Determination of boson spectrum from optical data in pseudogap phase of underdoped cuprates
DDetermination of boson spectrum from optical data in pseudogapphase of underdoped cuprates
Jungseek Hwang ∗ and J. P. Carbotte , Department of Physics, Sungkyunkwan University,Suwon, Gyeonggi-do 440-746, Republic of Korea Department of Physics and Astronomy, McMaster University,Hamilton, Ontario L8S 4M1 Canada and The Canadian Institute for Advanced Research, Toronto, ON M5G 1Z8 Canada (Dated: November 3, 2018)
Abstract
Information on the nature of the dominant inelastic processes operative in correlated metallicsystems can be obtained from an analysis of their AC optical response. An electron-boson spectraldensity can usefully be extracted. This density is closely related to the optical scattering rate.However, in the underdoped region of the high T c cuprate phase diagram a new energy scale (thepseudogap) emerges, which alters the optical scattering and needs to be taken into account in anyfit to data. This can influence the shape and strength of the recovered boson spectral function.Including a pseudogap in an extended maximum entropy inversion for optimally doped Bi-2212 ismore consistent with existing data than when it is left out as done previously. a r X i v : . [ c ond - m a t . s up r- c on ] O c t . INTRODUCTION Boson structures have been noted in the physical properties of the high critical temper-ature superconducting cuprates using various techniques. These include angular resolvedphotoemission (ARPES), infrared optical conductivity (IR), point contact, as well as scan-ning tunneling spectroscopy (STM), and Raman. Assuming that these structures can bedescribed approximately within a boson exchange formalism, they can provide valuable in-formation on the effective underlying electron-boson spectral density I χ ( ω ) related to theinelastic scattering or glue involved in their superconductivity. In such an approach a Kuboformula relates the spectral density of interest to the AC optical conductivity. For example,inversion of optical data then proceeds directly from the conductivity σ ( ω ) or from the op-tical scattering rate 1 /τ op ( ω ). A least square fit can be used to determine parameters in anassumed mathematical form for I χ ( ω ). A maximum entropy technique based on simplifiedyet quite accurate analytic forms for the conductivity derived by Allen have also been em-ployed. Such a technique has the advantage that there is no need for any assumption aboutthe particular form for I χ ( ω ). Instead it is obtained numerically in which case more finedetails may emerge.Much of the work so far has been restricted to cases in which the electronic structuredoes not develop a new energy scale of the same order of magnitude as the boson energieswe wish to probe. In principle, the assumption that the electronic density of state is en-ergy independent in the energy range of interest, is likely to be valid only for the optimaland overdoped region of the cuprate phase diagram. Modification can be expected in theunderdoped region when a pseudogap develops . Some analyses of data including a pseu-dogap have already appeared , in which parameters characterizing an assumed form forthe pseudogap density of states as well as for the spectral density are varied in a least squarefit. Here we consider how the maximum entropy technique is to be adapted to the case ofan energy dependent density of states.This work will be based on a generalized approximate, but still accurate, analytic form forthe relationship between the optical scattering rate and the spectral density which furtherincludes an electronic density of states factor ˜ N ( ω ). When this factor is assumed constantwe recover the equation given by Allen. The new equation at zero temperature T = 0was given by Mitrovic and Fiorucci and later generalized by Sharapov and Carbotte to2nclude finite temperatures. In the case of finite T but constant ˜ N ( ω ). The generalizedformula also reduces, as it should, to that given by Shulga et al. as the finite temperaturegeneralization of the original Allen equation. Section II is an introduction to the theoretical concepts on which this work is based,and section III is a summary of the maximum entropy inversion method (MEM) used here.Section IV establishes preliminary numerical MEM results which will guide us in inversionof real data which is found in section V.
II. THEORETICAL CONSIDERATIONS
The frequency dependent optical conductivity σ ( T, ω ) for a correlated electron system canusefully be written in terms of a frequency and temperature dependent optical self energyΣ op ( T, ω ) which plays a role in optics similar to the quasiparticle self energy of angularresolved photoemission (ARPES). Denoting the plasma energy by Ω p we can write σ ( T, ω ) = i π Ω p ω − op ( T, ω ) (1)The imaginary part of − op ( T, ω ) defines an optical scattering rate 1 /τ op ( T, ω ) and thereal part a renormalized optical effective mass m ∗ op ( T, ω ) /m with ω [ m ∗ op ( T, ω ) /m −
1] = − Re Σ op ( T, ω ). The optical mass enhancement λ op ( T, ω ) is defined as 1 + λ op ( T, ω ) = m ∗ op ( T, ω ) /m . In terms of 1 /τ op ( T, ω ) and λ op ( T, ω ) the conductivity takes on a Drude formwith its real part σ ( T, ω ) = Ω p π /τ op ( T, ω )[ ω (1 + λ op ( T, ω ))] + [1 /τ op ( T, ω )] (2)which differs from its simplest rendition only through energy and temperature dependence in 1 /τ op ( T, ω ) and mass renormalization λ op ( T, ω ). This energy and temperature dependencecarries the information on the inelastic scattering here, due to coupling to an effective bosonexchange mechanism. In conventional superconductors these lead to so called strong cou-pling corrections to conventional BCS theory. Of course additional corrections can alsoplay a role such as energy dependence in the density of electronic states and momentumanisotropies.
In general σ ( T, ω ) of eqn.(2) can be calculated from a Kubo formula A ( k, ω ) both at the same momentum k but displaced in energy ω by the photon energy Ω, neglecting vertex corrections. For a boson exchange modelwith electron-boson spectral density I χ ( ω ), Allen derived a very simple approximate, butanalytic, formula which relates 1 /τ op ( T, ω ) directly to I χ ( T, ω ). It was generalized to finitetemperature by Shulga et al. and by Sharapov and Carbotte to the case when there isimportant energy dependence in the effective electronic density of state ˜ N ( ω ) which needsto be taken into account. The formula of Sharapov and Carbotte is1 /τ op = πω ∞ (cid:90) d Ω I χ (Ω) + ∞ (cid:90) −∞ dz [ N ( z − Ω) + N ( − z + Ω)] × [ n B (Ω) + 1 − f ( z − Ω)][ f ( z − ω ) − f ( z + ω )] (3)where n B (Ω) and f (Ω) are respectively the Bose-Einstein and Fermi-Dirac distribution func-tions. This generalized formula properly reduces to the form given by Shulga el al. whenthe effective density of state ˜ N ( z ) is constant, and also to Allen’s form when temperature istaken to be zero. For zero temperature but a variable density of state, we get the formulagiven by Mitrovic and Fiorucci τ op ( T = 0 , ω ) ≡ τ op ( ω ) = 2 πω ω (cid:90) d Ω I χ (Ω) ω − Ω (cid:90) dz ˜ N ( z ) (4)where ˜ N ( ω ) is the symmetrized density of state [ N ( ω ) + N ( − ω )] /
2. When it is constant andequal to one we obtain the well known Allen formula and find that the second derivative of ω/τ op ( ω ) is I χ ( ω ) i.e. 12 π d dω (cid:104) ωτ op ( ω ) (cid:105) = I χ ( ω ) (5)While formula (5) is simple, even when the full Kubo formula for the conductivity in a bosonexchange model is used, this formula is known to reproduce accurately the spectral densityin the energy range in which it is non-zero. Above the cutoff in I χ ( ω ) the derivative on theleft hand side of eqn.(5) can become negative in the more complete theory , but this is ofno consequence here. 4 II. MAXIMUM ENTROPY INVERSION WITH ENERGY DEPENDENT ELEC-TRONIC DENSITY OF STATE
Equation (4) can be written in the general form1 τ op ( ω ) = (cid:90) d Ω I χ (Ω) K ( ω, Ω) (6)where the kernel K ( ω, Ω) can be read off eqn.(4) but for the present purpose can be leftgeneral and unspecified. For a general kernel, K ( ω, Ω), and input data, D in ( ω ), with D in ( ω ) = (cid:82) + ∞ K ( ω, Ω) I χ (Ω) d Ω the deconvolution of this equation to recover an effectivespectral density, I χ (Ω) is ill-conditioned and here we use a maximum entropy technique. The equation can be discretized D in ( i ) = (cid:80) j K ( i, j ) I χ ( j )∆Ω where ∆Ω is the differentialincrement on the integration over Ω j = j ∆Ω with j an integer. We define a χ by χ = N (cid:88) i =1 [ D in ( i ) − Σ( i )] (cid:15) i (7)where D in ( i ) is the input data, and Σ( i ) ≡ (cid:80) j K ( i, j ) I χ ( j )∆Ω is calculated from theknown kernel and a given choice of I χ ( j ), and (cid:15) i is the error assigned to the data D in ( i ).Constraints such as positive definiteness for the boson exchange function are noted and theentropy functional L = χ − σS (8)is minimized with the Shannon-Jones entropy , SS = ∞ (cid:90) (cid:104) I χ (Ω) − m (Ω) − I χ (Ω) ln (cid:12)(cid:12)(cid:12) I χ (Ω) m (Ω) (cid:12)(cid:12)(cid:12)(cid:105) d Ω (9)The parameter σ in eqn.(8) controls how close a fit to the data is obtained. The parameter m (Ω) is here taken to be some constant value on the assumption that there is no a prioriknowledge of the functional form of the electron-boson spectral density I χ (Ω). Whilethere is no guarantee that a boson exchange model can successfully reproduce consistently,quantitatively, and accurately all the details of optical data in highly correlated systems,it does produce important information. An important fact to note, and this has been well5ocumented and stressed in the review of Carbotte, Timusk, and Hwang , is that there is agreat deal of qualitative agreement between recovered spectrum using ARPES, IR, Ramanand STM tunneling. This provides confidence to go further and now include more rigorouslypseudogap features which here enter in eqn.(6) through the density of state factor ˜ N ( ω )with the necessary modifications due to the opening of a pseudogap ∆ pg . Here we do notwish to commit to a particular specific pseudogap model but instead take a parameterizedform for the effective symmetrized DOS ˜ N ( ω ) and vary parameters. Once this is fixed,maximum entropy will provide us with a spectral density I χ ( ω ) for a given set of datafor the optical scattering rate 1 /τ op ( ω ). This does not tell us anything about the actualorigin of the boson involved in the scattering of the charge carriers, and the origin of thesebosons remains controversial. A review of all available inversions based on optics as well ason Raman and angular resolved photo emission (ARPES) and other considerations givenin reference [1] led the authors to nevertheless conclude that the spin fluctuations play themajor role with possibly a small contribution at the 10 % level from the phonons. Shouldthe recently discovered novel magnetic modes associated with zero momentum ( q = 0)contribute significantly to the glue, they would also in principle, be included in the recoveredspectra. IV. PRELIMINARY NUMERICAL RESULTS FOR MAXIMUM ENTROPY IN-VERSIONS
Taking the second derivative of ω/τ op ( ω ) in eqn.(4) gives12 π d dω (cid:104) ωτ op ( ω ) (cid:105) = I χ ( ω ) ˜ N (0) − ω (cid:90) d Ω I χ (Ω) dd Ω [ ˜ N ( ω − Ω)] (10)which is quite different from the result of eqn.(5) for the constant density of state case.Here the first term does give I χ ( ω ) reduced by the factor ˜ N (0) ≡ N and the second isa correction. It is instructive to change the integral in eqn.(10) through an integration byparts to obtain12 π d dω (cid:104) ωτ op ( ω ) (cid:105) = I χ ( ω = 0) ˜ N ( ω ) + ω (cid:90) d Ω ˜ N ( ω − Ω) dd Ω [ I χ (Ω)] (11)6his form provides a first term for the second derivative of ω/τ op ( ω ) which is now theproduct of I χ ( ω ) at ω = 0 and ˜ N ( ω ) while the second term is a correction to this simplifiedresult. It is interesting to consider the case of a marginal fermi liquid (MFL) model for thespectral density I χ (Ω) which has the form A tanh(Ω /k B T ). For low temperature this formprovides a constant I χ ( ω ) = A and its derivative is zero. Thus for this particular case thesecond derivative of eqn.(11) gives the product of I χ ( ω = 0) ˜ N ( ω ) = I χ ( ω ) ˜ N ( ω ) and isto be contrasted with the eqn.(5). For a constant ˜ N ( ω ) we get I χ ( ω ) while for a constantspectral density we get the effective density of states ˜ N ( ω ) which includes the pseudogap.It is important to stress that this result is restricted to a constant spectral density and thesecond term in eqn.(11) will provide modifications in all other cases.In Fig. 1 we show results of our maximum entropy inversions of optical data generatedin a MFL model for I χ ( ω ), and square well model for ˜ N ( ω ) which is taken equal to 0.33below the pseudogap energy ω = ∆ pg = 20 meV with missing states piled up just abovethis energy and distributed equally in the range ω = ∆ pg to 2∆ pg . The input product of I χ ( ω ) × ˜ N ( ω ) is represented by the red dash-dotted curve, the maximum entropy inversionis the solid blue curve, and the second derivative technique yields the dashed green curve.Both agree quite well with the input product; our expectation that we should get to a goodapproximation to I χ ( ω ) × ˜ N ( ω ) is borne out by the numerical work. It is clear howeverthat we cannot get independent information on ˜ N ( ω ) and I χ ( ω ) from optics even in thisvery simplified case. V. APPLICATION OF MAXIMUM ENTROPY METHOD TO REAL DATA
In the top frame of Fig. 2 we present a model for the optical scattering rate 1 /τ op ( ω )serving as a convenient input for our maximum entropy inversions. The model is basedon data for Bi-2212 UD 69 at T = 70 K with intercept at ω = 0 set zero so as tosimulate a clean sample at zero temperature. Based on this realistic form we now study howmaximum entropy inversion at T = 0 works when there is a pseudogap in the system butthe corresponding details of the density of states ˜ N ( ω ) are not known. In the middle framewe show the recovered I χ ( ω ) for 5 cases. In all instances we have taken a pseudogap modelpreviously used with success by Hwang in his least square fit analysis of the Bi-2212. The7
20 40 60 80 1000.00.20.40.6 I χ ( ω ) input I χ ( ω ) x N( ω ) MEM, σ = 0.47 2nd dervative N = 0.33 ω (meV) FIG. 1. (Color online) The product (dash dotted red curve) of the input marginal fermi liquid(MFL) spectral density I χ ( ω ) multiplied by a density of state ˜ N ( ω ) which is 33 % its constantenergy value below ω = ∆ pg = 20 meV. Lost states in ˜ N ( ω ) are piled up above ∆ pg between 20and 40 meV. The solid blue curve is the spectrum recovered from a maximum entropy inversion ofthe scattering rate 1 /τ op ( ω ) and the dashed green curve is the second derivative of eqn.(10) and(11). model has the from. ˜ N ( ω ) = N + (1 − N ) (cid:16) ω ∆ pg (cid:17) for | ω | ≤ ∆ pg = 1 + 2(1 − N )3 for | ω | ∈ (∆ pg , pg )= 1 for | ω | ≥ pg . (12)This mathematical form is illustrated in the inset of Fig. 3 for a case N = 0 .
25 and ∆ pg =44meV. In Hwang’s previous work the electron-boson spectral density was modeled with twoanalytic curves I χ ( ω ) = A s ωω s + ω + A m ωω m + ω (13)which consists of an MMP piece (second term) representing coupling to spin fluctuationsas in the work of Millis, Monien and Pines (MMP). This provides a background extendingover several 100 meV with ω m a spin fluctuation frequency and A m an amplitude. An ad-ditional sharp peak (first term), possibly representing coupling to an optical resonance at ω s , is also included in the least square fit to the scattering rate which has six parameters.Here, however, we will not use the functional form eqn.(13) for I χ ( ω ) but instead employa maximum entropy technique; this in no way commits us to a particular form for I χ ( ω ).Such a technique applied to optical data in La . Sr . CuO produced a two peak structure8n the recovered electron-boson spectral density for example . Most recovered spectra ,however, show a low energy resonance structure superimposed on a background which ex-tends to energies as high as 300 meV or even higher. Such features are well representedqualitatively with the analytic form of eqn.(13). We note that the density of state modelused in reference , which we retain here, is similar to what is found in STM work of Renneret al. The I χ ( ω ) obtained by Hwang is shown as the grey dashed line in the bottomframe of Fig. 2 and will be discussed later. In the middle frame we show results of maximumentropy inversion of eqn.(6) with kernel given in eqn.(4) and the model ˜ N ( ω ) as in Hwang and previously in Hwang et al. where it is applied to the analysis of OrthoII YBCO. In allcases ∆ pg = 44 meV, but various value of N in eqn.(12) are employed, namely blue ( N = 0.1), pink ( N = 0 . N = 0.5), black ( N = 0.75), and red N = 1.0 whichcorresponds to the case of no pseudogap i.e. a flat density of state. In all instances good fitto 1 /τ op ( ω ) is obtained as shown in the top frame. We see that the recovered I χ ( ω ) howeverchanges significantly as N goes from 0.1 to 1.0. The peak moves to higher frequency andgenerally increases in height and more spectral weight is transferred from the ω ∼ = 0 regionwith increasing N . If there were a pseudogap in the system with N = 0 .
25 as found inthe least square fit approach of Hwang , and it were ignored in a maximum entropy fit, wesee that the resultant I χ ( ω ) given in the dashed grey curve of the lower frame would bevery different from its true value. It is clear from these results that, in an analysis of opticaldata in the underdoped region of cuprate phase diagram, we need to include the pseudogapif we are to obtain a reliable value of the spectral density and in particular get correctly theposition of its peak which represents coupling to a sharp resonance mode. The inset of thetop frame makes a similar point. It shows the real part of the optical conductivity basedon a model I χ ( ω ) of reference [6] (solid red) including a pseudogap and without (dashedblue). We see significant differences between these two curves. In particular the effectiveboson assisted incoherent Holstein sideband shows a sharp onset at the energy of the peakin our model I χ ( ω ), with onset is shifted to the right by the pseudogap energy as comparedwith the case without pseudogap. It is also reduced in magnitude.In the bottom frame of Fig. 2 we show that when maximum entropy is used for inversionwith the known pseudogap model, we get an excellent reproduction (solid purple curve) ofits least square fit determination (dashed grey curve). On the other hand if the maximumentropy inversion proceeds on the assumption of a constant density of state we get the9 T = 0 K
Model data based on Bi-2212 UD69 at T =70 K / τ op ( ω ) ( m e V ) ∆ pg = 44 meV N = 1.0N = 0.1 I χ ( ω ) N =0.10, σ = 1.2 N =0.25, σ = 2.0 N =0.50, σ = 2.5 N =0.75, σ = 2.0 N =1.0 (No PG), σ = 2.0 input I χ ( ω ) (N = 0.25) N =0.25, σ = 2.0 2nd derivative N =1.0 (No PG), σ = 2.0 I χ ( ω ) ω (meV) ω (meV) σ ( ω ) ( Ω - c m - ) FIG. 2. (Color online) Model optical scattering rate data 1 /τ op ( ω ) (solid black curve) for zerotemperature based on a Bi-2212 UD69 sample (top frame). The other curves are our maximumentropy (ME) fits. The middle frame gives the recovered electron-boson spectral density I χ ( ω )when our ME inversion includes a pseudogap ∆ pg = 44 meV with various values of N as noted inthe figure. The bottom frame is for fix N = 0 .
25. The grey dashed curve gives our input modelfor I χ ( ω ) and the solid purple curve the spectrum recovered from ME inversion including themodel DOS ˜ N ( ω ) with pseudogap. The solid red curve is the spectrum recovered when ˜ N ( ω ) isassumed constant i.e. N = 1 . / (2 π ) d /dω [ ω/τ op ( ω )]. In the inset we display the real part of the opticalconductivity σ ( ω ) with ( N = 0.0, solid red) and without ( N = 1.0, dashed blue) pseudogap fromthe work in reference [6]. solid red curve which peaks at higher energy than does the input I χ ( ω ). This agrees wellwith the second derivative result shown as the orange dash-dotted curve. Fig. 3 providesadditional results. The lower frame gives our MEM results for I χ ( ω ) when various valuesof ∆ pg itself are used (pink 0 meV, blue 10 meV, orange 20 meV, blue 30 meV and red44 meV, as before). The fixed parameter is the depth of the pseudogap well at ω = 0 i.e. N = 0 .
25 in all cases. The fits to the scattering rate data are given in the top frame. What10
T = 0 K
Model data based on Bi-2212 UD69 at T =70 K / τ op ( ω ) ( m e V ) ∆∆∆∆ pg = 44 meV ∆∆∆∆ pg = 0 meV N = 0.25 I χ ( ω ) ω (meV) ∆ pg = 0 meV, σ = 2.0 ∆ pg = 10 meV, σ = 2.0 ∆ pg = 20 meV, σ = 2.0 ∆ pg = 30 meV, σ = 2.0 ∆ pg = 44 meV, σ = 2.0 PG loss ∆ pg = 44 meV ω (meV) N ( ω ) FIG. 3. (Color online) As in Fig. 2 but now the depth of the well in the density of state ˜ N ( ω ) iskept fixed at N = 0 .
25 and the size of the pseudogap ∆ pg is varied as indicated in the figure. Thevarious lines in the top frame for the optical scattering rate 1 /τ op ( ω ) are the fits to the input data(black curve). The inset shows the model density of state used for the pseudogap and the shadedregion defines the spectral weight lost below ∆ pg due to pseudogap formation which we denote by P G loss . is clear from these data is that decreasing the value of the pseudogap pushes the peak inthe MEM I χ ( ω ) to higher energies, as the spectral density tries to compensate for the lossin scattering implied by a decrease in ∆ pg .Plotting the position of the peak in I χ ( ω ) obtained in all the cases considered in Fig. 2and Fig. 3, and additional ones for N = 0 .
25 in the upper frame of Fig. 4, shows that theyvary mainly with value of
P G loss defined as the area of the shaded region in the density ofstate shown in the inset of Fig. 3. This represents the area lost in the density of state belowthe pseudogap energy ω = ∆ pg as compared with its constant value (1.0 in our case). Itis also the area recovered in our model above ω = ∆ pg in the region to 2∆ pg . The almostlinear drop in the position of ω peak as a function of increasing P G loss is a useful observation11 λ ( ω c ) ω pea k ( m e V ) PG loss (meV) λ op ( ω c ) FIG. 4. (Color online) Microscopic parameters associated with our recovered I χ ( ω ) spectra as afunction of P G loss in meV. The top frame gives the energy of the peak in the spectral density. Themiddle frame gives the spectral mass enhancement λ ( ω c ) defined as twice the first inverse momentof I χ (Ω). The various points shown are based on the data in Fig. 2 and Fig. 3. The bottomframe gives the optical mass enhancement factor λ op ( ω ) at ω = 0 of equations (15) and (16), whichis also the same as its quasiparticle renormalization. Both differ from λ ( ω c ) when the electronicdensity of states varies with energy due to a finite pseudogap. because it can be employed, as we will elaborate below, to constrain parameters in theeffective density of state ˜ N ( ω ) when otherwise nothing is known about its variation with ω .However, before we address this issue we show in the middle frame of Fig. 4 correspondingresults for the derived mass enhancement parameter λ ( ω c ) defined in the usual way, as twicethe first inverse moment of the spectral function i.e. λ ( ω c ) = 2 (cid:82) ω c d Ω I χ (Ω) / Ω with acutoff on Ω set to 5000 cm − . We will refer to this quantity as the spectral lambda. By itsdefinition this is the electron-boson mass renormalization which enters many quantities suchas the critical temperature and quasiparticle, and optical mass in the case of a flat densityof electronic states. When ˜ N ( ω ) is not constant because there is a pseudogap, the optical12nd quasiparticle mass remain equal to each other, but are not given by the spectral lambda λ ( ω c ). In our model for the optical conductivity λ op ( ω ) is λ op ( ω ) = 2 ω ω c (cid:90) d Ω I χ (Ω) ∞ (cid:90) dω (cid:48) ˜ N ( ω (cid:48) ) ln (cid:104) ( ω (cid:48) + Ω) ( ω (cid:48) + Ω) − ω (cid:105) (14)and its zero energy limit ω → λ op ( ω = 0) = 2 ∞ (cid:90) dω (cid:48) ˜ N ( ω (cid:48) ) ω c (cid:90) d Ω I χ (Ω)( ω (cid:48) + Ω) (15)which is different from the spectral lambda λ ( ω c ) as discussed in reference and seen in thelower frame of Fig. 4. We can rewrite eqn.(15) for our ˜ N ( ω ) which is specified in eqn.(12). λ op ( ω = 0) = 2 ω c (cid:90) d Ω I χ (Ω) (cid:110) N (cid:16) −
1Ω + ∆ pg (cid:17) + (1 − N ) (cid:104) pg + 2Ω∆ pg ln (cid:12)(cid:12)(cid:12) ΩΩ + ∆ pg (cid:12)(cid:12)(cid:12) + (cid:16) Ω∆ pg (cid:17) (cid:16) −
1Ω + ∆ pg (cid:17)(cid:105) + (cid:104) − N ) (cid:105)(cid:16)
1Ω + ∆ pg −
1Ω + 2∆ pg (cid:17) + 1Ω + 2∆ pg (cid:111) (16)We see that the spectral renormalization λ ( ω c )increases with increasing P G loss (middle framein Fig. 4), by contrast the optical mass is nearly independent of pseudogap details. As shownin the top frame, there is a decrease in ω peak with increasing P G loss and this leads to anincreased contribution to λ ( ω c ) because of the term 1 / Ω in its definition. But in λ op ( ω = 0)the additional presence of the pseudogap has the opposite tendency, because it reduces theeffectiveness of small Ω below ∆ pg and both effects combined leave λ op (0) fairly constant.Armed with the observation that ω peak decreases with P G loss , and that this relationshipis robust and minimally dependent on the details of the energy variation assumed for ˜ N ( ω ),we turn to experiments. In the upper frame of Fig. 5 we reconsider the Bi-2212 UD69 firstanalyzed by Hwang . Here we assume that the sharp peak in I χ ( ω ) is due to couplingof the charge carriers to the spin one resonance observed in inelastic spin polarized neutronscattering following the law ω sr ∼ = 5 . k B T c where T c is the sample critical temperature.This allows us to fix the peak position in the spectral density as well as the value of P G loss in the pseudogap density of state at 16.1 meV read off the top frame of figure 4. This13
100 200 300 4000.00.20.40.60.80.00.51.01.52.02.5
MEM ∆ pg = 32 meV, N = 0.59; PG loss = 8.7 meV ∆ pg = 20 meV, N = 0.34; PG loss = 8.7 meV T = 102 K
Bi2212-OPT96 I χ ( ω ) ω ( meV ) MEM
T = 70 K ∆ pg = 54 meV, N = 0.55; PG loss = 16.1 meV ∆ pg = 35 meV, N = 0.30; PG loss = 16.1 meV Bi2212-UD69 I χ ( ω ) FIG. 5. (Color online) The electron-boson spectral density I χ ( ω ) as function of energy ω in meVrecovered from optical scattering rate data in Bi-2212. The upper frame is for an underdopedsample UD69 and the lower for optimally doped OPT96 from reference. The dash-dotted red andsolid blue curves result from a least square fit using a model spectral density consisting of an MMPbackground augmented with a sharp peak at ω s / (3 / ) (see eqn. 13) . This energy is taken tobe 5.4 k B T c and is the energy of the spin one resonance seen in the spin fluctuation spectrum byinelastic polarized neutron scattering. In addition a pseudogap is included in the density of statemodel shown in the inset of Fig. 3 with parameter N and ∆ pg which are also included in the leastsquare fit with fixed value of P G loss , 16.1 meV for UD69 and 8.7 meV for OPT96. The dashed bluecurve is for comparison and is the spectrum obtained in a flat band maximum entropy inversion ofthe same data. leaves a single parameter in the characterization of ˜ N ( ω ). Recently H¨ufner et al. haveprovided a summary of known pseudogap values as a function of doping ( p ) for a greatvariety of systems and from many different measurement techniques. They find that thepseudogap becomes zero only at the top of the superconducting dome and that it growsroughly linearly as doping ( p ) is decreased. We can use their average fit to the data toestimate the pseudogap value in the UD69 sample and obtain ∆ pg ∼ = 56 meV. This fixes our14seudogap density of state model completely and N = 0 .
55. The remaining parameters in I χ ( ω ) are then varied, and we get the solid blue curve in the upper frame of Fig. 5. Ifinstead we arbitrarily reduce ∆ pg to 35 meV but change N to a value of 0.3 to preserve P G loss at 16.1 meV, the dash-dotted red curve is obtained which shows that the recovered I χ ( ω ) of a least square fit to optical data is not very sensitive to the value of ∆ pg used,provided P G loss is left fixed.While we have presented here only the results of a least square fit with fixed model for˜ N ( ω ), we know from our results in the bottom frame of Fig. 2 that a maximum entropyinversion with this same fixed ˜ N ( ω ) would return the same I χ ( ω ) as the least square fitprocedure did. If, however, we had applied to the optical data a maximum entropy inversionassuming instead that the density of state is flat (no pseudogap structure), we would haveobtained the dashed blue curve shown in the upper frame of Fig. 5. As before, the peak inthis second function has been pushed upward as compared to the input function. When apseudogap forms, it depresses the scattering in the energy range below ω ≤ ∆ pg and if thisis assigned instead to the effect of the boson spectra, it effectively needs to be reduced inthat energy region. Further, for energies above ∆ pg it would need to be increased becauseof the recovery region in ˜ N ( ω ) from ω = ∆ pg to 2∆ pg where the DOS is larger than one.The two effects combine to reduce the spectra weight in I χ ( ω ) at low ω , compared with itsinput value, and to increase it in the region of the peak in the dashed blue curve.This new finding allows us to reassess the case of optimally doped B-2212 OPT96 invertedby maximum entropy in the work of Hwang et al. who assumed a flat density of state model.Returning to the curve given in H¨ufner et al. we estimate that the pseudogap ∆ pg for thissample has a value of 32 meV. A puzzle noted, but not resolved in Ref. [5], is that the peakposition in I χ ( ω ) obtained in that work, and shown here as the blue dashed curve in thelower frame of Fig. 5, was 60 meV while neutron scattering gives a smaller value of 45 meV.This is due to the existence of a pseudogap in Bi-2212 OPT96 which was not accounted forin the previous maximum entropy inversion. If we take ∆ pg = 32 meV then, reference tothe top frame of Fig. 4 tells us that we should take P G loss = 8.7 meV to get ω peak = 45meV which implies N = 0 .
59. This gives the solid blue curve for I χ ( ω ). Reducing ∆ pg to20 meV and keeping P G loss the same, leads to the same inverted I χ ( ω ) (dash-dotted redcurve) whether one uses a least square fit or maximum entropy.15 I. SUMMARY AND CONCLUSIONS
We have found that including a pseudogap in the inversion process to obtain an electron-boson spectral density from optical data can have a large influence on the shape of therecovered I χ ( ω ). This holds whatever the modality used, be it a maximum entropy tech-nique or a least square fit to a parameterized assumed functional form which represents thespectral density we wish to obtain. Conversely, inversions based on a constant density ofelectronic states in cases when a pseudogap exists will tend to move a peak associated, forexample, with coupling to a spin-1 resonance as measured in polarized inelastic neutronscattering experiments, to higher energies and effectively increase its spectral weight in theelectron boson function I χ ( ω ). Based on this finding we were led to reexamine the presentlyavailable inversion of data in optimally doped Bi-2212 OPT96 for which the optical reso-nance (a large peak in I χ ( ω )) was found at 60 meV which is considerably higher than theneutron resonance in this material found at 45 meV. This discrepancy, so far unresolved,here finds a natural explanation. Optimally doped Bi-2212 already has a significant pseu-dogap. Taking its value from the compilation provided by H¨ufner et al. and repeating theinversion, we get a new I χ ( ω ) with a large peak at 45 meV in agreement with neutrons.An important intermediate result is our finding that the detailed shape of the electronicdensity of state ˜ N ( ω ) including a pseudogap does not impact strongly on the position ω peak of the resonance in I χ ( ω ). What is most important is the number of states removed below ω = ∆ pg which are assumed to pile up above ∆ pg in a recovery region of order, ω (cid:39) pg . ACKNOWLEDGMENTS
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