Emergence of nearly flat bands through a kagome lattice embedded in an epitaxial two-dimensional Ge layer with a bitriangular structure

Ge atoms segregating on zirconium diboride thin films grown on Ge(111) were found to crystallize into a two-dimensional bitriangular structure, which was recently predicted to be a flat band material through an embedded kagome lattice. Angle-resolved photoelectron emission experiments together with theoretical calculations pointed out the existence of a nearly flat band in spite of non-negligible in-plane long-range hopping and interactions with the substrate. This provides an experimental verification for the fact that a flat band can emerge from the electronic coupling between atoms and not from the geometry of the atomic structure.

Figure 1

Spontaneous segregation of Ge atoms on an oxide-free ${\mathrm{ZrB}}_2$ thin film grown on a Ge(111) substrate. (a) and (b): Photoelectron spectra recorded in the $\mathrm{Zr}3d\text{−}\mathrm{B}1s$ and in the $\mathrm{Ge}3d$ core-level region, respectively, before (black) and after (red) annealing at $770{}^{\circ }\mathrm{C}$ for 2 h. The inset in panel (a) shows spectra recorded in the $\mathrm{O}1s$ region. Photon energies are indicated. The vertical bars at energies of 29.56, 29.40, 29.07, and 28.92 eV indicate calculated energies and the relative theoretical spectral weights of the $\mathrm{Ge}3d$ peaks of the structure of Figs. 3 and 3. (c) and (d): STM images ($150\times 100{\mathrm{nm}}^2$, $V=0.95\mathrm{V}$, $I=116\mathrm{pA}$ and $5\times 4.3{\mathrm{nm}}^2$, $V=0.55\mathrm{V}$, $I=56\mathrm{pA}$) recorded after annealing. The $(\surd 3\times \surd 3)$ unit cell is marked by a red rhombus. The inset shows LEED pattern recorded with electron energy of 70 eV. $(1\times 1)$ and $(\surd 3\times \surd 3)$ spots are indicated by green and red circles, respectively.

Figure 2

The Ge bitriangular lattice on ${\mathrm{ZrB}}_2$ thin films as determined from TRHEPD. (a): TRHEPD rocking curves plotted with open circles and calculated curves plotted with solid lines for various integer- and fractional-order diffraction spots (left: $\left[1\overline{1}00\right]$ incidence, right: $\left[11\overline{2}0\right]$ incidence). (b) and (c): Top and side views of the Ge bitriangular structure resulting from the fitting of the TRHEPD rocking curves. Top and bottom Ge atoms are light blue- and dark blue-colored, Zr and B atoms are in green and orange, respectively. The red rhombus emphasizes the ${\mathrm{ZrB}}_2$(0001)-$(\surd 3\times \surd 3)$ unit cell.

Figure 3

Band structure of the Ge bitriangular lattice on ${\mathrm{ZrB}}_2$. (a) and (b): Top- and side-views of the bitriangular structure determined from DFT calculations. (c): ARPES intensity plot obtained from spectra recorded along the $\mathrm{\Gamma }\text{−}M$ direction of the Brillouin zone of ${\mathrm{ZrB}}_2$(0001)-$(1\times 1)$ indicated by an arrow in the inset. Brillouin zones of the ${\mathrm{ZrB}}_2$(0001)-$(1\times 1)$ and of the bitriangular lattice are represented by green and red lines, respectively. (d): Calculated band structure of the Ge bitriangular lattice on ${\mathrm{ZrB}}_2$(0001) superimposed with the ARPES spectrum. The bands are plotted in pink and the weight of the contribution of the Ge $p_z$ orbitals is shown by the diameter of the circles. Bands discussed in the text are colored in orange. The energy region above Fermi level is darkened.

Figure 4

Comparison of calculated band structures for the free-standing Ge bitriangular lattice and the one in contact with the ${\mathrm{ZrB}}_2$ surface. (a): Band structure originating from the tight-binding model shown in (b) for $t_2^2=\varepsilon t_1+3t_1^2$ with $t_1=-0.281t$, $t_2=-0.7t$, and $\varepsilon =-0.9t$. The site energy of the bottom Ge atom is set to zero, which leads the energy of the flat band to $-3t_1=0.843t$. The flat band and the second band discussed in the text are colored in red and blue, respectively. (c)–(f): Contribution of the Ge ${\mathrm{p}}_{\mathrm{z}}$ orbitals to (c) the band structure and (d) the DOS of free-standing Ge bitriangular lattice and to (e) the band structure and (f) the DOS of epitaxial Ge bitriangular lattice on ${\mathrm{ZrB}}_2$(0001). Bands colored in red and blue in panels (c) and (e) indicate the bands that are derived from those indicated by the same colors in panel (a). (g): 2D plot of the band emphasized by a black line in (e) within the Brillouin zone of the bitriangular structure.