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```DEE4521
Semiconductor Device Physics
Lecture 2:
Band Structure in Semiconductors
Prof. Ming-Jer Chen
Department of Electronics Engineering
National Chiao-Tung University
09/20/2012
1
….Standing on the Shoulders of Giants…
-- Isaac Newton
With this in mind, then device physics can be
interesting and useful!
2
We see Semiconductors in x-y-z Space
but
Electrons and Holes and Phonons and Photons
also Live in another Space: kx-ky-kz Space
or Wavevector Space
or Momentum Space
Remember: Our EE’s Terminologies like V and I
want us to see Semiconductors in this additional
space as well.
3
SOP for Band Structure
1. Think a Photoelectric Effect Experiment by Einstein
 Potential Energy of Interest
2. Degree of Freedom (DOF) and Kinetic Energy
3. Combine Newton’s Mechanisms and De Broglie’s Hypothesis
Then, You have Conduction-Band Structure
Again think a Photoelectric Experiment with High Energy Photons
Or
A Photon Transparent Experiment through a Very Thin Sample
Finally, You Get Energy Gap and hence a Valence-Band Structure
4
by Analogy
1. De Broglie’s Wave and Particle Duality
2. Degree of Freedom (DOF) – Kinetic Energy
3. Potential Energy and its Reference
5
Electrons in Solid
A ball in the air
Ball’s Mass m in x direction
Electron Effective Mass mx* in x
direction
Ball’s Momentum mvx
Crystal Momentum ħkx
(kx: wave vector in x direction)
Effective Mass m*
Electron Momentum ħ(kx-kxo)
Ball’s Kinetic Energy mvx
2/2
Crystal momentum
Electron Kinetic
Energy Ek =
2
2
E = ħ kx /2m*
ħ2(kkx-kxo)2/2m
x*
1. kxo: a point in k space around
which electrons are likely found.
2. Crystal momentum (global) must
be conserved in k space, not Electron
6
Momentum (local).
Si Conduction-Band Structure in wave vector k-space
(Constant-Energy Surfaces in k-space)Effective mass approximation:
Kinetic energy
m* (to reflect electron
confinement in solid)
Ek = ħ2(ky – kcy)2/2m*
+ ħ2kx2/2m*
+ ħ2kz2/2m*
Ellipsoidal energy surface
(silicon)
E = Ek + Ec
6-fold valleys
Potential energy
total electron energy
Kcy  0.85 (2/a);
Longitudinal Effective Mass m* (or ml*)= 0.92 mo
Transverse Effective Mass m* (or mt*)= 0.197 mo
a: Lattice Constant
7
Effective Masses of Commonly Used Materials
(You may then find that these effective masses are far from
the rest mass. This is just one of the quantum effects.)
Electron and hole effective mass are anisotropic,
depending on the orientation direction.
Electron (not hole) effective mass
is isotropic, regardless of orientation.
Rest mass of electron mo
(by Prof. Robert F. Pierret)
= 0.9110-30 kg
Ge
Si
GaAs
ml*/mo
1.588
0.916
mt*/mo
0.081
0.190
me*/mo
0.067
mhh*/mo
0.347
0.537
0.51
mlh*/mo
0.0423
0.153
0.082
mso*/mo
0.077
0.234
0.154
8
Electron Energy E-k Relation in a Crystal
Zinc blende
a = 5.6533 Å
Diamond
a = 5.43095 Å
Quasi-Classical Approximation
Diamond
a = 5.64613 Å
(
3/2
)2/a
1
d 2 E 
2 

m* 
dK 2 
K 0
Bottom of valley
9
k-Space Definition
<001>
3-D View
(out-of-plane)
The zone center (Gamma at k = 0)
The zone end along <100>
On (001) Wafer
<100> (in-plane)
Length = 2/a (Gamma to X)
<010>
(in-plane)
Length =( 3/2 )2/a (Gamma to L)
(001)
The zone end along <111>
a: Lattice Constant
(Principal-axis x, y, and z coordinate system usually aligned to
match the k coordinate system)
10
Electron E-k Diagram
Indirect gap
Direct gap
EG: Energy Gap
11
Comparisons between Different Materials
Conduction Band
(Constant-Energy Surface)
8-fold valleys along <111>
(half-ellipsoid in Brillouin)
one valley at the zone center
(sphere)
6-fold valleys along <100>
(ellipsoid)
12
Valence Band Structure
13
Conduction-Band Electrons and
Valence-Band Holes
Hole: Vacancy of Valence-Band Electron
14
No Electrons in any Conduction Bands
All Valence Bands are filled up.
15
16
Work Function
E
 (Electron Affinity) (= 4.05 eV for Si)
Ec
x
17
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