Edge channels in graphene Fabry-P erot interferometer S. Ihnatsenka Department of Science and Technology Link oping University SE-60174 Norrk oping Sweden

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Edge channels in graphene Fabry-P´erot interferometer

S. Ihnatsenka

Department of Science and Technology, Link¨oping University, SE-60174, Norrk¨oping, Sweden∗

Quantum-mechanical calculations of electron magnetotransport in graphene Fabry-P´erot inter-

ferometers are presented with a focus on the role of spatial structure of edge channels. For an

interferometer that is made by removing carbon atoms, which is typically realized in nanolithogra-

phy experiments, the constrictions are shown to cause strong inter-channel scattering that establishes

local equilibrium and makes the electron transport non-adiabatic. Nevertheless, two-terminal con-

ductance reveals a common Aharonov-Bohm oscillation pattern, independent of crystallographic

orientation, which is accompanied by single-particle states that sweep through the Fermi energy for

the edge channels circulating along the physical boundary of the device. The interferometer con-

strictions host the localized states that might shorten the device or disrupt the oscillation pattern.

For an interferometer that is created by electrostatic conﬁnement, which is typically done in the

split-gate experiments, electron transport is shown to be adiabatic if the staggered potential is in-

troduced additionally into the model. Interference visibility decays exponentially with temperature

with a weaker dependence at low temperature.

PACS numbers: 72.80.Vp, 73.43.-f, 85.35.Ds, 73.23.-b

I. INTRODUCTION

Quantum Hall interferometers that operate on the

Aharonov-Bohm eﬀect have recently been demonstrated

in graphene, with high visibility and no Coulomb charg-

ing eﬀects.1,2 This suggests graphene-based interferom-

eters as a better platform for studying the exchange

statistics of anionic quasi-particles3in comparison to the

traditional GaAs-based counterpart.4,5 The conductance

oscillations that were measured in Refs. 1,2 were well-

described by a theoretical model that is based on an as-

sumption of idealized one-dimensional channels circulat-

ing along the edges of the device in the quantum Hall

eﬀect (QHE) regime.5–8 While good agreement between

experiments and the theory seemingly validates the cho-

sen model, or at least does not disprove it, the lack of

the spatial structure of the edge states and disregard for

electron scattering at the constriction regions in the phe-

nomenological modeling leaves an open question about

the physical mechanisms behind the electron interfer-

ence in the studied devices. The problem is evidenced

by strong electron scattering that occurs at the graphene

interfaces (i.e., the regions where either device size or

crystollographic orientation changes) that has been ob-

served in graphene nanoribbons,9constrictions10–12 and

other structures.13,14 As was already pointed out in Ref.

1, ”quantum Hall interferometer experiments require a

precise knowledge of the edge-channel conﬁguration”.

Therefore, getting this knowledge, particularly due to

massless Dirac fermions in the QHE regime, is necessary

for both the interpretation of the interferometry experi-

ments and for the foundation of theories such as those in

Refs. 1,2 and 5–7.

Previous studies of mesoscopic graphene devices

operating in the QHE regime have addressed en-

ergy structure, electronic states and transport in

nanoribbons,10,15–18 p-n heterojunctions,19,20 rings,21

and others.13,22,23 These studies have evidenced the ex-

istence of edge states,8which ﬂow in only one direction

along the physical edge of the sample. Edge states ﬂow-

ing in an opposite direction exist at the opposite edge,

and it is the absence of scattering between these two

edges that constitutes the fundamental reason for the

robustness of the quantization of QHE.24 In graphene,

the relativistic nature of charge carriers manifests in the

so-called anomalous QHE with Landau level (LL) present

at zero energy, which separates states with hole character

from states with electron character.25,26 The edge states

with the same index of propagating mode, following the

standard terminology,27 are referred to in this study as

an edge channel.

This manuscript will provide a microscopic theory of

edge channel transport in a graphene interferometer op-

erating on the Aharonov-Bohm eﬀect and will also eluci-

date the role of the spatial structure of the edge states in

electron quantum interference. To this end, the tight-

binding model of graphene placed in a perpendicular

magnetic ﬁeld is employed for numerical quantum trans-

port calculations. The interferometer’s geometry is cre-

ated from an inﬁnite graphene nanoribbon, either by re-

moving carbon atoms or by electrostatic conﬁnement in

such a way that a square central region is formed be-

tween two narrow constrictions, similarly to the Fabry-

P´erot device;1,2 see the inset in Fig. 1(b). These two

types of lateral conﬁnement correspond to a fabrication

technique based on etching nanolithography12,16,28 and

split-gates.1,2 For both cases, quantum transport calcu-

lations reveal a common Aharonov-Bohm (AB) interfer-

ence pattern1,2,4–6,20,24,27,29 in conductance, which is due

to edge channels circulating along device physical bound-

aries and scattered at the constrictions. Every conduc-

tance peak corresponds to the single-particle state sweep-

ing through the Fermi energy. Conductance oscillations

are independent of the crystallographic orientation of the

graphene lattice. In contrast to traditional GaAs-based

devices, where electron transport in the QHE regime is

arXiv:2210.15036v3 [cond-mat.mes-hall] 21 Oct 2023

adiabatic,4–8,24,27,29 AB interference in graphene inter-

ferometers that is made by removing carbon atoms oc-

curs because the edge states propagate non-adiabatically

and equilibrate locally at the constrictions (interface re-

gions). Relatively strong conﬁnement for Fermi electron

gas in graphene is found to cause electron localization

along the constriction and might cause a short circuit

or deviation in the AB interference signal. In the case

of electrostatic conﬁnement, for a device of the same

geometry and the staggered potential model, transport

is adiabatic, similar to that found in traditional GaAs-

based interferometers.4,5,7,29 The partial penetration of

the electron wave function into the potential barriers can

result in out-of-phase oscillations of the edge channels.

Interference visibility decays overall exponentially with

T, with weaker dependence at low T, in agreement with

recent experiments.1,20

This manuscript is organized as follows. The theo-

retical model is formulated in Sec. II. The results are

presented together with their interpretation and implica-

tions for experiment in Sec. III. The main conclusions

are summarized in Sec. IV.

II. MODEL

The model is based on the standard nearest-neighbor

tight-binding Hamiltonian on a honeycomb lattice

H=X

ϵia†

iai−X

⟨i,j⟩

tij (a†

iaj+ H.c.) (1)

where ϵiis the on-site energy, a†

i(ai) is the creation (de-

struction) operator of the electron on the site iand the

angle brackets denote the nearest neighbour indices. The

magnetic ﬁeld, B, is included via Peierls substitution

tij =texp(i2π

Φ0Zrj

A·dr),(2)

where A=B(−y, 0,0) is the vector potential in the Lan-

dau gauge, riis the coordinate of the site i, Φ0=h/|e|

is the ﬂux quantum, t= 2.7 eV. Hamiltonian (1) with

ϵi= 0 is known to describe the π-band dispersion

of graphene well at low energies,30 and has been used

in numerous studies of electron transport in graphene

nanostructures.13,14,17,21,26,31

Eﬀects due to the next-nearest neighbor hopping, spin,

electron-electron interactions are outside of the scope of

this study.

The Green’s function of the system connected at its

two ends to the semi-inﬁnite leads is written as24

G(ϵ)=[Iϵ −H−ΣL(ϵ)−ΣR(ϵ)]−1.(3)

Here, Hdescribes the scattering region that includes the

interferometer itself and a part of the leads, Iis the uni-

tary operator, ΣL(ϵ) is the self-energy due to the semi-

inﬁnite left lead at electron energy ϵ, and ΣR(ϵ) is sim-

ilarly for the right lead. The lead self-energies are ob-

tained from the surface Green’s functions by the method

given in Ref. 32. The system is supposed to be whole

graphene made, including the leads.

Having G(ϵ) calculated allows one to obtain ob-

servable quantities, like density of states (DOS) and

conductance.24 The local density of states (local DOS)

for the i-th site is given by the diagonal elements of the

Green’s function as

ρi(ϵ) = −1

πIm[Gii(ϵ)].(4)

The two-terminal (Hall) conductance Gof the system is

obtained from Landauer-B¨uttiker formula, which relates

conductance to the scattering properties of the system33

G=2e2

βα

tβα =2e2

βα

vβ

vα|sβα|2,(5)

where tβα is the transmission coeﬃcient from incoming

state αin the left lead to outgoing state βin the right

lead, sβα is the corresponding scattering amplitude, vα

and vβare the group velocities for those states, all at the

Fermi energy ϵF.sβα is obtained from the Green’s func-

tion that connects the ﬁrst and last slices of the scattering

region, see Appendix B. Another quantity of interest is

the probability of electron density |Ψα|2(the wave func-

tions modulus), which is obtained from the wave func-

tions in the leads, sβα and the Green’s function (3) by

applying Dyson equation as described in Appendix B.

III. RESULTS

The system studied is a graphene interferometer that

is made from a nanoribbon in an armchair or zigzag con-

ﬁguration by trimming (etching) carbon atoms away or

by applying electrostatic potential, see the inset in Fig.

1. Two (identical) constrictions deﬁne the central re-

gion similarly to an open quantum dot. For simplic-

ity, the results are presented for rectangular shaped con-

strictions; a smooth constriction will be commented on.

Adopting zigzag and armchair terminology from under-

lying nanoribbon structure, the interferometer is below

referred to as either a zigzag or armchair. The opera-

tion regime is chosen to support three channels for elec-

tron propagation within which electron can interfere. For

50 nm wide ribbon, which serves as an electron reser-

voir for the channels, this is achieved at ϵF= 0.2tand

B= 155..180 T.34 Appendix A elaborates on the prop-

agating states in the chosen regime. The temperature is

T= 0 K unless otherwise stated.

A. Edge channel interference

Numerical calculation of quantum electron transport,

described by Eqs. (1)-(5), reveals conventional conduc-

tance oscillations4,5,24,27,29 in a graphene AB interferom-

eter, irrespective of crystallographic orientation; see Figs.

G (2e2/h)

-10

-5

0.1

DOS

(arb.units)

(a) (b)

zigzag armchair

160 170 180

B (T)

(nm)

B (T)

(10-3 t)

(e) (f)

t2t3

FIG. 1: Conductance G, single-particle state spectroscopy

and edge channel displacement ξin graphene interferometers

with zigzag (left-hand panels) and armchair (right-hand pan-

els) orientation. (a,b) Goscillates as a function of magnetic

ﬁeld Bwith peaks matching the crossings of the resonant en-

ergy levels and the Fermi energy ϵFin (c,d). The energy levels

can be traced by enhanced DOS in (c,d), which is obtained by

integrating local DOS over the central region of interferome-

ter (a dot in between of two constrictions). Another set of the

energy levels, with the positive slope, is due to the states lo-

calized at the constrictions. The geometrical area of the dot,

reduced by ξ, whose evolution is plotted in (e,f), deﬁnes an

interfering area that is enclosed by the clockwise propagating

edge channel as illustrated in the inset between (a) and (b).

The dot geometry is a square with sides W= 50 nm. The

arrows in (a) mark Bfor which the edge states are shown in

Figs. 2 and 3. The dashed line in (e,f) is the magnetic length

lB; blue ﬁlled area denotes the standard deviation.

1(a),(b). The peaks in Gcorrespond to the resonant

states passing through ϵF, similar to what was found in

Refs. 5,29. These resonant states can be traced in Figs.

1(c),(d) as bright trenches with a negative slope. Slop-

ing downward with increasing Bimplies that degeneracy

of the occupied LLs increases via their edge states, pro-

vided by the geometry conﬁnement that makes LLs to

rise in energy on approaching the sample boundary.24,27

Downward sloping corroborates AB regime of interfer-

ence, as opposed to Coulomb dominated regime.4,7 For

each LL, the degree of degeneracy is quantiﬁed by the

number of states per unit area, which increases as B/Φ0.

Every resonant state is a result of the constructive in-

terference of the electron wave in the edge channels that

are backscattered at the two constrictions. The diﬀer-

ence between phases in two arms of the interferometer is

proportional to the total ﬂux Φ enclosed by the area S

of the interfering path;24 Φ = BS. Changing Φ by Φ0,

via applied B, causes the phase diﬀerence to accumulate

a value of 2πand Gto develop one oscillation period.5,24

(a)

(c)

(e)

B=164 T, G=1.84 G0

1231

B=165 T, G=2.72 G0

| 1|2

| 3|2

| 2|2

0 1

(b)

(d)

(f)

FIG. 2: The wave function modulus |Ψα|2of α-th incoming

state for Bmarked by the red arrows in Fig. 1(a). The left-

hand (right-hand) panels represent the minima (maxima) of G

oscillation. In (a), the arrow and cross illustrate schematically

the incoming state and forward scattering to another states

that occurs at the constriction bend. The insets in (c,f) show

the transmission coeﬃcients from incoming αto outgoing β

state. G0= 2e2/h.

One period, ∆B= 2.22 and 2.16 T for zigzag and arm-

chair conﬁguration, yeilds area, S= 1872 and 1924 nm2,

that is less than the geometrical area of the central re-

gion 2500 nm2. This discrepancy might be attributed

to a ﬁnite spatial extent of the edge channel, so the in-

terfering area is smaller then the geometrical one. Figs.

1(e),(f) substantiate this argument by showing an aver-

aged edge channel distance from the physical boundary

ξ= [⟨r|ψα⟩]α,x, where averaging is done over channels

and the straight segments along the boundaries. ξis

about the magnetic length lBand, interestingly, doesn’t

reveal any clear beat of ∆Bas it was argued to occur

in the Coulomb dominated regime.7Slightly larger ξfor

zigzag orientation, in comparison to armchair one, ex-

plains slightly smaller Sand larger ∆B. Transmission

coeﬃcients (dimensionless) for individual edge channels

tα, as shown in Fig. 1(a), reveal in-phase oscillations

of nearly equal amplitude. Therefore, Goscillations in

graphene AB interferometer are due to simultaneous in-

terference of the edge channels propagating at about lB

distance from the device boundaries.

Because zigzag and armchair interferometers reveal

qualitatively similar dependencies for Gand the struc-

ture of energy levels, below only zigzag conﬁguration is

considered.

Figure 2 shows the edge states characterizing conduc-

tance oscillation at its peak and dip values. The de-

tails on the electronic states entering and leaving the in-

terferometer, which are at the openings in these plots,

are given in the Appendix A. Edge channel visualization

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EdgechannelsingrapheneFabry-P´erotinterferometerS.IhnatsenkaDepartmentofScienceandTechnology,Link¨opingUniversity,SE-60174,Norrk¨oping,Sweden∗Quantum-mechanicalcalculationsofelectronmagnetotransportingrapheneFabry-P´erotinter-ferometersarepresentedwithafocusontheroleofspatialstructureofedgechannels....

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Edge channels in graphene Fabry-P erot interferometer S. Ihnatsenka Department of Science and Technology Link oping University SE-60174 Norrk oping Sweden

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