Temperature Dependence of Conductance and
Thermopower Anomalies of Quantum Point Contacts
O. A. Tkachenko∗ and V. A. Tkachenko

arXiv:1302.4025v1 [cond-mat.mes-hall] 17 Feb 2013

Rzhanov Institute of Semiconductor Physics SB RAS, 630090 Novosibirsk, Russia
(Dated: July 2, 2018)
It has been shown within the Landauer single-channel approach that the presence of the 0.7 anomaly in the
conductance of a ballistic microcontact and the respective plateau in the thermopower implies unusual pinning
of the potential barrier height U at a depth of kB T below the Fermi level EF . A simple way of taking into
account the effect of electron-electron interaction on the proﬁle and temperature dependence of a smooth onedimensional potential barrier in the lower spin degeneracy subband of the microcontact has been proposed. The
calculated temperature dependences of the conductance and Seebeck coefﬁcient agree with the experimental
gate-voltage dependences, including the emergence of anomalous plateaux with an increase in temperature.
PACS numbers: 71.10.Ay, 71.45.Gm, 73.23.Ad, 73.50.–h, 73.50.Lw, 73.63.Rt

I. INTODUCTION

Quantization of the conductance of submicron constriction in two-dimensional electron gas1 is described well by
the Landauer formula under the assumption of spin degeneracy of one-dimensional single-particle subbands in zero
magnetic ﬁeld.2–5 The same approach explains the alternation of zero plateaux and peaks of thermopower (Seebeck coefﬁcient S), which obeys the Mott formula, S M ∝
∂ ln G(Vg , T )/∂EF .6–10 However, the dependence G(Vg ) of
the conductance on the gate voltage exhibits a narrow region
of anomalous behavior, the 0.7 · 2e2/h plateau.11 This plateau
broadens with an increase in temperature,11–16 can disappear
at T → 0,12–16 but persists at a complete thermal spread of the
conductance quantization steps.14,15,17 The 0.7 conductance
anomaly is closely related to the anomalous plateau S = 0
of the thermopower,17 which implies violation of the Mott approximation S ∝ ∂ ln G(Vg , T )/∂Vg .9 There are dozens of
works attempting to explain the 0.7 anomaly (see Refs. 18, 19
and references therein). Numerous scenarios including spin
polarization, Kondo effect, Wigner crystal, charge-density
waves, and the formation of a quasi-localized state have been
suggested. Calculations that reproduce the unusual temperature behavior of the 0.7 conductance anomaly have been
performed so far only within phenomenological ﬁtting models with spin subbands,20 or beyond the Landauer formula.18
Although a particular mechanism of the appearance of the
anomalous plateaux in conductance and thermopower remains
unclear, their common reason is thought to be the electronelectron interaction, which should manifest itself most effectively at the onset of ﬁlling the ﬁrst subband, where the electron system is one-dimensional.
In this work, we start from the standard Landauer approach to the description of conductance and thermopower
of a single-mode ballistic quantum wire with spin degeneracy. This approach take into account interaction via the T dependent one-dimensional reﬂecting barrier. First, we show
that the appearance of the 0.7 anomaly implies pinning of the
barrier height U at a depth of kB T below the Fermi level
EF and that this pinning yields the plateau S = 0. Next,
motivated by description of Friedel oscillations surrounding a

delta-barrier in a one-dimensional electron gas,21 we suggest
a simple formula which reduces the T -dependent part of the
correction to the interaction-induced potential to the temperature dependence of the one-dimensional electron density.
The behavior of the conductance and Seebeck coefﬁcient
calculated with the corrected potential agrees with the published experimental results.

II. BASIC FORMULAS AND UNUSUAL PINNING

Conductance and Seebeck coefﬁcient of single mode ballistic channel can be written within the Landauer approach as
follows:6–8
2e2
h
2ekB
S=−
hG

∞

G=

D(E, U (x, Vg ))F (ǫ)dE,
0

(1)

∞

D(E, U (x, Vg ))ǫF (ǫ)dE,
0

where D is the transmission coefﬁcient, E is the energy
of ballistic electrons, U (x, Vg ) is the effective T -dependent
one-dimensional barrier, ǫ = (E − EF )/kB T , F (ǫ) =
4kB T cosh−2 (ǫ/2) is the derivative of the Fermi distribution
function with respect to −E. There is also the Mott approximation generalized for arbitrary temperatures T :10,17
SM = −

2
π 2 kB T ∂ ln G(Vg , T )
.
3e
∂EF

If the barrier U (x) in a one-dimensional channel is sufﬁciently wide (according to the three-dimensional electrostatic
calculations of gate-controlled quantum wires the barrier half>
width must be ∼ 200 nm), the step in the energy dependence
D(E) of the transmission coefﬁcient is abrupt and the respective transition is much narrower in the energy E than the thermal energy kB T , at which the 0.7 plateau occurs. This condition is satisﬁed in many experiments.12–16 Then, Eq. 1 for G,

2
S and Mott approximation gives
G=

2e2
(1 + e−η )−1 ,
h

kB
[(1 + e−η ) ln(1 + eη ) − η],
e
kB π 2
SM = −
(1 + eη )−1 ,
e 3

S=−

(2)

where η = (EF −U )/kB T and U = U (x = 0) is the height of
the barrier U (x, T, Vg ). Clearly, the dependences G(EF − U )
at different T are simply smooth steps of unit height with the
ﬁxed common point G = e2 /h. Curves S M (η) and S(η) are
numerically close to each other at the interval 0 < S < 2kB /e
(see Appendix A). The values G ≈ 0.7 · 2e2 /h in G(η)
are not particularly interesting except that they correspond to
η ≈ 1. However, in experiments, there appear plateaux of
G(Vg ) at these values, which implies pinning of U (Vg ) at a
depth of kB T below the Fermi level (see Appendix A). According to Eq.(2), the discovered pinning can be expected to
give the plateau S ≈ −0.8kB /e (S M ≈ −kB /e) in the curve
S(Vg ). Appendix A compares the calculated plateaux with
the experimental ones,17 and shows that parameter η obtained
from G is equal to that from S, which veriﬁes applicability
of Eq.(2) to the experiment. Notice that this pinning differs
from the pinning discussed earlier13,17,20,22–24 by unusual temperature dependence and single-channel transmission. The
pinning that we detected seems paradoxical and urges us to
suggest that a probe ballistic electron at the center of the barrier would “see” the potential U (T, Vg ), which is different
from the potential V (T, Vg ) computed self-consistently with
the electron density (see Appendix B). In fact, similar to our
previous calculations,4,5 we computed three-dimensional electrostatics of single-mode quantum wires using different kinds
of self-consistency between the potential and the electron density with24 and without the inclusion of exchange interaction
and correlations in the local approximation. These calculations show quite deﬁnitely that the one-dimensional electron
density nc in the center of the barrier is almost independent of
T and is linear in Vg starting from small nc0 values; i.e., the
electric capacitance between the gate and the quantum wire is
conserved at G > e2 /h. In addition, since the density of states
is positive, d(EF − V )/dnc > 0 and the dependence V (Vg )
of the self-consistent barrier height on the gate voltage does
not yield pinning even with the inclusion of the exchangecorrelation corrections in the local approximation (Appendix
B). Therefore, we suggest that the discovered pinning of the
reﬂecting barrier height is due to the nonlocal interaction.

III.

ESTIMATION OF NONLOCAL INTERACTION

It is well known in atomic physics and physics of metallic surfaces and tunneling gaps between two metals that the
potential seen by a probe electron in a low-density region
is different from the self-consistent potential found with the
inclusion of interaction in the local approximation.25–28 This
difference was attributed to nonlocal exchange and correla-

tions, i.e., to the attraction of the electron to an exchangecorrelation hole, which remains in a high-density region.29,30
Consideration of this phenomenon regarding a quantum point
contact is currently unavailable, despite the obvious analogy
between two metallic bars separated by a tunneling gap and
two-dimensional electron gas baths separated by a potential
barrier. We suggest that a ballistic electron coming to the
barrier region, where the density nc is low, gets separated
from its exchange-correlation hole, which is situated in the
region of dense electron gas. As a result, the local description of the correction to the potential becomes inadequate.
Although the hole has a complicated shape, it can be associated with the effective center. Then, a decrease in the potential for the ballistic electron in the center of the barrier
is U − V ≈ −γe2 /(4πǫǫ0 r), where r is the distance be<
tween the centers of the barrier and hole, whereas γ ∼ 1 takes
into account the shape of the hole and weakly depends on r.
In perturbation theory, we are interested in a small (i.e., T dependent) part of the correction:
δU ≈ −[e2 /(4πǫǫ0 ]γ(r(T )−1 − r(0)−1 ),

(3)

and the correction at T = 0 is thought to be already included in the independent variable, which is the initial barrier
U0 (x). Obviously, r decreases with an increase in nc , until
the electron and hole recombine and the local approximation
for the interaction term becomes valid in the center of the barrier. According to this tendency and the smallness of the T dependent correction, we can write γ(r(T )−1 − r(0)−1 ) ≈
(nc (T ) − nc (0))/(r∗ n∗ ), where nc (T ) and nc (0) are found
c
perturbatively from the single-particle wavefunctions in the
barrier U0 (x). In a certain range of the barrier height, we
can also neglect a change in the positive phenomenological parameter γr∗ n∗ . Under these assumptions, Eq.(3) is
c
formally a special case of the interaction-induced correction
δU (x) ∝ −αδn(x), here, δn(x) stands for Friedel density oscillations. This correction results from calculation of the propagation through the delta barrier in a one-dimensional electron
system,21 in which case α = α(0) − α(2kF ) is a result of the
competition between the exchange (α(0)) and direct (α(2kF ))
contribution to the interaction. A similar correction was used
to simulate multimode quantum wires.31 We can attempt to
extend the range of this correction, with the respective change
in the meaning of α to the entire ﬁrst subband of the quantum
wire, including the top of the smooth barrier.
IV.

CALCULATION OF 1D ELECTRON DENSITY

To ﬁnd this correction perturbatively, we ﬁrst compute the
complete set of wavefunctions for the bare smooth barrier
U0 (x) and ﬁnd the electron density n(x) at a given temperature:
n=

1
2πb

∞
0

|ψL (x, E)|2 + |ψR (x, E)|2
dE
, (4)
(EE0 )1/2
1 + e(E−EF )/kB T

where E0 = h2 /2m∗ b2 , b = 1 nm is the length scale,
¯
m∗ = 0.067me is the electron effective mass, and ψL (x, E),

3
0,005

V0=3 meV

0,12

V =8 meV
0

0,000

4

7

4.5

0,10

-0,005

5

5.5

)

6

-1

-0,015

n (nm

n (nm

-1

)

6
-0,010

5.5

0,08

7

0,06

8
0,04

5
-0,020

T, meV

-0,025

0.01
0,02

4.5

4
-0,030

0.21
0.41

3
-0,035

0.61
0,00
-500

T, meV

-400

-300

-200

-100

x (nm)

0

0.01

-0,040

-400

0.61
0

200

400

x (nm)

FIG. 2: Electron density correction δn = n(x) − n0 (x) at the
same parameters as in Fig. 1. Curves for different V0 are offset by
0.005 nm−1 for clarity.

is varied independently and, according to the electrostatic calculation (see Appendix B and Refs. 4, 5), there is a linear
relation between nc and Vg above some small nc0 value, so
that the temperature dependence of nc (Vg ) can be neglected.
According to Fig. 3, this contradiction is resolved under the
assumption that the quantity V0 actually depends on T at constant nc or Vg . The relation between Vg and the height of

0,03

T, meV

0.11

0,02

0.21

(nm

-1

)

0.01

0.31
0.41

c

ψR (x, E) is the wavefunction of electrons incident on the
barrier from the left (right). The amplitude of the incident wave is set to unity. The bare potential is speciﬁed
as U0 (x) = V0 / cosh2 (x/a). This form is quite appropriate for simulation of short ballistic channels, including escape to two-dimensional reservoirs.3,4 This potential does not
yield any features in the transmittance D(E) except the steps
of unit height.32 The values of the parameters were taken
to be typical for ballistic quantum wires in a GaAs/AlGaAs
two-dimensional electron gas. The calculated dependence
n(x, T ) is presented in Fig. 1. One can see that the density strongly changes with increasing temperature in the transition from the tunnel regime to the open one. At the lowest
temperature there are Friedel oscillations (FOs) in the tunnel
regime, while in the open regime they are suppressed. Calculated correction δn(x, T ) is a wide perturbation of density across the whole barrier (Figs. 1 and 2). At V0 > EF
one can see thermally activated increase n(x, T ) at the barrier top; this temperature behavior inverts in the open regime
V0 < EF . Details of this temperature behaviour are best seen
in δn = n(x) − n0 (x), where n0 (x = 0) = nc (T = 0), and
n0 (x = 0) = n(x, T = 0) is the density averaged over the
Friedel oscillations (Fig. 2).
The most interesting for the analysis of the consequences
of Eqs. (2), (3) is the dependence of the electron density nc
in the center of the barrier on the bare height V0 at various T .
Figure 3 shows quite clearly the details of the T -dependent
behavior of nc and δnc , when U0 (x) is an independent variable. In the experiment, on the contrary, the gate voltage Vg

-200

n

FIG. 1: 1D-electron density calculated with formula (4) at EF =
5 meV for potential U0 (x) = V0 / cosh2 (x/a), where the halfwidth a is ﬁxed a = 200 nm. Curves for different V0 are offset
by 0.01 nm−1 for clarity.

0.21

0.41

0,01

0.51

E

F

0,00
4

V

0

5

6

(meV)

FIG. 3: Electron density nc in the center of the barrier versus the
height V0 of the barrier U0 (x) = V0 / cosh2 (x/a) calculated according to Eq. (4) with a = 200 nm and EF = 5 meV.

4
the bare barrier V0 is mediated by the electron wavefunctions
and the Fermi distribution. This relation and n(x, T ) do not
depend, in the ﬁrst-order perturbation theory, on interaction.
However, they do contribute into it.
(meV)

According to Eq.(3) and similar to Ref. 21 the T -dependent
part of the interaction-induced correction to the bare potential
U0 (x) was calculated with the use of the phenomenological
formula:
δU (x) = −απ¯ vF δn(x, EF , T ),
h

(5)

where α = const > 0, (¯ vF )−1 is the one-dimensional denh
sity of states far from the barrier, δn = n(x) − n0 (x) where
n0 (x = 0) = nc (T = 0), and n0 (x = 0) = n(x, T =
0) is the density averaged over the Friedel oscillations. The
interaction-corrected potential U (x) at various V0 and T values is shown in Fig. 4. At high V0 values and low T values,
penetration of an incident electron to the classically forbidden region of the barrier is very low and the thermal perturbation of the electron density inside the barrier is negligible.
Therefore, the potential barrier remains almost unchanged. At
V0 = EF , the height of the barrier U (x) is lowered considerably with an increase in temperature. At constant V0 < EF ,
the barrier U (x) is raised with T , in contrast to the case of
V0 ≥ EF . The behavior of the barrier height U is shown in
more detail in Fig. 5. For T = 0 we have U = V0 , because
δn(x = 0) = 0 by deﬁnition. One can see that U becomes
independent of V0 near EF at T > 0.1 meV. There appears a
8

T, meV

U (meV)

0.01

0.21

0.41

6

0.61

EF

4

3

2

1

0

5

T, meV
0.01
0.11
0.21
0.31

4

0.41

EF

4

V

0

5

0.51

6

(meV)

FIG. 5: Corrected barrier height versus the original barrier height V0
at the same parameters as in Fig. 4.

plateau below EF . It becomes broader and deeper with an increase in T . The relative correction ∆U (x)/V0 at the parameters speciﬁed in the ﬁgure caption reaches 10%. This is close
to the limiting value for the present approximation, which implies that the correction is small. This limits the growth of T
and α in the model. Figure 5 in combination with Eq.(2) provides a qualitative understanding of the development of the
0.7 conductance anomaly and the thermopower plateau with
an increase in temperature. As is seen, the height U of the
temperature-dependent barrier near EF is stabilized at about
T below EF .

VI. CALCULATED TRANSPORT ANOMALIES

7

5

x=0

CORRECTED POTENTIAL

U

V.

6

-300

-200

-100

x (nm)

0

FIG. 4: Potential U0 + δU (V0 , T ) calculated from Eq. (5) with α =
0.2 for the case of U0 (x) = V0 / cosh2 (x/a), a = 200 nm and
V0 = 3, 4, 5, 6, 7, 8 meV.

Conductance as a function of the barrier height V0 was calculated with the aid of formulas (1), (4), (5). The result is
shown in Fig. 6. There is an usual conductance step with unit
height at T = 0.01 meV. However with increasing temperature additional 0.7-plateau is developed at V0 = EF . The
width of these plateaux well corresponds to temperature. Notice that the height of corrected barrier is almost not changed
at the lowest temperatures and only the distant Friedel oscillations can have inﬂuence on transmission. Indeed, similar
to Ref. 21, scattering off Friedel oscillations leads to a small
decrease in transmission coefﬁcient at low but ﬁnite temperatures and a shift of the conductance step to the lower values of
V0 . For comparison we show in Fig. 6b the curves G(V0 , T )
calculated for the bare potential U0 (x) = V0 / cosh2 (x/a).
Figure 7 shows that conductance behavior is somewhat universal for elevated T and α, and it strongly differs from that
for T → 0. Conductance G(T, α) for corrected potential is
plotted as a function of G(T, α = 0) in Figure 7a. These conductances are related to each other via common parameter V0 .
Conductance G calculated at EF = V0 as a function of interaction parameter α shows that the height of the 0.7-plateau is
saturated with increasing α (Fig. 7b).

5
2

1,0

a=700 nm

(-kB/e)

(a)

0,6

2

(2e /h)

0,8

G

0,2

=0.2

E

F

0,0

T,

1,0

T, meV

1,0

G (2e /h)

0.31

2

0.41
0.51

0,4

0.11

0,8

2

0.21

0,6

0.06

0.16

0.11

(b)

0.01
0

meV
0.01

0,8

(2e /h)

1

S

0,4

0,6

0,4

G

=0

0,2

0,2

E

V0

F

0,0
3,5

4,0

4,5

V

0

5,0

5,5

6,0

0,0

6,5

4,4

(meV)

F

The dependence of the Seebeck coefﬁcient on EF and T
(Fig. 8) was computed from Eqs. (4), (5), and (1). In this
case, we used a larger barrier half-width and a lower α value
than before. In agreement with the analysis within approximation (2) and similar to the experiment,17 S(EF ) exhibits
an anomalous step with a height of 0.9-1.0 of −kB /e. As
is clearly seen in Fig. 8, this step is formed with an increase
in T simultaneously with the 0.7 conductance anomaly. It is
noteworthy that such a combined evolution with temperature
has not yet been observed. Thus, we propose to make the respective experiment for the additional proof of the proposed
model.
1,0

(b)

0,6
T,

2

(2e /h)

=0.3

(a)

meV
0.01

0,4

G

0.11
0.21

0,2

0.31
0.41
0.51

0,0
0,0

0,2

0,4

G

0,6

0,8

1,0 0,3

4,8

E

FIG. 6: (a) Conductance G(V0 , T ) calculated for corrected potential U (x) = U0 (x) + δU (x) with U0 (x) = V0 / cosh2 (x/a),
a = 200 nm and interaction parameter α = 0.2. (b) G(V0 , T ) for
U (x) = U0 (x)

0,8

4,6

0,2

0,1

0,0

2

(2e /h)

FIG. 7: (a) Calculated G(Gα=0 , T ) for the same bare potentials as in
Fig. 6a, but for α = 0.3. (b) Calculated G(α) at V0 = EF = 5 meV
for the same bare potentials and different T : 0.01 ≤ T ≤ 0.51 meV.
The upper curve in (b) represents all the curves for T = 0.11 ÷ 0.51,
as they ﬁt within the indicated error bars.

5,0

5,2

(meV)

FIG. 8: Calculated thermopower and conductance of the onedimensional channel versus the Fermi energy EF at constant V0 (the
parameters are indicated in the ﬁgure).

Though dependences U (V0 ), G(V0 ), G(EF ) and G(Gα=0 )
are easy to calculate, they can hardly be measured. However,
we can compare with the experiment the respective dependences on the electron density nc in the center of the barrier,
see Fig. 9.
In the ﬁgure, dependences (a)–(c) show the plateaux which
appear and become more pronounced with increase of the
temperature. The shape of the computed plateaux is almost
the same as that of experimental ones (see Figs. 11 and
12 in Appendix A). The widths of the plateaux in Fig. 9,
∆nc ≈ 0.01 nm−1 and ∆Vg ≈ 0.01 − 0.02 V, agree in
several experiments (Figs. 10–12), within small variations of
the gate capacitance. Appendix B discusses the variations in
more details. If correction Eq.(5) is zero (α = 0), then calculated dependence G(V0 (nc , T )) is almost the same as for
self-consistent potential obtained in 3D-electrostatic potential
calculation (Fig. 13d).
We made a number of simplifying assumptions in our 1Dmodel. Therefore, a detailed ﬁt of the experimental data to the
calculated curves is hardly appropriate. For example, it was
checked that approximation (2) in the calculation of the conductance replaces quite well more general formula (1) (except
the case of ultimately low kB T values). Thus, the discovered
effect is unrelated to the details of the barrier proﬁle U (x) and
is induced merely by the dependence of the difference EF −U
on nc ; i.e., the correction δU (T ) in the center of the barrier,
where Eq. (3) presumably holds, yielding α > 0, is crucial.
It is difﬁcult to ﬁnd α from theoretical considerations. However, similarity of the experimental and calculated curves persists under a 50% variation of α (Fig. 7). It is noteworthy that
the effective value α = 0.2 corresponds to r∗ n∗ ≈ 1.5; i.e.,
c
the distance r between the probe electron in the center of the
barrier and the exchange-correlation hole is 1.5γ/nc. On the

6
1,0

Acknowledgements

(meV)

(a)

T, meV

0,5

This work was supported by the Presidium of the Russian Academy of Sciences, program no. 24, and the Siberian
Branch, Russian Academy of Sciences, project no. IP130. We
are grateful to Z.D. Kvon, M.V. Budantsev, A.P. Dmitriev, I.V.
Gornyi for fruitful discussions, and A. Safonov for translation.

0.11

E -U

0.21
0,0

0.31

F

=0.2

0.41
0.51

-0,5

Appendix A: Data processing

(b)

1

F

We tested the validity of Landauer formulas by the following way. Combining formulas in approximation (2) it
is easy to write S = −[(1 − G) ln(1 − G) + G ln G]/G,
S M = (π 2 /3)(1 − G) where thermopower S and conductance G are measured in units of −kB /e and 2e2 /h, respectively. Thus we can ﬁnd thermopower from conductance data
and compare it to the measured thermopower. We know about
only one paper,17 which reports anomalous plateaux for con-

0

=0.2

-1

1,0
0.01K

0,6

(c)

0,4

=0.2

0,2

2

(d)

2,0

(a)

S (-kB/e)

1,0

G (2e /h)

Sample C
T=0.3K

2,0

0,0

0,8

2,5

2,5

1,5

0.01K

1,5

1,0

=0

0,4

1,0

0,5

0,6

0,5

0,0
0,2

-2,40

2,0
n

c

-1

(nm

0,02

0,03

)

2,0

0.3K
0.7K (b)

1,5

S (-kB/e)

0,01

0,0
-2,28

Vg (V) -2,32

Sample B

0,0
0,00

-2,36

1,5

1,0

basis of the typical values nc ∼ 0.01 nm−1 (see Figs. 2 and
5), we can conclude that the conditions used to Eq. (3) and
Eq. (5) are fulﬁlled.

VII.

CONCLUSION

A simple model of anomalous plateaux in the conductance and thermopower of one-dimensional ballistic quantum
wires has been proposed on the basis of the Landauer approach with spin degeneracy. The key points of the model
are pinning of the effective one-dimensional barrier height
U at a depth of kB T below the Fermi level under a change
in the one-dimensional density in the center of the barrier
or the gate voltage and the inclusion of all (local and nonlocal) temperature-dependent interaction-induced corrections
via phenomenological formula (5).

1,0

0,5

0.7K

0,5

0.3K
0,0
-4,0

-3,9

Vg (V)

-3,7

0,0

(c)

Sample C, 0.3K
Sample B, 0.3K
Sample B, 0.7K

1

0

-3,8

S
M
S

2

S (-kB/e)

FIG. 9: (a,b) Calculated dependences EF − U , η(nc ) = (EF −
U )/kB T for EF = 5 meV, the interaction parameter α = 0.2, and
the bare potential U0 (x) = V0 / cosh2 (x/a) with a = 200 nm. (c,d)
Calculated conductance of the one-dimensional channel with the corrected (b) and the bare (c) potential versus nc .

G (2e2/h)

2

G (2e /h)

0,8

G (2e2/h)

(E -U)/T

2

-1

0

1

2

3

η = (EF-U)/kBT

4

5

FIG. 10: Data processing of the gate voltage dependences of conductance and thermopower from Ref. 17 for two samples with the same
geometry of metal gates and two temperatures. Measured S-values
are denoted by open blue and ﬁll red circles, or ﬁll olive square. Seebeck coefﬁcient S obtained from conductance is shown by green and
cyan line, S M obtained from G is shown by orange and yellow line
(a,b). Curves S(η) and S M (η) calculated by Eq.(2) and measured
S-values as a function of η-values obtained from measured G are
shown in panel (c).

7
about U being independent of T and EF −U being linear in Vg
do not work when the ﬁrst subband begins getting occupied in
the microcontact. Nevertheless, the spin degeneracy Landauer
approach and Mott approximation remain valid up to the temperature of 1K in the case of Fig. 10, including anomalous
plateaux.
We used Eq. (2) to study the behavior of the reﬂecting barrier U = EF + kB T ln(1/G(Vg , T ) − 1) in different quantum
point contacts (QPC). Figures 11 and 12 show that U (Vg ) is
pinned with increasing temperature. In addition, there is a
strong temperature dependence of the quantity U (Vg ), which
determines the transport. It is usual to assume that the main
temperature dependence of conductance at any ﬁxed Vg is deﬁned by the Fermi distribution, not the reﬂecting barrier. In
our case this assumption is not valid, though in Fig. 12b one
can notice common asymptotics U (Vg ) at different T < 4 K
when conductance approaches to 2e2 /h.

1,0
T=0.08K
0,8

2

G (2e /h)

ductance and thermopower simultaneously. We extracted S
and S M from G(Vg ) and ploted the reconstructed and measured points S(Vg ) (scaled to common unit −kB /e). The
values corresponding to the ﬁrst subband almost coincided
(Fig. 10a,b). One can see that the height of the anomalous
conductance plateau equals 0.7, and the corresponding height
of the thermopower plateau is the same for two samples and
agrees closely with values S ≈ 0.8 − 0.9, S M ≈ 1. Above
the ﬁrst subband the thermopower behaves in accordance with
the Mott law and with the calculations of the peak heights between zero plateaux.7,10 In Refs. 7 and 10 the height of the
ﬁrst peak was shown to be approximately equal to −0.5kB /e
if the conductance quantization plateaux are smoothed out, or
less than this value if the plateaux are pronounced. On the
other hand, experimental data in Ref. 17 was normalized to
the height of the ﬁrst peak. To deal with this, we reduced
the measured values of S from Ref. 17 by 2 times to plot the
curves in the units of −kB /e.
For additional veriﬁcation of the described interpretation of
the experimental data we extracted η-values from Eq. (2) and
measured G-values for the ﬁrst subband and plotted the points
(S, η) along with the calculated curves S(η) and S M (η) using
formulas (2). Figure 10c shows a good agreement between the
experimental data and universal curves S(η), S M (η), for different temperatures and devices. Therefore, we show that the
plateaux are present in the Vg dependences but not in the dependences on EF − U . This means that the usual assumptions

0,6

(a)

0.68
4.1K

0,4

0.4
0,2

0,0

1
1.1K
6.5K

0,4

1µ

0.21

0.43

0.82

1.6

4.1

(b)

0,0
0.08K

0

-0,2

4.2K

6.5K

2

(c)

F

1.1K

0.3K

(E -U)/k

0.37

0.24
0

B

T

1

F

E -U (meV)

4.1K
0.08

0,2

F

0.5µ

T, K

0,6

E -U

4.2K

0.66

(meV)

0,8

2

G (2e /h)

T=0.3K

1

0.75
0

-1

-1

-1,00

-0,98

Vg (V)

-0,96

1,24

1,28

1,32

Vg (V)

-0,49

-0,48

V

g

FIG. 11: Data processing of the gate voltage dependences of conductance from Refs. 15, 13. Top: Measured conductance for structures
with split metal gate15 (left) and in-plane side gate13 (right). The
insets show the shape of the gates and etching strips (gray regions)
forming the channel in 2DEG. Bottom: dependences EF − U (Vg )
extracted from the measured G(Vg ) with use of Eq. (2). The location
of the anomalies in G and EF − U is indicated with the dotted lines.

-0,47

-0,46

(V)

FIG. 12: Data processing of the gate voltage dependences of conductance from Ref. 14. (a) Measured conductance for a split metal
gate structure. (b,c) Dependences EF − U (Vg ) and η = (EF −
U (Vg ))/kB T extracted from the measured G(Vg ) with help of
Eq. (2). The dotted lines indicate the position of anomalies in G
and η.

8
Appendix B: Self-consistent calculations

0,02

(nm

-1

)

(a)

0,01

nc

1

0,00

(meV)

0,5

(b)

T,

0

0,0

K

E -V

0.06

F

0.58
1.16

-0,5

2.33
B

-V )/k T

2

(c)

0

1

(E

F

0

-1

1,0

K
0.06

2

(2e /h)

T,
(d)

0.58

0,5

G

1.16
2.33

0,0
-2,56

-2,54

V

g

-2,52

-2,50

(V)

FIG. 13: (a,b,c) Dependences nc (Vg ), EF − V0 (Vg ) and (EF −
V0 (Vg ))/kB T obtained from the solution of 3D-electrostatics of the
QPC for usual conﬁguration of split gate and heterostructure from
Ref. 5. (d) Dependence G(Vg ) obtained from Eq. (1) for the lowest
one-dimensional subband V (x) = E1 (x), calculated in 3D electrostatic modeling.

∗
1

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3
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6

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