158 3 Results for High–Tc Cuprates: Doping Dependence

Furthermore, when the phase diagram for hole–doped cuprates calculated one finds even better agreement with experiment in the underdoped regime. Interestingly, both of the temperatures Tc and Tc are renormalized to smaller values, and the region where preformed pairs are expected is also reduced.

3.5.2 Optical Conductivity and Electronic Raman Response

We come now to the discussion of the consequences of a dx2−y2 –wave pseudogap for the optical conductivity. While the c-axis conductivity in YBa2Cu3O7−δ shows a pseudogap with a size of approximately 300–400 cm−1 [166, 167], the size of the pseudogap extracted from the ab–plane conductivity is of the order of 600–700 cm−1 [168, 169]. This di erence cannot be attributed to the charge reservoir layers between the CuO2 planes, since recently it has been convincingly shown that the pseudogap seen in the c–axis conductivity indeed has its origin in the CuO2 planes [170]. In the following we shall demonstrate that the self–consistency of our calculation of the spin fluctuation interaction with the single–particle properties, especially the pseudogap itself, provided by the FLEX approximation leads to a natural understanding of the di erence in the size of the pseudogap seen in the ab– plane and c–axis conductivities. Also, the semiconducting behavior of the c–axis resistivity can be understood qualitatively [166, 167, 171].

We have calculated the c–axis and ab–plane conductivities in the presence of a temperature–independent but doping–dependent pseudogap as suggested by the work of Williams et al. [155], using ARPES data as an input. To do so, it is necessary also to take into account some scattering mechanism within the CuO2 planes. In particular, the self–consistently calculated interaction due to exchange of spin and charge fluctuations yields a quasiparticle scattering rate which varies linearly with frequency in the normal–state, and exhibits a gap-like suppression at lower frequencies in the superconducting state. At the same time, the e ective–mass ratio is increased at lower frequencies in the superconducting state. Thus, we can safely conclude that the FLEX approximation accounts for the damping and mass enhancement needed to extend the theory for thermodynamic quantities of Williams et al. [155] to dynamical quantities.

First, we discuss our normal–state results for the optical in–plane conductivity σab ≡ σ1(ω) in the presence of a d–wave pseudogap Eg (k). We have assumed for the pseudogap the simple but doping dependent form (introduced in (2.67))

Eg (k) = Eg [cos kx − cos ky ] ,

where Eg is temperature–independent and increases with decreasing doping level below the optimal doping level. The extended version of the generalized equations derived in Sect. 2.2.1 was solved self–consistently in the presence of this pseudogap, and yields the quasiparticle self–energy components Z(k, ω) and ξ(k, ω), as well as the superconducting gap φ(k, ω).

Fig. 3.38. (a) Optical conductivity σ1 ≡ σab(ω) for an amplitude Eg = 0.15t of the pseudogap, at temperatures T = 0.1, 0.050, and 0.030t. Inset: Re Z(ka , ω) for the same temperatures (increasing peaks at ω = 0 in this sequence). Dashed line, for Eg = 0 and T = 0.030t. (b) Quasiparticle damping, ω Im Z(ka , ω), at antinode ka, for an amplitude Eg = 0.15t of the pseudogap, at temperatures T = 0.1, 0.050, and 0.030t (decreasing values at ω = 0 in this sequence of temperatures).

Our results for σab(ω) are displayed in Fig. 3.38a, where one can see that the in–plane conductivity is coherent in character and shows a Drude peak at low frequencies even in the underdoped compounds [168, 172]. However, the size of the pseudogap structure seen in the ab–plane conductivity for underdoped YBCO has been measured to be 600–700 cm−1 [168, 169], while the gap extracted from the c–axis conductivity in the same compounds is of the order of 300–400 cm−1 [166, 167]. The corresponding quasiparticle damping, Γ (k, ω) = ω Im Z(k, ω), is highly anisotropic and exhibits, for k along the direction of the antinode of the gap and decreasing T , a gap of the order of Eg (see Fig. 3.38b, for Eg = 0.15t). At the same time, the e ective–mass ratio m /m = Re Z(k, ω) is enhanced at about ω 2Eg above its value at ω = 0, where Re Z ≈ 2 (see inset in Fig. 3.38a). In the absence of the pseudo-

160 3 Results for High–Tc Cuprates: Doping Dependence

Fig. 3.39. The coherent dynamical c–axis conductivity (2.170) for three temperatures T = 0.1t (solid lines), 0.05t (dashed lines), and 0.03t (dotted lines), (a) for a pseudogap amplitude Eg = 0, and (b) for Eg = 0.15t. The dashed–dotted line in (a) applies to the superconducting state for Eg = 0 (Tc = 0.023t) at T = 0.017t.

gap the scattering rate Γ varies linearly with frequency ω, as can be seen in Fig. 3.38b for higher temperatures. These results are in qualitative agreement with optical–conductivity data and the frequency–dependent scattering rate and e ective–mass spectra obtained from the complex optical conductivity for underdoped Bi 2212, YBCO, and LSCO compounds [173]. For example, for underdoped Bi 2212 with Tc = 67 K, the scattering rate 1/τ (ω) is linear in ω at T > T 200 K, and for T < T the low-frequency scattering rate is suppressed for ω < 500–700 cm−1 (62–87meV) [173]. This is in qualitative agreement with our results shown in Fig. 3.38b from which we estimate a crossover temperature T 0.1t 250 K and a threshold energy for the steep rise of about 0.3t 75 meV. It should be stressed that we have calculated the frequency-dependent scattering rate and mass enhancement with the help of the extended Eliashberg equations, while these quantities were obtained in [173] from theoretical expressions involving the complex conductivity.

Fig. 3.40. The incoherent dynamical c–axis conductivity (2.172) for three temperatures T = 0.1t (solid line), 0.05t (dashed line), and 0.03t (dotted line) and two values of Eg = 0 (upper three curves) and 0.15t (lower three curves).

Let us now turn to the c-axis conductivity. We shall see that, in our model, incoherent conductance gives a good description of the underdoped regime, while coherent conductance is more appropriate for the overdoped regime, confirming previous interpretations of the c–axis conductivity. In Fig. 3.39a, we show our results for the coherent c–axis conductivity σc(ω) calculated from (2.170) for di erent temperatures. Here we have taken Eg = 0. For decreasing temperature T , a coherent Drude peak develops at low frequencies. Such a development of a coherent Drude peak has been observed in overdoped cuprates [166, 170], where the pseudogap is absent or small. Thus, our coherent–conductance results account well for this observation in the overdoped compounds. In the superconducting state, a suppression of σc (ω) at intermediate frequencies sets in, as shown by the dashed–dotted line in Fig. 3.39a for T = 0.017t. Here, Tc = 0.023t. At the same time, the Drude peak continues to sharpen.

Figure 3.39b shows the normal–state coherent conductivity σc (ω) in the presence of a pseudogap with amplitude Eg = 0.15t. While the pseudogap leads to a suppression of σc(ω) at intermediate frequencies, the coherent Drude peak at low frequencies still remains and even sharpens, similarly to the superconducting state in Fig. 3.39a. These results are completely di erent from the experimental results for the normal state of underdoped cuprates, which show, instead of a coherent Drude peak, a gap–like suppression at low frequencies. Thus, it is not su cient to simply introduce a pseudogap in order to account for the c–axis conductivity in underdoped compounds! As has been noted earlier [167, 174], the c–axis conductance becomes incoherent in the underdoped regime the same time, and therefore it is necessary to

162 3 Results for High–Tc Cuprates: Doping Dependence

Fig. 3.41. The incoherent c–axis resistivity ρc as a function of temperature for Eg

= 0 (solid line), 0.1t (dashed line), 0.15t (dotted line), and 0.2t (dashed–dotted line).

upon lowering the temperature, while the conductivity stays almost constant

˜

for frequencies above this threshold energy. Here, Eg = 2Eg /Re Z(Eg ) is the renormalized amplitude of the d–wave pseudogap, and Re Z(ω) is the average mass renormalization at the Fermi surface, which is of the order of 2 for the parameters considered here. For Eg = 0, σcincoh is almost independent of frequency and temperature. These results are indeed in qualitative agreement with the measured interplane conductivity in underdoped YBCO compounds [166]. The gap in the c-axis conductivity σcincoh(ω) for frequencies ω below

˜ ˜

2Eg develops below a characteristic temperature T Eg /2. We want to stress that the temperature evolution of all physical quantities arises exclusively from the Fermi and Bose functions occurring in the FLEX equations in our real–frequency formulation [145] and in the expressions for the susceptibilities, since we have assumed that the pseudogap Eg (k) defined in (2.67) is temperature–independent. Physically, this means that above T the e ect of the pseudogap is smeared out such that the normal–state behavior (corresponding to the FLEX equations for Eg = 0) is recovered, while the e ect of the pseudogap on the quasiparticle and spin excitation spectra increases as T decreases below T towards Tc.

From our incoherent conductivity, we can extract the c-axis resistivity ρc = [σcincoh(ω = 0)]−1 in the presence of the pseudogap. The temperature dependence of ρc is shown in Fig. 3.41 for di erent values of the pseudogap amplitude Eg = 0, 0.1, 0.15, and 0.2t, corresponding to di erent doping levels. For Eg = 0, ρc is almost constant. For finite Eg , it starts to increase above the curve for Eg = 0 at lower temperatures. This “semiconducting” behavior of ρc is directly related to the depth of the pseudogap at zero frequency in σcincoh(ω = 0), which increases with increasing Eg and decreasing temperature (see Fig. 3.40). The results in Fig. 3.41 are consistent with the estimate

of the characteristic temperature given above, i.e. T ˜g g

E /2 = E /Re Z, because the steep rise of ρc for a given Eg appears approximately below

Fig. 3.42. The incoherent dynamical c-axis conductivity (2.172) for T = 0.03t and Eg = 0 (solid line), 0.1t (dashed line), 0.15t (dotted line), and 0.2t (dashed– dotted line). For comparison we also show the result in the superconducting state for Eg = 0 (Tc = 0.023t) at T = 0.017t (lower solid line).

T . Our results are in qualitative agreement with c–axis resistivity data for underdoped YBCO [166, 167]. Note that the definition of the characteristic

˜

temperature T Eg /2 corresponds to the scaling procedure in [155], where it was shown that the NMR Knight shift for a wide range of doping values follows closely a universal scaling curve if the data are plotted against a

˜

ferent values of Eg . From Fig. 3.42, we can see that the renormalized size 2Eg of the gap in σcincoh follows Eg . If one assumes that Eg changes strongly with doping level, as has been proposed in [155], the results shown in Fig. 3.42 cannot provide an explanation for the doping independence of the pseudogap seen in the c–axis conductivity of underdoped cuprates. This is an apparent inconsistency of the model of Williams et al., which can describe thermodynamic quantities well, but does not give a satisfactory account of the doping dependence of the ab–plane and c–axis conductivities, which are dynamical quantities. On the other hand, our results rather indicate that there are two independent energy scales involved in the pseudogap problem: first, the width

˜

of the pseudogap (here 2Eg ) as seen in the ω dependence of the c–axis and ab–plane conductivities, which are largely doping–independent, and second, the depth of the pseudogap as seen in the ω = 0 value of the incoherent c– axis conductivity, corresponding to the characteristic temperature T for the c–axis resistivity and the thermodynamic quantities, which increases upon lowering the doping level. Two such energy scales could be introduced into the problem by considering more complicated forms for the pseudogap than (2.67). For example, the pseudogap could have a frequency dependence or a momentum dependence which changes with temperature, as suggested by a

164 3 Results for High–Tc Cuprates: Doping Dependence

Fig. 3.43. Density of states N (ω) (solid line), incoherent c–axis conductivity σcincoh(ω) (dashed line), quasiparticle damping rate ω Im Z(ka, ω) (dotted line),

and ab–plane conductivity σab(ω) (dashed–dotted) as a function of frequency for Eg = 0.15t and T = 0.03t (arbitrary units). All four quantities show gap-like suppressions at low frequencies. The sizes of these gaps have a ratio of roughly 1:2:3:4. The ab–plane conductivity σab(ω) shows a strong Drude peak at low frequencies within the gap.

recent analysis of ARPES data [155, 156]. However, this is beyond the scope of this book.

In addition, the lower solid line in Fig. 3.42 shows the result for the incoherent c–axis conductivity in the superconducting state for T = 0.017t and Eg = 0 (Tc = 0.023t). This compares well with the experimental results for optimally doped YBCO [166], which show a suppression at low frequencies, similar to the suppression due to the normal-state pseudogap. In addition, a weak enhancement at the gap edge develops. Again, this behavior is completely di erent from the corresponding behavior of the coherent c–axis conductivity (see the dashed–dotted line in Fig. 3.39a). It is instructive to compare all of the results obtained for the optical conductivity described above with the corresponding density of states. In Fig. 3.43, we show our results for σab(ω) and σcincoh(ω) for Eg = 0.15t and T = 0.03t, along with the density of states N (ω) (2.171) and the quasiparticle damping rate ω Im Z(ka, ω) at the antinodal momentum ka at the Fermi surface (in arbitrary units). Here we see that the sizes of the pseudogap appearing in these four quantities are quite di erent. In fact, the gaps have an approximate ratio of 1:2:3:4 for the density of states, incoherent c–axis conductivity, quasiparticle damping rate, and ab–plane conductivity, respectively. In particular, we find that the

agreement with experiment. We believe that this relation among the gaps is a direct consequence of the electronic origin of the spin fluctuation scattering process and the self–consistency of the FLEX equations: the opening of the pseudogap leads to a suppression of the spin fluctuation interaction

Fig. 3.44. Density of states N (ω) for a d–wave pseudogap with amplitude Eg = 0.05t, in the normal state at T = 0.1t and 0.023t (solid lines), and in the superconducting state at T = 0.021t and 0.018t (T /Tc = 0.78 (dashed lines). The values N (0) decrease for this sequence of temperatures.

˜

via (2.62) and (2.61) at frequencies below 2Eg . This in turn results in a

˜

reduction of the self–energy below 3Eg and a corresponding structure at

˜g in σab(ω). However, in the incoherent c–axis conductivity, a gap of

E

4

˜

only 2Eg in size appears because of the momentum average of the spectral functions, resulting in a frequency convolution of the density of states with itself. The appearance of a gap of 4∆0 in the ab–plane conductivity in the superconducting state for an electronic pairing mechanism has been noted earlier in connection with marginal–Fermi liquid theory [175, 176, 177]. Here we suggest that a corresponding e ect may take place in the pseudogap state of underdoped high–Tc compounds.

The pseudogap structures in the curves in Fig. 3.43 appear to be washed out somewhat and show more complex behavior than a simple suppression at the e ective pseudogap. This is due to the fact that the pseudogap (2.67) is renormalized owing to self–energy e ects. The structures seen in the conductivity, density of states, and quasiparticle damping rate do not display a pure d–wave gap, but a renormalized one, similar to that in the superconducting state.

To summarize the results for the optical conductivity, we have investigated the influence of a normal–state d–wave pseudogap on the c–axis and ab–plane conductivities for spin fluctuation exchange scattering using an extension of the generalized Eliashberg equations and the self–consistent FLEX approximation as described in Sect. 2.2.1. We find that coherent conduction can describe the c–axis conductivity in the overdoped compounds, while it is necessary to consider incoherent c–axis conduction in the underdoped regime. Only incoherent conduction can account well for the dynamical c–axis conductivity and the c–axis resistivity in the underdoped compounds, which show “semiconducting” behavior. However, it is di cult to reconcile the doping

166 3 Results for High–Tc Cuprates: Doping Dependence

dependence of the amplitude of the pseudogap obtained from a microscopic theory with the doping–independent size of the pseudogap seen in the dynamical c–axis and ab–plane conductivities. This suggests that the pseudogap has a nontrivial momentum or frequency dependence, which changes with temperature. We find that the di erence between the sizes of the pseudogaps for ab–plane conductivity and c–axis conductivity finds a natural explanation

in the electronic origin of spin fluctuation scattering and its self–consistency

˜ with the single-particle properties. This leads to a gap structure of size 4Eg

in the ab–plane conductivity, while the gap seen in the incoherent c–axis con-

˜ ductivity has a size of only 2Eg .

In order to investigate the influence of a d–wave pseudogap on the Raman response, it is instructive to start with the density of states, which is the main factor that determines the pair–breaking peak in all scattering geometries.

for a pseudogap amplitude Eg = 0.05t which is assumed to be temperature– independent. One can see that with decreasing T , a typical d–wave gap develops in the normal state and that this spectrum merges continuously into the superconducting spectrum as T decreases through Tc = 0.022t. One can see that below Tc, a dip develops at negative ω below the quasiparticle peak and that the spectrum is quite asymmetric with respect to the Fermi energy at ω = 0. These results are quite similar to the measured tunneling spectra for underdoped Bi2212.

The results discussed in the previous sections encourage us to calculate the Raman response functions above and below Tc in the presence of this d–wave pseudogap Eg (k). The Raman response function, including vertex corrections, then reads

Im χγ (q = 0, ω)

In Fig. 3.45a, we show our results for the B1g symmetry, again for a gap amplitude Eg = 0.05t as in Fig. 3.44. Comparison with the results for Eg = 0 in Fig. 3.24 shows that the most prominent e ect of the pseudogap is to produce a broad peak at about a frequency ω 0.075t (3/2)Eg as T approaches Tc from above. This frequency is nearly the same as the frequency di erence between the quasiparticle peaks in the density of states in Fig. 3.44. We have also carried out calculations for larger values of the gap amplitude Eg corresponding to lower doping levels [10, 165], i.e. Eg = 0.075t and 0.1t [146]. There we find analogous results, namely, that with decreasing T in the