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Thermoelectric Effect in a Correlated Quantum Dot SideCoupled to Majorana Bound States
Nanoscale Research Letters volume 15, Article number: 79 (2020)
Abstract
We theoretically study the thermoelectric effect in a hybrid device composed by a topological semiconducting nanowire hosting Majorana bound states (MBSs) and a quantum dot (QD) connected to the left and right nonmagnetic electrodes held at different temperatures. The electronelectron Coulomb interactions in the QD are taken into account by the nonequilibrium Green’s function technique. We find that the sign change of the thermopower, which is useful for detecting the MBSs, will occur by changing the QDMBS hybridization strength, the direct overlap between the MBSs at the opposite ends of the nanowire, and the system temperature. Large value of 100% spinpolarized or pure spin thermopower emerges even in the absence of Zeeman splitting in the QD or magnetic electrodes because the MBSs are coupled to electrons of only one certain spin direction in the QD due to the chiral nature of the Majorana fermions. Moreover, the magnitude of the thermopower will be obviously enhanced by the existence of MBSs.
Introduction
The preparation and detection of zeroenergy Majorana bound states (MBSs) are of particular importance in modern condensed matter physics. Fundamentally, the MBSs are solid state counterpart of Majorana fermions and are associated with nonAbelian statistics that can enable topologically protected quantum information with potential applications in quantum computation free from decoherence [1–3]. Apart from this, the MBSs are also promising in design of highefficiency electronic devices, such as the spintronics [4]. Wellseparated MBSs can be prepared in various systems, of which the most important schemes include noncentrosymmetric superconductors [5], three or twodimensional topological insulators coupled to superconductors [6], electrostatic defects in topological superconductors [7], pwave superconductors [8], the semiconducting [9] or ferromagnetic [10] nanowires with native strong spinorbit interaction proximitization to a conventional swave superconductors, and Josephson junctions [11].
As for the detection of MBSs, it is also quite challenging because the Majorana fermions are their own antiparticles and chargeneutral due to their intrinsic particlehole symmetry. A variety of experiments have been carried out to verify the existence of MBSs through phenomena such as the 4π periodic Josephson current phase in junctions between topological superconductors [12], halfinteger conductance plateau at the coercive field in a hybrid structure composing of topological superconductors and topological quantum anomalous Hall insulator [13], tunneling spectroscopy using Rashba nanowires coupled to the bulk swave superconductors [14], and zero bias of the differential conductance at the edges of the wires [14, 15]. However, these phenomena have other possible physical origins except for MBSs, and alternative schemes then have been proposed. One of them is the hybridization of MBSs with other nanoscale structures, such as the zerodimensional quantum dot (QD) in which the energy levels, electronelectron Coulomb interactions, particle numbers, and coupling strength to external environment are all well controllable [16, 17]. At low temperature, a halfmaximum conductance when the energy level of the QD is aligned to the Fermi energy in the leads was theoretically predicted as a clear evidence of the formation of a pair of MBSs [18]. This result is completed unchanged by the adjusting of the QD energy level [19] and has successfully been observed in experiment in a QD coupled to an InAsAl nanowire [20]. Recently, optical schemes based on QD structure were also theoretically proposed to detect the MBSs with the help of optical pumpprobe technique. [21, 22] In ring or Tshaped QDbased systems, the quantum interference phenomena are drastically affected by the MBSs [23–25] and then can be used for the detection scheme with the help of, for example, the Fano effect [26–28].
Recently, there are also some work concerning detection of the MBSs via thermoelectric effect, which focuses on the conversion between electrical and thermal energies. This old research topic gains renewed attention due to the rapid progress of growth and fabrication of mesoscopic devices and nanostructures, in which the thermoelectric performances are obviously improved [29, 30]. Highefficiency energy harvesters based on QDs that are defined on such as a GaAs/AlGaAs interface twodimensional electron gas have recently been reported [31, 32]. Enhancement of the thermoelectric effect in them can be attributed to the considerable reduction of the thermal conductivity by boundary scattering and the optimization of the electrical transport properties unique in these lowdimensional systems [30–32]. The thermopower (Seebeck coefficient) is the central quantity in thermoelectric effect. It is the strength of a opencircuit voltage in response of a temperature gradient applied in a solid material with free electronic carriers. Hou et al. theoretically predicted that the thermopower between a QD and superconductor hosting a Majorana edge state satisfies the Mott formula and generically does not vanish by using the LandauerBüttiker formalism [33]. Based on such a property, one can infer the temperature of the Majorana edge state by measuring the differential conductance and the thermopower. Leijnse demonstrated theoretically that the coupling between a QD with tunable energy level and MBSs breaks particlehole symmetry, and the changes of thermopower provide a new way of proving the existence of Majorana states [34]. The thermoelectric properties in such a setup can also be used to detect the temperature of the superconductor and to extract information about the dissipative decay of MBSs [34]. In a structure with a QD coupled to two electrodes, López et al. showed that the thermopower will change its sign by changing the direct hybridization between the MBSs, a good evidence of the existence of MBSs [35]. The sign change of the thermopower was also subsequently found in systems of a QD with two [36] or three [37] electrodes. Moreover, it was demonstrated that the relationship between the shot noise and thermoelectric quantities may provide a purely electrical way to detect the chargeneutral MBSs [38, 39].
In the present paper, we propose a hybridized system composing of MBSs and a QD coupled to electrodes (see Fig. 1) to study the properties of the thermopower. In the nanosystem we considered, the strong Coulomb interaction in the dot, which has been neglected in previous works [18, 22–24, 34–39], is taken into account. Furthermore, we consider that only one spin component of the QD spin is coupled to the MBSs due to the chiral nature of the MBSs [40]. We find that the sign of the thermopower can be effectively reversed by changing the dotMBSs coupling strength, the direct hybridization between the MBSs, and the system temperature. The resulted large 100% spinpolarized and pure spin thermopower, which are the corresponding 100% spinpolarized and pure spin currents in closed circuit, are useful in spintronics. The coupling of both the two MBSs to the QD will further enhance the magnitude of the thermopower, but does not change the essential results when only one of the MBSs is coupled to the dot. Based on the presently advanced quantum transport measurements for the MBSs through QD coupled with topological superconducting nanowires, we believe our proposal could be experimentally tested in the future. Additionally, our proposal and findings in this work may provide an excellent way to detecting the formation of the MBSs in QD.
Model and Methods
The effective Hamiltonian of the QD coupled to MBSs and the left and right normal metal electrodes takes the following form [34, 35]:
where \(c_{k\beta \sigma }^{\dag } (c_{k\beta \sigma })\) creates (annihilates) an electron of momentum k, energy ε_{kβ} (its dependence on spin is neglected for normal metal electrode), and spin σ=↑,↓ in electrode β=L,R. For the QD, \(d_{\sigma }^{\dag } (d_{\sigma })\) is the creation (annihilation) operator of an electron with gate voltage tunable energy level ε_{d}, spin σ, and intradot Coulomb interaction U. The coupling strength between the QD and the leads is described by V_{kβ}. The last term H_{MBSs} in Eq. (1) stands for the zeroenergy MBSs located on opposite ends of the semiconducting nanowire and their coupling to the QD [18]:
in which δ_{M} is the overlap amplitude between the two MBSs with operator satisfying both \(\eta _{j}=\eta _{j}^{\dag } (j=1,2)\) and {η_{i},η_{j}}=δ_{i,j}. The hopping amplitude between MBSs and spin ↑ electrons in the QD is accounted by λ_{j}. It is helpful to write η_{j} in terms of the regular fermionic operators f as [18] \(\eta _{1}=(f^{\dag }+f)/\sqrt {2}\) and \(\eta _{2}=i(f^{\dag }f)/\sqrt {2}\), and then, H_{MBSs} is rewritten as:
We consider the system in linear response regime, i.e., under infinitely small bias voltage ΔV and temperature difference ΔT between the left and right leads, the electric and heat currents of each spin component are obtained as:
where e is the electron charge and T the system equilibrium temperature, and
where \(\hbar \) is the reduced Planck’s constant. We set the leads’ chemical potential μ=0 as the energy zero point. The Fermi distribution function is given by f(ε,μ)=1/{1+exp[(ε−μ)/k_{B}T]} with k_{B} being the Boltzmann constant. The transmission coefficient T_{σ}(ε) is calculated with the help of the retarded Green’s function as:
where \(\Gamma _{L(R)}=2\pi \sum _{k}V_{kL(R)}^{2}\delta [\varepsilon \varepsilon _{kL(R)}]\) is the linewidth function. We apply the standard equation of motion technique to obtain Green’s function. The higherorder Green’s functions are truncated by following scheme 2 in ref. [39], i.e., neglect the simultaneous tunneling of the electron of opposite spin. After some straightforward calculations, the spinup retarded Green’s function is given by:
where the MBSinduced selfenergies
and
with
in which
and ε_{±}=ε±ε_{d}+i(Γ_{L}+Γ_{R})/2. In the absence of dotMBSs hybridization (λ_{1}=λ_{2}=0), we have \(\Sigma ^{M}_{0,1}=0\) and \(G_{\uparrow }^{r}(\varepsilon)\) recovers that of ref. [39]. It is also the spindown retarded Green’s function by changing n_{↓} into n_{↑}. The occupation number is calculated selfconsistently from:
where f_{L/R}(ε) is the Fermi distribution function in the left/right electrode.
Once the transmission function is obtained from Green’s function, the electrical conductance and the thermopower (Seebeck coefficient) of each spin component are given by G_{σ}=e^{2}L_{0,σ} and S_{σ}=−L_{1,σ}/(eTL_{0,σ}), respectively.
Results and Discussions
In what follows, we assume symmetric coupling between the QD and electrodes, and set Γ=2Γ_{L}=2Γ_{R}=1 as the energy unit. The intradot Coulomb interaction is fixed as U=10Γ. We first study the case of the QD which is coupled to only MBS1 with different hybridization strength λ_{1} in Fig. 2 by setting λ_{2}=0. For λ_{1}=0, the conductance of each spin component in Fig. 2a develops two peaks located respectively at ε_{d}=−μ and −μ−U. Note now the QD is free from spin polarization induced by the MBS, and the conductance of the two spin component is equal to each other (G_{↑}=G_{↓}), accordingly. Turning on the hybridization between the MBS and the QD (λ_{1}≠0), the magnitude of G_{↑} is monotonously suppressed as shown in Fig. 2a, which is consistent with previous results [18, 34, 35]. The value of G_{↓}, however, is almost unchanged even the occupation number n_{↓} is changed by λ_{1} due to the presence of intradot Coulomb interaction (which is not shown in the figure). Meanwhile, the peaks’ position and width in G_{↑} are slightly modified by the value of λ_{1} due to the level renormalization by the dotMajorana coupling [18, 34, 35]. The configuration of the total conductance G=G_{↑}+G_{↓} in Fig. 2c resembles that of G_{↑}.
The thermopower S_{↑} in Fig. 2b shows the typical sawtooth configuration and has three zero points individually at ε_{d}=μ,−U/2, and μ−U [41, 42]. It develops a pair of sharp peaks with opposite signs at each of the two resonant states (ε_{d}=μ,μ−U) and changes sign whenever ε_{d} passes each zero points. In the absence of dotMBSs hybridization (λ_{1}=0) as indicated by the solid black line in Fig. 2b, S_{↑} is positive (negative) when ε_{d} is below (above) the zero point as the main carriers are electrons (holes). With increasing λ_{1}, the spindown thermopower S_{↓} is unchanged and the absolute value of S_{↑} firstly is suppressed and then enhanced. For sufficiently large λ_{1},S_{↑} changes its sign as shown in Fig. 2b. With further increased λ_{1}, the absolute value of S_{↑} exceeds that of S_{↓} and the total thermopower S=S_{↑}+S_{↓} also changes its sign. Such a phenomenon has also been previously found in the spinless model [35–37]. In fact, the sign change of the thermopower in QDbased device without MBSs was attributed to several causes, such as the system equilibrium temperature [29], magnetic momentum of the electrodes [43], Coulomb interaction [43, 44], coupling strength between the QDs, the applied magnetic field, quantum interference effect, or the magnetic flux penetrating through multipledot ones [45, 46]. The above mechanisms are quite different from the present case, and the sign change of the thermopower by changing the hybridization between the QD and the MBSs is helpful for detecting the MBSs [35–37].
Figure 3a, b shows the total conductance G and themopower S varying with the dot level ε_{d} for different values of the temperature T. The peak value of G is firstly enhanced and then suppressed by increasing temperature as shown in Fig. 3a. The magnitude of the thermopower in Fig. 3b, however, is mainly enhanced by increasing temperature, as there are more electrons (holes) excited above (below) the chemical potential. Moreover, S changes its sign for the cases of T=0.1 and 0.2 as indicated by the pink and green lines in Fig. 3b, which is similar to the case of thermoelectric effect in QDbased structure without MBSs. For T=0.2Γ, the peak value of S can reach as large as 2k_{B}/e, which is one order of larger than that of T=0.001. In fact, we have checked that the magnitude of the thermopower can be further enhanced by increasing the temperature. In the present paper, however, we focus on the sign change of S at relatively low temperature, which is usually the case of the MBSs formed in experiments. Figure 3c, d presents the conductance and the thermopower for different values of direct hybridization of the two MBSs at opposite ends of the nanowire at fixed T=0.025Γ. The peak value of the conductance in Fig. 3c is monotonously enhanced by increasing δ_{M}, which is in consistent with the results found by López et al. [35]. The thermopower in Fig. 3d changes its sign for 0.03Γ<δ_{M}<0.05Γ, which is larger than the temperature T=0.025Γ. In ref. [32], they found that the thermopower changes its sign at about δ_{M}≈k_{B}T in the spinless model. In the present paper, the sign change of S occurs at relatively larger δ_{M} as the MBSs are coupled to only one spin direction electrons. Moreover, the peak value of the thermopower can also be enhanced by increasing δ_{M}.
We show the spinresolved thermopowers individually as functions of λ_{1} and δ_{M} in Fig. 4. The spinup thermopower S_{↑} in Fig. 4a firstly increases, reaching a maximum and then decreases with increasing λ_{1}. At sufficiently large λ_{1}, it remains at a stable value. The value of spindown thermopower S_{↓} is unchanged by λ_{1} as expected. The behaviors of S_{↑} and S_{↓} bring about two interesting results: one is the 100% spinpolarized thermopower when S_{↑}=0 but S_{↓} has a finite value that can be used for filtering electron spin; the other is the finite pure spin thermopower S_{s}=S_{↑}−S_{↓} with zero charge thermopower S_{c}=S_{↑}+S_{↓}=0 which occurred when S_{↑}=−S_{↓} as shown by the dots in Fig. 4b. At closed circuit, the 100% spinpolarized and pure spin thermopowers are individually the corresponding currents, which are virtual in spintronic devices. Similar results are found in Fig. 4b, d, in which S_{↑} undergoes sign change by changing δ_{M}, whereas S_{↓} keeps unchanged. We emphasize that the present 100% spinpolarized and pure spin thermopowers emerge in the absence of magnetic field or magnetic materials in the QD.
In Fig. 5, we study the case of both the MBSs at the opposite ends of the nanowire which are coupled to the QD when the wire and the dot are close to each other enough with δ_{M}=0. Figure 5a shows that the total conductance G keeps the doublepeak configuration in the presence of λ_{2}. The peaks’ height will be suppressed by increasing λ_{2}. The lineshape of S is also unchanged by the value of λ_{2} as indicated by Fig. 5b. The peak’s value of S will be significantly enhanced since the thermopower is reversely proportional to the conductance. For λ_{2}∼0.2Γ, the magnitude of the thermopower can reach as large as 2 k_{B}/e. Moreover, we find that S will not change its sign by adjusting the value of λ_{2}. Figure 6 shows the total thermopower as a function of ε_{d} for different values of direct hybridization between the MBSs δ_{M} by fixing λ_{1}=λ_{2}=0.2Γ. It shows that both the magnitude and the sign can be effectively changed by tuning δ_{M}, which is similar to the case that only one of the MBSs is coupled to the QD. Finally, we briefly discuss the experimental realization of the present devices. The nanowire hosting the MBSs can be fabricated with InAs grown by molecular beam epitaxy with several nanometers of epitaxial Al layer [47]. It has been experimentally proven that a hard superconducting gap can be induced on such a kind of nanowires [47, 48] by applying a critical magnetic field exceeding 2 T along the wire axis [20]. A QD is formed in the bare InAs segment at the end of the wire due to density of state gradients at the edges of the Al shell [20, 47, 48].
Conclusions
In conclusion, we have studied the properties of the electrical conductance and thermopower in a quantum dot connected to the left and right normal metal electrodes with Coulomb interaction. The dot is also coupled to MBSs formed in a semiconducting nanowire. We find that the MBSs influence the conductance and thermopower of the spin component it only couples to, although the spinup and spindown electrons interact to each other via the Coulomb repulsion. The sign of the thermopower can be changed by adjusting the dotMBSs hybridization strength, the direction hybridization between the MBSs, and the system temperature. Large value of either 100% spinpolarized or pure spin themropowers can be obtained in nonmagnetic QD structure. The coupling between the dot and both the two MBSs can only change the magnitude of the thermopower, but not its sign. Our results may be useful in detecting the existence of the MBSs via thermoelectric technique.
Availability of Data and Materials
The datasets supporting the conclusions of this article are included within the article.
Abbreviations
 QD:

Quantum dot
 MBSs:

Majorana bound states
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Acknowledgments
This work was supported by the National Natural Science Foundation of China (grant no. 11675023). Fu acknowledges the Innovation Development Fund of China Academy of Engineering Physics (CAEP) (grant no. ZYCX192102) and the Presidential Foundation of CAEP (grant no. YZ2015014). Liu acknowledges the NSF of InnerMongolia (grant no.2017MS0112), and Science Foundation for excellent Youth Scholars of Inner Mongolia University of science and technology (grant no. 2017YQL06). Chi is supported by the Initial Project of UEST of China, Zhongshan Institute (415YKQ02), Science and Technology Bureau of Zhongshan (grant nos. 2017B1116, 2017B1016). This work is also supported by the Innovation Team of Zhongshan City (no. 170615151170710).
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Fu Z.G. and Chi F. contributed the ideas and derived the formulae in the paper. Chi F., Li K.M, and Liu J. performed the numerical calculations. Chi F. and Fu Z.G. wrote the original manuscript. All authors revised the paper. The authors read and approved the final manuscript.
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Chi, F., Fu, ZG., Liu, J. et al. Thermoelectric Effect in a Correlated Quantum Dot SideCoupled to Majorana Bound States. Nanoscale Res Lett 15, 79 (2020). https://doi.org/10.1186/s1167102003307y
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DOI: https://doi.org/10.1186/s1167102003307y
Keywords
 Thermoelectric effect
 Quantum dot
 Majorana bound states
 Thermopower