PHYSICS

Creating and probing electronwhispering-gallery modesin grapheneYue Zhao,1,2* Jonathan Wyrick,1* Fabian D. Natterer,1* Joaquin F. Rodriguez-Nieva,3*Cyprian Lewandowski,4 Kenji Watanabe,5 Takashi Taniguchi,5 Leonid S. Levitov,3

Nikolai B. Zhitenev,1† Joseph A. Stroscio1†

The design of high-finesse resonant cavities for electronic waves faces challenges due toshort electron coherence lengths in solids. Complementing previous approaches to confineelectronic waves by carefully positioned adatoms at clean metallic surfaces, wedemonstrate an approach inspired by the peculiar acoustic phenomena in whisperinggalleries. Taking advantage of graphene’s gate-tunable light-like carriers, we createwhispering-gallery mode (WGM) resonators defined by circular pn junctions, induced by ascanning tunneling probe. We can tune the resonator size and the carrier concentrationunder the probe in a back-gated graphene device over a wide range. The WGM-typeconfinement and associated resonances are a new addition to the quantum electron-opticstoolbox, paving the way to develop electronic lenses and resonators.

Charge carriers in graphene exhibit light-like dispersion resembling that of electro-magneticwaves. Similar to photons, electronsin graphene nanostructures propagate bal-listically over micrometer distances, with

the ballistic regime persisting up to room tem-peratures (1). This makes graphene an appeal-ing platform for developing quantum electronoptics, which aims at controlling electronwaves ina fully coherent fashion. In particular, gate-tunableheterojunctions in graphene can be exploited to

manipulate electron refraction and transmissionin the sameway that optical interfaces inmirrorsand lenses are used to manipulate light (2). Theseproperties have stimulated ideas in optics-inspiredgraphene electronics. First came Fabry-Pérot in-terferometers (3), which have been fabricatedin planar npn heterostructures in single-layergraphene (4) and subsequently in bilayer (5)and trilayer graphene (6). The sharpness of thepn junctions achievable in graphene can enableprecise focusing of electronic rays across the junc-

tion, allowing for electronic lensing and hyper-lensing (7–9).We report on electron whispering-gallerymode

(WGM) resonators, an addition to the electron-optics toolbox. TheWGMresonances are familiarfor classical wave fields confined in an enclosedgeometry—as happens, famously, in the whisper-ing gallery of St. Paul’s Cathedral. The WGM res-onators for electromagnetic fields are widely usedin a vast array of applications requiring high-finesse optical cavities (10–12). Optical WGMresonators do not depend on movable mirrorsand thus lend themselves well to designs witha high quality factor. This can render the WGMdesign advantageous over the Fabry-Pérot design,despite challenges in achieving tunability due totheir monolithic (single-piece) character [see (12)for a mechanically tunable optical WGM resona-tor]. Our system is free from these limitations,representing a fully tunable WGM resonator inwhich the cavity radius can be varied over a widerange by adjusting gate potentials. In contrast, thebest electronic resonators known to date—thenanometer-sized quantum corrals designed bycarefully positioning adatoms atop a clean me-tallic surface (13)—are not easily reconfigurable.

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1Center for Nanoscale Science and Technology, NationalInstitute of Standards and Technology, Gaithersburg, MD20899, USA. 2Maryland NanoCenter, University of Maryland,College Park, MD 20742, USA. 3Department of Physics,Massachusetts Institute of Technology, Cambridge, MA02139, USA. 4Department of Physics, Imperial CollegeLondon, London SW7 2AZ, UK. 5National Institute forMaterials Science, Tsukuba, Ibaraki 305-0044, Japan.*These authors contributed equally to this work. †Correspondingauthor. E-mail: nikolai.zhitenev@nist.gov (N.B.Z.); joseph.stroscio@nist.gov (J.A.S.)

pn

R

Vb

Vg

d

U(r)µ∞

µ0Weakly Confined Strongly Confined

–1 1Re(ψB)

Fig. 1. Confined electronic states in microscopic electron cavitiesdefined by pn junction rings in graphene. (A) The rings are induced by theSTM tip voltage bias (Vb) and back-gate voltage (Vg), adjusted so as to reversethe carrier polarity beneath the tip relative to the ambient polarity. The pnjunctions act as sharp boundaries giving rise to Klein scattering of electronicwaves, producing mode confinement via the whispering-gallery mechanism.The cavity radius and the local carrier density are independently tunable by thevoltages Vb and Vg. Electron resonances are mapped out by the STM spec-troscopymeasurements (see Fig. 2). Shown are the STM tip potentialU(r) and

the quantities discussed in the text: the STM tip radius (R), its distance fromgraphene (d), and the local (m0) and ambient (m1) Fermi levels with respect totheDirac point. n andp label the electron and hole regions. (B) Spatial profile ofWGM resonances. Confinement results from interference of the incident andreflected waves at the pn rings (dashed lines).The confinement is stronger forthe larger angular momentumm values, corresponding to more oblique waveincidence angles.This is illustrated for m = 5 (weak confinement) and m = 13(strong confinement). Plotted is the quantity ReðyBÞ, the real part of thesecond spinor component in Eq. 1.

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Further, although WGM resonators are ubiq-uitous in optics and acoustics (10–12, 14), only afew realizations of such resonators were obtainedin nonoptical and nonacoustic systems. Theseinclude WGM for neutrons (15), as well as forelectrons in organic molecules (16). In our measure-ments a circular electron cavity is created be-neath the tip of a scanning tunnelingmicroscope(STM), and we directly observe the WGM-typeconfinement of electronic modes. The cavity is de-fined by a tip-induced circular pn junction ring,at which the reflection and refraction of electron

waves are governed by Klein scattering (Fig. 1).Klein scattering originates from graphene’s lin-ear energy dispersion and opposite group veloc-ities for conduction and valence band carriers;Klein scattering at a pn junction features a strongangular dependence with a 100% probability fortransmission at normal incidence, as well as fo-cusing properties resembling negative refractiveindexmetamaterials (2, 7). AlthoughKlein scatteringis characterized by perfect transmission and noreflection for normal incidence, it gives rise tonearly perfect reflection for oblique incidence

occurring in the WGM regime (2). As illustratedin Fig. 1B, this yields excellent confinement andhigh-finesse WGM resonances for modes withhigh angular momentum m and a less perfectconfinement for non-WGM modes with lowerm values.Electron optical effects in graphene have so far

been explored using transport techniques, whichlack spatial and angular resolution that wouldbe indispensable for studying confined electronicstates and/or electron lensing. Our scanning probetechnique allows us to achieve nanometer-scalespatial resolution. The STM probe has a dualpurpose: (i) creating a local pn junction ring,which serves as a confining potential for electronicstates, and (ii) probing by electron tunneling theresonance states localized in this potential. Theplanar back gate and the STM tip, acting as acircular micrometer-sized top gate, can changeboth the overall background carrier density andthe local carrier density under the tip. As such,pn and np circular junctions centered underthe probe tip (Fig. 1A) can be tuned bymeans ofthe tip-sample bias Vb and the back-gate voltageVg [see fig. S4 (17)]. For the purpose of creatingresonant electronic modes inside the junction,this configuration gives us in situ, independentcontrol over the carrier concentration beneaththe STM tip and the pn ring radius. The tunnel-ing spectral maps from such a device show aseries of interference fringes as a function ofthe knobs (Vb,Vg) (Fig. 2). These fringes origi-nate from resonant quasi-bound states insidethe pn ring.The measured spacing between fringes (De)

can be used to infer the cavity radius (r). Usingthe formula De ¼ pℏvF=r (ℏ, Planck’s constant hdivided by 2p; vF ≈ 106 m/s) and an estimatefrom Fig. 2A (De ≈ 40meV), we obtain r ≈ 50 nm,a value considerably smaller than the STM tipradius (R ≈ 1 mm). This behavior can be under-stood froma simple electrostaticmodel of a chargedsphere proximal to the ground plane. When thesphere-to-plane distance d is small comparedwiththe sphere radiusR, the induced image charge den-sity cloud r(r) behaves as rðrÞº1=ðd þ r2=2RÞ,predicting a length scale

ffiffiffiffiffiffiffiffiffi2Rd

p≪ R. This crude

estimate is upheld, within an order of magni-tude, by a more refined electrostatic modeling(17), which also gives a length scalemuch smallerthan R.The experimental results were obtained on a

device consisting of a graphene layer on top ofhexagonal boron nitride, stacked on SiO2 with adoped Si back gate [see supplementarymaterialsfor details (17)]. Figure 2A shows a tunnelingconductance map as a function of back-gate volt-age (Vg) on the horizontal axis and sample bias(Vb) on the vertical axis. A series of interference-like fringes forming a curved fan (labeledWGM′)can be seen in the upper right of Fig. 2A. Thecenter of the fan defines the charge neutralitypoint. This point can be off (0,0) in the (Vg,Vb)plane due to impurity doping of graphene (shiftalong Vg) and the contact potential differencebetween the probe tip and graphene (shift inVb).As illustrated in fig. S5 (17), we are able to shift

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WGM'

WGM"

WGM'

WGM"

400

200

-400

-200

0

400

200

-400

-200

0

Vb

(mV

)dl

/dV

b (n

S)

dl/dVb (nS) dl/dVb (arb. units)

dl/d

Vb

(nS

)

Vg (V)

Vg (mV) Vb (mV)

Vg (V)

Vb

(mV

)

-40 -20 20 400 -40 -20 20 400

0 0.8

1'

2'

3'4'

Vb = 230 mV

Vg = –11 V

Vg = 16 V

x3

1"

1"

2"

2"3"

4'

3'

2'

1'

0.5

0.4

0.3

0.2

0.1-40 -20 0 20 40 -200 0 200 400

0.0

0.2

0.8

0.6

0.4

Fig. 2. Confined electronic states probed by STM measurements. (A) Differential tunnelingconductance (dI/dVb) for a single-layer graphene device, as a function of sample bias (Vb) and back-gate voltage (Vg).The gate map was obtained after increasing the probe-tip work function by exposure todeuterium to shift the interference fringes vertically downward (fig. S5) (17). The two fans of interferencefeatures, marked WGM′ and WGM′′, originate from WGM resonances in the DOS (see text). (B)Interference features in dI/dVb, calculated from the relativistic Dirac model. The features WGM′ andWGM′′ in the (Vg,Vb)map originate, respectively, from the conditions e = m0 and e = m0 + eVb (see text).Theboundaries of theWGM′ (andWGM′′) regions aremarked by dashed (and dotted) white lines, respectively.arb. units, arbitrary units. (C) dI/dVb spectra taken along the horizontal line in (A) at Vb = 230 mV. (D)dI/dVb spectra taken along the twovertical lines in themap in (A) atVg = 16V (red line) andVg =–11 V (blueline, scaled ×3 and offset for clarity) (see text for discussion).The four peaks at positive bias at Vg = 16 Varefit to Gaussian functions, with the fits shown in the lower right of the figure. The peaks labeled 1′′,2′′,3′′…correspond toWGM resonances probed at the energy e = m0 + eVb, whereas the peaks labeled 1′,2′,3′…, arethe sameWGMresonances probed at the Fermi level e = m0, giving rise to theWGM′′ andWGM′ fringes in thegate maps, respectively. The resonance spacing of order 40 mV translates into a cavity radius of 50 nm,using the relation De ¼ pℏvF=r (see text).

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the center point of the fan to lower Vb values bychanging the tip work function, for example,with D2 adsorption (18). Another interesting fea-ture in such conductance maps is a (some-what less visible) second fan of fringes (labeledWGM′′), which is crossing the primary WGM′fan. The fringes in the WGM′′ fan follow thetypical graphene dispersion with respect to theFermi energy,which varieswith doping asº

ffiffiffiffiffiffiffiffijVgjpfrom higher sample bias to lower as a functionof Vg. Examining the primary (WGM′) and sec-ondary (WGM′′) fringes more closely confirmsthat they originate from the same family of WGMresonances.Figure 2C shows nine oscillations in a line cut

across the WGM′ fan along the Vg axis. To un-derstand the origin of these oscillations, we ex-amine the two spectral line cuts along theVb axisin Fig. 2D. The first spectrum in Fig. 2D at Vg =–11 V (blue curve) contains a group of resonances(labeled 1′′ to 3′′) near the Fermi level (Vb = 0)with a spacing of 37.6 T 1.2 mV (19). In the mapin Fig. 2A, these resonances can be seen tomoveto lower energies approximately following thetypical Dirac point dispersion º

ffiffiffiffiffiffiffiffijVgjp . Takinga vertical cut at a higher back-gate voltage ofVg =16 V (red curve) shows resonances 1′′ and 2′′shifted down in energy in Fig. 2D. Focusing nowat slightly higher energies, theWGM′ resonancesappear at positive energies in Fig. 2D and arelabeled 1′ to 4′ for Vg = 16 V. These four reso-nances are fit to Gaussian functions and showndeconvolved from the background conductancein the bottom right of the figure. The averagespacing of these resonances is 116.9 T 7.5 mV (19).A close examination of Fig. 2A indicates the one-to-one correspondence between the WGM′′ reso-nances 1′′, 2′′,… and theWGM′ resonances 1′, 2′…,suggesting their common origin. We thereforeconclude that the WGM′′ resonances corre-spond to tunneling into the pn junction modesat energy e ¼ m0 þ eVb [m0, local Fermi level (seeFig. 1A)], whereas the WGM′ resonances reflectthe action of the STM tip as a top gate, allowingtunneling into the same resonance mode at e =m0 [see fig. S3 (17)]. For example, resonance 1′′seen at Vb ≈ –100 mV is now accessible at theFermi level by the tip-graphene potential differ-ence, as shown in fig. S3D (17), when tunnelinginto the WGM′ resonance 1′ at Vb = 82 mV inFig. 2A.To clarify the WGM character of these reso-

nances, we analyze graphene’s Dirac carriers inthe presence of a potential induced by the STMtip described by theHamiltonianH ¼ H0 þ UðrÞ,where H0 is the kinetic energy term and U(r)describes the STM tip potential seen by chargecarriers. Because relevant length scales—the elec-tron’s Fermiwavelength and thepn ring radius−aremuch greater than the atomic spacing, we fo-cus on the low-energy states. We linearize thegraphene electron spectrum near the K and K′points, bringingH0 to the massless Dirac form:eyðrÞ ¼ ½vFs ⋅ pþ U ðrÞ�yðrÞ, where p ¼ −iℏ∇rand s ¼ ðsx; syÞ are pseudospin Pauli matrices.We take the tip potential to be radially sym-metric, reflecting the STM tip geometry. Fur-

thermore, the distance from the tip to graphene(d) is considerably smaller than the electron’sFermiwavelength and the pn ring radius, both ofwhich are smaller than the STM tip radius. Wecan therefore use a parabola to approximate thetip potential, U ðrÞ ¼ kr2 (r, off-center displace-ment). The curvature k, which affects the energyspectrumofWGMresonances, can be tunedwiththe bias and gate potentials, as discussed in thesupplementary materials (17).The WGM states with different angular mo-

mentum can be described by the polar decom-position ansatz

ymðr; fÞ ¼1ffiffiffir

p uAðrÞeiðm−1ÞfuBðrÞeimf

� �ð1Þ

wherem is an integer angular momentum quan-tum number, f is the polar angle, and A, B labelthe two graphene sublattices. We nondimension-alize the Schrödinger equation using the charac-teristic length and energy scales (r� ¼

ffiffiffiffiffiffiRd

p,

e� ¼ ℏvF=ffiffiffiffiffiffiRd

p) to obtain the radial eigenvalue

equation of the two-component spinor u(r) withcomponents uA(r) and uB(r)

euðrÞ ¼ −isx∂r þmþ 1=2r

sy þ kr2� �

uðrÞ ð2Þ

Here r is in units of r�, e is in units of e�, andk is in units of k� ¼ e�=r2�. This equation issolved using a finite difference method [seesupplementary material (17)]. We can use this

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Dtotal

D1(ε)

D2(ε)

D3(ε)

D4(ε)

D5(ε)

Re(ψB)

r∗

-10 -5 0 5 10

DO

S (

arb.

uni

ts)

DO

S (

arb.

uni

ts)

–1 1

-4 -2 0 2 4

10

0

-10

Fig. 3. Contributions of the WGM resonances with different m to the DOS for the relativistic Diracmodel. (A) Colored curves represent partial-m contributions fromangularmomentumvaluesm= 1,2,3,4,5(see Eq. 3), evaluated for a confining potential UðrÞ ¼ kr2 with curvature value k ¼ k� ¼ e�=Rd. Differentcurves show the partial DOS contributions defined in Eq. 3, which are offset vertically for clarity.The insetshows the total DOS versus particle energy e and the curvature k (see text). The black curve shows thetotal DOS trace along the white line. (B) The Dirac wavefunction for different WGM states (see Eq. 1).Spatial structure is shown for several resonances in the partial DOS (black dashed circles mark the pnjunction rings).The quantity plotted, ReðyBÞ, is the same as in Fig. 1B.The length scale r� ¼

ffiffiffiffiffiffiRd

p(the same

in all panels) is marked. Note the confinement strength increasing with m.

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microscopic framework to predict the mea-sured spectral features. The tunneling current,expressed through the local density of states(DOS), is modeled as

I ¼ ∫m0þeVbm0

deTðe;VbÞ∑mDmðeÞ;

DmðeÞ ¼∑r;n

e−lr2=2junðrÞj2dðe − enÞ ð3Þ

which is valid for modest Vb values (20). Here m0is the Fermi energy under the tip, which in gen-eral is different from ambient Fermi energy m∞ asa result of gating by the tip (see below). Thetransmission function T(e,Vb), which depends onthe tip geometry, work function, and DOS, willbe taken as energy-independent. The quantityDðeÞ ¼ ∑mDmðeÞ represents the sum of partial-mcontributions to the total DOS beneath the tip,with n labeling eigenstates of Eq. 2 with fixed m.The weighting factor e−lr

2=2 is introduced to ac-count for the finite size of the region wheretunneling occurs, with the Gaussian halfwidthl−1=2 e r� [see supplementary materials (17)].The WGM resonances for different partial-m

contributionsDmðeÞ, which combine into the totalDOS (Fig. 3), reveal that individual WGM statesexhibit very different behavior depending on them value [see Figs. 1B and 3B]. Klein scattering atthe circular pn junction produces confinement cre-ating the WGMs, and the confinement is strongerfor the large-m modes and weaker for small-mmodes. The Klein reflection probability is stronglydependent on the angle of incidence q at the pninterface, growing as RðqÞ∼1 − exp½−xsin2ðqÞ�,where x is a characteristic dimensionless param-eter (21). The value of q growswithm as tan qºm.As a result, larger values of m must translate intolarger reflectivity and stronger confinement. Thistrend is clearly demonstrated in Figs. 1B and 3B.Also, as m increases, mode wavefunctions arebeing pushed away from the origin, becomingmore localized near the pn ring, in full accordwith the WGM physics.To understand how one family of WGM res-

onances gives rise to two distinct fans of inter-ference features seen in the data, wemust accountcarefully for the gating effect of the STM tip. Westart with recalling that conventional STM spec-troscopy probes features at energies en ¼ m0 þ eVb,where en are system energy levels. This correspondsto the family WGM′′ in our measurements. How-ever, as discussed above, the tip bias variationcauses the Fermi level beneath the tip to movethrough system energy levels en, producing anadditional family of interference features (WGM′)described by en ¼ m0 [see fig. S3 and accompany-ing discussion (17)]. To model this effect, weevaluate the differential conductance G = dI/dVbfrom Eq. 3, taking into account the dependencem0 versus Vb. This gives

Gºð1 − hÞDðm0 þ eVbÞ þ hDðm0Þ ð4Þwith h ¼ −∂m0=∂ðeVbÞ. The two contributions inEq. 4 describe the WGM′ and WGM′′ families.We note that the second family originates fromthe small electron compressibility in graphene,resulting in a finite h andwould not show up in a

system with a vanishingly small h (e.g., in a me-tal). We use Eq. 4 with a value h = 1/2 to generateFig. 2B. In doing so, we use electrostatic mod-eling [described in (17)] to relate the parameters(e,k) in the Hamiltonian, Eq. 2, and the experi-mental knobs (Vb, Vg). This procedure yields avery good agreement with the measured dI/dVb(Fig. 2, A and B).In addition to explaining how the two sets of

fringes, WGM′ and WGM′′, originate from thesame family of WGM resonances, our model ac-counts for other key features in the data. Inparticular, it explains the large difference in theWGM′ and WGM′′ periodicities noted above. Italso correctly predicts the regions where fringesoccur (Fig. 2B). The bipolar regime in which pnjunction rings and resonances in the DOS occur(see fig. S6 and S7) takes place for the probedenergies e of the same sign as the potential curva-ture. In the case of a parabola U ðrÞ ¼ kr2, thisgives the condition ek > 0, corresponding to theupper-right and lower-left quadrants in Fig. 3A,inset.However, under experimental conditions, thepotential is bounded byU ðjrj→ ∞Þ ¼ m0 − m∞ (seeFig. 1A), which constraints the regions in whichWGMs are observed (17). Accounting for the finitevalueU ðjrj→ ∞Þ yields the condition jej≤jm0−m∞j,with sgnðeÞ ¼ sgnðkÞ ¼ sgnðm0 − m∞Þ. This givestheWGM′ andWGM′′ regions in Fig. 2B boundedby white dashed and white dotted lines, respec-tively, and matching accurately the WGM′ andWGM′′ location in our measurements.The range of m values that our measurement

can probe depends on the specifics of the tun-neling region at the STM tip. We believe that awide range ofm values can be accessed; however,we are currently unable to distinguish differentpartial-m contributions, because the correspond-ing resonances are well aligned (Fig. 3). Differentm statesmay contribute if the tunneling center isnot the same as the geometric center of the tip,which is highly likely. As shown in (13), highermstates can be accessed by going off center by aslittle as 1 nm, which is likely in our real experi-ment due to a residual asymmetry of the STM tip[we model this effect by a Gaussian factor in Eq.3]. We note in this regard that different angularmomentum m values translate into different or-bital magnetic moment values, opening an op-portunity to probe states with different m byapplying a magnetic field.The explanation of the observed resonances in

terms of the whispering-gallery effect in circularpn rings acting as tunable electronic WGM res-onators has other notable ramifications. First, itcan shed light on puzzling observations of reso-nances in previous STM measurements (22–24),which hitherto remained unaddressed. Second,our highly tunable setup in which the electronwavelength and cavity radius are controlled in-dependently lends itself well to directly probingother fundamental electron-optical phenomena,such as negative refractive index for electronwaves,Veselago lensing (7), and Klein tunneling (2). Fur-ther,we envision probingmore exotic phenomenasuch as the development of caustics, where an inci-dent plane wave is focused at a cusp (25–27), and

special bound states for integrable classes of dy-namics, where the electron path never approachesthe confining boundary at perpendicular inci-dence (28). These advances will be enabled bythe distinct characteristics of graphene that allowfor electronic states to be manipulated at themicroscale with unprecedented precision andtunability, thus opening a wide vista of graphene-based quantum electron-optics.

REFERENCES AND NOTES

1. A. S. Mayorov et al., Nano Lett. 11, 2396–2399 (2011).2. M. I. Katsnelson, K. S. Novoselov, A. K. Geim, Nat. Phys. 2,

620–625 (2006).3. A. V. Shytov, M. S. Rudner, L. S. Levitov, Phys. Rev. Lett. 101,

156804 (2008).4. A. F. Young, P. Kim, Nat. Phys. 5, 222–226 (2009).5. A. Varlet et al., Phys. Rev. Lett. 113, 116601 (2014).6. L. C. Campos et al., Nat. Commun. 3, 1239 (2012).7. V. V. Cheianov, V. Fal’ko, B. L. Altshuler, Science 315,

1252–1255 (2007).8. C.-H. Park, Y.-W. Son, L. Yang, M. L. Cohen, S. G. Louie,

Nano Lett. 8, 2920–2924 (2008).9. J. R. Williams, T. Low, M. S. Lundstrom, C. M. Marcus,

Nat. Nanotechnol. 6, 222–225 (2011).10. K. J. Vahala, Nature 424, 839–846 (2003).11. A. B. Matsko, V. S. Ilchenko, IEEE J. Sel. Top. Quantum

Electron. 12, 3–14 (2006).12. M. Pöllinger, D. O’Shea, F. Warken, A. Rauschenbeutel,

Phys. Rev. Lett. 103, 053901 (2009).13. M. F. Crommie, C. P. Lutz, D. M. Eigler, Science 262,

218–220 (1993).14. L. Rayleigh, Philos. Mag. Ser. 6 20, 1001–1004

(1910).15. V. V. Nesvizhevsky, A. Y. Voronin, R. Cubitt, K. V. Protasov,

Nat. Phys. 6, 114–117 (2010).16. G. Reecht et al., Phys. Rev. Lett. 110, 056802

(2013).17. Additional supplementary text and data are available on

Science Online.18. K. Christmann, Surf. Sci. Rep. 9, 1–163 (1988).19. All uncertainties reported represent 1 SD from multiple

measurements.20. J. Tersoff, N. D. Lang, in Scanning Tunneling Microscopy,

J. A. Stroscio, W. J. Kaiser, Eds. (Academic Press, San Diego,CA, 1993), chap. 1.

21. V. V. Cheianov, V. I. Fal’ko, Phys. Rev. B 74, 041403(2006).

22. G. M. Rutter et al., Science 317, 219–222 (2007).23. S. Jung et al., Nat. Phys. 7, 245–251 (2011).24. J. Chae et al., Phys. Rev. Lett. 109, 116802 (2012).25. J. Cserti, A. Pályi, C. Péterfalvi, Phys. Rev. Lett. 99, 246801

(2007).26. R. L. Heinisch, F. X. Bronold, H. Fehske, Phys. Rev. B 87,

155409 (2013).27. N. Agrawal (Garg), S. Ghosh, M. Sharma, J. Phys. Condens.

Matter 26, 155301 (2014).28. J. Bardarson, M. Titov, P. Brouwer, Phys. Rev. Lett. 102,

226803 (2009).

ACKNOWLEDGMENTS

Y.Z. acknowledges support under the Cooperative ResearchAgreement between the University of Maryland and the NationalInstitute of Standards and Technology Center for NanoscaleScience and Technology, grant 70NANB10H193, through theUniversity of Maryland. J.W. acknowledges support from the NationResearch Council Fellowship. F.D.N. greatly appreciates supportfrom the Swiss National Science Foundation under projectnumbers 148891 and 158468. L.S.L. acknowledges support fromSTC CIQM/NSF-1231319. We thank S. Blankenship and A. Band fortheir contributions to this project and M. Stiles and P. First forvaluable discussions.

SUPPLEMENTARY MATERIALS

www.sciencemag.org/content/348/6235/672/suppl/DC1Supplementary TextFigs. S1 to S7References (29–33)

22 January 2015; accepted 31 March 201510.1126/science.aaa7469

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Creating and probing electron whispering-gallery modes in graphene

Taniguchi, Leonid S. Levitov, Nikolai B. Zhitenev and Joseph A. StroscioYue Zhao, Jonathan Wyrick, Fabian D. Natterer, Joaquin F. Rodriguez-Nieva, Cyprian Lewandowski, Kenji Watanabe, Takashi

DOI: 10.1126/science.aaa7469 (6235), 672-675.348Science

, this issue p. 672Sciencethe cavity size may provide a route for the manipulation of electrons in graphene and similar materials.patterns within the cavity were associated with an electronic wave version of whispering gallery modes. The tunability of

used that same principle to confine electrons in a nanoscale circular cavity in graphene. Periodicet al.the same. Zhao is−−reflection−−''whispering gallery'' circular chamber walls in St. Paul's Cathedral for sound, the principle of confinement Physical barriers are used to confine waves. Whether it is harbor walls for sea waves, a glass disk for light, or the

A circular route to confine electrons

ARTICLE TOOLS http://science.sciencemag.org/content/348/6235/672

MATERIALSSUPPLEMENTARY http://science.sciencemag.org/content/suppl/2015/05/06/348.6235.672.DC1

REFERENCES

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