Luttinger Hamiltonian Research Articles

Band parameters of AlAs, Ge and Si in the 34-band k ⋅ p model

A 34-band k ⋅ p model is used to compute the standard k ⋅ p band parameters at Γ, X and L valleys in indirect-band-gap bulk materials for Td (AlAs) and Oh (Si, Ge) group semiconductors. The values of the effective masses for electrons, heavy and light holes in the Γ, X and L valleys are in very good agreement with those reported in other publications. Satisfactory agreement with available experimental data is also obtained by the present model. Finally, our results show that the Luttinger parameters, the κ valence band parameter and the effective Landé factor g* are in good agreement with available experimental data. In particular, the adjustment of the k ⋅ p Hamiltonian parameters proved that the g* of AlAs, Ge and Si are respectively 1.50, −2.79 and 1.96, which are in good agreement with the experimental values of 1.52, −2.50 and 1.99.

Noise Correlations in One-Dimensional Systems of Ultracold Fermions

Time of flight images reflect the momentum distribution of the atoms in the trap, but the spatial noise in the image holds information on more subtle correlations. Using bosonization, we study such correlations in generic 1D systems of ultracold fermions. We show how pairing as well as spin and charge density wave correlations may be identified and extracted from time of flight images. These incipient orders manifest themselves as power-law singularities in the noise correlations, that depend on the Luttinger parameters, which suggests a general experimental technique to obtain them.

Current statistics of correlated charge tunnelling through an impurity in a 1D wire

We review recent advances in the full counting statistics of a generic transport model of a quantum Brownian particle. The model applies to charge transfer through an impurity embedded in a Luttinger liquid environment, to a coherent one-channel conductor in a resistive environment, and to a resistively shunted Josephson junction device. We derive analytic expressions for the cumulant generating function (CGF) at two particular values of the Luttinger parameter which are related by self-duality. We determine the self-duality relations between the CGFs and between the cumulants.

Spatial distribution of a hole localized on acceptor in deformed crystal

The influence of the host-crystal deformation on the acceptor wave-function in a cubic crystal is considered. The calculations are based on the numerical solution of the Luttinger Hamiltonian for a hole in a complex valence band. The deformation is taken into account using Bir-Pikus Hamiltonian for the deformation. The results can be used for the interpretation of the STM images in strained cubic crystals.

The quadrupole mechanism of hole spin relaxation

For spin-3/2 holes the anisotropic part of the instantaneous Luttinger Hamiltonian can be represented as an effective quadrupole coupling. We investigate the hole spin relaxation process induced by nonadiabatic fluctuations of this interaction. The obtained analytical solution of the stochastic Liouville equation describes the polarization decay of spin-3/2 holes in all regimes of momentum scattering: from collision-dominated to ballistic. Our results create the basis for quantitative interpretation of recent experiments and elucidate the striking difference between the hole spin relaxation process in bulk crystals and 2D semiconductor nanostructures.

Direct observation of transition from Tomonaga–Luttinger liquid states to superconductive phase in carbon nanotubes

We report the direct observation of the transition from the Tomonaga–Luttinger liquid (TLL) states to the superconductive (SC) phase in partially end-bonded multi-walled carbon nanotubes, in which the SC phase and the TLL states interplay. Two different transition behaviors are observed when the temperature decreases: (1) a drastic increase in the Luttinger parameter g toward g=1 in the power laws on non-normalized differential conductance (d l/d v) vs. voltages and (2) a deviation from the TLL formula in the low voltage-energy region of normalized d l/d v vs. eV/ kT. These findings shed light on understanding of one-dimensional electron correlations.

Explicit mapping between a two-dimensional quantum Hall system and a one-dimensional Luttinger liquid. I. Luttinger parameters

We study a simple model of a quantum Hall system with the electrons confined to a linear, narrow channel. The system is mapped to a 1D system which in the low-energy approximation has the form of a Luttinger liquid with different interactions between particles of equal and of opposite chiralities. We study this mapping at the microscopic level, and discuss the relation between the parameters of the 2D system and the corresponding 1D Luttinger liquid parameters. We focus in particular on how the parameters are renormalized by the electron interactions and show that the velocity parameter of the current is not modified by the interaction.

Zeeman splitting in cylindrical hole quantum wires

We have investigated the effect of one-dimensional (1D) confinement on the Zeeman splitting of holes in cylindrical quantum wires for magnetic fields applied parallel to the wire axis. Our theoretical study is based on the Luttinger–Hamiltonian description for the valence band in the spherical approximation. We obtain exact results for energies and Lande g-factors of hole states at the quasi-1D subband edges. The g-factor values are universal in the sense that they only depend on the subband index but not on the wire dimensions. Strong fluctuations are observed among the g-factors for low-lying subband edges, and their specific values are unexpected for a situation when the magnetic field is applied parallel to an axis of high symmetry that could naively be equated with the quantisation axis of total angular momentum. It appears that the requirement of a single-valued hole wave function associated with cylindrical symmetry induces a strong mixing between heavy and light holes, which results in the observed behaviour.

Effective spin-orbit Hamiltonian and field-induced spin polarization of two-dimensional holes

It is shown that the electric field applied in the plane of two-dimensional hole layers not only excites the intrinsic spin currents, but also leads to a homogeneous spin polarization. Theoretical study of this phenomenon, taking into account both the anisotropy of the Luttinger Hamiltonian and the bulk inversion asymmetry effects for holes, is carried out for the quantum wells grown along [ 0 0 1 ] crystallographic direction. The examples of GaAs and Si layers are considered.

Band structures of AlAs, GaP, and SiGe alloys: A 30 k×p model

The band structure of indirect-band gap semiconductors (AlAs, GaP) as well as indirect-band gap alloys semiconductors (GeSi) is described theoretically by using a 30×30 k×p model including the d far-level contribution. For all materials investigated, the resulting electronic band structure parameters are in good agreement with experimental values. The method also provides a good description of the second conduction band which is useful for transport modeling. Finally, our results show that Luttinger parameters, the κ valence band parameter, and the effective masses in the X and L valleys are in good agreement with available experimental data.

Single-particle states in sphericalSi∕SiO2quantum dots

We calculate the ground and excited electron and hole levels in spherical Si quantum dots inside $\mathrm{Si}{\mathrm{O}}_{2}$ in a multiband effective mass approximation. The Luttinger Hamiltonian is used for holes, and the strong anisotropy of the conduction electron effective mass in Si is taken into account. As the boundary conditions for the electron and hole wave functions, we use the continuity of the wave functions and the flux at the boundaries of the quantum dots.

Large-Scale Monte Carlo Study of a Realistic Lattice Model forGa1−xMnxAs

Mn-doped GaAs is studied with a real-space Hamiltonian on an fcc lattice that reproduces the valence bands of undoped GaAs. Large-scale Monte Carlo (MC) simulations on a Cray XT3, using up to a thousand nodes, were needed. Spin-orbit interaction and the random distribution of the Mn ions are considered. The hopping amplitudes are functions of the GaAs Luttinger parameters. At the realistic coupling J approximately 1.2 eV the MC Curie temperature and magnetization curves agree with experiments for x = 8.5% annealed samples. Mn-doped GaSb and GaP are also briefly discussed.

Valence-band mixing in first-principles envelope-function theory

This paper presents a numerical implementation of a first-principles envelope-function theory derived recently by the author [B. A. Foreman, Phys. Rev. B 72, 165345 (2005)]. The examples studied deal with the valence subband structure of $\mathrm{Ga}\mathrm{As}∕\mathrm{Al}\mathrm{As}$, $\mathrm{Ga}\mathrm{As}∕{\mathrm{Al}}_{0.2}{\mathrm{Ga}}_{0.8}\mathrm{As}$, and ${\mathrm{In}}_{0.53}{\mathrm{Ga}}_{0.47}\mathrm{As}∕\mathrm{In}\mathrm{P}$ (001) superlattices calculated using the local-density approximation to density-functional theory and norm-conserving pseudopotentials without spin-orbit coupling. The heterostructure Hamiltonian is approximated using quadratic-response theory, with the heterostructure treated as a perturbation of a bulk reference crystal. The valence subband structure is reproduced accurately over a wide energy range by a multiband envelope-function Hamiltonian with linear renormalization of the momentum and mass parameters. Good results are also obtained over a more limited energy range from a single-band model with quadratic renormalization. The effective kinetic-energy operator ordering derived here is more complicated than in many previous studies, consisting in general of a linear combination of all possible operator orderings. In some cases, the valence-band Rashba coupling differs significantly from the bulk magnetic Luttinger parameter. The splitting of the quasidegenerate ground state of no-common-atom superlattices has non-negligible contributions from both short-range interface mixing and long-range dipole terms in the quadratic density response.

Hall effect in strongly correlated low-dimensional systems

We investigate the Hall effect in a quasi-one-dimensional system made of weakly coupled Luttinger liquids at half filling. Using a memory function approach, we compute the Hall coefficient as a function of temperature and frequency in the presence of umklapp scattering. We find a power-law correction to the free-fermion value (band value), with an exponent depending on the Luttinger parameter ${K}_{\ensuremath{\rho}}$. At sufficiently high temperature or frequency, the Hall coefficient approaches the band value.

Fourier transform of the2kFLuttinger liquid density correlation function with different spin and charge velocities

We obtain a closed-form analytical expression for the zero-temperature Fourier transform of the $2{k}_{F}$ component of the density-density correlation function in a Luttinger liquid with different spin and charge velocities. For frequencies near the spin and charge singularities, approximate analytical forms are given and compared with the exact result. We find power-law-like singularities leading to either divergence or cusps, depending on the values of the Luttinger parameters, and compute the corresponding exponents. Exact integral expressions and numerical results are given for the finite-temperature case as well. We show, in particular, how the temperature rounds the singularities in the correlation function.

Evidence of Luttinger-liquid behavior in one-dimensional dipolar quantum gases

A strongly correlated Luttinger-liquid behavior is found to emerge well beyond the Tonks-Girardeau (TG) regime in a one-dimensional Bose gas with dipolar repulsions at $T=0$, persisting for a wide range of densities. After combining reptation quantum Monte Carlo and bosonization techniques, we provide a unifying theory of the underlying crossover physics, evolving from the TG gas at low density into a classical quasiordered state at high density. The density dependent Luttinger parameters extracted from the numerical data provide all that is needed to determine the low-energy behavior from analytical expressions. Our quantitative predictions, in the whole crossover, for measurable quantities such as the structure factor and the momentum distribution, are estimated to be accessible in underway experiments with ultracold polar molecules.

QUANTUM BEATS IN SEMICONDUCTORS

In this paper we present a new method to calculate the Faraday rotation using the Jones calculus applicable to light pulses. This result is used to calculate the quantum beats in a magnetic semiconductor generated by pump-probe experiments in Faraday geometry. Considering a six level system and neglecting band dispersion as well as Coulomb interaction we find the Luttinger parameter of the valence band to be the only band parameter contributing to the quantum beat frequency.

The eight-level k ⋅ p model for the conduction and valence bands of InAs, InP, InSb

We present a generalized theoretical description of the 30 × 30 k ⋅ p approach for determining the band structure of the direct-band-gap semiconductors (InAs, InP, InSb), including the d far-levels contribution. For all materials investigated, the resulting electronic band structure parameters are in good agreement with experimental values. This model gives access to the second conduction band which is useful for transport modelling. We finally show that this method also gives explicit expressions for the Luttinger parameters, the κ valence band parameter and the effective masses in the Γ valley.

Spontaneous Magnetization in Diluted Magnetic Semiconductor Quantum Wells

A diluted magnetic semiconductor (DMS) quantum well is an interesting system for exploring spintronic applications. We calculated the spontaneous magnetization (SM) in a 100-A Ga1−xMnxAs/Al0.35Ga0.65As quantum well. The Schrodinger equation was described by a 4×4 Luttinger Hamiltonian in the envelope function approximation with the exchange interaction between Mn ions and holes treated in the mean-field approximation. The Schrodinger-Poisson-DMS self-consistency was solved by using the finite element method. We studied how the SM depended on the hole concentration p, the temperature T , the effective Mn concentration xeff , the antiferromagnetic temperature TAF , and the exchange integral βNo. For T = 0 K, xeff = 0.05, TAF = 0.5 K, and βNo = − 1.2 eV, the SM begins to appear at p = 5 × 10 cm−3 and saturates around 3 × 10 cm−3. For p = 10 cm−3, the SM disappears around 5 K.

Exact numerical simulations of a one-dimensional trapped Bose gas

We analyze the ground-state and low-temperature properties of a one-dimensional Bose gas in a harmonic trapping potential using the numerical density matrix renormalization group. Calculations cover the whole range from the Bogoliubov limit of weak interactions to the Tonks-Girardeau limit. Local quantities such as density and local three-body correlations are calculated and shown to agree very well with analytic predictions within a local density approximation. The transition between temperature dominated to quantum dominated correlation is determined and it is shown that despite the presence of the harmonic trapping potential first-order correlations display over a large range the algebraic decay of a harmonic fluid with a Luttinger parameter determined by the density at the trap center.