Determination of the infrared complex magnetoconductivity tensor in itinerant ferromagnets from Faraday and Kerr measurements

We present measurement and analysis techniques that allow the complete complex magnetoconductivity tensor to be determined from midinfrared ($11–1.6\mu \mathrm{m}$; $100–800\mathrm{meV}$) measurements of the complex Faraday $\left({\theta }_F\right)$ and Kerr $\left({\theta }_K\right)$ angles. Since this approach involves measurement of the geometry (orientation axis and ellipticity of the polarization) of transmitted and reflected light, no absolute transmittance or reflectance measurements are required. Thick-film transmission and reflection equations are used to convert the complex ${\theta }_F$ and ${\theta }_K$ into the complex longitudinal conductivity ${\sigma }_{xx}$ and the complex transverse (Hall) conductivity ${\sigma }_{xy}$. ${\theta }_F$ and ${\theta }_K$ are measured in a ${\mathrm{Ga}}_{1-x}{\mathrm{Mn}}_x\mathrm{As}$ and $\mathrm{Sr}\mathrm{Ru}{\mathrm{O}}_3$ films. The resulting ${\sigma }_{xx}$ is compared to the values obtained from conventional transmittance and reflectance measurements, as well as the results from Kramers-Kronig analysis of reflectance measurements on similar films.

Figure 1

(Color online) Overall schematic of the optical path. The sample stick is rotated to select either the sample or reference. The polarization change induced by compressively straining a ZnSe slide is shown in the boxed inset on the left.

Figure 2

(Color online) The mount that allows the $700\mathrm{lb}$ magneto-optical cryostat (labeled 1 in figure) to be translated vertically and horizontally with $\sim 25\mu \mathrm{m}$ precision. The knob (9) attached to the lead screw (10) for horizontal translation is indicated in the left part of the photograph. Vertical adjustment is made by pulling on the timing belt (5), which rotates the four timing pulleys (4) simultaneously, causing the large brass lead screws (3) to move the support legs (2) up or down.

Figure 3

Optical path of light passing through a thick film (left) with thickness $d$ on a wedged substrate. The multiply reflected beams in the film are drawn at non-normal angles for clarity.

Figure 4

(Color online) Energy dependence of (a) ${\theta }_F$ and (b) ${\theta }_K$ from a $\mathrm{Sr}\mathrm{Ru}{\mathrm{O}}_3$ film with the sample fully magnetized perpendicular to the plane at $0\mathrm{T}$ and $10\mathrm{K}$. Since the measurement is at $H=0\mathrm{T}$, the OHE as well as background signals from the substrate and windows, which are linear in $H$, do not contribute to the signal. Note the strong energy dependence and the sign changes in $\mathrm{Re}\left({\theta }_F\right)$ and $\mathrm{Im}\left({\theta }_K\right)$ at $250\mathrm{K}$ and $130\mathrm{meV}$, respectively. The ${\theta }_F(E\to 0)$ plotted in (a) is determined using Eq. (13) with the four-probe dc measurements for the dc ${\theta }_H$ and ${\sigma }_{xx}$. Inset shows the relationship between $\mathrm{Re}\left({\theta }_F\right)$, $\mathrm{Im}\left({\theta }_F\right)$, and $\mathrm{Re}\left({\theta }_K\right)$ (which is multiplied by a factor of 5) at $117\mathrm{meV}$ and $10\mathrm{K}$.

Figure 5

(Color online) The longitudinal conductivity ${\sigma }_{xx}$ (a) and transverse (AHE) conductivity ${\sigma }_{xy}$ (b) for a $\mathrm{Sr}\mathrm{Ru}{\mathrm{O}}_3$ film as a function of probe energy. The solid symbols are obtained from ${\theta }_F$ and ${\theta }_K$ measurements at $10\mathrm{K}$ and $0\mathrm{T}$ with the sample fully magnetized out of plane. The smaller symbols with error bars in (a) are determined from conventional transmittance and reflectance measurements of the same $\mathrm{Sr}\mathrm{Ru}{\mathrm{O}}_3$ film. The heavy solid $\left[\mathrm{Re}\left({\sigma }_{xx}\right)\right]$ and dashed $\left[\mathrm{Im}\left({\sigma }_{xx}\right)\right]$ lines in (a) are from Kramers-Kronig analysis of reflectance measurements at $35\mathrm{K}$ and $0\mathrm{T}$ on a different $\mathrm{Sr}\mathrm{Ru}{\mathrm{O}}_3$ sample by Schlesinger and co-workers.

Figure 6

(Color online) Energy dependence of the AHE (a) ${\theta }_F$ and (b) ${\theta }_K$ from a ${\mathrm{Ga}}_{1-x}{\mathrm{Mn}}_x\mathrm{As}$ film with the sample fully magnetized perpendicular to the plane at $0\mathrm{T}$ and $10\mathrm{K}$. Note the strong energy dependence and the sign changes in both the real and imaginary parts of ${\theta }_F$ and ${\theta }_K$. ${\theta }_F$ and ${\theta }_K$ exhibit qualitatively similar behavior in this case.

Figure 7

(Color online) The longitudinal conductivity ${\sigma }_{xx}$ (a) and transverse (AHE) conductivity ${\sigma }_{xy}$ (b) from a ${\mathrm{Ga}}_{1-x}{\mathrm{Mn}}_x\mathrm{As}$ film as a function of probe energy. The measurements are at $10\mathrm{K}$ and $0\mathrm{T}$ with the sample fully magnetized out of plane. The smaller symbols with error bars in (a) are determined from conventional transmittance and reflectance measurements of the same ${\mathrm{Ga}}_{1-x}{\mathrm{Mn}}_x\mathrm{As}$ film. The $E=0$ results are from dc measurements by Dubon and co-workers on similar samples, which were also grown by them.