NONLINEAR CURRENT DENSITY IN A COMPOSITIONAL SEMICONDUCTOR SUPERLATTICE UNDER CROSSED ELECTRIC AND MAGNETIC FIELDS

: The dc electrical transport in a compositional semiconductor superlattice subjected to a crossed electric field and a magnetic field is studied theoretically. The electron - optical phonon interaction is taken into account at high temperature and strong magnetic field. We obtain the expression for nonlinear current density involving external (electric and magnetic) fields and characteristic parameters of the superlattice. The analytical result is numerically evaluated and graphed for GaAs/Al 0.3 Ga 0.7 As superlattice. The intra-and inter-subband magnetophonon resonance effect is observed when the dc electric field is absent. Resonance peaks shift to higher cyclotron energy region when the dc electric field is switched on. The current density increases linearly with small value region and nonlinearly with large value region of the dc electric field.


Introduction
In the late seventies and early eighties of the twentieth century, Carolyne M. van Vliet together with her coworkers in Montreal developed a many-body master equation, using the interactionpicture and projection operators in the Liouville space, thus extending earlier work by van Hove, Zwanzig, Kubo, and others [1][2][3][4]. The so-obtained Pauli-van Hove-van Vliet master equation has diagonal as well as non-diagonal terms, with irreversibility vested in the diagonal part. Using the formalism of second quantization for an electron gas in interaction with impurities or phonons, these researchers obtained a full quantum transport equation, applicable to both extended and localized electronic states. This quantum transport equation has proven to be a major tool for transport and conductivity calculations in modern submicron devices with extreme quantum confinement, such as quantum wells, quantum wires and quantum dots.
For the square wells with an infinite potential, Vasilopoulos et al. [9] presented the nonlinear electrical conduction in the presence of the strong electric and magnetic field where electrons are scattered by optical phonons and impurities. The authors replaced delta functions by Poisson summations in the expression of nonlinear current density to avoid the divergence. They also clearly showed the conditions for the appearance of the MPR effect. However, the results were calculated for the intrasubband transitions (n = n') only, and a graphical consideration was not carried out. In a recent work [10], we have applied the above-mentioned theory to study the nonlinear electrical conduction in a parabolic quantum well under crossed dc electric and magnetic fields. We derived the x-component of the nonlinear current density (NCD) taking into account the scattering of electrons by optical phonons at high temperatures. The analytical expression of the NCD was calculated for the arbitrary values of subband indices (both intra-subband and inter-subband transitions), and the numerical results were obtained for a specific parabolic quantum well with the help of a computational program.
It is well-known that the studying of the MPR effect is a powerful tool to understand transport phenomena in semiconductors, such as the determination of effective mass, effective charge, energy levels of carriers, etc. [11][12][13][14]. To our knowledge, the application of abovementioned theory to MPR effect problems in the low-dimensional electron systems has still been open for studying up to date. So, in the present work we calculate the NCD in a compositional semiconductor superlattice (CSSL) subjected to a crossed electric field and a magnetic field. The magnetic field is applied along to the growth direction of the CSSL. The present paper is organized as follows. In Sec. 2, we briefly describe the simple model of a CSSL in the crossed electric and magnetic fields. In the next section, we calculate analytically the NCD for electron-optical phonon interaction. Numerical results and discussion are given in Sec. 4. Finally, remarks and conclusions are shown briefly in Sec. 5.

A compositional semiconductor superlattice in crossed electric and magnetic fields
We consider the transport of an electron gas in a compositional semiconductor superlattice which is composed of 0 N layers of GaAs (the layer thickness is I d ) and 0 N layers of AlcGa1-cAs (the layer thickness is II d ) arranged alternatively. The energy-gap difference between these two materials is U , then the motions of electron gas along the growth direction (assumed the zdirection) are considered to be governed by the periodic superlattice potential, and the motions are nearly free along two other directions (assumed the x-direction and the y-direction). We assume that the difference of the electron effective mass between GaAs and AlcGa1-cAs can be neglected (  I II m m m ). If a static magnetic field B is applied in the z-direction, and a dc electric field E is applied in the x-direction, then the one-electron Hamiltonian ( 0 h ), its normalized eigenfunctions (  ), and the eigenvalues (   ) in the Landau gauge for the vector potential are, respectively, given by [9,15,16]   is the radius of the Landau orbit in the x-y plane; , z nk is the wave function in the z-direction. Also, in the tight-binding approximation, we have [16,17] , cos( ), 0,1,2,..., For the calculation of this paper, the following matrix elements will be used [10,15]    eq a r r , with r and eq r are, respectively, the position of carrier and its equilibrium position prior to the switching on of the external fields; x, y are the Cartesian components of r . Also, ,, n n z z z I k k q are assumed to be independent on the number of periods, and in this study, we will deal with the intrawell form factor only (neglecting the interwell form factor), then [20]   where Expression for the nonlinear current density For the above-mentioned model of CSSL, we consider the strong applied dc electric field and high lattice temperature, then the expression of the NCD is written as [9]  f are the occupancies of the eigenstates  and '  , respectively, given by Fermi-Dirac distribution functions; '  w is the binary transition rate (probability of transition) between two above eigenstates, given by the "golden rule", The first and the second terms of Eq. (14) stand for the absorption and emission of a phonon of wave vector q with energy  q , respectively. The notation the first term (second term) means that an electron at state  absorbs (emits) a phonon of wave vector q to change its state to '  . At high temperatures, the electrons system is non-degenerate and assumed to obey the Boltzmann's distribution function, then With above mentions, from Eqs. (6)- (14), the x-component of NCD can be written as x,x' 00   (20) where 0 A is the crossed section of CSSL. For simplicity in performing the integral over  q , we replace y q by , where x is a constant of the order of . This has been done in Ref. 9 and is equivalent to assuming an effective phonon momentum: , ' / I k k I k k q dq which will be numerically evaluated by the computational program, and This integral can be calculated immediately by using the formulas [7-9] This means that only the processes at the centre and the boundary of the first mini-Brillouin zone are included. We also replace the delta functions by Lorentzians to avoid the divergence as [6] , Expression (25)  which is difficult to find out the exact analytical result. This term will be numerically evaluated by the computational method in the next section where we give a deeper insight for the dependence of the NCD on the external fields and other parameters.

Numerical results and discussion
In this section, we present detailed numerical calculations of the x-component of the NCD,   In figure 2, we show the NCD versus cyclotron energy for two cases: absence (red-dashed curve) and presence (blue-dashed curve) of the dc electric field. The appearance of the peaks in the former case can be explained similarly as for figure 1. However, for the latter case the condition for the resonant peaks becomes here, for the second peak, and for the others. The condition (26) is generally called the intersubband MPR condition in the presence of a dc electric field. It was obtained in Refs. 5 -8 for square quantum wells without the dc electric field and '  nn . It is easily seen that the presence of the dc electric field shifts the resonant peaks to the right, and also the larger values of the magnetic field the smaller distance of the shifting. To explain this, we consider the condition (26) and recall that   1  xB . Hence, for a fixed value of the dc electric field E , the term  eE x decreases with the increase of the magnetic field. Namely, in the region of the strong magnetic field, the influence of the dc electric field on the peak shifting is weak. With the presence of the dc electric field, we also see that the first and the third peaks are not symmetrical to the second one. This is different compared with the case of absence of the electric field. In fact, if we denote 1 c  and 2 c  the cyclotron energies, respectively, at the first and the third peaks, the conditions for these peaks are  at different values of the temperature. Two ranges of the electric field are considered: small value (figure 3a) and large value (figure 3b). We can see that the NCD increases linearly in the small value range of the electric field. However, in the large value range of the electric field (larger than 5 -1 5 10 V.m  ), the NCD increases nonlinearly. Moreover, the NCD increases with increasing the temperature. This is a typical property of semiconductors.

Conclusions
In this work, we study the nonlinear electrical conduction in a CSSL in the presence of crossed electric and magnetic fields with the assumption that electron -optical phonon interaction is dominant at high temperatures. We obtain the analytical expression for the NCD which nonlinearly depends on the dc electric field. Numerical calculations are performed for a specific CSSL to clarify the theoretical results. We can summarize some main features of the obtained results as follows. All the resonant peaks arise under the condition