Topological Superconductivity and Majorana Fermions in Metallic Surface-States

0.61 MB·English

by  Andrew C. Potter

Topological Superconductivity and Majorana Fermions in Metallic Surface-States
Andrew C. Potter and Patrick A. Lee
Massachusetts Institute of Technology 77 Massachusetts Ave. Cambridge, MA 02139
Heavy metals, such as Au, Ag, and Pb, often have sharp surface states that are split by strong
Rashba spin-orbit coupling. The strong spin-orbit coupling and two-dimensional nature of these
surface states make them ideal platforms for realizing topological superconductivity and Majorana
fermions. In this paper, we further develop a proposal to realize Majorana fermions at the ends
of quasi-one-dimensional metallic wires. We show how superconductivity can be induced on the
metallic surface states by a combination of proximity effect, disorder, and interactions. Applying a
magneticfieldalongthewirecandrivethewireintoatopologicallynon-trivialstatewithMajorana
2 end-states. Unlike the case of a perpendicular field, where the chemical potential must be fined
1
tunedneartheRashba-bandcrossing,theparallelfieldallowsonetorealizeMajoranasforarbitrarily
0
large chemical potential. We then show that, despite the presence of a large carrier density from
2
the bulk metal, it is still possible to effectively control the chemical potential of the surface states

n by gating. The simplest version of our proposal, which involves only an Au(111) film deposited on
a a conventional superconductor, should be readily realizable.
J

0
1 The observation that ordinary s-wave superconductiv- heavyelementsInandSb,whichhavelargeatomicSOC,
ity (SC) coupled by proximity to a topological insulator the Rashba component of SOC is very small, because it

] can create an exotic topological superconductor possess- comes only from the breaking of inversion symmetry by
l
l ing non-Abelian Majorana states1 has inspired a large the wire-superconductor interface. Since these wires are
a
h bodyoftheoreticalwork,andspurredanewexperimental typically quite large (10’s-100’s of nm in diameter), the
- thrust to realize non-Abelian particles in the laboratory. electron wave-functions are spread over a large distance
s
While there are a variety of theoretical proposals1–8, and do not strongly feel the inversion asymmetry from
e
m a particularly promising class involve combining con- the interface.
ventionalsuperonductorswithtwo-dimensionalmaterials Small spin-orbit coupling is problematic for two main
.
t with Rashba-type spin-orbit coupling (SOC)2–6. reasons. Firstthesizeofthep-wavesuperconductinggap
a
m In these schemes, superconductivity (SC) is induced protecting Majorana end-states is limited by ∆p-wave <∼
in a spin-orbit coupled material by proximity to an or- ∆ 16, requiring one to work at very low-temperatures.
- so
d dinary s-wave superconductor. The two Rashba-split Second, small SOC renders the induced superconductiv-
n bands effectively convert the s-wave pairing into helical ity extremely vulnerable to even very small amounts of
o p+ip and p−ip pairing respectively. Applying a mag- disorder16–18. The extreme sensitivity to disorder for
c
netic field perpendicular to the plane opens up a Zee- small SOC may be problematic even though the bulk of
[
man gap in the Rashba spectrum. If the chemical po- semiconducting wires are typically quite clean; further-

1 tential is tuned within this Zeeman gap, then only one morethesuperconductinggapwillalsobesuppressedby
v
of the helically paired bands remains, leaving a single- roughness or inhomogeneity in the wire–superconductor
6
helicity, p + ip superconductor. Superconductors with interface.
7
1 p+ip pairing are known to be host non-Abelian Majo- The problems associated with small spin–orbit cou-
2 rana bound-states in vortex cores9, and at the ends of pling led us to propose building a topological super-
. one-dimensionaldomains10. Majoranaend-statesofone- conductor metallic surface–states12,13,16. Surface states
1
dimensional wires are particularly interesting, since they of heavy-metals are typically tightly bound to the sur-
0
2 can be manipulated by simple electrostatic gating5,11,12. face, with very small spatial extent. Consequently, the
1 Furthermore, networks of topological superconducting surface–stateelectronsarestronglyeffectedbytheinver-
: wiresprovideaneffecientplatformforbraidingMajorana sionasymmetryofthesurface–interface,generatinglarge
v
i fermions to probe their non-Abelian statistics11. Rashba spin-orbit couplings. For example, the Au(111)
X Sofar, thebulkoftheoreticalandexperimentalefforts surface hosts a well studied surface–state with Rashba
ar along these lines has focused on using semiconducting splittingof∆so ≈50meV19,ordersofmagnitudestronger
nanowireswithheavyelements,suchasInSb. Suchsemi- than the best ∆so available in semiconductor nanowires.
conductingnanowirescanbegrownwithveryfewdefects EvenlargerRashbasplittings,∆so ≈0.5eV,areavailable
and impurities. These materials can also have very large in the surface states of the Ag(111) surface alloyed with
electron g-factors (as high as ≈ 70)18, allowing one to Bi and Pb20.
create a large Zeeman splitting in the nanowire with an In this paper, we develop this proposal in greater de-
external magnetic field, without significantly impacting tail. The proposed setup is shown in Fig. 1a. A thin
the nearby superconductor5,6. However, semiconduct- metallic film is deposited on top of a convention super-
ing materials typically produce only very tiny Rashba conductor. Bytheproximityeffect,thebulkstatesofthe
SOC, ∆ . For example, the Rashba SOC in InSb nano– metal film will inherit some of the superconducting gap
so
wires is ∆ ≈ 1K4,5. Despite the presence of relatively ∆ from the nearby superconductor. If the metal film
so 0
2
N(ε) ε
k Surface States
STM Gate 2Δso μ
Dielectric
A u(111) C o n v e ntBio n al S C SurfacMSe uSetptaaetelr cTohnidnu Fcitlomr ΔS ε Projected Bulk BaεndFs
a) b) c) ΔB d) k
FIG.1. (a)Simplestpossibleversionoftheproposedsetup: astripofAu(111)thin-film(oranyothermetalwithRashba-split
surface state) is deposited on top a conventional superconductor. An external field is applied parallel to the wire, in order to
drivethesystemintoatopologicalSCstate. Majoranaend-statescanbedetectedbytunneling,e.g. withanSTMtip. (b)So
long as the surface–state survives the deposition of a gate–dielectric, the surface state chemical potential can be controlled by
a top–gate. (c) Tunneling density of states, N(ε), as a function of energy, ε; the full superconducting gap ∆ is induced on
B
the bulk states, by proximity effect. The surface gap develops a smaller gap ∆ due to indirect scattering from disorder and
S
interactions. (d)Sketchofband-structureofmetalwithaRashbaspin-orbitsplitsurfaceband. Bulkstatesareprojectedonto
the plane of the surface, and non-zero bulk projected density of states is indicated by gray shading. The surface-state band
forms within a region momentum space where there is no bulk states. The figure shows a one-dimensional cut through the
surfaceBrillouinzone. Thechemicalpotential,µ,isrepresentedbyadashedline. ThesurfaceFermi-energy,ε ,andspin-orbit
F
splitting at the Fermi–surface, ∆ , are indicated for the surface bands.
so
thickness is smaller than or comparable to the supercon- While the original proposals for creating topological
ducting coherence length, ξ , then the induced bulk gap, SC from Rashba SOC required applying a field perpen-
0
∆ ,willbelarge(∆ ≈∆ ). However,thesurfacestate dicular to the surface–plane3, it has since been pointed
B B 0
onthetopsurfaceofthemetalisnominallyisolatedfrom out5,6 that once the electron motion is confined along
the bulk states and does not couple directly to the su- a quasi-one-dimensional wire, a parallel field along the
perconductor(seetheFig. 1d). Instead,wemustrelyon wire can also create topological SC. Previous discussions
disorderandinteractionstoprovidesomemixingbetween alsoemphasizedthattheparallelfieldconfigurationisad-
thesurface–stateandbulkbandsinordertotransmitthe vantageousbecause,comparedtotheperpendicularfield,
bulkSCtothesurface–state. BecauseSCdevelopsinthe theadjacentsuperconductingfilmwillbelessaffectedby
surface–state only through indirect scattering processes, harmfulorbitaleffects5,6. Herewepointoutafurtherad-
the surface–pairing gap, ∆ will generically be smaller vantageoftheparallelfieldsetupformulti-channelwires:
S
than the bulk pairing gap, ∆ ≈ ∆ . In this case, the solongastheZeemansplitting–energyfromthefieldBis
B 0
surface–state SC can be revealed by tunneling measure- largerthan∆ ,thenthewirewillexhibitMajoranaend-
S
ments, which will show a coherence peaks at the edge of statesforarbitrarilylargechemicalpotentialµ. Thisisin
thebulkgap,andasmallersub-gapcorrespondingto∆ markedcontrasttotheperpendicularfieldcase,inwhich
S
(see Fig. 1c). topologically non-trivial regions were only available for
a small range chemical potentials near the Rashba-band
Once SC is established, one can pattern the metallic
crossing. Theabilitytooperateatarbitrarilylargechem-
film into a quasi-one dimensional wire. By applying a
ical potential frees one from fine-tuning the chemical po-
magneticfield, onecanremovethesub-banddegeneracy,
tential near the Rashba crossing, and allows one to work
and tune the chemical potential so that an odd-number
at much larger carrier-densities and spin-orbit couplings
of sub-bands is occupied. If the width of the wire is
(since ∆ grows with the Fermi-momentum).
comparable or smaller than ξ , then, occupying an odd so
0
number of sub-bands will result in Majorana end-states
The outline of the paper is as follows: we begin by
protected by the surface-state pairing gap ∆ 12–15.
S demonstrating that using a parallel field allows one to
In the simplest version of the proposed setup, shown achieve topological SC and Majorana end-states at ar-
in Fig. 5a, tuning to an odd number of sub-bands is bitrarily large chemical potentials in multi-band wires.
accomplished simply by applying an external magnetic Next,wedescribehowSCcanbeinducedonthesurface–
field, without gating. The simplicity of this setup, con- state by impurity scattering and interactions. Our anal-
sisting just of a metallic strip on a bulk superconductor, ysis suggests that it may be advantageous to artificially
makesitpromisingfortheinitialdetectionofMajoranas. disorder the surface in order to enhance ∆ . We then
S
Toperformmorecomplicatedexperiments,inwhichMa- confrontanoftenvoicedconcern21,22,thatthepresenceof
joranas are braided, it is necessary to control the local the nearby metal (superconductor) will make it impossi-
topological phase of different segments of the wire. For bletocontrolthechemicalpotentialofthesurface–states
this purpose, one could also add a top gate, as shown (nanowires) respectively. This concern has led to some
in 1b. In order for the top-gate geometry to work, one rather complicated proposals that attempt to avoid the
needs to check that the surface–state is not destroyed by perceivedgatingproblem21,22. Hereweshowthat,under
the presence of the gate dielectric. realistic experimental conditions, it should not actually
3
be difficult to tune the surface–state chemical potential spinsperpendiculartotheplaneliftsthedegeneracyand
over a wide range of ≈ 100meV. Therefore, conventional splits the energy spectrum into a series of individual
top-gatesshouldbesufficienttotunethewireintoatopo- (non-degenerate) sub-bands. If the sub-band splitting
logical state, and to manipulate Majoranas. Finally, we is sufficiently large compared to the induced supercon-
discuss a particularly promising candidate material, the ducting pairing, then it is possible to drive the system
Au(111) surface. intoatoplogicalnon-trivialstatebytuningthemagnetic
We believe that the simplicity and robustness of the field or chemical potential.
proposed setup make it a very promising route to real- For two-dimensional Rashba systems, the only way to
izing Majorana fermions. The large spin–orbit coupling achieve a chiral topological superconductor is to apply a
availableinmetallicsurface–statesallowforlargerintrin- Zeeman field perpendicular to the plane, and tune the
sic superconducting gaps, and render the topological SC chemical potential within the Zeeman gap. For small
effectively immune to disorder. spin-orbitcoupling,theZeemangapoccursatlowenergy,
forcing one to operate at low carrier densities and small
energy scales. By contrast, in a quasi-one-dimensional
I. SUB-BAND SPECTRUM AND wiretheelectronmotionoccurspredominantlyalongthe
TOPOLOGICAL PHASE DIAGRAM wire,andconsequentlyduetotheRashbaspin-orbitcou-
pling, the electron spins point predominantly perpendic-
TheBugoliobov-de-GennesHamiltonianforthesystem ular to the wire but in the plane. Unlike in 2D, applying
with Rashba SOC, Zeeman splitting, and induced SC is: a Zeeman-field along a wire also serves to split the sub-
band degeneracies. We will see below, that applying a
H =[ξ +α zˆ·(σ×k)]τ3−µ B·σ−∆ τ1 (1)
k k R 0 S field parallel to the wire allows one to operate at arbi-
trarily large chemical potential, well outside the regime
where{σ}arethespin-Paulimatrices,and{τ}arePauli
inwhichthebulk2Dsystemwouldbetopologicallynon-
matricesintheBCSparticle–holebasis. Hereξ = k2 −
k 2m trivial.
µ is the spin-indenpendent part of the dispersion, α is
R
theRashbavelocityrelatedtothespin-orbitcouplingby
∆ = α k (where k is the Fermi-momentum), B is
so R F F A. Out-of-plane Field
the magnetic field which couples to the spin with the
effective magnetic moment µ =gµ , and ∆ is the SC
0 B S The dispersion without superconductivity in the pres-
gap. The chemical potential µ, is measured with respect
ence of a perpendicular field is:
totheRashba-bandcrossingintheabsenceofB and∆ .
S
Furthermore,weconsiderelectronsconfinedtoaquasi- k2 (cid:113)
one-dimensional strip of width W along the y-direction εk,λ = 2m −µ+λ αR2k2+(µ0B)2 (2)
andlengthLalongthex-direction. Thewirewillexhibit
where λ = ±1. The resulting phase diagram for a su-
Majorana end-states under certain conditions, which are
perconducting wire obtained from numerical simulation
outlined below.
is shown in Fig. 2a. Topological phase transitions occur
Topologically non-trivial states arise only when 1) the
when the chemical potential coincides with the bottom
splitting between adjacent sub-bands is larger than the
of a transverse sub-band, so long as the transverse sub-
pairing gap ∆ and 2) an odd number of transverse
S
band spacing is larger than ∆ . For µ > 0,24 the sub-
sub-bands is occupied. Condition 1) ensures that pair- S
band splitting due to the applied field can be estimated
ing does not mix in states from neighboring sub-bands
by setting k =0 and k ≈±k in Eq. 2.
strongly enough to drive the system into the topologi- x y F
Consider the energy ε of the nth sub-band for B =
cally trivial state. Furthermore, it allows one to mean- n
0. For k = 0 there are four different states with
ingfully speak of the “number of occupied sub-bands”, x
ε =ε ,labeledbydifferentk . Inthewire,lin-
even though strictly speaking, this concept is well de- λ,(kx=0,ky) n y
ear superpositions of these four states are formed to sat-
fined only the absence of pairing. Condition 2) ensures
isfy the hard-wall boundary-conditions (which can only
that there are an odd number of Majorana end-states
be satisfied for a discrete set of energy values). In the
(one for each channel), which is guaranteed to leave one
absence of a magnetic field, the four k states at energy
decoupled Majorana mode at zero-energy. We empha- y
ε form two degenerate combinations related by time-
size, that while condition 2) is stated explicitly in terms n
reversal symmetry. Due to the Rashba SOC, the spin of
of number of sub-bands, the structure of the topological
eachofthek statesliesintheplane,andaperpendicular
phase diagram is qualitatively similer even for smoothly y
field does not directly mix the two states. Consequently,
meandering wires or wires with smooth spatial varations
in widths for which sub-bands are not well defined13. the sub-band splitting from the field occurs through vir-
tual admixture of higher energy states, and scales like
In the absence of the Zeeman field, µ B = 0, time-
0
reversalsymmetryisintactandsub-bandsoccurinpairs. ∆Esb ≈ ∆Bs2o where ∆so = αRkF. Inside the bulk Zee-
Generically, without breaking time-reversal symmetry it man gap, (|µ| < µ B), it is always possible to occupy
0
is impossible to occupy an odd number of sub-bands. an odd number of sub-bands. As chemical potential is
Applying a magnetic field perpendicular to the electron increased outside of the bulk Zeeman-gap, ∆ increases
so
4
1.0 1.0
0.8 0.8
0.6 0.6
μ/t μ/t
0.4 0.4
0.2 0.2
0.0 0.0
-0.2 -0.2 7
0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10
μ B/Δ μ B/Δ
a) 0 b) 0 c)
FIG. 2. (a) Numerical phase diagram for 40-site wide wire in perpendicular field (B∼zˆ), as a function of chemical potential
µ and magnetic field µ B. Black lines indicate sub-band bottoms in the normal state (without superconductivity), red filled
0
regionsindicatethepresenceofMajoranaend-states,whichoccurwhenthesub-banddegeneracyisremovedandthesub-band
splittingissufficientlylargerthanthepairinggap∆. Thesub-bandsareinitiallydegenerateforB =0,andsplitquadratically
asB isincreased. BlueshadedregionindicatestheZeemangapforafulltwo-dimensionalsample. Inthewire,thetopological
region extends slightly outside the Zeeman gap for sufficiently large B (for µ0B2/∆SO >∼∆). Simulation parameters: t=50,
α2/t = 10, ∆ = 1. (b) Same setup described in a) but with the magnetic field applied along the wire (B ∼ xˆ). Unlike
R S
the perpendicular field case, the wire always remains in the topological region so long as µ B > ∆ and an odd number of
0 S
sub-bandsisoccupied. Unliketheparallelfieldcase,theblacklinesthatindicatesub-bandbottomssplitlinearlyintheapplied
field,givingrisetoacriss-crossingdiamondpatternoftopologicalandnon-topologicalphases. (c)Parallelfieldphasediagram
for 10-site wide wire, topological regions occupy smaller fraction of the phase-diagram.
until ∆E < ∆ , at which point the topologically non- spacing is larger than the SC gap, ∆E > ∆ , corre-
sb S sb S
trivial regions stop occuring. sponding to W < ξ . Here, for small fields, the non-
0
topological regions occupy a larger fraction of the phase
diagram.
B. Field along the Wire Having ∆E ≈ ∆ is especially important for the
sb S
gateless setup shown in Fig.1a, where tuning sub-band
The dispersion without superconductivity in the pres- number is accomplished purely by changing µ0B. If
ence of a parallel field is: ∆Esb (cid:29) ∆S, then, without controlling µ, the wire is
most likely to be deep in the topologically trivial region.
(cid:115)
εk,λ =ξk+λαR kx2+(cid:18)ky+ µα0B(cid:19)2 (3) oTrhdiesrwtooutludnelikineltyo trheqeutiorepoalpogpilcyainlgphlaasreg.eIµn0cBon(cid:29)tra∆stS, foinr
R ∆E ≈∆ , the maximum require µ B is ≈∆ regard-
sb S 0 S
For µ > 0,24 the sub-band bottoms occur for k = 0. less of the initial µ, allowing one to readily tune to the
x
topological phase without controlling µ.
Unlike the perpendicular field case described above, the
initially degenerate sub-band bottoms are split linearly
by parallel B, ∆E ≈ µ B, independent of the spin-
sb 0
orbit coupling strength. The linear sub-band Zeeman– II. INDIRECTLY INDUCED SURFACE–STATE
splitting leads to the criss-crossing pattern of diamonds SUPERCONDUCTIVITY
shown in Fig.2b. So long as µ B >∆ , we expect to be
0 S
abletooccupyanoddnumberofsub-bandsandachievea Having described the advantages of applying a mag-
topologicallynon-trivialstatewithMajoranaend-states. netic field along the wire, we now address the issue of
The topological phase diagram obtained from numerical how superconductivity is induced on the surface–state.
simulations and shown in Fig. 2b,c bears out this ex- Considerathinfilmofaspin-orbitcoupledmetalwith
pectation, exhibiting topologically non-trivial phases for asurfacestate,depositedontopofaconventionals-wave
arbitrarily large chemical potential. superconductor. If the metal is in good contact with the
Two illustrative cases are shown in Fig. 2b,c. In Fig. superconductor and the film thickness does not greatly
2b the sub-band spacing is comparable to the SC gap, exceed the superconducting coherence length, ξ , then
0
∆E ≈ ∆ , corresponding to the metallic strip having nearly the full superconducting gap ∆ ≈ ∆ will be
sb S B 0
width comparable to the SC coherence length, W ≈ ξ . induced in the bulk-bands of the metal film.
0
In this case, the topological and non-topological phases However, in a pristine sample and in the absence of
occupy roughly equal portions of the phase diagram, al- interactions, the metal surface-state has no overlap with
lowingonetomoreeasilytuneintothetopologicalregion thebulkmetalbands(seeFig.1d). Consequentlywemust
by changing B or gate voltage. In Fig. 2c the sub-band rely on indirect scattering between the bulk and surface
5
bands to transmit the bulk superconductvity to the sur- 4τ2 (cid:0)∆2−ω2(cid:1). Forthelimitingcasesofstrongandweak
B 0 p
face states. This indirect scattering can occur either by disorder the induced gap reads:
elastic scattering off of static impurities, or by inelastic
(cid:26)(1−4∆2τ2)∆ ; ∆ τ (cid:28)1
scattering due to Coulomb interactions or phonons. Be- ∆ =ω = B B B B B (7)
imp p 1/2τ ; ∆ τ (cid:29)1
lowwediscussbothtypesofscattering,startingwiththe B B B
simpler case of elastic impurity scattering.
Forstrongdisorder,∆ τ (cid:28)1,theinducedgapisnearly
B B
equal to the full bulk gap, whereas for weak disorder,
∆ τ (cid:29)1onlyasmallfractionofthebulkgapistrans-
A. Surface-Bulk Mixing from Elastic Impurity B B
mitted to the surface state.
Scattering
Eq. 7 suggests that if the surface states are too well
isolated from the bulk bands, then it may actually be
Asasimplemodelofscreenedimpurities,weconsidera
advantageoustointroducesurfacedisordertoensuresuf-
random potential V (r) with zero average V (r) = 0
imp imp ficient mixing of the surface and bulk bands. However,
andshortrangecorrelations, Vimp(r)Vimp(r(cid:48))=W2δ(r− in order to drive the system into a topological supercon-
r(cid:48)). Here (...) indicates averaging over impurity config- ducting state one must apply an external magnetic field,
urations. The impurity scattering from the surface state in which case time-reversal symmetry is broken and dis-
to the bulk bands gives rise to the following self-energy, order is pair-breaking16–18. One might therefore worry
evaluatedwithintheself-consistentBornapproximation: thatincreasingdisordermaytendtosuppressratherthan
enhancesuperconductivity. However,thesizeofthepair-
Σimp(iω)=Vimp(r(cid:107),z =0)GB(r,0;r(cid:48)(cid:107),0)Vimp(r(cid:107),z =0) breaking component of disorder scattering was shown to
=−W2τ3(cid:88)ω2iω+−ξ2∆+0τ∆12 τ3 bpleinsgtro∆nsgolytodethpeenZdeeenmtaonnstphleitrtaintigoµo0fBth1e6.spInin-poarrbtiitcucloaur-,
k k B for very strong spin-orbit coupling, the pair–breaking ef-
=− 1 iω−∆0τ1 (4) fects of impurities is small.
(cid:112)
2τB ω2+∆2B For heavy metal materials with surface states ∆so is
commonly quite large, on the order of ≈100meV19,20,23.
Here ω is the Matsubara frequency corresponding Incontrast, thetypicalZeemansplittingneededisofthe
to Fourier transforming in imaginary time, τ = order µ B ≈2∆ ≈1meV. In this regime, the reduction
B 0 0
2πν (0)W2 is the elastic scattering time for bulk elec- of the surface pairing gap, δ∆ , due to disorder will be
B S
trons, ν (0) is the density of states at the bulk Fermi- quite small16:
B
surface, and G = 1/(iω−ξ τ −∆ τ ) is the Green’s
B k 3 B 1
function for bulk fermions with dispersion ξ and bulk (cid:18)δ∆ (cid:19) (cid:18)(µ B)2 1 (cid:19)2/3
k S ≈− 0
pairinggap∆B. Thisexpressionisvalidsolongaslocal- ∆S disorder ∆2so ∆SτS
izationcorrectionscanbeignoredinthebulk,i.e. solong
(cid:18) 1 (cid:19)2/3
as εF,BτB (cid:29)1, where εF,B is the bulk Fermi-energy. We ≈−10−3 (8)
emphasize that the impurity induced surface–bulk mix- ∆SτS
ing is sensitive only to impurities near the surface.
where τ is the elastic lifetime for surface-states due to
IncorporatingΣ intothesurfacestateGreen’sfunc- S
imp
tion yields:
E
Gap
(cid:20)(cid:16) (cid:17)−1 (cid:21)−1 (cid:68)
G (iω,k)= G(0) −Σ S
S S imp 1.0
Z(iω) 0.8
= (5)
iω−Z(iω)H −(1−Z(iω))∆ τ
0 B 1 0.6
whereG(S0)(iω)=[iω−H0]−1 isthebaresurfaceGreen’s 0.4 (cid:68)soΜ0B(cid:61)100
tfuhnecstuiornfa,cHe H0 a=m(il2tkmo2n−ianµ,−anαdRzˆ·(σ×k))τ3 −µ0Bσz is 0.2 (cid:68)(cid:68)ssoo(cid:144)ΜΜ0BB(cid:61)(cid:61)1500 1
0
(cid:144)
(cid:34) (cid:35)−1 0 2 4 6 8 10 (cid:68)SΤS
1/2τ (cid:144)
Z(iω)= 1+ B (6)
(cid:112)
∆2 +ω2
B
FIG. 3. Surface pairing gap, ∆ , for various ∆ /µ B. The
S so 0
reduction of the induced surface–gap due to disorder is very
is the surface quasi-particle residue.
weakfor∆ (cid:29)µ B. Plotisgeneratedfromthecalculations
The effective pairing gap from impurity induced so 0
of disorder induced pair breaking from Ref. 16.
surface–bulk mixing is given by smallest pole of G
S
(cid:16) (cid:17)2
which occurs at frequency ωp defined by: ∆ωBp −1 =
6
rowshowsinteractiondriven(virtual)tunnelingbetween
surfaceandbulkstatesaccompaniedbyavirtualparticle-
hole excitation.
Incontrasttothesurface–bulkmixing, whichdepends
S B S
only on the easily measurable quantities τ and ∆ , the
B B
inelasticsurface–bulkmixingisdifficulttoaccuratelyes-
timate. Doing so would require detailed knowledge of
screening properties, phonon dispersion, and electron-
B
phonon coupling matrix elements. These quantities are
highlynon-universal,anddifficulttomeasure. Therefore,
rather than attempting a detailed calculation, we simply
S S
illustrate that interaction driven processes can also con-
tribute to surface–state superconductivity.
B
III. GATING METALLIC SURFACE STATES
OneoftenstatedworryaboutproposalstorealizeMa-
S B S
joranas in nanowires with induced superconductivity, is
that, since the wire is necessarily in good contact with a
FIG. 4. Depiction of virtual scattering processes which superconductor, the chemical potential of the wire may
mix bulk and surface bands and generate surface supercon- be pinned to the Fermi-energy of the superconductor
ductivity (left column) along with representative Feynman–
making it impossible to gate the nanowire. This worry
diagrams (right column). In the diagrams, lines labeled by
wouldalsoapplytothesetupdiscussedhere,usingmetal-
’S’and’B’indicatesurface–stateandbulk–statepropagators
lic surface states.
respectively; propagators with left(right)arrowsare conven-
Here we address this worry, and demonstrate that
tional particle (hole) propagators, whereas propagators with
both left and right arrows are anomalous propagators due to the pinning of the surface chemical potential due to the
the Cooper-pair condensate. Each process shown in the left bulk Fermi-surface is not strong enough to prevent gat-
columnrepresentshalfofthecorrespondingdiagram(tocom- ing. Rather, under experimentally realistic assumptions
pletethediagram,theprocessisrepeatedinreverse). Thetop itshouldbestraightforwardtotunethesurfacechemical
rowdepictselasticscatteringfromimpurities,representeddi- potential across 100’s of sub-bands.
agrammaticallyby×’sconnectedbyadashedline(indicating Consider applying a voltage, V , to a gate separated
g
scatteringoffofthesameimpurity). Themiddleandbottom
from the surface of the grounded metal sample by a
rows show inelastic processes that generate surface pairing;
dielectric of dielectric constant (cid:15) and thickness d (see
wavy-lines represent either screened Coulomb interactions or
Fig. 5). The applied voltage induces a bulk screening
phonons. Themiddlerowshowsinelasticpair-scatteringfrom
charge density ρ (z) confined within a screening length,
surface–to–bulk, and the bottom row shows interaction in- B
(cid:113)
dcruecaetdionsuorffaaceb–ubluklkpatrutnicnlee–lihnoglewphaiicrh. is accompanied by the λTF = e2(cid:15)N0B,ofthesurface,andalsoinducesasurface–
state charge density ρ . For simplicity, we assume that
s
the extension of the surface-state into the bulk is much
(cid:113)
disorder. ThereductionoftheSCgapduetosurfacedis- smallerthanthescreeninglengthλTF = e2(cid:15)N0B,andap-
orderisshowninFig. 3,wherefor∆ /µ B ≈100wesee
so 0 proximatethesurfacestateasinfinitesimallythin. Incor-
almost no effect at all from disorder. Therefore, so long
porating a finite surface–state width is straightforward,
as spin-orbit coupling is large, it is possible to enhance
but does not substantially alter the results.
thesurface–statepairingbyaddingdisorderwithoutsup-
Within the Thomas-Fermi approximation the bulk
pressing the pairing gap by pair-breaking scattering. screeningchargeis: ρ (z)=−e2N φ(z)whereN isthe
B B B
bulk density of states, and the induced surface charge is
ρ (z) = −e2N φ(0)δ(z), where N is the surface density
s s s
B. Surface-Bulk Mixing from Inelastic Scattering of states, and φ(0) is the chemical potential at the metal
surface (z = 0). Solving Poisson’s equation we find for
the surface potential
Thesurface-stateandbulkbandsarealsomixedbyin-
elastic electron-electron scattering and electron-phonon λ V
φ(0)=(cid:15) TF g (9)
scattering. The middle and bottom rows of Fig. 4 il- R1+N /λ N d
S TF B
lustrate two processes that induce pairing in the sur-
face state. In the process shown in the middle row, a where(cid:15) istherelativepermittivityofthegatedielectric.
R
pairofsurface–electronsarevirtuallyscatteredintobulk Weseethattheconsequenceofapplyingthegatevolt-
states, where they develop pair correlations before re- age is to shift the chemical potential of the surface by
turningtothesurface. Theprocessshowninthebottom δµ = −eφ(0) compared to the bulk chemical potential.
S
7
IV. MINI–GAP
Bulk Screening
Charge
Surface Charge
WhilethegatelessgeometryofFig. 1a. isverysimple,
there are advantages to the top-gate geometry shown in
Vg Gate Dielectric Metal Fig. 1b. Forexample,ithasbeenshownthat,inthepres-
ence of multiply occupied sub-bands, the Majorana zero
d λ modes are accompanied by sub-gap fermion states local-
TF
ϕ(z) ized at the wire–ends12. These localized fermions have
energy spacing on the order of the so-called “mini-gap”
ϕ ∆mg < ∆S. Recently, it was shown that the maximal
0
mini-gap spacing occured when the wire-width was com-
parabletothesuperconductingcoherencelength,andfor
z perfect rectangular strips? , the optimal minigap scales
FIG.5. Electrostaticpotentialprofilefromappliedgatevolt- as ∆ ≈ ∆2/ε (cid:28) ∆ .26 We believe that the scaling
mg S F S
age(bottom) aligned with the proposed materials stack (top, ∆ ≈ ∆2/ε is partially an artifact of the assumption
shown here rotated 90◦ relative to Fig. 1b). The surface mg S F
of a perfectly rectangular geometry, which leads to Ma-
chemical potential is shifted by δµ =−eφ(0) relative to the
s joranaend-statesforeachbandthatarealmostperfectly
bulkchemicalpotential. Estimatesusingtypicalmaterialpa-
orthogonal, and therefore only very weakly mixed. For
rametersdemonstratethatonecanreadilytunethechemical
the more realistic case, where the wire-end is rounded
potentialby±100meV,despitethepresenceofalargedensity
(or otherwise distorted) on length-scales ≈ 1/k , then
of states from the metallic bulk. F
the end-states have randomized overlaps, leading to a
slightlymorefavorablemini-gapscalingthatshouldbeof
(cid:112)
the order ∆ ≈ ∆ ∆ /ε (see Appendix A. below).
mg S s F
For the case of Au, we have ε ≈ 0.5eV and optimisti-
F
cally one could use a large gap superconductor such as
Nb so that ∆ ≈1meV, giving ∆ ≈200mK, which is
S mg
For typical metals, v ≈ 1×106m/s, and the bulk and potentially resolvable in a dilution refrigerator.
F
surface band masses are comparable to the bare electron While these minigap states are known not to disrupt
mass, giving NS ≈ 4. Break-down fields for typi- topological operations involving spatially well separated
cal gate dielecλtTrFicNsB(e.g. SiO ) are of on the order of Majoranas27, the small mini-gap states complicate tun-
2
E ≈ 1V/nm, and typical screening metallic lengths neling based probes of the Majorana zero-modes unless
max
are λ ≈ 1˚A. For SiO , with (cid:15) = 4.9, this gives the temperature and resolution of the probe are lower
TF 2 R
δµ(smax) ≈ ±100meV. In comparison, for a metallic wire than ∆mg. The presence of a large number of mini-gap
withwidthoftheorderofthesuperconductingcoherence states can be avoided by selectively gating sections of
length, the typical sub-band spacing is ≈ ∆ ≈ 1meV, the wire so that the local sub-band number changes by
0
indicating that one could tune across hundreds of sub- at most ±1.26 Here we re-emphasize that this scheme
bands. Furthermore, using a higher-K dielectric such as does not rely on the existence of well-defined sub-bands,
HfO would allow one to tune the surface-potential over and that changing the average width by ±1 sub-bands
2
an even larger range. abruptly will trap a Majorana mode even for meander-
ing wires.
From simple electrostatic modeling, we have shown
that the close proximity to a metal does not substan-
tially impede the ability to tune the surface-state chemi- V. DISCUSSION AND CONCLUSION
calpotentialbyagatevoltage. Thisanalysisalsoimplies
thatonecoulduseatopgatetocontrolthechemicalpo- In summary, we believe that metallic thin-films with
tential of semiconducting nanowires placed on top of a Rashba split surface states offer a promising route to re-
superconductor. However, in order to get strong prox- alizing Majorana fermions. The large Rashba spin–orbit
imity induced SC, it is typically necessary to deposite couplingsinthesematerialsofferseveraladvantagesover
nanowiresonaninsulatingsubstrateandcoatthemwith similarproposalsinvolvingsemiconductingmaterials, al-
a superconductor. In this setup, one would need to em- lowingforsubstantiallylargerenergyscales,anddramat-
ploy a back-gate, which offers poor electrostatic control ically reducing the sensitivity to disorder.
(since the wire would be coated on three sides by super- Oneparticularlypromisingsurfacestateoccursonthe
conductor). Therefore, more complicated geometries are (111)surfaceofAu19. Thissurfaceisstableandhasbeen
required; for example, one could partially coat the wire well studied by ARPES. The surface bands have high
withsuperconductorandpartiallywithagate25. Incon- carrier density, ε ≈0.5eV, and large Rashba spin-orbit
F
trast, the metallic-surface state chemical potential can splitting ∆ ≈ 50meV. The first task towards creating
so
betunedusingjustasimpletop-gategeometry,substan- Majorana fermions in the Au(111) surface state, would
tially simplifying the fabrication requirements. beobservetheindirectlyinducedsurfacepairinggap∆ ,
S
8
which could be examined by planar tunneling or STM presenceofsub-gapstatesattheendofafullygappedsu-
tunneling (see Fig. 1b). If the surface–bulk band mixing perconducting wire. These sub-gap states would be con-
is insufficient to achieve large ∆ , the surface could be finedthewire-endandwoulddisappearwhenµ B <∆
S 0 S
intentionally disordered to improve the surface–pairing. giving a clear signature of topological superconductivity.
The measurements involved should be straightforward, We have shown that the parallel field geometry has the
andtoourknowledge,wouldconstitutethefirstobserva- advantage of allowing one to operate at arbitrarily large
tion of superconductivity induced onto a surface–state. chemicalpotential. Thisobservationisimportantforthe
The setup shown in Fig. 1a. is the simplest possible Au(111)surfacestate,becauseitslargeεF ≈0.5eVcould
version of our proposed scheme. By patterning a quasi- make it difficult to tune the chemical potential near the
one-dimensionalstripof Auontopofan ordinary super- Rashba crossing (which would be necessary for the per-
conductor, one can achieve a topological superconductor pendicular field setup).
withMajoranaend–statessimplybyapplyingamagnetic Finally, we have shown that, in contrast to semicon-
field parallel to the wire (without ever tuning the chem- ducting nanowire based proposals, it is possible to effec-
ical potential by gating). If the wire width is compa- tively control the metallic surface–state chemical poten-
rable to the coherence length, then only small magnetic tialwithasimpletop-gategeometry,despitethepresence
fields µ B ≈ ∆ are required to achieve Majorana end- of a large bulk-density of states. This obviates the need
0 S
states. As a concrete example, taking ε and k of the for more complicated gating geometries, or complicated
F F
Au(111) surface state measured in Ref. 19, and taking gatelesssetupssuchasthoseproposedinRef. 21and22.
∆ ≈5K givescoherencelength: ξ ≈5µm,corresponds Thisisincontrasttoproposalsinvolvingsemiconducting
S 0
to n≈500 occupied-subbands. nanowires,whichtypicallyneedtobecoatedwithsuper-
TheMajoranaend-statescouldbedetectedbytunnel- conductorinordertoinduceSCbyproximity. Forawire
ing measurements, e.g. by STM or by fabricated tunnel- coated with superconductor, a simple top-gate does not
ing contacts. Resonant Andreev reflection from a Majo- exert sufficient electrostatic control over the wire, and
rana fermion gives a distinctive quantized conductance: more complicated gating geometries are required.
G=2e2/h.28 As described above, in multichannel wires,
the Majorana zero-mode will coexist with other sub-gap
states localized to the end of the wire. These states have Acknowledgements – We thank J. Moodera and A.R.
energy spacing ≈ ∆ which is typically (cid:28) ∆ . If the Akhmerov for helpful conversations. This work was sup-
mg S
mini-gap spacing is too small to experimentally resolve ported by DOE Grant No. DEFG0203ER46076 (PAL)
by tunneling, it would still be interesting to show the and NSF IGERT Grant No. DGE-0801525 (ACP).
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9
Appendix A: A. Mini-Gap Scaling giving rise to random overlaps between Majoranas from
different sub-bands and leading. In this appendix, we
simulated strips with slightly distorted ends, and find a
Ref. 26examinedthescalingofthesizeofthemini-gap (cid:112)
more favorable mini–gap scaling ∆ ≈ ∆ ∆ /ε (cid:29)
to sub-gap fermionic states localized, along with Majo- mg s s F
∆2/ε . Specifically, we simulate numerically the tight-
ranas,totheendsofperfectlyrectangularp+ipsupercon- s F
binding model for a p+ip superconductor used in Refs.
ductingwires. Thereitwasfoundthatthemini-gap,∆
mg 12 and 26: H =H +H , of a single species of elec-
exhibitedamaximumforwireswithwidthW ≈ξ which t p-BCS
0 trons with p +ip BCS pairing:
scaled as ∆ ≈ ∆2/ε (cid:28) ∆ . Qualitatively speak- x y
mg s F s
ing, for perfectly rectangular wires, ∆mg is very small H =(cid:88)−t(cid:16)c†c +h.c.(cid:17)−(cid:88)µc†c
because each sub-band contributes Majorana end-states t i j j j
which are nearly orthogonal to each other, and therefore (cid:104)ij(cid:105) j
mix only very weakly15. H =(cid:88)∆ (cid:16)−ic† c†+c† c†(cid:17)+h.c (A1)
p-BCS s j+xˆ j j+yˆ j
j
where c† creates an electron on site j, t is the hopping
j
amplitude, µ is the chemical potential, ∆ is the p-wave
s
pairing amplitude, and we work in units where the lat-
tice spacing is unity. However, instead of rectangular
strips, we consider nearly rectangular strips with ellipti-
cal capped ends. When the length of the elliptical cap is
larger than the Fermi wavelength, 1/k , but still much
F
smaller than the coherence length, ξ , the ∆ is para-
0 mg
metrically enhanced.
Fig. 6 shows the optimal mini-gap scalings for wires
with straight and rounded ends as a function of ε ∼ t.
F
Inthesesimulations,thelengthofthewirewaschosento
beL=10ξ ,thewidthwaschosenasW ≈ξ tooptimize
0 0
the ∆ . The length of the rounded elliptical cap was
mg
5 lattice spacings, and the chemical potential was fixed
at µ = −2t. The surface–pairing gap was chosen to be
muchlessthenε (∆ (cid:28)t),sothatthecoherencelength
FIG.6. Log–logplotofscalingofmaximalmini-gap,∆ ,for F s
mg was much longer than the lattice spacing.
thetight-bindingmodelforp+ipsuperconductingstripsfrom
Qualitatively, we expect that the slightly rounded
Ref. 12 as a function of hopping strength, t. The maximal
mini-gap size occurs for strips of width W ≈ ξ . Best fit edges produce Majorana end-states for each sub-band
0
linesareshowninblack. Forperfectlyrectangularstripswith whichhavetheusualtransverseprofilealongthewidthof
straight ends (squares) ∆ ∼ ∆ (t/∆ )−1.052 as reported thewire,areconfinedtotheendofthewirewithcharac-
mg s s
in 26. In contrast wires with slightly rounded ends best–fit teristic size ξ , and are randomly oscillating with wave-
0
scales as ∆mg ∼∆s(t/∆s)−0.614. length ≈ kF along the length of the wire. The random
oscillations along the wire give rise to random overlaps
between different sub-bands, which based on the central
However, this near perfect orthogonality is special to √
limit theorem one would expect to scale as ≈ k ξ in
the case of perfectly rectangular sample geometry. For F 0
the limit 1/kF → ∞. The best-fit line in Fig. 6 has a
perfect ends, the Majorana modes contributed by each ξ0
sub-band are fine tuned to be almost exactly orthog- slightlydifferentexponent(≈0.6ratherthanthe0.5sug-
onal to each–other. In more realistic situations wire– gested by the above argument), which we expect is due
ends (of either self-assembled semiconducting nanowires to imperfect randomization by our choice of geometry as
or microfabricated metallic strips) will not be so precise, well as being limited to 1/kF ∼20−40.
ξ0