Electronic structure of d-wave superconducting quantum wires

0.67 MB·English

by  A. M. Bobkov

Electronic structure of d-wave superconducting quantum wires
A. M. Bobkov1,2, L.-Y. Zhu3, S.-W. Tsai3,4, T. S. Nunner3, Yu. S. Barash1,2 and P. J. Hirschfeld3
1Lebedev Physical Institute, Leninsky Prospect 53, Moscow 119991, Russia
2Institute of Solid State Physics, Chernogolovka, Moscow reg., 142432 Russia
5 3Department of Physics, University of Florida, Gainesville, FL 32611
0 4Department of Physics, Boston University, Boston, MA 02215
0
(Dated: February 2, 2008)
2
n We present analytical and numerical results for the electronic spectra of wires of a d-wave su-
a perconductor on a square lattice. The spectra of Andreev and other quasiparticle states, as well
J as the spatial and particle-hole structures of their wave functions, depend on interference effects
1 caused by the presence of the surfaces and are qualitatively different for half-filled wires with even
1 oroddnumberofchains. Forhalf-filledwireswithanoddnumberofchainsN at(110)orientation,
spectra consist of N doubly degenerate branches. By contrast, for even N wires, these levels are

] split,andallquasiparticlestates,eventheoneslyingabovethemaximalgap,havethecharacteristic
n properties of Andreev bound states. These Andreev states above the gap can be interpreted as a
o consequenceof aninfinitesequenceofAndreevreflections experiencedbyquasiparticles alongtheir
c
trajectories bounded by the surfaces of the wire. Our microscopic results for the local density of
-
r states display atomic-scale Friedel oscillations dueto the presence of the surfaces, which should be
p observablebyscanning tunnelingmicroscopy. Fornarrow wires theself-consistent treatment of the
u order parameter is found to play a crucial role. In particular, we find that for small wire widths
s
thefinitegeometry maydrivestrongfluctuationsorevenstablilize exoticquasi-1Dpairstateswith
.
t spin triplet character.
a
m
- I. INTRODUCTION whetherthenumberofelectronsontheislandareevenor
d
odd. Morerecently,superconductingwiresofwidthstens
n
o of nanometers [12] have also been fabricated. Although
Since the discovery of high temperature superconduc-
c they havenotyet been studied by STM orsimilar meth-
tivity (HTS), the origin of the pairing phenomenon in
[ ods, this should be technically feasible.
these materials has been the subject of intense debate,

2 Isolated nanoscale grains and wires of HTS material
and is still not clarified. Part of the unusual nature of
v have not been fabricated to our knowledge. While this
HTS which has hinderedtheoreticalanalysis is the short
7 may prove technically quite difficult to achieve due to
coherencelength,whichallowsshort-wavelengthfluctua-
0
the complexity of the crystal structure, there seems to
6 tionsofvarioustypesoflocalordertocoexistwithsuper-
be no fundamental obstacle in the long run. This ap-
2 conductivity. Mostprobesofthenatureofthesupercon-
0 ductingstatehavebeenrestricted,untilfairlyrecently,to plies as well to other superconductors thought to mani-
4 measurements of bulk properties and tunnelling through fest unconventionalsuperconducting order,where effects
0 offinite geometryshouldbe easierto see sincecoherence
relatively large areas. Following the pioneering work of
/ lengths tend to be larger.
t Hessetal.[1],itwasrealizedthatscanningtunnelingmi-
a
croscopy(STM) couldprovideanatomic-scalepictureof It is our purpose in this paper to study how d-wave
m
the superconducting state, particularly useful when ap- (e.g., HTS) and other unconventional symmetry super-
-
d pliedto inhomogeneoussituations likethe vortexlattice. conductorsbehaveinfinitegeometryattheatomicscale.
n Measurements of this type were subsequently performed Ziegler et al.[13] began the study of this problem in the
o onhigh-temperaturesuperconductors[2]. Inthepastfew case of (100) d-wave quantum wires, pointing out the
c years, scanning tunnelling microscopy on the surface of dependence of the Fermi level density of states on the
:
v HTS have compiled a novel and fascinating picture of parity of the wire width. This is a natural consequence
i the local electronic structure of a few of these materials ofthediscretizationoftheelectronicenergylevelsdueto
X
[3, 4, 5, 6, 7, 8, 9, 10]. In the Ba Sr CaCu O system the finite wire width in the d-wave state. While in the
2 2 2 8
r (BSCCO),onedramaticimplicationoftheseexperiments s-wave case all the interesting physics is tuned by the
a
is that even relatively high quality single crystals dis- level discretization, one expects a priori one fundamen-
playinhomogeneouselectronicstructureatthenanoscale tal difference in the d-wave case: for any geometry with
[7, 8, 9, 10]. surfacesmakinganarbitraryanglewiththecrystalaxes,
pair-breaking processes take place on a scale of the co-
In parallel to studies of the HTS materials, point con-
herencelength. The mostimportantconsequenceforthe
tact spectroscopy has been used to study the electronic
electronic structure should be the formation of Andreev
structure of ultrasmall conventional superconducting is-
surface states.
lands [11]. Among many fascinating consequences of the
nanoscale geometry are number parity effects, in which However, little is known about how Andreev surface
thequalitativeelectronicstructuredependssensitivelyon states behave when the size of the superconductor be-
2
comes comparable to ξ . The zero-energy states form reproduced by this technique, and then extend it to cal-
0
on surfaces with orientations different from the antin- culate new results on wires with other orientations. We
odal directions of a d-wave superconductor, due to the find thatthe results forelectronic spectraare verysensi-
signchangeofthe orderparameter. Inhigh-temperature tivetothenumberparityofthewirewidth,andthattrue
superconductors, such states manifest themselves as the zero-energy Andreev states can only exist in wires with
zero-bias conductance peak (ZBCP) in tunneling spec- oddnumbersofchains. Inevenwires,theAndreevstates
troscopy in the ab-plane [14, 15, 16, 17, 18, 19, 20, 21, are split, pushed away from the Fermi energy, and can
22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, have either surface or standing wave character. Finally,
37, 38, 39], the anomalous temperature behavior of the weshowthatforsmallerwiresself-consistencyeffectsbe-
Josephsoncriticalcurrent[40,41,42,43]andtheupturn come important and can even, within mean field theory,
in the temperature dependence of the magnetic penetra- lead to condensation of a fully gapped spin triplet state
tion depth [44, 45, 46] (see also review articles [47, 48]). instead of d-wave order.
The conventional description of Andreev surface states, Our results may also have some qualitative relevance
as well as the Andreev reflection itself, is based on the fortherelatedproblemofd-wavesuperconductinggrains,
quasiclassical approximation, a powerful tool in the the- where the geometry is constrained in 2 dimensions. In
oreticalstudyofvariouspropertiesofinhomogeneoussu- addition to artificially fabricated islands, some authors
perconducting systems. haveproposedthat the BSCCO-2212samples which dis-
The quasiclassical theory of superconductivity gives a play nanoscale inhomogeneity should be thought of as
so-called coarse-grained description of the phenomena, a collection of weakly coupled d-wave grains of roughly
averagedoverinteratomicdistances. Thishasbeenused, thesizeofthesuperconductingcoherencelengthξ0,ord-
for example, to calculate the local quasiparticle density wavegrainscoupledtograinsofanotherelectronicphase
of states (LDOS). However, these coarse-grained aver- [7, 9, 10, 52]. Infact, the structure of a general,possibly
aged results are not adequate to analyze atomic reso- irregular small grain of d-wave superconductor has not
lution measurements by STM and some other contem- been studied to our knowledge. Understanding how the
porary experimental techniques (e.g. atomic force mi- local density of states (LDOS) of these wires depend on
croscopy[49]). Toobtainthistypeofinformation,afully thewirewidthandtheorientation,aswellasonthedevi-
quantummechanicalatomic-scaleapproachgoingbeyond ation from half-filling, could provide important intuition
thequasiclassicalapproximationindescribinginhomoge- for the question of the electronic structure of the small
neous states of superconductors is required. We address irregular grains possibly present in BSCCO samples.
this problem in the present paper using a tight-binding The outline ofthe paperis asfollows. InSec.II, wein-
BCS-like model of d-wave superconductor on a square troduce the formalism for the problem. In Sec.III, we
lattice. discuss three special semi-infinite surface orientations:
(110), (210) and (100). In Sec.IVB2, we study the
Asafirststeptowardsunderstandingtheeffectofcon-
(110) superconducting wires with even and odd width
strained geometry, we study the simplest case of d-wave
and make an comparisonwith the discrete states in nor-
superconducting wires consisting of N parallel chains as
malmetalwires(Sec.IVB1)totrytoidentifythenature
thesystemsizeisreduced. Thequasiparticlespectrumof
of the true Andreev states. In Sec.IVB3, we study the
such systems is described both analytically, with an as-
effects of deviations from the half-filling. In Sec.V the
sumed spatially homogeneous order parameter, and also
results of fully self-consistent calculations are presented.
numerically with a fully self-consistent approach. In the
In particular, we allow the order parameter to vary spa-
limit N 1 the usual surface Andreev states in lattice
≫ tially and comment on the differences in our results. In
models[34,36,47],aswellasthesurfacestatesknownin
the case of narrow wires the self-consistent study shows
continuous models [23], can be recoveredat each surface
theappearanceofsomepeculiartypesofsuperconducting
of the wire. However,for sufficiently narrowwires,when
pairing. Finally, in Sec.VI, we present our conclusions.
the transverse wire dimension is the order of the super-
conductingcoherencelength,theAndreevstatesstrongly
interfere and give rise to qualitatively new effects. We
II. MODEL DESCRIPTION AND FORMALISM
showbelowthatonlythoseeffectswhichoccurforbands
sufficiently far from half-filling and relatively wide wires
can be described with the quasiclassicaltheory of super- TheHamiltonianforapuresingletsuperconductorcan
conductivity. Inaddition,wedemonstratehowandunder be written as:
what conditions one can recoverearlier quasiclassicalre-
= t c† c [µ U ]c† c
sults for Andreev states in d-wave superconducting films H − iσ jσ − − i iσ iσ
[50] from our microscopic approach. hiX,ji,σ Xi,σ
The microscopicmethodadoptedinthis papertocon- + ∆ c† c† +h.c. , (1)
{ ij i↑ j↓ }
strain the geometry involves introducing lines of impuri-
hXi,ji
ties of potential strength taken to infinity to bound the
wire. We show that earlier microscopic results on (110) where we have chosen a nearest neighbor tight-binding
surfaces [23, 34, 36, 47, 51] and (100) wires [13] can be band for simplicity; µ is the chemical potential. A
3
superconducting pairing is defined for nearest neigh- where the τα are the Pauli matrices and ξk =
bors ∆ = V c c on the bond i,j . The pa- 2t[cos(k a)+cos(k a)] µ. Calculating the T-matrix
ij j↓ i↑ a b
− h i { } − −
rameter t is of order 150 meV for high T materials, allows us to obtain the eigenenergies of the system from
c
and we consider both the particle-hole symmetric model the condition detTˆ−1(ω)=0.
µ = 0 and the more realistic case of finite µ. In The local spin-resolved quasiparticle density of states
non-self-consistent calculations the OP has the famil- is given as
iar k-space form ∆k = ∆0[cos(kaa) cos(kba)], where
∆0 = 21 ±(∆ii±ra −∆ii±rb) is ind−ependent of i, and ρ↑(↓)(r,ω) = −π−1Im G11,↑↑(↓↓)(r,r,ω) . (6)
is taken to be 0.2t. The lattice constant is denoted by
P After integration of the LDOS over energy, we should
a. The maximum gap is ∆ = 2∆ . We also present
max 0
obtain the total number of quasiparticle states per one
self-consistent calculations, in which case V is chosen to
site. Sinceeachsiteonthe latticepossesses2stateswith
yield this same value of ∆ far from wire edges.
0
opposite spins, the spin-resolvedLDOS normalization is
ItispossibletoconstrainthegeometryunderlyingEq.
(1) in several different ways. We present results here for
amethoddiscussed,forexample,inRefs.34,53inwhich ∞
the on-site potentials Ui are chosen to lie on the bound- dωρ↑(↓)(r,ω)=1. (7)
aryand their value is takento infinity to cut offelectron Z−∞
transportthrough the boundary. This technique has the
virtue that the strength of the barrier can in principle III. SURFACE CASE
beloweredtoallowdifferentdegreesoftransparencyand
the study of tunneling phenomena. In this work we re-
Upon a conventional reflection on the (110) surface
strict ourselves to impurity configurations and strengths
of a d-wave superconductor, the order parameter always
for which the constrained system is completely isolated
changessignasthedirectionofthequasiparticlemomen-
from its environment. This is equivalent to the assump- tum k is varied. This leads to Andreev reflection and,
tionofopenboundaries,whennohoppingandnopairing
eventually, to the formation of the dispersionless zero-
take place outside the region. No disorder is introduced
energy surface Andreev bound states. For other surface
inthesystemandweconsiderU as“impurity”potentials
i orientations the sign change does not take place for all
onlysoastoformthesurfacesofthesuperconductingre-
incoming momentum directions. It is important to no-
gion.
tice that the number of consecutive impurity lines that
The Hamiltonian of the surface term is taken to be
areneededinordertocutthesystemdependsontheori-
U =U c†c , (2) entation of the surface one wants to consider. For (100)
0 ℓ ℓ and(110)surfacesandnearestneighborhopping,oneline
ℓ
X of impurities is sufficient to cut communication between
where the set of sites ℓ is determined exclusively by the the two sides. For a (210) surface, the nearest neighbor
boundaries of the desired system (see below). The full hopping and pairing terms can still connect this particle
FourierspaceGreen’sfunctionforthesysteminthepres- with another across a single impurity line, so for a sim-
ence of these impurities is quite generally: ple tight-binding band, two consecutive lines are needed
to close the system (Fig. 1). Alternatively, if one in-
Gˇ(k,k′,ω) = Gˇ(0)(k,ω)δk,k′ + cludes a next-nearest neighbor hopping t′, even a (110)
+ Gˇ(0)(k,ω)Tˇ(k,k′,ω)Gˇ(0)(k′,ω) , (3) surface is not closed by a single line of infinite impuri-
ties; the system considered in Ref. [53] is therefore not
wheretheT-matrixcanbefoundfromthefollowingequa- a closed (impenetrable) surface. For a (hl0) surface in
tions: general, max(h,l) lines are needed for a model that in-
cludes nearest-neighbor terms only, and h+l lines are
Tˇ(k,k′,ω) = Uˇ(k,k′)+ needed for a model that includes next-nearest-neighbor
+ Uˇ(k,k′′)Gˇ(0)(k′′,ω)Tˇ(k′′,k′,ω).(4) terms. Clearly, the technique becomes cumbersome for
arbitrary angles, but one can nevertheless learn a good
Xk′′ deal by considering special cases.
In the presence of a surface of arbitrary orientation,
Here Gˇ and Tˇ take 4 4 matrix form in the four-
× the simplest way of applying Bloch’s theorem to this
dimensionalproductspaceofparticle-holeandspinvari-
discrete system is by using a surface-adapted Brillouin
ables. If we choose nonmagnetic on-site potentials and
zone[36, 51]. We define new coordinates (xˆ, yˆ), rotated
considersingletsuperconductors,the problemreducesto
with respect to the crystal axes (aˆ, ˆb), where xˆ is the
2 2 matrices in Nambu space. The Nambu retarded
× direction normal to the surface and yˆ is the direction
propagator for the pure d-wave superconductor is
along the surface. The system is periodic along the y-
direction and the crystal momentum component k of a
y
Gˆ(0)(k,ω)= ωτˆ0+ξkτˆ3+∆kτˆ1 , (5) quasiparticle is conserved. Instead of the usual square
(ω+i0)2−ξk2 −∆2k Brillouinzone ka =[−π,π], kb =[−π,π] (for unit lattice
4
The Fourier transform with respect to k of Eq. 3 is
x
Gˆ(n,n′,k ,ω)=Gˆ(0)(n n′,k ,ω) (14)
y y
− −
−1
Gˆ(0)(n,k ,ω) Gˆ(0)(0,k ,ω) Gˆ(0)( n′,k ,ω)
y y y
−
h i
Due to periodicity of the system along the y-direction,
calculation of the LDOS at site n will simply involve
a sum of G(n,n,k ,ω) over all values of k within the
y y
surface-adapted Brillouin zone and over two spin direc-
tions:
2 πd dk
ρ(r,ω)=ρ(n,ω)= Im yG (n,n,k ,ω)
11 y
FIG.1: (210) surfaceorwire. Twolinesofimpurities(closed −π 2πd
Z−πd
squares) areneeded toisolate thelattice sites(closed circles) (15)
with nearest neighbor hopping.
A. (100) surface
constant a = 1) we now use the surface-adapted Bril-
louinzonegivenbykx =[ π/d,π/d]andky =[ πd,πd]. Since(100)surfacesarenotpairbreakingind-wavesu-
− −
Here d=1/√h2+l2 is the distance between the nearest perconductors, we do not a priori expect to see interest-
chains (layers)alignedalong the surfaces. The momenta ing physics arising from Andreev states. On the other
inthetwocoordinatesystemsaresimplyrelatedthrough hand, the mere presence of a surface can induce surface
rotation of an angle θ =tan−1h/l. Tamm bands [54], decaying in the bulk on the atomic
Now we turn to the solution of the equation for the scale and experiencing the Friedel-like oscillations of the
T-matrix (Eq. 4) for the case of one line of impurities. LDOS. In our model Tamm states have nothing to do
We start with the ansatz with the superconductivity, although the possibility for
pairing of electrons occupying these surface states is not
Tˆ =T τˆ +T τˆ +T τˆ (8)
0 0 1 1 3 3 excluded[55]. Whileitisnotourintenttostudythesein
detail, we present some results to show that such states
and find, for arbitrary strength of impurity potential,
can be seen by the STM even in situations when sur-
G(0)(0,k ,ω) face Andreev states are absent. Fig.2 shows on upper
T0(ky,ω) = 0 y , (9) and lower panels the LDOS for various distances from
D
1
thesurfaceinthenormalmetalandthesuperconducting
T (k ,ω) = −G(10)(0,ky,ω) , (10) states, respectively.
1 y
D
1
(0)
c G (0,k ,ω)
T (k ,ω) = − 3 y (11)
3 y D B. (110) surface
1
whereD (k ,ω)=[c G(0)(0,k ,ω)]2 G(0)(0,k ,ω)2+
1 y − 3 y − 0 y Results for a (110) surface are expected to be quali-
G(10)(0,ky,ω)2, c=1/V0 andG(i0)(x,ky,ω)is the Fourier tatively different from the ones obtained for a (100) sur-
transform with respect to kx of the i-th Nambu compo- face. Thebulk orderparameterinthe coordinatesystem
nent of the bare Green’s function G(i0)(k,ω) ofthecrystalaxesaˆandˆbis∆k =∆0(coskxa−coskya),
but in the coordinates of the surface xˆ (perpendicular
d π/d to surface) and yˆ (parallel to surface) becomes ∆k =
G(i0)(n,ky,ω)= 2π G(i0)(kx,ky,ω)eikxnddkx .(12) 2∆0sinkxdsinkyd, with d = a/√2. In this case, an in-
Z cident particle with any non-zero k experiences a sign
kx=−π/d y
changeinthe orderparameterasitreflectsfromthe sur-
Thesiteindexncorrespondstox-coordinatex=nd. For face.
thecaseofinfinitelystrongimpuritypotentialconsidered Calculationofthe bare Green’s function Gˆ(0)(n,k ,ω)
here, c = 0. In this case the expression for Tˆ can be from Eq. 12 gives at µ=0, G (n,k ,ω)=0 for evyen n
1,3 y
written in the compact form and G (n,k ,ω)=0 for odd n. Explicitly,
0 y
−1
Tˆ(ky,ω)= Gˆ(0)(0,ky,ω) (13) Gˆ(0)(n=2m,k ,ω)=
− y
h i iω exp inz sgn ω(q2 ∆2)
and the poles of the T-matrix correspondto zeros of the
| | − | | − τˆ (16)
determinant of the Green’s function Gˆ(0)(0,ky,ω). − (cid:8)(ω2 ∆2)(q(cid:2)2 ω2) (cid:3)(cid:9) 0
− −
p
5
bulk
n=1 1
n=2 0.4
n=3 0.9
0.35
0.1 0.8
S 0.7 0.3
DO S0.6 OS 0.25
L DO0.5 LD 0.2
0.05 L0.4 0.15
0.3
0.1
0.2
0.05
0.1
0−10 −5 0 0−1 7 9 0−1 6 8 10
ω/∆max 0 ω 1 −11 3 5 0 ω 1 0 2 4
0.4
bulk
FIG. 3: Local density of states ρ(x,ω) in superconducting
n=1
0.3 n=2 state versus ω/∆max for the case of one (110) surface, ∆0 =
n=3 0.2t,µ=0. Left: chainsanodddistancen=x/dfromsurface
at x=0. Right: chainsan even distance from x=0.
S
O0.2
D
L
A2 ω2 <0,thebranch√A2 ω2 isgnω√ω2 A2,
0.1 for−eitherA=∆,q. InFig. 3−wesh→ow−theLDOSsp−ectra
on the different layers. Each layer is defined as an array
of sites parallelto the surface. Its index indicates its po-
0 sition; layern correspondsto sites ata distance ndaway
−1 0
ω/∆ from the surface.
max
The Andreev bound states are manifested as zero-
FIG. 2: Local density of states ρ(x,ω) for a (100) surface. energy peaks in the LDOS. For µ = 0, such states are
Normalstate(upperpanel)andsuperconductingstate(lower found in all odd layers, and are absent on even ones, as
panel), ρ(x,ω) vs. ω/∆max for various distances n = x/d seen in Figure 4. This even-odd effect can be easily un-
from surface; ∆max =0.4t, µ=0. derstoodfromtheformoftheT-matrixandGreen’sfunc-
tionsEqs. (13-17). Forevenn,thebareGreen’sfunction
6
and
Gˆ(0)(n=2m+1,k ,ω)= 5
y
e
= i∆exp −i|n|z sgn ω(q2−∆2) sgn n(q2−∆2) τˆ v stat4
(cid:2) (q2(cid:0) ∆2)(ω2 (cid:1)(cid:3)∆2) (cid:0) (cid:1) 1 dree
− − n
A3
+iqexp(cid:2)−i|np(|qz2s−gn∆(cid:0)2ω)((qq22−−∆ω22))(cid:1)(cid:3)sgn(ω)τˆ3 , (17) mplitude of 2
A
p
where
1
0
q2 ω2 0 1 2 3 4 5 6 7 8
z(ky) = tan−1 ω2−∆2 , (18) layer index
r −
∆(ky) = 2∆0sin(kyd) , (19) FIG. 4: Amplitude of Andreev state vs. distance n = x/d
q(ky) = 4tcos(kyd) , (20) from 110 surface for ∆0 =0.2t, µ=0 .
and ∆(ky) and q(ky) are the maximum gap ∆k and
single-particle spectrum ξk for fixed ky in 110 geometry. G(0)(n,ky,ω) is proportional to ω for low-frequencies.
Note that here the square root function takes positive ThentheT-matrixT 1/ωforsmallω. Eventhoughthe
∼
values for positive arguments, i.e. under the conditions T-matrixhasapoleatω =0,theproductoftwoGreen’s
∆(k ) < ω < q(k ) or q(k ) < ω < ∆(k ). For functionsintheanalogofEq. 3decreasesfaster,resulting
y y y y
| | | | | | | | | | | |
6
in zero LDOS at zero frequency. As seen from the right is conserved in a reflection event, and only conventional
panel of Fig. 3, the low-energy density of states on even specular reflection takes place. A tight-binding model
layers near the (110) surface is substantially less than in showsthat,generallyspeaking,this isnotthe case,since
the bulk,whereitvarieslinearlywithsufficientlylowen- the crystal momentum component k can also change in
y
ergy. Furthermore, the amplitude of the gap features in a reflection process by a reciprocal crystal vector along
the LDOS is noticeably suppressed with decreasing dis- the surface. Due to a difference between reciprocalcrys-
tance from the surface. These features of the LDOS can talvectorsatthesurfaceandinthebulk,themomentum
be understood based on a simple relation between the acquired by a quasiparticle in a reflection event can be
Green’s function Gˆ(n′ = n,k ,ω) of the half-space and physically distinguished in the bulk from that of specu-
y
the bare Green’s function Gˆ(0)(n n = 0,k ,ω), which larly reflecting quasiparticle. Hence, specific crystal pe-
y
takes place for even n=2m : − riodicity along a particularly oriented surface can result
inadditionalchannelsforquasiparticlereflection,ifthere
Gˆ(n′ =n,k ,ω)= 1 is areflectedstate kx(ky), ky onthe Fermisurfacecorre-
y
− sponding to the k surface Umklapp process [56].
y
n
exp 2inz sgn ω(q2 ∆2) Gˆ(0)(0,k ,ω).(21) In this case the Fermi surface, considered as a part of
y
− − | | − the surfaceadaptedBrillouinzone,shouldexhibitmulti-
(cid:2) (cid:0) (cid:1)(cid:3)o ple values ofoutgoingk fora fixedvalue ofk . This
ThefactorinthebracesinEq.(21)controlsthedeviation F,x F,y
situation is realized for the comparatively complicated
from the bulk behavior and diminishes with decreasing
multisheet structure of the Fermi surface in the surface-
even n. At low energies only narrow regions of k near
y
| | adaptedBrillouinzone for the (210)surface, asobtained
the center and the edge of the Brillouin zone contribute
inFig.9ofRef.36. Astrongdependence ofthe particular
to the LDOS. As a result, for even layers the density of
shapeofthesurface-adaptedBrillouinzone,aswellasthe
states goes as ω 3 at low energies, with the main con-
| | Fermisurfaceinthezone,onsurfaceorientationisanim-
tributions arising from k near the edge of the Brillouin
y
portantcharacteristicfeatureofthetight-bindingmodels
zone.
For odd layers, the τ and τ components of Gˆ(0)(n = [36, 51]. Within the quasiclassical theory, the shape of
1 3 the Fermi surface is usually considered as independent
2m+1,k ,ω) are the ones that are non-zero, and they
y of surface orientations relative to the crystal axes. We
approacha constantvalue as ω goes to zero. So the pole
note, that the zero-energy surface states, as a rule, do
in the T-matrix generates the peak in the LDOS, asso-
not exist within the quasiclassical approach, if multiple
ciated with the zero-energy Andreev surface states. For
channels for reflection of quasiparticles from an impene-
Andreev states, z is an imaginary quantity. The size of
trable surface are assumed. In this subsection, we now
thepeakdecreasesasthedistancetothesurfaceincreases
study the LDOS for quasiparticle spectra obtained with
duetothe e−|nImz| factorinEq.(17). Forlarge n,small
| | the tight-binding model at half-filling for (210) surface.
k dominates the integration over k in the LDOS and
y y Technically,wenowneedtosolvefortheGreen’sfunc-
we obtain the following zero-energy asymptotic behav-
2t tion in the presence of two impurity lines (Fig. 1). In
ior of the LDOS: ρ(ω) = δ(ω). The size of the thegeneralcaseoftwoparallelimpuritylines,wecutthe
π∆ n2
zero-energy peak in the LDOS0 n−2. crystalatanorientationgivenby(hl0)andintroducean
∝ impurity potential
U(r)=U δ(r R ) , (22)
C. (210) surface 0 − j
j
X
It is useful to study a case intermediate between the whereR arethepointsonthetwoimpuritylines,atthe
j
standard (100) and (110) surfaces to see what qualita- locationofthe boundaries. The firstboundaryisdefined
tively new features arise. From the usual quasiclassical to be located at x = 0 with impurity sites R = jd−1yˆ
j
viewpoint, the weight of the zero-energy Andreev (210)- and the other is parallel and located at x = (N +1)d
surface states should be finite, but smaller than for the with R =(N +1)dxˆ+(c+jd−1)yˆ, giving a total of N
j
(110) surface because the phase space for which the re- free chains. Here c is a shift along y-axis of sites on the
flectingquasiparticleexperiencesasignchangeofthe or- (N +1)-th chain relative to the sites on the 0-th chain.
derparameterisreduced. The tight-bindingmodelleads For the special case of the (210) surface, we need two
to a more complicated dependence of the weight of the adjacent lines at x = 0 and x = d, so formally this
−
zero-energy states on the surface orientations relative to corresponds to the case of N = 2.
−
the crystal axes. Thus, for the (210) surface the model TheequationfortheT-matrix(Eq. 4)forthisimpurity
showsathalf-fillingnozero-energyAndreevstatesatall. potential can be solved by choosing the ansatz
Weassociatethisdiscrepancy,inparticular,withthedif-
ference between reflection channels incorporated in the Tˆ(kx,kx′,ky) = tˆ0+tˆ1ei(N+1)dkx +tˆ2e−i(N+1)dkx′
two approaches. +tˆ3ei(N+1)d(kx−kx′) . (23)
Standardquasiclassicalconsiderationsimply that par-
allel to a smooth surface the momentum component k In the limit of infinitely strong impurity potential this
y
7
gives of the order parameter, while for the self-consistent spa-
tially dependent order parameter it lies slightly below
Gˆ(0)(0)−1 ∆(k ,0). It is associated with the surface Andreev
tˆ = − F,x
0 1 Gˆ(0)(0)−1Gˆ(0)( N 1)Gˆ(0)(0)−1Gˆ(0)(N +1) states and decays in the depth of the superconductor.
− − − These peaks have been theoretically found first in Ref.
Gˆ(0)(0)−1Gˆ(0)(N +1)Gˆ(0)(0)−1
tˆ = 23 with a continuous model and then also for the con-
1 1 Gˆ(0)(0)−1Gˆ(0)( N 1)Gˆ(0)(0)−1Gˆ(0)(N +1) ductance with a lattice model [36]. We notice that the
− − −
Gˆ(0)(0)−1 conductance spectrum shown in Fig.14 of Ref.36, is in
tˆ = agreement with the LDOS on the first chain (at x = d)
3 −1 Gˆ(0)(0)−1Gˆ(0)(N +1)Gˆ(0)(0)−1Gˆ(0)( N 1)
inFig.5. ThevariationsoftheLDOSfromchaintochain,
− − −
Gˆ(0)(0)−1Gˆ(0)( N 1)Gˆ(0)(0)−1 whichaccompany a large-scalebehavior,arethe Friedel-
tˆ = − −
2 1 Gˆ(0)(0)−1Gˆ(0)( N 1)Gˆ(0)(0)−1Gˆ(0)(N +1) like oscillations.
− − −
The Green’sfunctioninthis casecanthenbewrittenas:
IV. HALF-FILLED WIRES
Gˆ(n,n′)=Gˆ(0)(n n′) Gˆ(0)(n) Gˆ(0)(n N 1)
− − − − ×
We now consider wires where the second line of impu-
(cid:16) (cid:17)
ritiesconfinesthesystemtoafinitewidth,i.e. werestrict
Gˆ(0)(0) Gˆ(0)( N 1) −1 Gˆ(0)( n′) ourselvestothecaseswithsurfacenormalalongthe(100)
− − − ,(24) and(110)directions. Semiclassically,a quasiparticlewill
Gˆ(0)(N+1) Gˆ(0)(0) Gˆ(0)(N+1 n′)
(cid:18) (cid:19) (cid:18) − (cid:19) gothroughmultiplescatterings,bouncingbackandforth
between the two walls. Hence we expect that the inter-
where we have not written the dependence on k and ω
y playbetweenAndreevreflection,takingplaceduetosign
explicitly.
changeoftheorderparameter,andtheenergydiscretiza-
tion,duetofinitewirewidth, toyieldnovelfeatures. We
again model the surfaces by introducing impurities on
theappropriatesitestocompletelyisolatethewires. The
correspondingGreen’sfunctionisgivenbyEqs.(24). The
bound state energies are determined by the equation
0.35 Gˆ(0)(0) Gˆ(0)( N 1)
0.3 det − − =0 . (26)
0.25 Gˆ(0)(N +1) Gˆ(0)(0)
S (cid:18) (cid:19)
O0.2
D0.15 8
L
0.1
6 A. (100) wires
0.05
0 4
−1.5
−1 Ziegler et al.[13] investigated the problem of (100) d-
−0.5 2
0
0.5 ω wave quantum wires in some detail, and discovered the
1 1.5 0 existence of a number-parity effect as a function of the
widthN ofa mesoscopicd-wavewire: a finite totalDOS
FIG. 5: Local density of states ρ(x,ω) versus ω/∆max for a
is found at the Fermi level for odd N and zero DOS
closed(210) non-transparent surface at half-filling model and
(with afull gapinthe excitationspectrum, nota d-wave
∆0 =0.2t, µ=0. Each curve corresponds to a chain located
at a distance x/d to thesurface. like gap) is found for even N, at least for a simple tight-
bindingbandathalf-filling. Thedifferencesbetweeneven
andoddN wereshowntosurviveformoregeneralbands,
as well. For completeness, we reproduce, using our ap-
The equation for the bound state on (210) surface is
proach, some of their LDOS spectra in Fig. 6.
Gˆ(0)(0) Gˆ(0)(1)
det =0 . (25)
Gˆ(0)( 1) Gˆ(0)(0)
(cid:18) − (cid:19) B. (110) wires
Fig 5 displays the LDOS spectra calculated on several
layers for (210) surface. There is no zero-energy peak 1. Normal metal wires
observedonanychainatall,forthe reasonsexplainedin
thebeginningofthissubsection. ThepeaksintheLDOS, Basedonintuitionfromthesurfacecase,wewouldex-
seen close to the (210) surface at energies 0.5∆ , pectthat Andreevstates playanimportantrolein(110)
max
±
originatefromthe gapfeatures takenfor the momentum wires, with geometry shown in Figure 7. A crucial ques-
alongthesurfacenormal: ∆(k ,k =0)=0.5∆. The tion which arises in the following discussionis, how does
F,x F,y
peakatω =∆(k ,0)arisesforthehomogeneousmodel one identify a subgap state of true Andreev character?
F,x
8
N=4 N=5
0.4 0.4
pnu=r1e n=1
0.3 0.3 n=3
S
DO 0.2 0.2
L
0.1 0.1
0 0
−1 0 1 −1 0 1
0.4 0.4
n=2 n=2
0.3 0.3
S
DO 0.2 0.2
L
0.1 0.1
0 0
−1 0 1 −1 0 1
FIG.6: Localdensityofstatesρ(x,ω)vs. ω/∆maxfora(100)
wire of width N = 4 (left panels) and N = 5 (right panels), FIG.7: Geometryof110wire(sitesarefilledcircles)ofwidth
using ∆0 =0.2t, µ=0. Nboundedby2impuritylines(filledsquares). Wireisinfinite
in both y and −y directions.
BymerelymeasuringtheLDOSwithanSTM,forexam-
ple,onemayseeseveralpeakstructures,notallofwhich
and two special solutions (ν = 0 and N + 1) with
willberelatedtoAndreevreflectionsatthesurfaces. One
ω = q(k ) which we refer to as defining the ”effec-
setofcandidatestateswhichneedstobeinvestigatedfirst ± y
tive band edge”. One interesting feature is the exis-
is the set of discrete dispersive (with respect to k ) lev-
y tence in the normal metal wire of dispersionless zero-
els which arise simply because of the finite wire width.
energyquasiparticlestates (ZES)whichformthe branch
These are of course present already in the normal state
ν =(N+1)/2whenthewidthN ofthewireisodd. Since
wire.
thegroupvelocityvanishesfordispersionlessstates,they
The k -integrated bare normal state Green’s function
x are always localized. We note that the case N = 1 has
for an infinite lattice above T takes the form :
c onlythe obviousdispersionlessZESsince no transportis
G(101)(n,ky,ω) = 2dπ Z−ππ//dd (ωei+kxin0d)d−kxξk aeorlnlaoelwsNeud=chwodliotdhcacolainszleey,dintsetaaaprtepeset(adnroseuitgbhhlaybtodtrhehgeoreenpepisrianatgle.wiaInnysptheaxertagiccetlnley--
iexp[in arccos( ω/q(k ))]
y holespace)foranyfixedk . Adeviationfromhalffilling
= | | − ,(27) y
− q2(k ) ω2 shifts the zero-energy states to µ. Thus, for positive µ
y
− −
theyaretheholestates,while forµ<0-electronstates.
where q(k ) is defined in Eq.(2p0). The full Green’s func-
y
tionfora(110)wiremaythenbeobtainedbysolvingthe
T-matrix equation as Normal state Superconducting state
0.6 n=1 n=1
G (n,n′;k ,ω)= 2sin[nminz]sin[(nmax−N −1)z] n=3 0.3 n=3
11 y
q2(ky)−ω2sin[(N +1)z] (28) LDOS 0.3
p
where n,n′ = 1,2,...,N, z z(k ,ω) =
y
cos−1(−ω/q(ky)), nmin = min(n,n≡′), and nmax = 0−10 0 10 0 −1 0 1
max(n,n′).
0.4 0.2
For every value of k , there generally exists a series of n=2
y
eigenvalues which are the poles of the Green’s function. 0.3 n=2 0.15
They may be obtained by simply solving DOS 0.2 0.1
L
sin[(N +1)z(ky,ω)]=0 , (29) 0.1 0.05
which yields: 0 0
−10 0 10 −1 0 1
πν
ω (k )= q(k )cos , ν =0,1,...,N +1 . (30) FIG.8: Localdensityofstatesfora(110) wireofwidthN =
ν y y
− N +1
5, with ∆0 = 0.2t, µ = 0. Left: normal state, ρ(x,ω) vs.
This gives N branches of solutions for ν = 1,...,N, dis- ω/∆max; right: superconducting state, ρ(x,ω) vs. ω/∆max.
tributed symmetrically with respect to the Fermi level,
9
The contribution of each of these states to the LDOS k ) due to the finite width of the wire and transforms
y
canbeestimatedbyexaminingtheresiduenearthepole, into conventional continuous quasiparticle spectrum in
where the Green’s function can be approximated as: the massive normal metal in the limit N . The
→ ∞
momentumresolvedLDOSinthenormalmetalwirewith
Q (n,n′)
G(n,n′;k ,ω) ν , (31) discrete dispersive states takes the form
y
≈ ω ω (k )
ν y
− N
with the residue given by: ρ(n,k ,ω)= Q (n=n′)δ(ω ω (k )) . (34)
y ν ν y
−
ν=1
2sin nπν sinn′πν X
Q (n,n′)= N+1 N+1, (32) The integration over k gives the LDOS:
ν y
N +1
1 Q (n,n)
where n is the index of the layers. Note from (32) and ρ(n,ω)= ν , (35)
thespectralrepresentationoftheGreen’sfunctionneara 2πd dων
pole,itiseasytoreadoffthequasiparticlewavefunctions ων(Xky)=ω (cid:12)dky(cid:12)
as ψ (n) = 2/(N +1)sin[nπν/(N +1)], i.e. just the (cid:12) (cid:12)
ν wherethesumistakenoverthosek (cid:12)andν(cid:12),whichsatisfy
wave functions of a free particle confined to a box of y(cid:12) (cid:12)
p the equation ω (k )=ω.
width (N +1)d. ν y
The position of peaks in LDOS are then determined
0.5 by extrema of the dispersive branches (30), taking place
0.6 n=1 0.4 n=1 at ky =0:
0.5
OS 0.4 0.3 ω = 4tcos πν , ν =1,...,N . (36)
LD 0.3 0.2 ν,peaks − N +1
0.2
0.1
0.1 The peaks corresponding to the N normal metal wire
0−10 0 10 0 −1 0 1 states are seen clearly in Figs. 8 and 9 at the eigenfre-
0.4 0.4 quencies given by Eq. (36). It is easy to check that the
n=2 n=2
pure weights agree with Eq. (32). The LDOS for the normal
0.3 0.3
metal wires qualitatively differs from the LDOS for bulk
DOS 0.2 0.2 normalmetals with the nearestneighbor hopping on the
L
square. The zero-energy peak (the Van Hove singular-
0.1 0.1
ity) in the bulk metal is symmetric as a function of the
0 0 energy and its log singularity is much broader than the
−10 0 10 −1 0 1
δ-like peaks we find here for wires.
FIG. 9: LDOS for a (110) wire of width N = 4 with ∆0 =
0.2t, µ = 0. Left: normal state, ρ(x,ω) vs. ω/∆max; right:
superconductingstate, ρ(x,ω) vs. ω/∆max. 2. Superconducting wires
The T-matrixandGreen’s function equations are nec-
Itiseasytocheckthatforanyn,n′theresidueQ van-
ν essarily more complicated in the presence of supercon-
ishes everywhere for the ”states” at the effective band-
ductivity, but they are still tractable in the (110) case.
width ν =0 or N +1; we therefore do not discuss them
The bare Green’s functions Gˆ(0) are given by Eqs. (16)
further, but focus on the N branches with finite residue.
and (17), as before. One must then solve Eq. (26),
For these states the residue can vanish locally on chains
which applies to any situation which requires two lines
with number n, for which the quantity nν/(N +1) is an
ofimpurities,fortheeigenenergiesω. Similartothenor-
integer. The odd N ZES, for example, has a residue
mal metal case, there are special solutions of Eq. (26):
2(sinnπ)2 ω = ∆(ky), ω = q(ky) for any N and ω = 0 for even
Q(N+1)(n,n)= N +21 . (33) N. T±hey are pole±s of the T-matrix, but do not corre-
2
spond to the poles of the full Green’s function.
As is seen from Eq.(33), the probability density of the The behavior of subgap surface states on the narrow
ZES oscillates with a period 2d, taking finite values only superconducting wire differs qualitatively from the case
for odd n and vanishing on all nearest neighbor sites of the superconducting halfspace due to the interference
( where n is even ). This is also valid for the states with ofthewavefunctionsofthestatesonbothsurfaces. Since
energies µ in the case of the deviation from half-filling the zero-energypeak in the LDOS for the half-filled sur-
±
and ensures no transport with nearest neighbor hopping face vanishes on each even chain ( see Figs.(3), (4)), the
for quasiparticles with energies µ. spectrum of Andreev states on the (110) wires becomes
±
Allthesestatesarenotsurfaceboundstates,asonecan strongly dependent on the parity of the number N of
easily check that their amplitude does not decay across chains in the half-filled wire. This effect is quite pro-
thewire. Thequasiparticlespectrumisdiscrete(forfixed nouncedforwireswhosewidthistheorderoforlessthan
10
the superconducting coherence length. As we demon-
stratebelow,somenewqualitativefeaturesarisinginthe
superconductingstateofthe (110)wiresinthe quasipar-
ticle spectrum above ∆(k ) ( see Eq. (19)) can also be
y
strongly dependent on the parity of N.
Odd N. ForawirewithoddN,onlytheGreen’sfunc-
tions (16) with even arguments n = 0, (N +1) enter
±
Eq.(26). Since at small frequencies these Green’s func-
tion are ω, it is straightforwardto show that the only
∝
subgap state is a dispersionless ZES ω = 0. Dispersive
modes exist as well and take the form
πν πν
ω2(k )=q2(k )cos2 +∆2(k )sin2 ,
ν y y N +1 y N +1
N 1
ν =1,..., − .(37)
2
Thisexactlycoincideswiththe quasiparticlespectrum
in a bulk two-dimensional superconductor ω2(k ,k ) =
y x
ξ2(k ,k )+∆2(k ,k ), if the discrete values of momen-
y x y x
tum component across the wire k = πν/(N + 1)d
x,ν
are introduced. We note that the dispersionless zero-
energyAndreev states are the only true subgapstates in
the spectrum, since the energy ω (k ) of the dispersive
ν y
| |
states (37) lie above the respective value ∆(k ,k ) of
y x
| |
the order parameter, for any ν and k .
y
Thereare,asinthenormalstatecase,N 1dispersive
−
modes in additionto the ZES.They aredoubly degener-
ateduetotheparticle-holesymmetry. Thissimpleresult
canbeunderstoodasfollows. Forafixedk the problem
y
reduces to a one-dimensional problem. The correspond-
ing two-componentBogoliubov-DeGennes wave function
takes its values on N sites, resulting in 2N degrees of
freedom in the system. With this point of view it ap-
pearsnaturalthatforfixedk thetotalnumberoflevels,
y
which are twice degenerate, is N. The set of levels with
positive energies for N = 11 is represented in panel (a)
of Fig.10.
The LDOS for the superconducting wires with N=5
is shown in the right panel of Fig. 8. The dispersion-
less ZES results in a pronounced peak at zero energy on
FIG. 10: Quasiparticle spectra and the LDOS for supercon-
odd layers. Furthermore, each extremum of the disper-
ducting half-filled (110) wire with N = 11. (a) Dispersive
sive mode ω(k ) results in the peak in the LDOS at the
energy ω y. One series of peaks is associated with modes (kyd vs. ω/t) for N =11, ∆0 =0.2t, µ=0. (b) The
ky,extr blowupofthedispersivespectraneartheedgeoftheBrillouin
the extrema at k = 0. Since ∆(k = 0) = 0, the peaks
y y zone. (c) The LDOS for energies less than ∆max: n=2 (solid
lie at the same positions as in the normal metal wire
line), n=1 (dashed line), where a broadening δ = .005t has
(36). Although they are irrelevant to the superconduct- been used.
ing properties of the wire, some of these peaks can lie
at finite energies below the maximum of the gap func-
tion ∆ = 2∆ . For instance, the lowest position at
max 0
finite energies of the peaks of this series is 4tsin π . seriesof(N 1)quasiparticlepeaksinthe LDOSforthe
N+1 superconduc−ting wires, which lie below ∆ :
This can be both above or below ∆ , depending on max
max
the ratio ∆ /t and the wire width N. In contrast
max
with the normal metal wires, the dispersive quasiparti- πν N 1
cle modes Eq.(37) in the superconducting wires have ex- ωpeaks = ∆maxsin , ν =1,..., − . (38)
± N +1 2
trema also at the edge of the surface-adapted Brillouin
zone k = π/(2d) [57]. They are shown in panel (b) of Panel (c) of Fig.10 displays the series of peaks in the
y
±
Fig.10forthewirewithN =11. Thisleadstoadditional LDOS for N =11.