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Microscopic-macroscopic connection Valérie Véniard Laboratoire des Solides Irradiés Ecole Polytechnique, Palaiseau - France European Theoretical Spectroscopy Facility (ETSF) Lyon,10 Dec 2007 Microscopic-macroscopic connection Valérie Véniard Intro Macro Cubic Noncubic Conclu Outline 1 Introduction: Which quantities do we need? 2 Macroscopic average Definition Example 3 Dielectric tensor for cubic symmetries 4 Dielectric tensor for non-cubic symmetries Properties Principal axis 5 Summary Microscopic-macroscopic connection Valérie Véniard Intro Macro Cubic Noncubic Conclu Outline 1 Introduction: Which quantities do we need? 2 Macroscopic average Definition Example 3 Dielectric tensor for cubic symmetries 4 Dielectric tensor for non-cubic symmetries Properties Principal axis 5 Summary Microscopic-macroscopic connection Valérie Véniard Intro Macro Cubic Noncubic Conclu Linear response Perturbation theory For a sufficiently small perturbation, the response of the system can be expanded 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. 6= strong field interaction (intense lasers for instance) Example R R Density response function: n1 (r, t) = dt 0 dr0 χ(r, t, r0 , t 0 )V (r0 , t 0 ) R 0R 0 Dielectric tensor D(r, t) = dt dr (r, t, r0 , t 0 )E(r0 , t 0 ) Microscopic-macroscopic connection Valérie Véniard Intro Macro Cubic Noncubic Conclu Introduction: Which quantities do we need? Absorption coefficient i(Kr−ωt) The general solution of the Maxwell’s equations . √ is E(r, t) = E0 e Defining the complex refractive index as n = = ν + iκ, the electric field inside a medium is the damped wave: E (x, t) = E0 e iω c xn e −iωt = E0 e iω c νx ω e − c κx e −iωt ν and κ are the refraction index and the extinction coefficient and they are related to the dielectric constant as 1 = ν 2 − κ2 2 = 2νκ The absorption coefficient α is the inverse distance where the intensity of the field is reduced by 1/e ω2 α= νc (related to the optical skin depth δ). Microscopic-macroscopic connection Valérie Véniard Intro Macro Cubic Noncubic Conclu Introduction: Which quantities do we measure? Reflectivity Normal incidence reflectivity z Reflected beam R=| Transmitted beam x Incident beam Microscopic-macroscopic connection R= ER 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. Valérie Véniard Intro Macro Cubic Noncubic Conclu Introduction: Which quantities do we need? Energy loss by a fast charged particle Given an external charge density ρext , one can obtained the external potential Vext k 2 Vext (k, ω) = 4πρext (k, ω) (Poisson equation) 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 Vtot (k, ω) = 1 + 2 χ(k, ω) Vext (k, ω) = −1 (k, ω)Vext (k, ω) k Microscopic-macroscopic connection Valérie Véniard Intro Macro Cubic Noncubic Conclu Introduction: Which quantities do we need? Energy loss by a fast charged particle Charge density of a particle (e − ) with velocity v: ρext (r, t) = −eδ(r − vt) The total electric field is Etot (r, t) = −∇r Vtot (r, t) and the energy lost by the electron in unit time is Z dW = drj . Etot with the current density j = −evδ(r − vt) dt Z dW dk e2 ω We get =− 2 Im dt π k2 (k, ω) Electron Energy Loss Spectroscopy n o 1 −Im (k,ω) is called the loss function Microscopic-macroscopic connection Valérie Véniard Intro Macro Cubic Noncubic Conclu Outline 1 Introduction: Which quantities do we need? 2 Macroscopic average Definition Example 3 Dielectric tensor for cubic symmetries 4 Dielectric tensor for non-cubic symmetries Properties Principal axis 5 Summary Microscopic-macroscopic connection Valérie Véniard Intro Macro Cubic Noncubic Conclu Outline 1 Introduction: Which quantities do we need? 2 Macroscopic average Definition Example 3 Dielectric tensor for cubic symmetries 4 Dielectric tensor for non-cubic symmetries Properties Principal axis 5 Summary Microscopic-macroscopic connection Valérie Véniard Intro Macro Cubic Noncubic Conclu Maxwell’s equations Maxwell’s equations in the vaccum ∇.Eext (r, t) = 4πρext (r, t) ∇.Bext (r, t) = 0 1 ∂Bext (r, t) c ∂t 4π 1 ∂Eext (r, t) jext (r, t) + c c ∂t ∇ × Eext (r, t) = − ∇ × Bext (r, t) = ρext and jext are the external (free) charge and current density Microscopic-macroscopic connection Valérie Véniard Intro Macro Cubic Noncubic Conclu Maxwell’s equations Maxwell’s equations in a medium ∇.Etot (r, t) = 4πρtot (r, t) or ∇.D(r, t) = 4πρext (r, t) ∇.Btot (r, t) = 0 1 ∂Btot (r, t) c ∂t 4π 1 ∂Etot (r, t) jtot (r, t) + c c ∂t ∇ × Etot (r, t) = − ∇ × Btot (r, t) = with ρtot = ρext + ρind , and jtot = jext + jind . ρind and jind are the induced charge and current density. It is often more convenient to use D = Etot + 4πP instead of Etot , as D is very close to the external field (∇.D = ∇.Eext ) Microscopic-macroscopic connection Valérie Véniard Intro Macro Cubic Noncubic Conclu Macroscopic average Macroscopic quantities At long wavelength, external fields are slowly varying over the unit cells. 2π >> V 1/3 λ= q where V is the volume per unit cell of the cystal. 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 Microscopic-macroscopic connection Valérie Véniard Intro Macro Cubic Noncubic Conclu 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. Example Etot (r, t), jind (r, t), ρind (r, t),... Microscopic-macroscopic connection Valérie Véniard Intro Macro Cubic Noncubic Conclu Macroscopic average Measurable quantities One measures quantities that vary on a macroscopic scale. We have to average over distances large compared to the cell diameter small compared to the wavelength of the external perturbation Procedure Average over a unit cell whose origin is at point R Regard R as the continuous coordinate appearing in the Maxwell’s equations The differences between the microscopic fields and the averaged (macroscopic) fields are called the local fields Microscopic-macroscopic connection Valérie Véniard Intro Macro Cubic Noncubic Conclu Macroscopic average Procedure 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 V (r, ω) = X V (q + G, ω)e i(q+G)r qG It can be also written as V (r, ω) = X V (r; q, ω)e iqr q P where V (r; q, ω) = G V (q + G, ω)e iGr is a periodic function, with respect to the Bravais lattice. Microscopic-macroscopic connection Valérie Véniard Intro Macro Cubic Noncubic Conclu Macroscopic average Procedure Spatial average over a cell of the periodic part V (R, ω) = < V (r; q, ω) >R Z X 1 = dr V (q + G, ω)e iGr Ω G = V (q + 0, ω) The macroscopic average corresponds to the G = 0 component. ⇐⇒ Truncation that eliminates all wave vectors outside the first Brillouin zone. Macroscopic quantities have all their G components equal to 0, except the G = 0 component. Microscopic-macroscopic connection Valérie Véniard Intro Macro Cubic Noncubic Conclu Outline 1 Introduction: Which quantities do we need? 2 Macroscopic average Definition Example 3 Dielectric tensor for cubic symmetries 4 Dielectric tensor for non-cubic symmetries Properties Principal axis 5 Summary Microscopic-macroscopic connection Valérie Véniard Intro Macro Cubic Noncubic Conclu Macroscopic average A simple example: the longitudinal case All the fields can be expressed in terms of potentials (E = ∇V ) and the longitudinal dielectric function is defined as Vext (q + G, ω) = X GG0 (q, ω)Vtot (q + G0 , ω) G0 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 Microscopic-macroscopic connection Valérie Véniard Intro Macro Cubic Noncubic Conclu Macroscopic average The longitudinal case We have also 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 G00 GG00 (q, ω)G00 G0 (q, ω) = δGG0 ,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) Microscopic-macroscopic connection Valérie Véniard Intro Macro Cubic Noncubic Conclu Macroscopic average Macroscopic dielectric constant 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. Microscopic-macroscopic connection Valérie Véniard Intro Macro Cubic Noncubic Conclu Is it always meaningful? If the external applied field is not macroscopic (very short wavelength), the averaging procedure for the response function of the material has no meaning. When dealing with surfaces, the definition is unclear due to the lack of periodicity in the direction perpendicular to the surface. Microscopic-macroscopic connection Valérie Véniard Intro Macro Cubic Noncubic Conclu 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) macroscopic expression (classical scheme - Maxwell’s equations) Absorption ↔ Im {M } and EELS ↔ −Im Microscopic-macroscopic connection n 1 M o Valérie Véniard Intro Macro Cubic Noncubic Conclu Outline 1 Introduction: Which quantities do we need? 2 Macroscopic average Definition Example 3 Dielectric tensor for cubic symmetries 4 Dielectric tensor for non-cubic symmetries Properties Principal axis 5 Summary Microscopic-macroscopic connection Valérie Véniard Intro Macro Cubic Noncubic Conclu General properties Longitudinal fields ∇ × E(r) = 0 or (k) × E(k) = 0 E(k) propagates along k. Transverse fields ∇.E(r) = 0 or (k).E(k) = 0 E(k) propagates perpendicular to k. Longitudinal field : plasmon oscillations, screening, electron energy loss Transverse field : photons, optical properties of solids Microscopic-macroscopic connection Valérie Véniard Intro Macro Cubic Noncubic Conclu General properties Transverse - longitudinal decomposition Any vector field can be split into longitudinal and transverse components V(k) = VL (k) + VT (k) with k × VL (k) = 0 and k.VT (k) = 0 Macroscopic dielectric tensor ↔ The relation D(q, ω) = M (q, ω)Etot (q, ω) can written in terms of the transverse and longitudinal components L LL L M LT D Etot M = T T TL TT Etot D M M Microscopic-macroscopic connection Valérie Véniard Intro Macro Cubic Noncubic Conclu General properties Question How can we make the link between the microscopic dielectric tensor X↔ (q + G, q + G0 , ω)Etot (q + G0 , ω) D(q + G, ω) = G0 the macroscopic dielectric tensor ↔ D(q, ω) = M (q, ω)Etot (q, ω) Microscopic-macroscopic connection Valérie Véniard Intro Macro Cubic Noncubic Conclu Cubic symmetries Properties of Macroscopic tensor for cubic symmetries → D(q, ω) = ← M (q, ω)Etot (q, ω) Cubic symmetry with q → 0 No symmetry ↔ M (q, ω) = LL M TL M LT M TT M ↔ M LL M 0 0 TT M (ω) = Macroscopic quantities only: A longitudinal pertubation induces a longitudinal response only A transverse pertubation induces a transverse response only Microscopic-macroscopic connection Valérie Véniard Intro Macro Cubic Noncubic Conclu Cubic symmetries with q → 0 Longitudinal dielectric function LL M (ω) = lim q→0 1 1+ 4π χ (q, ω) q 2 ρρ 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)Vext (q, ω) Transverse dielectric function LL lim TT M (q, ω) = M (ω) q→0 Dielectric tensor The dielectric tensor is diagonal and contains only LL M (ω) Microscopic-macroscopic connection Valérie Véniard Intro Macro Cubic Noncubic Conclu Cubic symmetries with q 6= 0 Longitudinal dielectric function One can show that the relation 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 further approximations to be computed. Microscopic-macroscopic connection Valérie Véniard Intro Macro Cubic Noncubic Conclu 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) TT LL M (ω) = M (ω) = lim q→0 1 1+ 4π χ (q, ω) q 2 ρρ For q 6= 0, only LL M (q, ω) has a simple expression in terms of the response functions Microscopic-macroscopic connection Valérie Véniard Intro Macro Cubic Noncubic Conclu 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). Microscopic-macroscopic connection Valérie Véniard Intro Macro Cubic Noncubic Conclu Outline 1 Introduction: Which quantities do we need? 2 Macroscopic average Definition Example 3 Dielectric tensor for cubic symmetries 4 Dielectric tensor for non-cubic symmetries Properties Principal axis 5 Summary Microscopic-macroscopic connection Valérie Véniard Intro Macro Cubic Noncubic Conclu Outline 1 Introduction: Which quantities do we need? 2 Macroscopic average Definition Example 3 Dielectric tensor for cubic symmetries 4 Dielectric tensor for non-cubic symmetries Properties Principal axis 5 Summary Microscopic-macroscopic connection Valérie Véniard Intro Macro Cubic Noncubic Conclu Non-Cubic symmetries Properties - Macroscopic quantities LL M ↔ M (q, ω) = TL M LT M TT M Microscopic and macroscopic quantities Even for q → 0: A longitudinal pertubation induces a longitudinal and a transverse response A transverse pertubation induces a longitudinal and a transverse response Microscopic-macroscopic connection Valérie Véniard Intro Macro Cubic Noncubic Conclu Non-Cubic symmetries Dielectric tensor - General case ← → M (q, ω) = 1 + 4π α̃(q, q, ω) 1 + 4πq̂ q̂α̃(q, q, ω) 1 − 4π α̃LL (q, q, ω) But one can show that the relation holds also for the non-cubic symmetries. 1 LL M (q, ω) = 1 − 4π α̃LL (q, q, ω) R. Del Sole and E. Fiorino, Physical Review B, 29 4631 (1985). Quasipolarisability α̃: jind (q + G) = X α̃(q + G, q + G0 , ω)Epert (q + G0 , ω) G0 with α̃LL (q, q, ω) = − q12 χρρ (q, ω) Microscopic-macroscopic connection Valérie Véniard Intro Macro Cubic Noncubic Conclu Outline 1 Introduction: Which quantities do we need? 2 Macroscopic average Definition Example 3 Dielectric tensor for cubic symmetries 4 Dielectric tensor for non-cubic symmetries Properties Principal axis 5 Summary Microscopic-macroscopic connection Valérie Véniard Intro Macro Cubic Noncubic Conclu Non-cubic symmetries with q → 0 Principal axis One of the main general result concerning M is: M (q) is an analytic function of q The limit q → 0 does not depend on the q ⇒ M (ω) Depending on the symmetry of the system, one can use the 3 principal axis n1 ,n2 ,n3 , defining a frame in which M is diagonal. If Etot (q, ω) is parallel parallel to one of these axis ni ← → M (ω) : Etot (q, ω) = i (ω)Etot (q, ω) whatever the direction of q. i (ω) is an eigenvalue: ⇒ i can be seen as a longitudinal dielectric function i = LL M (ni , ω) but can also be used as a transverse dielectric function. Microscopic-macroscopic connection Valérie Véniard Intro Macro Cubic Noncubic Conclu Non-cubic symmetries with q → 0 Longitudinal and transverse dielectric functions Along these directions ni , a longitudinal perturbation induces a longitudinal response through the usual relation lim LL M (q, ω) = lim q→0 q→0 1 1+ 4π χ (q, ω) q 2 ρρ and if a transverse field is along ni , it will induce only a transverse response. Keep in mind that the principal frame is not always orthogonal and so q could be different from ni ! Consequences If the principal frame is known, on can deduce the optical properties from a longitudinal calculation performed in this frame. Microscopic-macroscopic connection Valérie Véniard Intro Macro Cubic Noncubic Conclu Outline 1 Introduction: Which quantities do we need? 2 Macroscopic average Definition Example 3 Dielectric tensor for cubic symmetries 4 Dielectric tensor for non-cubic symmetries Properties Principal axis 5 Summary Microscopic-macroscopic connection Valérie Véniard Intro Macro Cubic Noncubic Conclu Summary The key quantity is the dielectric tensor. Relation between microscopic and macroscopic fields. For cubic crystals, the longitudinal dielectric function LL M (ω) defines entirely the optical response in the long wavelength limit. For non-cubic crystals, the longitudinal 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: LL M (q, ω) defines only the longitudinal response Microscopic-macroscopic connection Valérie Véniard Intro Macro Cubic Noncubic Conclu Thank you for your attention Microscopic-macroscopic connection Valérie Véniard