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Extreme sensitivity of the spin-splitting and 0.7

anomaly to confining potential in one-dimensional

nanoelectronic devices

A. M. Burke,∗,† O. Klochan,† I. Farrer,‡ D.A. Ritchie,‡ A. R. Hamilton,† and A. P.

Micolich∗,†

School of Physics, University of New South Wales, Sydney NSW 2052, Australia, and Cavendish

Laboratory, University of Cambridge, CB3 0HE, U.K.

E-mail: amburke@unsw.edu.au; adam.micolich@nanoelectronics.physics.unsw.edu.au

Abstract

Quantum point contacts (QPCs) have shown promise as nanoscale spin-selective compo-

nents for spintronic applications and are of fundamental interest in the study of electron many-

body effects such as the 0.7×2e2/h anomaly. We report on the dependence of the 1D Landé

g-factor g∗ and 0.7 anomaly on electron density and confinement in QPCs with twodifferent

top-gate architectures. We obtaing∗ values up to 2.8 for the lowest 1D subband, significantly

exceeding previous in-planeg-factor values in AlGaAs/GaAs QPCs, and approaching that in

InGaAs/InP QPCs. We show thatg∗ is highly sensitive to confinement potential, particu-

larly for the lowest 1D subband. This suggests careful management of the QPC’s confine-

ment potential may enable the highg∗ desirable for spintronic applications without resorting

to narrow-gap materials such as InAs or InSb. The 0.7 anomaly and zero-bias peak are also

∗To whom correspondence should be addressed†School of Physics, University of New South Wales, Sydney NSW2052, Australia‡Cavendish Laboratory, University of Cambridge, CB3 0HE, U.K.

1

highly sensitive to confining potential, explaining the conflicting density dependencies of the

0.7 anomaly in the literature.

Keywords: one-dimensional system,g-factor, quantum point contact, nanoelectronics.

A current focus in nanoelectronics is the development of spintronic devices where spin is

used instead of charge for storage, transfer and processingof information.1 Non-magnetic spin-

tronic device elements are highly desirable;2 quantum point contacts (QPCs) have shown great

promise being used both as individual spin injectors and detectors,3–5 and in larger device struc-

tures for studying phenomena such as spin relaxation6 and ballistic spin resonance.7 The QPC is

the quintessential one-dimensional (1D) electron system,consisting of a narrow quasi-1D aperture

separating two regions of two-dimensional electron gas (2DEG) in a III-V semiconductor het-

erostructure. It is typically defined electrostatically byapplying a negative bias to nanoscale metal

gates on the heterostructure surface; its hallmark is a quantized electrical conductanceG = mG0,

whereG0 = 2e2/h, e is the electron charge,h is Planck’s constant, andm is the number of spin

degenerate 1D subbands beneath the Fermi energyEF of the adjacent 2DEG reservoirs.8,9 The

spin properties of QPCs are also of fundamental interest; one example is the conductance anomaly

at G ∼ 0.7G0,10 where the interplay between 1D confinement, quasi-bound state formation and

exchange-driven spin polarization are not yet fully understood.11 The combined influence of ex-

change and 1D confinement are also vital to remarkable behaviors such as spin-charge separation12

and the formation of electron liquid/solid states in 1D electron systems.13,14

An important quantity in considering the spin-properties of QPCs is the effective Landég-factor

g∗, the constant of proportionality between the Zeeman splitting of the 1D subbands and the applied

magnetic field. Theg-factor g∗m for each 1D subbandm is easily measured in QPCs.15 For spin

injection and detection, it is highly desirable to maximizethe lowest 1D subbandg-factorg∗1, which

sets the minimum field required to resolve the spin. Theg-factor is also a useful experimental probe

of the exchange interaction.16 The foundational work on the 0.7×2e2/h anomaly showed thatg∗m

increases from the bulk GaAs value of 0.44 at m = 25 to ∼ 1.15 at m < 4.10 This ‘exchange-

enhancement’ effect,17 also observed in InGaAs/InP QPCs,18,19 is central to the suggestion that

2

the 0.7 anomaly is caused by exchange-driven spontaneous spin-polarization within the QPC.10,17

Remarkably, after 15 years of study of the 0.7 anomaly, little more is known about the dependence

of g∗m on QPC confinement potential or the electron densityn of the 2DEG in which the QPC is

formed.11

Here we study howg∗m evolves withn for three samples featuring two different gate architec-

tures for enacting changes in density. We pay particular attention tog∗1 given its importance for

spintronics and the 0.7 anomaly. We obtaing∗1 values as high as 2.8. This exceeds previous reports

for the in-planeg-factor in GaAs QPCs,6,10,15,20,21and approaches both the perpendicularg-factor

recently demonstrated in GaAs QPCs22 and the lower-bound in-planeg∗ for InGaAs QPCs.18,19,23

The link betweeng∗m and density is not direct; for example, we seeopposite trends ing∗1 with n for

the two architectures. We find that theg-factorg∗m, andg∗1 in particular, is sensitive to the top-gate

configuration. This has important consequences for spintronic applications of QPCs; if highg∗1 can

be obtained in GaAs QPCs by careful management of the QPC’s electrostatic potential, it lessens

the need to use narrow band-gap materials (e.g., InGaAs, InSb) for which device fabrication is

more difficult. The second key result of our work arises from comparing the density dependence

of the 0.7 anomaly in the three samples with that ofg∗1 and the lowest 1D subband spacing∆E1,2,

a measure of the strength of the 1D confinement. The behavior of the 0.7 anomaly in our devices

is consistent with the density dependent spin-gap model24,25 if the spin-splitting rate is assumed

directly proportional to∆E1,2. This highlights the important role that confinement potential plays

in the 0.7 anomaly, and provides an explanation for the conflicting reports regarding the density

dependence of the 0.7 anomaly in earlier literature.17,24–32

We used two different device architectures in this experiment (see Fig. 1a/b), each fabricated

on the same heterostructure and featuring a pair of QPC gates(orange) biased atVg to define a 1D

channel, and a top-gate (yellow) biased atVt to independently varyn. The two devices differ in the

location of the top-gate, allowing us to study how the strength of the 1D confinement influences

g∗m. The bow-tie (BT) device (Fig. 1a) has a conformal top-gate with a length of∼ 60µm along the

transport direction. The polyimide (PI) device (Fig. 1b) has a 80×80 µm top-gate separated from

3

Figure 1: Top- and side-view schematics of (a) the bow-tie (BT) and (b) the polyimide (PI) devices.The side-views are sections along the green dot-dashed line. The 2DEG (blue dashed line) islocated 90 nm beneath the heterostructure surface (grey). In both architectures the QPC gates(orange) define a 300 nm long, 500 nm wide constriction. The top-gate (yellow) controls the2DEG densityn, and is insulated by a 140 nm thick polyimide layer (light blue) in the PI device.Gate/insulator structures are drawn to scale. PI was measured on two separate cool-downs withthe QPC gates trained whilst the top-gate was held atVt = 0 and+375 mV, referred to as PI-0and PI-375, respectively, to enableg∗m measurements for differentn ranges. (c) ac conductanceG versus QPC gate voltageVg for five differentVt settings from PI-0. ForVg > −0.25 V, G risessharply due to incomplete gate definition, limiting the density range over whichg∗m can be obtainedfor each 1D subbandm.

4

the QPC gates by a 140 nm polyimide layer. The PI device was measured in two separate cool-

downs, each with different top-gate ‘training’ to give a slightly differentn versusVt characteristic

(see Supplementary Fig. 1). Data is presented for training at Vt = 0 and+375 mV, referred to as

PI-0 and PI-375 hereafter, providing three separate ‘samples’ from the two device architectures. A

plot of n versusVt for each sample appears in Supplementary Fig. 2. The heterostructure used for

both devices (Cambridge W0191) features a 90 nm deep 2DEG, separated from the modulation

doping layer by a 40 nm undoped AlGaAs spacer. The 2DEG has a mobility of 2.7×106 cm2/Vs

at an ungated density of 1.8×1011 cm−2 and temperature of 4 K. Further device details appear

in the Supplementary Information. The devices were measured in a dilution refrigerator equipped

with a 15 T superconducting solenoid and a piezoelectric rotator33 for rotating the sample relative

to the applied magnetic fieldB without the device temperature exceeding 200 mK. The density n

was measured withB perpendicular to the 2DEG plane using a Fourier analysis of the Shubnikov-

de Haas oscillations. Measurements ofg∗m were obtained withB oriented in-plane and along the

QPC axis. To demonstrate device operation, Fig. 1c shows theac conductanceG versusVg for five

differentVt spanning the density range 1.07−1.71×1011 cm−2 for PI-0. Conductance quantization

is evident at eachVt , with pinch-off (i.e.,G= 0) occurring for smallerVg at more negativeVt , which

reduces the Fermi energyEF = π h̄2n/m∗ of the 2DEG reservoirs adjacent to the QPC. In each case,

G rises sharply forVg >−0.25 V due to loss of electrostatic depletion under the QPC gates. This

limits the number of quantized conductance plateaus observable for eachVt , and the accessible

density range over whichg∗m can be obtained for a given 1D subbandm.

We extractg∗m using the method developed by Patelet al15 to enable direct comparison with

the literature.10,15,18,19Two measurements are required to extractg∗m at eachn: The first is source-

drain bias spectroscopy; Fig. 2a shows a color-map of the ac transconductancedG/dVg against

source-drain biasVsd (x-axis) andVg (y-axis). The blue dotted vertical line in Fig. 2a corresponds

to the left-most (blue) trace in Fig. 1c. The bright regions indicate high transconductance and

correspond to the risers between plateaus, which occur whena 1D subband crosses the source/drain

chemical potentialµ = EF . With increasingVsd (i.e., moving right in Fig. 2a) the source and

5

Figure 2: ac transconductancedG/dVg vs (a)Vg (y-axis) and source-drain biasVsd (x-axis) and(b) Vg (y-axis) and in-plane magnetic fieldB‖ (x-axis) for PI-0 withVt = 0 V, corresponding tothe left-most (blue) trace in Fig. 1c. High transconductance (risers inG between plateaus) appearbright and indicate that a given 1D subband has crossed the chemical potentialµ. The respective1D subband indicesm are superimposed in both panels. The data in (a) allows us to measure thelowest 1D subband spacing∆E1,2 and the bias-splitting ratedVg/dVsd and (b) the Zeeman splitting∆Ez in units ofVg. We combine the latter two measurements to obtain theg∗m values in Fig. 3, with∆E1,2 used to characterize the confining potential in Fig. 4. The data in (a) has been symmetrizedaboutVsd = 0 to remove an asymmetric background artifact arising from instrumental issues in themeasurement.34

6

drain chemical potentials separate in energyµs − µd = eVsd producing a bifurcation of theVsd =

0 transconductance maxima (white dashed lines). The rising/falling bright line corresponds to

a given 1D subband coinciding withµd and µs, respectively. The 1D subband spacing∆E1,2

is obtained aseVsd at the crossing point between the lowest rising line and the second lowest

falling line (blue arrow in Fig. 2a). This provides an important measure of the ‘strength’ of the

transverse confinement at the center of the QPC; however, as we discuss later, it only provides

partial information about the overall confinement potential landscape of the QPC. The second

measurement is the Zeeman splitting of the 1D subbands; Fig.2b shows a color-map ofdG/dVg

against in-plane magnetic fieldB‖ (x-axis) andVg (y-axis). The blue dotted line corresponds to

the left-most (blue) trace in Fig. 1c. Each 1D subband splitswith increasingB‖ (white dashed

lines); however, this does not directly yield the Zeeman splitting ∆Ez because they-axis has units

of voltage not energy. To extract the Zeeman splitting, the splitting rates in Figs. 2a and b are

combined, viz:

∆Ez = e[dVg

dVsd]−1×

dVg

dB‖= e

dVsd

dB‖(1)

giving theg-factor asg∗ = ∆Ez/µBB‖, whereµB is the Bohr magneton. The two termsdVgdVsd

and

dVgdB‖

are obtained at the sameVg, making the confinement potential the same for both contributions

to Ez, and henceg∗m. Further details of the analysis appear in the Supplementary Information.

Figure 3: Plots of (a)g∗1, (b) g∗2, (c) g∗3 and (d)g∗4 versus 2DEG densityn. The black filledsquares, black half-filled squares and green circles correspond to data from PI-0, PI-375 and BT,respectively. The blue circles/error bars show the mean andstandard deviation for the set of datapoints in each of the four panels.

7

We now examine howg∗m evolves with densityn for the lowest four 1D subbands, with Fig. 3a-

d presentingg∗1, g∗2, g∗3 andg∗4 versusn for PI-0, PI-375 and BT. At each densityg∗m is obtained from

an individually measured pair of source-drain bias and fieldplots similar to those in Fig. 2. The

blue circle and error bar in each panel of Fig. 3 represents the mean and standard deviation for the

full set of data presented in that panel; comparing these forpanels a-d,g∗m clearly increases with de-

creasingm on average (see also Supplementary Fig. 3), consistent withprevious studies.10,15,18,19

The density-dependence ofg∗m is complex and evolves withm. We start first atm ≥ 2. Considering

each individual device on its own for a moment, in each panel (b-d) we see thatg∗m mostly in-

creases with decreasingn, as one would expect for exchange interactions.16 However, considering

the full three device data-set in each panel (b-d) there is noclear trend with density. Atm = 1 a

distinct difference in the density dependencies for the PI and BT devices emerges (Fig. 3a): asn

is reduced we observe increasingg∗1 for both PI samples but decreasingg∗1 for BT. Note also the

lack of overlap in the individualg∗1 versusn behavior for PI-0 and PI-375 in the common density

range 1.5−1.7×1011 cm−2. The findings above clearly point to a relationship betweeng∗ andn

that depends also on other parameters. It is worth noting that there is evidence for disorder effects

in BT, as discussed further below, which are more prevalent than for PI-0/PI-375. One possibility

that we cannot rule out at this stage is that the inherent disorder potential may also have an effect

in g∗(n). That said, given the identical heterostructure and QPC gate pattern for PI-0, PI-375 and

BT, a natural expectation is that the difference is due to thetop-gate; hence the logical next step is

to consider the influence of the QPC confinement potential.

Precise knowledge of QPC confinement potential is difficult,it not only depends on gate bias,

but also on the heterostructure doping profile and the self-consistent redistribution of charge in

the 2DEG.35–37The QPC is most commonly treated as a saddle-point potential,38 and while more

sophisticated self-consistent models yield a flat bottomedparabola for the transverse potential,36 a

simple parabola is usually sufficient, particularly in the small m limit. 30,39,40In a 1D parabolic well,

the energy level spacing is directly tied to the curvature; hence the 1D subband spacing∆Em,m+1 is

commonly used as a metric for the 1D confinement strength in QPCs.13,21,30,41–43Figure 4 shows

8

Figure 4: Plot of the lowest 1D subband spacing∆E1,2 vs densityn for PI-0, PI-375 and BT. Theletters (a-c) indicate the points corresponding to the datapresented in Fig. 7.

9

the measured∆E1,2 versusn for all three samples; we focus solely on∆E1,2 due to our interest in

g∗1 regarding both spintronic applications and the 0.7 anomaly. In each case∆E1,2 decreases asn

is reduced, indicating a softening 1D confinement potential, consistent with earlier studies using

similar device architectures.13,30

Comparing Figs. 3a and 4, an interesting but complex connection betweeng∗1, n and ∆E1,2

emerges. Considering PI-0 and PI-375 alone first; their∆E1,2 versusn trends almost overlap and

differ by at most 16% in the common density rangen = 1.4−1.8×1011 cm−2, indicating a similar

1D confinement strength. Despite this,g∗1 differs markedly (∼ 1.8 for PI-0 and∼ 2.5 for PI-375).

Turning to PI-0 and BT over the wider density rangen = 0.8− 1.8× 1011 cm−2, the different

top-gate implementation results in a∆E1,2 that is consistently∼ 160% larger for BT than PI-0,

indicating a harder, more square-well like confinement potential for BT. Yet, as Fig. 3a shows, not

only cang∗1 differ markedly between PI-0 and BT at a given density, but inthe former we observe

increasingg∗1 and the latter decreasingg∗1 with decreasingn. Finally, atn = 1.7×1011 cm−2 where

data exists for all three samples, we findg∗1(PI-375)> g∗1(BT)> g∗1(PI-0) whereas∆E1,2(BT)>>

∆E1,2(PI-0) ∼ ∆E1,2(PI-375). Hence while Koopet al21 suggest a clear and direct correlation

betweeng∗ and∆E1,2, our data suggests the connection is much more subtle. The subband spacing

∆E1,2 is only sensitive to the transverse quasi-parabolic 1D potential at the center of the QPC.

As such,∆E1,2 is a limited metric of the overall shape of the QPC potential landscape, which

depends on length, width, density36,44and realistically, the inherent disorder potential.45 Our data

suggests thatg∗1 is heavily dependent on the overall QPC potential, presumably via its influence

on exchange interactions within the QPC. Spin density functional theory (SDFT) calculations also

point to exchange effects being very sensitive to the precise geometry of the QPC as the device

approaches pinch-off.35,46,47The reduced variability ing∗ with increasingm ≥ 2 may be due to

improved screening arising from the higher electron density within the QPC (even with fixedn).

Considering the applied implications first, when using a QPCas a spin injector/detector, one

commonly applies a magnetic field to break the 1D subband degeneracy, and operates the QPC

at G < 0.5G0 so that transmission is dominated by the 1↓ subband.3,4 This makes maximizing

10

Figure 5: The precise conductanceG of the 0.7 anomaly vs densityn for Refs.26–29,31,32and ourdata from PI-0, PI-375 and BT in Fig. 6. The devices in Refs.26,29have a midline gate similar to BT(solid circles) and in Ref.31 has a polyimide-insulated top-gate similar to PI (half filled squares).No simple link between the location/evolution of the 0.7 plateau andn is evident suggesting the0.7 anomaly is heavily dependent on how the QPC is defined.

11

Figure 6: ConductanceG versus QPC gate voltageVg as top-gate voltageVt is changed for (a) PI-0,(b) PI-375 and (c) BT. TheVt ranges and increments are (a) 0 (left) to−400 mV (right) in stepsof 100 mV, (b)+375 (left) to−375 mV (right) in steps of 75 mV, and (c) 0 (left) to−210 mV(right) in steps of 7 mV. The purple dashed lines are guides tothe eye highlighting the evolutionof anomalous plateau-like structures atG < G0 with densityn. The red arrows at top/bottom of (c)indicate the right-most trace from whichg∗m data is extracted.

12

g∗1 of prime importance in reducing the magnetic field required for operation. As Fig. 3a shows,

we can achieveg∗1 as large as 2.8 approaching the high values obtained in InGaAs QPCs,18 and

exceeding the 0.75−1.5 typically reported for GaAs QPCs with the field applied in the plane of

the 2DEG.6,10,15,21We note thatg∗ ∼ 4 was recently reported for a GaAs QPC with the field ori-

ented perpendicular to the 2DEG by Rössleret al.22 Although substantially higherg∗ values can

be obtained in QPCs with the field perpendicular to the 2DEG,19 due to the strong confinement

in the heterostructure growth direction48 and exchange, this comes with associated problems of

cyclotron curvature, and at higher fieldsB > 1−2 T, Landau quantization. These can be problem-

atic for spintronic applications, such that in-plane fieldsare more commonly used; here the tight

confinement of the 2DEG allows fields exceeding 10 T to be applied before cyclotron issues arise,

which more than compensates for the lowerg∗ obtained for in-plane fields. An additional benefit of

using an in-plane field to break the spin-degeneracy is that an independently variable and smaller

perpendicular field component can be used to ‘steer’ a ballistic electron beam into a spin-polarized

collector QPC.4,49The trends in Fig. 3a are also important, because in additionto g∗1 values as high

as 2.8 we see values as low as 1.25. Hence the key to achieving and maintaining highg∗1 in QPCs is

very careful management of the QPC’s confinement potential and local electrostatic environment

(e.g., 2DEG density).

The evolution ofg∗1 and∆E1,2 with n in Figs. 3a and 4 also provides an opportunity to ad-

dress the conflict in the literature regarding the density dependence of the 0.7 anomaly. Briefly

reviewing the various observations: the 0.7 plateau was reported to gradually fall towards 0.5G0

with decreasing n in three devices: a modulation-doped midline-gated QPC26 (similar to BT), a

modulation-doped back-gated QPC27 and a modulation-doped QPC wheren was changed using

illumination/hydrostatic pressure.28 In contrast, theopposite dependence (i.e., 0.7 falls to 0.5G0

with increasingn) is observed in two other devices: an undoped QPC with a positively biased

bow-tie top-gate29 and a modulation-doped QPC with a mid-line gate.30 Finally, a 0.7 plateau that

fell towards 0.5G0 with both increasing and decreasingn was reported for an undoped QPC with

a polyimide-insulated top-gate,31 and a 0.7 plateau that is strong at low and highn and which

13

weakens whilst rising to 0.8G0 at intermediaten was reported for a modulation-doped back-gated

QPC.32 A summary of these results is presented in Fig. 5,1 where we plot theG at which the 0.7

plateau appears against densityn for the data in Refs.,26–29,31,32along with our data from Fig. 6.

There is no straightforward link between the location/evolution of the 0.7 anomaly and density in

Fig. 5, instead the behavior appears highly device-dependent. The QPC confinement potential is

also the crux of this problem, as we now show.

Figure 7: Differential conductanceG′(Vsd) vs source-drain biasVsd for a range ofVg for (a) BT atVt =−120 mV, (b) PI-0 atVt =−500 mV and (c) 0 mV. In each caseT < 100 mK andB = 0, with(a)n = 0.95×1011 cm−2 and∆E1,2 = 2.52 meV (b)n = 0.92×1011 cm−2 and∆E1,2 = 1.56 meV,and (c)n=1.70×1011 cm−2 and∆E1,2= 2.66 meV. The width and amplitude of the zero-bias peak(ZBP) is relatively constant withn for PI-0 and PI-375. ZBP suppression is a common feature forall n in BT. The data in each panel has been symmetrized aboutVsd = 0 to remove an asymmetricbackground artifact arising from instrumental issues in the measurement.34

Figure 6a-c showsg versusVg as a function ofVt for PI-0, PI-375 and BT, respectively. The

dashed purple lines highlight the evolution of the 0.7 anomaly withn. Before considering the

experimental data itself, we briefly digress to consider some predictions based on the density-

dependent spin-gap model introduced by Reillyet al,24,25since these will be vital to the discussion.

The one free parameter in this model, the opening rateγ = d∆E↑↓/dVg of the spin-gap∆E↑↓ with

QPC gate biasVg, is suggested to be linked to the potential mismatch betweenthe 1D channel

and 2D reservoirs.24,25 This mismatch is essentially a mode-matching effect,50,51 and should be

1Results by Leeet al30 are omitted from Fig. 5 as we are unable to precisely determinen values for this data.

14

dependent on the 1D confinement strength, i.e.,∆E1,2.37,52,53 Considering Fig. 3(b) of Ref.,25

the prediction is that for increasingγ the 0.7 anomaly will move lower inG and become more

pronounced. If we take the simplest assumptionγ ∝ ∆E1,2,25 two behaviors are expected given the

∆E1,2 data in Fig. 4: a) the 0.7 anomaly should be weak and at higherG for PI-0 and PI-375 and

be more pronounced and at lowerG in BT, and b) in each case the 0.7 anomaly should weaken and

tend to rise inG with decreasingn.

Considering the experimental data now; for both PI-0 and PI-375 we observe a weak inflection

at relatively highG < G0 (Fig. 6(a/b)). This inflection moves to higherG with decreasingn in both

cases, consistent with the corresponding reduction in∆E1,2 in Fig. 4. Because the plateau appears

as a weak inflection, its weakening with decreasingn is unfortunately difficult to distinguish. How-

ever, we note that a similar rise in conductance of the 0.7 anomaly is observed with reduced∆E1,2

by Lianget al (see Fig. 2 of Ref.43), and here the associated weakening of the 0.7 plateau is more

visible. It is also unclear whether the rise in the 0.7 feature with decreasingn is linear; a careful

look at Fig. 6(b) suggests it may not be (see also Fig. 6(c)). This would mean that the assumption

γ ∝ ∆E1,2 may only be approximate to the true functional relationshipbetweenγ and∆E1,2. For the

BT device, the 0.7 anomaly starts as a clear plateau atG ∼ 0.6G0 in the highn limit. The plateau

rises in conductance and weakens with decreasingn, ultimately merging in the lown limit with an

additional anomalous feature dropping down from higherG with decreasingn, as highlighted by

the dotted purple line. The observation of an additional plateau above the 0.7 anomaly is relatively

common26,30,54,55as are small ‘bumps’ on the 0.7 anomaly at higher density.24,30The appearance

of this structure in BT rather than PI is not unexpected – higher ∆E1,2 is associated with a more

sharply defined 1D channel, and this should enhance resonantfeatures in the conductance, as dis-

cussed by Kirczenow.56 While the BT data should be considered with care as disorder effects can

modify the behaviour of the 0.7 anomaly, the 0.7 feature behavior we observe for BT is entirely

consistent with expectations based on the density-dependent spin-gap model25 and the behavior

observed for PI-0/PI-375: The 0.7 plateau in BT is stronger and appears at a lower conductance

than for the PI samples, consistent with the much higher∆E1,2 in Fig. 4. The 0.7 plateau rises

15

and weakens with decreasingn in each case also, following expectations based on Fig. 4 andthe

density-dependent spin-gap model.

One possible explanation for our observations, in particular the link between the 0.7 anomaly

behavior and∆E1,2, but also the sensitivity ofg∗1, is the formation of a quantum-dot-like local-

ized charge state within the QPC due to the 1D-2D mismatch at the QPC openings. There is

strong experimental evidence that this can occur in QPCs including observations of Kondo-like

behavior,20,53,57–61Fano resonances62–64 and Fabry-Pérot oscillations52 on the integer conduc-

tance plateaus by other authors.2 Strong exchange effects that are very sensitive to the size,shape

and symmetry are a well-known feature of ultra-small quantum dots.65,66 We propose that varia-

tions in the size, shape and symmetry of a quantum-dot-like localized charge state within a QPC

might similarly affect the exchange interaction in QPCs, and thereby affect both theg-factor and

0.7 plateau behavior. Under this explanation, theg-factor and its sensitivity to confinement should

decrease with increasing subband index. Accordingly, the magnitude ofg∗ decreases with increas-

ing m in Fig. 4; however, a reduced sensitivity to confinement is not as apparent.

The mention of Kondo-like behavior above naturally leads tothe question of the zero-bias

peak in the differential conductanceG′(Vsd), a feature first discussed in detail by Cronenwettet

al,20 and commonly observed in QPCs.11 The role played by Kondo physics in QPCs is heavily

debated;11 some suggest that Kondo physics drives the weakening of the 0.7 plateau in the low

T limit, 20,67 others argue for the 0.7 plateau and Kondo-like physics in QPCs being separate and

distinct effects.59 In Fig. 7 we plotG′(Vsd) versusVsd for BT and PI-0, withVt settings chosen to

best isolate the effect of differences in∆E1,2 andn (the corresponding data points are indicated

(a-c) in Fig. 4). PI-0 shows a clear zero-bias peak (ZBP) overthe entire range 0< G < G0, this

ZBP behavior is consistent with most previous reports.11,20,57–59The ZBP amplitude and width for

a givenG are relatively independent of density (Fig. 7b/c); similarbehavior is found for PI-375. In

comparison, the ZBP for BT is heavily suppressed (Fig. 7a); whilst evident as a smaller amplitude

peak for 0.4G0 < G < 0.8G0 it vanishes in the limitsG → 0 andG → G0. This behavior also

2We also observe weak Fabry-Pérot-like structure alongVsd = 0 in some of our source-drain bias plots (see Sup-plementary Fig. 4); it is much stronger in Ref.,52 presumably due to the stronger 1D confinement (∆E1,2 ∼ 5 meV).

16

holds qualitatively as density is varied, but the amplitude, width and suppression vary slightly; an

in-depth study will be presented elsewhere. Thermally-induced suppression of the Kondo process,

and hence the ZBP, occurs as the temperatureT increases relative to the Kondo temperatureTK.68

In QPCs, this typically occurs forT > 0.5 K.20,57,59However, in our experimentT is fixed at

< 100 mK; hence it must beTK that differs between PI-0 and BT, withT BTK < T PI

K . This difference

in TK is not unexpected. In quantum dots,TK depends sensitively on the charging energyU , the

bound-state energyε0 and the couplingΓ to the reservoirs.68 These can be independently tuned in

dots,68 but not for a localized charge state within a QPC. Simulations predictΓ to be particularly

sensitive to QPC potential,37 but the influence of other factors such as QPC length, width, and

most notably 1D-2D mismatch, i.e., 1D confinement strength∆E1,2, are unknown. Note however

that the data in Fig. 6 is consistent with a Kondo-like scenario:20,37,67for BT where the ZBP is

suppressed, the 0.7 plateau is more evident and at a lower conductance than for PI, where the ZBP

is stronger. The behavior in Fig. 7 cannot be tied directly to∆E1,2 or n alone; we suggest that it

instead depends on the precise nature of the QPC confinement potential. The change in the ZBP

we observe may indicate a change in the coupling of a single localized state to the reservoirs, or

potentially to the emergence, loss, or interaction of multiple localized states within the QPC asG

is driven fromG0 to 0.35,69

In conclusion, we have studied the dependence of the 1D Landég-factorg∗ on density in QPCs

with two different top-gate architectures. We obtaing∗ values for the lowest 1D subband of up

to 2.8, approaching the high values obtained in InGaAs/InP QPCs,18 and significantly exceeding

previously reported values for the in-planeg-factor of AlGaAs/GaAs QPCs.6,10,15,20,21Careful

management of the QPC’s confinement potential appears key toobtaining highg∗1. This has im-

portant implications for using QPCs in spintronic applications. The appearance of the 0.7 plateau

is strongly linked to 1D confinement potential, explaining the conflicting density dependencies

reported in the literature.17,24–32 In particular, the 0.7 anomaly behavior in our devices is con-

sistent with predictions made using the density-dependentspin-gap model24,25 with the one free

parameter taken as directly proportional to the lowest 1D subband spacing∆E1,2.

17

Supporting Information. Extended details of methods used, device characterizationdata and

additional supporting data. This material is available free of charge via the Internet at http://pubs.acs.org.

Corresponding author. *E-mail: adam.micolich@nanoelectronics.physics.unsw.edu.au

Acknowledgement

This work was funded by the Australian Research Council (ARC) through the Discovery Projects

Scheme. APM acknowledges an ARC Future Fellowship (FT0990285). ARH acknowledges an

ARC Professorial Fellowship. IF and DAR acknowledge financial support from the EPSRC. We

thank T.P. Martin, U. Zülicke, D.J. Reilly and Y. Meir for helpful discussions.

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