Evolution of electron-boson spectral density in the underdoped region of Bi_2Sr_{2-x}La_xCuO_6
EEvolution of electron-boson spectral density in theunderdoped region of Bi Sr − x La x CuO 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, OntarioL8S 4M1 Canada The Canadian Institute for Advanced Research, Toronto, ON M5G 1Z8 CanadaE-mail: [email protected]
Abstract.
We use a maximum entropy technique to obtain the electron-boson spectral densityfrom optical scattering rate data across the underdoped region of the Bi Sr − x La x CuO (Bi-2201) phase diagram. Our method involves a generalization of previous work whichexplicitly include finite temperature and the opening of a pseudogap which modifiesthe electronic structure. We find that the mass enhancement factor λ associated withthe electron-boson spectral density increases monotonically with reduced doping andcloser proximity to the Mott antiferromagnetic insulating state. This observation isconsistent with increased coupling to the spin fluctuations. At the same time thesystem has reduced metallicity because of increased pseudogap effects which we modelwith a reduced effective density of states around the Fermi energy with the range ofthe modifications in energy set by the pseudogap scale.PACS numbers: 74.25.Gz,74.20.Mn a r X i v : . [ c ond - m a t . s up r- c on ] O c t volution of electron-boson spectral density in the underdoped region of Bi Sr − x La x CuO
1. Introduction
Our understanding of the interactions that most importantly determine the low energyproperties of the high critical temperature superconducting copper oxides remainsincomplete and a challenge to the field. Structures on a scale of the order ofphonon and/or spin fluctuation energies that are seen in various properties have foundinterpretation in terms of a phenomenological boson exchange model. In such anapproach an electron-boson spectral density plays a central role and an important aimis to determine its magnitude and frequency dependence[1]. In principle there willbe a different electron-boson spectral function for each electron involved in the inelasticscattering and so it would also depend on initial state momentum. On the other hand inoptics which is the case of interest here, all electrons are involved and the most relevantfunction is momentum averaged. For Raman scattering, the relevant average dependson the polarization of the incident light. For B g (B g ) the average involved is weightedmore in nodal (antinodal) direction in the CuO Brillouin zone. Of course anisotropiesare known to play a role even in conventional metals[2]. By contrast, angular resolvedphotoemission spectroscopy (ARPES) involves no momentum average but it is also asurface sensitive technique. Yet another average determines the critical temperature.In the gap channel the d-wave projection of the momentum discriminating spectraldensity would appear while in the renormalization channel it would be its average asin the normal state optical conductivity σ ( T, ω ) as a function of photon energy ω attemperature T .Mathematical techniques have been developed[3, 4] which make it possible toextract from optical data, an estimate of the underlying electron-boson spectraldensity I χ (Ω) involved in the inelastic scattering of the charge carriers. The opticalconductivity is related to microscopic quantities through the Kubo formula[5] and thisinvolves a two-particle propagator. The resulting expression for σ ( T, ω ) can be usefullyrewritten as a generalized Drude form in terms of a memory function. The memoryfunction plays a similar role for the conductivity as does the quasiparticle self-energyfor the single particle propagator that enters ARPES. In direct analogy, an opticalscattering rate 1 /τ op ( T, ω ) can be defined and taken as the input data for the inversionprocess whereby I χ (Ω) is extracted through the use of a maximum entropy technique.To make this possible it was necessary to obtain a simplified but still quite accuratelinear integral relationship between 1 /τ op ( T, ω ) and the unknown I χ (Ω). Allen[6] firstobtained the explicit form for the kernel in this integral equation. It applied at T =0 and was generalized to finite T by Shulga et al. [7]. Both derivations assumed thatthe band structure density of states could be taken constant over the energy range ofinterest. When this no longer applies because of pseudogap formation as is the casein the underdoped region of the high T c cuprates, Sharapov and Carbotte[8] provideda new formula valid at any temperature T which features in addition to I χ (Ω), theeffective self consistent electronic density of states ˜ N ( ω ). Their expression reduced, asit must, to the earlier formula of Mitrovic and Fiorucci[9] when the zero temperature volution of electron-boson spectral density in the underdoped region of Bi Sr − x La x CuO T → I χ (Ω) in the underdoped regionof Bi Sr − x La x CuO (Bi-2201) from the recently published optical data in this cupratefamily by Dai et al. [12]. An important aim is to investigate how I χ (Ω) evolves as thedoping ( p ) is reduced and the Mott antiferromagnetic insulating state is approached.Most of the previous work[3, 4], but not all[13], has not included effects of pseudogapformation[14] in the inversion process. Here we also generalized the technique to handlefinite temperature data.
2. Theoretical frame work
We begin with the generalized Drude formula for the temperature T and photon energy ω dependent optical conductivity[3] σ ( T, ω ) = i π Ω p ω − op ( T, ω ) (1)where Ω p is the plasma frequency and Σ op ( T, ω ) is the optical self-energy related to thememory function. It is conventional to write − Im Σ op ( T, ω ) as the optical scatteringrate 1 /τ op ( T, ω ) which is T and ω dependent. The real part − Re Σ op ( T, ω ) is writtenin terms of an optical effective mass m ∗ op as [ m ∗ op ( T, ω ) /m − ω where again it isdependent of both energy and temperature. The two functions τ op ( T, ω ) and m ∗ op ( T, ω )are not independent but are related by a Kramers-Kronig transform. Experimentalistsoften measure the reflectivity of their sample from which they obtain σ ( T, ω ). Manyalso provide the optical scattering rate which can be obtained from their knowledge ofthe conductivity. The relationship is1 τ op ( T, ω ) ≡ Ω p π Re (cid:104) σ ( T, ω ) (cid:105) . (2)Here we start with the data for 1 /τ op ( T, ω ) given by Dai et al. [12] in Bi-2201 andfor comparison we will also use similar data by Hwang et al. [15] for two samples ofBi-2212 namely UD69 (underdoped with T c = 69 K) and OPT96 (optimally dopedwith T c = 96 K). The relationship between optical scattering rate and the electron-boson spectral density I χ (Ω) which we wish to obtain is given by the Kubo formulawhich gives the optical conductivity in terms of the microscopic parameters of thematerials of interest. In general a two-particle correlation function is involved withvertex corrections. Simplifications are needed if one is to make progress. For the casewhen the electronic density of states ˜ N ( ω ) is approximately constant over the energyrange of interest in optical experiment P. B. Allen[6] derived a very simple, yet as it hasturned out, quite accurate formula that directly relates 1 /τ op ( ω ) to the electron-bosonspectral density I χ (Ω) which applies at T = 0. He found1 τ op ( ω ) = 2 πω (cid:90) ω d Ω I χ (Ω)( ω − Ω) . (3) volution of electron-boson spectral density in the underdoped region of Bi Sr − x La x CuO I χ (Ω) from theinformation on 1 /τ op ( ω )[3, 13]. Allen’s derivation is based on lowest order perturbationtheory (Fermi Golden Rule) with electron-boson spectral density I χ (Ω) appropriatefor transport. This function includes a weighting factor which emphasizes the enhancedeffect of backward scattering in depleting current. This weighting is absent in theequilibrium spectral density which enters quasiparticle properties. Formula (3) can alsobe obtained directly from the Kubo formula for the dynamic conductivity in a bosonexchange mechanism treated within an Eliashberg[16] formulation. In such an approacha weak coupling assumption is used to achieve the simplification involved in Eqn. (3)and vertex corrections are included at the level of changing the spectral density I χ (Ω)from its equilibrium to its transport value. An important conclusion of that work is thatformula (3) is quantitatively accurate. The same approach, based on the Kubo formula,was subsequently used by Sharapov and Carbotte[8] to include both a pseudogap andfinite temperature effects. In terms of the effective density of states N ( ω ), which nowhas an energy dependence, they found1 τ op ( ω, T ) = πω (cid:90) ∞ d Ω I χ (Ω) (cid:90) ∞−∞ dz [ N ( z − Ω) + N ( − z + Ω)] (4) × [ n B (Ω) + 1 − f ( z − Ω)][ f ( z − ω ) − f ( z + ω )]which is still linear in I χ (Ω) and a maximum entropy inversion technique still appliesbut the Kernel is now more complicated and depends both on temperature throughthe bose n B (Ω) and fermi f ( z ± Ω) thermal factors and on the self-consistent effectivedensity of states ˜ N ( z ) ≡ [ N ( z ) + N ( − z )] /
2. Note this quantity is symmetrized aboutthe Fermi energy at z = 0. In zero temperature limit Eqn. (4) was obtained previouslyby Mitrovic and Fiorucci[9] using a very different method. They generalized directlythe Fermi Golden Rule approach of Allen to the case of an energy dependent densityof electronic states. As discussed in Ref [13] Eqn. (4) at zero temperature can be usedto invert data on optical scattering rate in the standard way for a given model of ˜ N ( z )which must first be specified. In principle this function knows about the mechanism bywhich the pseudogap forms. But here we model it through a fit to experimental data.In this paper we have generalized our maximum entropy inversion codes to include thethermal factors of Eqn. (4) as well as our model for the effective density of states. Thisallows us to invert normal state data at any temperature T . In Fig. 1 (top frame) weshow the model we have used for ˜ N ( z ) in all our numerical works. ˜ N ( ω ) is taken tohave a value of N at ω = 0, it then increases as ω to values 1.0 at ω = ∆ pg abovethe pseudogap the state lost below ω = ∆ pg are distributed uniformly in the range from∆ pg to twice ∆ pg after which ˜ N ( ω ) becomes 1.0 (i.e. no changes over its constant baseband structure value). The ˜ N ( z ) can be written as˜ N ( z )= N +(1 − N ) (cid:16) z ∆ pg (cid:17) for | z | ≤ ∆ pg = 1 + 2(1 − N )3 for | z | ∈ (∆ pg , pg ) volution of electron-boson spectral density in the underdoped region of Bi Sr − x La x CuO
5= 1 for | z | > pg . (5)The parameters that can be varied are N related to the depth of the pseudogap welland the value of ∆ pg . The value of the pseudogap ∆ pg is known from the work ofH¨ufner et al. [14]. These authors have surveyed the results of many experiments whichgive information on the size of pseudogap, ∆ pg and have concluded that it rises linearlywith decreasing doping p , starting from zero at the upper side of the superconductingdome as shown (dashed red line) in the lower frame of Fig. 1. This is to be viewedas the behavior on average and a guide only. For the specific case of Bi-2201 not partof the H¨ufner et al. ’s survey, an early scanning tunnelling microscopy (STM) study[17]gave a small value of the pseudogap of order 10 meV. More recent data however givemuch larger values. Ma et al. [18] find a large gap of order ∼
27 meV for optimal dopingand ∼
10 meV for highly overdoped samples. Kurosawa et al. [19] find a large gap of30 meV for optimal and as large as 60 meV for highly underdoped ( p ∼ et al. [14] shownin Fig. 1 lower frame as the dashed red line. These values are also consistent with thesize of the pseudogap temperature T ∗ measured by NMR[20, 21] in Bi-2201 which are ofthe same order as measured in other high transition temperature cuprates[22] lendingfurther support for the phase diagram we have used in this study (lower frame of Fig.1). With the value of ∆ pg given, a single parameter remains to be specified in ourdensity of states model ˜ N ( ω ) namely N . In Ref. [13] we found that the inversionprocess was not very sensitive to the details of the density of state variation with energybut was mainly dependent on the number of lost states below ω = ∆ pg which arerecovered above this energy. Here we denote this quantity by P G loss and favor it asthe single measure of reduced metallic behavior brought about by the opening of thepseudogap.
3. Main results
First results are found in Fig. 2 which applies to UD11 (underdoped with T c = 11 K)sample of Dai et al. [12] at T = 100 K. The data for the optical scattering rate was readoff their figure (3b) and is reproduced in the top frame as the light solid black lines.For the top curve, no residual scattering was subtracted off the data while for other lowcurves we subtracted 1 /τ imp = 80, 120, and 140 meV (as marked in the figure) beforeproceeding with the maximum entropy inversion. In all cases temperature was set to 100K in the kernel of Eqn. (4). The solid blue curve is our fit to the top curve. Double dotdashed red curve is our fit to the data curves with 1 /τ imp = 140 meV. Fits to 1 /τ imp =80 meV (dashed green) and 120 meV (dash-dotted magenta) are also shown. Exceptfor the lowest curve the fits are equally good with ( σ = 6.0). Here σ is related to thequality of the maximum entropy fit to the data, as explained in reference [3], the defaultmodel for I χ ( ω ) is a constant.Turning next to the bottom frame of Fig. 2 we see that both solid blue and volution of electron-boson spectral density in the underdoped region of Bi Sr − x La x CuO Bi-2212
T = 100 KBi-2212 OPT96 ; N = 0.73, ∆ pg = 38 meV ; N = 0.60, ∆ pg = 38 meVBi-2212 UD69 ; N = 0.70, ∆ pg = 58 meV ; N = 0.60, ∆ pg = 58 meV ω (meV) Recent data from Ref. [17] & [18] ∆ pg ∆ pg ( m e V ) T c /T cmax T c / T c m a x hole doping, p Figure 1. (Color online) Top frame is the model used for the effective density ofstates ˜ N ( ω ) in the pseudogap phase. Parameters appropriate to Bi-2212 OPT96 withpseudogap ∆ pg = 38 meV and UD69 with ∆ pg = 58 meV are used. Two values of˜ N ( ω = 0) (denoted by N ) are considered N = 0.73 (dashed green) and 0.6 (dashed-double dotted brown) for OPT96 and 0.7 (dash-dotted magenta) and 0.6 (solid blue)curve. Bottom frame gives the phase diagram for Bi-2201 and Bi-2212. The lower dash-dotted (blue) curve gives T c /T maxc versus doping p (right hand scale). The straightdashed (red) line gives the value of the pseudogap size from the work of H¨ufner etal. [14] (left side scale in meV). The solid red circle symbols are from Ref. [18] and [19]. dashed green curves for I χ ( ω ) show unphysical upturns as ω → ω = 0 contribution to I χ ( ω ). We understand this as follows. In an electron-boson exchange formulation,impurity scattering can be simulate with a model I χ ( ω ) ∼ ω δ ( ω ). In all cases fromthis point on we first subtract a 1 /τ imp from the data of reference [12] chosen so that theunphysical upturn in I χ ( ω ) noted has just disappeared. Subtracting a larger elasticimpurity scattering only changes the small ω behavior of the spectral density and haslittle consequences on many of the average properties of I χ ( ω ) such as the area underit. But it does change the first inverse moment, which is related to the value of theelectron-boson mass renormalization. The weighting factor 1 /ω enhances the effect ofsmall ω in I χ ( ω ) but this is not very important when the unphysical rise in the spectraldensity at small ω has been removed. In all cases we take the value of residual scattering volution of electron-boson spectral density in the underdoped region of Bi Sr − x La x CuO / τ op ( ω ) - / τ i m p ( m e V ) τ imp = 0 meV; σ = 6.0 1/ τ imp = 80 meV; σ = 6.0 1/ τ imp = 120 meV; σ = 6.0 1/ τ imp = 140 meV; σ = 6.5 ω (meV)T = 100 K Bi2201 - UD11 I χ ( ω ) Figure 2. (Color online) Top frame gives the optical scattering rate in Bi-2201: UD11from the work of Dai et al. [12]. The data is in the normal state at T = 100 K (lightsolid black curves). We have subtracted from the raw data constant residual scatteringrates, 1 /τ imp , of 0 meV (solid blue), 80 meV (dashed green), 120 meV (dash-dottedmagenta) and 140 meV (dash double dotted red) from top to bottom. The fits tothe data are as labeled. The bottom frame gives the resulting electron-boson spectraldensities I χ ( ω ) in the same notation as the curves in the upper frame. Note the sharpunphysical rises in the spectral density at very low ω for no impurity scattering and80 meV subtracted cases. rate 1 /τ imp to be the value at which the unphysical divergence in I χ ( ω ) as ω → T = 100 K namely OPT96 lower curve and UD69 upper curve. The solidblack curve is data from reference[15, 23]. The dashed dark blue and dash-dotted pinkare our fits to the OPT96 sample, without ( N = 1.0) and with pseudogap ( N = 0.73).Here ∆ pg was taken to be 38 meV from reference[14]. In agreement with our previousfinding the peak in the case N = 1.0, which is at 61 meV, has shifted down to 53 meVwhen a pseudogap is accounted for (middle frame). If N is reduced to 0.60 the peak volution of electron-boson spectral density in the underdoped region of Bi Sr − x La x CuO UD69OPT96
T = 100 K
Bi-2212 / τ op ( ω ) - / τ i m p ( m e V ) OPT96
Maximum Entropy Fits : σ = 4.5, 1/ τ imp =20 meV no PG; Ω sf = 61 meV PG: Ω sf = 53 meV; N =0.73, ∆ pg = 38 meV PG: Ω sf = 47 meV; N =0.60, ∆ pg = 38 meV I χ ( ω ) UD69
Maximum Entropy Fits : σ = 5.0, 1/ τ imp = 0 meV no PG; Ω sf = 47 meV PG: Ω sf = 37 meV; N =0.70, ∆ pg = 58 meV PG: Ω sf = 34 meV; N =0.60, ∆ pg = 58 meV I χ ( ω ) ω (meV) Figure 3. (Color online) Top frame is the optical scattering rate (solid black curve)for Bi-2212 taken from the work of Hwang et al. [23]. The upper curve is for UD69 whilethe lower curve is for OPT96 with a residual scattering 1 /τ imp of 20 meV subtractedfrom the raw data. The dashed dark blue line is our maximum entropy fit assumingthere is no pseudogap. The dash-dotted (pink) curve includes a pseudogap of ∆ pg = 38meV and a well depth N = 0.73 while for the solid red curve N = 0.60. These are forOPT96 while for UD69, the purple dashed curve has no pseudogap, green dash-dottedhas ∆ pg = 58 meV with N = 0.70 and solid blue has N = 0.60. The two lowerframes show our results for the recovered electron-boson spectral density I χ ( ω ). Theline type and colors are the same as for the top frame. All spectra show a peak atΩ sf and a background extending to high energies. The value Ω sf that is obtained issmaller for the underdoped sample as compared with optimally doped. Including apseudogap shifts Ω sf to lower energies as does lowering the value of N . Ω sf , which is the spin-fluctuation scale, is shifted further to 47 meV (solid red curve).For the UD69 sample the pseudogap is larger, 58 meV (see Fig. 1 bottom frame).Its effect on the recovered I χ ( ω ) is shown in the bottom frame of Fig. 3. The solidblue curve is for N = 0.60, the dash-dotted green is for N = 0.70, and the dashedpurple is for N = 1.0 (i.e. no pseudogap is accounted for). Note that the peak energygoes from 47 meV ( N = 1.0) to 37 meV and 34 meV with N (cid:54) = 1 .
0. Including thepseudogap has a very significant effect on the recovered electron-boson spectral densityand this needs to be accounted for. In principle the emergence of a pseudogap changesthe electronic structure and hence should lead to corresponding modifications in theelectron-boson exchange spectral density I χ ( ω ). As our maximum entropy techniqueeffectively determines this function through a fit to the optical data itself, these effects volution of electron-boson spectral density in the underdoped region of Bi Sr − x La x CuO UD11 UD16 UD25 OPT33
T = 100 K
Bi-2201 / τ op ( ω ) - / τ i m p ( m e V ) With PG
UD11: σ = 6.0, N = 0.40 UD16: σ = 5.0, N = 0.43 UD25: σ = 7.0, N = 0.50 OPT33: σ = 5.0, N = 0.77 I χ ( ω ) UD11: σ = 6.0, 1/ τ imp = 120 meV UD16: σ = 5.0, 1/ τ imp = 70 meV UD25: σ = 7.0, 1/ τ imp = 100 meV OPT33: σ = 5.0, 1/ τ imp = 45 meV Without PG I χ ( ω ) ω (meV) Figure 4. (Color online)Top frame, the optical scattering rate (solid black curves)for Bi-2201 from the work of Dai et al. . The four samples with our maximum entropyfits are UD11 solid blue, UD16 dashed red, UD25 dash-dotted dark green and OPT33dash-double dotted brown curves. The recovered electron-boson spectral densities areshown in the middle and bottom frames where the line types and colors are the sameas for the top frame. The heavy curves (middle frame) include a pseudogap while thelight curves (bottom frame) do not. The parameter σ controls the quality of the fitwith default model a constant (see Ref. [11]), [3] and [13] are included automatically in the recovered I χ ( ω ) function.In Fig. 4 we show our results for the Bi-2201 series of reference[12]. The data (solidblack curve) was read off from their Fig. (3b). In all cases we are guided in our choice ofpseudogap well depth at zero energy (denoted here by N ) by the observation made inangular resolved photo emission (ARPES)[24] that the data are consistent with a Fermiarc model with arc length proportional to the reduced temperature t = T /T ∗ where T ∗ is the pseudogap temperature given in Ref [12]. As described by Hwang et al. [25]this implies in our density of state model that N also goes like T /T ∗ . This fixes thisparameter i.e. N = 100 K/ T ∗ where the pseudogap temperature T ∗ is taken from thedata of Ref [12]. We have also verified in our numerical work that small deviation fromthis chosen value makes no qualitative changes to our recovered I χ ( ω ) spectra. These N values are given in the middle frame of the figure. The pseudogaps (∆ pg ) used forfittings are 38, 49, 54, and 56 meV for OPT33, UD25, UD16 and UD11, respectively (seealso the lower frame of Fig. 1). In accordance with our previous finding we subtracted aresidual scattering contribution as noted in the figure before proceeding to the inversion. volution of electron-boson spectral density in the underdoped region of Bi Sr − x La x CuO Bi-2201 Bi-2212
T = 100 K λ / λ w i t hou t P G PG loss ∆ pg ω (meV) λ / λ OPT λ withoutPG / λ withoutPGOPT λ / λ OPT λ withoutPG / λ withoutPGOPT Bi-2212 Bi-2201 N o r m a li z ed λ o r λ w i t hou t P G PG loss (meV) Figure 5. (Color online) Top frame gives the ratio of the electron-boson massrenormalization factor ( λ ) including a pseudogap to its value without the pseudogap.The solid dark blue stars are for Bi-2201 and the empty red stars are for Bi-2212. ForBi-2212 with pseudogap you use N = 0.60 for both OPT96 and UD69 (see Fig. 3).On the horizontal axis we use as variable P G loss in meV, which measures the relativenumber of states below ω = ∆ pg (shaded region in the inset) which are transferred tothe region above ∆ pg in the effective density of states. The inset provides a sketchof this transfer of states. Bottom frame gives the ratio of λ normalized to the valueat optimum doping. The open red diamonds give twice the first inverse moment of I χ ( ω ) when a pseudogap is included in our maximum entropy inversion and the blueopen pentagons are the case when the pseudogap is not included. The solid symbolsare for Bi-2201 in contrast to the open symbols for Bi-2212 which are included forcomparison. Our fits to the data on the optical scattering rates are shown in the top frame, thehighest solid blue for UD11, dashed red for UD16, dashed-dotted dark green for UD25and dash-double dotted brown for OPT33. The middle and bottom frames give ourrecovered electron-boson spectral densities with and without pseudogap, respectively.Trends are similar to those found in Fig. 3. Other optical data in Bi-2201, some with Pbdoping, have appeared and have been inverted[4, 26] to recover a boson spectral density.A histogram is used for I χ ( ω ) with parameters determined through a least square fitto the conducting data. These works do not include a pseudogap and are based on anexpression for the dynamic conductivity which is obtained under the assumption thatthe electronic density of states is constant over the entire band taken to be infinite. Theresults are in reasonable agreement with those shown in the lower frame of Fig. 4 which volution of electron-boson spectral density in the underdoped region of Bi Sr − x La x CuO pg = 0. In particular the position in energy of the main peakagree. In both analysis the mass enhancement factor λ increases with decreasing doping p . The absolute value of λ is larger in Ref. [26] than what we find here. This may berelated to the different treatment of the residual scattering.In Fig. 5 we give results for the mass renormalization factor λ which has oftenbeen taken as the most important single measure of the associated renormalization. Bydefinition λ = 2 (cid:90) ∞ I χ (Ω)Ω d Ω (6)Because the I χ ( ω ) used in this equation was extracted from optical conductivity data,it is a transport spectral density and includes vertex corrections. Although closelyrelated, it is distinct from the equilibrium spectral density which enters ARPES dataand distinct from that which enters Raman which has its own vertex. A comparison ofthese various electron-boson exchange spectral densities is given in Ref [1] for the case ofoptimally doped Bi-2212. In each case the first inverse moment of I χ ( ω ) which definesthe single number λ in Eqn. (6) provides a useful measure of the magnitude of therenormalizations involved. What is plotted in Fig. 5 is the ratio λ/λ withoutP G which isseen to always be greater than one. On the horizontal axis we use the parameter P G loss which is defined in the inset. The area of the shaded region in meV defines
P G loss and provides a measure of the relative number of electronic states that are transferredfrom the region ω ≤ ∆ pg to the region ω > ∆ pg by the opening of the pseudogap.More explicitly P G loss = ∆ pg (1 − N ). As the ”metallicity” is reduced i.e. P G loss is increased, the mass enhancement parameter also increases monotonically over its nopseudogap value. We take this to mean that as the Mott insulating state is more closelyapproached by decreasing the doping p toward zero (half filling) the coupling to thebosons is also increased. If we assume that the major coupling represented in I χ ( ω ) isto spin fluctuation, this would make sense since the proximity to the antiferromagneticstate is also increased. In the bottom frame we provide additional information andcompare in each case the increases in λ over its value for optimum doping. For Bi-2212the increase is a factor of 2 while for Bi-2201 it is smaller but still of order 50 %. Evenwhen no pseudogap is included in the inversion process, blue symbols, this large increasein λ , as we more closely approach the antiferromagnetic state, remains.
4. Summary and conclusions
Structures or so called ”kinks” observed[27, 28, 29] in the dressed electronic dispersioncurves on the high T c copper oxides have been widely interpreted as due to interactionwith bosons. These structures are analyzed in terms of an electron-boson spectraldensity function I χ ( ω ) which provides a phenomenological description of the lowenergy scattering processes[1]. Such an approach is also at the basis of the nearlyantiferromagnetic Fermi liquid model of Pines and coworkers[30, 31, 32] whichhas enjoyed considerable success in understanding the measured properties of the volution of electron-boson spectral density in the underdoped region of Bi Sr − x La x CuO I χ ( ω )which all point to the same mechanism involving the exchange of spin fluctuations.Further, when comparison with inelastic spin-polarized neutron scattering data isavailable, agreement[39, 40, 41] between the shape of I χ ( ω ) and that of the localspin susceptibility χ ( ω ) is found. A particularly noteworthy case is for La − x Sr x CuO where the agreement with the detail spin susceptibility data of Vignolle et al. [42] isremarkable[40]. It has also been noted[1] that the scale difference found betweenthe optically derived spectral density and its ARPES counterpart is due to vertexcorrections present in transport but absent in equilibrium properties such as ARPES.The reader is referred to Ref [1] for more details. These facts support for the concept ofa boson exchange mechanism as a useful phenomenology that helps researchers betterunderstand the nature of the effective correlation in these materials. It also allows oneto correlate in a simple frame work, many of their observed properties.Here we used a maximum entropy technique to extract from optical data an estimateof the electron-boson spectral density which, for the first time, includes both finitetemperature effects and a model pseudogap. The pseudogap enters the formula for theoptical scattering rate only through the effective density of states symmetrized aboutthe Fermi energy ˜ N ( ω ). To model ˜ N ( ω ) we employ the known linear increase of ∆ pg with decreasing doping as well as an informed mathematical form for the ω dependenceof ˜ N ( ω ). This leaves a single unknown parameter, the depth of the pseudogap well ormore precisely its value at ω = 0. i.e. N . We set its value through consideration ofthe length of the Fermi arc measured by ARPES[24, 25] in the underdoped cuprates.Alternatively one can take the relative amount of spectral weight in the electronic densityof states which is transferred from the energy region below ∆ pg to the region above thisenergy.This same quantity( P G loss ) also provides a measure of the loss of metallicity dueto the pseudogap in as much as it can usefully be characterized by a single number.Applying the method to the data of Dai et al. [12] on the Bi-2201 family, provides aboson spectrum which is very similar in its main characteristic with that found beforefor Hg-1201[41] and Bi-2212[39]. These previous studies were restricted to overdopedand optimally doped samples as no pseudogap was included in the analysis. Here wefind that the trends with doping established previously, not only apply to Bi-2201 aswell, but also continue even into the highly underdoped regime. All spectra for Bi-2201have a pronounced peak at a frequency Ω sf which tends to decreases with decreasingdoping ( p ). This peak is superimposed on a background which extends to very highenergy as was observed in Bi-2212[39] and Hg-1201[41] and confirmed in Ref. [13]. A volution of electron-boson spectral density in the underdoped region of Bi Sr − x La x CuO I χ (Ω) continues to increase with decreasing doping p even in the most underdopedsample studied. This trend can easily be understood if we assume that the coupling tothe charge carriers is dominantly due to exchange of spin fluctuations which becomeslarger as the antiferromagnetic phase is more closely approached. At the same timethe pseudogap becomes larger and the material loses metallicity as the Mott insulatingstate is approached. Acknowledgments
JH acknowledges financial support from the National Research Foundation of Korea(NRFK grant No. 20100008552). JPC was supported by the Natural Science andEngineering Research Council of Canada (NSERC) and the Canadian Institute forAdvanced Research (CIFAR).
References [1] J. P. Carbotte, T. Timusk, and J. Hwang.
Reports on Progress in Physics , 74:066501, 2011.[2] H. K. Leung, J. P. Carbotte, D. W. Taylor, and C. R. Leavens.
Canadian Journal of Physics ,54:1585, 1976.[3] E. Schachinger, D. Neuber, and J. P. Carbotte.
Phys. Rev. B , 73:184507, 2006.[4] E. van Heumen, E. Muhlethaler, A. B. Kuzmenko, H. Eisaki, W. Meevasana, M. Greven, andD. van derMarel.
Phys. Rev. B , 79:184512, 2009.[5] J. P. Carbotte, C. Jiang, D. N. Basov, and T. Timusk.
Phys. Rev. B , 51:11798, 1995.[6] P. B. Allen.
Phys. Rev. B , 3:305, 1971.[7] S. V. Shulga, O. V. Dolgov, and E. G. Maksimov.
Physica C , 178:266, 1991.[8] S. G. Sharapov and J. P. Carbotte.
Phys. Rev. B , 72:134506, 2005.[9] B. Mitrovic and M. A. Fiorucci.
Phys. Rev. B , 31:2694, 1985.[10] E. Schachinger, M. G. Greeson, and J. P. Carbotte.
Phys. Rev. B , 42:406, 1990.[11] E. T. Jaynes.
Phys. Rev. , 106:620, 1957.[12] Y. M. Dai, B. Xu, P. Cheng, H. Q. Luo, H. H. Wen, X. G. Qiu, and R. P. S. M. Lobo.
Phys. Rev.B , 85:092504, 2012.[13] J. Hwang and J. P. Carbotte.
Phys. Rev. B , 86:094502, 2012.[14] S H¨ufner, M A Hossain, A Damascelli, and G A Sawatzky.
Reports on Progress in Physics ,71:062501, 2008.[15] J. Hwang, T. Timusk, and G. D. Gu.
Nature , 427:714, 2004.[16] F. Marsiglio, T. Startseva, and J. P. Carbotte.
Phys. Lett. A , 245:172, 1998.[17] M. Kugler, Ø. Fischer, Ch. Renner, S. Ono, and Yoichi Ando.
Phys. Rev. Lett. , 86:4911, 2001.[18] J.-H. Ma, Z.-H. Pan, F. C. Niestemski, M. Neupane, Y.-M. Xu, P. Richard, K. Nakayama, T. Sato,T. Takahashi, H.-Q. Luo, L. Fang, H.-H. Wen, Ziqiang Wang, H. Ding, and V. Madhavan.
Phys.Rev. Lett. , 101:207002, 2008.[19] T. Kurosawa, T. Yoneyama, Y. Takano, M. Hagiwara, R. Inoue, N. Hagiwara, K. Kurusu,K. Takeyama, N. Momono, M. Oda, and M. Ido.
Phys. Rev. B , 81:094519, 2010.[20] Guo qing Zheng, P. L. Kuhns, A. P. Reyes, B. Liang, and C. T. Lin.
Phys. Rev. Lett. , 94:047006,2005. volution of electron-boson spectral density in the underdoped region of Bi Sr − x La x CuO [21] Shinji Kawasaki, Chengtian Lin, Philip L. Kuhns, Arneil P. Reyes, , and Guo qing Zheng. Phys.Rev. Lett. , 105:137002, 2010.[22] T. Timusk and B. Statt.
Reports on Progress in Physics , 62:61, 1999.[23] J. Hwang, T. Timusk, and G. D. Gu.
J. Phys.: Condens. Matter , 19:125208, 2007.[24] A. Kanigel, M. R. Norman, M. Randeria, U. Chatterjee, S. Souma, A. Kaminski, H. M. Fretwell,S. Rosenkranz, M. Shi, T. Sato, Takahashi, Z. Z. Li, H. Raffy, K. Kadowaki, D. Hinks,L. Ozyuzer, and J. C. Campuzano.
Nat. Phys. , 2:447, 2006.[25] J. Hwang, J. P. Carbotte, and T. Timusk.
Euro. Phys. Lett. , 82:27002, 2008.[26] E. van Heumen, W. Meevasana, A. B. Kuzmenko, H. Eisaki, and D. van derMarel.
New Journalof Physics , 11:055067, 2009.[27] A. Lanzara, P. V. Bogdanov, X. J. Zhou, S. A. Kellar, D. L. Feng, E. D. Lu, T. Yoshida, H. Eisaki,A. Fujimori, K. Kishio, J.-I. Shimoyama, T. Noda, S. Uchida, Z. Hussain, and Z.-X. Shen.
Nature , 412:510, 2001.[28] A. Damascelli, Z. Hussain, and Z.-X. Shen.
Rev Mod. Phys. , 75:473, 2003.[29] S. V. Borisenko, A. A. Kordyuk, V. Zabolotnyy, J. Geck, D. Inosov, A. Koitzsch, J. Fink,M. Knupfer, B. B¨uchner, V. Hinkov, C. T. Lin, B. Keimer, T. Wolf, S. G. Chiuzb˘aian, L. Patthey,and R. Follath.
Phys. Rev. Lett. , 96:117004, 2006.[30] A. J. Millis, H. Monien, and D. Pines.
Phys. Rev. B , 42:167, 1990.[31] P. Monthoux and D. Pines.
Phys. Rev. B , 50:16015, 1994.[32] A. V. Chubukov, D. Pines, and J. Schmailian. in The Physics of Superconductors ed. K.-H.Bennemann, J. B. Ketterson (Springer, Berlin, 2002) , page 1349, 2002.[33] T. A. Maier, D. Poilblanc, and D. J. Scalapino.
Phys. Rev. Lett. , 100:237001, 2008.[34] E. Schachinger and J. P. Carbotte.
Phys. Rev. B , 77:094524, 2008.[35] J. M. Bok, J. H. Yun, H. Y. Choi, W. Zhang, X. J. Zhou, and C. M. Varma.
Phys. Rev. B ,81:174516, 2010.[36] Wentao Zhang, Jin Mo Bok, Jae Hyun Yun, Junfeng He, Guodong Liua, Lin Zhao, Haiyun Liu,Jianqiao Meng, Xiaowen Jia, Yingying Peng, Daixiang Mou, Shanyu Liu, Li Yu, Shaolong He,Xiaoli Dong, Jun Zhang, J. S. Wen, Z. J. Xu, G. D. Gu, Guiling Wang, Yong Zhu, XiaoyangWang, Qinjun Peng, Zhimin Wang, Shenjin Zhang, Feng Yang, Chuangtian Chen, Zuyan Xu,H.-Y. Choi, C. M. Varma, and X. J. Zhou.
Phys. Rev. B , 85:064514, 2012.[37] G. L. de Castro, C. Berthod, A. Piriou, E. Giannini, and Ø. Fischer.
Phys. Rev. Lett. , 101:267004,2008.[38] B. Muschler, W. Prestel, E. Schachinger, J. P. Carbotte, R. Hackl, Shimpei Ono, and Yoichi Ando.
J. Phys.: Cond. Matter , 23:375702, 2010.[39] J. Hwang, T. Timusk, E. Schachinger, and J. P. Carbotte.
Phys. Rev. B , 75:144508, 2007.[40] J. Hwang, E. Schachinger, J. P. Carbotte, F. Gao, D. B. Tanner, and T. Timusk.
Phys. Rev. Lett. ,100:137005, 2008.[41] J. Yang, J. Hwang, E. Schachinger, J. P. Carbotte, R. P. S. M. Lobo, D. Colson, A. Forget, andT. Timusk.
Phys. Rev. Lett. , 102:027003, 2009.[42] B. Vignolle, S. M. Hayden, D. F. McMorrow, H. M. Rønnow, B. Lake, C. D. Frost, and T. G.Perring.