Intrinsic Birefringence in Cubic Crystalline Optical Materials Eric L. Shirley,1 J.H. Burnett,2 Z.H. Levine3 (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: [email protected] Conceptual view of a solid Vibrational, valence electron & core electron degrees of freedom Li+ F− 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 Properties can be approached with LiF 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 } numerical & excitation spectra 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 » 2 ÿ electron-electron interaction effects) p + + ψ + 3 ′ Σ ′ ψ ′ = ψ … Vext VH Ÿ nk (r) — d r (r,r ; Enk ) nk (r ) Enk nk (r) 2m ⁄ electron wave function (n=band/core level, k=crystal momentum electron Light interacts with electrons (approximately) via the replacement, level energy p2 (p + eA / c)2 p2 e ≈ e2 ’ >2 → = + p⋅ A + ∆ A ⋅ A − ∇2 2m 2m 2m mc ∆ 2mc2 2m « The first term is the ordinary electron kinetic-energy operator. The second term couples electric fields to electron currents. -- absorption, emission The third term electron momentum p ↔ electron current couples to electron vector potential A ↔ electric field E ↔force on electrons density. --scattering Light coupling to electronic degrees of freedom Optical electronic excitation mechanisms p2 (p + eA / c)2 p2 e ≈ e2 ’ → = + ⋅ + ∆ ⋅ p A ∆ 2 A A 2m 2m 2m mc « 2mc Why are electronic excitations so hard to model? Electron-hole interaction or excitonic effects in excited state Connection between optical excitations and dielectric constant optical constants, ε = ε + ε = + 2 which depend on 1 i 2 (n ik) ε = 2 − 2 wave-vector q and 1 n k angular frequency ω: ε = 2 2nk D = ε ⋅E = E + 4πP (atomic units) index of refraction E = total electric field D = electric displacement index of absorption P = polarization of material P = Pion + Pval + Pcore Pval, Pcore= polarization because of val./core el. = * ⋅δ Pion Zi Ri i=ion (Born effective charge tensor Z* times displacement δR) 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 Hamiltonian Oˆ = perturbation Fermi's Golden Rule : ω = excitation frequency For I , E = initial state ′ = + ˆ − ω + I H H Oexp( i t) h.c., = F , EF final state have A = prefactor ω = ˆ 2δ +ω − S( ) A ƒ F O I (EI EF ) F A À + 1 = − Im I Oˆ Oˆ I π +ω − + η EI H 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, = ˆ + ˆ −1/ 2 ˆ → = v0 ( I O O I ) O I v0 v0 1 Establish seq. of vectors,{ v }in which H=H† is tri-diagonal, i NOTE: H v = a v + b v 0 0 0 1 1 Don’t need = + + H v1 b1 v0 a1 v1 b2 v2 to solve H. = + + Just need to H v2 b2 v1 a2 v2 b3 v3 4 act with H. Use structure And deduce spectrum (quickly!) from linear algebra... of H to speed this up. ω = −π −1 ˆ + +ω − + η −1 ˆ S( ) AIm I O (EI H i ) O I = −π −1 ˆ + ˆ +ω − + η −1 A I O O I Im v0 (EI H i ) v0 = −π −1 ˆ + ˆ +ω − + η − 2 +ω − + η − 2 2 −1 A I O O I Im{EI a0 i b1 /[EI a1 i b2 /( )]} continued fraction Incorporation of electron-hole interaction: Excited state = linear superposition of all states produced by a single electron excitation. Eel In each such electron-hole pair state, electron in band n′, with crystal momentum k+q. hole in [band/core-level] n, momentum with crystal momentum k, Call such a state |n n′ k(q)–, total crystal momentum q. Predictive electron “theory gap = expt. gap” curve band theory: Needs: Corrected * accurate band structure band gaps 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)– = [ Eel( n , k+q) Eel ( 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)– = [ Eel( n , k+q) Eel ( 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: Attractive “direct part” of Repulsive “exchange part” interaction: screened Coulomb of interaction: leads to attraction. Gives excitons, plasmons. 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 ¡ ¡ ¢ ¢ ¡ ¡ ¢ ¢ ¢ ¢ ¡ ¡ ¢ ¢ ¡ ¡ ¢ ¢ ¢ ¢ Bethe-Salpeter result: no spin-orbit, no central core-hole potential, no multipole interactions xtal-field splitting central core-hole pot. only spin orbit splitting 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 CaF2 cubic crystal (fluorite crystal structure) isotropic optical properties? Material problems extrinsic ~ 2 cm * index inhomogeneity ~ 2 cm DOF ~ 0.2 µ * stress-induced birefringence feature size ~ 65 nm (~ λ/3 for 157 nm) May 2001 announced an intrinsic To obtain resolution ~ 65nm (~ λ/3): birefringence and index anisotropy phase retardance for all rays d λ/8 ~11 × 10-7 over 10 × specs. index variation ~ 1 × 10-7 Cannot be reduced! Spatial-Dispersion-Induced Birefringence Origin of effect: Finite wave vector of light, q, breaks symmetry of light-matter interaction. History: H.A Lorentz (Lorentz contraction) considered this small symmetry-breaking effect in “regular crystals” in 1879, PRIOR to verified existence crystal lattices! (Laue 1912, Bragg 1913) Worked out simple theory by 1921 - measured in NaCl? H.A. Lorentz, “Double Refraction by Regular Crystals,” Proc. Acad. Amsterdam. 24, 333 (1921). First convincingly demonstrated by Pastrnak and Vedam in Si (1971). J. Pastrnak and K. Vedam, “Optical Anisotropy of Silicon Single Crystals,” Phys. Rev. B 3, 2567 (1971). Confirmed, extended by others, esp. Cardona & colleagues – academic curiosity Values “too” small to have implications for optics – Optics industry oblivious! We measured in CaF2, material for precision UV optics for 193 nm and 157 nm lithography, and worked out the implications for optics - alerted industry. J.H. Burnett, Z.H. Levine, E.L. Shirley, “Intrinsic birefringence in calcium fluoride and barium fluoride,” Phys. Rev. B 64, 241102 (2001). Wave Vector Dependence of the Index in Cubic Crystals spatial-dispersion-induced birefringence hν q Symmetry arguments “prove” natural birefringence forbidden in cubic crystals Isotropy “proof” assumes D linearly related to E by 2nd-rank tensor indept. of q Σ ε−1 ε−1 λ Ei = j ijDj ( ij inverse dielectric constant) - but assumes large! iq·r + . − . 2 … π λ Actually D = D0e = D0(1 iq r (q r) /2 + ) (q = 2 n/ ) . λ Cannot neglect (q r) terms if (aunit cell/ ) ~ 1 or equivalently (q/Kreciprocal lattice) ~ 1 Perturbation due to (q.r) terms: azimuthal symmetry about q For crystal axes w/ 3-fold or 4-fold symmetry (q.r) reduces isotropic to uniaxial NO birefringence for q || <111> or q || <001> Theory of Intrinsic Birefringence 0 −1 −1 ε ij (q,ω) = ε (0,ω)δ + γ (ω)q + α (ω)q q ( α respects cubic symmetry) ij k ijk k kl ijkl k l ijkl α α α α Cubic crystals (classes 43m,432,m3m) symmetry ijkl has 3 indep. comp. 11, 12 , 44 α α α 11 12 12 0 0 0 ’ ∆ ÷ α α α 0 0 0 ∆ 12 11 12 ÷ ∆α α α 0 0 0 ÷ α = 12 12 11 (same form as for piezo-optic tensor) ij ∆ α ÷ ∆ 0 0 0 44 0 0 ÷ ∆ α ÷ 0 0 0 0 44 0 ∆ α ÷ « 0 0 0 0 0 44 ◊ Using the 2 independent scalar invariants of a 4th rank tensor to separate terms: −1 −1 2 2 2 2 ε ij (q,ω) = »ε (0,ω) +α q ÿδ +α 2q l l + (α −α − 2α )5q δ l q 12 ⁄ ij 44 i j 11 12 44 ij i l3 |q| » ÿ 2 2 1 0 0 » l1 l1l2 l1l3 ÿ »l1 0 0 ÿ … Ÿ … 2 Ÿ … 2 Ÿ l2 …0 1 0 Ÿ …l2l1 l2 l2l3 Ÿ … 0 l2 0 Ÿ [010] … Ÿ …l l l l l 2 Ÿ … 0 0 l 2 Ÿ 0 0 1 ⁄ 3 1 3 2 3 ⁄ 3 ⁄ l1 isotropic longitudinal anisotropic [100] isotropic index shift isotropic L-T splitting • induces dir.