C.J. Olson Reichhardt, C. Reichhardt, D.L. Smith, and A.R. Bishop
Theoretical Division,
Los Alamos National Laboratory, Los Alamos, New Mexico 87545, U.S.A.
(Received 18 November 2002; published 1 July 2003)
We examine the effect of a field-effect transistor (FET) geometry on the charge ordering phase diagram of transition metal oxides using numerical simulations of a semiclassical model including long-range Coulomb fields, resulting in nanoscale pattern formation. We find that the phase diagram is unchanged for insulating layers thicker than approximately twice the magnetic correlation length. For very thin insulating layers, the onset of a charge "clump" phase is shifted to lower values of the strength of the magnetic dipolar interaction, and intermediate diagonal stripe and geometric phases can be suppressed. Our results suggest that, for sufficiently thick insulating layers, charge injection in an FET geometry can be used to experimentally probe the intrinsic charge ordering phases in these materials.
DOI: 10.1103/PhysRevB.68.033101 PACS number(s): 71.10.Hf, 73.50.-h
Charge ordering in doped transition metal oxides has attracted considerable recent interest, both in theory and experiment. Due to the competing long-range, e.g., Coulomb, repulsion and short range antiferromagnetic interactions in the charge system, a rich variety of phases can occur, including stripes 1,2, clumps 3,4, and liquid crystalline electron states 5. Simulation studies of the charge ordering phase diagram6,7 showed transitions among four phases depending on the hole density and the strength of a dipolar interaction induced by the holes: a Wigner crystal at low hole densities, a diagonal stripe phase, an intermediate geometric phase in which the charges form continuous filaments in a square checkerboard pattern, and a clump phase at larger dipole interaction strengths. The behavior of these phases is of interest not only for the charge ordered system, but also for their similarities to other pattern forming systems with coexisting short- and long-range interactions, including magnetic films,8 Langmuir monolayers, polymers, gels, and water-oil mixtures.9 A natural extension of this model is to consider charges interacting with a distortable charged membrane, which could be relevant to active membrane systems such as ion pumps.
To experimentally probe the charge ordering phase diagram, the hole doping of the material must be controlled. A recently proposed method of controllably varying the hole density is the use of a field effect transistor (FET) geometry to inject holes into the metal oxide plane. The geometry is illustrated in Fig. 1. An insulating layer is deposited on top of the metal oxide, and then a metallic gate is deposited on top of the insulator, forming a capacitive structure. By varying the gate voltage, holes move into or out of the metal oxide layer, allowing the sample to be conveniently tuned to the desired doping level10 Source and drain contacts can be used to probe the conductance properties of the structure. A potential drawback of this geometry for exploring the phase diagram is that the holes can interact with the gate layer, and complicate the intrinsic hole-hole interactions within the metal oxide which lead to the charge ordered phases: the phases could be distorted or disrupted by the presence of the FET geometry.
To asses the effect of an FET geometry on the charge ordering phases, we simulate a model of a single metal oxide layer interacting with a metallic gate layer that is offset by varying thicknesses of insulating material. We find that for a sufficiently thick insulating layer, the charge ordering is unaffected by the presence of the gate. When the insulating layer thickness approaches twice the magnetic correlation length in the metal oxide, however, we find a sharp downward shift in the onset of the clump phase as a function of dipole interaction strength. The diagonal stripe and geometric phase boundaries do not shift, but these phases can be suppressed by the intrusion of the clump phase as the insulating layer is made thinner. Our results show that, for sufficiently thick insulating layers, the FET geometry should provide a reliable probe of the charge ordering phase behavior.
We consider a sample constructed in an FET geometry, illustrated in Fig. 1. The metal oxide plane is parallel to the insulating layer and also to the metallic gate layer deposited on top of the insulator. Experimentally, a gate voltage would be used to tune the doping level present in the metal oxide layer. We simulate this effect by directly varying the hole density in our system, which is a rectangular computational box of size Lx x Ly, with Lx, Ly up to 100 unit cells in a CuO2 plane. At the beginning of each simulation, we place the holes at random and assign to each hole a magnetic dipole moment of constant size, but random direction. We find the minimum of the total potential in this system using the efficient Monte Carlo method described in Ref. 7.
Our model for the interactions between the charges is based on the spin density wave picture of the transition metal oxides. Full details of the model can be found in Refs. 6 and 7, but we summarize the essential details here. We use a numerical approach to simulate holes moving in an antiferromagnetic (AF) insulator in the presence of long-range Coulomb forces. We study a quasiclassical limit of the quantum problem of holes in an AF spin environment, using the model in Refs. 6 and 7, which incorporates the essential correlations for an effective hole-hole interaction. The AF background is integrated out, and the focus is on the charge subsystem. In the absence of disorder, this model produces four phases depending on the density of holes and the characteristic AF energy scales: a Wigner crystal, diagonal stripes, horizontal-vertical stripes (loops), and a clumped phase.
Our numerical approach utilizes the spin-density-wave (SDW)
picture of Schrieffer, Wen,
and Zhang,11
which is closely related to the semiclassical
approach to the model of Shraiman and Siggia
12
in which the interaction between doped holes stems from the spiral
distortion of the local Neel vector near a hole.
The SDW picture of layered transition metal oxides
has been successful in describing the stoichiometric insulating
AF phase of these systems at low
temperatures.11
In this
model, an electron moves in the self-consistent
staggered field of its spin. The electronic band is split into
upper and lower Hubbard bands13
separated by the
Mott-Hubbard gap, . At half filling the
upper band is empty and the lower one is full. We consider a system
at Tll
/kB
that is doped with holes with planar density
s, and focus
on the lower band which has a maximum at
ki=Q/2 = (
1,
1)
/2a,
where a is the lattice spacing.
A mobile carrier in an antiferromagnet produces a dipolar distortion
of the magnetic background as described
Appendix B of Ref. 7.
Thus, at
finite density, the holes interact via two different mechanisms: a uniform
short-range attractive force due to AF bond breaking, and a long-range
magnetic dipolar interaction.
The latter term is due to the long-range spiral distortion of the AF
background, which is a consequence of quasiparticles interacting with
soft (Goldstone) modes of the spin
system12,14. The
magnetic dipole moment associated with each hole is due to the coherent hopping
of holes between different sublattices and scales with the AF
magnetic energy.
The two-hole interaction Hamiltonian
has a Fourier transform, for quasiparticle momenta near ki, of
Here, ri is the coordinates of a hole in units of a,
r=|r1-r2|
is the relative hole-hole distance,
is a spin-density operator,
z(
)
are Pauli matrices, and
di is a unit vector in the direction of the dipole moment
of the hole.
In strictly two dimensions at finite T, the system is magnetically
disordered, characterized by a finite magnetic correlation length
(see Ref. 15),
and the range of the dipolar interaction between
the holes, mediated by the AF background, is also of order
.
Thus, at finite T the magnetic
dipolar interaction between the holes, mediated by the AF background, is
actually short-ranged. The holes also interact via the long-range Coulomb
interaction, which we take to be unscreened,
as appropriate at low doping where
rs=r0/a0 is
very large: r0 is the mean interparticle distance,
a0 is
the Bohr radius, and rs
8.
Each hole also carries a spin degree of freedom,
but it can be shown 6,7
that the overall spin energy is minimized in the
spin antisymmetric channel, as we assume here.
Hence we neglect the spin-symmetric channel and
only consider the charge channel with an effective (magnetic in
origin) interaction between two holes, 1 and 2, a distance
r apart in a single metal oxide plane, in the form
(2)
Here, q is the hole charge,
is the angle between r and
a fixed axis, and
1,2
are the angles of the magnetic dipoles
relative to the same fixed axis, which we assume can take an arbitrary
value.
A is the strength of the short-range anisotropic interaction,
and B is the strength of the magnetic dipolar interaction
[B
A/(2
)],
which we assume to be independent
variables.
In real cuprate oxide materials, B should be of order
~ 1 eV.
Throughout this work we take A=0.
Since the isotropic attractive interaction due to the second term in Eq. 2
is extremely short-range (in fact in an infinite
system it is a
function),
it is initially reasonable to
set A=0 and explore the behavior of the system as a function
of B.
The magnetic correlation length
is obtained from
neutron scattering measurements16, and here
we assume the approximate dependence
=3.8/
.
The doping level n is defined as the hole density measured
in units of the cuprate lattice spacing. Thus
n=1% corresponds
to 1 hole per 100a2, where
a
3.8
.
We have assumed for numerical convenience that
r can be
relaxed from a crystal lattice position to an arbitrary (continuous)
value.
The results presented here are for a system with 196 holes
with size ranging from
307.15
x
307.15
(for n=3%) to
137.4
x
137.4
(for n=15%).
The value of
then ranges from
21.9
for n=3% to
9.8
for n=15%. The sample is periodic in the x-y
plane17.
In the FET geometry, a metal oxide channel is created,
then an insulating layer of thickness d is deposited on top
of the channel, and finally a layer of metal is deposited to serve
as a gate. The interactions between
holes in the metal oxide layer is altered by the presence of image
charges in the gate layer,
(3)
giving the Coulomb energy between charges a distance d into
the material as
(4)
Thus, we modify the Coulomb interaction between the holes in our system
to this form. This introduces a new length scale 2d.
For comparison, we also ran simulations with the
unmodified Coulomb interaction, representing a bare metal oxide plane
without the FET gating.
In the absence of the FET interaction, we find a phase diagram
consistent with that observed in Ref.6, as illustrated
in Fig. 2(a).
For thick insulating layers,
when the FET interaction is included, the locations of the phases
are not affected and we obtain
the same phase diagram, as shown in Fig. 2(a).
As we decrease the thickness of the insulating layer, however, we find a
crossover thickness dc below which the phase boundaries begin
to change. Figure 2(b)
illustrates the phase diagram for the same system as in
Fig. 2(a)
but with the FET term added and with an insulating layer
of thickness d < dc
(d=14). The onset of the diagonal stripe phase
"S" is unaffected, but there is a
large shift downward to BC of the onset
of the cross-like
clump phase "C." The size of the downward shift,
B,
increases as the hole density n decreases. The geometric phase that
was present without the FET term is now completely suppressed.
The interplay of the correlation length and the
insulator thickness affects the onset
of the clump phase because only the clump phase possesses a characteristic
length
scale of order 2.
This is illustrated in Fig. 3, which shows
the clump
structure at n=13% and n=3%.
When the length
scale of the insulating layer is comparable with the length scale of
the clump structure, the interaction with the metal gate above the
insulating layer becomes comparable with the interaction with
neighboring clumps, and the transition to the clump phase is enhanced.
The effectiveness of the FET term extends only to
insulator thicknesses d that
are approximately twice the magnetic correlation
length,
d2
.
This is illustrated
in Fig. 4 for a sample with n=3% and varying insulator
thicknesses d. The arrow indicates the saturation of the clump
phase onset BC at
dc to the value of
B=BC0 observed in the absence of the FET
term. The other phase boundaries, Wigner to stripe
and stripe to geometric, shift very little
with d but instead the intermediate
phases are suppressed when BC moves below the
onset values for these phases. To assess
whether the magnetic correlation length
is responsible for the shift in the phases, we considered a
system at n=13%, which normally has a correlation length of
=10.54
,
and artificially changed the screening length to
=20
for
one series of runs, and to
=5
for a second series. In the
inset of Fig. 4, we show the cutoff insulator thickness
dc beyond which
the FET term has no effect as a function of correlation length
, both
for the normal screening lengths and for our two artificially changed
screening lengths. We indeed find a linear dependence for the
cutoff thickness on the correlation length, viz.
dc
1.7
.
In conclusion, we have found that the FET geometry does not affect the clump ordering phases unless the insulating layer is thin enough, namely less than approximately twice the magnetic correlation length. For thin insulating layers the onset of the clump phase is enhanced. This suggests that the FET geometry with a sufficiently thick insulating layer is suitable for studying the effects of hole concentration on the charge ordering phases because the presence of the FET does not alter the hole-ordered structures. Alternatively, FET geometries deliberately created with thin insulating layers can be used to probe the clump phase at higher hole densities, where the underlying value of B may preclude reaching the clump state in the bulk phase.
This work was supported by the U.S. Department of Energy under Contract No. W-7405-ENG-36.
Back to Home CJOR
Back to Home CR
Last Modified: 7/18/03