Wavefunction of Plasmon Excitations with Space Charge Effects
aa r X i v : . [ phy s i c s . p l a s m - ph ] J a n Wavefunction of Plasmon Excitations with Space Charge Effects
M. Akbari-Moghanjoughi Faculty of Sciences, Department of Physics,Azarbaijan Shahid Madani University, 51745-406 Tabriz, Iran (Dated: January 3, 2019)
Abstract
The one dimensional (1D) driven quantum coupled pseudoforce system governing the dynamicsof collective Langmuir electron oscillations is used in order to investigate the effects of variety ofspace charge distributions on plasmon excitations of a nearly free electron gas with arbitrary degreeof degeneracy and electron fluid temperature. A generalized closed form analytic expression for thegrand wavefunction of collective excitations in presence of an arbitrary space charge distributionis presented based on the stationary solutions of the driven coupled pseudoforce system which hasbeen derived from the Schr¨odinger-Poisson model. The wavefunction and electrostatic potentialprofiles for some especial cases such as the Heaviside charge distribution, Dirac charge sheet, impu-rity charge sheet in 1D plasmonic lattice and the Kroning-Penney Dirac charge distributions withparticular applications in plasmonics and condensed matter physics is investigated in this paper.It is remarkably found that two parallel Dirac charged sheets completely shield all interior plasmonexcitations with any given energy value from outside electrostatic fields and charge densities. It isalso found that the presence of even a weakly charged impurity layer within a perfect 1D plasmoniccrystal profoundly destroys the periodic electrostatic field of the crystal lattice, hence, the Blochcharacter of the wavefunction considered in band gap theory of solids. Current investigation ofelectron excitations in arbitrary degenerate electron gas in the presence of static charge distribu-tions may be used to develop analytic models for a variety of real physical situations. It also helpsin further developments of the rapidly growing fields of nanotechnology and plasmonics.
PACS numbers: 52.30.-q,71.10.Ca, 05.30.-d . INTRODUCTION Collective effects play a fundamental role in many body fermionic and bosonic systemslike solids [1, 2]. In fermionic systems like plasmas multi-ingredient aspects lead to variety ofinteresting linear and nonlinear properties due to colorful interactions among different species[3–6]. Therefore, a lot of analytical as well as numerical methods have been developed overthe years to investigate these interesting aspects of plasmas [7–11]. Almost all physicalproperties of condensed matter, biological and molecular systems, electron-hole dynamics insolids etc. are affected by complex collective electromagnetic interactions [12–16]. Dynamicproperties of such complex settings are usually dealt with kinetic and fluid models [17–20].In dense quantum systems with extra quantum potential interactions, on the other hand, thestatistical tools to study the field-density time evolution require a memory-time consuminglarge scale numerical simulations through well-known models like Hartree-Fock, Wigner-Poisson-Maxwell, magnetohydrodynamic and ground state density functional, etc. due tolarge degree of freedom and complex nature of particle interactions. Nowadays, however,very realistic simulations has been possible for kiloscale degree of freedom (DoF) systemswith the help of improved computing systems and algorithms which brings into view someinteresting statistical behavior of many body charged systems [21]. Recently, the generationof high-harmonic in metallic nanoparticles has been examined by numerical evaluation ofnonlinear electron dynamics using the quantum hydrodynamics model incorporating theself-consistent Hartree potential, in which collective excitations in 200 gold nanoparticleshas been examined [22]. In another work the extended quantum hydrodynamic model withHartree potential and exchange effect has been employed to numerically study the plasmonexcitations in a quantum trap [23].Plasmons are elementary collective excitations of free electron systems such as plasmas,metals, nano-metallic compounds, etc. which cause peculiar quantum features in physicalproperties like electric and thermal conductivities, optical properties and dielectric responsesetc. [24]. Electron-hole quantum transport and tunneling through potential barriers in inte-grated circuits are typical collective phenomenon understanding of which requires in depthknowledge of these elementary excitations [25–27]. Study of plasmon excitations has startedwith pioneering works of Bohm [28], Pines [29] and Levine [30]. There has been howevera renewed momentum in investigation of different aspects of these entities due to their2undamental contribution to the rapidly growing fields of nonotechnology, optoelectronics,plasmonics and low dimensional systems. For instance, there has been numerous recentinvestigations revealing many outstanding new quantum aspects of electron plasma oscilla-tions in collective charged environments using density functional, quantum hydrodynamicsand kinetic approaches [31–48]. In this paper we use the newly introduced concept of cou-pled pseudoforce [49, 50] based on the Schr¨odinger-Poisson to obtain the grand wavefunctionwhich includes the wavefunction together with the collective electrostatic potential of thefree electron gas with arbitrary degree of electron degeneracy and electron fluid temperaturein Sec. II. This approach is then extended to include the effect of static charge distributionas the driving pseudoforce in Sec. III. The later approach is then used to study the plasmonexcitations in Heaviside in Sec. IV and Dirac delta charge distributions and Kronig-Penneymodel in Sec. V. Conclusions are also presented in Sec. VI.
II. GRAND WAVEFUNCTION OF PLASMON EXCITATIONS
We would like to study plasmon excitations of finite temperature arbitrary degenerateelectron gas in presence of different electrostatic potential configurations solving for theplasmon wavefunction and the electrostatic potential function. Let us now consider a onedimensional isothermal electron gas with the chemical potential µ and fluid temperature T . Dynamics of electron fluid is described using the following effective Schr¨odinger-Poissonmodel which is related to the hydrodynamic formulation as described elsewhere [51] i ¯ h ∂ N ∂t = − ¯ h m ∂ N ∂x − ( eφ − µ ) N , (1a) ∂φ ∂x = 4 πen, (1b)in which N = p n ( x, t ) exp[ iS ( x, t )] is the time dependent electron fluid wavefunction with N N ∗ = n ( x, t ) and n ( x, t ) and u ( x, t ) = (1 /m ) ∂S ( x, t ) /∂x being the number density andfluid speed of the arbitrary degenerate electron gas, respectively. The isothermal electrongas is described via the following generalized equation of state n ( µ, T ) = 2 / m / π ¯ h Z + ∞ √ ǫdǫe β ( ǫ − µ ) + 1 , (2a) P ( µ, T ) = 2 / m / π ¯ h Z + ∞ ǫ / dǫe β ( ǫ − µ ) + 1 . (2b)3atisfying the thermodynamic identity ∂P ( µ, T ) /∂µ = n ( µ, T ) which has been used to re-place the electron statistical pressure with the chemical potential in the Schr¨odinger-Poissonmodel. Each electron in the system is described by a wavefucntion which is in the linearperturbation limit a combination of single particle wavefucntion and a collective electrostaticinteraction wavefunction as described elsewhere [52]. The later is attributed via the meanfield Hartree potential in the Schr¨odinger-Poisson model [23]. We consider the later modelas described in terms of the electron chemical potential instead of the scalar Fermi pressurewhich is assumed to remain constant to the linear order. Therefore assuming N = ψ ( x ) ψ ( t )the coupled pseudoforce model describing the stationary plasmon solution is as follows d Ψ( x ) dx + Φ( x ) = − E Ψ( x ) , (3a) d Φ( x ) dx − Ψ( x ) = 0 , (3b)in which assuming µ ≡ µ , we have used the normalized quantities in (1) as Ψ( x ) = ψ ( x ) / √ n where n is the equilibrium number density of the electron fluid, Φ( x ) = eφ ( x ), E = ( ǫ − µ ) / E p ( E p = ¯ hω p with ω p = p πe n /m being the quantized plasmon energy)and x = x/λ p with λ p = 2 π/k p and k p = p mE p / ¯ h being the characteristic plasmon wave-length. Also, the time dependent part of the wavefunction reads ψ ( t ) = exp( − iωt ) with ǫ = ¯ hω being the energy eigenvalues.The grand wavefunction G ( x, t ) characterizing the pseudoforce system 3(a) and 3(b) withthe boundary conditions Φ ′ (0) = Ψ ′ (0) = 0, Φ(0) = Φ and Ψ(0) = Ψ is G ( x, t ) = Φ( x )Ψ( x ) e − iωt = e − iωt α Ψ + k Φ − (Ψ + k Φ ) − (Φ + k Ψ ) Φ + k Ψ e ik x e ik x , (4)in which E = ¯ hω is the energy eigenvalue and k = √ E − α, k = √ E + α, α = √ E − . (5)Note that k = 1 /k holds for any energy eigenvalue E >
II. WAVEFUNCTION OF DRIVEN PLASMON
The following driven pseudoforce system [49] describes the plasmon excitations in thepresence of given static charge distribution due to ions or impurity charge in crystals d Ψ( x ) dx + Φ( x ) = − E Ψ( x ) , (6a) d Φ( x ) dx − Ψ( x ) = n i ( x ) . (6b)For the sinusoidal space charge distribution n i ( x ) = C exp( iKx ) (with C being the normal-ized charge density) the particular solution to time -independent system (3) isΦ p ( x ) = C α (cid:20) (1 + βk ) e ik x − (1 + βk ) e ik x − αβe iKx ( K − k ) ( K − k ) (cid:21) , (7a)Ψ p ( x ) = C α (cid:20) ( K − k ) e ik x − ( K − k ) e ik x − αe iKx ( K − k ) ( K − k ) (cid:21) . (7b)where β = K − E . Hence, the solution to system (6) is the combination of general (4) andparticular (7) solutions. It is however easy to give a generalized particular solution to thecase of arbitrary driven pseudoforce n i ( x ) = P C m exp ( iK m x ) with K m = mK as followsΦ p ( x ) = + ∞ X m = −∞ C m α (cid:20) (1 + βk ) e ik x − (1 + βk ) e ik x − αβe iK m x ( K m − k ) ( K m − k ) (cid:21) , (8a)Ψ p ( x ) = + ∞ X m = −∞ C m α (cid:20) ( K m − k ) e ik x − ( K m − k ) e ik x − αe iK m x ( K m − k ) ( K m − k ) (cid:21) . (8b)where C m = ( K/ π ) R n i ( x ) exp( − iK m x ) dx are the Fourier components of the charge dis-tribution. Note that (8) is the most general analytic particular solution to the plasmonwavefunction in the presence of an arbitrary one dimensional charge distribution n i ( x ). IV. HEAVISIDE CHARGE DISTRIBUTION
There are obviously physical instances in which an ambient free electron gas arrives ata uniform step charge density. However, it is valuable to derive expressions for probabilitycharge density and electrostatic field in the two regions for given energy of the plasmonexpiation. Assuming that the for there is an abrupt increase in the ion charge density in theregion x >
0, we have n i ( x ) = A Θ( x ) in which Θ( x ) is the Heaviside step function. Notethat the current case with charge density barrier is a new aspect of the pseudoforce model of5lasmon excitations which is fundamentally different from the case involving the potentialbarrier considered in many classical quantum problems of Schr¨odinger equation. It can beeasily confirmed that the following particular solutions satisfy the driven pseudoforce modelwith the Heaviside step density distribution, n i ( x ) = A Θ( x ).Φ p ( x ) = A Θ( x ) [4 αE − γ cos ( k x ) + γ cos ( k x )]2 α , (9a)Ψ p ( x ) = − A Θ( x ) [1 + γ − cos ( k x ) − γ cos ( k x )] , γ . (9b)where γ = 1 − Ek and γ = 1 − Ek .Figure 1 shows the numerical evaluation of the solution (9) for different values of thecharge density values. The initial values Φ = Ψ = 1 is chosen for all simulations in thispaper. However, the continuity of the solutions and the derivative of Ψ is found to becompletely independent of the initial values. Figure 1(a) shows the electrostatic potentialvariations of plasmon excitation in the region of space charge distribution for two differentvalues of negative (thin curves) and positive (thick curves) charge distributions and for agiven value of the plasmon energy. It is remarked that the average electrostatic potentiallevel of plasmon for positive/negative space charge distribution is positive/negative. More-over Fig. 1(b) shows the electrostatic potential variations for an elevated plasmon energyindicating that, while the spacial fluctuation frequency increases due to increase in plasmonenergy in both x < x > A = 0 .
5, these fluctuations fornegative value of space charge distribution A = − . x >
0. This is opposite to the case for electrostatic potential profiles in Fig.1(a). The same feature is also seen for Fig. 1(d) which depicts the wavefunction variationsfor an elevated plasmon energy. Another feature revealed by Fig. 1(d) is that increase inthe plasmon energy tends to further separate the average fluctuation levels correspondingto negative and positive space charge distribution values, while increasing the fluctuationfrequency for both negative and positive space charge distribution signs.6 . KRONIG-PENNEY PLASMONIC MODEL
The case of Dirac delta charge distribution, n i ( x ) = Aδ ( x ), may be realized as a coateduniform charge sheet sandwiched in a Josephson junction. The particular solution of drivencoupled pseudoforce system may be written as followsΦ p ( x ) = A [Θ( x ) − Θ(0)]2 α (cid:2) k sin ( k x ) − k sin ( k x ) (cid:3) , (10a)Ψ p ( x ) = A [Θ( x ) − Θ(0)]2 α [ k sin ( k x ) − k sin ( k x )] , (10b)in which Θ( x ) is the Heaviside theta function assuming Θ(0) = 1 /
2. The solution (10) maybe generalized to the case with one dimensional plasmonic crystal with n c ( x ) = B cos( Gx )( G = 2 π/a ) []. The initial values Φ and Ψ in this case are chosen so that the generalsolution satisfies the lattice periodicity condition. The complete, i.e., general+particularsolution for plasmonic crystal with a single Dirac delta charge at the origin is given belowΦ( x ) = − B ( G − k − k ) cos ( Gx )( G − k ) ( G − k ) + A [Θ( x ) − Θ(0)]2 α (cid:2) k sin ( k x ) − k sin ( k x ) (cid:3) , (11a)Ψ( x ) = − B cos ( Gx )( G − k ) ( G − k ) + A [Θ( x ) − Θ(0)]2 α [ k sin ( k x ) − k sin ( k x )] , (11b)Moreover, for a symmetric sheets located at x = ± b (the parallel charged sheet configu-ration) the solution for b > p ( x ) = A [Θ( b ) − Θ( b − x )]2 α (cid:8) k sin [ k ( x − b )] − k sin [ k ( x − b )] (cid:9) (12a) − A [Θ( b ) − Θ( b + x )]2 α (cid:8) k sin [ k ( x + b )] − k sin [ k ( x + b )] (cid:9) , (12b)Ψ p ( x ) = A [Θ( b ) − Θ( b − x )]2 α { k sin [ k ( x − b )] − k sin [ k ( x − b )] } (12c) − A [Θ( b ) − Θ( b + x )]2 α { k sin [ k ( x + b )] − k sin [ k ( x + b )] } . (12d)The particular solution for Dirac delta charge may be further generalized to the 1DKronig-Penney model which finds numerous applications in solid state physics. The partic-7lar solutions for the case of Dirac delta Kronig-Penney distribution ( b >
0) isΦ p ( x ) = + ∞ X m =1 A [Θ( mb ) − Θ( mb − x )]2 α (cid:8) k sin [ k ( x − mb )] − k sin [ k ( x − mb )] (cid:9) (13a) − + ∞ X m =0 A [Θ( mb ) − Θ( mb + x )]2 α (cid:8) k sin [ k ( x + mb )] − k sin [ k ( x + mb )] (cid:9) , (13b)Ψ p ( x ) = + ∞ X m =1 A [Θ( mb ) − Θ( mb − x )]2 α { k sin [ k ( x − mb )] − k sin [ k ( x − mb )] } (13c) − + ∞ X m =0 A [Θ( mb ) − Θ( mb + x )]2 α { k sin [ k ( x + mb )] − k sin [ k ( x + mb )] } . (13d)The initial values Φ and Ψ in the general solution should also be set in a way that the G ′ = 2 π/b is the reciprocal lattice vector of 1D Kronig-Penney plasmonic crystal, i.e.,Φ g ( x ) = − B ( G ′ − k − k ) cos ( G ′ x )( G ′ − k ) ( G ′ − k ) , (14a)Ψ g ( x ) = − B cos ( G ′ x )( G ′ − k ) ( G ′ − k ) . (14b)Figure 2 shows the plasmon excitations with given energy in the presence of a Diracdelta sheet at the origin. Figure 2(a) depicts the plasmon excitation in the presence of anextremely narrow negatively charged sheet at the origin. The thin curve shows the plasmonexcitation with the same energy in the absence of the charge sheet. It is clearly remarked thatthe charge sheet significantly modifies the plasmon electrostatic field variations in the wholespace around the sheet. However, the presence of Dirac charge sheet leads to a discontinuityof electric field at the place of sheet. Moreover, Fig. 2(b) depicts the plasmon excitation inspace around a positively charged sheet. It is clearly remarked that the plasmon electrostaticpotential around the positive sheet is relatively higher compared to the previous case withnegatively charged sheet. On the other hand, Figure 2(c) and 2(d) depict the plasmonwavefunction around the same sheets represented in Figs. 2(a) and 2(b), respectively. Forthe case of positive/negative charges the profile seems to be mirror reflected in places farfrom the origin where the sheet resides. However, the inspection of the values of Ψ ( x )around the charge sheet reveals that electrons are more localized around the positive sheetrather than that of negative charge.The quantum plasmon excitation profiles in driven pseudoforce model and in the presenceof double Dirac delta sheet is depicted in Fig. 3. In Fig. 3(a) and 3(b) It is seen that the8avefunction and electrostatic potential amplitudes inside the parallel charged sheets iscomparatively lower than those outside this region. It is remarked that the electrostaticpotential contributed from the charged sheets is only present outside of the sheet region. Inother words the plasmon excitations inside the parallel charged sheets is isolated from theoutside fields. This feature is found to be dominant even with the asymmetric charged sheetconfigurations (not shown here). Figure 3(c) shows the particular solution due to the chargedsheets only which reveals that this solution does not contribute to the plasmon excitationsinside the sheet region. Moreover, the plasmon excitations around the Dirac charged sheetsfor beating energy eigenvalue E ≃ a . Figure 4(a) shows the plasmon excitation withthe energy E = 5 in unit of the plasmon energy E p = ¯ hω p where ω p is the electron plasmaoscillation frequency, around a negative charge sheet at the origin. The thin curve representsthe period field of the lattice. It is remarked that driven plasmon excitations carries theperiodicity of the lattice modulated over a large amplitude due to the charge sheet. Thecurrent problem may represent a one dimensional impurity screening in a plasmonic crystal.Because of the one dimensional screening instead of realistic spherical problem the decayingdoes not take place. The plasmon excitation around a positive sheet with the same charge,lattice parameter and plasmon energy is seen in Fig. 4(b). The electrostatic potentialaround the sheet has become more positive in Fig. 4(b) due to the presence of positivelycharged sheet. Moreover, Fig. 4(c) shows the wavefunction profile quantum electron plasmaoscillation in a plasmonic crystal around the negatively charged Dirac sheet at the origin withsame parameters as in Fig. 4(a). It is remarked that, the electronic density is localized at theionic positions. However, it is shown that the wavefunction and electrostatic potential areout of phase everywhere in plasmonic lattice. In Fig. 4(d), on the other hand, the negativecharge sheet has been replaced with a positive sheet of same charge but with identical otherparameters. It is remarked that the free electronic charge in this case as compared to Fig.4(c) is more localized around the positive sheet.The time evolution of the plasmon excitation in lattice containing dirac charge sheetappears in Fig. 5. Figure 5(a) shows perfect periodic plasmon excitations in the absence of9harge distributions. As it is clear the Fig. 5(a) reflect a perfect harmonic pattern due to thelattice periodicity. The introduction of a weak negatively charged Dirac sheet at the origin,however, makes the clear periodic pattern somehow blurry but still the crystal periodicityis apparent from Fig. 5(b). With further increase in the charge of the Dirac sheet, on theother hand, the periodicity is further destroyed in Fig. 5(c). Figure 5(d) shows the timeevolution of plasmon excitation with elevated energy level as compared to Fig. 5(d). It isremarked that plasmon excitations with higher energy values feel less that those of lowerenergy the lattice periodicity in the presence of a Dirac charge sheet. VI. CONCLUSION
Using the driven coupled pseudoforce system derived from one dimensional Schr¨odinger-Poisson model the plasmon excitations of a free electron gas with arbitrary degree of degener-acy and electron fluid temperature is studied in the presence of various charge distributions.Closed forms of analytic functions is developed for plasmon excitations in the presence ofan arbitrary charge distribution. The developed model is shown to be completely flexible tomodel a variety of physical problems with wide range of applications such as the impurityscreening in 1D lattice, multi-charge distributed layers, etc. Current research help to study indetail the nature of quantum free electron excitations in periodic and quasi-periodic chargedenvironments like plasmonic lattice and in presence of arbitrary and random impurity dis-tributions. The model can also be further extended to include complex charge distributionscontaining arbitrary shaped charge regions fitting the desired physical situations.
VII. ACKNOWLEDGEMENT
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