Tuning thermoelectricity in a Bi₂Se₃ topological insulator via varied film thickness

The solid-state thermoelectric technologies, which enable the direct conversion of thermal and electrical energies with the advantages of being quiet, reliable, scalable and eco-friendly, are expected to play an important role in meeting the global energy and environmental challenges [1–4]. Widespread applications of thermoelectricity require the improvement in the efficiency of the thermoelectric devices. The performance of a thermoelectric material can be characterized by a dimensionless figure of merit defined as ZT = S²σT/κ, where S, σ, T and κ are the Seebeck coefficient (thermopower), electrical conductivity, absolute temperature and thermal conductivity, respectively. The quantity S²σ is called the power factor, which characterizes the ability of electrons in the thermoelectric material for energy output under a given temperature difference. The main approaches for enhancing ZT are either by maximizing the power factor and/or minimizing the thermal conductivity [5].

Quantum confinement introduced by low-dimensionality was proposed to be an efficient method for enhancing the power factor and suppressing the phonon thermal conductivity simultaneously [2, 6, 7]. It was predicted that in Bi₂Te₃ thin films, ZT can be significantly enhanced by reducing the film thickness [6]. In recent years, it was discovered that the best room temperature thermoelectric materials such as Bi₂Se₃, Bi₂Te₃ and Sb₂Te₃ are three-dimensional (3D) topological insulators (TI) with inverted bulk band structure and topologically protected metallic surface states [8–10]. Therefore, the topological surface states must be taken into account when considering the thermoelectric properties of TIs, especially in the thin limit. It has been proposed theoretically that in TI thin films, the thermoelectric contributions from the parallel bulk and surface channels tend to cancel each other because the bulk and surface ZTs are optimized at different Fermi levels (E_Fs). As the film becomes thinner, ZT would not be enhanced dramatically, but a decent increase could still be observed when the E_F lies in an optimal position [11].

The thermoelectric properties of Bi₂Te₃ family TI thin films have been experimentally investigated recently. By using the band structure engineering method to fine tune the E_F of TI, it was found that the thermopowers of (Bi_1−xSb_x)₂Te₃ ternary thin films do not show any improvement from the bulk value over a wide range of E_F positions. Instead, it exhibits a peculiar sign anomaly with the Hall effect induced by the drastically different transport properties of the bulk and surface states [12]. In a more recent study of p-type Sb₂Te₃ TI thin films [13], it was found that tuning the E_F position and film thickness has a significant impact on the total thermopower, also demonstrating the effect of topological surface states on the thermoelectric properties of TI.

In this work, we investigate the thermoelectric properties of the n-type Bi₂Se₃ TI thin films with different thicknesses grown by molecular beam epitaxy (MBE). As the film thickness is reduced from d = 30 QL (quintuple layer) to 5 QL, the thermopower as well as the power factor decreases systematically. This is mainly because the linear electron band dispersion of the Dirac-like surface state is not optimal for thermopower, which prefers an abrupt change of electron density of states (DOS) near the E_F [14]. With decreasing thickness, the growing contribution of the surface state reduces the total thermopower and the power factor. Another effect is the change of E_F position with film thickness, which mainly affects the bulk thermopower and the relative contribution of the surface states. These results can be explained quantitatively by theoretical calculations based on the realistic electronic band structures of Bi₂Se₃ TI thin films, and shed new light on future improvement of the thermoelectric performance of TI.

• A.  
  MBE sample growth: The Bi₂Se₃ thin films are grown on insulating SrTiO₃ (111) substrates (2 mm × 8 mm × 0.25 mm) in an ultra-high vacuum MBE system with a base pressure lower than 1 × 10⁻¹⁰ mbar. Before the sample growth, the SrTiO₃ substrates are degassed at 300 °C for 30 min. High-purity Bi (99.999%) and Se (99.999%) sources are evaporated from standard effusion cells. To reduce the number of Se vacancies, the growth is kept in an Se-rich condition with the substrate temperature kept at 180 °C.
• B.  
  ARPES measurements: The in situ angle-resolved photoemission spectroscopy (ARPES) measurements are carried out at 77 K by using a Scienta R4000 electron energy analyzer. A Helium discharge lamp with a photon energy of hν = 21.2 eV is used as the photon source. The energy resolution of the electron is 10 meV, and the angle resolution is 0.2°. All the spectra shown in the paper are taken along the K-Γ-K direction.
• C.  
  Transport measurements: Figure 1(a) shows the schematic setup of the transport measurements on the Bi₂Se₃ thin films. The resistance and thermopower measurements are performed on the same area in the central part of the Bi₂Se₃ thin film. Electrical transport properties are measured in an isothermal condition using the delta mode of the Keithley 6221 current source plus the 2182A nanovoltmeter. Thermoelectric measurements are carried out in the high-vacuum condition with pressure lower than 1 × 10⁻⁶ mbar. The temperature gradient is produced by a thin-film heater mounted on the right end of the substrate. A pair of fine-gauge thermocouples (type E, CHROMEGA/constantan) are connected in subtractive series and thermally anchored to the substrate to monitor the temperature difference across the sample. The dc voltages of the Seebeck effect and thermocouples are recorded by Keithley 2182A nanovoltmeters.
• D.  
  Theoretical calculations: First-principles electronic structure calculations were performed within the framework of density functional theory (DFT) as implemented in the Vienna ab initio simulation package [15]. The projector-augmented-wave potential, the plane-wave basis with an energy cutoff of 210 eV, and the Perdew–Burke–Ernzerhof [16] exchange-correlation functional in the generalized-gradient approximation were used in DFT calculations including the spin–orbit coupling effect. To study thermoelectric properties of Bi₂Se₃ thin films, band structure on a dense k grid of 200 × 200 × 1 was computed with the help of the Wannier-interpolation technique. The Wannier90 package [17] was used to construct the maximally localized Wannier functions (MLWFs) [18, 19] from DFT slab calculations, and then a tight-binding Hamiltonian was built from MLWFs to calculate band structure on a dense k grid at a much smaller computational cost [20]. Thermoelectric properties were studied by the Landauer transport approach [21, 22] with the constant mean-free-path model [21]. The electron transmission function is equal to the density of modes M(E) (i.e. the number of conduction channels per width) times the transmission probability. M(E) was calculated by counting conduction channels from the band structure in the whole Brillouine zone. The transmission probability is proportional to the mean free path (MFP) in the diffusive limit [21, 23, 24]. Two constant MFPs l_s and l_b are required to describe the TI surface and bulk states, respectively. In our calculations of the Seebeck coefficient, the only free parameter is the MFP ratio r_l = l_s/l_b, which was determined by fitting the experimental data. For Bi₂Se₃ TI thin films, M(E) can be separated into the surface and bulk parts according to the band dispersion, and the surface contribution M_s(E) was calculated from the Dirac-like bands.

To investigate the thickness dependence of the thermoelectric performance of TI thin films, Bi₂Se₃ thin films are prepared with varied thicknesses including d = 5, 7, 10, 15, 20 and 30 QL. Shown in figures 1(b) to (d) are the ARPES band maps of the 7, 10 and 20 QL Bi₂Se₃ thin films, respectively. In all three samples the E_F cuts through the bulk conduction bands (BCBs) due to the existence of electron-type bulk carriers donated by Se vacancies during sample growth. The Dirac-like topological surface states with linear band dispersion can be clearly observed in all samples. For the 7 QL sample, the E_F lies at around 0.11 eV above the bottom of BCB and 0.34 eV above the Dirac point of the surface state. With increasing film thickness, the E_F position is lowered towards the Dirac point due to the decrease of Se vacancy density in thicker films, which has a better crystalline quality as the Bi₂Se₃ layers move further away from the rather rough substrate surface. For d = 10 QL and 20 QL, the E_F positions are about 0.025 eV and 0.020 eV above the BCB minimum, which are nearly the same. Therefore the change of E_F position becomes much smaller for films thicker than 10 QL. This is very reasonable from the MBE sample growth perspective because when the film is sufficiently thick, the influence of the substrate becomes negligible.

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Figure 2 displays the temperature dependence of the two-dimensional (2D) sheet resistance R_□, the Seebeck coefficient S and the power factor S²σ for six Bi₂Se₃ thin films with different thicknesses. The resistances of all Bi₂Se₃ thin films (figure 2(a)) show a metallic behavior at high temperature and turn to weakly insulating at a very low temperature. This is the typical behavior of Bi₂Se₃ thin films in the 2D limit caused by the electron–electron interaction effect in the presence of disorders [25]. When the film thickness decreases from 30 QL to 5 QL, the R_□ value keeps rising and the insulating tendency is enhanced. On the one hand, this is due to the degradation of sample quality, hence electron mobility, as the film becomes thinner. On the other hand, this reflects the decrease of bulk contribution to transport in thinner films. As can be seen clearly in figure 3(b), the R_□ value at T = 300 K increases from 0.6 kΩ for 30 QL to 4.4 kΩ for 5 QL.

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The Seebeck coefficients of all the Bi₂Se₃ thin films (figure 2(b)) are negative, which is consistent with n-type bulk and surface charge carriers as revealed by the ARPES band maps. As expected for simple metals, the Seebeck coefficients show a quasi-linear relationship with T over the whole temperature range. With the decrease of film thickness, the general behavior of S remains qualitatively the same but the absolute value |S| decreases very systematically. Figure 3(a) summarizes the thickness dependence of S measured at T = 300 K of the six Bi₂Se₃ films. The value of S changes smoothly from −104.3 μV K⁻¹ for 30 QL to −64.6 μV K⁻¹ for 5 QL. As a consequence, the thermoelectric power factor S²σ shown in figure 2(c) drops significantly with reducing film thickness. Therefore, decreasing the film thickness, or lowering the dimensionality of the Bi₂Se₃ TI system, is actually detrimental to the thermoelectric performance.

To understand the experimental observations, we perform band structure and thermopower calculations for Bi₂Se₃ thin films with varied thicknesses. The existence of 2D Dirac-like surface bands within the insulating bulk gap is clearly visualized in figure 4(a) for 7 QL Bi₂Se₃. As the film thickness increases, the dispersion of surface bands remains essentially unchanged, but the number of bulk bands increases and their dispersion varies noticeably due to the quantum confinement effect. The thickness-dependent characteristics of the band structure is reflected in M(E), which is proportional to the DOS times carrier velocity [21]. As shown in figure 4(b), M(E) increases gradually with film thickness in the bulk-band region and becomes thickness independent in the bulk-gap region. A linear energy dependence of M(E) starting from the Dirac point is clearly shown within the bulk gap, which is a Hallmark of 2D surface states with linear dispersion.

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We can also make more quantitative comparisons between theory and experiment. The calculated BCB minimum is 0.24, 0.26 and 0.15 eV above the Dirac point for 7, 10 and 20 QL Bi₂Se₃, in good agreement with the ARPES results. In our calculations of the Seebeck coefficient, the E_F position was set to be 0.111, 0.025, 0.022 and 0.020 eV above the BCB minimum for 7, 10, 15 and 20 QL, respectively, as determined by the ARPES results. The MFP ratio r_l as the only free parameter was determined by comparing the calculated S as a function of temperature with the experimental data. We selected r_l to be 1.0, 2.0, 1.6 and 2.8 for 7, 10, 15 and 20 QL, respectively. As demonstrated in figure 4(c), there is an excellent agreement between theory and experiment for S over a wide range of temperatures.

In order to explain why S becomes smaller in thinner Bi₂Se₃ films, we need to consider both the surface and bulk contribution to S. Specifically, the total Seebeck coefficient S is a weighted contribution of the channels: S = (σ_s S_s + σ_b S_b)/(σ_s + σ_b), where σ_s and σ_b are the electrical conductivity of surface and bulk states, respectively. To analyze the influence of surface states, we artificially exclude the contribution of surface states in the calculation of S, and assign this Seebeck coefficient as S_b. S_b is noticeably larger than S for all the thicknesses (figures 4(c) and (d)), suggesting that the surface states suppress the total S. The degree of suppression, defined as |S_b − S|, at 300 K is relatively small for 7 QL (21 μV K⁻¹). For d = 10, 15 and 20 QL, the |S_b − S| values are 103, 82 and 72 μV K⁻¹, respectively, which decrease with increasing film thickness. Because the E_F positions are similar in these three samples, the more pronounced reduction of total S in thinner films can only be explained by the increased contribution of the topological surface states.

It is well known that strong electron-hole asymmetry is required to enhance S, hence S_s is small because of the gapless linear dispersion of surface states. In contrast, S_b could be greatly tuned by varying the E_F due to the existence of the bulk gap. The inclusion of S_s thus decreases the total S in general. The quantitative decrease of S depends crucially on the E_F position as well as the film thickness. For 7 QL, E_F lies well within the bulk bands. The metallic bulk states on the one hand introduce a small S_b with linear temperature dependence (figure 4(d)), and on the other hand lead to a small relative contribution of surface states to S. This explains the small |S_b − S| value in the 7 QL sample. For the d = 10, 15 and 20 QL thin films, their E_F positions are closer to the BCB minimum, hence the surface states have relatively large contributions to S. Therefore in this regime the suppressions of total S by the surface states becomes much more pronounced, leading to much larger |S_b − S| values. When the film thickness decreases from 20 to 10 QL, |S_b| increases due to the bulk quantum confinement effect, but the total |S| decreases due to the increase of |S_b − S| from 72 to 103 μV K⁻¹. The suppression of total S by the surface state thus grows with the increasing contribution of the surface state in thinner films, which further demonstrates the detrimental effect of the topological surface state on S. Our theoretical study thus indicates that the TI surface states play a crucial role in determining the thickness dependent properties of thermopower in the Bi₂Se₃ thin films.

In summary, we observe a systematic reduction of the Seebeck coefficient as well as a remarkable decrease of thermoelectric power factor in Bi₂Se₃ TI thin film when the thickness is reduced from 30 QL to 5 QL. These results can be understood by the combined bulk and surface state contribution to the thermoelectric properties of TI. The main origin of the thickness dependent behavior is the increasing contribution of the Dirac-like topological surface state, whose linear dispersion is not optimal to the thermopower. Another factor is the change of E_F position with film thickness, which affects the bulk thermopower value and the relative contribution of the surface states.

This work sheds important new light on the thermoelectric properties of 3D TIs, which are so far the best room temperature thermoelectric materials. Contrary to previous theoretical proposals [2, 6, 11], our results show that simply lowering the dimensionality of 3D TIs is not an effective way to enhance the thermoelectric performance. This is due to the existence of Dirac-like topological surface states, which were discovered only a few years ago. In order to improve the thermoelectric performance of 3D TI, it is perhaps necessary to reduce or completely remove the topological surface states. Possible approaches include further decreasing the film thickness to gap the surface state, as well as an optimization of the E_F position. For future investigations, we plan to tune the band structure of TI thin films by doping or gating to see the effect of the gapped topological surface state and the band topology on the thermopower, aiming to gain a deeper understanding of the thermoelectric properties of TIs and the methods to further improve them.

This work was supported by the National Natural Science Foundation of China and the Ministry of Science and Technology of China. YX acknowledges support from Tsinghua University Initiative Scientific Research Program. The calculations were done on the 'Explorer 100' cluster system of Tsinghua University and on the 'Tianhe-2' of National Supercomputer Computer Center in Guangzhou. SCZ is supported by the Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering, under contract DE-AC02-76SF00515. We also acknowledge the support from ENN Energy Holdings Limited.