# Resonant Electron-Plasmon Interactions in Drifting Electron Gas

M. Akbari-Moghanjoughi  Faculty of Sciences, Department of Physics, Azarbaijan
Shahid Madani University, 51745-406 Tabriz, Iran

###### Abstract

In this paper we investigate the resonant electron-plasmon interactions in a
drifting electron gas of arbitrary degeneracy. The kinetic corrected quantum
hydrodyanmic model is transformed into the effective Schrödinger-Poisson model
and driven coupled pseudoforce system is obtained via the separation of
variables from the appropriately linearized system. It is remarked that in the
low phase-speed kinetic regime the characteristic particle-like plasmon branch
is profoundly affected by this correction which is a function of the electron
number density and temperature. We also present an alternative explanation of
the quantum wave-particle duality as a direct consequence of resonant
electron-plasmon interaction (electron murmuration). In this picture drifting
electrons are resonantly scattered by spatial electrostatic energy
distribution, characterizing them by the de Broglie’s oscillations. The phase-
shift and amplitude of excitations in damped driven pseudoforce system is
derived and their variations in terms of normalized chemical potential and
electron temperature is studied. In particular we investigate the kinetic
correction effect on energy dispersion relation in the electron gas in detail.
It is revealed that only the low phase-speed branch of the dispersion curve is
significantly affected by the kinetic correction. It is also found that
increase in the electron number density leads to increase in effective mass
and consequently decrease in electron mobility while the increase in the
electron temperature has the converse effect. The kinetic correction also
significantly lowers the plasmon conduction band. Current model may be further
elaborated to investigate the beam-plasmon interaction and energy exchange in
multispecies quantum plasmas.

###### pacs:

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

##  I Introduction

Plasma oscillations play fundamental role in many physical phenomena
associated with collective interaction of charged species. Due to complex
nature of electromagnetic interactions between variety of charges in
multispecies plasmas a wide spectrum of interesting linear and nonlinear
phenomena such as different instabilities chen , wave-particle interaction
krall , solitons, double layer, shock waves drazin , etc. can exist in these
environments. Plasma theories such a kinetic and hydrodynamic models have been
developed over the past decade in order to study various interesting
collective effects stenf1 ; stenf2 ; stenf3 ; stenf4 . In plasmas the electron
fluid is almost the main ingredient of dielectric response to electromagnetic
waves and dynamic structure factor ichimaru1 ; ichimaru2 ; ichimaru3 ; sturm
leading to many interesting phenomena such as Thomson, Compton jung1 ; jung2 ;
jung3 , stimulated Raman and Brillouin son ; roz  scattering which are used as
efficient plasma diagnostic tools. Electrons are also responsible for other
important physical properties of plasmas such as charge shielding zhao ; mold0
; hong ; ydj1 ; ydj2 ; yoon ; elak , ionic bound states yu ; sahoo ; kar ,
electrical and energy transport phenomena, heat capacity akbheat  and many
others. Dielectric response of electron gas in solid state plasmas plays the
prominent role in optical properties of metals kit ; ash , semiconductors hu1
; seeg , nanometallic structures, low dimensional system and liquid crystals
lucio ; mark ; haug ; gardner ; man1 ; maier ; yofee . However, in order to
understand many of the physical properties of plasmas one has to have detailed
information on collective aspects of electron species in these environments.
In hydrodynamic plasma theories the electron dynamic is simply coupled to
electromagnetic fields through Maxwell equations. Therefore, hydrodynamic and
magnetohydrodynamic theories constitute a cost effective and analytic means to
investigate a wide range of plasma phenomena in a straightforward manner haas1
. These theories, if properly formulated, are also believed to be able to
capture many collective aspect such as collisionless or collisional damping
previously known to be purely kinetic effects mannew .

Due to evergrowing necessity of developments in miniaturized low-dimensional
semiconductor and nanostructured devices for futuristic electronic purposes, a
renewed effort has been devoted to discover physical properties of dense
quantum electron gas over the past few years. Although the theories of
collective quantum electron gas dielectric response has been fully developed
more than half a century ago madelung ; fermi ; hoyle ; chandra ; bohm ; bohm1
; bohm2 ; pines ; levine ; klimontovich ; lind ; stern ; fetter ; mahan , many
interesting properties of quantum electron fluid is yet to be discovered. Many
renewed and enhanced quantum kinetic and quantum hydrodynamic theories have
emerged over the recent years. Application of these theories predicts a very
insightful future for the science of quantum plasmas and have helped to
discover a broad range of new interesting collective phenomena of quantum
electron gas unknown up until the recent decade se ; sten ; ses ; brod1 ;
mark1 ; man3 ; mold2 ; sm ; hurst ; fhaas ; scripta ; kim . Application of
Schrödinger-Poisson system manfredi  and coupled pseudoforce method has shown
a double-tone wave-particle dynamics of the quantum electron gas in a unique
picture akbquant ; akbdirac ; akbgap ; akbtrans ; akbint ; akbfano ; akbdual .
We strongly believe that this dual-lengthscale theory of quantum plasmas
provides a better picture for wave-particle phenomena in collective
environments such as plasmas and fluids. The quantum fluid theory may also
provide appropriate answer to some of the most fundamental quantum mechanical
questions regarding the matter-waves and mysterious duality nature of
particles zm  and nonlocality effect bohmian . It has been recently shown that
the model of Bohmian mechanics and its relation to the quantum electron
diffraction potential appearing in the hydrodynamic formulation is more
fundamental than the single-particle Schrödinger wave equation atiq ; delph .
It was also concluded that the Bohm’s quantum potential makes a basis for the
Schrödinger equation instead of being a consequence of this equation. However,
the proof of de Broglie’s hypothetical matter-wave relation given in current
research is original in the sense that it does not rely on the fundamental
equation of the quantum mechanics.

In current research starting from quantum hydrodynamic theory of an electron
gas a one dimensional model of pseudoforce for arbitrary degenerate electron
beam is developed and the de Broglie’s hypothetical matter-wave equation is
mathematically described as a pseudoresonance effect between the particle-like
branch of the plasmon excitation in the beam and the electron drift. A
generalized dual wavenumber de Broglie equation is obtained which reduces to
the original equation the classical dilute electron beam limit. Some
interesting spatial matter-wave instability in the beam propagation is
introduced for the first time and discussed in terms of the average beam speed
and electron screening effect.

##  II The Quantum Fluid Model

Dynamic properties of a free electron gas can be studied through the
hydrodynamic model to a great extent. Quantum hydrodynamic model may be used
to study the dynamic properties of electron gas with arbitrary degree of
degeneracy. A closed set of conventional quantum hydrodynamic equations for
isothermal electron gas read

$\displaystyle\frac{{\partial n}}{{\partial t}}+\frac{{\partial nv}}{{\partial
x}}=0,$ (1a) $\displaystyle\frac{{\partial v}}{{\partial t}}+v\frac{{\partial
v}}{{\partial x}}=\frac{e}{m}\frac{{\partial\phi}}{{\partial
x}}-\frac{1}{m}\frac{{\partial\mu}}{{\partial
x}}+{\rm{}}\frac{{{\gamma\hbar^{2}}}}{{2{m^{2}}}}\frac{\partial}{{\partial
x}}\left({\frac{1}{{\sqrt{n}}}\frac{{{\partial^{2}}\sqrt{n}}}{{\partial{x^{2}}}}}\right),$
(1b) $\displaystyle\frac{{{\partial^{2}}\phi}}{{\partial{x^{2}}}}=4\pi
e\left({n-{n_{0}}}\right),$ (1c)

where the dependent variables $n$, $v$, $\mu$ and $\phi$ refer to the number
density, average speed, chemical potential and electrostatic potential,
respectively. The last term in the momentum equation is due to the Bohm
potential which is the origin of nonlocal effects in de Broglie-Bohm pilot
wave theory bohmian . The prefactor $\gamma$ is the kinetic correction to the
Bohm potential the origin of which has been the subject of intense
investigations during recent years bonitz1 ; sea1 ; bonitz2 ; sea2 ; bonitz3 .
The parametric equation of state (EoS) for isothermal electron gas with
arbitrary degree of nonrelativistic degeneracy is given in terms of the Fermi
integrals

$\displaystyle
n(\eta,T)=\frac{{{2^{7/2}}\pi{m^{3/2}}}}{{{h^{3}}}}{F_{1/2}}(\eta)=-\frac{{{2^{5/2}}{{(\pi
m{k_{B}}T)}^{3/2}}}}{{{h^{3}}}}{\rm{L}}{{\rm{i}}_{3/2}}[-\exp(\eta)],$ (2a)
$\displaystyle
P(\eta,T)=\frac{{{2^{9/2}}\pi{m^{3/2}}}}{{3{h^{3}}}}{F_{3/2}}(\eta)=-\frac{{{2^{5/2}}{{(\pi
m{k_{B}}T)}^{3/2}}({k_{B}}T)}}{{{h^{3}}}}{\rm{L}}{{\rm{i}}_{5/2}}[-\exp(\eta)],$
(2b)

where $\eta=\beta\mu$ with $\beta=1/k_{B}T$ and $F_{k}$ is the Fermi integral
of order $k$

${F_{k}}(\eta)=\int_{0}^{\infty}{\frac{{{x^{k}}}}{{\exp(x-\eta)+1}}}dx.$ (3)

The function ${\rm{Li}}_{k}$ is called the polylog function defined as

${F_{k}}(\eta)=-\Gamma(k+1){\rm{L}}{{\rm{i}}_{k+1}}[-\exp(\eta)],$ (4)

in which $\Gamma$ is the gamma function. Note that the isothermal EoS is not
the only choice and for fast processes the adiabatic EoS elak  may be
employed. However, for illustrative simplified purpose we have chosen to use
the isothermal electron EoS in current model. The kinetic correction has been
studied in many recent literature michta ; akbhd ; sm ; mold1 ; moldabekov ;
haas2016  and for electron gas with arbitrary degeneracy at finite temperature
may be written as haas2016

$\gamma=\frac{{{\rm{L}}{{\rm{i}}_{3/2}}\left[{-\exp(\beta{\mu_{0}})}\right]{\rm{L}}{{\rm{i}}_{-1/2}}\left[{-\exp(\beta{\mu_{0}})}\right]}}{3{{\rm{L}}{{\rm{i}}_{1/2}}{{\left[{-\exp(\beta{\mu_{0}})}\right]}^{2}}}},$
(5)

where, $\mu_{0}$ is the equilibrium chemical potential. It is well known that
the quantum hydrodynamic model (1) can be readily transformed manfredi  to the
following Schrödinger-Poisson system using the Madelung transformations
madelung

$\displaystyle i\hbar\frac{{\partial\cal N}}{{\partial
t}}=-\frac{\gamma{{\hbar^{2}}}}{{2m}}\frac{{\partial^{2}{{\cal
N}}}}{{\partial{x^{2}}}}-e\phi{\cal N}+\mu(n,T){\cal N},$ (6a)
$\displaystyle\frac{{\partial^{2}{\phi}}}{{\partial{x^{2}}}}=4\pi e(|{\cal
N}|^{2}-n_{0}),$ (6b)

where ${\cal N}=\sqrt{n(x,t)}\exp[iS(x,t)/\hbar]$ is the state function
characterizing spatiotemporal evolution of the electron gas with ${\cal
N}{\cal N^{*}}=n(x,t)$ being the number density and $v(x,t)=(1/m)\partial
S(x,t)/\partial x$ the electron fluid speed. Also, $n_{0}$ represents the
equilibrium electron gas number density. The coupled system (6) can be used to
study the propagation of an electron beam of arbitrary degeneracy with
arbitrary degree of degeneracy in an equilibrium temperature. Let us consider
an arbitrary degenerate electron beam drifting at constant fluid velocity,
$v$. The value of the electron beam chemical potential may vary depending on
the degeneracy degree and the temperature in a wide range. However, in a
classical regime it can have the negative values up to zero electronvolts and
in quantum regime it is positive few electronvolts such as for fully
degenerate typical metals. Therefore, the Schrödinger-Poisson system (6) can
model a wide range of phenomenon for electron beam excitations from classical
regime up to the fully degenerate quantum regime. For our case the beam of
electron gas drifting with constant speed $v$ leads to $S(x,t)=px+f(t)$ in
which $p=mv$ is the electron fluid beam momentum and $f(t)$ is an arbitrary
function of time. Therefore, the state function becomes ${\cal
N}(x,t)=\psi(x,t)\exp(ipx/\hbar)$ where
$\psi(x,t)=\sqrt{n(x)}\exp[if(t)/\hbar]$. It is interesting that in the limit
$\mu_{0}=0$ we have $\gamma=1/3$ while in the fully degenerate electron gas
limit it has the value $\gamma=1/9$.

To this end, let us now consider separable solution ${\cal
N}(x,t)={\psi}(x){\psi}(t)$. After the separation of spatial and temporal
parts in first equation (6), we arrive at

$\displaystyle\frac{\gamma{{\hbar^{2}}}}{{2m}}\frac{{{d^{2}}{\psi}(x)}}{{d{x^{2}}}}+e\phi(x){\psi}(x)-\mu(n,T){\psi}(x)={\epsilon}{\psi}(x)\exp(ipx/\hbar),$
(7a) $\displaystyle\frac{{d^{2}{\phi(x)}}}{{d{x^{2}}}}=4\pi
e(|{\psi}(x)|^{2}-n_{0}),$ (7b) $\displaystyle
i\hbar\frac{{d{\psi}(t)}}{{dt}}=\epsilon{\psi}(t),$ (7c)

where $\epsilon$ is the energy eigenvalue of the electronic system,
${\psi}(x)=\sqrt{n(x)}$ and ${\psi}(t)=\exp[if(t)/\hbar]$. Here we are
interested in finding the solution for spatial variations. Therefore, we have
the first two coupled equations in (7) to solve. The linear system for small
perturbations is akbquant

$\displaystyle\frac{\gamma{{d^{2}}\Psi(x)}}{{d{x^{2}}}}+\Phi(x)+E\Psi(x)=-E\exp(ik_{d}x),$
(8a) $\displaystyle\frac{{{d^{2}}\Phi(x)}}{{d{x^{2}}}}-\Psi(x)=0,$ (8b)

in which $k_{d}=\hbar\l_{p}/p$ is the normalized de Broglie’s wavenumber in
which $\l_{p}=\hbar/\sqrt{2m_{e}E_{p}}$ is the plasmon length and
$E_{p}=\hbar\omega_{p}$ being the plasmon energy and $\omega_{p}=\sqrt{4\pi
e^{2}n_{0}/m}$ the electron plasma frequency. Also, $E=\epsilon/E_{p}$ with
$\epsilon$ being the energy eigenvalue of the system. Note that in the
linerization of dependent quantities we have used
$\psi\simeq\psi_{0}+\psi_{1}$, $\phi\simeq 0+\phi_{1}$ and $p=0+p_{1}$ We have
also used the normalization scheme $\Psi(x)={\psi}(x)/{\psi_{0}}$
(${\psi_{0}}=\sqrt{n_{0}}$ is the unperturbed value of ${\psi}(x)$ and
$\Phi(x)=e\phi/E_{p}$. The spacial coordinate $x$ is normalized to the plasmon
wavelength $\lambda_{p}=2\pi/k_{p}$ with $k_{p}=\sqrt{2mE_{p}}/\hbar$ being
the characteristic plasmon wavenumber. On the other hand, the time is scaled
with respect to inverse plasmon frequency, $\omega_{p}=E_{p}/\hbar$. Note that
the solution for ${\cal N}(x,t)$ would be quasi-stationary for the beam since
$\psi(x,t)\psi^{*}(x,t)$ does not explicitly depend on time. Therefore, the
spacial distribution of electron density remains constant over the time
whereas the electron beam flows with uniform speed.

##  III Pseudoresonant Plasmon Excitations

Let us now study the solution to the coupled psuedoforced differential system
(8). The general solution to the homogenous system read

$\left[{\begin{array}[]{*{20}{c}}{{\Phi_{g}}(x)}\\\
{{\Psi_{g}}(x)}\end{array}}\right]=\frac{\gamma}{\alpha}\left[{\begin{array}[]{*{20}{c}}{{\Psi_{0}}+k_{2}^{2}{\Phi_{0}}}&{-\left({{\Psi_{0}}+k_{1}^{2}{\Phi_{0}}}\right)}\\\
{-\left({{\Phi_{0}}/\gamma+k_{1}^{2}{\Psi_{0}}}\right)}&{{\Phi_{0}}/\gamma+k_{2}^{2}{\Psi_{0}}}\end{array}}\right]\left[{\begin{array}[]{*{20}{c}}{\cos({k_{1}}x)}\\\
{\cos({k_{2}}x)}\end{array}}\right],$ (9)

in which $\Phi_{0}$ and $\Psi_{0}$ characterize the initial values where we
have assumed $d\Psi(x)/dx|_{x=0}=d\Phi(x)/dx|_{x=0}=0$, for simplicity. The
characteristic plasmon excitation wavenumbers $k_{1}$ and $k_{2}$ are given as

${k_{1}}=\sqrt{\frac{{E-\alpha}}{{2\gamma}}},\hskip
8.53581pt{k_{2}}=\sqrt{\frac{{E+\alpha}}{{2\gamma}}},\hskip
8.53581pt\alpha=\sqrt{{E^{2}}-4\gamma}.$ (10)

Note that the complementarity relation $k_{1}k_{2}=1$ always holds. It can be
seen the plasmon excitations satisfy the energy dispersion $E=(1+\gamma
k^{4})/k^{2}$ with a minimum at $(E_{m},k_{m})=(2\gamma^{1/2},\gamma^{1/4})$.
The particular solution, on the other hand, can be written in the following
simplified form

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

The kinetic correction in our normalization scheme takes the following form

$\gamma=\frac{{{\rm{L}}{{\rm{i}}_{3/2}}\left[{-\exp(2\sigma/\theta)}\right]{\rm{L}}{{\rm{i}}_{-1/2}}\left[{-\exp(2\sigma/\theta)}\right]}}{3{{\rm{L}}{{\rm{i}}_{1/2}}{{\left[{-\exp(2\sigma/\theta)}\right]}^{2}}}},$
(12)

in which $\sigma=\mu_{0}/E_{p}$, $\theta=T/T_{p}$ is the fractional electron
to plasmon energy $T_{p}=E_{p}/k_{B}$.

Figure 1: (a) Effect of normalized chemical potential $\sigma$ on the
characteristic plasmon wavenumbers for given normalized electron temperature
$\theta$. (b) Effect of normalized chemical potential $\sigma$ and normalized
electron temperature $\theta$ on the kinetic correction factor $\gamma$. (c)
Variations in plasmon energy of electron gas in terms of the normalized
chemical potential $\sigma$ for given normalized electron temperature
$\theta$. (d) Variations in plasmon length of electron gas in terms of the
normalized chemical potential $\sigma$ for given normalized electron
temperature $\theta$. The increase in thickness of curves indicate increase in
the varied parameter value above each panel.

Figure 1 depict various parameters involved in plasmon excitations. Figure
1(a) shows the characteristic wavenumber branches of plasmon excitations
(wave-like $k_{1}=1/k^{2}$ and particle-like $k_{2}=\gamma k^{2}$) for
different values of normalized chemical potential, $\sigma=\mu_{0}/E_{p}$ at
given normalized temperature $\theta=T/T_{p}$. Note that the plasmon
wavenumbers depend on the chemical potential and electron temperature via the
kinetic correction, $\gamma(\sigma,\theta)$. The lower/upper branches
correspond to wave/particle excitation in the electron gas. It is clearly
remarked that chemical potential has significant effect on particle branch at
higher kinetic energy values $E$. The parameter dependence of kinetic
correction factor is depicted in Fig. 1(b). It is shown that increase in the
normalized temperature increases the correction while increase in the
normalized chemical potential leads to decrease in this correction parameter.
It is seen that in the fully degenerate regime the kinetic correction
approaches the value $\gamma=1/9$ akbhd ; moldabekov . Moreover, Fig. 1(c)
shows the dependence of plasmon energy on chemical potential and temperature
of the electron gas. It is remarked that increase in values of normalized
chemical potential and temperature lead to increase in the plasmon energy.
Note that all these parameter depend on the number density of the electron
gas. The number density of electron gas in normalized for is given as

$n(\sigma,\theta)=\frac{{16{e^{6}}m_{e}^{2}{\theta^{6}}}}{{{\pi^{3}}{\hbar^{6}}}}{\rm{L}}{{\rm{i}}_{3/2}}{\left[{\exp\left({\frac{{2\sigma}}{\theta}}\right)}\right]^{4}}.$
(13)

The variations of plasmon length in micrometers is depicted in Fig. 1(d). It
is seen that the plasmon length at classical limit is few micrometers and
decreases rapidly to nanometer scale at degenerate regime. It is also remarked
that this characteristic lengthscale decreases significantly at higher
temperature only in classical regime.

The solution (11) becomes resonant solution at electron drift conditions
$k_{d}=\hbar/p=k_{1}/k_{p}$ and $k_{d}=\hbar/p=k_{2}/k_{p}$, in dimensional
units. Note that the later condition reduces to de Broglie’s hypothetical
assumption, hence, the particle branch of plasmon excitation becomes resonant
at condition $p=\hbar k_{2}$. In the single-particle limit or at elevated
kinetic energy values $E\gg 1$ the particle-like branch of plasmon excitations
becomes identical to the free electron branch and we obtain the relation
$p=\hbar k$. Note also that in such a limit the wave-like branch vanishes
according to Eqs. (10). However, current plasmon theory introduces the second
de Broglie wavenumber $k_{1}$ to the electron gas particles at high wavelength
limit. Due to the gradient correction to kinetic energy of the electron gas
the principal de Broglie’s wavenumber is affected strongly by the electron gas
density and temperature. However, the second or wave-like branch is not
affected by these parameters. These effects may be confirmed by different
experiments revealing the wave and particle aspects of a dense electron beam.
It can be seen that the de Broglie’s wavenumbers depend on chemical potential
and temperature of the electron gas via the kinetic correction factor.
However, the dependence on the chemical potential and temperature in fully
degenerate regime is insignificant according to Fig 1(b).

Figure 2: (a) The electrostatic energy distribution for given values of energy
eigenvalue, chemical potential, temperature and de Broglie wavenumber. (b) The
wavefuntion profile for given values of energy eigenvalue, chemical potential,
temperature and de Broglie wavenumber. (c) Effect of normalized electron
temperature on the plasmon excitation stability in arbitrary degenerate
electron gas. (d) Amplitude of driven plasmon excitations with dual resonance
in the absence of plasmon damping.

Figure 2(a) and 2(b) show the electrostatic energy and wavefunction profiles
for the driven plasmon excitations in drifting electron gas with given
normalized chemical potential, $\sigma=\mu_{0}/E_{p}$, and electron
temperature, $\theta=T/T_{p}$, values. The oscillations are evidently triple
mode consisting of dual plasmon mode in addition to the de Broglie,s mode
caused by the electron drift. The de Broglie parameter $k_{d}=v/v_{p}$ is the
normalized electron drift speed to that of the plasmon, $v_{p}=\hbar
k_{p}/m_{e}$. Note that $\Psi(x)\Psi^{*}(x)$ gives the normalized electron
number density distribution along the electron beam. Figure 2(c) shows the
stability region for plasmon excitations of arbitrary degenerate electron gas
as dashed areas for given normalized parameter values and for two different
electron temperatures. The excitations are stable for energy eigenvalues,
$E>2\sqrt{\gamma}$. It is remarked that the increase in chemical potential
gives rise to the stability of excitations, whereas, the increase in the
temperature leads to destabilization effect. Finally, Fig. 2(d) depicts the
dual pseudoresonance effect due to de Broglie’s wavenumber matching
$k_{d}=k_{1},k_{2}$, which is the plot of amplitude of plasmon excitations
driven by electron drift in terms of normalized drift speed for given other
normalized parameters. Therefore, for any given energy eigenvalue, $E$, the
wavenumbers $k_{1}$ and $k_{2}$ given by dispersion relation
$E(\sigma,\theta)=k_{1}^{2}+k_{2}^{2}$, are in fact the normal stationary
modes of plasmon excitations in the electron gas. In other words, an existing
stationary plasmon mode can be excited in the drifting electron gas by the de
Broglie’s wavenumber matching condition. Note also that for
$k_{d}=v/v_{p}=\gamma^{1/4}$ de Broglie’s condition coincides with the quantum
beating condition for which $k_{1}=k_{2}$.

The hypothetical wave nature of electrons proposed by de Broglie deb , which
was soon confirmed by many experiments, has been one of the most intriguing
phenomena for the life of quantum theory. The problem is the key to
understanding of the theory which has been called not well-understood by the
Nobel laureate Richard Feynman: ”I think I can safely say that nobody really
understands quantum mechanics”. The quantum nonlocality effects cause by the
wave-particle phenomenon has led to some unphysical interpretations of quantum
phenomena zm  such as the double-slit experiments and quantum measurments.
There has been early attempts to survive the deterministic nature of particle
dynamics in the quantum theory such as the proposal of de Broglie-Bohm pilot
wave theory bohmian . However, the later theory does not actually provide a
physical description of the pilot wave origin. Current hydrodynamic model of
plasmon excitation provides a clear picture of the phenomenon. The dual
lengthscale plasmonic excitations in an electron gas are caused by collective
effects due to the electrostatic interactions, we may call it the electron
murmuration phenomenon. On the other hand, it was shown that the de Broglie’s
wave-particle relation is a direct consequence of resonant electron-plasmon
interaction in the driven pseudoforce system (8). Therefore, it is seen that
electrons are piloted by electron murmuration wave which is produced by their
electrostatic interactions. However, the murmuration phenomenon is not the
actual cause of quantum wave. The quantum de Broglie wave is caused by the
resonant interaction of electron drift with the collective plasmon excitations
(the electron murmurations).

##  IV Transient and Persistent Plasmon Excitations

We will now consider the pseudodamped system in which the spacial damping of
the wavefunction and electrostatic energy may have various origins. One of the
cases is the screening effect around the positively charged ions in the gas.
Such screening is due to local changes in the chemical potential of the gas
which characteristic normalized wavenumber
$\xi(\sigma,\theta)=({E_{p}}/2)\partial
n/\partial\mu=(1/2\theta){\rm{L}}{{\rm{i}}_{1/2}}\left[{-\exp({2\sigma}/\theta)}\right]/{\rm{L}}{{\rm{i}}_{3/2}}\left[{-\exp({2\sigma}/\theta)}\right]$.
In a pure electron gas the plasmon damping may also occur due to various
channels such as the electron-neutral collisions and Landau damping.
Therefore, the appropriate model for plasmon excitations in electron gas with
screening effect is as follows

$\displaystyle\gamma\frac{{{d^{2}}\Psi(x)}}{{d{x^{2}}}}+2\gamma\xi\frac{{d\Psi(x)}}{{dx}}+\Phi(x)+{E}\Psi(x)=-E\cos(k_{d}x),$
(14a)
$\displaystyle\frac{{{d^{2}}\Phi(x)}}{{d{x^{2}}}}+2\xi\frac{{d\Phi(x)}}{{dx}}-\Psi(x)=0,$
(14b)

in which $\xi$ is the normalized pseudodamping parameter

$\displaystyle\begin{array}[]{l}{\Phi_{gd}}(x)=\frac{{\gamma{{\rm{e}}^{-\xi
x}}}}{\alpha}\left\\{{\begin{array}[]{*{20}{l}}{\left({k_{2}^{2}{\Phi_{0}}+{\Psi_{0}}}\right)\left[{\cos({\beta_{1}}x)+\frac{\xi}{{{\beta_{1}}}}\sin({\beta_{1}}x)}\right]-}\\\
{\left({k_{1}^{2}{\Phi_{0}}+{\Psi_{0}}}\right)\left[{\cos({\beta_{2}}x)+\frac{\xi}{{{\beta_{2}}}}\sin({\beta_{2}}x)}\right]}\end{array}}\right\\},\\\
{\Psi_{gd}}(x)=\frac{{\gamma{{\rm{e}}^{-\xi
x}}}}{\alpha}\left\\{{\begin{array}[]{*{20}{l}}{\left({{\Phi_{0}}/\gamma+k_{2}^{2}{\Psi_{0}}}\right)\left[{\cos({\beta_{2}}x)+\frac{\xi}{{{\beta_{2}}}}\sin({\beta_{2}}x)}\right]-}\\\
{\left({{\Phi_{0}}/\gamma+k_{1}^{2}{\Psi_{0}}}\right)\left[{\cos({\beta_{1}}x)+\frac{\xi}{{{\beta_{1}}}}\sin({\beta_{1}}x)}\right]}\end{array}}\right\\},\end{array}$
(15g)

where $\beta_{1}=\sqrt{k_{1}^{2}-\xi^{2}}$,
$\beta_{2}=\sqrt{k_{2}^{2}-\xi^{2}}$ are the characteristic pseudodamped
plasmon wavenumbers. The pseudodamped plasmon excitations follow the energy
dispersion $E=[1+\gamma(k^{2}+\xi^{2})^{2}]/2(k^{2}+\xi^{2})$. It is evident
that the general solution (15) is transient and does not extend to long
ranges. The particular solution, on the other hand, reads

$\displaystyle\Phi(x)=\frac{{E{{\rm{e}}^{-\xi
x}}}}{{\alpha{\eta_{1}}{\eta_{2}}}}\left[{{\eta_{1}}\left({k_{d}^{2}-k_{2}^{2}}\right)\cos({\beta_{2}}x)-{\eta_{2}}\left({k_{d}^{2}-k_{1}^{2}}\right)\cos({\beta_{1}}x)}\right]-$
(16a) $\displaystyle\frac{{E{{\rm{e}}^{-\xi
x}}}}{{\alpha{\eta_{1}}{\eta_{2}}}}\left[{\xi{\eta_{1}}\left({k_{d}^{2}+k_{2}^{2}}\right)\frac{{\sin({\beta_{2}}x)}}{{{\beta_{2}}}}-\xi{\eta_{2}}\left({k_{d}^{2}+k_{1}^{2}}\right)\frac{{\sin({\beta_{1}}x)}}{{{\beta_{1}}}}}\right]-\frac{E}{{\gamma{\eta_{1}}{\eta_{2}}}}\times$
(16b)
$\displaystyle\left\\{{\left[{\left({k_{d}^{2}-k_{1}^{2}}\right)\left({k_{d}^{2}-k_{2}^{2}}\right)-4k_{d}^{2}{\xi^{2}}}\right]\cos({k_{d}}x)+2{k_{d}}\xi\left({k_{1}^{2}+k_{2}^{2}-2k_{d}^{2}}\right)\sin({k_{d}}x)}\right\\},$
(16c) $\displaystyle\Psi(x)=-\frac{{E{{\rm{e}}^{-\xi
x}}}}{{\alpha{\eta_{1}}{\eta_{2}}}}\left[{{\eta_{1}}k_{2}^{2}\left({k_{d}^{2}-k_{2}^{2}}\right)\cos({\beta_{2}}x)-{\eta_{2}}k_{1}^{2}\left({k_{d}^{2}-k_{1}^{2}}\right)\cos({\beta_{1}}x)}\right]+$
(16d) $\displaystyle\frac{{E{{\rm{e}}^{-\xi
x}}}}{{\alpha{\eta_{1}}{\eta_{2}}}}\left[{\xi{\eta_{1}}k_{2}^{2}\left({k_{d}^{2}+k_{2}^{2}}\right)\frac{{\sin({\beta_{2}}x)}}{{{\beta_{2}}}}-\xi{\eta_{2}}k_{1}^{2}\left({k_{d}^{2}+k_{1}^{2}}\right)\frac{{\sin({\beta_{1}}x)}}{{{\beta_{1}}}}}\right]+\frac{E}{{\gamma{\eta_{1}}{\eta_{2}}}}\times$
(16e)
$\displaystyle\left\\{{\left[{1-\left({k_{1}^{2}+k_{2}^{2}-k_{d}^{2}}\right)\left({k_{d}^{2}+4{\xi^{2}}}\right)}\right]k_{d}^{2}\cos({k_{d}}x)+2{k_{d}}\xi\left({1+k_{d}^{4}-4{\xi^{2}}k_{d}^{2}}\right)\sin({k_{d}}x)}\right\\},$
(16f)

where $\eta_{1}=(k_{d}^{2}-k_{1}^{2})^{2}+4k_{d}^{2}\xi^{2}$ and
$\eta_{2}=(k_{d}^{2}-k_{2}^{2})^{2}+4k_{d}^{2}\xi^{2}$. The persistent
solution may be written in the forms $\Phi(x)=A\cos(k_{d}x-\phi)$ and
$\Psi(x)=-A\cos(k_{d}x-\psi)$ with amplitude and phase given as
$A={E}/{{\gamma{\eta_{1}}{\eta_{2}}}}$ and

$\tan\phi=\left[{\frac{{2\xi
k_{d}\left({k_{1}^{2}+k_{2}^{2}-2{k_{d}^{2}}}\right)}}{{\left({{k_{d}^{2}}-k_{1}^{2}}\right)\left({{k_{d}^{2}}-k_{2}^{2}}\right)-4{k_{d}^{2}}{\xi^{2}}}}}\right],\hskip
8.53581pt\tan\psi=\left[{\frac{{2(\xi/k_{d})\left({1+{k_{d}^{4}}-4{\xi^{2}}{k_{d}^{2}}}\right)}}{{1-\left({k_{1}^{2}+k_{2}^{2}-{k_{d}^{2}}}\right)\left({{k_{d}^{2}}+4{\xi^{2}}}\right)}}}\right].$
(17)

Note that the excitation amplitude and phases depend on the chemical potential
and the temperature of the electron gas via the kinetic correction and damping
parameters. It is also remarked that the presence of damping leads to a phase
shift between the electrostatic energy and density distribution in the system.

Figure 3: (a) Amplitude of high phase-speed damped driven plasmon excitations
in terms of normalized electron drift speed for different energy eigenvalues.
(b) Amplitude of low phase-speed damped driven plasmon excitations in terms of
normalized electron drift speed for different energy eigenvalues. (c)
Amplitude of low phase-speed damped driven plasmon excitations in terms of
normalized electron drift speed for different normalized chemical potential.
(d) Amplitude of low phase-speed damped driven plasmon excitations in terms of
normalized electron drift speed for different normalized electron temperature.
The increase in thickness of curves indicate increase in the varied parameter
value above each panel.

Figure 3 shows the amplitude of damped plasmon excitations (16) for various
normalized parameter values in terms of normalized electron drift. Figures
3(a) and 3(b) show that two set of resonant amplitude peaks occur for given
parameters. The long wavelength resonance in Fig. 3(a) are due to de Broglie’s
matching condition at wave-like branch while small wavelength onse in Fig.
3(b) are due to de Broglie’s matching condition at particle-like branch in
dispersion relation. It is revealed that increase in the energy eigenvalue of
excitations caused the amplitude height to decrease and move to smaller
wavenumbers for wave-like resonances, whereas, it causes the amplitude to move
to higher wavenumbers for particle-like resonances. It is noted that the wave-
like resonance amlitudes are relatively higher than that of particle-like
ones. Figure 3(c) shows that increases in the normalized chemical potential of
electron gas leads the particle-like resonance peak to sharply decrease and
shift to higher wavenumber values. However, the change in chemical potential
and electron temperature, which have not been shown, does not affect the wave-
like resonance peaks. Figure 3(d) shows that the effect of electron
temperature variation on the particle-like resonance peak is insignificant but
opposite to the case of chemical potential changes.

Figure 4: (a) The electrostatic energy distribution for given values of energy
eigenvalue, chemical potential, temperature, de Broglie wavenumber and damping
parameter. (b) The wavefuntion profile for given values of energy eigenvalue,
chemical potential, temperature, de Broglie wavenumber and damping parameter.
(c) Variation of the electrostatc energy distribution phase shift with respect
to the driving pseudoforce for different values of the normalized electron
temperature. (d) Variation of the electrostatc energy distribution phase shift
with respect to the driving pseudoforce for different values of the normalized
chemical potential. The increase in thickness of curves indicate increase in
the varied parameter value above each panel.

Figure 4 Depicts the electrostatic energy and wavefunction profiles of
persistent solution of resonant plasmon excitations for given parameters. The
presence of pseudodamping has led to filtration of other excitations than the
forcing oscillations. Therefore, resonant plasmon excitations became single-
tone with de Broglie wavelength distribution. In is remarkable that a drifting
single-momentum electron gas leads to a uniform periodic distribution of
electron density over the space wavelength of which is give by $\lambda=h/p$.
It is also seen that there is phase difference in oscillations between these
two profiles which caused by the damping (screening) effect. Figures 4(c) and
4(d) show the phase variations in the electrostatic energy in terms of the de
Broglie wavenumber. Effects of electron temperature and chemical potential is
shown respectively in Figs. 4(c) and 4(d). These variations have the
pronounced effect around the particle-like resonance and clearly have the
characteristic features of dual resonant wave phenomena. Note that when
$\phi=0$ the electric field spacial distribution becomes in-phase with driver
(electron flow). This parameter is therefore a key measure for the electron
beam-plasmon interaction quality. In classical resonance phenomenon this is
the point for which the maximum power in delivered to the system.

Figure 5: (a) The plasmon dispersion in the absence (dshed) and presence
(solid) of the kinetic correction. (b) The effect of normalized electron
chemical potential on the plasmon dispersion curve. (c) The effect of
normalized electron temperature on the plasmon dispersion curve. (d) The
effect of normalized electron chemical potential on the damped plasmon
dispersion curve. The increase in thickness of curves indicate increase in the
varied parameter value above each panel.

To this end, we study the effects of kinetic correction and electron gas
parameters on the energy dispersion of plasmon excitations. Figure 5(a)
depicts the energy disperion in the presence (solid curve) and the absence
(dashed curve) of kinetic correction for given chemical potential and electron
temperature parameter values. The effect of kinetic correction is huge on the
low phase-speed particle-like branch (which we call it the kinetic regime)
with almost no effect on the high phase-speed wave-like branch which are
separated at the quantum beating point, $k=2\sqrt{\gamma}$. The effect of
kinetic correction is evidently increases sharply as the energy eigenvalue
increases. The group speed of low phase-speed branch has significantly
decreased due to the kinetic correction. This feature has been previously
discussed in thermodynamic analysis of one component quantum plasmas akbhd ;
haas2016  where the kinetic correction only applies to low phas-speed plasma
phenomenon. It is remarkable that the kinetic correction profoundly lowers the
energy minimum (plasmon conduction levels) of excitations accessible to Fermi
electrons. Figure 5(b) reveals the effect of normalized chemical potential of
electron gas on the low phase-speed dispersion branch indicating that increase
of this parameter leads to lower group-speed in this branch. Note that the
second derivative of energy is a measure of effective mass and mobility of
electrons at a given wavenumber at plasmon level. It is therefore concluded
that increase in the chemical potential leads to increase in the effective
mass of electrons and decrease in their mobility. Moreover, Fig. 5(c) shows
the effect of electron temperature on the dispersion curve. It is evident that
increase in the electron gas temperature increases the electron mobility in
contrast to the case of chemical potential increase. Finally, Fig. 5(d) shows
the effect of chemical potential on dispersion curve of damped plasmons. The
similar effect as in Fig. 5(b) is also apparent in this case. However, the
presence of strong dampind (charge screening) leads to truncation of wave-like
branch.

##  V Conclusion

We studied the resonant interaction between drifting electrons and plasmon
excitations in an electron gas of arbitrary degenerate electron gas using the
effective kinetic corrected Schrödinger-Poisson system. It was found that the
kinetic correction leads to significant effect of electron number density and
temperature on the plasmon excitations. We gave a physical description of
hypothetical de Broglie wave-particle theory in terms of resonant drifting
electron iteration by the collective distribution of electrostatic charge in
space. We also found the solution of the damped driven pseudoforce system in
the presence of kinetic correction and calculated the amplitude and phase
shifts of electrostatic energy and wavefucntion relative to that of the driver
pseudoforce. The kinetic correction was found to profoundly affect the
dispersion curve in the low phase-speed (small wavelength) regime which we
call it the kinetic regime. It was further revealed that electron density
leads to increase in effective electron mass and decrease in its mobility in
plasmon conduction band while the increase in the temperature has the inverse
effect. Current mode of drifting quantum electron gas can be used to
investigate veriou other phenomena such as the beam-plasma interactions.

##  VI Data Availability

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

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