Non-homogeneous pairing in disordered two-band s-wave superconductors
NNon-homogeneous pairing in disordered two-band s-wave superconductors
Heron Caldas , S. Rufo , and M. A. R. Griffith Departamento de Ciˆencias Naturais,Universidade Federal de S˜ao Jo˜ao Del Rei,Pra¸ca Dom Helv´ecio 74, 36301-160,S˜ao Jo˜ao Del Rei, MG, BrazilandCentro Brasileiro de Pesquisas F´ısicas,Rua Dr. Xavier Sigaud, 150 - Urca,22290-180, Rio de Janeiro, RJ, Brazil (Dated: February 19, 2021)We investigate the effects of disorder in a simple model of a hybridized two-dimensional two-bands-wave superconductor. We take into account the situations in which these bands are formed byelectronic orbitals with angular momentum such that the hybridization V ( k ) among them is anti-symmetric, under inversion symmetry. The impurity potential is given by an independent randomvariable W which controls the strength of the on-site disorder. We find that while the random disor-der acts in detriment of superconductivity, hybridization proceeds favoring superconductivity. Thisshows that hybridization, which may be induced by pressure or doping, plays an important hole intwo-band models of superconductivity, making them eligible candidates to describe real materials.We also find that in moderate and strong disorder, the system is broken into islands, with correlatedlocal order parameters. These correlations persist to distances of several order lattice spacing whichcorresponds to the size of the SC-Islands. I. INTRODUCTION
The experimental discovery of high transition tem-perature in superconducting oxides [1] and the subse-quent discoveries of strontium ruthenate [2], magnesiumdiboride [3], and iron pnictides [4, 5] has motivated aintense theoretical investigation in the superconductingproperties of these materials. Experiments have indi-cated that one of these compounds, magnesium diboride(MgB ), with a transition temperature of ≈
40 K [6],has two distinct superconducting gaps [7–13] and, con-sequently, was classified as a two-band superconductor(SC) [14].Since in MgB the relevant coupling mechanism is ofintra-band character [15], two-band models have beenemployed to investigate materials possessing two pairinggaps. As pointed out in Ref. [16], the number of differentgaps that emerge in multi-orbital systems is a direct con-sequence of the hybridization among the orbitals presentin a given material. In fact, as shown in Ref. [17], theFermi surface (FS) of MgB is determined by three or-bitals, nevertheless only two different BCS gaps are ex-perimentally observed. This happens because two of thethree orbitals hybridize with each other forming one sin-gle band, responsible for a large superconducting gap onthe σ FS, while the non-hybridized orbital is related tothe smaller superconducting gap at the π band FS.It is worth to mention that besides intermetallic bi-nary superconductors such as MgB , angle-resolved pho-toelectron spectroscopy (ARPES) experiments on La-based cuprates (with the hybridization of d x − y and d z orbitals) provided direct observation of a two-band struc-ture in these compounds [18].It is well known that disorder has tremendous conse- quences on conductors and superconductors. In a metal,the effect of disorder is to induce the Anderson localiza-tion of the electrons transforming a metal into an insula-tor, while in superconductors pair-localization results insuperconductor-insulator transition (SIT). In very strongdisorder the Cooper pairs can get localized due to thefractal nature of the single-particle near-critical wavefunction [19–21]. Thus, the effects of disorder in a su-perconductor brings about a competition of long-rangephase coherence between electron pair states of the su-perconducting phase and the limitation of the spatial ex-tent of the wave functions due to localization [22]. Then,it is natural to expect a critical disorder at which super-conductivity is overcame. It was shown by Anderson [23]and Abrikosov and Gorkov (AG) [24] that nonmagneticimpurities have no significant effect on the superconduct-ing transition temperature. However, Anderson’s andAG theory are applicable only to weakly disordered sys-tems [25]. The problem of a (one-band) s -wave supercon-ductor in the presence of a random potential has beeninvestigated for weak and strong disorder in Ref. [26].They found that in the high-disorder regime, the systembreaks up into superconducting islands, separated by aninsulating sea.In this paper we investigate the effects of disorderin the superconducting properties of a simple two-bandmodel, subjected to the hybridization of two single bands,namely a and b . We consider superconducting (s-wave)interactions only inside each band, which will result inintra-band pairing gaps ∆ a and ∆ b , respectively, in thesebands. The immediate consequence of hybridization is totransfer the quasiparticles among the bands and the clearadvantage is that it can be adjusted experimentally byexternal factors like pressure or doping [27]. Indeed, ex-periments in cuprates have shown that its transition tem- a r X i v : . [ c ond - m a t . s up r- c on ] F e b perature is very sensitive to an applied (external) pres-sure, which is responsible for the hybridization betweenthe d orbitals of the cooper and the p orbitals of the oxy-gen [28, 29]. We appraise antisymmetric, k -dependenthybridization V ( k ), that in the position space is trans-lated to V ij = − V ji . This choice is due to the fact thatthe antisymmetric hybridization produces an odd-paritymixing between the a and b bands, and is responsible forthe p -wave nature of an induced inter-band gap [30]. 1Dmodels with a pairing gap possessing p -wave symmetryare highly desired for the investigation of the appearanceof Majorana zero-energy bound states [31–34].Using the Bogoliubov-de Gennes (BDG) mean-fieldtheory we show the distribution of the local pairing am-plitude P (∆) and density of states N ( ω ) for various hy-bridization and disorder strengths. We analyze how a(random) disorder modifies the behavior of the super-conducting two-band model. We have found that whilethe random disorder is detrimental to superconductivity,hybridization favors superconductivity. We also foundthat in the case of high disorder, the system breaks intocorrelated islands in a insulating sea. This fact is knownalready for a one-band system. The novelty here is thatthere is correlations between the a and b pairing regions,due to hybridization, and even the correlations between a and a pairing islands (and the same for b and b ) calcu-lated here depend on the hybridization.The rest of this paper is organized as follows: InSec. II we describe our two-band model for the disor-dered SC and describe the inhomogeneous BdG mean-field method. In this section we also obtain the BdGequations. Our results are shown in Sec. III. We con-clude in Sec. IV. II. MODEL HAMILTONIAN
We describe a generic two-band SC in two-dimensions(2 D ) by a Hubbard-type Hamiltonian, which is given by H = − (cid:88) ,σ t aij a † i,σ a j,σ − µ a (cid:88) i,σ a † i,σ a i,σ − (cid:88) ,σ t bij b † i,σ b j,σ − µ b (cid:88) i,σ b † i,σ b i,σ + (cid:88) ,σ V ij ( b † i,σ a j,σ + a † i,σ b j,σ ) − U a (cid:88) i n ai, ↑ n ai, ↓ − U b (cid:88) i n bi, ↑ n bi, ↓ , (1)where a † iσ ( a jσ ) and b † iσ ( b jσ ) are the fermionic creation(annihilation) operator at site r i for the a and b bands,respectively. The lattice parameter for the square latticeis a = 1, and spin σ = ↑↓ . µ a = µ + E a and µ b = µ + E b are the chemical potentials, where E a and E b are the bot-tom of the a and b bands. n ai,σ = a † iσ a iσ and n bi,σ = b † iσ b iσ are the density operators, t aij and t bij are the hopping in-tegrals between sites i and nearest neighbor j for the a and b electrons. U a ( U b ) is the onsite attractive potentialbetween the a ( b ) electrons. V ij is the ( k -dependent) near-est neigbours hybridization of the two bands, which maybe symmetric or antisymmetric, arising from a non-localcharacter of the mixing. Notice that in Hamiltonian inEq. (1) we have neglected the (rather involved) effects ofCoulomb repulsion [26]. However, as will be clear below,even with such a simplification, the hybridized two-bandmodel with attractive intra-band interactions and withdisorder shows very interesting results which were worthto be investigated. A. The Mean Field Theory
The interaction part of the Hamiltonian in Eq. (1) pos-sesses products of four fermionic operators, which can bedecoupled by the Hartree-Fock (HF) BCS decoupling (seeappendix A) of two-body terms [35, 36]. The decouplingprocedure generates the mean field Hamiltonian (MFH) H MF below H MF = − (cid:88) ,σ t aij a † iσ a jσ − (cid:88) i,σ (˜ µ a − W i ) a † iσ a iσ − (cid:88) ,σ t bij b † i,σ b jσ − (cid:88) i,σ (˜ µ b − W i ) b † iσ b iσ − (cid:88) ,σ V ij ( b † iσ a j,σ + a † iσ b jσ )+ (cid:88) i [∆ a,i a † i, ↑ a † i, ↓ + ∆ ∗ a,i a i, ↓ a i, ↑ ]+ (cid:88) i [∆ b,i b † i, ↑ b † i, ↓ + ∆ ∗ b,i b i, ↓ b i, ↑ ] , (2)where we have included the same local impurity potential W i in both a and b bands, and ˜ µ p = µ p + U p < n pi > / p ≡ a, b . Here < n pi > is the average of theoccupation number. The impurity potential is definedby an independent random variable W i uniformly dis-tributed over [ − W, W ], at each site r i . W thus controlsthe strength of the disorder. In this way, one can think ofeffective local chemical potentials given by µ effp,i ≡ ˜ µ p − W i that, as we will see below, are responsible, together withthe intra-band interaction U p and hybridization V ij , forthe rich physics and interesting phase diagrams we find.For simplicity, but without loss of generality, we assumethat t aij = t bij = t . Besides, the strength of the hybridiza-tion is given by V . Then, the MFH in Eq. 2 describesa hybridized two-band s-wave superconductor, under theinfluence of random nonmagnetic impurities, by meansof an attractive Hubbard model, pictorially described inFigure 1.In order to diagonalize H MF in Eq. (2) for n ( n sites along x direction and n sites along y FIG. 1. (Color online) Schematic representation of the hybridized two-band attractive Hubbard model in the presence ofdisorder. Electrons from a (blue) and b (red)-bands can hop between lattice sites (having lattice spacing a ) with a hoppingparameter t . The hybridization V ij destroys an electron of band a ( b ) and creates one in band b ( a ) (without spin-flip), in anearest neighbor site. Due to Pauli’s principle, both hopping and hybridization are only possible if the final lattice site is emptyor occupied with an electron with a different spin. The green marbles represent nonmagnetic impurities randomly distributedon the square lattice. Two electrons of band- a ( b ) with opposite spin localized at the same lattice site have an intra-band on-siteinteraction strength U a ( b ) . As an example, it is shown in the figure that with the action of V ij a spin-down electron from band- b has been annihilated and created in band- a in a nearest neighbor site, that has already a spin-up electron from band- a . This“blue pair” will now experience the interaction potential U a . direction) sites, we firstly express Eq. 2 in ma-trix form H MF = Ψ † H Ψ, such that Ψ T =( a † ↑ , ..., a † n ↑ , b † ↑ , ..., b † n ↑ , a ↓ , ..., a n ↓ , b ↓ ...b n ↓ ). TheMFH can be diagonalized by the transformation M † H M = diag ( − E ... − E N ; E N ...E ) for N = n ,see appendix B.The matrix elements of M can be related to coefficientsof the Bogoliubov-Valatin transformation [37]. Using theBogoliubov-Valatin transformation defined in appendixB, the local particle densities and local pairing gaps canbe expressed as (cid:104) n ai (cid:105) = 2 N (cid:88) n (cid:48) =1 [ | M i, N + n (cid:48) | f + | M i, N + n (cid:48) | (1 − f )] (3) (cid:104) n bi (cid:105) = 2 N (cid:88) n (cid:48) =1 [ | M i + N, N + n (cid:48) | f + | M i + N, N + n (cid:48) | (1 − f )](4)∆ a,i = U a N (cid:88) n (cid:48) =1 [ M i, N + n (cid:48) M i +2 N, N + n (cid:48) (1 − f )] (5)∆ b,i = U b N (cid:88) n (cid:48) =1 [ M i + N, N + n (cid:48) M i +3 N, N + n (cid:48) (1 − f )] , (6)where f ≡ f ( E n (cid:48) ) = 1 / ( e βE n (cid:48) + 1), with β = 1 /k B T , isthe Fermi function. The equations above form a systemof 4 N self-consistent equations. We numerically solved theses equations for a lattice with N = 144 and N = 400sites. The size of the matrix H is 4 N × N and, therefore,needed a huge computational effort.In the numerical simulations we have used an antisym-metric hybridization such that V ij = − V ji . This choiceis appropriate for our model to describe, for example,the s and p -orbitals, which hybridizes in different sub-lattices [38, 39]. III. RESULTSA. Clean systems
The solutions of Eqs. (3), (4), (5) and (6) provide N values for ∆ a , ∆ b , n a and n b . We defined a Distributionof the Local Pairing Amplitude (DOLPA), which tell ushow homogeneous will be the order parameters ∆ p in thewhole lattice.In Fig. 2 (a) and (b) we show DOLPA for ∆ a (blue)and ∆ b (red) for a clean system W/t = 0. Note that,when the disorder and hybridization are zero, see Fig. 2(a), all (∆ a , ∆ b ) possesses values equal to (0 . , . W/t = 0, when we turn on the hybridizationto
V /t = 2 .
0, see Fig. 2 (b), both the superconductingorder parameters ∆ a and ∆ b are reduced. On the otherhand, it is worth to comment that for a different set ofparameters our numerical analysis shows that the hy-bridization is also able to increase the order parameters (a) V/t = 0 . V/t = 2 . V/t = 0 . V/t = 2 . FIG. 2. (Color online) Distribution of the local pairing am-plitude DOLPA for a clean system
W/t = 0 and for
V /t = 0 . V /t = 2 .
0. We set N = 144, U a /t = 3 . U b /t = 3 .Blue curve is DOLPA for p = a , while the red curve for p = b .(a) Local pairing amplitude for V /t = 0 .
0, (b) Local pairingamplitude for
V /t = 2 .
0, (c) Density of states for
V /t = 0 . V /t = 2 . ∆ p (for instance U a /t = 3, U b /t = 4 and V /t = 0 . V ∗ such that thepairing gaps ∆ a ( V ∗ ) and ∆ b ( V ∗ ) are maximum [30, 40].In the next section, we will discuss that in both regimesthe hybridization term makes the systems robust againstthe disorder potential, regarding the case with zero hy-bridization. The superconducting phase is sensitive tolocal perturbations in both orbitals. Since the hybridiza-tion acts connecting electrons from orbitals a and b , thecorrelations < a † σ,i b σ,i > and < a ↑ ,i a ↓ ,i > and < b ↑ ,i b ↓ ,i > becomes strongly depending on the hybridization. Again,the DOLPA shows a homogeneity characteristic of sys-tems without local perturbations or disorder effects, seeFig. 2 (b).Finally, we show the local density of states (DOS) inFig. 2 (c) V /t = 0 and (d)
V /t = 2 .
0. There is a gapregion around the E = 0 in the DOS. Note that the gapregion decreases when the hybridization increases (i.e.∆ E = 0 . V /t = 0 while ∆ E almost vanishes for V /t = 2 . (a) W/t = 0 . W/t = 1 . W/t = 6 . W/t = 10 . FIG. 3. (Color online) Distribution of the local pairing am-plitudes DOLPA for various disorder strengths
W/t and for
V /t = 0. We set N = 144, U a /t = 3 . U b /t = 3. Theblue curve is DOLPA for p = a , while the red curve is for p = b . (a) W/t = 0 .
5, (b)
W/t = 1 .
0, (c)
W/t = 6 . W/t = 10 . B. Disordered systems
We now investigate the systems in which both a and b -bands have the same effective chemical potential and,consequently, are affected by the same sort of (random)disorder in each site, that is ˜ µ a = ˜ µ b = µ − W i .In Fig. 3 we show the distribution of the local pair-ing amplitudes for various disorder strengths W/t =0 . , . , . V /t = 0. Theblue curve is the distribution of the local pairing ampli-tude ∆ a , while the red curve is the distribution of thelocal pairing amplitude for ∆ b . Notice that as the disor-der strength increases, from ( a ) to ( d ), the pairing am-plitudes begin to concentrate around ∆ p = 0, showingthat strong disorder is deleterious for the superconduct-ing state also in a two-band decoupled model.In Fig. 4 we show the density of states for the dirty sys-tem without hybridization and several disorder strength W/t = 0 . , . , . W andfor hybridization V /t = 2 .
0. The blue curve is the distri- (a)
W/t = 0 . W/t = 1 . W/t = 6 . W/t = 10 . FIG. 4. (Color online) Density of states N ( ω ) for four disorderstrengths W/t and hybridization V /t = 0. Here, we have N = 144 sites, U a /t = 3 . U b /t = 3. Note that thespectral energy gap remains finite even at large W/t . (a)
W/t = 0 .
5, (b)
W/t = 1 .
0, (c)
W/t = 6 . W/t = 10 . bution of the local pairing amplitude ∆ a , while the redcurve is the distribution of the local pairing amplitude∆ b . Notice that for low values of disorder, (a) W/t = 0 . W/t = 1 .
0, the pairing amplitudes begin to spreadaround the same value, approximately ∆ p = 0 .
15. Inter-estingly, as the disorder keeps increasing, ( c ) W/t = 6 . d ) W/t = 10 .
0, both ∆ a and ∆ b remain nonzero,showing that the hybridization acts favoring supercon-ductivity against disorder (see in Fig. 3 c) and d) thatwhen V /t = 0, the majority of ∆ p is around zero ampli-tude). This could be expected, since antisymmetric hy-bridization may increase the pairing gaps of a two bandmodel (without disorder) for a certain range of the hy-bridization strength [30, 40]. We verified that even inthe range of the strength of the hybridization that de-creases the pairing gaps the system is robust against dis-order, when compared to same values of disorder in non-hybridized systems, i.e., in two single (decoupled) a and b bands. In addition, the results shown in Fig. 6 stillpresent finite energy gap at large values of W/t .After a numerical inspection in Fig. 4 and Fig. 6, onecan verify that the hybridization
V /t , disorder
W/t , andthe gap energy ∆ E possess a complex relationship. Inparticular, if W/t = 0 .
5, the energy gap is given by ∆ E =0 .
298 for
V /t = 0 .
0, while it is greatly diminished for
V /t = 2 . E = 0 . V /t and for afixed
W/t , we can deduce that the hybridization, at least (a)
W/t = 0 . W/t = 1 . W/t = 6 . W/t = 10 . FIG. 5. (Color online) Distribution of the local pairing am-plitude for various disorder strengths
W/t and for hybridiza-tion
V /t = 2 .
0. Here, we have N = 144 sites, U a /t = 3 . U b /t = 3. Blue curve is distribution of the local pairingamplitude for p = a , while the red curve is distribution ofthe local pairing amplitude for p = b . (a) W/t = 0 .
5, (b)
W/t = 1 .
0, (c)
W/t = 6 . W/t = 10 . in the analyzed values, tends to result in an energy gapdecreasing.In Fig. 7 (a) and (b) we show the phase diagram in the( W/t , V /t ) plane, while (c) and (d) in the (
W/t , U a /t ), forfixed µ/t = 0 . U b /t = 3 .
0. The contour plot indi-cates the percentage of sites with local pairing amplitude∆ a = 0 and ∆ b = 0. The percentage bar varies from 0%(white) for the pure superconducting state (SC) to 100%(black) for a fully insulating state (I), while intermediatepercentage values correspond to a kind of intermediatestate (yellow and red). Note that in the phase diagramsin Fig. 7 (c) and (d), obtained from our two-band model,the behaviors are consistent with the one obtained forone band in Ref. [26] (see Figure 15 inside this reference,where disorder is designated by V /t which is the same as
W/t in the present work). In both cases, we observed thepresence of a s-wave superconducting and gapped insulat-ing phases. In special, by comparing our phase diagramsin Fig. 7 (c) and (d) with the one in Ref. [26] (Fig. 15)one can identify additional regions that can support SC-Islands (intermediate phase in yellow and red) beyondthe pure superconducting and insulating phases (SC andI in white and black, respectively). Besides, our phase di-agrams in Fig. 7 (c) and (d) demonstrate the possibilityof the occurrence of a kind of reentrant superconductor-insulator transition [41] with decreasing U a /t [42]. (a) W/t = 0 . W/t = 1 . W/t = 6 . W/t = 10 . FIG. 6. (Color online) Density of states N ( ω ) for four dis-order strengths W/t and hybridization
V /t = 2 .
0. Here, wehave N = 144 sites, U a /t = 3 . U b /t = 3. Surprisingly,the increase of hybridization enhances the local superconduct-ing order parameters making the whole system more robustagainst the disorder and the energy gap remains finite evenat large W/t. (a) W/t = 0 .
5, (b)
W/t = 1 .
0, (c)
W/t = 6 . W/t = 10 . Notice that in Figures 7 (a) and (b), there is a puresuperconducting state for
W/t < . V /t analyzed, and for
W/t < . U a /t , showing the regions of applicability ofAnderson’s theorem.To have a better understanding of the competition be-tween the effects of disorder and hybridization in thewhole system, we present in Fig. 8 a heatmap plot of thelocal densities n a and n b ; local pairing amplitude ∆ a and∆ b as a function of hybridization, for a fixed W/t = 6 . µ/t = 0 . ×
20 lattices. (a) Local densities and (b) Local pair-ing amplitudes for hybridization
V /t = 0 .
1, where wehighlight the regions I-Insulating (∆ a = 0 and ∆ b = 0),II-mixed-superconducting (∆ a (cid:54) = 0 and ∆ b (cid:54) = 0), III-superconducting type b (∆ a = 0 and ∆ b (cid:54) = 0), IV andV-superconducting type a (∆ a (cid:54) = 0 and ∆ b = 0). (c)Local densities and (d) Local pairing amplitudes for hy-bridization V /t = 0 .
5. We then identify the presence of(local) superconducting islands (SC-Islands) in regionsII, IV, and V, separated by an insulating region. We ob-served that the presence of the SC-Islands is favored bythe increasing of U a /t and W/t and correspond to theintermediate phase in Fig. 7 (yellow and red regions).It is important to note, by comparing Fig. 8(b) and(d), that the increasing of the hybridization favors the II- mixed-superconducting region. For this reason the spa-cial fluctuation of the lattice is modified. These complexstructures of formation of superconducting and mixed is-lands and an insulating “medium” reveals a highly non-homogenous formation.Accordingly, as the disorder potential
W/t is randomlyassigned for each site we have uncorrelated local densi-ties n a and n b , as we can see in Fig. 8 (a) and (c) withthe absence of cluster formation [26]. Despite that, duethe self-consistent approach the local pairing amplitudes∆ a and ∆ b present a spatial correlation between the SC-islands (regions II, III, IV and V in Fig. 8 (b)). Thiscorrelation occurs from moderate to high disorder values,in this case W/t = 6 .
0. The typical size of a SC-Islandis of the order of the coherence length ξ which is sub-ject to the disorder potential W/t as well as the effectiveelectron attraction U a /t and U b /t .Thus, from Fig. 8 (b) and (d) we can see how the ap-pearance of disorder implies in a strong spatial fluctua-tion of the local order parameters ∆ p . In Fig. 9 (a) and(b) we show the spatial fluctuations for V /t = 0 . V /t = 0 .
5, respectively. The strong spatial fluctuationstructure reveals where the order parameter get a largeamplitude, blue for ∆ a , red for ∆ b and assuming a pur-ple color where the SC-Island are in superposition. TheseSC-Island are correlated and we present the results show-ing this fact in Fig. 9, as the disorder-averaged correla-tion functions ∆ a,i ∆ a,j (c), ∆ b,i ∆ b,j (d) and ∆ a,i ∆ b,j (e).The average is performed for each set of ∆ p,i ∆ p (cid:48) ,j thatshare the same spatial distance | r i − r j | . The thin curvescorrespond to the correlation function for V /t = 0 . V /t = 0 . V /t = 0 . V /t = 0 . a,i ∆ a,j , ∆ b,i ∆ b,j and∆ a,i ∆ b,j are closer. Besides, the coherent tunneling ofthe Cooper pairs between the SC-Islands is responsiblefor the establishment of correlation [43]. On the otherhand, the regions with a relative small ∆ p behave as aninsulating phase with unpaired Cooper pair electrons. Itis important to remark that we did not take into ac-count quantum fluctuations in the phase φ of the orderparameter (that would imply in gap parameters given by˜∆ p = e iφ ∆ p ), which are expected to destroy the long-range phase coherence between the small SC islands forvery large disorder [26, 43]. C. Experimental Consequences
It is also important to discuss the experimental as-pects of our two-band model subjected to disorder andhybridization. It is experimentally doable to grow homo-geneously disordered films [44] that are disordered bothon an atomic scale and granular films [45, 46]. These twotypes of films will essentially depend on the material,the substrate, and growth conditions [26] as, for exam-ple, a film of 99 .
99% Sn (or Pb) evaporated onto fire- Vt Wt SCI (a)∆ a for U a /t = 3 . U b /t = 3 . Vt Wt SCI (b)∆ b for U a /t = 3 . U b /t = 3 . U a t Wt I SC (c)∆ a for V/t = 1 . U b /t = 3 . U a t Wt ISC (d)∆ b for V/t = 1 . U b /t = 3 . FIG. 7. (Color online) Phase diagram for the hybridized two-band model of superconductivity in the presence of disorder
W/t for a fixed µ/t = 0 . U b /t = 3 . ×
20 lattices. The contour plot represents thepercentage of sites with local pairing amplitude ∆ a = 0 and ∆ b = 0. polished glass substrates [45], or a 99 . O ) samples evaporated onto a SiO substrate [47]. In these experiments [44, 46], the effectivedisorder was controlled by changing the film thickness.In the present case, the phase diagram of Fig. 7 couldbe useful as a guide to experimental measurements inorder to identify regions with SC-Islands features. Apossible experimental goal is the realization of the su-perconducting phase only of the type a with ∆ a , de-scribed by the correlation < a ↑ ,i a ↓ ,i > (or only of thetype b with ∆ b , described by the correlation < b ↑ ,i b ↓ ,i > )and then, the mixed-superconducting phase. In contrastwith the one-band model studied in Ref. [26], we remarkthat the mixed-superconducting phase appeared only inthe context of the highly hybridized two-band model,as explored here. We expect that landscapes showingthe regions with insulating, superconducting, and mixed-superconducting islands should be visible through scan-ning tunneling microscopy (STM), as the topography im- ages of a ‘ π (cid:48) -shaped Pb island sitting on top of a stripedincommensurate (SIC) surface [48]. IV. CONCLUSION
In this paper we investigated the effects of disorderin the superconducting properties of a simple hybridizedtwo-band model. We considered the situations in whichthese bands are formed by electronic orbitals with angu-lar momentum such that the hybridization V ( k ) amongthem is antisymmetric under inversion symmetry. Westudied the effect of both disorder and an antisymmetrichybridization in the cases where there is s-wave intra-band interactions only inside each of the two bands. Theimpurity potential is given by an independent randomvariable W which controls the strength of the disorder. Insome recent experiments, the effective disorder was con-trolled by varying the film thickness. On the other hand, I IIIIII IIIVIIIIVVV (a) (b) (c) (d) n A ( R ) A ( R ) n B ( R ) B ( R ) n a n b � a � b n a n b � a � b FIG. 8. (Color online) Local densities n a and n b ; local pairing amplitudes ∆ a and ∆ b as a function of hybridization, for afixed W/t = 6 . µ/t = 0 . ×
20 lattices. (a) Local densities and (b) Localpairing amplitudes for hybridization
V /t = 0 .
1, where we highlight the regions I-Insulating (∆ a = 0 and ∆ b = 0), II-mixed-superconducting (∆ a (cid:54) = 0 and ∆ b (cid:54) = 0), III-superconducting type b (∆ a = 0 and ∆ b (cid:54) = 0), IV and V-superconducting type a(∆ a (cid:54) = 0 and ∆ b = 0). (c) Local densities and (d) Local pairing amplitudes for hybridization V /t = 0 .
5. The regions II, IV andV denote the superconducting islands (SC-Islands). The SC-Islands corresponds to the intermediate phase in Fig. 7 (yellowand red regions). Note that the increase of the hybridization favors the II-mixed-superconducting region. the hybridization between the two-bands can be tunableby applying strain or by carrier doping. We found thatwhile (the random disorder) W acts in detriment of su-perconductivity (driving to zero the pairing gap param-eter through a suppression of the superconducting phasein favor of an insulating one), hybridization makes thesystem more robust against disorder effects.We also found that strong disorder implies in a promi-nent spatial fluctuation of the local order parameters ∆ a and ∆ b , which are correlated. These correlations perse-verate to distances of several order lattice spacing whichcorresponds to the size of the SC-Islands.An interesting theoretical extension of this work couldbe to consider this two-dimensional two-band hybridizedsystem in the presence of disorder under a static mag-netic field h parallel to the 2D plane [49]. Given thevariety of external effects, hybridization V , disorder W and magnetic field h , we hope that new and possibly ex-otic phases will appear. An investigation is this direction is intriguing and will be the subject of future work. V. ACKNOWLEDGMENTS
We thank D. Nozadze and N. Trivedi for enlighteningdiscussions. We wish to thank CAPES and CNPq, Brazil,for partial financial support. x1 10 20y 11020 (a)Spatial fluctuations of ∆ a (blue peaks) and ∆ b (redpeaks) for V/t = 0 . x1 10 20y 11020 (b)Spatial fluctuations of ∆ a (blue peaks) and ∆ b (redpeaks) for V/t = 0 . V/t = = (c)Disorder-averaged correlation function∆ a,i ∆ a,j . V/t = = (d)Disorder-averaged correlation function∆ b,i ∆ b,j . V/t = = (e)Disorder-averaged correlation function∆ a,i ∆ b,j . FIG. 9. (Color online). Spatial fluctuations of the local pairing amplitudes ∆ a and ∆ b in a 20 ×
20 lattice, for
V /t = 0 . V /t = 0 . W/t = 6 . µ/t = 0 .
1. The blue arrows denote the coherent tunneling between∆ a,i and ∆ a,j (blue peaks), the red arrows denote the coherent tunneling between ∆ b,i and ∆ b,j (red peaks) as well as thepurple arrows denote the coherent tunneling between ∆ a,i and ∆ b,j (blue and red peaks). We present the disorder-averagedcorrelation function ∆ a,i ∆ a,j (c), ∆ b,i ∆ b,j (d) and ∆ a,i ∆ b,j (e). The thin curves are obtained for V /t = 0 . V /t = 0 .
5. Note that the correlations persist to distances of several order lattice spacing which corresponds on theSC-Islands size. We observe that the increasing of the hybridization tends to “suspend” and equalize the correlations ∆ a,i ∆ a,j and ∆ b,i ∆ b,j . These results are normalized to be the unit for | r i − r j | = 0. Appendix A: Mean Field Decoupling for interactionterms ψ †↑ ψ †↓ ψ ↓ ψ ↑ = (A1) < ψ ↓ ψ ↑ > ψ †↑ ψ †↓ + < ψ †↑ ψ †↓ > ψ ↓ ψ ↑ − | < ψ ↓ ψ ↑ > | + < ψ †↑ ψ ↑ > ψ †↓ ψ ↓ + < ψ †↓ ψ ↓ > ψ †↑ ψ ↑ − < ψ †↑ ψ ↑ >< ψ †↓ ψ ↓ > − ( < ψ †↑ ψ ↓ > ψ †↓ ψ ↑ + < ψ †↓ ψ ↑ > ψ †↑ ψ ↓ ) , where ψ ↑ , ↓ = a ↑ , ↓ , b ↑ , ↓ . The “Fock condensates” are zerohere, so < ψ †↑ ψ ↓ > = < ψ †↓ ψ ↑ > = 0 in Eqs. (A2, A3 and A4below and consequently in Eq. (2). − U a (cid:88) i a † i, ↑ a † i, ↓ a i, ↓ a i, ↑ = − U a (cid:88) i [ < a i, ↓ a i, ↑ > a † i, ↑ a † i, ↓ + < a † i, ↑ a † i, ↓ > a i, ↓ a i, ↑ − | < a i, ↓ a i, ↑ > | (A2)+ < a † i, ↑ a i, ↑ > a † i, ↓ a i, ↓ + < a † i, ↓ a i, ↓ > a † i, ↑ a i,σ − < a † i, ↑ a i, ↑ >< a † i, ↓ a i, ↓ > ] , − U b (cid:88) i b † i, ↑ b † i, ↓ b i, ↓ b i, ↑ = − U b (cid:88) i [ < b i, ↓ b i, ↑ > b † i, ↑ b † i, ↓ + < b † i, ↑ b † i, ↓ > b i, ↓ b i, ↑ − | < b i, ↓ b i, ↑ > | (A3)+ < b † i, ↑ b i, ↑ > b † i, ↓ b i, ↓ + < b † i, ↓ b i, ↓ > b † i, ↑ b i,σ − < b † i, ↑ b i, ↑ >< b † i, ↓ b i, ↓ > ] . Defining ∆ a,i = − U a < a i, ↓ a i, ↑ > , ∆ b,i = − U b < b i, ↓ b i, ↑ > , < n ai,σ > = < a † i,σ a i,σ > and < n bi,σ > = < b † i,σ b i,σ > , theinteraction term reduces to H int = (cid:88) i ∆ a,i a † i, ↑ a † i, ↓ + ∆ ∗ a,i a i, ↓ a i, ↑ + | ∆ a,i | /U a − U a [ < n ai, ↑ > a † i, ↓ a i, ↓ + < n ai, ↓ > a † i, ↑ a i, ↑ − < n ai, ↑ >< n ai, ↓ > ]+ (cid:88) i ∆ b,i b † i, ↑ b † i, ↓ + ∆ ∗ b,i b i, ↓ b i, ↑ + | ∆ b,i | /U b − U b [ < n bi, ↑ > a † i, ↓ b i, ↓ + < n bi, ↓ > b † i, ↑ b i, ↑ − < n bi, ↑ >< n bi, ↓ > ] . (A4) Appendix B: Hamiltonian in real space
Defining the base Ψ T =( a † ↑ , ..., a † n ↑ , b † ↑ , ..., b † n ↑ , a ↓ , ..., a n ↓ , b ↓ ...b n ↓ ), wecan write HMF as H MF = Ψ † H Ψ, where H = H N × N for N = n . The Matrix H is given by H = A V ∆ aa V B bb ∆ aa − A − V bb − V − B , (B1) where A , V , B , ∆ ηη are matrices N × N . Here, A isformed by on-site energy and hopping terms betweenelectrons of the orbital a . The matrix B is formed byon-site energy and hopping terms between electrons ofthe orbital b . The matrix V is the subspace of the hy-bridization and ∆ aa,bb are formed by superconducting or-der parameters from orbitals a and b , respectively.In order to diagonalize H MF , we defined a matrix M .The matrix M acts in the basis Ψ asΨ = M Φ (B2)or, explicitly1 a ↑ ... a ↑ Nb ↑ ... b ↑ Na †↓ ... a †↓ Nb †↓ ... b †↓ N = M , . . . M ,N M ,N +1 . . . M , N M , N +1 . . . M , N M , N +1 . . . M , N ... ... ... ... ... ... ... ... ... ... ... ... MN, . . . MN,N MN,N +1 . . . MN, N MN, N +1 . . . MN, N MN, N +1 . . . MN, NMN +1 , . . . MN +1 ,N MN +1 ,N +1 . . . MN +1 , N MN +1 , N +1 . . . MN +1 , N MN +1 , N +1 . . . MN +1 , N ... ... ... ... ... ... ... ... ... ... ... ... M N, . . . M N,N M N,N +1 . . . M N, N M N, N +1 . . . M N, N M N, N +1 . . . M N, NM N +1 , . . . M N +1 ,N M N +1 ,N +1 . . . M N +1 , N M N +1 , N +1 . . . M N +1 , N M N +1 , N +1 . . . M N +1 , N ... ... ... ... ... ... ... ... ... ... ... ... M N, . . . M N,N M N,N +1 . . . M N, N M N, N +1 . . . M N, N M N, N +1 . . . M N, NM N +1 , . . . M N +1 ,N M N +1 ,N +1 . . . M N +1 , N M N +1 , N +1 . . . M N +1 , N M N +1 , N +1 . . . M N +1 , N ... ... ... ... ... ... ... ... ... ... ... ... M N, . . . M N,N M N,N +1 . . . M N, N M N, N +1 . . . M N, N M N, N +1 . . . M N, N α ↑ ... α ↑ Nα ↑ ... α ↑ Nα †↓ ... α †↓ Nα †↓ ... α †↓ N (B3) such that, the new base Φ T = ( α † ↑ , ..., α † N ↑ , α † ↑ , ..., α † N ↑ , α ↓ , ..., α N ↓ , α ↓ ...α N ↓ ) is a base where H MF become diagonal M † H MF M = diag ( − E ... − E N ; E N ...E ) (B4)We call attention to fact that the columns of M are the eigenvectors of H MF . We organized the eigenvalues of H MF from the lesser to greater values (i. e. − E ... − E N , E N ...E ) and therefore, note that, the last 2 N columns of M contain all eigenvectors that are related to positive energies of the Hamiltonian H MF .The Bogoliubov-Valatin transformations can be defined according the Eq. 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