]

# Strongly direction-dependent magnetoplasmons in mixed Faraday-Voigt
configurations

Afshin Moradi1∗ Martijn Wubs2,3,4 [<EMAIL_ADDRESS><EMAIL_ADDRESS>_1 Department of Engineering Physics, Kermanshah University of Technology,
Kermanshah, Iran
2Department of Photonics Engineering, Technical University of Denmark, DK-2800
Kgs. Lyngby, Denmark
3Center for Nanostructured Graphene, Technical University of Denmark, DK-2800
Kgs. Lyngby, Denmark
4NanoPhoton - Center for Nanophotonics, Technical University of Denmark,
DK-2800 Kgs. Lyngby, Denmark _

###### Abstract

The electrostatic theory of magnetoplasmons on a semi-infinite magnetized
electron gas is generalized to mixed Faraday-Voigt configurations. We analyze
a new type of electrostatic surface waves that is strongly direction-
dependent, and may be realized on narrow-gap semiconductors in the THz regime.
A general expression for the dispersion relation is presented, with its
dependence on the magnitude and orientation of the applied magnetic field.
Remarkably, the group velocity is always perpendicular to the phase velocity.
Both velocity and energy relations of the found magnetoplasmons are discussed
in detail. In the appropriate limits the known magnetoplasmons in the higher-
symmetry Faraday and Voigt configurations are recovered.

###### pacs:

41.20.Cv, 73.20.Mf

##  I Introduction

It is well known that a static magnetic field causes various important changes
in the electromagnetic behavior of different media S.A.H.G1158 ; Y.Y554 ;
S.B674 . Also, since the early works by Chiu and Quinn K.W.C4707 ; K.W.C1 , it
is known that on the surface of a semi-infinite magnetized cold electron gas,
a surface magnetoplasmon (SMP) can oscillate at constant frequencies only.
This infinite flat-band dispersion relation holds in the electrostatic
approximation Maier:2007a , and as long as spatial dispersion (a.k.a.
“nonlocal response”) can be neglected Raza:2015a ; Fitzgerald:2016a .

In particular, there are two configurations for which this constant SMP
frequency is given by
$\omega=\sqrt{\omega_{\mathrm{p}}^{2}+\omega_{\mathrm{c}}^{2}}/\sqrt{2}$
H.J.L755 , where $\omega_{\mathrm{p}}$ is the electron plasma frequency and
$\omega_{\mathrm{c}}=eB/m_{e}$ is the electron cyclotron frequency ($e$ and
$m_{e}$ are the elementary charge and the electron mass, respectively). The
first of these configurations is the Faraday configuration, when the applied
magnetic field is parallel both to the surface and to the direction of
propagation of the wave (see sketch in Fig. 1).

Figure 1: Sketch of a semi-infinite magnetized electron gas with static
magnetic field $\textbf{B}_{0}$ parallel to the surface. Here we choose the
direction of $\textbf{B}_{0}$ fixed along the $+\textbf{e}_{x}$ (red vector)
or $-\textbf{e}_{x}$ (blue vector). Note that $\textbf{e}_{x}$ is the unit
vector along the $x$-axis. Also, k is the wavevector of the propagating
surface wave, and $\theta$ (or $180^{\circ}-\theta$) is the angle between k
and $\textbf{B}_{0}$. The special cases $\theta=0$ or $180^{\circ}$ and
$\theta=90^{\circ}$ are called the Faraday and Voigt configurations,
respectively. Here we study magnetoplasmons for the general case of arbitrary
$\theta$, which we call mixed Faraday-Voigt configurations. Note that in the
Voigt configuration the magnetoplasmons travel in the $+\textbf{e}_{y}$
direction. The electron gas is bounded from above by a semi-infinite insulator
at $z=0$.

The other is the so-called perpendicular configuration, when the applied out-
of-plane magnetic field points perpendicularly both to the surface and to the
propagation direction of the wave (not shown in Fig. 1). Interestingly, when
the SMP in the Faraday configuration has strong surface-wave characteristics,
then the SMP in the perpendicular configuration acts like a bulk wave, and
vice versa A.M . In this sense, these Faraday and perpendicular configurations
are complementary.

The Voigt configuration is yet another high-symmetry configuration for which
the magnetoplasmon frequency depends on the magnetic field but again does not
depend on the wave vector. In the Voigt configuration, the surface wave
propagates perpendicularly to the (in-plane) external magnetic field that is
parallel to the interface. Then the plasmon frequency is given by
$\omega_{\mathrm{V}\pm}=\left(\sqrt{\omega_{\mathrm{c}}^{2}+2\omega_{\mathrm{p}}^{2}}\pm\omega_{\mathrm{c}}\right)/2$.
Here the $\pm$ solutions correspond to the Cartesian coordinate system shown
in Fig. 1: for a forward-going SMP, we have the $+$ solution and the SMP
frequency is blueshifted, while for a backward-going wave the frequency of SMP
is redshifted. Clearly, in the limit of vanishing magnetic fields, the plasmon
frequencies reduce to $\omega_{\mathrm{p}}$ in both configurations, as
expected.

Recently, Silveirinha et al. M.G.S022509  and Gangaraj et al. S.A.H.G201108
studied the propagation of SMPs on a semi-infinite magnetized electron gas
when the direction of propagation is oblique to the static magnetic field. In
the electrostatic approximation they found that the frequency of the SMPs does
not depend on the magnitude of the wavevector, as for the Voigt and Faraday
configurations. But now in the oblique configuration the SMP frequency does
depend on the angle of the wavevector with respect to the magnetic field
direction M.G.S022509 . Also, more recently, in the presence of a weakly
static magnetic field gradient, we found dispersive backward and forward
electrostatic waves on a cold magnetized electron gas half-space in the
Faraday configuration A.M054501 .

Here we study the surface magnetoplasmons on a semi-infinite magnetized
electron gas and in this oblique configuration (i.e., a mixed Faraday-Voigt
configuration) in more detail. As an interesting result, we show that new
electrostatic waves exist in such a structure due to differences between the
symmetry of the media in contact, just like the Dyakonov surface waves
M.I.D714 ; O.T126 ; O.T463001 . Note that these surface waves do not exist in
a semi-infinite gas plasma or a semi-infinite electron plasma in a metal
without a magnetic field, and may have application in signal processing with
electrostatic or slow electric waves.

##  II Basic equations for the surface magnetoplasmons

Here we derive conditions that surface magnetoplasmons in the metal-air
interface should satisfy, while in the next section we study the properties of
these SMPs. Consider a semi-infinite magnetized electron gas occupying the
half-space $z<0$ in Cartesian coordinates, as shown in Fig. 1. The plane $z=0$
is the electron gas-insulator interface. Without loss of generality, we assume
that the external magnetic-field vector $\textbf{B}_{0}=\pm
B_{0}\textbf{e}_{x}$ points parallel to the $x$-axis (plus and minus signs
refer to $\textbf{B}_{0}$ in the positive- and negative-$x$ directions). We
will investigate the propagation of a slow electric surface wave
($\textbf{E}\approx-\nabla\Phi$) whose wavevector k points along the interface
at an angle $\theta=\arctan(k_{y}/k_{x})$ (or $180^{\circ}-\theta$) with the
magnetic field $\textbf{B}_{0}=\pm B_{0}\textbf{e}_{x}$.

The electric potential may be represented in the form

Figure 2: Variation of (a) $\varepsilon_{\mathrm{FV}-}$, (b)
$\varepsilon_{\mathrm{FV}+}$, (c) $\varepsilon_{zz}$, and (d)
$\varepsilon_{xx}$, with respect to the dimensionless frequency
$\omega/\omega_{\mathrm{p}}$, when $\varepsilon_{1}=1$, and
$\omega_{\mathrm{c}}=0.5\omega_{\mathrm{p}}$.

$\Phi(x,y,z,t)=\tilde{\Phi}(z)\exp\left[i\left(k_{x}x+k_{y}y-\omega
t\right)\right]\;,$ (1)

where $\omega$ is the angular frequency of the wave and $k_{x}$ and $k_{y}$
are wavenumbers in the $x$\- and $y$-directions, respectively. For the present
geometry and the magnetic field $\textbf{B}_{0}=+B_{0}\textbf{e}_{x}$, the
relative dielectric tensor of the system has the form

$\underline{\varepsilon}(\omega)=\left(\begin{array}[]{ccc}\varepsilon_{xx}&0&0\\\
0&\varepsilon_{yy}&\varepsilon_{yz}\\\ 0&\varepsilon_{zy}&\varepsilon_{zz}\\\
\end{array}\right)\;,$ (2)

with tensor elements given by

$\displaystyle\varepsilon_{xx}$ $\displaystyle=$ $\displaystyle
1-\dfrac{\omega_{\mathrm{p}}^{2}}{\omega\left(\omega+i\gamma\right)}\;,$
$\displaystyle\varepsilon_{yy}=\varepsilon_{zz}$ $\displaystyle=$
$\displaystyle
1-\dfrac{\omega_{\mathrm{p}}^{2}\left(\omega+i\gamma\right)}{\omega\left[\left(\omega+i\gamma\right)^{2}-\omega_{\mathrm{c}}^{2}\right]}\;,$
$\displaystyle\varepsilon_{yz}=-\varepsilon_{zy}$ $\displaystyle=$
$\displaystyle\dfrac{i\omega_{\mathrm{c}}\omega_{\mathrm{p}}^{2}}{\omega\left[\left(\omega+i\gamma\right)^{2}-\omega_{\mathrm{c}}^{2}\right]}\;,$

where $\gamma$ is the damping constant. As mentioned in M.G.S022509 ;
S.A.H.G201108 , narrow-gap semiconductors such as InSb (with
$\omega_{\mathrm{p}}/2\pi\approx 4.9$THz, $\gamma/2\pi\approx 0.5$THz and
$0.25\omega_{\mathrm{p}}\leq\omega_{\mathrm{c}}\leq\omega_{\mathrm{p}}$ for a
bias magnetic field in the range of $1$ to $4$ Tesla have an optical response
analogous to Eq. (2), where for simplicity the contribution of bound electrons
to the permittivity response of InSb is disregarded, and its static (high-
frequency) permittivity is taken identical to unity. Also, for our purposes,
the limit of zero damping, i.e., $\gamma\to 0$ is sufficient. Small-size
limits where nonlocal response (neglected here) would start to play a role in
semiconductor plasmonics including in InSb are discussed in Refs. Maack:2017a
; Maack:2018a .

After substitution Eq. (1) into $\nabla\cdot\textbf{D}=0$, where
$\textbf{D}=\varepsilon_{0}\underline{\varepsilon}\cdot\textbf{E}$ (where
$\varepsilon_{0}$ is the electric permittivity of free space), we find

$\left(\dfrac{d^{2}}{dz^{2}}-\kappa^{2}\right)\tilde{\Phi}(z)=0\;,\qquad\mbox{for}\quad
z\leq 0,\;$ (3)

where

$\kappa=\sqrt{\dfrac{\varepsilon_{xx}k_{x}^{2}+\varepsilon_{yy}k_{y}^{2}}{\varepsilon_{zz}}}\;.$
(4)

The analogous equation for the wave in the insulator is

$\left(\dfrac{d^{2}}{dz^{2}}-k^{2}\right)\tilde{\Phi}(z)=0\;,\qquad\mbox{for}\quad
z\geq 0,$ (5)

with $k=\sqrt{k_{x}^{2}+k_{y}^{2}}$. Therefore, the combined solution of Eqs.
(3) and (5) has the familiar form for a surface wave

$\tilde{\Phi}(z)=\Phi_{0}\left\\{\begin{array}[]{clcr}\exp\left(-kz\right)\;,&\mbox{$z\geq
0$\;,}\\\ \exp\left(+\kappa z\right)\;,&\mbox{$z\leq 0$\;,}\\\
\end{array}\right.$ (6)

where $\Phi_{0}$ is the wave amplitude. For the present system, the
appropriate boundary condition at the separation surface $z=0$ is

$\varepsilon_{1}\dfrac{\partial\Phi_{1}}{\partial
z}\bigg{|}_{z=0}=\pm\varepsilon_{zy}\dfrac{\partial\Phi_{2}}{\partial
y}\bigg{|}_{z=0}+\varepsilon_{zz}\dfrac{\partial\Phi_{2}}{\partial
z}\bigg{|}_{z=0}\;,$ (7)

where subscripts 1 and 2 refer to outside and inside the electron gas,
respectively, $\varepsilon_{1}$ is the relative dielectric constant of the
insulator medium and plus and minus signs refer to $\textbf{B}_{0}$ in the
positive- and negative-$x$ directions, respectively.

##  III Dispersion relation and group velocities

On applying the mentioned boundary condition (7) at $z=0$, we find

$\varepsilon_{1}k+\varepsilon_{zz}\kappa\pm i\varepsilon_{zy}k_{y}=0\;,$ (8)

which leads to a relation between the wave-vector components $k_{x}$ and
$k_{y}$ and the frequency $\omega$,

$\dfrac{k_{y}^{2}}{k_{x}^{2}}=\varepsilon_{\mathrm{FV}\pm}(\omega)\;,$ (9)

with the frequency-dependent function $\varepsilon_{\mathrm{FV}\pm}(\omega)$
to be specified shortly. The dispersion relation Eqs. (9) is remarkable in
that it does not depend on the magnitude $k$ of the wavevector, but only on
the fraction $k_{y}/k_{x}$ of the wave-vector components $k_{x,y}$. In other
words, the problem has a cylindrical symmetry and in agreement with Ref.
M.G.S022509  we will find that the plasmon energies will only depend on the
angle $\theta$. Yet we are interested in propagation along and perpendicular
to the magnetic field and therefore will stick to Cartesian coordinates in
most of what follows.

The frequency-dependent function $\varepsilon_{\mathrm{FV}\pm}(\omega)$ of Eq.
(9) has the form

Figure 3: Dispersion curves of SMPs at a flat magnetized electron gas-vacuum
interface, as obtained from Eq. (9) for
$\omega_{\mathrm{c}}=0.5\omega_{\mathrm{p}}$. Panel (a): $k_{y}$ is constant
(positive). Panel (b): $k_{x}$ is constant (positive).

$\displaystyle\varepsilon_{\mathrm{FV}\pm}$ $\displaystyle=$
$\displaystyle\dfrac{-B\pm\sqrt{B^{2}-4AC}}{2A}\;,\qquad\mbox{with}$
$\displaystyle A$ $\displaystyle=$
$\displaystyle\left(\varepsilon_{1}^{2}-\varepsilon_{zy}^{2}-\varepsilon_{yy}\varepsilon_{zz}\right)^{2}+4\varepsilon_{1}^{2}\varepsilon_{zy}^{2}\;,$
$\displaystyle B$ $\displaystyle=$ $\displaystyle
2\left(\varepsilon_{1}^{2}-\varepsilon_{zy}^{2}-\varepsilon_{yy}\varepsilon_{zz}\right)\left(\varepsilon_{1}^{2}-\varepsilon_{xx}\varepsilon_{zz}\right)+4\varepsilon_{1}^{2}\varepsilon_{zy}^{2}\;,$
$\displaystyle C$ $\displaystyle=$
$\displaystyle\left(\varepsilon_{1}^{2}-\varepsilon_{xx}\varepsilon_{zz}\right)^{2}\;.$

It is clear that the frequency dependence of
$\varepsilon_{\mathrm{FV}\pm}(\omega)$ originates fully from the assumed
frequency dispersion of the dielectric functions.

We will now look for propagating-wave solutions in the $x$\- and
$y$-directions, so that the right-hand side of Eq. (9) must be positive-
valued. Working with the reduced variables $\omega/\omega_{\mathrm{p}}$ and
$\omega_{\mathrm{c}}/\omega_{\mathrm{p}}$, we depict in Fig. 2 the variation
of (a) $\varepsilon_{\mathrm{FV}-}$, (b) $\varepsilon_{\mathrm{FV}+}$, (c)
$\varepsilon_{zz}$, and (d) $\varepsilon_{xx}$, with respect to the
dimensionless frequency $\omega/\omega_{\mathrm{p}}$, when $\varepsilon_{1}=1$
and $\omega_{\mathrm{c}}=0.5\omega_{\mathrm{p}}$. From Fig. 2(a,b) it then
follows that only the lower-branch solution
$\varepsilon_{\mathrm{FV}-}(\omega)$ can lead to propagating-wave solutions in
Eq. (9), and only in a finite range of frequencies. Furthermore, for having a
surface wave, $\varepsilon_{zz}$ in panel (c) must be negative. There is
indeed a region below and above the SMP frequencies in the Voigt
configuration, i.e., $\omega_{\mathrm{V}-}=0.5\omega_{\mathrm{p}}$ and
$\omega_{\mathrm{V}+}=\omega_{\mathrm{p}}$, where $\varepsilon_{\mathrm{FV}-}$
is positive and $\varepsilon_{zz}$ is negative, and it is there that the
conditions for propagating SMPs are satisfied. In this region,
$\varepsilon_{xx}$ is also negative, as shown in panel (d). Note that the
existence of these SMPs is due to differences between the anisotropy of the
two media, similar to Dyakonov surface waves M.I.D714 ; O.T126 ; O.T463001 .
However, Dyakonov surface waves are a type of electromagnetic waves localized
at an interface between two transparent media: an isotropic medium and a
uniaxial crystal. In contrast to the Dyakonov surface waves, here SMPs are a
type of electrostatic (or, more accurately, quasi-electrostatic) waves
localized at an interface between an isotropic medium and an electric-
gyrotropic electron plasma A.M2947 .

In Fig. 3(a) we show the variation of $\omega/\omega_{\mathrm{p}}$ with
respect to $k_{x}/k_{y}$, for a positive constant value of $k_{y}$, while in
panel (b) we show the same as a function of $k_{y}/k_{x}$, for a positive
constant value of $k_{x}$. Two curves appear in each panel in agreement with
$\textbf{B}_{0}=\pm B_{0}\textbf{e}_{x}$. Note that for general dispersion
relations it would be important to state the value of the wavevector that is
kept constant, but not so for our dispersion relation where the dependence is
only on the fraction $k_{y}/k_{x}$ (or $k_{x}/k_{y}$). The panels in Fig. 3
give complementary information. Alternatively, one could plot the frequencies
as a function of the angle $\theta$ (not shown).

If we consider $k_{y}$ to be fixed and positive and take the symbol “$+$” in
Eq. (8) (i.e., $\textbf{B}_{0}=+B_{0}\textbf{e}_{x}$), then for $k_{x}>0$
($k_{x}<0$) there is a SMP with $v_{\mathrm{g}x}>0$ ($v_{\mathrm{g}x}<0$) in
the region below the line
$\omega=\omega_{\mathrm{F}}=\sqrt{\omega_{\mathrm{p}}^{2}+\omega_{\mathrm{c}}^{2}}/\sqrt{2}$
and above the line $\omega=\omega_{\mathrm{V}-}$ (see red curve in panel (a)
of Fig. 3). For $v_{\mathrm{g}x}>0$, we note that $k_{x}$ and $k_{y}$ are both
positive, while for $v_{\mathrm{g}x}<0$, we have $\left(-k_{x},+k_{y}\right)$.

If we consider $k_{y}$ to be fixed and positive and take the symbol “$-$” in
Eq. (8) (i.e., $\textbf{B}_{0}=-B_{0}\textbf{e}_{x}$), then for $k_{x}>0$
($k_{x}<0$) there is a SMP with $v_{\mathrm{g}x}<0$ ($v_{\mathrm{g}x}>0$) in
the region below the line $\omega=\omega_{\mathrm{V}+}$ and above the line
$\omega=\omega_{\mathrm{F}}$ (blue curve in panel (a) of Fig. 3). Again, for
$v_{\mathrm{g}x}<0$ one can find that $k_{x},k_{y}>0$, while for
$v_{\mathrm{g}x}>0$, we obtain $\left(-k_{x},+k_{y}\right)$.

If we consider $k_{x}$ to be fixed with positive sign and take the symbol
“$+$” in Eq. (8) (i.e., $\textbf{B}_{0}=+B_{0}\textbf{e}_{x}$), then there is
a SMP with $v_{\mathrm{g}y}<0$ ($v_{\mathrm{g}y}>0$ ) in the region below the
line $\omega=\omega_{\mathrm{F}}$ and above the line
$\omega=\omega_{\mathrm{V}-}$ (see red curve in panel (b) of Fig. 3) for
$k_{y}>0$ ($k_{y}<0$). For $v_{\mathrm{g}y}<0$, we have
$\left(+k_{x},+k_{y}\right)$, while $\left(+k_{x},-k_{y}\right)$ yields
$v_{\mathrm{g}y}>0$.

Finally, if we consider $k_{x}$ to be fixed with positive sign and take the
symbol “$-$” in Eq. (8) (i.e., $\textbf{B}_{0}=-B_{0}\textbf{e}_{x}$), then
there is a SMP with $v_{\mathrm{g}y}>0$ ($v_{\mathrm{g}y}<0$) in the region
below the line $\omega=\omega_{\mathrm{V}+}$ and above the line
$\omega=\omega_{\mathrm{F}}$ (blue curve in panel (b) of Fig. 3). For
$\left(+k_{x},+k_{y}\right)$ we see $v_{\mathrm{g}y}>0$, while it is clear
that $v_{\mathrm{g}y}$ is negative for $\left(+k_{x},-k_{y}\right)$ .

Figure 4: Group (energy) velocity curves of SMPs at a flat magnetized electron
gas-vacuum interface, as obtained from Eqs. (10) and (11), when $k_{x}$ and
$k_{y}$ are positive. Panel (a): variation of $v_{x}$ (i.e., $v_{\mathrm{g}x}$
or $v_{\mathrm{e}x}$), when $k_{y}$ is constant. Panel (b): variation of
$v_{y}$ (i.e., $v_{\mathrm{g}y}$ or $v_{\mathrm{e}y}$), when $k_{x}$ is
constant. Here the different curves refer to
$\omega_{\mathrm{c}}=0.5\omega_{\mathrm{p}}$ (solid lines), and
$\omega_{\mathrm{c}}=0.7\omega_{\mathrm{p}}$ (dashed lines).

To examine the conditions for the validity of these modes we calculate the
components of the group velocity of the new waves by using Eq. (8). We
differentiate the equation as it stands, first with respect to $k_{x}$ while
keeping $k_{y}$ constant, and then with respect to $k_{y}$ keeping $k_{x}$
constant. In doing so, we have to remember that $\omega$ is a function of both
$k_{x}$ and $k_{y}$. We find the group-velocity components

$v_{\mathrm{g}x}=\dfrac{k_{x}\left(\dfrac{\varepsilon_{1}}{k}+\dfrac{\varepsilon_{xx}}{\kappa}\right)}{\mp
ik_{y}\dfrac{d\varepsilon_{zy}}{d\omega}-\kappa\dfrac{d\varepsilon_{zz}}{d\omega}-\dfrac{k_{x}^{2}}{2\kappa}\left(\dfrac{d\varepsilon_{xx}}{d\omega}-\dfrac{\varepsilon_{xx}}{\varepsilon_{zz}}\dfrac{d\varepsilon_{zz}}{d\omega}\right)}\;,$
(10) $v_{\mathrm{g}y}=\dfrac{\mp
i\varepsilon_{zy}-k_{y}\left(\dfrac{\varepsilon_{1}}{k}+\dfrac{\varepsilon_{yy}}{\kappa}\right)}{\pm
ik_{y}\dfrac{d\varepsilon_{zy}}{d\omega}+\dfrac{\kappa}{2}\dfrac{d\varepsilon_{zz}}{d\omega}+\dfrac{1}{2\kappa}\left(k_{x}^{2}\dfrac{d\varepsilon_{xx}}{d\omega}+k_{y}^{2}\dfrac{d\varepsilon_{yy}}{d\omega}\right)}\;.$
(11)

To better understand the behavior of the SMPs, we show the variation of these
group-velocity components with respect to $\omega/\omega_{\mathrm{p}}$ in Fig.
4, using the same parameter values as for Fig. 2 (solid lines), when $k_{x}$
and $k_{y}$ are positive. The dashed lines show the result for the case
$\omega_{\mathrm{c}}=0.7\omega_{\mathrm{p}}$. Panel 4(a) shows the behavior of
the dimensionless variable $v_{\mathrm{g}x}k_{y}/\omega_{\mathrm{p}}$ and
panel (b) the variation of $v_{\mathrm{g}y}k_{x}/\omega_{\mathrm{p}}$. Both
group-velocity components change sign at $\omega=\omega_{\mathrm{F}}$, but in
a very different fashion: $v_{\mathrm{g}x}$ goes through zero continuously,
whereas $v_{\mathrm{g}y}$ makes a discontinous jump.

In an anisotropic medium the direction of group (signal) propagation differs
in general from the direction of phase propagation. Indeed, the phase velocity
points along the wave vector
$\textbf{k}=k_{x}\textbf{e}_{x}+k_{y}\textbf{e}_{y}$, whereas the energy
propagates in the direction of the Poynting vector and the signal velocity.
The magnitude of the phase velocity is given by $v_{\mathrm{ph}}=\omega/k$,
and the phase-velocity vector makes an angle
$\theta_{\mathrm{ph}}=\theta=\arctan\left(k_{y}/k_{x}\right)$ with
$\textbf{B}_{0}=+B_{0}\textbf{e}_{x}$ (or the $x$-axis). But what is the
direction of the group velocity vector $\nabla_{\bf k}\omega$ ? If we define
$\theta_{\mathrm{g}}$ to be the angle between the group-velocity and the
magnetic-field directions, then we can obtain $\tan(\theta_{\mathrm{g}})$ as
$v_{\mathrm{g}y}/v_{\mathrm{g}x}$, i.e. by dividing Eq. (11) by Eq. (10).
After using Eqs. (4) and (8), we find that

$\tan~{}\theta_{\mathrm{g}}=-\dfrac{k_{x}}{k_{y}}=-\dfrac{1}{\tan~{}\theta_{\mathrm{ph}}}.$
(12)

The phase and group velocities of the quasi-electrostatic magnetoplasmon waves
therefore have the remarkable property that they are perpendicular to each
other: energy flows in the direction perpendicular to the phase propagation.
This property is shown in Fig. 5 and reflects the dispersion relation (8) for
which $\omega(k,\theta)=\omega(\theta)$: the group velocity in the $\bf
k$-direction vanishes identically irrespective of the angle between $\bf k$
and the magnetic field, while the group velocity perpendicular to $\bf k$ is
finite. In optics the situation of group velocities being exactly opposite to
their phase velocities is well known to occur for negative-index materials
Veselago:1968a ; Pendry:2000a ; Smith:2004a . Perpendicular group and phase
velocities on the other hand for all wave vectors as found here are less well
known in optics. However, in fluid dynamics there is an interesting analogy
with so-called “internal waves” (or “(internal) gravity waves”). These are
well known to have perpendicular group and phase velocities whatever their
angle with the water surface, see for example Ref. Kundu:2016a .

Figure 5: Iso-frequency curves $(\omega=\mathrm{constant})$ of SMPs at a flat
magnetized electron gas-vacuum interface, as obtained from Eq. (9)
corresponding to the labeled points in Fig. 3, for $+B_{0}\textbf{e}_{x}$
(left figures) and $-B_{0}\textbf{e}_{x}$ (right figures). Panel (a): $k_{y}$
is constant. Panel (b): $k_{x}$ is constant. It is clear that the vectors
$\textbf{v}_{\mathrm{ph}}$ and $\textbf{v}_{\mathrm{g}}$ are always
perpendicular, and both are in the $xy$-plane. So the outer product of these
two vectors always points in the $\pm z$-direction. Note that for
$+B_{0}\textbf{e}_{x}$ ($-B_{0}\textbf{e}_{x}$), the outer product
$\textbf{v}_{\mathrm{ph}}\times\textbf{v}_{\mathrm{g}}$ points in the electron
plasma (vacuum) direction. Figure 6: Dispersion curve of SMPs at a flat
magnetized electron gas-vacuum interface, as obtained from Eq. (9), when (a)
$k_{y}$ is positive constant and (b) $k_{x}$ is positive constant. Here the
different curves correspond to $\omega_{\mathrm{c}}=0.2\omega_{\mathrm{p}}$
(solid lines), $\omega_{\mathrm{c}}=0.4\omega_{\mathrm{p}}$ (dashed lines),
and $\omega_{\mathrm{c}}=0.6\omega_{\mathrm{p}}$ (dotted lines).

We now turn to the effect of the magnetic-field strength through the cyclotron
frequency $\omega_{\mathrm{c}}$. Again, Fig. 6 is a plot of
$\omega/\omega_{\mathrm{p}}$ versus (a) $k_{x}/k_{y}$ and (b) $k_{y}/k_{x}$.
Here the different curves refer to
$\omega_{\mathrm{c}}=0.2\omega_{\mathrm{p}}$ (solid lines),
$\omega_{\mathrm{c}}=0.4\omega_{\mathrm{p}}$ (dashed lines), and
$\omega_{\mathrm{c}}=0.6\omega_{\mathrm{p}}$ (dotted lines). One can see that
changing the parameter $\omega_{\mathrm{c}}$ has a strong effect on the
dispersion curve of the SMPs. For weaker magnetic fields, the dispersion
curves are flatter and group velocities smaller. Thus group velocities of the
SMPs can be controlled with the static magnetic field as the control
parameter, see Fig. 4. From panel (a), it is clear that for the $x$-forward
mode (when $k_{x},k_{y}>0$), increasing $\omega_{\mathrm{c}}$ redshifts the
frequency of the mode for low values of $k_{x}/k_{y}$ and blueshifts the mode
frequency for high values of $k_{x}/k_{y}$. From the same panel (a), it
follows that for the $x$-backward mode and $k_{x},k_{y}>0$, increasing
$\omega_{\mathrm{c}}$ blueshifts the SMP frequency. Finally, as mentioned
before, one constant-frequency solution $\omega_{\mathrm{F}}$ exists in the
Faraday geometry, while two solutions $\omega_{\mathrm{V}\pm}$ exist in the
Voigt geometry. As can be seen in panel 6(a), in the limit
$k_{x}/k_{y}\rightarrow 0$, we indeed obtain the two frequencies of the Voigt
geometry, i.e., $\omega=\omega_{\mathrm{V}-}$, when we take
$+B_{0}\textbf{e}_{x}$; and $\omega=\omega_{\mathrm{V}+}$, when we consider
the $-B_{0}\textbf{e}_{x}$ solution. Also, from the limit
$k_{y}/k_{x}\rightarrow 0$ in Fig. 6(b), we indeed find back the result for
the Faraday geometry, i.e., $\omega=\omega_{\mathrm{F}}$.

##  IV Power flow of a surface magnetoplasmon

In this section, we calculate the $x$\- and $y$-components of the power flow
of the SMPs, i.e. along and perpendicular to the magnetic field, respectively.
Under the electrostatic approximation Maier:2007a ; A.M2947 ; A.M2976 , the
power flow associated with the SMPs of a semi-infinite magnetized electron gas
is given by

Figure 7: Normalized profile $S_{x}(z)$ of SMP modes of a flat magnetized
electron gas-vacuum interface, as obtained from Eq. (16) when $k_{x}/k_{y}=2$,
$k_{x}>0$, and $k_{y}$ is a positive constant, corresponding to the labeled
points in Fig. 3(a). Panel (a): $\textbf{B}_{0}=-B_{0}\textbf{e}_{x}$. Panel
(b): $\textbf{B}_{0}=+B_{0}\textbf{e}_{x}$.

$\textbf{S}=-\varepsilon_{0}\left\\{\begin{array}[]{clcr}\varepsilon_{1}\Phi_{1}\dfrac{\partial}{\partial
t}\nabla\Phi_{1}\;,&\mbox{$z>0$\;,}\\\ \Phi_{2}\dfrac{\partial}{\partial
t}\left[\underline{\varepsilon}(\omega)\cdot\nabla\Phi_{2}\right]\;,&\mbox{$z<0$\;,}\\\
\end{array}\right.$ (13)

where these vectors in the two media have components in the $x$-, $y$\- and
$z$-directions. The cycle-averaged $x$\- and $y$-components are

Figure 8: Normalized profile $S_{y}(z)$ of SMP modes of a flat magnetized
electron gas-vacuum interface, as obtained from Eq. (17) when
$k_{y}/k_{x}=0.5$, $k_{y}>0$, and $k_{x}$ has a positive constant value,
corresponding to the labeled points in Fig. 3(b). Panel (a):
$\textbf{B}_{0}=-B_{0}\textbf{e}_{x}$. Panel (b):
$\textbf{B}_{0}=+B_{0}\textbf{e}_{x}$.

$S_{x}=-\dfrac{\varepsilon_{0}}{2}{\rm
Re}\left\\{\begin{array}[]{clcr}\varepsilon_{1}\Phi_{1}\dfrac{\partial}{\partial
t}\dfrac{\partial}{\partial x}\Phi_{1}^{\ast}\;,&\mbox{$z>0$\;,}\\\
\varepsilon_{xx}\Phi_{2}\dfrac{\partial}{\partial t}\dfrac{\partial}{\partial
x}\Phi_{2}^{\ast}\;,&\mbox{$z<0$\;,}\\\ \end{array}\right.$ (14)
$S_{y}=-\dfrac{\varepsilon_{0}}{2}{\rm
Re}\left\\{\begin{array}[]{clcr}\varepsilon_{1}\Phi_{1}\dfrac{\partial}{\partial
t}\dfrac{\partial}{\partial y}\Phi_{1}^{\ast}\;,&\mbox{$z>0$\;,}\\\
\Phi_{2}\dfrac{\partial}{\partial
t}\left(\varepsilon_{yy}\dfrac{\partial}{\partial
y}\mp\varepsilon_{yz}\dfrac{\partial}{\partial
z}\right)\Phi_{2}^{\ast}\;,&\mbox{$z<0$\;,}\\\ \end{array}\right.$ (15)

in the complex-number representation, where ∗ denotes complex conjugation, and
${\rm Re}$ denotes taking the real part. Note that $\varepsilon_{xx}$ and
$\varepsilon_{yy}$ are real. After substitution of Eq. (6) into Eqs. (14) and
(15) and using Eq. (1), we obtain

$S_{x}=-\dfrac{1}{2}\varepsilon_{0}\omega
k_{x}\Phi_{0}^{2}\left\\{\begin{array}[]{clcr}\varepsilon_{1}e^{-2kz}\;,&\mbox{$z>0$\;,}\\\
\varepsilon_{xx}e^{+2\kappa z}\;,&\mbox{$z<0$\;,}\\\ \end{array}\right.\ $
(16) $S_{y}=-\dfrac{1}{2}\varepsilon_{0}\omega
k_{y}\Phi_{0}^{2}\left\\{\begin{array}[]{clcr}\varepsilon_{1}e^{-2kz}\;,&\mbox{$z>0$\;,}\\\
\left(\varepsilon_{yy}\mp
i\varepsilon_{yz}\dfrac{\kappa}{k_{y}}\right)e^{+2\kappa
z}\;,&\mbox{$z<0$\;,}\\\ \end{array}\right.\ $ (17)

We note that the distributions in Eqs. (16) and (17) are discontinuous at the
interface $z=0$. The total power flow densities (per unit width), associated
with the SMPs can be determined by an integration over $z=0$. We find

$\left\langle S_{x}\right\rangle=-\dfrac{1}{4}\varepsilon_{0}\omega
k_{x}\left[\dfrac{\varepsilon_{1}}{k}+\dfrac{\varepsilon_{xx}}{\kappa}\right]\Phi_{0}^{2}\;,$
(18) $\left\langle S_{y}\right\rangle=-\dfrac{1}{4}\varepsilon_{0}\omega
k_{y}\left[\dfrac{\varepsilon_{1}}{k}+\dfrac{1}{\kappa}\left(\varepsilon_{yy}\mp
i\varepsilon_{yz}\dfrac{\kappa}{k_{y}}\right)\right]\Phi_{0}^{2}\;,$ (19)

where $\left\langle\cdots\right\rangle\equiv\int_{-\infty}^{+\infty}\cdots
dz$. In Fig. 7, we calculate the normalized profiles of $S_{x}(z)$ of SMP
modes of a flat magnetized electron gas-vacuum interface, when
$k_{x}/k_{y}=2$, $k_{x}>0$, and $k_{y}$ is positive constant, corresponding to
the labeled points in Fig. 3(a). It can be seen that power flow densities are
largest at the boundary, and their amplitudes decay exponentially with
increasing distance into each medium from the interface. Comparing the curve
of case $\textbf{B}_{0}=-B_{0}\textbf{e}_{x}$ in Fig. 3(a) by the result in
Fig. 7(a), we conclude that the $x$-backward SMP mode in the region below the
line $\omega=\omega_{\mathrm{V}+}$ and above the line
$\omega=\omega_{\mathrm{F}}$ in panel (a) of Fig. 3 (blue curve) is an
acceptable mode with electric potential shown by Eq. (1). For the case
$\textbf{B}_{0}=+B_{0}\textbf{e}_{x}$ in the region below the line
$\omega=\omega_{\mathrm{F}}$ and above the line $\omega=\omega_{\mathrm{V}-}$
of panel (a) of Fig. 3 (red curve), the power flow in the EG region occurs in
the $+x$-direction, while in the vacuum region, the power flow occurs in the
$-x$-direction, i.e., opposite to the direction of phase propagation. Also,
the total power flow density (per unit width) is positive for the $x$-forward
SMP mode. This result is in agreement with the behavior of dispersion curve of
the $x$-forward SMP, shown in Fig. 3(a) and thus we have again an acceptable
mode.

In Fig. 8, by using Eq. (17), we calculate the normalized profiles of
$S_{y}(z)$ of SMP modes of a flat magnetized electron gas-vacuum interface,
when $k_{y}/k_{x}=0.5$, $k_{y}>0$, and $k_{x}$ is positive constant,
corresponding to the labeled points in Fig. 3(b). Here, one can see that for
the $y$-forward SMP mode (panel (a) of Fig. 8), the power flow in the EG
region occurs in the $+y$-direction, while in the vacuum region, the power
flow occurs in the $-y$-direction. Also, we find that the total power flow
density (per unit width) is positive for the $y$-forward mode. This result is
in agreement with the behavior of dispersion curve of $y$-forward SMP modes in
Fig. 3(b) for the case $\textbf{B}_{0}=-B_{0}\textbf{e}_{x}$. Furthermore,
comparing the lower curve in Fig. 3(b) for the case
$\textbf{B}_{0}=+B_{0}\textbf{e}_{x}$ by the result in Fig. 8(b), we conclude
that the $y$-backward SMP mode in the region below the line
$\omega=\omega_{\mathrm{F}}$ and above the line $\omega=\omega_{\mathrm{V}-}$
in panel (b) of Fig. 3 (red curve) is also an acceptable mode with electric
potential shown by Eq. (1). Finally, it is clear from Figs. 7 and 8 that the
power in the upper and lower half spaces flows in different directions, but
not for $S_{y}$ in the case $\textbf{B}_{0}=+B_{0}\textbf{e}_{x}$. Actually,
for the $y$-backward SMP mode the power flow in both media occurs in the
$-x$-direction.

##  V Energy distribution and energy velocities

Now, we first consider the energy distribution in the transverse direction.
For the cycle-averaged energy distribution associated with the SMPs of a semi-
infinite electron gas, we have, in the two media

$U=\dfrac{\varepsilon_{0}}{4}\left\\{\begin{array}[]{clcr}\varepsilon_{1}|\nabla\Phi_{1}|^{2}\;,&\mbox{$z>0$\;,}\\\
\nabla\Phi_{2}^{\ast}\cdot\left(\dfrac{d\left(\omega\underline{\varepsilon}\right)}{d\omega}\cdot\nabla\Phi_{2}\right)\;,&\mbox{$z<0$\;,}\\\
\end{array}\right.$ (20)

where losses are neglected. After substitution Eq. (6) into (20), we obtain

$U=\dfrac{1}{4}\varepsilon_{0}\Phi_{0}^{2}\\\
\times\left\\{\begin{array}[]{clcr}2\varepsilon_{1}k^{2}e^{-2kz}\;,&\mbox{$z>0$\;,}\\\
\left(k_{x}^{2}\dfrac{d\left(\omega\varepsilon_{xx}\right)}{d\omega}+k_{y}^{2}\dfrac{d\left(\omega\varepsilon_{yy}\right)}{d\omega}+\Xi\right)e^{+2\kappa
z}\;,&\mbox{$z<0$\;,}\\\ \end{array}\right.$ (21)

in the complex-number representation, where

$\Xi=\kappa^{2}\dfrac{d\left(\omega\varepsilon_{zz}\right)}{d\omega}\mp
2i\kappa k_{y}\dfrac{d\left(\omega\varepsilon_{yz}\right)}{d\omega}\;.$

From Eq. (21) we find that the contributions to the energy density of the two
half spaces are both positive. The total energy density associated with the
SMPs is again determined by integration over the out-of-plane coordinate $z$,
the energy per unit surface area being

$\left\langle U\right\rangle=\dfrac{1}{4}k\varepsilon_{0}\\\
\times\left[\varepsilon_{1}+\dfrac{1}{2k\kappa}\left(k_{x}^{2}\dfrac{d\left(\omega\varepsilon_{xx}\right)}{d\omega}+k_{y}^{2}\dfrac{d\left(\omega\varepsilon_{yy}\right)}{d\omega}+\Xi\right)\right]\Phi_{0}^{2}\;.$
(22)

In general, the energy velocity of the SMPs is given as the ratio of the total
power flow density (per unit width) and the total energy density (per unit
area). For our model of the SMPs, this leads to the energy-velocity components

$v_{\mathrm{e}x}=-\dfrac{\omega
k_{x}}{k}\dfrac{\dfrac{\varepsilon_{1}}{k}+\dfrac{\varepsilon_{xx}}{\kappa}}{\varepsilon_{1}+\dfrac{1}{2k\kappa}\left(k_{x}^{2}\dfrac{d\left(\omega\varepsilon_{xx}\right)}{d\omega}+k_{y}^{2}\dfrac{d\left(\omega\varepsilon_{yy}\right)}{d\omega}+\Xi\right)}\;,$
(23) $v_{\mathrm{e}y}=-\dfrac{\omega
k_{y}}{k}\dfrac{\dfrac{\varepsilon_{1}}{k}+\dfrac{1}{\kappa}\left(\varepsilon_{yy}\mp
i\varepsilon_{yz}\dfrac{\kappa}{k_{y}}\right)}{\varepsilon_{1}+\dfrac{1}{2k\kappa}\left(k_{x}^{2}\dfrac{d\left(\omega\varepsilon_{xx}\right)}{d\omega}+k_{y}^{2}\dfrac{d\left(\omega\varepsilon_{yy}\right)}{d\omega}+\Xi\right)}\;,$
(24)

The expression on the right-hand sides of Eqs. (23) and (24) are precisely
those obtained from the usual definition of the group velocity of SMPs in the
absence of damping, i.e., Eqs. (10) and (11), as were shown in Fig. 4. This
means that the net power flow is in the direction of the group velocity. Let
us note that, by contrast, in resonant multiply scattering media, the group
and transport velocities in general will differ A.L143 .

##  VI Conclusions

In summary, we have studied the propagation of SMPs on a semi-infinite
magnetized electron gas in the electrostatic approximation by consideration of
a mixed Faraday-Voigt configuration. We have shown that such a structure
permits propagation of new electrostatic waves that are strongly direction-
dependent and do not exist in a semi-infinite gas plasma or a semi-infinite
electron plasma in a metal. We have studied the dispersion relation, group
velocity and energy relations of the found magnetoplasmons in detail. In
particular, we found that the group velocities of the SMPs can be controlled
by the applied static magnetic field and that the phase and group velocities
are always perpendicular for these surface magnetoplasmons. Furthermore, we
analyzed situations in which power will flow in different directions in the
upper and lower half spaces, while we also discussed cases where in the upper
and lower half spaces the power will flow in the same directions.

## Funding

M.W. acknowledges support by the Danish National Research Foundation through
NanoPhoton - Center for Nanophotonics (grant number DNRF147) and Center for
Nanostructured Graphene (grant number DNRF103), and from the Independent
Research Fund Denmark - Natural Sciences (project no. 0135-004038).

## Conflicts of interest/Competing interests

The authors declare no conflicts of interest/competing interests.

## Authors’ contributions

A. M. proposed the idea and performed the initial calculations and analyzed
the initial numerical data. Both authors (A. M. and M. W.) have discussed the
results thoroughly and contributed to the writing and review of the
manuscript.

## Additional information

Correspondence and requests for materials should be addressed to A.M or M.W.

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