Planar Close-Packed Metals

Chapter 3. Magnon Dispersion, Spin Density Waves,

D. Ising Models for Simple Metals

1. Planar Close-Packed Metals

For a planar close-packed metal, each atom has six nearest neighbors. If this lattice is filled with Cu2+ (3d9) ions, then the atomic lattice is correct for describing the electronic structure of the hole states.

However, ifthis lattice is filled with Cu+ (3d10) ions andone valence electron per ion is added,the honeycomb (hexagonal) lattice definedby the triangular hollows — where eachlatticepoint has three nearest neighbors — should be used for describing the valence electronic structure.

For transition metals, such as Ni (3d94∙s1), the valence electronic lattice would interact with the atomic lattice, since the d electrons are atom-centered.

The honeycomb valence electronic lattice leads to the classical model of anti­ ferromagnetism — where the two sublattices consist of the {B} and {N} sites of graphitelike BN.

Since there aretwo triangular hollows peratom, for monovalent metal atoms, this leads to a half-filled valenceband for both the up-spin and the down-spin electrons, leading to a two-dimensional metal. For divalent simple metals (Be, Mn, Ca, etc.

and Zn, Cd, Hg), the up-spin and down-spin valence bands are filled, leading to two-dimensional insulators.

2o Hexagonal Close-Packed Metals

For hep metals, the “valence electronic lattice” consists of the centers of the tetra­ hedral hollows. For ⅛cp, each tetrahedral hollow shares a face (three edges) with another tetrahedral hollow, and the three remaining edges are shared with three other tetrahedral hollows. Hence, for hep the lattice of tetrahedral hollows corre-

sponds to the wurzite lattice (such as for hexagonal ZnS, where each lattice point has four neighbors).

The wurzite lattice leads to the classical model of antiferromagnetism ■— where the two sublattices consist of the {Zn} and {S} sites of hexagonal ZnS.

3. Face Centered Cubic Metals

For feemetals, the electronic latticealso consists of the centers of tetrahedralhollows.

For fee, each tetrahedral hollow shares an edge with six other tetrahedral hollows;

hence, for fee the lattice oftetrahedral hollows corresponds to the rock salt lattice (NaCl).

The rock salt lattice leads to the classicalmodelof antiferromagnetism — where the two sublattices consist ofthe {Na} and {C1} sites ofNaCl.

For both fee and hep,there are two tetrahedral hollows per atom. Hence, in each case the Ising model leadsto a half-filledenergy band for monovalent metalatoms and a completely filled energy band for divalent metal atoms. This isconsistent with the observation that the monovalent metals tend to have higher electrical conductivities than thedivalent metals.

IV. Summary

We optimized UHF and HF wavefunctions for all allowedmagnetizations of each of the Cuιo, Agβs Ag8, Agio, Auι0, Liβ, Lii0, Liι∙t, and Nai0 ring clusters. The results for these one-dimensional metals—and speculationsfor two-dimensional metalsand three-dimenstional metals — are summarized as follows.

Restricted Hartree-Fock leads to spurious results, such as (i) a charge density

wave for each of the Ag8, Li10, Li14, and Nai0 μ = 0 (low-spin) symmetric ring clusters,10,11 (it) charge and spin density waves for all intermediate magnetizations (0 < μ < 1) of each ring cluster, and (tit) ground states for Ag8, Naχo, and Li1o having 0.25, 0.6, and 1.0 unpaired electrons per atom, respectively. These spurious resultsindicate a fundamental flawinHF for these systems — the double occupation of valenceorbitals for describing magnetizations μ < 1.

For each Mpr ring cluster, unrestricted Hartree-Fock leads to an antiferromagnetic (μ = 0) ground state that does not have a charge density wave. However, for each magnetization μ < 1 (including μ = 0), UHF leads to a spin density wave (orbital localization). The UHFtotalenergyincreases monotonically with increasing magnetizationas long as spin densitywaves are allowed. The UHFmagnon spectrum for each of the various Mjf rings is described accurately in terms of a generalized Ising modelincludingonly effectivenearest-neighbor exchange interactions.

However, for these systems, the spin density waves resulting from the UHF- Ising description are due to an incomplete treatment of the electron correlation.11,13 Hence, generalized valence bond calculations11’13 lead to lower total energies and fully symmetrical charge and spin densities for all magnetizations 0 ≤ μ ≤ 1.

For these one-dimensional ring clusters, the UHF-Ising model is consistent with the classical two-sublattice model of antiferromagnetism,ιs where the principallattice is divided into two sublattices A and B — occupied by the up-spin and down-spin electrons, respectively— such that all nearest neighbors of one sublattice belong to the other sublattice. However, for these rings, the valence orbitalsarecentered at the bond midpoints; hence, the principal latticefor describing the valence electrons is the latticeofbond midpoints— not the atomic lattice.

For close-packed metals, the valence orbitals are centered at interstitial hollows

— triangular hollowsfortwo-dimensional metalsand tetrahedral hollows for three di­

mensional metals.34 For three cases (planar close-packed,feeand hep), weshow that the lattice defined by the interstitial hollows is consistent with the two-sublattice model of antiferromagnetism. Each of these three close-packed metals contains two interstitial hollows per atom, leading (for monovalent metal atoms) to metallic prop­ erties resultingfrom a half-filled valenceenergyband. Therefore, the common belief that the classical (two sublattice) model of antiferromagnetism cannot apply to the valence electrons of close-packed metals — because their atomiclattices containtri­

angles —- is incorrect.

Appendix A. Details of the Calculations

1.

The Frozen

Core

Approximation

The frozen core approximation can be described briefly as follows (more detail is given elsewhere).10

First, both the coreorbitals and the valenceorbitals are optimized simultaneously for the valence high-spin state (chosensincethe HF and UHF valencewavefunctions are identical for high-spin). The core orbitals optimized for the high-spin state are then used to construct the valence hamiltonian

« = sSosx + ∑ι(∙) + ∑-

i=l i>,∙ rij

i(i) _ -lv,1 + v(fi ) +vc°∞

Vcore = ∑(2Jc-⅛c) c

where (t) E%-ore includes the nuclear repulsion energy and all one-electron and two-electron energy terms involving only the core electrons {the [Ar]3d10, [Kr]4d10,