Wavevector-dependent optical properties from wavevector ...

arXiv:1708.06330v1 [physics.optics] 21 Aug 2017

Wavevector-dependent optical properties from wavevector-independent conductivity tensor R. Starkea , G. A. H. Schoberb , R. Wirnataa,∗, J. Kortusa a

Institute for Theoretical Physics, TU Bergakademie Freiberg, Leipziger Straße 23, 09599 Freiberg, Germany b Institute for Theoretical Solid State Physics, RWTH Aachen University, Otto-Blumenthal-Straße, 52074 Aachen

Abstract We discuss the standard ab initio calculation of the refractive index by means of the scalar dielectric function and show its inherent limitations. To overcome these, we start from the general, microscopic wave equation in materials in terms of the frequency- and wavevector-dependent dielectric tensor, and we investigate under which conditions the standard treatment can be justified. We then provide a more general method of calculating the frequencyand direction-dependent refractive indices by means of a (2 × 2) complexvalued “optical tensor”, which can be calculated from a purely frequencydependent conductivity tensor. Finally, we illustrate the meaning of this optical tensor for the prediction of optical material properties such as birefringence and optical activity. Keywords: electronic structure, electrodynamics in media, refractive index

Corresponding author. Email addresses: [email protected] (R. Starke), [email protected] (G. A. H. Schober), [email protected] (R. Wirnata), [email protected] (J. Kortus) ∗

Preprint submitted to

August 22, 2017

Contents 1 Introduction

2

2 Response functions and wavevector dependence

4

3 Wave equation and conductivity tensor

5

4 Optical properties from conductivity tensor

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1. Introduction The conventional treatment of optical properties in the ab initio community is based on the standard relation between the macroscopic dielectric function and the refractive index given by (see Refs. [1, Eq. (8.33)], [2, Eq. (2.17)], [3, Eqs. (18.26)], [4, Eq. (6.11)], [5, p. 534]) n2 (ω) = εr (ω) ,

(1.1)

where this macroscopic dielectric function is defined by the equation (see e.g. Refs. [1, Eqs. (8.25)], [2, Eq. (2.18)], [3, Eq. (18.22)], [6, p. 3]) εr (ω) = lim εr (k, ω) . |k|→0

(1.2)

Importantly, the dielectric function used in these equations is typically calculated from the density response function, and hence it corresponds to the longitudinal part of the dielectric tensor (see Refs. [7], [8, § 2.6.4], [9, Vol. 1, § 4]). This standard treatment has proven very valuable as it delivers sensible results for a huge variety of materials [10–21]. Recently, however, a meritorious article by D. Sangalli et al., Ref. [22], has drawn attention to the fact that in principle, optical properties should be calculated from the wavevector-dependent current response tensor. In accordance with this, let us shortly summarize the main conceptual problems of the standard treatment by means of the wavevector-independent dielectric function: 1. As a matter of principle, optical properties correspond to transverse electromagnetic waves in a material, and hence they should not be deduced from a longitudinal response function (at least not in a na¨ıve way). 2

2. The standard relation for the refractive index, Eq. (1.1), is only valid in the so-called optical limit, k → 0, whereas light waves definitely have a non-vanishing wavevector, i.e., k 6= 0. 3. In particular, for anisotropic media, the refractive index should at least depend on the direction of the wavevector k. 4. Moreover, for birefringent or optically active materials, there are polarization-dependent refractive indices. By contrast, Eq. (1.1) yields at best one refractive index, whose polarization cannot even be defined for vanishing wavevectors k → 0. These considerations make it clear that the calculation of optical material properties in general requires the wavevector to be taken into account, and this in turn requires a treatment based on the current response tensor (or equivalently the dielectric tensor). In fact, from the dielectric tensor— which naturally contains much more information than the scalar dielectric function—the density response function can be reconstructed by means of Universal Response Relations [23], while the converse is not true (see Refs. [1, 24–26]). In the limit k → 0, however, such response relations cannot be evaluated due to their singular behavior (see e.g. the relation (2.5) between the dielectric function and the density response function). The authors of Ref. [22] have therefore based their treatment on a full ab initio calculation of the wavevector-dependent response functions. While this is certainly the right approach in the most general case, the downside is that such wavevector-dependent calculations are in general extremely demanding, in particular if they are supposed to yield dispersion relations. In this article, we resume this problem in order to show that—at least in principle—it is possible to obtain wavevector-dependent optical properties without calculating wavevector-dependent response functions numerically, provided that the treatment is based on the conductivity tensor. Fittingly, a recent numerical study [27] has provided evidence that it may be precisely this quantity which turns out to be wavevector-independent, at least for optical wavelengths. Remarkably, however, this conclusion cannot simultaneously apply to other response functions such as the dielectric tensor (see the discussion below). This article is organized as follows: In § 2, we assemble some general wavevector-dependent relations between various response functions. In § 3, we then discuss the microscopic wave equation in materials in terms of the 3

conductivity tensor, and we investigate the conditions under which the standard treatment of optical material properties can be justified. This further allows us in § 4 to define the wavevector-dependent optical tensor in terms of the wavevector-independent conductivity tensor by an explicit and analytical formula. Finally, we shortly explain how optical properties such as birefringence and optical activity can be deduced from this optical tensor. 2. Response functions and wavevector dependence Fundamentally, the current response tensor χ is related to the conductivity tensor σ by means of a Universal Response Relation [23, § 6], which reads (see Refs. [8, Eqs. (2.177) and (2.198)], [24, Eq. (3.185)], [28], and for a gauge-independent derivation see Ref. [29, § 3.2.3]): ↔

↔

χ(x, x′; ω) = iω σ(x, x′; ω) .

(2.1)

The current response tensor on its side is the spatial part of the fundamental response tensor [25, § 7.4], χµν (x, x′ ) =

µ δjind (x) , ν δAext (x′ )

(2.2)

which—in the homogeneous limit—is of the following general form (see e.g. Refs. [1, 23, 25, 26]):   2 ↔ ↔ − ωc 2 kT χ(k, ω) k ωc kT χ(k, ω) . χµν (k, ω) =  (2.3) ↔ c ↔ − ω χ(k, ω) k χ(k, ω)

In these formulae, j µ = (cρ, A)T denotes the (induced) electromagnetic fourcurrent, and Aν = (ϕ/c, A)T the (external) four-potential. In particular, we read off that the density response function χ can be calculated from the current response tensor by means of [24, Eq. (3.175)] ↔

1 kT χ(k, ω)k δρind (k, ω) = 2 χ00 (k, ω) = − . χ(k, ω) ≡ δϕext (k, ω) c ω2

(2.4)

We remark that these response relations, i.e., Eqs. (2.1)–(2.4), analogously hold for the respective proper response functions, which relate the induced quantities to the total (external plus induced) quantities (see Ref. [30, § 2.3]). 4

In particular, the proper density response function is related to the dielectric function by [8, Eq. (2.172)] εr (k, ω) = 1 − v(k)e χ(k, ω) ,

(2.5)

where v(k) = 1/(ε0 |k|2 ) denotes the Coulomb interaction kernel in Fourier space. The reason for this is that the dielectric function actually coincides with the longitudinal part of the dielectric tensor (see [8, § 2.6.4]) given by ↔

kT ε r (k, ω)k εr (k, ω) ≡ εr,L(k, ω) = . |k|2

(2.6)

Generally, these considerations show that the microscopic current response tensor—or, equivalently, the microscopic conductivity tensor—already contains the complete information about all linear, electromagnetic response properties [23, § 5–6]. In particular, it is possible to reconstruct from the microscopic conductivity tensor the response with respect to strictly longitudinal external perturbations. Hereby, however, one has to stress that the corresponding response relations require knowledge of the response functions at finite wavevectors |k| > 0 (see Ref. [22]). In other words, these relations can in general not be evaluated na¨ıvely in the limit k → 0, but require precise knowledge of the response functions in the vicinity of the origin. Finally, we emphasize that the above response relations hold only for homogeneous materials, while they become more involved in the general (i.e., inhomogeneous) case [23, 31]. 3. Wave equation and conductivity tensor In Fourier space, the most general linear, electromagnetic wave equation for homogeneous media reads (see Refs. [30, 32–35]) ↔ ε r (k, ω)E(k, ω)

= 0,

(3.1)

↔

where ε r denotes the dielectric tensor. In the case of an isotropic medium, this general wave equation decouples into the equations εr,L(k, ω)EL(k, ω) = 0 ,

(3.2)

εr,T (k, ω)ET (k, ω) = 0 ,

(3.3)

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which are formulated in terms of the longitudinal and transverse dielectric functions. The resulting conditions for the existence of nontrivial solutions, εr,L(k, ωkL) = 0 ,

(3.4)

εr,T (k, ωkT ) = 0 ,

(3.5)

determine the respective dispersion relations, ωkL and ωkT , of the electromagnetic proper oscillations of the material [32, Eq. (2.34)]. In the longitudinal case described by Eq. (3.4), the resulting waves are conventionally called plasmons (see e.g. [24, Eq. (5.49)], [36, p. 262], [37, Eq. (4.92)]). The corresponding transverse proper oscillations determined by Eq. (3.5)—whose existence can already be deduced per analogiam—obviously describe the propagation of light in the medium. In the most general (i.e., not necessarily isotropic) case, both equations combine into the unified wave equation (3.1). It is now instructive to reformulate this general wave equation (3.1) in terms of the microscopic proper conductivity tensor [30, § 2.3]. As a matter of principle [30, § 2.5], this quantity is related to the dielectric tensor via [32, Eqs. (2.24)–(2.25)] ↔ ε r (k, ω)

= 1 − E (k, ω) ↔

↔

1 ↔ σ e (k, ω) , iω ε0

(3.6)

where the electric solution generator [30, § 2.4] is given in terms of the wellknown longitudinal and transverse projection operators [23, Eqs. (2.20) and (2.21)] by the concise formula 2

E (k, ω) = P L(k) + ω2 −ωc2 |k|2 P T(k) . ↔

↔

↔

(3.7)

We now restrict attention to transverse waves which per definitionem fulfill ↔

P L (k)E(k, ω) = 0 ,

↔

P T (k)E(k, ω) = E(k, ω) .

(3.8)

Acting with the transverse projection operator on Eq. (3.1) and using Eqs. (3.6)– (3.7), we obtain the wave equation ↔     ↔ ω2 ↔ σ e TT (k, ω) 2 − 2 1− + |k| 1 E(k, ω) = 0 , c iω ε0

6

(3.9)

which contains the transverse part (see Refs. [33, § 2.1] and [38, Sct. 2]) of the proper conductivity, ↔

↔

def ↔

↔

σ e TT (k, ω) = P T (k) σ e (k, ω) P T (k) .

(3.10)

Further defining the effective dielectric tensor as ↔ ↔ def ↔ ε r,eff (k, ω) = 1

−

σ e (k, ω) , iω ε0

we can rewrite Eq. (3.9) as   ↔ ω2 ↔ 2 P T (k) − 2 ε r,eff (k, ω) + |k| E(k, ω) = 0 . c

(3.11)

(3.12)

Apart from the transverse projection, this equation formally agrees with the standard wave equation used in ab initio materials physics (see e.g. Refs. [8, Eq. (2.203)] and [39, Eq. (2.2.9)]). We remark, however, that the effective dielectric tensor appearing in Eq. (3.12) is actually not identical, but only approximately equal to the real dielectric tensor (compare Eqs. (3.6) and (3.11)). Put differently, the phenomenological wave equation (3.12) combined with the approximate relation between the dielectric and the conductivity tensor, Eq. (3.11), is equivalent to the exact wave equation (3.1) in the form of Eq. (3.9). This explains, in particular, why the ab initio treatment of the refractive index is approximately correct although it does not explicitly start from the fundamental wave equation (3.1) (cf. Ref. [32, Eqs. (2.29)–(2.30) and (2.33)–(2.34)]). Furthermore, we stress that the equivalence of the phenomenological wave equation (3.12) and the fundamental wave equation (3.1) holds true only for transverse electric fields (which are typically encountered in optical phenomena). For the corresponding longitudinal oscillations, these equations are not equivalent, and this explains why in ab initio physics one usually works with two entirely different wave equations, both formulated in terms of the dielectric function: the above phenomenological wave equation (3.12) for optical, i.e. transverse oscillations, and the plasmon equation (3.2) for longitudinal oscillations. In actual fact, by using the exact relation (3.6) between the dielectric tensor and the conductivity tensor, both wave equations turn out to be of the same type; and in case that longitudinal and transverse oscillations do not decouple, they combine into the fundamental wave equation (3.1). 7

By contrast, the wave equation corresponding to the standard equation (1.1) for the refractive index reads   ω2 2 − 2 εr,L(ω) + |k| E(k, ω) = 0 . (3.13) c Surprisingly, this equation can also be justified from the fundamental wave equation in the isotropic case, i.e., from Eqs. (3.2)–(3.3): In the optical limit k → 0, it is plausible to assume that the longitudinal and transverse conductivities coincide in the sense of σ eL (k, ω) = σ eT (k, ω) .

(3.14)

With this relation, one shows directly from Eq. (3.6) that the longitudinal and transverse dielectric functions are related as follows [30, § 4.4]:   ω2 ω2 2 − 2 + |k| εr,T(k, ω) = − 2 εr,L(k, ω) + |k|2 . (3.15) c c Together with the fundamental wave equation (3.3), this implies     ω2 ω2 2 2 − 2 εr,L(k, ω) + |k| E(k, ω) = − 2 + |k| εr,T (k, ω)E(k, ω) = 0 . c c (3.16) Finally, by neglecting the k dependence of εr,L we arrive at the assertion, Eq. (3.13). On the other hand, as the condition (3.14) will not always be true, and as the k dependence of εr,L can not always be neglected, this corroborates again our initial statement (see § 1) that the deduction of optical properties should in the most general case not be based on the standard relation (1.1). 4. Optical properties from conductivity tensor The above considerations have shown already that in principle, optical properties have to be deduced from the general wave equation restricted to the transverse subspace, i.e., from Eq. (3.9), or from its standard form given by Eq. (3.12), which can also be written as ↔

↔

P T (k) ε r,eff (k, ω)E(k, ω) = 8

c2 |k|2 E(k, ω) . ω2

(4.1)

However, the solution of this equation requires the calculation of the full, i.e., frequency- and wavevector-dependent conductivity tensor, which is computationally very demanding. Moreover, even if the full conductivity tensor or the corresponding dielectric tensor is known, the dispersion relation ω = ωk has to be deduced from the implicit equation (4.1), where the frequency appears not only explicitly on the right-hand side but also implicitly as an argument of the dielectric tensor. Consequently, also the refractive index, which is defined by [40, Eq. (11.12)], c|k| nk = , (4.2) ωk is only given by an implicit equation (see Ref. [30, § 4.2]). We remark that instead of regarding Eq. (4.1) as an implicit equation for determining the frequency (or the refractive index) as a function of the wavevector, one may instead also fix the frequency ω and the direction of ˆ := k/|k|, and regard Eq. (4.1) as an implicit equation the wavevector, k for determining the modulus of the wavevector |k|. Correspondingly, one can also regard the refractive index as a function of the frequency and the ˆ ω). Even in this case, however, the direction of the wavevector, n = n(k, refractive index is given only implicitly by Eq. (4.1), since the modulus |k| does not only appear explicitly on the right-hand side of Eq. (4.1) but also ˆ |k|, ω). implicitly as an argument of the dielectric tensor, εr = εr (k, A way to overcome these difficulties is now the following: although the response relations are in general wavevector dependent, it is not contradictory to assume that one particular response function is actually wavevector independent (at least approximately), whereby it has to be stressed that this assumption cannot be upheld simultaneously for all response functions (see § 2, or the most general Universal Response Relations in Ref. [23, § 6]). Furthermore, the standard approach suggests that the assumption of wavevector independence actually applies to the proper conductivity tensor, or equivalently (see Eq. (2.1)), to the proper current response tensor. Note, however, that in this case the wavevector indepence of the conductivity tensor implies that the density response function and the dielectric tensor are definitely wavevector-dependent (see Eqs. (2.4) and (3.6)). We therefore assume in the following that for optical wavelengths, the transverse part of the proper conductivity tensor is wavevector independent

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in the sense that (cf. Eq. (3.10)) ↔ σ TT (k, ω)

↔

↔

↔

= P T (k) σ e (ω) P T (k) ,

(4.3)

where σ e(ω) ≡ σ e(k → 0, ω). Since the transverse projection operator on its side depends only on the direction (and not on the modulus) of the wavevector, ↔ ↔ kkT ↔ ˆ P T (k) = 1 − = P T (k) , (4.4) |k|2

the condition (4.3) implies that the effective dielectric tensor defined by Eq. (3.11) depends only on the frequency and the direction of the wavevecˆ ω). This in turn greatly simplifies the solution of the tor, i.e., εr,eff = εr,eff (k, wave equation (4.1), which now becomes an explicit equation for determining ˆ and ω. Consequently, the modulus of the wavevector |k| as a function of k the frequency- and direction-dependent refractive indices are now given by the explicit equation ↔

↔ ˆ ω)Eλ(k, ˆ ω) = n2 (k, ˆ ω)Eλ(k, ˆ ω) , P T (k) ε r,eff (k, λ

(4.5)

which in fact is just an eigenvalue problem of the effective dielectric tensor. Correspondingly, this equation can in general have several solutions which we enumerate by the index λ = 1, 2. Thus, the squared frequency- and direction-dependent refractive indices coincide with the eigenvalues of the effective dielectric tensor. We recall that by the condition (3.8), the above eigenvalue problem is actually restricted to the two-dimensional transverse subspace. Therefore, it is convenient to reformulate it in terms of the (2 × 2) optical tensor given by ! ↔ ↔ σ e (ω) ˆ ω) def (εr,opt)ij (k, = eT ekj , (4.6) 1− ki iω ε0

where eki (i = 1, 2) are two transverse (with respect to k) and orthonormal vectors in R3 , hence eki · k = 0 and eki · ekj = δij . Note that also the optical tensor depends only on the direction of k but not on its magnitude. With this, the eigenvalue problem (4.5) with the restriction to transverse fields is equivalent to the eigenvalue problem for the optical tensor, 2 X

ˆ ω) v j (k, ˆ ω) = n2 (k, ˆ ω) v i (k, ˆ ω) . (εr,opt )ij (k, λ λ λ

j=1

10

(4.7)

The eigenvalues (at most two for a given direction, here labeled by λ = 1, 2) are precisely the squared frequency- and direction-dependent refractive indices, while the eigenvectors determine via ˆ ω) = v 1 (k, ˆ ω) ek1 + v 2 (k, ˆ ω) ek2 Eλ (k, λ λ

(4.8)

the corresponding polarization vectors. Finally, the precise interpretation of the optical tensor in an ab initio framework, as well as the shear possibility of decoupling longitudinal and transverse oscillations, together require an investigation in its own right, to which we will dedicate a future work. It seems plausible, though, that one can distinguish at least the following four cases: (i) For the standard case of an isotropic material, the condition (4.3) implies that the transverse conductivity is wavevector independent, σ eT (k, ω) = σ eT (ω) ,

and thus the optical tensor is of the form   σ eT (ω) ˆ (εr,opt )ij (k, ω) = δij 1 − . iω ε0

(4.9)

(4.10)

In this case, any two orthonormal, transverse vectors ek1 and ek2 are eigenvectors of the effective dielectric tensor and can hence be regarded as polarization vectors. They share the same refractive index, which in this case does not even depend on the direction.

(ii) For a birefringent material, there are two, not necessarily orthogonal, real eigenvectors (v11 , v12 )T and (v21 , v22 )T of the optical tensor, which correspond to two real polarization vectors E1 and E2 . The corresponding eigenvalues, i.e., the refractive indices n1 and n2 , are different for these two polarizations. (iii) In an optically active material, we have two circular √ polarization vectors, i.e., eigenvectors of the form E± = (e1 ± ie2 )/ 2, and the refractive indices n± depend on the circular polarization. (iv) Since the optical tensor is in general not hermitean, only the existence of one eigenvector is guaranteed (by the fundamental theorem of algebra). If only one eigenvector exists, the material acts as a polarizer, i.e., it transmits light waves only with a particular polarization. 11

We emphasize again that the optical tensor can be calculated from the wavevector-independent conductivity tensor, but still allows one to deduce these wavevector-dependent optical material properties from first principles. Conclusion After shortly criticizing the standard calculation of the refractive index from the scalar, wavevector-independent dielectric function, we have shown how, on a fundamental level, optical material properties can be deduced from the general, microscopic wave equation in materials, Eq. (3.1). Subsequently, we have proposed an approximation scheme by which wavevector-dependent optical material properties, such as frequency- and direction-dependent refractive indices, can be calculated from the wavevector-independent conductivity tensor. The latter is a standard target quantity of any modern ab initio materials simulation code [41–44]. As shown in Ref. [45], the output of such DFT-based computer codes can in fact be directly identified with the proper conductivity tensor. In summary, the procedure to calculate optical material properties from first principles is the following: ↔

1. Calculate the frequency-dependent (proper) conductivity tensor σ e (ω).

ˆ ω) by means of Eq. (4.6). 2. From this, deduce the optical tensor εr,opt (k, ˆ of the wavevector.) (This quantity also depends on the direction k 3. Diagonalize this (2 × 2) tensor. The (non-zero) eigenvalues give the frequency- and direction-dependent refractive indices, and the corresponding eigenvectors give the polarizations (see Eqs. (4.7) and (4.8)).

Finally, the material can be classified by means of its optical properties using the criteria (i)–(iv) of § 4. Importantly, the above procedure requires as a numerical input only the frequency-dependent conductivity tensor, and from this all relevant optical properties can be obtained by simply diagonalizing a (2 × 2) matrix (for each frequency and direction). Thus, this work paves the way for an efficient ab initio calculation of optical material properties, with potential applications lying in the ab initio screening and functional design of new materials with unconventional optical properties.

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