Spin accumulation assisted by the Aharonov-Bohm-Fano effect of quantum dot structures
© Gong et al.; licensee Springer. 2012
Received: 11 April 2012
Accepted: 20 July 2012
Published: 17 September 2012
We investigate the spin accumulations of Aharonov-Bohm interferometers with embedded quantum dots by considering spin bias in the leads. It is found that regardless of the interferometer configurations, the spin accumulations are closely determined by their quantum interference features. This is mainly manifested in the dependence of spin accumulations on the threaded magnetic flux and the nonresonant transmission process. Namely, the Aharonov-Bohm-Fano effect is a necessary condition to achieve the spin accumulation in the quantum dot of the resonant channel. Further analysis showed that in the double-dot interferometer, the spin accumulation can be detailedly manipulated. The spin accumulation properties of such structures offer a new scheme of spin manipulation. When the intradot Coulomb interactions are taken into account, we find that the electron interactions are advantageous to the spin accumulation in the resonant channel.
KeywordsSpin accumulations Aharonov-Bohm-Fano effect quantum dot Coulomb interaction 73.63.Kv 71.70.Ej 72.25.-b
Quantum dot (QD), especially coupled-QD system (i.e., the QD molecule), is of fundamental interest in physics and possesses potential applications, such as quantum logic gates[1, 2]. As a result, many experimental and theoretical works have paid so much attention to the electron transport properties of various multi-QD systems in the past decades [3–10]. Besides, the progress of nanotechnology enables researchers to fabricate a variety of coupled-QD structures with sizes smaller than the electron coherence length . This also accelerates the development of researches on the coupled-QD characteristics.
With respect to the coupled-QD structures, the typical one is Aharonov-Bohm (AB) interferometer with one QD or whose individual arm is of one QD, respectively [12–38]. In such kind of structure, the AB phase can adjust the quantum interference, leading to abundant interesting results. Kobayashi et al. performed significant work to study the quantum interferences in the AB interferometers with embedded QDs [18–20]. According to their conclusions, the Fano effect, which manifests itself in the asymmetric lineshape of the transport spectrum, can be observed in such structures by constructing nonresonant and resonant channels for electron transmission. Moreover, they showed that the orientation of the Fano lineshape changes periodically with the magnetic flux. Due to this reason, in the AB interferometer with QDs, the AB-Fano interference attracted more attention and was further investigated [22, 23]. On the other hand, lots of theoretical investigations about electron transport behaviors of the AB interferometer have been reported. It was found that the interplay between the AB-Fano effect and the other mechanisms, e.g., Kondo physics and the spin-orbit interaction, indeed causes many interesting phenomena [24–38].
Electron not only has a charge but also spins with ; accordingly, the electron spin in the QD has been suggested as an ideal candidate for the qubit. Then, the coherent generation and control of electron spins in QDs has recently become one main subject in spintronics [39–42]. Various schemes have been proposed: by considering the spin transport and spin accumulation in QDs, based on the magnetic means, the spin-obit interaction, etc.[43–49]. Since in QD structures, the quantum interference contributes significantly to the electron motion properties, it is natural to question about the role of quantum interference on the spin accumulation. However, to our knowledge, little attention has been paid to such an issue so far. In this work, we choose the AB interferometer with embedded QDs and clarify the effect of a typical interference manner, i.e., the AB interference, on the spin manipulation in QDs. In doing so, we introduce a symmetric spin battery to the interferometer by considering the chemical potentials of the leads to be and [50–54]. We intend to investigate the role of quantum interference in adjusting the spin-bias-induced spin accumulation. ϵ F is the Fermi level of the system at the zero-spin-bias case, and V s is the magnitude of the spin bias. Due to the progress in experiment, such a scheme can be realized by injecting the charge current from a ferromagnetic source ( or a magnetic field ) into the leads of the QD structure [55–60]. Consequently, we find that to achieve the spin manipulation in the QDs of the AB interferometer, a finite magnetic flux and a nonresonant channel are prerequisites. Namely, the AB-Fano interference, not only the AB effect, is a necessary condition to realize the spin accumulation in the QDs. Also, the spin accumulation can be adjusted by varying the quantum interference of the interferometer. Therefore, we believe that such a structure is a promising candidate for spin manipulation.
Model and numerical results
H α (α=L,R) is the Hamiltonian in lead-α. H D is the Hamiltonian in the QDs, and the last term, H T , denotes electron traveling between the two leads. H α takes a form as , where is the creation (annihilation) operator corresponding to the basis in lead-α. ϵ αkσ is the single-particle level. Since we investigate the electron properties of two AB interferometers with one QD or two QDs, the expressions of H D and H T will be determined by the geometries of the interferometers.
The one-QD AB interferometer
Furthermore, by defining and 〈n s 〉=〈n ↑ 〉−〈n ↓ 〉, we can investigate the features of the charge and spin in the QD.
With the help of Equation 5, we investigate the average electron occupation number influenced by the structure parameters in Figure 1b,c,d. The system temperature is fixed at k B T=0.1. For the other parameters, we choose the spin bias e V s =1.0 and the QD-lead coupling strength |V α |=0.1. In Figure 1b, it is observed that at the case of ϕ=0.5Π, a spin-up electron can enter the QD only when the QD level decreases to the position of ϵ=−0.5. Instead, the QD is able to confine a spin-down electron if ϵ<0.5. By comparing the properties of opposite-spin electrons, we might as well consider that the spin-up and spin-down electrons are both in equilibrium, but they ‘feel’ the different ‘Fermi levels’, with the distance between them being the spin bias magnitude. Therefore, in such a structure, the striking spin accumulation can be realized in the QD. Next, in Figure 1c,d, by assuming ϵ=0, we present the influence of ϕ and W on the average occupation number of electron in the QD, respectively. It is observed that with the change of magnetic flux, the average occupation of different-spin electrons show opposite variation features. Different from the result of ϕ=0.5Π, when the magnetic flux is increased to ϕ=1.5Π, the QD confines a spin-up electron. Then, the spin accumulation in the QD can be completely adjusted. Alternatively, with W increased to W=0.3, the spin accumulation proportionally enhances; however, the further increase of W will lead to the suppression of spin accumulation.
Since the structure is relatively simple, we try to clarify the numerical result in an analytical way. Accordingly, we write out the expression of by using Equation 4, i.e., and with . When a local magnetic flux is applied, its effect on the quantum interference can be well defined by writing W LR =W e iϕ . Here, W is the strength of the lead-lead coupling, and the phase factor where Φ is the magnetic flux with the magnetic flux quantum . In the case of weak QD-lead coupling, e.g., |V α |=0.1, the analytical form of 〈n σ 〉 can be approximated as . χ=ΠρW, , , and with . The expression of 〈n c 〉 and 〈n s 〉 can then be obtained, i.e., and . We really find that the average charge occupation in the QD is independent of the presence of spin bias. However, in the case of finite spin bias, the factor Δ±ϕ, which is contributed by the local magnetic flux and direct lead-lead coupling, can adjust the value of 〈n s 〉. Also, the spin accumulation is an odd function of ϕ, so that the magnitude and sign of spin accumulation can be detailedly adjusted by the change of magnetic flux. Furthermore, we see that when ΠρW=±sinϕ=1, the expressions of 〈n dσ 〉 and 〈n s 〉 can be simplified. For the example of ΠρW=1 and , . Then, in such a case, the ‘Fermi level’ of the spin-σ electron is at the point of ϵ=μ Rσ , leading to the result that 〈n s 〉=f R↑ (ϵ)−f R↓ (ϵ). Next, when the magnetic flux is raised to , there will be 〈n dσ 〉=f Lσ (ϵ) and 〈n s 〉=f L↑ (ϵ)−f L↓ (ϵ). The property of the spin polarization is completely opposite to the case of . Based on such analysis, the spin accumulation in the QD is well understood.
The underlying physics being responsible for the above results is quantum interference. It is known that the interference in the QD ring structure is rather complicated. However, in such a structure, the quantum interference that affects the spin accumulation just occurs between two Feynman paths. This is because , where and with . It is evident that the phase difference between the two paths influences the magnitude of , hence changing the average electron occupation number in the QD. Via a simple calculation, the phase difference can be obtained, i.e., Δ θdL,σ=[θ R −ϕ] with θ α being the argument of . Similarly, the two transmission paths between lead-R and the QD can be given by and with Δ θdR,σ=[θ L + ϕ]. So, in the presence of finite magnetic flux, the amplitude of is different from that of . This leads to the different couplings between the QD and the leads. In the extreme case of ΠρW=1, the magnitudes of and are the same. Then, when , the destructive quantum interference between and causes to be equal to zero, which leads to the decoupling of the QD from lead-L. However, the quantum interference between and is constructive since Δ θdR,σ=0 in such a case. So, the QD only feels lead-R with 〈n σ 〉=f Rσ (ϵ). Oppositely, for the case of , only the property of lead-L influences the electron in the QD. So far, we have noted that the AB-Fano effect modulates the quantum interference that contributes to the electron distribution in the QDs.
The double-QD AB interferometer
where within the Hubbard approximation. Surely, Equation 5 is still suitable for evaluating the electron properties of this system.
Without loss of generality, we take QD-2 as an example to investigate the spin accumulation behaviors of such a structure. In the presence of magnetic flux, the coupling coefficients take the following form: VL 1=V1eiϕ/4, , VR 2=V2eiϕ/4, and . V j is the strength of the QD-lead coupling. The numerical results are shown in Figure 3b,c,d. In Figure 3b, by fixing , ϵ2=0, and V2=0.1, we plot the spectrum of spin accumulation in QD-2 vs ϵ1and V1. It is obvious that the increase of V1can efficiently enhance the spin accumulation in QD-2. This means that in the case of finite spin bias and magnetic flux, a nonresonant channel is necessary to realize the spin accumulation of such a structure. So, it is the AB-Fano effect, but not the AB effect, that promotes the spin accumulation. Besides, it shows that the level of QD-1 plays a nontrivial role in affecting the spin accumulation. To be precise, when the level of QD-1 is shifted around the zero energy point, there will be no spin accumulation in QD-2. When the level of QD-1 departs from the zero energy, finite spin accumulation emerges with its maximum approximately at the position where . The other important result is that the sign of 〈n2s〉 will change when the level ϵ1exceeds the zero energy point. Therefore, in comparison with the one-QD AB interferometer, we can find that the spin accumulation of this structure can be manipulated flexibly.
Next, we choose V1=0.6 and investigate the spin accumulation in QD-2 influenced by the change of QD levels, as shown in Figure 3c. We find that similar to the former structure, the spin accumulation occurs only when the corresponding QD level is located in the spin bias window. However, the characteristic of 〈n2s〉 lies where its sign ( + /−) is differentiated by the line of ϵ1=ϵ2, where the spin accumulation disappears. This result indicates that if the spin bias is large enough, at the point of ϵ1=0, the sign of 〈n2s〉 can be altered by the change of ϵ2. On the other hand, in Figure 3d, we investigate 〈n2s〉 as functions of ϕ and ϵ2. The QD-lead couplings are taken to V1=5V2=0.5, and the level of QD-1 is fixed at ϵ1=1. It is seen that the reversal of the magnetic flux direction can change the sign of spin accumulation, but in such a structure, the level of QD-2 tends to affect the maximum of spin accumulation, which appears around the points of ϵ2=0.25 and ϕ=±0.3Π. Thereby, we notice that the properties of the resonant channel, e.g., the level of QD-2, are also important factors to change the magnitude of the spin accumulation.
Despite the complicated quantum interferences among infinite paths, we try to clarify the quantum interference feature by calculating the phase differences between the lowest-order paths. This is because the quantum interference among lowest-order paths contributes mainly to the coupling between QD-2 and the leads. For instance, the three lowest-order paths between QD-2 and lead-L are , , and , and the phase differences are , , and , respectively, with θ j being the argument of . By a same token, we have the results that , , and ; and , , and . For a typical case of ω=0, ϵ1=1, and , we get the result that , , and . So, the destructive quantum interference among these paths leads to the decoupling of QD-2 from lead-L. In such a case, however, the quantum interference among , , and is constructive since , , and . Thus, the spin bias of lead-R determines the spin accumulation in QD-2. Surely, θ1 is dependent on ω, but one should understand that the quantum interference of ω=0 makes the main contribution to the spin accumulation. So, the accumulation of this structure. Meanwhile, note that only when the arm of QD-1 offers a nonresonant channel are the magnitudes of the paths close to one another, so that the quantum interference effect is clear.
Next, we demonstrate the effect of ϵ2on the value of 〈n2s〉. In Equations 9 to 10, one can find that in the higher-order paths, the two arms of the interferometer are visited repeatedly. Then, the properties of the two arms play an important role in affecting the quantum interference. In the study by Gong et al. , our calculations showed that when the levels of the two QDs are the same, the quantum interference between the two arms become weak, but only the nonresonant one determines the electron properties of this structure. As a consequence, in such a case, the interferometer can be considered as a single-channel structure, and then, the picture of quantum interference disappears. With this viewpoint, we understand the vanishment of the spin accumulation in the case of ϵ1=ϵ2.
In summary, we have studied the spin accumulation characteristics of two AB interferometers with QDs embedded in their arms by considering spin bias in the leads. It has been found that regardless of the configurations of the interferometers, the spin accumulations are strongly dependent on the quantum interference features of the interferometers. Namely, the nonresonant transmission ability between the leads and the local magnetic flux can efficiently adjust the spin accumulation properties of the QD. By analyzing the quantum interferences among the Feynman paths, it was seen that the quantum interferences can cause the QD in the resonant channel to be decoupled from one of the leads. Accordingly, the spin bias in one lead will drive the spin accumulation in such a QD. So, it is certain that the AB-Fano effect assists to manipulate the spin accumulation. Further analysis showed that the double-QD interferometer has advantages in manipulating the spin states in the resonant channel. In view of the obtained results, we propose the AB interferometers with QDs to be alternative candidates for spin manipulation in QD devices.
This work was financially supported by the National Natural Science Foundation of China (grant no. 10904010), the Natural Science Foundation of Liaoning Province (grant no. 201202085), the Fundamental Research Funds for the Central Universities (grant no. N110405010), and China Postdoctoral Science Foundation (grant no. 20100481206).
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