solid state electronics ec210 arab academy for science and...
TRANSCRIPT
Lecture 8 Band Theory: Kronig-Penny Model
and Effective Mass
Lecture Notes Prepared by: Dr. Amr Bayoumi, Dr. Nadia Rafat
Solid State Electronics EC210 Arab Academy for Science and Technology
AAST – Cairo Fall 2014
1
Lecture Notes Prepared by: Dr. Amr Bayoumi, Dr. Nadia Rafat
Solid State Electronics EC210, Fall 2014
Arab Academy for Science and Technology AAST – Cairo,
Lecture 8: Band Theory: Kronig-Penny Model and Effective Mass 2
These PowerPoint color
diagrams can only be used by
instructors if the 3rd Edition
has been adopted for his/her
course. Permission is given to
individuals who have
purchased a copy of the third
edition with CD-ROM
Electronic Materials and
Devices to use these slides in
seminar, symposium and
conference presentations
provided that the book title,
author and © McGraw-Hill are
displayed under each diagram.
Lecture Notes Prepared by: Dr. Amr Bayoumi, Dr. Nadia Rafat
Solid State Electronics EC210, Fall 2014
Arab Academy for Science and Technology AAST – Cairo,
Lecture 8: Band Theory: Kronig-Penny Model and Effective Mass 3
Pages
• Kasap: – P.355 (Kronig Penny)
– P.303-304, p. 454-455 (Effective Mass)
Lecture 8: Band Theory: Kronig-Penny Model and Effective Mass
Lecture Notes Prepared by: Dr. Amr Bayoumi, Dr. Nadia Rafat
Solid State Electronics EC210, Fall 2014
Arab Academy for Science and Technology AAST – Cairo,
Lecture 8: 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)
Lecture Notes Prepared by: Dr. Amr Bayoumi, Dr. Nadia Rafat
Solid State Electronics EC210, Fall 2014
Arab Academy for Science and Technology AAST – Cairo,
Lecture 8: Band Theory: Kronig-Penny Model and Effective Mass 5
Remember for the Hydrogen atom
r
erU
2
04
1)(
Lecture Notes Prepared by: Dr. Amr Bayoumi, Dr. Nadia Rafat
Solid State Electronics EC210, Fall 2014
Arab Academy for Science and Technology AAST – Cairo,
Lecture 8: Band Theory: Kronig-Penny Model and Effective Mass 6
Fig 5.48
The electron PE, V(x), inside the crsytal is periodic with the sameperiodicity as that of the crystal, a. Far away outside the crsytal, bychoice, V = 0 (the electron is free and PE = 0).
When N atoms arearranged to form thecrystal then there is anoverlap of individualelectron PE functions.
r
PE(r)
PE of the electron aroundan isolated atom
x
V(x)
x = Lx = 0 a 2a 3a
0
aaPE of the electron, V(x),inside the crystal isperiodic with a period a.
Surface SurfaceCrystal
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Lecture Notes Prepared by: Dr. Amr Bayoumi, Dr. Nadia Rafat
Solid State Electronics EC210, Fall 2014
Arab Academy for Science and Technology AAST – Cairo,
Lecture 8: Band Theory: Kronig-Penny Model and Effective Mass 7
Bloch’s Waves
If a periodic potential with period “𝑎” can be defined as: 𝑈 𝑥 + 𝑎 = 𝑈 𝑥
Then the wavefunction is periodic, and can be defined in terms of base function:
Ψ 𝑥 + 𝑎 = 𝑒𝑖𝑘𝑎 Ψ(𝑥)
Ψ(𝑥) = 𝑒𝑖𝑘𝑥𝑢(𝑥)
• 𝑎 can be replace by 𝑛𝑎
Lecture Notes Prepared by: Dr. Amr Bayoumi, Dr. Nadia Rafat
Solid State Electronics EC210, Fall 2014
Arab Academy for Science and Technology AAST – Cairo,
Lecture 8: Band Theory: Kronig-Penny Model and Effective Mass 8
Kronig-Penney Model
• Approximate crystal periodic Coulomb potential by rectangular periodic potential
ΨII ΨI
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Lecture Notes Prepared by: Dr. Amr Bayoumi, Dr. Nadia Rafat
Solid State Electronics EC210, Fall 2014
Arab Academy for Science and Technology AAST – Cairo,
Lecture 8: Band Theory: Kronig-Penny Model and Effective Mass 9
Wavefunction Periodic Boundary Conditions
Ψ𝐼 0 = Ψ𝐼𝐼 0
𝑑Ψ𝐼 0
𝑑𝑥=
𝑑Ψ𝐼𝐼 0
𝑑𝑥
Ψ𝐼 𝑎 = 𝑒𝑖𝑘(𝑎+𝑏)Ψ𝐼𝐼 −𝑏
𝑑Ψ𝐼 𝑎
𝑑𝑥= 𝑒𝑖𝑘(𝑎+𝑏)
𝑑Ψ𝐼𝐼 −𝑏
𝑑𝑥
ΨI ΨII
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Lecture Notes Prepared by: Dr. Amr Bayoumi, Dr. Nadia Rafat
Solid State Electronics EC210, Fall 2014
Arab Academy for Science and Technology AAST – Cairo,
Lecture 8: Band Theory: Kronig-Penny Model and Effective Mass 10
K-P Solution: Allowed Energies
Source: Dr. Fedawy’s Lecture notes
Fig 4.52
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Kronig-Penney Model
11
Source: Dr. M. Fedawy’s Lecture notes
Lecture Notes Prepared by: Dr. Amr Bayoumi, Dr. Nadia Rafat
Solid State Electronics EC210, Fall 2014
Arab Academy for Science and Technology AAST – Cairo,
Lecture 8: Band Theory: Kronig-Penny Model and Effective Mass 12
Fig 4.54
a
E
[11]k3
E
k1[10]
Band
Band
Energy gap
Energy gap
Band
BandFirst
Brillouin Zone
Second Brillouin
Zone
Second
Brillouin Zone
First
Brillouin Zone
a
The E-k behavior for the electron along different directions in the two
dimensional crystal. The energy gap along [10] is at /a whereas it is
at /a along [11].
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Lecture Notes Prepared by: Dr. Amr Bayoumi, Dr. Nadia Rafat
Solid State Electronics EC210, Fall 2014
Arab Academy for Science and Technology AAST – Cairo,
Lecture 8: Band Theory: Kronig-Penny Model and Effective Mass 13
Fig 4.55
(a) Metal: For the electron in a metal there is no apparent energy gapbecause the 2nd BZ (Brillouin Zone) along [10] overlaps the 1st BZ along[11]. Bands overlap the energy gaps. Thus the electron can always find anyenergy by changing its direction.(b) Semiconductor or insulator: For the electron in a semiconductor there isan energy gap arising from the overlap of the energy gaps along [10] and[11] directions. The electron can never have an energy within this energygap, Eg.
Energy gap
[11][10] Overlapped
energy gaps
Energy gap = Eg
1st BZ
band
2nd BZ
band
(b) Semiconductor and insulator
Energy gap
[11][10]Bands overlap
energy gaps
Energy gap
1st BZ
band
2nd BZ
&
1st BZ
overlapped
band
2nd BZ
band
(a) Metal
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Lecture Notes Prepared by: Dr. Amr Bayoumi, Dr. Nadia Rafat
Solid State Electronics EC210, Fall 2014
Arab Academy for Science and Technology AAST – Cairo,
Lecture 8: Band Theory: Kronig-Penny Model and Effective Mass 14
Fig 5.49
The E-k diagram of a direct bandgap semiconductor such as GaAs. The E-k curve consists of many discrete points each point corresponding to a possible state, wavefunction y k(x), that is allowed to exist in the crystal. The points are so close that we normally draw the E-k relationship as a continuous curve. In the energy range Ev to Ec there are no points (yk(x) solutions).
Ek
k/a-/a
Ec
Ev
CB
VB
Ec
Ev
The E-k Diagram The Energy BandDiagram
Empty k
Occupied k
h+
e-
Eg
e-
h+
h
VB
h
CB
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Lecture Notes Prepared by: Dr. Amr Bayoumi, Dr. Nadia Rafat
Solid State Electronics EC210, Fall 2014
Arab Academy for Science and Technology AAST – Cairo,
Lecture 8: Band Theory: Kronig-Penny Model and Effective Mass 15
Fig 5.51
(a) In the absence of a field, over a long time, average of all k values iszero, there is no net momentum in any one particular direction. (b) Inthe presence of a field E in the -x direction, the electron accelerates inthe +x direction increasing its k value along x until it is scattered to arandom k value. Over a long time, average of all k values is along the+x direction. Thus the electron drifts along +x.
E
k
CB
-k
k+k-
Latticescattering
kav = 0-x x
(a)
kav > 0-x x
E
k
CB
-k
E
k1-
Latticescattering
k1+k2+
k3+
(b)From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Lecture Notes Prepared by: Dr. Amr Bayoumi, Dr. Nadia Rafat
Solid State Electronics EC210, Fall 2014
Arab Academy for Science and Technology AAST – Cairo,
Lecture 8: Band Theory: Kronig-Penny Model and Effective Mass 16
Fig 5.52
(a) In a full valence band there is no net contribution to the current.There are equal numbers of electrons (e.g. at b and b') withopposite momenta. (b) If there is an empty state (hole) at b at thetop of the band then the electron at b' contributes to the current.
Ev
VB
E
k-k
bb'
(a)
VB
E
k-k
b'
(b)
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Effective Mass
17
F = q = moa
where mo is the electron mass
Fext = (-q)E
Fext + Fint= moa Fext = mn*a
where mn* is the electron effective mass
In vacuum In semiconductor
q
q
Lecture Notes Prepared by: Dr. Amr Bayoumi, Dr. Nadia Rafat
Solid State Electronics EC210, Fall 2014
Arab Academy for Science and Technology AAST – Cairo,
Lecture 8: Band Theory: Kronig-Penny Model and Effective Mass 18
Effective Mass
Group Velocity defined as the velocity of the wavefunction of the electrons (analogous to speed of sinusoidal wave ):
𝑣𝑔 =𝑑𝑥
𝑑𝑡=
𝑑ω
𝑑𝑘
𝜔 = 𝐸/ℏ → 𝑣𝑔 =1
ℏ
𝑑𝐸
𝑑𝑘
→ 𝑑𝐸 = 𝑣𝑔
ℏ 𝑑𝑘, 𝑑𝑥 = 𝑣𝑔
𝑑𝑡
𝑑𝐸 = 𝐹𝑒𝑥𝑡
𝑑𝑥 = 𝐹𝑒𝑥𝑡 𝑣𝑔 𝑑𝑡
𝐹𝑒𝑥𝑡 =1
𝑣𝑔
𝑑𝐸
𝑑𝑡 → 𝐹𝑒𝑥𝑡 = ℏ
𝑑𝑘
𝑑𝑡
18
Lecture Notes Prepared by: Dr. Amr Bayoumi, Dr. Nadia Rafat
Solid State Electronics EC210, Fall 2014
Arab Academy for Science and Technology AAST – Cairo,
Lecture 8: Band Theory: Kronig-Penny Model and Effective Mass 19
Effective Mass (2)
Acceleration:
𝑎 =𝑑𝑣𝑔
𝑑𝑡 =
𝑑
𝑑𝑡
1
ℏ
𝑑𝐸
𝑑𝑘=
1
ℏ
𝑑
𝑑𝑘
𝑑𝐸
𝑑𝑡=
1
ℏ
𝑑
𝑑𝑘
𝑑𝐸
𝑑𝑘
𝑑𝑘
𝑑𝑡
𝑎 =1
ℏ
𝑑2𝐸
𝑑𝑘2 𝑑𝑘
𝑑𝑡=
1
ℏ2
𝑑2𝐸
𝑑𝑘2ℏ
𝑑𝑘
𝑑𝑡=
1
ℏ2
𝑑2𝐸
𝑑𝑘2𝐹𝑒𝑥𝑡
Using 𝐹𝑒𝑥𝑡 = 𝑚∗𝑎
𝑚∗ =
1
ℏ2
𝑑2𝐸
𝑑𝑘2
−1
= ℏ2 𝑑2𝐸
𝑑𝑘2
−1
19