A combined experimental and theoretical study of Rashba-split surface states on the $(\sqrt{3}\times \sqrt{3})$ Pb/Ag(111)$R30{}^\circ$ surface

The STM experiments have been performed in a two-chamber UHV system equipped with a low-temperature STM working at 5 K. The system consists of separate chambers for sample preparation and surface analysis (base pressure $p \leqslant 1\times {{10}^{-10}}$ mbar). STM tips were electrochemically etched from a polycrystalline tungsten wire. The Ag(111) single-crystalline substrate was cleaned by cycles of $\text{A}{{\text{r}}^{+}}$-sputtering (E = 500 eV) and subsequent annealing ($T\approx 700\;\text{K}$). For preparation of the $(\sqrt{3}\times \sqrt{3})\text{Pb}/\text{Ag}(111)R30{}^\circ$ surface alloy a third of a Pb monolayer was evaporated from a home-built Knudsen cell onto the Ag(111) substrate held at $T\approx 520$ K. STS spectra and dI/dU maps for QPI measurements have been taken by means of lock-in technique ($\nu =789$ kHz).

The ARPES data were collected with a hemispherical electron analyzer (Scienta R4000) and a monochromated He discharge lamp (MB Scientific) operating at an excitation energy of 21.22 eV (He Iα). The energy and angular resolutions were 7 meV and 0.3°, respectively. ARPES measurements were performed at a base pressure $p<2\times {{10}^{-10}}$ mbar and a temperature of 25 K. The single-crystalline Ag(111) substrate was prepared by $\text{A}{{\text{r}}^{+}}$-sputtering (E = 500 eV) and annealing ($T\approx 900$ K). Pb was evaporated from a commercial Knudsen cell at a rate of approximately 0.03 ML min⁻¹. Before each evaporation procedure the Ag(111) substrate was mildly annealed and subsequently allowed to cool down during evaporation. The formation of the surface alloy was confirmed by LEED.

The calculation of the surface electronic structure is based on DFT employing the full-potential linearized augmented plane-wave method as implemented in the FLEUR code ⁵ . The surface was simulated in a thin film geometry with a nine-layer Ag(111) film terminated on one end by the $(\sqrt{3}\times \sqrt{3})$ Pb/Ag(111)$R30{}^\circ$ surface alloy reconstruction. For further specifications of the calculation, we refer the reader to [7, 13].

Figure 2(a) shows the sample surface of Pb/Ag(111) at a Pb coverage just above one third of a monolayer. Coexisting domains of the $(\sqrt{3}\times \sqrt{3})\;\text{Pb}/\text{Ag}(111)R30{}^\circ$ surface alloy (A) and the Pb wetting layer (B) can be found on the surface [22]. The wetting layer domains shows a characteristic moiré pattern which can also be recognized in the atomically resolved STM image of figure 2(b). It is preferentially found at step edges. Figure 2(c) shows an atomically resolved image of the $(\sqrt{3}\times \sqrt{3})\;\text{Pb}/\text{Ag}(111)R30{}^\circ$ surface alloy with a few defects. The visible modulation originates from QPI and will be discussed in more detail below. figure 2(d) displays the corresponding LEED pattern. It consists of an outer hexagon that belongs to the underlying Ag(111) substrate and an inner hexagon which is rotated by $30{}^\circ$ and reflects the $(\sqrt{3}\times \sqrt{3})\;R30{}^\circ$ symmetry of the surface alloy. figure 3 shows an averaged tunneling spectrum of the $(\sqrt{3}\times \sqrt{3})\;\text{Pb}/\text{Ag}(111)R30{}^\circ$ surface alloy. We can recognize two asymmetric peaks, the shape of which closely resembles the singularities described in [5]. The DOS of downwards dispersing Rashba-split bands is described by

$$\begin{eqnarray}\rho (E)=\left\{ \begin{matrix} \frac{|{{m}^{*}}|}{\pi {{\hbar }^{2}}}\sqrt{\frac{{{E}_{\text{R}}}}{{{E}_{0}}-E}}, & \quad \qquad {{E}_{0}}-{{E}_{\text{R}}}<E<{{E}_{0}} \\ \frac{|{{m}^{*}}|}{\pi {{\hbar }^{2}}}=\text{const}, & \quad \qquad E<{{E}_{0}}-{{E}_{\text{R}}} \\ 0, & \quad \qquad \text{else} \\ &&\end{matrix} \right.\end{eqnarray} \tag{ 1 }$$

where ${{m}^{*}}$ is the effective mass. The Rashba energy ${{E}_{\text{R}}}=\frac{{{\hbar }^{2}}k_{0}^{2}}{2\;{{m}^{*}}}$ describes the splitting strength. We have fitted the peaks following the algorithm described in [5]. Obviously, both peaks are very well described by the fits and can clearly be assigned to the $s,{{p}_{z}}$- and the ${{p}_{x}},{{p}_{y}}$-derived Rashba states as schematically represented in figure 1(c) (${{\Delta }_{z}}=0.67$) and described in [7]. For the ${{p}_{x}},{{p}_{y}}$-derived state our fit results in ${{E}_{1}}=(1580\pm 5)\;\text{meV}$ and $\Delta {{E}_{{{\text{R}}_{1}}}}=(130\pm 10)$ meV. For the $s,{{p}_{z}}$- derived band we obtain ${{E}_{2}}=(710\pm 5)$ meV and $\Delta {{E}_{{{\text{R}}_{2}}}}=(80\pm 10)$ meV, i.e. slightly further away from the Fermi level than reported previously [5].

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We would like to emphasize that in contrast to Bi/Ag(111), where we observed an obvious difference between the measured data and the fit for the occupied surface state due to a distinct shoulder [17], the agreement of the fit is very good for on Pb/Ag(111) at both peak positions. As has been pointed out in [17] the shoulder is ascribed to the opening of a hybridization gap which—as a result of the flat energy dispersion around the gap—leads to an enhanced DOS. Based on these observations we do not find any hint for hybridization of the two Rashba-split surface states on Pb/Ag(111), thereby favoring the model of figure 1(c).

To obtain the energy dispersion we measured two series of QPI maps by recording the spatial variation of the dI/dU signal in the bias range between $-600$ meV (occupied states) and 1600 meV (empty states) with an increment of 50 meV. Figure 4(a) shows the topography of the first sample area. Arrows indicate the crystallographic directions relative to the Pb overlayer, which are rotated by $30{}^\circ$ with respect to the Ag(111) substrate, i.e. ${{\overline{\Gamma \text{K}}}_{\text{Pb}}}\shortparallel {{\overline{\Gamma \text{M}}}_{\text{Ag}}}$ and ${{\overline{\Gamma \text{M}}}_{\text{Pb}}}\shortparallel {{\overline{\Gamma \text{K}}}_{\text{Ag}}}$. We can recognize atomically flat terraces which are separated by two step edges oriented along different crystallographic directions. Again, at the step edges we can identify small domains of the Pb wetting layer on Ag(111). Figures 4(b) and (c) show the interference patterns measured at this location at $U=-163\;\text{mV}$ and $U=-263\;\text{mV}$, respectively. Standing waves parallel to step edges and around defects are clearly visible. The corresponding FT QPI maps are shown in the insets. One can recognize a hexagonally shaped frame with cusps pointing into $\overline{\Gamma \text{K}}$ directions. Close inspection of the inset figure 4(c) reveals a second hexagon which is slightly larger and much less pronounced. The latter hexagon is rotated by $30{}^\circ$ with respect to the former, consequently pointing into $\overline{\Gamma \text{M}}$ directions. The QPI maps for determining the electronic properties of the ${{p}_{x}},{{p}_{y}}$-derived state have been measured on the same sample but at a different, second area, due to a tip change. The data obtained at the two positions have some energetic overlap confirming compliance of the observed wave vectors. The constant-current STM topograph of this area is shown in figure 5(a). Again, one can recognize atomically flat terraces which are separated by two step edges. The pattern observed in figure 5(b) is not as strong and pronounced as the interference patterns in figures 4(b) or (c). We speculate that the weaker interference pattern is due to the reduced lifetime of electronic states which are energetically further away from the Fermi level. As we discuss later, there are different scattering processes responsible for the two energy regions. This might also affect the appearance of the patterns. In the corresponding 2D FFT (inset of figure 5(b)) we again observe a hexagonally shaped frame with cusps pointing into the $\overline{\Gamma \text{M}}$ directions.

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The detailed analysis of the quasiparticle interference maps recorded over a wide range of bias voltages results in the extracted scattering vector q(E) displayed in figure 6(a). Clearly, we observe an energy dispersion of q(E) for both Rashba bands. Scattering within the ${{p}_{x}},{{p}_{y}}$-derived band (labeled ${{\text{D}}_{1}}$ in figure 6) starts at about 1500 meV. The parabolic evolution of scattering vectors confirms the dominant role of intra-band scattering in this energy range ($q=2k$ in this case). The dispersion was fitted with a cosine since it yields the best agreement to the experimentally observed scattering vectors. From this fit we obtain the band onset of the ${{p}_{x}},{{p}_{y}}$-derived state at ${{E}_{1}}=(1556\pm 44)$ meV. This value is in good agreement with our STS data presented in figure 3, thereby confirming the model given in figure 1(c). We obtain effective masses of $m_{\overline{\Gamma \text{K}}}^{*}=(-0.14\pm 0.01){{m}_{\text{e}}}$ and $m_{\overline{\Gamma \text{M}}}^{*}=(-0.15\pm 0.001){{m}_{\text{e}}}$ along the two high-symmetry directions.

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The analysis of the QPI maps taken in the energy range of the $s,{{p}_{z}}$-derived bands turned out to be not as straightforward. Surprisingly, we observe two and at some bias voltages even three scattering vectors, which are marked ${{\text{D}}_{2}}$, ${{\text{D}}_{3}}$ and ${{\text{D}}_{4}}$ in figure 6(a). The shorter scattering vector ${{\text{D}}_{2}}$ corresponds well to intra-band scattering events within the $s,{{p}_{z}}$-branches. Fitting ${{\text{D}}_{2}}$ with a parabolic function leads to ${{E}_{2}}=(708\pm 22)$ meV, which is in a good agreement with our STS data presented in figure 3. For the effective masses we obtain $m_{\overline{\Gamma \text{K}}}^{*}=(-0.166\pm 0.005){{m}_{\text{e}}}$ and $m_{\overline{\Gamma \text{M}}}^{*}=(-0.149\pm 0.004){{m}_{\text{e}}}$. These values are slightly different from data obtained in [5]. The signal ${{D}_{3}}$ in the coexistence area (labeled ${{\text{D}}_{3}}$ in figure 6(a)) shows the same anisotropy as the scattering events for the ${{p}_{x}},{{p}_{y}}$-derived state.

In order to compare the scattering vectors ${{\text{D}}_{2}}$–${{\text{D}}_{4}}$ obtained from QPI mapping with the occupied electronic band structure of Pb/Ag(111) we have performed ARPES measurements. The ARPES data are shown in figures 6(b)–(d). The spectrum in figure 6(b) shows the band dispersion along the $\overline{\Gamma \text{M}}$ direction. In agreement with previous works [4, 9] we identify three surface bands with negative effective mass ${{m}^{*}}$ which are labeled accordingly (we follow the notation of [9]). In addition we observe the bulk band u that is back-folded due to the surface reconstruction [4]. The two inner bands originate from one Rashba-split surface band of mainly $s,{{p}_{z}}$ orbital character with a momentum splitting of $2{{k}_{0}}=0.06(1)\;{{}^{-1}}$. The momentum separation between r1 and l2 and the corresponding r2 and l1 branches indeed matches with the scattering vectors ${{\text{D}}_{2}}$ determined by STM. The outer branches r3 and l3 correspond to another surface band of ${{p}_{x}},{{p}_{y}}$ character. For this band the ARPES data do not allow us to deduce a finite momentum splitting. Within the measured energy window the splitting $2{{k}_{0}}$ of the ${{p}_{x}},{{p}_{y}}$-band is smaller than $\sim 0.04\;{{}^{-1}}$ and thus considerably reduced compared to the $s,{{p}_{z}}$-band. This finding is in line with DFT calculations presented below that indicate only a very small splitting of the ${{p}_{x}},{{p}_{y}}$-band below ${{E}_{F}}$. The scattering vectors labelled ${{D}_{3}}$ can then be assigned to scattering events between the bands l3 and r1 or between the corresponding bands r1 and l3. The origin of ${{D}_{4}}$ remains unclear, as we cannot identify any potential scattering vector from the ARPES data that would quantitatively fit the periodicities observed in the QPI experiments.

More detailed information about the dispersion of the surface bands within the full k-plane is obtained from the constant energy cuts in figures 6(c) and (d) that depict the photoemission intensity as a function of both wave vector components ${{k}_{x}}$ and ${{k}_{y}}$ parallel to the surface at fixed energy. In figure 6(c) one can discern two inner contours that correspond to the split $s,{{p}_{z}}$-band. Note that the contour originating from the branches l2 and r2 shows a hexagonal deformation. The cusps of the hexagonal contour point along the $\overline{\Gamma \text{K}}$ direction, implying $m_{\overline{\Gamma \text{K}}}^{*}>m_{\overline{\Gamma \text{M}}}^{*}$ for the branches l2 and r2. The contour corresponding to the ${{p}_{x}},{{p}_{y}}$-band (branches l3 and r3) is best visible in figure 6(d). Also in this case we find a significant hexagonal warping with cusps of the contour pointing along the $\overline{\Gamma \text{M}}$ direction. Hence, for the ${{p}_{x}},{{p}_{y}}$-band one has $m_{\overline{\Gamma \text{M}}}^{*}>m_{\overline{\Gamma \text{K}}}^{*}$.

Obviously, ARPES finds anisotropies of the $s,{{p}_{z}}$- and the ${{p}_{x}},{{p}_{y}}$-band which are rotated by $30{}^\circ$ with respect to each other. This finding is in line with our STM results and DFT calculations [7]. Qualitatively, the same situation has also been found for other isostructural surface alloys, such as Bi/Cu(111) [12, 13], Sb/Cu(111) [12] and Sb/Ag(111) [10]. Within a tight-binding picture we can interpret the rotated anisotropy of the two bands by assuming that a smaller effective mass corresponds to a larger orbital overlap. For the ${{p}_{x}},{{p}_{y}}$-band the orbital overlap is then maximal along the direction connecting two neighbouring Pb atoms ([$1\bar{1}0$] corresponding to $\overline{\Gamma \text{K}}$). On the other hand, in the case of the $s,{{p}_{z}}$-band the orbital overlap is largest along the direction connecting a Pb atom with its neighbouring Ag atoms ([$11\bar{2}$] corresponding to $\overline{\Gamma \text{M}}$). Hence, for the ${{p}_{x}},{{p}_{y}}$-band and the $s,{{p}_{z}}$-band the lateral atomic bonding appears to be dominated by Pb–Pb and Pb–Ag nearest neighbour interaction, respectively.

With all parameters being determined we can exclude the band model schematically represented in figure 1(b). Instead, our data are consistent with the smaller relaxation model sketched in figure 1(c), where no hybridization between the two Rashba bands takes part.

In figure 7(a) we show the calculated spin-resolved band structure of the Pb-derived states around the Fermi level (${{E}_{\text{F}}}$) for a Pb relaxation of ${{\Delta }_{z}}=0.67$. It is clearly seen that the $s,{{p}_{z}}$ states with a band onset at about 0.5 eV show a strong Rashba-type spin polarization (as well as the ${{p}_{z}}$ states at 2.5 eV), while the ${{m}_{1/2}}$-bands, the band onset of which is at 1.5 eV, exhibit a more complex spin texture. In particular around the $\bar{\Gamma }$-point the polarization decreases rapidly and even exhibits a reversal of the spin direction at some $\mathbf{k}$-points . As the ${{m}_{1/2}}$-bands show no direct crossing with the $s,{{p}_{z}}$-bands (in contrast to the $\text{BiA}{{\text{g}}_{2}}$ case and the $\text{PbA}{{\text{g}}_{2}}$ with stronger relaxation), for larger k-values the spin direction reverses again and increases to larger momentum values.

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To shed light on the origin of this loss of spin polarization near the $\bar{\Gamma }$-point, we show the spin- and orbital-resolved band structure in figure 7(b) focusing on the ${{p}_{x}}$ and ${{p}_{y}}$ states. Apparently, near the Brillouin-zone the states consist of a mixture of ${{p}_{x}}$ with one spin-orientation and ${{p}_{y}}$ states of the opposite spin orientation. It is well known from hole-states in semiconductors that spin–orbit coupling at small momenta can lead to the formation of orbital-moment-carrying states (${{m}_{j}}=1/2$) [23], while Rashba-type spin–orbit coupling dominates at larger k–values and forces the spin direction in-plane. Additionally, crystal-field effects decouple the ${{p}_{x}}$ and ${{p}_{y}}$ states according to the momentum direction.

Although an apparent decrease of spin polarization for the ${{m}_{1/2}}$ states at small k-values is clearly visible in figure 7, we cannot directly identify the origin of this reduction. In principle, also an increased out-of-plane component of the spin moment may lead to the same behavior. We checked for this possibility and found no significant spin component along the surface normal, but rather a mixture of strongly in-plane polarized ${{p}_{x}}$ and ${{p}_{y}}$ orbitals with opposite spin polarization. We have to note here that, although the total spin polarization is dramatically reduced near the center of the Brillouin-zone, the scattering from $\mathbf{k}$ to $-\mathbf{k}$, i.e. between time-reversal partners, is still prohibited by time reversal symmetry [24].

Finally, we can turn to the question why the intra-band scattering—as indicated by the arrow in figure 7(b)—is allowed, despite the fact that initial and final states are dominantly of opposite spin character. Since the total momentum, $\mathbf{J}$, has to be conserved in the scattering process, the associated transfer of spin angular momentum has to be compensated by an equivalent transfer of orbital momentum, $\mathbf{L}$. Loosely speaking we can say that, if $\Delta \mathbf{J}=\Delta \mathbf{L}+1/2\Delta \mathbf{S}=0$, a $180{}^\circ$ reversal of spin $\mathbf{S}$ needs to be compensated by a $90{}^\circ$ rotation of the orbital moment, i.e. from a ${{p}_{x}}$ to a ${{p}_{y}}$ orbital or vice versa. This is exactly the process allowed by the spin-flip (off-diagonal) part of the spin–orbit coupling ⁶ [25].

Indeed, the scattering process indicated by the black arrow in figure 7(b) connects ${{p}_{x}}$ to ${{p}_{y}}$ orbitals of opposite spin and is, therefore, allowed. A forward scattering process, as could be imagined in the $\text{PbA}{{\text{g}}_{2}}$ surface alloy between different bands of the same spin, can be excluded since it involves scattering from a ${{p}_{x}}$ to a ${{p}_{y}}$ orbital without a spin-flip. The interpretation of scattering events in terms of spin polarization of the involved bands cannot be drawn from a plot of the spin polarization of the states (like figure 7(a)) alone, since it is not directly possible to draw conclusions about allowed or forbidden quasi-particle scattering events. Only in a simple case, i.e. when the involved states do not carry orbital moments ($s,{{p}_{z}},{{d}_{{{z}^{2}}}}$), the Rashba picture of spin polarization holds and selection rules can be directly inferred. In the case of the Bi(110) surface analyzed in this way [26] this was meaningful, since the surface states of this surface are of ${{p}_{z}}$ character. In the present system, however, such assignments are misleading, since—in addition to the spin—also the orbital character has to be taken into account. In the same spirit the STM data of [17] can be reconciled with the calculations presented in [7], as we checked by explicit calculations.