Spin injection efficiency from two adjoining ferromagnetic metals into a two-dimensional electron gas

In order to enhance spin injection efficiency from ferromagnetic (FM) metal into a two-dimensional electron gas (2DEG), we introduce another FM metal and two tunnel barriers (I) between them to investigate the current polarization in such ballistic FM/I/FM/I/2DEG junction. Our treatment is based on the free-electron scattering theory. It is found that due to quantum interference effect, the magnitude and sign of the current polarization exhibits periodical oscillating behavior with variation of the thickness of the middle FM metal layer or its exchange energy strength. For some suitable parameters, the spin injection efficiency may arrive over 80% in this junction and can also be controlled by the electron density of 2DEG. Our results may shed light on the development of new spin-polarized device 71.70.Ej, 73.21.-b 73.40.Sx Typeset using REVTEX 1 In the recent years there have been much theoretical and experimental work in the spin electronics (spintronics) field[1-3], in which the degrees of freedom of both electronic spin and charge are exploited. The magnetoelectronic device based on the spin-polarized transport in the semiconductors, which was first proposed by Datta and Das[4], has numerous potential applications in the information technology (IT) industry. The injection of spin-polarized carriers from ferromagnetic (FM) semiconductor into nonmagnetic semiconductor (SM)[5-6] has been achieved successfully with an efficiency ∼ 90%. Jonker et al.[7] even observed full polarized current by using an external magnetic field. Whereas spin injection from FM metal into SM is more attractive because FM metals such as Fe have a relatively high Cuire temperature, which makes them indispensable for the room temperature devices. However, the spin injection efficiency in this FM/SM junction are very low and moreover, there exist much debate on it[8]. As Schmidt et al.[9] pointed out, the basic obstacle for spin-polarized injection from FM metal into SM in the diffusive system results from the conductivity mismatch between them. Although many authors[10-12] have shown that this kind of conductivity mismatch could be improved by introduction of a tunnel barrier (I) between them, which can assume the tunnel conductance difference between two spin channels, the efficiency of spin injection still remain low in comparison with that from ferromagnetic-SM into SM. For instance, by interposing a tunnel barrier between FM metal and SM, Zhu et al.[13] have observed experimentally 2% efficiency of spin injection from Fe into n-GaAs at room temperature; Heersche et al.[14] theoretically calculated this ballistic FM/I/2DEG(two-dimensional electron gas)[15] junction and obtained ∼ 10% current polarization. A Schottky barrier formed at the Fe/AlGaAs interface by Hanbicki et al.[16] as a tunnel barrier can make the efficiency of spin injection ∼ 30% in this junction. In the present work, we show theoretically that the high efficiency of spin injection from FM metal into 2DEG might be achieved by introducing another FM material (FM metal or ferromagnetic SM) between them besides two tunnel barriers. In the ballistic approximation, we treat this FM/I/FM/I/2DEG junction with the free-electron scattering theory, which 2 has been widespread employed to deal with the interface scattering of electrons[17-18]. The first FM metal of the FM/I/FM/I/2DEG junction is a source of spin injection electrons (FM1), while the middle FM metal is taken as a resonant device to tune the tunnel current (FM2). Due to the quantum interference effect, the moderate thickness of the FM2 layer or its strength of spin exchange splitting energy may induce very high degree of current polarization. FM1 can even be a normal metal in our model since FM2 is crucial to cause the spin-polarized current. Increasing exchange energy of both FM1 and FM2 as well as the strength of two tunnel barriers would lead to enhancement of current polarization. The electron density of 2DEG affects the quantum interference effect so that it can also influence the degree of current polarization. In the free electron approximation, the Hamiltonian for the FM/I/FM/I/2DEG junction reads H = −h̄ 2 2m ∆ + V (x) + U1δ(x) + U2δ(x− L)− θ(−x)h1·σ − θ(x)θ(L− x)h2·σ, (1) where m is the effective electron mass, m = me in two FM metals for x < L and m = ms in 2DEG for x > L. Here we hypothesize that FM1 and FM2 have same effective electron masses. h1 and h2 are respectively the internal molecular fields of the FM1 and FM2 layer and σ denotes the Pauli spin operator. θ(x) is the step function. The two thin tunnel barriers are described by δ-type potentials, which does not lose generality. We wish to point out that even our two-dimensional model were replaced by three-dimensional one with different barrier shape such as rectangle one, or Schottky barrier between FM and SM, the qualitative results in this papers would not change. U1 at x = 0 and U2 at x = L are related with the barrier’s width and height. The potential energy V (x) is zero for x < L and EB for x > L. The schematic band structures of the FM/I/FM/I/2DEG junction is shown in Fig. 1. The spin quantum axis is taken along y direction and the magnetizations of two FM metals are assumed to be parallel for simplicity while the net tunnel current flows in the x direction. In the two-band model, the energy eigenvalues of a single electron with spin σ (↑ or ↓) are E ↑ = (h̄K ↑ )/2me and E ↓ = (h̄K ↓ )/2me + ∆1 in the FM1 layer, E ↑ = 3 (h̄K ↑ ) /2me and E fm2 ↓ = (h̄K fm2 ↓ ) /2me+∆2 in FM2 layer, and E sm σ = EB+(h̄K sm σ ) /2ms in 2DEG, where ∆1 = 2h1 and ∆2 = 2h2 are the exchange energies of FM1 and FM2, respectively. EB is the difference between the lower conduction-band edge of 2DEG and that of FM. In the two-dimensional system, the density of states in 2DEG is constant for the energy dispersion of free electrons and E σ = EB + πh̄ n2DEG/ms with n2DEG being the electron density of 2DEG. In the small bias approximation, only electrons near the Fermi energy (EF ) surface contribute greatly to the net tunnel current so that we can take E ↑ = E fm2 ↑ = E sm ↑ = EF and E fm1 ↓ = E fm2 ↓ = E sm ↓ = EF . Thus, the magnitude of Fermi wave vectors in three regions can be explicitly expressed as