Phenomenology of the 0.7 conductance feature

Abstract

We describe a phenomenological model for the conductance feature near $0.7 \times 2e^2/h$ that occurs in quantum point contacts. We focus on the transconductance at finite source-drain bias and contrast our model with the results expected from a single-particle picture. Good agreement is seen in comparing the model with experimental data, taken on ultra-low-disorder GaAs induced electron systems. Although simple, our phenomenology suggests important boundary conditions for an underlying microscopic theory. Comment: Proceedings of EP2DS-16 (July 2005). Comments welcome. reilly@fas.harvard.edu

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arXiv:cond-mat/0601135v1 [cond-mat.mes-hall] 7 Jan 2006
Phenomenologyofthe0.7conductancefeature
D. J. Reilly,∗Y. Zhang and L. DiCarlo
Department of Physics, Harvard University, Cambridge, 02138, USA
Abstract
We describe a phenomenological model for the conductance feature near 0.7×2e2/h that occurs in quantum point contacts.
We focus on the transconductance at finite source-drain bias and contrast our model with the results expected from a single-
particle picture. Good agreement is seen in comparing the model with experimental data, taken on ultra-low-disorder GaAs
induced electron systems. Although simple, our phenomenology suggests important boundary conditions for an underlying
microscopic theory.
Key words: 0.7, QPC, phenomenological, quantum wire, conductance
PACS:
1. Introduction
The conductance of a quantum point contact
(QPC), formed via geometric or electrostatic confine-
ment of a two-dimensional electron system (2DES), is
quantized in units of 2e2/h [1,2]. This quintessentially
quantum-mechanical phenomena is well explained in
terms of the single-particle discrete energy spectrum
associated with the confining potential [3]. QPCs also
exhibit unexpected non-quantized features that can-
not be explained within this non-interacting, single-
particle picture. A clear example is the feature occur-
ring below the first conductance plateau, in the range
0.5−0.7×2e2/h and at equivalent positions for higher
plateaus. This anomalous conductance structure was
observed in the earliest transport measurements on
ballistic QPCs [1,2] but has remained largely unex-
plained despite numerous experimental investigations
[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19] and is re-
∗Email: reilly@fas.harvard.edu
garded as a key outstanding problem in mesoscopic
physics [20].
A prime result, initially uncovered by Thomas et al.,
[4] is the evolution of the 0.7feature with increasing
parallel magnetic field into the well understood Zee-
man spin-split plateau at 0.5 × 2e2/h. This observa-
tion strongly suggests that the 0.7 structure at B = 0
is related to a many-body state in which symmetry is
broken between spin-up and spin-down electrons.
2. Phenomenological model
Here we describe a simple phenomenological model
that is in striking agreement with the experimental be-
havior of the conductance feature with temperature
T, magnetic field B, source-drain bias Vsd and poten-
tial profile. At the core of this model is the conjecture
that a density-dependent spin-gap opens in the one-
dimensional (1D) energy spectrum as the gate volt-
age is made less negative [13,21]. Fig. 1a captures the
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Fig. 1. a) Energy level configuration underlying our phe-
nomenological model. A spin-gap opens in the 1D energy
spectrum as a function of gate bias Vg. The Fermi-level
EF is parabolic with gate voltage because of the singular-
ity in the 1D density of states. b) Shows the difference be-
tween EF and the spin-up and spin-down band-edges. c)
Shows conductance calculations based on our model (red
and dashed line) together with experimental data (black
and solid line) taken at T=50mK. d) Shows calculated con-
ductance in the context of the model for increasing parallel
magnetic field B. Consistent with experiment we observe a
smooth evolution from 0.7 to 0.5×2e2/h. Panels c) and d)
are taken from Reference [21].
essence of the model. Before the first 1D sub-band is
occupied (below energy E1 in Fig. 1a) the spin-up and
spin-down sub-bands are degenerate. As the 1D states
begin to fill the spin-down band rapidly moves down in
energy with the available spin-up states moving higher
in energy above the Fermi-level (EF). The model as-
sumes a parabolic dependence of the Fermi-energy on
gate voltage (EF ∼ V2
E−1/2form of the 1D density of states [22]. For a de-
tailed discussion of this model we refer the reader to
Refs. [21,16].
Features in conductance are related to the occupa-
tion probability of the 1D sub-bands and are a func-
tion of temperature and the energy difference between
the sub-band edge and EF. Within the context of our
model, Fig. 1b shows the difference between the spin-
dependent band-edge energy and EF as a function of
gate voltage Vg. It is the functional form of this dif-
ference between the spin band-edges and EF that re-
lates to the experimental data (since it sets the con-
ductance) and of prime importance in suggesting possi-
ble ‘boundary conditions’ of an underlying microscopic
g) in connection with the usual
theory.Wepoint out that in contrast to a similar model
based on pinning of the band-edge [23], in our picture
the spin-down energy continues to move rapidly below
the Fermi-level as the gate voltage is swept.
Figure 1c (taken from Ref [21]) compares differen-
tial conductance g = di/dv calculations based on this
model with data taken on an ultra-low-disorder quan-
tum wire at T=50mK (See Refs [13,16,21] for exper-
imental details). We take the simplest approximation
and calculate conductance with a step transmission
function in the limit of zero source-drain bias. Impor-
tantly, the only free-parameter needed to fit the model
to the experimental data is the rate at which the spin-
gap opens with gate voltage γ = dE↑↓/dVg [21]. Note
the small discrepancy between model and experiment
at the top right of Fig. 1c which is due to the step
transmission function (no tunneling) that is used in the
calculation rather than a more realistic smoothly vary-
ing T(E) [3]. Fig. 1d (also taken from Ref. [21]) shows
conductance calculations as a function of increasing
parallel magnetic field, B?in which a smooth evolu-
tion of the conductance feature from 0.7 to 0.5×2e2/h
is observed, consistent with the experimental results
[4,15,24].
3. Single-particle results
We now focus on the transconductance (dg/dVg)
at finite Vsd which facilitates the study of transitions
between the conductance plateaus as a function of
chemical potential or gate voltage Vg. We firstly com-
pare transconductance plots based on our model to
the single-particle case at zero and finite magnetic
field. Fig. 2a shows a transconductance intensity plot
as a function of Vsd and Vg calculated using the
usual single-particle formalism [25,26]. We assume
the conductance at finite bias is approximated by the
weighted average of two zero Vsd conductances, one
for the chemical potential of the source (EF + βeVsd)
and the other for the drain (EF − (1 − β)eVsd), where
β characterizes the symmetry of the potential drop
across the constriction [26]. Centered about Vsd=0 are
the linear response integer-plateau diamonds (dark)
at 0,1 and 2 × 2e2/h. Symmetric either side of the
integer plateaus are the finite bias ‘half-plateaus’ that
occur when the chemical potential of the source (µs)
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Fig. 2. Single-particle (non-interacting) transconductance
dg/dVgcalculations for B=0 (a) and B?=8T (b). Transcon-
ductance is plotted on the intensity-axis in arbitrary units
as a function of souce-drain bias Vsdand gate voltage Vg.
Following Glazman and Khaetskii [25] we observe integer
plateau diamonds (dark areas) centered at Vsd=0 (labeled
0, 1 and 2...). Either side of the integer plateaus are the
finite bias half-plateuas (labeled 0.5 and 1.5...) that occur
when µsand µddiffer by two spin bands (a). In (b) the finite
B-field Zeeman spin splitting results in zero-bias plateaus
(labeled 0, 0.5, 1, 1.5 and 2...) and finite-bias plateaus (la-
beled 0.25, 0.5, 0.75, 1.25, 1.5, 1.75....).
or drain (µd) differ by one (spin degenerate) sub-band
(g = 0.5 and 1.5 × 2e2/h) [27].
With the application of an in-plane magnetic field
B?the spin degeneracy is lifted and additional Zeeman
spin-split plateaus appear at 0.5,1.5×2e2/h at Vsd=0
as shown in Fig. 2b. At finite bias and high magnetic
field new plateaus also appear at 0.25,0.75,1.25,1.75×
2e2/h. These are thefinitebias ‘half-plateaus’ that now
occur at quarter intervals with thelifting of thespin de-
generacy. The relative size of these ‘quarter-plateaus’
in comparison to the half-plateaus simply depends on
the relative Zeeman splitting to sub-band energy spac-
ing. In Fig. 2b we show calculations for B?=8T, setting
the in-plane g-factor to the bulk value of 0.44.
4. Comparison of model with experiment
Returning to our phenomenology, Fig. 3a shows
transconductance calculations based on the model. At
low gate voltage the sub-bands are spin-degenerate
and the transconductance looks similar to the B=0
single-particle case of Fig. 2a (in the range g <
0.5 × 2e2/h). With increasing 1D density (making the
gate voltage less negative) the degeneracy begins to
Fig. 3. Comparison of model and experiment. a) Transcon-
ductance dg/dVg calculations based on the phenomenolog-
ical model, for B=0 (transconductance is plotted on the
intensity-axis in arbitrary units as a function of souce-drain
bias Vsd and gate voltage Vg). Note the similarity with
the finite B-field case of Fig. 2b), but with an absence of
low-bias ‘quarter-plateaus’ at 0.25,1.25... × 2e2/h consis-
tent with a gate-dependent spin-gap. b) Shows transcon-
ductance data for an ultra-low-disorder QPC, formed using
induced-electron gating techniques [6,13] B=0, T=100mK.
Data is the numerical derivative with respect to Vg of data
in Fig. 5a, Ref. [21].
lift as a spin-gap opens in the 1D energy spectrum. At
g > 0.5 × 2e2/h the transconductance now resembles
Fig. 2b as the spin degeneracy is lifted and finite-bias
plateaus beginning at 0.75 × 2e2/h are observed.
Figure 3b shows data taken on an ultra-low-disorder
quantum wire at T=100mK. In these devices the elec-
trons in both the 1D and 2D regions are induced via
the application of positive bias to a surface gate [6].
Close examination of the experimental data reveals an
absence of finite bias plateaus at 0.25 × 2e2/h and the
presence of 0.75×2e2/h features, resembling our calcu-
lated results in Fig. 3a. This behavior is consistent with
a picture of a density-dependent spin gap opening with
gate voltage. Of particular interest is the occurrence of
weak features at 1.25 × 2e2/h, beyond the Vsdneeded
to produce the usual half-plateaus. These weak struc-
tures, which are present in both our calculations and
experimentaldata, connect to theusual0.7×2e2/h fea-
ture via the line that intersects the top half of the first
(2e2/h) diamond (either side of the label ‘1’). In the
context of the model these high Vsd1.25 features arise
from the chemical potential of the source and drain
differing by 3 spin sub-bands, as in the case where the
source is at 2 × 2e2/h (above 4 spin band-edges) and
the drain is at 0.5 × 2e2/h (in the gap between the
first spin-up and spin-down sub-bands). The presence
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Fig. 4. Transconductance calculations based on the model
for Vsd=0 as a function of gate bias Vg and in-plane mag-
netic field, B (traces are off-set). We use a bulk value for
the electron g-factor of 0.44. See main text and Ref. [24]
for detailed discussion.
of high bias1.25 features providesfurther evidencethat
a spin dependent energy gap remains open well below
the Fermi-level, continuing to increase (slightly) as the
1D density grows and higher sub-bands are populated.
5. 0.7 analogs
We now discuss our phenomenological model in the
context of a recent experiment by Graham et al., [24],
in which the sub-band energies can be made to cross at
very high in-plane magnetic fields. In addition to the
usual evolution of the 0.7 feature towards 0.5× 2e2/h,
Graham et al., discovered the appearance of similar
conductance structure at high B?evolving from 1.5 ×
2e2/h into the plateau at 2e2/h. This feature is termed
a ‘0.7-analog’. In Fig. 4 we show transconductance cal-
culations based on our modelas afunction of increasing
Zeeman energy (bottom to top). For simplicity we ig-
nore any changes in the electron g-factor related to cor-
relation or diamagnetic effects, taking the bulk value of
0.44. This likely accounts for the larger magnetic fields
required in our model to achieve sub-band crossing in
comparison to experimental observation.
Consistent with the experimental results of Graham
etal.,[24]weobservetheusual0.7featureatB=0,indi-
cated in Fig. 4 by the black region separating the spin-
up and spin-down transconductance lines that evolve
away from eachotherasB isincreased.Withoutadding
any new parameters to our model we also find good
agreement with the experimental results at magnetic
fields where the N=1 spin-up sub-band crosses the
N=2 spin-down sub-band. As in the case of the 0.7
structure at B=0, our model assumes that at high B?
as each spin-band is populated, the Fermi energy EF
is proportional to V2
g and the gate-dependent spin gap
(intrinsic) adds to the (extrinsic) Zeeman energy of the
B-field.
Following the crossing of sub-bands there is a dis-
continuous shift in δVg from the crossing point, mark-
ing the appearance of the 0.7 analog [24]. Our calcula-
tions also account for the appearance of higher order
0.7 analogs at the crossings of N=2 spin-up with N=3
spin-down sub-bands. The temperature dependence of
the 0.7 analogs follows the usual activated behavior of
the 0.7 feature in our model (not shown). Very recent
numerical work [28] using spin-density functional the-
ory also shows good agreement with the experimental
results of Graham et al.. How the functional form of
our phenomenology relates to these more detailed cal-
culations is an interesting question for future work.
The strong agreement between the transconduc-
tance calculations and the experimental data of Gra-
ham et al., provide further evidence that this model
captures the essence of the functional form underlying
the 0.7 feature. Despite this agreement however, the
observation of a zero bias anomaly in the bias spec-
troscopy and the low temperature restoration of the
conductance as observed by Cronenwett et al., [15]
cannot be explained without extending this model
to also include Kondo spin screening in the regime
T < TK, where TK is the Kondo temperature. Perhaps
a complete explanation will account for a density-
dependent gap that is screened by the formation of a
Kondo-like state at low temperatures and bias.
6. Conclusions
We have described a simple phenomenological model
for the 0.7 ×2e2/h conductance feature that occurs in
quantum point contacts. Our focus has been on the
transconductance at finite source-drain bias which em-
phasizestransitions between theconductanceplateaus.
The calculated results based on our model agree well
with data taken on both ultra-low-disorder induced
devices and more traditional split-gated heterostruc-
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tures. Without the inclusion of additional parameters
this model also accounts for the observation of 0.7
analogs at high magnetic field. Although largely em-
pirical, we believe the functional form underlying this
simple picture may prove useful in uncovering a de-
tailed microscopic theory of this effect.
7. Acknowledgments
The authors wish to thank C. M. Marcus, T. M.
Buehler, D. T. McClure, J. L. O’Brien, S. Das Sarma,
K.-F. Berggren, M. J. Biercuk, R. G. Clark, A. Dzurak
and A. R. Hamilton for useful discussions. D.J.R. is in-
debted to L. N. Pfeiffer and K. W. West for the excel-
lent heterostructure material used in the experiments.
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