Utilizing changes in the carrier distribution function by magnitude and momentum scaling of scattering rates, the author a number of interesting results concerning ionization coefficients and transient drift and diffusion. Starting with a general definition of the ionization coefficient which includes nonlocal effects, the behavior of this coefficient under scaling is determined and used to find a simple analytical expression in terms of physical parameters valid for all field strengths. When fit to data for silicon, surprisingly large but consistent high‐field, effective ionization energies are found for electrons (3.6 eV) and holes (5.0 eV). This expression can also relate ionization near an interface to that in the bulk. Rate scaling is also used to predict changes in velocity overshoot and diffusion‐limited rise times between bulk and interface behavior. These comparisons are facilitated by a novel relationship between the time dependence of the spacial diffusion of a carrier pulse and it spacial displacement in an applied field. A classification of scattering rates suggested by momentum scaling as well as alternative scaling procedures applicable in special cases are briefly included. The former provides information on the behavior under momentum scaling of the positional‐displacement correlation factor of surface‐roughness scattering. The latter shows how scattering‐rate scaling and momentum scaling can be made to transform into each another.

Applications of scaling to problems in high-field electronic transport

The paper presents a general approach to numerically simulate the noise behavior of bipolar solid-state electron devices through a physics-based multidimensional device model. The proposed technique accounts for noise sources due to carrier velocity and population fluctuations. The power and correlation spectra of the external current or voltage fluctuations are evaluated through a Green's function, linear perturbation theory equivalent to the classical Impedance Field Method for noise analysis and its generalizations. The numerical implementation of the method is performed through an efficient technique, which allows noise analysis to be carried out with negligible overhead with respect to the small-signal simulation. Some case studies are analyzed in order to compare the present approach with theoretical results from the classical noise theory of p-n junctions and bipolar transistors.

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An efficient approach to noise analysis through multidimensional physics-based models

Current LSI technology has progressed rapidly and is pushing toward fabrication of sub-micron dimensioned devices. Several authors have previously used static characteristics, power dissipation, noise, and packing density to look at limiting properties of small devices, although the actual device physics was not considered in detail. As devices become smaller, we expect that the temporal and spatial scales in these devices become sufficiently small that the semi-classical approach to transport theory, as expressed by the Boltzmann transport equation, becomes of questionable validity. In this paper, we address the question of whether our physical understanding of devices and their operation can be extrapolated to small space and time scales, and to what extent the essential quantum electronics prevents a down-scaling. We attempt to lay here a conceptual framework for an ultimate physics of small devices and the modeling necessary to characterize these devices. In this first paper, we work with a dimensional scale of l ∼ 2500 A , the medium small device, leaving a smaller scale to a subsequent work. Although this scale is marginally in a region where the semi-classical approach is valid, extensive modifications must be made to incorporate several new physical effects, including: intra-collision field effect, retarded spatial and temporal non-local effects, two-dimensional quantization, memory effects in the transport parameters, nonlinear screening/descreening, and multiple scattering effects.

On the physics and modeling of small semiconductor devices—I

A generalized current equation is presented for the operation of submicron devices by supplementing the drift and diffusion currents with gradient, rate and relaxation currents. The equation includes the most important features of velocity overshoot.

Current equations for velocity overshoot

Measurements of recombination along the tracks of 2‐MeV alpha particles and 0.7‐MeV protons in thermally grownsilicon dioxide are presented. The results agree quite well with the columnar recombination model at applied fields above about 1 MV/cm if one assumes an initial Gaussian column radius b=3.5 nm. The agreement between theory and experiment at high fields is accomplished even if the excess carriers are assumed to reach instantaneous thermal equilibrium with the lattice. At fields below 1 MV/cm, the agreement between the columnar model and the experiment is improved if one assumes that the carriers remain hot for a finite time before reaching thermal equilibrium with the lattice. The discussion of the details of the thermalization process is intended to be only rough and qualitative because the process is very difficult to model. However, the model and the experiment agree fairly well at low fields if one assumes that the carriers have an energy on the order of 1 eV for a time on the order of 10− 1 3 s. The sensitivity of the model calculations to these assumptions is discussed in some detail.

Recombination along the tracks of heavy charged particles in SiO2 films

The quantum theory of electronic transport phenomena in large electric fields in highly dissipative media is critically examined. Serious conceptual problems and computational difficulties arise because neither the field nor the dissipation can be treated as a perturbation. We review a decade-old calculation of the velocity acquired by an electron in a finite electric field in a polar crystal and subsequent work which expanded our understanding of our method and results. A key feature of the earlier work was that in a single curve of electric field vs velocity, all the expected phenomena appeared, including a threshold field for producing hot electrons, in quantitative agreement with experiment, and a decreasing rate of energy loss with velocity for very fast electrons. A more recently studied problem, that of electron acceleration below the threshold field will be discussed. This problem is very important since such acceleration is the necessary precursor of ionization and breakdown. The physical significance of dissipation processes far from thermal equilibrium will also be mentioned.

High-field electronic conduction in insulators

A general, theoretical example is discussed in which the spectral density S 1 of the thermal noise is 4 kT 2Re[ Z ( ω ) ] rather than 4 kTn Re[ Z ( ω ) ] for space-charge-limited current with I ∝ V ″. A second example is indicated in which again S v ≠ 4 kT ( V / I ). This establishes that in general the correlation between fluctuations in different regions of a device cannot be ignored.

Some consequences of spatial correlation on noise calculations

Equivalence of the Langevin method and the impedance-field method of calculating noise in devices

The structure of two widely used methods, the Langevin method and the impedance-field method, for calculating device noise is considered. The equivalence of these two methods follows directly from identifying and comparing in detail the basic steps one follows in carrying each out.

K.K. Thornber

Papers

High-field electronic conduction in insulators

Some consequences of spatial correlation on noise calculations

Equivalence of the Langevin method and the impedance-field method of calculating noise in devices

Structure of the Langevin and impedance-field methods of calculating noise in devices