Repulsive Centers Research Articles

Electron capture at the two acceptor levels of a zinc center in silicon

Electron-capture rates in zero electric field at both zinc acceptor levels have been directly and accurately measured by the transient-capacitance techniques for the first time. The data elucidate two major outstanding questions in recombination physics: (1) It has been suspected for two decades that the carrier capture rates associated with many centers with multiple charge states were due to a local Auger mechanism. We show that this postulate can be experimentally tested by the double-pulse method, and in the case of Zn in Si the Auger effect plays a negligible role. (2) While several workers have stressed that the electronic barrier may be more important than the vibronic barrier in determining the temperature dependence of carrier-capture rates at repulsive centers below room temperature, the charge effect was usually ignored in the literature. We show that in the case of Zn in Si, tunneling through the screened Coulomb barrier controls the temperature dependence below 200 K and above which tunneling through the vibronic barrier becomes important. Overall, the effect of the electronic barrier below room temperature is predominant and it gives a temperature-dependent capture rate that can be easily mistaken as the occurrence of a multiphonon-emission mechanism. Failure to recognize the charge effect may result in fitting of trap parameters inconsistent with physical reality.

Electron temperature dependences of nonradiative multiphonon hot-electron capture coefficients of deep traps in semiconductors—I: Small lattice relaxation

Generalizing the theory of nonradiative multiphonon capture of thermal electrons for cases where the effective temperature T e of conduction electrons differs from lattice temperature T L , we develop a corresponding theory of hot electron capture. Its principal novelty is due to an exponential decrease ∝ exp(− E/ k B τ) of the efficiency of this energy loss mechanism, at increasing electron energy E, for the usual regime of small lattice relaxation characterized by a corresponding “capture extinction temperature” τ. The resulting T e -dependences of hot electron capture coefficients at sufficiently low T e ( ▪ τ, i.e. in the “Sommerfeld factor regime”) are controlled by the net charge of the centre which means C( T e ) ∝ C(T e) ∝ T e − 1 2 , ∝ T e 0, or ∝ T e 2 3 exp [−3·(θ/T e) 1 3 ] in cases of attraction, neutrality, and repulsion, respectively. At high T e ▪ τ, i.e. in the “energy-loss factor regime”) these relations reduce to the form C( T e ) ∝ T e − 3 2 which is simply due to an increasing depopulation of the relevant region of low-lying band states. This theory is applied to electron capture by a singly charged repulsive gold centre and a doubly charged repulsive copper centre in germanium. The corresponding theoretical maximum values C max. (−1) = 5.2 · 10 −11 cm 3 s −1 and ifC ( rnmax.) (−2) = 2.4 · 10 −12 cm 3 s −1 of hot electron capture coefficients are in good agreement with experimen observations.

Effect of optical-phonon scattering on the energy distribution of hot electrons in semiconductors at low temperature

The variational method is used in finding the solution of the transport equation for a system of hot electrons inn-Ge atT=20 K in the presence of high electric field. The role of the emission of optical phonons by hot electrons together with the effect of electron capture by repulsive centres on the formulation of the distribution function are studied. It is shown that the emission of optical phonons plays a dominant role in the formulation of the distribution. The influence of electron capture is very small, it may become appreciable at higher trap concentrations. The obtained distribution function is then used in calculating the capture rate of electrons by negatively charged centres. It is shown that the capture rate increases with electric field.

Radiation-induced conductivity ofAl2O3: Experiment and theory

The steady-state radiation-induced conductivity (RIC) has been measured in single-crystal ${\mathrm{Al}}_{2}$${\mathrm{O}}_{3}$ samples maintained at elevated temperatures during continuous irradiation with 1.5-MeV electrons. As the temperature increases from 300 to 1300 K there are regions in which the RIC increases rapidly, with activation energies between 0.6 and 4.3 eV, and regions in which it slowly decreases with an activation energy of approximately 0.1 eV. A theoretical model is presented in which the rapid increases observed in the RIC are correlated with the thermal detrapping of electrons. In undoped Meller and Linde samples the RIC is controlled by trapping and detrapping from high concentrations of shallow (0.57 and 0.72 eV) electron traps. In 0.004- and 0.03-wt.%-${\mathrm{Cr}}_{2}$${\mathrm{O}}_{3}$-doped samples there is a sufficient concentration of 1.2-eV ${\mathrm{Cr}}^{3+}$ electron traps to prevent the RIC from increasing at low temperatures. To account for the decreases in the RIC, additional rate equations describing the thermal quenching of the conductivity by hole release are considered. The RIC data of the undoped samples indicate that hole release from 0.77-eV ${V}_{\mathrm{OH}}^{\ensuremath{-}}$ centers may be thermally quenching the conductivity. The results of an isochronal annealing study of the ${\mathrm{Cr}}^{4+}$ EPR signal intensity in a 0.004-wt.%-${\mathrm{Cr}}_{2}$${\mathrm{O}}_{3}$-doped sample and an undoped Linde sample are correlated with thermally-stimulated-current (TSC) measurements to show that electron release from the 0.72-eV trap at low temperatures is followed by hole release from the 0.77-eV ${V}_{\mathrm{OH}}^{\ensuremath{-}}$ centers at higher temperatures, in agreement with the assumptions of the thermal-quenching model. The RIC data of a 0.004-wt.%-${\mathrm{Cr}}_{2}$${\mathrm{O}}_{3}$-doped sample indicate that the bulk electron-hole recombination rate at room temperature is less than 9 \ifmmode\times\else\texttimes\fi{} ${10}^{\ensuremath{-}11}$ ${\mathrm{cm}}^{3}$/sec (electron-capture cross section 7 \ifmmode\times\else\texttimes\fi{} ${10}^{\ensuremath{-}18}$ ${\mathrm{cm}}^{2}$) assuming an electron mobility of 1 ${\mathrm{cm}}^{2}$/V sec. Since this recombination rate is significantly lower than the Langevin rate of 2 \ifmmode\times\else\texttimes\fi{} ${10}^{\ensuremath{-}7}$ ${\mathrm{cm}}^{3}$/sec for diffusion controlled bulk recombination, it is likely that bulk electron-hole recombination occurs predominantly at repulsive hole centers such as ${V}_{\mathrm{OH}}^{\ensuremath{-}}$ centers. There are several mechanisms other than thermal quenching which could account for the weak decreases in the RIC. These are the LO-phonon scattering of conduction electrons ("large-polaron" mobility) and bulk electron-hole recombination occurring through multiphonon emission.

A comparison of simple theoretical models for the photoionisation of impurities in semiconductors

A comparison of two simple theoretical models, the quantum-defect (QD) model with unit normalisation, and a new billiard-ball model (BB) recently proposed by Ridley (see ibid., vol.13, p.2015, 1980), is effected by relating to a third model which is based upon approximating the true impurity potential to a square-well core and, if the centre is charged, a Coulombic tail. This third model, which requires numerical computation, goes over to the QD or BB models under specific conditions, and so acts as a kind of compromise model. The photoionisation cross section for various depths of level, and for neutral, attractive and repulsive centres, is calculated for each model and conditions for the applicability of the QD and BB models are obtained. It is shown that the QD model predicts spectral dependences well, but underestimates magnitude because of incorrect normalisation. The QD model is expected to be reasonably good for shallow and moderately deep levels, provided the centres are not repulsive, but for very deep levels, and for all levels associated with repulsive centres, the BB model is expected to be more useful.

The interaction between interstitial impurities and substitutional solutes in vanadium

The nature of the interaction between interstitial impurities and substitutional solutes in vanadium is investigated experimentally by measuring the internal friction. The nature of the interaction can be explained in terms of the chemical affinity. Substitutional atoms, whose oxide or nitride formation energy is greater than that of vanadium act as attractive centers to oxygen or nitrogen. On the other hand, substitutional atoms with formation energy smaller than that of vanadium act as non-interactive or repulsive centers. Iron, copper, nickel, chromium, and manganese are non-interactive or repulsive to oxygen; iron, nickel, beryllium, copper, chromium, and manganese are non-interactive or repulsive to nitrogen. The region around the substitutional iron forbidden to oxygen is estimated to extend to the second nearest neighbour sites. Titanium, aluminum, and beryllium are attractive to oxygen and aluminum and titanium to nitrogen. Binding energies estimated are 0.26 ± 0.4 eV for titanium and oxygen, 0.14 eV for aluminum and oxygen, and 0.27 eV for beryllium and oxygen.

Relationships between the nonradiative multiphonon carrier‐capture properties of deep charged and neutral centres in semiconductors

AbstractOn the basis of a certain set of empirical trap‐structure parameters, an analytical description is given of nonradiative multiphonon capture and ejection processes (Shockley‐Read processes) occurring at deep traps in direct and indirect semiconductors. The mechanism‐specific connections between the capture properties of charged and neutral centres are represented in terms of suitably defined temperature‐averaged Sommerfeld factors. These factors are, within the approximation of an isotropic effective free‐carrier mass, calculated explicitly both for attractive and repulsive centres.

On the Photoionization of Deep Repulsive Impurity Centres in Semiconductors

AbstractThe photoionization spectrum is calculated for impurity centres having a repulsive Coulomb potential a t large distances. The short range potential is well approximated by a delta‐function. Effects of the electron‐phonon coupling are taken into account both to the acoustical and optical modes. Transitions to a higher lying band are considered. The results are in satisfactory agreement with the observed spectral dependence.

Recombination in low-resistivity n-type Zn-compensated silicon

Calculations of recombination times in low-resistivity n-type Zn-compensated silicon having a strongly compensated upper zinc level, show that this parameter is, in general, insensitive to trapping effects if the density of zinc atoms is much greater than that of the trapping centers. Measurements of resistivity and transient photocurrent decay in the range 100 to 300 °K on a sample containing approximately 6×1015 cm−3 Zn and 1.2×x 1016 cm−3 As, indicate strong correlation between the equilibrium electron concentration and the recombination time, whence the electron capture cross-section is calculated on the coulombically repulsive zinc center Zn− to have a value of 5×10–20 cm2 independent of temperature. The resistivity measurements also indicate that any traps with levels in the range 0.07 to 0.4 eV below the conduction band which may be present in the samples, have densities of much less than 5 × 1013 cm–3. [Russian Text Ignored]

Effects of temperature on current instabilities caused by recombination centers in semiconductors

With the carrier distribution function approximated by a displaced Maxwellian function the current-voltage characteristics for a semiconductor containing repulsive Coulomb recombination centers have been derived, and the current instabilities in n-GaAs as functions of temperature and applied field have been experimentally studied. The experimental results show that the threshold current for the onset of the current instabilities is not sensitive to temperature, but the corresponding threshold field decreases with increasing temperature; and that the current oscillation frequency increases and the amplitude decreases with increasing temperature. These results are in good agreement with the theory.

Bose-Einstein condensation in the presence of impurities

It is shown that, in three dimensions, a gas of noninteracting bosons in the presence of purely repulsive impurity centers undergoes a ``Bose-Einstein'' phase transition at sufficiently low temperature. In the course of the proof it is also shown that, as the size of the system approaches infinity, the lowest energy state approaches zero with probability one.

Rotational Temperature in an Underexpanded Jet

The axial rotational temperature distribution in an underexpanded nitrogen jet emitted from a sonic orifice has been investigated. The expansion is from room temperature and the gas possesses negligible vibrational energy. The observed difference between the rotational temperature measured with an electronbeam probe and the isentropic value can be described by a rotational relaxation analysis. The governing flow equations were integrated numerically by the classical fourth-order Runge-Kutta method. The rotational relaxation time developed by Lordi using Wang Chang and Uhlenbeck's formal kinetic theory and Parker's molecular model was employed. Both the easy and difficult energy transfer cases were investigated and the easy case was shown to underestimate nonequilibrium affects. The numerical solutions were very sensitive to the separation between the repulsive force centers, for both energy transfer cases. The rotational temperature distributions obtained in this study are in general agreement with existing experimental data for nitrogen.

Electron-hole recombination in cobalt-doped p-type germanium

The recombination properties of cobalt centers in p-type germanium containing cobalt in the concentration range 10 14 to 10 16 atoms/cm 3 have been investigated. The measurement of lifetime has been carried out by steady-state photoconductivity and photo-magneto-electric methods in the temperature range 145 to 300°K. The cross-sections S n o (electron capture cross-section at neutral centers). S n − (electron capture cross-section at singly negatively charged centers) and their temperature variations have been estimated by the analysis of the lifetime data on the basis of Sah-Shockley's multi-level formula. The value of S n o is (15±5).10 −16 cm 2 and is temperature independent. The value of S n − is ≈4·10 −16 cm 2 around 225°K and it increases with increase of temperature. The possible mechanisms for capture at neutral and repulsive centers are discussed and a summary of the capture cross-sections for cobalt centers is given. A comparison of the cross-section values of cobalt and their temperature variations with those of the related impurities-manganese, iron and nickel-in germanium has been made.

Low frequency oscillations in semi-insulating gallium arsenide†

Low frequency oscillations associated with trap-controlled moving domains in high resistance semiconductors have been studied by a number of investigators. The mechanism involved is (in increase of the capture cross section of a repulsive centre with increase in the applied electric field. Although many of the features involved have been established by experimental studies, the modelling and analysis available have so far been incomplete. The present paper shows that computer studies can make a contribution to understanding of the domain behaviour and the effects of illumination. Using a simple functional form for the capture coefficient of a repulsive centre in GaAs, the non-linear equation obtained can be shown to result in a dipole domain propagating with constant velocity for bulk material. The threshold for domain nucleation was 860 volts/cm. Domain velocities were found to increase with illumination level and applied voltage, and for the model considered ranged from 0-25 cm/sec in the dark to 125 cm...

Current oscillations caused by recombination centers in semiconductors

With the carrier distribution function approximated by a Maxwellian function and the scattering effect by a relaxation time, current-voltage characteristics have been derived taking into account the repulsive Coulomb recombination centers in semiconductors. The electron mobility as a function of applied electric field, and the threshold fields for oscillation have been computed for n-GaAs. These are discussed in the light of presently available experimental results.

Current saturation and negative resistance in evaporated CdSe layers

The observed current saturation and negative differential resistance of the I–V characteristics of evaporated CdSe layers may be explained by trapping of hot-charge carriers at repulsive centres located in the grain-boundaries.

Theoretical Investigations of Translation—Rotation Energy Transfer: (H2, He) and (D2, He) Systems

The classical analysis of translation—rotation energy transfer in (H2, He) and (D2, He) systems is reported. The accurate, a priori potential surface obtained by Krauss and Mies has been employed in a generalized, three-body Monte Carlo calculation to obtain, as a function of temperature, relaxation times and collision numbers for the 0→2 and the 2→0 rotational transitions in both H2 and D2. A negative dependence upon temperature is obtained in each case. The relaxation times for the 0→2 transition in H2 range from 7.725×10−8 sec at 100°K to 3.491×10−8 sec at 700°K. For the 2→0 deexcitation the corresponding values are 9.00×10−8 and 4.30×10−8 sec. When a van der Waals attraction term, whose magnitude is estimated from PVT data, is added to the Krauss—Mies surface, the values for the 0→2 transition decrease to 3.1×10−8 sec at 100°K and 2.0×10−8 sec at 700°K. Corresponding decreases occur for the 2→0 transition. Calculated relaxation times for D2 are 50%—100% smaller than those obtained for H2. These results are in satisfactory agreement with available experimental data and appear to be more accurate than previous theoretical results. The energy-transfer mechanism has been investigated and found to be smooth and continuous with transfer occurring in about 10−13 sec. The most favorable approach angle for transfer is found to be about 60°. Two other empirical potential surfaces have been employed to estimate the effect of potential well depth and the assumption of pairwise additivity of molecular potentials. It is found that as the well depth increases, the relaxation time decreases in a manner previously predicted by Parker. The assumption of pairwise additivity is found to yield results for the relaxation times which are low by about a factor of 2. The magnitude of this error, coupled with calculations employing off-center, pairwise potentials, indicates that the repulsive centers in H2 are located about 0.16 Å from the nuclei.

Impurity-Band Tails in the High-Density Limit. I. Minimum Counting Methods

A new approximate method is presented for calculating the density of states and one-electron Green's function in the low-energy tail of a high-density impurity band. Such states occur in regions of large attractive potential produced by unusually high (random) concentrations of attractive centers and/or unusually low concentrations of repulsive centers. These well-separated deep wells each have one bound state of lowest energy. The distribution of these lowest levels vields the low-energy tail. For any one well, a bound-state energy $E({\mathbf{x}}_{0})$ is estimated variationally using $\ensuremath{\psi}(\mathbf{x})=f(\mathbf{x}\ensuremath{-}{\mathbf{x}}_{0})$, where $f(\mathbf{x})$ has any fixed form. The best energy for any well is obtained by minimizing $E({\mathbf{x}}_{0})$ with respect to ${\mathbf{x}}_{0}$ in the vicinity of that well. The number of wells that contribute to the density of states at energy $E$ is given by the number of local minima in $E({\mathbf{x}}_{0})$ that occur at the level $E$. In the high-density limit, Gaussian statistics are adequate for treating the potential fluctuations and the expectation value for the density of states is easily calculated. At low energies the best choice for the function $f$ is that which maximizes the expected density of states. Application to an exactly soluble one-dimensional model, a Gaussian white noise potential, yields the correct asymptotic form $\ensuremath{\rho}(E)=\mathrm{const}|E|\mathrm{exp}[\ensuremath{-}(\frac{4}{3}){|2E|}^{\frac{3}{2}}]$. In three dimensions, the density of states in the Gaussian approximation is found to have the form $\ensuremath{\rho}(E)=[\frac{A(E)}{{\ensuremath{\xi}}^{2}}]\mathrm{exp}[\frac{\ensuremath{-}B(E)}{(2\ensuremath{\xi})}]$, where $\ensuremath{\xi}$ is proportional to the concentration of impurities and to the square of the strength of the impurity potential. For screened, charged impurities, $B(E)={E}_{{Q}^{2}}b(\ensuremath{\nu})$, $A(E)={Q}^{3}{E}_{{Q}^{3}}a(\ensuremath{\nu})$, where ${E}_{Q}=\frac{{\ensuremath{\hbar}}^{2}{Q}^{2}}{(2{m}^{*})}$, ($\frac{1}{Q}$) is the screening radius, and $\ensuremath{\nu}=\frac{({E}_{0}\ensuremath{-}E)}{{E}_{Q}}$ is the energy below the mean potential ${E}_{0}$ in units of ${E}_{Q}$. Computer-calculated curves are provided for the universal dimensionless functions $a(\ensuremath{\nu})$ and $b(\ensuremath{\nu})$. The exponent $b(\ensuremath{\nu})$ behaves like ${\ensuremath{\nu}}^{\frac{1}{2}}$ when $\ensuremath{\nu}$ is small (strong screening) and behaves like ${\ensuremath{\nu}}^{2}$ when $\ensuremath{\nu}$ is large (weak screening).

Erratum: “Theory of Recombination Processes in Semiconductors through Multielectron Centers with Many Excited States”

Excess carrier lifetimes are calculated under steady state conditions with explicit considerations for transitions between the various states of a multi-charged center. The results obtained are of nearly the same forms as in the usual theory in so far as a phenomenological treatment is concerned. Apparent capture probabilities of the center can be represented by probabilities of various elementary processes, and correspond to total conductances of a certain equivalent circuit. Basing upon these formulae, some studies of dynamical processes are made, and possibilities of relatively large capture cross section of repulsive centers as well as attractive ones are obtained.

Theory of Recombination Processes in Semiconductors through Multielectron Centers with Many Excited States

Excess carrier lifetimes are calculated under steady state conditions with explicit considerations for transitions between the various states of a multi-charged center. The results obtained are of nearly the same forms as in the usual theory in so far as a phenomenological treatment is concerned. Apparent capture probabilities of the center can be represented by probabilities of various elementary processes, and correspond to total conductances of a certain equivalent circuit. Basing upon these formulae, some studies of dynamical processes are made, and possibilities of relatively large capture cross section of repulsive centers as well as attractive ones are obtained.