An electron motion induced by magnetic field pulse in bi-layer quantum wire

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by  Tomasz Chwiej

An electron motion induced by magnetic field pulse in bi-layer quantum wire
T. Chwiej∗
AGH University of Science and Technology, al. A. Mickiewicza 30, 30-059 Cracow, Poland
Weconsider theoretically thepossibility of an electron acceleration in quantumwire byshort mag-
neticpulseslasted bewteen severaltofew tensof picoseconds. Weshowthat suchpossibility exists
provided that, the electron is initially localized in part of nanowire that consists of two vertically
aligned layers which are tunnel coupled. When a horizontally directed magnetic field, changeable
6 in time, is also perpendicular to the main axis of a wire, it generates a rotational electric field in
1
it which pushes the upper and the lower parts of the electron wavepacket in opposite directions.
0
We have found however, that for an asymmetric vertical confinement, the majority part of charge
2
densitystartstomoveinthedirectionoflocalelectricfieldinitslayerbutitalsodragstheminority

n partin thesame direction what resultsin coherentmotion ofan entirewavepacket. Wediscussthe
a dynamicsof thismotion in dependenceon thetime characteristics of themagnetic pulse.
J

5 PACSnumbers: 72.25.Dc,73.21.Hb
1 Keywords: quantum wire,ballistictransport,magneticfield

]
ll I. INTRODUCTION The action of temporary rotational electric field on the
a electronwavepacketconfinedinverticalbi-layernanowire
h
is schematically depicted in Fig.1. We have investigated
- The tunnel coupling between two quantum wires
s this mechanism of acceleration of an electron in nanos-
has a great impact on the single electron transport
e
m properties.1–5 If magnetic field penetrates such quantum tructure of this kind and have found that the assymetry
introduced to the vertical confinement enables an entire
systemitcanmodify themagnitude ofa tunnelcoupling
. electronwavepackettomoveinaparticulardirectionbut
t buttheextentofsuchmodificationsdependsonthemag-
a its further dynamics, i.e. after the pulse is finished, de-
m netic fieldstrentghandmutualarrangementofmagnetic
field and the wire axis.6,7 If magnetic field is parallel to pends mainly on the time length of magnetic pulse and
- onthat,whetherthemagneticfieldfinallyvanishesorits
d the wire axis, it squezees the wavefunctions of magneto-
value remains non-zero but is fixed for further times. In
n subbands within each layer what results in lower value
o of a tunnel factor.7,8 On the other hand, if it is set per- the latter case, which is an analogof switching the mag-
c pendicularilytowireaxisandtothelayerscoupledlater- netic field on, the electron motion in constant magnetic
[ field is governed by the magnetic force which for longer
ally or vertically, it can hybridize the magnetosubbands
times breaksthe coherencebetween the upper and lower
1 between layers. This modifies, to a large extent, the en-
v ergy dispersion relation E(k) since the pseudo-gaps are layers’parts of an electron wavepacket.
0 opened and the negative dispersion realation appear in Paper is organizedas follows. In Sec.II we present nu-
8 energy spectrum.3,9 As we haveshownin our last paper, mericalmodelofsolvingthetime-dependentSchr¨odinger
8 equation for an electron confined in bi-layer nanowire,
hybridizationwhichleadstoformationofpseudogapscan
3
be utilized for tuning the magnitude of spin polarization results of simulations are presented and discussed in
0
Sec.III.We endupconsiderationswithconclusionsgiven
. ofwire’sconductanceprovidedthatthewirehaslowden-
1 sity of defects.10 in Sec.IV.
0
Inpresentpaper we study the dynamics ofanelectron
6
1 motioninabi-layerquantumwiremadeofInGaAs/GaAs
: and GaAs/AlGaAs heterostructure which motion is in- II. THEORETICAL MODEL
v
ducedbythe magneticpulseonly. Magneticacceleration
i
X or deceleration of an electron in such nanostructure can We start our considerations with the single electron
r be conducted provided that: i) the wave functions origi- Hamiltonian H = (ppp + eAAA)2/2m∗ + V (rrr). Throuout
c
a
natedfromdifferentlayersaremutually hybridized,and, the paper we will use the time-dependent vector poten-
ii) the time duration of magnetic pulse is between a few tial in non-sybmmetribc form AAA(t) = [zB(t),0,0] which
andtensofpicoseconds. Atpresent,suchshortmagnetic gives the magnetic field piercing the layers horizontally
pulsescanbe generatedbyusingtheAuston’sphotocon- BBB = [0,B(t),0] and being perpendicular to the direc-
ductive switches11,12 or by the off-resonant magnetiza- tion of an electron motion. We assume an electron
tion of ferromagnetic thin films with the terahertz laser can move along the wire axis in x direction within the
pulses.13 Due to the Maxwell law i.e. ∂B~/∂t=−∇×E~, harmonic oscillator potential (V (x) = m∗ω2x2/2) and
1
if the time varying magnetic field is directed perpendic- can tunnel between two vertical layers which establish
ularilyto awire’saxisaswellastothe verticallyaligned a double-well potential in z direction (V (z)). Its mo-
2
transport layers, it generates the rotational electric field tion in y direction is frozen to the ground state, and is
which try to push two parts of the electron wavepacket neglected in further discussion. Sketch of a model con-
being localized in both layers in the opposite directions. finement potential is shown in Fig.1. Based on these as-
2
in quantum wire. For α = 0, basis wave functions f (z)
1
and f (z) have defined parities. Then, the third term
2
in Eq.1 disappears and only the secondterm mixes both
components ψ (x) and ψ (x) of an electron’s wave func-
1 2
tion through the off-diagonal elements H and H in
12 21
Eq.2 as the first term is pure diagonal. On the other
hand, when α > 0, the third term in Eq.1 also gives
contribution to off-diagonal elements in Eq.2 and to the
FIG. 1: (Color online) The cross-section of a model bi-layer diagonalonesaswell. Mixingofthesecomponentsiscru-
nanowireconsideredinpaper. Theelectronmotioninxdirec- cial for inducing the motion of an electron by the mag-
tion within harmonic oscillator potential (blue line) is stim- netic field pulse. If there is no mixing during the time
ulated by rotational electric field to be induced by the time
evolution, the electron permanently occupy the lowest
varyingmagneticfieldwhichispointedalongydirection,and,
state that is ψ (x,t) whereas occupation of ψ (x,t) has
providedthatboth,upperandlowerlayersaretunnelcoupled. 1 2
to be zero due to the condiction E(z) <E(z), otherwise,
1 2
mixing of these elements should change the dynamics of
sumptions, and additionally assuming that the confine- an electron in the wire. According to the definitions of
ment potential in x and z directions is separable that the off-diagonal elements given by the second and third
is V (x,z) = V (x)+V (z), we make the problem sim- terms in Eq.1, the mixing takes place only when B 6= 0
c 1 2
pler diagonalizingthe part of Hamiltonian dependent on thatisduringthemagneticpulse(∂B/∂t6=0)orforcon-
z variable only h = −(~2/2m∗)∂/∂z2 + V (z). From stantmagneticfieldbutprovidedthatanelectronmoves
z 2
the setof its all eigenstatesf (z)we chose only twolow- within the wire.
k
b
est eigenmodes to form the functions basis {f1,f2}. The Even though, the last two terms in Eq.1 can mix
rest lie much higher on energy scale and therefore their the wave functions ψ (x) and ψ (x), only the second
1 2
contributions to final solution are neglected. Next, cal- term may force an electron to change its position. The
culatingthematrixelementsofHamiltonianinthisbasis proofisstraightforward. First,considerthegroundstate
Hk,k′ =hfk|H|fk′i we get: of electron for B = 0, its wave function at t = 0 is
given then by Ψ(x,t) = [ψ (x,0),ψ (x,0)] = [g (x),0],
b b 1 2 0
Hk,k′ = Tx+V1(x)+Ek(z) δk,k′ with g0(x) being the normalized square integrable func-
(cid:16) (cid:17) tion. If we now skip in Hamiltonian in Eq.2 the term
b + ~ωbcZk(1,k)′kx+ m∗2ωc2Zk(2,k)′ (1) ~ωcZk(1,k)′kx and replace the time derivative with the
forward first-order finite difference formula, we obtain
b b
in first time step an approximated expressions for: i)
where Tx is kinetic operator for x direction, Ek(z) is a k- ψ (x,∆t) = (1 − i∆tm∗ω2E /~)ψ (x,0), where E is
th eigenenergy of Hamiltonian h , ω = eB/m∗ is the 1 c 1 1 1
b z c the total energy of electron at t = 0 and the contri-
cyclotron frequency, matrix elembents Zk(1,k)′ and Zk(2,k)′ are bution from ψ2 vanishes since ψ2(x,0) = 0, and, ii)
defined as Zk(m,k)′ =hfk|zm|fk′i and kx =px/~=−i∂/∂x. ψ2(x,∆t)=−(i∆tm∗ωc2Z2(21)/2)ψ1(x,0). In other words,
Hamiltonian given in Eq.1 depends on time through the part of g0(x) is moved from ψ1 to ψ2 and vice versa for
second and the third terms in abovbe equbation which in- furthertimestepsbuttheshapeof|Ψ(x,t)|2isretainedin
clude ω because magnetic field change its value during any further time instant. Simply, this Hamiltonian term
c
anevolutionofquantumsystem. Thisformofenergyop- is a diamagnetic energyshift whichis responsible for op-
eratorallowsustodefinethewavefunctionofanelectron timizingthewavefunctionshapeinzdirectionthatisfor
as a two-componentobject |Ψ(x,t)i= 2 ψ (x,t)|f i its strongerlocalizationin magnetic field. The dynamics
k=1 k k
forwhichtheeffectivetime-dependentSPchr¨odingerequa- of an electron during the time of simulation is governed
tion has the following form: bytheterm~ω Z(1) k whichincludesthederivativeover
c k,k′ x
x variable.
∂ ψ (x,t) H H ψ (x,t) b
i~ 1 = 11 12 1 (2) We simulate the motion of an electron in bi-layer
∂t(cid:20) ψ2(x,t) (cid:21) (cid:20)H21 H22 (cid:21)(cid:20) ψ2(x,t) (cid:21)
nanowire stimulated by the magnetic field pulse by find-
with matrix elements Hk,k′ defined by Eq.1. ing the solution of time-dependent Schr¨odinger equation
Todescribetheconfinementinverticaldirectionwithin [Eq.2]insubsequenttimeinstantsbymeansof4-thorder
two layers we use the following approximation5,14: Runge-Kutta method with time step ∆t = 10−4 ps. For
this purpose we have performeda set of numerical simu-
V (z)=V {sin[(1+z/b)π/2]+αsin[π(1+z/b)]} lationsonregularmeshofnodes inx direction. We limit
2 max
(3) thelengthofquantumwireto800nmandimposesmooth
which is symmetric for α = 0 and the upper layers be- confining potential in this direction as for the quantum
comes deeper than the lower one for α > 0. The latter oscillator V (x) = m∗ω2x2/2 with oscillator strength
1 0
case is studied in detail in next section. The shape of ~ω = 0.5 meV. This spatial constriction allows us to
0
V (z) has a great impact on the dynamics of an electron studyofanelectronmotionforquitelongtimesafterthe
2
3
magneticpulseisfinishedwhatisparticularilyimportant
ifaninfluenceofmagneticforce(B =const)onthecou-
y
plingstrengthbetweentheupperandlowerpartsofelec-
tron’s wavepacketis taken into account. The basis func-
tions {f (z),f (z)} were also found numerically with fi-
1 2
nitedifferencemethodafterdiscretizationofHamiltonian
h on spatial mesh with 200 nodes for z direction. The
z
number of nodes in x direction was set to 400. All simu-
b
lationswereprecedeedbydiagonalizationofHamiltonian
appearinginEq.2to preparethe startingwavepacketfor
t=0 i.e. to find Ψ(x,t=0)=[ψ (x,0),ψ (x,0)].
1 2
Value of parameter b which defines the extent of ver-
tical confinement in Eq.3 was equal 30 nm. The height
of tunnel barrier between the upper and lower layers de-
pends on magnitude of V . We set its value equal
max
to 150 meV for which the energy splitting between the
bounding (f ) and the antibounding (f ) basis states
1 2
equals E(z) = E(z) −E(z) = 2 meV for symmetric con-
21 2 1
(z)
finement(α=0). Whenαbecomesnon-zerothen∆E
21
growswhateventuallylowerstheextentofmixingψ1and FIG.2: (Color online) Expectation valueof electron position
ψ2 components of an electron’s wave function. This can (left column) and the density (right column) localized in the
notbe howeveromitted since for α=0, the wavepackets upper (red color) and in the lower layer (black color) during
localizedinsymmetriclayersmoveinoppositedirections the time of simulation for α = 0 (first row) and α = 0.005
[seeFig.1]andanelectronwavepacketmotionasawhole, (second and third rows). Inset in figures (a) and (e) shows
in an arbitrarily chosendirection, can be obtain only for zoomsofoscillationsofx¯andhρiremainedafterthemagnetic
α > 0 what will be shown below. In paper we consider pulse is ended. The inset in (d) shows C2 = hψ2|ψ2i that is
an occupation of second eigenstate for vertical quantization.
two cases, in first the electron is accelerated by single ∗
Calculations were performed for parameters: m =0.04 (In-
magnetic pulse, while in second, its motion is induced
GaAs)and∆E2z1 =9.83 meV(a)and∆E2z1 =9.9meV in(b)
when the magnetic field is switched on/off.
and (c).
III. RESULTS
oscillations remain stable for longer times. The beating
patternshowninthisinsetcomeintoexistenceduetothe
A. Acceleration of an electron by a single magnetic
pulse overlapof two kinds of oscilltions, one with period equal
to T = 8.3 ps and second with period T = 0.44 ps.
1 2
To determine their origin we first calculated the expec-
Westartpresentationofourresultsforacasetheelec-
tation value of electron position in upper layer accord-
tronis acceleratedby a single magnetic pulse defined as:
ing to formula hxi = hΨ|x·Θ(z)|Ψi. Exactly at the
B(t) = B sin(πt/t )·Θ(t)·Θ(t −t), where Θ(t) z>0
m imp imp moment the magnetic pulse ends (B = 0), the electron
is the Heaviside step function, the amplitude of mag-
wave function can be expressed as Ψ(x,y,z,t = t ) =
netic field equals B = 5T and the time duration of imp
m ψ (x,t )f (z)g (y). The part of wave function
a pulse is changed within the range t = 5− 50 ps. k k imp k 0
Figure 2(a) shows the time variationismpof the electron Pwhich is dependent on x variable was expanded within
the base constituted by Hermite polynomials i.e. the
positions in the upper and in the lower layers obtained
quantum oscillator eigenstates Φ :
according to Eqs. (5) and (6) [see description below] µ
for symmetric bi-layer nanowire (α = 0). The magnetic
pulse starts at t = 0 and ends at t = 20 ps. Within Ψ(x,y,z,t) = d Φ (x)
k,µ µ
thistime interval,thepartsofelectronwavepacketlocal- Xk Xµ
ibzuetdwinituhpoppeproasnitdeipnhlaosweesri.lea.yertsheosucpilplaetrepinarxt dstiraercttsiotno ·fk(z)g0(y)e−i(Eµx+Ekz+E0y)t/~ (4)
move to the right while the second one, localized in the
lower layer, to the left, and then, both change the di- Inaboveequation,dk,µ arethelinearcombinationcoeffi-
rections of their motions three times due to reflections cients, Φ (x) = C H (x)e−x2 are the normalized Her-
µ µ µ
within the regions of high confining potential. After the mite polynomials while three arguments appearing in
magnetic field eventually vanishes, these oscillations are phase factor are the eigenenergies for quantized motion
strongly dumped but do not disappear entirely. The in- of electron in x (Ex), y (Ey) and z (Ez) directions. Ac-
µ 0 k
set in Fig.2(a) shows that even though for t > t the cording to this definition, the position of electron in an
imp
amplitude of these oscillations become very small but upperlayer(dependenceong (y)andEy disappearafter
0 0
4
FIG. 3: (Color online) Dependence of the normalized expectation value of electron position (x¯) in bi-layer quantum wire on
timelengthofmagneticpulseandonthetimeofsimulation. Theinclineddashedlinesmarktherightborderofmagneticfield
pulse. TheupperrowdisplaystheresultsobtainedforInGaAsandeffectivemassm∗ =0.04whiletheloweroneforGaAs
InGaAs
∗
and effective mass m = 0.067. Parameter α which is shown on top of each subfigure represents an actual asymmetry in
GaAs
vertical confinement.
integration over y variable) reads: hρi if the terms hΦ |x|Φ i will be substituted by
z>0 µ ν
hΦ |Φ i=δ . Thissubstitutioneliminatesdependence
µ ν µ,ν
hxi = d∗ d hΦ |x|Φ ihf |Θ(z)|f i oftotal phase on∆Ex =0 but leavesit still dependent
z>0 k,µ m,ν µ ν k m µ,ν
Xk,mXµ,ν on ∆Ekz,m value. However in Fig.2(d) there are no oscil-
×ei∆Eµx,νt/~ei∆Ekz,mt/~ (5) lations at any time instant. For α = 0, the amounts of
electrondenistyconfinedinbothlayersareequal,simply
duetothe symmetryconstrictionimposedonthe confin-
Sincetheamountsofprobabilitydensitygatheredinboth
ing potential [see Fig.2(d)].
layers may differ from unity we normalized the expecta-
tion value of electron position in upper (or lower) layer: Whenα>0andtheconfinementenergyinlowerlayer
becomes higher than the one in an upper layer, then for
hxi t = 0 the larger part of electron density is gathered in
z>0
x¯ = (6)
z>0 upperlayer[seeFigs.2(e)and(f)]. Therefore,whenmag-
hΨ|Ψi
z>0
netic field starts to grow, the majority of wavepacket
The matrix elements hΦ |x|Φ i have non-zero values move to right but this time the rest of wavepacket is
µ ν
only if µ = ν ± 1 that is for ∆E = ±~ω and also pulled into this direction. In Figs. 2(b) and 2(c)
µ,ν 0
the coupling of these non-diagonal terms are responsi- we see that both densities move synchronously and in-
ble for the appearance of oscillations with longer period dependently on the time duration of magnetic pulse.
T1 = h/∆Eµ,µ±1 = 8.3 ps, provided that, ∆Ekzm van- Due to an asymmetry that has been introduced into the
ishes for k = m in Eq.5. Otherwise, when conditions confining potential in vertical direction, the amounts of
k 6=mandµ=ν±1arefulfilledthenthetotalphasefac- electron densities in both layers are smoothly changed
tor oscillates with period dependent on the sum of both when ∂B/∂t 6= 0 and for t > timp become fixed [Figs.
energy differences i.e. ∆E = ∆Ez +~ω = 10.33 meV 2(e)and2(f)] besidesthe smalloscillationsforfrequency
21 0 ′ ′
what gives the period equal to T = 0.44 ps. However ω = 2π/T visible in the inset in Fig.2(e). The length
this value is very close to value T2′ = h/∆E1 = 0.42 ps of2magneti2c pulse has great impact on the amplitude
2 21
what means that the coupling between vertical layers of the electron oscillation within quantum wire. For
[due to the terms hf |Θ(±z)|f i] is the main source of t = 20 ps the amplitude of expectation value of elec-
k m imp
highfrequencyoscillationsvisibleintheinsetinFig.2(a). tron position in wire reaches even x¯ = 100 nm while
max
Equation 5 can be further employed for calculation of fortwotimeslongerpulsetheamplitudeofelectronoscil-
5
lations falls to value x¯ = 2.3 nm. Despite the differ- which is modelled in our simulations by following for-
max
entlengthesofthesemagneticpulses,theperiodofoscil- mula:
lations in both cases is the same and equals T = 8.3 ps
1
that is the dynamics of wavepacket is governed by the B(t) = Bmsin(2πt/timp+βπ/2)·Θ(t)·Θ(timp−t)
energy levels structure of quantum harmonic oscillator + γ·B ·Θ(t−t ) (7)
m imp
established in x direction.
Analysis of the results showed in Fig.2(a-c) allows us to where: β = 0 and γ = 1 stand for the magnetic field
make a statement that the characteristics of the elec- growing in time [∂B/∂t > 0 for t ≤ timp] from 0 to Bm
tron motion induced by magnetic pulse in quantum bi- while β =1 and γ =0 for vanishing field [∂B/∂t<0 for
layer wire is dependent on at least two factors: i) the t≤timp] from Bm to 0.
time length of a magnetic pulse, and, ii) the degree of
asymmetryinverticalconfinementprovidedthattheam-
plitude of magnetic field is fixed. In order to study
the influence of these factors on dynamics of electron
motion in the wire we computed the expectation value
of the electron position as function of t length and
imp
on the time of simulation. Results for effective mass
m∗ =0.04(InGaAs)arepresentedinfirstrowinFig.3for
′
α = 0.005, 0.01, 0.02. These αs values give the energy
splittingsbetweentwolowesteigenstatesinverticaldirec-
tionequalto∆Ez =9.9 meV, 10.1 meV and 10.9 meV.
21
Forcomparison,secondrowinthisfiguredisplaysthere-
sultsform∗ =0.067(GaAs)andfortheenergysplittings
∆Ez = 2.3 meV ,3.1 meV and 5.1 meV. If an electron
21
effective mass is small [first row in Fig.3] then the in-
creaseofaconfiningpotentialassymetrybringstwomain
effects. First, the amplitude of x¯ grows [compare the FIG. 4: (Color online) Normalized expectation value of an
ranges of color scales in Figs.3(a-c)] if value of α is in- electronposition(x¯)inanupperlayerdependingonthetime
interval the magnetic field is switched on [∂B/∂t > 0 for
creased. For example, for α = 0.005 the amplitude of
electronoscillationsinxdirectionreachesx¯ =130 ps t < timp] and the time of simulation. Other parameters are
max given on thetop of each figure.
while for α = 0.02 it becomes two times larger. Second,
the time length of magnetic pulse which allows to ob-
Resultsforthefirstcase,whenmagneticfieldisturned
tain significant amplitudes systematically decreases for
larger α′s values. For α = 0.005 the electron motion in on, are presented in Fig.4. For InGaAs bi-layer wire,
the dynamics of an electron changes qualitatively and
quantum wire can not be induced by pulses longer than
quantitatively much as the imbalance in the confining
42.1 ps while for α = 0.02 it is limited only to 17.6 ps.
potential in the upper and in the lower layers raises [cf.
For longer pulses, oscillations are still possible but they
Figs. 4(a-c)]. For α=0.005 and α=0.01 the pattern of
might have much smaller amplitudes.
x¯isveryirregularforshortpulses(t<10ps)andbecomes
For larger effective mass [second row in Fig.3] the up-
regular that is with distinct single period of oscillations
per limit for the time length of magnetic pulse which
for longer pulses. However, it does not concern the case
still may induce large oscillations of electron position in
with α = 0.02 [Fig.4(c)] for which we may single out
bi-layer nanowire is lowered to several picoseconds but
onemainperiodbeingthesameforallvaluesoft . To
itssensitivitytopotentialasymmetryisnotsohighasin imp
explainthereasonsofthisirregularitywewillanalyzethe
thepreviouscase. Now,itsvaluedecreasesfrom14.8psto
changes in x¯ and in hρi in both layers for t = 5 ps,
12.7 ps whenthe value of α increasesfrom0.005to 0.02. α=0.005andm∗ =0.04. ResultsareshowniimnpFigs.5(a)
On the other hand, the largest amplitude of electron os-
and(b).
cillations becomes less dependent on potential asymme-
tryforheavierparticle,thechangeofα′svaluefrom0.005 In Fig.5(a) we see that the time characterictics of x¯
forbothpartsofwavepacketis similarfort<24.2psbut
to 0.02 increases the amplitude of x¯ from 210 nm to 250
thereafter their motions become decoupled even though
nm that is in a three times thinner range than for the
the denisty may still flow between the layers [Fig.5(b)].
smaller effective mass.
At t=0 the electron is at rest in the center of nanowire
[Fig.5(c)] and when the magnetic field starts raising the
rotational electric field is generated with opposite direc-
B. Electron motion induced by switching the tions in upper and lower layers. The upper layer con-
magnetic field on/off finesamajorityofelectrondensitywhichbeginstomove
to the right according to the electric field. Simultane-
In this section we present the results obtained for a ously, it pulls the minority of density in this direction
pulse generated by switching the magnetic field on/off but against the x-component of electric field in a lower
6
FIG.6: (Coloronline)Thetimechangesofexpectationvalues
of electron positions (a,c) and of the denisties (b,d) in the
upperand in thelower layers for timp =5 ps when magnetic
field is turned on. Black and red colors mark the results for
an upperand for a lower layer, respectively. Values of α and
of the effective masses are given on top of first and second
row.
stant t = 12.2 ps the decelerated motion of electron is
stopped on the left side of the nanowireand next the di-
rections of electron wavepacket motion and of magnetic
FIG.5: (Coloronline)Thetimevariationsoftheexpectation force are again reversed. For the right directed motion,
value of electron position (a) and of the elctron density in
theLorenzforcenoweasilyshiftsthepartofdensityfrom
the upper and in the lower layers (b) for timp =5 ps and for theuppertolowerlayer. Thus,theexistenceofmagnetic
magnetic field being switched on. Black and red colours in
force is crucial for the dynamics of the electron motion
(a) and (b) mark the results for the upper and lower layers,
in bi-layer nanosystem because when the electron oscil-
respectively. ∆ρc shows the density flow direction caused by
lates between the left and right turning points, the den-
thestrongerlocalization intheupperlayer,∆ρL indicatethe
densityflowduetomagneticforcewhile∆ρmarksthedensity sity may flow bewteen layers but with vertical velocities
flow caused by its unbalanced accumulation in layers. what eventually hinders the upper and lower parts of an
electronwavepacket. Forthatreasontheelectronmotion
is constantly slowing down on averageand both parts of
wavepacket,the upper and the lower ones, startto oscil-
layer. Even though the motion to the right implies the
late alongx axis in asynchronousmanner for t>24.2 ps
magnetic force is directed downwards to the lower layer,
[Fig.5(a)].
in fact, density flows upwards since the raising magnetic
fieldenhanceslocalizationofdensityindeeper,theupper If the vertical confinement becomes more asymmet-
well, for t < 5 ps. Then the strength of magnetic field ric (α = 0.02) or the effective mass is getting larger
is fixed (B = const) and for time instant t = 6.1 ps the (m∗ = 0.067) then an asynchronous motions of the up-
wavepacketis stopped andafter thatturned back due to per and lower electron densities also appear [see Figs.
its reflection of the high confinig potential on the right 6(a) and 6(c)]. In this case however, the disproportion
side [Fig.5(d)]. In the next stage the electron moves to between the amounts of densities gathered in layers is
the left, first being accelerated (6.1 ps<t<9.5 ps) and significantly larger e.g. hρi = 0.7 and hρi = 0.3 for
up lo
next whenit passesthe center ofnanowire(x=0)being t=0form∗ =0.04andα=0.02[Fig.6(b)]andincreases
decelerated (9.5 ps < t < 12.2 ps). Since the magnetic in time, what eventually has less impact on phase per-
fieldisstillappliedtothesystem,thedirectionofmagetic turbation in x¯ oscillations in upper layer. The inset in
forceisnowreversed[seeFigs. 5(d)and5(e)]whatforces Fig.6(a) shows that, the density flow between layers for
an additional part of electron density to flow from lower first few oscillations of x¯ lowers only the amplitude of
to upper layer what we notice in Fig.5(b). At time in- oscillations but motions of the upper and lower densi-
7
ties remain synchronous. But when the amplitude gets
smaller for t>100 ps, electron slows down what in con-
sequence lowersthe magnitude of magnetic force andfor
certain time period dumps the density flow between lay-
ers [comparethe amplitudes ofhρi atleft andrightsides
inFig.6(b)]. After that, the couplingbetweenupper and
lower denisties raises, both densities oscillate coherently
what eventualy makes the amplitude of x¯ to grow. This
periodic coupling and decoupling of both densities lead
eventually to formation of beating pattern in x¯(t) func-
tion if an electron is allowed to oscillate in nanowire for
longer times [Fig.6(a)]. Similar beating pattern for x¯
oscillations we also notice in Fig.6(c) for larger effective
massofanelectroni.e. form∗ =0.067. Inthiscase,only
a small assymetry in a confining potential (α=0.005)is
required to localize about 70% of total electron density
in upper layer. Unlike the previous case, now the mo-
tions of both parts of electron wavepacketare decoupled
for any time instant because as we see in Fig.6(c), the
oscillations of density in lower layer exhibit, to some ex-
tent, a chaotic behaviour. Besides the fact, the density
may still flow between layers [see Fig.6(d)], this chaotic
motionofdensityinlowerlayerdoesnotperturbthefre-
quency of x¯ oscillations in upper layer but it influences
their amplitude.
Inthelastpartofthissectionwepresenttheresultsof
FIG. 7: (Color online) (a,d) Dependence of x¯ on the time
simulations performed for the pulse being formed when
length of magnetic pulse and on the time of simulation for a
themagneticfieldisswitchoff[β =1andγ =0inEq.7].
casethemagneticfieldisswitchedoff[∂B/∂t<0fort<timp
PresentcaseisdifferentfromthatconsideredinSec.IIIA
and B =0 for t>timp]. (b,e) The time changes of densities
since now, the electron wavepacket is prepared at t = 0
gathered in upper layer (black color) and in lower layer (red
forB >0whatmeansthedensitiesconfinedinlayersare color). (c,f)Thenormalizedexpectationvalueofelectronpo-
magnetically coupled at the beginning of simulation and sition in upper (black color) and in lower (red color) layers -
their spatial localizations in vertical direction are low- they are the same. The results showed in first and in second
ered when magnetic filed vanishes in time. Results for column were obtained for α = 0.005 and α = 0.02, respec-
m∗ = 0.04 are presented in Fig.7. Unlike the preceding tively.
case with magnetic field growingin time and being fixed
afterwards,herewehaveB =0fort>t whatimplies
imp
the lack of an electron density flow between layers then. themagneticfieldisgettingsmaller,theelectricfieldgen-
Since the magnetic force is absent for t>t , densities erated in bi-layer wire pushes an upper, majority part
imp
inbothlayersoscillatecoherentlyalongthewireaxisand of density to the left what drags the lower one in this
the pattern of x¯ does not change in time of simulation direction too. Both densities start to move coherently
i.e. the period of oscillations is fixed [cf. Figs. 7(a) and [seeFig.7(c)]. However,sincethemagneticfieldbecomes
7(c)]. The degree of asymmetry in vertical confinement weaker,thedensityisnolongerstronglysqeezedinupper
has large impact on sensitivity of the system on a time layeranditstartstoflowtolowerlayerwhatisaddition-
length of magnetic pulse. For slightly asymmetric con- ally enhanced by the magnetic force which is directed
finement(α=0.005)theamplitudeofx¯inupperlayeris downwards. At t = 7.4 ps, when the upper and lower
slowly diminishing as t grows. For larger asymmetry densitiesareturnedback,aslargeas20%oftotaldensity
imp
(α = 0.02), the amplitude of x¯ gets smaller rapidly for has been transferred to the lower layer. After reversing
t >12 ps. This qualitative difference in amplitude of the direction of velocity of an electron, the lower part of
imp
oscillations of an electron position obviously stems from wavepacket is now accelerated by electric field while the
the different dynamics of the electron density in layers upper partis deceleratedby it. For this reason,the den-
for t < t that is when the layers are magnetically sitystillflowsfromanupperlayertoaloweroneuntilitis
imp
coupled throughthe non-vanishing off-diagonalelements almost equally distributed [hρi = 0.51 and hρi = 0.49]
u l
in Hamiltonian 2. We will analyze this difference for for t = 9.5 ps [see Fig.7(b)] what takes place just before
t = 12 ps for which the time variations of hρi and the electron will pass the center of a wire [cf. Figs. 7(b)
imp
x¯ are shown in Figs. 7(b,e) and 7(c,f), respectively. In and 7(c) for t = 0]. In next two picoseconds, the direc-
Fig.7(b) we notice that for α=0.005 about 80% of elec- tion of density flow is reversed, and when the magnetic
tron density is localized in upper layer at t = 0. When field is finally turned off about 56% of total density is
8
localized in upper layer and 44% in lower one. in the same direction against the electric field generated
If distortion in vertical confinement is larger (α = insecondlayer. Thedynamicsofanelectronwavepacket
0.02), even 90% of total density is gathered in upper becomesthusdependentoneffectivemassofanelectron,
layer at t = 0 and it drops to about 78% immediately degreeofasymmetryintheconfiningpotentialaswellas
it reaches the left turning point what one may notice in on the time characteristics of the magnetic pulse. It was
Fig.7(e). When the velocity of wavepacket is reversed, foundthatgenerallyonlythe shorttime magneticpulses
the majority part of denisty is strongly decelerated by with time duration of about several to maximally a few
an electric field what significantly diminishes the am- tens of picoseconds may significantly change the motion
plitude of oscillations of an electron position along the energy of an electron. Moreover, the coherent motion
wire’s axis [Fig.7(f)]. Therefore, we may conclude that of both the upper and lower parts of wavepacket in the
for longer time intervals needed for switching the mag- same direction can be realized only if the magnetic cou-
netic field off, the dynamics of the majority part of elec- pling between layers vanishes when the magnetic pulse
tron density for α=0.02 cannot adapt to the time scale is ended. This takes place when a single magnetic pulse
of the magnetic pulse what in consequence prevents the is applied to the system or a magnetic field is switched
electron from gaining the large values of x¯ [see Fig.7(d) off. In the opposite case, the magnetic force constantly
for t >12 ps]. changes the amounts of densities confined within layers
imp
what destroys their coherent motion in a nanowire.
We think that discussed here the effect of temporary
IV. CONCLUSIONS change of electron’s motion energy by picoseconds mag-
netic pulses can successfully be applied to the nanos-
The dynamics of an electron motion induced in quan- tructuresconsistingoftwotunnel coupledwiresallowing
tum wire by a magnetic pulse was theoretically studied e.g. to sample the dynamical properties of many-body
by means of computer simulations. We have shown that interactions15includingtheWignercrystals.16Moreover,
such motion can be realized in a semiconductor nanos- thiseffectcanbealsoutilizedinthenanostructureshold-
tructure whichconsistoftwoverticallystackedlayers. If ingatwo-dimensionalelectrongaswithinthesingle,wide
these layers are tunnel coupled then the wave functions quantum well established in growth direction7, allow-
associated with two lowest bounding and antibounding ing for studies of the dynamical properties of electron
eigenstatesfordirectionbeingperpendiculartoboth,the transportin QPC17,18 or temporary changes in the local
wire’s axis and layers, can be easily hybridized in mag- potential landscape in conjunction with external biased
netic field. In such case, the rotationalelectric field gen- gates.17
eratedduetotheMaxwelllawbyanon-zerotimederiva-
tive of B~ may accelerate the charge density localized in
theupper andlowerpartsofbi-layernanowire. Forsym-
Acknowledgements
metric confinement in vertical direction, the upper and
lower charge densities are forced to move always in op-
posite directions according to the local directions of the The work was financed by Polish Ministry of Science
electric field. However, it was proven that the electron and Higher Education (MNiSW)
wavepacket,as a whole, can be acceleratedin arbitrarily
chosendirectionifthe confinementinupperandinlower
layers becomes asymmetric. Then, the majority part of
charge density localized in one layer is pushed by the References
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