Ballisti Transport in Semiondutor Quantum
Wires in the Presene of Defets
A. R.Rohaand J. A. Brum
Laboratorio Naionalde LuzS
inrotron,ABTLuS,13083-970, Campinas,SP,Braziland
DFMC{InstitutodeF
isiaGlebWataghin,Uniamp,13.081-970Campinas,SP,Brazil
ReeivedonApril27,2001
Wepresentalulationsoftheondutaneinsemiondutorquasi-one-dimensionalsystemsusing
theLandauerformalism. Weonsidertheeetsonthetransportpropertiesinsidesemiondutor
quantumwaveguidesofdierentshapeswhenadefetisloatedeitherinthewireregionorinthe
quasi-two-dimensional region. Weobserve hangesonthe plateau'sthreshold whenthe defet is
plaedinsidethewireandloweringoftheondutaneplateausthemselvesbelowtheondutane
quantumG
0 =2e
2
/hwhenthedefetisoutsidethewire.
Introdution
The rst experimental evidenes of ondutane
quantization through one-dimensional (1D) quantum
systemsweremadebyvanWeesetal. [1℄andWharam
et al. [2℄, both in 1988[3℄. These experiments were
onduted on quantum point ontats (QPC) made
fromdopedsemiondutorheterostrutureswhihform
aquasi-two-dimensionaleletrongas(2DEG).The
sys-temisintheEletroniQuantum Limitandfromnow
onweassumeittobeapure2DEGthatis,weneglet
the sattering with the higher heterostruture levels.
The eletron ow through the point ontat is
on-strainedto1Dbybymeansofanegativepotentialbias
appliedonthemetalplates oftheontat.
Thefabriationof these struturesismade by
epi-taxialgrowth. Intheaseofthisstudy, weonsidered
GaAs-AlGaAs heterostrutures with a layerof
modu-lated n-doped AlGaAs. In general some of the
Alu-minumatoms will defuse through the interfae of the
juntion atingassatteringentersfortheeletrons.
The rst theoretial onsiderations onerning
de-fetsonQPC'sweremadebyNixonetal.[4℄,who
on-sideredrandomdistribution ofdefetsalongthe
stru-ture. Reently,Topinkaetal.[5℄,reordedeletronow
through a QPC under the inuene of an AFM tip,
plaed outsidethe point ontat and funtioning as a
probefortheeletronwavefuntion.
Ouraimwasto studytheoretiallytheeets that
the existene of these sattering entres might have
on the ondutane plateaus on two dierent shapes
of quantum waveguides. The rst one being a QPC
(oneonstrition)andtheseondoneanopenquantum
dot(OQD)struture(twonarrowonstritionsandone
wideonstrition)(seeFig.1).
Theoretial Modelling
Figure1. Shematiillustrationofthethetwostruturesthatwerestudiedandtheironningpotential1a)QPC1b)OQD.
Inorder to simulatethe twostrutures whih were
onsidered we used thevoltage proleshematized on
fetivemassapproximations[6℄. Theeigenstatesofthe
nitepotentialbarriersandlengthmuhgreaterthanthe
dimensionof theonstrition. Theonstritedregions
were depited as square wells with onstant potential
barrier V
0 .
The solutions inside the onstrited regions were
found by projeting the basis of the eletron bath
(jn
x
>) onto the Hamiltonian of the system.
There-fore,wewritethewavefuntions:
j fIg m >= N X n a mn jn x >j1 z > (1) the a mn
's are determined by projeting the
Hamilto-nian H narrow =T e +V 0 Y jxj L x 2 (2)
forthenarrowonstritionsonFig. 1a. and1b. and
H wide =T e +V 0 Y x+ L x 2 +V 0
Y[x ( L
x 2+Lw
x )℄
(3)
for the wide onstrition on Fig. 1b into Eq. 1 and
diagonalizingtheeigenvalueproblem.
The defet is simulated by a positive square
bar-rier of height V
def
plaed at dierent regions of the
waveguide. The ontribution to the Hamiltonian
is V def Y x x D LD 2 Y x x D + LD 2 ,
added to either the narrow or the wide onstrition
Hamiltoniandependingontheposition ofthedefet.
The eletrons were onsidered to be injeted with
anenergyE,whihmustbeonservedalongthe
stru-ture. Therefore,thewavevetorforeahregionisgiven
by: k fIg yi = r 2m ~ 2 E I i (4) where I i
is the energy of the i-th level of the well in
x-diretion fortheI-th region.
Hene,thetotalwavefuntionanbewriten,foreah
region:
j > 1 n 0=e ik 1 n 0 y jn 0 x >+ N X n r n 0 n e ik 1 n y jn x > (5)
j > fIg n 0 = N X j n 0 j e ik fIg j + n 0 j e ik fIg j j fIg j > (6)
j > 3 n 0 = N X n t n 0 n e ik 3 n y jn x > (7)
Byimposingtheonditionofontinuityofboththe
wavefuntionandtheux at eahinterfae,weobtain
asystemofnon-homogeneouslinearequationsthatan
be solvedfor the r's andt's (the reetionand
trans-missionoeÆients). Theondutaneanthenbe
al-ulated bytheLandauerformula[7℄:
G(E)= X 0 X k 3 i k 1 n 0 jt i j 2 (8)
ThisProedure isknowasMode Mathingandhas
beenwidelyusedforthiskindof problem[8,9℄.
Resultsand Disussion
On our alulations we onsidered a square defet
10
Ax10
A and 600meV high (the same potenial
esti-mated for the depleted region that dene the
stru-ture). Thisishigherthanastruturaldefetforwhih
V
def
v300meV,but simulatesadepletedareaindued
byaneletrostationtat. Theeetivemasswas
on-sideredto be0:067m
0
. Fortheasewith one
onstri-tion,theontatwastakentobe500 A(L x )wide700 A
long. For the ase of two onstritionsthe wire is as
wideastherstoneandeahonstritionis500
Along.
0
5
10
15
0,0
0,5
1,0
1,5
2,0
0
50
100
150
200
250
300
350
400
450
500
550
600
650
700
-250
-200
-150
-100
-50
0
50
100
150
200
250
Y [Angstrons]
X [Angstrons]
G/G
0
E [meV]
Figure2. a) Condutane as afuntionof initial eletron
energywiththedefetplaedonx=0
A.b)Projetiononthe
xyplaneof the EletronDensityinsidethe Point Contat
foraninitialenergyof3meV.
TheondutaneurvesshownonFig. 2weremade
by plaingthe defetinside the wire on two dierent
plaesalong they-axis fortheQPCase. Wean
ob-serve that the defet shifts the 1 st
plateau threshold
to higher energies. We an also see that the seond
plateauremainsunhanged.
On Fig. 3 the defet is kept at x = 130
Aand is
madetovary in theydiretion. Itisevidentthat the
oppositebehaviorofFig. 2nowours.
Theinterpretationofthisphenomenonreliesonthe
eletrondensities depited as density plots on Fig. 2
and3. Thelow energywavefuntion (rstondution
mode)presentsahigh eletrondensitylosertothe
y-axis, so the eet of the defet on the eletrons will
be higher when x 0. On theother hand, when the
energyis 11meV,thewavefuntionhasaknotandthe
hargedensityisonentratedparalleltothey-axis. In
thatasetheeletronsarenotinuenedbythedefet,
0
5
10
15
0,0
0,5
1,0
1,5
2,0
2,5
0
50
100
150
200
250
300
350
400
450
500
550
600
650
700
-250
-200
-150
-100
-50
0
50
100
150
200
250
Y [Angstrons]
X [Angstrons]
G/G
0
E [ meV ]
Figure 3. a) Condutane as a funtionof initial eletron
energywiththedefetplaedonx=130
A.b)Projetionon
the xy-planeofthe EletronDensity insidePoint Contat
foraninitialenergyof11meV.
Theoppositehappensfortheseondplateau,when
wehaveaombinedeetofthetworsteigenstatesof
theonstrition. Inthisase,thedefetdisplaedfrom
the wireentre showsastronger inuene onthe
se-ond plateauasaonsequeneofthe seondeigenstate
wavefuntionspatialdistribution.
0
5
10
15
0,0
0,5
1,0
1,5
2,0
G/G
0
E [ meV ]
Fig. 4.Condutaneasafuntionofinitialeletronenergy
withthedefetplaedoutsidethewire(aty=800
A).
WhenweplaethedefetoutsidetheQPC,asseen
in Fig. 4, there is no energy threshold shift.
How-ever,theondutaneplateausappearbelowG
0 dueto
thesatteringoftheeletronsbythedefet,preventing
someofthem to reah thedrain. Theinuene ofthe
defet'spositionhasthesameeletrondensity
interpre-tationthatwasgiventothephenomenainFig.2and3.
Fromtheanalysisoftheondutaneurvesforthe
OQDweanpointoutin Fig. 5thebound statethat
namesthisstruture[10℄. Thestatesignatureisapeak
on the ondutane due to resonanttunnelling of the
eletronsthroughthequantum dotgroundstate. The
greater onnement on the x diretion in the narrow
onstrition(outsidethedot)givesrisetoavirtual
on-ningpotentialin the ydiretion onbothsidesofthe
Figure5. Condutane asafuntionofinitialeletron
en-ergy with the defet plaed at x=0
Aforthe for the OQD
ase.
WeanalsoseeaFano-resonane-typeurvewhih
is due to the repulsion of the seond energy level of
theOQDandtherstbandoftheondutinghannel.
Hene,wehavetheformationofabandgapinthe
on-dutaneshowingthedestrutiveinterferenebetween
thehannels.
Byplaingthedefetinsidethedotthereisaraise
in the energy along thex diretion, thus blue-shifting
theboundstateand theFano-resonane. The
ondu-tane plateau a slight derease. Relying onwhat was
saidaboutFig. 4,weanimaginethat thedefetats
somehow asif it were on the2DEG and the lowering
oftheondutaneplateausisduetothesatteringon
thedefetand onthersttwointerfaesonly.
Insummary,wehaveseenthat theexisteneof
de-fets has an important role on ballisti transport. In
general, the defets will appear in far greater
num-berandrandomlydistributedthroughoutthestruture
and theirmaineet will be tolowertheondutane
plateaus.
Aknowledgments
We are grateful to CNPq and (Brazil) Fapesp
(Brazil)forthenanialsupport.
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