Energy States In Solids

The electronic states and direct interband absorption of light in quantum dot molecule (QDM) consisting 4 or 5 QDs are investigated. The electron energy levels and wave functions are calculated. It is shown that the energy levels are split in azimuth direction. In the case of the presence of the central QD in QDM an additional splitting in the radial direction is revealed. The suggested method can be used to calculate the electronic states and light absorption of arbitrary number QDM. The results are useful for the application of the QDM to photoelectric devices.

Previous quantum-transport theories relevant to the calculation of multiple-scattering contributions to the electron mobility of a doped semiconductor are re-examined. The Green's-function evaluations of the Kubo formula are extended to include contributions of second order in the impurity concentration in addition to the first-order terms included previously. The simple rigorous results obtained are used to justify and simplify the Kohn-Luttinger transport equations. Apparent difficulties with the impurity averaging are clarified, and the second-order Kohn-Luttinger equation is reduced to a Boltzmann form. For an isotropic solid a simple closed formula for the mobility is derived, which contains such quantum corrections to the Brooks-Herring formula as coherent scattering from pairs of impurities, higher Born approximations for the single scattering, and dressing effects of the impurities on the electron energy levels and wave functions. It is noted that a variational principle of the Kohler type exists, which should be of use in estimating multiple-scattering contributions in anisotropic solids.

Resonant coherent excitation of hydrogen-like ions planar channeled in a crystal; Transition into the first excited state

The theory of direct excitons in crystals of diamond and sphalerite type is generalized to the indirect exciton case. The problem is solved by the variational method taking into account the anisotropy and the nonparabolicity of the electron band. The comparison with the theoretical and experimental data known from literature is carried out.

Semiconductor devices often require band-gap engineering, which in turn requires the use of alloys and the tuning of their composition to achieve the required electrical or optical behavior. While their compositional fluctuations are generally treated as a perturbation in most conventional III-V semiconductors, the effects of alloy disorder are much more significant in the III-nitrides. Here, the effects of alloy disorder on carrier localization are compared for different III-V semiconductors, particularly for holes for which localization effects are more significant due to their heavier effective mass. This study is conducted using three-dimensional computation for III-V alloys with natural (random) compositional fluctuations. Given the complexity of the problem, we carry out the computations relying on a simplified Hamiltonian in the envelope wave-function approximation with a single heavy-hole valence band. We investigate the effects of compositional fluctuations on carrier localization using two methods. The first method, based on the localization landscape theory, is used to solve the localization landscape equations and obtain the effective potentials acting on carriers. This potential acts as a confining potential that predicts the regions of spatial localization of carriers, and thus allows a comparison of the effects of alloy fluctuations between the conventional III-V and the III-nitride semiconductors. We find that the effective potential of the III-nitride semiconductors exhibits much larger fluctuations compared to the other III-V semiconductors. This might point to a higher degree of carrier localization in the nitrides, particularly for holes. This is verified through the second method, solving Schr\"odinger's equation and obtaining the electron and hole wave functions. We find that for ${\mathrm{In}}_{x}{\mathrm{Ga}}_{1\ensuremath{-}x}\mathrm{N}$ the electron wave functions are delocalized even for the ground state, whereas the low-energy hole states are localized. This is in contrast with the behavior of holes in the common alloy ${\mathrm{In}}_{x}{\mathrm{Ga}}_{1\ensuremath{-}x}\mathrm{As}$, which are found to be always delocalized. Thus, our study shows the importance of accounting for alloy disorder in the nitrides.

The electronic structures of N quantum dot molecules (QDMs) are investigated theoretically in the framework of effective-mass envelope function theory. The electron and hole energy levels are calculated. In the calculations, the effects of finite offset and valence-band mixing are taken into account. The theoretical method can be used to calculate the electronic structures of any QDM. The results show that (1) electronic energy levels decrease monotonically and the energy difference between the N QDMs decreases as the quantum dot (QD) radius increases; (2) the electron energy level is lower and quantum confinement is smaller for the larger N QDM; (3) the hole ground state energy level is lower for the one dot QDM than N (greater 1) QDMs if the QD radius is larger than about 5nm due to the valence-band mixing. The results are useful for the application of the N QDM to photoelectric devices.

The existence of enormous strain fields in self-assembled quantum dots has led to the expectation of dramatic effects of piezoelectricity. However, only linear piezoelectric tensors were used in all previous calculations. We calculate the piezoelectric properties of self-assembled quantum dots using the linear and quadratic piezoelectric tensors derived from first-principles density functional theory. We find that the previously ignored quadratic term has similar magnitude as the linear term and the two terms tend to cancel each other. We show the effect of piezoelectricity on electron and hole energy levels and wave functions as well as on correlated absorption spectra.

Diffuse clusters of arbitrary numbers of helium atoms are treated in a mean-field theory of the many-body system. The nonlinear Hartree self-consistent-field equations are solved numerically to give their energy eigenvalues and eigenfunctions. Both isotopes 4He and 3He are treated according to their bosonic and fermionic statistics and symmetry requirements. Clusters of all sizes of 4He atoms are stable, but there is a minimum threshold of ∼70 atoms for the formation of 3He clusters. All the excited states in addition to the ground state were found. The wave functions are seen to have interesting differences from electronic wave functions in atoms.

We discuss applications of photogenerated strain in transient absorption experiments. These include measurements of sound velocity, phonon dispersion relations, and excited electronic state energy levels. In particular, we have identified a new type of spectroscopy in photoinduced absorption that can be used to detect both allowed and forbidden electronic transitions in thin semiconducting films. This technique was applied to a host of conducting polymers for which we detected energy levels of various excitonic states with odd and even symmetry.

Supersymmetry quantum mechanics method had been used to solve the time independent cosmic string equation for trigonometric non-central potential system. Variable separation method was applied to the time independent cosmic string equation, and three one-dimensional Schrodinger equation had been obtained. Radial wave functions, angular wave functions, azimuthal wave functions, and energy levels were analyzed using supersymmetry quantum mechanics method. The effect of the trigonometric non-central potential caused energy levels to increase. Energy levels for ≠ λ0 ≠ κ,0 ≠ ϖ,0, a ≠0, and b ≠0 had a higher value than energy levels for = λO = κ,0 = ϖ,0, a =0, and b = 0. The increased value of parameter λ, parameter κ, parameter ϖ, parameter a , parameter b , and the decreased of the cosmic string parameter a, caused an increase in energy levels.Supersymmetry quantum mechanics method had been used to solve the time independent cosmic string equation for trigonometric non-central potential system. Variable separation method was applied to the time independent cosmic string equation, and three one-dimensional Schrodinger equation had been obtained. Radial wave functions, angular wave functions, azimuthal wave functions, and energy levels were analyzed using supersymmetry quantum mechanics method. The effect of the trigonometric non-central potential caused energy levels to increase. Energy levels for ≠ λ0 ≠ κ,0 ≠ ϖ,0, a ≠0, and b ≠0 had a higher value than energy levels for = λO = κ,0 = ϖ,0, a =0, and b = 0. The increased value of parameter λ, parameter κ, parameter ϖ, parameter a , parameter b , and the decreased of the cosmic string parameter a, caused an increase in energy levels.

The envelope function approximation was generalized to calculate the electron subbands and wave functions of quasiperiodic semiconductor superlattices (QSS) within rational approximation, for nonzero K⊥ (the wave vector perpendicular to the axis of the QSS). For K⊥= 0, the electron subbands and wave functions are calculated for InAs/ GaSb QSS up to the generation number m =9 and for GaAs/AlxGa1-xAs QSS up to m = 6. For InAs/GaSb QSS, the influence of valence band states on conduction band state is strong. The dependences of the energies and band widths of electron subbands on K⊥ are obtained for m = 5 and 6. A theoretical method is proposed to calculate the intersubband collective excitations of the type-I QSS by means of the above results.

The engineering of the architecture of the quantum wires has shown a real challenge in the scientific community owing to their fascinating and auspicious application potential in the field of optoelectronics. The modulation of the morphology and structure of the quantum wires may give rise to the modulation of the energy levels and band offset positions to enhance the charge carriers transfer through any electronic device and improve the overall performance for the future application in the field of spintronics and photonics. Here, we proposed, for the first time, a novel rectangular architecture based AlxGa1-xAs and GaxIn1-xAs quantum wires to engineer the electron energy spectrum according to a wide range of applications in electronics and optical devices. The electron energy levels in rectangular AlxGa1-xAs and GaxIn1-xAs quantum wires with infinite potential barrier were calculated at different x values and different cross-section areas to explore the role of dopant and compared with the cylindrical shape. The calculations of the electron confinement energy in the first and second energy levels indicate that the energy value in cylindrical quantum wire is less than its value in rectangular one while for E3 the energy value in cylindrical quantum wire is larger than its value in rectangular one for all values of x. The confinement energy was found to be inversely proportional to the ratio of the doped material. The electron energy dispersion in AlxGa1-xAs and GaxIn1-xAs quantum wires of 100 nm2 cross-section area, x = 0.4 for E1, E2 and E3 with the wave vector value has been investigated. The calculations of the first and second energy levels indicated that the energy value in cylindrical quantum wire is less than its value in rectangular one for E1 and E2 while for E3 the energy value in cylindrical quantum wire is larger than its value in rectangular with a distinct value for each wave vector value for all x values. These unique features of the proposed novel architecture may open a new avenue for the future applications in photonics, spintronics and waveguides.

We have developed a variational formalism for calculations of the exciton binding energies in the GaAs-AlAs type-II quantum-well structures in the presence of a magnetic field applied along the growth axis. In contrast to some previous calculations, we take into account the finiteness of the potential barriers at the GaAs-AlAs interfaces and the mismatch of the effective masses of electron and hole across such potential barriers. The exciton wave function is expressed in terms of electron and hole subband wave functions and a multiterm Gaussian-type-orbital expansion, with variationally determined coefficients. The finite overlap of the electron and hole wave functions is found to increase exciton binding energies by up to 10--15 %, depending on thickness of the GaAs and AlAs layers. The application of the magnetic field compresses the excitonic wave functions and is found to enhance their binding energies.

An effective single-band Hamiltonian representing a crystal electron in a uniform magnetic field is constructed from the tight-binding form of a Bloch band by replacing $\ensuremath{\hbar}\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}$ by the operator $\stackrel{\ensuremath{\rightarrow}}{\mathrm{p}}\ensuremath{-}\frac{e\stackrel{\ensuremath{\rightarrow}}{A}}{c}$. The resultant Schr\"odinger equation becomes a finite-difference equation whose eigenvalues can be computed by a matrix method. The magnetic flux which passes through a lattice cell, divided by a flux quantum, yields a dimensionless parameter whose rationality or irrationality highly influences the nature of the computed spectrum. The graph of the spectrum over a wide range of "rational" fields is plotted. A recursive structure is discovered in the graph, which enables a number of theorems to be proven, bearing particularly on the question of continuity. The recursive structure is not unlike that predicted by Azbel', using a continued fraction for the dimensionless parameter. An iterative algorithm for deriving the clustering pattern of the magnetic subbands is given, which follows from the recursive structure. From this algorithm, the nature of the spectrum at an "irrational" field can be deduced; it is seen to be an uncountable but measure-zero set of points (a Cantor set). Despite these-features, it is shown that the graph is continuous as the magnetic field varies. It is also shown how a spectrum with simplified properties can be derived from the rigorously derived spectrum, by introducing a spread in the field values. This spectrum satisfies all the intuitively desirable properties of a spectrum. The spectrum here presented is shown to agree with that predicted by A. Rauh in a completely different model for crystal electrons in a magnetic field. A new type of magnetic "superlattice" is introduced, constructed so that its unit cell intercepts precisely one quantum of flux. It is shown that this cell represents the periodicity of solutions of the difference equation. It is also shown how this superlattice allows the determination of the wave function at nonlattice sites. Evidence is offered that the wave functions belonging to irrational fields are everywhere defined and are continuous in this model, whereas those belonging to rational fields are only defined on a discrete set of points. A method for investigating these predictions experimentally is sketched.

It is well known that in relativistic quantum theory there is not exact analytical formula for cross section of a charged particle elastic scattering on arbitrary electrostatic potential \(\varphi (\mathbf {r})\). Even in case of Coulomb field for which the exact solution of Dirac equation is known, relativistic “Coulomb wave functions”, nevertheless, because of its complex character it is impossible to obtain exact analytical expression for scattering cross section via these wave functions. The elastic scattering in relativistic domain is mainly described in approximations when the scattering potential can be considered as a perturbation (in opposite limits). These are the well-known Born and eikonal approximations corresponding to quantum perturbation theory by particle wave function (when the condition \(\left| U\right| \ll \hbar \mathrm {v}/a\) is satisfied; U is the potential energy, a is the space size of the range of effective scattering, \(\mathrm {v}\) is the particle initial velocity), and high-momentum approximation (if potential energy in the scattering field is much less than the particle initial energy: \(\left| U\right| \ll p\mathrm {v}\)), respectively. In case of Coulomb field there is also an approximation of large impact parameters (large momenta)—Farry-Sommerfeld-Maue (FSM) approximation , which describes well enough the scattering at the small angles. It is also known that the wave function in the eikonal approximation describes the particle state only in a limited space range of the scattering process (\(z\ll pa^{2}/\hbar \); z is the coordinate along the direction of particle initial momentum), i.e., the eikonal solution is not valid at large distances. The wave function of the Born approximation , in contrast to the eikonal one, describes the particle state at arbitrary point in the scattering range, particularly at asymptotically large distances. Nevertheless, the common region where both approximations under consideration are valid is very restricted. On the other hand, within the (small) potential range where both approximations are applicable, these wave functions describe the scattering by different accuracies. Practically, these wave functions describe the scattering at the opposite conditions: the eikonal wave function describes the particle state when the opposite condition of the Born approximation holds (e.g., for Coulomb field with a charge \(Z_{a}e\)—at the condition: \(Z_{a}e^{2}/\hbar \mathrm {v}\gtrsim 1\)). In these circumstances a natural question arises—is it possible to find out such an approximate solution of Dirac equation beyond the scope both perturbation theory and eikonal approximation—corresponding to more general wave function being applicable in both quantum and quasiclassical limits for relativistic potential scattering (including in particular the Born and eikonal approximations in corresponding limits—under its conditions of applicability)? Here we will try to derive such an approximate solution of the Dirac equation which satisfies the formulated requests. We will call it generalized eikonal approximation (GEA) for a spinor particle scattering on the arbitrary short-range or long-range electrostatic potential. Then we will clear up the relationship of this approximation to the known ones for electron elastic scattering on the short-range (Born and eikonal approximations) and long-range Coulomb potentials (Born, eikonal, and Farry-Sommerfeld-Maue approximations ). This GEA approximation is developed for electron inelastic scattering process too—to describe the relativistic stimulated bremsstrahlung in the field of strong and superstrong laser radiation with electrostatic potential fields of arbitrary form and finite or infinite effective radiuses (atoms, ions, etc.). The significance of GEA type wave function for description of such laser-assisted electron–atom–ion scattering processes, apart from the known restrictions of mentioned above approximations, is also conditioned with the fact that the wave functions in known approximations describe the particle state factorized by elastic scattering and induced radiation-absorption processes. In the result, we lose the phase relations at the description of electron quantum dynamics interacting with the both fields simultaneously, which have important role for induced process, in particular, for coherent part of interaction. Therefore, for description of strong and superstrong laser–matter (plasma) interaction processes, we need the electron wave function that takes into account the simultaneous influence of both scattering and radiation fields including dynamic phase relations. This induced GEA wave function may be applied specifically for the description of the above-threshold ionization (ATI) process of atoms/ions by superstrong laser fields in relativistic theory taking into account the photoelectron rescattering effect on the atomic remainder because of the action of long-range Coulomb field of atomic ion on the photoelectron final state (this process is considered in the next chapter of this book).