Fano resonances in hexagonal zigzag graphene rings under external magnetic flux

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Fano resonances in hexagonal zigzag graphene rings under external magnetic flux

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2015 J. Phys.: Condens. Matter 27 175301

(http://iopscience.iop.org/0953-8984/27/17/175301)

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Journal of Physics: Condensed Matter

J. Phys.: Condens. Matter 27 (2015) 175301 (6pp) doi:10.1088/0953-8984/27/17/175301

Fano resonances in hexagonal zigzaggraphene rings under externalmagnetic flux

D Faria1, R Carrillo-Bastos2,3,4, N Sandler2 and A Latge1

1 Universidade Federal Fluminense, Av. Litoranea sn, 24210-340 Niteroi, RJ, Brasil2 Ohio University, Athens, Ohio 45701-2979, USA3 Centro de Investigacion Cientıfica y de Educacion Superior de Ensenada, Apdo. Postal 360, 22800Ensenada, Baja California, Mexico4 Universidad Nacional Autonoma de Mexico, Apdo. Postal 14, 22800 Ensenada, Baja California,Mexico

E-mail: [email protected]

Received 7 January 2015, revised 16 February 2015Accepted for publication 2 March 2015Published 2 April 2015

AbstractWe study transport properties of hexagonal zigzag graphene quantum rings connected tosemi-infinite nanoribbons. Open two-fold symmetric structures support localized states thatcan be traced back to those existing in the isolated six-fold symmetric rings. Using atight-binding Hamiltonian within the Green’s function formalism, we show that an externalmagnetic field promotes these localized states to Fano resonances with robust signatures intransport. Local density of states and current distributions of the resonant states are calculatedas a function of the magnetic flux intensity. For structures on corrugated substrates we analyzethe effect of strain by including an out-of-plane centro-symmetric deformation in the model.We show that small strains shift the resonance positions without further modifications, whilehigh strains introduce new ones.

Keywords: graphene rings, electronic tranport, Fano resonances

(Some figures may appear in colour only in the online journal)

1. Introduction

The unique electronic properties of graphene [1] have guidedresearch combining confinement and interference effects onannular systems [2]. Closed ring geometries for example,have been extensively studied, rendering predictions of energyspectra for different choices of boundary conditions [3–7]and including the role of strain [8]. When a magnetic fluxis applied in these Aharonov-Bohm geometries, persistentcurrents with lifted-valley degeneracy appear [3, 9]. Modelswith different ring geometries reveal peculiar transportproperties: rectangular shapes exhibit resonant tunnelingphenomena [10], while hexagonal ones [11] allow for currentblocking mechanisms with the leads acting as valley filters.Samples with some of these characteristics have alreadybeen synthezised [12–17], with some fabrication techniquesable to produce rings with perfect hexagonal symmetry by

exploiting appropriate lattice orientations [18, 19]. Transportmeasurements in these open rings show peculiar conductanceoscillations in the presence of magnetic fields [16]. These newdevices present the opportunity to test theoretical predictionsand reveal new transport phenomena that may be used todevelop new technological applications.

Ring geometries are particularly useful to investigateinterference effects that may appear with precise fingerprints,such as those produced by Fano resonances [20–22]. Fanophysics is a rich phenomena produced by the coexistenceof resonant (localized) and nonresonant paths for scatteringwaves [23]. The transmission function exhibits an asymmetricline shape which depends on the coupling between resonantand extended states. In particular, Fano resonances have beenreported in semiconducting ring shaped structures under thepresence of an external magnetic field [24]. The resonancesarise as consequence of the broken time-reversal symmetry

0953-8984/15/175301+06$33.00 1 © 2015 IOP Publishing Ltd Printed in the UK

J. Phys.: Condens. Matter 27 (2015) 175301 D Faria et al

of localized states and their interference with extendedstates through the semiconducting ring. Fano resonanceswere also observed in similar semiconducting rings as aconsequence of broken parity of localized states due to spin–orbit interaction [25]. In semiconductors, emergent boundstates are produced by disorder that could create states in thegap (no observed Fano effect). It is important to mention thatclean semiconductor samples will not present Fano resonancesunless some symmetry is broken in the system. In graphenesystems, various interesting proposals for observing Fanoresonances have been advanced [26–29], with some ringgeometries exploiting a broken upper–lower arm symmetry toproduce the effect [27]. Transmission calculations in differenthexagonal configurations were explored and the conductancebands of zigzag rings were found to be considerably narrowerthan in the case of armchair edges, being less robust underperturbations [27].

In this paper, we carry out a study of transport propertiesof a graphene quantum ring connected to semi-infinite zigzagnanoribbons that exhibits signatures of Fano interference underexternal magnetic flux. The isolated structure possesses ahexagonal (six-fold) symmetry with zigzag inner and outeredges. Connecting the ring to leads renders a two-foldsymmetric setup that retains localized states as shown indensity of states (DOS) calculations (see below). We showthat an external magnetic field that fully pierces the ringregion produces signatures of Fano resonances in the DOSand strongly modifies the ring conductance in a wide rangeof energies. These resonances are a result of the pre-existing localized states that couple to the continuum bythe external flux. The addition of an out-of plane centro-symmetric deformation in the ring region, described by aneffective pseudo-magnetic field [30], modifies the transmissionproperties of the structure. The resulting Fano resonanceenergies are shifted from their original values for small strainswhile new resonances appear at energies precisely determinedby the curvature of the deformation.

2. Ring coupled to leads

A schematic representation of the model is shown infigure 1(a). The geometry of the ring is important as longas it respects the underlying crystalline symmetry of the 2Dgraphene lattice and it was chosen to avoind change of edgeorientations at the corners of the ring [27]. The ring andleads are defined by the number of zigzag chains, Nz andNy , respectively. We use a standard, single π -band nearest-neighbor tight-binding approximation to describe the centralstructure, with a Hamiltonian given by:

HC =∑〈i,j〉

tij c†i cj +

∑〈i,jL(R)〉

t0c†i cjL(R)

+ h.c., (1)

where the fermion operator c†i (ci) creates (annihilates) an

electron in the i-th site and t0 = −2.7 eV is the hoppingparameter [31]. The first term in equation (1) represents thedynamics in the disconnected ring, with indices i and j runningover all ring sites. The second term connects the ring to the

1

Ny

0.0 0.2 0.4 0.6 0.8 1.00

1

2

3

4

5

6 G nanoribbonG coupled ringDOS coupled ring

G(2

e2 /h)/

DO

S(a

.u.)

Energy(t0)

(a)

(b)

(x0, y0)

Nz

1

x

y

Figure 1. (a) Schematic view of the hexagonal ring connected tosemi-infinite zigzag nanoribbons (Nz = 6 and Ny = 8). Thecoordinate system is shown in the lower left part and the meanexternal radius (r ∼ 17a) of the ring is represented by the blackdashed line. The structure shown contains 680 atoms in the centralring. (b) Conductance (black) and DOS (green) of the coupledsystem as a function of the Fermi energy for � = 0. Theconductance for a perfect zigzag nanoribbon (Ny = 8) is alsoplotted for comparison (red). Because of particle-hole symmetry,data is shown only for positive energies.

leads, with jL and jR denoting sites on the left and right leads,respectively.

An external magnetic field B = Bz, permeating the entirering, is included via the Peierls substitution. As a result thehopping parameter acquires a complex phase [31], as follows:tij = t0ei�φij , with �φij = 2π(e/h)

∫ ri

rjA · dr and ri and rj

being the nearest neighbors positions. We choose the Landaugauge A=(0, Bx, 0), and measure the phase in units of themagnetic flux threading a single hexagonal plaquete, �/�0 =3a2

√3eB/2h, with a = 1.42 Å being the interatomic distance.

The DOS and Landauer conductance are calculated withthe Green’s function formalism [32, 33]. Reservoirs arerepresented by the self-energy �L(R), obtained from each leadGreen’s function calculated with real-space renormalizationtechniques [34, 35].

Results for a typical system in the absence of magneticfield are displayed in figure 1(b), where the energy dependenceof the DOS (green line) and the conductance (black line) areshown. The conductance for a zigzag nanoribbon of the samewidth as the leads is also drawn for comparison (red line). Asexpected, the low energy transport is caused by one-channeltransmission. The figure shows some of the sharpest DOSpeaks coinciding with minima in the conductance, thus notcontributing to transport. This effect is also present at higherenergies (see the second and third conductance plateaus forexample).

2

J. Phys.: Condens. Matter 27 (2015) 175301 D Faria et al

20 40 60 80 1000.00

0.02

0.04

0.06

0.08

0.10

Ene

rgy(

t 0)

/0(10-4)

0

0.1000

0.2000

0.3000

Figure 2. Contour plot of the total DOS as a function of magneticflux and Fermi energy. Color code represents DOS intensity.

We focus next in the energy range within the firstconductance plateau to analyze the nature of these sharp peaksin detail. Results for the calculation of the total DOS as afunction of the Fermi energy and the external magnetic fluxare shown in figure 2. The DOS is split into bands thatexhibit an oscillatory dependence on the external flux, andevolve towards fully developed Landau levels in the high-field regime [3, 6, 7]. Previous results [5] have shown asimilar dependence of the energy levels of the isolated ring,with this behavior being determined by the average radius. Incontrast to the energy spectrum of an isolated hexagonal ringthat exhibits six energy levels in each sub-band [5], the openstructure contains sub-bands with two levels only. These levelsdisplay different broadening, indicating different coupling tothe leads. Figure 2 shows that the reduced symmetry structure(from six-fold to two-fold) rendered by the connection to theleads, retains nevertheless those states that are compatible withboth symmetries. Thus it follows that some of the states in theclosed structure are not fundamentally altered by the symmetryreduction and persist in the open geometry.

3. Fano resonances

Figure 3 shows results for conductance (dash black line) andDOS (full green line) with and without the external magneticfield. The open ring structure is insulating at zero energy asshown by the null conductance in the absence and presence offlux. Panel a) corresponds to zero field and show the featuresobserved in figure 1 with greater detail. It is natural to associatethe sharp peaks that coincide with minima in the conductanceto localized states present in the open system.

Figure 3(b) shows results for a magnetic flux �/�0 =10−3. Note that the sharp peaks in the DOS of panel (a)are broadened. The conductance develops an asymmetricpeak-minima structure, with a zero value at the energiescorresponding to the originally sharp peaks. This asymmetricline shape is the characteristic fingerprint of a Fanoresonance [23]. Figure 3(c) shows the conductance with andwithout magnetic flux. In the presence of external flux thereis a doubling of conductance peaks, a finding consistent with

0.0

0.2

0.4

0.6

0.8

1.0

1.2

/0=10-3

G(2

e2 /h)

/DO

S(a

.u.)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

=0

G(2

e2 /h)

/DO

S(a

.u.)

(a)

(b)

0.00 0.04 0.08 0.12 0.160.0

0.2

0.4

0.6

0.8

1.0

1.2

=0

/0=10-3

G(2

e2 /h)

Energy(t0)

(c)

Figure 3. DOS (green solid line) and conductance (black dashedline) for: (a) � = 0 and (b) �/�0 = 10−3. (c) Conductancecomparison with and without external flux.

0 10 20 30 40 5002468

10121416

(10-4

t 0)

/0(10-4)

-2.6

-2.4

-2.2

-2.0

-1.8

-1.6

qpa

ram

eter

(a)

(b)

Figure 4. (a) Fano parameter q and (b) � of function G(ε)(equation (2)) as a function of the magnetic flux. Dots (red) are fitsfor conductance values at the lowest energy resonance. Full linecurve (blue) is a sinusoidal fitting of the data.

the existence of a newly formed Fano resonance. A numericalfit of the conductance curve is obtained with the renormalizedFano expression:

G(ε) = 1

1 + q2

(ε + q)2

1 + ε, (2)

being valid only if |ε| � 1, where ε = (E − E0)/� isa reduced energy, � is the resonance width, and E0 is the

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J. Phys.: Condens. Matter 27 (2015) 175301 D Faria et al

(a)

(c)

(b)

(d)

0

0.1

0.2

0.3

0.4

0.5

0

0.02

0.04

0.06

0.08

0.1

0

0.1

0.2

0.3

0.4

0.5

0

0.05

0.1

0.15

0.2

Figure 5. LDOS (left) and current density (right) mapping of the open ring for resonant (E = 0.0192t0) and antiresonant states(E = 0.0214t0). Here �/�0 = 10−3. (a) and (b) correspond to the conductance peak, while (c) and (d) refer to the conductance minima.Notice different scales optimized for each case.

resonance energy. The Fano fitting parameter q, a measure ofthe coupling between localized and extended states, acquires aperiodic dependence on the external flux as shown in figure 4.These results apply for all values of magnetic flux shownin figure 2 except in the region where sub-band states mix.In these ranges a convolution of Fano and Breit–Wigner(symmetric-broadened peak) expressions is used to determinethe asymmetry degrees of the resonances [36].

To provide further confirmation that a Fano resonanceis involved, we show results for current density patternsat energies corresponding to peaks and minima of theconductance [35]. Our numerical calculations show that inthe absence of magnetic field the current density splits equallyover the two arms of the ring, for all energies studied. Fora finite magnetic flux, in contrast, the current flows mostlythrough the upper or lower part of the ring, a consequence ofthe preferred circulation introduced by the field (not shown).Interestingly, the magnetic field generates two different currentpatterns for resonant (E = 0.0192t0) and antiresonant states(E = 0.0214t0) as shown in figure 5. Panel (a) shows theLDOS for the resonant energy that extends over the wholering, making possible perfect conductance. Panel (b) shows thecorresponding current that effectively exists in both arms witha unique sense of circulation. In contrast, at the antiresonantenergy (see figure 5(c)), there is an enhanced LDOS at thecentral part of upper and lower arms of the ring, consistent withthe two-fold symmetry of the open structure. The current flowbetween the two terminals is completely suppressed as shownin figure 5(d). Remarkably, there is a local charge circulationpattern at the central arms of the ring, that appears at a muchsmaller scale.

The existence of finite LDOS at the value of theconductance minima confirms the existence of extended statesinside the structure together with localized states at the upperand lower arms. These results provide additional evidencethat the asymmetric conductance minima is produced byinterference between extended and Fano resonant states.

4. Strained ring

The model analyzed in previous sections describe suspendedhexagonal graphene ring structures. However, structuresavailable in current settings are commonly deposited onsubstrates that introduce additional effects. In particular,smoothly corrugated substrates cause local out of planedeformations that produce strain in the system. To model itwe consider a centro-symmetric Gaussian bump described by:

h (ri) = Ae−(ri−r0)2/b2

, (3)

where ri represents an atomic site inside the ring withcoordinates ri = (xi, yi). The Gaussian center r0 = (x0, y0)

coincides with the geometric ring center (figure 1(a)). Thedeformation is included in equation (1) as a modification inthe hopping amplitude in the central structure [37]:

t ′ij = tij e−β

(l′ij /a−1

), (4)

where the new atomic distances l′ij are calculated usingelasticity theory up to linear order on strain [38–40] andβ = |∂ log t0/∂ log a| ≈ 3. The new first-neighbor vectors

4

J. Phys.: Condens. Matter 27 (2015) 175301 D Faria et al

0

1

2

3

4

5

6

7

DOS(a.u.)=0

b=14a

Am

plitu

de(a

)

0

1

2

3

4

5

6

70.00 0.05 0.10

G(2e2/h)/

0=10-3

b=14a

Energy(t0)

0.00 0.05 0.10Energy(t

0)

(a) (b)

Figure 6. Strain effects on (a) DOS (� = 0) and (b) conductance (�/�0 = 10−3), for different values of Gaussian deformation amplitude,and fixed FWHM b = 14a.

are given by �δ′ij = �δij .(I + ε), with I being the identity matrix

and εγ,λ = 12∂γ h∂λh the strain tensor [41]. We use the repeated

greek index summation convention and γ and λ representdirections on the 2D plane. The strain parameter α = (A/b)2

is defined in terms of the amplitude (A) and FWHM (b) of thebump. Notice that strain fields introduce an effective pseudo-magnetic field that competes with the externally applied one.

In figure 6(a) the DOS in the absence of external magneticflux is shown for a strained graphene ring with varyingGaussian amplitude and fixed FWHM (b = 14a). The maineffect of the deformation is to shift the position of variousnarrow peaks towards higher energy values. As an externalflux is added, the strain promotes the Fano resonances to higherenergies too, as shown in figure 6(b). These results highlightthe persistence of localized states in the open ring structure andthe robust effects of Fano resonances in the transmission in thepresence of strain fields. For higher strain values (α > 25%)other Fano resonances appear at low energies, even in theabsence of an external magnetic flux. These are caused by newlocalized states produced by the strain fields at the location ofmaximum curvature of the deformation [40].

5. Conclusions

We have studied a model for a hexagonal zigzag ring coupledto external contacts and showed that the structure containslocalized states. These are remnants of the discrete states ofthe isolated ring that exist on the open structure due to thecombination of the 2-fold and inversion symmetries of thegraphene lattice [42]. While these states do not contributeto the conductance, they can be detected by the application ofan external magnetic flux that mixes them with the continuum

background, generating Fano resonances. The Fano parameterq characterizing the asymmetric interference peak in theconductance shows a periodic dependence on the applied flux.

Contrarily to the Fano resonances previously addressed[27], that were provided by localized states induced by randomedge disorder and spread exactly along the edge defects, theFano resonance in our work are achieved in clean graphenerings, with no disorder. Interestingly, the kink formed at thecorners of the hexagon acts as a defect-like, localizing statesthat are originally extended along the edges. The confinementof these states appears along the sides of the ring geometry andnot at the position of the kinks.

For ring structures deposited on corrugated substrates weanalyzed the effect of strain on transport properties. Wefound that out-of plane deformations produce an overallshift in the energy of the resonances without affecting themotherwise. These results suggest that two terminal transportmeasurements in the presence of an external flux could be usedto characterize small strain patterns in samples by analyzingshifts in conductance minima. In the limit of large strains,the dependence of the conductance minima with the curvatureof the deformation suggests that transport could provide analternative way to characterize strain profiles on samples.

Acknowledgments

We thank useful discussions with F Adame, E Bastos, CLewenkopf, L Lima, F Mireles and P Orellana. This workwas supported by CNPq, CAPES (2412110) and DAAD (DF);FAPERJ E-26/101.522/2010 (AL); NSF No. DMR-1108285(DF, RC-B and NS), CONACYT, PAPIIT-DGAPA UNAMIN109911 (RC-B).

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J. Phys.: Condens. Matter 27 (2015) 175301 D Faria et al

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