8/28/2015 PHY 752 Fall 2015 -- Lecture 2 1 PHY 752 Solid State Physics 11-11:50 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
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.
Dec 13, 2015
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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
PHY 752 Fall 2015 -- Lecture 2 28/28/2015
PHY 752 Fall 2015 -- Lecture 2 38/28/2015
PHY 752 Fall 2015 -- Lecture 2 48/28/2015
Li2SnO3
PHY 752 Fall 2015 -- Lecture 2 58/28/2015
Li2SnS3
PHY 752 Fall 2015 -- Lecture 2 68/28/2015
Bloch theorem:
PHY 752 Fall 2015 -- Lecture 2 78/28/2015
Eigenvalue solutions to Kronig-Penny model
PHY 752 Fall 2015 -- Lecture 2 88/28/2015
PHY 752 Fall 2015 -- Lecture 2 98/28/2015
From wavefunction matching conditions:
Condition for non-trivial solution:
Simplified result:
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
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
PHY 752 Fall 2015 -- Lecture 2 128/28/2015
f
ka/p
Band gap
Band gap
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:
PHY 752 Fall 2015 -- Lecture 2 148/28/2015
( )L x ( )R x
Matching conditions for wavefunction coefficients:
PHY 752 Fall 2015 -- Lecture 2 158/28/2015
Solving for transfer matrix elements:
PHY 752 Fall 2015 -- Lecture 2 168/28/2015
Transmittance
22
22
1 R
L
At T tA m
PHY 752 Fall 2015 -- Lecture 2 178/28/2015
After some algebra:
PHY 752 Fall 2015 -- Lecture 2 188/28/2015
Electron transmission through a one-dimensional periodic barrier
Because of Bloch theorem:
PHY 752 Fall 2015 -- Lecture 2 198/28/2015
Bloch theorem & transmission coefficients:
R
R
A
B
Condition for solution:
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