Spin resonance spectroscopy with an electron microscope

Coherent spin resonance methods, such as nuclear magnetic resonance and electron spin resonance spectroscopy, have led to spectrally highly sensitive, non-invasive quantum imaging techniques. Here, we propose a pump-probe spin resonance spectroscopy approach, designed for electron microscopy, based on microwave pump fields and electron probes. We investigate how quantum spin systems couple to electron matter waves through their magnetic moments and how the resulting phase shifts can be utilized to gain information about the states and dynamics of these systems. Notably, state-of-the-art transmission electron microscopy provides the means to detect phase shifts almost as small as that due to a single electron spin. This could enable state-selective observation of spin dynamics on the nanoscale and indirect measurement of the environment of the examined spin systems, providing information, for example, on the atomic structure, local chemical composition and neighboring spins.

Modern-day transmission electron microscopy (TEM), with advanced techniques for aberration correction and cryogenic sample preparation [1,2], is a well-matured technology that employs wave properties of electrons to resolve structures at an atomic level.The development of ultra-fast transmission electron microscopy (< 1 ps) made it possible to investigate processes with both nanometer spatial resolution and sub-picosecond temporal resolution [3][4][5], nowadays in optimized interferometric setups [6][7][8][9][10].Most ultra-fast pump-probe experiments are based on a laser-triggered sample excitation (UV-IR) followed by a highly temporally resolved, but sparse electron probe.At the same time, bright electron sources with beam blanking systems (< 10 ns ) [11] and fast direct electron detectors (∼ 260 ps) [12] have been integrated into aberration-corrected TEMs.This facilitates the time-resolved probing of fast processes such as quantum spin dynamics.
The default method to probe the dynamics of spin samples is electron spin resonance (ESR) and nuclear magnetic resonance (NMR) spectroscopy.This non-invasive imaging technology [13,14] revolutionized not only medical diagnostics [15], biology [16,17], and chemistry [18,19], but also high-precision measurements for fundamental physics [20], including searches for dark energy and dark matter [21].Magnetic resonance spectroscopy is also used to characterize electrode materials for electrochemical energy storage [22,23], meeting the challenges of the green energy revolution.
In this paper, we propose a microwave-pump electronprobe method for ESR and NMR down to the level of individual spins, with coherence times on the order of 100 ns up to minutes or even hours [14,24].While standard ESR (NMR) techniques employ magnetic field gradients and optimized antennas to overcome the diffraction limit imposed by the microwave probe fields [25], we point out the surprisingly low technical challenge towards probing spin dynamics directly at the nanoscale with highly controlled electron beams in a TEM.
In the following, we present a basic electroninterferometric scheme that implements the proposed method.Therein, the magnetic moment of a spin sample imprints a phase onto the wave function of a passing electron.A tailored microwave pump pulse interaction causes a controlled change of that spin and thus a change in phase, which can then be detected by interferometric phase imaging [26][27][28].The microwave pulse thereby enables a differential measurement technique for separating the phase contribution of the spins.As an alternative to the interferometric read-out, one could measure the spatial gradient of the phase, which manifests in a deflection of the electron beam detectable in the far field, as in Differential Phase Contrast (DPC) imaging [27][28][29].
In magnetic resonance spectroscopy, one investigates the local resonance frequency of target spins by resonantly driving the transition of the two spin states with a tunable microwave pulse.The resonance can be controlled by an external magnetic bias field B 0 , but its precise value crucially depends on the spins' immediate surroundings-which one can thus infer from a measurement of the magnetic moment after the excitation pulse.
---FIG. 1. Model of the Electron Microscopic Spin Resonance Spectroscopy (EMSRS) scheme based on a Mach-Zehnder interferometer configuration.An electron beam is split coherently into two arms separated by the distance 2d, such that each individual electron can pass to the left side (path state |L⟩) and to the right side (|R⟩) of a small spin sample in the center.The sample is aligned with respect to a bias field B0, and is described by a magnetic moment µ(t), which interacts with the path-dependent magnetic field B(t) of the moving electron and thereby causes a small phase shift between the two arms.The phase shift is detected through the difference in electron counts upon recombination of the two arms at a second beam splitter.Analogous to ESR/NMR, microwave pulses may be applied to toggle between spin orientations of the sample and enable differential measurements to subtract systematic phase shifts.
Here, the change of the magnetic moment is monitored through free-space electrons.For their interferometric readout, consider the Mach-Zehnder interferometer (MZI) model configuration sketched in Fig. 1.A dilute beam of non-interacting electrons propagating with velocity v in the z-direction is coherently split into two distinct arms separated by ∆y = 2d, each passing by a magnetic sample on opposite sides.The sample is described by a magnetic moment µ(t) placed between the two arms in the yz-plane, with a time-dependent orientation.We assume that the electron beam is split much further away from the sample than the closest distance at which the electrons pass it.The sample will interact with the moving electron and thereby build up a relative phase between the interferometer arms, which will change the likelihood that the electron is detected in one of the output ports upon recombination of the two arms.This is the measurement signal by which we seek to monitor changes in the sample's magnetic moment after it is manipulated by a microwave pulse.
Treating the magnetic dipole moment as a classical source of a magnetic field, the differences of the phases imprinted on the electronic matter wave when passing on each side of the source is given to leading order by the magnetic flux through the interferometer plane [26,30], where e is the electron charge and ℏ is the reduced Planck constant.∆ϕ S is the well-known magnetic Aharonov-Bohm phase [31,32]; the integral over the vector potential A is performed on the closed curve given by the semiclassical electron trajectories corresponding to the two arms.
In the case of a point-like magnetic dipole moment of magnitude µ that is oriented in the e x -direction normal to the interferometer plane and placed exactly in the middle between the two arms of the MZI, we can approximate the vector potential by where r = x 2 + y 2 + z 2 and µ 0 is the magnetic permeability.This results in the phase shift (see Appendix A) A single electron spin exhibits a magnetic dipole moment of approximately one Bohr magneton, µ = −g s µ B /2 ≈ −eℏ/2m e .In TEM however, one normally analyzes samples with a thickness of around 10-300 nm, corresponding to atomic columns of up to 1000 spin-active atoms with their associated magnetic moments.For classical spherical sources, we obtain an additional geometrical factor of π/2 (see Appendix G).Table I compares the phase shift differences (between both interferometerplane-orthogonal spin orientations) induced by a single electron spin, a single nuclear spin, and spin columns when the interfering electrons pass at distances of 0.1 nm and 1 nm.As we will discuss below, state-of-the-art electron-phase sensitive methods are approaching the necessary resolution to detect such tiny phase shifts caused by single quantum spins-rendering our proposed method feasible.Such a level of sensitivity comes with an important side effect.If the magnetic moment is associated with a quantum system and if it is not orthogonal to the interferometer plane, the back-action on the spin system has to be included.An appropriate treatment can be given by reversing the perspective and describing the action of the electron's magnetic field on the quantum system.In this description, each electron is a moving charge that interacts with the magnetic moment represented by the dipole operator μ(t) through the magnetic field B(t) it creates at the sample location.We focus here on the quantum regime of a single spin-1/2 aligned with respect to a bias field and driven by a short microwave pulse in order to prepare a desired input state.To this end, we replace μ(t) by a spin operator, μ(t) = µ σ(t) that freely precesses at the Larmor frequency ω 0 = µB 0 /ℏ about FIG. 2. Phase shift ∆ϕS (red) and entanglement-induced drop in interferometric visibility V − 1 (green) due to a single electron spin in a pure state (on the surface of the Bloch sphere) at distance d = 0.1 nm in the Mach-Zehnder electron interferometer configuration in Fig. 1.The spin expectation value is oriented at varying angles β with respect to the normal axis of the interferometer plane.When the spin lies in plane (β = π/2, 3π/2), the phase shift vanishes and the visibility reaches its minimum.Orthogonal to the plane (β = 0, π), the phase shift is maximal at full visibility.Using microwave pulses to toggle between β = 0 and π, one can observe the differential phase shift 2∆ϕS ≈ 0.11 mrad.
the magnetic field axis n.Details can be found in Appendix D. We restrict our considerations to the stroboscopic regime where the passage of the electron through the interaction region with μ(t) is much shorter than the time scale of the free dynamics of the spin system (the inverse Larmor frequency or the life-time of the spin state).We may thus approximate μ(t) ≈ μ(t 0 ) over the duration of the single-electron pulse.
We neglect the energy loss or gain of the electron in the interaction with the spin system as well as flips of the electron spin (see Appendix C).The interaction is then approximately given by the magnetic field of the electron at the position of the spin system integrated over the semi-classical trajectory of the electron passing by the sample at distance d in the MZI corresponding to the two arm states |L⟩ and |R⟩.Due to the orientation of the setup (see Fig. 1), the integrated magnetic field is oriented in the x-direction and its effect on the spin system is given by x-projection of the spin operator.We consider the initial superposition state |ϕ⟩ = (|R⟩ + e iϕ |L⟩)/ √ 2 created by the first beam splitter of the MZI and an adjustable external phase shift ϕ.The interaction with the spin system leads to additional phase shifts and may entangle the electron with the spin system depending on the initial state of the latter.Ultimately, the electron is detected at one of the two output ports of the second beam splitter say, |+⟩.The likelihood for this to happen, where carries information about the sample's average spin-x component at the probe time, ⟨σ x (t 0 )⟩ = ⟨ψ|σ x (t 0 )|ψ⟩.
Here, we introduce the interaction strength θ = eµ 0 µ/2πℏd, and we denote expectation values with respect to the sample state by ⟨•⟩, noting that the results extend to classical mixtures of pure states.The effect of the sample on the sinusoidal interference signal described by p + (θ, ϕ; t 0 ) is two-fold: Firstly, it shifts the sinusoidal fringes by the net phase ∆ϕ S .Secondly, it reduces the fringe contrast, or visibility V.For ⟨σ x (t 0 )⟩ = ±1, we find ∆ϕ S = ±2θ at V = 100 % visibility.This resembles the ideal, backaction-free sensing of a fixed magnetic moment through the induced phase shift 2θ recovering ∆ϕ S in equation (3).
In the general case, the x-component ⟨σ x (t 0 )⟩ of the spin vector oscillates with the Larmor precession frequency ω 0 as a function of t 0 (unless the bias field B 0 points along n = e x ).Hence, by periodically probing the sample at a fixed value of ω 0 t 0 modulo 2π and recording the electron counts with adjustable external phase ϕ, one obtains sinusoidal interference fringes with the predicted phase shift ∆ϕ S and visibility V.In figure 2, we consider a pure state of the spin system and plot ∆ϕ S and V as functions of the angle β = arccos⟨σ x (t 0 )⟩ between the normal to the interferometer plane and the expectation value of the spin vector.There are two extreme cases: 1) β = (n + 1/2)π with n integer, where the phase vanishes and the visibility of the interference pattern reaches its minimum.This case corresponds to the situation of the spin vector lying in the interferometer plane.2) β = nπ, where the phase is maximal and visibility is full.In this case, the spin vector is orthogonal to the interferometer plane.
If we assume that we measure the current at the two output ports separately, we can infer the interaction strength θ from the mean current difference, which is proportional to 2p + (θ, ϕ; t 0 ) − 1 from (4).The measurement uncertainty, given in the limit of many repetitions by the variance of the current difference, translates accordingly --FIG.3. Electron interferometric schemes to investigate spin dynamics: A quantum spin is aligned with respect to a bias field B0 (on the Bloch sphere).A microwave pump pulse manipulates the spin and causes a change of the associated magnetic moment.The resulting variation of magnetic flux in x-direction is then measured with an interferometric electron probe in the yz-plane.(a) When the bias field is perpendicular to the interferometer plane, a π-pulse toggles between the two spin eigenstates and thus allows to detect a constant phase shift during the relatively long lifetime t life of the excited state.(b) For a bias field aligned in the plane, a π/2-pulse causes the spin to precess about the field axis, allowing to probe a fast varying phase shift with a precession time 2π/ω0 ≪ t life .Only an ultra-fast TEM might be able to resolve the phase shift.For slow electron detection, this varying phase shift will on average reduce the interference visibility.
to a mean-square error Var[θ].We conclude that the maximal precision that can be reached by the intensity measurement at the two ports is Var[θ] = 1/2 √ N e for N e non-interacting electrons in the interferometer (see Appendix E).
With the electron readout scheme at hand, we now discuss two exemplary spectroscopy protocols on the spin sample, as depicted in Fig. 3.In (a), we let the bias field point in e x -direction so that the interferometer probes the time-independent spin polarization ⟨σ x (t 0 )⟩ = ⟨σ x ⟩ = s x .This causes an average phase shift by arctan(s x tan 2θ), accompanied by a visibility drop if the spin state is mixed (|s x | < 1), for example, due to thermal excitation.After interaction with a microwave π-pulse, the spin and therefore the phase shift are reversed, allowing for a differential phase measurement.
In Fig. 3(b), we consider a strong bias field B 0 that sets the spin quantization axis as n = e z , such that σx (t 0 ) = σx cos ω 0 t 0 + σy sin ω 0 t 0 .An initially e z -aligned or antialigned spin state, or a mixture of these with ⟨σ z ⟩ = s z ∈ [−1, 1] and ⟨σ x,y ⟩ = 0, will cause a visibility drop by | cos 2θ| due to the entanglement between electron and spin state by the interaction, and there will be no net phase shift.If we then apply a π/2 microwave pulse, the spin will precess with ⟨σ x (t 0 )⟩ = s z sin ω 0 t 0 , resulting in an oscillating phase shift and visibility.In the optimal case of a pure spin state (s z = ±1), the phase periodically shifts as much as ±2θ at full visibility.These maximal absolute phase shifts are obtained for t 0 being an integer multiple of π/ω 0 , in other words, when the precessing spin vector is orthogonal to the interferometer plane.From this result, we can also conclude that the integrated signal from many exactly timed electron pulses that arrive with a periodicity defined by an angular frequency ω e will show a resonance at ω e = ω 0 .
In practice, our first spectroscopy protocol (Fig. 3a) may be realized by an electron microscope in the Lorentz mode ("in-plane" magnetic field [33]).In addition to the differential measurement, this geometry allows for a null measurement after a π/2−pulse that eliminates any magnetic flux through the interferometer area.The required temporal resolution for this pump-probe configuration is linked to the life-time of the spin state.This condition is easily met by fast direct electron detectors which exhibit temporal resolutions of 260 ps [12].Our second spectroscopy protocol may be realized by exploiting the strong magnetic field generated by the pole pieces of the electron microscope to align the sample's spin along the electron beam axis.Resolution of the spin precession could be achieved with an Ultrafast Transmission Electron Microscope (UTEM) which is precisely synchronized with the precession frequency.In this setup, even coherent manipulation of the spin state with timed electron interactions might become feasible [34].
The interferometric phase shift difference due to a differential measurement of the two spin states along the x-axis (⟨σ x ⟩ = ±1), 2∆ϕ S = 4θ = 2µ 0 eγ/(2πd), depends on the strength of the magnetic dipole moment, where γ = g s µ B /ℏ is the gyromagnetic ratio (e.g.2π •28 GHz/T for the electron spin, with life-times in the range of 100 ns; 2π • 42.6 MHz/T for the nuclear hydrogen spin, with life times of minutes or even hours).This leads to a phase shift difference of ∼ 1.1 × 10 −2 mrad due to a single electron spin interferometrically probed at a distance of 1 nm.Comparing to currently achievable phase sensitivities of ∼ 4.5 × 10 −2 mrad, [28] shows that single electron spins could soon become experimentally observable.Rather than the elementary case of a single spin, one could study coherent ensembles of N S spins.The resulting maximal phase shift is then ∆ϕ S = ±2N S θ.Details about the derivation of this result can be found in Appendix F. In transmission electron microscopy, samples with a thickness of about 10 − 300 nm are typically examined [26], which amounts to atomic columns consisting of up to N S ∼ 1000 spins.Realistic samples will not be completely coherent and point-like.In particular, finitetemperature effects will only lead to a partial polarization of the spin sample, and the finite size will lead to addi-TABLE I. Estimates of the difference 2∆ϕS of magnetic phase shifts an electron experiences due to the magnetic moment of a spin system at distance d for the two spin orientations orthogonal to the interferometer plane.We compare single electron and hydrogen nuclear spins to partially spin-polarized atomic columns of 1000 spins.For the electron spin column, we take the thermal polarization of 12% at a magnetic bias field of 1.8 T at a temperature of 10 K.The nuclear spin column is assumed to be 10% hyperpolarized [36][37][38]40].
tional geometric factors, as exemplified in Appendix G.In the high-temperature regime, k B T ≫ γ e ℏB 0 , we can estimate that a fraction γℏB 0 /2k B T of the number N S of spins will constitute the sample's net magnetization probed by the electrons.For example, at a bias field of B 0 =1.8 T, which is a standard magnetic field at the sample region in a TEM, and a temperature of 77 K, 1.6 % of electron spins will contribute to the phase signal.This goes up to 12 % at 10 K and leads to a phase shift of 1.4 mrad for d = 1 nm (see also Table I).Measuring this phase shift in a quantum projection noise limited scheme requires only ≲ 2.2 • 10 6 electrons while a typical beam current of 1.5 nA in scanning TEM amounts to ∼ 10 10 electrons/sec.These figures are an incentive to advance electron microscopy at dilution refrigerator temperatures in the sub-Kelvin range [35].
In such cryogenic environments or by using hyperpolarisation techniques [36][37][38] and summing over atomic "spin" columns, even the weak phase shifts due to nuclear spins should become detectable, especially since nuclear spins exhibit very long coherence times on the order of minutes to even hours [39].A short derivation of the phase shift due to a sample of I = 1/2 nuclear magnetic spins based on the model of classical magnetization can be found in Appendix G.
Owing to various technical improvements in aberration correction, fast direct electron detection and highly coherent electron sources, Differential Phase Contrast (DPC) measurements reach deflection sensitivities below 25 nrad (spatial resolution of ∼ 9 nm) [27].The deflection due to a single electron spin corresponding to the gradient of the single-arm phase ∆ϕ S /2 is α = ℏ∆ϕ S /(2mγ L vd), given the 1/d-dependence of ∆ϕ S in equation ( 3) and the Lorentz factor γ L (see Appendix B).For a probe beam distance of d = 0.1 nm and d = 1 nm and electron kinetic energies of 200 keV, we find α ∼ 110 nrad and α ∼ 1.1 nrad, respectively.This implies that measurements of the deflection due to single spins may be feasible in the near future with DPC.
These promising figures encourage combining noninvasive magnetic resonance techniques with electron microscopy with wide-ranging applications from characterizing spin dynamics in battery electrodes [41,42] to investigating biological systems [43,44] with NMR/ESR techniques on the nanoscale.In principle, since the electron only needs to pass adjacent to the region of interest, radiation damage could be strongly suppressed compared to the direct examination of samples.Moreover, for certain optimized geometries, the electron only acquires a phase and no energy is transferred to the quantum system.Realizing this non-invasive scheme to probe a quantum property previously inaccessible to electron microscopy techniques will provide deeper insights into the quantum nanoworld at the highest spectroscopic resolution.
Inserting (D5) leads to the simple expression (4), which exhibits sinusoidal interference fringes phase-shifted by ∆ϕ S and reduced in visibility by At a finite temperature T , the likelihood (D6) must be averaged over the thermal distribution of spin states.For n = e x and weak interaction strengths θ ≪ 1, this leads to the slightly reduced visibility k B T ≫ γ e ℏB 0 .In that case, we find for the magnetic susceptibility (with spin I = 1/2 and γ = γ e ) [47] χ = µ 0 n S γ 2 e ℏ 2 I(I + 1) 3k B T = µ 0 n S γ 2 e ℏ 2 4k B T , (G3) where n S = N S /V is the electron spin density, N S is the number of electron spins in the sphere and V is the volume of the sphere.We find for the magnetic flux with r e the classical electron radius and γ e = 2π • 28 GHz the electron gyromagnetic ratio.Besides the Boltzmann factor and the numerical factor π/2, we recover the results of the previous section.
Of course, similar calculations as for the interaction of a free electron with an electron spin can be made for the spin of a nucleus.We simply consider µ = µ N g I /2, where µ N = eℏ/(2m p ) is the nuclear magneton and g I is the total nuclear spin g-factor.For simplicity, we assume that the nuclear spin is also 1/2 and we use the representation in terms of the Pauli matrices as in the previous case writing μ = −µ N g I σ/2.Accordingly, we recover all above equations with the replacement of the coupling parameter θ by where γ N is the nuclear gyromagnetic ratio which depends on the atoms that are probed.The resulting phase shift is

Φ
≈ N S g S µ 0 µ B R γ e ℏB 0 8k B T .(G4)For the magnetic phase in the electron MZI, we find∆ϕ S = − e ℏ Φ ≈ −N S g S the above analysis, we can also calculate the phase shift due to a classical magnetization corresponding to nuclear spins for k B T ≫ ℏγ N B 0[47] χ = µ 0 n S γ 2 N ℏ 2 I(I + 1) 3k B T , (G8)and the maximal phase due to a magnetized sphere (if the electron interferometer encloses the whole sphere and just the sphere) becomes ∆ϕ S = − e θ I up to the Boltzmann factor and the numerical factor π/2.