Embedding Dynamical Mean-Field Theory for Superconductivity in Layered Materials and Heterostructures

0.48 MB·

by  Francesco Petocchi

Embedding Dynamical Mean-Field Theory for Superconductivity in Layered
Materials and Heterostructures
Francesco Petocchi1 and Massimo Capone1
1InternationalSchoolforAdvancedStudies(SISSA/ISAS)andCNR-IOMDemocritos,ViaBonomea265,34136,Trieste,Italy
Westudylayeredsystemsandheterostructuresofs-wavesuperconductorsbymeansofasuitable
generalization of Dynamical Mean-Field Theory. In order to reduce the computational effort, we
consider an embedding scheme in which a relatively small number of active layers is embedded in
aneffectivepotentialaccountingfortheeffectoftherestofthesystem. Weintroduceafeedbackof
the active layers on the embedding potential that improves on previous approaches and essentially
eliminatestheeffectsofthefinitenessoftheactiveslaballowingforcheapcomputationofverylarge
systems. We extend the method to the superconducting state, and we benchmark the approach
6
by means of simple paradigmatic examples showing some examples on how an interface affects the
1
superconducting properties. As examples, we show that superconductivity can penetrate from an
0
intermediate coupling superconductor into a weaker coupling one for around ten layers, and that
2
the first two layers of a system with repulsive interaction can turn superconducting by proximity

n effects even when charge redistribution is inhibited.
a
J
PACSnumbers: 71.10.Fd,71.10.-w,74.78.Fk,74.25.-q

3
1
I. INTRODUCTION count for the geometrical arrangement of these systems.

] The intrinsic difficulty to solve interacting systems be-
l
e yond the perturbative regime limits the number of ac-
- The advances in manufacturing and handling het-
cessible approaches and prompts for the use of suit-
r erostructures are in the forefront of solid state research.
t able approximations. Dynamical Mean-Field Theory
s Inparticularheterostructurebasedonoxideshaveahuge
. (DMFT)hasindeeddemonstratedtoaccuratelytreatthe
t potentialthankstotherichphysicsoftheirconstituents.
a competinginteractionscharacterizingoxideinterfacesin-
m Combining different oxides one can even engineer and cluding electron-electron interactions7, electron-phonon
tailor electronic and magnetic states which can be com-
- coupling8–11 and their interplay12–20 as well as for the
d pletelydifferentfromthoseofthebulkconstituents. The attractive Hubbard model21–28.
n possibility to control these emergent and intrinsic prop-
The extension of DMFT to treat surface and interface
o erties of the constituents opens an avenue towards the
c realization of new devices based on correlated electrons. effects has been pioneered by Potthoff and Nolting29,30
[ who introduced a layer generalization of DMFT and
One of the most studied examples of the novel physics applied it to a solid-vacuum interface in the presence
1
at oxide interfaces is the appearance of a high-mobility of short-range Coulomb interaction as described by the
v
2 electron gas at the interface between the band insu- Hubbard model. Including also long-range Coulomb in-
3 lator SrTiO3 (STO) and the Mott insulator LaTiO3 teractions Okamoto and Millis31,32 and Kancharla and
2 (LTO)1. This nearly two-dimensional metal can be eas- Dagotto33 have considered charge-transfer effects and
3 ily manipulated through gate voltages and turned into proposed that the charge leakage from one layer to an-
0 a superconductor2,3 which strikingly appears combining other is responsible for the metallic interface between
1. two non-superconducting materials. Superconductivity LTO and STO. As we discuss in more details in the fol-
0 has also been observed at interfaces between two band lowing, when DMFT is extended to inhomogeneous sys-
6 insulators such as STO and LaAlO34, while interfaces tems, the inclusion of more and more layers is the bot-
1 between different copper-based superconductors have a tleneck of the calculation. Therefore the main limitation
v: criticaltemperaturehigherthanthebulkconstituents5,6. of these approaches is the influence of finite-size effects
i Thesearejustexamplesofavarietyofphenomenainvolv- andtheslowconvergencetothebulklimit(infinitenum-
X
ing superconductivity in artificially crafted heterostruc- ber of layers). Ishida and Liebsch have proposed and
ar tures. The aim of this work is to develop a reliable for- implemented34,35 a strategy to overcome this limit. The
malism to study superconductivity in heterostructures ideaistoeffectivelydescribeasubstratewithanenergy-
beyond simplifying limits such as the Bardeen-Cooper- dependent embedding potential.
Schrieffer (BCS) approximation. In order to test the
In this work we extend the embedding potential to
method and to single out the effect of intermediate- and
study superconductivity with s-wave symmetry within
strong-coupling,weconsiderasimpleattractiveHubbard
eachlayerandweintroducea“feedback”effectwhichim-
model as a simple paradigmatic model for an s-wave su-
proves the performance of the embedding method. The
perconductor.
purpose of this paper is to demonstrate the feasibility
The theoretical description of interacting heterostruc- of this approach for superconducting state and to study
tures requires methods which are at the same time able the evolution of the physics as a function of the coupling
to treat the relevant interactions and to effectively ac- strength. Theextensiontoclustermethods,whichisnec-
2
essarytostudyd-wavesuperconductivityisconceptually Hubbard model, may be written as
simple but computationally demanding.
(cid:104) (cid:16) (cid:17)(cid:105)
The paper is organized as follows. In Sec. II we intro- H =(cid:80)ns ε cˆ† cˆ +V cˆ† dˆ +h.c.
ducethemodelandthegeneralconceptoflayeredDMFT imp lσ l lσ lσ l lσ oσ (2)
andtheembeddingapproach. Sec. IIIisdedicatedtothe +∆2l (cˆlσcˆlσ¯ +h.c.)(cid:3)−Unˆo↑nˆo↓−µ(nˆo↑+nˆo↓)
extension of the approach to superconducting systems
and to our recipe for the embedding potential. Sec. IV where cˆ† creates a particle in l − th level of a non-
lσ
describesourresultsfordifferentphysicalconfigurations, interacting bath which is parameterized by the energy
while Sec. V contains conclusions and perspectives. levels ε and the superconducting amplitudes ∆ and by
l l
the hybridizations V . The amplitudes ∆ give rise to
l l
an anomalous (superconducting) component of the hy-
bridization function between the impurity and the bath
II. MODEL AND METHOD
which is necessary to treat the superconducting phase.
Solving the impurity model and computing the normal
In this section we introduce our approach to extend and anomalous Green’s functions G = −(cid:10)Tc(τ)c†(cid:11) and
previous DMFT-based approaches to heterostructure to F = −(cid:10)Tc†(τ)c†(cid:11), we can obtain the impurity self-
allowforsuperconductivity. Forthesakeofclarity,inthe
energy as Σˆ = Gˆ−1 −Gˆ−1 , where the hat denotes
firstsubsectionwebrieflyreviewtheDMFTformalismin imp o imp
2× 2 matrices whose components are given by the nor-
the superconducting state and some aspects of the exact
mal and anomalous Greens’ functions
diagonalization (ED) solution of DMFT that we employ
in our practical implementation. (cid:18) (cid:19)
G(iω ) F (iω )
Gˆ = n n (3)
α F (iω ) −G(iω )(cid:63)
n n
A. Single-site DMFT and superconductivity and analogously for the two components of the “Weiss
field”, which coincide with the non-interacting Green’s
DynamicalMean-FieldTheoryisoneofthemostpop- functions of (2). Gˆo:
ular and successful theoretical methods to treat strongly
cmboyerarmenla-afipteepldidngaeplaepcrtlaorotatncihcestyomsttoehdmeeslq.ounatnIottuaemnxtdiemnydnpsaumrtithicyeamlcdloaodsmesliaciiannl Go−(111F)(o−i(ω112n))(=iωniω)n=+(cid:80)µnl−s|(cid:80)Vl|nl2sω|V2+l|∆ε22lω+n2∆i+ω2nε.2l++ε∆l 2l (4)
n l l
whichaninteractinglatticesiteishybridizedwithanon-
interacting bath which is self-consistently determined. The DMFT approximation is enforced re-
In this section we present the DMFT formalism for quiring that the local Green’s functions de-
superconducting solutions starting from the attractive fined above coincide with the local components
Hubbard model, which can be considered the simplest of the lattice Green’s function Gˆlat(iωn) =
model for an s-wave superconductor. However the same (cid:82) dερ(ε)(cid:104)iω 1(2)+(µ−ε)σˆ −Σˆ (iω )(cid:105)−1, be-
n 3 imp n
equation would be found for example for an electron-
ing ρ(ε) the non interacting density of states.
phonon model or even for models without an explicit
A practical implementation of DMFT requires to re-
source of pairing.
cursively solve the impurity model calculating G and F.
The Hamiltonian reads
Thisallowstocomputetheself-energymatrixandanew
H=−t (cid:88) (c† c +h.c.)−(cid:88)µ(n +n ) Weiss field Gˆo−1 = Σˆimp +Gˆ−la1t. The process is iterated
iσ jσ i↑ i↓ until the Weiss fields and the other quantities are con-
<ij>σ i
verged. A central issue in DMFT calculation is indeed
(cid:88)
−U n n , (1) the solution of the impurity model. Here we use an ex-
i↑ i↓
i act diagonalization “solver”, in which the groundstate of
the impurity Hamiltonian is found using a Lanczos algo-
where the sums run over the sites i and j of a lattice, rithm. In order to obtain a finite matrix, the sums over
ciσ(c†iσ) are annihilation (creator) operators for fermions l are truncated to a finite and small value Nb. Nonethe-
withspinσ onsitei,tisanearest-neighborhoppingam- less, small values of N have been shown to be sufficient
b
plitude, U is a positive energy measuring the strength to obtain converged results for thermodynamic observ-
of the on-site attractive interaction and µ is the chem- ables. TheEDsolutionofDMFTinvolvesonemorestep
ical potential. This model is known to have an s-wave with respect to the algorithm we described. After the
superconducting ground state for any value of the cou- self-consistencyconditionisusedtofindnewWeissfield,
pling U and it has been extensively studied by means of thesefunctionsneedtobecastintheform(4)withadis-
DMFT21–28. cretevalueofN . Thiscanbeachievedbyfittingthenew
b
Asmentionedabove,withinDMFTthelatticemodelis Weiss fields with Eq. (4) which has to be interpreted as
mapped onto an impurity model which, for an attractive a function of the “Anderson parameters” ε , V and ∆ .
l l l
3
B. Observables where 1(N) is the N-dimensional identity matrix and σˆ
3
is the third Pauli matrix. 2N×2N matrices are identified
To characterize the superconducting states of our lay- by a bar. The single-particle dispersion matrix is given
ered superconductor and its spatial dependence we use by:
several observables. The most direct evidence of the su-  ε(cid:107)σˆ ε⊥σˆ 0 
perconductingstateanditsstrengthisthelayer-resolved 1 3 12 3
zero-temperature pairing amplitude, simply obtained as ε¯k =ε⊥21σˆ3 ... ...  (9)
the integral of the anomalous part of the α −th layer 0 ... ε(cid:107)Nσˆ3
Green’ function
whose elements are: ε(cid:107) =−2t [cos(k )+cos(k )] and
αα α x y
∆α =T (cid:88)Fα(iωn). (5) ε⊥αβ = tz. The self-energy matrix is instead a block-
diagonal matrix
n
The nature of the superconducting state (for example Σˆ 0 0 
1
if the system is in an effective weak- or strong-coupling Σ¯(iωn)= 0 ... 0  (10)
regime)canbecharacterizeintermsofthedifferentcon- 0 0 Σˆ
N
tributiontothetotalenergy. Thelayer-resolvedpotential
energy is simply where each element is a 2 × 2 block with normal and
anomalous components corresponding to the local self-
Eα =U(cid:104)nˆ nˆ (cid:105) , (6) energy of an individual layer. The underlying approxi-
pot o↑ o↓ α
mation is that the self-energy remains local Σ¯ = δ Σ¯
ij ij i
while the kinetic energy reads: and it is uniform within each layer, while the different
layersareallowedtohavedifferentself-energiesΣ¯ , each
(cid:104)Eα(cid:105)=T (cid:88)(cid:90) d(cid:15)ρ((cid:15))Tr(cid:110)(cid:15)σˆ Gˆ ((cid:15),iω )(cid:111). (7) associated to a local effective impurity problem. αAn ex-
k 3 α n
plicitsolutionrequirestosolveasmanyimpuritymodels
n
as the number of layers, from which the individual self-
Notice that while the global order parameter and poten- energiesareobtainedandpluggedintoEq. (8). Summing
tial energy are simply obtained by summing the contri- over the momenta within each layer leads to a set of lo-
butions from the different layers, the bulk kinetic energy calGreen’sfunctionswhicharethenimposedtocoincide
also includes the contributions from the interlayer hop- with the impurity Green’s functions.
pings, which do not contribute to the above (cid:104)Ek(cid:105). Fi- Asopposedtosingle-siteDMFT,wearethereforelim-
nally we can compute the quasparticle weight, namely ited to a finite system along the z direction, which can
z = (cid:0)1−∂Σ11(iω )/∂(iω )(cid:1)−1. Where Σ11(iω ) is lead to finite-size effects, that are enhanced if we use an
α α n n α n
the normal component of the α − th layer self-energy, open slab, where the electrons on the outmost layers be-
which measures the coherence of the low-energy excita- come effectively more interacting because of the missing
tions. neighbors(Fig.(1)). Apossiblesolutiontoovercomethis
limitation is to sandwich the finite slab of N layers into
two media34,35 effectively accounting for the presence of
C. Superconducting DMFT applied to bulk layers.
heterostructures and embedding potentials Using the notations of Ref.36 we define the matrix in-
verse of (8)
In the previous subsections we introduced single-site
DMFT for bulk superconductors, in which full transla- AˆS(k,iωn)GˆS(k,iωn)=1(2N) (11)
tionalinvarianceisenforcedandanylatticesiteisequiv-
Partitioning the infinite three-dimensional system into a
alent. Inordertostudylayeredsystemsweneedtousea
slab (S) and two “bulk” samples (B and B ) we can
suitableextensionofDMFTabletotreatinhomogeneous R L
rewrite Eq. (11) as
systems with a layered geometry. We focus on a sim-
ple cubic lattice partitioned into N layers stacked along  Aˆ Aˆ 0  Gˆ Gˆ 0 
the (001) direction. Within each layer translational in- BL BL1 BL BL1
variance is assumed and the two-dimensional wavevector Aˆ1BL AˆS AˆBRN Gˆ1BL GˆS GˆBRN =1
k(cid:107) =(kx,ky) is a conserved quantity. 0 AˆNBR AˆBR 0 GˆNBR GˆBR
The Green’s function Gˆ of a slab made of N super- (12)
S
where the matrices are now (2N +4)×(2N +4). The
conducting layers can be expressed as a 2N×2N matrix
diagonal elements of the two matrices are the Green’s
corresponding to the two components of Nambu spinors
functions and the A functions for the slab (S) and the
and to the N layers
two semi-infinite substrate which embed the interacting
(cid:104)(cid:16) (cid:17) (cid:105)−1 slab. The non-zero off-diagonal elements describe the
G¯ (k,iω )= iω 1(2)+µσˆ ⊗1(N)−ε¯ −Σ¯(iω )
S n n 3 k n processes connecting the “left” efffective substrate with
(8) layer1oftheslabandthe“right”substratewithlayerN
4
of the slab. From this we can single out the equation for ∆
the slab Green’s function
0.865
(cid:16) (cid:17) Slab
Aˆ −Aˆ Aˆ−1Aˆ −Aˆ Aˆ−1Aˆ Gˆ =1(2N) Embedding
S 1BL BL BL1 NBR BR BRN S 0.86 Embedding and feeSdSba rcekf
(13)
0.855
whichshowshowthe1andNindicesareaffecteddirectly 0.8372
by the presence of the two semi-infinite bulks. The ex- 0.85
plicit result is 0.8368
0.845
Gˆ (k,iω )=(cid:2)(cid:0)iω 1(2)+µσˆ (cid:1)⊗1(N)−εˆ 0.8364
S n n 3 k 1 2 3 4 5 6 7 8 9 10
(cid:105)−1 0.84
−Σˆ(iω )−δ Sˆ (k,iω )−δ Sˆ (k,iω )
n α1 BL n αN BR n
0.835
(14)
1 2 3 4 5 6 7 8 9 10
wherewehavedefinedthecomplexembeddingpotentials
j
Sˆ (k,iω )=t2σˆ Gˆ (k,iω )σˆ (15)
BL,R n z 3 BL,R n 3 Figure1. (Coloronline)Validationofoutembeddingscheme
acting on the first and last layer only if the interlayer with feedback. We plot ∆ for a 10-layer embedded system
hoppingislimitedtonearestneighbors. Comparingwith with uniform U/t = −9 at half-filling. The red line with
(8) it is evident that the only difference is introduced by squares is for the open slab, the green with dots marks the
resultswiththeembeddingpotential,whilethebluelinewith
the embedding potential at the boundaries of the slab.
triangles denotes the data corrected with the feedback of the
slab. Black crosses on the two sides of the slab report the
bulk DMFT result. The embedding+feedback results are es-
D. Choice of the substrate Green’s functions
sentially uniform.
Here we introduce an optimized strategy to describe
heterostructures in terms of a few “active” layers em- wheretheGreen’sfunctiondependonthemomentaalong
bedded between two semi-infinite systems. The start- the layers k while the integral in the LHS is performed
(cid:107)
ing point is naturally the surface Green’s function of a over the transverse momentum. The self-energy in Eq.
semi-infinite system. We partition a bulk system in two (18) is determined self-consistently solving two more im-
semi-infinite halves along the direction of the layers of puritymodelscoupledwiththeslab. TheGˆ inEq. (17)
0
the heterostructure37. As in the rest of this work, the defines the left embedding potential in Eq. (14). The
Green’sfunctions areassumed tobetranslationalinvari- right potential is obviously identical.
ant along each layer and they are labelled according to Fig. 1 presents results for a ten-layer slab for uniform
the layer index. We denote the surface layer with 0 and, parameters U/t = −9 and half-filling. In the absence of
forthesakeofdefiniteness,wefocusonthe”left”system any embedding potential, the slab breaks translational
with negative layer index. symmetry and the order parameter ∆ becomes larger at
The relation between the Green function of the α−th the edges. Introducing the embedding potential accord-
layerintheleftsemi-infinitebulkGˆα andthesamelayer ingtothedescribedscheme, weobtaintheresultsshown
in the bulk crystal Gˆbαulk can be written as as a dotted green line with large dots in Fig. 1. Here we
Gˆ =Gˆbulk+(cid:88)GˆbulkVˆ Gˆ , (16) considercompletelyuniformparameters,andtheembed-
α α α αβ β ding potential strongly reduces the inhomogeneity, even
β if a minor enhancement of the order parameter is clear
where V are the hopping matrix elements connecting at the edges of the slab.
αβ
the right and left sides with indexes α ∈ 0,−1,−2,... In order to further reduce the effects of the finiteness
and β = 1,2,.... We are only interested in the surface of the slab, in this work we propose a simple strategy to
layer 0 which is connected only with the next layer 1 by improvethescheme,introducingafeedbackoftheslabon
the diagonal (in the Nambu space) hopping matrix Tˆ , thesemi-infinitebulks. Theideaissimplytodefineapo-
01
which leads to tentialcreatedbytheslabontothesemi-infinitebulkson
the two edges. As a matter of fact the equation amounts
(cid:16) (cid:17)−1
Gˆ =Gˆbulk 1(2)−GˆbulkTˆ , (17) to add a potential of the form (15) to the self-energy of
0 0 1 01
each semi-infinite systems
which requires the knowledge of the bulk Green’s func-
tion for the surface layer and for the first layer on the Sˆ (k,iω )=t2σˆ Gˆ (k,iω )σˆ . (19)
FBL,R n z 3 S1,N n 3
left, which can easily be computed within DMFT as
The data in Fig. (1) demonstrate that the feedback
further reduces the inhomogeneity and it allows to es-
(cid:90) dk eikα
Gˆbulk = , sentially reproduce the uniform bulk even with a very
α 2πiω 1(2)+(cid:0)µ−ε(cid:107)−2t cosk(cid:1)σˆ −Σˆ(iω ) limited number of layers.
n z 3 n
(18) In Fig. 2 we demonstrate that our feedback performs
5
∆(a) III. RESULTS
1
0.9
1. Weak/strong interacting superconductor
0.8
0.7
0.6 Inthissectionwepresentsomeresultsusingtheabove
0.5 defined embedding+feedback procedure for an attrac-
0.4 tive Hubbard model. In this work we limit ourselves to
0.3
paradigmatic situations and we postpone to future ap-
0.2
Layer plications more realistic set-ups corresponding to actual
0.1 SS cubic
SS square materialsandheterostructure. Wefixthelocaldensityto
0
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3
one electron per site on each layer by imposing particle-
U/t
hole symmetry. This obviously freezes charge redistri-
Z(b)
bution across the interface. We chose to start with this
1 situationtosingleouttheintrinsiceffectsduetotheprox-
imity from the effects due to charge transfer across the
0.95
interface,whichwouldobviouslyaffecttheresults. Inter-
0.9 estingly, we find important proximity effect even in this
case.
0.85
As a first example we consider the interface between
two semi-infinite systems with different values of the
0.8
Layer
SS cubic attractive interaction, considering ten active layers for
SS square
0.75 both systems. In Fig. 3 we present results in which
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4
U/t we fix the interaction at a relatively small interaction
D(b) U/t = −3 on the left side, while on the right side we
0.45 tune the interaction from U/t=−3 to a much larger at-
0.4 traction U/t = −7.5. We present layer-resolved pairing
amplitude∆ ,quasiparticleweightz ,doubleoccupancy
α α
0.35
D andin-planekineticenergy(cid:104)E (cid:105)asafunctionofthe
α kα
0.3 layer index α. On the right side of the figure the bulk
values are shown for reference.
0.25
We first observe that also in this case our embedding
0.2 Layer scheme provides the correct value of every observable in
SS cubic
SS square thelayersadjacenttotheleads. Theevolutionacrossthe
0.15
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 slabisrathersmooth,especiallyfortheorderparameter,
U/t
shown in panel (a), for which a significant proximity ef-
fect leads to an enhancement of the order parameter on
Figure 2. (a) Order parameter ∆, (b) quasiparticle weight
z, (c) double occupations expectation value D=(cid:104)nˆ nˆ (cid:105) vs the left side which penetrates for almost ten layers. Also
o↑ o↓
interactionstrengthU athalf-filling. Redandbluelinesrefers the right-side is substantially affected by the presence
respectivelytothesquareandcubiclattice,singlesiteDMFT of the weakly-coupled superconductor. Interestingly the
calculations. Greendotsrepresentstheresultsforthecentral spatial variation is not strongly dependent on the value
plane of an homogeneous system made of 7 layers, similar to of the interaction in the right half.
the blue one in Fig.(1). Thedoubleoccupancy, whichisalsorelatedtothepo-
tential energy has a similar evolution, but the proximity
effects are limited to a thinner slice of the slab of around
three layers. A similar behavior is shown by the layer
accurately for different observables and for any value of kinetic energy, which is negative and larger in amplitude
theparameters. Hereweplottheaverageovertheslabof on the left (weak-coupling side). Interestingly, the pres-
∆, Z and of the double occupancy D as a function of U ence of the stronger-coupling superconductors leads to a
and we compare with a bulk cubic lattice (which should slightincreaseofthemodulusofthekineticenergyinthe
be reproduced when the finite-slab effect are canceled) first layers of the weak-coupling side.
and,forreferencewithatwo-dimensionalcalculationcor- Finally, the quasiparticle weight, which can be used
respondingtoasinglelayer. Toillustratethegeneralva- to measure the coherence of the electronic excitations,
lidityofourapproachweconsiderbothanegativeU, for is slightly increased in the weak coupling side, and it
whichwefindsuperconductivity,andapositiveU model, decreases in the strong-coupling region, even if all these
in which s-wave superconductivity can not establish and variations are relatively small.
therefore represents the normal state. The three panels In Fig. 4 we present an analogous analysis in which
of Fig. 2 clearly show that for every value of U the three theleftsidehasaconstantU/t=−10whileontheright
observables coincide with their bulk counterparts. side the interaction goes from -10 to -3. The qualitative
6
∆(a) ∆(a)
0.8
0.9
0.7 -3/-3 0.8
-3/-4
-3/-5
0.6 -3/-6 0.7
-3/-7
0.5 SS ref 0.6 -10/-10
-10/-6
0.5 -10/-5
0.4 -10/-4
-10/-3
0.4 SS ref
0.3
0.3
0.2 0.2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
j j
Z(b) Z(b)
0.95
0.9
0.9
0.85 --33//--34 0.85
0.8 S---S333 ///r---e567f 0.8 -1---0111/000-///1---6540
0.75 -10/-3
SS ref
0.75 0.7
0.65
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
j j
D(c) D(c)
0.42
0.46
0.4 -3/-3 0.44
--33//--45 0.42
0.38 -3/-6
-3/-7 0.4 -10/-10
SS ref -10/-6
0.36 0.38 -10/-5
-10/-4
0.36 -10/-3
0.34 0.34 SS ref
0.32
0.32
0.3
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
j j
-<Ek>(d) -<Ek>(d)
0.65
0.65
0.6 -3/-3 0.6 -10/-10
-3/-4 -10/-6
-3/-5 0.55 -10/-5
-3/-6 -10/-4
0.55 -3/-7 -10/-3
0.5
0.5 0.45
0.4
0.45
0.35
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
j j
Figure 3. ∆ , z , D and −(cid:104)E (cid:105) in a 20-layer thick het- Figure 4. ∆ , z , D and −(cid:104)E (cid:105) in a 20-layer thick het-
α α α k α α α k
erostructure formed by two semi-infinite halves. The left erostructure. Thelefthalf(index≤10)iskeptatU/t=−10,
half (index ≤ 10) is kept at U/t = −3, while for the right whilefortherighthalf(index≥11)weusedsmallerorequal
half (index ≥ 11) we used different values of the attraction values of the attraction strength. The points outside the
with equal or larger absolute value. The points outside the heterostructure are those computed starting from the leads’
heterostructure are those computed starting from the leads’ Green’s functions used to compute the embedding potential
Green’s functions used to compute the embedding potential Gˆ . ThecrossesaretheresultsforabulkDMFTcalculation
B
Gˆ . ThecrossesaretheresultsforabulkDMFTcalculation for the cubic lattice.
B
for the cubic lattice.
7
results are similar to the previous even if the proximity ∆(a)
effects are reduced because of the stronger coupling on 0.7
theleftside, whichleadstoashortcoherencelengthand 0.6 4/0
the physics becomes more local. Still, a clear interme- 4/-3
0.5 4/-4
diate region in which the physical quantities smoothly 4/-5
4/-6
connect. 0.4 SS ref
0.3
0.2
2. Correlated metal/superconductor 0.1
0
We now move to a different situation where one of the 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
j
two halves of the system would not be superconducting
Z(b)
by itself. On the left side we consider a metal with a fi-
1.05
niterepulsionU/t=4,whichwouldleadtoamoderately
1 4/0
correlated metal in a bulk system, while on the right we 4/-3
0.95 4/-4
tune the attractive interaction from 0 to U = −6t. The 4/-5
0.9 4/-6
results,plottedinFig. 5,showthatdespitetherepulsive SS ref
interactionsuperconductivitycanpenetrateforafewlay- 0.85
ers of the metal, and that important effects are observed 0.8
on the superconducting side. This is a clear qualitative 0.75
violation of the local-density approximation even in the 0.7
absenceofchargeredistributionacrosstheinterface. The
0.65
effect on the order parameter is small but clearly visi- 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
j
ble, while the double occupancy is essentially unaffected
D(c)
bytheconnectionbetweenthetwosemi-infinitesystems.
The kinetic energy presents an interesting increase (in 0.4 4/0
4/-3
modulus) in the first layers of the metallic system, the 4/-4
0.35 4/-5
sameregionwheresuperconductivityisabletopenetrate 4/-6
SS ref
in the repulsive metal.
0.3
Theseresultsclearlydemonstratethattheapproachwe
have devised is able on one hand to reproduce the bulk 0.25
resultswhenwearesufficientlyfarfromtheinterfaceand
on the other hand to display non-trivial and interaction 0.2
dependentproximityeffects,whichcanleadtoimportant
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
effects in real systems. The effect is generally stronger
j
for the order parameter.
-<Ek>(d)
0.68
0.66
IV. CONCLUSION 0.64
0.62
In the present work we have introduced an effective 0.6 44/-/03
0.58 4/-4
extension of the embedding approach which allows to 4/-5
0.56 4/-6
studyheterostructureofinteractingsystemsbymeansof
0.54
asmallnumberofactivelayers. Ourextensionistwofold.
0.52
For the first time we extend the formalism to the super-
0.5
conducting state, and we also introduce a “feedback” of 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
j
theslabontotheembeddingpotentialdescribingtherest
of the system which reduces the inhomogeneity effects.
Figure 5. ∆ , z , D and −(cid:104)E (cid:105) in a 20 layer thick het-
α α α k
This feedback correction has been shown to dramati- erostructure. Here on the left side we have a correlate metal
cally reduce the effects of the finiteness of the slab and with a repulsive U/t=4, while on the right side we tune an
to produce essentially exact results for all the relevant attractiveinteraction. Thepointsoutsidetheheterostructure
layer-resolved observables observables when treating ho- arethosecomputedstartingfromtheleads’Green’sfunctions
mogeneous bulk systems within this approximation. used to compute the embedding potential GˆB. The crosses
are the results for a bulk DMFT calculation for the cubic
Wehavealsopresentedtwoapplicationsofthemethod
lattice.
to paradigmatic situations where a heterostructure is
formed out of two semi-infinite bulks. In particular we
consider a superconductor with different values of the
8
attractive strength connected with either a fixed weak- posing particle-hole symmetry and fixing every layer to
couplingsuperconductororametalwithintermediatere- be half-filled, we freeze the charge redistribution which
pulsiveinteractions. Wefindthatinthefirstcaseimpor- would naturally enhance the effects we describe.
tant proximity effects take place and stronger supercon-
ductorincreasesthesuperconductingorderparameterfor
around ten layers for a wide range of parameters. In the
ACKNOWLEDGEMENTS
secondcasesuperconductivitypenetratesintherepulsive
system for around two layers, qualitatively changing the
physics of the system. In both cases the strongest ef- We thank A. Amaricci and G. Giovannetti for useful
fects are seen on the order parameters, while the kinetic discussion. This work has been supported by the Euro-
andpotentialenergiesremainclosertotheresultsfortwo pean Union under FP7 ERC Starting Grant No. 240524
disconnected systems. It is worth mentioning that, im- “SUPERBAD”.
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