Quantum Hall effect in relaxed TMTSF2 ClO4

1
Quantum Hall effect in relaxed (TMTSF)2 ClO4
Danko Radic and Aleksa Bjeli Department
of Physics, Faculty of Science, University of
Zagreb, POB 162, Bijenicka 32, 10001 Zagreb,
Croatia
e-mail dradic_at_phy.hr
Abstract We present the analysis of quantum Hall
effect in the transversally dimerized
quasi-one-dimensional system, namely relaxed
(TMTSF)2ClO4. The distinct feature of presented
system is, according to the recent experiments
1, large dimerization gap in transverse
direction due to anion ordering. The solution of
one-electron problem is exact, obtained by
treating quasi-one-dimensional band, transverse
dimerization potential and magnetic orbital
effects on equal footing.4 The hybridization of
inter-band and intra-band correlations is
followed through rigorous matrix susceptibility
calculation within the framework of random phase
approximation. Two types of orderings in the
absence of magnetic field are possible SDW0
(inter-band channel) and SDW (intra-band
channel).3 There is a strong first order
transition between low field FISDW like SDW0
state into high field SDW state induced by
magnetic field that attains almost constant
plateau of critical temperature vs. magnetic
field.4 The analysis of the quantum Hall
effect, obtained here by using the generalized
Thouless-Kohomoto procedure 6 adapted to the
dimerized system, shows the odd-integered
quantization of Hall conductance in the field
induced cascade of SDW0 type, and non-quantized
oscillations of Hall conductance with increasing
magnitude in magnetic field.
I. Model quasi 1-dimensional system with
transverse dimerization
Crystal structure of (TMTSF)2ClO4
Fermi sheets
with anion dimerizing potential (V?0)
standard model (V0)
a, b molecular chains exposed to homogeneous
anion potential V, V vF Fermi velocity t a?b
inter-chain hopping p electron momentum A
vector potential GebH/h magnetic wave
vector ti Pauli matrices bond/antibond space
inter-band dimerization gap 2V ? magnetic
breakdown junction ? QUANTUM
OSCILLATIONS!
Anion ordering QAO(0,1/2,0) TAO24K
Relaxed (TMTSF)2ClO4 sample
1-e Hamiltonian
finite magnetic field H?0
III. Susceptibility
II. Magnetic breakdown (MB) solution
3
(I) Magnetic field 0
Interaction Hamiltonian
4
non-logarithmic part of c1
Spin density operators H homogeneous A
alternating
k electron momentum vF, kF Fermi velocity
and momentum wc vFG vFebH/h
magnetic energy N?Z
cij ltMiMjgt SDW susceptibility
RPA Susceptibility matrix
Splitting between - bands oscillates with
magnetic field due to oscillations of Floquet
parameter thus producing the quantum oscillations
in one-electron and other properties.
4
(II) Magnetic field ? 0
oscillations of d(H,tb,V)
Associating oscillations of d, periodic with H-1,
with rapid oscillations of magneto-transport and
thermodynamic properties observed in (TMTSF)2ClO4
(t300K, vF2105m/s, b7.710-10m), the best fit
for dimerizing potential is V/t0.8.
SDW instability criterion
wave vector of susceptibility maximum
IV. Phase diagram QHE
CONCLUSION The exact solution of quasi 1-D band
with transverse inter-band dimerization gap in
magnetic field, analogous to Stark electron
interferometer 5, is obtained using Floquet
theory and fundamental matrix method for Hills
problem with simple periodic coefficients.
Presence of quantum oscillations approximately
periodic with inverse magnetic field is shown at
the level of one-electron spectrum (without
closed electron orbits). Results are confronted
to semiclassical solution that turns to be valid
only in the limit of strong magnetic field and
weak dimerizing potential vs. transverse band
width. Solutions are applied to relaxed
(TMTSF)2ClO4 where rapid oscillations in
magneto-transport and thermodynamic quantities,
with frequency of 260T vs. inverse magnetic
field, are observed. We find the best fit for
dimerizing potential at 80 of electron
interchain hopping energy. Inter-band and
intra-band correlations introduced by electron
on-site energy dimerization, as well as their
hybridization, are treated through rigorous
matrix susceptibility approach within the
framework of random phase approximation. The
resulting phase diagram in the zero magnetic
field shows two types of orderings SDW0
(inter-band channel) for small values, and SDW
(intra-band channel) for large values of
dimerizing potential, while in the range of
intermediate dimerizing potential, of order of
interchain hopping, there is metal state stable
down to zero temperature. In finite magnetic
field the unordered state undergoes FISDW
instability, with present competition between
SDW0 and SDW orderings. The low field phase is
integer-quantized FISDW cascade of SDW0 type,
while the high field phase is nonquantized SDW
with strong first order transition between them
induced by magnetic field. The analysis of the
quantum Hall effect is done using the generalized
Thouless-Kohomoto procedure 6 adapted here to
the dimerized system. The result shows the
odd-integered quantization of Hall conductance in
the field induced cascade of SDW0 type. On the
other hand, within the high field SDW phase,
Hall conductance is not quantized any more, but
undergoes oscillations increasing in magnetic
field. Both of them are followed by the
corresponding behavior of longitudinal component
of SDW wavevector. Such type of behavor indicates
that SDW0 is a good candidate for the low
temperature subphase below 27T, where Hall
conductance is quantized, while the nonquantized
phase beyond 27T is of SDW type.
Experimental findings
1
Theoretical results
4
Hall effect
6
Thouless-Kohomoto procedure
nonquantized
SDW
(N even)
SDW0
(N odd)
integer-quantized
(a) Experimental phase diagram of (TMTSF)2ClO4,
the SDW critical temperature vs. magnetic field
after the low field FISDW state, the high field
final SDW state takes place with strong 1st order
transition at 7.5T. There are two (or even more)
subphases, i.e. SDW I and SDW II, enclosed within
the high field state, being separated by phase
boundary ending up at 27T for low
temperatures. (b) The Hall resistance is
quantized by integer number N1 in SDW I
subphase. In SDW II subphase, that takes place at
27T, Hall resistance is not quantized any more,
but undergoes vast oscillations increasing in
magnitude vs. magnetic field.
(a) The metal-SDW phase diagram calculated for
the best fitting choice of V/t0.85 (it is used
in other figures as well). FISDW cascade below 8T
is odd-integer quantized SDW0 state, while the
high filed state is SDW, not characterised by
integer quantization. The first order phase
transition from SDW0 cascade into SDW is induced
by magnetic field. (b) The oscillations of energy
splitting between sub-bands in one-electron
spectrum quantum oscillations appearing as
consequence of electron tunneling through the
dimerization gap. Inter-band processes are not
affected by energy splitting, while the
intra-band ones are, and thus are all quantities
connected with them, i.e. SDW state in high
field phase diagram and Hall effect within
it. (c) The longitudinal component of SDW
wavevector in SDW state shows oscillations
increasing in magnetic field, thus implying the
nonquantized Hall effect. (d) The longitudinal
component of SDW wavevector in SDW0 state is
linear in magnetic field, giving the
odd-integered quantization of Hall effect within
FISDW subphases. 7
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