# Collective Quantum Approach to Surface Plasmon Resonance Effect

M. Akbari-Moghanjoughi  Faculty of Sciences, Department of Physics, Azarbaijan
Shahid Madani University, 51745-406 Tabriz, Iran
<EMAIL_ADDRESS>
###### Abstract

In this research we present a theory of the surface plasmon resonance (SPR)
effect based on the dual length-scale driven damped collective quantum
oscillations of the spill-out electrons in plasmonic material surface. The
metallic electron excitations are modeled using the Hermitian effective
Schrödinger-Poisson system, whereas, the spill-out electron excitations are
modeled via the damped non-Hermitian effective Schrödinger-Poisson system
adapted appropriately at the metal-vacuum interface. It is shows that, when
driven by external field, the system behaves like the driven damped oscillator
in wavenumber domain, quite analogous to the driven damped mechanical
oscillation in frequency domain, leading to the collective surface spill-out
electron excitation resonance. In this model the resonance occurs when the
wavenumber of the driving pseudoforce matches that of the surface plasmon
excitations which can be either due to single-electrons or collective effects.
Current theory of SPR is based on longitudinal electrostatic excitations of
the surface electrons, instead of the polariton excitation parallel to the
metal-dielectric or metal-vacuum surface. Current theory may also be extended
to use for the localized surface plasmon resonance (LSPR) in nanometer sized
metallic surfaces in non-planar geometry. A new equation of state (EoS) for
the plasmon electron number density in quantum plasmas is obtained which
limits the plasmonic effects in high-density low-temperature electron gas
regime, due to small transition probability of electrons to the plasmon energy
band.

###### pacs:

52.30.-q,71.10.Ca, 05.30.-d

##  I Introduction

Plasmonics is a rapidly developing fundamental subject of research which
combines the theory and applications spanning from physics, chemistry,
engineering, biology, medicine and many other areas of active
interdisciplinary fields [1, 2, 3]. Plasmonics had a long tradition since the
discovery of surface plasmon polariton in 1950’s. It has attracted significant
attention after the discovery of the surface-enhanced Raman scattering (SERS)
effect in 1970’s [4] and continued to grow exponentially during the last
decade, due to explosive technological developments in nanofabrication and low
dimensional miniaturized semiconductor industry [5, 6]. Many new fields of
research, such as nanoplasmonics, nanophotonics [7, 8], nano-optics [9] are
emerging around the subject because of evergrowing applications of the
plasmonics in futuristic nanotechnology, ultrahigh sensitivity and terahertz-
range dielectric response of electrons to environmental changes. The surface
plasmon resonance (SPR) is versatile technology used in determining particular
substances called analytes. SPR biosensors are also used in label-free
detection of various important biomarkers [10]. Despite a huge experimental
advancement [11, 12, 13, 14, 15, 16] and success in the field, the theoretical
developments remain classical. The best investigation tools for consistent
results in plasmonic phenomena are numerical simulation and computational
algorithms, which are time consuming due to many body complexity from one hand
and prone to error, on the other. Furthermore, the classical Drude model and
numerical simulations do not provide fundamental insight into the underlying
atomic-scale physical phenomena at the quantum level.

Plasmonic systems, on the other hand, provide variety of interesting optical
properties depending on many parameters, such as, the shape, size, periodicity
and the material combination. The extreme sensitivity of electron oscillations
on material configurations provides a great chance for industry technicians to
invent vast variety of electronic devices by manipulation of involved
parameters. Plasmonic devices are also main components of biosensors, optical
modulators, photovoltaic technology etc. There are, however, technological
obstacle and insufficient theoretical developments which limit the harvesting
capacity of the plasmonic effects. By overcoming these limitations, plasmonic
devices can find their full potential in optical emitters [17], plasmon
focusing [18], nanoscale waveguiding [19], optical antennas [20],
communication device [21], solar cells [22], plasmonic sensors [23], modulator
[24] nanoscale swiches [25] and spasers [26]. Moreover, collective electron
excitations may become the best candidates for developing effective hot
electron generation [27] for solar energy conversion towards improved
photovoltaic and catalytic designs via the collective energy transport
phenomena. The new plasmonic energy conversion may require several important
considerations regarding the plasmonic materials, integrated architecture and
device fabrication methods [28]. Recent investigations reveal that guiding and
intensification of light by employing the right plasmonic geometries leads to
more effective solar-cell designs in which light is more efficiently absorbed
in a single quantum well, or in a single layer of quantum dots or molecular
chromophores [29].

Plasmonics is also an appealing theoretical field of research, since,
fundamental electronic excitations control almost every aspect of a solid,
such as conductivity, heat and electrical transport [30]. The electrons
uniquely interact with the lattice periodicity ans electromagnetic field to
produce complex energy band structure which determines the thermodynamic
behavior of a solid [31]. However, understanding the fundamental plasmonic
behavior of interacting electron gas requires the improvement of advanced
quantum many body [32, 33] and quantum electrodynamic [34] theories. Although,
the foundation of the theory has been laid by previous pioneering works [35,
36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51], the quantum
mechanical treatment of electron system are usually based on various limiting
approximations, such as, the single-electron energy dispersion and random-
phase approximation. Due to complex nature of plasmonic phenomenon and the
delicate interplay between the individual electrons and their collective
excitation, some important aspects of the phenomenon are underestimated when
the single-electron and collective behavior of the electron system are not
treated in equal footing. For instance, there has been an intense recent
debate [52, 53, 54, 55, 56, 57, 58, 59, 60] on the inconsistent prediction of
attractive force between static screened ions in quantum plasmas between the
linearized quantum hydrodynamic [61, 62, 63, 64, 65, 66, 67] and density
functional theory (DFT) [68] treatments . While these theories can be
reconciled when corrections are taken into account [69], the fundamental
theory of plasmonics must go well beyond these correction and approximations.
In an unpublished resent research the Lindhard theory is used with a more
general dual length-scale energy dispersion which includes both single-
electron as well as collective behavior [70]. It is seen that, the obtained
results of such adjustment deviates substantially from prediction of both
conventional quantum hydrodynamic and the Lindhard theory. The quantum
hydrodynamic theory which is based on the Wigner-Poisson system has shown many
success in predicting collective phenomena in dense electron environments [71,
72, 73, 74, 75, 76, 77, 78, 79, 80]. However, uses the independent single-
electron Hamiltonian to generalize to many body effects and therefore ignores
the energy band structure effect due to the electrostatic interactions. A more
elaborative treatment in a recent research the Schrödinger-Poisson model [81]
has been used to obtain the generalized energy dispersion relation from which
collective phenomena is studied in both single-electron and collective scales
at the same time. Some interesting new collective quantum effects have been
predicted within the Schrödinger-Poisson model [82, 83, 84, 85, 86, 87, 88].
In this research we develop a new theory of SPR based on the Schrödinger-
Poisson model based on the driven excitations in half-space collective quantum
excitations. The paper is organized in the following manner. The Linear
quantum excitations are modeled in Sec. II. The model for spill-out electron
excitations is provided in Sec. III. Driven collective excitations at the
surface is considered in Sec. IV and conclusions are drown in Sec V.

##  II Linear Collective Quantum Excitations

The elementary collective quantum excitations of the electron gas can be
described using various modeling such as kinetic, density functional and
hydrodynamic theories. Here we use the effective Schrödinger-Poisson system,
within the framework of quantum hydrodynamic model [89], in order to study
plasmonic excitations in vacuum-metal-vacuum configuration. The one-
dimensional collective dynamic state of the electrostatically interacting
metallic electrons follow the Hermitian coupled pseudoforce system [88]

$\displaystyle i\hbar\frac{{\partial N({\bf{r}},t)}}{{\partial
t}}=-\frac{{{\hbar^{2}}}}{{2m}}\Delta
N({\bf{r}},t)-e\phi({\bf{r}})N({\bf{r}},t)+\mu N({\bf{r}},t),$ (1a)
$\displaystyle\Delta\phi({\bf{r}})=4\pi
e\left[{|N({\bf{r}}){|^{2}}-{n_{0}}}\right],$ (1b)

in which ${\cal{N}}({\bf{r}},t)=\psi({\bf{r}},t)\exp[iS({\bf{r}},t)/\hbar]$
denotes the collective statefunction, in combination with the electrostatic
potential $\phi(x)$, characterizing the probability density of the electron
system which is related to the local electron number density via the
definition, $n({\bf{r}})=\psi({\bf{r}})\psi^{*}({\bf{r}})$ in which $n_{0}$
denotes the neutralizing positive background charge density in the metallic
slab region of length $L$ and the electron fluid momentum is given by,
${\rm{p(r,t)=}}\nabla{\rm{S(r,t)}}$. The electrostatic potential
$\phi({\bf{r}})$, plays the essential role of coupling the Schrödinger
equation to the Poisson’s relation.

In the slow electron gas compression limit, the chemical potential $\mu$ is
related to the electron number-density via the isothermal equation of state
(EoS) [69]

$\displaystyle{n_{e}(\mu,T)}=\frac{{{2^{1/2}}m{{}^{3/2}}}}{{{\pi^{2}}{\hbar^{3}}}}\int_{0}^{+\infty}{\frac{{\sqrt{{\varepsilon}}d{\varepsilon}}}{{{e^{\beta({\varepsilon}-\mu)}}+1}}},$
(2a)
$\displaystyle{P_{e}(\mu,T)}=\frac{{{2^{3/2}}m{{}^{3/2}}}}{{3{\pi^{2}}{\hbar^{3}}}}\int_{0}^{+\infty}{\frac{{{{\varepsilon}^{3/2}}d{\varepsilon}}}{{{e^{\beta({\varepsilon}-{\mu})}}+1}}.}$
(2b)

where $\beta=1/k_{B}T$ with $T$ being the equilibrium electron temperature and
$P_{e}$ the quantum statistical electron gas pressure. Note that the simple
thermodynamic identity, $n_{e}\nabla\mu=\nabla P_{e}(n_{e})$, holds.

For the one dimensional model in the stationary limit of equilibrium state,
$p=0$, the wavefunction $\psi(x,t)$ may be decomposed into separate variables
leading to following linearized system, after appropriate normalization of
variables [84]

$\displaystyle i\hbar\frac{{d\psi_{1}(t)}}{{dt}}=\varepsilon\psi_{1}(t),$ (3a)
$\displaystyle\frac{{d\Psi_{1}(x)}}{{dx}}+\Phi_{1}(x)+E\Psi_{1}(x)=0,$ (3b)
$\displaystyle\frac{{d\Phi_{1}(x)}}{{dx}}-\Psi_{1}(x)=0,$ (3c)

where $\mu_{0}$ is the equilibrium chemical potential and $\varepsilon$ is the
collective quantum-state energy eigenvalue. Note that the linear expansion
scheme, $\\{\psi^{0}=1,\phi^{0}=0,\mu^{0}=\mu_{0}\\}$ is employed to normalize
the functionals $\Psi({\bf{r}})=\psi({\bf{r}})/n_{0}$ and
$\Phi({\bf{r}})=e\phi({\bf{r}})/E_{p}$ and variables $x=x/l_{p}$ and
$t=t\omega_{p}$ in which $\omega_{p}=\sqrt{4\pi e^{2}n_{0}/m}$ is the
conventional plasmon frequency, $l_{p}=1/k_{p}$ is the characteristic plasmon
length with $k_{p}=\sqrt{2mE_{p}}/\hbar$ being the coresponding plasmon
wavenumber with $E_{p}=\hbar\omega_{p}$ being the plasmon energy. The
normalized energy $E=\epsilon-\sigma$ with $\epsilon=\varepsilon/E_{p}$ and
the $\sigma=\mu_{0}/E_{p}$ denotes the energy scale measured from the Fermi
energy level, $E_{F}=\hbar^{2}(3\pi n_{0})^{2/3}/2m$, of the degenerate
electron gas. The Fourier analysis of coupled pseudoforce system (3) admits
the generalized matter wave energy dispersion, $E=k^{2}+1/k^{2}$, with the
first and second terms characterizing the single-electron and collective
excitations, respectively. The presence of electrostatic coupling between
electrons leads to the dual length-scale plasmonic excitations of the electron
gas.

The real-valued solutions to the time-independent system (3) follows

$\displaystyle{\Phi_{1}}(x)=\frac{1}{\alpha}\left[{\left({{c_{1}}k_{21}^{2}+{c_{3}}}\right)\cos({k_{11}}x)-\left({{c_{1}}k_{11}^{2}+{c_{3}}}\right)\cos({k_{21}}x)}\right]$
(4a)
$\displaystyle+\frac{2}{\alpha}\left[{\frac{{\left({{c_{2}}k_{21}^{2}+{c_{4}}}\right)\sin({k_{11}}x)}}{{{k_{11}}}}-\frac{{\left({{c_{2}}k_{11}^{2}+{c_{4}}}\right)\sin({k_{21}}x)}}{{{k_{21}}}}}\right].$
(4b)
$\displaystyle{\Psi_{1}}(x)=1+\frac{1}{\alpha}\left[{\left({{c_{1}}+{c_{3}}k_{21}^{2}}\right)\cos({k_{21}}x)-\left({{c_{1}}+{c_{3}}k_{11}^{2}}\right)\cos({k_{11}}x)}\right]$
(4c)
$\displaystyle+\frac{2}{\alpha}\left[{{\frac{{\left({{c_{2}}+{c_{4}}k_{21}^{2}}\right)\sin({k_{21}}x)}}{{{k_{21}}}}-\frac{{\left({{c_{2}}+{c_{4}}k_{11}^{2}}\right)\sin({k_{11}}x)}}{{{k_{11}}}}}}\right],$
(4d)

where $c_{i}$ are the integration constants characterizing the boundary
conditions with $\alpha=\sqrt{E^{2}-4}$, $k_{11}=\sqrt{(E-\alpha)/2}$ and
$k_{21}=\sqrt{(E+\alpha)/2}$ being wavenumbers of the collective and single-
electron excitations which satisfy the complementarity-like relation,
$k_{11}k_{21}=1$.

Figure 1: (a) The variation in normalized plasmon temperature, $\theta$, with
electron concentration, for different values of the electron temperature. (b)
Variation of the plasmon length with the electron number density. (c)
Variation of normalized chemical potential $\sigma=\mu_{0}/E_{P}$ with free
electron concentration, for different values of the normalized plasmon
temperature. (d) Changes in the plasmon energy of the electron gas with
electron concentration. The increase in thickness of curves in plots (a) and
(c) refer to increases in the varied parameter above given frames.

Figure 1 shows variations in various plasmon parameters in terms of the
electron number density. Fig 2(a) depicts the variation of normalized electron
temperature, $\theta=T/T_{p}$, where $T_{p}=E_{p}/\hbar$ is the characteristic
plasmon temperature with electron concentration for different values of the
electron temperature, $T$. It is noted that the plasmon temperature, $T_{p}$,
despite its deceptive name has noting to do with real temperature and
analogous to the Fermi temperature depends solely on the electron number
density of the electron gas. It is remarked that $\theta$ decreases sharply
with increase in electron number density being larger for larger temperature
values. Variation the plasmon length $l_{p}$ is shown in Fig. 2(b) in terms of
the electron concentration. The plasmon length is seen to also decreases with
increase in the electron density with typical values of few tenth of a
nanometer in metallic electron density range. Figure 2(c) depicts variation in
the normalized chemical potential with electron concentration for different
values of $\theta$. The normalized chemical potential, $\sigma$ only varies
greatly with electron number density for lower $\theta$ values. Figure 2(d)
shows the dependence of plasmon energy on the electron number density. This
parameter in the fully degenerate electron gas regime has typical values of
few electron Volts and increases sharply as the electron number density is
increased.

An important note is to be made here regarding the standard definitions of
plasmon parameters in degenerate electron regime. Conventionally, the plasmon
frequency, hence, the plasmon energy is defined in terms of the equilibrium
number density of free electrons. However, at low temperature or equivalently
high electron number density regime, majority of metallic electrons are packed
below the Fermi level which prohibits them from collective contributions. It
will be shown that, in order to correctly describe the plasmonic phenomena in
metallic regime, these parameter definitions must be altered according to the
plasmon number density, $n_{p}$ based on transition probability of electrons
to the so-called plasmon energy band. According to the new energy band
definition for collective excitations, one is able to compute the spill-out
electron density at the surface of metals and nanoparticles, which is
considered in the following section.

##  III Spill-out Electron Excitations

The spill-out electrons in region 2 has been recently modeled through the
following non-Hermitian pseudoforce system [85]

$\displaystyle\frac{{{d^{2}}\Psi_{2}(x)}}{{d{x^{2}}}}+2\kappa\frac{{d\Psi_{2}(x)}}{{dx}}+\Phi_{2}(x)+E\Psi_{2}(x)=0,$
(5a)
$\displaystyle\frac{{{d^{2}}\Phi_{2}(x)}}{{d{x^{2}}}}+2\kappa\frac{{d\Phi_{2}(x)}}{{dx}}-\Psi_{2}(x)=0,$
(5b)

in which $\kappa$ is the normalized damping parameter characterizing the
spacial decay of the collective statefunctions in vacuum side of the metallic
sandwich. Although, this parameter may have a complex dependence on other
parameters, it can be directly measured using a Langmuir probe. This damping
parameter plays the analogous role of the tunneling parameter in the single-
electron quantum system. The only difference in this case is that a collective
tunneling of electrons leading to the electron spill-out effect has also an
oscillatory density behavior along the damping character. Different scenarios
of the damping are considered in the Ref. [84]. Recently it has been shown
that electrostatic interactions among the spill-out electrons in combination
of the single-electron excitations leads to a new photo-plasmonic phenomenon
and the collective tunneling of the electrons on the irradiated metallic
surface though the photo-plasmonic energy level [87].

The real-valued solutions to the coupled non-Hermitian system (5) follows

$\displaystyle{\Phi_{2}}(x)=\frac{{{{{e}}^{-\kappa
x}}}}{\alpha}\left[{\left({{d_{1}}k_{21}^{2}+{d_{3}}}\right)\cos({k_{22}}x)-\left({{d_{1}}k_{11}^{2}+{d_{3}}}\right)\cos({k_{12}}x)}\right]$
(6a) $\displaystyle+\frac{{{{{e}}^{-\kappa
x}}}}{{\alpha{k_{12}}}}\left[{\left({\alpha-E}\right)\left({{d_{1}}\kappa+{d_{2}}}\right)-2\left({{d_{3}}\kappa+{d_{4}}}\right)}\right]\sin({k_{12}}x)$
(6b) $\displaystyle+\frac{{{{{e}}^{-\kappa
x}}}}{{\alpha{k_{22}}}}\left[{\left({\alpha+E}\right)\left({{d_{1}}\kappa+{d_{2}}}\right)+2\left({{d_{3}}\kappa+{d_{4}}}\right)}\right]\sin({k_{22}}x).$
(6c) $\displaystyle{\Psi_{2}}(x)=\frac{{{{{e}}^{-\kappa
x}}}}{\alpha}\left[{\left({{d_{1}}+{d_{3}}k_{21}^{2}}\right)\cos({k_{22}}x)-\left({{d_{1}}+{d_{3}}k_{11}^{2}}\right)\cos({k_{12}}x)}\right]$
(6d) $\displaystyle+\frac{{2{{{e}}^{-\kappa
x}}}}{{\alpha{k_{22}}}}\left[{\kappa{d_{1}}+{d_{2}}+k_{21}^{2}\left({\kappa{d_{3}}+{d_{4}}}\right)}\right]\sin({k_{22}}x)$
(6e) $\displaystyle-\frac{{2{{{e}}^{-\kappa
x}}}}{{\alpha{k_{12}}}}\left[{\kappa{d_{1}}+{d_{2}}+k_{11}^{2}\left({\kappa{d_{3}}+{d_{4}}}\right)}\right]\sin({k_{12}}x),$
(6f)

where $d_{i}$ are integration constants set by the boundary conditions,
$k_{12}=\sqrt{k_{11}^{2}-\kappa^{2}}$ and
$k_{22}=\sqrt{k_{21}^{2}-\kappa^{2}}$ are the characteristic generalized
wavenumbers of the spill-out electron excitations admitting the new energy
dispersion relation, $E=(k^{2}+\kappa^{2})+1/(k^{2}+\kappa^{2})$, including
the pseudo-damping effect. The complementarity-like relation in this case it,
$k_{12}k_{22}=\sqrt{\kappa^{4}-E\kappa^{2}+1}$ which reduces to the undamped
plasmon energy dispersion in the limit $\kappa=0$.

The plasmonic gas has some distinct thermodynamic properties which can be
obtained quite analogous to the free electron gas. The number density of
plasmon mode is obtained via, $N(k)=4\pi k^{3}/3$, resulting in the plasmon
density of states (DoS), $g(k)=(dN/dk)/|dE/dk|$. Given that the electron
occupation function in plasmon mode is dictates the Pauli exclusion we have,
$F(k,\kappa,\theta)=1/[1+\exp(E/\theta)]$ in normalized form. Therefore, the
thermodynamic quantities such as the normalized plasmon electron number-
density $n_{p}(\theta,\kappa)$, the internal energy of plasmon electrons,
$U_{p}(\theta,\kappa)$ and heat capacity contribution from plasmonic
excitations, $c_{p}(\theta,\kappa)$, are given as

$\displaystyle{n_{p}}(\theta,\kappa)=\int\limits_{0}^{\infty}{\frac{{g(k,\kappa){d_{k}}E(k,\kappa)dk}}{{1+\exp[E(k,\kappa)/\theta]}}},\hskip
8.53581ptg(k,\kappa)=\frac{{2\pi
k{{\left({{k^{2}}+{\kappa^{2}}}\right)}^{2}}}}{{\left|{{{\left({{k^{2}}+{\kappa^{2}}}\right)}^{2}}-1}\right|}},$
(7a) $\displaystyle
U_{p}(\theta,\kappa)=\int\limits_{0}^{\infty}{\frac{{{g(k,\kappa)}{E(k,\kappa)}{d_{k}}{E(k,\kappa)}dk}}{{1+\exp[{E(k,\kappa)}/\theta]}}},\hskip
8.53581pt{E(k,\kappa)}=(k^{2}+\kappa^{2})+1/(k^{2}+\kappa^{2}),$ (7b)
$\displaystyle
c_{p}(\theta,\kappa)=\int\limits_{0}^{\infty}{g(k,\kappa)E(k,\kappa){d_{\theta}}F(k,\kappa,\theta){d_{k}}E(k,\kappa)dk},$
(7c)

As previously noted, there is a fundamental difference between plasmon
electron number density, $n_{p}$, and that of the free electrons, $n_{0}$,
appearing in (7) and the Fermi energy, respectively. The former quantity
refers to the excited electrons to the plasmon level, whereas, the later is
the total electron number density at the equilibrium temperature. In the fully
degenerate quantum plasmas only fraction of free electrons below the Fermi
energy are excited to the plasmonic level, depending on the plasma density-
temperature regime and contribute to collective oscillations.

Figure 2: (a) The plasmon energy dispersion profiles for different damping
parameter values. (b) The plasmon density of states (DoS) for different values
of the damping parameter, $\kappa$. (c) The plasmon energy band occupation
function for different values of the damping parameter. (d) The plasmon heat
capacity contribution for different values of $\kappa$. The dashed curves
indicate the corresponding curves for the free electron model. The increase in
thickness of curves in plots refer to increases in the varied parameter above
given frames.

In Fig. 2 we have shown the variations of thermodynamic quantities
corresponding to electron which are excited to the plasmon energy band. The
generalized plasmon energy dispersion for different values of the damping
parameter, $\kappa$, is depicted in Fig. 3(a). The dashed curve corresponds to
the free electron energy dispersion. For undamped case, $\kappa=0$, the
plasmon conduction band takes place at, $k=1$, (in plasmon wavenumber unit),
with an energy band gap for $0<E<2$ (in plasmon energy unit) in which the
plasmon excitations are unstable. For values of $E(k,0)>2$ the collective
excitations are of dual-tone nature, while, for $E\simeq 2$ the quantum
beating behavior takes place for which the collective and single-electron
excitation scale-lengths match approximately. The dispersion curve consists of
two distinct branches joining at $k=1$ with the left/right branch
corresponding to the wave-like/particle-like excitations. For the damped
plasmons, $\kappa\neq 0$, (as the case for spill-out electrons) the wave-like
oscillations become unstable above a critical wavenumber value,
$k_{c}=\kappa^{2}+1/\kappa^{2}$ and only single-electron excitations are
allowed. However, the damping parameter does not significantly affect the
particle-like oscillation branch. The plasmon electron DoS is shown in Fig.
2(b) for different values of the damping parameter. The dashed line
corresponds to the free electron model. There is a Van-Hove-like singularity
at the critical wavenumber value, $\sqrt{1-\kappa^{2}}$ which moves to lower
wavenumbers with increase of the damping parameter. This feature is quite
analogous to the well-known Van-Hove singularity in the DoS at the Fermi level
of most solids [30, 31]. It is clearly remarked that, at the small wavelength
limit the plasmon electron DoS approaches the value of the free electron gas.
The later is because at such limit the single electron excitations dominate
the collective ones. Figure 2(c) depicts the plasmon occupation number showing
a peak which coincides with $k=1$ for $\kappa=0$, with the dashed curve
corresponding to the free electron occupation function peaking at $k=0$. The
plasmon occupation number peak position, however, moves to lower $k$-values,
as the damping parameter is increased and for the total damping $\kappa=1$ the
occupation number peaks at $k=0$. Note that the long wavelength plasmon modes
become more occupied as the damping parameter increases and looks like the
single-electron dashed curve as expected due to domination of these
excitations. The variation of plasmonic heat capacity is shown in Fig. 2(d)
for different values of the damping parameter in terms of the normalized
temperature. The corresponding single-electron heat capacity (dashed curve)
shows increased contribution at lower temperature because the low temperature
limits the electron transitions to plasmon band are greatly diminished. It is
also remarked that plasmon damping has significant effect on the plasmon
density only at higher electron temperatures.

##  IV Surface Plasmon Resonance Effect

In order to study the surface plasmon resonance effect, we consider a more
generalized driven coupled pseudoforce system

$\displaystyle\frac{{{d^{2}}\Psi_{2}(x)}}{{d{x^{2}}}}+2\kappa\frac{{d\Psi_{2}(x)}}{{dx}}+\Phi_{2}(x)+E\Psi_{2}(x)=F\cos(qx),$
(8a)
$\displaystyle\frac{{{d^{2}}\Phi_{2}(x)}}{{d{x^{2}}}}+2\kappa\frac{{d\Phi_{2}(x)}}{{dx}}-\Psi_{2}(x)=0,$
(8b)

in which $F$ and $q$ are, respectively, the pseudoforce driving normalized
amplitude and wavenumber. The real-valued complementary solutions of the
system (8) reads

$\displaystyle\Phi(x)={\Phi_{2}}(x)-\frac{{F{e^{-\kappa
x}}}}{{\alpha{\eta_{1}}{\eta_{2}}}}\left[{{\eta_{1}}\left({{q^{2}}-k_{21}^{2}}\right)\cos({k_{22}}x)-{\eta_{2}}\left({{q^{2}}-k_{11}^{2}}\right)\cos({k_{12}}x)}\right]$
(9a) $\displaystyle+\frac{{F\kappa{e^{-\kappa
x}}}}{{\alpha{\eta_{1}}{\eta_{2}}}}\left[{{\eta_{1}}\left({{q^{2}}+k_{21}^{2}}\right)\frac{{\sin({k_{22}}x)}}{{{k_{22}}}}-{\eta_{2}}\left({{q^{2}}+k_{11}^{2}}\right)\frac{{\sin({k_{12}}x)}}{{{k_{12}}}}}\right]$
(9b)
$\displaystyle+\frac{F}{{{\eta_{1}}{\eta_{2}}}}\left\\{{\left[{\left({{q^{2}}-k_{11}^{2}}\right)\left({{q^{2}}-k_{21}^{2}}\right)-4{q^{2}}{\kappa^{2}}}\right]\cos(qx)+2q\kappa\left({k_{11}^{2}+k_{21}^{2}-2{q^{2}}}\right)\sin(qx)}\right\\},$
(9c) $\displaystyle\Psi(x)={\Psi_{2}}(x)+\frac{{F{{{e}}^{-\kappa
x}}}}{{\alpha{\eta_{1}}{\eta_{2}}}}\left[{{\eta_{1}}k_{21}^{2}\left({{q^{2}}-k_{21}^{2}}\right)\cos({k_{22}}x)-{\eta_{2}}k_{11}^{2}\left({{q^{2}}-k_{11}^{2}}\right)\cos({k_{12}}x)}\right]$
(9d) $\displaystyle-\frac{{F\kappa{e^{-\kappa
x}}}}{{\alpha{\eta_{1}}{\eta_{2}}}}\left[{{\eta_{1}}k_{21}^{2}\left({{q^{2}}+k_{21}^{2}}\right)\frac{{\sin({k_{22}}x)}}{{{k_{22}}}}-{\eta_{2}}k_{11}^{2}\left({{q^{2}}+k_{11}^{2}}\right)\frac{{\sin({k_{12}}x)}}{{{k_{12}}}}}\right]$
(9e)
$\displaystyle-\frac{F}{{{\eta_{1}}{\eta_{2}}}}\left\\{{\left[{1-\left({k_{11}^{2}+k_{21}^{2}-{q^{2}}}\right)\left({{q^{2}}+4{\kappa^{2}}}\right)}\right]{q^{2}}\cos(qx)+2q\kappa\left({1-{q^{4}}-4{\kappa^{2}}{q^{2}}}\right)\sin(qx)}\right\\},$
(9f)

where $\eta_{1}=(q^{2}-k_{11}^{2})^{2}+4q^{2}\kappa^{2}$,
$\eta_{2}=(q^{2}-k_{21}^{2})^{2}+4q^{2}\kappa^{2}$ with $\Phi_{2}$ and
$\Psi_{2}$ are the solutions of homogenous system, given in (5). The solutions
in the regions 1 (metallic) and 2 (vacuum) can be matched via appropriate
boundary conditions. Assuming, $c_{1}=c_{2}=c_{4}=0$ and $c_{3}=1$ without
loss of generality, we arrive at the following continuous solutions in the
region 1

$\displaystyle\Phi_{1}(x)=\frac{1}{\alpha}\left[{\cos({k_{11}}x)-\cos({k_{21}}x)}\right],$
(10a)
$\displaystyle\Psi_{1}(x)={1+\frac{1}{\alpha}\left[{k_{21}^{2}\cos({k_{21}}x)-k_{11}^{2}\cos({k_{11}}x)}\right]}.$
(10b)

with the normalized wavefunction defined as,
$\Psi_{N1}(x)=\Psi_{1}(x)/\Psi_{N}$, where $\Psi_{N}$ is the normalizing
factor, so that, $\int_{-L}^{0}{\left[{\Psi_{N1}^{2}(x)}\right]dx=1}$. The
normalization factor, then, reads

$\displaystyle{\Psi_{N}}=\left\\{{1+\frac{{a_{1}^{2}+a_{2}^{2}}}{2}+\frac{{a_{1}^{2}}}{{8L{k_{11}}}}\sin(2{k_{11}}L)+\frac{{a_{2}^{2}}}{{8L{k_{21}}}}\sin(2{k_{21}}L)}\right.$
(11a)
$\displaystyle{\left.{+\frac{{{a_{1}}}}{L}\left[{\frac{1}{{{k_{11}}}}+\frac{{{a_{2}}{k_{11}}\cos({k_{21}}L)}}{{k_{11}^{2}-k_{21}^{2}}}}\right]\sin({k_{11}}L)+\frac{{{a_{2}}}}{L}\left[{\frac{1}{{{k_{21}}}}-\frac{{{a_{1}}{k_{21}}\cos({k_{11}}L)}}{{k_{11}^{2}-k_{21}^{2}}}}\right]\sin({k_{21}}L)}\right\\}^{1/2}},$
(11b)

where $a_{1}=(\alpha-E)/2\alpha$ and $a_{2}=(\alpha+E)/2\alpha$. The $d_{i}$
coefficients of wavefunctions in the spill-out region, are given as

$\displaystyle{d_{1}}=\frac{{F\left\\{{\alpha\left({{q^{4}}+1}\right)+k_{21}^{2}{\eta_{1}}-k_{11}^{2}{\eta_{2}}-{q^{2}}\left[{{\eta_{1}}-{\eta_{2}}+\alpha\left({k_{11}^{2}+k_{21}^{2}+4{\kappa^{2}}}\right)}\right]}\right\\}}}{{2{\eta_{1}}{\eta_{2}}\left({k_{11}^{2}-k_{21}^{2}}\right)}},$
(12a)
$\displaystyle{d_{2}}=\frac{{F\kappa\left\\{{\alpha{k^{4}}\left({3k_{11}^{2}-3k_{21}^{2}-\alpha}\right)-\left({k_{11}^{2}-k_{21}^{2}+\alpha}\right)\left[{\left({\alpha+k_{21}^{2}{\eta_{1}}}\right)-k_{11}^{2}{\eta_{2}}}\right]}\right\\}}}{{\alpha{\eta_{1}}{\eta_{2}}\left({k_{11}^{2}-k_{21}^{2}}\right)}}$
(12b)
$\displaystyle+\frac{{F\kappa\left({k_{11}^{2}-k_{21}^{2}+\alpha}\right)\left\\{{2\alpha{\kappa^{4}}-{k^{2}}\left[{{\eta_{1}}-{\eta_{2}}+\alpha\left({k_{11}^{2}+k_{21}^{2}}\right)}\right]}\right\\}}}{{\alpha{\eta_{1}}{\eta_{2}}\left({k_{11}^{2}-k_{21}^{2}}\right)}},$
(12c) $\displaystyle
d_{3}=-\frac{{\left({k_{21}^{2}-k_{11}^{2}+\alpha}\right)}}{{2\left({k_{11}^{2}-k_{21}^{2}}\right)}}$
(12d)
$\displaystyle+\frac{{F\left\\{{k_{21}^{2}{\eta_{1}}\left({{q^{2}}-k_{21}^{2}}\right)-k_{11}^{2}{\eta_{2}}\left({{q^{2}}-k_{11}^{2}}\right)-{q^{2}}\alpha\left[{1-\left({k_{11}^{2}+k_{21}^{2}-{q^{2}}}\right)\left({{q^{2}}+4{\kappa^{2}}}\right)}\right]}\right\\}}}{{2{\eta_{1}}{\eta_{2}}\left({k_{11}^{2}-k_{21}^{2}}\right)}},$
(12e)
$\displaystyle{d_{4}}=\frac{{2\kappa\left[{\left({k_{21}^{2}+\alpha}\right)-k_{11}^{2}}\right]{\eta_{1}}{\eta_{2}}-F\kappa\left[{k_{11}^{4}+\alpha{q^{4}}\left({k_{11}^{2}+k_{21}^{2}-12{\kappa^{2}}}\right)}\right]}}{{{\eta_{1}}{\eta_{2}}\left({k_{11}^{2}-k_{21}^{2}}\right)}},$
(12f) $\displaystyle+\frac{{F\kappa\left\\{{k_{21}^{4}{\eta_{1}}+3\alpha
q-{q^{2}}\left[{\alpha+3\left({k_{21}^{2}{\eta_{1}}-k_{11}^{2}{\eta_{2}}}\right)+4\alpha{\kappa^{2}}\left({k_{11}^{2}+k_{21}^{2}}\right)}\right]}\right\\}}}{{{\eta_{1}}{\eta_{2}}\left({k_{11}^{2}-k_{21}^{2}}\right)}}.$
(12g)

Ignoring the transient part of the solutions in region 2, we arrive at the
following simple steady state solutions far beyond the metallic interface

$\displaystyle{\Phi_{ss}}(x)=-A\sin(qx+\gamma),\hskip
8.53581pt\gamma=\arctan\left[{\frac{{\left({{q^{2}}-k_{11}^{2}}\right)\left({{q^{2}}-k_{12}^{2}}\right)-4{q^{2}}{\kappa^{2}}}}{{2\kappa
q\left({k_{11}^{2}+k_{12}^{2}-2{q^{2}}}\right)}}}\right],$ (13a)
$\displaystyle{\Psi_{ss}}(x)=A\sin(qx+\delta),\hskip
8.53581pt\delta=\arctan\left[{\frac{{\left({k_{11}^{2}+k_{12}^{2}-{q^{2}}}\right)\left({{q^{2}}+4{\kappa^{2}}}\right)-1}}{{2(\kappa/q)\left({{q^{4}}+4{\kappa^{2}}{q^{2}}-1}\right)}}}\right].$
(13b)

where $A=F/\eta_{1}\eta_{2}$ is the amplitude of the steady state solutions.

Figure 3: (a) The dual-resonance amplitude profiles in terms of the driving
pseudoforce wavenumber for different values of the quasiparticle energy, $E$.
(b) The resonance amplitude profiles in terms of the driving pseudoforce
wavenumber for different values of the damping parametr, $\kappa$. (c) The
phase angle $\delta$ between steady state wavefunction and the driving
pseudoforce. (d) The phase angle $\gamma$ between the electrostatic potential
energy and the driving pseudoforce. The increase in thickness of curves in
plots refer to increases in the varied parameter above given frames.

Figure 3 depicts the amplitude of steady state solution as a function of the
input wavenumber for different values of the normalized quasiparticle energy
orbital, $E$. It is clearly remarked that there is a dual resonance feature
occurring due to both wave-like and particle-like wavenumber match with that
of the driving pseudoforce. However, the current plasmon resonance effect is
caused by the wavenumber matching unlike conventional cases which takes place
due to the frequency matching, hence, it is called the pseudo-resonance
effect. Figure 3(a) shows that the increase in quasiparticle energy leads to
further separation of the resonance peaks, whereas, the amplitude of both
peaks decreases. Conversely, increase in the damping parameter leads to sharp
decrease in peak magnitudes, whereas the their positions are left unaffected,
as shown in Fig. 3(b). Figure 3(c) depicts the phase angle between the steady
state solution and the driving pseudo-force as a function of the driving
wavenumber. It is remarked that the angle changes sign at resonant
wavenumbers, which is a known characteristic feature of the resonant coupled
systems. The phase angle variation between the driving force and the
electrostatic energy function is shown in Fig. 3(d). It is remarked that there
is a phase lag between the wavefunction and electrostatic energy for small
driving wavenumbers, whereas, for large wavenumbers no phase lag occurs.

Figure 4: (a) The wavefunction profile in metallic and vacuum region for given
parameters in the absence of driving pseudoforce. (b) The wavefunction profile
in metallic and vacuum region for given parameters in the presence of driving
pseudoforce. (c) The electrostatic energy profile in metallic and vacuum
region for given parameters in the absence of driving pseudoforce. (d) The
electrostatic energy profile in metallic and vacuum region for given
parameters in the presence of driving pseudoforce.

Figure 4 shows the local wavefunction and electrostatic energy in the metallic
and vacuum regions connected by the boundary conditions for given values of
the energy orbital, $E$, damping parameter, $\kappa$ and pseudoforce
magnitude, $F$. The dual-tone nature of oscillations is clearly apparent in
the metallic region. In the absence of driving pseudoforce, the wavefunction
in the spill-out region is of damped oscillatory type, as shown in Fig. 4(a).
However, as remarked from Fig. 4(b), in the presence of pseudoforce, the
oscillation extend further into the vacuum. Such plane-wave oscillations
beyond the metallic interface may be an indication of resonant collective
electron tunneling or amplified electromagnetic radiation with matched
wavenumber of quasiparticle energy orbital and the corresponding driving
psedoforce can be either an electron beam or the electromagnetic wave directed
into the metallic surface at the resonant angle. Figure 4(c) shows the
electrostatic energy distribution around the metal-vacuum interface. The
oscillation in metallic region are again dual-tone and strongly damp in vacuum
side. The Fig. 4(d) shows a similar feature as Fig. 4(b) for the case of
electrostatic potential energy in the presence of driving pseudoforce.

The solutions to psedoforce system, studied sofar, represent the pure
collective quantum states. The collective statistical quantities corresponding
to an equilibrium state are, however, mixed quantum states which can be
obtained via the standard averaging of the given quantity over all possible
quasiparticle energy orbital using the appropriate occupation function and the
Dos. For instance, the local electron distribution is given as

$\left\langle{{n}(x)}\right\rangle=\frac{{\int\limits_{2}^{\infty}{{{\left|{{\Psi_{N}}(x)}\right|}^{2}}g(E,\kappa)f(E,\theta)dE}}}{{\mathop{\smallint}\limits_{2}^{\infty}g(E,\kappa)f(E,\theta)dE}},$
(14)

where the density of state in the plasmonic energy band is given in terms of
the energy eigenvalues, $E$

$g(E,\kappa)=\frac{{\pi{{\left({E+\alpha}\right)}^{2}}\sqrt{E-2{\kappa^{2}}+\alpha}}}{{\sqrt{2}\left[{E\left({E+\alpha}\right)-4}\right]}}.$
(15)

On the other hand, the electrostatic energy distribution is

$\left\langle{\Phi(x)}\right\rangle=\frac{{\int\limits_{2}^{\infty}{\Phi(x)g(E,\kappa)f(E,\theta)dE}}}{{\int\limits_{2}^{\infty}{g(E,\kappa)f(E,\theta)dE}}}.$
(16)

Figure 5: (a) The quantum statistically averaged electron number density
distribution in metal-vacuum configuration for given normalized temperature
and damping parameter, in the absence of external driving pseudoforce. (b) The
quantum statistically averaged electron number density distribution in metal-
vacuum configuration for given normalized temperature and damping parameter,
in the presence of external driving pseudoforce. (c) The quantum statistically
averaged electrostatic potential energy distribution in metal-vacuum
configuration for given normalized temperature and damping parameter, in the
absence of external driving pseudoforce. (d) The quantum statistically
averaged electrostatic potential energy distribution in metal-vacuum
configuration for given normalized temperature and damping parameter, in the
presence of external driving pseudoforce.

Figure 5 shows the local equilibrium electron number density and electrostatic
potential energy distribution in the metal-vacuum junction for given plasma
parameters in the absence and presence of the driving pseudoforce. Figure 5(a)
reveals that in the metallic region density distribution is strongly
oscillatory with the amplitude of oscillations increasing toward the metallic
edge. Before the metal edge, in the metallic region, a strong electron
depletion occurs leading to a pronounced electric dipole, hence, an abrupt
change in the dielectric properties is expected at the metal-vacuum interface.
Beyond the metal interface, however, the sharp oscillatory decrease in the
electron concentration ocuurs, in the absence of driving pseudoforce. Figure
5(b) shows the electron number density in the presence of the pseudoforce with
given amplitude and driver wavenumber and similar other parameters as in Fig.
5(a). It is remarked that, presence of driving pseudoforce does not
significantly affect the plasmon excitations in the metallic region. However,
electron density of spill-out region extends to far beyond the interface. Note
that amplitude of resonant spill-out electron excitations depends on the
damping parameter and the temperature other than the driving force amplitude
and wavenumber. Figure 5(c) depicts the spacial electrostatic energy
distribution in the metallic and vacuum region in the absence of driving
pseudoforce. Similar features as in Fig. 5(a) are present in Fig. 5(c) showing
that the electrostatic potential energy drops sharply before the metallic edge
due to the electron depletion. It is also remarked that the electrostatic
energy penetrates into the vacuum even in the absence of a driving but quickly
fades further away from the interface. The spill-out of electrons has the same
oscillatory behavior as the static charge screening and can produce Friedel-
like feature at the metallic edge. The later feature can have fundamental
effect on scanning probe microscopy (SPM) measuring quantum device operations
such as the scanning tunneling microscope (STM).

Figure 6: (a) The density plot of electron distribution around the metal-
vacuum interface in the absence of driving pseudoforce showing the large pre-
interface electric dipole. (b) The density plot of electron distribution
around the metal-vacuum interface in the presence of driving pseudoforce for
given values of normalized temperature, chemical potential, damping parameter
and driving parameters. (c) The schematic plot of vacuum-metal-vacumm sandwich
in metal slab configuration. (d) Dynamics of spill-out electrons in the damped
plasmon energy band structure.

Figures 6(a) and 6(b) show the schematic density plot of the electron density
distribution in the absence and presence of driving pseudoforce (which is
assumed to be the radiation). In Fig. 6(a) the electron depletion has led to
the formation of strong post interface dipole. The surface plasmon resonance
is shown in Fig. 6(b) where the driving pseudoforce has been applied. It is
remarked that the surface charge oscillations is significantly manipulated by
the presence of pseudoforce. Figure 6(c) shows the schematic of the surface
plasmon resonance in vacuum-metal-vacuum sandwich. Note that vacuum side may
be replaced with a dielectric layer with the dielectric permittivity ratio
$\epsilon_{r}$. In that case the plasmon frequency definition replaces with
$\omega_{p}=\sqrt{4\pi e^{2}n_{0}/(\epsilon_{r}m_{e})}$. Each region in the
plot is characterized by two functions corresponding to the region. This is
contrary to the conventional quantum theory which uses only the wavefunction
to characterize the single electron behavior in a given potential. The
appropriately irradiated surface spill-out electrons cause the resonant
electron-photon interactions and a collective tunneling or amplified reflected
radiation intensity. Figure 6(d) shows the electron dynamics in the energy
band structure of the plasmonic device. At an equilibrium temperature finite
number of electrons are excited to the plasmon energy band in which collective
oscillations among the excited electrons takes place. The plasmon band plays
the role of a stirring room to produce hot electrons where the electrons
behave collectively analogous to the starling murmuration effect in nature.
Note that at low electron concentration or relatively higher temperatures
greater majority of electron excite to the plasmon band. However, in the
metallic dense electron regime only a fraction of electrons below the Fermi
energy level take part in the collective phenomena. Therefore, the effective
plasmon frequency of the quantum system, which is defined erroneously based on
the Fermi electron density, must be reformulated according to the new equation
of state (EoS) for plasmon electrons in an equilibrium temperature.

Statistical averages of other quantities, such as the (pseudo)resonance
amplitude, $A_{s}=\left\langle{A}\right\rangle$, and the (pseudo)resonance
power, $P_{s}$ can be obtained, accordingly. The pseudoresonance power may be
written as

$P=\frac{{{F^{2}}q}}{{{\eta_{1}}{\eta_{2}}}}\cos(qx)\cos(qx+\delta),$ (17)

Averaging over the complete cycle one gets

$\bar{P}=\frac{{{F^{2}}q\cos\delta}}{{2{\eta_{1}}{\eta_{2}}}},$ (18)

where $\cos\delta$ is the pseudoresonance power coefficient given by

$\cos\delta=\frac{{q\left[{\left({k_{11}^{2}+k_{21}^{2}-{q^{2}}}\right)\left({{q^{2}}+4{\kappa^{2}}}\right)-1}\right]}}{{\sqrt{{q^{2}}{{\left[{\left({k_{11}^{2}+k_{21}^{2}-{q^{2}}}\right)\left({{q^{2}}+4{\kappa^{2}}}\right)-1}\right]}^{2}}+4{\kappa^{2}}{{\left({{q^{4}}+4{q^{2}}{\kappa^{2}}-1}\right)}^{2}}}}}.$
(19)

The quantum statistically averaged pseudoresonance power over all energy
orbital leads to

$\left\langle{\bar{P}}\right\rangle=\frac{{\int\limits_{2}^{\infty}{\bar{P}f(E,\theta)g(E,\kappa)dE}}}{{\int\limits_{2}^{\infty}{f(E,\theta)g(E,\kappa)dE}}}.$
(20)

Figure 7: (a) The variation of psedoresonance amplitude with the driver
wavenumber for different values of damping paramter. (b) Temperature
dependence of the pseudoresonance amplitude for different values of driver
wavenumber. (c) The power coefficient of the pseudoresonance for various
damping level. (d) The power loss/gain in pseudo resonance in terms of driver
wavenumber for different damping values. The increase in thickness of curves
in plots refer to increases in the varied parameter above given frames.

Figure 7 shows the variation in quantum statistically averaged pseudoresonance
parameters of the driven spill-out electrons. The variation of steady state
damped driven solution amplitude is depicted in Fig. 7(a) for different values
of the damping parameter and fixed other parameters. It is remarked that the
amplitude maximized close to the plasmon wavenumber for small damping. The
amplitude decreases sharply and its maximum shifts toward smaller driving
wavenumbers as the damping increases. The temperature dependence of the
pseudoresonance amplitude is shown in Fig. 7(b). It is remarked that, close to
the plasmon wavenumber driving the resonance amplitude maximizes at small
temperatures. However, the later feature appears insignificant for wavenumbers
further away from the critical wavenumber. The variation of power coefficient,
$\cos\delta$ is shown with respect to the driving frequency for different
damping parameter values in Fig. 7(c). This quantity changes the sign at the
critical driving $q=1$, whereas, it maximizes at small $q$ and approximately
$q=1.5$. It is also remarked that this quantity is not significantly affected
by the change in damping parameter. In Fig. 7(d) the variation in
pseudoresonance power is depicted for different damping parameter values. The
quantity changes the sign close to the critical wavenumber indicating a power
loss/gain below/above the corresponding critical point. It is remarked that
the magnitude of power loss/gain strongly varies with the damping parameter.

Figure 8: The electron number density distribution of driven surface plasmon
resonance for different values the driving wavenumber and similar other
parameters. (a) Small under-critical regime $q=0.2$. (b) Under-critical regime
$q=0.8$. (c) Near-critical wavenumber regime $q=1$. (d) Over-critical
wavenumber regime $q=1.2$.

The electron density profiles for different wavenumber regimes are compared in
Fig. 8. It is remarked that the electron density profile close to the driven
metallic interface strongly depends on the driver wavenumber regime. For small
wavenumbers $q=0.2$, in Fig. 8(a), the resonance amplitude is very small with
the transient state extending to average distances beyond the interface. For
$q=0.8$ the same feature repeats with increased resonance amplitude. Close to
the critical wavenumber the transient surface density has increased
significantly. Finally, for overcritical wavenumber value $q=1.2$ the electron
accumulation in front of the metallic surface is further increased. It seems
that the sign of the pseudoresonance power has a keen relation to the momentum
transfer to the surface electrons.

Figure 9: (a) The plasmon energy density of states for different values of the
damping parameter. (b) The plasmon electron number density versus the total
electron number density of the electron gas for different values of the
electron temperatures. (c) Variation of the plasmon wavelength
$\lambda_{p}=2\pi/k_{p}$ with respect to temperature for different values of
the total electron number density. (d) Variation of the plasmon wavelength for
different dielectric replaces with the vacuum. The dashed line in (b)
indicates the plasmon saturation limit $n_{p}=n_{0}$. The increase in
thickness of curves in plots refer to increases in the varied parameter above
given frames.

In Fig. 9 we show some important quantities regarding the DoS of plasmon
energy band, the EoS of plasmon electrons, the real plasmon wavelength defined
in terms of number density of the excited electron to the plasmon band. As
already mentioned, the plasmon scale parameter $l_{p}$ depicted in Fig. 1 is
defined through the total electron concentrations in the metal. However, the
correct definition must be via the EoS of the excited electrons. The energy
DoS of the plasmon band as given in Eq. (15) is plotted in Fig. 9(a). It is
seen that the DoS diverges for minimum plasmon conduction energy, $E=2$, and
has minimum value which shifts toward this critical value as the damping
parameter increases. However, the increase in damping parameter leads to
overall decrease of plasmon DoS. Note that for large energies the DoS
approaches the free electron values $g_{0}(E)=2\pi\sqrt{E_{0}}$ with
$E_{0}=k^{2}$ is the normalized free electron energy dispersion. The number
density of plasmon electrons is shown in Fig. 9(b) in terms of the total
electron density, for different values electron temperature. The dashed line
in (b) indicates the plasmon saturation limit $n_{p}=n_{0}$. The plasmon
electron density is obtained using the following conventional definition

${n_{p}}(\sigma,\theta,\kappa)={n_{0}(\sigma,\theta)}\frac{{\int_{2}^{\infty}{g(E,\kappa)f(E,\theta)dE}}}{{\int_{0}^{\infty}{{g_{0}}(E_{0}){f}(E_{0},\theta)dE_{0}}}},$
(21)

where
$n_{0}={n_{c}}{\theta^{6}}{\rm{L}}{{\rm{i}}_{3/2}}[-{\rm{exp(}}\sigma{\rm{/}}\theta{\rm{)}}]^{4}$
is the total electron density given by (2) with
$n_{c}=16{e^{6}}{m^{3}}/{\pi^{3}}{\hbar^{6}}$ being the characteristic
electron number density, $\rm{Li}$ is the polylog function and
$f(E,\theta)=1/[1+\exp(E/\theta)]$, is the plasmon electron occupation
function. It is remarked that for a given electron temperature up to a lower
critical value of, $n_{0}$, we have $n_{p}=n_{0}$, that is, all the electrons
in the gas contribute to the collective phenomena, whereas, with increase of
$n_{0}$ beyond this critical value the $n_{p}$ sharply falls off and vanishes
for another upper critical, $n_{0}$. Hence for a given temperature there is a
cutoff plasmon electron density and a maximum value before this cutoff value.
The later phenomenon is an important feature of the quantum plasmon
excitations which is usually ignored in the plasmonic research. The cutoff
electron density increases with increase of the electron gas temperature.
Variations of plasmon wavelength $\lambda_{p}=2\pi/k_{p}$ (in nanometer unit)
is depicted in Fig. 9(c) with respect to total electron temperature (in $100$K
unit) for different total electron number density values. It is shown that for
larger electron concentrations the plasmon wavelength increases significantly
at lower temperatures and sharply decreases at higher temperatures. The
variation of $\lambda_{p}$ with the temperature is shown for the case where
the given semiconductor is replaced by the vacuum side. The dielectric values
are $\epsilon_{r}(AlN)\simeq 4.53$, $\epsilon_{r}(ZnSe)\simeq 5.4$,
$\epsilon_{r}(SiC)\simeq 6.7$ and $\epsilon_{r}(GaP)\simeq 9.09$, respectively
in the plot from thin to thick curves.

Figure 10: (a) The dependence of the resonace wavelength on total free
electron density for given temperature and damping parameter. (b) Dependence
of the resonance wavelength on the electron temperature for given electron
temperature and dampind parameter. (c) Dependence of the resonance on the
damping parameter for given electron density and temperature. (d) Dependence
of the resonance wanelength in the relative dielectric permittivity of
subsatnce replaced with the vacuum.

In Fig. 10 we have shown the variations of the surface plasmon resonance
wavelength in nanometers in terms of various electron gas parameters. Figure
10(a) shows that the resonance wavelength increases sharply with the total
electron number density of electrons falling in the visible range for electron
density $10^{19}$cm-3$<n_{0}<10^{20}$cm-3 at room temperature and
$\kappa=0.2$. Figure 10(b) reveals that $\lambda_{res}$ decreases sharply with
increase of the electron temperature for given electron concentration. In Fig.
10(c) variation in resonance wavelength is studied in terms of the damping
parameter for given values of electron density and temperature. It is remarked
that increase in this parameter leads to increase in the resonance wavelength.
Finally, Fig. 10(d) reveals that increase of the dielectric permittivity of
substance replaced with the vacuum leads to increase in the resonance
wavelength for given values of other parameters. Note that in current model of
the surface plasmon resonance we have assumed that the spill-out electrons
have the same normalized temperature as the metal electrons. This is far from
being exact because the plasmon temperature depends on the plasmon electron
density which itself is calculated from the normalized temperature. Therefore,
for correct theoretical modeling the plasmon spill-out electron density can be
measured, directly, using Langmuir probe and then a better estimate for the
plasmon energy of spill-out region can be obtained.

Figure 11: (a) The variation of plasmon energy with respect to the total
number density of electron gas as calculated based on the excited electron
number density. (b) The variation of plasmon wavelength (resonance scale-
length) with respect to the total number density of electron gas as calculated
based on the excited electron number density. (c) The plasmon pressure in
terms of electron temperature for different total electron number density. (d)
The plasmon pressure in terms of total electron number density for different
electron temperature. The dashed curves correspond to free electron energy
dispersion. The increase in thickness of curves in plots refer to increases in
the varied parameter above given frames.

Finally, in Fig. 11 we have shown the characteristic (not conventional) energy
and length scales variation with the total number density of electron gas
calculated based on the excited electrons to the plasmon energy band for given
electron temperature. The dashed curves correspond to the free electron
dispersion relation. It is remarked from Fig. 11(a) that the plasmon energy
has a maximum value depending of the electron temperature and the maximum
plasmon energy shifts to higher electron density as the temperature increases.
The overall plasmon energy value also increases with increase of the electron
temperature. It is noted that the plasmon energy of free electron gas (dashed
curve) increases monotonically. Figure 11(b) reveals that the plasmon
wavelength, which is a characteristic scale of surface plasmon resonance
effect, has a minimum value shifting higher electron densities as the
temperature increases. This minimum also decreases substantially with increase
of electron temperature. The plasmon length of free electron gas (dashed
curve) decreases monotonically. Figure 11(c) shows the plasmon electron
pressure in terms of the electron temperature for different total electron
concentrations. It can be shown that for large values of the temperature the
plasmon pressure obeys a simple relation of the form, $P_{p}=T+10^{\tau}$,
$\tau\simeq 1.1122$. It is also remarked that increase in the total electron
density lowers the plasmon pressure and that the free electron pressure
corresponding to the largest value (dashed curve) is even lower. Figure 11(d)
depict the plasmon pressure as a function of total electron number density for
different electron temperatures. It is shown that the plasmon pressure of an
arbitrary degenerate electron gas is higher for lower total electron number
density and higher temperatures. Pressure of the free electron gas (dashed
curve) indicates a slightly lower pressure corresponding to $T=800$K.

##  V Conclusion

In this research we developed a theory of surface resonance effect based on
the effective Schrödinger-Poisson system. The Hermitian and non-Hermitian
models were used to respectively, model the metallic and spill-out electron
excitations which are shown to be of dual-scale type. The quantum
statistically averaged resonance parameters were obtained and discussed in
terms of various plasmon parameters. Particularly, it was shown that the
conventional definitions of plasmon parameter should be corrected for dense
electron gas based on the electron transition probability into the plasmon
energy band. It was shown that the resonance wavelength falls into the visible
and infrared region for typical lower degenerate electron number densities at
room temperature. Current theory can further elucidate the underlying physical
mechanism involved in collective quantum resonance and unveils the dominant
effect of electrostatic interactions in collective phenomena.

##  VI Data Availability

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

##  VII References

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