Luttinger Hamiltonian Research Articles

Ab initio Studies of the Band Parameters of III–V and II–VI Zinc-Blende Semiconductors

Electronic band-structure calculations are performed for zinc-blende III–V (AlP, AlAs, AlSb, GaP, GaAs, GaP, InP, InAs, and InSb) and II–VI (ZnS, ZnSe, ZnTe, CdS, CdSe, and CdTe) semiconductors using an ab initio pseudopotential method within a local-density approximation (LDA). Lattice parameters, band gaps, Luttinger parameters, momentum matrix elements and effective masses are studied in detail. It is shown that LDA calculations cannot systematically give accurate band parameters. It is found that LDA band parameters calculated using experimentally determined lattice constants are more accurate than those using LDA lattice constants. We found that inclusion of the d electrons of Group-II atoms in the core gives more accurate band parameters.

Energy-band structure of Ge, Si, and GaAs: A thirty-bandk∙pmethod

A 30-band $\mathbf{k}\mathbf{∙}\mathbf{p}$ method taking into account spin-orbit coupling is used to describe the band diagram of Ge, Si, and GaAs over the whole Brillouin zone on an extent of $5\phantom{\rule{0.3em}{0ex}}\mathrm{eV}$ above and $6\phantom{\rule{0.3em}{0ex}}\mathrm{eV}$ under the top of the valence band. The band diagrams provide effective masses in agreement with experimental data both for direct gap semiconductors (GaAs) and for indirect gap semiconductors (Ge, Si). This method also gives explicit expressions for Luttinger parameters and effective masses in the $\ensuremath{\Gamma}$ valley.

Diluted magnetic semiconductor quantum dots: An extreme sensitivity of the hole Zeeman splitting on the aspect ratio of the confining potential

The valence band states confined in infinitely deep quantum dots made of diluted magnetic semiconductors (DMS) are considered theoretically. A complex anisotropic structure of the valence bands in DMSs with cubic symmetry described by the full Luttinger Hamiltonian is taken into account. It is found that the Zeeman splitting is very sensitive to the shape of the confining potential and, in particular, to its orientation relative to the direction of an external magnetic field. This sensitivity has its origin in a mixing of different spin components of a hole wave function which takes place for finite hole wave vectors $\mathbf{k}$. Several consequences of the effect are discussed, including a possibility to control the interdot tunneling by an external magnetic field. It is shown also that the polarizations of optical transitions in a single DMS quantum dot depend on details of geometry of its confining potential as well as on the strength of the magnetic field.

Spin symmetry and spin current of helicity eigenstates of the Luttinger Hamiltonian

A general spin symmetry argument is proposed for spin currents in semiconductors. In particular, due to the symmetry with respect to spin polarization of the helicity eigenstates of the Luttinger Hamiltonian for a hole-doped semiconductor, the spin polarized flux from a single helicity eigenstate induced by an external electric field, is canceled exactly when all such contributions from eigenstates that are degenerate in energy are summed. Thus, the net spin current predicted by Murakami et al. (Science 301 (2003) 1348), cannot be produced by such a Hamiltonian. Possible symmetry breaking mechanisms which may generate a spin current are discussed.

SU(2)non-Abelian holonomy and dissipationless spin current in semiconductors

Following our previous work [S. Murakami, N. Nagaosa, S. C. Zhang, Science 301, 1348 (2003)] on the dissipationless quantum spin current, we present an exact quantum mechanical calculation of this novel effect based on the linear response theory and the Kubo formula. We show that it is possibxle to define an exactly conserved spin current, even in the presence of the spin-orbit coupling in the Luttinger Hamiltonian of p-type semiconductors. The light- and the heavy-hole bands form two Kramers doublets, and an SU(2) non-abelian gauge field acts naturally on each of the doublets. This quantum holonomy gives rise to a monopole structure in momentum space, whose curvature tensor directly leads to the novel dissipationless spin Hall effect, i.e., a transverse spin current is generated by an electric field. The result obtained in the current work gives a quantum correction to the spin current obtained in the previous semiclassical approximation.

Optical properties of Ga 1− x In x As/GaAs(001) quantum well superlattices: Exciton and polariton dispersion curves

Wannier exciton wavefunctions and energies have been computed in superlattices of strained Ga1−xInxAs/GaAs(001) quantum wells (SLQWs) using Luttinger Hamiltonian and accurate variational envelope functions. Exciton dispersion curves of the SLQWs are then obtained by computing exciton energies for different K-points of the corresponding first Brillouin zone. Photon dispersion curves, due to the background dielectric constant modulation, and the polariton dispersion curves have been computed in the semiclassical self-consistent framework. The results are discussed for the case of exciton energies far from the photonic gaps. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Hole spin relaxation in quantum dots

We present results for relaxation of the spin of a hole in a cylindrical quantum dot due to acoustic phonon assisted spin flips at low temperatures with an applied magnetic field. The hole dispersion is calculated by numerical diagonalization of the Luttinger Hamiltonian and applying perturbation theory with respect to the magnetic field, and the hole-phonon coupling is described by the Bir-Pikus Hamiltonian. We find that the decoherence time for hole spins for dots ≤20 nm is on the order of 10 - 8 s. This is several orders smaller than the decoherence time due to phonon assisted processes for electron spins in similar dots and is comparable to the total decoherence time of an electron spin in a quantum dot, which is controlled by the hyperfine interaction with nuclei. We obtain the dependence of the relaxation rate of the hole spin on dot size and hole mass.

Competing density-wave orders in a one-dimensional hard-boson model

We describe the zero-temperature phase diagram of a model of bosons, occupying sites of a linear chain, which obey a hard-exclusion constraint: any two nearest-neighbor sites may have at most one boson. A special case of our model was recently proposed as a description of a ``tilted'' Mott insulator of atoms trapped in an optical lattice. Our quantum Hamiltonian is shown to generate the transfer matrix of Baxter's hard-square model. Aided by exact solutions of a number of special cases, and by numerical studies, we obtain a phase diagram containing states with long-range density-wave order with period 2 and period 3, and also a floating incommensurate phase. Critical theories for the various quantum phase transitions are presented. As a byproduct, we show how to compute the Luttinger parameter in integrable theories with hard-exclusion constraints.

Temperature dependence of the conductance in diluted magnetic semiconductors

Abstract The effect that the ferromagnetic order has on the electrical properties of the diluted magnetic semiconductors is investigated by analyzing the temperature dependence of the dielectric constant and the resistivity. We use a six band Kohn–Luttinger Hamiltonian, thermal fluctuations are treated in the mean-field approximation, carrier–carrier interaction in the random phase approximation, and transport properties using the relaxation time approximation. Resistivity changes near 20% when going from zero to Curie temperature are obtained in good agreement with experiment. We explain this change in the resistivity in terms of the variation of the Fermi surface and the transport scattering time when going from the ferromagnetic phase to the paramagnetic phase.

Luttinger parameter of carbon nanotubes investigated by shot noise experiment

Fermi liquid model with elemental charge of e*(e*=ge) is used to describe the electrical transport in carbon nanotubes, so the scattering theory can be used to calculate the zero-frequency shot noise in nanotubes. When the temperature is 0K, the nanotube with a strong barrier has the shot noise of 2geI. A new approach to measure the Luttinger g-factor of nanotubes is created: forming a strong barrier in nanotube, measuring its shot noise S, and then calculating the g-factor.

Bloch electrons in a magnetic field: Topology of quantum states and transport

Quantum states and Hall conductances of electrons in n-type heterojunctions and holes in p-type heterojunctions in a field of a lateral superlattice and a perpendicular magnetic field were studied. It is shown that the energy spectrum of magnetic subbands in a periodic potential without inversion center is not symmetric about the reversal of the quasi-momentum sign. The properties of wave functions and the related topological invariants determining the Hall conductance were examined. The method of calculating the magnetic Bloch states of holes was developed on the basis of the Luttinger Hamiltonian, allowing the spin and spin-orbit interactions to be taken into account in this problem. The Hall conductance quantization law was determined for 2D holes in a periodic superlattice potential.

Quantum Hall effect in ap-type heterojunction with a lateral surface superlattice

The quantization of Hall conductance in a p-type heterojunction with lateral surface quantum dot superlattice is investigated. The topological properties of the four-component hole wavefunction are studied both in r- and k-spaces. New method of calculation of the Hall conductance in a 2D hole gas described by the Luttinger Hamiltonian and affected by lateral periodic potential is proposed, based on the investigation of four-component wavefunction singularities in k-space. The deviations from the quantization rules for Hofstadter "butterfly" for electrons are found, and the explanation of this effect is proposed. For the case of strong periodic potential the mixing of magnetic subbands is taken into account, and the exchange of the Chern numbers between magnetic subands is discussed.

Heavy and light hole states in boron delta-doped diamond

Heavy and light hole states in B δ-doped diamond quantum wells are calculated within the effective mass approximation using a local density Thomas–Fermi approximation for the description of the band bending profile. The calculation starts with the solution of two-independent (hh+lh) hole bands model at the Brillouin zone center, and then, the quantized eigenstates at the Γ point are used as a basis to diagonalize the Luttinger–Kohn Hamiltonian, obtaining the different hole states at the non-Γ points. Different sets of Luttinger parameters for diamond reported in the literature are considered. The results could help to identify which of those sets describes the actual valence band structure in diamond.

Quantum dynamics in nonequilibrium strongly correlated environments

We consider a quantum point contact between two Luttinger liquids coupled to a mechanical system (oscillator). For non-vanishing bias, we find an effective oscillator temperature that depends on the Luttinger parameter. A generalized fluctuation-dissipation relation connects the decoherence and dissipation of the oscillator to the current-voltage characteristics of the device. Via a spectral representation, this result is generalized to arbitrary leads in a weak tunneling regime.

Numerical spurious solutions in the effective mass approximation

We have characterized a class of spurious solutions that appears when using the finite difference method to solve the effective mass approximation equations. We find that the behavior of these solutions as predicted by our model shows excellent agreement with numerical results. Using this interpretation we find a set of analytical expressions for conditions that the Luttinger parameters must satisfy to avoid spurious solutions. Finally, we use these conditions to check commonly used sets of parameters for their potential for generating this class of spurious solutions.

Valence band states in boron delta-doped diamond systems

Valence band ( hh and lh) states in B δ-doped diamond quantum wells are calculated within the effective mass approximation using a local density Thomas-Fermi approximation for the description of the band bending profile. Many-body Hartree, exchange and correlation terms are included in the potential energy well function. The diagonalization of the Kohn–Luttinger Hamiltonian is performed using different sets of Luttinger parameters for diamond reported in the literature. The results could be useful in identifying the actual valence band structure of diamond.

Quantum states and optics in ap-type heterojunction with a lateral surface quantum dot or antidot superlattice subjected to a perpendicular magnetic field

The studies of quantum states and optics in a {\it p}-type heterojunction with lateral surface quantum dot (antidot) superlattice and in the presence of perpendicular magnetic field are performed. For the first time the Azbel'-- Hofstadter problem is solved for holes in a complicated valence band described by the $4 \times 4$ Luttinger Hamiltonian. The set of magnetic subbands is obtained for separate hole levels in a wide interval of magnetic field. We found remarkable differences between hole spectra and the Hofstadter "butterfly" for electrons. The influence of the spin-orbit interaction onto wavefunctions and energy spectrum has been investigated. The probabilities of optical transitions between quantum states in the valence band and donors located in the monolayer inside the heterojunction are calculated. The set of parameters (superlattice periods, amplitude of periodic potential, magnitude of magnetic field, etc.) required for experimental observation of splitted hole Landau levels is determined.

Interedge tunneling in quantum Hall line junctions

We propose a scenario to understand the puzzling features of the recent experiment by Kang and coworkers on tunneling between laterally coupled quantum Hall liquids by modeling the system as a pair of coupled chiral Luttinger liquid with a point contact tunneling center. We show that for filling factors $\nu\sim1$ the effects of the Coulomb interactions move the system deep into strong tunneling regime, by reducing the magnitude of the Luttinger parameter $K$, leading to the appearance of a zero-bias differential conductance peak of magnitude $G_t=Ke^2/h$ at zero temperature. The abrupt appearance of the zero bias peak as the filling factor is increased past a value $ \nu^* \gtrsim 1$, and its gradual disappearance thereafter can be understood as a crossover controlled by the main energy scales of this system: the bias voltage $V$, the crossover scale $T_K$, and the temperature $T$. The low height of the zero bias peak $\sim 0.1e^2/h$ observed in the experiment, and its broad finite width, can be understood naturally within this picture. Also, the abrupt reappearance of the zero-bias peak for $\nu \gtrsim 2$ can be explained as an effect caused by spin reversed electrons, \textit{i. e.} if the 2DEG is assumed to have a small polarization near $\nu\sim2$. We also predict that as the temperature is lowered $\nu^*$ should decrease, and the width of zero-bias peak should become wider. This picture also predicts the existence of similar zero bias peak in the spin tunneling conductance near for $\nu \gtrsim 2$.

Doping-induced bandgap narrowing in Si rich n- and p-typeSi1−xGex

The shifts of the fundamental and optical bandgap energies as functions ofdopant concentration in heavily n-type and p-type doped Si1−xGex (x ≤ 0.3)have been investigated theoretically. The band structure ofthe intrinsic crystal was described by the k · p-perturbationmethod, where the Kohn–Luttinger parameters were determined froma first-principles and full-potential band-structure calculation. Thedoping-induced effects on the bandgap were thereafter calculated using azero-temperature Green function formalism within the random phaseapproximation and with a local field correction of Hubbard. We found onlysmall effects on the bandgap energies due to variation of compositionx. Thecalculated bandgap narrowing of Si and of Si0.82Ge0.18 were found to be in goodagreement with photoluminescence measurements.

Topological transitions in size-quantized heterostructures

An exact solution is obtained to the problem of the spectrum of holes, described by the Luttinger Hamiltonian, in a quantum well of finite depth under arbitrary uniaxial stresses in the well and the barrier. Conditions for the topological transitions accompanied by the variation in the connectivity of the isoenergetic surface are found. It is shown that, for certain values of model parameters, the effective mass of holes in the ground-state subband of size quantization becomes negative.