Microscopic-macroscopic connection

Transcription

1 icroscopic-macroscopic connection Valérie Véniard ETSF France Laboratoire des Solides Irradiés Ecole Polytechniue CEA-DS CNRS Palaiseau France

2 Outline 1. Introduction: which uantities do we need 2. acroscopic average - Definition - Examples Dielectric tensor for cubic symmetries Dielectric tensor for non-cubic symmetries - Properties - Principal axis Summary

3 Perturbation theory Linear response 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 interactionlaser 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 ' r t r' t' E r' t'

4 Which uantities do we need? Absorption coefficient The general solution of the axwell's euations for the electric field is E rt = E 0 e ikxt k = c " Defining the complex refractive index as field inside a medium is the damped wave: E r t the electric ν and κ are the refraction index and the extinction coefficient and they are related to the dielectric constant = 1 +i 2 as 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. n = = ν + iκ E e i xn c it = E0e e = 0 i νx c 2 2 = ν κ 2νκ 1 2 = = c "# e κx c e it = 2"# c = "$ 2 %c

5 E k Reflected beam Incident beam Which uantities do we need? z Transmitted beam x Reflectivity Normal incidence reflectivity R = E E T 1 ν R = 1 + ν i < 1 Using the continuity of the tangential component of the electric field at the surface 2 + κ 2 + κ The knowledge of the optical constant implies the knowledge of the absorption and of the reflectivity which can be compared with the experiment.

6 Which uantities do we need? Energy loss by a fast particle Given an external charge density ρ ext one can obtained the external potential V ext The response of the system is an induced density defined by the response function χ and the total induced + external potential acting on the system is 4 2 πρ k k V k ext ext = Poisson euation χ ρ k V k k ext ind = χ π k V k k V k k k V ext ext tot = = + k V k k V tot ext = Dielectric function:

7 Which uantities do we need? Energy loss by a fast particle Charge density of a particle e - with velocity v : ρ ext = e δ r vt The total electric field is Etot = rvtot r t and the energy lost by the electron in unit time is dw = dr j. E tot j = evδ r vt dt with the current density We get dw dt 2 e dr = Im 2 2 π k k Electron Energy Loss Spectroscopy: 1 Im k is called the loss function

8 Outline 1. Introduction: which uantities do we need 2. acroscopic average - Definition - Examples Dielectric tensor for cubic symmetries Dielectric tensor for non-cubic symmetries - Properties - Principal axis Summary

9 axwell s euations in vacuum. E ext rt = 4" ext rt. B ext rt = 0 " E ext rt = # 1 $ B ext c $t " B ext rt = 4 jext rt+ 1 c c $ E ext $t E: electric field B: magnetic field induction ρ ext and j ext are the external free charge and current density They can be arbitrary but they have to fulfill the continuity euation ρext. jext r t + = t 0

10 axwell s euations in a medium. E tot rt = 4" tot rt or. B tot rt = 0 " E tot rt = # 1 $ B tot c $t " B tot rt = 4 jtot rt+ 1 c c. D rt = 4" ext rt $ E tot $t It is often more convenient to use D = E tot + 4πP instead of E tot as D can be very close to the external field. D =. E ext see the previous slide With ρ tot =ρ ext +ρ ind and j tot =j ext +j ind ρ ind and j ind 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.

11 icroscopic spatial fluctuations Infinite crystals microscopic inhomogeneities atomic scale Semi-infinite crystals presence of the surface Desorded medium liuid Rough surfaces

12 acroscopic average We consider spatial fluctuations whose characteristic length scale is much smaller than the wavelength of light acroscopic uantities Quantities that are slowly varying over the unit cells. where V is the volume per unit cell of the crystal. >> V 1/3 Examples E ext r t A ext r t V ext r t Typical values: dimension of the unit cell for silicon a cell 0.5nm Visible radiation 400 nm < λ < 800 nm

13 acroscopic average icroscopic uantities 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 E tot r t j ind r t ρ ind r t...

14 acroscopic average easurable uantities One measures uantities 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

15 acroscopic average General definition: We have to define two operators ˆP and ˆPf a which extract the average component and the fluctuation component of any function F ˆP f = ˆ1 ˆP a F a = ˆP a F F f = ˆP f F ˆP a and ˆPf have the following properties: ˆP f 1 ˆP 2 a = ˆP ˆPf 2 a = ˆP f and ˆPa ˆPf = ˆP f ˆPa = 0 Projectors 2 ˆP a commutes with the time and space differential operators The average part of the field must obey the macroscopic axwell s euations

16 acroscopic average The differences between the microscopic fields and the averaged macroscopic fields are called the local fields Complexity of the problem: acroscopic 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 = d r ' r r 'E r ' Averaging: definition of which relates the average parts of D and E

17 acroscopic average Infinite crystals Functions having the crystal symmetries Vr+ R=Vr where R is any vector of the Bravais lattice can be represented by the Fourier series It can be also written as where = V + G e i + G r V r G V r = V r; e G V r; = V + G e 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. igr ir

18 acroscopic average Infinite crystals Spatial average over a cell of the periodic part V R = V r; R 1 = + Ω dr V G e G = V + 0 The macroscopic average corresponds to the G=0 component. Τruncation that eliminates all wave vectors outside the first Brillouin zone wave-vector truncation acroscopic uantities have all their G components eual to 0 except the G=0 component. igr Satisfies the two criteria previously defined

19 acroscopic 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 uantum mechanical sense beyond the scope of this lecture Exemples: X-ray spectroscopy very short wavelength EEL Spectroscopy with atomic resolution

20 Outline 1. Introduction: which uantities do we need 2. acroscopic average - Definition - Examples Dielectric tensor for cubic symmetries Dielectric tensor for non-cubic symmetries - Properties - Principal axis Summary

21 acroscopic average A simple example: the longitudinal case All the fields can be expressed in terms of potentials E=- V The longitudinal dielectric function is defined as V ext is a macroscopic uantity : V tot V + + G This is not the case for ext G = Vext δg0 acroscopic average of V ext : Vext 0G' Vtot + G' 00 V = G' V ext r = V ext + G = d r ' r r 'V tot r ' G' " + G + G'V tot + G' The average of the product is not the product of the averages tot Real space Reciprocal space

22 acroscopic average A simple example: the longitudinal case We have also ' 1 ' ' G V G V ext GG G tot + = + acroscopic average of V tot : where is -1 GGthe inverse dielectric function : ' 1 ' '' '' ' ' GG G G G GG δ = V ext is macroscopic 1 0 V G V ext G tot = V V ext tot =

23 acroscopic average A simple example: the longitudinal case V V tot ext = = acroscopic dielectric constant: Inversion of the full dielectric matrix We take the G=G=0 component of Interpretation All the microscopic components of the induced field will couple together to produce the macroscopic response 1 ' ' GG GG 1 ' GG

24 acroscopic average Another example: the Lorentz model For non interacting dipoles dilute media P = Nα If E loc =E tot we have E loc D = Etot =1+ 4π Nα α: polarisability of the dipole atoms or molecules N: number of dipoles per unit volume density E loc : local field acting on a given dipole D = E tot + 4 P = E tot + 4 N" E tot osotti 1850 Claisius 1879 Lorentz 1909

25 acroscopic average The Lorentz model In fact E loc E tot E loc Nα P = 4π 1 Nα 3 E 4 + π P 3 = in the Lorentz model tot for isotropic or cubic media E tot 4πNα = 1+ 4π 1 Nα 3 Based on classical physics does not include microscopic uantum-mechanical effects Increase of the static dielectric function Red shift of the transition freuency Not the case when including microscopic effects α = 1+ 2 f01 2 = πNf01 2 4π Nf

26 acroscopic average Summary We have defined microscopic and macroscopic fields icroscopic uantities have to be averaged to be compared to experiments The dielectric function has - a microscopic expression related to uantum mechanics - a macroscopic expression classical scheme - axwell's euations Absorption Im { } and EELS -Im {1/ }

27 Outline 1. Introduction: which uantities do we need 2. acroscopic average - Definition - Examples Dielectric tensor for cubic symmetries Dielectric tensor for non-cubic symmetries - Properties - Principal axis Summary

28 Dielectric tensor for cubic symmetries Useful definitions Longitudinal fields Transverse fields " E r = 0 or k E k = 0. E r = 0 or k. E k = 0 Ek propagates along k Examples: plasmon oscillations sceening electron energy loss Ek propagates perpendicular to k Examples: photons optical properties of solids Some definitions: E r E k with k = + G for crystals Fourier transform is in the first Brillouin zone and G is a reciprocical lattice vector

29 Dielectric tensor for cubic symmetries Transverse-longitudinal decomposition: Any vector field can be split into longitudinal and transverse components E = E L + E T with E L k = ˆk ˆk. E k " # k $ and ˆk = k " E L = 0 and. E T = 0 In real space the relations are nonlocal acroscopic dielectric tensor The relation D = E tot can be written in terms of the longitudinal and transverse components D D L T = LL TL LT TT E E L tot T tot

30 Dielectric tensor for cubic symmetries Question: How can we make the link between the microscopic dielectric tensor D + G = G' + G + G' E tot + G' icroscopic components of D and E tot the macroscopic dielectric tensor D = E tot acroscopic components of D and E tot

31 Dielectric tensor for cubic symmetries No symmetry " = # # " LL TL LT TT A longitudinal transverse perturbation induces longitudinal and transverse responses. $ & & % Cubic symmetry with 0 0 LL 0 = TT A longitudinal perturbation induces a longitudinal response only. A transverse perturbation induces a transverse response only. Independent of the direction of This holds only for macroscopic uantities The microscopic dielectric tensor has off-diagonal elements LT and TL

32 Cubic symmetries with 0 Longitudinal dielectric function LL = lim π χ 2 where χ ρρ is the density-density response function TDDFT defined as ρ = χ ρρ V ind ρρ ext Transverse dielectric function lim TT = " LL 0 Dielectric tensor The tensor is diagonal and contains only one uantity LL

33 Cubic symmetries with 0 Longitudinal dielectric function One can show that the relation LL " = holds also when # 2 1 $ %% " TT LL Transverse dielectric function TL LT We have also 0 0 depends on These uantities are much more complicated and need further approximation to be computed cannot be expressed in terms of TDDFT.

34 Cubic symmetries Summary We have defined the longitudinal and transverse components of the dielectric tensor. In the long wavelength limit 0 only one uantity is needed optical isotropy LL TT 1 = = lim 0 4π 1+ χ 2 ρρ For 0 only LL has a simple expression in terms of the response function.

35 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.. Pick in Advances in Physics Vol 19 p D. L. Johnson Physical Review B S. L. Adler Physical Review N. Wiser Physical Review R. Del Sole and E. Fiorino Phys. Rev. B W. L. ochan and R. Barrera Phys. Rev. B ; ibid

36 Outline 1. Introduction: which uantities do we need 2. acroscopic average - Definition - Examples Dielectric tensor for cubic symmetries Dielectric tensor for non-cubic symmetries - Properties - Principal axis Summary

37 Non-cubic symmetries Properties of the macroscopic uantities = TT TL LT LL

38 Non-cubic symmetries Dielectric tensor - General case But one can show that the relation : holds also for the non-cubic symmetries. + + = 4 1 ˆ ˆ πα α π πα LL πα LL LL = Quasipolarisability α : ' ' ' α G E G G G j pert G ind = + and we have 1 2 χ α ρρ LL = Longitudinal-longitudinal dielectric function

39 Outline 1. Introduction: which uantities do we need 2. acroscopic average - Definition - Examples Dielectric tensor for cubic symmetries Dielectric tensor for non-cubic symmetries - Properties - Principal axis Summary

40 Non-cubic symmetries 0 ain general result concerning : is an analytic function of The limit 0 does not depend on the direction of. We can define as " = lim " 0 is not analytic in the general case LL If Depending on the symmetry of the system one can define the 3 principal axis if they exist n 1 n 2 n 3 defining a frame in which is diagonal. E tot is parallel to one of these axis n i " E tot " = i " E tot " LL i can be calculated as a longitudinal dielectric function = i n i

41 Non-cubic symmetries 0 The distinction between longitudinal and transverse is not meaningful The only important direction is the direction of the electric field If 0 the fields do not propagate Existence of the principal frame? is symmetric but complex No general answer Use of geometrical arguments Symmetries Cubic Hexagonal Orthorombic onoclinic Triclinic

42 Shorter wavelength 0 Alternative: + + = 4 1 ˆ ˆ πα α π πα LL ' ' ' α G E G G G j pert G ind = + We can use where The induced current can be evaluated through the Time-Dependent-Density-Current Functional Theory TD-DCFT

43 Which uantities do we need? Electron Energy Loss Spectroscopy: Im In that case = " LL with " 1 # $ " % & ' V ext = " V tot Is this correct? The perturbation is longitudinal. What about the transverse response? One can show that E T = 2 D T and c c 2 v2 2 c 2 In the nonrelativistic approximation the transverse fields are negligible and the LL component of the dielectric tensor describes the energy loss of charged particles

44 Outline 1. Introduction: which uantities do we need 2. acroscopic average - Definition - Examples Dielectric tensor for cubic symmetries Dielectric tensor for non-cubic symmetries - Properties - Principal axis Summary

45 Summary The key uantity is the dielectric tensor. Relation between microscopic and macroscopic fields. For cubic crystals the longitudinal dielectric function defines entirely the optical response in the long wavelength limit 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: LL only can be defined in a simple way.