Noise In Semiconductors Research Articles

On the Interpretation of Noise in Thick-Film Resistors

Noise in thick-film resistors is interpreted by comparison to experimental and theoretical results previously reported for noise in metals and semiconductors. This comparison provides an explanation for the observed dependence of noise on the thick-film-resistor geometry and the 100-fold increase-in noise amplitude over that of metals and semiconductors This comparison indicates that mobile-charge interactions with traps are an inherent feature of charge transport in thick-film resistors.

Temperature fluctuations and [formula omitted] noise in semiconductors and metals

A theoretical description is given of the influence of equilibrium temperature fluctuations on the electrical resistivity of metals and semiconductors. It is found that in metals resistivity fluctuations due to temperature fluctuations exceed the 1 f noise at least for frequencies corresponding to the reciprocal thermal relaxation time. Resistivity fluctuations in semiconductors are found to be dominated by 1 f noise; equilibrium temperature fluctuations can here be neglected. These theoretical results are compared with experimental results.

Flicker (1f) noise: Equilibrium temperature and resistance fluctuations

We have measured the $\frac{1}{f}$ voltage noise in continuous metal films. At room temperature, samples of pure metals and bismuth (with a carrier density smaller by ${10}^{5}$) of similar volume had comparable noise. The power spectrum ${S}_{V}(f)$ was proportional to $\frac{{\overline{V}}^{2}}{\ensuremath{\Omega}{f}^{\ensuremath{\gamma}}}$, where $\overline{V}$ is the mean voltage across the sample, $\ensuremath{\Omega}$ is the sample volume, and $1.0\ensuremath{\lesssim}\ensuremath{\gamma}\ensuremath{\lesssim}1.4$. $\frac{{S}_{V}(f)}{{\overline{V}}^{2}}$ was reduced as the temperature was lowered. Manganin, with a temperature coefficient of resistance ($\ensuremath{\beta}$) close to zero, had no discernible noise. These results suggest that the noise arises from equilibrium temperature fluctuations modulating the resistance to give ${S}_{V}(f)\ensuremath{\propto}\frac{{\overline{V}}^{2}{\ensuremath{\beta}}^{2}{k}_{B}{T}^{2}}{{C}_{V}}$, where ${C}_{V}$ is the total heat capacity of the sample. The noise was spatially correlated over a length $\ensuremath{\lambda}(f)\ensuremath{\approx}{(\frac{D}{f})}^{\frac{1}{2}}$, where $D$ is the thermal diffusivity, implying that the fluctuations obey a diffusion equation. The usual theoretical treatment of spatially uncorrelated temperature fluctuations gives a spectrum that flattens at low frequencies in contradiction to the observed spectrum. However, the empirical inclusion of an explicit $\frac{1}{f}$ region and appropriate normalization lead to $\frac{{S}_{V}(f)}{{\overline{V}}^{2}}\ensuremath{\propto}\frac{{\ensuremath{\beta}}^{2}{k}_{B}{T}^{2}}{{C}_{V}}[3+2\mathrm{ln}(\frac{l}{w})]f$, where $l$ is the length and $w$ is the width of the film, in excellent agreement with the measured noise. If the fluctuations are assumed to be spatially correlated, the diffusion equation can yield an extended $\frac{1}{f}$ region in the power spectrum. We show that the temperature response of a sample to $\ensuremath{\delta}$- and step-function power inputs has the same shape as the autocorrelation function for uncorrelated and correlated temperature fluctuations, respectively. The spectrum obtained from the cosine transform of the measured step-function response is in excellent agreement with the measured $\frac{1}{f}$ voltage noise spectrum. Spatially correlated equilibrium temperature fluctuations are not the dominant source of $\frac{1}{f}$ noise in semiconductors and discontinuous metal films. However, the agreement between the low-frequency spectrum of fluctuations in the mean-square Johnson-noise voltage and the resistance fluctuation spectrum measured in the presence of a current demonstrates that in these systems the $\frac{1}{f}$ noise is also due to equilibrium resistance fluctuations.

Microcycle spectral estimates of 1/f noise in semiconductors

Many physical occurrences are characterized by extremely low spectral variations, the measurement and estimation of which has been invariably difficult. An estimate of the density of the power spectrum of very-low-frequency semiconductor 1/f noise is experimentally obtained from 10−6.3 to 1.0 cps with a greater accuracy than that achieved in previous similar attempts; it is concluded that the spectrum is 1/fα with α approximately 1.3 over most of the frequency range, but appearing to have a value of about 1.0 in the lowest decade. A peculiar form of stationarity seems to distinguish 1/f noise from other noise in semiconductors. Ten independent noise sources were time multiplexed and their spectral estimates were subsequently averaged. If the sources have similar spectra, this reduces the necessary data-taking time by a factor of 10 for a given accuracy. An estimator is derived for optimal spectral estimation based on a number of statistically independent noise sources. Other related topics considered are nonequidistant sampling, and a plausible mathematical model of such flicker noise. Finally, the variance of the spectral estimate obtained through the Blackman/Tukey algorithm is analyzed in great detail; the variance is shown to diverge for α≥1.0 in an assumed power spectrum of k/|f|α, unless the assumed spectrum is ``truncated''.

Random walk models of semiconductor noise

The ensemble averages of the spectral densities appropriate to various types of random walk are evaluated using a Monte Carlo method. The results are compared with theory. It is shown that a ``drifting'' walk cannot in itself explain the occurrence of 1/f noise in semiconductors.

Calculation on hot-electron noise in semiconductors

Under hot-electron conditions, noise arises both from fluctuations in the carrier velocity and the carrier collision time. The magnitudes of these two contributions are calculated by the Monte-Carlo method for InSb at 77°K.

Current noise in some transition-metal compounds

Measurements are reported on current noise in some single crystals of transition-metal compounds, namely: reduced Al-doped rutile (TiO 2), Li-doped NiO, Li-doped CoO and Ga-doped CdCr 2Se 4. Also results obtained with polycrystalline Ga-doped CdCr 2Se 4 are reported. The current-noise spectra of the materials mentioned above showed flicker noise with a frequency dependence of 1/f β, where the value of β is in the range 0.7 to 1.2. The magnitude of the flicker noise is compared with a phenomenological formula for flicker noise in semiconductors and metals. It turned out that the magnitude of the current noise in reduced rutile is smaller by a factor 10 3 than predicted by the phenomenological formula.

Some approximate solutions to the discrete master equation

Recent mathematical developments on approximate diffusionlike solutions to the master equation are summarized. The technique is applied to two master equations of physical interest-one that describes the phenomenon of superradiance and a second that characterizes generation-recombination noise in semiconductors. For this second case, some previously obtained equilibrium results are found and the extension of these results to finite times is given.

1/f Noise in Semiconductors

1/<i>f</i> Noise in Semiconductors

Current noise in vitreous semiconductors

Preliminary measurements of the current noise spectrum in the audio frequency range are reported for two glasses of the chalcogenide type, and for a copper phosphate glass of the type used by Drake et al. [1] in their studies of switching. In a glass of composition As2Te3Tl2Se, the current noise can be extremely low, – less, for example, than in a single crystal germanium filament. The other materials show a greater current noise, but in each case, the spectrum obeys an equation of the form noise ∼ Iαf−β, where I is the dc current, f the frequency, the constant α = 1.5 to 2.0, and β ≈ 1. Some tentative suggestions are made concerning the interpretation of the results. Es werden vorlaufige Messungen des Stromrauschens im Hor-Frequenzbereich an zwei Chalkogenidglasern und an einem Kupferphosphatglas des Typs, den Drake u. a. [1] fur das Studium von Schaltvorgangen benutzt haben, mitgeteilt. Bei Glas der Zusammensetzung As2Te3Tl2Se wurde ein extrem niedriges Stromrauschen gefunden – z. B. niedriger als an monokristallinen Germaniumschichten. Die beiden anderen Materialien weisen zwar ein hoheres Stromrauschen auf, aber in allen Fallen ist das Rauschspektrum durch eine Gleichung der Form Stromrauschen ∼ Iαf−β gegeben, wo I der Wert des Gleichstromes und f die Frequenz bedeuten und die Konstanten α = 1,5 bis 2,0 und β ≈ 1 sind. Vorlaufige Vorschlage fur die Interpretation der Resultate werden angegeben.

A proposal concerning the origins of [formula omitted] current noise in semiconductors

It is suggested that the dipole-dipole resonance interaction between impurity atoms in a semi-conducting crystal may be responsible for 1 ƒ current noise and results are given of a calculation using a statistical model based on this hypothesis.

On the Double Injection Current Noise in Solids

AbstractCurrent noise in dielectrics and semiconductors is investigated under the following conditions: double injection, negligible traps and recombination centre effects, space‐charge‐limited region, negligible diffusion and direct electron‐hole recombination. A method of calculating the spectral noise density based on the Lax approach and the Debye‐Hückel position correlations between carrier densities is used. Expressions for low and high frequencies are given in closed form; an enhancement effect is found at low frequencies. In the limit of the quasineutral (ambipolar) region, a small correction to the Hill‐van Vliet formula is obtained.

On the noise caused by diffusion and surface recombination

A critical discussion is given of various theories concerning noise in semiconductors resulting from surface generation and recombination of carriers and the associated diffusion to and from the surface. It is shown that critique by Champlin on the earlier papers by Hyde and by van Vliet and van der Ziel is unjustified and for the onedimensional geometry, used both by Champlin and by Lax and Mengert, complete agreement is established with the aforementioned earlier papers.

Irreversible Thermodynamics of Steady-State Processes

An irreversible thermodynamic theory of steady-state processes is derived using quantum-statistical perturbation techniques. The modification of the conventional fluctuation-dissipation theorem of Callen and his co-workers and of the Onsager reciprocity is examined for a system in the nonequilibrium steady state. The principal results are as follows: (1) The instantaneous fluctuation $\ensuremath{\Delta}J(t)$ of the current from its steady-state average is decomposed into a thermal contribution $\ensuremath{\Delta}\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{q}(t)$ characterizing the interaction of the system with a temperature reservoir and a contribution $\ensuremath{\Delta}j(t)$ characterizing the additional effects arising from the presence of a steady-state current. (2) The linear response $\ensuremath{\varphi}(t)$ of the system to an applied force, referred to the steady state, and the steady-state current fluctuation moment $〈\ensuremath{\Delta}J(0)\ensuremath{\Delta}J(t)〉$ are built up from different combinations of time correlation moments between $\ensuremath{\Delta}\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{q}(t)$ and $\ensuremath{\Delta}j(t)$. (3) A tentative identification is made of the various contributions to the current fluctuations with thermal noise, shot noise, and generation-recombination noise. (4) The underlying time and magnetic field reversal symmetry of the motion can be destroyed by the presence of a steady-state current, so that the Onsager reciprocity need not, in general, be satisfied. Illustrative applications of the formalism to generation-recombination noise and warm electrons in semiconductors are discussed.

1/f Noise from Vacuum-Cleaned Silicon

The removal of the adsorbed gases and the surface oxide on single-crystal, 100 Ω-cm silicon filaments by ion bombardment and annealing does not have an appreciable effect on either the magnitude or the frequency dependence of the 1/f noise from the filament. Since the slow surface states are commonly associated with gases adsorbed on the oxide, charge transfer between these states and the bulk cannot be the primary source of 1/f noise in semiconductors. The changes in the 1/f noise that were observed during the surface cleaning procedure can be correlated with the presence or absence of an inversion layer at the surface.

Recombination-generation noise theory for semiconductors in equilibrium

This paper offers a unified discussion of equilibrium recombination-generation noise in semiconductors, and of its relation to the steady-state and transient lifetimes. It extends many existing results and covers situations where certain types of Auger processes, as well as one-electron transitions, contribute to the noise. The cases of degenerate bands, and of centres each able to trap one or more electrons are included. The new general results are, however, on the whole less complicated in form than their already known special cases. This is in part due to the fact that the equations are here expressed in a form which is symmetrical in bands and traps, so that there is no need to distinguish between them in the formal part of the theory. The following procedure is used. The formalism of irreversible thermodynamics is applied to a certain class of systems of which a homogeneous semiconductor is an example. The kinetic equations are written in a form involving generalized resistance and capacitance matrices. An admittance matrix and a lifetime matrix are introduced, and expressions for the steady-state and transient lifetimes in terms of the latter are obtained. Results from semiconductor statistics are used to evaluate these matrices in terms of the equilibrium electron numbers in, and transition rates between, the various groups of states (bands and localized levels) in the semiconductor. It is shown that the transition rates due to a certain class of Auger processes have the same dependence on quasi-Fermi levels as the transition rates due to corresponding single-electron processes. The fluctuation-dissipation theorem is applied, relating the spectral densities and variances of the fluctuating electron numbers to the admittance matrix, and so to the equilibrium numbers and transition rates. Problems involving two or three groups of states are considered in detail, and connexions with previous results are established.

A unified statistical approach to recombination-generation noise in semiconductors

This paper outlines a unified discussion of equilibrium recombination—generation noise in semiconductors. It extends existing results in order to cover situations where certain types of Auger (electron collision) processes, as well as one-electron transitions, contribute to the noise. In order to achieve this, irreversible thermodynamics is used. The kinetic equations are written in a form involving generalized resistance and capacitance matrices. An admittance matrix and a lifetime matrix are also introduced. Results from semiconductor statistics are applied to evaluate these matrices in terms of the equilibrium electron numbers in, and equilibrium transition rates between, the various groups of states (bands and localized levels). The spectral densities and variances of the fluctuating electron numbers are related to the admittance matrix, and so finally to the equilibrium numbers and the transition rates, by means of the fluctuation-dissipation theorem. The spectral densities are in all cases superpositions of familiar terms of the form A k (1 +ω 2τ 2 k) , and the paper outlines a general method of obtaining the coefficients A k and the transient lifetimes τ k .

Generation-recombination noise and capture cross-sections in p-type gold-doped germanium

Generation-recombination noise in semiconductors arises from a modulation of the conductivity by random thermal fluctuations of the charge carriers between the valence (or conduction) band and the impurity levels. With certain impurities, such as gold in germanium, it is possible to have two interacting levels simultaneously contributing to the noise. We have studied two samples of gold-doped germanium in the temperature range 90°–100°K. In one sample, the Fermi level lies midway between the 0.041 eV donor level and the 0.145 eV acceptor level, so that the noise spectrum simultaneously exhibits relaxation times characteristic of both levels. In the second sample, the Fermi level is locked on the gold acceptor level. A straight-forward theory has been developed, capable of satisfactorily accounting, in general, for all of the observed noise in interacting two-impurity level systems of this type. Estimates of the total capture cross-sections of free holes by the 0.041 eV donor level and the 0.145 eV acceptor level in p-type gold-doped germanium of σ D = 3.7 × 10 −18 cm −2 and σ A = 1.5 × 10 −12 cm 2 have been obtained, in qualitative agreement with other independent measurements. Further, the results indicate some interesting possibilities regarding the existence of excited gold center states and their effect on the generation-recombination process.

Generation-recombination noise in semiconductors—The equivalent circuit approach

Generation-recombination noise in semiconductors in thermal equilibrium is treated from the standpoint of thermal fluctuations in equivalent electrical circuits. For the general volume recombination model, a method based on network reduction is presented which allows one to calculate the spectral density of the electron and hole fluctuations without solving for the spectra of the fluctuations in occupancy of the recombination centers and traps. The method is extended to a surface recombination model, thereby avoiding the ambiguities found in previous formal treatments. It is shown that the concept of ambipolar diffusion, the location and spectral density of the random sources, and the spatial correlation of Fourier coefficients of carrier density fluctuations all have simple significance in electrical terms. Using transmission line techniques, the generation-recombination (GR) spectrum is calculated for a two-level semiconductor where recombination occurs at opposite plane surfaces. This new result is examined in detail for the limiting cases approached when the recombination process is 1) volume-limited, 2) surface-limited, and 3) diffusion-limited. It is shown that in the first two cases, the spectrum is identical with that obtained from a zero-dimensional analysis provided the time constant is properly defined. For the diffusion-limited case, however, the spectrum varies as 1/ω3/2at high frequencies, and at low frequencies the noise is 5/6 that predicted by the simple theory. The new result is shown to compare favorably with measurements reported previously by Hill and van Vliet.

Noise in Semiconductors and Photoconductors

A survey is given of theory and experiments on noise in bulk semiconductors and photoconductors. This paper is divided into four parts, including generation-recombination (gr) noise in semiconductors, gr noise in photoconductors, 1/f noise in single crystals, and modulation noise in granular materials. In the first part an account is given of the appropriate analyses and the results are applied to extrinsic as well as intrinsic fluctuations, generated either in the bulk or at the surface. In the part about photoconductors the limiting sensitivity caused by photon noise is calculated and present infrared detectors are discussed. Next, a survey is given about present understanding of 1/f noise, and of its relation to the field effect as proposed by McWhorter and others. Finally, some remarks are made about 1/f noise in granular material and the proposed theories are briefly reviewed.