Tilting of the magnetic field in Majorana nanowires: critical angle and zero-energy differential conductance
aa r X i v : . [ c ond - m a t . m e s - h a ll ] O c t Tilting of the magnetic field in Majorana nanowires: critical angle and zero-energydifferential conductance
Stefan Rex and Asle Sudbø
Department of Physics, Norwegian University of Science and Technology, N-7491 Trondheim, Norway (Dated: June 11, 2018)Semiconductor nanowires with strong spin-orbit coupling and proximity-induced s -wave super-conductivity in an external magnetic field have been the most promising settings for approachestowards experimental evidence of topological Majorana zero-modes. We investigate the effect oftilting the magnetic field relative to the spin-orbit coupling direction in a simple continuum modeland provide an analytical derivation of the critical angle, at which the topological states disappear.We also obtain the differential conductance characteristic of a junction with a normal wire for differ-ent tilting angles and propose a qualitative change of the dependence of the zero-energy differentialconductance on the tunnel barrier strength at the critical angle as a new criterion for establishingthe topological nature of the observed signal. I. INTRODUCTION
Many decades after the prediction of Majoranafermions , with no direct and unequivocal experimen-tal evidence for their existence, the possibility of find-ing emergent Majorana modes of a topological nature incondensed matter systems has evoked considerable inter-est in a number of systems , partly because of theirexpected non-Abelian braiding statistics . Amongthe proposed systems, semiconductor nanowires withstrong spin-orbit coupling (SOC) and induced s -wave su-perconductivity in an external magnetic field (Majoranananowires) have become the most prominent setting.Here, suspected signatures of Majorana zero-modes havealready been measured . However, the experimentalfindings do not match the predictions precisely, and somepredictions therefore have been made for more realisticnanowire models . This includes, for instance, finitetemperature, finite-size effects and the three-dimensionalwire geometry. Still, further distinguishing criteria forthe existence of the topological states in experiment aredesirable.In the present work, we go back to a simple and analyt-ically accessible one-dimensional continuum model. Wefocus on the possibility of driving the topological phasetransition by changing the direction of the magnetic fieldrelative to the SOC direction, while the standard choiceis taking them orthogonal. It is immediately clear thatthe Majorana zero-modes cannot exist for arbitrary fielddirections. Some experiments have included a rotationof the external magnetic field, but there has been lim-ited quantitative analysis of the precise impact ofthe field direction on the Majorana zero-modes and themeasured quantity, namely the differential conductancein a junction of the Majorana nanowire with a normallead.In this paper, we carry out a detailed analysis of theeffect of rotating the magnetic field, with particular em-phasis on identifying features of the differential conduc-tance directly connected to the the topological characterof the zero-energy modes. In Sec. II, we formulate the Hamiltonian of the system. In Sec. III, we present away to analytically derive the allowed field directions interms of a critical angle, for which the system remains inthe topological phase. Our analytical results confirm thenumerically inspired results of Ref. 29. In Sec. IV, wecompute the differential conductance characteristics of anormal-Majorana nanowire junction for various angles ofthe Zeeman field relative to the spin-orbit coupling di-rection. In particular, we concentrate on the zero-energydifferential conductance and propose one further criterionfor testing the topological origin of the observed peak byvarying the tunnel barrier strength while tilting the fieldacross the critical angle. The main result is that belowsome critical tilting angle away from the direction wherethe Zeeman-field and the SOC are orthogonal, the valueof the zero-energy peak is quantized in units of 2 e h , where e is the electron charge and h is Planck’s constant, inde-pendent of the tunnel barrier of the junction, the valuebeing protected by topology. Beyond a certain angle, thisis no longer so, and the value of the zero-energy peak de-pends on the barrier potential. Conclusions are given inSec. V. II. MODEL HAMILTONIAN
We consider a one-dimensional semiconductornanowire with SOC strength α and a proximity-induced s -wave superconducting gap ∆. Thermal effects canbe taken into account in a simple way by taking intoaccount the temperature dependence of the gap ∆ inthe standard way, at least for temperatures not tooclose to the superconducting transition temperature. Inthis paper, we choose the nanowire to be aligned withthe x -axis, with the SOC in z -direction. We expressthe external magnetic field B in spherical coordinates,with the polar angle ϑ measured from the z -axis andthe azimuthal angle ϕ measured from the x -axis, andintroduce the Zeeman energy E Zee = gµ B B . A sketchof the system and the chosen coordinates can be found inFig. 1. The Bogoliubov-De Gennes (BdG) Hamiltonian FIG. 1. Schematic view of the system: The semiconductornanowire (yellow) is placed on a bulk s -wave superconductorand defines the x -axis of the coordinate system. The z -axisis parallel to the SOC direction (labeled S in the figure).The direction of the magnetic field B is represented by thetwo angles ϑ (tilting relative to the SOC) and ϕ (azimuthalrotation in the xy -plane). acting on spinors ψ = ( u ↑ , u ↓ , v ↑ , v ↓ ) T , where u, v referto the electron and hole part of a quasiparticle and ↑ , ↓ to the spin in z -direction, respectively, reads H ( k ) = (cid:18) h n ( k ) h sc ( k ) h † sc ( k ) − h Tn ( − k ) (cid:19) , (1a)with the normal part h n ( k ) = (cid:18) ξ k + E Zee cos ϑ + kα E Zee sin ϑe − iϕ E Zee sin ϑe iϕ ξ k − E Zee cos ϑ − kα (cid:19) (1b)and s -wave pairing h sc ( k ) = h sc = (cid:18) − ∆ 0 (cid:19) , (1c)where ξ k = ( ~ k ) / m − µ , m is the effective electronmass, and µ the chemical potential. III. CRITICAL ANGLE
It is well-known theoretically that the system harborsMajorana zero-modes in the topological phase, E Zee > p ∆ + µ , when B is orthogonal to the SOC di-rection ( ϑ = π ). If the field is tilted, on the other hand,the Majorana modes disappear at a critical angle ϑ c , where the energy gap closes. Figure 2 illustrates theeigenenergies of the BdG Hamiltonian Eq. (1) for par-allel and orthogonal field and at ϑ = ϑ c . We note thatlevel crossings happen only at ϑ = π , thus the gap closesonly indirectly at ϑ c . The second angle, ϕ , only givesa phase factor in the eigenstates and is irrelevant forthe eigenenergies and the discussion of topological states. −3 −2 −1 0 1 2 3k [2mα/ħ ]−4−2024 E FIG. 2. The four eigenenergies of the BdG HamiltonianEq. (1) as a function of momentum for ϑ = π (black dashedlines), at the critical angle (green solid lines), where thegap closes (here ϑ c ≈ . π ), and at ϑ = π (blue dottedlines). The orange line indicates zero energy. Parameters: m = 1 , ∆ = 1 . , E Zee = 1 . , α = p / , µ = 0. The critical angle was observed to follow a rule equiva-lent to cos ϑ c = ∆ /E Zee in numerical calculations . Inthis section, we provide the analytical derivation of thisrule.Technically, the task is to find the angle at which thelow-energy band first reaches zero energy. The calcula-tion of the eigenenergies is done via the characteristicpolynomial, p k ( E ) = det( H ( k ) − E ), which is of order8 in momentum. For E = 0, all odd powers of k van-ish, leaving a biquartic equation. With the substitution κ = k , it reads p ( κ ) = "(cid:18) ~ m κ − µ (cid:19) − α κ + ∆ − E +4 α (∆ − E cos ϑ ) κ . (2)As long as the band gap remains open, p k (0) will besolved only by complex momenta, whereas real solutionsappear when B is tilted beyond the critical angle. Thereal solutions of p k (0) lead to non-negative solutions of p ( κ ). To derive the critical angle, we will exploit thespecial form of Eq. (2), being the square of a quadraticpolynomial in κ , with one additional κ -linear term con-taining the dependence on ϑ . We analyze the quadraticexpression first, and find its zeros κ , = 12 (cid:18) m ~ (cid:19) (cid:20) ~ µm + α ± s(cid:18) ~ µm + α (cid:19) − (cid:18) ~ m (cid:19) ( µ + ∆ − E ) . (3)To allow for topological states at all, ( µ + ∆ − E )must necessarily be negative . Thus, Eq. (3) alwaysyields two real solutions, where κ > κ < −4 −2 0 2 4 ϰ [4m α /ħ ] −4−20246810 p ( ϰ ) FIG. 3. The characteristic polynomial p ( κ ) of the Hamilto-nian at zero energy as a function of κ = k for the tilting an-gles ϑ = 0 . π (blue dashed line), the critical angle ϑ c ≈ . π (green solid line), where positive solutions for κ appear first,and ϑ = 0 . π . Parameters: m = 1 , ∆ = 1 . , E Zee = 1 . , α = p / , µ = 0. semidefinite and will have precisely the same solutions,just two-fold degenerate each. If, however, the κ -linearterm is present with positive (negative) coefficient, thepoint-symmetry of p ( κ ) is lost and the solutions becomenon-degenerate, where the positive solution is split in twodistinct complex (real) values, cf. Fig 3. We concludefrom Eq. (2) that the system is in the topological phase,when ∆ − E cos ϑ >
0. Consequently, the criticalangle satisfies cos ϑ c = ± ∆ E Zee . (4)Thus, we have analytically confirmed the numerical re-sults obtained in Ref. 29. As the angle ϑ is increasedthrough the value ϑ c , topologically trivial zero-energystates will appear with the momentum ±√ κ . An al-ternative, but much more lengthy, derivation of the sameresult using the discriminant of the fourth-order poly-nomial p ( κ ), is also possible.The angle-resolved topological phase diagram is shownin Fig. 4. If the Zeeman energy is just slightly largerthan the superconducting gap, ϑ can be varied over awide range without destroying the Majorana zero-modes,whereas for large Zeeman energy the tilting angle is re-stricted to a narrow range about π . In that sense,a high field does not lead to a more stable topologi-cal phase, although E Zee > p ∆ + µ is a necessaryprerequisite . This is readily seen, since this lat-ter condition acts on the energy gap at zero-momentum,which does not depend on the direction of the field. Incontrast, if the phase transition is driven by ϑ , the gapcloses near the Fermi momentum at √ κ , cf. Fig. 2,where increasing the field strength pushes the low-energyband closer to zero. ∆/E Zee ππ ϑ trivial topologicaltrivial FIG. 4. The angle-resolved topological phase diagram of theMajorana nanowire.
IV. DIFFERENTIAL CONDUCTANCECHARACTERISTICS
In the remainder of this paper, we focus on the dif-ferential conductance characteristics of a junction of theMajorana nanowire with a normal lead and the impactof tilting B . To the best of our knowledge, the angulardependence of the differential conductance in such junc-tions has only been briefly discussed in Ref. 24 so far,based on numerical studies of a tight-binding model. Incontrast, we will analyze the current through the systemin a simple continuum model. In the following, we willfor simplicity set µ = 0.We assume infinite wire length and a tunnel barrierof strength V at the junction (located at x = 0). Thenormal ( x <
0) and superconducting ( x >
0) sections ofthe wire are modeled with the same Hamiltonian Eq. (1),where we just set ∆ = 0 in the normal state. For elec-trons impinging from the normal side onto the junctionwe investigate the coefficients of reflected and transmit-ted waves. To solve the scattering problem, we employa Blonder-Tinkham-Klapwijk (BTK) formalism , i.e.,matching of wavefunctions at the junction. The origi-nal BTK scheme is extended to account for the spin aswell.At a given energy E , we first obtain all possible mo-menta by solving p k ( E ) = 0 for the normal and the su-perconducting wire. Exact diagonalization of Eq. (1) ateach k (including complex) then yields plane-wave statesΨ k ( x ) = ψ k e ikx with four-component spinors ψ k . Theincident electron wave Ψ in k in is always chosen from thenormal low-energy band. All other states that corre-spond to incoming waves are discarded. The scatteringprocess comprises ordinary and Andreev reflection intothe normal lead, and transmission without ( k >
0) andwith ( k <
0) branch crossing into the superconductinglead. The corresponding scattering coefficients are de-noted a i , b i , c i , d i , respectively, where i ∈ { , } labelsthe pseudospin. The total wavefunctions on the normaland superconducting side of the junction are thenΨ n ( x <
0) = Ψ in k in + X i =1 , a i Ψ k a,i + b i Ψ k b,i , (5)Ψ sc ( x >
0) = X i =1 , c i Ψ k c,i + d i Ψ k d,i . (6)At the junction, we impose the boundary conditionsΨ n ( x → − ) − Ψ sc ( x → + ) = 0 (7) ∂ x Ψ n ( x → − ) − ∂ x Ψ sc ( x → + ) = 2 mV ~ Ψ(0) , (8)and solve the resulting linear system of equations to ob-tain all scattering coefficients. The probability current J = ~ m Im (cid:0) Ψ † ∂ x τ z Ψ (cid:1) + α ~ Ψ † σ z Ψ (9)carried by each outgoing wave, where we have taken intoaccount a contribution due to the SOC , is proportionalto the square of the absolute value of the respective co-efficient. Here, τ z and σ z denote Pauli matrices acting inparticle-hole and spin space, respectively. In the sub-gapregime, where the Majorana modes reside, the system iseffectively spinless, therefore we will relinquish the dis-tinction of states with different pseudospin for the dis-cussion of the scattering probabilities, denoted A, B, C, and D . Then, C , for instance, reads C = X i =1 , | c i | (cid:12)(cid:12)(cid:12) ψ † k c,i (cid:0) Re ( k c,i ) τ z + αm ~ σ z (cid:1) ψ k c,i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k in + αm ~ ψ † in σ z ψ in (cid:12)(cid:12)(cid:12) . (10)Note that for A and B the term ψ † k τ z ψ k gives always just − E through the junction at zero temperature is finallygiven by dIdE = 1 + A − B in units of e h , and insidethe gap, where C = D = 0, even simpler as dIdE = 2 A byconservation of probability ( A + B + C + D = 1).By this scheme, we obtain the scattering probabili-ties and the differential conductance profile dIdV ( E ) of thejunction for different field directions, cf. Fig. 5. Thescattering probability profiles indicate the features of thebandstructure at the respective respective angle, e.g. thegap width. In the topological phase, the conductancepeak at zero energy that signals the existence of Majo-rana zero-modes is clearly seen. The peak gets narroweras the tilting angle of the field approaches the criticalangle, and disappears in the trivial phase. As expected,the peak height exhibits the quantized value of 2 e h due to resonant Andreev reflection.Attempts at detecting emergent Majorana zero-modesexperimentally originally focused on the quantized valueof the zero-energy differential conductance as the hall-mark of such states. Under real conditions, however,only much smaller values are observed . Other, morequalitative and more robust distinguishing criteria are re-quired. We propose that sharp change in the zero-energy -4 -2 0 2 4E s c a tt e r a m p li t u d e ϑ =0.5π d I d E [ e h ] -4 -2 0 2 4E s c a tt e r a m p li t u d e ϑ =0.7π d I d E [ e h ] -4 -2 0 2 4E s c a tt e r a m p li t u d e ϑ =ϑ c ≈0.81π d I d E [ e h ] -4 -2 0 2 4E s c a tt e r a m p li t u d e ϑ =0.9π d I d E [ e h ] -4 -2 0 2 4E s c a tt e r a m p li t u d e ϑ =1.0π d I d E [ e h ] FIG. 5. The energy-resolved scattering coefficients (left scale)and differential conductance characteristic (right scale) of anormal-Majorana nanowire junction at different tilting an-gles of the magnetic field: Andreev reflection A (blue solidline), ordinary reflection B (green dash-dotted line), trans-mission without branch crossing C (red dotted line), trans-mission with branch crossing D (purple dashed line), anddifferential conductance (black bold solid line). Parameters: m = 1 , ∆ = 1 . , E Zee = 1 . , α = p / , V = 2 . π π ϑ c πϑ d I d E | E = [ e h ] V=0V=1V=2V=4V=10V=50
FIG. 6. The differential conductance at zero energy as a func-tion of the tilting angle of the field for different tunnel barrierstrengths V . Parameters: m = 1 , ∆ = 1 . , E Zee = 1 . , α = p / differential conductance peak at the critical tilting an-gle ϑ c of the field, provides an appropriate further qual-itative criterion for examining the topological nature ofmeasured signatures. In experiments, it may be diffi-cult to record the full conductance profiles as in Fig. 5with the required precision. Therefore, we propose tomeasure the zero-energy differential conductance for dif-ferent tilting angles of the field while varying the tunnelbarrier strength of the junction. The predicted behavioris shown in Fig. 6. A qualitative change of the depen-dence of dIdE (0) on V should be observed at the criticalangle upon entering the trivial phase, where the conduc-tance can be suppressed by increasing the tunnel bar-rier. In the topological state, the value of the zero-biasconductance peak is impervious to the change in barrierstrength, being protected by topology.At finite temperatures well below the superconduct-ing transition temperature, the impact on the results inFig. 5 is to slightly smear the sharp cusp at ϑ c . Themain change in qualitative behavior above and below ϑ c is robust. The main effect on the critical angle itself canbe accounted for by taking into account the temperature dependence of the gap in Eq. 4. Finite-size effects arealso present, in principle. A finite length of the Majoranananowire causes an overlap of the exponentially localizedtopological states at the ends of the wire . Thus, thetransition happens before the low-energy band reacheszero and the true topological regime is expected to beslightly narrower than predicted by ϑ c . Numerical datafrom Ref. 29 indicate, however, that this effect is notimportant. V. CONCLUSION
In this paper, we have studied semiconductornanowires with SOC and s -wave superconductivity inan external magnetic field with arbitrary direction inan analytically accessible continuum model. We havederived the critical tilting angle ϑ c of the field relativeto the SOC direction, at which the topological (Majo-rana) zero-modes disappear. Our result confirms recentnumerical findings . Furthermore, we have considerednormal-Majorana nanowire junctions and obtained thedifferential conductance characteristics at various angles,where, as expected, a stable peak at zero-energy withthe quantized value of 2 e h occurs as long as the field isnot tilted beyond the critical angle ϑ c . The peak disap-pears for fields aligned too much in the direction of theSOC and the value of the zero-energy differential conduc-tance becomes strongly dependent on the tunnel barrierstrength. We have pointed out the qualitative changeof the dependence on the barrier strength at the criticalangle and suggest it as further criterion to test the topo-logical nature of the experimentally observable signals,even if the theoretical quantized peak value may not bereached under realistic conditions.A.S. and S.R. acknowledge support from the Nor-wegian Research Council, Grants 205591/V20 and216700/F20. We thank Jacob Linder for helpful com-ments. E. Majorana, Nuovo Cimento , 171 (1937). N. Read and D. Green, Phys. Rev. B , 10267 (2000). A. Kitaev, Phys. Usp. , 131 (2001). S. Das Sarma, C. Nayak, and S. Tewari, Phys. Rev. B ,220502(R) (2006). Y. Tsutsumi, T. Kawakami, T. Mizushima, M. Ichioka,and K. Machida, Phys. Rev. Lett. , 135302 (2008). V. Gurarie, L. Radzihovsky, and A. V. Andreev, Phys.Rev. Lett. , 230403 (2005). L. Fu and C. L. Kane, Phys. Rev. Lett. , 096407 (2008). F. Wilceck, Nature Phys. , 614 (2009). J. Linder, Y. Tanaka, T. Yokoyama, A. Sudbø, and N. Na-gaosa, Phys. Rev. Lett. , 067001 (2010). J. Alicea, Rep. Prog. Phys. , 076501 (2012). Y. Tanaka, M. Sato, and N. Nagaosa, J. Phys. Soc. Jpn. , 011013 (2012). S. Nakosai, J. C. Budich, Y. Tanaka, B. Trauzettel, andN. Nagaosa, Phys. Rev. Lett. , 117002 (2013). Y. Asano and Y. Tanaka, Phys. Rev. B , 104513 (2013). D. A. Ivanov, Phys. Rev. Lett. , 268 (2001). A. Kiteav, Annals Phys. , 2 (2003). R. M. Lutchyn, J. D. Sau, and S. Das Sarma, Phys. Rev.Lett. , 077001 (2010). V. Mourik, K. Zuo, S. M. Frolov, S. R. Plissard, E. P.A. M. Bakkers, and L. P. Kouwenhoven, Science , 1003(2012). L. P. Rokhinson, X. Liu, and J. K. Furdyna, Nature Phys. , 795 (2012). A. Das, Y. Ronen, Y. Most, Y. Oreg, M. Heiblum, andH. Shtrikman, Nature Phys. , 887 (2012). A. D. K. Finck, D. J. Van Harlingen, P. K. Mohseni,K. Jung, and X. Li, Phys. Rev. Lett. , 126406 (2013). M. T. Deng, C. L. Yu, G. Y. Huan, M. Larsson, andP. Caroff, Nano Lett. , 6414 (2012). H. O. H. Churchill, V. Fatemi, K. Grove-Rasmussen, M. T.Deng, P. Caroff, H. Q. Xu, and C. M. Marcus, Phys. Rev.B , 241401(R) (2013). E. J. H. Lee, X. Jiang, M. Houzet, R. Aguado, C. M.Lieber, and S. De Franceschi, Nature Nanotechnol. , 79(2013). C.-H. Lin, J. D. Sau, and S. Das Sarma, Phys. Rev. B ,224511 (2012). J. S. Lim, R. Lopez, and L. Serra, New J. Phys. , 083020(2012). J. S. Lim, R. Lopez, and L. Serra, Europhys. Lett. ,37004 (2013). J. Osca and L. Serra, Phys. Rev. B , 144512 (2013). E. Prada, P. San-Jose, and R. Aguado, Phys. Rev. B ,180503(R) (2012). J. Osca, D. Ruiz, and L. Serra, Phys. Rev. B , 245405 (2014). Y. Oreg, G. Refael, and F. von Oppen, Phys. Rev. Lett. , 177002 (2010). With our choice of coordinates, the rule contains only oneinstead of two angles, and appears with a cosine instead ofa sine. E. L. Rees, The American Mathematical Monthly , 51(1922). G. E. Blonder, M. Tinkham, and T. M. Klapwijk, Phys.Rev. B , 4515 (1982). F. Bottegoni, H.-J. Drouhin, J.-E. Wegrowe, and G. Fish-man, J. Appl. Phys. , 07C305 (2012). K. Sengupta, I. Zutic, H. Kwon, V. M. Yakovenko, andS. Das Sarma, Phys. Rev. B , 144531 (2001). K. T. Law, P. A. Lee, and T. K. Ng, Phys. Rev. Lett. ,237001 (2009). K. Flensberg, Phys. Rev. B , 180516(R) (2010). M. Wimmer, A. R. Akhmerov, J. P. Dahlhaus, andC. W. J. Beenakker, New J. Phys.13