GLYPH<s45>GLYPH<s57>GLYPH<s48> GLYPH<s45>GLYPH<s54>GLYPH<s48> GLYPH<s45>GLYPH<s51>GLYPH<s48> GLYPH<s48> GLYPH<s51>GLYPH<s48> GLYPH<s54>GLYPH<s48> GLYPH<s57>GLYPH<s48> GLYPH<s48> GLYPH<s53> GLYPH<s49>GLYPH<s48> GLYPH<s49>GLYPH<s53> GLYPH<s49>GLYPH<s48> GLYPH<s49>GLYPH<s54>GLYPH<s32>GLYPH<s84> GLYPH<s67> GLYPH<s50> GLYPH<s67> GLYPH<s51> GLYPH<s32> GLYPH<s121> GLYPH<s120> GLYPH<s32> GLYPH<s40> GLYPH<s109> GLYPH<s32> GLYPH<s99> GLYPH<s109> GLYPH<s41> GLYPH<s32>GLYPH<s32>GLYPH<s40> GLYPH<s100> GLYPH<s101>GLYPH<s103> GLYPH<s41> GLYPH<s67> GLYPH<s50> 3 FIG. 4: (Color online) (a) SdH oscillations measured in the C 3 - C 1 plane. For better visibility, dρ xx /dB is calculated after a smooth background MR is subtracted, and the curves are shifted. (b) θ -dependences of the three fundamental frequen- cies obtained from the Fourier transform of the SdH oscilla- tions. Inset shows a schematic picture of the bulk Brillouin zone, in which the 2D FS is crossed by a plane perpendicular to the magnetic-field direction. FIG. 3: (Color online) “High-field” angular-dependent oscilla- tions of Δ ρ yx obtained by subtracting the expected R H B cos θ behavior from the ρ yx data shown in Fig. 1(b). Ticks mark the positions of distinguishable peaks. axis. This means that, in sufficiently high magnetic field, the 3D electrons will always remain in the quantum limit, while the 2D electrons can always be brought from the quantum limit back to the regime where higher Landau levels are populated. This will lead to the oscillating be- havior of ρ$_{xx}$ due to the 2D FS alone when the magnetic field is rotated toward the C 3 axis. Remembering that the resistivity oscillates as ρ$_{xx}$ ∼ cos[2 π ( F/B eff + γ )] with B eff = B sin θ , a maximum in ρ$_{xx}$ occurs when F/B eff + γ = n with integer n . Hence, one can understand that the Fermi level is crossed by the n -th Landau level of the 2D FS at the specific angle θ$_{n}$ given by f 3 ) are a = 1.2 × 10 6 cm − $^{1}$, b = 8.5 × 10 5 cm − $^{1}$, and c = 4.2 × 10 5 cm − $^{1}$. Note that the 2D FS resides on the plane perpendicular to the C 1 axis, while the 3D ellipsoids are located at the L points of the bulk Brillouin zone, with the longest axes in the binary plane and tilted from the C 1 axis by ∼ 5 ◦ . The divergence of f 1 at θ = 0 ◦ seen in Fig. 4(b) is of geometric origin and is a hallmark of the 2D FS, as can be seen in the inset of Fig. 4(b). The analysis of the SdH amplitude gives us the cy- clotron mass m$_{c}$ and the scattering time τ . For the field direction very close to the C 3 axis, the SdH oscillations coming from the 2D and 3D FSs can be easily sepa- rated, and their fits to the standard Lifshitz-Kosevich theory [27] at θ ≈ 3 ◦ yield m$_{c}$/m$_{e}$ of 0.13 and 0.033 for the 2D and 3D FSs, respectively ( m$_{e}$ is the free elec- tron mass). Note that the relatively large m$_{c}$ for the 2D FS originates from its characteristic angular depen- dence m$_{c}$ = m 0 $_{c}$/ sin θ , where m 0 c = 0 . 0057 m$_{e}$ is for the orbital motion in perpendicular fields ( B ‖ C 1 ), meaning that the 2D electrons are actually extremely light. Cor- responding Dingle plots (not shown) give Dingle temper- atures T$_{D}$ of about 5.5 K and 7 K for the 2D and 3D FSs, which imply the mean free paths of l $^{2}$D ≈ 150 nm and l $^{3}$D ≈ 16 nm, respectively [28]. This difference is not surprising, if one remembers the Dirac nature [29, 30] of the 2D surface states and that they are topologically protected against spin-conserving backscattering. We now discuss the origin of the “low-field” part of the angular-dependent oscillations in Bi $^{0}$.91 Sb $^{0}$.09 . A key insight comes from the recognition that the 3D FSs in our sample enter the quantum limit (where all the elec- trons condense into the first Landau level) with magnetic fields of only 1–3 T for any direction, while the necessary magnetic field to bring the 2D FS into the quantum limit diverges as the field direction is rotated toward the C 3 where F is the frequency and γ is the phase of the SdH os- cillations. According to this model, the outer-most peaks marked by arrows in Fig 2 are coming from the crossing of the Fermi level by the first Landau level ( n = 1) at a given magnetic-field strength, so their angles should cor- respond to θ 1 . Actually, the inset of Fig. 2 demonstrates that Eq. (1) fits the data reasonably well. If F is as- sumed to be 0.65 T based on the SdH result [31], the fit in Fig 2 yields γ of 0.25, suggesting a non-zero Berry phase associated with the 2D FS [32, 33]. Finally, let us discuss the origin of the “high-field” peaks shown in Figs. 1 and 3. Here, an important fea- ture is that the amplitude of these peaks weakens as the magnetic field is rotated away from the C 3 axis, which is somewhat reminiscent of the behavior of the ordinary AMRO in quasi-2D systems if the conduction planes lie perpendicular to the C 3 axis [21–23]. Note that this is different from what is observed for the “low-field” oscilla- tions, which are originating from the surface state resid- ing on the (2 ¯ 1 1) plane ( ⊥ C 1 ). Thus, it is probable that the “high-field” oscillations are coming from the (111) θ$_{n}$ = arcsin ( F/B n − γ ) , (1)