Table Of ContentRobust Transport Signatures of Topological Superconductivity in Topological
Insulator Nanowires
Fernando de Juan,1,2 Roni Ilan,2 and Jens H. Bardarson2,3
1Materials Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720
2Department of Physics, University of California, Berkeley, CA 94720, USA
3Max-Planck-Institut fu¨r Physik komplexer Systeme, N¨othnitzer Str. 38, 01187 Dresden, Germany
(Dated: September 10, 2014)
Finding a clear signature of topological superconductivity in transport experiments remains
an outstanding challenge. In this work, we propose exploiting the unique properties of three-
dimensional topological insulator nanowires to generate a normal-superconductor junction in the
4
single-mode regime where an exactly quantized 2e2/h zero-bias conductance can be observed over
1
a wide range of realistic system parameters. This is achieved by inducing superconductivity in half
0
2 of the wire, which can be tuned at will from trivial to topological with a parallel magnetic field,
while a perpendicular field is used to gap out the normal part, except for two spatially separated
p chiralchannels. ThecombinationofchiralmodetransportandperfectAndreevreflectionmakesthe
e
measurement robust to moderate disorder, and the quantization of conductance survives to much
S
higher temperatures than in tunnel junction experiments. Our proposal may be understood as a
8 variant of a Majorana interferometer which is easily realizable in experiments.
]
ll Atopologicalsuperconductorisaproposednovelphase a) b)
a
of matter with exotic properties like protected bound-
h
- ary states and emergent quasiparticles with non-Abelian
s statistics. Ifrealized,thesesuperconductorsareexpected
e
m toconstitutethemainbuildingblockoftopologicalquan-
. tum computers [1]. The prototypical example of this c) d)
t phase, the p-wave superconductor, has proven to be dif-
a
m ficult to find in nature, with superconducting Sr2RuO4
and,indirectly,theν =5/2fractionalquantumHallstate
-
d amongtheveryfewconjecturedcandidates. Whilemany
n experimentshavebeensuggestedandperformedonthese
o
systems,evidencefortheirtopologicalpropertiesremains
c
[ elusive. However,therecentrealizationthatap-wavesu- FIG. 1. a) An NS junction formed with a TI nanowire. b)
perconductorneednotbeintrinsic, butcanalternatively Schematicrepresentationofthemodesinvolvedintransport:
2 beengineeredwithregulars-wavesuperconductingprox- a chiral mode splits into two Majorana modes at the inter-
v face, recombines, and exits as a chiral mode again. c) The
imityeffectinstronglyspin-orbitcoupledmaterials[2–4],
2 Majorana interferometer proposed in Refs. 9 and 10. S and
3 has opened a promising new path in the search for topo-
D denote source and drain respectively. d) An unfolded rep-
8 logical superconductivity.
resentation of the setup in b).
5
. Aclassofthesenewtopologicalsuperconductorsispre-
1
dicted to be realized in one-dimensional (1D) systems
0
4 with broken time-reversal symmetry [5]. These systems
1 are characterized by Majorana zero-energy end states, a topological invariant [7, 8] that directly distinguishes
: which are responsible for a fundamental transport effect trivial from topological superconductors in a transport
v
i known as perfect (or resonant) Andreev reflection [6]: in experiment.
X
a junction between a normal contact that hosts a single A prominent example of a 1D topological supercon-
r propagatingmodeandatopologicalsuperconductor,this ductor is realized in semiconducting quantum wires in
a
modemustbeperfectlyreflectedasaholewithunitprob- the presence of a magnetic field [11, 12]. Recent trans-
ability, resulting in the transfer of a Cooper pair across port experiments with such wires aimed to demonstrate
the junction and an exactly quantized zero-bias conduc- the existence of this phase have reported a finite zero-
tance of 2e2/h. This effect does not depend at all on bias conductance across a NS junction [13, 14], but the
the details of the junction, and can be intuitively under- predictedquantizationhassofarremainedachallengeto
stood as resonant transport mediated by the Majorana observe. A possible reason is that these wires typically
end states [3, 6]. On the other hand, if the supercon- host several modes [15–20] and fine tuning the chemi-
ductor is trivial and hence has no Majorana state, in cal potential to the single mode regime can be difficult.
thesingle-moderegimetheconductanceexactlyvanishes. In the presence of several modes, either a tunnel bar-
The conductance in the single-mode regime is in fact rier [13, 14] or a quantum point contact [8] may be used
2
toisolatetheresonantcontribution,butthetemperature wavefunctions satisfy antiperiodic boundary conditions
required to resolve a zero-bias peak then becomes chal- insduetothecurvature-inducedπ Berryphase[22,23].
lengingly small. The optimal NS junction to probe this The eigenfunctions of H thus have the form
0
effect should therefore have a robust, easy to manipu-
late single-mode normal part smoothly interfaced with a ψk,n(x,s)=eikxeilnsχk,n, (2)
superconductor that can be controllably driven into the
with half-integer angular momentum l = n
topological phase. n
1/2 where n Z. The spectrum is E −=
In this work, we propose to realize such a junc- k,n
(cid:112) ∈
tion starting from an alternative route to 1D topolog- vF k2+(2π/P)2(ln+η)2, and is depicted in Fig. 2(a).
ical superconductivity, recently proposed by Cook and For η = 0 all modes are doubly degenerate, while for
Franz[21],basedontheuseofnanowiresmadefromthree η =1/2 the number of modes is always odd because the
dimensional topological insulators (TI). In the nanowire n=0 one is not degenerate.
geometry, the 2D surface states of a TI are resolved into By bringing the wire into contact with an s-wave su-
a discrete set of modes, with the property that when perconductor [21], as shown in Fig. 1, an s-wave pair-
a parallel flux of h/2e threads the wire, the number of ing potential ∆ is induced due to the proximity effect.
modes is always odd [22–24]. When a superconducting The Bogoliubov-de Gennes Hamiltonian can be written
gapisinducedonthesurfaceviatheproximityeffect,this as H = 1Ψ† Ψ with
2 H
guarantees that the system becomes a topological super-
(cid:18) (cid:19)
conductor [5, 21]. An NS junction can then be built be = H0 ∆(s) , (3)
proximitizing only part of the wire, where the supercon- H ∆∗(s) T−1H0T
−
ducting part can be tuned in and out of the topological
phase with the in-plane flux [21, 25]. where Ψ=(ψ↑,ψ↓,ψ↓†,−ψ↑†) is a Nambu spinor. The in-
In addition, our design simultaneously allows to drive duced pairing potential is ∆(s) = ∆0e−invs, where the
thenormalregionintothesingle-moderegimebyexploit- phaseof∆canwindaroundtheperimeterwithvorticity
ing the unique orbital response of TI surface states to n . For η = 0 the ground state has n = 0. Around
v v
magneticfields[26–30]. Whenaperpendicularfieldisap- η = 1/2, however, it should be energetically favorable
plied to the normal part of the wire, its top and bottom for ∆ to develop a vortex [39]. In an actual experiment,
regions become insulating because of the Quantum Hall n is expected to jump abruptly as η is ramped continu-
v
Effect. In between these regions, counter-propagating ously from zero to 1/2 [40]. For η around 1/2 and in the
chiral edge states are formed, which are protected from presenceofavortex, thenanowirebecomesatopological
backscattering due to their spatial separation. The re- superconductor for any µ within the bulk gap [21].
sulting NS junction, shown in Fig. 1, has a single chiral The presence of the vortex is essential in order to ob-
mode reflecting from the superconductor, and is ideal serveperfectAndreevreflectioninoursetup. Toseethis,
for probing conductance quantization. Moreover, all of consider the Hamiltonian in Eq. (3) in the presence of a
its components are readily available, as both surface NSinterfaceatx=0withn vortices. IntroducingPauli
v
transport in TI nanowires [31–34] and the contacting of matrices τ acting in Nambu space
i
bulk TI with superconductors [35–37] have already been
demonstrated experimentally. In the remainder of this (nv) =[ iσx∂x+σy( i∂s+η τz)2π/P µ]τz
H − − −
paper,weprovideadetailedstudyofthetransportprop- +∆ θ( x)e−iτznvsτ . (4)
0 x
erties of this system, demonstrating that conductance −
quantizationisachievableunderrealisticconditions, and For n =0, electron states in the normal part have fi-
v
discusstheadvantagesofoursetupoverotherproposals. niteangularmomentuml , seeEq.(2), whileholestates
n
To model the proposed device, we consider a rectan- have angular momentum l , independently of the value
n
gular TI nanowire of height h and width w (perimeter of η. Since angular mome−ntum must be conserved upon
P = 2h+2w). The surface of the wire is parametrized reflection, a single incoming electron can never be re-
withtwocoordinates(x,s),wheresisperiodics [0,2π] flected as a hole. For n = 1 rotational invariance ap-
∈ v
and goes around the perimeter of the wire, while x goes pears to be broken by the pairing term, but is explicitly
alongitslength. Wefirstconsideramagneticfieldparal- recovered after the gauge transformation Ψ eiτzs/2Ψ,
lel to the wire, B(cid:126) = (B(cid:107),0,0), described with the gauge which shifts η η 1/2. This transform→ation also
choice A(cid:126) = B (0, z/2,y/2). The dimensionless flux changes the bou→ndary−conditions to periodic, such that
(cid:107)
through the wire is−η = B hw/(h/e). The effective the- angularmomentatakeintegervaluesl(cid:48) =n. Asaresult,
(cid:107) n
ory for the surface states is the same as for a cylindrical the n = 0 electron state now has the same angular mo-
wire [22, 23], with s the azimuthal angle mentum as its conjugate hole state and can be reflected
into it.
H = iv [σ ∂ +σ (2π/P)(∂ +iη)], (1)
0 − F x x y s The NS conductance of the junction is computed from
whereweset(cid:126)=1andtakev =330meVnm[38]. The the Andreev reflection matrix, evaluated separately for
F
3
a) B(cid:166)(cid:61)0,Η(cid:61)0 B(cid:166)(cid:61)0,Η(cid:61)1(cid:144)2 B(cid:166)(cid:61)2T invariance is still preserved in the x direction
20 2π
(cid:76)meV 0 H =σx[−i∂x+eAx(s)]+ P σy(−i∂s+η). (7)
(cid:72)
E The vector potential in the surface coordinates is
(cid:45)20
b) 20 Η(cid:45)(cid:61)0.00k.50(cid:72)n,0mnVB(cid:45)1(cid:166)(cid:61)0(cid:76).(cid:61)0050T(cid:45)0.0k5(cid:72)n0m(cid:45)10(cid:76).05c)6 Η(cid:45)(cid:61)0.02.0(cid:45),0n.kV1B(cid:72)(cid:166)(cid:61)n0(cid:61)m0(cid:45)21T(cid:76)0.1 0.2 Ax(s)=B⊥P −2s2πs−π4r−4r+1434 −1311−−+4444+rrrr <<<< 2222ssssππππ <<<< 1133−−++4444rrrr , (8)
2(cid:144)(cid:76)eh Η(cid:61)0.5,nV(cid:61)1 2(cid:144)(cid:76)eh4 Η(cid:61)0.5,nV(cid:61)1 with r = ww+h. The profiles of Ax and B⊥ along the
(cid:72)S10 (cid:72)S s direction are shown in the inset of Fig. 3(b). Since
N N2
G G rotationalsymmetryisbroken, thedifferentnmodesare
0 0 mixed. In the angular momentum basis, Eq. (2), the
0 10 20 0 10 20
Μ(cid:72)meV(cid:76) Μ(cid:72)meV(cid:76) Hamiltonian is H =(cid:80)Nn,n(cid:48)=−Nχ†k,nHn,n(cid:48)(k)χk,n(cid:48), where
N is an angular momentum cutoff. The matrix element
is given by
FIG. 2. a) The spectrum of a wire of dimensions h=40 nm
and w = 160 nm for B = 0 and η = 0 (left), B = 0 and
⊥ ⊥
η =1/2 (center) and B⊥ =2T (right). Note that in the last Hnn(cid:48)(k)=[σxk+σy(2π/P)(n−1/2+η)]δn,n(cid:48)
casethespectrumisindependentofη. b)NSconductancefor
M
B⊥ = 0 and ∆0 = 0.25 meV as a function of µ, for η = 0, +σ (cid:88) eA(m)δ , (9)
n = 0 (dashed line) and η = 1/2, n = 1 (full line) c) The x x n,n(cid:48)+m
v v
m=−M
same for B =2T.
⊥
where A(m) = (cid:82)2π dse−imsA (s) =
x 0 2π x
B⊥P( 1)m2+1 sin(mπr/2)/m2π2 if m is odd and
−
every n, in a very similar way to Ref. 41. To com- vanishes otherwise, and M is a cutoff for the number of
pute it, we define incoming ψe− and outgoing ψe+ prop- Fourier components of Ax, with M N. The spectrum
n n ≤
agating electron states in the normal part, and similarly ofthewireonlychangesqualitativelywhenlB <w,with
for hole states ψh− and ψh+. Normalization is chosen lB =((cid:126)/eB⊥)1/2 themagneticlength,andLandaulevels
n n
such that all propagating states carry the same current, start to form in the top and bottom surfaces, which
J = ψ σ ψ =1. Thesearematchedtotheevanescent merge smoothly with dispersing chiral states localized
x x
states(cid:104)in|th|e s(cid:105)uperconductor ψS+ and ψS− by imposing in the sides. The spectrum in this regime, shown in
n n
continuity of the wavefunction at the junction (dropping Fig. 2(a), becomes independent of B(cid:107).
the label n for ease of notation) TheNSconductanceforfiniteB⊥ canbecomputedas
before with one important difference: in the basis states
forthenormalpart,theevanescentstates(withIm[k ]>
ψe−+r ψe++r ψh− =aψS++bψS−, (5) ev
ee he 0) must be included to obtain a well-defined matching
ψh−+rhhψh++rehψe− =a(cid:48)ψS++b(cid:48)ψS−. (6) condition. The incoming electron states, labelled now
by α = 1,...,Nprop, are ψαe− = e−ikx(cid:80)Nn=−Neilnsχen−,k,
The reflection matrix is defined as r = (ree reh), and is and similarly for ψe+, ψh− and ψh+. The evanescent
rhe rhh α α α
both unitary and particle-hole symmetric. The conduc- states are defined as ψN, with α(cid:48) = 1,...,N , with
α(cid:48) ev
tance is given by GNS = 2he2tr rehre†h, where the trace Nev +Nprop = 2N. Both propagating and evanescent
sums over all propagating modes. The resulting G for momentaandwavefunctionsareobtainedfromthetrans-
NS
∆ = 0.25 meV are shown in Fig. 2(b). When η = 1/2, fer matrix of the normal part [42–44]. We assume that
0
nv = 1 and in the range µ < π/P, a single mode is re- B⊥ is completely screened in the superconducting part
flected from a topological superconductor resulting in a ofthewire(seeFig.1), sothattheeigenstatesinthisre-
conductance of 2e2/h. gionremainunchanged. Continuityofthewavefunctions
at the interface
The conditions to observe conductance quantization
in this setup are not optimal yet, mainly because the
chemicalpotentialhastobetunedintoasmallgapπ/P. ψe−+N(cid:88)prop(cid:104)(r ) ψe++(r ) ψh−(cid:105)+ (10)
α ee αβ β he αβ β
Thislimitationcanbeovercomebytheadditionofaper-
β=1
pendicular field. Consider the Hamiltonian of the nor-
mal wire with B(cid:126) = (B(cid:107),B⊥,0) and a vector potential (cid:88)Nev (cid:104)c ψN,e+d ψN,h(cid:105)= (cid:88)N (cid:2)a ψS++b ψS−(cid:3).
A(cid:126) =B (z,0,0)+B (0, z/2,y/2)suchthattranslational αα(cid:48) α(cid:48) αα(cid:48) α(cid:48) αn n αn n
⊥ (cid:107) α(cid:48)=1 n=−N
−
4
3 B¶ s 3 Ax mbyeaasvuerreagoifngthoevdeirso1r0d3erdisstorrednegrthc.onOfiugrurdaattioaniss.oTbthaeinreed-
2(e/h)2 h w 2 0 Bp¶ 2p scuonltdsuacrteanshcoewonftihneFniogr.m3a.lIwnitrheeinsdinegedle-rmemoadienrseqguimanet,iztehde
N s
G1 1 to e2/h in the presence of moderate disorder, as long as
thechemicalpotentialisnotveryclosetozero. Thecon-
0 0
0 10 20 0 1 2 3 ductance for each disorder realization is also quantized.
µ(meV) B (T)
? A full characterization of the effects of disorder will be
presented in a future work [44].
FIG. 3. Disorder averaged conductance for a finite wire of
Discussion - An important feature of our proposal is
dimensions L = 400 nm, h = 40 nm and w = 160 nm for
different values of the disorder strength g. a) Conductance thatalleffectsinducedbythemagneticfieldareofpurely
as a function of chemical potential with fixed magnetic field orbital origin. The Zeeman coupling will be a small cor-
B =2 T. Inset: Cross section of the rectangular wire, with rection at the fields considered, and does not change our
⊥
thecoordinatesdepictedasadashedarrow. b)Conductance predictions qualitatively [44]. In our setup, a quantized
as a function of B at fixed chemical potential µ=10 meV.
⊥ conductancecanbeobtainedwithbothB =0andfinite
⊥
Inset: The vector potential A as given by Eq. (8), and its
x B , but the latter case has several advantages that are
associated magnetic field profile. ⊥
worth stressing. First, the single-mode regime remains
accessible for chemical potentials ranging up to values of
the order of the cyclotron frequency ω , rather than the
c
finite size gap π/P. Second, chiral mode transport in
For every value of α, we project into angular momen-
the normal part is robust against finite disorder due to
tum states with n = N,...,N. and since the spinors
− spatial separation of counter-propagating chiral modes.
have four components (spin and particle-hole degrees
Third, the spectrum of the normal part in the presence
of freedom) this yields a system of 8N equations with
ofB becomesindependentofB ,whichaffectsonlythe
2N + 2N + 4N = 8N coefficients. The system ⊥ (cid:107)
prop ev superconducting part. B thus becomes an independent
is solved numerically, and the conductance obtained is (cid:107)
knobdrivingthetransitionfromatrivialtoatopological
shown in Fig. 2(c). In the single-mode regime, at zero
superconductor,whilethechiralmodesremainintact. In
flux and n = 0 we have G = 0, but at η = 1/2 and
v NS this case, measuring G =0 would represent a genuine
n =1(whenthesuperconductoristopological),wehave NS
v consequence of reflection from a trivial superconductor,
G =2e2/h as expected.
NS as opposed to the B = 0 case where this value of G
⊥ NS
ThequantizationofG canbeunderstoodintuitively
NS couldresultfromaninsulatingnormalpart,seeFig.2(b).
intermsofa1Dlow-energymodel,depictedinFig.1(b),
Our proposal realizes a version of the Majorana inter-
similartotheonedescribingtheMajoranainterferometer
ferometerwithsomeimportantdifferences. Inoursetup,
proposedinRefs.9and10(seealsorelatedstudiesofMa-
instead of contacting the two chiral modes separately
jorana interferometry with chiral Majorana modes [45–
the source electrode contacts both channels and the su-
47] and Majorana bound states [48–57]). In this model,
perconductor is the drain [3], see Figs. 1(c-d). In addi-
an incoming chiral mode leaving the source is split into
tion, the original proposals use ferromagnets and a finite
two Majorana modes that appear at the interface be-
superconducting island to create the Majorana modes,
tweenthethesuperconductorandtheregionswithfinite
while our setup uses a bulk superconductor and a ho-
B [58]. In the absence of a vortex the two Majoranas
⊥ mogeneous magnetic field [58], making it experimentally
recombineasanelectronontheothersideofthewireand
morefeasible. Despitethesedifferences,thefinitevoltage
returntothesourcethroughthechannelofoppositechi-
and finite temperature behavior of G will be similar to
rality, yielding G =0. However, if a vortex is present, NS
NS thoseinRefs.9and10. Thisintroducesanimportantad-
the two Majoranas accumulate a relative phase of π and
vantage to our setup over current semiconducting wires,
recombine as a hole, while a Cooper pair is transferred
wherethetemperaturesrequiredtoobserveconductance
to the superconductor, yielding G =2e2/h.
NS quantizationareoftheorderofmK.Inoursetup,thelim-
Thequantizationoftheconductanceinoursetupisex-
itingtemperatureisdeterminedbytheproximityinduced
pected to be robust to disorder to some extent, because
gap[10]. Assuming∆ 0.1 0.25meV[13,14,60]this
0
transport in the normal part is mediated by spatially ≈ −
corresponds to 1-3 K.
separated chiral modes. In order to test this robustness
Finally, we note that screening B in the SC region
⊥
we introduce disorder into the Hamiltonian of a normal
requires the use of a superconductor with a high criti-
wire in the presence of B , and compute the two ter-
⊥ cal field. For example, the superconductor could be a
minal conductance G of a finite size wire numerically,
N Ti/Nb/Ti trilayer as the one used in the experiment in
following the method of Ref. 59. The disorder potential
Ref. 60, which was estimated to have H =2.5T.
hasacorrelator(cid:104)V(r)V(r(cid:48))(cid:105)=g(2(cid:126)πvFξD2)2e−|r−r(cid:48)|2/2ξD2,with WethankJ.Dahlhaus,J.E.Moore,A.cV1ishwanathand
ξ the disorder correlation length and g a dimensionless J.Analytisforusefuldiscussions. Weacknowledgefinan-
D
5
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