Intrinsic Birefringence in Cubic Crystalline Optical Materials Eric L. Shirley, 1 J.H. Burnett, 2 Z.H. Levine 3 (1) Optical Technology Division (844) (2) Atomic Physics Division (842) (3) Electron and Optical Physics Division (841) Physics Laboratory National Institute of Standards and Technology Gaithersburg, MD 20899-8441 Tel: 301 975 2349 FAX: 301 975 2950 email: eric.shirley@nist.gov
Conceptual view of a solid Vibrational, valence electron & core electron degrees of freedom F − − − − Li + Example: LiF EXCITATIONS Phonon excitations Valence excitations Core excitations
Optical properties throughout the spectrum * infrared absorption by phonons * absorption by inter-band transitions * absorption at x-ray edges Optical Constants: n = index of refraction k = index of absorption LiF Properties can be approached with theory. Theory is helpful when it is predictive or complementary to Plot taken from Palik. experiment. Goal: develop approach for unified ( n , k )-curve from far-IR to x-ray region.
A winding, sparsely Outline detailed trajectory circling between } • Introduction to optical excitations definitions of optical quantum constants • Model used to describe excitations mechanics (for electrons & excitation spectra numerical in solids) calculational techniques - developed in collab. with L.X. Benedict (LLNL), R.B. Bohn (ITL), and J.A. Soininen (U. Helsinki) • Sample ultraviolet (UV) & x-ray absorption spectra • Intrinsic birefringence in cubic solids
Photon interaction with electrons: coupling electron p to photon A Electron Schrödinger equation: self-energy (accounts for many-body � � electron-electron interaction effects) 2 p ′ ′ ′ + + ψ + 3 Σ ψ = ψ V V E E � � � ( r ) d r ( r , r ; ) ( r ) ( r ) H n n n n n ext k k k k k m 2 � � electron electron wave function (n=band/core level, k=crystal momentum level Light interacts with electrons (approximately) via the replacement, energy � � + 2 e c 2 2 e e 2 p ( p A / ) p � � → = + ⋅ + ⋅ p A A A � � 2 � m m m mc mc 2 − m ∇ 2 2 2 � � 2 2 2 The first term is the ordinary electron kinetic-energy operator. The second term couples electric fields to electron currents. -- absorption, emission The third term couples to electron electron momentum p ↔ electron current vector potential A ↔ electric field E ↔ force on electrons density. --scattering
Light coupling to electronic degrees of freedom Optical electronic excitation mechanisms � � 2 + e c 2 2 e e 2 p ( p A / ) p � � → = + ⋅ + ⋅ p A A A � � m m m mc mc 2 2 2 2 � � 2
Why are electronic excitations so hard to model? Electron-hole interaction or excitonic effects in excited state
Connection between dielectric constant optical excitations and optical constants, ε = ε + ε = + n k 2 i ( i ) which depend on 1 2 ε = − n 2 k 2 wave-vector q and 1 angular frequency ω : ε = nk 2 2 D = ε ε ε ⋅ ε ⋅ ⋅ E = E + 4 π P (atomic units) ⋅ index of refraction E = total electric field D = electric displacement index of absorption P = polarization of material P = P ion + P val + P core P val , P core = polarization because of val./core el. = * ⋅ δ � P Z R i i ion (Born effective charge tensor Z * = ion i times displacement δ R )
Example: empirical pseudopotential method * Non-interacting model * Optical absorption by electron inter-band transitions * Atomic pseudopotentials adjusted to match observed spectral features Samples of work by Marvin Cohen group (UCBerkeley):
Modeling excitation spectra (Standard time-dependent perturbation theory) = H Normal Hamiltonia n = ˆ O perturbati on Fermi' s Golden Rule : ω = excitation frequency For = I E , initial state I ′ = + − ω + H H O ˆ t exp( i ) h.c., = F E , final state F have = A prefactor 2 ω = � ˆ δ + ω − S A F O I E E ( ) ( ) I F F � � A 1 + = − ˆ ˆ I O O I � � Im π + ω − + η E H � � i I We use the Haydock recursion method, which expresses final expectation value as a continued fraction that depends on ω .
Haydock recursion method (a.k.a. Lánczos method): Introduce normalized vector, + − = 1 / 2 → = v I O ˆ O ˆ I O ˆ I v v ( ) 1 0 0 0 { } v in which H=H † is tri-diagonal, Establish seq. of vectors, i NOTE: = + H v a v b v 0 0 0 1 1 Don’t need = + + H v b v a v b v to solve H . 1 1 0 1 1 2 2 = + + H v b v a v b v Just need to 2 2 1 2 2 3 3 act with H . � Use structure And deduce spectrum (quickly!) from linear algebra... of H to speed this up. − + − ω = − π 1 + ω − + η 1 S A I O ˆ E H O ˆ I ( ) Im ( i ) I − + − = − π 1 + ω − + η 1 A I O ˆ O ˆ I v E H v Im ( i ) I 0 0 − + − = − π 1 + ω − + η − 2 + ω − + η − 2 1 A I O ˆ O ˆ I E a b E a b � Im{ i /[ i /( )]} I I 0 1 1 2 continued fraction
Incorporation of electron-hole interaction: Excited state = linear superposition of all states produced by a single electron excitation. E el In each such electron-hole pair state, electron in band n ′ , with crystal momentum k + q . hole in [band/core-level] n, with crystal momentum k , momentum Call such a state | n n ′ k ( q ) � , total crystal momentum q .
Predictive electron “theory gap = expt. gap” curve band theory: Needs: Corrected band gaps * accurate band structure methods (Schrödinger equation in solids) * many-body corrections to band energies Uncorrected band gaps GW self-energy of Hedin:
Bethe-Salpeter equation, motivation: In a non-interacting picture, one has H | n n ′ k ( q ) � = [ E el ( n ′ , k + q ) − E el ( n , k ) ] | n n ′ k ( q ) � . Thus, the states {| n n ′ k ( q ) � } diagonalize the Hamiltonian, H. In an interacting picture, one has H | n n ′ k ( q ) � = [ E el ( n ′ , k + q ) − E el ( n , k ) ] | n n ′ k ( q ) � + Σ n ′′ n ′′′ k ′ V ( n ′′ n ′′′ k ′ , nn ′ k ) | n ′′ n ′′′ k ′ ( q ) � , and the different states are coupled. Stationary states that diagonalize H are linear combinations of many electron-hole pair states. Resulting coupled, electron-hole-pair Schrödinger equation ( “Bethe-Salpeter” equation): difficult to solve, especially within a realistic treatment of a solid.
Interaction effects: Electron-hole interaction matrix-element: Repulsive “exchange part” Attractive “direct part” of of interaction: leads to interaction: screened Coulomb plasmons. attraction. Gives excitons, shifts spectral weight. Not included in a realistic framework until 1998.
Improved results: Incorporating effects of the electron-hole interaction in realistic calculations was made feasible and efficient through use of a wide variety of numerical & computational innovations. The outcome (e.g., GaAs): Meas. Calc. Besides affecting absorption spectra, index dispersion is greatly improved, especially in wide-gap materials.
Consistently Meas. better results Calc. results when incorporating electron-hole interaction effects.
MgO optical constants:
Core excitations in MgO Excitation of magnesium & oxygen 1s electrons Expt data from Lindner et al., 1986
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✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ � ✁� � ✁� � ✁� � ✁� � ✁� � ✁� � ✁� � ✁� Bethe-Salpeter result: no spin-orbit, no central core-hole potential, no multipole interactions ����������� ��������� central core-hole pot. only ����������� ��������� Ti 2p spin-orbit splitting only spin orbit and central core-hole pot. only banding-induced width included naturally spin-orbit, central core-hole pot, and multipole interactions higher-lying spectral features
157 nm Lithography Index Specifications CaF 2 cubic crystal (fluorite crystal structure) � isotropic optical properties? Material problems extrinsic * index inhomogeneity ~ 2 cm ~ 2 cm DOF ~ 0.2 µ * stress-induced birefringence May 2001 announced an intrinsic feature size ~ 65 nm (~ λ /3 for 157 nm) To obtain resolution ~ 65nm (~ λ λ λ /3): λ birefringence and index anisotropy × 10 -7 over 10 × ~11 × × × × × × specs. d λ λ /8 λ λ phase retardance for all rays d d d � index variation ~ 1 × � � � × 10 -7 × × Cannot be reduced!
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