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RF-quantum capacitance of the topological insulator Bi2Se3 in the bulk depleted regime for field-effect transistors

A Metal-dielectric-topological insulator capacitor device based on hBN-encapsulated CVD grown Bi2Se3 is realized and investigated in the radio frequency regime. The RF quantum capacitance and device resistance are extracted for frequencies as a high as 10 GHz, and studied as a function of the applied gate voltage. The combination of the superior quality hBN dielectric gate with the optimized transport characteristics of CVD grown Bi2Se3 (n~10^18cm-3 in 8 nm) allow us to attain a bulk depleted regime by dielectric gating. A quantum capacitance minimum is observed revealing a purely Dirac regime, where the Dirac surface state in proximity to the gate reaches charge neutrality, but the bottom surface Dirac cone remains charged, and couples capacitively to the top surface via the insulating bulk. Our work paves the way towards implementation of topological materials in RF devices.


Introduction
Topological phases of matter have emerged as a fundamental 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 bilayer 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] [22] 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 metalinsulator-metal capacitors ( Fig. 1(a)), in metal-insulator-graphene and metal-insulator-topological 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 n-doping renders a reliable detection of surface-state signatures difficult. [23] [24] [25] The first step in realizing an RFtransport 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 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 SiO2.
A Z-contrast scanning electron microscope image shown in Fig. 2   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 RF-waveguide 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. [  , similar to what was previously reported for graphene [17]. The total admittance total Y is given by: Where, the device length and width, respectively.  is the channel conductivity. C is the total device capacitance: Where geo c is the geometric capacitance and = 2 is quantum capacitance related to the compressibility = .
Here represents the chemical potential at the sample surface and n is a 2D carrier density.

V. Analysis of the quantum capacitance of top surface states
We next focus on the analysis of quantum capacitance. can be extracted using Eq. (3), by fixing = (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 is then determined by dividing the capacitance by the geometrical factors L and w. is shown in Fig. 5 (a). The grey lines in Fig. 5(a) delimit the propagated uncertainty on due to the uncertainty associated with . A powerful consequence of our measurements is the fact that it allows us to determine the local (top surface) chemical potential directly from experimental data via the Berglund integral, without a priori knowledge of the band structure. The Berglund integral is written as [39]: We use Eq. (4) to extract which we now plot as a function of Vg in Fig. 5(b) along with the propagated uncertainty associated with it. We can now plot the quantum capacitance versus the local chemical potential . This is shown in Fig. 5  We can additionally perform a linear fit to the versus 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, from the compressibility χ, which is related to : We find a Dirac velocity equal to ( Here, denotes the chemical potential of the top surface, is top surface quantum capacitance and is that of the bottom metallic surface. The geometric capacitance of the insulating bulk with a thickness 2 3 ≈ 8 is given by: When ( = 0)=0, an offset equal to (30 ± 5) fF/μm 2 has to result from the other term in Eq. 7).

VII. 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. (7).
The quantum capacitance of the top TSS is computed using Eq. (5), with ≈ 5 × 10 5 / . [25] Note that in Eq. (7) is the top surface chemical potential. Hence, the expression for that of the bottom TSS ( )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: The contribution of is shown in red in Fig. 6(a). The evolution of the effective capacitance of the bottom TSS in series with the insulating bulk as a function of top-surface chemical potential is shown in dashed magenta. For W ≈ 140 meV, the Eq. (9) 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 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.

VIII. THE BEHVIOR OF THE RESISTANCE CHANNEL
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.

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 2 : Here n2D is the surface carrier density, ≡ is the top surface chemical potential and ( )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. induced by the applied surface electrical potential 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: We can write: Again assuming V and tend to 0 at ∞, and we get: With ( = 0) by definition equal to 0 .
Then, the term in the quantum capacitance can be written as: Finally, combing this result with Eq. (A2) we get the quantum capacitance: From Gauss' law, we get: Integrating from 0 to infinity and assuming E tends to zero at infinity, yields: Here, the charge distribution is assumed to follow an exponential decay into the sample: where n0 is the charge at z=0 and λ is an effective screening length.
We get: According to (A3), we thus have: Note that here 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: Here is the bulk geometric capacitance, and ≡ 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: is the Fermi velocity.
The expression for the quantum capacitance of single Dirac cone at the bottom surface, is more challenging to extract: ( ) = 2 | ( )| 2 (ℏ ) 2 A knowledge of ( ) is required. This can be modelled by carefully studying electrochemical equilibrium in a system of two Dirac fluids in parallel, coupled by an insulating capacitive layer.
The electrochemical potential * imposed by the metallic contact ensures equilibrium and allows us to write: Here / and / 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 2 as a function of chemical potential for a non-spin degenerate Dirac surface states can be written as: The charge density is simply given by = − .
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: is the geometric top gate capacitance, and / is the carrier density of the top/bottom surface. 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 and determined from (B2) and that of and from (B3) we get: 2 We then set up a second degree equation to extract as a function of 4 (ℏ ) Only, the solution with the + sign satisfies > 0. Similarly for < 0, we get only one satisfactory solution: (1 − √ 1 + 2 ( + ) Finally, we have as a function of : This expression can then be used to find : We finally obtain the expression for the total quantum capacitance: Finally replacing by , to keep a coherent notation, we get: 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.

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. The quantum capacitance measurement is thus less precise in this sample.
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 and W the Fermi energy offset. 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 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 becoming smaller when the sample thickness increases (see Eq. (10)). This agrees with the fact the capacitance offset observed in HgTe (approximately 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.