Contrasting exchange-field and spin-transfer torque driving mechanisms in all-electric electron spin resonance

TL;DR

This paper analyzes all-electrical ESR-STM in a single-orbital Anderson impurity model, identifying two driving mechanisms: FLT from an exchange field that enables coherent spin control in the Coulomb-blockade regime, and STT from spin-polarized current that drives incoherent ESR above the CB thresholds. By mapping the quantum master equation to a spin dynamics framework, it derives Rabi rates for FLT and STT, energy dressing of the resonance, and the impact of homodyne detection on the ESR signal. The results show that the FLT regime supports long spin coherence (large $\Omega T_2$) and clear Rabi oscillations, while the STT regime yields strong spin polarization but limited coherence (small $\Omega T_2$), with observable ESR mainly through current readout. The study provides a unified, parameter-tunable picture linking transport, spin dynamics, and ESR signals, and highlights pathways for optimizing on-surface quantum control and sensing with electrically driven spins, while noting the need to incorporate cotunneling for a complete description.

Abstract

Understanding the coherent properties of electron spins driven by electric fields is crucial for their potential application in quantum-coherent nanoscience. In this work, we address two distinct driving mechanisms in electric-field driven electron-spin resonance as implemented in scanning tunneling spectroscopy. We study the origin of the driving field using a single orbital Anderson impurity, connected to polarized leads and biased by a voltage modulated on resonance with a spin transition. By mapping the quantum master equation into a system of equations for the impurity spin, we identify two distinct driving mechanisms. Below the charging thresholds of the impurity, electron spin resonance is dominated by a magnetically exchange-driven mechanism or field-like torque. Conversely, above the charging threshold spin-transfer torque caused by the spin-polarized current through the impurity drives the spin transition. Only the first mechanism enables coherent quantum spin control, while the second one leads to fast decoherence and spin accumulation towards a non-equilibrium steady-state. The electron spin resonance signals and spin dynamics vary significantly depending on which driving mechanism dominates, highlighting the potential for optimizing quantum-coherent control in electrically driven quantum systems.

Figures (11)

• Figure 1: a) Schematic of a QI connected a left (L) and right (R) lead with a bias voltage applied to the left lead. We consider the impurity to be more strongly coupled to the R lead. b) Energy scheme of the QI. The hollow arrows represent unoccupied levels of the QI at $\varepsilon+U$ as a product of charging the QI with one more electron to be doubly occupied with a total energy $E_2=2\varepsilon+U$. Meanwhile, the filled arrows, which defined the spin of the QI, are occupied single-electron states at $\varepsilon$. Only the left electrode is polarized in this example ($P_L>0$) and modulated ($t_L=t_L(t)$) being $t_L<t_R$. The applied positive bias ($eV_{\mathrm{DC}}=\mu_L-\mu_R>\varepsilon+U>0$) implies that the Rabi process $\Omega$ is dominated by STT, as for when ($eV_{\mathrm{DC}}<\varepsilon<0$). Contrary, if $\varepsilon < eV_\mathrm{DC} <\varepsilon+U$, the Rabi process is FLT-driven, as indicated on the right side of the figure. The transport through the QI in b)is the following: 50% of electrons up and down will tunnel from the right lead, overcoming the $\varepsilon+U$ energy, forcing the QI to be doubly populated. Since tip accepts more up electrons, the population of the up state in $\varepsilon+U$ will be depleted, leaving it down populated. Since spin-down SUMO state corresponds to a spin-up SOMO state, the QI ends up populated. The full process provides an incoherence resonance in the current when the L tunneling is modulated. c) Convention for the magnetic field direction with respect to the external coordinate system with the azimuthal angle $\phi=0$.
• Figure 2: Influence of the transport regime on the spin dynamics without AC driving. a) Time evolution of the impurity spin under a positive DC bias above $\varepsilon+U$ with a value $eV_\mathrm{DC}=-2\varepsilon>0$. In the steady state, the initial state is polarized towards the $|\mkern-5mu \uparrow\rangle$ state from the thermal initialization $\langle \mathbf{s} \rangle_\mathrm{thermal} = (-0.033, 0, -0.188)$ (red arrow), which mostly points at $|\mkern-5mu \downarrow\rangle$, since $\theta=\pi/18$. This means the quantum impurity rapidly torques when the DC bias is on and positive spin-polarized electrons are injected from the tip. b) Spin accumulation rate (Eq. \ref{['spin_acc']}) and its inverse as a function of the DC bias at constant tip-sample distance. The spin accumulation term dominates above the energy thresholds $\varepsilon$ and $\varepsilon+U$ but drastically drops off in the CB regime. The opposite behavior is observed when looking at its inverse (in blue), which diverges at zero voltage. The inset shows the transition region at positive voltages, where $\hbar/\langle\dot{s}_z\rangle_\mathrm{acc}$ saturates to 0.72 ns, indicating fast dynamics of the spin accumulation acting on the QI. c) Populations (left axis) and absolute value of the current (in blue, right axis, log scale) as functions of the normalized DC bias. Outside the CB regime, the current is large and constant ($\sim \mathrm{sign}(V_\mathrm{DC})\times 240$ pA), and higher‐energy states (doubly and empty QI states) can be slightly populated. Inside the CB regime, the population is fully governed by the singly occupied states, and the current rapidly decreases, exhibiting an $\arctan$ behavior, see Eq. \ref{['eq:current_analytical']}. The QI satisfies detailed balance ($\sum_l \rho_{ll}=1$) at all DC bias values.
• Figure 3: ESR outside of the CB regime; out of plane magnetic field. a) Bloch sphere representation of the spin components, on resonance with the AC field, in the lab frame for $\theta = \pi/18$ and positive $eV_\mathrm{DC}/|\varepsilon| = 2$. The spin is initialized thermally at $\langle \mathbf{s} \rangle_\mathrm{thermal}/{{\color{black}{\hbar}}\color{black} } = (-0.033, 0, -0.188)$ (red arrow). The positive DC bias causes the spin to flip to its spin polarization in the steady state as a consequence of the spin accumulation term. At the same time, the spin continuously performs Larmor oscillations even in the steady state. The Bloch sphere axes are reduced for clarity. b) Time-evolution of the current, Eq. \ref{['homo_current']}, from the thermal state, reaching the steady state after 2 ns. c) Corresponding ESR electric current signal, $\Delta I=I_L-I_\mathrm{BG}$, in the STT-driven regime in the frequency domain. The resonance frequency is slightly different to the bare one at 16.8 GHz, being now $f_0'=\Delta f+f_0=17.07$ GHz due to the energy dressing.
• Figure 4: ESR outside of the CB regime; in-plane magnetic field. a) Bloch sphere representation of the spin components, on resonance with the AC field, in the lab frame for $\theta = \pi/2$ and $eV_\mathrm{DC}/|\varepsilon| = 2$. The spin is thermally initialized at $\langle \mathbf{s} \rangle_\mathrm{thermal}/{{\color{black}{\hbar}}\color{black} } = (-0.19, 0, 0)$ (red arrow). Here, the spin accumulation forces alignment with the eigenbasis axis of the spin in the steady state, leading to continuous Larmor oscillations in the $zy$-plane. b) Time-evolution of the current, Eq. \ref{['homo_current']}, from the thermal situation. c) ESR electric current signal in the STT regime for the case of a) in the frequency domain, $\Delta I=I_L-I_\mathrm{BG}$. Since in-plane initialization leads to not dressing, the resonance occurs at the bare one $f_0=16.8$ GHz.
• Figure 5: ESR in the Coulomb Blockade regime; out of plane magnetic field. a) Bloch sphere representation of the spin components in the rotating frame for a magnetic field tilted by $\theta=\pi/18$ and a bias voltage of $eV_\mathrm{DC}/|\varepsilon|=1.275$, where the maximum quality factor $\Omega T_2 \approx 6$ is achieved. The spin is thermally initialized at $\langle \mathbf{s} \rangle_{\text{thermal}} = (-0.033, 0, -0.188)$ and experiences a torque due to the time dependent exchange field. This torque causes the spin to precess in a plane with the normal vector $\mathbf{n} = (-\sin(4\pi/9), 0, \cos(4\pi/9))$. b) Time evolution of the current, Eq. \ref{['homo_current']}, showing up to six Rabi oscillation cycles before decaying to the steady state in the rotating frame. c) ESR signal, $\Delta I=I_L-I_\mathrm{BG}$, DC contribution, featuring a peak with a subtle Fano profile. The resonance frequency is slightly tuned with respect to the bare one to be $f_0'=\Delta f+f_0=17.146$ GHz due to the energy dressing.