Symmetry of two-terminal, non-linear electric conduction
A. Löfgrena, C.A. Marlowb, I. Shorubalkoa, R.P. Taylorb, P. Omlinga, L. Samuelsona, and H. Linkeb,*
a Solid State Physics, Lund University, Box 118, S – 22100 Lund, Sweden
b Physics Department, University of Oregon, Eugene OR 97403-1274, USA
(Received # Month ####; published # Month ####)
The well-established symmetry relations for linear transport phenomena can not, in general, be applied in the non-
linear regime. Here we propose a set of symmetry relations with respect to bias voltage and magnetic field for the
non-linear conductance of two-terminal electric conductors. We experimentally confirm these relations using phase-
coherent, semiconductor quantum dots.
PACS numbers: 73.63.Kv, 73.23.Ad, 73.50.Fq
Symmetries with respect to the sign of a bias
voltage and the direction of an applied magnetic
field, B, are central to our understanding of electron
transport phenomena. In the linear response regime,
the Onsager-Casimir relations, σαβ(B) = σβα(-B),
describe these symmetries in terms of the local
conductivity tensor . These relations were derived
for macroscopic, disordered solid-state conductors
where the conductor boundaries are unimportant. In
mesoscopic samples, the characteristic length scales
for elastic and inelastic (phase-breaking) scattering
can exceed the dimensions of the device. In this limit
a local description of transport is not possible, and
the reciprocity theorem, R12,34(B) = R34,12(-B), must be
used [2, 3]. For two-terminal conductors, the
reciprocity theorem reduces to G12(B) = G12(-B),
where G12 is the conductance with the current flowing
from contact 1 to 2. The sign of the source-drain bias
voltage and the orientation of the measurement leads
are of no consequence in the linear response regime,
such that G12(B) = G21(B).
The reciprocity theorem breaks down in the
non-linear response regime [4-7]. In the general case,
where the conductor has no symmetry (e.g. due to
disorder), G12(V) ≠ G12(-V). This is because, if an
applied voltage modifies the asymmetric device
potential, the resulting device potential depends on
the voltage sign [6-8]. Similarly, no symmetries with
respect to magnetic field are expected for an
asymmetric device, that is, G12(V, B) ≠ G12(V, -B) (for
an illustration, see Fig. 1).
While the general breakdown of the
reciprocity theorem in the non-linear transport regime
is well known [4-7], a systematic evaluation of
surviving symmetries in this regime has not
previously been attempted. It is the point of the
present paper to establish a complete set of symmetry
relations for the non-linear conductance of two-
terminal conductors. One important motivation is that
the non-linear regime is fundamental to applications
of sub-micron electronic devices, for which linear
response is limited to very small voltages [4, 5].
In the following we consider conductors
without significant disorder. We first propose a set of
general symmetry relations with respect to bias
voltage and magnetic field for the non-linear electric
conductance. We will show that symmetry of non-
linear transport requires geometrical symmetry of the
conductor − a substantial experimental challenge in
terms of fabrication and material quality. Using
purposely-designed semiconductor quantum dots, we
then demonstrate that the symmetry relations are
experimentally observed, and that deviations from
perfect geometrical symmetry can be measured.
Without loss of generality we consider
triangular conductors because of their simple
geometrical shape. We refer to a device as left-right
(LR) symmetric when it possesses a symmetry axis
perpendicular to the current direction, and up-down
(UD) symmetric when it possesses a symmetry axis
parallel to the current direction. We consider the
symmetry of the non-linear electrical conductance
under reversal of voltage, magnetic field, and lead
orientation. In this context it is important to note that
in a real experimental set-up the reversal of voltage
(V → -V) is not generally equivalent to physically
interchanging the leads attached to the probes (G12 →
G21), because the circuit used to measure the
conductance may itself be asymmetric. For instance,
FIG. 1. Schematic electron trajectories for positive and
negative magnetic fields. In the absence of symmetry in a
mesoscopic device, the conductance is not expected to be
symmetric with respect to the direction of a magnetic
field, when a bias voltage defines a source and a drain
the gate voltage Vg used to electrostatically define the
conductor’s shape is usually set with respect to the
drain contact on one side of the device, breaking the
circuit symmetry. When appreciable source-drain
voltages are used, the resulting gradient in the local
electro-chemical potential along the conductor
deforms the device potential defined by the gate in a
way that depends on the voltage sign. This can lead
to circuit-induced asymmetry
conductance even when the device itself is LR-
symmetric [9-11]. In order to avoid CIA, special care
must be taken in the device design . Here we
focus on so-called “rigid” devices in which CIA is
not significant, and refer the reader to Ref.  for a
discussion of devices that are not rigid.
For rigid devices, regardless of their
symmetry, a voltage reversal is equivalent to
swapping source and drain leads, such that
G12(V, B) = G21 (-V, B) (rigid) (1)
This relation is illustrated in Fig. 2 (compare,
for instance, configurations A and G or D and F).
For the special case of rigid devices that are
LR-symmetric we expect
G12(V, B) = G12(-V, -B) (LR, rigid) (2)
Eq. (2) holds independent of whether or not
the device is UD-symmetric, but is not expected if
LR-symmetry is absent. This can be seen by
comparing, for instance, A and D or B and C in Fig.
UD-symmetry implies that, for a given
voltage, reversal of a magnetic field perpendicular to
the device plane should be of no consequence for
electron transport :
G12(V, B) = G12(V, -B) (UD)
This can be seen by comparing, for instance, A
and B in Fig. 2. Note, however, that the absence of
LR-symmetry implies that, in the non-linear regime
G12(V) ≠ G12(-V), regardless of the magnetic field
sign [5, 7, 8]. Eq. (3) does not involve a reversal of
lead orientation or voltage sign and is therefore valid
for both rigid and non-rigid devices.
Finally, we note that the conductance of a LR-
symmetric device (regardless of rigidity and UD-
symmetry) is expected to be invariant upon reversal
of lead orientation and of the external magnetic field:
G12(V, B) = G21(V, -B) (LR)
Relationships (1) – (4), the first main result of
our paper, are based on fundamental symmetry
arguments and are therefore expected to hold in both
the classical and quantum regimes of transport.
In order to test these relationships, we used ballistic
(CIA) of the
semiconductor devices defined by deep wet etching
in modulation-doped, 9 nm thick InP/GaInAs. The
devices were of equilateral-triangular shape with a
side length of 1 µm, smaller than the electrons’
elastic mean free path of 6.1 µm and smaller than the
phase-coherence length lφ = 3.5 µm at T = 230 mK
and V = 0 (lφ = 1.7 µm at T = 230 mK and V = 3 mV).
In this phase-coherent regime of electron transport,
the wave-like nature of the carriers leads to
conductance fluctuations (CF) as a function of an
applied magnetic field. Because of their origin in
wave-interference, and because of the short Fermi
wave-length (30 nm), details of the CF are known to
be sensitive to the exact shape of the potential
forming the device and to defects or impurities .
Phase-coherent measurements of CF are therefore
particularly well suited to test the influence of device
geometry and of disorder on the conductance
symmetry. Contact openings used to measure the
conductance were positioned such that either UD-
symmetric (Fig. 3) or LR-symmetric (Fig. 4)
quantum dots were formed. Two-terminal magneto-
conductance measurements were carried out in four-
point geometry. A small ac signal (rms amplitude 20
µV, comparable to kT ≈ 20 µeV) was added to a
tunable dc bias voltage V. The differential
conductance gij = dIij/dVij was measured using lock-in
techniques in order to reduce measurement noise. We
checked that there was no significant non-ohmic
behavior in the circuit.
Fig. 3 shows gij for a bias voltage |V| = 1 mV
≈ 50 kT/e as a function of a perpendicular magnetic
field for an UD-symmetric, triangular quantum dot
. The eight traces shown are individual
measurements taken over the course of two days in
the eight possible configurations of sign of the bias
voltage, direction of the magnetic field, and lead
orientation (see Fig. 2). As expected for a device
FIG. 2. Illustration of the symmetry relations expected for
a rigid device in the non-linear regime and at finite
magnetic field. The upper and lower rows show the two
possible lead configurations G12 and G21, distinguished by
the position of the grounding point relative to the device.
Different classical electron trajectories illustrate the
difference in transmission probability that results when
the potential depends on the sign of the voltage applied to
the source contact. Positive magnetic field is taken to be
into the page.
lacking LR symmetry, the CF are not symmetric in V,
as we note from a comparison of Figs. 3(a) and 3(b),
or Figs. 3(c) and 3(d) . However, for a rigid
device, Eq. (1) predicts that reversal of the leads and
bias voltage should lead to identical CF, regardless of
the device symmetry. The similarity of traces shown
in the same color (for instance A and G, or D and F)
qualitatively verifies Eq. (1), and shows that the
device used in Fig. 3 can be regarded as rigid .
According to Eq. (3), in the presence of
perfect UD symmetry conductance fluctuations
should be unaltered when the direction of the
magnetic field is reversed. This prediction can be
tested by comparing the pairs of traces in the
individual panels in Fig. 3 (e.g. A and B or C and D).
Again, striking similarities are observed.
In order to quantify the difference between
two magneto-conductance traces, say the difference
dAB between traces gA(B) and gB(B) measured in
configurations A and B, respectively, we determine
the root mean square (rms) of their difference, using
103 data points spaced by 0.5 mT between B = 0 and
Bmax = ± 0.5 T:
The value dAB = 0 would correspond to
identical traces. To calibrate the influence of
experimental noise and setup instabilities on d we use
two CF traces recorded two days apart in nominally
identical configurations (V = 0). Separately
evaluating d for the traces recorded for positive and
negative magnetic field and then averaging the
results, we find d0 = 2.99x10-2 e2/h, a value
comparable to experimental noise (≈ 0.5 %) of the
device conductance. In comparison, the four pairs of
traces that should be identical if the device is rigid (A
– G, B – H, C – E, D – F) yield an averaged value of
dCIA = (dAG + dBH + dCE + dDF )/4 = 3.10x10-2 e2/h and
dCIA/d0 = 1.04. A comparison of data sets that,
according to Eq. 2, should be the same if the device
was LR symmetric (A – D, B – C, E – H, F – G),
yields dLR/d0 = 3.05. In comparison, a test for UD
symmetry (A – B, C – D, E – F, G – H) yields the
averaged value dUD/d0 = 1.77. In other words, the
intentional absence of LR-symmetry in the device
Magnetic field (T)
FIG.3. Magneto-conductance fluctuations for an UD-
symmetric quantum dot measured in the eight possible
different configurations of lead orientation, sign of bias
voltage, and sign of magnetic field (capital letters refer
to the panels in Fig. 2): (a) shows g12(V = +1 mV, ±B),
(b) shows g12(V = -1 mV, ±B), (c) shows g21(V = +1
mV, ±B), and (d) shows g21(V = -1 mV, ±B). The lower
trace in each panel has been offset by – 0.5 e2/h for
clarity. The inset to (a) shows dLR/d0, dUD/d0, and
dCIA/d0 as a function of V (lines are guides to the eye).
FIG. 4. Magneto-conductance fluctuations g12(V, B) of a
LR-symmetric device for V = 0, V = +2 mV and V = –2
mV (capital letters in each measurement configuration
refer to the corresponding panel in Fig. 2). Data are offset
for clarity. Note that g12(V, B) ≠ g12(V, -B), while g12(V,
B) and g12(-V, - B) show very similar features, as
predicted by Eq. (2) (see, e.g. the marked features). Inset:
dLR/d0 and dUD/d0 as a function of V (d0 = 4.21x10-2 e2/h
for this device). Lines are guides to the eye.
geometry causes the largest conductance asymmetry, Download full-text
while unintentional deviations from UD-symmetry,
such as material and fabrication imperfections, have a
significantly smaller, but measurable effect. The
effect of CIA in our devices is not significant
compared to experimental noise, confirming that the
device is rigid. Note, however, that CIA can be
substantial in other devices, for instance in some
surface-gated devices [9, 10, 12].
The inset to Fig. 3(a) shows the quantified
asymmetries (normalized to d0) as a function of
increasing bias voltage. Consistent with a first order
non-linear effect, dLR increases approximately
linearly with bias voltage. On the other hand, dUD,
which is attributed to imperfections in the UD-
symmetry of the device, which are not expected to
change with voltage, increases only weakly with V.
At all voltages used, the influence of CIA remained
insignificant compared to the noise level (dCIA/d0 ≈
For comparison with the UD-symmetric
device discussed so far, in Fig. 4 we show CF for the
LR-symmetric device. Whereas at V = 0 (linear
regime) the conductance is symmetric in B, at finite V
(non-linear regime) each of the two data traces taken
is not symmetric in B, due to the absence of UD
symmetry. However, one can see by comparing the
marked conductance features that g12(V, B) ≈ g12(-V, -
B). This observation confirms Eq. (2) and indicates
that the device is rigid, consistent with our conclusion
about the UD-symmetric device. We therefore expect
that any dLR observed should be due to unintentional
deviations from LR symmetry. Indeed, at all bias
voltages dLR/d0 for this device (see inset to Fig. 4) is
substantially smaller than for the LR-asymmetric
device used in Fig. 3. As one would expect intuitively
from the symmetry of the device, for small bias dLR is
also smaller than dUD (inset to Fig. 4). Note, however,
that the values found for dUD/d0 in the UD and the
LR-symmetric devices are comparable, highlighting
an interesting open question: At present, no
theoretical prediction about the dependence of dUD on
disorder, magnetic field, or bias voltage is available.
Our data (see insets to Fig. 3(a) and Fig. 4) suggest a
sub-linear increase of dUD with V, and little sensitivity
to the amount of intentional asymmetry.
The symmetry relations demonstrated here
were predicted based on symmetry arguments. A
natural next step would be a rigorous theoretical
study along the lines of Ref. , and applicable to the
non-linear regime of transport.
Acknowledgments. P.E. Lindelof for useful
discussions, I. Maximov for lithography and W.
Seifert for crystal growth. Supported by an NSF
IGERT (C.A.M.), an NSF CAREER award (H.L.), a
Cottrell scholarship (R.P.T), the ONR, the Swedish
Foundation for Strategic Research, and the Swedish
*Corresponding author: firstname.lastname@example.org
1 L. Onsager, Phys. Rev. 38, 2265 (1931); H. B. G.
Casimir, Rev. Mod. Phys. 17, 343 (1945).
2 M. Büttiker, Phys. Rev. Lett. 57, 1761 (1986).
3 M. Büttiker, IBM J. Res. Developm. 32, 317 (1988).
4 B. L. Al'tshuler and D. E. Khmel'nitskii, JETP Lett.
42, 359 (1985); T. Christen and M. Büttiker,
Europhys. Lett. 35, 523 (1996).
5 R. Landauer, in Nonlinearity in Condensed Matter,
edited by A. R. Bishop, et al. (Springer Verlag,
6 R. A. Webb, S. Washburn, and C. P. Umbach, Phys.
Rev. B 37, 8455 (1988); S. B. Kaplan, Surf.Sci. 196,
93 (1988); P. G. N. de Vegvar, et al., Phys. Rev. B
38, 4326 (1988); R. Taboryski, et al., Phys. Rev. B
49, 7813 (1994); P. A. M. Holweg, et al., Phys. Rev.
Lett. 67, 2549 (1991); D. C. Ralph, K. S. Ralls, and
R. A. Buhrman, Phys. Rev. Lett. 70, 986 (1993).
7 H. Linke, et al., Europhys. Lett. 44, 341 (1998).
8 H. Linke, et al., Phys. Rev. B 61, 15914 (2000).
9 N. K. Patel, et al., Phys. Rev. B 44, 13 549 (1991).
10 A. Kristensen, et al., Phys. Rev. B 62, 10950 (2000).
11 A. Löfgren, et al., in preparation for Phys. Rev. B
12 CIA is important when the variation of the
conductance with gate voltage is comparable to
other non-linear effects, that is, when ∂G/∂Vg ≈
∂G/∂V [10, 11]. Our devices were defined by deep
wet-etching rather than
Furthermore, a Ti/Au top-gate used to tune the
carrier concentration was separated from the
quantum well by a 1 µm layer of insulating
polymer, and ∂G/∂Vg ≈ 0.6 (e2/h)/V was one to two
orders of magnitude smaller than in typical, surface-
gated GaAs/AlGaAs devices.
13 W. J. Skocpol, et al., Phys. Rev. Lett. 56, 2865
14 It can be shown in general that any symmetry
relation in V or B that holds for Gij must also hold
for gij .
15 The CF are superimposed onto a broad background
that is due to classical commensurability effects
. While the phase-sensitive CF show non-linear
behavior at very small voltages, significant classical
non-linear effects are not (in the absence of CIA)
expected until several mV, where electron orbits are
changed by the applied voltage. However, at those
higher voltages the sensitive CF are suppressed due
to heating effects. We therefore limit our study to
voltages up to 2 mV.
16 H. Linke, et al., Phys. Rev. B 56, 1440 (1997).
by surface gates.