Simulation of a hump structure in the optical scattering rate within a generalized Allen formalism and its application to copper oxide systems
SSimulation of a hump structure in the opticalscattering rate within a generalized Allen formalismand its application to copper oxide systems
Jungseek Hwang
Department of Physics, Sungkyunkwan University, Suwon, Gyeonggi-do 440-746,Republic of KoreaE-mail: [email protected]
Abstract.
We propose a possible way to simulate a hump structure in the opticalscattering rate. The optical scattering rate of correlated charge carriers can be definedwithin an extended Drude model formalism. When some electron and hole dopedcopper oxide systems are in the spin density or charge density wave phases they showhump structures in their optical scattering rates. The hump structures have not yetbeen simulated and understood clearly. We are able to simulate the hump structureby using a peak followed by a dip feature in the normalized density of states within ageneralized Allen formalism. We observe that reversing the order of the dip and peakgives completely different features in the optical scattering rate; a peak-dip (dip peak)results in a hump (a valley) in the scattering rate. We also obtain the real part ofthe optical conductivity and reflectance spectra from the simulated optical scatteringrate and compare them with published experimental spectra. From these comparisonswe conclude that the peak-dip order can give the hump structure, which is observedexperimentally in copper oxide systems. Finally we fit two published optical spectrawith our new model and discuss our results and the possible origin of the dip or peakfeatures in the normalized density of states.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 imulation of a hump structure in the optical scattering rate within a generalized Allen formalism and its application to copper oxide systems
1. Introduction
High temperature cuprates were discovered in 1986[1]. Since that time, manyspectroscopic studies with different experimental techniques have been done onthis intriguing material system[2]. However, the microscopic mechanism of thesuperconducting phenomenon of the material has not yet been figured out. The electrondoped cuprate system has been studied relatively less than the hole doped, because ofits lower transition temperature, even though its superconducting mechanism is believedto be the same as for the hole doped. Uderdoped electron cuprates show a spin densitywave (SDW) in mid infrared range, which appears as a partial suppression in thedensity of states[3, 4]. This SDW seems to be correlated to a hump structure in theoptical scattering rate[5, 6]. This hump feature has not yet been simulated and clearlyunderstood. Similar hump features have also been observed in the optical scatteringrate of underdoped hole cuprate systems when they contain the charge density wave(CDW) or charge stripes[7, 8] even though this have ever been pointed out explicitly sofar as the author knows.Here we propose a possible way to simulate the hump structure within a generalizedAllen formalism or the Sharapov-Carbotte formula[9, 10, 11], which can be applied tomaterial systems both at finite temperature and with non-constant density of states. Weperform model calculations to get the optical scattering rates by using the generalizedAllen formula with four differently shaped densities of states (peak only, dip only, peak-dip, and dip-peak cases) and two different electron-boson spectral functions (the Millis,Monien, and Pines (MMP) type antiferromagnetic spin fluctuation model[12] and theMMP + a sharp Gaussian mode). From these calculations we find that to get thehump structure in the scattering rate we must have both peak and dip structures in thenormalized density of states (DOS) and the peak and dip should be located at a finitefrequency. We also observe that the different orders of the peak and dip result in differentfeatures in the optical scattering rate; while a peak-dip order gives a hump structure, asobserved in some experiment, a dip-peak order gives a valley (or a suppression, whichis similar to the usual pseudogap[13] model except now the dip starts from a finitefrequency. We emphasize that the peak-dip order is a new model and different from thepseudogap model which has been used to extract the electron-boson spectral functionfrom the optical scattering rate of underdoped cuprates[14, 15]. While the pseudogapmodel gives a suppression, this new peak-dip model gives an enhancement (a hump) inthe scattering rate. We also obtained the optical conductivity and reflectance spectrafrom the calculated optical scattering rates and compare the spectra with experimentaldata. From these comparisons we conclude that the peak-dip model can be appliedto optical data with the hump structure. We apply this model to extract the electronboson spectral functions from the measured spectra of two copper oxide systems anddiscuss the results obtained.In section 2 we introduce the general formalism used in this study. In section 3we show our simulated results obtained using the formalism with various model density imulation of a hump structure in the optical scattering rate within a generalized Allen formalism and its application to copper oxide systems
2. General formalisms
Optical spectroscopy has been used to study strongly correlated electron systemsincluding high temperature superconducting materials[16]. Typically we measure thereflectance spectra, R ( ω ) of superconducting or metallic samples at various temperaturesabove and below some characteristic temperature including the superconductingtransition temperature. In general, the reflectance spectrum is quite a complicatedphysical quantity from the theoretical or analytical points of view. Therefore we needto extract simpler quantities from the measured reflectance spectrum. These simplerquantities are known as the optical constants, like the dielectric constant, the index ofrefraction, and so on. To get optical constants from the measured reflectance spectrum,we usually rely on a Kramers-Kronig (K-K) integral relation[17]. To perform the K-Kintegral we have to extrapolate the measured reflectance spectrum (which is known onlyin a finite spectral range because of experimental limitations) to both zero and infinitefrequency guided by appropriate physical models since the integral range in the K-Krelation is from 0 to ∞ . Once we get the corresponding phase of R ( ω ) through theK-K analysis we can calculate other optical constants using well-known relationshipsbetween them[17].For further analysis we rely on an extended Drude model formalism[18, 19, 20, 21].The extended Drude conductivity can be written as follows:˜ σ ( ω ) = i ω p π ω + [ − op ( ω ) + i /τ imp ] (1)where ω p is a square of the plasma frequency, which is proportional to the numberdensity of charge carriers in a conducting material, more explicitly ω p ≡ πn e e /m ∗ ,where n e is the number density of free charge carriers, e the unit charge and m ∗ the effective mass of the charge carrier, also known as the band mass. 1 /τ imp is theelastic scattering rate of charge carriers due to scattering off impurities. Here ˜Σ op ( ω )is the optical self energy, which is a complex function. This quantity carries importantinformation on correlations between the charge carriers. The correlation affects theoptical processes: transitions from filled to empty states. The optical self energy can berelated to the mass enhancement factor ( λ op ( ω )) and optical scattering rate (1 /τ op ( ω )),˜Σ op ( ω ) = Σ op ( ω ) + i Σ op ( ω ) = − [ ωλ op ( ω ) + i /τ op ( ω )] /
2. The real and imaginary partsof the optical self energy are also related through a Kramers-Knonig transformation, i.e. the complex optical self energy is a causal function.Since the optical self energy carries information on the interaction between chargecarriers, it can be related to the electron-boson spectral function[9], α F ( ω ), which imulation of a hump structure in the optical scattering rate within a generalized Allen formalism and its application to copper oxide systems − op ( ω, T ) ≡ τ op ( ω, T ) (2)= πω (cid:90) ∞ d Ω α F (Ω) (cid:90) ∞−∞ dz [ N ( z − Ω)+ N ( − z + Ω)] × [ n B (Ω) + 1 − f ( z − Ω)][ f ( z − ω ) − f ( z + ω )]where N ( z ) is the normalized density of states and f ( ω ) and n B ( ω ) are the Fermi-Dirac and Bose-Einstein distribution functions respectively. We note that we use asymmetrized DOS as an approximation. Here we neglect vertex corrections and wealso assume that α F (Ω) is independent of momentum. If retarded interactions amongelectrons contribute to the formation of the cooper pairs in exotic superconductorsincluding high temperature superconductors, the electron-boson function may play therole of glue for the pairing. But there is also experimental evidence which showsthat nonretarded interactions can also contribute to the pairing in cuprates[22]. Thisgeneralized Allen model can be applied to analyze correlated electron systems withnon-constant density of states like a pseudogap, which is a partial gap near the Fermienergy[20]. In cuprates the α F (Ω) is related to the antiferromagnetic spin fluctuations.The non-constant density of states features (peak, dip, or a combination of the two)in DOS and the spin fluctuations are interrelated in a feedback process and appearin the optical scattering rate as in Eqn. (2). We can obtain the electron-bosonspectral function from the measured optical scattering rate by solving this integralequation numerically. In this numerical process one has to know the exact shape of thepseudogap since the pseudogap affects the resulting electron-boson spectral functionsignificantly[15, 13]. This numerical process is known as an inversion process. There aretwo different numerical approaches to solve the integral equation: one is a least squarefitting method[23, 15] and the other is a maximum entropy method[24].
3. Simulation of hump structure in the optical scattering rate
Many inversion studies of the optical scattering rates of copper oxide superconductorshave been done[15, 24, 25, 26]. There is a theoretical literature, which shows humpstructures in the calculated optical scattering rate[27]. However, the hump structuresappear in the scattering rate because of finite band effects. Here we show that humpstructures can also be realized simply by manipulating the shape of the non-constantdensity of states within the generalized Allen formalism. From the integral equation,Eqn. (3), we see that the scattering rate is an increasing function of frequency if thedensity of states remains larger or equal to the normalized value, 1.0 at any energy. It candecrease only when the density of states becomes less than 1.0. We performed our model imulation of a hump structure in the optical scattering rate within a generalized Allen formalism and its application to copper oxide systems MMP + DOS (Dip)
Full dip Half dip No dip / τ op ( ω ) = - Σ op 2 ( ω ) ( c m - ) MMP α F ( ω ) N ( ω ) MMP + DOS (Peak)
Full Peak Half Peak No Peak N ( ω ) MMP + (Dip + Peak) Frequency (cm -1 ) Full Dip Recovered Half Dip Recovered No Dip / τ op ( ω ) = - Σ op 2 ( ω ) ( c m - ) Frequency (cm -1 ) MMP + (Peak + Dip) Full Peak Recovered Half Peak Recovered No Peak
Figure 1. (Color online) The electron-boson spectral function, α F ( ω ), consists ofthe MMP type spin fluctuation mode only (top frame). The input parameter functionsfor the electron-boson spectral function (top frame) and for two different normalizeddensity of states ( N ( z )) cases (the 2nd row) and the resulting optical scattering rate(the 3rd row). The input parameter functions for the electron-boson spectral function(top frame) and for two different N ( z ) cases (the 4th row) and the resulting opticalscattering rate (the last row). In this case the DOS loss (gain) in the dip (peak) isrecovered completely. calculations at 0 K for simplicity. At finite temperatures the qualitative properties willremain the same as those at 0 K. We used two different electron-boson spectral function( α F ( ω )) models: the MMP model[12] and the MMP + a sharp Gaussian model. Thesemodels are well-known typical shapes of the electron-boson spectral function in cupratesystems[28, 15].In Fig. 1 we show the optical scattering rates obtained using Eqn. (3) with theinput electron-boson spectral function α F ( ω ) (top frame) and the normalized density ofstates N ( ω ) (the 2nd and 4th rows). We use the MMP antiferromagnetic spin fluctuationmodel for the electron-boson function, i.e. α F ( ω ) = A s ωω + ω sf where A s = 300 cm − and ω sf = 500 cm − are the amplitude and the characteristic frequency of the spin fluctuationMMP mode, respectively. N ( ω ) has a rectangular dip (in left column of the 2nd row) or imulation of a hump structure in the optical scattering rate within a generalized Allen formalism and its application to copper oxide systems (MMP + Peak) + DOS (Dip) Full dip Half dip No dip / τ op ( ω ) = - Σ op 2 ( ω ) ( c m - ) MMP + Peak α F ( ω ) N ( ω ) Full Peak Half Peak No Peak (MMP + Peak) + DOS (Peak) N ( ω ) Full Dip Recovered Half Dip Recovered No Dip
Frequency (cm -1 ) (MMP + Peak) + (Dip + Peak) / τ op ( ω ) = - Σ op 2 ( ω ) ( c m - ) Full Peak Recovered Half Peak Recovered No Peak
Frequency (cm -1 ) (MMP + Peak) + (Peak + Dip) Figure 2. (Color online) The electron-boson spectral function, α F ( ω ), consists of theMMP type spin fluctuation mode and a sharp Gaussian peak (top frame and also seein the text). The input parameter functions for the electron-boson spectral function(top frame) and for two different normalized density of states ( N ( z )) cases (the 2ndrow) and the resulting optical scattering rate (the 3rd row). In the inset of the middleframe we show α F ( ω ) in low frequency region. The input parameter functions for theelectron-boson spectral function (top frame) and for two different normalized densityof states ( N ( z )) cases (the 4th row) and the resulting optical scattering rate (the lastrow). α F ( ω ) consists of the MMP type spin fluctuation mode and a sharp Gaussianpeak. In the inset of the middle frame we show α F ( ω ) in low frequency region. Thedensity of states loss (gain) in the dip (peak) is recovered completely. a rectangular peak (in right column of the 2nd row) with the same width ( W ) 1000 cm − in a spectral range between ω Low (= 500 cm − ) and ω High (= 1500 cm − ), where ω Low and ω High are, respectively, a lower frequency edge (or an onset frequency) and a higherfrequency one of the rectangular dip (or peak). We calculate the optical scattering ratesfor three different depths (heights) of the dip (peak): the black solid curve is for a fulldip/peak, the red dashed line for a half dip/peak, and the blue dot-dashed curve forno dip/peak ( i.e. a constant density of states). The dip (peak) in the normalized DOSintroduces a abrupt suppression (a sharp increase) in the optical scattering rate with its imulation of a hump structure in the optical scattering rate within a generalized Allen formalism and its application to copper oxide systems − ( (cid:39) ω sf + ω Low ) because of reduced (increased) amountof DOS involved in the scattering process. We note that the onset frequency of thesuppression or sharp increase is shifted by roughly the characteristic frequency ( ω sf ) ofthe MMP mode from the onset frequency ( ω Low ) of the dip or peak. In the 4th and lastrows the DOS loss (gain) in the dip (peak) is recovered completely by a peak (dip) rightabove the dip (peak) as shown in the the 4th row. The recovery in DOS ensures thatthe scattering rate at high frequencies is recovered. We observe that different ordersof the peak and the dip give different shapes for the resulting scattering rates; whilethe dip-peak order (in left column) gives a well-defined valley the peak-dip order (inright column) gives a well-defined hump in the scattering rate as compared with theflat DOS case, displayed as the blue dash-dotted curve. The minimum of the valley (orthe maximum of the hump) is located near 2000 cm − which is the sum of the threecharacteristic energy scales, i.e. (cid:39) ω sf + ω Low + W .In Fig. 2 we display similar results as in Fig. 1 with a different electron-bosonspectral function, α F ( ω ). Here we have both the MMP mode and a sharp Gaussianpeak (top frame) α F ( ω ) = A s ωω + ω sf + A √ π ( d/ . e − ( ω − ω ) / [2( d/ . ] where A s = 300 cm − , ω sf = 500 cm − are the amplitude and the characteristic frequency of the spin fluctuationMMP mode, respectively. A = 300 cm − , ω = 240 cm − , and d = 80 cm − are theamplitude, the center frequency, and the width of the Gaussian peak, respectively. TheGaussian peak can be modeled as the well-known magnetic resonance mode which wasobserved first in inelastic neutron scattering experiments[29, 30, 31]. Since the magneticresonance has strong momentum dependence current vertex correction may need to beincluded in the model. We ignore the vertex correction since it is none trivial to includethese in our model but this is not a major issue for this paper. In the 2nd and 3rd rows,we also observe that the rectangular dip (peak) in the normalized density of statesintroduces a abrupt suppression (a sharp increase) in the optical scattering rate. Thesharp Gaussian peak enhances these features in the scattering rate. The sharp Gaussianpeak introduces kinks ( i.e. sudden slope changes) in the scattering rate instead of thesmooth slope changes which we observed previously for the MMP only case. The kinksindicate the characteristic energy scales of the input parameter functions ( N ( ω ) and α F ( ω )) clearly. The onset frequency of the abrupt suppression (or the sharp increase)in the scattering rate appears near 740 cm − which is the sum of two characteristicenergy scales: the onset frequency ( i.e. ω Low = 500 cm − ) of the rectangular dip (orpeak) in the DOS and the Gaussian peak frequency ( i.e. ω = 240 cm − ) in α F ( ω ).In the 4th and last rows, DOS loss (gain) in the rectangular dip (peak) is recovered bya peak (dip) right above the dip (peak) as shown in the the 4th row. Because of therecovery in the density of states the scattering rates come together at high frequencies.We also see a valley (a hump) in the scattering rate for the dip-peak (peak-dip) case;as we have pointed out already we have a kink inside the valley or on top of the humpinstead of a smooth slope change because of the sharp Gaussian feature in α F ( ω ). Thefrequency of the kink on top of the hump (or bottom of the valley) in the scatteringrate is roughly 1740 cm − , which is the sum of three characteristic energy scales: the imulation of a hump structure in the optical scattering rate within a generalized Allen formalism and its application to copper oxide systems N ( ω ) rectangule triangle parabola / τ op ( ω ) = - Σ op 2 ( ω ) ( c m - ) MMP + (Peak + Dip)
Frequency (cm -1 )(MMP + Peak) + (Peak + Dip) Figure 3. (Color online) A shape sensitivity of the peak-dip feature in DOS to theresulting optical scattering rate. In the top frame three differently shaped peak-dipfeatures are shown. In the middle frame we display corresponding three scattering ratesfor the MMP electron-boson function case. In the bottom frame we show correspondingthree scattering rates for the MMP plus the sharp Gaussian electron-boson functioncase. onset frequency of the rectangular dip (peak) ( i.e. ω Low = 500 cm − ) in the DOS,the width ( i.e. W = 1000 cm − ) of the dip or peak and the Gaussian peak frequency( i.e. ω = 240 cm − ) in α F ( ω ). We see the characteristic energy scales clearly in thesimulated scattering rates. It may be possible to get these characteristic energy scales( ω Low , ω High , W , ω sf , and ω ) from the measured scattering rate, which would provideuseful information when one analyzes measured data using our model approach.We also checked the shape sensitivity of the peak-dip feature in the resulting opticalscattering rate. We simulate the optical scattering rate with three different shapes ofthe peak-dip structure (rectangle, triangle, and parabola) as shown in the top frameof Fig 3. The resulting scattering rates are shown in middle and bottom frame for theMMP bosonic mode and the MMP plus the sharp Gaussian mode cases, respectively.We can see in the resulting scattering rates that the shape dependence is negligible. imulation of a hump structure in the optical scattering rate within a generalized Allen formalism and its application to copper oxide systems α F( ω ): MMP Full Dip Recovered (Dip+Peak) Full Dip not Recovered (Dip) Flat DOS (no peak and no dip) σ ( ω ) ( Ω - c m - ) R e f l e c t an c e Frequency (cm -1 ) - Σ op 1 ( ω ) ( c m - ) Full Peak Recovered (Peak+Dip) Full Peak not Recovered (Peak) α F( ω ): MMP + Peak Frequency (cm -1 ) Figure 4. (Color online) (Left column) The calculated real part of the optical selfenergy (top frame), the real part of the optical conductivity (middle frame), andreflectance spectra (bottom frame) for the MMP electron-boson mode and five differentrepresentative normalized DOS cases. (Right column) The calculated spectra for theMMP plus the sharp Gaussian electron-boson mode and five different normalized DOScases.
4. Optical conductivity and reflectance spectra calculated from thesimulated scattering rates
We obtain the corresponding optical conductivity and reflectance spectra from thesimulated optical scattering rates of the previous section and compare them withpublished data. To get the reflectance from the calculated optical scattering rate wehave to go through a series of mathematical procedures. Once we know the opticalscattering rate ( i.e. the imaginary part of the optical self energy) we can calculate thereal part of the optical self energy using a Kramers-Kronig (K-K) transformation. Thenwe can calculate the optical conductivity using the extended Drude model, i.e.
Eqn.(2). More explicitly we can write down the real and imaginary parts of the opticalconductivity as a function of the real and imaginary parts of the optical self energy as imulation of a hump structure in the optical scattering rate within a generalized Allen formalism and its application to copper oxide systems Nd Ce CuO
10 K 190 K 295 K / τ op ( ω ) = - Σ op 2 ( ω ) ( c m - ) - Σ op 1 ( ω ) ( c m - ) σ ( ω ) ( Ω − c m - ) R e f l e c t an c e T = 0 K
MMP + (Peak + Dip)
Full Peak Recovered Half Peak Recovered Flat DOS La Ba CuO Frequency (cm -1 ) Figure 5. (Color online) Comparison of the simulated results (right column) withmeasured electron underdoped data of Nd . Ce . CuO (left column)[6] and withmeasured hole underdoped data of La . Ba . CuO (middle column)[32, 8]. In themiddle column the scales on the vertical axis are shown on the right side. follows: σ ( ω )= ω p π − op ( ω ) + 1 /τ imp [ − op ( ω ) + ω ] + [ − op ( ω ) + 1 /τ imp ] (3) σ ( ω )= ω p π − op ( ω ) + ω [ − op ( ω ) + ω ] + [ − op ( ω ) + 1 /τ imp ] We can easily obtain the reflectance spectrum from the complex optical conductivity[33].For these calculations we used 1 /τ imp = 100 cm − , ω p = 10,000 cm − , and (cid:15) ∞ =1.0. In practice we need to use an appropriate value for (cid:15) ∞ since we have to includecontributions from high energy absorption features such as the ionic background, theinterband absorptions and so on[33]. In that case (cid:15) ∞ is always larger than 1.0.We display three representative calculated quantities: the real part of the opticalself energy (top frames), the real part of the optical conductivity (middle frames), andreflectance (bottom frames) in Fig 4. Left column is for the MMP model of α F ( ω ) andfive representative different density of states (DOS) taken from the cases we discussed in imulation of a hump structure in the optical scattering rate within a generalized Allen formalism and its application to copper oxide systems − . While for the peak only case the reflectance is lower than for the flatDOS case at high frequencies, for the peak+dip case the reflectance recovers only slowlyto that of the flat case at high frequencies. This frequency dependent behavior resultsin a valley in the reflectance of the peak+dip case near 2000 cm − where the hump is inthe optical scattering rate (see the black solid curve in the left column of the last rowof Fig. 1). This feature in reflectance spectra of the peak+dip case looks similar that ofelectron underdoped cuprates and hole doped cuprates [7, 5, 34, 32, 6, 8] (see also Fig.5 for closer comparisons). In the right column we display the same quantities as in theleft column but with a different electron-boson spectral function α F ( ω ), which consistsof both the MMP and the sharp Gaussian modes. The quantities show qualitativelysimilar properties as those in the left column. However, the sharp Gaussian peak givesbetter defined features as we saw in the optical scattering rate in the previous section.Since charge carriers are strongly correlated we see clearly a boundary between thecoherent and incoherent parts of the conductivity near 240 cm − where the Gaussianpeak exists[35]. We also observe similar characteristic energy scales to those observedin the optical scattering rate in the previous section.In Fig. 5 we compare our resulting spectra (the optical scattering rate, the realpart of optical self energy, the optical conductivity, and reflectance ordered from top tobottom frames in the right column) from the simulation with measured optical spectraof an electron doped Nd . Ce . CuO (left column), which is non-superconducting[6]and has a spin density wave phase[5] at low temperatures. The simulated data areobtained with the MMP mode of the electron-boson spectral function and three differentpeak-dip normalized densities of states (full peak+full dip, half peak+half dip, and flatDOS). We modified the original scattering rates slightly in the low frequency region toremove possibly experimental noises. We note that while the experimental data are atfinite temperatures, the simulated ones are at the zero temperature. We can see thatthe two data sets are qualitatively very similar if we ignore finite temperature effectsin experimental data; the hump structure appears at low temperature and disappearsgradually as the temperature increases up to room temperature. We compare the samesimulated spectra (right column) with the measured spectra (middle column) of anunderdoped hole La . Ba . CuO , which has a charge ordered or stripe phase[36, 32]below 60 K. We see that the two data sets show similar qualitative properties; the humpstructure in the optical scattering rate grows gradually as we reduce the temperaturebelow the charge ordering transition temperature, 60 K. We emphasize that the peak- imulation of a hump structure in the optical scattering rate within a generalized Allen formalism and its application to copper oxide systems
5. Applications to two copper oxides systems
We apply the peak-dip model to analyze in more detail two selected experimental datasets[32, 6]: Nd . Ce . CuO (non superconducting) and La . Ba . CuO ( T c (cid:39) =2.4 K). Since both data sets are in the normal state we may use the generalized Allenformula, Eqn. 3 to analyze them. In the top frame of the left column of Fig. 6 wedisplay the optical scattering rates of the electron doped Nd . Ce . CuO at threedifferent temperatures (10, 190 and 295 K) and the reconstruction data obtained usinga maximum entropy method (MEM)[24, 13]. The fitting quality is quite good; we areclearly able to simulate the hump structure in the scattering rate. We note that when wefit the spectrum at 10 K we need to add a 35 meV impurity scattering rate to improvethe quality of the fit. In the middle frame we show the input normalized densities ofstates for three different temperatures. The characteristic energies of the peak and dipin the density of states are determined from the characteristic energies and shape ofthe hump in the optical scattering rate. The input DOS for each temperature is fixed.We note that we need to take a dip which is wider than the peak to fit to the datamore tightly. We also note that since the height of the peak and depth of the dip areonly roughly guessed in the fittings, the extracted α F ( ω )’s may not be determineduniquely. In the bottom frame we display the resulting electron-boson spectral functionobtained from our MEM analysis. The shape of the electron-boson spectral functionlooks like that of a typical underdoped hole cuprate[37]. The optical mass enhancementfactor, λ op ≡ (cid:82) ω c α F ( ω ) /ω dω , shows a typical temperature dependence as seen in theinset of the bottom frame; as temperature lowers this factor increases[37]. Here ω c is acutoff frequency, taken to be 600 meV. Results obtained from the same analysis on theunderdoped hole La . Ba . CuO at three different temperatures (5, 30 and 60 K) aredisplayed in the right column of Fig. 6. In the top frame we display the optical scatteringrates and corresponding fits. We note that the sharp features in the data between 10and 50 meV are ignored. We use a fixed density of states for each temperature as shownin the middle frame. The resulting electron-boson functions are displayed in the bottomframe. The function shows a strong temperature dependence; as temperature decreasesa large amount of spectral weight moves to lower frequency. This behavior causes themass enhancement factor to increase considerably on lowering the temperature as wecan see in the inset of the bottom frame; this is also the typical temperature dependentbehavior for the mass factor of hole underdoped cuprates[37]. However at our lowesttemperature 5 K, the factor is unusually large, i.e. Cu O . by inelasticneutron scattering experiment[38].We push the analysis a little further with some assumptions. We assume thatthe extracted α F ( ω ) contributes to the superconductivity even though the α F ( ω ) imulation of a hump structure in the optical scattering rate within a generalized Allen formalism and its application to copper oxide systems Nd Ce CuO
10 K 190 K 295 K fits / τ op ( ω ) = - Σ op 2 ( ω ) ( m e V ) α F ( ω ) Frequency (meV) N ( ω ) λ op ( T ) Temperature (K) La Ba CuO Frequency (meV) λ op ( T ) Temperature (K)
Figure 6. (Color online) (Left column) Application of our model to measuredelectron underdoped data of Nd . Ce . CuO [6] and the results. We use a maximumentropy method[42] to reconstruct the measured data by using the new model.(Right column) Application of the model to measured hole underdoped data ofLa . Ba . CuO [32, 8] and the results. is extracted from the optical scattering rate in the normal state. We also assumethat we can use the extended McMillan formula[39, 40, 41] to estimate fictitioussuperconducting transition temperatures. The McMillan formula can be written as k B T c ∼ = 1 . (cid:126) ω ln exp[ − (1+ λ op ) / ( gλ op )], where ω ln ≡ exp[2 /λ (cid:82) ω c ln ω α F ( ω ) /ω dω ] is thelogarithmically averaged boson frequency, and g is an adjustable parameter ( g ∈ [0 , T c ’s (when g = 1.0) are 29.8 K for Nd . Ce . CuO at10 K and 4.2 K for La . Ba . CuO at 5 K. The estimated T c ’s are much higher thantheir actual T c ’s. At least we can say that the extracted α F ( ω ) may be strong enoughto produce the superconductivity if the retarded electron-boson interaction contributesto the paring formation in the materials. We obtained some useful information from themeasured data by applying our proposed model. imulation of a hump structure in the optical scattering rate within a generalized Allen formalism and its application to copper oxide systems
6. Discussions and conclusions
We have shown that we could realize a hump structure in the optical scattering ratewithin a generalized Allen formalism, which is applicable to a system with non-constantdensity of states and finite temperatures. We had to introduce a peak followed bya dip structure in the density of states to realize the hump in the scattering rate.The order of the peak and the dip is important because the opposite order result incompletely different features in the scattering rate: while the peak-dip order producesa hump structure, the dip-peak order gives a valley. This valley structure in the opticalscattering rate can be understood as the normal pseudogap phenomena, which appearnear the Fermi energy. However the hump structure, which we can simulate with onlypeak-dip order, seems to be a completely new physical phenomenon. The physical originof the dip and/or the peak in the normalized density of states are not understood andclearly justified yet. The dip, which appears after the peak (see in the middle frame ofthe left column of Fig. 6), is a localized (partial) gap in finite frequency range which maybe related to the interband transition observed by recent angle resolved photoemissionspectroscopy study of electron doped Sm . Ce . CuO [3]. It may also be closely relatedto the charge or spin density waves in material systems[7, 5, 8] since those systems showhump structures in their optical scattering rates. Since the stripe phase seems to appearin a small doping and temperature region in the hole doped phase diagram, the peak-dipfeature in DOS is related to this special phase and is obviously different from the genericpesudogap phase, which is closely related to the dip-peak feature in the DOS.We also calculated corresponding optical conductivity and reflectance spectra fromthe simulated optical scattering rate. The resulting conductivity and reflectance spectraare similar to the observed conductivity and reflectance spectra of electron dopedcuprates[5, 34, 6] (see in the left column of Fig. 5) and hole doped copper oxides[7, 8] (seein the middle column of Fig. 5). We applied this peak-dip model to two measured opticalscattering rates with hump features and extracted a reasonable electron-boson spectralfunction (see in Fig. 6). Many researchers in the high-temperature superconductivitycommunity believe that the retarded electron-boson spectral function may carry theinformation on the glue for electron-electron cooper pairs, although others suggestthat nonretarded interaction also contribute to the paring interaction[22]. This studymakes it realistic to extract the electron-boson function out of the optical scatteringrate of copper oxides with a hump feature, which was not previously possible. Weexpect that this work may allow researchers to go one step further in figuring out thesuperconducting mechanism of cuprates. Acknowledgments
The author acknowledges financial support from the National Research Foundation ofKorea (NRFK Grant No. 20100008552 and No. 2012R1A1A2041150). The authorthanks J. P. Carbotte for his encouragement for this study, C. C. Homes for sharing his imulation of a hump structure in the optical scattering rate within a generalized Allen formalism and its application to copper oxide systems ++ code. References [1] T. G. Bednorz and A. Muller.
Z. Phys. B , 64:189, 1986.[2] J. P. Carbotte, T. Timusk, and J. Hwang.
Reports on Progress in Physics , 74:066501, 2011.[3] S. R. Park, Y. S. Roh, Y. K. Yoon, C. S. Leem, J. H. Kim, B. J. Kim, H. Koh, H. Eisaki, N. P.Armitage, and C. Kim.
Phys. Rev. B , 75:060501, 2007.[4] H. Yasuoka, T. Imai, and T. Shimizu. in Strongly Correlation and Superconductivity edited by H.Fukuyama, S. Maekawa, and A. P. Malozemoff . Springer-Verlag, 1989. (Note: Key materialon page 254).[5] Y. Onose, Y. Taguchi, K. Ishizaka, and Y. Tokura.
Phys. Rev. B , 69:024504, 2004.[6] N. L. Wang, G. Li, Dong Wu, X. H. Chen, C. H. Wang, and H. Ding.
Phys. Rev. B , 73:184502,2006.[7] M. Dumm, D. N. Basov, S. Komiya, Y. Abe, and Yoichi Ando.
Phys. Rev. Lett. , 88:147003, 2002.[8] C. C. Homes, M. Hucker, Q. Li, Z. J. Xu, J. S. Wen, G. D. Gu, and J. M. Tranquada.
Phys. Rev.B , 85:134510, 2012.[9] P. B. Allen.
Phys. Rev. B , 3:305, 1971.[10] B. Mitrovic and M. A. Fiorucci.
Phys. Rev. B , 31:2694, 1985.[11] S. G. Sharapov and J. P. Carbotte.
Phys. Rev. B , 72:134506, 2005.[12] A. J. Millis, H. Monien, and D. Pines.
Phys. Rev. B , 42:167, 1990.[13] J. Hwang and J. P. Carbotte.
Phys. Rev. B , 86:094502, 2012.[14] J. Hwang, J. P. Carbotte, and T. Timusk.
Phys. Rev. Lett. , 100:177005, 2008.[15] J. Hwang, J. Yang, T. Timusk, S. G. Sharapov, J. P. Carbotte, D. A. Bonn, R. Liang, and W. N.Hardy.
Phys. Rev. B , 73:014508, 2006.[16] D. N. Basov and T. Timusk.
Rev. Mod. Phys , 77:721, 2005.[17] Frederick Wooten.
Optical Properties of Solids . Academic, New York, 1972. (Note: Key materialon page 176).[18] J. W. Allen and J. C. Mikkelsen.
Phys. Rev. B , 15:2952, 1977.[19] A. V. Puchkov, D. N. Basov, and T. Timusk.
J. Phys.: Cond. Matter , 8:10049, 1996.[20] T. Timusk and B. Statt.
Reports on Progress in Physics , 62:61, 1999.[21] J. Hwang, T. Timusk, and G. D. Gu.
Nature , 427:714, 2004.[22] B. Mansart, J. Lorenzana, A. Mann, A. Odeh, M. Scarongella, M. Cherguib, and F. Carboneb.
PNAS , 110:4539, 2013.[23] S. V. Dordevic, C. C. Homes, J. J. Tu, T. Valla, M. Strongin, P. D. Johnson, G. D. Gu, and D. N.Basov.
Phys. Rev. B , 71:104529, 2005.[24] E. Schachinger, D. Neuber, and J. P. Carbotte.
Phys. Rev. B , 73:184507, 2006.[25] J. Hwang, T. Timusk, E. Schachinger, and J. P. Carbotte.
Phys. Rev. B , 75:144508, 2007.[26] 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.[27] A. Knigavko and J. P. Carbotte.
Phys. Rev. B , 72:035125, 2005.[28] E. Schachinger and J. P. Carbotte.
Phys. Rev. B , 62:9054, 2000.[29] J. Rossat-Mignod, L. P. Regnault, C. Vettier, P. Bourges, P. Burlet, J. Bossy, J. Y. Henry, andG. Lapertot.
Physica C , 185.[30] H. F. Fong, P. Bourges, Y. Sidis, L. P. Regnault, J. Bossy, A. Ivanov, D. L. Milius, I. A. Aksay,and B. Keimer.
Phys. Rev. B , 61:14772, 2000.[31] C. Stock, W. J. L. Buyers, R. Liang, D. Peets, Z. Tun, D. Bonn, W. N. Hardy, , and R. J.Birgeneau.
Phys. Rev. B , 69:014502, 2004.[32] C. C. Homes, S. V. Dordevic, G. D. Gu, Q. Li, T. Valla, and J. M. Tranquada.
Phys. Rev. Lett. ,96:257002, 2006. imulation of a hump structure in the optical scattering rate within a generalized Allen formalism and its application to copper oxide systems [33] J. Hwang, T. Timusk, and G. D. Gu. J. Phys.: Condens. Matter , 19:125208, 2007.[34] A. Zimmers, J. M. Tomczak, R. P. S. M. Lobo, N. Bontemps, C. P. Hilla, M. C. Barr, Y. Dagan,R. L. Greene, A. J. Millis, and C. C. Homes.
Europhys. Lett. , 70:225, 2005.[35] J. Hwang, J. Yang, J. P. Carbotte, and T. Timusk.
J. Phys. Condens. Matter , 20:295215, 2008.[36] J. M. Tranquada, H. Woo, T. G. Perring, H. Goka, G. D. Gu, G. Xu, M. Fujita, and K. Yamada.
Nature , 429:534, 2004.[37] J. Hwang.
Phys. Rev. B , 83:014507, 2011.[38] C. Stock, W. J. L. Buyers, Z. Yamani, C. L. Broholm, J.-H. Chung, Z. Tun, R. Liang, D. Bonn,and W. N. Hardy.
Phys. Rev. B , 73:100504, 2006.[39] W. L. McMillan.
Phys. Rev. , 167:331, 1968.[40] P. J. Williams and J. P. Carbotte.
Phys. Rev. B , 39:2180, 1968.[41] J. Hwang, E. Schachinger, J. P. Carbotte, F. Gao, D. B. Tanner, and T. Timusk.
Phys. Rev. Lett. ,100:137005, 2008.[42] J. Hwang and J. P. Carbotte.