Fabry-P\'erot resonances in a graphene/hBN Moir\'e superlattice

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by  Clevin Handschin

Fabry-P´erot Resonances in a Graphene/hBN Moir´e Superlattice
Clevin Handschin,1 P´eter Makk,1,∗ Peter Rickhaus,1 Ming-Hao Liu,2 K.
Watanabe,3 T. Taniguchi,3 Klaus Richter,2 and Christian Sch¨onenberger1
1Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland
2Institut fu¨r Theoretische Physik, Universita¨t Regensburg, D-93040 Regensburg, Germany
3National Institute for Material Science, 1-1 Namiki, Tsukuba, 305-0044, Japan
While Fabry-P´erot (FP) resonances and Moir´e superlattices are intensively studied in graphene
on hexagonal boron nitride (hBN), the two effects have not been discussed in their coexistence.
HereweinvestigatetheFPoscillationsinaballisticpnp-junctionsinthepresenceandabsenceofa
7 Moir´e superlattice. First, we address the effect of the smoothness of the confining potential on the
1 visibility of the FP resonances and carefully map the evolution of the FP cavity size as a function
0 ofdensities insideandoutsidethecavity intheabsenceof asuperlattice, when thecavityisbound
2 byregularpn-junctions. UsingasamplewithaMoir´esuperlattice,wenextshowthatanFPcavity

n canalsobeformedbyinterfacesthatmimicapn-junction,butaredefinedthroughasatelliteDirac
a point due to the superlattice. We carefully analyse the FP resonances, which can provide insight
J into the band-reconstruction due to the superlattice.

1
3
Clean graphene has shown to be an excellent platform though several aspects of FP cavities have been investi-

] toinvestigatevariouselectronopticalexperiments, rang- gated, such as the effect of the pn-junction smoothness
l
l ing from Fabry-P´erot (FP) resonances [1, 4–6], snake on the visibility of the FP resonances [4], the electronic
a
h states [3, 8], electron guiding [9], magnetic focusing tunability of the cavity size has not been studied. First,
- [10, 11], Veselago lensing or angle-dependent transmis- we will discuss samples with a large misalignment angle
s
sion studies of negative refraction [12, 13]. The pn- between the hBN and graphene. We find that the cav-
e
m junctions are basic building blocks of many of these ex- ity size is strongly dependent on carrier density in and
periments, which for example are used to confine elec- outside the cavity. Then, we turn to aligned samples
.
t
a trons into FP cavities, where the pn-junctions play the with a superlattice structure and take advantage of the
m roleofsemireflectivemirrors. ThestudyofFPresonances largetunabilityofthecavitysizetoobtainadditionalin-
- havealreadyrevealedthemselvesasapowerfultooltoin- formationonthemodulatedMoir´eband-structurewhich
d vestigate various aspects of graphene. Examples are the is yet not fully understood. Our findings are consistent
n
π-shift of the FP resonances at low magnetic field origi- with the electron-hole symmetry breaking of the Moir´e
o
natingfromKleintunnelling[26]insinglelayergraphene superlattice [29].
c
[ [1,2,7]ortheFPresonancesinultraclean,suspendedde- Thedevicegeometryofourtwo-terminalpnp-junction

vices thatprovedballistic transport over several microns is shown in Fig. 1a: a global back-gate and a local
1
v [4, 5]. top-gate is used to separately control the charge car-
1 Encapsulation of graphene in hexagonal boron nitride rier density in the inner (n ) and outer (n ) regions
in out
4 (hBN)[18]cannotonlyallowforquasi-ballistictransport thus forming pp’p- or pnp-junctions. In order to re-
1
(the mean free path is comparable or larger than the de- duce the cross-talk between the gates we used a thin
9
vice geometry), but can also be used for the creation of (∼30nm) hBN layer. Therefore n can be calculated
0 out
1. a superlattice in graphene due to the periodic potential accordingtonout =VBG·(1/CSiO2+1/ChBN,b)−1,where
modulation of the Moir´e pattern. Such a pattern stems C andC arethegeometricalcapacitancesofthe
0 SiO2 hBN,b
7 fromasmalllatticemismtachofgrapheneandhBN.The SiO2 and bottom hBN respectively, and VBG and VTG
1 modified bandstructure of graphene, which includes the the applied gate-voltages. In analogy, n is given by
in
: emergence of satellite Dirac peaks (satellite DPs), is ex- n =V ·(1/C +1/C )−1+V ·C where
v in BG SiO2 hBN,b TG hBN,t
i perimentally observable only for small misalignment an- ChBN,t is the top hBN capacitance. The Cr/Au contacts
X
gles (θ) between hBN and graphene [19]. The formation dope the graphene in its proximity n-type, independent
r ofaMoir´esuperlatticehasbeendemonstratedinvarious of V . The conductance is measured in a two-terminal
a BG
spectroscopy[19,22,30]andtransport[11,12,16,23,26– configurationasafunctionofV andV (whichcanbe
BG TG
30] experiments. Besides experiments, considerable ef- directly converted to n and n ), as shown in Fig. 1b.
in out
fort has been invested to calculate the band-structure of The overall device length is roughly 1µm, with a top-
grapheneundertheinfluenceofsuchaMoir´esuperlattice gate width of 230nm being centered in the middle of
[29, 33, 34]. the device. In the measured device, a high mobility of
Herewedemonstrateconfinementusingbandengineer- the electrons and holes (µe,h) was extracted from field
ing based on locally gated Moir´e superlattices and the effect measurements, yielding µe∼150000cm2 V−1 s−1
appearance of FP resonances defined by the main and and µh∼50000cm2 V−1 s−1 respectively.
satellite DPs playing the role of reflective barriers. Al- All structures were fabricated following the procedure
2
(a) (b) -1 0 1x1016 (c) (d)
d/dn G (e2/h)
V out
TG V 4x1016 h)50 ∆G/(2 G )~0.8% 400
AC 32 2G (e/3100-30 -15VB0G (V)15 30 L (nm) 230000 noutlhoiwgh
1 h)22 ∆G/(2 G )~4% 0 1 2 3x1016
-2n (m)in 0 2G (e/221910 (e)400 nin (m-2)
-12 -11 -10 -9
-1 VBG corr. (V) m) 300 ∆L= 30 nm
n
I -2 2G (e/h)111543∆G/(2 G )~11% L ( 200 L= 230 nm
-3
V
BG -1.5x1016 -1.0 -0.5 0.0 2.6 3.0 3.4 0 10 20 30 40 50
n (m-2) V (V) n /n
out TG in out
FIG. 1. Fabry-P´erot resonances measured in a two-terminal pnp-configuration. a, False-color SEM image with the
measurement configuration sketched. The contacts are shown in yellow, the top-gate in red and the graphene between the
two hBN-layers is shown in cyan. Scale-bar equals 500nm. b, Numerical derivative of the conductance as a function of global
back-gate (V ) and local top-gate (V ). Here it is replotted as a function of charge-carrier density in the outer (n ) and
BG TG out
inner cavity (n ). The three most important FP resonances present in the system are indicated. c, Visibilty ∆G/(cid:104)2G(cid:105) of
in
the FP resonances indicated in (b) in comparison. d, Extracted cavity length of the central cavity as a function of n , as
in
indicated with the cyan arrow in (b), for different values of n . e, The same data as shown in (d), but now plotted as a
out
function of n /n for experiment (red) and theory (blue, for transport simulation see Supporting Information). The black,
in out
dashedlinecorrespondstothewidthofthetop-gatemeasuredwiththeSEM.Theblue,dashedlinesareaguidetotheeyefor
the theoretical data and the theoretical data shifted by 30nm respectively.
given by Wang et al. [15] with some additional improve- sion/reflectionprobabilitiesofapn-interfacestronglyde-
ments. A detailed description of the fabrication is given pendsontheangleoftheincomingchargecarrier[28]. In
in the Supporting Information. The measurements were the case of a “sharp” pn-junctions (d(cid:28)λ , where d de-
F
done using low frequency lock-in technique in a vari- notesthedistanceoverwhichthedensitychangesandλ
F
able temperature insert (VTI) with a base-temperature is the Fermi wavelength), transmission of charge carriers
around T =1.6K. is possible up to large angles measured with respect to
The derivative of the differential conductance with re- thepn-junctionnormal. Incontrast,foravery“smooth”
spect to nin and nout are shown in Fig. 1b for nout < 0. pn-junction (d(cid:29)λF), only charge carriers at low angles
The visible fringes are FP resonances appearing in the are transmitted (in our device this is around ∼20◦, al-
pp’p regime (bottom part, n <0) and pnp regime (up- thoughdependsstronglyonthegatevoltages). Sinceour
in
perpart,n >0). Specificcutswithinanextended(red) conductancemeasurementisnotangleresolved,themea-
in
or limited (blue, green) gate-range are shown in Fig. 1c. suredsignalaveragesoverallpossibleanglesandleadsto
WhiletheFPresonancesintheinner(outer)cavitiesare smearing of the FP resonances. It turns out that for a
tuned predominantly by n (n ) as indicated with the smoothpn-junction,whichtransmitsanarrowerrangeof
in out
blue(green)arrowinFig.1b,theFPresonancesbetween angles, the highest FP visibilities can be observed [4].
the contacts depends on both densities as indicated by Inoursystemwediscriminatebetweentwotypesofpn-
the red arrow. The three types of FP resonances are junctions: i) the pn-junction created in proximity of the
very different in their visibility ∆G/(2(cid:104)G(cid:105)). Here ∆G is contacts for n <0 and ii) the pn-junction created us-
out
the difference between the conductance at constructive ingtheglobalback-gateandlocaltop-gate. Wenotethat
and destructive interference, and (cid:104)G(cid:105) denotes the mean (ii)ismuchsmoothercomparedto(i)(seeSupportingIn-
conduction in between oscillation maximum and mini- formation). Followingtheabovegivenargumentonesees
mum. The FP resonances between the contacts yield that the FP resonance visibility is highest/lowest when
the lowest visibility (∼ 1%), those in the outer cavi- the cavity is defined by two softer/sharper pn-junctions.
ties yield an intermediate visibility (∼ 4%) and FP res- Wewillnowextracttheeffectivecavitylength,Lfrom
onances in the inner cavity yield the highest visibility theFPoscillationsinthecentralcavity,whichdeviatesin
(∼11%). Inballisticgraphene,theFPvisibilitiesdepend most cases from the physical width of the top-gate. As-
on the transmission/reflection properties of the confin- suming a FP resonator with a hard-wall potential (fixed
ing boundaries, namely the pn-interface. The transmis- width of the cavity), the cavity width can be extracted
3
fromtheFPoscillations. Constructiveinterferenceforms Nowweturntographenewhichisalignedwithasmall
if the path-difference between directly transmitted and twist angle with respect to one of the hBN layers. The
twice reflected waves is equal to 2L = jλ , where λ resulting band-reconstruction includes additional satel-
F F
is the Fermi wavelength and j is an integer. The j-th lite DPs which are indicated with the purple and orange
√ √
FP resonance can be rewritten as L n = j π using triangles in Fig. 2a where the derivative of the conduc-
j
λ = 2π/k = 2(cid:112)π/n which is valid for single layer tanceisplotted. ThemainDPisindicatedwiththeblack
F F
graphene. For two neighboring peaks, for example j-th triangle. At low doping the semitransparent boundaries
peak at density n and (j+1)-th peak at density n : in the bipolar region are defined by the main DP (com-
j j+1
parable to Fig. 1b) and FP resonances within inner and
outercavityareobserved, markedbythegreenandcyan
√
π arrows, respectively.
L= √ √ . (1)
n − n Whereas in Fig. 1b,c the contacts are the only bound-
j+1 j
ariesleadingtoFPresonancesintheunipolarregime(red
We note here that in Equation 1 L is independent of arrow), additional semitransparent boundaries, formed
n which is an oversimplification of the problem. An by the satellite DPs, emerge at high doping in the pres-
out
alternative way to define the cavity length is to mea- ence of a superlattice. As a result, novel FP oscillation
sure the distance between the two zero-density points of are visible at high doping as indicated with the purple
the right and left pn-junctions. However, since the po- andorangearrowsinFig.2a,whichareabsentinFig.1b
sition of the pn-interface is experimentally not directly (noMoir´esuperlattice). ThisnewsetofFPoscillationre-
accessible we will use Equation 1 to deduce the cavity semblesagainthepatternknownfromthebipolarregime
size. More discussion on this is given in the Supporting and the transition from FP resonances across the whole
Information. The cavity length was extracted by taking sampletoFPresonanceswithintheinnerandoutercav-
variouslinecutscomparabletotheoneindicatedwiththe ity is a direct consequence of the satellite DPs forming
bluearrowinFig.1bandthenusingEquation1. Weob- these additional semitransparent boundaries. There are
serveanincreaseofLwithincreasingn (forfixedn ), alsochargecarriersthatbouncebetweenthecontactsand
in out
and a decrease of L with increasing n (for fixed n ) thesatelliteDPboundary,leadingtoweakresonancesas
out in
as shown in Fig. 1d. Consequently L does depend on a function of nout (see Supporting Information).
n , as expected. Surprisingly, by plotting L as a func- In the following, we compare the visibility be-
out
tionofn /n ,alldata-pointslieononeuniversalcurve tween FP resonances across the main DP and the
in out
whichisshowninFig.1e, independentoftheexactposi- satellite DP within the inner cavity for n > 0
in
tion within Fig. 1b from which they have been extracted and n < 0 separately (µ ∼100000cm2 V−1 s−1,
in e
from. Within the applied gate-range, L varies substan- µ ∼50000cm2 V−1 s−1. Charge carriers for n < 0
h in
tially, by up to 200nm, which corresponds to a shift of (hole side) bouncing between boundaries formed by the
around 100nm per pn-junction. The evolution of L as a satellite DPs (∆G ∼ 0.9%) show a visibility that is
function of n /n extracted from the experiment was 40% lower than the visibility of the main DP oscilla-
in out
compared with the one extracted from a transport simu- tion (∆G ∼ 1.5%), as shown in Fig. 2a. In contrast, for
lation (based on the method described in Ref. [4]) using n > 0 (electron side), the visibility is reduced by 85%
in
Equation 1. The latter reveals good qualitative agree- (∆G ∼ 0.4% at the satellite DP compared to ∆G ∼ 3%
mentwiththeexperimentasshowninFig.1e. Themost atthemainDP).Tointerprettheseobservations,wecon-
significantdifferencebetweenexperimentandtheoryare: sider a family of possible Moir´e minibands for graphene
i) an off-set of ∆L ∼ 30 nm from the theory to the ex- on hBN substrate, which was calculated by Wallbank et
periment and ii) a disagreement between the trends of al. [29] using a general symmetry-based approach. Be-
the two curves when approaching very low charge car- causetherelativelylargenumberofmodel-dependentpa-
rier densities in the outer cavity, corresponding to large rametersofthesymmetrybreakingpotentialcanstrongly
values of n /n (the same is true when depleting the influencetheobtainedMoir´eperturbation,thefocuswas
in out
innercavity,notshownhere). Intheexperimentthecav- on the generic features for different sets of parameters
ity length L saturates while it increases continuously in used. In Fig. 2b, the DOS for three different sets of pa-
the theory. We believe that the reason for this is that rameters are plotted. Details on the parameters can be
the measured sample bears a residual doping (n∗) which found in Ref. [29].
is not present in theory. The residual doping causes that For the case of n <0, when crossing the satellite DP
in
for values below n∗ the electrostatic gates are unable to ontheholeside,theDOSbetweeninnerandoutercavity
further deplete the graphene, thus for V → 0 the ef- decreasestozero(independentoftheparametersusedfor
BG
fective value of n /n (tuning L) remains fixed. The the calculations of the DOS as given in Ref. [29]) for all
in out
off-set of the two curves (i) might origin from a too nar- k , as indicated with the purple triangle in Fig. 2b.
x,y
row top-gate in theory if the top-gate in the experiment This result is experimentally supported by capacitance
was measured with an error of ∼30nm. spectroscopy of hBN-graphene-hBN heterostructures in
4
(a) -10 0 10 (b) (c) (d)
d/dn G (e2/h) -1 1x1015 -1 2x1015
in
E (a.u.) 7x1016 -3x1016
6x1016 ∆G/(2 G )~0.4% pp’p
4 ∆G/(2 G )~3.0% Model 1 -2n (m)in 5 -2n (m)in -5
DOS (a.u.) nn’n
2 3 -7
-2n (m)in0 pnp nn’n Model 2 (e) 1.5 no2u.t 0(m-2)2.5x1015 -1.5 no-u1t .(0m-2)-0.5x1015
-2 pp’p npn -2m)3x1015 theory (nout~1x1015)
DOS (a.u.) n (in 2
∆ 1
-4
∆G/(2 G )~1.5%
Model 3 -2m) 1.5
-6 ∆G/(2 G )~0.9% nl (out 0.7 x1015 nin>0 nin<0
-2 0 2 4x1016 DOS (a.u.) l 0 2 4 6x1016
n (m-2) ln l (m-2)
out in
FIG.2. Fabry-P´erotresonancesininthepresenceofaMoir´esuperlattice. a,Numericalderivativeoftheconductance
asafunctionofn andn . Thered,cyanandgreenlinesindicatetheregularFPresonancesdescribedinFig.1. Additional
in out
FP resonances emerge if n or n is tuned above or below a satellite Dirac peak. These FP resonances are indicated with
in out
orange and purple arrows. The position of the main and satellite DPs are indicated with black and purple/orange triangles
respectively. b, DOS of graphene as a function of energy in the presence of a Moir´e superlattice with hBN for three different
parameter sets used in the calculation performed by Wallbank et al. Figure adapted from Ref. [29]. Black, orange and
purpletrianglesindicatemainandsatelliteDirac-peaksrespectively. Theblue,dashedlineindicatestheDOSforunperturbed
graphene. c,d, High-resolution measurements (d/dn G (e2/h)) of the regions where the inner cavity is tuned beyond the
in
satelliteDPandtheadditionalFPresonancesarepresent. e,PositionoftheindividualFPresonancepeaks(lowerpanel)and
their relative spacing ∆n (upper panel) is plotted as a function of n . The black, dashed line indicates the values expected
in in
from theory. If the semitransparent boundaries are defined via the satellite DP the spacing is further increased. However, no
precise trend of ∆n (increasing or decreasing) as a function of n can be seen from the few data-points extracted.
in in
the presence of a Moir´e superlattice [30]. The vanishing individualFPresonancepeaks(|n |∼const.)andtheir
out
DOS leads to similar reflection/transmission coefficients relativespacing∆n asafunctionofn ,whichareplot-
in in
as for the main DP, resulting in a comparable visibility ted in the lower and upper panel of Fig. 2e. The latter
of the two FP resonances. A representative cut at fixed analysis is performed instead of the cavity length analy-
n <0 is shown in Fig. 2d. sissinceEquation1doesnotholdanymoreiftheFermi-
out
For the electron side with n >0, the significantly re- energy is tuned beyond the satellite DP. This is because
in
ducedvisibilityisinqualitativeagreementwiththeband- in Equation 1 a circular Fermi-surface and electron like
structuresshowninFig.2b,sincetheDOSatthesatellite dispersion is assumed, which does not hold in the pres-
DP (indicated with an orange triangle) is reduced (de- ence of a strong band-modulation. In contrast, mapping
pending on the parameters used in the calculation), but the FP resonance peak position as a function of charge-
never vanishes. A representative cut at fixed n >0 is carrier density is free of any assumptions and therefore
out
shown in Fig. 2c. A direct implication of the finite DOS independent of the band-structure. If the semitranspar-
is that only some charge-carriers have a nonzero reflec- ent interfaces are defined via the main DP, the evolution
tion coefficient, thus contributing to the FP resonances, of∆nin betweenneighboringpeaksasafunctionofnin is
while the remaining ones account for a background cur- ingoodagreementwiththevaluesextractedfromtheory
rent. It is worth noting, that even though the visibility (transport simulation in the absence of a Moir´e super-
of the FP resonances across the electron side satellite lattice), which is indicated with the black, dashed line
DP (n > 0, Fig. 2c) are significantly reduced, they in Fig. 2e. If the semitransparent interfaces are defined
in,out
seem to be more regular over a wider gate-range com- viathesatelliteDPs,thedensity-spacingbetweentheFP
pared to the FP resonances across the hole satellite DP resonancepeaksisfurtherincreasedonboth,theelectron
(n <0,Fig.2d). However,thiscouldbeaswelljust and hole side.
in,out
sample-specific since the mobility in the hole side was Although a precise extraction of the cavity size is not
significantly lower compared to the electron side. possible with Equation 1, an estimate can still be given.
We now compare the evolution of the position of the We have found a cavity size of L∼250nm to 310nm on
5
theelectronsideifthedensityismeasuredfromthemain describingtheexperimentalsituationbest. Furthermore,
DP, whereas setting the density to zero at the satellite formodel1,wheremultiplesatelliteDPsappearnearev-
DPgivesanunphysicalcavitysize(L∼80nmto200nm). erymainDP,Equation1wouldalsohavetobemultiplied
√
On the electron side the density of states seems almost byafactorof g ,whereg istheadditionaldegen-
SDP SDP
unaltered, as can be seen in Fig. 2b, however the band- eracy of the satellite Dirac cones. Although by including
structure is substantially modified, which can be seen additional degeneracies, and counting the density from
in Ref. [29]. The band-structure consists of two non- the satellite DP, the strong oscillations could give rea-
isotropic bands above the satellite DPs which makes the sonable cavity sizes, however, for the small oscillations,
situation rather complex, with different visibilities and theadditionaldegeneracieswouldresultintoolargecav-
angle dependent transmissions for the two bands. On ity sizes. Moreover, most experimental evidence points
the hole side the cavity size analysis yields cavity sizes to a single Dirac cone at the satellite DP. We also note
of L∼280nm to 360nm if the density is measured from here,thatthevisibilityoftheseresonancesarequitelow,
the main DP. At the satellite DP the density of states which could also introduce some experimental error in
decreases to zero, meaning that a real Dirac point is the analysis.
formed. Thereforeonemightexpect,thatthedensityfor In conclusion, we first analysed a pnp-junction in the
the FP oscillations should be measured from the satel- absence of a Moir´e superlattice where the varying visi-
lite DP. However by setting the charge-carrier density bility of the different types of FP resonances could be
to zero at the hole satellite DP, the analysis gives un- linked to sharp and smooth pn-junctions in the system.
physical results. We note, that close to the hole satellite Furthermore, the change of the effective cavity length
DP Equation 1 should be valid if counting the charge- depending on n and n could be mapped via the FP
in out
carrier density from the satellite DP, since the band- resonances. In a second sample, the presence of a Moir´e
structure is isotropic. Besides the most pronounced FP superlattice gave rise to satellite DPs and the modified
resonances indicated with the purple arrow in Fig. 2d, band-structure resulted in confinement of electronic tra-
a second set of resonances with a much shorter period jectories and in the appearance of FP resonances. Al-
seem to appear in the very vicinity of the hole satel- though the oscillations are formed in the Moir´e mini-
lite DP (see inset of Fig. 2d). For these resonances the bands, they can mostly be described as if they would
extracted cavity length leads only to reasonable values originate from the non-modified band-structures. Fur-
(L∼300nm) when counting the charge carrier density ther studies will be needed to explain these findings.
starting from the satellite DP. Unfortunately the resid- Our results show that confinement of electrons can be
ual doping (nres∼0.4×1015m−2 to 1×1015m−2) pre- obtainedusingminibandengineering. Futurestudiescan
ventsusfromresolvingmoreofthesefeaturesinthevery investigate the angle dependent transmission properties
vicinity of the satellite DP. A possible explanation for of such an interface. Moreover combining the studies of
the different behavior of the two sets of FP oscillation transverse magnetic focusing [16] and bias spectroscopy
(nout <0) might be the following: The small resonances studies of FP resonances or detailed studies of the
are only observed up to densities of ∆n<2×1015m−2 magnetic field dependence of FP resonances can reveal
(where∆nismeasuredfromthesatelliteDP),wherethe further details on the band reconstruction.
Moir´eminibandremainsclosetoDiraclike(lineardisper-
sionrelation). TheFermienergycorrespondingtotheses ACKNOWLEDGMENTS
smalloscillationsisconsequentlyexclusivelytunedinthe This work was funded by the Swiss National Science
linear part of the Moir´e miniband (see Supporting Infor- Foundation, the Swiss Nanoscience Institute, the Swiss
mation). Fromtheband-structureswewouldexpectthat NCCR QSIT, the ERC Advanced Investigator Grant
the small oscillations should be visible up to higher dop- QUEST, and the EU flagship project graphene. M.-H.L.
ing values, namely values which are larger by a factor andK.R.acknowledgefinancialsupportbytheDeutsche
of up to ∼ 10 depending on the model. This estimate Forschungsgemeinschaft (SFB 689). The authors thank
is based on the energy-spacing between the satellite DP John Wallbank, David Goldhaber-Gordon, Simon
and the first van Hove singularity at higher energies, ex- Zihlmann and L´aszlo Oroszl´any for fruitful discussions.
tracted from theory (see Supporting Information). Note
that the extracted energy-spacing corresponds to an up-
per bound within which the dispersion relation might be
considered as linear. The Fermi energy corresponding
to the stronger resonances indicated with the purple ar-
row, does partially reside outside of the linear region of ∗ [email protected]
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7
SUPPORTING INFORMATION
Low magnetic field measurements
(a) 8 14 (b) 20.0 23.5 (c) 21.5 23.5 (d) 31.9 32.3 (e) 42.5 43.0
G (e2/h) G (e2/h) G (e2/h) G (e2/h) G (e2/h)
0 -5 0 0 7 0 5.6 0
3.4
V) -3.0 V) V)
V (TG corr. V (V)TG V (V)TG V (TG corr. V (TG corr.
5
2.6 -4.2
-7 4.0
-100 0 100 -500 0 500 -500 0 500 -500 0 500 -500 0 500
B (mT) B (mT) B (mT) B (mT) B (mT)
d/dVTG corr.G0-20 20 -5 d/dVTG G0 -3 2 d/dVTG G0 -6 6 7d/dVTG corr.G0-0.4 0.4 5.6d/dVTG corr.G0-0.3 0.6
3.4
V (V)TG corr. V (V)TG V (V)TG-3.0 (V)TG corr. V (V)TG corr.
V
2.6 -4.2 5
-7 4.0
-100 0 100 -500 0 500 -500 0 500 -500 0 500 -500 0 500
B (mT) B (mT) B (mT) B (mT) B (mT)
-0.8 0.8 -0.2 0.2 -0.2 0.2 -0.025 0.025 -0.04 0.04
∆G (e2/h) ∆G (e2/h) ∆G (e2/h) ∆G (e2/h) ∆G (e2/h)
osc -5 osc osc osc osc
7 5.6
3.4
V (V)TG corr. (V)VTG V (V)TG-3.0 (V)TG corr. V (V)TG corr.
V
2.6 -4.2 5
-7 4.0
-100 0 100 -500 0 500 -500 0 500 -500 0 500 -500 0 500
B (mT) B (mT) B (mT) B (mT) B (mT)
FIG.S1. Low magnetic field measurements of the FP resonances across the main and satellite DPs. First rowis
thenetconductance,second rowthenumericalderivativealongthegate-axisandthird rowthenetconductanceoscillation.
Columns(a)isthemeasurementsacrossthemainDPwithn >0andn ∼−0.3·1016m−2,column(b,c)aremeasurements
in out
acrossthesatelliteDPwithn <0wheren ∼−0.7·1016m−2 andn ∼−2.8·1016m−2,column(d,e)aremeasurements
in out out
across the satellite DP with n >0 where n ∼2.1·1016m−2 and n ∼2.8·1016m−2.
in out out
Fabry-Perot (FP) resonances disperse towards higher doping (k values) with increasing magnetic field [1–6] as seen
in Fig. S1a-c. This results from the resonance condition ∆θ +Φ = const. where ∆θ is the Wentzel-
WKB AB WKB
Kramers-Brillouinphaseacquiredonthepathofthecharge-carrierstrajectory,andΦ istheAharonovBohmphase
AB
[3, 7].
Furthermore one can determine the Berry’s phase acquired by a charge carrier encircling the origin in k-space. For
singlelayergraphenewithastaggeredsublatticepotential,whichbreaksitsinversionsymmetry[8],theBerry’sphase
becomes exactly π as the gap approaches zero [9]. This has been observed experimentally in FP measurements [1]
and quantum Hall measurements [10]. In Fig. S1 we show low magnetic field measurements for FP oscillations across
the main DP (a) and the satellite DP in the hole (b,c) and electron (d,e) doped region. Each of the measurements
was plotted in three different ways showing the net conductance, the numerical derivative of the net condctance
and the net oscillation. The black/red lines in Fig. S1 were extracted numerically and track the peak-position of
the FP oscillations with magnetic field. For the FP resonances across the main DP we see the π-shift at roughly
B =30−50mT as shown in Fig. S1a. Following the previously given argument, one might expected as well a π-shift
for the FP resonances across the satellite DP on the hole side, since there the DOS drops to zero comparable to the
main DP. However, measurments at different n , shown in Fig. S1b,c, did not allow a conclusive statement whether
out
suchaπ-shiftispresentornot. Itseemshoweverthatforbothoftheherepresenetedsituationsanintriguingpattern
appears at low magnetic fields (i.e. B < 200 mT). At higher field, the FP oscillations disperse constantly. On the
electron side, i.e. n >0, the situation is less clear. The pattern in the middle exhibits less strong variations.
in
8
Bias-spectroscopy of Fabry-P´erot resonances
(a) 21.05 21.25 (b) 57.5 57.8
G (e2/h) G (e2/h)
10 10
∆E ∆E

5 5
V) V)
m m
V (SD 0 V (SD 0
-5 -5
-10 -10
-5.7 -5.45 -5.2 3.6 4.1 4.6
V (V) V (V)
TG TG
FIG. S2. Bias spectroscopy on FP resonances a, Measurement of FP resonances across the main DP (n <0, n >0)
in out
where an energy spacing corresponding to L = 300nm is indicated. b, Comparable measurement for the satellite DP with
n ,n >0. Energy spacing corresponds to L=280nm.
in out
Bias spectroscopy of the FP resonances can be used as an alternative way to extract the cavity length. The
resonance condition in a cavity is given by L=j·λ/2, or k =jπ/L, where j is an integer. For single-layer graphene
F
in the absence of a Moir´e superlattice, the energy spacing ∆E between two consecutive FP resonances is equidistant
and given by
π π
∆E =(cid:126)v [(j+1)−j] =(cid:126)v (2)
F L FL
where (cid:126) is the reduced Planck constant and v ∼106m/s is the Fermi velocity. The size of the diamond corresponds
F
to twice the level spacing, 2∆E, leading to:
π
L=(cid:126)v . (3)
F∆E
ThemeasurementofthebiasspectroscopyacrossthemainDPisinqualitativeagreementwiththeextractedcavity
length using Equation (4). In Fig. S2a, an energy spacing corresponding to L ∼ 300nm according to Equation (3)
is indicated. Similar results are found for the FP resonances across the satellite DP with n >0, shown in Fig. S2b
in
where an energy spacing corresponding to L = 280nm is indicated in the plot. Note that Equation (3) is based on
the energy-dispersion of unperturbed graphene, neglecting a modification of the band structure in the presence of a
Moir´e superlattice. On the hole side, the pattern was too complex to extract any reasonable information.
9
Quantum Hall Measurements
=-2 =-18 =2 =-14 =6 =-10 =10 =-6 =14 =-2=-18 =18 =2 =-14 =6 =-10 =10 =-6 =14=-2 =18
ν ν ν ν ν ν ν ν ν ν ν ν ν ν ν ν ν ν ν ν ν
8 15
6
h)
2e/
T) G (
B (4 G corr.
B
V
d
d/
2
0 -15
-60 -40 -20 0 20 40 60
V (V)
BG corr.
FIG.S3. Quantum Hall measurement in high magnetic field. Numericalderivativeoftheconductanceasafunctionof
the global back-gate and magnetic field reveals the expected filling-factors emerging at the main DP (red) and at the satellite
DPs (cyan and green) as observed in Ref. [11, 12]
MeasurementofthequantumHallEffectintheunipolarregimerevealstheemergenceofthefilling-factorsassociated
with graphene from the main DP and the two satellite DPs, and similar results have been reported in Ref. [11, 12].
Thermal annealing of hBN-graphene-hBN heterostructures
(a) 20 30 40 (b) 20 30 40 (c) 20 30 40 (d)
G (e2/h) G (e2/h) G (e2/h)
5
4.8
4.6
4
4.4
V)
(V)TG 3 (G corr.44..20
V 2 VT
3.8
1 3.6
3.4
0
-30 -20 -10 -30 -20 -10 -30 -20 -10 27 28 29
V (V) V (V) V (V) G (e2/h)
BG BG BG
FIG.S4. Self-cleaningpropertiesofhBN-graphene-hBNheterostructuresuponthermalannealing. a,Conduction
map with FP resonances clearly visible in the bipolar region. b, Same map after degradation of the sample where the FP
resonancesarenearlyabsent. c,MostoftheFPresonancesarerestoredafterthermalannealingat200◦Cfor20min(outside
the cryostat). d, Linecuts as indicated in (a)-(c) in comparison.
As a result of applying high gate voltages the device quality degraded over a longer time period (2 weeks). The
degradation over time and the subsequent improvement upon thermal annealing was observed in several samples
independently. In between the maps shown in Fig. S4a and S4b, the global and local gates were swept over a wide
10
range (e.g. ±60V for the global back-gate during the QHE measurement) for an extended time period (2 days). The
sample shown in the main text was annealed at least 6 times outside the cryostat. No significant difference between
annealing on a hotplate in air at 200−250◦C or in a rapid thermal annealer under forming-gas atmosphere (Ar/H )
2
at 300◦C could be found. An example before and after thermal annealing is shown in Fig. S4a-c. In Fig. S4d an
identical linecut of the original (red), degraded (green) and annealed (blue) sample are shown in comparison. We
speculate that the decrease of the device quality might be due to the migration of contaminations when continuously
sweeping the gates to high voltages. By applying a thermal annealing step, these contaminations are very likely to
aggregate again in pockets resulting in larger areas of clean graphene [13, 14].
Temperature dependence of Fabry-P´erot resonances
(a) (b)
T (K) 57.0 T (K)
15
18 18
2e/h) 14 14 2e/h)56.8 14
G (0 10 G (0 10
56.6
6 6
13
2 56.4 2
2.6 3.0 3.4 3.5 4.5 5.5
V (V) V (V)
(c) TG corr. (d) TG
T (K) 1.0
23.0
2G (e/h)02222..50 111840 T)/G(T)1.7 K000...864 SMSaaatti..n DD DPPP ((nniinn><00))
G(
21.5 6 0.2
0.0
2
-6.0 -5.5 -5.0 2 4 6 8 10 12 14 16 18
V (V) T (K)
TG
FIG. S5. Temperature dependence of the FP resonances FP resonances across the a, main DP and the two satellite
DPs where b, n < 0 and c, n > 0 as a function of temperature. d, Oscillation amplitude as a function of temperature
in in
renormalized by the value measured at base temperature (T ∼ 1.7K). While the FP resonances remains very fainth even at
T ∼20K for (a) and (b), it disappears already around T ∼10K in the case of (c).
The temperature dependence of the various FP resonances are shown in Fig. S5. Across the main DP and the
satellite DP on the electron side (Fig. S5a,b), the temperature where the resonances disappear are in the order of
T ∼ 20K (E ∼ 2meV), which is on the same order of the level spacing obtained from bias spectroscopy. The
FP resonances across the satellite DP on the hole side seem to vanish at slightly lower temperatures (T ∼ 10K,
E ∼1meV).