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University of Cambridge > Department of Physics. Cavendish Laboratory > Theory of Condensed Matter > CoMePhS |
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Manganites Phenomenological Theory of Doped
Manganites We start with the free
energy of the commensurate-incommensurate phase transition
With the order parameters,
ρ(r) the charge density, Ф(r) the incommensurate part of
the charge density. These come from
the combined order parameter If we make the phase
modulation only approximation, that is where ρ(r) is constant, this leads to solutions of the form
for n>2. This is the sine-gordon equation of a
forced harmonic oscillator. The
solutions are jacobi amplitude functions which have jumps (solitions). The
jumps are as a result of the comptetition between the elastic and umklapp,
the last two terms in the free energy respectively All this can be seen in
many sources, for example the book Tolèdano and Tolèdano “The Landau Theory
of Phase Transitions” (World Scientific 1987). Including Magnetisation To include mangetisation
we must add to the free energy
And must allow the charge
ordering to interact with magnetic ordering and so we must include some
coupling terms to the free energy, To
lowest order these are
It is worth noting that
the second of these terms is linear,
that is because the sign of The new free energy can be
minimized numerically and if one sets the magnetic coupling to the free
energy to be small, so d1
<< d2 and am and bm much less than the
corresponding parameters in the charge density we find a very good agreement
with the transitions observed in experiment.
Figure 1a shows the
experimentally measured incommensurate wave vector, scaled by the reciprocal
lattice vector against temperature for different dopings of two different
manganites. The antiferromagnetic
transition temperature and charge ordering temperatures are shown as TAF
and TCO respectively.
This data was gathered from a number of experimental sources which are
acknowledged.
The inset graph shows the
low temperature incommensurate wave vector as a function of the doping, That
is the x in La1-x Cax MnO3 or the Pr1-x Cax MnO3.
The phenomenalogical model
reproduces the experimental data, as can be seen in Figure 1b. TL represents the “lock-in”
temperature, that is the temperature at which the incommensurate-commensurate
transition takes place. TM
is the temperature at which the magnetisation goes to zero, which we
associate with the antiferromagnetic transition in the experimental
data. The lines a to e on the
graph correspond to different dopings and hence different values of x. These lines are also shown on Figure 2
which shows the theoretical phase diagram with temperature and doping, so
each line a-e corresponds to a vertical line on the phase diagram. The inset graph in Figure
1b sumarises the low temperature incommensurate wave vector as a function of
doping and can be seen to be in very good agreement with the experimental
summary inset in Figure 1a. The complete theoretical
phase diagram is presented below:
Figure 2 Beyond the Phase Modulation Only
Approximation
One can use the same free
energies with the additional enhancement of allowing magnetisation and the
charge modulation to vary spatially.
Rich textures and interplay between these parameters can be
found. In example (a) we see a
magnetic domain wall. The
magnetisation, green, changes sign. A
consequence of the phenomenalogical model is that the charge modulation will
be enchanced as can be seen in the rise in the red line. So we expect a rise in charge density in a
magnetic domain wall which may explain their anomalously high resistance, In (b) we see a
discomensuration. The blue line
corresponding to the incommensurate part of the wave function undergoes a
jump from one value in phase with the lattice to another. As the discommensuration occurs we can see
that the magnetisation is enhanced and the charge modulation amplitude
surpressed. For those interested more
detail can be found in the paper GC Milward, MJ Calderon & PB Littlewood Electronically
soft phases in manganites; Nature 433
(Feb 2005) 607-610 |
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