Magic Doping Fractions for High-Temperature Superconductors

We report hole-doping dependence of the in-plane resistivity ${\rho }_{ab}$ in a cuprate superconductor ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x\mathrm{Cu}{\mathrm{O}}_4$, carefully examined using a series of high-quality single crystals. Our detailed measurements find a tendency towards charge ordering at particular rational hole-doping fractions of $1/16$, $3/32$, $1/8$, and $3/16$. This observation appears to suggest a specific form of charge order and is most consistent with the recent theoretical prediction of the checkerboard-type ordering of the Cooper pairs at rational doping fractions $x=(2m+1)/2^n$, with integers $m$ and $n$.

Figure 1

Temperature dependences of ${\rho }_{ab}$ for a series of LSCO single crystals. The $x$ values shown are the actual Sr contents measured by the inductively coupled plasma atomic-emission spectroscopy.

Figure 2

(a) $x$ dependence of the inverse mobility ${\mu }^{-1}$ ($=ne{\rho }_{ab}$) at representative temperatures. Inset shows the zero-resistivity $T_c$ versus $x$. (b) $x$ dependence of ${\rho }_{ab}(T)/{\rho }_{ab}(300\mathrm{K})$ at $T=200$, 100, and 50 K. The hole motion tends to be hindered at low temperature at $x\simeq 0.06$, 0.09, 0.13, and 0.18. The magic doping fractions expected for the checkerboard order and the 1D stripes are shown by dashed lines and dotted lines, respectively; dash-dotted lines show the fractions both models predict.

Figure 3

Hierarchical construction of the checkerboard-type ordering of the hole pairs at magic doping fractions $(2m+1)/2^n$, where $m$ and $n$ are integers. Following the construction of Ref. 8, the original $\mathrm{Cu}{\mathrm{O}}_2$ lattice is grouped into nonoverlapping plaquettes, which can be represented by the squares on a checkerboard. A checkerboard can be alternately colored black and white; in our case, each black square contains four sites and no holes, while each white square contains four sites and two holes in the form of a Cooper pair. Such a state has hole-doping density $x=1/4$ (a), as represented at the highest level of the hierarchy (e). (Electrons are denoted by black dots, and holes are denoted by open dots; since we only address the charge ordering here, electron spin is not explicitly indicated.) At the next level of the hierarchy, consider the lattice of white squares only, and alternately color half of them black. Such a state has hole-doping density $x=1/8$ (b). At one further level down in the hierarchy, one can either consider the lattice of the white squares, and alternately color half of them black, thus obtaining a state with $x=1/16$ (c), or one can consider the lattice of the newly colored black squares and alternately color half of them white, thus obtaining a state with $x=3/16$ (d). This hierarchy construction can be obviously iterated ad infinitum, generating a binary tree of magic doping fractions as shown in (e).