Critical parameters for the disorder-induced metal-insulator transition in FCC and BCC lattices
Andrzej Eilmes1, Andrea M. Fischer2 and Rudolf A. Römer2
1Department
2Department
of Computational Methods in Chemistry, Jagiellonian University, Ingardena 3, 30-060 Krakow, Poland
of Physics and Centre for Scientific Computing, University of Warwick, Coventry, CV4 7AL, United Kingdom
BCC
Summary
Introduction
We use a simple localised spin model to study how the magnetic
Compositional
disorder is an
important
of real We
crystalline
properties
of ferromagnets
depend
uponproperty
temperature.
derive
solids.
It provides
a mechanism
for the localisation
valence
the
temperature
dependence
of magnetisation
usingofboth
the
electrons,Heisenberg
without which
insulating
states
wouldmechanical
not be possible.
classical
model
and the
quantum
Ising
The disorder-induced
metal-insulator
transition (MIT) anisotropy
has been
model.
For a system with
uniaxial magnetocrystalline
studied, we
extensively
for over
forty years.
To theitsbest
of our
(MCA)
adopt a mean
field approach
to calculate
anisotropy
knowledge
critical parameters
have only
for
energy
(thethe
difference
in free energy
whenbeen
it iscalculated
magnetised
the simple
(SC) latticeto[1].
surprising,
it is the
the
parallel
andcubic
perpendicular
its This
easyisaxis).
We since
see that
body centred
cubic (BCC)
(FCC)
lattices,
single-ion
anisotropy
model and
doesface
not centred
give thecubic
correct
temperature
which are prevalent
nature. To
fill thisforgap
we usetemperature
a transferdependence
of the in
anisotropy
energy
a large
matrix method
(TMM)
andthe
finite
size scaling
(FSS)ifto
calculate
the
range.
However,
we find
correct
dependence
the
anisotropy
critical
parameters
for the MITconstant
in the BCC
and FCC
latticesthe
[2]. two
is
present
in the exchange
instead.
Allowing
different kinds of anisotropy to favour different directions of
magnetisation, enables us to model a temperature induced spin
Themain
model
reorientation transition. Our
effort involves extending our
model to more than one layer, seeking agreement with results for
We use the standard Anderson Hamiltonian:
layers of cobalt obtained using an extension of the relativistic
disordered local moments scheme [1].
H    i i i   tij i j .
Data analysis – finite size scaling
The MIT is characterised by a divergent correlation length, so at
fixed energy E,
We performed the TMM calculations for system
sizes up to M=15, though we were restricted to odd
M for the BCC lattice, due to singular connectivity
matrices.
Fig.3. Reduced localisation length versus
disorder for the BCC lattice. System sizes M are
3(h),5(z),…,(15). The curves are the best fit for
the model n  3, n  2, m  3, m  1.
r
i
r
i
Results of the TMM calculations for the BCC lattice
can be seen in Fig.3.. Since the curves do not cross
at a single point, irrelevant scaling is required i.e.
ni , mi  0 in (3) and (5). Fig.5. shows the scaling
results for nr  3, ni  2, mr  3,It m
has
two branches
i  1.
for extended and localised states as expected for the
MIT. The inset shows the scaling parameter which
diverges at critical disorder. Averaging over the
models providing the best fits gives the critical
parameters as:
(1)

Fig.5. Scaling function (solid line) and scaled
data points for the BCC lattice and nr  3, ni  2,
mr  3, mi  1.
Averaging over the best fit models gives the critical
parameters:
Wc  26.72  0.01,   1.53  0.01.
Fig.4. Reduced localisation length versus
disorder for the FCC lattice. System sizes
M are 3(h), 4(z),…,15(). The curves are
the best fit for the model nr  2, mr  2.
Concentrating on the first of these cases, we define the reduced
localisation length,  M (W ) :  (W ) / M . The one-parameter
scaling law states  M (W )  f M /  (W ) (3) [3], which means
that the reduced localisation lengths calculated for different system
sizes can be made to collapse onto a single curve. The FSS
obtains the scaling curve such that deviations of the data from it
are minimised. The critical parameters can then be obtained by
fitting
. We use a slightly altered version of (3) namely
~
transfer  matrix
Fig.6. Scaling function (solid line) and scaled
data points for the FCC lattice and nr  2, mr  2.
ni
M    M
Hi
Ci
i
 2D Hamiltonian of the i th slice.
 connectivity matrix describing the connections between
the i th and i  1th slices.
 coefficient of i in the expansion of the wavefunction
in the orthonormal basis.
The eigenvalues of the product of transfer matrices
 L : TLTL1...T2T1
Fig.7. Phase diagram for BCC lattice.
converge to the Lyapunov exponents as L   . We use the
localisation length  of the electronic wavefunction to determine
whether a state is localised or extended and this can be obtained
as the inverse of the smallest Lyapunov exponent.
Fig.7. and Fig.8. show the phase diagrams for the BCC and FCC lattices
respectively. Originally a grid of W versus E values was created with low
accuracy data and each point was assigned the status ‘localised’ or
‘extended’ by comparing the localisation lengths for system sizes M=7 and
M=9. We obtained curves by averaging over the extended and localised
points nearest to the boundary separately and using a spline fit. The region
enclosed by these two curves should thus contain the phase boundary.
Points ‘inside’ the phase boundary are extended and those ‘outside’ are
localised. For low values of disorder, there are more fluctuations in the
Lyapunov exponents, so we did not obtain data accurate enough to use in
this region. The points calculated with more accurate data (see sections
above and below) have been included as well as points that can be derived
analytically at zero disorder. An interesting feature is the symmetry around
the E=0 axis for the BCC lattice, which is absent for the FCC lattice. This is
due to the fact that the BCC lattice is bipartite, whereas the FCC lattice is
non-bipartite.
i 0
r
~

fn r M
ny
1

,
1

(5)
(6)
can also be expanded:
n 1
mi
i (w)   cn wn
(7)
n 0
with
and
.
are
chosen
but are kept
nr , low
ni , to
mr minimise
and mi the
b1 toc0 give
 1 the wbest
 (Wfit,
c  W ) / Wc
number of parameters. The non-linear fit is performed using the
Levenberg-Marquardt method [4].
Conclusion
Fig.8. Phase diagram for FCC lattice.
The results obtained for the critical exponent 
are in
agreement with previous results obtained for the SC lattice [1].
This was expected since they are in the same orthogonal
universality class.
At the barycentre there is an increase in the critical disorder from
the SC to the BCC to the FCC lattices. This is due to the number
of nearest neighbours, which are 4, 8 and 12 respectively. It is
intuitive that the greater the number of paths available for
transport, the higher the disorder required to localise an electron.
Calculations were also performed away from the band centre, fixing the
disorder and allowing the energy to vary.
The TMM calculations took a lot longer to converge to the desired
accuracy.
References
1. K. Slevin and T. Ohtsuki, Phys. Rev. Lett. 82, 382 (1999).
Ec  10.96  0.03,   1.47  0.09.
2. Andrzej Eilmes, Andrea M. Fischer, Rudolf A. Römer, submitted
to Phys. Rev. B (2008), arXiv: 0801.3338
For W=20 they are
Fig.9. Results for BCC lattice with
W=17.5, nr  2, mr  1 and no irrelevant
scaling. Symbols have same meaning as
in Fig.3.
Fig.2. FCC lattice structure.
i
i
r
 r (w)   bn wn ,
The results for the BCC lattice for W=17.5 can be seen in Fig.9. For
W=17.5 the critical parameters obtained are
Fig.1. BCC lattice structure.
and
mr
Critical parameters away from the band centre
The lattice structures
nr
n
i
f n   ani  M .
Phase Diagrams
Ti
(4).
and
 i and provide non-linear corrections.
respectively
also
curves
accounts for a shift in the point at which the
cross.
i
Taylor expanding (4) gives:
 M (W )
~
 i 1   C ( E1  H i )  C Ci   i 


  

.
(
2
)




0  i 1 
 i  1



r

(W
)
are the relevant Mand irrelevant scaling variables
n 0
1
i 1
1

However y < 0, so as M becomes larger (2) approaches the form
of (3). Close to the transition
is approximately linear.
The transfer-matrix
e method
1
i 1

 M (W )  f  r (W ) M ,  i (W ) M y
im e
We use a quasi-1D bar of width M and length L>>M. The
Schrödinger equation with the Hamiltonian given in (1) can be
written in the TMM form:

are the critical disorder/ critical energy at which a transition occurs
and  is the critical exponent.
TMM results for the FCC lattice are shown in Fig.4..
Here the curves cross at roughly the same point, so
irrelevant scaling is unnecessary. Fig. 6. shows the
scaling results for the case
nr  2, mr  2.
tij    ij .  i is the random onsite potential distributed uniformly
and at fixed disorder W,  ( E )  E  Ec
,
 (W )  W  Wc
where  is the correlation length for an infinite system, Wc / Ec
Wc  20.85  0.01,   1.61  0.01.
The set of orthonormal states i correspond to Wannier orbitals
for atoms located at different sites i in the lattice. t ij is the hopping
integral; we consider nearest neighbour hopping only, taking
between –W/2 and W/2. Thus the size of W provides a measure
of the disorder strength.
FCC
Critical parameters at the band centre
i j
i
Results
Ec  11.18  0.06,   1.49  0.11.
The results for the FCC lattice with W=18 can be seen in Fig.10.. The
critical parameters obtained are
Ec  8.684  0.002,   1.63  0.02.
Fig.10. Results for FCC lattice with W=18,
nr  1, mr  2 and no irrelevant scaling.
Symbols have same meaning as in Fig.4.
3. D.J. Thouless, Phys. Rep. 13, 93 (1974).
4. W.H. Press, B.P. Flannery, S.A. Teukolsky and W.T. Vetterling,
Numerical Recipes in FORTRAN, 2nd ed. (Cambridge University
Press, Cambridge, 1992).