1. Introduction

It has been proposed to employ magneto-optical (MO) effects in bismuth and gallium substituted iron garnet films, (YBiLuPr)₃(FeGa)₅O₁₂, (Bi-YIGaG) epitaxially grown on optically transparent gadolinium gallium garnet Gd₃Ga₅O₁₂ (GGG) substrates as probes for contactless and non-invasive metrology of microwave (mw) currents in integrated circuits with diffraction limited resolution [1–3]. The mw currents generate fringing magnetic fields which cause the magnetization in an adjacent Bi-YIGaG film to precess. The precessional angle is proportional to the magnitude of the mw magnetic field [4]. The magnetization component perpendicular to the film surface can be detected with linear (in magnetization) magneto-optical (MO) effects leading to linearity between the currents and MO response. As the current-carrying circuits are optically non-transparent, it is more convenient to perform the MO detection in reflection. The operation of such a MO probe requires applied in-plane static magnetic flux density fields, B_appl, biasing garnet films close to the ferromagnetic resonance at a desired microwave frequency. With the present semiconductor chips the bandwidth of which exceeds 100 GHz, B_appl may reach several Tesla.

The cost, total probe thickness and requirements for B_appl can be reduced by using ultrathin ferromagnetic iron layer with the saturation magnetization, µ₀M_s = 2.158 Tesla [5], instead of Bi-YIGaG films (about 3 µm thick) on 500 µm GGG substrates where µ₀M_s is more than two orders in magnitude smaller [1,6]. Indeed, according to Kittel [7], for a plate with in-plane µ₀M_s the required in-plane B_appl for the ferromagnetic resonance (FMR) frequency, f, is given by

 

Fig. 1 Ferromagnetic resonance frequency, f, vs. magnetic flux density field, B_appl, applied in-plane on the plates with the saturation magnetizations µ₀M_s = 0.0160 Tesla (gallium substituted yttrium iron garnet, GaYIG) and µ₀M_s = 2.158 Tesla (Fe).

Download Full Size | PDF

While the chemically inert refractory garnet oxides can be exposed to the ambient, the ultrathin iron layer requires sandwiching between protecting layers. In addition to their protecting function, the sandwiching layers must enable efficient transfer of the MO signal to the detector. Consequently, they should be transparent at the operating laser wavelength and the sandwiched Fe layer must be deposited on an appropriate reflector. The material choice for the sandwiching layers is restricted to dielectrics, preferably oxygen free. The dielectrics must be compatible with Fe and the reflector as for chemical inertness and deposition temperature. Optically thick Au, Ag, Cu, Pt, or Al layer can be used as the reflectors. Substrates for the probe must be reasonable flat and stable. Commercially available Si or GaAs chips represent a suitable choice. Adding a noble metal (Au) capping layer on top of the probe improves the protection against ambient effects. The probe then may consist of a sequence of five layers (Fig. 2 ).

 

Fig. 2 Diagram of the multilayer system operating as a magneto-optic probe.

Download Full Size | PDF

Two multilayers were grown and their MO response was tested experimentally. One system designed for the operating wavelength of 810 nm (Ti:sapphire laser) employed the layer sequence Au/FeF₂/Fe/FeF₂/Ag/GaAs [6,13]. Note that FeF₂ required the protection with an Au capping. The second one designed for the operating wavelength of 410 nm employed the layer sequence AlN/Fe/AlN/Cu/Si [14]. Prior the deposition, the MO response was evaluated using a transfer matrix model [15,16]. The degree to which the desired MO performance was achieved in the grown sandwiches was evaluated with MO spectroscopy. The sandwiching of the ultrathin iron with dielectrics transformed the smooth MO polar Kerr effect spectrum with broad lines of bulk Fe [6,13,14] to a MO spectrum with sharp peaks in both MO polar Kerr rotation and ellipticity located close to a desired laser wavelength.

In the past, quadri-layers formed by a magnetic layer sandwiched between dielectrics on an Al reflector received attention as media for high capacity MO recording disks [17–20]. The magnetic layer was most often an amorphous 3d-4f alloy with magnetic perpendicular anisotropy and a strong temperature dependence of coercivity. Two opposite orientations of the local magnetization on the disk represented logic zeros and ones. The magnetic and MO properties had to be uniform in the whole disk area. There were studies on the enhancement of MO effects in magnetic sandwiches motivated by other applications [21,22]. Here we discuss similar layer sequences, but with different requirements and objectives.

Based on previous experimental studies [6,13,14], the present work provides guidelines for the optimization of the MO response in reflection of magnetic nanostructure probes starting from the analytical expressions for a five-layer system with abrupt interfaces. The optimization leads to a maximal MO reflected wave amplitude polarized perpendicular to the incident linearly polarized wave. For given laser wavelength, dielectrics and reflector, the maximum can be achieved with a proper choice of the thicknesses of all layers forming the probe. Section 2 summarizes the analytical expressions for the corresponding Jones reflection and transmission matrices. Illustrating examples with numerical evaluation of the optical and MO characteristics at a wavelength of 632.8 nm in the structures consisting of an ultrathin Fe (typically 15 nm thick) sandwiched between AlN dielectrics and deposited on an Au reflector are given in Section 3. The discussion of the results is given in the last Section 4.

2. Figure of merit

We look for a set of individual layer thicknesses in a sandwich consisting of five layers Au(1)/dielectrics(2)/Fe(3)/dielectrics(4)/reflector(5) at which the Fe saturation magnetization z-component normal to the sandwich surface provides the maximal MO response (Fig. 2). Upon reflection at normal light incidence, an incident wave propagating parallel to the z-axis and linearly polarized parallel to the x-axis is partially reflected with the same x-polarization, partially absorbed in Fe, and partially transmitted into the optically thick reflector. A fraction of electron transitions responsible for the absorption in Fe are MO active. They produce the desirable mode conversion and generate transmitted and reflected waves with linear y-polarization.

The efficiency of the MO mode conversion in a magnetic layer is evaluated with |r_yx|, defined by Mansuripur as MO figure of merit (FOM) [23]. Here r_yx, an off-diagonal element of the Cartesian Jones reflection matrix,

The matrices relate the electric field x- and y-components, E_x and E_y of the incident (i) and reflected (r) or transmitted (t) waves propagating parallel to the z-axis.

An ideal |r_yx| corresponds to the situation where the incident x-polarized wave is absorbed in Fe (r_xx→0) and converted to y-polarized wave with no power transmitted into the reflector. Mansuripur estimated this ideal upper limit for the optical frequency (ω) dependent |r_yx(ω)| assuming linear dependence on the thickness d^(m) of the ultrathin ferromagnetic layer (m) of both |r_yx| and A, the absorbed power [23]

The criterion depends exclusively on the material parameters of the magnetic layer but due to the approximations involved overestimates the upper limit for FOM. A more realistic upper limit for |r_yx(ω)| would account for the nonlinear dependence on d ^(m) of both |r_yx| and A. We show below that with our restriction to the structures with five layers at most, we cannot completely eliminate the power leak into the reflector.

3. Reflected and transmitted waves

The Helmholtz equation for the optical wave electric field vector $E$

The four values for the propagation vector, $\widehat{z}\frac{\omega }{c}{N}_z^{(m)},$ i.e., $\pm \widehat{z}\frac{\omega }{c}{N}_{+}^{(m)}$and $\pm \widehat{z}\frac{\omega }{c}{N}_{-}^{(m)}$of the four circularly polarized (CP) waves follow from the eigen value equation$N_{\pm }^{(m)2}={\epsilon }_{xx}^{(m)}\pm \text{j}{\epsilon }_{xy}^{(m)}$.Their vector electric fields are given by [24]

With the Fresnel reflection and transmission coefficients

The absolute value $\left|r_{yx}^{(05)}\right|$ represents the MO reflected wave amplitude y-polarized perpendicular to the incident linearly x-polarized wave. We give also the off-diagonal element of the Cartesian Jones transmission matrix, $t_{yx}^{(05)},$ in the same approximation, i.e., to first order in $\left(N_{+}^{(3)}-N_{-}^{(3)}\right),$

We observe that the linearized $r_{yx}^{(05)}$in Eq. (17) consists of four factors. The optical and MO characteristics of the ultrathin Fe layer at the operating wavelength are given as$F_0=\frac{{\epsilon }_{yx}^{(3)}}{4{\epsilon }_{xx}^{(3)}}$. The factor $F_1=t^{(03)}{t}^{(30)}$depends on d⁽¹⁾ and d⁽²⁾. Note that the product $t^{(03)}{t}^{(30)*}$of $t^{(03)}$ with the complex conjugated $t^{(30)}$provides the power flux delivered across the front interface of an optically thick magnetic layer (3) per unit incident power flux. The factor $F_2=4j{\beta }^{(3)}{e}^{-2j{\beta }^{(3)}}{r}^{(35)}+\left(1-e^{-2j{\beta }^{(3)}}\right)\left(1+r^{(35)2}{e}^{-2j{\beta }^{(3)}}\right)$ depends on d⁽³⁾ and d⁽⁴⁾. At the other parameters fixed, there exist d⁽³⁾ of the order of the penetration depth for which the absolute value of this factor reaches its maximum. This is due to the differences in the d⁽³⁾ dependence of the terms $4j{\beta }^{(3)}{e}^{-2j{\beta }^{(3)}}{r}^{(35)}$ and $\left(1-e^{-2j{\beta }^{(3)}}\right)\left(1+r^{(35)2}{e}^{-2j{\beta }^{(3)}}\right)$. Note that $r^{(35)}$ in Eq. (20) characterizes the reflection at the back interface of the magnetic layer. The denominator factor $F_3={\left(1+r^{(30)}{r}^{(35)}{e}^{-2j{\beta }^{(3)}}\right)}^2$ depends on the thicknesses of all layers. The structure of the linearized $t_{yx}^{(05)}$can be understood in a similar way.

In the next section we apply the analytical formulae to the evaluation of the MO response of several sandwich configurations at the wavelength, λ = ω/(2πc), of 632.8 nm. We take the Jones matrix element, $r_{yx}^{(05)},$as the most important parameter for the evaluation of the probe performance. There are however other useful parameters which will be considered, i.e., the power reflection and transmission coefficients, $R={\left|r_{xx}^{(05)}\right|}^2,$ and $T=\Re \left[t_{xx}^{(05)}{t}_{xx}^{(50)\ast }\right]$, respectively. Here T represents the power flux traversing the front interface of the reflector (per unit incident power flux). As observed in Eq. (11) or Eq. (14), T cannot be made vanishingly small in the structures with ultrathin Fe or Fe and Au. The power absorbed in the sandwich becomes $A=1-R-T.$ The upper limit for the maximal figure of merit, ${\left|r_{yx}\left(\omega \right)\right|}_{max},$defined in Eq. (4), corresponds to the situation where the x-polarized wave is completely absorbed in the magnetic layer, i.e., to A = 1, R = T = 0.

The azimuth rotation, ${\theta }_r^{(05)},$and ellipticity, ${\epsilon }_r^{(05)},$ in reflection are defined with help of the MO ellipsometric ratio in reflection ${\chi }_r^{(05)}=r_{yx}^{(05)}/r_{xx}^{(05)}$as [27]

4. Response at 632.8 nm

The procedure leading to the maximal value of the MO response, $r_{yx}^{(05)},$ will be now illustrated on a sandwich Au(d⁽¹⁾)/AlN(d⁽²⁾)/Fe(d⁽³⁾)/AlN(d⁽⁴⁾)/Au(d⁽⁵⁾) at the wavelength, λ = 632.8 nm. The thickness of the capping Au will be set to d⁽¹⁾ = 0 for dielectrics inert in the ambient. In the other cases, it is assumed that a reasonable protection against ambient effects can be achieved with d⁽¹⁾ = 5 nm. We set the Au substrate optically thick with d⁽⁵⁾ = 1000 nm. At λ = 632.8 nm, we characterize AlN as a nonabsorbing dielectrics with a real index of refraction, N^(AlN) = ε^(AlN)1/2 = 2.00. The actual value of N^(AlN) in sandwiches depends on the deposition conditions [28]. In Table 1 , we summarize the optical and magneto-optical parameters of Fe [10–12], i.e., the complex relative permittivity, ε_xx^(Fe)/ε^(vac), the complex index of refraction, N^(Fe) = ε_xx^(Fe)1/2/ε^(vac)1/2, the penetration depth, d^(Fe)_penetr, reflectivity R^(Fe) at d^(Fe) = 1000 nm, and ${\left|r_{yx}\right|}_{max}\approx 0.0188$computed from Eq. (4). The corresponding optical parameters of Au [29] are summarized in Table 2 .

Table 1. Optical and magneto-optical parameters of Fe

View Table | View all tables in this article

Table 2. Optical parameters of Au

View Table | View all tables in this article

We first consider the situations where d⁽¹⁾ = 0 and start the search for the maximal $\left|r_{yx}^{(05)}\right|$with an AlN layer on an Au substrate. The maximal reflectivity, R = ${\left|r_{xx}^{(05)}\right|}^2$, corresponding to the minimal transmission, T, is achieved for d⁽²⁾ + d⁽⁴⁾ = 133 nm and arg$\left(r_{xx}^{(05)}\right)=\pi$(Table 3 ). We insert a Fe layer and look for the maximal $\left|r_{yx}^{(05)}\right|,$ which we expect at d⁽³⁾≈d^(Fe)_penetr and reduce d⁽²⁾ + d⁽⁴⁾ by $\left[\Re \left(N^{(\text{Fe})}\right)/N^{(\text{AlN})}\right]d^{(3)}$ to account for the phase contribution of the Fe layer to the total thickness (d⁽²⁾ + d⁽³⁾ + d⁽⁴⁾). In the next step, we look for the position of the Fe layer between the AlN sandwiching layers, keeping the new d⁽²⁾ + d⁽⁴⁾ fixed, giving maximal $\left|r_{yx}^{(05)}\right|.$ In this way, we arrive at a sandwich AlN(63 nm)/Fe(19 nm)/AlN(42 nm)/Au(1000 nm) with a reasonable $\left|r_{yx}^{(05)}\right|=\mathrm{0.01823.}$ The structure is distinguished by two desirable features, high A and low T. A further refinement of d⁽²⁾, d⁽³⁾, and d⁽⁴⁾ led to the sandwich AlN(50 nm)/Fe(14 nm)/AlN(26 nm)/Au(1000 nm) with $\left|r_{yx}^{(05)}\right|=0.018344$ close to the highest $\left|r_{yx}^{(05)}\right|=0.0183467$ (Table 3).

Table 3. Structures AlN/Fe/AlN/Au with d⁽¹⁾ = 0

View Table | View all tables in this article

This is a considerable improvement with respect to $\left|r_{yx}^{(05)}\right|=0.006413$ at a planar interface between air and optically thick Fe where the incident power is nearly equally distributed between the R and A parts (Table 3). The figure of merit of the thick Fe can be improved in two ways, either with an ultrathin Fe layer (d⁽³⁾ = 23 nm) on an Au substrate with $\left|r_{yx}^{(05)}\right|=0.00958$, or better by depositing an AlN layer (d⁽²⁾ = 61 nm) on the optically thick Fe $\left|r_{yx}^{(05)}\right|=0.0126$. The combination of these remedies in the structure AlN(57 nm)/Fe(23 nm)/Au(1000 nm) with the Fe layer situated below the AlN dielectrics provides $\left|r_{yx}^{(05)}\right|=0.01797,$ a value rather close to the maximal $\left|r_{yx}^{(05)}\right|$ achieved in the sandwiches. With the Fe layer situated above the AlN dielectrics, i.e., with the Fe layer exposed to the ambient, in the structure Fe(6 nm)/AlN(56 nm)/Au(1000 nm), the figure of merit reaches $\left|r_{yx}^{(05)}\right|=0.01804$. The details are listed in Table 3 along with other remarkable examples of the sandwich structures with $\left|r_{yx}^{(05)}\right|>0.0183$.

We now return to the sandwich structure AlN(50 nm)/Fe(14 nm)/AlN(26 nm)/Au(1000 nm) with $\left|r_{yx}^{(05)}\right|=0.018344$ and consider the effect of d⁽²⁾, d⁽³⁾, and d⁽⁴⁾ on the probe performance. Figure 3 shows the effect of d⁽³⁾ (thickness of Fe layer) on the parameters of the structure AlN(50 nm)/Fe(d⁽³⁾)/AlN(26 nm)/Au(1000 nm).

 

Fig. 3 Effect of the thickness of the Fe layer, d⁽³⁾, on the parameters of the structure AlN(50 nm)/Fe(d⁽³⁾)/AlN(26 nm)/Au(1000 nm): (a) power reflection (R), transmission (T), and absorption (A) coefficients, (b) off-diagonal amplitude reflection coefficient$r_{yx}^{(05)}=\left|r_{yx}^{(05)}\right|\exp \left[{\text{j}}^{}\text{arg}\left(r_{yx}^{(05)}\right)\right]$, (c) diagonal amplitude reflection coefficient$r_{xx}^{(05)}=\left|r_{xx}^{(05)}\right|\exp \left[{\text{j}}^{}\text{arg}\left(r_{xx}^{(05)}\right)\right]$, and (d) magneto-optic azimuth rotation, ${\theta }_r^{(05)}$, and ellipticity, ${\epsilon }_r^{(05)}$at 10% magnetic saturation.

Download Full Size | PDF

In Figs. 4 and 5 , we illustrate the effect of d⁽²⁾ (thickness of upper AlN layer) and d⁽⁴⁾ (thickness of lower AlN layer) on the parameters of the structures AlN(d⁽²⁾)/Fe(14 nm)/AlN(26 nm)/Au(1000 nm) and AlN(54 nm)/Fe(14 nm)/AlN(d⁽⁴⁾)/Au(1000 nm), respectively. From the inspection of Table 3 and Figs. 3-5 we observe that $\left|r_{yx}^{(05)}\right|$ in the structure AlN(d⁽²⁾)/Fe(d⁽³⁾)/AlN(d⁽⁴⁾)/Au(1000 nm) is enhanced by a factor of 2.86 with respect to air/Fe(1000 nm) interface. In the region of the maximal$\left|r_{yx}^{(05)}\right|$, R can be made vanishingly small, but T cannot be reduced below 1.5%. As a result, the computed minimum of R, R_min≈2 × 10⁻⁵, at {d⁽²⁾, d⁽³⁾, d⁽⁴⁾} = {54nm, 16 nm, 25 nm} is displaced from the position {50 nm, 14 nm, 25 nm} of the maximal $\left|r_{yx}^{(05)}\right|$. At R_min (“antireflection condition”), i.e., at the minimum of $\left|r_{xx}^{(05)}\right|$, where arg$\left(r_{xx}^{(05)}\right)$ changes rapidly, ${\theta }_r^{(05)}$and ${\epsilon }_r^{(05)}$display strongly localized maxima (minima). On the other hand, in the region of the maximal$\left|r_{yx}^{(05)}\right|$ both$\left|r_{yx}^{(05)}\right|$ and $\arg \left(r_{yx}^{(05)}\right)$ are slowly varying functions of {d⁽²⁾, d⁽³⁾, d⁽⁴⁾} which makes the tolerances for the sandwich deposition easily achievable. If required, a fine tuning of d⁽²⁾, d⁽³⁾, d⁽⁴⁾ or λ can shift the structure to zero ${\epsilon }_r^{(05)}$ and/or to maximal ${\theta }_r^{(05)}$ at negligible reduction of $\left|r_{yx}^{(05)}\right|$ with respect to its maximal value.

 

Fig. 4 Effect of the thickness of the upper AlN layer, d⁽²⁾, on the parameters of the structure AlN(d⁽²⁾)/Fe(14 nm)/AlN(26)/Au(1000 nm): (a) power reflection (R), transmission (T), and absorption (A) coefficients, (b) off-diagonal amplitude reflection coefficient$r_{yx}^{(05)}=\left|r_{yx}^{(05)}\right|\exp \left[{\text{j}}^{}\text{arg}\left(r_{yx}^{(05)}\right)\right]$, (c) diagonal amplitude reflection coefficient$r_{xx}^{(05)}=\left|r_{xx}^{(05)}\right|\exp \left[{\text{j}}^{}\text{arg}\left(r_{xx}^{(05)}\right)\right]$, and (d) magneto-optic azimuth rotation, ${\theta }_r^{(05)}$, and ellipticity, ${\epsilon }_r^{(05)}$at 10% magnetic saturation.

Download Full Size | PDF

 

Fig. 5 Effect of the thickness of the lower AlN layer, d⁽⁴⁾, on the parameters of the structure AlN(50 nm)/Fe(14 nm)/AlN(d⁽⁴⁾)/Au(1000 nm): (a) power reflection (R), transmission (T), and absorption (A) coefficients, (b) off-diagonal amplitude reflection coefficient $r_{yx}^{(05)}=\left|r_{yx}^{(05)}\right|\exp \left[{\text{j}}^{}\text{arg}\left(r_{yx}^{(05)}\right)\right]$, (c) diagonal amplitude reflection coefficient$r_{xx}^{(05)}=\left|r_{xx}^{(05)}\right|\exp \left[{\text{j}}^{}\text{arg}\left(r_{xx}^{(05)}\right)\right]$, and (d) magneto-optic azimuth rotation, ${\theta }_r^{(05)}$, and ellipticity, ${\epsilon }_r^{(05)}$at 10% magnetic saturation.

Download Full Size | PDF

In the considered structures, the deposition of the protecting Au layer, d⁽¹⁾ = 5 nm, reduces maximal $\left|r_{yx}^{(05)}\right|$ (Table 4 ). In Fig. 6 , we illustrate the effect of d⁽³⁾ (thickness of Fe layer) on the parameters of the structure Au(5 nm)/AlN(75 nm)/Fe(d⁽³⁾)/AlN(20 nm)/Au(1000 nm). The effect of changing thickness of the upper or lower AlN, d⁽²⁾ or d⁽⁴⁾, respectively, is shown in Figs. 7 and 8 . The trends with the changing d⁽²⁾, d⁽³⁾, and d⁽⁴⁾ correspond to those observed for d⁽¹⁾ = 0 in Figs. 3-5. The optimization of $\left|r_{yx}^{(05)}\right|$required significant modifications in d⁽²⁾ and/or d⁽⁴⁾ of AlN but only smaller adjustments of d⁽³⁾ for Fe layers. The maximal 
$\left|r_{yx}^{(05)}\right|\approx 0.0172$ was achieved in the structure Au(5 nm)AlN(73 nm)/Fe(15 nm)/AlN(20 nm)/Au(1000 nm).

Table 4. Structures Au/AlN/Fe/AlN/Au with d⁽¹⁾ = 5 nm

View Table | View all tables in this article

 

Fig. 6 Effect of the thickness of the Fe layer, d⁽³⁾, on the parameters of the structure Au(5 nm)/AlN(73 nm)/Fe(d⁽³⁾)/AlN(20 nm)/Au(1000 nm): (a) power reflection (R), transmission (T), and absorption (A) coefficients, (b) off-diagonal amplitude reflection coefficient$r_{yx}^{(05)}=\left|r_{yx}^{(05)}\right|\exp \left[{\text{j}}^{}\text{arg}\left(r_{yx}^{(05)}\right)\right]$, (c) diagonal amplitude reflection coefficient$r_{xx}^{(05)}=\left|r_{xx}^{(05)}\right|\exp \left[{\text{j}}^{}\text{arg}\left(r_{xx}^{(05)}\right)\right]$, and (d) magneto-optic azimuth rotation, ${\theta }_r^{(05)}$, and ellipticity, ${\epsilon }_r^{(05)}$at 10% magnetic saturation.

Download Full Size | PDF

 

Fig. 7 Effect of the thickness of the Fe layer, d⁽³⁾, on the parameters of the structure Au(5 nm)/AlN(d⁽²⁾)/Fe(15 nm)/AlN(20 nm)/Au(1000 nm): (a) power reflection (R), transmission (T), and absorption (A) coefficients, (b) off-diagonal amplitude reflection coefficient$r_{yx}^{(05)}=\left|r_{yx}^{(05)}\right|\exp \left[{\text{j}}^{}\text{arg}\left(r_{yx}^{(05)}\right)\right]$, (c) diagonal amplitude reflection coefficient$r_{xx}^{(05)}=\left|r_{xx}^{(05)}\right|\exp \left[{\text{j}}^{}\text{arg}\left(r_{xx}^{(05)}\right)\right]$, and (d) magneto-optic azimuth rotation, ${\theta }_r^{(05)}$, and ellipticity, ${\epsilon }_r^{(05)}$at 10% magnetic saturation.

Download Full Size | PDF

 

Fig. 8 Effect of the thickness of the Fe layer, d⁽³⁾, on the parameters of the structure Au(5 nm)/AlN(73 nm)/Fe(15 nm)/AlN(d⁽⁴⁾)/Au(1000 nm): (a) power reflection (R), transmission (T), and absorption (A) coefficients, (b) off-diagonal amplitude reflection coefficient$r_{yx}^{(05)}=\left|r_{yx}^{(05)}\right|\exp \left[{\text{j}}^{}\text{arg}\left(r_{yx}^{(05)}\right)\right]$, (c) diagonal amplitude reflection coefficient $r_{xx}^{(05)}=\left|r_{xx}^{(05)}\right|\exp \left[{\text{j}}^{}\text{arg}\left(r_{xx}^{(05)}\right)\right]$, and (d) magneto-optic azimuth rotation, ${\theta }_r^{(05)}$, and ellipticity, ${\epsilon }_r^{(05)}$at 10% magnetic saturation.

Download Full Size | PDF

From the inspection of Tables 3 and 4, we observe an enhancement of $\left|r_{yx}^{(05)}\right|$ by a factor of 2.64 with respect to air/Fe(1000 nm) interface, a value slightly reduced with respect to that achieved in the absence of the protecting Au layer (d⁽¹⁾ = 0). The addition of the protecting Au layer may considerably reduce R_min, corresponding to “antireflection,” and enhance in this way magneto-optic azimuth rotation, ${\theta }_r^{(05)}$, and ellipticity, ${\epsilon }_r^{(05)}.$ The situation is illustrated in Table 4 for the structure Au(5 nm)/AlN(75 nm)/Fe(18 nm)/AlN(12 nm)/Au(1000 nm). The modifications of d⁽³⁾ and d⁽⁴⁾ led to $\left|r_{yx}^{(05)}\right|$ reduced from the maximal value of 0.0172 by less than 1% to $\left|r_{yx}^{(05)}\right|\approx 0.0171,$ only. A comparison of the structures AlN(61 nm)/Fe(1000 nm)/Au(1000 nm) and AlN(57 nm)/Fe(23 nm)/Au(1000 nm) in Table 3 with Au(5 nm)/AlN(82 nm)/Fe(1000 nm)/Au(1000 nm) and Au(5 nm)/AlN(78 nm)/Fe(26 nm)/Au(1000 nm) in Table 4, respectively, represent another examples. A similar enhancement of the MO azimuth rotation was reported by Querishi et al. on Au/SiO₂/Ni structures [21].

5. Discussion

The figure of merit, $\left|r_{yx}^{(05)}\right|,$ was evaluated for the case of a complete saturation of ultrathin Fe perpendicular to the interfaces, which would require the perpendicular field, $B_{\perp }\ge {\mu }_0{M}_s.$ On the other hand, the MO probes are considered to detect much lower $B_{\perp }$ at which both precessional angles of FMR and MO angles ${\theta }_r^{(05)}$ and ${\epsilon }_r^{(05)}$remain within the range of linearity in $B_{\perp }$. In the ideal case, the maximal figure of merit ${\left|r_{yx}^{(05)}\right|}_{\text{max}}$of Eq. (4) is achieved where an incident x-polarized wave $\left({\chi }_i^{(05)}\to 0\right)$ is completely converted to a MO reflected y-polarized wave $\left({\chi }_r^{(05)}\to \infty \right)$ corresponding to $\left|r_{xx}^{(05)}\right|\to 0,$ i.e., the maximum of $\left|r_{yx}^{(05)}\right|$ and the minimum of $\left|r_{xx}^{(05)}\right|$ coincide and the corresponding MO azimuth rotation reaches 90 degrees. There would be no reflected wave with the incident polarization (R = 0) and there would be no power leak, into the optically thick reflector (T = 0). Because of our restriction to five-layer structures T≠0 and the positions of the maximal $\left|r_{yx}^{(05)}\right|$and those of the minimal $\left|r_{xx}^{(05)}\right|$as well as those of the maximal A are slightly displaced. The maximum of $\left|r_{yx}^{(05)}\right|$is broad, while the minimum of $\left|r_{xx}^{(05)}\right|$ is sharp. Around the minimum of $\left|r_{xx}^{(05)}\right|$, i.e., around the “antireflection,” R_min, the ratio${\chi }_r^{(05)}$varies rapidly with d⁽²⁾, d⁽³⁾, or d⁽⁴⁾ producing sharp maxima and minima of ${\theta }_r^{(05)}$ and ${\epsilon }_r^{(05)}$ often exceeding the region of linearity in Eq. (23) and Eq. (24). This illustrates the important effect of $r_{xx}$ on the observed quantities ${\theta }_r^{(05)}$and ${\epsilon }_r^{(05)}$which should be accounted for even in the ultrathin film limit where $\left|r_{yx}\right|$ can be assumed as linear in the thickness of the magnetic layer [24]. In multilayer structures with ultrathin Fe, the roughness and mixing at interfaces, layer uniformity, etc., do not allow to achieve R_min as deep as predicted by the calculations [6,13,14]. The MO azimuth rotation of 90 degrees were observed on single crystals only where a nearly perfect atomic layering may produce antireflection and ${\chi }_r=r_{yx}/r_{xx}>1$ [30]. For the present purpose, it was then sufficient to evaluate the d⁽²⁾, d⁽³⁾, or d⁽⁴⁾ dependence of the parameters, including $r_{xx},$ in 1 nm steps and evaluate ${\theta }_r^{(05)}$, and ${\epsilon }_r^{(05)}$at 10% saturation magnetization using the proportionality ${\chi }_r^{(05)}\propto r_{yx}^{(05)}\propto {\epsilon }_{yx}^{(3)}\propto {\mu }_0{M}_s^{\text{(Fe)}}.$Our focus was not on ${\theta }_r^{(05)}$ and ${\epsilon }_r^{(05)}$ but on the figure of merit, $\left|r_{yx}^{(05)}\right|$, as the most important parameter for the probe operation. This can alternatively be evaluated using the first order expression of Eq. (17) with a practically sufficient precision. The corresponding expression for transmission,$\left|t_{yx}^{(05)}\right|$, given in Eq. (22), can be used in the same way.

The search for structures with optimized MO response in terms of $\left|r_{yx}^{(05)}\right|$ was originally motivated by mapping of mw currents. The present approach is potentially useful in a variety of other situations, where contactless and non-invasive MO detection of currents, magnetic fields or field gradients may be of interest, e.g., in the MO magnetometry of magnetic multilayers which may find applications in magnetic recording or spintronics [31–33]. The expressions for a five-layer system in Eqs. (10)-(22) were given here for the case of normal light incidence. This is not only the simplest but also the most important configuration as the response at the normal to the surface (polar) magnetization becomes here maximal. Equations (10)-(22) provide valuable results even at non-zero small angles of incidence, at least below 10 degree and can be easily extended to structures with up to five magnetic layers. The MO transmission case of Section 3 may be applied, e.g., in the design of a MO probe deposited on the tip of an optical fiber. The expressions can be extended to include an arbitrary orientation of magnetization and arbitrary angles of incidence [34].

Due to the chemical instability of pure Fe, special measures must be used to imbed an ultrathin Fe into a dielectric resonator. The present results (Table 3 and 4) indicate that the resort to structures with a single dielectric layer and ultrathin Fe may already provide a reasonable performance with $\left|r_{yx}^{(05)}\right|$ reduced by a few percent only with respect to that achieved in five-layer sandwiches. For example, in the structure AlN(57 nm)/Fe(23 nm)/AlN(0 nm)/Au(1000 nm), $\left|r_{yx}^{(05)}\right|=0.0172$ and in the structure with the Fe layer exposed to the ambient, AlN(0 nm)/Fe(6 nm)/AlN(56 nm)/Au(1000 nm), even $\left|r_{yx}^{(05)}\right|=0.0180$ (Table 3). The latter case is of less practical use but it suggests that a replacement of pure Fe with a chemically stable alloy, e.g., FePt, represents a promising solution. In addition, the reduced theoretical $\left|r_{yx}^{(05)}\right|$can be compensated by a better growth perfection achievable in simpler structures.

The illustrations were performed at the wavelength λ = 632.8 nm. The probe performance can be equally well evaluated in any other wavelength, e.g., in blue region with a better diffraction limited resolution. Here the ultrathin Fe with low ${\left|r_{yx}\right|}_{max}\approx 0.008$at λ = 310 nm would be better replaced, e.g., with an ultrathin FePt alloy.