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Microscopic-macroscopic connection Valérie Véniard ETSF France Laboratoire des Solides Irradiés, Ecole Polytechnique, CEA-DSM, CNRS, 91128 Palaiseau, France Outline 1. Introduction: which quantities do we need 2. Macroscopic average - Definition - Examples • Dielectric tensor for cubic symmetries • Dielectric tensor for non-cubic symmetries - Properties - Principal axis • Summary Linear response Perturbation theory For a sufficiently small perturbation, the response of the system can be extended into a taylor series, with respect to the perturbation. We will consider only the first order (linear) response, proportional to the perturbation. The linear coefficient linking the response to the perturbation is called a response function. It is independent of the perturbation and depends only on the system. ≠ Strong field interaction(laser field for instance) Examples Density response function Dielectric tensor n1 (r , t ) = ∫ dt ' ∫ dr ' χ (r , t , r ' , t ' )V (r ' , t ' ) D(r , t ) = ∫ dt ' ∫ dr ' ε (r , t , r ' , t ' ) E(r ' , t ' ) Which quantities do we need? Absorption coefficient The general solution of the Maxwell's equations for the electric field is ! ! ! i(kx!!t ) E(r, t) = E0 e Defining the complex refractive index as n = field inside a medium is the damped ωwave: k= ! " c ε = ν + iκ , the electric i xn −iωt i ωνx −ω κx −iωt E(r , t ) = E0e c e = E0e c e c e ν and κ are the refraction index and the extinction coefficient and they are related to the dielectric constant (ε=ε1+iε2) as 2 ε1 = ν − κ 2 ε 2 = 2νκ The absorption coefficient α is the inverse distance where the intensity of the field is reduced by 1/e (related to the optical skin depth δ). c != "# 2"# "$ 2 != = c %c Which quantities do we need? ! E ! k Reflected beam Reflectivity Normal incidence reflectivity z R= Transmitted beam x Incident beam ET Ei 2 <1 Using the continuity of the tangential component of the electric field at the surface (1 −ν ) 2 + κ 2 R= (1 +ν ) 2 + κ 2 The knowledge of the optical constant implies the knowledge of the absorption and of the reflectivity, which can be compared with the experiment. Which quantities do we need? Energy loss by a fast particle Given an external charge density ρext, one can obtained the external potential Vext 2 (Poisson equation) k Vext (k , ω) = 4πρext (k , ω) The response of the system is an induced density, defined by the response function χ ρind (k , ω ) = χ (k , ω )Vext (k , ω ) and the total (induced + external) potential acting on the system is 4π −1 Vtot (k , ω ) = 1 + 2 χ (k , ω Vext (k , ω ) = ε (k , ω )Vext (k , ω ) k Vext (k , ω) = ε (k , ω)Vtot (k , ω) Dielectric function: Which quantities do we need? Energy loss by a fast particle Charge density of a particle (e-) with velocity v : ρext = eδ (r − v t ) Etot = −∇rVtot (r , t ) The total electric field is and the energy lost by the electron in unit time is dW = ∫ dr j .Etot with the current density dt j = ev δ (r − v t ) ω dW e dr = − 2 ∫ 2 Im dt π k ε (k , ω ) 2 We get 1 − Im Electron Energy Loss Spectroscopy: ε (k , ω ) is called the loss function Outline 1. Introduction: which quantities do we need 2. Macroscopic average - Definition - Examples • Dielectric tensor for cubic symmetries • Dielectric tensor for non-cubic symmetries - Properties - Principal axis • Summary Maxwell’s equations in vacuum ! ! ! ! !. Eext (r, t) = 4!"ext (r, t) ! ! ! !. Bext (r, t) = 0 ! ! ! ! 1 $Bext ! " Eext (r, t) = # c $t ! ! ! ! 4! ! ! 1 $Eext ! " Bext (r, t) = jext (r, t) + c c $t ρext and jext are the (external) free charge and current density They can be arbitrary, but they have to fulfill the continuity equation E: electric field B: magnetic field (induction) ∂ρ ext ∇. jext (r , t ) + =0 ∂t Maxwell’s equations in a medium ! ! ! ! ! ! ! ! It is often more !. Etot (r, t) = 4!"tot (r, t) or !. D(r, t) = 4!"ext (r, t) convenient to use ! ! ! !. Btot (r, t) = 0 D = Etot + 4πP ! ! ! ! 1 $Btot ! " Etot (r, t) = # instead of Etot , as D can c $t be very close to ! ! ! ! 4! ! ! 1 $Etot the external field ! " Btot (r, t) = jtot (r, t) + c c $t ∇.D = ∇.Eext (see the previous slide) With ρtot =ρext+ρind and jtot =jext+jind , ρind and jind are the induced charge and current density: they are not arbitrary, but reflect the spatial structure of the medium on a microscopic level and the motion of the particle in it, including also the response to an external field. Microscopic spatial fluctuations • Infinite crystals → microscopic inhomogeneities (atomic scale) • Semi-infinite crystals → presence of the surface • Desorded medium → liquid • Rough surfaces Macroscopic average We consider spatial fluctuations whose characteristic length scale is much smaller than the wavelength of light Macroscopic quantities Quantities that are slowly varying over the unit cells. where V is the volume per unit cell of the crystal. Examples Eext(r, t), Aext(r, t), Vext(r, t),… Typical values: • dimension of the unit cell for silicon acell ≈0.5nm • Visible radiation 400 nm < λ < 800 nm ! >> V 1/3 Macroscopic average 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. Examples Etot(r, t), jind(r, t), ρind(r, t),... Macroscopic average 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 Macroscopic average General definition: We have to define two operators P̂a and P̂f which extract the average component and the fluctuation component of any function F P̂f = 1̂! P̂a Fa = P̂a F Ff = P̂f F P̂a and P̂f have the following properties: 1) 2) P̂a2 = P̂a P̂f2 = P̂f and P̂a P̂f = P̂f P̂a = 0 P̂f ⇒ Projectors P̂a commutes with the time and space differential operators The average part of the field must obey the macroscopic Maxwell’s equations Macroscopic average The differences between the microscopic fields and the averaged (macroscopic) fields are called the local fields 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) ! ! ! !! ! ⇒ Definition of the microscopic dielectric tensor D(r, ! ) = ! dr '! (r, r ', ! )E(r ', ! ) • Averaging: definition of εM which relates the average parts of D and E Macroscopic average Infinite crystals Functions having the crystal symmetries V(r+ R)=V(r), where R is any vector of the Bravais lattice, can be represented by the Fourier series i ( q +G ) r V (r , ω ) = ∑ V (q + G, ω )e qG It can be also written as iqr V ( r , ω ) = ∑ V ( r ; q , ω )e q where iGr V (r ; q , ω ) = ∑ V ( q + G , ω )e G is a periodic function, with respect to the Bravais lattice. Varies strongly even if the original wave is a long wave and nearly constant within each cell (contains all the G-harmonics of the field). Macroscopic average Infinite crystals Spatial average over a cell of the periodic part V ( R, ω ) = V ( r ; q , ω ) R 1 iGr = ∫ dr ∑ V ( q + G , ω )e Ω G = V (q + 0, ω ) The macroscopic average corresponds to the G=0 component. ⇔ Τruncation that eliminates all wave vectors outside the first Brillouin zone (wave-vector truncation) Macroscopic quantities have all their G components equal to 0, except the G=0 component. → Satisfies the two criteria previously defined Macroscopic average • If the external applied field is not macroscopic, this averaging procedure for the response function of the material has no meaning. • One has to consider an average procedure based on the statistical and quantum mechanical sense (beyond the scope of this lecture) Exemples: X-ray spectroscopy (very short wavelength) EEL Spectroscopy with atomic resolution Outline 1. Introduction: which quantities do we need 2. Macroscopic average - Definition - Examples • Dielectric tensor for cubic symmetries • Dielectric tensor for non-cubic symmetries - Properties - Principal axis • Summary Macroscopic average A simple example: the longitudinal case All the fields can be expressed in terms of potentials (E=-∇V) ! ! !! ! (Real space) V ( r, ! ) = d r ' ! ( r, r ', ! )V ( r ', ! ) ! ext tot The longitudinal dielectric function is defined as ! ! ! ! ! ! ! ! Vext (q + G, ! ) = !" (q + G, q + G ', ! )Vtot (q + G ', ! ) G' Vext is a macroscopic quantity : Vext (q + G, ω ) = Vext (q, ω )δ G 0 This is not the case for Vtot (q + G, ω ) (Reciprocal space) Macroscopic average of Vext : Vext (q, ω) = ∑ ε 0G ' (q, ω)Vtot (q + G' , ω) ≠ ε 00 (q, ω)Vtot (q, ω) G' The average of the product is not the product of the averages Macroscopic average A simple example: the longitudinal case We have also −1 Vtot (q + G, ω ) = ∑ ε GG ' (q , ω )Vext (q + G ' , ω ) G' where is ε-1GG the inverse dielectric function : ∑ε GG'' (q, ω)ε G−1''G' (q, ω) = δGG' G '' Macroscopic average of Vtot : Vext is macroscopic ⇒ Vtot (q + G, ω ) = ε G−10 (q, ω )Vext (q, ω ) −1 Vtot (q , ω ) = ε 00 (q , ω )Vext (q, ω ) Macroscopic average A simple example: the longitudinal case Macroscopic dielectric constant: Vext (q , ω ) = ε M (q , ω )Vtot (q , ω ) ⇒ ε M (q , ω ) = 1 −1 ε 00 (q , ω ) −1 • Inversion of the full dielectric matrix ε GG ' (q , ω ) → ε GG ' (q , ω ) −1 • We take the G=G =0 component of ε GG ' (q , ω ) Interpretation All the microscopic components of the induced field will couple together to produce the macroscopic response Macroscopic average Another example: the Lorentz model For non interacting dipoles (dilute media) α: polarisability of the dipole (atoms or molecules) N: number of dipoles per unit volume (density) P = Nα Eloc Eloc: local field acting on a given dipole If Eloc=Etot, we have ! ! ! ! ! D = Etot + 4! P = Etot + 4! N" Etot D = ε M Etot ⇒ ε M = 1+ 4π Nα Mosotti (1850) Claisius (1879) Lorentz (1909) Macroscopic average The Lorentz model In fact, Eloc≠Etot P= 4 Eloc = Etot + π P 3 Nα Etot 4π 1− Nα 3 εM in the Lorentz model (for isotropic or cubic media) 4πNα = 1+ 4π 1− Nα 3 Based on classical physics, does not include microscopic (quantum-mechanical) effects Increase of the static dielectric function Red shift of the transition frequency Not the case, when including microscopic effects f 012 α (ω ) = 2 ⇒ ω01 − ω 2 4πNf 012 ε M = 1+ 4π 2 2 ω01 − ω − Nf 012 3 Macroscopic average Summary • We have defined microscopic and macroscopic fields • Microscopic quantities have to be averaged to be compared to experiments • The dielectric function has - a microscopic expression (related to quantum mechanics) - a macroscopic expression (classical scheme - Maxwell's equations) • Absorption ↔ Im {εM} and EELS ↔ -Im {1/εM} Outline 1. Introduction: which quantities do we need 2. Macroscopic average - Definition - Examples • Dielectric tensor for cubic symmetries • Dielectric tensor for non-cubic symmetries - Properties - Principal axis • Summary Dielectric tensor for cubic symmetries Useful definitions Longitudinal fields ! ! ! ! " E(r ) = 0 or k × E (k ) = 0 E(k) propagates along k Examples: • plasmon oscillations • sceening • electron energy loss Some definitions: with Transverse fields ∇.E (r ) = 0 or k .E (k ) = 0 E(k) propagates perpendicular to k Examples: • photons • optical properties of solids E (r ) ↔ E (k ) (Fourier transform) k = q + G for crystals (q is in the first Brillouin zone and G is a reciprocical lattice vector) Dielectric tensor for cubic symmetries Transverse-longitudinal decomposition: Any vector field can be split into longitudinal and transverse components ! ! L !T E =E +E with ! !L ! ! ! k E (k ) = k̂ !"k̂. E(k )#$ and k̂ = k ! !L !"E = 0 and ! !T !. E = 0 (In real space, the relations are nonlocal) Macroscopic dielectric tensor The relation D(q,ω) = εM(q,ω) Etot(q,ω) can be written in terms of the longitudinal and transverse components D L ε MLL T = TL D ε M ε MLT EtotL TT T ε M Etot Dielectric tensor for cubic symmetries Question: How can we make the link between • the microscopic dielectric tensor D(q + G, ω ) = ∑ ε (q + G, q + G' , ω ) Etot (q + G' , ω ) Microscopic components of D and Etot G' • the macroscopic dielectric tensor D(q, ω) = ε M (q, ω) Etot (q, ω) Macroscopic components of D and Etot Dielectric tensor for cubic symmetries No symmetry ! LL LT ! ! ! ! ! M (q, " ) = # MTL MTT # ! " M !M Cubic symmetry with q→0 $ & & % A longitudinal (transverse) perturbation induces longitudinal and transverse responses. ε MLL ε M (q, ω ) = 0 0 TT εM • A longitudinal perturbation induces a longitudinal response only. • A transverse perturbation induces a transverse response only. • Independent of the direction of q This holds only for macroscopic quantities The microscopic dielectric tensor has off-diagonal elements ! LT and ! TL Cubic symmetries with q→0 Longitudinal dielectric function ε MLL (ω ) = lim q →0 1 1+ 4π χ ( q ,ω) ρρ 2 q where χρρ(q,ω) is the density-density response function (TDDFT), defined as ρ ind (q , ω ) = χ ρρ (q , ω )Vext (q , ω ) Transverse dielectric function ! lim ! MTT (q, ! ) = " MLL (! ) q!0 Dielectric tensor The tensor is diagonal and contains only one quantity ε MLL (ω ) Cubic symmetries with q≠0 Longitudinal dielectric function One can show that the relation ! ! (q, " ) = 1 LL M 1+ ! q !0 holds also when ! depends on q Transverse dielectric function TL ! 4# $ ( q, ") %% 2 q LT ε MTT (q, ω ) ≠ ε MLL (q, ω ) We have also ε M (q, ω ) ≠ 0 ε M (q, ω ) ≠ 0 These quantities are much more complicated and need further approximation to be computed (cannot be expressed in terms of TDDFT). Cubic symmetries Summary • We have defined the longitudinal and transverse components of the dielectric tensor. • In the long wavelength limit q→0 ,only one quantity is needed (optical isotropy) ε MLL (ω ) = ε MTT (ω ) = lim 1 4π χ ( q ,ω) ρρ 2 q • For q≠0, only εMLL has a simple expression in terms of the response function. q →0 1+ Cubic symmetries Some references 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. D. L. Johnson, Physical Review B, 12 3428 (1975). S. L. Adler, Physical Review, 126 413 (1962). N. Wiser, Physical Review, 129 62 (1963). R. Del Sole and E. Fiorino, Phys. Rev. B 29, 4631 (1984). W. L. Mochan and R. Barrera, Phys. Rev. B 32 4984 (1985); ibid 4984 (1985). Outline 1. Introduction: which quantities do we need 2. Macroscopic average - Definition - Examples • Dielectric tensor for cubic symmetries • Dielectric tensor for non-cubic symmetries - Properties - Principal axis • Summary Non-cubic symmetries Properties of the macroscopic quantities LL LT ε M (q , ω ) ε M (q , ω ) ε M (q, ω ) = TL TT ε ( q , ω ) ε ( q , ω ) M M Non-cubic symmetries Dielectric tensor - General case qˆα (q , q , ω ) ε M (q , ω ) = 1 + 4πα (q , q , ω ) 1 + 4π qˆ LL 1 − 4 πα ( q , q , ω ) Quasipolarisability α : jind (q + G, ω ) = ∑ α (q + G, q + G' , ω ) E pert (q + G' , ω ) G' But one can show that the relation : holds also for the non-cubic symmetries. 1 and we have α LL (q, q, ω ) = − 2 χ ρρ (q, q, ω ) q ε MLL (q, ω ) = 1 1 − 4πα (q, qω ) LL Longitudinal-longitudinal dielectric function Outline 1. Introduction: which quantities do we need 2. Macroscopic average - Definition - Examples • Dielectric tensor for cubic symmetries • Dielectric tensor for non-cubic symmetries - Properties - Principal axis • Summary Non-cubic symmetries q→0 ! q Main general result concerning εM : εM is an analytic function of The limit q→0 does not depend on the direction of q. ! ! ( " ) = ! ( q, lim M " ) ⇒ We can define εM(ω) as M ! q!0 ! MLL is not analytic in the general case Depending on the symmetry of the system, one can define the 3 principal axis, if they exist, (n1, n2,n3) defining a frame in which εM(ω) is diagonal. ! ! ! ! If Etot is parallel to one of these axis ni ! M (" )Etot (" ) = !i (" )Etot (" ) εi (ω) can be calculated as a longitudinal dielectric function ε i (ω ) = ε MLL (ni , ω ) Non-cubic symmetries q→0 The distinction between longitudinal and transverse is not meaningful The only important direction is the direction of the electric field If q→ 0, the fields do not propagate Existence of the principal frame ? εM is symmetric but complex … No general answer Use of geometrical arguments Symmetries • Cubic • Hexagonal • Orthorombic • Monoclinic • Triclinic Shorter wavelength q≠0 Alternative: ˆ qα (q , q , ω ) ε M (q , ω ) = 1 + 4πα (q , q , ω ) 1 + 4π qˆ We can use LL 1 − 4 πα ( q , q , ω ) j ( q + G , ω ) = α ( q + G , q + G ' , ω ) E ( q + G' , ω ) where ind ∑ pert G' The induced current can be evaluated through the (Time-Dependent)-Density-Current Functional Theory (TD-DCFT) Which quantities do we need? " 1 % Electron Energy Loss Spectroscopy: ! Im # ! & $ ! (q, " ) ' ! LL ! In that case, ! (q, ! ) = " M ! ! ! (q, ! ) with Vext (q, ! ) = " (q, ! )Vtot (q, ! ) Is this correct? The perturbation is longitudinal. What about the transverse response? One can show that !T ! !2 !T ! E (q, ! ) = 2 2 D (q, ! ) and cq !2 v2 ! 2 2 2 cq c In the nonrelativistic approximation, the transverse fields are negligible and the LL component of the dielectric tensor describes the energy loss of charged particles Outline 1. Introduction: which quantities do we need 2. Macroscopic average - Definition - Examples • Dielectric tensor for cubic symmetries • Dielectric tensor for non-cubic symmetries - Properties - Principal axis • Summary Summary The key quantity is the dielectric tensor. Relation between microscopic and macroscopic fields. For cubic crystals, the longitudinal dielectric function εM(ω) defines entirely the optical response in the long wavelength limit (q→0). For non-cubic crystals, the dielectric functions calculated along the principal axis can be used to define entirely the optical response in the long wavelength limit. For non-vanishing momentum, the situation is not so simple: εMLL (q,ω) only can be defined in a simple way.