Spin and orbital magnetic response in metals: Susceptibility and NMR shifts

A DFT-based method is presented which allows the computation of all-electron NMR shifts of metallic compounds with periodic boundary conditions. NMR shifts in metals measure two competing physical phenomena. Electrons interact with the applied magnetic field (i) as magnetic dipoles (or spins), resulting in the Knight shift, and (ii) as moving electric charges, resulting in the chemical (or orbital) shift. The latter is treated through an extension to metals of the gauge-invariant projector augmented wave developed for insulators. The former is modeled as the hyperfine interaction between the electronic spin polarization and the nuclear dipoles. NMR shifts are obtained with respect to the computed shieldings of reference compounds, yielding fully ab initio quantities which are directly comparable to experiment. The method is validated by comparing the magnetic susceptibility of interacting and noninteracting homogeneous gas with known analytical results, and by comparing the computed NMR shifts of simple metals with experiment.

Figure 1

(Color online) Spin and orbital susceptibility per unit volume of the interacting-electron homogeneous gas with respect to $r_s∕a_0=(3∕4\pi \rho {)}^3$. Solid lines represent analytical results while dots represent results computed with this code at $0.4\mathrm{eV}$ smearing. The susceptibilities are dimensionless. The exchange correlation is modeled with the PBE functional.

Figure 2

(Color online) Convergence with respect to smearing $\left(\sigma \right)$ of the spin susceptibility of aluminum [green (triangles)], lithium [red (squares)], and copper [blue (crosses)]. For comparison purposes, the spin susceptibility of each metal is normalized to its value at the lowest achieved smearing. The $x$ axis represents smearing in eV.

Figure 3

(Color online) Convergence with respect to smearing $\left(\sigma \right)$ of the orbital susceptibility of aluminum [green (triangles)], lithium [red (squares)], and copper [blue (crosses)]. For comparison purposes, the orbital susceptibility of each metal is normalized to its value at the lowest achieved smearing. The $x$ axis represents smearing in eV.

Figure 4

(Color online) Convergence with respect to smearing $\left(\sigma \right)$ of the Knight shift (not including the spin susceptibility) of aluminum [green (triangles)], lithium [red (squares)], and copper [blue (crosses)]. For comparison purposes, the Knight shift of each metal is normalized to its value at the lowest achieved smearing. The $x$ axis represents smearing in eV.

Figure 5

(Color online) Convergence with respect to smearing $\left(\sigma \right)$ of the orbital shift (not including the orbital susceptibility) of aluminum [green (triangles)], lithium [red (squares)], and copper [blue (crosses)]. For comparison purposes, the orbital shift of each metal is normalized to its value at the lowest achieved smearing. The $x$ axis represents smearing in eV.

Figure 6

(Color online) Band structure of lithium. The width of the line is representative of its contribution to the isotropic Knight shift. The Fermi energy is set to zero on the energy scale. Computations were done with a smearing of $0.2\mathrm{eV}$, for which the $\frac{1}{T}$ dependence is obvious at the Fermi energy. The “divergence” in $T$ disappears with the Brillouin zone integration.