Multiple Chiral Majorana Fermion Modes and Quantized Transport
We propose a general recipe for chiral topological superconductor (TSC) in two dimensions with multiple chiral Majorana fermion modes from a quantied anomalous Hall insulator in proximity to an -wave superconductor with nontrivial band topology. A concrete example is that a chiral TSC may be realized by coupling a magnetic topological insulator and the ion-based superconductor such as FeTeSe (). We further propose the electrical and thermal transport experiments to detect the Majorana nature of three chiral edge fermions. A unique signature is that the electrical conductance will be quantized into in a quantized anomalous Hall-TSC junction, which is due to the fully random edge mode mixing of chiral Majorana fermions and is distinguished from possible trivial explanations.
Majorana fermions have attracted intense interest in both particle physics and condensed matter physics Wilczek (2009); Elliott and Franz (2015). The chiral Majorana fermion, a massless fermionic particle being its own antiparticle, could arise as a one-dimensional (1D) quasiparticle edge state of a 2D topological states of quantum matter Moore and Read (1991); Read and Green (2000); Mackenzie and Maeno (2003); Kitaev (2006); Fu and Kane (2008); Sau et al. (2010); Alicea (2010); Qi et al. (2009). The propagating chiral Majorana fermions could lead to non-abelian braiding Lian et al. () and may be useful in topological quantum computation Kitaev (2003); Nayak et al. (2008). A simple example hosting chiral Majorana fermion mode (CMFM) is the chiral topological superconductor (TSC) with a Bogoliubov-de Gennes (BdG) Chern number , which can be realized from a quantum anomalous Hall (QAH) insulator with a Chern number in proximity to a conventional -wave superconductor Qi et al. (2010); Chung et al. (2011); Wang et al. (2015a). The quantum transport in a QAH-TSC-QAH (QTQ) junction is predicted to exhibit a half quantized conductance plateau induced by a single CMFM Chung et al. (2011); Wang et al. (2015a); Lian et al. (2016), which has been recently observed in Cr(Bi,Sb)Te (CBST) thin film QAH system in proximity with Nb superconductor He et al. (2017).
Physically, the chiral TSC emerges in the neighborhood of the QAH plateau transitions, where the superconducting pairing gap exceeds QAH gap Qi et al. (2010), and it can be driven by an external magnetic field or electric field in magnetic topological insulators (MTIs) Wang et al. (2014); Kou et al. (2015); Feng et al. (2015); Wang (2016). The magnetic field at coercivity inevitably introduces random domains, making MTIs to be strongly disorderd Yasuda et al. (2017). The single CMFM in this system is robust against disorder Lian et al. (2018). However, alternative explanations of the half plateau without CMFM under strong disorders have been proposed Ji and Wen (2018); Huang et al. (2018), which arises from incoherence due to disorder. The noise and interferences measurement may distinguish chiral Majorana fermion from the disorder-induced metallic phases Chung et al. (2011); Strübi et al. (2011); Li et al. (); Fu and Kane (2009); Akhmerov et al. (2009); Law et al. (2009) .
In this Letter, we propose a general recipe for a higher odd Chern number chiral TSC which supports multiple CMFMs. The random edge mode mixing of chiral Majorana fermions lead to novel quantized transport. In sharp contrast to the previous proposal that chiral TSC is achieved near the QAH plateau transition Qi et al. (2010); Wang et al. (2015a), where strong disorders accompany in the system. Here the system we proposed is homogenous, which provides an ideal platform for studying the exotic physics of chiral Majorana fermions.
Model. The basic mechanism for 2D chiral TSC is to introduce -wave superconductivity (-SC) and ferromagnetism (FM) into a strong spin-orbit coupled (SOC) system, such as the spin-helical surface states (SSs) of TIs Hasan and Kane (2010); Qi and Zhang (2011). Instead of inducing superconductivity into a MTI for chiral TSC, one can introduce the FM proximity effect into superconducting Dirac SSs, where the CMFM exists at the boundary between FM and superconductor Fu and Kane (2009). The latter one is more practical for a homogenous system, since FM exchange coupling is usually much larger than conventional -SC proximity. Therefore, it is natural to ask whether exotic topological states exist in the heterostructure of topological FM insulator and -SC with nontrivial band topology dubbed as topological -SC (T-SC) shown in Fig. 1. The T-SC has a fully bulk pairing gap and -SC gap on the single spin-helical Dirac SS. The prototype T-SC materials are the ion-based superconductors such as FeTeSe (FTS) Zhang et al. (2018). The general theory presented here for chiral TSC is generic for the higher Chern number QAH insulator Wang et al. (2013) and T-SC. We would like to start with a simple model describing the QAH in MTIs Chang et al. (2013) for concreteness. The low energy physics of the heterostructure is described by four Dirac SSs only, for the bulk states in MTI and T-SC are gapped. The generic form of the 2D effective Hamiltonian without superconducting proximity effect is
Here describes the T-SC SSs with proximity effect from MTI, the bulk metallic states in T-SC are neglected since they are gapped with superconducting pairing, describes the QAH in MTI film,
with the basis of , (), where and denote the top and bottom SSs and and represent the spin up and down states, respectively. and () are Pauli matrices acting on spin and layer, respectively. is the Fermi velocity, which have opposite signs in FTS and MTI Zhang et al. (2018); Wang et al. (2015b); Wu et al. (2016). (The relative sign of velocities doe not affect the results). is the FM exchange field along the axis which can be tuned by a magnetic field. The short-range FM proximity effect only affects the bottom SS of T-SC and . is the hybridization between top and bottom SSs in MTI. guarantees QAH state in MTI. is the energy band alignment between two Dirac cones. For simplicity, we set , , and neglect the inversion symmetry breaking in each material. is the hybridization between the bottom T-SC and top MTI surfaces at interface, where , is real constant.
With superconducting proximity effect, a finite pairing amplitude is induced in MTI and T-SC SSs. The BdG Hamiltonian becomes , with , and
Here is the chemical potential relative to the Dirac cone in MTI, and with the Pauli matrix in Nambu space. , , and are pairing gap function in SSs of T-SC, top, and bottom MTI. All are chosen as independent, since they are induced by the -SC proximity effect, for example from the bulk hole pocket at point in FTS. Usually . Here we set and , which is realistic in superconducting proximity effect between BiTe thin film and FTS with short coherence length Chen et al. ().
Phase diagram. The BdG Hamiltonian in Eq. (3) can be classified by the Chern number . Since the topological invariants cannot change without closing the bulk gap, the phase diagram can be determined by first finding the phase boundaries as gapless regions in parameter spaces, and then calculate of the gapped phases. To start, we first consider the phase diagram in the limit , in which case the system is decoupled into two BdG models and ,
Here is the superconducting Dirac SSs of T-SC with only the bottom SS in proximity to FM. The top and bottom SSs in T-SC are further decoupled. The energy spectrum of the top SS is , and , which resembles that of the spinless superconductor but respects time-reversal symmetry Fu and Kane (2008). The excitation spectrum of the bottom SS is , with the gap closing point at and . For , the bottom SS is adiabatically connect to the top SS in the limit, so they are topologically equivalent. Therefore, the whole T-SC SS possesses non-trivial topology, but there is no chiral edge state, since there is no geometric edge to the 2D surface of a 3D bulk. For , the bottom SS is adiabatically connected to FM with which is topologically trivial, so there exists a single CMFM at the edge domain boundary at T-SC bottom, and is the sign of . is the superconducting proximity coupled QAH insulator, which has been studied in Ref. Wang et al. (2015a). For , for (which vanishes when ), for , and for . Here . A finite enlarges the odd TSC phase. The total Chern number of the heterostructure without is . The phase diagram with parameters is shown in Fig. 2(a), where the different chiral TSC phases are denoted by the corresponding Chern numbers.
Next, we study the effect of at interface. Similar to the case, we determine the phase boundaries by the bulk gap closing regions in Eq. (3), which is always at point. As show in Fig. 2(b), when the term is turned on, it makes the chiral TSC phase with the same Chern numbers simply connected. Meanwhile, it shrinks the phase and enlarges the phase, and further pushes the phase boundary between and towards a larger . For a given exchange field, will drive the system into TSC phases with smaller . Therefore, one optimal condition for TSC is , which corresponds to undoped QAH system. This is just the opposite to the optimal condition for obtaining the TSC phase from the QAH plateau transition Wang et al. (2015a). As shown in Fig. 2(c), enlarges the phase and shrinks all other phases. This can be understood from the band crossing at the interface. The single-particle Hamiltonian at interface is , with the energy spectrum . splits the two copies of Dirac bands up and down in energy. Whenever the Dirac band edge crosses the chemical potential, reduces by one. As shown in Fig. 2(d), enlarges the trivial even TSC, and shrink the nontrivial odd TSC towards larger . Similarly, enlarges TSC and shrinks TSC. Thus is preferred for higher TSC. For a simple case and infinitesemal , a simple sum rule for Chern number of the heterostucture is
In general, the coupling will strongly modify the Chern number of the heterostructure from that of the decoupled systems. The chiral TSC with higher odd Chern numbers requires a large enough exchange field, and is simply obtained by growing multilayer heterostructure or using higher Chern number QAH following the above recipe.
Transport. To probe the multiple neutral CMFMs, we consider the electrical and thermal transports in chiral TSC. The Hall bar device we shall disucss is a QTQ junction as shown in Fig. 3, which has been studied for and chiral TSCs. Both the left and right QAH regions have Chern number , and thus have a charged chiral fermion mode on their edges with vacuum. The charge chiral fermion mode can be equivalently written as two CMFMs and as shown in Fig. 3, and the electron annihilation operators on the left (right) bottom (top) QAH edges are locally related to the CMFMs as . There exists a third CMFM on the vertical edges between QAH and TSC, which merges with and on the top and bottom TSC edges.
We shall assume the electrical current is only applied at terminals 1, 2 and 3, while terminals 4 to 7 are only used as voltage leads. Lead on electrode 3 is connected to the TSC bulk, while all the other leads are on the edge. The electrical transport of the superconducting junction is governed by the generalized Landauer-Büttiker formula Anantram and Datta (1996); Entin-Wohlman et al. (2008); Chung et al. (2011); Wang et al. (2015a), which takes the form among leads 1-3 as
where and are the inflow current and voltage of lead , is the voltage of the TSC, and we have assumed the contact resistance vanishes between lead and the TSC bulk, which is appropriate when the electrodes are good metals. Here and are the normal reflection, Andreev reflection, normal transmission and Andreev transmission probabilities between leads and , respectively, which satisfy .
To examine the normal and Andreev probabilities, consider the charged chiral fermion mode incident from lead . When propagating on the bottom TSC edge A-B, it could randomly mix with due to unavoidable edge disorders. Therefore, when the incident charge mode reaches corner B, it has the normal and Andreev probabilities and to become and , but also has a remaining probability to propagate as . The mode will then circulate along the TSC edge, and has a propagation probability into charge modes , (or , ) whenever it reaches corner D (or B), thus contributing probabilities and (or and ) during its -th lap. Summing over then yields the total and . Such a summation is difficult. However, since is charge neutral, its propagation probability into electron and hole states will always be equal, so we conclude for any , and for all . Therefore, we find , and , which are the only quantities needed in the Landauer-Büttiker formula of Eq. (6).
Next we calculate . The Majorana basis will generically undergo a random SO(3) transformation after propagating along the TSC edge A-B, due to the unavoidable random chemical potential and pairing term , where is a random real antisymmetric matrix. This results in a random SO(3) transformation , where is path order and is the average Majorana velocity Levin et al. (2007); Lee et al. (2007); Lian and Wang (2018). We assume there is no magnetic vortices tunnelling across the edges since they cost energy, so the transformation is only SO(3) instead of O(3). The average normal and Andreev transmissions are thus given by the mean value over all SO(3) matrices :
where is the electron annihilation operator under Majorana basis. A straightforward calculation yields , and , and thus . This leads to the two-terminal conductances
Here is defined for current applied between leads and (with ), while is defined for current applied between leads and (with ). Similarly, one can get the resistance matrix measured from other leads as , and , where with current applied between leads and .
The exchange field can be tuned by either a perpendicular or an in-plane external magnetic field. Therefore, the TSC phases will experience the BdG Chern number variation as decreases in the hysteresis loop. Meanwhile, the QAH phase will experience the Chern number change , and in terms of . In general, will exhibit the plateau transition as shown in Fig. 4(a). Since the system in the magnetized state without external magnetic fields is homogenous in the sense of weak disorder without percolation transition, the unique quantized conductance plateau manifests the TSC.
Finally, we discuss the thermal transport. The chiral TSC exhibits a quantized thermal Hall conductance in units of , where is the Boltzmann constant and is temperature. Moreover, the QTQ junction will exhibit quantized thermal resistances resembling the electric resistances of a filling factor 2-3-2 integer quantum Hall junction Williams et al. (2007). For a heat current applied between leads and , the thermal resistances are given by , and . Generically, phonons and magnons also contribute to the thermal conductance, which will deviate from the quantized value. However, their contribution can be well distinguished from the temperature dependence Banerjee et al. (2017).
Discussion. We discuss the experimental feasibility of higher odd Chern number chiral TSC. The key point is to invert the bands by a large exchange field, while keeping the QAH insulating. The hybridization between top and bottom SSs in QAH better to be small. For QAH in magnetic TIs, the exchange field meV in CBST Lee et al. (2015), and is meV in V-I codoped TI Qi et al. (2016). vanishes when film thickness exceeds five quintuple layers. For T-SC in FTS, meV and meV below K Zhang et al. (2018). The work function in FTS grown on SrTiO is around eV, which is in the same range for that in (Bi,Sb)Te thin film on SrTiO about eV. Therefore, can be tuned to be small. is unknown, but can be tuned by inserting an insulating ultrathin layer between T-SC and QAH. Other possible T-SC materials include ion-based superconductor such as BaFeAs, LiFeAs zha (), and the superconducting doped TIs such as CuBiSe, TlBiTe Hor et al. (2010); Fu and Berg (2010); Wang et al. (2016). Recently, the QAH with higher Chern numbers has been realized in a multilayer of MTI he2 (). Such experimental progress on the material growth and rich material choice of MTI and T-SC makes the realization of the higher odd chiral TSC feasible. The transport of QTQ junctions of other higher chiral TSC will be studied in future work.
Acknowledgements.Acknowledgments. We thank Yang Feng, Xiao Feng, Tong Zhang and Ke He for helpful discussions. J.W. is supported by the Natural Science Foundation of China through Grant No. 11774065; the National Key Research Program of China under Grant No. 2016YFA0300703; the Natural Science Foundation of Shanghai under Grant No. 17ZR1442500; the National Thousand-Young-Talents Program; the Open Research Fund Program of the State Key Laboratory of Low-Dimensional Quantum Physics, through Contract No. KF201606; and by Fudan University Initiative Scientific Research Program. B.L. is supported by the Princeton Center for Theoretical Science at Princeton University.
- Wilczek (2009) F. Wilczek, Nat. Phys. 5, 614 (2009).
- Elliott and Franz (2015) S. R. Elliott and M. Franz, Rev. Mod. Phys. 87, 137 (2015).
- Moore and Read (1991) G. Moore and N. Read, Nucl. Phys. B 360, 362 (1991).
- Read and Green (2000) N. Read and D. Green, Phys. Rev. B 61, 10267 (2000).
- Mackenzie and Maeno (2003) A. P. Mackenzie and Y. Maeno, Rev. Mod. Phys. 75, 657 (2003).
- Kitaev (2006) A. Kitaev, Ann. Phys. 321, 2 (2006).
- Fu and Kane (2008) L. Fu and C. L. Kane, Phys. Rev. Lett. 100, 096407 (2008).
- Sau et al. (2010) J. D. Sau, R. M. Lutchyn, S. Tewari, and S. Das Sarma, Phys. Rev. Lett. 104, 040502 (2010).
- Alicea (2010) J. Alicea, Phys. Rev. B 81, 125318 (2010).
- Qi et al. (2009) X.-L. Qi, T. L. Hughes, S. Raghu, and S.-C. Zhang, Phys. Rev. Lett. 102, 187001 (2009).
- (11) B. Lian, X.-Q. Sun, A. Vaezi, X.-L. Qi, and S.-C. Zhang, arXiv:1712.06156 (2017).
- Kitaev (2003) A. Kitaev, Ann. Phys. 303, 2 (2003).
- Nayak et al. (2008) C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma, Rev. Mod. Phys. 80, 1083 (2008).
- Qi et al. (2010) X.-L. Qi, T. L. Hughes, and S.-C. Zhang, Phys. Rev. B 82, 184516 (2010).
- Chung et al. (2011) S. B. Chung, X.-L. Qi, J. Maciejko, and S.-C. Zhang, Phys. Rev. B 83, 100512 (2011).
- Wang et al. (2015a) J. Wang, Q. Zhou, B. Lian, and S.-C. Zhang, Phys. Rev. B 92, 064520 (2015a).
- Lian et al. (2016) B. Lian, J. Wang, and S.-C. Zhang, Phys. Rev. B 93, 161401 (2016).
- He et al. (2017) Q. L. He, L. Pan, A. L. Stern, E. C. Burks, X. Che, G. Yin, J. Wang, B. Lian, Q. Zhou, E. S. Choi, K. Murata, X. Kou, Z. Chen, T. Nie, Q. Shao, Y. Fan, S.-C. Zhang, K. Liu, J. Xia, and K. L. Wang, Science 357, 294 (2017).
- Wang et al. (2014) J. Wang, B. Lian, and S.-C. Zhang, Phys. Rev. B 89, 085106 (2014).
- Kou et al. (2015) X. Kou, L. Pan, J. Wang, Y. Fan, E. S. Choi, W.-L. Lee, T. Nie, K. Murata, Q. Shao, S.-C. Zhang, and K. L. Wang, Nat. Commun. 6, 8474 (2015).
- Feng et al. (2015) Y. Feng, X. Feng, Y. Ou, J. Wang, C. Liu, L. Zhang, D. Zhao, G. Jiang, S.-C. Zhang, K. He, X. Ma, Q.-K. Xue, and Y. Wang, Phys. Rev. Lett. 115, 126801 (2015).
- Wang (2016) J. Wang, Phys. Rev. B 94, 214502 (2016).
- Yasuda et al. (2017) K. Yasuda, M. Mogi, R. Yoshimi, A. Tsukazaki, K. S. Takahashi, M. Kawasaki, F. Kagawa, and Y. Tokura, Science 358, 1311 (2017).
- Lian et al. (2018) B. Lian, J. Wang, X.-Q. Sun, A. Vaezi, and S.-C. Zhang, Phys. Rev. B 97, 125408 (2018).
- Ji and Wen (2018) W. Ji and X.-G. Wen, Phys. Rev. Lett. 120, 107002 (2018).
- Huang et al. (2018) Y. Huang, F. Setiawan, and J. D. Sau, Phys. Rev. B 97, 100501 (2018).
- Strübi et al. (2011) G. Strübi, W. Belzig, M.-S. Choi, and C. Bruder, Phys. Rev. Lett. 107, 136403 (2011).
- (28) Y.-H. Li, J. Liu, H. Liu, H. Jiang, Q.-F. Sun, and X. C. Xie, arXiv:1804.10872 (2018).
- Fu and Kane (2009) L. Fu and C. L. Kane, Phys. Rev. Lett. 102, 216403 (2009).
- Akhmerov et al. (2009) A. R. Akhmerov, J. Nilsson, and C. W. J. Beenakker, Phys. Rev. Lett. 102, 216404 (2009).
- Law et al. (2009) K. T. Law, P. A. Lee, and T. K. Ng, Phys. Rev. Lett. 103, 237001 (2009).
- Hasan and Kane (2010) M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010).
- Qi and Zhang (2011) X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. 83, 1057 (2011).
- Zhang et al. (2018) P. Zhang, K. Yaji, T. Hashimoto, Y. Ota, T. Kondo, K. Okazaki, Z. Wang, J. Wen, G. D. Gu, H. Ding, and S. Shin, Science (2018), 10.1126/science.aan4596.
- Wang et al. (2013) J. Wang, B. Lian, H. Zhang, Y. Xu, and S.-C. Zhang, Phys. Rev. Lett. 111, 136801 (2013).
- Chang et al. (2013) C.-Z. Chang, J. Zhang, X. Feng, J. Shen, Z. Zhang, M. Guo, K. Li, Y. Ou, P. Wei, L.-L. Wang, Z.-Q. Ji, Y. Feng, S. Ji, X. Chen, J. Jia, X. Dai, Z. Fang, S.-C. Zhang, K. He, Y. Wang, L. Lu, X.-C. Ma, and Q.-K. Xue, Science 340, 167 (2013).
- Wang et al. (2015b) Z. Wang, P. Zhang, G. Xu, L. K. Zeng, H. Miao, X. Xu, T. Qian, H. Weng, P. Richard, A. V. Fedorov, H. Ding, X. Dai, and Z. Fang, Phys. Rev. B 92, 115119 (2015b).
- Wu et al. (2016) X. Wu, S. Qin, Y. Liang, H. Fan, and J. Hu, Phys. Rev. B 93, 115129 (2016).
- (39) M. Chen, X. Chen, H. Yang, Z. Du, and H.-H. Wen, arXiv:1804.09319 (2018).
- Anantram and Datta (1996) M. P. Anantram and S. Datta, Phys. Rev. B 53, 16390 (1996).
- Entin-Wohlman et al. (2008) O. Entin-Wohlman, Y. Imry, and A. Aharony, Phys. Rev. B 78, 224510 (2008).
- Levin et al. (2007) M. Levin, B. I. Halperin, and B. Rosenow, Phys. Rev. Lett. 99, 236806 (2007).
- Lee et al. (2007) S.-S. Lee, S. Ryu, C. Nayak, and M. P. A. Fisher, Phys. Rev. Lett. 99, 236807 (2007).
- Lian and Wang (2018) B. Lian and J. Wang, Phys. Rev. B 97, 165124 (2018).
- Williams et al. (2007) J. R. Williams, L. DiCarlo, and C. M. Marcus, Science 317, 638 (2007).
- Banerjee et al. (2017) M. Banerjee, M. Heiblum, A. Rosenblatt, Y. Oreg, D. E. Feldman, A. Stern, and V. Umansky, Nature 545, 75 (2017).
- Lee et al. (2015) I. Lee, C. K. Kim, J. Lee, S. J. L. Billinge, R. Zhong, J. A. Schneeloch, T. Liu, T. Valla, J. M. Tranquada, G. Gu, and J. C. S. Davis, Proc. Natl. Acad. Sci. USA 112, 1316 (2015).
- Qi et al. (2016) S. Qi, Z. Qiao, X. Deng, E. D. Cubuk, H. Chen, W. Zhu, E. Kaxiras, S. B. Zhang, X. Xu, and Z. Zhang, Phys. Rev. Lett. 117, 056804 (2016).
- (49) P. Zhang et al., arXiv:1803.00846 (2018).
- Hor et al. (2010) Y. S. Hor, A. J. Williams, J. G. Checkelsky, P. Roushan, J. Seo, Q. Xu, H. W. Zandbergen, A. Yazdani, N. P. Ong, and R. J. Cava, Phys. Rev. Lett. 104, 057001 (2010).
- Fu and Berg (2010) L. Fu and E. Berg, Phys. Rev. Lett. 105, 097001 (2010).
- Wang et al. (2016) Z. Wang, A. A. Taskin, T. FrÃ¶lich, M. Braden, and Y. Ando, Chem. Mater. 28, 779 (2016).
- (53) K. He, private communication.