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     

Conductivity variations in composites of α-zirconium phosphate and fumed silica

Composite proton-conducting solid electrolytes have been formed from α-zirconium phosphate (α-Zr(HPO₄)₂·H₂O, α-ZrP) and high surface area non-porous fumed silica in varying mole ratios. Conductivity variations as a function of temperature and relative humidity have been characterised and compared to that for a similarly prepared delaminated ZrP (no silica). Conductivities increase with relative humidity and E_a decreases at high silica mole fractions. Conduction is via exposed lamellar surfaces and other microstructural imperfections in the composites.     

Linewidths of HCl broadened by CO2 and N2 and CO broadened by CO2

High resolution measurement of the linewidths of HCl broadened by CO₂ and N₂ and CO broadened by CO₂ have been performed in both the 1-0 and 2-0 bands of HCl and the 2-0 band of CO. The data were analyzed by the direct and the peak absorption methods. Values of the linewidths obtained by the two methods are in good agreement. For |m| ≤ 3, for the case of HCl + CO₂, the agreement is good for the values obtained in both bands of HCl. However for |m| > 3, the HCl + CO₂ linewidths in the 1-0 band are smaller than the corresponding lines in the 2-0 band by as much as 11% for |m| = 9. Lines (|m| ≤ 3) of the 1-0 and 2-0 bands of HCl broadened by CO₂ were also analyzed in terms of the super-Lorentzian line profile proposed by Varanasi, S. K. Sarangi, and G. D. T. Tejwani (J. Quan. Spectr. Radiative Transfer12, 857 (1972)) and the Lorentzian profile. The results indicate that near the line center (within 3γ), the shape of HCl + CO₂ lines are Lorentzian.     

Microwave spectrum of 2-chloropyridine

The microwave spectrum of 2-chloropyridine, C₅H₄NCl, has been studied in the frequency range from 26.5–40.0 GHz. The spectrum is characterized by strong parallel type transitions of a near-prolate asymmetric top. The assigned transitions have been used to evaluate the ground state rotational constants of the two chlorine isotopes. The rotational constants are (in MHz): A = 5872.52, B = 1637.83, C = 1280.48 for the ³⁵Cl isotopic species and A = 5872.16, B = 1591.76, C = 1252.17 for the ³⁷Cl isotopic species. The small inertial defect indicates the molecule is planar. In addition an excited vibrational state of C₅H₄N³⁵Cl has been observed and analyzed. The chlorine quadrupolar coupling constants were determined for the ground state and are: χ_aa = ?71.9 MHz for ³⁵Cl and χ_aa = ?54.9 MHz for ³⁷Cl. By assuming the pyridine ring structure the CCl bond length is found to be 1.72 Å.     

Self-adjoint algebras of unbounded operators

Unbounded *-representations of *-algebras are studied. Representations called self-adjoint representations are defined in analogy to the definition of a self-adjoint operator. It is shown that for self-adjoint representations certain pathologies associated with commutant and reducing subspaces are avoided. A class of well behaved self-adjoint representations, called standard representations, are defined for commutative *-algebras. It is shown that a strongly cyclic self-adjoint representation of a commutative *-algebra is standard if and only if the representation is strongly positive, i.e., the representations preserves a certain order relation. Similar results are obtained for *-representations of the canonical commutation relations for a finite number of degrees of freedom.Work supported in part by U.S. Atomic Energy Commission under Contract AT(30-1)-2171 and by the National Science Foundation.Alfred P. Sloan Foundation Fellow.     

Young modules and filtration multiplicities for Brauer algebras

We define permutation modules and Young modules for the Brauer algebra B _k (r,δ), and show that if the characteristic of the field k is neither 2 nor 3 then every permutation module is a sum of Young modules, respecting an ordering condition similar to that for symmetric groups. Moreover, we determine precisely in which cases cell module filtration multiplicities are well-defined, as done by Hemmer and Nakano for symmetric groups. Supported by the European Community through Marie Curie fellowship MCFI 2002-01325 Supported by EPSRC grant GR/S18151/01     

Asymptotics of eigenvalue clusters for Schrödinger operators on the Sierpinski gasket

In this note we investigate the asymptotic behavior of spectra of Schrödinger operators with continuous potential on the Sierpinski gasket . In particular, using the existence of localized eigenfunctions for the Laplacian on we show that the eigenvalues of the Schrödinger operator break into clusters around certain eigenvalues of the Laplacian. Moreover, we prove that the characteristic measure of these clusters converges to a measure. Results similar to ours were first observed by A. Weinstein and V. Guillemin for Schrödinger operators on compact Riemannian manifolds.

     

Endpoints of set-valued dynamical systems of asymptotic contractions of Meir–Keeler type and strict contractions in uniform spaces

In this paper, we introduce the concepts of the set-valued dynamical systems of asymptotic contractions of Meir–Keeler type and set-valued dynamical systems of strict contractions in uniform spaces and we present a method which is useful for establishing conditions guaranteeing the existence and uniqueness of endpoints of these contractions and the convergence to these endpoints of all generalized sequences of iterations of these contractions. The result, concerning the investigations of problems of the set-valued asymptotic fixed point theory, include some well-known results of Meir and Keeler, Kirk and Suzuki concerning the asymptotic fixed point theory of single-valued maps in metric spaces. The result, concerning set-valued strict contractions (in which the contractive coefficient is not constant), is different from the result of Yuan concerning the existence of endpoints of Tarafdar–Vyborny generalized contractions (in which the contractive coefficient is constant) in bounded metric spaces and provides some examples of Tarafdar–Yuan topological contractions in compact uniform spaces. Definitions and results presented here are new for set-valued dynamical systems in uniform, locally convex and metric spaces and even for single-valued maps. Examples show a fundamental difference between our results and the well-known ones.     

Comparison of Boolean algebras

In this work, some results related to superatomic Boolean interval algebras are presented, and proved in a topological way. Let x be an uncountable cardinal. To each I x, we can associate a superatomic interval Boolean algebra B _I of cardinality x in such a way that the following properties are equivalent: (i) I I x, (ii) B _I is a quotient algebra of B _J, and (iii) there is an homomorphism f from B _J into B _I such that for every atom b of B _I, there is an atom a of B _J satisfying f(a)=b. As a corollary, there are 2 ^x isomorphism types of superatomic interval Boolean algebras of cardinality x. This case is quite different from the countable one.