Non-Kondo zero-bias anomaly in quantum wires
ABSTRACT It has been suggested that a zero-bias conductance peak in quantum wires signifies the presence of Kondo spin-correlations, which might also relate to an intriguing 1D spin-effect known as the 0.7 structure. These zero-bias anomalies (ZBA) are strongly temperature dependent, and have been observed to split into two peaks in magnetic field, both signatures of Kondo correlations in quantum dots. We present data in which ZBAs in general do not split as magnetic field is increased up to 10 T. A few of our ZBAs split in magnetic field but by significantly less than the Kondo splitting value, and evolve back to a single peak upon moving the 1D constriction laterally. The ZBA therefore does not appear to have a Kondo origin, and instead we propose a simple phenomenological model to reproduce the ZBA which is in agreement mostly with observed characteristics.
arXiv:0903.3203v2 [cond-mat.mes-hall] 17 Apr 2009
Non-Kondo zero-bias anomaly in quantum wires
T.-M. Chen,∗A. C. Graham,∗M. Pepper, I. Farrer, and D. A. Ritchie
Cavendish Laboratory, J J Thomson Avenue, Cambridge CB3 0HE, United Kingdom
(Dated: April 17, 2009)
It has been suggested that a zero-bias conductance peak in quantum wires signifies the presence
of Kondo spin-correlations, which might also relate to an intriguing one-dimensional (1D) spin effect
known as the 0.7 structure. These zero-bias anomalies (ZBA) are strongly temperature dependent,
and have been observed to split into two peaks in magnetic field, both signatures of Kondo cor-
relations in quantum dots. We present data in which ZBAs in general do not split as magnetic
field is increased up to 10 T. A few of our ZBAs split in magnetic field but by significantly less
than the Kondo splitting value, and evolve back to a single peak upon moving the 1D constriction
laterally. The ZBA therefore does not appear to have a Kondo origin, and instead we propose a
simple phenomenological model to reproduce the ZBA which is in agreement mostly with observed
PACS numbers: 73.21.Hb, 73.23.Ad, 72.10.Fk
The varied manifestations of the Kondo effect , in
systems as diverse as carbon nanotubes , semiconduc-
tor quantum dots , two-dimensional (2D) molecular
systems  and low-density mesoscopic 2DEGs , have
continued to interest experimentalists and theoreticians
alike. Hence, when the Kondo effect was suggested as a
possible explanation  of a many-body phenomenon in
quantum wires known as the 0.7 structure , this cre-
ated great interest in the physics of quantum wires. A mi-
croscopic explanation of the 0.7 structure has proved to
be a difficult theoretical challenge [8, 9, 10, 11, 12, 13, 14].
The suggested link with Kondo physics answered some
questions, such as why the 0.7 structure rises in conduc-
tance and disappears with decreasing temperature [7, 15],
but raised others: could Kondo physics possibly occur in
an open system[16, 17]? Why does a dc-bias strengthen
the 0.7 structure and increase its conductance [15, 18],
when Kondo correlations are quickly destroyed by finite
bias ? Very recently, a channel with a variable open
dot geometry shows that the Kondo effect and the 0.7
structure are completely separate. It was also found
that the 0.7 structure is not associated with backscat-
tering of electron waves and reduction in transmission
probability. As more and more studies imply the
Kondo effect is not linked to the 0.7 structure, this topic
requires further studies.
The key evidence linking the 0.7 structure and the
Kondo effect, is the presence of a ‘zero-bias anomaly’
(ZBA), or conductance peak at zero dc-bias, in the re-
gion of the 0.7 structure ; the Kondo effect causes a
similar feature in quantum dots . This ZBA was found
to broaden and disappear with increasing temperature,
and to split in two with increasing magnetic field, both of
which can be interpreted as manifestations of the Kondo
Here, in contrast, we present evidences that ZBA char-
acteristics in quantum wires are inconsistent with spin-
one-half Kondo physics. We observe the single ZBA peak
up to an in-plane magnetic field of 10 T, and at conduc-
tances less than e2/h such that only a single spin-type can
be present. Such a fully spin-polarised regime prevents
the spin-flips occurring which are essential to the spin-
one-half Kondo effect. Thus the measured ZBA does not
relate to Kondo effect as it is blocked in this regime. We
have also found a few ZBAs splitting in a magnetic field,
but all of these evolve back to a single peak when the
1D channel is laterally shifted. It implies that the split-
ting of the ZBA is related to disorder rather than Kondo
splitting. We propose instead a simple phenomenolog-
ical model whereby small increases in the 1D subband
energies as a function of dc bias can directly reproduce
dimensional electron gas formed at 96 nm below the
surface, has a mobility of 8.04 × 105cm2/Vs and a
carrier density of 1.44 × 1011cm−2in the dark. All the
devices used in this work have a width of 0.8 µm, and
lengths from 0.3 µm to 1.5 µm and were measured in a
dilution refrigerator at 60 mK.
Typical differential conductance data exhibiting a
strong zero-bias anomaly and 0.7 structure are shown
in Fig. 1(a). With increasing magnetic field B, the ZBA
weakens, but does not split in two, and a single peak
is still visible at B = 10 T, as shown by the more
detailed evolution in Fig. 1(c). A ZBA of Kondo ori-
gin should have been split by |2g∗µBB| = 1.2 meV at
B = 10 T [19, 22], where the effective g-factor |g∗| = 1.06
is obtained from the spin gap measured by dc-bias spec-
troscopy as well as the technique reported in Ref. .
Note that the g-factor of 1.06 measured in this work
is consistent with the g value reported in previous 1D
studies[6, 7]. Furthermore, Fig. 1(c) (B = 10 T) exhibits
a strong bunching of traces at e2/h (i.e., a plateau) in-
dicating complete spin-polarisation which would prevent
Kondo spin-flip scattering from occurring. The 0.7 ana-
log and the weakening 1.5×(2e2/h) plateau at B = 9.6 T
(bold trace) in Fig. 1(b) indicates that a crossing of
FIG. 1: A) Differential conductance vs dc-bias for fixed split-
gate voltages, at B = 0 and B = 7.6 T, both showing a
clear zero-bias anomaly (ZBA). B) Differential conductance
vs split-gate voltage for B = 0 → 16 T. C) DC-bias data
showing the evolution of the ZBA with increasing field.
spin-split subbands has already occurred by B = 10T,
further implying a completely spin-polarised system.
Thus, the evidence that the ZBA still occurs in a com-
pletely spin-polarised regime, in which the Kondo effect
is blocked, and does not split in magnetic field implies
that Kondo is unlikely to be the cause of the ZBA in
Another argument to link the ZBA with the Kondo
state is the Kondo-like temperature dependence of the
equilibrium conductance. However, this equilibrium
conductance in the steeply-rising region in 1D sys-
tems is also expected to have an activated tempera-
ture dependence. The ZBAs weaken and disappear
with increasing temperature in both zero (Fig.2A) and
finite magnetic fields (not shown).
show that the temperature dependence of the ZBAs is
well described by both the empirically modified Kondo
function, GK= (2e2/h)[0.5(1+(21/s−1)(T/TK)2)−s+
0.5] with s = 0.22, and the activation model, GA=
(2e2/h)[1 − CAe−TA/T]. Note that this modified Kondo
function can only describe the ZBA between 0.5(2e2/h)
and 1 × (2e2/h), since the lower limit of GK equals
0.5(2e2/h); however, in experiments the ZBA is observed
continuously from 0 to 2e2/h. The Kondo temperature,
TK, and the activation temperature, TA, extracted from
the fits of these two functions are furthermore shown in
Fig. 2(d). The values of TKmeasured for different sam-
ples in this work and previous study are all close to
FIG. 2: (Color online) A) Temperature dependence of differ-
ential conductance vs dc-bias at B = 0. B) Linear conduc-
tance as a function of scaled temperature T/TK at various Vg,
where the Kondo temperature TK is the fit parameter in the
modified Kondo function. C) The same experimental data
shown in an Arrhenius plot: ln((1−G(T))/CA) as a function
of scaled temperature TA/T, where the activation tempera-
ture TA and CA are the fit parameters in Arrhenius function.
D) TK(solid symbols) and TA(open symbols) extracted from
the fits of both models, along with the conductance at 60 mK
(solid), 230 mK (dashed), and 800 mK (dotted).
each other, and also rule out a possible reason for the dis-
appearance of the splitting of ZBAs, i.e., kBTK> g∗µB
. Note that the equivalence of Kondo-like and acti-
vated fitting of the same data was also found by Cronen-
wett et al. (Footnote 25 in Ref., and Ref.). It ap-
pears that the temperature dependence of the ZBA does
not favor the Kondo model over the activation model. It
is worth emphasizing that the ZBAs reported here and
also in previous studies[6, 24] can not be fitted to the
usual Kondo function .
Although most of our samples show only a single peak
at all B, two samples do exhibit splitting of the ZBA
into two peaks with increasing B [Fig.3(b), with conduc-
tance vs gate voltage characteristics in (a)]. However,
the peaks are split by 0.17 meV at B = 4 T, which is
only one-third of the value expected for Kondo splitting,
2g∗µBB = 0.48 meV. It was suggested that the Kondo
splitting could be less than |2g∗µBB| and have a min-
imum value of about |4
3g∗µBB| due to the interaction
between the impurity spin and the lead electrons.
However, the splitting of 0.17 meV at B = 4T is still
less than the minimum value4
The strength of the split peaks can be enhanced by
shifting the quantum wire laterally through the 2DEG
, by setting a difference between the voltages on each
split-gate of dVg = +0.3 V [Fig. 3(d)]. However, if the
3g∗µBB = 0.32 meV for a
FIG. 3: A) Differential conductance vs split-gate voltage at
B = 0,2,4 and 6 T. B) Differential conductance vs dc-bias
for fixed split-gate voltages, at B = 0,2,4 and 6 T, showing
a clear ZBA which splits into two with increasing B. C), D)
Same as A) and B) but with a difference of dVg = +0.3 V
between the voltages on the two split-gates — the splitting of
the ZBA with increasing B is even clearer. E), F) Same as
A) and B) but for dVg = −0.6 V — the ZBA is stronger for
all B, but no longer exhibits any splitting.G), H) Same as A)
and B) but for dVg = +0.6 V — these data are very different
to E) and F), indicating that disorder strongly affects ZBAs.
wire is shifted in the other direction, by applying a dif-
ference of −0.3 V, then the form of the ZBA changes to
an asymmetric single peak (not shown), which becomes
almost symmetric and very pronounced at dVg= −0.6 V
[Fig. 3(f)]—in Fig. 3(f) at B = 4 T, the ZBA is a
single peak, clearly not two broad peaks separated by
2g∗µBB = 0.48 meV. In contrast, the observed ZBA
which does not split with magnetic fields at dVg = 0
remains as a single peak when the wire is shifted later-
The difference in our data for dVg= +0.6 V [Fig. 3(f)],
compared to dVg= −0.6 V [Fig. 3(h)], is evidence that
the ZBA is strongly affected by a disordered confining
potential, which changes as the wire is moved laterally; in
the absence of disorder, the conductance characteristics
should be the same for dVg= ±0.6 V. The recurring of a
symmetric single ZBA peak from a split ZBA by laterally
moving the 1D channel clearly suggests that disorder is
related to the splitting of the ZBA in a magnetic field, as
opposed to Zeeman splitting of a Kondo resonance.
It has been shown that the differential conductance of
ES (Vsd ,T) =
ES0 + γ(T)Vsd
ES (Vsd) =
ES0 + γVsd
-0.2 0 0.2
ES (Vsd) = ES0
ES (Vsd) =
ES0 + γVsd
-0.2 0 0.2
-0.2 0 0.2
-0.2 0 0.2
Subband energy (meV)
Subband energy (meV)
-0.2 -0.1 0 0.1 0.2
-0.1 0 0.1
FIG. 4: (Color online) A) Calculated differential conductance
vs dc-bias, assuming subband energy, Es = Es0, is fixed w.r.t.
the average of the source and drain chemical potentials, where
Es0is the subband energy at zero dc-bias. B) Differential con-
ductance vs dc-bias, calculated in the same way, but including
a linear increase, E(Vsd), in the subband energy Es, as a func-
tion of dc-bias. A zero-bias anomaly is present. C)Variation
in subband energy required to keep density fixed as function
of DC-bias, for kT = 0.02 − 0.16 meV. Curves at y > 0
(y < 0) are for the subband sitting above (below) Ef at
Vsd = 0. D)The temperature variation of the characteris-
tics in B). E) Same as D), but now using a linear increase
E(Vsd,T) in subband energy, which decreases with increasing
temperature. F) Comparison of subband energy used in E) to
calculate dI/dVsd (bottom) to curves for fixed density (top)
taken from C).
an asymmetric electric constriction, caused by disorder in
our case, is strongly affected by magnetic field and bias
voltage . Moreover, impurity causes resonant interfer-
ence of the electrons in the wire; electron interference is
also expected to change with dc bias and magnetic field,
as diamagnetic shift, Zeeman splitting and localisation
can all alter the energy spectrum. All these provide a
reason why zero-bias anomalies in a few devices exhibit
splitting-like behaviour with magnetic field.
Some simple simulations were performed to demon-
strate how a ZBA might occur using a very general phe-
nomenology, which does not include spin. Figure 4(a)
shows differential conductance characteristics assuming
subband energies Es stay fixed as a function of dc-
bias, with respect to the average of the source and
drain chemical potentials (µs+ µd)/2 — as expected,
there is no ZBA. In contrast, Fig. 4(b) shows character-
istics where the subband energy rises linearly at a small
rate (γ = 0.33 meV per mV) with increasing dc-bias
(Es(Vsd) = Es(0) + γVsd) — a sharp ZBA is now present.
We have used a linear increase in energy with Vsd for
simplicity, but other increasing functions of Vsdalso give
ZBAs. Thus, a zero-bias anomaly occurs in 1D whenever
the subband energy rises slightly with increasing dc-bias,
irrespective of the precise functional form of this rise in
There is a fundamental reason for our phenomenologi-
cal model wherein the subband energy rises upward with
increasing dc-bias. In 1D systems, a fixed subband en-
ergy, with respect to (µs+µd)/2, ensures that 1D density
must change as a function of the square root of dc bias,
which is unlikely to be energetically favourable because
of the cost in Coulomb energy. Therefore, simply min-
imizing the Coulomb energy alone gives a reason why
subband energy should change with dc-bias. The curves
in Fig. 4(c) show how the 1D subband energy must vary
in order to keep the 1D density constant with increas-
ing dc bias. It further suggests an upward shift rate of
γ = 0.33 meV/mV when the subband is near the chem-
ical potential to give rise ZBAs. Note that completely
fixed density is not required to reproduce ZBAs; simply
reducing changes in density also causes subbands to in-
crease in energy with Vsd, giving a ZBA as well.
Although the zero-bias anomalies arising from a linear
increase in subband energy with Vsdbroaden and weaken
with increasing temperature [Fig. 4(d)], for the ZBA to
disappear completely at high temperatures, as in experi-
ment, it is necessary that the linear increase in subband
energy be temperature dependent, tending to zero at high
temperatures, as shown in Fig. 4(e) — γ = 0.33 meV per
mV at the lowest temperature (kBT = 0.02 meV) gives
a strong narrow ZBA, which broadens and disappears at
the highest temperature (kBT = 0.16 meV) for which
γ = 0.1 meV/mV.
There is a fundamental justification for introducing
this temperature-dependent correction to give the cor-
rect phenomenology. Figure 4(c) clearly demonstrates
that the variations in subband energy with Vsd due to
conservation of density are temperature-dependent and
disappear with increasing temperature, just as is required
in our phenomenological model [Fig. 4(e)] in order to re-
produce a ZBA that disappears at high T. Fig. 4(f) shows
that the energy shifts used to give a ZBA which disap-
pears at high T in our phenomenological model (lower
curves), are similar in magnitude to those required to
keep 1D density fixed (upper curves).
Here we do not claim that complete or partial conser-
vation of density is necessarily the cause of the exper-
imentally observed ZBAs, as our simple model is only
of qualitative significance. Since Coulomb and exchange
interactions vary with density, they are also expected
to have an impact on the shift of subband energy with
Vsd and require further investigations. It is important
to stress that this simple model demonstrates that the
ZBAs in quantum wires, unlike those in quantum dots,
do not need to have the Kondo mechanism as the non-
linear conductance in 1D changes easily when subband
energy varies with source-drain bias. This could provide
a basis for future theoretical study in anomalous nonlin-
ear conductance features, such as the ZBA and its two
significant side peaks which cannot be explained by the
The model presented here, though simple, is consis-
tent with the temperature characteristics of the ZBAs
which appears to have an activated behaviour. In addi-
tion, it explains why ZBAs do not split with increasing
magnetic fields and why ZBAs still occur in a completely
spin-polarised phase in which the Kondo spin-flip is pro-
hibited. However, the model cannot interpret the sup-
pression of the ZBAs with increasing magnetic field and
will need a field-dependent correction. A more sophis-
ticated theoretical study with Coulomb and exchange-
correlation interactions taken into account is thus highly
To conclude, we have found zero-bias anomalies in gen-
eral suppress with increasing magnetic fields without be-
ing accompanied with splitting. Some of ZBAs persist up
to a very large field when only one spin species occupies
the quantum wire, wherein the Kondo effect is not ex-
pected to occur. While two samples exhibit split ZBAs
in magnetic fields, the splitting value is not consistent
with the the Kondo model and, more importantly, the
split ZBA evolves back into a single ZBA peak when the
1D constriction is moved laterally. The measurement of
ZBAs when quantum wires are laterally shifted strongly
suggests that the splitting of ZBAs is related to disorder,
as opposed to Kondo splitting. A simple phenomenolog-
ical model is suggested to explain how a ZBA can occur
when the 1D subband energy rises with increasing source-
drain bias, clearly showing that the Kondo mechanism is
not required for zero-bias anomalies to occur in quantum
We acknowledge useful discussions with F. Sfigakis, K.
Das Gupta and K.-F. Berggren.
ported by EPSRC (UK). T.M.C. acknowledges an ORS
award and financial support from the Cambridge Over-
seas Trust. A.C.G. acknowledges support from Em-
manuel College, Cambridge.
This work was sup-
∗These authors contributed equally to this work
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