Quantum Transport through a Triple Quantum Dot System in the Presence of Majorana Bound States

Transcription

1 Commun. Theor. Phys. 65 (016) 6 68 Vol. 65, No. 5, May 1, 016 Quantum Transport through a Triple Quantum Dot System in the Presence of Majorana Bound States Zhao-Tan Jiang ( ), Zhi-Yuan Cao ( ì ), and Cheng-Cheng Zhong ( ) School of Physics, Beijing Institute of Technology, Beijing , China (Received December, 015; revised manuscript received January 7, 016) Abstract We study the electron transport through a special quantum-dot (QD) structure composed of three QDs and two Majorana bound states (MBSs) using the nonequilibrium Green s function technique. This QD-MBS ring structure includes two channels with the two coupled MBSs being Channel 1 and one QD being Channel, and three types of transport processes such as the electron transmission (ET), the Andreev reflection (AR), and the crossed Andreev reflection (CAR). By comparing the ET, AR, and CAR processes through Channels 1 and, we make a systematic study on the transport properties of the QD-MBS ring. It is shown that there appear two kinds of characteristic transport patterns for Channels 1 and, as well as the interplay between the two patterns. Of particular interest is that there exists an AR-assisted ET process in Channel, which is different from that in Channel 1. Thus a clear X pattern due to the ET and AR processes appears in the ET, AR, and CAR transmission coefficients. Moreover, we study how Channel affects the three transport processes when Channel 1 is tuned in the ET and CAR regimes. It is shown that the transport properties of the ET, AR and CAR processes can be adjusted by tuning the energy level of the QD embedded in Channel. We believe this research should be a helpful reference for understanding the transport properties in the QD-MBS coupled systems. PACS numbers: b, 73.1.La, c Key words: electron transport, quantum dot, Andreev reflection 1 Introduction In recent years, the exploration of Majorana fermions, as well as the manipulation of the Majorana bound states (MBSs), has drawn extensive attention because they are their own antiparticles obeying the non-abelian statistics and have great possible application in the field of topological quantum information. [1 3] It has been reported that the MBSs can be formed in many solid state systems, such as the vortex core in a p-wave superconductor [4 5] and superfluid, [6 7] and the ends of a semiconductor wire with strong spin-orbit coupling, which is placed in proximity to an s-wave superconductor. [8 10] Correspondingly, many schemes used to verify the existence of the MBSs, such as the noise measurements, [11 1] the resonant Andreev reflection, [13] the Majorana Josephson currents, [14] and so on, have been proposed. Note that an important way to detect the MBSs is to study the transport properties through a quantum dot (QD) structure coupled with the MBSs. Usually, the MBSs can be coupled to the QD system as a side part or an embedded one. In 011 Liu et al. studied the conductance of a QD structure side-coupled with the end of a p-wave superconducting nanowire and found that the conductance may be viewed as a probe of the existence of the MBSs. [15] Then Cao et al. found that a subtraction of the source and drain currents can expose the essential feature of the MBSs by considering the Majorana s dynamic aspect of the QD coupled with the MBSs. [16] In 013 Wang et al. used a QD-MBSs-QD structure to study the nonlocal quantum nature of the MBSs, which provides another evidence for the existence of the Majorana fermions. [17] Moreover, Zocher and Rosenow studied the charge transport through the QD-MBSs-QD structure, which shows that the nonlocality of the MBSs opens the possibility of the crossed Andreev reflection (CAR). [18] Furthermore, Liu et al. reported that the two nonlocal processes including the CAR and the electron tunneling (ET) can be directly controlled by the energy levels of the QDs. [19] Also, Li et al. [0] and Wang et al. [1] studied the quantum transport through the double QD system coupled with the MBSs. Moreover, Gong et al. studied the transport through a transverse T-shaped QD array with the side MBSs, [] and Jiang et al. studied the tunable quantum transport through the horizontal QD array with the MBS side-coupled to the QDs, indicating the feasibility to manipulate the current by means of the QD-MBS coupling. [3] These researches indicate that the MBSs make serious influences on the transport properties Supported by National Natural Science Foundation of China under Grant No , and by the Program for New Century Excellent Talents in University under Grant No. NCET Corresponding author, jiangzhaotang@hotmail.com c 016 Chinese Physical Society and IOP Publishing Ltd

2 No. 5 Communications in Theoretical Physics 63 of the QD systems. A deep investigation on the transport properties of the QDs coupled with the MBSs is needed for comprehensive understanding the nature of the MBSs. Motivated by these researches, we design a new QD- MBS ring structure including the MBS arm and the QD arm, as shown in Fig. 1. In this QD-MBS ring, the MBS arm serves as Channel 1, and the QD arm as Channel. Therefore, it can be anticipated that this QD-MBS ring can show more interesting properties than that with only one arm. Based on the Green s function method, we study the transmission coefficients of the QD-MBS ring with the emphasis on the competition between the two arms. We find that clear X patterns due to the ET and AR processes can be observed in the ET, Andreev reflection (AR), and CAR transport coefficients. Also, we study the transport properties when the MBS arm is tuned in the ET regime and the CAR regime. It is shown that the transport processes can be manipulated by adjusting the energy level of QD. This study exposes lots of interesting transport mechanisms in the ET, AR, and CAR transport processes of the QD-MBS ring. Fig. 1 (Color online) Schematic illustration of the QD- MBS ring structure including the MBS arm (Channel 1) and the QD one (Channel ). The two MBSs η 1 and η are coupled to QD1 and QD3 by λ 1 and λ, respectively, while QD is coupled to them via the interdot couplings t 1 and t. By the tunneling couplings Γ L and Γ R the QD-MBS ring is connected to the left and right leads. The rest of the paper is organized as follows. In Sec., the model Hamiltonian is presented, and the Green s functions as well as the current formula and the transmission coefficients are derived. In Sec. 3, we numerically investigate the transmission coefficients through the QD-MBS ring. Finally, a brief conclusion is given in Sec. 4. Model The QD-MBS ring structure including two MBSs and three QDs sandwiched between the left and right leads is schematically shown in Fig. 1. The lower arm is a semiconductor nanowire with the strong Rashba spin-orbit interaction and the proximity-induced s-wave superconductivity, and the upper one is just QD. The entire QD-MBS ring system can be expressed by the Hamiltonian [19 3] H = H D + H L + H M + H LD + H MD. (1) Here H D represents the Hamiltonian of the three coupled QDs H D = i ε i d i d i + (t 1 d 1 d + t d d 3 + H.c.), () where d i and d i are the creation and annihilation operators of the electron in QDi (i = 1,, 3), respectively. ε i is the QD energy level for QDi, and the coefficient t 1 (t ) the tunneling coupling between QD1 (QD3) and QD. H L = kα=l,r ε αkc αk c αk describes the left and right leads, and the couplings between the leads and the related QDs are expressed as H LD = k (V L d 1 c Lk + V R d 3 c Rk + H.c.), (3) where c αk is the annihilation operator of the electron at the continuous state k in Lead α, and V L (V R ) is the tunneling coupling between QD1 and Lead L (QD3 and Lead R). The low-energy effective Hamiltonian H M = iε M η 1 η (4) describes the paired MBSs formed at the ends of the nanowire and coupled to each other by an energy ε M e l/ξ, in which l is the length of the nanowire, and ξ the superconducting coherence length. Also, η 1 and η denote the corresponding Majorana fermion operators. The tunneling coupling H MD between the QDs and the MBSs takes the following form H MD = (λ 1 d 1 λ 1d 1 )η 1 + i(λ d 3 + λ d 3 )η, (5) where λ 1 represents the coupling strength between QD1 and the MBS on the left, and λ the coupling strength between QD3 and the MBS on the right. For convenience, the Majorana fermion operators should be replaced by the conventional fermion operators through the relations η 1 = (f + f)/ and η = i(f f)/, where f and f are the regular fermion operators satisfying the anticommutative relation {f, f} = 1. Thus, in the new representation the related Hamiltonians H M and H MD become H M = ε M (f f 1/), (6) H MD = (λ 1 d 1 λ 1d 1 )(f + f)/ (λ d 3 + λ d 3 )(f f)/. (7) Then we make use of the nonequilibrium Green s function technique to study the quantum transport through the QD-MBS ring structure. Based on the equation of motion method, the retarded Green s function of the entire system can be derived in the Nambu representation as

3 64 Communications in Theoretical Physics Vol. 65 G R = g t λ 1 λ 1 0 g t λ 1 λ 1 t 1 0 g 1 0 t t 1 0 g t 0 g t 0 0 λ λ t 0 g 1 3 λ 1 λ λ λ λ λ g 1 M 0 λ 1 λ λ λ 0 g 1 M 1. (8) /, g 1 1 = ω + ε 1 + iγ h L /, 3 = ω ε 3 + iγ e R /, g 3 1 = ω + ε 3 + iγ h R /, g 1 M = ω ε M, and g 1 M = ω + ε M with the linewidth functions Γ e,h L,R π V L,R ρ e,h L,R (ω). Here g 1 1 = ω ε 1 + iγ e L g 1 = ω ε, g 1 = ω + ε, g 1 According to the time-dependent evolution of the electron number in the left lead, as well as the standard derivation procedures, we can obtain the current flowing in the left lead as J L = e/h dω[t E (f e L f e R) + T A (f e L f h L) + T H (f e L f h R)]. (9) Here T E = Γ e L Gr 15 Γe R Ga 51 is the electron transmission coefficient, T A = Γ e L Gr 1 Γh L Ga 1 the local AR transmission coefficient, and T H = Γ e L Gr 16Γ h R Ga 61 the CAR transmission coefficient. G r,a ij are the matrix elements of the retarded and advanced Green s functions G R,A. fα e,h = 1/[1 + e (ω±µα)/kbt ] is the Fermi function of the electron or hole in Lead α. Within the wide-band limit approximation, we have Γ e α = Γ h α = Γ α. 3 Results and Discussions In this section we will perform a numerical investigation on the quantum transport properties of the QD-MBS ring. Here the temperature T is chosen to be zero in the calculation, and the unit of the related parameters is selected to be 10 mev as adopted in the previous research works. [18,] For clarity, our study is focused on the weak coupling case with Γ L = Γ R = Note that two channels including the MBS arm (Channel 1) and the QD arm (Channel ) can be used for the electron to transport from Leads L to R. 3.1 Symmetrical Case with ε 1 = ε 3 = 0 First of all, we consider the electron transmission coefficients T E, T A, and T H of the QD-MBS ring with ε 1,3 = 0, t = t 1, = 0.01, and λ = λ 1, = 0.01 in Fig.. In the weak coupling case with ε M = 0.0, this structure can be viewed as a triple QD system including a left QD (with three energy levels E 0,± ), a mediate QD (with energy level ε ), and a right QD (with three energy levels E 0,± ). Here E 0 = 0 and E ± = ± λ denote the molecular energy levels of the single QD coupled with the MBS. When ε is aligned with E 0,±, the electron transport will be more unblocked. Figures (a), (d), (g) show the three kind transport processes corresponding to ε M = 0, where only Channel is open. In Fig. (a) we can find that the ET transmission coefficient T E shows the typical characteristic pattern of the isolated serial triple QD structure, where the resonant tunneling appears whenever E = ε, while the linear transmission coefficient T E (E = 0) is suppressed for all different ε of QD which is due to the couplings with the side MBSs. In sharp contrast to T E, the corresponding T A and T H show observable values only at E 0. For ε ε 1,3, T A (E 0) dominates the transmission while T H (E 0) does when ε ε 1,3. This should be reasonable since it is advantageous for the electron to transport from QD1 to QD3 via QD when ε is aligned with ε 1,3, which strengthens the CAR process and simultaneously suppresses the AR one. Furthermore, we turn to the case of the nonzero inter-mbs coupling ε M 0, which means that both Channels 1 and are open. By comparing Figs. (b), (c) with Fig. (a), we can find that the T E (E = 0) = 0 region becomes disappeared and two extra ET transport peaks are observed near E = ±ε M. Also, the transmission coefficients of T A and T H near E = 0 are seriously suppressed and two transmission regions appear at E = ±ε M, as shown in Figs. (e), (f) and (h), (i). Actually, for ε M 0 the coupling λ 1 with the energy level of QD1 will form three energy levels as E 0 = 0 and E ± = ± ε M + λ 1 which is approximately equal to ±ε M when ε M = 0.1, 0. and λ 1 = When the incident electron is injected with the energy E 0,±, the ET, AR, CAR processes will be strengthened, inducing the patterns as shown in Fig.. However, at the position

4 No. 5 Communications in Theoretical Physics 65 near ε = ε M, all the three processes for the incident electron with E = E ± are prohibited surprisingly, as depicted in Figs. (c), (f), (i). This is because the couplings t 1, between QDs 1, 3 and QD push the coupled molecular energy levels away from E ±, thus preventing the electron coming from Lead L from entering into QD1 and leading to the suppression of the three transport processes. Fig. (Color online) The transmission coefficients T E, T A, and T H are shown in panels (a) (i) in the case of λ 1, = 0.01 with ε M = 0.0, 0.1, and 0., respectively. The other parameters are selected to be Γ L,R = 0.03, ε 1,3 = 0.0, and t 1, = The color scale bars of (a) (d) are that with scale from 0 to 1, and those of (e) (i) are that with scale from 0 to 0.3. The dashed rectangles in (e) and (h) are just the guidance for your attention. Fig. 3 (Color online) The transmission coefficients T E, T A, and T H are shown in panels (a) (i) in the case of λ 1, = 0.1 with ε M = 0.0, 0.1, and 0., respectively. The other parameters are selected to be Γ L,R = 0.03, ε 1,3 = 0.0, and t 1, = 0.01.

5 66 Communications in Theoretical Physics Vol. 65 Then we consider the transport properties in the case of a stronger MBS-QD coupling λ = 0.1 in Fig. 3. Obviously, the increase of λ, which strengthens Channel 1 to adjust the competition between two channels, makes the serious variations of the transmission patterns. Under this condition, the molecular energy levels of the single QD coupled with the MBS are E 0 = 0 and E ± = ± ε M In Figs. 3(a), 3(d), 3(g) we can see that clear X patterns appear in the case of ε M = 0. For the ET process, a resonant ET peak will be observed when the incident electron is injected with energy E = ε, which is the normal ET process forming the E = ε branch in the X pattern. However, the appearance of the other branch with E = ε is unexpected. We attribute this branch to the AR-assisted electron tunneling between the two leads. In this process, a Cooper pair at the right end of the nanowire can be broken up into two electrons, and then one electron with energy E = ε enters the right lead and the other with energy ε enters QD. Furthermore this electron in QD may combine with the electron with energy ε injected from Lead L to form a new Cooper pair at the left end of the nanowire. Therefore, this unusual ET process can be observed when the energy of the incident electron is equal to ε, which produces the E = ε branch in the X pattern. As λ increases, the E = ε branch is suppressed while the E = ε is strengthened. In the same way, the X for the CAR process can be understood easily. For example, when the electron from Lead L is injected into QD1 with the energy E = E, two transport peaks can be found at ε = E and ε = E. When ε = E, this electron is transported to QD3 via QD and then combined with the electron from Lead R with energy E = E to form a Cooper pair at the η terminal. However, for the case of ε = E, this electron is combined to form a Cooper pair at the η 1 terminal with the electron from Lead R to QD1 via QD. As a response or compensation, the AR process will be suppressed in the area where the ET and CAR processes are strengthened, inducing the X pattern for the AR process. With the increase of ε M, new transport patterns of T E, T A, and T H through Channel 1 appear, as shown in Figs. 3(b), 3(e), 3(h), 3(c), 3(f), 3(i). Obviously, the transport patterns through Channel 1 is clearly strengthened and the two patterns for Channels 1 and coexist. 3. Symmetrical Case with ε 1 = ε 3 = ε M 0 Furthermore, we intend to study the transport properties through the QD-MBS ring in the symmetrical case of ε 1 = ε 3 = ε M with ε M = 0.. For the isolated Channel 1, the QD1-MBSs-QD3 structure can be tuned in the ET regime when ε 1 = ε 3 = ε M and the CAR regime when ε 1 = ε 3 = ε M. For clarity, we will discuss how Channel affects Channel 1 in these two special regimes separately. Fig. 4 (Color online) The transmission coefficients T E, T A, and T H are shown in panels (a) (i) in the case of ε M = 0., ε 1 = ε M, and ε 3 = ε M with t = t 1, = 0.01, 0.05, and 0.1, respectively. The other parameters are selected to be Γ L,R = 0.03 and λ 1, = The color scale bars of (a) (c) are that with scale from 0 to 1, those of (e), (f), and (h), (i) are that with scale from 0 to 0.3, and those of (d) and (g) are that with scale from 0 to 0.0. The dashed ellipses are just the guidance for your attention.

6 No. 5 Communications in Theoretical Physics 67 Figure 4 shows the transmission coefficients T E, T A, and T H for the different interdot couplings t. In the weak coupling case of t = 0.01, by comparing Figs. 4(a), 4(d), 4(g) with Figs. (c), (f), (i), we can find that the T E peak region is moved to the proximity of E = ε 1,3, while the T A and T H peak regions are kept invariant except the suppression of their heights. It is clear that the E = ε peak region, induced by Channel, is superimposed on the E = ε M line in Fig. 4(a). With the increase of t, Channel is more favorable for the electron transport than Channel 1, which makes the typical ε -dependent characteristic pattern of the ET transmission coefficient much clearer. Also, the AR and CAR processes through Channel 1 are further suppressed, while those by Channel are strengthened, which shows the peaks at the positions of E = ±ε M and ε ε M. This clearly demonstrates the competition between Channels 1 and for Channel 1 in the ET regime. 3.3 Asymmetrical Case with ε 1 = ε 3 = ε M 0 Finally, we consider how Channel affects Channel 1 when Channel 1 is in the CAR regime in Fig. 5. It is clear that the CAR process shows a peak line near E = ε M through Channel 1 as shown in Fig. 5(g). When ε is tuned to be close to ε 1,3, the couplings t 1, will move the coupled molecular energy level far away from ε 1,3, thus preventing the CAR process to form the suppression of T H near ε = ε 1,3. Clearly, this suppression becomes much stronger as t increases. Due to the same reason, transport forbidden regions through the QD-MBS ring near ε = ε 1,3 for the ET process are also observed. For the AR process, the transport shows different behaviours from those for the CAR process. Obviously, the AR process is weakened near ε = ε M, while that is strengthened near ε = ε M. Why? In the case of ε = ε M, the energy level of QD is close to that of QD1, which on one hand blocks the ET and CAR processes, and on the other hand extracts the electron in QD1 to QD to suppress the AR process. However, in the case of ε = ε M, the electron tunneling from QD1 to QD is prohibited, which strengthens the AR process. This clearly shows the influence of Channel on Channel 1 when Channel 1 is in the CAR regime. Fig. 5 (Color online) The transmission coefficients T E, T A, and T H are shown in panels (a) (i) in the case of ε M = 0., ε 1 = ε M, and ε 3 = ε M with t = t 1, = 0.01, 0.05, and 0.1, respectively. The other parameters are selected to be Γ L,R = 0.03 and λ 1, = The color scale bars of (a) (c) and (g) (i) are that with scale from 0 to 1, those of (d) (f) are that with scale from 0 to The dashed circle and ellipse are just the guidance for your attention. 4 Conclusion In conclusion, we have studied the transport properties of the QD-MBS ring by using the Green s function method. For clarity, our calculation is focused on the weak coupling case. We first study the transport properties of the QD-MBS ring for the different couplings between the two MBSs in the symmetrical case with QD1 and QD3 having the same zero energy levels. The competition among the transport coefficients via the two Channels is investigated in detail, which shows that a clear X pattern due to the ET and AR processes will appear. Then we discuss the transport properties for the

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