arXiv subjects

Wlodek Zawadzki

Publications and source records attributed to Wlodek Zawadzki.

At least 19 recordsLinked to original sources

arXiv2021

The Pauli Exclusion Operator: example of Hooke's atom

The Pauli Exclusion Operator (PEO) which ensures proper symmetry of the eigenstates of multi-electron systems with respect to exchange of each pair of electrons is introduced. Once PEO is added to the Hamiltonian, no additional constraints on multi-electron wave function due to the Pauli exclusion principle are needed. For two-electron states in two dimensions ($2D$) the PEO can be expressed in a closed form in terms of momentum operators, while in the position representation PEO is a non-local operator. Generalizations of PEO for multi-electron systems is introduced. Several approximations to PEO are discussed. Examples of analytical and numerical calculations of PEO are given for isotropic and anisotropic Hooke's atom in~$2D$. Application of approximate and kernel forms of PEO for calculations of energies and states in~$2D$ Hooke's atom are analyzed. Relation of PEO to standard variational calculations with the use of Slater determinant is discussed.

quant-ph↗

arXiv2020

Exact Green's function approach to RKKY interactions

The Green's function~(GF) of two localized magnetic moments embedded in the electron gas is calculated exactly. The electrons are treated in the effective mass approximation and the magnetic moments are coupled with electrons by a delta-like~$s-d$ interaction. The resulting GF is obtained by the exact summation of the Born series using a generalization of the method developed by Slater-Koster and Ziman to non-commuting spin operators with the use of the Woodbury identities. The exact GF crucially depends on the value of the one-electron GF at the origin, denoted as~$g_0$. The Born series is convergent only if~$g_0$ is finite, which holds for electrons in parabolic energy bands in~$1D$, but not in~$2D$ and~$3D$. In the general case, the exact GF includes nonlinear combination of localized spins operators. A method of calculating matrix elements of these operators is given. For spins~$S_a=S_b = 1/2$ the exact GF is expressed as a linear combination of components of~$\hS_a, \hS_b$, and the exact range function~${\cal J}(r)$ is obtained as a double integral over analytical expression. For electron energy~$E=0$ and~$J g_0/2 = 2$ or~$J g_0/2=-2/3$ the range function and GF are singular. Poles of GF occur in the vicinities of singularity points and the resulting energies of bound states are calculated. There are three regimes of~$J$. For small $J$ the range function resembles RKKY one: it has the same period~$π/k_F$, the same decay character and a slightly different amplitude, usually within a few percent. This regime occurs for most frequently in the nature. For~$|J|$ comparable to~$|g_0|^{-1}$ the exact range function differs qualitatively from RKKY one,. For large $|J|$ the exact range function oscillates with the same period and power-like decay as the usual RKKY function but it has much lower amplitude decaying with growing~$|J|$.

cond-mat.str-el↗

arXiv2019

The Klein paradox in a magnetic field: effects of electron spin

Reflection and transmission of electrons scattered by a rectangular potential step in the presence of an external magnetic field parallel to the electron beam is described with the use of the Dirac equation. It is shown that in addition to the known effects present in the so called Klein paradox, the presence of magnetic field gives rise to electron components with reversed spin in the reflected and transmitted beams. The spin-flip scattering processes are caused by the spin-orbit interaction activated by electric field of the potential step and transverse momentum components of electron motion induced by the magnetic field. The contemporary understanding of the Klein paradox, consisting in the finite transmission even when the potential height tends to infinity, is generalized to the presence of magnetic field and spin-reversed electron beams. The spin-reversed beams are shown to occur also for electrons moving above the step. It is proposed that, accounting for the anomalous value of the electron spin $g$-factor related to radiation corrections, the reflection and transmission scattering from the potential step can be used as an electron spin filter.

quant-ph↗

arXiv2019

Wave and uncertainty properties of electrons in crystalline solids

Relations between particle and wave properties for charge carriers in periodic potentials of crystalline metals and semiconductors are derived. The particle aspects of electrons and holes in periodic potentials are considered using properties of quasimomentum (QM), while the wave aspects are described employing wave packets of Bloch waves. The two aspects are combined in the derivation of QM-wavelength relations for energy bands of arbitrary nonparabolicity and nonsphericity. An effective mass relating electron QM to its average velocity for spherical energy bands is defined and used to calculate energy dependences of wavelengths for electrons in narrow gap semiconductors, graphene and surface states of topological insulators. An uncertainty relation between electron quasimomentum and its spatial coordinate in periodic potentials is derived. It is emphasized that the described properties apply to the average (not instantaneous) electron behavior. Analogies between the wave and uncertainty properties of electrons in crystalline solids and those in vacuum are traced.

cond-mat.mes-hall↗

arXiv2018

Theory of Friedel oscillations in monolayer graphene and group-VI dichalcogenides in a magnetic field

Friedel oscillations (FO) of electron density caused by a delta-like neutral impurity in two-dimensional (2D) systems in a magnetic field are calculated. Three 2D cases are considered: free electron gas, monolayer graphene and group-VI dichalcogenides. An exact form of the renormalized Green's function is used in the calculations, as obtained by a summation of the infinite Dyson series and regularization procedure. Final results are valid for large ranges of potential strengths $V_0$, electron densities $n_e$, magnetic fields $B$ and distances from the impurity $r$. Realistic models for the impurities are used. The first FO of induced density in WS$_2$ are described by the relation $Δn(\vec{r}) \propto \sin(2πr/T_{FO})/r^2$, where $T_{FO} \propto 1/\sqrt{E_F}$. For weak impurity potentials, the amplitudes of FO are proportional to $V_0$. For attractive potentials and high fields the total electron density remains positive for all $r$. On the other hand, for low fields, repulsive potentials and small $r$, the total electron density may become negative, so that many-body effects should be taken into account.

cond-mat.mes-hall↗

arXiv2017

On calculation of RKKY range function in one dimension

The effect of strong singularity in the calculation of range function for the RKKY interaction in 1D electron gas is discussed. The method of handling this singularity is presented. A possible way of avoiding the singularity in the Ruderman-Kittel perturbation theory in 1D is described.

cond-mat.mes-hall↗

arXiv2017

Semirelativity in Semiconductors: a Review

Analogy between behavior of electrons in narrow-gap semiconductors (NGS) and relativistic electrons is reviewed. Energy bands in NGS correspond to special relativity, the latter is analogous to two-band k.p description for NGS. Maximum electron velocity in NGS is u=(okolo)1x 10^8 cm/s corresponding to the light velocity. An effective mass of electrons in semiconductors is introduced relating their velocity to quasimomentum. This mass depends on energy similarly to the mass of relativistic electrons. In Hg_(1-x)Cd_(x)Te alloys one can reach vanishing energy gap at which electrons and light holes are 3D massless Dirac fermions. Wavelength lam_z is defined for NGS, in analogy to the Compton wavelength, lam_z is around tens of Angstroms in semiconducting materials, in agreement with tunneling experiments. Interband electron tunneling in NGS is in close analogy to tunneling between negative and positive energies of the Dirac equation. Relativistic analogy holds for orbital and spin properties of electrons in an external magnetic field. The spin magnetic moment of both NGS electrons and relativistic electrons approaches zero with increasing energy. Electrons in crossed electric and magnetic fields are described. It is the semirelativistic two-band description that gives a correct account of experiments in this situation. A transverse Doppler shift is observed in crossed fields indicating that there exists a time dilatation between an electron and an observer. Phenomenon of Zitterbewegung (ZB, trembling motion) for semiconductor electrons follows an analogy to free relativistic electrons. Graphene, carbon nanotubes, topological insulators illustrate extreme semirelativistic regime. Approximations and restrictions of the relativistic analogy are emphasized. It is often easier to observe semirelativistic effects in semiconductors than relativistic effects in vacuum.

cond-mat.mtrl-sci↗

arXiv2016

Green's functions of electrons in group-VI dichalcogenides in a magnetic field

Closed expression for the Green's function of the stationary two-dimensional Schrodinger equation for an electron in group-VI dichalcogenides in the presence of a magnetic field is obtained in terms of the Whittaker functions. The resulting Green's function operator is a~$8 \times 8$ matrix consisting of four block-diagonal~$2\times 2$ matrices, each of them characterized by different values of valley index and electron spin. The obtained results are used to calculate local density of states induced by a neutral delta-like impurity in the presence of a magnetic field within the lowest Born approximation. The spatial electron density experiences either the Friedel-like oscillations or an exponential decay as a function of the distance from the impurity.

cond-mat.mes-hall↗

arXiv2016

Conduction electrons in acceptor-doped GaAs/GaAlAs heterostructures: a review

We review magneto-optical and magneto-transport effects in GaAs/GaAlAs heterostructures doped in GaAlAs barriers with donors, providing two-dimensional electron gas in GaAs quantum wells, and additionally doped with smaller amounts of acceptors (mostly Be atoms) in the vicinity of 2DEG. One may also deal with residual acceptors (mostly C atoms). The behavior of such systems in the presence of a magnetic field differs appreciably from those doped in the vicinity of 2DEG with donors.

cond-mat.mtrl-sci↗

arXiv2014

Zitterbewegung of electrons in carbon nanotubes created by laser pulses

We describe a possibility of creating non-stationary electron wave packets in zigzag carbon nanotubes (CNT) illuminated by short laser pulses. After disappearance of the pulse the packet experiences the trembling motion (Zitterbewegung, ZB). The band structure of CNT is calculated using the tight-binding approximation generalized for the presence of radiation. Employing realistic pulse and CNT parameters we obtain the ZB oscillations with inter-band frequencies corresponding to specific pairs of energy bands. A choice of optimal parameters is presented in order to observe the phenomenon of ZB experimentally. The use of Gaussian wave packets to trigger the electron Zitterbewegung, as used in the literature, is critically reexamined.

cond-mat.mes-hall↗

arXiv2014

Two-photon echo method for observing electron Zitterbewegung in carbon nanotubes

The phenomenon of Zitterbewegung (ZB, trembling motion) of electrons is described in zigzag carbon nanotubes (CNT) excited by laser pulses. The tight binding approach is used for the band structure of CNT and the effect of light is introduced by the vector potential. Contrary to the common theoretical practice, no a priori assumptions are made concerning electron wave packet, the latter is determined as a result of illumination. In order to overcome the problem of various electron phases in ZB, a method of two-photon echo is considered and described using the density function formalism. The medium polarization of CNT is calculated by computing exact solutions of the time-dependent electron Hamiltonian. The signal of two-photon echo is extracted and it is shown that, using existing parameters of CNT and laser pulses, one should be able to observe the electron trembling motion. Effects of electron decoherence and relaxation are discussed.

cond-mat.mes-hall↗

arXiv2014

Oblique spin-flip reflection of relativistic electrons from a potential barrier

The Dirac equation is used to describe oblique spin-conserving and spin-flip reflections of relativistic electrons from a one-dimensional potential barrier in a vacuum. When an electron hits the barrier from an oblique direction, its spin-up and spin-down states have different energies due to the spin-orbit interaction and the fact that the system has a structural inversion asymmetry. When the electron is reflected in an elastic spin-conserving process, incoming and outgoing angles are the same. If an elastic reflection is a spin-flip process, the outgoing angle is different from the incoming one. As a result, a beam of incoming spin-polarized electrons is split after the reflection into two beams. Angles and intensities of the reflected beams are calculated. Spin splitting of energies for relativistic electrons due to the spin-orbit interaction in an asymmetric quantum well is described.

quant-ph↗

arXiv2014

Spin-flip reflection of electrons from a potential barrier in heterostructures

Spin-conserving and spin-flip opaque reflections of electrons from a potential barrier in heterostructures are described. An electric field of the barrier is considered to be the only source of energy spin splitting in the presence of spin-orbit interaction and its form is calculated in the three-level {\kp} model for a nontrival case of unbound electrons. Reflection angles and amplitudes are calculated for oblique incoming directions. It is shown that the system can serve as a source or filter of spin-polarized electron beams. Two unexpected possibilities are pointed out: a) non attenuated electron propagation in the barrier whose height exceeds the energies of incoming electrons, b) total reflection of electrons whose energies exceed barrier's height.

cond-mat.mes-hall↗

arXiv2014

Damping of electron Zitterbewegung in carbon nanotubes

Zitterbewegung (ZB, trembling motion) of electrons in semiconductor carbon nanotubes is described taking into account dephasing processes. The density matrix formalism is used for the theory. Differences between decay of ZB oscillations due to electron localization and that due to dephasing are discussed.

cond-mat.mes-hall↗

arXiv2013

Multi-mode behavior of electron Zitterbewegung induced by an electromagnetic wave in graphene

Electrons in monolayer graphene in the presence of an electromagnetic (or electric) wave are considered theoretically. It is shown that the electron motion is a nonlinear combination of Zitterbewegung (ZB, trembling motion) resulting from the periodic potential of graphene lattice and the driving field of the wave. This complex motion is called "Multi-mode Zitterbewegung". The theory is based on the time-dependent two-band Hamiltonian taking into account the graphene band structure and interaction with the wave. Our theoretical treatment includes the rotating wave approximation and high-driving-frequency approximation for narrow wave packets, as well as numerical calculations for packets of arbitrary widths. Different regimes of electron motion are found, depending on relation between the ZB frequency $ω_Z$ and the driving frequency $ω_D$ for different strengths of the electron-wave interaction. Frequencies and intensities of the resulting oscillation modes are calculated. The nonlinearity of the problem results in a pronounced multi-mode behavior. Polarization of the medium is also calculated relating our theoretical results to observable quantities. The presence of driving wave, resulting in frequencies directly related to $ω_Z$ and increasing the decay time of oscillations, should facilitate observations of the Zitterbewegung phenomenon.

cond-mat.mes-hall↗

arXiv2012

Electron dynamics in crystalline semiconductors

Electron dynamics in crystalline semiconductors is described by distinguishing between an instantaneous velocity related to electron's momentum and an average velocity related to its quasi-momentum in a periodic potential. It is shown that the electron velocity used in the theory of electron transport and free-carrier optics is the average electron velocity, not the instantaneous velocity. An effective mass of charge carriers in solids is considered and it is demonstrated that, in contrast to the "acceleration" mass introduced in textbooks, it is a "velocity" mass relating carrier velocity to its quasi-momentum that is a much more useful physical quantity. Among other advantages, the velocity mass is a scalar for spherical but nonparabolic energy bands $ε(k)$, whereas the acceleration mass is not a scalar. Important applications of the velocity mass are indicated. A two-band ${\bm k}\cdot {\bm \hp}$ model is introduced as the simplest example of a band structure that still keeps track of the periodic lattice potential. It is remarked that the two-band model, adequately describing narrow-gap semiconductors (including zero-gap graphene), strongly resembles the special theory of relativity. Instructive examples of the "semi-relativistic" analogy are given. The presentation has both scientific and pedagogical aspects.

cond-mat.mtrl-sci↗

arXiv2012

Non-locality of energy separating transformations for Dirac electrons in a magnetic field

We investigate a non-locality of Moss-Okninski transformation (MOT) used to separate positive and negative energy states in the 3+1 Dirac equation for relativistic electrons in the presence of a magnetic field. Properties of functional kernels generated by the MOT are analyzed and kernel non-localities are characterized by calculating their second moments parallel and perpendicular to the magnetic field. Transformed functions are described and investigated by computing their variances. It is shown that the non-locality of the energy-separating transformation in the direction parallel to the magnetic field is characterized by the Compton wavelength $λ_c=\hbar/mc$. In the plane transverse to magnetic field the non-locality depends both on magnetic radius $L=(\hbar/eB)^{1/2}$ and $λ_c$. The non-locality of MO transformation for the 2+1 Dirac equation is also considered.

quant-ph↗

arXiv2012

Spatial control of electron spins by electric and magnetic fields in double quantum wells

A system of two quantum wells (QW), one made of HgCdTe and the other of HgCdMnTe, subjected to electric and magnetic fields $F$ and $B$ parallel to the growth direction, is proposed and described theoretically. It is shown that in such a system the spin $g$ factor of mobile electrons strongly depends on the sign and magnitude of electric field. Adjusting $F$ at a constant $B$ one can transfer almost all electrons into one or the other QW and polarize their spins along the desired orientation. Changing $B$ at a constant $F$ can produce a similar transfer and polarization effect. Possible applications of the spatial reservoirs filled with spin-polarized electrons are discussed.

cond-mat.mtrl-sci↗