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Journal ArticleDOI

Anticrossing of spin-split subbands in quasi-one-dimensional wires.

06 Jun 2008-Physical Review Letters (American Physical Society)-Vol. 100, Iss: 22, pp 226804

TL;DR: The results indicate that both anticrossings and magnetic phase transitions are also possible in quasi-1D quantum wires in an in-plane B field, Bparallel, and imply that the well-known 0.7 structure may evolve into a spin-hybridized state in finite dc bias.

AbstractIn quantum Hall systems, both anticrossings and magnetic phase transitions can occur when opposite-spin Landau levels coincide. Our results indicate that both processes are also possible in quasi-1D quantum wires in an in-plane $B$ field, ${B}_{\ensuremath{\parallel}}$. Crossings of opposite-spin 1D subbands resemble magnetic phase transitions at zero dc source-drain bias, but display anticrossings at high dc bias. Our data also imply that the well-known 0.7 structure may evolve into a spin-hybridized state in finite dc bias.

Topics: Quantum spin Hall effect (58%), Quantum phase transition (57%), Quantum Hall effect (56%), Landau quantization (55%), DC bias (52%)

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Summary

  • In quantum Hall systems, both anticrossings and magnetic phase transitions can occur when oppositespin Landau levels coincide.
  • The authors results indicate that both processes are also possible in quasi-1D quantum wires in an in-plane B field, Bk. Crossings of opposite-spin 1D subbands resemble magnetic phase transitions at zero dc source-drain bias, but display anticrossings at high dc bias.
  • The authors data also imply that the well-known 0.7 structure may evolve into a spin-hybridized state in finite dc bias.

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Anticrossing of Spin-Split Subbands in Quasi-One-Dimensional Wires
A. C. Graham, M. Y. Simmons,
*
D. A. Ritchie, and M. Pepper
Cavendish Laboratory, J. J. Thomson Avenue, Cambridge, CB3 OHE, United Kingdom
(Received 26 February 2008; published 6 June 2008)
In quantum Hall systems, both anticrossings and magnetic phase transitions can occur when opposite-
spin Landau levels coincide. Our results indicate that both processes are also possible in quasi-1D
quantum wires in an in-plane B field, B
k
. Crossings of opposite-spin 1D subbands resemble magnetic
phase transitions at zero dc source-drain bias, but display anticrossings at high dc bias. Our data also imply
that the well-known 0.7 structure may evolve into a spin-hybridized state in finite dc bias.
DOI: 10.1103/PhysRevLett.100.226804 PACS numbers: 72.25.Dc, 71.70.d, 73.21.Hb, 73.23.Ad
The varied and complex physics of the quantum Hall
ferromagneta 2D electron or hole system tuned to bring
two Landau levels into coincidence has been extensively
studied both theoretically [1,2] and experimentally [3,4],
and depends delicately on interaction strength, carrier
density, the forms of the wave functions and spins of the
coincident levels. In contrast, the magnetic properties of
quasi-1D systems in the vicinity of crossings of spin-split
1D subbands [59] are still poorly understood, as is the
case with much of the interesting physics associated with
these strongly interacting quasi-1D systems.
In this Letter, we present experimental evidence that 1D
subbands of opposite spin can undergo both magnetic
phase transitions and anticrossings, depending on whether
or not the subbands coincide near the Fermi energy E
F
.It
has previously been shown that crossings of Zeeman-split
subbands in high B
k
exhibit nonquantized conductance
structures, known as analogs, which have the same tem-
perature B
k
and dc bias dependences as the 0.7 structure
[5,10]. We demonstrate that although the conductance
features of the crossings at zero dc bias imply an abrupt
change in the magnetic polarization of the quantum wire,
the finite-dc-bias features continuously evolve from one
spin type to the other with increasing B
k
, while maintain-
ing a finite energy gap, which is more reminiscent of a
hybridized state or anticrossing, than an abrupt magnetic
phase transition. In a finite dc bias, the lower (drain)
chemical potential
d
can provide information about a
subband after it has populated and is well below the upper
(source) chemical potential
s
. This allows us to study the
coincidence of subbands which are heavily populated and
far below
s
, whereas in zero dc bias, we can only study a
coincidence of subbands if it occurs at E
F
. We argue that
this explains the difference in observed behavior in the low
and high-bias regimes.
We begin by presenting quantum wire conductance data
which exhibit crossings of spin-split subbands as a function
of B
k
. This is to aid interpretation of the more complex
finite dc bias conductance data which we go on to present,
taken at five B
k
through the crossing region. The zero-bias
B
k
data exhibit an anomalous discontinuity in the crossing
region; nevertheless, we will show that at all fields the
features can be unambiguously labeled with one spin
type. In contrast, although the features in the finite-dc-
bias data remain clearly defined at all B
k
, we demonstrate
that there is a field range in which these features cannot be
labeled with a particular spin type, but are inherently
ambiguous in spin character. We conclude by discussing
mechanisms that could be responsible for this surprising
spin effect, and possible implications for the origin of the
0.7 structure [11] and analogs [5,10].
Our samples comprise split-gate devices on a
GaAs=Al
0:33
Ga
10:33
As heterostructure. Samples used in
this Letter have a length 0:4 m and width 0:6 m, but we
observe the same effects with other sample dimensions.
The 292 nm deep two-dimensional electron gas has a
mobility of 1:1 10
6
cm
2
=Vs and a carrier density of
1:15 10
11
cm
2
. B
k
was applied perpendicular to the
current direction, but we observe the same features in a
parallel field. Hall measurements indicate the sample
alignment was better than 0.5
. The measurement tempera-
ture was 100 mK.
Applying an in-plane B field causes 1D electric sub-
bands to Zeeman split. This is apparent in a gray-scale plot
of the derivative of the differential conductance with re-
spect to split-gate-voltage (Fig. 1). Left-moving (right-
moving) dark lines correspond to the population of
lower-energy spin-down subbands (higher-energy spin-up
subbands). Also visible at 9Tare the first crossings of
opposite-spin subbands (from left to right, 1 " with 2 # , 2 "
with 3 # , etc.), and second crossings at 12 –14 T (from left
to right, 1 " with 3 # , 2 " with 4 # , etc.).
Going from higher to lower subband indices (from right
to left) the crossings display increasingly pronounced dis-
continuities, indicating that opposite-spin subbands
abruptly rearrange as they populate. As we have previously
noted [9], this is reminiscent of the magnetic phase tran-
sitions predicted to occur for opposite-spin Landau levels.
Apart from these discontinuities, however, the crossings of
the lower subbands broadly resemble those of higher sub-
bands. Since the general features of Fig. 1 do not differ
greatly from a noninteracting picture, we can assign
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spin types to the features accordingly; we have replotted
the data in Fig. 2(a) and superimposed red or gray lines
(blue or dark gray lines) on features associated with higher-
energy spin-up (lower-energy spin-down) subbands. This
is to aid identification of features in the finite-dc-bias data
in Fig. 2 which are the focus of our Letter.
We now demonstrate that although there is no apparent
ambiguity in the spin types of features at zero dc bias, in
finite dc bias two branchlike features continuously evolve
from one spin type to the other with increasing B
k
. These
two finite-bias features do not display any obvious discon-
tinuity as a function of B
k
, or any other indication that a
change of spin type has occurred it is only by following
the features at zero dc bias that this change in spin type
becomes apparent. At B
k
5T, the data broadly resemble
a noninteracting picture. As illustrated in Fig. 2(b), each
dark point at zero dc bias, corresponding to a sharp in-
crease in differential conductance due to the populating of
a 1D subband, splits into a pair of V-shaped dark branches
with increasing dc bias; the left (right) branch corresponds
to the subband intercepting the higher-energy source
chemical potential
s
(lower-energy drain chemical po-
tential
d
). We therefore assume that the spin type of a
branch in finite dc bias will be the same as the spin type of
the feature at zero dc bias that the branch has split from
[see Fig. 2(b)]. Thus, since Fig. 2(a) shows that the second-
and third-from-left features at 5 T are spin up (1 " subband)
and spin down (2 # subband), respectively, we have labeled
the right-moving branch A second from left in Fig. 2(c),
B 5T, as spin up, and the right-moving branch B
third from left as spin down.
We will follow the evolution of branches A and B in
particular as B
k
is increased and subbands 1 " and 2 # cross.
From 5 to 6.6 T, the branches have only moved slightly, and
the spin types of the features are not yet ambiguous.
Although, again, by 7.8 T, branches A and B have slightly
FIG. 2 (color online). dc bias spectroscopy in the
crossing region of 1 " and 2 # . Lower-energy spin-down features
are marked blue or dark gray, and higher-energy spin-up features
are marked red or gray. (a) Gray-scale plot, as in Fig. 1, show-
ing the evolution of quantum wire conductance characteristics
in an in-plane B field. Green or light gray lines indicate the fields
at which dc bias data in (c) were taken. (b) Schematic illustrat-
ing how we have assigned spin types to features in the dc bias
data at a fixed B
k
, we assume that a spin-up (spin-down)
feature at zero dc bias splits into two spin-up (spin-down)
features in finite-bias. (c) Gray scale of the derivative of the
differential conductance, as a function of gate voltage, for dc
biases from 0:5 to 1.4 mV, at B
k
marked with green or light
gray lines in (a)the spin types of the features along a green or
light gray line in (a) enable us to identify the spin types of the
finite-bias features. Data on the left are reproduced without the
annotation on the right for clarity. The spin types at B 7:8T
are ambiguous, however, and imply that an anticrossing is
occurring.
FIG. 1. Gray scale of the derivative of the differential con-
ductance, as a function of gate voltage, for B
k
016 T. Dark
lines correspond to populating a spin-split 1D subband. White
regions correspond to conductance plateaux features are
labeled accordingly.
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changed position compared to 6.6 T, the data still look
broadly similar. Note, however, that although these fea-
tures at zero bias have moved closer together by 7.8 T
because we are approaching the crossing of 1 " and 2 # ,
branches A and B in finite dc bias are almost as far apart in
gate voltage as they were at 5 T. Assigning the spins
associated with these branches at 7.8 T does not appear
problematic, as they have only moved slightly from 6.6 T.
However, the dc bias data taken at 9 T, which is in the
crossing region, reveals a dramatic complication. At zero
dc bias, branch B has split apart from the left branch it was
joined to at 7.8 T, and now has no accompanying left
branch, the significance of which was discussed in
Ref. [10]; furthermore, the feature at zero dc bias that
branch B evolves from is now spin up instead of
spin down, according to Fig. 2(a) at 9 T. Therefore,
although the form of branch B changes very little from 5
to 9 T, at 5 T it is associated with the 2 # subband, but at 9 T
it is associated with the 1 " subband. Thus, assigning
spin types to the features at 7.8 T is not actually straight-
forward they are ambiguous so are marked with both
colors.
By 11 T, the dc bias data once again broadly resemble a
noninteracting picture each feature at zero dc bias splits
into two branches at finite dc bias, and there is no apparent
difficulty in assigning a subband and spin index to the
branches. At this field it is now clear that branch A must
be associated with the 2 # subband instead of the 1 " sub-
band, because it splits from a spin-down feature at zero dc
bias, so it too has changed its spin type between 5 and 11 T.
At higher fields (not shown), branch A remains unchanged,
but branch B takes part in a second crossing, this time with
3 # , and reverses its spin again. In summary, at zero dc bias
[Figs. 1 and 2(a)], features of opposite spin move very
close to each other in gate voltage as B
k
increases, and for
lower subband indices, give rise to a crossing with an
abrupt discontinuity that resembles a magnetic
phase transition; however, in finite dc bias [Fig. 2(c)],
although features A and B of opposite spin maintain a
large gap in gate voltage between them at all B
k
, they
appear to evolve continuously into the opposite spin type
as they pass through the crossing regionthere is no
discontinuity in the gate-voltage position of branches A
and B with increasing B
k
. This is more reminiscent of an
anticrossing than a phase transition.
Both magnetic phase transitions and anticrossings be-
tween opposite-spin Landau levels can occur in quantum
Hall systems and have now been thoroughly characterized,
theoretically and experimentally. In contrast, crossings
between opposite-spin 1D subbands in quantum wires are
poorly understood, and as yet, no rigorous theoretical
framework exists to describe them. Hence, we cannot
definitively explain the nature of these 1D subband cross-
ings in zero and finite bias at present. However, our data
imply that magnetic phase transitions and anticrossings
can both occur in quantum wires, with magnetic
phase transitions occurring for zero or low-dc bias, and
anticrossings in the high-dc bias regime. We will proceed
by identifying some differences and similarities between
1D-subband crossings and Landau-level crossings, and
then use these to discuss why anticrossings between 1D
subbands of opposite spin in a quantum wire might be
observed specifically in the finite-bias regime.
Whereas a Landau level contains a finite number of
states which are filled completely as it passes below E
F
,
a 1 D subband has states which extend to infinite ener-
giesno matter how far below E
F
the subband edge is,
there will still be states in that subband available at E
F
.On
the one hand, if E
F
in a quantum Hall system is set at 2
filling factor, the system can remain completely unpolar-
ized with equal numbers of spin-up and spin-down elec-
trons in n 0 # and n 0 " , until a magnetic
phase transition occurs, which will then give a completely
polarized system with only spin-down electrons in n 0 #
and n 1 # . In a 1D system, however, as B
k
is increased
from zero towards the crossing of the 1 " and 2 # subbands,
the system becomes increasingly polarized, because 1 "
continuously depopulates while 2 # increasingly populates.
For this reason alone, it is no surprise that a 1D-subband-
crossing is qualitatively different to a Landau-level-
crossing.
A second key difference between the two systems is that
density and confining potential are set to some fixed values
in a quantum Hall system and opposite-spin Landau levels
are made to cross by varying B
k
alone, using different tilt
angles. In quantum wires, however, both density and con-
fining potential are varied along with B
k
in order to induce
crossings. Given that the nature of Landau-level crossings
delicately depends on density and confinement, this is
likely to be true of 1D subband crossings alsoit is
possible that the character of the subband crossing changes
as the gate voltage is swept through the crossing region.
Additionally, the finite-bias regime in a quantum wire
has no analogy in a quantum Hall system. So far, there has
yet to be any theoretical study of this regime in a realistic
interacting quasi-1D electron system. It is, however, be-
coming increasingly apparent that dc bias spectroscopy
provides powerful insight into electron interactions in
quantum wires. This is because in a finite bias, the lower
(drain) chemical potential provides information about a
subband even after it has populated and is well below the
upper (source) chemical potential.
We suggest that the key difference between the crossings
we observe in the zero- and finite-bias regimes is this: at
zero dc bias, both subbands taking part in the crossing are
very close to E
F
, and therefore do not have a large number
of occupied states, whereas at a ‘right branch’ in
finite dc bias, both of the subbands are near the drain
chemical potential, and therefore have many occupied
states. In the case of coincident Landau levels, the nature
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of the crossing greatly depends on whether one, both, or
neither of the levels are occupied, and whether they are
near E
F
. If both or neither are occupied and they are not
near E
F
when they cross, then an anticrossing is likely.
However, if they are near E
F
, and only one is occupied,
then a paramagnetic-ferromagnetic phase transition be-
comes possible, with all the electrons abruptly emptying
from, for example, the n 0 " spin-up level, and filling the
n 1 # Landau level instead, to create a completely spin-
polarized system. There is no energy saving to be made by
the levels rearranging in this way if both are well below E
F
or well above it when they coincidein this situation, the
levels hybridize instead and anticross. Therefore, in the
case of 1D subbands near the drain chemical potential at
large dc bias, they are so far below the source chemical
potential when they coincide that a phase transition is no
longer possible. However, in the presence of exchange
interactions or spin-orbit coupling, mixing of spins can
occur, and the levels will instead anticross.
The large enhancement of spin splitting which allows
crossings to be observed at around 10 T instead of around
50 T for bare Zeeman splitting, is good evidence that
exchange interactions in quantum wires are strong, and
aided by the electron-electron interaction could mix the
opposite-spin subbands. Additionally, the numerous differ-
ences which have been observed in the conductance char-
acteristics of spin-up and spin-down subbands in finite B
k
[10,12,13] constitutes further evidence of strong exchange
interactions.
It is less clear whether spin-orbit coupling will play a
role here. Although the Dresselhaus contribution to spin-
orbit coupling for electrons in GaAs (due to crystal inver-
sion asymmetry) is usually negligibly small, typical
Rashba contributions to spin-orbit coupling (due to electric
fields associated with confining potentials) are consider-
ably larger [14]. Since quantum wires are electrostatically
confined in two directions, it has been argued that Rashba
spin-orbit coupling is particularly important in these sys-
tems [15]. Irrespective of the strength of spin-orbit cou-
pling, the electron interaction coupled with exchange could
mix spins and give the observed anticrossing between 1D
subbands of opposite spins.
Lastly, we note that the crossing region displays all of
the same characteristics as the lowest plateau in zero B
k
they both exhibit nonquantized conductance structures, the
0.7 structure and analogs, which evolve into quantized
structures with increasing B
k
. These structures weaken
and rise in conductance with decreasing temperature, and
strengthen and rise in conductance under finite bias.
Furthermore, the dc bias gray scale at B 9T in
Fig. 2(c) shows that the right-branch labeled B has no
accompanying left branch, unlike the data at lower and
higher B
k
; such anomalous behavior also typifies the dc
bias dependence of the 0.7 structure at zero B
k
, as was
discussed in Ref. [10]. This similarity of the 0.7 structure to
the crossing region implies that it may also evolve into a
spin-hybridized state under a finite dc bias clearly fur-
ther investigation is required, in order to establish whether
this is indeed the case.
In conclusion, we have presented experimental evidence
that 1D subbands of opposite spin may be able to hybrid-
ize, creating an anticrossing when they are tuned to coin-
cide in a large in-plane B field. We only observe
anticrossings in high dc biases, however. At zero-to-low
dc bias, subbands of opposite spin appear to abruptly
rearrange, giving a discontinuity when they cross; this
does not resemble an anticrossing, but rather, a magnetic
phase transition. We suggest that the difference between
the low and high dc bias regime is that at low dc biases, we
can only observe subbands coinciding if this occurs at
E
F
this happens to be the regime in which a magnetic
phase transition is possible. At high dc biases, however, the
lower-energy drain chemical potential allows us to study
the coincidence of subbands which have populated at much
higher energies. Since magnetic phase transitions [16] are
only expected to occur for subbands near E
F
, subbands
which coincide near the drain chemical potential in high dc
biases may instead anticross, as is implied by our data. This
result emphasizes the wealth of interesting physics which
can occur in quasi-1D quantum wires.
We acknowledge useful discussions with C. H. W.
Barnes, C. J. B. Ford, M. Kataoka, F. Sfigakis, T.-M.
Chen, J. P. Griffiths, K.-F. Berggren, and B. Spivak. This
work was supported by EPSRC, U.K. A. C. G. acknowl-
edges support from Emmanuel College, Cambridge.
*Present address: University of New South Wales, School
of Physics, Sydney, NSW 2052, Australia.
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PRL 100, 226804 (2008)
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Q1. What are the contributions in "Anticrossing of spin-split subbands in quasi-one-dimensional wires" ?

In this paper, it was shown that crossings of opposite-spin 1D subbands resemble magnetic phase transitions at zero dc source-drain bias, but display anticrossings at high dc bias.