Microscopic theory of equilibrium properties of F/S/F trilayers with weak ferromagnets

0.39 MB·English

by  R. Mélin

Microscopic theory of equilibrium properties of
F/S/F trilayers with weak ferromagnets
R. M´elin∗
Centre de Recherches sur les Tr`es Basses Temp´eratures (CRTBT)†,
Boˆıte Postale 166, F-38042 Grenoble Cedex 9, France
4
0
The aim of this paper is to explain the non monotonic temperature dependence of the self-
0
consistentsuperconductinggapofferromagnet/superconductor/ferromagnet (F/S/F)trilayerswith
2
weak ferromagnets in the parallel alignment (equivalent to F/S bilayers). We show that this is

n due to Andreev bound states that compete with the formation of a minigap. Using a recursive
a algorithmwediscussindetailtherolesofvariousparameters(thicknessesofthesuperconductorand
J
ferromagnets, relative spin orientation of the ferromagnets, exchange field, temperature, disorder,

3 interface transparencies).
2
PACSnumbers: 74.78.Na,74.45.+c,74.50.+r
]
l
l
a I. INTRODUCTION netic layers15,16,17,18,19. The bad quality of interfaces
h
may however also play a role in these experiments20,21.
-
s Direct evidences of the π-coupling have been obtained
e Inconventionalsuperconductivitythe attractiveinter-
recently22,23,24,25. The proximity effect at F/Sinterfaces
m action mediated by phonons binds electrons into Cooper
isrelatedtotheformationoftheso-calledAndreevbound
. pairs that condense in the BCS groundstate with a zero
t states. Andreev bound states were first discussed by
a temperaturegap∆tothefirstquasiparticleexcitations1.
de Gennes and Saint-James26 and Andreev27 for normal
m In ferromagnetism electron interactions generate a spin
metal(N)/Sinterfaces. Severaltheoreticalinvestigations
- symmetry breaking that can be described by the Stoner ofAndreevboundstatesatF/Sinterfaceshavebeenpre-
d modelinwhichelectronssubjecttoanexchangefieldh
n ex sentedrecently21,28,29 as wellasnumericalinvestigations
acquire a Zeeman energy.
o of the proximity effect at F/S interfaces, based on simu-
c Many physical phenomena are involved at the inter- lations of the Bogoliubov-de Gennes equations30.
[ faces between superconductors (Ss) and ferromagnets
The interest in the inverse proximity effect (the prop-
2 (Fs). For instance it was shown in the early 1970’s that ertiesofthesuperconductor)atF/Sinterfacesdatesback
v the Fermi surface spin polarization of a ferromagnetic tothe1960’s31,32,33whereitwasshownthatwithinsulat-
9 metal could be measured by spin-resolved tunneling be-
ing ferromagnets an exchange field is induced in the su-
2 tween a ferromagnet and a superconducting film in the
0 presence of Zeeman splitting2. The Fermi surface spin perconductingelectrodeofaF/S/Ftrilayerinthe paral-
2 lelalignment. Thetheoreticalpredictionbyde Gennes31
polarization was measured more recently3,4 by Andreev
1 was further confirmed by experiments in the late 1960’s
reflection at F/S interfaces5 with highly transparent in-
3 with metallic32 and insulating33 ferromagnets. Recent
0 terfaces, not in thin film geometries. The non equilib- experimentswerecarriedoutwithmetallicferromagnets,
/ riumspinpopulationintheferromagnetplaysalsoarole
t whichconfirmedthe effect onthe criticaltemperature34.
a in Andreev reflection at F/S interfaces6,7,8. Andreev re-
It was shown recently35,36 that an exchange field in the
m flection with Zeeman splitting in a thin film geometry
superconductor exists also with metallic ferromagnets
- could be another way to probe the Fermi surface spin
but the sign of the magnetization in the superconduc-
d polarization9.
n tor is opposite to the magnetization in the ferromagnet.
o These experiments can be interpreted without dis- An exchange field in a superconductor is pair breaking.
c cussing the proximity effect (the pair correlations in- As a consequence the superconducting transition tem-
v: duced in the normal metal or ferromagnet) that has fo- perature in the parallel alignment is smaller than in the
i cused an important interest recently. It is well estab- antiparallel alignment.
X
lished both theoretically and experimentally that the
With metallic ferromagnets the proximity effect may
r pair amplitude induced in a ferromagnetic metal oscil-
a influence the value of the superconducting gapandtran-
lates in space and can become negative10,11,12,13,14, giv-
sition temperature. It was realized recently36,37,38,39,40
ing rise to the π-coupling. The π-coupling generates
that the proximity effect in F/S/F trilayers is quite
oscillations of the critical temperature of F/S multi-
special since under some conditions the zero temper-
layers as a function of the thickness of the ferromag-
ature superconducting gap in the parallel alignment
can be larger than in the antiparallel alignment. This
was established36,37 within a model of multitermi-
nal hybrid structure originally proposed for transport
∗E-mail: [email protected]
†U.P.R.5001duCNRS,Laboratoireconventionn´eavecl’Universit´e properties41,42,43,44,45 and is related to spatially sepa-
JosephFourier ratedsuperconducting correlationsamong the two ferro-
2
magnets. The same behavior was found within a model D
of F/S/F trilayer with atomic thickness and half-metal
ferromagnets39, which was finally extended to Stoner
ferromagnets40. H
(a)
F S F
y
Other physical effects take place with weak ferromag-
x
nets for which the exchange field is smaller than the su-
perconducting gap, as indicated by the reentranceof the z L L
0 F L F
critical temperature of F/S superlattices as a function S
of the exchange field46,47, and by the reentrance of the
critical temperature of F/S bilayers as a function of the
thickness of the ferromagnet38,48,49. The full tempera-
(b) W S F
ture dependence of the self-consistent superconducting
gap was calculated in Ref. 40 and it was shown that for
F/SbilayersorF/S/Ftrilayerswithatomicthicknessthe
superconductinggapcanhaveanonmonotonictempera-
d’
ture dependence: withina givenrangeofparametersthe d
superconducting gap first increases as temperature T is
reduced,reachesa maximum, anddecreasesto zero asT FIG. 1: (a) Schematic representation of the F/S/F trilayer.
D and H are sent to to infinity (infinite planar geometry).
is further reduced. Within a narrow range of interface
transparencies there exists also a reentrant behavior of Theferromagnets(superconductor)aremadeofLF (LS)lay-
ers stacked in the lateral (z) direction. (b) Schematic repre-
the superconducting gap at low temperature40.
sentation of the 2D F/S interface model29 considered in sec-
tion VIIB. d′ is sent to infinity.
The purpose of this article is to show that this behav-
ior is related to the formation of Andreev bound states Ferromagnet Superconductor Ferromagnet
that can compete with the formation of a superconduct- L =4 L =3 L =4
F S F
ing minigap. We find a systematic correlation between
Andreev boundstatesatthe Fermilevelanda reduction
of the low temperature self-consistent superconducting
gap, which constitutes the main result of this article.
y
The article is organized as follows. Preliminaries are
given in section II in which we discuss the models and Layer a’ Layer a Layerα Layer β Layer b Layer b’
the Green’s functions formalism. A recursive algorithm,
0 z
more efficient that the direct inversion of the Dyson
matrix40, is presented in section III. Different regimes
FIG. 2: Cut in the (z,y) plane of the F/S/F trilayer with
canbeobtaineddependingonhowthelateraldimensions
L of the superconductor and L of the ferromagnets (LS/a0,LF/a0) = (3,4), where a0 is the interplane spacing.
S F The ferromagnets and the superconductor are infinite in the
compare respectively (i) to the zero temperature lateral
x and y directions. The left ferromagnet ends at the layers
superconducting coherence length ξ(⊥) = 2t a /π∆28, a′ and a. The right ferromagnet ends at the layers b and b′.
S 0 0
where t is the lateral hoping amplitude, and a is equal The superconductorends at layers α and β.
0 0
to the interatomic distance in the tranverse direction;
and (ii) to the zero temperature lateral exchange length
ξ(⊥) = 2t a /πh 28. We present a detailed investiga-
F 0 0 ex II. PRELIMINARIES
tion of the different regimes L ,L ξ(⊥) (section IV);
F S ≪ S
L . ξ(⊥) and L ξ(⊥) (section V); L ξ(⊥) and
LS & ξS(⊥) (sectioFn V≪I).FWe cannot carrySou≪t a Ssystem- A. The models
F F
atic study of the regime L ξ(⊥) , L & ξ(⊥) which is
S ∼ S F F
too demanding from a computational point of view. In We suppose that the two ferromagnets of the F/S/F
section VII we provide a comparison with other models trilayer have the same thickness L . We note L the
F S
proposed recently to describe Andreev bound states in thickness of the superconductor (see Fig. 1). The three
F/S hybrids28,29. Concluding remarks are given in sec- electrodesaremadeof2Dplanesstackedalongthez axis
tion VIII. (see Fig. 2). Eachsuperconducting plane is describedby
3
the BCS Hamiltonian B. Green’s functions
(2D) = ǫ(k)c+ c (1)
HBCS k,σ k,σ 1. Zero temperature Green’s functions
k,σ
X
+ ∆ c+ c+ +c c , The Green’s functions of an isolated superconduc-
k,↑ −k,↓ k,↓ −k,↑
tor can be gathered in a 4 4 matrix in the spin
Xk (cid:16) (cid:17) × ⊗
Namburepresentationbutintheabsenceofnoncollinear
where the wave vector k is parallel to the 2D layer. The magnetizations44,50,51,52 the quantization axis can be
freeelectrondispersionrelationwithinonelayerisǫ(k)=
chosen parallel to the exchange field so that the 4 4
~2k2/2m. The ferromagnets are described by the Stoner ×
Green’s functions reduce to two separate 2 2 matrices,
model ×
one in each spin sector. For practical purpose we work
in the spin-up sector. The Green’s function is given by
(2D) = ǫ(k)c+ c (2)
HStoner Xk,σ k,σ k,σ gˆx,y(t,t′)=−i (5)
h c+ c c+ c , T c (t),c+ (t′) T (c (t),c (t′))
− ex k,↑ k,↑− k,↓ k,↓ h t x,↑ y,↑ i h t x,↑ y,↓ i ,
Xk (cid:16) (cid:17)  T (cid:16)c+ (t),c+ (t′)(cid:17) T c+ (t),c (t′) 
h t x,↓ y,↑ i h t x,↓ y,↓ i
where hex is the exchange field. The coupling between  (cid:16) (cid:17) (cid:16) (cid:17) 
where x and y are two arbitrary sites in the supercon-
twoplanesbelongingtothesamesuperconductingorfer-
ductor and T is the usual T-product53. The “11” com-
romagnetic electrode is given by t
ponent describes the propagation of a spin-up electron,
= =t c+ c +c+ c , the “22”component describes the propagationofa spin-
WF−F WS−S 0 xL,σ xR,σ xR,σ xL,σ down hole and the “12” and “21” components describe
xXL,σ(cid:0) (cid:1) superconductingcorrelations. AfterFouriertransforming
(3)
weobtainastandardexpressionforthedifferentelements
whereas the coupling between the ferromagnets and the
of the Green’s function53:
superconductor is given by
u2 v2
WF−S =t c+xL,σcxR,σ+c+xR,σcxL,σ , (4) gα1,,1α(ξ,ω) = ω−Ekk+iη + ω+Ekk−iη (6)
xXL,σ(cid:0) (cid:1) f1,2(ξ,ω) = ∆ . (7)
where the summation runs over all sites at the inter- α,α −[ω Ek+iη][ω+Ek iη]
− −
face (xL corresponds to a site on the left side and xR The variable ξ is related to the kinetic energy: ξk =
is the corresponding site on the right side). The pa- ~2k2/(2m) ǫ whereǫ =~2k2/2mistheFermienergy.
rameters are such that the coherence length within one E = ∆2−+Fξ2istheqFuasiparFticleenergyandu2 =(1+
lξa(y⊥e)r=ξS(2kt) =a /~πvF(∆k)/a∆realanrdgetrhtehtarnantvheerwseidctohheLrenocfetlheengstuh- fξakkc/tEor)sp/.2 and vk2k = (1−ξk/E)/2 are the BCS cokherence
S 0 0 S
perconductor. Themodelshouldstricktlyspeakingapply The Green’s function of a spin-up ferromagnet is di-
to ballistic systems whereas real samples are usually in agonal in Nambu space. The “11” component is given
the diffusive regime. However we can include disorder by
in one particular case (the F/S/F trilayer with atomic
1
thickness, see section IVC). In this case the qualitative g1,1(ξ,ω)= , (8)
a,a [ω ξ+h +iη sgn(ξ h )]
behavior is robust against increasing disorder. Fabry- − ex − ex
Perot resonances generate parity effects for small values andtheGreen’sfunctionsofaspin-downferromagnetare
ofL andL inthe ballistic model. These parityeffects obtained by changing h into h .
are Snot expFected to occur in the diffusive regime, but The Green’s functionesxGˆx,y−of etxhe connected trilayer
do not occur either in our simulations if LS and LF are aregivenbytheDysonequationGˆ =gˆ+gˆ Σˆ Gˆ,where
sufficiently large. By increasing LF and LS we obtain ina compactnotationΣˆ is the self-energy⊗corr⊗esponding
a cross-over between the regime LF,LS . λF and the to the tunnel Hamiltonian (4) and corresponds to a
regime λ . L ,L ξ(⊥). We find the same quali- summation over spatial variables and⊗a convolution over
F F S ≪ S
tative effects in the two regimes. Moreover we obtain in time variables. We look for non perturbative solutions
sectionVanonmonotonicvariationoftheself-consistent of the Dyson equations suitable for describing Andreev
superconducting gap as a function of the exchange field. bound states.
This is compatible with the non monotonic variation of
the critical temperature as a function of the exchange
field38,48,49 obtained in the context of linearized Usadel 2. Finite temperature Green’s functions
equations for disordered conductors. The compatibility
between the two behaviors indicates the validity of our Finite temperature Green’s functions are obtained
approach. through the analytic continuation ω iω and by sum-
→
4
ming over the Matsubara frequencies ωn = (2n+1)πT B. Green’s functions of the connected trilayer
where T is the temperature53. The superconducting gap
is determined from the BCS self-consistency equation53 We note Gˆ the Green’s functions of the connected
F/S/F trilayer (see Fig. 2). Layers a and α are con-
d2k nected by a tunnel amplitude t = t and layers b
∆ =λT G1,2(k,iω ), (9) a,α α,a
x (2π)2 x,x n and β are connected by a tunnel amplitude t = t .
b,β β,b
n Z
X We use the notation t=t =t and denote by t the
a,α b,β 0
tunnelamplitudewithineachlayer(seeEq.3andEq.4).
whereλisthestrengthoftheattractiveelectron-electron
The definition of the tunnel Hamiltonian that we use
interaction. To evaluate (9) we change variable to ξ =
~2k2/(2m) ǫ and restrict the integral to ξ < ω . for the F/S/F trilayer is slightly different from the con-
F D
− | | ventional definition given by (4):
To avoidintroducing new parameters we use ω =ǫ =
D F
~2k2/2m. We note ∆ the superconducting gap of an t
F 0 = c+ c +c+ c (12)
isolated 2D layer. All energy scales will be compared to WFSF √2 x,y,α,σ x,y,a,σ x,y,a,σ x,y,α,σ
x,σ
∆0. X(cid:0) (cid:1)
t
+ c+ c +c+ c .
√2 x,y,β,σ x,y,b,σ x,y,b,σ x,y,β,σ
Xx,σ(cid:16) (cid:17)
III. RECURSIVE ALGORITHM FOR THE With the factors √2 the F/S/F trilayer with L /a = 1
S 0
F/S/F TRILAYER in the parallelalignment is equivalent to the F/S bilayer
with L /a =1 and a tunnel amplitude equal to t. The
S 0
A. Green’s functions of an isolated electrode correspondence between the bi and trilayer is useful for
checking the numerical simulations. For L /a 2 we
S 0
≥
carry out the simulations of the F/S/F trilayer but we
We aim to describe F/S/F trilayers with a fi-
notethatintheparallelalignmentthequalitativepredic-
nite thickness of all electrodes. In this respect the
tions are valid also for F/S bilayers as long as the thick-
(L /a ,L /a ) = (1,1) trilayer is just viewed as a toy-
S 0 F 0 ness of the superconductor is smaller than the coherence
model: we use a mean field approach that does not in-
length.
corporate the phase fluctuations of the order parameter
We note Kˆ = hˆ tˆ hˆ tˆ , Kˆ =
andwe do notconsiderpossible instabilities suchasspin α,α α,α α,a a,a a,α β,β
or charge density wave. hˆβ,βtˆβ,bhˆb,btˆb,β, Kˆα,β = hˆα,βtˆβ,bhˆb,btˆb,β, Kˆβ,α =
The ferromagnetic and superconducting electrodes of hˆβ,αtˆα,ahˆa,atˆa,α. The Green’s function Gˆα,α is given by
the F/S/F trilayer consist of a finite number of layers −1
−1
stacked along the z axis and labeled from 1 to L (see Gˆ = Iˆ Kˆ Kˆ Iˆ Kˆ Kˆ (13)
α,α α,α α,β β,β β,α
− − −
Fig.2). Twoconsecutivelayersnandn+1arecoupledby (cid:20) h i (cid:21)
atunnelamplitudet . WeuseGreen’sfunctionsthatare −1
n ˆh +Kˆ Iˆ Kˆ ˆh ,
parametrized by the wave vector in the (x,y) direction × α,α α,β − β,β β,α
and by the spatial coordinate in the lateral direction. (cid:20) h i (cid:21)
and Gˆ is given by
We note hˆ(L) the Green’s functions of the system of L β,α
i,j
layers in a given electrode and gˆi,i the Green’s function Gˆ = Iˆ Kˆ −1 hˆ +Kˆ Gˆ . (14)
oftheisolatedlayernumberi. TheGreen’sfunctionˆh(L) β,α − β,β β,α β,α α,α
L,L h i h i
canbecalculatedrecursivelythroughamatrixcontinued TheGreen’sfunctionGˆ isdeducedfromGˆ through
α,β β,α
fraction: the relation Gˆτ1,τ2 = Gˆτ2,τ1, where τ and τ are the
α,β β,α 1 2
ˆh(L) = Iˆ gˆ tˆ hˆ(L−1) tˆ −1gˆ . (10) Nambu indexes. Gˆβ,β is given by
L,L − L,L L−1 L−1,L−1 L−1 L,L −1
h i Gˆβ,β = Iˆ Kˆβ,β hˆβ,β +Kˆβ,αGˆα,β . (15)
−
The local Green’s functions of the system of L stacked h i h i
WededucethevaluesofGˆ ,Gˆ ,Gˆ andGˆ aswell
layers are obtained through a,a b,b a,b b,a
as G in the superconductor:
i,i
hˆ(L) =ˆh(L−1)+hˆ(L−1)tˆ ˆh(L)tˆ hˆ(L−1), (11) Gˆ = hˆ +hˆ tˆ Gˆ tˆ hˆ +hˆ tˆ Gˆ tˆ hˆ
i,i i,i i,L−1 L−1 L,L L−1 L−1,i i,i i,i i,α α,a a,a a,α α,i i,α α,a a,b b,β β,i
+ hˆ tˆ Gˆ tˆ ˆh +hˆ tˆ Gˆ tˆ hˆ . (16)
i,β β,b b,a a,α α,i i,β β,b b,b b,β β,i
where hˆ(L) and ˆh(L) are calculated recursively through
i,L L,i To obtain the pair amplitude in the ferromagnets and
the relations hˆ(LL,i) = hˆ(LL,L)tˆL−1hˆL(L−−11,i) and hˆ(i,LL) = superconductor we first calculate recursively the ˆh’s and
hˆi(,LL−−11)tˆL−1hˆL(L,L). The computation time required to ob- next evaluate Gˆ1,2. The evaluation of the self-consistent
superconducting gap is done either by dichotomy if
tain hˆ(L) is proportional to L whereas it is proportional
L,L LS/a0 =1,2orbyiterationsoftheself-consistencyequa-
to L2 if one wants to calculate all the ˆh(L). tion (9) if L /a 3.
i,i S 0 ≥
5
1
0.75 ttt /// ∆∆∆t 000/ ===∆ 0000 =...578 1706 t tttt/ ////∆ ∆∆∆∆0 0000= ==== 1 57891.....70371 0.5 (att //) ∆∆00 == 10..47 (ctt //) ∆∆00 == 10..47 (et /)t ∆/ 0∆ =0 =2 .08
0.5
(LS,LF) = (1,1) (LS,LF) = (1,2) s) 0.25
0.25 nit
0 (a) (b) u
∆∆(T) / 0.705 tttt //// ∆∆∆∆0000 ==== 0011....89123715 ttttt ///// ∆∆∆∆∆00000 ===== 24578.....82603 OS (arb. 0.05 (btt //) ∆∆00 == 22..81 (dtt //) ∆∆00 == 22..81 (ftt )// ∆∆00 == 85..36
0.5 D
(LS,LF) = (1,3) (LS,LF) = (1,4) L
0.25
0.25
(c) (d)
0 0
0 0.25 0.5 0.75 0 0.25 0.5 0.75 1
-2.5 0 2.5-2.5 0 2.5-2.5 0 2.5
T / ∆
0 E / ∆
0
FIG.3: Temperaturedependenceoftheself-consistentsuper-
FIG.4: Energydependenceofthespin-upLDOS(inarbitrary
conducting gap for the F/S/F trilayer in the parallel align-
units) in the superconducting layer of the F/S/F trilayer in
ment with (LS/a0,LF/a0) = (1,1) (a), (1,2) (b), (1,3) (c),
(1,4) (d), with weak ferromagnets (hex/∆0 = 0.83). LF the parallel alignment with (LS/a0,LF/a0) = (1,3). The
superconducting gap, equal to ∆ , is not self-consistent. (a)
and LS are small compared to ξS(⊥,0) = 2t0a0/π∆0 = 17.7a0, and(b)correspondtoh /∆ =00,ξ(⊥) =∞(N/Sinterface).
ξF(⊥) = 2t0a0/πhex = 21.2a0 and λF = 6.28a0. We use (c)and(d)correspondetxoh 0/∆ =F0.83, ξ(⊥) =21.2a (F/S
∆0/ǫF =0.014 and t0/ǫF =0.4. interface with weak ferromeaxgnet0s). (e) anFd (f) corre0spond
to h /∆ = 13.9, ξ(⊥) = 1.3a (F/S interface with strong
ex 0 F 0
IV. F/S/F TRILAYERS WITH LS,LF ≪ξS(⊥),ξF(⊥) f8e.r3r×om1a0g−n3e,tλs)F. =We6.u28sea0∆.0/ǫF =0.014. t0/ǫF =0.4, η/∆0 =
In this section we consider the regime L ,L
S F
≪
ξ(⊥),ξ(⊥), establishaconnectionbetweenthe LDOSand nant case (see Figs. 3-(a) and (c)). This is because the
S F
the self-consistent superconducting gap, and show that effective couplingbetweenthe superconductorandferro-
the regime h /∆ 1 is characterized by Andreev magnets is much weaker in the off-resonant case due to
ex 0
bound states compet∼ing with the formation of a mini- the differences in the density of states.
gap.
2. Role of the exchange field
A. Self-consistent superconducting gap for weak
ferromagnets in the parallel alignment We repeated the simulations for the F/S/F trilayer
with (L /a ,L /a ) = (2,2) but with different values
S 0 F 0
1. Role of the thicknesses of the electrodes of h /∆ . For the smallest value of h /∆ (h /∆ =
ex 0 ex 0 ex 0
0.28)we obtaineda monotonicdecreaseof the supercon-
ductinggapasafunctionoftemperatureforallvaluesof
We have shownonFig. 3 the temperature dependence
the tunnel amplitude. Forh /∆ =0.56,0.83,1.11,1.39
of the superconducting gap for values of (L ,L ) such ex 0
S F
that L ,L ξ(⊥),ξ(⊥). We used L /a = 1 on Fig. 3 weobtainedanonmonotonictemperaturedependenceof
S F ≪ S F S 0 the superconductinggapsimilarto Fig.3-(a)andFig.3-
but similar results were obtained with L /a = 2. As
S 0 (b). Thenonmonotonicvariationofthesuperconducting
expected from Appendix A the breakdown of supercon-
gap occurs typically for h being a fraction of ∆ up to
ductivity in the resonant F/S/F trilayer in the parallel ex 0
values slightly above ∆ .
alignment resembles the case (L /a ,L /a ) = (1,1)40: 0
S 0 F 0
as temperature is reduced the superconducting gap in-
creases, reaches a maximum and decreases to zero. We
B. Relation between the local density of states and
obtain a reentrant behavior in a narrow range of inter-
the self-consistent superconducting gap
face transparencies (not shown on Fig. 3). By contrast
we obtain a monotonic behavior for off-resonant values
of (L /a ,L /a ), as expected from Appendix A. 1. Local density of states in the parallel alignment
S 0 F 0
The values of the tunnel amplitude needed to destroy
superconductivity for off-resonant trilayers (see Figs. 3- The spin-up LDOS ρ (ω) in the superconductor is
↑
(b) and (d)) is almost ten times larger than in the reso- shown on Fig. 4 for the F/S/F trilayer in the paral-
6
lel alignment, for (LS/a0,LF/a0) = (1,3). We ob- 0.3
tained similar results for (LS/a0,LF/a0) = (2,2). The TT/∆/∆0 = = 0 0.0.1576
spin-down LDOS ρ (ω) is obtained through the relation T/∆0 = 0.28
↓ ) 0
s
ρ↓(ω) = ρ↑(−ω). The zero temperature superconduct- nit
ing gap ∆ is fixed to the BCS value ∆0 of an isolated u 0.2

superconducting layer. b.
r
The caseof a N/S interface is shownonFig. 4-(a) and a
(

Fig. 4-(b). The LDOS is symmetric with respect to a S
O 0.1
change of sign in energy, as expected in the absence of
D
an exchange field. There exists a minigap at the Fermi L
energy so that superconductivity is robust against in-
creasing the tunnel amplitude t. Increasing t gives rise 0
-2.5 0 2.5
topairsofAndreevboundstatesatoppositeenergies(see E/∆
Fig.4-(a)). Eachpeakcorrespondstoaminiband,which 0
is visible on Fig. 4-(b) obtained with larger values of t.
FIG. 5: Energy dependence of the finite temperature
The formation of the minibands is related to the infinite
spin-up LDOS (in arbitrary units) in the superconducting
planar geometry: the layers are infinite in the x and y
layer of the F/S/F trilayer in the parallel alignment with
directions (see Fig. 1) and there is thus a degeneracyas- (LS/a0,LF/a0) = (1,3). The zero temperature LDOS cor-
sociated to the position of the Andreev bound state in responds to Fig. 4-(c) with t/∆ =1.4.
0
the (x,y) plane. The bound state can delocalize in the
(x,y) plane and thus acquire a dispersion which would
not occur if the dimensions H and D (see Fig. 1) were temperature LDOS is found to be
smallcomparedtotheBCScoherencelength,asituation
considered in Ref. 28. ρ(ω′)
ρ (ω)= dω′ , (17)
For weak ferromagnets with t, h and ∆ having the T 4T cosh2 ω′−ω
ex Z 2T
same order of magnitude (panel (c) on Fig. 4) we obtain
(cid:0) (cid:1)
one Andreev bound state miniband inside the gap and where ρ(ω) is equal to the zero temperature LDOS.
one resonant scattering state above the gap. The An- We calculated the finite temperature LDOS of the
dreev bound state miniband moves to the Fermi energy (L /a ,L /a )=(1,3)trilayerintheparallelalignment
S 0 F 0
as the tunnel amplitude t increases (see Fig. 4-(c)) and (see Fig. 5). Increasing temperature tends to reduce the
appears at a positive energy for larger values of t (see intensity of the Andreev bound state at the Fermi en-
Fig. 4-(d)). ergyon Fig.4-(c) so that the LDOS atthe Fermienergy
For larger values of the exchange field the Andreev decreases if T increases. For relatively large values of
bound state miniband disappears from the LDOS. For T the peak structure has almost disappeared from the
h /∆ 1 the induced exchange field is opposite to LDOS but there remains a minimum associated to the
ex 0
the magn≫etization in the ferromagnets35,36. superconducting gap that is significantly filled. As T in-
creases there is thus a first cross-overwhere the peak at
There are thus two qualitatively different depairing
theFermienergydisappearsandasuperconductingmini-
mechanisms for strong and weak ferromagnets. For
gapis restored,andasecondcross-overwherethe super-
strong ferromagnets with h /∆ 1 (see Figs. 4-(e)
ex 0
≫ conducting gap disappears. The first cross over occurs
and 4-(f)) the breakdown of superconductivity is due
atatemperature equalto the bandwidth ofthe Andreev
mainlytoZeemansplitting. Thecaseofveryweakferro-
bound state miniband and the second cross-over occurs
magnets (h /∆ 1) resembles that case h /∆ = 0:
ex 0 ≪ ex 0 at a temperature comparable to the zero temperature
non perturbative Andreev bound states are generatedin
superconducting gap. The finite temperature LDOS is
the superconducting gap approximately at opposite en-
thus in a qualitative agreement with the non monotonic
ergies. In the case h /∆ 1 the Andreev bound state
ex 0
∼ self-consistent superconducting gap.
minibandcanbeatzeroenergy,thereforedestructingthe
Reentrance obtained in a narrow range of interface
minigap.
transparencies40canalsobeexplainedbythisqualitative
picture: if the Andreev bound state miniband is narrow
and located at a slightly positive energy like on Fig. 4-
(c) then the LDOS at the Fermi energy is vanishingly
2. Finite temperature local density of states
small at T = 0 since the Fermi energy is not in the An-
dreev bound state miniband. By increasing T the width
The spin-up LDOS at a finite temperature T is re- of the Andreev bound state miniband increases so that
latedtheconductanceofascanningtunnelingmicroscope the density of states at the Fermi energy increases. By
(STM)inwhichthetipismadeofahalf-metalferromag- further increasing T the intensity of the Andreev bound
net. The finite temperature currentatan arbitraryvolt- stateminibandisreduced,andthedensityofstatesatthe
age is obtained through Keldysh formalism. The finite Fermienergydecreases. Thisbehavioriscompatiblewith
7
1
nits) 0.5 t( at/ ∆/) ∆0 0= = 0 1.7.40 t( b/t ∆/) ∆0 0= = 0 .17.04 t( ct/ /∆) ∆0 0= = 1 81..31 0.75 tt // ∆∆00 == 00..7907 ttt /// ∆∆∆000 === 001...792075
u
b.
ar 0.25 0.5
(
S
O 0.25
D 0 (a) (b)
L 0 -2.5 0 2.5-2.5 E 0/ ∆0 2.5-2.5 0 2.5 ∆∆(T) / 0.705 tttt //// ∆∆∆∆0000 ==== 0111....92587531 t ttt/ ///∆ ∆∆∆0 000= === 1 5821....6381
FIG. 6: Energy dependence of the spin-up LDOS (in arbi-
traryunits)inthesuperconductinglayeroftheF/S/Ftrilayer 0.5
in the antiparallel alignment with (LS/a0,LF/a0) = (1,3).
The superconductinggap, equalto ∆ , is not self-consistent. 0.25
0
(a) and (b) correspond to h /∆ = 0.83, ξ(⊥) = 21.1a (c) (d)
ex 0 F 0
(weak ferromagnets). (c) corresponds to h /∆ = 2.78, 0
ex 0 0 0.25 0.5 0.75 0 0.25 0.5 0.75 1
tξ0F(⊥/ǫ)F==6.04.a40, η(s/t∆ro0n=gf8e.r3ro×m1a0g−n3etasn).dWλFeu=se6d.2∆8a00/.ǫF =0.014, T / ∆0
FIG. 7: Temperature dependence of the self-consistent su-
perconductinggapoftheF/S/Ftrilayerintheparallelalign-
a reentrantbehavior of the self-consistent superconduct-
mentwith(LS/a0,LF/a0)=(1,1)inthepresenceofdisorder
ing gap. The correlation between the low temperature
in the superconducting layer. Without vertex corrections we
superconducting gap and the LDOS at the Fermi energy use h /∆ = 0.83, δ(0) = 6.2 (a), δ(0) = 7.6 (b), δ(0) = 9.8
ex 0 S S S
is further established in section VI. (c). (d) corresponds to δ(0) =9.8 with vertex corrections. In
S
all cases we use δ(0) =0.
F
3. Local density of states in the parallel alignment
ferromagnets. We obtain a significant effect of disorder
The spin-up LDOS in the antiparallel alignment is for relatively large values of δ(0) and δ(0).
shown on Fig. 6. The LDOS is symmetric with respect S F
to a change of sign of energy, but not equivalent to the
LDOS of a N/S interface (see Fig. 4-(a) and Fig. 4-(b)).
1. Disorder in the superconducting and ferromagnetic
Thereexistsawell-definedminimumattheFermienergy
layers
corresponding to the superconducting minigap. The en-
ergy dependence of the LDOS in the antiparallel align-
The temperature dependence of the self-consistent su-
ment shows that superconductivity is stronger than in
perconducting gap in the presence of disorder in the su-
the parallel alignment, both for weak and strong ferro-
perconductor is shown on Fig. 7 for (L /a ,L /a ) =
magnets. This is expected because of the exchange field S 0 F 0
(1,1) in the parallel alignment. In the absence of disor-
induced in the superconductor in the antiparallel align-
ment is reduced compared to the parallel alignment31. derFig.7correspondstoFig.3-(a). Theeffectofdisorder
is to reduce the effect of the tunnel amplitude40 so that
Theself-consistentsuperconductinggaphasamonotonic
if disorder increases a larger value of t is needed to de-
temperature dependence in this case which is because a
stroysuperconductivity. ComparingFig.3andFig.7we
well-defined minigap is obtained in the LDOS.
see that the variations of ∆(T) are affected by a weak
disorder especially if the F/S/F trilayer in the parallel
alignment is close to the breakdown of superconductiv-
C. Role of disorder for weak ferromagnets and for
ity. In this case the dimensionless parameter controlling
the F/S/F trilayer with atomic thickness
(0)
the strength of disorder is δ = δ ∆ /∆(T) which can
S S 0
Disorder plays an important role in the proximity ef- be much larger than δ(0).
S
fect. Diffusive N/S interfaces are described by quasi-
classical theory54. A small disorder can be incorporated
in our description based on microscopic Green’s func- 2. Vertex corrections
tions like in Ref. 53 (see Appendix B). The strength
of disorder in the superconductor is characterized by
There exist two perturbative series: one in the hop-
δS(0) = nαu2α/∆0, where nα is the concentration of im- pingamplitudet andthe otherinthe disorderscattering
purities and uα is the scattering potential, and we use potential u. Vertex corrections arise from diagrams that
p
a similar parameter δ(0) to characterize disorder in the mix the two series (see Appendix B2). The tempera-
F
8
ature dependence of the self-consistent superconducting
0.75 hex/∆0 = 0.56 gap (not shown on Fig. 8).
0 0.5
∆
∆ / 0.25 FERRVOI.MAFGINNIETTEST(HLSIC≪KξNS(⊥E)SASNINDTLHF E&ξF(⊥))
(a) T/∆0 = 0.14 (b)
0 We discuss now the regime L ξ(⊥) and L &ξ(⊥).
0 0.5 1 1.5 0 0.25 0.5 S ≪ S F F
h / ∆ T / ∆ We find Andreev bound states at the Fermi energy in
ex 0 0
theparallelalignmentforstrongferromagnets,correlated
FIG. 8: (a) Variation of the self-consistent superconduct- withnonmonotonictemperaturedependencesoftheself-
ing gap calculated at T/∆ = 0.14 in the parallel (P) align- consistent superconducting gap.
0
ment (open symbols) and in the antiparallel (AP) align- The regime L & ξ is characterized by oscillations
F F
ment (filled symbols) as a function of hex/∆0. (b) Vari- of the self-consistent superconducting gap, critical tem-
ation of the self-consistent superconducting gap calculated perature and pair amplitude as a function of L . We
F
withhex/∆0 =0.56intheparallelalignment,asafunctionof obtainboundstateswithin thesuperconducting gapand
T/∆0. Weusedinbothcases(LF/a0,LS/a0)=(11,11). The resonant scattering states outside the superconducting
Fermi wave length is λF =6.28a0. We use t/∆0 =5.6 ((cid:3) in gap. We calculated systematically the LDOS and the
thePalignmentand(cid:4)intheAPalignment),t/∆ =8.3(◦in
0 self-consistentsuperconductinggapfor L /a =1,2and
thePalignmentand•intheAPalignment),andt/∆ =11.1 S 0
0
(△ in the Palignment and N in theAPalignment). LF/a0 =1,...,100, in the parallel and antiparallel align-
ments (see Fig. 9). The systematic calculation of the
finite temperature LDOS given by Eq. (17) is too de-
manding from a computational point of view. Instead
ture dependence of the self-consistent superconducting
we calculate the LDOS at zero temperature with η =T.
gap with the vertex corrections is shown on Fig. 7-(d).
Theconsistencybetweenthetwocalculationswasverified
The critical temperature is larger if vertex corrections
in a few cases. Depending on the interface transparen-
are included. We obtain a non monotonic temperature
cies∆(L /a )/∆ tendseithertoafinitevalueortozero
dependence ofthe superconductinggapeveninthe pres- F 0 0
in the limit L /a + . We concentrate on the first
ence of vertexcorrections. The role ofvertex corrections F 0 → ∞
case only. For 1 L /a 40 we see on Fig. 9 that
increasesift increasessince the vertexcorrectiontermis F 0
≤ ≤
proportional to n t2u2. on average the superconducting gap is smaller when the
α α LDOS at the Fermi energy is larger. For L /a =1 and
S 0
t/∆ = 2.8 (see Fig. 9-(b)) we obtain ∆(T)/∆ = 0
0 0
for four values of L /a (L /a = 4,6,11,13). For
F 0 F 0
V. FINITE THICKNESS IN THE
three other values of L /a (L /a = 16,23,25)we ob-
SUPERCONDUCTOR (LS .ξS(⊥) AND LF ≪ξF(⊥)) tain a non monotonicFvari0atioFn of0∆(T)/∆ . For two
0
other values of L /a (L /a = 32 and L /a = 35)
F 0 F 0 F 0
In this section we include a finite thickness in the su- ∆(T)/∆ is monotonic but far from the BCS variation.
perconductor for LF ≪ ξF(⊥). For weak ferromagnets in These ni0ne values of LF/a0 with an anomalous temper-
theregimehex/∆0 1weobtainnonmonotonictemper- ature dependence of the self-consistent superconducting
∼
aturedependences ofthe self-consistentsuperconducting gap have large values of the LDOS at the Fermi energy
gap in the parallel alignment, therefore confirming sec- (ρ(ω =0)>0.045onFig.9-(b),inarbitraryunits). Like
tion IV. in section IVB1 Andreev bound states near the Fermi
We have shown on Fig. 8-(a) the variation of the self- energycorrelatewithunconventionaltemperaturedepen-
consistent superconducting gap in the middle of the su- dences of the superconducting gap.
perconductor as a function of the reduced exchangefield For 40 L /a 100 corresponding to L ξ(⊥),
h /∆ . As expected from the variation of the critical ≤ F 0 ≤ F ≫ F
ex 0 we obtain a cloud of points with a small dispersion and
temperature38,47,48 weobtainaminimumiftheexchange
withnocorrelationbetweenthesuperconductinggapand
fieldiscomparabletothesuperconductinggap. Moreover
the LDOS at the Fermi energy. This corresponds to the
we obtain a minimum also in the antiparallel alignment
cross-overto the F/S/F trilayer with bulk ferromagnets.
thatisrelatedtotheminigapformedinbetweenAndreev
We obtain bound states within the superconducting
bound states at opposite energies (see section IVB3).
gapand unconventionaltemperature dependences of the
The full temperature dependence of the self-consistent
superconductinggapwithstrongferromagnetsinthepar-
superconducting gap in the parallel alignment is shown
allelalignment. Bycontrastwithstrongferromagnetswe
onFig.8-(b)forLS/a0 =11(comparabletoξS(⊥)/a0)and obtainaconventionaltemperaturedependenceofthe su-
L /a = 11 (much smaller than ξ(⊥)/a ). We obtain a perconducting gap for the F/S/F trilayer with smaller
F 0 F 0
maximum in the variation of ∆(T)/∆ in the parallel values of (L /a ,L /a )40.
0 S 0 F 0
alignment. We carried out the same simulation in the We carriedout the same simulation in the antiparallel
antiparallel alignment and found a monotonic temper- alignment and found that ∆(T)/∆ is close to the BCS
0
9
0.15 1
(a) (c) (e)
0.1 0.5
∆
nits)0.05 E / 0
u
b. 0 -0.5
ar (a) (b)
S ( (b) (d) (f) -1
O 0.1 0 1000 2000 3000 4000 0 1000 2000 3000 4000 5000
D LF/a0
L
0.05
FIG.10: ReducedenergyE/∆oftheAndreevboundstatesas
afunction of thenumberof sites LF/a0 in theferromagnetic
0 chain for weak ferromagnets (h /∆ = 0.5). We used the
0 0.5 1 0 0.5 1 0 0.5 1 ex
∆/∆ parameters ∆/t = ∆0/tF = 0.01, t/tS = 0.01 , LS = 105a0.
0
The entire spectrum is kept in (a). Only the levels with a
residue larger than 2 are kept in (b). The superconducting
FIG. 9: Correlation between ∆/∆ and the LDOS at the
0 gap is not self-consistent.
Fermi energy (in arbitrary units) for LS/a0 = 1 and LF/a0
between 1 and 40. The temperature is T/∆ = 0.14. In the
0
calculation of theLDOSthe superconductinggap is not self-
consistent. We use strong ferromagnets with h /∆ = 13.9
ex 0
and ξF(⊥) = 1.3a0. We use LS/a0 = 1 and t/∆0 = 2.1 (a);
LS/a0 = 1 and t/∆0 = 2.8 (b); LS/a0 = 2 and t/∆0 = 3.5 A. A one dimensional model
(c); LS/a0 = 2 and t/∆0 = 4.2 (d); LS/a0 = 1 and t/∆0 =
2.8 (e); LS/a0 = 1 and t/∆0 = 3.3 (f). (a), (b), (c), (d)
correspond to the parallel alignment. (e) and (f) correspond
Wesupposethatthesuperconductorandferromagnets
totheantiparallelalignment. Thesolidlinein(b)corresponds
to ρ(ω=0)=0.045 (see text). are described by 1D chains with open boundary condi-
tions with L /a sites in the superconductor and L /a
S 0 F 0
sites in the ferromagnet. We denote by t (t ) the hop-
S F
pingamplitudeinthesuperconductor(ferromagnet)and
temperature dependence of the superconducting gap for we use t = t . The energy level spacing in the super-
S F
all values of LS/a0 between 1 and 100. A larger density conductorismuchsmallerthanthesuperconductinggap
of states at the Fermi energycorrelateswith smaller val- ∆.
ues of the self-consistentsuperconducting gaplike in the
parallel alignment (see Fig. 9-(e) and (f)). Because of
We have shown on Fig. 10 the evolution of the energy
the minigap that can be formed since the bound states
of Andreev bound states as a function of the length of
are at opposite energies (see Fig. 6) we do not obtain
theferromagneticchainobtainedwiththealgorithmpre-
the points with large values of the density of states at
sented in Appendix C. The spectrum is symmetric un-
the Fermi energy like in the parallel alignment. The ab-
derachangeofsigninenergyifwe keepallenergylevels
sence of large values of the LDOS at the Fermi energy
intoaccount. Andreevboundstatesarisingfromthegap
is related to the fact that the Andreev levels in the an- edges move to the Fermi energy as L /a increases28.
F 0
tiparallelalignmentdo notcrossthe Fermienergyasthe
ThereislevelrepulsionbetweenAndreevboundstateand
interface transparency is increased.
oscillationsof the energylevels with a periodof order10
times ξ = 2t a /πh . These oscillations do not exist
F F 0 ex
in Ref. 28. Among all Andreev bound states obtainedat
a fixed L /a some have a spectral weight much larger
F 0
thantheother. Wekeeponlythelevelshavingaspectral
VII. COMPARISON WITH OTHER MODELS
weight larger than a given cut-off. The evolution of the
remaining Andreev bound states as a function of L /a
F 0
Thediscussioninthe precedingsectionswasrestricted is then in agreement with Ref. 28.
tothe infinite planargeometry. NowwediscussAndreev
bound states in other geometries without imposing self- In connection with the discussion in section IVB1 we
consistencyonthesuperconductinggap. InsectionVIIA see that Andreev bound states near the Fermi energy
we improve the discussion of a model proposed by re- occur only if the ferromagnetic chain is long enough,
cently Vecino et al.28 in which a ferromagnetic chain is largerthanapproximately10timesξ =2t a /π∆. This
S S 0
connected to a superconductor55. In section VIIB we should be contrasted with the LDOS in the infinite pla-
compare the LDOS to a model discussed recently by narlimit discussedin sectionIVB1,andwith the model
Cserti et al.29. discussed in section VIIB.
10
a minigap, which explains why the self-consistent super-
s) 300 (a) (b) conducting gap is weakened by the formation of an An-
nit dreev bound state miniband near the Fermi energy. In-
u
b. 200 Z = 1000 Z = 40 creasing temperature decreases the intensity of the An-
ar Z = 200 Z = 8 dreev bound state peak in the LDOS which correlates
∆E/) ( 100 wsuiptheracornedduuccttiinogngoafpt.he low temperature self-consistent
ρ( We found non monotonic temperature dependences
0 of the superconducting gap with weak ferromagnets for
-1 0 - 11 0 1
E/∆ L ,L ξ(⊥),ξ(⊥) as well as for L ξ(⊥) and
F S ≪ F S S ∼ S
L ξ(⊥). In the regime L ξ(⊥) and L & ξ(⊥)
FIG.11: DensityofAndreevboundstates(inarbitraryunits) F ≪ F S ≪ S F F
we obtain non monotonic temperature dependences for
ρ(E/∆) as a function of reduced energy E/∆ for Z = 1000
some values of L /a for strong ferromagnets.
and Z = 200 (a); Z = 40 and Z = 8 (b). We used the F 0
FortheF/S/Ftrilayerintheparallelalignmentwefind
parameters ∆/ǫF =0.02, ∆ =0.01, hex/∆=0.5, d=λF/2,
W ≃15900λF where λF =2π/kF is theFermi wave length. pairs of Andreev bound states at opposite energies but
there exists a minigap so that the self-consistent super-
conducting gap decreases monotonically with tempera-
ture. However at a fixed temperature the self-consistent
B. Bogoliubov-de Gennes equations
superconducting gap is non monotonic as a function of
theexchangefieldwhichisduetopairsofAndreevbound
WeconsidernowthemodelproposedbyCsertietal.29
states at opposite energies in the regime h /∆ 1
ex 0
inwhicha2Dferromagneticdotwitharectangularshape ∼ ≪
t/∆ .
of dimensions (d,W) is connected to a superconductor 0
Concerning the induced exchange field in the F/S/F
(see Fig.1-(b)). The superconductorhas a width W but
trilayer in the parallel alignment we find that for weak
isinfiniteintheotherdirection(d′ =+ onFig.1). Us-
∞ ferromagnets the exchange field is in the same direction
ing the solution of this model29 based on Bogoliubov-de
asintheferromagneticelectrodeswhereasitisintheop-
Gennes equations we generatedthe set E of Andreev
{ n} posite direction for strong ferromagnets. The change of
boundstateenergies. LikeforthedeGennes-SaintJames
sign is visible both in the LDOS and in the magnetiza-
model26 the density of bound states is not equal to the
tion profile. As the exchange field in the ferromagnets is
LDOSinthesuperconductor28. Theevolutionoftheden-
increased we find a first order transition to the normal
sityofAndreevboundstatesasafunctionoftheinterface
state, as for paramagnetically limited superconducting
transparency is shown on Fig. 11. The interface trans-
films in an applied magnetic field.
parency is parametrized by the dimensionless coefficient
Z, equal to the repulsive interface potential in units of
the Fermi energy56. For large values of Z (correspond-
ACKNOWLEDGMENTS
ing to tunnel interfaces) we obtain an Andreev bound
state miniband around E/∆ = h /∆ . As the inter-
0 ex 0
face transparency decreases the miniband broadens and The author wishes to thank D. Feinberg for fruitful
splits into two separate minibands (see Fig. 11). This is discussions and useful comments on the manuscript.
inaqualitativeagreementwithsectionIVB1(seeFig.4-
(c) and Fig. 4-(d)) where we obtain also Andreev bound
state minibands that evolve inside the superconducting APPENDIX A: RESONANCES IN LATERAL
gap and can generate a large density of states at the CONFINEMENT
Fermienergyforsomevaluesoftheinterfacetransparen-
cies. In this Appendix we account for the differences be-
tweentheresonantandoff-resonantF/S/Ftrilayers. The
wavefunctioninthelateraldirectionwithinagivenelec-
VIII. CONCLUSIONS trode is described by the tight binding Hamiltonian
=t [z+1 z + z z+1], (A1)
We havepresentedadetailedanalysisofF/S/Ftrilay- H 0 | ih | | ih |
erswithweakferromagnetsbasedonmicroscopicGreen’s Xz
functions. The exchange field induced in the supercon-
where t is the hopping between neighboring layers,and
0
ductorintheregimeh /∆ 1generatesZeemansplit-
ex 0 z/a isanintegerbetween0toL/a 1. Theeigenstates
≫ 0 0
tingoftheLDOStheregimeh /∆ 1ischaracterized −
ex 0 with open boundary conditions are given by
≪
byAndreevboundstates. Inthenonperturbativeregime
t/∆ h /∆ 1 the Andreev bound state miniband
0 ∼ ex 0 ∼ 2a L/a0−1 z+a
crosses the Fermi energy as the interface transparency ψ = 0 sin nπ 0 z , (A2)
n
is increased therefore competing with the formation of | i rL+a0 z=0 (cid:18) L+a0(cid:19)| i
X