High-Sensitivity Characterization of Ultra-Thin Atomic Layers using Spin-Hall Effect of Light

Engineering spin-orbit coupling (SOC) and interfacial symmetry breaking in materials via magnetic - nonmagnetic interfacing are emerging trends in spintronics. The capability of SOC to enhance and tune the magnetic effects at the surface aids in the study of fundamental interactions at such surfaces and is also of significant interest in applicationsDayen et al. (2020); Hellman et al. (2017); Žutić et al. (2004). Interfacing soft magnetic materials with atomically thin layers has been shown to improve the coercivity via SOC and strong short-range correlations between electronic spin and orbital degrees of freedomHellman et al. (2017). Understanding the modified magnetic properties at such interfaces is of paramount importance to adopt and tailor material for varied applications. In this context, the development of a simple but highly sensitive characterization tool for studying surface magnetic and dielectric properties holds importance.

At present, vibrating sample magnetometry, magnetic force microscopy, and magneto-optic Kerr effect (MOKE) are widely used experimental methods for the investigation of magnetic properties of surface dominated atomic layersBonilla et al. (2018); Kazakova et al. (2019); Lan et al. (2020). However, limitations arising due to large substrate contributions and measurement sensitivity are opening up opportunities for the development of alternate techniques de Lima Bernardo (2014); Qiu et al. (2014); Du et al. (2020); Li et al. (2020); You et al. (2021); Li et al. (2019). Optical methods to characterize the magnetic properties of ultra-thin samples, through light-matter interaction, hold significant advantages including simple operation, non-contact probing, and high-sensitivity. One such promising method, proposed and demonstrated recently is the spin-Hall effect of light (SHEL)Hosten and Kwiat (2008); Ling et al. (2017), based on the weak measurement (WM) protocol of Aharanov et al.Aharonov et al. (1988). The SHEL is a polarization-dependent transverse shift in the center of gravity of the optical beam trajectory arising due to spin-orbit interaction. By the appropriate selection of a pre and post-interaction state of polarization (SOP) pointer states of a paraxial beam of light, the WM method can significantly enhance the measurable signal beyond the system noise levelsHosten and Kwiat (2008); Yin et al. (2013); He et al. (2009). The SHEL is an archetypical example of the WM method, arising due to spin-orbit interaction (SOI) with underlying geometric phaseOnoda et al. (2004), and has revolutionized optical measurements, enabling extremely high precision measurementsChen et al. (2020); Zhou et al. (2018); Ling et al. (2017). Recently it was shown that SHEL measurements offer a new method to probe magnetic ordering in finite-sized 2D crystals Sahoo et al. (2020). We expand on the basic idea to show that WM-based investigation is also suitable to measure the complex optical beam-shift to characterize and quantify the magnetization, coercivity, spin-orbit coupling strength, and the optical conductivity arising at the surface and interface of ultra-thin film samples. Although a few efforts have been madeDu et al. (2020); Qiu et al. (2014); Li et al. (2019, 2020); You et al. (2021) in employing WM to study magnetic ordering in thin films, the concept was limited to certain techniques and orientations. A generalized theory was needed to understand and extract relevant material information from these experiments. We present here a generalized concept to understand the WM-based magneto-optical effects.

We demonstrate here our ability to tune the coercivity of ultra-thin films of permalloy (Py) via SOC by interfacing it with an atomically thin layer of molybdenum disulphide (MoS₂). The MoS₂ atomic layers are known to be high SOC material but lacking in magnetic ordering. The MoS₂, in proximity to the ferromagnetic Py film, modifies the magnetic ordering at the interface due to possible charge/spin transferYin et al. (2015); Bansal et al. (2019). The changes in interfacial magnetic ordering, in turn, change the SOP of the reflected light beam. Monitoring the changes in the SOP using the WM technique allows us an accurate and sensitive measurement of the interaction-dependent changes. This technique is referred to as the SHEL-MOKE method in this article. This SHEL-MOKE technique allows sensitive characterization of the enhancement in coercivity in polar and longitudinal MOKE configurations. The technique can be extended to different MOKE configurations too, leading to the possibilities of complete analyses of ultrathin magnetic films without using expensive magnetometers or other surface probing techniques. In addition, we also show that by careful designing of the SHEL-MOKE experiment, one can get information on the complex Kerr rotation – the Kerr angle and the Kerr ellipticity of the sample.We show both theoretically and experimentally how the SHEL-MOKE method can be employed to characterise ultra-thin films to extract not only their magnetic properties but their other optical properties too.

When a paraxial beam of light is reflected or transmitted at an air-dielectric interface (including the magnetic thin film interface presented here), the phase-polarization characteristics in the beam cross-section get modified significantly. These changes depend on the SOP of the incident beam, angle of incidence, interface characteristics, and on the externally applied field Ling et al. (2017). Monitoring the phase-polarization changes in the interacted beam via the weak measurement method provides us with invaluable information about the interface characteristics up to a level that is much below the sensitivity of most existing methods Hosten and Kwiat (2008); Yin et al. (2013); He et al. (2009). To understand in detail the interface characteristics and changes arising in it due to the light-matter interaction, we consider a monochromatic linearly-polarized Gaussian beam of frequency $\omega$, incident on the interface between a lossless non-magnetic and magnetic media. The dielectric properties of magnetic thin film are effectively described by the tensor $\varepsilon$ given byQiu and Bader (2000); Yang and Scheinfein (1998):

$$\varepsilon=\epsilon\begin{bmatrix}1&&-iQ_{z}&&iQ_{y}\\ iQ_{z}&&1&&-iQ_{x}\\ -iQ_{y}&&iQ_{x}&&1\end{bmatrix}$$

(1)

Here, $\epsilon$ is the dielectric constant of the material, and $Q_{x,y,z}$ represents the magneto-optic Voigt constants of the magnetic material in x,y, and z directions respectively. In the right-handed Cartesian coordinate system (X, Y, Z), the (X, Y) plane forms the interface at Z=0, with the YZ plane taken as the plane of incidence Fig. 1(a). The light beam is described in a coordinate system defined by ($x^{a},y^{a},z^{a}$), where superscript a= (i), (r) denotes the incident and reflected light respectively. In this coordinate system $y^{a}$-axis coincides with the Y-axis and $z^{a}$ is directed along the wave vector of the beam’s central plane corresponding to k${}^{a}_{\circ}$. A linearly polarized Gaussian beam propagating in this coordinate system can be expressed in momentum space representation as:

$$\ket{\Psi(\textbf{k})}^{a}=\ket{\psi(\textbf{k})}^{a}\phi^{a}_{k}$$

(2)

with

	$\displaystyle\ket{\psi}^{a}$	$\displaystyle=$	$\displaystyle E_{H}^{a}\ket{H}^{a}+E_{V}^{a}\ket{V}^{a}$		(3)
	$\displaystyle\phi^{a}_{k}$	$\displaystyle=$	$\displaystyle\frac{\omega_{0}}{\sqrt{2\pi}}\exp{-(\frac{iz}{2k_{o}}+\frac{\omega^{2}_{0}}{4})((k^{a}_{x})^{2}+(k^{a}_{y})^{2})}$

Here, $\ket{\psi}^{a}$ refers to the polarization state while $\phi^{a}_{k}$ describes the Gaussian profile of the incident (reflected) beam with $\omega_{0}$ and z representing the beam waist and propagation distance respectively. Since, we mostly work in k-space, explicit mention of k is dropped in the function representations. The coordinate space functions are represented with a tilde. $k^{a}_{x}$ and $k^{a}_{y}$ represent the transverse spread of wave vectors corresponding to $x^{a}$ and $y^{a}$ coordinate axis. $\ket{H}^{a}(\ket{V}^{a})$ represents the horizontal (vertical) polarization state of light directed along $(\hat{x}^{a}-\frac{k_{x}^{a}}{k_{\circ}^{a}}\hat{z}^{a})$ and $(\hat{y}^{a}-\frac{k_{y}^{a}}{k_{\circ}^{a}}\hat{z}^{a})$ axisBliokh and Bliokh (2007); Hosten and Kwiat (2008), which corresponds to transverse magnetic mode (transverse electric mode) of light and $E^{a}_{H}(E^{a}_{V})$ is the corresponding electric field amplitude. A Gaussian beam with spherical wavefront can be described by an infinite set of plane waves with wave vectors $\textbf{k}^{a}=k^{a}_{x}\hat{x}^{a}+k^{a}_{y}\hat{y}^{a}+k^{a}_{\circ}\hat{z}^{a}$ enclosing the central wave vector $k^{a}_{\circ}\hat{z}^{a}$ Bliokh and Bliokh (2007). Under the paraxial approximation, the spread of the beam perpendicular to the central propagation direction given by $k^{a}_{x},k^{a}_{y}$, are in general much smaller than the central wave vector $k^{a}_{\circ}=k_{\circ}=\frac{2\pi}{\lambda}$. Fresnel reflection coefficients relate the incident and reflected plane wave vector and describe the action of the interface. Thus, the reflection of Gaussian beam can be effectively modeled by using Fresnel reflection coefficients acting on individual incident plane wave vectors. The electric field components of H and V in the incident and reflected Gaussian beam can be related using reflection matrix R as $\big{(}\begin{smallmatrix}E^{r}_{H}\\ E^{r}_{V}\end{smallmatrix}\big{)}=R\big{(}\begin{smallmatrix}E^{i}_{H}\\ E^{i}_{V}\end{smallmatrix}\big{)}$, where R can be obtained using the approach as mentioned in the references Luo et al. (2011); Bliokh and Aiello (2013) by relating incident and reflected Gaussian beam using Fresnel reflection matrix for a magnetic medium. R is a $2\crossproduct 2$ matrix given by,

$$R=\begin{bmatrix}r^{pp}-\frac{k^{r}_{y}(r^{ps}-r^{sp})\cot{\theta_{i}}}{k_{\circ}}&&r^{ps}+\frac{k^{r}_{y}(r^{pp}+r^{ss})\cot{\theta_{i}}}{k_{\circ}}\\ r^{sp}-\frac{k^{r}_{y}(r^{pp}+r^{ss})\cot{\theta_{i}}}{k_{\circ}}&&r^{ss}-\frac{k^{r}_{y}(r^{ps}-r^{sp})\cot{\theta_{i}}}{k_{\circ}}\end{bmatrix}.$$

(4)

where, $r^{pp},r^{ss},r^{sp},r^{ps}$ are the Fresnel reflection coefficient of the sample [S3-S6]. These coefficients are, in general, derived by applying boundary conditions on the electric and magnetic field components of the electromagnetic wave at the material interface. For reflection from magnetic material interface one can easily find these components by enforcing the boundary conditions as specified by Zak et al. Zak et al. (1990). Considering the fact that Gaussian beam also has a small spread of wave vectors in the plane of incidence, the Fresnel coefficients can be Taylor expanded to first-order approximation asBliokh and Bliokh (2007); Pan et al. (2013): $r^{ss}=r_{\circ}^{ss}-\frac{k^{r}_{x}}{k_{\circ}}\frac{\partial{r^{ss}}}{\partial{\theta_{i}}}$, $r^{pp}=r_{\circ}^{pp}-\frac{k^{r}_{x}}{k_{\circ}}\frac{\partial{r^{pp}}}{\partial{\theta_{i}}}$, $r^{sp}=r_{\circ}^{sp}-\frac{k^{r}_{x}}{k_{\circ}}\frac{\partial{r^{sp}}}{\partial{\theta_{i}}}$ and $r^{ps}=r_{\circ}^{ps}-\frac{k^{r}_{x}}{k_{\circ}}\frac{\partial{r^{ps}}}{\partial{\theta_{i}}}$. Relating the incident and reflected wavevectors using $\ket{\psi}^{r}=R\ket{\psi}^{i}$, the reflection matrix R in equation (4) can now be rewritten as,

$$R=\left(\begin{smallmatrix}r^{pp}_{\circ}-\frac{k^{r}_{x}}{k_{\circ}}\frac{\partial{r^{pp}}}{\partial{\theta_{i}}}-\frac{k^{r}_{y}(r^{ps}_{\circ}-r^{sp}_{\circ})\cot{\theta_{i}}}{k_{\circ}}&&r^{ps}_{\circ}-\frac{k^{r}_{x}}{k_{\circ}}\frac{\partial{r^{ps}}}{\partial{\theta_{i}}}+\frac{k^{r}_{y}(r^{pp}_{\circ}+r^{ss}_{\circ})\cot{\theta_{i}}}{k_{\circ}}\\ r^{sp}_{\circ}-\frac{k^{r}_{x}}{k_{\circ}}\frac{\partial{r^{sp}}}{\partial{\theta_{i}}}-\frac{k^{r}_{y}(r^{pp}_{\circ}+r^{ss}_{\circ})\cot{\theta_{i}}}{k_{\circ}}&&r^{ss}_{\circ}-\frac{k^{r}_{x}}{k_{\circ}}\frac{\partial{r^{ss}}}{\partial{\theta_{i}}}-\frac{k^{r}_{y}(r^{ps}_{\circ}-r^{sp}_{\circ})\cot{\theta_{i}}}{k_{\circ}}\end{smallmatrix}\right).$$

(5)

In the spin basis, the horizontal and vertical polarization of light can be written as $\ket{H}^{a}=\frac{\ket{+}^{a}+\ket{-}^{a}}{\sqrt{2}}$ and $\ket{V}^{a}=\frac{i(\ket{-}^{a}-\ket{+}^{a})}{\sqrt{2}}$, where $\ket{+}^{a}$ and $\ket{-}^{a}$ denotes the left and right circular polarization state respectively. The components of circular polarization of reflected light can be related to incident light in linear polarization as $\big{(}\begin{smallmatrix}E^{r}_{+}\\ E^{r}_{-}\end{smallmatrix}\big{)}=\frac{R^{c}}{\sqrt{2}}\big{(}\begin{smallmatrix}E^{i}_{H}\\ E^{i}_{V}\end{smallmatrix}\big{)}$, where, $R^{c}$ takes care of the transformation from linear to circular polarization basis of the incident beam as well as the reflection:

$$R^{c}=\left(\begin{matrix}r_{\circ}^{pp}[1+ik^{r}_{y}\delta_{H}-k^{r}_{x}\zeta_{H}-i\theta_{H}[1-k^{r}_{x}\zeta_{H}^{\prime}+ik^{r}_{y}\delta_{H}^{\prime}]]&&-ir_{\circ}^{ss}[1+ik^{r}_{y}\delta_{V}-k^{r}_{x}\zeta_{V}+i\theta_{V}[1-k^{r}_{x}\zeta_{V}^{\prime}+ik^{r}_{y}\delta^{\prime}_{V}]]\\ r_{\circ}^{pp}[1-ik^{r}_{y}\delta_{H}-k^{r}_{x}\zeta_{H}+i\theta_{H}[1-k^{r}_{x}\zeta_{H}^{\prime}-ik^{r}_{y}\delta_{H}^{\prime}]]&&ir_{\circ}^{ss}[1-ik^{r}_{y}\delta_{V}-k^{r}_{x}\zeta_{V}-i\theta_{V}[1-k^{r}_{x}\zeta_{V}^{\prime}-ik^{r}_{y}\delta^{\prime}_{V}]]\end{matrix}\right).$$

(6)

Here, $\delta_{H}=\frac{\cot{\theta_{i}}}{k_{\circ}}(1+r_{\circ}^{ss}/r_{\circ}^{pp})$, $\delta^{\prime}_{H}=\frac{\cot{\theta_{i}}}{k_{\circ}}(1-r_{\circ}^{ps}/r_{\circ}^{sp})$, $\delta_{V}=\frac{\cot{\theta_{i}}}{k_{\circ}}(1+r_{\circ}^{pp}/r_{\circ}^{ss})$, $\delta^{\prime}_{V}=\frac{\cot{\theta_{i}}}{k_{\circ}}(1-r_{\circ}^{sp}/r_{\circ}^{ps})$, $\zeta_{H}=\frac{1}{k_{\circ}r^{pp}_{\circ}}\frac{\partial{r^{pp}}}{\partial{\theta_{i}}}$, $\zeta_{H}^{\prime}=\frac{1}{k_{\circ}r^{sp}_{\circ}}\frac{\partial{r^{sp}}}{\partial{\theta_{i}}}$, $\zeta_{V}=\frac{1}{k_{\circ}r^{ss}_{\circ}}\frac{\partial{r^{ss}}}{\partial{\theta_{i}}}$, $\zeta_{V}^{\prime}=\frac{1}{k_{\circ}r^{ps}_{\circ}}\frac{\partial{r^{ps}}}{\partial{\theta_{i}}}$ $\theta_{H}=r^{sp}_{\circ}/r^{pp}_{\circ}$, and $\theta_{V}=r^{ps}_{\circ}/r^{ss}_{\circ}$Zak et al. (1990); Qiu et al. (2014). In equation (6), $(1\pm ik^{r}_{y}\delta_{H,V)}$ represents the spin orbit interaction (SOI) of light due to the diagonal component of Fresnel reflection coefficients. This SOI term is the main contributor to the initial spin dependent shift in SHEL. On the other hand, $(1\pm ik^{r}_{y}\delta^{\prime}_{H,V})$ represent the SOI of light due to off-diagonal Fresnel reflection coefficients. This term along with, $\theta_{H}$ and $\theta_{V}$, represents the optical activity of the material and mostly acts as perturbation to the initial spin dependent shift.

In general, for magnetic materials displaying the Kerr effect, the complex Kerr angle for H and V polarized incident light are represented by $\theta_{H}$ and $\theta_{V}$, respectively. The real ($\theta_{r}$) and imaginary ($\theta_{e}$) part of the complex Kerr angle represents the Kerr rotation of polarization and Kerr ellipticity change due to change in sample magnetization. Using equations S5 and S6, it is apparant that $\delta^{\prime}_{(H/V)}$ vanishes for polar and transverse magnetization direction whereas it has finite contribution in spin separation for longitudinal magnetization direction.

The SHEL shift of the reflected beam is obtained by calculating the expectation value of the transverse position operator. However, in real scenarios the actual shift is usually very small, approximately a fraction of a wavelength, which cannot be detected using conventional photodetectors or cameras. To efficiently detect the spin-dependent shift and the effect of sample magnetization on this shift, the principles of weak measurements are used to obtain an amplified spin-dependent shift. The amplified shift can be easily detected using a regular imaging camera. The ideal way of using the weak measurement to amplify spin-dependent shift lies in using H or V linear polarization as the initial state and projecting to the nearly orthogonal polarization state for detection. The incident polarization state $\ket{\psi}^{i}$ after reflection evolves to $\ket{\psi_{evl}}_{(H/V)}=(E^{r}_{+}\ket{+}^{r}+E^{r}_{-}\ket{-}^{r})_{(H/V)}$ in the circular basis. Here the subscript H and V represents the initial incident light polarization in H or V direction i.e., $E^{i}_{H}=1,E^{i}_{V}=0$ for H polarization and $E^{i}_{H}=0,E^{i}_{V}=1$ for V polarization. This evolved state is then observed using an analyzer in nearly orthogonal polarization configuration $\ket{\psi_{post}}$. Using the expressions for $E^{r}_{+}$ and $E^{r}_{-}$, the evolved light can be written as:

	$\displaystyle\ket{\psi_{evl}}_{(H)}=$	$\displaystyle\frac{r^{pp}_{\circ}}{\sqrt{2}}\Big{[}\big{(}1+ik^{r}_{y}\delta_{H}-k^{r}_{x}\zeta_{H}-i\theta_{H}[1-k^{r}_{x}\zeta_{H}^{\prime}+ik^{r}_{y}\delta_{H}^{\prime}]\big{)}\ket{+}^{r}$		(7)
		$\displaystyle+\big{(}1-ik^{r}_{y}\delta_{H}-k^{r}_{x}\zeta_{H}+i\theta_{H}[1-k^{r}_{x}\zeta_{H}^{\prime}-ik^{r}_{y}\delta_{H}^{\prime}]\big{)}\ket{-}^{r}\Big{]}$
	$\displaystyle\sim$	$\displaystyle r^{pp}_{\circ}\left[\hat{\mathds{1}}+ik^{r}_{y}\delta_{H}\hat{A}-k^{r}_{x}\zeta_{H}\hat{\mathds{1}}\right.$
		$\displaystyle-i\theta_{H}(\hat{A}-k^{r}_{x}\zeta_{H}^{\prime}\hat{A}+ik^{r}_{y}\delta_{H}^{\prime}\hat{\mathds{1}})\left.\right]\ket{\psi_{pre}}_{(H)}.$

	$\displaystyle\ket{\psi_{evl}}_{(V)}=$	$\displaystyle\frac{-ir^{ss}_{\circ}}{\sqrt{2}}\Big{[}\big{(}1+ik^{r}_{y}\delta_{V}-k^{r}_{x}\zeta_{V}+i\theta_{V}[1-k^{r}_{x}\zeta_{V}^{\prime}+ik^{r}_{y}\delta^{\prime}_{V}]\big{)}\ket{+}^{r}$		(8)
		$\displaystyle-\big{(}1-ik^{r}_{y}\delta_{V}-k^{r}_{x}\zeta_{V}-i\theta_{V}[1-k^{r}_{x}\zeta_{V}^{\prime}-ik^{r}_{y}\delta^{\prime}_{V}]\big{)}\ket{-}^{r}\Big{]}$
	$\displaystyle\sim$	$\displaystyle r^{ss}_{\circ}\left[\hat{\mathds{1}}+ik^{r}_{y}\delta_{V}\hat{A}-k^{r}_{x}\zeta_{V}\hat{\mathds{1}}\right.$
		$\displaystyle+i\theta_{V}(\hat{A}-k^{r}_{x}\zeta_{V}^{\prime}\hat{A}+ik^{r}_{y}\delta_{V}^{\prime}\hat{\mathds{1}})\left.\right]\ket{\psi_{pre}}_{(V)}.$

Here, $\ket{\psi_{pre}}_{(H)}=\frac{1}{\sqrt{2}}(\ket{+}+\ket{-})$, $\ket{\psi_{pre}}_{(V)}=\frac{i}{\sqrt{2}}(\ket{-}-\ket{+})$, and $\hat{A}=\big{(}\begin{smallmatrix}1&&0\\ 0&&-1\end{smallmatrix}\big{)}$. To take advantage of the weak value amplification, post-selection is performed on the evolved state of light. A quarter wave plate (QWP) and an analyzer that selects orthogonal polarization to the initial state perform the post selection. Their action can be represented in the circular basis as Zhou et al. (2014):

$$\begin{split}\ket{\psi_{post}}_{(H)}&=i\big{[}(e^{i\beta}\cos{(\frac{\pi}{4}+\alpha)}\ket{-}-e^{-i\beta}\sin{(\frac{\pi}{4}+\alpha)}\ket{+}\big{]}.\\ \ket{\psi_{post}}_{(V)}&=e^{-i\beta}\cos{(\frac{\pi}{4}+\alpha)}\ket{+}+e^{i\beta}\sin{(\frac{\pi}{4}+\alpha)}\ket{-}.\end{split}$$

(9)

where $\alpha$ is small deviation of the QWP optic axis from the initial polarization direction and $\beta$ is the small angular deviation from the perfect orthogonal position of the analyser optic axis. There actions on the light beam are represented on the Bloch sphere in Fig.1(b). Hence, the final state of reflected light is represented by the projection of $\ket{\psi_{evl}}_{(H/V)}$ on the post selected polarization state $\ket{\psi_{post}}_{(H/V)}$. Mathematically, the weak measurement on evolved state of light i.e., $\ket{\psi_{final}}_{(H)}=\ket{\psi_{post}}\bra{\psi_{post}}\ket{\psi_{evl}}_{(H)}$ and $\ket{\psi_{final}}_{(V)}=\ket{\psi_{post}}\bra{\psi_{post}}\ket{\psi_{evl}}_{(V)}$ can be written in momentum space as shown in equation (10). In the coordinate space, it is given by the angular spectrum representation as in equation (12):

	$\displaystyle\ket{\psi_{final}}_{(H)}=$	$\displaystyle\ \gamma_{(H)}(1+ik^{r}_{y}\delta_{H}A_{(H)}^{W}-k^{r}_{x}\zeta_{(H)}$		(10)
	$\displaystyle-i\theta_{(H)}($	$\displaystyle A_{(H)}^{W}-k^{r}_{x}\zeta_{H}^{\prime}A_{(H)}^{W}-ik^{r}_{y}\delta^{\prime}_{H}))\ket{\psi_{post}}\bra{\psi_{post}}\ket{\psi_{pre}}_{(H)}$

	$\displaystyle\ket{\psi_{final}}_{(V)}=$	$\displaystyle\ \gamma_{(V)}(1+ik^{r}_{y}\delta_{V}A_{(V)}^{W}-k^{r}_{x}\zeta_{(V)}$		(11)
	$\displaystyle+i\theta_{(V)}($	$\displaystyle A_{(V)}^{W}-k^{r}_{x}\zeta_{V}^{\prime}A_{(V)}^{W}-ik^{r}_{y}\delta^{\prime}_{V}))\ket{\psi_{post}}\bra{\psi_{post}}\ket{\psi_{pre}}_{(V)}$

		$\displaystyle\ket{\tilde{\Psi}_{final}}_{(H/V)}=$		(12)
		$\displaystyle\iint_{-\infty}^{\infty}\exp{i(k^{r}_{x}x^{r}+k^{r}_{y}y^{r}+k_{\circ}z^{r})}\ket{\psi_{final}}_{(H/V)}\phi^{r}_{k}\,dk^{r}_{x},\,dk^{r}_{y}.$

Here $\gamma_{(H/V)}=r^{pp}_{\circ}\ \textrm{or}\ r^{ss}_{\circ}$ for H or V pre-selected state respectively and the analytical expression for corresponding weak value amplification factor $A_{(H/V)}^{W}$ is given by:

$$A_{(H/V)}^{W}=\frac{\bra{\psi_{post}}\hat{A}\ket{\psi_{pre}}_{(H/V)}}{\bra{\psi_{post}}\ket{\psi_{pre}}_{(H/V)}}\sim\frac{\pm\sin{2\alpha}+i\sin{2\beta}\cos{2\alpha}}{2\absolutevalue{\bra{\psi_{post}}\ket{\psi_{pre}}_{(H/V)}}^{2}}.$$

(13)

From the above relation, it is obvious that a large weak value can be obtained when the pre and post-selection states are nearly orthogonal. Using equation (12) and equation (13) we can model the amplified spin shift by calculating the expectation value of the transverse position operator using,

$$\Delta^{mod}_{(H/V)}=\frac{\bra{\tilde{\Psi}_{final}}y\ket{\tilde{\Psi}_{final}}_{(H/V)}}{\bra{\tilde{\Psi}_{final}}\ket{\tilde{\Psi}_{final}}_{(H/V)}}.$$

(14)

The dependence of modified shift on complex Kerr angle is obtained by solving equation (14) and is provided in equation S1 and equation S2 in supporting information [SI] for initial H and V polarized light respectively. Considering the demagnifying factor (F) introduced by the use of a second lens in order to contain the image within the camera area, the observed amplified spin shift is obtained by multiplying F to equation (14). In the absence of magnetic field dependence and for samples in which the refractive index is purely real, the modified shift is reduced to:

$$\Delta^{mod}_{H/V}=F\frac{2\gamma_{H/V}(r^{pp}_{\circ}+r^{ss}_{\circ})z\cot{\beta}\cot{\theta_{i}}}{(k_{\circ}\omega_{0}\gamma_{H/V})^{2}+k_{\circ}^{2}(\gamma_{H/V})^{2}\zeta^{2}_{H/V}+(r^{pp}_{\circ}+r^{ss}_{\circ})^{2}\cot^{2}{\beta}\cot^{2}{\theta_{i}}}.$$

(15)

For a V-polarized incident light, Fig.2(a) shows the plot of the expected modified shift as a function of analyzer angle away from orthogonal position ($\beta$) for a system with real refractive index (n=1.45), in the absence of magnetic field. When the system does not have any remnant Kerr rotation or ellipticity in zero field, the shift is as given by the dashed black line in the plot. If we introduce a zero field polarization rotation $\theta_{r}=1\times 10^{-3}$ deg , the curve moves along the $\beta$ axis without any change in slope (red curve, Fig.2(a)). The presence of a small rotation leads to a large change in the analyzer angle for the crossed position, $\beta_{0}$. The presence of zero field ellipticity on the other hand changes the slope of the curve without any change in the zero shift position (blue curve, Fig.2(a)). The small change in slope is accompanied by a reduction in the maximum observed shift. However, the behavior in the case of materials with a complex refractive index (n=1.45 + i 1) is quite different for the same condition, as shown in Fig.2(b). A zero field polarization rotation causes change in $\beta_{0}$ corresponding to zero shift position (red curve, Fig.2(b)) similar to the earlier case, but zero field ellipticity gives rise to change in $\beta_{0}$ as well as asymmetry about the $\beta$ axis(blue curve, Fig.2(b)). The maximum values of positive and negative shifts are not the same, but there is no change in the slope of the curve. In the presence of equal zero field rotation and ellipticity, the change along the $\beta$ axis is compensated, showing that contributions of both to the centroid shift are in opposite directions. Fig.2(c) shows the dependence of centroid shift on the angle of incidence for materials with a real value of the refractive index at $\beta=0.1$ deg. For confirmation, we performed SHEL measurement on Py samples which have a complex refractive index at two different saturation magnetization fields. The plots are shown in Fig. 2(d). The curves for the positive and negative saturation magnetizations have the same slope, but the analyzer angle for zero shift changes due to the modification in the SOP of light, depicting the presence of both rotation and ellipticity. To study the material’s magnetic properties, SHEL-MOKE has been performed and the results are compared with MOKE. The details of experimental methods and the results are discussed in the following sections.

To determine the improved measurement capabilities of the SHEL-MOKE and to compare its performance with the standard MOKE setup, permalloy thin films (Py, Ni₈₀Fe₂₀), a well studied ferromagnetic material, and molybdenum disulphide (MoS₂)/Py samples (MSPy) are prepared and used to study the effects arising due to spin-orbit coupling (SOC). We prepared five different samples of Permalloy (Py) thin films. Among them, one sample contains molybdenum disulphide (MoS₂)/Py interface (MSPy). The five samples are refereed as Py10, Py20, Py30, Py45, and MSPy45, where the number at the end represents Py thickness in nanometer (nm). The details of the sample preparation are given below:

Initially, we performed SHEL measurement on Si/SiO₂ substrate, which we treat as the reference sample. The Si/SiO₂ substrate data is used to confirm our theoretical model developed in section 2. Fig. 3(a) shows the experimentally observed SHEL shift vs change in analyser angle for Si/SiO₂ substrate. As can be seen from the plot, the theoretical model agrees well with the experimentally observed variation in the SHEL shift, indicating a strong possibility of characterizing the magnetic and dielectric properties of the sample using this method.

Now, on replacing the Si/SiO₂ substrate with substrates coated with ferromagnetic thin film samples, say Py5, the SHEL shift vs analyzer angle changes dramatically as shown in Fig.3(b) and reveals the high sensitivity of the SHEL shift to accurately characterize the material properties. Since the centroid shift has a linear dependence on the analyzer angle in the region close to the orthogonal analyzer angle with a steep slope, the sensitivity of the system for polarization rotation is highly enhanced. We make use of this to perform sensitive Kerr rotation measurements on our samples.

To study the magnetic properties of the samples, both L-MOKE and P-MOKE measurements are performed using conventional MOKE and SHEL-MOKE setup. The hysteresis loops are recorded for all the samples by periodically reversing the magnetization direction as shown in Fig.3(c) and Fig.4 . In Fig.3(c), keeping analyzer angle at position 2 as denoted in Fig.3(a), the changes in centroid shift as a function of applied magnetic field are plotted. On sweeping the magnetic field, the sample magnetization changes and imparts optical polarization rotation on the reflected light, which in turn leads to a centroid shift. The formation of a typical soft magnetic hysteresis loop is evidence from the figure (L-MOKE configuration). The inset shows the intensity profile at two different saturation magnetizations of the Py system. The variation in the intensity of LCP and RCP light at the two extreme magnetic field are evident from Fig.3(c).

The plots in Fig.4(a) and (b) depict the hysteresis loop of Py20 obtained using conventional L-MOKE experiment, while, Fig.4(c) is the hysteresis loop of the corresponding SHEL-MOKE experiment. In the conventional MOKE experiment, we measure the ellipticity and rotation separately, while the SHEL-MOKE signal depends on both simultaneously. The L-MOKE coercivity is about 1-2 G. In the P-MOKE, the sample shows higher ellipticity change and no hysteresis in the rotation data. The hysteresis in SHEL-MOKE would also be determined by ellipticity change with the magnetic field. To obtain the values of rotation and ellipticity from the SHEL data, we would have to compare the data with theory. The low coercivity seen in L-MOKE data suggests that the easy axis of magnetization lies in the sample plane.

Fig.5(a) and (b) show the SHEL shift for different thicknesses of Py samples in L-MOKE and P-MOKE configuration, i.e., $\theta_{i}$ is 70^∘ and 45^∘, respectively at zero magnetic field. The systematic changes in the centroid shift variation with the thickness of the Py films are evident from the figures. The changes in the analyzer angle corresponding to zero centroid shift with the thickness of films in the two different configurations are shown in Fig.4(c) and (d), further strengthening the case for the SHEL-based measurement method to accurately and sensitively measure changes in polarization rotation angles in fractions of a degree. Variations in the coercivity with sample thickness observed using SHEL-MOKE are shown in Fig.S4 and Fig.S5 [SI]. The observed coercivity variation in L-MOKE configuration is in the range of 1-3 Gauss for different sample thicknesses. The observance of coercivity in P-MOKE and its variation with sample thickness, as shown in Fig.S6 [SI] proves the soft magnetic nature of Py thin films.

Interfacing soft magnetic materials with atomically thin layers to improve the coercivity via spin-orbit coupling is an emerging trend in spintronics due to their fundamental and applied interest Ahn (2020). We demonstrate our ability to tune the coercivity of thin films of Py via spin-orbit coupling by interfacing with atomically thin layers of MoS₂. The MSPy films developed are now investigated for the changes arising due to SOC by leveraging on the weak amplification-assisted SHEL-MOKE technique. MoS₂, being a high SOC material, in proximity to the ferromagnetic materials, modulates the magnetic ordering at the interface due to possible charge/spin transferSahoo et al. (2020). Using SHEL-MOKE, we observe this modulation in the magnetic ordering of the permalloy in proximity to MoS₂ as a significant enhancement in the observed coercivity in the P-MOKE configuration. A comparison of the coercivity of MSPy samples and Py thin film of similar thickness is shown in Fig.6. The origin of this enhanced coercivity has been traced to electronic spin and charge transfer between MoS₂ and ferromagnets. Hence, magnetic hardening is observed in such an interface.

Furthermore, from the theoretical model, we know that the centroid shift is dependent on the Kerr rotation and Kerr ellipticity introduced by the sample. The zero field SHEL measurement would directly tell us about the rotation and ellipticity contribution to the shift. Permalloy has a complex-valued refractive index, hence the symmetry of the SHEL data in Fig.5 implies that the ellipticity contribution to the shift is negligible. In the case of samples with Kerr ellipticity, the asymmetry observed in the SHEL data would help us estimate the ellipticity contribution. The SHEL data can then be modeled to extract the value of ellipticity and rotation. To study the change in ellipticity as a function of the magnetic field, one would have to measure centroid shift versus analyzer angle at various magnetic fields and evaluate the value of ellipticity from the asymmetry for complex refractive index materials. Thus SHEL-MOKE can in general be used to measure both Kerr rotation and ellipticity of samples having different dielectric and magnetic properties such as magnetic insulators, non-magnetic insulators, magnetic conductors, non-magnetic conductors, etc.

In addition, the SHEL measurement described above can also be used for sensitive characterisation of the optical conductivity of non-magnetic 2D materials like monolayer MoS₂. Existing methods to determine the optical conductivity and susceptibility of MoS₂, like the slab approach Li et al. (2014); Hsu et al. (2019) or the surface conductivity approach Morozov and Kuno (2015) do not consider the non-planar nature of finite beams and hence cannot give the most accurate charactersation of surface optical properties. Our measurement technique based on SHEL on the other hand, includes the inevitable effect of beam curvature and is a very sensitive technique to determine optical response of MoS₂ interfaces for applications in optoelectronic devices.

Fig.7 (a) and (b) show the observed SHEL shift of SiO₂ and monolayer MoS₂ grown over SiO₂ substrate obtained using 633 nm and 780 nm laser respectively. The SHEL shift for SiO₂ with 633 nm are fitted with Eq.(15) considering Eq.S7, S8 for fresnel reflection coefficients and shows a good agreement with the theoritical model. The experimental parameters like the beam waist at focus ($\omega_{0}$) obtained from fitting of SiO₂ SHEL shift is used to obtain the optical conductivity and dielectric sucseptibility of monolayer MoS₂ at the given wavelength. From fitting, the optical conductivity of monolayer MoS₂ at 633 nm is found to be $(1.9\crossproduct 10^{-4}\pm 3.3\crossproduct 10^{-5})\Omega^{-1}$. The obtained value is found in close agreement with the reported value Jayaswal et al. (2018); Elliott et al. (2020). From the figure, we note that the ellipticity introduced by MoS₂ monolayer, given by the change in slope of SHEL shift is greater than that of bare SiO₂ at 633 nm and lesser than SiO₂ at 780 nm. This wavelength dependent ellipticity variation is an unexplored field of the SHEL technique, both theoretically and experimentally, and will be explored in the near future.

Thus, the veracity of the theoretical model is established and it is shown that it can be a powerful method for analysing ultra-thin samples’ magnetic, dielectric, and optical properties with a high spatial resolution.

With a detailed theoretical analysis and simulations provided here, we establish the potential of SHEL-MOKE as a high-sensitivity simple surface probing tool for studying materials’ magnetic and dielectric properties. We experimentally demonstrated the versatility of the SHEL-MOKE method by characterizing the surface and interface magnetic properties of ultra-thin film samples of permalloy (Py) and MoS₂-Py interfaced samples. The ability to characterize the real and imaginary parts of the complex Kerr angle is expected to further improve the applicability of the method, and enable researchers to characterize accurately the effects arising at the magnetic-nonmagnetic interface. The coercivity measurement done using the SHEL-MOKE provides a direct and non-perturbative measurement technique to probe SOC-induced effects in material interfaces. Variations in the surface-interface conductivity, anticipated by the theoretical calculations, are currently being investigated, which adds value to the proposed methodology. The capability to extract surface-interface magnetization, coercivity, and conductivity from a single SHEL-MOKE scan with significantly improved sensitivity combined with simplicity will no doubt make the method a potential successor to several existing and widely used methods. At the end, the method has been extended to other ultra-thin films such as monolayer MoS₂, where its optical conductivity is extracted, indicating the potential of this method in analysing thin films of different kinds.

Authors from TIFR thank the support of the Department of Atomic Energy, Government of India, under Project Identification No. RTI 4007. GR. TNN and NKV thank SERB-SUPRA for suppport under grant no. SPR/2020/000220. NKV thanks SERB-CRG and IOE-UOH for financial support to this area of research. JJP acknowledges Sushree S. Sahoo and Parswa Nath for useful discussions.

See Supplementary Information [SI] for supporting details about theory, material characterization, and additional data.

The data underlying this study are available in the published article and its online supplementary material.