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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.
672 8 MAY 2015 • VOL 348 ISSUE 6235 sciencemag.org SCIENCE
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:
[email protected] (N.B.Z.); [email protected]
(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
SCIENCE sciencemag.org 8 MAY 2015 • VOL 348 ISSUE 6235 673
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
674 8 MAY 2015 • VOL 348 ISSUE 6235 sciencemag.org SCIENCE
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.
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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
SCIENCE sciencemag.org 8 MAY 2015 • VOL 348 ISSUE 6235 675
RESEARCH | REPORTSon June 8, 2021
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Creating and probing electron whispering-gallery modes in
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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
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electronic wave version of whispering gallery modes. The tunability
of
used that same principle to confine electrons in a nanoscale
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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
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