Abstract
We theoretically propose a double quantum dots (QDs) ring to filter the electron spin that works due to the Rashba spin–orbit interaction (RSOI) existing inside the QDs, the spindependent interdot tunneling coupling and the magnetic flux penetrating through the ring. By varying the RSOIinduced phase factor, the magnetic flux and the strength of the spindependent interdot tunneling coupling, which arises from a constant magnetic field applied on the tunneling junction between the QDs, a 100% spinpolarized conductance can be obtained. We show that both the spin orientations and the magnitude of it can be controlled by adjusting the abovementioned parameters. The spin filtering effect is robust even in the presence of strong intradot Coulomb interactions and arbitrary dotlead coupling configurations.
Keywords:
Quantum dots; Spin filter; Rashba spin–orbit interaction; Spindependent interdot couplingIntroduction
With the rapid progress in miniaturization of the solidstate devices, the effect of carriers’ spin in semiconductor has attracted considerable attention for its potential applications in photoelectric devices and quantum computing [1,2]. The traditional standard method of spin control depends on the spin injection technique, with mainly relies on optical techniques and the usage of a magnetic field or ferromagnetic material. Due to its unsatisfactory efficiency in nanoscale structures [1,3,4], generating and controlling a spinpolarized current with allelectrical means in mesoscopic structures has been an actively researched topic in recent years. The electric field usually does not act on the spin. But if a device is formed in a semiconductor twodimensional electron gas system with an asymmetricalinterface electric field, Rashba spin–orbit interaction (RSOI) will occur [5]. The RSOI is a relativistic effect at the lowspeed limit and is essentially the influence of an external field on a moving spin [6,7]. It can couple the spin degree of freedom to its orbital motion, thus making it possible to control the electron spin in a nonmagnetic way [8,9]. Many recent experimental and theoretical works indicate that the spinpolarization based on the RSOI can reach as high as 100% [7,10] or infinite [1113], and then attracted a lot of interest.
Recently, an AharnovBohm (AB) ring device, in which one or two quantum dots (QDs) having RSOI are located in its arms, is proposed to realize the spinpolarized transport. The QDs is a zerodimensional device where various interactions exist and is widely investigated in recent years for its tunable size, shape, quantized energy levels, and carrier number [1416]. A QDs ring has already been realized in experiments [17] and was used to investigate many important transport phenomena, such as the Fano and the Kondo effects [18,19]. When the RSOI in the QDs is taken into consideration, the electrons flowing through different arms of the AB ring will acquire a spindependent phase factor in the tunnelcoupling strengths and results in different quantum interference effect for the spinup and spindown electrons [10,13,20,21].
In this article, we focus our attention on the 100% spinpolarized transport effect in a double QDs ring. As shown in Fig. 1, the two QDs embedded in each arms of the ring are coupled to the left and the right leads in a coupling configuration transiting from serial (λ = 0) to symmetrical parallel (λ > 0) geometry. We assume that the RSOI exists only in the QDs and the arms of the ring and the leads are free from this interaction. Furthermore, the two dots are assumed to couple to each other by a spinpolarized coupling strength where t_{c} is the usual tunnel coupling strength, σΔt may arises from a constant magnetic field applied on the junction between the QDs [22], and the phase factor ϕ_{R} is induced by the RSOI in the QDs.
Figure 1. System of a double QDs ring connected to the left and the right leads with different coupling strengths
Model and Method
The secondquantized form of the Hamiltonian that describes the doubledot interferometer can be written as [20,21]
where is the creation (annihilation) operator of an electron with momentum k, spin index or ±1, and and energy ɛ_{kα} in the αth (α = LR) lead; creates (annihilates) an electron in dot i with spin σ and energy ɛ_{i};U_{i} is the Coulomb repulsion energy in dot i with being the particle number operator, in the following we set U_{1} = U_{2} = U for simplicity; t_{σ} describes the dot–dot tunneling coupling and the matrix elements t_{α}iσ are assumed to be independent of k for the sake of simplicity and take the forms of and The phase factor ϕ_{Ri} arises from the RSOI in dot i, which is tunable in experiments [20,23,24]. In fact, the RSOI will also induce a interdot spinflip, which has little impact on the current and is neglected here [25]. The spindependent tunnelcoupling strength (linewidth function) between the dots and the leads is defined as Γ_{ijσ}^{α} = 2π∑_{k}t_{αiσ}t_{αjσ}^{*}δ(ɛ−ɛ_{kασ}), (α = LR). According to Fig. 1, the matrix form of them read (here we set t_{L1} = t_{R2} = t and t_{R1} = t_{L2} = λt)
where the spindependent phase factor ϕ_{σ} = φ−σϕ_{R}, with ϕ_{R} = ϕ_{R1}−ϕ_{R2}, this indicates that the tunnelcoupling strength only depends on the difference between ϕ_{R1} and ϕ_{R2}, and then one can assume that only one QD contains the RSOI, making the structure simpler and more favorable in experiments. The phaseindependent tunnelcoupling strength is Γ = Γ^{L} + Γ^{R}, with Γ^{α} = 2πt^{2}ρ_{α}, and ρ_{α} is the density of states in the leads (the energydependence of ρ_{α} is neglected).
The general current formula for each spin component through a mesoscopic region between two noninteracting leads can be derived as [26,27]
where is the Fermi distribution function for lead α with chemical potential μ_{α}. The 2 × 2 matrices and G^{r(a)}(ɛ) are, respectively, the lesser and the retarded (advanced) Green’s function in the Fourier space. We employ the equation of motion technique to calculate both the retarded and the lesser Green’s functions by adopting the HartreeFock truncation approximation, and arrive at the Dayson equation form for the retarded one [28]:
where the retarded selfenergy The diagonal matrix elements of Green’s function g_{σ}^{r}(ɛ) for the isolated DQD are
and the offdiagonal matrix elements are t_{c}. The advanced Green’s function G_{σ}^{a}(ɛ) is the Hermitian conjugate of G_{σ}^{r}(ɛ). The occupation number <n_{iσ}> in Eq. 6 needs to be calculated selfconsistently; its selfconsistent equation is Within the same truncating approximation as that of the retarded Green’s function, the expression of can be simply written in the Keldysh form The matrix elements of the lesser selfenergy are In general and thus Eq. 6 of the current is reduced to the LandauerBüttiker formula for the noninteracting electrons [27]
and then the total transmission T_{σ}(ɛ) for each spin component can be expressed as The linear conductance G_{σ}(ɛ) is related to the transmission T_{σ}(ɛ) by the Landauer formula at zero temperature [28], G_{σ}(ɛ) = (e^{2}/h)T_{σ}(ɛ).
Results and Discussion
In the following numerical calculations, we set the temperatureT = 0 throughout the article. The local density of states in the leads ρ is chosen to be 1 andt = 0.4 so that the corresponding linewidth is set to be the energy unit.
Figure 2a–c shows the dependence of the conductance G_{σ} and spin polarization on the Fermi level ɛ for λ = U = 0 and various Δt. The two dots now are connected in a serial configuration and the conductance of each spin component is composed of two BreitWigner resonances peaked at respectively [18,21]. Since the phase factors originating from both the magnetic flux and the RSOI do not play any role, the device is free from their influences. When Δt = 0, the spinup and spindown conductances are the same and the spin polarization p = 0 as shown by the solid lines in the three figures. With increasing Δt, the distance between the spinup resonances is enhanced whereas that between the spindown ones is shrunk because of as shown in Fig. 2a, b. Meanwhile, the spin polarization p increases accordingly. If Δt is set to be Δt = t_{c}, the spinup and spindown interdot tunneling coupling strengths are and respectively. Then the spinup conductance has a finite value but meanwhile as the conduction channel for the spindown electrons breaks off, which is shown by the dotdashed lines in Fig. 2a, b. The spin orientation of the nonzero conductance can be readily reversed by tuning the direction of the magnetic field, which is applied on the tunnel junction between the dots, to set Δt = −t_{c}.
Figure 2. Spindependent conductance G_{σ} and spin polarization p as functions of the Fermi level ɛ with λ = U = 0 and various Δt. In this and all following figures, the normal interdot tunneling coupling t_{c} = 1 and the dots ’ levels are ɛ_{1} = ɛ_{2} = 0
We now study how the dotlead coupling configuration influences the spin filtering effect in Fig. 3 by varying the value of λ. It is found that if the parameters are set to be Δt = t_{c} and φ = ϕ_{R} = π/2, the spindown conductance remains to be zero for any λ, and then only is plotted. For nonzero λ, the transmission T_{σ}(ɛ) is
where ɛ_{0} = ɛ_{1} = ɛ_{2}. Since and the spindown transmission regardless of the choice of λ. The spinup conductance is composed of one broad BreitWigner and one asymmetric Fano resonance centered, respectively, at the bonding and antibonding states [18,21]. Detail investigation of this spindependent Fano lineshape can be found in our previous papers and we do not discuss it anymore here. It should be indicated that the spin orientation of the nonzero conductance can be reversed by setting Δt = −t_{c} and φ = −π/2 + 2nπ with n is an integer.
It is known that the Coulomb interaction in the QDs plays an important role and we now study if the spin filtering effect survives in the presence of it. Figure 4a shows that the conductance of the spin down electrons is still zero and that of the spin up shows typical Fano resonance. Due to the existence of the intradot Coulomb interaction, two resonances emerge in higher energy region. Moreover, the positions of the bonding and antibonding states can be readily exchanged by tuning the magnetic flux as shown in Fig. 4b, where φ is changed from π/2 to 5π/2. Since the Fano effect is a good probe for quantum phase coherence in mesoscopic structures, the tuning of its resonance position and the asymmetric tail direction is an important issue. To date, much works have been devoted to this topic concerning both the charge and the spindependent Fano effect. But most previous works about the Fano effect in QDs ring ignored the Coulomb interaction [18], especially when the spin degree of freedom is considered [20,21], and this limitation is supplemented here.
Figure 4. Spindependent conductance G_{σ} as a function of the Fermi level for fix ϕ_{R} = π/2,U = 4, Δt = t_{c} and different φ. In this and the following figure, the solid and the dashed lines are for the spinup and spindown electrons, respectively
In fact, to realize the RSOI in a tiny device such as the QDs is somewhat difficult, and then we study if the spin filtering effect can be found in the absence of it. In Fig. 5, we set ϕ_{R} = 0 and plot the two spin components conductance by varying Δtand the magnetic fluxinduced phase factor φ. Figure 5a shows that when Δt = t_{c}and φ = π + 2nπ withnis an integer, the conductance of the spinup electrons still has finite value whereas that of the spindown electrons is exactly zero. Moreover, to swap the spin direction of the nonzero conductance, one can simply tune Δtfromt_{c}to −t_{c}with unchanged magnetic flux as shown in Fig. 5b. The peaks’ width and position of the nonzero conductance in Fig. 5a, b are the same, indicating that one can flip the electron spin in the bonding and antibonding states without affecting its sate properties.
Figure 5. Spindependent conductance G_{σ} as a function of the Fermi level for fix ϕ_{R} = 0,U = 4, φ = π and different Δt
Conclusion
In conclusion, we have investigated the spin filtering effect in a double QDs device, in which the two dots are coupled to external leads in a configuration transiting from serialtoparallel geometry. We show that by properly adjusting the spindependent interdot tunneling coupling strengtht_{σ}, a net spinup or spindown conductance can be obtained with or without the help of the RSOI and the magnetic flux. The spin direction of the nonzero conductance can be manipulated by varying the signs oft_{σ}. The above means of spin control can be fulfilled for a fixed RSOIinduced phase factor, and then the QDs in the present system can be either a gated or a selfassembly one, making it easier to be realized in current experiments.
Acknowledgment
This work was supported by the National Natural Science Foundation of China (Grant Nos. 10647101 and 10704011).
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