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Microscopicmacroscopic connection Valérie Véniard Microscopic-macroscopic connection Valérie Véniard Laboratoire des Solides Irradiés Ecole Polytechnique, Palaiseau - France European Theoretical Spectroscopy Facility (ETSF) Microscopicmacroscopic connection Outline Valérie Véniard Intro Macro 1 Introduction: Which quantities do we need? Definition Example Procedure Infinite crystal Calculation of ˆM cubic symmetries Non-cubic symmetries 2 Macroscopic average Definition Example Procedure 3 Infinite crystal Calculation of ˆM cubic symmetries Non-cubic symmetries Microscopicmacroscopic connection Valérie Véniard Intro Macro Definition Example Procedure Infinite crystal Calculation of ˆM cubic symmetries Non-cubic symmetries Introduction Microscopicmacroscopic connection Valérie Véniard Introduction: Which quantities do we need? Intro Macro Definition Example Procedure Absorption coefficient √ Defining the complex refractive index as n = = ν + iκ, the electric field inside a medium is a damped wave: Infinite crystal Calculation of ˆM cubic symmetries Non-cubic symmetries E (x, t) = E0 e iω c νx ω e − c κx e −iωt ν=refraction index κ =extinction coefficient They are related to the dielectric constant The absorption coefficient α is the inverse distance where the intensity of the field is reduced by 1/e α= ω2 νc Microscopicmacroscopic connection Valérie Véniard Introduction: Which quantities do we measure? Intro Macro Definition Example Procedure Reflectivity Normal incidence reflectivity Infinite crystal Calculation of ˆM cubic symmetries Non-cubic symmetries z Reflected beam R=| Transmitted beam x Incident beam R= ET 2 | <1 Ei (1 − ν)2 + κ2 (1 + ν)2 + κ2 The knowledge of the optical constant implies the knowledge of the reflectivity, which can be compared with the experiment. Microscopicmacroscopic connection Introduction: Which quantities do we need? Valérie Véniard Intro Macro Definition Example Procedure Infinite crystal Calculation of ˆM cubic symmetries Non-cubic symmetries Energy loss by a fast charged particle The energy lost by the electron in unit time is Z dW e2 dk ω =− 2 Im dt π k2 (k, ω) Electron Loss Spectroscopy o n Energy −Im 1 (k,ω) is called the loss function Microscopicmacroscopic connection Valérie Véniard Intro Macro Definition Example Procedure Infinite crystal Calculation of ˆM cubic symmetries Non-cubic symmetries Macroscopic average Microscopicmacroscopic connection Maxwell’s equations Valérie Véniard Intro Macro Definition Example Procedure Infinite crystal Calculation of ˆM cubic symmetries Non-cubic symmetries Maxwell’s equations can be written either in the medium or in vaccum Etot (r, t), Btot (r, t), jtot (r, t),ρtot (r, t) with ρtot = ρext + ρind , and jtot = jext + jind Eext (r, t), Bext (r, t), jext (r, t),ρext (r, t) It is often more convenient to use D = Etot + 4πP instead of Etot , as D is very close to the external field (∇.D = ∇.Eext ) ρ and j are not independent. Continuity equation : ∇j + ∂ρ ∂t = 0 Microscopicmacroscopic connection Valérie Véniard Intro Linear response For a sufficiently small perturbation, the response of a system can be expanded into a taylor series, with respect to the perturbation. Macro Definition Example Procedure Infinite crystal Calculation of ˆM cubic symmetries Non-cubic symmetries The first order (linear) response is proportional to the perturbation. jind (r, ω) = − e 1 < ρ̂ > Apert (r, ω) − mc c Z dr0 χjj (r, r0 , ω)Apert (r0 , ω) Z + dr0 χjρ (r, r0 , ω)Vpert (r0 , ω) Z Z 1 ρind (r, ω) = − dr0 χρj (r, r0 , ω)Apert (r0 , ω) + dr0 χρρ (r, r0 , ω)Vpert (r0 , ω) c For simplicity, we will write e 1 < ρ̂ > Apert − χjj Apert + χjρ Vpert mc c 1 = − χρj Apert + χρρ Vpert c jind = − ρind Microscopicmacroscopic connection Linear response Valérie Véniard Intro Macro Definition Example Procedure The four response functions are not independent and using gauge invariance, one has iωχjρ (r, r0 , ω) = Infinite crystal Calculation of ˆM cubic symmetries Non-cubic symmetries ∂ e ∂ < ρ̂ > δ(r − r0 ) − 0 χjj (r, r0 , ω) ∂r0 mc ∂r ∂ χρj (r, r0 , ω) = −iωχρρ (r, r0 , ω) ∂r0 Only two quantities are needed, in terms of the electric field ie i < ρ̂(r) > Epert + χjj Epert mω ω i = χρj Epert ω jind = ρind Depending on the approximation, these response functions can be evaluated (IPA, RPA ...). Microscopicmacroscopic connection Valérie Véniard Linear response Microscopic dielectric tensor Intro Macro Definition Example Procedure Z D(r, ω) = dr0 ˆ(r, r0 , ω)Etot (r0 , ω) Infinite crystal Calculation of ˆM cubic symmetries Non-cubic symmetries In a perfect crystal ˆ(r + R, r0 + R, ω) = ˆ(r, r0 , ω) and Fourier components must satisfy D(q + G, ω) = X ˆ(q + G, q + G0 , ω)Etot (q + G0 , ω) G0 q lies in the first Brillouin zone and G is a reciprocal lattice vector Microscopicmacroscopic connection Microscopic spatial fluctuations Valérie Véniard Intro Macro Definition Example Procedure Infinite crystal Calculation of ˆM cubic symmetries Non-cubic symmetries • Infinite crystals →microscopic inhomogeneities (atomic scale) • Semi-infinite crystals → presence of the surface • Desorded medium →liquid • Rough surfaces Microscopicmacroscopic connection Micro and macro Valérie Véniard Intro Macro Definition Example Procedure Infinite crystal Calculation of ˆM cubic symmetries Non-cubic symmetries Macroscopic quantities in the context of condensed matter At long wavelength, external fields are slowly varying over the unit cells. 2π λ= >> V 1/3 q where V is the volume per unit cell of the crystal. Example Eext (r, t), Aext (r, t), Vext (r, t),... Typical values: • dimension of the unit cell for silicon acell ' 0.5nm • Visible radiation 400nm ≤ λ ≤ 800nm • X-ray range? Microscopicmacroscopic connection Micro and macro Valérie Véniard Intro Macro Definition Example Procedure Infinite crystal Calculation of ˆM cubic symmetries Non-cubic symmetries Microscopic quantities Total and induced fields are rapidly varying. They include the contribution from electrons in all regions of the cell. The contribution of electrons close to or far from the nuclei will be very different. ⇒ Large and irregular fluctuations over the atomic scale. Example Etot (r, t), jind (r, t), ρind (r, t),... Microscopicmacroscopic connection Macroscopic average Valérie Véniard Intro Macro Definition Example Procedure Infinite crystal Calculation of ˆM cubic symmetries Non-cubic symmetries Measurable quantities One measures quantities that vary on a macroscopic scale. In the long wavelength limit, the macroscopic neighbourhood contains many particles We have to average over distances : • large compared to the cell diameter • small compared to the wavelength of the external perturbation Microscopicmacroscopic connection Macroscopic average Valérie Véniard Intro Macro Definition Example Procedure If we assume that the system is homogeneous ˆ(r, r0 , ω) = ˆ(r − r0 , ω) one has ˆ(q + G, q + G0 , ω) = δGG0 ˆ(q + G, ω) and Infinite crystal Calculation of ˆM cubic symmetries Non-cubic symmetries D(q + G, ω) = ˆ(q + G, ω)E(q + G, ω) Measurable quantities : vary on a macroscopic scale → the system is considered as homogeneous. One has to transform X D(q + G, ω) = ˆ(q + G, q + G0 , ω)Etot (q + G0 , ω) G0 into D(q, ω) = ˆ(q, ω)Etot (q, ω) Microscopicmacroscopic connection Macroscopic average Valérie Véniard Intro The differences between the microscopic fields and the averaged (macroscopic) fields are called the local fields. Macro Definition Example Procedure Infinite crystal Calculation of ˆM cubic symmetries Non-cubic symmetries Complexity of the problem: • Macroscopic external field → induced fields • The macroscopic procedure must take into account the fact that all the components of the induced fields will create the response. Procedure: • model for the system expressed in terms of an hamiltonian • microscopic response of the system (linear-response theory for instance) → R Definition of the microscopic dielectric tensor D(r, ω) = dr(r, r0 , ω)E(r0 , ω) • Averaging: definition of M which relates the average parts of D and E Microscopicmacroscopic connection Macroscopic average Valérie Véniard Intro Macro Definition Example Procedure Infinite crystal Calculation of ˆM cubic symmetries Non-cubic symmetries A simple example: the longitudinal case All the fields can be expressed in terms of potential (E = ∇V ) and the longitudinal dielectric function is defined as X Vext (q + G, ω) = GG0 (q, ω)Vtot (q + G0 , ω) G0 Macroscopic average : cut-off of high wave vector Vext is a macroscopic quantity : Vext (q + G, ω) = Vext (q, ω) δG 0 This not the case for Vtot (q + G, ω). Macroscopic average of Vext : Vext (q, ω) = X 0G0 (q, ω)Vtot (q + G0 , ω) 6= 00 (q, ω)Vtot (q, ω) G0 The average of the product is not the product of the averages Microscopicmacroscopic connection Macroscopic average Valérie Véniard Intro The longitudinal case Macro We have also Definition Example Procedure Infinite crystal Calculation of ˆM cubic symmetries Non-cubic symmetries Vtot (q + G, ω) = X 0 −1 GG0 (q, ω)Vext (q + G , ω) G0 where −1 GG0 (q, ω) is the inverse dielectric function: P −1 00 0 G00 GG (q, ω)G00 G0 (q, ω) = δGG ,ω Macroscopic average of Vtot : Vext is macroscopic ⇒ Vtot (q + G, ω) = −1 G0 (q, ω)Vext (q, ω) Vtot (q, ω) = −1 0O (q, ω)Vext (q, ω) Microscopicmacroscopic connection Macroscopic average Valérie Véniard Intro Macroscopic dielectric constant Macro Definition Example Procedure Infinite crystal Calculation of ˆM cubic symmetries Non-cubic symmetries Vext (q, ω) = M (q, ω)Vtot (q, ω) ⇒ M (q, ω) = 1 −1 00 (q, ω) • Inversion of the full dielectric matrix GG0 (q, ω) → −1 GG0 (q, ω) • We take the G = G0 = 0 component of −1 GG0 (q, ω) Interpretation All the microscopic components of the induced field will couple together to produce the macroscopic response. Microscopicmacroscopic connection Average quantities : procedure Valérie Véniard Intro Procedure proposed in : W. L. Mochan and R. G. Barrera, Phys. Rev. B 32 4984 (1985); Phys. Rev. B 32 4989 (1985) Macro Definition Example Procedure Infinite crystal Calculation of ˆM cubic symmetries Non-cubic symmetries General definition: There are different averaging procedures, according to the specific characteristics of the system as well as the length scales involved. We formally define two operators P̂a and P̂f which extract the average component and the fluctuation component of any function F: Fa = P̂a F and Ff = P̂f F . P̂a and P̂f have the following properties: • P̂a2 = P̂a . • Thus P̂f2 = P̂f and P̂a P̂f = P̂f P̂a = 0. • P̂a commutes with the time and space differential operators: the average part of the fields must obey the Maxwell’s equations P̂a and P̂f are projector operators Microscopicmacroscopic connection Average quantities Valérie Véniard Intro Macro Definition Example Procedure Specific examples : 1 Infinite crystal Calculation of ˆM cubic symmetries Non-cubic symmetries 2 3 P Ensemble-average Fa (λ) = c Pc Fc (λ) Pc : probability of finding the system in the configuration c of the ensemble Fc : value of the function in this configuration. R Spatial average Fa (r) = dr0 Pa (r − r0 )F (r0 ) Pa (r) : weight function Wave-vector truncation Fa (q) = Pa (q)F (q) Pa (q): cut-off for the high wave-vector components Microscopicmacroscopic connection Average quantities Valérie Véniard Intro Notations: „ F = Fa + Ff → Macro Definition Example Procedure Infinite crystal Calculation of ˆM cubic symmetries Non-cubic symmetries Fa Ff « Microscopic Dielectric Tensor : D = ˆ Etot Da Df = ˆaa ˆaf ˆfa ˆff Ea Ef Ôαβ = P̂α Ô P̂β The problem of finding the macroscopic dielectric tensor is to decouple Da and Ea , by finding a relationship between Df and Ef , to get Da = ˆM Ea Note: we suppose that we know the full dielectric tensor Microscopicmacroscopic connection Valérie Véniard Average quantities Going back to the Maxwell’s equations, Intro ∇×∇× E= Macro Definition Example Procedure Infinite crystal Calculation of ˆM cubic symmetries Non-cubic symmetries 4iπω ω2 jext + 2 D 2 c c If the external current has no microscopic fluctuation (jext,f = 0) : [ˆ ff − Solving for Ef , one has c2 ˆ ˆ )ff ]Ef = −ˆ (∇ × ∇× fa Ea ω2 Ef = −[ˆ ff − ˆM = ˆaa − ˆaf [ˆ ff − c2 ˆ ω 2 (∇ ˆ )ff ]−1 ˆfa Ea × ∇× c2 ˆ ˆ )ff ]−1 ˆfa (∇ × ∇× ω2 This exact result shows that the macroscopic dielectric response is the sum of the average microscopic response and a correction due to the coupling between the average and the fluctuating fields. Microscopicmacroscopic connection Longitudinal and transverse fields Valérie Véniard Intro Longitudinal fields Macro Definition Example Procedure ∇ × E(r) = 0 or k × E(k) = 0 Infinite crystal Calculation of ˆM cubic symmetries Non-cubic symmetries E(k) propagates along k. In the Coulomb gauge, E(r) is related to a scalar potential V Transverse fields ∇.E(r) = 0 or k.E(k) = 0 E(k) propagates perpendicular to k. In the Coulomb gauge, E(r) is related to a vector potential A(r) Microscopicmacroscopic connection Microscopic spatial fluctuations Valérie Véniard Intro Macro Definition Example Procedure Infinite crystal Calculation of ˆM cubic symmetries Non-cubic symmetries Characteristic length scale of the spatial fluctuations l << λ (wavelength of the radiation) Transverse projector ˆ ×∇ ˆ −2 ∇× ˆ P̂ T = −∇ Longitudinal projector ˆ∇ ˆ −2 ∇ ˆ P̂ L = ∇ P̂ L and P̂ T are also projectors and commute with P̂a and P̂f ˆ = iq and ∇ ˆ −2 = −1/q 2 in q-space ∇ 2 ˆ × ∇× ˆ )ff ]−1 ˆfa onto the Projection of ˆM = ˆaa − ˆaf [ˆ ff − ωc 2 (∇ longitudinal and transverse subspaces c2 ˆ ˆ )ff ]−1 = [ˆ ff − 2 (∇ × ∇× ω ˆLL ff ˆTL ff ˆTT ff ˆLT ff 2 ˆ 2 P̂ T P̂f − ωc 2 P̂f ∇ −1 Microscopicmacroscopic connection Microscopic spatial fluctuations Valérie Véniard Intro ˆ −2 P̂f || ≈ l 2 /λ2 Expansion in terms of ||ω 2 /c 2 ∇ Macro Definition Example Procedure Infinite crystal Calculation of ˆM cubic symmetries Non-cubic symmetries [ˆ ff − c2 ˆ ˆ )ff ]−1 = (∇ × ∇× ω2 „ −1 (ˆ LL ff ) 0 0 0 Keeping only the first term LL LL −1 ˆaf ˆLT (ˆ ff ) af ˆM = ˆaa − TT 0 ˆTL ˆ af af « + 0 0 ω 2 ˆ −2 ∇ P̂f c2 ˆLL fa ˆTL fa one gets ˆM −1 = ˆaa − ˆaf (ˆ LL ˆfa ff ) The same calculation can be done with ˆ−1 M : ˆ−1 M −1 −1 = ˆ−1 ˆ−1 −1 )TT ˆfa aa − ff ) af ((ˆ „ ... ... ... ... ˆLT fa ˆTT fa « Microscopicmacroscopic connection Interpretation Valérie Véniard Intro Macro Definition Example Procedure −1 ˆM = ˆaa − ˆaf (ˆ LL ˆfa ff ) Infinite crystal Calculation of ˆM cubic symmetries Non-cubic symmetries The expansion corresponds to λ2 /l 2 >> ||ˆ ||. In case of a resonance , the procedure can be questioned (J. E. Sipe and J. van Kranendonk, Phys. Rev. A 9 1806 (1974).) The neglect of terms of the order of l 2 /λ2 is equivalent to the neglect of the transverse fluctuating electric field EfT . Microscopicmacroscopic connection Valérie Véniard Intro Macro Definition Example Procedure Infinite crystal Calculation of ˆM cubic symmetries Non-cubic symmetries Infinite crystal Microscopicmacroscopic connection Valérie Véniard Intro Macro Infinite crystal Invariance : ˆ(r + R, r0 + R) = ˆ(r, r0 ) X D(q + G) = ˆ(q + G, q + G0 )E(q + G0 ) Definition Example Procedure Infinite crystal Calculation of ˆM cubic symmetries Non-cubic symmetries G0 Longitudinal projector : P̂ L (q + G, q + G0 ) = q+G q+G 0 |q+G| . |q+G| δGG Average projector= truncation eliminating all wave vectors outside the first Brillouin zone P̂a (q + G, q + G0 ) = δG0 δG0 0 ˆM = ˆ00 − X −1 ˆ0G (ˆ LL ˆG 0 0 r )GG0 GG0 6=0 ˆ−1 ˆ−1 00 − M = X GG0 6=0 0 h i−1 −1 TT ˆ−1 (ˆ ) ˆ−10 r 0G 0 G 0 GG Meaning of the subscript r: suppress the G = 0 component before doing the inversion Microscopicmacroscopic connection Infinite crystal Valérie Véniard Intro Macro Definition Example Procedure Infinite crystal Calculation of ˆM cubic symmetries Non-cubic symmetries The procedure can be simplified by defining the scalar operator ˆcc : ρext = ˆcc ρ −1 (ˆ LL = (ˆ LL )−1 cc )−1 M) 00 = (ˆ 00 q̂q̂ ˆcc can be calculated directly from the response of the system in the absence of transverse electric field. Well-known result obtained first by Adler (1962) and Wiser (1963) using the random-phase-approximation(RPA) and neglecting longitudinal - transverse coupling • does not depend on the microscopic theory used to obtain • Valid even in case of longitudinal - transverse coupling Microscopicmacroscopic connection Calculation of ˆM Valérie Véniard Intro Macro Definition Example Procedure Infinite crystal Calculation of ˆM cubic symmetries Non-cubic symmetries Starting point : microscopic dielectric tensor obtained in the linear response theory. The hamiltonian H is divided into two parts: • a nonperturbed hamiltonian H0 • a time-dependent perturbation Hpert The induced current and induced density can be computed in terms of the perturbing field and the response functions are written in terms of the eigenfunctions of the unperturbed hamiltonian. Microscopicmacroscopic connection Calculation of ˆM Valérie Véniard Intro Macro Definition Example Procedure Infinite crystal Calculation of ˆM cubic symmetries Non-cubic symmetries Example: Etot = Eext + ELind + ET ind • Epert = Etot ⇒ P = α̂(0) Etot ⇒ ˆ = 1 + 4πα(0) • Epert = Eext ⇒ P = α̂Eext and h i−1 h i ˆ ×∇ ˆ × + 12 ∂ 22 ˆ ×∇ ˆ × + 12 ∂ 22 + 4π2 ∂ σ̂ Etot Eext = ∇ ∇ c ∂ c ∂t c ∂t i h i h 2 −1 ˆ ×∇ ˆ × + 12 ∂ 22 + 4π2 ∂ σ̂ ˆ ×∇ ˆ × + 12 ∂ 2 ∇ ˆ = 1 + 4π α̂ ∇ c ∂ c ∂t c ∂t σ̂: conductivity and we have • α̂(0) = independent particle polarisability • α̂ =full polarisability ⇒ Leads to the inversion of large matrices Microscopicmacroscopic connection Calculation of ˆM Valérie Véniard Intro Macro Definition Example Procedure Infinite crystal Calculation of ˆM cubic symmetries Non-cubic symmetries Example: Etot = Eext + ELind + ET ind • Epert = Eext + ET ind The former results cannot be directly applied: Epert is not solution of any Maxwell’s equations. The formalism can be extended (R. Del Sole and E. Fiorino, Phys Rev B 29, 4631 (1984).) qq α̃(q, q, ω) M (q, ω) = 1 + 4π α̃(q, q, ω). 1 + 4π q q 1 − 4π α̃LL (q, q, ω) with P = α̃ Epert , α̃ is the quasi-polarisability. • Same assumption: E T (q + G) ≈ 0 for G 6= 0 • Easier for practical calculation Microscopicmacroscopic connection Calculation of ˆM : EELS Valérie Véniard Intro The charge density associated to an external classical charge e, with velocity v is given by ρext (r, t) = eδ(r − vt) Macro Definition Example Procedure ρext (k, ω) = e δ(ω − kv) V Infinite crystal Calculation of ˆM cubic symmetries Non-cubic symmetries We assume that the potential created by the external charge is sufficiently smooth, so that it does not contain any microscopic component : Eext (q + G, ω) = Eext (q, ω)δG0 with Eext (q, ω) = −iqVext (q, ω) We know that the only nonvanishing component of the tranverse field corresponds to G = 0, and we have ET (q) = ω2 DT (q) c 2 |q|2 Microscopicmacroscopic connection Infinite crystal: EELS Valérie Véniard Intro With ω = qv and in the nonrelativistic regime, v << c : Macro Definition Example Procedure ω2 c 2 |q|2 ≈ v2 << 1 c2 Infinite crystal Calculation of ˆM cubic symmetries Non-cubic symmetries The transverse field ET (q) = Ei,T (q) is negligible. With Epert = Eext + ET ind , the perturbing field is equal to the external longitudinal field and using q q E(q, ω) = 1 − 4π . α̃(q, q, ω) EP (q, ω) |q| |q| the total field is also longitudinal. The calculation of the energy loss can be done using only scalar field and the loss function is Im −1/LL M (q, ω) Microscopicmacroscopic connection Cubic symmetries with q → 0 Valérie Véniard Intro Longitudinal dielectric function Macro LL M (ω) = lim Definition Example Procedure Infinite crystal Calculation of ˆM cubic symmetries Non-cubic symmetries q→0 1 1+ 4π q 2 χρρ (q, ω) where χρρ (q, ω) is the density-density response function (TDDFT), relating the induced density to the external potential ρind (q, ω) = χρρ (q, q, ω)V ext (q, ω) Transverse dielectric function LL lim TT M (q, ω) = M (ω) q→0 Dielectric tensor The dielectric tensor is diagonal (in terms of L and T) and contains only LL M (ω) Microscopicmacroscopic connection Cubic symmetries with q 6= 0 Valérie Véniard Intro Longitudinal dielectric function Macro One can show that the relation Definition Example Procedure Infinite crystal Calculation of ˆM cubic symmetries Non-cubic symmetries LL M (q, ω) = 1 1+ 4π χ (q, ω) q 2 ρρ holds also when q 6= 0. Transverse dielectric functions LL TT M (q, ω) 6= M (q, ω) TL We have also LT M (q, ω) 6= 0 and M (q, ω) 6= 0 These quantities are much more complicated and need, in principle, further approximations to be computed. Microscopicmacroscopic connection Non-cubic symmetries with q → 0 Valérie Véniard Intro Macro Definition Example Procedure Infinite crystal Calculation of ˆM cubic symmetries Non-cubic symmetries • Main general result concerning ˆM (q, ω) : ˆM is an analytic function of q (R. Del Sole and E. Fiorino, Phys Rev B 29, 4631 (1984).) =⇒ The limit q → 0 does not depend on the direction of q. • We can define ˆM as ˆM (ω) = limq→0 ˆM (q, ω) ˆLL M (q, ω) is not analytic in the general case • Depending on the symmetry of the system, one can define 3 principal axis (if they exist) (n1 , n2 ,n3 ) defining a frame in which ˆM (ω) is diagonal. If E is parallel to one of these axis : ˆM (ω)E = i (ω)E i can be calculated as LL i (ni , ω) Microscopicmacroscopic connection Non-cubic symmetries with q → 0 Valérie Véniard Intro Macro Definition Example Procedure The distinction between longitudinal and transverse is not meaningful The only important direction is the direction of the electric field Infinite crystal Calculation of ˆM cubic symmetries Non-cubic symmetries If q → 0 , the fields do not propagate Calculation of ˆLL ˆTT M and M should lead to the same information. i can be seen as a longitudinal dielectric function i = LL M (ni , ω) but can also be used as a transverse dielectric function. Microscopicmacroscopic connection Non-cubic symmetries with q → 0 Valérie Véniard Intro Macro Definition Example Procedure Infinite crystal Calculation of ˆM cubic symmetries Non-cubic symmetries Existence of the principal frame ? • M is symmetric but complex ⇒ No general answer • To determine the principal axis, one needs ˆM but i can only be calculated in the optical (principal) frame. Use of geometrical arguments • cubic • orthorombic • hexagonal The optical axis are given by the symmetry • monoclinic • triclinic The number of symmetries is too low to get 3 optical axis Microscopicmacroscopic connection Non-cubic symmetries with q → 0 Valérie Véniard Intro Dielectric tensor Macro As q → 0, ˆM (ω) is a tensor and in the (x,y,z)-basis has 6 independent component. aa is calculated as a longitudinal response (TDDFT) through the relation aa (ω) = limqa →0 1+ 4π χ 1(q ,q ,ω) with qa along a, Definition Example Procedure Infinite crystal Calculation of ˆM cubic symmetries Non-cubic symmetries qa2 a = x, y, z ρρ a a Off-diagonal elements ab is defined as ab = a.ˆ M .b With the expression obtained for M , one gets ab = 4π a.α̃(qa , qa ).b 1 − 4π α̃LL (qa , qa ) or ab = 4π a.α̃(qb , qb ).b 1 − 4π α̃LL (qb , qb ) Microscopicmacroscopic connection Non-cubic symmetries with q → 0 Valérie Véniard Intro Macro Definition Example Procedure Off-diagonal elements One gets in terms of the usual response functions Infinite crystal Calculation of ˆM cubic symmetries Non-cubic symmetries ab = 4πaa a.χjj (qa , qa ).b ω2 or ab = 4πbb a.χjj (qb , qb ).b ω2 Using the relations between the response functions ab = 4πaa χρj (qa , qa ).b ωqa or ab = Up to now, it cannot be related to χρρ 4πbb a.χjρ (qb , qb ) ωqb Microscopicmacroscopic connection Non-cubic symmetries with q → 0 Valérie Véniard Intro Macro Definition Example Procedure Infinite crystal Calculation of ˆM cubic symmetries Non-cubic symmetries Off-diagonal elements Another way to get the off-diogonal terms M (a + b) = (a + b)ˆ M (a + b) = aa + bb + 2ab Three independent calculations, based on a purely longitudinal approach, give access to ab Microscopicmacroscopic connection References Valérie Véniard Intro Macro Definition Example Procedure Infinite crystal Calculation of ˆM cubic symmetries Non-cubic symmetries S. L. Adler, Phys. Rev. 126 413 (1962); N. Wiser, Phys. Rev. 129 62 (1963.) H. Ehrenreich, in the Optical Properties of Solids, Varenna Course XXXIV, edited by J. Tauc (Academic Press, New York, 1966), p. 106 R. M. Pick, in Advances in Physics, Vol 19, p. 269 J. E. Sipe and J. van Kranendonk, Phys. Rev. A 9 1806 (1974). D. L. Johnson, Physical Review B, 12 3428 (1975). R. Del Sole and E. Fiorino, Phys. Rev. B 29 4631 (1984) W. L. Mochan and R. G. Barrera, Phys. Rev. B 32 4984 (1985) W. L. Mochan and R. G. Barrera, Phys. Rev. B 32 4989 (1985)
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