Global strain-induced scalar potential in graphene devices
Lujun Wang, Andreas Baumgartner, Péter Makk, Simon Zihlmann, Blesson S. Varghese, David I. Indolese, Kenji Watanabe, Takashi Taniguchi, Christian Schönenberger
GGlobal strain-induced scalar potential in graphene devices
Lujun Wang,
1, 2, ∗ Andreas Baumgartner,
1, 2
P´eter Makk,
1, 3
Simon Zihlmann, Blesson S. Varghese, David I. Indolese, Kenji Watanabe, Takashi Taniguchi, and Christian Sch¨onenberger
1, 2 Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland Swiss Nanoscience Institute, University of Basel,Klingelbergstrasse 82, CH-4056 Basel, Switzerland Department of Physics, Budapest University of Technology and Economics and Nanoelectronics MomentumResearch Group of the Hungarian Academy of Sciences, Budafoki ut 8, 1111 Budapest, Hungary Research Center for Functional Materials, National Institute for Material Science, 1-1 Namiki, Tsukuba, 305-0044, Japan International Center for Materials Nanoarchitectonics,National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan
By mechanically distorting a crystal lattice it is possible to engineer the electronic and opticalproperties of a material. In graphene, one of the major effects of such a distortion is an energyshift of the Dirac point, often described as a scalar potential. We demonstrate how such a scalarpotential can be generated systematically over an entire electronic device and how the resultingchanges in the graphene work function can be detected in transport experiments. Combined withRaman spectroscopy, we obtain a characteristic scalar potential consistent with recent theoreticalestimates. This direct evidence for a scalar potential on a macroscopic scale due to deterministicallygenerated strain in graphene paves the way for engineering the optical and electronic properties ofgraphene and similar materials by using external strain.
Graphene is a model system on which a large varietyof new and prominent physical phenomena have beendiscovered [1–4]. A particularly promising topic is thecontrol of its electronic properties by external strain,which has been extensively studied theoretically. Thepredicted strain effects in the low-energy band structureof graphene can be summarized as changes in the mag-nitude and isotropy of the Fermi velocity and thus inthe density of states (DoS) [5–8], shifts in the energyof the Dirac point, which is typically incorporated as ascalar potential [6, 8, 9], and changes in the position ofthe Dirac cone in the two-dimensional Brillouin zone, of-ten described by a pseudo-vector potential acting on thevalley degree of freedom [9–14]. Previous experimentsexplored some of these strain effects on a local scale us-ing scanning tunneling microscopy [15–23], Kelvin probeforce microscopy [24, 25], or angle-resolved photoemis-sion spectroscopy [26]. However, studying strain effectsin transport measurements and on a global scale is stillchallenging due to the lack of in situ strain tunabil-ity [21, 27, 28] or ambiguities resulting from simultaneouschanges in the gate capacitance [29–31].Here, we demonstrate the formation of a scalar poten-tial generated by systematically tuning the strain in amicrometer sized graphene electronic device and inves-tigate its effects on two fundamental electron transportphenomena, quasi-ballistic transport and the quantumHall effect (QHE). We find that all investigated transportcharacteristics are shifted systematically in gate voltage,qualitatively and quantitatively consistent with the ex-pectations for the scalar potential generated by the ap-plied strain, where the strain values are confirmed byseparate Raman spectroscopy experiments.The work function (WF) of a material, i.e. the en- ( a ) E vac increasing tensile strain W G0 ~ W G0 E D ~ E D ( b) ( c ) hBN/Gr/hBNCr/Au polyimide ∆ zcounter supports p h o s p h o r b r o n z e pushing wedge FIG. 1. (a)
Illustration of the strain induced shift of the Diracpoint. The vacuum level is labeled with E vac , and the Diracpoint and the work function of unstrained (strained) graphenewith E D ( e E D ) and W ( f W ), respectively. (b) Schematicsof the three-point bending setup and (c) the encapsulatedgraphene device. The displacement ∆ z of the pushing wedgecontrols the bending of the substrate and thus the inducedstrain in the graphene. ergy required to remove an electron from the material,is defined as the difference between the vacuum level E vac and the Fermi level E F of the material [32]. Forundoped graphene, E F coincides with the Dirac pointenergy E D [1], therefore the WF of undoped graphene is W = E vac − E D . A strain-induced scalar potential shifts E D , which therefore leads to a change in W . With in-creasing tensile strain, the scalar potential shifts E D tolower values, resulting in an increase in W [6, 8], as il- a r X i v : . [ c ond - m a t . m e s - h a ll ] S e p lustrated in Fig. 1(a). Quantitatively, strain shifts E D to e E D = E D + S , where S is the scalar potential, and canbe written as [6, 8, 9]: S ( x, y ) = − s · ( ε xx + ε yy ) , (1)with ε xx and ε yy the diagonal components of the straintensor, and s a constant defined for small strain val-ues. The value of s is not well established and theoret-ical values are reported in the range between 2 . . × . × . z [31]. The schematics of the device configuration isshown in Fig. 1(c). The edge contacts to graphene actas clamps for the strain generation and at the same timeas electrical contacts for transport experiments [31]. Ametallic global bottom gate is used to tune the chargecarrier density in the device. The on-chip hBN encapsu-lation ensures that the geometrical capacitance betweenthe gate and the graphene is not changed in the strain-ing process. Here, we investigate strain effects on deviceswith a rectangular geometry, which results in an essen-tially homogeneous uniaxial strain field. Details of thedevice fabrication and the strain field pattern are dis-cussed in [31].In our devices, the grounded graphene sheet and themetallic gate essentially form a plate capacitor. The de-tailed diagram of energy level alignment and its modifi-cation by strain are given in the Supplemental Material.The strain-induced scalar potential shifts the Dirac point,resulting in a systematic change in the charge carrier den-sity of the device at a given gate voltage, which we detectin transport experiments.To investigate the strain effect, we perform transportexperiments at liquid helium temperature ( T ≈ . G = dI/dV of asquare device is measured as a function of V g for differentbendings ∆ z of the substrate. An overview measurementis plotted in Fig. 2(a), on the scale of which no signifi-cant strain effects can be observed. The charge neutral-ity point (CNP) occurs at a positive gate voltage. Froma linear fit near the CNP we find a field effect chargecarrier mobility of ∼
130 000 cm V − s − , independent of∆ z , suggesting a high device quality and that randomstrain fluctuations are probably not dominating scatter-ing processes here [33]. The additional conductance min-imum at V g ≈ − . FIG. 2. (a)
Two-terminal differential conductance G as afunction of the gate voltage V g for different ∆ z . Inset: opticalimage of the measured device, scale bar: 2 µ m. (b) Zoom-inon the CNP. Inset: position of the CNP ( V CNP ) as a functionof ∆ z . V CNP is extracted as the gate voltage of minimumconductance. Red line is a linear fit with a slope of about −
10 mV / mm. (c) Zoom-in on the CNP for ∆ z = 0 and∆ z = 0 . (d) Same data as in (c) with the ∆ z = 0 . V g . both the top and the bottom hBN layers are aligned tothe graphene lattice [35].The zoom-in to the CNP plotted in Fig. 2(b) showsvery regular oscillations in conductance, which we ten-tatively attribute to Fabry-P´erot resonances in the re-gions near the electrical contacts with a different dopingcompared to the graphene bulk [36–39] (see Supplemen-tal Material for a detailed discussion). With increasing∆ z and therefore increasing tensile strain, these conduc-tance oscillations are shifted systematically to lower gatevoltages. This effect is fully reversible with deceasing∆ z , which is demonstrated in the Supplemental Mate-rial. The strain-induced shift is best seen by following theCNP: in the inset of Fig. 2(b) we plot the gate voltage ofminimum conductance, V CNP as a function of ∆ z , whichshows a linear decrease with increasing ∆ z , consistentwith the picture described in the Supplemental Material.To demonstrate that the complete conductance curvesare shifted with strain, we plot in Fig. 2(c) the two curveswith the lowest (∆ z = 0) and the highest (∆ z = 0 . z = 0 . V g . Wefind that all conductance curves merge to the same curveas at ∆ z = 0 (blue) when shifted by a constant gatevoltage offset. This shift we attribute to a strain-inducedscalar potential in the graphene sheet. We note that thiseffect is very different from bending-induced changes inthe gate capacitance found in suspended samples, wherethe V g axis is rescaled by a constant factor [31]. !" $ % & ’ ( ) "*+,"*+""* -"* . / $%.) %0) / $%.)12$%33)$"$"*4 %5) $6(789&54-, " $ % & ’ ( ) +*" *: *"!*:!*""*: . / $%.) $$$ %;) $< ,$< 4$< !" $ % & ’ ( ) "*+-"*+,"*+ "*+""* 4"* -"* ,"* . / $%.) %=) FIG. 3. (a)
Two-terminal differential conductance as a func-tion of gate voltage at three different magnetic fields for dif-ferent ∆ z . (b) Zoom-in to a small region in (a) , showing theshift of the curves in V g with increasing ∆ z . (c) Same as in (b) for ∆ z = 0 and ∆ z = 0 . (d) Same as in (c) withthe ∆ z = 0 . V g . To demonstrate that this is a general effect, indepen-dent of the device or the physical origin of the transportcharacteristics, we have investigated more than 5 devices,all showing similar effects (another example is provided in the Supplemental Material). Here, we now focus on theimpact of homogeneous uniaxial strain on the QHE in thesame device, and perform a similar analysis as for the zerofield measurements. Figure 3(a) shows the two-terminaldifferential conductance as a function of the gate volt-age for three different quantizing magnetic fields, B , andfor different ∆ z values. Typical quantum Hall plateausof graphene can be observed on the electron side, withsmall deviations of the plateau conductances from thequantized values 2, 6, 10 e /h due to the contact resis-tance. The plateaus at the filling factors ν = 0 and ν = 1are well developed alreday at B = 2 T, and more brokensymmetry states and fractional quantum Hall states canbe observed at B = 8 T [40–42], again highlighting thevery good device quality. In contrast, the plateaus onthe hole side are not well developed (see SupplementalMaterial) presumably due to a p-n junction forming nearthe contacts [43, 44]. Comparing the measurements fordifferent ∆ z on this scale shows no clear strain effects.However, in the data near the CNP shown in Fig. 3(b),we again find a systematic shift in V g with increasing ∆ z .The clear offset between the ∆ z = 0 . z = 0 curve (blue) is shown in Fig. 3(c). Shift-ing the red curve by 8 mV, as shown in Fig. 3(d), thetwo curves are virtually identical, in the same mannerand with the same shift as discussed for Fig. 2 with thedevice at zero magnetic field. Since the QHE is quite adifferent transport regime than quasi-ballistic transport,the observed effect is very general and we attribute it toa strain-induced scalar potential. tensile strainno strain ~~ ~ W M W M -eV CNP W G0 E F(M) E F(M) E D E D E vac E vac Metallic gate W G0 E F(G) ~Metallic gate e ∆ V CNP ∆ W G0 -eV CNP ~ FIG. 4. Schematic energy level diagram of the device at theCNP for unstrained (left) and strained (right, green shaded)graphene. The Fermi levels of the graphene and the metallicgate are denoted E ( G )F and E ( M )F ( e E ( M )F ), respectively. TheWF of the metallic gate is denoted W M , assumed to be aconstant. In our measurements, the graphene is grounded andtherefore E ( G )F is fixed. The gate voltages tuning the grapheneto the CNP for the unstrained and strained cases are denoted V CNP and e V CNP , respectively. The WF difference of undopedgraphene between with and without strain is denoted ∆ W . We now extract the scalar potential from the transportexperiments by evaluating the shift between the mini-mum (∆ z = 0) and maximum strain (∆ z = 0 . z = 0 . S ∆ z =0 . = − ∆ W = − e ∆ V CNP ≈ − . (2)To determine s in Eq. 1, we need to estimate theapplied strain. This we achieve using spatially resolvedRaman spectroscopy at room temperature on the samedevice [31, 45, 46]. For small uniaxial strain, a singleLorentzian describes the graphene Raman 2D peak, withthe center frequency ω redshifting linearly with in-creasing tensile strain. Figure 5 shows the mean centerfrequency ¯ ω averaged over the entire device area as afunction of ∆ z . With increasing ∆ z , ¯ ω shifts to lowervalues, indicating an increasing average strain in thegraphene [31]. Since the displacement ∆ z is much smallerthan the length of the substrate, the strain increaseslinearly with ∆ z , with a slope of ∼ . − / mm ex-tracted by linear fitting. Using ∂ω D /∂ε = −
54 cm − / %from the literature [47], we obtain a value for the in-duced tensile strain of ε = ε xx + ε yy ≈ .
21% at∆ z = 0 . s = − S/ε ≈ . !" ’ ( ) * ) + , - . / ./ =-- FIG. 5. Spatially averaged center frequency ¯ ω of the Raman2D peak plotted as a function of ∆ z . Black circles are datapoints and the red line is a linear fit. The linear decrease withincreasing ∆ z indicates an increasing average strain. predicted by theory [6, 8, 9] and is consistent with themost recent calculations [8].In conclusion, we have demonstrated how large scalehomogeneous strain in a graphene electronic device re-sults in a scalar potential, which we detect using trans-port experiments in two different regimes. Combinedwith strain values extracted from Raman spectroscopyon the same device, we report the first systematicallymeasured characteristic number for the scalar potentialstrength, consistent with the most recent theoretical cal-culations. This in situ strain tuning and the combina-tion of transport and Raman measurements thus con-firms the scalar potential as the origin of the observedstrain effects. Our study forms the basis to investigatestrain effects in transport experiments, which is crucialfor future strain engineering in graphene and related 2Dmaterials, such as generating a strain-induced in-planeelectric field for observing the phenomenon of the Lan-dau level collapse [8], realizing graphene quantum straintransistors [48], or creating a pseudo-magnetic field witha non-uniform strain field [9, 11]. Author contributions
L.W. fabricated the devices, performed the measure-ments and did the data analysis. A.B., P.M., S.Z. andC.S. helped to understand the data. B.V. performedparts of the Raman measurements. D.I. supported thesample fabrication. K.W. and T.T. provided the high-quality hBN. C.S. initiated and supervised the project.L.W. and A.B. wrote the paper and all authors discussedthe results and worked on the manuscript. All datain this publication are available in numerical form at:https://doi.org/10.5281/zenodo.4017429.
ACKNOWLEDGMENTS
This work has received funding from the SwissNanoscience Institute (SNI), the ERC project Top-Supra (787414), the European Union Horizon 2020 re-search and innovation programme under grant agree-ment No. 785219 (Graphene Flagship), the Swiss Na-tional Science Foundation, the Swiss NCCR QSIT, To-pograph, FlagERA network and from the OTKA FK-123894 grants. P.M. acknowledges support from theBolyai Fellowship, the Marie Curie grant, TopographFlagera network and the National Research, Develop-ment and Innovation Fund of Hungary within the Quan-tum Technology National Excellence Program (ProjectNo. 2017-1.2.1-NKP-2017-00001). K.W. and T.T. ac-knowledge support from the Elemental Strategy Initia-tive conducted by the MEXT, Japan, Grant NumberJPMXP0112101001, JSPS KAKENHI Grant NumbersJP20H00354 and the CREST(JPMJCR15F3), JST. 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1, 2, ∗ Andreas Baumgartner,
1, 2
P´eter Makk,
1, 3
Simon Zihlmann, Blesson S. Varghese, David I. Indolese, KenjiWatanabe, Takashi Taniguchi, and Christian Sch¨onenberger
1, 21
Department of Physics, University of Basel,Klingelbergstrasse 82, CH-4056 Basel, Switzerland Swiss Nanoscience Institute, University of Basel,Klingelbergstrasse 82, CH-4056 Basel, Switzerland Department of Physics, Budapest University of Technology and Economics andNanoelectronics Momentum Research Group of the Hungarian Academy of Sciences,Budafoki ut 8, 1111 Budapest, Hungary Research Center for Functional Materials,National Institute for Material Science,1-1 Namiki, Tsukuba, 305-0044, Japan International Center for Materials Nanoarchitectonics,National Institute for Materials Science,1-1 Namiki, Tsukuba 305-0044, Japan a r X i v : . [ c ond - m a t . m e s - h a ll ] S e p nergy level alignment diagram The energy level diagram of a graphene electronic device at a gate voltage V g is shown inFig. S1, where the band alignment between the metallic gate and the graphene connectedto the grounded reservoirs is given by E ( M )F − E ( G )F = − eV g , with E ( G )F and E ( M )F the Fermilevels of the graphene and the metallic gate, respectively. Since the graphene is groundedvia the metallic contacts, E ( G )F is fixed throughout the measurements. The work function(WF) difference between the metallic gate ( W M ) and undoped graphene ( W ) results inan electrostatic potential difference φ with a corresponding charge carrier density in thegraphene. Due to the low density of states in graphene, one also needs to take into accountthe corresponding kinetic energies in the form of a finite chemical potential µ ch , where E D labels the Dirac point energy. Charged impurities in the surrounding, e.g. trapped chargesin the dielectrics or adsorbed molecules nearby, induce an additional offset of the grapheneWF, which we term ∆W a [1, 2]. This offset then leads to a modification of µ ch . In the end,the actual WF of the graphene ( W G ) in the device is W G = ∆W a + W − µ ch . For theunstrained case, the fundamental relation between the relevant quantities can be directlyfound from Fig. S1: W M − eV g = W G − eφ = ( ∆W a + W − µ ch ) − eφ. (1)For tensile strain, the effect of the scalar potential is shown in the right part (shaded ingreen) of the diagram in Fig. S1, where we label the resulting changed quantities by symbolswith a tilde and assume that W M is a constant (due to the very large density of states inmetals) and ∆W a is not affected by strain. The strain-induced scalar potential shifts theDirac point energy, leading to an increase in the intrinsic WF of undoped graphene, from W to f W . This change has two effects in the device at a given gate voltage: a shift inthe electrostatic potential difference, and a shift in the chemical potential. This results in achange in the charge carrier density, which can be detected in transport experiments. Forthe strained case at e V g , the corresponding quantities in Eq. 1 should be replaced by thosewith a tilde, leading to W M − e e V g = f W G − e e φ = ( ∆W a + f W − e µ ch ) − e e φ. (2)2 ~~ ~~ ~~~ W M -eV g W G0 W G W G W G0 - e φ - e φ E F(M) E D E D E vac µ ch µ ch Metallic gate W M -eV g E F(M) E vac Metallic gate E F(G) ∆ W a ∆ W a tensile strainno strain FIG. S1. Schematic energy level diagram of the device for unstrained graphene at gate voltage V g (left) and for strained graphene at gate voltage e V g (right, green shaded). The symbols are definedin the text. In our measurements, the graphene is grounded through the metallic contacts andtherefore E ( G )F is fixed. To extract the changes in intrinsic WF of undoped graphene induced by strain, andtherefore the strain-induced scalar potential, we observe the evolution of a given conduc-tance feature, which we assume to occur at a specific charge carrier density, n . A fixed n corresponds to a fixed chemical potential ( µ ch = e µ ch ) in the graphene and a fixed electrostaticpotential difference ( φ = e φ ), assuming no changes in the gate capacitance by straining [3].Under these conditions, the strain-induced scalar potential S can be directly extracted fromEq. 1 and 2: S = − ( f W − W ) = e ( e V f g − V f g ) , (3)where V f g and e V f g are the gate voltages for a given conductance feature for the unstrainedand strained cases, respectively. This directly demonstrates that all conductance featuresare shifted by the same amount in gate voltage for a given global homogeneous straining, asobserved in the experiments in the main text.3 eversibility Here we present the measurements of the device shown in the main text for decreasing∆ z at zero magnetic field. The two-terminal differential conductance G as a function of gatevoltage V g is shown in Fig. S2(a) for different ∆ z . No significant changes can be seen on thisscale. An enlargement of the CNP is shown in Fig. S2(b), where the original ∆ z = 0 curve(black) is also added. With decreasing ∆ z , the curve gradually reverts back to the originalposition, which demonstrates the full reversibility of the strain-induced scalar potential. !" & ’ ( ) % * + , !-% / ’(.,(0, 12’(33,’!-"’!- & ’ ( ) % * + ,
64 6% 6 ! % 4. / ’(., 12’(33,’!-"’!- FIG. S2. (a)
Two-terminal differential conductance G plotted as a function of gate voltage V g fordecreasing ∆ z . “b” stands for “backwards” in ∆ z . (b) Zoom-in to the CNP with the original∆ z = 0 curve added as black. iscussion of the conductance oscillations Cr/Au hBN/Gr/hBN400 nm 400 nm
FIG. S3. Schematic cross section and optical microscope image of the device. The 400 nm overlapat each contact is intended for mechanical reinforcement.
Very regular oscillations in conductance are observed around the CNP, which stay unaf-fected by strain, as shown in Fig. S2(b). These features are reminiscent of the quantized con-ductance originating from quantum point contacts [4, 5]. However, the corresponding Fermiwavelengths at the carrier densities of these oscillations are on the order of a few hundred nm,estimated with λ F = 2 π/k F = 2 π/ √ πn and n = α ( V g − V CNP ) with α ≈ . × m − V − ,which are an order of magnitude smaller than the device width ( ∼ . µ m). Therefore, thepossibility of the quantized channel conductance as the origin of these oscillations is ruledout. Another possibility could be the Fabry-P´erot resonances due to a region near eachcontact with a doping level different to the bulk of the device [6–9]. This is possible sincethe electrical contacts to the graphene are below the metallic leads and the overlap region isabout 400 nm, as shown in Fig. S3. This design is intended for mechanical reinforcement ofthe contacts [3]. From the conductance oscillations in the measurement, the cavity length L can be estimated with L = √ π/ ( √ n j+1 − √ n j ), where √ n j+1 and √ n j are the correspondingcarrier densities of two consecutive oscillations [9]. A cavity length of ∼
450 nm is extractedfrom our measurements, which matches well the ∼
400 nm overlap near the contacts. Wetherefore tentatively attribute these oscillations to Fabry-P´erot resonances.5 eversibility in quantum Hall regime !" $ % & ’ ( ) "*+,"*+-"*+ "*+""* ."* ,"* -"* / $%/) %1) 23$%44)$"*.$"*,1$"*-1$"* 1$"1 $" !5!"5" $ % & ’ ( )
6+ 6 6! " ! +/ $%/)23$%44)$"*.$"*,1$"*-1$"* 1$"1%7) $8 -$8 .$8 FIG. S4. (a)
Two-terminal differential conductance G plotted as a function of the gate voltage V g for decreasing ∆ z at B = 2 T, 4 T and 8 T. “b” stands for “backwards” in ∆ z . (b) Zoom-inaround the ν = 1 quantum Hall plateau with the original ∆ z = 0 curve added as black. Figure S4(a) shows the measured data from the same device as in the main text, butfor decreasing strain values, at three different quantizing magnetic fields. No significantchanges are found on this scale, except for the very left part. The quantum Hall plateausin this region are not well developed due to a p-n junction forming near the contacts, whichaffects the electrical coupling of the graphene to the metallic contact. The changes observedhere might be due to a small change in the effective contact resistance with strain. An6nlargement around the ν = 1 plateau is shown in Fig. S4(b), where the shift of the curvesin gate voltage due to strain-induced scalar potential can be seen. The curve graduallyreverts back with decreasing ∆ z , indicating the reversibility of the observed strain effect.The small mismatch between the ∆ z = 0(b) curve (blue) and the original ∆ z = 0 curve(black) can be attributed to a slight mechanical hysteresis of the bending setup. Second monolayer device showing scalar potential
FIG. S5. Two-terminal differential conductance G plotted as a function of gate voltage V g fordifferent ∆ z at (a) B = 0 and (b) B = 4 T. The inset is an optical image of the measureddevice. Scale bar: 2 µ m. The corresponding enlargements near the CNP are shown in (c) and (d) ,respectively. Data for a second monolayer device showing the strain-induced scalar potential are givenhere as another example. The conductance curves without magnetic field and in the quantum7all regime are plotted in Fig. S5(a) and (b), respectively. No significant effects are observedwith increasing ∆ z on this scale, except for the very left part of Fig. S5(a), where theconductance is limited by the p-n junction near the contact. The changes seen in this regionmight be attributed to a small change in the contact resistance, which does not affect theinterpretation of the strain effect observed near the CNP, as shown in Fig. S5(c) and (d). Ashift of ∼ V g is observed for ∆ z = 0 . Fabrication and Raman measurements
The hBN/graphene/hBN heterostructures were first assembled using the standard pick-up technique with a PDMS/PC stamp and then deposited onto the metallic gate structureprefabricated on a polyimide-coated phosphor bronze plate. The typical thickness for the top(bottom) hBN is ∼
20 nm ( ∼
30 nm). The graphene flake was exfoliated from natural graphite.One-dimensional edge contacts [10] (Cr/Au, 5 nm/110 nm) were made to electrically connectthe graphene. A controlled etching recipe was employed to stop in the middle of the bottomhBN and the remaining hBN acts as the insulating layer between the metallic leads and thebottom gate [3], as can be seen in Fig. S3.The Raman measurements at room temperature were performed to determine the strainafter the low-temperature transport measurements. A commercially available confocal Ra-man system WiTec alpha300 was used. The Raman spectra were acquired using a linearlypolarized green laser (532 nm) with a power of 1 . ∗ [email protected][1] G. Giovannetti, P. A. Khomyakov, G. Brocks, V. M. Karpan, J. van den Brink, and P. J.Kelly, Phys. Rev. Lett. , 026803 (2008).[2] Y.-J. Yu, Y. Zhao, S. Ryu, L. E. Brus, K. S. Kim, and P. Kim, Nano Lett. , 3430 (2009).[3] L. Wang, S. Zihlmann, A. Baumgartner, J. Overbeck, K. Watanabe, T. Taniguchi, P. Makk,and C. Schnenberger, Nano Lett. , 4097 (2019).[4] B. J. van Wees, H. van Houten, C. W. J. Beenakker, J. G. Williamson, L. P. Kouwenhoven, . van der Marel, and C. T. Foxon, Phys. Rev. Lett. , 848 (1988).[5] B. Terr´es, L. A. Chizhova, F. Libisch, J. Peiro, D. Jrger, S. Engels, A. Girschik, K. Watanabe,T. Taniguchi, S. V. Rotkin, J. Burgdrfer, and C. Stampfer, Nature Communications (2016).[6] A. F. Young and P. Kim, Nature Physics , 222 (2009).[7] P. Rickhaus, R. Maurand, M.-H. Liu, M. Weiss, K. Richter, and C. Schnenberger, NatureCommunications (2013).[8] A. L. Grushina, D.-K. Ki, and A. F. Morpurgo, Applied Physics Letters , 223102 (2013).[9] C. Handschin, P. Makk, P. Rickhaus, M.-H. Liu, K. Watanabe, T. Taniguchi, K. Richter, andC. Sch¨onenberger, Nano Lett. , 328 (2017).[10] L. Wang, I. Meric, P. Y. Huang, Q. Gao, Y. Gao, H. Tran, T. Taniguchi, K. Watanabe, L. M.Campos, D. A. Muller, J. Guo, P. Kim, J. Hone, K. L. Shepard, and C. R. Dean, Science , 614 (2013)., 614 (2013).