Model of orbital populations for voltage-controlled magnetic anisotropy in transition-metal thin films

Voltage-controlled magnetic anisotropy (VCMA) is an efficient way to manipulate the magnetization states in nanomagnets and is promising for low-power spintronic applications. The underlying physical mechanism for VCMA is known to involve a change in the $d$ orbital occupation on the transition-metal interface atoms with an applied electric field. However, a simple qualitative picture of how this occupation controls the magnetocrystalline anisotropy (MCA) and even why in certain cases the MCA has the opposite sign remains elusive. In this paper, we exploit a simple model of orbital populations to elucidate a number of features typical for the interface MCA, and the effect of the electric field on it, for $3d$ transition-metal thin films used in magnetic tunnel junctions. We find that in all considered cases, including the Fe(001) surface, clean $\mathrm{F}{\mathrm{e}}_{1-x}\mathrm{C}{\mathrm{o}}_x\left(001\right)/\mathrm{MgO}$ interface, and oxidized Fe(001)/MgO interface, the effects of alloying and the electric field enhance the MCA energy with electron depletion, which is largely explained by the occupancy of the minority-spin $d_{xz,yz}$ orbitals. However, the hole-doped Fe(001) exhibits an inverse VCMA in which the MCA enhancement is achieved when electrons are accumulated at the Fe (001)/MgO interface with the applied electric field. In this regime, we predict a significantly enhanced VCMA that exceeds 1 pJ/Vm. Realizing this regime experimentally may be favorable for the practical purpose of voltage-driven magnetization reversal.

Figure 1

(a) Schematic of the minority $3d$ orbital splitting by the crystal field of tetragonal symmetry. Broadening of the orbital levels mimics the energy bands. The vertical black dotted lines represent the Fermi energies of Fe, Co, and Ni. (b) MAE as a function of orbital band filling $n$ for different values of the $T_{2g}$ level splitting Δ and broadening δ.

Figure 2

(a) MgO/Fe/MgO supercell structure (vacuum layer is not shown). (b) and (c) Contributions from different Fe sites (monolayers) to MAE (b) and to the change in MAE (ΔMAE) in applied electric field $E_{\mathrm{vac}}=2\mathrm{V}/\mathrm{nm}$ (c).

Figure 3

Orbital resolved DOS at the interfacial Fe atom for clean Fe(001)/MgO (a) and oxidized Fe(001)/FeO/MgO (b) interfaces. Top and bottom panels correspond to majority- and minority-spin contributions, respectively. Fermi energy lies at zero energy, as indicated by the dashed line.

Figure 4

Results of DFT calculations for Fe(001)/MgO (a) and Fe(001)/FeO/MgO (b) interfaces. MAE as a function of applied electric field $E_{\mathrm{MgO}}$ in MgO (red curves) and $E$ field-induced changes in the orbital contribution to MAE (green and blue curves). In the plot, the electric field $E_{\mathrm{MgO}}$ is scaled according to the experimental dielectric constant of MgO $\varepsilon =9.5$.

Figure 5

Calculated electrostatic potential energy (black curve) and excess charge distribution (green curve) across the MgO (3 ML)/Fe (3 ML)/MgO (3 ML)/vacuum (2 nm) supercell structure for $n_e=-0.025$.

Figure 6

Calculated MAE (a) and VCMA (b) as a function of the excess number of valence electrons $n_e$ (holes for negative $n_e$) at the Fe(001)/MgO interface. The inset shows the MAE as a function of the electric field in MgO. The electric field is estimated from $E_{\mathrm{MgO}}=n_e/\varepsilon {\varepsilon }_0{a}^2$. The positive field is assumed to be pointing from Fe to MgO and corresponds to electron depletion [45].

Figure 7

MAE of the $\mathrm{F}{\mathrm{e}}_{1-x}{M}_x/\mathrm{MgO}$ interface ($M=\mathrm{Co},\mathrm{Cr}$) versus a number of doped electrons $n_e$ (holes for negative $n_e$) calculated by the KKR-CPA method.