Collective Free Electron Excitations in Half-Space Configuration
aa r X i v : . [ phy s i c s . p l a s m - ph ] O c t Hot Electrons and Bound State Potential at Plasmon Boundaries
M. Akbari-Moghanjoughi Faculty of Sciences, Department of Physics,Azarbaijan Shahid Madani University, 51745-406 Tabriz, Iran ∗ (Dated: October 23, 2020) Abstract
Current research presents a model of half-space plasmon excitations for electron gas of arbitrarydegeneracy in a homogenous ambient neutralizing positive background. The half-space plasmonexcitations in current linearized Schr¨odinger-Poisson model reveals some interesting features. Thisis due to the fact that it benefits an effective dual length-scale character of quantum plasmonexcitations, simultaneously, taking into account the high phase-speed collective excitations due toelectrostatic interactions in the electron fluid and low phase-speed phenomenon caused by singleelectron excitations. The mutual coupling between these length scales is remarked to lead to theformation of well defined miniature periodic density fringes in the excited electron fluid whichare modulated over the envelop density and potential patterns. The later interaction leads toformation of a hot electron packet outside the physical jellium boundary of the system. Thepotential energy exterior to surface gives rise to a bound state potential valley in parametricdensity-temperature region relevant to the metallic surfaces, which resembles those of the surfacebinding potentials. This effect may be appropriate to explain the Casimir-Polder-like forces betweenparallel metallic plates in vacuum and metallic nanoparticle interactions. We also use the modelto describe the hot electron generation in plasmonic band structure. Current model can be furtherextended to explore characteristic features of plasmon excitations in other geometries like sphericalfor investigation of short range interactions among nanoparticles and to better understand thephotocatalytic properties of nanometals.
PACS numbers: 52.30.-q,71.10.Ca, 05.30.-d ∗ Corresponding author: [email protected] . INTRODUCTION Plasmonics [1, 2] is a new interdisciplinary field with important applications in nano-electronics [3, 4], optoelectronics [5], and semiconductor integrated circuit industry [6–11].Plasmons are elementary collective excitation of quantum plasmas which play an inevitablerole in nature [12–18]. These entities provide ideal platform for fast THz device commu-nications [19] and beyond where the conventional wired communications fail to operate.Plasmonic energy conversion is an alternative way of solar power extraction in future photo-voltaic and catalytic designs due to their much higher efficiency compared to semiconductortechnologies [20–27]. The state of the art nanocatalitic engineering for energy extractionis based on conversion of energy of collective surface electron plasma oscillations caused bylocalized electromagnetic radiation into hot electron generation in plasmonic devices whichoperates in ultraviolet-visible (UV-VIS) frequency range. However, the conventional solarcell devices are based on low energy interband electron-hole transitions in semiconductorswhich operate at visible-near infrared (VIS-NIR) low frequency range. However, plasmonicdevices operating in ultraviolet spectrum in the wavelength range (10 − T iO in Schottky config-uration [30]. Studies also reveal that hot electron extraction efficiency is strongly dependenton the size and geometry of nanoparticles as well as their composition [31].Quantum plasma regime ranges from doped semiconductors [32, 33] with moderately highelectron number density at low temperature and high density metallic compounds up to high-temperature and density called warm dense matter (WDM) [34, 35]. Quantum effect arisenaturally when the interspecies distances exceeds the thermal de Broglie wavelength [36].In inertial confinement fusion experiment quantum effects vary in strength due to change inequation of state (EoS) of matter during shock compression causing increase in temperatureand number density [37]. The change of EoS has one fundamental quantum effect on the faithof compact stellar object setting distinct limit on the mass of such entities [38]. Another2uantum feature of charged particle ensemble is their complex collective electromagneticinteractions even with their average interparticle distances below the quantum limit. Thelater is because of nonlocal nature of interactions acting via the Bohm’s quantum diffractionpotential. This is why a dilute electron gas interaction with crystal lattice can lead todistinct quantum effects in Bragg diffraction phenomenon [39, 40]. The development ofquantum plasmas had a long history with many pioneering contributions over the pastcentury [41–50]. Because of the dominant quantum effects caused by the EoS of a Fermigas, semiclassical plasma theories which incorporate the quantum statistical pressure effectsbut ignore the electron nonlocality still lead to many interesting features [51–56]. However,many old quantum plasma theories based on the standard Thomas-Fermi assumptions whichignore the von Weizsacker gradient corrections to kinetic energy [57] as the main root tocollective nonlocal behavior of an electron gas, fail to capture the full essence of quantummany-body effects.There has been an increasing interest in the quantum plasma theories over the recentdecade due to their effective description of collective quantum features [58–69]. Quantumkinetic [70], quantum (magneto)hydrodynamic [71] and gradient corrected density functionaltheory (DFT) has been used to investigate different aspects of many particle interactionsin plasmas. Application of the simplest one, i.e. quantum hydrodynamic approach, hasled to discovery of many fascinating collective properties of plasmas which has not beendiscovered in old theories due to ignorance of quantum electron diffraction phenomenon. Thequantum hydrodynamic theory may also be cast into the form of well-known single particleMadelung fluid theory using the appropriate transformations on statistical quantities. Theuse of such framework, so-called Schr¨odinger-Poisson model [72], has the power to studyvariety of interesting linear and nonlinear features of plasmonic environments [73–75] withthe least complexity and computational cost. Recently, the pseudoforce system derived fromlinearized Schr¨odinger-Poisson model has been used to study some novel features of plasmonexcitations [76]. This new model has a fundamental property of taking into account thedual lengthscale nature of single-particle as well as collective interactions in a single frame.The use of linearized quantum hydrodynamic model has recently led to some controversieson the novel attractive potential between quantum screened ions due to the wavenumberscale mismatch in conventional model [77–86]. It has been shown that energy dispersion ofplasmon excitations consists of two main branches each representing different high and low3hase-speed phenomena in plasmonics [87–93]. The attractive quantum screening potentialobtained in linearized quantum hydrodynamic framework has also been shown to be theground state plasmonic effect in Schr¨odinger-Poisson model. II. THE MATHEMATICAL MODEL
In order to study the plasmon excitations in an arbitrary degenerate electrostatically in-teracting electron gas we assume an ambient jellium-like neutralizing background of positiveions and use the following one-dimensional Schr¨odinger-Poisson model [76] i ¯ h ∂ N ∂t = − ¯ h m ∂ N ∂x − eφ N + µ ( n, T ) N , (1a) ∂ φ∂x = 4 πe ( |N | − n ) , (1b)in which N = p n ( x, t ) exp[ iS ( x, t ) / ¯ h ] is the state function of the electron gas in thejellium model with N N ∗ = n ( x, t ) characterizing the number density and v ( x, t ) =(1 /m ) ∂S ( x, t ) /∂x the speed of fluid. Also, φ is the collective electrostatic field due toall charges in the system and n is the background charge density. Moreover, µ ( n, T ) is thechemical potential of the gas defined through the generalized equation of state (EoS) forisothermal electron gas with arbitrary degree of degeneracy.For our case the electron EoS relates the fundamental thermodynamic quantities such asthe statistical pressure and the number density to the chemical potential µ and the electrontemperature T as n = −D Li / [ − exp ( βµ )] , P = − D β Li / [ − exp ( βµ )] , (2)in which β = 1 /k B T and D is so-called the effective electron density of states [32] D = 2Λ = 2 (cid:18) m πβ ¯ h (cid:19) / , (3)with the parameter Λ being the electron thermal de Broglie wavelength. The EoS (2) is alsoconveniently written in a more compact structure P = nβ Li / [ − exp( βµ )]Li / [ − exp( βµ )] , (4)where the polylogarithm function is defined in terms of the Fermi functionsLi ν ( − e z ) = − ν ) ∞ ∫ x ν − exp( x − z ) + 1 d x, ν > , (5)4nd Γ being the ordinary gamma function.Note that the EoS can be expanded in terms of the limiting cases of non-degenerate(classical) and fully degenerate limits. For the fully degenerate isothermal electron gas, z ≫
1, one obtains the form lim z →∞ Li ν ( − e z ) = − z ν / Γ( ν + 1) while in the extreme oppositenon-degenerate limit, z ≪ −
1, one gets Li ν ( − e z ) ≈ − e z . It is evident that in the completedegeneracy limit (such as in metals) the equation of state becomes P = (2 / nk B T F in which T F = E F /k B is the electron Fermi temperature with E F = ¯ h (3 π n ) / / (2 m ) being the Fermienergy of the completely degenerate (zero temperature) electron gas where ¯ h denotes thescaled Planck constant, whereas, in the non-degenerate limit we retain P = nk B T for aclassical dilute electron gas.Here we intend to use the model for metallic compounds with highly degenerate elec-trons and strongly coupled ions (jellium model). The defining parameter for degeneracy is η = T /T F in which T and T F are respectively the electron fluid and Fermi temperatures.Therefore, the parametric regime η < η >
1) denotes the degenerate(nondegenrate) elec-tron fluid. For fully degenerate metals and metallic nanoparaticles we have the extremecase of η ≪ T F ≫ T . In elemental metals at room temperature the Fermi tempera-ture amounts to about 10 K or higher depending on their electron concentrations. In thislimit the chemical potential equals the Fermi energy and all electrons reside below the welldefined Fermi level. On the other hand, ion dynamic properties are characterized via theion coupling parameter, Γ = Ze /dk B T i , in which d is the average inter-ion distance and T i is the ion temperature being much lower compared to that of electron fluid T i ≪ T e dueto the large fractional mass ratio. A good criteria for the weak and strong coupling casesare Γ ≪ ≫
1, respectively [94]. Therefore, metallic compounds with r s = 2 − r s = d/r B is the Bruckener parameter and r B is the Bohr radius, are categorized asstrongly coupled material possessing ionic crystal lattice. However, in current jellium ionmodel we ignore the effect of lattice vibrations (phonons) on electronic properties of the elec-tron gas. Moreover, the electron fluid with r s ≪ − is regarded as nonrelativistic electrongas while semiconductors have the parameter value of r s >
25. Because for semiconductorsthe Fermi temperature is close to that of the electron fluid, they are regarded as partiallydegenerate electron fluid. The electron degeneracy starts approximately at number densityof n ≃ cm − . However, in doped semiconductors the electron density can be much lowerthan this critical value. In fully degenerate elemental metals the electron concentration is5ypically in the range (10 − )cm − .We intend to study the system (1) in the linear limit for simplicity in current analysis.Such a simplified model ignoring other minor interaction effects such as the electron exchangeand correlations still provides many interesting collective features of the electron system.Appropriately normalized closed model constitute of a coupled time-independent system ofcoupled differential equation d Ψ( x ) dx + Φ( x ) + 2 E Ψ( x ) = 0 , (6a) d Φ( x ) dx − Ψ( x ) = 0 , (6b)where it has been assumed that N ( x, t ) = ψ ( x ) ψ ( t ) for the purpose of variables separationand Ψ( x ) = ψ ( x ) / √ n and Φ( x ) = eφ ( x ) / E p with E p = ¯ h p πe n /m being the plasmonenergy of the system. Also, E = ( ǫ − µ ) / E p with ǫ being the eigenenergy of the system isdefined through ǫψ ( t ) = ¯ hωψ ( t ) = i ¯ hdψ ( t ) /dt with ω being the eigenfrequency of plasmonoscillations. Note that the space and time variables are also normalized, respectively, to theplasmon length 1 /k p with k p = p mE p / ¯ h being the plasmon wavenumber and ¯ h/E p . Notealso that we have used the Thomas-Fermi assumption for the chemical potential in fullydegenerate limit where this potential is supposed to be constant throughout the systemin the linear approximation. However, for the Bose-Einstein condensate or in a nonlinearcharged system this quantity may vary considerably, so that the local electrostatic potentialmay not cancel the variations in the chemical potential. The dual-tone solution to (6) is Φ g ( x )Ψ g ( x ) = 12 α Ψ + k Φ − (Ψ + k Φ ) − (Φ + k Ψ ) Φ + k Ψ cos( k x )cos( k x ) , (7)where Φ and Ψ denote the wave functional values at the origin and the characteristicwavenumbers k and k are given as k = √ E − α, k = √ E + α, α = √ E − . (8)The interesting complementarity-like relation k k = 1 characterizes the connection betweenthe wave and particle aspects of plasmon excitations. The solution (8) leads to the gener-alized energy dispersion E = (1 + k ) / k in which E and k are normalized to 2 E p andplasmon wavenumber k p , respectively. The solution (7) reminds the de Broglie’s doublesolution proposed of the pilot-wave theory [95] in which a single electron may be assumed6o be guided by the collective electrostatic interactions in the electron gas. The pilot wavetheory has gained a renewed interest with many experimental support [96] over the past fewyears.The one-dimensional model of plasmon excitations in electron gas (6) has been generalizedto the damped pseudoforce system in order to include the charge screening effect or othersimilar many-body effects [89]. d Ψ( x ) dx + 2 ξ d Ψ( x ) dx + Φ( x ) + 2 E Ψ( x ) = 0 , (9a) d Φ( x ) dx + 2 ξ d Φ( x ) dx − Ψ( x ) = 0 , (9b)where ξ = k sc /k p with the normalized screening parameter ξ = ( E p / n ) ∂n/∂µ =(1 / θ )Li / [ − exp(2 µ/θ )] / Li / [ − exp(2 µ/θ )] being the one-dimensional screeningwavenumber in the finite temperature Thomas-Fermi model [89] with θ = T /T p be-ing the normalized temperature of the electron gas and T p = E p /k B being the characteristicplasmon temperature. The solution to damped pseudoforce system (9) isΦ d ( x ) = e − ξx α ( k Φ + Ψ ) h cos( β x ) + ξβ sin( β x ) i − ( k Φ + Ψ ) h cos( β x ) + ξβ sin( β x ) i , (10a)Ψ d ( x ) = e − ξx α (Φ + k Ψ ) h cos( β x ) + ξβ sin( β x ) i − (Φ + k Ψ ) h cos( β x ) + ξβ sin( β x ) i , (10b)where β = p k − ξ and β = p k − ξ . The initial values Ψ ( x ) and Φ ( x ) are relatedthrough a universal relation Φ = γ Ψ with γ = 2¯ h p π/m at plasmon boundaries betweentwo environments [92].Figure 1 shows variations of the characteristic plasmon parameters for the electron gasof arbitrary degeneracy. Figure 1(a) depicts the variations of normalized one-dimensionaldamping parameter ξ ( µ, θ ) in terms of the normalized chemical potential µ for various valuesof the normalized plasmon temperature, θ . It is remarked that with increase in the chemicalpotential of the gas the damping parameter decreases substantially. In the model of one-dimensional screened surface it might be interpreted that with increase in the degeneracydegree at given electron temperature the screening length increases, while it seems contra-dictory, since when plasma density increases the Debye length always decreases. However,for the normalized screening parameter, ξ , with the scaling parameter as k p which is directly7 .0 0.2 0.4 0.6 0.80.60.70.80.91.01.11.21.31.4 μ E p ) ( a ) μ , θ ) θ Log n cm - (cid:0) b ) E p e V ) Plasmon Energy E p ℏω p
18 19 20 21 220.00.10.20.30.40.5
Log n cm (cid:1) (cid:2) c ) θ T 200, 400, 800 ( K )
16 17 18 19 200200400600800
Log n cm (cid:3) (cid:4) d ) l p n m ) Plasmon Length l p = / k p FIG. 1: 1(a) The variation of normalized screening (damping) parameter as a function of normalizedchemical potential for different values of normalized electron temperature, θ . 1(b) Variation inthe plasmon energy in terms of electron number density in logarithmic scale. 1(c) Variations ofnormalized electron temperature in terms of electron number density for different values of thenormalized electron temperature. 1(d) The plasmon length variation in terms of the electronnumber density. The increase in the thickness of curves in each plot is meant to represent anincrease in the varied parameters above each panel. ξ over the chemicalpotential range decreases. The figure also shows that for the case of complete degeneracythe dependence of the normalized damping parameter to θ becomes insignificant. Moreover,Fig. 1(b) shows the variation in plasmon energy in terms of electron number density in alogarithmic scale. It is remarked that the increase in plasmon energy becomes sharp as thedegeneracy limit, ( n ≃ cm − ), sets in. For typical metals the plasmon energy amountsa few electron Volts. For instance, for cesium with the plasmon energy as low as E p ≃ . E p ≃ E CsF = 1 . E AlF = 11 . θ as a functionof the electron number density for various electron fluid temperatures. It is seen that fora given electron temperature value the normalized electron temperature decreases signifi-cantly with increase in the electron number-density. The variation in this parameter getseven more significant when the electron temperature increases. It is remarked that in thecomplete degeneracy limit dependence of this parameter to electron temperature becomesinsignificant. Figure 1(d) shows the variation in the plasmon length in terms of the electronnumber density in nanometer scale. It is remarked that this characteristic length decreasessharply with increase in the electron number density with values as high as few tenth ofmicrometers for semiconductors to values as low as few nanometers in metallic densities. III. MODEL OF HALF-SPACE PLASMON EXCITATIONS
To this end, let us consider the case in which the electron gas bounded in a half-space( x < x = 0. For the half-space x > x < x ) = Ψ < ( x ) + Ψ > ( x ) and Φ( x ) = Φ < ( x ) + Φ > ( x ) where Ψ > ( x ) = Ψ d (Φ > ( x ) = Φ d ) and Ψ < ( x ) = Ψ g (Φ < ( x ) = Φ g ).Figure 2 shows the variations in normalized perturbed electron number-density, n ( x ) =9 lectron Gaswith positive Background - - x ( l p ) ( a ) | Ψ ( x ) E = - - x ( l p ) ( b ) | Ψ ( x ) E = - - -
20 0 20 40 - - - x ( l p ) ( c ) Φ ( x ) E = - - -
20 0 20 40 - - - x ( l p ) ( d ) Φ ( x ) E = Vacuum
FIG. 2: Profiles of normalized perturbed electron number-density and electrostatic potential energyfor electron gas-vacuum half-space plasmon excitations at given energy eigenvalue shown above eachpanel. The boundary at x = 0 separates the electron gas (shaded area) from the vacuum (unshadedregion). Ψ( x )Ψ ∗ ( x ), and electrostatic potential energy, Φ( x ), for the electron gas region for differentvalues of the energy eigenvalues. It is clearly evident that the values and derivatives ofstate functions must match at the half-space boundary. As remarked in Figs. 2(a) and2(b), for x < x ),10nd consequently in electron number density. Meanwhile, in the region x > x < x ) is regular as comparedto that of Ψ( x ). The plasmonic oscillations are of double-wavenumber character with thesmaller wavelength scale corresponding to the single particle behavior and the larger oneto collective behavior in the electron gas. It is therefore seen that modulated oscillationamplitude in electrostatic energy of the electron gas caused by single particle effects iscomparably lower and is shown as small variations modulated over a larger oscillatory patterncause by the collective behavior of the gas. However this is not the case for the probabilitydensity Ψ( x ) in which the modulated oscillations are profound as compared to those of theelectrostatic energy profile. On the other hand, for x > x > x ) and Φ( x ) provide sufficient information in order to calculatethermodynamic quantities at an equilibrium state. We need however to know the statisticalplasmon energy distribution and plasmon density of states (DoS) in order to obtain thestatistically averaged quantities h Φ( µ, θ, x ) i = ∞ R Φ( E, µ, θ, x ) f ( E, µ, θ ) D ( E ) dE ∞ R f ( E, µ, θ ) D ( E ) dE , (11a) h Ψ( µ, θ, x ) i = ∞ R Ψ( E, µ, θ, x ) f ( E, µ, θ ) D ( E ) dE ∞ R f ( E, µ, θ ) D ( E ) dE , (11b) h n ( µ, θ, x ) i = |h Ψ( µ, θ, x ) i| , (11c)where f ( E ) = [exp(2 E/θ ) − − is the Bose-Einstein occupation number and D ( E ) is the11lasmon energy DoS given as [87] D ( E ) = q E (4 E −
3) + (4 E − √ E − π √ E − . (12)Note that the DoS used here is different from that in Refs. [87, 89]. This is because incurrent calculations we use the modified plasmon dispersion, E = (1 + k ) / k in which E isnormalized to 2 E p , whereas, in Refs. [87, 89] the dispersion E = (1 + k ) /k has been usedin which the energy is normalized to E p . Consequently, the lower integration limits differ inlater cases.Figure 3 depicts the spatial distribution of normalized perturbed electron number-densityand electrostatic energy for given normalized chemical potential and temperature. Figure3(a) shows the electron number density profile for given values of µ = 0 . θ = 0 .
2. Thisfigure illustrates some interesting features of half-space plasmon excitations at equilibriumtemperature. It is remarked that electrons are significantly depleted and consequently aplasmon boundary forms just before the physical edge due to the accumulation of electronsbefore the boundary. The second feature which is quite unique to this model is the well-defined periodic density structure in the electron gas region ( x <
0) which is obviously dueto resonant interaction between single electron and collective excitations. Such feature hasbeen shown to be also characteristics of an electron gas confined in an infinite potential well[76]. Finally, another remarkable feature is that well beyond the jellium edge ( x > θ increases. Note the formation of a well-defined surface dipole just before the jellium edge.It is also remarked that electron halo density increases significantly due to increase of thenormalized temperature. Quite similar trend appears for the electrostatic energy profile inelevated temperature values, θ .Figure 4 depicts the scaled variations of equilibrium density and electrostatic energybeyond the physical boundary, x >
0. Figure 4(a) shows that the density profile has adual peak structure with the first smaller peak due to electron spill-out which is mergedinto the boundary while the second larger peak (hot electron halo) is almost detached fromboundary and extends towards infinity. The increase of the parameter, θ , leads to significantincrease in the amplitude of the hot electron halo as well as the smaller peak maximum,with almost no effect on the position of the maximum amplitudes. The scaled electrostaticenergy profile for the same parameter variations as in Fig. 4(a) is depicted in Fig. 4(b). Itis remarked that the electrostatic energy profile has a flat-top structure near the boundarywith surface value strongly dependent on the fractional temperature. Figures 4(c) and 4(d)show these profiles for the variations in normalized chemical potential of the electron gas.It is remarked that increase in the value of the chemical potential with, θ , being fixedleads to an increase in the amplitude of hot electron halo density shifting the maximumamplitude close to the jellium boundary. However. such variations in the chemical potentialof the electron gas has insignificant effect on the smaller density peak. It is interestingthat for higher values of µ , which coincides with that of the metallic compounds, Friedel-like oscillations in density [40] appears in front of the hot electron halo. This feature isquite similar to the positive charge screening in quantum plasmas. Effect of the change innormalized chemical potential on electrostatic energy profile for similar parameter values asin Fig. 4(c) is shown in Fig. 4(d). It is seen that increase in the chemical potential of thegas leads to significant increase in potential energy decay while for fully degenerate electrongas, with higher chemical potential values, a pronounced bound state valley forms whichhas a shape quite similar to the Lennard-Jones attractive potential with a distances few tento hundred nanometers away from the boundary. This feature also is quite similar effect aspreviously studied quantum charge screening in plasmas [77, 83]. The existence of boundstate beyond the perfectly conducting metallic surfaces together with the surface dipole13ffect may be appropriate to explain the Casimir effect and less understood Casimir-Polderforces [99–101].Figure 5 shows the detailed variations of the electrostatic potential valley profile withthe change in fractional temperature and normalized chemical potential. Figure 5(a) showsthat increase of the normalized temperature for fixed value of the chemical potential leads tosignificant decrease of bound potential energy valley without affecting the minimum position.On the other hand, Fig. 5(b) indicates that the increase of the chemical potential for fixednormalized electron gas temperature leads to increase in depth of the attractive potentialvalley, moving the potential minimum closer to the boundary. Figures 5(c) and 5(d) show thevariations of θ and µ with respect to the electron number density of the electron gas. In termsof θ and µ the electron number density can be written as n ( µ, θ ) = n p θ Li / [ − exp(2 µ/θ )] with n p = 16 e m e / ( π ¯ h ) ≃ . × cm − being the characteristic plasmon number-density. According to Fig. 5(c) almost independent of the value of θ the metallic density isuniversal in µ > θ the electron density approaches the value for fully degenerategas where the bound potential should be present.Current model may also be used for alternative description of the hot electron generationmechanism via surface plasmonic excitations in nanometallic compounds. The hot electrongeneration via LSPR phenomenon is a new efficient method of electricity production fromlocalized high energy UV-VIS electromagnetic radiation [20]. The schematic diagram of hotelectron generation is depicted in Fig. 6(a). The Fermi energy (zero chemical potential)level coincides with E = 0 and the Fermi sea of electrons remain under this energy level.Figure 6(a) also shows the normalized free electron ( E = k /
2) dispersion (in red) andthe plasmon ( E = (1 + k ) / k ) dispersion (in blue). The dual scale-length character ofplasmon excitations is evident from the corresponding curve for given energy E > E p .The plasmon dispersion curve consists of two distinct branches of low and high phase-speedexcitations with respectively positive and negative group velocities. The plasmon dispersionapproaches asymptotically to the free electron curve in the low phase-speed limit. Forthe purpose of hot electron generation the high energy electromagnetic radiation on themetallic surface can excite electrons of the Fermi sea to finite width plasmonic band which14s called the hot electron band. Note that, presence of such a finite lifetime plasmonicband structure is necessary for high energy collective stir up of electromagnetically excitedelectrons. It is noted that only radiation with energies ¯ hω ≥ E p is able to excite electronsto hot electron band with effective maximum width, E F . For conventional nanometallicmaterials such as silver and gold the corresponding plasmon energies are E p = 9 . E p = 9 . hω ≃ λ ≃ . s [20]. However, forultrahigh energy electron extraction the hot electron plasmonic band formation is a necessarycondition.Collective excitations may decay through various channels in metallic nanoparticle LSPR[27]. These decays are of the radiative and nonradiative types [20]. In a radiative type decaythe electromagnetically excited electrons reemit radiation which is either absorbed by thematerial causing the heat production or due to finite width of hot electron band leads tothe photochromism effect [103]. The electromagnetic decay causing the photochromism oc-cur at timescale of the femtosecond order [104]. The electron-electron and electron phononscattering and Landau damping phenomenon are of the most effective decay types. Theelectron-electron scattering are strongly lowered in metallic compound by the well-knownPauli-Blocking [70]. Moreover, the electron-phonon scattering are rare in highly orderedcrystalline material but can play dominant role in glasses [32]. In current model electron-phonon coupling is ignored due to the fact that ion dynamics is not considered in the jelliummodel. However, in recent investigations, consideration of dynamic ions shows its fundamen-tal contributions to the low energy plasmon band structure [105]. It is therefore possiblefor low energy phono-assisted plasmon excitations in the presence of dynamic ions. TheLandau damping is one of well studied phenomena in physics [106]. It is known to occur dueto wave-particle interactions in electron plasma oscillations. A classical interpretation of theLandau damping follows that particles with speed slightly less than the plasmon phase-speedextract energy from the wave and viceversa. Since the population of low speed electrons islarger than those with high speed ones in the distribution, the plasmon excitation is rapidlydamped. The Landau damping rate is directly related to the hot electron extraction inplasmonic materials which is an important parameter in photovoltaic device efficiency. The15onradiative decay of hot electrons into the free electron band, consequently, leads to escapeof hot electrons with energies exceeding that of the Schottky barrier potential, as depicted inFig. 6(a). The energy distribution of hot electrons in plasmonic band is shown in Fig. 6(b)for different values of the normalized temperature, as given by dn ( E ) /dE ≃ D ( E ) f B ( E )with D ( E ) being the plasmon density of states and f B ( E ) is the Maxwell-Boltzmann dis-tribution function for dilute hot electrons in plasmonic band. The hot electron energy bandextends from E min ≥ hω . In case the hot electronband completely takes place inside the plasmon band, we have E min = ¯ hω − E F . Therefore,the number density of hot electrons (normalized to the equilibrium electron number densityof Fermi gas) in the band is given by n = ¯ hω Z E min f B ( µ, θ ) D ( E ) dE, (13)which is the area under the curves in Fig. 6(b). It is however remarked that with in-crease in the electron gas temperature the number of hot electrons increases significantly.As remarked before, the decay time is a determining parameter in hot electron genera-tion of photovoltaic devices. It has been shown that the landau damping rate in sphericalnanometallic particles has inverse dependence to the particle radius ( R ) [31]. The dampingrate Γ( R ) = Γ ∞ + Γ L ( R ) consists of the resistive Drude damping (Γ ∞ ) and Landau wave-particle damping Γ L ( R ) = Av F /R contributions, in which A is a constant dependent toparticle geometry and v F is the Fermi speed [31]. It is evident that in the large particle limit R → ∞ the Landau damping vanishes and the dielectric properties of metallic compoundsshows Drude-like behavior. It is seen that smaller metallic nanoparticles are more efficientfor hot electron regeneration because of higher plasmonic damping character. Therefore,the efficient plasmonic photovoltaic devices should be composed of nanometallic particleswith smaller size but higher electron concentration. It is interesting that these devices op-erate more efficient at higher temperatures unlike their semiconductor counterparts. Ourmodel may also be appropriate for description of plasmon excitations in a two dimensionalelectron gas (2DEG) and high electron concentrated gapped graphene. However, the collec-tive excitations of massless Dirac hot electron generation in two dimensional materials likeintrinsic graphene and silicene requires further development of appropriate hydrodynamicmodels. For this purpose one needs to use the appropriate kinetic theory to obtain pseudo-16orce model from hydrodynamic equations of doped (metallic) graphene. One such a modelof Dirac electron hydrodynamic has been recently derived in Ref. [107]. IV. CONCLUSION
In current research we developed a theoretical model to study the half-space plasmonicexcitations in an electron gas of arbitrary degeneracy. We used the linearized Schr¨odinger-Poisson model and reduced the system it to the coupled pseudoforce type second-order dif-ferential equations, solutions of which with and without damping effects is used to constructthe appropriate state functions for half-space electron gas excitations in different spatialregions. Current dual-tone theory of plasmon excitations may fully captures essential elec-trostatic interactions between single electrons and the collective entity. Some novel featuresof half-space plasmonic excitation were coined in this research and its relevance to plasmonichot electron generation was discussed in detail. A Lennard-Jones-like bound electrostaticpotential energy valley forms beyong the physical boundaries in fully degenerate electrongas which may characterize the Casimir effects. Current study may also be extended tostudy the half-space plasmon excitations in other geometries such as spherical geometry inorder to study the microscopic forces among metallic nanoparticles.
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20 00.00.20.40.60.81.01.21.4 x ( l p ) ( a ) n / n - ∞ θ = μ = - - - -
20 0 - - - x l p ) (cid:5) b ) Φ / Φ - ∞ θ = μ = - - - -
20 0 - - - x l p ) (cid:6) d ) Φ / Φ - ∞ θ = μ = E l e c t r on H a l o Plasmon Boundary - - - -
20 00.00.51.01.5 x l p ) (cid:7) c ) n / n - ∞ θ = μ = Surface Dipole J e lli u m E dge FIG. 3: 3(a) and 3(c) Show statistically averaged values of normalized perturbed electron number-density. 3(b and 3(d)) Show normalized perturbed electrostatic potential energy for electron gas-vacuum half-space plasmon excitations at a thermal equilibrium condition for given parametervalues of θ and µ . The boundary at x = 0 separates the uniform positive beckground charge(shaded area) from the vacuum (unshaded region) in plots 3(a) and 3(c). x ( l p ) ( a ) n / n - ∞ ( × - ) θ = μ = x ( l p ) ( b ) Φ / Φ - ∞ ( × - ) θ = μ = x ( l p ) ( c ) n / n - ∞ ( × - ) θ = μ = x ( l p ) ( d ) Φ / Φ - ∞ ( × - ) θ = μ = H(cid:12)(cid:13) (cid:14)(cid:15)(cid:16)(cid:17)(cid:18)(cid:19)(cid:20)(cid:21) (cid:22)(cid:23)(cid:24)(cid:25)D(cid:26)(cid:27)(cid:28)(cid:29)(cid:30)(cid:31) !"
Attractive PotentialFriedel - Like Oscillations