Collective nuclear spin excitations, called nuclear spin waves or magnons, are enabled in strongly magnetic materials by the hyperfine coupling of the nuclear and electronic spins in an atom and the exchange interaction between electronic spins of neighboring atoms. Nuclear spin waves attracted the interest of theoretical and experimental researchers worldwide about four to five decades ago and then waned. Very recently, two experimental reports of nuclear spintronic effects in the canted antiferromagnet MnCO_{3} have shown that spin currents can be generated using nuclear spin states, bridging two quite separate worlds, one of nuclear spin excitations and the other of spintronics. In this Tutorial, we briefly review the basic concepts and properties of nuclear spin waves in ferro- and antiferromagnetic (AF) materials and present a few significant experimental results obtained some time ago with the uniaxial anisotropy AF MnF_{2} and the cubic anisotropy AF RbMnF_{3} and compare them with theory. We also briefly present the recent experimental observations of the nuclear spin pumping effect and the nuclear spin Seebeck effect in the canted antiferromagnet MnCO_{3}. Other possible AF candidates for studies of nuclear spintronic effects are discussed.

## I. INTRODUCTION

As is well-known, the atomic nuclei of some isotopes of several elements have a magnetic moment $\mu \u2192N$ and a spin angular momentum $\u210fI\u2192$ which are parallel and proportional to each other so that one can write

where $\gamma N$ is the nuclear gyromagnetic ratio. Its value for a proton is $\gamma N1=2.68\xd7104s\u22121Oe\u22121$ and for Mn^{55} it is $\gamma N55=6.6\xd7103s\u22121Oe\u22121$, which are three orders of magnitude smaller than the electronic gyromagnetic ratio $\gamma e=1.76\xd7107s\u22121Oe\u22121$. When subjected to a static magnetic field with intensity *H*_{0} and properly perturbed, the nuclear moment precesses about the field with frequency

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 Rabi^{1} 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 Purcell^{2} 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 materials, the NMR frequency is a result of various interactions, the most important of which is the coupling between nuclear and electronic spins that is called hyperfine interaction. This originates in the classical magnetic dipole–dipole coupling and in the quantum contact interaction. In the simplest form, the hyperfine interaction is represented by the Hamiltonian

where $I\u2192$ and $S\u2192$ are, respectively, the nuclear and electronic spin operators and *A* is the hyperfine constant. This hyperfine coupling changes the NMR frequency to

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 Shulman^{3} in antiferromagnetic MnF_{2}, Suhl^{4} and Nakamura^{5} 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 MnF_{2}, and the other with cubic anisotropy, such as RbMnF_{3} and KMnF_{3}, 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.

## II. NUCLEAR SPIN WAVES IN FERROMAGNETS

We consider initially the collective nuclear spin excitations in ferromagnets. As mentioned in Sec. I, in each magnetic ion, or atom, of a magnetic material, the spin of the nucleus interacts with the electronic spin through the hyperfine coupling. In this way, the exchange interaction between the electronic spins of neighboring ions provides an indirect mechanism for the coupling of the nuclear spins, as illustrated in Fig. 1(a). The Hamiltonian of the systems consisting of contributions from Zeeman, exchange, and hyperfine interactions can be written in the form

where we have assumed the exchange interactions only between the nearest electronic spins at lattice sites *i* and *i* + *δ* with the exchange parameter *J*. Notice that we have taken the electronic Zeeman term with the positive sign because the electronic magnetic moment is opposite to the spin, while for the nuclei the moment and the spin are in the same direction. The orientation of the spins in the ground state is shown in Fig. 1(b). Considering the field applied in the *z*-direction of a Cartesian coordinate system, the Hamiltonian (5) can be written in terms of the components of the spin operators as

where $Si\xb1=Six\xb1Siy$ and $Ii\xb1=Iix\xb1Iiy$are electronic and nuclear spin-raising and lowering operators that obey the well-known commutation relations.^{44} In the spin wave approximation, we can introduce spin-deviation operators^{45,46}

where $ai\u2020$, $ai$ and $bj\u2020$, $bj$ are the creation and destruction operators for the nucleus and electron spin deviations, which satisfy the boson commutation rules $[ai,ai\u2032\u2020]=\delta ii\u2032$, $[ai,ai\u2032]=0$, $[bj,bj\u2032\u2020]=\delta jj\u2032$, and $[bj,bj\u2032]=0$. The use of $\u27e8Iz\u27e9$ instead of *I* is due to the fact that even at low temperatures, the upper Zeeman levels of the nuclear spin systems are reasonably well populated because their spacing is small. The use of the average is well justified because it does lead to meaningful collective excitations of the system.^{6} Also, although $\u27e8Iz\u27e9$ is small at not too low temperatures, the use of the Holstein–Primakoff transformations in Eq. (7) is justified by the good agreement of the end results with experiments. Inserting the transformations (7) into Eq. (6) and neglecting terms with four operators, we obtain the Hamiltonian in terms of the spin-deviation operators

This equation shows that in the presence of the hyperfine interaction neither the pure electronic magnons nor the simple nuclear spin excitations can be eigenstates of the Hamiltonian. The next step to obtain the new normal modes of the coupled system consists of introducing a transformation from the localized field operators to collective boson operators *a _{k}* and

*b*that satisfy the commutation rules $[ak,ak\u2032\u2020]=\delta kk\u2032$, $[ak,ak\u2032]=0$, $[bk,bk\u2032\u2020]=\delta kk\u2032$, $[bk,bk\u2032]=0$, defined by

_{k}where *N* is the number of spins in each sublattice and $k\u2192$ is a wave vector, and we have the orthonormality condition

Substitution in Eq. (8) gives

where $\omega k=\gamma e(H0+Dk2)$ is the frequency of the electronic magnons as a function of the wave number *k*, $D=2JSa2/\gamma e\u210f$ is the exchange parameter, *a* is the lattice parameter for a cubic crystal, and *ω _{N}* is the nuclear magnetic resonance frequency given by Eq. (4). With the Hamiltonian (11) in terms of the boson operators, we shall calculate the frequency of the nuclear magnons using two approaches, perturbation theory and diagonalization of the Hamiltonian.

### A. Perturbation theory: The Suhl–Nakamura interaction

Notice that if the hyperfine interaction is null, the last term in the Hamiltonian (11) vanishes and the eigenstates of the system are the direct products of the independent states representing the electronic and nuclear magnons. These can be generated by the successive application of the creation operators on the vacuum state $|0\u27e9$and are given by

where $nke$ and $nkN$ are the number of electronic and nuclear magnons. For a finite hyperfine interaction, we consider the last term in Eq. (11) as a perturbation, namely,

Thus, the energy of a state $|nkN,nke\u27e9$ calculated with perturbation theory up to second order due (13) is

where the zero-order term is

The first-order term in (14) is zero because the operators in the Hamiltonian (13) change the numbers by one unit, generating orthogonal states. To calculate the effect of the hyperfine coupling on the nuclear spin excitations, we consider that the electronic system is in its vacuum state, i.e., $|nke\u27e9=|0\u27e9$. Therefore, the energy correction for the nuclear spin state is, in second order,

Next, we assume that the nuclear energies in the denominator are negligible compared to the electronic magnon energy, a condition that is comfortably satisfied in usual cases, and note that the only magnon state that contributes to the sum is $|nk\u2032e\u27e9=|1\u2212ke\u27e9=a\u2212k\u2020|0\u27e9$. Thus,

which can be reduced further using the completeness relation$\u2211nk\u2032N|nk\u2032N\u27e9\u27e8nk\u2032N|=1$. Hence, the energy becomes

plus a constant term. The result (18) means that the effect of the electronic magnons on the nuclear excitations is to create an effective interaction between nuclear spins given by

which is called the Suhl–Nakamura Hamiltonian. Equation (19) shows that the energy eigenvalues depend on the wavevector $k\u2192$ such that the nuclear spin excitations in a crystal have a collective nature. The total Hamiltonian for the nuclear system becomes

Since usually $\gamma NH0<<AS$, $\omega N\u2248AS$, the nuclear magnon frequency becomes

where $HN=A\u27e8Iz\u27e9/\gamma e$ is the hyperfine field seen by the electronic spins. The second term in Eq. (21) is the fractional frequency pulling of the NMR frequency due to the Suhl–Nakamura interaction. We can gain insight into the nature of this interaction by rewriting the Hamiltonian (19) in terms of the nuclear spin operators using Eqs. (6) and (9),

which can be written as

where

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 $Ii\u2032\u2212$ at site *i’* and the absorption of this magnon by another spin flip $Ii+$ 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 distance through which the Suhl–Nakamura interaction is effective is determined by the range of the function *B _{ii’}*. This function can be cast in a convenient form that shows its spatial dependence. For this, we use in Eq. (23b) $\omega k=\gamma e(H0+Dk2)$ and replace the sum in

*k*with an integral in the usual manner so that

where $R\u2192=r\u2192i\u2032\u2212r\u2192i$, $\alpha =(H0/D)1/2$, and Ω is the volume of the unit cell, which is *a*^{3} for a cubic crystal. The parameter *α* can also be written as

where $HE=2zJS/\gamma e\u210f$ is the effective exchange field on the electron spin and *z* is the number of nearest neighbors. By extending the upper limit of the integral in Eq. (24) to infinity, the integration can be performed analytically, giving

where

The result (26) has the form of a Yukawa potential, showing that the interaction between nuclear spins decays exponentially with distance, having range $a(HE/zH0)1/2$, 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.

### B. Diagonalization of the nuclear–electronic Hamiltonian

Another approach to obtain the frequencies of the nuclear and electronic spin excitations in the presence of the hyperfine interaction that gives further insight into the nature of the collective modes consists in diagonalizing the boson Hamiltonian (11). This is done with the same formalism used by Holstein and Primakoff to obtain the spin wave modes in the presence of the dipolar interaction.^{45,46} For this, we rewrite Eq. (11) as

where the new parameters are

The Hamiltonian (27) can be diagonalized by means of canonical transformations from the collective boson operators $ak\u2020,ak,bk\u2020,bk$ into creation and annihilation operators for the normal mode excitations $\alpha k\u2020,\alpha k,\beta k\u2020,\beta k$, given by

Substituting these expressions in Eq. (27) and imposing that the Hamiltonian be cast in the diagonal form

where $\omega \alpha k$ and $\omega \beta k$ are the frequencies of the two magnon modes, one can find the frequencies and the transformation coefficients.^{46} The frequencies are given by

and the coefficients are

In usual cases, *ω _{k}* is in the microwave range,

*ω*is one order of magnitude smaller, and $C/\u210f$ is even smaller due to the factor $\u27e8Iz\u27e91/2$. Thus, we can consider $C/(Ak+B)<<1$ so that Eqs. (31) and (32) can be simplified. With the binomial approximations of Eqs. (31) and (32) and the parameters defined in (28), we obtain for the transformation coefficients

_{N}and for the coupled mode frequencies

Equations (33) and (34) show that the mode *α _{k}* (≈

*a*) involves essentially the excitation of electronic spins, with a frequency that is slightly different from the pure electronic magnon frequency

_{k}*ω*. On the other hand, the mode

_{k}*β*(≈

_{k}*b*) corresponds to collective excitations of the nuclear spins with a frequency that is lower than the usual NMR frequency

_{k}*ω*. Note that the two frequencies have opposite signs because the electronic and nuclear spins precess with opposite senses. The variation with the wave number

_{N}*k*of the absolute values of the two frequencies is shown in Fig. 2(a). The hyperfine field

*H*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.

_{N}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 Nd_{2}CuO_{4}, a parent compound of the electron-doped superconducting compound Nd_{2−x}Ce* _{x}*CuO

_{4}. Hyperfine induced nuclear spin ordering had been reported

^{48}earlier in Nd

_{2}CuO

_{4}at temperatures below 200 mK associated with the isotopes

^{143}Nd and

^{145}Nd 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 Nd

_{2}CuO

_{4}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).

## III. NUCLEAR MAGNONS IN ANTIFERROMAGNETS

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 MnF_{2}, cubic crystalline anisotropy as in KMnF_{3} and RbMnF_{3}, biaxial anisotropy as in CsMnF_{3}, and Dzyaloshinskii–Moriya interaction as in MnCO_{3}. 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 Mn^{2+} ions, because they have the ^{55}Mn magnetic isotope with 100% abundance with electronic ground state configuration 3d^{5}(^{6}S_{5/2}), which has no orbital angular momentum, so that it has very small single-ion anisotropy. The origin of the magnetic anisotropy of MnF_{2} relies mainly on the dipolar interaction,^{49} and the relatively small anisotropy is an important requirement for having low electronic magnon frequencies. FeF_{2} that is another well-studied AF material with the rutile structure has no trace of nuclear magnons because the magnetic ^{57}Fe 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 CoCO_{3}, where the isotope ^{59}Co is 100% abundant, and FeBO_{3}, 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 MnF_{2}, has crystalline uniaxial anisotropy, the perovskite AFs, such as KMnF_{3} and RbMnF_{3}, 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.

### A. Easy-axis antiferromagnets: AF phase

We consider that the electronic spins $S\u2192i,j$ of two sublattices *i*,*j* point in the direction $\xb1z$ of the symmetry axis, as in Fig. 3(a), and that in each sublattice the nuclear spins $I\u2192i,j$ point in the opposite direction, as in Fig. 1(b). The magnetic properties are described very well by a Hamiltonian consisting of contributions from Zeeman, exchange, uniaxial magnetic anisotropy, and hyperfine energies in the form

where we have considered only the inter-sublattice exchange *J* and uniaxial anisotropy with parameter *D*. As in Sec. II, the components of the local spin operators are expressed in terms of creation and annihilation operators of spin deviations. For the up-electronic spin sublattice, the raising spin operator is related to the operator $ai$ that destroys a spin deviation, while for the down-sublattice it is related to the operator $bj\u2020$that creates a spin deviation^{46,50}

Similarly, for the *i* and *j* sublattices, we have for the nuclear spin operators

The next step consists of introducing the transformation from the localized field operators to collective boson operators, as in Eqs. (9) and (10), so that the Hamiltonian becomes

where $HE=2zJS/\gamma e\u210f$ is the effective exchange field on the electron spin, $HA=2SD/\gamma e\u210f$ is the anisotropy field, $\gamma k$ is a structure factor defined by $\gamma k=(1/z)\u2211\delta exp(ik\u2192.\delta \u2192)$, $\delta \u2192$ are the vectors connecting the *z* nearest neighbors in opposite sublattices, and $\omega N\u2191,\u2193=\xb1\gamma NH0+AS$ are the two unpulled NMR frequencies. The next step consists of performing canonical transformations from the four collective boson operators into four magnon creation and annihilation operators. This can be done with the method developed by White, Sparks, and Ortenburger that generalizes the Bogoliubov transformation for diagonalizing quadratic Hamiltonians.^{50,51} The calculation of the transformation coefficients and normal mode frequencies is quite lengthy and can be found in the literature.^{14,15,18} For the purposes of this Tutorial, we consider that the hyperfine field $A\u27e8Iz\u27e9$ is negligible compared to all other fields acting on the electron spin. Thus, the frequencies of the two electronic magnon modes are approximately the same as with no coupling with the nuclear modes and are given by^{46,50}

where

Using the body centered tetragonal structure of MnF_{2} the vectors connecting nearest neighbors $\delta \u2192=\xb1x^(a/2)\xb1y^(a/2)\xb1z^(c/2)$, the geometric structure factor is

The two nuclear magnon modes have frequencies given approximately by^{6,23}

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 MnF_{2}. Figure 4(a) shows variation of the *k* = 0 down-mode electronic frequency, or antiferromagnetic resonance frequency (AFMR), measured in MnF_{2} 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 $Hc=[2HEHA+HA2]1/2$, which is approximately 92.94 kOe, a value consistent with the effective fields $HE=526$ kOe and $HA=8.2$ kOe in MnF_{2}.^{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 *H*_{0} = 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 *H*_{0} = 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 *H*_{0} = 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 *H*_{0} = 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.

### B. Antiferromagnets with cubic anisotropy: Spin-flop phase

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 KMnF_{3} and RbMnF_{3}. 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 KNiF_{3} is also a well-studied AF material,^{56} but has no sign of nuclear magnons because the magnetic ^{61}Ni 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.

In this section, we shall direct attention mostly to RbMnF_{3}, because this is the perovskite AF that shows the most pronounced effects of the collective nuclear spin excitations. The equilibrium spin configurations for various directions of the applied magnetic field and the frequencies of the coupled electronic–nuclear excitations have been calculated in detail by Ince.^{18} Here, we shall only make some considerations about the equilibrium spin configurations and derive the spin wave frequencies in the spin-flop configuration. The Hamiltonian for the cubic anisotropy interaction is

where the coordinate system *x,y,z* has the axes along the $\u27e8100\u27e9$ direction of the crystal unit cell, as in Fig. 3. The anisotropy parameter *K* is positive in RbMnF_{3} so that the anisotropy energy is minimum for the spins along $\u27e8111\u27e9$ directions. Thus, if there is no external magnetic field, the spins of opposite sublattices are antiparallel to each other in the AF phase, pointing along a $\u27e8111\u27e9$, that characterizes the direction of the Néel vector $n^=(S\u2192i\u2212S\u2192j)/2S$. If a field *H*_{0} is applied along the [001] direction, the electronic spins in the two sublattices tilt slightly away from the Néel vector and toward the field. As the field increases, the Néel vector rotates and reaches the direction perpendicular to the field, in the spin-flop configuration shown in Fig. 5, for $H0=Hc=(3HEHA/2)1/2$, where $HA=8SK/3\gamma e\u210f$ is the anisotropy field.

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 $S\u2192i,S\u2192j$ of the two sublattices have directions at an angle *θ* with the field and in the (110) plane, while the nuclear spins $I\u2192i,I\u2192j$ 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 $x^=y^\xd7z^$.

Now we express the electronic spin components in the coordinate systems for the two sublattices, shown in Fig. 5, in terms of spin deviation operators, as follows:

Similarly, for the *i* and *j* sublattices, we have for the nuclear spin operators

As in Sec. III A, the next step consists of introducing the transformation from the localized field operators to collective boson operators. Since we know that the effect of the hyperfine interaction on the electronic modes is very small, we write the Hamiltonian in terms of the collective boson operators as a sum of three parts, namely,

where the electronic part is

and the parameters are

where, for small anisotropy, the angle of the spins with the field is approximately $\theta =cos\u22121\u2061(H0/2HE)$ and $HA=8SK/3\gamma e\u210f$. The nuclear Hamiltonian is

where we have neglected the effect of the applied field because of the spins are nearly perpendicular to the field. Finally, the term Hamiltonian that represents the coupling between the electronic and nuclear spins is

As in the case of the uniaxial anisotropy AF treated in Sec. III A, the diagonalization of the full Hamiltonian (45)–(49) is cumbersome and can be found in the literature.^{14,15,18} Thus, again, we consider that the hyperfine field is very small compared to all other fields acting on the electron spin and obtain the frequencies of the two electronic magnon modes assuming no coupling with the nuclear modes. In the SF phase, for $ka\u226a1$, they are given by^{15,18}

Note that the frequency *ω*_{2k} does not depend on the magnetic field. The mode with frequency *ω*_{1k}, which corresponds to the down-going mode in the AF phase, has minimum frequency at the field $Hc=(3HEHA/2)1/2$, at which there is a transition from the AF to the SF phase. The frequencies of the two “nuclear-modes” are given by^{15,18}

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 RbMnF_{3} 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*= 816 kOe, and

_{E}*H*= 4.59 Oe.

_{A}^{18}These values give for the spin-flop field $Hc=(3HEHA/2)1/2=2.37kOe$, 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).

## IV. SPINTRONICS WITH NUCLEAR MAGNONS

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 MnCO_{3}, in which nuclear spin waves were studied some time ago.^{9,10} In MnCO_{3}, a ^{55}Mn 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

*H*

_{0}= 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 MnCO

_{3}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 MnCO_{3} 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 ^{4}He 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 MnCO_{3} 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 process^{71} due to the hyperfine coupling between nuclear spins in the MnCO_{3} and conduction-electron spins in the attached Pt is taken into consideration.

## V. SUMMARY AND PERSPECTIVES

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 MnF_{2}, and the other with cubic anisotropy, such as RbMnF_{3} and KMnF_{3}, 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 MnCO_{3}, 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 MnF_{2} 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 KMnF_{3} and RbMnF_{3} 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}

## ACKNOWLEDGMENTS

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).

## AUTHOR DECLARATIONS

### Conflict of Interest

The author has no conflicts to disclose.

### Author Contributions

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

## DATA AVAILABILITY

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

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