar semiconductors. We find that the previous zero-photon resonance peaks are suppressed by the irradiation of the terahertz field, while many new peaks, which may be related to multiphoton absorption and emission processes, emerge and can become quite distinct, at moderately strong radiation field. Received 17 May 1999     

Nonparabolic multivalley balance-equation approach to impact ionization: Application to wurtzite GaN

Extended nonparabolic multivalley balance equations including impact ionization (II) process are presented and are applied to study electron transport and impact ionization in wurtzite-phase GaN with a , L-M, and conduction band structure at high electric field up to 1000kV/cm. Hot-electron transport properties and impact ionization coefficient are calculated taking account of the scatterings from ionized impurity, polar optical, deformation potential, and intervalley interactions. It is shown that, for wurtzite GaN when the electric field approximately equals 530kV/cm, the II process begins to contribute to electron transport and results in an increase of the electron velocity and a decrease of the electron temperature, in comparison with the case without the II process. Similar calculations for GaAs are also carried out and quantitative agreement is obtained between the calculated II coefficients by this present approach and the experimental data. Relative to GaAs, GaN has a higher threshold electric field for II and a smaller II coefficient. Received: 27 April 1998 / Revised: 17 July 1998 / Accepted: 13 August 1998     

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.     

Spectral statistics in a quantum optical model

In this paper, we have investigated spectral statistics in a quantum optical model, the condition for the random matrix theory is analysed and illustrated with numerical results.