Fermi surface of Mo(112) and indirect interaction between adsorbed atoms

A comprehensive examination of the Fermi surface of Mo(112) is presented. The Fermi surface contours for the Mo(112) surface, obtained by density functional theory calculations, agree well with the direct observations via angle-resolved photoemission spectroscopy and indicate the existence of flattened segments in the Fermi contours perpendicular to the direction of the atomic furrows. Both the calculation and the experiment indicate significant surface weight for these states. Such flattened Fermi contours at the surface can give rise to long-range charge density waves (CDW) and long-range indirect lateral interactions, especially in the case of adsorption of electropositive atoms. When mediated by the surface electrons, exhibiting flattened Fermi contours, the oscillatory potential of the indirect interaction between adsorbed atoms decays very slowly (∼1/$r$) in the direction along the furrows, which can explain the formation of long-period chain structures of electropositive adsorbates on the furrowed surface of Mo(112).

Figure 1

The schematic representation of the Mo(112) surface. Crystallographic directions are labeled in relation to the Cartesian coordinate axes used in our discussion.

Figure 2

The ARPES band mapping of Mo(112) along the $\overline{\Gamma }-\overline{\mathrm{X}}$ direction in $p$-polarization geometry using 50-eV photon energy. The energy versus wave-number plot (a), taken shortly after the cleaning, clearly shows the surface-weighted band, labeled as “Surface,” with $k_F$ = 0.59 ± 0.02 ${\AA }^{-1}$ that is estimated to have more than 78$\%$ charge localization within the first two layers.[62] In the energy versus wave-number plot (b), taken after adsorption of small amount of contamination (largely CO), the previously visible band “Surface” (a) is absent. The panel (c) shows the MDCs at the Fermi level taken with the incident photon energies of 22, 40, 50, and 70 eV and with $p$-polarization geometry.

Figure 3

The Fermi surface of Mo(112), calculated for the 7-layer (a) and 11-layer (b) Mo(112) slabs. The overlaid contour lines of the Fermi surface in the (112) plane calculated for the 7-layer (green), 9-layer (blue), and 11-layer (red) Mo(112) slabs in (c). Since the number of the bands crossing Fermi level depends on the thickness of the slab, the contours pertinent to particular thickness may be understood as representatives of the projection of a bulk Fermi surface, while those remaining essentially at the same position (so that they nearly coincide and thus do not depend on the thickness) are indicative for the Fermi contours with strong surface weights.

Figure 4

The Fermi surface of Mo(112) obtained at the photon energy of 50 eV with (a) $p$- and (b) $s$-polarization geometry as described in detail in the experimental section. In both polarization geometries, the sample was rotated by the increment of 3° to obtain the presented image. The contours of the Fermi surface calculated for the 11-layer Mo(112) slab are overlaid. The calculated Fermi contour lines with dots indicate the Fermi contours with strong surface weight.

Figure 5

The schematic of the effective potential for the lateral surface interaction along the furrows for Sr/Mo(112), resulting in widely spaced quasi-one-dimensional chains of Sr running perpendicular to the surface corrugation of Mo(112). (a) The potentials for Sr adatoms due to the surface-weighted states crossing $E_F$ (or the sheets of the Fermi surface crossing the $\overline{\Gamma }-\overline{\mathrm{X}}$ line) at $k_F$ = 0.58 ${\AA }^{-1}$ and ∼0.8 ${\AA }^{-1}$, calculated using Eq. (1) with appropriate phase δ, are shown by a black dotted and blue solid curves, respectively. The effective potential, including the both surface-weighted states and the dipole-dipole interaction between Sr adatoms (green dashed line), is shown as a red/ solid curve. (b) Including considerations for the potential relief of the substrate, (the depth of the relief along Mo(112) furrows for Sr, calculated by Kiejna and Nieminen,[42] is 0.08 eV) the adatom separation of 9$a$ becomes more favorable than that of 5$a$ because of the closer match of the minima between the effective potential (red) and potential relief (black) of Mo(112).

Figure 6

Snapshots of adsorbate ordering, abstracted from a sequence of Monte Carlo simulations for characteristic Sr coverages on Mo(112), and these are provided together with the related model LEED patterns, seen to be in agreement with experiment.[18] The $p$(9×1) structure is formed at coverages below and equal to 1/9 monolayer (a). With increasing coverage, the phase transition from the $p$(9×1) to $p$(5×1) structure reveals itself in characteristic coupling of LEED reflections (b). The formation of the p(5×1) structure is completed at the coverage of 1/5 monolayer (c).

Figure 7

The $p$(9×1) Sr structure on Mo(112) surface (a) is compared to the model scanning tunneling (STM) image (b). The induced CDW should be revealed in the modulation of the brightness of the image of Mo atoms of the rows, i.e., modulation of the intensity along the direction of the substrate corrugation.