Transcript 2013.9.23

DEE4521
Semiconductor Device Physics
Lecture 2:
Band Structure
Prof. Ming-Jer Chen
Department of Electronics Engineering
National Chiao-Tung University
09/24/2013
1
Electron Distribution Function f(x, y, z, kx, ky, kz, t)
According to Heisenberg’s Uncertainty Principle,
We have a 6 dimensionality space at a time for a
Semiconductor in a Real x-y-z Space; and at
each point (x,y.z),
Electrons, Holes, Phonons, and Photons
are all better dealt with in another space:
kx-ky-kz Space
or Wavevector Space
or Momentum Space
2
by Analogy
• De Broglie’s Wave and Particle Duality
• Degree of Freedom (DOF) – Kinetic Energy
• Potential Energy and its Reference
3
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
4
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
5
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
6
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
7
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)
8
Electron E-k Diagram
Indirect gap
Direct gap
EG: Energy Gap
9
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)
10
Valence-Band Structure
11
Conduction-Band Electrons and
Valence-Band Holes and Electrons
Hole: Vacancy of Valence-Band Electron
12
No Electrons in Conduction Bands
All Valence Bands are filled up.
13
14
Work Function
E
 (Electron Affinity) (= 4.05 eV for Si)
Ec
x
15