Quantum interference effect in electron tunneling through a quantum-dot-ring spin valve
© Ma et al; licensee Springer. 2011
Received: 12 September 2010
Accepted: 28 March 2011
Published: 28 March 2011
Spin-dependent transport through a quantum-dot (QD) ring coupled to ferromagnetic leads with noncollinear magnetizations is studied theoretically. Tunneling current, current spin polarization and tunnel magnetoresistance (TMR) as functions of the bias voltage and the direct coupling strength between the two leads are analyzed by the nonequilibrium Green's function technique. It is shown that the magnitudes of these quantities are sensitive to the relative angle between the leads' magnetic moments and the quantum interference effect originated from the inter-lead coupling. We pay particular attention on the Coulomb blockade regime and find the relative current magnitudes of different magnetization angles can be reversed by tuning the inter-lead coupling strength, resulting in sign change of the TMR. For large enough inter-lead coupling strength, the current spin polarizations for parallel and antiparallel magnetic configurations will approach to unit and zero, respectively.
Manipulation of electron spin degree of freedom is one of the most frequently studied subjects in modern solid state physics, for both its fundamental physics and its attractive potential applications [1, 2]. Spintronics devices based on the giant magnetoresistance effect in magnetic multi-layers such as magnetic field sensor and magnetic hard disk read heads have been used as commercial products, and have greatly influenced current electronic industry. Due to the rapid development of nanotechnology, recent much attention has been paid on the spin injection and tunnel magnetoresistance (TMR) effect in tunnel junctions made of semiconductor spacers sandwiched between ferromagnetic leads. Moreover, semiconductor spacers of InAs quantum dot (QD), which has controllable size and energy spectrum, has been inserted in between nickel or cobalt leads[4–6]. In such a device, the spin polarization of the current injected from the ferromagnetic leads and the TMR can be effectively tuned by a gate nearby the QD, and opens new possible applications. Its new characteristics, for example, anomalies of the TMR caused by the intradot Coulomb repulsion energy in the QD, were analyzed in subsequent theoretical work based on the nonequilibrium Green's function method .
The TMR is a crucial physical quantity measuring the change in system's transport properties when the angle φ between magnetic moments of the leads rotate from 0 (parallel alignment) to arbitrary value (or to φ in collinear magnetic moments case). Much recent work has been devoted to such an effect in QD coupled to ferromagnetic leads with either collinear[4–13] or noncollinear[14–16] configurations. It was found that the electrically tunable QD energy spectrum and the Coulomb blockade effect dominate both the magnitude and the signs of the TMR[4–16].
On the other hand, there has been increasing concern about spin manipulation via quantum interference effect in a ring-type or multi-path mesoscopic system, mainly relying on the spin-dependent phase originated from the spin-orbit interaction existed in electron transport channels[17–20]. Many recent experimental and theoretical studies indicated that the current spin polarization based on the spin-orbital interaction can reach as high as 100%[21–23] or infinite[24–29]. Meanwhile, large spin accumulation on the dots was realized by adjusting external electrical field or gate voltages to tune the spin-orbit interaction strength (or equivalently the spin-dependent phase factor)[27–30]. Furthermore, there has already been much very recent work about spin-dependent transport in a QD-ring connected to collinear magnetic leads[31–34]. Much richer physical phenomena, such as interference-induced TMR enhancement, suppression or sign change, were found and analyzed[31–34].
Model and Method
where is the creation (annihilation) operator of the electrons with momentum k, spin-σ and energy ε kβσ in the β th lead (β = L, R); creates (annihilates) an electron in the QD with spin σ and energy ε d ; t βd and t LR describe the dot-lead and inter-lead tunneling coupling, respectively; U is the intradot Coulomb repulsion energy. φ denotes the angle between the magnetic moments of the leads, which changes from 0 (parallel alignment) to π (antiparallel alignment).
where the Keldysh Green's function G(ε) is the Fourier transform of G(t - t') defined as , . In our present case, it is convenient to write the Green's function as a 6 × 6 matrix in the representation of (|L ↑〉, |R ↓〉, |d ↑〉, |L ↓〉, |R ↓〉, |d ↓〉). Thus the lesser Green's function G < (ε) and the associated retarded (advanced) Green's function G r(a)(ε) can be calculated from the Keldysh and the Dayson equations, respectively. Detail calculation process is similar to that in some previous works [20, 30], and we do not give them here for the sake of compactness. Finally, the ferromagnetism of the leads is considered by the spin dependence of the leads' density of states ρ βσ . Explicitly, we introduce a spin-polarization parameter for lead β of P β = (ρ β↑- ρ β↓ )/(ρ β↑+ ρ β↓), or equivalently, ρ β↑(↓)= ρ β (1 ± P β ), with ρ β being the spin-independent density of states of lead β .
Result and Discussion
In the following numerical calculations, we choose the intradot Coulomb interaction U = 1 as the energy unit and fix ρ L = ρ R = ρ 0 = 1, t Ld = t Rd = 0.04. Then the line-width function in the case of p L = p R = 0 is Γ β ≡ 2πρ β |t βd |2 ≈ 0.01, which is accessible in a typical QD[41–43]. The bias voltage V is related to the left and the right leads' chemical potentials as eV = μ L - μ R , and μ R is set to be zero throughout the paper.
When the relative angle between the leads' magnetic moments φ. rotates from 0 to π, a monotonous suppression of the electric current appears, which is known as the typical spin valve effect. The suppression of the current can be attributed to the increased spin accumulation on the QD[14–16]. Since the line-width functions of different spin orientations are continuously tuned by the angle variation, a certain spin component electron with smaller tunneling rate will be accumulated on the dot, and furthermore prevents other tunnel processes. As shown in Figure 2(b), the current spin polarizations in the bias ranges of eV < ε d and eV > ε d + U are constant and monotonously suppressed by the increase of the angle, which changes the spin-up and spin-down line-width functions. In the Coulomb blockade region of ε d < eV < ε d + U, the difference between the current spin polarizations of different values of φ is greatly decreased, which is resulted from the Pauli exclusion principle. The current spin polarizations also have small dips and peaks respectively near eV = ε d and eV = ε d + U, where new transport channel opens. The most prominent characteristic of the TMR in Figure 2(c) is that its magnitude in the Coulomb blockade region depends much sensitively on the angle than those in other bias ranges. The deepness of the TMR valleys are shallowed with the increasing of the angle. Meanwhile, dips emerge when the Fermi level crosses ε d and ε d + U. In the antiparallel configuration (φ = π), the magnitude of the TMR is larger than those in other bias voltage ranges.
When the inter-lead coupling is turned on as shown in Figure 2(d)-(f), both the studied quantities are influenced. Since the bridge between the leads serves as an electron transport channel with continuous energy spectrum, the system electric current increases with in-creasing bias voltage [Figure 2(d)]. For the present weak inter-lead coupling case of t LR < t βd , the transportation through the QD is the dominant channel with distinguishable Coulomb blockade effect. The current spin polarizations for different angles in the voltage ranges out of the Coulomb blockade one now change with the bias voltage value, but their relative magnitudes somewhat keep constant. The difference between the current spin polarization magnitude of different angle is enlarged by the interference effect brought about by the inter-lead coupling. Comparing Figure 2(f) with 2(c), the behavior of TMR is less influenced by the bridge between the leads in the present case.
We have studied the characteristics of tunneling current, current spin polarization and TMR in a quantum-dot-ring with noncollinearly polarized magnetic leads. It is found that the characteristics of these quantities can be well tuned by the relative angle between the leads' magnetic moments. Especially in the Coulomb blockade and strong inter-lead coupling strength range, the currents of larger angles are larger than those of smaller ones. This phenomenon is quite different from the usual spin-valve effect, of which the current is monotonously suppressed by the increase of the angle. The TMR in this range can be suppressed even to negative, and the current spin polarizations of parallel and antiparallel configurations individually approach to unit and zero, which can then serve as a effective spin filter even for usual ferromagnetic leads with 0 < P β < 1.
This work was supported by the Education Department of Liaoning Province under Grants No. 2009A031 and 2009R01. Chi acknowledge support from SKLSM under Grant No. CHJG200901.
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