Abstract
We investigate the spin accumulations of AharonovBohm 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 AharonovBohmFano effect is a necessary condition to achieve the spin accumulation in the quantum dot of the resonant channel. Further analysis showed that in the doubledot 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.
Keywords:
Spin accumulations; AharonovBohmFano effect; quantum dot; Coulomb interaction; 73.63.Kv; 71.70.Ej; 72.25.bBackground
Quantum dot (QD), especially coupledQD 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 multiQD systems in the past decades [310]. Besides, the progress of nanotechnology enables researchers to fabricate a variety of coupledQD structures with sizes smaller than the electron coherence length [11]. This also accelerates the development of researches on the coupledQD characteristics.
With respect to the coupledQD structures, the typical one is AharonovBohm (AB) interferometer with one QD or whose individual arm is of one QD, respectively [1238]. 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 [1820]. 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 ABFano 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 ABFano effect and the other mechanisms, e.g., Kondo physics and the spinorbit interaction, indeed causes many interesting phenomena [2438].
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 [3942]. Various schemes have been proposed: by considering the spin transport and spin accumulation in QDs, based on the magnetic means, the spinobit interaction, etc.[4349]. 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 [5054]. We intend to investigate the role of quantum interference in adjusting the spinbiasinduced spin accumulation. ϵ_{F} is the Fermi level of the system at the zerospinbias 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 [5560]. 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 ABFano 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
The Hamiltonian that describes the electron motion in the AB interferometer can be written as
H_{α}(α=L,R) is the Hamiltonian in leadα. 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 leadα. ϵ_{αkσ} is the singleparticle 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 oneQD AB interferometer
We first focus on the AB interferometer of one QD, whose schematic is shown in Figure 1a. Then in such a case, and are the creation (annihilation) operator of electron in the QD, and ϵis the energy level of QD. U is the intradot electron interaction strength. W_{LR} denotes the direct transmission between the leads, and V_{α} represents the coupling between the QD and leadα.
Figure 1. The AB interferometer of one QD. (a) Schematic of an AB interferometer with an embedded QD. (b, c, d) The average electron occupation number and spin accumulation in QD affected by the structure parameters. The relevant parameters are taken to be ρ = 1, V_{α}= 0.1, and eV_{s} = 1.0.
The electron properties can be evaluated by using the nonequilibrium Green function technique. In the Green function space, the average electron occupation number of the QD is denoted as [61,62]
G^{<} is the lesser Green function, which can be obtained from the Dyson equation
G^{r} and G^{a} are the retarded and advanced Green functions, respectively. Due to the presence of electron interaction, the Green function is difficult to solve. However, if the system temperature is higher than the Kondo temperature, the electron interaction term can be included by using the HubbardI approximation [6163]. In this work, we would like to consider the case of weak electron correlation; then, the retarded Green function can be analytically solved within the HubbardI approximation, i.e.,
where is the Green function of the isolated leadα. Due to the continuum states in the leads, we write with ρ(ω) being the density of states of the leads. , the Green function of the isolated QD, can be written as in the HubbardI approximation. In this structure, can be expressed as , where is the lesser Green function of the isolated leadα with the Fermi distribution function . When the bandwidth of the leads is large enough, the density of states can be viewed as a constant. Accordingly, we have . As a result, 〈n_{σ}〉 can be reexpressed as
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 eV_{s}=1.0 and the QDlead coupling strength V_{α}=0.1. In Figure 1b, it is observed that at the case of ϕ=0.5Π, a spinup 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 spindown electron if ϵ<0.5. By comparing the properties of oppositespin electrons, we might as well consider that the spinup and spindown 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 ϕand 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 differentspin 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 spinup 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}=We^{iϕ}. Here, W is the strength of the leadlead coupling, and the phase factor where Φ is the magnetic flux with the magnetic flux quantum . In the case of weak QDlead 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 leadlead 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 spinσ 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., Δθ_{dL,σ}=[θ_{R}−ϕ] with θ_{α} being the argument of . Similarly, the two transmission paths between leadR and the QD can be given by and with Δθ_{dR,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 leadL. However, the quantum interference between and is constructive since Δθ_{dR,σ}=0 in such a case. So, the QD only feels leadR with 〈n_{σ}〉=f_{Rσ}(ϵ). Oppositely, for the case of , only the property of leadL influences the electron in the QD. So far, we have noted that the ABFano effect modulates the quantum interference that contributes to the electron distribution in the QDs.
In the following, we incorporate the electron interaction into the calculation. In the case of weak QDlead coupling, 〈n_{σ}〉 can be expressed in an analytical way, i.e.,
In Figure 2, by assuming W=0.3 and , we show the spin accumulation in the QD in the cases of , 1.0, and 2.0, respectively. As shown in Figure 2a, for the cases of U≤eV_{s}, the energy region where the spin accumulation emerges is directly widened, and in the whole region of , the spin polarization is robust. Thus, the intradot Coulomb interaction benefits the spin accumulation in the QD. Such a result can be explained in the following way: When , the QD only ‘couples to’ leadR in which and . Consequently, when the QD level ϵis shifted below μ_{R↓}, a spindown electron will occupy such a level, but at this time, the level ϵ + U is unoccupied. Next, when the level ϵ + Uis below μ_{R↓}, the Pauli exclusion principle makes it empty, so the spin accumulation in the QD is equal to −1 approximately. Only when the level ϵ + U decreases to the position of μ_{R↑}does a spinup electron have an opportunity to occupy it, and then, the spin accumulation disappears. However, with regard to the case of U=2.0, we see that with the decrease of ϵto −0.5, the magnitude of spin accumulation goes down deeply, and around the region of ϵ=−1.0, the value of 〈n_{s}〉 almost encounters its zero. However, by a further adjustment of ϵ_{0}to ϵ_{0}=−1.5, such a spin accumulation then gets close to 1 again. We can understand this phenomenon as follows. Since the strong Coulomb interaction, the levels ϵand ϵ + U will not be located in the spin bias window simultaneously. Then, they respectively contribute to the spin accumulation. So, in the regions of −0.5<ϵ<0.5 and −0.5<ϵ + U<0.5, there emerges an apparent spin accumulation. However, when the level is shifted around the point of ϵ=−1.0, the level ϵ + U is above μ_{R↓}, so it is unoccupied. Then, the level ϵ, which is below μ_{R↑}, will confine the differentspin electrons with the same ability. So far, we have known the role of electron interaction in adjusting the spin accumulation in such a structure.
Figure 2. The influence of Coulomb interaction on the properties of 〈n_{σ}〉 (a) and 〈n_{s}〉 (b). The relevant parameters are taken to be W = 0.3, V_{α} = 0.1, and .
The doubleQD AB interferometer
The AB interferometer with one QD in each of its arm (see Figure 3a) is another typical structure in studying the electron transport behaviors modified by the AB phase. For such a structure, both H_{D} and H_{T} have alternative forms as , and . is the creation (annihilation) operator of electron in QDj. ϵ_{j} is the corresponding QD level, and U_{j} denotes the intradot electron interaction strength. V_{αj} represents the coupling between QDj and leadα.
Figure 3. The AB interferometer with two QDs. (a) Schematic of an AB interferometer with one QD in each of its arm. (b) The spin accumulation in QD2 influenced by the properties of the other arm. (c) The influence of the QD levels on the spin accumulation in QD2. (d) The spin accumulation in QD2 affected by the local magnetic flux. The spin bias is fixed with eV_{s} = 1.0.
Here, we would like to know whether the Fano interference manner is also necessary to achieve the spin accumulation of such a structure. If so, how do the properties of nonresonant channel affect the spin accumulation? Based on such an idea, we begin to analyze the average electron occupation number of QDj by the formula . The lesser Green function G^{<}(ω) also obeys the relationship in Equation 3, and the retarded Green function G^{r} can be expressed as
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 QD2 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: V_{L1}=V_{1}e^{iϕ/4}, , V_{R2}=V_{2}e^{iϕ/4}, and . V_{j} is the strength of the QDlead coupling. The numerical results are shown in Figure 3b,c,d. In Figure 3b, by fixing , ϵ_{2}=0, and V_{2}=0.1, we plot the spectrum of spin accumulation in QD2 vs ϵ_{1}and V_{1}. It is obvious that the increase of V_{1}can efficiently enhance the spin accumulation in QD2. 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 ABFano effect, but not the AB effect, that promotes the spin accumulation. Besides, it shows that the level of QD1 plays a nontrivial role in affecting the spin accumulation. To be precise, when the level of QD1 is shifted around the zero energy point, there will be no spin accumulation in QD2. When the level of QD1 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 〈n_{2s}〉 will change when the level ϵ_{1}exceeds the zero energy point. Therefore, in comparison with the oneQD AB interferometer, we can find that the spin accumulation of this structure can be manipulated flexibly.
Next, we choose V_{1}=0.6 and investigate the spin accumulation in QD2 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 〈n_{2s}〉 lies where its sign ( + /−) is differentiated by the line of ϵ_{1}=ϵ_{2}, where the spin accumulation disappears. This result indicates that if the spin bias is large enough, at the point of ϵ_{1}=0, the sign of 〈n_{2s}〉 can be altered by the change of ϵ_{2}. On the other hand, in Figure 3d, we investigate 〈n_{2s}〉 as functions of ϕ and ϵ_{2}. The QDlead couplings are taken to V_{1}=5V_{2}=0.5, and the level of QD1 is fixed at ϵ_{1}=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 QD2 tends to affect the maximum of spin accumulation, which appears around the points of ϵ_{2}=0.25 and ϕ=±0.3Π. Thereby, we notice that the properties of the resonant channel, e.g., the level of QD2, are also important factors to change the magnitude of the spin accumulation.
For such a structure, it is difficult for us to write out the analytical expression of 〈n_{2s}〉. So, we can only present a qualitative discussion to explain the above results by analyzing the quantum interference that contributes to the spin accumulation. Obviously, the expression of can be written as the summation of two Feynman paths, i.e., . Then, the quantum interference feature determines the coupling strength between QD2 and the leads. However, it is found that
where and with . So, the coupling between QD2 and leadL is determined by the quantum interference among infiniteorder Feynman paths, different from that in the oneQD structure. This inevitably leads to the complicated features of the spin accumulation. Similarly, the three transmission paths between leadR and QD2 can be given by
Despite the complicated quantum interferences among infinite paths, we try to clarify the quantum interference feature by calculating the phase differences between the lowestorder paths. This is because the quantum interference among lowestorder paths contributes mainly to the coupling between QD2 and the leads. For instance, the three lowestorder paths between QD2 and leadL 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, ϵ_{1}=1, and , we get the result that , , and . So, the destructive quantum interference among these paths leads to the decoupling of QD2 from leadL. In such a case, however, the quantum interference among , , and is constructive since , , and . Thus, the spin bias of leadR determines the spin accumulation in QD2. Surely, θ_{1} 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 QD1 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 ϵ_{2}on the value of 〈n_{2s}〉. In Equations 9 to 10, one can find that in the higherorder 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. [63], 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 singlechannel structure, and then, the picture of quantum interference disappears. With this viewpoint, we understand the vanishment of the spin accumulation in the case of ϵ_{1}=ϵ_{2}.
In Figure 4, by choosing V_{1}=0.5,V_{2}=0.1, and , we investigate the influence of the intradot Coulomb interactions on the spin accumulation in QD2. From Figure 4a,b,c, we clearly find that the manybody effect in QD2 (i.e., the resonantchannel QD) on the spin accumulation is similar to that in the singleQD AB interferometer. Namely, in the case of U_{2}≤eV_{s}, e.g., U_{2}=1.0, the energy region where the spin accumulation emerges is directly widened. As a result, the spin polarization is always robust in the whole region of . In the case of strong Coulomb interaction, e.g., U_{2}=3.0 in Figure 4c, the spectrum of 〈n_{2s}〉 vs ϵ_{2} is divided into two groups, which are analogous to each other. This result is easy to understand in terms of the analysis about the manybody effect in the above subsection. Alternatively, in Figure 4a,b,c we see that the Coulomb interaction with the nonresonant channel plays a more significant role in modifying the spin accumulation. First, a nonzero U_{1} causes the energy region where the positive spin accumulation appears to shift to the lowenergy direction, and only when varying ϵ_{1}to ϵ_{1} + U_{1}≤0 can one see the positive spin accumulation. Secondly, with the further increase of U_{1}, in the middleenergy region where ϵ_{1}<0<ϵ_{1} + U_{1}, finite spin accumulation in QD2 is also observed. For instance, for the case of U_{1}=3.0, positive 〈n_{2s}〉 comes up around the point of ϵ_{1}=−0.5, whereas the negative 〈n_{2s}〉 occurs in the vicinity of ϵ_{1}=−2.5. We explain this result as follows. A finite U_{1}will lead to ϵ_{1}splitting into ϵ_{1}and ϵ_{1} + U_{1}. Accordingly, two nonresonant channels contribute to the quantum interference. When U_{1}=1.0, the two nonresonant channels have the opportunity to simultaneously act on the quantum interference. In the case of −1.0<ϵ_{1}<0.0, the sign of ϵ_{1} + U_{1} is greater than zero. Then, the electron waves in the two channels are phaseopposite, which significantly weakens the quantum interference and suppresses the spin accumulation in QD2. However, for a strong Coulomb interaction in QD1, when ϵ_{1} is tuned below the zero energy point, the level ϵ_{1} + U_{1}is still much greater than zero. Then in such a case, the Coulombinduced level contributes little to the quantum interference, and a singleelectron interference picture remains. Due to this reason, we find the positive spin accumulation in QD2 when ϵ_{1}=−0.5 in Figure 4b,c. On the contrary, when ϵ_{1}=−2.5, ϵ_{1} + U_{1}=0.5. Then, the coupling between ϵ_{1} + U_{1} and the leads provides a channel for the quantum interference, leading to the appearance of negative spin accumulation in QD2. The further decrease of ϵ_{1} will cause ϵ_{1} + U_{1} to be less than zero. Compared with the result of ϵ_{1}=−2.5, the change of the sign of ϵ_{1} + U_{1}brings about the positive 〈n_{2s}〉. In addition, note that when , one can obtain the result of ϵ_{1}=−ϵ_{1} + U_{1}. Then, the oppositephase electron waves in the two nonresonant channels contribute zero to the quantum interference, so in such a case, no spin accumulation occurs in QD2.
Figure 4. The influence of Coulomb interactions on the spectra of 〈n_{2s}〉. The relevant parameters are taken to be V_{1} = 0.5, V_{2} = 0.1, and . (a) The spectrum of 〈n_{2s}〉 vs ϵ_{1} and ϵ_{2}. The Coulomb interactions are taken to be U_{1} = U_{2} = 1.0. (b) The spectrum of 〈n_{2s}〉 with U_{1} = 3.0 and U_{2} = 1.0. (c) The curve of 〈n_{2s}〉 vs ϵ_{2} with U_{1} = U_{2} = 3.0. The level of QD1 is taken to be ϵ_{1} = 0 and ±0.5, respectively.
Summary
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 ABFano effect assists to manipulate the spin accumulation. Further analysis showed that the doubleQD 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.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
WJG designed the theoretical model, deduced the relevant formula, and drafted the manuscript. YH carried out the numerical calculations. GZW participated in the analysis about the results. AD improved the manuscript. All authors read and approved the final manuscript.
Acknowledgements
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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