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Surface Science 229 ( 1990) 16-20
North-Holland
SPIN-REVERSED
GROUND STATE AND ENERGY GAP
IN THE FRACTIONAL QUANTUM HALL EFFECT
Tapash CHAKRABORTY
Max-Planck-Institut ftir Festkijrperforschung, Heisenbergstrasse I, D- 7000 Stuttgart SO, Fed. Rep. of Germany
Received
11 July 1989; accepted
for publication
14 September
1989
The role of reversed spins in the fractional quantum Hall effect is briefly reviewed. The ground-state
spin configurations
for various
tilling fractions and the spin-reversed
quasi-particles
are discussed. Some new results for the spin-reversed
excitations are presented and
compared with recent experimental
results.
1. Introduction
In this paper, I shall review the current status of
one of the interesting aspects of the fractional quantum Hall effect (FQHE) - the role of reversed spins
in the incompressible many-electron ground state and
in the quasi-particle (QP )-quasi-hole (QH ) energy
gap [ 11. Some new results obtained recently in collaboration with Pietilginen will also be presented [ 21.
Although the theoretical
work on this important
problem began soon after the original experimental
discovery
[ 31 of FQHE and its explanation
by
Laughlin [4], the experimental
support of the theoretical predictions
on reversed spins [ 5-71 came
only recently [ 8,9].
In the standard theory (for a review, see refs.
[ 1,5] ) developed to explain the FQHE at u= 1/m
filling of the lowest Landau level (m is an odd integer) [ 4 1, electrons were considered to be in a fully
spin-polarized
state (here V= 27&p is the filling factor of the Landau level, p is the area1 density of twodimensional
electrons, and lo = (fic/&) ‘I2 is the
magnetic length). Considering the high magnetic field
involved at these filling fractions, such a choice was
quite reasonable. For the higher-order
filling fractions observed in the experiments, this approach requires modifications.
For example, at + filling of the
lowest Landau level, Halperin [ 51 proposed a spinunpolarized
ground-state
wave function. The small
g-factor in GaAs prompted Halperin to predict that,
when the magnetic field is not too large, such a state
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might be energetically preferred. Subsequent calculations revealed that in the absence of Zeeman energy, this state has indeed the lowest energy [ 1,6].
It was also found that, even for a fully spin-polarized
ground state, one can obtain spin-reversed
excitations [ 71, which might manifest itself in the linear
behaviour of the activation energy observed in the
experiment
[ lo] (the Zeeman energy is linear in
magnetic field B, while the Coulomb energy goes as
B”‘).
2. Ground state
The results to be described below are based on the
calculations done for finite electron systems in a periodic rectangular geometry [ 111, suitably modified
to deal with spin [ 6 1. The total energy of the twodimensional
interacting electron system consists of
a Coulomb term (which depends on the spin quantum number S, and the component of magnetic field
normal to the electron plane), and also the Zeeman
term. The ground state energy is calculated for different S by numerically diagonalizing
the Hamiltonian. The spin state having the lowest energy will be
the preferred ground state. In table 1, I present the
ground state energy (in units of e2/&, E being the
background dielectric constant ) for a few filling fractions. Similar results for other filling fractions are
available in our earlier work [ 6 1. The Zeeman energy is not included in these results.
T. Chakraborty/Spin-reversed
Table 1
Potential energy per particle for four-electron systems (for f and
$) and six-electron systems (for 4 and 3) for various values of spin
polarization; the Zeeman energy is not included
v Potential
s=o
f -0.4135
Ground
energy
S=l
s=2
s=3
state
f
!
-0.5331
-0.4464
-0.4120
-0.5291
-0.4410
-0.4152
-0.5257
-0.4403
-0.5232
-
Polarized
Unpolarized
Unpolarized
3
-0.5074
-0.5096
-0.5044
-0.50104
Partially polarized
1-l
ground state and energy gap in FQHE
It is interesting to note that even allowing spin degrees of freedom, the ground state at Y= 4 is fully
spin-polarized
(the inclusion of the Zeeman energy
would make this state even more favourable relative
to the other spin states). This is in line with the fully
antisymmetric
wave function proposed by Laughlin
[ 41 for Y= f . For the other tilling fractions, v=p/q
with q odd and p> 1, spin-reversed
states are found
to be energetically favourable over the fully spin-polarized states in the absence of Zeeman energy
[ 6,121. A pictorial representation
of the low magnetic field spin assignments
obtained theoretically
for various tilling fractions can be given as
netic field is tilted from the direction normal to the
electron plane [ 8,9]. Dramatic changes occurring at
different filling factors with tilt angle are ascribed [ 81
to various spin polarizations
consistent
with the
above predictions.
In the other experiment
[ 9 1, a
sharp change in the dependence of the activation energy on tilt angle was observed for v=g (electronhole symmetric to v=$). This is described as a transition from a spin-unpolarized
ground state at small
angles to a polarized state at larger angles. The linear
behaviour of the activation energy at the two ground
states were identified with the Zeeman energy, and
they appear because of the presence of spin-reversed
quasi-particles
and quasi-holes [ 15 1, to be discussed
below.
In fig. 1, I present the results for the ground-state
energy (per particle) for (a) v=$ and (b) v=$,
-0.4445
-’
-0.4455
-
\
I
I
Ground
v=2/5
‘\
\,4-electrons
‘\
EG
-0.4465
-
-0.4475
- (a)
‘\
‘\
State
I
I
~‘~~---_!_;
I
‘\\
,l
I
30
I
I
I
I
I
40
50
6.0
70
80
B ITI
-0.528
(here t?, # and 1 corresponds to fully spin-polarized, spin-unpolarized
and partially spin-polarized
states respectively).
The spin assignments
for
2 > v> 1 can be obtained from the electron-hole
symmetry which, in the absence of Landau level
mixing is between v and 2- v [ 131. The absence of
electron-hole
symmetry between f and 3 was first
noticed in the tilted-field measurements
of FQHE by
Haug et al. [ 141. The different spin polarization
at
different filling factors has been explained in terms
of competition
between Coulomb
repulsion,
exchange and interaction of unlike spins [ 12 1.
The spin assignments for various filling fractions
discussed above have received strong support from
two recent experiments
on FQHE where the mag-
-0.537
6-electrons
v=2/3
t
- (b)
L-L
5.0
I
1
15.0
10.0
I
20.0
1
25.0
B (T)
Fig. 1. Ground-state
energy per article
B (Tesla)
for (a) v=:
and (b)
polarizations.
(in units of e2/clo ) versus
v=f
for different
spin
18
T. ChakrabortyLYpin-reversed ground state and energy gap in FQHE
where the Zeeman energy (g= 0.4) contributions
are
also included. In both cases, we observe a cross-over
point in the magnetic field below which the spin unpolarized state (SC 0) is energetically favoured. The
fully spin-polarized
state (S= NJ2, N, is the electron number) is favoured for magnetic fields beyond
the cross-over point. In the case of f and f, the
ground-state energy is remarkably insensitive to the
system size [ 111. Therefore, a similar situation in
the present case is expected. From these results, it is
clear that, when we study the QP-QH gap, we should
consider the two different ground states for two regions of the magnetic field.
For the spin-unpolarized
ground state, the spin-reversed QP-QH excitation gap has the lowest energy.
For the polarized ground state, the gap for spin-reversed QP (or QH) and the other partner spin-polarized, is close to zero. The energy gap for spin-reversed QP and QH is even negative. At first sight,
this is quite disturbing. However, we should remember that, in the absence of Zeeman energy, the fully
spin-polarized
ground state is not the preferred
ground state. The Zeeman term will add a positive
contribution
to the energy gap thereby increasing its
value.
The spin-reversed
QP-QH gaps for two different
ground states (spin-unpolarized
and fully spin-polarized) at v=$ are depicted in fig. 2a. They are the
3. Energy gap
It was first pointed out by Laughlin [4] that the
elementary charged excitations in the quantized Hall
states are quasi-particles with fractional charge, more
precisely, the charge &e/q for a filling fraction of
ZJ=p/q. The energy required to create a pair of quasiparticles of opposite charge well separated from each
other is then related to the discontinuity
of the slope
of the energy by [ 16 ]
0.0100
;&b-_&G@
,.’
0.0025
.x
0
(1)
E~=f(El”+-iFl”_).
Defining
the chemical
potential
_ 9.p. 1 + q.h 1
v=2/5
4 - electrons
,’
,’
,’
l’
‘I.
,’
--I_
-.._
2..
(al
I
I
30
4.0
I
50
I
/
60
I
70
80
I
90
B (T)
as
p=~o(~)+~a~o/a~,
6-electrons
(2)
where Eo( v) is the energy per particle at V, the QPQH gap is
P& =-&+tl/lq)(E~
-J%) l(vk
-v) .
(3)
Here v?=NJ(N,T
1) and E,=E,(v,),
and N, is
the number of flux quanta related to the filling fraction [ 11. The spin-reserved
QP and QH excitation
energy gaps are obtained by evaluating Ei in a system where the spin difference from the ground state
is unity (spin-l excitations).
The ground-state
energy (per particle) at v is taken to be spin-polarized
or spin-reversed, depending on the magnetic field region of fig. 1.
Let us first consider the case of v=$, and also the
situation where the Zeeman energy is not included.
Ground
State
s=3
0
5.0
15.0
10.0
20.0
B(T)
Fig. 2. Lowest energy quasi-particle-quasi-hole
gap (in units of
e’/tl,) ) as a function of magnetic field B (in Tesla) for (a)
u=: and (b) v= 5. Above and below the cross-over point, two
different ground states form fig. I, are considered.
The gap is
nonexistent in the shaded region, as discussed in the text.
T. ChakrabortyBpin-reversed
ground state and energy gap in FQHE
lowest energy excitations in the presence of Zeeman
energy. The results are qualitatively
similar to the
observed behaviour at v=: [ 91. In comparing the
present results with those of ref. [ 91, it should be remembered that here I assume the magnetic field to
be perpendicular
to the electron plane, while in the
experiment,
the magnetic field was tilted from the
direction normal to the plane. The tilted field is however expected to change primarily the Zeeman energy, but the Coulomb energy is likely to remain unchanged.
Our theoretical
result therefore
is in
agreement with the model proposed in ref. [ 91 that
the observed sharp change in the slope of the activation energy corresponds
to spin-reversed
excitations from the different ground states. From the results discussed above , we identify the excitations as
a spin-reversed
QP-QH pair [ 15 1.
The results for the QP-QH gap for v= 3 is depicted in fig. 2b. Just as in fig. 2a, we plot only the
lowest energy excitations as a function of magnetic
field, with the Zeeman energy included. In this case,
the situation is clearly different from that of v=$
Below the cross-over point, the preferred ground state
is spin-unpolarized.
The lowest energy excitations in
this state involve a spin-polarized
QP-spin-reversed
QH pair. The energy gap decreases rapidly and vanishes before the cross-over point is reached. From
this point onward, the discontinuity
in the chemical
potential is in fact negative, indicating that FQHE is
unstable in this region of magnetic field. Beyond the
cross-over point, the spin-unpolarized
state is no
longer the ground state and the energy gap is to be
calculated from the fully spin-polarized
ground state.
In this case, spin-reversed
QP-spin-polarized
QH
pair excitations have the lowest energy. The energy
gap steadily increases with the magnetic field. Therefore, between the two ground states exhibiting FQHE,
there is a gapless domain, where the FQHE state is
not stable. Such a gapless domain is not present at
v=$.
A somewhat similar situation has indeed been observed for v=! (electron-hole
symmetric to 3 ) in
the experiments
of ref. [ 8 1. The activation energy
for this filling fraction is found to decrease rapidly
with increasing magnetic fields. The gap then vanishes for a small region of magnetic field. The gap reappears with increasing magnetic fields. In our calculation, the effect is purely a result of the Zeeman
19
energy, which stabilizes different ground states in two
regions of magnetic field.
The absence of such a gapless domain for 3, both
in the experimental
observation
and in our calculation is noteworthy. I tend to take the novel feature
of the energy gap for 3 in our present calculation as
a physical effect, and not an artifact of our finite-size
calculation. Another interesting result for 3 is that
the spin-reserved
QP seems to persist for a rather
large magnetic field.
Having made the comparison of our theoretical results with the experimental observations in refs. [ 8,9 ]
I wish to make a few cautionary remarks. In our earlier work, we noticed that, while the ground-state energy is insensitive to the system size, the QP-QH gap
is in fact, system-size dependent
[ 1,7]. Therefore,
the absolute value of the different QP-QH gap discussed above, might change with increasing system
size. The finite-thickness correction [ 1 ] is also known
to influence the size of the gap. Larger system sizes
are difficult to handle numerically, especially in the
presence of reversed spins. The physical characteristics of the energy gap in the presence of spin reversal observed here are however, expected to remain unchanged when these corrections are made.
We have made similar calculations for other filling
fractions (v=? and :) [ 21. The gapless domain was
also found to appear in these cases. For these two
fractions, we observed several other interesting features in the energy gap. Besides the usual transition
from one ground state to the other as seen here, we
also observed in the QP-QH gap, transitions
from
one spin state to the other in the same ground state
121. While the current experimental
results for
v=$ (electron-hole
symmetric to 2) seem to indicate the presence of such transitions
[ 8 1, more experimental
work is needed to clarify the situation.
Verification
of such transitions
would provide a
means to distinguish between the spin reversal of the
quasi-particles
and quasi-holes from activation energy measurements.
Finally, just in the case of
v=$, the energy gap for : was found to vanish at a
finite magnetic: field. Experimental
attempts to observe its possible re-appearance
would be very
interesting.
In closing, the tilted-field experiments of ref. [ 8,9 ]
have brought in a wealth of information
about the
various spin-reversed ground states and excitations.
20
T. ChakrabortyLYpin-reversed ground state and energy gap in FQHE
They have provided convincing evidence for the existence of spin-reversed
quasi-particles
introduced
theoretically in ref. [ 7 1. Finite-size systems are found
to be very effective in explaining several interesting
features observed in these experiments.
Acknowledgements
Most of the work described here has been done in
collaboration
with Pekka Pietilainen
(Oulu, Finland). The work on v= 3 was inspired by a conversation with Bob Clark (Oxford), during the QHE
meeting in Schleching, Germany.
References
[ I] Tapash
Chakraborty
and P. Pietillinen,
The Fractional
Quantum Hall Effect (Springer, Heidelberg, 1988).
[ 2 ] Tapash Chakraborty and P. Pietillinen, to be published.
[ 3 ] D.C. Tsui, H.L. Stormer and A.C. Gossard, Phys. Rev. Lett.
48 (1982) 1559.
[4] R.B. Laughlin,
Phys. Rev. Lett. 50 (1983) 1395.
Helv. Phys. Acta 56 (1983) 75.
[6] Tapash Chakraborty
and F.C. Zhang, Phys. Rev. B 29
(1984) 7032; B 30 (1984) 7320.
[ 71 Tapash Chakraborty, P. Pietillinen and F.C. Zhang, Phys.
Rev. Lett. 57 ( 1986) 130;
Tapash Chakraborty
and P. Pietillinen,
Phys. Ser. T 14
(1986) 58;
see also: Tapash Chakraborty, Phys. Rev. B 34 ( 1986) 2926;
R. Morfand B.I. Halperin, Phys. Rev. B 33 ( 1986) 222 1.
[ 81 R.G. Clark, S.R. Haynes, A.M. Suckling, J.R. Mallet& J.J.
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[9] J.P. Eisenstein, H.L. Stiirmer, L. Pfeiffer and K.W. West,
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[ 131 F.C. Zhang and Tapash Chakraborty,
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[ 141 R.J. Haug, K. von Klitzing, R.J. Nicholas, J.C. Maan and
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[ 51B.I. Halperin,