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### $L$-gap surface resonance at Pt(111): Influence of atomic structure, $d$ bands, and spin-orbit interaction

##### Fabian Schöttke, Peter Krüger, Lutz Hammer, Tilman Kißlinger, M. Alexander Schneider, and Markus Donath

##### Phys. Rev. Research **6**, 023314 – Published 24 June 2024

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#### Abstract

Pt(111) hosts a surface resonance with peculiar properties concerning energy vs momentum dispersion and spin texture. At variance with the free-electron-like behavior of the $L$-gap Shockley-type surface states on the fcc(111) surfaces of Au, Ag, and Cu, it splits into several branches with distinct spin polarization around the center of the surface Brillouin zone $\overline{\mathrm{\Gamma}}$. Theoretical predictions based on density-functional theory vary depending on the particular functionals used. To clarify this issue, we investigate the atomic structure of Pt(111) by low-energy electron diffraction and the unoccupied electronic structure by spin- and angle-resolved inverse photoemission. The experimental results are backed by theoretical studies using different functionals, which show that the characteristics of the surface band depend critically on the lattice constant. From the analysis of the energy-dependent low-energy electron diffraction intensities, we derive structural parameters of the Pt(111) surface relaxation with high accuracy. In addition, we give an unambiguous definition of the nonequivalent mirror-plane directions $\overline{\mathrm{\Gamma}}\phantom{\rule{0ex}{0ex}}\overline{\text{M}}$ and $\overline{\mathrm{\Gamma}}\phantom{\rule{0ex}{0ex}}{\overline{\text{M}}}^{\prime}$ at fcc(111) surfaces, which is consistent with band-structure calculations and inverse-photoemission data. Concerning the surface resonance at the bottom of the $L$ gap, we identified a delicate interplay of several contributions. Lattice constant, hybridization with $d$ bands, and the influence of spin-orbit interaction are critical ingredients for understanding the peculiar energy dispersion and spin character of the unoccupied surface resonance.

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- Received 29 February 2024
- Accepted 24 May 2024

DOI:https://doi.org/10.1103/PhysRevResearch.6.023314

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

#### Physics Subject Headings (PhySH)

- Research Areas

Electronic structureSpin-orbit couplingSurface states

- Physical Systems

Elemental metalsFace-centered cubic

- Techniques

Current-voltage low-energy electron diffractionDensity functional theoryGGAInverse photoemission spectroscopyLDA

Condensed Matter, Materials & Applied Physics

#### Authors & Affiliations

Fabian Schöttke^{1,*}, Peter Krüger^{2}, Lutz Hammer^{3}, Tilman Kißlinger^{3}, M. Alexander Schneider^{3}, and Markus Donath^{1,†}

^{1}Physikalisches Institut, Universität Münster, Wilhelm-Klemm-Straße 10, 48149 Münster, Germany^{2}Institut für Festkörpertheorie, Universität Münster, Wilhelm-Klemm-Straße 10, 48149 Münster, Germany^{3}Lehrstuhl für Festkörperphysik, Universität Erlangen-Nürnberg, Staudtstraße 7, 91058 Erlangen, Germany

^{*}Contact author: fabian.schoettke@uni-muenster.de^{†}Contact author: markus.donath@uni-muenster.de

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#### Images

###### Figure 1

(a)Ball model of the Pt(111) surface in side view ($z$ scale exaggerated) with relaxation of layer spacings (in percent of the bulk layer distance) derived from the LEED-

*I(V)*analysis (red) compared with the predictions of DFT (LDA: green; PW91-GGA: blue). (b)Selection of experimental*I(V)*spectra for various beams with their counterparts calculated for the best fit structural model. The $R$ factor values denote the quality of the single beam correspondence. (c)Error curves displaying the $R$ factor increase with a vertical shift of single layers from their best fit position. The range below the $R+\text{var}\left(R\right)$ line is an estimate for the statistical error of the determined vertical positions of layers.###### Figure 2

(a)Real-space sketch of the fcc(111) surface with first (yellow) and second (red) layer, and crystallographic directions. (b)The fcc reciprocal structure (purple truncated octahedron) with projection to the (111) surface Brillouin zone (yellow). (c)LEED pattern of Pt(111) with $\left(1\right|0)$ and $\left(0\right|1)$ diffraction spots and overlayed surface Brillouin zone. (d)Measured LEED-

*I(V)*spectra of the $\left(1\right|0)$ and $\left(0\right|1)$ diffraction spots [see Fig.1]. The dashed red line marks the energy of the LEED pattern in (c), indicating the observed intensity difference.###### Figure 3

Angle-resolved IPE spectra of Pt(111) along the high-symmetry directions (a)$\overline{\mathrm{\Gamma}}\phantom{\rule{0ex}{0ex}}\overline{\text{M}}$, (b)$\overline{\mathrm{\Gamma}}\phantom{\rule{0ex}{0ex}}{\overline{\text{M}}}^{\prime}$, and (c)$\overline{\mathrm{\Gamma}}\phantom{\rule{0ex}{0ex}}\overline{\text{K}}$. The blue and purple lines connect spectral features attributed to transitions into the surface resonance SR and image-potential state IS, respectively. The black lines indicate features from transitions into bulk states B1–B8. The wide gray lines indicate non-

**k**-conserving transitions to $d$ bands.###### Figure 4

Energy vs ${\mathbf{k}}_{\parallel}$ dispersion of electronic states at Pt(111) along $\overline{\mathrm{\Gamma}}\phantom{\rule{0ex}{0ex}}{\overline{\text{M}}}^{\prime}$ (left) and $\overline{\mathrm{\Gamma}}\phantom{\rule{0ex}{0ex}}\overline{\text{M}}$ (right). Squares represent peak positions of experimental spectra from Figs.3, 3. The golden-shaded areas display the calculated surface-projected bulk band structure. The brown dots represent calculated and ${k}_{\perp}$-selected transitions to bulk states with vector potential along ${A}_{\left[111\right]}$ (see text for details). Yellow dots represent transitions with ${A}_{\left[\overline{2}11\right]}$ and green dots transitions with ${A}_{\left[0\overline{1}1\right]}$. The size of the dots is proportional to the transition probability.

###### Figure 5

(a)IPE spectra of the image-potential state IS at Pt(111) for various angles of electron incidence $\theta $ along ${\overline{\text{M}}}^{\prime}\overline{\mathrm{\Gamma}}\phantom{\rule{0ex}{0ex}}\overline{\text{M}}$. Black lines are fits to the data (see text for details.) (b)$E\left({\mathbf{k}}_{\parallel}\right)$ dispersion of IS with a parabolic fit to the data resulting in an effective mass of ${m}^{*}/{m}_{\text{e}}=1.21\pm 0.05$ and a binding energy ${E}_{\text{B}}={E}_{\text{V}}-E=0.55\pm 0.07$eV. For comparison, the dashed gray line shows the free-electron parabola (${m}^{*}/{m}_{\text{e}}=1$).

###### Figure 6

Theoretical data from DFT calculations in (a)PW91-GGA and (b)LDA. Gray lines represent results from the slab calculation and golden-shaded areas the surface projected bulk band structure. The $L$-gap surface feature is shown with its up and down spin polarization as red and blue dots. The symbol size is proportional to the expectation value of the Rashba spin-polarization direction as defined in the inset of (b). The paths for IPE measurements at $\theta =\pm {2}^{\circ}$ are shown as thin black lines.

###### Figure 7

Theoretical data from DFT calculations in (a)PW91-GGA and (b)LDA as a function of the lattice constant $a$. Displayed are the energetic positions of the SR branches (with their spin polarization) and of the $d$ bands at a ${k}_{\parallel}$ value that corresponds to $\theta ={2}^{\circ}$ along $\overline{\mathrm{\Gamma}}\phantom{\rule{0ex}{0ex}}\overline{\text{M}}$. Symbols and colors have the same meaning as in Fig.6.

###### Figure 8

(a)–(c)Simplified sketches of the $L$-gap surface states and $d$ bands at (a)Au(111) [5, 7], (b)Ni(111) [64], and (c)Pt(111) [data from Fig.6]. Red and blue lines in (a)and (c)denote the Rashba-type spin polarization as defined in Fig.6. Red and green colors in (b)indicate minority and majority spin directions, respectively. (d)Spin- and angle-resolved IPE spectra of Pt(111) along $\overline{\mathrm{\Gamma}}\phantom{\rule{0ex}{0ex}}\overline{\text{M}}$. Red (blue) triangles indicate spin up (down). The spectra for $\theta =+{2}^{\circ}[\theta =-{2}^{\circ}]$ are enlarged in (e) [(f)] and presented with the corresponding spin asymmetry.