Evolution of the Fermi surface of a doped topological insulator with carrier concentration

In an ideal bulk topological insulator (TI) conducting surface states protected by time-reversal symmetry enfold an insulating crystal. However, the archetypical TI, Bi${}_2$Se${}_3$, is actually never insulating; it is in fact a relatively good metal. Nevertheless, it is the most studied system among all the TIs, mainly due to its simple band structure and large spin-orbit gap. Recently, it was shown that copper intercalated Bi${}_2$Se${}_3$ becomes superconducting and it was suggested as a realization of a topological superconductor. Here we use a combination of techniques that are sensitive to the shape of the Fermi surface (FS): the Shubnikov-de Haas effect and angle-resolved photoemission spectroscopy to study the evolution of the FS shape with carrier concentration, $n$. We find that as $n$ increases, the FS becomes two-dimensional-like. These results are of crucial importance for understanding the superconducting properties of Cu${}_x$Bi${}_2$Se${}_3$.

Figure 1

Transport and ARPES characterization. (a) Longitudinal resistivity versus temperature for Cu-doped Bi${}_2$Se${}_3$with $n\simeq {10}^{20}{\mathrm{cm}}^{-3}$. Metallic type behavior as well as supercoductivity below $\simeq$3 K is observed. Different samples may exhibit various superconducting volume fractions and $T_c$ variations. (b) Transverse resistivity versus magnetic field at 2 K for three samples. The solid lines are linear fits from which we extract the carrier concentration $n$. Inset: The same for $n\simeq {10}^{19}{\mathrm{cm}}^{-3}$. (c) Typical ARPES data from a highly doped sample measured with 20-eV photon energy. The detector image shows the dispersion along the $\Gamma$-K direction. One can see that both the surface-state and the bulk-band dispersion coexist. (d) Surface-state dispersion as measured using ARPES for eight different samples with carrier concentrations ranging from $4\times {10}^{17}{\mathrm{cm}}^{-3}$ (green curve) to $4\times {10}^{20}{\mathrm{cm}}^{-3}$ (gray curve).

Figure 2

SdH data and analysis. (a) Longitudinal resistance versus magnetic field applied parallel to the C3 axis $(\theta =0)$ at 4.2 K for $n\simeq {10}^{20}$ cm${}^{-3}$ (sample A). Inset: Fast Fourier transform (FFT) of these data plotted versus $\frac{1}{{\mu }_{0}H}$ after subtracting a smooth polynomial background. The sharpness of the FFT peak indicates a well-defined frequency. Its full width at half maximum is used as an upper limit for the uncertainty in determining the frequency. A drawing of the sample configuration used in this experiment is also shown. (b) Resistance versus $\frac{1}{{\mu }_{0}H}$ after subtraction of a smooth polynomial background at various temperatures [4.2 K data are taken from Fig. 2a]. The field is applied parallel to the C3 axis. Inset: Effective mass is extracted by following the oscillation amplitude at high field as a function of the temperature. The solid line is a fit to the Dingle formula,[14] yielding m*$\simeq 0.24m_e$ for this sample ($n\simeq {10}^{20}$ cm${}^{-3}$). (c) The frequency as determined from the FFT versus tilt angle $\theta$ between the magnetic field and the C3 axis for three carrier concentrations. Solid lines are fits for an ellipsoidal FS ($n\simeq {10}^{17}$, 10${}^{19}$ cm${}^{-3}$) and for a cylindrical FS [$F\propto \frac{1}{\cos \left(\theta \right)}$] for $n\simeq {10}^{20}$ cm${}^{-3}$ (sample B).

Figure 3

Evolution of the Fermi surface with carrier concentration. [(a) and (b)] Calculated ellipsoidal FS from the SdH data in Fig. 2c, for $n\simeq {10}^{17},{10}^{19}$ cm${}^{-3}$, respectively. Detailed profile view of the Fermi surfaces is shown. (c) Calculated FS using tight binding corrugated cylinder model fit to the SdH data in Fig. 2c (sample B, $n\simeq {10}^{20}$ cm${}^{-3}$) (see Appendix for more information). (D) The Brillouin-zone momenta axes. [(e)–(g)] The Fermi surfaces of (a)–(c), respectively, plotted to scale with respect to the Brillouin zone.

Figure 4

Photon energy dependence of the ARPES data. We show normal emission data for three different samples: (a) $n=4\times {10}^{17}$ cm${}^{-3}$, (b) $n=4\times {10}^{20}$ cm${}^{-3}$, and (c) $n=2\times {10}^{20}$ cm${}^{-3}$. The white dots represent the bottom of the bulk band. For the low-carrier-density sample the bulk band is seen only around 19 eV ($\Gamma$ point), and for the the high-carrier-density samples the bulk band is visible at the entire photon-energy range measured. In panel (b) we show low-photon-energy data, believed to be more bulk sensitive. We find that the bulk band is visible at the entire photon-energy range measured, which covers a momentum range larger than the $\Gamma$-Z separation.

Figure 5

Effective mass of the bulk band. (a) Dispersion of the bulk band around the $\Gamma$ point, where the dispersion of the bulk is clear and allows an accurate measurement of the effective mass. The dashed line represents the parabolic fits to the dispersion. (b) MDCs for the 19-eV photon-energy data. The red points are the maxima of the MDCs. These maxima are used to extract the dispersion. (c) Summary of the fit results. The red points (black circles) represent $k_F$ (effective mass) as a function of the photon energy. (d) A close view of the FS calculated in Fig. 3c. The color code corresponds to the effective mass calculated for the whole momentum range using the fit in Fig. 5c. The cylindrical-type shape has a corrugation ratio of $\simeq$1.05. This corrugation ratio together with the enhancement of the effective mass at the zone boundary explain the absence of a second frequency and the amplitude attenuation at high tilt angles as detailed in the text.

Figure 6

The energy position of the Dirac point and the value of $k_F$ for the different samples appearing in Fig. 1d of the main text. Error bars are 95% confidence levels.

Figure 7

Finding the inner potential $V_0$: The high symmetry points are identified from the photon energy scans, similarly to what is shown in Fig. 4 of the main text. The $k_z$ axis separation between the data points is taken to be the $\Gamma -Z$ distance, and the error bars show the uncertainty in the position. The dashed line is the equation of free-electron final-state approximation (see text) plotted with $V_0=10.3$ eV.

Figure 8

Dispersion of the bulk band along the $k_z$ axis. The plot contains data from five different samples with varying Fermi energies (noted by the horizontal dash lines) as measured from the bottom of the bulk band at the $\Gamma$ point.

Figure 9

Carrier density as a function of $k_F$. The carrier densities are obtained from Hall measurements and they span nearly three orders of magnitude. The $k_F$ (blue dots) was obtained from SdH data, measured with magnetic field parallel to the C3 axis. The $k_F$ (hollow black circles) is that of the surface state obtained from the ARPES data, and it represents an upper bound on the $k_F$ of the bulk. The solid line is the expected carrier density for a spherical Fermi surface. The dashed line is the carrier density for which the radius of the Fermi sphere equals the $\Gamma$-$Z$ distance, that is, the Fermi surface reaches the edge of the first Brillouin zone.

Figure 10

Sample with $n3D\simeq {10}^{17}$. (a) Longitudinal resistance versus magnetic field at various tilt angles between the C3 axis and the magnetic field. (b) Frequency versus tilt angle. The solid line is a fit for an ellipsoidal Fermi surface model. Inset: FFT analysis of each tilt angle.

Figure 11

Sample with $n3D\simeq {10}^{19}$. (a) Longitudinal resistance versus magnetic field at various tilt angles. 0${}^{\circ }$ corresponds to magnetic field parallel to the C3 axis. (b) Frequency versus tilt angle. The solid line is a fit for an ellipsoidal Fermi surface model. Inset: FFT analysis of each tilt angle.

Figure 12

Sample with $n3D\simeq {10}^{20}$. (a) Longitudinal resistance versus magnetic field at various tilt angles. (b) Frequency versus tilt angle. The solid line is a fit for a cylindrical Fermi surface model [$F\propto \frac{1}{\cos \left(\theta \right)}$]. Inset: FFT analysis for each tilt angle.