arXiv:1606.06239v4 [cond-mat.mes-hall] 23 Oct 2017 n na xoeta rwho h oprpiigampli- pairing Cooper the of growth exponential tude resu an sea, Fermi in the ing destabilize could electrons the attractive between short-range tiny a even where perconductivity, ain feetosssandb ogrneCuobforce surfaces. Coulomb metal along long-range propagating by and sustained undu- density electrons – of (SPWs) waves lations instabi plasma surface This by metals. caused is of o ity surfaces amount the the on manifest of accumulated increase is charges oscillatory and while an frequency exponential discuss finite an we a by Here at transition. occurs phase that a instability as up shows and bility em e a eoeunstable are become they can provided sea sea, Fermi Fermi independent the and as free known space, mentum .Rthei 97ivsiae h nrylse felectro of foils losses metal energy through the passing investigated when 1957 ago in century a Ritchie half R. over begun SPWs of studies tematical fSW ntefloigdcd rso or decade following properties the fundamental in many SPWs of of accomplished soon was ing aino lcrnhl ar ln ol oeasu a pose would exci- alone with pairs associated electron-hole damping of Landau tation the that argued even prto fsae lsoi nlg flaser of analogy plasmonic a - t spaser deterring of in operation plasmonics sub-wavelength in factor adverse yeeg ossdet lcrnccollisions electronic hampered to prac- are due some applications losses of SPW-based energy be Many by also may interest. situation tical This hydrodynam the fails. which description for that ballistically, assuming move by electrons perspective the complementary a take we work in areas of re- plethora decades and two a past in oriented the made application been have largely progresses become markable have SPWs on udwti h yrdnmcDueapproach a hydrodynamic-Drude like the move SPWs within underpinning fluid electrons the that presumed n uueeprmns ept h neet alsi SP ballistic interest, the gui in Despite value great experiments. of future be ing then wit underpinned should SPWs electrons of ballistic theory such a and extent bein certain are to quality listic high of materials sought losses, actively such reduce to der Pscnttt bqiosett nmtloptics metal in entity ubiquitous a constitute SPWs tlwtmeaue lcrn eiei peei h mo- the in sphere a in reside electrons temperatures low At 3 uecnutvt ersnsatemdnmcinsta- thermodynamic a represents Superconductivity . ASnmes 51.10. numbers: PACS seri no poses therefore and overcompensated wisd always common shown with is contrast In unveiled. is conversion energy otevr rsneo ufc.Teapicto aeo thi of rate amplification e The the and surface. conservation energy of presence very the to fteisaiiyi lrfida rsn rmnvle novel from arising as clarified Fermi is the la instability the limit the for of opens collision-less which the oscillations, plasma in surface against that show we formalism 19–21 edrv eei omls o tdigteeeg conver energy the studying for formalism generic a derive We osbeisaiiyo h em e gis ufc plasm surface against sea Fermi the of instability Possible nteemtraseetoscudb bal- be could electrons materials these In . .INTRODUCTION I. 11,12 1 pntrigo hi neatos the interactions, their on turning Upon . oee,ms xsigsuishave studies existing most However, . 8,9 colo hsc,Uiest fEee,E44LEee,Uni Exeter, 4QL EX4 Exeter, of University Physics, of School opeesv understand- comprehensive A . + 2 aiireapei su- is example familiar A . ,5.5D,5.7A,7.0M,73.22.Lp 73.20.Mf, 52.27.Aj, 52.25.Dg, y, 10 ff ic hnstudies then Since . cso ossaedsusd npriua,teuiu role unique the particular, In discussed. are losses of ects 18 thsbeen has It . 13–16 18 ffi nthis In . 4–7 nor- In . ciently Sys- . a-a Deng Hai-Yao force Ws he ed lt- ns d- ic l- h g s f ff csrsligfo h rnlto yer raigdue breaking symetry translation the from resulting ects tra nrni efapicto hne.Teorigin The channel. self-amplification intrinsic an tter iprt opnns hc ecall we which components, disparate h lcrn hnasraei rsn.Mr pcfial,w density specifically, current More electrical the present. from that is find energy surface draw a to when SPWs electrons allow the motions These subse- motion surface. is electronic the instability ballistic the between interplay that show the We to supported SIM. quent a SPWs of ballistic surface o the in picture on involved detailed conversion a provide energy we the work, this In understanding. an instability scenario. furnishes the plification also of explanation but transparent approach physically motion a of com- equation only not the picture conversion plements electro energy the an Such from waves. transferred the actually to is mechani energy physical the the which into by directly penetrate to us allow not vr h qaino oinapoc sdi Ref. in used approach How- motion surface. of the equation to the due ever, interactio consequen breaking Coulomb symmetry a long-range translation obviously the and is between instability interplay This the of instab latter. an the signifying kinet of thereby sea, the ity Fermi of an the decrease in conserved, a stored be implies energy must energy energy potential total of the increase that Now ener potential system. electrostatic the the of growing-up a indicates ssoni i.1() h ea a ecie ytejellium the by described was metal The model (a). 1 Fig. in shown as ea lsadtesm cnrooccurs extended scenario been same the also and has in films theory amplify metal The spontaneously can limit. and collision-less SPWs den- unstable the that charge are discovered system the we this theory, for in Boltzmann’s motion on of based equation sity an constructing By ytercreain ihteeeti field electric the with correlations exampl for their ways, by many in discriminated are They spectively. ytecag density charge the by h afspace half occupyi the (SIM) metal semi-infinite a namely, system, typical aes a eevdltl attention. little received far so have e ihu ufcs eadeso h au ftethermal the of value the rate collision of electronic regardless surfaces, without tem J yrdnmcmdlbtLna apn.However, the damping. in Landau captured but partially model be hydrodynamic would which component, bulk the a osml eainwith relation simple no has b ( madi iewt bevtos adudamping Landau observations, with line in and om h anproeo h rsn oki ofilti a of gap this fill to is work present the of purpose main The Recently hne saayial vlae ntebssof basis the on evaluated analytically is channel s x u susi u-aeeghplasmonics. sub-wavelength in issues ous , ∗ e fmtl hudeprec ninstability an experience should metals of sea t eae to relates ) 24 inpoessi one eas sn this Using metals. bounded in processes sion n ne-adtastosaeacrigyneglected. accordingly are transitions inter-band and 22 esuidblitcSW na da e proto- yet ideal an in SPWs ballistic studied we z ≥ E ihagoercsraelctdat located surface geometric a with 0 ( x e Kingdom ted , t ρ ntesm aso si uksys- bulk a in as fassion same the in ) ( τ x − , oscillations a lydb h ufc in surface the by played 1 t o hsreason, this For . ftesse.I un u that out turns It system. the of ) E ( x , t n sttlyasn from absent totally is and ) J J ( x b , ( 23 x t a eslti two in split be can ) h amplification The . , t and ) E J ( b x ( , x t , generated ) J t scalled is ) s 22,23 ( x J , z t am- d s yof gy and s ,re- ), ( does = x sm , ng ns il- ce to 0, ic t n e e f ) 2

(a) (b) 40

Metal 30

z

20

s

k

10

0 -6 0 6

Vacuum Ez /ρs Ex /ρs

FIG. 1. Surface plasma waves (SPWs) supported on a semi-infinite metal (SIM) and the associated electric field. The SIM surface is located on the plane z = 0. In (a), the arrows indicate the electric field generated by the charges which are indicated by the colors, of the SPWs. In 2 2 (b), a plot of the electric field is displayed according to Eq. (4) with ρq 1/(ωb ωs ), where ωb is the bulk plasma wave dispersion relation 2 2 ∝ − and thus a function of k + q and ωs is the SPW frequency. The exact form of ρq is not necessary for the calculations performed in the present work. The apparentp oscillation appearing in Ez(z) is due to numerical inaccuracy. ks = ωs/vF , where vF denotes the Fermi velocity of the metal.

the hydrodynamic description. What is peculiar with Js(x, t) transfer of energy would happen between the electrons and is that, it would wholly disappear without the surface and thus waves. This is so only for the presence of a surface. On the represents genuine surface effects. We call it the surface com- contrary, the surface component always imparts an amount of ponent. We calculate the rate of growth of the electrostatic po- energy from the electrons to the waves and thus makes an in- tential energy and equate it with the work done per unit time trinsic gain for SPWs. More interestingly, this gain always on the electrical currents by the electric field E(x, t) to obtain overcompensates for the Landau damping and only competes the self-amplification rate γ0. This calculation is in spirit sim- with the loss due to thermal electronic collisions. The intrin- ilar to Dawson’s evaluation of the rate of Landau damping25 sic amplification rate is calculated in Sec. VII. In Sec. VIII, but is more subtle due to the surface. We have then developed we discuss the result and summarize the paper. Finally, an a generic formalism for the energy conversion processes in- appendix is provided to illustrate some conceptual point. volving surfaces. We find that, in spite of Landau damping, Js(x, t) imparts a net amount of kinetic energy of the electrons to the waves and is responsible for the instability. The present II. ENERGY CONVERSION WITH A SURFACE formalism can also serve a general framework for discussing the losses due to inter-band transitions and radiation. The system to be studied is a SIM described in our previous work22, see Fig. 1 (a). In accord with the jellium model, we In the next section, we introduce a rigorous framework for treat it as a free electron gas embedded in a static background studying the energy conversion in the presence of a surface. of uniformly distributed positive charges and confined to the A generic equation of energy balance is established and em- half space z 0. In equilibrium it is neutral everywhere. Per- ployed to prove the instability of the Fermi sea of a SIM in turbing the system≥ by for example a beam of light leads to a later sections. The critical role of the surface in energy con- variation in the concentration of electrons and the appearance version, which has so far not been recognized, is unveiled and of a charge density ρ(x, t). With no regard to the underlying highlighted. In Sec. III, we prescribe the electronic distri- dynamics of the charges, be it classical or quantum mechani- bution function, whose structure is analyzed in Sec. IV. We cal, the equation of continuity must hold, namely, split this function into a bulk component and a surface com- 1 ponent, the definitions of which are quantitatively established. (∂t + τ− )ρ(x, t) + ∂x j(x, t) = 0, In Secs. V and VI, we evaluate the work done by the electric · field on the electrons via the bulk and the surface components, where j(x, t) stands for the electrical current density solely due respectively. It is shown that, the only effect of the bulk com- to the presence of an electric field E(x, t), τ denotes the re- ponent is to bring about Landau damping; otherwise, no net laxation time, which by definition approaches infinity in the 3 collision-less limit. Throughout we write x = (x, y, z) and re- The electrostatic potential energy of the system is given by serve r = (x, y) for planar coordinates. In the continuity equa- tion, we have included a damping term ρ(x, t)/τ to account 1 3 Ep(t) = d x ρ(x, t)φ(x, t), for the microscopic thermal electrical currents− that arise from 2 Z electronic collisions that tend to equilibrate the system. These where φ(x, t) is the electrostatic potential satisfying currents correspond to the collision integral in Boltzmann the- ory and have nothingto do with the macroscopicfields appear- ∂2φ(x, t) + 4πρ(x, t) = 0. ing in the equation26. x

The effects of a surface are two-fold. Firstly, the surface One may be tempted to think that the rate of change of Ep(t) scatters and redistributes the electrons, an aspect to be dis- can be directly calculated as the negative of the work done per coursed in the next section within Boltzmann-Fuchs formal- unit time by the electric field E(x, t) = ∂xφ(x, t) on the elec- − ism. Secondly, the surface prevents any electrons from es- trons, which is, however, not true. This is because Ep(t) does caping the metal and j(x, t) must identically vanish for z < 0. not count all the potential energy in the system. Specifically, Thus, we write j(x, t) = Θ(z)J(x, t), where Θ(z) denotes the it does not include the surface potential energy, which may be Heaviside step function. With this prescription the equation written of continuity becomes 3 Es(t) = d x ρ(x, t)φs(x), 1 (∂t + τ− )ρ(x, t) + ∂x J(x, t) = δ(z)Jz(x , t), (1) Z · − 0 where φs(x) denotes the surface potential. For an ideal sur- x = r, δ z Here 0 ( 0) denotes a point on the surface, ( ) is the face, φs(x) should vanish in the metal but rise to infinity ev- Dirac function peaked on the surface and Jz(x, t) denotes the erywhere on the surface so that no electrons can escape the z-component of J(x, t). Equation (1) can also be derived other metal. In the electrostatic and collision-less limit, energy con- ways (see Appendix ) and serve as the equation of motion servation dictates that for ρ(x, t) if we express J(x, t) as a functional of ρ(x, t) by means of Maxwell’s equations. As discussed in Ref.22,23, bulk 3 E˙ p(t) = E˙s(t) E˙k(t) = d xJ(x, t) Es(x) Pb(t), plasma waves, for which Jz(x0, t) = 0, are governed only by − − Z · − the left hand side of this equation, while SPWs are described where the over-dot takes the time derivative, and Es(x) = by solutions with non-vanishing Jz(x0, t). If Jz(x0, t) identi- ∂xφs(x) as well as cally vanishes, the surface will be completely severed from − the rest of the metal. In this case, by whatever external stim- 3 uli, e.g. a grazing charged particle, no charges can build up on Pb(t) = d xJ(x, t) E(x, t). Z · the surface and SPWs can not be excited.

We prescribe all field quantities in the form of a plane wave This relation explains why E˙ p is not given by Pb(t). The − propagating along positive x direction with a complex fre- details of φs(x) and Es(x), however, can hardly be known and i(kx ωt) quency ω = ωs + iγ. We write ρ(x, t) = ρ(z)e − ′, could vary greatly from one sample to another. Notwithstanding, since the surface effects have been totally J(x, t) = J(z)ei(kx ωt) ′ and E(x, t) = E(z)ei(kxh ωt) ′, wherei − − incorporated in the continuity equation, we can deduce a com- k 0 denotesh the wavei number and a primeh takes thei real part plete equation of energy balance from it. To this end, we mul- of≥ a quantity. As ρ(z) exists only in half of the space, we also tiply Eq. (1) by φ(x, t) and integrate it over x. As both ρ(x, t) introduce a cosine Fourier transform like this, i(kx ωt) and φ(x, t) evolve by the factor e − , we have

∞ = 3 ρq dz cos(qz)ρ(z). d x φ(x, t)∂tρ(x, t) = E˙ p(t). Z0 Z

23 For SPWs, ρq may be taken real-valued and it only weakly Further, by integration by parts, depends on q for not so large q. We then put ρq ρs, where ≈ 3 ρs may be called the surface charge density. A cut-off qc has d x φ(x, t)∂x J(x, t) = Pb(t). to be imposed on q to reflect on the fact that ρ(z) can not vary Z · 1/3 significantly over the mean inter-particle spacing n− ; oth- Similarly, from the right hand side of Eq. (1) it arises erwise, the jellium model would break down. Thus,∼ we take 1/3 27 ρq>qc 0 and qc n . Here n denotes the mean con- 3 ≈ ∼ Ps(t) = d x φ(x, t)δ(z)Jz(x0, t) centration of electrons. With this prescription, ρ(z) spreads Z over a layer of thickness of the order of 1/qc within the sur- face, as required in the complete theory22 and seen in Fig 1. which signifies genuine surface effects. As far as we are con- In terms of the characteristic plasma frequency of the metal, cerned, this term has not been noticed in existing work. It can 2 be rewritten ωp = 4πne /m, with e being the charge and m the mass of an electron,p we have ωs ωp/ √2 as an approximation. We 2 ≈ Ps(t) = d r Jz(x0, t)φ(x0, t). can calculate γ by the principle of energy balance. Z 4

Combining the above expressions, we arrive at Generally the charge density ρ(z) spreads over a layer a few 1 multiples of vF/ωp = k−p thick within the surface; see the 2 example displayed in Fig. 1 (a). One can show that Ex(z) + ∂t Ep(t) = Pb(t) Ps(t), (2) kz ≈ τ ! − − Ez(z) E(z) = 2πρse− outside the layer, while in the layer ≈ Ex(z) and Ez(z) are distinctly different and take opposite signs. which is the energy balance equation of the system. Nonetheless, this layer makes a negligible contribution, of the Let us write the areal density of Ep(t) and Pb,s(t) as p(t) order of κ = k/k , to the electrostatic potential energy E . E p p and b,s(t), respectively. We can perform the integration over This point becomes clear if we write E (t) = 1 d3x E2(x, t). P p 8π r to get As an approximation, we may neglect the contributionR of this layer and obtain e2γt e2γt b,s(t) = b,s, p(t) = p, (3) 2 P 2 P E 2 E πρs p , (11) E ≈ k where b,s are given by P Upon substituting this expression in Eq. (8), we end up with an equation for γ , since are functions of γ . In order to = dz J (z) E (z) + J (z) E (z) , (4) 0 b,s 0 b ′ ′ ′′ ′′ find out γ , what remainsP to be done is to work out J(x, t) and P Z h · · i 0 use it to calculate b,s. s = Jz′(0)φ′(0) + Jz′′(0)φ′′(0). (5) P As an illustrationP of Eq. (8), let us apply it to the Drude 2 Here J′(z) denotes the real part of J(z) and J′′(z) the imaginary model, by which ρ(z) = ρsδ(z) and J(z) = (i/ω¯ )(ωp/4π)E(z), part, similarly for E(z) and other quantities. Analogously, we whereω ¯ = ωs + iγ0 and E(z) ( i, 1)E(z) outside the surface ≈ − 2 obtain layer. Assuming γ0/ωs 1 and then i/ω¯ γ0/ωs + i/ωs, 2 2 ≪ ≈ ωp/ωs 1 we find J(z) = 4π ωs iγ0, iωs + γ0 E(z). With this we p = dz ρ′(z)φ′(z) + ρ′′(z)φ′′(z) . (6) ω2 − ω2 γ0 p 2 p
E 2 Z find b = 2 ∞ dz E (z) = γ0 2 p, where p = ρsφ(0)/2. h i 2π ωs 0 ωs P 2 E E R ωp Taking ρ(z) to be real, the terms involving the imaginary parts Similarly, we find s = γ0 2 p. Thus, b + s = 0 and P − ωs E P P are then all gone. We thus obtain γ0 = 0 for the Drude model, agreeing with the equation of motion approach for the same model. Even for this simple b = dz J′(z) E′(z) + J′′(z) E′′(z) , model, the conventionalpicture of SPWs is incorrect. Accord- P Z h · · i ing to this picture, one would wrongly assume that the SPW 1 damping is due to energy transfer between the electrons and s = Jz′(0)φ(0), p = dz ρ(z)φ(z). (7) P E 2 Z the waves, by way of b,b. The present calculation, however, shows that there is noP net transfer of energy and the damping Equation (2) can now be transformed in the following form is solely caused by the presence of thermal currents that drives 1 the system toward thermodynamic equilibrium. 2γ0 = ( b + s)/ p, γ0 = + γ . (8) − P P E τ ! This is a key equation of the present paper. We shall show that III. THE ELECTRONIC DISTRIBUTION FUNCTION γ0 is always non-negative for surface plasma waves. By the laws of electrostatics, we find the potential given by We ignore inter-band transitions and use Boltzmann’s the- ory to study the electrical responses of the system. Surface 2π k z z′ scatters electrons. In principle, such scattering can be han- φ(z) = dz′e− | − |ρ(z′), k Z dled with a microscopic surface potential φs(x); see Appendix. However, φs(x) variesfromone sample to anotherandis rarely from which it follows that known in practice. Alternatively, those effects may be dealt with using phenomenological boundary conditions.28–32 This 2π ∞ kz φ(0) = ξ, ξ = dz e− ρ(z). is possible because φs(x) acts only on the surface and in the k Z0 bulk the electronic distribution function f (x, v, t) obtained as solutions to Boltzmann’s equation can be written down with- As for the electric field, we write it as E(z) = iEx(z), Ez(z) . In terms of ρ , we have − out explicitly referring to the surface. A few parameters shall q
occur in the solutions and their values reflect on surface scat- kz tering. In the present paper, we will follow this approach to Ex(z) ∞ 4kρq 2 cos(qz) e− = dq − kz . (9) study the electrical responses of a SIM. Ez(z)! Z k2 + q2 2(q/k) sin(qz) e− ! 0 − As usual we divide the distribution function in two terms,

Thus, E′(z) = 0, Ez(z) and E′′(z) = Ex(z), 0 . With this we f (x, v, t) = f ε(v) + g(x, v, t), − 0 can rewrite

m 2
where ε(v) = 2 v is the kinetic energy of an electron while f (ε) is the Fermi-Dirac function giving the equilibrium dis- = 0 b dz Jz′(z)Ez(z) Jx′′(z)Ex(z) . (10) tribution and g(x, v, t) denotes the deviation. Let us write P Z h − i 5

i(kx ωt) g(x, v, t) = Re g(v, z)e − . In the regime of linear re- IV. DECOMPOSITION INTO BULK AND SURFACE sponse, Boltzmann’sh equationi reads COMPONENTS AND THE POSITIVENESS OF γ0

∂g(v, z) 1 v E(z) The distribution function provided in Eqs. (13) - (17) pos- + λ− g(v, z) + e f0′(ε) · = 0, (12) ∂z vz sesses a notable structure, which we reveal in this section. To this end, we substitute the expression of E(z) given by Eq. (9) where λ = ivz/ω˜ withω ˜ = ω¯ kvx andω ¯ = ω + i/τ, and − into (13) - (17) and perform the integration over z′. We find f ′(ε) = ∂ε f0(ε). The general solution is given by 0 that g(v, z) can be split into two parts, one denoted by gb(v, z) z while the other by gs(v, z). They are given by z e f0′v z′ g(v, z) = e− λ C(v) dz′ e λ E(z′) , (13)  v  ∞ 4ρq − z · Z0 gb(v, z) = e f ′ dq   − 0 Z k2 + q2 ×   0 where C(v) = g(v, 0) is the non-equilibrium deviation on the kz 2F+ cos(qz) + 2iF sin(qz) F0e− , (19) surface to be determined by boundary conditions. We require −  −  g(v, z) = 0 distant from the surface, i.e. z . For electrons where we have introduced the following functions, →∞ moving away from the surface, vz > 0, this condition is auto- matically fulfilled. For electrons moving toward the surface, 1 K v K v F (K, ω,¯ v) = · · − , K = (k, 0, q). vz < 0, it leads to ± 2 "ω¯ K v ± ω¯ K v # − · − · − (20) e f ′v / 0 ∞ z′/λ Note that F+/ is an even odd function of vz. In addition, C(v) = dz′ e E(z′), vz < 0, (14) − vz · Z0 k∗ v F (k, ω,¯ v) = · , k∗ = (k, 0, ik). (21) yielding 0 ω¯ k v − ∗ · Moreover, we have e f ′v z z 0 ∞ ′− g(v, z) = dz′ e λ E(z′), vz < 0. (15) ω¯ z ρ vz · Z i ∞ 4 q z g v, z =Θ v e f e vz dq s( ) ( z)( 0′) 2 2 (22) − Z0 k + q × To determine C(v) for vz > 0, the boundary condition at z = 0 F0(k, ω,¯ v) pF0(k, ω,¯ v ) + 2(p 1)F+(K, ω,¯ v) . has to be used, which, whoever, dependson surface properties. − − − We adopt a simple picture first conceived by Fuchs28, accord- At this stage, it is clear that γ 0; otherwise, the factor 0 ≥ ing to which a fraction p (Fuchs parameter varying between exp(iω˜ z/vz) exp( γ0z/vz) contained in gs(v, z) would di- zero and unity) of the electrons impinging on the surface are verge far away∝ from− the surface for any electrons departing specularly reflected back, i.e. the surface. This is nothing but a consequence of the causality principle. We shall confirm this by direct calculation of the g(v, z = 0) = pg(v , z = 0), v = (vx, vy, vz), vz 0. energy conversion. − − − ≥ (16) Hereafter we call gb(v, z) the bulk component and gs(v, z) Note that this condition is identical with the condition used in the surface component, for their disparate dependences on the Ref.22,23 at p = 0 but differs otherwise. It follows that presence of surface. For a bulk metal without the surface, i.e. if we send the surface to or in other words replace z by e f ′v ∞ z′ −∞ 0 λ z + z0 with z0 , gs(v, z) will disappear identically whereas C(v) = p − dz′ e− E(z′), vz 0. (17) →∞ − vz · Z0 ≥ gb(v, z) does not, meaning that gs(v, z) signifies genuine sur- face effects while gb(v, z) gives the electrical responses of a Equations (13) - (17) fully specify the distribution function for bulk system. Actually, gb(v, z) has exactly the same form as the electrons. The electrical current density is calculated as the distribution function for a bulk metal in the presence of an electric field given in Eq. (9). One might think that gs(v, z) J(z) = (m/2π~)3 d3v evg(v, z). (18) is insignificant. However, to the contrary it is indispensable Z in ensuring the boundary conditions (16) and hence consti- It should be pointed out that the charge density is not given by tutes an integral part of the complete electrical responses of a bounded system. We note that only electrons departing the surface contribute 3 i(kx ωt) 3 ρ˜(x, t) = (m/2π~) Re e − d v eg(v, z) , to g (v, z). Those electrons either thermally emerge from " Z # s the surface or have been bounced back (with probability p). iϕ(z) which differs from the actual density ρ(x, t) by a part localized We also note that gs(v, z) possess a phase factor e , where on the surface. Actually, J(x, t)andρ ˜(x, t) obey the equation ϕ(z) = ωsz/vz. Physically, this phase is acquiredwhen an elec- tron leaves the surface and travels to the depth z without suf- (∂t + 1/τ)˜ρ(x, t) + ∂x J(x, t) = 0, fering a collision. It is exactly these ballistic motions that are · totally beyond the scope of the hydrodynamic-Drude model. of which no SPWs are admitted, rather than the equation of Accordingly the current density J(z) also splits into a bulk continuity [c.f. Eq. (1)]. and surface part, denoted by Jb(z) and Js(z) respectively. They 6 are defined via Eq. (18) with g(v, z) replaced by gb/s(v, z). If As we have discussed in Sec. II, the Drude current makes no we discard Js(z), expand F (K, ω,¯ v) in a series of K v/ω¯ net contributions. Moreover, as JL,z(0) 0 from Eq. (29), we ± · ≡ and similarly F0(k, ω,¯ v) in k∗ v/ω¯ , and retain only the first conclude that s,b = 0. We then obtain terms in the series, we will then· recover the current density P expected of the hydrodynamic-Drude model. As Js(z) is es- ,b = L,b := dz JL′ ,z(z)Ez(z) JL′′,x(z)Ex(z) . (30) sential in gratifying the boundary condition (16), the hydro- P P Z h − i dynamic model is inadequate. In the next two sections, we kz On using Ex(z) Ez(z) 2πρse− outside the layer of sur- calculate their contributions to b/s and show that, while Jb(z) ≈ ≈ P face charges, this expression can be broughtinto the following produces what is expected of the hydrodynamic-Drude model form, apart from Landau damping, Js(z) warrants a positive γ0 and leads to an incipient instability of the system. 3 K v K v K v ′′ L,b = 2πρs ˜ q v · · · . (31) P − Z D D k2 + q2 ω¯ ω¯ K v! − ·

V. ELECTRICAL WORK VIA Jb Here we have introduced another shorthand 3 4ρ ˜ 3 m ∞ q 3 2 q v... = dq d v e f0′ ... Our purpose here is to calculate the contribution of Jb(z) Z D D 2π~ Z k2 + q2 Z −   −∞   to b/s and show that this contribution would vanish if Lan- P In this shorthand, ρq<0 := ρ q has been implicitly understood. dau damping was excluded. Let us denote the contribution by − Note that L,b is always positive and it brings about damp- ,b = b,b + s,b, where s,b = Jb′,z(0)φ(0) and P P P P P ing even in the limit γ 0+. This is so because the integrand 0 → has a pole located at ωs = K v. For infinitesimal positive γ0, b,b = dz J′ (z)Ez(z) J′′ (z)Ex(z) , (23) · P Z b,z − b,x one may take 1 ′′ πδ(ω K v). In this limit, the h i ω¯ K v s − · ≈ − − · Using Eq. (19) and the parity of F with respect to vz, we damping corresponds  to nothing but the usual Landau damp- obtain ± ing24. As is well known, the rate of Landau damping incurred by SPWs is of the order of kvF . This statement is easily con- 3 kz Jb,x(z) = q v vx 2F+ cos(qz) F0e− . (24) firmed with Eq. (31). Performing the integration with ρq ρs Z D D − ≈ h i yields for infinitesimal positive γ0 the following where we have defined 2 3 ωp m 3 4ρq / kv . 3 ∞ 3 2 L,b p 2 F (32) q v... = dq d v e f ′ ... P E ≈ 2 ω 2π~ k2 + q2 0 s Z D D   Z0 Z −  Had we ignored J , we would reach by means of this equa- as a shortcut. Similarly, s tion and Eq. (8) the well known result for SPW damping in the hydrodynamic-Drude model with inclusion of the Landau J (z) = q 3v v 2iF sin(qz) F e kz . (25) b,z z 0 − damping, i.e. Z D D h − − i 2 By expansion of F0 in a series of k∗ v/ω¯ and assuming that 1 3 ωp ω¯ be real, one can show that d3v·f v F is real whereas γHD = kvF . 0′ x 0 − τ − 4 ω2 3 R s d v f0′vzF0 is imaginary. With this, it follows from Eqs. (24) and (25) that J′′ (z) and J′ (z) would vanish ifω ¯ were real. This formula has often been used to estimate the electronic R b,x b,z 1 As such, b/s,b would also vanish under this assumption. collision rate τ− by measurement of the line width of electron P However,ω ¯ is not real but with an imaginary part γ0. To energy loss spectra (EELS) due to the excitation of SPWs. As calculate b/s,b under this circumstance, we find it instructive to be seen in what follows, including the contribution of Js P D L D kvx D qvz calls into question the validity of this procedure. to rewrite F = F + F , with F+ = ω¯ , F = ω¯ , and ± ± ± − We should point out that, the as-established Landau damp- 2 2 L 1 (K v) (K v ) ing would become a Landau gain if we assumed an infinitesi- F = · · − . (26) ± 2 1 K v/ω¯ ± 1 K v /ω¯  mal negative γ0. This is, of course, in violation of the causality  − · − · −  principle and unphysical33. Also in view of this, we must have   Further taking F0  k∗ v/ω¯ , which is valid at long wave- γ positive always. ≈ · 0 lengths kvF /ωp < 1, we can split Jb(z) into a Drude term and an extra term as follows, 2 VI. ELECTRICAL WORK VIA Js i ωp J (z) = E(z) + J (z), (27) b ω¯ 4π L Now we consider the contribution of Js(z) to b/s and show P where JL(z) is given by that it would result in an instability of the system if not for thermal electronic collisions. Let us call the contribution 3 L JL,x(z) = q v vxF+(K, ω,¯ v) cos(qz), (28) ,s = b,s + s,s, where s,s = J (0)φ(0) and Z D D P P P P s′,z 3 L JL,z(z) = i q v vzF (K, ω,¯ v) sin(qz). (29) b,s = dz Js′,z(z)Ez(z) Js′′,x(z)Ex(z) , (33) Z D D − P Z h − i 7

(a) (b)

0.13 p = 0 0.12 k = 0.07kp p 0.12 p 0.1 / / 0 0 0.08 0.11 0.06 0.1 0.02 0.06 0.1 0 0.5 1 p k/k p

FIG. 2. The rate of gain γ0/ωp belonging with the intrinsic channel plotted against (a) the wavenumber k/kp and (b) the Fuchs parameter p. Here kp = ωp/vF . Dots: γ0 obtained by numerically solving Eq. (44). Dashed line: a linear fitting as given in Sec. VII while also serves as a guide for the eye.

kz Again using Ex(z) Ez(z) 2πρse for z outside the layer the same manner as we evaluated L,b assuming infinitesimal ≈ ≈ − P of surface charges, this expression can be rewritten as positive γ0, we find 2 3(p 1) ωp = Ξ Ξ , Ξ = πρ dz J z e kz. / kv , b,s z′ ′′x 2 s s( ) − (34) L,s p − 2 F (39) P − Z P E ≈ 4 ωs Using Eq. (22), we find which is negative, i.e. it counteracts Landau damping rather than reinforces it. The last term in Eq. (37) contributes a 3 ivzv higher order term in kvF/ωp. Because of the presence of vx Ξ = 2πρs q vΘ(vz) Z D D ω¯ k∗ v × in the integrand, the contribution of that term after divided by − · − 2 F0(k, ω,¯ v) pF0(k, ω,¯ v ) + 2(p 1)F+ .(35) p goes like (kvF/ωp) and may be neglected in the first order − − − Eapproximation. As such, we establish that   For small kv /ω and assuming ρ ρ , it follows that F p q s 2 ≈ 3(3 + p) ωp 2 2 b,s/ p kvF . (40) 3 + p πρ 3kv ωp P E ≈− 8 ω2 = s F + p πρ s b,s 2 (1 )2 s P − 2 k 4 ωs − × It should be pointed out that this term alone already wins over 3 2F+ ′ 2F+ ′′ Landau damping, as L,b + b,s 0 always [c.f. Eq. (32)]. q vΘ(vz)vz vx + vz .(36) P P ≤ Z D D  ω¯ ! ω¯ !  The inclusion of b,s in the present calculation supplements   P 22,23   the calculation reported in Ref. in the following technical   22,23 By virtue of the separation F = FD + FL as defined in matter. In previous work , we omitted the there-named + + + M Eq. (26), we obtain matrix; see Appendix B in Ref.22. Theeffects of this matrix in the equation of motion for the charge density are tantamount 2 2 1 + 3p πρs 3kvF ωp to those of b,s in energy conversion. b,s = L,s + (1 p)2πρs P P P − 2 k 4 ω2 − × The calculation of s,s can be performed in a straightfor- s ward manner. We obtainP 3 vzvx K v K v ′ ˜ q vΘ(vz) · · , (37) 2 Z D D ω¯ ω¯ ω¯ K v! 1 + p ωp s,s/ p = 2Γ(γ ) + 2bγ , b = < 1, (41) − · P E − 0 0 4 ω2 where s where Γ(γ0) is a function of γ0 given by L,s = (1 p)2πρs P − × 1 3 K v K v ′ 2 ′′ Γ = ˜ Θ 3 vz K v K v (γ0) (1 p)ρ−s q v (vz)vz · · . ˜ q vΘ(vz) · · , (38) − Z D D ω¯ ω¯ K v! Z D D  ω¯ ω¯ ω¯ K v − · (42)  − ·    Note that Γ is positive and it dominates all the contributions which has a similar form as Eq. (31). One might think that from other parts of b/s. Actually, we have this should give the Landau damping stemming from Js(z). P However, instead of damping, it actually represents a gain. In 2γL = ( L,b + b,s)/ p (kvF /ωp) Γ. (43) − P P E ∼ 8

Combining Eqs. (32), (40) and (41), we can recast the energy if not for Landau damping, the surface component transfers balance equation (8) as follows it to the SPWs. This picture of energy conversion is summa- rized in Eq. (44), which we now solve to demonstrate that an Γ(γ0) + γL (1 + b)γ0 = 0. (44) instability of the Fermi sea might take place under certain cir- − cumstances. We note that γL may be regarded as the net contribution to γ0 Before we numerically solve Eq. (44), let us note that al- from the usual electrical work, i.e. b. P though the formalism derived of the energy conversion in this If we use Eqs. (32) and (40) as an estimate of L,b and b,s P P work is exact, equation (44) does not provide a complete pic- respectively, we obtain ture of SPWs on its own. The reason is because it does not

2 afford a means of evaluating ωs and ρq at the same time. In ωp 3(1 p) 22 γ kv , the complete description established in Ref. , these quanti- L 2 − F (45) ≈ ωs 16 ties were determined self-consistently. In the present work, we have provided them in a reasonable yet ad hoc manner. which is always non-negative. From this we may conclude These provisions are consistent with the complete description that even if s,s is totally ignored, the Landau damping as P and the details of them are not important in the energy conver- caused by Jb(z) will still be overcompensated by the gain due sion process in question here. to Js(z). As such, in contrast to what is claimed by J. Khur- We obtain γ0 by first numerically evaluating both γL and Γ gin et al. and in agreement with observations, Landau damp- as functions of γ0 and then inserting them in Eq. (44), which ing does not constitute an unsurmountable loss barrier in sub- is further solved outright. The results are displayed in Fig. 2. wavelength plasmonics. The long-standing puzzle, as high- In panel (a), the k dependence is shown at fixed p, where lighted in the book by Raether10, of the apparent weak cou- we see that γ0 linearly decreases as k increases. Roughly, pling of SPWs to single particle excitations – the origin of γ0(p, k)/ωp 0.16 0.25k/kp for p = 0, in good agree- Landau damping – becomes thus explicable: such coupling is ment with what≈ was− found in our previous work? . It should not weak but just overshadowed by surface effects. be noted that this relation is universal in the sense that it is In the next section, we shall solve Eq. (44) and show that regardless of the material parameters, which – in the jellium γ > 0 invariably, in agreement with what is expected of 0 model – are signified by ωp and kp only. In panel (b), we show the causality principle [c.f. the remark below Eq. (22)]. By the p dependence of γ0 at fixed k. Again a linear dependence Eq. (45), one might wrongly think that γ should vanish for 0 develops here, γ0(p, k)/ωp 0.12 0.066p for k = 0.07kp. p = 1. This is not true, because Eq. (45) is based on the Combined, we may fit the numerical≈ − solutions by the follow- estimate of L,b by Eq. (32), which assumes an infinitesimal P ing function γ0. For actual γ0, equation (32) only gives an overestimate of L,b. Thus, γL is always positive even for specularly reflecting γ0(p, k)/ωp 0.16 0.25k/kp 0.066p. surfaces.P Now that the net damping rate is ≈ − − As explained in the remarks made in the last paragraph of the γ = ω′′ = 1/τ γ0, preceding section, γ0 remains finite even for p = 1. See that − − − the p dependence disagrees with what was found in previous the positiveness of γ0 then implies that the system, or more work. This discrepancy occurs because in this work we have precisely the Fermi sea, is unstable if τ is sufficiently large. In used a different boundary condition for the electronic distri- the meanwhile, if we approach the instability point from the bution function [c.f. Eq. (16)]. In Ref.22, the corresponding side where the system is stable, we can in principle make γ condition assumes p as the probability of specular reflection − as small as required provided that 1/τ can be tuned below γ0. only in the absence of the normal component of the electric This observation calls into question the practice of identifying field. In the present work, p is the probability for any electric 1 τ− with the line width of the peak due to excitation of SPWs field and thus more realistic. Nevertheless, these two condi- in for example EELS, which is γ. tions are identical for diffuse boundaries. − In closing this section, we discuss some experimental evi- dences and repercussions for the results. In the preceding sec- VII. INSTABILITY AND AMPLIFICATION tion, we have noted that the apparent experimental absence of the coupling between SPWs and single particle excitations, We have demonstrated that the electrical current in a semi- which is supposed to give rise to pronounced Landau damp- infinite metal generally consists of two components, which we ing, is well explicable in our theory; see the remarks follow- call the bulk and the surface components, respectively. The ing Eq. (45). Apart from this, other evidence in support of bulk componentdescribes bulk electrical responses of the sys- the existence of the intrinsic channel of gain may be found by tem and naturally extends the hydrodynamictheory to account comparing these two rates: the directly measured relaxation for Landau damping. The surface component, however, arises rate 1/τ at SPW frequencies and the directly measured SPW only in the presence of a surface and hence describes purely damping rate γ. We expect a substantial difference between surface effects. It is totally beyond the scope of the hydro- them if the intrinsic− channel is not suppressed. For materials dynamic model. The bulk and surface components play dis- in which inter-band transitions can be neglected at both the parate roles in energy conversion. While the bulk component frequencies of SPWs and bulk plasma waves, we may take the would basically preserve the kinetic energy of the electrons bulk wave damping rate as a measure of 1/τ. In such case, 9 the SPW damping rate should be substantially less than the tally or ab initio. Then the rate of inter-band absorption can / ∞ dz σ ω E2 z πσ ω bulk wave damping rate. At least for two alkali metals, potas- be estimated as (1 2 p) 0 ′p( ) ( ) ′p( s). De- 34 E ≈ sium (K) and cesium (Ce), this proposition is confirmed : pending on whether ωs staysR close to the inter-band transition the damping rates for SPWs and bulk waves in K are 0.1eV threshold or not, this rate can be significant or negligible. De- and 0.24eV, respectively, while those in Ce are 0.23eV and spite this uncertainty, in a future paper we will show that inter- 0.75eV, respectively. However, the situation with other mate- band absorption, though capable of diminishing the intrinsic rials remains unclear. gain, can not erase it in total. Several factors may contribute to suppress the intrinsic Radiative losses occur due to the transmutation of a plas- channel. Firstly, there is a size effect. In Ref.23, we showed mon into a propagating photon. In the case of an ideally flat that for metal films γ0 decreases quickly to zero as the film metal surface, this process can not happen because of the con- thickness decreases below the SPW wavelength. Secondly, servation of both energy and momentum. With a non-flat sur- inter-band transitions and extra losses due to surface scatter- face, the momentum conservation is lifted and the transmuta- ing and radiation can also reduce the value of γ0; see discus- tion takes place in the form of optical scattering. By means of sions in the next section. In addition, if the metal is in contact a dimensionalanalysis and some simple physical arguments10, 2 2 with a dielectric rather than the vacuum, γ may also be af- the associated loss rate may be determined as ωs(k σδ) , 0 ∼ 0 fected. Some of these effects have recently been addressed in where k0 is the wavenumber of emitted light of frequency ωs Ref.35 and the intrinsic channel of gain survives. A detailed while σ and δ are the mean squared height fluctuation and discussion of them is beyond the scope of the present paper. the variation, which characterize the profile of a Gaussian sur- face. This expression also gives an estimate of the losses due to SPW scattering by surface roughness. These losses simply 1 VIII. SUMMARY addto1/τ. Considering that k0− 100nm and σ δ 0.1nm typically, we may safely ignore them∼ in most cases.∼ ∼ In summary, we have derived a generic formalism for On the basis of the continuity equation and the Boltzmann studying energy conversion processes in systems with bound- theory, we have presented a systematic analysis of the energy aries. We have applied it to SPWs and shown that the Fermi conversion in SPWs supported in semi-infinite metals. An sea of metals is unstable thanks to the interplay between bal- important role played by the surface is revealed in the con- listic electronic motions and the presence of a surface. version process. We find that ballistic motions could desta- We hope this work will stimulate more interest in this new bilize the system and lend SPWs an intrinsic amplification aspect of SPWs from both the theoretical and experimental channel with a rate γ given as the solution to Eq. (44). Via 0 communities. this channel, SPWs can extract energy from the Fermi sea and amplify themselves if the loss channels are sufficiently sup- Acknowledgement – This work had commenced when pressed. The positiveness of γ is actually warranted by the 0 the author was affiliated with Kwansei Gakuin University, principle of causality. Japan, where he was a JSPS Research Fellow supported by In the present work we have explicitly considered the losses the International Research Fellowship of the Japan Society due to Landau damping and thermal electronic collisions. The for the Promotion of Science (JSPS). Landau damping has been shown to be a higher order effect in comparison with the intrinsic gain and negligible at long wavelengths, while electronic collisions directly counteracts Appendix: The surface term in Eq. (1) the gain. For any real materials, of course there are additional losses such as inter-band absorption and radiative losses. We The continuity equation, Eq. (1) may also be understood briefly discuss these losses in what follows. from Boltzmann’s equation in the relaxation time approxima- Inter-band transitions not only give rise to losses but also tion, which is written as modify the SPW frequency ωs. A systematic treatment of these effects can only be achieved by self-consistently solv- 1 F F ∂t + τ− + v ∂x g(x, v, t) + ∂v f (v) = ∂vg(x, v, t), ing the basic equation of motion for the charge density, as · m · 0 −m · we did in Ref.22. In regard to energy conversion, however,   where f and g are the equilibrium and non-equilibrium part one may obtain a qualitative appreciation by resorting to a 0 of the Boltzmann distribution function, respectively, m is the simple picture. Inter-band effects stem from the electrical electron mass and F is the total force (excluding the part taken responses of inner-shell electrons of the atoms in the metal, care of by the τ 1 term) acting on the electrons. We may write namely, the valence electrons. These electrons are usually − tightly bound to their host atoms and less susceptible to the F = eE + Fs, presence of atomic surroundings and sample boundaries. As such, one may reasonably assume a spatially non-dispersive where e is the electron charge and Fs is the force that prevents response function to describe the motions of these electrons the electrons from escaping the metal. We then write under an an electric field. The current from such motions Fs = eEs = e∂xφs, can then be written as Jp(z) = σp(ω)E(z). Note that the − conductivity σp(ω) can be related to a dielectric function with φs denotes the surface potential. For an ideal surface ǫp(ω) = 4πiσp(ω)/ω, which can be determined experimen- Fs should vanish everywhere except on the surface and point 10 normal to the surface. Keeping g to the first order in E, we that the equation of continuity holds, i.e. can write 3 e(m/2π~) dv(Fs/m) ∂vg = δ(z)Jz(x, t). · (F/m) ∂vg = (Fs/m) ∂vg. Z · · This then leads to Eq. (1). This derivation makes it clear that Now multiplying the equation by e(m/2π~)3 and integrating it this term can be traced back to the force exerted by the sur- over v, we find face. From Boltzmann’s equation, one can show that the total energy Ek + Es + Ep is conserved for τ , where →∞ 1 3 (∂t + τ− )ρ + ∂x J = e(m/2π~) dv(Fs/m) ∂vg. · − Z · 3 2 Ek(t) = (m/2π~) dx dv(m/2)v f0(v) + g(x, v, t) Z Z
We cannot proceed further without knowing Fs, whose details are generally difficult to know and could vary greatly from one is the kinetic energy and Es(t) = dxφs(x)ρ(x, t) as well as sample to another. Despite this, we can fix it by demanding Ep(t) = (1/2) dxφ(x, t)ρ(x, t). R R

19 ∗ [email protected] F. Javier Garca de Abajo et al., Faraday Discussions 178, 123 1 N. W. Ashcroft and N. D. Mermin, Solid State Physics (Saunders, (2015). Philadelphia, 1976). 20 P. R. West, S. Ishii, G. V. Naik, N. K. Emani, V. M. Shalaev and 2 C. J. Pethick and D. G. Ravenhall, Ann. Phys. 183, 131 (1988). A. Boltasseva, Laser & Photonics Reviews 4, 795 (2010) 3 P. Philip, Advanced Solid State Physics (Cambridge University 21 Charles A. Downing and Guillaume Weick, Phys. Rev. B 95, Press, 2012). 125426 (2017). 4 T. W. Ebbesen, C. Genet and S. I. Bozhevolnyi, Phys. Today 61, 22 H.-Y. Deng, K. Wakabayashi and C.-H. Lam, Phys. Rev. B 95, 44 (2008). 045428 (2017). 5 E. Ozbay, Science 311, 189 (2006). 23 H.-Y. Deng, Phys. Rev. B 95, 125442 (2017). 6 A. V. Zayats, I. S. Igor and A. A. Maradudin, Phys. Rep. 408, 131 24 D. Pines, Elementary excitations in solids (Perseus, New Edition, (2005). 1999). 7 W. L. Barnes, A. Dereux and T. W. Ebbesen, Nature 424, 824 25 J. Dawson, Phys. Fluids 4, 869 (1961). (2003). 26 L. P. Pitaevskii and E. M. Lifshitz, Physical Kinetics (Butterwoth- 8 R. H. Ritchie, Phys. Rev. 106, 874 (1957). Heinemann, Oxford, 1981). 9 ~2k2 R. A. Ferrell, Phys. Rev. 111, 1214 (1958). 27 ~ 2πne2~ F 3 2 If we take ωs and εF = as well as kF = 6π n, we 10 H. Raether, Surface plasmons on smooth and rough surfaces and ≈ q m 2m 1/6 1/6 6 on gratings (Springer Berlin Heidelberg, 1988). ~ωs √8π n0 n0 e √m obtain ε = 2 2/3 n 0.5 n , where n0 = ~ 11 B. Rothenh¨ausler and K. Wolfgang, Nature 332, 615 (1988). F (6π ) ≈   ≈ 1024cm 3 comparable ton in magnitude.  Thus, ~ω ε and k 12 S. A. Maier, Plasmonics: fundamentals and applications − s F s k n1/3. As such, we have q k . In numerical∼ calculations,∼ (Springer Science & Business Media, 2007). F c s ∼ q = . k ∼ 35 13 J. B. Pendry, Y. Luo and R. Zhao, Science 348, 521 (2015); O. we use c 1 5 p in this paper as in previous work. In Ref. , we have demonstrated that the values of ω + iγ as solutions to Schinitzer, V. Giannini, S. A. Maier and R. V. Craster, Proc. R. s 0 the fundamental equation of SPWs [c.f. Eq. (18) in Ref.22] are Soc. A 472, 20160258 (2016). independent of the choice of q as long as the latter is sufficiently 14 J. M. Pitarke, V. M. Silkin, E. V. Chulkov and P. M. Echenique, c large – convergence is quickly achieved for q beyond k . Rep. Prog. Phys. 70, 1 (2007). c p 28 34 ∼ 15 M. L. Brongersma and P. G. Kik, Surface plasmon nanophotonics K. Fuchs, Proc. Camb. Phil. Soc. , 100 (1938). 29 195 (Springer, 2007). G. E. H. Reuter and E. H. Sondheimer, Proc. R. Soc. Lond. A , 338 (1948). 16 A. L. Fetter, Ann. Phys. (NY) 81, 367 (1973); Phys. Rev. B 33, 30 J. M. Ziman, Electrons and Phonons: the theory of transport phe- 3717 (1986). nomena in solids (Oxford University Press, 2001). 17 J. B. Khurgin, A. Boltasseva, MRS Bulletin 37, 768 (2012); J. B. 31 M. I. Kaganov, G. Y. Lyubarskiy and A. G. Mitina, Phys. Rep. Khurgin, Nat. Nanotechnology 10, 2 (2015). 288 18 J. B. Khurgin, Faraday Discussions 178, 109 (2015); Nanopho- , 291 (1997). 32 The physical properties of thin tonics 1, 3 (2012). G. P. Zhigal’skii and B. K. Jones, metal films (Taylor & Francis, London, 2003). 33 L. Landau, J. Phys. USSR X, 25 (1946). 34 D. E. Beck, Phys. Rev. B 43, 12611 (1991). 35 Hai-Yao Deng, arXiv 1706.03404 (2017).