Electromagnetic Response Tensors for Normal Conducting Materials

Article pubs.acs.org/JPCC

Electromagnetic Response Tensors for Normal Conducting Materials Maturi Renuka and Amrendra Vijay* Department of Chemistry, Indian Institute of Technology Madras, Chennai 600036, India ABSTRACT: Maxwell’s equations along with the space-time symmetry in normal conducting materials reveal that the electric permittivity and magnetic permeability tensors of rank 2 are fully determined from the conductivity tensor of rank 2, and therefore one needs a dynamical model only for the latter. We elucidate this unification of response tensors and study its ramifications by explicit constructions for the cubic, tetragonal, and orthorhombic classes of crystals. We next use the Boltzmann transport equation in conjunction with a semiclassical model for the statistics of fermionic charge carriers in the material to obtain an analytical expression for the wave vector and frequency-dependent conductivity tensor and thence the permittivity and permeability tensors for metals possessing spherical and nonspherical Fermi surface. We find, inter alia, the spatial dispersion in various response tensors as an important component for a realistic description of optical properties of metals. We ascertain the efficacy of the present theory by computing frequency and wavevectordependent response tensors and other optical properties of a model metallic system, as a representative example. Final expressions for response tensors are suitable for immediate use in Maxwell’s equations to study problems, for example, in subwavelength and near-field optics, plasmonic devices, photonic crystals, and surface-enhanced spectroscopies.

I. INTRODUCTION In a phenomenological description of the interaction of classical electromagnetic fields with complex materials (involving polarizable charges and currents), it is often convenient to introduce various response tensors (functions of the frequency ω and the wave vector k)⃗ that connect the field strengths (E⃗ and B⃗ ) with the excitation fields (D⃗ , H⃗ , and J)⃗ .1−4 One then naturally seeks a mathematical expression (in an analytically closed form) for the tensor elements that may be used, in conjunction with the field equations, to address the problems of optics, plasma physics, and other sciences. Response tensors are conceived, through the works of Lorentz and others,5−8 to be statistically related to the dynamics of fundamental charge carriers in the matter, and at present, in the context of normal metals, the phenomenon of electron conduction is generally conceptualized along the line of early works of Drude and Sommerfeld.9 Since then, there have been a number of important studies, and as a result we now have an enriched understanding of the electromagnetic responses of normal metals.10−13 Despite this, we do not yet possess a full set of response tensors (even in the linear response regime), exhibiting the necessary spatial and frequency dispersions, for immediate use in the studies of metal optics. Recent works14−16 on optical properties of metals have utilized the Drude model9 along with its hydrodynamic extension.17,18 Effects of magnetic response have mostly been ignored. While such simplified models do provide important insight, they serve only a limited purpose. In this Article, we advance a theoretical framework based on formal transport theory11,12 and the classical Maxwell’s equations,1,3 and obtain a simple closed-form expression for linear response (electrical, magnetic, and conductivity) tensors ⃗ suitable (functions of the frequency ω and the wave vector k), © 2014 American Chemical Society

particularly for electromagnetic studies in surface-enhanced spectroscopies,19−21 near-field optics,22−25 plasmonics,26−33 subwavelength optics of metals,34−36 and periodic heterostructures and interfaces,37,38 research directions of considerable interest today. Tensors beyond the linear response regime such as those given in refs 39−41 are not considered here. This Article consists of two parts. First, we ask: Given the conductivity tensor, σ (experimentally measured or theoretically determined) of rank 2, how do we determine (a) electric permittivity and inverse magnetic permeability tensors (ε and ζ respectively) and (b) effective conductivities, suitable for various crystal classes of normal conducting materials? Inherent in this question is the viewpoint of the present investigation, that is, the conductivity tensor σ, within the linear response regime, encodes sufficient information on the dynamics of charged particles (in the presence of external fields) in the matter, and hence ε and ζ, to be conceptualized here as derived tensors, do not require independent mathematical models. As we will later see, the answer fundamentally lies within Maxwell’s equations, which we decipher by taking into consideration the invariance of physical tensors due to the spatial symmetry of various crystal classes42 and the time reversal symmetry expressed in the form of the reciprocity relations of Onsager.43 Next, we address the question of theoretical determination of σ from the viewpoint of the statistical dynamics of the fundamental charge carriers in the matter. For this, we use a simplified model of the Boltzmann transport equation44 and treat the fermionic charge carriers (bare electrons or quasiparticles) in metals as statistically independent but Received: January 7, 2014 Revised: March 3, 2014 Published: March 6, 2014 7018

dx.doi.org/10.1021/jp5001732 | J. Phys. Chem. C 2014, 118, 7018−7031

The Journal of Physical Chemistry C

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satisfying the spin-statistics theorem.45 This finally yields a mathematically closed-form expression for σ that involves two, easily available, physical variables (Fermi momentum and the collision time of the charge carriers) defining the nature of the conducting materials of interest. The use of Boltzmann transport theory in the study of electron conduction in metals has a rich history,10−13,46,47 and the present investigation fundamentally shares the philosophical outlook with an early work of Lindhard,46 but differs significantly in the execution of the transport theory and hence the final expressions for various response tensors. In fact, the expressions for various response tensors available in the literature,46−49 that we are aware of, depend only on the magnitude of the Fermi momentum ℏkf⃗ (hence limiting its utility for the system with spherical Fermi surface), and also they are not valid beyond the cubic symmetry of the metal. The present work does not suffer from these limitations. A key finding of the present investigation is that the spatial dispersion in the conductivity tensor, which is absent in the well-known Drude−Sommerfeld model, is an important ingredient for a realistic description of optical properties of metals. This Article is organized as follows. In section II, we derive a fundamental identity that connects the conductivity tensor with the electric permittivity and the inverse magnetic permeability tensors. We next invert this identity and obtain the effective conductivity and the permittivity and permeability tensors (in terms of the fundamental conductivity tensor σ), suitable for metals with cubic, tetragonal, and orthorhombic crystal symmetries. In section III, we outline a semiclassical transport theory framework to discuss the electron conduction in normal metals and determine a closed-form expression for the frequency and wave vector-dependent conductivity tensor and thence the permittivity and permeability tensors, considering the Fermi surface in the metal to be spherical as well as nonspherical. We apply the present formalism to a model metallic system as a representative example and discuss important features of the conductivity, permittivity, and permeability tensors. In section IV, we present the numerical results for the frequency-dependent reflectivity and skin depth for the model metallic system as an example and suggest the possibility of experiments to precisely locate the plasma edge in the metals. In section V, we conclude with a brief outlook on future directions.

3

Ji ̃ (k ⃗ , ω) =

where Jĩ in eq 3 refers to the ith component of the total current vector. For notational brevity, we will hereafter suppress the dependence of the field variables and response tensors on ω and k⃗ and assume that to be implicit, unless stated otherwise. In what follows, we establish a fundamental relation among the elements of the electric permittivity, inverse magnetic permeability, and conductivity tensors. First, the second rank response tensors, as elucidated by Onsager,43 are invariant to the permutation of indices (a consequence of the microscopic time reversal symmetry), and therefore we extract, if so required, only the symmetric part of the tensors to be used in eqs 1−3. We now consider the dynamics of the electromagnetic ⃗ fields, which, in the (k,ω) space, according to Maxwell’s equations, can be stated as follows.1

ωB⃗ − k ⃗ × E ⃗ = 0

(5)

i ⎡ (n) (m)⎤ ⎣kmαl − knαl ⎦ ω

(6)

i ⎡ (n) (l)⎤ ⎣klαl − knαl ⎦ ω

(7)

σlm ̃ = iωεlm −

(m) ̂ ,⃗ α(m) where kl = l·k = k⃗ l , is the lth component of the vector α⃗ (m) (⃗ m) ̂ ̂ ⃗ × ζ and ζ = lζlm + m̂ ζmn + nζnm. Before we proceed further, it is important to realize that the response tensor σ̃ij(k,⃗ ω) in eq 3 and hence eqs 6 and 7 does not refer to the induced current vector, and therefore eqs 6 and 7 are not immediately useful. In fact, the ith component of the induced current Ji is obtained by subtracting a background current vector J0ĩ from eq 3 as given below.

3

Ji = Ji ̃ − Ji0̃ =

3

∑ (σij̃ − σij̃ 0)Ej = ∑ σijEj j=1

j=1

(8)

In the present study, we consider the background to be the vacuum, and hence we determine σ̃0ij in eq 8 by substituting εij = δijε0 and ζij = δijζ0 in eqs 6 and 7. The final results are as follows. i σll̃ 0(k ⃗ , ω) = iωε0 − (km2 + kn2)ζ0 ω

(9)

i klkmζ0 ω

(10)

σlm ̃ 0 (k ⃗ , ω) = (1)

Here, ε0 and ζ0 are, respectively, the electric permittivity and the inverse magnetic permeability in a vacuum. Thus, the desired conductivity tensor σ referring only to the induced current in eq 8 is obtained using eqs 6, 7, 9, and 10 as given below.

3

∑ εij(k ⃗ , ω)Ej(k ⃗ , ω) j=1

(4)

σll̃ = iωεll −

3

Di(k ⃗ , ω) =

→ ∼⎯ J − i(k ⃗ × H⃗ + ωD⃗ ) = 0

The other two equations of Maxwell essentially define the constraints on the field variables, which are not of immediate significance here. We now substitute the constitutive relations from eqs 1−3 into eq 4, use eq 5 to express B⃗ in terms of E⃗ , and then equate the coefficients of the components of E⃗ to finally obtain the following identities among various tensor elements.

∑ ζij(k ⃗ , ω)Bj(k ⃗ , ω) j=1

(3)

j=1

II. RESPONSE TENSORS: UNIFICATION In the present work, we assume all of the field variables and response tensors to possess convergent Fourier transformations, and henceforth we will work in the wave vector k⃗ and frequency ω space. To set the notations, we introduce a complete set of minimal linear constitutive relations connecting the field strengths (E⃗ and B⃗ ) and the excitation fields (D⃗ , H⃗ , and J)⃗ of the classical Maxwell’s equations as follows. Let l,̂ m̂ , and n̂ constitute an orthogonal triad of unit vectors so that l ̂ · ⃗ ⃗ H⃗ (k,ω) = Hl(k,ω), and so forth, we have Hi(k ⃗ , ω) =

∑ σij̃ (k ⃗ , ω)Ej(k ⃗ , ω)

(2) 7019


The Journal of Physical Chemistry C σll = iω(εll − ε0) − +

2 (z) 1/2 then eq 15 reduces to σeff = σ(z) = ((σ(z) ∥ ) + (σ⊥ )) , the (z) magnitude of σ⃗ . B. Tetragonal. Here, the electric permittivity and the inverse magnetic permeability tensors are diagonal, and the crystal symmetry demands: εxx = εyy = εx, εzz = εz, ζxx = ζyy = ζx, and ζzz = ζz, and therefore eqs 11 and 12 transform to the following forms.

i ⎡ (n) (m)⎤ ⎣kmαl − knαl ⎦ ω

i 2 2 (km + kn )ζ0 ω

σlm = iωεlm −

Article

(11)

i ⎡ (n) (l) ⎤ ⎣klαl − knαl + klkmζ0 ⎦ ω

(12)

Aside from the constraints due to the spatial symmetry of the physical systems of interest, expressions 11 and 12, ipso facto, constitute a fundamental unification principle that must be respected in a physically meaningful theory of electromagnetic responses for conducting materials. To our knowledge, this important point has not been recognized in the literature. The exposition of Lindhard on electromagnetic responses,46 as we have verified here, does conform to the principle behind eqs 11 and 12, albeit in a bare minimum form. We will say more on this later. Now, recognizing the fact that the induced current J ⃗ is the physical quantity that may potentially be measured in an experiment, we regard the tensor σ as the fundamental observable, and hence ε and ζ, from eqs 11 and 12, will here stand for the derived tensors. Therefore, the objective here is to invert eqs 11 and 12 and determine ε and ζ in terms of σ; this is indeed possible, as we will see below, if we take cognizance of the spatial symmetry inherent to the physical system of interest. Thus, we admit the idea of invariance of the physical tensors arising due to the spatial symmetry of the crystals42 and consider here, for simplicity, only those classes for which the axes of the Bravais cells constitute an orthogonal triad of Cartesian vectors. In the discussion below, σ⃗(i) (=x̂σxi + ŷσyi + ẑσzi, i = x,y,z) stands for the response of the system, induced by the external electric field in the ith direction. Subscripts ∥ and ⊥ refer to the components of the vector, parallel and perpendicular to ẑ, respectively, and σ(z) ∥ is the magnitude of σ⃗(z) . || A. Cubic. This is a fully isotropic situation, which demands εij = εδij and ζij = ζδij, and hence eqs 11 and 12 reduce to the following expressions. ε = ε0 −

i ⎛ (z) k⊥ (z)⎞ ⎜⎜σ + σ⊥ ⎟⎟ k ω⎝ ⎠

i ⎛ (z) k⊥ (z)⎞ ⎜⎜σ + σ⊥ ⎟⎟ k ω⎝ ⎠

(17)

ζx = ζ0 −

iω (z) σ⊥ k k⊥

(18)

ζz = ζ0 −

iω σxy kxk y

(19)

That is, one needs to measure the induced currents, parallel and perpendicular to the external fields in any two independent directions (presently x̂ and ẑ), to extract, using eqs 16−19, the necessary elements of the electric permittivity and the inverse magnetic permeability tensors. For tetragonal crystals, we need two effective conductivities satisfying: Jx/y = σeff ⊥ Ex/y and Jz = eff eff σeff ∥ Ez. A simple calculation, using eq 8, yields σ⊥ and σ∥ in terms of σ as follows. σ⊥eff (k ⃗ , ω) =

1/2 E ⎡ (x) ̂ 2 (y) 2⎤ ⎣(σ ⃗ ·E) + (σ ⃗ ·E)̂ ⎦ E⊥

⎛E⎞ σ eff (k ⃗ , ω) = ⎜⎜ ⎟⎟σ ⃗(z)·E ̂ ⎝E ⎠

(20)

(21)

As in the case of cubic symmetry, the direction of the electric field Ê , in eqs 20 and 21, is arbitrary. C. Orthorhombic. Here, the symmetry demands εij = δijεi and ζij = δijζi, and this allows us to invert eqs 11 and 12 to obtain ε and ζ in the following form. εl(k ⃗) = ε0 −

(14)

i ⃗ (l) k ·σ ⃗ ωkl

(22)

iω ζl(k ⃗) = ζ0 − σmn kmkn

As eqs 13 and 14 reveal, we need to measure (by experimental or theoretical means) the induced currents that are parallel and perpendicular to the external field in any one direction (presently ẑ) to finally determine the electric permittivity and the inverse magnetic permeability of matter. For cubic symmetry, we also demand σij = σeffδij that satisfies the generalized Ohm’s law: J ⃗ = σeffE⃗ , implying σeff = J/E. Recognizing J as the magnitude of J ⃗ and Ji = σ⃗(i)·E⃗ from eq 8, we obtain the following expression for σeff in terms of the conductivity tensor. ⎡ ⎤1/2 (i) ̂ 2 ⎥ ⃗ ⎢ σ (k , ω) = ∑ (σ ⃗ ·E) ⎢⎣ i ⎥⎦

(16)

εz = ε0 −

(13)

iω (z) ζ = ζ0 − σ⊥ k⊥k

eff

(x) ⎞ k i ⎛ k⊥⃗ ·σ⊥⃗ ⎜⎜ + σ⊥(z)⎟⎟ k⊥ ω ⎝ kx ⎠

εx = ε0 −

(23)

That is, a full conductivity tensor of rank 2 determines the diagonal electric permittivity and the inverse magnetic permeability tensors as revealed from eqs 22 and 23. The effective conductivities satisfying Ji = σeff i Ei for orthorhombic symmetry are determined using eq 8 as given below. ⎛E⎞ σieff (k ⃗ , ω) = ⎜ ⎟σ ⃗(i)·E ̂ ⎝ Ei ⎠

(i = x , y , z ) (24)

Here too, the direction of Ê is arbitrary. The discussion so far has been concerned with the Cartesian tensors, which, if required, can be transformed to an orthogonal curvilinear coordinate system using the standard relation:42 σij′ = ∑klLikLjlσkl, where Lij = êi·x̂j with êi and x̂i standing for the curvilinear and Cartesian unit vectors, respectively. We also note that the constitutive relation in eq 1 is sometimes

(i = x , y , z ) (15)

The direction of the electric field, Ê , in eq 15 is arbitrary, which may be chosen on physical considerations (for example, the radial direction of a sphere). If we subject the field E⃗ along ẑ, 7020



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⃗ ⃗ ⃗ expressed as Bi(k,ω) = ∑jμij(k,ω)H j(k,ω), where μij is an element of the magnetic permeability tensor. It is then clear ⃗ ⃗ that the tensor μ(k,ω) is the inverse of ζ(k,ω). For the cubic, tetragonal, and orthorhombic symmetry classes, the tensors are ⃗ ⃗ diagonal, and hence μii(k,ω) = 1/ζii(k,ω). We have thus shown that the mathematical relations among the elements of various response tensors of rank 2 are direct consequences of the symmetry principles and Maxwell’s equations. Still, the above deductions remain subject to final experimental verifications. We now briefly describe the experimental context that may be fruitfully utilized here. As an example, we consider the case of cubic symmetry, representative examples of which are copper (Cu), silver (Ag), and gold (Au) metals. The relevant relations among the elements of response tensors are given in eqs 13 and 14, the validity of which, for one, may directly be ascertained if we have (z) an independent experimental determination of σ(z) ∥ , σ⊥ , ε, and ζ. Let us now consider the speed of light, which, inside the conducting matter, can be expressed as follows:50 vmatter(k ⃗ , ω) =

ζ /(ε + iσ eff /ω)

the external fields is governed by the Lorentz force law.1,7 We also assume the existence of impurity centers in metals, acting as potential barriers, that lead to scattering of charge carriers, and, as it is well-known,10 the collision time (relaxation time), τ, well approximates the cumulative effect of such scattering events. Under the collision time approximation, the linear transport equation, obtained after neglecting terms that are of second and higher orders in external fields, takes the following well-known form.10−12,46 ∂f1 ∂t

+

f p⃗ ·∇r⃗ f1 + eE ⃗ ·∇p⃗ f0 = − 1 m τ

(26)

Here, f1(r,⃗ p⃗,t) = f(r,⃗ p⃗,t) − f 0(p⃗) is the fluctuation of the charge distribution from its equilibrium value, f 0(p⃗), p⃗ is the momentum vector, m is the mass of the particle, eE⃗ is the external force driving the electrons, and τ is the relaxation (collision) time. The term f1/τ in eq 26 physically represents the net number of charge carriers scattered out of an unit volume of the phase space around (r,⃗ p⃗) in a unit time interval. The induced macroscopic current is then given as follows (assuming eq 26 has been solved for the nonequilibrium distribution function, f1).

(25)

⃗ Using now eqs 13, 14, and 15, we can express vmatter(k,ω) fully (z) in terms of σ(z) and σ . That means the knowledge of ∥ ⊥ longitudinal and transverse conductivities unambiguously determines the speed of light inside the conducting matter, which is either isotropic or forms a cubic lattice. This deduction, although logically consistent, needs an experimental verification. Similar deductions may be obtained for other classes of crystals. To complete the present program on electromagnetic responses of metals so that it can be utilized in various applications, we need an explicit expression for the conductivity tensor. We develop such a theory in the following section.

J ⃗ (k ⃗ , ω) =

2 ⎛⎜ e ⎞⎟ ℏ3 ⎝ m ⎠

∫ d3p f1 (k ⃗ , p⃗ , ω)p⃗

(27)

⃗ ,ω) is the Fourier where ℏ is the Planck constant, f1(k,p⃗ transform of f1(r,⃗ p⃗,t), and (e/m) is the ratio of the electronic charge and its mass. The factor 2 in eq 27 accounts for two electrons filling a ℏ3 volume element. We finally obtain the expression for σij by a formal identification of eq 8 with eq 27 for the components of E⃗ . This completes the formalism (semiclassical) of electron conduction used in the present study. We note the physical picture behind eq 26, although apparently very simple, continues to be an important theme in the modern discourse on nonequilibrium electron transport in normal metals.10 To proceed further, we need an appropriate expression for the equilibrium distribution function, f 0(p⃗), to be used in eq 26. In fact, f 0(p⃗), to be consistent with the conceptual picture of the metal in the present study, essentially defines the shape of the Fermi surface of the metal, and hence we, in the following, consider the cases when the metal is characterized by either a spherical or a nonspherical Fermi surface. A. Nonspherical Fermi Surface. The consideration of the Fermi surface in the metal to be nonspherical ensures that the final results for σij are adaptable to various symmetry classes of metals, and therefore we choose f 0(p⃗) to be a simple boxcar function (neglecting temperature effect) as shown below.

III. CONDUCTIVITY TENSOR That response tensors, in essence, encode the statistically averaged dynamics of the charge carriers in the presence of electromagnetic fields, a corollary of Lorentz’s program on classical macroscopic electrodynamics,5−8 is the fundamental thesis upon which we here construct a theory of conductivity tensor. To obtain σij, as evident from eq 8, what we need is a formalism for computing the macroscopic induced current J,⃗ and for this we use the ideas from formal transport theory (Boltzmann),11,12 relevant to the electron conduction in metals, and formally recognize the macroscopic current to be an average of the microscopic current j ⃗ over an appropriate nonequilibrium distribution function: that is, J(⃗ r,⃗ t) = ∫ d3p j ⃗ f(r,⃗ p⃗,t), where f(r,⃗ p⃗,t) satisfies the Boltzmann equation: (∂/∂t + v·⃗ ∇⃗ r + F⃗·∇⃗ p)f = (∂f/∂t)coll. Now, for the J(⃗ r,⃗ t) to have a closedform expression, the averaging integral must be fully analytical, and for that to happen the Boltzmann equation must yield f(r,⃗ p⃗,t) in a sufficiently simple mathematical form. As it is wellknown, the source of complication here is, inter alia, the complex nature of the collision integral, (∂f/∂t)coll. While there have been notable developments in this field,52,53 the analytical solution for the full Boltzmann equation remains rare. Thus, we must look for a simple, but sufficiently realistic, model for the collision integral. To fulfill this objective, we conceptualize the metal as a degenerate system that consists of noninteracting fermionic charge carriers (each delocalized within a h3 volume of the phase space) and satisfy the spin-statistics theorem.45 The system is overall neutral. Interaction of charge carriers with

f0 (p ⃗ ) = π −p0 , p0(px )π −p0 , p0(py )π −p0 , p0(pz ) x

x

y

y

z

z

with π −p0 , p0(pz ) = 1 (−pz0 < pz < pz0 ) z

z

= 1/2 (pz = ±pz0 ) = 0 (otherwise)

(28)

where p0x is the x-component of the Fermi momentum and so forth. The form of equilibrium distribution function f 0(p⃗) in eq 28 is appropriate for the metal, conceptualized as a degenerate Fermi system. We now use eq 28 in eq 26 to solve (using ⃗ ,ω) with the external field E⃗ Fourier transformation) for f1(k,p⃗ pointing in a particular direction, and then use eq 27 to 7021



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frequency in the unit of 1014 s−1, and hence we choose ωs to be 1014 s−1. Furthermore, we measure the collision time τ, the wave vector k⃗ , and the Fermi velocity v0⃗ in the units of 10−14 s, 108 m−1, and 106 m s−1, respectively. Equations 30 and 31 in the units of 0.7322 × 106 S m−1 are then given as follows.

determine the induced currents that are longitudinal and transverse to the external field. An identification of the components of E⃗ with eq 8 then finally determines the desired elements of the conductivity tensor. As an example, we obtain the following, easy to evaluate, integral for an element of σ. σxy = −i

×

2e 2 h3

∫

dpx dpy dpz

px m

π −p0 , p0(pz )π −p0 , p0(px ) z

z

x

x

[δ(py + py0 ) − δ(py − py0 )] ω−

p ⃗ ·k ⃗ m

+ i/τ

(29)

A number of integrals of the type 29 finally yield the following exact expression for the σ. ⎛ 2k v 0 ⎞ σll(k ⃗ , ω) = iC ⎜ l3 l ⎟ωs(z 0 + zl − zm − zn) ⎝ k̃ ⎠

(30)

⎛k ⎞ σlm(k ⃗ , ω) = iC ⎜⎜ m3 ⎟⎟ωs2(y0, l + ym , n ) ⎝ klk ̃ ⎠

(31)

2 2

3

where C = (m e /h ), = jth-component of the Fermi velocity ̂ ). ⃗ Various terms appearing in eqs 30 v0⃗ , and k3̃ = klkmkn (kl = l·k and 31 are defined below. (32)

y0, l = ω0−ωl− ln(ω0−/ωl−) − ω0+ωl+ ln(ω0+/ωl+)

(33)

ym , n = ωm−ωn+ ln(ωn+/ωm−) + ωn−ωm+ ln(ωm+/ωn−)

(34)

ωj± = ωext ± ωjint

(35)

ωext = (ω + i/τ )/ωs

(36)

ωjint = [v ⃗ 0·k ⃗ − 2(1 − δj0)vj0kj]/ωs

(37)

(38)

⎛k ⎞ σlm(k ⃗ , ω) = i⎜⎜ m3 ⎟⎟(y0, l + ym , n ) ⎝ klk ̃ ⎠

(39)

where all of the quantities on the right-hand side of eqs 38 and 39 are now dimensionless numbers (real or complex). We now discuss the limiting cases (related to various time scales present in the system) for σ that would be of interest in some applications, and also they will provide a deeper insight into the problem at hand. To establish a clear perspective, a few remarks on the physical content of eqs 30−37 are in order here. In fact, we notice that there are three fundamental frequency scales involved in the problem at hand. The first is that determined by the collision time τ of the charge carriers, and the second is the external frequency ω as defined in eq 36, which, in a scattering problem, would refer to the frequency at which the electromagnetic wave is incident on the material of interest. For electronic processes in metals, the relaxation (collision) time τ is usually larger than the time scale of the external field ω. The last one is the internal frequency ωint j , which, as eq 37 reveals, is related to the Fermi velocity v0⃗ and the wave vector k⃗ that defines the length-scale of the physical system. It is thus evident that the ωint j values , akin to the plasma frequency of the isotropic conducting medium (consisting of free charge carriers), are the defining characteristics of the system. What is also clear from eq 37 is that the fermionic plasma in a metallic system will, in general, possess a range of fundamental characteristic frequencies (belonging to different directions of the cluster of wave vectors centered ⃗ in place of just one (plasma frequency) that has around k), commonly been used to discuss the plasma oscillations in metals.54−56 In any event, given that the relaxation time τ is long (as compared to 1/ω), we may consider the limits when ext the ωint j is either greater or smaller than the magnitude of ω . ext < ω , the distance covered by the charge In the limit ωint j carriers during one period (T = 2π/ω) of the field would be shorter than the wavelength of the field, and hence they will experience the full time dependence of the field. In the other ext limit (ωint j > ω ), the path, during one period of the field, would be larger, and therefore the charge carriers will essentially experience as if they have been subjected to static external fields. To obtain the conductivity tensor in these limits,, we just need to expand the logarithms in eqs 32−34 in ext powers of xj (with ωint j = xjω ) and retain only low order terms. Carrying this program, we have the following convergent expressions (infinite series) for zj in eq 32.

v0j

zj = ωj− ln(ωj−) + ωj+ ln(ωj+)

⎛ 2k v 0 ⎞ σll(k ⃗ , ω) = i⎜ l3 l ⎟(z 0 + zl − zm − zn) ⎝ k̃ ⎠

where ωs is an arbitrary scale (frequency) that has been introduced to render zj, y0,l, and ym,n dimensionless. This completes the derivation for the conductivity tensor, the evaluation of which, as evident from eqs 30−37, involves simple algebraic operations and requires two physical quantities (Fermi velocity v0⃗ and the relaxation time τ for the charge carriers) that are the characteristics of the metallic systems of interest. Finally, we can extract the symmetric component of σ from eqs 30 and 31 and use it in the formalism discussed in section II to determine the electric permittivity and the inverse magnetic permeability in various symmetry situations. An important remark concerning the spatial symmetry inherent in eqs 30 and 31 is in order here. It is easy to deduce ⃗ ⃗ from eqs 30−37 that the relation σij(k,ω) = σij(−k,ω) holds for real k.⃗ That means the conductivity tensor in the real space, σij(r,⃗ ω), will be invariant to the inversion operation and hence centrosymmetric, if the wave vector k⃗ is real-valued. Still, it is sometimes useful to introduce inhomogenous plane waves for the representation of electromagnetic fields, wherein the wavevector k⃗ is necessarily complex.51 In such situations, the space inversion symmetry of the conductivity tensor in eqs 30 and 31 will be spontaneously broken. To facilitate numerical applications, we evaluate fundamental constants in eqs 30 and 31 as follows. First, we measure the

⎡ zj = ωext⎢2 ln(ωext) + ⎢⎣ (|xj| = 7022

|ωjint /ωext|

∞

∑ r=1

< 1)

⎤ ⎥ r(2r − 1) ⎥⎦ (xj)2r

(40)



Article

⎡ 1 zj = ωjint⎢iπ + iπ + 2⎡⎣1 + ln(ωjint)⎤⎦ − xj ⎢⎣

(

∞

∑ r=1

The physical content of eqs 49 and 50 is revealing and significant. To get an insight, let us consider a particular column (say, lth) of the tensor σ, carrying elements σll, σml, and σnl. These elements signify the response (in the directions l,̂ m̂ , and n̂) of the system due to the disturbing field in the lth direction. ext Thus, in the nontrivial lowest order limit of ωint j < ω , the direct response σll, the longitudinal conductivity, is independent of the wave vector and therefore exhibits the system to be effectively homogeneous, as eq 49 reveals. The transverse components (σml and σnl), on the other hand, show spatial dispersion at all nontrivial orders, the implication of which will ext be seen shortly. We thus conclude that, in the limit ωint j < ω , the spatial dispersion in the effective conductivity of the metal, as defined in eqs 15, 20, 21, and 24, arises substantially due to the inhomogeneity in the transverse response. In a similar manner, we can obtain the conductivity tensor σ in the ext situation when ωint j > ω using eqs 41, 43, and 46, and the final results in the lowest order of ωext/ωint j are as follows.

)

⎤ 1 ⎥ r(2r + 1)(xj)2r + 1 ⎥⎦

(|xj| = |ωjint /ωext| > 1) (41)

The element y0,l in eq 33, obtained as an infinite series, is given as follows. ⎡ y0, l = 2(ωext)2 ⎢(xl − x0) + ⎢⎣

∞

⎤

∑ sr(l)⎥ r=1

⎥⎦

(|xl| = |ωlint /ωext| < 1)

(42)

∞ ⎤ ⎡ ⎛ ω int ⎞ y0, l = 2ω0intωlint⎢(xl + x0) ln⎜ lint ⎟ + (xl − x0) +∑ sr(l)⎥ ⎢⎣ ⎥⎦ ⎝ ω0 ⎠ r=1

⎛ 4v 0k ⎞⎡ σll = iC ⎜ l3 l ⎟⎢iπ ( −vl0kl + vm0 km + vn0kn) ⎝ k ̃ ⎠⎢⎣

(|xl| = |ωext /ωlint| < 1) (43)

⎤ 2r xl 1 ⎡ x0 − ⎢ ⎥x 0 2r ⎣ (2r + 1) (2r − 1) ⎦ ⎤ 2r xl 1 ⎡ x0 + ⎢ − ⎥x l 2r ⎣ (2r − 1) (2r + 1) ⎦

⎛ (v 0 k − v 0k )2 − (v 0k )2 ⎞⎤ l l ⎟⎥ + (ωsωext) ln⎜ m0 m 2 n n0 0 2 ⎝ (vl kl) − (vmkm + vn kn) ⎠⎥⎦

with sr(l) =

⎛ 2k ⎞ σlm = iC ⎜⎜ m3 ⎟⎟⎡⎣−iπ {(vl0kl)2 − (vm0 kn − vn0km)2 ⎝ klk ̃ ⎠

(44)

Similarly, the series expansion for the element ym,n in eq 34 is as given below. ym , n

⎡ = (ωext)2 ⎢2(xm + xn) − ⎢⎣

⎧ ⎡ (v 0 k )2 − (v 0k − v 0k )2 ⎤ − (ωsωext)2 } + ⎨vm0 km ln⎢ m0 m 0 n2 n 0l l 2 ⎥ ⎣ (vl kl + vn kn) − (vmkm) ⎦ ⎩ ⎡ (v 0k )2 − (v 0k − v 0 k )2 ⎤⎫⎤ l l m m 0 ⎥⎬⎥ + vn kn ln⎢ n0 n 0 2 ⎣ (vl kl + vmkm) − (vn0kn)2 ⎦⎭⎥⎦ (53) ⎪

⎤ ∑ sr(m,n) + sr(n,m)⎥ ⎥⎦ r=1 ∞

⎪

⎪

(|xm| = |ωmint /ωext| < 1)

(45)

⎪

⎡ ym , n = −ωmintωnint⎢2πi + (am , n + an , m) ⎢⎣ ∞ − ∑ sr(m , n)+sr(n , m) ⎤⎦ r=1

(

It is immediately clear from eqs 52 and 53 that all of the components of σ in the limit ωint > ωext remain spatially j dispersive at the lowest nontrivial order expansion. We now consider the conductivity tensor in eqs 30 and 31 in the limit of vanishingly small wavevector; that is, |k|⃗ ≪ ωext/v0 ⃗ ext may be completely ignored in favor of unity. such that v0|k|/ω This is the situation when the material may be considered to be spatially nondispersive (because k⃗ effectively goes to zero, and consequently the resulting quantity becomes the average over the material volume), and we thus define an effectively homogeneous medium. To understand this limit, it is sufficient to examine the case when all of the components of the Fermi velocity and the wavevector are numerically identical; that is, v0l = v0m = v0n = v0/√3 and k0l = k0m =k0n = k0/√3. The expression 35 for ω±j then simplifies as follows.

)

(|xm| = |ωext /ωmint| < 1) with am , n = xm[2 ln(exn/xm) − iπxn] and sr(m , n) =

xn ⎤ 1 ⎡ xm 2r + ⎥(xm) ⎢ r ⎣ 2r + 1 2r − 1 ⎦

(46) (47)

(48)

Consider now the conductivity tensor in eqs 30 and 31 in the ext ext int limit ωint j < ω . We find that the terms of the order (ωj /ω ) in eqs 40, 42, and 45 yield a null tensor for σ and are hence ext 2 trivial. Retaining terms of the order (ωint j /ω ) , we obtain the following simplified expression for the conductivity tensor. 16 0 0 0 σll = iC vl vmvn ωsωext

(49)

3 0 32 km ⎛ vl kl ⎞ 0 0 ⎜ ⎟ (vmvn kmkn) σlm = iC 3 klk 3̃ ⎝ ωsωext ⎠

(50)

σlm + σml = iC

(52)

0 2 0 2 32 klkm⎣⎡(vl ) + (vm) ⎦⎤ 0 0 0 vl vmvn 3 ext 3 (ωsω )

ωj± = ωext ± (2δj0 + 1)

v0k 3ωs

(j = 1, 2, 3)

(54)

It is now easy to show that the ratios ω±0 /ω±j and ω±m/ω±n ⃗ ext tends to zero, implying a approach unity when v0|k|/ω vanishingly small transverse conductivity in eq 31. A vanishingly ⃗ k→0 small σ⊥(k,ω)| and hence the vanishing transverse current ⃗ may then also be used, in practical situations, as a definite limit criterion for the material to be defined as practically ⃗ k→0 homogeneous. Furthermore, a vanishingly small σ⊥(k,ω)| ⃗ also means, as eqs 14, 18, 19, and 23 reveal, the inverse magnetic permeability of the material will approach the vacuum

(51) 7023



Article

value, ζ0. We thus conclude if a conducting material that possesses a nonspherical Fermi surface is spatially homogeneous, it will not support an experimentally measurable transverse current, and as a consequence the material will appear to be practically nonmagnetic. This connection between the spatial dispersion of the conductivity and the magnetic property of the conducting material appears to be a fundamental law, as the present theory unequivocally suggests. An experimental verification would hence be highly desirable here. Furthermore, as we will later see, the transverse ⃗ k→0 conductivity σ⊥(k,ω)| does not vanish if the metal possesses ⃗ a spherical Fermi surface, and therefore a vanishingly small ⃗ k→0 σ⊥(k,ω)| may also be taken as a signature for the Fermi ⃗ surface of the conducting material to be nonspherical. On the other hand, as eq 54 reveals, ω±0 ≠ ω±j (j = 1,2,3), and hence the longitudinal conductivity in eq 30 will be nonvanishing and it will approach a definite asymptotic value as |k|⃗ tends to zero. In practical terms, it means that the electromagnetic responses of the conducting material, if spatially homogeneous, will remain faithfully represented if we simply set the inverse magnetic permeability to its vacuum value, ζ0. We now present, as an example, representative numerical results on various response tensors for a model metallic system, the Bravais lattice of which forms the cubic symmetry (section II.A). It is important to recall that the computation of conductivities in eqs 30 and 31 requires information on the collision time τ, the internal frequency ωint j , expressed in terms of Fermi velocity v0⃗ and the wave vector k⃗ in eq 37, for the metal in question. For the model metallic system to represent a somewhat realistic physical situation, we here choose the numerical values for τ and v0⃗ as those obtained from the free electron gas description of the Ag metal, as given in the standard text.57 Admittedly, for a more realistic description, we need a much better estimate on the values of collision time and the Fermi velocity of the metal. We will say more on this later. 0 We thus choose the value of ωint 0 ω = v k in eq 37 to be 2 eV, which sets the internal frequency (equivalently, the plasma frequency) for the metal in the optical range around 4.83 × 1014 s−1. The Fermi velocity for the electrons in the metal is taken to be 1.39 × 106 m/s as given in ref 57. For simplicity, we take all of the components of k⃗ to be identical; that is, kx = ky = 0 8 −1 ⃗ kz |k|/√3 and hence |k|⃗ = ωint 0 /v = 3.48 × 10 m . The numerical value of the collision time for the electron in the metal is taken to be 2.8 × 10−14 s, which is the Drude relaxation time, representing the Ag metal.57 The longitudinal direction of the external field E⃗ is aligned along the crystallographic ẑ (=l)̂ axis. On substituting the values of v0⃗ , τ, and k in eqs 38 and 39, we obtain the following expression for the longitudinal and transverse conductivities (in the units of 107 S m−1). σ (z) = i

0.254 × 1.39 (z 0 − z l ) (k)2

(55)

σ⊥(z) = i

0.538 (y + ym , n ) (k)3 0, l

(56)

Figure 1. The longitudinal conductivity σ(z) ∥ (ω) at a fixed wavevector (|k|⃗ = 3.48 × 108 m−1) for the model metallic system (see text) in the unit of 107 S/m. The solid (red) and dotted (blue) lines correspond, respectively, to the real and imaginary parts of the conductivity. The vertical line at 2 eV refers to the location of the plasma edge in the metal.

⃗ Figure 2. The transverse conductivity σ(z) ⊥ (ω) at a fixed wavevector (|k| = 3.48 × 108 m−1) for the model metallic system (see text) in the unit of 107 S/m. The solid (red) and dotted (blue) lines correspond, respectively, to the real and imaginary parts of the conductivity. The vertical line at 2 eV refers to the location of the plasma edge in the metal.

4, respectively. Also included in Figure 3, for comparison, is the experimentally derived permittivity (real part)59 for the Ag metal that the model system considered here represents. We note that the experimental determination of ε(ω) in ref 59 involved a fitting of differential inelastic mean free path data to the Drude−Lindhard oscillator model for the complex dielectric function. As the present theory of ε(ω) is fundamentally distinct and goes beyond the Drude-type description for the metallic system, a definite and quantitative comparison of the present ε(ω) with the experimentally derived one is not totally feasible. It is however gratifying to note that the experimental Re ε(ω), as Figure 3 reveals, qualitatively follows the present theoretical prediction, notwithstanding the simplicity of the model for the Fermi surface and a rudimentary description of the relaxation time used here. Furthermore, it is clear from Figures 1−4 that the location of plasma edge (2 eV, in the present study), which separates the relaxation and the transparent regimes of the

where k is equal to 3.48. What we immediately observe is that the present formulation correctly predicts the order of magnitude (107 S m−1), which is typical for the conductivities (z) of Ag metal.58 σ(z) ⊥ and σ∥ , computed using eqs 55 and 56, as a function of frequency are shown in Figures 1 and 2, respectively. We next use eqs 55 and 56 in eqs 13 and 14 to compute the electric permittivity and the inverse magnetic permeability, the results of which are displayed in Figures 3 and 7024



Article

that there is a critical frequency (ω = 0.884 eV), in the relaxation regime, at which Re ε(ω) changes its sign. Such a behavior of Re ε(ω) is consistent with the analysis based on Drude’s model of frequency dispersion.60 Significantly, the experimentally determined Re ε(ω) also shows such a change in sign, as Figure 3 shows. The high frequency behavior of the response tensors presented here is as expected for a normal metal. Figures 1−4 also reveal that all of the response quantities exhibit a characteristic transition (a peak at 1.36 eV in the Re σ(z) ⊥ curve, for example), just before the plasma edge. This can potentially be verified by experiments. The predictive power of the present formulation will be strengthened if there is an independent and reliable determination (either by experiments or by a detailed quantum mechanical study) of the plasma edges of the metals. We now consider the spatial dispersion of the conductivity tensor at the fixed frequency. As it is well-known, the nonlocal optical properties of metals are described by the wavevector dependence of the response function, and therefore we here present representative numerical results for the model metallic system, under study here. For this, we set the frequency ω in eq (z) 36 to be equal to 2 eV and compute σ(z) ∥ and σ⊥ in eqs 55 and ⃗ 56 as a function of k = |k| and show the results in Figures 5 and

Figure 3. The electric permittivity ε(ω) at a fixed wavevector (|k|⃗ = 3.48 × 108 m−1) for the model metallic system (see text) in the unit of 10−7 F/m. The solid (red) (a) and dotted (blue) (b) lines correspond, respectively, to the real and imaginary parts of the permittivity. The vertical line at frequency ω = 0.884 eV shows the location at which the real part changes sign. The vertical line at 2 eV refers to the location of the plasma edge in the metal. The experimentally determined Re ε(ω) for the Ag metal is displayed by curve (c). Curve (c) has been multiplied by a factor of 2 for the clarity of presentation.

Figure 4. The inverse magnetic permeability ζ(ω) at a fixed wavevector (|k|⃗ = 3.48 × 108 m−1) for the model metallic system (see text) in the unit of 105 m/H. The solid (red) and dotted (blue) lines correspond, respectively, to the real and imaginary parts of the permeability. For display here, a value of 7.9 × 105 m/H has been subtracted from the real parts of the permeability. The vertical line at 2 eV refers to the location of the plasma edge in the metal.

Figure 5. The longitudinal conductivity σ(z) ∥ (k) at a fixed frequency (ω = 2 eV) for the model metallic system (see text) in the unit of 107 S/ m. The solid (red) and dotted (blue) lines correspond, respectively, to the real and imaginary parts of the conductivity. The vertical line at 0.348 nm−1 refers to the location of the characteristic wavevector, kcenter, of the metal (see text). The inset in the figure is the longitudinal conductivity σ(z) ∥ (k) for large values of k. For display here, a factor of 10 × 107 S/m has been multiplied by the imaginary part of the conductivity in the inset.

metal, plays a decisive role toward the behavior of various response quantities. At higher frequencies (beyond the plasma (z) edge), as Figures 1 and 2 reveal, Re σ(z) ∥ and σ⊥ asymptotically (z) go to zero, but the Im σ∥ attains a nonzero constant value (3.74 × 105 S m−1 at 6.83 eV). To understand the meaning of this, we reconsider eqs 13 and 14. Beyond the plasma edge, σ(z) ⊥ → 0 and therefore ζ → ζ0 as eq 14 reveals, and consequently (ε − ε0) → Im σ(z) ∥ /ω from eq 13; that is, the inverse magnetic permeability ζ assumes the vacuum value, and the asymptotic value of Im σ(z) ∥ plays the role of the electric permittivity ε of the metal. This high frequency behavior of ε and ζ is clearly reflected in Figures 3 and 4. In the high frequency limit, metals will thus exhibit no dissipation and consequently behave essentially as a dielectric. Furthermore, we find from Figure 3

6, respectively. To understand the results, we first note that there is a characteristic length scale involved in the problem at hand. This length scale, called here as lmaterial, is approximately given by the ratio of the Fermi velocity v0 and the frequency ω (z) of eq 36. It is expected that the conductivity (both σ(z) ⊥ and σ∥ ) will show a broad maximum centered around the wavevector of magnitude kcenter ≈ 1/lmaterial, which is a characteristic of the material. For the model metallic system under study here, we have ω = 2 eV and v0 = 1.39 × 106 m s−1, and therefore kcenter = 0.348 × 109 m−1. It is seen from Figures 5 and 6 that the (z) conductivity (both σ(z) ⊥ and σ∥ ) falls rapidly as we move far from the wavevector of magnitude kcenter, which is shown as a vertical line in Figures 5 and 6. Figures 5 and 6 also reveal that (z) both σ(z) approach zero as |k|⃗ tends to infinity. ⊥ and σ∥ 7025



Article

Figure 6. The transverse conductivity σ(z) ⊥ (k) at a fixed frequency (ω = 2 eV) for the model metallic system (see text) in the unit of 107 S/m. The solid (red) and dotted (blue) lines correspond, respectively, to the real and imaginary parts of the conductivity. The vertical line at 0.348 nm−1 refers to the location of the characteristic wavevector, kcenter, of the metal (see text). The inset (a) in the figure is the transverse conductivity σ(z) ⊥ (k) for large values of k. For display here, a factor of 10 × 107 S/m has been multiplied by the imaginary part of the conductivity. The inset (b) in the figure is the transverse conductivity σ(z) ⊥ (k) for small values of k.

⃗ Figure 7. Longitudinal conductivity σ(z) ∥ (ω) at a fixed wavevector (|k| = 3.48 × 108 m−1) and a fixed Fermi velocity (|v0⃗ | = 1.39 × 106 m s−1) for the model metallic system (see text) in the unit of 107 S/m for three different relaxation time τ. Curves (a) and (b) represent the real −14 s. and imaginary parts of σ(z) ∥ (ω), respectively, for τ = 1.4 × 10 Curves (c) and (d) represent the real and imaginary parts of σ(z) ∥ (ω), respectively, for τ = 2.8 × 10−14 s. Curves (e) and (f) represent the real −14 s. The and imaginary parts of σ(z) ∥ (ω), respectively, for τ = 4.9 × 10 vertical line at 2 eV denotes the plasma edge in the metal.

relaxation times in the Boltzmann equation may be required for further improvements in the theory here. We now consider the sensitivity of σ∥(ω) toward the Fermi velocity. In Figure 8, we thus present σ∥(ω) for the model metallic system considered here at a fixed wave vector (kx = ky = kz = 2.01 × 108 m−1) and the relaxation time (τ = 2.8 × 10−14 s−1) for three different values of the Fermi velocity. As Figure 8 reveals, the complex-valued σ∥(ω) as expected increases with the magnitude of the Femi velocity; however, the relatively

However, as |k|⃗ → 0, → 0, but approaches an 7 asymptotic value (Re σ(z) = 0.125 × 10 S m and Im σ(z) ∥ ∥ = 9.3 4 −1 × 10 S m for the model metallic system being studied here). These observations are consistent with the theoretical analysis presented above (see the discussion following eq 54). In ⃗ particular, a vanishing σ(z) signals that the metal is ⃗ ⊥ (k,ω)|k→0 effectively homogeneous and hence nonmagnetic. Finally, Figures 5 and 6 also show that the conductivity tensor is band-limited (non-negligible only up to a finite wavevector), indicating a finite support in the real space, and, as a ⃗ consequence, the Fourier transformations of σ(z) ∥ (k,ω) and (z) ⃗ σ⊥ (k,ω) will be unambiguously convergent. We now address the question: how sensitive is the present model of electric conductivity to the Fermi velocity v0⃗ and the relaxation time τ of the charged carrier? For a definite comparison, we choose the longitudinal conductivity σ(z) ∥ (ω) of the model metallic system considered here as a representative example. We thus present in Figure 7 the complex-valued σ∥(ω) at a fixed wave vector (kx = ky = kz = 2.01 × 108 m−1) and the Fermi velocity (v0x = v0y = v0z = 0.802 × 106 m s−1) for three different values of the relaxation time τ. Salient features, as Figure 7 reveals, are as follows. We first notice that the Re σ∥(ω) falls off but the Im σ∥(ω) increases with the increase of τ; however, the relatively sharp (but continuous) transition near the plasma edge that separates the relaxation and the transparent regimes of the metal remains a robust feature. At higher frequencies, σ∥(ω) reaches an asymptotic value, which is independent of τ. In any event, the qualitative behavior of σ∥(ω) remains fairly invariant to τ. This is possibly the limit of the single relaxation time for the charged carriers used here. We parenthetically note that the collision operator of the linearized Boltzmann equation admits a range of relaxation times in the form of its eigenvalues. Accordingly, the use of a single relaxation time is expected to be only a primitive description of the relaxation kinetics of the charged carriers toward equilibrium. Thus, the use of multiple σ(z) ⊥

σ(z) ∥ −1

⃗ Figure 8. Longitudinal conductivity σ(z) ∥ (ω) at a fixed wavevector (|k| = 3.48 × 108 m−1) and relaxation time (τ = 2.8 × 10−14 s) for a model metallic system (see text) in the units of 107 S/m for three different Fermi velocities |v⃗0|. Curves (a) and (b) represent the real and 0 6 −1 imaginary parts of σ(z) ∥ (ω), respectively, for |v⃗ | = 1.7 × 10 m s , and the corresponding plasma edge in the metal is shown by the vertical line at 2.44 eV. Curves (c) and (d) represent the real and imaginary 0 6 −1 parts of σ(z) ∥ (ω), respectively, for |v⃗ | = 1.39 × 10 m s , and the corresponding plasma edge in the metal is shown by the vertical line at 2 eV. Curves (e) and (f) represent the real and imaginary parts of 0 6 −1 σ(z) ∥ (ω), respectively, for |v ⃗ | = 1.0 × 10 m s , and the corresponding plasma edge in the metal is shown by the vertical line at 1.44 eV. 7026



Article

sharp (but continuous) transition of σ ∥ (ω) near the corresponding plasma edges, as noted earlier, remains a robust feature here too. Qualitative behavior of σ∥(ω) here also remains fairly invariant to the magnitude of the Fermi velocity. We note that the present numerical test has assumed the three components of the Fermi velocity v0⃗ to be identical. An anisotropic description of v0⃗ is expected to show a qualitatively different behavior of σ∥(ω). Finally, a comparison of Figures 7 and 8 indicates that the numerical values of σ∥(ω) are more sensitive to the magnitude of the Fermi velocity than the relaxation time. B. Spherical Fermi Surface. Metals with spherical Fermi surface belong to the cubic symmetry, and hence they are characterized by the response tensors of the form εij = εδij, ζij = ζδij, and σij = σeffδij (see section II.A). There results further simplify if we also assume that the response function ε, ζ, and σeff depend only on the magnitude k of the vector k.⃗ This was the situation originally studied by Lindhard.46 This may be of interest in some applications, and hence we present here for completeness, from the viewpoint of present investigation. We now choose f 0 in eq 26 to be the Heaviside function, θ(p0−p), where p0 is the magnitude of the Fermi momentum. Applying the electric fields that are parallel and perpendicular to k⃗ in eq 26, evaluating the integral in eq 27 in the spherical polar coordinates, and finally using the definition of the induced currents in eq 8, we obtain (with x = (ω + i/τ)/(v0k), v0 = the ⃗ the following expressions for the Fermi velocity, and k = |k|) direct and transverse (with respect to the wave vector k)⃗ conductivities. (57)

⎛ x + 1⎞ 1 3 ⎟ − σ⊥(k , ω) = i C ln⎜ σ (k , ω) ⎝ x − 1⎠ 4 2

(58)

ωp2 ω + i/τ

where J ⃗ (⊥) = σ and J ⃗ = (k ·̂ J ⃗ )k ;̂

and

ωp2 =

2 2 3 8π m e v0 3 h3ε0

⃗

(⊥)E (⊥)

J⊥⃗ = J ⃗ − (k ·̂ J ⃗ )k ̂

(62)

⎡ ⎤ k2 J⊥⃗ = iω⎢(ε − ε0) − 2 (ζ − ζ0)⎥E⊥⃗ ⎣ ⎦ ω

(63)

Finally, eq 60 along with eqs 62 and 63 yield the electric permittivity and the inverse magnetic permeability in terms of direct and transverse conductivities as follows. ε = ε0 −

ζ = ζ0 +

iσ (64)

ω iω(σ⊥ − σ ) k2

(65)

Expressions in 64 and 65 are, mutatis mutandis, transverse and longitudinal dielectric constants of Lindhard.46,47 Thus, we find that Lindhard’s theory is consistent with the fundamental viewpoint of the present study; that is, the conductivity tensor fully determines the electric permittivity and the inverse magnetic permeability of metals. For cubic symmetry, we also need an effective conductivity that satisfies J ⃗ = σeffE⃗ . Although Lindhard and others46−49 have not evaluated σeff, if we use eqs 57 and 58 to define a new tensor σij = δijσ⊥ + (σ∥ − σ⊥)kikj/k2 and use eq 15, we will obtain the correct expression for σeff. This is evaluated here, which can be computed using eqs 57 and 58 as follows. σ eff (k ⃗ , ω) =

⎡ x ⎛ x + 1 ⎟⎞⎤ σ (k , ω) = −i3Cx 2⎢1 − ln⎜ ⎥ ⎣ 2 ⎝ x − 1 ⎠⎦

C = ε0

J ⃗ = iω(ε − ε0)E ⃗

k=

1 (k σ )2 + (k⊥σ⊥)2 k

k 2 + k⊥2

where (66)

The longitudinal and transverse conductivities as given in eqs 57 and 58 are valid for a spatially dispersive material and hence useful for studying nonlocal optical behavior of the conducting matters (possessing spherical Fermi surface). To get some insight, we now present representative numerical results for the same set of parameters approximating the Ag metal as in section III.A. σ∥ and σ⊥ in eqs 57 and 58 as a function of frequency (for a fixed k = 3.48 × 108 m−1) are shown in Figures 9 and 10, respectively. The electric permittivity and the inverse magnetic permeability in eqs 64 and 65 are shown in Figures 11 and 12, respectively. At frequencies beyond the plasma edge, as Figures 9 and 10 reveal, Re σ∥ → 0, Re σ⊥ → 0, and Im σ∥ → Imσ⊥ constant (∼0.105 × 107 S m−1 at 6.83 eV). This implies, from eq 65, ζ → ζ0 and Im σ∥ play the role of the electric permittivity from eq 64. This behavior of ε and ζ, at large frequencies, is clearly reflected in Figures 11 and 12, respectively. Similar to what we noted in section III.A (see Figure 3) in reference to nonspherical Fermi surface, Figure 11 here reveals that there is a critical frequency (ω = 1.567 eV in the figure), in the relaxation regime, at which Re ε changes its sign, which is consistent with the analysis based on Drude’s model of frequency dispersion60 as well as the experimentally derived59 Re ε(ω) as shown in Figure 3. Here also, we find all response functions displaying a characteristic transition (a peak at 1.78 eV in the Re σ∥ curve, for example) near the plasma edge of the metal. We now examine the limit of vanishingly small wavevector k, that is, the situation when the material may be considered, in practical terms, as spatially uniform. Expanding the logarithm in eqs 57 and 58, we obtain

(59) (60) (61)

where ε0 is the electric permittivity in a vacuum (the numerical value is 8.854187 × 10−12 F/m), h is the Planck constant, and e/m is the charge/mass of the electron. The value of C in eq 59 is equal to 6.13407804 × 106 v30/(ω + i/τ) S m−1, if we measure the Fermi velocity v0, the collision time τ, and the frequency in the units of 106 m s−1, 10−14 s−1, and 1014 s−1, respectively. In eq 59, ωp stands for the plasma frequency of the metal, wherein the free electron gas expression for the electron concentration in terms of the Fermi velocity has been used. Expressions 57−61 are the central semiclassical results, originally due to Lindhard.46,47 To obtain the electric permittivity and the (z) inverse magnetic permeability in terms of σ(z) ∥ and σ⊥ , we here adopt a direct approach and proceed as follows. We first decompose eq 4 in the directions that are parallel and perpendicular to k,⃗ apply the constitutive relations that are valid for the cubic symmetry (see section II.A), and then use eq 5 to eliminate B⃗ , to finally obtain the following results (after subtracting for the background current as discussed in section II) for the longitudinal and transverse currents. 7027



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Figure 11. The electric permittivity ε(ω) at a fixed wavevector (|k|⃗ = 3.48 × 108 m−1) for the model metallic system (see text) in the unit of 10−7 F/m. The solid (red) and dotted (blue) lines correspond, respectively, to the real and imaginary parts of the permittivity. The real part crosses zero at the frequency ω = 1.567 eV. The vertical line at 2 eV refers to the location of the plasma edge in the metal.

Figure 9. The longitudinal conductivity σ∥(ω) at a fixed wavevector (| k|⃗ = 3.48 × 108 m−1) for the model metallic system (see text) in the unit of 107 S/m. The solid (red) and dotted (blue) lines correspond, respectively, to the real and imaginary parts of the conductivity. The vertical line at 2 eV refers to the location of the plasma edge in the metal.

Figure 12. The inverse magnetic permeability ζ(ω) at a fixed wavevector (|k|⃗ = 3.48 × 108 m−1) for the model metallic system (see text) in the unit of 105 m/H. The solid (red) and dotted (blue) lines correspond, respectively, to the real and imaginary parts of the permeability. For display here, a value of 7.75 × 105 m/H has been subtracted from the real part of the permeability. The vertical line at 2 eV refers to the location of the plasma edge in the metal.

Figure 10. The transverse conductivity σ⊥(ω) at a fixed wavevector (| k|⃗ = 3.48 × 108 m−1) for the model metallic system (see text) in the unit of 107 S/m. The solid (red) and dotted (blue) lines correspond, respectively, to the real and imaginary parts of the conductivity. The vertical line at 2 eV refers to the location of the plasma edge in the metal.

⎡ σ = iC ⎢1 + ⎢⎣ ⎡ σ⊥ = iC ⎢1 + ⎢⎣

∞

∑ n=1 ∞

∑ n=1

⎛ v0k ⎞2n⎤ 3 ⎜ ⎟ ⎥ (2n + 3) ⎝ ω + i/τ ⎠ ⎥⎦

(67) 2n ⎤

⎛ v0k ⎞ 3 ⎜ ⎟ ⎥ (2n + 1)(2n + 3) ⎝ ω + i/τ ⎠ ⎦⎥

(71)

ζ(ω) = ζ0

(72)

and σ eff (ω) = σDrude

(73)

(68)

In the limit k → 0, eqs 67 and 68 become degenerate and thus yield the following well-known Drude’s conductivity (using eq 59). σ0 σ = σ⊥ = iC = = σDrude (69) 1 − i ωτ where σ0 = (τε0)ωp2

⎛ ⎞ ωp2 ⎟ ε(ω) = ε0⎜⎜1 + ω(ω + i/τ ) ⎟⎠ ⎝

It is evident from the above deductions that the conducting matter, if spatially nondispersive (that is, k → 0), is necessarily nonmagnetic, for the inverse magnetic permeability, as eq 72 shows, assumes a value in the vacuum. The reason as to why this is so can be traced to the fact that the σ∥(k,ω)|k→0 and σ⊥(k,ω)|k→0 become identical. In contrast, as we have pointed out earlier (see the discussion following eq 54), the matter with nonspherical Fermi surface, in the limit k → 0, becomes nonmagnetic because it cannot sustain a transverse current

(70)

Consequently, in the limit k⃗ → 0, eqs 64−66 transform to the following forms: 7028



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anymore. This suggests that, the measurement of transverse current, may be used as a distinctive measure to determine if the metal possesses a spherical or a nonspherical Fermi surface. Finally, the complete set of response quantities in eqs 71−73 is the one that unambiguously characterizes what may be called as Drude−Sommerfeld’s metal, we stress. Notably, the importance of eq 71 has been recognized in a recent work.61 We shall now examine the speed of light inside the matter when k tends to zero. We can obtain an estimate by using eq ⃗ k→0ε0/ζ0 = c 25, which reveals using eqs 71−73 that vmatter(k,ω)| (the velocity of light in a vacuum). This is a significant result for it shows that Drude−Sommerfeld’s metal can not scatter light and thence change its speed. In a sense, then, the spatial inhomogeneities in the conductivity of the medium, phenomenologically speaking, act as the driving force in the light− matter interaction, and therefore a realistic mathematical expression for electromagnetic response tensors must necessarily be spatially dispersive, we stress. Earlier studies on this subject have used a hydrodynamic15,17,18 extension of the Drude model, which, although useful, provides only an ad hoc supplement to the spatial dispersion problem. The present treatment offers, we believe, a pedagogically complete solution to the problem.

Figure 13. Frequency-dependent reflectivity R(ω) at a fixed wavevector (|k|⃗ = 3.48 × 108 m−1) for the model metallic system (see text). The solid (red) and dotted (blue) lines correspond to the nonspherical and the spherical Fermi surfaces, respectively. The vertical line at 2 eV refers to the location of the plasma edge in the metal.

IV. REFLECTIVITY AND SKIN DEPTH The reflectivity R(ω) of a material system is defined as the ratio of the time averaged energy flux reflected from the surface to the incident flux, and this quantity, for the situation of normal incidence, is given as follows.62 R(ω) =

(1 − n)2 + k 2 (1 + n)2 + k 2

(74)

where n and k define the effective complex index of refraction as ñ = (n + ik). For metals, ñ can be written as follows. n(̃ ω) = c

ε + iσ (eff)/ω ζ

(75) Figure 14. Frequency-dependent skin depth δ(ω) at a fixed wavevector (|k|⃗ = 3.48 × 108 m−1) for the model metallic system (see text) in the unit of 10−6 m. The solid (red) and dotted (blue) lines correspond to the nonspherical and the spherical Fermi surfaces, respectively. The vertical line at 2 eV refers to the location of the plasma edge in the metal.

where c is the velocity of light. The skin depth δ, on the other hand, measures a characteristic length scale for the attenuation of the electromagnetic waves as the distance over which the value of the field reduces by the factor 1/e (e = 2.718). The expression for δ is given as follows.62 c c = δ= ωk ω Im n(̃ ω) (76)

light inside the Drude−Sommerfeld metal we made earlier (vide supra). That is, the Drude−Sommerfeld metal is completely transparent to the light, and this is physically unrealistic. We parenthetically note that the hydrodynamic extension of the Drude−Sommerfeld model,17 which incorporates the spatial dispersion (albeit in a tentative fashion), will not suffer from such artifacts. The present investigation also reveals, as is evident from Figure 13, that the plasma edge in the metal, instead of being a sharply defined point, is a fuzzy extension connecting the deep relaxation and transparent regimes, insofar as the reflectivity is concerned. We also find in Figure 13 that the reflectivity computed using the nonspherical model for the Fermi surface falls off rapidly as compared to that obtained using a spherical model. Metals are certainly expected to be fully transparent at high frequencies, but the final arbitration on the rate at which metals would transform from being a perfect mirror (R = 1.0) to a fully transparent object (R = 0.0) can be fully reached only

where ñ(ω) is as given in eq 75. To obtain the numerical values for ñ(ω), we have used the information from Figures 1−4 and Figures 9−12 for the nonspherical and spherical Fermi surfaces, respectively. The expression for σeff in eq 75 is given in terms of longitudinal and transverse conductivities as follows: σ(eff) = 2 (z) 2 1/2 ((σ(z) ∥ ) + (σ⊥ ) ) . Reflectivity and skin depth computed using the same set of parameters for the model metallic system as discussed in section III.A are shown in Figures 13 and 14, respectively. We find from Figure 13 that the frequencydependent reflectivity falls off smoothly as we traverse from the relaxation regime to the transparent regime. This differs significantly from the prediction of the Drude model, as defined by the set of response quantities in eqs 71−73. In fact, eq 75 along with eqs 71−73 yields n̂(ω) = c(ε0/ζ0)1/2 = 1, which means that the reflectivity as given in eq 74 is simply zero, and this is consistent with the observation on the speed of 7029



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matter.63 Extension of the present work for the determination of nonlinear responses tensor will also be desirable. Finally, the success of the present theory of electromagnetic responses critically hinges upon our ability to make an a priori and accurate estimate of the Fermi velocity v0⃗ and the collision time τ for the conducting materials of interest. At a purely phenomenological level, one may employ experimental protocols (such as the angle-resolved photoemission spectroscopy and the resistivity measurements) to arrive at a realistic estimate of v0⃗ and τ. A first principle quantum mechanical determination of the Fermi velocity v0⃗ may be accomplished, for example, by differentiating the ab initio dispersion curve, ⃗ obtained from a detailed electronic structure calculations ε(k), on metals, at the Fermi surface. A theoretical estimate of the collision time τ may be obtained by computing the weighted scattering cross section for the electron−impurity collisions in the metal. In conclusion, we have presented a complete set of frequency- and wave vector-dependent linear response tensors, which provides an unambiguous setting to address various problems in metal optics.

with definite experimental information. On the other hand, the skin depth, as Figure 14 reveals, shows a rather sharp transition at the location of plasma edge in the metal, which seems to be independent of the model for the Fermi surface. We hence believe that the experimental determination of frequencydependent skin depth can indeed locate the existence of plasma edge in the metal with full precision; this will strengthen the predictive power of the present theoretical formulation of the electromagnetic responses of the metal. We also find from Figure 14 that the magnitude of the skin depth computed using the nonspherical model for the Fermi surface is larger than that obtained with the spherical model. Here also, only a definite experiment can accomplish the final arbitration.

V. CONCLUDING REMARKS We now briefly describe how the present formalism of response tensor may be used in the studies of optical properties of metals. In fact, Maxwell’s equations, in the context of electromagnetic wave propagation in matter, may be cast in the form of a vector Helmholtz equation, a Cartesian component of which (for cubic symmetry, as an example) may be written as:3,50 [∇2 + Veff]Al = 0. Here, Veff essentially encodes the material property, which, in the context of metals, may be expressed in simple situations as follows.50 eff

Veff ( r ⃗ , ω) = ωμ( r ⃗)[ωε( r ⃗) + iσ ( r ⃗)]

■

AUTHOR INFORMATION

Corresponding Author

*Phone: +91 44 22574234. Fax: 91 44 22574202. E-mail: [email protected].

(77)

Notes

where the electric permittivity function ε(r,⃗ ω) is the Fourier ⃗ transform of ε(k,ω) and so forth. The expressions for various response tensors obtained in the present work fully determine Veff(r,⃗ ω) in eq 77 and hence pave the way for a realistic optical studies of conducting materials. We will report such works in the future. We now comment on possible further extensions of the present work on response theory. First, we note that the present study has considered the metal to be a degenerate Fermi system, described by a uniform equilibrium distribution function as shown in eq 28. In a more realistic model, fermionic charge carriers in metal would be distributed in various energy bands. As the information on band structures has not been included here, the response tensors (nonlocal and frequency dependent) obtained in the present work will not be able to differentiate the contribution of interband transitions (a local material property) from that due to the intraband transitions (a nonlocal material property). As the intraband transitions are mostly responsible for both the plasmonic optical responses of metals and the nonlocal effects,15 the response tensors (nonlocal and frequency dependent) obtained here would be appropriate for such optical studies. An extension of the present work that incorporates the information on energy band structure of the metal into the equilibrium distribution function as shown in eq 28 would reveal the explicit mechanism of interband transitions and their impact on the electromagnetic responses of normal metals. It would also be of interest to implement an improved treatment of the collision integral in the Boltzmann transport equation, while ensuring a closed form expression for the conductivity tensor as in section III. The present work has been concerned with the linear response tensors, and therefore the external force, in eq 26, driving electrons in metal has been restricted to be electrical in nature. As it is well-known, introduction of magnetic term (v⃗ × B⃗ ) in the external force leads to nonlinear optical responses in

The authors declare no competing financial interest.

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REFERENCES

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