Luttinger Hamiltonian Research Articles

Hole states in boron delta-doped diamond

Hole states in B δ-doped diamond quantum wells are calculated in the Brillouin zone center within the effective mass approximation using a local density Thomas–Fermi approximation for the description of the band bending profile. The calculation assumes a two-independent (hh+lh) hole bands model, and considers some of the different sets of Luttinger parameters for diamond reported in the literature. The results for the hole energy states allow to propose an indirect way of determining the actual set of valence band parameters: the measurement of optical absorption involving transitions from hole energy states in diamond systems with δ-doping are performed.

Luttinger parametergfor metallic carbon nanotubes and related systems

The random phase approximation (RPA) theory is used to derive the Luttinger parameter g for metallic carbon nanotubes. The results are consistent with the Tomonaga-Luttinger models. All metallic carbon nanotubes, regardless if they are armchair tubes, zigzag tubes, or chiral tubes, should have the same Luttinger parameter g. However, a (10,10) carbon peapod should have a smaller g value than a (10,10) carbon nanotube. Changing the Fermi level by applying a gate voltage has only a second order effect on the g value. RPA theory is a valid approach to calculate plasmon energy in carbon nanotube systems, regardless if the ground state is a Luttinger liquid or Fermi liquid. (This paper was published in PRB 66, 193405 (2002). However, Eqs. (6), (9), and (19) were misprinted there.)

Hartree and Exchange Effects in the Calculation of Hole Levels in p-Type Delta-Doped Diamond Systems

Γ-point hole states in boron δ-doped diamond quantum wells are calculated with the use of a local density Thomas–Fermi–Dirac approximation for the potential energy function and the solution of an effective mass equation. The approach includes Hartree and exchange contributions in the hole gas. The calculation assumes a two-independent (hh + lh) hole band model, and different sets of Luttinger parameters for diamond, reported in the literature, are used. It is shown that the hole spectrum is highly sensitive to the values of the valence effective masses. The results for the hole energy states would help to clarify the actual set of valence band parameters if measurements of optical absorption involving transitions from hole energy states in diamond systems with delta doping are performed.

Haldane-gapped spin chains as Luttinger liquids: Correlation functions at finite field

We study the behavior of Heisenberg, antiferromagnetic, integer-spin chains in the presence of a magnetic field exceeding the attendant spin gap. For temperatures much smaller than the gap, the spin chains exhibit Luttinger liquid behavior. We compute exactly both the corresponding Luttinger parameter and the Fermi velocity as a function of magnetic field. This enables the computation of a number of correlators from which we derive the spin conductance, the expected form of the dynamic structure factor relevant to inelastic neutron scattering experiments, and NMR relaxation rates. We also comment upon the robustness of the magnetically induced gapless phase both to finite temperature and finite couplings between neighbouring chains.

Strain-dependence of the gain saturations in InGaAsP/InP quantum-well gain media

Numerical analyses of polarization-dependent optical gain saturations are given for quantum-well (QW) lasers in the presence of strain in the well regions in order to investigate the strain dependence of polarization-bistable operations. Gain saturation coefficients are obtained from nonlinear susceptibilities calculated in the perturbative analyses of density matrices. Band dispersions and dipole matrix elements, which are put into the density matrices, are calculated by diagonalizing Luttinger's Hamiltonian, including valence band mixing. The strain induces a change in band dispersions and wavefunctions, leading to strain-dependent saturation coefficients. The self-saturation coefficients and the cross-saturation coefficients (with orthogonal optical polarizations) pertinent to InGaAsP/InP QW vertical-cavity surface-emitting lasers are calculated. We find that the relative magnitudes of self- and cross-saturation coefficients are strongly dependent on the strain; in the presence of compressive strain, the cross-saturation coefficients are larger than the self-saturation in the wide range of the linear gain spectra, especially in the vicinity of the gain peak, indicating that the compressively strained structure is more favorable for the polarization-bistable operations.

Polarization Dependence of the Optical Interband Transition Defined by the Spatial Variation of the Valence p-Orbital Bloch Functions in Quantum Wires

We develope a method to express wave functions of hole states in semiconductor quantum wire (QWR) structures based on spatial variation of the valence p-orbital Bloch functions, to show how envelope wave functions relate to polarization-dependent interband transition. A wave function of a hole state is obtained solving the Schrödinger equation based on the 4×4 Luttinger Hamiltonian, and then recomposed by means of six bases of three p-orbital Bloch functions with two spin components. As a result, the hole wave function is expressed by six envelope wave functions for the six bases. Then, interband optical transition matrix elements with x-, y-, and z-polarizations are separately given by overlap integrals between envelope wave functions of holes for px, py, and pz orbitals and those of electrons. We also calculate the wave functions for a modeled ridge QWR structure with mirror symmetry as well as for an asymmetric structure, and discuss the polarization dependence of the optical transition.

First-principles calculations of the effective mass parameters of Al xGa1-xN and Zn xCd1-xTe alloys

First-principles calculations of electronic band structures of the ordered cubic alloys Al xGa1-xN and Cd xZn1-xTe are carried out. The band structures are used to provide e ective masses and Luttinger parameters which are useful in the parametrization of theories based on e ective hamiltonians.

Spin-Flip Effect in Narrow-Gap Semiconductor Quantum Wells

The 8 × 8 k · p Kane–Weiler model has been extended to study narrow-gap semiconductor quantum wells in the presence of a magnetic field. It is shown that the study of the inherent symmetry in this Hamiltonian model permits a suitable separation of the solution into two orthogonal Hilbert sub-spaces. The electronic properties are evaluated as function of Kane–Luttinger parameters, magnetic field strength, and Landau level number. The selection rules and anomalies in the conduction Zeeman splitting observed in the interband magneto-optical absorption spectrum are analyzed for the Faraday and Voigt geometry configurations. Due to strong intra- and interband admixtures in narrow-gap materials, “spin-flip-like” transitions are predicted to occur in the magneto-optical spectrum of HgCdTe/CdTe quantum well solid solutions.

Band structure ofCdGeAs2near the fundamental gap

First-principles band structure calculations were carried out for the chalcopyrite semiconductor ${\mathrm{CdGeAs}}_{2}$ using the linear muffin-tin orbital method, including spin-orbit coupling. The emphasis of the analysis is on the band gaps and energy band splittings near the fundamental gap. The gap underestimate due to the local-density approximation is corrected using information on quasiparticle calculations for the parent compound GaAs. The experimental information on optical transitions near the gap is reviewed critically in the light of our calculations. The polarization dependence and the pseudodirect nature of some of the transitions is discussed. The effective masses of the conduction band and valence bands are derived from the calculated band structure. A generalization of the Luttinger Hamiltonian for chalcopyrite is presented and its parameters determined.

Optical polarization anisotropy of quantum wells induced by a cubic anisotropy of the host material

The optical polarization anisotropy of quantum wells (QW) in structures fabricated from materials with cubic symmetry due to the anisotropic character of the Luttinger Hamiltonian for the valence band is studied. The matrix elements of interband optical transitions are shown to be different for different orientations of the linear polarization of light, oriented in the plane of a structure, provided that the structure growth axis does not coincide with any of the main crystallographic directions, i.e., with either [1 0 0],[0 1 0],[0 0 1] or [1 1 1] . The degree of this in-plane anisotropy of the polarization is calculated theoretically within an appropriate model for arbitrary structure growth orientation. The polarization anisotropy is experimentally detected for the fundamental 1 hh–1e transition by studying the reflectivity from [1 2 0] -oriented Cd 0.8Mn 0.2Te/CdTe/Cd 0.8Mn 0.2Te QWs. At the same time the effect is shown to be absent in the case of QW grown along [1 0 0] direction. The measured polarization anisotropy in the former case shows a predicted π-periodicity as the polarization direction is rotated within the plane of the QW. The magnitude of the effect is also in qualitative agreement with the model. Quantitatively, the measured values tend to be some what larger than those predicted by the model. Surprisingly, the polarization anisotropy of the reflection by the barrier excitons is also detected. For the [1 2 0] -oriented structures this anisotropy has, approximately, the same magnitude as the one of the exciton transitions in the QWs. In the structure grown along [1 0 0] , the anisotropy of the reflectivity of the barrier exciton transitions is also observed, although it is much smaller in magnitude. Some hypothetical explanations of these observations are put forward.

NOVEL APPROACH TO FANO RESONANCE OF EXCITONS IN SEMICONDUCTOR QUANTUM WELLS

A new approach to Fano resonance of excitons in quantum-confined systems is presented based on the 4×4 Luttinger Hamiltonian. This method, consisting of two stages, namely, the adiabatic expansion and the R-matrix propagation, allows to implement high-resolution calculations of Fano profiles having natural spectral widths without any empirical broadening parameter which is indispensable in conventional methods. As a demonstration, this approach is applied to Fano resonance of an exciton in a wide quantum well, where hole-subband mixing is substantial and a complicated energy-structure is expected. Resulting spectral profiles show rich fine structures due to overlap with adjacent resonances and the hole-subband mixing.

Overlap structure of Fano-resonance profiles of excitons in a semiconductor quantum well

The Fano resonance of an exciton in a quantum well of ${\mathrm{G}\mathrm{a}\mathrm{A}\mathrm{s}/\mathrm{A}\mathrm{l}}_{0.3}{\mathrm{Ga}}_{0.7}\mathrm{As}$ belonging to the ${\ensuremath{\Gamma}}_{7}^{+}$-irreducible representation is investigated based on the excitonic $4\ifmmode\times\else\texttimes\fi{}4$ Luttinger Hamiltonian. Multichannel scattering problems for the resonance states are solved by virtue of the adiabatic expansion and the R-matrix propagation method, which enable us to implement high-resolution calculations without introducing any empirical broadening parameters. Absorption spectra for excitons are calculated in 100--$500\ensuremath{-}\AA{}$-thick quantum wells, and detailed Fano-resonance profiles for both optically active and inactive exciton states are revealed. Specifically, it is noticed that complicated interference between Fano resonances pertaining to different subbands occurs in a quantum well of more than $350 \AA{}.$ A resulting composite profile manifests itself as overlap resonance, accompanying marked changes in a peak height, a width, and an asymmetry pattern from the corresponding isolated profiles. Such changes are evaluated qualitatively by use of Fano's model for one open-channel and two closed-channels. Moreover, quantitative interpretations are also made by employing a time delay given by an eigenphase sum.

Band parameters for III–V compound semiconductors and their alloys

We present a comprehensive, up-to-date compilation of band parameters for the technologically important III–V zinc blende and wurtzite compound semiconductors: GaAs, GaSb, GaP, GaN, AlAs, AlSb, AlP, AlN, InAs, InSb, InP, and InN, along with their ternary and quaternary alloys. Based on a review of the existing literature, complete and consistent parameter sets are given for all materials. Emphasizing the quantities required for band structure calculations, we tabulate the direct and indirect energy gaps, spin-orbit, and crystal-field splittings, alloy bowing parameters, effective masses for electrons, heavy, light, and split-off holes, Luttinger parameters, interband momentum matrix elements, and deformation potentials, including temperature and alloy-composition dependences where available. Heterostructure band offsets are also given, on an absolute scale that allows any material to be aligned relative to any other.

Structural, electronic, and effective-mass properties of silicon and zinc-blende group-III nitride semiconductor compounds

The electronic band structures of silicon and the zinc-blende-type III-N semiconductor compounds BN, AlN, GaN, and InN are calculated by using the self-consistent full potential linear augmented plane wave method within the local-density functional approximation. Lattice constant, bulk modulus, and cohesive energy are obtained from full relativistic total-energy calculations for Si and for the nitrides. Band structures and total density of states (DOS) are presented. The role played by relativistic effects on the bulk band structures and DOS is discussed. In order to provide important band structure-derived properties, such as effective masses and Luttinger parameters, the ab initio band structure results are linked with effective-mass theory. Electron, heavy-, light-, and split-off-hole effective masses, as well as spin-orbit splitting energies are extracted from the band-structure calculations. By using the Luttinger-Kohn $6\ifmmode\times\else\texttimes\fi{}6$ effective-mass Hamiltonian we derive the corresponding Luttinger parameters for the materials. A comparison with other available theoretical results and experimental data is made.

Scaling exponents in the incommensurate phase of the sine-Gordon and U(1) Thirring models

In this paper we study the critical exponents of the quantum sine-Gordon and U(1) Thirring models in the incommensurate phase. This phase appears when the chemical potential $h$ exceeds a critical value and is characterized by a finite density of solitons. The low-energy sector of this phase is critical and is described by the Gaussian model (Tomonaga-Luttinger liquid) with the compactification radius dependent on the soliton density and the sine-Gordon model coupling constant $\beta$. For a fixed value of $\beta$, we find that the Luttinger parameter $K$ is equal to 1/2 at the commensurate-incommensurate transition point and approaches the asymptotic value $\beta^2/8\pi$ away from it. We describe a possible phase diagram of the model consisting of an array of weakly coupled chains. The possible phases are Fermi liquid, Spin Density Wave, Spin-Peierls and Wigner crystal.

Commensurate-incommensurate transition in two-coupled chains of nearly half-filled electrons

We investigate the physical properties of two coupled chains of electrons, with a nearly half-filled band, as a function of the interchain hopping t_\perp and the doping. We show that upon doping, the system undergoes a metal-insulator transition well described by a commensurate-incommensurate transition. By using bosonization and renormalization we determine the full phase diagram of the system, and the physical quantities such as the charge gap. In the commensurate phase two different regions, for which the interchain hopping is relevant and irrelevant exist, leading to a confinement-deconfinement crossover in this phase. A minimum of the charge gap is observed for values of t_\perp close to this crossover. At large t_\perp the region of the commensurate phase is enhanced, compared to a single chain. At the metal-insulator transition the Luttinger parameter takes the universal value K_\rho^*=1, in agreement with previous results on special limits of this model.

Evaluation of an Equivalent Hole Effective Mass for Si/SiGe Structures

We have studied the carrier profile and calculated a unique effective mass for a two-dimensional hole gas confined in a SiO2/Si/Si1_xGex/Si structure when an external voltage is applied. For this purpose, we have solved the system of differential equations provided by the 6 x 6 Luttinger Hamiltonian [1] simultaneously with the Poisson equation, applying an iterative process until convergence was achieved. This enabled us to incorporate the warping and the strong coupling among the subbands that constitute the band structure [2].From the calculated density of carriers we were able to evaluate an average effective mass. Although our calculations indicate that this effective mass varies notably with the hole confinement level, in general, we observed a decrease in the calculated effective mass as the molar fraction of Ge increases, which might justify the use of this kind of device.

Measuring Fractional Charge in Carbon Nanotubes

The Luttinger model of the one-dimensional Fermi gas is the cornerstone of modern understanding of interacting electrons in one dimension. In fact, the enormous class of systems whose universal behavior is adiabatically connected to it are now deemed Luttinger liquids. Recently, it has been shown that metallic single-walled carbon nanotubes are almost perfectly described by the Luttinger Hamiltonian. Indeed, strongly non-Fermi liquid behavior has been observed in a variety of DC transport experiments, in very good agreement with theoretical predictions. Here, we describe how fractional quasiparticle charge, a fundamental property of Luttinger liquids, can be observed in impurity-induced shot noise.

Valence-band structure of undoped and p-doped cubic GaN/InGaN multiple quantum wells

Valence band structure calculations of cubic GaN/In x Ga 1− x N quantum wells and superlattices are presented, in which coupling between the heavy-, light-, and spin–orbit-split-hole bands, strain effects due to lattice mismatch, and the difference in the parameters due to the materials forming the heterostructures are considered. The calculations are performed within a self-consistent approach to the k· p theory by means of a full six-band Luttinger–Kohn (LK) Hamiltonian. The effects of different materials are taken into account explicitly through the inclusion of different sets of Luttinger parameters and by considering the corresponding additional LK matrix elements, not present in the one-material based heterostructures. Our results show that strain effects and the inclusion of the split-off band play a fundamental role in a realistic description of the valence band structure in these systems. Due to the large difference in hole effective masses, for p-doped systems we find that only heavy-hole bands are occupied in the well.

Magnetospectroscopy of acceptors in “blue” diamonds

Naturally occurring, nitrogen-free, p-type diamond—now known to be boron-doped—as well as man-made diamonds deliberately doped with boron display an electronic Raman transition, Δ′, originating in the 1 s( p 3/2) : Γ 8 ground state of the acceptor and terminating in its 1 s( p 1/2) : Γ 7 spin–orbit partner. With magnetic field B along [0 0 1] , [1 1 1] , or [1 1 0] , the electronic Raman spectrum displays eight Zeeman transitions and four Raman lines ascribed to transitions between the Zeeman sublevels of Γ 8 (Raman-electron-paramagnetic-resonance: Raman-EPR). They exhibit polarizations expected from the polarizability tensors formulated in terms of the Luttinger parameters γ 1, γ 2 , and γ 3 characterizing the top of the valence band. The selection rules and the relative intensities of the Zeeman components as well as of the Raman-EPR lines, observed in diverse polarization configurations and scattering geometries, have led to: assignments of magnetic quantum numbers; the level ordering of the Zeeman sublevels, or equivalently, the magnitudes and signs of g 1 and g 2, the orbital and spin g-factors of the acceptor-bound hole; the extreme mass anisotropy as reflected in the ratio ( γ 2/ γ 3)=0.08±0.01. Magnetic-field-induced mixing of zero field states, time reversal symmetry, and the diamagnetic contributions which characterize the different sublevels are fully taken into account in the interpretation of the experimental results. These include the striking mutual exclusion of the Stokes spectrum from its anti-Stokes counterpart in specific polarization configurations.