Anisotropy of quadratic magneto-optic effects in reflection

Abstract

Quadratic or second-order magneto-optic effects in reflection significantly effect in-plane magnetization measurements. While the magneto-optic effects linear in magnetization are independent of orientation of cubic crystal axes, the amplitude and sign of the quadratic effects change significantly under crystal rotation. Theoretical formulas for the magneto-optic effects have been derived using a permittivity tensor including terms quadratic in magnetization. A method for separation of the diagonal and off-diagonal quadratic magneto-optic tensor components (G11−G12) and 2G44 is proposed. The theory was completed by an experimental observation of the quadratic effect anisotropy in an epitaxial Fe layer prepared on a MgO substrate. The influence of the magnetization components on the magneto-optic vector magnetometry is discussed for an interface, a single layer, and exchange coupled bilayer system for a general magnetization direction including the quadratic magneto-optic effect anisotropy.

Thickness dependence of linear and quadratic magneto-optical Kerr effects in ultrathin Fe(001) films

“…They depend on the sum of −K 2 / n 2 and G 11 − G 12 − ⌬G cos 2 ͑2␥͒. 13 In some cases the first term is much smaller than the second term; however for our system and experimental setup both terms have a comparable order of magnitude. Therefore, errors in the determination of n and K affect the accuracy of the G values.…”

“…By contrast, the quadratic coefficients are due to a combined effect of the linear and quadratic MO couplings and are anisotropic; i.e., they depend on the relative orientation of the sample with respect to the plane of incidence. For cubic systems the resulting q coefficients have been found to have the form 13 q 1 = q 001 + ͑q 011 − q 001 ͒sin 2 ͑2␥͒, ͑4͒…”

“…A wrong sign of the q 1 value will not simply lead to wrong signs, but to wrong values of the second-order MO-coupling constants. On the other hand, the q 2 coefficients stem purely from the G parameters, namely, from ⌬G / 2 sin͑4␥͒, 13 and vanish if the plane of incidence is parallel to the symmetry directions ͓001͔ and ͓011͔.…”

“…The parameter ⌬G = ͑G 11 − G 12 ͒ −2G 44 is a measure of the anisotropy strength. 13 The q 1 coefficients have an isotropic and an anisotropic contribution. They depend on the sum of −K 2 / n 2 and G 11 − G 12 − ⌬G cos 2 ͑2␥͒.…”

“…To our knowledge the only published value of a second-order MO-coupling constant is the imaginary part of G 44 / K of Ni of about Im͑G 44 / K͒ = −0.02 at the wavelength = 514 nm, which was experimentally determined from Brillouin light-scattering data by Giovannini et al 15 The second-order magnetooptical parameters, which are G 11 , G 12 , and G 44 for systems with cubic symmetry, give rise to MO effects quadratic and even in the magnetization M. These effects are known as quadratic magneto-optical Kerr effect ͑QMOKE͒ or Voigt effect in reflection, and have recently received a lot of attention. 1,[11][12][13][14] Effects quadratic in M also turn out to be important in magnetization-dependent second-harmonic generation ͑MSHG͒, 16 x-ray magnetic linear dichroism, and the closely related x-ray Voigt effect. 17 All the more it is surprising that to our knowledge no one has yet determined a full set of corresponding material parameters G to describe the QMOKE.…”