Luttinger Hamiltonian Research Articles

Wave packet dynamics in hole Luttinger systems

For hole systems with an effective spin 3/2 we analyzed analytically and numerically the evolution of wave packets with the different initial polarizations. The dynamics of such systems is determined by the $4\times 4$ Luttinger Hamiltonian. We work in the space of arbitrary superposition of light- and heavy-hole states of the "one-particle system". For 2D packets we obtained the analytical solution for the components of wave function and analyzed the space-time dependence of probability densities as well as angular momentum densities. Depending on the value of the parameter $a=k_0d$ ($k_0$ is the average momentum vector and $d$ is the packet width) two scenarios of evolution are realized. For $a>>1$ the initial wave packet splits into two parts and the coordinates of packet center experience the transient oscillations or {\it Zitterbewegung} (ZB) as for other two-band systems. In the case when $a<<1$ the distribution of probability density at $t>0$ remains almost cylindrically symmetric and the ripples arise at the circumference of wave packet. The ZB in this case is absent. We evaluated and visualized for different values of parameter $a$ the space-time dependence of angular momentum densities, which have the multipole structure. It was shown that the average momentum components can precess in the absence of external or effective magnetic fields due to the interference of the light- and heavy hole states. For localized initial states this precession has a transient character.

Transient effects in the backscattered current of a Luttinger liquid

We study the backscattered current in a Luttinger liquid in the presence of a point like weak impurity switched on at finite time, taking into account finite-temperature effects. We show how the well-known results for a static impurity are distorted. We derive a dimensionless parameter $\tau_{R}$ as function of the electron-electron interaction and the temperature, such that for $\tau_{R} < 1$ ($> 1$) the switching process is relevant (irrelevant). Our results suggest the possibility of determining the value of the Luttinger parameter $K$ through time measurements in transport experiments at fixed voltage.

Motion-enhanced magnetic moments of excitons in ZnSe

Recent studies of excitons in wide quantum wells have shown that the magnetic properties are strongly affected as the excitons acquire kinetic energy. In the center of mass approximation, these motion-induced changes are ascribed to mixing between the $1S$ exciton ground state and the higher lying $nP$ states. The origin of the mixing is due to the dispersion curves for the valence band not being of simple parabolic form. Detailed previous studies of excitons in CdTe have resulted in excellent agreement between experiment and the predictions of this model. One consequence of the mixing is that the magnetic moment of the exciton is not simply the sum of the magnetic moments of the electron and hole, but contains motion-induced contributions, which can easily dominate the contributions from the individual charge carriers. To confirm the validity of the model, we have carried out detailed investigations of the magnetic properties of center of mass excitons in a second semiconductor, ZnSe, for which the magneto-optical properties of the individual charge carriers are completely different from those of CdTe. Excellent agreement is obtained between theory and experiment with a choice of the Luttinger parameter ${\ensuremath{\gamma}}_{3}=0.98$, in close agreement with the value determined independently by two-photon magnetoabsorption experiments. The success of the model when applied to both materials provides strong evidence that motion-induced changes in magnetism are a universal feature in zinc-blende semiconductors.

Design of a GaN White Light-Emitting Diode Through Envelope Function Analysis

In this paper, we present an envelope function analysis in order to design the emission spectra of a white quantum well light emitting diode. The nanometric heterostructure that we are dealing with is a multiple quantum well, consisting periods of three single quantum wells with various well thicknesses. With the aid of 6x6 Luttinger Hamiltonian, we employ the combination of two methods, k.p perturbation and transfer matrix method, to acquire electron and hole wavefunctions analytically. The envelope function approximation was considered to obtain these wavefunctions for a special basis set. While adjacent valence subbands have been studied exactly, the conduction bands are approximated as parabolic. The effect of Stokes shift has been also taken into account. The dipole moment matrix elements for interband atomic transitions are evaluated via correlation between electron and hole envelope functions, for both orthogonal polarizations. This has simplified the calculation of photoluminescence intensity. Spatial variations in hole/electron wavefunctions have been examined with the introduction of piezoelectric and spontaneous polarizations internal field. We theoretically establish the possibility of a highly efficient InGaN red emitter, resulting in a uniform luminescence in red, green and blue emissions from the while light emitting diode, through adjusting material composition, potential slope, and thickness.

Exciton spin dynamics in spherical CdS quantum dots

Exciton spin dynamics in quasi-spherical CdS quantum dots is studied in detail experimentally and theoretically. Exciton states are calculated using the 6-band k.p Hamiltonian. It is shown that for various sets of Luttinger parameters, when the wurtzite lattice crystal field splitting and Coulomb interaction between the electron-hole pair are taken into account exactly, both the electron and hole wavefunction in the lowest exciton state are of S-type. This rules out the spatial-symmetry-induced origin of the dark exciton in CdS quantum dots. The exciton bleaching dynamics is studied using time- and polarization-resolved transient absorption technique of ultrafast laser spectroscopy. Several samples with a different mean size of CdS quantum dots in different glass matrices were investigated. This enabled the separation of effects that are typical for one particular sample from those that are general for this type of material. The experimentally determined dependence of the electron spin relaxation rate on the radius of quantum dots agrees well with that computed theoretically.

Complete series for the off-diagonal correlations of a one-dimensional interacting Bose gas

We develop a generalized harmonic-fluid approach, based on a regularization of the effective low-energy Luttinger-liquid Hamiltonian, for a one-dimensional Bose gas with repulsive contact interactions. The method enables us to compute the complete series corresponding to the large-distance, off-diagonal behavior of the one-body density matrix for any value of the Luttinger parameter K. We compare our results with the exact ones known in the Tonks-Girardeau limit of infinitely large interactions (corresponding to K=1) and, different from the usual harmonic-fluid approach, we recover the full structure of the series. The structure is conserved for arbitrary values of the interaction strength, with power laws fixed by the universal parameter K and a sequence of subleading corrections.

Accurate band structures and effective masses for InP, InAs, and InSb using hybrid functionals

The band structures of $\text{In}\text{ }X$ ($X=\text{P}$, As, and Sb) are calculated using the hybrid HSE06 functional and $GW$ with spin-orbit coupling effects included up to second order. Conventional local or semilocal density functionals predict an incorrect band ordering for InAs and InSb when spin-orbit coupling is included. We show that inclusion of one quarter of the exact exchange allows us to predict very accurate band gaps for InP, InAs, and InSb, i.e., 1.48, 0.42, and 0.28 eV, respectively, in good agreement with recent experiments. Furthermore, calculated effective masses for the conduction band (electron) and the valence band (heavy-hole, light-hole, and split-off band) are in fairly good agreement, whereas the values obtained using semilocal functionals deviate significantly from experiment even for InP. The calculated Luttinger parameters are also in reasonable agreement with experiment, although a tendency toward underestimation is observed with increasing anion mass. This underestimation is shown to be partially related to a tendency to overestimate the band gaps for the heavier elements using the particular hybrid functional used in this study. By adjusting the screening parameter in the hybrid functional, agreement with the experimental band gap can be achieved, but even then the Luttinger parameters are 15% smaller than the experimental values for InAs and InSb.

Attractively bound pairs of atoms in the Bose-Hubbard model and antiferromagnetism

We consider a periodic lattice loaded with pairs of bosonic atoms tightly bound to each other via strong attractive on-site interaction that exceeds the inter-site tunneling rate. An ensemble of such lattice-dimers is accurately described by an effective Hamiltonian of hard core bosons with strong nearest-neighbor repulsion which is equivalent to the $XXZ$ model with Ising-like anisotropy. We calculate the ground-state phase diagram for a one-dimensional system which exhibits incompressible phases, corresponding to an empty and a fully filled lattice (ferromagnetic phases) and a half-filled alternating density crystal (anti-ferromagnetic phase), separated from each other by compressible phases. In a finite lattice the compressible phases show characteristic oscillatory modulations on top of the anti-ferromagnetic density profile and in density-density correlations. We derive a kink model which provides simple quantitative explanation of these features. To describe the long-range correlations of the system we employ the Luttinger liquid theory with the relevant Luttinger parameter $K$ obtained exactly using the Bethe Ansatz solution. We calculate the density-density as well as first-order correlations and find excellent agreement with numerical results obtained with density matrix renormalization group (DMRG) methods. We also present a perturbative treatment of the system in higher dimensions.

Effects of impurities in spin Bose-metal phase on a two-leg triangular strip

We study effects of nonmagnetic impurities in a Spin Bose-Metal (SBM) phase discovered in a two-leg triangular strip spin-1/2 model with ring exchanges (D. N. Shenget al, arXiv:0902.4210). This phase is a quasi-1D descendant of a 2D spin liquid with spinon Fermi sea, and the present study aims at interpolating between the 1D and 2D cases. Different types of defects can be treated as local energy perturbations, which we find are always relevant. As a result, a nonmagnetic impurity generically cuts the system into two decoupled parts. We calculate bond energy and local spin susceptibility near the defect, both of which can be measured in experiments. The Spin Bose-Metal has dominant correlations at characteristic incommensurate wavevectors that are revealed near the defect. Thus, the bond energy shows a static texture oscillating as a function of distance from the defect and decaying as a slow power law. The local spin susceptibility also oscillates and actually {\it increases} as a function of distance from the defect, similar to the effect found in the 1D chain [S. Eggert and I. Affleck, Phys. Rev. Lett. {\bf 75}, 934 (1995)]. We calculate the corresponding power law exponents for the textures as a function of one Luttinger parameter of the SBM theory.

Spin Bose-metal phase in a spin-12model with ring exchange on a two-leg triangular strip

Recent experiments on triangular lattice organic Mott insulators have found evidence for a two-dimensional (2D) spin liquid in close proximity to the metal-insulator transition. A Gutzwiller wave function study of the triangular lattice Heisenberg model with a four-spin ring exchange term appropriate in this regime has found that the projected spinon Fermi sea state has a low variational energy. This wave function, together with a slave particle-gauge theory analysis, suggests that this putative spin liquid possesses spin correlations that are singular along surfaces in momentum space, i.e., ``Bose surfaces.'' Signatures of this state, which we will refer to as a ``spin Bose metal'' (SBM), are expected to manifest in quasi-one-dimensional (quasi-1D) ladder systems: the discrete transverse momenta cut through the 2D Bose surface leading to a distinct pattern of 1D gapless modes. Here, we search for a quasi-1D descendant of the triangular lattice SBM state by exploring the Heisenberg plus ring model on a two-leg triangular strip (zigzag chain). Using density matrix renormalization group (DMRG) supplemented by variational wave functions and a bosonization analysis, we map out the full phase diagram. In the absence of ring exchange the model is equivalent to the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg chain, and we find the expected Bethe-chain and dimerized phases. Remarkably, moderate ring exchange reveals a new gapless phase over a large swath of the phase diagram. Spin and dimer correlations possess singular wave vectors at particular ``Bose points'' (remnants of the 2D Bose surface) and allow us to identify this phase as the hoped for quasi-1D descendant of the triangular lattice SBM state. We use bosonization to derive a low-energy effective theory for the zigzag spin Bose metal and find three gapless modes and one Luttinger parameter controlling all power law correlations. Potential instabilities out of the zigzag SBM give rise to other interesting phases such as a period-3 valence bond solid or a period-4 chirality order, which we discover in the DMRG. Another interesting instability is into a spin Bose-metal phase with partial ferromagnetism (spin polarization of one spinon band), which we also find numerically using the DMRG.

Exciton–photon interaction in a quantum dot embedded in a photonic microcavity

We present a detailed analysis of exciton–photon interaction in a microcavity made out of a photonic crystal slab. Here we have analysed a disc-like quantum dot where an exciton is formed. Excitonic eigen functions in addition to their eigen energies are found through direct matrix diagonalization, while wavefunctions corresponding to unbound electron and hole are chosen as the basis set for this procedure. In order to evaluate these wavefunctions precisely, we have used the 6 × 6 Luttinger Hamiltonian in the case of hole while ignoring bands adjacent to the conduction band for electron states. After analysing excitonic states, a photonic crystal-based microcavity with a relatively high quality factor mode has been proposed and its lattice constant has been adjusted to obtain the prescribed resonant frequency. We use a finite-difference time-domain method in order to simulate our cavity with sufficient precision. Finally, we formulate the coupling constants for the exciton–photon interaction both where intra-band and inter-band transitions occur. By evaluating a sample coupling constant, it has been shown that the system can be in a strong-coupling regime and Rabi oscillations can occur.

Spin-orbit mixing in III–V semiconductors at the Γ point

The consequences of the theory of spin-orbit mixing in III–V semiconductors at the Γ point have been analyzed within the five-band model for the first time. A formula that describes the optical matrix element relating the Γ7 and Γ8 bands and expresses it through the known Luttinger parameter q is derived. Until recently, it was believed that this optical matrix element is equal to zero and that the corresponding transition is forbidden. The role of this transition in the experiment on photon absorption by free holes in the p-GaSb compound is discussed.

Cyclotron resonance of carbon-doped two-dimensional hole systems: From the magnetic quantum limit to low magnetic fields

We report on cyclotron resonance (CR) experiments of high-quality C-doped two-dimensional hole systems in GaAs/AlGaAs quantum wells covering the microwave and far-infrared regimes. At high magnetic field $B$ we observe a strong $B$ dependence of the cyclotron mass ${m}_{c}$ which can be reproduced by calculations based on a $4\ifmmode\times\else\texttimes\fi{}4$ Luttinger Hamiltonian in the axial approximation. At small $B$ one main and up to three secondary CRs are found, which continuously evolve with smooth magnetic-field dependences. This has not been observed and predicted so far. In the regime of the magnetic quantum limit a further CR is observed, which can be attributed to a transition between Landau levels originating from the two different spin-split heavy- and light-hole subbands.

Measurements on quantum capacitance of individual single walled carbon nanotubes

We report measurements of the quantum capacitance of individual semiconducting and small band gap single walled carbon nanotubes (SWNTs). The observed quantum capacitance, 82 aF/μm for a semiconducting SWNT with chiral index (16,8) and 10.3 aF/μm for a small band gap SWNT upon Fermi level lying at the first subband are remarkably smaller than those originating from the density of states. We attribute the discrepancy to a strong electron correlation in SWNTs and derive the Luttinger parameter g of 0.25–0.3 for the (16,8) SWNT and of 0.32 for a small band gap SWNT.

Wave functions and energy levels of acceptors and excitons in GaAs-type semiconductors

A spherical model for shallow acceptors or excitons in a GaAs-type semiconductor is considered. The valence band is described by the Luttinger Hamiltonian (matrix 4 × 4). The integral equations for the wavefunctions in momentum representation are numerically solved. The functions of the ground state of the acceptor are expanded in a series over the Laguerre polynomials. The dependences of the coupling energy and the wavefunctions of the acceptor on the ratio of light and heavy masses of a hole are calculated. The effective constants of deformation potentials of the acceptor in a weakly strained crystal are determined.

Corner Junction as a Probe of Helical Edge States

We propose and analyze interedge tunneling in a quantum spin Hall corner junction as a means to probe the helical nature of the edge states. We show that electron-electron interactions in the one-dimensional helical edge states result in Luttinger parameters for spin and charge that are intertwined, and thus rather different from those for a quantum wire with spin rotation invariance. Consequently, we find that the four-terminal conductance in a corner junction has a distinctive form that could be used as evidence for the helical nature of the edge states.

High-field splitting of the cyclotron resonance absorption in strainedp-InGaAs/GaAsquantum wells

We report a systematic study of the cyclotron resonance (CR) absorption of two-dimensional holes in strained InGaAs/GaAs quantum wells (QWs) in the quantum limit. The energies of the CR transitions are traced as a function of magnetic field up to 55 T. A remarkable CR line splitting was evidenced when the resonant field exceeds 20 T. We analyze our data with a $4\ifmmode\times\else\texttimes\fi{}4$ Luttinger Hamiltonian including strain and QW potentials using two different methods to calculate Luttinger parameters for ternary alloys. We found excellent agreement with the experiment when linear interpolation of the Luttinger parameters is used.

Electronic structure of Mn-doped III-V semiconductor quantum dots

The electronic structure of GaAs and InAs quantum dots (QDs) containing a single substitutional Mn impurity is investigated in the envelope-function formalism. The Mn impurity in these compounds is known to be a shallow acceptor in the configuration ${d}^{5}+h$ and characterized by a strong antiferromagnetic $sp\text{\ensuremath{-}}d$ exchange interaction between the hole $(h)$ and the Mn ion. Our model for the hole states is based on the Luttinger Hamiltonian and the Coulomb potential with a central-cell correction that accounts for the observed binding energy and the effective $g$ factor in the bulk. The binding energy as well as the exchange contribution is found to increase with decreasing QD size. However, in contrast with the case of spherical nanocrystals (NCs), the binding energy in lens-shaped self-assembled QDs in the low-confinement limit is lower than that in the bulk because of their highly anisotropic shape. With an on-center impurity, NCs retain the bulk ${T}_{d}$ symmetry and the ground state is a $j=3/2$-like ${\ensuremath{\Gamma}}_{8}({T}_{d})$ level. In self-assembled QDs it splits into two doublets: ${\ensuremath{\Gamma}}_{6}$ $(|{j}_{z}|=1/2)$ and ${\ensuremath{\Gamma}}_{7}$ $(|{j}_{z}|=3/2)$ of ${D}_{2d}$, which mix in the presence of in-plane asymmetry, both belonging to ${\ensuremath{\Gamma}}_{5}$ of the reduced symmetry ${C}_{2v}$. The order and the splitting between the doublets depend on the degree of confinement and the strain-induced separation between the light- and heavy-hole valence bands. In lattice-matched GaAs/(Ga,Al)As QDs the ground-state doublet is $|{j}_{z}|=3/2$-like in the low-confinement limit. As the lateral size decreases there is a rapid crossover to a $|{j}_{z}|=1/2$-like ground state in QDs of typical sizes. On the other hand, in strained InAs/GaAs QDs the ground state is always $|{j}_{z}|=3/2$-like and the splitting relatively large. The $sp\text{\ensuremath{-}}d$ coupling with the Mn spin $S=5/2$ finally leads to a splitting of the ground-state doublet into six doubly-degenerate levels. The components are close to one another as the effective exchange parameters are an order of magnitude smaller than in the bulk. Our results thoroughly contradict the previously adopted picture based on treating the confinement potential as a small perturbation to the bulk impurity levels. We also consider the lowest two-hole states: the ground state in InAs/GaAs QDs is a singlet almost uncoupled to the Mn spin. We deduce the zero-field fine structure of the excitonic transitions and compare the results with the recently reported photoluminescence spectra.

Motion-dependent magnetic properties of excitons in CdTe

Very large changes in the Zeeman splittings and in the diamagnetism of excitons as they acquire kinetic energy in wide quantum wells of CdTe are reported. The changes are found to be functions of the translational wave vector ${K}_{z}$ of the exciton in the growth direction of the well, irrespective of the width of the well, and are also found to be strong functions of the direction of the magnetic field. The behavior is accounted for by a model in which mixing occurs between the hydrogenic states which describe the exciton in the center of mass or adiabatic approximation. The mixing is ascribed to terms which arise in the Luttinger Hamiltonian when it is extended to describe excitons. Excellent quantitative agreement with experiment, including the results of changing the strain in the wells, is obtained by using Luttinger parameters close to those previously reported. The model is applicable to wide quantum wells made from any zinc-blende semiconductor and confirms that the huge motion induced changes in magnetic properties, observed here for CdTe and previously also for ZnSe and GaAs, should be universal for such materials.

Band structures of GaAs, InAs, and InP: A 34 k⋅p model

The band structure of direct-band gap semiconductors (GaAs, InAs, InP) is described theoretically by using a 34×34 k⋅p model. We extend the sp3d5 basis functions by the inclusion of sV∗ orbitals. We find that the sp3d5s∗ k⋅p model is sufficient to describe the electronic structure of all materials investigated over a wide energy range, obviating the use of any d valence orbitals. Finally, our results show that Luttinger parameters, the κ valence band parameter, the effective Landé factor g∗, and the effective-masses in the X and L valleys are in good agreement with available experimental data. In particular, the adjustment of the k⋅p Hamiltonian parameters proved that g∗ of GaAs, InAs, and InP are, respectively, −0.41, −15.82, and 1.35, which are in good agreement with the experimental values of −0.44, −14.90, and 1.26.