Abstract

The optical Hall effect is a physical phenomenon that describes the occurrence of magnetic-field-induced dielectric displacement at optical wavelengths, transverse and longitudinal to the incident electric field, and analogous to the static electrical Hall effect. The electrical Hall effect and certain cases of the optical Hall effect observations can be explained by extensions of the classic Drude model for the transport of electrons in metals. The optical Hall effect is most useful for characterization of electrical properties in semiconductors. Among many advantages, while the optical Hall effect dispenses with the need of electrical contacts, electrical material properties such as effective mass and mobility parameters, including their anisotropy as well as carrier type and density, can be determined from the optical Hall effect. Measurement of the optical Hall effect can be performed within the concept of generalized ellipsometry at an oblique angle of incidence. In this paper, we review and discuss physical model equations, which can be used to calculate the optical Hall effect in single- and multiple-layered structures of semiconductor materials. We define the optical Hall effect dielectric function tensor, demonstrate diagonalization approaches, and show requirements for the optical Hall effect tensor from energy conservation. We discuss both continuum and quantum approaches, and we provide a brief description of the generalized ellipsometry concept, the Mueller matrix calculus, and a 4×4 matrix algebra to calculate data accessible by experiment. In a follow-up paper, we will discuss strategies and approaches for experimental data acquisition and analysis.

© 2016 Optical Society of America

1. INTRODUCTION

In this paper, we review and provide a tutorial on an emerging optical concept and methodology: the optical Hall effect [123]. 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 β-gallia crystal structure (β phase [35,36]) of Ga2O3 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 βGa2O3, which is estimated at 8MVcm1, 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 βGa2O3 for use in high-power switches and transistors [38,39]. Details about free charge carrier properties in βGa2O3 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 [4247]), 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 [5064]. Measurement of magnetic field induced polarization rotation [65,66], such as Faraday rotation (normal incidence transmission configuration, e.g., [6769]) 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 ω is equivalent to approximately the cyclotron energy, ωc=q|B|m, ωωc, provided that the plasma broadening parameter γp is small compared with ωc. This picture is correct for isotropic materials, where the cyclotron frequency is proportional to the magnitude of the magnetic field B and inversely proportional to the free charge carrier effective mass m. Typical frequencies of ωc in semiconductors with free charge carriers are within the microwave region, where absorption features can be observed for ωωc. Such experiments are typically performed at low temperatures to meet the condition ωcγp [66]. Hence, measurement of cyclotron resonance absorption in reflection and/or transmission can provide ωc from which the effective mass can be obtained. From broadening of the resonance, the parameter γp 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 [55,7173] or continuous femtosecond-laser-pumped THz time-domain spectroscopy (TDS) systems [7476]. 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 [7779]. In combination with static or pulsed high-value magnetic fields [80], TDS permits spectroscopic cyclotron resonance measurements [8084]. 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 [8595]. 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 [103106]. 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 [71,103,111115]. 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(ω). 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 [14,21,116], grants access to effective mass parameters [1,7,8,10,12,1417], and can be used to study quantum mechanical effects [6,20,117].

 figure: Fig. 1.

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 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.

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 figure: Fig. 2.

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 PFCC produces PHall due to the Lorentz force, oriented perpendicular to B and the incident electric field vector (analogous to the transverse Hall voltage). PFCC+PHall 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.

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The optical Hall effect can be measured in terms of the so-called Mueller matrix [118120], which characterizes the transformation of an electromagnetic wave’s polarization state [121]. Experimentally, the Mueller matrix is measured by generalized ellipsometry [1,46,47,122133]. 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 3cm1 to 7000cm1 (0.1–210 THz or 0.4–870 meV) by combining MIR (6007000cm1), FIR (30650cm1), and THz (3.333cm1) 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=1.4K 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 3×3 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(B). ϵ is a measure for the optical response of a medium and can be defined by the electric displacement field 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

D=ϵ0E+P=ϵ0E+χE=ϵ0(I+χ)E=ϵ0ϵE,
where ϵ0, E, P, and χ denote the electric vacuum permittivity, electric field vector, electric polarization vector, and electric susceptibility tensor of the medium, respectively [135]. If the optical response of the material is linear, the total dielectric tensor can be written as the sum of electric susceptibility tensors:
ϵ=I+χ=I+kχk,
where each χk may describe an independent mechanism of polarization within the medium: for example, long wavelength active phonon modes or electronic band-to-band transitions [136]. The electric susceptibility and dielectric tensor are second-rank tensors. In Cartesian coordinates (x,y,z), the dielectric tensor takes the form
ϵ(x,y,z)=(ϵxxϵxyϵxzϵyxϵyyϵyzϵzxϵzyϵzz)=I+(χxxχxyχxzχyxχyyχyzχzxχzyχzz).

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:

ϵij=δij+12(χij+χji)+12(χijχji),
where δij is the Kronecker symbol. Processes that are symmetric in time and space only produce contributions to the symmetric part. Processes that involve symmetry breaking in time such as by magnetic fields, or symmetry breaking in space such as by chiral arrangement of matter produce nonsymmetric contributions. In this work we ignore phenomena due to chiral arrangement of matter.

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:

ϵ(±B)=I+χB=0+χ±B.

The term χB=0 may comprise all contributions in Eq. (2) that are not affected by a given magnetic field: for example, lattice vibrations. The term χ±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 χ±B is composed of symmetric and antisymmetric parts. The antisymmetric part of χ±B vanishes for B=0. The symmetric part of χ±B may not necessarily vanish for B=0 and is only distinguishable from χB=0 at B0. These symmetry properties inspire procedures for measurement of the optical Hall effect tensor where data are obtained at B=0 and ±B. Combinations of these data allow us to differentiate between the symmetric and antisymmetric changes in ϵ with 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 χξk,χηk,χςk. 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 A(R) and decompositions using circularly (A(C)), elliptically (A(E)), or generally (A(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 (ξ,η,ς) into the laboratory coordinate system (x,y,z) may exist for each of the contributions to the electric susceptibility tensor, ϵξ,η,ςAϵx,y,z:

ϵx,y,z=A1ϵξ,η,ςA=I+kA1(χξk000χηk000χςk)A,
where k denotes the index for each independent mechanism of polarization within the medium, A is the invertible transformation matrix with (x,y,z)=A(ξ,η,ς)T, and χξk,χηk,χςk are the electric susceptibilities, or eigenvalues, of the kth independent mechanism of polarization in the corresponding eigensystem. A special case is optically isotropic materials with χξk=χηk=χςk (for all k). Because the transformation matrices are invertible, the dielectric tensors take the same shape in the laboratory coordinate system and the eigensystem ϵx,y,z=ϵξ,η,ς. Therefore, the dielectric tensor can in this case be replaced by the scalar dielectric function ϵ, with ϵ=ϵI.

1. Spatial Rotations

An explicit presentation of spatial rotations is given here using the zxz convention. In the zxz convention, the first rotation is performed around the z axis by the Euler angle ϕ, the coordinate system is then rotated by the Euler angle θ around the new x axis, and finally a rotation by the Euler angle ψ around the new z axis is performed:

Aϕ,θ,ψ(R)=(cosψsinψ0sinψcosψ0001)(1000cosθsinθ0sinθcosθ)(cosϕsinϕ0sinϕcosϕ0001).

A rotation Aϕ,θ,ψ(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, 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 Aϕ,θ,ψ(R), always results in a fully symmetric electric susceptibility and therefore a fully symmetric dielectric tensor. For real-valued arguments of Aϕ,θ,ψ(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 B is pointing in the z direction, the magnetic-field-induced contribution to the displacement phasor field vector P can be expressed by a pair of electric susceptibility functions, χ+ and χ [6,9]:

PC=(χ+000χ0001)EC.

 figure: Fig. 3.

Fig. 3. Left-handed [E+, Figs. 3(a) and 3(c)] and right-handed [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 B. The field B is collinear with the wave propagation direction. The displacement field phasors P± are proportional to complex-valued, frequency-dependent response functions χ±(B). Symmetry requires switch of indices upon reversal of the magnetic field: χ±(B)=χ(B). The latter statement originates from the assumption that P± does not depend on propagation direction of E± but only on the course of the electric field phasor at a given plane within the material. Functions χ± then determine the symmetric and antisymmetric parts in the optical Hall effect tensor [Eq. (11)].

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In Fig. 3 and Eq. (8) the electric field E is given by [142]

EC=(E+EEz)=(12(ExiEy)12(Ex+iEy)Ez).

A transformation matrix can be found that projects from the eigensystem of PC into the Cartesian laboratory system [6,9,143]:

A(C)=12(1i01i0002).

Hence, the electric susceptibility tensors χ can be transformed into the Cartesian laboratory system using A(C):

χ(±B)=12((χ++χ)i(χ+χ)0±i(χ+χ)(χ++χ)0000).

Equation (11) reveals important properties of χ(±B), which affect symmetry properties of the optical Hall effect data:

  • χ is completely symmetric when electric susceptibilities for left- and right-handed circularly polarized light are equal, i.e., when χ+=χ.
  • χ is completely antisymmetric when electric susceptibilities for left- and right-handed circularly polarized light differ only in their sign, i.e., when χ+=χ.
  • χ changes sign of its skew symmetric part upon magnetic field inversion.

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

A(E)=1aa1(a10a11000aa1),
where a=tanΨEeiΔE, and parameters ΨE and ΔE characterize the ellipticity of the orthogonal modes of the elliptic eigensystem, a is the complex conjugate of a [144]. Such situations occur, for example, when free charge carrier properties are anisotropic within the plane perpendicular to the magnetic field. The magnetic-field-induced contribution to the dielectric polarization vector P can then be expressed again by a pair of electric susceptibility functions χ±.

General eigenvector decomposition (G): In general, three different magneto-optic susceptibility functions may exist λ1, λ2, λ3, 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 λ1, λ2, λ3 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 λ1, λ2, λ3 may all differ from unity. One can make use of the eigenvectors Ξϵ obtained from the eigenvalue equation of the optical Hall effect tensor ϵ(B):

ϵΞk=λkΞk,k=1,2,3;
the electric field can be written on the basis of orthogonal eigenvectors:
EG=a1Ξ1+a2Ξ2+a3Ξ3.

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

A(G)=1det(A(G))(Ξ1,xΞ1,yΞ1,zΞ2,xΞ2,yΞ2,zΞ3,xΞ3,yΞ3,z).

C. Optical Hall Effect Tensor Energy Conservation Conditions

The introduction of the eigensystem polarizations λ1, λ2, λ3 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 λ1, λ2, λ3 are positive or negative, in particular, is not immediately obvious. To begin with, we show that the product of λ1, λ2, λ3 is equal to the product of the dielectric tensor eigenvalues within the laboratory coordinate system:

0=det(ϵ)λ1λ2λ3=det(AϵA1)λ1λ2λ3=det(A)det(ϵ)det(A1)λ1λ2λ3=det(ϵ)λ1λ2λ3.

Note that the equivalence holds for both real and imaginary part of the product λ1λ2λ3. 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, Im(ϵ)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 ut in an anisotropic media with dielectric tensor ϵ, exposed to a monochromatic electromagnetic plane wave with angular frequency ω, electric field E=E0eiωt, and magnetic field H=H0eiωt, has to be positive:

ut=Re[divS]=ED˙*+E*D˙+HB˙*+H*B˙=ωiEϵ*E*iE*ϵE+iHμ*H*iH*H=ωIm[E0*ϵE0]+ωIm[H0*μH0]0,
where the field amplitudes E0 and H0 carry the spatial dependency, S is the Poynting vector, ϵ is the dielectric tensor, and μ is the magnetic permeability tensor [145]. Assuming a nonmagnetic material with μ=μ0I (μ0: magnetic vacuum permeability) H0*μH0 becomes real valued, and the second term vanishes. Replacing the electric field with its orthogonal eigenvectors Ξk basis presentation, and with ω0 and because of (akΞk)(akΞk)*0(k=1,2,3), the inequality in Eq. (17) is fulfilled if all imaginary parts of the eigenvalues of the optical Hall effect tensor, λ1,λ2,λ3, are non-negative [146]:
Im[λk]0,
and the optical Hall effect tensor does not violate the conservation of energy.

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]:

mx¨+mγx˙+mω02x=qE+q(x˙×B),
where x is the carrier’s spatial coordinate. The remaining constitutive parameters are the effective mass tensor (m), the Newtonian friction tensor (γ), the eigen-resonance frequency tensor of the undamped system without external magnetic field (ω0), and the carrier’s charge (q). A useful abbreviation is the optical mobility tensor:
μopt=qm1γ1.

With the Ansatz of a time harmonic electromagnetic plane wave with an electric field EEexp(iωt) (phasor) with angular frequency ω, the time derivative of the spatial displacement of the charge carrier x˙=v also becomes time harmonic vvexp(iωt), where v stands for the drift velocity of the charge carrier. Introducing the current density, j=Nqv, Eq. (19) reads:

E=1Nq[imqω(ω02ω2Iiωγ)j+(B×j)],
where N is the charge density parameter. With the Levi–Cevita symbol ϵijk, the conductivity tensor σ, the dielectric constant ϵ0, and using E=σ1j and ϵ=I+σiϵ0ω, the contribution χD to the dielectric tensor of free charge carriers subject to the external magnetic field B can be expressed as [147]
χikLD=Nq2ϵ0[mik(ω0,ik2ω2iωγik)iωϵijkqBj]1.

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 ϵL can be approximated using Eq. (22) with B=0. Hence, χ+=χ, and χ±BL=0. When assuming isotropic constitutive parameters ({Xii=XjjXij=0}ij, for X={m,ω0,γ}), the result is a simple harmonic oscillator function with Lorentzian-type broadening, ϵL [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

ϵL=I+χL=(ϵxL000ϵyL000ϵzL).

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

ϵkL=1+χkL=ϵ,kj=1lω2+iωγLO,k,jωLO,k,j2ω2+iωγTO,k,jωTO,k,j2,
where χkL, ϵ,k, ωLO,k,j, γLO,k,j, ωTO,k,j, and γTO,k,j denote the k={x,y,z} component of the electric susceptibility of polar lattice vibration, the high-frequency dielectric constant, the frequency and broadening parameters of the jth longitudinal optical (LO), and transverse optical (TO) phonon modes, respectively, while the index j runs over l modes [8,149154].

Free charge carriers (extended Drude model): For free charge carriers, no restoring force is present, and the eigenfrequency tensor of the system is ω0=0. For isotropic effective mass and conductivity tensors, and magnetic fields aligned along the z axis, Eq. (22) can be written in the form

χD=χB=0D+χ±BD,
with the Drude contribution to the dielectric tensor for B=0
χB=0D=ωp2ω(ω+iγ)I=χDI,
where the isotropic plasma frequency is defined as
ωp=Nq2mϵ0,
and χD is the isotropic Drude contribution to the dielectric function for B=0. Using Eq. (11), the magneto-optic contribution χ±BD to the dielectric tensor for isotropic effective masses and conductivities can be expressed through susceptibility functions for right- and left-handed circularly polarized light:
χ±=χD1ωcω+iγ,
where the isotropic cyclotron frequency is defined as
ωc=q|B|m.

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|=|n|±1 for transitions between levels with numbers n and n [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 T0 each Landau level possesses a mean lifetime τk=1/γk. 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 χ± in Eq. (11) can be expressed by

χ±=e±iϕkAkω0,k2ω2iγkω,
where Ak, ω0,k, and γk are the amplitude, energy, and broadening parameters of the kth transition, respectively, which in general depend on the magnetic field. The phase factor ϕ was introduced empirically here to describe the experimentally observed line shapes of the optical Hall effect in graphite and graphene. For example, transitions in graphite or bi-layer graphene are best described by ϕ=π/4, and ϕ=0 for transitions in single-layer graphene [20,157]. Note that for ϕ=0, the polarizabilities for left- and right-handed circularly polarized light are equal (χ+=χ), and χ±BLL is diagonal [158]. The Hamiltonian for conduction band electrons in graphite with effective mass m [159], situated in the parabolic energy bands, at a magnetic field strength B perpendicular to the plane of confinement, is equivalent to the Hamiltonian of the quantum mechanical oscillator. Therefore, the resulting eigenvalues of the Hamiltonian are linear in B:
EGraphiteLL(n,B)=ωc(n+n0),
where e, , n, n0=12, and ωc=emB denote the elementary charge, the reduced Planck constant, the Landau level number, the factor for the zero-point energy, and the cyclotron frequency, respectively [145,148,160]. The eigenvalues of the Hamiltonian of massless fermions in single-layer graphene [161], decoupled graphene sheets [162], and the mono-layer-like branch of the eigenvalues of the Hamiltonian of Bernal-stacked N-layer graphene with an odd number of layers [60] depend on |B|:
ESLGLL(n,B)=sign(n)E0|n|,
with E0=c˜2e|B|, where c˜ is the average Fermi velocity. The bi-layer-like branch of the eigenvalues of the Hamiltonian of Bernal-stacked N-layer graphene follows a sublinear behavior in B [57,60,163]:
ENBLGLL(n,μ,B)=sign(n)12[(λNγ)2+(2|n|+1)E02+μ(λNγ)4+2(2|n|+1)E02(λNγ)2+E04]1/2,
with an inter-mono-layer coupling constant γ, layer number parameter λN [60], and where μ=1,+1 corresponds to the higher and lower subbands in the limit of zero magnetic field, respectively [60].

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 J for the electric field vectors [164] Ein and Eout:

Eout=JEin.

The matrix J, called Jones matrix [165], is a dimensionless, complex-valued 2×2 matrix, and can be written as

J=(jppjpsjspjss).

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

rpp=(EpoutEpin)Esin=0,rps=(EsoutEpin)Esin=0,rsp=(EpoutEsin)Epin=0,rss=(EsoutEsin)Epin=0.
where Epin, Esin, Epout, and Esout are the projections of the electric field vectors into the plane parallel (p) and perpendicular (s) to the plane of incidence of the incoming (in) and outgoing (out) wave. For completely polarized light and nondepolarizing interactions, the Jones matrix represents a complete mathematical description of any nondepolarizing transformation of the polarization state of a plane electromagnetic wave, e.g., on a surface [46,166,167].

 figure: Fig. 4.

Fig. 4. Wave vector kin of the incoming electromagnetic plane wave and the sample normal n define the angle of incidence Φ and the plane of incidence. The amplitudes of the electric field of the incoming Ein and the reflected Eout plane wave can be decomposed into complex field amplitudes Epin, Esin, Epout, and Esout, where the indices p and s stand for parallel and perpendicular to the plane of incidence, respectively.

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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, 4×1 Stokes vector 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

(S1S2S3S4)=(EpEp*+EsEs*EpEp*EsEs*EpEs*+Ep*Esi(EpEs*Ep*Es))=(Ip+IsIpIsI+45I45Iσ+Iσ)
and, thus, provides experimental access through quantities Ip, Is, I+45, I45, Iσ+, and Iσ, which denote the intensities for the p-, s-, +45°, 45°, right- and left-handed circularly polarized light components, respectively [46,169]. The Mueller matrix M is a convenient mathematical construct that transforms Stokes vectors S [45,46,118,170]:
Sout=MSin,
where Sout and Sin denote the Stokes vectors of the electromagnetic plane wave before and after the transformation. The Mueller matrix is a dimensionless, real-valued, 4×4 matrix:
M=(M11M12M13M14M21M22M23M24M31M32M33M34M41M42M43M44).

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]

Mij=12Tr(JσiJσj),
where J is the Hermitian conjugate of the Jones matrix, and σi is a set of 2×2 matrices comprising of the unity matrix and the Pauli matrices [167]:
σ1=(1001),σ2=(1001),
σ3=(0110),σ4=(0ii0).

The resulting Mueller matrix can be expressed as the sum of two matrices M=Mis+Man, with Mis including only terms independent of jps and jsp. With the reflection case as an example,

Mis=(12(rpprpp*+rssrss*)12(rpprpp*rssrss*)0012(rpprpp*rssrss*)12(rpprpp*+rssrss*)0000Re(rpprss*)Im(rpprss*)00Im(rpprss*)Re(rpprss*)),
and Man including all terms dependent on rps and rsp,
Man=(12(rpsrps*+rsprsp*)12(rpsrps*rsprsp*)Re(rpprps*+rssrsp*)Im(rpprps*rssrsp*)12(rpsrps*rsprsp*)12(rpsrps*+rsprsp*)Re(rpprps*rssrsp*)Im(rpprps*+rssrsp*)Re(rpprsp*+rssrps*)Re(rpprsp*rssrps*)Re(rpsrsp*)Im(rpsrsp*)Im(rpprsp*rssrps*)Im(rpprsp*+rssrps*)Im(rpsrsp*)Re(rpsrsp*)).

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 [M13M14M23M24] and [M31M32M41M42] only deviate from zero if p- and s-polarization mode conversion appears, that is, rps0 and rsp0. The matrix elements in the two on-diagonal blocks [M11M12M21M22] and [M33M34M43M44] 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

MB=M(ϵB0(k)).

Explicit expressions for the elements of MB 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 MB. The decomposition M=Mis+Man 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., [56,62,68,86,103,114,171173], 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 MijB:

φFK=12arctan[M31B+M32Bcos[2β]+M33Bsin[2β]tan[2β](M21B+M22Bcos[2β]+M23Bsin[2β])M21B+M22Bcos[2β]+M23Bsin[2β]±tan[2β](M31B+M32Bcos[2β]+M33Bsin[2β])],
where β is the angle of the input polarization with respect to the p direction of the coordinate system into which MB is cast. The upper or lower signs in Eq. (46) stand for Faraday or Kerr angles, respectively. Equation (46) is the most general description of Faraday and Kerr rotation angles. It covers the possibilities that the sample itself is anisotropic. The equation also is valid for arbitrary orientation of the magnetic field. Note further that it is often assumed for a Faraday or Kerr rotation measurement that the sample itself is not anisotropic. Faraday and Kerr rotation measurements convolute the information from multiple Mueller matrix elements MijB into one result. A deconvolution is difficult, in general, and Faraday or Kerr rotation measurements may provide insufficient insight into the cause of a particular optical Hall effect. An example is discussed further below for the occurrence of Faraday rotation at Landau level transitions in graphene [86].

B. 4×4 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 4×4 matrix formalism. Extending and generalizing the work by Berreman [174], a 4×4 matrix formalism was introduced [124], which enables fast computational modeling of generalized ellipsometry parameters for arbitrary anisotropic media [4547,175]. Quintessential to Schubert’s version of the 4×4 formalism is the replacement of the first-order differential equation:

Ψz=iωcΔΨ,
for the electromagnetic fields components Ψ=(Ex,Ey,Hx,Hy)T within a plane (x, y) at arbitrary z, by the transfer matrix equation:
(EpIEsIEpREsR)=L(EpTEsTEpBEsB),
for the electric field amplitudes Ep (Es) parallel (perpendicular) to the plane of incidence, of the incoming (I), reflected (R), transmitted (T), and backward-traveling (B) electromagnetic waves (Fig. 5). The medium in which the reflected electromagnetic plane wave travels shall be called R (complex index of refraction nR), the medium in which the transmitted wave travels T (nT). Between medium R and T m layers with parallel interfaces and homogenous optical properties are embedded. For optically isotropic media R and T, the complex-valued 4×4 transfer matrix L can be expressed as the product
L=LR1(k=1mLPk)LT.

 figure: Fig. 5.

Fig. 5. Schematic presentation, under the incoming angle Φ, of the electromagnetic wave EI, and the reflected ER, transmitted ET, and backward-traveling electromagnetic waves EB used in the 4×4 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 EB 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.

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The elements of the incident and exit matrix (LR,T)i,j=(ΞR,T)ji are composed of eigenvectors ΞR,T of matrices ΔR,T for incident and exit mediums, respectively. Matrix Δ is the characteristic matrix of a given homogeneous medium, defined through Eq. (47):

Δ=(kxϵzxϵzzkxϵzyϵzz01kx2ϵzz0010ϵyx+ϵyzϵzxϵzzkx2ϵyy+ϵyzϵzyϵzz0kxϵyzϵzzϵxxϵxzϵzxϵzzϵxyϵxzϵzyϵzz0kxϵxzϵzz),
where kx=nRsinΦ is the projection of the wave vector of the incoming electromagnetic plane wave onto the x axis. For isotropic materials,
LR1=12(01(nRcosΦ)1001(nRcosΦ)10(cosΦ)100nR1(cosΦ)100nR1),
LT=(00cosΦTcosΦT1100nTcosΦTnTcosΦT0000nTnT).

The angle ΦT under which the electromagnetic plane wave is transmitted into medium T is given by

cosΦT=1(nR/nT)2sin2Φ.

The partial transfer matrices of the layers k with thickness dk are obtained from a serial expansion of the matrix Δk for layer k:

LPk=exp(iωcΔkdk)=j=03βjkΔkj.

The complex scalars βjk(j=03) are defined by (the index k is dropped) [176]

βn=j=03αnexp(iωqj(d)/c)(qjqk)(qjql)(qjqm),
with the parameters
α0=qkqlqm,α1=qkql+qkqm+qlqm,α2=(qk+ql+qm),α3=1,
with {k,l,m}={0,1,2,3}{j} [177]. The four complex-valued eigenvalues of the matrix Δ for layer k are denoted as qj (j=03). Two eigenvalues qj, associated with the eigenmodes Ξj within each layer k, have positive real parts and correspond to the transmitted (forward traveling) electromagnetic plane. Accordingly, the two eigenvalues with negative real parts belong to backward-traveling electromagnetic waves. The eigenvalues qj (j=03) are the key to using the matrix formalism for calculating optical Hall effect Mueller matrix elements. Explicit solutions for these eigenvalues were provided in [124] when the dielectric tensor elements are symmetric (no magneto-optic effects) and in [178] for dielectric tensor elements, which are fully nonsymmetric (including magneto-optic effects). The eigenvalues qj (j=03) are further key to calculating matrix elements (LR,T)i,j when either or both incident and exit medium consist of materials, which reveal magneto-optic properties. For example, the case of a substrate-ambient situation is calculated by Eq. (49) with all partial transfer matrices replaced by unity matrices, and matrix LT contains any of the dielectric tensor models described above via characteristic matrix Δ and its subsequently derived eigenvectors (ΞT)ji.

The complex Fresnel reflection coefficients [Eq. (36)] [179] are calculated from the elements Lij of the transfer matrix L in Eq. (49):

rpp=L11L43L13L41L11L33L13L31,rps=L33L41L31L43L11L33L13L31,rsp=L11L23L13L21L11L33L13L31,rss=L33L21L31L23L11L33L13L31.

Using the 4×4 matrix formalism, it can be shown that, if the dielectric tensors of all k layers have diagonal shape ϵlm=0 (with lm), the off-diagonal elements of the Jones matrix vanish, i.e., rps=rsp=0. Thus, isotropic media and anisotropic media in special measurement configurations (crystallographic or magnetic field orientations) exhibit no p- or s-polarization mode conversion. Therefore, the p- and s-polarization mode conversion describing Jones matrix ellipsometry parameters vanish Ψps=Ψsp=0. Accordingly, all elements in Man vanish.

C. Example: The Optical Hall Effect at Normal Incidence in Graphene

Explicit expressions for description of the Mueller matrix elements MB are lengthy and may be cumbersome to obtain. However, it is insightful to derive such expressions, in particular for comparatively simple cases. An example is included here for description of the optical Hall effect on a 2D gas of free charge carriers. A nearly ideal realization of such system is single-layer graphene. Depending on the level of the Fermi energy, the system is composed of free electrons or free holes and may translate laterally under the influence of electric fields with large scattering times and hence small plasma broadening. If brought into an external magnetic field, quantization into Landau levels occurs. As a result, carriers can only uptake energy by transitions into higher Landau levels, separated by energy quanta, which reveal, for example, the linear dispersion of holes and electrons in graphene in the vicinity of the Fermi energy. It is insightful to use the above-described model system and derive explicit expressions for the Faraday rotation angle. The 4×4 matrix algorithm is exploited for this purpose where one partial transfer matrix is evaluated to represent the single-layer graphene. When conducted in the far- to mid-infrared spectral range, the thickness dgraphene of the transfer matrix model layer is much smaller than the wavelength λ. Therefore the partial transfer matrix is most easily obtained by simple linearization in Eq. (54). Matrices LR,T are as shown above.

A key question is which model system to select. Ideally, symmetric bands for holes and electrons should permit Landau level transitions in single-layer graphene with equal probability for left- and right-handed circularly polarized light. As discussed in Section 2.D.2, the polarizabilities for left- and right-handed circularly polarized light are equal (χ+=χ--), and χ±BLL is diagonal. Accordingly, at normal incidence, no Faraday or Kerr rotation should be observable in single-layer graphene. This is intriguing because clear experimental evidence was recently shown by Crassee et al. [86]. On the other hand, oblique angle of incidence optical Hall effect measurements provided clear evidence that Landau-level transitions in single-layer graphene are independent on polarization, as shown by Kühne et al. [20]. Indeed, if the Landau-level model described in Section 2.D.2 is implemented for the dielectric function tensor of the graphene layer, at normal incidence, all elements of Man vanish, and the Faraday/Kerr rotation is zero regardless of B. However, as detected and discussed by Kühne et al. [20], a certain amount of free charge carriers remains unaffected by confinement into Landau levels, and a Drude term needs to be augmented to the model system. Then, M21=M31=0, M22=±M33, and M32=M23, where the upper stands for transmission through (Faraday) and the lower for reflection from (Kerr) the sample. The Faraday/Kerr angle can then be expressed as

φ=12arctan[M23M33].

A sheet carrier density can be introduced Ns=Ndgraphene where N is the volume carrier density, and the meaning of the plasma frequency parameter can be redefined as that of a 2D sheet (ωpωpdgraphene). As a result, in the limit for dgraphene0, the Faraday/Kerr rotation angle for single-layer graphene at normal incidence can be expressed as

φ=±12arctan[2ωcRe[χBeff(1+nTiχLL*)]γ|χBeff|2|1+nT+i((ω+iγ)χBeff+χLL)|2|χBeff|2],
where nT is the index of refraction of the isotropic substrate and
χBeff=ωcω+iγ(ω+iγ)2ωc2,χB=0D=ωp2c1(ω+iγ)2ωc2.

It is then clear that the Landau-level transitions in single-layer graphene contribute to the Faraday/Kerr rotation only because of the presence of free charge carriers. Because only one type of free charge carrier can be available in graphene (set by the location of the Fermi level), at long wavelength a small rotation angle is measurable. The rotation angle vanishes when the sheet density is zero. The Landau contributions only enter as product χBeffχLL. This is an interesting observation because it appears as if the species contributing to Landau transitions couple with the species that contribute to the plasma motion, while no coupling was explicitly introduced, except for merely adding the two contributions. Such addition is frequently used to render the actual physical coupling of longitudinal-optical phonon modes with plasmon modes in polar semiconductors with free charge carriers [40,148]. A similar coupling mechanism can be described here when inspecting the determinant of the inverse of the dielectric function tensor and which shall be the subject of future work.

4. SUMMARY

We provided a rationale for the inception of the optical Hall effect as a physical phenomenon, which describes the occurrence of magnetic-field-induced dielectric displacement at optical wavelengths analogous to the static electrical Hall effect. We presented an overview of approaches to model the optical Hall effect suitable for complex layered semiconductor materials. The optical Hall effect dispenses with the need for electrical contacts, and electrical material properties such as effective mass and mobility parameters, including their anisotropy as well as carrier type and density, can be obtained. We provided a review on the concept of generalized ellipsometry, which permits measurement of the optical Hall effect. In a forthcoming report, we will describe approaches and strategies in data acquisition and data analysis. We will review previous cases and discuss that spectroscopic data taken over large regions of the wavelength spectrum and at multiple angles of incidence, combined with variations of magnetic field strength and direction, for example, can provide unique sensitivity to volume or sheet charge density, optical mobility, effective mass, and signature (hole, electron) of free charge carriers. We believe that the physical model approach presented in this paper will stimulate development of further theories for emerging materials and device structures. We also envision the optical Hall effect to become a useful and widespread technique analogous to the electrical Hall effect.

Funding

National Science Foundation (NSF) (CMMI 1337856, DMR 1420645, EAR 1521428, EPS 1004094); Vetenskapsrådet (VR) (2010-3848, 2013-5580); Swedish Governmental Agency for Innovation Systems (2011-03486, 2014-04712); Swedish Foundation for Strategic Research (SSF) (FFL12-0181, RIF14-055); J. A. Woollam Foundation.

REFERENCES AND NOTES

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134. Conceptually, a magneto electric optical Hall effect also may exist, where a current driven by the time-harmonic electric field component, under the influence of an external magnetic field, produces in addition to, or separately from a magneto-optic dielectric displacement, a magnetization response.

135. The dielectric tensor is considered nonlocal in time but local in space, that is, frequency dependent but not wave vector dependent. A charged compressible fluid model resulting in a dielectric tensor for a nonlocal spatial response is described by Weiglhofer. In principle, the optical Hall effect should be observable in semiconductors with very large carrier concentrations where nonlocal spatial effects may need to be considered.

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141. A common requirement in theoretical studies of the electromagnetic response of matter consists in the imposition that a specific medium should be Lorentz reciprocal. For a dielectric medium (the magnetic susceptibility tensor being diagonal and unity), this means that the dielectric tensor is equal to its transposed form. The magnetized plasma and more general types of gyrotropic mediums belong to the most prominent representatives of nonreciprocal mediums. A gyrotropic material is a material in which left- and right-rotating elliptical polarizations can propagate at different speeds. The gyrotropic effect caused by a quasi-static magnetic field breaks the time-reversal symmetry as well as the Lorentz reciprocity. For more information see, for example, [102].

142. Corresponding expressions for arbitrary orientations of the magnetic field are given by EC=Aφ,θ,ψ(R)EC, with the Euler angles given by B|B|=Aϕ,θ,ψ(R)(0,0,1)T.

143. Corresponding transformation matrices for arbitrary orientations of the magnetic field are given by (A(C))=(Aϕ,θ,ψ(R))1A(C)Aϕ,θ,ψ(R), with the Euler angles given by B|B|=Aϕ,θ,ψ(R)(0,0,1)T.

144. For ΨE=π/4 and ΔE=π/2, the elliptic eigensystem is equivalent to the circular eigensystem A(E)=A(C).

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146. Note that Eq. (17) must hold for any E0; thus, Eq. (17) can be stated for each λk separately.

147. In the following equation the Einstein notation is used, and the covariance and contravariance are ignored because all coordinate systems are Cartesian. The summation is only executed over pairs of lower indices.

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151. M. Schubert, T. Hofmann, C. M. Herzinger, and W. Dollase, “Generalized ellipsometry for orthorhombic, absorbing materials: dielectric functions, phonon modes and band-to-band transitions of Sb2S3,” Thin Solid Films 455–456, 619–623 (2004).

152. A. S. Barker, “Transverse and longitudinal optic mode study in MgF2 and ZnF2,” Phys. Rev. 136, A1290–A1295 (1964). [CrossRef]  

153. T. Hofmann, V. Gottschalch, and M. Schubert, “Far-infrared dielectric anisotropy and phonon modes in spontaneously CuPt-ordered Ga0.52In0.48P,” Phys. Rev. B 66, 195204 (2002).

154. A. Kasic, M. Schubert, S. Einfeldt, D. Hommel, and T. E. Tiwald, “Free-carrier and phonon properties of n- and p-type hexagonal GaN films measured by infrared ellipsometry,” Phys. Rev. B 62, 7365–7377 (2000). [CrossRef]  

155. M. L. Sadowski, G. Martynez, M. Potemski, C. Berger, and W. A. D. Heer, “Magneto-spectroscopy of epitaxial graphene,” Int. J. Mod. Phys. B 21, 1145–1154 (2007). [CrossRef]  

156. W. W. Toy, M. S. Dresselhaus, and G. Dresselhaus, “Minority carriers in graphite and the H-point magnetoreflection spectra,” Phys. Rev. B 15, 4077–4090 (1977). [CrossRef]  

157. P. J. C. Kühne, “The optical Hall effect in three- and two-dimensional materials,” Ph.D. dissertation (University of Nebraska, 2014).

158. Note that the constant but generally complex amplitude parameter in Eq. (30) also may be augmented with a frequency-dependent imaginary part in order to represent the effect of an harmonic coupling. See also [150].

159. The same formalism can be used in case of holes but with a different effective mass parameter.

160. L. Landau, “Diamagnetismus der Metalle,” Z. Phys. 64, 629–637 (1930).

161. A. Geim and K. Novoselov, “The rise of graphene,” Nat. Mater. 6, 183–191 (2007). [CrossRef]  

162. M. Orlita, C. Faugeras, P. Plochocka, P. Neugebauer, G. Martinez, D. K. Maude, A.-L. Barra, M. Sprinkle, C. Berger, W. A. de Heer, and M. Potemski, “Approaching the Dirac point in high-mobility multilayer epitaxial graphene,” Phys. Rev. Lett. 101, 267601 (2008). [CrossRef]  

163. M. Orlita, C. Faugeras, R. Grill, A. Wysmolek, W. Strupinski, C. Berger, W. A. de Heer, G. Martinez, and M. Potemski, “Carrier scattering from dynamical magnetoconductivity in quasi-neutral epitaxial graphene,” Phys. Rev. Lett. 107, 216603 (2011). [CrossRef]  

164. Note that electric field vectors E contain four independent pieces of information if the plane wave is fully coherent and time harmonic. Representing time averages over infinite observation times, the four parameters can be used to characterize the electric field amplitude, absolute phase, ellipticity, and orientation of the polarization ellipse.

165. R. C. Jones, “A new calculus for the treatment of optical systems,” J. Opt. Soc. Am. 31, 488–493 (1941). [CrossRef]  

166. A. Gerrard and J. Burch, Introduction to Matrix Methods in Optics, Dover Books on Physics (Dover, 1994).

167. D. Goldstein, Polarized Light, 3rd ed. (CRC Press, 2011).

168. Note the four independent pieces of information contained in the Stokes vector. The four parameters can be used to characterize the total light intensity, degree of polarization, ellipticity, and orientation of the polarization ellipse.

169. A. Röseler, Infrared Spectroscopic Ellipsometry (Akademie-Verlag, 1990).

170. K. Järrendahl and B. Kahr, “Hans Mueller (1900-1965),” Woollam Annual Newsletter 2011(11), 8–9 (2011).

171. B. Rheinländer, “Infrarot-Faraday-Effekt an Halbleitern,” Master’s thesis (Universität Leipzig, 1965).

172. Y. Ikebe and R. Shimano, “Characterization of doped silicon in low carrier density region by terahertz frequency Faraday effect,” Appl. Phys. Lett. 92, 012111 (2008). [CrossRef]  

173. T. Morimoto, M. Koshino, and H. Aoki, “Faraday rotation in bilayer and trilayer graphene in the quantum Hall regime,” Phys. Rev. B 86, 155426 (2012). [CrossRef]  

174. D. W. Berreman, “Optics in stratified and anisotropic media: 4 × 4-matrix formulation,” J. Opt. Soc. Am. 62, 502–510 (1972). [CrossRef]  

175. M. Schubert, “Theory and application of generalized ellipsometry,” in Handbook of Ellipsometry, E. Irene and H. Tompkins, eds. (William Andrew, 2004).

176. H. Wöhler, G. Haas, M. Fritsch, and D. A. Mlynski, “Faster 4 × 4 matrix method for uniaxial inhomogeneous media,” J. Opt. Soc. Am. A 5, 1554–1557 (1988). [CrossRef]  

177. For example, j=2{k,l,m}={0,1,3}.

178. W. Xu, L. Wood, and T. Golding, “Optical degeneracies in anisotropic layered media: Treatment of singularities in a 4 × 4 matrix formalism,” Phys. Rev. B 61, 1740–1743 (2000). [CrossRef]  

179. For example, explicit expressions for the complex Fresnel transmission coefficients can be found in [45].

References

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  142. Corresponding expressions for arbitrary orientations of the magnetic field are given by EC′=Aφ,θ,ψ(R)EC, with the Euler angles given by B|B|=Aϕ,θ,ψ(R)(0,0,1)T.
  143. Corresponding transformation matrices for arbitrary orientations of the magnetic field are given by (A(C))=(Aϕ,θ,ψ(R))−1A(C)Aϕ,θ,ψ(R), with the Euler angles given by B|B|=Aϕ,θ,ψ(R)(0,0,1)T.
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  159. The same formalism can be used in case of holes but with a different effective mass parameter.
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  164. Note that electric field vectors E contain four independent pieces of information if the plane wave is fully coherent and time harmonic. Representing time averages over infinite observation times, the four parameters can be used to characterize the electric field amplitude, absolute phase, ellipticity, and orientation of the polarization ellipse.
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2016 (4)

M. Schubert, R. Korlacki, S. Knight, T. Hofmann, S. Schöche, V. Darakchieva, E. Janzén, B. Monemar, D. Gogova, Q.-T. Thieu, R. Togashi, H. Murakami, Y. Kumagai, K. Goto, A. Kuramata, S. Yamakoshi, and M. Higashiwaki, “Anisotropy, phonon modes, and free charge carrier parameters in monoclinic β-gallium oxide single crystals,” Phys. Rev. B 93, 125209 (2016).
[Crossref]

A. Parisini and R. Fornari, “Analysis of the scattering mechanisms controlling electron mobility in β−Ga2O3 crystals,” Semicond. Sci. Technol. 31, 035023 (2016).
[Crossref]

B. F. Spencer, W. F. Smith, M. T. Hibberd, P. Dawson, M. Beck, A. Bartels, I. Guiney, C. J. Humphreys, and D. M. Graham, “Terahertz cyclotron resonance spectroscopy of an AlGaN/GaN heterostructure using a high-field pulsed magnet and an asynchronous optical sampling technique,” Appl. Phys. Lett. 108, 212101 (2016).
[Crossref]

J. A. Curtis, T. Tokumoto, A. T. Hatke, J. G. Cherian, J. L. Reno, S. A. McGill, D. Karaiskaj, and D. J. Hilton, “Cyclotron decay time of a two-dimensional electron gas from 0.4 to 100 K,” Phys. Rev. B 93, 155437 (2016).
[Crossref]

2015 (3)

L. Wu, W.-K. Tse, M. Brahlek, C. Morris, R. V. Aguilar, N. Koirala, S. Oh, and N. Armitage, “High-resolution Faraday rotation and electron-phonon coupling in surface states of the bulk-insulating topological insulator Cu0.02Bi2Se3,” Phys. Rev. Lett. 115, 217602 (2015).
[Crossref]

Z. Jin, A. Tkach, F. Casper, V. Spetter, H. Grimm, A. Thomas, T. Kampfrat, M. Bonn, M. Kläui, and D. Turchinovich, “Accessing the fundamentals of magnetotransport in metals with terahertz probes,” Nat. Phys. 11, 761–766 (2015).
[Crossref]

S. Knight, S. Schöche, V. Darakchieva, P. Kühne, J.-F. Carlin, N. Grandjean, C. M. Herzinger, M. Schubert, and T. Hofmann, “Cavity-enhanced optical Hall effect in two-dimensional free charge carrier gases detected at terahertz frequencies,” Opt. Lett. 40, 2688–2691 (2015).
[Crossref]

2014 (5)

M. Higashiwaki, K. Sasaki, A. Kuramata, T. Masui, and S. Yamakoshi, “Development of gallium oxide power devices,” Phys. Stat. Solidi A 211, 21–26 (2014).
[Crossref]

P. Kühne, C. M. Herzinger, M. Schubert, J. A. Woollam, and T. Hofmann, “An integrated mid-infrared, far-infrared and terahertz optical Hall effect instrument,” Rev. Sci. Instrum. 85, 071301 (2014).
[Crossref]

Y. Lubashevsky, L. Pan, T. Kirzhner, G. Koren, and N. Armitage, “Optical birefringence and dichroism of cuprate superconductors in the THz regime,” Phys. Rev. Lett. 112, 147001 (2014).
[Crossref]

J. Lloyd-Hughes, “Terahertz spectroscopy of quantum 2D electron systems,” J. Phys. D 47, 374006 (2014).
[Crossref]

O. Arteaga, M. Baldrìs, J. Antó, A. Canillas, E. Pascual, and E. Bertran, “Mueller matrix microscope with a dual continuous rotating compensator setup and digital demodulation,” Appl. Opt. 53, 2236–2245 (2014).
[Crossref]

2013 (7)

N. Ubrig, I. Crassee, J. Levallois, I. O. Nedoliuk, F. Fromm, M. Kaiser, T. Seyller, and A. B. Kuzmenko, “Fabry–Perot enhanced Faraday rotation in graphene,” Opt. Express 21, 24736–24741 (2013).
[Crossref]

T. Nagashima, M. Tani, and M. Hangyo, “Polarization-sensitive THz-TDS and its application to anisotropy sensing,” J. Infrared Millim. Terahertz Waves 34, 740–775 (2013).
[Crossref]

K. Sasaki, M. Higashiwaki, A. Kuramata, T. Masui, and S. Yamakoshi, “MBE grown Ga2O3 and its power device applications,” J. Cryst. Growth 378, 591–595 (2013).
[Crossref]

S. Schöche, P. Kühne, T. Hofmann, M. Schubert, D. Nilsson, A. Kakanakova-Georgieva, E. Janzén, and V. Darakchieva, “Electron effective mass in Al0.72Ga0.28N alloys determined by mid-infrared optical Hall effect,” Appl. Phys. Lett. 103, 212107 (2013).
[Crossref]

P. Kühne, V. Darakchieva, R. Yakimova, J. D. Tedesco, R. L. Myers-Ward, C. R. Eddy, D. K. Gaskill, C. M. Herzinger, J. A. Woollam, M. Schubert, and T. Hofmann, “Polarization selection rules for inter-landau-level transitions in epitaxial graphene revealed by the infrared optical Hall effect,” Phys. Rev. Lett. 111, 077402 (2013).
[Crossref]

S. Schöche, T. Hofmann, V. Darakchieva, N. B. Sedrine, X. Wang, A. Yoshikawa, and M. Schubert, “Infrared to vacuum-ultraviolet ellipsometry and optical Hall-effect study of free-charge carrier parameters in Mg-doped InN,” J. Appl. Phys. 113, 013502 (2013).
[Crossref]

M.-H. Kim, T. Tanaka, C. T. Ellis, A. Mukherjee, G. Acbas, I. Ohkubo, H. Christen, D. Mandrus, H. Kontani, and J. Cerne, “Infrared anomalous Hall effect in CaxSr1−xRuO3 films,” Phys. Rev. B 88, 155101 (2013).
[Crossref]

2012 (9)

N. A. Goncharuk, L. Nádvorník, C. Faugeras, M. Orlita, and L. Smrčka, “Infrared magnetospectroscopy of graphite in tilted fields,” Phys. Rev. B 86, 155409 (2012).
[Crossref]

B. N. Szafranek, G. Fiori, D. Schall, D. Neumaier, and H. Kurz, “Current saturation and voltage gain in bilayer graphene field effect transistors,” Nano Lett. 12, 1324–1328 (2012).
[Crossref]

T. Hofmann, P. Kühne, S. Schöche, J.-T. Chen, U. Forsberg, E. Janzén, N. B. Sedrine, C. M. Herzinger, J. A. Woollam, M. Schubert, and V. Darakchieva, “Temperature dependent effective mass in AlGaN/GaN high electron mobility transistor structures,” Appl. Phys. Lett. 101, 192102 (2012).
[Crossref]

D. Molter, G. Torosyan, G. Ballon, L. Drigo, R. Beigang, and J. Léotin, “Step-scan time-domain terahertz magneto-spectroscopy,” Opt. Express 20, 5993–6002 (2012).
[Crossref]

C. M. Morris, R. V. Aguilar, A. V. Stier, and N. P. Armitage, “Polarization modulation time-domain terahertz polarimetry,” Opt. Express 20, 12303–12317 (2012).
[Crossref]

D. K. George, A. V. Stier, C. T. Ellis, B. D. McCombe, J. Černe, and A. G. Markelz, “Terahertz magneto-optical polarization modulation spectroscopy,” J. Opt. Soc. Am. B 29, 1406–1412 (2012).
[Crossref]

M. Neshat and N. P. Armitage, “Terahertz time-domain spectroscopic ellipsometry: instrumentation and calibration,” Opt. Express 20, 29063–29075 (2012).
[Crossref]

O. Arteaga, J. Freudenthal, B. Wang, and B. Kahr, “Mueller matrix polarimetry with four photoelastic modulators: theory and calibration,” Appl. Opt. 51, 6805–6817 (2012).
[Crossref]

T. Morimoto, M. Koshino, and H. Aoki, “Faraday rotation in bilayer and trilayer graphene in the quantum Hall regime,” Phys. Rev. B 86, 155426 (2012).
[Crossref]

2011 (10)

K. Järrendahl and B. Kahr, “Hans Mueller (1900-1965),” Woollam Annual Newsletter 2011(11), 8–9 (2011).

M. Orlita, C. Faugeras, R. Grill, A. Wysmolek, W. Strupinski, C. Berger, W. A. de Heer, G. Martinez, and M. Potemski, “Carrier scattering from dynamical magnetoconductivity in quasi-neutral epitaxial graphene,” Phys. Rev. Lett. 107, 216603 (2011).
[Crossref]

K. Yatsugi, N. Matsumoto, T. Nagashima, and M. Hangyo, “Transport properties of free carriers in semiconductors studied by terahertz time domain magneto-optical ellipsometry,” Appl. Phys. Lett. 98, 212108 (2011).
[Crossref]

S. Schöche, J. Shi, A. Boosalis, P. Kühne, C. M. Herzinger, J. A. Woollam, W. J. Schaff, L. F. Eastman, M. Schubert, and T. Hofmann, “Terahertz optical-Hall effect characterization of two-dimensional electron gas properties in AlGaN/GaN high electron mobility transistor structures,” Appl. Phys. Lett. 98, 092103 (2011).
[Crossref]

T. Hofmann, A. Boosalis, P. Kühne, C. M. Herzinger, J. A. Woollam, D. K. Gaskill, J. L. Tedesco, and M. Schubert, “Hole-channel conductivity in epitaxial graphene determined by terahertz optical-Hall effect and midinfrared ellipsometry,” Appl. Phys. Lett. 98, 041906 (2011).
[Crossref]

P. Kühne, T. Hofmann, C. M. Herzinger, and M. Schubert, “Terahertz frequency optical-Hall effect in multiple valley band materials,” Thin Solid Films 519, 2613–2616 (2011).

T. Hofmann, C. M. Herzinger, J. L. Tedesco, D. K. Gaskill, J. A. Woollam, and M. Schubert, “Terahertz ellipsometry and terahertz optical-Hall effect,” Thin Solid Films 519, 2593–2600 (2011).
[Crossref]

B. Kumara and S.-W. Kim, “Recent advances in power generation through piezoelectric nanogenerators,” J. Mater. Chem. 21, 18946–18958 (2011).
[Crossref]

I. Crassee, J. Levallois, D. van der Marel, A. L. Walter, T. Seyller, and A. B. Kuzmenko, “Multicomponent magneto-optical conductivity of multilayer graphene on SiC,” Phys. Rev. B 84, 035103 (2011).
[Crossref]

T. Hofmann, D. Schmidt, A. Boosalis, P. Kühne, R. Skomski, C. M. Herzinger, J. A. Woollam, M. Schubert, and E. Schubert, “Thz dielectric anisotropy of metal slanted columnar thin films,” Appl. Phys. Lett. 99, 081903 (2011).
[Crossref]

2010 (5)

D. Molter, F. Ellrich, T. Weinland, S. George, M. Goiran, F. Keilmann, R. Beigang, and J. Lèotin, “High-speed terahertz time-domain spectroscopy of cyclotron resonance in pulsed magnetic field,” Opt. Express 18, 26163–26168 (2010).
[Crossref]

X. Wang, D. J. Hilton, J. L. Reno, D. M. Mittleman, and J. Kono, “Direct measurement of cyclotron coherence times of high-mobility two-dimensional electron gases,” Opt. Express 18, 12354–12361 (2010).
[Crossref]

Y. Ikebe, T. Morimoto, R. Masutomi, T. Okamoto, H. Aoki, and R. Shimano, “Optical Hall effect in the integer quantum Hall regime,” Phys. Rev. Lett. 104, 256802 (2010).
[Crossref]

I. Crassee, J. Levallois, A. L. Walter, M. Ostler, A. Bostwick, E. Rotenberg, T. Seyller, D. van der Marel, and A. B. Kuzmenko, “Giant Faraday rotation in single- and multilayer graphene,” Nat. Phys. 7, 48–51 (2010).
[Crossref]

T. Hofmann, C. M. Herzinger, A. Boosalis, T. E. Tiwald, J. A. Woollam, and M. Schubert, “Variable-wavelength frequency-domain terahertz ellipsometry,” Rev. Sci. Instrum. 81, 023101 (2010).
[Crossref]

2009 (1)

M. Orlita, C. Faugeras, G. Martinez, D. Maude, J. Schneider, M. Sprinkle, C. Berger, W. de Heer, and M. Potemski, “Magneto-transmission of multi-layer epitaxial graphene and bulk graphite: a comparison,” Solid State Commun. 149, 1128–1131 (2009).
[Crossref]

2008 (8)

O. L. Berman, G. Gumbs, and Y. E. Lozovik, “Magnetoplasmons in layered graphene structures,” Phys. Rev. B 78, 085401 (2008).
[Crossref]

M. Koshino and T. Ando, “Magneto-optical properties of multilayer graphene,” Phys. Rev. B 77, 115313 (2008).
[Crossref]

T. Hofmann, C. von Middendorff, V. Gottschalch, and M. Schubert, “Optical Hall effect studies on modulation-doped AlxGa1−xAs:Si/GaAs quantum wells,” Phys. Stat. Solidi C 5, 1386–1390 (2008).
[Crossref]

T. Hofmann, V. Darakchieva, B. Monemar, H. Lu, W. Schaff, and M. Schubert, “Optical Hall effect in hexagonal InN,” J. Electron. Mater. 37, 611–615 (2008).
[Crossref]

T. Hofmann, C. M. Herzinger, C. Krahmer, K. Streubel, and M. Schubert, “The optical Hall effect,” Phys. Stat. Solidi A 205, 779–783 (2008).
[Crossref]

W. Weber, S. Seidl, V. Bel’akov, L. Golub, S. Danilov, E. Ivchenko, W. Prettl, Z. Kvon, H.-I. Cho, J.-H. Lee, and S. Ganicheva, “Magneto-gyrotropic photogalvanic effects in GaN/AlGaN two-dimensional systems,” Solid State Commun. 145, 56–60 (2008).
[Crossref]

M. Orlita, C. Faugeras, P. Plochocka, P. Neugebauer, G. Martinez, D. K. Maude, A.-L. Barra, M. Sprinkle, C. Berger, W. A. de Heer, and M. Potemski, “Approaching the Dirac point in high-mobility multilayer epitaxial graphene,” Phys. Rev. Lett. 101, 267601 (2008).
[Crossref]

Y. Ikebe and R. Shimano, “Characterization of doped silicon in low carrier density region by terahertz frequency Faraday effect,” Appl. Phys. Lett. 92, 012111 (2008).
[Crossref]

2007 (9)

A. Geim and K. Novoselov, “The rise of graphene,” Nat. Mater. 6, 183–191 (2007).
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M. L. Sadowski, G. Martynez, M. Potemski, C. Berger, and W. A. D. Heer, “Magneto-spectroscopy of epitaxial graphene,” Int. J. Mod. Phys. B 21, 1145–1154 (2007).
[Crossref]

T. Hofmann, M. Schubert, G. Leibiger, and V. Gottschalch, “Electron effective mass and phonon modes in GaAs incorporating Boron and Indium,” Appl. Phys. Lett. 90, 182110 (2007).
[Crossref]

M. Krames, O. Shchekin, R. Mueller-Mach, G. O. Mueller, L. Zhou, G. Harbers, and M. Craford, “Status and future of high-power light-emitting diodes for solid-state lighting,” IEEE J. Disp. Technol. 3, 160–175 (2007).
[Crossref]

D. S. L. Abergel and V. I. Fal’ko, “Optical and magneto-optical far-infrared properties of bilayer graphene,” Phys. Rev. B 75, 155430 (2007).
[Crossref]

M. L. Sadowski, G. Martinez, M. Potemski, C. Berger, and W. A. de Heer, “Infrared magnetospectroscopy of two-dimensional electrons in epitaxial graphene,” AIP Conf. Proc. 893, 619–620 (2007).

M. Tonouchi, “Cutting-edge terahertz technology,” Nat. Photonics 1, 97–105 (2007).
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H. Sumikura, T. Nagashima, H. Kitahara, and M. Hangyo, “Development of a cryogen-free terahertz time-domain magnetooptical measurement system,” Jpn. J. Appl. Phys. 46, 1739–1744 (2007).
[Crossref]

X. Wang, D. J. Hilton, L. Rein, D. M. Mittleman, J. Kono, and J. L. Reno, “Coherent THz cyclotron oscillations in a two-dimensional electron gas,” Opt. Lett. 32, 1845–1847 (2007).
[Crossref]

2006 (6)

O. Morikawa, A. Quema, S. Nashima, H. Sumikura, T. Nagashima, and M. Hangyo, “Faraday ellipticity and Faraday rotation of a doped-silicon wafer studied by terahertz time-domain spectroscopy,” J. Appl. Phys. 100, 033105 (2006).
[Crossref]

M. Schubert, “Another century of ellipsometry,” Ann. Phys. 15, 480–497 (2006).
[Crossref]

T. Hofmann, T. Chavdarov, V. Darakchieva, H. Lu, W. J. Schaff, and M. Schubert, “Anisotropy of the Γ -point effective mass and mobility in hexagonal InN,” Phys. Stat. Solidi C 3, 1854–1857 (2006).
[Crossref]

T. Hofmann, U. Schade, K. C. Agarwal, B. Daniel, C. Klingshirn, M. Hetterich, C. M. Herzinger, and M. Schubert, “Conduction-band electron effective mass in Zn0.87Mn0.13Se measured by terahertz and far-infrared magnetooptic ellipsometry,” Appl. Phys. Lett. 88, 042105 (2006).
[Crossref]

T. Hofmann, U. Schade, W. Eberhardt, C. M. Herzinger, P. Esquinazi, and M. Schubert, “Terahertz magnetooptic generalized ellipsometry using synchrotron and black-body radiation,” Rev. Sci. Instrum. 77, 063902 (2006).
[Crossref]

G. E. Jellison, J. D. Hunn, and C. M. Rouleau, “Normal-incidence generalized ellipsometry using the two-modulator generalized ellipsometry microscope,” Appl. Opt. 45, 5479–5488 (2006).
[Crossref]

2005 (1)

T. Hofmann, M. Schubert, C. von Middendorff, G. Leibiger, V. Gottschalch, C. M. Herzinger, A. Lindsay, and E. O’Reilly, “The inertial-mass scale for free-charge-carriers in semiconductor heterostructures,” AIP Conf. Proc. 772, 455–456 (2005).
[Crossref]

2004 (3)

M. Schubert, T. Hofmann, and C. M. Herzinger, “Far-infrared magnetooptic generalized ellipsometry: determination of free-charge-carrier parameters in semiconductor thin film structures,” Thin Solid Films 455–456, 563–570 (2004).
[Crossref]

M. Schubert, T. Hofmann, C. M. Herzinger, and W. Dollase, “Generalized ellipsometry for orthorhombic, absorbing materials: dielectric functions, phonon modes and band-to-band transitions of Sb2S3,” Thin Solid Films 455–456, 619–623 (2004).

Y. Ino, R. Shimano, Y. Svirko, and M. Kuwata-Gonokami, “Terahertz time domain magneto-optical ellipsometry in reflection geometry,” Phys. Rev. B 70, 155101 (2004).
[Crossref]

2003 (5)

M. Schubert, T. Hofmann, and C. M. Herzinger, “Generalized far-infrared magneto-optic ellipsometry for semiconductor layer structures: determination of free-carrier effective-mass, mobility, and concentration parameters in n-type GaAs,” J. Opt. Soc. Am. A 20, 347–356 (2003).
[Crossref]

T. Hofmann, M. Schubert, C. M. Herzinger, and I. Pietzonka, “Far-infrared-magneto-optic ellipsometry characterization of free-charge-carrier properties in highly disordered n-type Al0.19Ga0.33In0.48P,” Appl. Phys. Lett. 82, 3463–3465 (2003).
[Crossref]

T. Hofmann, M. Grundmann, C. M. Herzinger, M. Schubert, and W. Grill, “Far-infrared magnetooptical generalized ellipsometry determination of free-carrier parameters in semiconductor thin film structures,” Mater. Res. Soc. Symp. Proc. 744, M5.32.1–M5.32.16 (2003).

J. F. Wager, “Transparent electronics,” Science 300, 1245–1246 (2003).
[Crossref]

A. Kasic, M. Schubert, J. Off, B. Kuhn, F. Scholz, S. Einfeldt, T. Böttcher, D. Hommel, D. J. As, U. Köhler, A. Dadgar, A. Krost, Y. Saito, Y. Nanishi, M. R. Correia, S. Pereira, V. Darakchieva, B. Monemar, H. Amano, I. Akasaki, and G. Wagner, “Phonons and free-carrier properties of binary, ternary, and quaternary group-III Nitride layers measured by infrared spectroscopic ellipsometry,” Phys. Stat. Solidi C 0, 1750–1769 (2003).
[Crossref]

2002 (4)

R. Shimano, Y. Ino, Y. P. Svirko, and M. Kuwata-Gonokami, “Terahertz frequency Hall measurement by magneto-optical Kerr spectroscopy in InAs,” Appl. Phys. Lett. 81, 199–201 (2002).
[Crossref]

T. Hofmann, M. Schubert, and C. M. Herzinger, “Far-infrared magneto-optic generalized ellipsometry determination of free-carrier parameters in semiconductor thin film structures,” Proc. SPIE 4779, 90–97 (2002).
[Crossref]

T. Hofmann, V. Gottschalch, and M. Schubert, “Far-infrared dielectric anisotropy and phonon modes in spontaneously CuPt-ordered Ga0.52In0.48P,” Phys. Rev. B 66, 195204 (2002).

M. Schubert, A. Kasic, T. Hofmann, V. Gottschalch, J. Off, F. Scholz, E. Schubert, H. Neumann, I. J. Hodgkinson, M. D. Arnold, W. A. Dollase, and C. M. Herzinger, “Generalized ellipsometry of complex mediums in layered systems,” Proc. SPIE 4806, 264–276 (2002).
[Crossref]

2001 (3)

M. A. Zudov, R. R. Du, J. A. Simmons, and J. L. Reno, “Shubnikov-de Haas-like oscillations in millimeter wave photoconductivity in a high-mobility two-dimensional electron gas,” Phys. Rev. B 64, 201311 (2001).
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Y.-F. Wu, D. Kapolnek, J. Ibbetson, P. Parikh, B. Keller, and U. K. Mishra, “Very-high power density AlGaN/GaN HEMTs,” IEEE Trans. Electron Devices 48, 586–590 (2001).
[Crossref]

L. F. Eastman, V. Tilak, J. Smart, B. Green, E. Chumbes, R. Dimitrov, H. Kim, O. Ambacher, N. Weimann, T. Prunty, M. Murphy, W. J. Schaff, and J. Shealy, “Undoped AlGaN/GaN HEMTs for microwave power amplification,” IEEE Trans. Electron Devices 48, 479–485 (2001).
[Crossref]

2000 (4)

O. Morikawa, M. Tonouchi, and M. Hangyo, “A cross-correlation spectroscopy in subterahertz region using an incoherent light source,” Appl. Phys. Lett. 76, 1519–1521 (2000).
[Crossref]

T. E. Tiwald and M. Schubert, “Measurement of rutile TiO2 dielectric tensor from 0.148 to 33 μm using generalized ellipsometry,” Proc. SPIE 4103, 19–29 (2000).
[Crossref]

A. Kasic, M. Schubert, S. Einfeldt, D. Hommel, and T. E. Tiwald, “Free-carrier and phonon properties of n- and p-type hexagonal GaN films measured by infrared ellipsometry,” Phys. Rev. B 62, 7365–7377 (2000).
[Crossref]

W. Xu, L. Wood, and T. Golding, “Optical degeneracies in anisotropic layered media: Treatment of singularities in a 4 × 4 matrix formalism,” Phys. Rev. B 61, 1740–1743 (2000).
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1999 (1)

O. Morikawa, M. Tonouchi, and M. Hangyo, “Sub-THz spectroscopic system using a multimode laser diode and photoconductive antenna,” Appl. Phys. Lett. 75, 3772–3774 (1999).
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1998 (1)

G. E. Moore, “Cramming more components onto integrated circuits,” Proc. IEEE 86, 82–85 (1998).

1997 (3)

W. Knap, S. Contreras, H. Alause, C. Skierbiszewski, J. Camassel, M. Dyakonov, J. L. Robert, J. Yang, Q. Chen, M. A. Khan, M. L. Sadowski, S. Huant, F. H. Yang, M. Goiran, J. Leotin, and M. S. Shur, “Cyclotron resonance and quantum hall effect studies of the two-dimensional electron gas confined at the GaN/AlGaN interface,” Appl. Phys. Lett. 70, 2123–2125 (1997).
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G. E. Jellison and F. Modine, “Two-modulator generalized ellipsometry: experiment and calibration,” Appl. Opt. 36, 8184–8189 (1997).
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G. E. Jellison and F. A. Modine, “Two-modulator generalized ellipsometry: theory,” Appl. Opt. 36, 8190–8198 (1997).
[Crossref]

1996 (3)

M. Schubert, B. Rheinländer, J. A. Woollam, B. Johs, and C. M. Herzinger, “Extension of rotating-analyzer ellipsometry to generalized ellipsometry: determination of the dielectric function tensor from uniaxial TiO2,” J. Opt. Soc. Am. A 13, 875–883 (1996).
[Crossref]

M. Schubert, “Polarization-dependent optical parameters of arbitrarily anisotropic homogeneous layered systems,” Phys. Rev. B 53, 4265–4274 (1996).
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Y. J. Wang, R. Kaplan, H. K. Ng, K. Doverspike, D. K. Gaskill, T. Ikedo, I. Akasaki, and H. Amono, “Magneto-optical studies of GaN and GaN/AlxGa1−xN: donor Zeeman spectroscopy and two dimensional electron gas cyclotron resonance,” J. Appl. Phys. 79, 8007–8010 (1996).
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1994 (1)

D. Some and A. Nurmikko, “Real-time electron cyclotron oscillations observed by terahertz techniques in semiconductor heterostructures,” Appl. Phys. Lett. 65, 3377–3379 (1994).
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1993 (1)

J. R. Meyer, C. A. Hoffman, F. J. Bartoli, D. A. Arnold, S. Sivananthan, and J. P. Faurie, “Methods for magnetotransport characterization of IR detector materials,” Semicond. Sci. Technol. 8, 805–823 (1993).
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1988 (1)

1978 (1)

1977 (2)

W. W. Toy, M. S. Dresselhaus, and G. Dresselhaus, “Minority carriers in graphite and the H-point magnetoreflection spectra,” Phys. Rev. B 15, 4077–4090 (1977).
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D. A. Platts, D. D. L. Chung, and M. S. Dresselhaus, “Far-infrared magnetoreflection studies of graphite intercalated with bromide,” Phys. Rev. B 15, 1087–1092 (1977).
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1974 (3)

K. Suzuki and J. C. Hensel, “Quantum resonances in the valence bands of germanium,” Phys. Rev. B 9, 4184–4218 (1974).
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F. Gervais and B. Piriou, “Anharmonicity in several-polar-mode crystals: adjusting phonon self-energy of LO and TO modes in Al2O3 and TiO2 to fit infrared reflectivity,” J. Phys. C 7, 2374–2386 (1974).

R. F. Wallis, J. J. Brion, E. Burstein, and A. Hartstein, “Theory of surface polaritons in anisotropic dielectric media with application to surface magnetoplasmons in semiconductors,” Phys. Rev. B 9, 3424–3437 (1974).
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1973 (1)

J. J. Brion, R. F. Wallis, A. Hartstein, and E. Burstein, “Interaction of surface magnetoplasmons and surface optical phonons in polar semiconductors,” Surf. Sci. 34, 73–80 (1973).
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1972 (3)

J. J. Brion, R. F. Wallis, A. Hartstein, and E. Burstein, “Theory of magnetoplasmons in semiconductors,” Phys. Rev. Lett. 28, 1455–1458 (1972).
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D. W. Berreman, “Optics in stratified and anisotropic media: 4 × 4-matrix formulation,” J. Opt. Soc. Am. 62, 502–510 (1972).
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B. Rheinländer, H. Neumann, P. Fischer, and G. Kühn, “Anisotropic effective masses of electrons in AlAs,” Phys. Stat. Solidi B 49, K167–K169 (1972).
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1971 (1)

B. Rheinländer and H. Neumann, “Faraday rotation in n-type AlAs,” Phys. Stat. Solidi B 45, K9–K13 (1971).
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1969 (2)

B. Rheinländer, “Der Einfluss der Energieverteilung der Ladungsträger in Halbleitern auf die Bestimmung ihrer effektiven Masse aus der Infrarot-Plasma-Reflexion,” Phys. Lett. 29A, 420–421 (1969).
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R. H. Muller, “Definitions and conventions in ellipsometry,” Surf. Sci. 16, 14–33 (1969).
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1968 (2)

D. W. Berreman and F. C. Unterwald, “Adjusting poles and zeros of dielectric dispersion to fit reststrahlen of PrCl3 and LaCl3,” Phys. Rev. 174, 791–799 (1968).
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P. R. Schroeder, M. S. Dresselhaus, and A. Javan, “Location of electron and hole carriers in graphite from laser magnetoreflection data,” Phys. Rev. Lett. 20, 1292–1295 (1968).
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1965 (2)

M. E. Brodwin and R. J. Vernon, “Free-carrier magneto-microwave Kerr effect in semiconductors,” Phys. Rev. 140, A1390–A1400 (1965).
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H. H. Tippins, “Optical absorption and photoconductivity in the band edge of β−Ga2O3,” Phys. Rev. 140A, A316–A319 (1965).
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1964 (2)

N. Hindley, “Cyclotron resonance and the free-carrier magneto-optical properties of a semiconductor,” Phys. Stat. Solidi B 7, 67–80 (1964).
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A. S. Barker, “Transverse and longitudinal optic mode study in MgF2 and ZnF2,” Phys. Rev. 136, A1290–A1295 (1964).
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1961 (3)

M. Cardona, “Electron effective masses of InAs and GaAs as a function of temperature and doping,” Phys. Rev. 121, 752–758 (1961).
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E. D. Palik, S. Teitler, and R. F. Wallis, “Free carrier cyclotron resonance, Faraday rotation, and Voigt double refraction in compound semiconductors,” J. Appl. Phys. 32, 2132–2136 (1961).
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G. B. Wright and B. Lax, “Magnetoreflection experiments in intermetallics,” J. Appl. Phys. 32, 2113–2117 (1961).
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1958 (1)

P. Nozières, “Cyclotron resonance in graphite,” Phys. Rev. 109, 1510–1521 (1958).
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1957 (1)

W. G. Spitzer and H. Y. Fan, “Determination of optical constants and carrier effective mass of semiconductors,” Phys. Rev. 106, 882–890 (1957).
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1956 (1)

J. K. Galt, W. A. Yager, and H. W. Dail, “Cyclotron resonance effects in graphite,” Phys. Rev. 103, 1586–1587 (1956).
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1955 (1)

G. Dresselhaus, A. F. Kip, and C. Kittel, “Cyclotron resonance of electrons and holes in silicon and germanium crystals,” Phys. Rev. 98, 368–384 (1955).
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1952 (1)

R. Roy, V. G. Hill, and E. F. Osborn, “Polymorphism of Ga2O3 and the system Ga2O3,” J. Am. Chem. Soc. 74, 719–722 (1952).
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1948 (1)

1947 (1)

1941 (1)

1930 (1)

L. Landau, “Diamagnetismus der Metalle,” Z. Phys. 64, 629–637 (1930).

1900 (2)

P. Drude, “Zur Ionentheorie der Metalle,” Physikal. Z. 1, 161–165 (1900).

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1890 (1)

P. Drude, “Bestimmung der optischen Constanten der Metalle,” Ann. Phys. Chem. 275, 481–554 (1890).
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1888 (1)

P. Drude, “Beobachtungen über die Reflexion des Lichtes am Antimonglanz,” Ann. Phys. 270, 489–531 (1888).
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1887 (1)

P. Drude, “Üeber die Gesetze der Reflexion und Brechung des Lichtes an der Grenze absorbirender Krystalle,” Ann. Phys. 268, 584–625 (1887).
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1879 (1)

E. H. Hall, “On a new action of the magnet on electric currents,” Amer. J. Math. 2, 287–292 (1879).
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Abergel, D. S. L.

D. S. L. Abergel and V. I. Fal’ko, “Optical and magneto-optical far-infrared properties of bilayer graphene,” Phys. Rev. B 75, 155430 (2007).
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Acbas, G.

M.-H. Kim, T. Tanaka, C. T. Ellis, A. Mukherjee, G. Acbas, I. Ohkubo, H. Christen, D. Mandrus, H. Kontani, and J. Cerne, “Infrared anomalous Hall effect in CaxSr1−xRuO3 films,” Phys. Rev. B 88, 155101 (2013).
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Agarwal, K. C.

T. Hofmann, U. Schade, K. C. Agarwal, B. Daniel, C. Klingshirn, M. Hetterich, C. M. Herzinger, and M. Schubert, “Conduction-band electron effective mass in Zn0.87Mn0.13Se measured by terahertz and far-infrared magnetooptic ellipsometry,” Appl. Phys. Lett. 88, 042105 (2006).
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Aguilar, R. V.

L. Wu, W.-K. Tse, M. Brahlek, C. Morris, R. V. Aguilar, N. Koirala, S. Oh, and N. Armitage, “High-resolution Faraday rotation and electron-phonon coupling in surface states of the bulk-insulating topological insulator Cu0.02Bi2Se3,” Phys. Rev. Lett. 115, 217602 (2015).
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C. M. Morris, R. V. Aguilar, A. V. Stier, and N. P. Armitage, “Polarization modulation time-domain terahertz polarimetry,” Opt. Express 20, 12303–12317 (2012).
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Akasaki, I.

A. Kasic, M. Schubert, J. Off, B. Kuhn, F. Scholz, S. Einfeldt, T. Böttcher, D. Hommel, D. J. As, U. Köhler, A. Dadgar, A. Krost, Y. Saito, Y. Nanishi, M. R. Correia, S. Pereira, V. Darakchieva, B. Monemar, H. Amano, I. Akasaki, and G. Wagner, “Phonons and free-carrier properties of binary, ternary, and quaternary group-III Nitride layers measured by infrared spectroscopic ellipsometry,” Phys. Stat. Solidi C 0, 1750–1769 (2003).
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Y. J. Wang, R. Kaplan, H. K. Ng, K. Doverspike, D. K. Gaskill, T. Ikedo, I. Akasaki, and H. Amono, “Magneto-optical studies of GaN and GaN/AlxGa1−xN: donor Zeeman spectroscopy and two dimensional electron gas cyclotron resonance,” J. Appl. Phys. 79, 8007–8010 (1996).
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Alause, H.

W. Knap, S. Contreras, H. Alause, C. Skierbiszewski, J. Camassel, M. Dyakonov, J. L. Robert, J. Yang, Q. Chen, M. A. Khan, M. L. Sadowski, S. Huant, F. H. Yang, M. Goiran, J. Leotin, and M. S. Shur, “Cyclotron resonance and quantum hall effect studies of the two-dimensional electron gas confined at the GaN/AlGaN interface,” Appl. Phys. Lett. 70, 2123–2125 (1997).
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Amano, H.

A. Kasic, M. Schubert, J. Off, B. Kuhn, F. Scholz, S. Einfeldt, T. Böttcher, D. Hommel, D. J. As, U. Köhler, A. Dadgar, A. Krost, Y. Saito, Y. Nanishi, M. R. Correia, S. Pereira, V. Darakchieva, B. Monemar, H. Amano, I. Akasaki, and G. Wagner, “Phonons and free-carrier properties of binary, ternary, and quaternary group-III Nitride layers measured by infrared spectroscopic ellipsometry,” Phys. Stat. Solidi C 0, 1750–1769 (2003).
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Ambacher, O.

L. F. Eastman, V. Tilak, J. Smart, B. Green, E. Chumbes, R. Dimitrov, H. Kim, O. Ambacher, N. Weimann, T. Prunty, M. Murphy, W. J. Schaff, and J. Shealy, “Undoped AlGaN/GaN HEMTs for microwave power amplification,” IEEE Trans. Electron Devices 48, 479–485 (2001).
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Amono, H.

Y. J. Wang, R. Kaplan, H. K. Ng, K. Doverspike, D. K. Gaskill, T. Ikedo, I. Akasaki, and H. Amono, “Magneto-optical studies of GaN and GaN/AlxGa1−xN: donor Zeeman spectroscopy and two dimensional electron gas cyclotron resonance,” J. Appl. Phys. 79, 8007–8010 (1996).
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Ando, T.

M. Koshino and T. Ando, “Magneto-optical properties of multilayer graphene,” Phys. Rev. B 77, 115313 (2008).
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Aoki, H.

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M. Schubert, T. Hofmann, C. M. Herzinger, and W. Dollase, “Generalized ellipsometry for orthorhombic, absorbing materials: dielectric functions, phonon modes and band-to-band transitions of Sb2S3,” Thin Solid Films 455–456, 619–623 (2004).

T. Hofmann, M. Grundmann, C. M. Herzinger, M. Schubert, and W. Grill, “Far-infrared magnetooptical generalized ellipsometry determination of free-carrier parameters in semiconductor thin film structures,” Mater. Res. Soc. Symp. Proc. 744, M5.32.1–M5.32.16 (2003).

T. Hofmann, M. Schubert, C. M. Herzinger, and I. Pietzonka, “Far-infrared-magneto-optic ellipsometry characterization of free-charge-carrier properties in highly disordered n-type Al0.19Ga0.33In0.48P,” Appl. Phys. Lett. 82, 3463–3465 (2003).
[Crossref]

M. Schubert, T. Hofmann, and C. M. Herzinger, “Generalized far-infrared magneto-optic ellipsometry for semiconductor layer structures: determination of free-carrier effective-mass, mobility, and concentration parameters in n-type GaAs,” J. Opt. Soc. Am. A 20, 347–356 (2003).
[Crossref]

M. Schubert, A. Kasic, T. Hofmann, V. Gottschalch, J. Off, F. Scholz, E. Schubert, H. Neumann, I. J. Hodgkinson, M. D. Arnold, W. A. Dollase, and C. M. Herzinger, “Generalized ellipsometry of complex mediums in layered systems,” Proc. SPIE 4806, 264–276 (2002).
[Crossref]

T. Hofmann, M. Schubert, and C. M. Herzinger, “Far-infrared magneto-optic generalized ellipsometry determination of free-carrier parameters in semiconductor thin film structures,” Proc. SPIE 4779, 90–97 (2002).
[Crossref]

M. Schubert, B. Rheinländer, J. A. Woollam, B. Johs, and C. M. Herzinger, “Extension of rotating-analyzer ellipsometry to generalized ellipsometry: determination of the dielectric function tensor from uniaxial TiO2,” J. Opt. Soc. Am. A 13, 875–883 (1996).
[Crossref]

Hetterich, M.

T. Hofmann, U. Schade, K. C. Agarwal, B. Daniel, C. Klingshirn, M. Hetterich, C. M. Herzinger, and M. Schubert, “Conduction-band electron effective mass in Zn0.87Mn0.13Se measured by terahertz and far-infrared magnetooptic ellipsometry,” Appl. Phys. Lett. 88, 042105 (2006).
[Crossref]

Hibberd, M. T.

B. F. Spencer, W. F. Smith, M. T. Hibberd, P. Dawson, M. Beck, A. Bartels, I. Guiney, C. J. Humphreys, and D. M. Graham, “Terahertz cyclotron resonance spectroscopy of an AlGaN/GaN heterostructure using a high-field pulsed magnet and an asynchronous optical sampling technique,” Appl. Phys. Lett. 108, 212101 (2016).
[Crossref]

Higashiwaki, M.

M. Schubert, R. Korlacki, S. Knight, T. Hofmann, S. Schöche, V. Darakchieva, E. Janzén, B. Monemar, D. Gogova, Q.-T. Thieu, R. Togashi, H. Murakami, Y. Kumagai, K. Goto, A. Kuramata, S. Yamakoshi, and M. Higashiwaki, “Anisotropy, phonon modes, and free charge carrier parameters in monoclinic β-gallium oxide single crystals,” Phys. Rev. B 93, 125209 (2016).
[Crossref]

M. Higashiwaki, K. Sasaki, A. Kuramata, T. Masui, and S. Yamakoshi, “Development of gallium oxide power devices,” Phys. Stat. Solidi A 211, 21–26 (2014).
[Crossref]

K. Sasaki, M. Higashiwaki, A. Kuramata, T. Masui, and S. Yamakoshi, “MBE grown Ga2O3 and its power device applications,” J. Cryst. Growth 378, 591–595 (2013).
[Crossref]

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R. Roy, V. G. Hill, and E. F. Osborn, “Polymorphism of Ga2O3 and the system Ga2O3,” J. Am. Chem. Soc. 74, 719–722 (1952).
[Crossref]

Hilton, D. J.

Hindley, N.

N. Hindley, “Cyclotron resonance and the free-carrier magneto-optical properties of a semiconductor,” Phys. Stat. Solidi B 7, 67–80 (1964).
[Crossref]

Hodgkinson, I. J.

M. Schubert, A. Kasic, T. Hofmann, V. Gottschalch, J. Off, F. Scholz, E. Schubert, H. Neumann, I. J. Hodgkinson, M. D. Arnold, W. A. Dollase, and C. M. Herzinger, “Generalized ellipsometry of complex mediums in layered systems,” Proc. SPIE 4806, 264–276 (2002).
[Crossref]

Hoffman, C. A.

J. R. Meyer, C. A. Hoffman, F. J. Bartoli, D. A. Arnold, S. Sivananthan, and J. P. Faurie, “Methods for magnetotransport characterization of IR detector materials,” Semicond. Sci. Technol. 8, 805–823 (1993).
[Crossref]

Hofmann, T.

M. Schubert, R. Korlacki, S. Knight, T. Hofmann, S. Schöche, V. Darakchieva, E. Janzén, B. Monemar, D. Gogova, Q.-T. Thieu, R. Togashi, H. Murakami, Y. Kumagai, K. Goto, A. Kuramata, S. Yamakoshi, and M. Higashiwaki, “Anisotropy, phonon modes, and free charge carrier parameters in monoclinic β-gallium oxide single crystals,” Phys. Rev. B 93, 125209 (2016).
[Crossref]

S. Knight, S. Schöche, V. Darakchieva, P. Kühne, J.-F. Carlin, N. Grandjean, C. M. Herzinger, M. Schubert, and T. Hofmann, “Cavity-enhanced optical Hall effect in two-dimensional free charge carrier gases detected at terahertz frequencies,” Opt. Lett. 40, 2688–2691 (2015).
[Crossref]

P. Kühne, C. M. Herzinger, M. Schubert, J. A. Woollam, and T. Hofmann, “An integrated mid-infrared, far-infrared and terahertz optical Hall effect instrument,” Rev. Sci. Instrum. 85, 071301 (2014).
[Crossref]

S. Schöche, T. Hofmann, V. Darakchieva, N. B. Sedrine, X. Wang, A. Yoshikawa, and M. Schubert, “Infrared to vacuum-ultraviolet ellipsometry and optical Hall-effect study of free-charge carrier parameters in Mg-doped InN,” J. Appl. Phys. 113, 013502 (2013).
[Crossref]

P. Kühne, V. Darakchieva, R. Yakimova, J. D. Tedesco, R. L. Myers-Ward, C. R. Eddy, D. K. Gaskill, C. M. Herzinger, J. A. Woollam, M. Schubert, and T. Hofmann, “Polarization selection rules for inter-landau-level transitions in epitaxial graphene revealed by the infrared optical Hall effect,” Phys. Rev. Lett. 111, 077402 (2013).
[Crossref]

S. Schöche, P. Kühne, T. Hofmann, M. Schubert, D. Nilsson, A. Kakanakova-Georgieva, E. Janzén, and V. Darakchieva, “Electron effective mass in Al0.72Ga0.28N alloys determined by mid-infrared optical Hall effect,” Appl. Phys. Lett. 103, 212107 (2013).
[Crossref]

T. Hofmann, P. Kühne, S. Schöche, J.-T. Chen, U. Forsberg, E. Janzén, N. B. Sedrine, C. M. Herzinger, J. A. Woollam, M. Schubert, and V. Darakchieva, “Temperature dependent effective mass in AlGaN/GaN high electron mobility transistor structures,” Appl. Phys. Lett. 101, 192102 (2012).
[Crossref]

P. Kühne, T. Hofmann, C. M. Herzinger, and M. Schubert, “Terahertz frequency optical-Hall effect in multiple valley band materials,” Thin Solid Films 519, 2613–2616 (2011).

T. Hofmann, C. M. Herzinger, J. L. Tedesco, D. K. Gaskill, J. A. Woollam, and M. Schubert, “Terahertz ellipsometry and terahertz optical-Hall effect,” Thin Solid Films 519, 2593–2600 (2011).
[Crossref]

S. Schöche, J. Shi, A. Boosalis, P. Kühne, C. M. Herzinger, J. A. Woollam, W. J. Schaff, L. F. Eastman, M. Schubert, and T. Hofmann, “Terahertz optical-Hall effect characterization of two-dimensional electron gas properties in AlGaN/GaN high electron mobility transistor structures,” Appl. Phys. Lett. 98, 092103 (2011).
[Crossref]

T. Hofmann, A. Boosalis, P. Kühne, C. M. Herzinger, J. A. Woollam, D. K. Gaskill, J. L. Tedesco, and M. Schubert, “Hole-channel conductivity in epitaxial graphene determined by terahertz optical-Hall effect and midinfrared ellipsometry,” Appl. Phys. Lett. 98, 041906 (2011).
[Crossref]

T. Hofmann, D. Schmidt, A. Boosalis, P. Kühne, R. Skomski, C. M. Herzinger, J. A. Woollam, M. Schubert, and E. Schubert, “Thz dielectric anisotropy of metal slanted columnar thin films,” Appl. Phys. Lett. 99, 081903 (2011).
[Crossref]

T. Hofmann, C. M. Herzinger, A. Boosalis, T. E. Tiwald, J. A. Woollam, and M. Schubert, “Variable-wavelength frequency-domain terahertz ellipsometry,” Rev. Sci. Instrum. 81, 023101 (2010).
[Crossref]

T. Hofmann, C. von Middendorff, V. Gottschalch, and M. Schubert, “Optical Hall effect studies on modulation-doped AlxGa1−xAs:Si/GaAs quantum wells,” Phys. Stat. Solidi C 5, 1386–1390 (2008).
[Crossref]

T. Hofmann, V. Darakchieva, B. Monemar, H. Lu, W. Schaff, and M. Schubert, “Optical Hall effect in hexagonal InN,” J. Electron. Mater. 37, 611–615 (2008).
[Crossref]

T. Hofmann, C. M. Herzinger, C. Krahmer, K. Streubel, and M. Schubert, “The optical Hall effect,” Phys. Stat. Solidi A 205, 779–783 (2008).
[Crossref]

T. Hofmann, M. Schubert, G. Leibiger, and V. Gottschalch, “Electron effective mass and phonon modes in GaAs incorporating Boron and Indium,” Appl. Phys. Lett. 90, 182110 (2007).
[Crossref]

T. Hofmann, U. Schade, W. Eberhardt, C. M. Herzinger, P. Esquinazi, and M. Schubert, “Terahertz magnetooptic generalized ellipsometry using synchrotron and black-body radiation,” Rev. Sci. Instrum. 77, 063902 (2006).
[Crossref]

T. Hofmann, T. Chavdarov, V. Darakchieva, H. Lu, W. J. Schaff, and M. Schubert, “Anisotropy of the Γ -point effective mass and mobility in hexagonal InN,” Phys. Stat. Solidi C 3, 1854–1857 (2006).
[Crossref]

T. Hofmann, U. Schade, K. C. Agarwal, B. Daniel, C. Klingshirn, M. Hetterich, C. M. Herzinger, and M. Schubert, “Conduction-band electron effective mass in Zn0.87Mn0.13Se measured by terahertz and far-infrared magnetooptic ellipsometry,” Appl. Phys. Lett. 88, 042105 (2006).
[Crossref]

T. Hofmann, M. Schubert, C. von Middendorff, G. Leibiger, V. Gottschalch, C. M. Herzinger, A. Lindsay, and E. O’Reilly, “The inertial-mass scale for free-charge-carriers in semiconductor heterostructures,” AIP Conf. Proc. 772, 455–456 (2005).
[Crossref]

M. Schubert, T. Hofmann, and C. M. Herzinger, “Far-infrared magnetooptic generalized ellipsometry: determination of free-charge-carrier parameters in semiconductor thin film structures,” Thin Solid Films 455–456, 563–570 (2004).
[Crossref]

M. Schubert, T. Hofmann, C. M. Herzinger, and W. Dollase, “Generalized ellipsometry for orthorhombic, absorbing materials: dielectric functions, phonon modes and band-to-band transitions of Sb2S3,” Thin Solid Films 455–456, 619–623 (2004).

T. Hofmann, M. Grundmann, C. M. Herzinger, M. Schubert, and W. Grill, “Far-infrared magnetooptical generalized ellipsometry determination of free-carrier parameters in semiconductor thin film structures,” Mater. Res. Soc. Symp. Proc. 744, M5.32.1–M5.32.16 (2003).

T. Hofmann, M. Schubert, C. M. Herzinger, and I. Pietzonka, “Far-infrared-magneto-optic ellipsometry characterization of free-charge-carrier properties in highly disordered n-type Al0.19Ga0.33In0.48P,” Appl. Phys. Lett. 82, 3463–3465 (2003).
[Crossref]

M. Schubert, T. Hofmann, and C. M. Herzinger, “Generalized far-infrared magneto-optic ellipsometry for semiconductor layer structures: determination of free-carrier effective-mass, mobility, and concentration parameters in n-type GaAs,” J. Opt. Soc. Am. A 20, 347–356 (2003).
[Crossref]

T. Hofmann, M. Schubert, and C. M. Herzinger, “Far-infrared magneto-optic generalized ellipsometry determination of free-carrier parameters in semiconductor thin film structures,” Proc. SPIE 4779, 90–97 (2002).
[Crossref]

M. Schubert, A. Kasic, T. Hofmann, V. Gottschalch, J. Off, F. Scholz, E. Schubert, H. Neumann, I. J. Hodgkinson, M. D. Arnold, W. A. Dollase, and C. M. Herzinger, “Generalized ellipsometry of complex mediums in layered systems,” Proc. SPIE 4806, 264–276 (2002).
[Crossref]

T. Hofmann, V. Gottschalch, and M. Schubert, “Far-infrared dielectric anisotropy and phonon modes in spontaneously CuPt-ordered Ga0.52In0.48P,” Phys. Rev. B 66, 195204 (2002).

T. Hofmann, “Far-infrared spectroscopic ellipsometry on semiconductor heterostructures,” Ph.D. thesis (University of Leipzig, 2004).

Hommel, D.

A. Kasic, M. Schubert, J. Off, B. Kuhn, F. Scholz, S. Einfeldt, T. Böttcher, D. Hommel, D. J. As, U. Köhler, A. Dadgar, A. Krost, Y. Saito, Y. Nanishi, M. R. Correia, S. Pereira, V. Darakchieva, B. Monemar, H. Amano, I. Akasaki, and G. Wagner, “Phonons and free-carrier properties of binary, ternary, and quaternary group-III Nitride layers measured by infrared spectroscopic ellipsometry,” Phys. Stat. Solidi C 0, 1750–1769 (2003).
[Crossref]

A. Kasic, M. Schubert, S. Einfeldt, D. Hommel, and T. E. Tiwald, “Free-carrier and phonon properties of n- and p-type hexagonal GaN films measured by infrared ellipsometry,” Phys. Rev. B 62, 7365–7377 (2000).
[Crossref]

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W. Knap, S. Contreras, H. Alause, C. Skierbiszewski, J. Camassel, M. Dyakonov, J. L. Robert, J. Yang, Q. Chen, M. A. Khan, M. L. Sadowski, S. Huant, F. H. Yang, M. Goiran, J. Leotin, and M. S. Shur, “Cyclotron resonance and quantum hall effect studies of the two-dimensional electron gas confined at the GaN/AlGaN interface,” Appl. Phys. Lett. 70, 2123–2125 (1997).
[Crossref]

Humphreys, C. J.

B. F. Spencer, W. F. Smith, M. T. Hibberd, P. Dawson, M. Beck, A. Bartels, I. Guiney, C. J. Humphreys, and D. M. Graham, “Terahertz cyclotron resonance spectroscopy of an AlGaN/GaN heterostructure using a high-field pulsed magnet and an asynchronous optical sampling technique,” Appl. Phys. Lett. 108, 212101 (2016).
[Crossref]

Hunn, J. D.

Ibbetson, J.

Y.-F. Wu, D. Kapolnek, J. Ibbetson, P. Parikh, B. Keller, and U. K. Mishra, “Very-high power density AlGaN/GaN HEMTs,” IEEE Trans. Electron Devices 48, 586–590 (2001).
[Crossref]

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Y. Ikebe, T. Morimoto, R. Masutomi, T. Okamoto, H. Aoki, and R. Shimano, “Optical Hall effect in the integer quantum Hall regime,” Phys. Rev. Lett. 104, 256802 (2010).
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Y. Ikebe and R. Shimano, “Characterization of doped silicon in low carrier density region by terahertz frequency Faraday effect,” Appl. Phys. Lett. 92, 012111 (2008).
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Y. Ino, R. Shimano, Y. Svirko, and M. Kuwata-Gonokami, “Terahertz time domain magneto-optical ellipsometry in reflection geometry,” Phys. Rev. B 70, 155101 (2004).
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R. Shimano, Y. Ino, Y. P. Svirko, and M. Kuwata-Gonokami, “Terahertz frequency Hall measurement by magneto-optical Kerr spectroscopy in InAs,” Appl. Phys. Lett. 81, 199–201 (2002).
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M. Schubert, R. Korlacki, S. Knight, T. Hofmann, S. Schöche, V. Darakchieva, E. Janzén, B. Monemar, D. Gogova, Q.-T. Thieu, R. Togashi, H. Murakami, Y. Kumagai, K. Goto, A. Kuramata, S. Yamakoshi, and M. Higashiwaki, “Anisotropy, phonon modes, and free charge carrier parameters in monoclinic β-gallium oxide single crystals,” Phys. Rev. B 93, 125209 (2016).
[Crossref]

S. Schöche, P. Kühne, T. Hofmann, M. Schubert, D. Nilsson, A. Kakanakova-Georgieva, E. Janzén, and V. Darakchieva, “Electron effective mass in Al0.72Ga0.28N alloys determined by mid-infrared optical Hall effect,” Appl. Phys. Lett. 103, 212107 (2013).
[Crossref]

T. Hofmann, P. Kühne, S. Schöche, J.-T. Chen, U. Forsberg, E. Janzén, N. B. Sedrine, C. M. Herzinger, J. A. Woollam, M. Schubert, and V. Darakchieva, “Temperature dependent effective mass in AlGaN/GaN high electron mobility transistor structures,” Appl. Phys. Lett. 101, 192102 (2012).
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Johs, B.

Jones, R. C.

Kahr, B.

Kaiser, M.

Kakanakova-Georgieva, A.

S. Schöche, P. Kühne, T. Hofmann, M. Schubert, D. Nilsson, A. Kakanakova-Georgieva, E. Janzén, and V. Darakchieva, “Electron effective mass in Al0.72Ga0.28N alloys determined by mid-infrared optical Hall effect,” Appl. Phys. Lett. 103, 212107 (2013).
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Z. Jin, A. Tkach, F. Casper, V. Spetter, H. Grimm, A. Thomas, T. Kampfrat, M. Bonn, M. Kläui, and D. Turchinovich, “Accessing the fundamentals of magnetotransport in metals with terahertz probes,” Nat. Phys. 11, 761–766 (2015).
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Y. J. Wang, R. Kaplan, H. K. Ng, K. Doverspike, D. K. Gaskill, T. Ikedo, I. Akasaki, and H. Amono, “Magneto-optical studies of GaN and GaN/AlxGa1−xN: donor Zeeman spectroscopy and two dimensional electron gas cyclotron resonance,” J. Appl. Phys. 79, 8007–8010 (1996).
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Y.-F. Wu, D. Kapolnek, J. Ibbetson, P. Parikh, B. Keller, and U. K. Mishra, “Very-high power density AlGaN/GaN HEMTs,” IEEE Trans. Electron Devices 48, 586–590 (2001).
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A. Kasic, M. Schubert, J. Off, B. Kuhn, F. Scholz, S. Einfeldt, T. Böttcher, D. Hommel, D. J. As, U. Köhler, A. Dadgar, A. Krost, Y. Saito, Y. Nanishi, M. R. Correia, S. Pereira, V. Darakchieva, B. Monemar, H. Amano, I. Akasaki, and G. Wagner, “Phonons and free-carrier properties of binary, ternary, and quaternary group-III Nitride layers measured by infrared spectroscopic ellipsometry,” Phys. Stat. Solidi C 0, 1750–1769 (2003).
[Crossref]

M. Schubert, A. Kasic, T. Hofmann, V. Gottschalch, J. Off, F. Scholz, E. Schubert, H. Neumann, I. J. Hodgkinson, M. D. Arnold, W. A. Dollase, and C. M. Herzinger, “Generalized ellipsometry of complex mediums in layered systems,” Proc. SPIE 4806, 264–276 (2002).
[Crossref]

A. Kasic, M. Schubert, S. Einfeldt, D. Hommel, and T. E. Tiwald, “Free-carrier and phonon properties of n- and p-type hexagonal GaN films measured by infrared ellipsometry,” Phys. Rev. B 62, 7365–7377 (2000).
[Crossref]

Keilmann, F.

Keller, B.

Y.-F. Wu, D. Kapolnek, J. Ibbetson, P. Parikh, B. Keller, and U. K. Mishra, “Very-high power density AlGaN/GaN HEMTs,” IEEE Trans. Electron Devices 48, 586–590 (2001).
[Crossref]

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W. Knap, S. Contreras, H. Alause, C. Skierbiszewski, J. Camassel, M. Dyakonov, J. L. Robert, J. Yang, Q. Chen, M. A. Khan, M. L. Sadowski, S. Huant, F. H. Yang, M. Goiran, J. Leotin, and M. S. Shur, “Cyclotron resonance and quantum hall effect studies of the two-dimensional electron gas confined at the GaN/AlGaN interface,” Appl. Phys. Lett. 70, 2123–2125 (1997).
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T. Hofmann, U. Schade, K. C. Agarwal, B. Daniel, C. Klingshirn, M. Hetterich, C. M. Herzinger, and M. Schubert, “Conduction-band electron effective mass in Zn0.87Mn0.13Se measured by terahertz and far-infrared magnetooptic ellipsometry,” Appl. Phys. Lett. 88, 042105 (2006).
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[Crossref]

Knight, S.

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M. Schubert, R. Korlacki, S. Knight, T. Hofmann, S. Schöche, V. Darakchieva, E. Janzén, B. Monemar, D. Gogova, Q.-T. Thieu, R. Togashi, H. Murakami, Y. Kumagai, K. Goto, A. Kuramata, S. Yamakoshi, and M. Higashiwaki, “Anisotropy, phonon modes, and free charge carrier parameters in monoclinic β-gallium oxide single crystals,” Phys. Rev. B 93, 125209 (2016).
[Crossref]

S. Knight, S. Schöche, V. Darakchieva, P. Kühne, J.-F. Carlin, N. Grandjean, C. M. Herzinger, M. Schubert, and T. Hofmann, “Cavity-enhanced optical Hall effect in two-dimensional free charge carrier gases detected at terahertz frequencies,” Opt. Lett. 40, 2688–2691 (2015).
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P. Kühne, C. M. Herzinger, M. Schubert, J. A. Woollam, and T. Hofmann, “An integrated mid-infrared, far-infrared and terahertz optical Hall effect instrument,” Rev. Sci. Instrum. 85, 071301 (2014).
[Crossref]

S. Schöche, T. Hofmann, V. Darakchieva, N. B. Sedrine, X. Wang, A. Yoshikawa, and M. Schubert, “Infrared to vacuum-ultraviolet ellipsometry and optical Hall-effect study of free-charge carrier parameters in Mg-doped InN,” J. Appl. Phys. 113, 013502 (2013).
[Crossref]

P. Kühne, V. Darakchieva, R. Yakimova, J. D. Tedesco, R. L. Myers-Ward, C. R. Eddy, D. K. Gaskill, C. M. Herzinger, J. A. Woollam, M. Schubert, and T. Hofmann, “Polarization selection rules for inter-landau-level transitions in epitaxial graphene revealed by the infrared optical Hall effect,” Phys. Rev. Lett. 111, 077402 (2013).
[Crossref]

S. Schöche, P. Kühne, T. Hofmann, M. Schubert, D. Nilsson, A. Kakanakova-Georgieva, E. Janzén, and V. Darakchieva, “Electron effective mass in Al0.72Ga0.28N alloys determined by mid-infrared optical Hall effect,” Appl. Phys. Lett. 103, 212107 (2013).
[Crossref]

T. Hofmann, P. Kühne, S. Schöche, J.-T. Chen, U. Forsberg, E. Janzén, N. B. Sedrine, C. M. Herzinger, J. A. Woollam, M. Schubert, and V. Darakchieva, “Temperature dependent effective mass in AlGaN/GaN high electron mobility transistor structures,” Appl. Phys. Lett. 101, 192102 (2012).
[Crossref]

T. Hofmann, C. M. Herzinger, J. L. Tedesco, D. K. Gaskill, J. A. Woollam, and M. Schubert, “Terahertz ellipsometry and terahertz optical-Hall effect,” Thin Solid Films 519, 2593–2600 (2011).
[Crossref]

T. Hofmann, A. Boosalis, P. Kühne, C. M. Herzinger, J. A. Woollam, D. K. Gaskill, J. L. Tedesco, and M. Schubert, “Hole-channel conductivity in epitaxial graphene determined by terahertz optical-Hall effect and midinfrared ellipsometry,” Appl. Phys. Lett. 98, 041906 (2011).
[Crossref]

P. Kühne, T. Hofmann, C. M. Herzinger, and M. Schubert, “Terahertz frequency optical-Hall effect in multiple valley band materials,” Thin Solid Films 519, 2613–2616 (2011).

S. Schöche, J. Shi, A. Boosalis, P. Kühne, C. M. Herzinger, J. A. Woollam, W. J. Schaff, L. F. Eastman, M. Schubert, and T. Hofmann, “Terahertz optical-Hall effect characterization of two-dimensional electron gas properties in AlGaN/GaN high electron mobility transistor structures,” Appl. Phys. Lett. 98, 092103 (2011).
[Crossref]

T. Hofmann, D. Schmidt, A. Boosalis, P. Kühne, R. Skomski, C. M. Herzinger, J. A. Woollam, M. Schubert, and E. Schubert, “Thz dielectric anisotropy of metal slanted columnar thin films,” Appl. Phys. Lett. 99, 081903 (2011).
[Crossref]

T. Hofmann, C. M. Herzinger, A. Boosalis, T. E. Tiwald, J. A. Woollam, and M. Schubert, “Variable-wavelength frequency-domain terahertz ellipsometry,” Rev. Sci. Instrum. 81, 023101 (2010).
[Crossref]

T. Hofmann, C. M. Herzinger, C. Krahmer, K. Streubel, and M. Schubert, “The optical Hall effect,” Phys. Stat. Solidi A 205, 779–783 (2008).
[Crossref]

T. Hofmann, V. Darakchieva, B. Monemar, H. Lu, W. Schaff, and M. Schubert, “Optical Hall effect in hexagonal InN,” J. Electron. Mater. 37, 611–615 (2008).
[Crossref]

T. Hofmann, C. von Middendorff, V. Gottschalch, and M. Schubert, “Optical Hall effect studies on modulation-doped AlxGa1−xAs:Si/GaAs quantum wells,” Phys. Stat. Solidi C 5, 1386–1390 (2008).
[Crossref]

T. Hofmann, M. Schubert, G. Leibiger, and V. Gottschalch, “Electron effective mass and phonon modes in GaAs incorporating Boron and Indium,” Appl. Phys. Lett. 90, 182110 (2007).
[Crossref]

T. Hofmann, U. Schade, W. Eberhardt, C. M. Herzinger, P. Esquinazi, and M. Schubert, “Terahertz magnetooptic generalized ellipsometry using synchrotron and black-body radiation,” Rev. Sci. Instrum. 77, 063902 (2006).
[Crossref]

T. Hofmann, T. Chavdarov, V. Darakchieva, H. Lu, W. J. Schaff, and M. Schubert, “Anisotropy of the Γ -point effective mass and mobility in hexagonal InN,” Phys. Stat. Solidi C 3, 1854–1857 (2006).
[Crossref]

M. Schubert, “Another century of ellipsometry,” Ann. Phys. 15, 480–497 (2006).
[Crossref]

T. Hofmann, U. Schade, K. C. Agarwal, B. Daniel, C. Klingshirn, M. Hetterich, C. M. Herzinger, and M. Schubert, “Conduction-band electron effective mass in Zn0.87Mn0.13Se measured by terahertz and far-infrared magnetooptic ellipsometry,” Appl. Phys. Lett. 88, 042105 (2006).
[Crossref]

T. Hofmann, M. Schubert, C. von Middendorff, G. Leibiger, V. Gottschalch, C. M. Herzinger, A. Lindsay, and E. O’Reilly, “The inertial-mass scale for free-charge-carriers in semiconductor heterostructures,” AIP Conf. Proc. 772, 455–456 (2005).
[Crossref]

M. Schubert, T. Hofmann, and C. M. Herzinger, “Far-infrared magnetooptic generalized ellipsometry: determination of free-charge-carrier parameters in semiconductor thin film structures,” Thin Solid Films 455–456, 563–570 (2004).
[Crossref]

M. Schubert, T. Hofmann, C. M. Herzinger, and W. Dollase, “Generalized ellipsometry for orthorhombic, absorbing materials: dielectric functions, phonon modes and band-to-band transitions of Sb2S3,” Thin Solid Films 455–456, 619–623 (2004).

T. Hofmann, M. Grundmann, C. M. Herzinger, M. Schubert, and W. Grill, “Far-infrared magnetooptical generalized ellipsometry determination of free-carrier parameters in semiconductor thin film structures,” Mater. Res. Soc. Symp. Proc. 744, M5.32.1–M5.32.16 (2003).

M. Schubert, T. Hofmann, and C. M. Herzinger, “Generalized far-infrared magneto-optic ellipsometry for semiconductor layer structures: determination of free-carrier effective-mass, mobility, and concentration parameters in n-type GaAs,” J. Opt. Soc. Am. A 20, 347–356 (2003).
[Crossref]

T. Hofmann, M. Schubert, C. M. Herzinger, and I. Pietzonka, “Far-infrared-magneto-optic ellipsometry characterization of free-charge-carrier properties in highly disordered n-type Al0.19Ga0.33In0.48P,” Appl. Phys. Lett. 82, 3463–3465 (2003).
[Crossref]

A. Kasic, M. Schubert, J. Off, B. Kuhn, F. Scholz, S. Einfeldt, T. Böttcher, D. Hommel, D. J. As, U. Köhler, A. Dadgar, A. Krost, Y. Saito, Y. Nanishi, M. R. Correia, S. Pereira, V. Darakchieva, B. Monemar, H. Amano, I. Akasaki, and G. Wagner, “Phonons and free-carrier properties of binary, ternary, and quaternary group-III Nitride layers measured by infrared spectroscopic ellipsometry,” Phys. Stat. Solidi C 0, 1750–1769 (2003).
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M. Schubert, A. Kasic, T. Hofmann, V. Gottschalch, J. Off, F. Scholz, E. Schubert, H. Neumann, I. J. Hodgkinson, M. D. Arnold, W. A. Dollase, and C. M. Herzinger, “Generalized ellipsometry of complex mediums in layered systems,” Proc. SPIE 4806, 264–276 (2002).
[Crossref]

T. Hofmann, M. Schubert, and C. M. Herzinger, “Far-infrared magneto-optic generalized ellipsometry determination of free-carrier parameters in semiconductor thin film structures,” Proc. SPIE 4779, 90–97 (2002).
[Crossref]

T. Hofmann, V. Gottschalch, and M. Schubert, “Far-infrared dielectric anisotropy and phonon modes in spontaneously CuPt-ordered Ga0.52In0.48P,” Phys. Rev. B 66, 195204 (2002).

A. Kasic, M. Schubert, S. Einfeldt, D. Hommel, and T. E. Tiwald, “Free-carrier and phonon properties of n- and p-type hexagonal GaN films measured by infrared ellipsometry,” Phys. Rev. B 62, 7365–7377 (2000).
[Crossref]

T. E. Tiwald and M. Schubert, “Measurement of rutile TiO2 dielectric tensor from 0.148 to 33 μm using generalized ellipsometry,” Proc. SPIE 4103, 19–29 (2000).
[Crossref]

M. Schubert, B. Rheinländer, J. A. Woollam, B. Johs, and C. M. Herzinger, “Extension of rotating-analyzer ellipsometry to generalized ellipsometry: determination of the dielectric function tensor from uniaxial TiO2,” J. Opt. Soc. Am. A 13, 875–883 (1996).
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M. Schubert, “Polarization-dependent optical parameters of arbitrarily anisotropic homogeneous layered systems,” Phys. Rev. B 53, 4265–4274 (1996).
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M. Schubert, “Generalized ellipsometry,” in Introduction to Complex Mediums for Optics and Electromagnetics, W. S. Weiglhofer and A. Lakhtakia, eds. (SPIE, 2003), pp. 677–710.

M. Schubert, Infrared Ellipsometry on Semiconductor Layer Structures: Phonons, Plasmons and Polaritons, Vol. 209 of Springer Tracts in Modern Physics (Springer, 2004).

M. Schubert, “Theory and application of generalized ellipsometry,” in Handbook of Ellipsometry, E. Irene and H. Tompkins, eds. (William Andrew, 2004).

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S. Schöche, T. Hofmann, V. Darakchieva, N. B. Sedrine, X. Wang, A. Yoshikawa, and M. Schubert, “Infrared to vacuum-ultraviolet ellipsometry and optical Hall-effect study of free-charge carrier parameters in Mg-doped InN,” J. Appl. Phys. 113, 013502 (2013).
[Crossref]

T. Hofmann, P. Kühne, S. Schöche, J.-T. Chen, U. Forsberg, E. Janzén, N. B. Sedrine, C. M. Herzinger, J. A. Woollam, M. Schubert, and V. Darakchieva, “Temperature dependent effective mass in AlGaN/GaN high electron mobility transistor structures,” Appl. Phys. Lett. 101, 192102 (2012).
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Other (44)

B. Rheinländer, “Infrarot-Faraday-Effekt an Halbleitern,” Master’s thesis (Universität Leipzig, 1965).

M. Schubert, “Theory and application of generalized ellipsometry,” in Handbook of Ellipsometry, E. Irene and H. Tompkins, eds. (William Andrew, 2004).

Note that electric field vectors E contain four independent pieces of information if the plane wave is fully coherent and time harmonic. Representing time averages over infinite observation times, the four parameters can be used to characterize the electric field amplitude, absolute phase, ellipticity, and orientation of the polarization ellipse.

A. Gerrard and J. Burch, Introduction to Matrix Methods in Optics, Dover Books on Physics (Dover, 1994).

D. Goldstein, Polarized Light, 3rd ed. (CRC Press, 2011).

Note the four independent pieces of information contained in the Stokes vector. The four parameters can be used to characterize the total light intensity, degree of polarization, ellipticity, and orientation of the polarization ellipse.

A. Röseler, Infrared Spectroscopic Ellipsometry (Akademie-Verlag, 1990).

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Note that the constant but generally complex amplitude parameter in Eq. (30) also may be augmented with a frequency-dependent imaginary part in order to represent the effect of an harmonic coupling. See also [150].

The same formalism can be used in case of holes but with a different effective mass parameter.

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A common requirement in theoretical studies of the electromagnetic response of matter consists in the imposition that a specific medium should be Lorentz reciprocal. For a dielectric medium (the magnetic susceptibility tensor being diagonal and unity), this means that the dielectric tensor is equal to its transposed form. The magnetized plasma and more general types of gyrotropic mediums belong to the most prominent representatives of nonreciprocal mediums. A gyrotropic material is a material in which left- and right-rotating elliptical polarizations can propagate at different speeds. The gyrotropic effect caused by a quasi-static magnetic field breaks the time-reversal symmetry as well as the Lorentz reciprocity. For more information see, for example, [102].

Corresponding expressions for arbitrary orientations of the magnetic field are given by EC′=Aφ,θ,ψ(R)EC, with the Euler angles given by B|B|=Aϕ,θ,ψ(R)(0,0,1)T.

Corresponding transformation matrices for arbitrary orientations of the magnetic field are given by (A(C))=(Aϕ,θ,ψ(R))−1A(C)Aϕ,θ,ψ(R), with the Euler angles given by B|B|=Aϕ,θ,ψ(R)(0,0,1)T.

For ΨE=π/4 and ΔE=π/2, the elliptic eigensystem is equivalent to the circular eigensystem A(E)=A(C).

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Note that Eq. (17) must hold for any E0; thus, Eq. (17) can be stated for each λk separately.

In the following equation the Einstein notation is used, and the covariance and contravariance are ignored because all coordinate systems are Cartesian. The summation is only executed over pairs of lower indices.

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Conceptually, a magneto electric optical Hall effect also may exist, where a current driven by the time-harmonic electric field component, under the influence of an external magnetic field, produces in addition to, or separately from a magneto-optic dielectric displacement, a magnetization response.

The dielectric tensor is considered nonlocal in time but local in space, that is, frequency dependent but not wave vector dependent. A charged compressible fluid model resulting in a dielectric tensor for a nonlocal spatial response is described by Weiglhofer. In principle, the optical Hall effect should be observable in semiconductors with very large carrier concentrations where nonlocal spatial effects may need to be considered.

C. Klingshirn, Semiconductor Optics (Springer-Verlag, 1995).

The electrical Hall effect is well known and has been described in many textbooks. It is beyond the scope of this paper to provide an in-depth review of the electrical Hall effect. The primary limitation in the electrical Hall effect is the physical requirement of ohmic contacts. Due to the fact that proper electric contact formation requires precise knowledge of surface potential functions for any given material and appropriate technological procedures, it is commonly difficult to provide equal contacts to multiple layered structures. Usually, contacts are made to the surface or bottom layer. Often, the actual passage, which the driving currents will take within the sample structure, is difficult to ascertain and hampers accurate data analysis.

P. Yu and M. Cardona, Fundamentals of Semiconductors (Springer, 1999).

W. S. Weiglhofer and A. Lakhtakia, Introduction to Complex Mediums for Optics and Electromagnetics (SPIE, 2003).

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Figures (5)

Fig. 1.
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 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.
Fig. 2.
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 P FCC produces P Hall due to the Lorentz force, oriented perpendicular to B and the incident electric field vector (analogous to the transverse Hall voltage). P FCC + P 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.
Fig. 3.
Fig. 3. Left-handed [ E + , Figs. 3(a) and 3(c)] and right-handed [ 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 B . The field B is collinear with the wave propagation direction. The displacement field phasors P ± are proportional to complex-valued, frequency-dependent response functions χ ± ( B ) . Symmetry requires switch of indices upon reversal of the magnetic field: χ ± ( B ) = χ ( B ) . The latter statement originates from the assumption that P ± does not depend on propagation direction of E ± but only on the course of the electric field phasor at a given plane within the material. Functions χ ± then determine the symmetric and antisymmetric parts in the optical Hall effect tensor [Eq. (11)].
Fig. 4.
Fig. 4. Wave vector k in of the incoming electromagnetic plane wave and the sample normal n define the angle of incidence Φ and the plane of incidence. The amplitudes of the electric field of the incoming E in and the reflected E out plane wave can be decomposed into complex field amplitudes E p in , E s in , E p out , and E s out , where the indices p and s stand for parallel and perpendicular to the plane of incidence, respectively.
Fig. 5.
Fig. 5. Schematic presentation, under the incoming angle Φ , of the electromagnetic wave E I , and the reflected E R , transmitted E T , and backward-traveling electromagnetic waves E B used in the 4 × 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 E 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.

Equations (60)

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D = ϵ 0 E + P = ϵ 0 E + χ E = ϵ 0 ( I + χ ) E = ϵ 0 ϵ E ,
ϵ = I + χ = I + k χ k ,
ϵ ( x , y , z ) = ( ϵ x x ϵ x y ϵ x z ϵ y x ϵ y y ϵ y z ϵ z x ϵ z y ϵ z z ) = I + ( χ x x χ x y χ x z χ y x χ y y χ y z χ z x χ z y χ z z ) .
ϵ i j = δ i j + 1 2 ( χ i j + χ j i ) + 1 2 ( χ i j χ j i ) ,
ϵ ( ± B ) = I + χ B = 0 + χ ± B .
ϵ x , y , z = A 1 ϵ ξ , η , ς A = I + k A 1 ( χ ξ k 0 0 0 χ η k 0 0 0 χ ς k ) A ,
A ϕ , θ , ψ ( R ) = ( cos ψ sin ψ 0 sin ψ cos ψ 0 0 0 1 ) ( 1 0 0 0 cos θ sin θ 0 sin θ cos θ ) ( cos ϕ sin ϕ 0 sin ϕ cos ϕ 0 0 0 1 ) .
P C = ( χ + 0 0 0 χ 0 0 0 1 ) E C .
E C = ( E + E E z ) = ( 1 2 ( E x i E y ) 1 2 ( E x + i E y ) E z ) .
A ( C ) = 1 2 ( 1 i 0 1 i 0 0 0 2 ) .
χ ( ± B ) = 1 2 ( ( χ + + χ ) i ( χ + χ ) 0 ± i ( χ + χ ) ( χ + + χ ) 0 0 0 0 ) .
A ( E ) = 1 a a 1 ( a 1 0 a 1 1 0 0 0 a a 1 ) ,
ϵ Ξ k = λ k Ξ k , k = 1,2 , 3 ;
E G = a 1 Ξ 1 + a 2 Ξ 2 + a 3 Ξ 3 .
A ( G ) = 1 det ( A ( G ) ) ( Ξ 1 , x Ξ 1 , y Ξ 1 , z Ξ 2 , x Ξ 2 , y Ξ 2 , z Ξ 3 , x Ξ 3 , y Ξ 3 , z ) .
0 = det ( ϵ ) λ 1 λ 2 λ 3 = det ( A ϵ A 1 ) λ 1 λ 2 λ 3 = det ( A ) det ( ϵ ) det ( A 1 ) λ 1 λ 2 λ 3 = det ( ϵ ) λ 1 λ 2 λ 3 .
u t = R e [ div S ] = E D ˙ * + E * D ˙ + H B ˙ * + H * B ˙ = ω i E ϵ * E * i E * ϵ E + i H μ * H * i H * H = ω I m [ E 0 * ϵ E 0 ] + ω I m [ H 0 * μ H 0 ] 0 ,
I m [ λ k ] 0 ,
m x ¨ + m γ x ˙ + m ω 0 2 x = q E + q ( x ˙ × B ) ,
μ opt = q m 1 γ 1 .
E = 1 N q [ i m q ω ( ω 0 2 ω 2 I i ω γ ) j + ( B × j ) ] ,
χ i k L D = N q 2 ϵ 0 [ m i k ( ω 0 , i k 2 ω 2 i ω γ i k ) i ω ϵ i j k q B j ] 1 .
ϵ L = I + χ L = ( ϵ x L 0 0 0 ϵ y L 0 0 0 ϵ z L ) .
ϵ k L = 1 + χ k L = ϵ , k j = 1 l ω 2 + i ω γ LO , k , j ω LO , k , j 2 ω 2 + i ω γ TO , k , j ω TO , k , j 2 ,
χ D = χ B = 0 D + χ ± B D ,
χ B = 0 D = ω p 2 ω ( ω + i γ ) I = χ D I ,
ω p = N q 2 m ϵ 0 ,
χ ± = χ D 1 ω c ω + i γ ,
ω c = q | B | m .
χ ± = e ± i ϕ k A k ω 0 , k 2 ω 2 i γ k ω ,
E Graphite L L ( n , B ) = ω c ( n + n 0 ) ,
E SLG LL ( n , B ) = sign ( n ) E 0 | n | ,
E N B L G LL ( n , μ , B ) = sign ( n ) 1 2 [ ( λ N γ ) 2 + ( 2 | n | + 1 ) E 0 2 + μ ( λ N γ ) 4 + 2 ( 2 | n | + 1 ) E 0 2 ( λ N γ ) 2 + E 0 4 ] 1 / 2 ,
E out = JE in .
J = ( j p p j p s j s p j s s ) .
r p p = ( E p out E p in ) E s in = 0 , r p s = ( E s out E p in ) E s in = 0 , r s p = ( E p out E s in ) E p in = 0 , r s s = ( E s out E s in ) E p in = 0 .
( S 1 S 2 S 3 S 4 ) = ( E p E p * + E s E s * E p E p * E s E s * E p E s * + E p * E s i ( E p E s * E p * E s ) ) = ( I p + I s I p I s I + 45 I 45 I σ + I σ )
S out = MS in ,
M = ( M 11 M 12 M 13 M 14 M 21 M 22 M 23 M 24 M 31 M 32 M 33 M 34 M 41 M 42 M 43 M 44 ) .
M i j = 1 2 Tr ( J σ i J σ j ) ,
σ 1 = ( 1 0 0 1 ) , σ 2 = ( 1 0 0 1 ) ,
σ 3 = ( 0 1 1 0 ) , σ 4 = ( 0 i i 0 ) .
M is = ( 1 2 ( r p p r p p * + r s s r s s * ) 1 2 ( r p p r p p * r s s r s s * ) 0 0 1 2 ( r p p r p p * r s s r s s * ) 1 2 ( r p p r p p * + r s s r s s * ) 0 0 0 0 R e ( r p p r s s * ) I m ( r p p r s s * ) 0 0 I m ( r p p r s s * ) R e ( r p p r s s * ) ) ,
M an = ( 1 2 ( r p s r p s * + r s p r s p * ) 1 2 ( r p s r p s * r s p r s p * ) R e ( r p p r p s * + r s s r s p * ) I m ( r p p r p s * r s s r s p * ) 1 2 ( r p s r p s * r s p r s p * ) 1 2 ( r p s r p s * + r s p r s p * ) R e ( r p p r p s * r s s r s p * ) I m ( r p p r p s * + r s s r s p * ) R e ( r p p r s p * + r s s r p s * ) R e ( r p p r s p * r s s r p s * ) R e ( r p s r s p * ) I m ( r p s r s p * ) I m ( r p p r s p * r s s r p s * ) I m ( r p p r s p * + r s s r p s * ) I m ( r p s r s p * ) R e ( r p s r s p * ) ) .
M B = M ( ϵ B 0 ( k ) ) .
φ F K = 1 2 arctan [ M 31 B + M 32 B cos [ 2 β ] + M 33 B sin [ 2 β ] tan [ 2 β ] ( M 21 B + M 22 B cos [ 2 β ] + M 23 B sin [ 2 β ] ) M 21 B + M 22 B cos [ 2 β ] + M 23 B sin [ 2 β ] ± tan [ 2 β ] ( M 31 B + M 32 B cos [ 2 β ] + M 33 B sin [ 2 β ] ) ] ,
Ψ z = i ω c Δ Ψ ,
( E p I E s I E p R E s R ) = L ( E p T E s T E p B E s B ) ,
L = L R 1 ( k = 1 m L P k ) L T .
Δ = ( k x ϵ z x ϵ z z k x ϵ z y ϵ z z 0 1 k x 2 ϵ z z 0 0 1 0 ϵ y x + ϵ y z ϵ z x ϵ z z k x 2 ϵ y y + ϵ y z ϵ z y ϵ z z 0 k x ϵ y z ϵ z z ϵ x x ϵ x z ϵ z x ϵ z z ϵ x y ϵ x z ϵ z y ϵ z z 0 k x ϵ x z ϵ z z ) ,
L R 1 = 1 2 ( 0 1 ( n R cos Φ ) 1 0 0 1 ( n R cos Φ ) 1 0 ( cos Φ ) 1 0 0 n R 1 ( cos Φ ) 1 0 0 n R 1 ) ,
L T = ( 0 0 cos Φ T cos Φ T 1 1 0 0 n T cos Φ T n T cos Φ T 0 0 0 0 n T n T ) .
cos Φ T = 1 ( n R / n T ) 2 sin 2 Φ .
L P k = exp ( i ω c Δ k d k ) = j = 0 3 β j k Δ k j .
β n = j = 0 3 α n exp ( i ω q j ( d ) / c ) ( q j q k ) ( q j q l ) ( q j q m ) ,
α 0 = q k q l q m , α 1 = q k q l + q k q m + q l q m , α 2 = ( q k + q l + q m ) , α 3 = 1 ,
r p p = L 11 L 43 L 13 L 41 L 11 L 33 L 13 L 31 , r p s = L 33 L 41 L 31 L 43 L 11 L 33 L 13 L 31 , r s p = L 11 L 23 L 13 L 21 L 11 L 33 L 13 L 31 , r s s = L 33 L 21 L 31 L 23 L 11 L 33 L 13 L 31 .
φ = 1 2 arctan [ M 23 M 33 ] .
φ = ± 1 2 arctan [ 2 ω c R e [ χ B eff ( 1 + n T i χ LL * ) ] γ | χ B eff | 2 | 1 + n T + i ( ( ω + i γ ) χ B eff + χ LL ) | 2 | χ B eff | 2 ] ,
χ B eff = ω c ω + i γ ( ω + i γ ) 2 ω c 2 , χ B = 0 D = ω p 2 c 1 ( ω + i γ ) 2 ω c 2 .

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