==== Front
Nano Lett
Nano Lett
nl
nalefd
Nano Letters
1530-6984
1530-6992
American Chemical Society

37289250
10.1021/acs.nanolett.3c00499
Letter
Specular Electron Focusing between Gate-Defined Quantum Point Contacts in Bilayer Graphene
https://orcid.org/0000-0001-9179-1570
Ingla-Aynés Josep *†
Manesco Antonio L. R. †
https://orcid.org/0000-0002-3490-5356
Ghiasi Talieh S. †
Volosheniuk Serhii †
https://orcid.org/0000-0003-3701-8119
Watanabe Kenji ‡
https://orcid.org/0000-0002-1467-3105
Taniguchi Takashi §
https://orcid.org/0000-0002-5385-0282
van der Zant Herre S. J. †
† Kavli Institute of Nanoscience, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands
‡ Research Center for Functional Materials, National Institute for Materials 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
* Email: J.InglaAynes@tudelft.nl.
08 06 2023
28 06 2023
23 12 54535459
08 02 2023
02 06 2023
© 2023 The Authors. Published by American Chemical Society
2023
The Authors
https://creativecommons.org/licenses/by/4.0/ Permits the broadest form of re-use including for commercial purposes, provided that author attribution and integrity are maintained (https://creativecommons.org/licenses/by/4.0/).

We report multiterminal measurements in a ballistic bilayer graphene (BLG) channel, where multiple spin- and valley-degenerate quantum point contacts (QPCs) are defined by electrostatic gating. By patterning QPCs of different shapes along different crystallographic directions, we study the effect of size quantization and trigonal warping on transverse electron focusing (TEF). Our TEF spectra show eight clear peaks with comparable amplitudes and weak signatures of quantum interference at the lowest temperature, indicating that reflections at the gate-defined edges are specular, and transport is phase coherent. The temperature dependence of the focusing signal shows that, despite the small gate-induced bandgaps in our sample (≲45 meV), several peaks are visible up to 100 K. The achievement of specular reflection, which is expected to preserve the pseudospin information of the electron jets, is promising for the realization of ballistic interconnects for new valleytronic devices.

ballistic transport
bilayer graphene
quantum point contact
trigonal warping
European Commission 10.13039/100010665 101027187 Nederlandse Organisatie voor Wetenschappelijk Onderzoek 10.13039/501100003958 016.Vidi.189.180 Ministry of Education, Culture, Sports, Science and Technology 10.13039/501100001691 21H05233 Ministry of Education, Culture, Sports, Science and Technology 10.13039/501100001691 20H00354 Ministry of Education, Culture, Sports, Science and Technology 10.13039/501100001691 19H05790 European Commission 10.13039/100010670 863098 document-id-old-9nl3c00499
document-id-new-14nl3c00499
ccc-price
==== Body
pmcElectronic devices with well-defined ballistic electron trajectories have triggered extensive research,1−4 and to exploit their full potential, specular reflection of electron jets is a major requirement. Electrostatically defined geometries are optimal platforms to realize the specular reflection, as shown by transverse electron focusing (TEF) measurements.5−16 In this context, the exceptional electronic properties of graphene make it an ideal candidate for a wide variety of gate-defined devices where Klein tunneling enables new functionalities.2,17−20 However, the absence of a bandgap complicates the creation of collimated beams and specular mirrors in graphene. The former has been realized by etching high-mobility graphene devices in absorptive pinhole collimators.21 The latter has been improved by recent fabrication progress, leading to the observation of multiple focusing peaks.10,13 However, the reflection induced by disordered graphene edges is not specular.22 This is a fundamental limitation that in TEF experiments results in a decrease of the peak amplitude as the number of reflections at the edge increases9,10,12,13 and randomizes the valley degree of freedom.22 An alternative approach has been implemented in the quantum Hall regime, where the gaps between Landau levels have been used to create gate-defined interferometers23−25 and quantum point contacts (QPCs).25 However, the effective confinement of carriers at zero magnetic field in monolayer graphene remains a challenge.

In contrast, bilayer graphene (BLG) is a tunable-bandgap semiconductor with a trigonally distorted Fermi surface.26−30 It has recently been introduced as an ideal system for the realization of gate-defined QPCs31−39 capable of transmitting valley-polarized electron jets40 and of hosting quantum dots with controllable spin and valley polarizations.41,42 Even though BLG hosts extraordinary properties, such as chirality-assisted cloaking43,44 or anti-Klein tunneling,45 experiments on gate-defined BLG devices have so far focused on the characterization of QPCs,31−33,36−38,40 quantum dots,41,42,46 quantum interference effects,47 and topological edge channels.48−52

In this work, we exploit the electrically tunable bandgap of BLG to create ballistic multiterminal BLG devices and measure TEF between gate-defined QPCs. We observe up to eight focusing peaks with comparable amplitudes, which is a clear indication of specular reflection at the gate-defined edges. Temperature-dependent measurements show that the TEF signal persists at up to 100 K.

We fabricated two double-gated, boron nitride (hBN)-encapsulated BLG heterostructures on few-layer graphene back gates, each containing multiple devices using the dry transfer technique described in refs (53 and 54). The electrodes were defined by using conventional e-beam lithography. The BLG flakes were connected to Ti/Au electrodes (brown rectangles in Figure 1a) after using a CHF3/O2 plasma to etch the upper hBN and BLG layers at the contact area.55 The top gates, which are dark yellow in Figure 1a, were deposited on the top hBN (see the Supporting Information (SI) Section S1 for the fabrication details). The side and top view images of a typical QPC are shown in Figure 1a. Here we discuss the results on the first heterostructure (sample 1); the results on sample 2 are shown in SI Section S9.

Figure 1 Gate-defined QPCs in the BLG at 1.8 K. (a) Side (left) and top (right) view of the fabricated device. The top view is a false-color AFM image. The separation between the split top gates (TG1 and TG2) is approximately 50 nm, and their width is 580 nm. At the side view (left panel), the hBN layers are green, and the BLG and the few-layer graphene back gate (BG) are black. In both panels, the contacts to 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 the transverse electron focusing experiments as a function of Vbg and Vtg. Vtg is the same for TG1 and TG2. (c) Point contact conductance obtained along the diagonal line in panel b that follows Vtg ≈ Vtg0 – βVbg, corresponding to the charge neutrality point of the double-gated regions. Vtg0 ≈ −1.08 V is the charge neutrality point at Vbg = 0, and β is the ratio between the back and top gate capacitances. The TEF measurements are also performed along this line.

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 a function of the top gate voltage (Vtg) and the back gate voltage (Vbg). As shown in Figure 1b, three features can be distinguished from this result: The first one is a vertical line at Vbg ≈ 0, which corresponds to the charge neutrality point (CNP) of the non-top-gated BLG channel. The CNP does not occur at exactly Vtg = 0 due to small hole doping. The second feature is a faint vertical line at Vbg ≈ −1 V. Four-terminal measurements (see SI Section S3) indicate that it corresponds to the CNP of the BLG near the Ti/Au contacts, where the top hBN 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 CNP of the regions under TG1 and TG2. Because both Vbg and Vtg influence the carrier density (n) at these regions, the introduction of electrons by Vbg to the BLG channel must be counteracted by an opposite Vtg to keep the channel charge neutral. We use the slope of this line to obtain the ratio between the top gate (Ctg) and back gate (Cbg) capacitances: β = Cbg/Ctg = −ΔVtg/ΔVbg ≈ 1.22. This value is consistent with the factor of 1.22 obtained from the ratio between the hBN-flake thicknesses extracted from AFM imaging (see SI Section S1). Even though the electric field applied by the gates opens a bandgap in the double-gated BLG regions which increases with |Vbg|,26−29 the resistance along the diagonal line does not increase with |Vbg|. This is due to the small gap between TG1 and TG2 (Figure 1a). In this region the carrier density is not zero, leading to the formation of a Vbg-controlled QPC with tunable carrier density.

To determine if the QPC conductance (G) is quantized, we have determined its resistance by taking, for each Vbg, the difference between the maximal and the minimal R. This operation allows us to subtract the resistances of the Ti/Au contacts and the BLG regions that are not affected by Vtg. The results are shown in Figure 1c. For negative Vbg, G shows values higher than 7 × 4e2/h and changes in a monotonic way with small oscillations. In contrast, for positive Vbg, G shows four steps at G = N × 4e2/h with N = 1, 2, 3, and 4. This behavior, which is reproduced in five of the six QPCs characterized, indicates the formation of a spin and valley-degenerate QPC.33,36,37,56 Note that the sharp increase of G near Vbg = 0 is a consequence of the extraction method when there is no bandgap under the double-gated regions, and R shows very small changes with Vtg. Even though the reason for the electron–hole asymmetry is not clear, we believe that one possibility may be a residual doping of the double-gated regions caused by the fabrication. Because the QPC region is not affected by this process, the potential landscape could become asymmetric to the sign reversal of the gate voltages, modifying the confinement potential. This could make the QPC narrower for electron than hole doping or modify its carrier density. In addition, the electric field applied on the BLG at the CNP changes sign with Vbg, leading to opposite layer polarizations57 which could also enhance the asymmetry.

When a magnetic field (B) is applied perpendicular to the plane of a ballistic BLG device, electrons deviate from their straight trajectories by the Lorentz force. If the Fermi surface is circular, they follow circular orbits with radius rc = ℏkF/eB, where ℏ is the reduced Planck constant and kF is the Fermi wavevector . As a consequence, the transmission between different contacts connected at a distance L from each other shows maxima at magnetic fields (Bf) given by5,71

where θ is the angle at which the electron flow departs from the emitter, L = 2 μm is the injector–detector distance, and p = 1, 2, 3, ..., n is an integer which accounts for the p – 1 reflections that occur at the device edge between the contacts (Figure 2a).

Figure 2 Transverse electron focusing between gate-defined QPCs in the BLG at 1.8 K. (a) Measurement geometry. The nonlocal voltage (Vnl) is measured as a function of B while applying current I between the right QPC and a reference lead. The ballistic trajectories are sketched for the three 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 been introduced for clarity. Vtg is tuned to follow the charge neutrality line of the top-gated regions (diagonal line in Figure 1b). (c) Focusing spectra extracted 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 the normalized 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. The vertical 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 line is 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 crystallographic direction. The black curves were obtained by adding the K and K′ valley-resolved spectra. The insets show the trigonally warped trajectories corresponding to the average incidence angles for valleys K and K′, and the dark yellow rectangles represent the gate-defined edges.

TEF measurements were performed using configuration C1, which is shown in Figure 2a. A current (I) is applied to the right QPC to generate electron flow into the ballistic BLG channel that is steered using the out-of-plane B field. To detect the ballistic skipping orbits, the nonlocal voltage (Vnl) is measured between the left QPC and a reference electrode connected to the left of the BLG channel. To avoid voltage Vnl offsets, we used a differential DC measurement technique to obtain the TEF spectra in Figures 2 and 3.

Figure 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 to C3. The scale bar is 5 μm. The BLG edges are indicated by black dashed 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 linear fits to illustrate the trend.

The results from such measurements performed for different Vbg values are shown in Figure 2b. Note that, to ensure that the charge transport occurs only through the QPCs, we have adjusted Vtg to keep the double-gated regions charge neutral (diagonal line in Figure 1b). We first consider the Vbg = −4 V case. For B < 0, the signal is zero (dashed lines) or smaller than the noise level of the measurement, which is 2 Ω, consistent with the fact that the ballistic electron stream deviates toward the right and does not generate a signal on the detector. In contrast, when B > 0, five clear focusing peaks are observed, indicating that even though the QPC conductance is not quantized for Vbg < 0 (Figure 1c), the hole trajectories are well-defined and reflection at the gate-defined edge between both QPCs is smooth. As Vbg approaches zero, n in the BLG channel decreases, and the distance between the peaks becomes smaller. At Vbg > 0 peaks occur for B < 0, consistent with the fact that the carriers have changed from holes to electrons.9,10,13

For a more detailed comparison, in Figure 2c we show the Vbg = ±3 V spectra, where the bandgap is approximately 45 meV.29 Two clear differences can be distinguished: (i) The p = 1 peak is 2 times higher for Vbg = +3 V. This is most likely due to the lower G at Vbg = +3 V, which converts the collector current (Ic) into the measured Vnl = Ic/G. As shown in Figure 1c, G is roughly 2 times larger for Vbg = −3 V than for Vbg = +3 V, explaining most of the measured asymmetry in the p = 1 peak magnitude. (ii) The peak amplitude decays with p much faster at Vbg = −3 V. To quantify the TEF signal decay with p and correct for a small contact magnetoresistance (see SI Section S4), we calculated the area under the TEF peaks10 normalized by the two-terminal resistance (see SI Section S5b). The result is shown in the inset of Figure 2c with a linear fit excluding the p = 1 peak (which has the smallest area). The obtained peak areas are fairly constant from p = 2 up to p = 8 (including the p = 4 peak, which occurs between 0.75 and 1 T and 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 terms of a change of the QPC width (W). The finite W of the detector poses an upper bound of the maximum number of peaks that can be measured. In particular, if rc cos θ ≤ W/2, all electrons will enter the detector and extra peaks cannot be detected,7 leading to B ≤ 2 T for W = 200 nm, cos θ = 1, and a circular trajectory. In contrast, for W = 100 nm, we obtain B ≤ 3.9 T. As shown in Figure 1c, G is almost 8 times smaller for electrons than for holes, indicating that a significant electron–hole asymmetry in the QPC width is plausible. Additionally, decreasing the injector W is known to lead to electron jets with improved collimation.21,58 Because the focusing length of a trajectory depends on its injection angle, the differences between focusing lengths of different trajectories increase with p. Thus, a narrow angular distribution is expected to help maintain a constant peak amplitude, even after several edge reflections.

In Figure 2b, for Vbg = 1.5 V (red curve), additional oscillations similar to those in refs (7 and 25) can be observed on top of the focusing spectrum. The amplitude of these oscillations decreases with increasing Vbg, a result that is consistent with quantum interference between the different electron paths contributing to the TEF signal because the Fermi wavelength increases with decreasing n. For completeness, Rnl near Vbg = 0 V is shown in SI section S7.

To gain more insight into the measured TEF spectra, we analyzed the positions of the focusing peaks (Bf) as a function of p. In particular, we determined Bf and fit it to Bf = B0 + (dB/dp) × p, where B0 and dB/dp are constants accounting for the magnet remanence and the average spacing between the peaks, respectively. In Figure 2d we show |dB/dp| and compare it with the result from eq 1 for normal incidence (θ = 0). The similarity between both curves further confirms that our signal is due to TEF. Even though the small underestimation of Bf by eq 1 could be due to trigonal warping,9 the orientation dependence of Bf expected from ref (9) is not observed here (see SI Section S6), preventing a conclusive statement.

The results shown in Figure 2c at Vbg = +3 V show features resembling a beating pattern. In particular, all the peaks except p = 1 and 4 can be decomposed into two narrower peaks, and the latter, which has a dip where one would expect a peak, can be decomposed into three well-separated peaks. Additionally, the Fourier transform of the TEF spectrum (see SI Section S5b for details) also indicates the presence of a beating pattern, implying a periodic modulation. Even though there may be a combination of impurities or irregularities at the confinement potential that could explain this effect, there is a fundamental reason to expect such features in the TEF spectra. BLG is known for showing trigonal warping; i.e., its Fermi surface is not circular. In this case, the emission of electrons by the QPCs occurs in jets that depend on the crystallographic orientation of the QPCs on the BLG. If the QPCs are slightly misaligned with respect to a crystallographic direction, the valley-polarized jets will be emitted with slightly different |θ|, leading to two different Bf values for the peaks in valleys K and K′.40

Semiclassical calculations considering the effect of trigonal warping (see SI Section S9 for details) are shown in Figures 2e and 2f for the perfectly aligned and the small misalignment (0.05 rad ≈ 3°) cases, respectively. The trajectories are shown in the insets. In the latter, a beating pattern arises that is compatible with the measured data.

To show the robustness of the TEF measurements and explore the role of the BLG crystallographic orientation in the TEF spectra, we have patterned QPCs in different directions on the same BLG flake. The relative angle between the QPC sets is 30° to compare the armchair with the zigzag crystallographic directions. As shown in Figure 3a, C2 is aligned parallel to the longest BLG straight edge with an accuracy of ∼5°. Because we expect the flake edge to be aligned with a crystallographic direction with an uncertainty of a few degrees,59,60 we assume that the C2 QPCs are aligned to a crystallographic direction with an accuracy of less than 10°. Thus, the 30° rotated C1 QPCs, are expected to be along the other. We compare the TEF spectra in Figure 2c with the TEF spectra obtained using configurations C2, C3, and C4 from Figure 3a, shown in Figures 3b, 3c, and 3d, respectively. The results show several features: (i) The TEF peaks decay faster with p for holes than for electrons in all the geometries. (ii) For C3 and C4, which contain hornlike QPCs not showing size quantization (see SI Section S6 for details), the decay in peak amplitude for electrons is more pronounced than for C1 and C2 where G is quantized. As a consequence, six peaks can be distinguished instead of eight. (iii) The width of the p = 1 peak is significantly smaller than that of the p = 2 peak in all the configurations, for both electron and hole doping. Observations (i) and (ii) show a correlation between G and the TEF peak amplitude decay, further indicating that the QPC width plays a relevant role in the peak amplitude decrease.

Because the occurrence of a beating pattern is very sensitive to a tiny misalignment, the lack of a clear beating pattern in Figure 3b is not inconsistent with the Fermi surface being warped.

To characterize the scattering sources in BLG, we have measured Rnl vs B at different temperatures (T) at Vbg = ±3 V. At 2 K, the peak height is the highest, and as T increases, the background becomes more pronounced and the focusing signal gets smaller. Comparing the 2 K with the 10 K measurements, the 2 K spectra contain extra features at positive and negative B-fields. A fast decay when increasing T indicates that these features are likely due to quantum interference, as the phase-coherence length is known to drop within this range.61

To extract the T dependence of the scattering rate (τp–1) from Figures 4a and 4b, we have followed ref (10) (see SI Section S5 for details). The result obtained using the area under the p = 2 peak is shown in Figure 4c. Here, the dots correspond to the values extracted from Figure 4a,b, and the lines are fits to τp–1 = aT2 + bT + c. Assuming a hard-wall confinement potential, a quadratic T dependence of τp–1 is associated with electron–electron interactions.10,62,63 In contrast, 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 SI Section S5 for the fitting parameters and a more detailed discussion. We suspect that the background signals in Figure 4 are caused by a small miscalibration of Vtg.

Figure 4 Temperature dependence of TEF in BLG. B dependence of Rnl 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 background corrected. (c) Scattering rate estimated using the spectra in panels a and b (dots) and its fit to a parabola (lines). The inset shows the T dependence of the QPC resistance.

To conclude, we have measured the TEF in hBN-encapsulated BLG devices where QPCs are defined in different directions using electrostatic gating. Our results show eight focusing peaks with similar amplitudes together with quantum interference features. By comparing TEF spectra with semiclassic simulations, we identify a periodic modulation of the peak size that is consistent with the effect of trigonal warping. Moreover, the TEF temperature dependence shows that the signal persists up to 100 K. Our results are promising for future valleytronic devices.

Data Availability Statement

All the data and code associated with the analysis and theoretical simulations are available free of charge from ref (65).

Supporting Information Available

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.nanolett.3c00499.Details on the fabrication and measurement techniques, characterization of the backgate capacitance, electronic mobility, two-terminal magnetoresistance, extraction of the area under the focusing peaks, the complete TEF data set, the results obtained in sample 2, and numerical simulations (PDF)

Supplementary Material

nl3c00499_si_001.pdf

Author Contributions

J.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. wrote the manuscript with inputs from all authors. H.S.J.v.d.Z. supervised the project.

The authors declare no competing financial interest.

Acknowledgments

We thank Prof. K. Ensslin and K. Vilkelis for insightful discussions. This project received funding from the European Union Horizon 2020 research and innovation program under grant agreement no. 863098 (SPRING). J.I.A. acknowledges support from the European Union’s Horizon 2020 research and innovation programme for a Marie Sklodowska–Curie individual fellowship No. 101027187-PCSV. ALRM work was supported by VIDI Grant 016.Vidi.189.180. K.W. and T.T. acknowledge support from JSPS KAKENHI (Grants 19H05790, 20H00354, and 21H05233).
==== Refs
References

Bøggild P. ; Caridad J. M. ; Stampfer C. ; Calogero G. ; Papior N. R. ; Brandbyge M. A two-dimensional Dirac fermion microscope. Nat. Commun. 2017, 8 , 1–12. 10.1038/ncomms15783.28232747
Wang K. ; Elahi M. M. ; Wang L. ; Habib K. M. ; Taniguchi T. ; Watanabe K. ; Hone J. ; Ghosh A. W. ; Lee G.-H. ; Kim P. Graphene transistor based on tunable Dirac fermion optics. Proc. Natl. Acad. Sci. U. S. A. 2019, 116 , 6575–6579. 10.1073/pnas.1816119116.30877246
LaGasse S. W. ; Cress C. D. Unveiling Electron Optics in Two-Dimensional Materials by Nonlocal Resistance Mapping. Nano Lett. 2020, 20 , 6623–6629. 10.1021/acs.nanolett.0c02443.32787176
Heinrich A. J. ; Oliver W. D. ; Vandersypen L. M. ; Ardavan A. ; Sessoli R. ; Loss D. ; Jayich A. B. ; Fernandez-Rossier J. ; Laucht A. ; Morello A. Quantum-coherent nanoscience. Nat. Nanotechnol. 2021, 16 , 1318–1329. 10.1038/s41565-021-00994-1.34845333
Tsoi V. S. Focusing of electrons in a metal by a transverse magnetic field. JETP Lett. 1974, 19 , 114–116.
Tsoi V. ; Bass J. ; Wyder P. Studying conduction-electron/interface interactions using transverse electron focusing. Rev. Mod. Phys. 1999, 71 , 1641 10.1103/RevModPhys.71.1641.
Van Houten H. ; Beenakker C. ; Williamson J. ; Broekaart M. ; Van Loosdrecht P. ; Van Wees B. ; Mooij J. ; Foxon C. ; Harris J. Coherent electron focusing with quantum point contacts in a two-dimensional electron gas. Phys. Rev. B 1989, 39 , 8556 10.1103/PhysRevB.39.8556.
Heremans J. ; Santos M. ; Shayegan M. Observation of magnetic focusing in two-dimensional hole systems. Appl. Phys. Lett. 1992, 61 , 1652–1654. 10.1063/1.108441.
Taychatanapat T. ; Watanabe K. ; Taniguchi T. ; Jarillo-Herrero P. Electrically tunable transverse magnetic focusing in graphene. Nat. Phys. 2013, 9 , 225–229. 10.1038/nphys2549.
Lee M. ; Wallbank J. R. ; Gallagher P. ; Watanabe K. ; Taniguchi T. ; Fal’ko V. I. ; Goldhaber-Gordon D. Ballistic miniband conduction in a graphene superlattice. Science 2016, 353 , 1526–1529. 10.1126/science.aaf1095.27708100
Morikawa S. ; Dou Z. ; Wang S.-W. ; Smith C. G. ; Watanabe K. ; Taniguchi T. ; Masubuchi S. ; Machida T. ; Connolly M. R. Imaging ballistic carrier trajectories in graphene using scanning gate microscopy. Appl. Phys. Lett. 2015, 107 , 243102 10.1063/1.4937473.
Bhandari S. ; Lee G.-H. ; Klales A. ; Watanabe K. ; Taniguchi T. ; Heller E. ; Kim P. ; Westervelt R. M. Imaging cyclotron orbits of electrons in graphene. Nano Lett. 2016, 16 , 1690–1694. 10.1021/acs.nanolett.5b04609.26845290
Berdyugin A. I. ; Tsim B. ; Kumaravadivel P. ; Xu S. G. ; Ceferino A. ; Knothe A. ; Kumar R. K. ; Taniguchi T. ; Watanabe K. ; Geim A. K. ; Grigorieva I. V. ; Fal’ko V. I. Minibands in twisted bilayer graphene probed by magnetic focusing. Science Advances 2020, 6 , eaay7838 10.1126/sciadv.aay7838.32494602
Sonntag J. ; Li J. ; Plaud A. ; Loiseau A. ; Barjon J. ; Edgar J. ; Stampfer C. Excellent electronic transport in heterostructures of graphene and monoisotopic boron-nitride grown at atmospheric pressure. 2D Materials 2020, 7 , 031009 10.1088/2053-1583/ab89e5.
Bachmann M. D. ; Sharpe A. L. ; Barnard A. W. ; Putzke C. ; König M. ; Khim S. ; Goldhaber-Gordon D. ; Mackenzie A. P. ; Moll P. J. Super-geometric electron focusing on the hexagonal Fermi surface of PdCoO2. Nat. Commun. 2019, 10 , 1–8. 10.1038/s41467-019-13020-9.30602773
Rao Q. ; Kang W.-H. ; Xue H. ; Ye Z. ; Feng X. ; Watanabe K. ; Taniguchi T. ; Wang N. ; Liu M.-H. ; Ki D.-K. Ballistic transport spectroscopy of spin-orbit-coupled bands in monolayer graphene on WSe2, 2023, 2303.01018. arXiv. https://arxiv.org/abs/2303.01018 (accessed 2023-06-01).
Novoselov K. S. ; Geim A. ; et al. The rise of graphene. Nat. Mater. 2007, 6 , 183–191. 10.1038/nmat1849.17330084
Cheianov V. V. ; Fal’ko V. ; Altshuler B. The focusing of electron flow and a Veselago lens in graphene pn junctions. Science 2007, 315 , 1252–1255. 10.1126/science.1138020.17332407
Lee G.-H. ; Park G.-H. ; Lee H.-J. Observation of negative refraction of Dirac fermions in graphene. Nat. Phys. 2015, 11 , 925–929. 10.1038/nphys3460.
Chen S. ; Han Z. ; Elahi M. M. ; Habib K. M. ; Wang L. ; Wen B. ; Gao Y. ; Taniguchi T. ; Watanabe K. ; Hone J. ; et al. Electron optics with pn junctions in ballistic graphene. Science 2016, 353 , 1522–1525. 10.1126/science.aaf5481.27708099
Barnard A. W. ; Hughes A. ; Sharpe A. L. ; Watanabe K. ; Taniguchi T. ; Goldhaber-Gordon D. Absorptive pinhole collimators for ballistic Dirac fermions in graphene. Nat. Commun. 2017, 8 , 1–6. 10.1038/ncomms15418.28232747
Walter E. ; Rosdahl T. ; Akhmerov A. ; Hassler F. Breakdown of the law of reflection at a disordered graphene edge. Phys. Rev. Lett. 2018, 121 , 136803 10.1103/PhysRevLett.121.136803.30312101
Wei D. S. ; van der Sar T. ; Sanchez-Yamagishi J. D. ; Watanabe K. ; Taniguchi T. ; Jarillo-Herrero P. ; Halperin B. I. ; Yacoby A. Mach-Zehnder interferometry using spin-and valley-polarized quantum Hall edge states in graphene. Sci. Adv. 2017, 3 , e1700600 10.1126/sciadv.1700600.28835920
Veyrat L. ; Jordan A. ; Zimmermann K. ; Gay F. ; Watanabe K. ; Taniguchi T. ; Sellier H. ; Sacépé B. Low-magnetic-field regime of a gate-defined constriction in high-mobility graphene. Nano Lett. 2019, 19 , 635–642. 10.1021/acs.nanolett.8b02584.30654611
Ronen Y. ; Werkmeister T. ; Haie Najafabadi D. ; Pierce A. T. ; Anderson L. E. ; Shin Y. J. ; Lee S. Y. ; Lee Y. H. ; Johnson B. ; Watanabe K. ; et al. Aharonov–Bohm effect in graphene-based Fabry–Pérot quantum Hall interferometers. Nat. Nanotechnol. 2021, 16 , 563–569. 10.1038/s41565-021-00861-z.33633404
Castro E. V. ; Novoselov K. ; Morozov S. ; Peres N. ; Dos Santos J. L. ; Nilsson J. ; Guinea F. ; Geim A. ; Neto A. C. Biased bilayer graphene: semiconductor with a gap tunable by the electric field effect. Phys. Rev. Lett. 2007, 99 , 216802 10.1103/PhysRevLett.99.216802.18233240
Oostinga J. B. ; Heersche H. B. ; Liu X. ; Morpurgo A. F. ; Vandersypen L. M. Gate-induced insulating state in bilayer graphene devices. Nat. Mater. 2008, 7 , 151–157. 10.1038/nmat2082.18059274
Zhang Y. ; Tang T.-T. ; Girit C. ; Hao Z. ; Martin M. C. ; Zettl A. ; Crommie M. F. ; Shen Y. R. ; Wang F. Direct observation of a widely tunable bandgap in bilayer graphene. Nature 2009, 459 , 820–823. 10.1038/nature08105.19516337
Icking E. ; Banszerus L. ; Wörtche F. ; Volmer F. ; Schmidt P. ; Steiner C. ; Engels S. ; Hesselmann J. ; Goldsche M. ; Watanabe K. ; et al. Transport Spectroscopy of Ultraclean Tunable Band Gaps in Bilayer Graphene. Adv. Electron. Mater. 2022, 8 , 2200510 10.1002/aelm.202200510.
Seemann L. ; Knothe A. ; Hentschel M. Gate-tunable regular and chaotic electron dynamics in ballistic bilayer graphene cavities. Phys. Rev. B 2023, 107 , 205404 10.1103/PhysRevB.107.205404.
Allen M. T. ; Martin J. ; Yacoby A. Gate-defined quantum confinement in suspended bilayer graphene. Nat. Commun. 2012, 3 , 1–6. 10.1038/ncomms1945.
Goossens A. S. M. ; Driessen S. C. ; Baart T. A. ; Watanabe K. ; Taniguchi T. ; Vandersypen L. M. Gate-defined confinement in bilayer graphene-hexagonal boron nitride hybrid devices. Nano Lett. 2012, 12 , 4656–4660. 10.1021/nl301986q.22906072
Overweg H. ; Eggimann H. ; Chen X. ; Slizovskiy S. ; Eich M. ; Pisoni R. ; Lee Y. ; Rickhaus P. ; Watanabe K. ; Taniguchi T. ; et al. Electrostatically induced quantum point contacts in bilayer graphene. Nano Lett. 2018, 18 , 553–559. 10.1021/acs.nanolett.7b04666.29286668
Kraft R. ; Mohrmann J. ; Du R. ; Selvasundaram P. B. ; Irfan M. ; Kanilmaz U. N. ; Wu F. ; Beckmann D. ; von Lohneysen H. ; Krupke R. ; Akhmerov A. ; Gornyi I. ; Danneau R. Tailoring supercurrent confinement in graphene bilayer weak links. Nat. Commun. 2018, 9 , 1722 10.1038/s41467-018-04153-4.29712916
Knothe A. ; Fal’ko V. Influence of minivalleys and Berry curvature on electrostatically induced quantum wires in gapped bilayer graphene. Phys. Rev. B 2018, 98 , 155435 10.1103/PhysRevB.98.155435.
Overweg H. ; Knothe A. ; Fabian T. ; Linhart L. ; Rickhaus P. ; Wernli L. ; Watanabe K. ; Taniguchi T. ; Sánchez D. ; Burgdörfer J. ; et al. Topologically nontrivial valley states in bilayer graphene quantum point contacts. Phys. Rev. Lett. 2018, 121 , 257702 10.1103/PhysRevLett.121.257702.30608777
Kraft R. ; Krainov I. ; Gall V. ; Dmitriev A. ; Krupke R. ; Gornyi I. ; Danneau R. Valley subband splitting in bilayer graphene quantum point contacts. Phys. Rev. Lett. 2018, 121 , 257703 10.1103/PhysRevLett.121.257703.30608811
Velasco J. Jr. ; Lee J. ; Wong D. ; Kahn S. ; Tsai H.-Z. ; Costello J. ; Umeda T. ; Taniguchi T. ; Watanabe K. ; Zettl A. ; et al. Visualization and control of single-electron charging in bilayer graphene quantum dots. Nano Lett. 2018, 18 , 5104–5110. 10.1021/acs.nanolett.8b01972.30035544
Lane T. L. ; Knothe A. ; Fal’ko V. I. Semimetallic features in quantum transport through a gate-defined point contact in bilayer graphene. Phys. Rev. B 2019, 100 , 115427 10.1103/PhysRevB.100.115427.
Gold C. ; Knothe A. ; Kurzmann A. ; Garcia-Ruiz A. ; Watanabe K. ; Taniguchi T. ; Fal’ko V. ; Ensslin K. ; Ihn T. Coherent jetting from a gate-defined channel in bilayer graphene. Phys. Rev. Lett. 2021, 127 , 046801 10.1103/PhysRevLett.127.046801.34355933
Eich M. ; Herman F. ; Pisoni R. ; Overweg H. ; Kurzmann A. ; Lee Y. ; Rickhaus P. ; Watanabe K. ; Taniguchi T. ; Sigrist M. ; et al. Spin and valley states in gate-defined bilayer graphene quantum dots. Physical Review X 2018, 8 , 031023 10.1103/PhysRevX.8.031023.
Banszerus L. ; Frohn B. ; Epping A. ; Neumaier D. ; Watanabe K. ; Taniguchi T. ; Stampfer C. Gate-defined electron–hole double dots in bilayer graphene. Nano Lett. 2018, 18 , 4785–4790. 10.1021/acs.nanolett.8b01303.29949375
Gu N. ; Rudner M. ; Levitov L. Chirality-assisted electronic cloaking of confined states in bilayer graphene. Phys. Rev. Lett. 2011, 107 , 156603 10.1103/PhysRevLett.107.156603.22107310
Lee K. ; Lee S. ; Eo Y. S. ; Kurdak C. ; Zhong Z. Evidence of electronic cloaking from chiral electron transport in bilayer graphene nanostructures. Phys. Rev. B 2016, 94 , 205418 10.1103/PhysRevB.94.205418.
Katsnelson M. ; Novoselov K. ; Geim A. Chiral tunnelling and the Klein paradox in graphene. Nat. Phys. 2006, 2 , 620–625. 10.1038/nphys384.
Kurzmann A. ; Overweg H. ; Eich M. ; Pally A. ; Rickhaus P. ; Pisoni R. ; Lee Y. ; Watanabe K. ; Taniguchi T. ; Ihn T. ; et al. Charge detection in gate-defined bilayer graphene quantum dots. Nano Lett. 2019, 19 , 5216–5221. 10.1021/acs.nanolett.9b01617.31311270
Iwakiri S. ; de Vries F. K. ; Portolés E. ; Zheng G. ; Taniguchi T. ; Watanabe K. ; Ihn T. ; Ensslin K. Gate-defined electron interferometer in bilayer graphene. Nano Lett. 2022, 22 , 6292–6297. 10.1021/acs.nanolett.2c01874.35880910
Martin I. ; Blanter Y. M. ; Morpurgo A. Topological confinement in bilayer graphene. Phys. Rev. Lett. 2008, 100 , 036804 10.1103/PhysRevLett.100.036804.18233021
San-Jose P. ; Prada E. ; McCann E. ; Schomerus H. Pseudospin valve in bilayer graphene: towards graphene-based pseudospintronics. Phys. Rev. Lett. 2009, 102 , 247204 10.1103/PhysRevLett.102.247204.19659043
Li J. ; Wang K. ; McFaul K. J. ; Zern Z. ; Ren Y. ; Watanabe K. ; Taniguchi T. ; Qiao Z. ; Zhu J. Gate-controlled topological conducting channels in bilayer graphene. Nat. Nanotechnol. 2016, 11 , 1060–1065. 10.1038/nnano.2016.158.27570941
Lee J. ; Watanabe K. ; Taniguchi T. ; Lee H.-J. Realisation of topological zero-energy mode in bilayer graphene in zero magnetic field. Sci. Rep. 2017, 7 , 1–6. 10.1038/s41598-017-06902-9.28127051
Li J. ; Zhang R.-X. ; Yin Z. ; Zhang J. ; Watanabe K. ; Taniguchi T. ; Liu C. ; Zhu J. A valley valve and electron beam splitter. Science 2018, 362 , 1149–1152. 10.1126/science.aao5989.30523108
Zomer P. ; Guimarães M. ; Brant J. ; Tombros N. ; Van Wees B. Fast pick up technique for high quality heterostructures of bilayer graphene and hexagonal boron nitride. Appl. Phys. Lett. 2014, 105 , 013101 10.1063/1.4886096.
Purdie D. ; Pugno N. ; Taniguchi T. ; Watanabe K. ; Ferrari A. ; Lombardo A. Cleaning interfaces in layered materials heterostructures. Nat. Commun. 2018, 9 , 1–12. 10.1038/s41467-018-07558-3.29317637
Wang L. ; Meric I. ; Huang P. ; Gao Q. ; Gao Y. ; Tran H. ; Taniguchi T. ; Watanabe K. ; Campos L. ; Muller D. ; et al. One-dimensional electrical contact to a two-dimensional material. Science 2013, 342 , 614–617. 10.1126/science.1244358.24179223
Van Wees B. ; Van Houten H. ; Beenakker C. ; Williamson J. G. ; Kouwenhoven L. ; Van der Marel D. ; Foxon C. Quantized conductance of point contacts in a two-dimensional electron gas. Phys. Rev. Lett. 1988, 60 , 848 10.1103/PhysRevLett.60.848.10038668
McCann E. ; Koshino M. The electronic properties of bilayer graphene. Rep. Prog. Phys. 2013, 76 , 056503 10.1088/0034-4885/76/5/056503.23604050
Molenkamp L. ; Staring A. ; Beenakker C. ; Eppenga R. ; Timmering C. ; Williamson J. ; Harmans C. ; Foxon C. Electron-beam collimation with a quantum point contact. Phys. Rev. B 1990, 41 , 1274 10.1103/PhysRevB.41.1274.
You Y. ; Ni Z. ; Yu T. ; Shen Z. Edge chirality determination of graphene by Raman spectroscopy. Appl. Phys. Lett. 2008, 93 , 163112 10.1063/1.3005599.
Kim K. ; Artyukhov V. I. ; Regan W. ; Liu Y. ; Crommie M. ; Yakobson B. I. ; Zettl A. Ripping graphene: preferred directions. Nano Lett. 2012, 12 , 293–297. 10.1021/nl203547z.22149252
Kozikov A. ; Horsell D. ; McCann E. ; Fal’Ko V. Evidence for spin memory in the electron phase coherence in graphene. Phys. Rev. B 2012, 86 , 045436 10.1103/PhysRevB.86.045436.
Bandurin D. ; Torre I. ; Kumar R. K. ; Ben Shalom M. ; Tomadin A. ; Principi A. ; Auton G. ; Khestanova E. ; Novoselov K. ; Grigorieva I. ; et al. Negative local resistance caused by viscous electron backflow in graphene. Science 2016, 351 , 1055–1058. 10.1126/science.aad0201.26912363
Bandurin D. A. ; Shytov A. V. ; Levitov L. S. ; Kumar R. K. ; Berdyugin A. I. ; Ben Shalom M. ; Grigorieva I. V. ; Geim A. K. ; Falkovich G. Fluidity onset in graphene. Nat. Commun. 2018, 9 , 1–8. 10.1038/s41467-018-07004-4.29317637
Hwang E. ; Sarma S. D. Acoustic phonon scattering limited carrier mobility in two-dimensional extrinsic graphene. Phys. Rev. B 2008, 77 , 115449 10.1103/PhysRevB.77.115449.
Ingla-Aynés J. ; Manesco A. ; Ghiasi T. S. ; Volosheniuk S. ; van der Zant H. S. J. Data underlying the publication: Specular electron focusing between gate-defined quantum point contacts in bilayer graphene. 2023, 10.4121/21977237.