Enhanced pairing mechanism in cuprate-type crystals

Using a BCS mean-field approach, we show how the interplay between low-momentum optical phonons and Jahn-Teller-type lattice distortions can open an attractive channel that allows the formation of pairs with the corresponding density exhibiting characteristic features of a pair-density wave. We demonstrate this numerically on a copper-oxide-type lattice.

Figure 1

Two-dimensional ${\mathrm{CuO}}_2$ cubic lattice.

Figure 2

An example of a Jahn-Teller-type distortion to $\mathrm{Cu}{\mathrm{O}}_2$. On the left, a unit cell with displacements ${\delta }_{x/y}$ from the positions of the oxygen atoms at the symmetry points $(a/2,0)$ and $(0,a/2)$ along the respective axes is shown.

Figure 3

The deformed lattice and bond structure resulting from the distortion of Fig. 2

Figure 4

Dispersion relation in the tight-binding model with $t_1=1.5\mathrm{eV}, t_2=0.825\mathrm{eV}$. For the BCS model, we consider only the lowest branch (bottom).

Figure 5

The effective electron-electron potential $V$ for the lowest electron branch in the first Brillouin zone, shown in units of ${\hslash \left|pv\right|}^2/{\omega }_0^2$ for distortion parameter $\delta =\rho =0.05a$.

Figure 6

Absolute value of the gap function $\mathrm{\Delta }$ in a tight-binding model with parameters $\rho =0.05a, \delta =0.05a, t_1=1.5\mathrm{eV}, t_2=0.8\mathrm{eV}$. The Fermi surface with $\mu =0.8\mathrm{eV}$ is indicated in red on the first gap plot.

Figure 7

Pair-wave density Eq. (19) in configuration space for order parameter $\alpha \left(\mathbf{k}\right)=\left|\alpha \right(\mathbf{k}\left)\right|$ from Fig. 6 at $T\approx 0.9T^{*}$, calculated via Riemann sums with $N=101$ support points in both coordinate directions.

Figure 8

Pair-wave density Eq. (19) with additional $d$-wave-like phase $\alpha \left(\mathbf{k}\right)=\sigma \left(\mathbf{k}\right)\left|\alpha \right(\mathbf{k}\left)\right|$ at $T\approx 0.9T^{*}, \sigma \left(\mathbf{k}\right):=\mathrm{sgn}(k_x^2-k_y^2)$.

Figure 9

The kernel maximum function $M\left(\mathbf{k}\right):={max}_{k^{\prime }\in \text{FBZ}}(-L_{{\beta }^{*}}(\mathbf{k},{\mathbf{k}}^{\prime })V(\mathbf{k},{\mathbf{k}}^{\prime }))$ from the linear gap equation at $\mu =0.85\text{eV}$, where ${\beta }^{*}=100{\text{eV}}^{-1}$.

Figure 10

The plot of the linear gap kernel $-\left(LV\right)({\mathbf{k}}_{\text{max}},{\mathbf{k}}^{\prime })$ as a function of ${\mathbf{k}}^{\prime }$ at one of the global maxima ${\mathbf{k}}_{\text{max}}$ of $M$ shows that exactly the two pairings $({\mathbf{k}}_{\text{max}},{\mathbf{k}}_{\text{max}})$ and $({\mathbf{k}}_{\text{max}},-{\mathbf{k}}_{\text{max}})$ emerge at the critical temperature. All other parameters are as in Fig. 9.

Figure 11

Largest eigenvalue of $-V{L}_{\beta }$ for ${\omega }_0=0.5\text{eV}$ as function of $\mathbf{K}$ (other parameters as described in the text). Here and in the following figures, we will indicate the Fermi surface for $\mu =0.85\text{eV}$ in red. The boundary points of the discretization will always only be included on the positive sides of the corresponding axes. The plot meshes are from now on matched to the discretization.

Figure 12

Absolute square of the wave function for $\mathbf{K}=0$ in Fig. 11 as function of $\mathbf{p}$.

Figure 13

Largest eigenvalue for ${\omega }_0=0.33\text{eV}$ as function of $\mathbf{K}$ (other parameters as described in the text). The eigenvalues at A, B, and at other similar peaks are dominating over the eigenvalue at the origin $\mathbf{K}=0$.

Figure 14

Absolute square of the wave function for $\mathbf{K}=(\frac{3}{10}\pi ,\pi )$ (point A in Fig. 13) as function of $\mathbf{p}$. Solid and dashed red lines show the Fermi surface for the two electron momenta $\mathbf{K}+\mathbf{p}$ and $\mathbf{K}-\mathbf{p}$, respectively. The energy difference between the two peaks is proportional to ${\omega }_0$.

Figure 15

Absolute square of the wave function for $\mathbf{K}=(-\frac{7}{10}\pi ,\pi )$ (point B in Fig 13) as function of $\mathbf{p}$, showing that the wave function is concentrated near $(\pm \pi ,0)$.