Anomalous quantum transport in a thin film

0.1 MB·

by  Santanu K. Maiti

Anomalous quantum transport in a thin film
Santanu K. Maiti†,‡,∗
†Theoretical Condensed Matter Physics Division, Saha Institute of Nuclear Physics,
9 1/AF, Bidhannagar, Kolkata-700 064, India
0 ‡Department of Physics, Narasinha Dutt College, 129, Belilious Road, Howrah-711 101, India
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2 Abstract

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a We present a numerical study of electron transport in a thin film of varying disorder strength with the
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distance from its surface. A simple tight-binding model is used to describe the system, in which the

4 film is attachedto two metallic electrodes and the coupling of this film to the electrodes is illustrated by

the Newns-Andersonchemisorptiontheory. Quite interestingly we observethat, in the smoothly varying
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l disordered film current amplitude increases with the increase of disorder strength in the strong disorder
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a regime, while it decreases in the weak disorder regime. This behavior is completely opposite from a
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conventional disordered film, where current amplitude always decreases with the increase of disorder
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s strength.
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t PACS No.: 73.23.-b;73.63.Rt; 73.21.Ac
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Keywords: Thin film; Conductance; Current; Disorder
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∗Corresponding Author: Santanu K. Maiti
Electronic mail: [email protected]
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1 Introduction
regions. Thesesystems exhibitseveralpeculiarfea-
tures in electron transport. For example, in a very
recentworkYanget al.21 haveshowna localization
Improvements of nanoscience and technology have
toquasi-delocalizationtransitioninedgedisordered
stimulatedustoinvestigateelectricalconductionon
graphene nanoribbons by varying the strength of
mesoscopic/nanoscopic scale in a very tunable en-
edge disorder. Quite similar in behavior has also
vironment. The transport properties of quantum
been observed in a shell-doped nanowire,22 where
systems attached to electrodes have been studied
the electron dynamics undergoes a localization to
extensively over the last few decades both theoret-
quasi-delocalizationtransitionbeyondsomecritical
ically as well as experimentally due to their possi-
doping. In the shell-doped nanowire, the dopant
bletechnologicalapplications. In1974,Aviramand
atoms are spatially confined within a few atomic
Ratner1firstdevelopedatheoreticalformulationfor
layers in the shell region. This is completely op-
the description of electron conduction in a molec-
posite to that of a traditional disordered nanowire,
ular electronic device. Later many experiments2−6
where the dopant atoms are distributed uniformly
have been performed in several bridge systems to
throughoutthesystem. Fromthenumerousstudies
justify the basic mechanisms underlying the elec-
of electron transport in many such systems, it can
tron transport. From the numerous studies of elec-
be emphasized that the surface reconstructions,24
tron transport, exist in the literature, we can able
surface scattering25 and surface states26 may be
to understand different features, but yet the com-
quite significant to exhibit several diverse trans-
plete knowledge of the conduction mechanism in
port properties. Motivated with these investiga-
this scale is not well established even today. Many
tions, here we consider a particular kind of thin
significant factors are there which control the elec-
filminwhichdisorderstrengthvariessmoothlyfrom
tron transport in a molecular bridge, and all these
layer to layer with the distance from its surface. In
effects have to be taken into account properly to
this system,our numericalstudy predicts a strange
reveal the transport. For our illustrative purpose,
behaviorofelectrontransportwherecurrentampli-
here we mention some of them as follows. In order
tudeincreaseswiththeincreaseofdisorderstrength
torevealthedependence ofthemolecularstructure
in the strong disorder limit, while it decreases in
on the electron transport, Ernzerhof et al.7 have
the limit of weak disorder. On the other hand,
performed few model calculations and predicted
for a traditional disordered thin film current am-
some interesting results. The molecular coupling8
plitude always decreases with the increase of disor-
to the side attached electrodes is another impor-
der strength. An analytic approach based on the
tantparameterthatcontrolsthe electrontransport
tight-binding modelis usedto incorporatethe elec-
in a significant way. The most significant issue
trontransportinthefilm,andweadopttheNewns-
is probably the effects of quantum interferences of
Andersonchemisorptionmodel27−29 todescribethe
electron waves in different pathways, and several
sideattachedelectrodesandtheinteractionofthese
studies8−16 are available in the literature describ-
electrodes with the film.
ing these effects. In addition to these, dynamical
Our organization of this article is as follows. In
fluctuations provide an active role in the determi-
Section2,wedescribethemodelandthetheoretical
nation of molecular transport which can be man-
formulationforourcalculations. Section3discusses
ifested through the measurement of shot noise, a
the significant results, and at the end, we summa-
direct consequence of the quantization of charge.
rize our results in Section 4.
Thiscanbeusedtoobtaininformationonasystem
whichisnotdirectlyavailablethroughconductance
measurements, and is generally more sensitive to
2 Model and the theoretical
theeffectsofelectron-electroncorrelationsthanthe
average conductance.17,18 description
In this article, we concentrate ourselves on a
different aspect, related to the effect of disorder, The system of our concern is depicted in Fig. 1,
of quantum transport than the above mentioned where a thin film is attached to two metallic elec-
issues. The characteristic properties of electron trodes, viz, source and drain. In this film, disorder
transport in conventional disordered systems are strength varies smoothly from the top most disor-
well established in the literature. But there are dered layer (solid line) to-wards the bottom layer,
some special type of nano-scale materials, where keeping the lowest bottom layer (dashed line) as
the charge carriers are scattered mainly from their disorder free. The electrodes are symmetrically at-
surface regions19−23 and not from the inner core tached at the two extreme corners of the bottom
2
layer. Using the Green’s function formalism30 and nearest-neighborsitesbothforthelongitudinaland
singlechannelLandauerconductanceformula,30 we transversedirectionsofthefilm. Nowtoachieveour
calculate the transmissionprobability (T), conduc- concernedsystem,we choosethe site energies(ǫ ’s)
i
tance (g) and current (I) through the film. randomly from a “Box” distribution function such
Atlowtemperatureandbiasvoltage,theconduc- that the top most front layer becomes the highest
tance g of the film is obtained from the Landauer disordered layer with strength W and the strength
conductance formula,30 ofdisorderdecreasessmoothlyto-wardsthebottom
layer as a function of W/(N −m), where N gives
2e2 l l
g = T (1) the totalnumber oflayersandm representsthe to-
h
talnumberoforderedlayersfromthebottomsideof
where the transmission probability T can be writ- thefilm. While,intheconventionaldisorderedthin
ten in terms of the retarded and advanced Green’s filmallthelayersaresubjectedtothesamedisorder
functions (Gr and Ga) of the film as30 strength W. In our present model, the electrodes
F F
are described by the standardtight-binding Hamil-
T =Tr[ΓSGrFΓDGaF] (2) tonian, similar to that as prescribed in Eq.(4), and
parametrized by constant on-site potential ǫ and
The parametersΓ and Γ describe the couplings 0
S D nearest-neighbor hopping integral v.
Assuming the entire voltage is dropped across
the film-electrode interfaces,31 the current passing
Drain
throughthe film, whichis regardedasa singleelec-
tron scattering process between the reservoirs, can
be expressed as,30
e ∞
I(V)= (f −f )T(E)dE (5)
π¯hZ S D
−∞
Source
wheref =f E−µ givestheFermidistri-
S(D) S(D)
bution function w(cid:0)ith the ele(cid:1)ctrochemical potential
Figure1: Schematicviewofasmoothlyvaryingdis- µS(D) = EF ±eV/2. In this article, we focus our
orderedthinfilmattachedtotwometallicelectrodes study on the determination of the typical current
(source and drain). The top most front layer (solid amplitude which is obtained from the relation,
line)isthehighestdisorderedlayerandthedisorder
strength decreases smoothly to-wards the bottom I = <I2 > (6)
typ W,V
q
layerkeeping the lowestbottom layer (dashed line)
asdisorderfree. Twoelectrodesareattachedatthe where W and V correspond to the impurity
two extreme corners of the bottom layer. strength and the applied bias voltage respectively.
Throughout this presentation, all the results are
of the film with the source and drain respectively. computed at absolute zero temperature, but they
The Green’s function of the film becomes, should valid even for finite temperature (∼ 300 K)
as the broadening of the energy levels of the film
G =(E−H −Σ −Σ )−1 (3)
F F S D duetoitscouplingwiththeelectrodeswillbemuch
larger than that of the thermal broadening.30 For
where E is the energy of the injecting electron and
simplicity, we take the unit c = e = h = 1 in our
H correspondstotheHamiltonianofthefilm. The
F
present calculations.
parametersΣ andΣ denotetheself-energiesdue
S D
to the coupling of the film with the source and
drain respectively, and are described by the use of 3 Results and discussion
Newns-Andersonchemisorptiontheory.27−29 In the
tight-binding framework, the Hamiltonian can be
All the numerical calculations we do here are per-
expressed within the non-interacting picture as,
formed for some particular values of the different
H = ǫ c†c + t c†c +c†c (4) parameters, but all the basic features in which we
F X i i i X (cid:16) i j j i(cid:17) are interested in this particular study remain also
i <ij>
invariant for the other parametric values. The val-
Hereǫ givestheon-siteenergyofanelectronatsite ues of the different parameters are as follows. The
i
iandtrepresentsthehoppingstrengthbetweentwo coupling strengths of the film to the electrodes are
3
taken as τ = τ = 1.5, the nearest-neighbor hop- (H − H )ψ = Eψ and (H − H )ψ = Eψ .
S D 0 1 0 0 d 2 d d
pingintegralinthefilmisfixedtot=1. Theon-site Here H and H representthe sub-Hamiltonians of
0 d
potentialandthehoppingintegralintheelectrodes the ordered and disordered regions of the film re-
are set as ǫ = 0 and v = 2 respectively. In ad- spectively, and ψ and ψ are the corresponding
0 0 d
dition to these, here we also introduce three other eigenfunctions. The terms H and H in the above
1 2
parameters N , N and N to specify the system two expressions are the most significant and they
x y z
size of the thin film, where they correspond to the
16
total number of lattice sites along the x, y and z
directionsofthe film respectively. Inournumerical
L
calculations, the typical current amplitude (Ityp) is HItyp 12
determined by taking the average over the disor- e
d
deredconfigurationsandbias voltages(seeEq.(6)). plitu 8
Since in this particular model the site energies are m
a
chosen randomly, we compute Ityp by taking the nt
e
averageoveralargenumber(60)ofdisorderedcon- urr 4
figurationsineachcasetogetmuchaccurateresult. C
On the other hand, for the averaging over the bias
0
voltage V, we set the range of it from −10 to 10. 0 2 4 6 8 10 12 14 16 18 20 22
In this presentation, we focus only on the systems DisorderstrengthHWL
with small sizes since all the qualitative behaviors
remain also invariant even for the large systems. Figure2: Ityp vs. W forsomethin filmswithNx =
10, N = 8 and N = 5. Here we set m = 1. The
y z
Figure 2 illustrates the variation of the typical solidanddottedcurvescorrespondtothesmoothly
current amplitude (Ityp) as a function of the disor- varying and complete disordered films respectively.
der (W) for some thin films with N =10, N =8
x y
and N =5. Here we set m=1, i.e., only the low- can be expressedas: H =H (H −E)−1H and
z 1 od d do
est bottom layer is free from any disorder for these H =H (H −E)−1H . H andH correspond
2 do o od od do
films. The solid and dotted curves correspond to to the coupling betweenthe orderedregionandthe
the results of the smoothly varying and complete disordered region. From these mathematical ex-
disordered thin films respectively. A remarkably pressions, the anomalous behavior of the electron
differentbehaviorisobservedforthesmoothlyvary- transportinthefilmcanbedescribedclearly. Inthe
ing disorderedfilm comparedto the film with com- absenceofanyinteractionbetweenthe orderedand
plete disorder. In the later system, it is observed disordered regions, we can assume the full system
that I decreases rapidly with W and eventually as a simple combination of two independent sub-
typ
it drops to zero for the higher value of W. This re- systems. Therefore, we get all the extended states
duction of the current is due to the fact that the in the orderedregion,while the localized states are
eigenstates become more localized32 with the in- obtainedinthedisorderedregion. Inthissituation,
crease of disorder, and it is well established from themotionofanelectroninanyoneregionisnotaf-
the theory of Anderson localization.33 The appre- fectedbytheother. Butforthecoupledsystem,the
ciable changeinthe variationofthe typicalcurrent motionofthe electronis no moreindependent, and
amplitude takes place only for the unconventional we have to take the combined effects coming from
disordered film. In this case, the current ampli- boththetworegions. Withtheincreaseofdisorder,
tude decreases initially with W and after reaching the scattering effect becomes dominated more, and
to a minimum atW =W (say), it againincreases. thus the reduction of the current is expected. This
c
Thus the anomalous behavior is observed beyond scattering is due to the existence of the localized
the critical disorder strength W , and we are inter- eigenstates in the disordered regions. Therefore, in
c
ested particularly in this regime where W > W . the case of strong coupling between the two sub-
c
In order to illustrate this peculiar behavior, we systems, the motion of the electron in the ordered
consider the smoothly varying disordered film as a region is significantly influenced by the disordered
coupled system combining two sub-systems. The regions. Now the degree of this coupling between
coupling exists between the lowest bottom ordered the two sub-systems solely depends on the two pa-
layer and the other disordered layers. Thus the rametersH andH ,thoseareexpressedearlier. In
1 2
system can be treated, in other way, as a coupled thelimitofweakdisorder,thescatteringeffectfrom
order-disorderseparatedthinfilm. Forthiscoupled both the two regions is quite significant since then
system we can write the Schr¨odinger equations as: the terms H and H have reasonably high values.
1 2
4
4 Concluding Remarks
With the increase of disorder, H decreases gradu-
1
allyandforaverylargevalueofW itbecomesvery
small. Hence the term (H −H ) effectively goes To summarize, we have done a numerical study
0 1
to H in the limit W →0, whichindicates that the to show the anomalous behavior of electron trans-
0
ordered region becomes decoupled from the disor- portinaunconventionaldisorderedthinfilmwhere
deredone. Therefore,inthe higherdisorderregime the disorder strength varies smoothly from its sur-
face. A simple tight-binding model has been used
to describe the system, where the coupling of the
20
film to the electrodes has been described by the
L Newns-Anderson chemisorption theory. We have
p
HIty 15 calculated the typical current amplitudes by us-
e
d ing the Green’s function formalism, and our re-
u
plit 10 sults have shown a remarkably different behavior
m
a for the unconventionaldisorderedfilm comparedto
ent the traditional disordered film. In the smoothly
urr 5 varying disordered film, the typical current ampli-
C
tude decreases with W in the weakdisorderregime
0 (W <W ), while it increasesin the strong disorder
c
0 2 4 6 8 10 12 14 16 18 20 22
regime (W > W ). On the other hand for the con-
DisorderstrengthHWL c
ventionaldisorderedfilm, the currentamplitude al-
waysdecreaseswithdisorder. In this presentinves-
Figure3: I vs. W forsomethinfilms withN =
typ x
tigations,wehavealsostudiedthe finite sizeeffects
12, N = 10 and N = 6. Here we set m = 2. The
y z
of the film and our numerical results have shown
solid and dotted curves correspond to the identical
thatthetypicalcurrentamplitudestronglydepends
meaning as in Fig. 2.
on the size of the film. Quite similar feature of
anomalousquantumtransportcanalsobeobserved
the scattering effect becomes less significant from
in some other lower dimensional systems like, edge
the ordered region, and it decreases with W. For
disordered graphene sheets of single-atom-thick, fi-
thelowregimeofW,theeigenstatesofboththetwo
nite width rings with surface disorder, nanowires,
effectiveHamiltonians,(H −H )and(H −H ),are
0 1 d 2 etc. Our study has revealed that the carrier trans-
localized. WiththeincreaseofW,H graduallyde-
1 port in an order-disorder separated mesoscopic de-
creases,resultinginmuchweakerlocalizationinthe
vice may be tailored to desired properties through
statesof(H −H ),whilethestatesof(H −H )be-
0 1 d 2 doping for different applications.
comemorelocalized. AtacriticalvalueofW =W
c
(say) (≃ band width of H ), we get a separation
0
between the much weaker localized states and the References
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