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Zawadzki W.,Polish Academy of Sciences | Rusin T.M.,Orange Customer Service Sp. Z O. O.
Journal of Physics Condensed Matter | Year: 2011

We review recent research on Zitterbewegung (ZB, trembling motion) of electrons in semiconductors. A brief history of the subject is presented, the trembling motion in semi-relativistic and spin systems is considered and its main features are emphasized. ZB of charge carriers in monolayer and bilayer graphene as well as in carbon nanotubes is elaborated in some detail. We describe the effects of an external magnetic field on ZB using monolayer graphene as an example. The nature of electron ZB in crystalline solids is explained. We also review various simulations of the trembling motion in a vacuum and in semiconductors, and mention ZB-like wave phenomena in sonic and photonic periodic structures. An attempt is made to quote all the relevant literature on the subject. © 2011 IOP Publishing Ltd.


Rusin T.M.,Orange Customer Service Sp. Z O. O. | Zawadzki W.,Polish Academy of Sciences
Journal of Physics Condensed Matter | Year: 2014

We describe a possibility of creating non-stationary electron wave packets in zigzag carbon nanotubes (CNT) illuminated by short laser pulses. After the disappearance of the pulse the packet experiences a trembling motion (Zitterbewegung, ZB). The band structure of CNT is calculated using the tight-binding approximation generalized for the presence of radiation. By employing realistic pulse and CNT parameters we obtain the ZB oscillations with interband 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. © 2014 IOP Publishing Ltd.


Rusin T.M.,Orange Customer Service Sp. Z o. O. | Zawadzki W.,Polish Academy of Sciences
Semiconductor Science and Technology | Year: 2014

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 (2PE) 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 2PE 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. © 2014 IOP Publishing Ltd.


Rusin T.M.,Orange Customer Service Sp. Z O. O. | Zawadzki W.,Polish Academy of Sciences
Journal of Physics A: Mathematical and Theoretical | Year: 2012

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 = /mc. In the plane transverse to the magnetic field, the non-locality depends both on magnetic radius L = (/eB) 1/2 and λ c. The non-locality of MOT for the 2+1 Dirac equation is also considered. © 2012 IOP Publishing Ltd.


Rusin T.M.,Orange Customer Service Sp. Z O. O. | Zawadzki W.,Polish Academy of Sciences
Physica E: Low-Dimensional Systems and Nanostructures | Year: 2015

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. © 2015 Elsevier B.V. All rights reserved.


Rusin T.M.,Orange Customer Service Sp. Z O. O. | Zawadzki W.,Polish Academy of Sciences
Physical Review A - Atomic, Molecular, and Optical Physics | Year: 2011

Nonlocalities of Foldy-Wouthuysen and related transformations, which are used to separate positive- and negative-energy states in the Dirac equation, are investigated. Second moments of functional kernels generated by the transformations are calculated, and the transformed functions and their variances are computed. It is shown that all the transformed quantities are smeared in the coordinate space by an amount comparable to the Compton wavelength λ c= /mc. © 2011 American Physical Society.


Rusin T.M.,Orange Customer Service Sp. Z O. O. | Zawadzki W.,Polish Academy of Sciences
Physical Review A - Atomic, Molecular, and Optical Physics | Year: 2012

The Klein-Gordon equation is used to calculate the Zitterbewegung (ZB, trembling motion) of spin-zero particles in the absence of fields and in the presence of an external magnetic field. Both Hamiltonian and wave formalisms are employed to describe ZB and their results are compared. It is demonstrated that if one uses wave packets to represent particles, then the ZB motion has a decaying behavior. It is also shown that the trembling motion is caused by an interference of two subpackets composed of positive- and negative-energy states, which propagate with different velocities. In the presence of a magnetic field, the quantization of the energy spectrum results in many interband frequencies contributing to ZB oscillations and the motion follows a collapse-revival pattern. In the limit of nonrelativistic velocities, the interband ZB components vanish and the motion is reduced to cyclotron oscillations. The exact dynamics of a charged Klein-Gordon (KG) particle in the presence of a magnetic field is described on an operator level. The trembling motion of a KG particle in the absence of fields is simulated using a classical model proposed by Morse and Feshbach-it is shown that a variance of a Gaussian wave packet exhibits ZB oscillations. © 2012 American Physical Society.

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