Quasi-stationary states of electrons interacting with strong electromagnetic field in two-barrier resonance tunnel nano-structure

An exact solution of non-stationary Schrodinger equation is obtained for a one-dimensional movement of electrons in an electromagnetic field with arbitrary intensity and frequency. Using it, the permeability coefficient is calculated for a two-barrier resonance tunnel nano-structure placed into a high-frequency electromagnetic field. It is shown that a nano-structure contains quasi-stationary states the spectrum of which consists of the main and satellite energies. The properties of resonance and non-resonance channels of permeability are displayed.


Introduction
An intensive investigation of resonance tunnel structures (RTSs) is caused by their utilization in nanodevices having unique physical characteristics [1][2][3][4][5][6][7], and are widely used in medicine, environment monitoring and communication systems. The knowledge of the properties of RTS is urgent both from the applied and fundamental physics point of view.
The theory of ballistic and non-ballistic transport of electrons through the RTS was developed in detail mainly within the approximation of a small amplitude of high-frequency electromagnetic field [8][9][10][11][12][13]. Also, only linear terms over the intensity of an electric field were preserved in the Hamiltonian and wave functions. This approximation did not permit to quit the frames of the first order of the perturbation theory depending on time.
In order to clarify the effect of strong electromagnetic fields on the spectra of electrons and their tunnel through the RTS, an approximated iterating method was used for the two-level model of a nanosystem [14,15] and a numeric method was used for a multi-level model of periodical structures [16,17].
In the majority of papers, in order to use the well developed method of analytical calculations [18,19], the Hamiltonian of electron-electromagnetic field interaction is written not as the one proportional to the product of electron kinetic momentum on the vector potential of the field but as a term proportional to the product of field intensity on the respective electron coordinate. The latter Hamiltonian is correct (as proven in reference [20]) at the assumption that the electron interacts with the electromagnetic field inside the RTS which is not very strong. If the electromagnetic field is strong, its interaction with an electron should be taken into account in the whole space (outside the RTS too). It means that not only linear but also square terms over the kinetic momentum and vector potential must be present in a complete Hamiltonian.
In the paper, we propose an exact analytical solution of one-dimensional non-stationary Schrodinger equation obtained for the first time with the Hamiltonian of a system containing linear and square terms both over the electron kinetic momentum and vector potential of electromagnetic field. Using it, we develop a theory of transport of a mono-energetic electronic current through a two-barrier RTS placed into a high-frequency electromagnetic field with an arbitrary intensity and frequency. It is shown that the interaction between electrons and electromagnetic field is the reason why, besides the main quasi-stationary states (QSSs), there appear satellite states (QSSs) with the energies multiplied by field.

Permeability coefficient for a two-barrier RTS placed into a high-frequency electromagnetic field
We study the symmetric two-barrier RTS (figure 1) in a homogeneous high-frequency electromagnetic field: E (t ) = 2E cos(ωt ) with an arbitrary electric field intensity E and frequency ω. The mono-energetic electronic current with the energy E moving perpendicularly to the planes of RTS gets in it from the left hand side. The electron wave function has to satisfy a complete onedimensional Schrodinger equation Using the known expressions for a kinetic momentum operatorp z and vector potential A z written in Coulomb calibra- The complete Hamiltonian contains an electron kinetic energy operator the potential energy of electron in two-barrier RTS written within the typical δ-barrier approximation, references [8][9][10] and the potential energy of interaction between the electron and electromagnetic field where e, m are the electron charge and mass; U , ∆ are the height and width of potential barriers; a is the width of potential well. The equation (2) has two exact linearly independent solutions for all parts of a nano-structure describing the incident and the reflected waves with quasi-momentum k 0 = ħ −1 2mE .
A complete wave function, being a linear combination of both solutions (6), taking into account the expansion of all periodical functions into Fourier range and considering the super positions with all electromagnetic field harmonics, is written as follows: where Here, convenient denotations are with the evident physical sense: α is the kinetic energy of an electron with quasi-momentum k a , β is the potential energy of an electron interacting with the electromagnetic field written in the units of electromagnetic field energy Ω. All unknown coefficients: B ± (0,1,2),p are obtained from the conditions of a wave function and its density of current continuity at RTS interfaces at any moment of time t : Also, B + 0,p 0 = B − 2,p = 0 because the mono-energetic electronic current gets in RTS only along the main channel (p = 0) and there are no incident currents along the other channels (p 0).
Thus, calculating the densities of forward J + and backward J − electronic currents getting in RTS (z = 0) and coming out of RTS (z = a), according to reference [21], the permeability coefficient is expressed through partial terms Using it, one can obtain the resonance energies and widths of the main and arbitrary number of satellite QSSs for the electrons interacting with a high-frequency electromagnetic field. The developed theory proves that this interaction causes renormalization of "pure" electron QSSs and, besides, the appearance of satellite QSSs corresponding to all possible electromagnetic field harmonics. Consequently, the respective maxima of permeability coefficient can be observed when a satellite QSS of a certain electron state should resonate with the main or satellite state of the other QSS, producing complex QSSs.

Properties of quasi-stationary spectrum of electron-electromagnetic field system in a two-barrier RTS
It is well known, references [15,22], that when there is no electromagnetic field, a quasi-stationary electron spectrum is characterized by resonance energies E n , with the magnitudes determined by the 33703-3 It means that the sum of indexes for each complex QSS in an anti-crossing pair is the same. The sizes of all anti-crossings in certain series (the first at Ω = Ω 21 , the second at Ω = E 2 = E 1 + Ω 21 ) and the distances between its neighbouring anti-crossings are the same too. The distance between the neighbouring anticrossings of the first series is, naturally, smaller than that of the second one. The sizes of anti-crossings of the first series are much bigger than those of the second series.

33703-5
In figure 3, the dependencies of permeability coefficient D on the electron energy E are presented in the vicinity of degeneration (points a, b, c) and in the vicinity of anti-crossings (d, e, f) for three different magnitudes of the field energy written in the figure. The latter are chosen in such a way that the points of degeneration or minimal magnitudes of anti-crossings are enveloped from both sides. We should also note that the characters near the points of degeneration (a, b, c) and near the anti-crossings (d, e, f) in figure 2 correspond to those in figure 3.
In figure 3 (a)

Conclusions
1. Using an exact solution of non-stationary one-dimensional Schrodinger equation for electrons interacting with a high-frequency electromagnetic field and passing through the two-barrier RTS we established the existence of resonance and non-resonance channels of permeability. It is proven that there are observed the main satellite and double complexes of QSSs producing different channels of two-barrier RTS permeability.
2. In the vicinity of electromagnetic field energies that resonate with the difference of the energies of the main QSSs, a series of anti-crossings arise both between the main states with the field satellites of the other main states and between the satellites of different states with each other.
3. As far as the permeability of the both channels in a pair complex, producing an anti-crossing of the main and satellite states, is rather big, it can essentially effect the operation of nano-lasers and nano-detectors the basic element of which is a nano-RTS.