Domain walls in superconductors: Andreev bound states and tunneling features

2.7 MB·English

by  S. P. Mukherjee

Domain walls in superconductors: Andreev bound states and tunneling features
S. P. Mukherjee and K. V. Samokhin
Department of Physics, Brock University, St. Catharines, Ontario, Canada L2S 3A1
Domainwallscanbeformedinsuperconductorswithadiscretedegeneracyofthegroundstate,for
instance,duetothebreakingoftimereversalsymmetry. Westudyall caseswheretheformation of
domainwallsispossibleinatetragonalsuperconductorwiththepointgroupD4h. Wediscussboth
5 triplet and mixed singlet order parameters. It is found that in all cases the domain walls support
1 subgap Andreev bound states, whose energies strongly depend on the direction of semiclassical
0 propagation. The bound state contribution to the density of quasiparticle states exhibits peculiar
2 features, which can be observed in tunnelingexperiments.

r
a PACSnumbers: 74.20.-z,74.55.+v
M
8 I. INTRODUCTION pairing.8Inbothcasesthegroundstateistwofolddegen-
1 erate and, therefore, DWs can exist.

] The order parameter inhomogeneities in unconven- Other examples of superconducting systems which
n tional superconductors and superfluids, such as the admit the existence of DWs are d + is and d + id
o
Abrikosovvortices,twinandgrainboundaries,interfaces, states. In these cases the pairing is singlet and pre-
c
- magnetic impurities, and domain walls (DWs) are the dominantly d-wave, with an admixture of a subdomi-
r breeding ground for gapless localized quasiparticles with nant component of a different symmetry, s-wave or an-
p
u distinct topological properties.1 On the other hand, the other d-wave, in the bulk. The possibility of such fully
s verypresenceofDWsisadirectevidenceofanunconven- gappedstates,whichbreakbothtime reversalanda dis-
t. tionalnatureofthepairing,becausetheDWscanappear crete lattice symmetry, had been first discussed theoret-
a only in a superconductor with two or more distinct de- ically for high-Tc cuprates,9 but it was later ruled out
m
generategroundstates,whichtransformoneintoanother basedonphase-sensitivetestsofpairingsymmetry.10 Re-
- by some discrete symmetry operations, e.g. by time re- cently, there has been a renewal of interest in the d+is
d
versal. This is possible if the superconducting order pa- statesinthecontextofiron-basedsuperconductors,both
n
o rameter has more than one component, i.e. corresponds theoretically11andexperimentally.12VariousotherTRS-
c to either a multi-dimensional irreducible representation breaking states have also been discussed for URu2Si2
[ of the crystal point group or to a mixture of different, (Ref. 13), Ba1−xKxFe2As2 (Ref. 14), UPt3 (Ref. 15),
e.g., one-dimensional (1D), representations. dopedgraphene(Ref. 16), undopedbilayersilicene(Ref.
2
v One of the best-studied examples of superconductors 17), SrPtAs (Ref. 18), NaxCoO2·yH2O (Ref. 19), and
0 or superfluids which can support DWs is a spin-triplet PrOs4Sb12 (Ref. 20).
7
chiral p-wave state, which might be realized in Sr RuO Motivated by the growing evidence of the existence of
5 2 4
1 (Ref. 2) or in thin films of superfluid 3He-A (Ref. 3). multiply degenerate, in particular TRS-breaking, states
0 The order parameter in the chiral p-wave state corre- in unconventional superconductors, in this paper we
. sponds to the odd two-dimensional (2D) representation, study how the DWs separating different ground states
1
E , of the tetragonal point group D and is propor- affect the spectrum of fermionic quasiparticles. The or-
0 u 4h
5 tional to kx ±iky. DWs separate the opposite chirality der parameter variation across a DW creates a quasi-
1 states, k +ik and k −ik , which break time rever- particle bound state, resulting in characteristic features
x y x y
: sal symmetry (TRS) and have the same energy in the in the density of states (DOS), which can be detected,
v
absence of external magnetic field. One possible mech- e.g., in tunneling measurements. We focus on the quasi-
i
X anism of the DW formation is that domains of opposite 2Dtetragonalcase,whichis applicableto bothSr RuO
2 4
r chirality appear spontaneously in different parts of the andtheiron-basedmaterials. Whilemostoftheprevious
a
system upon cooling across the phase transition, due to worksconsideredonly the DWbound state spectrumfor
the sample inhomogeneity. Alternatively, an increase in the chiral p-wavestate,21,22 we also study the d+is and
the gradientenergy near the DW might be compensated d+id states, as well as nonchiral p-wave states. While
bythecreationoflow-energyquasiparticlesboundtothe we consider mixed pairing states in the bulk, another
DW, which is particularly effective in one-dimensional possibility is thata subdominantorderparameteris sta-
systems.4 Experimentally, there is evidence of DWs in bilized only near a surface, in which case the spectrum
Sr RuO , both from the magnetic field modulation of of the surface bound states was found in Ref. 23. The
2 4
the single-face Josephson critical current5 and from the paper is organized as follows. In Sec. II, we derive a
anomalous hysteresis of voltage-current characteristics,6 generalexpressionforthe DWboundstate energyinthe
and also in slabs of superfluid 3He (Ref. 7). Nonchiral semiclassical (Andreev) approximation. In Sec. III, we
states with the order parametersproportionalto k ±k list all possible cases which admit the DW formation for
x y
ork ,k arealsophenomenologicallypossible for p-wave the point group D . In Secs. IV and V, we discuss the
x y 4h
2
DW eikFr andaslowlyvaryingenvelopefunctionΨ(x),which
satisfies the equation
y
k
F −ivF,x∇x ∆kF(x) Ψ(x)=EΨ(x), (2)
∆ − ∆ + (cid:18) ∆∗kF(x) ivF,x∇x (cid:19)
θ where v = v cosθ and v is the Fermi velocity. Us-
F,x F F
x
ing the sharp DW model, Eq. (1), we find that the
bound-state wave function has the form Ψ (x)∼e∓κ±x
±
at x > 0 (x < 0), where κ = Ω /|v |, with Ω =
± ± F,x ±
|∆ |2−E2 ≥ 0. In the general pairing case, the en-
±
velopewavefunctionhas fourcomponents,but inallthe
p
modelsweconsiderbelowthespinchannelsdecoupleand
the equations can be reduced to a two-component form.
FIG. 1: (Color online) Schematic diagram showing the DW,
Substitutingthewavefunctionsintotheboundarycon-
thesemiclassicaltrajectoryofaBogoliubovquasiparticle,and
dition Ψ(−0)= Ψ(+0), we arrive at the following equa-
theorder parameters ∆± on both sides.
tion for the bound state energy:
∆ ∆
+ −
triplet and mixed singlet cases, respectively, and calcu- = , (3)
E−iv κ E+iv κ
late the spectrum of the bound states as well as their F,x + F,x −
contribution to the quasiparticle DOS. Finally, we sum-
which can be represented in the form
marize our results in Sec. VI. Throughout the paper we
use the units in which ~=e=c=1. E˜+iΩ =γ(E˜−iΩ ), (4)
− +
where E˜ = Esgn(v ) and γ = ∆ /∆ = γ +iγ .
F,x − + R I
II. ANDREEV BOUND STATES Taking the real and imaginary parts of Eq. (4), we ob-
tain:
We consider a quasi-2D superconductor with a cylin-
1−γ |γ|2−γ
drical Fermi surface and assume that the DW is along Ω = RE˜, Ω = RE˜. (5)
+ −
the y-axis, so that the superconducting domains are ex- γI γI
tended on both sides of the DW to infinity along the x-
The first of these equations gives
axis. The orderparametervariationtakesplace within a
trieogniowneoufsweitdhtehAξdndnreeaervtahpepDroWxi.mTahtiroonu,g2h4,o25utinouwrhdicehritvhae- E˜ =sgn(E˜) 1 |∆+|, (6)
1+β2
Bogoliubov quasiparticles propagate along semiclassical
trajectories characterized by the Fermi-surface wavevec- where p
tors k = (k ,k ) = k (cosθ,sinθ). This approxi-
F F,x F,y F
1−γ
mation is justified since the scales of variation of the su- R
β = . (7)
perconductingorderparameter,includingtheDWwidth γI
ξ , are much greater than the Fermi wavelength k−1.
d F The sign function that appears on the right-handside of
For a given direction of semiclassical propagation, the
Eq. (6) can be assigned using the condition that Ω ≥
gap function takes the form ∆k (x) =∆(θ,x), with the +
F 0, from which we find sgn(E˜) = sgn(β). Therefore, we
θ-dependence determined by the pairing symmetry, see
finallyobtainfortheAndreevboundstate(ABS)energy:
Sec. III below. The unavailability of an exact analytical
expressionforthex-dependenceoftheDWorderparam-
1
eterleadstovariousapproximationschemes,seeRefs. 26 E (θ)=|∆ (θ)| sgn[β(θ)cosθ]. (8)
b +
1+β2(θ)
and 27 for the chiral p-wave case. We use a simple, an-
alytically treatable, model in which the DW is assumed
Thereisanadditionalcponstraintcomingfromthesecond
to be a sharp boundary of zero width located at x = 0.
of the expressions (5). Since Ω ≥0, we have
Then the gap function has the following form: −
sgn(|γ|2−γ )sgn(1−γ )=1. (9)
∆kF(x)=∆+(θ), x>0, R R
(1)
∆kF(x)=∆−(θ), x<0, Thus,foreachdirectionofsemiclassicalpropagationsat-
isfying the condition (9), there is one ABS with the en-
i.e. it is characterizedby two different complex numbers ergy given by Eq. (8). For singlet pairing, the gap func-
along each semiclassical trajectory, see Fig. 1. tion is even in momentum and ∆ (θ + π) = ∆ (θ),
± ±
In the Andreev approximation, the quasiparticle wave while for triplet pairing, the gap function is odd and
function is a product of a rapidly oscillating plane wave ∆ (θ+π) = −∆ (θ). It follows from Eqs. (7) and (8)
± ±
3
that β(θ+π)=β(θ) and,therefore, E (θ+π)=−E (θ)
b b
TABLE I: Superconducting states corresponding to the irre-
in all cases.
The ABS wave function is exponentially localized on duciblerepresentationEu ofthepointgroupD4h withstrong
spin-orbit coupling. The first column lists the possible su-
bothsidesoftheDW,withthecharacteristicscalesgiven
perconducting classes, where R stands for time reversal (the
by κ−1(θ) = |v |/ |∆ |2−E2. The sharp DW ap-
± F,x ± b notations are the same as in Ref. 26), the second and third
proximation is valid for those directions of semiclassical columns show the number of degenerate ground states and
p
propagation for which the DW width ξd is smaller than thespecificformsoftheorderparameterinthetwodomains,
κ−1. This condition is strongly angle-dependent and, in and thevector d(k) is given in the last column.
±
particular, fails for “grazing” trajectories, for which θ
D4h degeneracy η d(k)
is close to ±π/2, therefore, v → 0 and the Andreev
F,x
approximation itself is inapplicable. However, for most D4(E) 2 (1,i) zˆ(kx+iky)
directions of k , one can use the following rough esti- (1,−i) zˆ(kx−iky)
F
mate: κ−±1 & vF/∆0 ∼ ξ, where ∆0 is a characteristic D4(C2)×R 2 (1,0) zˆkx
valueofthe gapandξ isthe superconductingcorrelation (0,1) zˆky
length. Therefore,the sharpDWmodelisquantitatively D2(C2′)×R 2 (1,1) zˆ(kx+ky)
justified if ξd .ξ .κ−±1. (1,−1) zˆ(kx−ky)
TheABS’scontributetothequasiparticleDOS,which
can be probed in tunneling experiments. The momen-
tum along the y-axis is conserved and the bound-state which the formation of DWs is possible. Note that only
DOSper unit lengthof the DW is givenby the following the states with η ∝ (1,±i) break TRS and do not have
expression: gap nodes in the bulk, while the other two states pre-
serve TRS but break some discrete crystal symmetries
kF dk and have bulk gap nodes.
N (E)= F,y δ[E−E (k )],
b 2π b F In the singlet case and quasi-2D geometry (the latter
Z−kF kXF,x implies kz-independent basis functions), TRS breaking
and DWs are possible only in the states formed by a
where kF,x =± kF2 −kF2,y. Making the change of vari- combinationoftwopurestatesbelongingtotwodifferent
ables kF,x =kFqcosθ, kF,y =kF sinθ, we obtain: 1D irreducible representations.28 The semiclassical gap
function can be written as
2π
Nb(E)=NFvF dθ|cosθ|δ[E−Eb(θ)], (10) ∆kF(x)=ψ1(x)ϕ1(kˆF)+ψ2(x)ϕ2(kˆF), (12)
Z0
where ψ are the order parameters and ϕ are the
1,2 1,2
where NF is the Fermi-surface DOS in two dimensions. basis functions of the 1D representations, satisfying
ϕ (kˆ ) = ϕ (−kˆ ). We consider the following three
1,2 F 1,2 F
possibilities: a mixture of B and B representations,
1g 2g
III. NONUNIFORM STATES IN A
for which
TETRAGONAL SUPERCONDUCTOR
ϕ =kˆ2 −kˆ2 , ϕ =2kˆ kˆ , (13)
1 F,x F,y 2 F,x F,y
The point group D has ten irreducible representa-
4h a mixture of B and A representations,for which
1g 1g
tionsintotal(consideringevenaswellasoddrepresenta-
tions),ofwhicheightare1Dandtwoare2D.Theforma- ϕ1 =kˆF2,x−kˆF2,y, ϕ2 =1, (14)
tion of DWs is possible only for those superconducting
anda mixture ofB and A representations,for which
classes which are degenerate with respect to some dis- 2g 1g
crete symmetry.8,26 We consider the cases of singlet and ϕ =2kˆ kˆ , ϕ =1. (15)
1 F,x F,y 2
triplet pairing separately.
The superconducting state in the bulk is given by
Triplet order parameter is described by a spin vector
d(k). We assume that the spin-orbit coupling fixes the (ψ1,ψ2) ∝ (∆1,±i∆2), where ∆1 and ∆2 characterize
direction of d(k) along the z axis. Since the 1D rep- the strengths of the individual pairing components. The
resentations cannot support DWs, we consider the odd first of the above states is referred to as a dx2−y2 ±idxy
irreducible representation E , which corresponds to p- state, while the last two are dx2−y2 ± is and dxy ± is
u
states, respectively. All of these states are fully gapped
wave pairing. The semiclassical gap function takes the
in the bulk.29
following form:
∆kF(x)=η1(x)kˆF,x+η2(x)kˆF,y, (11)
where η = (η ,η ) is the two-component order param- IV. TRIPLET STATES
1 2
eter, which transforms like a vector in the xy plane,
and the basis functions of Eu are given by kˆF,x = cosθ, In this section we discuss the ABS spectra and the
kˆ = sinθ. In Table I we show all stable states for corresponding DOS for the three triplet p-wave states
F,y
4
1 χ=0 1.5 χ=0
0.8
0.6
0.4
∆0 0.2 1
E / b-0.20 Nb
-0.4
0.5
-0.6
-0.8
-1
1 χ=π/4
0
0.8 1.5
χ=π/4
0.6
0.4
∆0 0.2
E / b-0.20 1
-0.4 Nb
-0.6
0.5
-0.8
-1
1 χ=π
0.8
0.6 0
0.4 1.5 χ=π
∆0 0.2
E / b-0.20 1
-0.4
-0.6 Nb
-0.8
-10 0.2 0.4 0.6 0.8 1 0.5
θ/2π
FIG.2: (Color online) Thevariation of theABSenergy with 0-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
the direction of the quasiparticle propagation for domains E / ∆
0
with η∝(1,±i), for different values of χ.
FIG. 3: (Color online) The energy dependence of the ABS
contributiontotheDOS(inarbitraryunits)fordomainswith
listed in Table I. η∝(1,±i),for different values of χ.
A. η∝(1,±i) all θ and χ. For the ABS energy we then obtain:
E (θ) χ χ
In the chiral p-wave state, the order parameters on b =cos θ− sgn cosθsin θ− , (17)
both sides of the DW have the following form: ∆0 2 2
(cid:16) (cid:17) h (cid:16) (cid:17)i
seealsoRef. 22. InFig.2weplotthis lastexpressionfor
η =∆ (1,i), x<0,
0 (16) three representative values of the common phase differ-
η =∆0(1,−i)eiχ, x>0, ence χ. The ABS energy has discontinuities at the spe-
cialdirectionsofsemiclassicalpropagation: atθ =±π/2,
where ∆ is the gap magnitude and χ is the common
0 i.e. for the quasiparticles moving parallel to the DW, in
(Josephson) phase difference between the domains. The
which case the Andreev approximation is not applica-
latter has to be included in order to satisfy the current
ble, and also at θ = χ/2 and θ = χ/2+π, for which
conservationacrosstheDW.Itsvaluedependsonthemi-
∆ = ∆ and, therefore, the DW is “invisible” to the
+ −
croscopicdetailsofthesystem,seeAppendixA.Without
quasiparticles.
loss of generality we assume 0≤χ≤π.
Next, we calculate numerically the ABS contribution
It follows from Eqs. (1), (7), and (11) that ∆ (θ) =
− to the DOSper unitlengthofthe DWby usingEq. (10)
∆ eiθ, ∆ (θ)=∆ eiχe−iθ, and
0 + 0 and plot it in Fig. 3. One notable feature of them is the
χ presence of van Hove singularities at the gap edge, i.e.
β =tan θ− . at E = ±∆ for χ 6= π. These singularities are due to
2 b 0
(cid:16) (cid:17) the vanishing of the slope of Eb(θ) at θ = χ/2 and θ =
Note that the condition (9) is automatically satisfied, χ/2+π. For χ=π the slope vanishes for θ =π/2,3π/2,
since |γ|2 −γ = 1−γ , and Eq. (8) can be used for wherecosθ =0andthevanHovesingularitiesareabsent.
R R
5
1 6
χ=0 χ=0
0.8
0.6 5
0.4
∆0 0.2 4
E / b-0.20 Nb3
-0.4
2
-0.6
-0.8
1
-1
0.6 χ=π/4 0
6
χ=π/4
0.4
5
0.2
∆E / 0b 0 4
-0.2 Nb3
-0.4
2
-0.6
1
0.4
0
6
0.2 χ=π/2
∆0 5
E / b 0 4
-0.2
Nb3
-0.4 χ=π/2
2
0 0.2 0.4 0.6 0.8 1
θ/2π
1
FIG.4: (Color online) Thevariation of theABSenergy with 0-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
the direction of the quasiparticle propagation for domains E / ∆
0
with η ∝ (1,0) and (0,1), for different values of χ. Regions
boundedbyblueverticaldottedlinesandnotcontainingany
FIG. 5: (Color online) The energy dependence of the ABS
bound states correspond to the directions of propagation for
contributiontotheDOS(inarbitraryunits)fordomainswith
which thecondition (9) is not satisfied. η ∝(1,0) and (0,1), for different values of χ. For χ=0 the
DOShas a delta-function singularity at E =0.
In fact, in the latter case, the DOS is independent of
energy.
and, therefore, there are no bound states. The case of
χ=π/2, when γ =0, is an exception: the ABS’s exist
R
at all θ.
The ABScontributiontothe DOSis plottedin Fig.5.
B. η∝(1,0) or (0,1)
For χ = 0, the DOS has a delta-function singularity at
zeroenergyduetotheflatnessofthecorrespondingABS
The order parameters in this nonchiral p-wave state
dispersion. This can be understood as follows. At χ =
have the form
0, the gap functions on both sides of the DW are real,
∆ (θ) = ∆ cosθ and ∆ (θ) = ∆ sinθ. If sin2θ < 0
η =∆ (1,0), x<0, − 0 + 0
0 (18) then ∆ and ∆ have opposite signs. In this case, the
η =∆ (0,1)eiχ, x>0, + −
0 Andreev equations are equivalent to the Witten model
ofsupersymmetricquantummechanicsandthe ABScan
therefore, ∆ (θ)=∆ cosθ, ∆ (θ)=∆ eiχsinθ, and
− 0 + 0 be shown to have exactly zero energy, see Ref. 25 and
cosχ−tanθ the references therein and also Ref. 30. The condition
β = sgn(∆ ∆ )=−1issatisfiedonlyatπ/2<θ <π andat
sinχ + −
3π/2<θ <2π, where the ABS energy is zero, as shown
We calculate numerically the ABS spectra using Eq. (8) in the top panel of Fig. 4.
for different representative values of χ and present the Adding the phase χ makes the gap functions ∆ in-
±
results in Fig. 4. Note that for some directions of semi- trinsically complex, in which case the analogy with the
classical propagation the condition (9) is not satisfied supersymmetric quantum mechanics breaks down and a
6
1 3
χ=0 χ=0
0.8
0.6
0.4
2
∆0 0.2
E / b-0.20 Nb
-0.4
1
-0.6
-0.8
-1
1
χ=π/4 0
0.8 3
χ=π/4
0.6
0.4
∆E / 0b-00..220 2
Nb
-0.4
-0.6
1
-0.8
-1
0.8
χ=π/2
0.6
0
0.4 3 χ=π/2
0.2
∆0
E / b-0.20 2
-0.4 Nb
-0.6
-0.80 0.2 0.4 0.6 0.8 1 1
θ/2π
FIG.6: (Color online) Thevariation of theABSenergy with 0-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
the direction of quasiparticle propagation for domains with E / ∆
0
η ∝ (1,±1), for different values of χ. Regions bounded by
blueverticaldottedlinesandnotcontaininganyboundstates
FIG. 7: (Color online) The energy dependence of the ABS
correspondtothedirectionsofpropagationforwhichthecon-
contributiontotheDOS(inarbitraryunits)fordomainswith
dition (9) is not satisfied. η∝(1,±1),fordifferentvaluesof χ. Forχ=0theDOShas
a delta-function singularity at E =0.
simple physical explanation of the spectra is no longer
available. Our results show that as χ increases,the ABS therefore ∆ (θ) = ∆ (cosθ + sinθ), ∆ (θ) =
− 0 +
bands acquire curvature and also the range of the ABS ∆ eiχ(cosθ−sinθ), and
0
existencegetsbroaderuntil,eventually,theinternalgaps
completely disappear for χ = π/2, see the middle and cosχ−tan(π/4−θ)
β = .
bottom panels of Fig. 4. At 0<χ < π/2, there are two sinχ
setsofthevanHovesingularitiesintheDOS,correspond-
ing to the vanishing of the dispersion slope at the ABS The ABS spectra and contribution to the DOS are cal-
band centers, and also two internal gaps, corresponding culated numerically for different representative values of
totheabsenceoftheABSatcertainenergies. Finally,at χ and plotted in Figs. 6 and 7, respectively.
χ=π/2, the four van Hove singularities merge into two. Similarly to the previous case, the condition (9) is not
satisfied for some values of θ (except at χ = π/2), re-
sulting in the spectra consisting of several disconnected
parts and the gaps in the DOS. The van Hove singu-
larities are again present, degenerating for χ = 0 into
C. η∝(1,±1)
a delta-function singularity at zero energy. The latter’s
origin is explained by the fact that at χ = 0 the gap
The order parameters in this nonchiral p-wave state functions ∆ and ∆ are real and have opposite signs
+ −
have the form if cos2θ < 0, in which case the ABS has zero energy for
π/4θ < 3π/4 and 5π/4θ < 7π/4, as shown in the top
η =∆0(1,1), x<0, (19) panelofFig. 6. Atnonzeroχ, ∆+ and∆− become com-
η =∆ (1,−1)eiχ, x>0, plex,whichresultsintheABSbandsacquiringdispersion
0
7
1 8
χ=0 χ=0
0.8
7
0.6 ρ=0.2 ρ=0.2
0.4 ρ=0.8 6 ρ=0.8
max0.2 5
∆E / b -0.20 Nb4
-0.4 3
-0.6
2
-0.8
-1 1
1 χ=π/2
0
0.8 8
χ=π/2
0.6 ρ=0.2
ρ=0.8 7
0.4
ρ=0.2
max0.2 6 ρ=0.8
∆E / b -0.20 5
-0.4 Nb4
-0.6 3
-0.8
2
-1
0.8 χ=π 1
0.6
0
0.4 ρρ==00..82 8 χ=π
max0.2 7
∆E / b -0.20 6 ρρ==00..28
5
-0.4 Nb4
-0.6
3
-0.80 0.2 0.4 0.6 0.8 1
2
θ/2π
1
FIG.8: (Color online) Thevariation of theABSenergy with 0-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
the direction of quasiparticle propagation for the domains E / ∆
dx2−y2±idxy,fordifferentvaluesofχandρ. Thinredlineis max
for ρ = 0.2 and thick blue line is for ρ = 0.8. For χ = 0 the
FIG. 9: (Color online) The energy dependence of the ABS
curvessuperpose on each other.
contribution totheDOS(in arbitrary units) for thedomains
dx2−y2±idxy,fordifferentvaluesofχandρ. Thinredlineis
for ρ=0.2 and thick blue line is for ρ=0.8. For χ=0 they
superpose on each other.
andspreading to other anglesof propagationuntil filling
the entire angular range at χ=π/2.
A. dx2−y2±idxy
In this case, we obtain from Eqs. (1), (12), and (13):
V. MIXED SINGLET STATES ∆ (θ)=∆ cos2θ+i∆ sin2θ,
− 1 2 (21)
∆ (θ)=(∆ cos2θ−i∆ sin2θ)eiχ.
+ 1 2
In this section we discuss the ABS spectra for TRS-
We see that the gap magnitude is the same in both do-
breaking mixtures of different singlet states. In all three
mains and given by
cases,see Sec. III, the orderparameterson both sides of
a sharp DW can be written as
|∆ |=|∆ |=∆ (θ)= ∆2cos22θ+∆2sin22θ,
+ − bulk 1 2
q
(ψ1,ψ2)=(∆1,i∆2), x<0, (20) which attains its maximum value, ∆max = ∆1, at θ =
(ψ ,ψ )=(∆ ,−i∆ )eiχ, x>0, nπ/2, and its minimum value, ∆min =∆2, at θ =(2n+
1 2 1 2
1)π/4, where n = 0,1,2,3. Thus, the bulk gap has an
anisotropicmagnitudewithoutnodes,whilethephasesof
where ∆ ,∆ > 0 are constant amplitudes of the two
1 2 the gapfunction are differentin the twodomains: ∆− =
singlet channels (without loss of generality, we assume ∆ eiφ and ∆ =∆ eiχe−iφ, where
bulk + bulk
that ∆ ≤ ∆ ) and χ is the common phase difference,
2 1
see Appendix B. φ(θ)=arctan(ρtan2θ)
8
and 1 χ=0
0.8
ρ= ∆2, 0≤ρ≤1. (22) 0.6 ρρ==00..82
0.4
∆
1 max0.2
From Eq. (7) we obtain: ∆E / b -0.20
χ -0.4
β =tan φ− , (23) -0.6
2
-0.8
(cid:16) (cid:17)
-1
therefore,the expression(8)for the ABSspectrumtakes 1
χ=π/2
the following form: 0.8
0.6 ρ=0.2
ρ=0.8
E (θ) χ 0.4
∆b = ρ2+(1−ρ2)cos22θcos φ− 2 max0.2
max p ×sgn cosθsin φ(cid:16)− χ (cid:17), (24) ∆E / b -0.20
2 -0.4
h (cid:16) (cid:17)i
-0.6
which is shown in Fig. 8 for different representative val-
-0.8
uesofρ andχ. Althoughthe ABSexistforalldirections
-1
of semiclassical propagation, the spectra are not contin-
0.6 χ=π
uous, due to the presence of the sign functions in Eq.
(24). Insomecases,relativelysimpleexplicitexpressions 0.4 ρρ==00..82
for the ABS energy can be obtained: for χ=0, we have max0.2
∆ 0
Eb(θ) E / b-0.2
=cos2θsgn(sinθ),
∆max -0.4
-0.6
which does not depend on ρ, and for χ=π,
0 0.2 0.4 0.6 0.8 1
θ/2π
E (θ)
b
=−ρsin2θsgn(cosθcos2θ).
∆
max FIG.10: (Coloronline)ThevariationoftheABSenergywith
the direction of quasiparticle propagation for the domains
TcohrereDspOoSn,dsihnogwtnointhFeigv.a9n,ishhaisngthoefvathneHdoivsepseirnsgiounlasrliotipees., dx2−y2 ±is, for different values of χ and ρ. Thin red line
is for ρ=0.2 and thickblue line is for ρ=0.8.
For χ = 0, the singularities occur at |E| = 1, while for
χ=π – at |E|=ρ.
and ρ is given by Eq. (22). Therefore, we obtain for the
ABS spectrum:
B. dx2−y2±is
E (θ) ρ2+cos22θ χ
b
= cos φ−
According to Eqs. (1), (12), and (14), the supercon- ∆max s ρ2+1 2
(cid:16) (cid:17)
ductingdomainsaredescribedbythefollowinggapfunc- χ
×sgn cosθsin φ− , (26)
tions: 2
h (cid:16) (cid:17)i
∆−(θ)=∆1cos2θ+i∆2, (25) which is plotted in Fig. 10. One can see that the ABS
∆ (θ)=(∆ cos2θ−i∆ )eiχ. exist for all θ, but their spectra are not continuous. The
+ 1 2
DOSisshowninFig. 11. Inparticular,inthecaseofχ=
The gap magnitudes are the same in both domains: π, we find from Eq. (26) that |E |/∆ = ρ/ ρ2+1,
b max
i.e. does not depend on θ, which results in the delta-
|∆+|=|∆−|=∆bulk(θ)= ∆21cos22θ+∆22. function singularities in the DOS. p
q
The bulk gap magnitude is anisotropic, without nodes.
It attains its maximum value, ∆ = ∆2+∆2, at
max 1 2
θ = nπ/2, and the minimum value, ∆ = ∆ , at θ =
min 2
p
(2n+1)π/4, where n=0,1,2,3. C. d ±is
xy
The parameter β is given by Eq. (23), where
ρ According to Eqs. (1), (12), and (15), the supercon-
φ=arctan ductingdomainsaredescribedbythefollowinggapfunc-
cos2θ
(cid:16) (cid:17)
9
8 χ=0 1 χ=0
0.8
7
ρ=0.2 0.6 ρ=0.2
6 ρ=0.8 0.4 ρ=0.8
5 max0.2
Nb4 ∆E / b -0.20
3 -0.4
-0.6
2
-0.8
1 -1
1
0 χ=π/2
8 0.8
χ=π/2
0.6 ρ=0.2
7 0.4 ρ=0.8
ρ=0.2
6 ρ=0.8 max0.2
∆ 0
5 E / b-0.2
Nb4 -0.4
3 -0.6
-0.8
2
-1
1
0.6 χ=π
08 χ=π 0.4 ρρ==00..28
7 max0.2
65 ρρ==00..28 ∆E / b -0.20
Nb4 -0.4
3 -0.6
0 0.2 0.4 0.6 0.8 1
2 θ/2π
1
0-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 FIG. 12: (Color online) The variation of the ABS energy
E / ∆ with the direction of quasiparticle propagation for the do-
max mains dxy±is, for different values of χ and ρ. Thin red line
is for ρ=0.2 and thickblue line is for ρ=0.8.
FIG. 11: (Color online) The energy dependence of the ABS
contribution totheDOS(in arbitrary units) for thedomains
dx2−y2 ±is, for different values of χ and ρ. Thin red line is
for ρ= 0.2 and thick blue line is for ρ= 0.8. For χ =π the following form:
DOS has delta-function singularities.
E (θ) ρ2+sin22θ χ
b
= cos φ−
∆max s ρ2+1 2
tions: (cid:16) (cid:17)
χ
×sgn cosθsin φ− , (28)
2
∆ (θ)=∆ sin2θ+i∆ ,
− 1 2 (27) h (cid:16) (cid:17)i
∆ (θ)=(∆ sin2θ−i∆ )eiχ. which is shown in Fig. 12. Again, the ABS exist for
+ 1 2
all θ and their spectra are not continuous. The DOS is
The gap magnitudes are the same in both domains: shown in Fig. 13. In particular, in the case of χ = π,
we find from Eq. (28) that |E |/∆ =ρ/ ρ2+1, i.e.
b max
|∆+|=|∆−|=∆bulk(θ)= ∆21sin22θ+∆22. sdionegsunlaortitdieespeinndthoenDθ,OwSh.ichresultsinthedpelta-function
q
The bulk gap magnitude is anisotropic, without nodes.
It attains its maximum value, ∆ = ∆2+∆2, at
max 1 2 VI. CONCLUSION
θ = (2n+1)π/4, where n = 0,1,2,3, and the minimum
p
value, ∆ =∆ , at θ =nπ/2, where n=0,1,2,3.
min 2
We have shown that the DWs separating degener-
The parameter β is given by Eq. (23), where
ategroundstatesinunconventionalsuperconductorscan
ρ trap fermionic quasiparticles and create Andreev bound
φ=arctan
sin2θ states with energies inside the bulk gap. Using as an
(cid:16) (cid:17) example a quasi-2D tetragonal superconductor, we have
and ρ is given by Eq. (22). The ABS spectrum has the studied both chiral and nonchiral p-wave states, as well
10
8 χ=0 ateddelta-functionpeaksinthe DOSisexplainedbythe
7 signchangeofthe realgapfunctionalongthe semiclassi-
ρ=0.2 cal trajectory. In the mixed singlet cases, the spectra of
6 ρ=0.8
d+isstatescanhavefourdelta-functionpeaksatnonzero
5
energies due to the Andreev bands with flat dispersion,
Nb4 which never happens in the d+id state. We hope that
3 theabovementionedfeaturescanbedetectedintunneling
2 experiments,thus providingadirectevidence ofthe DW
presence and shedding light on the pairing symmetry of
1
unconventional superconductors.
0
8 χ=π/2
7
ρ=0.2
6 ρ=0.8
5
Nb4
3
2
1
0
8 χ=π
7
6 ρ=0.2
ρ=0.8
5
Nb4
3
2
1
0
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
E / ∆
max
FIG. 13: (Color online) The energy dependence of the ABS
contribution totheDOS(in arbitrary units) for thedomains
dxy±is, for different values of χ and ρ. Thin red line is for
ρ=0.2andthickbluelineisforρ=0.8. Forχ=π theDOS
has delta-function singularities.
as the mixed singlet states of d+id and d+is symme-
tries. Time reversal symmetry is broken in each case,
except the nonchiral p-wave states.
We derivedageneralexpressionfortheABSenergyin
the semiclassical approximation, for a sharp DW model.
The bound states exist only for certain directions of the
quasiparticle propagation, and there is only one bound
state for each “allowed” direction. The ABS spectrum
is highly anisotropic, forming the ABS “bands” as the
directionoftheFermiwavevectorvaries,whichresultsin
some prominent features in the quasiparticle DOS, such
as internal gaps, abrupt steps, and the van Hove singu-
larities.
Some of the DOS features are specific to certain pair-
ing symmetries and can help to distinguish them. For
instance, the ABS spectrum in the chiral p-wave state
has neither internal gaps nor delta-function peaks, in
contrast to the nonchiral ones. In the latter cases, the
origin of the flat zero-energy ABS bands and the associ-