Strain-induced doping and zero line mode at the fold of twisted Bernal-stacked bilayer graphene

The folding of Bernal-stacked bilayer graphene leads to electronic devices that can be understood as combinations of a twisted double-bilayer graphene and a fold. In magnetotransport experiments contributions of the two different parts can be identified. For the twisted double-bilayer graphene Landau fan diagrams with satellite fans depending on twist angle are observed. The fold gives rise to a local minimum in conductance which does not shift with applied perpendicular magnetic field. Regardless of twist angle the fold favors electron doping attributed to compressive strain at the kink geometry. The curvature of the folded structure provides for a systematic explanation, which is also in agreement with the observed correlation between twist angle and interlayer distance. Finally, the appearance of the topological zero line mode formed at the fold is discussed.

Given that folding can provide for a sharp interface overcoming limits of conventional lithography, it can be adopted to realize new topological states. A representative phenomenon is the so-called zero line mode (zLM) in Bernal-stacked BLG where chiral edge states originate from different valleys [24][25][26]. It has been reported that the zLM can be formed by folding of Bernal-stacked BLG [27,28]. Therefore, double bilayer graphene (DBLG) formed by folding of Bernal-stacked BLG becomes an important platform to investigate (a) superlattice related phenomena from stacked region and (b) topological phenomena from folded region.
Here, we investigate the magnetotransport properties of folded DBLG. As in tBLG Landau fan diagrams depending on twist angle between the upper and lower BLG are obtained. In addition, a magneticfield-independent feature as an electrical signature of folding is observed, exhibiting electron doping behavior and topological transport related with zLM. Combining structural and electrical studies, we reveal (a) the origin of the electron doping in terms of lattice deformation, compressive strain, and (b) a structural correlation between twist angle and interlayer distance, which can provide for an additional base in the emerging research field of twistronics [29].

Experimental section
The folded DBLGs discussed here were obtained during the mechanical exfoliation process. The redfiltered contrast in optical microscopy helps to identify the constituent bilayer graphene. In addition, we used a Raman microscope (LabRam, Horiba) with a laser at an excitation wavelength of 532 nm and ambient conditions to pin down the Bernal stacking. The devices have been encapsulated by hexagonal boron nitride (h-BN from HQ Graphene) [30] and the entire heterostructures are located on SiO 2 (330 nm)/Si substrates. For all heterostructures studied each h-BN layer has been intentionally misaligned to exclude any coupling between the graphene layers and the h-BNs. We have annealed all devices with forming gas (5% H 2 and 95% N 2 ) at 350 • C and performed atomic force microscopy (AFM) measurement to select bubble-free regions. A SF 6 reactive ion etching process was performed to allow for metal electrodes with atomic edge-contact configuration. By applying a back-gate voltage (V bg ) to the backside of the device the charge carrier densities (n) of the graphene heterostructures are modulated. Top-gate (TG) voltages were kept to be ground during all measurements. Electrical experiments have been carried out by low frequency AC or DC methods. All transport measurements have been done in a variabletemperature cryostat equipped with a superconducting magnet.  figure 1(a). An external magnetic field (B) is applied perpendicular to the planar surface and the electrical current flows along both the stacked and the folded region. The external magnetic field effectively acts as being perpendicular to the stacked region and being parallel to the folded region as shown in the lower panel of figure 1(a). Furthermore, a uniform field applied to the folded layers is equivalent to opposite fields on spatially distinct regions as shown in figure 1(b) giving origin to snake states [15,16] under applied magnetic fields and zLMs [24][25][26][27][28] under applied electric fields (E). Such a folded region should provide a sharp interface on a scale of few nanometers being much sharper than typical interfaces which can be generated by lithographical methods.

Results and discussion
We present here results about two representative devices, one device with a large twist angle (device 1) and one device with a small twist angle (device 2). An AFM image of device 1 is shown in the lower panel of figure 2(a) together with a height profile giving a height of 2.4 nm. A corresponding schematic image is depicted in the upper panel of figure 2(a), where the blue triangular region indicates the folded DBLG. We used the relation θ = 2φ between twist angle (θ) and folding angle (φ) (see the upper panel of figure 2(a)) to estimate the twist angle to θ ≈ 28 • [2]. Having folded Bernal-stacked graphene was confirmed by the 2D peak of the Raman spectrum (see inset of figure 2(b)) [31], which is fitted by four Lorentzians (gray curves). The 2D peak stems from double-resonance processes, which can be used as a finger-print for the number of layers and the stacking order [31,32]. MLG or tBLG with large twist angle exhibits only a single Lorentzian peak, while for Bernal-stacked graphene four Lorentzian peaks are typically seen. Figure 2(b) also shows the gate-voltage dependent conductance (G) of device 1 at 0 T. A charge neutrality point (CNP) appears as a large dip at V bg ≈ −10 V (blue square) and an additional small dip is present at V bg ≈ −4 V (red-dotted arrow). The two conductance minima exhibit different dependencies on magnetic field. The position of the conductance minimum originating from the fold is more or less independent on magnetic field (red-dotted arrows in figure 2(c)), whereas from the CNP a Landau fan originates (blue square). Blue-dashed lines in figure 2(c) correspond to total filling factors v = ±4, ±8, where the observed sequences of v can be understood by the asymmetric doping between upper and lower layers [4,13,14,22,[33][34][35] due to the application of only back-gate voltages. When two layers are symmetrically doped, the total filling factor (v upper + v lower ) is supposed to exhibit 8 (4 + 4), 16 (8 + 8), 24 (12 + 12),··· for DBLG. However, the sequence of 4, 8 is expected to originate from (0 + 4), (4 + 4) because the upper layer is relatively less doped due to screening by the lower layer. Satellite dips from superlattice structures are not observed and are also not expected in case of such a large twist angle. Figure 3(a) shows an AFM image of the second device (device 2) with a height profile giving a height of 2.0 nm for this folded DBLG. A clear bump-like structure as commonly observed in the tBLGs [36] does not appear at the fold of our folded DBLG similar to device 1. Compared to the tBLG, the DBLGs seem to be more consistent with the simple geometry without bumps hinting towards DBLGs being more rigid than MLG. The gate-voltage dependent conductance (G) of device 2 at 0 T is shown in figure 3(b). Local conductance minima which are identified by their magnetic field dependence are seen at However, the multiples of four in filling factors for device 2 can be attributed to the influence of the Moiré potential; in other words, the folded DBLG with small twist angle can be regarded as a single sys-tem which possesses a nontrivial Berry curvature of the flat bands [37]. The satellite features occur when the superlattice (Moiré) unit cell is fully occupied. As a result, we extract a twist angle θ ≈ 0.7 • , from the superlattice density, n = 4/A SL , unit cell area A SL = √ 3/2λ 2 SL , and wavelength λ SL = a/2sin (θ/2), where a is the lattice constant of AB-BLG, 0.246 nm. Furthermore, weak features for filling factors (multiples of v = 4) are observed for the two satellite fans in figures 3(c) and (d). It is noteworthy that correlation phenomena such as superconductivity and Mott insulator are not observed in our folded graphene layers in spite of similar twist angle as recently reported twisted DBLG [38][39][40]. Even though a lower mobility and therefore stronger disorder are mainly considered to explain the absence of these correlation phenomena, the influence of the folded structure might be also of importance. A recent study has reported that correlated phenomena are sensitively affected by the interlayer interaction (e.g. interlayer spacing) tuned with external pressure [41]. Indeed, the observation that the interlayer distance is connected to the twist angle in folded graphene [36] may reveal the significant role of folding. Namely, the folded structure can play a role as interlayer coupling parameter, which needs to be further investigated.
In the following we will discuss the electronic properties of the fold in more detail. As in device 1 and 2 where the folds showed up at V bg ≈ −4 and −40 V, interestingly in the previous observation of a fold for tBLG also a negative value of V bg ≈ −25 V was measured [2]. The fact that the folded signatures are always located at negative values of V bg (being independent of the residual doping of the twisted layers which lead to slight variations in the gate-voltage positions of the CNPs) hints towards a preferential electron doping of the folded region. Since folding produces structural deformations within the layers, the electronic properties of the fold will differ from the properties of the stacked regions. In view of the fact that compressive strain is present at the folded region [42], the electron doping can be explained with the increase of the Fermi level due to a compressed unit cell [43,44]. In addition to such compressive strain electron accumulation due to the kink geometry is another factor. From electrostatics it is well known that the electron density in a conductor is higher near a kink geometry [45]. The curvature of the folded structure should be an appropriate parameter to estimate the degree of structural deformation. Using the simple geometry for the folded graphene as shown in the lower panel of figure 1(a), the curvature can be described by κ ≡ 1/R = 2/h, where R and h are the radius of the folded region and the interlayer distance, respectively. The values for the heights h of the DBLGs are given by the height-profiles as obtained from the AFM measurements.
According to Rode et al [36], the twist angle and the interlayer distance of a folded structure are correlated. In our simple model the interlayer distance determines the curvature of the folded structure. Therefore, we compare the amount of electron doping (n folded ), interlayer distance (h), curvature (κ), and twist angle (θ). Note that n folded is calculated by a parallel capacitance model using V bg value of the folded signature. The doping concentrations n folded found for device 1 and 2 are 2.5 × 10 11 cm -2 and 17 × 10 11 cm -2 , respectively. Device 1 (θ ≈ 28 • ) and 2 (θ ≈ 0.7 • ) exhibit h of 2.4 and 2.0 nm. Our observation here is in agreement with the correlation between interlayer distance (i.e. curvature of the folded region) and twist angle as found for tBLGs [36]. Even though the interlayer distance is expected to vary non-monotonically as a function of twist angle, two twist angles in this work show consistent results. From the different interlayer distances calculated curvatures κ are given as 0.83 nm -1 (device 1) and 1 nm -1 (device 2). One sees that the curvature being higher the more electrons are doped in the folded region. In addition, one sees that the smaller twist angle has the higher curvature, which results in a higher doping tendency for smaller twist angles. We note that this trend maintains validity within the tBLG case as previously reported (n folded ≈ 16 × 10 11 cm -2 for θ ≈ 1.1 • ) [2].
Considering the fact that the stacked and folded regions contribute to the entire conductance in parallel in our device, we can extract the electrical properties of the folded structure by subtracting from the two-terminal conductance G the background conductance G background , where G background might be approximately determined by its bulk part (i.e. the stacked region). Figures 4(a) and (b) show temperature and magnetic-field dependences of the extracted conductance ∆G of the fold of device 2 in such a way. The depth of the minimum observed in ∆G is around e 2 /h inspiring quantized transport via a topological zero-energy mode (or zLM). The minimal conductance seems to be only weakly dependent on temperature, whereas the width of the minimum in ∆G is broadened with increasing temperature hinting towards thermal broadening. While the gate-voltage position of the folded signature is fixed as function of magnetic field, the minimal value of ∆G increases with magnetic field (i.e. decreased depth) as shown in figure 4(b). The suppression of the depth shows that the conductance contributed from the folded region to the total conductance of the device increases. Such an increase in conductance is commonly observed for transitions to chiral edge state transport as it appears for the topological zero-energy mode. Nevertheless, the fact that the electrical signature for the zLM is pinned at certain V bg and has a nearly quantized value indicates the topological robustness of the edge channel emerging in the folded region. Furthermore, quantum interference effects such as weak localization and universal conductance fluctuation might be involved in electrical transport causing. The positive magnetoconductance of the folded region might be influenced by a suppression of weak localization in the regions adjacent to the fold. Even though the direction of the magnetic field is effectively parallel for the kink of the fold, perpendicular components can be locally present. Indeed, a similar positive magnetoconductance was observed in the case of ripples [46] which could represent multiple folds. Also, the weak oscillatory features overlapped on ∆G are suppressed by either elevating temperature (figure 4(a)) or applying magnetic field ( figure 4(b)), which may imply the contribution of universal conductance fluctuation.
It is noteworthy that the zLM is expected to be observed regardless of twist angle. Even though we obtained a signature of the fold for device 1 as shown in figure 2(c), a distinction from the Landau fan of the bulk twist system is difficult. Namely, the electrical signatures for the folded region and twisted bulk regions in device 1 are closely located in terms of V bg , preventing further analysis for the zLM of device 1. Because the V bg values are determined by the curvature of the fold and the unintentional doping for folded and twisted bulk regions, respectively, certain conditions for twist angle and unintentional doping could provide for a better observation even in the case of large twist angles.
For tBLGs a clear signature of the fold was achieved within a separate-contact configuration [2], where the electrical current is strongly guided through the folded structure. In the work presented here the stacked and folded regions are connected in parallel, which turns out to be sufficient for detecting transport due to the fold, similar to the recently reported observation of zLM [28]. In order to explore snake states or zLMs in more detail an insulating layer such as h-BN would help to guide all the current through the fold.

Conclusions
In conclusion, we have studied the magnetotransport properties of folded DBLGs with large and small twist angles. In magnetotransport the electronic properties of the stacked and folded regions can be distinguished. The stacked region results in twistangle dependent Landau fan diagrams. Satellite fans are observed for the small twist angle. This twist-angle dependence hints towards similar new rich physics as already found for tBLGs. The folded region is always electron doped regardless of twist angle. The amount of doping is correlated with the twist angle and the distance between the layers and is attributed to the additional compressive strain due to the folding. In addition, the conductance of the fold which might be related to transport via topological zLMs is studied as function of temperature and magnetic field.

Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files).