Coherent magnetotransport spectroscopy in an edge-blocked double quantum wire with window and resonator coupling

Abstract

We propose an electronic double quantum wire system that contains a pair of edge blocking potential and a coupling element in the middle barrier between two ballistic quantum wires. A window and a resonator coupling control between the parallel wires are discussed and compared for the enhancement of the interwire transfer processes in an appropriate magnetic field. We illustrate the results of the analysis by performing computational simulations on the conductance and probability density of electron waves in the window and resonator coupled double wire system.

Full text

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arXiv:cond-mat/0608027v1 [cond-mat.mes-hall] 1 Aug 2006
Coherent magnetotransport spectroscopy in an edge-blocked double quantum wire
with window and resonator coupling
Chi-Shung Tang1and Vidar Gudmundsson2
1Physics Division, National Center for Theoretical Sciences, P.O. Box 2-131, Hsinchu 30013, Taiwan
2Science Institute, University of Iceland, Dunhaga 3, IS-107 Reykjavik, Iceland
We propose an electronic double quantum wire system that contains a pair of edge blocking po-
tential and a coupling element in the middle barrier between two ballistic quantum wires. A window
and a resonator coupling control between the parallel wires are discussed and compared for the en-
hancement of the interwire transfer processes in an appropriate magnetic field. We illustrate the
results of the analysis by performing computational simulations on the conductance and probability
density of electron waves in the window and resonator coupled double wire system.
PACS numbers: 73.23.-b, 73.21.Hb, 75.47.-m, 85.35.Ds
I.INTRODUCTION
Coherent transport spectroscopy allows us to ex-
plore localized resonance processes when states interact
through a coupling element. Earlier experimental consid-
erations include the tunneling transfer of states between
electron waveguides through a thin tunneling barrier,1
and the window coupling between diffusive wires.2Later
on, theoretical studies were performed on noninteger con-
ductance steps in a gapped double waveguide,3and tun-
neling conductance between two coupled waveguides.4
Very recently, Rashba spin-orbit effect in parallel quan-
tum wires has also been studied.5Of particular inter-
est are the dynamics of the transfer processes for single-
energy electron spectroscopy in coupled quantum states
with either tunneling6or window coupling.7However, the
optimal transfer conditions in the coupled electronic sys-
tems have not been investigated.
In the presence of magnetic field, the energy spectra
have been studied pointing out the complex structure of
the evanescent states of the system in a homogeneous
double wire,8in a disordered double wire with boundary
roughness,9,10and in spatially coincident coupled elec-
tron waveguides.11Recently, magnetotransport in a par-
allel double wire coupled through a potential barrier was
studied by Shi and Gu.12They have shown that the step-
wise conductance increasing and decreasing features can
be changed by the applied magnetic field and the height
of the potential barrier.
In this paper, we investigate how a resonator in the
coupling window and a uniform perpendicular magnetic
field affect the electronic transport characteristics in a
ballistic two-terminal double wire system. In order to
enhance the interwire transfer coupling, a pair of edge-
blocking potentials are also included. The potential land-
scapes of the controlled coupled systems are depicted in
Fig. 1.
We shall demonstrate coherent magnetotransport
properties in an edge blocked double wire system us-
ing a rigorous Lippmann-Schwinger formalism in a
momentum-coordinate space,13and transforming the
two-dimensional magnetotransport equation into a set
FIG. 1: (Color online) Schematic illustration of edge-blocked
double quantum wire with (a) window coupling and (b) res-
onator coupling between the wires. The color scale on the
right shows the potential height in meV. The effective con-
finement length aw = 33.7 nm in zero magnetic field.
of coupled one-dimensional integral equations for the T-
matrix.14The characteristicsin conductance and electron
probability density of the system show the dynamics of
forward and backward interwire transfer by properly ad-
justing the strength of the magnetic field, the window
size, and the resonator.

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II.EDGE-BLOCKED DOUBLE WIRE
The system under investigation is composed of a lat-
erally parallel double quantum wire with transverse con-
fining potential
Vconf(y) =1
2m∗Ω2
0y2+ Vδ(y) + V00
(1)
with a symmetric deviation
Vδ(y) = − Vd1exp?−β1(y − y0)2?
+ Vd0exp(−β0y2)
− Vd1exp?−β1(y + y0)2?
for the sake of forming a parallel double wire from the
parabolic confinement, in which V00denotes a global shift
of potential to avoid negative energy in the wire. The
typical parameters for the confining potentials have the
values: ¯ hΩ0= 1.0 meV, V00= 2.0 meV, Vd0= 2.0 meV,
Vd1= 6.0 meV, β0= 4.0 × 10−3nm−2, β1= 7.0 × 10−4
nm−2, and y0= 100 nm.
The scattering potential
(2)
Vsc(x,y) = Vblock(x,y) + Vcoup(x,y) (3)
contains two terms, namely an edge blocking potential
Vblock(x,y) = V1exp(−γxx2)?1 − exp(−γyy2)?,
and a coupling element
(4)
Vcoup(x,y) = V0exp(−αx2− βy2).
These scattering potentials could be implemented in an
experimental system by means of depositing Schottky
front gates.15Based on the scattering potentials pre-
sented above we discuss two different coupling elements:
(i) the edge blocking with simple window coupling if
V0= −Vd0, and (ii) the edge blocking with resonator cou-
pling if V0= −2Vd0. First, with the choice V0= −Vd0,
α = αW= 2.0 × 10−4nm−2, and β = β0, a simple win-
dow coupling between the parallel wires can be made with
the edge blocking strength V1= 6 meV. Further, with the
choice V0= −2Vd0, α = αR= 0.2αW, and β = 0.05β0,
a window resonator coupling between the parallel wires
can be made with the increased edge blocking strength
V1= 8 meV. For both cases, the other parameters for the
edge blocking potential are: γx= β1and γy= γx/15.9.
Using a momentum-coordinate representation,13the
lateral component Hamiltonian for the conduction elec-
trons in a double wire system can be written in the form
(5)
H(q,y) =
?
+¯ h2q2
2m∗
p2
2m∗+1
y
2m∗Ω2
w(y − yq)2
?
1 −ω2
c
Ω2
w
?
+ Vδ(y)
?
, (6)
where yq = qa2
shift. The effective magnetic length aw = ¯ h/(m∗Ωw)
wωc/Ωw is the effective parabolic center
with Ω2
dimensional cyclotron frequency. The wave functions in
the considered double wire system away from the scat-
tering region can be generally written in the expansion
form
w= Ω2
0+ ω2
cand ωc= eB/(m∗c) being the two-
ΨE(q,y) =
?
n
ϕn(q)Φn(q,y)(7)
containing the eigenfunctions Φn(q,y) of the double wire
confinement Eq. (1), which can be expanded in terms of
the eigenfunctions for the parabolic confinement13
Φn(q,y) =
?
n
cnm(q)φm(q,y), (8)
where φm(q,y) is an eigenfunction for the parabolic wire
with finite magnetic field. The coefficients cnm(q) are in-
dependent of the energy E of the incident electron waves,
and can be obtained by diagonalizing separately the de-
viated Hamiltonian in each q-subspace. Then we reduce
the Lippmann-Schwinger equation into a set of coupled
one-dimensional integral equations for the T-matrix.14
The matrix elements of the scattering potential are of
the integral form
Vnn′(q,p) =
?
?
dyΦ∗
n(q,y)Vsc(q − p,y)Φn′(p,y)
=
ls
c∗
nl(q)cn′s(p)Vls(q,p), (9)
where Vls(q,p) is given by
Vls(q,p) =
?
dyφ∗
l(q,y)V (q − p,y)φs(p,y). (10)
In the asymptotic regions of the double wire, the propa-
gating electrons can be described by
ΨE(q,y) = 2πδ [q − kn(E)]Φn(q,y).
The corresponding energy subbands En(q) for the incom-
ing propagating states are represented by
(11)
En(q) = E0
n(q) +(qaw)2
2
(¯ hΩ0)2
¯ hΩw
, (12)
where E0
the parabolic confinement E0
correction ǫ(n,q) due to the deviation potential Vδ(y). It
should be noted that the deviation energy ǫ(n,q) makes
the subbands generally not equidistant in energy.
To achieve numerical accuracy, we notice that the
evanescent modes are in general not orthogonal and cen-
tered around y = 0, whereas the propagating modes shift
along the y-direction and their number of nodes corre-
lates to the subband index n. This fact leads us to expand
the evanescent modes in terms of the unshifted eigenfunc-
tions, but for the case of finite magnetic field, namely the
complete basis
n(q) = E0
n+ ǫ(n,q) contains contributions from
n= ¯ hΩw(n + 1/2) and the
φ0
n(y) =
exp
?2n√π n! aw
?
−y2
2aw
?
Hn
?y
aw
?
.(13)

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Using this basis and keeping the real part of the en-
ergy spectrum for the evanescent modes, we find that the
evanescent-mode energy spectrum is consistent with the
results of nonparabolic confinement by Barbosa et al.,8
and Korepov et al.9On the other hand, we would like
to mention that the character of the evanescent modes
require a larger basis than the expansion for propagating
modes to obtain sufficient numerical accuracy.
Due to the nonparabolic double wire confinement, each
electron subband provides more than one propagating
mode corresponding to the poles of the retarded scatter-
ing Green function
Gn
E(q) =?(kn(E)aw)2− (qaw)2+ i0+?−1.
Here kn(E) is independent of q, which is the Fourier vari-
able with no connection to E, and the index i labels the
modes in subband n. The subband momentum ¯ hkn(E)
can be determined by
(14)
[kn(E)aw]2= 2?E − E0
In contrast to the parabolic confinement measuring the
kinetic energy from the subband bottom at zero q, the
electron kinetic energy for a double wire system is mea-
sured from E0
n(q) with finite q values. For an electron
incident in mode mj with momentum ¯ hkmj, the trans-
mission amplitude in mode ni with momentum ¯ hkniis
given by14
n(q)? ¯ hΩw
(¯ hΩ0)2. (15)
tnimj(E) = δnimj−i?(kmj/kni)
2(kmjaw)
טTnimj(kni,kmj),
?¯ hΩ0
¯ hΩw
?2
(16)
where the positive subindexes j and i count, respectively,
the incident and forward scattering modes for a given
subband index m and n, as is illustrated in Fig. 2. Two
different values of the incoming energy are labeled for the
active transport modes. The conductance, according to
the framework of Landauer-B¨ uttiker formalism,16,17can
be written as
G(E) = G0Tr[t†(E)t(E)], (17)
where G0= 2e2/h and t is evaluated at the Fermi energy.
III. NUMERICAL RESULTS AND DISCUSSION
To investigate the magnetotransport properties of the
edge-blocked double wire system with either window or
resonator coupling, we select a typical magnetic field
strength B = 0.5 T such that the effective confinement
length aw = 29.3 nm and the effective parabolic sub-
band separation ¯ hΩw = 1.32 meV. We then correlate
the conductance for a certain incoming energy, and seek
information from the electron probability density of the
1
2
3
4
5
-4 -3 -2 -1 0 1 2 3 4
E (meV)
qaw
??
??
??
??
0
−1
??
??
??
??
1
−1
??
??
??
??
????
??????
−1
??
??
??
??
?
?
?
?
−2
0000
01
+2
+1
+1
+1
FIG. 2: (Color online) The energy spectrum of the propa-
gating electronic states (solid red) vs. the Fourier parameter
q, and the energy spectrum of the evanescent states (dotted
blue) vs. iq, in the double wire away from the coupling element
with magnetic field B = 0.5 T. The active transport modes
are labeled for two energies with the notation n±i, where the
+i and −i indicate, respectively, the number of the right- and
left-going active modes in the subband n.
various modes active at that incoming energy. All the
calculations presented below are carried out under the
assumption that the electrons have an effective mass of
0.067me, which is appropriate to the AlGaAs/GaAs in-
terface. Numerical accuracy is assured by comparing the
data obtained from a larger basis set.
In the absence of magnetic field, the deviation poten-
tial causes a near degeneration between the lowest two
subbands, and hence the first plateau of the quantized
conductance would be 2G0. However, in the presence of
magnetic field, the Lorentz force may push the electrons
away from the center of the system in ratio to their lon-
gitudinal momenta and then destroy the parity of the
electron waves. The lowest two subbands are no longer
degenerate but form an energy gap ∆01= 0.16 meV be-
tween the n = 0 subband top and the n = 1 subband
bottom at q = 0, shown in Fig. 2, reducing the con-
ductance to G0. For a homogeneous spatially separated
double wire without coupling element, the conductance
plateaus are not monotonically increased with increasing
incident electron energy due to the rich subband struc-
ture of the system.
A.Window coupling
To study the window coupled transport behavior of
the edge-blocked double wire, we select αW= 2.0×10−4
nm−2corresponding to the effective window length LW=
70 nm, V0 = −Vd0, β = β0, and the edge blocking
strength V1= 6 meV. The numerical results of the con-
ductance spectroscopy and its correspondingenergy spec-
trum are depicted in Fig. 3.

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0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
1.0 1.5 2.0 2.5 3.0 3.5 4.0
G(E)/G0
0
1
2
3
4
1.0 1.5 2.0 2.5 3.0 3.5 4.0
qaw
E (meV)
FIG. 3: (Color online) Upper subfigure: Conductance of an
edge-blocked double quantum wire without coupling element
(dashed green) and with a simple window coupling between
the wires (solid red). Lower subfigure: The energy spectrum
of the propagating electronic states (solid red) vs. the Fourier
parameter q, and the energy spectrum of the evanescent states
(dotted blue) vs. iq, in the asymptotic regions of the system.
B = 0.5 T, α = 2.0 × 10−4nm−2, and V0 = −2.0 meV.
Without a coupling element, electrons with energy
greater than the pinch off energy E = 1.33 meV, the dou-
ble conductance step G = 2G0, shown by dashed green
line in the upper subfigure of Fig. 3, indicates the pres-
ence of both 0+1 and 0+2 incident modes in the lowest
subband n = 0 caused by the degenerate subband bot-
tom with finite q. With a window coupling, the transport
features of the 0+1and 0+2modes are affected. Increas-
ing the incident energy, the 0+2 mode is pushed away
the central barrier by the Lorentz force. Differently, the
0+1 mode demonstrates an electron-like propagation in
the low kinetic energy regime, whereas it exhibits hole-
like transport behavior in the high kinetic energy regime.
Both propagation types are steered by the Lorentz force.
Moreover, conductance oscillations are induced for the
incident electron waves with higher energies just below
the lowest subband top E = 2.95 meV, as is depicted by
solid red curve in the upper subfigure of Fig. 3. It has
been shown that conductance oscillation could be useful
to obtain information about the amplitude of the nuclear
spin polarization.18
In the low kinetic energy regime, the conductance
rises slowly due to the efficient blocking of the incoming
modes. When the electron energy is increased, the quan-
tum interference becomes significant, and hence induces
oscillation behavior. The valley structure at E = 2.64
meV with G ≈ G0is due to the near total reflection of
the 0+1 mode around the middle barrier and the reso-
nant transmission of the 0+2outer mode induced by the
blocking potential. The hump structure in conductance
at around the higher energy E = 2.82 meV indicates
resonant transmission feature of the two incident modes.
Figure 4(a) shows that the 0+1 mode makes only one
local resonant state in the window and then prefers reso-
nant transmission through and above the central region
of the double wire system. Figure 4(b) shows that the
outer 0+2 transport mode prefers transmission slightly
perturbed by the edge blocking potential. The simple
electron probability feature shown in Fig. 4 implies a neg-
ligible coupling between these two propagating modes.
FIG. 4: (Color online) The electron probability density at
E = 2.82 meV for the ni = (a) 0+1 and (b) 0+2 modes,
corresponding to the solid red curve in the upper subfigure of
Fig. 3. B = 0.5 T and aw = 29.3 nm.
In the first energy subband gap region 2.95 < E < 3.11
meV, the conductance G is a bit smaller than G0, which
implies that only a single edge transport mode goes
through the upper wire, which is slightly reflected by
the edge blocking potential Vblock. Just above E = 3.11
meV, the second subband n = 1 becomes active in
the transport and the total conductance of the system
reaches 1.6G0, which is less than 2G0due to the block-
ing effect of edge potential. For electrons with energies
3.24 < E < 3.42 meV, there are four incident modes:
1+1, 1+2, 1+3, and 0+1due to the complicated structure
of the second subband n = 1. In this energy regime,
the highest conductance G = 3.3G0is at E = 3.32 meV,
which is less than 4G0mainly because of the strong re-

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flection effect in the coupling window for the 1+2mode.
FIG. 5: (Color online) The electron probability density at
E = 3.51 meV for the ni = (a) 0+1 outer and (b) 1+1 in-
ner modes, corresponding to the solid red curve in the upper
subfigure of Fig. 3. B = 0.5 T and aw = 29.3 nm.
In the second energy subband gap region 3.42 < E <
3.77 meV, there are two incident modes allowed to prop-
agate. The edge blocking potential together with the
appropriate strength of magnetic field steers the electron
waves to the window region and then enhances multi-
ple scattering resulting in scarring of wave functions and
interwire transfer effects. The dip structure in conduc-
tance at E = 3.51 meV demonstrates clearly such inter-
esting transport dynamics. Figure 5(a) shows that the
0+1 outer mode propagates along the edge of the up-
per wire and the coupling to the lower wire is visible.
Interestingly, Fig. 5(b) shows that due to the smaller q
of the 1+1 mode, it is able to reveal stronger interwire
transverse resonant features and then manifest resonant
quasibound state with coupling to the lower wire. We
note in passing that the probability density of the lowest
subband 0+1 mode usually exhibits simple pattern, the
complex probability pattern shown in Fig. 5(a) implies a
mechanism of inter-mode transition.
Similar features can be found in another dip structure
in conductance at higher energy E = 3.70 meV. Local-
ized interwire resonant states can be found for the case
of window coupling. To demonstrate the possibility of
extended interwire resonant transfer, below we shall dis-
cuss the case when the transfer window is embedded with
a transversely coupled resonator.
B.Resonator coupling
In order to improve the interwire coupling, we investi-
gate the window resonator coupled transport character-
istics of the edge-blocked double wire. To this end, we
select the physical parameters V0= −2Vd0, αR= 0.2αW,
and β = 0.05β0to form a deeper and broader Gaussian
potential and then create a longer window LW= 158 nm
with a transversely coupled double open-dot resonator,
as illustrated in Fig. 1(b). The longer window can more
efficiently interfere with the wave with the help of the
Lorentz force. The strength of the edge blocking poten-
tial is V1= 8 meV. Below we shall demonstrate coherent
interwire magnetotransport features for different values
of magnetic field for comparison.
1.B = 0.5 T
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
1.0 1.5 2.0 2.5 3.0 3.5 4.0
G(E)/G0
0
1
2
3
4
1.0 1.5 2.0 2.5 3.0 3.5 4.0
qaw
E (meV)
FIG. 6: (Color online) Upper subfigure: Conductance of an
edge-blocked double quantum wire without coupling element
(dashed green) and with a resonator coupling between the
wires (solid red). Lower subfigure: The energy spectrum of
the propagating electronic states (solid red) vs. q, and the
energy spectrum of the evanescent states (dotted blue) vs.
iq, in the asymptotic regions of the system.
V0 = −2Vd0, αR = 0.2αW, β = 0.05β0, and V1 = 8 meV.
B = 0.5 T,
In the upper subfigure of Fig. 6, we show the nu-
merical results for the case of B = 0.5 T, the window
resonator coupling leads to a richer conductance spec-
trum. The lowest subband is conducting up to E = 2.95
meV with an inner 0+1and an outer 0+2incoming trans-
port modes. In the low kinetic regime, the conductance

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G < G0 at E < 2.46 meV implies an efficient blocking
of the 0+2mode. The sharp peaks at energies 1.73 meV
(G ≈ G0) and 2.14 meV (G ≈ 0.8G0) imply stronger
inter-mode mixing. In addition, both propagating modes
can form well defined localized states due to the trans-
versely coupled resonator in the window. In the high ki-
netic regime, the electrons propagate demonstrating dif-
ferent dynamics: The conductance oscillation peaks have
values G > G0 implying that one of the active modes
prefers forward transmission made possible by interwire
forward transfer.
In the second subband gap region 3.42 < E < 3.77
meV, when the in-state energy is below 3.67 meV, the
0+1outer mode is well resisted by the edge-blocking po-
tential. Hence, the conductance is decreased to G ≈ G0.
For in-state energy higher than 3.67 meV, the electronic
conductance is increased approaching 2G0. This means
that the outer mode has sufficient kinetic energy to pass
the edge-blocking potential.
FIG. 7: (Color online) The electron probability density at
E = 2.48 meV for the ni = (a) 0+1 and (b) 1+1 modes,
corresponding to the solid red curve in the upper subfigure of
Fig. 3. B = 0.5 T and aw = 29.3 nm.
In Fig. 7(a), we show the electron probability density
for the 0+1 transport mode, which propagates close to
the central barrier manifesting strong multiple scatter-
ing and performing a symmetric quasibound-state pat-
tern along the transport direction. On the other hand,
the outer 0+2 transport mode, shown in Fig. 7(b), has
higher energy to perform resonant transmission to pass
the edge blocking and then make more visible interwire
coupling in the lower right lead. However, the entangle-
ment between the propagating modes in the upper and
lower wires dominates the transport feature.
2.B = 0.8 T
In order to improve interwire transfer, we select a
stronger magnetic field B = 0.8 T such that the effective
magnetic confinement length aw= 25.8 nm. The accu-
racy of these high magnetic field results has been checked
by increasing all relevant grid or basis set sizes used in
the numerical calculation. The other physical parameters
remain the same: V0= −2Vd0, αR= 0.2αW, β = 0.05β0,
and the strength of the edge blocking potential V1 = 8
meV.
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5
G(E)/G0
Without coupling
With coupling
Inner mode
Outer mode
Mode mixing
0
1
2
3
4
5
1.0 1.5 2.0 2.5 3.0
E (meV)
3.5 4.0 4.5 5.0 5.5
qaw
FIG. 8: (Color online) Upper subfigure: Conductance of an
edge-blocked double wire without coupling element (dashed
green) and with a resonator coupling between the wires (solid
red); and transmission probabilities of inner mode (dashed
blue), outer mode (dotted peach), and the mode mixing
(dash-dotted watchet) of the two modes. Lower subfigure:
The energy spectrum of the propagating electronic states
(solid red) vs. q, and the energy spectrum of the evanescent
states (dotted blue) vs. iq, in the asymptotic regions of the
system. B = 0.8 T, other parameters are in the text.
The conductance for the case of B = 0.8 T is shown
by solid red curve in the upper subfigure of Fig. 8. In
comparison with the lower subfigure of Fig. 8, we see
that the conduction electrons in the lowest subband at
the energy region 1.44 < E < 3.60 meV exhibit higher
transmission than the case of B = 0.5 T. In this energy
region, the transmission of the 0+2outer mode increases
almost monotonically indicating its increasing ability to
pass the edge-blocking potential, as is illustrated by the

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dotted peach curve in the upper subfigure of Fig. 8. More
interestingly, the strong conductance oscillation of the
0+1inner mode (dashed blue curve) implies rich and en-
ergy sensitive resonant transport features and provides
the possibility of efficient interwire transfer. The mode
mixing of the two lowest modes shown by the dash-dotted
watchet curve is generally not active but is slightly per-
turbed for the energy E > 4.5 meV. For the first subband
gap region at 3.60 < E < 3.69 meV, the ideal behavior of
conductance G = G0 indicates the perfect transmission
of the 0+1outer mode away from the central barrier.
FIG. 9: (Color online) The electron probability density at
E = 1.69 meV for the ni = (a) 0+1 inner and (b) 0+2 outer
modes, corresponding to the solid red curve in the upper sub-
figure of Fig. 3. B = 0.8 T such that aw = 25.8 nm.
The electron probability density shown in Fig. 9 corre-
sponds to the electron modes at incident energy E = 1.69
meV with conductance maximum G ≈ 0.9G0. Figure
9(a) demonstrates the transport properties of the 0+1low
q mode incident from the left lower channel. The elec-
trons are steered by the Lorentz force into the resonator
coupling region and exhibit resonant features in the win-
dow. It turns out that the Lorentz force fits the win-
dow size and provides efficient electron interwire forward
transfer to the right upper channel. In Fig. 9(b), we see
that the 0+2high q incident mode is strongly affected by
the Lorentz force, and thus incident from the left upper
channel. The electrons entering the resonator coupling
element region exhibit weak coupling to the lower wire.
On the other hand, the electrons also perform resonant
transmission through the upper blocking potential with
more completed cyclotron motion. The periodic peak fea-
tures of the probability density in the right upper chan-
nel indicates a signature of electron propagation in the
sufficient low kinetic energy regime with strong multiple
scattering in the transverse direction.
FIG. 10: (Color online) The electron probability density at
E = 2.07 meV for the ni = (a) 0+1 inner and (b) 0+2 outer
modes, corresponding to the solid red curve in the upper sub-
figure of Fig. 3. B = 0.8 T such that aw = 25.8 nm.
In comparison with the conductance maximum at in-
cident energy E = 1.69 meV, we see the case of a little
higher incident energy E = 2.07 meV with conductance
minimum G ≈ 0.6G0, the corresponding probability den-
sity is shown in Fig. 10. Figure 10(a) demonstrates the
transport property of the 0+1 mode. We see that the
increasing of the kinetic energy suppresses a little the in-
terwire forward transfer with negligible mode mixing. In
Fig. 10(b), we show the probability density of the 0+2
mode incident from the left upper channel. This outer
mode makes stronger resonant transmission in the upper
wire, and the interwire coupling is enhanced to exhibit
interwire backward transfer to the left lower channel. We
note in passing that even for such a simple transfer mech-
anism, all possible intermediate states have to be taken
into account in the calculation, which can not be obtained
using a low order perturbation theory.19
It should be noticed that when the electron Fermi en-

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8
ergy is above the lowest subband local top 3.60 meV,
the lowest active inner mode is changed to be 1+1mode,
and the lowest outer mode would be 0+1mode. For the
second subband gap region 4.48 < E < 5.31 meV, the
conductance plateau G ≤ G0indicates that the 1+1in-
ner mode is efficiently backscattered by the window res-
onator. In addition, the 0+1outer mode manifests near
ideal transmission due to its high kinetic energy.
We would like to mention that only the larger q evanes-
cent modes exist in the second gap region. This fact
leads to stronger localized bound states in the window
resonator with longer dwell time. Especially, there are
three small conductance peaks at energies E = 4.54,
4.94 and 5.28 meV in the second subband gap region.
By checking the inner mode and the outer mode contri-
butions to the conductance shown in the upper subfigure
of Fig. 8, we see that both of them manifest small sharp
change in conductance at these energies. We have tested
the case of E = 4.94 meV to see that both modes exhibit
amazingly similar scarring resonant state patterns in the
window covering the upper and the lower wire, which is
an example of persistence of a scarring wave function in
an open system.20
IV. CONCLUDING REMARKS
In this report we have investigated theoretically to
what extend the coherent magnetotransport properties
in an edge-blocked lateral double wire system with possi-
ble resonant interwire coupling in the presence of a uni-
form perpendicular magnetic field. The complex subband
structure, due to the non-parabolic double wire confine-
ment in the magnetic field, causes the appearance of ir-
regular steps in the conductance as a function of the
energy of the incoming electron wave. Even with the
coupling element in the tunneling regime, a usual per-
turbation theory is not valid once the length of the cou-
pling region exceeds a characteristic length scale.21Our
numerical method employed allows for a wide variety of
wire shape and scattering potentials.
To enhance the interwire forward or backward trans-
port, we have shown that not only appropriate window
size but transversely coupled resonator and the proper
magnetic field strength are required. The electron prob-
ability density shown in this report should be detectable
by using high-resolution scanning-probe images.22,23,24
We have also demonstrated that the mode mixing in
energy regions containing two active modes is relatively
weak, we expect that the conductance oscillations could
be clearly observable. An efficient interwire forward
transfer can be achieved by the inner mode with neg-
ligible mode mixing in a properly applied magnetic field,
whereas the outer mode exhibits interwire backward
transfer. The magnetic field manipulated window and
resonator coupling features in a double wire system are
important to understanding magnetotransport properties
of other open quantum structures.
Acknowledgments
The authors acknowledge the financial support by the
Research and Instruments Funds of the Icelandic State,
the Research Fund of the University of Iceland, and the
National Science Council of Taiwan. C.S.T. is grateful to
inspiring discussions with P.G. Luan and S.A. Gurvitz,
and the computational facility supported by the National
Center for High-Performance Computing of Taiwan.
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