Measuring interfacial Dzyaloshinskii-Moriya interaction in ultrathin magnetic films

The Dzyaloshinskii-Moriya interaction (DMI), one of the origins of chiral magnetism, is currently attracting considerable attention in the research community focusing on applied magnetism and spintronics. For future applications, an accurate measurement of its strength is indispensable. Here the state of the art of measurement techniques involving the coefficient of the Dzyaloshinskii-Moriya interaction, the DMI constant $D$, is reviewed, with a focus on systems where the interaction arises from the interface between two materials (i.e., the interfacial DMI). An overview of the experimental techniques, as well as their theoretical background and models for the quantification of the DMI constant, is given. The measurement techniques are divided into three categories: (a) domain-wall-based measurements, (b) spin-wave-based measurements, and (c) spin-orbit torque-based measurements. The advantages and disadvantages of each method are analyzed, and $D$ values at different interfaces are compared. The review aims to obtain a better understanding of the applicability of the different techniques to various stacks and of the origin of apparent disagreements among literature values.

Figure 1

Diagram of a system possessing an interfacial DMI. The exchange coupling between the spins ${\overrightarrow{S}}_i$ and ${\overrightarrow{S}}_j$ in a ferromagnetic (FM) layer is mediated by a heavy metal (HM) atom with large spin-orbit coupling. In this case, the local DMI vector with magnitude $D_{ij}$ is perpendicular to the plane formed by the two atoms of the ferromagnetic layer and the one of the heavy metal layer.

Figure 2

Bloch and Néel configurations for a domain wall in a perpendicularly magnetized material. The Néel wall has left-handed chirality ($\uparrow \leftarrow \downarrow$), considering the positive $Z$ direction parallel and fixed to the normal, which is depicted by the arrow $\widehat{n}$ and defined pointing from the substrate to the magnetic film.

Figure 3

Sketch of the collective coordinates used to describe DW dynamics in different types of 1D models. $q$ is the domain-wall position, $\mathrm{\Phi }$ is the angle of projection of the magnetization $\overrightarrow{M}$ in the $x$-$y$ plane, $\mathrm{\Delta }$ is the width, and $\chi$ is the tilting angle of the DW.

Figure 4

Sketch of the dynamics of DWs driven by current in (a) the absence or (b) the presence of an applied in-plane field $H_x$. The white arrows indicate DW velocity. The strength of $H_x$ applied in (b) matches that of the DMI field $H_{\mathrm{DMI}}$, thus stopping the motion of ${\mathrm{DW}}_1$. (c) Sketch of the velocity for the two DWs as a function of applied $H_x$.

Figure 5

Dependence of the DW velocity $v_{\mathrm{DW}}$ as a function of an increased applied field (schematic only). (a) Display of the three regimes: creep, depinning, and flow. The depinning transition, which is abrupt at zero temperature, shows some rounding due to thermal effects. In (b), two distinct regimes can be distinguished within the flow regime, namely, steady and precessional flow. The drop in the velocity is called Walker breakdown and occurs at the Walker field $H_W$. In experiments, the breakdown is visible if $H_W\gg H_{\mathrm{dep}}$, with two flow regimes as in (c), while it is hidden by the thermal regime when $H_W\ll H_{\mathrm{dep}}$ as in (d). Adapted from [210].

Figure 6

First observation by the magneto-optical Kerr effect (MOKE) of the asymmetric expansion of a bubble under an in-plane magnetic field in a continuous film of $\mathrm{Pt}/\mathrm{Co}/\mathrm{Pt}$. Arrows show the in-plane field directions. From [133].

Figure 7

Dynamics of bubble DWs driven by a perpendicular field ${\mu }_0{H}_z=3\mathrm{mT}$ in (a) the absence (or (b) the presence of an in-plane field ${\mu }_0{H}_x=50\mathrm{mT}$ in a $\mathrm{Ta}(5\mathrm{nm})/\mathrm{Pt}(2.5\mathrm{nm})/\mathrm{Co}(0.3\mathrm{nm})$ thin film with PMA. Each image is obtained by adding four sequential images with a fixed time step (0.4 s), which are captured using a magneto-optical Kerr effect microscope. The blue box in (b) designates the field of measurement. From [129]. (c) A typical symmetric profile of the DW moving left and right as a function of the in-plane field $H_x$ in a $\mathrm{Ta}(5\mathrm{nm})/{\mathrm{Co}}_{20}{\mathrm{Fe}}_{60}{\mathrm{B}}_{20}(0.8\mathrm{nm})/\mathrm{MgO}(2\mathrm{nm})$ thin film. Here the velocity minima occur at $|H_x|=H_{\mathrm{DMI}}$. From [142].

Figure 8

Example of asymmetry of the domain-wall velocity profile as a function of the in-plane field (red dots) for a set of multilayers with nominal structure $\mathrm{Si}/{\mathrm{SiO}}_2/X(t)/{\mathrm{Co}}_{20}{\mathrm{Fe}}_{60}{\mathrm{B}}_{20}(1\mathrm{nm})/\mathrm{MgO}(2\mathrm{nm})/\mathrm{Ta}(1\mathrm{nm})$, with four underlayers $X(t)$: (a) W (2 nm), (b) W (3 nm), (c) TaN (1 nm), and (d) Hf (1 nm). All cases are highly asymmetric in the velocity profile, even showing a local maximum, as in (d). Note that the velocity in the direction orthogonal to the in-plane field (black squares) remains symmetric. From [260].

Figure 9

MFM domain patterns for (a) an out-of-plane and (b) an in-plane demagnetized multilayer composed of $[\mathrm{Pt}(1\mathrm{nm})/\mathrm{Co}(0.8\mathrm{nm})/{\mathrm{Al}}_2{\mathrm{O}}_3(1\mathrm{nm}){]}_{20}$. From [175].

Figure 10

Annihilation field of two homochiral walls as a function of the DMI value calculated from micromagnetic simulations for two different anisotropy values. Inset: magnetic configuration of the two Néel walls squeezed together by the perpendicular field. From [25].

Figure 11

(a) Theoretical nucleation field as a function of the magnetic field applied perpendicularly to the edge of the sample for edge and bubble (film) nucleation as a function of the field. (b) Experimental nucleation field as a function of the magnetic field applied perpendicularly to the edge for edge and bubble (film) nucleation. (c) Schematics of the nucleation of bubble domain (red area) showing the orientation of the wall core parallel and antiparallel to the applied field for in-plane fields below the DMI field. (d) Calculated domain-wall energy as a function of the in-plane field for different DMI fields. (a),(b) From [233]. (c),(d) From [156].

Figure 12

Schematic of the stray field measurement using a NV magnetometer. The internal DW magnetization angle $\psi$ ($\mathrm{\Phi }$ in our notation) depends on the DMI constant, resulting in different profiles of the stray field component $B_x^{\psi }(x)$. From [108].

Figure 13

(a) Simulations of SPSTM images based on Monte Carlo calculations ([118]) of a monolayer of Mn on $\mathrm{W}(001)$ at 13 K. Inset: Fourier transform. (b) Detailed spin structure for the white square indicated in (a), where the helical spinstructure can be directly seen in the illustration. From [94].

Figure 14

Evolution of bubble domain shapes in an applied in-plane magnetic field in $\mathrm{Co}/\mathrm{Ni}$ multilayers measured by the perpendicular Kerr effect. From [171].

Figure 15

Schematics of spin-polarized scanning tunneling microscope using a magnetically coated scanning tunneling microscope tip. (a) Geometry of the experimental setup including the magnetization axes of tip ${\mathbf{M}}_{\mathrm{T}}$ and sample ${\mathbf{M}}_{\mathrm{S}}$. (b) Electronic structure of the tip and sample. In the case of a perfect parallel alignment of ${\mathbf{M}}_{\mathrm{T}}$ and ${\mathbf{M}}_{\mathrm{S}}$, only majority electrons can tunnel into unoccupied majority states, and only minority electrons can tunnel into unoccupied minority states. From [118].

Figure 16

Schematics of a SPLEEM microscope. Spin-polarized electrons, photoemitted from a GaAs photocathode, are injected into a spin manipulator where azimuthal and polar orientation of the polarization is adjusted. The electron beam then passes through an illumination column before being decelerated in the objective lens. Electrons finally hit the surface with normal incidence. Electrons that are backscattered elastically are collected in an imaging column and focused on a phosphorous screen, where a magnified image of the surface is obtained. The incoming and reflected electron beams are separated in a magnetic beam splitter using the Lorentz force. From [239].

Figure 17

(a) Sketch of a spin wave in the Damon-Eshbach configuration propagating toward the positive $x$ direction, with a $+k$ wave vector, in the FM film. All the individual magnetic moments (blue arrows) are precessing counterclockwise around the $z$ axis, i.e., around the direction of the static magnetization and of the external field $H$. Owing to the phase delay from one spin to the next (moving from left to right), the chirality associated with the SW is clockwise, that is, favored by a positive DMI (magenta $D$ vector). In thin bilayer $\mathrm{FM}/\mathrm{HM}$ films, the latter couples two neighboring spins via a three-site exchange mechanism with the underlying atom of the HM (white atoms). As a consequence, the absolute frequency of $+k$ spin waves is down-shifted in frequency. (b) SWs propagating along the negative $x$ direction, i.e., with a $-k$ wave vector, are characterized by an counterclockwise chirality, that is disfavored by a positive $D$, resulting in an up-shift of their absolute frequency.

Figure 18

Schematic diagram of the used geometry and notation. $k$ is the wave vector and indicates the propagation direction of the spin wave, $H$ is the applied in-plane field, and $M$ is the equilibrium magnetization, pointing slightly out of plane. The DMI frequency depends on the angles ${\phi }_k$ and ${\phi }_M$, as given in Eq. (3.12).

Figure 19

DMI-induced asymmetric spin-wave propagation in the small-$k$ limit. (a) Dispersion relation. (b) Attenuation length as a function of the frequency $f$. From [215].

Figure 20

Schematic diagram of the BLS interaction geometry. $k_i^{\text{light}}$ and $k_s^{\text{light}}$ represent the wave vectors of the incoming and the backscattered light. Owing to the conservation of the wave vector component parallel to the surface of the specimen, the length of the wave vector of the SW involved in the scattering process is $2k_i^{\text{light}}\sin (\theta )$, as shown in the upper right panel. SWs propagating along the $+k$ ($-k$) direction correspond to those involved in the Stokes (anti-Stokes) process, i.e., in the generation (annihilation) of a magnon. Bottom panels: typical BLS spectra in the absence of a DMI (dashed line) and with a DMI (solid line) for positive or negative values of $D$ (i.e., of $f_{\mathrm{DMI}}$), respectively.

Figure 21

Scheme of a time resolved magneto-optic Kerr effect experiment. The pulse train of a femtosecond laser system is synchronized to the sinusoidal excitation field that is generated by sending a microwave current through a coplanar wave guide. The change of polarization of the reflected light is analyzed in an optical bridge and sent to a lock-in analyzer synchronized to a microwave mixer or chopper.

Figure 22

Scheme of the PSWS measurement. The two antennas are emitting and receiving HF signals from a VNA, which generates spin waves with an antenna specific $k$-vector distribution. Applying an in-plane magnetic field parallel to the antenna and focusing on spin waves with $k$ perpendicular to it, one observes an absorption in the signal due to the Damon-Eshbach (surface) modes that travel preferentially on one of the film surfaces contributing with different anisotropies ($K_s^{\mathrm{bot}}$ and $K_s^{\mathrm{top}}$). From [106].

Figure 23

Left panel: frequency shift with respect to the applied magnetic field. The black dashed line is the calculated Damon-Eshbach frequency with $D=0$. Right panel: shift of the mutual inductance spectra with $D$. The plot was obtained from numerical calculations based on the Landau-Lifshitz-Gilbert approach with the use of the surface permeability ([92]) on a magnetic film (saturation magnetization ${\mu }_0{M}_s=1.70\mathrm{T}$, damping $\alpha =0.012$, film thickness $d=10\mathrm{nm}$, and $D=\pm 0.5\mathrm{mJ}/{\mathrm{m}}^2$) for an antenna with an excitation maximum for a wave vector of $k_{\max }=10.4\mu {\mathrm{m}}^{-1}$.

Figure 24

(a),(b) Mutual inductance spectra of Py/Pt bilayers measured at ${\mu }_{0}H=37\mathrm{mT}$ for spin waves with $k<0$ and $k>0$. The frequency of the left-moving spin wave is shifted with respect to the frequency of the right-moving one with a difference $\delta f_{\mathrm{NR}}$. (c) Frequency nonreciprocity as a function of the Py thickness. The dashed and dash-dotted lines are the frequency nonreciprocity contributions induced by the DMI and magnetic anisotropy asymmetry, respectively. From [105].

Figure 25

(a) Scheme of the experimental setup, the sample, and the coordinate system of the magnetoacoustic spin-wave (MASW) measurement. The transmission coefficients $S_{21}$ and $S_{12}$ for the oppositely propagating surface acoustic waves (SAWs) with wave vectors $k_{S21}$ and $k_{S12}$ are measured by a VNA. (b) SAW transmission signal of CoFeB(2 nm). (c) SAW transmission signal of $\mathrm{CoFeB}(2\mathrm{nm})/\mathrm{Pt}$. $\mathrm{CoFeB}(2\mathrm{nm})/\mathrm{Pt}$ shows nonreciprocities both in the transmission amplitude at resonance and in the resonant magnetic field. From [168].

Figure 26

(a) Schematics of the current-induced effective magnetic field $H_{\mathrm{eff},z}$ at the Néel chiral domain wall in a HM/FM bilayer with perpendicular magnetic anisotropy. In the absence of an external magnetic field, $H_{\mathrm{eff},z}$ at the DW points in the opposite direction ($H_{\mathrm{eff},z}=0$). Thus, the motion of a domain wall $v_{\mathrm{DW}}$ is in the $-x$ direction, and there is no domain expansion. (b) In the presence of an external magnetic field $H_x$ large enough to align the Néel DW alongside $H_x$, the induced $H_{\mathrm{eff},z}$ points in the same direction, which leads to domain expansion due to the domain-wall motion in opposite directions. (a),(b) From [226]. (c) In the absence of $H_x$, there is no shift of the anomalous Hall loop. (d) However, a shift of the anomalous Hall loop is expected with the application of $H_x$, which provides a measure of the DMI and the spin-torque efficiency.

Figure 27

(a) Schematics of the measurement of the DMI field utilizing the anomalous Hall effect (AHE). (b) AHE loops for $\mathrm{Pt}(4\mathrm{nm})/\mathrm{Co}(1\mathrm{nm})/\mathrm{MgO}(2\mathrm{nm})$ with $I_e=\pm 6\mathrm{mA}$ [note that for consistency in the main text we use $I_e$ ($J_e$) instead of the $I_{\mathrm{dc}}$ used in the figure] and an in-plane bias magnetic field $H_x=0.25\mathrm{T}$. For clarity, slight offsets are introduced for both AHE loops. (c) Switching fields $H_{\mathrm{SW}}$ for down-to-up (red triangles) and up-to-down (blue circles) magnetization reversal as a function of $I_e$. $H_0=H_{\mathrm{eff},z}$ represents the average value of the switching fields. (d) Charge-to-spin conversion efficiency as a function of the in-plane bias field for $\mathrm{Pt}(4\mathrm{nm})/\mathrm{Co}(1\mathrm{nm})/\mathrm{MgO}(2\mathrm{nm})$. The blue squares and red circles represent the data obtained with an in-plane magnetic field along the $x$ and $y$ directions, respectively. Note that we use ${\chi }_{\mathrm{DL}}$ in the text instead of ${\chi }_{\mathrm{SHE}}$. From [226].

Figure 28

Literature data of the DMI constant $D_s$ for the $\mathrm{Pt}/\mathrm{Co}/X$ thin films, where the top layers $X=(\mathrm{MgO},\mathrm{GdO},{\mathrm{AlO}}_x,\mathrm{Al},\mathrm{Ta},\mathrm{Au},\mathrm{Ir},\mathrm{Pt},\mathrm{W})$ can be metals (light green) or oxides (light yellow). The blue-green and orange-red palettes are for experimental methods where the signs are known (negative and positive, respectively), while the gray palette is for data where only the magnitude of $D_s$ is available. The thickness of the bottom layer of Pt varies between 0.8 and 30 nm, with about 70% of the data between 3 and 5 nm. The top layer thickness (in nm) is shown as internal ticks (up to 6 nm). The large circles are the scale of $D_s$ (up to $3\mathrm{pJ}/\mathrm{m}$).

Figure 29

Literature data of the DMI constant $D_s$ for the $X/\mathrm{CoFeB}/\mathrm{MgO}$ thin films, where the bottom layers $X=(\mathrm{IrMn},\mathrm{Hf},\mathrm{Pt},\mathrm{Ta},\mathrm{TaN},\mathrm{W})$ are the most popular metals in the literature. The blue-green and orange-red palettes are for experimental methods where the signs are known (negative and positive, respectively), while the gray palette is for data where only the magnitude of $D_s$ is available. Most of the data ($>70\%$) are for a MgO thickness of 2 nm, with nearly 20% for 1 nm. The bottom layer thickness (in nm) is shown as internal ticks (up to 6 nm). The large circles are the scale of $D_s$ (up to $1.5\mathrm{pJ}/\mathrm{m}$).