Edge states interferometry and spin rotations in zigzag graphene nanoribbons

Gonzalo UsajCentro Atómico Bariloche and Instituto Balseiro, Comisión Nacional de Energía Atómica, 8400 S. C. de Bariloche, Argentina

and Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET), Argentina�Received 1 July 2009; published 31 August 2009�

An interesting property of zigzag graphene nanoribbons is the presence of edge states, extended along itsborders but localized in the transverse direction. Here we show that because of this property, electron transportthrough an externally induced potential well displays two-path-interference oscillations when subjected eitherto a magnetic or a transverse electric field. This effect does not require the existence of an actual “hole” in thenanoribbon’s geometry. Moreover, since edge states are spin polarized, having opposite polarization on oppo-site sides, such interference effect can be used to rotate the spin of the incident carriers in a controlled way.

DOI: 10.1103/PhysRevB.80.081414 PACS number�s�: 72.25.�b, 73.23.�b, 81.05.Uw, 85.75.�d

Graphene, a two-dimensional array of carbon atoms in ahoneycomb lattice, is a very interesting material with un-usual electronic properties.1–4 It has attracted much attentionsince its first experimental realization5,6 as it offers great po-tential for technological applications while, at the same time,it has led to the observation of new physical phenomena suchas an anomalous quantization of the Hall effect,2,7 observableat room temperature, or the manifestation of the Klein tun-neling paradox in transport,1,8,9 among others. The key tounderstand graphene’s peculiarities relies on its band struc-ture: electronic excitations around the Fermi energy �EF� canbe described by an effective Hamiltonian that mimics theDirac equation for massless chiral fermions where the spin isreplaced by a pseudospin �the two inequivalent sites of thehoneycomb lattice� and the speed of light by the Fermivelocity.4,10 The actual spin plays no crucial role in bulksamples.

A novel effect unique to graphene appears in graphenenanoribbons �GNRs�: when the termination of the GNR cor-responds to a zigzag ordering of the carbon atoms �see Fig.1�a�� the system presents edge states.11–13 That is, there areeigenfunctions that are extended along the zigzag nanoribbon�ZGNR�, but that decay exponentially away from the edgestoward the center of the ZGNR. These states have recentlybeen observed in graphite surfaces near monoatomic stepedges.14,15 From the theoretical point of view, they can beeasily obtained from either a discrete tight-binding model forthe honeycomb lattice11,12 or a low-energy effective Hamil-tonian �Dirac equation�.13 If only nearest-neighbor hoppingis considered in the former, the edge states have an exponen-tially small group velocity vg, which leads to a high densityof states near the EF of the undoped material. These stateshave been studied in detail by several authors �see Ref. 4 andreferences therein� including the recent proposal of a novelquantum spin Hall effect in the presence of spin-orbitcoupling.16 When next-to-nearest-neighbor hopping is takeninto account, the edge states become dispersive—they ac-quire a finite vg—and more stable.17 In addition, electron-electron interactions lead to a magnetic ordering of the edgestates11,18 and the appearance of an energy gap in the bandstructure.18 Since the resulting edge states are then spin po-larized, several groups have proposed to use them for spin-tronics applications such as creating pure spin currents19 orinducing half-metallic behavior with electric fields.18

Here, we analyze electron transport through a ZGNR with

a potential well �PW� created by external gates20 and tunedin such a way that transport inside the well is governed onlyby the edge states. In this case, while the current flow isessentially homogeneous outside the PW region, it flowsalong the edges inside it. We show then that the system be-haves as a two-path interferometer even though the ZGNR isstructurally homogeneous, an effect unique to the ZGNRband structure. Interference between the two paths can betested by either using a magnetic21,22 or a transverse electricfield to tune the orbital phase difference between the twobranches.

Furthermore, since the ground state corresponds to an an-tiferromagnetic ordering of the polarization of the two edges,each path corresponds to a different spin orientation. Then, ifthe spin polarization of the incoming electron, set for in-stance by a ferromagnetic contact, is perpendicular to theintrinsic spin-quantization axis of the ribbon, the two-pathinterference leads to a rotation of the carrier’s spin. Its anglecan be controlled externally, offering an interesting potentialfor spintronics.

FIG. 1. �Color online� �a� Scheme of a ZGNR. The energy of a32-ZGNR as a function of the wavevector along the x axis is shownfor: �b� Ba=0; �c� �BBa= t� /2; �d� �BBa= t�; and �e� �BBa=1.2t�.The bands connecting the two nonequivalent Dirac points corre-spond to the edge states.

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We describe the ZGNR in the tight-binding approxima-tion. The Hamiltonian then reads as H=HGNR

0 +Hext+Hint,where

HGNR0 = − t �

�i,j�,�bj�

† ai� − t� ���i,j��,�

�ai�† aj� + bi�

† bj�� + H.c. �1�

describes the ribbon. Here, ai�† �bi�

† � creates an electron on aWannier orbital centered at site ri of the sublattice A �B� withspin �, t�2.8 eV and t��−0.1t are the nearest- and next-to-nearest-neighbor hopping parameters, respectively.4 Thesymbols � . . . � and �� . . . �� restrict the sum to the correspond-ing neighboring sites. The borders contain A sites on oneedge and B sites on the other. Hext, which describes the ac-tion of external gates, is defined below. Finally, Hint de-scribes the electron-electron interaction. Because of the highdensity of states induced by the edge states, the system ismagnetically unstable. Density functional theory andHartree-Fock calculations18,23 show that the ground state cor-responds to an antiferromagnetic ordering of the sublattices’magnetization. Since the latter is mainly localized at theedges, we take such interaction into account by introducingan effective magnetic field only at the edges sites,

Hint = − �BBa���

�a��† a�� − �BBb�

��

�b��† b��, �2�

where ���� labels the top �bottom� edge. We take this field tobe perpendicular to the plane of the ZGNR �z axis�. In theground state the two edges have opposite magnetizations,Bb=−Ba. The value of Ba should, in principle, be determinedby a self-consistent calculation. Since its precise value de-pends on the chemical passivation of the edges,24–27 and inorder to discuss different situations, we take it here as a freeparameter.28

Figure 1 shows the energy dispersion of a 32-ZGNR �Ref.29� for different values of Ba. Several bands originated fromthe quantization along the y axis are clearly visible. Thebands in the range kxa� �2� /3,4� /3� that are close to theDirac point, E3t�, are the ones that correspond to the edgestates with a characteristic localization length ��kx��−3a0 /2 ln2 cos�kxa /2�.17 Here, a=�3a0 is the lattice pa-rameter with a0 the C-C bond length. For Ba=0 �Fig. 1�b��,there are two of those bands �for each spin orientation� thatare almost degenerated; there is an exponentially small split-ting between them. They essentially correspond to the sym-metric and antisymmetric combinations of the exponentiallydecaying solutions of each individual edge.4,13,17 For Ba�0�Figs. 1�c�–1�e��, both the spatial and the spin degeneraciesare broken. For each spin orientation, each band now corre-sponds to states localized on a different edge. The energydispersion is approximately given by E�kx��3t�+ �t���BBa��2 coskxa+1�. Note that it is nonzero due to thenonzero value of either t� or Ba.17,30 The key point is to noticethat, for a given energy, the states with opposite spin polar-ization in the z direction are localized on opposite edges ofthe ZGNR.

Let us now consider the transport properties of a ZGNR inthe presence of an electrostatic potential created by externalgates,

Hext = �i,�

Vgf�xi��ai�† ai� + bi�

† bi�� , �3�

where f�x� is a smooth function describing a PW of height Vg�see Fig. 2�a��. For simplicity, we use a sum of Fermi func-tions, with the parameter playing the role of the tempera-ture, to set the spatial profile of f�x� �which depends only onx�. We assume that EF3t�0 far from the PW which en-sures that the current carrying states in that region are ex-tended throughout the entire width of the ribbon. On theother hand, Vg0 can be tuned in such a way that EF−Vgcorresponds to the energy of an edge state. For the sake ofsimplicity, we discuss first the conceptually simpler case Ba=0.30 Then, if f�x� changes smoothly, the electrons’ wavefunction will adiabatically change from extended to local-ized, while keeping its band index and having a positiondependent wave vector kx�x�. Correspondingly, the chargeflow will “split” in two paths inside the well and merge againafterwards, creating a “hole” in its spatial distribution �Fig.2�a��. In this way, we have created an interferometer, whichcan be tested by introducing a relative phase difference be-tween the two paths.

As the Aharonov-Bohm �AB� effect provides the simplestway to do this, we introduce a magnetic field B� perpendicu-lar to the ZGNR �via a Peierls substitution in the hoppings�and calculate the zero-temperature conductance using theLandauer approach.31 For that, we separate the system into acentral region �containing the PW� and the leads’ regions anduse the standard recursive method to obtain the latticeGreen’s functions31,32 and the transmission coefficient fromthem. Figure 2�b� shows the conductance G of a 32-ZGNRas a function of the Vg for different values of B�. It is ap-

0

0.5

1

1.5

2

2.5

3

-0.4 -0.35 -0.3 -0.25 -0.2 -0.15

G[2

e2/h

]

Vg/t

(b)

0.99

1

0 0.5 1 1.5 2 2.5 3 3.5 4

G[2

e2/h

]

B [T]

FIG. 2. �Color online� �a� Schematics of the proposed setup andthe potential profile �which depends only on x�. The two-path char-acter of the current flow allows for interference effects to manifest;�b� conductance of a 32-ZGNR �W=45a0� as a function of Vg fordifferent values of B� and EF=−0.6 t, L=2400a, =30a, and Ba

=Bb=0. G only changes in the presence of edge states. Inset: G asa function of B� for Vg / t=−0.384 ���, −0.356 ���, and −0.31 ���.

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parent that G changes with B� only when Vg is below thethreshold where the edge states participate on transport �in-dicated by the arrow�. The inset shows the oscillatory behav-ior of G as function of B� for three different values of Vg.The period is roughly �0 /A��1.3 T with �0 the flux quan-tum and A��LeffW with Leff��L−4�3.5�. An incrementof the period, due to the reduction in the effective “hole”area, is difficult to see since the visibility of the oscillationsis rapidly lost. In addition, and despite this seemly simplepicture, the behavior of the conductance is more involved asit shows pronounced narrow dips when B��0. This is re-lated to the fact that bonding and antibonding bands aremixed by B� �recall that for Ba=0 the splitting is exponen-tially small� and then both bands get involved in transport,which in turns leads to Fano-like interference between themand a reduction in the visibility of the AB oscillations.33

A more interesting situation occurs for Ba�0. As wementioned above, in this case, both the spatial and the spindegeneracies are broken. Therefore, an incoming electronwith its spin quantize along the z axis, will follow either theupper or lower path �colored arrows in Fig. 2�a�� dependingon whether its spin is “up” or “down.” Clearly, in this casethere is no interference and the conductance is independentof B�. Nevertheless, it can be readily verified that if theincoming electron is polarized in the n=cos x+sin ydirection—its spin state being denoted by n↑�—it will berotated

in� =z↑� + ei z↓�

�2→ out� =

z↑� + ei� +��z↓��2

, �4�

where � is the relative phase of the transmission amplitude ofthe two paths. Due to the symmetry of the setup, the spinprojection remains on the plane of the ZGNR. The probabil-ity for an electron to keep its spin orientation is cos2�� /2�and so we expect the conductance between two collinearferromagnetic leads34 to oscillate as a function of �. Note thatwe have assumed that L�Lcorr, where Lcorr is the spin-correlation length of the ferromagnetic order along eachedge.35

Figure 3 shows the spin-resolved transmission probabilityT�+ for an incident electron with spin +�= x↑� to be trans-mitted with spin �=� �in the same axis� as a function of Vgand B�. The relative phase of the two paths is �=2�� /�0,where �=B�Aeff is the magnetic flux enclosed by the currentflow and Aeff=LeffWeff is the effective area. For our geometry,the latter depends mainly on the effective width Weff�Vg�,which is a function of Vg through the energy dependence of��kx� �Weff�W�coth�W /��−� /W� for � /W�1�. As ex-pected, the transmission is a simple oscillatory function ofB�. Note that the shorter period corresponds to the maxi-mum effective area, �0 / �LeffW��1.5 T and that for Vg�Vg

��−0.5t �threshold for the participation of the edgestates� there are no oscillations. The total transmission T=T+++T−+ is constant, implying that the effect of the field isto produce a pure spin rotation. The features that are appar-ent in the figure for Vg�−0.54t, are related to the presenceof the “w-shaped” edge states band �see discussion below�. Itis worth pointing out that B� cannot be too large to avoid atransition to a ferromagnetic state �B��2 T �9 T� for a 32-ZGNR �16-ZGNR��.36

Interestingly enough, there is also a way to produce acontrolled spin rotation using an all-electrical setup. The keyis to change � by inducing a difference between the wavevectors of the two paths, and therefore changing their relativeplane-wave phase. This can be achieved by applying a smalltransverse electrical field that changes the energy of the twopaths, and then the wave vectors, in a small fraction and inopposite directions. Note that only a change �kx�2� /L isrequired. The transverse potential is described by adding aterm VT��yi−W /2�2 /W�f�xi� to Vgf�xi� in Eq. �3�. Figure 4shows the spin-dependent transmission for this setup. As forthe previous case, there are clear oscillations indicating therotation of the spin of the carriers, even for a very smalltransverse field ET=2VT /W ��2 �V /Å for VT�5�10−5t�.Again, the rotation disappears for Vg�Vg

�. The period of theoscillations is in good agreement with the estimated value

a)

-0.56 -0.54 -0.52 -0.5Vg/t

0

1

2

3

4

5

B⊥

[T]

b)

-0.56 -0.54 -0.52 -0.5Vg/t

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

0 1 2 3

Tra

nsm

issi

on

B⊥ [T]

c)

FIG. 3. �Color online� �a� Density plot of the spin-resolvedtransmission T++ as a function of the depth of the potential well Vg

and the perpendicular magnetic field B� for a 32-ZGNR and�BBa= t� /2, EF=−0.8t, L=2400a, and =30a; �b� same for T−+; �c�magnetic field dependence of T++ �open symbols�; and T−+ �filledsymbols� for Vg / t=−0.53 ��,��, −0.515 ��,��, and −0.51 ��,��.

a)

-0.56 -0.54 -0.52 -0.5Vg/t

0

4

8

12

16

VT

/tx

10

-5

b)

-0.56 -0.54 -0.52 -0.5Vg/t

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

0 4 8 12 16

Tra

nsm

issi

on

VT /t x 10-5

c)

FIG. 4. �Color online� �a� and �b� Same as Fig. 3 but as afunction of the transverse electrostatic potential VT. �c� Transverseelectric field dependence of T++ �open symbols� and T−+ �filledsymbols� for Vg / t=−0.544 ��,��, −0.53 ��,��, and −0.515 ��,��.

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�= �kx+−kx

−�Leff where kx� is the wave vector of the edge states

with energy EF−Vg+��VT� and �VT� is the average value ofthe transverse potential in the corresponding edge state��VT��VT�coth�Na /��−� /Na� for � /W�1�.

Adiabatic transport is not possible for �BBa� t� as elec-trons reach a point where vg�0 before they penetrate thewell and are then reflected. However, transport is still pos-sible due to a resonant mechanism that involves the upper“w-shaped” edge states band �Fig. 1�. This involves aLandau-Zener-like transition between bands so that the widthof the resonances increases as the potential profile is moreabrupt. Some of those resonances are already apparent inFigs. 3 and 4. We note that in Fig. 3, they present a period of2�0 /Aeff. This is also present when the incident electron hasits spin direction in z, where we would have naively expectedno dependence with B� as the electrons in that case follow asingle path. This is, however, not true as each minimum ofthe “w-shaped” band involves edge states for electrons mov-ing in one direction but extended states for those moving in

the opposite direction— another unique characteristic of theZGNR band structure. The phase difference is then related tohalf the area of the PW. A detailed analysis33 shows that thespin rotation is still possible for some of the resonances,showing that the effect is robust against the precise value ofBa.

In summary, we showed that ZGNRs present interestinginterference phenomena in the presence of a PW. Moreover,the spin-dependent structure of the edge states allows for acontrolled rotation of the spin of the carriers by either mag-netic or electric fields. Since the characteristic of the zigzagtermination seems to be generic37 and robust againstdisorder,19 we expect these effects to manifest in less idealsamples, opening an alternative for spintronics in graphene.

We thank C. A. Balseiro for useful discussions and P. S.Cornaglia for a careful reading of the manuscript. We ac-knowledge financial support from ANPCyT Grant No.483/06 and CONICET PIP Grant No. 5254/05.

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