Magnetothermoelectric properties of Bi${}_2$Se${}_3$

We present a study of entropy transport in Bi${}_2$Se${}_3$ at low temperatures and high magnetic fields. In the zero-temperature limit, the magnitude of the Seebeck coefficient quantitatively tracks the Fermi temperature of the three-dimensional Fermi surface at the $\Gamma$ point as the carrier concentration changes by two orders of magnitude (10${}^{17}$ to 10${}^{19}$ cm${}^{-3}$). In high magnetic fields, the Nernst response displays giant quantum oscillations indicating that this feature is not exclusive to compensated semimetals. A comprehensive analysis of the Landau level spectrum firmly establishes a large $g$ factor in this material and a substantial decrease of the Fermi energy with increasing magnetic field across the quantum limit. Thus, the presence of bulk carriers significantly affects the spectrum of the intensively debated surface states in Bi${}_2$Se${}_3$ and related materials.

Figure 1

Electrical and entropy transport measurements of Bi${}_2$Se${}_3$ for the samples A${}_1$ [$n\left(A_1\right)$ $\approx$1e19 cm${}^{-3}]$ and B${}_1$ [$n\left(B_1\right)$ $\approx$ 1e17 cm${}^{-3}$] (a) ${\rho }_{xx}$ (in blue) and ${\rho }_{yx}$ (in red) of A${}_1$ as a function of the magnetic field for $T$ = 1.9 K. (b) $S$ (in blue) and $N$ (in red) of A${}_1$ as a function of the magnetic field for $T$ = 2.6 K (the insert shows a zoom of $S$ and $N$ between 8 and 12 T). (c) ${\rho }_{xx}$ (in blue) and ${\rho }_{yx}$ (in red) of B${}_1$ as a function of the magnetic field for $T$ = 350 mK. (d) $S$ (in blue) and $N$ (in red) of B${}_1$ as a function of the magnetic field for $T$ = 3.1 K.

Figure 2

(a) and (b) $\frac{\nu }{T}=\frac{N}{TB}$ as a function of B${}^{-1}$ for the sample A${}_1$ and B${}_2$ (sample different of B${}_1$ with a similar bulk concentration) for the various temperatures explored. (c) Temperature dependence of the amplitude of the oscillations of the sample A${}_1$ (in red) and B${}_2$ (in blue). The line corresponds to a fit using the Lifshitz-Kosevich formula (see the text).

Figure 3

(“Dingle plot”) $\ln \left(\frac{A}{R_T}\right)$ as a function of B${}^{-1}$. $A$ corresponds to the amplitude of the oscillations as observed in $\frac{\nu }{T}$ and $RT=\frac{X}{\sin h\left(X\right)}$ where $X=\frac{14.69m^{*}T}{B}$ ($m^{*}=0.2$m${}_0$ in the case of A${}_1$).

Figure 4

(a) $\frac{S}{T}$ as a function of the temperature for A${}_2$ (same batch as A${}_1$ with a slightly lower concentration) and B${}_1$. (b) $\frac{\nu }{T}$ at $B$ = 1 T as a function of the temperature for A${}_1$,A${}_2$, B${}_1$, and B${}_2$.

Figure 5

(a) Field dependence of the Nernst effect at $T$ = 3 K, ${\alpha }_{xy}*B$($T$ = 3 K), $\Delta {\rho }_{xx}$ and $\Delta {\sigma }_{xx}={\sigma }_{xx}\left(0.25\mathrm{K}\right)-{\sigma }_{xx}(4.2$ K), respectively, in red, green, blue, and dark orange. (b) Landau-level fan diagram for oscillations in $N$ and ${\alpha }_{xy}*B$, respectively, red and green. The red line corresponds to a linear fit of the red points. As seen in (a), the peak positions in $N$ and ${\alpha }_{xy}*B$ are almost merging.

Figure 6

Nernst effect (red line) of the sample B${}_1$ at $T$ = 3 K on top of calculated Landau level energies and Fermi energy $E$${}_F$ (black open circle) with different parameters: $M=0,1,2$, and 3 which, respectively, correspond to (a), (b), (c), and (d). The calculation was performed with a carrier concentration of $n=1.08\times {10}^{17}{\mathrm{cm}}^{-3}$ for (a), (b), and (d) and $n=1.25\times {10}^{17}{\mathrm{cm}}^{-3}$ for (b), a Dingle temperature of $T$${}_D=8$ K and $m_1=m_2=0.14m_0$ and $m_3=0.24m_0$.

Figure 7

$N$ (in red lines) of the sample B${}_1$ at $T$ = 3 K on top of a calculated Landau level energies and Fermi energy $E$${}_F$ (black open circle) with $M$ = 0 for (a) and $M$ = 3 for (b). In a case of (a) the carrier concentration has been chosen in order that the last resolved peak corresponds to the LL 1${}^{\pm }$ ($n=1.8\times {10}^{17}{\mathrm{cm}}^3$). For (b), the carrier concentration has been chosen in order that the last resolved peak matches the LL 2${}^{-}$ ($n=2.9\times {10}^{17}{\mathrm{cm}}^3$). The calculation was performed with a Dingle temperature $T$${}_D=8$ K and $m$${}_1=m_2=0.14m_0$ and $m_3=0.24m_0$.

Figure 8

In red line is the Nernst effect, $N$, as a function of B${}^{-1}$ at $T$ = 5 K for the sample B${}_3$ (same as B${}_1$ few months after the (a) experiment) on (a) and (b). The evolution of the Fermi energy ($E$${}_F$) is plotted in black points. Each time that the chemical potential is crossing a Landau level (colors lines), there is a kink due to the carrier variation. The best parameter to describe the peak position as measured in $N$ is found to be for $M$ = 1 (with a bulk carrier concentration $n=1.8\times {10}^{17}$ cm${}^{-3}$) on (a) and $M$ = 2 (with a bulk carrier concentration $n=1.5\times {10}^{17}$ cm${}^{-3}$) on (b).

Figure 9

(a) Field dependance ${\rho }_{yx}$ for B${}_1$ ($T$ = 1.5 K up to 12 T), B${}_2$ ($T$ = 0.35 K up to 17 T), and B${}_3$ ($T$ = 1.6 K up to 30 T). (b) Field dependance ${\rho }_{yx}$ for A${}_1$ and A${}_2$ at $T$ = 1.6 K up to 12 T.