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ChemistryChemistry-251 Questions for Monday the 14th March 8, 2004 • Describe the d-levels in a tetrahedrally coordinated transition metal compound I Cr3+ [Cr(I)6]3- Matter and Light • Optical properties of matter • Properties of waves in matter • Some applications • Dielectric function Figure 21.26 The d Orbitals in a Tetrahedral Arrangement of Point Charges C M S M a s t e r s : Electronic Structure of Solids: Light Optical Properties © B.Dam2005 The spectrum Wavelength mm Energy eV Frequency Hz • The optical phenomena reflect the electronic properties of a material Visible spectrum wavelength 0.4 µm • Human perception of solids depends on their interaction with visible light (1.5-3 eV): – why are metals shiny and 'metallic' ? – why are diamonds transparent ? – why is white paint white ? 1 Angstrom 1 nanometer Blue 0.5 µm Green 0.6 µm Orange Yellow 1 micrometer 1 millimeter 1 meter C M S M a s t e r s : Electronic Structure of Solids: Light Violet 0.7 µm Red 1 kilometer © B.Dam2005 1 Optical Properties Translucency • Reflection – Scattering at interfaces of materials with different index of refraction n – The larger ∆n, the larger the scattering • Scattering may occur at: • grain boundaries in poly-crystalline materials • fine pores in ceramics • The boundaries of a phase mixture – Translucency • Light scattered at interfaces within a transparant material Privacy window with polymer dispersed liquid crystals C M S M a s t e r s : Electronic Structure of Solids: Light Sol-gel mixture © B.Dam2005 Absorption Absorption by electronic excitation • Transition between well-defined electronic levels http://www.nhn.ou.edu/~kieran/reuhome/vizqm/figs/hydrogen.gif C M S M a s t e r s : Electronic Structure of Solids: Light © B.Dam2005 Energy scales Absorption in a gas or liquid • 26 meV ~ 300 K • 1eV/atom~ 96 kJ/mol k = 2π/λ = 2πν k = 2π/λ = 2πν = E/hc • Vibration spectra of weakly interacting molecules C M S M a s t e r s : Electronic Structure of Solids: Light © B.Dam2005 • λ = 1 µm ν = ??? cm-1 • λ = 1 µm Ε = ??? eV C M S M a s t e r s : Electronic Structure of Solids: Light © B.Dam2005 B.Dam2004 2 Impurities give large bandgapbandgap-crystals color Impurity absorption in sapphire • Corundum • Al2O3 • Ruby, sapphire, topaz, amethyst http://home.achilles.net/~jtalbot/glossary/photopumping.gif http://www.valleydesign.com/images/sapp.jpg – Colorless in pure form – Impurities result in different colors • A similar technique is used to colour glasses or pottery glaze by adding impurities into the molten state: The Cr levels give strong absorptions at 400 nm (green) and 600 nm (blue) leaving only red to be transmitted – Cu2+: blue-green, Cr3+: green – Co2+: blue-violet, Mn2+: yellow Small particles show size dependent absorption Absorption • Transition between well-defined electronic levels • Electron excitation across the band gap • Electron excitation to defect levels in the band gap • Electronic Polarization C M S M a s t e r s : Electronic Structure of Solids: Light Absorption by electronic excitation Band Gap Energy and Color optical spectroscopy can give an idea of band-gap Color that corresponds to band gap energy Apparent color of material (unabsorbed light) ultraviolet colorless Absorption Bandgap energy (eV) 4 3 violet © B.Dam2005 yellow blue green 2 1 C M S M a s t e r s : Electronic Structure of Solids: Light © B.Dam2005 orange yellow red infrared red . shiny C M S M a s t e r s : Electronic Structure of Solids: Light © B.Dam2005 3 Absorption Absorption by electronic excitation • Transition between well-defined electronic levels optical spectroscopy can give an idea of band-gap • Electron excitation across the band gap • Electron excitation to defect levels in the band gap • Electronic Polarization Conduction Band Valence Band insulator = gap in VUV semicon. semicon. = gap at border IR/vis IR/vis.. C M S M a s t e r s : Electronic Structure of Solids: Light Absorption by electronic excitation Band gaps & excitons (non(non-metals) excitons peaks above threshold: exciton Eg electron: -ve Eopt hole: +ve insulator = gap in VUV © B.Dam2005 Frenkel or Wannier excitons.... ......details come in Ch. 7 semicon. semicon. = gap at border IR/vis IR/vis.. C M S M a s t e r s : Electronic Structure of Solids: Light © B.Dam2005 C M S M a s t e r s : Electronic Structure of Solids: Light © B.Dam2005 Band gaps & excitons (non(non-metals) Absorption excitons • Transition between well-defined electronic levels mean that the optical spectra don't give the true band-gap Eopt • Electron excitation to defect levels in the band gap photoconductivity or • Electronic Polarization PES - IPES: N -1 N+1 • Electron excitation across the band gap PES-IPES EgPES= 3.5 eV 4 Absorption by polarization Matter and Light • The electronic polarizability of large atoms leads to absorption • Optical properties of matter • This also leads to a reduction of the speed of light! • Properties of waves in matter • Some applications • Dielectric function The electronic polarization increases for larger atoms •The addition of Pb to glass causes •a slightly darker color (absorption) •an enhanced refractivity Linked? Electromagnetic Wave in vacuum C M S M a s t e r s : Electronic Structure of Solids: Light © B.Dam2005 Electromagnetic Wave in a medium ε = permittivity µ = permeability o - in a vacuum r - relative Speed of light is related to electric permittivity and magnetic permeability c= 1 ε o µo C M S M a s t e r s : Electronic Structure of Solids: Light Electromagnetic Waves εµ = ε οµ o n=c = v = 3 x 108 m / s E ( x ) = E0e − iωt eikx E ( x ) = E0e − iωt eikx In a medium: In a medium: n ≅ εr © B.Dam2005 Electromagnetic Waves In vacuo: ω 2π k= = λ c nω 2nπ = c λ n = n'+in' ' n ≅ εr εr µ r C M S M a s t e r s : Electronic Structure of Solids: Light © B.Dam2005 In vacuo: ω 2π k= = λ c k= 1 v= 1 ε οµ o εµ c= nω 2nπ = c λ n = n'+in' ' k= n ≅ εr E ( x ) = E0e −iωt e iω ( n ' + in '') x c ⎛ iωn ' x ⎞ −iωt ⎟ − ωn '' x ⎜ c ⎠ c E ( x ) = E0 e ⎝ E ( x ) = E0e e ⎛ i 2πn ' x ⎞ −iωt ⎟ −2πn '' x ⎜ ⎝ λ ⎠ λ e 5 Absorption coefficient Electromagnetic Waves ⎛ i 2πn ' x ⎞ −iωt ⎟ −2πn '' x ⎜ λ ⎠ λ In vacuo: ω 2π k= = c λ E ( x ) = E0 e ⎝ E ( x ) = E0e − iωt eikx −4πn '' x I ( x ) = EE* = E02e In a medium: Decay of wave amplitude nω 2nπ k= = c λ n = n'+in' ' e α= λ = I 0e −αx 4πn(ω )' ' λ Phase velocity ⎛ i 2πn ' x ⎞ −iωt ⎟ −2πn '' x ⎜ λ ⎠ λ n ≅ εr E ( x ) = E0e⎝ e Absorption coefficients of various semiconductors Complex index of refraction Photon energy (eV) 5 4 3 1 2 0.9 0.8 0.7 108 Ge In0.7Ga0.3As0.64P0.36 In0.53Ga0.47As Si 106 GaAs InP m α((m-1-1)) 107 ⎛ I ( x) ⎞ ⎟⎟ αx = − ln⎜⎜ ⎝ I0 ⎠ 105 Drastic increase in α at energies larger than the bandgap except for Si! a-Si:H ⎛ iσ ⎞ ⎟ = ε ' + iε ' ' ε r = ⎜⎜ ε 'r + ωε ⎟ n = µ rε r ≈ ε r ⎝ 0 ⎠ Note the removal of the subscript If εr has imaginary parts, the refractive index is: complex: n=n’-in” and ω-dependent: n(ω ) 104 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 n(ω ) = n'+in" n ≅ εr 103 1.8 Wavelength (µm) n'2 −n"2 = ε ' 2n' n" = ε " Fig. 9.19: Absorption coefficient (α) vs. wavelength (λ) for various semiconductors (Data selectively collected and combined from various sources.) Vacuum/dielectric interface Vacuum/dielectric interface T+A+R = 1 ER ET EI I0 = e-αz α = (4πn”)/λ n2 n1 = n1’ + in1” ET n2 n1 = n1’ + in1” n = n1 /n2 = n’ + in’’ n = n1 /n2 = n’ + in’’ C M S M a s t e r s : Electronic Structure of Solids: Light ER From matching the tangential E and Hcomponents across the EI boundary one finds for the reflection coefficient R = ER/EI ⎛ 1 − n ⎞ (n'−1) + n" R=⎜ ⎟ = 2 2 ⎝ 1 + n ⎠ (n'+1) + n" 2 © B.Dam2005 2 2 6 Absorption and reflectivity Why are Metals Shiny? proportion of light reflected by a solid, R : 2 ⎛ 1 − n ⎞ (n'−1) + n" R=⎜ ⎟ = 2 2 ⎝ 1 + n ⎠ (n'+1) + n" 2 2 R is large when: n' >> 1 or n' << 1 n" >> 1 n ' = normal refraction n '' = absorption • Metals Eg = 0 eV • All light (with λ above X-ray wavelengths) absorbed by continuous number of unoccupied states above Ef. • Light is reemitted with exact energy of absorption as electrons fall back into lowest state. Metals appear reflective as the light we see is re-emitted. Si: Si: Eg = 1.1 eV in vis. vis. light it's reflective and 'metallic'..... Ef occurs for E > Eg (non(non-metal) absorption emission Switchable Absorbing Mirrors: 3 of a kind Matter and Light • Optical properties of matter 1st generation: • Properties of waves in matter RE-Hx (YHX) Huiberts et al, VU • Some applications 2nd generation: • Dielectric function GdMgHX Vd Sluis Philips 3rd generation: Black state Mg2NiHX Giebels + Lohstroh VU The MetalMetal-Insulator transition in YHx Hcp fcc hcp 10 1 .0 YH0 1 YH3 YH2 0 .8 0 .6 0 .4 0 .1 0 .2 0 500 1000 1500 Transmission (a.u.) © B.Dam2004 Resistivity (mΩ cm) C M S M a s t e r s : Electronic Structure of Solids: Light 2000 T im e (s ) Huiberts et al. Nature (1996) 7 Visualisation of H diffusion in Y Den Broeder, van der Molen et al, Nature 394 (1998) 656 H Y2O3 Pd Y 3 H/Y 2 hcp- γ hcp- γ β metal insulator 1 α 0 The MI transition depends on the HH-concentration only 10 Van Gogh et al. PRL (1999) 10 YH3 2.90 2.86 1 101 0.1 10-1 0.01 -2 0.1 ρ 0.2 0.08 0 500 1.9 1.0 0.8 0.6 0.4 YH2 0.2 1000 1500 Transmission (a.u.) 1 2.93 102 Resistivity (mΩ cm) 10 ρ (mΩcm) x=H/Y YHx YDx ρ (mΩcm) 100 Switchable mirror applications • • • • 3 Switchable mirrors for office building windows Switchable absorbers for solar collectors Optical hydrogen sensors Search for new hydrogen storage materials – Complex, light-weigth metal hydrides appear to be insulating. – Their bandgap can be used for identification 2000 Time (s) 1.8 2.0 2.2 2.4 2.6 2.8 3.0 10 x = H(D)/Y 1 10 T (K) 100 The biggest problem in storage is weight! 45 40 MJ/kg MJ/ltr 35 System energy densities 30 25 C M S M a s t e r s : Electronic Structure of Solids: Light Hoekstra, Rosenbaum et al (2001) The energy density of hydrogen in a MgH2 based hydrogen tank (6 wt % of hydrogen) is 6x less than that of gasoline © B.Dam2004 The AJA sputter system 3” Turbo 20 15 Argon box TSP 10 5 0 Li battery MgH2 LH2 in tank 700atm gasoline H2 gas P < 10-8 mbar 6 DC/RF Turbo 8 Optical screening for hydrogen storage properties Mg rich Ni-rich Matter and Light • Optical properties of matter • Properties of waves in matter Mg rich MgH2 Ni-rich • Some applications 3 CCD Camera • Dielectric function Mg2NiH4 ? Loading cell 300 oC / 10 bar H2 Light source C M S M a s t e r s : Electronic Structure of Solids: Light Absorption: simple model Absorption: Lorentz model absorbing mode: electron on a spring Now put the oscillators in the solid, with N per unit volume force constant: vibrational frequency ω0 interaction with surroundings: decay time τ ω0 ω0 ε’ © B.Dam2004 ε’ ε” ε” damping, = plasma frequency Single absorbtion line at ω0 ε (ω) has real (ε ') and imaginary (ε ") parts Width ~1/τ Absorption: Lorentz model Absorption: Lorentz model strongest absorption Now put the oscillators in the solid, with N per unit volume ε ' (ω ) = 1 + ω0 ε’ ε(ω) ε” ε ' ' (ω ) = ω p 2 (ω0 2 − ω 2 ) strongest absorption (ω0 − ω 2 ) 2 + ω 2 / τ 2 2 ω p 2ω / τ (ω0 − ω 2 ) 2 + ω 2 / τ 2 2 ε’ ω0 ε” ε (ω ) = 1 + ωp2 ω 0 2 − ω 2 + iω / τ Shift is small if τ >> ω0 ω p ω /τ 2 (ω0 − ω ) + ω / τ 2 2 2 2 Width of ε’’ ~ 1/τ 2 ω p (ω0 − ω 2 ) 2 ε '= 1+ n’ n” Neglect local fields 2 ε ''= (ω0 − ω 2 ) 2 + ω 2 / τ 2 2 9 Absorption: Lorentz model ε ' (ω ) = 1 + ε ' ' (ω ) = ω p 2 (ω0 2 − ω 2 ) strongest absorption (ω0 − ω ) + ω / τ 2 2 2 Absorption: Lorentz model 2 ω p 2ω / τ (ω0 − ω ) + ω / τ 2 2 2 2 2 ε ' ' (ω ) = ωp ε’ ε” ω 2 n' (ω ) ≈ 1 − 2 ω p2 /τ ω3 ε ' ' (ω ) = ω0 ω > ω0 ε ' (ω ) = 1 − ε ' (ω ) = 1 + 2 n' ' (ω ) ≈ ωp 2 2ω 2 ε ' (ω ) = 1 + ε ' ' (ω ) = 3 ε ' ' (ω ) = ω p 2 (ω0 2 − ω 2 ) strongest absorption ω p 2ω / τ (ω0 − ω ) + ω / τ 2 2 n(ω ) = n'+in" n'2 −n"2 = ε ' 2n' n" = ε " ωp 2 ω0 2 ω0 n' ' (ω ) < 4 ε ' ' (ω ) = ω0 ε’ ε” ε(ω) ωω p 2 / τ 2ω0 4 ω p 2 (ω0 2 − ω 2 ) strongest absorption (ω0 − ω 2 ) 2 + ω 2 / τ 2 ω0 ω p 2ω / τ (ω0 − ω ) + ω / τ 2 2 2 2 ε’ 2 n'2 −n"2 = ε ' 2n' n" = ε " ε” ε(ω) Trm. Abs. Refl. Trm n(ω ) = n'+in" n(ω) Gross structure (for solid with a single absorption mode): increasing ω or E 2ω 2 1 Absorption and reflectivity n(ω) ω ⎛ 1 − n ⎞ (n'−1) + n" R=⎜ ⎟ = 2 2 ⎝ 1 + n ⎠ (n'+1) + n" 2 2 2 T+A+R=1 I. The best white pigment....? White: no absorption: gap > 3 eV strongest absorption high transmission at low E High reflectivity in visible ω0 n' (thus e’(ω)) as big as possible ε’ absorption edge just above 3eV ε” high reflectivity transmission again ωp 1 2 2 ω absorption band ω0 ε’ ε” n' (ω ) ≈ 1 + ωω p 2 / τ ε ' (ω ) = 1 + (ω0 − ω 2 ) 2 + ω 2 / τ 2 2 2 (ω0 − ω 2 ) 2 + ω 2 / τ 2 Absorption: Lorentz model 2 2 ω p 2ω / τ 2 ω < ω0 Absorption: Lorentz model ε ' (ω ) = 1 + strongest absorption (ω0 − ω 2 ) 2 + ω 2 / τ 2 1 ω p2 /τ 2ω ω p 2 (ω0 2 − ω 2 ) 2 Transm. Abs. Refl. Transm n’ n” strong scatterer e.g powder rutile: TiO2 ! also cheap and non-toxic 10 MgO QM and the oscillator model? Oscillator frequency equals energy between quantum levels The optical behaviour of MgO (continuous lines) can roughly be understood on the basis of R single oscillater model (broken line) Decay time τ (inverse of the damping γ) is related to the transition probabilities to all available quantum states ω0 ~ 14 eV 2 γ= 1/τ = 8 eV n strongest absorption MgO ω0 ε’ ε” The gap is 7.6 eV, the peak just below is due to an exciton 1 Transm. Abs. Refl. Transm k n’ 20 10 Absorption and reflectivity: metals Absorption and reflectivity: metals reflects A metal has no gap: absorption frequency ω0=0 ωp2 ω 2 + iω / τ ωp 2 ωp ω2 ⎛ Ne 2 ⎞ ⎟⎟ = ⎜⎜ ⎝ ε 0m ⎠ n purely imaginary ε” 2 ε (ω ) = 1 − photon frequency (ω) ω0 ε’ transmits dielectric const. ε(ω) negative Also the damping can be neglected ε (ω ) = 1 − n” ⎛ Ne 2 ⎞ ⎟⎟ ω p 2 = ⎜⎜ ⎝ ε 0m ⎠ n'=0 N is density of conduction electrons n’ n” n = n ' - i n '' for E < plasma frequency, metals reflect High reflection already at ω=0 Plasma frequency depends on electron density, N Absorption and reflectivity: metals From optical measurements one deduces the plasma frequency for most metals, ωp is in the far UV → high reflectivity in visible = 'shiny' l Metal Li Na K Al Ag E (calculated) 8.04 6.05 4.40 15.8 8.98 E (measured) 6.64 5.94 4.15 14.7 3.93 Due to ‘bound’ charges 11 Silver Aluminium In the dielectric function of silver you find an anomaly at ωp which is due to interband transitions This metal has only little bound charges at 2 eV, resulting in a slightly reduced reflectivity Accordingly, a dip in the reflectance is found at ~4 eV The reflectanc remains low for E > ~9eV Thus the spectrum is composed of a free electron part εf and a bound electron part εb Plasma frequency Plasma frequency Sn1-xSbxO2 (x=3%) depends on electron density, N So the carrier density is a new parameter to play with! for most metals, ωp is in the far UV → high reflectivity in visible = 'shiny' ωp in IR: transparent low N metal: e.g. doped metal oxide Sn1-xSbxO2 (x=3%) EELS Reflection in the IR (due to the low electron density): good IR-mirror. It is a good conductor too The bandgap is >3eV, transparant in the visible II. Switchable Absorbing Mirrors The switching of complex Mg2NiH4 c 1st generation: a RE-Hx (YHX) z Huiberts et al, VU x z x y y 2nd generation: b GdMgHX Vd Sluis Philips 3rd generation: Mg2NiHX Giebels + Lohstroh VU c b a Black state Spacegroup P6222 g = 120 o a = 0.52 nm b = 0.52 nm c = 1.32 nm Mg2Ni Symmetry Stacking Bonding 30 % volume Spacegroup C2/c Angle b = 113.5o a = 1.43 nm b = 0.64 nm c = 0.65 nm Mg2NiH4 12 Reflection of Mg2NiHx The ‘impossibility of the black state 3.0 2.5 Photon 2.0 1.5 energy Lo 1.0 (eV) 3nm Pd RT / up to 1bar H2 Lambert-Beer For a 220 nm film we need n’’ > 2 to explain T < 0.0001 T ≅e 4.0 0.6 (n'−n0 )2 + n' '2 R≅ (n'+ n0 )2 + n' '2 Reflectance Reflectance 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 -0.1 -0.2 -0.3 -0.4 -0.5 1000 2000 3000 s) e ( Isidorsson et al. 4000 im APL80 (2002) 2350 5000 gt n i 220nm Mg2.17Ni 6000 ad 3.5 3.0 ω − 2 n ''d c 2.5 2.0 1.5 0.4 0.4 0.3 0.2 0.2 0.1 0.0 4.0 0.1 T 3.5 Lohstroh PRL 89 (2004) 197404 3.0 2.5 2.0 1.5 1.0 0.5 0.0 0.0 Photon energy (eV) Film Surface Mg2Ni columns 20 x 200nm Mg2NiH4 substrate palladium Interface Substrate Mg2Ni 0.5 then R > 0.4 ! Mg 22NiH Mg NiH0.30.3 Pd Mg2NiH0.3 0.0 0.6 0.3 Pd caplayer palladium substrate Pd 0.5 Thin film grain size and loading behaviour of Pd/Mg2Ni The black state viewed from two sides Pd 1.0 R 0.5 Mg2NiH4 Mg<2εNiH > 4 Substrate • Smallest grains near interface • Smallest grains have lowest activation energy for transition • Surface energy confines the phases in two layers Westerwaal MH2004 Black state solar absorption Mg1.7NiHx Area Solar Spectrum Area absorption The emissivity at around 100 oC is less than 16% 1238.1 959.4 0,9 700 0,8 600 0,7 500 Total absorption 78 % 400 300 200 Black body power output 5 < λ < 50 µm Solar power input within 0.25 < λ < 2 µm 0,6 0,5 0,4 0,3 Atmospheric transmittance around 10 µm 0,2 100 0 0,0 Absorption Solar spectrum SS x Absorption 800 1,0 Power input/output 0,1 0,5 1,0 1,5 2,0 2,5 3,0 Energy [eV] 3,5 4,0 0,0 4,5 Sensitivity of the human eye 0.4 > λ < 0.7 µm 13 Project goal Solar contrast: reflective/absorbing state Find a layer which is adapting its optical properties according to solar input and heat output Solar absorbance Reflecting state ~ 40% VAREM Solar collector Solar absorbance Black state ~ 80% H2O VAREM: Variable absorbing and reflecting mirror which switches between two states: 1)Absorb the solar spectrum, without emitting or absorbing heat 2)Reflect the solar spectrum Multiple absorptions In many structures both electronic and vibronic absorption is seen Multiple absorptions strongest absorption Added contribution due to slow lattice vibrations ε ' (ω ) = 1 + εopt is related to the bandgap!! Ge (0.7 eV) => ε = 16 NaCl (8.5 eV)=> ε = 2.3 ωe average excitation energy ω0 ε’ ε” ω <<n’ω0 strongest absorption Added contribution due to slow lattice vibrations ω0 ε’ In many structures both electronic and vibronic absorption is seen ω << ωvn’ n” ωp2 ωe 2 ε” εs related to lattice vibrations Ge (0.7 eV) => ε = 16 NaCl (8.5 eV)=> ε = 2.3 ε 's (ω ) = ε opt (ω ) + A n” NeT Mε 0ωv 2 ωv vibrational frequency 14