Hole Hamiltonian Research Articles

Inter- and intraband Coulomb interactions between holes in silicon nanostructures

We present a full derivation of the interaction Hamiltonian for holes in silicon within the six-band envelope-function scheme, which appropriately describes the valence band close to the $\boldsymbol{\Gamma}$ point. The full structure of the single-hole eigenstates is taken into account, including the Bloch part. The scattering processes caused by the Coulomb interaction are shown to be both intraband and interband, the latter being mostly short-ranged. In the asymptotic long-range limit, the effective potential tends to the screened Coulomb potential, and becomes purely intraband, as assumed in previous models. We apply our model to compute the excitation spectra of two interacting holes in prototypical silicon quantum dots, taking into account different dielectric environments. It is shown that, in the highly screened regime, short-range interactions (both intra- and inter-band) can be very relevant, while they lose importance when there is no screening other than the one proper of the bulk silicon crystal. In the latter case, we predict the formation of hole Wigner molecules.

Strain engineering the charged-impurity-limited carrier mobility in phosphorene

We investigate, based on the tight-binding model and in the linear deformation regime, the strain dependence of the electronic band structure of phosphorene, exposed to a uniaxial strain in one of its principle directions, the normal, the armchair and the zigzag directions. We show that the electronic band structure of strained phosphorene, for the experimentally accessible carrier densities and the uniaxial strains, is well described by a strain-dependent decoupled electron–hole Hamiltonian. Then, employing the decoupled Hamiltonian, we consider the strain dependence of the charged-impurity-limited carrier mobility in phosphorene, for both types of carriers, arbitrary carrier densities and in both armchair and zigzag directions. We show that a uniaxial tensile (compressive) strain in the normal direction enhances (weakens) the anisotropy of the carrier mobility, while a uniaxial strain in the zigzag direction acts inversely. Moreover applying a uniaxial strain in the armchair direction is shown to be ineffective on the anisotropy of the carrier mobility. These will be explained based on the effects of the strains on the carrier effective masses.

Magnetic field implementation in multiband k⋅p Hamiltonians of holes in semiconductor heterostructures

We propose an implementation of external homogeneous magnetic fields in k⋅p Hamiltonians for holes in heterostructures, in which we make use of the minimal coupling prior to introducing the envelope function approximation. Illustrative calculations for holes in InGaAs quantum dot molecules show that the proposed Hamiltonian outperforms the standard Luttinger model (Luttinger 1956 Phys. Rev. 102 1030) describing the experimentally observed magnetic response. The present implementation culminates our previous proposal (Planelles et al 2010 Phys. Rev. B 82 155307).

Photogenerated Metallic States in Charge-Ordered Insulators in (BEDT-TTF)2X

We theoretically investigate the physical properties of the photoexcited states in the charge-ordered states in α-(BEDT-TTF) 2 I 3 and θ-(BEDT-TTF) 2 RbZn(SCN) 4 . We adopt the 1/4-filled extended Hubbard Hamiltonian for holes on the two-dimensional anisotropic triangular lattice. The photoexcited states are numerically calculated by solving the time-dependent Schrodinger equation subject to pump light. In the lower-energy region of pump light photons, an extremely small number of energy eigenstates dominate the transition moment from the ground state. These energy eigenstates are collective excited states that have the charge order, and they are insulating as a result of the charge order. In the higher-energy region, the density of energy eigenstates that contribute to the light absorption spectrum, is much larger than that in the lower-energy region, and these energy eigenstates are metallic and have no charge order. The present results show that the metallic states are directly generated in the early s...

Binary spinning black hole Hamiltonian in canonical center-of-mass and rest-frame coordinates through higher post-Newtonian order

The recently constructed Hamiltonians for spinless binary black holes through third post-Newtonian order and for spinning ones through formal second post-Newtonian order, where the spins are counted of zero post-Newtonian order, are transformed into fully canonical center-of-mass and rest-frame variables. The mixture terms in the Hamiltonians between center-of-mass and rest-frame variables are in accordance with the relation between the total linear momentum and the center-of-mass velocity as demanded by global Lorentz invariance. The various generating functions for the center-of-mass and rest-frame canonical variables are explicitly given in terms of the single-particle canonical variables. The no-interaction theorem does not apply because the world-line condition of Lorentz covariant position variables is not imposed.

Valence-band of cubic semiconductors: Clifford algebra approach II

Application of Clifford algebra in the analysis of valence-band spin properties in semiconductors is considered. In the first part (Dargys A 2009 Phys. Scr.80 065701), for this purpose the isomorphism between multivectors and their matrix representations was used to transform the problem to Clifford algebra. Here equivalence rules are established between the spinors of Hilbert space and basis elements of the five-dimensional Clifford algebra Cl4, 1. Then, the rules are applied to the total angular momentum components and the two-band hole Hamiltonian. The resulting biquaternionic Schrödinger equation for hole spin is solved as an example.

Confined electrons and holes in Si nanocrystals: Theoretical modeling of the energy spectrum and radiative transitions

We construct the theory of carriers confined in spherical Si quantum dots with finite energy barriers for electrons and holes in the framework of Luttinger Hamiltonian for holes and taking into account the strong anisotropy of the conduction electron effective mass in Si. As a boundary condition for the electron and hole wave functions we use continuity of the wave functions and the current at the boundary of the nanocrystals. We apply this theory for the case of the SiO2 matrix surrounding Si quantum dots. We show that for experimentally relevant quantum dots energy spacings between neighbouring electron and hole levels are of the order of hundreds of meV. Therefore the relaxation of excited electrons and holes is damped.Theoretical calculations of probabilities of various radiative transitions are presented.

Quantum-confined Stark effect in interdiffused quantum dots

The authors theoretically investigate the effects of isotropic composition interdiffusion on the optical transition energy of quantum-dot (QD) structures under an electric field. Their three-dimensional QD calculation is based on coupled InAs∕GaAs QDs arranged periodically in a tetragonal superlattice, taking into account the effects of finite band offset, valence-band mixing, strain, effective mass anisotropicity, and different QD shapes. The electron and hole Hamiltonians with the interdiffusion effect are solved in the momentum space domain. The enhanced Stark shifts and the reduced built-in dipole moment have been found for the interdiffused QD structures.

Electronic structure and optical properties of quantum rods with wurtzite structure

The Hamiltonian of the wurtzite quantum rods with an ellipsoidal boundary is given after a coordinate transformation. The energies, wave functions, and transition possibilities are obtained as functions of the aspect ratio e with the same method we used on spherical dots. With an overall consideration of both the transition matrix element and the Boltzmann distribution we explained why the polarization factor increases with increasing e and approaches a saturation value, which tallies quite well with the experimental result. When e increases more and more ${S}_{z}$ states are mixed into the ground, second, and third states of ${J}_{z}=1/2,$ resulting in an increase of the emission of z polarization. It is just the linear terms of the momentum operator in the hole Hamiltonian that cause the mixing of S and P states in the hole ground state. The effects of the crystal field splitting energy, temperature, and transverse radius to the polarization are also considered. We also calculated the band gap variation with the size and shape of the quantum rods.

Systematic corrections to the equivalent core model

The widely used equivalent core model (ECM) describes core hole states in systems with atomic charge Z by considering corresponding states with fully occupied core in systems with increased charge Z+1. When calculating energies of Z-core hole states, the valence energy of these states often has been assumed to equal the valence energy of the (Z+1) ground state. This approach misses several points: most importantly, the different spin symmetry of the corresponding states. The behavior of core hole states is governed by an effective 2×2 matrix Hamiltonian due to the two possible spin states of the core hole. A recently introduced diagonalization gives rise to a scalar core hole Hamiltonian. Both the ECM and the core hole Hamiltonian act in valence space. This allows establishment of a connection between these two approaches. By expressing the core hole Hamiltonian in the (Z+1) orbital basis, we systematically derive corrections to the ECM. Those corrections, including the one arising because of the different spin symmetry of the corresponding states, are presented in second order of Møller–Plesset perturbation theory (MP2). Hence, they can be implemented very easily so that ground-state calculations in a (Z+1) system may directly provide the core hole state energy in the original Z system.

Relationship between Feshbach's and Green's Function Theories of the Nucleon–Nucleus Mean Field

We clarify the relationship and difference between theories of the optical-model potential which had previously been developed in the framework of Feshbach's projection operator approach to nuclear reactions and of Green's function theory, respectively. For definiteness, we consider the nucleon–nucleus system but all results can readily be adapted to the atomic case. The effects of antisymmetrization are properly taken into account. It is shown that one can develop along closely parallel lines the theories of “hole” and “particle” mean fields. The hole one-body Hamiltonians describe the single-particle properties of the system formed when one nucleon is taken away from the target ground state, for instance in knockout or pickup processes. The particle one-body Hamiltonians are associated with the system formed when one nucleon is elastically scattered from the ground state, or is added to it by means of stripping reactions. An infinite number of particle, as well as of hole, Hamiltonians are constructed which all yield exactly the same single-particle wave functions. Many “equivalent” one-body Hamiltonians can coexist because these operators have a complicated structure: they are nonlocal, complex, and energy-dependent. They do not have the same analytic properties in the complex energy plane. Their real and imaginary parts fulfill dispersion relations which may be different. It is shown that hole and particle Hamiltonians can also be constructed by decomposing any vector of the Hilbert space into two parts which are not orthogonal to one another, in contrast to Feshbach's original theory; one interest of this procedure is that the construction and properties of the corresponding hole Hamiltonian can be justified in a mathematically rigorous way. We exhibit the relationship between the hole and particle Hamiltonians and the “mass operator.” The latter is associated to the time-ordered Green's function, rather than to its advanced and retarded parts separately as the hole and particle Hamiltonians. Similarities and differences between the hole and particle Hamiltonians and the mass operator are exhibited by constructing their explicit expressions in the case of nuclear matter, in the framework of second-order perturbation theory. Particular attention is paid to the connection of the mass operator and the various hole and particle Hamiltonians with observables which can be extracted from stripping, pickup and knockout reactions, in particular the spectroscopic factors and the spectral function. Since many different one-body Hamiltonians exist which all yield the same single-particle wave functions, their relative merits and drawbacks need to be discussed, with particular attention to their relationship to empirical shell- and optical-model potentials and to the possibility of developing practical approximation schemes.

Electronic Structure and Phonon-Assisted Luminescence in Self-Assembled Quantum Dots

We present photoluminescence (PL) measurements on an ensemble of InAs/GaAs self-assembled quantum dots embedded in GaAs. We observe a transition from an inhomogeneously broadened photoluminescence band under non-resonant excitation into up to five phonon-assisted bands under selective excitation. We interpret the phonon-assisted PL as being indicative of an enhanced electron–phonon interaction. We also perform theoretical calculations of the single-particle energy spectrum of self-assembled quantum dots, in the framework of the single-band effective-mass approximation for electrons and using the Luttinger Hamiltonian for holes. Finally, by taking advantage of the computed wave functions we evaluate the Huang-Rhys parameter: We find an enhancement of the electron–phonon interaction that partially accounts for the experimental results.

Variational tight-binding theory of excitons in compositionally modified semiconductor superlattices

We present results for the binding energy of an exciton formed when an electron–hole pair is photoexcited within a single, compositionally modified layer of a semiconductor superlattice, for example by adding a small percentage of In atoms to a single GaAs layer of a GaAs/AlGaAs system. Such a system could serve as the basis for spatially-selective photoexcitation, a process whereby a laser pulse would create electron–heavy-hole pairs exclusively in the modified layer. We first derive an effective one-dimensional (1D) Hamiltonian for an electron, by averaging the 3D electron–hole Hamiltonian using a one-parameter trial wavefunction, which is dependent on the in-plane relative coordinates, as well as a normalized Wannier orbital for a single hole. The exciton binding energy is then obtained by computing the lowest bound-state energy of the effective 1D electron Hamiltonian in the nearest-neighbor tight-binding approximation. As a demonstration of the effectiveness of our approach, we find that for periodic superlattices our results for the exciton binding energy are in very good agreement both with experiment and the results of other theoretical calculations.

Exact-diagonalization study of electron-lattice coupling in the effective two-bandt−Jmodel

The effective two-band $t\ensuremath{-}J$ Hamiltonian for holes in Cu-O planes of cuprates is obtained from the three-band Hubbard Hamiltonian. In this model a doped oxygen hole together with two neighboring copper holes is treated as a composite spin-$\frac{1}{2}$ three-sites particle. The model is studied by exact diagonalization in a ${\mathrm{Cu}}_{16}{\mathrm{O}}_{32}$ square cluster. Both binding energy and hole-hole correlation function indicate that two holes bind together for $J/t>0.11$ and are predominantly located on the next-nearest oxygen sites in the same Cu-O chain. Electron-phonon interaction is studied in the adiabatic phonons approximation. Copper (\ensuremath{\pi},0) ``half-breathing'' mode is shown to couple strongly to doped holes and increase the hole-hole correlation as well as the binding energy. These results are compared with experimental data.

Effective Hamiltonians for holes in antiferromagnets: a new approach to implement forbidden double occupancy

A coherent state representation for the electrons of ordered antiferromagnets is used to derive effective Hamiltonians for the dynamics of holes in such systems. By an appropriate choice of these states, the constraint of forbidden double occupancy can be implemented rigorously. Using these coherent states, one arrives at a path integral representation of the partition function of the systems, from which the effective Hamiltonians can be read off. We apply this method to the t-J model on the square lattice and on the triangular lattice. In the former case, we reproduce the well-known fermion-boson Hamiltonian for a hole in a collinear antiferromagnet. We demonstrate that our method also works for non-collinear antiferromagnets by calculating the spectrum of a hole in the triangular antiferromagnet in the self-consistent Born approximation and by comparing it with numerically exact results.

Asymptotics of the Interband Light Absorption Coefficient near the Band Edge for an Alloy-Type Model

We find the asymptotics of the interband light absorption coefficient of an alloy-type model in the case when the ground-state energies of the electron and the hole Hamiltonians are finite.

Relationship between Feshbach′s and Green′s Function Theories of the Nucleon-Nucleus Mean Field

We clarify the relationship and difference between theories of the optical-model potential which had previously been developed in the framework of Feshbach′s projection operator approach to nuclear reactions and of Green′s function theory, respectively. For definiteness, we consider the nucleon-nucleus system but all results can readily be adapted to the atomic case. The effects of antisymmetrization are properly taken into account. It is shown that one can develop along closely parallel lines the theories of "hole" and "particle" mean fields. The "hole" one-body Hamiltonians describe the single-particle properties of the system formed when one nucleon is taken away from the target ground state, for instance in knockout or pickup processes. The particle one-body Hamiltonians are associated with the system formed when one nucleon is elastically scattered from the ground state, or is added to it by means of stripping reactions. An infinite number of particle, as well as of hole, Hamiltonians are constructed which all yield exactly the same single-particle wave functions. Many "equivalent" one-body Hamiltonians can coexist because these operators have a complicated structure: they are nonlocal, complex, and energy-dependent. They do not have the same analytic properties in the complex energy plane. Their real and imaginary parts fulfill dispersion relations which may be different. It is shown that hole and particle Hamiltonians can also be constructed by decomposing any vector of the Hilbert space into two parts which are not orthogonal to one another, in contrast to Feshbach′s original theory; one interest of this procedure is that the construction and properties of the corresponding hole Hamiltonian can be justified in a mathematically rigorous way. We exhibit the relationship between the hole and particle Hamiltonians and the "mass operator." The latter is associated to the time-ordered Green′s function, rather than to its advanced and retarded parts separately as the hole and particle Hamiltonians. Similarities and differences between the hole and particle Hamiltonians and the mass operator are exhibited by constructing their explicit expressions in the case of nuclear matter, in the framework of second-order perturbation theory. Particular attention is paid to the connection of the mass operator and the various hole and particle Hamiltonians with observables which can be extracted from stripping, pickup and knockout reactions, in particular the spectroscopic factors and the spectral function. Since many different one-body Hamiltonians exist which all yield the same single-particle wave functions, their relative merits and drawbacks need to be discussed, with particular attention to their relationship to empirical shell- and optical-model potentials and to the possibility of developing practical approximation schemes.

Consistent low-energy reduction of the three-band model for electrons and holes in copper oxides to the effective t-J model

A full three-band model for the CuO 2 plane with inclusion of all essential interactions - Cu-O and O-O hopping, repulsion at the copper and oxygen and between them - is considered. A general procedure of the low-energy reduction of the primary Hamiltonian to the Hamiltonian for holes and electrons is developed. For the purpose of fixing of the t- J model parameters we calculate the values of the superexchange constant J and the charge-transfer gap E gap in the framework of the three-band model. Fitting values of J and E gap to the experimental data allows to narrow the uncertainty region of the three-band model parameters. As a result the ratio ( t J ) of the t- J model is fixed in the range 2.4÷ 2.7 for holes and 2.5 ÷ 3.0 for electrons.

One-dimensional spin model of phase separation

A one-dimensional spin model having a fraction of strong ferromagnetic bonds on the antiferromagnetic background is considered, in which the former are connected with mobile holes. Spin variables can be eliminated, leading to an exact effective hole Hamiltonian. It is shown that the nature of the Hamiltonian-hole attractive or hole repulsive-depends on the model of the bond strength. For some bond strength models phase separation arises even in one dimension.

Hole pairing and dependence of transition temperature on carrier concentration in high- Tc superconductors

An expression for the superconducting gap function for high- T c superconductors is derived using a Hubbard-like tight binding Hamiltonian for holes and following the Hubbard treatment of electron correlations. The dependence of T c on the hole concentration n is studied. The results are in excellent agreement with experiments. Our results support the hole pairing mechanism as a source of superconductivity in high- T c superconductors as well as in transition metal superconductors.