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Optical Properties of Insulators 1 (B) F 3 or R center (A) V K center 21st September 2003 c 2003, Michael Marder
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Optical Properties of Insulatorsvojtat/class_481/Lecture_Notes/... · 2008. 1. 15. · Optical Properties of Insulators 1 (B) F3 or R center (A) VK center 21st September 2003 c 2003,

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  • Optical Properties of Insulators 1

    ��

    ��

    ��

    ��

    � �

    ��

    (B)

    ��

    F3 or R center

    (A)

    VK center

    21st September 2003c� 2003, Michael Marder

  • Definitions 2

    ☞ Polarization

    ☞ Optical Modes

    ☞ Polaritons

    ☞ Polarons

    ☞ Point Defects

    ☞ Color Centers

    ☞ Electron Spin Resonance

    ☞ Franck–Condon Effect

    ☞ Urbach Tails

    21st September 2003c� 2003, Michael Marder

  • Polarization 3

    (A) (B)

    Figure 1: Ambiguity of dielectric

    21st September 2003c� 2003, Michael Marder

  • Ferroelectrics 4

    J

    Figure 2: Measuring the spontaneous electric polarization of a sample.

    21st September 2003c� 2003, Michael Marder

  • Clausius–Mossotti Relation 5

    E ��

    E0�

    4 �

    3

    P� (L1)

    � � � ��� � � �

    E0

    E

    E0

    E0

    E0

    Figure 3: A dielectric sphere placed in a uniform electric field

    E0.

    21st September 2003c� 2003, Michael Marder

  • Clausius–Mossotti Relation 6

    E1 � � �

    R ��� 0

    �� �R

    �p� �RR3

    ��

    R ��� 03

    R ��

    R� �p �

    R5

    �pR3

    (L2)

    �p � � �Ecell �p � ��

    E0� (L3)

    P � n � �E0� (L4)

    E � 4 � PE

    E0�

    4 �

    3P � 4 � P

    E0�4 �

    3P

    (L5)

    � � 3 � 8 � n �3� 4 � n � � (L6)

    E ��

    E0� ��

    P (L7)21st September 2003

    c� 2003, Michael Marder

  • Clausius–Mossotti Relation 7

    �Ecell �

    E0� ��

    P �

    4 �

    3

    P ��

    E �

    4 �

    3

    P (L8) �Ecell �

    4 �

    3

    � � 2

    �� 1

    P (L9)� 4 �

    3� � 2

    �� 1n ��

    Ecell (L10)

    � � 34 � n

    �� 1� � 2 (L11)

    � � 3 � 8 � n �

    3� 4 � n � � (L12)

    21st September 2003c� 2003, Michael Marder

  • Optical Modes in Ionic Crystals 8

    �u � �u1� �u2 (L13)

    �� � 2 �

    M where M

    � M1M2

    � M1 � M2 �� (L14)

    M��u � � M �� 2 �u� M

    ��u ��� � e�

    Ecell� (L15) �u � � e

    M � � 2��� 2 � i � ��� �

    Ecell� (L16)�p � e �u � �� �Ecell� (L17)

    P � n

    � e � 2M �

    �� 2� � 2� i � ��� �� �

    Ecell (L18)

    34 �

    � � � �� 1

    � � � � � 2

    � n

    � e � 2M �

    �� 2� � 2� i � ��� �� ��

    � (L19)

    21st September 2003c� 2003, Michael Marder

  • Optical Modes in Ionic Crystals 9

    �� � 34 � n

    ��� � 1

    ��� � 2 � (L20)

    �u � e

    �EcellM

    �� 2 (L21)� e � 2 �

    9M�� 2

    4 � n

    � 0� ���

    � � 0 � 2 � � ��� � 2 �� (L22)

    � � � � � ��� �

    ��� � � 0

    � 2

    �� 2 � i

    �� �� 2

    � 0 � 2��� � 2

    � 1

    � (L23)

    21st September 2003c� 2003, Michael Marder

  • Polaritons 10

    0 200 400 600

    -50

    0

    50

    100

    � � 2 � c (cm � 1)

    ������

    1 � � 2 � �

    0

    Figure 4: Dielectric function for CdS, deduced from reflection data by Balkanski (1972).

    21st September 2003c� 2003, Michael Marder

  • Polaritons 11

    � 2T �

    �� 2 ���

    � 2

    � 0 � 2 (L24)

    � 2L � � 2T

    � 0

    ��� � (L25) � � � � � ��� �

    � 2 � i � ��� � � 2L

    � 2 � i � ��� � � 2T � (L26)

    � 2 � � � �c2

    � q2 (L27)� � � � � 0� (L28)

    21st September 2003c� 2003, Michael Marder

  • Polaritons 12

    0 500 1000 1500 20000.0

    0 � 5 � 1013

    1 � 0 � 1013

    1 � 5 � 1013

    2 � 0 � 1013

    0 500 1000 1500 2000

    Re � q �� cm

    � 1 � Im � q � � cm

    � 1 �

    �(H

    z)

    � T

    � L

    Figure 5: Frequency � of transverse waves as a function of complex wave vector q.

    21st September 2003c� 2003, Michael Marder

  • Polaritons 13Compound � � � 0 � T

    2 � c

    � L2 � c

    m�

    m

    � p m

    m

    � 1 � � p6

    m�

    pol

    m

    (cm

    � 1) (cm � 1)LiF 1.93 8.50 318 667LiH 3.60 12.90 590 1116

    NaF 1.75 4.73 262 431NaI 3.08 6.60 124 182

    KF 1.86 5.11 202 334KI 2.68 4.68 102 144 0.325 2.51 0.461 0.540

    RbF 1.94 5.99 163 286RbI 2.61 4.55 76 108 0.368 3.16 0.562 0.720

    CsF 2.17 7.27 134 245CsCl 2.67 6.68 107 168CsBr 2.83 6.38 78 118CsI 3.09 6.32 66 94 0.420 3.67 0.677 0.960

    GaAs 10.90 12.83 273 296 0.066 0.07 0.067 0.066GaSb 14.40 15.69 231 240 0.047 0.03 0.047 0.047GaP 8.46 10.28 365 403 0.338 0.20 0.349 0.350InAs 11.80 14.61 219 243 0.023 0.05 0.023 0.023InSb 15.68 17.88 185 197 0.014 0.02 0.014 0.013

    CdS 5.27 8.42 244 308 0.155 0.53 0.169 0.170CdSe 6.10 9.30 174 214 0.130 0.46 0.140 0.140CdTe 7.21 10.23 141 168 0.091 0.32 0.096 0.096ZnS 5.14 8 282 352 0.280 0.65 0.310 0.313ZnSe 5.90 8.33 207 246 0.171 0.43 0.183 0.184

    ZnTe 7.28 9.86 177 205 0.160 0.33 0.169 0.169ZnO 4 8.15 414 591 0.240 0.85 0.274 0.279PbS 18.50 190 67 214 0.082 0.32 0.086 0.087PbSe 25.20 280 44 147 0.047 0.21 0.049 0.049PbTe 36.90 450 32 110 0.034 0.15 0.035 0.035

    21st September 2003c� 2003, Michael Marder

  • Polarons 14

    P � n

    e �u � �

    Ecell

    � (L29)

    E � � 4 ��

    P (L30)

    �Ecell �

    23

    E � � 8 �

    3

    P (L31) �P � ne

    1 � n � 8 � � 3

    �u� (L32)

    ne

    1 � n � 8 � � 3

    � n

    9M �� 24 � n

    � 0� ���

    � � 0 � 2 � � ��� � 2 �

    1 � 2 � � � � 1 � � � ��� � 2 �

    (L33)

    � M �

    2Ln

    4 �

    1

    ���� 1

    � 0 � (L34)

    21st September 2003c� 2003, Michael Marder

  • Polarons 15

    P � ��u (L35)

    � �

    M � 2Ln4 �

    1

    ���� 1

    � 0 � (L36)

    �Uel� phon � e d �r�

    P � �r� �� � �r�

    1

    ��

    R� �r� �� (L37)

    Uel� phon � e � d

    �r�

    �h

    2M � LN �k

    kk

    ��

    � �r�

    1

    ��r� �

    R � �

    ei

    k� �r� �a�k � e

    � i�k� �r� �a� �

    k �

    (L38)

    � � e � d

    �r�

    h2M � LN �

    k

    i�

    k� �kk

    1

    ��r� �

    R ��

    ei

    k� �r� �a�k� e

    � i�k� �r� �a� �

    k �� (L39)

    Uel� phon � e � 4 � i �

    k

    h2M � LN

    1k �

    e

    � i�k� �R �a� �k

    � ei

    k� �R �a�k �� (L40)

    21st September 2003c� 2003, Michael Marder

  • Polarons 16� p �

    e2

    22m � L�

    h1

    h � L1

    ���� 1

    � 0

    � 1� 44� 108 1���� 1

    � 0m � m

    � L� s

    � (L41)�

    Uel� phon � i 4 � � p1

    ��

    h5 � 3L2m

    1 � 4

    k

    1k �

    e

    � i�k� �R �a� �

    k

    � ei

    k� �R �a�k �� (L42)�

    Uel� phon � �q �q��c� �q� �

    �q� ��

    Uel� phon ��q �

    �c �q (L43)

    Uel� phon � i 4 � � p1

    � �

    h5 � 3L2m

    1 � 4�q� � �k

    1k �

    �c� �q� � � �k

    �c �q� � �a� �k��c� �q� � � �k

    �c �q� � �a�k �� (L44)

    � �� 2 �� � �q�

    � � �q � �� 0 ��

    Uel� phon �� � � � �q� � � 2

    � � �q � 0 �� � � �q� � � �� (L45)

    21st September 2003c� 2003, Michael Marder

  • Polarons 17� �� 2 � 4 � � p

    1

    h5 � 3L2m �q�

    1

    ��q� �q� � 2 �

    1

    h2q2

    2m�

    h2q� 2

    2m ��

    h � L

    (L46)� 4 � � p

    1�

    h5 � 3L2m dq

    � d � cos � �

    � 2 � � 32 ��

    h2q2

    2m�

    h2 ��q� � �q � 2

    2m ��

    h � L(L47)

    � � p

    h5 � 3L2m

    1

    � 1ds

    �0

    dq�

    1

    h2q2

    2m�

    h2 � q� 2 � q2 � 2qq� s �

    2m ��

    h � L(L48)

    � � � p m �h � 3L

    � 2q

    sin

    � 1�

    h2q2

    2m �h � L� (L49)

    � �� 2 � � � p �h � L� � p

    h2q2

    12m � (L50)21st September 2003

    c� 2003, Michael Marder

  • Polarons 18

    m polm

    � 1 �� p

    6

    � Table of data.� �� 2 � � � p m �h � 3L

    � 2q

    ��

    � � 2 � icosh

    � 1

    h2q2

    2m �h � L

    ��

    � (L51)

    exp�

    � i�h � �

    � 0

    � � �� 2 � t

    (L52)

    exp

    2�

    hIm � � �� 2 � t

    �� (L53)

    2 � p m �h � 3L

    � 2�

    hqcosh

    � 1�

    h2q2

    2m �h � L� (L54)

    21st September 2003c� 2003, Michael Marder

  • Vacancies 19

    Crystal Cohesive Energy � � N Vacancy Energy(eV) (eV)

    Na 1.16 0.42

    Au 3.8 0.97

    Al 3.4 0.76

    Pt 5.3 1.4

    Ne 0.021 0.020

    Kr 0.11 0.077

    Ge 3.9 2.0

    21st September 2003c� 2003, Michael Marder

  • F Centers 20

    � �

    � �

    ��

    (A) (B)

    Electron trapped in vacancy

    Figure 6: The F center is a halogen ion vacancy that has trapped an electron.

    21st September 2003c� 2003, Michael Marder

  • F Centers 21

    Compound � abs (eV) � em (eV) Compound � abs (eV) � em (eV)NaF 3.72 1.67 RbCl 2.05 1.09

    NaCl 2.77 0.98 RbBr 1.86 0.87

    KF 2.85 1.66 RbI 1.71 0.81

    KCl 2.31 1.22 CsF 1.89 1.42

    KBr 2.06 0.92 CsCl 2.17 1.26

    KI 1.87 0.83 CsBr 1.96 0.91

    RbF 2.43 1.33 CsI 1.68 0.74

    21st September 2003c� 2003, Michael Marder

  • Electron Spin Resonance and ElectronNuclear Double Resonance 22

    2.5 3.0 3.5 4.0Magnetic field H0 along [110] (kG)

    d�

    2

    dH

    Figure 7: Electron spin resonance in RbCl F centers at a temperature of 90 K. [Source:

    Pick (1972)]

    B ��

    B0 �

    l

    Bl (L55)

    21st September 2003c� 2003, Michael Marder

  • Electron Spin Resonance and ElectronNuclear Double Resonance 23

    0.0 0.5 1.0 1.5 2.0 2.5 3.0

    1

    10� 6

    10� 4

    10� 2

    Distance r � d

    Ele

    ctro

    nde

    nsity

    ���� r��

    �2 d3

    Figure 8: Electron density versus distance from vacancy center [Seidel and Wolf (1968)]

    21st September 2003c� 2003, Michael Marder

  • Color Centers 24

    � �

    � �

    ��

    � �

    � �

    � �

    ��

    � �

    � �

    ��

    � �

    � �

    ��

    (A) (B)

    Figure 9: The F2 or M center.

    21st September 2003c� 2003, Michael Marder

  • Color Centers 25

    ��

    ��

    ��

    ��

    � �

    ��

    (B)

    ��

    F3 or R center

    (A)

    VK center

    Figure 10: F3 center [Lüty (1961)]

    21st September 2003c� 2003, Michael Marder

  • Franck–Condon Effect 26

    �� F � F0 � � � 0 � F0 � � 0 (L56a)

    �� F � F1 � � � 1 � F1 �� (L56b)

    �� ion �

    P2

    2M

    M � 2i2

    �x2� (L57)

    �� int � g

    �x

    �� F (L58)

    ��

    � F � 1 � g �x � ��

    � ion � �� � � � tot �� �� (L59)

    � l � x � � � x � l �� �� (L60)

    � l � 1 � gx � �� �h2 � 2

    2M

    M � 2i2

    x2 � l � x � � � tot � l � x �� (L61)

    21st September 2003c� 2003, Michael Marder

  • Franck–Condon Effect 27

    � l �� lg

    M � 2i

    � (L62)

    � l �� �h2 � 2

    2M

    M � 2i2 � � x � � l �

    2� � 2l

    � l � x � � � tot � l � x �� (L63)

    � l � n � � l ��

    h � i � n � 12 �� 12 � 2l M � 2i� (L64)

    21st September 2003c� 2003, Michael Marder

  • Franck–Condon Effect 28

    A

    B

    C

    D

    Ene

    rgy

    Configuration coordinatex

    h � ph

    Figure 11: Franck–Condon effect21st September 2003

    c� 2003, Michael Marder

  • Franck–Condon Effect 29

    final

    � � � tot � final� � tot � 0��

    h � � � � � 0 ��

    Uint �� final� � 2 (L65)

    h � � � 1 � n �h � i� 12 � 21M � 2i (L66)

    � dx � 0 � x � � n � x � � 1 � � 2� (L67)

    x0 �

    �h

    M � i� � 1 (L68)

    1 � � 1g�hM � 3i

    � (L69)

    � 1 � x0 (L70)

    21st September 2003c� 2003, Michael Marder

  • Franck–Condon Effect 30

    dx � 0 � x � � n � x � � 1 � (L71)� d �

    1

    � 2nn�

    e� � 1 � x0� � � 1 � x0 � 2 � 2 �� 1 � ndn

    d � ne

    � � 2 (L72)

    � d �

    1� 2nn� �

    � 1

    x0 �

    ne� � 1 � x0� � � 1 � x0 � 2 � 2e� � 2 (L73)

    � 12nn� �

    � 1

    x0 �

    ne

    � � � 1 � x0 � 2 � 4� (L74)

    n � 12 � � 1 � x0 � 2� (L75)

    21st September 2003c� 2003, Michael Marder

  • Urbach Tails 31

    200 220 240 260 280 300 320 34010 � 2

    1

    102

    104

    106

    20 K

    200 K

    300 K

    400 K

    500 K

    600 K

    700 K

    800 K

    900 K

    Wavelength ( � m)

    Abs

    orpt

    ion

    coef

    fici

    ent

    �(mm

    �1�

    Figure 12: Urbach tails [Haupt (1959)]

    �� exp

    �� � � g�

    h � �

    kBT � (L76)

    21st September 2003c� 2003, Michael Marder