1. INTRODUCTION

In this paper, we review and provide a tutorial on an emerging optical concept and methodology: the optical Hall effect [1–23]. By the first statement, the optical Hall effect measures the analogue of the quasi-static electric-field-induced electrical Hall effect at optical frequencies in conductive and complex structured materials. Advances in materials preparation and understanding of materials’ physical properties define today’s abilities in electrical devices: for example, in power generation [24], in power electronics and diagnostics components for the advanced electrical grid [25], in manufacturing [26], in numeric computation and processing [27], in solid-state lighting [28], and in 3D flash data storage [29]. In many if not all contemporary materials applications, properties of free charge carriers are crucial for choices of materials in device design and operation. The most prominent example, computer technology, has seen rapid development over the last few decades. The progress in this field is mainly based on two pillars: the miniaturization of transistor structures and the increase of processor speed. One of these pillars, the increase of processor frequencies, has only seen marginal improvements in the last 5 to 10 years. This is a consequence of the free charge carrier properties of the core material in semiconductor industries: silicon. Future increases of the clock rates of electronic device structures up to terahertz frequencies might only become possible by employing new materials with high breakdown voltages, large charge carrier saturation velocities, and high thermal stability [27]. Currently, two groups of materials, which are conveniently accessible through manufacturing, satisfy these requirements: group-III nitride semiconductor alloys [27,30] and graphene [31]. On the other hand, miniaturization of silicon-based structures is still following Moore’s law [32], reaching structure sizes of a few nanometers [33,34]. A side effect of the ongoing miniaturization is that, in devices with structure sizes of a few nanometers, free charge carriers will become more and more subject to quantum mechanical phenomena (particle in a box). Consequently, a better understanding of the high-frequency behavior of free charge carriers in continuum and quantum regimes in semiconductors such as silicon, in group-III nitride semiconductor, and in emerging 2D material such as graphene is essential for future development in computer technology. A key element is measurement of the free charge carrier parameters, effective mass, mobility, and density. In another example, group-III sesquioxides have regained interest as wide bandgap semiconductors with unexploited physical properties. The stable but highly anisotropic monoclinic $\beta$-gallia crystal structure ($\beta$ phase [35,36]) of ${\mathrm{Ga}}_2{\mathrm{O}}_3$ is of particular interest due to its large bandgap energy of 4.85 eV, lending promise for applications in short wavelength photonics and transparent electronics [37]. The high electric breakdown field value of $\beta -{\mathrm{Ga}}_2{\mathrm{O}}_3$, which is estimated at $8{\mathrm{MVcm}}^{-1}$, exceeds those of contemporary semiconductor materials such as Si, GaAs, SiC, group-III nitrides, or ZnO [38]. Recent reports on device characteristics indicate potential of $\beta -{\mathrm{Ga}}_2{\mathrm{O}}_3$ for use in high-power switches and transistors [38,39]. Details about free charge carrier properties in $\beta -{\mathrm{Ga}}_2{\mathrm{O}}_3$ are beginning to emerge [40,41].

Nondestructive and noninvasive measurement of the free charge carriers is not only vital for making progress in modern materials and device design but also constitutes a challenge. While the underlying principles of the optical Hall effect, the motion of free charge carriers within external magnetic fields, is not new, the interaction of electromagnetic waves with free charge carriers within conducting and semiconducting materials when subjected to arbitrarily oriented external magnetic fields offers vast new opportunities for investigating charge carriers in continuum and quantum regimes. The optical Hall effect can be conveniently exploited to characterize the electrical properties of materials. Brought upon the nature of the basic underlying measurement principle (ellipsometry [42–47]), the optical Hall effect can be studied in complex structured systems. Thereby free charge carrier properties become accessible in structures with 3D densities (bulk materials), 2D or sheet densities (ultra-thin layers), 1D densities (wires), or 0D densities (quantum dots). Furthermore, the optical Hall effect is capable of differentiating directionally dependent free charge carrier properties in structures made from anisotropic materials or structures that induce anisotropic properties by ordered arrangement of nanoscopic building blocks [48]. Anisotropy is inherent to many modern device architectures: for example, the family of electronic and optoelectronic devices fabricated from wurtzite-structure group-III nitrides [49], high-power electronic devices based on hexagonal silicon carbide [23], or the envisioned class of high-voltage high-power devices based on monoclinic gallium oxide and related compounds [39,40]. In such materials, coupling with anisotropic longitudinal optical lattice modes and directionally dependent plasmon modes causes electrical transport characteristics, which depend on the direction of free charge carrier motion. The optical mobility parameter, a crucial element in device design for high-frequency operation, is affected primarily by phonon scattering, which is anisotropic. Above all, and aside from knowledge about densities and type of free charge carriers, the optical Hall effect offers, as the only available technique as of today, access to the determination of the effective mass tensor of the free charge carriers without a priori knowledge of intrinsic major axes of carrier displacement within a given material.

Traditionally, optical determination of the free charge carrier properties, particularly the effective mass parameter, has been performed by measuring the magneto-optic reflectance and/or transmittance at long wavelengths, as reported for example in [50–64]. Measurement of magnetic field induced polarization rotation [65,66], such as Faraday rotation (normal incidence transmission configuration, e.g., [67–69]) or Kerr rotation (normal incidence reflection configuration, e.g., [70]) can provide accurate information, but the approaches are limited to simple sample structures. Faraday rotation can only be measured in spectral regions of sufficient sample transparency. Cyclotron resonance occurs when the incident photon energy $\hslash \omega$ is equivalent to approximately the cyclotron energy, $\hslash {\omega }_{\mathrm{c}}=\frac{\hslash q|B|}{m}$, $\hslash \omega \approx \hslash {\omega }_c$, provided that the plasma broadening parameter ${\gamma }_p$ is small compared with ${\omega }_c$. This picture is correct for isotropic materials, where the cyclotron frequency is proportional to the magnitude of the magnetic field $\mathbf{B}$ and inversely proportional to the free charge carrier effective mass $m$. Typical frequencies of ${\omega }_c$ in semiconductors with free charge carriers are within the microwave region, where absorption features can be observed for $\omega \approx {\omega }_c$. Such experiments are typically performed at low temperatures to meet the condition ${\omega }_c\gg {\gamma }_p$ [66]. Hence, measurement of cyclotron resonance absorption in reflection and/or transmission can provide ${\omega }_c$ from which the effective mass can be obtained. From broadening of the resonance, the parameter ${\gamma }_p$ can be obtained, which can be related to the carrier scattering time. Terahertz (THz) measurements of cyclotron resonance in static or pulsed magnetic fields can be performed at fixed excitation frequencies from multiple microwave or laser line sources [55,71–73] or continuous femtosecond-laser-pumped THz time-domain spectroscopy (TDS) systems [74–76]. Laser-based THz TDS is a time-domain technique, which employs optical delay paths and laser-switched THz and far-infrared wave generation and detection, respectively, using photoconductive antenna configurations and nonlinear photosensitive detection materials [77–79]. In combination with static or pulsed high-value magnetic fields [80], TDS permits spectroscopic cyclotron resonance measurements [80–84]. The use of polarizing elements permits determination of the complex-valued Faraday and Kerr responses and access to THz-induced magneto gyrotropic and photoinduced conductivity effects [85–95]. TDS magneto-optic ellipsometric instrumentation and use for measurement of the optical Hall effect has been reported [96,97]. TDS optical Hall effect studies were reported on charge carrier systems in quantum Hall regimes [98,99] and using cavity coupling enhancement effects [23,100,101].

The optical Hall effect is introduced here as the physical phenomenon whereby the occurrence of magnetic field-induced anisotropy is observed, caused by the nonreciprocal [102] magneto-optic response of mobile electric charges [103–106]. The magneto-optic anisotropy observed in the optical Hall effect is produced by the motion of the free charge carriers and is thereby dependent on the strength and direction of the external magnetic field. This is conceptually different from anisotropy caused by spatially anisotropic molecular arrangements with (achiral) or without (chiral) mirror symmetry. The term “optical Hall effect” originates from its analogy to the electrical Hall effect [107]. Discovered by Edwin Herbert Hall (November 7, 1855–November 20, 1938) in 1879, the electrical Hall effect describes the observation of a potential difference across an electrical conductor, transverse to an electric current in the conductor and a magnetic field perpendicular to a current (Fig. 1) [108]. The electrical Hall effect and certain cases of the optical Hall effect phenomenon can be explained by extensions of the classic model for the transport of electrons in matter (metals) developed by Paul Drude [109,110]. Hence, we have adopted the term “optical Hall effect” for this associated optical phenomenon. An example of the effect of the induced anisotropy in the optical Hall effect is depicted in Fig. 2. An incident electromagnetic plane wave with linear polarization parallel to the surface of a sample subjected to an external magnetic field causes displacement of the free charge carriers along the direction of the electric field oscillation. The Lorentz force acts on this movement, which is zero at the time of the maximum amplitudes of the driving electric field and strongest at the reversal point. As a result, the motion of the free charge carrier deviates from a straight line and adopts a small circular component. The circular component only depends on the effective mass and the Fermi velocity of the free charge carrier and can be brought into resonance, which is then known as cyclotron resonance. However, for cyclotron resonance, the time between scattering events of the free charge carrier must be small compared with the turnaround time within the cyclotron orbit. Hence, cyclotron resonance is often measured in low-defect density materials to reduce impurity potential scattering and at very low temperatures to reduce phonon potential scattering [71,103,111–115]. Regardless of resonance conditions, and even for very short free charge carrier scattering times, the reflected (or transmitted) electromagnetic wave in the example in Fig. 2 now contains a small fraction of circularly polarized light. The strength and handedness of this component are directly analogous to the transverse potential difference measured in the electrical Hall effect. This example can be conceptually repeated for any polarization state of an incident electromagnetic wave as well as for any orientation and strength of the magnetic field. Thereby, magneto-optic anisotropy induced for all Cartesian directions as well as for all conceivable phases (left or right handed elliptical or circular polarizations) can be detected. In this manner, the optical Hall effect extends the electrical Hall effect to a truly 3D phenomenon and dispenses with the requirement of an ideal sheet with infinitesimal small thickness. As will be discussed further below, for the optical Hall effect, the classic Drude model is extended by a magnetic field and frequency dependency, describing a free charge carrier’s momentum and motion under the influence of the Lorentz force. As a result, an antisymmetric contribution to the dielectric polarizability density, whose sign depends on the type of the free charge carrier (electron or hole), is then augmented onto the dielectric tensor $ϵ(\omega )$. The nonvanishing off-diagonal elements of the dielectric tensor reflect the frequency-dependent magneto-optic anisotropy, which lead to conversion of $p$-polarized into $s$-polarized electromagnetic waves and vice versa. The magnitude and dispersion of this $p$- and $s$-polarization mode conversion is a precise fingerprint of the density, mobility, and effective mass properties of the free charge carriers in a given sample. Thus, analysis of optical Hall effect data provides insight into the high-frequency properties of free charge carriers in complex layered samples [1–4,21,116], grants access to effective mass parameters [1,7,8,10,12,14–17], and can be used to study quantum mechanical effects [6,20,117].

 

Fig. 1. Schematic of the electrical Hall effect in a thin conducting sheet. Free charge carriers produce a transverse voltage by charge separation under the influence of the Lorentz force due to a magnetic field $\mathbf{B}$ when driven by a DC current through the sheet. Accordingly, the longitudinal voltage required to drive the DC current differs with and without the magnetic field. Note that, ideally, the sheet should be infinitesimally thin [107]. The electric Hall voltage is characteristic of the sheet material and depends on the free charge carrier types and density properties that constitute the current leading to the Hall voltage. Note that transport must occur homogeneously across the entire sheet, which makes analyses of the electric Hall voltage measured across complex sheet structures with multiple constituents difficult.

Download Full Size | PDF

 

Fig. 2. Schematic of the optical Hall effect in a conducting sample consisting of single or multiple conducting layers and sheets, in reflection configuration. Free charge carriers produce a dielectric polarization following the electric field of an incident electromagnetic field (analogous to the longitudinal Hall voltage), here for example parallel to the surface. The induced polarization ${\mathbf{P}}_{\mathbf{FCC}}$ produces ${\mathbf{P}}_{\mathbf{Hall}}$ due to the Lorentz force, oriented perpendicular to $\mathbf{B}$ and the incident electric field vector (analogous to the transverse Hall voltage). ${\mathbf{P}}_{\mathbf{FCC}}+{\mathbf{P}}_{\mathbf{Hall}}$ are the source of the reflected light and contain a small circular polarization component, which provides information on the type of charge carrier, its density, mobility, and effective mass properties [1]. The physical motion of the charges remains local within the lattice of the material, describing pathways that depend on the Fermi velocity, their average scattering time, the frequency of the incident light, and the magnetic field direction. If multiple layers are thin enough against the skin depth at long wavelengths, light interacts with multiple layers and reveals, for example, free carrier properties within buried layers otherwise inaccessible to direct electrical measurements. Hence, the optical Hall effect can be measured across complex sheet and layer structures.

Download Full Size | PDF

The optical Hall effect can be measured in terms of the so-called Mueller matrix [118–120], which characterizes the transformation of an electromagnetic wave’s polarization state [121]. Experimentally, the Mueller matrix is measured by generalized ellipsometry [1,46,47,122–133]. During a generalized ellipsometry measurement, different polarization states of the incident light are prepared, and their change upon reflection from or transmission through a sample is determined. Thus, an optical Hall effect instrument is an instrument capable of conducting generalized ellipsometry measurements with the samples exposed to high, quasi-static magnetic fields, and detects magnetic-field-induced changes of the Mueller matrix [13,22]. So far optical Hall effect instruments are not commercially available. Ellipsometry instrumentation for the terahertz spectral range recently became commercially available (2012), while instruments at far-infrared spectral range are not commercially available. Therefore, the relatively new optical Hall effect technique [1] is still exotic. We reported recently on an optical Hall effect instrument covering the spectral range from $3{\mathrm{cm}}^{-1}$ to $7000{\mathrm{cm}}^{-1}$ (0.1–210 THz or 0.4–870 meV) by combining MIR ($600–7000{\mathrm{cm}}^{-1}$), FIR ($30–650{\mathrm{cm}}^{-1}$), and THz ($3.3–33{\mathrm{cm}}^{-1}$) magneto-optic generalized ellipsometry in a single instrument. This optical Hall effect incorporates a commercially available, closed-cycle refrigerated, superconducting 8 Tesla magneto cryostat, with four optical ports, providing sample temperatures between $T=1.4\mathrm{K}$ and room temperature. The ellipsometer subsystems were built in-house and operate in the rotating-analyzer configuration, which is capable of determining the normalized upper $3\times 3$ block of the sample Mueller matrix [22].

2. OPTICAL HALL EFFECT IN MATERIALS

In this section we will showcase simple models that can explain the occurrence of the optical Hall effect in materials. These theories address changes in the dielectric function tensor and their wavelength dependencies under the influence of an external magnetic field. Without loss of generality, we address materials whose free charge carriers may interact with polar lattice vibrations or with internal electric fields, for example. We only consider here the dielectric optical Hall effect, that is, magnetic-field-induced anisotropy within the dielectric tensor [134].

A. Optical Hall Effect Tensor Definition

The optical Hall effect tensor may be defined as the dielectric tensor $ϵ$ under the influence of an external magnetic field, $ϵ(\mathbf{B})$. $ϵ$ is a measure for the optical response of a medium and can be defined by the electric displacement field $\mathbf{D}$, which is an auxiliary quantity used in the Maxwell equations. The electric displacement field describes the electric flux density at the surface of a medium and can be written as

In general, the dielectric function tensor, which comprises then all linear dielectric responses of the material, is composed of its symmetric part and its antisymmetric part:

It is often desirable to identify the physical mechanisms that cause an optical Hall effect observation. For this purpose, identification of the parts of $ϵ$ that depend on the external field and those that do not can be useful:

The term ${\chi }_{\mathbf{B}=0}$ may comprise all contributions in Eq. (2) that are not affected by a given magnetic field: for example, lattice vibrations. The term ${\chi }_{\pm \mathbf{B}}$ comprises then all contributions in Eq. (2) that are affected by a given magnetic field, for example, due to polarization caused by free charge carriers or by electronic level transitions. It is important to note that ${\chi }_{\pm \mathbf{B}}$ is composed of symmetric and antisymmetric parts. The antisymmetric part of ${\chi }_{\pm \mathbf{B}}$ vanishes for $\mathbf{B}=0$. The symmetric part of ${\chi }_{\pm \mathbf{B}}$ may not necessarily vanish for $\mathbf{B}=0$ and is only distinguishable from ${\chi }_{\mathbf{B}=0}$ at $\mathbf{B}\ne 0$. These symmetry properties inspire procedures for measurement of the optical Hall effect tensor where data are obtained at $\mathbf{B}=0$ and $\pm \mathbf{B}$. Combinations of these data allow us to differentiate between the symmetric and antisymmetric changes in $ϵ$ with $\mathbf{B}$.

B. Optical Hall Effect Tensor Diagonalization

For the optical Hall effect, it is often useful to find the eigenvalues of the optical Hall effect tensor. The eigenvalues are functions of frequency and are rendered by complex-valued response functions. Examples will be given further below. Note that we use the convention of positive notation for the imaginary part of the complex-valued eigenfunctions ${\chi }_{{\xi }_k},{\chi }_{{\eta }_k},{\chi }_{{\varsigma }_k}$. This choice results in positive imaginary parts of the four complex-valued indices of refraction [45,137,138]. The eigenvalues and the symmetry properties of the optical Hall effect tensor often hint at the mechanisms that may cause the observed optical Hall effect. Two transformations must be discussed: spatial rotations ${\mathbf{A}}^{(\mathrm{R})}$ and decompositions using circularly (${\mathbf{A}}^{(\mathrm{C})}$), elliptically (${\mathbf{A}}^{(\mathrm{E})}$), or generally (${\mathbf{A}}^{(\mathrm{G})}$) polarized eigenvectors. The goal is to diagonalize by transformation, representing the optical Hall effect tensor in an appropriate coordinate system (eigensystem). Conceptually, such transformation from one eigensystem $(\xi ,\eta ,\varsigma )$ into the laboratory coordinate system $(x,y,z)$ may exist for each of the contributions to the electric susceptibility tensor, ${ϵ}_{\xi ,\eta ,\varsigma }\stackrel{\mathbf{A}}{\to }{ϵ}_{x,y,z}$:

1. Spatial Rotations

An explicit presentation of spatial rotations is given here using the $z-x^{\prime }-z^{\prime \prime }$ convention. In the $z-x^{\prime }-z^{\prime \prime }$ convention, the first rotation is performed around the $z$ axis by the Euler angle $\varphi$, the coordinate system is then rotated by the Euler angle $\theta$ around the new $x^{\prime }$ axis, and finally a rotation by the Euler angle $\psi$ around the new $z^{\prime \prime }$ axis is performed:

A rotation ${\mathbf{A}}_{\varphi ,\theta ,\psi }^{(\mathrm{R})}$ to diagonalize $ϵ$ can always be found for symmetric tensors. A necessary condition for the underlying structure to represent an orthogonal system of electric susceptibilities, $\mathbf{A}$ must be wavelength independent. Major dielectric functions only can be obtained for materials with cubic, hexagonal, trigonal, tetragonal, and orthorhombic crystal systems [139]. Such functions can no longer be meaningfully defined for materials with monoclinic and triclinic crystal systems; instead, one must consider the major dielectric polarizability functions and their eigenvectors [40,47]. A coordinate transformation of an electric susceptibility tensor from its diagonal form, using ${\mathbf{A}}_{\varphi ,\theta ,\psi }^{(\mathrm{R})}$, always results in a fully symmetric electric susceptibility and therefore a fully symmetric dielectric tensor. For real-valued arguments of ${\mathbf{A}}_{\varphi ,\theta ,\psi }^{(\mathrm{R})}$, the rotation corresponds to a true physical rotation, such as an azimuthal rotation of a sample between successive measurements, or to represent the actual surface orientation of an anisotropic material in an optical Hall effect experiment.

2. Decompositions Using Magneto-optic Eigenvectors

An ad hoc assumption for the form of the dielectric tensor of a material subjected to a static magnetic field is that of a nonreciprocal medium. As will be shown below, a nonreciprocal response leads to anisotropic optical properties [47,140,141].

Circular eigenvector decomposition (C): The magneto-optic anisotropy can be modeled by assuming different interactions for right- and left-handed circularly polarized electromagnetic plane waves within a material, traveling parallel to the magnetic field orientation [1,13] (Fig. 3). In this Ansatz, and without loss of generality, if the quasi-static magnetic field $\mathbf{B}$ is pointing in the $z$ direction, the magnetic-field-induced contribution to the displacement phasor field vector $\mathbf{P}$ can be expressed by a pair of electric susceptibility functions, ${\chi }_{+}$ and ${\chi }_{-}$ [6,9]:

 

Fig. 3. Left-handed [${\mathbf{E}}_{+}$, Figs. 3(a) and 3(c)] and right-handed [${\mathbf{E}}_{-}$, Figs. 3(b) and 3(d)] circularly polarized electromagnetic plane waves interact with a dielectrically polarizable material under the influence of an external quasi-static magnetic field $\mathbf{B}$. The field $\mathbf{B}$ is collinear with the wave propagation direction. The displacement field phasors ${\mathbf{P}}_{\pm }$ are proportional to complex-valued, frequency-dependent response functions ${\chi }_{\pm }(\mathbf{B})$. Symmetry requires switch of indices upon reversal of the magnetic field: ${\chi }_{\pm }(\mathbf{B})={\chi }_{\mp }(-\mathbf{B})$. The latter statement originates from the assumption that ${\mathbf{P}}_{\pm }$ does not depend on propagation direction of ${\mathbf{E}}_{\pm }$ but only on the course of the electric field phasor at a given plane within the material. Functions ${\chi }_{\pm }$ then determine the symmetric and antisymmetric parts in the optical Hall effect tensor [Eq. (11)].

Download Full Size | PDF

In Fig. 3 and Eq. (8) the electric field $\mathbf{E}$ is given by [142]

A transformation matrix can be found that projects from the eigensystem of ${\mathbf{P}}_{\mathrm{C}}$ into the Cartesian laboratory system [6,9,143]:

Hence, the electric susceptibility tensors $\chi$ can be transformed into the Cartesian laboratory system using ${\mathbf{A}}^{(\mathrm{C})}$:

Equation (11) reveals important properties of $\chi (\pm \mathbf{B})$, which affect symmetry properties of the optical Hall effect data:

Elliptical eigenvector decomposition (E): In more general cases, $\chi$ may no longer be diagonalized using an eigensystem of circular polarization ${\mathbf{A}}^{(\mathrm{C})}$. Instead, an eigensystem of two orthogonal elliptically polarized electromagnetic waves has to be chosen:

General eigenvector decomposition (G): In general, three different magneto-optic susceptibility functions may exist ${\lambda }_1$, ${\lambda }_2$, ${\lambda }_3$, which may fully characterize the physical origin of the optical Hall effect. The circular and elliptical decomposition discussed above is possible for as long as there is no coupling between polarization processes parallel and perpendicular to the magnetic field and, hence, one of the functions ${\lambda }_1$, ${\lambda }_2$, ${\lambda }_3$ is unity. However, in situations with anisotropic materials (for example, when charge carrier effective mass and/or mobility are rendered by tensors instead of isotropic scalars), these parameters can differ along certain axes of a given material. The external magnetic field can further take an arbitrary orientation relative to these axes and coupling of displacement within the plane perpendicular to the magnetic field and the direction along the magnetic field occurs. Then the three functions ${\lambda }_1$, ${\lambda }_2$, ${\lambda }_3$ may all differ from unity. One can make use of the eigenvectors ${\mathrm{\Xi }}_{ϵ}$ obtained from the eigenvalue equation of the optical Hall effect tensor $ϵ(\mathbf{B})$:

A transformation matrix can then be constructed that projects from the magneto-optic eigensystem into the Cartesian laboratory system:

C. Optical Hall Effect Tensor Energy Conservation Conditions

The introduction of the eigensystem polarizations ${\lambda }_1$, ${\lambda }_2$, ${\lambda }_3$ requires consideration of energy conservation. Energy conservation in linear optics is commonly assured by requiring that the imaginary parts of the response functions are positive. Whether the imaginary parts of ${\lambda }_1$, ${\lambda }_2$, ${\lambda }_3$ are positive or negative, in particular, is not immediately obvious. To begin with, we show that the product of ${\lambda }_1$, ${\lambda }_2$, ${\lambda }_3$ is equal to the product of the dielectric tensor eigenvalues within the laboratory coordinate system:

Note that the equivalence holds for both real and imaginary part of the product ${\lambda }_1{\lambda }_2{\lambda }_3$. Thermodynamically, the dielectric tensor has to comply with the law of energy conservation. For example, in case of an isotropic medium in thermodynamic equilibrium, the dielectric function $ϵ$ has to have a positive imaginary part, $\mathfrak{I}\mathrm{m}(ϵ)\ge 0$, to ensure that the energy of absorbed light is converted into heat [145]. With restrictions to monochromatic electromagnetic waves, a more general expression can be derived for the optical Hall effect tensor. In general, the time-averaged change of the electromagnetic energy density $⟨\frac{\partial u}{\partial t}⟩$ in an anisotropic media with dielectric tensor $ϵ$, exposed to a monochromatic electromagnetic plane wave with angular frequency $\omega$, electric field $\mathbf{E}={\mathbf{E}}_0{e}^{-\mathrm{i}\omega t}$, and magnetic field $\mathbf{H}={\mathbf{H}}_0{e}^{-\mathrm{i}\omega t}$, has to be positive:

D. Optical Hall Effect Tensor Models

1. Lorentz-Drude Model (Classical Mechanics Approach)

Charge carriers, subject to a slowly varying magnetic field, obey the classical Newtonian equation of motion (Lorentz–Drude model) [115]:

With the Ansatz of a time harmonic electromagnetic plane wave with an electric field $\mathbf{E}\to \mathbf{E}\exp (-\mathrm{i}\omega t)$ (phasor) with angular frequency $\omega$, the time derivative of the spatial displacement of the charge carrier $\dot{\mathbf{x}}=\mathbf{v}$ also becomes time harmonic $\mathbf{v}\to \mathbf{v}\exp (-\mathrm{i}\omega t)$, where $\mathbf{v}$ stands for the drift velocity of the charge carrier. Introducing the current density, $\mathbf{j}=Nq\mathbf{v}$, Eq. (19) reads:

Polar lattice vibrations (Lorentz oscillator): For conveniently achievable magnetic field strengths (10–15 T), the mass of the vibrating atoms of polar lattice vibrations render the field leading term in Eq. (22) small compared with the mass leading term and can be neglected. Therefore, the dielectric tensor of polar lattice vibrations ${ϵ}^{\mathrm{L}}$ can be approximated using Eq. (22) with $B=0$. Hence, ${\chi }_{+}={\chi }_{-}$, and ${\chi }_{\pm \mathbf{B}}^{\mathrm{L}}=0$. When assuming isotropic constitutive parameters ($\{X_{ii}=X_{jj}\wedge X_{ij}=0\}\forall i\ne j$, for $X=\{m,{\omega }_0,\gamma \}$), the result is a simple harmonic oscillator function with Lorentzian-type broadening, ${ϵ}^{\mathrm{L}}$ [66,115,148].

For materials with orthorhombic crystal system, the effective mass, eigenfrequency, and mobility tensors typically have the same eigensystem. The dielectric tensor can in this case be diagonalized to

If this class of materials possesses multiple optical excitable lattice vibrations, the diagonal elements ${ϵ}_{\mathrm{k}}^{\mathrm{L}}$ ($k=\{x,y,z\}$) can be expressed by [149,150]

Free charge carriers (extended Drude model): For free charge carriers, no restoring force is present, and the eigenfrequency tensor of the system is ${\mathit{\omega }}_0=0$. For isotropic effective mass and conductivity tensors, and magnetic fields aligned along the $z$ axis, Eq. (22) can be written in the form

2. Landau Level Model (Quantum Mechanics Approach)

Absorption of light by free charge carriers changes their momentum and is affected by the Lorentz force in the presence of a magnetic field. If the free charge carrier scattering time is high enough, cyclotron orbits of electrons (holes) in 2D confinement quantize into Landau levels. Such levels are characterized by certain allowed orbits within momentum space. Absorption of light can only occur by transitions between Landau levels and must obey optical selection rules: for example, $|n^{\prime }|=|n|\pm 1$ for transitions between levels with numbers $n$ and $n^{\prime }$ [60,155,156]. Landau level quantization can, for example, appear in decoupled graphene mono-layers, coupled graphene mono-layers, and graphite at low temperatures. The absorption of light due to the transition of an electron between discrete energy levels, e.g., inter-Landau level transitions, can be described by Fermi’s golden rule. At a given temperature $T\ne 0$ each Landau level possesses a mean lifetime ${\tau }_k=1/{\gamma }_k$. Therefore, the spectral function describing the absorption process of a series of inter-Landau level transitions can be written as a sum of Lorentz oscillators. The quantities ${\chi }_{\pm }$ in Eq. (11) can be expressed by

3. OPTICAL HALL EFFECT IN SAMPLES WITH PLANE INTERFACES

Measurement of the optical Hall effect will be discussed in a forthcoming second part of this paper. Here we introduce the concepts required for setup of optical Hall effect instrumentation and data analysis. The interaction of light with a specimen subjected to magnetic fields, internal or external to a given specimen, can be described by the Maxwell’s postulates. Conveniently accessible experimental conditions involve plane electromagnetic waves and samples with plane surfaces and interfaces. The interaction of the light can then be cast into either a field-phasor description (Jones vector approach) or an intensity description (Stokes vector approach). The concept that permits both experimental and theoretical access to sample descriptive parameters (Jones or Mueller matrix elements) is magneto-optic generalized ellipsometry. For samples with plane surfaces and interfaces, the optical Hall effect can be measured in reflection or transmission and at oblique and/or normal incidence. The normal incidence situations are identical to the traditional Faraday (normal transmission) and Kerr (normal reflection) magneto-optic configurations. The Faraday and Kerr configurations can therefore be regarded as special cases of the optical Hall effect. At oblique incidence one gains two advantages: first, the equality between $p$ and $s$ polarization is removed providing added information. Second, light propagation at various directions can be imposed within the sample. Thereby, the tensor elements of the optical Hall effect, which relate to polarization properties perpendicular to the sample surface, can be measured.

A. Jones and Mueller Matrix Calculus

Two conceptually different mathematical approaches are useful to connect experimental optical Hall effect data with model calculations. The Jones calculus and the Mueller–Stokes calculus assume that all electromagnetic interactions with optical instrument components and the sample are linear in the electromagnetic field amplitudes.

1. Jones Formalism

Linear interactions of a plane electromagnetic wave with an object, such as a sample (Fig. 4), can be characterized by a matrix $\mathbf{J}$ for the electric field vectors [164] ${\mathbf{E}}^{\mathrm{in}}$ and ${\mathbf{E}}^{\mathrm{out}}$:

The matrix $\mathbf{J}$, called Jones matrix [165], is a dimensionless, complex-valued $2\times 2$ matrix, and can be written as

The four matrix elements $j_{ij}$ are better known as the Fresnel coefficients [121] for polarized light. For example, the individual (e.g., reflection, Fig. 4) coefficients are defined by

 

Fig. 4. Wave vector ${\mathbf{k}}^{\mathrm{in}}$ of the incoming electromagnetic plane wave and the sample normal $\mathbf{n}$ define the angle of incidence $\mathrm{\Phi }$ and the plane of incidence. The amplitudes of the electric field of the incoming ${\mathbf{E}}^{\mathrm{in}}$ and the reflected ${\mathbf{E}}^{\mathrm{out}}$ plane wave can be decomposed into complex field amplitudes $E_p^{\mathrm{in}}$, $E_s^{\mathrm{in}}$, $E_p^{\mathrm{out}}$, and $E_s^{\mathrm{out}}$, where the indices $p$ and $s$ stand for parallel and perpendicular to the plane of incidence, respectively.

Download Full Size | PDF

2. Mueller–Stokes Formalism

Instead of electric field amplitudes, the Mueller–Stokes formalism describes the transformation of the polarization state based on time-averaged polarized intensities. The polarization state is determined by the real-valued, $4\times 1$ Stokes vector $\mathbf{S}$ [168]. The Stokes vector can be obtained from time averages over products of the electric field components in terms of the $p$- and $s$-coordinate system

3. Jones to Mueller Matrix Transformation

Any Jones matrix can be converted into a Mueller matrix; the inversion, however, is not possible in all cases. Individual Mueller matrix elements can be calculated from the Jones matrix by [167]

The resulting Mueller matrix can be expressed as the sum of two matrices $\mathbf{M}={\mathbf{M}}_{\mathrm{is}}+{\mathbf{M}}_{\mathrm{an}}$, with ${\mathbf{M}}_{\mathrm{is}}$ including only terms independent of $j_{ps}$ and $j_{sp}$. With the reflection case as an example,

Equations (43) and (44) display that the Mueller matrix can be decomposed into four sub-matrices, where the matrix elements of the two off-diagonal blocks $\left[\begin{array}{cc}{M}_{13}& M_{14}\\ M_{23}& M_{24}\end{array}\right]$ and $\left[\begin{array}{cc}{M}_{31}& M_{32}\\ M_{41}& M_{42}\end{array}\right]$ only deviate from zero if $p$- and $s$-polarization mode conversion appears, that is, $r_{ps}\ne 0$ and $r_{sp}\ne 0$. The matrix elements in the two on-diagonal blocks $\left[\begin{array}{cc}{M}_{11}& M_{12}\\ M_{21}& M_{22}\end{array}\right]$ and $\left[\begin{array}{cc}{M}_{33}& M_{34}\\ M_{43}& M_{44}\end{array}\right]$ are typically different from zero and contain information about $p$- and $s$-polarization mode conserving processes.

4. Optical Hall Effect Mueller Matrix

The Mueller matrix of a sample consisting of multiple ($k$) constituents of dielectric materials with dielectric function tensor $ϵ$ and subjected to a magnetic field may be written as

Explicit expressions for the elements of ${\mathbf{M}}^{\mathbf{B}}$ are complex and intricate and depend on many parameters and experimental circumstances. A matrix formalism is described further below, which allows for convenient calculation of ${\mathbf{M}}^{\mathbf{B}}$. The decomposition $\mathbf{M}={\mathbf{M}}_{\mathrm{is}}+{\mathbf{M}}_{\mathrm{an}}$ is conceptually important when inspecting changes of the elements upon field reversal. For example, when the optical Hall effect tensor is diagonal, the field-induced changes in the off-diagonal blocks are zero.

5. Faraday and Kerr Rotations

In the literature, magneto-optic effects are often quantified in terms of the Faraday or Kerr rotation in case of transmission- or reflection-type experiments, respectively (see, e.g., [56,62,68,86,103,114,171–173], and references in Section 1). The Faraday and Kerr rotations are the simplest cases where, experimentally, magneto-optic effects can be accessed and quantified. These cases establish the optical Hall effect at normal incidence. In both cases a sample is exposed to a homogeneous quasi-static magnetic field, and linear polarized light is sent onto the sample. After interaction with the sample, the light becomes elliptically polarized due to the magneto-optic birefringence. The Faraday or Kerr angle is defined as the angle a linear polarizer must be oriented in the reflected/transmitted beam with respect to the incoming polarization direction in order to detect maximum signal. The incoming polarization direction can be arbitrarily chosen but must remain fixed during the procedure of finding the Faraday or Kerr angle. In the Mueller matrix formalism, this angle can be expressed generally for both Faraday and Kerr rotation configurations, by elements of the optical Hall effect Mueller matrix $M_{ij}^{\mathbf{B}}$:

B. $\mathbf{4}\times \mathbf{4}$ Matrix Formalism

The Jones and Mueller matrix formalisms describe the changes of polarization, observable by magneto-optic generalized ellipsometry from the external perspective to a given sample. The sample internal processes leading to the external change in the polarization state can be treated conveniently by a $4\times 4$ matrix formalism. Extending and generalizing the work by Berreman [174], a $4\times 4$ matrix formalism was introduced [124], which enables fast computational modeling of generalized ellipsometry parameters for arbitrary anisotropic media [45–47,175]. Quintessential to Schubert’s version of the $4\times 4$ formalism is the replacement of the first-order differential equation:

 

Fig. 5. Schematic presentation, under the incoming angle $\mathrm{\Phi }$, of the electromagnetic wave ${\mathbf{E}}^{\mathrm{I}}$, and the reflected ${\mathbf{E}}^{\mathrm{R}}$, transmitted ${\mathbf{E}}^{\mathrm{T}}$, and backward-traveling electromagnetic waves ${\mathbf{E}}^{\mathrm{B}}$ used in the $4\times 4$ matrix formalism. The medium into which the wave is reflected (transmitted) is labeled R (T). Between the media R and T, $n$ slabs of parallel layers with homogenous optical properties may be located. Backward-traveling waves ${\mathbf{E}}^{\mathrm{B}}$ in the medium T are permitted. Plane ($x$, $y$) is parallel to the interfaces/surfaces; $z$ points into the surface. The surface is the interface against which the incoming beam is directed.

Download Full Size | PDF

The elements of the incident and exit matrix ${({\mathbf{L}}_{\mathrm{R},\mathrm{T}})}_{i,j}={({\mathrm{\Xi }}_{\mathrm{R},\mathrm{T}})}_{ji}$ are composed of eigenvectors ${\mathbf{\Xi }}_{\mathrm{R},\mathrm{T}}$ of matrices ${\mathbf{\Delta }}_{\mathrm{R},\mathrm{T}}$ for incident and exit mediums, respectively. Matrix $\mathbf{\Delta }$ is the characteristic matrix of a given homogeneous medium, defined through Eq. (47):

The angle ${\mathrm{\Phi }}_{\mathrm{T}}$ under which the electromagnetic plane wave is transmitted into medium T is given by

The partial transfer matrices of the layers $k$ with thickness $d_k$ are obtained from a serial expansion of the matrix ${\mathbf{\Delta }}_k$ for layer $k$:

The complex scalars ${\beta }_{jk}(j=0\dots 3)$ are defined by (the index $k$ is dropped) [176]