Introduction

In recent years, the discovery of a new class of topological states of matter, known as time-reversal invariant Z2 topological insulators (TIs), has generated considerable excitement in condensed matter physics. These materials are characterized by a bulk insulating energy gap and gapless edges or surface states which are topologically protected because an electron's spin is locked to its momentum due to a strong spin-orbit interaction1,2,3,4,5,6,7,8,9,10,11,12,13,14. Theory predicts that the unique surface states hosting Dirac fermions are robust and immune to defects, non-magnetic impurities and other perturbing influences in their environment, leading to potential non-dissipative applications3,14,15,16. To experimentally determine the macroscopic properties of the Dirac fermions and intentionally gain a control of the helical states, substantial efforts have been made in improving material quality using topological thin films17,18,19,20,21,22, bulk crystals5,13,23 and nanostructures24,25,26 and in developing sensitive approaches for revealing surface transport and spin textures using angle-resolved photoemission (ARPES)6,13,17,21,27,28, scanning tunneling microscopy (STM)8,29, low-temperature transport5,23,30,31,32,33 and optical polarizations10. While significant progress has been achieved in identifying Dirac fermions via Shubnikov-de Haas (SdH) oscillations5,31 and ambipolar field effects30 at low temperatures, experimental realization of exotic new physical phenomena arising from the surface states has been hindered by parallel bulk conductions due to naturally occurring defects and residual carrier doping13,21,28. Thus, developing new methods of separating the contributions from the bulk and surface states becomes one of the most important tasks in the field, particularly in the high temperature regime where bulk carriers are activated25.

Quantum capacitance has advantages over traditional transport measurements because it allows us to directly probe the surface density of states (DOS), whereas the latter are more complicated and sensitive to scattering details34. Quantum capacitance is described by the equation CQ = e2D(E), where D(E) represents the DOS34,35,36. Thus, a simple measurement of quantum capacitance provides a quantitative description of the DOS at the Fermi energy34,36,37,38,39,40,41,42,43. Despite its capability of directly probing electronic compressibility at high temperatures34, this concept has never been applied to TIs, which is in part attributed to the lack of high-quality crystals, on which a large atomically flat surface is required for detectable quantum capacitance. As the material quality has been progressively improved recently5,17,18,21,22,23,24,30,31,32,44, the quantum capacitance measurements become especially important as they can potentially solve today's intriguing challenge of high-temperature surface-state detection and may subsequently lead to the exploration of quantized anomalous and quantum spin Hall effects1,14,15,16,45.

Here, we report on an observation of quantum capacitance using a topological capacitor device fabricated on Bi2Se3 thin films. Under external magnetic fields, by varying gate voltages, we observed pronounced capacitance oscillations associated with the creation of quantized Landau levels. Remarkably, by changing the ac frequency during the measurements, the amplitudes of the oscillations can be systematically tuned. As a result, high-frequency excitations are observed to sufficiently resolve the surface states up to 60 K. Our theoretical simulations further suggest that the different frequency responses of carriers from the surface and bulk states may account for the distinction of the surface states at high frequencies.

Results

Growth of Bi2Se3 thin films

Bi2Se3 thin films were grown on Si (111) substrates using an ultra-high vacuum molecular-beam epitaxy (MBE) system. Figure 1a presents a large-scale atomic force microscopy (AFM) image of an as-grown Bi2Se3 film with a thickness of 10 nm, exhibiting terraces over 500 nm in size. The surface consists of triangle-shaped terraces and steps (~1 nm in height, Supplementary Fig. S1), indicative of a hexagonal crystal structure along the (111) direction18. Real-time reflection high-energy electron diffraction (RHEED) was used to monitor the in-situ growth dynamics with the electron beam incident to the direction. The sharp streaky lines indicate a 2D growth mode and a flat surface morphology (Fig. 1b). To verify the existence of the surface states, high-resolution ARPES experiments along the direction were performed at 64 K (Fig. 1c). A nearly linear E-k dispersion is observed near the Dirac point. The inset presents a curvature analysis of the raw data for a clear visualization of the band dispersion. The Fermi level is located ~0.4 eV above the Dirac cone, suggesting a very high bulk concentration. Such a large binding energy of the Dirac point in the Bi2Se3 thin film is believed to originate from the creation of Se vacancies by Se out-diffusion22,31,46 during the pre-annealing process, as the film was annealed at 200°C for 2 hours before the ARPES experiments. Other widely observed effects such as band bending22,31,47 and photoemission-induced gating48 may not play significant roles in our film because of the large bulk density after annealing. The surface states must also be examined via other techniques such as low-temperature transport and STM, where a heating process and period can be minimized during sample preparation. As anticipated, the transport experiments exhibit clear SdH oscillations (Supplementary Fig. S2). Unlike the ARPES results, the Fermi level was estimated to be 53±5 meV above the Dirac point, consistent with the value of 70±20 meV obtained from the STM experiments (Supplementary Fig. S3). The location of the Fermi level relative to the Dirac point can be further verified using the quantum capacitance measurements, as discussed later.

Figure 1
figure 1

Bi2Se3 growth, an ARPES spectrum and a topological capacitor device.

(a) An AFM image of an MBE-grown Bi2Se3 thin film with terrace size exceeding 500 nm. (b) A streaky RHEED pattern along the direction of the as-grown surface of Bi2Se3, indicating a 2-D growth mode. (c) An ARPES spectrum of the film along the direction at T = 64 K. The inset shows the curvature of the raw data for better visualization of the band dispersion. The Fermi Level lies ~0.4 eV above the Dirac point. (d) A capacitor device consisting of a 10 nm-thick Bi2Se3 thin film as the channel layer. The source-drain contacts were composed of Fe/Al (20 nm/100 nm), while the gate electrode consists of Ti/Au, 10 nm/100 nm. (e) An optical microscope image of the capacitor device structure. The effective area of the capacitor approaches 1600 μm2, which enables a precise detection of quantum capacitance from the Bi2Se3 top surface.

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Capacitor device structure

To perform accurate capacitance measurements, an appropriate design of the device structure is essential. Capacitance measurements are extremely intricate because they involve multiple components that respond to different frequencies. To minimize the undesired components, the structure of the devices must be as simple as possible. Figure 1d presents a schematic drawing of a capacitor using a 10 nm-thick Bi2Se3 thin film as the channel layer. A metal stack of Fe/Al (20/100 nm) is deposited on the top surface of the Bi2Se3 thin film to act as the source and drain contacts. A thin layer of Al2O3 (15 nm) is used as the gate oxide.

The total gate capacitance (CT) can be modeled as a series of oxide capacitance (COX), quantum capacitance (CQ), depletion/accumulation capacitances (CD/A) and bulk capacitance (CB), that is, 1/CT = 1/COX+1/CQ+1/CD/A+1/CB. Note that there are other parasitic capacitances associated with the source and drain; however, by performing a measurement on a device with identical geometry but without the topological insulator channel, these parasitic capacitances are negligible compared with the top-gate capacitance. To a first approximation, we may thus neglect the effect due to the parasitic capacitances. The total gate capacitance is primarily dominated by the smallest capacitance component in the system, i.e., the smallest magnitude among COX, CQ, CD/A and CB. To gain further insight into the underlying physics, the concepts of CQ and CD/A must be clarified. CQ represents the quantum capacitance associated with the topological surface DOS, while CD/A describes a depletion (CD) or an accumulation capacitance (CA) during device operation. Under a depletion mode, electrons are depleted within a depth of w. Depending on the applied voltage, the depletion capacitance CD changes its magnitude owing to the dynamic variation of w (ref. 49). Thus, in this scenario, a direct comparison of the magnitude of the different capacitive components is challenging without constructing numerical simulation models. However, in contrast, under an accumulation mode, the capacitance CA becomes much larger than the other components. In particular, as the gate bias increases, a steady rise of the carrier density produces a large CA according to the equation (refs. 37,49,50), where εTI is the dielectric constant of Bi2Se3, λn is the extrinsic Debye length, Ns is the carrier density in the inversion layer and ND is the density of ionized donors. Consequently, as the device enters a saturation regime, the total capacitance can be simplified as 1/CT = 1/COX+1/CQ+1/CB with the bulk capacitance CB being an unknown parameter. By systematically increasing the measurement frequency, the CB component can be easily eliminated based upon different frequency response of the surface and bulk carriers49,51,52,53,54, as elaborated in the Simulation and Discussion section. Therefore, the extraction of CQ becomes immediately feasible under the accumulation mode, where the oxide capacitance can be calculated by COX = εrε0/d (d is the thickness of Al2O3).

Figure 1e shows an optical microscope image of the capacitor device structure. We performed the Hall and capacitance measurements with a Hall bar geometry. The effective area of the capacitor is measured to be approximately 1600 μm2. The large area ensures a high signal-to-noise ratio and a good estimation of the quantum capacitance from the Bi2Se3 top surface. The cross-section of the practical Hall bar devices was examined using a high-resolution transmission electron microscopy (HRTEM). As illustrated in Fig. 2b, the device consists of 15 nm of Al2O3 and 10 nm of Bi2Se3. The interfaces at the top and bottom of the Bi2Se3 thin film both have atomically sharp heterojunctions (Fig. 2a and 2c). The lattice spacing is observed to be 0.96 nm, consistent with other reports18,24.

Figure 2
figure 2

Cross-section TEM images of a Bi2Se3 capacitor device.

(a) A high-resolution TEM picture of the top surface of Bi2Se3 with a thickness of approximately10 nm. Al2O3–Bi2Se3 has an atomically sharp interface, which helps to preserve the topological states. The lattice spacing between the (0003) planes is measured to be 0.96 nm. (b) A low-magnification TEM image of the entire cross-section structure of the Bi2Se3 capacitor device. A thin and high-quality Al2O3 film of 15 nm is used to increase the oxide capacitance, making it possible to observe quantum capacitance from the Bi2Se3 top surface. (c) A high-resolution TEM for the bottom interface between Bi2Se3 and Si. An atomically sharp interface is also observed.

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Quantum capacitance in Bi2Se3 thin films

Figure 3 plots the total capacitance as a function of gate voltage, magnetic field and measurement frequency in a Bi2Se3 capacitor at 25 K. In general, two regimes are observed for the gate voltage ranging from 0 to 12 V. In regime I, the total capacitance is dominated by the depletion capacitance; however, the quantum capacitance cannot be extracted because of the uncertainty of the depletion component49. When the gate voltage further increases, the total capacitance tends to saturate (Fig. 3a and 3c), the quantum capacitance not only contributes a small correction to the total gate capacitance but also begins to dominate over the other components in this regime. The system can be then simplified as 1/CT = 1/COX+1/CQ (the term CB is eliminated at high frequencies as shown in Fig. 4); here, the oxide capacitance is calculated to be 0.46 μF/cm2 by taking d = 15 nm and εr = 7.8 (extracted from the control sample, Supplementary Fig. S4).

Figure 3
figure 3

Extraction of quantum capacitance in Bi2Se3 capacitors.

(a) The total capacitance as a function of gate voltage at T = 25 K, B = 0 T and f = 100 KHz. The capacitance values in regimes I (yellow) and II (light purple) are dominated by the bulk and the TI top surface, respectively. The continuing increase in the total capacitance in regime II suggests the presence of quantum capacitance (CQ), which can be directly extracted based on a series capacitor model where 1/CT = 1/COX+1/CQ. The resultant CQ is shown in the inset. (b) The total capacitance as a function of gate voltage at T = 25 K, B = 7 T and f = 100 KHz. Capacitance oscillations were observed under an external magnetic field of 7 T, suggesting the creation of Landau levels from the surface states. The inset presents the extracted oscillatory CQ. (c) The total capacitance as a function of gate voltage at T = 25 K, B = 7 T and f = 400 Hz. As compared with (b) under low-frequency excitation, the quantum oscillations tend to diminish. These results indicate that detection of the surface states can be achieved by applying high frequencies during the C-V measurements.

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Figure 4
figure 4

Quantitative analysis of quantum capacitance in Bi2Se3.

(a, b) Magnetic-field- and frequency-dependent capacitance at T = 25 K and f = 100 KHz. (e, f) and (i, j) The same measurements when the temperature is increased to 35 and 50 K, respectively. CQ demonstrates a strong frequency dependence at all three temperatures. (c) Temperature-dependent capacitance at B = 7 T and f = 100 KHz. Three oscillation peaks are clearly identified, denoted as P1, P2 and P3. (d) The first derivative of the total capacitance as a function of temperature. P1, P2 and P3 correspond to the three valleys of the peaks because of a 90° phase shift compared with those in (c). (g) Extraction of the cyclotron effective mass by taking the temperature-dependent oscillation amplitude in (d). Three effective masses for peaks P1–P3 were obtained in the inset. (h) Fermi vector kF and Fermi energy EF as a function of gate voltage. The inset describes the calculated positions of the Fermi level relative to the Dirac point. (k) The total capacitance as a function of the measurement angles. The inset depicts the measurement setup, in which the device was rotated from 0 to 50°. (i) The first derivative of the total capacitance as a function of the rotation angle. The traces of arrows (in purple) and dots (in green) suggest the change of the oscillation frequencies. The curves in all the figures except for (g), (h) and (k) are vertically shifted for clarity.

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For a traditional MOS capacitor, under the accumulation mode, the capacitance should saturate at a constant value corresponding to the oxide capacitance (Supplementary Fig. S4, a regular MOS capacitor without Bi2Se3). For a Bi2Se3 capacitor, however, the total capacitance increases continuously as the gate scans from 6 to 12 V (Fig. 3a). The gradual increase of the total capacitance must originate from an additional capacitive component (Supplementary Fig. S5), which could be attributed to the quantum capacitance of the surface states. To verify this presumption, we applied a magnetic field of 7 T perpendicular to the sample surface to observe the possible quantization of Landau levels. Indeed, significant capacitance oscillations were resolved (Fig. 3b), suggesting that the quantum capacitance becomes dominant and changes significantly with varying gate voltage and magnetic field34. It is speculated that during the capacitance-voltage (C-V) measurements, the Fermi level is shifted because of the gating effect33: as it moves across each discrete Landau level, the quantum capacitance becomes oscillatory (Fig. 3b inset, CQ vs. Vg). This analysis is essentially the same as for the case of SdH oscillations (Rxx vs. B, where Rxx represents longitudinal resistance)42,55 and the capacitance results are nearly identical to the transport measurements but with a much higher temperature. By subtracting the oxide capacitance, CQ from the topological surface is retrieved, as depicted in all three insets of Fig. 3. The obtained CQ is on the order of 2 μF/cm2, reasonably near the theoretical calculation (Supplementary Fig. S5). To explore the effect of frequency on the surface states, the excitation frequency was reduced from 100 KHz to 400 Hz. The quantum oscillations, however, diminished rapidly, exhibiting a noisy spectrum without evident traces of oscillations (Fig. 3c inset). These observations imply that the detection of the surface states can be achieved by applying high frequency excitations during the C-V measurements.

Figure 4 shows a comprehensive analysis of the quantum capacitance in Bi2Se3 for different experimental settings. Figure 4a demonstrates the magnetic field dependence of the total capacitance at 100 KHz and 25 K. When the magnetic field increases from 3 to 9 T, the low-index Landau levels with large interleave spacing progressively move across the Fermi level, reminiscent of typical SdH oscillations from transport measurements42,55 (see also Supplementary Fig. S2). Figure 4b demonstrates a clear trend of frequency-dependent quantum oscillations, where high frequency excitations resolve the surface oscillations quite well. Figures 4(e,f) and 4(i,j) present similar results but at higher temperatures of 35 and 50 K, respectively. Note that the frequency can have a significant effect on the distinction of the surface from the bulk49,51,52,53,54, presumably because of the dramatic difference in carrier mobility: in the bulk, the electron mobility is approximately 380 cm2/V s, while on the surface the electron mobility reaches over 10,000 cm2/V s (Supplementary Figs. S2 and S6). Such a large difference enables a fast response from the surface states, analogous to the fast response of high-mobility electrons in graphene (in a gigahertz range)56; as a comparison, the bulk carriers cannot follow the high frequency excitations, resulting in a “filtering” effect49,51. More importantly, the surface states can be clearly distinguished at approximately 60 K (Fig. 4c and Supplementary Fig. S7), much beyond the capability of the low-temperature transport, where movable carriers travel along horizontal channels and are extremely sensitive to bulk properties (scattering details)34. In addition, quantum capacitance probes both the extended and localized states, whereas electron transport can only investigate the former one34.

To better understand the physics behind this phenomenon, we performed a quantitative analysis of the SdH oscillations. Figure 4c demonstrates the temperature dependence of the total capacitance when the magnetic field and frequency are set to 7 T and 100 KHz, respectively. The surface states are persistent up to 60 K. Three peaks are identified, corresponding to the generation of three Landau levels in the system. The first derivative of the total capacitance was also taken to remove the background parasitic capacitances (Fig. 4d). The three peaks from P1 to P3 have a common 90° phase shift compared with those in Fig. 4c. Now, we can take the amplitude of the capacitance oscillations − note that the Al2O3 capacitance COX is removed − and apply , where the thermal factor is given by (refs. 5,23). Here, mcycl denotes the cyclotron mass, kB is Boltzmann's constant and is the reduced Plank's constant. By taking the conductivity oscillation amplitude and performing the best fit to the equation of ΔC(T)/ΔC(0) (Fig. 4g), mcycl is calculated to be 0.079 m0, 0.091 m0 and 0.102 m0 at gate voltages of 5.9 V (P1), 8 V (P2) and 10.9 V (P3), respectively (Fig. 4g (inset)). These values are reasonably close to those extracted from the transport measurements (0.07 m0, Supplementary Fig. S2). Here, m0 is the electron rest mass. Because the cyclotron mass is given by (ref. 5), where EF and VF are the Fermi level and the Fermi velocity, both EF and VF can be precisely obtained given (ref. 23). It is noted that the Fermi level increases from 61.5 to 79.4 meV when the gate voltage is varied from +5.9 to +10.9 V (Fig. 4h), exhibiting the clear gate modulation behavior of the surface states. The kF value can also be obtained in the range of 0.025–0.033 Å−1, consistent with the transport results (Supplementary Fig. S2).

We performed a series of sample rotations under a constant magnetic field of 7 T to identify the nature of the quantum oscillations (Fig. 4k and 4i)5,23. As the system is rotated from 0 to 50°, the amplitude of the oscillations dampens and the oscillation frequency (peak interval) changes, depending on the magnitude of (Fig. 4k inset), which signifies the typical 2-D characteristics of the topological surface states5. The relationship between the carrier density n and the Fermi vector kF is the same for both quadratic (2-DEG) and linear (TI surface state) E-k dispersion relations and is given by (refs. 39,57). Accordingly, the period of SdH oscillations for both 2-DEG and TI surface states also exhibit similar behavior given the Onsager relation of (ref. 5). Therefore, it is difficult to distinguish the surface states from 2-DEG through the conventional magneto-transport measurements, which can only probe the carriers that transport through the channel. However, unlike the transport approach, the quantum capacitance method can circumvent this challenge by providing a direct extraction of an important quantity, dn/dVa (ref. 39, Va − the voltage dropped on the TI surface), which explicitly describes the surface states. By carefully examining their E-k relations and DOS39,57, i.e., & for the surface states, & for 2-DEG, one can theoretically derive and simulate their capacitances, as demonstrated in Supplementary Figs. S5 and S8. Here, g = gsgv is the degeneracy factor, considering both the spin and energy band degeneracies for the TI surface states. The quantum capacitance associated with the surface states strongly depends on the density of states, similar to that of graphene34,36,37. By contrast, for 2-DEG the capacitance is saturated at a constant value irrespective of the change in Va(Supplementary Fig. S8). Therefore, the observed quantum oscillations in Figs. 3 and 4 are attributed to the surface states instead of 2-DEG, which is also verified by an ambipolar field effect in quantum capacitance (Supplementary Fig. S11).

Discussion

To understand the frequency dependence of quantum oscillations, we carried out qualitative simulations using Technology Computer Aided Design (TCAD). A capacitor device was constructed by considering the experimental conditions. Figure 5 shows that the total capacitance changes with the frequency of the ac signals. In essence, the capacitance for the system depends on the total available charges under specific measurement frequencies49,51. In the low-frequency regime (f < 1 KHz, Fig. 5), both the bulk and surface carriers can respond to the external ac signals and the system can be represented by a series capacitor model (1/CT = 1/COX+1/CQ+1/CB). The bulk interference exists in the system and the quantum capacitance can hardly make itself distinguishable. In the high-frequency regime (f > 1 KHz, Fig. 5), however, the available charges accounting for the total capacitance primarily originate from the surface states because the bulk carriers do not have adequate time to respond to the high ac frequencies51,52,53,54. Thus, the elimination of the bulk term leads to a significant manifestation of the surface quantum capacitance (1/CT = 1/COX+1/CQ). By taking two different surface carrier densities (Fig. 5, red and blue curves), we can verify the effect of the available surface charges on the total capacitance. In short, our simulations provide sensible explanations for the observed frequency dispersion of the total capacitance, although a detailed theoretical treatise is still needed. The discovery of the quantum capacitance in topological insulators may facilitate future progress toward the creation of topological devices by probing high-temperature surface states and incorporating gated structures.

Figure 5
figure 5

Simulated frequency-dependent total capacitance.

A device model from the inset is constructed in accordance with the experimental setup (Figs. 1d and 1e). During simulations, only the top surface of Bi2Se3 is considered. At low frequencies, the total capacitance exhibits a strong dependence on frequency, primarily attributed to the sufficient responses for the bulk and the surface carriers. When the frequency is set beyond 1 KHz, however, a desirable surface-dominant regime is attained, in which the bulk interference is reduced at elevated frequencies. The inset depicts the device structure used for the simulations.

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Methods

MBE growth

Thin film growth was carried out with an ultra-high vacuum Perkin Elmer MBE system. Si (111) substrates were cleaned using the standard Radio Corporation of America (RCA) procedure before being transferred into the growth chamber. High-purity Bi (99.9999%) and Se (99.99%) elementary sources were evaporated by conventional effusion cells. During growth, the Bi and Se cells were kept at 490 and 200°C, respectively, while the Si (111) substrate was kept at 150°C (growth temperature). The epitaxial growth was monitored using the in-situ RHEED technique and the surface was observed to be atomically flat, as evidenced by the streaky RHEED patterns (Fig. 1a). Digital images of the RHEED were captured using a KSA400 system made by K-space Associate, Inc.

Characterizations

(1) ARPES. High-resolution ARPES experiments were performed at beam line 12.0.1 of the Advanced Light Source at Lawrence Berkeley National Laboratory, using 100 eV and 52 eV photons to measure the core levels and surface states, respectively. (2) TEM. High-resolution TEM experiments were performed on a Philips Tecnai F20 (S)TEM operating at 200 KV. The digital images were recorded by a Gatan® 2k×2k CCD camera. (3) Capacitance measurements. Frequency-dependent capacitance measurements were conducted with an HP/Agilent 4274A multi-frequency LCR meter. The ac frequency could be tuned from 100 Hz–100 KHz, while a dc bias can be applied from 0 to ± 35 V. The LCR meter was connected to a physical property measurement system (PPMS), where the devices were located. We were also able to systematically vary several experimental variables such as the temperature, magnetic field, measurement frequency and external gate bias. Multiple lock-in-amplifiers and Keithley source meters were also connected to the PPMS system, enabling comprehensive and high-sensitivity transport measurements for the top-gated Hall bar and capacitor devices.

Device fabrication

The MBE-grown Bi2Se3thin film (~10 nm) was patterned into a micron-scale Hall bar geometry using conventional optical photolithography and a subsequent CHF3dry-etching of 18 s. A 15-nm-thick, high-κ Al2O3 dielectric layer was deposited using atomic layer deposition (ALD). Hall channel contacts were defined by e-beam evaporation after etching away Al2O3 in the contact areas. A metal stack of Fe/Al (20 nm/100 nm) was directly deposited onto the exposed Bi2Se3 surface. A top-gate metal scheme of Ti/Au (10 nm/90 nm) was achieved using a second step of photolithography and e-beam evaporation.

Device simulation

We used a visual TCAD software package (version 1.7.3) from Cogenda.com. The simulated device structure is depicted in Fig. 5 inset. The topological surface is assumed to have a depth of 1 nm and carrier densities of 1×1018 cm−3 and 5×1018 cm−3 (two scenarios). Bulk Bi2Se3 has a carrier density of 6.5×1019–1.3×1020 cm−3, attributed to the formation of an impurity band as elaborated in Supplementary Fig. S6. Our simplified model describes the physics of frequency-dependent capacitance in a qualitative manner. The absolute values of the total capacitance can be varied by altering the above parameters; however, the underlying physics remains the same, which forms the basis of our simulations.