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Optical Properties of Solids: Lecture 5 Stefan Zollner New Mexico State University, Las Cruces, NM, USA and Institute of Physics, CAS, Prague, CZR (Room 335) [email protected] or [email protected] NSF: DMR-1505172 http://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.

Optical Properties of Solids: Lecture 5 Stefan Zollner · Stefan Zollner, February 2019, Optical Properties of Solids Lecture 5 18. Drude-like εabove 1 eV. The dielectric function

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Page 1: Optical Properties of Solids: Lecture 5 Stefan Zollner · Stefan Zollner, February 2019, Optical Properties of Solids Lecture 5 18. Drude-like εabove 1 eV. The dielectric function

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.

Page 2: Optical Properties of Solids: Lecture 5 Stefan Zollner · Stefan Zollner, February 2019, Optical Properties of Solids Lecture 5 18. Drude-like εabove 1 eV. The dielectric function

Optical Properties of Solids: Lecture 5+6Lorentz and Drude model: Applications1. Metals, doped semiconductors2. InsulatorsSellmeier equation, Poles, Cauchy dispersion

Al

NiO

Page 3: Optical Properties of Solids: Lecture 5 Stefan Zollner · Stefan Zollner, February 2019, Optical Properties of Solids Lecture 5 18. Drude-like εabove 1 eV. The dielectric function

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).

Page 4: Optical Properties of Solids: Lecture 5 Stefan Zollner · Stefan Zollner, February 2019, Optical Properties of Solids Lecture 5 18. Drude-like εabove 1 eV. The dielectric function

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).

Page 5: Optical Properties of Solids: Lecture 5 Stefan Zollner · Stefan Zollner, February 2019, Optical Properties of Solids Lecture 5 18. Drude-like εabove 1 eV. The dielectric function

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).

Page 6: Optical Properties of Solids: Lecture 5 Stefan Zollner · Stefan Zollner, February 2019, Optical Properties of Solids Lecture 5 18. Drude-like εabove 1 eV. The dielectric function

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

Page 7: Optical Properties of Solids: Lecture 5 Stefan Zollner · Stefan Zollner, February 2019, Optical Properties of Solids Lecture 5 18. Drude-like εabove 1 eV. The dielectric function

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

Page 8: Optical Properties of Solids: Lecture 5 Stefan Zollner · Stefan Zollner, February 2019, Optical Properties of Solids Lecture 5 18. Drude-like εabove 1 eV. The dielectric function

Metals

Stefan Zollner, February 2019, Optical Properties of Solids Lecture 5 8

Page 9: Optical Properties of Solids: Lecture 5 Stefan Zollner · Stefan Zollner, February 2019, Optical Properties of Solids Lecture 5 18. Drude-like εabove 1 eV. The dielectric function

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.

Page 10: Optical Properties of Solids: Lecture 5 Stefan Zollner · Stefan Zollner, February 2019, Optical Properties of Solids Lecture 5 18. Drude-like εabove 1 eV. The dielectric function

(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.

Page 11: Optical Properties of Solids: Lecture 5 Stefan Zollner · Stefan Zollner, February 2019, Optical Properties of Solids Lecture 5 18. Drude-like εabove 1 eV. The dielectric function

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.

Page 12: Optical Properties of Solids: Lecture 5 Stefan Zollner · Stefan Zollner, February 2019, Optical Properties of Solids Lecture 5 18. Drude-like εabove 1 eV. The dielectric function

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

Page 13: Optical Properties of Solids: Lecture 5 Stefan Zollner · Stefan Zollner, February 2019, Optical Properties of Solids Lecture 5 18. Drude-like εabove 1 eV. The dielectric function

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.

Page 14: Optical Properties of Solids: Lecture 5 Stefan Zollner · Stefan Zollner, February 2019, Optical Properties of Solids Lecture 5 18. Drude-like εabove 1 eV. The dielectric function

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

Page 15: Optical Properties of Solids: Lecture 5 Stefan Zollner · Stefan Zollner, February 2019, Optical Properties of Solids Lecture 5 18. Drude-like εabove 1 eV. The dielectric function

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).

Page 16: Optical Properties of Solids: Lecture 5 Stefan Zollner · Stefan Zollner, February 2019, Optical Properties of Solids Lecture 5 18. Drude-like εabove 1 eV. The dielectric function

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

Page 17: Optical Properties of Solids: Lecture 5 Stefan Zollner · Stefan Zollner, February 2019, Optical Properties of Solids Lecture 5 18. Drude-like εabove 1 eV. The dielectric function

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

Page 18: Optical Properties of Solids: Lecture 5 Stefan Zollner · Stefan Zollner, February 2019, Optical Properties of Solids Lecture 5 18. Drude-like εabove 1 eV. The dielectric function

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

Page 19: Optical Properties of Solids: Lecture 5 Stefan Zollner · Stefan Zollner, February 2019, Optical Properties of Solids Lecture 5 18. Drude-like εabove 1 eV. The dielectric function

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.

Page 20: Optical Properties of Solids: Lecture 5 Stefan Zollner · Stefan Zollner, February 2019, Optical Properties of Solids Lecture 5 18. Drude-like εabove 1 eV. The dielectric function

Band structure of Ni; Interband transitions

Stefan Zollner, February 2019, Optical Properties of Solids Lecture 5 20

Lina Abdallah, Ph.D. thesis (2014)

Page 21: Optical Properties of Solids: Lecture 5 Stefan Zollner · Stefan Zollner, February 2019, Optical Properties of Solids Lecture 5 18. Drude-like εabove 1 eV. The dielectric function

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)

Page 22: Optical Properties of Solids: Lecture 5 Stefan Zollner · Stefan Zollner, February 2019, Optical Properties of Solids Lecture 5 18. Drude-like εabove 1 eV. The dielectric function

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.

Page 23: Optical Properties of Solids: Lecture 5 Stefan Zollner · Stefan Zollner, February 2019, Optical Properties of Solids Lecture 5 18. Drude-like εabove 1 eV. The dielectric function

Stefan Zollner, February 2019, Optical Properties of Solids Lecture 5 23

Lina Abdallah, Ph.D. thesis (2014)

Total DOS Ni3Pt Projected DOS

EF EF

Page 24: Optical Properties of Solids: Lecture 5 Stefan Zollner · Stefan Zollner, February 2019, Optical Properties of Solids Lecture 5 18. Drude-like εabove 1 eV. The dielectric function

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.

Page 25: Optical Properties of Solids: Lecture 5 Stefan Zollner · Stefan Zollner, February 2019, Optical Properties of Solids Lecture 5 18. Drude-like εabove 1 eV. The dielectric function

Semiconductors

Stefan Zollner, February 2019, Optical Properties of Solids Lecture 5 25

Page 26: Optical Properties of Solids: Lecture 5 Stefan Zollner · Stefan Zollner, February 2019, Optical Properties of Solids Lecture 5 18. Drude-like εabove 1 eV. The dielectric function

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

Page 27: Optical Properties of Solids: Lecture 5 Stefan Zollner · Stefan Zollner, February 2019, Optical Properties of Solids Lecture 5 18. Drude-like εabove 1 eV. The dielectric function

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)

Page 28: Optical Properties of Solids: Lecture 5 Stefan Zollner · Stefan Zollner, February 2019, Optical Properties of Solids Lecture 5 18. Drude-like εabove 1 eV. The dielectric function

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.