Commensurate and Incommensurate Checkerboard Charge Ordered States
C. Reichhardt, C.J. Olson Reichhardt, and A.R. Bishop
Theoretical Division and Center for Nonlinear Studies,
Los Alamos National Laboratory, Los Alamos, New Mexico 87545
Available online 19 April 2007
Abstract
We consider a system of charged particles interacting through Coulomb
repulsion with an additional short range attraction to model charge
ordering in cuprate superconductors. The particles move over a
background periodic substrate. In the absence of the substrate, we
find a checkerboard type ordering of the charges along with a variety
of other crystalline states as the charge density is varied. In the
presence of a substrate, there are additional commensurate-incommensurate
grain boundary structures that form within the checkerboard state. We
compare our results to recent scanning tunneling experiments.
Competing interactions between holes
in metal oxide materials arise from a combination of the repulsive
Coulomb interaction and a dipolar attraction created when the holes distort
the spins in the antiferromagnetic background.
In such systems,
the competing interactions can produce various types of
pattern formation, including checkerboard type orderings [3].
In physical systems, any checkerboard pattern formed by the holes will also
interact with the periodic substrate created by the underlying atomic
lattice. In this work
we investigate the effect of an underlying periodic substrate on
checkerboard orderings.
We start with a phenomenological model for the holes
which is given in full detail in Refs. [1,2].
We adjust the doping level n of the system by altering
the hole density in a computational box of size L ×L.
The interaction between two holes a distance r apart is
given by
V(r) =
q2
r
− Ae−r/a−Bcos(2θ−ϕ1−ϕ2)e−r/ξ.
(1)
Here, q=1 is the hole charge, θ is the angle between r and
a fixed axis, and ϕ1,2 are the angles of the magnetic dipoles
relative to the same fixed axis, which we assume can take an arbitrary
value.
B is the strength of the magnetic dipolar interaction, and we set the
short-range anisotropic interaction A=0. We take the magnetic
correlation length ξ = 3.8/√n Å.
At intermediate densities, filamentary patterns form, which assume a
checkerboard-like appearance in the presence of thermal fluctuations
[3]. Here, we study the interaction of these patterns with the
periodicity imposed by the simulation cell size. We fix the density of
the system to n=0.1 and increase the system size so that the number of
holes at fixed density increases from N=25 to N=961. For each system
size, we measure the average length of a side of the square box that appears
in the checkerboard hole pattern.
We denote this quantity Lp. We also count the
number of boxes in the checkerboard pattern across the system in the
x direction.
Figure 1:
Length of the side of a single box in the pattern, Lp, versus the
number of boxes that tile the x direction of the system. The data
were obtained by simultaneously increasing the number of holes N and
the box size L to maintain a fixed hole density n=0.1.
Figure 2:
Images of the holes in the presence of thermal fluctuations for a
system with N=625 holes at increasing hole density. Lines indicate
the trajectories of the holes over a fixed period of time.
(a) n=0.1, (b) n=0.11, (c) n=0.12, (d) n=0.125.
In Fig. 1 we plot
Lp versus the number of boxes present in the system along the x
direction. We find that the box size remains close to an average size that is
determined by the hole density n, but that it fluctuates in order to maintain
an integer number of boxes in the pattern. These fluctuations are
particularly pronounced at smaller N when the number of boxes present
in the pattern is not large. This result indicates that the pattern is
locking into the periodic structure provided by the simulation cell.
As the number of holes becomes large, the locking becomes ineffective and
we begin to observe noninteger numbers of boxes per side. In these larger
sized systems, the box size is determined by the intrinsic parameters of
the simulation and not by the simulation cell. In order to accommodate any
discrepancy between the simulation cell and an integer number of boxes, the
system develops grain boundaries within the pattern.
Examples of grain boundaries in the presence of thermal fluctuations are
illustrated in Fig. 2 for several hole densities in a system with
N=625 holes. This system is large enough to allow the native box size
to emerge. We find a variety of different configurations for the grain
boundaries. As shown in Fig. 2(c,d), at higher densities we find
coexistence between the box pattern and an incipient anisotropic Wigner
crystal pattern.
Recent
scanning tunneling measurements have observed checkerboard type
structures [4]. These
checkerboard patterns do not show complete ordering throughout the
sample but occur in patches.
It is not known whether this disordering is simply due to an
incommensuration effect with
the substrate or to additional random pinning.
Our results suggest that incommensurations
can produce domain wall type disorder.
Future directions will be to characterize these
domain walls and to add the effects of random point disorder.
This work was supported by the U.S. DoE under Contract No.
W-7405-ENG-36.
