Solid State Electronics EC210
Arab Academy for Science and Technology
AAST – Cairo
Spring 2015
Lecture 9
Band Theory:
E-K Diagram, Energy Gaps and Effective Mass
Lecture Notes Prepared by:
Dr. Amr Bayoumi, Dr. Nadia Rafat
1
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Electronic Materials and
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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
Lecture 9:E-K Diagram, Band Gap,
Effective Mass
Solid State Electronics EC210, Spring 2015
Arab Academy for Science and Technology
AAST – Cairo,
2
Pages
• Kasap:
– P.355 (Kronig Penny)
– P.303-304, p. 454-455 (Effective Mass)
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
Lecture
Lecture
8: 9:E-K
Band Diagram,
Theory: Kronig-Penny
Band Gap,
Model and Effective
EffectiveMass
Mass
Solid State Electronics EC210, Spring 2015
Arab Academy for Science and Technology
AAST – Cairo,
3
E-K Diagram
usingKronig-Penney Model
Source: Dr. M. Fedawy’s Lecture notes
Fig 4.52
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
4
Energy Gap (Bandgaps, Eg)
E
Band
Second
Brillouin Zone
Energy gap
Band
First
Brillouin Zone
Band
Second Brillouin
Zone
k1[10]
E
Energy gap
From Principles of
Electronic Materials
and Devices, Third
Edition, S.O. Kasap (©
McGraw-Hill, 2005)
Band
 
a a
First
Brillouin Zone
k3[11]
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
Lecture 9:E-K Diagram, Band Gap,
Effective Mass
Solid State Electronics EC210, Spring 2015
Fig 4.54
Arab Academy for Science
and Technology
AAST – Cairo,
5
Energy Gaps (Eg)
Energy gap
2nd BZ
band
2nd BZ
&
1st BZ
overlapped
band
Energy gap
1st BZ
band
[10]
[11]
2nd BZ
band
Energy gap = Eg
Energy gap
1st BZ
band
Bands overlap
energy gaps
(a) Metal
[10]
[11]
Overlapped
energy gaps
(b) Semiconductor and insulator
(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 From Principles of
Electronic Materials
[11] directions. The electron can never have an energy within this energy and Devices, Third
Edition, S.O. Kasap (©
gap, Eg.
McGraw-Hill, 2005)
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
Lecture 9:E-K Diagram, Band Gap,
Effective Mass
Solid State Electronics EC210, Spring 2015
Fig 4.55
Arab Academy for Science
and Technology
AAST – Cairo,
6
EE
Ev
b'
b
(a)
VB
-k
k
E
b'
(b)
From Principles of
Electronic Materials and
Devices, Third Edition,
S.O. Kasap (© McGrawHill, 2005)
VB
k
-k
(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.
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
Lecture 9:E-K Diagram, Band Gap,
Effective Mass
Solid State Electronics EC210, Spring 2015
Fig 5.52
Arab Academy for Science
and Technology
AAST – Cairo,
7
The E-k Diagram
Ek
The Energy Band
Diagram
CB
CB
Empty k
e-
e-
Ec
Ec
h
Eg
h+
VB
From Principles of
Electronic Materials
and Devices, Third
Edition, S.O. Kasap (©
McGraw-Hill, 2005)
h
Ev
Ev
Occupied k
h+
VB
-/a
/a
k
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).
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
Lecture 9:E-K Diagram, Band Gap,
Effective Mass
Solid State Electronics EC210, Spring 2015
Fig 5.49
Arab Academy for Science
and Technology
AAST – Cairo,
8
E
E
CB
Lattice
scattering
k1-
k-
k3+
k1+k2+
k+
-k
From Principles of
Electronic Materials
and Devices, Third
Edition, S.O. Kasap (©
McGraw-Hill, 2005)
CB
Lattice
scattering
-x
E
kav = 0
k
-k
x
-x
k
kav > 0
(a)
x
(b)
(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.
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
Lecture 9:E-K Diagram, Band Gap,
Effective Mass
Solid State Electronics EC210, Spring 2015
Fig 5.51
Arab Academy for Science
and Technology
AAST – Cairo,
9
Effective Mass
In vacuum
In semiconductor
q
q
F = q e = moa
where
mo is the electron mass
10
Fext = (-q)E
Fext + Fint= moa
Fext = mn*a
where
mn* is the electron effective mass
Effective Mass
Ex
Ex
CRYSTAL
VACUUM
e-
a=
Fext
F int
Fext
a=
me
Fext
m e*
x
x
(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.)
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
From Principles of Electronic Materials and
Solid State Electronics EC210, Spring 2015
Lecture Notes Prepared by:
Lecture 9:E-K Diagram, Band Gap,
Devices, Third Edition, S.O. Kasap (©
Fig 4.19
Arab Academy for Science
and Technology
Dr. Amr Bayoumi, Dr. Nadia Rafat
Effective Mass
McGraw-Hill, 2005)
AAST – Cairo,
11
Example of Effective Mass
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
Lecture 9:E-K Diagram, Band Gap,
Effective Mass
From Principles of Electronic
Solid State ElectronicsMaterials
EC210, Spring
and2015
Devices, Third
Arab Academy for ScienceEdition,
and Technology
S.O. Kasap (©
AAST – Cairo,
12
Effective Mass
Group Velocity defined as the velocity of the wavefunction of the
electrons (analogous to speed of sinusoidal wave ):
𝑑𝑥 𝑑ω
𝑣𝑔 =
=
,
𝑑𝑥 = 𝑣𝑔 𝑑𝑡
𝑑𝑡 𝑑𝑘
𝐸
𝜔 =
ℏ
1 𝑑𝐸
→ 𝑣𝑔 =
ℏ 𝑑𝑘
𝑑𝐸 = 𝐹𝑒𝑥𝑡 𝑑𝑥 = 𝐹𝑒𝑥𝑡 𝑣𝑔 𝑑𝑡 → 𝐹𝑒𝑥𝑡
→ 𝑑𝐸 = 𝑣𝑔 ℏ 𝑑𝑘
1 𝑑𝐸
𝑑𝑘
=
→ 𝐹𝑒𝑥𝑡 = ℏ
𝑣𝑔 𝑑𝑡
𝑑𝑡
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Lecture
13 Notes Prepared by:
Dr. Amr Bayoumi, Dr. Nadia Rafat
Lecture 9:E-K Diagram, Band Gap,
Effective Mass
Solid State Electronics EC210, Spring 2015
Arab Academy for Science and Technology
AAST – Cairo,
13
Effective Mass (2)
Acceleration:
𝑑𝑣𝑔
𝑑 1 𝑑𝐸
1 𝑑 𝑑𝐸
1 𝑑 𝑑𝐸 𝑑𝑘
𝑎=
=
=
=
𝑑𝑡
𝑑𝑡 ℏ 𝑑𝑘
ℏ 𝑑𝑘 𝑑𝑡
ℏ 𝑑𝑘 𝑑𝑘 𝑑𝑡
1 𝑑2𝐸 𝑑𝑘
1 𝑑2𝐸 𝑑𝑘
1 𝑑2𝐸
𝑎=
= 2 2ℏ
= 2 2 𝐹𝑒𝑥𝑡
2
ℏ 𝑑𝑘 𝑑𝑡 ℏ 𝑑𝑘 𝑑𝑡 ℏ 𝑑𝑘
Using 𝐹𝑒𝑥𝑡 = 𝑚∗ 𝑎
1 𝑑2 𝐸
∗
𝑚 = 2 2
ℏ 𝑑𝑘
−1
= ℏ2
𝑑2 𝐸
𝑑𝑘2
−1
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Lecture
14 Notes Prepared by:
Dr. Amr Bayoumi, Dr. Nadia Rafat
Lecture 9:E-K Diagram, Band Gap,
Effective Mass
Solid State Electronics EC210, Spring 2015
Arab Academy for Science and Technology
AAST – Cairo,
14