Jun Su

Tao Zhu

Thierry Pauporté

Ilaria Ciofini

Frédéric Labat
email: frederic.labat@chimie-paristech.fr

Improving the Heterointerface in Hybrid Organic-Inorganic Perovskite Solar Cells by Surface Engineering: Insights from Periodic Hybrid Density Functional Theory Calculations

A periodic hybrid density functional theory computational strategy is presented to model the heterointerface between the methylammonium lead iodide (MAPI) perovskite and titanium dioxide (TiO 2 ), as found in perovskite solar cells (PSC), where the 4-chlorobenzoic acid (CBA) ligand is used to improve the stability and the band alignment at the interface.

The CBA ligand acts as a bifunctional linker to efficiently connect the perovskite and the oxide moieties, ensuring the stability of the interface through Ti-O and Pb-Cl interactions.

The computed density of states reveals that the perovskite contributes to the top of the valence band while the oxide contributes to the bottom of the conduction band with a direct bandgap of 2.16 eV, indicating a possible electron transfer from MAPI to TiO 2 . Dipole moment analysis additionally reveals that the CBA ligand can induce a favorable effect to improve band alignment and thus electron transfer from MAPI to TiO 2 . This latter has been quantified by calculation of the spin density of the reduced MAPI/CBA/TiO 2 system and indicates an almost quantitative (99.94%) electron transfer from MAPI to TiO 2 for the surface engineered system, together with an ultrafast electron injection time in the femtosecond timescale.

Overall, the proposed DFT-based computational protocol therefore indicates that surface engineering and the use of a bifunctional linker can lead to a better stability, together with improved band alignment and electron injection in PSC systems.

INTRODUCTION

With excellent optoelectronic properties and potential for low-temperature device fabrication, organometal halide perovskites (AMX 3 ) are extensively studied materials. [1] Since their very first combination with mesoporous TiO 2 photoanode for visible-light conversion in photovoltaic cells by Myasaka et al., [2] organometal halide perovskites AMX 3 have attracted considerable attention as efficient light harvesters. In particular, organic-inorganic hybrid perovskite solar cells (PSC) have appeared as excellent potential candidates to produce clean energy, with easy fabrication, long charge carrier diffusion lengths and lifetimes, high absorption coefficient, solution processability and high photoconversion efficiencies (PCE). [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20] These perovskites are composed of an organic cation A, a divalent metal cation M generally belonging to the fourth main group and a halide anion X, which are arranged in a rigid three-dimensional network of corner-sharing MX 6 octohedra. By varying the different possible combinations between these cationic and anionic species, bandgap tuning of the perovskite material is possible, thus allowing to readily modulate its light absorption for photovoltaic application. Although the M and X species can in principle be easily tuned in these structures, the A cation on the other hand fills the 12-fold coordinated cubo-octahedral space created by the inorganic metal halide octahedra, hence making its choice limited to either monovalent cations such as Cs + , Rb + or K + , or to the small organic methylammonium (MA + ) and formanidinium (FA + ) ions. [16] In particular, among the most studied hybrid perovskites for PSC application, the pioneering methylammonium lead iodide hybrid perovskite (MAPbI 3 , referred to as MAPI) still remains actively considered nowadays. [2,7,21] In a PSC, the perovskite layer [22] ensures light absorption and generates electron-hole charge carrier pairs which then dissociate into free charge carriers. The free excited electrons and holes diffuse within electron and hole transport layers (ETL and HTL) and are then collected by corresponding electrodes. [16] By optimizing the different components of PSC as well as by finely tuning their combination, their PCE has increased from 3.8 [2] to 25.2% [23] during the last decade. [5,[7][8][9][10][11][12][13][14][15] Although PSCs are then now generally considered as one of the most promising devices for photovoltaic energy production, several points still remain critical however for real-world application, including mainly toxicity and long-term stability. For example, MAPI hydrolyzes in the presence of water, breaking the weak interaction between the inorganic and organic networks. [24][25][26] In addition, the commonly-used organic HTL for high PCE such as spiro-OMeTAD, P3HT or PTAA are expensive and sensitive to humidity, [27][28][29] and various alternatives such as transition-metal oxides or copper compounds have been explored. [30][31][32][33][34] The choice of the HTL and ETL is also critical for cell performances, since PSC rely on different interfaces which should all be adequately designed and matched to achieve high PCE, by reducing charge recombination losses and maximizing the open-circuit voltage, [35] hence implementing interfacial and surface engineering is particularly important in these systems. [16,36] One general possible strategy for surface engineering is the use of facile solution-based processes considering self-assembled monolayers (SAMs) of organic molecules [37][38][39] which can form thin, dense and stable layers able to modify the electronic structure of the metal oxide surface they are chemically attached to. In the PSC field, SAMs have already been proposed since they can potentially induce various favorable effects such as morphology modification, band alignment with the energy level of the photoexcited active layer, trap passivation and delay of charge recombination. [40,41] Among the available modeling tools, density functional theory (DFT) has already been widely used and shown to be a useful tool to predict the structural and electronic properties of key interfaces found in sensitized solar cells, [42][43][44][45][46][47][48][49][50][51] including the one formed between the MAPI perovskite and TiO 2 in PSC. [52][53][54][55] While the works devoted to the investigation of the isolated components of this interface are numerous, [56][57][58][59] much fewer have focused on the investigation of the full interface, probably due to its complexity and to the associated high computational cost. For example, Yoshitaka et al. recently investigated the structural stability and the electronic structures of a series of low-indices surfaces of tetragonal MAPI, highlighting that the (110) and (001) terminations are expected to be stable, with shallow surface states that can act as efficient intermediates for hole transport to HTL. [60,61] The tetragonal MAPI (001) surface was also recently investigated, comparing the two possible MAI and PbI 2 terminations, indicating that the MAI termination is relatively more favorable than the PbI 2 one in equilibrium growth. [62] On the other hand, for the ETL, TiO 2 is usually considered due to its wide bandgap and fairly transparent nature. [63] Although the rutile polymorph is the most stable one for large particles, nanoparticles preferentially form with the anatase variant which is commonly used in photovoltaic materials. [64,65] More precisely, the anatase (101) surface is thought to be the most stable surface orientation in such systems. [66][67][68][69] Finally, among the works devoted to the modeling of the full MAPI/TiO 2 interface, [45,50,53,[70][71][72] De Angelis [53] et al. proposed the (110) surface of MAPI to preferentially grow on TiO 2 due to better lattice matching, with the chloride atoms concentrating at the TiO 2 interface. [70] In this article, we present a theoretical investigation of a modified MAPI/TiO 2 interface that was first optimized experimentally. Indeed, we have studied the effect of SAMs of a series of para-substituted benzoic acid derivatives as TiO 2 /perovskite interfacial modifiers. [41] Among them, we have found that 4-chlorobenzoic acid (CBA) was the best candidate which improves markedly the power conversion efficiency (PCE) of the solar cells. Interfacial engineering with CBA enabled us to achieve a PCE of 21.3 % to be compared to 20.3% for the untreated cells (see Tables S1 andS2, Supporting Information). However, while interfaces play a key role in such cells, their experimental characterization at the structural and electronic levels still remains difficult. Consequently, modeling of such interfaces with appropriate computational methods is relevant for a better understanding of the basic working principles of PSC, and thus for the design of novel systems. We show here that the linking of perovskite and TiO 2 moieties is enhanced by using the bifunctional CBA ligand. A model of the MAPI/CBA/TiO 2 interface (Figure 1) has been built and its structural and electronic properties have been fully characterized at the periodic hybrid DFT level. The key electron injection process found at this interface has also been addressed. We show that SAMs can be efficiently used for surface engineering of the heterointerface found in PSC between MAPI and TiO 2 , due its bifunctional nature, acting both as a suitable spacer at the interface to improve its stability, but also imposing a favorable dipole moment to improve band alignment and thus electron injection in such systems. In the discussion, computational details are first presented in Section 2 while results obtained are discussed in Section 3. Finally, some conclusions are drawn in Section 4.

COMPUTATIONAL DETAILS

All calculations were carried out with the periodic Crystal17 code, which is based on a linear combination of atom-centered Gaussian orbitals and solves self-consistently both Hartree-Fock (HF) and Kohn-Sham equations. [73][74][75] Exchange and correlation energies were described using the global hybrid PBE0 functional, containing 25% of HF exchange. [76] Previous studies have shown that this functional can indeed provide reliable geometric and electronic properties of both TiO 2 [77] and MAPI [46] , that are the two building blocks of the interface targeted. Additional dispersion effects were accounted for through the empirical D3

correction. [78] All-electron basis sets have been used for the CBA ligand and the methylammonium (MA + ) moieties. More precisely, for MA atoms, ( to 4 and 6 explicit electrons, respectively, were chosen. [79,80] Finally, for Pb and I, small core fully relativistic pseudopotentials developed at the Stuttgart University were selected, with 22 explicit electrons described using a (8811/1188/611/1) contraction scheme for Pb, and 25 explicit electrons described using a (8811/661/61) contraction scheme for I. [81] All basis sets have been previously validated to describe bulk MAPI [46] as well as the adsorption of CBA on a TiO 2 surface. [41] Reciprocal space was sampled using a shrinking factor of 2, corresponding to 4 k points in the irreducible Brillouin zone. Numerical DFT integration was performed considering 99 radial points and 1454 angular points, ensuring an error on the integrated electron density lower than 10 -5 |e-| per unit cell. The Coulomb and exchange series were truncated with threshold values of 10 -7 , 10 -7 , 10 -7 , 10 -9 , and 10 -27 .

Cell parameters of the lattice as well as atomic positions of a selected fragment of atoms (see below) were relaxed during geometry optimizations. Convergence was determined from the root-mean-square and absolute value of the largest component of the forces and displacements, considering default values. [75] Interface formation was energetically characterized by considering two quantities:

-adhesion energy, which is the energy gained by forming the interface, computed

as:

E ads = E(MAPI/CBA) + E(TiO 2 ) -E(MAPI/CBA/TiO 2 ) (1)
where E(MAPI/CBA) and E(TiO 2 ) are the total energies of the optimized MAPI/CBA slab and TiO 2 supercell slab models, respectively, and E(MAPI/CBA/TiO 2 ) is the total energy of the MAPI/CBA/TiO 2 interface -specific adhesion energy, which is the adhesion energy per unit surface area, calculated as:

β ads = E ads /S ( 2
)
where S is the surface area of the MAPI/CBA/TiO 2 interface model. [82] The electron injection time was evaluated by considering a simplified orbital-based model derived from the original Newns-Anderson proposal for adsorbates on surfaces, [83] which was already successfully applied to dye-sensitized solar cells. [51,84] In this approach, only the evaluation of the broadening (Δ, in meV) of the donor level of the perovskite sensitizer when it is adsorbed on the TiO 2 surface is required to compute the injection time (, in fs) according to:

τ=658/Δ (3)
More details regarding this methodology are available in Ref. 85 for instance. [85]

RESULTS AND DISCUSSION

The models:

A MAPI/CBA/TiO 2 interface was built starting from converged surface models of its two For MAPI, although both MAI-and PbI 2 -terminated (110) surfaces can exist, the former is known to be stable [61] and is expected to more easily bind to the TiO 2 surface than the latter. A resulting in an interface model with P1 symmetry, 328 atoms and 3876 atomic orbitals per unit cell. The lattice was then fully-relaxed, and all atoms but the lower 6 bottom layers of the TiO 2 substrate were allowed to relax. In addition, to more clearly highlight the role of the TiO 2 substrate in the MAPI/CBA/TiO 2 interface, and especially on its electronic properties, we also considered adsorption of CBA on a MAI-terminated MAPI (110) slab with 3-PbI 2 layers, considering a 1:1 substitution ratio between I -and Cl atoms. We note here that the CH 3 NH 2 layer formed in contact with the CBA ligands is relevant for PSC application, since the CH 3 NH 3 + ion in MAPI is well-known to readily deprotonate in the presence of water. In addition, chlorine atoms have already been shown to tend to concentrate at the TiO 2 interface. [71] Figures 2c and 2d present resulting structures of both models after geometry optimization. the same surface orientations, [70] as well with those obtained with other orientations. [72]

Structural and energetic features: Table 1 summarizes the main structural features of the optimized lattices of the different building blocks and interface models considered, while

Electronic properties:

Figure 3 presents the computed band structures, as well as the total and projected density of states (TDOS and PDOS) of the MAPI/CBA/TiO 2 and MAPI/CBA interfaces.

For MAPI/CBA, band dispersion can be noticed at the top of the valence band (VB) and at the bottom of the conduction band (CB). This can be related to the interaction of the CBA ligand with MAPI through its Cl atoms. In addition, the top of the VB and the bottom of the CB are mainly composed by the MAPI perovskite in which I -has been substituted with Cl in a 1:1 ratio. Compared to the clean MAPI (110) surface therefore, this leads to a band gap increase from 2.97 to 3.39 eV. When considering the MAPI/CBA/TiO 2 interface on the other hand, the band gap decreases to 2.16 eV mainly due to the TiO 2 contribution at the bottom of the CB. In fact, the band structure and DOS analysis of the MAPI/CBA/TiO 2 interface reveals that the top of the VB is mainly due to MAPI while the bottom of the CB is mainly due to TiO 2 , suggesting a possible electron transfer from the perovskite to the oxide from an energetic viewpoint, which is in line with the working principles of PSC mentioned above.

Furthermore, it should be noted that the CBA ligand does not contribute to the valence or conduction bands close to the band gap region. Here, we note that no spin-orbit coupling effects have been taken into account in these calculations, but these have previously been quantified and shown to further reduce the above-mentioned band gap values of about 1.1

eV. [46] Since neither the CBA ligand nor the TiO 2 oxide substrate aren't expected to contribute significantly to this effect, we can reasonably estimate the influence of the spin-orbit coupling to be qualitatively similar at the interface. Furthermore, both band structures present direct band gaps at , highlighting a possible favorable and efficient photoexcitation of the heterointerface.

Band edge movements upon interface creation can also be qualitatively rationalized from the dipole moments components along the normal direction to the surface plane of the interface according to the Poisson equation [41] :

where N s is the surface dipole density, μ the dipole moment, θ a the tilt angle of the dipole with respect to the surface normal. ε r is the dielectric constant, and ε 0 is the permittivity of free space. A step in the local vacuum level ΔV can be obtained and qΔV, with q the elementary charge, can then modify the work function and the band edges of the interface. In particular, a positive value of ΔV would enhance the charge transfer to TiO 2 while a negative one would reduce it. From the values reported in Table 3, for CBA/TiO 2 , the positive value suggests a charge transfer to the TiO 2 surface. [41] As for MAPI/CBA, the value of the dipole moment pointing towards MAPI in the chemistry notation is negative, indicating that CBA may block electron reflow to MAPI. Furthermore, it can be noticed that a positive value is also obtained for MAPI/CBA/TiO 2 , thus highlighting that CBA can potentially enhance the electron transfer from MAPI to TiO 2 . To better clarify this point, electron injection from MAPI to TiO 2 has been analyzed using an approach that we have developed notably for dye-sensitized solar cells [42,43] and quantum-dot sensitized solar cells [44] . More precisely, the spin density of the reduced MAPI/CBA/TiO 2 system has been computed and the corresponding isodensity plot is shown in Finally, based on the Newns-Anderson model and the computed electronic structure of the MAPI/CBA/TiO 2 system, the electron injection time is estimated to be 24 fs, indicating an ultrafast electron injection. This result is in good agreement with a previously-reported estimate based on a coupling matrix elements approach for the bare MAPI/TiO 2 interface [86] where an injection in the femtosecond time scale has been obtained. We note that controversial experimental results are available regarding the electron injection time with values reported both in the picosecond [START_REF] Marchioro | Physical Chemistry of Interfaces, Nanomaterials[END_REF] or sub-picosecond [START_REF] Ponseca | [END_REF]89] timescales, preventing a more rigorous comparison between experimental and computed data.

CONCLUSIONS

In this article, a periodic hybrid density functional theory computational strategy has been presented to model the heterointerface between the MAPI perovskite and TiO 2 , as found in PSC, in which the CBA ligand has been used to improve the stability and the band alignment at the interface. The CBA ligand has been selected to act as a bifunctional linker to efficiently link the perovskite and the oxide moieties, ensuring stability of the interface through Ti-O and Pb-Cl interactions. The computed density of states revealed that the perovskite contributes to the top of the valence band while the oxide contributes to the bottom of the conduction band with a direct bandgap of 2.16 eV, indicating a possible electron transfer from MAPI to TiO 2 . Dipole moment analysis additionally revealed that the CBA ligand can induce a favorable effect to improve band alignment and thus electron transfer from MAPI to TiO 2 . This latter has been quantified by calculation of the spin density of the reduced MAPI/CBA/TiO 2 system and indicated an almost quantitative electron transfer from MAPI to TiO 2 for the surface engineered system, together with an ultrafast injection time in the femtosecond timescale. Overall, the proposed DFT-based computational protocol therefore indicated that by surface engineering with a bifunctional linker such as CBA, a better stability, together with improved band alignment and electron injection, could be obtained for PSC systems. Work is in progress to design novel PSC systems with enhanced efficiency based on these guidelines.

Figure 1 :

 1 Figure 1: Schematic of the MAPI/CBA/TiO 2 interface.

  building blocks: the TiO 2 oxide and the MAPI perovskite. A (2x2) supercell model of the anatase TiO 2 -(101) surface with 8 layers, corresponding to 8 O-Ti-O trilayers with 96 atoms and lattice parameters a=14.981 Å, b=11.088 Å and =109.7 was selected for the oxide part.

3 -

 3 PbI 2 layers (110) slab was therefore obtained from the tetragonal MAPI bulk system with space group I 4 /mcm, resulting in a 2D lattice with lattice parameters a, b and  of 12.343 Å, 12.555 Å and 90.0, respectively. Consequently, a large mismatch between the lattice parameters of the 2D lattices of the TiO 2 and MAPI slabs was obtained. The bifunctional nature of the CBA ligand was exploited by considering its carboxylate part to bind to TiO 2 in a bridging bidentate mode, while its Cl atom was oriented towards the Pb atoms of MAPI close to the TiO 2 substrate. By substituting all I -of the last layer of the MAPI surface with Cl atoms, Pb-Cl bonds were then created and the charge neutrality of the resulting unit cell was ensured by removing all H + of the last layer of MA moieties of MAPI. The final interface model was obtained by modifying the MAPI slab lattice to the values of the TiO 2 supercell,

Figure 2 :

 2 Ball and stick representation of the (a) MAPI(110) surface, (b) the MAPI-slab with all iodide atoms on one surface replaced by chloride atoms, (c) MAPI/CBA unit cell and (d) MAPI/CBA/TiO 2 interface unit cell. The solid yellow line represents the unit cell. Red, light grey, white, green, purple, dark grey, blue and grey spheres correspond to O, Ti, H, Cl, I, Pb, N and C atoms, respectively. The relaxed and frozen parts considered for the geometry optimization of the MAPI/CBA/TiO 2 interface are indicated in Figure 2(d).

Figure 3 :

 3 Figure 3: (a) and (c) Band structures and (b) and (d) total and projected density of states (TDOS and PDOS) of the MAPI/CBA (top) and MAPI/CBA/TiO 2 (bottom) interfaces.k-points used for the band structure are:  (0, 0, 0), Y (0, 1/2, 0), M (1/2, 1/2, 0) and X (1/2, 0, 0).

Figure 4 .

 4 It is clear that the computed spin density is mainly delocalized over the TiO 2 part, with a 99.94% contribution of the added electron on the TiO 2 substrate. This value is a quantification of the injection efficiency of the photogenerated electrons and indicates an almost quantitative electron transfer from MAPI to TiO 2 through the CBA ligand.

Figure 4 :

 4 Figure 4: Spin density of the reduced MAPI-CBA-TiO 2 interface system (excess of alpha electron is indicated in yellow), isosurface contour value: |0.00271| a.u.

Table 2

 2 reports selected resulting distances. The optimized a and b cell parameters of the MAPI/CBA/TiO 2 interface model are close to those of the TiO 2 supercell considered, with values of 14.821 Å and 11.003 Å, together with a γ angle of 108.0°. Compared to the TiO 2 supercell lattice, a small shrinking of all

lattice parameters is therefore obtained when going from the TiO 2 supercell to the MAPI/CBA/TiO 2 interface, with shrinking values of -1.07, -0.77 and -1.55 %, respectively for the a, b and γ parameters. From the data reported in

Table 2 ,

 2 both Ti-O and Pb-Cl bonds can be evidenced, the CBA ligand acting with two complementary features to ensure interface stability between the perovskite and the oxide. The first feature corresponds to its bifunctional linker nature: a bridging bidentate mode is involved between the CBA carboxylate moiety and TiO 2 , while strongly distorted PbI 5 Cl octohedra can be evidenced between MAPI and CBA.Here, we note that Pb atoms both interact with Cl atoms and the deprotonated CH 3 NH 3 + groups in the layer in contact with CBA. The same conclusion can be drawn in the MAPI/CBA case. The second feature is the ability of CBA ligands to lay on the TiO 2 substrate, with computed tilting angles with respect to the surface normal of 30.0, 38.6, 38.6 and 46.9° for the four CBA molecules of the MAPI/CBA/TiO 2 interface model unit cell for instance. In particular, favorable H + / interactions between the dissociated H + of the carboxylate groups of CBA adsorbed on the TiO 2 substrate and the benzene ring of the CBA ligands tend to further improve interface stability by building a favorably organized CBA-TiO 2 interface for the binding of MAPI. This can be related to the rigid nature of the CBA ligand. The computed adhesion energy of the MAPI/CBA/TiO 2 model further confirms these conclusions, with a computed value of 3.45 eV, corresponding to a specific adhesion energy of 0.02 eV/Å 2 , indicating a stable interface formation. We note that this value is in line with the 0.04 eV/Å 2 value recently reported by Mosconi et al. for a similar interface based on

Table 1 :

 1 Lattice

	also given.
		MAPI (110)	CBA/MAPI (110) TiO 2 (101) MAPI (110)/CBA/TiO 2 (101)
	a	12.343	12.343	14.981	14.821
	b	12.555	12.555	11.088	11.003
	γ	90.00	90.00	109.74	108.03
	E g	2.97	3.39	5.03	2.16
	E ad	--	--	--	3.45
	β	--	--	--	0.02

parameters a, b (in Å) and γ (in degrees) as well as computed band gaps (E g , in eV) for selected systems involved in the interface model construction. Adhesion energy (E adh , in eV) and specific adhesion energy (, in eV•Å -2 ) of the MAPI/CBA/TiO 2 interface are

Table 2 :

 2 Selected distances (in Å) between Pb, N and Cl atoms of the MAPI/CBA and MAPI/CBA/TiO 2 systems.

		d N/Pb	d Cl/Pb	d Ti/O
	MAPI/CBA	3.358 3.710 4.032 5.472 2.627 2.632 4.749 5.566 -	-
	MAPI/CBA/TiO 2 3.457 3.957 4.686 5.424 2.517 2.553 5.063 5.669 2.176 2.100

Table 3 :

 3 Dipole moments components along the surface normals ( z , in Debye) of the CBA/TiO 2 , MAPI/CBA and MAPI/CBA/TiO 2 systems.

	Systems	 z
	CBA/TiO 2	0.89

ACKNOWLEDGMENTS J. S. and T. Z. acknowledge the CSC for Ph.D. scholarship. The authors acknowledge the use of computational resources at Très Grand Centre de Calcul (TGCC) provided by GENCI through project A0050810135. IC and FL thank the ITN program COSINE (Grant agreement ID 765739) for support.


E Boukhicha

J Bocquillon

K Palomo

Fève

A Inhofer

J Duffy

M Boukhicha

E Bocquillon

J Palomo

K Watanabe

T Taniguchi

I Estève

J M Berroir

G Fève

B Plaçais

B A Assaf

RF-Quantum Capacitance of the Topological Insulator Bi 2 Se 3

come

Introduction

Topological phases of matter have emerged as a novel paradigm in the study of condensed matter physics. [1] [2] [3] [4] Topological insulators (TI) are essentially material realizations that stem from this new theoretical paradigm. [4] [5] [6] [7] [8] They are interesting from both the fundamental and applied perspective. Typically, a topological insulator is a material that has an inverted orbital band ordering in the 3D bulk, which leads to the existence of Dirac cones at the surface of the material, at symmetric points in the Brillouin zone. [4] [5] These Dirac fermions are spin-momentum locked and highly robust to backscattering. From the fundamental perspective, a number of novel states of matter have so far been realized in topological insulators. The quantum anomalous Hall state, [9] [10] the Majorana fermion [11] and the quantized Faraday and Kerr effects [12] [13] are examples of such realizations. From the applied perspective, implementation in spintronic data storage devices and high frequency transistors are envisaged. Highly efficient spin-torque switching and spin injection have already been demonstrated in ferromagnet/TI devices thus establishing potential use in data storage, [14] [15] [16] however, studies aimed at realizing high frequency transistors still lack.

An important step to realize high frequency transistors is to characterize the capacitive response of the TI at radio frequency (RF) and establish its Dirac-like nature. The RF transport regime has already been significantly studied in graphene [17] [18] [19] [20] and more recently in HgTe TIs. [21] [START_REF] Inhofer | Probing AC Compressibility of 3D HgTe and Bi2Se3 Topological Insulators at High Electric Fields: Evidence for Massive Surface States[END_REF] In this regime, one can simultaneously measure the quantum capacitance of the Dirac states and the conductivity of the material. Contrary to the constant capacitance-voltage characteristic typical of metal-insulator-metal capacitors (Fig. 1(a)), in metal-insulator-graphene and metal-insulatortopological insulator capacitors (MITI-CAP), the capacitance is a function of the applied voltage (Fig 1(b)). The quantum capacitance being related to the compressibility or the density of states, then allows one to directly measure those quantities. [17] This is, however, only possible for materials that have a low carrier density and a good mobility. For this particular reason, RF capacitance studies have remained highly challenging in Bi-based 3D-TIs. In the case of Bi2Se3 for example, residual bulk ndoping renders a reliable detection of surface-state signatures difficult. [START_REF] Analytis | [END_REF] [24] [25] The first step in realizing an RF-transport device based on Bi2Se3 is a solution to the issue of material quality. Motivated by recent positive results on the growth of Bi2Se3 by chemical vapor deposition (CVD) on mechanically exfoliated hexagonal boron nitride (hBN), [26] , [27] we undertake a similar procedure.

We first grow Bi2Se3 by CVD on high-quality hBN, [28] [29] and then transfer a second layer of hBN on top of the grown Bi2Se3 to realize a capacitor device. The excellent dielectric properties of the hBN used in this work, and the improved transport characteristics of CVD grown Bi2Se3 on hBN, allow us to observe clear signatures of Dirac surface-states in the RF transport regime. We are able to simultaneously measure both the quantum capacitance and the channel resistance of the device. The capacitance exhibits a linear variation and a minimum versus chemical potential characteristic of Dirac fermions. The resistance shows a strong increase with decreasing voltage in the depleted regime. It does not reach a maximum at the capacitance dip. We argue that this is due to the contributions of the bottom surface in the bipolar regime. Our work provides a first quantitative analysis of the compressibility of Bi2Se3 in the RF regime and establishes the Dirac nature of the RF response in TIs.

Growth and characterization

Bi2Se3 nanoflakes are grown by catalyst free chemical vapor deposition (CVD) using a three zone tube furnace following a procedure similar to what is reported by Xu et al. [26] All growths are performed on high-resistivity Si/SiO2 substrates having an oxide thickness of 300nm, on which we mechanically exfoliate h-BN. The furnace tube is initially pumped down to 8x10 -2 mbar. A powder source of high purity (99.99%) Bi2Se3 is placed in the hot zone (A) of the furnace in a stream of Argon gas (99.999%) flowing at 200 sccm (Fig. 2(a)). The substrate (Fig. 2(b)) is placed downstream from the source in the colder zone (B). Zones (A) and (B) are initially heated up to 300⁰C in 30 min. (A) is then heated up to 600⁰C while (B) is only heated up to 400⁰C in 30 min. These temperatures are maintained for 60 seconds. Both zones are finally cooled down to 200⁰C in 80 min. An absolute pressure of 3.9 mbar is maintained during the entire process.

An optical microscope image of a characteristic sample is shown in Fig. 2(b,c), before and after the growth respectively. A layer of Bi2Se3 is seen to coat the hBN flakes, but does not nucleate on the Three flakes (6QL, 8QL, 13QL) having different thicknesses obtained from identical growths are selected for device implementation. The 8QL sample is studied in detail. The results from the other three samples are shown in the appendix section. An AFM image of the 8QL sample is shown in Fig. 3(a). We measure the Bi2Se3 thickness across a line shown in the inset of Fig. 3(a). We find a thickness varying between 8 and 10 quintuple layers (QL). A small peak-to-peak surface roughness of (1-2)QL is detected. A hBN flake is then transferred on top of the grown Bi2Se3 using the standard dry transfer method, already proven successful for graphene [31] [32] and 2D semiconductors.

[33] [34] [35] [36] [37] Optical images of the sample are shown in Fig. 3(b,c) before and after the transfer respectively. The transfer is performed in air, therefore exposing the top surface of Bi2Se3 to atmosphere. The transferred hBN layer is chosen to be thinner than 10 nm (8 nm in this case), in order to maximize its geometric capacitance and render the quantum capacitance of the Bi2Se3 experimentally visible. A single e-beam lithography step then allows us to pattern the gate and drain electrodes, as well as a co-planar waveguide. A metallic bilayer of Ti(5 nm)/Au(150 nm) is then deposited. Note that prior to depositing the Ti layer, light argon etching (<10 s) is performed in-situ to remove any native oxides, and to minimize contact resistance. The device processing sequence is summarized in Fig. 3(d). The hBN-encapsulated Bi2Se3 capacitor device embedded in an RFwaveguide is shown Fig. 4 (a,b).

RF-transport measurements

The devices are characterized using radio-frequency (RF) transport measurements for frequencies between 0.03 and 10 GHz using a variable network analyzer, in a cryogenic RF-probe station, as detailed in our previous works. [17] [22] [21] A standard short-open-load-through calibration is performed before the measurement. The S-matrix components are extracted by measuring the reflected and transmitted wave intensity through the device as a function of frequency for different gate voltages using a variable network analyzer. The complex admittance (inverse impedance) is then extracted from the S-matrix components. The real and imaginary parts of device admittance are quantified versus frequency and gate voltage. Proper care is taken to de-embed [START_REF] Pozar | Microwave Engineering[END_REF] parasitic capacitive and inductive contributions resulting from the device geometry by measuring a dummy device having identical contact geometry without the Bi2Se3 flake in between, as well as a conductive through-line.

Such measurements also rule out any parasitic RF signals stemming from the substrate Si/SiO2 substrates. All measurements are made at 10K. In what follows, we will focus on measurements made on the 8QL device shown in Fig. 2 and3.

    contact RC total R Y Y      1 1 , (1)
Where,

     / ) / tanh( C j L C j L CLw j Y RC  (2)
Here, f

  2  is the frequency, 1   j and m L  3 . 4  and m w  8 . 12  are
the device length and width, respectively.  is the channel conductivity. C is the total device capacitance:

1 1 1      Q geo c c C (3)
Where as a function of gate voltage between -6 V and 1V. The capacitance decreases progressively for decreasing voltage (0 to -5 V), goes through a minimum at about -5 V, then increases again between -5 V and -6 V. With a capacitance dip of 5%, our results agree with previous low frequency capacitance measurements on similar Bi2Se3 reporting a capacitance dip of 6%. [26] The resistance exhibits a continuous increase that accelerates near the capacitance minimum.

No resistance maximum is observed. A fixed contact resistance Rc≈20Ω was included in the curve fit to the admittance data. Allowing this contact resistance to vary yielded negligible variation compared to the 5-fold increase observed in the channel resistance. We can thus confidently claim, that our two-point RF measurement yields a reliable simultaneous measurement of the quantum capacitance and the channel resistance.

Analysis of the quantum capacitance of top surface states

We next focus on the analysis of quantum capacitance. 𝑐 𝑄 can be extracted using Eq. ( 3), by fixing 𝑐 𝑔𝑒𝑜 = (124 ± 1) fF, the value at which the capacitance is seen to saturate in Fig. 4(e). The measured geometric capacitance is slightly lower than what is expected for hBN having κ=3.2, possibly as a result of the rough surface of Bi2Se3. The capacitance per unit area 𝑐 𝑄 is then determined by dividing the capacitance by the geometrical factors L and w. 𝑐 𝑄 is shown in Fig. 5 (a). The grey lines in Fig. 5(a) delimit the propagated uncertainty on 𝑐 𝑄 due to the uncertainty associated with 𝑐 𝑔𝑒𝑜 . A powerful consequence of our measurements is the fact that it allows us to determine the local (top surface) chemical potential 𝜀 𝑓 directly from experimental data via the Berglund integral, without a priori knowledge of the band structure. The Berglund integral is written as [START_REF] Berglund | [END_REF]:

𝜀 𝑓 = ∫ 𝑑𝑉 (1 - 𝐶(𝑉) 𝑐 𝑔𝑒𝑜 ) 𝑉 𝑔 0 (4)
We use Eq. ( 4) to extract 𝜀 𝑓 which we now plot as a function of Vg in Fig. 5 (assumed parabolic with m*=0.14m0). [41] This yields a slight native bulk n-doping of 2×10 18 cm -3 . This is a significant improvement compared to pristine quality Bi2Se3 where typically 𝑛 > 10 We can additionally perform a linear fit to the 𝑐 𝑄 versus 𝜀 𝑓 curve. This is shown as the solid red line, fit to the data in Fig. 5(c). The dashed lines are fit extrema to the edges of the shaded area, and allow us to determine the uncertainty on the extracted slope. The slope allows us to extract the Dirac velocity of the topological surface states, near the Dirac point. This follows from the expression of the compressibility χ, which is related to 𝑐 𝑄 :

𝑐 𝑄 = 𝑒 2 𝜒 = 𝑒 2 𝜀 𝑓 2𝜋(ℏ𝑣 𝑓 ) 2 (5)
We find a Dirac velocity equal to (5.8±1.4)×10 5 m/s. This agrees within experimental uncertainty with previously reported velocities that vary between 5×10 5 m/s and 4.5×10 5 m/s. [40] [50] [25] It is worthwhile highlighting that while we can perfectly account for variation of the quantum capacitance versus Fermi energy shown in Fig. 5(c), this analysis cannot account for the finite 𝑐 𝑄 offset observed at the minimum. From Eq. ( 5) it is evident that 𝑐 𝑄 should go to zero at 𝜀 𝑓 = 0. Although this has never been observed experimentally, the typical capacitance offset at the Dirac point observed in graphene, for example, is an order of magnitude lower than what is observed here. [51] Our data suggests the presence of additional capacitive contributions in parallel with that of the top Dirac surface states. A thorough understanding of these capacitive contributions from other transport channels must thus be provided.

The origin of the capacitance offset: capacitive coupling of top and bottom surface channels

First, it is simple to rule out reminiscent bulk carriers as the source of this offset, simply by computing the screening length for Bi2Se3 using the measured quantum capacitance. In order to extract the screening length from experimental data, we need to develop and expression that relates the screening length to the 2D density-of-states and to the quantum capacitance. This is discussed in detail in appendix A. The end result is a screening length that scales linearly with the inverse of the 2D quantum capacitance of bulk states 𝑐 𝑄 𝑏𝑢𝑙𝑘 :

𝜆 = 𝜅𝜀 0 𝑐 𝑄 𝑏𝑢𝑙𝑘 ( 7
)
Note that here 𝑐 𝑄 𝑏𝑢𝑙𝑘 is a quantum capacitance, related to a surface charge accumulation or depletion that screens the electric field over a finite length. 𝜅 ≈ 100 is the generally accepted static dielectric constant of Bi2Se3. [52] [53] [54] [55] [56] 𝜀 0 = 8.85 × 10 -12 𝐹/𝑚 is the permittivity of free space. If

we assume that all measured quantum capacitance contributions are due to screening, then for 𝑐 𝑄 𝑏𝑢𝑙𝑘 < 100 fF/μm 2 , 𝜆 exceeds 10 nm and the thickness of the sample (8nm); implying that full bulk depletion is possible. Near the capacitance minimum, it is thus highly unlikely that reminiscent charge carriers from the bulk contribute to the transport.

Consequently, we can consider the situation of a fully depleted insulating bulk that contributes a geometric capacitance, between two metallic surfaces. The Dirac character of the top metallic surface is proven by the quantum capacitance observed in Fig. 5. However, the character of the bottom surface is not evident in our experiments. Experimentally, we can still determine the net quantum capacitance contribution of this bottom metallic layer, assuming it couples capacitively in parallel with the top surface via the insulating bulk:

𝑐 𝑄 = 𝑐 𝑄 𝑇𝑇𝑆𝑆 (𝜀 𝑓 ) + ( 1 𝑐 𝑔 𝐵𝑢𝑙𝑘 + 1 𝑐 𝑄 𝐵𝑜𝑡𝑡𝑜𝑚 (𝜀 𝑓 ) ) -1 (8)
Here, 𝜀 𝑓 denotes the chemical potential of the top surface, 𝑐 𝑄 𝑇𝑇𝑆𝑆 is top surface quantum capacitance and 𝑐 𝑄 𝐵𝑜𝑡𝑡𝑜𝑚 is that of the bottom metallic surface. The geometric capacitance of the insulating bulk with a thickness 𝑑 𝐵𝑖2𝑆𝑒3 ≈ 8 𝑛𝑚 is given by:

𝑐 𝑔 𝐵𝑢𝑙𝑘 = 𝜅𝜀 0 𝑑 𝐵𝑖2𝑆𝑒3 ≈ 110 fF/μm 2 (9)
When 𝑐 𝑄 𝑇𝑇𝑆𝑆 (𝜀 𝑓 = 0)=0, an offset equal to (30 ± 5) fF/μm 2 has to result from the other term in Eq. ( 8). This yields 𝑐 𝑄 𝐵𝑜𝑡𝑡𝑜𝑚 = (40 ± 10)fF/μm 2 . It can be easily shown that the quantum capacitance expected from a quadratically dispersing Bi2Se3 interfacial 2DEG significantly exceeds this value:

ℎ 2 𝜋 𝑒 2 𝑚 * = 94fF μm 2 > 𝑐 𝑄 𝐵𝑜𝑡𝑡𝑜𝑚
using the effective mass of Bi2Se3, 𝑚 * = 0.14𝑚 0 . [41] This is therefore an unlikely scenario.

The last scenario to consider is that the bottom Dirac TSS having a finite Fermi energy at the charge neutrality point of the top TSS, yields the offset in the quantum capacitance. Using Eq. ( 5), we estimate, for a chemical potential close to (170±40)meV above the Dirac point (with 𝑣 𝑓 ≈ 5 × 10 5 𝑚/𝑠), a Dirac quantum capacitance 𝑐 𝑄 𝐵𝑜𝑡𝑡𝑜𝑚 = (40 ± 10)fF/μm 2 . We conclude that the most likely explanation to the observed offset is presence of the bottom TSS that couples capacitively to the top TSS via the insulating bulk.

Charging curve of a bulk-depleted topological insulator

In order to get further insight on the charging mechanism expected in such as situation, we develop in Fig. 6(a,b) a model that describe the charging of both the top and bottom TSS coupled via the bulk, as a function of a top gate voltage. This capacitance model is summarized by the circuit shown in Fig.

6(b)

. The total quantum capacitance corresponding to this circuit is given by Eq. ( 8).

The quantum capacitance of the top TSS 𝑐 𝑄 𝑇𝑇𝑆𝑆 is computed using Eq. ( 5), with 𝑣 𝑓 ≈ 5 × 10 5 𝑚/ 𝑠. [25] Note that in Eq. ( 8) 𝜀 𝑓 is the top surface chemical potential. Hence, the expression for that of the bottom TSS 𝑐 𝑄 𝐵𝑜𝑡𝑡𝑜𝑚 (𝜀 𝑓 )is not simply given by Eq. ( 5). We show in the appendix, that the effective capacitance of the bottom TSS that is capacitively coupled via the insulating bulk is given by:

( 1 The contribution of 𝑐 𝑄 𝑇𝑇𝑆𝑆 is shown in red in Fig. 6(a). The evolution of the effective capacitance of the bottom TSS 𝑐 𝑄 𝐵𝑇𝑆𝑆 in series with the insulating bulk as a function of top-surface chemical potential 𝜀 𝑓 is shown in dashed magenta. For W ≈ 140 meV, the Eq. ( 10) is in good agreement with the data as seen in Fig. 6(a). The quantum capacitance associated to the bottom surface is seen to flatten out at large values of 𝜀 𝑓 due to an enhanced screening of the electric field at large charge carrier density by the top-TSS. Note that this treatment is only valid when the bulk is fully depleted (region (I) in Fig. 6(a)); the quantum capacitance from populated bulk bands should be taken into account at larger Fermi energies (region II in Fig. 6(a)), however our data is not precise enough to provide any quantitative analysis in this region.

𝑐 𝑔 𝐵𝑢𝑙𝑘 + 1 𝑐 𝑄 𝐵𝑇𝑆𝑆 (𝜀 𝑓 ) ) -1 = 𝑐 𝑔 𝐵𝑢𝑙𝑘 [ 1 - 1 ( √ 1 + 𝑒 2 |𝜀 𝐹 + 𝑊| 𝜋(ℏ𝑣 𝑓 ) 2 𝑐 𝑔 𝐵𝑢𝑙𝑘 ) ] (10)

The behavior of the channel resistance

Next, we qualitatively discuss the variation of the channel resistance. Near the capacitance minimum, a fast increase of the channel resistance is observed, likely corresponding to the depletion of surface carriers on the top and bottom surfaces, but no resistance maximum is reached. Since the Fermi energy is likely higher above the Dirac point at the bottom surface, bottom Dirac electrons provide a parallel conduction channel that has a much lower resistivity. The effective resistance from both top and bottom parallel channels is hence dominated by the resistivity of the bottom channel. Therefore, one does not expect to observe an ambipolar resistance maximum as long as the bottom Dirac cone remains heavily occupied.

Lastly, going up in Fermi energy away from the capacitance minimum, the resistance curve flattens out and eventually increases at large positive values of Vg (Fig. 4(e)). We point out, that this increased scattering at high charge carrier density in top TSS is likely associated to sub-band scattering as reported in Ref. [22] [21]. It is worth noting that any issue with the contact resistance can be ruled out.

First, the Au/Bi2Se3 interface is not under the gate stack and therefore remains ohmic throughout the entire experiment, since Bi2Se3 remains n-doped away from the gate stack. Second, even when the top surface is very close to neutrality, the bottom surface of the sample remains carrier doped, thus ensuring a good connection with the source contact. This highlights the strength of our measurement and the local nature of the quantum capacitance in our experiment.

Conclusion

In conclusion, we have realized implementation of CVD grown Bi2Se3 in RF capacitor devices and reported the observation of the quantum capacitance of the top Dirac surface state and its variation versus gate voltage. The reduced electron doping of Bi2Se3 grown in these conditions, and the use of a high quality hBN dielectric allows us to quantify the quantum capacitance of Bi2Se3 and observe its minimum resulting from the top topological surface Dirac point. A detailed analysis of the field effect mechanism in thin Bi2Se3 flakes shows that a bulk depleted regime can be reached at an accessible gate voltage in hBN-encapsulated Bi2Se3 allowing to investigate Dirac physics. We have lastly modeled the capacitance-voltage curve of a TI slab consisting of two surface states separated by an insulating bulk, and confirmed its correspondence with our data. As an outlook, a dual-gated device might allow to electrostatically compensate the chemical doping asymmetry between the two surfaces. [57] Topological materials with even larger dielectric constants will also be interesting to investigate. [60] [61] Overall, our work establishes hBN encapsulated Bi2Se3 as a promising platform to motivate future work on implementation in high frequency transistors. Ref. [49] (1-3)×10 18

At

MBE

Ref. [56] (5-10)×10 18 10 MBE

Ref. [46] (1-2)×10

Appendix A. Quantum capacitance and dielectric screening of electron-doped 3D semiconductor

The quantum capacitance is directly related to the compressibilitythe variation of the 2D electrical charge density per unit chemical potential

𝜕𝑛 2𝐷 𝜕𝜇 𝑇 : 𝑐 𝑄 = 𝑒 2 𝜕𝑛 2𝐷 𝜕𝜇 𝑇 = 𝑒 2 𝜕 𝜕𝜇 𝑇 [∫ 𝜌(𝑧) 𝑒 𝑑𝑧 𝑑 0 ] (𝐴1)
Here n2D is the surface carrier density, 𝜇 𝑇 ≡ 𝜀 𝐹 is the top surface chemical potential and 𝜌(𝑧)the 3D charge density. The z-axis is chosen to be perpendicular to the surface of the material and therefore parallel to the electric field. The top BN-Bi2Se3 interface is at z=0.

From the conservation of electrochemical potential we get: 𝜇 𝑇 = -𝑒𝑉 0 .

𝑉 0 = 𝑉(𝑧 = 0) is the electric potential at the dielectric-semiconductor interface. From this we get:

𝑐 𝑄 = 𝑒 2 𝜕𝑛 2𝐷 𝜕𝜇 𝑇 = - 𝜕 𝜕𝑉 0 [∫ 𝜌(𝑧)𝑑𝑧 ∞ 0 ]
Poisson's law relates the charge to the electric potential:

𝜌(𝑧) = 𝜅𝜀 0 𝜕 2 𝑉 𝜕𝑧 2
𝜅 is the dielectric constant and 𝜀 0 = 8.85 × 10 -12 𝐹/𝑚 is the permittivity of free space.

Plugging this into 𝑐 𝑄 and performing the integration gives:

𝑐 𝑄 = -𝜅𝜀 0 𝜕 𝜕𝑉 0 ([ 𝜕𝑉 𝜕𝑧 ] 𝑧→∞ -[ 𝜕𝑉 𝜕𝑧 ] 𝑧=0 )
We assume [ induced by the applied surface electrical potential 𝑉 0 . Given in this form, the quantum capacitance from a 3D semiconductor is conceptually simple to understand but not straight forward to quantify experimentally. We will next relate this quantum capacitance to the screening length.

Using the Poisson equation again we can show that since:

𝜌(𝑧) 𝜅𝜀 0 = 𝜕 2 𝑉 𝜕𝑧 2
We can write:

∫ 𝜌(𝑧) 𝜅𝜀 0 𝜕𝑉 𝜕𝑧 ∞ 0 𝑑𝑧 = ∫ 𝜕 2 𝑉 𝜕𝑧 2 𝜕𝑉 𝜕𝑧 𝑑𝑧 ∞ 0 ∫ 𝜌(𝑉)𝜕𝑉 𝜅𝜀 0 0 𝑉(0) = - 1 2 ( 𝜕𝑉 𝜕𝑧 ) 𝑧=0 2
Again assuming V and

𝜕𝑉 𝜕𝑧

tend to 0 at ∞, and we get:

Or, √2 ∫ 𝜌(𝑉)𝜕𝑉 𝜅𝜀 0 𝑉(𝑧=0) 0 = [ 𝜕𝑉 𝜕𝑧 ] 𝑧=0
With 𝑉(𝑧 = 0) by definition equal to 𝑉 0 .

Then, the term

𝜕 𝜕𝑉 0 [ 𝜕𝑉 𝜕𝑧 ] 𝑧=0
in the quantum capacitance can be written as:

𝜕 𝜕𝑉 0 [ 𝜕𝑉 𝜕𝑧 ] 𝑧=0 = 𝜕 𝜕𝑉 0 [√2 ∫ 𝜌(𝑉)𝜕𝑉 𝜅𝜀 0 𝑉 0 0 ] = 𝜕 𝜕𝑉 0 ∫ 𝜌(𝑉)𝜕𝑉 𝑉 0 0 √2𝜅𝜀 0 ∫ 𝜌(𝑉)𝜕𝑉 𝑉 0 0 𝜕 𝜕𝑉 0 [ 𝜕𝑉 𝜕𝑧 ] 𝑧=0 = 𝜌(𝑉 0 ) √2𝜅𝜀 0 ∫ 𝜌(𝑉)𝜕𝑉 0 𝑉 0 Or, 𝜕 𝜕𝑉 0 [ 𝜕𝑉 𝜕𝑧 ] 𝑧=0 = 𝜌(𝑉 0 ) 𝜅𝜀 0 [ 𝜕𝑉 𝜕𝑧 ] 𝑧=0 = 𝜌(𝑉 0 ) 𝜅𝜀 0 𝐸(0)
Finally, combing this result with Eq. (A2) we get the quantum capacitance:

𝑐 𝑄 = 𝜌(𝑉 0 ) 𝐸(0) (𝐴3)
From Gauss' law, we get:

𝜕𝐸 𝜕𝑧 = 𝜌 𝜅𝜀 0
Integrating from 0 to infinity and assuming E tends to zero at infinity, yields:

𝜅𝜀 0 𝐸(0) = -𝑒 ∫ 𝑛 0 𝑒 - 𝑧 𝜆 ∞ 0 𝑑𝑧 = 𝑒𝑛 0 𝜆
Here, the charge distribution is assumed to follow an exponential decay into the sample: 𝜌(𝑧) = -𝑒𝑛 0 𝑒 -𝑧 𝜆 , where n0 is the charge at z=0 and λ is an effective screening length. This gives: 𝜆 = 𝜅𝜀 0 𝐸(0) 𝑒𝑛 0 = 𝜅𝜀 0 𝐸(0) 𝜌(𝑉 0 ) According to (A3) this gives:

𝜆 = 𝜅𝜀 0 𝑐 𝑄

𝑏𝑢𝑙𝑘

Note that here 𝑐 𝑄 𝑏𝑢𝑙𝑘 is a quantum capacitance related a 2D compressibility from a charge accumulation/depletion profile. It is related to the 2D density of states of this charging profile. The screening length λ is not the Thomas-Fermi screening length, but rather a screening length parameter that varies with the chemical potential and depends on the 2D quantum capacitance of a surface charging/depletion layer. We have thus derived the dependence of the screening length on the quantum capacitance that we measured experimentally.

Appendix B. Quantum capacitance from two Dirac surfaces states coupled via an insulating bulk

A capacitance model accounting for two Dirac surface states coupled via an insulating bulk (Fig. 6(b)) results in the following expression:

𝑐 𝑄 (𝜇 𝑇 ) = 𝑐 𝑄 𝑇𝑇𝑆𝑆 (𝜇 𝑇 ) + ( 1 𝑐 𝑔 𝐵𝑢𝑙𝑘 + 1 𝑐 𝑄 𝐵𝑇𝑆𝑆 (𝜇 𝑇 ) ) -1
Here 𝑐 𝑔 𝐵𝑢𝑙𝑘 is the bulk geometric capacitance, and 𝜇 𝑇 ≡ 𝜀 𝐹 is the top surface chemical potential.

The expression for the quantum capacitance of a single Dirac cone at the top surface (Eq. 5 in manuscript) is straight forward to determine:

𝑐 𝑄 𝑇𝑇𝑆𝑆 (𝜇 𝑇 ) = 𝑒 2 |𝜇 𝑇 | 2𝜋(ℏ𝑣 𝑓 ) 2 (𝐵1)
𝑣 𝑓 is the Fermi velocity.

The expression for the quantum capacitance of single Dirac cone at the bottom surface, is more challenging to extract:

𝑐 𝑄 𝐵𝑇𝑆𝑆 (𝜇 𝑇 ) = 𝑒 2 |𝜇 𝐵 (𝜇 𝑇 )| 2𝜋(ℏ𝑣 𝑓 ) 2 A knowledge of 𝜇 𝐵 (𝜇 𝑇 ) is required. This can be modelled by carefully studying electrochemical equilibrium is a system of two Dirac fluids in parallel, coupled by an insulating capacitive layer.

The electrochemical potential 𝜇 * imposed by the metallic contact ensures equilibrium and allows us to write:

𝜇 * = 𝜇 𝑇 -𝑒𝑉 𝑇 + 𝑊 = 𝜇 𝐵 -𝑒𝑉 𝐵 (B2)

Here 𝜇 𝑇/𝐵 and 𝑉 𝑇/𝐵 are respectively the chemical potential and the electric potential at the top/bottom surface, and W is a work function term that includes contributions that allows band misalignment such as band bending and surface charging due to impurities which essentially lead to a band offset between the top and bottom surface Dirac point.

The 2D carrier density 𝑛 2𝐷 as a function of chemical potential 𝜇 for a non-spin degenerate Dirac surface states can be written as:

𝑛 = 𝑠𝑖𝑔𝑛(𝜇) 𝜇 2 4𝜋(ℏ𝑣 𝑓 ) 2 (𝐵3)
The charge density is simply given by 𝜌 = -𝑛𝑒.

We now apply Gauss' law to find an expression for the charge density at the top and bottom interfaces:

For the bottom surface we have: We highlight the simplicity of this model from an experimental viewpoint, since most parameters can be determined independently from previous measurements. The only adjustable parameter is the band offset W.

𝑐 𝑔 𝐵𝑢𝑙𝑘 (𝑉 𝐵 -𝑉 𝑇 ) = -𝑒𝑛 𝐵 (

Appendix C. Thickness dependence

We have also measured two additional MITI-devices having respectively thinner and thicker Bi2Se3. The device characteristics are summarized in the Table II. Both devices exhibit a capacitance that varies with voltage. The quantum capacitance is extracted in each case using the analysis described in the manuscript. We did not observe a capacitance minimum in those devices. The quantum capacitance versus gate voltage curve for the 6QL and 14QL samples are shown in Fig. 7(a) and compared to that of the 8QL sample. The experimental error bars are quite large in the 6QL near V=0 since the top hBN used for this sample is quite thick (25nm) and has a small geometrical capacitance. This renders quantum capacitance measurements less precise.

This comparison allows us to get an idea about how the capacitance offset depends on thickness. Recall in Eq. ( 10), the offset is shown to depend on both 𝑐 𝑔 𝐵𝑢𝑙𝑘 and W the Fermi energy offset. 𝑐 𝑔 𝐵𝑢𝑙𝑘 is inversely proportional to the Bi2Se3 thickness. In Fig. 7(b) we compare the smallest quantum capacitance values measured for each sample. The capacitance minima are plotted versus sample thickness, and compared to the variation of Eq. ( 10) with sample thickness and W. The data suggests that varying thickness yields a changing 𝑐 𝑔 𝐵𝑢𝑙𝑘 and W. In the thicker sample, even though the bottom surface is further decoupled from the top gate, its influence on the quantum capacitance is also smaller. This is due to 𝑐 𝑔 𝐵𝑢𝑙𝑘 becoming smaller when the sample thickness increases (see Eq. ( 10)).

This agrees with the fact the capacitance offset observed in HgTe (~70nm) [21] is smaller than that measured in Bi2Se3 8QL.

While we cannot draw more conclusions from this analysis, we motive further work on the thickness dependence and the question of capacitive top-bottom coupling.

FIG 1 .

 1 FIG 1. (a) Linear charge (Q) vs voltage (V) characteristic of a metal-insulator-metal (MIM) device. The capacitance is constant in this case (b) Non-linear Q vs V curve characteristic of voltage dependent quantum capacitance in a MITI-CAP. The quantum capacitance is due to finite change in density (eΔn) and chemical potential at the surface of the TI. The arrow in C-V graph indicates the Dirac point.

  SiO2. A Z-contrast scanning electron microscope image shown in Fig. 2(d) confirms nucleation of Bi2Se3 on the hBN flake. The dark spots observed on the light grey flake indicate the presence of heavy atoms such as Bi or Se. This growth mechanism is consistent with previous reports on CVD synthesis of Bi-based TI on hBN. [26] [27]

FIG 2 .

 2 FIG 2. (Color Online) (a) Schematic of the CVD growth tube showing the Bi2Se3 source in hot zone A, and substrates in colder zone B, down stream in the Argon flow direction. (b) Optical microscopy image showing hBN exfoliated flakes on SiO2 prior to growth. (c) The same hBN flakes coated with 90QL of Bi2Se3 after the growth. (d) Z-contrast SEM image of Bi2Se3 growth (dark patches) on hBN flake. (e) Microscopic Raman spectroscopy of Bi2Se3 flakes of different thicknesses on hBN. The 2-4QL flakes are shown in (f). Three Raman active peaks are observed in (e) corresponding to the three vibrational modes shown in (g), namely, the 1 1g A and 2 1g A out-of-plane modes and the 2 g E in-plane mode.

Fig. 2 (FIG 3 .

 23 Fig. 2(e) shows Raman spectra obtained using an excitation wavelength of 532nm on Bi2Se3 flakes having thicknesses ranging from 2QL to 90QL. The thinnest flakes studied in Raman spectroscopy (2QL-4QL) all nucleate on the same BN flake, shown in Fig. 2(f). Three characteristic Raman active phonon peaks (Fig. 2(g)) are observed in Fig. 2(e) between 50 and 200 cm -1 , confirming the presence of a Bi2Se3 layer on the exfoliated hBN. A blueshift of the 2 1g A mode and a redshift of the 2 g E and 1 1g A modes are observed with decreasing thickness in agreement with previous reports on Raman spectroscopy on Bi2Se3. [26] [30]

FIG 4 .

 4 FIG 4. (Color Online) (a) Optical microscope image of the finished device with the capacitor shown embedded inside a coplanar waveguide (W). G and D denote the gate and drain contacts respectively. (b) Zoomed in microscope image showing the capacitor device. (c) Real (red) and Imaginary (blue) parts of the RF-admittance as a function of frequency for three typical gate voltages. (d) Schematic of a distributed RC-line model in series with a contact resistance. The capacitance is split into two components the geometric capacitance Cgeo and the quantum capacitance CQ. (e) Total device capacitance and channel resistance extracted by fitting the model corresponding to the circuit shown in (d) to the data in (c). Curve fits in (c) are shown as dashed lines.

  capacitance and 𝑐 𝑄 = 𝑒 2 𝜒 is quantum capacitance related to the compressibility 𝜒 = 𝜕𝑛 𝜕𝜀 𝐹 . Here 𝜀 𝐹 represents the chemical potential at the sample surface and n is a 2D carrier density. The curve fit of    total Y allows us to separate C and  for different gate voltages. As seen in Fig. 4(c), the model yields an excellent fit to the data up to 10 GHz. Results for C and R from curve fits up to 10 GHz are shown in Fig. 4(e)

  (b) along with the propagated uncertainty associated with it. We can now plot the quantum capacitance 𝑐 𝑄 versus the local chemical potential 𝜀 𝑓 . This is shown in Fig. 5(c). The minimum in 𝑐 𝑄 defines the chemical potential origin, as it is associated with the position of the Dirac point. This allows us to determine the Fermi energy at zero applied potential to be close to (200±40) meV above the Dirac point. This corresponds to a surface Dirac carrier density of (3±1)×10 12 cm -2 . Assuming the Dirac point occurs at 200meV below the bottom-most bulk conduction band, as seen in ARPES, [40] [23] this would corresponds to a Fermi level position of at most 40 meV above conduction band bottom of Bi2Se3

FIG 5 .

 5 FIG 5. (Color Online) (a) Quantum capacitance versus gate voltage at 10K. (b) Surface chemical potential (Fermi energy) versus gate voltage. (c) Quantum capacitance versus Fermi energy. Blue line represents data, grey lines show uncertainty extrema, red lines are linear fits using Eq. 5.

  𝑐 𝑄 𝐵𝑇𝑆𝑆 (𝜀 𝑓 ) is the bottom TSS Dirac quantum capacitance as a function of the top surface chemical potential. W is the band offset between the top and bottom TSS. It is the only adjustable parameter in the model. The W term allows the top and bottom TSS to have a different local chemical potential (Fig. 6(c)), as has been reported in previous studies. [57] [58]

FIG 6 .

 6 FIG 6. (Color Online) (a) Quantum capacitance versus surface chemical potential data (blue) compared to model of topological insulator shown in (b). Gray curves represent upper and lower bounds of experimental data, black solid is the total quantum capacitance resulting from this model, the dashed red line is that of the top TSS, the dashed magenta line is that of the bottom TSS. (b) MITI-CAP model of a topological insulator in the bulk depleted regime where the top TSS (𝑐 𝑄 𝑇𝑇𝑆𝑆 ) is

  quantum capacitance measures the changing local (top surface) depletion/accumulation profile [

FIG 7 . 2 .

 72 FIG 7. (a) Quantum capacitance versus gate voltage for the 3 Bi2Se3 samples. (b) Smallest cQ measured in each respective sample. A full square indicates the sample for which we observe the Dirac point feature. An empty square is used for the other. The thickness (horizontal) error bar accounts for sample roughness. Solid lines represent the variation of Eq. (9) versus thickness for different values of W.

Table I .

 I Comparison of the Bi2Se3 sample studied in this work to those previously reported.

		19	10	MBE
	Ref. [47]	7×10 19	10	MBE
	Ref. [44]	2×10 19	20	MBE
	Ref. [42]	2×10 19	8	MBE
	Ref. [52]	>10 19	10QL pristine	Exfoliated

  B4)𝑐 𝑇𝐺 is the geometric top gate capacitance, and 𝑛 𝑇/𝐵 is the carrier density of the top/bottom surface. 𝑉 𝐺 is the gate voltage. We can now proceed and compute the variation of the quantum capacitance as a function of the chemical potential of the top surface.By plugging into eq. (B4) the expression for 𝑉 𝑇 and 𝑉 𝐵 determined from (B2) and that of 𝑛 𝑇 and 𝑛 𝐵 from (B3) we get: This allows us to setup a second degree equation to extract 𝜇 𝐵 as a function of 𝜇 𝑇 4𝜋(ℏ𝑣 𝑓 )Finally, we have 𝜇 𝐵 as a function of 𝜇 𝑇 :We finally obtain the expression for the total quantum capacitance:Finally replacing 𝜇 𝑇 by 𝜀 𝐹 , to keep a coherent notation, we get:

		𝜇 𝐵 =	2𝜋(ℏ𝑣 𝑓 ) 2 𝑒 2	𝑐 𝑔 𝐵𝑢𝑙𝑘 (1 -√ 1 +	𝑒 2 (𝜇 𝑇 + 𝑊) 𝜋(ℏ𝑣 𝑓 ) 𝑐 𝑔 𝐵𝑢𝑙𝑘 2	)
	𝜇 𝐵 (𝜇 𝑇 ) = ±	2𝜋(ℏ𝑣 𝑓 ) 2 𝑒 2	𝑐 𝑔 𝐵𝑢𝑙𝑘 (1 -√ 1 +	𝑒 2 (𝜇 𝑇 + 𝑊) 𝜋(ℏ𝑣 𝑓 ) 𝑐 𝑔 𝐵𝑢𝑙𝑘 2	)
	This expression can then be used to find 𝑐 𝑄 𝐵𝑇𝑆𝑆 :
	𝑐 𝑄 𝐵𝑇𝑆𝑆 (𝜇 𝑇 ) = 𝑐 𝑔 𝐵𝑢𝑙𝑘 (-1 + √ 1 +	𝑒 2 (𝜇 𝑇 + 𝑊) 𝜋(ℏ𝑣 𝑓 ) 𝑐 𝑔 𝐵𝑢𝑙𝑘 2	) > 0
		𝑐 𝑄 (𝜇 𝑇 ) = 𝑐 𝑄 𝑇𝑇𝑆𝑆 (𝜇 𝑇 ) +	𝑐 𝑔 𝐵𝑢𝑙𝑘 𝑐 𝑄 𝐵𝑇𝑆𝑆 (𝜇 𝑇 ) 𝑐 𝑔 𝐵𝑢𝑙𝑘 + 𝑐 𝑄 𝐵𝑇𝑆𝑆 (𝜇 𝑇 )
	𝑐 𝑄 (𝜇 𝑇 ) = 𝑐 𝑄 𝑇𝑇𝑆𝑆 (𝜇 𝑇 ) + 𝑐 𝑔 𝐵𝑢𝑙𝑘	1 -	1
						[	( √ 1 +	𝑒 2 (𝜇 𝑇 + 𝑊) 𝜋(ℏ𝑣 𝑓 ) 2 𝑐 𝑔 𝐵𝑢𝑙𝑘	)	]
	𝑐 𝑄 (𝜇 𝑇 ) =	𝑐 𝑔 𝐵𝑢𝑙𝑘 (𝜇 𝐵 -𝜇 𝑇 -𝑊) = -𝑒 2 𝑠𝑖𝑔𝑛(𝜇 𝐵 ) 𝑒 2 |𝜀 𝐹 | [ ( √ 1 + 𝜋(ℏ𝑣 𝑓 ) 4𝜋(ℏ𝑣 𝑓 ) 𝜇 𝐵 2 2 𝑐 𝑔 𝐵𝑢𝑙𝑘 ) 𝑒 2 |𝜀 𝐹 + 𝑊| 2𝜋(ℏ𝑣 𝑓 ) 2 + 𝑐 𝑔 𝐵𝑢𝑙𝑘 1 -1	]	(𝐵5)
				2
		𝑒 2		𝑐 𝑔 𝐵𝑢𝑙𝑘 (𝜇 𝐵 -𝜇 𝑇 -𝑊) + 𝑠𝑖𝑔𝑛(𝜇 𝐵 )𝜇 𝐵 2 = 0
	We get four solutions, two of which are inconsistent with the sign of 𝜇 𝐵 :
	For 𝜇 𝐵 > 0:
		𝜇 𝐵 =	2𝜋(ℏ𝑣 𝑓 ) 2 𝑒 2	𝑐 𝑔 𝐵𝑢𝑙𝑘 (-1 ± √ 1 +	𝑒 2 (𝜇 𝑇 + 𝑊) 𝜋(ℏ𝑣 𝑓 ) 𝑐 𝑔 𝐵𝑢𝑙𝑘 2	)
	Only, the solution with the + sign satisfies 𝜇 𝐵 > 0.
	Similarly for 𝜇 𝐵 < 0, we get only one satisfactory solution:

2

Acknowledgements. We acknowledge Michael Rosticher's valuable assistance with cleanroom equipment and the Pascal Morfin for his assistance with growth equipment setup. We also acknowledge discussions with H. Graef. BAA and JD acknowledge funding from ANR-LabEx grant ENS-ICFP ANR-10-LABX-0010/ANR-10-IDEX-0001-02PSL. JD is partially supported by the Northeastern University Coop fellowship. The scanning electron microscope (SEM) facility of the Institut de Minéralogie et de Physique des Milieux Condensés is supported by Région Ile-de-France grant SESAME 2006 N°I-07-593/R, INSU-CNRS, INP-CNRS, University Pierre et Marie Curie -Paris 6, and by the French National Research Agency (ANR) grant no. ANR-07-BLAN-0124-01. Growth of hexagonal boron nitride crystals was supported by the Elemental Strategy Initiative conducted by the MEXT, Japan and JSPS KAKENHI Grant Numbers JP15K21722.