8/28/2015PHY 752 Fall 2015 -- Lecture 21 PHY 752 Solid State Physics 11-11:50 AM MWF Olin 103 Plan...
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Transcript of 8/28/2015PHY 752 Fall 2015 -- Lecture 21 PHY 752 Solid State Physics 11-11:50 AM MWF Olin 103 Plan...
![Page 1: 8/28/2015PHY 752 Fall 2015 -- Lecture 21 PHY 752 Solid State Physics 11-11:50 AM MWF Olin 103 Plan for Lecture 2: Reading: Continue reading Chapter 1 in.](https://reader030.fdocuments.in/reader030/viewer/2022032722/56649f4d5503460f94c6d6a4/html5/thumbnails/1.jpg)
PHY 752 Fall 2015 -- Lecture 2 18/28/2015
PHY 752 Solid State Physics11-11:50 AM MWF Olin 103
Plan for Lecture 2:
Reading: Continue reading Chapter 1 in GGGPP; Electrons in one-dimensional periodic potentials
1. Review of Bloch’s Theorem
2. Kronig-Penny model potential from the viewpoint of scattering or tunneling
![Page 2: 8/28/2015PHY 752 Fall 2015 -- Lecture 21 PHY 752 Solid State Physics 11-11:50 AM MWF Olin 103 Plan for Lecture 2: Reading: Continue reading Chapter 1 in.](https://reader030.fdocuments.in/reader030/viewer/2022032722/56649f4d5503460f94c6d6a4/html5/thumbnails/2.jpg)
PHY 752 Fall 2015 -- Lecture 2 28/28/2015
![Page 3: 8/28/2015PHY 752 Fall 2015 -- Lecture 21 PHY 752 Solid State Physics 11-11:50 AM MWF Olin 103 Plan for Lecture 2: Reading: Continue reading Chapter 1 in.](https://reader030.fdocuments.in/reader030/viewer/2022032722/56649f4d5503460f94c6d6a4/html5/thumbnails/3.jpg)
PHY 752 Fall 2015 -- Lecture 2 38/28/2015
![Page 4: 8/28/2015PHY 752 Fall 2015 -- Lecture 21 PHY 752 Solid State Physics 11-11:50 AM MWF Olin 103 Plan for Lecture 2: Reading: Continue reading Chapter 1 in.](https://reader030.fdocuments.in/reader030/viewer/2022032722/56649f4d5503460f94c6d6a4/html5/thumbnails/4.jpg)
PHY 752 Fall 2015 -- Lecture 2 48/28/2015
Li2SnO3
![Page 5: 8/28/2015PHY 752 Fall 2015 -- Lecture 21 PHY 752 Solid State Physics 11-11:50 AM MWF Olin 103 Plan for Lecture 2: Reading: Continue reading Chapter 1 in.](https://reader030.fdocuments.in/reader030/viewer/2022032722/56649f4d5503460f94c6d6a4/html5/thumbnails/5.jpg)
PHY 752 Fall 2015 -- Lecture 2 58/28/2015
Li2SnS3
![Page 6: 8/28/2015PHY 752 Fall 2015 -- Lecture 21 PHY 752 Solid State Physics 11-11:50 AM MWF Olin 103 Plan for Lecture 2: Reading: Continue reading Chapter 1 in.](https://reader030.fdocuments.in/reader030/viewer/2022032722/56649f4d5503460f94c6d6a4/html5/thumbnails/6.jpg)
PHY 752 Fall 2015 -- Lecture 2 68/28/2015
Bloch theorem:
![Page 7: 8/28/2015PHY 752 Fall 2015 -- Lecture 21 PHY 752 Solid State Physics 11-11:50 AM MWF Olin 103 Plan for Lecture 2: Reading: Continue reading Chapter 1 in.](https://reader030.fdocuments.in/reader030/viewer/2022032722/56649f4d5503460f94c6d6a4/html5/thumbnails/7.jpg)
PHY 752 Fall 2015 -- Lecture 2 78/28/2015
Eigenvalue solutions to Kronig-Penny model
![Page 8: 8/28/2015PHY 752 Fall 2015 -- Lecture 21 PHY 752 Solid State Physics 11-11:50 AM MWF Olin 103 Plan for Lecture 2: Reading: Continue reading Chapter 1 in.](https://reader030.fdocuments.in/reader030/viewer/2022032722/56649f4d5503460f94c6d6a4/html5/thumbnails/8.jpg)
PHY 752 Fall 2015 -- Lecture 2 88/28/2015
![Page 9: 8/28/2015PHY 752 Fall 2015 -- Lecture 21 PHY 752 Solid State Physics 11-11:50 AM MWF Olin 103 Plan for Lecture 2: Reading: Continue reading Chapter 1 in.](https://reader030.fdocuments.in/reader030/viewer/2022032722/56649f4d5503460f94c6d6a4/html5/thumbnails/9.jpg)
PHY 752 Fall 2015 -- Lecture 2 98/28/2015
From wavefunction matching conditions:
Condition for non-trivial solution:
Simplified result:
![Page 10: 8/28/2015PHY 752 Fall 2015 -- Lecture 21 PHY 752 Solid State Physics 11-11:50 AM MWF Olin 103 Plan for Lecture 2: Reading: Continue reading Chapter 1 in.](https://reader030.fdocuments.in/reader030/viewer/2022032722/56649f4d5503460f94c6d6a4/html5/thumbnails/10.jpg)
PHY 752 Fall 2015 -- Lecture 2 108/28/2015
1/ 22
cos( ) cosinh 0.687 1
1 11.7 ( )4 1
1( ) tanh 0.687
s
2t
2 1an 1
ff f
f fk
ff
a
ff f
22
0 20
2 49 16Details for ; ;
2 16 17
EV w a f
m w V
![Page 11: 8/28/2015PHY 752 Fall 2015 -- Lecture 21 PHY 752 Solid State Physics 11-11:50 AM MWF Olin 103 Plan for Lecture 2: Reading: Continue reading Chapter 1 in.](https://reader030.fdocuments.in/reader030/viewer/2022032722/56649f4d5503460f94c6d6a4/html5/thumbnails/11.jpg)
PHY 752 Fall 2015 -- Lecture 2 118/28/2015
1/ 22
cos( ) cosinh 0.687 1
1 11.7 ( )4 1
1( ) tanh 0.687
s
2t
2 1an 1
ff f
f fk
ff
a
ff f
v
v
Forbidden states
Forbidden states
f
1/ 22sinh 0.687 1
1 11.7 ( )4 1
( ) cosf
f ff f
X f
![Page 12: 8/28/2015PHY 752 Fall 2015 -- Lecture 21 PHY 752 Solid State Physics 11-11:50 AM MWF Olin 103 Plan for Lecture 2: Reading: Continue reading Chapter 1 in.](https://reader030.fdocuments.in/reader030/viewer/2022032722/56649f4d5503460f94c6d6a4/html5/thumbnails/12.jpg)
PHY 752 Fall 2015 -- Lecture 2 128/28/2015
f
ka/p
Band gap
Band gap
![Page 13: 8/28/2015PHY 752 Fall 2015 -- Lecture 21 PHY 752 Solid State Physics 11-11:50 AM MWF Olin 103 Plan for Lecture 2: Reading: Continue reading Chapter 1 in.](https://reader030.fdocuments.in/reader030/viewer/2022032722/56649f4d5503460f94c6d6a4/html5/thumbnails/13.jpg)
PHY 752 Fall 2015 -- Lecture 2 138/28/2015
Treatment of same problem in terms of transmission and reflection from a periodic barrier. First consider a single barrier:
( )L x ( )R x
Matrix for determining coefficients:
![Page 14: 8/28/2015PHY 752 Fall 2015 -- Lecture 21 PHY 752 Solid State Physics 11-11:50 AM MWF Olin 103 Plan for Lecture 2: Reading: Continue reading Chapter 1 in.](https://reader030.fdocuments.in/reader030/viewer/2022032722/56649f4d5503460f94c6d6a4/html5/thumbnails/14.jpg)
PHY 752 Fall 2015 -- Lecture 2 148/28/2015
( )L x ( )R x
Matching conditions for wavefunction coefficients:
![Page 15: 8/28/2015PHY 752 Fall 2015 -- Lecture 21 PHY 752 Solid State Physics 11-11:50 AM MWF Olin 103 Plan for Lecture 2: Reading: Continue reading Chapter 1 in.](https://reader030.fdocuments.in/reader030/viewer/2022032722/56649f4d5503460f94c6d6a4/html5/thumbnails/15.jpg)
PHY 752 Fall 2015 -- Lecture 2 158/28/2015
Solving for transfer matrix elements:
![Page 16: 8/28/2015PHY 752 Fall 2015 -- Lecture 21 PHY 752 Solid State Physics 11-11:50 AM MWF Olin 103 Plan for Lecture 2: Reading: Continue reading Chapter 1 in.](https://reader030.fdocuments.in/reader030/viewer/2022032722/56649f4d5503460f94c6d6a4/html5/thumbnails/16.jpg)
PHY 752 Fall 2015 -- Lecture 2 168/28/2015
Transmittance
22
22
1 R
L
At T tA m
![Page 17: 8/28/2015PHY 752 Fall 2015 -- Lecture 21 PHY 752 Solid State Physics 11-11:50 AM MWF Olin 103 Plan for Lecture 2: Reading: Continue reading Chapter 1 in.](https://reader030.fdocuments.in/reader030/viewer/2022032722/56649f4d5503460f94c6d6a4/html5/thumbnails/17.jpg)
PHY 752 Fall 2015 -- Lecture 2 178/28/2015
After some algebra:
![Page 18: 8/28/2015PHY 752 Fall 2015 -- Lecture 21 PHY 752 Solid State Physics 11-11:50 AM MWF Olin 103 Plan for Lecture 2: Reading: Continue reading Chapter 1 in.](https://reader030.fdocuments.in/reader030/viewer/2022032722/56649f4d5503460f94c6d6a4/html5/thumbnails/18.jpg)
PHY 752 Fall 2015 -- Lecture 2 188/28/2015
Electron transmission through a one-dimensional periodic barrier
Because of Bloch theorem:
![Page 19: 8/28/2015PHY 752 Fall 2015 -- Lecture 21 PHY 752 Solid State Physics 11-11:50 AM MWF Olin 103 Plan for Lecture 2: Reading: Continue reading Chapter 1 in.](https://reader030.fdocuments.in/reader030/viewer/2022032722/56649f4d5503460f94c6d6a4/html5/thumbnails/19.jpg)
PHY 752 Fall 2015 -- Lecture 2 198/28/2015
Bloch theorem & transmission coefficients:
R
R
A
B
Condition for solution: