Wannier-Stark Quantization Research Articles

Analytic Theory of Wannier–Stark Quantization in Arbitrary‐Size Atomic Square Lattices

The theory of electron energy quantization in an atom long chain affected by a homogeneous electric field is extended to the 2D case of atom long and atom wide conductor with the lattice constant a. The conductor is modeled by the standard one‐parameter tight‐binding Hamiltonian (hopping integral t) of an atomic rectangular square lattice, where the change of electron site energy from atom to atom across the conductor equals (in units) to electric field parameter (efp). Each field‐provoked level (transverse μ‐mode energy) is shown giving rise to the μth subband of field‐independent levels of extended states due to the electron resonant transfer in the longitudinal direction perpendicular to the electric field. The level spacing and, hence, the width of μ subbands, is dictated by the conductor width and hopping integral t. The spectra of arbitrary‐size atomic square lattices are classified in terms of subband center spacing (SBCS) within the transverse‐mode bands of extended states, edge‐localized states, and Wannier–Stark states. We identified special cases of integer and fractional SBCS quantization in efp‐portions and found the exact and/or highly accurate explicit approximate expressions for the weighted number of states within these bands. The presented novel results clarify and correct their analogs reported previously.

Electron transport in silicon carbide natural superlattices under the Wannier-Stark quantization conditions: Basic issues and application prospects

Hot-electron transport in silicon carbide natural superlattices was investigated. Almost all of the theoretically predicted effects related to the Wannier-Stark localization, such as Bloch oscillations, Stark-phonon resonances, miniband-state localization, and resonance tunneling between the minibands, were observed for the first time. In n+-n−-n+ structures optimized for the measurements at microwave frequencies, the formation of the mobile electric domain was observed in the electric-field range corresponding to the Bloch oscillation conditions; thus, the onset of the microwave oscillations in the 6H-SiC natural superlattice can be stated with a high degree of confidence.

Evolution of energy levels of a GaAs/AlGaAs superlattice under the influence of a strong magnetic field

We present photocurrent spectra of the Wannier–Stark ladder of a GaAs/AlGaAs superlattice in a magnetic field up to 9 T and oriented along the growth direction of the superlattice. The Landau and Wannier–Stark quantization directions are perpendicular to each other. The salient observation is the evolution of Landau fans for the different Wannier–Stark transitions, with the Landau fan assigned to the spatially direct Wannier–Stark-ladder transitions being very pronounced quite in contrast to the Landau fans for the spatially indirect Wannier–Stark transitions which are weak.

Cyclotron-Stark-phonon resonances in semiconductor superlattices under terahertz irradiation

The influence of strong terahertz irradiation on quantum transport in semiconductor superlattices under parallel magnetic and dc electric fields is studied within a quantum-kinetic approach. The consideration is focused on field strengths at which Wannier-Stark and Landau quantization occur. The current density comprises two different contributions. One is due to scattering-induced delocalization of carriers. The other one is negative, and results mainly from photon absorption processes. Within a microscopic as well as a simple relaxation-time approach, the electric- and magnetic-field dependencies of combined cyclotron-Stark-phonon and Stark-photon resonances are studied.

High-field miniband transport in semiconductor superlattices under the influence of a transverse magnetic field

The miniband transport in semiconductor superlattices under the influence of high electric and magnetic fields aligned parallel to the superlattice axis is treated within a rigorous quantum-mechanical approach. Intra-collisional field effects due to the electric and magnetic field are taken into account. Pronounced resonances are predicted to occur in the current density due to Landau and Wannier–Stark quantization of the electronic states and due to their scattering on polar-optical phonons. The combined electric and magnetic field effects are treated by a non-equilibrium lateral electron distribution function, which is the solution of a quantum-kinetic equation. The field-dependent coupling between longitudinal and transverse degrees of freedom via the lateral distribution function leads to a non-separable dependence of the current on the electric and magnetic fields. Wannier–Stark current oscillations are strongly enhanced by high magnetic fields. The experimentally detected crossover in the temperature dependence of the current is reproduced. Finally, the influence of an electric field on magneto-phonon resonances is studied.

Wannier-Stark resonances under strong localization conditions in natural silicon-carbide superlattices

A detailed experimental investigation is made of the electronic transport under conditions of Wannier-Stark localization of carriers in a natural superlattice of hexagonal polytypes of silicon carbide. The 4H and 6H polytypes, which possess different superlattice and miniband spectrum parameters, are employed. Direct measurements of the electronic current versus the average electric field in the active region of the sample revealed a series of regions of negative differential conductivity in fields ranging from 500 to 2100 kV/cm. Analysis of the results shows that the observed current resonances are associated with the development of the Wannier-Stark quantization process and are due to conduction mechanisms such as hopping conduction, induced between the levels of a Wannier-Stark ladder by a resonant electron-phonon interaction, and the resonant interminiband tunneling from the first into the second miniband.

A multiband quantum well with an external electric field

The effect of an external electric field ( S) on the eigen energies ( E) and the probability profiles (| ψ( z)| 2) of a mult quantum well is studied. The quantum well is represented by a Kronig-Penney structure with a finite number of cells. In the weak field regime, the quantum confined Stark effect occurs, and the energy shifts of the excited states are opposite to that of the ground state. The displacement of the probability profiles of the ground and excited states are opposite too. When the electric field increases, the gradual transition to the Wannier-Stark quantization is observed, and the field-induced interactions between the first-band Stark ladder and the second-band levels appear. The anticrossing gaps between the electron and hole levels increase as the electric field increases or the quantum-well width decreases.

Magnetic susceptibility of a semiconductor superlattice under parallel electric and magnetic fields

We have calculated the magnetic susceptibility chi of the degenerate electron gas in a semiconductor superlattice placed in parallel quantizing electric (E) and magnetic (H) fields directed along the superlattice growth axis. In this case the electrons have a purely discrete energy spectrum consisting of Landau and Wannier-Stark levels. This leads to: (i) the dependence of the monotonic part of the diamagnetic susceptibility of the electron gas on the electric field; (ii) the rise of a new oscillation period of chi as a function of H different from that of the de Haas-van Alphen oscillations; and (iii) the oscillatory behaviour of chi as a function of E with a periodicity in square root E. It is shown that in weak magnetic fields the Wannier-Stark quantization results in a step-like dependence of the paramagnetic susceptibility of the electron gas on the electric field strength.

Wannier–Stark quantization by internal field in the HgTe/CdTe superlattice

The Stark ladder can be formed in a semiconductor superlattice by an applied electric field. The localization of electrons by an external electric field is known as Wannier–Stark quantization. We have performed the Shubnikov–de Haas measurements with a tilted magnetic field and the photoluminescence measurement on a HgTe/CdTe superlattice in the absence of an external electric field. From the observation of a two-dimensional electron gas and a blueshift of the photoluminescence spectrum, we conclude that the Stark ladder exists in the HgTe/CdTe superlattice and it is formed due to the Wannier–Stark quantization by the internal electrostatic field.

Limits of semiclassical transport in narrow miniband GaAs/AlAs superlattices

We address in this work the electronic conduction properties of superlattices (SL) along the growth axis, in the limit of narrow minibands. Experimental results on a series of GaAs/AlAs SLs are compared to earlier data on wide miniband samples, and to theoretical velocity-field characteristics obtained from the solution of the Boltzmann equation. Negative differential velocity is systematically observed, and a remarkably regular variation of the peak velocity with the miniband width is found, in excellent agreement with calculations including realistic scattering. Semiclassical miniband conduction therefore appears to be an adequate description of transport between ∼ 77K and 400K, down to miniband widths of 4 meV in our samples. This unexpected result is discussed with respect to disorder localization and Wannier-Stark quantization.

Coexistence of Wannier–Stark localization and negative differential velocity in superlattices

We show the coexistence of Wannier–Stark localization as monitored by photoconduction experiments, and of negative differential velocity (NDV) for electrons in a perpendicularly biased GaAs/AlAs superlattice. The critical electric field for the onset of NDV, and that for the appearance of localization as monitored optically are nearly the same, which exemplifies the fundamental link between Wannier–Stark quantization [Phys. Rev. 117, 432 (1960)], and Esaki–Tsu nonlinear transport [IBM J. Res. Dev. 14, 61 (1970)].

Electron minibands and Wannier-Stark quantization in an In0.15Ga0.85As-GaAs strained-layer superlattice.

We have investigated the electronic properties of an ${\mathrm{In}}_{0.15}$${\mathrm{Ga}}_{0.85}$As-GaAs strained-layer superlattice using photoluminescence excitation and photocurrent spectroscopies. Flatband spectra show transitions at the center and edge of the Brillouin minizone, and photocurrent spectra at finite bias show, for the first time in this system, the effects of Wannier-Stark quantization. The heavy-hole transitions evidence the importance of the excitonic interaction between spatially separated carriers. The light-hole transitions show a qualitatively different behavior resulting from their weak confinement in the GaAs layers. Our data agree with a numerical calculation of the electro-optical absorption spectra.

Wannier-Stark quantization and its applications to electro-optical modulation

An external electric field applied parallel to the growth axis of a semiconductor superlattice produces a localization of the eigenstates and a quantization of the energy spectrum. These effects, which we have named Wannier-Stark quantization, are not only remarkable from the point of view of fundamental physics, they also have a good potential for various applications in opto-electronics. Indeed, the superlattice electro-absorption shows both a blue shift associated with the “decoupling” of the quantum wells and, at very low field, a red shift associated with the existence of transitions “oblique in real space” between states localized near adjacent quantum wells. Both effects have been observed at room temperature, and have led to outstanding device performances.

Electric Field Effects on the Optical Properties of InxGa1-xAs/GaAs Strained Quantum Wells and Superlattices

AbstractWe have used photoluminescence excitation and photocurrent spectroscopy to investigate the electronic properties of InxGa1-xAs/GaAs strained layer quantum wells and superlattices. In quantum wells, sharp excitonic transitions between discrete energy levels are observed both in excitation and near flatband photocurrent spectra whereas superlattices show heavy-hole to conduction miniband transitions at the Brillouin mini-zone centre and edge, directly giving the electron miniband width. Applying a longitudinal electric field to the quantum wells produces a red shift of the excitons due to the quantum confined Stark effect, while in superlattices, photocurrent spectra at finite applied electric fields show for the first time in this system, the effects of Wannier-Stark quantization. The analysis of the spectra provides a precise determination of the band offset.

Observation of the Wannier-Stark quantization in a semiconductor superlattice.

When an electric field $F$ is applied parallel to the growth axis of a semiconductor superlattice, the eigenstates are quantized and strongly localized. This Wannier-Stark quantization manifests itself as an effective blue shift of the optical-absorption edge and is accompanied by a new type of electro-optical oscillation of the absorption coefficient which is periodic in ${F}^{\ensuremath{-}1}$. We report experimental evidences for these theoretical predictions from the study of the low-temperature electroreflectance of a GaAs-AlGaAs superlattice.