Neutron Diffraction and Reflectometry: A Review on Unique Role in Functional Magnetic Materials

Abstract

Magnetic materials either in bulk or thin films are ubiquitous in our daily life. Technology based on magnetic materials range from chunk of bulk ferromagnet to thin film spintronics. In solid state bulk materials, information about its magnetic structure together with crystal structure is absolutely necessary to manipulate them in applications. Neutron diffraction is an important tool to determine atomic magnetic moments and its directions at the lattice site in the magnetic unit cell. It also investigates the type of magnetic ordering in conventional as well as new exotic materials. Magnetic thin films are engineered materials in which nanometer or sub-nanometer thickness scale films are grown. At such thicknesses nanoscale magnetic properties are fundamentally different than its bulk counterpart. Neutron reflectometry is a unique tool to investigate nano-magnetism in thin films. Moreover, in multilayer thin films generally used for spintronics, polarized neutron reflectometry is indispensable characterizing tool which investigates the magnetic properties in different layers and at the interfaces. In this chapter, we will introduce how neutron diffraction and reflectometry techniques play unique role in the investigation of magnetic structure and magnetic properties of functional bulk and nano-scale thin films.

1. Introduction

Technological development owing to magnetic materials are present everywhere in our daily life, just to name few, power generation and transmission, Hard Disk Drive (HDD), Magnetic Random Access Memory (MRAM), Sensors, Magnetic Resonance Imaging (MRI) and drug delivery [1, 2, 3, 4]. Magnetic materials have some intrinsic (magnetic moment, magnetic order, exchange interaction, and magnetic anisotropy) and extrinsic (magnetization, coercivity, and domain wall) properties which are exploited for its use in the technology [5]. There are only few traditional magnetic materials in the periodic table. However, advancement of sophisticated material synthesis techniques and characterization tools have enabled the preparation of new kind of magnetic materials with appealing functionality. For example, Colossal Magnetoresistance (CMR) in bulk materials, Giant Magnetoresistsnce (GMR), coexistence of superconductivity and magnetism, spin dependent quantum confinement at the interface in thin films, and manipulation of electronic and nuclear spin for quantum computing utilizing quantum entanglement [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17].

Microscopic understanding of the synthetic magnetic materials are necessary for its better utilization in technology and commercial products. Investigation of magnetic properties in such materials require an understanding of the interplay between structure and magnetism at atomic scale [18]. Neutron scattering is a unique non destructive probe which comprehensively characterize both structure (using diffraction) and dynamics (using spectroscopy) of magnetic materials. Such investigations are possible because neutron is deeply penetrating neutral but magnetic particle [19, 20]. In this chapter, we will introduce basics of neutron scattering followed by unpolarized and polarized neutron diffraction and reflectometry techniques.

2. Neutron scattering basics

Fundamental properties of neutrons are given in Table 1. Energy of neutrons for materials research using neutron scattering techniques lies in the range of 0.1 meV to 100 meV. These energies are achieved by moderating high energy neutrons produced at reactor or spallation sources. Corresponding wavelength of neutrons range from $\lambda$ $\approx$ 28 Å to 0.9 Å. Thermal (0.9 Å $<$ $\lambda$ $\le$ 4 Å) and cold (4 Å $<$ $\lambda$ $<$ 28 Å) neutrons cover most of the length scales to be investigated using diffraction instruments. Thermal energies of neutrons are comparable to elementary excitation (e. g. phonon and spin dynamics) in the materials and can be investigated using spectroscopy instruments. The fact that neutron has spin ($\frac{1}{2}$) and magnetic moment (−1.913 ${\mu }_N$, ${\mu }_N$ is nuclear magneton), it becomes a unique probe to investigate magnetism at atomic scale. Neutrons interact with materials through short range nuclear interaction. The strength of the nuclear interaction also depends on the relative orientation of neutron and nuclear spins [19, 21].

Property	value
Mass, mn	9.27 × 10^−27 kg
Charge	0
Spin, S	$\frac{1}{2}$
Magnetic Moment, ${\mu }_n$	−1.913 ${\mu }_N$
Life Time, $\tau$	886 ± 1
Wavelength, $\lambda$	0.9–4 Å for thermal neutrons
	4–28 Å for cold neutrons

Table 1.

Fundamental properties of neutrons.

Properties of the materials under investigation are obtained by measuring and analyzing the scattered neutrons from the sample. In scattering experiments, scattered neutrons as a function of scattering vector are measured. Neutron scattering cross section for monoatomic system (chosen for simplicity) can be expressed in the following form [19, 22, 23]

$$\frac{{\partial }^2\mathrm{\Sigma }}{\mathrm{\partial \Omega \partial }\omega }=\frac{k_f}{k_i}\eta \left[\left(⟨b^2⟩-{⟨b⟩}^2\right)S_{\mathit{inc}}(\mathbf{Q},\omega )+{⟨b⟩}^2{S}_{\mathit{coh}}(\mathbf{Q},\omega )\right]$$E1

where $\eta$ is number of atoms, $K_i$ and $k_f$ are magnitudes of initial and final wave vectors, respectively, $b$ is neutron scattering length of the atom which changes in an irregular way with atomic number. Isotopes of the same element has different $b$. Moreover, $b$ also depends on the nucleus spin state ($I$) as a result has two different values $b^{+}$ (for $I+\frac{1}{2}$) and $b^{-}$ (for $I-\frac{1}{2}$) corresponding to two different spin states of neutrons. $S_{\mathit{inc}}\left(\mathbf{Q},\omega \right)$ and $S_{\mathit{coh}}\left(\mathbf{Q},\omega \right)$ are the incoherent and coherent scattering functions.

2.1 Incoherent scattering

Incoherent scattering represent the space–time correlation between an atom at time $t=0$ and that of the same atom at time $t=t$, sum over the sample volume. It arises due to the disorder of the scattering length of chemically identical atoms and non zero nuclear spin. Incoherent scattering function $S_{\mathit{inc}}\left(\mathbf{Q},\omega \right)$ is a Fourier transform in space and time of the self correlation function $G_s\left(\mathbf{r},t\right)$ given in the following form:

$$\begin{array}{l}{S}_{\mathit{inc}}\left(\mathbf{Q},\omega \right)=\frac{1}{2\pi \mathrm{\hslash }}\int G_s\left(\mathbf{r},t\right)e^{i\left(\mathbf{Q}.\mathbf{r}-\mathit{\omega t}\right)}{d}^{3}r\mathit{dt}\end{array}$$E2

Amorphous/liquid structure of materials is obtained using incoherent elastic neutron scattering and atomic diffusion is measured using incoherent inelastic neutron scattering.

2.2 Coherent scattering

Coherent scattering represent the space–time correlation between an atom at time $t=0$ and that of the another atom at time $t=t$, sum over the sample volume. It arises from the average of the scattering length of the atoms giving rise to interference. Coherent scattering function $S_{\mathit{coh}}\left(\mathbf{Q},\omega \right)$ is a Fourier transform in space and time of the pair correlation function $G\left(\mathbf{r},t\right)$ given in the following form:

$$\begin{array}{l}{S}_{\mathit{coh}}\left(\mathbf{Q},\omega \right)=\frac{1}{2\pi \mathrm{\hslash }}\int G\left(\mathbf{r},t\right)e^{i\left(\mathbf{Q}.\mathbf{r}-\mathit{\omega t}\right)}{d}^{3}r\mathit{dt}\end{array}$$E3

Crystal and magnetic structure through the Bragg scattering measurements are obtained using coherent elastic neutron scattering and phonons (lattice vibrations) and magnons (spin wave) are measured using coherent inelastic scattering. Figure 1 shows the schematic diagram of the popular experimental techniques based on neutron scattering. This chapter is focused on neutron diffraction and reflectometry techniques to investigate magnetic structure and magnetic properties of the materials. In both of these methods intensity of the scattered neutron as a function of detector angle is measured via coherent elastic neutron scattering i. e. Intensity ($I$) $\propto$ $\frac{\mathrm{\partial \Sigma }}{\mathrm{\partial \Omega }}$.

Figure 1.

Schematic diagram of neutron scattering experiments, their contribution and information that can be obtained.

3. Neutron diffraction

Neutron diffraction experiments are performed to obtain crystallographic structure and associated magnetic structure of atomic arrangement in the unit cell [24, 25]. Three dimensional periodic repetition of unit cells forms crystal. During the experiments intensity of neutrons is measured at an angle 2$\theta$ with respect to incident beam falling on the crystal. Atoms, in crystals, sit on crystallographic planes. The orientation of these planes are represented by Miller indices ($\mathit{hkl}$). One can imagine that the set of $\mathit{hkl}$ planes act like mirrors as shown in the Figure 2.

Figure 2.

An illustration of Braggs’ law for the reflection of neutron beam from ($h,k,l$) planes.

Neutron beam reflected from adjacent planes interfere to produce diffraction pattern. When the path difference between the interfering beam is an integer multiple of wavelength, constructive interference is formed. The condition for constructive interference is given by Braggs’ law:

$$\begin{array}{l}2d_{\mathit{hkl}}\mathit{\text{sinθ}}=\mathit{n\lambda }\end{array}$$E4

where $d_{\mathit{hkl}}$ is interplaner spacing (also known as lattice spacing) and $\lambda$ is wavelength of neutrons. The integer $n$ is known as order of diffraction. The $d_{\mathit{hkl}}$ is a geometrical function of the unit cell. Its value depends on the size (unit cell lengths $a$, $b$ and $c$) and shape (unit cell angles $\alpha$, $\beta$ and $\gamma$). These six parameters are known as lattice parameters. The detection of $d_{\mathit{hkl},\mathit{min}}$ is limited by wavelength of the beam. Hence, wavelength of the beam should be in the same order of magnitude of unit cell lengths.

Diffraction geometry is described through the concept of reciprocal space and Ewald’s sphere [26]. Each lattice point $\mathit{hkl}$ in reciprocal space refers to a set of ($\mathit{hkl}$) planes in real space. Ewald’s sphere provides geometrical condition for Braggs’ diffraction in reciprocal space i.e. $\overrightarrow{K_f}=\overrightarrow{K_i}+\overrightarrow{Q}$ where $\mid \overrightarrow{K_f}\mid =\mid \overrightarrow{K_i}\mid =\frac{2\pi }{\lambda }$ and $\mid \overrightarrow{Q}\mid =\frac{4\mathit{\text{πsinθ}}}{\lambda }$. In diffraction experiments reflection peak from ($\mathit{hkl}$) planes in real space is observed when the corresponding reciprocal lattice point $\mathit{hkl}$ lies on the surface of Ewald’s sphere. This visualization for reflection peaks fit for single crystal diffraction. However, for polycrystalline or powder samples it is assumed that there are random orientation of crystallites. Hence, randomly oriented crystallites produce reciprocal lattice point in all possible orientation. As a result single reflection corresponding to ($\mathit{hkl}$) planes of a single crystal as shown in the Figure 3a turns into continuous reflection making a cone at 2$\theta$ for polycrystalline or powder samples as shown in the Figure 3b.

Figure 3.

Illustration of Ewald construction in reciprocal space showing diffraction condition for (a) single crystal (b) polycrystalline (powder) samples.

Phase, phase transformation and unit cell parameters can be obtained by performing neutron diffraction experiments. Quantitative analysis is done by determining $d_{\mathit{hkl}}$ and corresponding peak intensities. Diffraction data is usually analyzed with the help of Rietveld refinement [27]. Intensities of the Braggs’ peak due to the neutron scattering from atoms in the unit cell and its arrangement is given by [28, 29, 30]

$$\begin{array}{c}I\propto \frac{\mathrm{\partial \Sigma }}{\mathrm{\partial \Omega }}\propto {\left|F_{\mathit{hkl}}\right|}^2\\ F_{\mathit{hkl}}=\sum _{j=1}^n{b}_j{T}_j{e}^{2\mathit{\pi i}\left({\mathit{hx}}_j+{\mathit{ky}}_j+{\mathit{lz}}_j\right)}\end{array}$$E5

$F_{\mathit{hkl}}$ is structure factor for Braggs’ reflection from ($\mathit{hkl}$) planes, $n$ is number of atoms in the unit cell, $b_j$ is scattering length of neutron for atom $j$ and $T_j$ is Debye Waller factor for atom $j$. Scattering length $b$ for adjacent elements in the periodic table can deffer significantly [31]. Neutron-nucleus interaction behaves as point like scattering thus $b$ is independent of Bragg angle. Nevertheless, magnetic scattering length ($b_m$), due to the fact that unlike nuclear form factor magnetic form factor is not constant, is strongly $\theta$ dependent which reduces with increasing Bragg angle [32]. Debye Waller factor $T_j$ reduces the Bragg peak intensity due to the thermal motion of atoms. It should be noted that effects such as absorption, extinction, polarization and the Lorentz factor also attenuate the intensity of the peak [29]. When the magnetic moment associated with atoms in the the crystal have magnetic ordering then coherent magnetic scattering occurs. The structure factor for magnetic scattering is given by [28, 29, 30]

$$\begin{array}{r}{\mathbf{F}}_{\mathit{hkl}}^{\mathit{mag}}=\sum _{j=1}^n{b}_{m,j}{\mathbf{q}}_j{T}_j{e}^{2\mathit{\pi i}\left({\mathit{hx}}_j+{\mathit{ky}}_j+{\mathit{lz}}_j\right)}\\ {\mathbf{q}}_j={\widehat{\mu }}_j-\left(\widehat{Q}.{\widehat{\mu }}_j\right)\widehat{Q}\text{with}\mid {\widehat{\mathbf{q}}}_j\mid =\mathit{\text{sinα}}\end{array}$$E6

where $b_{m,j}$ is magnetic scattering length of neutron for atom $j$, ${\mathbf{q}}_j$ is magnetic interaction vector, ${\widehat{\mu }}_j$ is unit vector along the magnetic moment of atom $j$, $\alpha$ is angle between scattering vector $Q$ and ${\widehat{\mu }}_j$. For Bragg reflection, scattering vector $Q$ is perpendicular to ($\mathit{hkl}$) planes. The summation in Eq. (6) run over the atoms having magnetic moment. Magnetic scattering depend on the neutron polarization $P$ and its direction $\widehat{P}$ with respect to $\mu$. In a simplest case of coherent scattering where unpolarized neutron beam scatters from magnetic sample, polarization dependent magnetic scattering will average out to be zero. The differential scattering cross section will depend on nuclear, nuclear-magnetic, magnetic interaction term

$$\begin{array}{l}\frac{d\mathrm{\Sigma }}{d\mathrm{\Omega }}=F_{\mathit{nuc}}^2\left(\mathbf{Q}\right)+2\left(\widehat{P}.\widehat{\mu }\right)\mid F_{\mathit{nuc}}\left(\mathbf{Q}\right)\Vert F_{\mathit{mag}}\left(\mathbf{Q}\right)\mid +F_{\mathit{mag}}^2\left(\mathbf{Q}\right)\end{array}$$E7

3.1 Unpolarized neutron diffraction

In unpolarized neutron diffraction measurements incident neutron beam polarization $P=0$. Hence, the nuclear-magnetic interaction term in Eq. (7) average out to be zero. The first neutron diffraction experiment was performed on MnO powder sample [33, 34]. MnO is a paramagnetic material but it turns into antiferromagnet (AFM) below 120 K. Magnetic moments associated with Mn atoms order itself as shown in the Figure 4a. MnO has NaCl type structure and does not change its crystallographic phase when temperature is cooled below its magnetic transition temperature. Shull et al. performed neutron diffraction experiments on MnO powder sample (Figure 4b) at room temperature (RT) and at 80 K (below its magnetic transition temperature) [34]. One can see additional Bragg peaks together with nuclear (chemical) Bragg peaks in neutron diffraction data obtained at 80 K. These peak positions are forbidden according to nuclear structure factor analysis. It was analyzed that the additional peaks coming from the magnetic unit cell which is twice in size as compared to chemical unit cell. Magnetic moment direction of Mn atoms sitting along the cubic axis are oriented antiferromagnetically.

Figure 4.

(a) Schematic diagram of magnetic structure of MnO. Only Mn atoms are shown in the unit cell. Size of the magnetic unit cell is twice the size of the chemical unit cell. (b) Neutron diffraction pattern of MnO measured below and above the magnetic transition temperature. Figure (b) is adapted from Shull et al. [34].

Neutron diffraction technique has been very useful method in the study of the magnetic oxides such as perovskites and transition metal oxides. ABO₃ (A = La, Sr., B=Fe, Co, Ni, Cu, Mn, Ti) type perovskites are interesting material because of its use in solid oxide fuel cell [35, 36, 37]. Properties of solid state fuel cell largely depend on the oxygen deficiency. Neutron powder diffraction is a very effective tool to determine the oxygen vacancy concentration because the sensitivity of neutron to oxygen is comparable to other atoms [38, 39]. Neutron diffraction has been also very useful in studying multiferroic materials. For example YMn₂O₅ magnetic and ferroelectric properties are linked to structural transitions at low temperatures [40, 41]. Moreover, commensurate to incommensurate phase transitions and magnetic structure to ferroelectric order were further studied in these materials using neutron diffraction [42, 43].

3.2 Polarized neutron diffraction

In polarized neutron diffraction (PND) either the incident beam is polarized using polarizer, flipper and guide field or the diffracted beam polarization is analyzed using analyzer or both are done for full magnetic analysis of the material. PND can unambiguously separate magnetic and nuclear structure factor. Magnetic structure factor (Eq. 6) is a vector quantity. Hence, PND allows to find out the different directional component of the magnetic structure factor.

3.2.1 Flipping ratio measurement

In this method usually incident neutron beam is polarized and no polarization analysis is performed on diffracted beam. The ratio of the differential scattering cross section for spin up (’+’, polarization of neutron parallel to the guide field) and spin down neutrons (’-’, polarization of neutron antiparallel to the guide field) is known as flipping ratio ($\mathit{FR}$). By measuring $\mathit{FR}$, information of the $F_{\mathit{mag}}$ for the magnetic material can be obtained. Here we introduce the $\mathit{FR}$ measurement for a collinear magnetic material. Differential cross sections for spin up and spin down neutrons and $\mathit{FR}$ are given by [29]

$$\begin{array}{l}{I}^{+}\sim {\left(\frac{d\mathrm{\Sigma }}{d\mathrm{\Omega }}\right)}^{+}={\left[F_{\mathit{nuc}}\left(\mathbf{Q}\right)+\mathit{\text{sinα}}{F}_{\mathit{mag}}\left(\mathbf{Q}\right)\right]}^2\\ I^{-}\sim {\left(\frac{d\mathrm{\Sigma }}{d\mathrm{\Omega }}\right)}^{-}={\left[F_{\mathit{nuc}}\left(\mathbf{Q}\right)-\mathit{\text{sinα}}{F}_{\mathit{mag}}\left(\mathbf{Q}\right)\right]}^2\\ \mathit{FR}=\frac{{\left[1+\mathit{\text{sinα}}\frac{F_{\mathit{mag}}\left(\mathbf{Q}\right)}{F_{\mathit{nuc}}\left(\mathbf{Q}\right)}\right]}^2}{{\left[1-\mathit{\text{sinα}}\frac{F_{\mathit{mag}}\left(\mathbf{Q}\right)}{F_{\mathit{nuc}}\left(\mathbf{Q}\right)}\right]}^2}={\left[\frac{1+\mathit{\text{sinα}}\mathrm{\Upsilon }}{1-\mathit{\text{sinα}}\mathrm{\Upsilon }}\right]}^2\end{array}$$E8

If the magnetization is perpendicular to scattering vector ($\alpha =\frac{\pi }{2}$) for a known crystal structure, one can easily calculate $F_{\mathit{mag}}\left(\mathbf{Q}\right)$ (= ${\mathrm{\Upsilon }}^{\mathit{exp}}{F}_{\mathit{nuc}}\left(\mathbf{Q}\right)$). Spin density distribution in the sample can also be obtained applying Fourier transform to $F_{\mathit{mag}}\left(\mathbf{Q}\right)$ [44, 45, 46]. Gukasov et al. performed polarized neutron flipping ratio measurements on terbium stannate, Tb₂Sn₂O₇, pyrochlore compound [47]. They found that on the application of magnetic field on Tb₂Sn₂O₇ leads to dramatic changes in the diffraction pattern as shown in the Figure 5. Diffraction peaks at different temperature and field are strongly polarization dependent. Flipping ratio measurement allowed Gukasov et al. to obtain the information about anisotropy of the local magnetic susceptibility at different magnetic sites [47].

Figure 5.

Polarized neutron diffraction pattern from ordered dipolar spin ice Tb₂Sn₂O₇ at (a) 2 K and 1 T (b) 100 K and 5 T. Adapted from Gukasov et al. [47].

3.2.2 Uniaxial (longitudinal) polarization analysis

In this method polarization of scattered neutron beam is analyzed with respect to incoming beam polarization. During the experiments four differential scattering cross sections are measured i.e. two non-spin flip ${\left(\frac{d\mathrm{\Sigma }}{d\mathrm{\Omega }}\right)}^{++}$, ${\left(\frac{d\mathrm{\Sigma }}{d\mathrm{\Omega }}\right)}^{--}$ and two spin flip ${\left(\frac{d\mathrm{\Sigma }}{d\mathrm{\Omega }}\right)}^{+-}$, ${\left(\frac{d\mathrm{\Sigma }}{d\mathrm{\Omega }}\right)}^{-+}$ keeping $\widehat{P}\parallel \widehat{Q}$ or $\widehat{P}\perp \widehat{Q}$. First ‘+ (−)’ sign indicates the polarization of incident neutrons and second ‘+ (−)’ sign shows the polarization of scattered neutrons. It should be noted that nuclear scattering is always non-spin flip. However, magnetic scattering contributes in both non-spin flip and spin flip intensities. For $\widehat{P}\perp \widehat{Q}$, non-spin flip scattering occurs from the sample magnetization parallel to incident neutron beam polarization and spin flip occurs from in-plane perpendicular component of magnetization. For $\widehat{P}\parallel \widehat{Q}$, only the spin flip magnetic scattering occurs. Polarization analysis allows to separate nuclear and magnetic scattering, magnons and phonons scattering, and coherent and spin incoherent scattering by measuring four differential cross sections with $\widehat{P}\parallel \widehat{Q}$ or $\widehat{P}\perp \widehat{Q}$ [48, 49, 50, 51, 52, 53].

4. Neutron reflectometry

Neutron diffraction, discussed in the previous section, is generally known as bulk probe for the investigation of magnetic structures. However, its interface sensitivity makes it a unique tool to investigate magnetic properties of under surface, buried layers and across the interfaces in thin films and multilayers. In reflectivity, unlike diffraction, incident angle is low (usually less than 5°). In thin films and multilayers, the thickness of one dimension along out of plane direction is in nanometer or subnanometer scale. Interface sensitivity comes from the fact that their wavelength projection in out of plane direction matches the layer thickness and that the neutron wave field get distorted near surface and interfaces [54]. Neutron reflectivity (NR) measures the nuclear density profile, roughness and roughness correlation. For magnetization profile, neutron beam is polarized and analyzed before and after scattering, respectively. Polarized neutron reflectivity (PNR) allows the investigation of collinear and noncollinear interlayer exchange coupling [55].

NR follows optical principle and the neutron refractive index ($n$) of a sample is given by $n=1-\delta -\mathit{i\beta }$ where $\delta$ and $\beta$ account for dispersion and absorption of the medium, respectively [56]. These parameters are defined as

$$\begin{array}{l}\delta =\frac{{\lambda }^2}{2\pi }\left({\mathit{Nb}}_n\pm {\mathit{Nb}}_m\right)=\frac{{\lambda }^2}{2\pi }\left({\rho }_n\pm {\rho }_m\right)\text{and}\beta =\frac{\lambda }{4\pi }{\sigma }_{\mathit{abs}}N\end{array}$$E9

where $\lambda$ is the wavelength of the neutrons, $N$ is the number density of atoms (molecules) in the sample, $b_n$ ($b_m$) is nuclear (magnetic) scattering length for neutrons, ${\rho }_n$ (${\rho }_m$) is the nuclear (magnetic) scattering length density (SLD), and ${\sigma }_{\mathit{abs}}$ is the absorption cross section of neutrons for the sample material. The ${}^{\text{'}}{\pm }^{\prime }$ sign in the scattering term shows that the total SLD of neutrons for a magnetic material is enhanced or reduced depending on the orientation of the neutron spin with respect to the sample’s magnetization. For a nonmagnetic or demagnetized magnetic material, ${\rho }_m$ is zero. ${\sigma }_{\mathit{abs}}$ is very small for most materials, with a few exceptions (e.g. $B$, $\mathit{Cd}$, $\mathit{Gd}$), and the absorption component $\beta$ can be neglected in most of the cases.

Figure 6a and b show the general scattering geometry for reflectometry and geometry for the specular PNR. In specular reflectivity, where incidence angle is equal to reflected angle and scattering plane is perpendicular to the sample surface plane, neutron beam is incident on the sample and reflected neutrons are measured as a function of out of plane momentum transfer vector ($Q_z$).

Figure 6.

(a) General scattering geometry for reflectometry. Specular reflectivity is performed for out of plane depth profiling. Off-specular reflectivity and diffuse scattering are performed to study in-plane nanostructures such as magnetic and non-magnetic nanoparticles. (b) Geometry for specular polarized neutron reflectometry. Neutron spins are polarized in the guide field and polarization analysis after scattering gives the information about layer magnetization of each layer and interlayer coupling.

$$\begin{array}{l}{Q}_z=\frac{4\mathit{\text{πsinθ}}}{\lambda }\end{array}$$E10

Reflected intensity usually denoted as $R\left(Q_z\right)$ is unitary upto a value $Q_z$ = $Q_c$ known as total external reflection also known as critical edge. $R\left(Q_z\right)$ above $Q_c$ drops rapidly and usually measured upto 5–6 order of dynamical range. Figure 7 shows the simulated polarized neutron reflectivity pattern of 100 nm magnetic Ni thin film. The roughness of the film was kept zero during the simulation. Two critical edges appearing around $Q_z$ = 0.02 Å⁻¹ for neutron spin up and neutron spin down reflectivity can be obtained by

Figure 7.

Simulated polarized neutron reflectivity of 100 nm thick Ni film.

$$\begin{array}{l}{Q}_c^{\pm }\approx \sqrt{16\pi \left({\rho }_n\pm {\rho }_m\right)}\end{array}$$E11

Two critical edges for spin up and spin down neutrons can be clearly seen in Figure 7. Critical edge depends on the scattering length density which is a characteristic of chemical composition of the material. The oscillation in the reflectivity pattern is known as Kiessig oscillations. Slope of reflectivity depends on the roughness ($\sigma$) of the film. A rough estimate of layer thickness and roughness ($R^{\mathit{\text{rough}}}\left(Q_z\right)=R^{\mathit{\text{flat}}}\left(Q_z\right)e^{Q_z^2{\sigma }^2}$) can be obtained from the difference between consecutive Kiessig oscillations and the reflectivity slope, respectively [56]. Precise information of individual layer thickness (in case of multilayer), interface roughness and interdiffusion at the interface can be obtained by fitting the PNR pattern.

It is important to note that PNR method measures the magnetization of the film but it does not distinguish contribution of magnetization due to orbital and spin magnetic moments. Moreover, PNR is insensitive to the magnetization component of the sample parallel to the scattering vector. Practically, one can measure 4 different differential cross sections in PNR measurement i. e. $R^{++}$, $R^{--}$, $R^{+-}$ and $R^{-+}$. $R^{++}$ and $R^{--}$ are known as non-spin flip, whereas $R^{+-}$ and $R^{-+}$ are known as spin flip reflectivity. The first and second sign represents the polarization state of the neutrons before and after scattering from the sample, respectively. Non-spin reflectivity probes the in-plane magnetization component along the direction of the magnetic field, whereas spin flip reflectivity is sensitive to the magnetization component perpendicular to the magnetic field. Combining non-spin flip and spin flip reflectivities, it is possible to construct depth dependent magnetization and its direction in magnetic thin films and multilayers.

4.1 Depth profiling in thin films and multilayers

Magnetic properties of thin film of a magnetic material may be very different than when the material is in bulk form. Generally, air sensitive thin films are capped with noble metals to protect the surface of the film but capping may alter the behavior due to interface effect [57]. PNR is a unique method to investigate magnetization within the layer of a film and at the interfaces. Figure 8 shows PNR pattern of 30 Å Co thin film deposited on Pt(205 Å)/MgO(001). $R^{+}$ ($R^{++}+R^{+-}$) and $R^{-}$ ($R^{--}+R^{-+}$) reflectivities were measured under ultra high vacuum (UHV) condition (Figure 8a) and in ambient air (Figure 8b) [58]. Magnetic and nuclear SLDs were obtained by fitting the measured data as shown in Figure 8c and d for PNR measured in UHV and in air, respectively. It can be clearly seen that Co film exposed to air forms an oxide layer and average magnetic moment at the interface is reduced. Banu et al. deposited Co thin film on Si(111) substrate [59]. PNR measurements performed on the samples reveal that super dense nonmagnetic Co film are formed at the interface. Kreuzpaintner et al. performed in-situ PNR on Fe films during its growth [60]. They used PNR to investigate evolution of structural and magnetic properties of ultrathin Fe film. It was found that ultrathin Fe film forms face centered cubic ($\mathit{fcc}$) structure which otherwise cannot be stabilized in bulk at room temperature.

Figure 8.

PNR curves of Co film (a) measured under UHV condition (b) measured in ambient air (c) and (d) neutron SLD profiles obtained from fitting for the measurements shown in (a) and (b), respectively. Adapted from Syed Mohd et al. [58].

Discovery of GMR effect in magnetic multilayers due to interlayer exchange coupling opened up new era for technological development in the field of magnetoelectronics and spintronics. Professor Alber Fert of Université Paris-Sud, France and Professor Peter Grünberg of Forschungszentrum Jülich, Germany got the Nobel Prize in 2007 [1]. PNR turned out to be unique technique to investigate magnetization profile of multilayers showing GMR effect. For an example, layer-by-layer magnetization configuration of Fe/Cr multilayers were obtained by measuring non-spin flip and spin flip reflectivities [61]. Figure 9 shows the multilayer structure and magnetization vector in ferromagnetic Fe layers through the multilayer.

Figure 9.

(a) Schematic presentation of [Cr(9 $\mathring{\mathrm{A}}$/Fe(67 $\mathring{\mathrm{A}}$]₁₂ multilayer deposited on Cr(68 $\mathring{\mathrm{A}}$/Al₂O₃(110) (b) configuration of magnetization in Fe layer in the magnetic field $\mathbf{H}$ applied in plane along the easy axis; dashed lines mark the hard axis (c) shows the SLD profiles for magnetization component parallel to hard axis ($\parallel$) and easy axis ($\perp$). Part of the figure (b) and (c) are adapted from Lauter-Pasyuk et al. [61].

4.2 Interface induced magnetic phenomenon at heterostucture interface

At the interface of hetrostuctures, there is an interplay between charge, spin, orbital and lattice degrees of freedom [13]. Interfaces therefore are important because its engineering and manipulation create new type of emergent state due to spin orbit coupling, broken symmetry, quantum confinement, strain and electronic reconstruction. In complex oxide interfaces it has been found that interplay at the interface may create magnetism in non magnetic layers [62, 63, 64, 65]. PNR technique has been very successful tool in resolving the issue of origin of magnetism in such samples. Conventional techniques like vibrating sample magnetometer (VSM) and superconducting quantum interference device (SQUID) magnetometer are sensitive to macroscopic magnetization of the whole sample which limits its role in heterostructures. Grutter et al. deposited LaNiO₃ (LNO)/CaMnO${}_3$ (CMO) superlattices where LNO is paramagnetic metal and CMO is antiferromagnet (AFM) insulator [65]. PNR results as shown in the Figure 10 reveal that ferromagnetism is induced at both the interfaces of CMO below 70 K which is also metal to insulator transition temperature for LNO. PNR results were key to propose the mechanism of double exchange interaction mediated by LNO e_g band.

Figure 10.

(a) Nuclear and magnetic SLD profile of CaMnO₃ (antiferromagnetic insulator)/LaNiO₃ (paramagnetic metal)superlattice. (b) Schematic presentation of induced ferromagnetic layer at both the interface of CaMnO₃ (width of the layers shown in the schematic are not to scale). Figure (a) is adapted from Grutter et al. [65].

Bulk LaCoO₃ (LCO) shows paramagnetic bahaviour. However, when grown as a thin film, ferromagnetism emerges at low temperatures [66, 67, 68, 69]. The exact mechanism behind the origin of ferromagnetism is not well established. Guo et al. performed PNR and quantitatively measured the magnetization profile in LCO films [70]. They found reduced magnetization at the interface which attribute to the symmetry mismatch at the interface. Here, PNR helped to provide unique insight for understanding the emergence of ferromagnetism in LCO thin films.

PNR is being successfully used in new emerging functional materials like chiral magnets [71] and topological insulators [72, 73] as well as conventional subject like superconductivity [74]. In chiral magnets it is believed that lack of inversion symmetry induces Dzyaloshinski-Moriya interaction leading to chiral spin structure. PNR technique has played a crucial role in understanding of noncollinear magnetic order in such materials. Topological materials are new kind of materials and there are efforts to make use of these materials in future electronics and quantum computing [75, 76, 77]. PNR has been used to see the proximity effect in topological insulator/ferromagnetic thin films to use it in magnetic functionality [78, 79, 80]. In recent years quantum computer based on superconductor/insulator/superconductor type Josephson junctions allowed Google to demonstrate 54-qubit system [15, 81]. PNR is the tool to investigate such type of junctions [82].

5. Conclusions

In this chapter fundamentals of neutron scattering, neutron scattering techniques and its applications are briefly introduced. The focus of the chapter was neutron diffraction and neutron reflectivity from magnetic materials. Neutron diffraction is an incomparable tool to enhance fundamental understanding of magnetic structure. The importance of both unpolarized and polarized neutron diffraction have been shown using examples. Polarized neutron diffraction probes magnetic structure associated with the atomic arrangement in the unit cell. It allows us to investigate magnetic ordering, its magnetic coupling and orientation of magnetic moments in complex and new functional magnetic materials. Neutron diffraction is the tool which directly probes spin waves and allows dynamics of crystalline magnetic material investigation.

Magnetic thin films and multilayers exhibit novel fundamental properties. But, such layered nanostructures have reduced scattering volume. Hence, it is difficult to probe magnetic properties using conventional methods. It has been introduced that neutron reflectivity plays a unique role in the investigation of two dimensional magnetic thin films. It is also explained how neutron reflectometry technique and measurement strategies probe the surface, interfaces and buried layer by measuring reflected neutrons incident at glancing angles on the sample. Polarized neutron reflectivity proved to be a very important tool in improving and investigating magnetic thin films for spintronics applications. Its unique capabilities helped in resolving many outstanding issues in the field of complex oxide heterostructures. PNR is being regularly used and has played a pivotal role in the exploration and optimization of quantum materials and heterostructures based on topological insulators.