Nonequilibrium spin distribution in single-electron transistor

Single-electron transistor with ferromagnetic outer electrodes and nonmagnetic island is studied theoretically. Nonequilibrium electron spin distribution in the island is caused by tunneling current. The dependencies of the magnetoresistance ratio δ on the bias and gate voltages show the dips which are directly related to the induced separation of Fermi levels for electrons with different spins. Inside a dip δ can become negative. 73.23.Hk; 75.70.Pa; 73.40.Rw Typeset using REVTEX On leave from Nuclear Physics Institute, Moscow State University, Moscow 119899, Russia 1 Magnetoresistance of tunnel structures is currently an attractive topic for both experimental and theoretical studies (see, e.g., Refs. [1–5]). For tunnel junctions made of ferromagnetic films, the difference as high as 26% at 4.2 K (up to 18% at room temperature) between the tunnel resistances for parallel and antiparallel film magnetization has been observed [2], that allows their application as magnetic sensors. The low temperature values agree well with the theoretical result [1] ∆R/R = 2P1P2/(1 + P1P2) where P1 and P2 are spin polarizations of tunneling electrons in two films, that proves the good achieved quality of junctions. As an example, the polarization is about 47% for CoFe, 40% for Fe, and 34% for Co [6,2]. With the decrease of the tunnel junction area, the single-electron charging [7] becomes important leading to new physical effects. The study of tunnel magnetoresistance in this regime is a rapidly growing field [8–14] (see also Refs. [3] and [5]). For example, the enhancement of the magnetoresistance ratio ∆R/R due to Coulomb blockade has been discussed in Refs. [8,13]. The magnetic field dependence of the tunneling current between two Co electrodes via the layer of few-nm Co clusters has been measured in Ref. [10]. In Ref. [12] the model of small ferromagnetic double tunnel junction (i.e. single-electron transistor [15] (SET-transistor) without gate electrode) has been considered, in which the tunnel resistances of junctions are different for parallel and antiparallel magnetizations of electrodes, thus changing the current through the system. A similar SET-transistor has been also studied theoretically in Ref. [14]. The very interesting effect of magneto-Coulomb oscillations in SET-transistor has been observed and explained in Refs. [8,9]. Strong magnetic field H causes the Zeeman shift of two spin bands and their repopulation. Since in ferromagnets the densities of states of these bands are different, the Fermi level moves with H leading to magneto-Coulomb oscillations. In the present letter we consider a SET-transistor which has ferromagnetic outer electrodes and nonmagnetic central island (see inset in Fig. 1a). When the coercive fields of two ferromagnetic electrodes are different, the standard technique of the magnetic field sweeping (see, e.g., Ref. [2]) easily allows to obtain parallel or antiparallel polarizations of outer 2 electrodes. In the first approximation the current through SET-transistor does not depend on these polarizations because the island is nonmagnetic (the Zeeman splitting is negligible because of small H). However, if the electron spin relaxation in the island is not too fast (estimates are discussed later), then the tunneling of electrons with preferable spin orientation creates the nonequilibrium spin-polarized state of the island (similar to the effect discussed in Refs. [16,17] in absence of the Coulomb blockade). This in turn affects the tunneling in each junction and leads to different currents Ip and Ia through SET-transistor in the parallel and antiparallel configurations. We will calculate the dependence of the relative current change δ = (Ip − Ia)/Ip on the bias and gate voltages (we call δ magnetoresistance ratio despite for finite voltage this terminology could be misleading). Nonzero δ is already the evidence of the nonequilibrium spin state in the island. Moreover, the voltage dependence of δ shows the dips, the width of which directly corresponds to the energy separation between Fermi levels of electrons with different spins in the island. We consider SET-transistor consisting of two tunnel junctions with capacitances C1 and C2. Induced background charge Q0 as usual [15] describes the influence of the gate voltage (general equivalence relations for finite gate capacitance are discussed, e.g., in Ref. [18]). We assume that the voltage scale related to the magnetic polarization of ferromagnetic electrodes [4] and the voltage scale of the barrier suppression [19] are large in comparison with the single-electron charging energy (that is a typical experimental situation). Then the polarization of outer electrodes can be taken into account by the difference between the tunnel resistances R 1,2 and R d 1,2 for electrons with “up” and “down” spins. The total junction resistances R1 = (1/R u 1 + 1/R d 1) −1 and R2 = (1/R u 2 + 1/R d 2) −1 do not depend on the magnetic polarizations P1 and P2 of electrodes, while “partial” resistances are given by R i = 2Ri/(1 + Pi) and R d i = 2Ri/(1 − Pi) (similar to the model of Ref. [1]). Notice that Pi describes the polarization of tunneling electrons [6] which is different (typically even in sign) from the total electron polarization at Fermi level (the latter one determines, e.g., the period of magneto-Coulomb oscillations [9]). 3 We assume that the energy relaxation of electrons in the island is much faster than the spin relaxation. So, we characterize the nonequilibrium spin state by the difference ∆EF between Fermi levels for “up” and “down” spins while both distributions are determined by the thermostat temperature T . The spin diffusion length is assumed to be much larger than the island size (that is a typical experimental situation – see Ref. [16]), so the spin distribution is uniform along the island. The equations of the “orthodox” theory for single-electron transistor [7,15] (we assume Ri ≫ RK = h/e ) should be modified in our case. The energy gain W u(d)± i for tunneling to (+) or from (-) the island through ith junction is different for “up” and “down” electrons, W u± i (n) = W ± i ∓ ∆EF 2 , W d± i (n) = W ± i ± ∆EF 2 (1) W i = e CΣ [ ∓(ne +Q0)∓ (−1) iV C1C2 Ci − e 2 ]