This document contains lecture notes on band theory, E-K diagrams, energy gaps, and effective mass in solid state electronics. It includes diagrams and explanations of how the energy-momentum (E-K) diagram represents allowed electron states in crystals and how energy gaps differentiate metals, semiconductors and insulators. It also discusses how the effective mass of electrons inside crystals is different than in vacuum due to interactions with the lattice, and how this affects electron acceleration under an applied electric field.

1. Lecture 9
Band Theory:
E-K Diagram, Energy Gaps 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
Spring 2015
1

2. Lecture Notes Prepared by:
Dr. Amr Bayoumi, Dr. Nadia Rafat
Solid State Electronics EC210, Spring 2015
Arab Academy for Science and Technology
AAST – Cairo,
Lecture 9:E-K Diagram, Band Gap,
Effective Mass 2
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
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3. Lecture Notes Prepared by:
Dr. Amr Bayoumi, Dr. Nadia Rafat
Solid State Electronics EC210, Spring 2015
Arab Academy for Science and Technology
AAST – Cairo,
Lecture 9:E-K Diagram, Band Gap,
Effective Mass 3
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Pages
• Kasap:
– P.355 (Kronig Penny)
– P.303-304, p. 454-455 (Effective Mass)
Lecture 8: Band Theory: Kronig-Penny
Model and Effective Mass

4. Fig 4.52
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
E-K Diagram
usingKronig-Penney Model
4
Source: Dr. M. Fedawy’s Lecture notes

5. Lecture Notes Prepared by:
Dr. Amr Bayoumi, Dr. Nadia Rafat
Solid State Electronics EC210, Spring 2015
Arab Academy for Science and Technology
AAST – Cairo,
Lecture 9:E-K Diagram, Band Gap,
Effective Mass 5
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Energy Gap (Bandgaps, Eg)
Fig 4.54

a
E
[11]
k3
E
k1
[10]
Band
Band
Energy gap
Energy gap
Band
Band
First
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)

6. Lecture Notes Prepared by:
Dr. Amr Bayoumi, Dr. Nadia Rafat
Solid State Electronics EC210, Spring 2015
Arab Academy for Science and Technology
AAST – Cairo,
Lecture 9:E-K Diagram, Band Gap,
Effective Mass 6
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Energy Gaps (Eg)
Fig 4.55
(a) Metal: For the electron in a metal there is no apparent energy gap
because the 2nd BZ (Brillouin Zone) along [10] overlaps the 1st BZ along
[11]. Bands overlap the energy gaps. Thus the electron can always find any
energy by changing its direction.
(b) Semiconductor or insulator: For the electron in a semiconductor there is
an energy gap arising from the overlap of the energy gaps along [10] and
[11] directions. The electron can never have an energy within this energy
gap, 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)

7. Lecture Notes Prepared by:
Dr. Amr Bayoumi, Dr. Nadia Rafat
Solid State Electronics EC210, Spring 2015
Arab Academy for Science and Technology
AAST – Cairo,
Lecture 9:E-K Diagram, Band Gap,
Effective Mass 7
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
E-
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') with
opposite momenta. (b) If there is an empty state (hole) at b at the
top of the band then the electron at b' contributes to the current.
Ev
VB
E
k
-k
b
b'
(a)
VB
E
k
-k
b'
(b)
From Principles of
Electronic Materials and
Devices, Third Edition,
S.O. Kasap (© McGraw-
Hill, 2005)

8. Lecture Notes Prepared by:
Dr. Amr Bayoumi, Dr. Nadia Rafat
Solid State Electronics EC210, Spring 2015
Arab Academy for Science and Technology
AAST – Cairo,
Lecture 9:E-K Diagram, Band Gap,
Effective Mass 8
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
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 Band
Diagram
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)

9. Lecture Notes Prepared by:
Dr. Amr Bayoumi, Dr. Nadia Rafat
Solid State Electronics EC210, Spring 2015
Arab Academy for Science and Technology
AAST – Cairo,
Lecture 9:E-K Diagram, Band Gap,
Effective Mass 9
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Fig 5.51
(a) In the absence of a field, over a long time, average of all k values is
zero, there is no net momentum in any one particular direction. (b) In
the presence of a field E in the -x direction, the electron accelerates in
the +x direction increasing its k value along x until it is scattered to a
random 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-
Lattice
scattering
kav = 0
-x x
(a)
kav > 0
-x x
E
k
CB
-k
E
k1-
Lattice
scattering
k1+
k2+
k3+
(b)
From Principles of
Electronic Materials
and Devices, Third
Edition, S.O. Kasap (©
McGraw-Hill, 2005)

10. Effective Mass
10
F = q e = 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

11. Lecture Notes Prepared by:
Dr. Amr Bayoumi, Dr. Nadia Rafat
Solid State Electronics EC210, Spring 2015
Arab Academy for Science and Technology
AAST – Cairo,
Lecture 9:E-K Diagram, Band Gap,
Effective Mass 11
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Effective Mass
Fig 4.19
From Principles of Electronic Materials and
Devices, Third Edition, S.O. Kasap (©
McGraw-Hill, 2005)
e-
Fext
VACUUM
x
Fext
me
a =
(a)
Ex
Fint
CRYSTAL
x
Fext
me
*
a =
(b)
(a) An external force Fext applied to an electron in vacuum results in an
acceleration avac = Fext / me . (b) An external force Fext applied to an
electron in a crystal results in an acceleration acryst = Fext / me*. ( Ex is the
electric field.)
Ex

12. Lecture Notes Prepared by:
Dr. Amr Bayoumi, Dr. Nadia Rafat
Solid State Electronics EC210, Spring 2015
Arab Academy for Science and Technology
AAST – Cairo,
Lecture 9:E-K Diagram, Band Gap,
Effective Mass 12
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Example of Effective Mass
From Principles of Electronic
Materials and Devices, Third
Edition, S.O. Kasap (©

13. Lecture Notes Prepared by:
Dr. Amr Bayoumi, Dr. Nadia Rafat
Solid State Electronics EC210, Spring 2015
Arab Academy for Science and Technology
AAST – Cairo,
Lecture 9:E-K Diagram, Band Gap,
Effective Mass 13
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
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
, 𝑑𝑥 = 𝑣𝑔 𝑑𝑡
𝜔 =
𝐸
ℏ
→ 𝑣𝑔 =
1
ℏ
𝑑𝐸
𝑑𝑘
→ 𝑑𝐸 = 𝑣𝑔 ℏ 𝑑𝑘
𝑑𝐸 = 𝐹𝑒𝑥𝑡 𝑑𝑥 = 𝐹𝑒𝑥𝑡 𝑣𝑔 𝑑𝑡 → 𝐹𝑒𝑥𝑡 =
1
𝑣𝑔
𝑑𝐸
𝑑𝑡
→ 𝐹𝑒𝑥𝑡 = ℏ
𝑑𝑘
𝑑𝑡
13

14. Lecture Notes Prepared by:
Dr. Amr Bayoumi, Dr. Nadia Rafat
Solid State Electronics EC210, Spring 2015
Arab Academy for Science and Technology
AAST – Cairo,
Lecture 9:E-K Diagram, Band Gap,
Effective Mass 14
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
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
𝑑𝑘2
ℏ
𝑑𝑘
𝑑𝑡
=
1
ℏ2
𝑑2𝐸
𝑑𝑘2
𝐹𝑒𝑥𝑡
Using 𝐹𝑒𝑥𝑡 = 𝑚∗
𝑎
𝑚
∗
=
1
ℏ2
𝑑2𝐸
𝑑𝑘2
−1
= ℏ2
𝑑2𝐸
𝑑𝑘2
−1
14