Dispersion and damping of potential surface waves in a degenerate plasma
Yu. O. Tyshetskiy, D. Williamson, R. Kompaneets, S. V. Vladimirov
aa r X i v : . [ phy s i c s . p l a s m - ph ] N ov Dispersion and damping of potential surface waves in adegenerate plasma
Yu. Tyshetskiy, ∗ D. Williamson, R. Kompaneets, and S.V. Vladimirov
School of Physics, The University of Sydney, NSW 2006, Australia (Dated: November 7, 2018; Received)
Abstract
Potential (electrostatic) surface waves in plasma half-space with degenerate electrons are studiedusing the quasi-classical mean-field kinetic model. The wave spectrum and the collisionless dampingrate are obtained numerically for a wide range of wavelengths. In the limit of long wavelengths,the wave frequency ω approaches the cold-plasma limit ω = ω p / √ ω p being the plasmafrequency, while at short wavelengths, the wave spectrum asymptotically approaches the spectrumof zero-sound mode propagating along the boundary. It is shown that the surface waves in thissystem remain weakly damped at all wavelengths (in contrast to strongly damped surface waves inMaxwellian electron plasmas), and the damping rate nonmonotonically depends on the wavelength,with the maximum (yet small) damping occuring for surface waves with wavelength of ≈ πλ F ,where λ F is the Thomas-Fermi length. PACS numbers: ∗ Electronic address: [email protected] . INTRODUCTION It has long been known [1–3] that bounded plasmas support a special type of collectiveelectrostatic and electromagnetic excitations – the surface plasma waves – whose field andenergy are concentrated near, and propagate along, the plasma boundaries. The surfacewaves (SW) in various (classical) plasmas have been extensively studied, and they havefound many applications (see Ref. [4] and references therein). Yet recently, due to a re-markable progress in nanotechnology, the interest in surface waves supported by variousnanostructures (especially metallic structures such as thin films and tiny metallic particles),and in their interaction with light, has been revived. It is believed that light-induced surfaceexcitations in such structures may offer a route to faster, smaller, and more efficient electron-ics, as well as new technology [5]. In particular, one could note such recent advents in thenew and promising area of quantum nanoplasmonics as the development of the concept ofspaser [6] followed by its further development into a lasing spaser [7], and the experimentaldemonstration of a spaser-based nanolaser [8, 9].In view of these developments, understanding the properties of surface waves in variousmetallic (and semiconductor) structures, bounded by vacuum or dielectric, is thus impor-tant. Such understanding requires using models describing the dynamic response of suchstructures to self-consistent electromagnetic fields, that also appropriately take into accountthe relevant quantum effects arising from quantum nature of free charge carriers in suchstructures and, in general, from their (quantum) interaction with each other and with theunderlying ion lattice. These quantum effects may significantly alter the properties of thesurface waves; see, e.g., Refs. [10, 11].Recently, the dispersion relation of surface waves in one of the basic structures modelinga nanoplasmonic device – a semi-bounded collisionless quantum plasma with degenerateelectrons – was obtained in Ref. [12] using the quantum fluid theory (QFT) approach [13].In the electrostatic limit, the authors of Ref. [12] obtained for the frequency of surface wavesin this structure: Ω = 1 √ r K k q H K k ! , (1)where Ω = ω/ω p , K k = k k λ F , H = ~ ω p / mv F , ω p = (4 πe n e /m e ) / is the electron plasmafrequency, v F = ~ √ π n e /m e is the electron Fermi velocity, λ F = v F / √ ω p is the Thomas-Fermi length, ω is the SW frequency, k k is the SW wave vector along the plasma boundary, e ,2 e and n e are electron charge, mass and number density, respectively, and ~ is the reducedPlanck constant. However, the validity of the dispersion relation (1) obtained in Ref. [12](as well as of the similar dispersion relation obtained in Ref. [11]) is limited by the validityof the QFT approach itself [13, 14], and is restricted only to long waves, K k ≪
1. Moreover,the QFT approach by its nature completely ignores the purely kinetic effect of collisionlessdamping of surface waves, which is known to be significant, e.g., for potential SW in warmelectron plasma at short wavelengths [15]. To overcome these limitations, a kinetic approachto the problem of SW in semi-bounded plasma with degenerate electrons is needed.In this paper, we study potential surface waves in semi-bounded collisionless quantumplasma with Fermi-degenerate electrons, using the initial value problem solution for the semi-classical Vlasov-Poisson system [3]. We obtain the dispersion and collisionless damping rateof these waves, valid for both long and short wavelengths, and report on a surprising resultthat these waves remain weakly damped for all values of K k (i.e., for all wavelengths), unlike,e.g., in plasma with Maxwellian electrons in which the SWs are weakly damped only for K k ≪ K k increases. We also report a nontrivialnonmonotonic dependence of the damping rate on K k , featuring a maximum damping at K k ≈ . ≈ πλ F ). II. METHODA. Model and Assumptions
We consider a uniform nonrelativistic quantum plasma consisting of mobile electronswith charge e , mass m e and number density at equilibrium n e = n , and immobile uniformbackground of singly charged ions with number density n that neutralizes the electroncharge at equilibrium. The plasma is assumed to be confined to the region x < x = 0. We will be interested in evolution of an initialperturbation to the equilibrium state of the system’s electronic component applied at time t = 0.In general, this system of many interacting quantum particles (electrons) can be de-scribed by the quantum analog of the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY)hierarchy [16, 17] of equations for the electron j -particle quantum distribution functions3 j ( r j , p j , t ) (also called the j -particle Wigner functions), where j = 1 , , ..., N , r j and p j are the 3 j -dimensional vectors denoting the sets of coordinates and canonical momenta ofsystem particles, and N is the total number of electrons in the system. In this hierar-chy, each of the equations for the j -particle quantum distribution function contains the( j + 1)-particle distribution function, making the whole set of N equations coupled, andthus prohibitively large to solve. In practice, however, this hierarchy can be truncatedby making a physically justified assumption about correlation of particles due to their in-teraction. In particular, for a system of weakly interacting particles, with a small plasmacoupling parameter Q = U int /ǫ kin ≪ U int is the characteristic potential energy of par-ticle interaction, and ǫ kin is the characteristic kinetic energy of plasma particles), the two-and higher-order particle correlations can be ignored, leading to the collisionless mean-fieldapproximation [18] involving only one equation for the one-particle quantum distributionfunction W ( r , p , t ), where r and p are now the 3-dimensional vectors of particle coordinateand momentum. In the quasi-classical approximation, with the effect of quantum recoilignored (see Sec. III C), this equation reduces to the Vlasov equation for the one-particleclassical distribution function f ( r , v , t ), where v = p /m e is electron velocity. For a systemof particles with electrostatic interaction, in the chosen geometry, the Vlasov equation for f ( r , v , t ) = f ( x, r k , v x , v k , t ) (where r k and v k are, respectively, the components of r and v parallel to the boundary) reads ∂f∂t + v x ∂f∂x + v k · ∂f∂ r k − em e (cid:18) ∂φ∂x ∂f∂v x + ∂φ∂ r k · ∂f∂ v k (cid:19) = 0 , (2)where the electrostatic potential φ ( x, r k , t ) is defined by the Poisson’s equation − ∇ φ = 4 πe (cid:20)Z f ( r , v , t ) d v − n (cid:21) . (3)In the absence of fields, the equilibrium distribution function of plasma electrons f ( v ) isdefined by the Pauli’s exclusion principle for fermions, and for low electron temperatures T e /ǫ F ≪ T e is the electron temperature, ǫ F = m e v F / π ~ n e ) / / m e is theelectron Fermi energy) it becomes f ( v ) = n πv F = m e (2 π ~ ) if v ≤ v F , v > v F , (4)corresponding to fully degenerate electron distribution.4he condition of specular reflection of plasma electrons off the boundary at x = 0 implies f ( x = 0 , r k , − v x , v k , t ) = f ( x = 0 , r k , v x , v k , t ) . (5) B. Initial value problem
We now introduce a small initial perturbation f p ( x, r k , v x , v k , t = 0) to the equilibriumelectron distribution function f ( v ), | f p ( x, r k , v x , v k , t = 0) | ≪ f ( v ), and study the resultingevolution of the system’s charge density ρ ( x, r k , t ) = e (cid:2)R f ( x, r k , v , t ) d v − n (cid:3) , and hence ofthe electrostatic potential φ ( x, r k , t ) defined by (3). Introducing the dimensionless variablesΩ = ω/ω p , K = k λ F , V = v /v F , X = x/λ F , R k = r k /λ F , λ F = v F / √ ω p , and followingGuernsey [3], the solution of the formulated initial value problem for ρ ( X, R k , T ) with theboundary condition (5) is ρ ( X, R k , T ) = en ˜ ρ ( X, R k , T ) , (6)where ˜ ρ ( X, R k , T ) = 1(2 π ) Z + ∞−∞ dK x e iK x X Z d K k e i K k · R k ˜ ρ k ( T ) , (7)˜ ρ K ( T ) = 12 π Z iσ + ∞ iσ −∞ ˜ ρ (Ω , K ) e − i Ω T d Ω , with σ > . (8)The integration in (8) is performed in complex Ω plane along the horizontal contour thatlies in the upper half-plane Im(Ω) = σ > ρ (Ω , K ).The function ˜ ρ (Ω , K ), defined as the Laplace transform of ˜ ρ K ( T ):˜ ρ (Ω , K ) = Z ∞ ˜ ρ K ( T ) e i Ω T dT , (9)is given by˜ ρ (Ω , K ) = iε (Ω , K ) Z d V G ( V , K )Ω − √ K · V + iK k πζ (Ω , K k ) (cid:20) − ε (Ω , K ) (cid:21) Z + ∞−∞ dK ′ x K ′ ε (Ω , K ′ ) Z d V G ( V , K ′ )Ω − √ K ′ · V , (10)where the Fourier transforms G ( V , K ) and G ( V , K ′ ) of the (dimensionless) initial pertur-bation are defined by G ( V , K ) = Z + ∞−∞ dX e − iK x X ˜ g ( X, V x , V k , K k ) , (11) G ( V , K ′ ) = Z + ∞−∞ dX e − iK ′ x X ˜ g ( X, V x , V k , K k ) , (12)5ith ˜ g ( X, V x , V k , K k ) = Z d R k e − i K k · R k ˜ f p ( X, R k , V x , V k , , (13)˜ f p ( X, R k , V x , V k ,
0) = v F n f p ( X, R k , V x , V k , , (14)where K k = | K k | , K = | K | , K = ( K x , K k ), K ′ = | K ′ | , and K ′ = ( K ′ x , K k ). The functions ε (Ω , K ) and ζ (Ω , K k ) in (10) are defined (for Im(Ω) >
0) as follows: ε (Ω , K ) = 1 − √ K Z K · ∂ ˜ f ( V ) /∂ V Ω − √ K · V d V , (15) ζ (Ω , K k ) = 12 + K k π Z + ∞−∞ dK x K ε (Ω , K ) , (16)with ˜ f ( V ) = v F n f ( V ) = v F n f ( v ) | v = v F V . For fully degenerate plasma with electron distribution (4), the function ε (Ω , K ) becomes [15,19]: ε (Ω , K ) = 1 + 1 K " − Ω2 √ K ln Ω + √ K Ω − √ K ! , (17)where ln( z ) is the principal branch of the complex natural logarithm function.Note that the solution (10) differs from the corresponding solution of the transformedVlasov-Poisson system for infinite (unbounded) uniform plasma only in the second terminvolving ζ (Ω , K k ); indeed, this term appears due to the boundary at x = 0.The definition (9) of the function ˜ ρ (Ω , K ) of complex Ω has a sense (i.e., the integral in(9) converges) only for Im(Ω) >
0. Yet the long-time evolution of ˜ ρ k ( T ) is obtained from(8) by displacing the contour of integration in complex Ω plane from the upper half-planeIm(Ω) > ≤ ρ (Ω , K )to be extended to the lower half-plane, Im(Ω) ≤
0, by analytic continuation of (10) fromIm(Ω) > ≤
0. Hence, the functions I (Ω , K ) ≡ Z d V G ( V , K )Ω − √ K · V , (18) ε (Ω , K ), and ζ (Ω , K k ) that make up the function ˜ ρ (Ω , K ), must also be analytically con-tinued into the lower half-plane of complex Ω, thus extending their definition to the wholecomplex Ω plane. With thus continued functions, the contributions to the inverse Laplacetransform (8) are of three sources [3]: 6. Contributions from the singularities of I (Ω , K ) in the lower half of complex Ω plane(defined solely by the initial perturbation G ( V , K )); with some simplifying assump-tions about the initial perturbation [3] these contributions are damped in a few plasmaperiods and can be ignored.2. Contribution of singularities of 1 /ε (Ω , K ) in the lower half of complex Ω plane, oftwo types: (i) residues at the poles of 1 /ε (Ω , K ), which give the volume plasma os-cillations [3], and (ii) integrals along branch cuts (if any) of 1 /ε (Ω , K ) in the lowerhalf-plane of complex Ω, which can lead to non-exponential attenuation of the volumeplasma oscillations [21, 22].3. Contribution of singularities of 1 /ζ (Ω , K k ) in the lower half of complex Ω plane, oftwo types: (i) residues at the poles of 1 /ζ (Ω , K k ), corresponding to the surface wavesolutions of the initial value problem in the considered system [3], and (ii) integralsalong branch cuts (if any) of 1 /ζ (Ω , K k ) in the lower half-plane of complex Ω, whichwill be discussed elsewhere.In this paper, we will only consider the surface wave solutions due to the contribution of theresidues at the poles of 1 /ζ (Ω , K k ). The dispersion and damping properties of these surfacewave solutions are defined by the dispersion relation for plasma surface waves [3] ζ (Ω , K k ) = 0 , (19)which in case of stable plasma (with velocity distribution of electrons having only one max-imum) only has non-growing solutions Ω = Ω( K k ) ∈ C with Im(Ω) ≤ C. Weakly damped surface waves
Out of all complex-valued solutions of (19), Ω( K k ) = Ω s ( K k ) + i Γ s ( K k ) (here Ω s ∈ R isthe dimensionless frequency, and Γ s ≤ weakly damped surface waves, with | Γ s ( K k ) / Ω s ( K k ) | ≪ | Γ s ( K k ) / Ω s ( K k ) | ≪ (cid:2) ζ (Ω s , K k ) (cid:3) = 0 , yielding Ω s = Ω s ( K k ) ∈ R , (20)Γ s = − Im (cid:2) ζ (Ω s , K k ) (cid:3) ∂ Re (cid:2) ζ (Ω s , K k ) (cid:3) /∂ Ω s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Ω s =Ω s ( K k ) , (21)which only involve ζ (Ω , K k ) as a function of real Ω. Thus, instead of performing analyticcontinuation of ζ (Ω , K k ) from Im(Ω) > ≤
0, it suffices to only analyticallycontinue ζ (Ω , K k ) from Im(Ω) > ∈ R . This analytic continuation of ζ (Ω , K k ), required for the numericalsolution of Eqs (20)–(21), is discussed in Appendix A. III. RESULTS AND DISCUSSION
The numerical solution of Eqs (20) and (21) for a given K k >
0, with ζ (Ω , K k ) definedfor Ω ∈ R as discussed in Appendix A, yields the dispersion Ω s ( K k ) and damping Γ s ( K k ) ofsurface waves; below we discuss them in greater detail. A. Dispersion
The dispersion curve of surface waves Ω s ( K k ) is shown in Fig. 1. The SW frequencyincreases monotonically with K k , starting from the well-known long-wavelength limit ofΩ s = 1 / √ K k → s ( K k ) ≈ √ (cid:0) . K k (cid:1) (22)for K k ≪
1. We note the discrepancy between (22) and the corresponding small- K k asymptote of Lazar et al.’s result (1) of Ref. [12]: Ω s ( K k ) ≈ (1 / √ p / K k ) ≈ (1 / √ . K k ). This discrepancy can not be attributed to the effect of quantumrecoil, which is ignored here but included in the QFT model of Ref. [12], as the quan-tum recoil only gives the higher-order contribution ∝ H K k to the dispersion (1) at small K k . In fact, the mentioned discrepancy is due to the error in the coefficient of the clas-sical pressure gradient term used in Ref. [12]: instead of the three-dimensional pressuregradient term (3 / v F ∇ ( n e /n ) [24, 25], the one-dimensional term v F ∇ ( n e /n ) was used.8 .2 0.4 0.6 0.8 1.0 1.2 1.4 K ° W s H K ° L FIG. 1: (Color online) The dispersion curve of surface waves Ω s ( K k ) (solid blue line), and theasymptotes: the long-wave limit 1 / √ With this error corrected, the QFT model of Ref. [12] yields the small- K k asymptote ofΩ s ( K k ) = (1 / √ p / K k ) that matches our asymptote (22).At short wavelengths, K k ≫
1, the SW dispersion can be approximated as (see Ap-pendix B) Ω s ≈ √ K k (cid:0) (cid:2) − − K k (cid:3)(cid:1) , (23)and quickly approaches the line Ω = √ K k that corresponds to the dispersion of volume zerosound mode in degenerate Fermi gas [19, 26] propagating along the boundary, with K = K k .Note that for K k ≫ s becomes practically indistinguishable from thecorresponding frequency Ω v of the volume plasma wave propagating along the boundary(with K = K k ) [19]: Ω v = √ K k (cid:0) (cid:2) − − K k (cid:3)(cid:1) . . Damping The K k -dependence of the surface wave damping rate (cid:12)(cid:12) Γ s ( K k ) (cid:12)(cid:12) is shown in Fig. 2. In thelong-wave limit K k ≪
1, the damping rate increases linearly with K k as (cid:12)(cid:12) Γ s ( K k ) (cid:12)(cid:12) ≈ . √ · − K k , (24)similar to the damping rate of SW in plasma with Maxwellian electrons [3, 15]. However,as K k increases, an important difference between SW damping in plasma with degenerateelectrons and in plasma with Maxwellian electrons becomes obvious. In Maxwellian plasma,the SW damping rate increases monotonically with K k , quickly exceeding the SW frequency,so that the SW become strongly damped at short wavelengths [3, 15]. In degenerate electronplasma, however, the SW damping rate has a non-monotonic dependence on K k , as seen inFig. 2: at small K k , the damping rate increases almost linearly with K k , reaching a distinctmaximum of | Γ s | ≈ . · − at K k ≈ .
4, and then decreases monotonically with K k for K k > .
4, approaching zero at large K k .Most importantly, as seen from Fig. 2, the SW damping rate in degenerate electronplasma remains small ( | Γ s / Ω s | ≪
1) at all values of K k , and hence the electrostatic surfacewaves in degenerate collisionless electron plasma are weakly damped at all wavelengths (yetthe surface waves with K k = k k λ F ≈ . K k when their phasevelocity greatly exceeds the electron thermal velocity [3, 15]. The reason is that, as seenfrom Eq. (16), the surface wave field is essentially a result of plasma response to a collectionof “virtual” plasma waves [i.e., having frequency Ω and wave vector K , but not havinga dispersion relation Ω = Ω( K ), unlike the “real” plasma waves with dispersion relationΩ = Ω( K ) following from ε (Ω , K ) = 0], with the same frequency Ω = Ω s but with differentwave vectors K with absolute values K = q K x + K k ranging from K = K k (for K x = 0) to K → ∞ (for K x → ∞ ). Each of these virtual waves with its own K interacts resonantly withplasma electrons whose velocities along K is close to the phase velocity of the virtual wave(and such electrons always exist in Maxwellian plasma), and thus is subject to a non-zero10 .02 0.05 0.10 0.20 0.50 1.00 K ° ´ - ´ - ´ - È G s H K ° LÈ FIG. 2: (Color online) The absolute value of surface wave damping rate (cid:12)(cid:12) Γ s ( K k ) (cid:12)(cid:12) (solid blue line),and the long-wave asymptote (24) (dashed line), as functions of K k . Landau damping. As a result, the surface wave, consisting of both weakly damped virtualwaves (with Ω < K , where for Maxwellian plasma K = kλ D , λ D = v T e /ω p is the electronDebye length) and strongly damped virtual waves (with Ω > K ), interacts resonantly with asignificant part of the plasma electron distribution, and hence is strongly damped, comparedto the “real” plasma waves with the same wavelength.In the degenerate plasma considered here, the difference in damping of surface and the“real” volume plasma waves is even more striking. Indeed, the “real” volume waves in suchplasma, defined by the dispersion relation ε (Ω , K ) = 0 with ε (Ω , K ) given by Eq. (17), arenot subject to Landau damping at all, as their phase velocity exceeds the maximum possiblevelocity of degenerate plasma electrons, Ω > √ K (here K = kλ F , λ F = v F / √ ω p is theelectron Fermi length), for all values of K [19]. The “virtual” plasma waves are, however,still subject to Landau damping by plasma electrons if their phase velocity is less than theelectron Fermi velocity, Ω s < √ K , i.e., if their K x is large enough, K x > q Ω s / − K k ;such virtual waves always give a contribution to the surface wave field, as K x in (16) attainsarbitrarily large values. This also results in the finite, albeit small (yet infinitely largecompared to the zero damping of the “real” volume plasma waves) Landau damping ofsurface waves in degenerate plasma. 11 . On applicability of the used approximations The collisionless mean-field approximation used here is justified for a system of weakly in-teracting particles, when two-particle correlations (and higher-order particle correlations aswell) can be ignored. Physically, this corresponds to a system in which the collective effectsdominate over the effects of particle collisions, which happens when the plasma couplingparameter Q = U int /ǫ kin is small (see Sec. II A). With U int ∼ e n / and ǫ kin ∼ ǫ F for degen-erate electrons, this implies that the collisionless mean-field approximation for degenerateelectron plasma is justified when Q ∼ e n / ǫ F ∼ (cid:18) ~ ω p ǫ F (cid:19) ≪ . (25)Moreover, when the condition (25) is satisfied, the effect of quantum recoil on the dispersionproperties of electrostatic plasma waves is also negligible, at least for K . Q ∼ e n / /ǫ F scales as n − / andthus decreases as the plasma density n increases; hence the condition (25) is satisifed forsufficiently dense plasma, with n ≫ ( e m ∗ / ~ ) , where m ∗ is the effective electron mass insuch plasma. However, as n increases, the electron Fermi velocity increases as n / and maybecome comparable to the speed of light c , in which case the relativistic effects may becomeimportant; to avoid this, we also require that n ≪ ( m ∗ c/ ~ ) . Thus the nonrelativisticmean-field quasi-classical approximation (2) is valid for quantum plasmas with degenerateelectrons with densities (cid:18) e m ∗ ~ (cid:19) ≪ n ≪ (cid:18) m ∗ c ~ (cid:19) . (26)The condition (26) may be satisfied for some semiconductors, as pointed out in Ref. [22].For electron gas in metals, however, the condition (25) (as well as the first part of thecondition (26)) is not satisfied (in fact in metals Q & good agreement with the corresponding theoretical formula that follows from the dis-persion relation of electron plasma waves (excited by the energetic electrons traversing themetallic film), derived using the mean-field kinetic model of free electrons in metal, despitethis model being formally unjustified in metals due to violation of (25) there. Moreover, theeffect of quantum recoil, although detectable, is small compared to the effect of Fermi-Diracstatistics in the whole range of measured scattering angles, especially at small scattering an-gles (corresponding to small values of plasma wave number). These facts can be perceivedas an experimental evidence supporting the claim that the mean-field quasi-classical kineticmodel can work well even in metals, despite there being Q ∼ K ), which becomes:Ω = 1 + 95 K − QK , (27)where the third term on the right is due to the exchange correlations, while the second termis due to the quantum statistics resulting in the Fermi-Dirac distribution (4) of electronvelocities in degenerate plasma. We see from (27) that the term due to exchange correlationsis small compared to the term due to quantum statistics even for Q ∼
1. This suggests thatin metals the exchange correlations of electrons only lead to minor modification of plasmawave spectra, and neglecting them does not lead to a serious error, while greatly simplifyingthe model. As for the quantum recoil – it modifies the spectrum (27) by adding the higher-13rder term (1 / QK . Even for Q ∼
1, the term due to quantum recoil remains smallcompared with the term due to quantum statistics for K .
1, and can be safely neglected.Finally, it has beem pointed out in Ref. [13] that in plasmas with low electron temper-atures, T e /ǫ F ≪ ν ee for metals at room temperature is of order ν ee ∼ s − [13], which is many orders ofmagnitude smaller than the typical plasma oscillation frequency ω p ∼ s − ; hence theelectron-electron collisions are not expected to play a significant role for processes occuringat the characteristic collective plasma time scale τ p ∼ ω − p (such as the surface waves stud-ied here, with ω s ∼ ω p ), and can thus be neglected. Indeed, the dimensionless surface wavedamping rate due to the electron-electron collisions, Γ ees = γ ees /ω p with γ ees ∼ ν ee , is of theorder Γ ees ∼ − , which is much smaller than the characteristic collisionless damping rateΓ s ∼ − obtained in this work (see Fig. 2), and thus can be safely neglected.Beside colliding with each other, the electrons can also collide with the ions of metal lat-tice, yet these collisions were also neglected in our model. To justify this approximation, letus estimate the electron-ion collision frequency ν ei in typical metals used in plasmonic appli-cations, and compare the characteristic surface wave damping rate Γ eis associated with thesecollisions with the collisionless damping rate Γ s ∼ − obtained above. The experimentallymeasured electric resistivity of metals such as gold and aluminium at room temperature isof order ρ ∼ − Ohm · m[30]. Using the definition of resistivity ρ = E/j , where E is theapplied electric field, and j is the current density in metal (which is assumed to be entirelydue to the free electrons of conductivity), and the Ohm’s law j = ( ε ω p /ν ei ) E for electrons,where ε = 8 . · − F m − is the dielectric permittivity of vacuum, we have (in SIunits): ν ei ∼ ε ω p ρ. (28)For metals, ω p ∼ s − , and (28) yields ν ei ∼ s − , which is small compared to thecharacteristic surface oscillation frequency, and thus is not expected to have a significanteffect on the dispersion of surface waves. On the other hand, the dimensionless surface wavedamping rate due to electron-ion collisions, Γ eis = γ eis /ω p with γ eis ∼ ν ei , is of order Γ eis ∼ ν ei /ω p ∼ − , which is comparable to the collisionless damping rate Γ s ∼ − obtainedhere (recall that the maximum collisionless damping rate at K k ≈ . s ≈ . · − ).Thus the electron-ion collisions, neglected in our model, may lead to additional collisional14amping of surface waves that is comparable to the collisionless damping obtained from thecollisionless kinetic model.The above discussion suggests that the quasi-classical mean-field kinetic model used hereis adequate for describing surface plasma waves even in moderately coupled plasmas with Q ∼ Q , a comparison of the model’s predictions with experiments invarious quantum plasmas, including the electron plasma in metals, is needed. IV. CONCLUSION
In this paper, electrostatic surface waves in semi-bounded plasma with degenerate elec-trons were studied using the nonrelativistic collisionless mean-field kinetic model. The dis-persion relation for the waves is obtained from the initial value problem, and its solutioncorresponding to weakly damped surface waves is presented, yielding dispersion and col-lisionless damping of the waves for an arbitrary wave number K k . It is shown that thesurface waves in the semi-bounded plasma with degenerate electrons are weakly damped atall wavelengths, and their damping rate exhibits nonmonotonic dependence on K k , linearlyincreasing with K k at K k ≪
1, then reaching maximum at K k ≈ .
4, then falling off rapidlyto zero as K k increases. This is in contrast with the strong damping of surface waves insemi-bounded plasma with Maxwellian electrons, and is the consequence of the effect ofquantum statistics (leading to Fermi-Dirac velocity distribution) for plasma electrons. Thiswork, using the more general kinetic model, extends the results of Ref. [12] obtained usingthe quantum fluid theory, in two ways: (i) the range of the dispersion relation Ω s ( K k ) isextended from K k ≪ K . Acknowledgments
This work was supported by the Australian Research Council. R.K. acknowledges thereceipt of a Professor Harry Messel Research Fellowship funded by the Science Foundationfor Physics within the University of Sydney.15 ppendix A: Analytic continuation of ζ (Ω , K k ) To perform the required analytic continuation of ζ (Ω , K k ) onto the real axis of the complexΩ plane, we start from the definition (16) of ζ (Ω , K k ), with ε (Ω , K ) given by (17), forIm(Ω) > ζ (Ω , K k ) is an analytic function of Ω), and then reduce Im(Ω) downto zero, taking the limit Im(Ω) →
0+ while ensuring that the analyticity of ζ (Ω , K k ) ispreserved in the process. For Im(Ω) > ζ (Ω , K k ) is defined in terms of the integral Z + ∞−∞ dK x K ε (Ω , K ) , (A1)where K k > K x ∈ R , K = q K x + K k , and ε (Ω , K ) is given by Eq. (17). The function[ K ε (Ω , K )] − under the integral in (A1) is an elementary function of K = q K x + K k ,which in turn is an elementary function of K x . Thus the function [ K ε (Ω , K )] − can beextended to complex K x plane by analytic continuation from the real axis Im( K x ) = 0 of thecomplex K x plane, which is achieved by taking the principal branches of the complex squareroot function K = q K x + K k and of the complex logarithm function ln[(Ω + √ K ) / (Ω −√ K )] in (17), considered as functions of complex K x . The resulting function [ K ε (Ω , K )] − of complex K x has the following singularities in the complex K x plane:1. Branch cut of the complex square root q K x + K k , taken along the negative real axis ofthe argument K x + K k . This branch cut maps into two branch cuts of [ K ε (Ω , K )] − in the complex K x plane, given by two parametric equations: K x = ± i q K k + τ , with K k > , τ ∈ [0 , + ∞ ) . (A2)2. Branch cut of the complex logarithm in (17), taken along the negative real axis ofthe argument (Ω + √ K ) / (Ω − √ K ). This branch cut maps into two branch cuts of[ K ε (Ω , K )] − in the complex K x plane, given by two parametric equations: K x = ± i s Ω (cid:18) τ + 1 τ − (cid:19) − K k , with Ω ∈ C , K k > , τ ∈ [0 , + ∞ ) . (A3)3. Two poles K x = ± iK k ( K k >
0) at the roots of K = 0, lying symmetrically aboveand below the real axis of the complex K x plane.4. Two poles ± K rx ∈ C at the roots of ε (Ω , K ) = 0. Note that for any K ∈ R , ε (Ω , K ) = 0does not have roots with Im(Ω) >
0, if the plasma equilibrium is stable [23], which is16 m H K x L Re H K x L ä K ÈÈ -ä K ÈÈ Re H W L < W I K ÈÈ M Im( W )>0 - - - - ä K ÈÈ -ä K ÈÈ Re H W L < W I K ÈÈ M Im( W ) ®
0+ Im H K x L Re H K x L - - - - FIG. 3: (Color online) Singularities (branch cuts and poles) of the function (cid:2) K ε (Ω , K ) (cid:3) − in thecomplex K x plane, and their modification from Im(Ω) > → < Re(Ω) < Ω ( K k ), with Ω ( K k ) defined from (A4). The branch cuts of thesquare root and the logarithm are shown with the black dashed lines and the blue dot-dashedlines, respectively. The poles ± K rx at the roots of ε (Ω , K ) = 0 are shown with the filled circles.The arrows show the direction of motion of the singularities in the process of Im(Ω) → K x in (16) is shown with the solid red line, with the arrow showing thedirection of the contour. the case considered here; therefore, for any Im(Ω) > ± K rx are located away from the real axis of the complex K x plane, and thus do not lie on the integrationcontour in (A1).The location of these singularities in the complex K x plane is shown in Figs 3–5. ForIm(Ω) > K ε (Ω , K )] − lieon, or intersect with, the real axis Im( K x ) = 0 of the complex K x plane, along whichthe integration in (A1) is carried out; the function (A1) and, consequently, the function ζ (Ω , K k ) are thus analytic functions of Ω for Im(Ω) >
0. However, as Im(Ω) →
0+ in theprocess of analytic continuation of ζ (Ω , K k ) to the real axis of complex Ω plane, some ofthe singularities of [ K ε (Ω , K )] − move about in the complex K x plane as shown in Figs 3–5, and may collapse onto, or cross with the real axis, thus requiring deformation of theintegration contour in (A1) to avoid crossing these singularities and to preserve analyticity17 K ÈÈ -ä K ÈÈ Re H W L > W I K ÈÈ M Im( W )>0 Im H K x L - - - - ä K ÈÈ -ä K ÈÈ Re H W L > W I K ÈÈ M Im( W ) ®
0+ Im H K x L Re H K x L - - - - FIG. 4: (Color online) Same as Fig. 3, but for Re(Ω) > Ω ( K k ) >
0. The branch cuts of thesquare root and the logarithm are shown with the black dashed lines and the blue dot-dashedlines, respectively. The poles ± K rx at the roots of ε (Ω , K ) = 0 are shown with the filled circles.The arrows show the direction of motion of the singularities in the process of Im(Ω) → K x in (16) is shown with the solid red line, with the arrow showing thedirection of the contour. of ζ (Ω , K k ). Below we consider three cases: (i) | Re(Ω) | < | Ω ( K k ) | , (ii) | Re(Ω) | > | Ω ( K k ) | ,and (iii) Re(Ω) = ± Ω ( K k ), where ± Ω ( K k ) ∈ R are found from the equation ε (Ω , K ) | K x =0 = ε (Ω , K k ) = 0 , for Ω , K k ∈ R . (A4)For | Re(Ω) | < | Ω ( K k ) | , the logarithm branch cuts (A2) collapse onto the real axis, whilethe poles ± K rx collapse onto the imaginary axis of the complex K x plane as Im(Ω) → K x is shown in Fig. 3.For | Re(Ω) | > | Ω ( K k ) | , both the logarithm branch cuts (A2) and the poles ± K rx collapseonto the real axis of the complex K x plane as Im(Ω) → K x is displaced to avoid the poles ± K rx , as shown in Fig. 4.Finally, for Re(Ω) = ± Ω ( K k ) (the boundary between the above two cases), the twopoles ± K rx both collapse towards zero as Im(Ω) decreases, as shown in Fig. 5, “squeezing”the K x integration contour between them. In the limit Im(Ω) → ± K rx mergeat K x = 0, and the integration contour passes through both of them, resulting in ζ (Ω , K k )18 K ÈÈ -ä K ÈÈ Re H W L = W I K ÈÈ M Im( W )>0 Im H K x L Re H K x L - - - - Ζ I W , K ÈÈ M is singular ä K ÈÈ -ä K ÈÈ Re H W L = W I K ÈÈ M Im( W ) ®
0+ Im H K x L Re H K x L - - - - FIG. 5: (Color online) Same as Fig. 3, but for Re(Ω) = Ω ( K k ) >
0. The branch cuts of thesquare root and the logarithm are shown with the black dashed lines and the blue dot-dashed lines,respectively. The poles ± K rx at the roots of ε (Ω , K ) = 0 are shown with the filled circles. Thearrows show the direction of motion of the singularities in the process of Im(Ω) → K x in (16) is shown with the solid red line, with the arrow showing the directionof the contour; for Ω = ± Ω ( K k ), this contour is “squeezed” between the poles ± K rx , leading to asingularity in ζ (Ω , K k ) for Im(Ω) → − K rx and K rx merge. being singular (non-analytic) at the point Ω = ± Ω ( K k ).The function ζ (Ω , K k ), defined for { Ω , K k } ∈ R as described above, was calculated nu-merically by performing integration over K x along the appropriate one of the integrationcontours shown in Figs 3–4. The characteristic Ω-dependence of thus defined ζ (Ω , K k ) isshown in Fig. 6, for a fixed value of K k > Appendix B: Dispersion of surface waves for K k ≫ Consider the function ζ (Ω , K k ) defined by Eq. (16), with ε (Ω , K ) given by (17), whosedefinition is extended to Ω ∈ R as described in Appendix A. For any given Ω ∈ R , onecan find K k large enough so that | Ω | < | Ω ( K k ) | , where Ω ( K k ) is defined by Eq. (A4) (thisfollows from the fact that Ω( K k ) is a monotonically growing function of K k ). Hence forsufficiently large K k , all poles of the function 1 /K ε (Ω , K ) under the integral in (16) lie on19 I K ÈÈ M W- - Ζ FIG. 6: (Color online) The characteristic Ω-dependence of ζ (Ω , K k ) analytically continued toIm(Ω) = 0, for a fixed K k = 0 .
1. Re[ ζ (Ω , K k )] and Im[ ζ (Ω , K k )] are shown with the solid blue anddashed purple lines, respectively. The frequency Ω ( K k ) defined by Eq. (A4), at which ζ (Ω , K k ) issingular, is shown with the dotted vertical line. the imaginary axis of the complex K x plane, symmetrically above and below K x = 0, asshown in Fig. 3. Therefore, the “projection” of 1 /K ε (Ω , K ) onto the integration contouralong the real axis of complex K x plane is peaked at K x = 0 (the point on the integrationcontour closest to the poles) and decreases away from K x = 0 (as the distance from thepoles increases). At K k ≫ K x = ± iK k of 1 /K are located closer to the realaxis of complex K x plane than the poles K x = ± K rx of 1 /ε (Ω , K ), hence the variation of theintegrand in ζ (Ω , K k ) with K x is mainly defined by the variation of 1 /K = 1 / ( K x + K k )along the real axis of complex K x plane. Neglecting the variation of 1 /ε (Ω , K ) with K x (andusing the peak value of 1 /ε (Ω , K ) at K x = 0), we can thus approximate the integral over K x as Z + ∞−∞ dK x K ε (Ω , K ) ≈ Z + ∞−∞ dK x ( K x + K k ) ε (Ω , K k ) = πε (Ω , K k ) K k , (B1)with ε (Ω , K k ) defined by (17) with K = K k . The corresponding approximation for ζ (Ω , K k )is then ζ (Ω , K k ) ≈ " K k − Ω2 √ K k ln Ω + √ K k Ω − √ K k ! − , K k ≫ . (B2)Substituting (B2) into the dispersion equation (20) for weakly damped surface waves, we20ave Ω2 √ K k ln (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Ω + √ K k Ω − √ K k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 2 + 1 K k . (B3)Assuming the solution of (B3) to be of the formΩ = √ K k [1 + δ Ω( K k )] , with lim K k →∞ | δ Ω( K k ) | = 0 , (B4)we obtain for δ Ω( K k ): δ Ω( K k ) ≈ (cid:2) − K k (cid:3) , which tends to zero at large K k , in agreement with the assumption (B4). Thus we arrive atthe approximation (23) for the frequency of surface waves at K k ≫ [1] R. H. Ritchie, Phys. Rev. , 874 (1957).[2] A. W. Trivelpiece and R. W. Gould, J. Appl. Phys. , 1784 (1959).[3] R. L. Guernsey, Phys. Fluids , 1852 (1969).[4] S. V. Vladimirov, M. Y. Yu, and V. N. Tsytovich, Phys. Rep. , 1 (1994).[5] M. L. Brongersma and V. M. Shalaev, Science , 440 (2010).[6] D. J. Bergman and M. I. Stockman, Phys. Rev. Lett. , 027402 (2003).[7] N. I. Zheludev, S. L. Prosvirnin, N. Parasimakis, and V. A. Fedotov, Nature Photon. , 351(2008).[8] M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout,E. Herz, T. Suteewong, and U. Wiesner, Nature , 1110 (2009).[9] F. J. Garcia-Vidal and E. Moreno, Nature , 604 (2009).[10] S. V. Vladimirov, Phys. Scr. , 625 (1994).[11] M. Marklund, G. Brodin, L. Stenflo, and C. S. Liu, Europhys. Lett. , 17006 (2008).[12] M. Lazar, P. K. Shukla, and A. Smolyakov, Phys. Plasmas , 124501 (2007).[13] G. Manfredi and F. Haas, Phys. Rev. B , 075316 (2001).[14] S. V. Vladimirov and Y. O. Tyshetskiy, Phys. Usp., in press (2011).[15] A. F. Alexandrov, L. S. Bogdankevich, and A. A. Rukhadze, Principles of Plasma Electrody-namics (Springer-Verlag, 1984).[16] Y. L. Klimontovich,
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