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.     

Instability of nonlinear bound states

We establish a sharp instability theorem for the bound states of lowest energy of the nonlinear Klein-Gordon equation,u _tt–u+f(u)=0, and the nonlinear Schrödinger equation, –iu _t–u+f(u)=0.Supported in part by NSF Grants MCS 81-21487 and MCS 82-01599     

Nonlinear transport in {\sf \beta}-Na 0.33V 2O 5

Transport properties of the charge ordering compound β-Na_0.33V₂O₅ are studied in the temperature range from 30 K to 300 K using current driven DC conductivity experiments. It is found that below the metal-insulator transition temperature ( ) this material shows a nonlinear charge density modulation behavior. The observed conductivity is discussed in terms of a classical domain model for charge density modulation transport.     

Multi-exponential analysis of DLTS

Deep-level transient spectroscopy (DLTS), which is widely used to characterize deep impurity centers in semiconductors, assumes a single exponential wave form for the transient junction capacitance. When there are several closely spaced energy levels this assumption is no more valid, and the conventional DLTS may lead to errorneous results. To overcome this difficulty we propose here a novel method which we call the multi-exponential DLTS(MEDLTS). The transient wave form of the junction capacitance is directly analysed into multi-exponential compouents using the nonlinear least-squares analysis program DISCRETE developed by Provencher. The resolved time constants of these components are then displayed in the form of aT ²–1/T plot. According to the results of simulation with various parameters MEDLTS is shown quite effective to resolve closely spaced energy levels which can not be resolved by the conventional DLTS. As an example of the application of this method deep levels in Si: Au were investigated. The results have shown that a single peak in conventional DLTS actually consists of two adjacent levels with activation energies and capture cross-sectionsE _B1=0.49 eV, _B1=1.1×10^–14cm² andE _B2=0.46 eV, _B2=1.3×10^–15 cm² and with amplitude ratio 11.     

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     

Stochastic Schrödinger operators and Jacobi matrices on the strip

We discuss stochastic Schrödinger operators and Jacobi matrices with wave functions, taking values in ^l so there are 2l Lyaponov exponents ₁..._l0 _l+1..._2l =–₁. Our results include the fact that if ₁=0 on a set positive measure, thenV is deterministic and one that says that {E|exactly 2j 's are zero} is the essential support of the a.c. spectrum of multiplicity 2j.Research partially supported by USNSF under grant DMS-8416049     

Selectively dopedn-AlxGa1−xAs/GaAs heterostructures with high-mobility two-dimensional electron gas for field effect transistors

The transport properties of warm and hot electrons in selectively dopedn-Al _x Ga_1–x As/GaAs heterostructures created by electric fields up to 500 V/cm were studied by Hall effect, conductivity, and Shubnikov-de Haas measurements at lattice temperatures from 4.2 to 300 K. Hall measurements revealed a substantial decrease of electron mobility and also of sheet electron concentration at 77 K with enhanced electric field. The accelerated 2D electrons are partly scattered into the low-mobility first excited (E ₁) subband, and they are partly trapped in immobile states located in the Al_xGa_1–xAs near the interface. Consequently, two differentv(E) characteristics were obtained at 77 K. The 2D electrons populating only the lowest (E ₀) subband exhibit a velocity of v-2×10⁷ cm/s at 500 V/cm, while the averaged velocity due to all electrons reaches a value of v-1.5×10⁷cm/s at 500 V/cm. The analysis of the Shubnikov-de Haas oscillations and Fast Fourier transformation of the data manifested that the 2D electrons are very rapidly accelerated at 4.2 K and achieve electron temperatures much higher than the lattice temperature at electric fields as low as 1 V/cm. The major cooling process for these electrons is scattering into the low-mobilityE ₁ subband.     

Hydrogenated amorphous silicon films obtained by a low pressure dc glow discharge

Films of a-Si: H have been deposited by means of a dc hot cathode discharge of SiH₄ with electrostatic confinement at a pressure as low as 0.4 Pa. The plasma used is quite quiescent as has been observed by means of reproducible Langmuir probe measurements. Substrates have been placed at different locations in between the electrodes, some of them facing the anode and the others facing the cathode.Films deposited on substrates facing the cathode present a granular, non-columnar, structure, an IR spectrum with only SiH absorption peaks, and a very low photoresponse. Films deposited on substrates facing the anode have a similar IR spectrum but are homogeneous, have lower hydrogen content, and present a high photoresponse. The optical absorption coefficient shows in all samples thenE=C(E–E₀) ^x behaviour, but with exponentx=3 and notx=2 as is usually considered in a-SiH.     

The influence of the transformation of waves on the threshold currents of the beam instability

In a hot anisotropic plasma with an inhomogeneity across the magnetic field, eigenmodes are a superposition of long waves of cold plasma and the short-wave Bernstein modes. As a consequence of the fact that the Bernstein modes with non-zero wave numbers are damped more intensively than the waves of cold plasma, the threshold currents of the beam instability increase correspondingly compared with the case of homogeneous plasma. For frequencies between the first and second harmonics of the electron cyclotron frequency the threshold current is from ₀/ _T to ( ₀/ _T )³ times greater, where ₀ is the beam velocity and _T is the thermal velocity of plasma electrons.It is the author's pleasant duty to express his thanks to the theorists of the Institute of Plasma Physics, especially to ing. K. Jungwirth CSc, for valuable discussions he has had with them.     

Kinetic equation in the kinetic region of the dilute and nonuniform electron plasma

Mori's scaling method is used to derive the kinetic equation for a dilute, nonuniform electron plasma in the kinetic region where the space-time cutoff (b, t _c) satisfies _Dbl _f , _D t _{c f} , with _D the Debye length, _D ^–1= _p the plasma frequency, andl _f and _f the mean free path and time, respectively. The kinetic equation takes account of the nonuniformity of the order ofl _f and _D for the single-and the two-particle distribution function, respectively. Thus the Vlasov term associated with the two-particle distribution function is retained. This kinetic equation is deduced from the kinetic equation in the coherent region obtained by Morita, Mori, and Tokuyama, where the space-time cutoff of the coherent region satisfies _Dbr ₀, _Dt _{c 0}, withr ₀ the Landau length and ₀ the corresponding time scale.     

A “hydrodynamic” theory of surface reaction rates

The Brownian motion of adsorbed particles is described in terms of the first four velocity moments of the distribution function (number density, momentum density, energy density and energy current density). The resulting hydrodynamic equations turn out to be sufficient for a simple derivation and extension of Kramers' results for chemical reaction rates in terms of the friction constant of an underlying Fokker-Planck equation. An interpolation formula is obtained for() containing Kramers' results for small and large as limiting cases. For temperaturesT small compared to the well depthV ₀ one finds a large regionT/V ₀/v ₀V ₀/T in which Eyring's absolute rate theory is approximately valid.On leave of absence from Physikdepartment der TUM, München-Garching     

Microwave dielectric properties of Ti4+/MgO and Ni2+/MgO

The dielectric properties of titanium-doped magnesium oxide (Ti/MgO) and nickel-doped magnesium oxide (Ni/MgO) single crystals have been measured in the range of temperature from 300 to 450 K at the microwave frequency of 9.31 GHz. For both crystals the dielectric properties are found similar. From the conductivity data, the activation energy in the measured temperature region has been estimated to be 0.15 eV. The values of the temperature dependence (–1)^–1(+2)^–1 (/T) _p have been calculated. The data confirms the Bosmann and Havinga postulate that, for materials in which the dielectric constant; is less than 20 the temperature dependence should be positive.     

Scaling of Mandelbrot sets generated by critical point preperiodicity

Letzf (z) be a complex holomorphic function depending holomorphically on the complex parameter . If, for =0, a critical point off ₀ falls after a finite number of steps onto an unstable fixed point off ₀, then, in the parameter space, near 0, an infinity of more and more accurate copies of the Mandelbrot set appears. We compute their scaling properties.On leave from the University of Geneva     

Continuity of type-I intermittency from a measure-theoretical point of view

We study a simple dynamical system which displays a so-called type-I intermittency bifurcation. We determine the Bowen-Ruelle measure and prove that the expectation (g) of any continuous functiong and the Kolmogoroff-Sinai entropyh() are continuous functions of the bifurcation parameter. Therefore the transition is continuous from a measure-theoretical point of view. Those results could be generalized to any similar dynamical system.     

Shallow acceptor population and free hole concentration in Si: In and Si:Ga with IR-photoexcitation

Hall measurements at low temperaturesT<50 K have been performed on Si:In (N _In10¹⁷ cm^–3) and Si:Ga (N _Ga10¹⁸ cm^–3) with infrared photoexcitation of holes into the valence band. It is shown in quantitative agreement with a theoretical model that the population of shallow acceptors, e.g. B and Al, which are present as impurities in concentrations ofN _B,Al10¹²-10¹⁴ cm^–3 strongly affects the photoexcited hole concentration. Photo-Hall measurements can, therefore, serve as a tool for the determination of low impurity acceptor concentrations in the case of high In- or Ga-doping. Hole capture coefficientsB _In=6×10^–4 (T/K)^–1,8 cm³ s^–1 andB _Ga=2×10^–4 (T/K)^–1 cm³ s^–1 have been determined.     

Instability in degenerate two-photon running wave laser with detuning

The previous stability analysis of the degenerate two-photon running wave laser is extended to the inclusion of detuning between frequencies of cavity and atoms. We derive the analytical equation for the critical pumping and prove analytically that for the special case ofr (/) being unity, there is no Hopf bifurcation instability for the bad cavity. The good cavity case is analysed numerically. The role played by detuning is to raise the critical pumping. In the case ofk (or <K) where there is no Hopf bifurcation instability for the perfect tuning case, the large detuning can give rise to self-pulsing instability.     

Quantum Hamiltonian for the Rigid Rotator

An excess term exists when using hermitian form of Cartesian momentum p _i (i = 1,2,3) in usual kinetic energy 1/(2) p ² _i for the rigid rotator, and the correct kinetic energy turns to be 1/(2) (1/f _i ) p _i f _i p _i where f _i are dummy factors in classical mechanics and nontrivial in quantum mechanics.     

Differential Systems for Biorthogonal Polynomials Appearing in 2-Matrix Models and the Associated Riemann–Hilbert Problem

We consider biorthogonal polynomials that arise in the study of a generalization of two–matrix Hermitian models with two polynomial potentials V ₁ (x), V ₂ (y) of any degree, with arbitrary complex coefficients. Finite consecutive subsequences of biorthogonal polynomials (windows), of lengths equal to the degrees of the potentials V ₁ and V ₂ , satisfy systems of ODEs with polynomial coefficients as well as PDEs (deformation equations) with respect to the coefficients of the potentials and recursion relations connecting consecutive windows. A compatible sequence of fundamental systems of solutions is constructed for these equations. The (Stokes) sectorial asymptotics of these fundamental systems are derived through saddle-point integration and the Riemann-Hilbert problem characterizing the differential equations is deduced. Work supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC) and the Fonds FCAR du Québec     

The uniformly driven kink in the damped φ4-chain

The motion and the linear modes of a uniformly driven kink (domain wall) in the damped ⁴-chain are examined in classical continuum approximation. The model-specific component of the Büttiker-Thomas formula (yielding the velocity-field characteristic of a kink driven between two different domains of a multistable system) is calculated explicitly. Similarly to the free ⁴-kink, the driven ⁴-kink is linearly stable. We find, however, that the (single) bound state of the free kink can only survive in the driven and damped system below a critical strengthF _c of the driving field, where (i.e. forF<F _c<F _max) it is splitted off in two localized relaxation modes. The only localized modes present in the whole existence domainF<F _max of the driven kink are (1) the zero frequency translation mode and (2) the inertia mode, which is a universal smooth relaxation mode of the driven kinks in all the linearly damped multistable systems. Further key words: Landauer formula (mobility), nerve impulses, Schlögl model, Rosen-Morse potential.Work supported by the Swiss National Science Foundation     

Hierarchy of Chaotic Maps with an Invariant Measure

We give hierarchy of one-parameter family (, x) of maps at the interval [0, 1] with an invariant measure. Using the measure, we calculate Kolmogorov-Sinai entropy, or equivalently Lyapunov characteristic exponent of these maps analytically, where the results thus obtained have been approved with the numerical simulation. In contrary to the usual one-parameter family of maps such as logistic and tent maps, these maps do not possess period doubling or period-n-tupling cascade bifurcation to chaos, but they have single fixed point attractor for certain values of the parameter, where they bifurcate directly to chaos without having period-n-tupling scenario exactly at those values of the parameter whose Lyapunov characteristic exponent begins to be positive.