Chapter 11: Dielectric Properties of Materials


 Jessie Richards
 11 months ago
 Views:
Transcription
1 Chapter 11: Dielectric Properties of Materials Lindhardt January 30, 2017 Contents 1 Classical Dielectric Response of Materials Conditions on ɛ Kramer s Kronig Relations Absorption of E and M radiation Transmission Reflectivity Model Dielectric Response The Freeelectron gas 18 4 Excitons 20 1
2 Electromagnetic fields are essential probes of material properties IR absorption Spectroscopy The interaction of the field and material may be described either classically or Quantum mechanically. We will first do the former. 1 Classical Dielectric Response of Materials Classically, materials are characterized by their dielectric response of either the bound or free charge. Both are described by Maxwells equations and Ohm s law E = 1 c B t, H = 4π c j + 1 c D t (1) j = σe. (2) Both effects may be combined into an effective dielectric constant ɛ, which we will now show. For an isotropic medium, we have 2
3 x e x 0 x << λ E(x,t) E(x,t) 0 λ < G ε(k,ω) ε(ω) k < Figure 1: If the average excursion of the electron is small compared to the wavelength of the radiation < x > λ, then we may ignore the wavevector dependence of the radiation so that ɛ(k, ω) ɛ(ω). D(ω) = ɛ(ω)e(ω) (3) where E(t) = dωe iωt E(ω) H(t) = D(t) = dωe iωt D(ω) B(t) = dωe iωt H(ω) (4) dωe iωt B(ω) (5) Then H = 4π c j + 1 c E(ω) = E ( ω) E(t) R (6) D t dωe iωt H(ω) = 4π c dωe iωt j(ω)+ 1 c t dωe iωt D(ω) (7) 3
4 dωe iωt { H(ω) 4π c j(ω) 1 c ( iω)d(ω) } = 0 (8) H = 4π c j iω c D(ω) = 4πσ c E iω c ɛe i ω c = E ( ɛ c ω ) 4πσ c = iω c E ɛ 4π c E ( σ c iω 4π c ɛ) = 4π c E σ (9) Thus we could either define an effective conductivity σ = σ iωɛ 4π which takes into account dielectric effects, or an effective dielectric constant ɛ = ɛ + i 4πσ ω, which accounts for conduction. 1.1 Conditions on ɛ From the reality of D(t) and E(t), one has that E(+ω) = E ( ω) and D(ω) = D ( ω), hence for D = ɛe, ɛ(ω) = ɛ ( ω) (10) 4
5 Additional constraints are obtained from causality D(t) = dωɛ(ω)e(ω)e iωt dt = dωɛ(ω)e iωt 2π eiωt E(t ) = 1 dtdω (ɛ(ω) 1 + 1) E(t )e iω(t t ) 2π then we make the substitution χ(ω) ɛ 1 4π (11) (12) D(t) = 2 dtdω ( ) χ(ω) + 1 4π E(t )e iω(t t ) D(t) = E(t) + 2 dtdωχ(ω)e(t )e ω(t t ) (13) Define G(t) 2 dωχ(ω)e iωt, then D(t) = E(t) + dt G(t t )E(t ) (14) Thus, the electric displacement at time t depends upon the field at other times; however, it cannot depend upon times t > t by causality. Hence G(τ) 0 τ < 0 (15) 5
6 no poles τ < 0 X poles X Figure 2: If χ(ω) is analytic in the upper half plane, then causality is assured. This can be enforced if χ(ω) is analytic in the upper half plane, then we may close G(τ) = 2 dωχ(ω)e iωτ = 2 dωχ(ω)e iωτ 0 (16) contour in the upper half plane and obtain zero for the integral since χ is analytic within and on the contour. 1.2 Kramer s Kronig Relations One may also derive an important relation between the real and imaginary parts of the dielectric function ɛ(ω) using this analytic property. From the Cauchy integral formula, if χ is 6
7 analytic inside and on the contour C, then χ(z) = 1 χ(ω ) (17) 2πi C z ω dω Then taking the contour shown in Fig. 3 and assuming that ω ω. Figure 3: The contour (left) used to demonstrate the Kramer s Kronig Relations. Since the contour must contain ω, we must deform the contour so that it avoids the pole 1/(ω ω ). χ(ω) < 1 ω for large ω (in fact Reχ < 1 ω and Imχ < 1 ω 2 ) we may ignore the large semicircle. Let z = ω + i0 +, so that χ(ω + i0 + ) = 1 2πi 2πiχ(ω + i0 + ) = P χ(ω )dω ω ω i0 + (18) χ(ω )dω + iπχ(ω) (19) ω ω χ(ω) = 1 iπ P dω χ(ω ) (20) ω ω Then let 4πχ(ω) = ɛ(ω) 1 = ɛ 1 + iɛ 2 1 and we get ɛ 1 (ω) + iɛ 2 (ω) 1 = 2 π P 7 iɛ 1 (ω ) + ɛ 2 (ω ) + 1 ω ω dω (21)
8 or ɛ 1 (ω) 1 = 1 π P ɛ 2 (ω) = 1 π P ɛ 2 (ω ) ω ω dω (22) ɛ 1 (ω ) 1 ω ω dω (23) which are known as the Kramer s Kronig relations. Many experiments measure ɛ 2 and from Eq. 22 we can calculate ɛ 1! 2 Absorption of E and M radiation 2.1 Transmission iω(tnx/c) ~ ξ = ξ e 0 n ~ = 1 n ~ = 1 ~ n = n + iκ = ε(ω) ε ~ 2 n = n +2inκ  κ 2 = ε + iε ε = n  κ 1 2 ε = 2nκ Figure 4: In transmission experiments a laser beam is focused on a thin slab of some material we wish to study. Imagine that a laser beam of known frequency is normally 8
9 incident upon a thin slab of some material (see Fig. 4), and we are able to measure its transmitted intensity. Upon passing through a boundary, part of the beam is reflected and part is transmitted (see Fig. 5). In the case of normal incidence, it is easy to calculate the related coefficients from the conditions of continuity of E and H = B (µ = 1) B 0 ξ 0 X X ξ µ = 1 µ = 1 B X ξ B Figure 5: The assumed orientation of electromagnetic fields incident on a surface. E = 1 c E 0 + E = E (E 0 E ) = ne B t, ik E = iω c B (24) ne = B (25) t = 2 n + 1 (26) In this way the other coefficients may be calculated (see Fig. 6). Accounting for multiple reflections the total transmitted field is 9
10 t 1 r 1 r 2 t 2 t = 2 1 ~ t 2 = r = n + 1 2n ~ ~ n + 1 ~ n ~ n + 1 Figure 6: Multiple events contribute the the radiation transmitted through and reflected from a thin slab. E = E 0 t 1 t 2 e ikd + E 0 t 1 r 2 r 2 t 2 e 3ikd +, where k = ñω c (27) E = E 0t 1 t 2 e idñω/c 1 r2 2 (28) e2idñω/c If ñ 1 and d is small, then E E 0 e idñω/c (29) I I 0 e idω(ñ ñ )/c = I 0 e dω2κ/c (30) κ = ɛ 2 2 (31) ɛ 1 ( I I 0 e (ɛ2ω/c)d I 0 1 ɛ ) 2ω c d (32) If the thickness d is known, then the quantity ωɛ 2 (ω) c = 4πσ 1(ω) c 10 (33)
11 may be measured (the absorption coefficient). The real part of the dielectric response, ɛ 1 (ω) may be calculated with the Kramers Kronig relation ɛ 1 (ω) = π P ɛ 2 (ω ) ω ω dω (34) According to Young Kim, this analysis works for sufficiently thin samples in the optical regime, but typically fails in the IR where ɛ 2 becomes large. 2.2 Reflectivity Of course, we could also have performed the experiment on a very thick slab of the material, and measured the reflectivity R However, this is a much more complicated experiment since R I depends upon both ɛ 1 and ɛ 2 (ω), and is hence much more difficult to analyze [See Frederick Wooten, Optical Properties of Solids, (Academic Press, San Diego, 1972)]. To analyze these experiments, one must first measure the reflectivity, R(ω) over the entire frequency range. We then write R(ω) = r(ω)r (ω) = (n 1)2 + κ 2 (n + 1) 2 (35) + κ 2 11
12 ~ n = 1 R I = ~ n  1 ~ n +1 2 ~ n Figure 7: The coefficient of reflectivity (the ratio of the reflected to the incident intensities) R/I depends upon both ɛ 1 (ω) and ɛ 2 (ω), making the analysis more complicated than in the transmission experiment. where ñ = n + iκ and r(ω) = ρ(ω)e iθ (36) so that R(ω) = ρ(ω) 2. If ρ(ω) 1 and θ 0 fast enough as the frequency increases, then we may employ the Kramers Kronig relations replacing χ with ln r(ω) = ln ρ(ω) + iθ(ω), so that θ(ω ) = 2ω π P ln R(ω) dω (37) 0 ω 2 ω 2 Thus, if R(ω) is measured over the entire frequency range where it is finite, then we can calculate θ(ω). This complete knowledge of R is generally not available, and various extrapolation and fitting schemes are used on R(ω) so that the integral above may be completed. 12
13 We may then use Eq. 35 above to relate R and θ to the real and imaginary parts of the refractive index, n(ω) = 1 R(ω) 1 + R(ω) 2 R(ω) cos θ(ω) (38) κ(ω) = 2 R(ω) sin θ(ω) 1 + R(ω) 2 R(ω) cos θ(ω), (39) and therefore the dielectric response ɛ(ω) = (n(ω) + κ(ω)) 2 and the optical conductivity σ(ω) = iωɛ 4π. 2.3 Model Dielectric Response We have seen that EM radiation is a sensitive probe of the dielectric properties of materials. Absorption and reflectivity experiments allow us to measure some combination of ɛ 1 or ɛ 2, with the remainder reconstructed by the KramersKronig relations. In order to learn more from such measurements, we need to have detailed models of the materials and their corresponding dielectric properties. For example, the electric field will interact with the moving charges associated with lattice vibrations. At the simplest level, we can model this as the interaction of iso 13
14 lated dipoles composed of bound damped charge e and length s p = e s (40) The equation of motion for this system is µ s = µγṡ µω0s 2 + e E (41) where the first term on the righthand side is the damping force, the second term is the restoring force, and the third term is the external field. Furthermore, the polarization P is P = N V e s + N V αe (42) where α is the polarizability of the different molecules which make up the material. It is to represent the polarizability of rigid bodies. For example, (see Fig. 8) a metallic sphere has a E = 0 Figure 8: We may model our material as a system harmonically bound charge and of metallic spheres with polarizablity α = a 3. α = a 3 (43) 14
15 If we F.T. these two equations, we get and µω 2 s = iωµrs µω 2 0s + e E(ω) (44) s(ω) { ω 2 0 ω 2 iωγ } = e E(ω) µ (45) P (ω) = ne s(ω) + nαe(ω) (46) or { ne 2 } /µ P (ω) = E(ω) ω0 2 ω2 iωγ + nα (47) The term in brackets is the complex electric susceptibility of the system χ = E/P. χ(ω) = nα + ɛ(ω) = 1 + 4πnα + 4πne 2 /µ ω 2 0 ω2 iγω, ne 2 /µ ω0 2 ω2 iωγ = 1 (ɛ(ω) 1) (48) 4π Or, introducing the high and zero frequency limits where ω2 p = 4πne 2 µ (49) ɛ = 1 + 4πnα (50) ɛ 0 = 1 + 4πnα + ω2 p ω 2 0 = ɛ + ω2 p ω 2 0 ɛ(ω) = ɛ + ω2 0(ɛ 0 ɛ ) ω 2 0 ω2 iγω (51) (52) 15
16 For our causality arguments we must have no poles in the upper complex half plane. This is satisfied since γ > 0. In addition, we need χ = 1 (ɛ(ω) 1) 0, as ω (53) 4π This means that ɛ = 1 + 4πnα = 1. Of course α is finite. The problem is that in making α = constant, we neglected the electron mass. Ie., for our example of a metallic sphere, α < a 3 for very high ω! (See Fig. 9) due to the finite electronic mass. a ξ(ω) Figure 9: For very high ω, α < a 3 since the electrons have a finite mass and hence cannot respond instantaneously to changes in the field. To analyze experiments ɛ is separated into real ɛ 1 and imaginary parts ɛ 2 ɛ 1 (ω) = 4π ω σ 2 = ɛ + (ɛ 0 ɛ )ω 2 0(ω 2 0 ω 2 ) (ω 2 0 ω2 ) 2 + γ 2 ω 2 (54) 16
17 ɛ 2 (ω) = 4π ω σ 1 = (ɛ 0 ɛ )ω 2 0γω (ω 2 0 ω2 ) 2 + γ 2 ω 2 (55) Thus a phonon mode will give a roughly Lorentzianlike line ε (ω) 1 ε (ω) 2 ε 0 γ ε ω 0 ω 0 ω Figure 10: Sketch of the real and imaginary parts of the dielectric response of the harmonically bound charge model. shape in the optical conductivity σ 1 centered roughly at the phonon frequency. In addition, this form may be used to construct a model reflectivity R = ñ 1 2 / ñ + 1 2, with ñ = ɛ which is often appended to the high end of the reflectivity data, so that the KramersKronig integrals may be completed. 17
18 3 The Freeelectron gas Metals have a distinct feature in their optical conductivity σ 1 (ω) which may be emulated by the freeelectron gas. The equation of motion of the freeelectron gas is In steady state γṡ = nee; however nm s = γṡ nee (56) neṡ = j = σe = ne2 τ m E (57) in the relaxation time approximation, so ṡ = eτ m E = ne γ E or γ = nm τ. (58) Thus, the equation of motion is nm s = nm τ ṡ nee (59) If we work in a Fourier representation, then mω 2 s(ω) = iωm s(ω) ee(ω) (60) τ However, since P = ens ( mω 2 + iωm ) P = ne 2 E(ω) (61) τ 18
19 or P = ne2 /m ω 2 + iω/τ E = χe = 1 (ɛ 1)E (62) 4π ɛ = 1 4πn2 e 2 { } 1 mω ω + i/τ = 1 ω2 p ω i/τ (63) ω ω 2 + 1/τ 2 ɛ 1 = 1 ω2 p ω { ω ω 2 + 1/τ 2 }, ɛ 2 = ω2 p 1 (64) τω ω 2 + 1/τ 2 However, recall that ɛ 1 = 4πσ 2 ω and ɛ 2 = 4πσ 1 ω, so that σ 1 (ω) = ω2 p 4 1/τπ ω 2 + 1/τ 2. (65) σ (ω) 1 Drude peak (electronic) σ (ω) dω = ω 2 /4 1 p 1/τ phonons Figure 11: A sketch of the optical conductivity of a metal at finite temperatures. The lowfrequency Drude peak is due to the coherent transport of electrons. The higher frequency peak is due to incoherent scattering of electrons from phonons or electronelectron interactions. 19
20 4 Excitons One of the most dramatic effects of the dielectric properties of semiconductors are excitons. Put simply, an exciton is a hydrogenic bound state made up of a hole and an electron. The Hamiltonian for such a system is H = 1 2 m hv 2 h m ev 2 e e2 ɛr As usual, we will work in the center of mass, so (66) K electron hole E g R r O Figure 12: Excitons are hydrogenic bound states of an electron and a hole. In semiconductors with an indirect gap, such as Si, they can be very long lived, since a phonon or defect must be involved in the recombination process to conserve momentum. For example, in ultrapure Si ( impurities per cc) the lifetime can exceed τ Si 10 5 s; whereas in directgap GaAs, the lifetime is much shorter τ GaAs 10 9 s and is generally limited by surface states. On the right, the coordinates of the exciton in the center of mass are shown. 1 2 m hv 2 h m ev 2 e = 1 2 (m h + m e)ṙ2 + µṙ 2 (67) 20
21 R = m e x e + m h x h m h +, r = x e x h (68) m e The eigenenergies may be obtained from the Bohr atom solution E n = µ(ze 2 ) 2 /(2 h 2 n 2 ) by making the substitutions Ze 2 e2 ɛ (69) µ µ = m h m e m h + m e In addition there is a cm kinetic energy, let hk = P, then E nk = (70) h 2 K 2 2(m h + m e) µ e 4 ɛ 2 2 h 2 n 2 + E g (71) The binding energy is strongly reduced by the dielectric effects, since ɛ 10. (it is also reduced by the effective masses, since typically m < m). Thus, typically the binding energy is a small fraction of a Ryberg. Similarly, the size of the exciton is much larger than the hydrogen atom a ɛµ m e a 0 (72) This in fact is the justification for the hydrogenic approximation. Since the orbit contains many sites the lattice may be approximated as a continuum and thus the exciton is well approximated as a hydrogenic atom. 21
22 e + 10A e Figure 13: An exciton may be approximated as a hydrogenic atom if its radius is large compared to the lattice spacing. In Si, the radius is large due to the reduced hole and electron masses and the enhanced dielectric constant ɛ
Macroscopic dielectric theory
Macroscopic dielectric theory Maxwellʼs equations E = 1 c E =4πρ B t B = 4π c J + 1 c B = E t In a medium it is convenient to explicitly introduce induced charges and currents E = 1 B c t D =4πρ H = 4π
More information9 The conservation theorems: Lecture 23
9 The conservation theorems: Lecture 23 9.1 Energy Conservation (a) For energy to be conserved we expect that the total energy density (energy per volume ) u tot to obey a conservation law t u tot + i
More information9 The conservation theorems
9 The conservation theorems 9.1 Energy Conservation (a) For energy to be conserved we expect that the total energy density (energy per volume ) u tot to obey a conservation law t u tot + i S i tot = 0
More informationTime dependent perturbation theory 1 D. E. Soper 2 University of Oregon 11 May 2012
Time dependent perturbation theory D. E. Soper University of Oregon May 0 offer here some background for Chapter 5 of J. J. Sakurai, Modern Quantum Mechanics. The problem Let the hamiltonian for a system
More informationOptical Properties of Semiconductors. Prof.P. Ravindran, Department of Physics, Central University of Tamil Nadu, India
Optical Properties of Semiconductors 1 Prof.P. Ravindran, Department of Physics, Central University of Tamil Nadu, India http://folk.uio.no/ravi/semi2013 Light Matter Interaction Response to external electric
More informationchiral m = n Armchair m = 0 or n = 0 Zigzag m n Chiral Three major categories of nanotube structures can be identified based on the values of m and n
zigzag armchair Three major categories of nanotube structures can be identified based on the values of m and n m = n Armchair m = 0 or n = 0 Zigzag m n Chiral Nature 391, 59, (1998) chiral J. Tersoff,
More informationPhysics of Condensed Matter I
Physics of Condensed Matter I 11004INZ`PC Faculty of Physics UW Jacek.Szczytko@fuw.edu.pl Dictionary D = εe ε 0 vacuum permittivity, permittivity of free space (przenikalność elektryczna próżni) ε r relative
More informationAtomic cross sections
Chapter 12 Atomic cross sections The probability that an absorber (atom of a given species in a given excitation state and ionziation level) will interact with an incident photon of wavelength λ is quantified
More informationChapter 2 Optical Transitions
Chapter 2 Optical Transitions 2.1 Introduction Among energy states, the state with the lowest energy is most stable. Therefore, the electrons in semiconductors tend to stay in low energy states. If they
More informationFourier transform spectroscopy and the study of the optical quantities by farinfrared Reflectivity measurements
1 Fourier transform spectroscopy and the study of the optical quantities by farinfrared Reflectivity measurements Pradeep Bajracharya Department of Physics University of Cincinnati Cincinnati, Ohio 45221
More informationOverview in Images. S. Lin et al, Nature, vol. 394, p , (1998) T.Thio et al., Optics Letters 26, (2001).
Overview in Images 5 nm K.S. Min et al. PhD Thesis K.V. Vahala et al, Phys. Rev. Lett, 85, p.74 (000) J. D. Joannopoulos, et al, Nature, vol.386, p.1439 (1997) T.Thio et al., Optics Letters 6, 1971974
More information5.74 Introductory Quantum Mechanics II
MIT OpenCourseWare http://ocw.mit.edu 5.74 Introductory Quantum Mechanics II Spring 9 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. Andrei Tokmakoff,
More informationKeywords: Normal dispersion, Anomalous dispersion, Absorption. Ref: M. Born and E. Wolf: Principles of Optics; R.S. Longhurst: Geometrical
28 Dispersion Contents 28.1 Normal dispersion 28.2 Anomalous Dispersion 28.3 Elementary theory of dispersion Keywords: Normal dispersion, Anomalous dispersion, Absorption. Ref: M. Born and E. Wolf: Principles
More informationJaroslav Hamrle. October 21, 2014
Magnetooptical Kerr effect (MOKE) Jaroslav Hamrle (jaroslav.hamrle@vsb.cz) October 21, 2014 Photonphoton spectroscopies (absorption) I: Type of investigations (polarized light x nonpolarized light,
More informationE E D E=0 2 E 2 E (3.1)
Chapter 3 Constitutive Relations Maxwell s equations define the fields that are generated by currents and charges. However, they do not describe how these currents and charges are generated. Thus, to find
More informationNONLINEAR OPTICS. Ch. 1 INTRODUCTION TO NONLINEAR OPTICS
NONLINEAR OPTICS Ch. 1 INTRODUCTION TO NONLINEAR OPTICS Nonlinear regime  Order of magnitude Origin of the nonlinearities  Induced Dipole and Polarization  Description of the classical anharmonic oscillator
More informationOSE5312 LightMatter Interaction. David J. Hagan and Pieter G. Kik
OS5 LightMatter Interaction David J. Hagan and Pieter G. Kik This document represents the course notes for CROL course OS5. The text is a work in progress, so if you have any comments, please let us know!
More information9 Atomic Coherence in ThreeLevel Atoms
9 Atomic Coherence in ThreeLevel Atoms 9.1 Coherent trapping  dark states In multilevel systems coherent superpositions between different states (atomic coherence) may lead to dramatic changes of light
More informationSet 5: Classical E&M and Plasma Processes
Set 5: Classical E&M and Plasma Processes Maxwell Equations Classical E&M defined by the Maxwell Equations (fields sourced by matter) and the Lorentz force (matter moved by fields) In cgs (gaussian) units
More informationOptical Properties of Metals. Institute of Solid State Physics. Advanced Materials  Lab Intermediate Physics. Ulm University.
Advanced Materials  Lab Intermediate Physics Ulm University Institute of Solid State Physics Optical Properties of Metals Burcin Özdemir Luyang Han April 28, 2014 Safety Precations MAKE SURE THAT YOU
More informationOptical Lattices. Chapter Polarization
Chapter Optical Lattices Abstract In this chapter we give details of the atomic physics that underlies the Bose Hubbard model used to describe ultracold atoms in optical lattices. We show how the ACStark
More informationPlasmons, Surface Plasmons and Plasmonics
Plasmons, Surface Plasmons and Plasmonics Plasmons govern the high frequency optical properties of materials since they determine resonances in the dielectric function ε(ω) and hence in the refraction
More informationSolidstate spectroscopy
Solidstate spectroscopy E.V. Lavrov Institut für Angewandte Physik, Halbleiterphysik Contents 1 Electromagnetic waves in media 4 1.1 Electromagnetic spectrum........................... 4 1.2 Maxwell equations...............................
More informationLecture 2: Dispersion in Materials. 5 nm
Lecture : Disersion in Materials 5 nm What Haened in the Previous Lecture? Maxwell s Equations Bold face letters are vectors! B D = ρ f B = E = t Curl Equations lead to E P E = με + μ t t Linear, Homogeneous,
More informationH ( E) E ( H) = H B t
Chapter 5 Energy and Momentum The equations established so far describe the behavior of electric and magnetic fields. They are a direct consequence of Maxwell s equations and the properties of matter.
More informationTypical anisotropies introduced by geometry (not everything is spherically symmetric) temperature gradients magnetic fields electrical fields
Lecture 6: Polarimetry 1 Outline 1 Polarized Light in the Universe 2 Fundamentals of Polarized Light 3 Descriptions of Polarized Light Polarized Light in the Universe Polarization indicates anisotropy
More information3. THE COMPLEX DIELECTRIC FUNCTION AND REFRACTIVE INDEX
3. THE COMPLEX DIELECTRIC FUNCTION AND REFRACTIVE INDEX 3.1 Conductivity and dielectric constant In this this chapter I ll define the complex dielectric function as I ll use it for the remainder of the
More informationOPTICAL PROPERTIES of Nanomaterials
OPTICAL PROPERTIES of Nanomaterials Advanced Reading Optical Properties and Spectroscopy of Nanomaterials Jin Zhong Zhang World Scientific, Singapore, 2009. Optical Properties Many of the optical properties
More informationLightmatter Interactions Of Plasmonic Nanostructures
University of Central Florida Electronic Theses and Dissertations Doctoral Dissertation (Open Access) Lightmatter Interactions Of Plasmonic Nanostructures 2013 Jennifer Reed University of Central Florida
More informationLinear Response and Onsager Reciprocal Relations
Linear Response and Onsager Reciprocal Relations Amir Bar January 1, 013 Based on Kittel, Elementary statistical physics, chapters 3334; Kubo,Toda and Hashitsume, Statistical Physics II, chapter 1; and
More informationNote on Group Velocity and Energy Propagation
Note on Group Velocity and Energy Propagation Abraham Bers Department of Electrical Engineering & Computer Science and Plasma Science & Fusion Center Massachusetts Institute of Technology, Cambridge, MA
More informationPolynomial Chaos Approach for Maxwell s Equations in Dispersive Media
Polynomial Chaos Approach for Maxwell s Equations in Dispersive Media Prof. Nathan L. Gibson Department of Mathematics Applied Mathematics and Computation Seminar March 15, 2013 Prof. Gibson (OSU) PCFDTD
More informationMicroscopic Ohm s Law
Microscopic Ohm s Law Outline Semiconductor Review Electron Scattering and Effective Mass Microscopic Derivation of Ohm s Law 1 TRUE / FALSE 1. Judging from the filled bands, material A is an insulator.
More informationEnhancing the Rate of Spontaneous Emission in Active CoreShell Nanowire Resonators
Chapter 6 Enhancing the Rate of Spontaneous Emission in Active CoreShell Nanowire Resonators 6.1 Introduction Researchers have devoted considerable effort to enhancing light emission from semiconductors
More informationvan Quantum tot Molecuul
10 HC10: Molecular and vibrational spectroscopy van Quantum tot Molecuul Dr Juan Rojo VU Amsterdam and Nikhef Theory Group http://www.juanrojo.com/ j.rojo@vu.nl Molecular and Vibrational Spectroscopy Based
More informationElectronphonon scattering (Finish Lundstrom Chapter 2)
Electronphonon scattering (Finish Lundstrom Chapter ) Deformation potentials The mechanism of electronphonon coupling is treated as a perturbation of the band energies due to the lattice vibration. Equilibrium
More informationInteraction of Molecules with Radiation
3 Interaction of Molecules with Radiation Atoms and molecules can exist in many states that are different with respect to the electron configuration, angular momentum, parity, and energy. Transitions between
More informationElectrons in metals PHYS208. revised Go to Topics Autumn 2010
Go to Topics Autumn 010 Electrons in metals revised 0.1.010 PHYS08 Topics 0.1.010 Classical Models The Drude Theory of metals Conductivity  static electric field Thermal conductivity Fourier Law WiedemannFranz
More information( ) /, so that we can ignore all
Physics 531: Atomic Physics Problem Set #5 Due Wednesday, November 2, 2011 Problem 1: The acstark effect Suppose an atom is perturbed by a monochromatic electric field oscillating at frequency ω L E(t)
More informationPhysics 442. ElectroMagnetoDynamics. M. Berrondo. Physics BYU
Physics 44 ElectroMagnetoDynamics M. Berrondo Physics BYU 1 Paravectors Φ= V + cα Φ= V cα 1 = t c 1 = + t c J = c + ρ J J ρ = c J S = cu + em S S = cu em S Physics BYU EM Wave Equation Apply to Maxwell
More informationChapter 3 Properties of Nanostructures
Chapter 3 Properties of Nanostructures In Chapter 2, the reduction of the extent of a solid in one or more dimensions was shown to lead to a dramatic alteration of the overall behavior of the solids. Generally,
More informationTheorie der Kondensierten Materie II SS Scattering in 2D: the logarithm and the renormalization group
Karlsruher Institut für Technologie Institut für Theorie der Kondensierten Materie Theorie der Kondensierten Materie II SS 2017 PD Dr. B. Narozhny Blatt 2 M. Sc. M. Bard Lösungsvorschlag 1. Scattering
More information6. Molecular structure and spectroscopy I
6. Molecular structure and spectroscopy I 1 6. Molecular structure and spectroscopy I 1 molecular spectroscopy introduction 2 lightmatter interaction 6.1 molecular spectroscopy introduction 2 Molecular
More informationTime Resolved Faraday Rotation Measurements of Spin Polarized Currents in Quantum Wells
Time Resolved Faraday Rotation Measurements of Spin Polarized Currents in Quantum Wells M. R. Beversluis 17 December 2001 1 Introduction For over thirty years, silicon based electronics have continued
More informationPhysics 221B Spring 2018 Notes 34 The Photoelectric Effect
Copyright c 2018 by Robert G. Littlejohn Physics 221B Spring 2018 Notes 34 The Photoelectric Effect 1. Introduction In these notes we consider the ejection of an atomic electron by an incident photon,
More informationTesting the validity of THz reflection spectra by dispersion relations
Testing the validity of THz reflection spectra by dispersion relations K.E. Peiponen *, E. Gornov and Y. Svirko Department of Physics, University of Joensuu, P. O. Box 111, FI 80101 Joensuu, Finland Y.
More informationPH575 Spring Lecture #26 & 27 Phonons: Kittel Ch. 4 & 5
PH575 Spring 2009 Lecture #26 & 27 Phonons: Kittel Ch. 4 & 5 PH575 Spring 2009 POP QUIZ Phonons are: A. Fermions B. Bosons C. Lattice vibrations D. Light/matter interactions PH575 Spring 2009 POP QUIZ
More informationOptical Spectroscopy of Advanced Materials
Phys 590B Condensed Matter Physics: Experimental Methods Optical Spectroscopy of Advanced Materials Basic optics, nonlinear and ultrafast optics Jigang Wang Department of Physics, Iowa State University
More informationOverview  Previous lecture 1/2
Overview  Previous lecture 1/2 Derived the wave equation with solutions of the form We found that the polarization of the material affects wave propagation, and found the dispersion relation ω(k) with
More informationCrystals. Peter Košovan. Dept. of Physical and Macromolecular Chemistry
Crystals Peter Košovan peter.kosovan@natur.cuni.cz Dept. of Physical and Macromolecular Chemistry Lecture 1, Statistical Thermodynamics, MC26P15, 5.1.216 If you find a mistake, kindly report it to the
More informationI e i ω t. Figure 3.0 Simple Dipole Antenna driven by the current source Ie ω
6. ANTENNAS AND RADIATED FIELDS Now that Maxwell's equations for the radiated field have been developed a practical result is appropriate. When Maxwell's equations are written in terms of E and B, they
More informationHomework 2: Solutions
Homework : Solutions. Problem. We have approximated the conduction band of many semiconductors with an isotropic, parabolic expression of the form: E(k) h k m. () However,this approximation fails at some
More informationNonlinear Optics (WiSe 2015/16) Lecture 12: January 15, 2016
Nonlinear Optics (WiSe 2015/16) Lecture 12: January 15, 2016 12 High Harmonic Generation 12.1 Atomic units 12.2 The three step model 12.2.1 Ionization 12.2.2 Propagation 12.2.3 Recombination 12.3 Attosecond
More informationElectromagnetic optics!
1 EM theory Electromagnetic optics! EM waves Monochromatic light 2 Electromagnetic optics! Electromagnetic theory of light Electromagnetic waves in dielectric media Monochromatic light References: Fundamentals
More informationNanophotonics and Metamaterials*
*)Krasnoyarsk Federal University, Russia lectures available on www.ece.purdue.edu/~shalaev Nanophotonics and Metamaterials* Professor Vladimir M. Shalaev 5 nm Overview of the Course Part I: a) Light interaction
More informationDr. Tao Li
Tao Li taoli@nju.edu.cn Nat. Lab. of Solid State Microstructures Department of Materials Science and Engineering Nanjing University Concepts Basic principles Surface Plasmon Metamaterial Summary Light
More informationPhysikalisches Praktikum für Fortgeschrittene 1 Dielectric Constant Assistent: Herr Dr. M. Ghatkesar by Therese Challand theres.
Physikalisches Praktikum für Fortgeschrittene 1 Dielectric Constant Assistent: Herr Dr. M. Ghatkesar by Therese Challand theres.challand stud.unibas.ch Contents 1 Theory 1 1.1 Introduction............................
More informationSpins and spinorbit coupling in semiconductors, metals, and nanostructures
B. Halperin Spin lecture 1 Spins and spinorbit coupling in semiconductors, metals, and nanostructures Behavior of nonequilibrium spin populations. Spin relaxation and spin transport. How does one produce
More informationIntroduction to Nonlinear Optics
Introduction to Nonlinear Optics Prof. Cleber R. Mendonca http://www.fotonica.ifsc.usp.br Outline Linear optics Introduction to nonlinear optics Second order nonlinearities Third order nonlinearities Twophoton
More informationSemiclassical Electron Transport
Semiclassical Electron Transport Branislav K. Niolić Department of Physics and Astronomy, University of Delaware, U.S.A. PHYS 64: Introduction to Solid State Physics http://www.physics.udel.edu/~bniolic/teaching/phys64/phys64.html
More informationInteraction Xrays  Matter
Interaction Xrays  Matter Pair production hν > M ev Photoelectric absorption hν MATTER hν Transmission Xrays hν' < hν Scattering hν Decay processes hν f Compton Thomson Fluorescence Auger electrons
More informationLecture 1: The Equilibrium Green Function Method
Lecture 1: The Equilibrium Green Function Method Mark Jarrell April 27, 2011 Contents 1 Why Green functions? 2 2 Different types of Green functions 4 2.1 Retarded, advanced, time ordered and Matsubara
More informationand the radiation from source 2 has the form. The vector r points from the origin to the point P. What will the net electric field be at point P?
Physics 3 Interference and Interferometry Page 1 of 6 Interference Imagine that we have two or more waves that interact at a single point. At that point, we are concerned with the interaction of those
More informationCrystal Properties. MS415 Lec. 2. High performance, high current. ZnO. GaN
Crystal Properties Crystal Lattices: Periodic arrangement of atoms Repeated unit cells (solidstate) Stuffing atoms into unit cells Determine mechanical & electrical properties High performance, high current
More informationOptomechanically induced transparency of xrays via optical control: Supplementary Information
Optomechanically induced transparency of xrays via optical control: Supplementary Information WenTe Liao 1, and Adriana Pálffy 1 1 MaxPlanckInstitut für Kernphysik, Saupfercheckweg 1, 69117 Heidelberg,
More informationLaser Diodes. Revised: 3/14/14 14: , Henry Zmuda Set 6a Laser Diodes 1
Laser Diodes Revised: 3/14/14 14:03 2014, Henry Zmuda Set 6a Laser Diodes 1 Semiconductor Lasers The simplest laser of all. 2014, Henry Zmuda Set 6a Laser Diodes 2 Semiconductor Lasers 1. Homojunction
More informationPenning Traps. Contents. Plasma Physics Penning Traps AJW August 16, Introduction. Clasical picture. Radiation Damping.
Penning Traps Contents Introduction Clasical picture Radiation Damping Number density B and E fields used to increase time that an electron remains within a discharge: Penning, 936. Can now trap a particle
More informationMolecular spectroscopy
Molecular spectroscopy Origin of spectral lines = absorption, emission and scattering of a photon when the energy of a molecule changes: rad( ) M M * rad( ' ) ' v' 0 0 absorption( ) emission ( ) scattering
More informationIn order to describe the propagation of the optical laser pulses in semiconductors, we
In order to describe the propagation of the optical laser pulses in semiconductors, we begin with the wave equation in its most general form. Taking into account both the laser fieldinduced polarization
More informationThe Formation of Spectral Lines. I. Line Absorption Coefficient II. Line Transfer Equation
The Formation of Spectral Lines I. Line Absorption Coefficient II. Line Transfer Equation Line Absorption Coefficient Main processes 1. Natural Atomic Absorption 2. Pressure Broadening 3. Thermal Doppler
More informationFluctuations of Time Averaged Quantum Fields
Fluctuations of Time Averaged Quantum Fields IOP Academia Sinica Larry Ford Tufts University January 17, 2015 Time averages of linear quantum fields Let E (x, t) be one component of the electric field
More informationTopical Review: optics of excitonplasmon nanomaterials. Maxim Sukharev 1 and Abraham Nitzan 2
1 Topical Review: optics of excitonplasmon nanomaterials Maxim Sukharev 1 and Abraham Nitzan 2 1 College of Integrative Sciences and Arts, Arizona State University, Mesa, Arizona 85212, USA 2 Department
More informationarxiv:physics/ v3 [physics.genph] 2 Jan 2006
A Wave Interpretation of the Compton Effect As a Further Demonstration of the Postulates of de Broglie arxiv:physics/0506211v3 [physics.genph] 2 Jan 2006 ChingChuan Su Department of Electrical Engineering
More informationExperimental Determination of Crystal Structure
Experimental Determination of Crystal Structure Branislav K. Nikolić Department of Physics and Astronomy, University of Delaware, U.S.A. PHYS 624: Introduction to Solid State Physics http://www.physics.udel.edu/~bnikolic/teaching/phys624/phys624.html
More informationOptical and Photonic Glasses. Lecture 37. NonLinear Optical Glasses I  Fundamentals. Professor Rui Almeida
Optical and Photonic Glasses : NonLinear Optical Glasses I  Fundamentals Professor Rui Almeida International Materials Institute For New Functionality in Glass Lehigh University Nonlinear optical glasses
More informationModule I: Electromagnetic waves
Module I: Electromagnetic waves Lecture 3: Time dependent EM fields: relaxation, propagation Amol Dighe TIFR, Mumbai Outline Relaxation to a stationary state Electromagnetic waves Propagating plane wave
More informationPhysics Scholarship Examination Questions
Physics Scholarship Examination Questions Mark Allen September 8, 2012 Contents 1 Thermodynamics 2 2 Optics 9 3 Oscillations 15 4 Electromagnetism 21 1 1 Thermodynamics 1. TODO: 2012 Q 1 2. TODO: 2012
More informationEUF. Joint Entrance Examination for Postgraduate Courses in Physics
EUF Joint Entrance Examination for Postgraduate Courses in Physics For the second semester 2015 14 April 2015 Part 1 Instructions Do not write your name on the test. It should be identified only by your
More informationQuantum transport in nanoscale solids
Quantum transport in nanoscale solids The Landauer approach Dietmar Weinmann Institut de Physique et Chimie des Matériaux de Strasbourg Strasbourg, ESC 2012 p. 1 Quantum effects in electron transport R.
More information221B Lecture Notes Scattering Theory II
22B Lecture Notes Scattering Theory II Born Approximation Lippmann Schwinger equation ψ = φ + V ψ, () E H 0 + iɛ is an exact equation for the scattering problem, but it still is an equation to be solved
More informationExciton spectroscopy
Lehrstuhl Werkstoffe der Elektrotechnik Exciton spectroscopy in wide bandgap semiconductors Lehrstuhl Werkstoffe der Elektrotechnik (WW6), Universität ErlangenNürnberg, Martensstr. 7, 91058 Erlangen Vortrag
More informationThermal and Statistical Physics Department Exam Last updated November 4, L π
Thermal and Statistical Physics Department Exam Last updated November 4, 013 1. a. Define the chemical potential µ. Show that two systems are in diffusive equilibrium if µ 1 =µ. You may start with F =
More informationLecture 0. NC State University
Chemistry 736 Lecture 0 Overview NC State University Overview of Spectroscopy Electronic states and energies Transitions between states Absorption and emission Electronic spectroscopy Instrumentation Concepts
More informationGreen s functions for planarly layered media
Green s functions for planarly layered media Massachusetts Institute of Technology 6.635 lecture notes Introduction: Green s functions The Green s functions is the solution of the wave equation for a point
More informationFermi polaronpolaritons in MoSe 2
Fermi polaronpolaritons in MoSe 2 Meinrad Sidler, Patrick Back, Ovidiu Cotlet, Ajit Srivastava, Thomas Fink, Martin Kroner, Eugene Demler, Atac Imamoglu Quantum impurity problem Nonperturbative interaction
More informationNanooptics of surface plasmon polaritons
Physics Reports 408 (2005) 131 314 www.elsevier.com/locate/physrep Nanooptics of surface plasmon polaritons Anatoly V. Zayats a,, Igor I. Smolyaninov b, Alexei A. Maradudin c a School of Mathematics and
More informationPHYS 110B  HW #5 Fall 2005, Solutions by David Pace Equations referenced equations are from Griffiths Problem statements are paraphrased
PHYS 0B  HW #5 Fall 005, Solutions by David Pace Equations referenced equations are from Griffiths Problem statements are paraphrased [.] Imagine a prism made of lucite (n.5) whose crosssection is a
More informationNonlinear Optics II (Modulators & Harmonic Generation)
Nonlinear Optics II (Modulators & Harmonic Generation) P.E.G. Baird MT2011 Electrooptic modulation of light An electrooptic crystal is essentially a variable phase plate and as such can be used either
More informationChapter 2 Undulator Radiation
Chapter 2 Undulator Radiation 2.1 Magnetic Field of a Planar Undulator The motion of an electron in a planar undulator magnet is shown schematically in Fig. 2.1. The undulator axis is along the direction
More informationRadiating Dipoles in Quantum Mechanics
Radiating Dipoles in Quantum Mechanics Chapter 14 P. J. Grandinetti Chem. 4300 Oct 27, 2017 P. J. Grandinetti (Chem. 4300) Radiating Dipoles in Quantum Mechanics Oct 27, 2017 1 / 26 P. J. Grandinetti (Chem.
More information1 Interaction of radiation and matter
Physics department Modern Optics Interaction of radiation and matter To describe the interaction of radiation and matter we seek an expression for field induced transition rates W i j between energi levels
More informationMetals: the Drude and Sommerfeld models p. 1 Introduction p. 1 What do we know about metals? p. 1 The Drude model p. 2 Assumptions p.
Metals: the Drude and Sommerfeld models p. 1 Introduction p. 1 What do we know about metals? p. 1 The Drude model p. 2 Assumptions p. 2 The relaxationtime approximation p. 3 The failure of the Drude model
More information24/ Rayleigh and Raman scattering. Stokes and antistokes lines. Rotational Raman spectroscopy. Polarizability ellipsoid. Selection rules.
Subject Chemistry Paper No and Title Module No and Title Module Tag 8/ Physical Spectroscopy 24/ Rayleigh and Raman scattering. Stokes and antistokes lines. Rotational Raman spectroscopy. Polarizability
More informationApplied Nuclear Physics (Fall 2006) Lecture 19 (11/22/06) Gamma Interactions: Compton Scattering
.101 Applied Nuclear Physics (Fall 006) Lecture 19 (11//06) Gamma Interactions: Compton Scattering References: R. D. Evans, Atomic Nucleus (McGrawHill New York, 1955), Chaps 3 5.. W. E. Meyerhof, Elements
More informationON THE SCATTERING OF ELECTROMAGNETIC WAVES BY A CHARGED SPHERE
Progress In Electromagnetics Research, Vol. 109, 17 35, 2010 ON THE SCATTERING OF ELECTROMAGNETIC WAVES BY A CHARGED SPHERE J. Klačka Faculty of Mathematics, Physics and Informatics Comenius University
More informationResonance Raman scattering in photonic bandgap materials
PHYSICAL REVIEW A, VOLUME 63, 013814 Resonance Raman scattering in photonic bandgap materials Mesfin Woldeyohannes, 1 Sajeev John, 1 and Valery I. Rupasov 1,2 1 Department of Physics, University of Toronto,
More information1. Reminder: EDynamics in homogenous media and at interfaces
0. Introduction 1. Reminder: EDynamics in homogenous media and at interfaces 2. Photonic Crystals 2.1 Introduction 2.2 1D Photonic Crystals 2.3 2D and 3D Photonic Crystals 2.4 Numerical Methods 2.5 Fabrication
More informationThe Dielectric Function of a Metal ( Jellium )
The Dielectric Function of a Metal ( Jellium ) Total reflection Plasma frequency p (10 15 Hz range) Why are Metals Shiny? An electric field cannot exist inside a metal, because metal electrons follow the
More informationMOLECULAR SPECTROSCOPY
MOLECULAR SPECTROSCOPY First Edition Jeanne L. McHale University of Idaho PRENTICE HALL, Upper Saddle River, New Jersey 07458 CONTENTS PREFACE xiii 1 INTRODUCTION AND REVIEW 1 1.1 Historical Perspective
More informationElectromagnetic fields and waves
Electromagnetic fields and waves Maxwell s rainbow Outline Maxwell s equations Plane waves Pulses and group velocity Polarization of light Transmission and reflection at an interface Macroscopic Maxwell
More information