Quantum capacitive coupling between large-angle twisted graphene layers

Alina Mrenca-Kolasinska,1, 2, ∗ Peter Rickhaus,3 Giulia Zheng,3

Klaus Richter,4 Thomas Ihn,3 Klaus Ensslin,3 and Ming-Hao Liu (劉明豪)1, †

1Department of Physics, National Cheng Kung University, Tainan 70101, Taiwan2AGH University of Science and Technology, Faculty of Physics and Applied Computer Science, al. Mickiewicza 30, 30-059 Krakow, Poland

3Solid State Physics Laboratory, ETH Zurich, CH-8093 Zurich, Switzerland4Institut fur Theoretische Physik, Universitat Regensburg, D-93040 Regensburg, Germany

(Dated: October 5, 2021)

Large-angle twisted bilayer graphene (tBLG) is known to be electronically decoupled due to the spatial sep-aration of the Dirac cones corresponding to individual graphene layers in the reciprocal space. This mechanismalso leads to the decoupling in multilayer graphene systems including twisted double bilayer graphene, whichare just a small subset of a broad class of systems consisting of graphene layers and other materials, decoupledby the twist or separated by dielectrics. For the former, the close spacing between the layers causes strongcapacitive coupling, opening possibilities for new applications in atomically thin devices. Here, we present aself-consistent quantum capacitance model for the electrostatics of decoupled graphene layers, and further gen-eralize it to deal with decoupled tBLG at finite magnetic field and large-angle twisted double bilayer grapheneat zero magnetic field. We probe the capacitive coupling through the conductance, showing good agreementbetween simulations and experiments for all these systems considered. Our model can be extended to sys-tems composed of decoupled graphene multilayers as well as non-graphene systems, opening a new realm ofquantum-capacitively coupled materials.

Recently, there has been an increasing interest in thin vander Waals heterostructures [1, 2], including twisted bilayergraphene (tBLG). In tBLG [Fig. 1(a)], the Brillouin zones ofthe two layers are rotated against each other [Fig. 1(b)], and alarge twist angle leads to the separation of the Dirac cones ofboth layers [3–7]. This suppresses interlayer scattering due tothe large momentum difference, making the two layers essen-tially electronically decoupled [8–12]. However, their atomi-cally thin layer spacing allows them to couple electrostaticallybecause the electric charge on one layer causes an effectivegating of the other layer. This mechanism enables realizationof atomically thin devices composed of decoupled layers, withthe large twist being an alternative to isolating the layers withdielectrics [13–17]. However, the strong quantum capacitivecoupling makes precise electrostatic modeling indispensablefor simulation of these devices [18, 19].

In this work, we present the self-consistent quantum ca-pacitance model used in Ref. 7 for decoupled tBLG at zeromagnetic field and generalize it considerably to deal with de-coupled tBLG in the presence of magnetic field, decoupledtwisted double Bernal-stacked bilayer graphene (tdBLG), anddecoupled multilayer graphene systems. The quantum con-ductance of such layered structures depends strongly on thecapacitive coupling and can be used as a sensitive probe ofthe latter. We show quantitatively good agreement with ourown experimental results for a dual-gated two-terminal tBLGdevice sketched in Figs. 1(c) and (d), showing strong reliabil-ity of our model. For tdBLG, our transport simulations agreewell with the experimental findings [20], despite the strongcomplication due to the gate-tunable band gap [21–26]. Ourmodels can be in general applied to electronically decoupledmaterials that are quantum-capacitively coupled to each other,including topological insulator surface states [27], but is notlimited to alike layers, being adaptable to hybrid systems con-

sisting of different materials hosting two-dimensional electrongas [28].

Self-consistent quantum capacitance model for decoupledtBLG. Reference 7 investigated dual-gated two-terminal de-vices consisting of decoupled large-angle tBLG samples,schematically shown in Fig. 1(c) for a perspective top viewand in Fig. 1(d) for its side view. In the following discus-sion we focus on a device fabricated with a top gate of length`= 320nm and sample width W ≈ 2.9 µm. Details of the de-vice fabrications are given in Ref. 7.

To model the decoupled tBLG device, we assume two lay-ers of graphene described by the linear Dirac dispersion rela-tion E = ±hvF k, where h is the reduced Planck constant andvF ≈ 106 m · s−1 is the Fermi velocity of graphene. For com-putational convenience, we adopt hvF ≈ 3

√3/8eV ·nm. The

two electronically decoupled single-layer graphene (SLG)flakes are tightly spaced (assuming the spacing to be dG =0.12nm found in Ref. 7) such that a tiny shift of the Fermi

(a) (b)

(c)

(d)

θ

θ∆K

`

W

xy

(Vt)top gate

hBN

vacuum

hBN

(Vb)backgate

dt

dG

db

Ct

Cb

Cg

FIG. 1. Schematics of (a) a tBLG lattice composed of two graphenelayers twisted by an angle θ and (b) their corresponding Dirac conesin reciprocal space. The dual-gated two-terminal decoupled tBLGdevice considered in the transport experiment and simulations issketched in (c) for a perspective top view and (d) for its side view.

arX

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110.

0090

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2

energy of the first layer causes an appreciable gating effect onthe second layer, which in turn acts as a back gate of the first,and so forth. The whole process can be efficiently iterated us-ing the formulas derived in [29]. Applying the results of [29],the carrier density

n = nC +∆n (1)

of a SLG free of intrinsic doping and subject to two gates atvoltages V1 and V2 is composed of the classical carrier density

nC = ∑i=1,2

CiG

eVi , (2)

where e > 0 is the elementary charge and CiG is the capac-itance (per unit area) between gate i and graphene, and thecorrection

∆n = sgn(nC)nQ

(1−

√1+2

|nC|nQ

)(3)

accounting for the quantum nature of the finite density ofstates of the conducting plate. The correction (3) depends onthe classical density (2) as well as

nQ =π

2

(hvF

eC1G +C2G

e

)2

(4)

that arises solely from the quantum capacitance [18, 30].Moreover, the corresponding electric potential of the graphenesheet is given by

VG =− e∆nC1G +C2G

. (5)

Decoupled tBLG without magnetic field. To apply Eqs.(1)–(5) to our dual-gated decoupled tBLG device sketched inFig. 1(d), we consider the top graphene layer (upper dashedline) to be dual-gated by the top gate at voltage Vt and bottomgraphene layer (lower dashed line) at electric potential VGb.Substituting V1 = Vt , V2 = VGb, C1G = Ct , and C2G = Cg intoEqs. (2)–(5), we obtain the electric potential VGt of the topgraphene layer, given Vt and VGb, the former being a fixed in-put while the latter to be self-consistently iterated. Similarly,the bottom graphene layer is dual-gated by the top graphenelayer at potential VGt and back gate at voltage Vb. SubstitutingV1 =VGt , V2 =Vb, C1G =Cg, and C2G =Cb into Eqs. (2)–(5),we obtain the electric potential VGb of the bottom graphenelayer, given VGt and Vb, the former being just computed andthe latter being a fixed input. The newly obtained VGb is usedto compute VGt and vice versa iteratively, until VGt and VGbboth converge to a satisfactory precision. The converged VGtand VGb can be used to obtain the individual carrier densitynt for the top layer and nb for the bottom layer, using Eqs.(1)–(4). Multiplied by the electron charge, −eVGt and −eVGbare the onsite energies entering the tight-binding Hamiltonianfor transport calculations to be explained later. Note that due

to the finite size of the top gate, Ct = Ct(x), the above intro-duced self-consistency needs to be achieved for all relevant x,leading all of nt ,nb,VGt ,VGb to depend on x. However, the it-eration process is found to converge still very rapidly thanksto the equations (1)–(5). For a local gate of a capacitanceC = C(x,y), the iteration is required for all relevant x and ybut is still expected to converge well.

To simulate real tBLG devices, we express the two-terminalconductance as G = (Rc +G−1

0 )−1, where Rc is the contactresistance and G0 is the ballistic conductance calculated us-ing the real-space Green’s function approach [31]. To speedup calculations, the hopping parameter t0 and lattice spac-ing a0 approximated by 3eV and 1/4

√3nm, respectively,

are scaled to t0/s f and s f a0 [32] using s f = 4. Because ofthe device geometry [Fig. 1(c)] and its width of nearly 3 mi-crons, the ballistic conductance can be efficiently computedby using the method of periodic boundary hopping [33, 34]:G0 = (W/3πs f a0)(gb +gt), where g j = (e2/h)

∫ kF−kF

T (ky)dky(kF being the Fermi momentum) [35] is the normalized con-ductance of the bottom (top) graphene layer for j = b ( j = t).The transmission function T that depends on the transversemomentum ky is computed from the Green’s function ap-proach which requires the tight-binding Hamiltonian

H j = H0− e∑n

VG j(xn)c†ncn , (6)

where j = t,b is the layer index, H0 is the clean part of theminimal tight-binding model for bulk graphene [34], and theoperator cn (c†

n) annihilates (creates) an electron on site n lo-cated at (xn,yn). It is the second term in Eq. (6) for the on-site energy where the electric potential VGb and VGt , foundfrom the self-consistent electrostatic model for the decoupledtBLG, enter the transport calculations.

Figure 2(a) shows the computed two-terminal conductancesimulated for the considered decoupled tBLG device sketchedin Figs. 1(c)–(d) as a function of Vt and Vb, using Rc =0.005h/e2 as a reasonable parameter for the contact resis-tance. The diagonal charge neutrality line splits into twowhich is a signature of the decoupling of the two graphenelayers. The splitting as well as the superimposed Fabry-Perot(FP) interference fringes [36, 37] are better seen by mean ofnumerical differentiation. We show dg/dVt as an overlaid in-set on Fig. 2(a), where the horizontally aligned dots mark thescan with 3V ≤ Vt ≤ 5V at Vb = −10V that we are goingto focus on for the rest of the discussions of the decoupledtBLG device. Along this Vt scan, the carrier density profilesnb(x) and nt(x) are shown in Fig. 2(b) and (c), respectively.The Vt dependence of nb(0) and nt(0) is shown in Fig. 2(d),where three regions can be clearly seen: Both graphene lay-ers in unipolar ppp for Vt . 3.6V, top layer in pnp but bottomlayer remaining in ppp for 3.6V .Vt . 4.5V, and both layersin pnp for Vt & 4.5V. These regions are characterized by noFP fringes, one set of FP fringes and two sets of FP fringes,respectively.

Decoupled tBLG with magnetic field. We next go beyondRef. 7 to account for magnetotransport in the same decou-

3

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(a) (b)

(c)

(d)

(e) (f)

(g) (h)

(i) (j)

FIG. 2. (a) Calculated two-terminal conductance G as a function oftop gate voltage Vt and back gate voltage Vb at zero magnetic fieldB = 0. The overlay at the bottom right corner shows the numericalderivative of the normalized conductance g with respect to Vt . Car-rier density profiles nb(x) and nt(x) of the bottom and top graphenelayer are shown in (b) and (c), respectively, subject to gate voltageconfigurations marked in (a) with the line and dot colors correspond-ing to each other. (d) nb(x = 0) and nt(x = 0) marked by 5 and 4as a function of Vt at fixed Vb = −10V corresponding to panel (b)and (c), respectively. Shaded areas distinguish three regions: bothlayers in ppp (light green), top layer in pnp but bottom layer in ppp(white), and both layers in pnp (pink). (e) Measured and (f) sim-ulated two-terminal conductance G as functions of Vt and B up to0.3T. Fabry-Perot interference fringes of dG/dVt from the exper-iment and dg/dVt from the simulations are shown in (g) and (h),respectively. Calculated normalized conductance g for the (i) bottomand (j) top graphene layer. Color bars are in units of e2/h for (e), (f),(i), and (j), and e2/hV−1 for (g) and (h).

pled tBLG device, where the uniform magnetic field B is ap-plied along z perpendicular to the graphene layers. When Bis weak, the Dirac linear dispersion remains valid, and theabove introduced self-consistent model can be directly ap-plied. Figure 2(e) shows the experimentally measured two-terminal conductance G as a function of Vt restricted to therange marked in Fig. 2(a) and B up to 0.3T. Our simulatedG shown in Fig. 2(f) exhibits a similar profile, despite the dif-ferent magnitude of G. To better compare the details withour simulation, we mirror the experimental data about the Vtaxis and show dG/dVt in Fig. 2(g). It exhibits complex FPfringes that are satisfactorily consistent with our computeddifferentiated normalized conductance dg/dVt shown in Fig.2(h). Closer inspection of the region with Vt & 4.5V showsthat there are two sets of FP fringes superimposed, one dis-persing with B slower and the other faster. The slower (faster)set is expected to come from the top (bottom) graphene layerbecause of the higher (lower) gating efficiency; the layer with

lower gating efficiency needs a larger gate voltage to com-pete with the B-dependent Aharanov-Bohm phase picked upby the interfering electron within the FP cavity [38]. This isconfirmed by showing the individual contribution gb and gt inFig. 2(i) and (j), respectively, which sum up to g = gb +gt .

Next we turn to strong external magnetic field ac-counting for the Landau quantization of the densityof states, D(E,Bz) = 4eBz

h ∑nL

δ (E − EnL), where EnL =

sgn(nL)√

2eBzhv2F |nL| and nL = 0,±1,±2, . . . . The carrier

density is given by

n(E,Bz) =∫ E

0D(E ′,Bz)dE ′. (7)

To account for the Landau level (LL) broadening, we approxi-mate the Dirac delta by a Lorentzian function, and the integra-tion (7) can be done analytically. The resulting carrier densityis quantized in energy and magnetic field. The carrier densitygiven by Eq. (7) is equal to the sum of the gate-induced doping

n(eVG,Bz) =C1G

e(V1−VG)+

C2G

e(V2−VG). (8)

We solve Eq. (8) for VG numerically. Then, for two decoupledgraphene layers in strong external magnetic field, the calcu-lation of the electric potential VGt and VGb is done in a simi-lar self-consistent way as for the linear dispersion relation butsubstituting Eqs. (1)–(5) with the numerical solution of Eq. (8)(for details see the Supplemental Materials [39]).

We next perform strong-magnetic-field quantum transportcalculations, considering a zigzag ribbon of width 400 nm.For the transport modeling we use the wave-function match-ing method [40] for graphene with the scaling factor s f =

0.3

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150

2 6 10 14 18 22 26 30 34 38

3 3.5 4 4.5

low high

3 3.5 4 4.5 5

(a)

(b)

(c)

(d)

(e)

(f)

FIG. 3. Numerical derivative of the (a) measured and (b) calculatedtwo-terminal conductance as a function of top gate voltage Vt andmagnetic field at Vb = −10 V. (c), (d) Individual conductances cal-culated for the top and bottom layer, respectively. (e) dnt/dVt and(f) dnb/dVt values show a clear correspondence to the conductanceplateaus kinks.

4

8. At zero-temperature the conductance is calculated usingthe Landauer formula G(B) = 2e2

h T (B), with T (B) being thetransmission summed over all modes.

Figure 3(a) shows the experimentally measured andFig. 3(b) the calculated transconductance obtained at Vb =−10 V as a function of the top-gate voltage and magneticfield. In the transconductance map, two sets of Landau lev-els are visible, emerging from the two split charge neutralitypoints, marked by red arrows in Fig. 3(a) and (b). The individ-ual top and bottom layer conductances are shown in Fig. 3(c)and (d), respectively, confirming that the entire LL spectrumconsists of two layers’ superimposed Landau fans, and thatthe layers remain electrically decoupled at strong magneticfield. The Landau fans are dramatically different from thecommonly observed ones in graphene, and exhibit ”kinks”at the crossing between the Landau levels of the two layers.Their origin can be understood by comparing the conductancemap to the top and bottom layer density gradient with respectto Vt in Fig. 3(e), and (f), respectively.

We first focus on the 2nd, 3rd, and 4th LL of the bottomlayer marked by squares in Fig. 3(d). The density of states(DoS) is high at the LL, as is the density per gate voltage.Thus, the dnb/dVt value is high, [Fig. 3(f)] and the dnt/dVtvalue [Fig. 3(e)] is low as the total carrier density inducedby the top gate is conserved. On the other hand, the pointsmarked with circles in Fig. 3(d), (e) and (f) are along the toplayer 0th LL, and the top layer DoS is high. Based on the ar-gument above, here the dnt/dVt (dnb/dVt ) value is high (low).Therefore, we expect the slope of the LLs to change, and inparticular at the points marked with circles, the bottom layerLLs slope becomes smaller. Recent experiment [12] reportedsimilar effects. Note that this feature is only recovered in theLL-quantized-density model. For the result obtained with thelinear dispersion relation see [39]. The good qualitative agree-ment between the experimental and theoretical results showsthat the self-consistent model is accurate for other than lineardispersion relations.

Decoupled tdBLG. We next consider large-angle tdBLG[Fig. 4(a)], where, similar to the tBLG case, the two BLGsare decoupled electronically by the large momentum separa-tion of the Dirac cones of the two BLGs [Fig. 4(b)]. We firstdevelop a quantum capacitance model for an individual dual-gated BLG based on Refs. [41, 42]. With the quantum ca-pacitance model for BLG at hand, we can further extend it totdBLG. To this end, we consider two stacked BLGs, coupledcapacitively to external top and bottom gates and to each other[see Fig. 4(c)].

The problem can be then solved self-consistently as fortBLG; however, we found treating the problem as a set of cou-pled nonlinear equations more efficient (see [39]).

The gate capacitances Ct , Cb are obtained from a finite el-ement electrostatic simulation for a sample geometry adoptedfrom Ref. [20], with the hBN thicknesses db = 90 nm, dt = 60nm, placed between a global back gate and narrow top gatewidth 400 nm. The interlayer capacitance within an individ-ual BLG is assumed to be Cg = 7.4 µFcm−2 [7], whereas

θ

(a) (b)

θ∆K

bottom gate (Vb)hBN

vacuumvacuumvacuum

hBNtop gate (Vt)

db

dtCt

Cb

Cg

Cg

Cm

(c)

5 0 5Vt (V)

8

6

4

2

0

2

4

6

8

Vb

(V) npn

npn

npnnnn

pnpppp

pnppnp

nnnnnn

pppppp

5 0 5dG/dVt

(d)

0.00

0.05

onsit

een

ergy

top

0.01

0.00

onsit

een

ergy

bottom

500 0 500x (nm)

0.01

0.00

onsit

een

ergy

(e)

(f)

(g)

FIG. 4. (a) Sketch of the Bernal stacked large-angle tdBLG and (b)its low-energy bandstructure. (c) Schematic of the dual-gated tdBLGsystem. (d) Numerical derivative of two-terminal conductance as afunction of top gate and back gate voltage in tdBLG. The onsite en-ergy profiles at the voltage configuration marked by / are shown in(e) for the top BLG and (f) and (g) for the bottom BLG.

the value of the capacitance between the BLG layers is Cm =3.5 µFcm−2 [20]. For the transport calculations we assumethe system is translationally invariant along the lateral direc-tion, allowing us to use the model with periodic hopping [34].The elementary cell is a zigzag bilayer graphene nanoribbonof width w = 3s f a0 with s f = 2.

In the self-consistent model for tdBLG, we include the ef-fect of the crystal field [43–46] which was shown in Ref. [20]to open a bandgap even without gate voltage. In the tdBLGsample the inner and outer graphene layers see a different en-vironment, and thus feel an unequal electrostatic potential,which effectively creates an intrinsic bias. This induces asmall negative charge in the inner layers. We can include thiseffect in our model by assuming a constant density difference∆n0 between the bottom and top layer of a BLG. From themeasured values of the displacement field needed to close thebandgaps [20] we estimate ∆n0,1 = 13×1011cm−2 for the up-per BLG and ∆n0,2 =−14×1011cm−2 for the lower BLG.

Figure 4(d) shows the transconductance as a function of Vtand Vb, which recovers the key features observed in Ref. [20].The FP oscillations occur when a bipolar junction is formedin the top or bottom BLG. Interestingly, for negative Vb theoscillations occur for the top BLG only (the slope highlightedby black solid lines). Conversely, for 0 < Vb < 4 V the FPoscillation is present only for the bottom BLG (highlightedby red lines); at higher Vb only a faint oscillation for the topBLG can be spotted, when the n-p-n junction is formed in theupper BLG [see labels in Fig. 4(d)]. This difference in the

5

visibility of the oscillations can be explained as due to a largebandgap across the device, which strongly reduces the trans-mission through the cavity when the p-n interface is smooth.For example, the oscillation in the top BLG is hardly visibleat Vb > 0 where the bandgap at the p-n interface happens to belarge [see the onsite energy profile in Fig. 4 (e)], but in the bot-tom BLG the bandgap is reduced by the applied displacementfield [Fig. 4 (f)].

Another feature which our model captures in good quali-tative agreement with experiment is the bandgaps in the top-gated region that are opened even at low applied gate volt-ages [shown in Fig. 4(d) by black dashed lines] and closed at(2.2,−10.6)V for the upper BLG and (Vt ,Vb)≈ (−7.7,3.4)Vfor the lower BLG [see Fig. 4 (g)]. A feature not taken intoaccount by the model is the difference in the electron and holeeffective mass leading to a kink of the charge neutrality linein the experiment [20].

Decoupled multi-layer graphene. The iterative processcan also be applied to a system composed of more graphenelayers, provided that each one is twisted by a large angle suchthat all the layers are electrically decoupled. Such systemshave been realized experimentally [47, 48]. For more detailson the carrier density in n-layer graphene see [39].

In summary, we developed self-consistent methods for theelectrostatic and transport calculations for electronically de-coupled graphene multilayers. The good agreement with theexperimentally measured conductance for tBLG and tdBLGconfirms their applicability for a broad class of systems con-sisting of decoupled conducting layers. The self-consistentmethod is suitable to other materials hosting Dirac carriers aswell as described by other band structures [27], opening a newarea of capacitively-coupled materials.

Acknowledgments This work was supported by the Min-istry of Science and Technology grant MOST 109-2811-M-006-544. KR acknowledges funding through the DeutscheForschungsgemeinschaft (DFG, German Research Founda-tion) Project-ID No. 314695032–SFB 1277 (subproject A07).This research was supported in part by PL-Grid Infrastructure.

∗ alina.mrenca@fis.agh.edu.pl† minghao.liu@phys.ncku.edu.tw

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