Landau diamagnetism and determination of the longitudinal and transverse kinetic energy components of a degenerate nonrelativistic electron gas

The Fermi energy, average total kinetic energy density, and also average kinetic energy of finite diamagnetic motion of an electron gas of specified concentration are calculated. The kinetic energy of continuous longitudinal motion of the electrons along the direction of an external magnetic field H is determined. It is found that in the quantum limit, when the maximum Landau quantum number N _m =0, the kinetic energy of continuous longitudinal motion tends to zero with increase in the external magnetic field strength. If the maximum Landau quantum number is greater than zero, the longitudinal and transverse kinetic energy components of the degenerate electron gas change only insignificantly. Byelorussian State University; Brest State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 31–35, July, 1999.     

Landau Diamagnetism,Pauli Paramagnetism,and Determination of Longitudinal and Transverse Kinetic Energy Components of a Degenerate Nonrelativistic Electron Gas

The Fermi energy, density of average kinetic energy, and average density of kinetic energy of the transverse finite motion of an electron gas of a specified concentration are calculated taking into account Landau diamagnetism and Pauli paramagnetism. The kinetic energy of a longitudinal continuous electron motion along the direction of the external magnetic field H is estimated. It is shown that the kinetic energy of the longitudinal continuous motion vanishes with increase in the external magnetic field strength in the quantum limit where the maximum Landau quantum number N _m = 0. For N _m > 0, the longitudinal kinetic energy component of a degenerate electron gas somewhat increases with magnetic field strength. The cause of the erroneous result is discussed.     

A Mathematical Representation of Quantizing the Motion of Relativistic Electrons in a Magnetic Field

It is shown that the quadratic component of the kinetic energy of continuous longitudinal motion of relativistic electrons in the external magnetic field is varied continuously between 0 and 2(2m _e c ²_B H) within each Landau energy level, undergoing an abrupt change at the boundaries of the levels. This results in the fact that in the quantum limit of a superstrong magnetic field where all electrons are at the zero Landau level, the maximum quadratic component of the kinetic energy of free longitudinal electron motion along the direction of the magnetic field is twice as high as the maximum quadratic component of the kinetic energy of its bound transverse motion.     

A Mathematical Representation of Quantizing the Motion of Electrons in a Magnetic Field

It is shown that the concept of a continuous variation in the kinetic energy of a free longitudinal electron motion along the direction of an external magnetic field results in a physically incorrect conclusion on a zero phase volume in the system where the zero Landau lever is realized. Since the kinetic energy of a longitudinal continuous motion is varied in the same way between 0 and 2_B H within each Landau energy level, this problem is circumvented.     

Effects of rotation on Landau states of electrons on a spherical shell

Based on a previously observed analogy between electromagnetic and non-inertial effects, we investigate the competition between magnetic field and rotation in the quantum motion of an electron constrained to the surface of a sphere. We solve numerically the Schrödinger equation of the problem for the energy eigenvalues and the eigenfunctions and compare the effects of the magnetic field and rotation. We obtain that, for a weak magnetic field, an electron can not distinguish between magnetic field and rotation, since they lead to equivalent behavior. But this is no longer true for strong magnetic fields. However, surprisingly, even though the rotation and magnetic fields play different roles in the electronic properties of the system, when together, each influence of the magnetic field on the energy levels can be separately balanced by rotation. We also show that no matter the intensity of the magnetic field, it is always possible to destroy the Landau levels in the sphere by rotation.     

Two-dimensional electron gas in a strong magnetic field. II

The quantum mechanical problem on the motion of electrons in a strong (quantizing the motion) magnetic field is considered, taking into account the Coulomb interaction between charges. It is shown how, on the basis of the solutions found, it is possible to obtain the fractional filling factor of the Landau levels that enters into the expression for the Hall conductivity of a two-dimensional electron gas.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 3, pp. 25–29, March, 1990.     

A Similar Change in the Kinetic-Energy Components of Longitudinal Continuous Motion of Magnetized Electrons within the Landau Levels and Determination of the Characteristics of a Degenerate System

Account is taken of a similar change in the kinetic-energy components of a longitudinal electron motion along the magnetic field within each Landau energy level. The maximum electron energy is found as well as the kinetic energy of a transverse electron motion in the magnetic field. The electron concentrations are determined depending on the magnetic-field strength, whose intrinsic magnetic moments are directed along and opposite the field. The magnetic-field strengths for which a certain number of Landau quantum levels is realized in an electron degenerate system of a specified concentration are also found.__________Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 3, pp. 18–22, March, 2005.     

Quantum-mechanical substantiation of a similar change in the kinetic energy component of an electron along the magnetic field within each Landau level

A detailed solution to the Schr?dinger equation for an electron in the presence of a quantizing magnetic field is considered using the method of separation of variables. The discrete energy levels occurring in this case are regarded as the potential barriers of finite height, with an electron moving normal to them along the magnetic-field direction. A similar change in the difference between the energy values within the neighboring Landau levels allows us to show, basing on the general properties of one-dimensional electron motion in the magnetic field, that within each quantum energy level, the kinetic-energy component of a longitudinal electron motion continuously is varied between 0 and 2μ_BH, whereas the probability density of particle location is practically damped into the potential-barrier depth as it moves along the magnetic-field direction. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 3–7, October, 2006.     

Microscopic calculation of the magnetic field of neutron stars

Based on the Dirac equation describing an electron moving in a uniform and cylindrically symmetric magnetic field which may be the result of the self-consistent mean field of the electrons themselves in a neutron star, we have obtained the eigen solutions and the orbital magnetic moments of electrons in which each eigen orbital can be calculated. From the eigen energy spectrum we find that the lowest energy level is the highly degenerate orbitals with the quantum numbers pZ=0, n=0, and m≥0. At the ground state, the electrons fill the lowest eigen states to form many Landau magnetic cells and each cell is a circular disk with the radius λfree and the thickness λe, where λfree is the electron mean free path determined by Coulomb cross section and electron density and λe is the electron Compton wavelength. The magnetic moment of each cell and the number of cells in the neutron star are calculated, from which the total magnetic moment and magnetic field of the neutron star can be calculated. The results are compared with the observational data and the agreement is reasonable.     

The physics of helical electron beam in a uniform magnetic field as a testing ground of gauge principle

According to Bliokh et al., allowing free propagation along the direction of a uniform magnetic field, the familiar Landau electron state can be regarded as a non-diffracting version of the helical electron beam propagating along the magnetic field. Based on this observation, they argued that, while propagating along the magnetic field, the Landau electrons receive characteristic rotation with three different angular velocities, depending on the eigen-value m of the canonical OAM operator, which is generally gauge-variant, and this splitting was in fact experimentally confirmed. Through complete analyses of highly mysterious m-dependent rotational dynamics of the quantum Landau states, we try to make clear how and why their observation does not contradict the widely-believed gauge principle.     

Principles of Adaptive Optics, 3rd edn., by Robert K. Tyson

A review is made of the quantum effects which are observed in the transport coefficients of semiconductors. Quantization of the free carriers in semiconductors is produced whenever an external potential acts on an otherwise uniform and perfect crystal. Typical examples are a magnetic field, an electric field or the physical boundaries of the sample. A magnetic field quantizes the electron and hole states into a ladder of equally spaced Landau levels. This gives rise to the Shubnikov–de Haas, magnetophonon and magneto-impurity effects, where the positions of the Landau levels resonate with the Fermi, phonon, or impurity energies present in the sample. A series of oscillations in the magneto-resistance of many different types of materials results. Electric fields applied to the surface of metal oxide semiconductor (MOS) devices result in a set of quantum levels for motion perpendicular to the surface. At low temperatures the charge carriers are bound to the surface and behave as if they were two-dimensional. This is shown to give rise to very dramatic oscillatory metal–insulator behaviour in high magnetic fields. Quantization is also shown to occur in very thin layers of semiconductors which act like a simple square well potential, the energy levels of which can be studied as a function of layer thickness. The carriers are confined within the layers, and also show two-dimensional behaviour.     

Electromagnetic radiation from positive-energy bound electrons in the Coulomb field of a nucleus at rest in a strong uniform magnetic field

A classical analysis is presented of the electromagnetic radiation emitted by positive-energy electrons performing bound motion in the Coulomb field of a nucleus at rest in a strong uniform magnetic field. Bounded trajectories exist and span a wide range of velocity directions near the nucleus (compared to free trajectories with similar energies) when the electron Larmor radius is smaller than the distance at which the electron-nucleus Coulomb interaction energy is equal to the mechanical energy of an electron. The required conditions occur in magnetic white dwarf photospheres and have been achieved in experiments on production of antihydrogen. Under these conditions, the radiant power per unit volume emitted by positive-energy bound electrons is much higher than the analogous characteristic of bremsstrahlung (in particular, in thermal equilibrium) at frequencies that are below the electron cyclotron frequency but higher than the inverse transit time through the interaction region in a close collision in the absence of a magnetic field. The quantum energy discreteness of positive-energy bound states restricts the radiation from an ensemble of bound electrons (e.g., in thermal equilibrium) to nonoverlapping spectral lines, while continuum radiative transfer is dominated by linearly polarized bremsstrahlung.     

Landau diamagnetism and determination of the longitudinal and transverse components of the average kinetic energy of a nonrelativistic, nondegenerate electron gas

With allowance for Landau diamagnetism for a nondegenerate, nonrelativistic electron gas, the average components of the kinetic energies of longitudinal free motion of electrons and of their finite motion in the plane perpendicular to the direction of the external magnetic field are determined. It is shown that with increasingH the component of the average kinetic energy of longitudinal free motion of electrons decreases to zero, while the component of the average kinetic energy of finite transverse motion of the electrons increases linearly. Brest State University, Brest. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, Vol. 42, No. 2, pp. 50–55, February, 1999.     

点缺陷扶手型石墨烯量子点的电子性质研究

基于有限差分方法, 数值求解了Dirac方程, 研究了垂直磁场下的点缺陷扶手型 石墨烯 量子点的能谱结构, 分析了尺寸大小对带隙的影响. 与无磁场时具有一定带隙 (带隙的大小与半径成反比) 的量子点相比, 在外加有限磁场下, 能谱中出现朗道能级, 最低朗道能级能量为零并与磁场强度无关, 并且朗道能级的简并度随着磁场的增加而增加. 进一步的计算表明, 最低朗道能级的简并度与磁场成线性关系, 与半径的平方成线性关系. 本文工作对基于石墨烯量子点的器件设计具有一定的指导意义.     

New polaron effect in magnetooptic phenomena in a quantum well

It is predicted that resonance coupling between two discrete electron energy levels corresponding to different size-quantization quantum numbers and different Landau quantum numbers can occur in a quantum well in a quantizing magnetic field. The resonance coupling is due to the interaction of an electron with LO phonons and results in the formation of polaron states of a new type. It is shown that for a certain value of the magnetic field, which depends on the splitting of the electron size-quantization levels, the absorption peak and the two-phonon resonance Raman scattering peak split into two components, the separation between which is determined by the electron-phonon coupling constant. The resonance coupling between size-quantization levels with the same Landau quantum numbers is also studied. The splitting of the peaks in this case is virtually independent of the magnetic field and can be observed in much weaker fields. The experimental observation of the effect will make it possible to determine the relative position of the electronic levels and the electron-phonon coupling constant. Pis’ma Zh. éksp. Teor. Fiz. 65, No. 7, 511–515 (10 April 1997)     

System of quantum wells in a parallel magnetic field

The energy spectrum and quantum states of electrons in a system of quantum wells in a strong magnetic field parallel to the heterogeneous boundaries are studied. The combined effect of the quantizing magnetic field and the potential of the system of quantum wells leads to a radical change in the electron dispersion relation owing to the appearance of one-dimensional Landau bands. The neighborhoods of the anticrossing points of the different bands correspond to an effective redistribution of the electron envelope functions, which becomes stronger as the magnetic field is raised. The character of the electron-state density in the size-quantization subbands is examined qualitatively in connection with the change in the system of isoenergy contours when a magnetic field is applied. Fiz. Tverd. Tela (St. Petersburg) 40, 1719–1723 (September 1998)     

Influence of the intrinsic magnetic moment of an electron on the filling of energy levels of a nonrelativistic degenerate electron gas in a quantizing magnetic field

The effect of a quantizing magnetic field on the filling of energy subbands of a degenerate nonrelativistic electron gas, whose magnetic moments are parallel and antiparallel to the direction of the field, is studied. Calculations are made under the assumption that the Fermi kinetic energy of the electron system increases as a result of Pauli's paramagnetism as compared to the kinetic energy calculated without taking this effect into account. Numerical values of the electron concentration are calculated as a function of the magnetic moment direction for magnetic fields under which the electrons are in states with given numbers of the Landau quantum level.A. S. Pushkin State Education Institute, Brest. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 6, pp. 9–13, June, 1992.     

Photodetachment of H- near an elastic surface in a magnetic field

This paper investigates the photodetachment of the negative hydrogen ion H near an elastic wall in a magnetic field.The magnetic field confines the perpendicular motion of the electron,which results in a real three-dimensional well for the detached electron.The analytical formulas for the cross section of the photodetachment in the threedimensional quantum well are derived based on both the quantum approach and closed-orbit theory.The magnetic field and the elastic surface lead to two completely different modulations to the cross section of the photodetachment.The oscillation amplitude depends on the strength of the magnetic field,the ion-wall distance and the photon polarization as well.Specially,for the circularly polarized photon-induced photodetachment,the cross sections display a suppressed(E E th) 1/2 threshold law with energy E in the vicinity above Landau energy E th,contrasting with the(E E th) 1/2 threshold law in the presence of only the magnetic field.The semiclassical calculation fits the quantum result quite well,although there are still small deviations.The difference is attributed to the failure of semiclassical mechanics.     

Free electron laser study of the suppression of non-radiative scattering processes in semiconductors

A brief review is given of pump–probe studies of far infrared inter-sub-level relaxation between conduction band states in doped `quasi' quantum dots (created by the application of a magnetic field along the growth direction of an InAs/AlSb quantum well) and of mid-infrared (MIR) interband recombination in narrow gap semiconductors, using the free electron laser at FOM-Rijnhuizen (FELIX). In the former case, the longitudinal optic (LO) phonon scattering rate is shown to be suppressed by a factor of about 100 when the Landau level separation is off-resonance with the optical phonon energy; in the latter case, Auger recombination is shown to be substantially suppressed in the lead salts due to their `mirror' energy band structure.     

Impurity states of electrons in quantum dots in external magnetic fields

The influence of isolated impurity atoms on the electron energy spectrum in a parabolic quantum dot in quantizing magnetic field is studied. The impurity potential is approximated by a Gaussian separable operator which allows one to obtain the exact solution of the problem. We demonstrate that in the electron energy spectrum there is a set of local levels which are split from the Landau zone boundaries in the upward or downward direction depending on the impurity type. We have calculated the local level positions, the wave functions of electrons in bound states, and the residues of the electron scattering amplitudes by impurity atoms at the poles.