Shedding light on non-Ising polar domain walls: Insight from second harmonic generation microscopy and polarimetry analysis

Topological structures¹ including domain walls (DW), vortices, Bloch points, Bloch lines, bubbles, or skyrmions have attracted great attention owing to their ability to be easily created and controlled in ordered materials. When nanoscale topological structures develop in ferroic materials such as ferromagnetic, ferroelectric, or ferroelastic systems, they may represent a great playground for novel functional device concepts.^2–4 Applications based on magnetic DWs and skyrmions have been incrementally developed by the magnetism community over the past 20 years,^5–7 whereas the interest in topological and chiral structures in multiferroics^2,8 and ferroelectric systems^9–12 is comparatively new, as highlighted in recent review articles.^13,14 Only recently, for example, ferroelectric skyrmionic structures have been predicted to exist,^15–17 and their first observation was reported in ferroelectric/dielectric heterostructures.¹⁸ These newly discovered aspects are expected to pave the way toward applications, similar to the developments achieved in magnetism, which further motivates an intensified research in this field.

In this context, fundamental researchers and technology developers have joined forces to explore DWs.¹⁹ An early thermodynamic study reported by Lajzerowicz and Niez²⁰ predicted the existence of a phase transition within ferroelectric DWs, while the adjacent domains remain unchanged. This has laid the foundation for the discovery of unusual polar internal structures at DWs with non-Ising^21–24 internal structures, which are often predicted to show chiral^25–27 or bichiral^28,29 configurations. Nowadays, DWs are regarded as individual functional topological structures^30–32 presenting physical properties that can be fundamentally different from those of the material in which they develop. They can be polar in non-polar materials,^33,34 show conductivity^35–44 or even superconductivity⁴⁵ in otherwise insulating materials, and present mechanical properties different from those of the adjacent domains.⁴⁶ 

These fascinating properties of polar DWs have inspired the design of original electronic device concepts and solid-state memories in which ferroelectric DWs are key elements.^31,47 Next-generation DW-based technology is being developed after the concept of DW nanoelectronics was settled in the last decade.⁴⁸ This comprises DW memristors,⁴⁹ field effect transistors,⁵⁰ multi-level non-volatile memory cells,^51,52 and DW-based nano-diodes.⁵³ Original approaches consisting of DW-based networks are also anticipated, laying the basis for atomic-scale electronics.⁵⁴ The emergence of a polar state at twin boundaries in SrTiO₃^55,56 has inspired the design of an original memory device based on the wall chirality.⁵⁷ In a recent review article, Ekhard Salje explains that multifunctionality can be achieved in ferroelastic materials through a smart combination of the intrinsic functionality of polar twins with the physical processes emerging from their dynamics.⁵⁸ 

The genuine nanoscopic scale of polar DWs, which are only a few atomic cells wide, makes their observation and study extremely challenging. Owing to its atomic resolution, transmission electron microscopy (TEM) has emerged as the method of choice for the observation of DWs.^59–62 Nonetheless, this technique can hardly be applied in a routine manner since the sample preparation for TEM measurements is in general difficult and invasive. For instance, the interaction of the sample with the intense electron beam may alter the domain structure.⁶³ Scanning probe microscopy methods can, to some extent, present excellent alternatives through piezoresponse force microscopy (PFM) and conductive atomic force microscopy (C-AFM).⁶⁴ While C-AFM has been proven to be highly suited for probing the local conduction at DWs, PFM measurements can hardly distinguish between an internal polarization structure of the DW and an electromechanical response arising from its displacement or deformation.⁶⁵ Non-contact methods in which specimen–probe interactions are reduced should therefore be preferred for the study of the internal structure of DWs. Polarized soft x-ray photoemission electron microscopy combined with low energy electron microscopy allows for the non-invasive study of domain structures in ferroic materials with a lateral resolution better than 50 nm,⁶⁶ which is sufficient to map the local electrostatic potential at the domain boundary regions.^67–69 Nevertheless, this method probes only the topmost surface, and it requires a thorough surface treatment in ultrahigh vacuum conditions prior to the measurements in order to reduce surface contamination and external charge screening which strongly affects the quality of the detected signal. In contrast to this, in optical studies, the experiments are usually conducted in an ambient atmosphere and no substantial sample preparation is required. Although optical methods have long been shunned because of their limited lateral resolution, they have made it possible to evidence important properties such as DWs birefringence⁷⁰ and three-dimensional (3D) polarization structures,⁷¹ accumulation of defects,^72,73 as well as local strain, electric fields, and unusual internal structures.⁷⁴ The modern literature on this topic includes several articles and text books discussing optical studies on domains and DWs, like the recent review by Nataf and Guennou,⁷⁵ a book chapter on three-dimensional optical studies of ferroelectric DWs,⁷⁶ as well as different articles addressing the complementarity between scanning probe microscopy, spectro-microscopy, and optical methods.^77–79 

Non-linear optical methods have emerged as powerful, user-friendly, and non-invasive means to study polar nano- and microstructures showing time or space inversion symmetry breaking, through a frequency doubling process known as second-harmonic generation (SHG).⁸⁰ When the non-linear optical signal is collected by means of near field, wide field, or far field microscopes, spatial distribution of the SHG can be accessed. This provides the possibility to probe ferroic domains in magnetic,^81,82 ferroelectric,^83,84 and complex multiferroic systems.^85–87 Recent advances on SHG studies in oxide thin films⁸⁸ involve operando measurements of functioning devices⁸⁹ as well as in situ experiments in which the domain structure can be precisely controlled during the film growth.⁹⁰ Remarkably, SHG microscopy allows for the imaging of small objects with sizes far below the optical diffraction limit, thereby making it possible to observe not only domains but also domain boundary regions.^91–98 Recently, SHG microscopy with polarimetry analysis has been employed for the direct observation of polar twin boundaries in non-polar centrosymmetric materials.^99–102 The method has furthermore been used to demonstrate the existence of Néel-type domain walls in PbZrTiO₃ thin films,^23,24 and to observe chiral Bloch walls and Bloch lines, a particular type of ferroelectric topological structures, in LiTaO₃.²⁴ 

This Tutorial highlights the capabilities of SHG microscopy to probe polar DWs and their internal structure. It aims at providing the necessary guidance to explore ferroelectric DWs with a Bloch or Néel-type configuration. After an overview of fundamental aspects related to the local symmetry, the non-Ising character, and the properties of DWs, we present the basics of the SHG method and emphasize its ability to probe the local symmetry down to the nanoscale. We explain how insight into the non-Ising character of polar DWs can be obtained by combining SHG polarimetry analysis and modeling. A general method is presented to investigate the internal structure of DWs with arbitrary orientation. To conclude, we discuss the validity of the presented method and review the still unsettled questions regarding non-Ising DWs.

Ferroic systems,¹⁰³ such as magnetic, ferroelectric, or ferroelastic materials, are crystalline solids characterized by the existence of an order parameter inherent to the material.¹⁰⁴ In the case of ferroelectrics this is the polarization, a vectorial quantity whose magnitude at zero external fields is a material-specific constant. Although more formal definitions of the polarization can be used,¹⁰⁵ in the case of a system of finite size we can simply think of it as the density of dipole moments, described by a vector field.¹⁰⁶ The thermodynamic potential of a ferroic material, its free energy, depends on the structure of the order parameter, i.e., on the spatial distribution of the vector field. The order parameter can be manipulated with externally applied fields and it typically possesses hysteretic properties such that the distribution of the vector field is not a unique function of the externally applied field. Rather, in equilibrium, the order parameter can adopt a variety of complex structures minimizing the energy. In particular, one usually observes a spontaneous subdivision into domains, i.e., regions within the crystal in which the orientation of the order parameter is homogeneous. The orientation of the polarization changes from one domain to another. The shape and size of the domains are often irregular. On a larger length scale, however, the ensemble of domains can display some kind of order in terms of size, orientation, and shape. The existence of domains was first postulated by Weiss¹⁰⁷ in ferromagnetic materials and it was thereafter evidenced in ferroelectric and ferroelastic systems.^108,109 Domains are typically very large compared to the atomic lattice constant, but smaller than the sample size (else the sample is said to be in a single-domain state). The size of domains is also much larger than that of the regions separating them, where the order parameter undergoes a transition from one orientation to another. Those transition regions between the domains are the DWs, which are at the core of this Tutorial.

The quest for miniaturization in modern nanotechnology has motivated scientists to focus their research on ferroic systems with reduced dimensionalities such as thin films and nanostructures. In this context, the importance of domains and DWs have tremendously increased in virtue of the Landau–Lifshitz–Kittel law^110,111 that lays the theoretical basics for the formation of domains (stripe and closure domains) and demonstrates the variation of their density (or size) with the thickness. It shows, in particular, a larger domain density in very thin films and a growth of the domain size with the square root of the film thickness. Soon after its derivation for magnetic systems, the Landau–Lifshitz–Kittel law has been extended to ferroelectric domains and DWs.¹⁰⁸ More recently, this law has been updated and modernized.^112,113 A thorough discussion on domains in ferroic crystals and thin films can be found in the book by Tagantsev et al.,¹¹⁴ as well as in various articles on nano- and microdomain engineering.^115–117 

In the following, we briefly discuss the analytic form of one-dimensional profiles describing the transition between two neighboring domains with opposite polarization such that the polarization changes its direction by 180° within the DW.

The theoretical analysis of the profile of domain walls in ferroelectrics is based on the energy terms dominating the structure of the polarization in the material, combined with boundary conditions enforcing the formation of the structures. In a single-component representation, the Gibbs free energy density of a ferroelectric material can be written in terms of even powers of the order parameter $Pz$⁠,¹⁰⁶ 

The Gibbs energy is a function of the electric field E and the strain $σ$⁠. From the equation of state it follows that, at constant strain $σ$⁠, the changes in the energy caused by an electric field are

and hence,

The solution describing a spontaneous, i.e., non-zero polarization $P0$ at $E=0$ is given by

The representation of the thermodynamic potential according to Eq. (1) describes the properties of a homogeneous ferroelectric material. In the case of spatially varying polarization structures, $Pz=Pz(x)$⁠, a further term is required in order to account for the additional energy related to these inhomogeneities. Because changes in the polarization are usually related to a change in the crystal structure, inhomogeneous structures of the polarization are generally accompanied by a local strain that increases the energy of the system. As a result, ferroelectric materials have a tendency to preserve a homogeneous polarization structure. Regardless of its microscopic origin, this tendency can be captured in a phenomenological approach by including gradient terms of the energy term, which penalizes any inhomogeneity of the polarization. Such an approach is also well known in the case of ferromagnets, where the exchange energy tends to prevent inhomogeneities of the magnetization structure. An energy term combining various spatial derivatives of the magnetization is obtained with coefficients that must be compatible with certain symmetry properties.¹¹⁸ In ferroelectrics, a similar gradient term in the energy is known as the correlation energy. Using a single-component representation of P, this inhomogeneity energy density can be written as follows:

In the one-dimensional case of a 180° domain wall, the polarization changes sign in the transition from one domain to the other. This transition leads to the boundary conditions,

Removing the background energy term $Φ0$ that is independent of $Pz$⁠, the total energy of the one-dimensional system is

We seek the domain wall profile $Pz(x)$ which minimizes Eq. (7) under the conditions (6). The Euler–Lagrange equation of the energy functional can be used to derive this profile,

which yields

This equation has a kink-type solution¹¹⁹ 

that fulfills the boundary conditions (6).

The parameter,

represents the scale of x in Eq. (10), and it is, thus, a measure for the width of the domain wall. Since domain walls form a continuous transition between two domains, the definition of their width is not unique. The commonly used definition of the width w is

This definition of the domain wall width can be interpreted as the distance between the intersection points of the slope of the polarization profile at the center of the domain wall with the asymptotically approached value of the polarization in the domains, as shown in Fig. 1.

The DW described by Eq. (10) is known as Ising-type 180° DW. It is characterized by a single component of the polarization, which changes sign along the transition from one domain to the other through a gradual change of $Pz$⁠, as shown in Fig. 1. In Sec. III, we will discuss other DW types, which involve changes in more than one polarization component. These other types of DWs correspond to those forming in ferromagnets. Although the Ising-type DW profile discussed here has been derived by using the specific energy density of a ferroelectric material, the “kink”-type transition in the form of a $tanh$-type profile is a general solution that is found also in other types of one-dimensional DWs, whose formation is dominated by other energy types.

Beyond its specific interpretation as a domain wall width parameter, $λ$ can more generally be regarded as a characteristic length scale on which inhomogeneities of the polarization develop within a ferroelectric material. The role of a length scale can equivalently be assigned to the correlation radius, defined as $rc=λ/2$⁠.¹¹⁴ This interpretation of the parameter $λ$ is analogous to the role played by the exchange lengths in micromagnetism.¹²⁰ The magnetic exchange lengths, too, are derived from the calculation of one-dimensional domain wall profiles, and they are used to describe more generally the characteristic size of inhomogeneous magnetization structures, such as vortices or more complicated three-dimensional domain wall types. Exchange lengths also play an important practical role in simulation studies, where they represent estimates for the required size of the discretization cells used in numerical calculations. Within the framework of Landau’s theory, the correlation length $λ$ in ferroelectrics depends sensitively on the temperature due to $α=α0/(T−T0)$⁠, where $T0$ is the Curie temperature. This dependence of the domain wall width on the temperature in ferroelectrics is different from the situation in ferromagnets, where exchange lengths are usually considered to be material-specific parameters that are not significantly affected by external parameters.

It is worth pointing out that the value of $λ$⁠, and hence the width of the domain wall, is determined by the outcome of competing interactions. As discussed before, the energy term proportional to $γ$⁠, the correlation energy, tends to avoid inhomogeneities of the polarization. Therefore, if a change of sign has to occur between two domains, then this term favors a transition on a scale that is as large as possible such that the gradient of P remains small. In contrast to this, the other terms in Eq. (7) which are proportional to $Pz2$ favor maximization of $|Pz|$ at temperatures below $T0$⁠. If the boundary conditions impose a change $−P0→P0$ between the domains, then these “anisotropy” terms would be minimized by shrinking the transition region as far as possible. If a spatial variation of P by 180° is enforced by the boundary conditions, then the resulting DW profile represents a compromise between these opposite tendencies.

For a long time, DWs in ferroelectrics were believed to be exclusive of Ising type. In this ideal configuration, the polarization decreases to zero at the DW core, and it reverses its direction while remaining parallel to the polarization of the adjacent domains (i.e., without rotation in the transition region). On the other hand, several theoretical studies have predicted the existence of polar DWs showing a clear polarization rotation^{21,25–27,29,121–128} with a non-zero planar component parallel to the DW (Bloch type) or perpendicular to it (Néel-type). These predictions were followed by experimental studies, which have confirmed the existence of non-Ising type DWs in ferroelectrics. These non-Ising DWs were named by Lee et al.¹²⁵ according to the corresponding fundamental types of magnetic DWs. Studies on polar DWs are currently a very active topic of research.¹⁹ In most cases, it can be assumed that non-Ising type DWs are not “pure” Bloch or Néel, but rather a mixed form of DW types combining, e.g., the characteristics of a Bloch wall and an Ising wall.¹²⁵ 

A fair amount of knowledge on ferroelectric DWs is based on the comparison with the well-established concepts of ferromagnetic DWs. In this context, it is useful to emphasize the differences and commonalities between the domains forming in these ferroic materials. Compared to ferroelectric domain walls, the mathematical description of magnetic domain walls has an additional complication related to the presence of a non-linear constraint: the magnetization can only change its direction not its magnitude. Therefore, unlike the polarization in Ising walls, the value of the magnetization cannot drop to zero in the center of a magnetic domain wall. Instead, it must rotate by 180° along with the transition from one domain to the other. There are two fundamental one-dimensional 180° DW types in ferromagnetism, which differ by the plane in which the magnetization rotates as it changes direction: the Bloch wall and the Néel wall. In the following, we describe these domain profiles in the case of a transition of the polarization P(x) in a ferroelectric material in the same way as a magnetic domain wall profile M(x) in a ferromagnet, even though the vector field P is not subject to the same constraints as M.

The Bloch wall is characterized by a rotation of the order parameter around the normal of the domain wall. Like in the Ising wall, the spatial dependence of the z component is described by a “kink”-type profile, proportional to $tanh(x/λB)$⁠, but in the case of the Bloch wall, the y component also changes in the transition region, such as to preserve the norm of the local polarization vector $P(x)$⁠. Since the energy of the system does not depend on the sense of rotation of $P$ with respect to the domain wall normal, the Bloch wall can appear in two different flavors with different handedness. The Bloch wall is, thus, a chiral domain wall type.

In the case of the Néel wall, the rotation of the order parameter occurs on a plane that is perpendicular to the domain wall, leading to a non-zero x component in the domain wall region. The kink-type profile of the z component proportional to $tanh(x/λN)$ is found also in this case. In Néel walls, the rotation can also occur in two equivalent and opposite directions, resulting in two variants with opposite signs of the x component. In nominally neutral 180° DWs, the Bloch wall transition does not contain divergences $(∇⋅P=0)$⁠, and, therefore, this wall type is said to be uncharged. This is in contrast to the Néel wall, which generates regions of opposite bound charge density $ρb=−∇⋅P$ near the center of the domain wall.

Idealized profiles of an Ising wall, a Bloch wall, and a Néel wall are displayed in the bottom row of Fig. 2. In spite of the similarities in the shape of their profile, the three domain wall types generally have different widths since their respective DW parameters $λN,λI,andλB$ have different functional forms. In magnetism, the parameters $λB$ and $λN$ are related to two different types of exchange lengths. It is obvious that the three domain wall types shown in Fig. 2 have different symmetry properties. Each of these DWs has local symmetry properties that are not only different from those of the other DW types but also from those of the parent material within the domains. As will be discussed in detail in Sec. III C, this fact can be exploited by experimental techniques that are particularly sensitive to changes in the local symmetry to both detect the DWs and to obtain indirect information on the internal structure of the polarization within the walls. In particular, this can be achieved with optical investigations based on SHG. Even though the DWs are too small to be resolved directly by such optical methods, their presence leads to characteristic changes in the local symmetry which can be detected and analyzed by SHG polarimetry.

Non-linear optics refers to the field of research studying the interaction of light with matter in the regime where the response of the material to the applied electromagnetic wave is non-linear. In this regime, the light–matter interaction can induce optical fields (i.e., a light emission) at frequencies (or wavelengths) different from those with which the material is irradiated, corresponding to harmonics of the incident wave. The principles of the optical harmonic generation process have long been known. The process was already anticipated in the 1930s with the theoretical work conducted by Maria Goeppert Mayer¹²⁹ during her Ph.D. However, the low-power light sources available at that time did not allow for the experimental demonstration of this effect. The era of modern non-linear optics came after the invention of powerful light sources in the form of lasers (intense and coherent light sources) and the experimental evidence of the second-harmonic generation process by Frankel et al.⁸⁰ Non-linear optics has since become an important field of research with high impact on the photonics industry, where such processes are exploited for frequency conversion and power amplification. Extensive information on the basic principles of non-linear optics can be found in history books published in the mid-1960s^130,131 as well as in the recent literature.^132,133

In the following, after introducing fundamental aspects of the frequency doubling process, we provide a hands-on method to determine the internal structure of polar DWs based on SHG polarimetry measurements (i.e., experiments recording the variation of the SHG intensity as a function of the laser polarization and analyzer angle). We first derive the analytic form of the SHG and its dependence on light polarization. This will include a detailed description of the non-linear optical susceptibility tensor, which is a key element in the SHG process. The modeling of the SHG polarimetry response is then obtained by assuming the symmetry of the parent material, the local symmetry reduction at the DWs, and their non-Ising character (i.e., Bloch or Néel-type transitions). The validity of the model is finally assessed by fitting the experimental data with the as-derived analytic form of SHG. Note that, for didactic reasons, the modeling of the SHG signal presented in this Tutorial is simplified. A more detailed description of advanced numerical modeling of SHG microscopy and polarimetry can be found in the specialized literature (see, e.g., Ref. 134 and references therein). In particular, detailed information on field distributions in strongly focused laser beams can be found in the text books by Gu¹³⁵ or Novotny and Hecht.¹³⁶ Furthermore, optical effects related to phase shift and birefringence^137,138 or to the light depolarization induced by a strong focusing of the incident laser beam in polarized laser microscopy have been discussed by different authors.^139–141 Some of these aspects can be implemented in the model, depending on the measurement geometry and the complexity of the studied system. For instance, Spychala et al.¹⁴² have recently used vectorial modeling of the second-harmonic emission at DWs to account for the effect of strong field focusing occurring when objective lenses with particularly large numerical apertures (0.95) are used.

Light traveling through a dielectric material at a given frequency induces, at the atomic level, a charge separation creating local electrical dipoles, i.e., a rapidly varying polarization P, that is proportional to the electric field E of the incoming wave. At low intensities, the induced polarization is linearly proportional to the oscillating electric field of the incident beam, and the radiating dipoles produce an outgoing beam at the same frequency. However, at high incident intensities, the oscillations of the induced dipoles do not follow the frequency of the incoming wave, and different frequency components can be contained in the radiated wave. The optical response of the material can be accounted for by expanding the polarization in a power series of the incident electric field,

The first term in Eq. (13) represents the linear polarization and the higher-order terms correspond to the non-linear polarization response. $ε0$ is the vacuum permittivity, D is a degeneracy factor, $χ(m)$ is the mth order optical susceptibility tensor, and the indices i,j,k refer to the Cartesian laboratory coordinates (x,y,z). The second term represents the second-harmonic generation process occurring in non-centrosymmetric materials such as ferroelectrics. In a microscopic picture, it involves the coupling of two incident photons at frequency ω that produce a polarization oscillating with the double frequency P(2ω),

This non-linear process arises in non-centrosymmetric crystals in which the second-order non-linear optical susceptibility χ⁽²⁾ is non-zero. The intrinsic permutation symmetry allows the susceptibility tensor to be replaced by a contracted d-tensor following the Voigt notation: $2dij=χikl(2)$⁠. The degeneracy factor is D⁽²⁾= ½ for indistinguishable fields involved in SHG and optical rectification. The number of the non-vanishing d_ij elements is further reduced when considering the point group symmetry of the material according to Neumann's principle. The SHG intensity is given by $ISHG=|P(2ω)|2=∑i(Pi(2ω))2$⁠, where

The intensity of the second-harmonic emission varies quadratically with the power of the incident laser, in addition to the frequency doubling process. Therefore, non-linear materials, which allow for both signal amplification and efficient energy transfer from one wave to another, are key elements in the photonic industry. Figure 3 shows that ferroelectric domains imaged in an SHG microscope exhibit a localized emission at the DWs. The intensity at the DWs is found to increase quadratically with the laser power [Fig. 3(d)] and the emission wavelength corresponds to half of the fundamental wavelength (i.e., the double of the frequency). This is an unambiguous signature of SHG. Such spectral analysis can be easily implemented in standard optical microscopes in which the operative mode can be switched from spectroscopy to imaging by means of a movable mirror. Note that, in principle, photon counting allows for quantitative analysis of the SHG intensity. This is however a delicate task since it requires a rigorous comparative study with a reference sample. Moreover, the SHG intensity can depend on the sample form (bulk or the thin film) and on the detection geometry involving forward, backward, or reflected backward SHG signals.¹⁴³ We present in the following the SHG intensity in arbitrary units and use the polarimetry analysis to obtain the internal polar structure of the DWs (i.e., the orientation of the polarization) rather than quantitative information on their optical properties.

The susceptibility tensor in SHG is a third-rank tensor connecting a cause given by a product of a vector (a second-order term that essentially represents the square of the electric field of the incident wave) to an effect represented by a vector (the induced polarization of the material). It is an optical property of the material that reflects its structural symmetry. This comprises not only the point group symmetry of the crystal but also information about the ferroic order. In multidomain ferroelectric materials, the SHG polarimetry response depends on the crystal symmetry (which determines the non-zero $dij$ tensor elements), the orientation of the domains (crystal coordinate system), and the measurement geometry (laboratory frame). The modeling of the optical response requires, therefore, the transformation of either the susceptibility tensor to the laboratory system or the electric field vector to the crystal coordinates system. Although these two methods are perfectly equivalent (see, e.g., Denev et al.⁸³), we choose the first method for its simplicity. It requires only the knowledge of a reference susceptibility tensor $d0$ defined in the crystallographic reference frame (X,Y,Z). Any “new” tensor element corresponding to an arbitrary coordinate system (X′,Y′,Z′) with a particular ferroelectric polarization of the underlying material can be deduced from the general rotation matrix transformations as follows:

where A is the rotation matrix resulting from the product of three individual rotations of Euler angles: ϕ, θ, ψ counterclockwise about Z, X′ and Z″,¹⁴⁴ 

The elements of the transformation matrix α⁻¹ are functions of the directional cosines,

Note that the representation of the susceptibility tensor in an arbitrary coordinate system requires only the application of the rotation matrix of Eq. (17). The transformation matrix α⁻¹ is only required if the result shall be presented in the laboratory coordinate system.

This method is often used to obtain information on the domain structure of ferroelectric¹⁴⁵ and complex multiferrioc systems^86,87,146 based on the SHG polarimetry analysis. A precise polarimetry analysis of the domain structure was reported by Gopalan and co-workers^147,148 in studies on the evolution of the domain structure during phase transitions in KNbO₃ and Bi₄Ti₃O₁₂. This method was then improved by Mishina et al.¹⁴⁹ to account for both coherent and incoherent contributions in inhomogeneous films. SHG polarimetry has since become a routine method to study ferroelectric domain structures. The question of whether this approach can be used to investigate the polar structure of DWs, instead of the domains, arises naturally. As a starting point, we can consider the tensor $d0$ corresponding to the symmetry of the parent material with an ideal Ising-type DW structure. Any deviation from the Ising configuration can be characterized by the rotation angles $ϕ,θ,ψ$ leading to a new tensor $dNew$ defined by Eq. (16). While this method is expected to give qualitative information on the local anisotropy and the possible existence of a non-Ising configuration (i.e., non-zero averaged polarization at the DWs), the local SHG at DWs cannot be correctly modeled based on the susceptibility tensor of the parent material (i.e., based solely on the symmetry and optical response of the adjacent domains). This is because symmetry operations (e.g., tensor rotation used to model domains with different orientations) preserve the original symmetry of the susceptibility tensor, while a reduction of symmetry is expected at the domain boundary regions, as will be discussed in Sec. III C. Thus, deriving the local susceptibility tensor at DWs is indispensable. This is arguably the most challenging task in DW studies with SHG polarimetry.

Besides being particularly suited for the study of non-centrosymmetric materials, SHG experiments with polarimetry analysis provide important information on symmetry.¹⁵⁰ According to the Neumann–Minnigerode–Curie principle,¹⁵¹ the physical properties of the object of interest, for instance, a DW, can be derived from its symmetry. It should, thus, be possible to identify the physical properties of DWs based on symmetry arguments. A local property tensor was derived by Přívratská and Janovec¹⁵² for nonferroelastic DWs by determining the symmetry of DWs using the crystallographic layer group¹⁵³ and by accounting for the symmetry of the adjacent domains. This approach has shown, e.g., that a spontaneous polarization or magnetization can be present at DWs in materials that do not display such ordering in their homogeneous form.¹⁵⁴ Alternatively, the DW symmetry can be deduced from the primary transition order parameter of the adjacent domains, and by taking into account a possible reduction of the parent order parameter space at the domain boundary region.¹⁵⁵ Schranz et al.¹⁵⁶ have recently combined the analysis of the order parameter across the walls with the layer group analysis developed by Janovec to derive the local symmetry and the related properties at polar twin boundaries.

Based on Landau potential calculations describing phase transitions at DWs, Bul’bich and Gufan¹⁵⁷ have evidenced a symmetry lowering at DWs corresponding to a transition of a wall in which the order parameter varies only in magnitude (i.e., Ising wall) to a wall in which the change in magnitude is accompanied by a rotation of the order parameter (i.e., non-Ising wall). Nowadays, it has become clear that a DW holds its own symmetry and properties which usually differ from those of the adjacent domains. In this situation, the following question arises: can we probe the local symmetry of DWs using SHG polarimetry experiments combined with simulations to gain insight into their properties? Based on the recently reported SHG polarimetry studies of twins,¹⁰⁰ phase boundaries,¹⁵⁸ as well as ferroelectric DWs,^23,159 we confirm that the internal structure of the DWs in terms of chirality and non-Ising characters²⁴ can indeed be accessed.

Polar twin boundaries in ferroelastic systems have been proven to exhibit either a 2 or m point group symmetry^99,101,160 (except for LaAlO₃ that exhibits 3m symmetry¹⁰⁰). This result applies also to non-Ising type ferroelectric DWs with Néel and Bloch internal structures. Figure 4 displays the internal structure and symmetry of Bloch and Néel-type DWs located in the zy-plane. In this case, Bloch-type walls exhibit a point group symmetry 2, Néel-type walls show a point group symmetry m, while Ising-type DWs are centrosymmetric with a point group symmetry 2/m. It is also worth noting that, even though the two non-Ising DW types appear in two variants, only the Bloch-type DW is chiral in the sense that it does not display a mirror symmetry.

In the following, the susceptibility tensor corresponding to Bloch and Néel-type ferroelectric DWs is derived assuming point group symmetry 2 or m. For the sake of simplicity, we assume an averaged polarization oriented along with the DW in the Bloch case and perpendicular to it for Néel-type DWs instead of the hyperbolic tanh profile. The orientation of the DWs is given by the δ angle (δ is taken from the x axis in the laboratory coordinate system) and their susceptibility accounting for their non-Ising charter is defined by the Euler angles as follows: $ϕ=0,θ=0,ψ=90°−δ$. The resulting non-Ising DW susceptibility tensor with arbitrary orientation [defined by the δ angle taken from the x axis, where the laboratory coordinate system (x,y,z) coincides with the crystallographic axes (X,Y,Z)] takes the following form:

The observation of ferroelectric DWs by means of SHG experiments has long been known. They can be revealed in different measurement geometries such as far field,⁹³ near field,⁹² as well as in collinear⁹¹ and noncollinear⁹⁴ (Čerenkov) SHG experiments. In these experiments, the DWs can either appear as dark lines or as bright regions. This SHG contrast variation from dark to bright results primarily from the SHG emission of the adjacent domains rather than from the measurement geometry. Fiebig et al.¹⁶¹ explain the dark contrast at DWs in RMnO₃ (R = Y, Ho) by destructive interference from the oppositely polarized neighboring domains. This is due to the large lateral size of the probe beam with respect to the DW, which induces an overlap of the SHG waves emitted by the adjacent domains. Their phase shift results in destructive interference at the 180° DW, making them appear as dark lines. A local SHG emission that can be used to gain insight into the physical properties of DWs can only be observed if the emission from the domains is minimal. Therefore, SHG is particularly suited for the study of polar twin boundaries owing to the absence of SHG at the centrosymmetric neighboring domains (see Yokota et al.⁹⁹). In ferroelectric materials, the SHG arising from the domains may overshadow the DW signal. The orientation of the sample and the measurement geometry should be carefully chosen in this case such as to optimize the DW signal. In Sec. IV D, we will see that confocal SHG microscopy is particularly helpful in this sense, as it allows, e.g., to focus the study on the core of bulk materials where the emission from the domains is low.¹⁶² 

A detailed description of all possible measurement geometries can be found in Ref. 83 and references therein. Here, we present more specifically scanning confocal microscopy in the backreflection geometry [see Fig. 5(a)] which ensures the overall non-invasive character of the method and allows for a better lateral resolution with respect to other configurations. It is furthermore particularly adapted for the study of ferroelectric thin films grown on nontransparent substrates. This type of instrument typically includes a laser source with sufficient output power to produce a non-linear response (e.g., a pulsed fs laser with 100 fs pulses and a repetition rate of 80 MHz) and a fundamental wavelength in the range of 750 nm–1000 nm. The laser beam is focused through an objective lens and directed at normal incidence to the sample, which allows us to use the same objective for the imaging. The SHG images are obtained by scanning the sample with respect to the incoming beam using computer-controlled stepping motors. The output wave is spectrally filtered and transferred to a photomultiplier for imaging. When a spectral analysis of the emission signal is necessary, a mirror is inserted to deflect the emitted signal toward a spectrometer. Polarimetry measurements are performed by recording SHG images at different polarizer and analyzer angles. Objectives with high numerical apertures would normally improve the lateral resolution of the microscope. However, the use of moderate numerical apertures is recommended because the paraxial approximation is satisfied in this case. When the numerical aperture became large (e.g., in the order of 0.95 as discussed in Ref. 141), many effects such as apodization, depolarization, and aberrations can occur and must be included in the theoretical models.¹³⁵ Note that all the SHG measurements presented in this Tutorial have been obtained in the case of medium focusing (numerical aperture in the order of 0.7) for which the paraxial approximation holds.

In SHG polarimetry experiments, image series [see Fig. 5(c)] are recorded as a function of the analyzer angles α at a given laser polarization φ or vice versa (α fixed and φ rotating). The study of the SHG polarimetry anisotropy can also require measurements in which both the polarizer and analyzer are rotated (either parallel or perpendicular to each other). The integration of the intensity over a selected area of the image sequence yields the local SHG polarimetry response within the region of interest. The data are then normalized by the number of pixels of the selected region to eliminate the effect of the selected area size on the intensity. The SHG intensity variation is traditionally presented as polar plots [see Fig. 5(c)]. The fitting of the polar plots based on the analytic form of the SHG provides information on the local symmetry and polarization order.

The SHG signal is described by the fundamental equations of the second-order non-linear process given by Eq. (15). In the measurement geometry depicted in Fig. 5(a), both the fundamental wave (FW) and the SHG emission propagate along the z axis (normal incidence), and the detection is in the reflection geometry. In this case, the electric field of the FW components lies within the (xy) plane $(Ex=E0cosφ,Ey=E0sinφ,Ez=0)$⁠, where $φ$ is the polarization angle of the FW measured from the x axis. The full SHG polarimetry response is given by

The polarization signal after traversing the linear output analyzer is obtained as a function of the orientation of the analyzer angle α by using the Jones formalism for a linear polarizer with a transmission angle α,

Equations (21) and (22) provide a complete description of the SHG polarimetry response $ISHG(φ,α)=|P2ω(φ,α)|2$⁠. This general approach is adapted for the study of the internal structure of polar DWs by taking into account its specific local symmetry and assuming a Néel-type or Bloch-type internal structure, i.e., using the susceptibility tensors given by Eqs. (19) and (20). The comparison between the data obtained in the experiment and the expected data by assuming different analytic models allows us to infer the structure of the observed DWs.

Figure 6 shows a detailed SHG polarimetry analysis at DWs surrounding c-domains (i.e., domains with out-of-plane polarization pointing either upward c⁺ or downward c⁻ on a background with opposite polarity). Two different systems are considered: tetragonal PbZrTiO₃ (point group 4mm) thin films showing triangular shape domains and LiTaO₃ bulk crystals (point group 3m) with hexagonal shape domains. If we focus on horizontal DWs, i.e., DWs parallel to the x axis (δ = 0°), we observe that the local polar plots display a maximum along with the DW in LiTaO₃ [see Fig. 6(b)], while the maximum is perpendicular to the DW in PbZrTiO₃ [see Fig. 6(d)]. This polarimetry response suggests a Néel-like character in PbZrTiO₃ and a Bloch-like response in LiTaO₃ hexagons. This result is confirmed by the simulation of the SHG polar plots at DWs with different angles [see Figs. 6(c)–6(e)] and by the good agreement of the fit of the experimental data using a Néel model PbZrTiO₃ and a Bloch model in LiTaO₃.

In order to confirm the ability of SHG polarimetry to distinguish between Bloch-like and Néel-like characters, advanced 2D simulations of the SHG images have been conducted. In this model, the DW region is subdivided into discrete regions, in which the ferroelectric polarization is allowed to rotate. The susceptibility tensor is then calculated at each rotation angle. This allows deriving the expected SHG intensity at any position (pixel) in both the domains and DW regions for given polarizer and analyzer angles. The result is displayed in Fig. 7 for a square shape domain. The isotropic SHG image (i.e., without polarization analysis) shows localized SHG emission at non-Ising DWs, in agreement with the experiments [see, e.g., Fig. 3(a)]. Moreover, the polarimetry response (characterized by the variation of contrast in SHG images with the analyzer and polarizer angles) shows clear and distinct signatures for Néel [Fig. 3(b)] and Bloch-type [Fig. 3(c)] characters.

The existence of Néel-type ferroelectric DWs is currently debated among theoreticians and experimentalists. In fact, Néel DWs are intrinsically charged, even in nominally neutral 180° DWs, which makes them energetically less favorable than Bloch-type walls. Yet, a Néel-like configuration is more often observed in experimental studies than the Bloch type (observed so far only in nearly stoichiometric LiTaO₃²⁴). Tetragonal PbZrTiO₃ (often used as a stand-in system for pure PbTiO₃) has been reported to exhibit Néel-type DWs in experiments conducted at room temperature, while Bloch-type walls have been predicted in PbTiO₃.^123,163 This discrepancy between theory and experiments can be related to a difference in the temperature at which the studies are conducted. A recent work combining theory and experiments has evidenced a dependence of the polar order with the magnitude on the configurational entropy.¹⁶⁴ The increased entropy at the DW regions was directly linked to the Néel-like component in LiNbO₃. Two other effects may also change the internal structure of the DWs in tetragonal PbZrTiO₃ thin films: (i) the tilt angle of the walls with respect to the polar axis²³ and (ii) the 3D polar structure of the DW in the volume, which could be Bloch- or Ising-like in the volume and of Néel-type at the topmost surface, as recently suggested by Li et al.¹⁶⁵ 

Confocal SHG microscopy is particularly adapted for the mapping of the SHG response in three dimensions.^166,167 Both collinear (presented in this Tutorial) and non-collinear SHG known as Čerenkov SHG can be conducted for this purpose.⁷⁶ The use of galvanometric scanners to control the laser beam position in Čerenkov type SHG provides fast imaging of the DW morphology in three dimensions. This has permitted, e.g., establishing direct correlations between the 3D morphology of the walls in lithium niobate,¹⁶⁸ their charged character,^169,170 and their enhanced conductivity. Owing to the fast acquisition of the images, this method is often applied to study DW kinetics as a function of electric¹⁷¹ or temperature¹⁷² stimuli. These temperature- or field-driven kinetic processes are known to yield exotic domain patterns¹⁷³ with closely packed charged DWs showing, e.g., zig-zag shapes. The study of the local SHG emission in such complex systems is rather challenging since it involves interfering signals including different DWs or a domain wall and a background SHG signal resulting from the adjacent domain or interface effects as discussed by Kämpfe et al.¹⁷⁴ In fact, a background signal with a characteristic polarimetry response often results from the electric field discontinuity occurring at surfaces and interface^175,176 in thin films. Even if the intensity of this surface-induced second-harmonic emission is small, the subtraction of such anisotropic background may be difficult, especially when this signal interferes with that of phase-shifted SHG nanoemitters. Optimum measurement conditions are thus obtained in the case of isolated DWs and minimum background signal.

In collinear SHG [e.g., in the measurement geometry displayed in Fig. 5(a)], the sample is usually scanned with respect to the laser beam. This makes the acquisition of the images rather slow with a typical scan rate of 100 nm/20 ms. Nevertheless, this method allows the combination of precise local polarimetry analysis with high-resolution imaging that is necessary to study topological structures such as non-Ising and chiral DWs.²⁴ Collinear SHG microscopy is optimized for 2D imaging. Yet, a 3D image can be constructed based on a stack of images measured at different depths of the sample (i.e., by measuring series of images at different distances between the objective lens and the sample). Figure 8 shows such an example of a 3D reconstruction of stacked images, displaying the 3D profile of a Bloch-type DW in nearly stoichiometric LiTaO₃. The modulation of the SHG signal in the 3D profile of the wall suggests a complex 3D wall structure involving both the wall distortion and a chirality transition. This structure is reminiscent of topological features known from magnetism. Bloch-type walls may exist in two variants since the ferroelectric polarization between c-domains of the opposite sign can rotate either clockwise or counterclockwise across the wall. The absence of mirror symmetry makes Bloch walls chiral. This property becomes particularly interesting if the chirality changes not only from one wall to another but also within the domain wall, as it is known to be possible in magnetic systems.¹⁷⁷ The transition regions separating Bloch walls with opposite chirality represent topological defects in the form of lines within the bulk of the magnetic crystals. Such line defects are known as Bloch lines. These 1D topological structures have inspired the design of original solid-state magnetic memory devices^178,179 and have stimulated numerous studies among the magnetism community.^180–183 It has been recently predicted that similar structures should also exist in ferroelectric¹⁸⁴ and ferroelastic^57,185 systems. SHG does not allow for the discrimination between DW segments with antiparallel polarization because of its quadratic dependence on the susceptibility. However, the region between the two wall segments can show a Néel-like transition or zero polarization (called Ising line¹⁸⁴) leading to a different contrast with respect to the walls. This makes it possible to visualize Bloch lines in 3D SHG measurements.²⁴ 

The use of light, in particular, through the second-harmonic generation process, is a powerful means to detect and investigate local properties of polar DWs even though their size is far below the resolution limit of optical methods. The convolution of the probe beam with the comparatively zero size of the DWs makes it possible to observe SHG at non-centrosymmetric DWs. Their non-Ising character produces a unique spectral signature in SHG polarimetry experiments,^23,24 which allows distinguishing between Néel-type and Bloch-type configurations.

The experimental studies on the internal structure of polar DWs are obviously inspired and supported by tremendous theoretical work that has been conducted on the subject. Several aspects predicted by theory still remain to be explored. Future experimental studies could, e.g., focus on the impact of the flexoelectric field on the chirality and the internal structure of polar twin boundaries^186,187 and ferroelectric^27,29,122 DWs, and attempt to evidence the link between the conductivity of ferroelectric DWs and their internal structure, as predicted by Morozovska⁴⁰ and Eliseev et al.¹⁸⁸ This would require the combination of different experiments with SHG microscopy polarimetry, like C-AFM, micros-diffraction, PFM, TEM, etc.

Experimental studies on the internal structure of DWs are also expected to assess theoretical models predicting their occurrence by exploring the range of validity of the models in terms of boundary conditions, temperature, pressure (strain), and surface vs bulk (3D) structure. Systematic experimental studies in which the effect of the external parameters on the internal structure of the walls are necessary to explain apparent discrepancies between theoretical predictions and experimental results and could be employed to specifically address the questions that are still unsettled in this context.

The recent development of new imaging modes to probe the non-linear optical response at the nanoscale shows great promise for the exploration of complex polar nanostructures. Rendón-Barraza et al.¹⁸⁹ have recently reported an SHG polarimetry study unraveling local crystalline aspects—at the sub-diffraction limit—in BaTiO₃ nanoparticles. This method can be adopted to resolve exotic polar topological structures predicted to arise in ferroelectric nanoparticules¹⁹⁰ such as vortices¹⁹¹ and Bloch points.¹⁹² 

This work was supported by the French National Research Agency (ANR) under Contract No. ANR-18-CE92-0052 through the TOPELEC project cofounded by the DFG under Grant No. EN-434/41-1. S.C.-H. and C.V. acknowledge funding by the LabEx NIE (ANR-11-LABX-0058_NIE) in the framework of the Interdisciplinary Thematic Institute QMat (ANR-17-EURE-0024), as part of the ITI 2021-2028 program supported by the IdEx Unistra (ANR-10-IDEX-0002-002) and SFRI STRATUS (ANR-20-SFRI-0012) through the French Programme d'Investissement d'Avenir. S.C.-H is grateful to Gregory Taupier, Patrycja Paruch, Katia Gallo, and Lukas Eng for their close collaboration. This article summarizes results that have been published elsewhere, as cited in the text. Some of these previous results have been adapted and reused for didactic purposes.

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