Theory of surface spectroscopy for noncentrosymmetric superconductors

6.3 MB·

by  N. Wennerdal

Theory of surface spectroscopy for noncentrosymmetric superconductors
Niclas Wennerdal1 and Matthias Eschrig1
1Department of Physics, Royal Holloway, University of London, Egham, Surrey TW20 0EX, United Kingdom
(Dated: 14 December 2016)
WestudynoncentrosymmetricsuperconductorswiththetetrahedralT ,tetragonalC ,andcubic
d 4v
pointgroupO. Theorderparameteriscomputedself-consistentlyinthebulkandnearasurfacefor
severaldifferentsinglettotripletorderparameterratios. Itisshownthatasecondphasetransition
below T is possible for certain parameter values. In order to determine the surface orientation’s
c
effect on the order parameter suppression, the latter is calculated for a range of different surface
7 orientations. For selected self-consistent order parameter profiles the surface density of states is
1 calculatedshowingintricatestructureoftheAndreevboundstates(ABS)aswellasspinpolarization.
0 The topology’s effect on the surface states and the tunnel conductance is thoroughly investigated,
2 andatopologicalphasediagramisconstructedforopenandclosedFermisurfacesshowingasharp
transition between the two for the cubic point group O.
n
a
J
I. INTRODUCTION Thelackofacenterofinversionintheunitcellallowsthe
0
gradientofthepotentialtobelargethroughouttheBril-
2
louin zone (BZ), and thus the SOC cannot be neglected.
Non-centrosymmetric materials lack a center of inver-
] The above-mentioned property, that diagonal elements
n sion in their crystal lattice. They have attracted in-
of the SOC in a Bloch basis are in general non-vanishing
o creasing attention in recent years due to the fact that
innon-centrosymmetricmaterials,allowstostudytheef-
c
- spin-orbit interaction has a strong effect on their phys- fectsoftheSOCinaminimalone-bandmodel,8 whichis
r ical properties.1–7 In crystals with a center of inver-
not possible in centrosymmetric materials.
p
sion the band-diagonal elements of the spin-orbit in-
u InthispaperwetheoreticallystudyNCSswiththeem-
teraction in a Bloch basis, L (k), vanish by symme-
s nn phasis on self-consistent superconducting order parame-
. try. This is not the case for non-centrosymmetric ma-
t ters for various surface orientations, as well as for all the
a terials, where these diagonal elements can be non-zero
topological phases of the crystal point groups T , C ,
m andindeedlarge(30-300meV).8 Anderson, indiscussing d 4v
and O with a closed Fermi surface. The SOC vector is
- heavy fermion materials, used group classification to
expandedintermsofharmonicfunctions,constrainedby
d
study the possibilities for spin-triplet superconductivity
n the symmetries of the point group, to second order. The
inspin-orbitcoupledmaterials.9Experimentalsignatures
o relative weight of the first and second order terms is pa-
ofspin-triplet(aswellasspin-singlet)pairingwerefound
c rameterized by g . Second order terms are investigated
[ in the non-centrosymmetric heavy-fermion superconduc- 2
1 mtoorreCneoPnt-3cSein,trdoissycmovmereetdricinsup2e0r0c4o.n10duSctinocrse(NthCenSs)mhaanvye gfo2rfothreCp4voianntdgrtohurepesfCor4vOa.nBdesOid:eosntehengoanp-zpeerdotvoapluoelogo-f
v ically trivial phases all point groups have one non-trivial
been identified, including Y C ,11 Li (Pd Pt ) B,12
4 2 3 2 1 x x 3 gapless phase, and O has, for a closed Fermi surface,
4 CeIrSi3,13 UIr,14, BiPd,15, and PbTaSe2,−16 amongst four non-trivial gapped phases, and we have chosen val-
7 others.17–24 These materials show signs of both spin-
ues of g to correspond to these phases. In the litera-
5 singlet and spin-triplet supconductivity to a varying de- 2
ture the point group C with g = 0 has been studied
0 gree. The system Li (Pd Pt ) B has been studied in 4v 2
1. more detail,25,26 ind2icatinxg t1h−axt3the difference between etoxtoeunrsivcheloyi,c3e6–o40f agsw=el0l.a7s.4O1–4w3itAhllvarelusuesltosfwge2 epqruesivenaltenint
0 thetwoendcompounds, x=0andx=1, canatleastin 2
this paper are self-consistent, and for parameter combi-
7 partbeexplainedbyadominatingtripletcomponentfor
nations not discussed so far in the literature. Non-self-
1
x = 0, i.e. Li Pt B, whereas Li Pd B seems to have
: 2 3 2 3 consistent results for surface spectra for various point
v a dominating s-wave singlet component, indicated by
groupsandsurfaceorientationswereobtainedinRef. 40,
i the rather low value of the upper critical magnetic field
X andsubsequentlyinRef. 43. Topologicalaspectswerein
extrapolated to zero temperature. Some systems, like
the focus of attention in Ref. 44, whereas in Ref. 45 the
r
LaNiC and LaNiGa , are candidates for a non-unitary
a 2 2 possibility of a surface instability was suggested.
spin-triplet pairing state.27
Furthermore, it has been shown that, as spin-orbit in-
teraction is time-reversal invariant, these superconduc-
II. THEORY
tors can be topologically non-trivial.2,4,28–35 The topol-
ogy and the singlet-triplet admixture are both a con-
A. Normal state band dispersion
sequence of the spin-orbit coupling (SOC) term in the
Hamiltonian of these materials, which is derived from
thenon-relativisticlimitoftheDiracequationandispro- Within an effective one-band model, the SOC term in
portional to the gradient of the crystal lattice potential. the Hamiltonian is given by HSO = αl σ, where α is
k k ·
2
the SOC strength, σ = (σ ,σ ,σ ) is a vector of spin B. Superconducting state
1 2 3
Pauli matrices, and l is the SOC vector which is real,
k
invariant under crystal point group operations g, Superconductivity is modeled within the Nambu-
l l(k)=gl(g 1k), (1) Gor’kov formalism. Under the canonical transfor-
k ≡ − mation defined above the Nambu spinor Cˆ =
k
and odd in k, l = l . We normalize the SOC vector
smuacTxhhkte∈hBkaZitni|etlktsi|mc=pa−xa1kri.mt uofm−thmkeangnoritmuadle-swtaittheinHtahmeiBltoZniisaunncitayn, (adcliekan↑gt,(cUBkˆk↓k,,Uc=†−∗kk(↑b),,kc+a†−n,kbd↓k)−tTh,ebt†−r”akhn+as,tf”bo†−rdmke−sn)oTitnet≡soNUiˆtaksmCˆhbkeulwicsiattrlhuecUqˆtuukirv≡e-.
It is straig−htforward to construct 4 4 helical Green
thus be written as
Hˆk =k(cid:88)αβc†kα(ξkσ0+αlk·σ)αβckβ (2) ftu2)n(cid:104)c{tBiˆokn1s(,t1e).,gB.ˆk†2t(hte2)r}e(cid:105)tHa,rdwehderGˆeRkΘ1k2is(t1t×h,te2)He=av−isiidΘe(ts1te−p
function, denotes a grand canonical average, ,
with ξk = (cid:15)k − µ, where (cid:15)k is the band dispersion in is an antic(cid:104)o•m(cid:105)Hmutator, and Bˆk(t) a Heisenberg oper{a•to•r}.
the absence of SOC (we will use for simplicity a nearest-
The quasiclassical propagator is obtained by integrating
neighbor tight-binding dispersion), µ is the chemical po-
outfastoscillationsfromthefullGreenfunctions. Inthe
tential,andckα (c†kα)arefermionannihilation(creation) case when the magnitude of the SOC is much smaller
operators for a quasiparticle with spin α , . We than the Fermi energy, α E , it suffices to integrate
will study simple cubic (CUB) and body∈ce{n↑te↓r}ed cu- over ξ and treat the SO(cid:28)C teFrm perturbatively. For
k
bic (BCC) lattices. The corresponding nearest-neighbor this case, in Wigner coordinates the quasiclassical prop-
tight binding dispersions are agator is given by gˇ(k ,R,(cid:15),t) = (cid:82) dξ τˆ Gˇ(k,R,(cid:15),t),
F k 3
(cid:15)CkUB =t1[cos(kx)+cos(ky)+cos(kz)] (3) with k parameterized by (ξk,kF), ξk = vF ·(k−kF),
τˆ = (τˆ ,τˆ ,τˆ ) are Pauli matrices in particle-hole space,
1 2 3
and
and the ”check” denotes Keldysh matrix structure.
(cid:15)BCC =8t cos(k /2)cos(k /2)cos(k /2), (4) The SOC term enters the transport equations as a
k 1 x y z
source term. Within this approximation the Eilenberger
where t is the hopping integral.
1 equation49 forthequasiclassicalGreenfunctiontakesthe
The point groups considered here are the cubic point
following form in the helicity basis
group O, relevant for e.g. Li Pd Pt ;12,25,26,46,47 the
2 x 3 x
tetragonal point group C4v, relevant f−or e.g. CePt3Si;10 ivF RgˆR,A,M+[zτˆ3 ∆ˆ vˆSO,gˆ]R,A,M =ˆ0 (7)
·∇ − −
and the tetrahedral point group T , relevant for e.g.
d
Y2C3.48 We use dispersion (3) for the cubic point group, with z = i(cid:15)n = iπT(2n + 1) for Matsubara, and z =
O,andforsakeofsimplicityalsoforthetetragonalpoint (cid:15) i0+ for retarded (advanced), quantities. [ , ] is a
± • •
group, C4v, whereas dispersion (4) will be used for the commutator, theSOCtermisvˆSO =α|lkF|σ3τˆ0, andthe
tetrahedral point group T . The SOC vectors are ob- gap has the form
d
(cid:80)
twahinereedRasnlaarteticBeraFvoauirsielartstiecreievs,ecltkor=s, anndlwnshienr(ekt·hRe nin)-, ∆ˆ =(cid:18)∆0˜ ∆0(cid:19) (8)
variance under point group operations, Eq. (1), leads to
restrictions on the l .8
n where the ”tilde operation” is defined as the particle-
TheHamiltonian,Eq.(2),isdiagonalizedandbrought hole conjugate, Q˜(k ,R,z,t) Q ( k ,R, z ,t).
F ∗ F ∗
to the so-called helicity basis by the canonical transfor- ≡ − −
Eq. (7) is supplemented by the normalization condition
mation Uk(lk·σ)Uk† =|lk|σ3, where (gˆR,A,M)2 = π2ˆ1. In order to simplify notation, we will
Uk =(cid:18) eciφolss(cid:0)inθ2l(cid:0)(cid:1)θl(cid:1) e−icφolss(cid:0)inθl(cid:0)(cid:1)θ2l(cid:1)(cid:19) , (5) thuemnc;eafollrtmhodm−roenptathienstuhbesqcuriapsticFlasasitcatlhtehFeoerrymiarmeoFmeremn-i
− 2 2 momenta. The subscript will be written out only when
(cid:113)
with φ = tan 1(l /l ) and θ = tan 1( l2 +l2/l ) it is necessary to avoid confusion. We consider time-
l − y x l − x y z
independentsituations,suchthatthetimevariabletwill
being the spherical angles of the SOC vector, l =
k be dropped from here on.
(l ,l ,l )T, yielding
x y z The lack of a center of inversion allows for an admix-
(cid:88) (cid:88)
Hˆk = ξkλb†kλbkλ, bkλ = Ukλαckα (6) ture of spin-singlet and spin-triplet pairing.37 The spin-
triplet vector is set to be parallel to the SOC vector in
kλ α
ordertomaximizeT .38Inspinbasistheorderparameter
where the helical index takes the values λ = +, , c
{ −} is written
and the helical band dispersion is given by ξk± = ξk ±
αis|lakn|t.isNymotmeetthraict.ξTkλh=isξi−sλka ceovnensetqhuoeungche otfheEqS.O(C2)vbeecitnogr ∆(k)=Yk(∆s+∆tlk·σ)iσ2, (9)
time-reversal invariant. Furthermore, the quasiparticle where is a crystal basis function corresponding to ir-
k
Y
spinisfixedwithrespecttoitsmomentumoneachband, reducible representation of the dominant pairing chan-
being parallel (λ=+) or antiparallel (λ= ) to l . nel, and ∆ and ∆ are referred to as the singlet and
k s t
−
3
triplet component, respectively. In the helicity ba- of step functions in the middle between the desired grid
sis the order parameter takes the form ∆(k) = points. Each step is solved analytically.51 Parameteriz-
k
Y ·
diag(∆ (k)t (k),∆ (k)t (k)), where ing the path as R = R +ρv and writing the order
+ + 0 F
− − parameter ∆(ρ) = ∆ +Θ(ρ)(∆ ∆ ) at one of these
0 1 0
∆ (k)=∆s ∆t lk , (10) steps, γ(ρ) with ρ>0 is given by −
± ± | |
and the phase factors are given by γ(ρ)=γ +eiΩ1ρδ (cid:0)eiΩ2ρ+C(ρ)δ (cid:1)−1 (15)
h 0 0
t±(k)=−e∓iφl(k), φl(k)=tan−1(ly/lx). (11) with δ0 = [γ0−γh], where γ0 ≡ γ(0) is the initial value
and γ is the homogeneous solution for ρ > 0, Ω =
Note that t ( k)= t (k). h 1
Eq. (7) ±can−be pa−ra±meterized in terms of coherence z−γh∆˜ andΩ2 =−z+∆˜γh,andC(ρ)=C0eiΩ1−eiΩ2C0,
where C is the solution to C Ω Ω C = ∆˜. The
functions, γ(k,R,z) and γ˜(k,R,z),50 in such a way as 0 0 1 − 2 0
solution for γ˜(ρ) is completely analogous.
to automatically fulfill the normalization condition,
The reflection at the surface is in leading approxima-
gˆR,A,M ≡(cid:18)g˜f g˜f(cid:19)R,A,M =∓iπ(cid:2)N−1G(cid:3)R,A,M twioitnh(tahseαm(cid:28)omEeFn)tucmonsciodmerpedonteonbtepsapreaclluellartointhspeinsusrpfaaccee,,
k , conserved. Writing the momentum for incoming tra-
(cid:18) (cid:19)
= (σ0−γγ˜) 0 je(cid:107)ctories k = (k ,k ) this gives the momentum for out-
N 0 (σ0 γ˜γ) going trajectorie⊥s as(cid:107) k = ( k ,k ). Following Ref. 50,
(cid:18)(σ +γγ˜) −2γ (cid:19) incoming (outgoing) quant−itie⊥s ar(cid:107)e written with lower-
= 0 (12)
G 2γ˜ (σ0+γ˜γ) case (uppercase) symbols and the surface boundary con-
− − ditions become
wherethetop(bottom)signcorrespondstogˆR (gˆA),and
in the case of gˆM, to positive (negative) Matsubara fre- Uk†Γ(k,ε)U∗k =Γs(k,ε)=γs(k,ε)=Uk†γ(k,ε)U∗k
− −
quencies. With this, Eq. (7) transforms into two decou- (16)
pled Riccati differential equations,
and
(iv +2z)γ =γ∆˜γ+[αl σ ,γ] ∆, (13)
F R k 3
(ivF ··∇∇R−2z)γ˜ =γ˜∆γ˜+[α||lk||σ3,γ˜]−−∆˜ . (14) U−TkΓ˜(k,ε)Uk =Γ˜s(k,ε)=γ˜s(k,ε)=U−Tkγ˜(k,ε)U(k1,7)
In the homogeneous case, i.e. in the bulk, the solution
is γ = diag(γ (k)t (k),γ (k)t (k)) with the ab- wherethessuperscriptindicatesthatthecoherencefunc-
h k + +
breviatioYns·γ = ∆ /(z+i(cid:112)− ∆− 2 z2). For this tions are expressed in the spin basis.
k
case the SOC±term−dr±ops out. |Y ±| −
The surface problem is treated by solving Eqs. (13)-
(14) along classical trajectories parallel to v , using the C. Gap equation
F
homogeneoussolutionsasinitialconditionsatasufficient
distance from the surface. This is done by discretizing The pairing potential in spin space can be written as
the path and treating the order parameter as a series a sum of singlet, triplet, and a mixture term37
V (cid:8)
Vs1s2s3s4(k,k(cid:48))= 2YkYk∗(cid:48) vs(iσ2)s1s2(iσ2)†s3s4 +vt(lk·σiσ2)s1s2(lk(cid:48) ·σiσ2)†s3s4+
(cid:2) (cid:3)(cid:9)
vm (lk·σiσ2)s1s2(iσ)†s3s4 +(iσ)s1s2(lk(cid:48) ·σiσ2)†s3s4
(18)
where v , v , and v are free parameters that describe the relative coupling strength of each term, respectively, V is
s t m
the overall pairing potential strength, and is the basis function of the irreducible representation with the highest
k
Y
T . To avoid ambiguity, we normalize the relative pairing strengths according to
c
v2+v2+v2 =1 (19)
s t m
and for later reference introduce spherical coordinates
(v ,v ,v )=(cos(φ )sin(θ ),sin(φ )sin(θ ),cos(θ )) .
s t m v v v v v
(20)
In helicity space the pairing potential takes the form
(cid:18) (cid:19)
V(k,k(cid:48))= V2YkYk∗(cid:48) vvss+vvtt|llkk||llkk(cid:48)(cid:48)|+−vvmmll+ vvss−+vvtt|llkk||llkk(cid:48)(cid:48)|−+vvmmll−+ (21)
− | || | − | || |
4
twhiethMla±ts=ub|alrka|±for|lmk(cid:48)a|l.ismThies esexlpf-rceosnsesdistienntceyrmeqsuoaftiFoenrmini ∆th−e)sealnf-dco∆nsti|sltke|nc=y e21q(u∆a+tio−n∆fo−r)t,htehesinimglpetlicaintdfotrrmipleotf
surface averages , defined as components of the order parameter reads
(cid:104)•(cid:105)
1 (cid:90) d2k (cid:90) d2k
F F
= ( ), N = .(22)
(cid:104)•(cid:105) NF (2π)3|vF| • F (2π)3|vF| (cid:18)∆s(cid:19)=T |(cid:15)n(cid:88)|<(cid:15)cN V (cid:28)A (cid:18)f+(cid:19)(cid:29) , (25)
∆ F k f
With this, the self-consistency equation takes the form t (cid:15)n −
(cid:18)∆ (k)(cid:19) |(cid:15)n(cid:88)|<(cid:15)c(cid:28) (cid:18)f (k,(cid:15) )(cid:19)(cid:29)
+ =TN V(k,k) + (cid:48) n (23) where
∆ (k) F (cid:48) f (k,(cid:15) )
− (cid:15)n − (cid:48) n k(cid:48)
(cid:18) (cid:19)
where f± are defi(cid:18)nfed(kb,y(cid:15) )t (k) 0 (cid:19) Ak = 12Yk∗ vvst|−lk|v−m|vlkm| −vvst+|lkv|m−|lvkm| . (26)
f(k,(cid:15) )= + n + , (24)
n 0 f (k,(cid:15) )t (k)
n
− − AftereliminationofthecutoffandthepairingstrengthV
the phase factors are defined in Eq. (11), and (cid:15) is the in favor of the superconducting transition temperature,
c
BCS technical cutoff. Using the relations ∆ = 1(∆ + one obtains
s 2 +
(cid:18)∆s(cid:19)=(cid:20)ln(cid:18)T(cid:104)Lk(cid:105)(cid:19)(cid:21)−1T (cid:88)(cid:28)A (cid:18)f+(cid:19) π L (cid:18)∆s(cid:19)(cid:29) , (27)
∆t Tcλmax (cid:15)n k f− − |(cid:15)n| k ∆t
where the matrix exponent in the logarithm, T(cid:104)Lk(cid:105), is can be seen in Fig. 1. When both eigenvalues are posi-
taken element-wise, i.e. tivetherearetwopossiblenucleationchannels, thedom-
inant and the subdominant one. The dominant chan-
(cid:104) (cid:105)
T(cid:104)Lk(cid:105) T(cid:104)[Lk]ij(cid:105) , (28) nel is responsible for the transition to superconductiv-
ij ≡ ity due to its larger critical temperature. The domi-
nant channel also determines the singlet to triplet or-
and with
der parameter ratio, ∆ /∆ , and their relative sign. The
s t
(cid:18) (cid:19) subdominant channel nucleates at a lower temperature
v 2 v l 2
Lk = −vsm|Y|Yk|k|2 −vtm|Y|Yklkk|k2| . (29) Ttucsrueb.o≤f siTncg.letWainthd atrifipnleittecmomixpinong,envtms i(cid:54)=s o0b,taainneadd.mFioxr-
certain choices of the parameters (v ,v ,v ) it is possi-
Furthermore, λ max λ ,λ , and λ are the s t m
max ≡ { 1 2} 1,2 ble to achieve a cross-over from dominating singlet com-
eigenvalues of the matrix L . We follow Ref. 52 in
(cid:104) k(cid:105) ponent at T = T to a dominating triplet component
eliminatingthecut-offdependenceinclosevicinitytothe c
surface as well. For details on the numerical procedure
to achieve self-consistency see appendix A. # Nucleation Channels
2
D. Bulk superconducting phase 1.5
0
π
/ 1
At T =Tc the self-consistency equation reduces to φv 1
ln(cid:18)2πeTγ(cid:15)cc(cid:19)(cid:104)Lk(cid:105)(cid:18)∆∆st(cid:19)= NF1V (cid:18)∆∆st(cid:19) (30) 0.5 2
0
0 0.5 1
where γ = 0.5772... is the Euler-Mascheroni constant. θ /π
v
The number of positive eigenvalues of L determines
k
(cid:104) (cid:105)
the number of nucleation channels, with T determined FIG. 1. Dependence of the number of nucleation channels,
c
bythelargesteigenvalueλ . UsingEq.(20),theeigen- i.e. positiveeigenvaluestothematrixdefinedinEq. (29),on
max
(cid:112)
values can be mapped onto the unit sphere. How the theanglesφv =tan−1(vt/vs)andθv =tan−1( vs2+vt2/vm).
number of nucleation channels depends on the spherical The ovals are given by 2cot(θv) sin(2φv). The number of
anglesφ =tan 1(v /v )andθ =tan 1((cid:112)v2+v2/v ) channels is independent of the SO≤C as long as it is finite.
v − t s v − s t m
5
g rbulk where
2 ∆
O 0.0 0 0.26 0.38 0.50 0.62 0.74 0.86 0.98 1.1
N(j)(k,(cid:15)) (cid:26) 1 (cid:27)
O 0.7 0 0.20 0.33 0.46 0.59 0.72 0.84 0.97 1.1 =ReTr σ [σ γs(k,(cid:15))Γ˜s(k,(cid:15))] 1 σ .
s j 0 − j
O 1.03 0 0.09 0.23 0.38 0.52 0.67 0.81 0.96 1.1 N − − 2
F
O 2.5 0 0.17 0.30 0.44 0.57 0.70 0.83 0.97 1.1 (33)
C4v 0.0 0 0.14 0.28 0.41 0.55 0.69 0.83 0.96 1.1 Note that all quantities in Eq. (33) are expressed in the
C4v 4.0 0 0.14 0.28 0.41 0.55 0.69 0.83 0.96 1.1 spin basis. Using a non-self-consistent order parameter,
T N/A 0 0.14 0.28 0.41 0.55 0.69 0.83 0.96 1.1
d with the bulk solution all the way to the surface, it is
straightforwardtoshowthattherearetwoclassesoftra-
TABLE I. The scaled bulk singlet to triplet ratios, rbulk
∆ ≡ jectories giving rise to Andreev bound states (ABS) at
∆ /(∆ max l(k )), chosen for the different point groups,
s t | F | zero energy (see appendix B for details). Introducing
O, C , and T , and their respective g values used in this
4v d 2 the notation Υ sign[ ∆ (k)] the first class of tra-
work. jectories is simpkly≡givenYbky Υ− = Υ = 0. With the
k k
− (cid:54)
spherical angles (φ ,θ ) and (φ ,θ ) corresponding to l
l l l l k
at T = 0. An example in the single-channel region is andlk respectively, thesecondclassisgivenbysolutions
(v ,v ,v )=(1,0,a/( l 2 a2)) (ignoring normaliza- to
s t m k
(cid:104)| | (cid:105)−
tion) with the parameter a being slightly larger than the
maximum value of the SOC vector on the Fermi surface, F(φl,θl,φl,θl)= 1, (34)
−
e.g. a = 1.01max l . This means that the topology of
| k| with the definition F(φ ,θ ,φ ,θ ) = cos(θ )cos(θ ) +
the system can be sensitive to its sub-critical tempera- l l l l l l
cos(φ φ )sin(θ )sin(θ ), provided that (Υ ,Υ ) =
ture. l − l l l k k
(0, 1), (Υ ,Υ ) = ( 1,0), or (Υ ,Υ ) = ( 1, 1).
In certain parameter ranges for (vs,vt,vm) it is possi- Thi−ssecondkclasksofbou−ndstatesariskesdkuetoth−ep−hase
bletoconstructaconfigurationwithtwoactivechannels
factors t (k) defined in Eq. (11), which can yield an ex-
inwhichthesubdominantchannelhasalowerfreeenergy
tra phas±e shift of π. These results remain true for self-
at T =0, thus inducing a second phase transition below
consistent order parameters as long as the gap does not
T . The simplest way to get a second phase transition
c completely close at some distance from the surface.
is to choose (v ,v ,v ) in such a way as to get a domi-
s t m
nantchannelwithalargetripletcomponent, aswellasa
rather large subdominant critical temperature. A exam-
F. Point contact spectra
ple for such a choice is (v ,v ,v ) = (0.999 l 2 ,1,0)
s t m k
(cid:104)| | (cid:105)
(ignoring normalization) giving a dominant pure triplet
The point contact conductance between a normal
channel, and a subdominant pure singlet channel. The
subdominant critical temperature is Tsub. = 0.996T for metal and an NCS is computed using the following as-
c c
sumptions: the size of the point contact is much smaller
the point groups and SOC vectors in table I (assuming
thanthecoherencelengthbutmuchlargerthantheFermi
= 1). The condensation energy at zero temperature,
k
Y wavelength,theFermisurfacesonbothsidesoftheinter-
assuming the same density of states on both Fermi sur-
face are considered to be equal, and the proximity effect
facesheets(whichistheapproximationemployedhereas
is ignored. The normal metal having index 1, and the
the splitting is small), is given by
NCS index 2, the scattering matrix of the interface in
δΩ= NF (cid:0)∆ 2+2∆ ∆ l + ∆ 2 l 2 (cid:1) , the spin/helicity basis is given by
− 2 | s| | s t|(cid:104)| kF|(cid:105) | t| (cid:104)| kF| (cid:105)
(cid:18) (cid:19) (cid:18) (cid:19)
(31) S= S11 S12 = rσ0 tUk† (35)
aanlldthyrieeeldpsoainlotwgeroruvpasluaenfdorStOhCesvuebctdoormsicnoannstidcehraendnienl,thfoisr S21 S22 t∗U−k −rUkUk†
work, with this choice of (v ,v ,v ). with the transmission amplitude
s t m
t cos(α )
0 k
t(α )= (36)
E. Angle-resolved density of states k (cid:113)
1 t2sin2(α )
− 0 k
The angle-resolved surface density of states (DOS) is where t is the tunneling parameter and α is the angle
0 k
given by N(k,(cid:15)) = (2π)−1NFImTrλ[gR(k,(cid:15))], or ex- betweenthesurfacenormalandtheFermivelocityofthe
−
plicitly in terms of coherence functions outgoingtrajectoriesinthenormalmetal. Thereflection
(cid:26) (cid:27) amplitudeisgivenbyr =√1 t2. Thezero-temperature
N(k,(cid:15)) =ReTr [σ γ(k,(cid:15))Γ˜(k,(cid:15))] 1 1σ . tunnel conductance is given b−y40
λ 0 − 0
N − − 2
F (cid:68) (cid:104) (cid:105)(cid:69)
(32) G(eV)= n v B((cid:15)) 2 S A ((cid:15)) 2
· F1 || || −|| 12 2 || out
The spin-resolved DOS along the quantization axis j (cid:28) (cid:12)(cid:12) (cid:12)(cid:12)2(cid:29)
∈ + n v (cid:12)(cid:12)B( (cid:15))γ ( (cid:15))S˜ (cid:12)(cid:12) (37)
x,y,z is given by N(j)(k,(cid:15)) = N(k,(cid:15)) N(j)(k,(cid:15)), · F1(cid:12)(cid:12) − 2 − 21(cid:12)(cid:12)
{ } ± ± out
6
where the expression is evaluated at (cid:15) = eV, v is the We calculate ν numerically using the procedure in ap-
F1
Fermi velocity in the normal metal, indicates that pendix C.
out
(cid:104)•(cid:105)
theaverageisonlyforoutgoingtrajectoriesinthenormal Nodal systems are classified by calculating the 1D
metal, B((cid:15))=S (σ +A ((cid:15))S ), winding number which is defined as
12 0 2 22
(cid:73)
A2((cid:15))=(cid:16)σ0−γ2((cid:15))S˜22γ˜2((cid:15))S22(cid:17)−1γ2((cid:15))S˜22γ˜2((cid:15)) (38) NL = 2dπliTr(cid:2)q−1∇lq(cid:3) (43)
L
taanndce||,•G||2,≡is s21iTmrp(cid:2)l(y•)o(b•t)a†i(cid:3)n.edThbey nseotrtminagl tshtaetecochoenrdenucce- wdihreecretiolnpaalragmraedtieernizteasltohneglothoipsLloionp.thTehBeZl,oaonpd∇claisnnthoet
N L
functions to zero. pass through nodes of the order parameter, but is other
than that arbitrary. The 1D Hamiltonian for this loop
is in general not time-reversal invariant and is thus of
G. Topology symmetry class AIII.53 In order to characterize a nodal
phase the loop needs to be constructed in such a way as
Wecharacterizethetopologyofasystembycomputing to always encircle a line node of ∆ (k−F) for any Fermi
surface geometry. −
three topological invariants. The starting point is the
With increasing singlet to triplet ratio the first nodes
Bogolioubov-de Gennes (BdG) Hamiltonian
(cid:18) h(k) ∆(k) (cid:19) acrpepaesainrgat∆tsh/∆e tpofuinrttshewrhtehree n∆osd/a∆ltri=ngsmcionn|tl(inku−Fe)|t.oIbne-
H(k)= ∆ (k) hT( k) (39) positioned around these points until they connect with
†
− − one another. At this stage the nodal rings become posi-
obeying time-reversal symmetry, , particle-hole sym- tionedaroundthepointswheretheyeventuallydisappear
me=tryi, C ., aTsheweBlldGasHthaemiclotomnbiainnTedis’tchhuirsalo’fsythmemseytmry- ∆thsr/o∆ugth=thmeapxoi|nl(tks−Fw)h|.erTehthues naogdeanlerrianlgsloaopppsehaorualnddpdaisss-
mSetryTclCass DIII.53 It anticommutes with and in the appear. This is accomplished by the loop
S
basis where is block diagonal H becomes block off-
diagonal, H¯ =S VHV†. The flat-band block off-diagonal L : Γ→min|l(k−F)|→∂BZ→max|l(k−F)|→Γ (44)
Hamiltonian Q(k) is constructed by projecting all bands
where ∂BZ is the BZ boundary, and the arrows do not
above (below) the gap to +1 ( 1)
− necessarily imply straight lines.
(cid:18) (cid:19) In order to study the topology’s effect on the sur-
0 q(k)
Q(k)= (40) face states the 1D winding number is also computed for
q (k) 0
† straight noncontractible loops, i.e. loops traversing one
or several of the three circles making up the BZ torus
where q(k) is a 2 2 matrix in the one-band model (we
× T3 =S1 S1 S1,thatareperpendiculartothesurface.
set for simplicity Yk =1) Writingt×hem×omentumk=(k ,k )andthesurfacenor-
1 mal n=(l,m,n) the 1D windi(cid:107)ng⊥number is written
q(k)= [Al λ +B λ ]σ +
k 1 k 2 0
2 | | (cid:90)
1[Alk λ2+Bkλ1] lk σ (41) N(lmn)(k(cid:107))= d2kπ⊥iTr(cid:2)q−1∇⊥q(cid:3) . (45)
2 | | l ·
k
| |
Restricting ourselves to time-reversal invariant non-
wλan−+itd1h+tAhuλ=s−−q1α(,k+w),hi∆iesrtei,llBλ-d±ke=fi=nξe|kdA+|flokir∆|n±so,dBλak1l|.=orNdλe−+ort1ep−atrλhaa−−mt1e,Qtλe(2rks=)., cdoenfitnreadc.tiNblaemloeloypsthaenZot2hienrvatroipaonltogical invariant can be
Fully gapped systems are classified by calculating the (cid:89) Pf[iσ2qT(K)]
W (K )= (46)
3D winding number which is defined as (lmn) (cid:107) (cid:112)det[iσ qT(K)]
K 2
ν =(cid:90) d3k εabcTr(cid:2)(q 1∂ q)(q 1∂ q)(q 1∂ q)(cid:3) (42) where K are time-reversal invariant momenta on the
24π2 − a − b − c loop, and Pf[ ] denotes the Pfaffian of an antisymmet-
BZ
•
ric matrix . The 1D Hamiltonian for this loop is of the
where Einstein summation is implied, εabc is the Levi- symmetry c•lass DIII.53
Civita pseudo-tensor, a,b,c kx,ky,kz , and the inte- The singlet (triplet) component is said to be domi-
∈ { }
gisraclleiasrotvheartthνeisenotnirlyefiwresltl-BdeZfi.nFerdomif tthheeodredfienritpiaornaomfeqteirt nWaintthifatdhoeminienqaunatlistiyng∆lest/c∆otm>pomnaexnt|lt(hke−Fm)|aistetrriuael(isfaflusell)y.
onthenegativehelicalFermisurfacedoesnothavenodes, gapped. Increasing ∆ and/or decreasing ∆ the mate-
s t
i.e. ∆−(k−F)(cid:54)=0. Therearetwowaysthiscanbetrue;ei- rial becomes nodal and eventually fully gapped again if
ther sign[∆−(k−F)] = +1 ∀k−F =⇒ ∆s/∆t > max|l(k−F)|, min|l(k−F)| > 0. As is shown below the dominance of
or sign[∆−(k−F)] = −1 ∀k−F =⇒ ∆s/∆t < min|l(k−F)|. either component is temperature dependent.
7
H. Surface band structure compared to the Fermi-surface splitting in order for the
approximation of equal Fermi surfaces for both helicities
ThesurfacebandstructureiscomputedbyfirstFourier to be valid, and µ<0 must smaller than t1 α in order
−
transforming the BdG Hamiltonian in the relative mo- fortheFermisurfacetobeclosed. Thechosenvaluesare
mentum coordinate k in the direction of the surface consistent with the approximation of an approximately
normal n, ⊥ spherical Fermisurface. The SOC term enters the trans-
port equations as a source term. In the following, we
H(k ,k ,R) H(k ,ρ,R). (47) restrict our discussion to the maximally symmetric basis
(cid:107) ⊥ → (cid:107) function corresponding to the irreducible representation
The helical dispersion, ξλ, contains for the tight-binding A , i.e. =1.
k 1 k
Y
approximation we use trigonometric functions whose
Fourier transform give rise to a series of delta functions
A. The Cubic Point Group O
(cid:88) 1
H(k ,ρ,R)= H (k ,R+ ρn)δ(j ρ/ρ ) (48)
j 0
(cid:107) (cid:107) 2 −
j Tonext-nearestneighborsinthesumoverBravaislat-
tice sites8 the SOC vector corresponding to the cubic
winhdeerxe,Han−dj(ρk(cid:107),isRt−he12lρenng)t=h Honj†e(kn(cid:107)e,eRds+to21ρmno)v,ejailsoanglatyheer point group O, takes the form
0
direction of the surface normal in order to return to a  
sin(k )[1 g (cos(k )+cos(k ))]
translation-equivalent point in the lattice unit cell. The x − 2 y z
sum has a finite number of terms, i.e. there exist a num- lk =sin(ky)[1−g2(cos(kz)+cos(kx))] (50)
sin(k )[1 g (cos(k )+cos(k ))]
ber j such that H = 0 : j > j . The terms H with z − 2 x y
c j c j
| |
j = 0 can be interpreted in terms of hopping across the
lay(cid:54) ers. Discretizing the center-of-mass coordinate R in whereg2isafreeparameterwhichdeterminestherelative
weight between the first and second order contributions.
steps of ρ , the Schr¨odinger equation for L layers can be
0 Its magnitude and direction is illustrated in Fig. 2.
written
An important property of the SOC vector correspond-
j(l) (cid:18) (cid:19) ing to the cubic point group is its lack of line nodes in
(cid:88) 1
H k ,nρ (l+ j) ψ (k )=E (k )ψ (k ), the BZ, it only vanishes at specific points. With g = 0
j 0 j l l 2
(cid:107) 2 (cid:107) (cid:107) (cid:107) these points are simply Γ, X, M, and R [for the nota-
j= j(l)
− (49) tion see Fig. 3(a)]. A finite value of g2 brings about two
morepoints. Withg >0theyarepositionedsomewhere
2
wherel=0,1,...,L 1andj(l)=min j ,l whichtakes on the paths Γ R, and Γ M, and with g < 0 on
− { c } → → 2
care of the boundary conditions, i.e. no hopping across Γ R, and X R, in Fig. 3 (a). The exact positions,
→ →
theboundary. Eq.(49)canbewrittenmorecompactlyas
a matrix equation H (k )ψ(k )=E(k )ψ(k ), and the
eff
band structure is given b(cid:107)y the(cid:107)eigenvalu(cid:107)es of(cid:107)Heff, Non- (a) O,g2 = 1.03
trivial topology gives rise to zero-energy ABS. We are
1
therefore mainly interested in the band structure close
0.25
to zero energy. This allows us to avoid diagonalizing
0.5
H , and instead only compute the smallest magnitude
eff
eigenvalues using the Lanczos method. Note that the | 0.2
F
order parameter is suppressed at both surfaces. k 0
|
/ 0.15
2
k
−0.5
III. NUMERICAL RESULTS 0.1
InthisworktheSOCstrengthenteringthequasiclassi- −1 0.05
calcalculationsisconsideredtobemuchsmallerthanthe −1 0 1
k
Fermi energy, α E . In this case the Fermi surface is k /| |
(cid:28) F 1 F
only weakly split. Ignoring this splitting, and the Fermi
velocityrenormalisation,thequasiparticleswithopposite
FIG. 2. The magnitude (color) and direction (arrows) of the
helicity are assigned to a single, common Fermi surface, SOC vector corresponding to the point group O defined in
andmovecoherentlyalongclassicaltrajectories. Inaddi- Eqs. (50), with the g = 1.03. The SOC is shown upon
2
tion, for the quasiclassical part of the numerical calcula- thesphericalFermisurfacedefinedbytheaverageFermimo-
tions, the Fermi surface is approximated to be spherical, mentum given by ξ(kF) = 0, where ξ is the correspond-
with k being equal to the average of the Fermi sur- ing tight-binding dispersion in the absence of SOC with
F
face d|efin|ed by ξ(kF) = 0 with (t1,µ) = ( 40α, 50α). (t1,µ) = (−40α,−50α). The Fermi surface is seen from the
− − k=(1,1,1) direction.
Here, t determines the bandwidth, which must be large
1
CHAPTER7. ZEROENERGYSURFACESTATESANDTOPOLOGY
CHAPTER7. ZEROENERGYSURFACESTATESANDTOPOLOGY
(a)Td (b)C4v,µ=−50α
1 1
(a)Td (b)C4v,µ=−50α
0.8 0.8
1 1
∆t0.6 ∆t0.6
0.8 0.∆8/s ∆/s
∆t0.6 ∆t0.6 0.4 0.4
/ / 8
∆s ∆s 0.2 0.2 8
0.4 0.4
0 0
(a) BZ0.2 High Symmetry 0.2 (b)−5m0 in−|4l0(k−−)3µ0|/α −20 −10 0 −5 g02 5
0 0 F
−50 −40 −3µ0/α −20 −10 0 0 −5 (c)g0O2 ,µ=−20α 05.1+ (d)O,µ=−50α
10 10
R(c)O,µ=−20α −10 1 −2(d )O,µ=−50α 1
0.8 0.8
1 −20 1
∆t00..68 ΓM X µα/−30 ∆to00p..∆∆68/estn00.. 46FS 2 0.05 ∆∆/st00..46
∆/s −40 ∆/sclose0d.2 FS 0.2
0.4 0.4 −1
−50 1 0 0
0.2 −5 0.2 −5 0 g02 5 0 5 −1 −0.5 0 0.5 g12 1.5 2 2.5 3
−05 0 5 −01 −0.5 0g20.5−51 71.5 2 2.5 3
g2 g2
(c) Open Fermi Surface (d) ClosedFermi Surface
Figure7.2: ThetopologicalphasediagramforthepointgroupsTd,C4vandO. The1Dwinding
Figure7.12: Thetopologicaµlp/αha=se−d2ia0nWgu rhmaimtbeefroaNrretLah1seisipncodailincctuagtlaertoeaudpgsaalTpodnp,µgeCd/tα4hvpe=hal−naosdo5epO0w�. itT!hhetmr1iivDnia|wll(iktn�Fodp)in|ogl!ogyB,Z(Nbou,n⌫d)a=ry!(0,m0)a;xg|rl(eky�Fa)|n!od�a.l
nWuhmi∆tbteer00a..N68reLasisincdaliccuatlaeteadgaalponpgedthpehlaopsohepaws�eit!∆whtittm00hr..i68ivnNia|lLl(kt=�Fop)1|o;l!ocgoyBl,oZu(Nbreodu,n⌫ad)rae=arys !(g0a,pm0p)a;exdg|rln(eokyn�F-a)t|rni!voida�al.lphases Lwith NL = 0 and ⌫ taking
/ the val/ues (black, red, cyan)L= (+10, 2,+2) in (c), and (green, blue, yellow, magenta) =
phas∆esw0.i4th NL = 1; coloured area(s+g1a,pp∆5es,d+0n.74o,n-1tr)ivinial(dp)h.aNseostewtithheNsmLal�=l p0haasnedat⌫gt2ak=ing5 in (c), with ⌫ = +10 and is thus
t(ch+oel1o,uv�rael50ud,.02e+bs1l7a(0,bc�k la.1c)kT,ihnree(ddp−,)a.cr2yaN amnoe)ttee1=rts0h(ae +rse21cSmo0(O a↵l,oCl�,lut�2rp1ve,)ehd+ca=t2sb0oe).lr(02a1aicin,�stk.ng1(42ocT 0)r=,)mheiaa−�nnlpi5d5staee ridr(anmgmar(sesec7et)omne ,fr,awskbxBialt|urTlhkeec−,|⌫(i=y↵1n=e, l1atlo1+lilw)n1p,=0tlmohat(easn1g,B,de�arnini4stldla0ot))uhthuii=n�nesztoernme.s of kBTc in all plots, and the
−5 0 5 −1 � 0 1 2 3
SOC vector is normalisge2d as max|lk|=1 in the Brillouin zonge2.
approximation.
FIG.3. (a)ThehighsymmetrypointsandaxesintheBZforasimplecubiccrystal. (b)TheminimaoftheSOCvectoronthe
For the simple cubic lattice there is the possibility of open orbits, i.e. Fermi surfaces
FneIgGa.t3intvo.eepg(oahalto)eivglTieicchahaellelpFhicheiagarlsmheFaisedpyrsipmmauFrgrimorofsaaxruemcitrmetfr,ahfyaocewtrepi,iaosotwninhimn.iotttpphs1leeat=n1ncad=−undba4�xi0cc4eαl0os.l↵saie.nNtdtNotiFchtoSeeteeitBtsthhshZeehceroofteotwrnrraanninsaneisscnstiitttih(emiicoeod)nnpptalbeobnoedesctestwua(widbbeceeh)iicelnirnotecaytsraphycoeelscocrftltsaoieaovlsd.tpeelde(atybhnn.a)edWnoTBodrhphbZioeteintepbsmeao,Fnriueenirn.aFimemsde.iarianmsrFduyoiir.efcfrsaaTtmuctheheriefeaaasStccugOehaµrpeCfaa=mptcevieµtdec1sca.p=tlhoTparthso1oee.tneTntthhieaeldetermineswhetherthis
topolowgiitchaltrpivhiaalsetodpioalgorgaym,(Nfor,⌫a)n=op(0en,0a);ngdrecyloasendodFaSlpishassheowwinthinN(c=)a1nwdi(thd)arleosoppecdteifivneelyd.bWyEhqit.e(a44re);ascoilnoduirceadtaeraeagsagpappepdedphase
with tnroivni-atrlivtioaploplhoagsye,cso(Nwnniteh,cνt⌫L)ed=tatk(oi0n,eg0a)tch;hegorvteahyluearesan(tobdtlhaacelkp,BhrZeadsb,eiocswuytanihtndhe)aNrc=yaL.s(e+T=,1ho01er,wcifh2ite,thm+he2aic)FaloilSnoppi(osctd)ce,elnfioatnsniedaedld.(dgberAyeteeEcnrlm,qob.silneu(de4es,4F)yw;Sehl�cleootwlihos,remaerdlawtghaaeirynsestaaso)bg=taapipneedd with µ < t1�↵ for this
n(+on1-,t−r(i+5v,1ia+,l�7p5,,−h+a17s)e,s�in1w)(idictisrn)hy.t(shνdLte)a.tlcaNsaktsoirentue,gcottthruherieefs.mvthaTallehluupeFshsSas(seibestltaacictnlkogg,s2eµr=de.d=�,A5cic54syrticyn1aclsnho=(t)socae)sl�=,desnw5LtF(i0rft+uS↵ohc�r1⌫tp0tuiu,h=s�r−teesa+.2Blu1,wTCs0+ah.wCy2u)sesllaolisntbebtttei(taclcioien)nw,,geyadµtinhewd=lidsiit(ln54hgimtgr1eµicet=ln.<o,s�Tebthd51lu0e�F↵ese,a↵rpymmuefletiolsorsvwuuatr,slhufmiawesceaelgslebfnoetrlao)bwo=tthh.islimit. Thesamevalue
on the negativeisheclhicoaslenFSfo,rmtihne|lB(kC�FC)|,laisttnicoet,zyeireold.inTghecFlousretpdhreFersmesriomornei,sdutehrpfeaebcneadsnsdfoowrnidbttohhtehs.hsuorufladcebeormieuncthatliaorngetrhethoarndtehregap. WiththeSOCstrength
kgi∗v,enodtmfhbeiteypnheiSemnOsdueCemnppcoeiasirnoaztfmesmrceodhitneeoarps|Flleoeg(unnkn2rg�Ftdtihso)ce|oserbhonrmentoawot↵tihrnnheee,=ilnitvcnhhakFeeelBsuimgbTei.ainccn,3oadtfal(hwlpblgisio)2dg.t,pateThpnaahrtsnsaiehadcmloaScbeuaalOhctenlreudoCledorlsawbetnee6dmt|pf(witoan1u1aicin,d|rctbe1ahhiae,ccmn0nhall↵ad)eooatrterpoge8=msptre|etasar1r(kli|1ttdssB,5ho,,eTs1apcfioht,cnoonrb,o1mlawe)etoachydpgliilennui�bcggtggaey(aastd0lopplh+E,yus.a1fqtt,on�cW.2raott)thl(hnicta4ee<-uh4tlcrr)apth9i(a.htav1↵oneeti,sdg,ha0eSelw,niO0nnnhop)Coif.cacd=hhrsAdaaatlirmsip↵s(epte1anehewr,tgream0eetnrs,5lhetle0va,)oassNbuluu!errLeey-s=�ars1e,+co�nstis<ten9t↵., which is well
soiΓnppaet→chnee.aRBnTdZ:h.ckeTlo∗lhsi=needesebcFaeoltiSlsno,T−µweis1h.=e6ei(cid:18).n|tttt2Fo11h1g|piemg2ao.F(cid:19)nlao3Srdg(k(i1is8csb,|a)ttt1lh1aa,|pen,l1shg)tsoeaTrhnasmoetn,wastdriiotnikaigtogthnhrteaehbXmabet-topcw(ut⌫ao5henni1eend=)ntb-cThe0ho,sesegeo�tfoetronrfonneissuppTueytpirrlohnaehf.nledftr/oefiooai(gsgrffrdm�ius!a.ceaeptsteaerl7luiptalfeo.ren-r2nfc,pcr.eo,hthomsinvwssanva-aidshtosaplxeruiinellsisu|orevdltt!etsekteiiithaanFsnwaelegc|td,or)hrtefraNioiiatmcgnsgretLo2diatonr,Fcpe=hesanirgperigns!a�ib1.etopsmubd,enclanke4aaser/tltlnasei.ry(oendmd�saepudg)nretois2fcht.u-li,aeoe∈nrrt(gfwdeacfii{hcioc)gt0s.eaelr.o,rilc0vules7Tyaii.r.na72rhlecgl�,s,den,uul1weorlsr.ftan.0huetg3-ept⌘etori,-orde2inv.ws5fioh}agri,lateppreegdionnosni-ntrdiivciaatle, t⌫ri6=via0l,,
aries of fully ga⌫pp=ed0r,e(cid:18)ggir1oenysnwoid(cid:19)thaldni↵oner-etnritvivaall,uNes of=ttohp1e,olaongdypprhtreheassesspieocenocwtloiiivstuehrsleeydea.nBrcetleogocisaobeunedssteFhgteaehrplemaprSeigdOessCuntroffvnoae-rcctertti,ohvreiia.feslou,.rr⌫fTtahd=ceed0noc,oeorslmonraoeltdcroen-tain the parameter g2 the
ΓX3afoDnr→dwa(niMdnR)doipi::nnekkgnw∗∗,nhu==µicmhtt=(cboobotepp,hsr�ooπ−e⌫ll2,oo1t.0bggo↵)TyypT,,oghrls2ai,eoshnsg−boipdiswce=d1accnleltocmipivson(heeos1adlnp−,ys,sl1.1eotµ,rt(cid:18)Bd0ai(=te)aa1eTcg)dar�,u,+aion5smne01fil↵(cid:19)iytgs,hsLd.s.eFhe((3eStrop55roawO(e23mtpcn))nCi)ioodlsov�geoyscnnbgso/,tciino�atos=aAerhhsnltleeo,f(asocodw1oncwcr,inthngn1biiTdtev,utihmudF1nehlc)keiiadtcpcgaohasl.gsnoealaiisctnmv3npeegn(noe,nal(oteecd)petltot,).na1mc.ott.tTohinopanTTlhuloytththeafaresediiindsnsepeudtlpwrvirtesafethatiahnnteldchuedrseopiaestsEntasoiiooinqerroanani.ge,rfmln(oteetr3htoetr∆ba7set/utth)enfoil,crrookihoniwntgsmee∈rs2mni6pptvtla[ihliohac0enteaFel,ttul1teSzead.pe�1srbo]oilt,nwo-eefwintItth.iihtaheolpaenndotrhbeitssinisglseetatmoletsrsipfloert
sfuulrl→fyacgeaprpesepdeactnidrvaettloyiop.o�Wlosgh/iic�taetll,iynwdtiritcihvaitaaels,g⌫tivh=eagnt20tt,h1w.ehTseyrhseetaestmrtahinessitioFnigtIo.n/4for(roedm)e-ra(thFo)S.i�nFvoweriststhiinggoalpteeetnthooortwbriipttslheeitsrosaretdaioemsrliepnsastrhfaoemrinetteerr-sup-
The lcaoclkouorfedlinreegnioondse(sexmceluadnisngthgarteyi)tiinsdeicaastyettohactonthsterusycst- vparlesmsiionn|lkdFe|p<en�dss/�ont <thmeasxu|rlfkaFc|eveo9rry7ielnatragteioznerot-hbeiasorder
aFermtemisuisrffauclleyfgoarpwpehdicahntdhteompoilnoigmicuamllyvnaolnu-etroivfitahl,e⌫S=OC0. cpoanrdaumcteatnerceiissseceonmfporuatelldsufrofarceaorrieanntgaetioonfsedxicffeeprtetnhte sur-
tdohneepteShnGNOedrLCeTneney=hpcgeeiaa1nstr,odeiawvfilmcfei-matcethhoitenneeslolsr|iiaolcsg(ptak2teslon−FidFpts)eoS|fiosl,ohrondgmneoiedwcritannbhlplye|yalin(rEcknahqFmo−Fe.nim)eg(-|tt4.,ier4crii3)avs.ilis(anpbloco)tna.tloeczTdnuearhtlaloieat.pelSdhTa6OanhfsCodeer, tTctf(oawih1foh9ca,oni7etn1sshh,g.iei0nesgs)Fohdsrous→umrsipegyaanpmt(llro1lsmeu,,otspe1htstot,hieronr1eryaner)crabes→itxnuflehegreisenfc(,agt0ocrni,oeaun1nt=too,ior2ftti(o)re1hrarn,→e∆bjt0etua,hcltp0kte(ioa)o/s1retnr,aihs∆es0nsus,udtr0nhffr.o)fni,ars.c=w=wieAshh(sonei1(rcro1,aihete1,n,m0�trt0,ha∆se�)0ue-.a)rsf.u→r≡e
minimfouumrdii↵sezreenrotvaallounesgocfegr2t,aninamleinlyesg2in2{t0h,i0s.7p,a1r.0a3m,2e.t5e}r, c∆asssuer,f.i/n(c∆lustduirnf.gmtahxe|lhkiFgh|)siysmtmheetrsycaalxedis snur=fac(e1,s1i,n1g)l.et to
soppaecneop.anhnTeadshefcoelwrolisietnehadecahFatcSdl,oµissit.e=iedn.ctFtt1hergemmaFpaipSrskeuidsrsfattthcaoeenp,gotiel.roenag.tnicsttahiotleliytochnoenloobXnue--rttpewrdoieviernieant-l Ndtpreroigepteselnseiettorhnaratateits,iaosla,lenedilnisntetphsoleofbtoztereerdtwoh-hbeiniicalhasFrig�cgoe.sns/td�4uftoct(raa<nt)hcem-ei(sinsdu|)rlz.fkeaFrco|Teahfnoreeorrsmupa-l
inthegsciBoanlZes.diTnbhuFelkisge.silni3ngel(edst)i.ntoTFthirgiisp.lie3st(drboa)ntieaol,fsoorrbmunlkianrekv[ta0hl,ue1e.b1s]o,oufwntidhth-e �ns=/�(t1,>1,m1)ax|lkF|. Furthermore, the surface suppres-
aries of fully gapped regions with differ�ent2values of the sioAnldounegttohseelsf-acmonesipstaetnhcyofdoseusrfnaocteao↵reicetnttahteiozenrso-tbhiaeszero-
one active channel. These values are shown in table I.
conductance. Thisreflectsthefactthatthegapdoesnot
3Dwindingnumberν. Thisisdemonstratedinfigs. 3(c) bias conductance, computed with Eq. (37), is plotted in
In order to investigate how the order parameter sup- go to zero at some distance inwards from the surface for
and (d) in which the topological phase diagram is shown Fig. 4 (e) - (h). For singlet to triplet ratios in the inter-
for an open, µ = 20α, and closed, µ = 50α, Fermi val min l < ∆ /∆ < max l very large zero-bias
− − | kF| s t | kF|
surface respectively. White indicates that the system is conductanceisseenforallsurfaceorientationsexceptthe
fullygappedandtopologicallytrivial,ν =0,whereasthe two high symmetry axes, n = (1,0,0) and n = (1,1,0).
coloredregions(excludinggrey)indicatethatthesystem This is due to there being no trajectories for which ∆
isfullygappedandtopologicallynon-trivial,ν =0. Grey changes sign upon reflection for these surface orienta−-
(cid:54)
9
(a)OPSuppression,g2=0 (b)OPSuppression,g2=0.7 (c)OPSuppression,g2=1.03 (d)OPSuppression,g2=2.5
1 1 1 1
CHAPTER5. THESELF-CONSISTENTORkDERPARAMCHEATPETRER5. THESELF-CONSISTENTOkRDERPARACHMAEPTTEERR5. THESELF-CONSISTENTORkDERPARACMHAEPTTEERR5. THESELF-CONSISTENTORkDERPARAMETER
ul 0.8 ul 0.8 ul 0.8 ul 0.8
b∆ b∆ b∆ b∆
PoinCCtTG44dvvroup Ng04/2A 000 000...111444surf.r/r000∆...222000888...246000...444111 r000�b...555ul555kPoi000n...CC666tTG99944dvvro000u...888p333 N000g...04/9992A666 111000...111000...111444surf.r/r∆000...000222888...246 000...444111 r000�b...555uPl555koin000CCtT...G66644dvv999rou000p...888333Ng04/0002A...999666 000111...111000...111444surf.r/r∆000...222000888...246000...444111 r000�b...555uPl555koin000CCtT...G66644dvv999rou000p...888333Ng00004/2...A999666 000111...111000...111444surf.r/r000∆...222000888...246000...444111 r000�b...555ul555k 000...666999 000...888333 000...999666 111...111
O 0 0 0.26 0.38 0.50 0.62 0.7O4 0.86 0.098 10.10.26 0.38 0.50 0.62 0O.74 0.86 00.98 01.10.26 0.38 0.50 0.62 0.O74 0.86 00.98 01.10.26 0.38 0.50 0.62 0.74 0.86 0.98 1.1
O 0.7 0 0.20 0.3300.46 0.59 0.7O2 0.84 00..977 10.10.20 0.330 0.46 0.59 0O.72 0.84 00.7.97 01.10.20 0.3300.46 0.59 0.O72 0.84 00.7.97 01.10.20 0.3300.46 0.59 0.72 0.84 0.97 1.1
OO 12..053 00 00..0197 00..2330[10000..34]84 [100S..155u027r]fa00c..[671eOO701N1]00o..88r13m[01a10022l....099]5367 [11100..011000]..0197 00..23[3010000]..3484[1S001u..055r27]fac00[OO1..e67170N1]o00..r88m[13011a22.00l.05]..39967 [001110..11000]..0197 00..2330[10000..]3484 [1S001..u55027r]fa00c[..OO1e67701N1]00o..r8813m[011a2.002.0l5..]39967 00[111..011000..]0197 00..2330[10000..34]84 [100S..155u027r]fa00c..[671e701N1]00o..88r13m[01a002..l99]67 [111..0110]
Table5.1:ThescaledsinglettotripletratiosfortThaebdleiff5e.r1e:ntTphoeinsctagleroduspins.glFetortoTdtraipnldetCr4avtiosforTtahbeledi5ff.e1r:enTthpeosicnatlegdrosuinpgs.letFotorTtrdipalnetdrCat4ivosforTtahbeledi5ff.e1r:enTthpeosicnatlegdrosuinpgs.letFotroTtrdipalnedtrCa4tviosforthedifferentpointgroups. ForTdandC4v
tpeapThqtaaheurrelaaiesldmmaienisseeevttttsaaeeolnrrcuntoecrelrysaordtmeosiasofpppt,aroaor�cbnn�beupedudlknolskitinf,nte2toaatnt[fhn0oude,rirn1etiaeh.n1tuthe]nse,qrorownuvduoiahedgtlleiehrsmrsoet�bsaaiuusnsgtlsktaft0)/Galopy2hNnriysws[O11p12gti05atha·thcpehmemspedfiiaisenntpeapTxe.d(vh(qitiaa|h|gtalleurrktkehellaaoFieFusltdmmpaise|(e|n)i)ilsosseniee,=evttnltt)1gs0oaoaeeeo..lglfn5srr1ecuZnit3t]otcecrcel. ,BhraysaoorelbdtmTreoCsirpuasofbphepptht,,suairoasaopr�cbldngsnk�boeupioee2sdudlnknomsslsdkdiit=oinnfi,inoten2fntgoaanaOglt0nt[efethn0oudittn(e,riironns1tetgeiaoehfi.ne1otutnthe]trnrsse,irdqiretorpiownueecpvdlurotiaehlerdiegttlloet�bite,horsmnursoetloo�bskaahi7urrusns.addgtals1kt0)/Gvaeref)talNoeerrpy2.hnriy11sws[Op0512gtiatha·thcpehmemspetpeapTdfiiaihqsetnaanxeh.deu(v(rriie|l|gaatalilesktldkehmm(laieoFFunitfssps|eee|evt))t)ilttossania,=eenlolZ1ng0oorrcune..ltgfo4s1eeBcrelir1ystao]tccr. ,ChdtmaoeosiealbsoTrfpp,ptrpu,arbhoaeohgtr�cbsunin�sabeup2pleddudlskoknoiole=ssknitmsisnf,ndted2itonioaai0onfntgnt[fnah.n0Ogould7eete,riritnt1etn(ioiaeh.nsntg1teuofithe]neose,tnqrotrrrsowniurdvdietupioiaheecepdgltrltleelirehiersmtrot�sboett,�obsnuaaiuulsoonskgthl7srrkta.afdd0)/Gatal1opvy2hNeren)reerri.ysws[O11p12gti05atha·thcpehmemspedfitpeapTiaisheqntnaaxeh.d(veu(irri|e|lgtalaaliekts(keldhlmmaioFeFugntispss|e|eeev))t)itlosttnisa,=aeenlo1gZl0noorrcuen..lgtfo1s1eBecreli7rty]saotcc.r ,hCdtamoeosielbaTrsofpp,rptpu,bahreoaohgtr�cbsuinnsa�pbe2upldedsukdloknoioel=ssknitmssinf,dndite2otnioaia1onfngtnt[fna.hOgn0olud0eete,ririttn31etn(ioiaehns.tng1teuofithe]neose,tnqrotrrrsowinurdivdetpuioieeahcpedlgtrtlleelriehiertsmort�bstoe,to�bnsuaaiuuloosknshgt7lsrrkta.addfatal10)/Gopvy2herenN)reerri.ysws[O11p12gti05atha·thcpehmemspedfiiaisennxe.d(v(ii||gtallktkehloFFu(tps|he|))ilosni,=)nl1g0ooe.Z.lgf3s1ei4t]Btcc. ,haoCelbTrrpubh,ehtsuigsapldsk2oioesnmss=ddioniionfng2naOgleet.5ittn(ionstgeofieotntrrsirdietpieecplrtelrietot�bt,o123456789nulookh7rr.adda1vere)eerr.
G( 5 G( 5 G( 5 G( 5
isplottedfordifferentsurfaceorientations.Thesurfisacpelontotremdafolrndiisffsewreenpttsaulrofnagceaopraietnhtadteifionnesd.Thesuirsfapcloetnteodrmfoarldniffisersewnetpstuarfloancegoaripeantthatdioefinns.edThesuirsfapcloetnteodrmfoarldniffisersewnetpstuarlfoancegoarpieantthatdieofinns.edThesurfacenormalnissweptalongapathdefined
bfisuyrsrftnatch=eino(rg1ie,tn0ot,a0nt)oiot!en.is(T1th,h1ea,ts0ur)p�su!prrf.es(s1i,or1n�b,u1al)kff,e!catnsd(0g,[tl1a0h10n,e02csi)]nubfisg!puyrpsrtftrrn[(ae1a1tcsS1jh=,see0u0iicono],tr(nrg0o1fi)rea,itins0eo(ct[s,is1eag0net1tn)oehiN1onote!]renoitm.niorsg(oTm1bst[hn,th0e,1eaoa1,hrdtls20miuu]gr)epah�su!plltiryrosf.ea[sts(1tehs1i0oni,eo0nrs1mn]i�b),t.u1iablv)kffeTe,ei!nhcatogtensd(0g[1,t0lah10n,e02c]sbfisi)nuuyrg!psrftpnat[rr1tc(eahS=e11sji0,useno0i](crgor1,tinfe0o,tan0)roiitc,se[(a0n1esist)go1ietnotN1eh!enon]er.oiistm(nrTo1mgtoh,h[bs01enatae,1,tos0l2uhrdr)m]piu�sgu!peahrrlflti.eyoss([a1s1stti0,ehoir1o0nen�b,n]smu1ia)lt).kffibv,e!eTecaitnnhtsogd(e0g[,tl10ah10n,e02cs]i)bfisnuuyrg!psrftpntar[rtc(1eah=1eSs1ji,sen0ou0ic(or]g,rt1in0oef,tn)ra0oiits,ec(a[0nsi1estg)oei1tnotehN!1eonne]r.ioitmsn(Tor1gtomh,hbs[1en0tae,a,ot1s0hrdul2r)miup]�sgu!epahrrlflti.eyoss(a[s11stti,eh0ioro1nen0�bn,sm]u1i)atl.)kffibv,e!eTecaitnnhtsogd(e0g,t[l1ah10n0,e2c0si)]nug!pptrr[(ea11sSj1,se00iuco,]trn0of)raiise(cs[is1eget1nehN1oner]iotmnorgombs[nte0,oa1hrdl2miug]eahlltiyosa[stt1ehi0one0nsmi])t.ibveTeinhtoge
longerandthushavingmorespacetoadjusttothelocnhgaenrgainndgothrduesrhpaavrianmgemteorr.eTsphaeccehtooicaedojufsttotlhoengcheranagnidngthoursdehrapvianrgammeotreer.spTahceecthooaicdejuosfttothloencghearnagnidngthourdsehrapvainragmmeoterer.spTahceecthooaicdejuosfttothechangingorderparameter.Thechoiceof
r�bulWkviathluetwsoneaetdtsraecxtpivlaeinchinagn.nIenlsfitgh.e5Fss.u8urIrfpGa�bcupel.krr=4aets.0iosi,rsPi�broue�sllxunWkorcfvl..tiu,atsdhliTuseetd(nwshaofonoti)ersaneotde-itbcsrsvea(esicodxsdtaupiovr)sliaelnriyensceahihsnsmaogfonn.aonwslIle.renlrsafitttghhh.raeen5as.u8ntqrhrfgeua�becuaelknor=aftt0iirdot�bi,ysuirlffWke�sxurevrcifat∆slr.hu,luuediertsnewsfd.nton/ofeotasrertdun∆tbosrerubacevceflxtskiapoisvalucae=rseiinlcryhiennaa[sg∆snom.onnraIesnslmll.s/efirt∆gaht.ehl5tsas.]nu8sarutrflha�broceufeln.kr=·gat[0i∆tro�bhi,sutrleeWk�s/xuvrcp∆ifatl.u,hlaudsiettsew]sdhnbonofueotnaelrtkdntoser=baceavecxstisposi(vaul1aerasii,nlcryhie0mnaasg,snm.oen0naIeansl)ll.sesfirutght.reh5eas(.nu81ortrf,ha�fbecu1eltk,hr=0aet)0ioi,osrre�sxdurc(fel.u,1rdi,se1pdn,oafot1rrn)aoebcmvesioseautrs(iel0ryre,ass1msoun,asl2rl.ef)rathcaenthe
ratiointhebulk.Andonehasthepossibilityofaphraatsieotirnanthsietibounlkto.Athnedsuobnedohmasintahnetpcohsasnibnieliltyofaprahtaioseintrtahnesibtiuolnk.toAtnhdeosnuebdhoamstihneanptocshsiabnilniteylofaprahtaisoeintrathnesibtiuol1nk.toAtnhdeosnuebdhoamstinhae→nptocshsiabninlietylofaph→asetransitionto→thesubdominant→channel
atthesurface.Forsimplicity’ssakeal(l1ca,lc0u,la0ti)on.satiIntnhtehpesuflrooflaltocwse.in(Fgeorc)hsai-mptpe(lrihsci)htya’vtsehsbaekeeenzaeldlorcnoael-cublaitaiosantsctiohnenthsdueruffoaclcleot.waFinnogrccsehim,appctlieocristmyh’aspvseuakbteeeeandlldcowanlceiutlahtiotna0tstinh=ethsue1rf0foal−cleo.w2iFn,ogricsshimaspphtleicoristwyh’asnvseafkboeeeranlltdcohanleceulsataiomnseinsthuerffoallocweingncohrapmteraslhsa.veTbehenedone
forvaluesof(vs,vt,vm)yieldingonlyonnuemactbiveerchsfaonrinnveal.ltuheseofl(evsg,evtn,vdm)hyioellddingfoonrlyaonlleapctliovetscfhoaranvnnaeldul.escoofr(vrse,vstp,vomn)ydieltdoingtohnleyocnoelauctmivencfhsoarnivnnaellu.teasobfl(evs,Ivts,vhmo)wyieilndigngtonhlyeosnecaaclteivdecshainnngell.et to triplet ratios.
tions. For all other surface orientations this is not the thespinpolarizationaxisisfoundtobedependentonthe
case, including the high symmetry axis n = (1,1,1). singlettotripletratio, inadditiontosurfaceorientation.
Note that all lines for which ∆s/∆t < min|lkF| are The momentum-resolved zero-energy ABS for n =
degenerate, and the zero-bias conductance is zero for (1,1,1) are shown in Fig. 5 (b), computed with the bulk
∆s/∆t > max|lkF|. Furthermore, the surface suppres- value of the order parameter all the way to the surface,
sion due to self-consistency does not affect the zero-bias assuming rbulk =0.67. The tunneling parameter was set
∆
cgoontdouzcetraoncaet.sTomhiesrdeiflsteacntscethinewfaacrtdsthfarotmthtehgeaspudrfoaecsenfoort to t0 = 10−21 (or t20 = 0.1, making sure to be in the
tunneling regime), and the broadening of the energies,
the obtained gap profiles. (cid:15) (cid:15)+iδ, with δ =10 3. The disk is the projection of
Th65e Andreev bound states (AB6S5) of NCSs have intri- th←e65spherical Fermi surf−ace ont6o5 the slab surface. Black
cate structures and are spin polarized.36 This is a con- indicatesthattherearenoABSforthosemomenta,green
sSetqauteesncceororfestphoenSdOinCg tboeidnigffearnetnitsyAmnmdertereivc,bloku=nd−slt−atke. i(nΥdic,aΥtes)A=B(Sf1o,r+w1h).ichFo(rΥtkh,iΥskc)ho=ic(e+o1f,s−u1r)fa,caendoryieenlltoaw-
k k
−
branches have opposite spin polarization, and this spin tionandsinglettotripletratiothesetwotypesoftrajec-
polarization changes sign for reversed trajectories. As a toriesaretheonlyonesyieldingABS.Thisisnotthecase
result, the Andreev states carry spin current along the for lower singlet to triplet ratios, other g values, and/or
2
interface.36 The existence of a surface spin current is a othersurfaceorientations. Thentherecanexistsolutions
direct consequence of the spin-orbit coupling in the sys- to Eq. (34). Indeed, for ∆ /∆ <min l they are the
s t | kF|
tem. only solutions yielding ABS. For ∆ /∆ > max l no
s t | kF|
As an example, the momentum angle-resolved and zero-energy ABS are seen.
spin-resolvedlocaldensityofstates,N(z)(φ,(cid:15)),computed Point contact conductance spectra for g = 1.03,
2
withEq. (33), isplottedinFig. 5(a)formomentainthe t0 = 10−21 and n = (1,1,1) are shown in Fig. 5 (c).
xy-plane (parameterized by the azimuthal angle φ, the A small energy broadening (cid:15) (cid:15)+iδ with δ = 10 2
−
→
polar angle is θ = π/2), at the surface with surface nor- was used for the plot, except close to zero energy, where
mal n=(1,0,0), for g =1.03 and a self-consistent pure δ = 10 5 was used (and 2.5 times as many momen-
2 −
tripletorderparameter. Anenergybroadening(cid:15) (cid:15)+iδ tum directions in the momentum average) in order to
→
with δ = 10 2 was used, and the self-consistent order show the sharp zero-bias conductance peak. The trans-
−
parameter was computed at T =0.2Tc. Red (blue) indi- mission parameter is set to t0 = 10−21. Furthermore,
cate relative polarization for spin up (down) quasiparti- max(∆) ∆bulk +∆bulkmax(l ), and the plots are
≡ s t | kF|
cles. The spin polarization axis is along the z-axis and shifted 0.2 upwards from each other for the sake of vis-
N(x) =N(y) =0. This is true for all values of g with a ibility. The order parameters used are computed self-
2
pure triplet order parameter. However, the ABS struc- consistentlyatT =0.2T ,andwithonlyoneactivechan-
c
ture is very different for the four g values. Furthermore nel. The point contact conductance spectra differ widely
2
10
(a) Spin Polarisation (c) Tunnel Conductance (e) Band Structure
0.5 2+ 4 1
1
2
3
3 4 0.5
N 5
π/ 0 0 )/G2 678 /Tc 0
φ V 9 E
e
(
G1 −0.5
−0.5 −1 −0.5 0 0.5 1 −2− 0 −1 −0.5 0 0.5 1 −−11 −0.5 0 0.5 1
ǫ/max(∆) eV/max(∆) kk/|k |
2 F
(b) Andreev Bound States (d) Topological Invariants (f) Lowest Positive Band
1 1 1 0.2+
0.5 0.5 0.5
| (−1,+1) | (−1) |
F F F
/k| 0 /k| 0 (0) k/| 0 0.1
k∥2 (+1,−1) k∥2 (+1) kk2
−0.5 −0.5 −0.5
−1 −1 −1 0
−1 0 1 −1 0 1 −1 0 1
k1∥/|kF| k1∥/|kF| k1k/|kF|
FIG. 5. All plots are for the cubic point group O with g = 1.03. (a) N(z)(k,(cid:15)), defined in Eq. (33), which is a measure
2
of the spin polarization along the z-axis. It is shown for a self-consistent pure triplet order parameter and for momentum
directions in the xy plane (i.e. θ = π/2), at the surface with the surface normal n = (1,0,0). (b) Momentum-resolved ABS
at zero energy computed assuming a constant order parameter with rbulk = 0.67. The disk is the projection of the Fermi
∆
surface onto the slab surface with n=(1,1,1). Green regions corresponds to ABS for which (Υ ,Υ )=(+1, 1), and yellow
k k
−
regions to (Υ ,Υ ) = ( 1,+1). Momenta of trajectories not yielding ABS are colored black. (c) Point contact conductance
k k
−
spectra along n = (1,1,1) for self-consistent order parameters (the numbers refer to columns for rbulk in table I), and with
∆
t0 = 10−12. (d) The topological invariant N(111), with r∆bulk = 0.67, where light green/blue corresponds to N(111) = ±1, and
white to trivial topology. (e) The surface band structure with k1(cid:107) =0, and r∆bulk =0.67. (f) The lowest positive eigenvalues of
H for self-consistent order parameter with rbulk =0.67. Black regions correspond to zero energy. Dashed circles in (d) and
eff ∆
(f) show for comparison the projection of the spherical Fermi surface used in the quasiclassical calculations.
between surface orientations and the values of g , in ad- non-trivial values. In Fig. 5 (e) the surface band struc-
2
dition to the less pronounced difference between singlet ture is shown for r∆bulk = 0.67 along the k2(cid:107)-axis with
to triplet ratios. The most striking difference is the ap-
pearance of zero-bias conductance peaks (ZBCPs) which k1(cid:107) = 0, and L = 1.3·104 layers. N(111) (cid:54)= 0 gives rise
to singly degenerate zero-energy flat bands, one on each
are present for all singlet to triplet ratios in the inter-
surface, with the corresponding wavefunctions decaying
val min l < ∆ /∆ < max l provided there are
| kF| s t | kF| exponentiallyintothebulk. Thesurfacemomentaofthe
trajectories with sign[∆ (k)]= sign[∆ (k)].
− − − zero-energyflatbandsaregivenbyN(111)(k )=0,which
In Fig. 5 (d) the topological invariants N(111) and can be seen in Fig. 5 (f) where the lowest p(cid:107)osi(cid:54)tive eigen-
W(111) areplottedforr∆bulk =0.67. However,W(111) =1, value of Heff [see Eq. (49)] is plotted for self-consistent
i.e. trivial, for this choice of parameters, and trivial order parameter. Note that the zero-energy flat-bands
topology is colored white. Light green/blue corresponds are given by the projection of non-trivial values of the
to N(111) = 1. The dashed circle indicates the projec- 1D winding number.
±
tion of the spherical Fermi surface used in the quasiclas-
sical calculations, i.e. Fig. 5 (a) - (c). Even though the
Fermi surface is not spherical, it is clear that the zero-
energy ABS are directly related to the topology. As is
shownforthetetragonalpointgroupC below,theABS
4v
given by solutions to Eq. (34), for the relevant values of
B. The Tetragonal Point Group C
(Υ ,Υ ), is directly related to the Z invariant being 4v
k k 2
non-trivial (i.e. W = 1).
(111)
−
Zero-energy states are present in the band structure Tonext-nearestneighborsinthesumoverBravaislat-
whenevertheaforementionedtopologicalinvariantshave tice sites8 the SOC vector corresponding to the tetrago-