Light-Matter Interactions Light-Matter Interactions Peter Oppeneer - PowerPoint PPT Presentation

Light-Matter Interactions Light-Matter Interactions Peter Oppeneer Department of Physics and Astronomy Uppsala University, S-751 20 Uppsala, Sweden 1

Outline – Lecture I – General Overview  Introduction – phenomenology  Electronic information vs. structural information • Electronic structure picture of materials  Theory/understanding of light-matter interactions • The classical fields’ description • Quantum theory with classical fields • Complete quantum field theory 2

Outline – Lecture II – Light-magnetism Interaction  Phenomenology of magnetic spectroscopies  Electronic structure theory, linear-response theory  Theory/understanding of magnetic spectroscopies • Optical regime • Ultraviolet and soft X-ray regime 3

Phenomenology – Types of light-matter interactions X-ray light Transmitted, absorbed, Emitted e- diffracted light reflected light Gain information on two main information areas: electronic & magnetic structure and structural information (with many subdivisions each) 4

Pure structural information X-ray diffraction Positions (crystallography, biomolecules, phonons, etc) Often very hard x-rays ! 5

Electronic structure information Excitation by photon of one electronic state to another one, provides information on the materials’ electronic structure E Can measure in Free electron photon-in / photon-out states set-up, or photon-in / electron-out photon Information on binding E energies, unoccupied F states, spin- and orbital Valence band properties, electron distributions, quasi-particles etc. Core level 6

Detailed electronic structure information High-resolution Angular Resolved Photoemission Spectroscopy (ARPES) Observation of shadow bands FeSe/STO Rebec et al, PRL 118 , 067002 (2017) Superconducting gap FeSe/SrTiO 3 Lee et al, Nature 515 , 245 (2014) 7

Further example: X-ray absorption Transmission Absorption coefficient m L 3 L 2 Sample Beer-Lambert law 3 d Detailed understanding Resonant excitation of dipole L allowed transitions at edges 3 2 p 3 / 2  2 p ; 2 p 3 d 2 p 1 / 2 3 / 2 1 / 2 L 2 8

Core-level absorption edges Basic electronic structure: Positions of the core levels (here of a 4p element) Spin-orbit split states ~16 eV (3d element) SO splitting of core states:      j s 1 2 1       j s 1 2 0  Increase in absorption at each edge 9

Magnetic information - XAS of ferromagnetic materials        i / c ( n z ) i t X-ray magnetic circular dichroism E ( z , t ) ( e i e ) e   x y XMCD Provides a powerful tool to measure element-selectively the atomic magnetic moment 10

Fundamentals of Light – Matter Theory Theory/understanding of light-matter interactions – 3 levels • The classical fields’ description • Quantum theory with classical fields • Complete quantum field theory First level: Maxwell theory and Fresnel theory (classical fields), macroscopic materials’ quantities (no quantum physics) Second level: Maxwell theory and Fresnel theory (classical fields), materials’ quantities given by quantum theory for materials Third level: Quantized photon fields, coupled to quantum theory for materials (i.e., 2 nd quantization of photon fields) 11

First level: Maxwell-Fresnel theory To describe the interaction between matter and the E-M wave field there are several ingredients: (1) eigenwaves in vacuum & material and (2) the boundary conditions Both (1) & (2) follow from Materials equations are also needed: the Maxwell equations: D : displacement field E : electrical field B : magnetic induction H : magnetic field (in CKS units!) j : current density r : charge density 12

Materials relations Just as important are the materials relationships : With the material specific(!) tensors: e : permittivity tensor m : permeability tensor s : conductivity tensor And: P : electrical polarization M : magnetization Note: we use here e 0 =1, m 0 =1 Note: materials fields are These equations are valid not uniquely defined. for constant e , m , and s . This is usually not the case! 13

A closer look at the materials relationships If we don ´ t have constant material ´ s tensors, things become nastier when we consider the full dependence on the space and time coordinates: (homogeneous approximation!) But, going to reciprocal space makes life easy again ! With the material specific(!) tensors: e : permittivity tensor m : permeability tensor And: s : conductivity tensor     j ind . P t 14

Consequences of Maxwell equations Solutions of the M.E. for isotropic medium: transverse plane E-M waves: Light is a transverse E-M wave 15

Index of refraction The plane-wave solution is possible under the condition: em em em k   2 2       2 2 ( k k ) E E , ( k k ) B B 2 2 2 c c c Dispersion relation c k    em  Index of refraction: n n , n c v  Remarks: 1) For materials e , m are complex n is complex & vector m =1 2) The ”spins cannot follow the rapid moving H field”   em  e  n ( ) ( ) (Dispersion relation) Nonetheless, all magnetic information is acounted for (see later) 16

Measured relative permeability m (  ) Ni Kittel, Phys. Rev. 70 , 281 (1946) 1 eV = 0.25 10 15 Hz Arguments 1) no unique separation between D and H in the Maxwell equations 2) physically: „spins cannot follow the rapidly varying B field“     ( k 2 0 ) m (  ) => 1 at optical frequencies, 17

Energy dispersion of optical constants In the x-ray regime, n is close to one and complex:         n ( ) 1 ( ) i ( )  can be positive or negative ! Also,  and  do depent on the magnetization ! Eventhough  and  are small they can be measured accurately at modern synchrotrons 18

Fresnel equation for the material A combination of the M-E leads to the following wave equation in the material : This is similar to the equation for the isotropic, constant e case Substitute: Gives us the Fresnel equation: The solution gives 2 n in the  ( n : n ) n n material and the eigen modes E 0 ij i j Note: we used m =1 Written in full (SI), it would be: 19

Fresnel equation, continued The symmetry of e tensor is an important ingredient for solving the Fresnel equation. In short, one needs to know about the crystallographic and magnetic symmetry of the material ! Some examples for non-magnetic materials: e e e         xx xx xy Cubic: e  e e  e e     Monoclinic: xx xy yy     e e (1 quantity)     (4 quantities) xx zz (biaxial) e     e e e Tetragonal   xx   xx xy xz e  e hexagonal   e  e e e   Triclinic: xx xy yy yz     trigonal e e e e     (6 quantities) zz xz yz zz (2 quantities) (uniaxial) (biaxial) 20

Example of Fresnel equation for magnetic medium Magnetic medium, M||z : Dielectric tensor: z E y k x Why? Consequence of magnetism! M Look at s tensor:  4 i e  s s  , 0 because of the magnetism! xy xy xy  Hall current, s xy (SI units: ) 21

Examples magnetic Fresnel equation, continued Solve Fresnel equation: M z n 2  e xx    e xy   n 2  e xx  e xy  0   y n 2  e zz  n 2     x  e zz [( n 2  e xx ) 2  e xy 2 ]  0 Hall current s xy  ( n 2  e xx ) 2   e xy 2  n 2  e xx  i e xy Note: s xy = - s yx e   e Thus: There are two solutions: yx xy   e  e 2 2 n n i  1 , 2 xx xy 22

Fresnel equation, magnetic case Eigenmodes:  e  e     i E   xy xy x    E    i 1    x   e  e       i E 0 0 & E 0 ,         xy xy y   z E 1 i         y  e E     zz z   E   1 1  x      (normalized eigenmodes!)      E i 2     y  Solutions are circularly polarized 1     i / c ( n r ) i t    E ( r , t ) ( e i e ) e  waves (in the material):  x y 2 One circularly polarized wave with helicity + corresponds to n + , the other one with helicity - to n -  c 2    This situation is called ”magnetic circular dichroism”, i.e. 2 colors n  (will apply this to XAS/XMCD in Lecture II) 23

Materials ´ boundary conditions R Experiments always require at least two different media Next to the Fresnel equation (1) M we must also know the „matching“ conditions (2) at the boundaries ! T These will follow (again) from the Maxwell equations Continuity of temporal and spacial wave parts at interface 1) Snell ´ s law 2) reflection/transmission coefficients Convenient: Jones vector formulation: E s   E   2-dim. vector s  E   E   E p p 24