⇦Chinese Physics Letters, 2022, Vol. 39, No. 9, Article code 097302 Effect of Boundary Scattering on Magneto-Transport Performance in BN-Encapsulated Graphene Lijun Zhu (朱丽君)1,2,3, Lin Li (李林)1,2,3*, Xiaodong Fan (范晓东)1,2,3, Zhongniu Xie (谢忠纽)1,2,3, and Changgan Zeng (曾长淦)1,2,3* Affiliations 1CAS Key Laboratory of Strongly Coupled Quantum Matter Physics, and Department of Physics, University of Science and Technology of China, Hefei 230026, China 2International Center for Quantum Design of Functional Materials (ICQD), Hefei National Research Center for Physical Sciences at the Microscale, University of Science and Technology of China, Hefei 230026, China 3Hefei National Laboratory, University of Science and Technology of China, Hefei 230088, China Received 21 June 2022; accepted manuscript online 24 August 2022; published online 3 September 2022 *Corresponding authors. Email: lilin@ustc.edu.cn; cgzeng@ustc.edu.cn Citation Text: Zhu L J, Li L, Fan X D et al. 2022 Chin. Phys. Lett. 39 097302    Abstract For conductors in the ballistic regime, electron-boundary scattering at the sample edge plays a dominant role in determining the transport performance, giving rise to many intriguing phenomena like low-field negative magnetoresistance effect. We systematically investigate the magneto-transport behaviors of BN-encapsulated graphene devices with narrow channel width $W$, wherein the bulk mean free path $L_{\rm mfp}$ can be very large and highly tunable. By comparing the magnetoresistance features and the amplitude of $L_{\rm mfp}$ in a large parameter space of temperature and carrier density, we reveal that the boundary-scattering-dominated negative magnetoresistance effect can still survive even when the ballistic ratio ($L_{\rm mfp}/W$) is as low as 0.15. This striking value is much smaller than the expected value for achieving (quasi-) ballistic transport regime ($L_{\rm mfp}/W \ge 1$), and can be attributed to the ultra-low specularity of the sample edge of our graphene devices. These findings enrich our understanding of the effects of boundary scattering on channel transport, which is of vital importance for future designs of two-dimensional electronic devices with limited lateral sizes.
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 DOI:10.1088/0256-307X/39/9/097302 © 2022 Chinese Physics Society Article Text In high-quality two-dimensional (2D) electronic systems, the bulk mean free path of carrier ($L_{\rm mfp}=\hslash \mu \sqrt {\pi n} /e$, where $\hslash$ is Planck's constant, $\mu$ is the carrier mobility, $n$ is the carrier density, and $e$ is the elementary charge) determined by inelastic scatterings from impurities and phonons can be very large, and typical (quasi-) ballistic transport behaviors will emerge when $L_{\rm mfp}$ is comparable to the sample size.[1] In such (quasi-) ballistic systems, the charge carrier has a higher scattering probability at channel boundaries than in the bulk region. When boundary scattering is diffusive, applying a magnetic field $B$ induces bending of electron trajectories and suppresses the probability of boundary scattering of electrons, resulting in negative magnetoresistance (MR) instead of classical positive MR.[2-5] In past years, numerous experimental and theoretical studies have revealed a number of transport features originating from the high ballisticity of the conducting channel in systems represented by semiconductor quantum wells and graphene.[2-9] Many of these features can be attributed to the modulation of electron-boundary scattering under external fields. For a narrow sample whose channel width $W$ is smaller than its length, a high ballistic ratio of $L_{\rm mfp}/W$ ($\ge 1$) is widely recognized as an essential requirement for (quasi-) ballistic transport.[3,8] However, there are several studies demonstrating that characteristic boundary-scattering-induced negative MR effects can still be observed even when $L_{\rm mfp}$ is smaller than $W$.[4] In this study, we further explore the limiting condition for the contribution of boundary scattering to channel transport using BN-encapsulated graphene as a model system, in which a large and highly tunable $L_{\rm mfp}$ can be achieved.[6,10,11] By varying temperature and carrier density over a wide range, the transition from the typical quasi-ballistic regime to the normal diffusive one is observed. Strikingly, we find that the lower limit of the ballistic ratio for inducing typical negative MR can be as low as 0.15, which can be attributed to the ultra-low specularity of the channel edge. Figure 1(a) shows the schematic of the BN-encapsulated single-layer graphene device (BN/SLG/BN). Mechanically exfoliated SLG flake was encapsulated by two hBN flakes via a typical van der Waals assembly technique[11] and then stacked on a SiO$_{2}$/Si substrate. After shaping the device into a hall-bar geometry (sample width $W = 2$ µm) by electron-beam lithography and reactive ion etching, Cr/Pd/Au with the thicknesses of 1/7/45 nm were deposited as electrode materials by electron beam evaporation. In addition to the BN/SLG/BN devices, we fabricated control samples by transferring a bare SLG flake directly onto a SiO$_{2}$/Si substrate, which are denoted as SLG/SiO$_{2}$ [see Fig. 1(d)]. Transport measurements were performed in a $^{4}$He cryostat. Figure 1(b) shows the longitudinal resistance ($R_{xx}$) as a function of back-gate voltage ($V_{\rm BG}$) for the BN/SLG/BN device utilizing 300-nm-thick SiO$_{2}$ as the dielectric, from which a typical ambipolar field effect is observed. The Dirac point occurs at $V_{\rm D} \sim -6$ V, indicating slight electron doping. By performing Hall measurements at various $V_{\rm BG}$, corresponding carrier density $n$ and mobility $\mu$ can be extracted, while the latter one was calculated by using the expression $\mu =\frac{\sigma}{ne}$ with $\sigma$ being the conductivity at zero magnetic field. The results plotted in Fig. 1(c) show that the carrier mobility increases when the Fermi level is tuned towards the Dirac point, with the value ranging from 61000 cm$^{2}$/V$\cdot$s to 142000 cm$^{2}$/V$\cdot$s. These values are comparable with those reported previously.[6,10-12] As a comparison, the fabricated SLG/SiO$_{2}$ device possesses a high level of hole doping [see Fig. 1(e)], and its mobility is almost two orders of magnitude lower than that of the BN/SLG/BN device [see Fig. 1(f)].
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Fig. 1. Basic transport performance of the two graphene devices. [(a), (d)] Schematic illustration of the BN/SLG/BN and the SLG/SiO$_{2}$ devices, respectively. [(b), (e)] Longitudinal resistance ($R_{xx}$) as a function of back-gate voltage ($V_{\rm BG}$) for these two devices. Insets: optical microscope images of the two devices. The scale bars are both 5 µm. [(c), (f)] Extracted carrier density and Hall mobility versus $V_{\rm BG}$ for the BN/SLG/BN and the SLG/SiO$_{2}$ devices, respectively. All of the measurements were made at $T=1.5$ K.
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Fig. 2. Low-temperature magneto-transport behaviors: (a) $R_{xx}$ and Hall resistance $R_{xy}$ as functions of magnetic field $B$ for the SLG/SiO$_{2}$ device measured at $V_{\rm BG} = 20$ V and $T = 1.5$ K. (b) Enlarged view of $R_{xx}$ vs $B$ shown in (a) in the low-field regime, and the corresponding fitting result using the typical WL formula for graphene (red curve). Inset: schematic illustration of the WL effect originating from the interference between a pair of time-reversed closed loops. (c) $R_{xx}$ and $R_{xy}$ as functions of $B$ for the BN/SLG/BN device measured at $V_{\rm BG} = 50$ V and $T = 1.5$ K. Quantized Hall plateaus are clearly seen, and the corresponding filling factors are indicated. (d) Enlarged view of $R_{xx}$ vs $B$ shown in (c) in the low-field regime. Inset: schematic carrier trajectories in a narrow 2D channel under magnetic fields with different magnitudes.
Next, we focused on the magneto-transport performance under a perpendicular magnetic field $B$. Representative results for the SLG/SiO$_{2}$ device clearly show Shubnikov-de Haas (SdH) oscillations in $R_{xx}$ [Fig. 2(a)]. However, the low carrier mobility hinders the ability to reach quantized Hall resistance ($R_{xy}$) plateaus. Conversely, for the BN/SLG/BN device with nearly the same carrier density [see Fig. 2(c)], well-defined quantized Hall plateaus with filling factor $v = 22$, 26, 30,$\ldots$, accompanied by the minima of $R_{xx}$, are clearly seen, which further demonstrate its higher quality.[13,14] Another attractive difference between the two devices is the magnetoresistance (MR) behavior in the low-field regime. For the SLG/SiO$_{2}$ device, a sharp resistance peak of $R_{xx}$ occurs at zero field, together with a clear negative MR effect at $B < 0.1$ T [see Fig. 2(b)]. Similar results have been widely obtained in previous graphene devices and can be attributed to the weak localization (WL) effect [inset of Fig. 2(b)].[15,16] As demonstrated in Fig. 2(b), the low-field data can be well fitted using the typical WL formula for graphene[17,18] (as detailed in Section I in the Supplemental Material). The as-extracted phase coherence length $L_{\varPhi}$ ($L_{\varPhi} = 381$ nm) is much higher than the inter-(intra-)valley scattering length $L_{i}(L_{\ast}$) ($L_{i} \sim L_{\ast} = 73$ nm), indicating the presence of strong elastic inter- and intra-valley scatterings. In the BN/SLG/BN device with substantially enhanced quality, negative MR behavior is also clearly seen in the low field regime [see Fig. 2(d)]. However, as compared with that observed in the SLG/SiO$_{2}$ device, the negative MR effect here spans a dramatically wider range of $B$, and the magnitude (defined as MR$(B)=\{[R_{xx}(B)-R_{xx}(0)]/R_{xx}(0)\}\times 100{\%})$ is more than one order of magnitude larger (e.g., 53% versus 3% at 0.5 T). Furthermore, it is also failed to fit the MR data by the typical WL formula for graphene,[17,18] indicating its origins other than the WL effect. Actually, the resistance increases at first with increasing $B$ and then is followed by the negative MR effect described above, giving rise to a side peak of resistance occurring at $B_{{\max}} \sim 0.07$ T. The resulting double-peak feature is strongly reminiscent of the characteristic MR behavior of quasi-ballistic transport in narrow conducting channels of a two-dimensional electron gas (2DEG), wherein scattering of electrons at channel boundaries plays an essential role.[2-5,7,8] This is possible because the bulk mean free path $L_{\rm mfp}$ (1.64 µm for $n = 3.9 \times 10^{12}$ cm$^{-2}$) is comparable to the sample width $W$ (2 µm). In addition, the peak field of $B_{{\max}}$ scales with the ratio of $W$ to the cyclotron radius $r_{\rm c}$ of carriers as $\alpha =\frac{W}{r_{\rm c}}=\frac{WeB_{{\max}}}{\hslash \sqrt {\pi n}}\mathrm{\backsim 0.6}$, consistent with quasi-ballistic transport performance in semiconductor 2DEG and graphene systems.[2,4,5,7] The low-field MR behavior of the BN/SLG/BN device can be better interpreted using the framework of quasi-ballistic transport.[2,3,8] At smaller $B$, the carrier cyclotron orbit induced by the Lorentz force is much larger than the sample width, i.e., $r_{\rm c}=\frac{\hslash }{eB}\sqrt {\pi n} > W$. In this scenario, diffusive scattering between the channel boundaries gives rise to an increase in the backscattering probability of carriers [solid lines in the inset of Fig. 2(d)] and thus higher resistance. A higher $B$ leads to a smaller cyclotron radius $r_{\rm c}$, which, on the contrary, decreases the probability of electron-boundary scattering and results in a decrease in resistance. This trend with increasing $B$ slows down when 2$r_{\rm c}$ is smaller than $W$ because the carriers scattered by one boundary cannot reach the other unless they are scattered within the bulk [dashed lines in the inset of Fig. 2(d)].
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Fig. 3. Lower-limit of the bulk mean free path for inducing boundary-scattering-dominated negative MR effect in the BN/SLG/BN device. [(a), (b)] MR curves measured at different temperatures for $V_{\rm BG} = 50$ V and $V_{\rm BG} = 10$ V, respectively. (c) Extracted values of $L_{\rm mfp}$ under different $V_{\rm BG}$ as a function of temperature. The black line serves as a guide to the eyes. (d) Relation between the MR value at $B = 0.5$ T and the extracted $L_{\rm mfp}$. Corresponding data were obtained from the electron side at various $V_{\rm BG}$ and temperatures.
Figure 3(a) presents the evolution of MR behavior across a range of temperature (for more data, see Figs. S1 and S2 in the Supplemental Material). As the temperature increases, the magnitude of negative MR decreases monotonically and the side peak at $B_{{\max}}$ weakens. Nevertheless, in contrast to the WL effect that normally appears at relatively low temperatures,[19-21] the boundary-scattering-dominated negative MR effect observed here is quite robust, persisting even at temperatures of up to 300 K at high doping levels ($n = 3.9\times 10^{12}$ cm$^{-2}$ at $V_{\rm BG} = 50$ V). Noticeably, for low $n$ cases with smaller negative MR at the base temperature (see Fig. 3(b) for $n = 1.1\times 10^{12}$ cm$^{-2}$ at $V_{\rm BG} = 10$ V), thermally induced suppression is more evident, manifested as a classical positive MR at elevated temperatures. This temperature effect can be largely attributed to the reduction of $L_{\rm mfp}$ due to the enhancement of electron-phonon scattering [see Fig. 3(c)],[10,22-24] which results in a transition from quasi-ballistic to diffusive transport. To better demonstrate the intimate relationship between $L_{\rm mfp}$ and the boundary scattering-dominated negative MR effect, we plot the dependence of MR (at $B = 0.5$ T) on $L_{\rm mfp}$ at various temperatures and gate voltages [Fig. 3(d)]. It is clear that the negative MR effect gets overall weakened with decreasing $L_{\rm mfp}$, and a positive MR regime eventually emerges when $L_{\rm mfp}$ is smaller than a certain value [see Fig. 3(d)]. This value corresponds to the lower limit of $L_{\rm mfp}$, or rather, the upper-limit of bulk scattering rate, above which boundary scattering contributes to the channel transport. Unexpectedly, this value can be as small as 0.3 µm, that is, only 0.15 times the sample width. Consistent results have also been obtained for the hole side of this BN/SLG/BN device, and another BN-encapsulated bilayer graphene device, as shown in Figs. S2 and S3 in the Supplemental Material. These results are seemingly inconsistent with previous study, since boundary scattering is taken into consideration only in the typical (quasi-) ballistic regime, when $L_{\rm mfp}$ is larger than $W$ or at least comparable with it.[2,3,5,7,8] This intriguing result may be related to the reflection properties of sample edge.[8,9] For ideal specular boundary scattering (the specularity parameter $p = 1$), carriers are elastically reflected at the edge without momentum relaxation, which results in negligible backscattering probability. However, practically there are always inevitable factors that increase the edge roughness and thus the diffusiveness of boundary scattering. Accordingly, carriers are scattered at a random angle with respect to the normal of sample edge, leading to extra backscattering. The negative MR effect described above arises from the suppression of backscattering under a magnetic field. In BN-encapsulated graphene devices, the observation of negative MR at a small ballistic ratio can thus be attributed to the ultra-low specularity of the edge, since etching graphene flakes into Hall-bar geometry will introduce dopants and defects at the edge and increase the roughness. Note that in addition to the reduction of the negative MR magnitude, increasing temperature also leads to the broadening of the overall resistance peak. Based on previous theoretical calculations,[8] this result cannot be simply interpreted by a decrease in ballisticity. Enhanced electron-electron scatterings, with an effective scattering rate proportional to $T^{2}$,[8] play a more significant role.[25-27] From the dashed line in Fig. 3(d), it is clear that the lower limit of $L_{\rm mfp}$ becomes larger as $V_{\rm BG}$ decreases on the electron side (i.e., the carrier density decreases). To understand this behavior, we plot the 1.5 K MR curves measured at different $V_{\rm BG}$ in Fig. 4(a) (see Fig. S5 for typical data for the hole side). As the carrier density decreases (indicated by the arrows), the magnitude of negative MR decreases and the side peak at $B_{{\max}}$ becomes less obvious, consistent with previous observations of quasi-ballistic transport.[2,3,5] For the low $n$ case, e.g., at $V_{\rm BG} = 0$ V, the MR curve in the small field regime evolves from a double- to single-peaked structure, and typical SdH oscillation behavior occurs as the field decreases to 0.6 T.
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Fig. 4. Evolution of MR effect with varying $V_{\rm BG}$: (a) MR curves for different $V_{\rm BG}$ measured at 1.5 K. (b) MR value at $B = 0.5$ T (black line), the extracted $L_{\rm mfp}$ (red line) and $L_{\rm long}$ (blue line) as functions of $V_{\rm BG}$ at 1.5 K. The corresponding values of carrier density for each $V_{\rm BG}$ can be found in Fig. 1(c).
Based on Fig. 4(b), we find that the negative MR value at $B = 0.5$ T (black line) is not strictly positively correlated with the extracted $L_{\rm mfp}$ (red line) at different $V_{\rm BG}$. Although bulk mean free path is determined by various types of carrier scattering, in graphene, short-range scattering (e.g., due to neutral impurities) and long-range charged-impurity scattering are assumed to be the main contributors.[3,28] As detailed in Section II in the Supplemental Material, the mean free path for long-range scattering ($L_{\rm long}$) and for short-range scattering ($L_{\rm short}$) can be extracted. $L_{\rm long}$ is smaller than $L_{\rm short}$ across a wider range of $V_{\rm BG}$ [see Fig. S4(c)], demonstrating the dominant role of charged-impurity scattering within the bulk region. As shown in Fig. 4(b), the amplitude of negative MR increases monotonically with increasing $L_{\rm long}$ at different $V_{\rm BG}$, indicating that the boundary scattering effect is more pronounced when charged-impurity scattering in the bulk region get reduced, while the latter one is induced by the increasing screening effect in the cases with higher doping. This may explain why the lower limit of the ballistic ratio varies under different gate voltages. In summary, using high-quality BN-encapsulated graphene as a model system, we have systemically investigated the evolutions of quasi-ballistic transport behaviors of a 2D conductor over a large parameter space of temperature and carrier density. We find that the typical negative MR effect dominated by electron-boundary scattering occurs even when the ballistic ratio is as low as 0.15, which can be attributed to the ultra-low specularity of the sample edge. These findings lead us to a deeper understanding of the role of boundary scattering in transport performance of mesoscopic systems. Further optimizing the negative MR effect with high temperature robustness may have important applications in 2D electronic devices. Acknowledgments. This work was supported by the National Natural Science Foundation of China (Grant Nos. 92165201 and 11974324), the Innovation Program for Quantum Science and Technology (Grant No. 2021ZD0302800), the Strategic Priority Research Program of Chinese Academy of Sciences (Grant No. XDC07010000), the Anhui Initiative in Quantum Information Technologies (Grant No. AHY170000), the Hefei Science Center CAS (Grant No. 2020HSC-UE014), and the Fundamental Research Funds for the Central Universities (Grant No. WK3510000013). References Boundary scattering in quantum wiresBoundary Scattering in Ballistic GrapheneEvidence for hydrodynamic electron flow in PdCoO2Visualizing Poiseuille flow of hydrodynamic electronsMicrometer-Scale Ballistic Transport in Encapsulated Graphene at Room TemperatureMagnetotransport in heterostructures of transition metal dichalcogenides and grapheneManifestations of classical size effect and electronic viscosity in the magnetoresistance of narrow two-dimensional conductors: Theory and experimentAnomalous Angle-Dependent Magnetotransport Properties of Single InAs NanowiresBoron nitride substrates for high-quality graphene electronicsOne-Dimensional Electrical Contact to a Two-Dimensional MaterialElectrically tunable transverse magnetic focusing in grapheneExperimental observation of the quantum Hall effect and Berry's phase in grapheneTwo-dimensional gas of massless Dirac fermions in grapheneInelastic scattering in a monolayer graphene sheet: A weak-localization studyWeak Localization in Graphene FlakesWeak-Localization Magnetoresistance and Valley Symmetry in GrapheneWeak Antilocalization in Epitaxial Graphene: Evidence for Chiral ElectronsLimitations to Carrier Mobility and Phase-Coherent Transport in Bilayer GrapheneProximity-induced spin-orbit coupling in graphene/ Bi 1.5 Sb 0.5 Te 1.7 Se 1.3 heterostructuresLocalization to delocalization probed by magnetotransport of hBN/graphene/hBN stacks in the ultra-clean regimeIntrinsic and extrinsic performance limits of graphene devices on SiO2Acoustic phonon scattering limited carrier mobility in two-dimensional extrinsic grapheneGiant Intrinsic Carrier Mobilities in Graphene and Its BilayerNegative Magnetoresistance in Viscous Flow of Two-Dimensional ElectronsHydrodynamic Electron Flow and Hall ViscosityBallistic and hydrodynamic magnetotransport in narrow channelsElectronic transport in two-dimensional graphene
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