Quantum Interference of Rashba-Type Spin-Split Surface State Electrons

We studied the quantum interference of electrons in the Bi $(p_x,p_y)$ orbital-derived $j=1/2$ spin-split surface states at $\mathrm{Bi}/\mathrm{Ag}(111)\sqrt{3}\times \sqrt{3}$ surfaces of 10 monolayer thick Ag(111) films on Si(111) substrates. Surface electron standing waves were observed clearly at the energy ($E$) below the intersection of the two spin-split downward dispersing parabola bands ($E_x$). The $E$ dependence of the standing wave pattern reveals the dispersion as the average of the two spin-split surface bands due to the interference between $|(k+\Delta ),\uparrow ⟩$ and $|-(k-\Delta ),\uparrow ⟩$ [or $(|(k-\Delta ),\downarrow ⟩)$ and $|-(k+\Delta ),\downarrow ⟩$] states. In contrast, it was impossible to deduce the dispersion from the standing wave pattern at $E\ge E_x$ because the surface electron cannot find its backscattered state with the same spin polarization.

Figure 1

(a) Topographic STM image of Bi-induced $\sqrt{3}\times \sqrt{3}$ reconstruction in a 10 ML thick Ag(111) film on a Si(111) substrate. The inset in the lower right corner is a magnified image of the $\sqrt{3}\times \sqrt{3}$ lattice at the terrace. (b)–(h) $dI/dV$ images of the $\mathrm{Bi}/\mathrm{Ag}(111)\sqrt{3}$ surface. $V_s=+0.15\mathrm{V}$ (b), $+0.20\mathrm{V}$ (c), $+0.25\mathrm{V}$ (d), $+0.30\mathrm{V}$ (e), $+0.35\mathrm{V}$ (f), $+0.40\mathrm{V}$ (g), and $+0.45\mathrm{V}$ (h), respectively. The imaging size was $31.2\times 31.2{\mathrm{nm}}^2$ for (a)–(h), and $2.5\times 2.5{\mathrm{nm}}^2$ for the inset in (a). The image size was calibrated with respect to the $7\times 7$ unit cell at the Si(111) surfaces. The tunneling current was 0.5 nA.

Figure 2

Dispersion of the $m=1/2$ surface states along the $\Gamma \mathrm{\text{−}}M$ and $\Gamma \mathrm{\text{−}}K$ directions at a $\mathrm{Bi}/\mathrm{Ag}(111)\sqrt{3}\times \sqrt{3}$ surface. Light gray (red) and dark gray (blue) lines indicate theoretically calculated $|\uparrow ⟩$ and $|\downarrow ⟩$ states, respectively, from Ref. 24. The solid black convex curve represents an average of the theoretically calculated $|\uparrow ⟩$ and $|\downarrow ⟩$ states. The red squares indicate experimentally deduced dispersion from the surface standing wave patterns. The errors in the estimation of $k$ are indicated by red bars. The theoretical dispersions were shifted by $-0.1\mathrm{eV}$.

Figure 3

$dI/dV$ images of $\mathrm{Bi}/\mathrm{Ag}(111)\sqrt{3}$ reconstructed surfaces. The image size is $52.0\times 52.0{\mathrm{nm}}^2$. $V_s$ was (a) $+0.40\mathrm{V}$, (b) $+0.50\mathrm{V}$, (c) $+0.55\mathrm{V}$, and (d) $+0.60\mathrm{V}$. The tunneling current was 0.55 nA.

Figure 4

DOS of the $m=1/2$ spin-split surface state bands at a $\mathrm{Bi}/\mathrm{Ag}(111)\sqrt{3}$ surface. (a) Experimentally obtained $dI/dV$ curve. (b) Black line:  Calculated DOS for the theoretically reported dispersions of the gray lines in Fig. 2. The DOS was broadened by instrumental functions with widths of $1.73\times V_{pp}$ [dark gray (blue) curve] and $3.00\times V_{pp}$ [light gray (red) curve].