Tuning MgB${}_2$(0001) surface states through surface termination

Surface state localization and hybridization on B, Mg, and Li terminated MgB${}_2$(0001) surfaces are studied within density functional theory (DFT). For the B terminated surface we find a low energy B ${\sigma }_1$ surface state, a $s{p}_z$ surface state, and B ${\sigma }_2$ and ${\sigma }_3$ quantum well states, which are $90\%$ localized in the topmost B layer. Our results demonstrate that by charging the B atomic layer, either by changing the surface termination or through electro-chemical doping, the B $\sigma$ surface states are shifted down in energy, filled, and delocalize through hybridization as they cross the bulk MgB${}_2$ bulk bands. On the other hand, the $s{p}_z$ surface state should be shifted up in energy, emptied, and gain an increasingly metallic $s$ character by adding a Mg, Mg${}^{+1}$, or Li terminating atomic layer. These results clearly show both the robust nature of MgB${}_2$(0001) surface states, and how their localization and energy range may be tuned by surface termination and charging, with implications for the superconducting and plasmonic behavior of MgB${}_2$.

Figure 1

Schematic of the structurally optimized MgB${}_2$(0001) surface unit cells with B, Mg, and Li termination, repeated twice in each direction.

Figure 2

Projected density of surface and subsurface states ${\varrho }_{n,\mathbf{k}}\left(z\right)$ in the B terminated MgB${}_2$(0001) surface vs position normal to the surface plane $z$, and Kohn-Sham eigenenergies ${\varepsilon }_{n,\mathbf{k}}$ relative to the Fermi level ${\varepsilon }_F$. Results from plane wave PWscf calculations before (—) and after (– –) structural optimization are compared with real-space GPAW calculations for the relaxed structure ($\cdots$). Solid and dotted vertical lines denote positions of the B and Mg atomic layers, respectively. Inserts show ${\varrho }_{n,\mathbf{k}}\left(z\right)$ from plane wave PWscf calculations for a relaxed extended supercell model with six bulk unit cells added in the center of the slab.

Figure 3

Band structure (lines) and energies of the surface and subsurface states shown in Fig. 2 (symbols) for the B terminated MgB${}_2$(0001) surface in eV relative to the Fermi level ${\varepsilon }_F$. Results from plane wave PWscf calculations before (—; $$) and after (– – ; $█$) structural optimization are compared with real-space GPAW calculations for the relaxed structure ($\cdots$; $▵$).

Figure 4

GPAW overlap-based band structure of relaxed B terminated MgB${}_2$(0001) surface, with ${\psi }_{n,\mathbf{k}}\to {\psi }_{n^{\prime },\mathbf{k}+\Delta \mathbf{k}}$ for $n^{\prime }$ which maximize $|{\langle {\psi }_{n,\mathbf{k}}|{\psi }_{n^{\prime },\mathbf{k}+\Delta \mathbf{k}}\rangle }_{\mathcal{V}}|$. Surface states with more than two thirds of their weight in the first vacuum and first two bulk layers, $s_{n,\mathbf{k}}\gtrsim 0.66$, are shown separately as thicker lines.

Figure 5

Projected density of surface and subsurface states ${\varrho }_{n,\mathbf{k}}\left(z\right)$ in the Mg terminated MgB${}_2$(0001) surface vs position normal to the surface plane $z$, and Kohn-Sham eigenenergies ${\varepsilon }_{n,\mathbf{k}}$ relative to the Fermi level ${\varepsilon }_F$. Results from plane wave PWscf calculations before (—) and after (– –) structural optimization are compared with real-space GPAW calculations for the relaxed structure ($\cdots$). Solid and dotted vertical lines denote positions of the B and Mg atomic layers, respectively. Inserts show ${\varrho }_{n,\mathbf{k}}\left(z\right)$ from plane wave PWscf calculations for a structurally optimized extended supercell model with six bulk unit cells added in the center of the slab.

Figure 6

Band structure (lines) and energies of the surface and subsurface states shown in Fig. 5 (symbols) for the Mg terminated MgB${}_2$(0001) surface in eV relative to the Fermi level ${\varepsilon }_F$. Results from plane wave PWscf calculations before (– ; $$) and after (– – ; $█$) structural optimization are compared with real-space GPAW calculations for the relaxed structure ($\cdots$; $▵$).

Figure 7

GPAW overlap-based band structure of relaxed Mg terminated MgB${}_2$(0001) surface, with ${\psi }_{n,\mathbf{k}}\to {\psi }_{n^{\prime },\mathbf{k}+\Delta \mathbf{k}}$ for $n^{\prime }$ which maximizes $|{\langle {\psi }_{n,\mathbf{k}}|{\psi }_{n^{\prime },\mathbf{k}+\Delta \mathbf{k}}\rangle }_{\mathcal{V}}|$. Surface states with more than two thirds of their weight in the first vacuum and first two bulk layers, $s_{n,\mathbf{k}}\gtrsim 0.66$, are shown separately as thicker lines.

Figure 8

Projected density of surface and subsurface states ${\varrho }_{n,\mathbf{k}}\left(z\right)$ in the Li terminated MgB${}_2$(0001) surface vs position normal to the surface plane $z$, and Kohn-Sham eigenenergies ${\varepsilon }_{n,\mathbf{k}}$ relative to the Fermi level ${\varepsilon }_F$. Results from plane wave PWscf calculations before (—) and after (– –) structural optimization are compared with real-space GPAW calculations for the relaxed structure ($\cdots$). Solid, dotted, and dashed vertical lines denote positions of the B, Mg, and Li atomic layers, respectively. Insets show ${\varrho }_{n,\mathbf{k}}\left(z\right)$ from plane wave PWscf calculations for a structurally optimized extended supercell model with six bulk unit cells added in the center of the slab.

Figure 9

Band structure (lines) and energies of the surface and subsurface states shown in Fig. 8 (symbols) for the Li terminated MgB${}_2$(0001) surface in eV relative to the Fermi level ${\varepsilon }_F$. Results from plane wave PWscf calculations before (– ; $$) and after (– – ; $█$) structural optimization are compared with real-space GPAW calculations for the relaxed structure ($\cdots$; $▵$).

Figure 10

GPAW overlap-based band structure of relaxed Li terminated MgB${}_2$(0001) surface, with ${\psi }_{n,\mathbf{k}}\to {\psi }_{n^{\prime },\mathbf{k}+\Delta \mathbf{k}}$ for $n^{\prime }$ which maximizes $|{\langle {\psi }_{n,\mathbf{k}}|{\psi }_{n^{\prime },\mathbf{k}+\Delta \mathbf{k}}\rangle }_{\mathcal{V}}|$. Surface states with more than two thirds of their weight in the first vacuum and first two bulk layers, $s_{n,\mathbf{k}}\gtrsim 0.66$, are shown separately as thicker lines.

Figure 11

Surface and subsurface state energies at the $\overline{\Gamma }$ point, ${\varepsilon }_{n,\overline{\Gamma }}$ vs surface termination and doping of the MgB${}_2$(0001) surface by B, Mg$\to$Li$\sim$Mg${}^{+1}$, Li, and Mg, in eV relative to the Fermi level ${\varepsilon }_F$, for (a) B ${\sigma }_2$ and ${\sigma }_3$, (b) $s{p}_z$, and (c) B ${\sigma }_1$ states. Results from plane wave PWscf calculations before ($$) and after ($█$) structural optimization are compared with real-space GPAW calculations for the relaxed structures ($▵$). The B ${\sigma }_{2,3}$ and B ${\sigma }_1$ bulk band regions are shaded gray. Fits to the surface state positions (– –) are provided as guides to the eye.