Berry Phase Transition in Twisted Bilayer Graphene

The electronic dispersion of a graphene bilayer is highly dependent on rotational mismatch between layers and can be further manipulated by electrical gating. This allows for an unprecedented control over electronic properties and opens up the possibility of flexible band structure engineering. Here we present novel magnetotransport data in a twisted bilayer, crossing the energetic border between decoupled monolayers and coupled bilayer. In addition a transition in Berry phase between pi and 2pi is observed at intermediate magnetic fields. Analysis of Fermi velocities and gate induced charge carrier densities suggests an important role of strong layer asymmetry for the observed phenomena.

Additionally the energetic range between vHs features two effectively decoupled systems in closest possible vicinity, associated with phenomena like excitonic condensation, Coulomb drag or quantum capacitive screening of charge [14][15][16][17][18][19]. To date, TBG have been extensively studied by scanning tunneling microscopy resolving angle dependent moiré superstructures of wavelength ( ) = /(2 • sin ( 2  )) (1) and confirming the predicted vHs in spectroscopy measurements [6,8,[11][12][13]. Another powerful tool of investigation lies in magnetotransport experiments which provide access to many of graphene´s unique features [20][21][22][23]: In magnetic fields applied perpendicular to the sample plane, the Landau level spectrum for TBG is predicted to be divided into two 3 regimes [9,[24][25][26][27]: Below the vHs, assuming uniform carrier density in the two decoupled layers, ) • 4 (with + as nonzero integer) like in a Bernal stacked bilayer [22] (=0°). This scenario works in the extended zone scheme and neglects commensuration effects [24]. Cyclotron orbits around and θ merge into one above the vHs (same for , and θ , ), now enclosing two Dirac points, which corresponds to = ±2 and a Berry phase of  = 2π [28]. The distinguishing experimental factor for one [9,27] or the other manner of coupling and quantization [24,25]  there has been one report on Bernal-bilayer-like Quantum Hall data in a TBG, which is in line with the second above described model [24,25]. We here present further evidence for the according scenario, witnessing the corresponding Berry phase transition within a primary Landau fan for the first time.  Graphene samples are prepared by mechanical exfoliation of natural graphite onto a substrate of SiO2. Some flakes fold over during this procedure, yielding twisted layers which are processed and contacted for electrical measurements as sketched in Fig. 1a. Fig. 1b shows atomic force microscopy (AFM) topography data over the step between TBG and the uncovered monolayer, revealing a height difference of 6.2 ± 0.2 Å as evident in the histogram in Fig. 1c, fit by a double Gaussian distribution. Note that this value is larger than the interlayer spacing in graphite, which is ascribed to the different stacking arrangements [10,29,[32][33][34]. Fig. 1d shows the torsional signal of an AFM scan on the twisted bilayer under investigation, which reveals a periodic structure of 5.7 ± 0.2 nm wavelength fit by an overlain honeycomb pattern. Using eq. 1 the corresponding twist angle  can be calculated as 2.5 ± 0.1 °.    8 An important theoretical prediction for the low energy dispersion between vHs is a twist angle dependent reduction in Fermi velocity following with F red and F 0 as reduced and native Fermi velocity respectively [5,8,13]. For = 2.5°, eq. 2 yields a renormalization factor of 0.62 with the commonly found parameters F 0 = 1 • layer, we ascribe this discrepancy to electron-hole asymmetry. Like in the present case, stronger reduction in Fermi velocities on the hole side has been found in other TBG [8,13,30] and is ascribed to enhanced next-nearest-neighbor hopping [8]. As tot goes across the border of region III, Fermi velocity starts to rise, indicating changes to the dispersion. Because region II oscillations are confined to low magnetic fields only however, further velocity data could not be reliably acquired for region II. High density data points in the red area stem from high magnetic field oscillations with  = 2π (region I) and center around a constant value of 0.94 • 10 6 ms −1 near the one of native graphene. Note that the lack of a slope in Fermi velocity over energy is 9 indicative of massless carriers and a linear dispersion. This clearly sets our region I data apart from a Bernal stacked bilayer and its parabolic dispersion, commonly associated with a Berry phase of 2.    layers. Such a symmetric offset in Fermi energy may also be caused by an inherent shift due to breaking of electron-hole symmetry in the TBG [5,38]. The fits to the decoupled layers´ densities 12 are used to determine a total charge carrier density tot = b + t , extrapolating the TBG´s capacitive coupling to the backgate away from overall charge neutrality.
In addition to the discussed modelling and data for b , t and tot in the layer-decoupled region III, Fig. 5 shows charge carrier concentrations extracted at higher energies. Gray dots indicate concentrations extracted from low magnetic field data at  = π (region II), red dots in high magnetic fields at  = 2π (region I). Solid lines are linear fits sharing an absolute slope of 6.59 ± 0.18 · 10 14 m −2 V −1 which is in good agreement with the backgate´s calculative capacitive coupling constant = 6.53 · 10 14 m −2 V −1 and slope of tot over BG .

13
This suggests all of the induced charge carriers filling up the examined high-energetic Landau levels, which indicates quantization of a coupled system in the corresponding ranges. Said behavior partly conforms to theory as beyond a certain energy vHs , layers should merge in a single system [5,[24][25][26]. The most important prediction for this layer-coupled case is a quantization at Berry phase  = 2π due to a topologically protected zero mode [24][25][26]. comply with theory while regarded on their own, the persistence of Berry phase  at low magnetic fields as well as deviation from tot in both 2π and π constitute an interesting deviation from the predicted scenario. We attribute this to strong layer asymmetry in our system, which is not accounted for in the predicted Landau quantization for TBG [24,25]. In the following we will provide a self-consistent qualitative explanation for the observed deviations from the layer-symmetric case: An important peculiarity lies in the fact, that the transition from region III to II takes place at a charge carrier concentration π close to b on the electron side ( Fig. 5 at around VBG~50V) and close to t on the hole side ( Fig. 5 at around VBG~20V), while the opposite layer´s density is small in comparison. Note that firstly, the transition to π at only the dominant layer´s density b (or t respectively) is consistent with the Berry phase of  in the according oscillations, as a Berry phase of 2 would require the inclusion of both layers´ zero modes [24,25]. Secondly, exclusion of the other layer´s charge may be linked to localization due to strongly reduced Fermi velocities, when interlayer bias renders ∆ eff small (compare ∆ 1,2 in Fig. 4a) and the corresponding energy scale eff 0 becomes comparable to θ (see Fig. 6). Note that the excluded layer´s calculated Fermi velocity at the transition point (Fig. 4b) is much 14 smaller than the dominant one´s, and even close to zero on the hole side (hole-branch of bottom layer at VBG~20V). Another interesting cohesion can be found at the II-I transition. Figure 6 shows a schematic picture of the calculated dispersion at the triple point between regions I, II and III on the electron side (compare Fig. 5). The Fermi energy still lies below the vHs and, in the absence of a magnetic field, in the regime of electron conduction for the bottom and hole conduction for the top layer. Around the II-I transition at a magnetic field ≈ 6.75 T (see Fig. 2 After submission of this manuscript, very recent experimental indications [39] for the more rigorous backfolding scheme with a change of effective carrier polarity around the vHs [9,27] came to our notice. A different shaping of the superlattice due to a smaller angle as well as 16 encapsulation of the TBG device is likely to be responsible for the manifestation of the corresponding coupling scenario [9,27] as opposed to the one evidenced in our present work [24,25].

ASSOCIATED CONTENT
Supporting Information. Supplements on sample preparation and conduction of measurements; determination of Berry phase and Fermi velocities; account of screening model and implementation of dynamic asymmetry.

Author Contributions
The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript.

Funding Sources
This work was supported by the DFG within the priority program SPP 1459, graphene and by the School for Contacts in Nanosystems.

S1
Supporting information for: Twisted bilayer graphene (TBG) of a desired angular range can be selected based on an estimated interlayer twist derived from sample geometry S1 . A more accurate measure of the two layer´s configuration is then obtained via resolution of the moiré pattern between the stacked lattices using an atomic force microscope.
To enable transport measurements, electrical contacts were fabricated via e-beam lithography and evaporation of Cr/Au in a longitudinal four-probe setup. A substrate of highly doped silicon, capped with 330 nm of silicon dioxide, served as a backgate to adjust the Fermi level.
Measurements were carried out in a 4 He evaporation cryostat at temperatures down to 1.5 K in perpendicular magnetic fields up to 13 T. At a constant DC current of 500 nA, the sample resistance was measured in four-probe configuration.

DATA ANALYSIS: BERRY PHASE
Beside the graphic approach presented in the main paper, the Berry phase  of Shubnikov-de Haas (SdH) oscillations can be derived by a linear fit to the inverse magnetic field position of the Nth oscillation minimum (N + 0.5th maximum) over a range of N S2,S3 . Fig. s1a,b shows two exemplary oscillations across regions I and II (see main paper for partition of measured range).

S2
Extrema are determined and plotted as their index vs. position in inverse magnetic field for the two presented and three further oscillations at different charge carrier densities for region I and II respectively (Fig. s1c,d).

Decoupled range
To analyze temperature damping and consecutively effective masses and Fermi velocities from the two superimposed SdH oscillations, we have used the following method: The resistance data R in dependence on inverse magnetic field B -1 has to be separated into three different contributions of the first and second oscillation and a background magnetoresistance. This is achieved by fitting the data with: The polynomial of second order in magnetic field B with coefficients 0−2 accounts for background magnetoresistance S4,S5 . Two cosine functions of frequency , damped by the exponential Dingle factor S3,S6 with coefficient , account for SdH oscillations with Berry phase  = in top (i=t) and bottom (i=b) layer respectively.  . Lines: Fits of eq. s2 to data. S5 To extract effective masses for the separate layers, eq. s2 can be conveniently fit to oscillation values ∆ at constant filling factor  = • h/(  • e) = 4(2) for the bottom (top) layer vs.
the composite variable •  −1 . According data and fits based on the oscillations in Fig. s3a,b are shown in Fig. s3c  The above described procedure is repeated for a range of charge carrier densities. Thusly extracted effective masses are presented in Fig. s4. In single layer graphene the Fermi velocity F relates to the effective mass * as S6 with h as Planck constant and n as charge carrier density in the Dirac cones. A fit of eq. s3 to b * vs. b (see Fig. s4b) yields F b = 6.84 • 10 5 ± 0.14 • 10 5 ms -1 for the bottom layer. This is a clearly reduced value with respect to the Fermi velocity ≈ 1 • 10 6 ms -1 observed in pristine monolayer graphene, as marked by the dashed black line in Fig. s4b.
The top layer data scatter more strongly and are extracted over a small range of t which has a flat progression in the examined region (see fig. 4c of main paper). The extracted top layer´s Fermi velocity of around F t = 4 • 10 5 ms -1 can therefore only be seen as a rough estimate.
Nevertheless it can be surely stated, that F t is also reduced with respect to the pristine graphene value.

Coupled range
To determine and remove background resistance, the data in region I are fit by eq. s1 with only one damped cosine term and a Berry phase of  = 2 . Fig. s5a shows thusly isolated oscillations for five different temperatures. A fit of eq. s2 to ∆ T at fixed filling factor  = 14 (see example in Fig. s5b) for three different total charge carrier concentrations yields values between 2 * = 3 • 10 −32 and 4 • 10 −32 , presented as black squares in Fig. s5c.
As an alternative method of extraction, the simple problem of solitary oscillations in region I (as opposed to superposition in region III) allows for a global fitting procedure yielding effective masses based on temperature dependence over the whole range in −1 as opposed to a single filling factor. Accordingly extracted values (red circles, Fig. s5c) show good agreement with the aforementioned fixed-filling-factor results (black squares in Fig. s5c) and are used for further analysis in region I. S7 Extracted effective masses 2 * clearly rise with charge carrier density tot (Fig. s5c) S8

Calculation of
The charge density t induced by the backgate in the top layer gives rise to an energetic shift between top and bottom layer´s dispersion by with e as elementary charge, = ε 0 as interlayer capacitance (ε 0 as dielectric constant, d as interlayer distance) and δ as doping charge in the top layer S7-S9 . This way, the carrier density in the top layer t can be calculated in dependence on variable E and free parameters d and n.

Calculation of Fermi velocities
The position of intersection between two rotationally displaced Dirac cones (positioned at = ± ∆ 2 with k=0 in the middle of a straight connection between Dirac points S10 ), dK is obtained by equating two linear slopes and solving to Fermi velocities are now calculated, using ∆ 1,2 = ∆ ± 2 • d as described in the main text.

Calculation of
The Fermi energy with respect to the top layer´s Dirac point is calculated as with F t as top layer´s Fermi velocity in the half-cone, crossing the Fermi level.
The Fermi energy with respect to the bottom layer´s Dirac point can now be calculated as The bottom layer´s charge carrier density follows as with F b as bottom layer´s Fermi velocity in the half-cone, crossing the Fermi level.

Assignment of
In the model of a parallel plate capacitor, the total charge carrier density tot induced via an electrical gate with dielectric material of relative permittivity r and thickness L couples with to the gate voltage BG . The used wafers feature a dielectric of SiO2 with L=330 nm and r=3.9 which translates to a coupling constant of = 6.53 • 10 14 m −2 V −1 .
The free parameter of overall charge neutrality in gate voltage, BG 0 is adjusted by fitting tot = • ( BG − BG 0 ) to the data. S10