One-dimensional spin-polarized surface states: A comparison of Bi(112) with other vicinal bismuth surfaces

Vicinal surfaces of bismuth are unique test beds for investigating one-dimensional (1D) spin-polarized surface states that may one day be used in spintronic devices. In this paper, such states have been observed for the (112) surface when measured using angle- and spin-resolved photoemission spectroscopy, and also when calculated using a tight-binding model and with density functional theory. The surface states appear as elongated Dirac-cones which are 1D and almost dispersionless in the ${\mathbf{k}}_{\text{y}}$ direction, but disperse with energy in the orthogonal ${\mathbf{k}}_{\text{x}}$ direction to form two $\times$-like features centered at the ${\mathbf{k}}_{\text{y}}$ line through $\overline{\mathrm{\Gamma }}$. Unlike many materials considered for spintronic applications, their 1D nature suggests that conductivity and spin-transport properties are highly dependent on direction. The spin polarization of the surface states is mainly in plane and parallel to the 1D states, but there are signs of a tilted out-of-plane spin-vector component for one of the features. The Bi(112) surface states resemble those found for other vicinal surfaces of bismuth, strongly indicating that their existence and general properties are robust properties of vicinal bismuth surfaces. Furthermore, differences in the details of the states, particularly related to their spin polarization, suggest that the electronic band structure may be engineered simply by precise cutting and polishing of the crystal.

Figure 1

The atomic and electronic structure of Bi(112). (a) Sketch showing the atomic structure of the vicinal Bi(112) surface from the side. Red, green, and blue spheres represent the uppermost, second, and third layers of the crystal, respectively. (b) Top view schematic of the Bi(112) surface with the unit cell indicated (green). The $x/{\mathbf{k}}_{\text{x}}, y/{\mathbf{k}}_{\text{y}}$, and $z/{\mathbf{k}}_{\text{z}}$ axes correspond to the crystallographic directions [1-10], [12.7 12.7 1], and [112], respectively. (c) Low-energy electron diffraction (LEED) pattern of the Bi(112) surface, measured with an electron energy of 40 eV. The green rectangle shows the first Brillouin zone (BZ). $b_2$ is the distance between the streaks, and corresponds to the length $a_2$ of the unit cell along the atomic chains. (d) The two-dimensional Fermi surface of Bi(112). The first and second BZs are indicated by green and purple rectangles, respectively. (e) Volumetric representation of the measured Bi(112) band structure. (f) Measured band structure ($E_{\text{B}}$ vs ${\mathbf{k}}_{\text{x}}$) at ${\mathbf{k}}_{\text{y}}=0$, i.e., orthogonal to the one-dimensional surface states.

Figure 2

(a) An overview of the band structure near the ${\overline{\mathrm{\Gamma }}}_1$ point ($E_{\text{B}}$ vs ${\mathbf{k}}_{\mathbf{\text{x}}}$) at ${\mathbf{k}}_{\mathbf{\text{y}}}=0$. The dark blue and light blue bands represent the two measured $\times$-like features at ${\overline{\mathrm{\Gamma }}}_1$, where the dark blue bands are clearest in (b) and (f)–(h), and the light blue bands clearest in (c) and (d). The arrows from ${\times }_{E_{\text{F}}}$ and ${\times }_{0.16\text{eV}}$ indicate the crossing points of the $\times$-like features. The green and pink/brown bands are bulk and surface states, respectively, calculated using a tight-binding (TB) model. The dark pink bands are thought to correspond to ${\times }_{E_{\text{F}}}$ and the light pink bands to ${\times }_{0.16\text{eV}}$ seen in the measurements. The yellow and purple circles show the measured (dark blue cross light blue) and calculated (light pink cross dark pink) crossing of ${\times }_{E_{\text{F}}}$ and ${\times }_{0.16\text{eV}}$, respectively. (b)–(d), (f)–(h) The band structure along $\overline{\mathrm{X}}-{\overline{\mathrm{\Gamma }}}_1-\overline{\mathrm{X}}$ measured with photoexcitation energies $h\nu =21\mathrm{eV}$ (b); $h\nu =32\mathrm{eV}$ (c), (d); $45\mathrm{eV}$ (f); $70\mathrm{eV}$ (g); and $80\mathrm{eV}$ (h). In (b)-(c) the TB calculations are overlaid on the measurements. The yellow/purple circles and purple arrow indicate where the $\times$-like features cross each other in the measurements and calculations. In (d), which is measured inside the orange region in (c), the left-hand side is measured with $p$-polarized (partly out-of-plane) light, and the right-hand side with $s$-polarized (in-plane) light. The direction of the light polarization with respect to the sample surface is sketched in (e).

Figure 3

Spin polarization from tight-binding calculations. (a)–(c) Band structure between $\overline{\text{X}}-{\overline{\mathrm{\Gamma }}}_1-\overline{\text{X}}$. In (a) and (b), the color scale shows the spin-polarization ${\mathbf{S}}_{\text{y}}$, the spin-vector component in plane and along the 1D lines, (a), and ${\mathbf{S}}_{\text{z}}$, the out-of-plane component of the spin vector, (b). Note that the color scale is different between (a) and (b). In (c), the colors label the surface bands. (d), (e) Spin-polarization of the spin-vector component ${\mathbf{S}}_{\text{y}}$ (d) and ${\mathbf{S}}_{\text{z}}$ (e), respectively. The color shows which band in (c) the spin-polarization value belongs to.

Figure 4

Measured spin texture. (a), (b) Constant energy surfaces at binding energies $E_{\text{B}}=0.12\mathrm{eV}$ (a) and $E_{\text{B}}=0.72\mathrm{eV}$ (b). The first Brillouin zone is substituted by the in-plane spin polarization along the ${\mathbf{k}}_{\text{y}}$ direction visualized by red and blue arrows. (c), (f) $E_{\text{B}}$ vs ${\mathbf{k}}_{\text{x}}$ plot measured with photoexcitation energies of $h\nu =32\mathrm{eV}$ and $h\nu =21\mathrm{eV}$. The in-plane spin-polarization along ${\mathbf{k}}_{\text{y}}$ was measured along the dashed white lines and has been overlaid. (d), (e), (g) In-plane spin polarization along ${\mathbf{k}}_{\text{y}}$ at $E_{\text{B}}=0\mathrm{eV}$ (d) and $E_{\text{B}}=0.63\mathrm{eV}$ (e) in (c), and $E_{\text{B}}=0.36\mathrm{eV}$ (g) in (f).