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Out-of-plane dielectric susceptibility of graphene in twistronic and Bernal bilayers

Out-of-plane dielectric susceptibility of graphene in twistronic and Bernal bilayers Sergey Slizovskiy, Aitor Garcia-Ruiz, Alexey I. Berdyugin, and Na Xin National Graphene Institute, University of Manchester, Booth St.E., M13 9PL, Manchester, UK and Dept. of Physics & Astronomy, University of Manchester, Manchester M13 9PL, UK Takashi Taniguchi and Kenji Watanabe National Institute for Materials Science, 1-1 Namiki, Tsukuba, 305-0044, Japan Andre K. Geim National Graphene Institute,, University of Manchester, Booth St.E., M13 9PL, Manchester, UK and Department of Physics & Astronomy, University of Manchester, Manchester M13 9PL, UK Neil D. Drummond Department of Physics, Lancaster University, Lancaster LA1 4YB, UK Vladimir I. Fal'ko National Graphene Institute, Booth St.E., M13 9PL, Manchester, UK Department of Physics & Astronomy, University of Manchester, Manchester M13 9PL, UK and Henry Royce Institute for Advanced Materials, Manchester, M13 9PL, UK (Dated: June 15, 2021) We describe how the out-of-plane dielectric polarizability of monolayer graphene in uences the electrostatics of bilayer graphene { both Bernal (BLG) and twisted (tBLG). We compare the polar- izability value computed using density functional theory with the output from previously published experimental data on the electrostatically controlled interlayer asymmetry potential in BLG and data on the on-layer density distribution in tBLG. We show that monolayers in tBLG are described well by polarizability = 10:8 A and e ective out-of-plane dielectric susceptibility  = 2:5, in- exp z cluding their on-layer electron density distribution at zero magnetic eld and the inter-layer Landau level pinning at quantizing magnetic elds. Bilayer graphene [1{3] is a two-dimensional (2D) ma- an electric eld oriented perpendicular to the 2D crys- terial with electronic properties tuneable over a broad tal. Here, we compute the e ective dielectric suscepti- range. The manifestations of the qualitative change bility,  , of graphene monolayer using ab initio density of electronic characteristics of both Bernal (BLG) and functional theory (DFT) and implement the estimated twisted (tBLG) bilayer graphene, produced by electro- DFT value of  in the self-consistent description of (a) static gating [3] and inter-layer misalignment [4, 5], were tBLG electrostatics in tBLG with twist angles outside the observed in numerous experimental studies of the elec- magic angle [5, 8] range and (b) conditions for the inter- tronic transport in graphene-based eld-e ect transis- layer Landau level pinning in FET with a 30 -twisted tor (FET) devices. These versatile electronic proper- bilayer, with the results of modelling showing favourable ties make FETs based on BLG and tBLG an attractive quantitative agreement with the available and new exper- hardware platform for applications tailored [6{8] for var- imental data. Then we take into account the out-of-plane ious quantum technologies. While, over the recent years, dielectric susceptibility of a single graphene layer in the the fundamental electronic properties of bilayer graphene analysis of BLG inter-layer asymmetry gap, in partic- have been intensively studied, a mundane but practical ular, its dependence on the vertical displacement eld, characteristic of this material related to the out-of-plane (D), comparing the results with the measured exciton dielectric susceptibility of graphene layers largely escaped spectroscopy in gapped BLG [14]. attention of those investigations of BLG and tBLG in For the theoretical modelling of the out-of-plane dielec- FETs, despite several already recorded indications [9{ tric susceptibility, we employ the CASTEP plane-wave- 13] of its relevance for the quantitative modelling of the basis DFT code [15] with ultra-soft pseudopotentials. We operation of such devices. use a 53×53×1 k-point grid, a large plane-wave cut-o The out-of-plane dielectric susceptibility of a single of 566 eV, and a variety of interlayer distances c along graphene layer stems from the polarisability of its carbon the z-axis to compute the total energy, E , of graphene orbitals, that is, from the mixing of  and  bands by in a saw-tooth potential, Dz= , centered on the car- bon sites of the graphene layer (D being the displace- ment eld and c=2 < z < c=2). Then we determine [16] the polarizability in each cell of length c using on leave from NRC \Kurchatov Institute" PNPI, Russia the relation E = E D =(2 ), with E being the vac- 0 0 0 [email protected] uum energy. As the arti cial periodicity, introduced in arXiv:1912.10067v3 [cond-mat.mes-hall] 14 Jun 2021 2 ers of a bilayer, each with the electron density n , has b=t the form, D 1 +  n n b t u  U U = e e d: (1) t b 2 2 0 z 0 This expression is applicable to the description of both BLG and tBLG in a FET, improving on the earlier- published studies [3, 20, 21] where the out-of-plane di- electric susceptibility of graphene layers was missed out in the self-consistent band structure analysis. To describe a twisted bilayer with an interlayer twist angle , we use the minimal tBLG Hamiltonian, [4, 5], [H + u] 0 T 0 t k ;k k ;k H(k ;k) = ; (2) T [H u] 0 b k ;k k;k 2 t=b; H =~v ; t=b t=b; = k + i(k  K sin ); t=b; x y FIG. 1. (a) Sketches illustrating how dielectric polarizability 2 2 X i j 1 e of each monolayer enters in the electrostatics analysis of bilay- T =  (j); k ;k 0 i j k ;k+g 3 e 1 ers in Eq. (1). (b) Characteristic electron dispersion in tBLG j=0 (here,  = 3 ; u = 100 meV). Electron state amplitude on 2j 2j (j) the top/bottom layer is shown by red/blue. (c) Mini-valley g =  sin ; 1 cos 2K sin : carrier densities n 0 in a single-gated tBLG calculated for 3 3 2 various misalignment angles outside the magic angle range, in comparison with the densities corresponding to SdHO mea- Here, v = 10:2  10 m=sec is Dirac velocity in mono- sured [10] in a tBLG ake with an unknown twist angle (black layer graphene. Equation (2) determines [4] charac- dots). teristic low-energy bands, illustrated in Fig. 1(b) for 1    =~vK  2 (away from the small magic an- gles  1 ). This spectrum features two Dirac minivalleys 0 0 at  and  (j  j = 2K sin ), each described by its the DFT code, leads to a systematic error in the po- own Fermi energy, larizability,  (c) / c , we t the obtained DFT data with (c) = + a=c + b=c and nd  (c ! 1 DFT 3 E 0  (1 3)~v jn 0j sign n 0; F = = = 1) = 11:0 A per unit cell of graphene with the Perdew{ E E 0  (1 4)u; (3) Burke{Ernzerhof (PBE) functional and = 10:8 A F  F DFT with the local density approximation (LDA). These val- and carrier density n 0 , determined by the minivalley ues are close to the DFT-PBE polarizability reported in 3 3 area encircled by the corresponding Fermi lines (as in Ref. 17, = 0:867 4 A = 10:9 A , and when recalcu- Fig. 1(b)) and measurable using Shubnikov - de Haas os- lated into an e ective 'electronic thickness' =A, where cillations [9, 10] or Fabry-Perot interference pattern [12]. A = 5:2 A is graphene's unit cell area, we get 2:1 A, The above expression was obtained using expansion up comparable to the earlier-quoted `electronic thickness' of 1 2 to the linear order in  = [ ]  1. We also ~vj j graphene [12, 18]. Furthermore, for the analysis of bilay- note that, due to the interlayer hybridisation of electronic ers' electrostatics in this paper, the e ective out-of-plane 1 wave functions, the on-layer charge densities in Eq. (1) dielectric susceptibility is  = [1 =Ad] , where z DFT di er from the minivalley carrier densities, d is the distance between the carbon planes in the bilayer (d = 3:35 A for BLG and d = 3:44 A for tBLG, as in tur- n  n 0  2 n 0 n + 0:07 j  j ; (4) bostratic graphite [19]). This gives  = 2:6 for BLG, b=t = ~v and  = 2:5 for tBLG. For the electrostatic analysis of bilayers built into which makes the results of the self-consistent analysis FETs, we note that out-of-plane polarisation of carbon of tBLG electrostatics slightly dependent on the twist orbitals in each graphene monolayer is decoupled from angle, . We illustrate this weak dependence in Fig. 1(c) the charges hosted by its own -bands, because of mirror- by plotting the relation between the values of n and n symmetric charge and eld distribution produced by the in a single-side-gated tBLG computed using Eqs. (3), (4) latter, see in Fig. 1(a). As a result, the di erence between and (1) with  = 2:5 and d = 3:44 A. On the same plot, the on-layer potential energies in the top and bottom lay- we also show the values of n and n 0 recalculated from FIG. 2. (a) Resistance map for a double-gated tBLG with a 30 twist angle, computed with  = 2:5 and d = 3:44 A (left) and measured (right) as a function of the total carrier density, n and vertical displacement eld, D, at B = 0 and T = 2 K. (b) Computed density of states of pinning LLs (left) and the measured resistance,  (right) in a 30 tBLG at B = 2 Tesla, plotted xx as a function of displacement eld and lling factor. Bright regions correspond to the marked N =N LL pinning conditions. t b the earlier measured SdHO [10] in tBLG devices with an termines [22] the Fourier form-factor of the scatterers, unknown twist angle (seemingly, 10 15 ). 2 2 e =2  e = 0 0 = ; r = ; k = jn j; q s Fi i We also compared the results of the self-consistent q + r (k + k ) ~v s Ft Fb tBLG analysis with the measurements of electronic trans- port characteristics of a double-side gated multi-terminal and the corresponding momentum relaxation rate of FET based on a tBLG with  = 30 . The latter Dirac electrons [22], choice provides us with the maximum misalignment, cor- t=b 2 2 responding to  ! 0 in Eqs. (3) and (4), hence n = b=t = hj j sin 'i n : ' c 2k sin('=2) t=b Ft=b 2~ n 0 . In the experimentally studied device, tBLG was encapsulated between hBN lms on the top and bot- Then, in Fig. 2(a), we compare the computed and tom, thus providing both a precise electrostatic control measured tBLG resistivity. As in monolayer graphene of tBLG for B = 0 measurements and its high-mobility, [22], density of states, , cancels out from each t=b enabling to observe quantum Hall e ect at a magnetic 2 2 = 2 =(e v ), making the overall result,  = t=b t=b xx t=b eld as low as B = 2 Tesla. =[ +  ], dependent on the carrier density only t b t b For a quantitative comparison of the measured and through the wave-number transfer, 2k sin('=2), and Ft=b modelled tBLG characteristics, we assumed elastic scat- screening. This produces ridge-like resistance maxima at tering of carriers from residual Coulomb impurities in k = 0 or k = 0, that is, when Ft Fb the encapsulating environment with a dielectric constant 5, with an areal density n , screened jointly by the ~v jnj sign n e n(1 +  ) c z z D= = + : (5) carriers in the top and bottom layers. The screening de- ed 4 0 4 Lines corresponding to the above relation are laid over the experimentally measured resistivity map for a direct comparison. We nd an even more compelling coincidence be- tween the theory and experiment when studying the Landau level pinning between two electronically inde- pendent by electrostatically coupled graphene monolay- ers in a 30 tBLG. In a magnetic eld, graphene spec- trum splits into Landau levels (LLs) with energies E = v 2~je B Nj signN . In a twisted bilayer, in nite degen- eracy of LLs gives a leeway to the interlayer charge trans- fer which screens out displacement eld and pins par- tially lled top/bottom layer LLs, N and N , to each t b other and to their common chemical potential, , so that p p v 2~ejBj( N N ) = u, as in Fig. 2(b). This LL pin- t b ing e ect also persists for slightly broadened LLs. Tak- ing into account a small Gaussian LL broadening, , we write, u E 4eB n = erf ;   1; (6) t=b 2~ N = FIG. 3. Inter-layer asymmetry potential (dashed lines) and band gap (solid lines) in an undoped BLG, self-consistently solve self-consistently Eq. (1), and compute the total computed with various values of  = 1 (green), 2.6 (blue) density of states (DoS) in the bilayer. The computed and 2.35 (red) and compared to the experimentally measured DoS for B = 2 Tesla and  0:5 meV) is mapped in gate-tunable optical gap [14] (circles) and transport gap [23] Fig. 2(b) versus displacement eld and tBLG lling fac- 6 (crosses). Here, we use [24, 25] v = 10:2  10 m=sec, = 5 4 tor,  = hn=eB. Here, the `bright' high-DoS spots tot 0:38 eV, v = 1:23  10 m/s, v = 4:54  10 m/s,  = 22 meV, 3 4 indicate the interlayer LL pinning conditions, whereas and d = 3:35 A. Additionally, dotted lines show the values of the `dark' low-DoS streaks mark conditions for incom- the gap computed with = 0:35 eV and the same all other pressible states in a tBLG. We compare this map with parameters. Sketch illustrates 4 BLG bands (1 and 2 below / 3 and 4 above the gap) highlighting a small di erence between (D;  ) measured in the quantum Hall e ect regimes xx tot u and . (similar to the ones observed earlier [9, 10] in other tBLG devices), where the high resistivity manifests mutual pin- ning of partially lled LLs, whereas the minima corre- spond to the incompressible states. To mention, the com- gap between bands = 1; 2 and = 3; 4) are puted pattern broadly varies upon changing  , whereas 2 3 the value of  = 2:5 gives an excellent match between 2 X X d k 1 the computed and measured maps in Fig. 2(b). 4 5 n = j j ; (8) t=b ;t=b;k Finally, we analyse the electrostatically controlled =1;2 =A=B asymmetry gap [2] in Bernal bilayer graphene, taking into account out-of-plane polarizability of its constituent The on-layer potential energy di erence, u, and a band monolayers. In this case, we use Eq. (1) with  = z gap,  in the BLG spectrum (see inset in Fig. 3), com- [1 =Ad]  2:6, recalculated from polarizability puted using self-consistent analysis of Eqs. (7), (8) and using d = 3:35 A, and the BLG Hamiltonian [2], DFT (1) with  = 1 (as in Refs. 3, 20, and 21) and with = 2:6, are plotted in Fig. 3 versus displacement eld, 0 1 v~ v ~ v ~ D. On the same plot, we show the values of activa- 4 3 B C tion energy in lateral transport [26] and the IR 'optical v~  + v ~ 1 4 B C H = ; (7) @ A gap' - interlayer exciton energy [14], measured in vari- v ~  v~ 4  1 ous BLG devices. The di erence between those two data v ~ v ~ v~ 3 4 sets is due to that the single-electron 'transport' gap is enhanced by the self-energy correction [27] due to the which determines the dispersion and the sublattice (A/B) electron-electron repulsion, as compared to the `electro- amplitudes, , in four ( = 1 4) spin- and ;t=b;k static' value, u, whereas that enhancement is mostly can- valley-degenerate bands, E . Here,   k + ik , celled out by the binding energy of the exciton [14, 27], x y k = (k ; k ) is the electron wave vector in the valleys an optically active electron-hole bound state. As one x y K = (4=3a; 0),  = . The on-layer electron densi- can see in Fig. 3, u and  computed without taking ties in an undoped gapped BLG (with Fermi level in the into account monolayer's polarizability ( = 1) largely z 5 overestimate their values. At the same time, the val- ability of the monolayer, = 10:8 A , accounts very well ues of u and  obtained using  = 2:6 appear to be less z for all details of the electrostatics of twisted bilayers, in- than the exciton energy measured in optics, for interlayer cluding the on-layer electron density distribution at zero coupling across the whole range 0:35 eV < < 0.38 eV 1 magnetic eld and the inter-layer Landau level pinning at covered in the previous literature [24, 25, 28{32]. This quantizing magnetic elds. For practical applications in discrepancy may be related to that the interaction terms modeling of FET devices based on twisted bilayers, the in the electron self-energy are only partially cancelled by polarizability of monolayer graphene can be converted the exciton binding energy [27]. It may also signal that to its e ective dielectric susceptibility,  = 2:5, which the out-of-plane monolayer polarizability, , is reduced should be used for the self-consistent electrostatic analy- by  10% when it is part of BLG, as the values of  sis of tBLG using Eq. (1) of this manuscript. computed with  = 2:35 and = 0:35 eV agree very This work was supported by EPSRC grants z 1 well with the measured optical gap values. EP/S019367/1, EP/S030719/1, EP/N010345/1, ERC In summary, the reported analysis of the out-of-plane Synergy Grant Hetero2D, Lloyd Register Foundation dielectric susceptibility of monolayer graphene shows Nanotechnology grant, and the European Graphene that the latter plays an important role in determin- Flagship Core 3 Project. Computing resources were ing the electrostatics of both Bernal and twisted bilayer provided by Lancaster HEC cluster and Manchester graphene. We found that the DFT-computed polariz- SCF. [1] K. S. Novoselov, E. McCann, S. V. Morozov, V. I. Fal'ko, pearing in the arti cial triangular well of the saw-tooth M. I. Katsnelson, U. Zeitler, D. Jiang, F. Schedin, and potential, which sets the limits for the applicability of the A. K. Geim, Nature Physics 2, 177 (2006). DFT method we used. Also, we nd that is sensitive [2] E. McCann and V. I. Fal'ko, Phys. Rev. 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Out-of-plane dielectric susceptibility of graphene in twistronic and Bernal bilayers

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Abstract

Sergey Slizovskiy, Aitor Garcia-Ruiz, Alexey I. Berdyugin, and Na Xin National Graphene Institute, University of Manchester, Booth St.E., M13 9PL, Manchester, UK and Dept. of Physics & Astronomy, University of Manchester, Manchester M13 9PL, UK Takashi Taniguchi and Kenji Watanabe National Institute for Materials Science, 1-1 Namiki, Tsukuba, 305-0044, Japan Andre K. Geim National Graphene Institute,, University of Manchester, Booth St.E., M13 9PL, Manchester, UK and Department of Physics & Astronomy, University of Manchester, Manchester M13 9PL, UK Neil D. Drummond Department of Physics, Lancaster University, Lancaster LA1 4YB, UK Vladimir I. Fal'ko National Graphene Institute, Booth St.E., M13 9PL, Manchester, UK Department of Physics & Astronomy, University of Manchester, Manchester M13 9PL, UK and Henry Royce Institute for Advanced Materials, Manchester, M13 9PL, UK (Dated: June 15, 2021) We describe how the out-of-plane dielectric polarizability of monolayer graphene in uences the electrostatics of bilayer graphene { both Bernal (BLG) and twisted (tBLG). We compare the polar- izability value computed using density functional theory with the output from previously published experimental data on the electrostatically controlled interlayer asymmetry potential in BLG and data on the on-layer density distribution in tBLG. We show that monolayers in tBLG are described well by polarizability = 10:8 A and e ective out-of-plane dielectric susceptibility  = 2:5, in- exp z cluding their on-layer electron density distribution at zero magnetic eld and the inter-layer Landau level pinning at quantizing magnetic elds. Bilayer graphene [1{3] is a two-dimensional (2D) ma- an electric eld oriented perpendicular to the 2D crys- terial with electronic properties tuneable over a broad tal. Here, we compute the e ective dielectric suscepti- range. The manifestations of the qualitative change bility,  , of graphene monolayer using ab initio density of electronic characteristics of both Bernal (BLG) and functional theory (DFT) and implement the estimated twisted (tBLG) bilayer graphene, produced by electro- DFT value of  in the self-consistent description of (a) static gating [3] and inter-layer misalignment [4, 5], were tBLG electrostatics in tBLG with twist angles outside the observed in numerous experimental studies of the elec- magic angle [5, 8] range and (b) conditions for the inter- tronic transport in graphene-based eld-e ect transis- layer Landau level pinning in FET with a 30 -twisted tor (FET) devices. These versatile electronic proper- bilayer, with the results of modelling showing favourable ties make FETs based on BLG and tBLG an attractive quantitative agreement with the available and new exper- hardware platform for applications tailored [6{8] for var- imental data. Then we take into account the out-of-plane ious quantum technologies. While, over the recent years, dielectric susceptibility of a single graphene layer in the the fundamental electronic properties of bilayer graphene analysis of BLG inter-layer asymmetry gap, in partic- have been intensively studied, a mundane but practical ular, its dependence on the vertical displacement eld, characteristic of this material related to the out-of-plane (D), comparing the results with the measured exciton dielectric susceptibility of graphene layers largely escaped spectroscopy in gapped BLG [14]. attention of those investigations of BLG and tBLG in For the theoretical modelling of the out-of-plane dielec- FETs, despite several already recorded indications [9{ tric susceptibility, we employ the CASTEP plane-wave- 13] of its relevance for the quantitative modelling of the basis DFT code [15] with ultra-soft pseudopotentials. We operation of such devices. use a 53×53×1 k-point grid, a large plane-wave cut-o The out-of-plane dielectric susceptibility of a single of 566 eV, and a variety of interlayer distances c along graphene layer stems from the polarisability of its carbon the z-axis to compute the total energy, E , of graphene orbitals, that is, from the mixing of  and  bands by in a saw-tooth potential, Dz= , centered on the car- bon sites of the graphene layer (D being the displace- ment eld and c=2 < z < c=2). Then we determine [16] the polarizability in each cell of length c using on leave from NRC \Kurchatov Institute" PNPI, Russia the relation E = E D =(2 ), with E being the vac- 0 0 0 [email protected] uum energy. As the arti cial periodicity, introduced in arXiv:1912.10067v3 [cond-mat.mes-hall] 14 Jun 2021 2 ers of a bilayer, each with the electron density n , has b=t the form, D 1 +  n n b t u  U U = e e d: (1) t b 2 2 0 z 0 This expression is applicable to the description of both BLG and tBLG in a FET, improving on the earlier- published studies [3, 20, 21] where the out-of-plane di- electric susceptibility of graphene layers was missed out in the self-consistent band structure analysis. To describe a twisted bilayer with an interlayer twist angle , we use the minimal tBLG Hamiltonian, [4, 5], [H + u] 0 T 0 t k ;k k ;k H(k ;k) = ; (2) T [H u] 0 b k ;k k;k 2 t=b; H =~v ; t=b t=b; = k + i(k  K sin ); t=b; x y FIG. 1. (a) Sketches illustrating how dielectric polarizability 2 2 X i j 1 e of each monolayer enters in the electrostatics analysis of bilay- T =  (j); k ;k 0 i j k ;k+g 3 e 1 ers in Eq. (1). (b) Characteristic electron dispersion in tBLG j=0 (here,  = 3 ; u = 100 meV). Electron state amplitude on 2j 2j (j) the top/bottom layer is shown by red/blue. (c) Mini-valley g =  sin ; 1 cos 2K sin : carrier densities n 0 in a single-gated tBLG calculated for 3 3 2 various misalignment angles outside the magic angle range, in comparison with the densities corresponding to SdHO mea- Here, v = 10:2  10 m=sec is Dirac velocity in mono- sured [10] in a tBLG ake with an unknown twist angle (black layer graphene. Equation (2) determines [4] charac- dots). teristic low-energy bands, illustrated in Fig. 1(b) for 1    =~vK  2 (away from the small magic an- gles  1 ). This spectrum features two Dirac minivalleys 0 0 at  and  (j  j = 2K sin ), each described by its the DFT code, leads to a systematic error in the po- own Fermi energy, larizability,  (c) / c , we t the obtained DFT data with (c) = + a=c + b=c and nd  (c ! 1 DFT 3 E 0  (1 3)~v jn 0j sign n 0; F = = = 1) = 11:0 A per unit cell of graphene with the Perdew{ E E 0  (1 4)u; (3) Burke{Ernzerhof (PBE) functional and = 10:8 A F  F DFT with the local density approximation (LDA). These val- and carrier density n 0 , determined by the minivalley ues are close to the DFT-PBE polarizability reported in 3 3 area encircled by the corresponding Fermi lines (as in Ref. 17, = 0:867 4 A = 10:9 A , and when recalcu- Fig. 1(b)) and measurable using Shubnikov - de Haas os- lated into an e ective 'electronic thickness' =A, where cillations [9, 10] or Fabry-Perot interference pattern [12]. A = 5:2 A is graphene's unit cell area, we get 2:1 A, The above expression was obtained using expansion up comparable to the earlier-quoted `electronic thickness' of 1 2 to the linear order in  = [ ]  1. We also ~vj j graphene [12, 18]. Furthermore, for the analysis of bilay- note that, due to the interlayer hybridisation of electronic ers' electrostatics in this paper, the e ective out-of-plane 1 wave functions, the on-layer charge densities in Eq. (1) dielectric susceptibility is  = [1 =Ad] , where z DFT di er from the minivalley carrier densities, d is the distance between the carbon planes in the bilayer (d = 3:35 A for BLG and d = 3:44 A for tBLG, as in tur- n  n 0  2 n 0 n + 0:07 j  j ; (4) bostratic graphite [19]). This gives  = 2:6 for BLG, b=t = ~v and  = 2:5 for tBLG. For the electrostatic analysis of bilayers built into which makes the results of the self-consistent analysis FETs, we note that out-of-plane polarisation of carbon of tBLG electrostatics slightly dependent on the twist orbitals in each graphene monolayer is decoupled from angle, . We illustrate this weak dependence in Fig. 1(c) the charges hosted by its own -bands, because of mirror- by plotting the relation between the values of n and n symmetric charge and eld distribution produced by the in a single-side-gated tBLG computed using Eqs. (3), (4) latter, see in Fig. 1(a). As a result, the di erence between and (1) with  = 2:5 and d = 3:44 A. On the same plot, the on-layer potential energies in the top and bottom lay- we also show the values of n and n 0 recalculated from FIG. 2. (a) Resistance map for a double-gated tBLG with a 30 twist angle, computed with  = 2:5 and d = 3:44 A (left) and measured (right) as a function of the total carrier density, n and vertical displacement eld, D, at B = 0 and T = 2 K. (b) Computed density of states of pinning LLs (left) and the measured resistance,  (right) in a 30 tBLG at B = 2 Tesla, plotted xx as a function of displacement eld and lling factor. Bright regions correspond to the marked N =N LL pinning conditions. t b the earlier measured SdHO [10] in tBLG devices with an termines [22] the Fourier form-factor of the scatterers, unknown twist angle (seemingly, 10 15 ). 2 2 e =2  e = 0 0 = ; r = ; k = jn j; q s Fi i We also compared the results of the self-consistent q + r (k + k ) ~v s Ft Fb tBLG analysis with the measurements of electronic trans- port characteristics of a double-side gated multi-terminal and the corresponding momentum relaxation rate of FET based on a tBLG with  = 30 . The latter Dirac electrons [22], choice provides us with the maximum misalignment, cor- t=b 2 2 responding to  ! 0 in Eqs. (3) and (4), hence n = b=t = hj j sin 'i n : ' c 2k sin('=2) t=b Ft=b 2~ n 0 . In the experimentally studied device, tBLG was encapsulated between hBN lms on the top and bot- Then, in Fig. 2(a), we compare the computed and tom, thus providing both a precise electrostatic control measured tBLG resistivity. As in monolayer graphene of tBLG for B = 0 measurements and its high-mobility, [22], density of states, , cancels out from each t=b enabling to observe quantum Hall e ect at a magnetic 2 2 = 2 =(e v ), making the overall result,  = t=b t=b xx t=b eld as low as B = 2 Tesla. =[ +  ], dependent on the carrier density only t b t b For a quantitative comparison of the measured and through the wave-number transfer, 2k sin('=2), and Ft=b modelled tBLG characteristics, we assumed elastic scat- screening. This produces ridge-like resistance maxima at tering of carriers from residual Coulomb impurities in k = 0 or k = 0, that is, when Ft Fb the encapsulating environment with a dielectric constant 5, with an areal density n , screened jointly by the ~v jnj sign n e n(1 +  ) c z z D= = + : (5) carriers in the top and bottom layers. The screening de- ed 4 0 4 Lines corresponding to the above relation are laid over the experimentally measured resistivity map for a direct comparison. We nd an even more compelling coincidence be- tween the theory and experiment when studying the Landau level pinning between two electronically inde- pendent by electrostatically coupled graphene monolay- ers in a 30 tBLG. In a magnetic eld, graphene spec- trum splits into Landau levels (LLs) with energies E = v 2~je B Nj signN . In a twisted bilayer, in nite degen- eracy of LLs gives a leeway to the interlayer charge trans- fer which screens out displacement eld and pins par- tially lled top/bottom layer LLs, N and N , to each t b other and to their common chemical potential, , so that p p v 2~ejBj( N N ) = u, as in Fig. 2(b). This LL pin- t b ing e ect also persists for slightly broadened LLs. Tak- ing into account a small Gaussian LL broadening, , we write, u E 4eB n = erf ;   1; (6) t=b 2~ N = FIG. 3. Inter-layer asymmetry potential (dashed lines) and band gap (solid lines) in an undoped BLG, self-consistently solve self-consistently Eq. (1), and compute the total computed with various values of  = 1 (green), 2.6 (blue) density of states (DoS) in the bilayer. The computed and 2.35 (red) and compared to the experimentally measured DoS for B = 2 Tesla and  0:5 meV) is mapped in gate-tunable optical gap [14] (circles) and transport gap [23] Fig. 2(b) versus displacement eld and tBLG lling fac- 6 (crosses). Here, we use [24, 25] v = 10:2  10 m=sec, = 5 4 tor,  = hn=eB. Here, the `bright' high-DoS spots tot 0:38 eV, v = 1:23  10 m/s, v = 4:54  10 m/s,  = 22 meV, 3 4 indicate the interlayer LL pinning conditions, whereas and d = 3:35 A. Additionally, dotted lines show the values of the `dark' low-DoS streaks mark conditions for incom- the gap computed with = 0:35 eV and the same all other pressible states in a tBLG. We compare this map with parameters. Sketch illustrates 4 BLG bands (1 and 2 below / 3 and 4 above the gap) highlighting a small di erence between (D;  ) measured in the quantum Hall e ect regimes xx tot u and . (similar to the ones observed earlier [9, 10] in other tBLG devices), where the high resistivity manifests mutual pin- ning of partially lled LLs, whereas the minima corre- spond to the incompressible states. To mention, the com- gap between bands = 1; 2 and = 3; 4) are puted pattern broadly varies upon changing  , whereas 2 3 the value of  = 2:5 gives an excellent match between 2 X X d k 1 the computed and measured maps in Fig. 2(b). 4 5 n = j j ; (8) t=b ;t=b;k Finally, we analyse the electrostatically controlled =1;2 =A=B asymmetry gap [2] in Bernal bilayer graphene, taking into account out-of-plane polarizability of its constituent The on-layer potential energy di erence, u, and a band monolayers. In this case, we use Eq. (1) with  = z gap,  in the BLG spectrum (see inset in Fig. 3), com- [1 =Ad]  2:6, recalculated from polarizability puted using self-consistent analysis of Eqs. (7), (8) and using d = 3:35 A, and the BLG Hamiltonian [2], DFT (1) with  = 1 (as in Refs. 3, 20, and 21) and with = 2:6, are plotted in Fig. 3 versus displacement eld, 0 1 v~ v ~ v ~ D. On the same plot, we show the values of activa- 4 3 B C tion energy in lateral transport [26] and the IR 'optical v~  + v ~ 1 4 B C H = ; (7) @ A gap' - interlayer exciton energy [14], measured in vari- v ~  v~ 4  1 ous BLG devices. The di erence between those two data v ~ v ~ v~ 3 4 sets is due to that the single-electron 'transport' gap is enhanced by the self-energy correction [27] due to the which determines the dispersion and the sublattice (A/B) electron-electron repulsion, as compared to the `electro- amplitudes, , in four ( = 1 4) spin- and ;t=b;k static' value, u, whereas that enhancement is mostly can- valley-degenerate bands, E . Here,   k + ik , celled out by the binding energy of the exciton [14, 27], x y k = (k ; k ) is the electron wave vector in the valleys an optically active electron-hole bound state. As one x y K = (4=3a; 0),  = . The on-layer electron densi- can see in Fig. 3, u and  computed without taking ties in an undoped gapped BLG (with Fermi level in the into account monolayer's polarizability ( = 1) largely z 5 overestimate their values. At the same time, the val- ability of the monolayer, = 10:8 A , accounts very well ues of u and  obtained using  = 2:6 appear to be less z for all details of the electrostatics of twisted bilayers, in- than the exciton energy measured in optics, for interlayer cluding the on-layer electron density distribution at zero coupling across the whole range 0:35 eV < < 0.38 eV 1 magnetic eld and the inter-layer Landau level pinning at covered in the previous literature [24, 25, 28{32]. This quantizing magnetic elds. For practical applications in discrepancy may be related to that the interaction terms modeling of FET devices based on twisted bilayers, the in the electron self-energy are only partially cancelled by polarizability of monolayer graphene can be converted the exciton binding energy [27]. It may also signal that to its e ective dielectric susceptibility,  = 2:5, which the out-of-plane monolayer polarizability, , is reduced should be used for the self-consistent electrostatic analy- by  10% when it is part of BLG, as the values of  sis of tBLG using Eq. (1) of this manuscript. computed with  = 2:35 and = 0:35 eV agree very This work was supported by EPSRC grants z 1 well with the measured optical gap values. 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Condensed Matter – arXiv (Cornell University)

Published: Dec 20, 2019

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