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Josep Ingla-Aynés, Antonio L. R. Manesco, Talieh S. Ghiasi, Serhii Volosheniuk, [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), Herre S. J. van der Zant

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[Specular Electron Focusing between Gate-Defined Quantum Point Contacts in Bilayer Graphene](https://mdr.nims.go.jp/datasets/5ef2327c-ab69-47c5-8fad-41975a570212)

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Specular Electron Focusing between Gate-Defined Quantum Point Contacts in Bilayer GrapheneSpecular Electron Focusing between Gate-Defined Quantum PointContacts in Bilayer GrapheneJosep Ingla-Aynés,* Antonio L. R. Manesco, Talieh S. Ghiasi, Serhii Volosheniuk, Kenji Watanabe,Takashi Taniguchi, and Herre S. J. van der ZantCite This: Nano Lett. 2023, 23, 5453−5459 Read OnlineACCESS Metrics & More Article Recommendations *sı Supporting InformationABSTRACT: We report multiterminal measurements in a ballistic bilayergraphene (BLG) channel, where multiple spin- and valley-degeneratequantum point contacts (QPCs) are defined by electrostatic gating. Bypatterning QPCs of different shapes along different crystallographicdirections, we study the effect of size quantization and trigonal warping ontransverse electron focusing (TEF). Our TEF spectra show eight clearpeaks with comparable amplitudes and weak signatures of quantuminterference at the lowest temperature, indicating that reflections at thegate-defined edges are specular, and transport is phase coherent. Thetemperature dependence of the focusing signal shows that, despite thesmall gate-induced bandgaps in our sample (≲45 meV), several peaks arevisible up to 100 K. The achievement of specular reflection, which isexpected to preserve the pseudospin information of the electron jets, is promising for the realization of ballistic interconnects for newvalleytronic devices.KEYWORDS: ballistic transport, bilayer graphene, quantum point contact, trigonal warpingElectronic devices with well-defined ballistic electrontrajectories have triggered extensive research,1−4 and toexploit their full potential, specular reflection of electron jets isa major requirement. Electrostatically defined geometries areoptimal platforms to realize the specular reflection, as shownby transverse electron focusing (TEF) measurements.5−16 Inthis context, the exceptional electronic properties of graphenemake it an ideal candidate for a wide variety of gate-defineddevices where Klein tunneling enables new functional-ities.2,17−20 However, the absence of a bandgap complicatesthe creation of collimated beams and specular mirrors ingraphene. The former has been realized by etching high-mobility graphene devices in absorptive pinhole collimators.21The latter has been improved by recent fabrication progress,leading to the observation of multiple focusing peaks.10,13However, the reflection induced by disordered graphene edgesis not specular.22 This is a fundamental limitation that in TEFexperiments results in a decrease of the peak amplitude as thenumber of reflections at the edge increases9,10,12,13 andrandomizes the valley degree of freedom.22 An alternativeapproach has been implemented in the quantum Hall regime,where the gaps between Landau levels have been used to creategate-defined interferometers23−25 and quantum point contacts(QPCs).25 However, the effective confinement of carriers atzero magnetic field in monolayer graphene remains a challenge.In contrast, bilayer graphene (BLG) is a tunable-bandgapsemiconductor with a trigonally distorted Fermi surface.26−30It has recently been introduced as an ideal system for therealization of gate-defined QPCs31−39 capable of transmittingvalley-polarized electron jets40 and of hosting quantum dotswith controllable spin and valley polarizations.41,42 Eventhough BLG hosts extraordinary properties, such as chirality-assisted cloaking43,44 or anti-Klein tunneling,45 experiments ongate-defined BLG devices have so far focused on thecharacterization of QPCs,31−33,36−38,40 quantum dots,41,42,46quantum interference effects,47 and topological edge chan-nels.48−52In this work, we exploit the electrically tunable bandgap ofBLG to create ballistic multiterminal BLG devices and measureTEF between gate-defined QPCs. We observe up to eightfocusing peaks with comparable amplitudes, which is a clearindication of specular reflection at the gate-defined edges.Temperature-dependent measurements show that the TEFsignal persists at up to 100 K.We fabricated two double-gated, boron nitride (hBN)-encapsulated BLG heterostructures on few-layer graphene backgates, each containing multiple devices using the dry transferReceived: February 8, 2023Revised: June 2, 2023Published: June 8, 2023Letterpubs.acs.org/NanoLett© 2023 The Authors. Published byAmerican Chemical Society5453https://doi.org/10.1021/acs.nanolett.3c00499Nano Lett. 2023, 23, 5453−5459Downloaded via NATL INST FOR MATLS SCIENCE (NIMS) on June 30, 2023 at 23:54:31 (UTC).See https://pubs.acs.org/sharingguidelines for options on how to legitimately share published articles.https://pubs.acs.org/action/doSearch?field1=Contrib&text1="Josep+Ingla-Ayne%CC%81s"&field2=AllField&text2=&publication=&accessType=allContent&Earliest=&ref=pdfhttps://pubs.acs.org/action/doSearch?field1=Contrib&text1="Antonio+L.+R.+Manesco"&field2=AllField&text2=&publication=&accessType=allContent&Earliest=&ref=pdfhttps://pubs.acs.org/action/doSearch?field1=Contrib&text1="Talieh+S.+Ghiasi"&field2=AllField&text2=&publication=&accessType=allContent&Earliest=&ref=pdfhttps://pubs.acs.org/action/doSearch?field1=Contrib&text1="Serhii+Volosheniuk"&field2=AllField&text2=&publication=&accessType=allContent&Earliest=&ref=pdfhttps://pubs.acs.org/action/doSearch?field1=Contrib&text1="Kenji+Watanabe"&field2=AllField&text2=&publication=&accessType=allContent&Earliest=&ref=pdfhttps://pubs.acs.org/action/doSearch?field1=Contrib&text1="Takashi+Taniguchi"&field2=AllField&text2=&publication=&accessType=allContent&Earliest=&ref=pdfhttps://pubs.acs.org/action/doSearch?field1=Contrib&text1="Takashi+Taniguchi"&field2=AllField&text2=&publication=&accessType=allContent&Earliest=&ref=pdfhttps://pubs.acs.org/action/doSearch?field1=Contrib&text1="Herre+S.+J.+van+der+Zant"&field2=AllField&text2=&publication=&accessType=allContent&Earliest=&ref=pdfhttps://pubs.acs.org/action/showCitFormats?doi=10.1021/acs.nanolett.3c00499&ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.nanolett.3c00499?ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.nanolett.3c00499?goto=articleMetrics&ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.nanolett.3c00499?goto=recommendations&?ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.nanolett.3c00499?goto=supporting-info&ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.nanolett.3c00499?fig=tgr1&ref=pdfhttps://pubs.acs.org/toc/nalefd/23/12?ref=pdfhttps://pubs.acs.org/toc/nalefd/23/12?ref=pdfhttps://pubs.acs.org/toc/nalefd/23/12?ref=pdfhttps://pubs.acs.org/toc/nalefd/23/12?ref=pdfpubs.acs.org/NanoLett?ref=pdfhttps://pubs.acs.org?ref=pdfhttps://pubs.acs.org?ref=pdfhttps://doi.org/10.1021/acs.nanolett.3c00499?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-ashttps://pubs.acs.org/NanoLett?ref=pdfhttps://pubs.acs.org/NanoLett?ref=pdfhttps://creativecommons.org/licenses/by/4.0/https://creativecommons.org/licenses/by/4.0/https://acsopenscience.org/open-access/licensing-options/technique described in refs 53 and 54. The electrodes weredefined by using conventional e-beam lithography. The BLGflakes were connected to Ti/Au electrodes (brown rectanglesin Figure 1a) after using a CHF3/O2 plasma to etch the upperhBN and BLG layers at the contact area.55 The top gates,which are dark yellow in Figure 1a, were deposited on the tophBN (see the Supporting Information (SI) Section S1 for thefabrication details). The side and top view images of a typicalQPC are shown in Figure 1a. Here we discuss the results onthe first heterostructure (sample 1); the results on sample 2 areshown in SI Section S9.The two-terminal resistance of the QPC, defined as R = V/I,where V and I are the measured voltage and applied current,respectively (Figure 1a, right panel), has been recorded as afunction of the top gate voltage (Vtg) and the back gate voltage(Vbg). As shown in Figure 1b, three features can bedistinguished from this result: The first one is a vertical lineat Vbg ≈ 0, which corresponds to the charge neutrality point(CNP) of the non-top-gated BLG channel. The CNP does notoccur at exactly Vtg = 0 due to small hole doping. The secondfeature is a faint vertical line at Vbg ≈ −1 V. Four-terminalmeasurements (see SI Section S3) indicate that it correspondsto the CNP of the BLG near the Ti/Au contacts, where the tophBN and BLG have been etched.The last feature is a diagonal line that has a negative slope(Vtg decreases as Vbg increases) that corresponds to the CNPof the regions under TG1 and TG2. Because both Vbg and Vtginfluence the carrier density (n) at these regions, theintroduction of electrons by Vbg to the BLG channel must becounteracted by an opposite Vtg to keep the channel chargeneutral. We use the slope of this line to obtain the ratiobetween the top gate (Ctg) and back gate (Cbg) capacitances: β= Cbg/Ctg = −ΔVtg/ΔVbg ≈ 1.22. This value is consistent withthe factor of 1.22 obtained from the ratio between the hBN-flake thicknesses extracted from AFM imaging (see SI SectionS1). Even though the electric field applied by the gates opens abandgap in the double-gated BLG regions which increases with|Vbg|,26−29 the resistance along the diagonal line does notincrease with |Vbg|. This is due to the small gap between TG1and TG2 (Figure 1a). In this region the carrier density is notzero, leading to the formation of a Vbg-controlled QPC withtunable carrier density.To determine if the QPC conductance (G) is quantized, wehave determined its resistance by taking, for each Vbg, thedifference between the maximal and the minimal R. Thisoperation allows us to subtract the resistances of the Ti/Aucontacts and the BLG regions that are not affected by Vtg. Theresults are shown in Figure 1c. For negative Vbg, G showsvalues higher than 7 × 4e2/h and changes in a monotonic waywith small oscillations. In contrast, for positive Vbg, G showsfour steps at G = N × 4e2/h with N = 1, 2, 3, and 4. Thisbehavior, which is reproduced in five of the six QPCscharacterized, indicates the formation of a spin and valley-degenerate QPC.33,36,37,56 Note that the sharp increase of Gnear Vbg = 0 is a consequence of the extraction method whenthere is no bandgap under the double-gated regions, and Rshows very small changes with Vtg. Even though the reason forthe electron−hole asymmetry is not clear, we believe that onepossibility may be a residual doping of the double-gatedregions caused by the fabrication. Because the QPC region isnot affected by this process, the potential landscape couldbecome asymmetric to the sign reversal of the gate voltages,modifying the confinement potential. This could make theQPC narrower for electron than hole doping or modify itscarrier density. In addition, the electric field applied on theBLG at the CNP changes sign with Vbg, leading to oppositelayer polarizations57 which could also enhance the asymmetry.When a magnetic field (B) is applied perpendicular to theplane of a ballistic BLG device, electrons deviate from theirstraight trajectories by the Lorentz force. If the Fermi surface iscircular, they follow circular orbits with radius rc = ℏkF/eB,where ℏ is the reduced Planck constant and kF is the Fermiwavevector =k n( )F . As a consequence, the transmissionbetween different contacts connected at a distance L from eachother shows maxima at magnetic fields (Bf) given by5,7=Bp keL2 cosfF(1)where θ is the angle at which the electron flow departs fromthe emitter, L = 2 μm is the injector−detector distance, and p= 1, 2, 3, ..., n is an integer which accounts for the p − 1reflections that occur at the device edge between the contacts(Figure 2a).Figure 1. Gate-defined QPCs in the BLG at 1.8 K. (a) Side (left) andtop (right) view of the fabricated device. The top view is a false-colorAFM image. The separation between the split top gates (TG1 andTG2) is approximately 50 nm, and their width is 580 nm. At the sideview (left panel), the hBN layers are green, and the BLG and the few-layer graphene back gate (BG) are black. In both panels, the contactsto the BLG flake are brown, and the top gates (TG) are dark yellow.(b) Two-terminal resistance (R) of one of the contacts used for thetransverse electron focusing experiments as a function of Vbg and Vtg.Vtg is the same for TG1 and TG2. (c) Point contact conductanceobtained along the diagonal line in panel b that follows Vtg ≈ Vtg0 −βVbg, corresponding to the charge neutrality point of the double-gatedregions. Vtg0 ≈ −1.08 V is the charge neutrality point at Vbg = 0, and βis the ratio between the back and top gate capacitances. The TEFmeasurements are also performed along this line.Nano Letters pubs.acs.org/NanoLett Letterhttps://doi.org/10.1021/acs.nanolett.3c00499Nano Lett. 2023, 23, 5453−54595454https://pubs.acs.org/doi/suppl/10.1021/acs.nanolett.3c00499/suppl_file/nl3c00499_si_001.pdfhttps://pubs.acs.org/doi/suppl/10.1021/acs.nanolett.3c00499/suppl_file/nl3c00499_si_001.pdfhttps://pubs.acs.org/doi/suppl/10.1021/acs.nanolett.3c00499/suppl_file/nl3c00499_si_001.pdfhttps://pubs.acs.org/doi/suppl/10.1021/acs.nanolett.3c00499/suppl_file/nl3c00499_si_001.pdfhttps://pubs.acs.org/doi/10.1021/acs.nanolett.3c00499?fig=fig1&ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.nanolett.3c00499?fig=fig1&ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.nanolett.3c00499?fig=fig1&ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.nanolett.3c00499?fig=fig1&ref=pdfpubs.acs.org/NanoLett?ref=pdfhttps://doi.org/10.1021/acs.nanolett.3c00499?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-asTEF measurements were performed using configuration C1,which is shown in Figure 2a. A current (I) is applied to theright QPC to generate electron flow into the ballistic BLGchannel that is steered using the out-of-plane B field. To detectthe ballistic skipping orbits, the nonlocal voltage (Vnl) ismeasured between the left QPC and a reference electrodeconnected to the left of the BLG channel. To avoid voltage Vnloffsets, we used a differential DC measurement technique toobtain the TEF spectra in Figures 2 and 3.The results from such measurements performed for differentVbg values are shown in Figure 2b. Note that, to ensure that thecharge transport occurs only through the QPCs, we haveadjusted Vtg to keep the double-gated regions charge neutral(diagonal line in Figure 1b). We first consider the Vbg = −4 Vcase. For B < 0, the signal is zero (dashed lines) or smaller thanthe noise level of the measurement, which is 2 Ω, consistentwith the fact that the ballistic electron stream deviates towardthe right and does not generate a signal on the detector. Incontrast, when B > 0, five clear focusing peaks are observed,indicating that even though the QPC conductance is notquantized for Vbg < 0 (Figure 1c), the hole trajectories are well-defined and reflection at the gate-defined edge between bothQPCs is smooth. As Vbg approaches zero, n in the BLGchannel decreases, and the distance between the peaksbecomes smaller. At Vbg > 0 peaks occur for B < 0, consistentwith the fact that the carriers have changed from holes toelectrons.9,10,13For a more detailed comparison, in Figure 2c we show theVbg = ±3 V spectra, where the bandgap is approximately 45meV.29 Two clear differences can be distinguished: (i) The p =1 peak is 2 times higher for Vbg = +3 V. This is most likely dueto the lower G at Vbg = +3 V, which converts the collectorcurrent (Ic) into the measured Vnl = Ic/G. As shown in Figure1c, G is roughly 2 times larger for Vbg = −3 V than for Vbg = +3V, explaining most of the measured asymmetry in the p = 1peak magnitude. (ii) The peak amplitude decays with p muchfaster at Vbg = −3 V. To quantify the TEF signal decay with pand correct for a small contact magnetoresistance (see SISection S4), we calculated the area under the TEF peaks10normalized by the two-terminal resistance (see SI SectionS5b). The result is shown in the inset of Figure 2c with a linearfit excluding the p = 1 peak (which has the smallest area). Theobtained peak areas are fairly constant from p = 2 up to p = 8(including the p = 4 peak, which occurs between 0.75 and 1 Tand is split in two), indicating specular reflection. In contrast,for Vbg = −3 V, the peak area decays with increasing p.The faster peak decay for Vbg < 0 can be explained in termsof a change of the QPC width (W). The finite W of thedetector poses an upper bound of the maximum number ofpeaks that can be measured. In particular, if rc cos θ ≤ W/2, allelectrons will enter the detector and extra peaks cannot bedetected,7 leading to B ≤ 2 T forW = 200 nm, cos θ = 1, and acircular trajectory. In contrast, for W = 100 nm, we obtain B ≤3.9 T. As shown in Figure 1c, G is almost 8 times smaller forelectrons than for holes, indicating that a significant electron−hole asymmetry in the QPC width is plausible. Additionally,decreasing the injectorW is known to lead to electron jets withimproved collimation.21,58 Because the focusing length of atrajectory depends on its injection angle, the differencesbetween focusing lengths of different trajectories increase withp. Thus, a narrow angular distribution is expected to helpmaintain a constant peak amplitude, even after several edgereflections.Figure 2. Transverse electron focusing between gate-defined QPCs in the BLG at 1.8 K. (a) Measurement geometry. The nonlocal voltage (Vnl) ismeasured as a function of B while applying current I between the right QPC and a reference lead. The ballistic trajectories are sketched for thethree first focusing peaks, which involve 0, 1, and 2 reflections with the gate-defined edge and assuming no trigonal warping. The scale bar is 2 μm.(b) Nonlocal resistance (Rnl = Vnl/I) as a function of B for different Vbg values. The dashed lines show the spectra offsets, which have beenintroduced for clarity. Vtg is tuned to follow the charge neutrality line of the top-gated regions (diagonal line in Figure 1b). (c) Focusing spectraextracted from panel b at Vbg = ±3 V. A small offset in B was added to correct for the magnet remanence. The inset shows the evolution of thenormalized area under the peaks (Ap/A1) with p (dots), and the lines are fits to illustrate the trends. (d) Peak separation as a function of Vbg. Thevertical error bars are the uncertainties from the Bf vs p linear fit, and the horizontal ones account for a 0.1 V uncertainty of the CNP. The black lineis the result from eq 1 assuming normal incidence from the QPCs (θ = 0). The gray area corresponds to the experimental error from determining n(14%) and L (10%). Simulated TEF signal for perfectly aligned (e) and 3° misaligned (f) QPCs with respect to the armchair crystallographicdirection. The black curves were obtained by adding the K and K′ valley-resolved spectra. The insets show the trigonally warped trajectoriescorresponding to the average incidence angles for valleys K and K′, and the dark yellow rectangles represent the gate-defined edges.Nano Letters pubs.acs.org/NanoLett Letterhttps://doi.org/10.1021/acs.nanolett.3c00499Nano Lett. 2023, 23, 5453−54595455https://pubs.acs.org/doi/suppl/10.1021/acs.nanolett.3c00499/suppl_file/nl3c00499_si_001.pdfhttps://pubs.acs.org/doi/suppl/10.1021/acs.nanolett.3c00499/suppl_file/nl3c00499_si_001.pdfhttps://pubs.acs.org/doi/10.1021/acs.nanolett.3c00499?fig=fig2&ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.nanolett.3c00499?fig=fig2&ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.nanolett.3c00499?fig=fig2&ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.nanolett.3c00499?fig=fig2&ref=pdfpubs.acs.org/NanoLett?ref=pdfhttps://doi.org/10.1021/acs.nanolett.3c00499?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-asIn Figure 2b, for Vbg = 1.5 V (red curve), additionaloscillations similar to those in refs 7 and 25 can be observed ontop of the focusing spectrum. The amplitude of theseoscillations decreases with increasing Vbg, a result that isconsistent with quantum interference between the differentelectron paths contributing to the TEF signal because theFermi wavelength increases with decreasing n. For complete-ness, Rnl near Vbg = 0 V is shown in SI section S7.To gain more insight into the measured TEF spectra, weanalyzed the positions of the focusing peaks (Bf) as a functionof p. In particular, we determined Bf and fit it to Bf = B0 + (dB/dp) × p, where B0 and dB/dp are constants accounting for themagnet remanence and the average spacing between the peaks,respectively. In Figure 2d we show |dB/dp| and compare it withthe result from eq 1 for normal incidence (θ = 0). Thesimilarity between both curves further confirms that our signalis due to TEF. Even though the small underestimation of Bf byeq 1 could be due to trigonal warping,9 the orientationdependence of Bf expected from ref 9 is not observed here (seeSI Section S6), preventing a conclusive statement.The results shown in Figure 2c at Vbg = +3 V show featuresresembling a beating pattern. In particular, all the peaks exceptp = 1 and 4 can be decomposed into two narrower peaks, andthe latter, which has a dip where one would expect a peak, canbe decomposed into three well-separated peaks. Additionally,the Fourier transform of the TEF spectrum (see SI Section S5bfor details) also indicates the presence of a beating pattern,implying a periodic modulation. Even though there may be acombination of impurities or irregularities at the confinementpotential that could explain this effect, there is a fundamentalreason to expect such features in the TEF spectra. BLG isknown for showing trigonal warping; i.e., its Fermi surface isnot circular. In this case, the emission of electrons by theQPCs occurs in jets that depend on the crystallographicorientation of the QPCs on the BLG. If the QPCs are slightlymisaligned with respect to a crystallographic direction, thevalley-polarized jets will be emitted with slightly different |θ|,leading to two different Bf values for the peaks in valleys K andK′.40Semiclassical calculations considering the effect of trigonalwarping (see SI Section S9 for details) are shown in Figures 2eand 2f for the perfectly aligned and the small misalignment(0.05 rad ≈ 3°) cases, respectively. The trajectories are shownin the insets. In the latter, a beating pattern arises that iscompatible with the measured data.To show the robustness of the TEF measurements andexplore the role of the BLG crystallographic orientation in theTEF spectra, we have patterned QPCs in different directionson the same BLG flake. The relative angle between the QPCsets is 30° to compare the armchair with the zigzagcrystallographic directions. As shown in Figure 3a, C2 isaligned parallel to the longest BLG straight edge with anaccuracy of ∼5°. Because we expect the flake edge to bealigned with a crystallographic direction with an uncertainty ofa few degrees,59,60 we assume that the C2 QPCs are aligned toa crystallographic direction with an accuracy of less than 10°.Thus, the 30° rotated C1 QPCs, are expected to be along theother. We compare the TEF spectra in Figure 2c with the TEFspectra obtained using configurations C2, C3, and C4 fromFigure 3a, shown in Figures 3b, 3c, and 3d, respectively. Theresults show several features: (i) The TEF peaks decay fasterwith p for holes than for electrons in all the geometries. (ii) ForC3 and C4, which contain hornlike QPCs not showing sizequantization (see SI Section S6 for details), the decay in peakamplitude for electrons is more pronounced than for C1 andC2 where G is quantized. As a consequence, six peaks can bedistinguished instead of eight. (iii) The width of the p = 1 peakis significantly smaller than that of the p = 2 peak in all theconfigurations, for both electron and hole doping. Observa-tions (i) and (ii) show a correlation between G and the TEFpeak amplitude decay, further indicating that the QPC widthplays a relevant role in the peak amplitude decrease.Because the occurrence of a beating pattern is very sensitiveto a tiny misalignment, the lack of a clear beating pattern inFigure 3b is not inconsistent with the Fermi surface beingwarped.To characterize the scattering sources in BLG, we havemeasured Rnl vs B at different temperatures (T) at Vbg = ±3 V.At 2 K, the peak height is the highest, and as T increases, thebackground becomes more pronounced and the focusing signalgets smaller. Comparing the 2 K with the 10 K measurements,the 2 K spectra contain extra features at positive and negativeB-fields. A fast decay when increasing T indicates that thesefeatures are likely due to quantum interference, as the phase-coherence length is known to drop within this range.61To extract the T dependence of the scattering rate (τp−1)from Figures 4a and 4b, we have followed ref 10 (see SISection S5 for details). The result obtained using the areaunder the p = 2 peak is shown in Figure 4c. Here, the dotscorrespond to the values extracted from Figure 4a,b, and thelines are fits to τp−1 = aT2 + bT + c. Assuming a hard-wallconfinement potential, a quadratic T dependence of τp−1 isFigure 3. TEF along different crystallographic directions at 1.8 K. (a)False-color AFM image with the QPCs involved in C1 (Figure 2), C2,C3, and C4 delimited by white dashed lines. C2 is rotated an angle β= 30° with respect to C1 and C4 is rotated β = −30° with respect toC3. The scale bar is 5 μm. The BLG edges are indicated by blackdashed lines. (b−d) TEF in configurations C2−C4 at Vbg = ±3 V.The insets show the normalized peak area vs p and the lines are linearfits to illustrate the trend.Nano Letters pubs.acs.org/NanoLett Letterhttps://doi.org/10.1021/acs.nanolett.3c00499Nano Lett. 2023, 23, 5453−54595456https://pubs.acs.org/doi/suppl/10.1021/acs.nanolett.3c00499/suppl_file/nl3c00499_si_001.pdfhttps://pubs.acs.org/doi/suppl/10.1021/acs.nanolett.3c00499/suppl_file/nl3c00499_si_001.pdfhttps://pubs.acs.org/doi/suppl/10.1021/acs.nanolett.3c00499/suppl_file/nl3c00499_si_001.pdfhttps://pubs.acs.org/doi/suppl/10.1021/acs.nanolett.3c00499/suppl_file/nl3c00499_si_001.pdfhttps://pubs.acs.org/doi/suppl/10.1021/acs.nanolett.3c00499/suppl_file/nl3c00499_si_001.pdfhttps://pubs.acs.org/doi/suppl/10.1021/acs.nanolett.3c00499/suppl_file/nl3c00499_si_001.pdfhttps://pubs.acs.org/doi/10.1021/acs.nanolett.3c00499?fig=fig3&ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.nanolett.3c00499?fig=fig3&ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.nanolett.3c00499?fig=fig3&ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.nanolett.3c00499?fig=fig3&ref=pdfpubs.acs.org/NanoLett?ref=pdfhttps://doi.org/10.1021/acs.nanolett.3c00499?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-asassociated with electron−electron interactions.10,62,63 Incontrast, a linear dependence is associated with phonon-dominated scattering.9,64 The fits indicate that electron−electron interactions may play a relevant role in the T-dependent scattering for electrons, but not for holes; see SISection S5 for the fitting parameters and a more detaileddiscussion. We suspect that the background signals in Figure 4are caused by a small miscalibration of Vtg.To conclude, we have measured the TEF in hBN-encapsulated BLG devices where QPCs are defined in differentdirections using electrostatic gating. Our results show eightfocusing peaks with similar amplitudes together with quantuminterference features. By comparing TEF spectra with semi-classic simulations, we identify a periodic modulation of thepeak size that is consistent with the effect of trigonal warping.Moreover, the TEF temperature dependence shows that thesignal persists up to 100 K. Our results are promising for futurevalleytronic devices.■ ASSOCIATED CONTENTData Availability StatementAll the data and code associated with the analysis andtheoretical simulations are available free of charge from ref 65.*sı Supporting InformationThe Supporting Information is available free of charge athttps://pubs.acs.org/doi/10.1021/acs.nanolett.3c00499.Details on the fabrication and measurement techniques,characterization of the backgate capacitance, electronicmobility, two-terminal magnetoresistance, extraction ofthe area under the focusing peaks, the complete TEFdata set, the results obtained in sample 2, and numericalsimulations (PDF)■ AUTHOR INFORMATIONCorresponding AuthorJosep Ingla-Aynés − Kavli Institute of Nanoscience, DelftUniversity of Technology, 2628 CJ Delft, The Netherlands;orcid.org/0000-0001-9179-1570; Email: J.InglaAynes@tudelft.nlAuthorsAntonio L. R. Manesco − Kavli Institute of Nanoscience, DelftUniversity of Technology, 2628 CJ Delft, The NetherlandsTalieh S. Ghiasi − Kavli Institute of Nanoscience, DelftUniversity of Technology, 2628 CJ Delft, The Netherlands;orcid.org/0000-0002-3490-5356Serhii Volosheniuk − Kavli Institute of Nanoscience, DelftUniversity of Technology, 2628 CJ Delft, The NetherlandsKenji Watanabe − Research Center for Functional Materials,National Institute for Materials Science, Tsukuba 305-0044,Japan; orcid.org/0000-0003-3701-8119Takashi Taniguchi − International Center for MaterialsNanoarchitectonics, National Institute for Materials Science,Tsukuba 305-0044, Japan; orcid.org/0000-0002-1467-3105Herre S. J. van der Zant − Kavli Institute of Nanoscience,Delft University of Technology, 2628 CJ Delft, TheNetherlands; orcid.org/0000-0002-5385-0282Complete contact information is available at:https://pubs.acs.org/10.1021/acs.nanolett.3c00499Author ContributionsJ.I.A. and H.S.J.v.d.Z. conceived the experiment. J.I.A.fabricated the devices with help from T.S.G. and S.V. J.I.A.performed the measurements with help from T.S.G. A.R.L.M.performed the tight-binding simulations. A.R.L.M. and J.I.A.performed the semiclassical simulations. K.W. and T.T.synthesized the hexagonal boron nitride crystals. J.I.A. wrotethe manuscript with inputs from all authors. H.S.J.v.d.Z.supervised the project.NotesThe authors declare no competing financial interest.■ ACKNOWLEDGMENTSWe thank Prof. K. Ensslin and K. Vilkelis for insightfuldiscussions. This project received funding from the EuropeanUnion Horizon 2020 research and innovation program undergrant agreement no. 863098 (SPRING). J.I.A. acknowledgessupport from the European Union’s Horizon 2020 researchand innovation programme for a Marie Sklodowska−Curieindividual fellowship No. 101027187-PCSV. ALRM work wassupported by VIDI Grant 016.Vidi.189.180. K.W. and T.T.acknowledge support from JSPS KAKENHI (Grants19H05790, 20H00354, and 21H05233).Figure 4. Temperature dependence of TEF in BLG. B dependence ofRnl for T = 2, 10, 20, ..., 100 K at (a) Vbg = +3 V and (b) Vbg = −3 V.The insets correspond to the 100 K data with a smooth backgroundcorrected. (c) Scattering rate estimated using the spectra in panels aand b (dots) and its fit to a parabola (lines). The inset shows the Tdependence of the QPC resistance.Nano Letters pubs.acs.org/NanoLett Letterhttps://doi.org/10.1021/acs.nanolett.3c00499Nano Lett. 2023, 23, 5453−54595457https://pubs.acs.org/doi/suppl/10.1021/acs.nanolett.3c00499/suppl_file/nl3c00499_si_001.pdfhttps://pubs.acs.org/doi/10.1021/acs.nanolett.3c00499?goto=supporting-infohttps://pubs.acs.org/doi/suppl/10.1021/acs.nanolett.3c00499/suppl_file/nl3c00499_si_001.pdfhttps://pubs.acs.org/action/doSearch?field1=Contrib&text1="Josep+Ingla-Ayne%CC%81s"&field2=AllField&text2=&publication=&accessType=allContent&Earliest=&ref=pdfhttps://orcid.org/0000-0001-9179-1570https://orcid.org/0000-0001-9179-1570mailto:J.InglaAynes@tudelft.nlmailto:J.InglaAynes@tudelft.nlhttps://pubs.acs.org/action/doSearch?field1=Contrib&text1="Antonio+L.+R.+Manesco"&field2=AllField&text2=&publication=&accessType=allContent&Earliest=&ref=pdfhttps://pubs.acs.org/action/doSearch?field1=Contrib&text1="Talieh+S.+Ghiasi"&field2=AllField&text2=&publication=&accessType=allContent&Earliest=&ref=pdfhttps://orcid.org/0000-0002-3490-5356https://orcid.org/0000-0002-3490-5356https://pubs.acs.org/action/doSearch?field1=Contrib&text1="Serhii+Volosheniuk"&field2=AllField&text2=&publication=&accessType=allContent&Earliest=&ref=pdfhttps://pubs.acs.org/action/doSearch?field1=Contrib&text1="Kenji+Watanabe"&field2=AllField&text2=&publication=&accessType=allContent&Earliest=&ref=pdfhttps://orcid.org/0000-0003-3701-8119https://pubs.acs.org/action/doSearch?field1=Contrib&text1="Takashi+Taniguchi"&field2=AllField&text2=&publication=&accessType=allContent&Earliest=&ref=pdfhttps://orcid.org/0000-0002-1467-3105https://orcid.org/0000-0002-1467-3105https://pubs.acs.org/action/doSearch?field1=Contrib&text1="Herre+S.+J.+van+der+Zant"&field2=AllField&text2=&publication=&accessType=allContent&Earliest=&ref=pdfhttps://orcid.org/0000-0002-5385-0282https://pubs.acs.org/doi/10.1021/acs.nanolett.3c00499?ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.nanolett.3c00499?fig=fig4&ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.nanolett.3c00499?fig=fig4&ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.nanolett.3c00499?fig=fig4&ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.nanolett.3c00499?fig=fig4&ref=pdfpubs.acs.org/NanoLett?ref=pdfhttps://doi.org/10.1021/acs.nanolett.3c00499?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-as■ REFERENCES(1) Bøggild, P.; Caridad, J. 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