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Electron transport through a quantum interferometer withside-coupled quantum dots: Green’s function approach

Santanu K. Maiti1,2,∗

1Theoretical Condensed Matter Physics Division, Saha Institute of Nuclear Physics,1/AF, Bidhannagar, Kolkata-700 064, India

2Department of Physics, Narasinha Dutt College, 129 Belilious Road, Howrah-711 101, IndiaAbstract

We study electron transport through a quantum interferometer with side-coupled quantum dots. Theinterferometer, threaded by a magnetic flux φ, is attached symmetrically to two semi-infinite one-dimensional metallic electrodes. The calculations are based on the tight-binding model and the Green’sfunction method, which numerically compute the conductance-energy and current-voltage characteristics.Our results predict that under certain conditions this particular geometry exhibits anti-resonant states.These states are specific to the interferometric nature of the scattering and do not occur in conventionalone-dimensional scattering problems of potential barriers. Most importantly we show that, such a simplegeometric model can also be used as a classical XOR gate, where the two gate voltages, viz, Va and Vb, areapplied, respectively, in the two dots those are treated as the two inputs of the XOR gate. For φ = φ0/2(φ0 = ch/e, the elementary flux-quantum), a high output current (1) (in the logical sense) appears if one,and only one, of the inputs to the gate is high (1), while if both inputs are low (0) or both are high (1),a low output current (0) appears. It clearly demonstrates the XOR gate behavior and this aspect maybe utilized in designing the electronic logic gate.

PACS No.: 73.23.-b; 73.63.Rt.

Keywords: Quantum interferometer; Conductance; I-V characteristic; XOR gate; Anti-resonant state.

∗Corresponding Author: Santanu K. MaitiElectronic mail: santanu.maiti@saha.ac.in

1

1 Introduction

Electronic transport in quantum confined systemslike quantum rings, quantum dots, quantum wires,etc., has become a very active field both in thetheoretical and experimental research. The presentprogress in nanoscience and technology has enabledus to use such quantum confined geometric modelsin electronic as well as spintronic engineering sincethese simple looking systems are the basic buildingblocks of designing nano devices. The key idea ofmanufacturing nano devices is based on the con-cept of quantum interference effect which is gen-erally preserved throughout the sample of muchsmaller sizes, while, it disappears for larger sys-tems. A mesoscopic normal metal ring is one suchpromising example where electronic motion is con-fined, and with the help of such a ring we canconstruct a quantum interferometer. In this arti-cle we will explore the electron transport proper-ties through a quantum interferometer with side-coupled quantum dots, and show how such a simplegeometric model can be used to design an XOR logicgate. To reveal this phenomenon we make a bridgesystem where the interferometer is attached sym-metrically to two external electrodes, the so-calledelectrode-interferometer-electrode bridge. The the-oretical progress of electron transport in a bridgesystem has been started after the pioneering workof Aviram and Ratner [1]. Later, many excellentexperiments [2, 3, 4, 5, 6, 7] have been done in sev-eral bridge systems to justify the actual mechanismsunderlying the electron transport. Though in liter-ature many theoretical [8, 9, 10, 11, 12, 13, 14, 15,16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27] as wellas experimental papers [2, 3, 4, 5, 6, 7] on electrontransport are available, yet lot of discrepancies arestill present between the theory and experiment.The electronic transport through the interferome-ter significantly depends on the interferometer-to-electrodes interface structure. By changing thegeometry one can tune the transmission probabil-ity of an electron across the interferometer. Thisis solely due to the quantum interference effectamong the electronic waves traversing through dif-ferent arms of the interferometer. Furthermore, theelectron transport through the interferometer canbe modulated in other way by tuning the mag-netic flux, the so-called Aharonov-Bohm (AB) flux,that threads the interferometer. The AB flux canchange the phases of the wave functions propa-gating along the different arms of the interferom-eter leading to constructive or destructive inter-

ferences, and accordingly the transmission ampli-tude changes [28, 29, 30, 31, 32]. Beside these fac-tors, interferometer-to-dots coupling is another im-portant issue that controls the electron transportin a meaningful way. All these are the key fac-tors which regulate the electron transmission in theelectrode-interferometer-electrode bridge and theseeffects have to be taken into account properly toreveal the transport mechanisms.

The purposes of this paper are twofold. In thefirst part we explore the appearance of unconven-tional anti-resonant states for our model. Thesestates are specific to the interferometric nature ofthe scattering and do not appear in ordinary scat-tering problems. While, the second part addressesthe XOR gate response in this simple geometry. Inour model, the interferometer, threaded by a mag-netic flux φ, is directly coupled to two quantumdots, and two gate voltages Va and Vb, are applied,respectively (see Fig. 1) in these dots. These gatevoltages are treated as the two inputs of the XORgate. Here we adopt a simple tight-binding modelto describe the system and all the calculations areperformed numerically. We narrate the XOR gatebehavior by studying the conductance-energy andcurrent-voltage characteristics as functions of theinterferometer-to-dots coupling strengths, magneticflux and gate voltages. Our study reveals that fora particular value of the magnetic flux, φ = φ0/2,a high output current (1) (in the logical sense) isavailable if one, and only one, of the inputs to thegate is high (1), while if both the inputs are low (0)or both are high (1), a low output current (0) ap-pears. This phenomenon clearly demonstrates theXOR gate behavior which may be utilized in man-ufacturing the electronic logic gate. To the bestof our knowledge the XOR gate response in such asimple system has not been described earlier in theliterature.

The scheme of the paper is as follow. Follow-ing the introduction (Section 1), in Section 2, wepresent the model and the theoretical formulationsfor our calculations. Section 3 discusses the signifi-cant results, and finally, we conclude in Section 4.

2 Model and the synopsis of

the theoretical background

Let us start by referring to Fig. 1. A quantuminterferometer, threaded by a magnetic flux φ, isattached symmetrically to two semi-infinite one-dimensional (1D) metallic electrodes, viz, source

2

and drain. Two quantum dots a and b (designedby filled blue circles) are directly coupled to theatomic sites 2 and 3 of the interferometer, respec-tively. These two quantum dots are subjected tothe gate voltages Va and Vb, respectively, those areregarded as the two inputs of the XOR gate. Thegate voltages in the dots are given via the gateelectrodes, viz, gate-a and gate-b. These gate elec-trodes are ideally isolated from the dots and can beregarded as two parallel plates of a capacitor. Theactual scheme of connections with the batteries forthe operation of the XOR gate is clearly presentedin the figure (Fig. 1), where the source and the gatevoltages are applied with respect to the drain.

To calculate the conductance of the interferom-eter with side-coupled quantum dots, we use theLandauer conductance formula [33, 34]. At very low

41Φ

2

3

b

a

Source Drain

Gate−a

Gate−b

Figure 1: (Color online). The scheme of connec-tions with the batteries for the operation of theXOR gate. A quantum interferometer with side-coupled quantum dots (filled blue circles), threadedby a magnetic flux φ, is attached symmetrically totwo semi-infinite 1D metallic electrodes. The gatevoltages Va and Vb, those are variable, are appliedin the dots a and b, respectively. The source andthe gate voltages are applied with respect to thedrain. Filled red circles correspond to the positionof the atomic sites in the interferometer.

temperature and bias voltage, the conductance gcan be expressed in terms of the transmission prob-ability T of an electron through the interferometeras,

g =2e2

hT (1)

This transmission probability can be represented interms of the Green’s function of the interferome-ter including the dots and its coupling to the twoelectrodes by the relation [33, 34],

T = Tr [ΓSGrIΓDGa

I ] (2)

where GrI and Ga

I are respectively the retarded andadvanced Green’s functions of the interferometerwith the side-attached quantum dots including theeffects of the two electrodes. The factors ΓS andΓD describe the coupling of the interferometer tothe source and drain, respectively. For the com-plete system i.e., the interferometer with the cou-pled quantum dots, source and drain, the Green’sfunction is defined as,

G = (E −H)−1

(3)

where E = E + iη. E is the injecting energy of thesource electron and η gives an infinitesimal imag-inary part to E . To Evaluate this Green’s func-tion, the inversion of an infinite matrix is neededsince the full system consists of the interferome-ter with four atomic sites and two coupled quan-tum dots, and the two semi-infinite 1D electrodes.However, the entire system can be partitioned intosub-matrices corresponding to the individual sub-systems and the Green’s function for the interfer-ometer with side-coupled quantum dots can be ef-fectively written as,

GI = (E −HI − ΣS − ΣD)−1

(4)

where HI corresponds to the Hamiltonian of theinterferometer with the two dots. Within the non-interacting picture it can be expressed in the form,

HI =∑

i

(ǫi + Vaδia + Vbδib) c†i ci

+∑

<ij>

t(

c†i cjeiθ + c†jcie

−iθ)

+ ta

(

c†ac2 + c†2ca

)

+ tb

(

c†bc3 + c†3cb

)

(5)

In this Hamiltonian ǫi’s are the site energies forall the sites i except the sites i = a and b wherethe gate voltages Va and Vb are applied, those arevariable. These gate voltages can be incorporatedthrough the site energies as expressed in the aboveHamiltonian. c†i (ci) is the creation (annihilation)operator of an electron at the site i and t is the hop-ping integral between the nearest-neighbor sites ofthe interferometer. θ = πφ/2φ0 is the phase factordue to the flux φ threaded by the interferometer.

3

The factors ta and tb correspond to the couplingstrengths of the quantum dots a and b to the atomicsites 2 and 3 of the interferometer, respectively. Forthe two semi-infinite 1D perfect electrodes, a simi-lar kind of tight-binding Hamiltonian is also used,except the phase factor θ, where the Hamiltonianis parametrized by constant on-site potential ǫ′ andnearest-neighbor hopping integral t′. The hoppingintegral between the source and the interferometeris τS , while it is τD between the interferometer andthe drain. The parameters ΣS and ΣD in Eq. (4)represent the self-energies due to the coupling ofthe interferometer to the source and drain, respec-tively, where all the information of the coupling areincluded into these two self-energies [33].The current (I) passing through the interferome-

ter is depicted as a single-electron scattering processbetween the two reservoirs of charge carriers. Thecurrent can be evaluated as a function of the appliedbias voltage by the relation [33],

I(V ) =e

πh

EF+eV/2∫

EF−eV/2

T (E) dE (6)

where EF is the equilibrium Fermi energy. Herewe make a realistic assumption that the entire volt-age is dropped across the interferometer-electrodeinterfaces, and it is examined that under such anassumption the I-V characteristics do not changetheir qualitative features.In this presentation, all the results are computed

at absolute zero temperature. These results are alsovalid even for some finite (low) temperatures, sincethe broadening of the energy levels of the interfer-ometer with side-coupled quantum dots due to itscoupling with the electrodes becomes much largerthan that of the thermal broadening [33]. On theother hand, at high temperature limit, all these phe-nomena completely disappear. This is due to thefact that the phase coherence length decreases sig-nificantly with the rise of temperature where thecontribution comes mainly from the scattering onphonons, and accordingly, the quantum interferenceeffect vanishes. Throughout the calculations we setEF = 0, and choose the unit c = e = h = 1.

3 Results and discussion

We describe our results in two parts. In the firstpart, we narrate the existence of the anti-resonantstates in this particular geometry those are specificto the interferometric nature of the scattering and

do not occur in traditional one-dimensional scat-tering problems of potential barriers. On the otherhand, in the second part, we demonstrate how thissimple geometric model can be used as an XORgate. The key controlling parameter for the wholeoperation of the XOR gate is the magnetic flux φthreaded by the interferometer.

As illustrative examples, in Fig. 2 we present thevariation of the conductance g (red curves) andthe density of states ρ (black curves) as a func-tion of the energy E for the interferometer withside-coupled quantum dots, considering the differ-

-8 0 8E

0

1

2

gHEL

HcL

-8 0 8E

0

1

2

gHEL

HdL

-8 0 8E

0

1

2

gHEL

HaL

-8 0 8E

0

1

2

gHEL

HbL

Figure 2: (Color online). g-E (red color) and ρ-E(black color) curves for the interferometer with side-coupled quantum dots. (a) ta = tb = 0, (b) ta =tb = 1, (c) ta = tb = 2 and (d) ta = tb = 3. Otherparameters are, t = 3, τS = τD = 2.5, φ = 0, andthe on-site potential and the hopping integral in theelectrodes are set as ǫ′ = 0 and t′ = 4, respectively.Here we set ǫ1 = ǫ2 = ǫ3 = ǫ4 = 0 and Va = Vb = 0.

ent values of the side-coupling strengths ta and tb,respectively. Here we set the site energies of all theatomic sites of the interferometer including the twodots as zero. Our results predict that, for some par-ticular energies the conductances show resonancepeaks (red curves), associated with the density ofstates (black curves). These energies are the so-called resonant energies, and the associated statesare defined as the resonant states. The g-E spec-tra predict that, though the resonance peaks areavailable for some particular energies, but the elec-

4

tron conduction from the source to drain throughthe interferometer is possible almost for all otherenergies. This is due to the overlap of the twoneighboring resonance peaks, where the contribu-tion for the spreading of the resonance peaks comesfrom the imaginary parts of the self-energies ΣS andΣD, respectively [33]. At the resonances, the con-ductance g approaches the value 2, and therefore,for these energies the transmission probability Tgoes to unity, since the relation g = 2T is satis-

-8 -6 -4 -2 0 2 4 6 8E

0

1

2

gHEL

Figure 3: (Color online). g-E curves for the in-terferometer with side-coupled quantum dots. Thered, green, blue and black curves correspond to theresults for the cases when the site energies of all thesites of the interferometer including the two dots areidentically set to 1, 2, 3 and 4, respectively. Otherparameters are, t = 3, ta = tb = 3, τS = τD = 2.5,φ = 0, ǫ′ = 0 and t′ = 4.

fied from the Landauer conductance formula (seeEq. (1) with e = h = 1). Now we interpret thedependences of the interferometer-to-dot couplingstrengths on the electron transport. In Fig. 2(a),when there is no coupling between the interferom-eter and the two dots, the conductance shows non-zero value for the entire energy range. The situa-tion becomes much more interesting as long as thecoupling of the two dots with the interferometer isintroduced. To illustrate it, in Figs. 2(b)-(d) weplot the results for the three different choices of thecoupling strengths ta and tb, respectively. The in-troduction of the side-coupling provides the anti-resonant state. For all these three different choicesof ta and tb, the conductance spectra (Figs. 2(b)-(d)) show that the conductance drops exactly tozero at the energy E = 0. At this particular en-ergy, the density of states has a sharp peak, and itis examined that the height of this particular peakincreases very rapidly as we decrease the imaginarypart η to the energy E . It reveals that the electronconduction through this state is no longer possible,

and the state is the so-called anti-resonant state.With the increase of the side-coupling strength, thewidth of the resonance peaks gradually decreases, asclearly observed from these figures. Both these res-onant and anti-resonant states are associated withthe energy eigenvalues of the interferometer includ-ing the two side-coupled dots, and thus, we cansay that the conductance spectrum reveals itself theelectronic structure of the interferometer includingthe two dots.

To examine the dependence of the anti-resonantstate on the site energies of the interferometer andthe two side-attached quantum dots, in Fig. 3 weplot the g-E characteristics for the four differentcases of these site energies. The red, green, blueand black curves correspond to the results whenthe energies of the six atomic sites (four sites of the

-8 0 8E

0

1

2

gHEL

HcL

-8 0 8E

-1

0

1gHEL

HdL

-8 0 8E

-1

0

1gHEL

HaL

-8 0 8E

0

1

2

gHEL

HbL

Figure 4: (Color online). g-E curves for the in-terferometer with side-coupled quantum dots. (a)Va = Vb = 0, (b) Va = 2 and Vb = 0, (c) Va = 0and Vb = 2 and (d) Va = Vb = 2. Other parametersare, ǫ1 = ǫ2 = ǫ3 = ǫ4 = 0, t = 3, ta = tb = 3,τS = τD = 2.5, φ = 0.5, ǫ′ = 0 and t′ = 4.

interferometer and the two sites of the dots) areidentically set to 1, 2, 3 and 4, respectively. Quiteinterestingly we see that the anti-resonant state sit-uates at these respective energies. Our critical in-vestigation also shows that no anti-resonant statewill appear if the site energy of anyone of these sixsites is different from the other sites. The similarbehavior will be also observed for the case if any-

5

one of the three hopping strengths t, ta and tb isdifferent from the other two. Thus it can be em-phasized that the anti-resonant state will appearonly when the site energies of all the six sites areidentical to each other as well as the strengths ofthe three different hopping parameters (t, ta andtb) are same. In this context it is also importantto note that, though all the results presented aboveare done only for the flux φ = 0, but the positions ofthese anti-resonant states will not change at all inthe presence of φ and no new significant feature willbe observed for the description of the anti-resonantstates.

Next we concentrate our study on the XOR gateresponse exhibited by this geometric model. Forthe operation of the XOR gate, we set the mag-netic flux φ at φ0/2 i.e., 0.5 in our chosen unit

-8 0 8V

-6

0

HcL6

I

-8 0 8V

-6

0

HdL6

I

-8 0 8V

-6

0

HaL6

I

-8 0 8V

-6

0

HbL6

I

Figure 5: (Color online). I-V curves for the in-terferometer with side-coupled quantum dots. (a)Va = Vb = 0, (b) Va = 2 and Vb = 0, (c) Va = 0and Vb = 2 and (d) Va = Vb = 2. Other parametersare, ǫ1 = ǫ2 = ǫ3 = ǫ4 = 0, t = 3, ta = tb = 3,τS = τD = 2.5, φ = 0.5, ǫ′ = 0 and t′ = 4.

c = e = h = 1. As representative examples, inFig. 4 we show the variation of the conductanceg as a function of the injecting electron energy E,where (a), (b), (c) and (d) represent the results forthe different cases of the gate voltages Va and Vb

applied in the two side-attached quantum dots, re-spectively. When both the two inputs Va and Vb areidentical to zero, i.e., both the two inputs are low,

the conductance becomes exactly zero for the entireenergy range (see Fig. 4(a)). This reveals that theelectron conduction through the interferometer isnot possible for this particular case. Similar behav-ior is also observed for the typical case when boththe two inputs are high i.e., Va = Vb = 2. In this sit-uation also the electron conduction from the sourceto drain through the interferometer is not possiblefor the whole energy range (see Fig. 4(d)). On theother hand, for the rest two cases i.e., when any oneof the two inputs is high and other one is low i.e.,either Va = 2 and Vb = 0 (Fig. 4(b)) or Va = 0and Vb = 2 (Fig. 4(c)), the conductance exhibitsresonances for some particular energies. Thus forboth these two cases the electron conduction takesplace across the interferometer. Now we justify thedependences of the gate voltages on the electrontransport for these four different cases. The prob-ability amplitude of getting an electron across theinterferometer depends on the quantum interferenceof the electronic waves passing through the upperand lower arms of the interferometer. For the sym-metrically connected interferometer i.e., when thetwo arms of the interferometer are identical witheach other, the probability amplitude is exactly zero(T = 0) for the flux φ = φ0/2. This is due to theresult of the quantum interference among the two

Table 1: XOR gate response of the quantum inter-ferometer with side-coupled dots. The current I iscomputed at the bias voltage 6.02.

Input-I (Va) Input-II (Vb) Current (I)0 0 02 0 4.5680 2 4.5682 2 0

waves in the two arms of the interferometer, whichcan be obtained through the few simple mathemat-ical steps. Thus for the cases when both the twoinputs (Va and Vb) are either low or high, the trans-mission probability drops to zero. While, for thetwo other cases, the symmetry of the two arms ofthe interferometer is broken by applying the gatevoltage either in the dot a or in b, and therefore,the non-zero value of the transmission probabilityis achieved which reveals the electron conductionacross the interferometer. Thus we can predict thatthe electron conduction takes place across the inter-ferometer if one, and only one, of the two inputs to

6

the gate is high, while if both the inputs are lowor both are high the conduction is no longer pos-sible. This feature clearly demonstrates the XORgate behavior.The XOR gate response can be much more clearly

noticed by studying the I-V characteristics. Thecurrent I passing through the interferometer is com-puted from the integration procedure of the trans-mission function T as prescribed in Eq. (6). Thetransmission function varies exactly similar to thatof the conductance spectrum, differ only in magni-tude by the factor 2 since the relation g = 2T holdsfrom the Landauer conductance formula Eq. (1).As representative examples, in Fig. 5 we displaythe variation of the current I as a function of theapplied bias voltage V for the four different casesof the two gate voltages Va and Vb. In the particu-lar cases when both the two inputs are identical toeach other, either low (Fig. 5(a)) or high (Fig. 5(d)),the current is zero for the complete range of thebias voltage V . This behavior is clearly understoodfrom the conductance spectra, Figs. 4(a) and (d),since the current is computed from the integrationmethod of the transmission function T . For theother two cases when only one of the two inputs ishigh and other is low, a high output current is ob-tained which are clearly described in Figs. 5(b) and(c). From these I-V curves the behavior of the XORgate response is nicely observed. To make it muchclear, in Table 1, we present a quantitative estimateof the typical current amplitude, computed at thebias voltage V = 6.02. It shows that, I = 4.568only when any one of the two inputs is high andother is low, while for the other cases when eitherVa = Vb = 0 or Va = Vb = 2, the current achievesthe value 0.

4 Concluding remarks

To summarize, we have studied electron trans-port in a quantum interferometer with side-coupledquantum dots. The interferometer, threaded by amagnetic flux φ, is attached symmetrically to twosemi-infinite 1D metallic electrodes and two gatevoltages, viz, Va and Vb, are applied, respectively,in the two dots those are treated as the two inputsof the XOR gate. The system is described by thetight-binding Hamiltonian and all the calculationsare done in the Green’s function formalism. Wehave numerically computed the conductance-energyand current-voltage characteristics as functions ofthe interferometer-to-dots coupling strengths, mag-netic flux and gate voltages. We have described

the essential features of the electron transport intwo parts. In the first part, we have addressed theexistence of the anti-resonant states, those are notavailable in the traditional scattering problems ofpotential barriers. On the other hand, in the secondpart, we have explored the XOR gate response forthis particular model. Very interestingly we havenoticed that, for the half flux-quantum value of φ(φ = φ0/2), a high output current (1) (in the logicalsense) appears if one, and only one, of the inputs tothe gate is high (1). On the other hand, if both thetwo inputs are low (0) or both are high (1), a lowoutput current (0) appears. It clearly demonstratesthe XOR gate behavior, and, this aspect may beutilized in designing a tailor made electronic logicgate.

The importance of this article is concerned with(i) the simplicity of the geometry and (ii) thesmallness of the size. To the best of our knowl-edge the XOR gate response in such a simple low-dimensional system has not been addressed earlierin the literature.

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