Andreev bound states and tunneling characteristics of a non-centrosymmetric superconductor

0.46 MB·English

by  C. Iniotakis

Andreev bound states and tunneling characteristics of a
non-centrosymmetric superconductor
C. Iniotakis,1 N. Hayashi,1 Y. Sawa,2 T. Yokoyama,2 U. May,1 Y. Tanaka,2 and M. Sigrist1,3
1Institute for Theoretical Physics, ETH Zurich, 8093 Zurich, Switzerland
7 2Department of Applied Physics, Nagoya University, Nagoya, 464-8603, Japan
0 3Department of Physics, Kyoto University, Kyoto 606-8502, Japan
0 (Dated: February 4, 2008)
2
The tunneling characteristics of planar junctions between a normal metal and a non-centro-
n
symmetric superconductor like CePt3Si are examined. It is shown that the superconducting phase
a
with mixed parity can give rise to characteristic zero-bias anomalies in certain junction directions.
J
The origin of these zero-bias anomalies are Andreev bound states at the interface. The tunneling
6
characteristics for different directions allow to test the structureof the parity-mixedpairing state.
2

PACSnumbers: 74.20.Rp,74.70.Tx,74.45.+c
]
n
o
c Since the early sixties tunneling spectroscopy has to have additional proof for this kind of state through a
- playedanimportantrole in gatheringinformationabout phase sensitive test. The characteristics of quasiparticle
r
p the gap function of conventionalsuperconductors [1]. In tunneling may provide such a means. Despite the fact
u the context of unconventional superconductivity tunnel- that the parity mixing A -phase has the full symmetry
1
s ing appearedas a tool to probe the internal phase struc- of the system, it gives rise to characteristic non-trivial
.
t ture of the Cooper pair wave functions. Surface states features in the tunneling spectra (cf. also [17]).
a
m with sub-gap energy, known as Andreev bound states, In the present study, we address the problem of
provide channels for resonant tunneling leading to so- possible Andreev bound states at the boundaries of a
-
d calledzero-biasanomalies. Zero-biasanomaliesobserved non-centrosymmetric superconductor like CePt Si and
3
n
in high-temperature superconductors showed the pres- demonstrate that quasiparticle tunneling could give im-
o
ence of zero-energy bound states at the surface, giving portant information on the gap structure of such a ma-
c
[ strong evidence for d-wave pairing [2, 3, 4, 5, 6]. Simi- terial. The theoretical framework for our calculations is

larlythetunnelingspectrumobservedinSr2RuO4iscon- givenbythequasiclassicalEilenbergertheory[18,19,20].
1
sistent with the existence of chiral surface states as ex- Our starting point is the quasiclassical bulk propagator
v
3 pectedforachiralp-wavesuperconductor[7,8,9,10,11]. gˇB in a non-centrosymmetric superconductor with an-
4 Thus quasiparticle tunneling has emerged as an impor- tisymmetric Rashba-type spin-orbit coupling, which has
6 tant phase sensitive probe for unconventional supercon- already been derived in Ref. [16]:
1
ductors.
0 gˆ ifˆ
07 FeTrmhieorneccoenmtpdoiuscnodveCreyPotf3sSuiphearscmonodtuivcattivedityinitnentsheeehxepaevry- gˇB =−iπ −ifˆ¯ −gˆ¯!, (1)
/ imental and theoretical studies, since its crystal symme-
t where the upper two components are given as
a try is lacking a center of inversion [12, 13, 14, 15, 16]. It
m wasshownthat the presence ofantisymmetric spin-orbit gˆ = ωn σˆ++ ωn σˆ− (2)
- coupling in such a material could be responsible for a ωn2 +|∆+|2 ωn2 +|∆−|2
d
numberofintriguingphenomena,suchastheratherhigh
on upper criticalfield exceeding the standardparamagnetic fˆ = p ∆+ σˆ++p ∆− σˆ− iσˆy(3)
c limit [12, 13]. The absence of inversion symmetry yields ωn2 +|∆+|2 ωn2 +|∆−|2 !
v: aclassificationschemeofthe pairingsymmetry,different and similarprelations hold forpthe lower components fˆ¯
Xi from the standard distinction between even- and odd- and gˆ¯. On the right hand side of Eqs. (2) and (3), ωn =
parity states. Actually, the pairing states in these non-
(2n+1)πk T denotestheMatsubarafrequency,andboth
r B
centrosymmetricsuperconductorscanbeviewedasstates
a the gap amplitudes ∆± and the spin matrices σˆ± are kˆ-
of mixed parity which do not have a definite spin-singlet
dependent. For a spherical Fermi surface parametrized
or spin-triplet configuration,either. For CePt Si various
3 in the standard way by kˆ =(cosφsinθ,sinφsinθ,cosθ),
experimentscanbe interpretedina waythatthe pairing
the gap amplitudes are
state has the highest possible symmetry, but would de-
veloplinenodesinthegapdue toamixingofthes-wave ∆± =ψ±∆sinθ, (4)
and a p-wave component. This state belongs to the A -
1
while the spin matrices may be written as
representation of the generating tetragonal point group
Cgo4ovdfoerviCdeePnct3eSfio.rWthhiislebtehhearvmioord,yintawmoiculqdubanetidteiessiragbivlee σˆ± = 21 ±i1eiφ ∓ie1−iφ . (5)
(cid:18) (cid:19)
2
The variables ψ and ∆ are taken to be real and denote For an impenetrable interface, the coherence functions
the gap amplitudes of the s- and p-wave components of γ ,γ˜ atthe surfaceofthe superconductoraregivenan-
S S
theparity-mixedsuperconductingstate. Altogether,this alytically in terms of the bulk coherence functions:
result is interpreted as a superconducting phase on two
splitsphericalFermisurfaces,eachofthemhavinganon- γS(kˆout)=γB(kˆin) γ˜S(kˆout)=γ˜B(kˆout). (10)
trivial gap function with amplitudes given by Eq. (4)
Here kˆ and kˆ denote the incoming and outgoing di-
and complex spin structures according to Eq. (5). For in out
rectionsofthe wavevector,whichareconnectedthrough
technical simplicity we assume here that the splitting is
the condition of specular scattering. Now the full quasi-
small, such that both Fermi surfaces have identical size
classical propagator gˇ at the boundary is easily gen-
and a descriptionby a one-bandformalismis possible as S
erated from the coherence functions according to Eq.
described by Eqs. (1)-(5). A generalization to a really
(10). From gˇ we obtain, for example, the angular
splitFermisurfaceisstraightforward,butrequiresamore S,11
resolved local density of states at the boundary of the
extended notation without giving additional insights.
non-centrosymmetric superconductor:
In order to finally include the boundary effects, it is
convenienttousetheRiccati-parametrizationofthequa- N(E,kˆ) (11)
siclassicaltheory[21, 22]. Within this technique, the full
1
quasiclassicalpropagatorgˇincombinedparticle-holeand = N Re tr (1−γ γ˜ )−1(1+γ γ˜ ) ,
02 S S S S iωn→E+i0+
spinspacemaybewrittenintermsoftwocoherencefunc-
h (cid:0) (cid:1)(cid:12) i
tions γ and γ˜ in the following way [22]: where E denotes the quasiparticle energy(cid:12)and N0 is the
density ofstates ofthe normalstate. With h...i denoting
(1−γγ˜)−1 0 1+γγ˜ 2γ an average over the Fermi surface, the local density of
gˇ=−iπ 0 (1−γ˜γ)−1 −2γ˜ −1−γ˜γ states is just given as N(E) = hN(E,kˆ)i. Note, that
(cid:18) (cid:19)(cid:18) (cid:19)
(6) at the boundary it is sufficient to average over half of
The coherence functions γ,γ˜ themselves are 2×2-spin the Fermi sphere only, i.e. we can restrict ourselves to
matrices, which contain local information about the quasiparticles traveling in kˆ directions.
out
particle-hole coherence for a given kˆ-direction. Partic- Eventually, we also obtain the normalized differen-
ularly, γB and γ˜B in the bulk of a non-centrosymmetric tial tunneling conductance for a normal metal/non-
superconductor can be obtained just by comparing the centrosymmetricsuperconductorjunctionatlowtemper-
knownresultofEqs. (1)-(3)withthegeneralforminEq. atures T ≪ T . To first order of the interface transmis-
c
(6), leading to: sion probability, it can be generated from the angular
resolved local density of states of Eq. (11) in the follow-
γ = i(1+gˆ)−1fˆ (7a)
B ing way [25]:
γ˜ = ifˆ−1(1−gˆ), (7b)
B G(eV)=hcosα D(α)N(eV,kˆ)ihcosα D(α)i−1. (12)
which can be written as
The function D(α) denotes the transmission coefficient
γB = −(γ+σˆ++γ−σˆ−)σˆy (8a) of the interface depending on the angle of incidence α,
which is related to the normal vector n of the boundary
γ˜B = σˆy(γ+σˆ++γ−σˆ−), (8b) by cosα = n·kˆ . In accordance with the result for
out
where we define an interface of the δ-function type [2], the transmission
coefficient is taken to be
∆±
γ± = . (9) D cos2α
ωn+ ωn2 +|∆±|2 D(α)= 0 , (13)
1−D sin2α
0
The advantage of this Riccpati formalism is, that spatial
wheretheparameterD specifiesthemaximumtransmis-
inhomogeneitiessuchastheboundariescanbetakeninto 0
sion coefficient for the angle of normal incidence α=0.
account in a straightforwardway [22, 23, 24].
For the surface with normal vector n k z the rela-
In the vicinity of the boundary the pair potential can
tion between kˆ and kˆ needed to evaluate γ (kˆ )
develop a spatial dependence and deviate from the bulk in out B in
in Eq. (10) is given by φ = φ and θ = π −θ ,
form. Since we focus mainly on the qualitative aspects, in out in out
and the angle of incidence is α = θ . In this situa-
however,weignorethiseffectinthefollowingandassume out
the gapis constantthroughoutthe superconductor. The tion, we find immediately that ∆±(kˆin)=∆±(kˆout) and
usual procedure within the Riccati formalism, namely σ±(kˆin) = σ±(kˆout), which directly yields γS(kˆout) =
solving differential equations of the Riccati type numer- γB(kˆout). Hence, both γS and γ˜S are finally given by
ically for given kˆ-direction, is unnecessary in this case. the bulk coherence functions for the same kˆ-direction,
The presence of the surface enters through rather sim- and therefore the resulting angular resolved density of
ple boundaryconditionsforthe coherencefunctions [22]. states N(E,kˆ) is identical to the one of the bulk. No
3
GHeVL
5 q=0
nþz q=0.5
q=1.0
q=1.5
4
3
2
1
0 eV(cid:144)D
-3 -2 -1 0 1 2 3
FIG. 2: (color online) Typical angular resolved density of
FIG.1: NormalizedlowtemperatureconductanceG(eV)fora statesN(E,kˆ)attheboundaryofanon-centrosymmetricsu-
normal metal/non-centrosymmetric superconductor junction perconductorwithnormalvectornky. Forthisexample,we
with boundary orientation n k z. The ratio q of the gap set q = 0.5 and θ = π/2, i.e. we deal with kˆ-vectors of the
amplitudes is varied from 0 to 1.5 in steps of 0.1. Between
x,y-plane only. Bright colors represent high spectral weight.
two subsequent curves, a vertical offset of 0.2 has been used
The branches of low-energy Andreev bound states according
forclarityoftheplot. Thetransparencyoftheinterfaceisset
to Eq. (14) are clearly visible.
to D0 =0.1.
contrast to n k z, low-energy Andreev bound states can
sub-gap Andreev bound states are generated by surface
exist for this orientation. By examinationof the angular
scattering. For small but finite energies the shape of the resolvedlocaldensityofstatesN(E,kˆ)giveninEq. (11)
localdensity of states is, however,determined by the ac-
we find the conditionthat these bound states only occur
tualnodaltopologyofthegapfunctionsgiveninEq. (4). for kˆ-vectors with ∆− < 0, i.e. the contributions come
Thisbehavioristhenpassedontotheresultingtunneling
fromthosepartsoftheFermisurface,wherethep-waveis
conductances, which are presentedin Fig. 1 for different
stronger than the s-wave (sinθ >q). Actually, for every
valuesoftheratioq =ψ/∆. Startingwithq =0,wehave
angle θ with ∆− < 0 there is a whole branch of bound
a pure p-wavestate with point nodes on the poles of the
states living between the energies ±|∆−| determined by
Fermisurface. Thisleadstoa parabolicbehaviorforlow
bias voltage reflecting the E2-dependence of the density E2−∆+∆−− ∆2+−E2 ∆2−−E2
o(4f)stbaetecso.mWeshfeunllqyisgainpcpreedas(eadl,btehitegaanpisfoutnrocptiiocnal∆ly+),owfhEiqle. cosφ=± E(∆q+−∆−) q . (14)
∆− exhibits immediately line nodes developing around Zero-energyAndreevboundstatesoccurforquasiparticle
the top and bottom of the Fermi sphere. Accordingly, directions with φ = π/2 which corresponds to vanishing
a linear dependence can be observed for low bias. For transversemomentumalongthex-direction. Anexample
q = 1 the line node of ∆− has moved to the equator of ofthe low-energyAndreev boundstatesinthe planeθ =
the Fermi sphere. This removes the linear low-bias be- π/2 for q =0.5 is provided by Fig. 2.
havior in G, since the nodal quasiparticles traveling par- The normalized low-temperature conductance for this
allelto the surfacecannotcontribute tothe conductance situation is presented in Fig. 3. We find an extended
Eq. (12) any longer. As soon as q >1 however, also the zero-bias conductance peak for q = 0, analogous to the
gap function ∆− is nodeless, which results in the open- one for a pure p-wave superconductor with a cylindrical
ing of a corresponding gap in the conductance situated Fermi surface [7, 8]. This peaked structure at low bias
between the normalized energies E/∆=±|q−1|. voltages persists as long as q < 1 but becomes progres-
Nextweturntothe boundarywithnormalvectorn⊥ sivelynarrowerforq →1. Thesestructuresareeasilyun-
zwherewecanchoosefreelynky. Here,therelationbe- derstoodinthecontextoftheAndreevboundstates. For
tween incoming and outgoing quasiparticle directions is agivenq <1thereisalwaysaregionaroundtheequator
determinedbyφin =−φout andθin =θout. Furthermore, oftheFermispherewithsinθ >q,where∆− <0. Quasi-
the angle of incidence is α = arccos(sinφ sinθ ). In particles from those parts of the Fermi surface support
out out
4
has also been anticipated that the ratio q could be tem-
GHeVL
perature dependent. Thus it is possible that the tunnel-
5 n¦z qq==00.5 ing spectrum displays qualitative changes as a function
q=1.0 of temperature.
q=1.5
4
We would like to thank D.F. Agterberg, P.A. Frigeri,
K. Wakabayashi and Y. Yanase for helpful discussions.
This work was financially supported by the Swiss Na-
3
tionalfond and the NCCR MaNEP.
2
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3