Majorana zero mode assisted spin pumping

Abstract

We present a theoretical investigation of Majorana zero mode (MZM) assisted spin pumping which consists of a quantum dot (QD) and two normal leads. When the coupling between the MZM and the QD is absent, d.c. pure spin current can be excited by a rotating magnetic field where low energy spin down electrons are flipped to high energy spin up electrons by absorbing photons. However, when the coupling is turned on, the d.c. pure spin current vanishes, and an a.c. charge current emerges with its magnitude independent of the coupling strength. We reveal that this change is due to the formation of a highly localized MZM assisted topological Andreev state at the Fermi level, which allows only the injection of electron pairs with opposite spin into the QD. By absorbing or emitting photons, the electron pairs are separated to opposite spin electrons, and then return back to the lead again, generating an a.c. charge current without spin polarization. We demonstrate the switching from d.c. pure spin current to a.c. charge current based on both Kitaev model and a more realistic topological superconductor nanowire. Although this switching can also be induced by partially separated Andreev bound state (ps-ABS) in the topological trivial phase, it is extremely unstable and highly sensitive to the Zeeman field, which is different from the switching induced by MZM. Our result suggests that quantum spin pumping may be a feasible local transport method for detecting the presence of MZMs at the ends of a superconducting nanowire.

Graphical abstract

Keywords

Majorana zero mode / spin pumping

1 Introduction

As a crucial component of topological materials, the topological superconductor (TSC) has attracted increasing attention both theoretically and experimentally in recent years [1-13]. TSC represents a class of superconductors with bulk pairing gaps and a pair of degenerate gapless Majorana zero mode (MZM) at the boundary or near a topological defect. An intriguing property of MZM is its non-locality along with non-Abelian braiding, which is the key to realize fault-tolerant topological quantum computing. The Kitaev model, based on a spinless p-wave superconducting chain with time reversal symmetry breaking, offers the most representative early model describing MZM [14,15]. However, since stable p-wave superconductors with spin triplet states are rare in nature [9,16,17], Fu and Kane [18] proposed a more realistic implementation scheme based on the combination of a traditional s-wave superconductor and a topological insulator to achieve an unpaired Majorana fermion. Lutchyn et al. [19] and Oreg et al. [20] provided an experimentally feasible setting to realize one-dimensional (1D) TSC by breaking the time reversal symmetry of an atomic chain with spin−orbit coupling (SOC) due to the superconducting proximity effect. Signatures of Majorana fermions have already been found in semiconducting nanowires with strong SOC in proximity to a superconductor [21-27], and at the end of atomic iron chains on the surface of a superconductor [28,29]. These studies provide effective methods and ideas for experimentally realizing and exploring MZMs in condensed matter physics, stimulating the rapid development of TSC in the past two decades [30-32].

Tunneling spectroscopy is a widely-used method for detecting and manipulating MZM in TSC systems. A notable feature is the appearance of a zero bias conductance peak (ZBCP), which was predicted to be 2e^2/h at the interface of a 1D normal metal and TSC nanowire [33]. Although the ZBCP has been observed in various semiconductor−superconductor hybrid systems [21,28,34], it remains a controversial criterion for confirming the presence of MZM due to alternative causes such as disorder [35-37], weak antilocalization [38] and coupling to a quantum dot [39,40]. Recently, a 2e^2/h ZBCP was observed in an InSb nanowire covered with superconducting Al, demonstrating robustness to external parameters such as the Zeeman field and tunnel barrier height, thus providing definitive evidence of MZM presence [41]. Subsequently, Moore et al. [42,43] pointed out that the 2e^2/h ZBCP may also be induced by ps-ABSs, which exists generically in the topological trivial phase of semiconductor−superconductor nanowire in the presence of Zeeman field. The authors suggested that only a non-local charge tunneling detection can distinguish true MZMs from ps-ABSs. However, another study found that the absence of ZBCP does not necessarily indicate the absence of MZM [44].

Since local d.c. detection of MZM always encounters difficulties, can it be detected from the perspective of a.c. transport to obtain at least some useful information? As a typical a.c. transport behavior, spin pumping has been recognized as an efficient method for generating and injecting spin into ferromagnetic/nonmagnetic material contacts [69]. The precession of magnetization caused by ferromagnetic resonance [46,47] or acoustic waves [48-50] can drive the spin angular momentum of the ferromagnetic metal into the nonmagnetic material, creating a pure spin current without any accompanying charge current in the contact. Since spin injection relies on dynamic spin exchange rather than charge transfer, it is not affected by the impedance mismatch across the ferromagnetic/nonmagnetic material interface, and thus has a much higher efficiency than regular electron transport. Thus far, spin pumping has been extensively used for spin injection in spin sink materials, such as nonmagnetic metals [51-53], semiconductors [54,55], and two-dimensional materials [56,57]. Recently, spin pumping based on the Kitaev chain has been theoretically studied [58-60]. The authors have shown that MZM can be found in the absorbed power of an a.c. magnetic field at low frequencies in the resonance approximation. Unlike the parametric excitations of spin waves in ferromagnets, spin pumping in 1D topological superconductors does not produce any parametric instability. Moreover, in the dynamical regime, the frequencies of Rabi oscillations for MZM are much higher than those for gapped extended states. However, in the steady-state regime, these Rabi oscillations are blurred by relaxation processes. Very recently, spin pumping in single-subband ferrromagnetic insulator−superconducting nanowires has been theoretically investigated. It was found that the spin pumping is robustly quantized when the hybrid nanowire is in the topologically nontrivial phase. The authors suggested that observation of correlated and quantized peaks in the conductance, entropy change and spin pumping would provide strong evidence of MZMs [61].

In this study, we investigated the spin pumping effect in a two terminal quantum dot system which is coupled to MZMs at the termini of a superconducting nanowire. A constant pure spin current can be induced in a spin pumping device without coupling to TSC by a rotating magnetic field, where low energy spin down electrons can be flipped to high energy spin up electrons by absorbing photons in the QD [62]. However, when one end of a TSC nanowire is coupled to the QD, a highly localized topological Andreev state is formed at the Fermi level even with weak coupling strength, which dominates the quantum transport of the spin pumping system. In this case, only electron pairs with opposite spin can be injected into the QD. By absorbing or emitting photons, this pair of electrons is separated and excited into opposite spin electrons, and then returns back to the lead again, generating an a.c. charge current without spin polarization. The switching from constant pure spin current to a.c. charge current suggests that spin pumping may be an effective method for detecting MZM in a TSC nanowire.

2 MZM involved spin pumping

We consider a spin-resolved three-probe system as shown in Fig.1 where a QD is Ohmic-contacted by two semi-infinite leads and coupled to one end of a 1D topological superconductor nanowire. A rotating magnetic field is applied to the QD producing a spin resolved pumped current. The effective Hamiltonian of the system can be written as [39,62]

(1)H=H_P+H_D+H_{DP}+H_B+H_M+H_{DM},

where H_P = \sum _{\alpha k}{ϵ}_{\alpha k}{c}_{\alpha k\sigma }^{†}{c}_{\alpha k\sigma } describes the spin resolved Hamiltonian of the left (\alpha =L) and right (\alpha =R) lead with \sigma =\uparrow ,\downarrow indicating the spin index. H_D = \sum _{\sigma }{ϵ}_d{d}_{\sigma }^{†}{d}_{\sigma } represents spin degenerate single energy level of the isolated QD, where {ϵ}_d can be adjusted by a gate voltage. H_{DP}=\sum _{\alpha k\sigma }{t}_{\alpha k\sigma }{c}_{\alpha k\sigma }^{†}{d}_{\sigma }+H.c. refers to the coupling between the QD and two metallic leads. H_B describes a time-dependent driving potential due to a rotating magnetic field \mathit{B}(t)=B_0(\sin \theta \cos \omega t\mathit{i}+\sin \theta \sin \omega t\mathit{j}+\cos \theta \mathit{k}), where B_0 is the magnitude of magnetic field, \theta is the angle between directions of the magnetic field and the rotation axis, and \omega is the rotating frequency. The rotating magnetic field is crucial to generate a pure spin current. For example, a counter-clockwise rotating field allows a spin down electron to flip to spin up electron by absorbing a photon, but does not allow a spin up electron to flip to spin down electron by absorbing a photon, and thus the angular momentum is conserved. H_B is assumed to have the following form [62]

H_B=\kappa [\text{exp}(-\mathrm{i}\omega t)d_{\uparrow }^{†}{d}_{\downarrow }+\text{exp}(\mathrm{i}\omega t)d_{\downarrow }^{†}{d}_{\uparrow }]

with \kappa =B_0\sin \theta defining the effective magnetic field perpendicular to the rotation axis. These four terms give a minimal model of spin pumping as described in the previous study [62]. H_M=\mathrm{i}{ϵ}_M{\gamma }_1{\gamma }_2/2 is the Hamiltonian of the topological superconductor nanowire described by Kitaev model, where {\gamma }_1 and {\gamma }_2 are two Majorana operators obeying {\gamma }_1{\gamma }_2=-{\gamma }_2{\gamma }_1 and {\gamma }_m^2=1. {ϵ}_M\sim {\mathrm{e}}^{-l/\xi } is the coupling strength between {\gamma }_1 and {\gamma }_2 with l the chain length and \xi the superconducting coherence length. H_{DM}=\sum _{\sigma }({\lambda }_{\sigma }{d}_{\sigma }-{\lambda }_{\sigma }^{\ast }{d}_{\sigma }^{†}){\gamma }_1 is the coupling between the QD and the adjacent MZM, where {\lambda }_{\uparrow }=\lambda and {\lambda }_{\downarrow }=-\mathrm{i}\lambda indicate the coupling strength. The assumption of a real {\lambda }_{\uparrow } and an imaginary {\lambda }_{\downarrow } in this work is not arbitrary, which represents the same coupling strength and a different coupling phase [40]. By defining two new ordinary fermion operators f = ({\gamma }_1+\mathrm{i}{\gamma }_2)/2 and f^{†} = ({\gamma }_1-\mathrm{i}{\gamma }_2)/2, H_M can be written as H_M = {ϵ}_M(f^{†}f-\frac{1}{2}), and H_{DM} can be written as H_{DM} = \lambda (d_{\uparrow }-d_{\uparrow }^{†})(f+f^{†})-\mathrm{i}\lambda (d_{\downarrow }+d_{\downarrow }^{†})(f+f^{†}).

In the adiabatic regime when \omega is small, spin resolved current induced by rotating magnetic field from lead \alpha to the QD can be calculated by [62]

(2)I_{\alpha \sigma }=q{\displaystyle \int \frac{\mathrm{d}E}{2\pi }(-{\mathrm{\partial }}_{E}f)\mathrm{T}\mathrm{r}{\left[{\mathrm{\Gamma }}_{\alpha }{G}^r\mathrm{d}{H}^{\mathrm{\prime }}{G}^a\right]}_{\sigma \sigma },}

where q is the charge quantity, {\mathrm{\Gamma }}_{\alpha } is the linewidth function describing the coupling between the leads and the QD, and \mathrm{d}{H}^{\mathrm{\prime }} = \mathrm{d}{H}_B/\mathrm{d}t. G^r is the non-equilibrium Green’s function of the system, and the matrix elements G_{\sigma {\sigma }^{\prime }}^r related to the quantum transport are expressed as

(3)G_{\sigma {\sigma }^{\prime }}^r={\displaystyle \frac{1}{|A|}}{\left(\begin{array}{cc}\left(E^2-{\displaystyle \frac{{ϵ}_M^2}{4}}\right)(E-{ϵ}_2)-2{\lambda }^{2}E& -2\mathrm{i}{\lambda }^{2}E+\kappa {\mathrm{e}}^{-\mathrm{i}\omega t}\left(E^2-{\displaystyle \frac{{ϵ}_M^2}{4}}\right)\\ 2\mathrm{i}{\lambda }^{2}E+\kappa {\mathrm{e}}^{\mathrm{i}\omega t}\left(E^2-{\displaystyle \frac{{ϵ}_M^2}{4}}\right)& \left(E^2-{\displaystyle \frac{{ϵ}_M^2}{4}}\right)(E-{ϵ}_1)-2{\lambda }^{2}E\end{array}\right)}_{\sigma {\sigma }^{\prime }},

where {ϵ}_{1,2}={ϵ}_d\pm B_0\cos \theta -\mathrm{i}\mathrm{\Gamma }/2 with \mathrm{\Gamma }={\mathrm{\Gamma }}_L+{\mathrm{\Gamma }}_R, and A=[(E-{ϵ}_1)(E-{ϵ}_2)-{\kappa }^2](E^2-{ϵ}_M^2/4)-2E(2E-{ϵ}_1-{ϵ}_2){\lambda }^2- 4{\lambda }^{2}E\kappa \sin (\omega t). By substituting G_{\sigma {\sigma }^{\prime }}^r into Eq. (2), in the low temperature limit, Eq. (2) is reduced to

(4)\begin{array}{rl}{I}_{\alpha \sigma }(t)=& \frac{q{\mathrm{\Gamma }}_{\alpha }\omega \kappa }{2\pi |A_{\mu }{|}^2}\{[4{\lambda }^2\mu \cos (\omega t)(\mu -{ϵ}_d)+2\sigma {\lambda }^2\mu \sin (\omega t)\mathrm{\Gamma }]\\ & \times \left({\mu }^2-\frac{{ϵ}_M^2}{4}\right)-8{\lambda }^4{\mu }^2\cos (\omega t)+\sigma \kappa \mathrm{\Gamma }{\left({\mu }^2-\frac{{ϵ}_M^2}{4}\right)}^2\},\end{array}

where \mu is the chemical potential of the leads and A_{\mu }\equiv A(E\to \mu ). \sigma labels spin polarization \uparrow /\downarrow and stands for \pm 1 for electron energy. In the rest of this work, I_{\alpha \sigma } is abbreviated as I_{\sigma } for simplicity.

Case I. We first consider the case of \lambda =0, where Eq. (4) is reduced to

(5)I_{\uparrow }\equiv -I_{\downarrow }={\displaystyle \frac{q\omega {\mathrm{\Gamma }}_{\alpha }\mathrm{\Gamma }{\kappa }^2}{2\pi |(\mu -{ϵ}_1)(\mu -{ϵ}_2)-{\kappa }^2{|}^2}},

which is accordant with that obtained in Ref. [62]. Obviously, the total charge current I_c (=I_{\uparrow }+I_{\downarrow }) is zero and a pure spin current I_s (=I_{\uparrow }-I_{\downarrow }) is generated. The pure spin current is time-independent, and its magnitude depends on the rotating magnetic field and the coupling strength between the QD and the leads.

Case II. We then consider the case of \lambda \ne 0 and {ϵ}_M=0. When |\mu |\gg \lambda, Eq. (4) is reduced to Eq. (5) meaning that spin pumping is not influenced by the MZM. However, when \lambda \gg \mu which is easily satisfied if \mu =0, Eq. (4) is simplified as

(6)I_{\uparrow }(t)\equiv I_{\downarrow }(t)=-{\displaystyle \frac{q\omega {\mathrm{\Gamma }}_{\alpha }\kappa \cos (\omega t)}{4\pi [({ϵ}_d-\kappa \sin (\omega t){)}^2+{\mathrm{\Gamma }}^2/4]}},

which has two obvious differences compared with that of \lambda =0. Firstly, I_{\uparrow } is always equal to I_{\downarrow } meaning a switching from pure spin current to charge current, and the charge current is independent of the magnitude of nonzero \lambda. Secondly, I_{\uparrow } varies periodically with time, and the period is 2\pi /\omega. Fig.2(a) shows I_{\uparrow } (solid curves) and I_{\downarrow } (dash curves) as functions of \mu with different \lambda. When \lambda =0, I_{\uparrow } is always equal to -I_{\downarrow } meaning a pure spin current. When \lambda \ne 0, I_{\uparrow } falls and I_{\downarrow } rises near \mu =0. In particular, I_{\uparrow } and I_{\downarrow } reach the same value at \mu =0 indicating a switching from pure spin current to charge current. The half widths of I_{\uparrow } and I_{\downarrow } peaks increase with \lambda, but the heights remain unchanged. It means that I_c is independent of the magnitude of \lambda at \mu =0 as long as \lambda \ne 0. Fig.2(b) plots the time evolution of I_c under different magnetic field. We see that I_c increases firstly in the first half period and then decreases to the initial value in the second half period showing an inversion of current direction. The average current in one period is confirmed to be zero.

Case III. Finally, we consider the case of \lambda \ne 0 and {ϵ}_M\ne 0. When \mu \to 0, Eq. (4) is reduced to Eq. (5) which is independent of the magnitude of {ϵ}_M. It means that the a.c. charge current with {ϵ}_M=0 is switched to a d.c. pure spin current again with {ϵ}_M\ne 0. The same results can be obtained when |\mu |\gg \lambda and |\mu |\gg {ϵ}_M. However, when \mu =\pm {ϵ}_M/2, Eq. (4) is reduced to

(7)I_{\uparrow }\equiv I_{\downarrow }=-{\displaystyle \frac{q\omega {\mathrm{\Gamma }}_{\alpha }\kappa \cos (\omega t)}{4\pi [({ϵ}_d-\kappa \sin (\omega t)\mp {ϵ}_M/2{)}^2+{\mathrm{\Gamma }}^2/4]}}.

Obviously, an a.c. charge current occurs when \mu =\pm {ϵ}_M/2. When {ϵ}_M\to 0, Eq. (7) is reduced to Eq. (6). I_{\sigma } versus \mu with different \lambda is given in Fig.2(c) when {ϵ}_M\ne 0. Compared with Fig.2(a) where {ϵ}_M=0, the sharp peaks of I_{\sigma } move from \mu =0 to \mu =\pm {ϵ}_M/2. These two sharp peaks show similar information to Fig.2(a), except that the peak width is halved. These features indicate that the MZM can always be detected by observing the switching from pure spin current to charge current, except that the chemical potential has to be shifted to \pm {ϵ}_M/2.

To better understand the switching between pure spin current and charge current in the MZM coupled spin pumping system, we introduce a unitary transformation U={\mathrm{e}}^{\mathrm{i}\omega t/2(d_{\uparrow }^{†}{d}_{\uparrow }-d_{\downarrow }^{†}{d}_{\downarrow })} to redefine the Hamiltonian of the system in the rotating reference as follows:

(8)\begin{array}{rl}{H}_0^{RF}=& U^{†}{H}_{0}U+\mathrm{i}\frac{\mathrm{d}{U}^{†}}{\mathrm{d}t}U\\ =& \left(\begin{array}{cccc}{\widetilde{ϵ}}_{d\uparrow }& -\tilde{\lambda }& \kappa & -\tilde{\lambda }\\ -{\tilde{\lambda }}^{\ast }& {ϵ}_M/2& \mathrm{i}{\tilde{\lambda }}^{\ast }& 0\\ \kappa & -\mathrm{i}\tilde{\lambda }& {\widetilde{ϵ}}_{d\downarrow }& -\mathrm{i}\tilde{\lambda }\\ -{\tilde{\lambda }}^{\ast }& 0& \mathrm{i}{\tilde{\lambda }}^{\ast }& -{ϵ}_M/2\end{array}\right),\end{array}

where H_0=H_D+H_B+H_M+H_{DM} represents the Hamiltonian of the MZM coupled QD with rotating magnetic field, and the matrix is defined with the basis set [d_{\uparrow },f,d_{\downarrow },f^{†}]. {\widetilde{ϵ}}_{d\sigma }={ϵ}_{d\sigma }+\sigma \omega /2 and \tilde{\lambda }=\lambda {\mathrm{e}}^{-\mathrm{i}\omega t/2}. Obviously, an energy shift \pm \omega /2 occurs in the rotating reference for the spin up and spin down states of the QD, respectively. The influence from magnetic field becomes time-independent, while the coupling between the QD and the MZM evolves with time. The change of coupling explains the periodic oscillation of the charge current. Note that this transformation is not a strict derivation, as it is not started from the BdG representation. It serves only as a heuristic understanding to the oscillation of the charge current. The left panels of Fig.3 illustrate the density of states (DOS) of the QD in a rotating reference under different conditions. The corresponding physical pictures of spin pumping are shown in the right panels of Fig.3. When \lambda =0, two symmetrical spin states can be found at E=\pm \omega /2 in the QD, as depicted in Fig.3(a). Despite equal coupling strengths between the lead and these two states, electrons from the left lead can only enter the low energy spin down state but not the high energy spin up state due to the selection rule of the rotating magnetic field. The spin down electrons are flipped to spin up electrons by absorbing photons, as demonstrated in Fig.3(b), and then exit to the left lead again. The incoming spin down electrons and outgoing spin up electrons give rise to a pure spin current in the left lead. Considering of the symmetric coupling between the QD and two leads, the identical process occurs in the right lead and thus only the transport process in the left lead is presented in the right panels of Fig.3. When \lambda \ne 0 but {ϵ}_M=0, a highly localized topological Andreev state is formed at E=0 as shown in Fig.3(c). This state dominates quantum transport of the spin pumping device when the incident electron’s energy is zero. In this scenario, only electron pairs with opposite spin can access the zero energy level in the QD. The spin down electrons are excited to spin up electrons at the high energy level by absorbing photons, whereas the spin up electrons are flipped to spin down electrons at the low energy level by emitting photons as shown in Fig.3(d). Eventually, the spin up and spin down electrons flow back to the lead again. As the coupling strength between the QD and MZM is time-dependent (\tilde{\lambda }=\lambda {\mathrm{e}}^{-\mathrm{i}\omega t/2}), the number of incoming electron pairs and outgoing electrons pairs also evolve with time, leading to the generation of an a.c. charge current [see Fig.2(b)]. When the two MZMs are coupled ({ϵ}_M\ne 0), the topological zero energy state disappears indicating a phase transition of the nanowire from a nontrivial TSC to an ordinary superconductor, and the highly localized topological Andreev state at E=0 is split into two states at \pm {ϵ}_M/2 as shown in Fig.3(e). As the superconducting gap near \mu =0 does not participate in electron transport, the device is reduced to the case of \lambda =0, resulting in only pure spin current [see Fig.3(f)]. As \mu approaches the band edges of the superconducting gap \pm {ϵ}_M/2, the Andreev bound states dominate the quantum transport again and charge current occurs as shown in Fig.2(c).

To achieve the switching from pure spin current to charge current, it is essential to have equal coupling strength between the quantum dot and spin-resolved MZMs, i.e., |{\lambda }_{\uparrow }|=|{\lambda }_{\downarrow }|. However, due to the spin polarization of MZMs, this requires precise parameter tuning in experimental setups [63,64]. To overcome this challenge, we couple the quantum dot to two MZMs at both ends of the superconducting nanowire simultaneously. Since the spin polarization of the MZM at the left end of the nanowire is opposite to that of the MZM at the right end, the spin up coupling strength |{\lambda }_{\uparrow }| is equal to the spin down coupling strength |{\lambda }_{\downarrow }| when the two MZMs are coupled to the quantum dot equally. Considering of the expression of H_{DM} in Eq. (1), it is clear that coupling to two MZMs with opposite spin polarization is equivalent to coupling to a spin-degenerate MZM with coupling strength doubled.

To investigate whether an ordinary fermion can also produce the switching from pure spin current to charge current, we coupled an ordinary fermion instead of a Majorana quasi-particle to the QD of the spin pumping device. The Hamiltonian of a minimal model is given by H_{D^{\prime }}=\sum _{\sigma }{ϵ}_{d\sigma }^{\prime }{b}_{\sigma }^{†}{b}_{\sigma }+\beta b_{\sigma }^{†}{b}_{\overline{\sigma }}, and the coupling between two QDs is described by H_{D{D}^{\prime }}=\sum _{\sigma }\lambda d_{\sigma }^{†}{b}_{\sigma }+H.c.. When \beta ={ϵ}_{d\sigma }^{\prime }=0, we find that the spin resolved current is given by I_{\sigma }=\eta q\omega {\mathrm{\Gamma }}_{\alpha }\mathrm{\Gamma }{\kappa }^2{\mu }^4/(2\pi |C{|}^2), where C=[(\mu -{ϵ}_d+\mathrm{i}\mathrm{\Gamma }/2)\mu -{\lambda }^2{]}^2-{\kappa }^2{\mu }^2. Interestingly, I_{\uparrow } is equal in magnitude to -I_{\downarrow } and remains time independent. However, when \lambda \gg \mu \to0, both I_{\uparrow } and I_{\downarrow } approach to zero, indicating the absence of charge and spin current in the system [black dashed and solid curves in Fig.2(d)]. As \beta increases, pure spin current gradually appears [blue and red solid curves in Fig.2(d)], but no charge current is observed at any time. It is evident that the transport property produced by ordinary fermion-assisted spin pumping is entirely distinct from that produced by Majorana quasi-particle-assisted spin pumping. Hence, we conclude that the transition from pure spin current to charge current in quantum spin pumping provides an effective method to detect the presence of MZM.

3 Realistic TSC nanowire assisted spin pumping

To better comprehend the resilience of the MZM assisted spin pumping in actual physical devices, we replace the individual MZM with a more realistic TSC nanowire, as outlined in Ref. [32],

(9)\begin{array}{rl}& H_{wire}=H_0+H_{soc}+H_{sc},\\ & H_0=\sum _{i,\sigma }(-\mu +\eta V_z)C_{i,\sigma }^{†}{C}_{i,\sigma }-\sum _{i,\sigma }t{C}_{i,\sigma }^{†}{C}_{i+1,\sigma }+H.c.,\\ & H_{soc}=i{\alpha }_R\sum _{\mathrm{i},\sigma ,{\sigma }^{\prime }}{C}_{i,\sigma }^{†}(\hat{x}\times \mathit{\sigma }{)}_z^{\sigma {\sigma }^{\prime }}{C}_{i+1,{\sigma }^{\prime }}+H.c.,\\ & H_{sc}=\mathrm{\Delta }\sum _i{C}_{i,\uparrow }^{†}{C}_{i,\downarrow }^{†}+H.c..\end{array}

H_0 describes the Hamiltonian of a finite-length nanowire. C_{i,\sigma }^{†}(C_{i,\sigma }) generates (annihilates) an electron at site i with spin \sigma. \mu is the chemical potential, and t is the coupling strength between the neighborhoods. V_z describes the Zeeman splitting regulated by an external magnetic field perpendicular to the nanowire. H_{soc} describes the Rashba spin−orbit coupling in the nanowire, where {\alpha }_R represents the strength of spin−orbit interaction and \mathit{\sigma } is the Pauli matrix. H_{sc} describes the s-wave paring term with superconducting order parameter \mathrm{\Delta } arising from the superconducting proximity effect. The Bogoliubov−de Genes equation is formulated in the standard Nambu representation. When V_z>\sqrt{{\mathrm{\Delta }}^2+{\mu }^2}, the superconducting nanowire enters the topological nontrivial phase accompanied by two MZMs at both ends of the nanowire. When , the superconducting nanowire is in the topological trivial states. In our calculation, \mu is fixed equal to zero, and a topological phase transition occurs at V_z=\mathrm{\Delta }. Moreover, the nanowire is sufficiently long to ensure the decoupling between two MZMs.

Fig.4(a) and (b) display I_s and I_c as functions of \mu, respectively, for the superconducting nanowire in its topological nontrivial phase (red curves and blue curves with V_Z>\mathrm{\Delta }) and trivial phase (black curves with ). A weak coupling strength between the QD and the MZM is assumed to be \lambda =t/50. When , I_s remains constant with respect to \mu, and I_c remains zero. Conversely, when V_Z>\mathrm{\Delta }, both I_s and I_c oscillate near \mu =0. Moreover, I_s tends to zero while I_c remains finite when \mu =0. For \mu away from zero, I_s tends to constant, while I_c approaches zero. These results agree with those in Fig.2, which illustrate that a side-coupled TSC nanowire can facilitate the switching from pure spin current to charge current in the spin pumping device. In Fig.4(c) and (d), we present the evolution of I_c and I_s with different V_z and B_0 when \mu =0. When the superconducting nanowire is in its topologically nontrivial phase with V_Z>\mathrm{\Delta }, I_c oscillates with time for different B_0 with zero average current per cycle, while I_s remains perpetually zero, corroborating the findings in Fig.2(a). Conversely, when , I_c matches zero, and I_s oscillates against time. The differences in current information for V_Z>\mathrm{\Delta } and can serve to distinguish the topological phenomenon of a superconducting nanowire and verify the existence of MZM in 1D TSC.

4 ps-ABSs assisted spin pumping

Previous studies have demonstrated that ps-ABSs in the superconducting nanowire-coupled QD systems can also exhibit the same 2e^2/h ZBCP behavior as induced by MZM in local quantum transport detection [42,43]. When a nonuniform chemical potential emerges in the nanowire, a pair of ps-ABSs with quasi-zero energy can form at one end and the interior, even in the topologically trivial phase. When the QD is coupled to the ps-ABSs located at the end of the nanowire, ZBCP can also arise, which is robust in a wide range of Zeeman field and tunnel barrier height, potentially misleading the presence of MZM. In light of this, it is essential to clarify the influence of ps-ABSs on spin pumping. We consider a representative case to generate the ps-ABSs in a superconducting nanowire, as used in Ref. [42], where an additional long-range parabolic potential is included throughout the nanowire. The maximum value is set equal to \mathrm{\Delta } in the middle, and the minimum value is fixed to be zero at both ends of the nanowire, as depicted by the light-yellow filled image in Fig.5(c) and (d).

Fig.5(a) illustrates the eigenvalues of the superconducting nanowire including the parabolic potential with increasing Zeeman splitting. As V_Z/\mathrm{\Delta } increases, the band gap gradually closes at approximately 1.3, and quasi-zero modes emerge in a wide range from around 1.3 to 2.6. Each quasi-zero mode corresponds to a pair of ps-ABSs consisting of two overlapping MBSs with separation on the order of the Majorana decay length \xi, where the system remains in the topological trivial state. With further increase of V_Z/\mathrm{\Delta } beyond 2.6, the system enters the topological nontrivial states, and MZMs appear at the ends of the nanowire. It is important to note that the criterion of topological phase transition is no longer the simple relation between V_z and \sqrt{{\mathrm{\Delta }}^2+{\mu }^2} due to the presence of the non-uniform potential. The details of the quasi-zero modes are amplified in Fig.3(b), and each cross corresponds to a pair of ABSs at E=0, consistent with the finding in Refs. [42,43]. Correspondingly, the topological trivial ps-ABSs at V_Z/\mathrm{\Delta }=1.5 and topological nontrivial MZMs at V_Z/\mathrm{\Delta }=2.8 are depicted in Fig.5(c) and (d), respectively, where the ps-ABSs overlap at one end of the nanowire, while two MZMs appear at both ends.

Fig.5(e) plots the pumping current I_c and pure spin current I_s against V_Z/\mathrm{\Delta } from the topological trivial to nontrivial region. When , I_c remains at zero, and a d.c. pure spin current I_s is present, consistent with the case of topological trivial phase without the non-uniform potential, as shown in Fig.2. As V_Z/\mathrm{\Delta } changes from 1.3 to 2.6, a series of sharp peaks in I_c appear, accompanied by disappearance of I_s. To see the peaks more clearly, one of them is amplified in Fig.5(f). We observe that I_c tends to zero with finite I_s away from the peak, while I_s is equal to zero with finite I_c at the peak, signifying a switch from pure spin current to charge current. Moreover, this switch only occurs at the cross points in Fig.5(b) where the energy of the ps-ABSs is precisely zero. With further increase of V_Z/\mathrm{\Delta } beyond 2.6, a plateau of I_c appears, accompanied by I_s=0. The plateau remains stable over a wide range of V_Z/\mathrm{\Delta } with value equal to that of the peaks, corresponding to a topological phase transition from trivial to nontrivial states of the superconducting nanowire, or in other words, the presence of MZMs. We also investigated other nonuniform potentials, such as a short parabolic potential as used in Ref. [42], and found that only the ps-ABSs with precisely zero energy can switch the pure spin current to charge current like MZMs. However, this switch is extremely unstable and highly sensitive to the Zeeman field, providing a method to distinguish between zero-energy ps-ABSs and MZMs. This is different from the case of ZBCP, where the induced ZBCP by ps-ABSs and MZMs is indistinguishable over a wide range of Zeeman field in local charge tunneling experiments [42]. Our investigation suggests that spin pumping could potentially be an effective method for detecting the presence of MZMs at the ends of a TSC nanowire. However, the proposed scheme is upon a simple theoretical model, and there are still many issues to be addressed, such as model simplification, experimental feasibility, and the potential impact of other factors on the spin pumping process. Further theoretical and experimental studies are required to fully elucidate this concept.

5 Expending discussion

Next, we give a brief discuss to this study from the perspective of decoherence, shot noise and experimental implementation, aiming to gain a deeper understanding and broader application of the MZM assisted spin pumping.

Decoherence is a fundamental phenomenon resulting from inelastic scattering in quantum transport, which can be effectively modeled using a phenomenological approach by involving a hypothetical probe [65]. It is equivalent to appending an additional virtual self-energy term {\mathrm{\Sigma }}_v^r=\mathrm{i}\eta into G^r of the system, where \eta represents the dephasing parameter. Notably, the presence of this virtual self-energy term solely modifies the value of {\mathrm{\Gamma }}_{\alpha } in Eq. (4), without altering other parameters. In essence, decoherence exclusively modifies the magnitude of the spin-resolved current, leaving the characteristic behavior of MZM assisted spin pumping unaltered, where the pure spin current transitions to charge current.

Shot noise originates from the particle nature of electrons, which has been thoroughly explored in quantum transport systems [66-68]. By quantifying shot noise, we can identify the quasi-particle’s spin or charge unit. In quantum spin pumping, both cross- and auto-correlation measurements are crucial for characterizing spin current noise. Despite the absence of a net charge current, a shot noise in charge current persists due to the counterbalancing flow of spin up and spin down electrons. The emergence of MZM transforms spin correlations into charge correlations, potentially offering richer quantum insights. Consequently, a thorough examination of spin noise in MZM assisted spin pumping systems is imperative in future research.

Although this work provides a theoretical framework for exploring the MZM assisted spin pumping, we maintain an optimistic perspective for its experimental realization. Spin pumping devices have been successfully demonstrated across various platforms, including GaAs/AlGaAs two-dimensional electron gas systems that utilize electron-beam patterned CrAu depletion gates and nonmagnetic PtAuGe Ohmic contacts [69]. However, the successful fabrication of the 1D TSC remains a notable challenge in experimental settings. Despite this, given the firmly established theory of MZMs, we continue to anticipate that the unambiguous observation of MZMs in experiments, along with the achievement of integrated measurements with spin pumping devices, will be feasible with the advancements in experimental technology.

6 Summary

In summary, we investigated the spin pumping effect influenced by MZMs at the end of a TSC nanowire. Even a very weak coupling between the QD and the adjacent MZM can change the d.c. pure spin current in the pumping system to time-dependent charge current when the chemical potential is equal to zero. However, when the superconducting nanowire is at the topological trivial state, the a.c. charge current reverts to d.c. pure spin current again. Although the ps-ABSs with precise zero energy can also switch the pure spin current to charge current, it is extremely unstable and highly sensitive to the Zeeman field, which is different from the MZMs assisted switching. Our investigation suggests that spin pumping could potentially be an effective method for detecting the presence of MZMs at the ends of a TSC nanowire.

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