The Two Component Optical Conductivity in the Cuprates: A Necessary Consequence of Preformed Pairs
aa r X i v : . [ c ond - m a t . s up r- c on ] S e p The Two Component Optical Conductivity in the Cuprates: A NecessaryConsequence of Preformed Pairs
Dan Wulin , Hao Guo , Chih-Chun Chien and K. Levin James Franck Institute and Department of Physics,University of Chicago, Chicago, Illinois 60637, USA Department of Physics, University of Hong Kong, Hong Kong, China and Theoretical Division, Los Alamos National Laboratory, MS B213, Los Alamos, NM 87545, USA (Dated: November 25, 2018)We address how the finite frequency real conductivity σ ( ω ) in the underdoped cuprates is affectedby the pseudogap, contrasting the behavior above and below T c . The f-sum rule is analyticallyshown to hold. Here we presume the pseudogap is associated with non-condensed pairs arising fromstronger-than-BCS attraction. This leads to both a Drude and a mid infrared (MIR) peak, the latterassociated with the energy needed to break pairs. These general characteristics appear consistentwith experiment. Importantly, there is no more theoretical flexibility (phenomenology) here thanin BCS theory; the origin of the two component conductivity we find is robust. PACS numbers: BHR1204
The behavior of the in-plane ac conductivity σ ( ω ) inthe underdoped high temperature superconductors hasraised a number of puzzles [1] for theoretical scenariossurrounding the origin of the mysterious pseudogap. Atthe same time, there has been substantial recent progressin establishing experimental constraints on the inter-playof the pseudogap and σ ( ω )[2] . A key feature of σ ( ω )is its two component nature consisting of a “coherent”Drude like low ω feature followed by an approximately T -independent mid-infrared (MIR) peak [1–3]. The lat-ter “extends to the pseudogap boundary in the phasediagram at T ∗ . Moreover a softening of the MIR bandwith doping [scales with] the decrease in the pseudogaptemperature T ∗ ” [2]. Crucial to this picture is that “high T c materials are in the clean limit and that ... the MIRfeature is seen above and below T c ”[4]. Thus, it appearsthat this feature is not associated with disordered super-conductivity and related momentum non-conserving pro-cesses, but rather it is “due to the unconventional natureof the [optical] response” [1].It is the purpose of this paper to address these relatedobservations in the context of a preformed pair Gor’kovbased theory that extends BCS theory to the strong at-traction limit [5]. Our expressions for σ ( ω ) are equiva-lent to their BCS analogue when the pseudogap vanishes.This approach is microscopically based and the level ofphenomenological flexibility [6, 7] is no more than thatassociated with transport in strict BCS superconductors.Alternative mechanisms for the two component opticalresponse include Mott related physics [8] and d-densitywave [9] approaches, which have acknowleged inconsis-tencies [10], as well as approaches that build on inhomo-geneity effects [11]. Distinguishing our approach is itsvery direct association with the pseudogap. In an evi-dently less transparent way, a two component response arises numerically [8] in the presence of Mott-Hubbardcorrelations above T c . However, experiments show howthe MIR feature must persist in the presence of super-conductivity, suggesting that pseudogap physics affectssuperconductivity below T c , as found here.Unique is our capability to address both the normal(pseudogap) and superconducting phases. Moreover, weare also able to establish [6, 7] compatibility with thetransverse f-sum rule without problematic negative con-ductivity [8] contributions. Finally, our approach is tobe distinguished from the phase fluctuation scenario thatappears problematic in light of recent optical data relatedto imaginary THz conductivity [12]. In experimental sup-port of our scenario is the claim based on σ ( ω ) data [13]that the “doping dependence suggests a smooth transi-tion from a BCS mode of condensation in the overdopedregime to a different mode in underdoped samples, [as]in the case of a BCS to Bose-Einstein crossover.”Our analysis leads to the following physical picture:the presence of non-condensed pairs both above and be-low T c yields an MIR peak. This peak occurs aroundthe energy needed to break pairs and thereby create con-ducting fermions. Its position is doping dependent, andonly weakly temperature dependent, following the weak T dependence of the excitation gap ∆( T ). The relativelyhigh frequency spectral weight from these pseudogap ef-fects, present in the normal phase, is transferred to thecondensate as T decreases below T c , leading to a nar-rowing of the low ω Drude feature, as appears to be ex-perimentally observed. Even relatively poor samples arein the clean limit [1, 4], so that an alternative pair cre-ation/annihilation contribution associated with brokentranslational invariance cannot be invoked to explain theobserved MIR absorption.Before doing detailed calculations, it is possible to an-ticipate the behavior of σ ( ω ) at a physical level. In ad-dition to the ω ≡ ω = 0conductivity consists of two terms, the more standard oneassociated with scattering of fermionic quasiparticles andthe other associated with the breaking of the pairs. Theterm associated with the scattering of fermionic quasi-particles gives rise to the usual Drude peak. In the pres-ence of stronger than BCS attraction, we observe thissecond contribution, a novel pair breaking effect of thepseudogap. It reflects processes that require a minimalfrequency of the order of 2∆( T ). We associate this termwith the MIR peak. Sum rule arguments imply that thelarger this MIR peak is, the smaller the ω ≈ σ dc [7]. This transfer of spectral weightcan be understood as deriving from the fact that whennon-condensed pairs are present, the number of fermionsavailable for scattering is decreased; these fermions aretied up into pairs.We have derived the optical conductivity σ ( ω ) in pre-vous work [5, 6, 14]. The current-current correlationfunction is χ ↔ JJ = P ↔ + n ↔ m − C χ , where C χ is associatedwith collective modes, which do not enter above T c norin the transverse gauge below T c .For notational convenience we define E ≡ E k ≡ p ξ k + ∆ as the fermionic excitation spectrum, ξ k isthe normal state dispersion, f ≡ f ( E ) is the Fermi dis-tribution function, and the pairing gap ∆ = ∆ + ∆ is found [5, 14] to contain both condensed ( sc ) and non-condensed ( pg ) terms. In the d-wave case, we write ∆ k =∆ ϕ k , ξ k = − t (cos k x + cos k y ) − µ , and E k = p ξ k + ∆ k ,where ϕ k = (cos k x − cos k y ) / P ↔ ( Q ) ≈ X K ∂ξ k + q / ∂ k ∂ξ k + q / ∂ k h G K G K + Q + F sc,K F sc,K + Q − F pg,K F pg,K + Q i (1)where Q = ( q , i Ω m ), i Ω m is a bosonic Matsubara fre-quency, and the three forms of propagators, introducedin earlier work [7] are G ( K ) = (cid:16) iω n − ξ k + iγ − ∆ pg, k iω n + ξ k + iγ − ∆ sc, k iω n + ξ k (cid:17) − F sc ( K ) ≡ − ∆ sc, k iω n + ξ k iω n − ξ k − ∆ k iω n + ξ k F pg ( K ) ≡ − ∆ pg, k iω n + ξ k + iγ G ( K ) (2)where K = ( k , iω n ) and iω n is the fermionic Matsubarafrequency. The real part of the conductivity can be ex-tracted from P ↔ ( Q ) using the definition Re σ ( ω = 0) ≡− lim q → Im P xx ( i Ω m → ω + i + , q ) /ω . Here γ repre-sents the damping associated principally with the inter-conversion of fermions and bosons. The first equation representing the full Green’s function is associated with aBCS self energy ( ∝ ∆ sc ) and a similar contribution fromthe non-condensed pairs ( ∝ ∆ pg ). The latter is fairlystandard in the literature [15] and importantly was de-rived microscopically in our earlier work [16]. Above, F sc represents the usual Gorkov-like function associated withcondensed pairs and we can interpret F pg as their non-condensed counterpart. The full excitation gap ∆( T )does not have a strong temperature dependence in theunderdoped regime; below T c this is because of a conver-sion of non-condensed to condensed pairs as T is reduced.We may rewrite P ↔ ( Q ) in the regime of very weak dis-sipation ( γ ≈
0) where the behavior is more physicallytransparent. For simplicity we will illustrate this resultfor s -wave pairing P ↔ ( ω, q ) = X k kk m h E + + E − E + E − (cid:0) − f + − f − (cid:1) × E + E − − ξ + ξ − − δ ∆ ω − ( E + + E − ) − E + − E − E + E − × E + E − + ξ + ξ − + δ ∆ ω − ( E + − E − ) (cid:0) f + − f − (cid:1)i , (3)where f ± = f ( E ± ) and δ ∆ = ∆ − ∆ , ξ ± = ξ k ± q / ,and E ± = E k ± q / . Importantly, for this weak dissipationlimit, one can analytically show that [7] the transversesum rule is precisely satisified. This sum rule is inti-mately connected to the absence above T c (and presencebelow) of a Meissner effect. The proof depends on thesuperfluid density, which at general temperatures is givenby n s = (2 / /m ) P k k /E (cid:16) (1 − f ) / E + ∂f /∂E (cid:17) .In addition, the total number of particles can be writtenas n = P k (cid:0) − ξ (1 − f ) /E (cid:1) . In this way, it is seen [7]that Re σ ( ω →
0) = ( πn s /m ) δ ( ω ). Since ∆ sc = ∆ − ∆ pg ,one can see that pseudogap effects, through ∆ pg , actto lower the superfluid density; the excitation of thesenon-condensed pairs provides an additional mechanism,beyond the fermions, for depleting the condensate withincreasing temperature.We introduce a transport lifetime τ = γ − intoEq.3 via the replacement δ ( ω − ( E + k ± E − k )) =lim τ →∞ π τ ( ω − ( E + k ± E − k )) + τ , to yield (for the more gen-eral d -wave case) Reσ ( ω = 0) = X k k x t (cid:16) ∆ ( T ) ϕ k E k − f ( E k )2 E k × (cid:2) τ ω − E k ) τ + τ ω + 2 E k ) τ (cid:3) − E k − ∆ ϕ k E k ∂f ( E k ) ∂E k τ ω τ (cid:17) (4)where we have dropped a small term associated with thederivative of the d-wave form factor ϕ k . Here ∆ sc, ± =∆ sc ( T ) ϕ k ± q / and ∆ pg, ± = ∆ pg ( T ) ϕ k ± q / . Because of Figure 1: Upper panel (top) curve plots
Re σ ( ω ) for T =1 . T c , while the shaded (red) area labelled “PG” shows thetransfer of spectral weight from low to higher ω associatedwith non-condensed pairs. Inset shows the dc resistivity.Lower panel plots σ ( ω ) at different indicated temperatures.Normalization is σ = σ (0) at 1 . T c . The inset shows the dif-ference of spectral weight between 1 . . T c normalizedby the difference in superfluid densities. The present theory(red) is contrasted with a BCS-like case (blue) where all ex-plicit ∆ pg contributions are dropped. their complexity, we do not include self consistent impu-rity effects which, due to bosonic contributions, will re-quire a modification of earlier work [17] predicting d -wavefermionic quasi-particles in the ground state. Moreover,it seems plausible that non-condensed pairs may also beassociated with these impurity effects, thereby leadingto incomplete condensation and finite ∆ pg in the groundstate. In general, our calculations tend to underestimatethe very low T spectral weight away from ω = 0.The upper panel in Fig.1 displays a decomposition ofthe normal state conductivity vs ω . The top curve is Re σ ( ω ) while the shaded (red) region labelled “PG” in-dicates the contribution from non-condensed pairs arisingfrom the F pg terms in Eq. (1). This figure shows clearlywhat is implicit in Eq.(4), namely that these pseudogapeffects transfer spectral weight from low to high ω . Herethe inset plots the resistivity as a function of T .The lower panel in Fig.1 plots the real part of the op-tical conductivity versus ω at the four different temper- atures T /T c = 1 . , . , . .
2. There are two peakstructures in these plots, the lower Drude-like peak, fromthe quasi-particle scattering contribution and the upperpeak associated with the breaking of pre-formed pairs.The “PG” contribution disappears at the lowest temper-atures, as all pairs go into the condensate. Thus one seesin the figure once the condensate is formed below T c , thelow frequency peak narrows and increases in magnitude.Conversely, the proportion of the spectral weight residingat high energies on the order of 10 cm − increases withtemperature.To more deeply analyze this redistribution of spec-tral weight, the difference of the frequency integratedconductivity between 1 . T c and 0 . T c of the presenttheory is plotted as a function of ω/t in the in-set of the bottom panel in Figure 1. Here we de-fine W ( ω, T ) = (2 /π ) R ω dω ′ σ ( ω ′ , T ) and ∆ W ( ω ) = W ( ω, . T c ) − W ( ω, . T c ). For comparison, we plot acounterpart “BCS-like” spectral weight change which isderived by effectively neglecting the terms involving ∆ pg in Eq. (4). Both conductivities are normalized by theirindependently calculated change in superfluid densities,∆ n s /m . The present theory leads to the full integrated(normalized) spectral weight by ω ≈ eV , while theBCS-like curve counterpart corresponds to ω ≈ meV .One can see that the presence of non-condensed pairs re-distributes an appreciable amount of spectral weight tohigher energies. Experimentally, there have been claimsthat very high energy scales ranging from 1 . − eV maybe needed to satisfy the sum rule. This figure shows howpseudogap contributions can be, at least partly, respon-sible for these high energy scales.We present a more detailed set of comparisons betweentheory and experiment in Fig. 2, where, for the latter,we reproduce the y = 6 .
75 plots in Fig. 4 from Ref. 2in panels (a)-(c) and the bottom panel of Fig. 5 fromthe same work in panel (d). Panels (e)-(g) in Fig. 2 areassociated with
T /T c = 1 . , . , and 0 . T decreases. The MIR peak posi-tion is relatively constant, (as seen experimentally) andin the theory roughly associated with 2∆, the value ofwhich is identified in each figure (e)-(g). That ∆( T )is roughly constant through the displayed temperaturerange, reflects the inter-conversion of non-condensed tocondensed pairs.It should be noted, however, that the height of theMIR peak in the data is more temperature independentthan found in theory. This would seem to suggest thatthere are non-condensed pair states at T = 0 perhaps as-sociated with inhomogeneity or localization [17] effects.This interpretation of the optical data appears consistentwith our previous studies [18] of angle resolved photoe-mission (ARPES) data from which we have inferred thatthe ground state in strongly underdoped samples may not Figure 2: The left column consists of figures reproduced fromRef. 2 and is to be contrasted with the corresponding theo-retical results in the right column. Panels (a)-(c) show theoptical conductivity for decreasing temperature and (d) plotsthe MIR peak location ω mid and T ∗ as a function of doping.The theoretical results in (e)-(g) show the optical conductiv-ity for T /T c = 1 . , . , and 0 . σ ( ω ) is normalized by σ = σ (0) at T = 1 . T c . The final panel (h) displays ω mid and T ∗ as functions of doping. A dashed line in plots (d) and (h)indicates the insulator boundary, which represents the limitof validity for our theory. be the fully condensed d-wave BCS phase. Rather theremay be some non-condensed pair or pseudogap effectswhich persist to T=0. In ARPES experiments one couldattribute this persistence to the fact that the T = 0 gapshape is distorted relative to the more ideal d -wave formfound in moderately underdoped systems [19]. Similarobservations are made from STM experiments [20].We show in Fig. 2(h) a plot of the MIR peak location ω mid as a function of T ∗ as calculated in our theory; thisplot suggests that the MIR peak position scales (nearlylinearly) with the pairing gap or equivalently with T ∗ .This observation is qualitatively similar (within factorsof 2 or 3) to Fig.2(d), reproduced from Ref.2. Finally, westress that we have investigated the effects of varying γ as well as its T dependence and find that our results inFig. 2 remain very robust.At the core of interest in the optical conductivity iswhat one can learn about the origin of the pseudogap.We earlier discussed problematic aspects of alternativescenarios for the two component optical response. Wereiterate that the observed tight correlation with the twocomponent optical response and the presence of a pseu-dogap [2] is natural in the present theory, where theMIR peak is to be associated directly with the break-ing of meta-stable pairs. Such a contribution does notdisappear below T c , until all pairs are condensed. Insummary, our paper appears compatible with the veryimportant experimental conclusion in Ref. 2 that “Ourfindings suggest that any explanation [of the MIR peak]should take into account the correlation betwen the for-mation of the mid IR absorption and the development ofthe pseudogap.”This work is supported by NSF-MRSEC Grant0820054 and we thank Vivek Mishra for helpful conver-sations. C.C.C. acknowledges the support of the U.S.Department of Energy through the LANL/LDRD Pro-gram. Appendix A: Further numerical studies
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