Optical Properties of Insulatorsvojtat/class_481/Lecture_Notes/... · 2008. 1. 15. · Optical...
Transcript of Optical Properties of Insulatorsvojtat/class_481/Lecture_Notes/... · 2008. 1. 15. · Optical...
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Optical Properties of Insulators 1
��
��
��
��
�
� �
�
�
�
��
(B)
�
��
�
�
�
�
F3 or R center
(A)
VK center
21st September 2003c� 2003, Michael Marder
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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
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Polarization 3
(A) (B)
Figure 1: Ambiguity of dielectric
21st September 2003c� 2003, Michael Marder
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Ferroelectrics 4
J
Figure 2: Measuring the spontaneous electric polarization of a sample.
21st September 2003c� 2003, Michael Marder
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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Color Centers 24
�
�
� �
� �
�
�
��
� �
� �
� �
��
� �
� �
�
�
�
��
� �
� �
�
�
�
��
(A) (B)
Figure 9: The F2 or M center.
21st September 2003c� 2003, Michael Marder
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Color Centers 25
��
��
��
��
�
� �
�
�
�
��
(B)
�
��
�
�
�
�
F3 or R center
(A)
VK center
Figure 10: F3 center [Lüty (1961)]
21st September 2003c� 2003, Michael Marder
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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
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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
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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
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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
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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
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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