Superconducting proximity effect and Majorana flat bands in the surface of a Weyl semimetal

2.3 MB·

by  Anffany Chen

Superconducting proximity effect and Majorana flat bands at the surface of a Weyl
semimetal
Anffany Chen and M. Franz
Department of Physics and Astronomy, University of British Columbia, Vancouver, BC, Canada V6T 1Z1 and
Quantum Matter Institute, University of British Columbia, Vancouver BC, Canada V6T 1Z4
(Dated: May 30, 2016)
We study the proximity effect between an s-wave superconductor (SC) and the surface states of
a Weyl semimetal. An interesting two-dimensional SC forms in such an interface with properties
6 resembling in certain aspects the Fu-Kane superconductor with some notable differences. In a
1 Weyl semimetal with unbroken time reversal symmetry the interface SC supports completely flat
0 Majorana bands in a linear Josephson junction with a π phase difference. We discuss stability of
2 these bands against disorder and propose ways in which they can be observed experimentally.

y
a
M Interfacing topological materials with conventionally a) ky b) ky c)
ordered states of matter, such as magnets and supercon- SC SC

6 ductors, has led to important conceptual advances over �(x)
2 the past decade. Notable examples of this approach in- kx kx

clude the Fu-Kane superconductor [1] that occurs in the
] '=0 '=⇡
n interface of a 3D strong topological insulator (STI) and
o aconventionals-waveSC,the“fractional”quantumHall
c effect that arises when STI is interfaced with a mag- FIG. 1: Surface Fermi arcs in a minimal model of a Weyl
- semimetal with (a) and without (b) time reversal symmetry
r netic insulator [2–4], as well as many interesting phe-
p T. Weylnodesarerepresentedbycircleswithpositive(nega-
nomenathatoccurwhenbothSCandmagneticdomains
u tive) chirality; green arrows indicate the possible direction of
are present [5, 6]. Rich physics, including Majorana zero
s electron spin along the arcs. Panel (c) shows the π junction
t. modes, alsoresultswhentheedgeofa2Dtopologicalin- setup on the surface of a Weyl semimetal.
a sulator is interfaced with magnets and superconductors
m
[7–9]. More recently various exotic phases of quantum
- matter have been predicted to emerge based on these has to be another pair of T-conjugate Weyl nodes with
d
same ingredients in strongly interacting systems [10–20]. an opposite chirality. We begin by discussing the SC
n
o InthisLetterweexplorethephysicsoftheinterfacebe- proximity effect in this minimal case with N = 4. We
c tween a Weyl semimetal and an s-wave superconductor. note that although the currently known Weyl semimet-
[ Surface states of a Weyl semimetal exhibit characteristic als exhibit larger number of nodes (N =24 in the TaAs
2 Fermi arcs that terminate at surface projections of the family) recent theoretical work identified materials that
v bulk Weyl nodes [21, 22]. Such “open” Fermi surfaces couldrealizeinstanceswithsmallerN [29,30], including
7 are fundamentally impossible in a purely 2D system and the minimal case with N = 4 predicted in MoTe2 [31].
2 we thus expect the resulting SC state to also be anoma- ThegenericsurfacestateofsuchaminimalT-preserving
7
lous. In this respect the situation is similar to the Fu- Weyl semimetal is depicted in Fig. 1a.
1
0 Kane superconductor [1] whose existence hinges on the Interfacing such a surface with a spin singlet s-wave
. odd number of Dirac fermions present on the surface of SC one expects formation of a paired state from time-
1
0 an STI. As we shall see there are several notable dif- reversed Bloch electrons at crystal momenta k and −k
6 ferences between STI/SC and Weyl/SC interfaces which alongthearcs. Aminimalmodeldescribingthissituation
1 make the latter a distinct and potentially more versatile isdefinedbythesecondquantizedBogoliubov-deGennes
v: platform for explorations of new phenomena. (BdG) Hamiltonian H=(cid:80)kΨˆ†khBdG(k)Ψˆk where Ψˆk =
Xi Nondegenerate Weyl points can occur in crystals with (ck↑,ck↓,c†−k↓,−c†−k↑)T is the Nambu spinor and
broken time reversal symmetry T or broken bulk inver- (cid:18) (cid:19)
ar sion symmetry P. Recent experimental work reported hBdG = h0(k∆)†−µ −syh∗(−∆k)sy+µ , (1)
convincing evidence for Weyl nodes and surface Fermi 0
arcs in T-preserving noncentrosymmetric crystals in the andµisthechemicalpotential. Thenormalstatesurface
TaAs family of semimetals [23–28]. We therefore focus HamiltonianfortwoparallelFermiarcswithelectronspin
our discussion on the SC proximity effect in this class locked perpendicular to its momentum as in Fig. 1a can
of materials. When a crystal respects T the minimum be written as
number of Weyl points N is 4. This is because under
h (k)=vszk −η, for |k |<K. (2)
T a Weyl point at crystal momentum Q maps onto a 0 x y
Weyl point at −Q with the same chirality. Since the to- Here sα are Pauli matrices acting in electron spin space.
tal chiral charge in the Brillouin zone must vanish there One can easily check that Eq. (2) produces two parallel
2
Fermi arcs of length 2K along k separated by distance exponentially localized near the junction.
y
η/v in the k direction when µ = 0. It is possible to We shall demonstrate below by an exact numerical di-
x
make this model more generic by allowing the velocity agonalization of a 3D lattice model that the Majorana
v and parameter η to depend on k and the cutoff K flatbandsremainrobustbeyondtheminimallowenergy
y
on k (which would curve and shift the arcs similar to surface theory. Their stability however can be deduced
x
Fig. 1a) but the minimal model defined above already fromthesurfacetheoryaloneandreliesonthetimerever-
captures the essential physics we wish to describe. The sal symmetry T and the bulk-boundary correspondence
surface model (2) does not capture the bulk bands that presentintheWeylsemimetal. Themostgeneralsurface
become gapless near the Weyl points. The model based Hamiltonian consistent with these requirements is of the
on Eq. (1) is therefore valid only away from the surface formh(k)=(v ·s)k −η for|k |<K whereboththe
ky x ky y
projections of the Weyl points, specifically when v||k |− velocity vector and η are even functions of k . Time
y ky y
K|>∆, and in the absence of large momentum transfer reversalsymmetryfurtherpermitsaterm(u ·s)k but
ky y
disorder scattering that would mix the surface and bulk this is not consistent with h(k) describing the surface
low-energy states. state of a Weyl semimetal [33]. For each k one can now
y
If we define another set of Pauli matrices τα in the perform anSU(2) rotation in spinspace to bring h(k) to
Nambu space we can write Eq. (1) for each |k |<K as the form indicated in Eq. (2). Because the pairing term
y
intheBdGHamiltonian(3)isnotaffectedbythistrans-
h =(vszk −µ˜)τz+∆ τx−∆ τy, (3)
BdG x 1 2
formation the calculation demonstrating the zero modes
where µ˜ = µ+η and ∆ = ∆1+i∆2. For spatially uni- proceeds as before, with the result (5) modified in two
form ∆ the spectrum of hBdG is fully gapped. Its topo- minor ways: v and µ˜ may now depend on ky and the
logical character can be exposed by noting that for each spinor structure reflects the SU(2) rotation. The exact
allowed value of ky Eq. (3) coincides with the Hamilto- zeromodespersistforallallowedvaluesofky inthismore
nian describing the edge of a 2D TI in contact with a general case.
SC.ItiswellknownthatunpairedMajoranazeromodes It is easy to see that breaking T lifts the zero
(MZMs) exist in such an edge at domain walls between modes: forinstanceinterpolatingbetween−∆ and+∆
0 0
SCandmagneticregions[8]. ThisisbecauseT-breaking through complex values of ∆(x) moves MZMs to finite
also opens up a gap in 2D TI edge and MZMs occur at energies. Flat Majorana bands are thus protected by
boundaries between two differently gapped regions. In T and by translation symmetry that is necessary to es-
ourpresentcaseoftheWeylsemimetalbreakingT gener- tablish the underlying momentum space Weyl structure.
ically does not open a gap. However, we can achieve a We argue below that our results are in addition robust
similar result by creating a domain wall in the complex against weak nonmagnetic disorder.
order parameter ∆(x). Specifically, we show below that To ascertain the validity and robustness of the above
a π junction (i.e. a boundary between two regions whose analyticresultswenowstudytheproblemusingalattice
SC phase ϕ differs by π, see Fig. 1c) hosts a pair of pro- model for the Weyl semimetal. We consider electrons in
tectedMZMs(oneforeacharcandforeachallowedvalue a simple cubic lattice with two orbitals per site and the
ofky), separatedbyagapfromtherestofthespectrum. following minimal Bloch Hamiltonian,
As a function of k , then, the band structure of such a π
y (cid:88)
junctionexhibitsaflatMajoranabandpinnedtozeroen- hlatt(k)=λ sαsinkα+σysyMk. (6)
ergy. We argue below that such flat Majorana bands are α=x,y,z
experimentallyobservableandrepresentagenericrobust HereM =(m+2−cosk −cosk )andσα arePaulima-
k x z
property of any T-preserving Weyl/SC interface.
tricesactingintheorbitalspace. TheHamiltonian(6)is
To exemplify this property within the effective surface inspiredbyRef.[34]andadaptedtorespectT (generated
theory (3) we replace kx →−i∂x and assume ∆(x) to be here by isyK with K the complex conjugation). It has a
purely real with a soliton profile such that ∆(x)→±∆
0 simple phase diagram. For m > λ (taking λ positive) it
as x→±∞. We next note that Hamiltonian (3) can be
describes a trivial insulator. At m=λ two Dirac points
brought to a block diagonal form hBdG = diag(h+,h−) appearatk=(0,±π/2,0)whichthensplitintotwopairs
by a unitary transformation that exchanges its second
ofWeylnodespositionedalongthek axis. Thesepersist
y
and third rows and columns. Here h± are 2×2 matrices as long as |m|<λ. In this phase surfaces parallel to the
(cid:18)−ivs∂ −µ˜ ∆(x) (cid:19) y crystal axis exhibit two Fermi arcs terminating at the
h = x , (4)
s ∆(x) ivs∂ +µ˜ surface projections of the bulk Weyl nodes illustrated in
x
Fig. 2.
ands=±. Itiseasytoshowthatforeachallowedvalue
Weperformedextensivenumericalcomputationsbased
of k and each s the above Hamiltonian (4) supports an
y on h (k) defined above focusing on the slab geometry
latt
exact Jackiw-Rossi zero mode [32] with a wave function
with surfaces parallel to the x-y plane and thickness of
ψ (x)=A(cid:18)i(cid:19)exp(cid:26)−1(cid:90) xdx(cid:48)[∆(x(cid:48))−isµ˜](cid:27) (5) Lz sites. The proximity effect is studied using the BdG
s s v 0 Hamiltonian (1) with h0 replaced by hlatt and ∆ taken
3
1 1 1
a) b) c) d)
ky Ek Ek Ek
E 0 E 0 E 0
normal SC SC+junction
−1 −1 −1
kx −π 0 ky π −π 0 ky π −π 0 ky π
ky ky ky
1 1 1
e) f) g) h)
ky Ek Ek Ek
E 0 E 0 E 0
Out[9]=
SC SC+junction SC+junction
−1 −1 −1
kx −π k0y ky π −π k0y ky π −π k0x kx π
FIG.2: TightbindingsimulationsoftheSC/Weylproximityeffect. Panelsa)ande)showthesurfacespectralfunctionA(k,ω)
for the tight binding Hamiltonian h +δh and ω = 0.15. In all panels we use m = 0.5, λ = 1,, L = 50 and L = 40;
latt latt x z
while (µ,(cid:15)) = (0,0) and (0.1,1.0) for top and bottom row respectively. Panel b) shows the normal state spectrum with flat
bands representing the surface states. The effect of the SC proximity effect with ∆ =0.5 is indicated in panel c) while panel
0
d) shows the effect of two parallel equidistant π junctions (protected zero modes indicated in red). The bottom row displays
ourresultsforrotatedarcs;panelf)theuniformSCsurfacestate,panelsg)andh)twoparallelπ junctionsalongy andxaxes,
respectively. The small gap at Weyl points in panels c) and f) is a finite size effect – the gap closes as L →∞.
z
tobenon-zerointhesurfacelayersonly[35]. Toprowin The lattice Hamiltonian (6) has high symmetry with
Fig. 2 summarizes our results. The normal state shows Weylnodesconfinedtolieonthek axis. Itisimportant
y
gapless Fermi arcs (Fig. 2a,b) while uniform SC order is to verify that the phenomena discussed above are not
seen to gap out the Fermi arc states except in the vicin- dependent on such a fine tuned lattice symmetry. To
ity of the Weyl points where they merge into the gapless this end we perturb h by adding to it
latt
bulk continuum (Fig. 2c). Presence of the π junctions
δh (k)=(cid:15) σysx(1−cosk −cosk ), (7)
(which we define parallel to the y crystal axis) generates latt y z
perfectlyflatbandsatzeroenergybetweentheprojected which respects T but breaks the C rotation symmetry
4
Weyl nodes, Fig. 2d. We note that due to the periodic around the y axis thus allowing Weyl nodes to detach
boundary conditions adopted in both x and y directions fromk . For(cid:15)(cid:54)=0thearcsrotateandcurveasillustrated
y
ournumericsbynecessityimplementtwoparallelπ junc- inFig.2e. Majoranaflatbandshoweverremainrobustly
tionswhichresultsintwicethenumberofflatbandscom- present in this low symmetry case (Fig. 2g). They now
paredtoasinglejunction. Wealsofindthat,remarkably, also appear for a junction parallel to the x crystal di-
the bands remain completely flat even when the system rection (Fig. 2h) because the Weyl points project onto
size Lx is small along the direction perpendicular to the distinct kx momenta in the boundary BZ.
junctions(inwhichcaseonewouldnaivelyexpectalarge We now address the stability of Majorana bands
overlapbetweentheboundstatewavefunctionsresulting against nonmagnetic disorder that will inevitably be
insignificantenergysplitting). Thispropertycanbeun- present in any real material. Scaling arguments [36–
derstood by noting that wavefunctions (5) remain exact 39] and numerical simulations [40–42] show that dis-
zero modes of the Hamiltonian (1,2) for periodic bound- order is a strongly irrelevant perturbation in a Weyl
ary conditions along x as long as ∆(x) averages to zero semimetal: electron density of states (DOS) at low en-
over all x and µ˜ = (2πv/Lx)n with n integer. These ergies D(ω) ∼ ω2 remains unchanged up to a critical
conditions are satisfied for two equally spaced junctions disorder strength U at which point a transition occurs
c
and µ = 0 used in Fig. 2. We checked that violating into a diffusive regime with finite DOS at ω = 0. The
these conditions indeed leads to zero mode splitting that Fermi arcs likewise remain stable in the weak disorder
depends exponentially on the junction distance. Impor- regime. These theoretical results are confirmed by ex-
tantly, the fact that this detailed property of the simple perimental studies which show clear evidence for Weyl
model (2) is borne out in a more realistic lattice model points and Fermi arcs in real materials using momen-
gives us confidence that the low energy theory provides tumresolvedprobessuchasARPES[23–28]andFT-STS
correctdescriptionofthephysicalsurfacestateofaWeyl [43,44],inagreementwiththepredictionsofmomentum
semimetal. spacebandtheory. InT-preservingWeylsemimetalsone
4
furthermoreexpectsthesurfacesuperconductingorderto nodes but expect their signatures to survive in materials
be stable against non-magnetic impurities due to Ander- with larger N. In this case each Kramers pair of Fermi
son’s theorem [47]. Finally, stability of Majorana flat arcswillproduceaMajoranabandatzeroenergy. Inthe
bands can be argued as follows. In the clean system presence of weak disorder there may be some additional
our calculations show MZMs localized in the vicinity of broadening due to interband scattering but given that
thejunctionateachmomentumk betweentheprojected Fermi arcs are quite easy to resolve experimentally [23–
Weylpoints. Exceptinthevicinityofthelattertheseare 28]evenwhenN =24weexpectnosignificantdifficulties
separated by a gap ∼ ∆ from the excited states in the to arise when N >4.
system. TurningonweakrandompotentialofstrengthU Majorana surface states, including flat bands, have
willcausemixingbetweenMZMsatdifferentmomentak been theoretically predicted to occur in various 2D and
as well as with the bulk modes. We expect the former to 3Dtopologicalandnodalsuperconductors[53–63]. With
beamoreimportanteffect(exceptforMZMsintheclose the exception of high-T cuprates, where dispersionless
c
vicinityoftheWeylpoints)becauseofthelowbulkDOS edgestatesareknowntoexist(andwerereinterpretedre-
and the assumption of predominantly small momentum cently as Majorana flat bands [51]), these have not been
scattering. Importantly, the disorder averaged spectral observed experimentally because of the general paucity
function of the system must evolve continuously with oftopologicalsuperconductors. Majoranaflatbandsdis-
the disorder strength U. For weak disorder, therefore, cussedinthisLetteronlyrequireingredientsthatarecur-
the Majorana band cannot abruptly disappear; instead rently known to exist – a T-preserving Weyl semimetal
disorder will produce a lifetime broadening, giving the interfaced with an ordinary superconductor – and are
δ-function peak present in the clean spectral function a thus promising candidates for experimental detection.
finite width proportional to U. This behavior is indeed
The authors are indebted to T. Pereg-Barnea and D.I.
observed in numerical simulations of disorder in Majo-
Pikulin for discussions, and thank NSERC, CIfAR and
rana flat bands at the edges of 2D topological supercon-
Max Planck - UBC Centre for Quantum Materials for
ductors [48]. We expect the broadened Majorana band
support. M.F. acknowledges KITP Santa Barbara for
toremainobservableuntilitswidthbecomescomparable
hospitality during the initial stages of this project.
to the gap ∆ implying a significant range of stability in
a weakly disordered sample.
Majorana flat bands should be directly observable in
tunneling spectroscopy of the junction region by a tech-
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