Current modulation at nano-scale level: A theoretical study

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by  Santanu K. Maiti

Current modulation at nano-scale level: A theoretical study
Santanu K. Maiti1,2
1Theoretical Condensed Matter Physics Division,
Saha Institute of Nuclear Physics, 1/AF, Bidhannagar, Kolkata-700 064, India
2
Department of Physics, Narasinha Dutt College, 129 Belilious Road, Howrah-711 101, India
We explore the possibilities of current modulation at nano-scale level using mesoscopic rings. A
single mesoscopic ring or an array of such rings is used for current modulation where each ring is
threaded by a time varying magnetic flux φ which plays the central role in the modulation action.
0
1 Within a tight-binding framework, all the calculations are done based on the Green’s function for-
0 malism. Wepresent numericalresults for thetwo-terminal conductanceand current which support
2 theessential featuresof currentmodulation. Theanalysismay behelpfulinfabricating mesoscopic
or nano-scale electronic devices.
n
a PACSnumbers: 73.63.-b,73.63.Rt,81.07.Nb
J

3
1 I. INTRODUCTION Furthermore, electron transport in the ring can also be
modulatedinotherwaybytuning themagneticflux,the
]
l Electron transport in low-dimensional systems has so-called Aharonov-Bohm (AB) flux, penetrated by the
l
a drawn much attention in the field of theoretical as well ring.
h Aimofthepresentpaperistoillustratethepossibilities
as experimental research due to flourishing development
- of current modulation at nano-scale level using simple
s innanotechnologyandnano-scaledevicemodeling. Low-
e dimensionalmodelquantumsystemsarethe basicbuild- mesoscopic rings. To achieve current modulation we de-
m signanelectroniccircuitusingasinglemesoscopicringor
ing blocks for future generation of nano-electronic de-
. vices. Several exotic features are observed in this length aclusterofsuchrings,whereeachringispenetratedbya
t
a scale owing to the effect of quantum interference. This timevaryingmagneticfluxφwhichplaysthecentralrole
m for the modulation action. For a constant DC voltage,
effect is generally observed in samples with size much
currentthroughthe circuitshows oscillatorybehavioras
- smaller or comparable to phase coherence length L ,
φ
d a function of time t depending on the phase of the mag-
while the effect disappears for larger systems. A nor-
n
netic flux φ passing throughthe ring. Therefore,current
mal metal mesoscopic ring is a very good example to
o
modulation can be achieved simply by tuning the phase
c study the effect of quantum interference. Current trend
[ of fabricating nano-scale devices has resulted much in- of magnetic flux φ threaded by the ring. Within a tight-
bindingframework,asimpleparametricapproach[21–29]
terest in characterization of ring type nanostructures.
1
is given and all the calculations are done through single
There are severalmethods for preparationof mesoscopic
v
particleGreen’sfunctiontechniquetorevealtheelectron
4 rings. Forinstance,goldringscanbedesignedusingtem-
transport. Herewepresentnumericalresultsforthetwo-
8 plates of suitable structure in combination with metal
1 deposition through ion beam etching [1, 2]. In a recent terminalconductance and currentwhich clearly describe
2 experiment, Yan et al. have proposed how gold rings the essential features of current modulation. Our exact
. analysismaybehelpfulfordesigningmesoscopicornano-
1 canbe preparedbyselectivewettingofporoustemplates
0 using polymer membranes [3]. With such rings we can scaleelectronicdevices. Tothebestofourknowledgethe
0 fabricate nano-scale electronic circuits which can be uti- modulationactionusingsuchsimplemesoscopicringshas
1 not been addressed earlier in the literature.
lized for the operation of current modulation. To ex-
v: plore this phenomenon the ring is coupled to two elec- Theschemeofthepresentpaperisasfollows. Withthe
i trodes,to formanelectrode-ring-electrodebridge,where brief introduction (Section I), in Section II, we describe
X
the ringispenetratedbyatimevaryingmagneticflux φ. the model and theoretical formulations for our calcula-
ar Electron transport through a molecular bridge system tions. Section III presents the significant results, and
was first studied theoretically by Aviram and Ratner [4] finally, we conclude our results in Section IV.
during 1970’s. Following this pioneering work, several
experiments have been done using different bridge sys-
tems to reveal the actual mechanism of electron trans- II. MODEL AND SYNOPSIS OF THE
port. Though, to date a lot of theoretical [5–16] as well THEORETICAL FORMULATION
as experimental works [17–20] on two-terminal electron
transport have been done addressing several important In the forthcoming two sub-sections we focus on two
issues, yet the complete knowledge of conduction mech- different circuit configurations, for our illustrative pur-
anism in nano-scalesystems is still unclear to us. Trans- poses, those are used for current modulation. Here we
port properties are characterized by several significant trytoillustratehowasinglemesoscopicringortwosuch
factors like quantizationof energy levels,quantum inter- rings, where each ring is penetrated by a time varying
ference effect, ring-to-electrode interface geometry, etc. magneticfluxφ,underaDCbiasvoltagecansupportan
2
oscillating output current. A single mesoscopic ring can
provideanoscillatingcurrentwithaparticularfrequency, Φ Φ
while in the case of two rings, oscillating currents with 1 2
SOURCE DRAIN
other frequencies can be obtained. These ideas may be
generalized further to produce oscillating currents with RING−1 RING−2
other frequencies by considering more number of rings.
+ −
A. Circuit configuration I
FIG.2: (Coloronline). Actualschemeofconnectionwiththe
Let us start by referring to Fig. 1. A mesoscopic ring, battery where two mesoscopic rings, subject to time vary-
penetratedbyatimevaryingmagneticfluxφ,isattached ing magnetic fluxesφ1 and φ2 are attached symmetrically to
symmetrically to two semi-infinite one-dimensional (1D) source and drain. The blue arrow indicates current direction
metallicelectrodes,namely,sourceanddrain. Thesetwo in the circuit.
electrodes are directly coupled to the positive and nega-
tive terminals of a battery, a source of constant voltage.
where, δ refers to constant phase difference between the
The time varyingmagnetic flux passingthroughthe ring
two fluxes φ and φ . Using this circuit configuration
1 2
also oscillating current in the output can be achieved,
butinthiscasethefrequencyofthecurrentgetsmodified
Φ
depending on the phase shift δ.
SOURCE DRAIN
RING
C. Theoretical formulation
In this sub-section we will describe the basic theoret-
+ −
ical formulation for calculation of conductance and cur-
rent through a single mesoscopic ring, penetrated by a
FIG. 1: (Color online). Actual scheme of connection with
magnetic flux φ, attached to two source and drain. This
the battery where a mesoscopic ring, subject to a time vary-
similar theory is also used to study electron transportin
ing magnetic flux φ, is attached symmetrically to source and
an array of mesoscopic rings.
drain. The blue arrow indicates current direction in the cir-
Using Landauer conductance formula [30, 31] we de-
cuit.
termine two-terminal conductance (g) of the mesoscopic
can be expressed mathematically in the form, ring. At much low temperatures and bias voltage it (g)
can be written in the form,
φ
0
φ(t)= sin(ωt) (1)
2 2e2
g = T (4)
where, φ =ch/eis the elementaryflux-quantum, ω cor- h
0
responds to the angular frequency and t represents the
where, T corresponds to the transmission probability of
time. This electronic circuit provides an oscillating cur-
anelectronacrossthe ring. Intermsofthe Green’sfunc-
rent in the output though a constant DC input signal is
tion of the ring and its coupling to two electrodes, the
applied which we will describe in the forthcoming sec-
transmission probability can be expressed as [30, 31],
tion. The frequency of the current is identical to that of
the applied flux φ(t). T =Tr[Γ GrΓ Ga] (5)
1 R 2 R
where,Γ andΓ describethecouplingoftheringtothe
S D
B. Circuit configuration II sourceanddrain,respectively. Here, Gr andGa arethe
R R
retardedandadvancedGreen’sfunctions,respectively,of
In Fig. 2 two such mesoscopic rings those are directly the ring considering the effects of the electrodes. Now,
coupled to each other, penetrated by time varying mag- for the full system i.e., the mesoscopic ring, source and
netic fluxes φ and φ are attachedsymmetrically to the drain, the Green’s function is expressed as,
1 2
electrodes, viz, source and drain. A DC voltage source
G=(E−H)−1 (6)
is connected to these two electrodes. The time varying
magnetic fluxes are expressed mathematically as,
where, E is the energy of the source electron. Evalua-
φ tion of this Green’s function needs the inversion of an
0
φ (t) = sin(ωt) (2)
1 2 infinite matrix, which is really a difficult task, since the
φ full system consists of the finite size ring and two semi-
0
φ2(t) = sin(ωt+δ) (3) infinite 1D electrodes. However, the full system can be
2
3
partitioned into sub-matrices corresponding to the indi- results consideringaring with N =20. The upper panel
vidualsub-systemsandthe effective Green’sfunctionfor presents the time variation of magnetic flux with ampli-
the ring can be written in the form [30, 31], tude φ /2 whose mathematical form is given in Eq. (1).
0
The variationof conductance g as a function of time t is
GR =(E−HR−ΣS −ΣD)−1 (7) illustrated in the middle panel. Here we determine the
typical conductance for a particular energy E = 2.5. It
where,H describestheHamiltonianofthering. Within
R shows that the conductance oscillates periodically as a
the non-interacting picture, the tight-binding Hamilto-
function of ωt exhibiting π periodicity and it gets the
nian of the ring can be expressed like,
amplitude g = 2. This reveals that the transmis-
max
sionamplitudeT becomesunitysincewegettherelation
H = ǫ c†c + v c†c eiθ+c†c e−iθ (8)
R i i i i j j i g = 2T from the Landauer conductance formula in our
Xi <Xij> (cid:16) (cid:17) chosen unit c=e=h=1. Now we try to justify the os-
where,ǫ andvcorrespondtothesiteenergyandnearest-
i
neighbor hopping strength, respectively. c† (c ) is the Φ
i i
0
creation(annihilation)operatorofanelectronatthesite
i and θ =2πφ/Nφ is the phase factor due to the flux φ
0
enclosed by the ring consists of N atomic sites. A simi- g
larkindoftight-bindingHamiltonianisalsoused,except
the phase factor θ, to describe the electrodes where the 0
Hamiltonian is parametrized by constant on-site poten-
tialǫ′ andnearest-neighborhoppingintegralt′. Thehop-
I
ping integral between the ring and source is τ , while it
S
isτ betweentheringanddrain. InEq.(7),Σ andΣ
D S D 0
aretheself-energiesduetothecouplingoftheringtothe 0 Π 2Π 3Π 4Π 5Π 6Π
source and drain, respectively, where all the information ®Ωt
of the coupling are included into these self-energies.
To determine current, passing throughthe mesoscopic FIG. 3: (Color online). Responses in circuit configuration
ring, we use the expression [30, 31], I. Upper, middle and lower panels describe the time depen-
dencesoffluxφ,conductanceg andcurrentI asafunctionof
∞ time t. Conductance is calculated at the energy E =2.5 and
2e
currentisdeterminedatthetypicalbiasvoltageV =2.5. The
I(V)= (f −f )T(E) dE (9)
S D
h Z ringsizeisfixedatN =20. Theamplitudesare: φmax =0.5,
−∞ gmax =2 and Imax =2.54.
where, f = f E−µ gives the Fermi distribu-
S(D) S(D) cillating behaviorof conductancewith time t. The prob-
tion function with(cid:0)the electroc(cid:1)hemicalpotential µS(D) = ability amplitude of getting an electron from the source
E ±eV/2 and E is the equilibrium Fermi energy. For
F F to drain across the ring depends on the quantum inter-
the sake of simplicity, we take the unit c= e=h=1 in
ferenceeffectofthe electronicwavespassingthroughthe
our present calculations.
upper and lower arms of the ring. For a symmetrically
connected ring (upper and lower arms are identical to
eachother),penetrated by a magnetic flux φ, the proba-
III. NUMERICAL RESULTS AND DISCUSSION
bilityamplitudeofgettinganelectronacrosstheringbe-
comesexactlyzero(T =0)forthetypicalflux,φ=φ /2.
To illustrate the numerical results, we begin our dis- 0
This vanishing behavior of transmission probability can
cussion by mentioning the values of different parameters
be shownveryeasilybysimple mathematicalcalculation
usedforourcalculations. Inthe mesoscopicring,theon-
as follows.
site energy ǫ is fixed to 0 for all the atomic sites i and
i Forasymmetricallyconnectedring,thewavefunctions
nearest-neighbor hopping strength v is set to 3. While,
passingthroughtheupperandlowerarmsoftheringare
for the side-attached electrodes the on-site energy (ǫ′)
given by,
andnearest-neighborhoppingstrength(t′) arechosenas
0 and 4, respectively. The hopping strengths τS and τD h¯iec A~.d~r
are set as τS = τD = 2.5. The equilibrium Fermi energy ψ1 = ψ0e γR1
E is fixed at 0.
F h¯iec A~.d~r
ψ = ψ e γR2 (10)
2 0
A. Responses in circuit configuration I where, γ and γ are used to indicate the two different
1 2
paths of electron propagation along the two arms of the
The modulation action for the circuit configuration I ring. ψ denotesthewavefunctioninabsenceofmagnetic
0
is clearly illustrated in Fig. 3, where we compute all the fluxφanditissameforbothupperandlowerarmsasthe
4
ring is symmetrically coupled to the electrodes. A~ is the as in ring-1 with a phase shift π/2. Therefore, ring-1
vector potential associated with the magnetic field B~ by and ring-2 enclose φ0/2 flux alternatively in the interval
therelationB~ =∇~ ×A~. Hencetheprobabilityamplitude ωt=π/2, andaccordingly,zero transmissionprobability
of finding the electron passing through the ring can be is achieved at this interval. In the same footing, here we
calculated as, alsodescribethevariationcurrentI withtimet(seelower
panelofFig.4)tosupporttheoscillatoryactionobserved
2πφ in this circuit configuration II. The current is computed
|ψ +ψ |2 =2|ψ |2+2|ψ |2cos (11)
1 2 0 0 (cid:18) φ (cid:19) at the typical bias voltage V = 4. The variation of cur-
0
rent shows π/2 periodicity with an amplitude I = 2
max
where, φ= A~.d~r = B~.d~s is the flux enclosedby the andthisperiodicnatureiswellunderstoodfromthecon-
ductancespectrum. Fromtheseconductanceandcurrent
ring. H R R
From Eq. (11) it is clearly observed that at φ = φ /2
0
the transmissionprobabilityof anelectrondropsexactly Φ
i
to zero. On the other hand, for all other values of φ 0
i.e., φ 6= φ /2, electron transmission through the ring
0
takes place which provides non-zero value of conduc-
tance. Thus, for the particular cases when φ(t) becomes g
maximum (+φ /2) or minimum (−φ /2), conductance
0 0
drops to zero which is clearly shown from the conduc- 0
tance spectrum (middle panel of Fig. 3). Hence, chang-
ingthefrequencyoftimedependentfluxφ(t),periodicity I
in conductance can be regulated. To visualize the oscil-
latory action more prominently we present the variation 0
0 Π 2Π 3Π 4Π
ofcurrentasafunctionofωtinthelowerpanelofFig.3.
®Ωt
ThecurrentI throughtheringisobtainedbyintegrating
overthe transmissionfunction T (see Eq. ( 9)). Here we FIG. 4: (Color online). Responses in circuit configuration
computethecurrentforthetypicalbiasvoltageV =2.5. II. Upper, middle and lower panels describe the time depen-
Following the conductance pattern, the oscillatory be- dences of two fluxes φ1 (orange line) and φ2 (magenta line),
havior of the current is clearly understood, and like the conductance g and current I as a function of time t. Con-
conductancespectrumcurrentexhibitsπperiodicitypro- ductance is calculated at the energy E = 2.5 and current is
viding the amplitude I = 2.54. All these character- determined at the typical bias voltage V = 4. In each ring,
max
total number of atomic sites N is fixed at 8 and we choose
istic features suggest that an oscillatory response in the
output is obtained though the ring is subject to a DC δ = π/2. The amplitudes are: φmax = 0.5, gmax = 1.9 and
Imax =2.
bias voltage.
spectraitismanifestedthatinthetworingsystemwhich
is subject to a DC bias voltage, the oscillatory response
B. Responses in circuit configuration II
canbe modulated very easily by tuning the phase differ-
ence δ between two time varying magnetic fluxes.
Next, we concentrate on the responses obtained in
Finally, we can say that extending this idea to an ar-
the circuit configurationII. The results are illustratedin
ray of multi-ring system in which different rings subject
Fig.4,wheretotalnumberofatomicsitesN ineachring
to time varying magnetic fluxes in different phases, os-
isfixedat8. Intheupperpanel,weplotthe timedepen-
cillatoryresponsescanbeachievedwithπ/nfrequencies,
dent fluxes φ (t) (orange line) and φ (t) (magenta line)
1 2 where n corresponds to an integer. Our exact analysis
those passthroughtwo differentrings. A constantphase
may provide some significant insights in designing nano-
shiftδ existsbetweenthesetwofluxesasmathematically
electronic circuits.
expressedinEqs.(2)and(3). Herewesetδ =π/2. Inthe
middlepanel,wedescribethetimedependenceofconduc-
tance g with amplitude g = 1.9, where conductance
max
IV. CONCLUDING REMARKS
is evaluated at the typical energy E =2.5. Conductance
showstheoscillatorybehaviorasafunctionofωtprovid-
ing π/2 periodicity. Thus, for this circuit configuration In a nutshell, we have addressed the possibilities of
II, periodicity becomes exactly half comparedto the cir- current modulation at nano-scale level using mesoscopic
cuit configuration I. The explanation of π/2 periodicity rings enclosing a time varying magnetic flux. We have
is as follows. For this two ring system, the transmission shown that a single mesoscopic ring or two such rings,
probability depends on the combined effect of quantum subject to a DC bias voltage, can support an oscillat-
interferences in the two rings. In ring-1, φ (t) is sinu- ing output current. A single mesoscopic ring can ex-
1
soidal in form as described mathematically in Eq. (2), hibit an oscillating current with a particular frequency
while in ring-2, the variation of flux φ (t) is the same associated with the flux φ(t), while the frequency of the
2
5
currentcan be regulatedin the case of two rings by tun- exactly invariant. To be more specific, it is important to
ing the phase difference δ between the fluxes φ (t) and notethat,inrealsituationtheexperimentallyachievable
1
φ (t). The whole modulation actionis basedon the cen- ringshavetypicaldiameterswithintherange0.4-0.6µm.
2
tral idea of quantum interference effect in presence of In such a small ring, unrealistically very high magnetic
flux φ in ring shaped geometries. We adopt a simple fields are required to produce a quantum flux. To over-
tight-binding framework to illustrate the model and all come this situation, Hod et al. have studied extensively
the calculations are done using single particle Green’s and proposed how to construct nanometer scale devices,
function formalism. Our exact numerical results provide based on Aharonov-Bohm interferometry, those can be
two-terminal conductance and current which clearly de- operated in moderate magnetic fields [32–35].
scribe the essential features of current modulation. Our
In the present paper we have done all the calcula-
analysis can be used in designing tailor made nano-scale
tionsbyignoringtheeffectsofthetemperature,electron-
electronic devices.
electron correlation, etc. Due to these factors, any scat-
Throughoutourwork,wehavedescribedalltheessen-
teringprocessthatappearsinthe mesoscopicringwould
tial features of current modulation for two different ring
haveinfluenceonelectronicphases,and,inconsequences
sizes. In circuit configuration I, we have chosen a ring
can disturb the quantum interference effects. Here we
with total number of atomic sites N =20. On the other
have assumed that, in our sample all these effects are
hand, in circuit configurationII, we have consideredtwo
too small, and accordingly, we have neglected all these
identical rings, where each ring contains 8 atomic sites.
factors in this particular study.
In our model calculations, these typical numbers (20 or
2×8 = 16) are chosen only for the sake of simplicity. Theimportanceofthisarticleismainlyconcernedwith
Though the results presented here change numerically (i) the simplicity of the geometry and (ii) the smallness
with the ring size (N), but all the basic features remain of the size.
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