Checkerboard and stripe inhomogeneities in cuprates

We systematically investigate charge-ordering phases by means of a restricted and unrestricted Gutzwiller approximation to the single-band Hubbard model with nearest- $\left(t\right)$ and next-nearest- $\left(t^{\prime }\right)$ neighbor hopping. When $\mid t^{\prime }∕t\mid$ is small, as appropriate for ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x\mathrm{Cu}{\mathrm{O}}_4$, stripes are found, whereas in compounds with larger $\mid t^{\prime }∕t\mid$ (such as ${\mathrm{Ca}}_{2-x}{\mathrm{Na}}_x\mathrm{Cu}{\mathrm{O}}_2{\mathrm{Cl}}_2$ and ${\mathrm{Bi}}_2{\mathrm{Sr}}_2\mathrm{Ca}{\mathrm{Cu}}_2{\mathrm{O}}_{8+\delta }$) checkerboard structures are favored. In contrast to the linear doping dependence found for stripes the charge periodicity of checkerboard textures is locked to $4\text{unit}$ cells over a wide doping range. In addition we find that checkerboard structures are favored at surfaces.

Figure 1

(Color online) CO textures for $U∕t=8$. The length of arrows is proportional to the spin density whereas the squares are proportional to the hole density minus the hole density in the insulator. (a) Bond-centered stripe texture with charge spacing $d=4$, doping $n_h=1∕8$, and $t^{\prime }∕t=-0.2$; (b) SF checkerboard with $d=4$ at $x=0.06$ and $t^{\prime }∕t=-0.5$; (c) same as (b) with $n_h=1∕8$; (d) LF checkerboard with $d=4$ and $t^{\prime }∕t=-0.5$ at $n_h=1∕8$.

Figure 2

(Color online) Energy per site as a function of doping for the different textures studied and for $t^{\prime }∕t=-0.2$ (a) and $t^{\prime }∕t=-0.5$ (b). For clarity we subtracted the energy of the AFM solution and the line ${\mu }_0{n}_h$ with ${\mu }_0=-1.6t$. Different choices of ${\mu }_0$ correspond to different choices of the origin of the energy of the single-particle states and do not change the physics. In each curve for the stripes the charge periodicity perpendicular to the stripe is fixed and takes the values (from left to right) $d=10,8,6,5,4,3$. In panel (b) we also show the energies allowing for spin canting and AFM-SFC mixing (solid circles) and Maxwell construction (MC) for AFM-SFC phase separation and for SFC and LFC phase separation.

Figure 3

(Color online) (a) Energy per site vs $-t^{\prime }∕t$ for $U∕t=8,10$ for stripes and checkerboard at $n_h=0.0625,0.187$. (b) Mixed AFM-SFC solution at $n_h=0.047$ and allowing for spin canting.