Effect of Dynamic Ions on Band Structure of Plasmon Excitations
aa r X i v : . [ phy s i c s . p l a s m - ph ] S e p Effect of Dynamic Ions on Band Structure of Plasmon Excitations
M. Akbari-Moghanjoughi Faculty of Sciences, Department of Physics,Azarbaijan Shahid Madani University, 51745-406 Tabriz, Iran (Dated: September 18, 2020)
Abstract
In this paper we develop a new method to study the plasmon energy band structure in multi-species plasmas. Using this method, we investigate plasmon dispersion band structure of differentplasma systems with arbitrary degenerate electron fluid. The linearized Schr¨odinger-Poisson modelis used to derive appropriate coupled pseudoforce system from which the energy dispersion struc-ture is calculated. It is shown that the introduction of ion mobility, beyond the jellium (staticion) model with a wide plasmon energy band gap, can fundamentally modify the plasmon dis-persion character leading to a new form of low-level energy band, due to the electron-ion bandstructure mixing. The effects ionic of charge state and chemical potential of the electron fluid onthe plasmonic band structure indicate many new features and reveal the fundamental role playedby ions in the phonon assisted plasmon excitations in the electron-ion plasma system. Moreover,our study reveals that ion charge screening has a significant impact on the plasmon excitationsin ion containing plasmas. The energy band structure of pair plasmas confirm the unique role ofions on the plasmon excitations in many all plasma environments. Current research helps to betterunderstand the underlying mechanisms of collective excitations in charged environment and theimportant role of heavy species on the elementary plasmon quasiparticles. The method developedin this research may also be extended for complex multispecies and magnetized quantum plasmasas well as to investigation the surface plasmon-polariton interactions in nanometallic structures.
PACS numbers: 52.30.-q,71.10.Ca, 05.30.-d . INTRODUCTION Theories of electron gas excitations are of primary importance in physical properties ofsolids [1]. Because of inertialess feature of electrons almost all the electronic propertiesof matter is influenced by dielectric response of the electron gas to external perturbations[2–4]. These properties range from optical and mechanical to transport phenomena [5–8].In crystalline solids due to complex nature of energy bands the electromagnetic responseof the electrons are fundamentally dependent on the energy dispersion and the structureof energy bands in a specific direction [9, 10]. One of the well-known features caused byfree electrons in metals is the existence of sharp plasmon edge leading to their distinguishedoptical properties. Plasmons are elementary collective excitations of electrons with manyapplications in newly emerged interdisciplinary fields such as opto- and nanoelectronics[11, 12], plasmonics [13, 14] and miniaturized integrated semiconductor circuit industry [15]etc. due to their ultrafast terahertz scale electromagnetic response feature [16–18]. Plasmonsalso find applications in optical surface engineering due to production of appealing visualeffects on nanocoated materials [19]. On the other hand, electrons have the first role in manyquantum effects in metals and semiconductors such as the Landau quantization leading tothe integer quantum Hall and de Haasvan Alphen effects [20]. Moreover, the role playedby electrons in linear and nonlinear wave phenomenon of complex plasmas is tremendous[21–27]. Study of electronic properties of solid within the free electron model has a limitedapplications due to the fact that it ignores interactions between the crystal lattice and theelectron gas. Due to strong ionic coupling in solids, in the jellium model, ions usually makethe neutralizing positive background and do not contribute to the electronic properties.However, the electrostatic potential of the ionic lattice is the most important source ofinteraction between electrons and the lattice potential. Moreover, manybody theories such asthe density functional theory (DFT) [28–31] and Hartree-Fock perturbation [32] take furtherstep in accounting for interactions among electrons in expense of considerable computationalefforts [33–35].In plasma theories, on the other hand, different plasma species are coupled through thecommon electromagnetic fields that is caused by these species. Therefore, the many bodyinteraction effects is the main building block of the plasma theories. However, in specificoscillation regimes the dynamics effects due to one or more species can be ignored where2hese species are considered as the jellium in the plasma model. For instance, in a plasmamodel of dust acoustic excitations both electron and ion dynamics are ignored due to theirinertialess character as compared to dust fluid. Despite the long history of pioneering de-velopments [36–45], recent advancements in quantum kinetic and (magneto)hydrodynamictheories [46–48] have opened new opportunities to explore the fields of complex dense plas-mas with arbitrary degree of electron degeneracy and ion coupling strength, ranging fromstrongly coupled nanometallic compounds and warm dense matter (WDM) [49, 50] up tothe astrophysical dense objects such as planetary cores and white dwarf stars [51]. Applica-tion of quantum plasma theories have revealed a large number of interesting new linear andnonlinear aspects of collective interactions among different plasma species [52–65] and hasled do discovery of many new phenomena such as resonant shift and electron spill-out [66],novel quantum screening [67–70] and plasmon-soliton [71], to name a few. Application oflinearized Schr¨odinger-Poisson system for eigenvalue problem has shown many interestingnew aspects of quantum plasmas due to the unique dual lengthscale character of plasmonexcitations [72–79]. In current research we would like to extend our previous results ofpseudoforce approach to the plasmon energy band structure in complex plasma environ-ments. This may lead to a better understanding of collective effects on electron transportand plasmonic properties of the fourth state of matter which constitutes at least 90 percentof the visible universe.
II. THE THEORETICAL MODEL
In order to investigate the energy dispersion relation for one-dimensional plasmon exci-tations we use the linearized Schr¨odinger-Poisson system which is cast into the followingnormalized coupled pseudoforce model representing the eigenvalue problem for plasmon en-ergies in an ensemble of arbitrary degenerate electron gas in an ambient jellium-like positivebackground [75] d Ψ( x ) dx + Φ( x ) + E Ψ( x ) = 0 , (1a) d Φ( x ) dx − Ψ( x ) = 0 , (1b)in which E = ( ǫ − µ ) /E p with E p = ¯ h p πe n /m e where n and µ are, respectively,the equilibrium number density and chemical potential of the electron gas. Moreover, m e
3s the electron mass and ǫ is the kinetic energy of electrons in the band structure. Also,the normalized functions Ψ( x ) = ψ/ √ n and Φ( x ) = eφ/E p represent the local probabilitydensity and electrostatic energy of the gas, so that, n ( x ) = ψ ( x ) ψ ∗ ( x ) is the local numberdensity functional. Note that in the Thomas-Fermi approximation the chemical potentialremains constant throughout the gas for linear perturbations. At thermal equilibrium forisothermal processes at temperature T the electron number density and chemical potentialare connected via a simple equation of state (EoS) [69] n = − (cid:18) m e πβ ¯ h (cid:19) / Li / [ − exp ( βµ )] , P = − Nβ Li / [ − exp ( βµ )] , (2)in which P is the statistical pressure satisfying the thermodynamic identity n = dP/dµ and β = 1 /k B T . The polylogarithm function Li ν is given in terms of the Fermi functionsLi ν ( − exp[ z ]) = − ν ) ∞ ∫ x ν − exp( x − z ) + 1 d x, ν > , (3)where Γ is the ordinary gamma function. The system (1) has simple general solutiondiscussed elsewhere [72]. The linear plasmon dispersion relation is obtained assumingΨ( x ) = Ψ exp( ikx ) and Φ( x ) = Φ exp( ikx ) which together with Eq. (1) leads to E − k − − k Ψ Φ = . (4)which leads to the simple plasmon dispersion E = (1 + k ) /k where E and k are normalizedto E p and k p = p m e E p / ¯ h , respectively. The system (1) can be generalized for particle ofarbitrary mass M and charge Zγ d Ψ( x ) dx − Z Φ( x ) + E Ψ( x ) = 0 , (5a) d Φ( x ) dx + Z Ψ( x ) = 0 , (5b)where γ = m e /M is the fractional mass. The system (5) the energy dispersion relation E =( Z + γk ) /k which in the limit M = m e and Z = ± Z = −
1) and positron ( Z = +1)gases are identical. The plasmon energy dispersion has two distinct lengthscales. For k ≫ E ≃ γk whilefor k ≪ E ≃ Z /k . The duallength scalecharacter of plasmon excitations, which is evidently due to both single-particle and collective4ehavior, has shown to produce many fundamental properties in an unmagnetized electrongas [73–79]. The dispersion curve has a minimum value at k m = γ / √ Z and E m = 2 Z √ γ and for E > E m has two characteristic wavenumbers k ± = s E ± p E − γZ γ , (6)in which k + and k − are particle- and wave-like branches satisfying the complementarity-like relation k + k − = | Z | . For electron or positron gas we simply have E m = k m = 1 and k + = 1 /k − .In Fig. 1(a) we have shown the values of plasmon energies for different values of thechemical potential of an arbitrary degenerate electron gas for different value of electron gastemperature. In the completely degenerate gas, µ ≫
1, or in dilute classical limit, µ ≪ − n ≃ cm − . Furthermore,Fig. 1(c) the region in temperature-chemical potential for which the thermal energy in thegas is comparable to twice as much as the plasmon energy (which is the plasmon gapstudied in the following). It is remarked that for dilute classical electron gas µ < E p even for very large electron temperatures. The later is because theincrease in the temperature of the gas also increases its plasmon energy. However, in theclassical region µ < E T ≥ E p . Figure1(d) shows the variation of the chemical potential versus the electron number density fordifferent values of temperature. It ia again remarked that for fully degenerate gas the effectof temperature becomes insignificant. However, for the classical electron gas for a fixedchemical potential increase of the temperature significantly increases the electron numberdensity.Figure 2 shows the variations in plasmon energy dispersion curves for various parameters.5 T > E p - - μ eV ) ( a ) E p e V ) E p = (cid:1)(cid:0) p , T = , 2 × ( K )
15 16 17 18 19 20 21 220.00.51.01.52.02.53.0
Log n cm - (cid:2) b ) E p e V ) Plasmon Energy E p = (cid:3)(cid:4) p - - - - - μ eV ) (cid:5) c ) T K ) Thermal Region E T > E p - - μ eV ) (cid:6) d ) L og [ n ( c m - ) ] T = , 2 × , 3 × ( K ) FIG. 1: (a) Variations of electron plasmon energy in terms of the chemical potential of arbitrarydegenerate electron gas for different values of electron gas temperature. (b) Variations in plasmonenergy in terms of equilibrium electron gas number density. (c) Parametric region in which thethermal energy of electrons exceed twice the plasmon energy of the electron gas. (d) Variations ofelectron number density versus the chemical potential of the electron gas. The thickness of curvesin plots are intended to charcterize the increasing of the values in varied parameter above eachpanel. ero Temperature Electron Sea ℏω (cid:7) (cid:8) k ( k p ) ( b ) E ( E p ) Z = (cid:9) γ (cid:10) μ e (cid:11) μ , μ + μ + - - -
20 0 20 40 600.000.020.040.060.080.10 k ( k p ) ( c ) E ( E p ) A =
3, Z =
1, 2, 3, μ i = ( Ion ) - - -
20 0 20 40 600.000.050.100.150.200.250.30 k ( k p ) ( d ) E ( E p ) A =
1, 2, 3, Z = μ i = ( Ion ) Mainplasmonenergyband - - - k ( k p ) ( a ) E ( E p ) Z = - γ = ( Electron ) FIG. 2: (a) The plasmon energy band struture of plasmon excitations electron gas of arbitrary de-generacy in Thomas-Fermi approximation in jellium model. (b) Variations in the electron plasmonband structure with changes in the normalized chemical potential. (c) Plasmon dispersion struc-ture for heavy particle and the effects of particle charge state on the band structure. (d) Effect ofatomic number of ion species on the energy band structure of neutralized ion fluid. The thicknessof curves in plots are intended to charcterize the increasing of the values in varied parameter aboveeach panel. E = 0 corresponds to top of the electron see at zero temperature ( ǫ = µ ). At thefull degeneracy limit ( µ = ǫ F with ǫ F being the Fermi energy of the gas) all available electronenergies are packed in the area E ≤ ǫ < ǫ F ( ǫ F ≃ . E >
2. Thedashed curve, on the other hand, shows the free electron dispersion. The origin of plasmonband gap is in fact a resonant quantum scattering between free electron-like ( E = k )propagations by their collective excitation with the characteristic dispersion of E = 1 /k ,shown as dashed curves in Fig. 1(a). At low temperature where thermal energy of electronsis low (room temperature thermal energy ≃ . ǫ > E p ( E p ≃ . k . For excitations with k ≫ m b = m ∗ /m e =2( d E/dk ) − = k / ( k +3) (in the normalized form) which is definitely positive in the wholeplasmon band approaches that of the free particle limit, i.e. m ∗ ≃ m e . On the other hand,for k ≪ m b →
0. Therefore, to distinct regimes of fast wave-like ( ω > ω p and k < k p ) and slow particle-like ( ω > ω p and k > k p ) electron response to externalperturbations are possible in the plasmon band. Note that in room temperature there is avery small probability for plasmon excitations at the Fermi surface of metals due to limitingplasmon occupation number factor f ( E, θ ) = [exp(2
E/θ ) + 1] − ath thermal equilibrium,where θ = T /T p with T p = E p /k B being the plasmon temperature.Figure 2(b) shows the effect of chemical potential on the electron plasmon dispersion. Itis remarked that as the electron gas become dense at constant temperature E m increases but k m does not change. For a classical dilute electron gas the chemical potential can becomenegative and the plasmon energy can have small values comparable to room temperaturethermal energy in which case the bottom of plasmon conduction band sink into the electronsea. The later is the reason for why plasmon oscillations of electron gas are dominant for8lassical rather than degenerate electron gas (with fixed jellium-like positive background).Figure 2(c) shows plasmon dispersion curves for classical gass of charged particle with atomicnumber A ( γ ≃ / Am p with m p being the proton mass) and different charge values, Z .The values of E m ≪ E p and k m ≫ k p are due to very small γ values of ions comparedto the electron. The in the very high wavenumber limit the dispersion approaches theparticle-like branch γk . It is remarked that as the charge value increases k m decreases but E m increases significantly. Such changes in the charge alters the wave-like branch but doesnot significantly modify the particle-like branch of the dispersion. It is concluded that theions charge state significantly affects the wave-like behavior of plasmon excitations. On theother hand, Fig. 2(d) shows the effect of changes in the atomic number of ionic gas on thedispersion pattern. It is seen that in this case the effect is prounanced on the particle-likebranch while the wavelike branch stays almost intact. It is also remarked that with in creasein the value of A , k m increases while E m decreases as opposed to the variations in Fig. 2(c). III. PLASMONS IN DYNAMIC-ION ENVIRONMENTS
The model in Sec. III is known as the jellium model in which neutralizing background ionsare static. In this section we will explore how the ion dynamics even negligible compared tothat of electrons can fundamentally modify the plasmon dispersion character in electron-ionplasma. To this end, we consider the pseudoforce system d Ψ ( x ) dx + Φ( x ) + E Ψ ( x ) = 0 , (7a) γ d Ψ ( x ) dx − Z Φ( x ) + ( E + µ )Ψ ( x ) = 0 , (7b) d Φ( x ) dx − Ψ ( x ) + Z Ψ ( x ) = 0 , (7c)where Ψ and Ψ respectively denote the probability density of electrons and ions in thesystem and we assumed the ions are classical µ i ≃
0. Solving the system for plasmoneigenenergy values leads to the following dispersion relation E ± = k h λ ± p λ + 4 k ( k − λ − γ + 1) + 4 µk i λ = (1 + γ ) k − µk + Z + 1 , (8)It is evident that the dispersion relation has two distinct branches due to dual species gasand reduces to dispersion relation for electron gas in jellium background for γ = Z = 0. The9 lectron Sea Conduction like Conduction likeNewBandFermiSea - - - k ( k p ) ( c ) E ( E p ) A =
3, Z =
1, 2, 3, μ e = ( Electron - Ion ) - - - - k ( k p ) ( d ) E ( E p ) A =
3, Z = μ e = ( Electron - Ion ) BandGapValence like - - - k ( k p ) ( a ) E ( E p ) A =
3, Z = μ e = ( Electron - Ion ) - -
20 0 20 40 - k ( k p ) ( b ) E ( E p ) A =
3, Z = μ e = ( Electron - Ion ) FIG. 3: (a) Plasmon energy band for electron-ion plasma which dynamic ions and arbitrary de-generate electron gas. (b) Phonon assisted plasmon conduction band in a wider scale for givenplasma parameters. (c) Effect of dynamic ion charge on the plasmon band structure of electron-ionplasma. (d) Effect of the electron gas chemical potential on the energy dispersion on electron ionplasmas. The thickness of curves in plots are intended to charcterize the increasing of the valuesin varied parameter above each panel. plasmon dispersion of electron-ion plasma gives rise to upper and lower excitation bands.This is shown in Fig. 3(a) together with the parabolic free electron dispersion k E = ( Z + γk ) /k − µ shown as the dashed asymptotic curves.Note that in each plot the point E = 0 denotes ǫ = µ where ǫ is the normalized kinetic10nergy of electrons and E = − µ denotes the zero kinetic energy point. It is remarkablethat introduction of mobile ions fundamentally alters the plasmon band structure adding alower band overlapping the electron sea. It is evident that the new band originates due tothe coupling of free electron and ion plasmon excitations coupling (crossing of the dasheddispersion curves). Physically, this is interpreted as the resonant scattering of electronsby the wavelike branch of ion plasmon excitations. The latter effect is completely differentfrom the band gap structure caused by the resonant scattering of free electrons by the latticepotential of static ions in crystalline materials. Moreover, the main band is slightly liftedso that the height of the band gap remains almost unchanged. The great resemblance ofcurrent band structure to that in crystal lattices [1] is remarkable. The new band containsa valence-like ( m b <
0) structure around k ≃ m b > k < . k > .
5. The lower energy band can have a significant effect onlow temperature variation of macroscopic quantities such as the specific heat and currentdensity. Note that electrons in valence- and conduction-like band response oppositely tothe external field due to the relation a = q E /m e where a and E refer to the accelerationand external electric field respectively. Note also that valence bands contain k ranges forwhich the plasmon group-speed v g = (1 / ¯ h ) dE/dk can be either positive or negative. Dueto different curvature values of the conduction-like valleys the electron at lower k valuesrespond faster compared to those in large k plasmon conduction electrons. It is also notedthat for very low-wavenumber (very large wavelength) plasmon excitations electrons almostfeel free. Because the new band sinks into the electron sea, the electrons need not to bethermally excited in order to contribute collectively. However, there is zero-temperature cut-off wavenumber range 0 . < k < µ = µ / E p = 0 . A = 3 and Z = 2. Figure 3(b) shows the plasmon dispersionband structure of electron-ion plasma at a larger wavenumber scale. It is seen that the newenergy band approaches the particle-like branch of ion E = γk − µ shown as dashed curve inFig. 3(b). Note that in large wavenumber plasmon excitation in conduction band electronseffective mass equal nearly that of the ions. The later phonon-coupled plasmon excitationsare less effective in the electronic properties of the plasma compared to the large wavelengthplasmon excitations. Figure 3(c) and 3(d) reveal the effects of ion charge state and elctronfluid chemical potential on the new band structure in electron-ion plasma. It is seen thatwith increase in the ion charge the wavenumber cut-off range decreases and more and more11lectrons in the Fermi gas contribute to the plasmon excitations at lower temperatures.Figure 3(d) on the other hand reveals that for very low values of µ = 0 . µ (consequently the degree of degeneracy)increases the portion of electrons and wavenumbers in the plasmon excitations. The lateris one of the fundamental feature of metal behavior formation in solids which is beautifullyillustrated in current plasmon band structure of dense electron-ion plasmas. IV. EFFECT OF ION CHARGE SCREENING
Let us now study the effect of charge screening on the plasmon band structure of electron-ion plasmas. In doing this we consider the following generalized pseudoforce system d Ψ ( x ) dx + 2 ξ d Ψ ( x ) dx + Φ( x ) + E Ψ ( x ) = 0 , (9a) γ d Ψ ( x ) dx + 2 γξ d Ψ ( x ) dx − Z Φ( x ) + ( E + µ )Ψ ( x ) = 0 , (9b) d Φ( x ) dx + 2 ξ d Φ( x ) dx − Ψ ( x ) + Z Ψ ( x ) = 0 , (9c)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 [75]. The eigenenergy equationfor the system (9) may be found using the transformations Ψ ( x ) = ψ ( x ) exp( − ξx ),Ψ ( x ) = ψ ( x ) exp( − ξx ) and Φ ( x ) = φ ( x ) exp( − ξx ) and using ψ ( x ) = ψ exp( ikx ), ψ ( x ) = ψ exp( ikx ) and φ ( x ) = φ exp( ikx ). Then we find E − k − ξ E + µ − γ ( k + ξ ) − Z − Z − k − ξ ψ ψ φ = . (10)The dispersion relation is then given as E ± = K h Λ ± p Λ + 4 K ( K − Λ − γ + 1) + 4 µK i Λ = (1 + γ ) K − µK + Z + 1 ,K = p k + ξ . (11)It is obvious to find out that all the dispersion relations we have calculated so far do notdepend on the sign of the ion charge. The Eq. (11) reduce to that of the unscreened jellium12 - - k ( k p ) ( a ) E ( E p ) A =
3, Z = ξ = μ e = ( Electron - Ion ) - - k ( k p ) ( b ) E ( E p ) A =
3, Z = ξ = μ e = - - - - k ( k p ) ( c ) E ( E p ) A =
3, Z = ξ = μ e = - -
20 0 20 40 - k ( k p ) ( d ) E ( E p ) A =
3, Z = ξ = μ e = FIG. 4: (a) The plasmon energy band structure of electron-ion plasma with screened ions. (b) Theeffect of normalized charge screening parameter on the main plasmon conduction band for givenplasma parameters. (c) Effect of normalized charge screening parameter on lower plasmon conduc-tion band for high wavelength excitations. (d) Effect of normalized charge screening parameter onlower plasmon conduction band for small wavelength excitations. The thickness of curves in plotsare intended to charcterize the increasing of the values in varied parameter above each panel. model for γ = ξ = Z = 0. Note also that in the limit ξ = 0 we retain the dispersion forunscreened electron-ion plasma (8). It is evident that there are two distinct bands in thedispersion structure similar to the previous case.13igure 4 shows the energy dispersion of plasmon excitations in screened electron-ionplasma. Figure 4(a) as compared to Fig. 3(a) reveals that introduction of charge screeningreduces electron population in small wavenumber range of plasmon conduction band. Thedetails of dispersion curve variations for each band are shown in Figs. (c)-(d). It is seenfrom Fig. 4(b) that the in screened plasma the wave-like branch in main dispersion band islimited. It is also remarked from Fig. 4(c) that increase in the value of screening parameterleads to shift of low wavenumber conduction valley to higher values. It is obvious that sucha shift significantly lowers the plasmon states in the lower energy band which contributesthe most to the plasmon excitations in electron-ion plasmas. Moreover, Fig. 4(d) showsthat the charge screening has insignificant effect on the large wavenumber conduction valleyin the lower plasmon band. V. PLASMON DISPERSION IN PAIR PLASMAS
For the sake of completeness we would like to investigate the plasmon band structure inpair plasmas. For instance in a electron-pair-ion plasma the pseudoforce system reads d Ψ ( x ) dx + Φ( x ) + E Ψ ( x ) = 0 , (12a) γ d Ψ ( x ) dx − Z Φ( x ) + ( E + µ )Ψ ( x ) = 0 , (12b) γ d Ψ ( x ) dx + Z Φ( x ) + ( E + µ )Ψ ( x ) = 0 , (12c) d Φ( x ) dx − Ψ ( x ) + Z Ψ ( x ) − Z Ψ ( x ) = 0 . (12d)The dispersion relation then reads E = γk − µE ± = k h λ ± p λ − k ( γk + 2 Z + γ ) + 4 k (1 + k ) i λ = (1 + γ ) k − µk + 2 Z + 1 . (13)It is evident that there are three bands corresponding to three species in the plasma.The dispersion band structure for (13) is shown in Fig. 5(a). It is remarked that thenew band E = γk which is the parabolic dispersion of ion now appears as the new wideband. Due to very large effective mass of electrons associated with this new conduction bandit does not significantly contribute to electronic properties of electron plasmon excitations.14 lectron Sea P(cid:12)(cid:13)(cid:14)(cid:15)(cid:16)(cid:17)(cid:18) (cid:19)(cid:20)(cid:21) N - Type Silicon n = × cm - - - - k ( k p ) ( a ) E ( E p ) A =
3, Z = μ e = - - - k ( k p ) ( c ) E ( E p ) μ e = - μ p = - - - k ( k p ) ( d ) E ( E p ) A =
3, Z = μ e = - μ p = - - k ( k p ) ( b ) E ( E p ) β e = β h = μ e = - E g = FIG. 5: (a) Plasmon energy dispersion of pair-ion electron-ion plasmas. (b) Plasmon dispersionstructure of electron-hole pair with static ions in intrinsic semiconductor plasmas. Plasmon exci-tation band structure of electron-positron plasma (c) without and (d) with dynamic ion species.
Another instance of pair plasma is the electron-hole plasma relevant to semiconductors. Thepseudoforce system reads γ e d Ψ e ( x ) dx + Φ( x ) + E Ψ e ( x ) = 0 , (14a) γ h d Ψ h ( x ) dx − Φ( x ) + ( E − E g )Ψ h ( x ) = 0 , (14b) d Φ( x ) dx − Ψ e ( x ) + Ψ h ( x ) = 0 , (14c)15here Ψ e , Ψ h and Ψ i refer to probability density of electron, hole and static ion species.Also, γ e = m e /m ∗ e , γ h = m e /m ∗ h and E g = µ h − µ e [80] is the normalized (to E p ) gap energyof semiconductor [80]. The characteristic eigenenergy equation in this case follows E − γ e k E − E g − γ h k − − − k ψ e ψ h φ = . (15)Note that the ions have chosen to be static in this case. The corresponding plasmon energydispersion reads E ± = k (cid:26) λ ± q [ E g + ( γ h − γ e ) k ] k + 4 (cid:27) ,λ = ( γ e + γ h ) k + E g k + 2 , (16)where β e = 0 .
26 = 1 /γ e and β h = 0 .
39 = 1 /γ h are the effective-mass ratios for electrons andholes in the silicon semiconductor, respectively, which we used in Fig. 5(b). The dispersioncurve is shown in the figure for an electron doped silicon with room temperature number-density of n ≃ × cm − ( E p ≃ µ e ≃ − . ǫ g ≃ . E = E g + µ e with E = 0 denoting the Fermi energy level which is closer to theconduction band for N-type semiconductors, as is the case. For the electron-positron plasmawe have d Ψ e ( x ) dx + Φ( x ) + E Ψ e ( x ) = 0 , (17a) d Ψ p ( x ) dx − Φ( x ) + ( E + 2 µ e )Ψ p ( x ) = 0 , (17b) d Φ( x ) dx − Ψ e ( x ) + Ψ p ( x ) = 0 , (17c)where we have used µ e + µ p ≃
0. This leads to the dispersion relation E ± = ( k ± p µ e k + 1 + 1) /k . (18)16he plot of dispersion branches shows that in electron-positron pair plasma there is sig-nificant overlap between the electron sea and the small wavenumber band available forelectron-positron plasmon excitations. This pair plasma is zero band gap plasmonic systemwith nearly a free electron parabolic dispersion conduction band. In the following it maybe illustrative to consider a more realistic electron-positron pair plasma in the presence ofa dynamic ion species. The pseudoforce system in this case is d Ψ e ( x ) dx + Φ( x ) + E Ψ e ( x ) = 0 , (19a) d Ψ p ( x ) dx − Φ( x ) + ( E + 2 µ e )Ψ p ( x ) = 0 , (19b) γ d Ψ i ( x ) dx − Z Φ( x ) + ( E + µ e )Ψ i ( x ) = 0 , (19c) d Φ( x ) dx − Ψ e ( x ) + Ψ p ( x ) + Z Ψ i ( x ) = 0 , (19d)with a long expression for the dispersion solution which is avoided here for simplicity. Thecharacteristic eigenenergy equation in this case follows E − k E − k + 2 µ e −
10 0 E − γk + µ e − Z − Z − k ψ e ψ p ψ i φ = . (20)The dispersion curve is shown in Fig. 5(d). The complex structure shows multiple bandstructure with the superposition of asymptotic previously studied dispersion curves, namely,the electron-positron (Fig. 5(c)) and ion (Fig. 3(a)). It is remarked that introduction ofdynamic ions in this case has led to double conduction band in the middle and has signif-icantly increased the electron population in conduction band. Moreover, small wavelengthexcitations appear naturally as phonon assisted plasmon excitation in this system. VI. CONCLUSION
In this research we developed a new theory of plasmon band structure in plasmas. Weused the linearized Schr¨odinger-Poisson model and the pseudoforce method to study theenergy band structure of plasmon excitations in different plasmas with arbitrary degree ofdegeneracy. Our study reveals the fundamental role played by dynamic ions on the band17tructure and available energy levels for electrons in the conduction bands. Many valenceand conduction-like structure is found to be present in the band structure of the electron-ion plasma with fascinating resemblance to band structure of solids. We also investigatedthe effect of charge screening effect on the energy band and revealed its many importantimpact on quantum dense plasmas. The significant effects of ion charge state and electronchemical potential on the plasmon excitations was revealed and band structure of differentpair plasmas were studied as well.
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