Single spin resonance driven by electric modulation of the $g$-factor anisotropy

We address the problem of electronic and nuclear spin resonance of an individual atom on a surface driven by a scanning tunneling microscope. Several mechanisms have been proposed so far, some of them based on the modulation of exchange and crystal field associated with a piezoelectric displacement of the adatom driven by the radio frequency (RF) tip electric field. Here we consider another mechanism, where the piezoelectric displacement modulates the $g$-factor anisotropy, leading both to electronic and nuclear spin flip transitions. We discuss thoroughly the cases of hydrogenated Ti ($S=1/2$) and Fe $(S=2)$ on MgO, relevant for recent experiments. We model the system using two approaches. First, an analytical model that includes crystal field, spin orbit coupling, and hyperfine interactions. Second, we carry out density-functional-based calculations. We find that the modulation of the anisotropy of the $g$ tensor due to the piezoelectric displacement of the atom is an additional mechanism for scanning tunneling microscopy (STM)-based single spin resonance that would be effective in $S=1/2$ adatoms with large spin orbit coupling. In the case of hydrogenated Ti on MgO, we predict a modulation spin resonance frequency driven by the DC electric field of the tip.

Figure 1

Scheme of the STM-EPR experimental setup [3, 4, 5, 6, 7, 8, 9, 10, 11, 12] with the atomic structure of Ti-H on the oxygen site of MgO. $\vec{B}$ is the external magnetic field applied during the experiment (with an angle ${\theta }_B$ measured with respect to the surface) and ${\vec{n}}_T$ shows the direction of the magnetic moment of the tip. Red balls represent O atoms, blue balls are Mg atoms, the violet ball is for the Fe atom, the green ball indicates the H atom, and the orange one is for the Ti atom.

Figure 2

(a) DFT calculations for the spin-unpolarized density of states projected over $d$ and $s$ orbitals of Ti and the $s$ orbital of hydrogen. The gray shadow shows the total density of states. (b) Energy level cartoon.

Figure 3

Dependence of $g_x$ and $g_z$ on spin orbit coupling $\lambda$, for Ti-H on oxygen, obtained in two ways: solution of full model Eq. (7) with $D=-255$ meV and $F=14\mathrm{meV}$ (symbols) and using analytical results ignoring ${\ell }_z\ne 2$ manifolds [Eqs. (15) and (22)] (lines). DFT results are shown for reference as dashed lines.

Figure 4

Dependence of $g_x$ and $g_z$ on crystal field parameters $F$ (a) and $D$ (b) for Ti-H on oxygen, obtained from full model Eq. (7) (solid points in both panels), DFT (dashed lies), and analytically (solid lines) [Eqs. (15) and (22)]. In both panels, we take $\lambda =10\mathrm{meV}$. (a) $D=-255\mathrm{meV}$. (b) $F=14\mathrm{meV}$.

Figure 5

Schematic showing a magnetic atom (orange) on MgO and an Fe atom (violet) attached to the apex of the STM tip. The solid (orange and violet) arrows indicate the orientations of the respective magnetic moments ($\vec{S}$ and $\overrightarrow{S_{\mathrm{tip}}}$). The exchange interaction $J$ is indicated by the red curve and the piezoelectric displacement is represented with a red arrow. A radio-frequency voltage drives the ESR of the magnetic atom. The external magnetic field $B_{\mathrm{ext}}$ is applied out of the plane of the substrate (with a small angle).

Figure 6

(a) Rabi coupling due to $g$-factor modulation as a function of the angle ${\theta }_B$ between $\vec{B}$ and surface calculated with two methods: Eqs. (6) and (27). $\lambda =10\mathrm{meV}, F=1.4\lambda =14\mathrm{meV}, D=-255\mathrm{meV}, B=1T, V_F=20\mathrm{meV}$, STM-surface distance $d=5\AA , k=290\mathrm{eV}/{\mathrm{nm}}^2$. (b) Dependence of ${\mathrm{\Omega }}_g$ on $\lambda$, keeping all the other constants the same and ${\theta }_B=45$ deg.

Figure 7

Breakdown of Rabi coupling for Fe on MgO as a function of in-plane magnetic field $B_x$. The calculation were performed for $B_z=0.2\mathrm{T}, k=600\mathrm{eV}/{\mathrm{nm}}^2, d=0.6\mathrm{nm}, V_{\mathrm{RF}}=8\mathrm{mV}, D=-290\mathrm{meV}, F=-10\mathrm{meV}, \lambda =35\mathrm{meV}$, and $q=2e$.

Figure 8

Shift of the resonance frequency for $B=1\mathrm{T}, \lambda =10\mathrm{meV}, D=-255\mathrm{meV}, F=1.4\lambda \mathrm{meV}, k=290\text{eV}/{\mathrm{nm}}^2$, and $d=5\text{Å}$ for three different magnetic field angles.