We consider theoretically tunneling characteristic of a junction between a
normal metal and a chain of coupled Majorana bound states generated at
crossings between topological and non-topological superconducting sections, as
a result of, for example, disorder in nanowires. While an isolated Majorana
state supports a resonant Andreev process, yielding a zero bias differential
conductance peak of height 2e^2/h, the situation with more coupled Majorana
states is distinctively different with both zeros and 2e^2/h peaks in the
differential conductance. We derive a general expression for the current
between a normal metal and a network of coupled Majorana bound states and
describe the differential conductance spectra for a generic set of situations,
including regular, disordered, and infinite chains of bound states.Comment: 6 pages, 4 figure

Flensberg, Karsten

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Karsten Flensberg

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Tunneling characteristics of a chain of Majorana bound states

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u n i ve r s i t y  o f  co pe n h ag e n  Københavns UniversitetTunneling characteristics of a chain Majorana boundFlensberg, KarstenPublished in:Physical Review B Condensed MatterDOI:10.1103/PhysRevB.82.180516Publication date:2010Document versionEarly version, also known as pre-printCitation for published version (APA):Flensberg, K. (2010). Tunneling characteristics of a chain Majorana bound. Physical Review B CondensedMatter, 82(18), 180516 (R). https://doi.org/10.1103/PhysRevB.82.180516Download date: 02. Feb. 2020Tunneling characteristics of a chain of Majorana bound statesKarsten FlensbergDepartment of Physics, Harvard University, Cambridge, Massachusetts 02138, USAand Niels Bohr Institute, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen, DenmarkReceived 17 September 2010; published 22 November 2010We consider theoretically tunneling characteristic of a junction between a normal metal and a chain ofcoupled Majorana bound states generated at crossings between topological and nontopological superconduct-ing sections, as a result of, for example, disorder in nanowires. While an isolated Majorana state supports aresonant Andreev process, yielding a zero-bias differential conductance peak of height 2e2 /h, the situation withmore coupled Majorana states is distinctively different with both zeros and 2e2 /h peaks in the differentialconductance. We derive a general expression for the current between a normal metal and a network of coupledMajorana bound states and describe the differential conductance spectra for a generic set of situations, includ-ing regular, disordered, and infinite chains of bound states.DOI: 10.1103/PhysRevB.82.180516 PACS numbers: 74.45.c, 74.25.F, 74.55.v, 72.10.BgTopological materials are of large current interest, in partbecause of their potential for topological quantum computingand their interesting non-Abelian quasiparticles.1 One variantof this is topological superconductors where the low-energyquasiparticles in addition are Majorana fermions.2,3 Cur-rently, there is an active search for materials that can hostsuch particles, either in certain p-wave superconductors,4–8or semiconductors with proximity-induced superconductivityand strong spin-orbit coupling.9–12 Because it takes two Ma-jorana fermions to form a usual fermion that can couple toother degrees of freedom, detection of the state of the Majo-rana fermion system requires nonlocal measurements orinterferometry.13–15 In contrast, a local tunnel current, beingindependent of the parity of the topological superconductor,does not reveal information about the state of the Majoranafermions. Nevertheless, a tunneling probe could detect thepresence of a Majorana bound state MBS Refs. 12, 16, and17 and the detection of Majorana bound states is the firstmajor challenge in this field. Tunneling contact to an isolatedMajorana state give rise to a resonant Andreev process thatgives a zero-bias conductance peak of 2e2 /h.16 With twocoupled Majorana states cross correlations of the current intoeach could also show their existence and nonlocalcharacter.18,19However, because of material difficulties, isolated Majo-rana states might be rare. Rather it is to be expected thatdensity fluctuations will generate a random configuration oftopological/nontopological boundaries, at which Majoranastates will be located. For strong disorder the distances be-tween these states are sufficiently short for the MBS to over-lap and therefore it is important to understand how a networkof coupled Majorana fermions maps onto the tunneling char-acteristic. This problem was recently considered in Ref. 20 inthe weak-coupling regime, using a renormalization group toreduce a chain to a sum of single Majorana pairs on a loga-rithmic energy scale.In this Rapid Communication, a theory for tunneling be-tween a metallic probe and a collection of coupled Majoranastates in the strong-coupling regime is developed, and ex-perimentally relevant situations are addressed. This is donein the limit where the voltage eV, the tunneling broadenings, and the hopping matrix elements between any two MBSall are much smaller than the superconducting energy gap Eg.The regime of large Eg is well suited for characterization anddetection of the MBS because in absence of Majorana statesthe Andreev conductance is on the order of21 e2 /h /Eg2and thus much smaller than the resonant Andreev currentcarried by the Majorana states. Examples for different num-ber and configurations of Majorana states are given and thecase of a uniform infinite chain is solved exactly. Finally,disordered Majorana chains are addressed. Disorder is intro-duced as random nearest-neighbor couplings and it is shownto reduce to a finite chain, truncated by the first weak linkquantified below in the chain.The borders of the topological superconductor segmentsgive rise Majorana bound states. These states are zero-energysolutions to the Boguliubov-de Gennes equations for the ge-ometry in question. The general form of a MBS isSTS TS TSNMBSDensityPositionΓ t2,3FIG. 1. Color online Illustration of a semiconductor with in-duced superconducting order parameter from an adjacent supercon-ductor s. Spatial variations in density or superconducting orderparameter create crossings between TS and nontopological segmentwhen the density crosses the critical density vertical red lines.MBS circles are located at each crossing point and the distancesbetween them determine the coupling matrix elements tij of theresulting Majorana network. A tunneling contact N probes thenetwork by tunneling into the end Majorana mode, with tunnelingdensity of states .PHYSICAL REVIEW B 82, 180516R 2010RAPID COMMUNICATIONS1098-0121/2010/8218/1805164 ©2010 The American Physical Society180516-1i =  dxf,ixx + f,i x†x . 1The Majorana Fermion has the properties that i=i† andi2=1. The superconductor Hamiltonian, describing thecoupled Majorana state network, isHS =i2ij tiji j , 2where itij is the matrix element between Majorana boundstates i and j. For the examples presented in the plots in thisRapid Communication we assume that tij includes nearest-neighbor coupling only. This is a good approximation whenthe correlation length of the disorder is larger than the typicaldecay length of the Majorana bound states.The tunnel Hamiltonian between the normal metal and thesuperconductor isHT = k dxtkxck† x + H.c. ,where ck, are lead-electron annihilation operators and xthe superconductor electron field operator. As explainedabove, for eV ,Eg the Majorana states contribution tothe current dominates. Using Nambu representation, = ↑ ,↓ ,↓†,↑†, the projection of the field operator onto the manifold of Majorana states is xiif↑,ix , f↓,ix , f↓,i x , f↑,i x, which then leads to theeffective tunnel Hamiltonian describing the coupling be-tween the lead and the Majorana statesHT = kiVk,i ck†− Vk,icki, 3where Vk,i=dxf,ixtkx. The current operator is given bythe rate of change in the number of electrons in the normalleadI = − eN˙ = − ieHT,N/ =2eRekiVk,i Gi,k	 0 , 4where the lesser Green’s function is defined as Gi,k	 t= i	ck† it, which is written asGi,k	 t = jGijVk,jGk0	, 5where Gij ,=−i	Ti j is the full Keldysh time-ordered Green’s functions for the Majorana operators, andGk0 ,=−i	Tckck† 0 is the unperturbed normallead Green’s function. By choosing the chemical potential ofthe superconductor as a reference, the general current for-mula is derived to be22I =eh dMf−  + eV − f − eV 6with f being the Fermi-Dirac distribution andM = TrGR− GA . 7Here the retarded Majorana Green’s function isGR = 2 − 2it + i + −  −  −− −1, 8where t is an antisymmetric matrix while the Hermitian ma-trices  and  areij = 2kVk,iVk,j  − k , 9ij = P d2 ij −  . 10If the coupling matrix respects particle-hole symmetry,=−, the current is antisymmetric IV=−I−V.22The expression in Eq. 6 is a general finite temperatureexpression for the current into a Majorana state network interms of matrices describing the coupling to the normal leadand the Majorana network. The general formula is straight-forwardly extended to the case with more normal-metal con-tact connected to the network.23The remainder of this Rapid Communication deals withthe case where only a single Majorana bound state is coupledto the lead, i.e., iji,1 j,1. Moreover, assuming an energyindependent  so-called wide-band limit, which leads toij =0, the differential conductance reduces todIdV=2e2h  d ImG11R eVdf − eVd  11withGR = 2 − 2it + i2−1. 12For a single isolated Majorana state with tunnel broaden-ing coupling the Green’s function is: GiiR=2 / + i2 and thezero-temperature differential conductance is easily obtainedasdIdV=2e2h42eV2 + 42, 13which confirms that the resonant Andreev tunneling withzero-bias conductance G=2e2 /h.16 With two Majoranascoupled by tunneling t and only one of them coupled to thelead the differential conductance isdIdV=2e2h2eV2eV2 − 4t22 + 2eV2, 14which has a dip at zero voltage and peaks at eV=2t, wherethe conductance again reaches 2e2 /h. In fact, a very generalstatement holds for tunneling into the end of a chain, namely,that with an odd number of coupled Majorana states thezero-bias conductance is 2e2 /h, and with an even number thezero-bias conductance is zero. This can be shown by theinversion in Eq. 12 setting =0 and for an arbitrary chainmatrix tij. Moreover, for a cluster with n MBS the differen-tial conductance versus bias voltage has n−1 zeros and nvoltages where dI /dV=2e2 /h. If the normal metal electrodeoverlaps with more than one MBS these conclusions change,as shown in Fig. 2, where the conductance for some ex-amples is plotted.With many coupled MBS in the chain the conductanceoscillates between 2e2 /h and 0, as seen in Fig. 2. As theKARSTEN FLENSBERG PHYSICAL REVIEW B 82, 180516R 2010RAPID COMMUNICATIONS180516-2number of sites in the chain is increased, the period of theoscillations decreases. If the period is smaller than tempera-ture the conductance will average to a value between the twoextremes. The same occurs for an infinite homogeneouschain, which is considered next.With an infinite chain of Majorana states with nearest-neighbor couplings, tij, the Green’s function for the firstMBS is G11R=2g11, whereg11 =1g110 −1 − 4t122g˜22, 15and where g˜22 is the Green’s function for the network startingwith site 2, decoupled from site 1, and where g110 −1=+2i. An illustrative example is a homogeneous chain, i.e.,with all couplings identical tij = t. Then the Dyson equationfor g˜22 isg˜22 =1 + i+4t2 + ig˜33g˜22. 16Since all connections are equal g˜22= g˜33, and hence4t2 /g˜222− g˜22+1 /=0, which gives can be solved for g˜22.Choosing the correct branch cuts24 and setting this into theGreen’s function 15, the differential conductance is derivedto bedIdV=2e2h 44 + 4t2 − eV2eV2 + 4 + 4t2 − eV22, eV	 4t ,42eV + eV2 − 4t22 + 42, eV 4t .17This is an interesting expression with a line shape thatstrongly depends on the ratio t /, which is shown in Fig. 3.Furthermore, for eV=0 it reduces to dIdVV=0 = 2e2h2t + 2, 18which shows that for small tunnel broadening compared tothe bandwidth of the chain the zero-bias conductance is2e2 /h, which was expected because it corresponds to tunnel-ing into an effectively isolated Majorana state.As the last situation, which might also be the most experi-mentally relevant, we now discuss different realizations oflong disordered chains, sampled, for example, by scanningthe average density. The tunneling coupling between twoneighboring MBS depends both on the distance betweenthem and the deviation from the critical value for thetopological/nontopological transition, with exponential de-pendence on both, as was shown by Shivamoggi et al.20 for aspecific example. The distribution of tunneling couplings isthus a complicated convolution of amplitude fluctuations andthe level-crossings statistics,25 in this case with two cross-ings. The resulting distribution of tunneling couplings is aninteresting problem in itself and several regimes can be iden-tified depending on the ratio of the amplitude to thetopological/nontopological threshold and the ratio of the cor-relation length to the decay length of the Majorana boundstates. For example, for large correlation length and for thedisorder amplitude either close to or much larger than theS-TS threshold, the Majorana bound states would tend tocome in pairs. In all other regimes, there is no correlationbetween the MBS couplings. Here, we focus on the genericbehavior expected for a given configurations of the tunnelingcouplings in the chain, also taking the temperature broaden-ing into account.As we learned for the infinite chain, different behaviorsoccur depending on the ratio of  to the tunneling couplingstij. For the random chain the ratio of  to the spread oftunneling couplings turns out to be crucial. Clearly, if thespread in tunneling couplings is much smaller than , theaverage conductance resembles that of the homogenous infi-nite chain, which we have verified by numerical simulation.In contrast, with large fluctuations in the tunneling cou-plings, the infinite chain will be effectively truncated into afinite chain, where one of the tunneling coupling happens to10 5 0 5 100.0.51.1.52.dIdVe2 h10 5 0 5 100.0.51.1.52.10 5 0 5 100.0.51.1.52.eVdIdVe2 h10 5 0 5 100.0.51.1.52.eVFIG. 2. Color online The conductance for tunneling into theend of a chain with one, two, five, and eight Majorana statescoupled by t=, illustrated with a small icon over each plot. The ncoupled Majoranas give rise to a n states inside the gap, illustratedby the insets. The number of zeros and peaks with conductance2e2 /h is also determined by n, see text.10 0 100.0.51.1.52.eVdIdVe2 h. . .|t|/Γ323212313110FIG. 3. Color online The conductance for tunneling into theend of an infinite Majorana chain with identical tunneling cou-plings. The parameters t / ranges from 5 to 1/8, from outside in.TUNNELING CHARACTERISTICS OF A CHAIN OF… PHYSICAL REVIEW B 82, 180516R 2010RAPID COMMUNICATIONS180516-3be much smaller than kBT. To see this, invert the matrix inEq. 12 and pull out the dependence on the weak link andwrite is asGR11, =2D2,D1, − 4tn,n+12 D1,nDn+1, + iD2,, 19where tweak= tn,n+1 is the weak link, and where Di,j is thedeterminant of the matrix −2t for the isolated chain be-tween sites i and j, but with tn,n+1=0. The weak link has twoeffects, 1 it gives small shifts of the existing resonancesand 2 it creates new resonances. The new resonances, how-ever, have widths that scale with the square of the weakcoupling weak tweak2 / 	t, where 	t denotes typical cou-plings in the first part of the chain. Therefore the new reso-nances introduced by the chain after the weak link is notresolved if kBtweak. An example of this is shown in Fig. 4,where the conductance of a truncated chain is compared withthat of a full chain.In summary, the differential conductance for a junctionbetween a normal metal and topological superconductorhosting a network of Majorana bound states has been stud-ied. Different configurations of the interacting network ofbound states give rise to distinct tunneling spectra. Longchains with fluctuating tunneling couplings is truncated intoa finite chain once a coupling becomes smaller than a certaincritical value, determined by temperature.C. M. Marcus and M. Leijnse are gratefully acknowl-edged for stimulating discussions. Research funded in partby The Danish Council for Independent Research  NaturalSciences and by Microsoft Corporation Project Q.1 C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. DasSarma, Rev. Mod. Phys. 80, 1083 2008.2 A. Y. Kitaev, Phys. Usp. 44, 131 2001.3 L. Fu and C. L. Kane, Phys. Rev. Lett. 100, 096407 2008.4 S. Fujimoto, Phys. Rev. B 77, 220501 2008.5 M. Sato and S. Fujimoto, Phys. Rev. B 79, 094504 2009.6 P. Lee, arXiv:0907.2681 unpublished.7 A. Potter and P. Lee, arXiv:1007.4569 unpublished.8 X. Qi and S. Zhang, arXiv:1008.2026 unpublished.9 Y. Oreg, G. Refael, and F. V. Oppen, Phys. Rev. Lett. 105,177002 2010.10 J. Alicea, Phys. Rev. B 81, 125318 2010.11 R. M. Lutchyn, J. D. Sau, and S. Das Sarma, Phys. Rev. Lett.105, 077001 2010.12 J. Sau, S. Tewari, R. Lutchyn, T. Stanescu, and S. Das Sarma,arXiv:1006.2829 unpublished.13 L. Fu and C. L. Kane, Phys. Rev. Lett. 102, 216403 2009.14 A. R. Akhmerov, J. Nilsson, and C. W. J. Beenakker, Phys. Rev.Lett. 102, 216404 2009.15 L. Fu, Phys. Rev. Lett. 104, 056402 2010.16 K. T. Law, P. A. Lee, and T. K. Ng, Phys. Rev. Lett. 103, 2370012009.17 J. Linder, Y. Tanaka, T. Yokoyama, A. Sudbø, and N. Nagaosa,Phys. Rev. Lett. 104, 067001 2010.18 C. J. Bolech and E. Demler, Phys. Rev. Lett. 98, 237002 2007.19 J. Nilsson, A. R. Akhmerov, and C. W. J. Beenakker, Phys. Rev.Lett. 101, 120403 2008.20 V. Shivamoggi, G. Refael, and J. E. Moore, Phys. Rev. B 82,041405 2010.21 G. E. Blonder, M. Tinkham, and T. M. Klapwijk, Phys. Rev. B25, 4515 1982.22 See supplementary material at http://link.aps.org/supplemental/10.1103/PhysRevB.82.180516 for more details of the derivationof Eqs. 6–1023 by generalizing the k-sum in Eq. 9 to run over the differentcontacts as well and inserting a matrix that specifies the mea-surement contact in the trace in Eq. 7.24 For 	4t this is g˜22= − i4t2−2 /8t2 whereas for 4t the result is g˜22= −sgn2− 4t2 /8t2.25 I. Blake and W. Lindsey, IEEE Trans. Inf. Theory 19, 2951973.5 0 50.0.51.1.5eVdIdVe2 h5 0 50.0.51.1.5eVFIG. 4. Color online Both panels show the conductance for achain with ten sites and one weak link with tunnel coupling tweak=0.25 while the other links have t=. The weak links are the 1–2and 5–6 connections for the left/right panel, respectively. The thickblue curve is the conductance for kT=0.1, the thin gray curve forkT=0 while the crosses show the conductance for the chain truncateat the weak link also with kT=0.1. It is clearly seen that thetruncated approximation works well, because the chain after thetruncations leads to structure barely resolvable, because kBT is notmuch smaller than the width 0.125 of the additional resonances.It should be noted that the situation with only a single deviating linkis not the generic situation but it is chosen here to illustrate thepoint, that the first weak link determines the effective length of thechain.KARSTEN FLENSBERG PHYSICAL REVIEW B 82, 180516R 2010RAPID COMMUNICATIONS180516-4

Tunneling characteristics of a chain Majorana bound

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Tunneling characteristic of a chain of Majorana bound states

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