Introduction to nuclear spin waves in ferro- and antiferromagnets

For fields produced by resistive magnets, this frequency lies in the range 10–100 MHz so that the nuclear spins can be set in precession by radio-frequency waves, similarly to electronic spins driven by microwaves in the gigahertz range. This was first demonstrated by Rabi¹ in 1938 using a molecular beam of LiCl as in the Stern-Gerlach experiments, but he did not coin the name nuclear magnetic resonance (NMR) by which the technique would later be known. Rabi used the technique to measure the nuclear moment directly with great precision, and he was awarded the Physics Nobel Prize in 1944 for this work. In 1946, Bloch and Purcell² improved the technique for use on liquids and solids, a major achievement for which they shared the Nobel Prize in Physics in 1952. With continuing improvements, NMR became an important routine technique for the chemical analysis of materials and for advanced medical tomography techniques, such as in magnetic resonance imaging (MRI).

In paramagnetic materials, in which the electronic spins are not ordered, the nuclear spins are independent of each other hence, they do not have collective excitations. However, to explain the large NMR linewidth observed by Jaccarino and Shulman³ in antiferromagnetic MnF₂, Suhl⁴ and Nakamura⁵ proposed in 1958 that in strongly magnetic materials the nuclear spins can interact indirectly. The Suhl–Nakamura interaction, which is mediated by the hyperfine coupling with electronic spins and the exchange interaction between them, was shown by de Gennes et al.⁶ in 1963 to give rise to collective excitations of the nuclear spins. In the following years, nuclear spin waves attracted considerable interest both theoretically and experimentally, and several manifestations of their presence were observed in various ferromagnetic and antiferromagnetic materials.^7–30 

In recent years, there was a renewal of interest in nuclear spin waves for their possible applications in quantum information science and in antiferromagnetic spintronics. Indeed, a nuclear spin is a promising candidate for a quantum bit because of the long nuclear spin decoherence times.^31,32 Also, hybrid quantum computing schemes can benefit from multiple quantum degrees of freedom through the coherent transfer of quantum information between them.^32–34 In this case, the fact that the nuclear magnetic moment is three orders of magnitude smaller than an electron spin enables a hybrid system for quantum computing in which the electron spin qubit (which is more readily polarized and more quickly manipulated) is used for initialization and processing, while the nuclear spin is used as a memory. Also, the fact that nuclear spin waves have been studied in detail mainly in antiferromagnetic materials comes hand-in-hand with the emergence of antiferromagnetic spintronics.^35–41 The very recent observations of spintronic phenomena produced by nuclear spin waves in antiferromagnets reported by the group of Eiji Saitoh^42,43 have provided new motivations for the study of nuclear spin waves and mark the beginning of a new research field called nuclear spintronics.

This Tutorial is devoted to review the basic concepts and properties of nuclear spin waves, which are the collective nuclear spin excitations in magnetic materials, and to present recent developments in the new field of nuclear spintronics. Similar to electronic spin waves, nuclear spin waves are quantized, and their quanta are called nuclear magnons. We start the review presenting the nuclear spin excitations in ferromagnets because they are simple and contain the basic physics involved in the indirect coupling between nuclear spins in neighboring atoms, the Suhl–Nakamura interaction. In the treatment of antiferromagnets (AFs), we restrict attention to simple AFs with only two spin-sublattices, one class with uniaxial magnetic anisotropy, such as MnF₂, and the other with cubic anisotropy, such as RbMnF₃ and KMnF₃, in which the nuclear spin waves have been studied experimentally. Finally, we briefly present very recent observations of spintronic phenomena produced by nuclear spin waves, which mark the beginning of a new research field called nuclear spintronics, and discuss possible developments in the field.

The Hamiltonian (23) represents an effective interaction between nuclear spins that resembles the exchange interaction between electronic spins. It is an indirect interaction that takes place by the emission of a virtual electronic magnon by a nuclear spin flip $I i ′ −$ at site i’ and the absorption of this magnon by another spin flip $I i +$ at site i. This process is analogous to the indirect interaction between nuclear spins via conduction electrons in metals, the Ruderman–Kittel interaction.⁴⁷ 

The result (26) has the form of a Yukawa potential, showing that the interaction between nuclear spins decays exponentially with distance, having range $a ( H E / z H 0 ) 1 / 2$⁠, which is on the order of 10–30 lattice spacings. This is actually quite a long range compared to the exchange interaction between electronic spins. This is the interaction between nuclear spins proposed by Suhl and Nakamura to explain the broad NMR lines observed in magnetic materials.

Equations (33) and (34) show that the mode α_k (≈a_k) involves essentially the excitation of electronic spins, with a frequency that is slightly different from the pure electronic magnon frequency ω_k. On the other hand, the mode β_k (≈b_k) corresponds to collective excitations of the nuclear spins with a frequency that is lower than the usual NMR frequency ω_N. Note that the two frequencies have opposite signs because the electronic and nuclear spins precess with opposite senses. The variation with the wave number k of the absolute values of the two frequencies is shown in Fig. 2(a). The hyperfine field H_N is of the order of a few oersteds at low temperatures so that the fractional frequency pulling of the nuclear mode in ferromagnets is quite small.

Only recently, the nuclear magnon dispersion relation was measured in a ferromagnetic material. Chatterji et al.³⁰ used inelastic neutron scattering to investigate the nuclear spin excitation in ferromagnetic Nd₂CuO₄, a parent compound of the electron-doped superconducting compound Nd_2−xCe_xCuO₄. Hyperfine induced nuclear spin ordering had been reported⁴⁸ earlier in Nd₂CuO₄ at temperatures below 200 mK associated with the isotopes ¹⁴³Nd and ¹⁴⁵Nd that have nuclear spin I = 7/2 and natural abundances 12.18% and 8.29%, respectively. Using neutron-spin-echo spectroscopy at millikelvin temperatures, the authors of Ref. 48 managed to measure the frequency of the nuclear spin excitation as a function of the wave number. Figure 2(b) shows the dispersion relation measured in Nd₂CuO₄ at T = 40 mK for the wave vector in the [111] direction, which is in very good qualitative agreement with the dispersion curve for the nuclear excitation in Fig. 2(a).

The basic concepts underlying the origin of nuclear magnons in AFs are the same as in ferromagnets, as presented in Sec. II. In each magnetic ion that has a magnetic nucleus, the nuclear spin interacts with the electronic spin by the hyperfine interaction and indirectly interacts with the nuclear spin of another ion through the electronic spin exchange coupling. Like in ferromagnets, this results in a Suhl–Nakamura interaction between nuclear spins, with a strength that increases with decreasing electronic magnon frequency. However, since AF materials have two or more sublattices of electronic spins, they also have two or more sublattices for nuclear spins. Thus, the calculation of the frequencies of the coupled electron–nuclear spin excitations requires the solutions of four or more coupled equations.

As will be shown in this section, in antiferromagnetic (AF) materials that have electronic spin wave frequencies in the gigahertz range, the coupling with the collective nuclear excitations is stronger than in ferromagnets. For this reason, these materials have been in the center stage for the studies of nuclear magnons. However, only few AF materials exhibit nuclear magnon effects, because these require magnetic ions with abundant magnetic isotopes and low electronic magnon frequencies. Besides the exchange interaction, the electronic spins in most studied AF materials have a variety of other interactions, such as uniaxial anosotropy as in MnF₂, cubic crystalline anisotropy as in KMnF₃ and RbMnF₃, biaxial anisotropy as in CsMnF₃, and Dzyaloshinskii–Moriya interaction as in MnCO₃. A consequence of these interactions is that the electronic spin configuration depends on the temperature and applied magnetic field and exhibits various phases, each with a different field dependence of the electronic magnon frequencies. The most studied AFs have Mn²⁺ ions, because they have the ⁵⁵Mn magnetic isotope with 100% abundance with electronic ground state configuration 3d⁵(⁶S_5/2), which has no orbital angular momentum, so that it has very small single-ion anisotropy. The origin of the magnetic anisotropy of MnF₂ relies mainly on the dipolar interaction,⁴⁹ and the relatively small anisotropy is an important requirement for having low electronic magnon frequencies. FeF₂ that is another well-studied AF material with the rutile structure has no trace of nuclear magnons because the magnetic ⁵⁷Fe isotope has natural abundance of only 2.25%, and also the electronic magnon frequencies are very high because of the non-zero orbital angular momentum of Fe⁺² that results in large crystal field anisotropy. Other studied AFs include CoCO₃, where the isotope ⁵⁹Co is 100% abundant, and FeBO₃, that has magnon frequency in the gigahertz range.²⁸ 

The calculation of the frequencies of the electron–nuclear spin excitations can be carried out either with a semi-classical treatment using the Landau–Lifshitz equation or with a quantum treatment such as the one described in Sec. II. Here, we present the main steps of the quantum formulation for antiferromagnetic fluorides with the rutile structure, shown in Fig. 3(a), and with the perovskite structure, shown in Fig. 3(b). While the rutile AF, such as MnF₂, has crystalline uniaxial anisotropy, the perovskite AFs, such as KMnF₃ and RbMnF₃, have cubic anisotropy. Depending on the direction and intensity of the applied magnetic field, the spins can be aligned with the field or in a canted configuration, resulting in quite different behavior of the excitation frequencies.

This result shows that similarly to ferromagnets, the frequencies of the collective nuclear spin excitations in antiferromagnets are pulled down by the electronic modes and have a dependence on the wave number. However, since the second term in Eq. (41) contains the exchange field that is three orders of magnitude larger than the hyperfine field in Eq. (21), the frequency pulling in antiferromagnets can be much larger than in ferromagnets.

Another feature of AFs that is unique for studies of the nuclear excitations is the fact that one of the electronic magnon frequencies decreases with increasing field, making possible to tune the strength of the coupling between neighboring nuclear spins. This has been clearly demonstrated in MnF₂. Figure 4(a) shows variation of the k = 0 down-mode electronic frequency, or antiferromagnetic resonance frequency (AFMR), measured in MnF₂ at 4.2 K with the field parallel to the easy-axis. The frequency decreases linearly with increasing field, as in Eq. (39a), and tends to zero at the critical field value $H c= [ 2 H E H A + H A 2 ] 1 / 2$⁠, which is approximately 92.94 kOe, a value consistent with the effective fields $H E=526$ kOe and $H A=8.2$ kOe in MnF₂.^49,52 This is the field intensity above which the spins flip to directions nearly perpendicular to the field direction in the so-called spin-flop phase. The four panels in Fig. 4(b) show clearly the effect of the increasing strength of the interaction between nuclear spins with increasing field. For H₀ = 75 kOe, the AFMR frequency is high, nearly 50 GHz, so that the coupling is very small. In this case, the NMR spectrum shows the five lines due to the quadrupole splitting, typical of uncoupled nuclear spins as in nonmagnetic materials.⁵³ For H₀ = 88.5 kOe, the AFMR frequency has decreased to 12.5 GHZ, and the intensity of the line corresponding to transitions between the two quadrupole-split levels has greatly increased relative to all others while the splitting is unchanged.

For H₀ = 91 kOe, the strongest line has become Lorentzian in shape, a clear indication of the collective nuclear spin excitation, analogous to ferromagnetic resonance lines that have no hyperfine splitting. The mechanism for the disappearance of the quadrupole split lines is the exchange narrowing, well-known in paramagnetic electronic spin resonance.⁵³ Note that the line has the shape of a Lorentzian derivative because the measurements are made with frequency modulation. Finally, for H₀ = 92.2 kOe, the spectrum measured with frequency modulation exhibits a (210) nuclear magnetostatic mode (MSM), in addition to the main uniform (110) mode. The inset shows the numerous MSM that appear on the low-frequency side of the (110) mode. These are analogous to the electronic magnetostatic modes that correspond to the resonances of spin waves with wavelengths comparable to the sample dimensions, well-known in ferromagnetic resonance experiments in ferrites.⁵⁴ This result represents a clear demonstration of the wave nature of the collective nuclear spins coupled by the Suhl–Nakamura interaction, namely, the nuclear spin waves.

Another class of materials that exhibits pronounced effects of the collective nuclear spins excitations is formed by antiferromagnets with the perovskite structure, shown in Fig. 3(b), in which the magnetic ion is Mn⁺², such as KMnF₃ and RbMnF₃. Since in the cubic arrangement the total dipole field is zero, these materials have very small anisotropy, resulting from the residual single-ion origin.⁵⁵ Thus, the spin-flop field is small compared to most AFs, and the electronic magnon frequencies are low, making possible to study nuclear magnons with magnetic fields of few kOe. Note that KNiF₃ is also a well-studied AF material,⁵⁶ but has no sign of nuclear magnons because the magnetic ⁶¹Ni isotope has natural abundance of only 1.25%, and also the electronic magnon frequencies are high because of the non-zero orbital angular momentum of Ni⁺² that results in large crystal field anisotropy.

The solutions of the coupled electronic–nuclear excitations in the AF phase are similar to the ones in Sec. III A, except close to the spin-flop (SF) transition. Thus, here we restrict attention to the modes in the SF phase, where the electronic and nuclear spins have the configuration illustrated in Fig. 5. The electronic spins $S → i, S → j$ of the two sublattices have directions at an angle θ with the field and in the (110) plane, while the nuclear spins $I → i, I → j$ are antiparallel to them. We shall study the spin wave excitations in the SF phase using two different coordinate systems, one for each sublattice, as shown in Fig. 5. In each system, the z-axis is chosen to point along the spin equilibrium direction in that sublattice, the y-axes are in directions perpendicular to the field and to the plane of the spins, and the x-axes are determined by $x ^= y ^× z ^$⁠.

Thus, the frequency ω_4k of one of the nuclear modes does not vary with magnetic field, while the frequency ω_3k of the other nuclear mode is strongly dependent on the field.

The dependencies of the frequencies on the applied field have been measured in detail in RbMnF₃ by Ince and Morgenthaler.^16,18 The measurements, shown in Fig. 6, were made with fixed signal frequency and scanning field intensity, with the field applied along the [001] axis of the sample. Two ranges of frequencies were used, one in the microwave range, from 4 to 12.5 GHz, to measure the AFMR absorptions, and one in the UHF range, from 280 to 700 MHz, to measure the NMR absorptions. As shown in the insets of Fig. 6, in each field scan two absorption peaks are observed, one corresponding to a magnon mode in the AF phase and the other in the SF phase. In this way, each scan gives two data points for the figure. The solid lines represent the results of calculations made with γ_e = 2.8 GHz/kOe, H_E = 816 kOe, and H_A = 4.59 Oe.¹⁸ These values give for the spin-flop field $H c= ( 3 H E H A / 2 ) 1 / 2=2.37 kOe$⁠, the field for which the AFMR frequency is minimum and the frequency pulling of the nuclear mode is maximum, in agreement with the results expressed in Eqs. (50) and (51).

The active field of spintronics is devoted to the investigation of basic phenomena and their applications in devices for transport, storage, and processing of information, in which the main physical entity is the electron spin. A key concept in spintronics is the spin current that expresses the flow of spin angular momentum in a material. In metals, the spin current is carried by the spins of the conduction electrons, whereas in magnetic insulators the spin current is transported by magnons. Three of the most important phenomena in spintronics are the spin Hall effects, direct (SHE) and inverse (ISHE),^57,58 the electric spin pumping effect (SPE),^59,60 and the spin Seebeck effect.⁶¹ These phenomena have been intensively studied in structures with ferromagnetic insulators, which have one spin current channel, and in two-sublattice antiferromagnetic insulators, which have two spin current channels. Very recently, the group of Eiji Saitoh has reported SPE and SSE involving the nuclear magnon channels in an AF insulator, opening new possibilities in spintronics by combining the advantages of the long-lived nuclear spin states with the information technology readily available through spintronic devices.^42,43

The spin pumping process in a bilayer made of a ferromagnetic (FM) material and a metallic layer (ML) consists in the emission of a spin current by the precession of the magnetization in the FM driven in ferromagnetic resonance and its injection in the ML. The spin pumping was conceived in 2002 as a mechanism for the magnetization damping in FM/NM bilayers.⁶² However, a more interesting effect of the spin pumping is the generation of an electric voltage in the ML layer, due to the conversion of the injected spin current into a charge current by means of the inverse spin Hall effect.^58–60 This electric spin pumping effect has been investigated in a variety of material structures driven by microwave radiation. The work of Saitoh's group has opened a new era in spintronics with the demonstration of the SPE generated by nuclear spin waves.

In the investigation of Shiomi et al.,⁴² the magnetic material in the bilayer is the weakly anisotropic antiferromagnet MnCO₃, in which nuclear spin waves were studied some time ago.^9,10 In MnCO₃, a ⁵⁵Mn nucleus (spin I = 5/2) has a strong hyperfine interaction and a 100% abundance of the magnetic isotope. Due to the weak anisotropy and Dzyaloshinskii–Moriya interaction, the electron spins in the two sublattices order in a canted arrangement at temperatures below T_N = 35 K. The frequencies of the two electronic modes are non-zero and similar at H₀ = 0, and as the field increases, one frequency increases continuously with field while the other does not vary.⁹ Since the spin excitations are the same as in the spin-flop phase, the two nuclear modes have frequencies given by Eq. (50). Very interestingly, Shiomi et al.⁴² found that a nuclear spin wave excited by a radio wave in insulating MnCO₃ single crystals pumps a spin current into an attached 5-nm-thick Pt film through the electronic spin dynamics created by driving the nuclear spin dynamics. The electronic spin current injected into the Pt film is converted into a charge current by the ISHE, producing a voltage at the ends, as in the usual SPE experiments.

The measurements were performed at a fixed magnetic field, by applying continuous radio waves in a perpendicular driving configuration. At each magnetic field, a clear dip structure indicating the NMR absorption appears in the radio-wave reflection spectra, as shown in Fig. 7(a). With increasing magnetic field, the NMR peak-frequency f_NMR increases because the frequency of one of the electronic modes increases so that the frequency pulling decreases. Figure 7(b) shows the NMR frequency as a function of the applied field at several temperatures, resembling the data in Fig. 6(b) for fields above the spin-flop value. The solid curves in Fig. 7(b) show that a theoretical calculation of the field dependence of the NMR frequency using the Landau–Lifshitz equation for the four-sublattice system is in good agreement with the data.⁴²  Figure 7(c) shows the spin pumping DC voltage measured in the Pt layer at a constant field of 0.3 T, as a function of radio-wave-frequency, at various temperatures. At 3.0 K, no voltage signal is observed. However, as the temperature decreases to 2.4 K, a clear voltage peak appears around the NMR frequency (600 MHz). Voltage lines broader than those of the NMR spectra shown in Fig. 7(a) result from slower frequency-sweep rates used in the voltage measurements. The intensity of the voltage peak increases with decreasing temperature and reaches approximately 8 nV at 1.52 K.

As Mewes states,⁶³ “With their elegant experiment, Shiomi et al.⁴² have shown that pure spin currents can be generated using nuclear spin states, bridging two mostly separate worlds: the world of nuclear spin excitations and the world of spintronics. These fascinating results have the potential to drive new developments in nuclear spintronics by combining the advantages of the long-lived nuclear spin states with the information technology readily available through spintronic devices.”

Saitoh's group has also recently reported the observation of another important spintronic effect produced by nuclear spin excitations, namely, the nuclear spin Seebeck effect (SSE). The spin Seebeck effect in a bilayer made of a ferromagnetic insulator (FMI) and a metallic layer (ML) consists in the generation of a spin current in the FMI by a temperature gradient and its injection in the ML. In the ML, the spin current is converted into a charge current by means of the inverse spin Hall effect that produces a voltage at the ends of the ML.⁶¹ The insulator-based SSE device exhibits unconventional characteristics potentially useful for thermoelectric applications, such as simple structure, device-design flexibility, and convenient scaling capability.⁶⁴ 

Two mechanisms have been proposed for the SSE, one based on an interfacial phenomenon,^65,66 and one based on bulk spin waves.^67,68 The interfacial mechanism relies on the temperature difference between the phonon–electron system in the metal and the magnons in the insulator at the interface. This temperature difference produces a spin pumping by thermal magnons resulting in a spin current that is injected in the ML where it is converted into a charge current.^65,66 The bulk mechanism considers that the thermal gradient across the FMI layer produces a flow of magnons that transports a magnonic spin current, which is injected into the ML where it is converted into a charge current.^67,68 Recently, Jiménez-Cavero et al.⁶⁹ have quantified the two contributions and compared to experimental data in a FMI/ML device and shown that for FMI layer thickness larger than 100 nm the bulk mechanism accounts for over 90% of the measured SSE voltage. The bulk magnonic spin current mechanism also explains measurements of the SSE in bilayers made of antiferromagnetic insulators and metallic layers.⁷⁰ 

Kikkawa et al.⁴³ observed the nuclear SSE in heterostructure devices composed of a crystal of the easy-plane canted antiferromagnetic MnCO₃ and a Pt film, similar to the ones used in the studies of the nuclear SPE. The Pt strip acts as a heater as well as a spin-voltage converter for measuring the nuclear SSE. By applying an alternating current (AC) to the Pt layer, the heat generated produces a spin current by the SSE that can be measured by the second harmonic voltage in the Pt layer with a lock-in technique, allowing the selective detection of the SSE-ISHE voltage. The experiments were conducted in a ⁴He cryostat and a dilution refrigerator for ultralow-temperature experiments down to 100 mK with an applied magnetic field that produces the canted spin alignment.

Figure 8(a) shows a schematic illustration of the nuclear spin Seebeck effect in a hybrid structure made with the canted antiferromagnet MnCO₃ in contact with a Pt film. By heating the Pt film with an AC, the temperature gradient across the device produces a spin current J_s that is converted into a charge current via the ISHE in the Pt strip generating a voltage that is measured by the second harmonic component. Figure 8(b) shows the SSE voltage normalized with the heating electric power as a function of the applied magnetic field, measured at several sample temperatures. At temperatures of 20 K or above, no voltage signal appears with the application of the field. However, as the temperature decreases, a voltage signal appears with sign that changes by reversing the field direction, typical of the SSE.^61,64,67 The signal intensity scales linearly with the heat power, demonstrating that its origin is thermoelectric.

The experimental results are quantitatively explained by Kikkawa et al.⁴³ using a theory for the nuclear SSE in terms of the interfacial thermal spin pumping rather than bulk spin transport, in which the Korringa process⁷¹ due to the hyperfine coupling between nuclear spins in the MnCO₃ and conduction-electron spins in the attached Pt is taken into consideration.

In this Tutorial, we have presented a brief review of the basic concepts and properties of the collective nuclear spin excitations, or nuclear spin waves, in ferro- and antiferromagnetic materials. The treatment in ferromagnets contains the basic physics involved in the indirect coupling between nuclear spins in neighboring atoms, the Suhl–Nakamura interaction. The treatment of antiferromagnets is restricted to simple AFs with only two sublattices, one class with uniaxial magnetic anisotropy, such as MnF₂, and the other with cubic anisotropy, such as RbMnF₃ and KMnF₃, in which the nuclear spin waves have been observed experimentally. The theoretical results are compared with the experimental data reported for these materials. Finally, we briefly presented very recent experimental observations of two main spintronic phenomena produced by nuclear spin waves in the canted antiferromagnet MnCO₃, the nuclear spin pumping effect and the nuclear spin Seebeck effect. These recent observations set the stage for a new research field, called nuclear spintronics. Certainly, in the near future, we shall see detailed theoretical and experimental studies of these two effects and other nuclear spintronic phenomena in other antiferromagnets. One strong candidate is the uniaxial AF MnF₂ in which one electronic magnon frequency can be lowered to the microwave range by the application of a large magnetic field,²³ and that has shown to exhibit the magnonic spin Seebeck effect.⁷² Other candidates are KMnF₃ and RbMnF₃ that have small anisotropy and consequently low electronic magnon frequencies, and in which the nuclear spin waves have been studied experimentally in detail.^7,8,16,18

The author is grateful to José Diego Marques de Lima for helpful discussions. This research was supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES), Financiadora de Estudos e Projetos (FINEP), and Fundação de Amparo à Ciência e Tecnologia do Estado de Pernambuco (FACEPE).

The author has no conflicts to disclose.

Sergio M. Rezende: Conceptualization (lead); Writing – review & editing (lead).

The data that support the findings of this study are available from the corresponding author upon reasonable request.