Spin-resolved band structure of a densely packed Pb monolayer on Si(111)

Monolayer structures of Pb on Si(111) attracted recently considerable interest as superconductivity was found in these truly two-dimensional (2D) structures. In this study, we analyzed the electronic surface band structure of the so-called striped incommensurate Pb phase with $\frac{4}{3}$ ML coverage by means of spin-resolved photoemission spectroscopy. Our results fully agree with density functional theory calculations done by Ren et al. [Phys. Rev. B 94, 075436 (2016)]. We observe a local Zeeman-type splitting of a fully occupied and spin-polarized surface band at the ${\overline{\text{K}}}_{\sqrt{3}}$ points. The growth of this densely packed Pb structure results in the formation of imbalanced rotational domains, which triggered the detection of $C_{3v}$ symmetry forbidden spin components for surface states around the Fermi energy. Moreover, the Fermi surface of the metallic surface state of this phase is Rashba spin split and revealed a pronounced warping. However, the 2D nesting vectors are incommensurate with the atomic structure, thus keeping this system rather immune against charge density wave formation and possibly enabling a superconducting behavior.

Figure 1

Structure of the SIC phase of Pb/Si(111) comprising a coverage close to $\frac{4}{3}$ MLs of Pb. (a) Large-scale STM image ($20\mathrm{nm}\times 20\mathrm{nm}$, +1.5 V, 1 nA, 200 K, Ref. [37]). (b) SPA-LEED pattern with $E=80$ eV. (c), (d) High-resolution STM images of the H3 and T4 geometry ($2\mathrm{nm}\times 2\mathrm{nm}$, +100 mV, 0.6 pA, 300 K), respectively. (e) A zoom into the SPA-LEED pattern shows a splitting of the $(\sqrt{3}\times \sqrt{3})$ reconstruction spots [cf. red rectangle in (b)]. (f) Top-view schematic for a domain-wall structure (superstructure) with $(\sqrt{3}\times \sqrt{3})$ reconstructions in H3 and T4 geometries separated by quasi-$(\sqrt{7}\times \sqrt{3})$-domain walls [$(13\times \sqrt{3})$ superstructure with 1.308 ML coverage marked by the orange rectangle]. The unit cells for the local $(\sqrt{3}\times \sqrt{3})$ reconstruction in real and reciprocal space are indicated by blue-dashed hexagons.

Figure 2

(a) CEM of the SIC phase taken 178 meV below $E_{\text{F}}$. The black and red lines represent the borders of the first SBZs of the Si(111) and $(\sqrt{3}\times \sqrt{3})$ reconstruction, respectively. The blue dashed lines (1–4) denote directions at which MDCs and band maps were taken. (b) Schematic of the SBZs for the Si(111)-$(1\times 1)$ substrate (black) and the $(\sqrt{3}\times \sqrt{3})$ reconstruction (red). The $k_x$ and $k_y$ directions are along the $\left[\overline{1}\overline{1}2\right]$ and $\left[1\overline{1}0\right]$ directions, respectively. The dashed green lines denote the mirror planes of the $(\sqrt{3}\times \sqrt{3})$ reconstruction in reciprocal space.

Figure 3

Band structure of the SIC phase along the $\overline{\mathrm{\Gamma }\text{K}}$ direction [direction (1) in Fig. 2]. The red dotted line denotes the position of the ${\overline{\text{M}}}_{\sqrt{3}}$ point. (a) Spin-integrated MDC taken at $E=95$ meV [see bluish area in panels (b) and (c)]. (b) The band map shows clearly the metallic surface band S1 crossing $E_{\text{F}}$ at $k_y\approx 0.7 {\AA }^{-1}\mathrm{in}$ the second SBZ of the $(\sqrt{3}\times \sqrt{3})$ reconstruction. (c) Map of the second derivative of the intensity towards energy $d^{2}I/d{E}^2$ to highlight details of the band structure. Also, the S1 band in the first SBZ of the $(\sqrt{3}\times \sqrt{3})$ reconstruction is shown. The yellow and green dashed lines indicate the Rashba splitting $\mathrm{\Delta }{k}_y$ of the S1 state. The horizontal yellow and blue lines show the energy, where the CEM and MDC have been taken, respectively. Only indications of the S2 band are visible. VBM denotes the Si-valence band maximum.

Figure 4

Band structure of the SIC phase along the $\overline{\mathrm{\Gamma }\text{M}}$ direction [direction (2) in Fig. 2]. The red (black) dotted line denotes the position of the ${\overline{\text{K}}}_{\sqrt{3}}$ point ($\overline{\text{M}}$ point). (a) Spin-integrated MDC taken at $E=95$ meV [see bluish area in (b) and (c)]. (b) The band map shows the metallic surface band S1 and the fully occupied surface band S2. (c) $d^{2}I/d{E}^2$ map of (b) to highlight details of the band structure. The magenta-shaded box denotes the phase space where spin-resolved measurements (shown in Fig. 6) have been taken, and the horizontal yellow line indicates the cut of the CEM in Fig. 2. The progression of the S2 state is taken from Ref. [13] and is in good agreement with the experiment. The Rashba splitting of S1 is $\mathrm{\Delta }{k}_x=0.04 {\AA }^{-1}$. S1* denotes the projection of the S1 states originating from the other two equivalent ${\overline{\text{K}}}_{\sqrt{3}}{\overline{\text{M}}}_{\sqrt{3}}$ directions.

Figure 5

Band structure of the SIC phase parallel to the $\overline{\text{KMK}}$ direction at $k_x=0.90 {\AA }^{-1}$[cf. with line (4) in Fig. 2]. (a) The MDC taken 95 meV below $E_{\text{F}}$ [see bluish area in right part of (b)] clearly shows a splitting due to the Rashba effect. (b) The band map on the left part and the $d^{2}I/d{E}^2$ map on the right part show the metallic S1 state as well as the spin-split state S2 at higher binding energies. The horizontal yellow line indicates the cut of the CEM in Fig. 2.

Figure 6

Spin-resolved MDCs (a), (c) and corresponding spin-polarization vector components (b), (d) along the $k_y$ direction through the ${\overline{\text{K}}}_{\sqrt{3}}$ point [(a), (b), cf. direction (3) in Fig. 2] and along the $\overline{\mathrm{\Gamma }\text{M}}$ direction [(c), (d), cf. direction (2) in Fig. 2] taken 145 meV below $E_{\text{F}}$. The shaded areas in (b) and (d) account for the uncertainty of the simultaneous fitting of all four data sets, i.e., $S_x, S_y, S_z$ and the MDC. Details of the fit parameters for the individual peaks (labeled by a-f) are summarized in Tables 1 and 2.

Figure 7

Modeling of the band structure and spin texture. (a) CEM taken 178 meV below $E_{\text{F}}$. The spin-polarized constant energy contours (red and blue lines) of the S1 states are deduced from nonlinear Rashba and warping terms. The threefold-symmetric boomerang-shaped isoenergy lines originate from the S2 state. (b), (c) Spin texture of the $S{1}^{+}$ and $S{1}^{-}$ states around the $\overline{\mathrm{\Gamma }}$ point. The arrows indicate the orientation and strength of the in-plane components, while $S_z$ is color coded. The hexagonally warped lines show the isoenergy lines at $E_{\text{F}}$. Values at certain $(k_x,k_y)$ points of the S1 state are reported in Tables 1 and 2 and are compared with the experiment. (d) In-plane spin texture around the ${\overline{\text{K}}}_{\sqrt{3}}$ points of the S1/S1* state at $E_{\text{F}}$ showing opposite spin polarization of the inner and outer ellipses, respectively. (e) Orientation of the effective magnetic field of the S2 state around the ${\overline{\text{K}}}_{\sqrt{3}}$ point, which generates the spin textures shown in (f) and (g). (f), (g) Spin texture of the $S{2}^{+}$ and $S{2}^{-}$ states around the ${\overline{\text{K}}}_{\sqrt{3}}$ point. The circles are isoenergy lines at 300 and 400 meV binding energy, respectively. The modeling of the spin texture has been done with a spin polarization of $S=100$%. The $k_x$ values are given with respect to the ${\overline{\text{K}}}_{\sqrt{3}}$ point.