Quantum effects in linear and non-linear transport of T-shaped ballistic junction
J. Wr´ obel,1P. Zagrajek,1M. Czapkiewicz,1M. Bek,2D. Sztenkiel,1K. Fronc,1R. Hey,3K. H. Ploog,3and B. R. Bu? lka2
1Institute of Physics, Polish Academy of Sciences, al Lotnik´ ow 32/46, 02-668 Warszawa, Poland
2Institute of Molecular Physics, Polish Academy of Sciences,
ul. M. Smoluchowskiego 17, 60-179 Pozna´ n, Poland
3Paul Drude Institute of Solid State Electronics, Hausvogteiplatz 5-7, D-10117 Berlin, Germany
(Dated: December 10, 2009)
We report low-temperature transport measurements of three-terminal T-shaped device patterned
from GaAs/AlxGa1−xAs heterostructure. We demonstrate the mode branching and bend resis-
tance effects predicted by numerical modeling for linear conductance data. We show also that the
backscattering at the junction area depends on the wave function parity. We find evidence that
in a non-linear transport regime the voltage of floating electrode always increases as a function of
push-pull polarization. Such anomalous effect occurs for the symmetric device, provided the applied
voltage is less than the Fermi energy in equilibrium.
PACS numbers: 73.21.Nm, 73.23.Ad, 85.35.Ds
Recently, nanotechnology advances have led to a grow-
ing interest in electrical transport properties of the so-
called three-terminal ballistic junctions (TBJs). As the
name indicates, such structures consist of three quantum
wires connected via a ballistic cavity to form a Y-shaped
or T-shaped current splitter. One motivation is that in
principle such systems can operate at high speed with a
very low power consumption. Therefore, interesting and
unexpected nonlinear transport characteristics of TBJs
are intensively investigated due to possible applications
as high frequency devices or logic circuits[1, 2].
Another reason for the increased number of studies de-
voted to TBJs are quantum mechanical aspects of car-
rier scattering, which dominate at low temperatures in
the linear transport regime. This applies especially to
T-shaped splitters. For example, it is expected that a T-
branch switch, made of materials with a significant spin-
orbit interactions, can act as an effective spin polarizer
. Also, for such geometry an ideal splitting of electrons
from a Cooper pair is expected, provided the lower part
of the letter T is made of a superconducting material
. Both effects rely very strongly on the perfect shape
of the devices and high enough transparency of individ-
ual wires. Unfortunately, experimental data available for
the lithographically perfect T-branch junctions are lim-
ited mostly to a non-linear transport regime . Quan-
tum linear transport is usually studied for less symmetric
structures, typically consisting of short point contact at-
tached to a side wall of a wider channel .
In this work we report on fabrication and low tempera-
ture transport measurements of T-shaped three-terminal
devices, for which we take a special care to preserve the
perfect symmetry and reduce the geometrical disorder.
By comparing our data to conductance modeling by the
recursive Green-function method, we find out that quan-
tum effects dominate up to source-drain voltages equal
to the Fermi energy. In particular, we show that the
non-linear response of symmetric TBJ behaves in a non-
FIG. 1: (Color online) Currents Iij vs gate voltage Vg at
temperature T ≈ 0.3 K. Iij is defined as current flowing from
contact j when voltage Vi is applied to terminal i (see the
measurement scheme). Upper inset shows scanning electron
micrograph of the T-junction device, top metal gate is not
classical way and is highly tunable with carrier density.
The three-terminal ballistic junctions are made of a
GaAs/AlGaAs:Si heterostructure with electron concen-
tration n2D = 2.3 × 1011cm−2and carrier mobility
µ = 1.8×106cm2/Vs. The interconnected wires of equal
length L = 0.6 µm and lithographic width Wlith= 0.4 µm
are patterned by e-beam lithography and shallow-etching
techniques to form a T-shaped nanojunction (see inset
to Fig.1). The physical width of all branches is simulta-
neously controlled by means of a top metal gate which
is evaporated over the entire structure. The differential
conductances have been measured in a He-3/He-4 dilu-
tion refrigerator, by employing a standard low-frequency
lock-in technique. We have also studied non-linear trans-
port in the typical for TBJs, so-called push-pull bias
arXiv:0912.2004v1 [cond-mat.mes-hall] 10 Dec 2009
FIG. 2: (Color online) Gij = Iij/Vi plotted vs gate voltage
at T ≈ 0.3 K. (a) G23 and G21. (b) G32 and G13, here both
conductances involve transmission to side terminal 3. Inset:
comparison between G23and G13oscillations, a smooth back-
grounds have been removed from the original data (∆G is in
2e2/h units, Vg is in volts).
regime, when equal but opposite in sign dc voltages are
simultaneously applied to the opposite input contacts.
The application of a metal gate over the active region of
the device helps to symmetrize transmission coefficients
by smoothing the confinement potential . Neverthe-
less, even a perfectly shaped and gated junction may re-
main disordered at low electron densities, when screening
effects are weak. Figure 1 shows linear currents flowing
from each of three terminals for negative gate voltages
close to the threshold regime. The data indicate clearly
that there is a weak asymmetry between contacts – chan-
nels open at slightly different Vg. Additionally, small re-
producible wiggles are visible above threshold voltage.
All investigated structures show similar behavior and
we attribute it to the presence of quasi-localized states,
formed in the central part of the device. In this paper we
present data for the sample which has a lowest disorder
and highest degree of symmetry.
Although channel 2 → 1 opens last, at higher elec-
tron densities I12is larger than I23and I13, as predicted
by Baranger  for the ideal T-shaped quantum split-
ter. Figure 2 presents the conductances Gij as a func-
tion of gate voltage up to +0.12 V. For Vg > −0.05 V
the regular oscillations corresponding to the successive
population of electric sub-bands in each of the three ter-
minals are visible. Since magnetic field is zero, we expect
Gij= Gjiand this is indeed observed in the experiment.
For example, curves G23 and G32 are almost identical.
Larger differences are noticed for G13 and G23 curves
FIG. 3: (Color online) (a) Local current intensity (upper
panel) and transmission coefficients Tij vs Fermi energy EF
(below). Lines A, B and C mark energy values for which the
local current densities have been calculated. Black color in
density plot corresponds to zero current and bright areas to
maximal current intensity. (b) Conductance G1 = G12+ G13
vs gate voltage, T ≈ 0.3 K. Only oscillating part is shown,
a smooth background has been removed.
subfigures indicate backscattering at even mode numbers.
Arrows on both
which should be equal for the perfectly shaped device.
Relevant data are presented in the inset to Fig.2 where
oscillating parts of G23and G13are compared. On aver-
age G13is smaller and oscillate less regularly than G23.
Nevertheless, maxima and minima on both curves are
close to each other and for Vg> 0.05 V they oscillate ex-
actly in phase. It means that starting from a disordered
structure at the threshold voltage, for Vg? 0 the device
becomes more symmetrical and experimental data can be
compared with the theory of ballistic transport.
We model TBJ by three semi-infinite strips of “atoms”
and the square coupling region. Calculations have been
performed at temperature T = 0, using a tight-binding
approach and a recursive Green functions technique .
To determine a local current intensity inside the junction
we have incorporated parts of each wire to the coupling
region and used a newly developed, so-called knitting al-
gorithm . Results of this modeling are presented in
Fig.3(a).Transmission coefficients Tij between j-th
and i-th electrode are calculated for disorder free and
symmetric device with rounded corners in the coupling
region. Note that the value of T21increases almost mono-
tonically as a function of energy, whereas T32oscillates
strongly. This is the co-called bend resistance effect. T32
reaches maximum when the upper, just populated sub-
band, is almost fully transmitted to the terminal 3 (see
intensity plot A). For higher kinetic energies, however,
coupling becomes weaker and as a result T32decreases,
leading to the non-monotonic behavior as a function of
Fermi energy EF.
Presented calculations are consistent with the experi-
mental data obtained at electron densities high enough.
For Vg > 0 the curve G21 is similar to T21 and rather
smooth as compared to G23, which (like T23) shows
deeper minima due to the bend resistance effect (see
Fig. 2). Note also, that calculated energy dependence
of transmission coefficients differ for odd and even chan-
nel numbers. For example, the backscattering for N = 2
and N = 4 channels is stronger, as indicated with arrows
in Fig. 3. This effect was already predicted for a perfect
T coupler  and is apparently enhanced by the round-
ing of the “corners” in a junction area. For even parity
modes electron has high probability density at the center
of the device and therefore is more likely transmitted (to
see this compare density plots B and C). We believe that
such conductance dependence on wave function parity is
also observed in the experiment. It is especially well re-
solved for the total conductance G1= I1/V1= G12+G13.
Relevant data are presented in Fig. 3(b).
Next we consider the measurement scheme where stub
terminal (3) acts as a floating voltage probe (I3= 0). For
a classical device we have V3= (V1− V2)/2. This simple
formula should be modified for ballistic transport, where
it takes form V3/V1= T31/(T31+ T32) with V2= 0 for
simplicity. If T31= T32then classical result V3/V1= 1/2
Conductance data shown in Fig. 2(b) indicate that
on average G31is smaller than G32. Therefore, to imi-
tate the real sample, we rounded the junction “corners”
of a model device in such a way that T31 < T32. The
shape of the coupling area and results of calculations are
shown in Fig. 4(a). Ratio V3/V1 is on average below
1/2 but oscillates as energy increases. Very similar de-
pendence is observed in the experiment. The measured
value of V3/V1ratio reaches maximum, each time a new
one-dimensional level becomes occupied. Interestingly,
theory also predicts the occurrence of additional asym-
metric and very narrow resonances when a new conduc-
tion channel opens to transport in stub terminal. They
are probably related to the so-called Wigner singulari-
ties, which exist when the energies of quantized levels in
a side probe differ from those in the rest of the device.
Similar features are also visible in the experiment, espe-
cially for −0.1 < Vg< 0, but their possible connection to
Wigner resonances requires further studies.
Now let us turn to the non-linear transport regime
where the probabilities of transmission from input termi-
nals to a floating contact may differ, even for a perfect de-
vice. In such case, when V1is large enough and positive,
then V3/V1 is less then 1/2. Equivalently, if V1 = Vpp
FIG. 4: (Color online) (a) V3/V1ratio vs energy, calculated for
a device with asymmetrically rounded corners in the coupling
region (see inset). (b) V3/V1 data obtained as a function of
gate voltage at T ≈ 0.2 K for V1 = 50 µV (measurement
scheme is shown above). Arrows correspond to minima on
and V2 = −Vpp (push-pull bias regime) then V3 = VC
is always negative, as it was predicted in  and then
proved experimentally . Using the quantum scatter-
ing approach Csontos and Xu  extended the calcula-
tion range to a low temperature regime. They showed
that VC may be also positive, provided ∂T31/∂EF =
∂T32/∂EF< 0 and kT ? EF. To our knowledge, how-
ever, the predictions of Ref.  have not been confirmed
Figure 5(a) shows measurement schematics and cor-
responding VC data obtained when |Vpp| < 15 mV. VC
is not a symmetric function of Vpp, yet above a certain
threshold, data — as expected — bend towards nega-
tive values of VC. Such behavior is often observed in
experiments  because T31?= T32due to imperfections
which are always present in the real devices. Apart from
such asymmetry, however, data reported here behave in
an anomalous way. When a linear trend has been re-
moved, VCfirst increases with |Vpp|, and then goes down
reaching maximum at ∼ 7 mV. To investigate this ef-
fect in more detail we have used a modulation method to
measure the switching parameter β = ∂VC/∂Vppdirectly
with a better voltage resolution. Figure 5(a) explains the
measurement idea and Fig. 5(b) shows values of param-
eter βs= β − βaas a function of Vppfor a different gate
voltages. Here βais the mean value of switching parame-
ter calculated at each Vgfor |Vpp| < 15 mV. Subtracting
βais equivalent to removing a linear trend from the dc
data and therefore reduces the influence of the T31vs T32
To compare the experimental findings with theory we
FIG. 5: (Color online) (a) Stub voltage VC vs push-pull polarization Vpp at Vg = 0 (dotted line). The same data with a linear
trend removed are also shown (solid line). Below: experimental setup; small ac voltage (50 µV) is inductively coupled to Vpp ,
β = ∂VC/∂Vpp is measured directly using a low-frequency lock-in technique. (b) Variation in βs = β − βa with the applied Vpp
for Vg of 0.09 , 0.04, 0 and −0.11 V; here βa is the mean value of β and equals −0.18, −0.15, −0.12, and 0.01 respectively. (c)
Variation in βswith gate voltage for Vppof −10 and −5 mV. All experimental results at T = 0.8 K. (d) Nonlinear transport data
calculated for an ideal T-shaped junction. Solid line: EF = −1.55, ∂T31/∂EF < 0. Dashed line: EF = −1.15, ∂T31/∂EF > 0.
EF, VC and Vpp in arbitrary units.
calculated VCand β for an ideal T-shaped junction from
the energy dependence of a transmission coefficients. Re-
sults are consistent with the explanation of Xu , as
it follows from Fig. 5(d). If ∂T31/∂EF < 0 then VC
increases with |Vpp| and β has a positive slope in this
voltage range. When ∂T31/∂EF> 0 stub voltage is neg-
ative and switching parameter behaves “normally”. In-
terestingly, when experimental VCdata are compared to
linear conductance G3= G31+ G32, no such correlation
can be found. For example at Vg = 0, 0.04, and 0.09
V, derivative ∂G3/∂Vgis negative, positive and approx-
imately zero, but switching parameter does not change
its shape and sign as would be expected from modeling.
Results indicate that an anomalous data range, where β
has a positive slope, always exists — only its width de-
creases with EF. This fact can be used to tune switching
parameter with the gate voltage. Figure 5(c) shows βs
as a function of Vg for the two values of Vpp. Remark-
ably, not only amplitude but also the sign of βscan be
changed. We conclude that the behavior of VC in Fig.
5 cannot be explained by a single particle transmission
approach. Probably, as suggested in , the non-linear
transport regime requires a self-consistent calculations.
In summary, we have shown that linear transport in T-
shaped ballistic junction can be successfully described by
quantum scattering effects and weak disorder in a cavity
area. We have shown for the first time, that stub voltage
can increase as a function of push-pull polarization in a
non-linear transport regime, however, the energy depen-
dence of such non-equilibrium effect is inconsistent with
the standard single-particle picture of electron transmis-
sion. Nevertheless, novel applications of symmetric TBJ
structure, for example as the component of a multilogic
device, are still possible.
This work was funded by grant No. 107/ESF/2006
and MNiSW projects N202/103936 and N202/229437.
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