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

ferromagnet—a 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 [5–9] 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

PRL 100, 226804 (2008)

PHYSICAL REVIEW LETTERS

week ending

6 JUNE 2008

0031-9007=08=100(22)=226804(4) 226804-1 © 2008 The American Physical Society

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

0–16 T. Dark

lines correspond to populating a spin-split 1D subband. White

regions correspond to conductance plateaux —features are

labeled accordingly.

PRL 100, 226804 (2008)

PHYSICAL REVIEW LETTERS

week ending

6 JUNE 2008

226804-2

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 region—there 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-

gies—no 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 also—it 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

PRL 100, 226804 (2008)

PHYSICAL REVIEW LETTERS

week ending

6 JUNE 2008

226804-3

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 coincide—in 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.

[1] G. F. Giuliani and J. J. Quinn, Phys. Rev. B 31, 6228

(1985).

[2] T. Jungwirth and A. H. MacDonald, Phys. Rev. B 63,

035305 (2000).

[3] W. Desrat et al., Phys. Rev. B 71, 153314 (2005).

[4] A. J. Daneshvar et al., Phys. Rev. Lett. 79, 4449 (1997).

[5] A. C. Graham et al., Phys. Rev. Lett. 91, 136404 (2003).

[6] A. J. Daneshvar et al., Phys. Rev. B 55, R13 409 (1997).

[7] G. Grabecki et al. , Physica (Amsterdam) 13E, 649 (2002).

[8] K.-F. Berggren, P. Jaksch, and I. Yakimenko, Phys. Rev. B

71, 115303 (2005).

[9] A. C. Graham et al., Solid State Commun. 131, 591

(2004).

[10] A. C. Graham et al., Phys. Rev. B 75, 035331 (2007).

[11] K. J. Thomas et al., Phys. Rev. Lett. 77, 135 (1996).

[12] A. C. Graham et al., Phys. Rev. B 72, 193305 (2005).

[13] A. C. Graham et al., Physica (Amsterdam) 22E, 264

(2004).

[14] T. Hassenkam et al., Phys. Rev. B 55, 9298 (1997).

[15] A. V. Moroz, K. V. Samokhin, and C. H. W. Barnes, Phys.

Rev. Lett. 84, 4164 (2000).

[16] S. M. Girvin, Phys. Today 53, No. 6, 39 (2000).

PRL 100, 226804 (2008)

PHYSICAL REVIEW LETTERS

week ending

6 JUNE 2008

226804-4