Magnetoswitching of current oscillations in diluted magnetic semiconductor nanostructures

Strongly nonlinear transport through Diluted Magnetic Semiconductor multiquantum wells occurs due to the interplay between confinement, Coulomb and exchange interaction. Nonlinear effects include the appearance of spin polarized stationary states and self-sustained current oscillations as possible stable states of the nanostructure, depending on its configuration and control parameters such as voltage bias and level splitting due to an external magnetic field. Oscillatory regions grow in size with well number and level splitting. A systematic analysis of the charge and spin response to voltage and magnetic field switching of II-VI Diluted Magnetic Semiconductor multiquantum wells is carried out. The description of stationary and time-periodic spin polarized states, the transitions between them and the responses to voltage or magnetic field switching have great importance due to the potential implementation of spintronic devices based on these nanostructures.


I. INTRODUCTION
Spin injection is one of the aims of spintronics 1 thanks to the potential applications of injectors as spin LED devices, etc. Also quantum state transfer from spin electrons to photons by interband transitions is actively investigated 2,3,4 . One of the most efficient ways of spin injection to date 5,6 is the use of II-VI dilute magnetic semiconductors (DMSs) that exhibit the giant Zeeman effect 7 : they have a conductivity comparable to that of nonmagnetic semiconductors, and can boast spin polarizations close to 100% at a small applied magnetic field. However, spin-injection experiments in semiconductors enter easily the regime of nonlinear response 8 . Different effects could contribute to nonlinear transport and therefore to nonlinear spin injection. For example, band bending effects 9 in nanostructures give rise to a nonlinear current due to the interplay between Coulomb interaction and electron tunnel in these confined systems, which have quasi-discrete states.
Other physical mechanisms inherent to these systems explain their current-voltage characteristics: for instance a large Zeeman level splitting ∆ in an applied magnetic field B. Recently, spin transport through DMS diodes 10 and multi-quantum well structures (MQWS) has been analyzed 11,12,13,14 . These works study nonlinear features of the current (hysteresis, multistability) as a function of the external voltage. Under strong dc voltage bias V , electric field domains are formed in MQWS due to the interplay between electron-electron interaction and resonant tunneling 11 . In other sample configurations or, for different doping density, there are spin polarized self-sustained current oscillations (SSCOs) and the system could behave as a spin oscillator 12,13 . To tailor the properties of these spin oscillators or injectors, it is important to perform a systematic analysis of the transition from stationary to time dependent current, in terms of sample configuration, external magnetic field, doping density, etc.
In this paper we analyze the response to voltage (V ) or magnetic (B) switching in a n-doped dc voltage biased semiconductor MQWS having its first quantum well (QW) doped with Mn. Both spin polarized stationary states (SSs) and SSCOs are possible stable states of the MQWS for different values of the parameters. Stationary states field profiles consist of two electric field domains separated by a domain wall which is a charge monopole 14 .
Magnetic field switching requires knowing phase diagrams of the current density J and the applied voltage V versus the level splitting ∆ (due to the magnetic field B), and these diagrams are among the results of this paper. The phase diagram of V versus ∆ shows regions of stable SSCOs embedded in others of stable SSs. The extension of the SSCO regions increases with the number of QWs in the structure. Sudden changes of V or B may switch or disconnect SSCOs from an initial stable SS or force the domain wall to change its location. The SSCOs are due to periodic triggering of charge dipoles at the Mn-doped well and their motion towards the collector 13 . Large level splitting induced by B due to the exchange interaction provides DMS MQWSs with a new degree of freedom which is absent in conventional III-V weakly coupled n-doped semiconductor MQWSs 15 . Another important difference is that, in the latter, both charge dipoles and monopoles may be triggered at the injector (depending on its current-field characteristics: its conductivity if the relation between current and field is linear) and both may cause SSCOs 15,16 . In these materials and for moderate conductivity of the injecting contact, switching the voltage V between different SSs involves either upward monopole motion or a dipole-tripole mechanism 15,17,18 .
For sufficiently large conductivity of the injector, the dipole-tripole mechanism ceases to exist and voltage switching involves injection of a charge monopole that moves towards the collector until it reaches the QW corresponding to the final stable SS 19 . Voltage or magnetic switching in II-VI MQWSs always involves dipole nucleation at the Mn-doped QW.

II. MODEL
Our sample configuration consists of an n-doped ZnSe/(Zn,Cd,Mn)Se weakly coupled MQWS. The spin for the magnetic ion Mn ++ is S=5/2 and the exchange interaction between the Mn local moments and the conduction band electrons is ferromagnetic in II-VI QWs.
The energy spectrum corresponding to N isolated QWs comprising our weakly coupled MQWS has the form E j +h 2 k ⊥ /(2m * ), where m * is the effective mass, k ⊥ is the in-plane wave vector orthogonal to the growth direction and j = 1, . . . , N is the QW subband index.
In the weak magnetic fields considered here, we disregard Landau-level formation and k ⊥ is a continuous variable 11 . Using the virtual crystal and mean field approximations, the exchange interaction causes the subband energies to depend on spin in those QWs containing Mn ions: where for spin s = ±1/2, and B S , J sd , N M n and T eff are the Brillouin function, the exchange integral, the density of magnetic impurities and an effective temperature which accounts for Mn interactions, respectively 11,20 .
We model spin-flip scattering coming from spin-orbit or hyperfine interaction by a phenomenological scattering time τ sf , which is larger than impurity and phonon scattering times: τ scat < τ sf . Vertical transport in the weakly coupled MQWS is spin-independent sequential tunneling between adjacent QWs, so that when electrons tunnel to an excited state they instantaneously relax by phonon scattering to the ground state, with the same spin polarization 11 . Lastly, electron-electron interaction is considered within the Hartree mean field approximation.
The equations governing the model are 13,14 : the discrete Poisson equation relating the two-dimensional spin-up and spin-down electron densities, n + i , n − i , respectively, to the average electric field −F i at the ith MQWS period (of lengh l), and the rate equations for n ± i , for i = 1, . . . , N. For numerical convenience, we have introduced here a smoothed form of the scattering term used in 11 , given by where γ µ is a small smoothing parameter (smaller than γ =h/τ scatt or than the thermal energy) such that, as γ µ → 0, otherwise, which was used by Sánchez et al 11 In these expressions, µ ± i is the chemical potential at the ith MQWS period and E ± j,i are the spin-dependent subband energies (measured from the bottom of the ith well): E ± j,1 = E j ∓ ∆/2, and E ± j,i =E j for i =1. Also, N D and ε are the 2D doping density at the QWs and the average permittivity.
In weakly coupled MQWS, tunneling between adjacent QWs can be treated in leading order perturbation theory. Since elastic and inelastic scattering times in the QWs are shorter than any other time scale of the problem, we can assume that the electrons in each well are in quasiequilibrium between succesive tunneling events and that their temperature is that of the lattice. We ignore interwell spin-flip processes, so that currents are carried between wells by the two spin subsystems in parallel. Then, as in the case of non-magnetic MQWSs, the tunneling current densities across the ith barrier J ± i→i+1 can be calculated by the Bardeen Transfer Hamiltonian method 21,22,23,24 . See the detailed derivation for non-magnetic MQWSs in Ref. 19 . The well known resulting expression 19,25 can be approximated by the formula 14 : i = 1, . . . , N − 1, provided that scattering-induced broadening of energy levels is much smaller than subband energies and chemical potentials; see Appendix A of Ref. 25 . The spindependent "forward tunneling velocity", v (f )± , is a sum of Lorentzians of width 2γ, with γ =h/τ scatt (the same value for all subbands, for simplicity), centered at the resonant field where T i is proportional to the transmission coefficient of the ith barrier 25 . For electrons with spin ±1/2, the chemical potential µ ± i and the electron densities n ± i are related by The voltage bias condition can be written as Defining J i→i+1 = J + i→i+1 + J − i→i+1 , the total current density J(t) can be calculated as Then, time-differencing the Poisson equation, inserting the rate equations for n ± i in the result, and assuming a constant applied voltage (dV /dt = 0), we obtain the following equation relating F i (t), J i→i+1 (t) and J(t) for i = 0, . . . , N: Boundary tunneling currents for i = 0 and i = N are determined by using tunneling currents with n ± 0 = n ± N +1 = N D /2 (identical normal contacts) 11 . As initial conditions, we set n ± i = N D /2 (normal QWs) and F i = φF M , where F M is a reference field corresponding to the first local maximum (F M , J M ) of the tunneling current 14 , and φ is a dimensionless average field defined by (N + 1)φ is the dimensionless voltage across the MQWS.

III. RESULTS
We have considered barrier and QW widths of 10 and 5 nm, respectively, τ sf =10 −9 s (normal QW) and 10 −11 s (magnetic QW), m * =0.16m 0 , N D =10 10 cm −2 , ε=7.1ε 0 , T =5 K,   For fixed φ, a sudden increment of B from a stable SS region to a SSCO region (horizontal arrow in Fig. 1) induces SSCOs, as shown in Fig. 4(c-d). The transient stage between the SS and SSCOs after switching B is due to the formation of a high field domain at the first QW which travels towards the collector. After the domain reaches the MQWS end, a new high field domain is formed at the first QW and the same situation is periodically repeated.

IV. DISCUSSION
In this paper, we have systematically analyzed the transition from stationary states to selfsustained current oscillations through a dilute magnetic semiconductor multi-quantum well structure. Switching suddenly a control parameter as the (dimensionless) applied voltage φ or the external magnetic field B may force the system to move between stable oscillatory and stationary states through the transition region. Since self-sustained current oscillations are caused by triggering high field domains at the magnetic quantum well, we expect our results not to change qualitatively with the contact boundary condition. We have used two other conditions to check this: (i) n ± 0 = n ± N +1 = κN D /2 in the tunneling currents for normal contacts, where κ is a positive constant.
(ii) The electric field at the injector F 0 is calculated by using the Ohm's law: instead of the tunneling current formulas 14 with known n ± 0 .
The resulting phase diagrams for (i) with different values of κ (from 0.5 to 1.5) and for While it is feasible to list all possible oscillation types in terms of contact parameter values (see 28,30 ), these values cannot be modified once the Gunn diode has been made. Similarly, in a conventional III-V weakly coupled n-doped semiconductor superlattice (SL), the boundary condition at the injector, the SL configuration and the doping density at the QWs determine The situation is different in the case of a dilute magnetic semiconductor multi-quantum well structure: the magnetic QW plays the role of a "tunable doping density notch". In principle, any self-oscillations that may appear are due to triggering of dipoles at the magnetic QW. However, by changing the external magnetic field we can select either stable stationary states or SSCOs as the DMS multi-quantum well response.
Our results show how to design a device operating a spin injector and a spin oscillator by tuning the Zeeman splitting and the parameters determining the sample configuration.
R.E. thanks the Spanish Ministry of Education Ramón y Cajal Program.