Theory of electromagnetic wave propagation in ferromagnetic Rashba conductor
We present a comprehensive study of various electromagnetic wave propagation phenomena in a ferromagnetic bulk Rashba conductor from the perspective of quantum mechanical transport. In this system, both the space inversion and time reversal symmetries are broken, as characterized by the Rashba field and magnetization , respectively. First, we present a general phenomenological analysis of electromagnetic wave propagation in media with broken space inversion and time reversal symmetries based on the dielectric tensor. The dependence of the dielectric tensor on the wave vector and are retained to first order. Then, we calculate the microscopic electromagnetic response of the current and spin of conduction electrons subjected to and , based on linear response theory and the Green’s function method; the results are used to study the system optical properties. Firstly, it is found that a large enhances the anisotropic properties of the system and enlarges the frequency range in which the electromagnetic waves have hyperbolic dispersion surfaces and exhibit unusual propagations known as negative refraction and backward waves. Secondly, we consider the electromagnetic cross-correlation effects (direct and inverse Edelstein effects) on the wave propagation. These effects stem from the lack of space inversion symmetry and yield -linear off-diagonal components in the dielectric tensor. This induces a Rashba-induced birefringence, in which the polarization vector rotates around the vector . In the presence of , which breaks time reversal symmetry, there arises an anomalous Hall effect and the dielectric tensor acquires off-diagonal components linear in . For , these components yield the Faraday effect for the Faraday configuration , and the Cotton-Mouton effect for the Voigt configuration (). When and are noncollinear, - and -induced optical phenomena are possible, which include nonreciprocal directional dichroism in the Voigt configuration. In these nonreciprocal optical phenomena, a “toroidal moment,” , and a “quadrupole moment,” , play central roles. These phenomena are strongly enhanced at the spin-split transition edge in the electron band.
Momentum-dependent spin-orbit coupled systems such as topological insulators, Weyl semimetals, and Rashba conductors having broken symmetries have recently attracted widespread attention. These media exhibit electromagnetic cross-correlation effects due to the momentum-dependent spin-orbit interaction and, thus, a rich variety of optical phenomena can be expected Tse-MacDonald10 (); Ohnoutek16 (); Grushin12 (); Burkov12 (); Tewari13 (); Franz13 (); Kargarian15 (); Ma15 (); Zhong16 (); STKT16 ().
Among them, optical phenomena induced by the spatial dispersion and/or magnetization in media lacking space inversion and/or time reversal symmetries are important for elucidating the electromagnetic responses of electrons. Such phenomena have been the subject of numerous studies performed in gasses, liquids, and solids Barron82 (); Agranovich84 (). These optical phenomena are well described by the Maxwell equations and governed by the symmetry properties of the dielectric tensor of the medium Agranovich84 (); Landau (). Here, and are the wave vector and (angular) frequency of the electromagnetic wave, respectively, and is either the spontaneous magnetization, which exists in a ferromagnetic medium, or an applied static magnetic field. Microscopically, is related to the quantum mechanical transport coefficients and satisfies the Onsager reciprocity relation
as a consequence of microscopic reversibility. Various optical phenomena caused by the absence of spatial inversion and/or time reversal symmetries (or the presence of ) can be deduced from the expansion of with respect to and Portigal71 (); Rikken98 ():
where , , , and are the tensor coefficients and functions of . Note that, in the above expression and hereafter, repeated indices in a single term imply summation. The Onsager relation (1) requires that the tensors and are symmetric with respect to and , and that and are antisymmetric. That is, , , , and .
Each term of the right-hand side of Eq. (2) has the following significance with regard to optical phenomena. The first term, , determines the fundamental properties of the electromagnetic wave propagation, such as the dispersion relation. The second term, , can only exist when the medium has no spatial inversion symmetry as, otherwise, would be an even function of . Generally, the -dependence of is termed the “spatial dispersion”; however, the term is more specialized in that it is odd in , generating two types of rotation of the polarization vector of linearly polarized light. These two behaviors are known as natural optical activity (Fig. 1(a)) and linear birefringence (Fig. 1(b)) Barron82 (); Raab05 (). In this paper, we refer to these optical phenomena as spatial-dispersion-induced phenomena, or simply, -induced phenomena. The third term, , can exist only when the system breaks the time reversal symmetry, or more specifically, in the presence of magnetization or an applied (static) magnetic field (i.e., ). This term yields magneto-optical phenomena known as the Faraday effect (Fig. 1(c)) and the Cotton-Mouton effect (Fig. 1(d)) Faraday (); Cotton-Mouton (), in which is rotated around . We call these behaviors -induced phenomena. For the - and -induced phenomena such as magneto-chiral birefringence and dichroism (Fig. 2(a)), and nonreciprocal directional dichroism (Fig. 2(b)) HS68 (); Barron84 (); Rikken97 (); Wagniere98 (); Wagniere99 (); Krichevtsov00 (); Vallet01 (); Train08 (); Tokura11 (); Mochizuki13 (); Furukawa14 (); Tomita14 (), the system is required to simultaneously break the space inversion and time reversal symmetries. These phenomena are described by the fourth term of Eq. (2), , which is bilinear in and (i.e., linear in both and ).
The - and -induced phenomena described above have a common aspect of rotation; however, there is an essential difference in the reciprocity. That is, the wave propagations in the -induced phenomena are reciprocal, i.e., the polarization vectors () of the linearly polarized forward () and backward () waves rotate in mutually opposite directions. In contrast, the wave propagations in the magnetization-induced phenomena are nonreciprocal, i.e., rotates in the same direction.
In this paper, we study - and/or -induced optical phenomena in a medium with low symmetry based on each term in Eq. (2) for . To do so, we evaluate each term microscopically and investigate the electromagnetic wave propagations by solving the Maxwell (wave) equation. As a concrete microscopic model, we focus on a free electron model with Rashba-type spin-orbit coupling and (or a static external magnetic field), e.g., a ferromagnetic bulk Rashba conductor STKT16 ().
The remainder of this paper is organized as follows. In the first half of the paper (Secs. II–IV), general aspects of electromagnetic wave propagation are described based on a phenomenological symmetry argument. In detail, in Sec. II, a brief overview of the ferromagnetic Rashba conductor is provided. In Sec. III, we derive the wave equation for an electric field propagating in a ferromagnetic and electromagnetically cross-correlated material. The relations between , Eq. (2), and various transport coefficients are also given. In Sec. IV, we solve the wave equation by considering each term in Eq. (2) for , and discuss possible optical phenomena.
In the second half of the paper (Secs. V–IX), we present microscopic analyses by considering a ferromagnetic Rashba conductor specifically. In Sec. V, after defining the model, we formulate current and spin responses to an electromagnetic field based on linear response theory. Calculated results for the transport properties for a nonmagnetic Rashba conductor and ferromagnetic bulk Rashba conductor are presented in Secs. VI and VII, respectively. These results are then used in Sec. VIII to demonstrate various wave propagations in a nonmagnetic bulk Rashba conductor, which include negative refraction, backward waves, and Rashba-induced birefringence. In Sec. IX, optical phenomena in the ferromagnetic Rashba conductor are studied, including the Faraday and Kerr effects and the nonreciprocal directional dichroism. Finally, the findings of this study are summarized in Sec. X. Various calculation details are given in the Appendices.
Ii Ferromagnetic Rashba conductor
Recently, momentum-dependent giant spin splitting was found in the electron band of the BiTeI polar semiconductor Ishizaka11 (); Lee-Tokura11 (); Tokura12 (). This behavior, called the “Rashba effect,” is ascribed to the Rashba spin-orbit interaction (RSOI), which is expected in systems without space inversion symmetry Rashba60 (). The BiTeI polar semiconductor has a layered structure stacked along the axis with a trigonal crystal symmetry. The spin splitting is , which corresponds to Å, where is the Rashba spin-orbit field specifying the strength and direction of the RSOI:
where is a momentum operator and is a vector of Pauli spin matrices. Furthermore, it was proposed that provides a highly anisotropic property to the medium and that the bulk Rashba conductor can be regarded as a kind of hyperbolic material STKT16 ().
To date, hyperbolic media have been realized through artificial engineering in metal-dielectric multilayer systems having metallic in-plane and insulating inter-plane properties Podolskiy05 (); Hoffman07 (); Liu08 (); Hoffman09 (); Harish12 (); JOpt12 (); APN12 (); Poddubny13 (); Esslinger14 (); Narimanov15 (); PQE15 (). Such materials have hyperbolic dispersion surfaces for electromagnetic waves in a certain frequency range and, thus, exhibit unusual electromagnetic responses such as negative refraction and backward waves Veselago (); Pendry96 (); Pendry00 (); Smith00 (); Lindell01 (); Smith03 (); Belov03 (). This hyperbolic frequency region has also been found in natural materials, e.g., tetradymites (, , and ) Poddubny13 (); Esslinger14 (); Narimanov15 (). Interestingly, these behaviors are already described by the first term of Eq. (2) (see Secs. IV-A and VIII-A).
The Rashba effect has also attracted attention in the field of spintronics, because of interesting electromagnetic cross-correlation effects. One is the Edelstein effect Edelstein (), in which a nonequilibrium spin accumulation
is induced by an external electric field Obata08 (); Manchon2008 (); Matos2009 (); Manchon12 (); Kim12a (); MacDonald12 (); Duine12 (); Titov15 (). Here, is a frequency-dependent coefficient, is a unit vector, and is the Fourier amplitude of the external electric field. Such a spin accumulation is observable as a torque on , which is called the spin-orbit torque Brataas14 (), through the exchange interaction between the electron spin and magnetization, such that
Indeed, magnetization reversal and magnetic domain wall motion on the surface of a ferromagnetic ultrathin metal sandwiched between a heavy metal layer and an oxide layer have been proposed and performed experimentally Miron08 (); Miron10 (); Miron11a (); Miron11b ().
As a reciprocal cross-correlation effect, the inverse Edelstein effect has also been studied intensively Fert-Nat-Comm-13 (); Nomura15 (); Sangiao15 (); Zhang15 (); Isasa16 (). In this effect, a non-equilibrium spin current is pumped into a heavy metal layer (such as Bi/Ag bilayer) from a ferromagnetic layer (such as NiFe) by the ferromagnetic resonance, and is converted to a charge Hall current through the RSOI Fert-Nat-Comm-13 (). The generation of motive force by the dynamics of magnetization in a ferromagnetic Rashba metal has also been studied theoretically in Refs Kim12b, and Nakabayashi-Tatara13, . Theoretically, the polarization current induced by the inverse Edelstein effect is given by Raimondi14 (); STKT16 ()
where is a time-dependent external magnetic field and is the transport coefficient, which is related to through the Onsager relation STKT16 ()
where is the gyromagnetic ratio (see Sec. VI). Thus, the total current induced by the electromagnetic field is given by
where is a -induced effective magnetic field (see Sec. VI). Thus, a momentum-dependent spin-orbit coupling yields an electromagnetic response that involves a Hall effect due to , which induces a rotation of around .
The corresponding optical conductivity , defined by , is read from Eq. (9), such that
where is the completely antisymmetric tensor with . Thus, in a Rashba conductor, the electromagnetic cross-correlation effects contribute to the antisymmetric (hence, off-diagonal) part of ,
where is the vacuum permittivity. This is linear in and induces linear birefringence (see Sec. IV-B).
In the presence of , a Rashba conductor may exhibit an anomalous Hall effect Inoue (); Kovalev (); BRV11 (); Titov16 (). This effect is maximal when and are parallel or antiparallel, and the induced current has the form
where is the anomalous Hall conductivity (see Sec. VII). This effect exists even at and contributes to the third term of Eq. (2) (see Sec. VII-C):
These off-diagonal components, which are linear in , yield the Faraday and Kerr effects for the Faraday configuration (), and the Cotton-Mouton effect for the Voigt configuration (); see Sec. IV-C. Recently, a large Kerr effect was observed in BiTeI under a static magnetic field Tokura12 ().
Finally, the bilinear term in Eq. (2), , which describes the -induced spatial dispersion, can be deduced from the magnetoresistance effect Potter75 () due to and . The relevant current density may be written as
where is a tensor symmetric under . If (neglecting other possible terms), where is a frequency-dependent coefficient, a “Doppler shift” term Kawaguchi-Tatara16 () (with ) appears in the diagonal components of , with
As reflects the broken inversion symmetry, it is a polar vector similar to an electric polarization vector . Thus, the vector is an analog of the toroidal moment discussed in the context of multiferroics Spaldin08 (); TSN14 (). Recently, a giant nonreciprocal directional dichroism induced by the toroidal moment was observed in Arima08-1 (). Such optical phenomena are described by a term linear in both and and in the diagonal component of . These points are pursued further in Sec. VII.
Iii Wave equation
In this section, we derive the wave equation for the electric field that propagates in electromagnetically cross-correlated materials with broken space inversion and time reversal symmetries. This derivation clarifies the connections of the various correlation functions to .
In general, the Fourier components of the electric and magnetic fields are expressed as
where and are the permittivity and magnetic permeability of free space, respectively. and are the electric displacement and magnetic field intensity, respectively, which are related to and through the electric polarization and magnetization of the medium that includes the conduction electrons. The Fourier representation of Faraday’s law and the Maxwell-Ampére equation in the absence of external electric currents are respectively expressed as
where is an induced current density that consists of the polarization current density and the magnetization current density . That is,
Microscopically, the and induced by electromagnetic fields are evaluated on the basis of linear response theory, as
where , , , and are the current-current, current-spin, spin-current, and spin-spin correlation (or response) functions, respectively. The second and first terms on the right-hand sides of Eqs. (22) and (23) represent the electromagnetic cross-correlation effects, which are mutually related through the Onsager reciprocity relation
Similarly, the other two response functions satisfy
Proof of these relations under certain conditions is given in Appendix A.
Using Faraday’s law [Eq. (18)] to eliminate the magnetic field, we obtain
where is the optical conductivity, with
is the dielectric tensor. It is apparent that the optical properties of cross-correlated materials are governed by all types of correlation functions through the given in Eq. (III). From Eqs. (III) and (32), each term of in Eq. (2) is expressed in a microscopic sense as
Note that explicit evaluation of these expressions is performed in Secs. V and VI. Here, it is sufficient to calculate to first order in and , and and to first order in at . As for , the last term of Eq. (III) can be dropped as it is already second-order in . However, we note that all correlation functions for materials with momentum-dependent spin-orbit coupling involve , because of the anomalous velocity that contains a spin operator.
Iv Phenomenological analysis of wave propagation
In this section, we study electromagnetic wave propagations in low-symmetry media based on the wave equation (31), by considering each term of Eq. (2) for phenomenologically. Various optical phenomena, as listed in Table I, are classified according to the form of . By choosing the wave propagation direction to be in the - plane, we have , where is a unit vector in the -direction. Then, we can write Eq. (31) as
where . Various optical phenomena expected in electromagnetically cross-correlated media are contained within this wave equation.
|Dielectric tensor||I||T||Optical phenomenon||Ferromagnetic Rashba conductor|
|Negative refraction and backward wave|
|(Faraday: )||Faraday and Kerr effects|
|(Voigt: )||Cotton-Mouton effect|
|(Faraday)||MChD and MChB|
Magneto-chiral dichroism, Magneto-chiral birefringence, Nonreciprocal birefringence, Nonreciprocal directional dichroism
iv.1 Effects of anisotropy in
For an isotropic metal or semiconductor, the first term in Eq. (2) describes the conventional symmetric tensor, which takes the diagonal form of , with and being a complex function of and the Kronecker delta in three dimensions, respectively. In this case, the wave equation (37) becomes
The plane wave solution exists if and only if and satisfy the characteristic equation
which yields the dispersion relations and , where
For , the eigen vector is given by with being a scalar. This solution represents the linearly polarized wave. On the other hand, for , the eigen vector is given by , where and are components of the electric field vector satisfying the orthogonality condition . This wave is linearly polarized in the - plane. Thus, in the case of an isotropic medium, it is apparent that conventional wave propagation is obtained in the frequency region satisfying . However, if the medium is anisotropic in nature, the property of the wave propagation changes dramatically. In the case of a uniaxially anisotropic medium for which the optic axis is the -axis, the dielectric tensor has the form
where and are the dielectric constants in the perpendicular and parallel directions, respectively, with respect to the anisotropy (optic) axis. Substituting this tensor into the wave equation (37), we have
The characteristic equation is given by
which yields two types of wave propagation, i.e., ordinary and extraordinary waves, respectively. For the ordinary wave, the dispersion relation is and the eigen vector is . This wave is linearly polarized in the -direction and can propagate in a conducting medium for . On the other hand, for an extraordinary wave, the dispersion relation is given by
and the eigen vector is given by , which satisfies
Thus, the orthogonality condition is not satisfied, i.e., . This means that the wavefront propagation direction and the Poynting vector are not parallel. As noted above, such a wave is called an extraordinary wave. When and , the equifrequency contour of Eq. (44) is elliptical in the -plane, the wave propagation of which is conventional Landau () (Fig. 3(a,b)), where and are refracted to the positive side. On the other hand, when and have opposite sign, the equifrequency dispersion curve becomes hyperbolic, as illustrated in Fig. 3(c, d). Both figures indicate that the transverse component of can have the opposite sign to that of . This indicates that the energy flow of an obliquely incident wave is refracted to the negative side with respect to the interface normal of the medium. This unusual optical phenomenon is called “negative refraction” Lindell01 (); Smith03 (); Belov03 (). On the other hand, although the energy flow direction should be positive, it is possible for the vertical component of the wave vector to be negative. Such an optical phenomenon is called a “backward wave” Lindell01 (); Smith03 (); Belov03 ().
Recently, materials possessing hyperbolic dispersion and known as “hyperbolic materials” have become the focus of research attention JOpt12 (); Poddubny13 (); PQE15 (). These materials consist of a metal-dielectric multilayer Hoffman07 (); Liu08 (); Hoffman09 (); Harish12 () and natural materials Poddubny13 (); Esslinger14 (); Narimanov15 (). In the case of a Rashba conductor, the existence of implies that the system has, at least, a uniaxial anisotropy. The dielectric tensor takes the uniaxial form, . In Sec. VIII, we demonstrate that enhances the anisotropy, suggesting that a medium with large Rashba spin-split bands becomes a kind of hyperbolic material. Such a medium is expected to exhibit unusual electromagnetic wave propagation behavior, as demonstrated in Sec. VIII.
iv.2 Effects of caused by broken space inversion symmetry
When the system breaks the space inversion symmetry, the second term in Eq. (2), , appears in the expression for . Recall that is antisymmetric under . Such -linear off-diagonal components originate from the electromagnetic cross-correlation effects and generate two types of -induced optical phenomena, known as “natural optical activity” Barron82 (); Agranovich84 () and “linear birefringence” Raab05 ().
The term “optical activity” refers to the phenomenon in which the of the linearly polarized light is rotated around (Fig. 1(a)) Barron82 (). This indicates that the refractive indexes of the left- and right-handed circularly polarized waves differ. This difference occurs when the antisymmetric part of the expression takes the form
where is a complex function of . To examine this behavior more closely, we consider the case of and . Thus, the dielectric tensor is given by
This setup yields the wave equation
which in turn yields
As , the plane wave solution exhibits a transverse wave propagating in the -direction. Solving for , we obtain the dispersion relations
Substituting this expression into Eq. (48), we then find the eigen vector
which represents the left- () and right-handed () circularly polarized waves, respectively. In this study, we state that a circularly polarized plane wave is left-handed (right-handed), , if the electric field vector rotates counter-clockwise (clockwise) at a fixed point when viewed from the wave propagation directionJackson ().
Let us consider the optical rotation of the of a linearly polarized wave and evaluate the rotation angle. If an incident wave is initially polarized in the -direction in a vacuum, with , the wave passing through the conductor is given by
Thus, the rotation angle following passage through a conductor of thickness is given by . Note that this optical rotation is reciprocal, i.e., the of the wave propagating backward () in the sample rotates in the opposite direction () to that of the forward wave (). This reciprocity originates from the fact that the present effect comes from the -linear term in the off-diagonal component of the dielectric tensor. Indeed, the electric current induced by this -linear term has the form , indicating that the electric field vector rotates around the vector.
The imaginary part of represents the difference in absorption between left- and right-handed circularly polarized waves, which is called “(optical) circular dichroism.” Thus, the absorption rate per unit length is given by in Eq. (55), which is proportional to the imaginary part of .
Linear birefringence is a phenomenon known as “double refraction,” which is due to two types of linearly polarized waves, i.e., an ordinary and extraordinary wave. The polarization plane of the latter, which is defined by and vectors, is rotated by the -linear term in the off-diagonal component of (Fig. 1(b))Raab05 (). This phenomenon occurs when the antisymmetric part of takes the form
where is a complex function of real and is a unit vector pointing to the optic axis of the medium. Setting and , we obtain
Thus, the wave equation is given by
There are two types of solution, i.e., and , where
The former represents the ordinary wave, the eigen vector of which is , and the latter represents the extraordinary wave, , with
which yields . Thus, the off-diagonal components of induce a longitudinal component of the electric field. Hence, the direction of (the energy flow) of the extraordinary wave is not parallel to .
The degree of birefringence, , is defined by the difference in the refractive indexes of the ordinary and extraordinary waves Raab05 (), such that
The second equality follows when . On the other hand, the tilt angle of the polarization plane (spanned by and ) for the extraordinary wave, which is denoted by