On the interpretation of Hahn echo measurements in electron spin resonance scanning tunneling microscopy

Abstract

Electron spin resonance scanning tunneling microscopy (ESR-STM) has become a powerful tool for probing spin dynamics and coherence of individual atoms and molecules on surfaces. In this work, we perform Rabi oscillation and Hahn echo pulse protocols on individual iron phthalocyanine (FePc) molecules on MgO/Ag(001) using ESR-STM. While Hahn echo protocols are widely used to extract spin coherence times, we show that in ESR-STM they are particularly susceptible to misinterpretation due to tunneling electrons generated by the applied radio-frequency (RF) voltage. The RF voltage not only drives the spin, but simultaneously probes and relaxes it, which consequently leads to an exponential decay that reflects spin relaxation rather than intrinsic phase coherence. We moreover show that varying both delay times in the refocusing pulse sequence is a reliable way to ensure a coherent nature of the echo signal. The extracted decay for the latter protocol suggests that T2 is approximately 30 ns and is thus closer to the decoherence time observed in Rabi oscillation measurements. This is significantly shorter than values reported in previous echo measurements. Our findings underscore the need for caution in interpreting T2 times from Hahn echo and Carr-Purcell protocols in ESR-STM and provide practical criteria for distinguishing true spin echoes from tunneling-induced relaxometry signals.

Figures (5)

• Figure 1: Rabi oscillation measurements of FePc on MgO/Ag(001).(a) Schematic of the ESR-STM setup. An RF voltage $V_{\mathrm{RF}}$ is applied to the STM junction in addition to a constant bias $V_{\mathrm{DC}}$, while an out-of-plane magnetic field $B$ defines the quantization axis. (b) STM topography of an FePc molecule adsorbed on a bilayer MgO/Ag(001) surface ($I_{\mathrm{set}}=50~\mathrm{pA}$, $V_{\mathrm{DC}}=100~\mathrm{mV}$). (c) Pulse sequence used in the A and B cycle of the lock-in scheme for Rabi oscillation measurements. The spin is coherently driven by $V_{\mathrm{RF}}$ for a pulse duration $\tau$, while a background voltage $V_{\mathrm{DC}}$ is constantly applied. (d) Experimental Rabi oscillation measurements: Change in current $\Delta I$ as a function of $\tau$ for different $V_{\mathrm{RF}}$ amplitudes (30--90 mV). The traces show oscillatory behavior characteristic of coherent spin rotations. Solid lines represent fits to $\Delta I(t)=I_0 \sin(\Omega t+\phi)\exp(-t/T_2^{\mathrm{Rabi}})$. Traces are vertically offset for clarity. A linear background due to current rectification ($k_{\mathrm{lin}}$) has been subtracted ($I=5~\mathrm{pA}$, $V_{\mathrm{set}}= 40~\mathrm{mV}$, $B=536~\mathrm{mT}$, $f_0=14.772~\mathrm{GHz}$, $\tau_{\mathrm{cycle}}=800~\mathrm{ns}$). (e) Colormap of the data shown in (d). (f) Extracted Rabi frequency $\Omega$ as a function of $V_{\mathrm{RF}}$ from the data in (d) with a linear dependence (black line; slope: $0.47~\mathrm{MHz/mV}$).
• Figure 2: Hahn echo and Carr--Purcell (CP) measurements.(a) Pulse sequence used for Hahn echo, consisting of two $\pi/2$ pulses separated by a $\pi$ pulse and a background DC voltage (blue). (b) Change in tunneling current $\Delta I$ for a Hahn echo sequence as a function of pulse separation $\tau$ for different tunneling currents ($V_{\mathrm{DC}}=60~\mathrm{mV}$, $V_{\mathrm{RF}}=60~\mathrm{mV}$, $B=523~\mathrm{mT}$, $f_0=13.813~\mathrm{GHz}$, $\tau_\pi=15.9~\mathrm{ns}$, $\tau_{\mathrm{cycle}}=2500~\mathrm{ns}$). Solid lines represent exponential fits of the form $\Delta I(\tau)=I_0 \exp\!\left(-\tau/T_2^{\mathrm{Echo}} \right)$. Datasets are offset for clarity. (c) Extracted echo decay rates $T_2^{-1}$ as a function of tunneling current $I$, showing a linear dependence. (d) CP-measurements with increasing numbers of refocusing $\pi$ pulses (CP-1 to CP-16). The inset illustrates the CP-$N$ sequence for $N=2$ composed of an initial $\pi/2$ pulse, $N$ evenly spaced $\pi$ pulses with delay time $\tau/N$, and a final $\pi/2$ pulse ($I=7~\mathrm{pA}$, $V_{\mathrm{DC}}=50~\mathrm{mV}$, $V_{\mathrm{RF}}=60~\mathrm{mV}$, $B=536~\mathrm{mT}$, $f_0=13.991~\mathrm{GHz}$, $\tau_\pi=8.6~\mathrm{ns}$. $\tau_{\mathrm{cycle}}=0.8 \text{--} 7.6~\mathrm{\mu s}$). (e) Extracted coherence time $T_2$ as a function of the number of $\pi$ pulses in the CP sequence, revealing an apparent linear increase in $T_2$ with pulse number.
• Figure 3: Control experiments for Hahn echo measurements.(a) Hahn echo sequence recorded on-resonance (black, $f=14.772~\mathrm{GHz}$) and off-resonance (gray, $f=15.4~\mathrm{GHz}$). A pronounced signal is observed only at the resonance frequency. The on-resonance trace yields an apparent echo decay time $T_2^{\mathrm{Echo}}=(85\pm22)~\mathrm{ns}$ from an exponential fit (red line), while the off-resonant dataset shows no signal ($I=5~\mathrm{pA}$, $V_{\mathrm{DC}}=40~\mathrm{mV}$, $V_{\mathrm{RF}}=60~\mathrm{mV}$, $B=536~\mathrm{mT}$, $\tau_\pi=19~\mathrm{ns}$, $\tau_{\mathrm{cycle}}=2000~\mathrm{ns}$). (b--e) Control measurements testing the robustness of the exponential decay against imperfections in the pulse sequence. (b) unequal spacing between the two $\pi/2$ pulses and $\pi$ pulse; (c) use of two $\pi$ pulses instead of a $\pi/2$--$\pi$--$\pi/2$; (d) incorrect pulse length for the final $\pi/2$ pulse; (e) three pulses of equal length $2\pi/3$. Insets illustrate the respective pulse schemes. [(b-d): $I=5~\mathrm{pA}$, $V_{\mathrm{DC}}=40~\mathrm{mV}$, $V_{\mathrm{RF}}=60~\mathrm{mV}$, $B=536~\mathrm{mT}$, $f=14.772~\mathrm{GHz}$, $\tau_\pi=19~\mathrm{ns}$, $\tau_{\mathrm{cycle}}=2000~\mathrm{ns}$; (e): $I=5~\mathrm{pA}$, $V_{\mathrm{DC}}=60~\mathrm{mV}$, $V_{\mathrm{RF}}=60~\mathrm{mV}$, $B=523~\mathrm{mT}$, $f_0=13.813~\mathrm{GHz}$, $\tau_\pi=15.8~\mathrm{ns}$, $\tau_{\mathrm{cycle}}=2500~\mathrm{ns}$]. (f) Extracted $T_2^{\mathrm{Echo}}$ times from the measurements shown in (a)-(e). All datasets exhibit similar exponential decay behavior.
• Figure 4: Schematic illustration of how Hahn echo pulse sequences can effectively act as two consecutive $T_1$ relaxation probes (orange: RF voltage; blue: DC voltage): The initial pulse rotates the spin system, while subsequent pulses -- intended for spin refocusing -- simultaneously probe and rotate the spin population. Each pulse thus partially measures the remaining magnetization, which produces an apparent exponential decay. This decay reflects $T_1$ relaxation rather than true spin decoherence ($T_2$). This leads to an echo-like decay signal whose exponential decay can be misinterpreted as a coherent process.
• Figure 5: Coherent spin control in an Fe--FePc S=1/2 molecular complex.(a) ESR-STM spectrum of the Fe--FePc complex, showing a distinct resonance peak near $14.15~\mathrm{GHz}$. The inset displays an STM topography of the complex ($I=4~\mathrm{pA}$, $V_{\mathrm{DC}}= -60~\mathrm{mV}$, $V_{\mathrm{RF}}=30~\mathrm{mV}$, $B=461~\mathrm{mT}$). (b) Rabi oscillation measurements ($I=4~\mathrm{pA}$, $V_{\mathrm{DC}}= -60~\mathrm{mV}$, $V_{\mathrm{RF}}=60~\mathrm{mV}$, $B=461~\mathrm{mT}$, $f_0=14.15~\mathrm{GHz}$, $\tau_\pi=15.9~\mathrm{ns}$, $\tau_{\mathrm{cycle}}=300~\mathrm{ns}$, $\tau_{\mathrm{Probe}}=150~\mathrm{ns}$, $V_{\mathrm{Probe}}=-60~\mathrm{mV}$). The red curve represents a sinusoidal fit to the data yielding $\Omega=(48\pm4)~\mathrm{MHz}$. The inset illustrates the used pulse schemes in A and B cycle with a DC readout pulse instead of a constant DC background. (c) Two-dimensional Hahn echo measurements performed by varying the first delay time $\tau_1$ (10--60 ns) while sweeping the second delay $\tau_2$. The corresponding pulse schemes for lockin A and B cycles are shown at the top. A dip in signal is observed when $\tau_1$ and $\tau_2$ are equal, i.e. under echo conditions ($I=4~\mathrm{pA}$, $V_{\mathrm{DC}}= -60~\mathrm{mV}$, $V_{\mathrm{RF}}=60~\mathrm{mV}$, $B=461~\mathrm{mT}$, $f_0=14.26~\mathrm{GHz}$, $\tau_\pi=11~\mathrm{ns}$, $\tau_{\mathrm{cycle}}= 450~\mathrm{ns}$, $\tau_{\mathrm{Probe}}=150~\mathrm{ns}$, $V_{\mathrm{Probe}}=-60~\mathrm{mV}$). The traces were vertically shifted for clarity. (d) Colormap of the full $\tau_1$--$\tau_2$ dataset, revealing the evolution and decay of the interference pattern characteristic for a coherent echo response. (e) Extracted echo position $\tau_2$ as a function of the first delay $\tau_1$, showing a near-linear relationship. Gray line is a guide to the eye with $\tau_1=\tau_2$