Hybrid density functional calculations of the surface electronic structure of GdN

Rare-earth nitrides are a promising class of materials for application in spintronics, with GdN a particularly well-studied example. Here we perform band-structure calculations employing a hybrid density functional, which enables the band gap to be more accurately predicted through the inclusion of short-range exact exchange. The sensitivity of the band gap to the exchange term is demonstrated. The surface electronic structure is simulated through the use of slab models of the GdN(111) surface, which provide a consistent description of metallic surface states in the majority-spin channel.

Figure 1

Bulk band structure of GdN at the HSE06-optimized lattice constant. The left-hand figure shows the results of spin-polarized hybrid-DFT calculation, while on the right-hand site we have included spin-orbit coupling. For the spin-polarized case, local projections on the Gd and N atoms with a local contribution of at least 75% are also shown. In both cases, majority- and minority-spin bands are indicated as full and dotted black lines, respectively, while spin-orbit coupled bands are indicated in red.

Figure 2

Band gap (bottom) and averaged position of the occupied $f$ bands with respect to the Fermi energy (top) for calculations using a screened hybrid functional based on HSE06, including varying fractions of short-range exact exchange. The band-gap values are calculated at the $X$ point for the direct majority [E(MAJ)] and minority [E(MIN)] gap while E(INDIR) indicates the value for the indirect $\left(\mathrm{\Gamma }\to X\right)$ gap. The average $f$-electron energy is taken at the $\mathrm{\Gamma }$ point. Note that a fit for the minority-spin gap, ignoring the value obtained for zero exact exchange, gives $E=0.72+4.60EXX$. On the right-hand side, two band structures, calculated employing short-range exact exchange fractions of 17% (top) and 44% (bottom), are also shown. These represent the results of the linear fits for a zero direct $(X$-point) band gap in the majority-spin channel (17%) as well as the experimentally observed XPS peak corresponding to the ${\mathrm{Gd}}^{3+} 4f^6$ final state multiplet at $E-E_F=-7.8$ eV [43] (44%). Here, majority- and minority-spin bands are shown as full-blue and dotted-red lines, respectively.

Figure 3

Schematic representation of the structural models used for slab calculations in the case of Gd-terminated (a),(b) and N-terminated slabs (c). The atoms being relaxed are shown in red (Gd) and blue (N), respectively, while atoms being fixed during structure optimization are indicated in gray. Relaxed interlayer distances are indicated for a selected set of relaxed layers. The unrelaxed interlayer distance is also shown for the fixed layers.

Figure 4

Surface band structure for asymmetric and symmetric Gd-terminated GdN slabs as shown in Fig. 3. Majority and minority spin bands are shown as full-blue and dotted-red lines, respectively.

Figure 5

Surface band structure for a symmetric/asymmetric, Gd-terminated slab as shown in Fig. 3. The figure also shows the local contribution by two (top and bottom) surface layers as defined by the first Gd/Gd+N atomic layers as indicated above the plots. Subplots (a) and (b) show different degrees of Gd contribution for the symmetric slab, while (c) shows a summed (Gd+N) contribution for the same case. Subplot (d), on the other hand, shows the Gd contribution for the asymmetric case. Blue circles (squares) indicate majority (minority) -spin bands where the sum of local contributions is greater than or equal to the percentage indicated in the parentheses. Red symbols, on the other hand, correspond to an analogous projection for the asymmetric, Gd-terminated slab. In all cases, majority- and minority-spin bands are again shown as full and dotted lines, respectively.

Figure 6

Surface band structure for a symmetric, Gd-terminated GdN slab as shown in Fig. 3. Blue symbols indicate bands with summed local contributions by the two central Gd atoms $\ge 40\%$. On the right-hand side, a 3D plot of the partial charge density corresponding to the band forming the electron pocket at the $M$ point is also shown. Note that the partial charge density is not integrated over all of $k$ space and corresponds only to the $M$ point itself. Green isosurfaces have been drawn at $1.0\times {10}^{-4}e/a_0^3$.