Wannier-Stark Levels Research Articles

Photocurrent spectroscopy of low-electric-field anticrossings in semiconductor superlattices

We have investigated the anticrossings of Wannier-Stark levels at low electric fields in a superlattice with wide quantum wells by photocurrent spectroscopy. The measurements were performed at low temperatures in electric and magnetic fields parallel to the growth direction. The diamagnetic shift of the excitonic Wannier-Stark levels was measured for constant electric fields smaller and higher than the resonance electric field. By comparing the diamagnetic shift of the various transitions, the character of the states involved in the anticrossing was determined and the interchange of the states during the low-field anticrossing could be verified. Furthermore, the interwell-to-intrawell ratio of the diamagnetic shift is compared to theoretical calculations of the exciton radius.

Transformation of quantum size levels into virtual levels at the boundary between p-GaAs and an AlAs/GaAs superlattice

The transformation of quantum size levels into virtual levels upon a change in the electric field in an AlAs/GaAs superlattice located in the i region of a p-i-n structure is studied experimentally and theoretically. It is shown that an interfacial state at the boundary between the superlattice and the p-GaAs contact layer results in a resonant increase in the probability of photoelectron tunneling from the contact into the superlattice via Wannier-Stark levels.

Combined description for semiclassical and quantum transport in superlattices

The dc current in strongly coupled superlattices can either be described in a semiclassical picture in terms of miniband transport or in a quantum mechanical theory based on hopping transitions between the localized wave functions of the Wannier-Stark ladder. We demonstrate the equivalence of both models in the field regime of Bloch-oscillating electrons. Combining these two pictures we calculate the superlattice drift velocity resulting from microscopic phonon scattering between the Wannier-Stark levels. Our results show that previous discussions of the temperature dependence of the drift velocity have to be revised drastically.

Degenerate four-wave-mixing signals from a dc- and ac-driven semiconductor superlattice

Within the one-dimensional tight-binding model and $\ensuremath{\chi}\ensuremath{-}3$ approximation, we have calculated four-wave-mixing (FWM) signals for a semiconductor superlattice in the presence of both static and high-frequency electric fields. When the exciton effect is negligible, the time-periodic field dynamically delocalizes the otherwise localized Wannier-Stark states, and accordingly quasienergy band structures are formed, and manifest in the FWM spectra as a series of equally separated continua. The width of each continuum is proportional to the joint width of the valence and conduction minibands and is independent of the Wannier-Stark index. The realistic homogeneous broadening blurs the continua into broad peaks, whose line shapes, far from the Lorentzian, vary with the delay time in the FWM spectra. The swinging range of the peaks is just the quasienergy bandwidth. The dynamical delocalization (DDL) also induces significant FWM signals well beyond the excitation energy window. When the Coulomb interaction is taken into account, the unequal spacing between the excitonic Wannier-Stark levels weakens the DDL effect, and the FWM spectrum is transformed into groups of discrete lines. Strikingly, the groups are evenly spaced by the ac field frequency, reflecting the characteristic of the quasienergy states. The homogeneous broadening again smears out the line structures, leading to the excitonic FWM spectra quite similar to those without the exciton effect. However, all these features predicted by the dynamical theory do not appear in a recent experiment [Phys. Rev. Lett. 79, 301 (1997)], in which, by using the static approximation the observed Wannier-Stark ladder with delay-time-dependent spacing in the FWM spectra is attributed to a temporally periodic dipole field, produced by the Bloch oscillation of electrons in real space. The contradiction between the dynamical theory and the experiments is discussed. In addition, our calculation indicates that the dynamical localization coherently enhances the time-integrated FWM signals. The feasibility of using such a technique to study the dynamical localization phenomena is shown.

Photocurrent resonances in short-period AlAs/GaAs superlattices in an electric field

The photocurrent was measured as a function of the external electric field in short-period AlAs/GaAs superlattices for various photon energies. Transport resonances, whose positions do not depend on the photon energy, were observed in these dependences together with optical resonances due to interband transitions in Wannier-Stark levels. It is shown that the transport resonances are due to tunneling of photoelectrons from the p-GaAs contact region into the first level in GaAs wells located 2–5 lattice periods from the contact layer.

Interaction of Wannier–Stark levels in a wide well superlattice

We have investigated the energy spectrum of a superlattice with wide quantum wells under the bias of an electric field perpendicular to the superlattice layers. By using photocurrent spectroscopy, transitions of Wannier–Stark levels for the various electron and hole states are observed, and at low fields, further structures corresponding to miniband edge transitions are found. Various anticrossings could be observed at higher and lower electric fields. The anticrossings at high electric fields are due to energy alignment of different electronic sublevels in adjacent wells. The anticrossing structures at low fields could be interpreted as resonances between intrawell and interwell excitonic Wannier–Stark states with equal sublevel states, where the anticrossing is caused by differences in exciton binding energy. Fitting of transitions and anticrossings was done by using a semi-empirical model and we have extracted relevant fitting parameters like the quantum-confined Stark coefficient, binding energies for the excitonic Wannier–Stark levels and the resonant coupling strength for states involved in the various anticrossing transitions. Finally, insight into the excitonic influences on the coupling of the WS states could be obtained by comparing the fitted parameters for the various transitions.

Determination of the roughness of heteroboundaries from photocurrent spectra of short-period AlAs/GaAs superlattices

Photocurrent spectroscopy is used to study the nature of the roughness of heteroboundaries in (AlAs)m/(GaAs)n short-period superlattices (m=3−5, n=10−13) grown by molecularbeam epitaxy. The formation of minibands broadens the optical spectra of superlattices in comparison with isolated quantum wells; therefore to analyze the degree of perfection of the boundaries we used the decay of the minibands into a series of discrete Wannier-Stark levels in an electric field parallel to the superlattice axis. Exciton lines were observed in the photocurrent spectra in an electric field corresponding to direct and indirect (in space) transitions between the Wannier-Stark levels. Comparison of experimental data with calculation indicates that even in the better structures, in addition to monotonic variation of the thickness of the layers over area, roughnesses in the heteroboundaries one monolayer in height are present with characteristic lateral dimension not exceeding 10 nm.

Ground state energy of the negatively charged exciton X− in bidimensional semiconductors in a steady electric field

The ground state energy of a negatively charged exciton X − (trion) in a bidimensional (2D) semiconductor under a steady parallel electric field has been calculated variationally for different values of the electron to hole effective mass ratio. It is found that the binding energy W, defined as the difference between the trion and exciton energies, increases with a weak electric field. The results are compared with those we have previously obtained in bulk (3D) semiconductors. It is shown that charged excitons bound states can exist in 2D semiconductors for larger values of the field than in the 3D case. It is also found that the ratio W 2 D W 3 D decreases at increasing fields for all values of the mass ratio. The quantization of the center of mass motion gives rise to Wannier-Stark levels.

Acoustic-phonon Raman scattering from Wannier-Stark levels in GaAs/AlAs superlattices

We study the electronic structure of a GaAs/AlAs superlattice in an electric field by acoustic-phonon Raman scattering. Resonantly enhanced disorder-induced processes (continuous emission) occur whenever the exciting laser energy coincides with interband transitions between heavy hole and electron Stark levels with $\ensuremath{\Delta}n=0,\ifmmode\pm\else\textpm\fi{}1$. We also observe oscillations of the scattering intensity due to the delocalization of localized excitons induced by resonant interaction either with emerging Stark states of neighboring quantum wells in the same miniband or with Stark states from the next higher electron miniband.

Hopping transport in superlattices

We calculate the d.c.-current of a semiconductor superlattice in the hopping conduction picture. Electronic transport in this regime is described by hopping transitions between the partially localized Wannier–Stark states. In our numerical model we start from the exact wavefunctions of the superlattice and include both impurity and acoustic phonon scattering. We then obtain the electron drift velocity by considering the thermal in-plane distribution function of the Wannier–Stark levels and by summing over all possible hopping processes. Our results deviate significantly from those of the Esaki–Tsu model and the solutions of the Boltzmann equation. We can identify three different regimes of field dependence of the drift velocity, obeying power lawsF−n. For low to moderate fieldsnis apparently 1, in agreement with the high field limit obtained from the models based on a description in momentum space. For intermediate fieldsnis large, resulting from increasing field dependent localization of the wavefunctions. In the high field limit the wavefunctions become equivalent to the Wannier functions of the superlattice and the localization saturates. Therefore, a lower exponentnis obtained.

Excitons in superlattices: Absorption asymmetry, dimensionality transition, and exciton localization

We calculate the field-dependent excitonic absorption spectra of semiconductor superlattices by diagonalizing the full Hamiltonian of the problem. Our model therefore includes both bound and continuum states of the excitons. The theoretical results are compared with experiments. We first show that the Coulomb interaction does not destroy the field-induced transition from the Franz-Keldysh to the Wannier-Stark regime, but causes an absorption asymmetry in the spectra, i.e., a dominance of Wannier-Stark transitions with negative indices. This effect, which has been observed before, is explained by a Coulomb-induced shift of oscillator strength from higher to lower energies in a simple model involving an effective Coulomb potential. We further address the problem of the transformation of the quasi-two-dimensional excitons of the Wannier-Stark levels to the three-dimensional miniband excitons obtained for zero field. For finite electric fields the quasibound miniband exciton states are explained as resonant states of the various Wannier-Stark excitons. We show that in contradiction to previous assumptions the Wannier-Stark ladder transitions are never suppressed by exciton localization. The observability of the transitions only depends on the relation between the zero-field miniband width and the linewidth of the transitions, but not on the ratio between miniband width and exciton binding energy.

Wannier-Stark Localization in Superlattices

We present experimental and theoretical evidence for Wannier-Stark (WS) oscillations in the DC electric current through reverse-biased highly doped p-i-n GaAs diodes. The intrinsic region in the diode contained seven (GaAs)5/(AlAs)2 multi-quantum wells. Carrying out these transport experiments at low temperatures, we found periodic WS oscillations in the second derivative of the Zener current, which is in qualitative agreement with theoretical predictions based on a realistic multiband and multichannel scattering theory. These findings resolve long-standing controversies about the existence of Wannier-Stark levels in the Zener tunneling current.

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 states in nanostructures

We study the effects of a longitudinal-external-electric field on the electronic properties of an open-quantum-dot superlattice. These structures present the formation of two different gaps, one associated with the periodicity of the structure imposed upon the ID mode and the other associated to the destructive interference between the ID mode and the quantum-dot states. The transmission coefficients between two quasi-two dimensional reservoirs through these structures are calculated. The electric field presents different regimes: at weak fields the transmission peaks associated to the tunneling through the confined dot states show an energy shift and a progressive decline of the transmission. At moderate fields the transmission bands become sharp and the transmission peaks to one. As the electric field increases the anti-resonance gaps widen. Finally, at large electric fields, the Wannier-Stark levels are completely formed and the transmission is characterized by a series of relatively equally spaced peaks.

Density of states and Wannier-Stark levels of superlattices in an electric field.

The density of states is calculated for superlattices in the presence of an electric field, using the concept of a generalized transmission probability. Various ${\mathrm{Ga}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$${\mathrm{Al}}_{\mathit{x}}$As-based quantum-well structures with up to 200 periods are examined for electric fields ranging from 4\ifmmode\times\else\texttimes\fi{}${10}^{2}$ V/cm to 1\ifmmode\times\else\texttimes\fi{}${10}^{6}$ V/cm. Different field effects are observed in the below- and above-barrier regions of the density-of-states spectra. In the below-barrier region the spectra embody the occurrence of Wannier-Stark localization. The above-barrier region is dominated by two kinds of structures, one similar to Franz-Keldysh oscillations, and another resembling a Wannier-Stark-ladder behavior.

Wannier-Stark levels in the valence band of semiconductor multiple quantum wells.

We report on calculations of the in-plane dispersion of semiconductor multiple quantum wells subjected to a longitudinal electric field. At the in-plane zone center the decoupled heavy- and light-hole subbands split into evenly spaced Wannier-Stark ladders. Their in-plane dispersion relations display anticrossing behaviors which correspond either to intrawell interactions between the zone center light- and heavy-hole-states or to interactions between hole levels which are mainly localized in adjacent wells.

Temperature dependence of the wannier‐stark level width in a semiconductor in a strong electric field

AbstractAnalytical expressions for the level width of the Wannier‐Stark ladder caused by an electron‐phonon interaction are derived. They allow to investigate the dependence of the width on temperature, external field, and semiconductor band parameters. It turns out that with increase in temperature the width associated with acoustic phonons is drastically increased whereas the width associated with optical phonons rises much more slowly. It seems that almost all experiments for finding the Wannier‐Stark ladder had been carried out at too high temperatures when the interaction of an electron with acoustic phonons eliminates the ladder.

Further evidence for the existence of Wannier-Stark levels and electron-phonon resonance states in zinc sulphide

Further evidence for the existence of Wannier-Stark levels and electron-phonon resonance states in zinc sulphide

Quantitative evidence for the existence of Wannier-Stark levels

Measurements of the conductivity characteristics of CuxS-ZnS:Mn,Cu film heterojunctions provide quantitative evidence for the existence of Wannier-Stark levels. Corresponding peaks in the electroluminescence characteristics are also ascribed to the electron-phonon resonance states by Wannier (1960). It is suggested that temperature effects may be relatively unimportant.

Electron–Phonon Interaction of Crystal Electrons in High Electric Fields

AbstractThe electron–phonon interaction in high electric fields is investigated. The calculation of the electron lifetime shows that Wannier–Stark levels may exist. It is furthermore shown that there is a field dependent phonon production. It can possibly be measured by Raman scattering and serve to demonstrate the existence of the Wannier–Stark levels.