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arXiv:0808.0947v1 [cond-mat.mes-hall] 7 Aug 2008

Suppressed conductance in a metallic graphene nano-junction

Haidong Li, Lin Wang, and Yisong Zheng∗

Department of physics, Jilin University, Changchun 130023, China

(Dated: August 7, 2008)

Abstract

The linear conductance spectrum of a metallic graphene junction formed by interconnecting

two gapless graphene nanoribbons is calculated. A strong conductance suppression appears in

the vicinity of the Dirac point. We found that such a conductance suppression arises from the

antiresonance effect associated with the edge state localized at the zigzag-edged shoulder of the

junction. The conductance valley due to the antiresonance is rather robust in the presence of the

impurity and vacancy scattering. And the center of the conductance valley can be readily tuned

by an electric field exerted on the wider nanoribbon.

PACS numbers: 81.05.Uw, 73.40.Jn, 73.23.-b, 72.10.-d

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Graphene, an atomically thin layer of graphite, is regarded as a perspective base for the

post-silicon electronics since its first experimental realization.1Motivated by possible device

applications, the electronic and transport properties of various graphene nano-structures

have been studied both experimentally and theoretically,2,3,4,5,6,7,8,9,10,11,12Among these struc-

tures, graphene nanoribbon(GNR) is the basic element to carry the current. Band structure

calculations indicate that the zigzag-edged graphene nanoribbons are always metallic while

the armchair-edged ones are either metallic or semiconducting, depending on the width

of the armchair GNR.8A graphene junction can be formed by interconnecting two semi-

infinite GNRs with different widths. In such a graphene nanostructure, a traveling carrier

is scattered by the junction interface, which causes a finite junction conductance. Recently,

the conductance spectrum of a graphene metal-semiconductor junctions has been studied

in details.10,11,12It has been found that the presence of the lattice vacancy can efficiently

enhance the junction conductance, because that a vacancy makes the coupling between the

electron states of the two GNRs at the junction interface stronger.

Apart from the metal-semiconductor junction, a metallic graphene junction can be con-

structed by interconnecting two metallic armchair GNRs with different widths. In this letter,

we investigate the conductance spectrum of such a metallic graphene junction, the lattice

structure of which is depicted in Fig.1(a). Unlike a metal-semiconductor junction,11,12the

gapless subband structures of the two metallic armchair nanoribbons shown in Fig.1(b) in-

dicates that electronic transmission through the metallic junction with an arbitrary energy

is formally allowed. In particular, the electronic transmission via the linear subbands is

reflectionless even in the presence of a long-range scattering potential, due to pseudospin

conservation.13Therefore, a plausible anticipation is that the electronic transmission prob-

ability near the Dirac point should be close to unity. However, our calculation gives the

opposite result: the junction conductance at the Dirac point is equal to zero, and a conduc-

tance valley appears around this point. This means a strong conductance suppression in the

vicinity of the Dirac point. Further analysis demonstrates that the conductance suppression

arises from the antiresonance due to the existence of an edge state localized in the shoulder

region of the junction.

To calculate the linear conductance of the metallic graphene junction G(E) as a function

of the incident electron energy E, we adopt the Landauer-B¨ utikker formula in the discrete

lattice representation.14,15It gives G(E) =2e2

hTr(Γ1GΓ2G†), where G = [E+i0+−Hd−Σ1−

Σ2]−1is a retarded Green function, and Hdis the tight-binding Hamiltonian of the device

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region in the nearest-neighbor approximation. The contributions of the two semi-infinite

leads are incorporated by the two self-energy terms Σ1(2) which are associated with the

coupling functions Γ1(2)by Γ1(2)= i[Σ1(2)− Σ†

1(2)]. The two self-energy terms are obviously

the key quantities for calculating the conductance, which can be evaluated by the recursive

method.15In what follows we use the lattice constant a and the hopping energy t between

the nearest neighbor atoms as the units of the length and energy.

The calculated conductance spectra(G vs E) for some typical square junctions(θ = 90◦)

are shown in Fig.2(a). All these conductance spectra exhibit the staircase-like structures,

which can be readily explained by matching of the subband structures of the two component

ribbons as shown in Fig.1(b). Another point to note in Fig.2(a) is that the conductance

spectrum shows a notable suppression in the vicinity of the Dirac point when the difference

of the widths of the two GNRs is larger than 3. In particular, a zero conductance occurs

at the Dirac point. However, from Fig.1(b) we can readily find that the linear subbands

of the two ribbons always match each other to provide an electron transmission mode, and

hence a nonzero conductance at the Dirac point, in contradiction with the calculated zero

conductance. As shown in Fig.2(b), we find that the conductance suppression near the

Dirac point is tightly associated with the zigzag edge of the shoulder of the junction. On

the contrary, when the edge of shoulder is of an armchair type(θ = 120◦), the conductance

spectrum no longer shows any suppression. From such a result we infer that the nature of

the conductance suppression is the antiresonance effect, the detail of which is as follows.

In the vicinity of the Dirac point, only the linear subbands are relevant to the electron

transmission. Hence the two metallic GNRs can be viewed as single mode quantum wires

coupling to each other directly. However, the zigzag-edged shoulder induces a localized edge

state with the eigen-energy equal to the Dirac energy. Such a localized state couples to the

linear subband of the wider GNR. Consequently, when the electronic transport is limited

to the vicinity of the Dirac point, the graphene junction is equivalent to the T-shaped

quantum dot structure as shown in Fig.2(c). The linear conductance of such a model has

been extensively studied16,17,18and can be expressed in terms of the model parameters

G(E) =2e2

h(1 + ξ)2

2ξ[E − ε]2

(E − ε)2+ [

Γ

2(1+ξ)]2, (1)

where Γ = 2πρ2τ2and ξ = π2ρ1ρ2v2with ρ1(2)being the electron density of the states in two

leads. This expression presents a zero conductance at the quantum dot level ε, which is called

the antiresonance effect. The antiresonance is in fact a result of quantum interference. The

lateral quantum dot introduces new Feynman paths with a phase shift π. As a result, the

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destructive quantum interference occurs among electron Feynman paths.17,18In the metallic

graphene junction, the edge state attached to the zigzag-edged shoulder of the junction plays

a role of laterally coupled quantum dot, which results in the antiresonance at the Dirac point.

When the widths of the two GNRs are fixed, the geometry of the junction can be changed

by shifting downwards the narrower ribbon. In Fig.2(d) the conductance spectra are com-

pared for differently shaped junctions. We can see that the width of conductance valley

depends on the junction shape sensitively. To be more specific, in terms of the valley width

the spectra are classified into three groups, each of which appears periodically whenever

the narrower ribbon shifts downwards by three multiples of the lattice constant. This phe-

nomenon can be explained with the help of the above quantum dot model. The parameter

v in the quantum dot model is a relevant quantity to the width of the antiresonance valley.

From Fig.1(a) we can see that v is proportional to the product of the electron probability

amplitudes of the A and B atoms interconnecting directly at the junction interface. And

the probability amplitudes can be obtained by solving the Dirac equation8. In such a way

we work out the following relation v ∝?

j∈oddsin(2jπ/3)sin[2(j + n0)π/3], where j is any

odd number within the range from 1 to N1, and n0denotes the displacement of the narrower

ribbon with respect to the upper edge of the wider ribbon. From this relation we can readily

understand the periodic feature of the valley width of the conductance spectra shown in

Fig.2(d).

The antiresonance picture of the conductance suppression in the metallic graphene junc-

tion is further demonstrated by the calculated spectra of the local density of states (LDOS)

at some lattice points near the junction interface. From Fig.3 we can see that only for the

lattice points at the zigzag edge of the shoulder of junction(θ = 90◦and 150◦), the LDOS

spectrum exhibits a very sharp peak at the Dirac point. This indicates the existence of a

localized state in the shoulder region of the junction. An exception occurs for the junction

with a very short shoulder(N1 = 20 and N2 = 23 in Fig.3(a)). The LDOS spectrum of

the lattice point at the zigzag-edged shoulder does not show a notable peak. This result is

consistent with the previous work9, which argued that a zigzag edge of the width smaller

than three lattice constant can not induce any localized edged state.

We now proceed on to discuss the influence of the possible scatterers in an actual graphene

junction on the antiresonance valley. At first, we consider the individual effect of an impurity

appearing at distinct positions. In the numerical calculation, an impurity is simulated by the

deviation of the on-site energy of an specific lattice point where the impurity appears. The

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calculated conductance spectra with an individual impurity positioned at different lattice

points are compared in Fig.4(a). We can see that only when the impurity appears in the

shoulder region, the variation of the conductance spectrum is notable. But the effect of

such an impurity is not to destroy the antiresonance at the Dirac point. Instead it causes

another conductance zero near the Dirac point. This result implies that the edge state is

intricately affected by an impurity in the shoulder region. On the contrary, the impurity at

other lattice points can not modify the antiresonance valley notably since it is irrelevant to

the edge state. Fig.4(b) shows the conductance spectrum in the presence of many impurities

distributed randomly in the device region with fluctuating strengths. We can see that the

antiresonance is rather robust even if the impurity concentration and strength are nontrivially

large. Fig.4(c) shows the effect of vacancies positioned uniformly at the zigzag edge of the

shoulder of a square junction. A vacancy is simulated in the numerical calculation by simply

cutting off a carbon atom of type A at the edge of the shoulder. If we use the notations M1

and M2to denote the atom number of type A belonging to the edge of the shoulder and

the number of the vacancy in this edge respectively, the ratio r = M2/M1can be viewed

as the concentration of vacancy. From Fig.4(c) we can see only when the concentration of

the vacancies is about r = 1/3, the zero conductance at the Dirac point can be completely

eliminated. Higher or lower concentrations of vacancies can not destroy the conductance

valley around the Dirac point. Our calculation also indicates that such a conclusion is

independent of the size of the shoulder. This result implies the complicated effect of the

edge defect on the edge state. Such an interesting topic is left for our study in the future.

Finally, we can apply a step-like potential in the right hand side of the junction to tune

the position of antiresonance, which can be simulated by shifting the on-site energy of all

the lattice points of the wider GNR. The result is shown in Fig.4(d). We can see that

the step-like potential simply shifts the antiresonance point, without drastically altering the

lineshape of the conductance spectrum. What is noteworthy is that the conductance at the

Dirac point can be easily tuned from zero to unity by a step-like potential. This suggests a

possible device application of a nanoswitch based on such a metallic graphene junction.

This work was financially supported by the National Nature Science Foundation of China

under Grant NNSFC10774055.

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