Lecture 11
Band Theory: Kronig-Penney Model
Original Lecture Notes Prepared by:
Dr. Amr Bayoumi, Dr. Nadia Rafat Solid State Electronics EC210
Arab Academy for Science and Technology AAST – Cairo
Spring 2016
1
Original Lecture Notes Prepared by:
Dr. Amr Bayoumi, Dr. Nadia Rafat
Solid State Electronics EC210, Spring 2016 Arab Academy for Science and Technology
AAST – Cairo,
Lecture 11: Band Theory: Kronig-
Penny Model and Effective Mass 2
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Original Lecture Notes Prepared by:
Dr. Amr Bayoumi, Dr. Nadia Rafat
Solid State Electronics EC210, Spring 2016 Arab Academy for Science and Technology
AAST – Cairo,
Lecture 11: Band Theory: Kronig-
Penny Model and Effective Mass 3
Pages
• Kasap:
– P.355 (Kronig Penny)
– P.303-304, p. 454-455 (Effective Mass)
Original Lecture Notes Prepared by:
Dr. Amr Bayoumi, Dr. Nadia Rafat
Solid State Electronics EC210, Spring 2016 Arab Academy for Science and Technology
AAST – Cairo,
Lecture 11: Band Theory: Kronig-
Penny Model and Effective Mass 4
Particle in a Crystalline Solid (Periodic Potential )
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (©
McGraw-Hill, 2005)
Original Lecture Notes Prepared by:
Dr. Amr Bayoumi, Dr. Nadia Rafat
Solid State Electronics EC210, Spring 2016 Arab Academy for Science and Technology
AAST – Cairo,
Lecture 11: Band Theory: Kronig-
Penny Model and Effective Mass 5
Remember for the Hydrogen atom
r r e
U
2
4 0
) 1
(
Original Lecture Notes Prepared by:
Dr. Amr Bayoumi, Dr. Nadia Rafat
Solid State Electronics EC210, Spring 2016 Arab Academy for Science and Technology
AAST – Cairo,
Lecture 11: Band Theory: Kronig-
Penny Model and Effective Mass Fig 5.48 6
The electron PE, V(x), inside the crsytal is periodic with the same periodicity as that of the crystal, a. Far away outside the crsytal, by choice, V = 0 (the electron is free and PE = 0).
When N atoms are arranged to form the crystal then there is an overlap of individual electron PE functions.
r PE(r)
PE of the electron around an isolated atom
x V(x)
x = L x = 0 a 2a 3a
0
a a
PE of the electron, V(x), inside the crystal is periodic with a period a.
Surface Crystal Surface
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (©
McGraw-Hill, 2005)
Original Lecture Notes Prepared by:
Dr. Amr Bayoumi, Dr. Nadia Rafat
Solid State Electronics EC210, Spring 2016 Arab Academy for Science and Technology
AAST – Cairo,
Lecture 11: Band Theory: Kronig-
Penny Model and Effective Mass 7
Kronig-Penney Model
• Approximate crystal periodic Coulomb potential by rectangular periodic potential
Ψ II Ψ I
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (©
McGraw-Hill, 2005)
Original Lecture Notes Prepared by:
Dr. Amr Bayoumi, Dr. Nadia Rafat
Solid State Electronics EC210, Spring 2016 Arab Academy for Science and Technology
AAST – Cairo,
Lecture 11: Band Theory: Kronig-
Penny Model and Effective Mass 8
Crossing a Potential Barrier
Region I (Tunneling):
Free particle: 𝑘 2 = 2𝑚𝐸
ℏ 2
Ψ I 𝑥 = 𝐴𝑒 𝑗𝑘𝑥 + 𝐵𝑒 −𝑗𝑘𝑥
Region II (Change of Kinetic Enegry):
If 𝐸 < 𝑈 𝑜 :
𝛼 2 = 2𝑚(𝑈𝑜 − 𝐸)/ℏ 2 Ψ II 𝑥 = 𝐶𝑒 α𝑥 + 𝐷𝑒 −α𝑥 If 𝐸 > 𝑈 𝑜 :
𝑘 𝐼𝐼 2 = −𝛼 2 = 2𝑚(𝐸 − 𝑈𝑜)/ℏ 2 Ψ II 𝑥 = 𝐶𝑒 𝑗𝑘 𝐼𝐼 𝑥 + 𝐷𝑒 −𝑗𝑘 𝐼𝐼 𝑥
E Uo
I II
0 x U
Uo E
I II
0 x
U
Original Lecture Notes Prepared by:
Dr. Amr Bayoumi, Dr. Nadia Rafat
Solid State Electronics EC210, Spring 2016 Arab Academy for Science and Technology
AAST – Cairo,
Lecture 11: Band Theory: Kronig-
Penny Model and Effective Mass 9
Kronig-Penney Solution:
Allowed Energies
Source: Dr. Fedawy’s Lecture notes
• The wavefunctions have solutions only in some allowed
“continuous” ranges “or “Bands”of 𝑘:
→ 𝑬 = 𝒌 𝟐 ℏ 𝟐
𝟐𝒎 is allowed
• The solution is NOT allowed in other ranges of 𝑘:
→ Energies corresponding to these 𝑘 are “Forbidden”. i.e.
form “Energy Gaps” in the electron energy range
Original Lecture Notes Prepared by:
Dr. Amr Bayoumi, Dr. Nadia Rafat
Solid State Electronics EC210, Spring 2016 Arab Academy for Science and Technology
AAST – Cairo,
