3 the Mn is missing the Ga 4 p valence electron, it intro- duces a hole in the system, simultaneously acting as an acceptor and a source of magnetic moments. In bulk, the holes introduced by many Mn impurities form itin- erant carriers that mediate the ferromagnetic coupling between localized moments. Fig. 1 shows a schematic drawing of what happens when a Mn substitutes a Ga in the GaAs lattice. 11 The Mn 3 d 5 up states (for example) that lodge in the valence band are exchange-split from the down states that end up in the conduction band. The crystal field imposes the tetrahedral host symmetry, resulting in a further split into a doublet of e$_{g}$ symme- try and a triplet of t$_{2}$$_{g}$ symmetry. 13 The e$_{g}$ states couple only weakly to the host. The t$_{2}$$_{g}$ states on the other hand, hybridize with the As nearest neighbor dangling- bond p states at the top of the valence band, forming bonding and antibonding combinations. Effectively, the neighboring As p spins at the top of the valence band that are parallel to the Mn spin are shifted up in energy, whereas p spins that are antiparallel to the Mn spin are shifted down. Therefore, the main effect of a single sub- stitutional Mn with spin up, is to introduce three spin- polarized (”up”) levels above the top of the valence band and three spin-polarized (”down”) levels below. These six states are predominantly of As p character. The hole introduced by a single Mn will occupy the highest of the three antibonding states (indicated by the empty circle in Fig. 1). In this way, the hybridization mechanism gives rise to an antiferromagnetic coupling between itinerant and localized spins by what is known as Zener’s kinetic- exchange mechanism. 14–17 We account for this p - d in- direct exchange by introducing a classical Mn vector of magnitude 5/2, which couples to the neighboring As p states. There are no explicit d electrons in our model; they are accounted for only implicitly via the exchange interaction between the localized moments and the As p spins on neighboring sites. Our tight-binding kinetic exchange Hamiltonian takes the following form: The first term reproduces the band-structure of bulk GaAs 18 and contains the near-neighbor hopping and on- site energies in terms of the Slater-Koster parameters 19,20 t ij µµ ′ for the s and p electrons of Ga and As. In (1) i and j are atomic indices, µ and µ ′ are orbital indices and σ denotes spin. In simulating the (110) surface, buckling is accounted for by rescaling 21,22 the tight-binding param- eters and modifying the direction cosines appropriately. The second term implements the p - d exchange mecha- nism described above. It couples the unit spin vector of Mn atom m : ˆ Ω$_{m}$, to the p orbitals of the nearest neigh- bor As atoms n [ m ]: ⃗ S$_{n}$ = 1 2 ∑ πσσ ′ a † $_{nπσ}$⃗ τ$_{σσ}$ ′ a$_{nπσ}$ ′ , where ⃗ τ is a vector of Pauli matrices. The value of J$_{pd}$ has been inferred by theory 23 and experiment 24 to approx- imately J$_{pd}$ = 1 . 5 eV. The three p - d hybridized levels, spin-polarized in the direction of the Mn moment ˆ Ω$_{m}$ are split from the three levels polarized in the opposite direction by an energy of order J$_{pd}$ . The third term in Eq. (1) accounts for SO interactions in an atomic approximation with the renormalized spin- orbit splittings 18 λ$_{Ga}$ = 2 λ$_{Mn}$ = 58 meV and λ$_{As}$ = 140 meV. Spin-orbit interaction causes the band energy to depend not only on the relative angles between different Mn spin directions, but also on spin-orientations relative to the lattice. The fourth term of the Hamiltonian accounts for a long-range repulsive Coulomb interaction in the presence of a Mn impurity, which attracts a weakly bound hole and repels electrons. At the surface, we crudely account for the weaker dielectric screening by reducing the bulk dielectric constant ε$_{r}$ in half. The Coulomb behavior at short distances is parametrized in V$_{corr}$, which is a Mn central cell correction term used as a parameter to tune the Mn acceptor level to the experimental position. 25–28 It contains an on-site V$_{on}$ = 1 . 0 eV (that is, acting on the Mn) and an off-site term V$_{off}$ = 2 . 4 eV. The off- site Coulomb correction affects the nearest neighbor As atoms and together with J$_{pd}$ controls the amount of p - d hybridization in the system and the range of the acceptor wave function. FIG. 1: Schematic drawing of the level dynamics leading to a polarized acceptor at the top of the valence band. The Mn 3 d 5 up and down electrons are split by the atomic exchange in accordance with Hund’s first rule. The levels are further split by the crystal field into a doublet of e$_{g}$ and a triplet of t$_{2}$$_{g}$ -symmetry. The t$_{2}$$_{g}$ -levels hybridize with nearest neighbor dangling bond As p -states, such that the levels at the top of the valence band with spin parallel to the Mn spin are pushed up in energy. Spin up is denoted by + and spin down by -. H = ∑ ij,µµ $^{′}$,σ t ij µµ ′ a † $_{iµσ}$a$_{jµ}$ $^{′}$σ + J$_{pd}$ ∑ m ∑ n [ m ] ⃗ S$_{n}$ · ˆ Ω$_{m}$ + ∑ i,µµ $^{′}$,σσ ′ λ$_{i}$ 〈 µ, σ | ⃗ L · ⃗ S | µ $^{′}$, σ $^{′}$〉 a † $_{iµσ}$a$_{iµ}$ $^{′}$σ ′ + e 2 4 πε$_{0}$ε$_{r}$ ∑ m ∑ iµσ a † $_{iµσ}$a$_{iµσ}$ | ⃗ r$_{i}$ − ⃗ R$_{m}$ | + V$_{corr}$. (1) Mn 3d As 4p + t$_{2g}$ - t$_{2g}$ e$_{g}$ + e$_{g}$ - + - Conduction Band Valence  Band Atomic Exchange Crystal  Field Hybridization Dangling Bonds