# Quantum effects in linear and nonlinear transport of T-shaped ballistic junction patterned from GaAs/Al_ {x} Ga_ {1− x} As heterostructures

**ABSTRACT** 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 resistance 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 nonlinear 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.

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**ABSTRACT:**We present studies of ballistic transport in three terminal T-shaped junction in a linear and non-linear regime. The floating electrode acts as a scatterer and modifies the conductance in a direct channel (between source and drain electrode). In the low voltage limit, the conductance shows the Wigner threshold effect and the bend resistance. A specific shape of the Wigner singularities can be changed by applied voltage to the floating electrode as well as by a shift of the Fermi level. The system also exhibits filtering properties with current distribution between different modes propagating in the junction. Back action of current flowing in the direct channel on changes of the voltage in the floating electrode is considered in the non-linear regime.12/2011; - SourceAvailable from: O. A. Tkachenko[Show abstract] [Hide abstract]

**ABSTRACT:**We report the observation of the Fermi energy controlled redirection of the ballistic electron flow in a three-terminal system based on a small (100 nm) triangular quantum dot defined in a two-dimensional electron gas (2DEG). Measurement shows strong large-scale sign-changing oscillations of the partial conductance coefficient difference G(21) - G(23) on the gate voltage in zero magnetic field. Simple formulas and numerical simulation show that the effect can be explained by quantum interference and is associated with weak asymmetry of the dot or inequality of the ports connecting the dot to the 2DEG reservoirs. The effect may be strengthened by a weak perpendicular magnetic field. We also consider an additional three-terminal system in which the direction of the electron flow can be controlled by the voltage on the scanning gate microscopy (SGM) tip.Nanotechnology 03/2012; 23(9):095202. · 3.84 Impact Factor

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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

[3]. 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

[4]. 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 [5]. 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 [6].

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-

−0.14

−0.12

−0.1

−0.08

0

1

2

3

Vg(V)

Iij(nA)

0.6 µm

I13

I23

I12

Vg

3

V2

1

2

A

A

I23

I21

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

visible here.

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

Page 2

2

G32

G23

G21

G13

(a)

(b)

0

3

1

2

3

Gij(2e2/h)

−0.15

−0.1

−0.05

0

0.05 0.1

0

1

2

Vg(V)

3

1

2

0

0.1

−0.2

0

0.2

Vg

∆G

G23

G13

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 [7]. 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 [8] 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

−4

−2

0

24

6

0

1

2

3

EF(a. u.)

Tij

A

A

B

C

B

C

T12

T32

(a)

N =2

N =4

−0.1

−0.05

0

0.05

−0.2

0

0.2

Vg(V)

∆G (2e2/h)

N =2

4

6

810

G1=G12+G13

(b)

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 [9].

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 [10]. 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

Page 3

3

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 [8] 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

is recovered.

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[9].

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

−0.1

−0.05

0

0.05

0.45

0.5

0.55

Vg(V)

V3/V1

5 101520

0.45

0.5

0.55

E (a.u.)

0

(a)

(b)

V3

I3=0

V2=0

V1

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

G23 curve.

and V2 = −Vpp (push-pull bias regime) then V3 = VC

is always negative, as it was predicted in [11] and then

proved experimentally [12]. Using the quantum scatter-

ing approach Csontos and Xu [13] 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. [13] have not been confirmed

experimentally.

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 [12] 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

asymmetry.

To compare the experimental findings with theory we

Page 4

4

−100 10

−2

−1

0

1

2

Vpp(mV)

V2=−Vpp

2

VC(mV)

Vg=0

∂VC

I3=0

∼

3

∂Vpp

∼

V1=+Vpp

1

−0.08

−0.04

0

VC

−0.4

0

0.4

−0.2

0

0.2

Vpp(a.u.)

β

−0.2

0

0.2

βs

Vg=0.09 V

0

Vpp(mV)

0.04 V

−0.2

0

0.2

Vg=0

−505

−0.11 V

−55

(a)

(b)(d)

−0.1

0

0.1

−0.2

0

0.2

0.4

Vg(mV)

βs

Vpp=

−10 mV

−5 mV

(c)

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 [11], 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 [14], 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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