optical properties of solids: lecture 5 stefan zollner · stefan zollner, february 2019, optical...
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Optical Properties of Solids: Lecture 5
Stefan ZollnerNew Mexico State University, Las Cruces, NM, USA
and Institute of Physics, CAS, Prague, CZR (Room 335)[email protected] or [email protected]
NSF: DMR-1505172http://ellipsometry.nmsu.edu
These lectures were supported by • European Union, European Structural and Investment Funds (ESIF) • Czech Ministry of Education, Youth, and Sports (MEYS), Project IOP
Researchers Mobility – CZ.02.2.69/0.0/0.0/0008215
Thanks to Dr. Dejneka and his department at FZU.
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Optical Properties of Solids: Lecture 5+6Lorentz and Drude model: Applications1. Metals, doped semiconductors2. InsulatorsSellmeier equation, Poles, Cauchy dispersion
Al
NiO
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New Mexico State University
References: Dispersion, Analytical Properties
Stefan Zollner, February 2019, Optical Properties of Solids Lecture 5 3
Standard Texts on Electricity and Magnetism:• J.D. Jackson: Classical Electrodynamics• L.D. Landau & J.M. Lifshitz, Vol. 8: Electrodynamics of Cont. Media
Ellipsometry and Polarized Light:• R.M.A. Azzam and N.M. Bashara: Ellipsometry and Polarized Light• H.G. Tompkins and E.A. Irene: Handbook of Ellipsometry
(chapters by Rob Collins and Jay Jellison)• H. Fujiwara, Spectroscopic Ellipsometry• Mark Fox, Optical Properties of Solids• H. Fujiwara and R.W. Collins: Spectroscopic Ellipsometry for PV (Vol 1+2)• Zollner: Propagation of EM Waves in Continuous Media (Lecture Notes)• Zollner: Drude and Kukharskii mobility of doped semiconductors extracted
from FTIR ellipsometry spectra, J. Vac. Sci. 37, 012904 (2019).
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New Mexico State UniversityNew Mexico State University
Question: Inhomogeneous Plane WavesPlane waves do not solve Maxwell’s equations, if Im(ε)≠0.
Inhomogeneous plane wave (aka generalized plane waves):
Allow complex wave vector:
The amplitude of the plane wave decays in the medium due to absorption.
Snell:sin 𝜃𝜃1sin𝜃𝜃2
=𝑛𝑛1𝑛𝑛2
𝐸𝐸 𝑟𝑟, 𝑡𝑡 = 𝐸𝐸0 exp 𝑖𝑖 𝑘𝑘 𝑟𝑟 − 𝜔𝜔𝑡𝑡
𝑘𝑘 = 𝑘𝑘1 + 𝑖𝑖𝑘𝑘2 = 𝑘𝑘1𝑢𝑢 + 𝑖𝑖𝑘𝑘2𝑣
𝐸𝐸 𝑟𝑟, 𝑡𝑡 = 𝐸𝐸0 exp −𝑘𝑘2 𝑟𝑟 exp 𝑖𝑖 𝑘𝑘1 𝑟𝑟 − 𝜔𝜔𝑡𝑡Attenuation Propagation
Stefan Zollner, February 2019, Optical Properties of Solids Lecture 5 4
Mansuripur, Magneto-Optical Recording, 1995Stratton, Electromagnetic Theory, 1941/2007
Landau-Lifshitz§63, Jackson, ClemmowDupertuis, Proctor, Acklin, JOSA 11, 1159 (1994).
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Stefan Zollner, February 2019, Optical Properties of Solids Lecture 5 5
Drude and Lorentz Models: Free and Bound Charges
𝜀𝜀 𝜔𝜔 = 1 +𝜔𝜔𝑃𝑃2
𝜔𝜔02 − 𝜔𝜔2 − 𝑖𝑖𝑖𝑖𝜔𝜔
𝜔𝜔𝑃𝑃2 =𝑛𝑛𝑏𝑏𝑞𝑞2
𝑚𝑚𝜀𝜀0𝜔𝜔02 =
𝑘𝑘𝑚𝑚
plasma frequency
resonance frequency
H. Helmholtz, Ann. Phys 230, 582 (1875)
𝐸𝐸 𝑡𝑡 = 𝐸𝐸0 exp −𝑖𝑖𝜔𝜔𝑡𝑡
q
Lorentz:Bound Charges
q qx
qE
bv
v
𝐸𝐸 𝑡𝑡 = 𝐸𝐸0 exp −𝑖𝑖𝜔𝜔𝑡𝑡
Drude:Free Charges
𝜀𝜀 𝜔𝜔 = 1 −𝜔𝜔𝑃𝑃2
𝜔𝜔2 + 𝑖𝑖𝑖𝑖𝜔𝜔
𝜔𝜔𝑃𝑃2 =
𝑛𝑛𝑓𝑓𝑒𝑒2
𝑚𝑚𝜀𝜀0𝜔𝜔02 = 0
plasma frequency
resonance frequency
P. Drude, Phys. Z. 1, 161 (1900).
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Stefan Zollner, February 2019, Optical Properties of Solids Lecture 5 6
Drude-Lorentz Model: Free and Bound Charges
𝐸𝐸 𝑡𝑡 = 𝐸𝐸0 exp −𝑖𝑖𝜔𝜔𝑡𝑡
qq qx
qE
bv
v
𝐸𝐸 𝑡𝑡 = 𝐸𝐸0 exp −𝑖𝑖𝜔𝜔𝑡𝑡
𝜀𝜀 𝜔𝜔 = 1 −𝑖𝑖
𝜔𝜔𝑃𝑃,𝑖𝑖2
𝜔𝜔2 + 𝑖𝑖𝑖𝑖𝐷𝐷,𝑖𝑖𝜔𝜔+
𝑖𝑖
𝐴𝐴𝑖𝑖𝜔𝜔0,𝑖𝑖2
𝜔𝜔0,𝑖𝑖2 − 𝜔𝜔2 − 𝑖𝑖𝑖𝑖0,𝑖𝑖𝜔𝜔
ωP (unscreened) plasma frequency of free chargesω0 resonance frequency of bound chargesγD, γ0 broadenings of free and bound chargesA amplitude of bound charge oscillations (density, strength)
𝜔𝜔𝑃𝑃2 =
𝑛𝑛𝑓𝑓𝑒𝑒2
𝑚𝑚𝜀𝜀0Discuss plasma frequency trends.
Lorentz:Bound Charges
Drude:Free Charges
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Stefan Zollner, February 2019, Optical Properties of Solids Lecture 5 7
Drude-Lorentz Model: Free and Bound Charges
𝜀𝜀 𝜔𝜔 = 1 −𝑖𝑖
𝜔𝜔𝑃𝑃,𝑖𝑖2
𝜔𝜔2 + 𝑖𝑖𝑖𝑖𝐷𝐷,𝑖𝑖𝜔𝜔+
𝑖𝑖
𝐴𝐴𝑖𝑖𝜔𝜔0,𝑖𝑖2
𝜔𝜔0,𝑖𝑖2 − 𝜔𝜔2 − 𝑖𝑖𝑖𝑖0,𝑖𝑖𝜔𝜔
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Metals
Stefan Zollner, February 2019, Optical Properties of Solids Lecture 5 8
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Atomic Radius
Stefan Zollner, February 2019, Optical Properties of Solids Lecture 5 9
H He
Li Be B C N O F Ne
Na Mg Al Si P S Cl Ar
K Ca Ga Ge As Se Br Kr
Rb Sr In Sn Sb Te I Xe
Cs Ba Tl Pb Bi Po At Rn
atomic radius decreases
Atomic radius decreases from K to Ca to Cu.
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(Unscreened) Plasma Frequency
Stefan Zollner, February 2019, Optical Properties of Solids Lecture 5 10
𝜔𝜔𝑃𝑃2 =
𝑛𝑛𝑓𝑓𝑒𝑒2
𝑚𝑚𝜀𝜀0
Fox, Table 7.1Valency determined by row in period table.Atomic radius decreases from K to Ca to Cu.
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Free-Carrier Reflection/Absorption in Metals
Stefan Zollner, February 2019, Optical Properties of Solids Lecture 5 11
Photon Energy (eV)0 2 4 6 8
ε 1
ε2
-8
-6
-4
-2
0
2
0
20
40
60
80
100
ε1ε2
Photon Energy (eV)0 2 4 6 8
n k
01234567
01234567
nk
Dielectric function ε Refractive index n+ik=√ε
𝜀𝜀 𝜔𝜔 = 1 −𝜔𝜔𝑃𝑃2
𝜔𝜔2 + 𝑖𝑖𝑖𝑖𝜔𝜔ωp=3 eV, γ=1 eV
𝑅𝑅90 𝜔𝜔 =𝑛𝑛 + 𝑖𝑖𝑘𝑘 − 1𝑛𝑛 + 𝑖𝑖𝑘𝑘 + 1
2
ε1<0 below 3 eV
Metals reflect below ωP(plasma edge)
Fox, Fig. 7.1
R=1 if n is purely imaginary (γ=0) below ωP.
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Stefan Zollner, February 2019, Optical Properties of Solids Lecture 5 12R.W. Wood, Phys. Rev. 44, 353 (1933)
Fox, Table 7.2
λ (Å)
K
ωP=4.4 eV (280 nm)
U.S. Whang et al., PRB 6, 2109 (1972)
Transparent Alkali Metals above ωP
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Bands of Total Reflection
Stefan Zollner, February 2019, Optical Properties of Solids Lecture 5 13
Drude model:
Small damping (γ`ωP):
Small frequency (ω<ωP):
Refractive index (ω<ωP):
Reflectance at 90Ø (ω<ωP):
𝜀𝜀 𝜔𝜔 = 1 −𝜔𝜔𝑃𝑃2
𝜔𝜔2 + 𝑖𝑖𝑖𝑖𝜔𝜔
𝜀𝜀 𝜔𝜔 = 1 −𝜔𝜔𝑃𝑃2
𝜔𝜔2(real, negative)
𝜀𝜀 𝜔𝜔 < 0
𝑛𝑛 𝜔𝜔 = 𝜀𝜀 𝜔𝜔 = 𝑖𝑖𝑘𝑘
𝑅𝑅90 𝜔𝜔 =𝑛𝑛 + 𝑖𝑖𝑘𝑘 − 1𝑛𝑛 + 𝑖𝑖𝑘𝑘 + 1
2
=𝑖𝑖𝑘𝑘 − 1𝑖𝑖𝑘𝑘 + 1
2
=𝑖𝑖𝑘𝑘 − 1 −𝑖𝑖𝑘𝑘 − 1𝑖𝑖𝑘𝑘 + 1 −𝑖𝑖𝑘𝑘 + 1
= 1
(purely imaginary)
Occur below plasma frequency and between TO/LO energies.Increased sensitivity to weak absorption processes.
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Free-Carrier Reflection in Ag and Al
Stefan Zollner, February 2019, Optical Properties of Solids Lecture 5 14
Ag is a noble metal.Filled 4d-shell, 5s1
High reflectance below ωP=9 eV (138 nm)Sharp drop above ωP. Damping.
silver
Al
Al has three electrons (3s2, 3p1)High reflectance below ωP=16 eV (78 nm)Sharp drop above ωP.Damping, interband absorption.
ωP
Fox, Optical Properties of Solids
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Free-Carrier Reflection in Al
Stefan Zollner, February 2019, Optical Properties of Solids Lecture 5 15
Al
Al has three electrons (3s2, 3p1)High reflectance below ωP=16 eV (78 nm)Sharp drop above ωP.Damping, interband absorption.
Al
Interband transitions at W cause absorption band at 1.5 eV, lowers reflectivity.
Fox, Optical Properties of SolidsSee also: G. Jungk, Thin Solid Films 234, 428 (1993).
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Free-Carrier Reflection in Cu
Stefan Zollner, February 2019, Optical Properties of Solids Lecture 5 16
Noble metal, 4s1, ωP=10.8 eVTransitions from 3d to 4s at 2 eV (near L and X). Similar for Ag, Au.
Fox, Optical Properties of Solids
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Gold is not always yellow.Nanoparticle radius a<λ
m: metal, d: dielectricEnhance molecular absorption.
Plasmon resonance in gold nanoparticles
Stefan Zollner, February 2019, Optical Properties of Solids Lecture 5 17
20 to 100 nm
ln(α
t)
𝛼𝛼 = 4𝜋𝜋𝑎𝑎3𝜀𝜀𝑚𝑚 − 𝜀𝜀𝑑𝑑𝜀𝜀𝑚𝑚 + 𝜖𝜖𝑑𝑑
Fox, Optical Properties of SolidsLittle, APL 98, 101910 (2011)
Ag
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Dielectric function of transition metals (Pt)
Stefan Zollner, February 2019, Optical Properties of Solids Lecture 5 18
Drude-like ε above 1 eV
The dielectric function of Pt deviates from the Drude model below 1 eV due to d-interband transitions.Pt is not a noble metal, partially filled d-shell.
S. Zollner, phys. stat. solidi (a) 177, R7 (2000)
E Fermi
Photon Energy (eV)0 2 4 6 8
ε 1
ε2
-8
-6
-4
-2
0
2
0
20
40
60
80
100
ε1ε2
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Dielectric function of transition metals (Ni)
Stefan Zollner, February 2019, Optical Properties of Solids Lecture 5 19
Farzin Abadizaman (unpublished)
Photon Energy (eV)0 1 2 3 4 5 6 7
Ψ in
deg
rees
27.5
30.0
32.5
35.0
37.5
40.0
42.5
45.0
Model Fit Exp E 65°Exp E 70°Exp E 75°Exp E 65°Exp E 70°Exp E 75°
Photon Energy (eV)0 1 2 3 4 5 6 7
∆ in
deg
rees
40
60
80
100
120
140
160
180Model Fit Exp E 65°Exp E 70°Exp E 75°Exp E 65°Exp E 70°Exp E 75°
Low frequency:ψ→0, ∆→180°Ni, 300 K
σDC=143,000/ΩcmEven at 30 meV, the optical σis still much smaller than σDC.
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Band structure of Ni; Interband transitions
Stefan Zollner, February 2019, Optical Properties of Solids Lecture 5 20
Lina Abdallah, Ph.D. thesis (2014)
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Thickness dependence of dielectric function (Ni)
Stefan Zollner, February 2019, Optical Properties of Solids Lecture 5 21
50 Å not metallic
σ1 with treduced grain boundary scattering in thicker films
Ola Hunderi, PRB, 1973
Lina Abdallah, Ph.D. thesis (2014)
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Difference between Ni and Pt
Stefan Zollner, February 2019, Optical Properties of Solids Lecture 5 22
Lina Abdallah, Ph.D. thesis (2014)
Ni 3d states are more localized.Pt 5d states are broader, more dispersive.
Ni-Pt alloys have broader transitions than pure Ni.• Alloy broadening: Potential fluctuations• Initial Pt 5d states broader than Ni 3d states.
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Stefan Zollner, February 2019, Optical Properties of Solids Lecture 5 23
Lina Abdallah, Ph.D. thesis (2014)
Total DOS Ni3Pt Projected DOS
EF EF
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Optical conductivity of Ni-Pt alloys
Stefan Zollner, February 2019, Optical Properties of Solids Lecture 5 24
Lina Abdallah, Ph.D. thesis (2014)
Si CMOS32 nm
(~10% Pt)
Interband transitions broader in Ni-Pt alloys than in pure Ni.
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Semiconductors
Stefan Zollner, February 2019, Optical Properties of Solids Lecture 5 25
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Free-Carrier Reflection in doped semiconductors
Stefan Zollner, February 2019, Optical Properties of Solids Lecture 5 26
Fox, Optical Properties of Solids
Doped semiconductors behave just like a metal, except for the lower carrier density; plasma frequency in infrared region.
Carrier density in m-3
InSb
Reflectance minimum near plasma frequency
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Why infrared ellipsometry ?
Stefan Zollner, February 2019, Optical Properties of Solids Lecture 5 27
Advantages:• Measures amplitude ψ and phase ∆.• Direct access to complex ε (no Kramers-Kronig transform).• Modeling may contain depth information.• No need to subtract substrate reference data.• Anisotropy information (off-diagonal Jones and MM data)• Possible measurements in a magnetic field (optical Hall effect)• Obtain plasma frequency and scattering rate (B=0)• Obtain carrier density, scattering rate, effective mass (B≠0).Disadvantages:• Time-consuming (15 FTIR reflectance spectra)• Requires polarizing elements (polarizer, compensator)• Requires large samples (no focusing), at least 5 by 10 mm2
• Requires modeling for thin layer on substrate.• Commercial instruments only down to 30 meV (250 cm−1)
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Summary
Stefan Zollner, February 2019, Optical Properties of Solids Lecture 5 28
• Drude model explains optical response of metals.• High reflectance below the plasma frequency.• Interband transitions overlap with Drude absorption.
• Doped semiconductors have infrared plasma frequencies.
• Lorentz model explains infrared lattice absorption.• TO/LO modes result in reststrahlen band.• Multiple modes for complex crystal structures.