Isotope effect on the superconducting critical temperature of cuprates in the presence of charge order

Figures 3 and 4 show α versus T_c and δ, respectively, for V = 0.06 and $V=0.10,$ and two phonon frequencies. ${\omega }_{0}=0.1$ and 0.2 corresponds to the buckling and half-breathing phonon modes in YBCO. The solid points are experimental results from [18]. They all lie in the region between the curves calculated with parameter values representative for cuprates. The above choice of parameter values for phonons is, of course, unproblematic. More controversial may be the employed values for the EP coupling constants. First principles calculations of total EP constants have been described in [21]. The results are given in terms of dimensionless coupling constants ${\lambda }_{s}$ and ${\lambda }_{d}$ for the s- and d-wave channel, respectively. For each channel, λ and V are related by $\lambda ={VN}(0),$ where N(0) is the density of electronic states at the Fermi energy in the corresponding symmetry channel. LDA calculations yield for YBa₂Cu₃O₇ ${\lambda }_{s}\sim 0.24$ and ${\lambda }_{d}\sim 0.022$ [21]. These values are rather small, particularly, the value for ${\lambda }_{d}.$ On the other hand there is good evidence from several experiments that such small values are not unreasonable: Angle-resolved photoemission data in LSCO [15, 28] yielded ${\lambda }_{s}\sim 0.4.$ Similarly, superconductivity-induced shifts of zone center phonons are in good agreement with calculated LDA values [29, 30] and therefore with such small EP coupling constants. In figures 3 and 4 only EP coupling constants in the d-wave channel enter, for which we used V = 0.06 and 0.10, which are slightly larger than the LDA values. They describe the experimental points somewhat better than the bare LDA values. One should keep in mind that such adjustments (from ${\lambda }_{d}\sim 0.02$ to ${\lambda }_{d}\sim 0.04$) should be considered as minor, because we always stay in the region of very small EP couplings.

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It is worth remarking that the observed increase of α in the underdoped region of YBCO can be quantitatively explained not only by employing such small values for the EP coupling constants but by noting that a reasonable agreement between experiment and theory requires them. As discussed above the large isotope effect found in underdoped cuprates has been interpreted as evidence for a strong EP coupling in these systems. From the above analysis the conclusion is quite different: The large observed isotope shifts in underdoped cuprates are the result of a competition of superconductivity with another ground state which produces the pseudogap. They can be explained using the small d-wave EP coupling constant obtained in the LDA calculations. Figures 3 and 4 demonstrate this for the case of a d CDW state as a competing state, but we expect similar results for other ground states, as long as they are associated with a gap or a pseudogap.

The above calculations indicate that a pseudogap and the associated shift of the density of states from low to high energies are responsible for the strong increase of α in the underdoped regime. It is, however, clear that a reduction of ${N}_{d}(0)$ alone cannot increase α as long as ${N}_{d}(\omega )$ is constant on the scale of ${\omega }_{0}.$ It is therefore interesting to analyze the above equations in more detail and to find out what exactly causes the increase in α. For the following analytical results we will assume that the density of states ${N}_{d}(\omega )$ can be considered either as constant or that T_c is sufficiently low; i.e., we will consider the overdoped or the strongly underdoped regions. It also will be sufficient to consider the expression (16) for α, which is valid for a weak EP coupling.

The derivative in the denominator of (16) may be approximated as

$$\begin{eqnarray}&&\displaystyle \frac{\partial {F}_{11}}{\partial {T}_{c}}=\displaystyle \frac{\tilde{J}}{{T}_{c}}{\displaystyle \int }_{0}^{\infty }{\rm{d}}\epsilon \displaystyle \frac{{N}_{d}(2\epsilon {T}_{c})}{{\mathrm{cosh}}^{2}(\epsilon )}\approx \displaystyle \frac{\tilde{J}}{{T}_{c}}{N}_{d}(0).\end{eqnarray} \tag{ 17 }$$

The term F₁₂ can be written as

$$\begin{eqnarray}{F}_{12} & = & -2\sqrt{V\tilde{J}}{\displaystyle \int }_{0}^{\infty }{\rm{d}}\epsilon {N}_{d}(\epsilon )\left[{T}_{c}\displaystyle \sum _{n}\displaystyle \frac{1}{{\omega }_{n}^{2}+{\epsilon }^{2}}\right.\\ & & -\left.{T}_{c}\displaystyle \sum _{| {\omega }_{n}| \gt {\omega }_{0}}\displaystyle \frac{1}{{\omega }_{n}^{2}+{\epsilon }^{2}}\right].\end{eqnarray} \tag{ 18 }$$

Let us write F₁₂ as

$$\begin{eqnarray}&&{F}_{12}=\sqrt{V\tilde{J}}({F}_{12}^{(1)}+{F}_{12}^{(2)}),\end{eqnarray} \tag{ 19 }$$

with

$$\begin{eqnarray}&&{F}_{12}^{(1)}={\displaystyle \int }_{0}^{{\omega }_{0}}{\rm{d}}\epsilon \displaystyle \frac{{N}_{d}(\epsilon )}{\epsilon }\mathrm{tanh}(\displaystyle \frac{\epsilon }{2{T}_{c}}),\end{eqnarray} \tag{ 20 }$$

$$\begin{eqnarray}{F}_{12}^{(2)} & = & {\displaystyle \int }_{{\omega }_{0}}^{\infty }{\rm{d}}\epsilon \displaystyle \frac{{N}_{d}(\epsilon )}{\epsilon }\mathrm{tanh}(\displaystyle \frac{\epsilon }{2{T}_{c}})\\ & & -{\displaystyle \int }_{0}^{\infty }\displaystyle \frac{{N}_{d}(\epsilon )}{{\omega }_{0}}\mathrm{arctan}(\displaystyle \frac{{\omega }_{0}}{\epsilon }).\end{eqnarray} \tag{ 21 }$$

In the last term in (21) the zero-temperature limit has been taken. The derivative $\frac{\partial {F}_{12}}{\partial {\omega }_{0}}$ acts only on the last term in ${F}_{12}^{(2)},$ yielding

$$\begin{eqnarray}&&\displaystyle \frac{\partial {F}_{12}^{(2)}}{\partial {\omega }_{0}}=-{\displaystyle \int }_{0}^{\infty }{\rm{d}}\epsilon \displaystyle \frac{{N}_{d}(\epsilon )}{{\omega }_{0}^{2}+{\epsilon }^{2}}.\end{eqnarray} \tag{ 22 }$$

Inserting the above results into (16) gives

$$\begin{eqnarray}&&\alpha =-\displaystyle \frac{{\omega }_{0}V}{{N}_{d}(0)}\displaystyle \frac{{F}_{12}}{\sqrt{V\tilde{J}}}{\displaystyle \int }_{0}^{\infty }{\rm{d}}\epsilon \displaystyle \frac{{N}_{d}(\epsilon )}{{\omega }_{0}^{2}+{\epsilon }^{2}}.\end{eqnarray} \tag{ 23 }$$

Numerical evaluation of (18) shows that F₁₂ is practically constant as a function of doping above optimal doping and only very slowly increasing towards lower dopings. Thus we may consider F₁₂ in (23) as a constant. As a result we obtain for the increase of α relative to its value ${\alpha }_{0}$ at optimal doping, i.e., at the onset of the CDW,

$$\begin{eqnarray}&&\alpha /{\alpha }_{0}=\displaystyle \frac{2}{\pi }{\displaystyle \int }_{0}^{\infty }{\rm{d}}\epsilon \displaystyle \frac{{\omega }_{0}}{{\omega }_{0}^{2}+{\epsilon }^{2}}\cdot \displaystyle \frac{{N}_{d}(\epsilon )}{{N}_{d}(0)}.\end{eqnarray} \tag{ 24 }$$

The above formula allows us to understand the large increase of α in the underdoped regime. Near optimal doping $N(\epsilon )$ depends only weakly on frequency so that the density ratio ${N}_{d}(\omega )/{N}_{d}(0)$ and, therefore, also $\alpha /{\alpha }_{0}$ is near 1. Below optimal doping the spectral weight is shifted from low to high frequencies, which produces the pseudogap. As a result the ratio ${N}_{d}(\omega )/{N}_{d}(0)$ is large around the pseudogap, leading to large values for $\alpha /{\alpha }_{0}.$ Since ${N}_{d}(0)$ is roughly proportional to δ, $\alpha /{\alpha }_{0}$ increases monotonically with decreasing doping, yielding values which may exceed by far the canonical BCS value of 1/2. Taking the limit ${\omega }_{0}\to 0$ in (24), the first factor under the integral becomes a delta function, and we obtain $\alpha /{\alpha }_{0}=1.$ The absence of an enhancement of α for small ${\omega }_{0}$ can easily be understood: The main part in the integral in (24) comes from the region of small , well below the pseudogap, where ${N}_{d}(0)$ is small. The contribution from electronic spectral density shifted to the frequency region around the gap is missing, and no substantial enhancement of α can occur. This case also shows that the reduction of ${N}_{d}(0)$ due to the formation of the gap does not cause an increase of α by itself. Instead the shift of spectral weight from low to high frequencies near the pseudogap is responsible for the increase of α. For large phonon frequencies, $\alpha /{\alpha }_{0}$ decreases with increasing ${\omega }_{0}.$ Thus one expects that $\alpha /{\alpha }_{0}$ as a function of ${\omega }_{0}$ first increases, passes then through a maximum, and finally decreases. The curves in figure 5, calculated with the full expressions for the functions F, illustrate nicely this behavior.

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In the overdoped region one may assume that the density of states ${N}_{d}(\epsilon )$ is constant. The approximations leading to (24) imply then $\alpha ={\alpha }_{0}$ throughout the overdoped region. A better approximation in this region is obtained by using ${\omega }_{0}$ as the cutoff in the energy integration in F₁₂. The two terms in ${F}_{12}^{(2)}$ cancel then exactly, and ${F}_{12}^{(1)}$ can be evaluated as in the usual BCS theory, yielding,

$$\begin{eqnarray}&&\alpha =\displaystyle \frac{\pi {\lambda }_{d}}{2}\ \mathrm{log}(\displaystyle \frac{1.14{\omega }_{0}}{{T}_{c}}).\end{eqnarray} \tag{ 25 }$$

A similar result has been first derived in [31]. For optimal doping where T_c is highest, α shows a minimum with a value which for ${T}_{c}\gt 1.14{\omega }_{0}$ is weakly negative. For more realistic parameter values the logarithmic factor is about 1, and thus $\alpha \approx {\lambda }_{d}.$ The observed small values ${\alpha }_{0}\sim 0.05$ in cuprates with the highest T_c do require similar values for ${\lambda }_{d}$ in good agreement with the values obtained from the LDA calculations. Our analytical expressions for α in the under- and overdoped region used ${\omega }_{0}$ as a cutoff, but one time along the imaginary and one time along the real axis. Taking the cutoff always along the real axis is not suitable in the underdoped region, because if the density peak in ${N}_{d}(\omega )$ coincides with the phonon frequency, one obtains a spurious peak in the curve α versus δ. Using the cutoff along the imaginary axis we never found such an artifact and therefore used this choice of cutoff in all our numerical calculations.

The inset of figure 5 shows α versus δ, calculated without approximations, over a large doping region. In the overdoped region this function increases roughly as predicted by (25). A closer look reveals, however, that both the analytic expressions (24) for the underdoped regime and, to a lesser degree, (25) for the overdoped regime do not well agree with the numerically evaluated curves. One reason is that our model ${N}_{d}(\omega )$ in equation (17) varies in the energy interval $[0,2{T}_{c}]$ considerably so that the approximation proposed in that equation is problematic. Band structure effects thus play a role in our model, even at low energies, producing fluctuations in the curve α versus δ if calculated with (24). In the numerically exact calculated curves in figures 3–5, such fluctuations are absent due to a properly carried out energy integration in (17). Though (24) and (25) are thus not suitable to obtain accurate values, they nevertheless explain correctly the curves α versus δ at low and high dopings.

Our theory can also be applied to Sr- and Ba- doped La₂CuO₄. La${}_{2-x}$Ba_xCuO₄ shows a variety of phase transitions near the doping $\delta =1/8$ [32]. T_c exhibits a dip between ${\delta }_{1}=0.155$ and ${\delta }_{2}=0.095,$ which nearly touches zero at $\delta =0.125$ [32, 33]. Between these limiting dopings a leading-phase CO with charge stripe order extends towards higher temperatures above the superconducting phase in a domelike manner, touching T_c at the end points. There are additional phases of spin stripe order or of orthorhombic or tetragonal symmetry present, which will be disregarded in the following. Above ${\delta }_{1}$ or below ${\delta }_{2},$ T_c is decreasing with increasing distance from these points, reaching very small values near $\delta =0.25$ and 0.05, respectively. The phase diagram of the sister compound LSCO does not contain long-ranged phases around the doping 1/8. Nevertheless it is probable that superconductivity competes with phases in the particle–hole channel near this doping, because T_c shows a pronounced dip in this region [5, 6]. Also angle resolved photoemission experiments find in LSCO a pseudogap that sets in at around $\delta =0.20,$ and exists and increases in magnitude towards lower dopings. The isotope coefficient in LSCO (see figure 6) is very small at the large doping value of 0.20, where the pseudogap forms. With decreasing δ, α first increases slightly and then very rapidly, reaching a maximum near the doping 1/8 with a value of about 1. Decreasing δ further, α decreases but settles down in the region of about 0.5.

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In view of the above phase diagrams and the behavior of α in LSCO, it is natural to assume that the doping dependence of T_c and α are caused by gaps or pseudogaps similar to those in YBCO. In order to transfer our results from YBCO to La${}_{2-x}$Ba_xCuO₄ and LSCO, we consider the calculated α in YBCO as a function of ${T}_{c}/{T}_{c,0},$ where ${T}_{c,0}$ is the optimal T_c for a doping near the onset of the pseudogap. Treating first La${}_{2-x}$Ba_xCuO₄, we can read off from figure 2 of [33] the doping as a function of ${T}_{c}/{T}_{c,0},$ where ${T}_{c,0}$ is the value of T_c near the dopings ${\delta }_{1}$ or ${\delta }_{2}.$ Identifying the two ratios for the reduction of T_c, one obtains α as a function of doping in La${}_{2-x}$Ba_xCuO₄. The same procedure can be applied to LSCO. From figure 3 in [6] one can read off δ as a function of ${T}_{c}/{T}_{c,0},$ where ${T}_{c,0}$ is the largest transition temperature near $\delta =0.15.$ Comparing this reduction ratio with the case of YBCO, one finds the doping dependence of α in LSCO. Figure 6 contains the obtained curves, calculated for V = 0.1 and ${\omega }_{0}=0.1,$ together with experimental values for LSCO from [5]. Our calculation suggests the following interpretation of the experimental α in LSCO. One has to distinguish between two pseudogaps in LSCO. The first one is the usual pseudogap, which is observed by angle-resolved photoemission and which exists below $\delta \sim 0.20$ [34]. This pseudogap corresponds to that occurring in underdoped cuprates and develops at ${T}^{*}.$ At low doping, superconductivity competes with the phase underlying the pseudogap, leading to the overall decrease of T_c down to very low values near $\delta =0.05.$ At the same time this decrease of T_c is associated with a monotonic increase of α similar to that in YBCO. The second pseudogap is located between $\delta =0.10$ and $\delta =0.15$ and is due to static (in La${}_{2-x}$Ba_xCuO₄) or fluctuating (in LSCO) stripes. As a result T_c is strongly (in La${}_{2-x}$Ba_xCuO₄) or slightly (in LSCO) suppressed. Correspondingly, the increase of α in figure 6 is large for the Ba- and small for the Sr- doped systems. It is interesting to see that the calculated curve for LSCO shows only a small peak at doping 1/8, quite in contrast to the experimental curve. The reason for this is that the employed experimental T_c curve exhibits only a small dip near the doping 1/8. One prediction of our calculation is that α in Ba- doped La₂CuO₄ should show a large peak near doping 1/8. Though our calculation for α in LSCO is not able to get quantitative agreement with experiment, we think that the shift of the spectral weight from low to higher energies associated with the formation of the pseudogap plays also a role in LSCO.

Finally we would like to mention that strong renormalizations (softenings and broadenings) of phonons have been observed in underdoped cuprates (see [35] and references therein). These anomalies occur for wave vectors along the crystalline axis with a length of about 0.3 and are seen both in acoustical [36] and optical bond-stretching phonons [37]. They are probably related to the recently observed charge order in these systems [36]. On the other hand, extensive LDA calculations, performed for YBa₂Cu₃O₇, did not yield unusual softenings [21, 38] and cannot explain these phonon anomalies. However, our main result is independent of the validity of the LDA in cuprates and shows only that the observed large values of α in underdoped cuprates are at least compatible with the LDA values for the EP coupling constants. The existence of the above phonon anomalies suggests that it is not possible to conclude quite generally from these and other successful LDA results [29, 30] that the EP coupling is necessarily small in the cuprates. Thus, the correct explanation of the phonon anomalies is presently an open problem and beyond the scope of our paper, similarly to their influence on T_c.