Quantum Interference Mapping of Rashba-Split Bloch States in $\mathrm{Bi}/\mathrm{Ag}(111)$

We report on low-temperature scanning tunneling spectroscopy investigations of the ($\sqrt{3}\times \sqrt{3}$) $\mathrm{Bi}/\mathrm{Ag}(111)R30°$ surface alloy which provides a giant Rashba-type spin splitting. We observed spectroscopic features that are assigned to two Rashba-split bands. Quantum interference mapping shows that backscattering is not only allowed below but also above the Rashba energy. We argue that the observed behavior can be understood within the Bloch picture where $k$ refers to the crystal momentum and the velocity of an electronic state is defined as $v_n(E)=\frac{1}{\hslash }{\nabla }_k{E}_n(k)$. The analysis of the energy dispersion of scattering channels reveals a conventional Rashba splitting for the unoccupied Rashba bands, while hybridization is observed in the occupied states.

Figure 1

Schematic representation of the $E(k_x)$ dispersion of Rashba-type spin-split bands and the resulting DOS in the case of two downwards dispersing surface states. Electrons in the red (blue) parabola exhibit spin polarization along $\pm \widehat{y}$. (a) Within a conventional Rashba model the two parabolas are shifted along $k_x$ by $\pm k_0$. (b) Density-functional theory (DFT) calculations have predicted a more complicated spin topology for the $(\sqrt{3}\times \sqrt{3})$ $\mathrm{Bi}/\mathrm{Ag}(111)R30°$ surface alloy [12]. (c) The theoretical DOS with singularities at the onset of the surface state bands. The opening of a hybridization gap leads to an enhanced DOS resulting in a shoulder at the asymmetric peak [hatched line]. (d) Illustration of the spin-dependence of quantum interference. Eigenstates of electrons with opposite spin polarization are orthogonal and cannot interfere (top). An interference pattern is expected for electrons with parallel spins (bottom).

Figure 2

Tunneling spectrum (red dots) of the $(\sqrt{3}\times \sqrt{3})$ $\mathrm{Bi}/\mathrm{Ag}(111)R30°$ surface alloy showing peaks at $E_1=725\mathrm{meV}$ and $E_2=-110\mathrm{meV}$ (set point: $I_{\mathrm{set}}=1\mathrm{nA}$). The blue lines represent fits to the peaks (see text for details).

Figure 3

(a) Constant-current image of the $(\sqrt{3}\times \sqrt{3})$ Bi/Ag(111)$R30°$ surface alloy. Inset: Atomic scale image. (b) $dI/dU$ map measured at $U=137\mathrm{meV}$ and (c) at $U=587\mathrm{meV}$. (d) Corresponding two-dimensional Fourier transform of (b). The hexagonal shape depicts an anisotropic dispersion for $\overline{\Gamma K}$ and $\overline{\Gamma M}$ directions.

Figure 4

Dispersion of scattering vector $q$ as extracted from a series of $dI$/$dU$ maps. Data measured along the $\overline{\Gamma K}$ and $\overline{\Gamma M}$ directions are plotted in light blue and dark blue color, respectively. Red lines are guides for the eye only.

Figure 5

Proposed spin topology of the $(\sqrt{3}\times \sqrt{3})$ $\mathrm{Bi}/\mathrm{Ag}(111)R30°$ surface alloy. Above the Fermi level $E_F$ the upper band exhibits a conventional Rashba splitting. The spin topology of the lower band is more complicated as it hybridizes with the inner branch of the upper Rashba band. Arrows indicate the two scattering channels observed in the occupied states.