Excitation of collective modes in a quantum flute
ABSTRACT We use a generalized master equation (GME) formalism to describe the
nonequilibrium timedependent transport of Coulomb interacting electrons
through a short quantum wire connected to semiinfinite biased leads. The
contact strength between the leads and the wire is modulated by outofphase
timedependent potentials which simulate a turnstile device. We explore this
setup by keeping the contact with one lead at a fixed location at one end of
the wire whereas the contact with the other lead is placed on various sites
along the length of the wire. We study the propagation of sinusoidal and
rectangular pulses. We find that the current profiles in both leads depend not
only on the shape of the pulses, but also on the position of the second
contact. The current reflects standing waves created by the contact potentials,
like in a wind musical instrument (for example a flute), but occurring on the
background of the equilibrium charge distribution. The number of electrons in
our quantum "flute" device varies between two and three. We find that for
rectangular pulses the currents in the leads may flow against the bias for
short time intervals, due to the higher harmonics of the charge response. The
GME is solved numerically in small time steps without resorting to the
traditional Markov and rotating wave approximations. The Coulomb interaction
between the electrons in the sample is included via the exact diagonalization
method. The system (leads plus sample wire) is described by a lattice model.
 Citations (19)
 Cited In (0)

Article: Applied Physics Letters
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ABSTRACT: A narrowband terahertz signal generated by a unitravelling carrier photodiode (UTCPD) interfaced with a dualmode FabryP\'{e}rot laser diode is demonstrated. A beat tone corresponding to the free spectral range is generated on the UTCPD, and radiated by a transverseelectromagnetichorn antenna. A terahertz signal at a frequency of 372 GHz, featuring a linewidth of 17 MHz is recorded by a subharmonic mixer coupled to an electrical spectrum analyzer. All components involved in this experiment operate at room temperature. The linewidth and the frequency of the emitted terahertz wave are analyzed, along with their dependency on DCbias conditions applied to laser diode.Applied Physics Letters 01/2010; 96:241106. · 3.79 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: A quantized current in a lateral quantum dot, defined by metal gates in the twodimensional electron gas (2DEG) of a GaAs/AlGaAs heterostructure, was observed. By modulating the tunnel barriers in the 2DEG with two phaseshifted RF signals and employing the Coulomb blockade of electron tunneling, quantized current plateaus in the currentvoltage characteristics were produced at integer multiples of ef, where f is the RF frequency. This demonstrates that an integer number of electrons pass through the quantum dot each RF cycle.Physical Review Letters 10/1991; 67(12):16261629. · 7.73 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: We present numerical results of an examination of the current mirror effect in a singleelectron turnstile capacitively coupled to a onedimensional (1D) array of small tunnel junctions. We deal with a 20junction 1D array and 4junction turnstile coupled at their center electrodes via a coupling capacitor. A coupling parameter, Qc, and a current mirror index are introduced to define the strength of coupling and to evaluate the quality of the current mirror effect, respectively. Numerical results show that a finite gate charge of e/2 to the center electrodes of both the 1D array and the turnstile enhances the current mirror effect, where −e is the charge of an electron. To investigate the enhanced current mirror effect, charge propagation in the 1D array under a finite gate charge of e/2 is simulated. It is found that electrons temporarily trapped at the center electrode in the 1D array play an important role in the enhancement of the current mirror effect. © 2003 American Institute of Physics.Journal of Applied Physics 09/2003; 94(7):44804484. · 2.21 Impact Factor
Page 1
arXiv:1202.0566v1 [condmat.meshall] 2 Feb 2012
Excitation of collective modes in a quantum flute
Kristinn Torfason,1,2Andrei Manolescu,1Valeriu Molodoveanu,3and Vidar Gudmundsson2
1School of Science and Engineering, Reykjavik University, Menntavegur 1, IS101 Reykjavik, Iceland
2Science Institute, University of Iceland, Dunhaga 3, IS107 Reykjavik, Iceland
3National Institute of Materials Physics, P. O. Box MG7, BucharestMagurele, Romania
(Dated: February 6, 2012)
We use a generalized master equation (GME) formalism to describe the nonequilibrium time
dependent transport of Coulomb interacting electrons through a short quantum wire connected to
semiinfinite biased leads. The contact strength between the leads and the wire is modulated by
outofphase timedependent potentials which simulate a turnstile device. We explore this setup by
keeping the contact with one lead at a fixed location at one end of the wire whereas the contact with
the other lead is placed on various sites along the length of the wire. We study the propagation of
sinusoidal and rectangular pulses. We find that the current profiles in both leads depend not only on
the shape of the pulses, but also on the position of the second contact. The current reflects standing
waves created by the contact potentials, like in a wind musical instrument (for example a flute),
but occurring on the background of the equilibrium charge distribution. The number of electrons in
our quantum ”flute” device varies between two and three. We find that for rectangular pulses the
currents in the leads may flow against the bias for short time intervals, due to the higher harmonics
of the charge response. The GME is solved numerically in small time steps without resorting to
the traditional Markov and rotating wave approximations. The Coulomb interaction between the
electrons in the sample is included via the exact diagonalization method. The system (leads plus
sample wire) is described by a lattice model.
PACS numbers: 72.15.Nj, 73.23.Hk, 78.47.da, 85.35.Be
I. INTRODUCTION
The control of transient transport properties of open
nanodevices subjected to timedependent signals is nowa
days considered as the main tool for charge and spin ma
nipulation. Pumpandprobe techniques allow the indi
rect measurement of tunneling rates and relaxation times
of quantum dots in the Coulomb blockade1. Quantum
point contacts and quantum dots submitted to pulses
applied only to the input lead generate specific output
currents2–4. Single electrons pumping through a double
quantum dot defined in an InAs nanowire by periodic
modulation of the wire potential has been observed5, as
well as nonadiabatic monoparametric pumping in Al
GaAs/GaAs gated nanowires6.
Modeling such shorttime processes is a serious task
because even if the charge dynamics is imposed by the
timedependent driving fields, the geometry of the sam
ple itself and the Coulomb interaction play also an impor
tant role. A wellestablished approach to timedependent
transport relies on the nonequilibrium Greens’ function
(NEGF) formalism, the Coulomb effects being treated
either via densityfunctional methods7or within many
body perturbation theory8. Alternatively, equation of
motion methods were used for studying pumping in fi
nite and infiniteU Anderson singlelevel models9. The
numerical implementation of these formal methods in the
interacting case requires extensive and costly computa
tional work if the sample accommodates more than few
electrons; as a consequence, accurate simulations for sys
tems having a more complex geometry and/or complex
spectral structure are not easily obtained.
Recently we reported transport calculations for a two
dimensional parabolic quantum wire in the turnstile
setup10, neglecting the Coulomb interaction between
electrons in the wire. The latter is connected to semi
infinite leads seen as particle reservoirs. Let us remind
here that the turnstile setup was experimentally real
ized by Kouwenhoven et al.11.
a timedependent modulation (pumping) of the tunnel
ing barriers between the finite sample and drain and
source leads, respectively. During the first half of the
pumping cycle the system opens only to the source lead
whereas during the second half of the cycle the drain
contact opens. At certain values of the relevant parame
ters an integer number of electrons is transferred across
the sample in a complete cycle. More complex turnstile
pumps have been studied by numerical simulations, like
onedimensional arrays of junctions12or twodimensional
multidot systems13.
It essentially involves
In this work we perform a similar study for an inter
acting onedimensional quantum wire coupled to an in
put (source) lead at one end, while the output (drain)
lead can be plugged at any point along its length. Both
contacts are modulated by periodic pulses (sinusoidal or
square shaped). Our study is motivated by the possibil
ity to control the transient currents through the variation
of the drain contact. We shall see in fact that the flexi
bility of the drain contact allows us to capture different
responses of the sample to local timedependent pertur
bations which can lead to transient currents with specific
shapes. In some sense, our system works like a ‘quan
tum flute’, this fact being revealed when analyzing the
distribution of charge within the wire. In particular, we
Page 2
2
calculate and discuss the deviation of the charge density
from the mean value for each site of the quantum wire
and observe the onset of standing waves.
The effect of the electronelectron interaction is in
cluded in the sample via the exact diagonalization
method while the timedependent transport is performed
within the generalized master equation (GME) formal
ism as it is described in Ref. 14. The implemented GME
formalism can be used to describe both the initial tran
sient regime immediately after the coupling of the leads
to the sample and the evolution towards a steady state
achieved in the long time limit. The GME formalism
captures the transient charging of manybody states and
Coulomb blockade effects14.
To the best of our knowledge these are the first numer
ical simulations of electronic transport through an inter
acting quantum turnstile which is not a quantum dot.
We emphasize that most of the studies on pumping in
interacting systems are focused on singlelevel quantum
dots15,16in the Kondo regime. Here we consider a sys
tem with spatial extension where the charge distribution
plays an important role in the transport processes.
We discuss for the first time the effect of contacts’ lo
cation on the transient currents. More precisely, we show
that if the drain lead is attached to different regions of
the quantum wire the currents in both leads are consid
erably affected.
The paper is organized as follows: The model and the
methodology are described in Section II, the numerical
results are presented in Section IV, and the conclusions
in Section V.
II. THE QUANTUM FLUTE MODEL
The physical system consists in a sample connected to
two leads acting as particle reservoirs. We shall adopt
a tightbinding description of the system: the sample is
a short quantum wire and the leads are 1D and semi
infinite. In this work we consider a sample of 10 sites.
This number optimizes the computational time and the
physical phenomenology which we intend to describe. A
sketch is given in Fig. 1. The left lead (or the source,
marked as L) is contacted at one end of the sample and
the right lead (or the drain, marked as R) may be con
tacted on any other site. The Hamiltonian of the coupled
and electrically biased system reads as
H(t) =
?
ℓ
Hℓ+ HS+ HT(t) = H0+ HT(t), (1)
where HSis the Hamiltonian of the isolated sample, in
cluding the electronelectron interaction,
HS=
?
n
End†
ndn+1
2
?
mn
m′n′
Vmn,m′n′d†
md†
ndm′dn′ . (2)
The (noninteracting) singleparticle basis states have
wave functions {φn} and discrete energies En. Hℓ, with
χL
χR
11234567789 10
µL
µR
FIG. 1. (Color online) A sketch of the system under study.
A 1D lattice with 10 sites (“the sample”) is connected to two
semiinfinite leads via tunneling. The left lead is connected to
the left end of lattice, while the position of the right lead can
be changed. The contacts (χL, χR) are modulated in time.
{ℓ} = (L,R), is the Hamiltonian corresponding to the left
and the right leads. The last term in Eq. (1), HT de
scribes the timedependent coupling between the single
particle basis states of the isolated sample and the states
{ψqℓ} of the leads:
HT(t) =
?
n
?
ℓ
?
dq χℓ(t)(Tℓ
qnc†
qℓdn+ h.c.). (3)
The function χℓ(t) describes the timedependent switch
ing of the samplelead contacts, while d†
ate/annihilate electrons in the corresponding single
particle states of the sample or leads, respectively. The
coupling coefficient
nand cqℓ cre
Tℓ
qn= V0ψ∗
qℓ(0)φn(iℓ), (4)
involves the two eigenfunctions evaluated at the contact
sites (0,iℓ), 0 being the site of the lead ℓ and iℓthe site
in the sample14. In our present calculations we keep the
left lead connected to the site iL= 1, while the position
of the right lead iR is varied. The parameter V0 plays
the role of a coupling constant between the sample and
the leads.
We will ignore the Coulomb effects in the leads, where
we assume a high concentration of electrons and thus
strong screening and fast particle rearrangements. The
Coulomb electronelectron interaction is considered in de
tail only in the sample, where Coulomb blocking effects
may occur. The matrix elements of the Coulomb poten
tial in Eq. (2) are given by,
Vmn,m′n′ =
?
d? xd? x′φ∗
m(? x)φ∗
n(? x′)
uC
? x − ? x′φm′(? x)φn′(? x′).
(5)
We calculate the manyelectron states (MES) in the
sample by incorporating the Coulomb electronelectron
interaction following the exact diagonalization method,
i. e. without any mean field approximation. The MES
are calculated in the Fock space built on noninteracting
singleparticle states14. Since the sample is open the
number of electrons is not fixed, but the Coulomb in
teraction conserves the number of electrons. With 10
lattice sites we obtain 10 singleparticle eigenstates and
Page 3
3
thus 210= 1024 elements in the Fock space spanned by
the occupation numbers. The Coulomb effects are mea
sured by the ratio of a characteristic Coulomb energy
UC= e2/(κa) and the hopping energy ts= ?2/(2meffa2).
Here a denotes the intersite distance (the lattice con
stant of the discretized system), while κ and meff are
material parameters, the dielectric constant and the elec
tron effective mass, respectively. In our calculations we
use the relative strength of the Coulomb interaction,
uC= UC/ts, which is treated as a free parameter.
III. THE TRANSPORT FORMALISM
The equation of motion for the our system is the quan
tum Liouville equation,
i?˙W(t) = [H(t),W(t)] ,W(t < t0) = ρLρRρS. (6)
W(t), the statistical operator is the solution of the equa
tion and completely determines the evolution of the sys
tem. At times before t0the systems are assumed to be
isolated and W(t) is simply the product of the density
operator of the sample and the equilibrium distributions
of the leads.
Following the NakajimaZwanzig technique17we define
the reduced density operator (RDO), ρ(t), by tracing out
the degrees of freedom of the environment, the leads in
our case, over the statistical operatorof the entire system,
W(t)
ρ(t) = TrLTrRW(t),ρ(0) = ρS. (7)
The initial condition corresponds to a decoupled sample
and leads when the RDO is just the statistical opera
tor of the isolated sample ρS. For a sufficiently weak
coupling strength (V0) one obtains the nonMarkovian
integrodifferential master equation for the RDO
˙ ρ(t) = −i
?[HS,ρ(t)]
?
−1
?2
?
ℓ
dq χℓ(t)
?
[Tqℓ,Ωqℓ(t)] + h.c.
?
,
(8)
where the operators Ωqℓand Πqℓare defined as
Ωqℓ(t) = e−itHS
?t
0
T†
ds χℓ(s)Πqℓ(s)ei(s−t)εqℓeitHS,
Πqℓ(s) = eisHS?
qℓρ(s)(1 − fℓ) − ρ(s)T†
qℓfℓ
?
e−isHS,
and fℓis the Fermi function of the lead ℓ describing the
state of the lead before being coupled to the sample. The
operators Tqℓand T†
qℓdescribe the ’transitions’ between
two manyelectron states (MES) α? and β? when one
electron enters the sample or leaves it:
(Tqℓ)αβ=
?
n
Tℓ
qn?αd†
nβ?.(9)
The GME is solved numerically by calculating the ma
trix elements of the RDO in the basis of the interacting
MES, in small time steps, following a CrankNicolson al
gorithm. More details of the derivation of the GME can
be found in Ref. 18. The calculation of the interacting
MES is described in Ref. 14
Mean values of observables can by obtained by tak
ing the trace of product of the corresponding opera
tor and the RDO. The time dependent charge den
sity is obtained from the particledensity operator,
n(x) =?
particle wave functions,
l,mφ∗
l(x)φm(x)d†
ldm, where φl,m(x) are single
?Q(t,x)? =
?
αβ
ραβ(t)
?
lm
φ∗
l(x)φm(x)?βd†
ldmα?. (10)
The total time dependent charge in the sample is found
by integrating over x or by using the number operator
N =?
?Q(t)? = eTr{ρN} = e
?
N
md†
mdm:
N
?
αN
?αNρ(t)αN?, (11)
where αNdenotes the (Coulomb interacting) MESs with
fixed number of electrons N. Remark that one can also
calculate the partial charge accumulated on Nparticle
MESs.
The currents in the system are then found by taking
the derivative of Eq. (11) with respect to time,
?J(t)? = JL(t)−JR(t) = e
?
N
N
?
αN
?αN ˙ ρ(t)αN?. (12)
The time derivative of the RDO can be substituted by
the righthand side of the GME [Eq. (8)] and so it is
possible identify the currents in each lead,
?Jℓ(t)? = −1
?2
?
N
N
?
αN
?
dq χℓ(t)?αN[Tqℓ,Ωqℓ(t)]αN?
+ h.c.
(13)
We also introduce a pindexed period average for the cur
rents (the pth period covers the interval [tp−1,tp]):
jp=1
T
?tp
tp−1
dt Jℓ(t),(14)
which in the periodic phase, i. e. sufficiently long after
the initial transient stage, does not depend on k and on
the lead. T is the period of the pulses, and Qp = Tjp
is the total charge transferred through the sample within
the period p.
The switchingfunctions in Eq. (3) act on the contact
regions shaded blue in Fig. 1 and are used to mimic
potential barriers with time dependent height. In the
present study we use two kinds of switchingfunctions.
The first switchingfunction used in the study is a sine
function,
χℓ(t) = A
?
1 + sin(ω(t − s) + φℓ)
?
,(15)
Page 4
4
where A = 0.5 controls the amplitude and ω = 0.105 the
frequency. The phase shift between the leads is π, φL= 0
and φR= π. The last parameter s = 15 is used to shift
the functions as needed.
The second switching function corresponds to quasi
rectangular pulses, and it is made by combining two quasi
Fermi functions that are shifted relatively to each other,
χℓ(t) = 1 −
1
et−γℓ
s−δ+ 1−
1
e−(t−γℓ
s)+(Tℓ
p+δ)+ 1,
t ∈ [0, 2Tℓ
p],
(16)
where γℓ
leads (ℓ = L,R) and Tℓ
in the two leads. The pulses are not built with perfect
rectangles for reasons related to numerical stability. The
parameter δ controls the shape of the pulse and is fixed
at the value δ = 10. The time unit used is ?/ts.
The time dependent contact functions are graphed at
the bottom of Figs. 7 and 3. The frequency of the func
tions in Eq. (15) and (16) were chosen to be similar. The
initial values are χL,R(0) = 0, i. e. the leads and the
sample are initially disconnected.
s= {0, TL
p} defines the phase shift between the
p= 30 is the pulse length, the same
IV.RESULTS
We will use the relative Coulomb energy uC= 1.0. For
a material like GaAs this value would correspond to a
sample length of 9a ≈ 45 nm. Although quite short, this
is an experimentally attainable length. We believe our
results are also valid for longer samples, but the set of our
parameters is also restricted by the computational time
spent in solving the GME which grows very fast with the
number of MES. (A typical calculation took several days
of CPU.) The time unit is ?/ts≈ 0.029 picoseconds. The
leadsample coupling parameter is also constant, V0= 1.0
(units of ts).
A. Energy spectrum
The MESs of the sample are characterized by the chem
ical potentials µ(i)
ergy of the interacting MBS number i containing N par
ticles, i = 0 indicating to the ground state and i > 0 the
excited states. In Fig. 2 we show the chemical potential
diagram for our system. The strength of the Coulomb
interaction is uC = 1. For the singleparticle states
(N = 1) the chemical potentials are in fact the single
particle energies. The effect of the Coulomb interaction
is clearly visible for N > 2.
chemical potential for N = 2, µ(0)
absence of Coulomb interaction it is equal to µ(1)
We select the bias window ∆µ = µL− µR such that
it includes the ground state with N = 3 electrons,
µL = 3.20 and µR = 2.98. The bias window includes
N:= E(i)
N− E(0)
N−1, where E(i)
Nis the en
For example the lowest
≈ 2.58, whereas in the
2
1
≈ 2.32.
12
N
3
2.0
2.5
3.0
3.5
4.0
µ(i)
N[ts]
N = 1
N = 2
N = 3
µL = 3.37
µR = 3.15
FIG. 2. (Color online) The energy diagram. The blue circles
(
) correspond to singleparticle states, the red squares
(
) to twoparticle states and the brown diamonds (
to threeparticle states. The green solid line (
window ∆µ = µL−µR = 3.20−2.98 = 0.22. The bias window
includes the three particle ground state, but also excited one
and twoparticle states. So the expected number of electrons
in the steady state is slightly below three.
)
) is the bias
also excited single and twoparticle states. The single
particle state is well below the top of the bias and so
the population of this state will be very small, and the
number of electrons in the sample is expected to be some
where between 2 and 3. Therefore we also expect only
two and threeparticle states to be involved in the trans
port of electrons through the sample.14
B. Sinusoidal pulses
We begin the timedependent calculations with N = 3
electrons in the sample, initially assumed in the ground
state. This is done by initializing the diagonal density
matrix element of the sample corresponding to this state
to one and all the other matrix elements to zero. The left
lead (L) is permanently in contact with the left end of the
sample, i. e. site 1. The right lead (R) is placed on various
other sites, as indicated in Fig. 1. The time evolution is
then followed in short time steps, by using the contact
functions χL,R(t). The charge accumulated in the sample
and the currents in the two leads are calculated at each
time step. In Fig. 3 we show the results with sinusoidal
pulses, corresponding to Eq. (15), and with two different
placements of the R lead: on sites 10 and 3.
We first observe the charge in the sample and its time
evolution shown in the upper panels of Fig. 3. After the
contacts begin to operate the initial charge of N = 3
electrons changes in time. Part of the charge flows into
Page 5
5
the leads, depending on which one is accessible, and the
average charge drops, until a periodic regime is estab
lished. The lower panels of the figure show the contact
functions. In Fig. 3(a) the right contact is placed at
site 10. The charge in the sample has maxima and min
ima at the time points when the contact functions are
equal. In between these time points, the charge increases
when the left contact opens further and the right one
closes, and decreases otherwise. The population of the
threeparticle and twoparticle states are also shown (the
population of the singleparticle state being negligible)
and they oscillate in antiphase, i. e. the gain of one is
partly compensated by the loss of the other one.
The currents in the leads are shown in 3(b), and they
have similar shape as the contact functions, except in
the initial transient phase, before the periodic regime is
stabilized. In the first cycle the current in the left lead
is initially negative. The sign rule is that positive cur
rents correspond to charge flow from the left to the right
lead, and negative currents correspond to the opposite
direction. The initial negative current in the left lead in
dicates initial charge flow from the sample into that lead
during the first cycle as long as the contact to the right
lead is closed. The main impression of these results is
that the periodic regime qualitatively corresponds to a
linear response of the charge and currents to the contact
strength.
The situation may change when the right contact is
placed on another site, for example on site 3, as shown
in 3(cd). In this case the oscillations of the charge and
currents are weaker. The current in the left lead is no
longer sinuslike. This shows now a nonlinear behav
ior of the charge response to the same pulses as before.
Some sort of standing waves are created in the sample
and the right contact creates a local perturbation of the
charge fluctuations at that point. Negative currents in
the left lead may occur now during more pulse cycles as
before. This is somewhat surprising, since such currents,
although very small, are actually driven against the bias.
Let’s mention that in the absence of a bias (∆µ = 0)
the currents in both leads oscillate between positive and
negative values, but with zero average, such that no real
pumping effect is obtained in this setup, irrespectively of
the placement of the leads (not shown).
The currents in the leads reflect the charging or dis
charging of the sample, but these are actually complex
processes, because different states may be occupied with
different time constants, related to the tunneling matrix
elements, and thus the charging and the currents may
have shorttime fluctuations. The fine structure of the
currents is thus a complicated issue, which will be dis
cussed further.
C.Charge distribution in the sample
The charge distribution inside the sample is shown in
the Fig. 4 and it is far from homogeneous. The charge is
0
1
2
3
Q [e]
Left 1, Right 10
(a)
Left 1, Right 3
(c)
−0.005
0
0.005
0.01
I [ets/¯ h]
(b)
(d)
0 100 200 300 400
0
1
t [¯ h/ts]
χℓ(t)
40100 200 300 400
Left leadRight lead
Charge
N2
N3
FIG. 3. (Color online) Charge and current for µL = 3.37,
µR = 3.15, uC = 1.0 and χℓ(t) ∝ sin(ωt).
brown solid line ( ), charge for two particle states black
dashed (
), for three particle states violet dotted (
Current for the left lead blue dashed (
red solid ( ). We consider two locations of the right lead.
(a) Charge, left lead 1, right lead 10. (b) Current, left lead
1, right lead 10. (c) Charge, left lead 1, right lead 3. (d)
Current, left lead 1, right lead 3.
Total charge
).
), for the right lead
averagedin time over an entire period of the contact func
tions when the system is in a periodic regime. In the case
shown the right contact is placed on site 10. The charge
distribution does not qualitatively change for other place
ments of the right contact (not shown). The distribution
is symmetric along the sample, in spite of the presence
of the bias window, which shows that the contacts be
tween the sample and the leads are actually weak in our
case. We can say that the charge distribution follows
the geometrical extend of those singleparticle states that
contribute to the active two and threeparticle MBS.
Next, in Figs. 5 and 6, we show the deviation of the
charge density from the mean value, on each lattice site,
for selected time moments during half a cycle. For the
other halfcycle the reverse motion occurs.
placements of the right lead ate again selected at sites
10 and 3. Standing waves are clearly seen.
contact configuration L1R10 (Fig. 5) the standingwave
pattern shows something between two and three wave
lengths. Nodes and antinodes can be distinguished, and
also a global up and down motion mode seems to occur.
But it is clear that the amplitude of the charge oscilla
tions at the contact sites is quite large.
When the right contact is on site 3 (Fig. 6) only about
two wavelengths may be seen, at least for t < 3T/8, but
also the charge seems to oscillate with different ampli
The two
For the
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