Table of contents

Foreword Some key Fundamental Effects in Semiconductor Systems were discussed at a Nordita Conference bearing this title and held at Nordita, Copenhagen, on 6th to 8th August 1986. This volume contains the record of the proceedings. Most of the 16 invited talks and some of the contributed posters are here represented either by a paper or at least by an Abstract.

I should like to thank the participants for their help with the reviewing of the papers and the authors for their unfailing cooperation which made it possibe to produce this issue of Physica Scripta rather quickly.

Peter Landsberg, Honorary Guest Editor

Introduction A few years ago now, Karl Berggren approached me with the suggestion that Nordita organize a satellite to the International Conference on Semiconductors to be held in Stockholm in August 1986. The idea was that we should focus on problems related to semiconductors and, more generally, advanced materials technology, which at the same time lay within Nordita's domain of fundamental theoretical physics. So we began to make plans for what came to be known as the Nordita Conference on Fundamental Effects in Semiconductor Systems. We were very lucky in getting Mike Pepper and Gordon Thomas to join us on the organizing committee, and it is very largely thanks to them that we achieved the success we did. The scientific program of the conference is essentially their creation, and they must get the credit for persuading so many key speakers to participate.

In the early planning stages, it became clear to us that the area defined by the title of the conference was far too broad to hope to cover in a meeting of three days, so three subjects of intense current activity and interest were chosen: the quantum Hall effect, localization, and what Gordon suggested we call "glitches". This is something of a catch-all designation for topics like chaos and the newly-discovered universal sample-to-sample conductance fluctuations. In the course of the conference, we heard much about the concept of mesoscopic physics, i.e., systems which are big enough that statistical methods must be applied to their description, yet still exhibit properties traceable to their quantum mechanical level structure. This theme pervaded much of the work reported on both of the last two subjects, and I came away with the impression that this area will be very active in the next few years.

The title of the conference may appear somewhat misleading to a hard-core semiconductor physicist, and we are aware that there are many fundamental problems in semiconductors which we have not touched on, but we feel we have been true to the original intent of focusing on some truly fundamental physics that comes up in current materials science.

We decided early on in the planning to encourage participants, both invited speakers and contributors of posters, to submit their contributions to a published conference proceedings. However, we felt that this should be optional rather than a condition for participation, in order not to deter potential participants from coming to the meeting. We do include here, however, abstracts of all the contributions. We have also been very fortunate in having the help of Dr. Peter Landsberg, who has served as editor of these proceedings.

I would like to thank all the participants for their contributions – talk, posters, discussions, and finally these proceedings. And finally, special thanks are due to the Nordita administrative staff, especially Ellen Pedersen, who did so many of the practical arrangements for all of us.

John Hertz, Copenhagen, October 1986

Fluctuations of the current-voltage (IV) characteristic of a mesoscopic sample are considered. Such a sample can be realized as a junction of two normal metals. The voltage dependence of current is a random function, and the IV-characteristic has a "blades of grass" form on a usual ohmic background. The scale of each "blade" in voltage is of order V_c ~ ℏ/τ_fe, where τ_f is the time of flight through the junction and e is the electron charge. The scale in current δI depends on the voltage V and temperature T, and of order (e²/ℏ)(VV_c)^1/2, if V ≫ V_c, T/e. As a result, provided the voltage is sufficiently large, there appear regions of negative differential resistance.

Doped semiconductors, being a physical realization of an ensemble of one-electron (hydrogenic) atoms distributed randomly in space, are in a sense the simplest disordered system. Our current understanding of their magnetic properties is reviewed, for densities on both sides of a critical density n_c at which the system undergoes a transition (at zero temperature) from an insulating phase (n < n_c) to a metallic phase (n > n_c). It is argued that the insulating phase is well modeled in terms of a disordered Heisenberg antiferromagnet, and quantitative agreement with experiment can be obtained. In contrast, the metallic phase just beyond n_c, is not as well understood, and a number of possible candidate models are described. Finally, the issue of the effect of magnetic properties on the metal-insulator transition is addressed.

The quasi two-dimensional resistivity and magnetoresistance has been measured on Si(100) MOSFET electron inversion layer as a function of carrier density, temperature and magnetic field. The magnetoresistance is interpreted by the weak localization theory, and we determine the phase relaxation time. The phase relaxation rate has three components, of which two can be identified as the electron-electron scattering rate and the so-called Nyquist (or impurity mediated electron-electron) scattering rate. A third contribution becomes important at low carrier densities and suggests itself as being related to the screened impurity relaxation rate, which however is normally considered to be elastic and therefore should not contribute to the phase relaxation. The screened impurity relaxation rate and the Nyquist scattering rate are the dominating contributions to the temperature dependent mobility of our samples. The screened impurity relaxation rate is compared to recent calculations by Gold and Dolgopolov. At magnetic fields intermediate between the weak-localization-regime and the Landau-level-dominated-regime the weak localization magneto-resistance has been measured and compared to current theories.

We report the first calculation for the critical concentration N_c of the MNM transition in double-donors Si:P, As and Si:P, Sb. From an interpretation of the Hubbard model for the transition we found the result for Si:P, As in good agreement with recent experimental finding [Newman and Holcomb, Phys. Rev. Lett. 51, 2144 (1983)]. We also find that the model predicts value of N_c for Si:P, Sb in the following order N_c^Sb < N_c^P,Sb < N_c^P < N_c^P,As < N_c^As.

Results of magnetoconductivity and magnetospectroscopic measurements of semimagnetic semiconductors with impurity concentrations close to that corresponding to the Anderson-Mott transition are discussed. The effect of the s-d exchange spin-splitting, fluctuations of magnetization and alloy composition, as well as the RKKY interaction and magnetic polarons are considered in the reference to the data for n-Cd_1-xMn_xSe. It is concluded that quasi-delocalized states appear to exist on the insulating side of the transition while quasi-static s-spins seem to persist into the metallic phase.

The Burstein-Moss effect is observed for thin samples of Hg_1-xCd_xTe with x = 0.165 d = 10 μm, x = 0.170, d = 9 μm and x = 0.194, d = 24 μm, and for one In-doped degenerate sample x = 0.190, n = 7 × 10¹⁷ cm^-3. The optical energy gap obtained from measurements of the intrinsic absorption spectra agrees well with the calculated Fermi level.

The density of states of the two-dimensional electron gas at high magnetic fields has been measured via magnetization. These experiments, along with recent specific heat, magneto-capacitance and activated transport measurements of the density of states, will be reviewed and compared. The common result, that a significant DOS exists in the Landau gap, will be discussed in the framework of current theoretical work.

A theory of the Fractional Quantum Hall Effect is constructed based on magnetic flux fractionization, which lead to instability of the system against selfcompression. A theorem is proved stating that arbitrary potential fail to lift a specific degeneracy of the Landau level. For the case of 1/3 fractional filling a model 3-particles interaction is constructed breaking the symmetry. The rigid 3-particles wave function plays the role of order parameter. In a BCS type of theory the gap in the single particles spectrum is produced by the 3-particles interaction. The mean field critical behaviour and critical parameters are determined as well as the Ginsburg-Landau equation coefficients. The Hall conductivity is calculated from first principles and its temperature dependence is found. The simultaneous tunneling of 3, 5, 7 etc. electrons and quantum interference effects are predicted.

The elementary charged excitations at 1/3 Landau level filling have been studied for various spin polarizations for finite electron systems in a periodic rectangular geometry. In the absence of Zeeman energy, the spin-reversed excitations, studied for three to six electron systems show significant reduction of the energy gap, compared to the fully spin-polarized quasiparticle-quasihole gap, studied for upto seven electron systems. In the presence of Zeeman energy, the lowest energy excitations involve spin-reversed quasiparticles and spin-polarized quasiholes for low magnetic fields. With increasing magnetic fields, a crossover point is reached (in the region B < 10 T), beyond which a fully spin polarized quasiparticle-quasihole state will be energetically favored. The Zeeman energy due to spin-flip explains qualitatively the linear magnetic field dependence of the activation energies observed experimentally.

Systems with large amounts of disorder have some remarkable properties induced by localisation of the electronic wavefunctions [1, 2]. Current formulations of quantum transport have limited relevance to the case of strong disorder and require the use of scaling theories to apply them to these systems [3]. We have been developing a first principles theory of quantum transport that addresses the problem of localisation directly without resort to scaling [4-6]. The theory reproduces well known results for the d.c. conductivity, but goes beyond that to describe the bandwidth of localised structures when used to convey electrical signals, as well as the non-linearity in the d.c. conductivity implicit in these systems. We have also been able to show that the localised states behave as extremely long lived traps with a distribution of lifetimes that leads to the generation of 1/f noise at surfaces and interfaces.

Two sets of experimental results are given to demonstrate how modern nonlinear dynamics can be used to understand the dynamics of extrinsic Ge photoconductors. In the first, we show how complex and apparently random chaotic broadband noise is physically produced in devices which have nearly ideal stochastic noise performance in the absence of drive. In the second, we demonstrate that the circle map, a simple model of periodically-driven active oscillators with universal predictions, quantitatively describes the dynamics of a driven spatially-dependent high field instability in Ge.

We have studied the nonlinear dynamical properties of a simple periodically-driven rate equation model describing electrical conduction in extrinsic photoconductors. Extensive numerical simulations analysed using phase portraits, Poincaré sections, approximate one-dimensional mappings, and nonlinear phase diagrams show a range of nonlinear dynamical behavior including chaotic oscillation, transitions to intermittently chaotic oscillation, and phase locking behavior for large drive amplitudes. In addition, we have fully explored the strong dependence of observed nonlinear dynamical behavior on parameter variation in the model.

Bifurcation routes to chaos of a driven current filament in semiconductors have been investigated by numerical computation. Ths S-shaped current-voltage (J-E) characteristic is described by the modified Crandall's model. The catastrophic J-E curve induced by impact ionization of neutral donors is driven by an alternating electric field superimposed on d.c. bias E₀. As a function of E₀, the prototype of period-doubling bifurcation (Feigenbaum scenario) and intermittency are simulated. However, under the influences of joule heating and slow response of space charge, breakdown of the Feigenbaum scenario is found.

Two systems of recent interest are compared and the dual nature of the predicted quantum oscillations is emphasized. A small one-dimensional normal loop with an Aharonov-Bohm flux through the hole of the loop, exhibits a persistent current with period hc/e in the presence of a time-independent flux and an oscillatory Josephson-like current if the flux is increased linearly in time. A small capacitance Josephson junction, coupled capacitively to an external circuit, exhibits a currentless potential drop across the junction, with period e in the presence of a time-independent induced charge, and exhibits an oscillatory voltage if the induced charge is increased linearly in time. The role of the Aharonov-Bohm flux in the loop is taken by the induced charge in the junction, and the oscillatory behavior of the current of the loop is dual to the oscillatory behavior of the voltage of the junction. The current-voltage characteristic of the two systems is compared, and the sample specific nature of the effects is discussed.

A prototype model of nonlinear driven oscillators with a limit cycle is studied. In the fast-relaxation limit dynamics can be reduced to a one-dimensional mapping. Mode-locking with rational winding numbers takes place for weak external force, while the order of occurence of the orbits is governed by the U-sequence of unimodal mappings for strong external force. In the intermediate region new sequences of periodic orbits occur.

Experimental measurements have been made on Au-Al tunnel junctions in which the gold electrode was semicontinuous. The gold film was analysed by TEM and harmonic analysis was used to obtain the small signal conductance and its derivative at low temperatures. Periodic variations and linear characteristics were found in the conductance versus junction bias potential curves. These results are compared with similar results obtained from gold-island films. The overall effects are discussed both in terms of a Coulomb charging model and localization in a film consisting of a random network of grains.

The resistance of two-dimensional electron systems such as thin disordered films shows deviations from Boltzmann theory, which are caused by quantum corrections and are called "weak localization". Theoretically, weak localization is originated by the Langer-Neal graph in the Kubo formalism. In this review the physics of weak localization is discussed. It represents an interference experiment with conduction electrons split into pairs of waves interfering in the back-scattering direction. The intensity of the interference (integrated over the time) can be easily measured by the resistance of the film. A magnetic field introduces a magnetic phase shift in the electronic wave function and suppresses the interference after a "flight" time proportional to 1/H. Therefore the application of a magnetic field allows a time-of-flight experiment with conduction electrons. Spin-orbit scattering rotates the spin of the electrons and yields an observable destructive interference. Magnetic impurities destoy the coherence of the phase. The experimental results as well as the theory is reviewed. The role of the spin-orbit scattering and the magnetic scattering are discussed. The measurements give selected information about the inelastic lifetime of the conduction electrons in disordered metals and raise new questions in solid state physics. Future applications of the method of weak localization are considered and expected.

Studies have been made of two types of small localized 2D systems on silicon surfaces. In one type the samples were long (~12 μm) and narrow (~0.05 μm). These samples showed variable range (1D) hopping above about 300 mK. Below strong structure occurs as a function of chemical potential because of the small number of states within 10 kT (10-100). Because single hops are involved, conduction in the peaks is simply activated and rectifying. The second type of sample was short (0.5 μm) and conduction was dominated by tunnelling at low temperatures (<200 mK). The peaks observed did not vary in width or length below about 80 mK but broadened and decreased at higher temperatures as expected of tunneling. Temporal fluctuations and substructure were seen that might reasonably be caused by fluctuations in the occupation of other sites. The long samples were studied with R A Webb, J Wainer and A Hartstein. The tunneling samples were studied with G Timp, R A Webb and J Wainer.

Despite the fact that one is dealing with fermions and not bosons, there are close analogies between the fractional quantum Hall effect (FQHE) and superfluidity in helium films. A theory has been developed [1] for the collective excitation spectrum in the FQHE which is closely analogous to Feynman's theory of superfluid ⁴He. The physical content of the theory is the notion that the continuum of single-particle excitations is quenched by the magnetic field, leaving a single collective mode (per Landau level) which absorbs nearly all of the available oscillator strength. Thus, despite the Fermi statistics, the system behaves much like a boson superfluid. At long wavelengths the collective-mode is a phonon (densitywave) but unlike the case of ⁴He, the phonon has a large gap. At the characteristic wave vector associated with the interparticle spacing, the collective-mode energy suffers a deep "magneto-roton" minimum analogous to the roton minimum in helium. The single mode approximation is quantitatively quite accurate and allows one to rather easily and accurately compute experimentally relevant quantities such as the AC conductivity and the static susceptibility in addition to the mode energy itself.

Pursuing the superfluidity analogy further shows that the analogs of quantized vortices are Laughlin's fractionally charged quasi-particles. The existence of a finite rather than divergent vortex energy (as occurs in helium) is associated with the absence of a gapless Goldstone mode via a remarkable analog of the Anderson-Higgs mechanism which leads, not to flux quantization, but rather to (fractional) charge quantization. Some first steps towards a Landau-Ginsburg theory incorporating these ideas will be presented.

Transport through a small disordered wire was considered at low temperatures where quantum interference effects are important. Sample specific versus ensemble average properties were emphasized. As examples, the h/e Aharonov-Bohm oscillations in a ring and the universal conductance fluctuations in a wire were discussed. The strengths of these effects was elucidated by introducing the concept of effective conductance channel and appealing to the theory of random matrices.

It is shown how a simple model for chaos in semiconductors results from a three-fold argument:

(a) One starts with a Chapman-Kolmogorov equation for the generation-recombination processes. These are assumed in the simplest case (considered here) to be uniform in space and to involve only holes. A recurrence relation for the average hole concentration at successive discrete time intervals is derived by averaging the Chapman-Kolmogorov equation.

(b) One sets up a generation-recombination rate for a specific model.

(c) The possibility of chaos then follows by combining (a) and (b) and hence finding recurrence relations similar to those known to produce chaos, e.g., X_{k + 1} = rX_k(1 - X_k).

Barely insulating and barely metallic Si: As samples have been studied in the temperature range 4.2 K to 50 mK. Metallic samples show a new high-field magnetoresistance behavior explainable in terms of magnetic tuning of the critical density N_c. Insulating samples at fixed fields show Mott variable-range-hopping behavior and a strong increase in the characteristic temperature T₀(N, H) with H. The ratio T₀(N, H)/T₀(N, 0) at fixed field shows quasicritical behavior increasing by a large field-dependent factor as N → N_c. Magnetocapacitance measurements on an 0.87N_c sample show an initial quadratic decrease with H followed by a flattening toward a linear decrease above 4 T. Both T₀(N, H) and ε'(N, H) - ε_h can be explained in terms of the localization length decreasing with H, however the interpretation of the field-dependent decrease depends somewhat on the theoretical model employed. The decrease in ξ(N, H) cannot be explained in terms of N_c-tuning, but also seems to require some field-induced tuning in the localization length exponent which is qualitatively consistent with Hikani's theoretical prediction.

Every conductance measurement on metallic systems is affected to some degree by random quantum interference arising from the particular disordered microscopic arrangement of the electron scatterers. We have measured the conductance variations of submicron inversion layer segments in silicon MOSFETs, systematically changing the length, width, inelastic diffusion length, gate voltage, magnetic field and temperature. The statistical properties of the observed conductance variations agree quantitatively with the theory of universal conductance fluctuations, demonstrating that random quantum interference causes rms conductance changes of magnitude e²/h in each phase-coherent subunit of each conducting segment. In small, narrow devices, the fractional conductance variations are of order unity, resulting in large resistance changes. By studying the effects of trapping single electrons at individual interference traps, we demonstrate that the random quantum interference can be extremely sensitive to changing a single scatterer.

The conductance of any metallic sample is shown to fluctuate as a function of chemical potential, magnetic field, or impurity configuration by an amount of order e²/h independent of sample size and degree of disorder at low temperature. Fluctuations of this size are present as long as kT and the inelastic scattering rate are less than the inverse time to diffuse across the sample; at higher T they are reduced as a weak power law of T. This fluctuation effect means that weak localization theory may fail to describe experimental systems in this regime of size and temperature. This provides a quantitative explanation for the observability of sample-specific h/e Aharonov-Bohm oscillations in normal metal rings. The conductance of a given sample is found to be extremely sensitive to very small changes in impurity configuration. This result may provide the basis for a quantitative microscopic theory of low-frequency noise in dirty metals. It also suggests that the sample-specific magnetoresistance patterns ("magnetofingerprints") may provide a useful tool for probing microscopic processes associated with defects in dirty metals. The theory is in good agreement with experiments on small wires, rings and FETs, and also has implications for experiments studying classical wave motion in random media.