Effect of orbital relaxation on the band structure of cuprate superconductors and implications for the superconductivity mechanism

Where the doped holes reside in cuprate superconductors has crucial implications for the understanding of the mechanism responsible for their high-temperature superconductivity. It has been generally assumed that doped holes reside in hybridized $\mathrm{Cu}{d}_{x^2-y^2}\text{−}\mathrm{O}p\sigma$ orbitals in the ${\mathrm{CuO}}_2$ planes, based on results of density functional band-structure calculations. Instead, we propose that doped holes in the cuprates reside in $\mathrm{O}p\pi$ orbitals in the plane, perpendicular to the $\mathrm{Cu}\text{−}\mathrm{O}$ bond, that are raised to the Fermi energy through local orbital relaxation, that is not taken into account in band-structure calculations that place the bands associated with these orbitals well below the Fermi energy. We use a dynamic Hubbard model to incorporate the orbital relaxation degree of freedom and find in exact diagonalization of a small ${{\mathrm{Cu}}_4\mathrm{O}}_4$ cluster that holes will go to the $\mathrm{O}p\pi$ orbitals for relaxation energies comparable to what is expected from atomic properties of oxygen anions. The bandwidth of this band becomes significantly smaller than predicted by band-structure calculations due to the orbital relaxation effect. Within the theory of hole superconductivity the heavy hole carriers in this almost full band will pair and drive the system superconducting through lowering of their quantum kinetic energy.

Figure 1

Unit cell in the $\mathrm{Cu}\text{−}\mathrm{O}$ plane with five electronic orbitals: $\mathrm{Cu}{d}_{x^2-y^2}$ and $\mathrm{O}{p}_{\sigma },p_{\pi }$ orbitals. There are two oxygen atoms in the unit cell denoted by ${\mathrm{O}}_1,{\mathrm{O}}_2$.

Figure 2

Band structure in the $\mathrm{Cu}\text{−}\mathrm{O}$ planes in the $\Gamma \text{−}X$ direction [(0,0) to $(\pi ,\pi )\text{]}$ from a tight-binding calculation with five orbitals per unit cell (see Fig. 1). Parameters used are $t_{\pi \pi }=t_{\sigma \sigma }\equiv t_1=0.65,t_{\pi \sigma }\equiv t_2=0.35,t_{d\sigma }\equiv t_d=1.75,{\epsilon }_d=-5.2,{\epsilon }_{p\sigma }=-5.5$, and ${\epsilon }_{p\pi }=-4.7$ (see text).

Figure 3

Weights of the different orbitals in the band states. Bands 1–5 are numbered from lowest to highest energy. Band 2 (not shown) has only contributions from $p\sigma$ (mostly) and $p\pi$ and none from $d$ for all values of $k$.

Figure 4

Atomic orbitals in the ${\mathrm{CuO}}_2$ unit cell for the undoped case (left) and when an electron is removed (hole is added) (right). If the hole goes into the $p\pi$ orbital, it will shrink as shown schematically with the dashed and full lines in the right panel of the figure.

Figure 5

States with one and two electrons in an atomic $p$ orbital. $E\left(1\right)$ and $E\left(2\right)$ are the energies of the lowest-energy states with one and two electrons, respectively. The orbital expands when the second electron is added. $\overline{E}\left(1\right)$ is the energy of one electron in the expanded orbital, and $\overline{E}\left(2\right)$ is the energy of two electrons in the unexpanded orbital.

Figure 6

Effect of orbital relaxation on the band structure (b) (schematic) compared to the case when orbital relaxation is not taken into account (a). The energies of the states of band 4, arising from overlap of oxygen $p\pi$ orbitals, are lifted by an amount given by the orbital relaxation energy ${\epsilon }_R$ with respect to the energies when orbital relaxation is not taken into account. In addition, the bandwidth of band 4 becomes significantly smaller due to the modulation of the hopping amplitude by the overlap matrix elements.

Figure 7

Eight-site cluster for the exact diagonalization calculation. There is one electronic $d$ orbital at each of the four Cu sites, a $p\sigma$ and a $p\pi$ orbital at each of the four O sites, and an auxiliary spin degree of freedom at each O site to describe the expansion/contraction of the $p\pi$ orbitals indicated by the dotted lines.

Figure 8

Occupation of the orbitals in the cluster of Fig. 7 in the ground state when there is one hole in the cluster as a function of the orbital relaxation energy ${\epsilon }_R$ (in eV) for $S=0.333$.

Figure 9

Energy eigenvalues for the cluster of Fig. 7 with one hole in the absence of orbital relaxation and when orbital relaxation is included with the dynamic Hubbard model with $S=0.333$ and ${\epsilon }_R=3$. The four $p\pi$ states in the cluster are moved up in energy by several eV when orbital relaxation is included. The numbers next to the symbols indicate the $p\pi$ occupation for the state. Note also that the spread in energy of the $p\pi$ states becomes much smaller in the presence of orbital relaxation.