Impurity induced bound states and proximity effect in a bilayer exciton condensate
aa r X i v : . [ c ond - m a t . m e s - h a ll ] D ec Impurity induced bound states and proximity effect in a bilayer exciton condensate
Yonatan Dubi and Alexander V. Balatsky , Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA and enter for Integrated Nanotechnologies, Los Alamos National Laboratory, Los Alamos, NM 87545 The effect of impurities which induce local interlayer tunneling in bilayer exciton condensatesis discussed. We show that a localized single fermion bound state emerges inside the gap for anystrength of impurity scattering and calculate the dependence of the impurity state energy and wavefunction on the potential strength. We show that such an impurity induced single fermion stateenhances the interlayer coherence around it, and is similar to the superconducting proximity effect.As a direct consequence of these single impurity states, we predict that a finite concentration ofsuch impurities will increase the critical temperature for exciton condensation.
PACS numbers: 73.21.-b,73.20.Hb
Introduction . – The Bose condensation of electron-holepairs (excitons) in semiconductors is an old idea [1, 2]which received renewed attention, mostly due to the pos-sible experimental realization of such a condensate insemiconductor bilayers [3, 4]. In these systems, two quan-tum wells are separated by an insulating barrier, whichprevents fast recombination of the excitons and allowsfor a coherent exciton condensate (EC) to develop. Thelack of direct tunneling between the layers is thus a cru-cial component in the existence of an EC. The role ofinterlayer tunneling has been studied some time ago [5],and it was shown that interlayer tunneling is not nec-essarily detrimental to the EC, although it may inducefinite dissipation in the current flow, which may explainthe failure to observe pure dissipationless current flow inthese systems.The key issue in the experimental verification of an ECis the identification of a clear signature that provides aconvincing proof of EC. Indirect evidence for exciton con-densation has been provided by tunneling experiments[6, 7], vanishing Hall resistance [8], photoluminescence[9, 10] and pattern formation [11] in photoexcited indi-rect excitons to name the few. Yet, in the absence ofdirect evidence of dissipationless supercurrent, it is im-portant to devise other methods in which the propertiesof the EC may be probed.In this paper we suggest that the presence of an ECmay be determined by studying its response to local im-purities. This notion, of studying impurities to deter-mine the structure of an underlying condensate structure,was suggested in the context of excitonic condensate [12]and d-wave superconductors [13–15], and was expandedto various systems such as bilayer cuprate superconduc-tors [16], inhomogeneous cuprates [17], iron-based super-conductors [18], exotic superconductors [19], and variousGraphene systems [20, 21]. While in these cases the im-purities are either scattering or magnetic impurities, aswe will show below the bilayer exciton system will sup-port impurities of another kind, somewhat analogous tonegative-U impurities in superconductors [22–24].Consider a bilayer system, composed of two quantum wells placed one on top of another (with an insulatingbarrier in between), in which at a certain point defect inspace the two layers become close enough to allow greaterdirect interlayer tunneling. This local defect is not a holein tunneling barrier but a region of weaker tunneling gap.We call this point the tunneling impurity (Fig. 1(a)). ++- - +- +- loc. statee-layerh-layer xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
Fermi level xxxx imp. levelimp. level xxxxxxxxxxxxxxxxxxxxxxxxxx E g xxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxx (a) (b) FIG. 1: (a) Real-space schematic representation of the localtunneling impurity. Far from the impurity electron-hole pairsform the exciton condensate, but at the tunneling impuritylocalized bound states are formed. (b) Energy-space repre-sentation of our results, showing the energy position of theimpurity-induced bound states. New spectroscopic featureswill emerge as a result of these bound states, depending ontheir occupation. A few photoluminescence processes due tothe bound states are illustrated.
Clearly, if there are too many such tunneling im-purities, the excitons will recombine before the EC isachieved. Here we wish to study the case of either a singletunneling impurity or a finite (yet small) concentrationof such impurities. Our main results are as follows: • For a single tunneling impurity, we find that a sub-gap bound state is formed, at an energy ω + β = ± ∆ √ − α q − Γ1+Γ (Fig. 1(b)), where α and β de-fine the band structure and Γ is defines the impuritystrength (see Eq.(8-9)). The spatial extent of thebound state is given by a length-scale ξ ∼ q Γ1+Γ .This result is non-perturbartive, and applies to anytunneling strength. • In the vicinity of the tunneling impurity, the inter-layer coherence is enhanced . • Following, a finite concentration of tunneling impu-rities should result in an increase of the EC criticaltemperature.We suggest to test our predictions by deliberately in-troducing such impurities into the bilayer systems (forinstance by ion bombardment). Our results should ap-ply to both EC formed in quantum hall bilayer and inphotoexcited exciton systems. There is evidence in opti-cal measurements of bilayer systems that such impuritiesare formed in the growth process [4].
Single-impurity bound state . –The starting point for this calculation is the usualmean-field description of the bilayer system [2, 25, 26] H MF = X αk ε kα c † kα c kα + ∆ X k (cid:16) ∆ k c † k + c k − + h.c. (cid:17) (1)where +( − ) refers to the upper (lower) layer, and ε k + = ~ m + k , ε k − = − ~ m − k − E g (the chemical potentials canbe absorbed into E g ). The order parameter ∆ k should inprinciple be determined self consistently, but for the sakeof allowing for an analytic calculation we will assume itsvalue is known. Moreover, we will assume that it takesa similar form to that of the superconducting gap, i.e. itis finite (and uniform) within some range from the Fermienergy. This is not a bad approximation when the ECis of a BCS-like nature [25]. We also point that the realspin of the electrons has been disregarded, as it plays nosignificant role in the situation we describe here.The hamiltonian of Eq. (1) is similar to the BCS hamil-tonian, and it is thus useful to follow the formulationused to study single impurity states in superconductors[14]. We define Nambu-like operators, ψ k = (cid:18) c k + c k − (cid:19) , forwhich the Green’s function may be written as a 2 × G k ( t ) = h ψ † k ( t ) ψ k (0) i = (cid:18) G k, + F k F † k G k, − (cid:19) . (2)In the absence of impurities, the electron (+), hole (-)and anomalous Green’s functions (in energy domain) aregiven by g + = ω − E k − ε k ( ω − E k ) − ε k − ∆ , g − = ω − E k + ε k ( ω − E k ) − ε k − ∆ f = ∆( ω − E k ) − ε k − ∆ , (3)where E k = ( ε k + + ε k − ) , ε k = ( ε k + − ε k − ), and the ex-plicit dependence on k has been omitted for convenience.We note that, as opposed to the BCS case, the electronand hole Green’s functions are not symmetric, due to theunequal masses (and hence the different band structure).We now turn to the local tunneling impurity. In realspace, one can imagine it as a point in which the layers are closer to each other, and hence tunneling there isamplified (Fig. 1(a)). Thus, the impurity hamiltonian is H imp = − λψ (0) † + ψ (0) − + h.c. = − λ X kk ′ c † k + c k ′ − + h.c.(4)The sign of λ defines the nature of the coupling betweenthe single-particle states of the two layers, with a positive(negative) λ corresponding to bonding (anti-bonding).The first is the more natural situation, but one can imag-ine an anti-bonding situation if, for instance, the inter-layer tunneling is mediated by non-s-wave orbitals in thelayer separating the two quantum wells. As will be ev-ident from the results, the sign of λ does not have anysignificant effect on the final outcome.In the language of the Nambu operators the impurityhamiltonian takes the form H imp = − λ X kk ′ ψ † k τ ψ k ′ (5)where τ i , i = 0 , ..., λ describes the local tunneling amplitude betweenthe layers.To continue, we follow the prescription used by Shibaand others [14, 27] and introduce the T -matrix, definedvia the Dyson equation for the Green’s function in thepresence of the impurity,ˆ G = ˆ g + ˆ g ˆ T ˆ g , (6)where ˆ g is the bare (Nambu) Green’s function. For a per-fectly local impurity (as assumed here), the interactionvertex does not depend on momentum and is given byˆ U = − λτ . The T -matrix is determined by the equation T ( ω ) = ˆ U + ˆ U X k ˆ G k T ( ω ) (7)andˆ X = X k ˆ G k = 2 πiN p ( ω + β ) − ( α − ω + βα − − ∆∆ − ω + βα +1 ! , (8)where α = m − − m + m − m + , β = m + m − + m + Eg , and N is the two-dimensional density of states with the reduces mass. Itis now a matter of straight-forward algebra to evaluatethe T -matrix. The position of the single-particle levelinduced by the impurity potential is determined from theposition of the poles of the T -matrix, which are given by ω + β = ± ∆ p − α r − Γ1 + Γ , Γ = 4 π λ N − α . (9)The real-space length scale associated with the impu-rity state may be found by evaluating the real-space de-pendence of the single-particle Green’s function. Thisamounts to performing the inverse Fourier transform ofthe Green’s function (Eq. (6) with the help of the solu-tion of Eq. (7), and we find that the real-space structurehas an exponential decay around the impurity site (lo-cated at r = 0) with a length scale ξ ∼ q . As Γ → ξ .There are several reasons why these impurity-inducedsingle-fermion bound states are important. First, sincethey are optically active (i.e. one can optically excitethem and induce transition between them and the reg-ular excitations), they should be in principal observableto spectroscopy experiments. Second, they point out tothe fact that a simple mean-field approach to disorderin bilayer systems [12] may not be enough to adequatelycharacterize the effect of disorder. Finally, as we showbelow, they induce interlayer coherence in their vicinityand thus may increase the critical temperature. In addi-tion, since they are experimentally achievable and due tothe analogy with superconducting negative-U impurities,they may shed light on the physics of the latter, whichare not experimentally accessible. Finite impurity concentration . – Next we turn to the ef-fect of a finite impurity concentration. The usual treat-ment of this case dates back to Abrikosov and Gorkov[28], yet it involves averaging over impurity positions, andthus fails to produce the single impurity physics which weare interested in. For that reason, we choose a real-spaceapproach, by solving the bilayer problem numerically ona square lattice. The Hamiltonian is given by H = − t X h i,j i ,α c † iα c j,α + X i,α E α c † i,α c i,α + X i,j (cid:16) ∆ i,j c † i, + c j, − + h.c. (cid:17) − λ X j (cid:16) c † j, + c j, − (cid:17) , (10)where again α = ± corresponds to the electron and holelayers, with E ± = ± E g / t is the usual tight-binding (intralayer) hopping parameter, and t = 1 setsthe energy scale hereafter. The order-parameter ∆ i,j iscalculated self-consistently via∆ i,j = U | r i − r j | e −| r i − r j | /ξ h c † i, + c j, − i , (11)where U is the strength of the coulomb interaction, | r i − r j | is the distance between the two sites labeled i and j (including the interlayer separation d ), and ξ issome screening length for the Coulomb interaction, whichin principal should be determined from the intralayerCoulomb screening. We have tested our results for dif-ferent values of ξ and found no qualitative difference be-tween them. However, small ξ allowed for better numer-ical convergence, and thus the results presented below were performed with ξ = 1 (in units of lattice spacing).In the last sum of Eq. (10) λ is the interlayer tunnelingstrength, and the sum is performed over a randomly cho-sen set of sites { j } which comprises a fraction p of theentire lattice.The numerical calculations were performed until lo-cal self-consistency was achieved for both the order pa-rameter ∆ i,j and the local density (which we kept at n + = n − = 0 .
46, i.e. slightly below half-filling). From∆ i,j we define a local order parameter ∆ i = P j ∆ i,j . Wehave performed our numerical calculations with variousparameters (i.e. lattice size, interaction strength, impu-rity concentration) and have found similar results in allof them.In Fig. 2 we show the average local order parameter¯∆ = P ′ i ∆ i as a function of temperature, for differentvalues of the tunneling amplitude λ = 0 , . , ..., .
5. Thetunneling impurity concentration is p = 0 . ×
25 lattice sites, interlayer distance(in units of the lattice spacing) d = 0 .
5, and interactionstrength U = 1. One clearly sees that a finite impurityconcentration results in an increase in T c and an apparentsmoothing of the transition. Both these effects should beobservable in experiment. In the inset of Fig. 2 we plotthe real-space structure of the order parameter along onedirection in a system with the same parameter exceptthe presence of only one impurity (with λ = 0 . T = 0 .
01 o T = 0 . T c for λ = 0 there is a clear jump in the order parameter,yet it remains finite on sites in the vicinity of the tun-neling impurity. This strongly resembles the behavior ofthe superconducting order-parameter in the vicinity of anegative-U impurity [24], i.e. a proximity effect. Inter-estingly, we found from our numerical calculations thatthe spatial dependence of the order parameter (as a func-tion of its distance from the impurity) is approximatelygiven by ∆( r ) ∼ exp( − (cid:16) rr (cid:17) / ), with the length-scale r independent of the tunneling amplitude λ . Summary and Discussion . – In this work we studiedthe properties of an exciton condensate in the presenceof an impurity which induces local tunneling between thelayers. It was shown that the impurities induce sub-gapsingle-particle bound states (Eq. 9). It is worth pointingout that for strong tunneling the impurity states crossthe Fermi level, and a phase-transition occurs, since theground-state will now have an occupied single-particlefermionic state in it, in similarity to strong magnetic scat-tering in superconductors [14]. This transition shouldhave clear spectroscopic features, since the allowed tran-sitions between the bands and the impurity levels, as well x D T D l =0 l =0.5 inreasing T FIG. 2: The order parameter, average over cites without atunneling impurity, as a function of temperature for differentvalues of the impurity tunneling amplitude λ = 0 , . , ..., . T c and a smearing of the transition are clearlyseen (see text for numerical parameters). Inset: the real-space dependence of the order parameter around a tunnelingimpurity, exhibiting a proximity-effect. as the transitions between the two impurity levels them-selves, will depend on their occupation.In addition, it was demonstrated that around theimpurity the condensate order parameter is enhanced(Fig. 2). This is a unique situation, and to see this it isuseful to compare our system to a superconductor witha magnetic impurity and with a negative-U impurity. Inthe first case, a single-particle bound-state is formed, butthat state disrupts the order parameter in its vicinity,since it acts as a pair-breaker. In the second case the or-der parameter is enhanced, but there is no single-particlebound state. The tunneling impurity in bilayers com-bines both these effects. This is due to the unique orderparameter of the EC, which corresponds to the interlayertunneling amplitude.In the case where the impurity concentration is verylarge, it is well established that the EC long-range coher-ence would vanishes due to fluctuations [5]. The detailedmanner at which the long-range coherence vanishes withincreasing impurity concentration is beyond the mean-field level of arguments presented here, and requires cal-culations in the presence of the order-parameter phasefluctuations (i.e. Kostelitz-Thouless phase fluctuationsand the presence of supercurrents). There is preliminaryindication that for a small impurity concentration, thesupercurrents simply avoid the impurity [29]. How ex-actly they behave in the presence of a large impurityconcentration is left for future studies.The authors acknowledge valuable discussions with J.-J. Su, M. Lilly and J. Zaanen. 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