Fall 2006
Fundamentals of
Semiconductor Physics
万 歆
Z hejiang I nstitute of M odern P hysics xinwan@zimp.zju.edu.cn
http://zimp.zju.edu.cn/~xinwan/

In memory of Prof. Xie Xide (1921-2000)

Chapter 1. Fundamentals
1.1 Bonds and bands
1.2 Impurities and defects
1.3 Statistical distribution of charge carriers
1.4 Charge transport
Total 12 hours.

Resistivity of Conductors,
Semiconductors & Insulators

Semiconductor-forming
Elements
B C N O
Al Si P S
Zn Ga Ge As Se
Cd In Sn Sb Te

Chapter 1. Fundamentals
1.1 Bonds and bands
– Crystal structures
– Bond picture
– Band picture
• “Nearly free” electron model
• Tight-binding model (LCMO)
• k·p perturbation
1.2 Impurities and defects
1.3 Statistical distribution of charge carriers
1.4 Charge transport

An Apparently Easy Problem
Nuclei
Solid
In principle, by solving Schrödinger's equation interaction
Electrons

Lattice & Unit Cell
Crystal = periodic array of atoms
Crystal structure = lattice + basis
Unit cell: Parallelpiped spanned by a,b,c
The choice of lattice as well as its axes a,b,c is not unique.
But it is usually convenient to choose with the consideration of symmetry

Cubic Lattices
Q
: How many atoms are there in each of the unit cells?

Miller Indices

Elemental Semiconductors
Si, Ge: 4 valence electrons
14 Si: 1s 2 2s 2 2p 6 3s 2 3p 2 32 Ge: [Ar]3d 10 4s 2 4p 2

Covalent Bond

Complex Lattices
(a) Diamond: Si, Ge
(b) Zinc blende: GaAs, ZnS
Interpenetrating fcc lattices

ZnS
Wurtzite Lattice

Band: An Alternative Picture
From hydrogen atom to molecule
H H
H
2 antibonding bonding

Band Formation
N degenerate levels evolve into an energy band

Silicon
14 Si: 1s 2 2s 2 2p 6 3s 2 3p 2

Resistivity
Conduction band , valence band & band gap
  constant, T

0
  
, T

0

Formal Treatment

Approximation 1
Separation of electrons into valence electrons and core electrons
Ion cores = core electrons + nuclei e.g. Si: [Ne] 3s 2 3p 2

Approximation 2
Born-Oppenheimer or adiabatic approximation
To electrons, ions are essentially stationary.
Ions only see a time-averaged adiabatic electronic potential. ionic motion
 Lattice vibration (phonons) electronic motion electron-phonon interaction

Separation of Motion

Approximation 3
Mean-field approximation : Every electron experiences the same average potential V(r)
V(r): by first principle ( ab initio ), or by semi-empirical approach

Can we calculate everything?
• Yes.
– First principle band calculations
– Slater, …
• No.
– Disordered & strongly correlated systems
– Mott, Anderson, …

Reciprocal Lattice
• Definition: The set of all wave vectors K that yield plane waves with the periodicity of a given lattice is known as its reciprocal lattice .
R
 ma
 nb
 pc e i K
 
 e i K r e i K R 
1
Try to verify the following b i the reciprocal lattice of the lattice spanned by a i
.
form
In 1D, we have simply K = 2n p
/ a , where a is the lattice constant.

FCC  BCC

FCC Lattice & its Brillouin Zone
Lattice (fcc): real space
Diamond structure
Reciprocal lattice (bcc)

Construct the Brillouin Zone

A Digression on Group Theory
Ref.: Yu & Cardona, Section 2.3

Rotational Symmetry
Ref.: Yu & Cardona, Section 2.3

Translational Symmetry
• Define operator T
R
( )

(

R )
R
• Discrete translational symmetry
( )

(

R )

V x
R
 
R

0
• Bloch’s theorem:
 nk
( )
 e u nk
( )
T u
R nk
( )
 u nk
( )

Example: 1D Empty Lattice
• V  0:

2 2 k
,

( )
 e ikx
2 m
• We assume an imaginary periodicity of a. Define the reciprocal lattice constant G = 2 p
/ a . We can therefore restrict k within the range of [ -G/2, G/2 ].
 nk
( k
 nG )
 e
 
E ( ) nk

 ikx e u nk
( ), u nk
( )
 e inGx
2
( k
 nG )
2
2 m

V  0:
Free Electrons in 1D
E ( ) nk

2
 k
 nG

2
,
2 m
 nk
( )
 e
 

Comments
• The wave vector k is not momentum p/

, since Hamiltonian does not have complete translational invariance. Rather,
 k is known as crystal momentum (quantum number characteristic of the translational symmetry of a periodic potential).
• The wave vector k can be confined to the first Brillouin zone.
• More in A/M Chapter 8.

“Nearly Free” Electrons
( )

(

)
 
U e k iKx
K





2
2 2 k m
U
K
U
K
2

K
 k

2




2 m

Nearly Free & Pseudopotential
Si

Overlap of Molecular Orbitals

Tight-binding or LCMO hopping
  

H t c c . .
i j

E
 
2 t cos( k a x
)
 cos( k a y
)
E(k)
2D: z = 4 nearest neighbors k

Si – Ge – a
-Sn

Comparison
Nearly free e ’s + pseudopotential
• Electrons nearly free
• Wave functions approximated by plane waves
• Electrons in conduction band are delocalized, so can be approximated well by nearly free electrons
Tight-binding or LCMO approach
• e ’s tightly bound to nuclei
• Linear combination of atomic wave functions
• Valence electrons are concentrated mainly in the bonds and so they retain more of their atomic character.

Band Diagram
Conduction band
E
E g
Allowed states k
Forbidden band gap
Conduction band
Allowed states
Valence band
E

2 2 k
2 m *
Valence band
1 m *

1
2


2 d E dk 2


E c
E i
E v

Direct Bandgap GaAs

Band Structure: Si & Ge

Electrons and Holes
J cb
   i
 cb ev i
J vb
   i
 vb ev i
0
J vb
  i
 vb
 ev i



 i
 filled band
 ev i



 


 i
 empty state
 ev i



  i
 empty state ev i

The k ·p Method

Conduction Band (Nondegenerate)

Comment on k ·p Method
• Band structure over the entire BZ can be extrapolated by the zone center energy gaps and optical matrix elements.
• One can btain analytical expression for band dispersion and effective mass around high-symmetry points.
• Nondegenerate perturbation is applicable to the conduction band minimum in direct-bandgap semiconductors (zinc-blende, wurtzite); degenerate perturbation to top valence band (diamond, zinc-blende, wurtzite). Ref.: Yu & Cardona, Section 2.6
• The trend of m* can be explained.

Chapter 1. Fundamentals
1.1 Bonds and bands
1.2 Impurities and defects
– Classification of defects
– Point defects
– Shallow (hydrogenic) impurities
1.3 Statistical distribution of charge carriers
1.4 Charge transport

Classification of Defects
• Point defects
• Line defects
• Surface states

Point Defects
V
A
I
A
C
A

More Classifications
• Intrinsic vs extrinsic
– Intrinsic: native, such as vacancies or antisite defects
– Extrinsic: foreign, Si:P
• Shallow vs deep – “effective mass approximation”
• Donors, double donors, isovalent center
– Examples: Si:P, Si:Se, Si:C

Si:P
Shallow Impurity States
H

H cryst
 e
2
 r impurity potential
B C N O
Al Si P S
Zn Ga Ge As Se
Cd In Sn Sb Te
Screened Coulomb potential
• Break translational symmetry
• No Bloch’s theorem!?
Effective mass approximation

Hydrogenic Wave Function

( )
  k
0
Bloch wave
Hydrogenic envelope
E



1
 
2 m
* e
2
 r


( )

( )
2s
1s
2p
Hydrogenic bound states 1s, 2s, 2p, ...
+ continuum
(conduction band)
“Rydberg”
“Bohr radius” m
* e
4
2


2  2
 2 m
* e
2
~
~
10 meV (Ge)
30 meV (Si)
50 Å (Ge)
20 Å (Si)

Band Diagram
E
C. B.
V. B.
Donor
(Si:P, Ge:As) k
Acceptor
(Si:B)
Allowed states
Forbidden band gap
Allowed states
Conduction band
Valence band
E c
E d
E i
E a
E v

Effective Mass Approximation
1. Introduce Wannier Functions (indexed by lattice vector in real space): Fourier transforms of
Bloch functions
For very localized electrons, Wannier functions are roughly atomic orbitals.

Effective Mass Approximation
2.
Express H in the basis of Wannier functions
Assume
3.
Parabolic, isotropic, nondegenerate

Envelope Wave Function
Approximation valid for large effective Bohr radius a*(small
D k).

Comment on EMA
• The net effect of the crystal potential on the donor electron inside the crystal is to change the electron mass from the value in free space to the effective mass m* and also to contribute a dielectric constant
 of the host crystal.
• Only conduction band states over a small region of reciprocal space around the band minimum contribute to the defect wave function if the effective Bohr radius a* is much larger than the lattice constant a
0
.

Heavy Doping
• Light doping: impurity atoms do not interact with each other
 impurity level
• Heavy doping: perturb the band structure of the host crystal  reduction of bandgap
E
E c
E d
E v
E g

(E)

Metal-Insulator Transition
• Average impurity-impurity distance = Bohr radius
• Mott criterion
N a d
3
B
 1
64

Chapter 1. Fundamentals
1.1 Bonds and bands
1.2 Impurities and defects
1.3 Statistical distribution of charge carriers
– Thermal equilibrium
– Mass-action law
– Fermi level
1.4 Charge transport

Thermal Equilibrium
• Thermal equilibrium is a dynamic situation in which every process is balanced by its inverse process.
• Thermal equilibrium means that time can run toward the past as well as into the future.
E
2
E
1

Mass-Action Law
• Electron-hole pairs: generation rate = recombination rate
• Generation: G = f
1
(T) f
1
: determined by crystal physics and T
• Recombination: R = npf
2
(T)
– Electrons and holes must interact to recombine
• At equilibrium, G = R npf T
2
 f T
1
( ) np
 f T
1 f T
2
( )
 f T
3
( )
 n i
2
• Intrinsic case (all carriers result from excitation across the forbidden gap): n = p = n i

Fermi Level
• Fermi-Dirac distribution
D


E
1

E f

/ k T
• Boltzmann distribution
B
( )
 exp




• Density of electrons

E

E f


 k T
B p n

D
( ) c
( ) cb

 
1

F E
D

N E dE v
( ) vb

E

E f k T
B
(not too heavily doped)

Intrinsic Semiconductors

A Parabolic Band
E

E c

2 2 k
 dE

2 k m * dk
( )

( )
3
N E dE N k d k

2
 
3
4 p
2 k dk
 n
 

4 p



2 p h m
2
*


 3/ 2
  c

1 / 2
( ) ( )
C ex p




E
C

E
F k
B
T



2 p

0
 d 


 c k T
B


  ex p 


1 k

B
T
C 


 k

B
T c
 1/ 2

 n

N
C exp



E
C

E
F k T
B


, N
C

2
2 p m k T
C B h 2
 3/ 2

Comment
• Multivalley, such as Si:
– additional factor of number of valleys
Density of states effective mass
• Anisotropic band
– Effective mass: Geometrical average of mass components
• Valence band: N
V
– Sum of heavy hole, light hole (neglecting split-off band)
– replace m* by hole effective mass

Intrinsic Carrier Concentration n

N
C exp



E
C

E
F k T
B p

N
V exp



E
F

E
V k T
B



 n i
2  np

N N
C V exp



E
C

E
V k T
B



N N
C V exp



E g k T
B

 p

N
V exp



E
F

E
V k T
B


, N
V

2
2 p m k T
V B h
2
3/ 2

Doped Semiconductors
• Assuming full ionization, charge neutrality
• With intentional doping, typically for n-type
N d

N a n i n

N d

N a p
 n i
2
N d

N a
N d
 
N a
 n n
   n n i
2

N d

N a n

N d

N a
2






N d

N a
2
2
 n i
2 

 1/ 2
• Majority carriers
• Minority carriers
• Compensation

Extrinsic Semiconductors n
E i i


N
C exp
E
C

E
V
2





E
C

E i k T k T
B
2
B ln



N
V

E
C

E
V
N 2
C n
 n i exp 

 p
 n i
E f
B
 k T
E exp 


E i

E f k T
B i 





E
F
Conduction band
E c
E d
E i
E a
E v
Valence band

Carrier Concentration vs T

Nonmonotonic behavior
Fermi Energy in Si
E g
↓

Occupation of Impurity level
0 n d

N d
D  
,
E d
E d
1
 e
 
E d

E
F
D 
/
1
2 e
 
E d

E
F

/ k T e
 
E d

E
F
D 
/ n d
N d

1
1
2 e
 
E d

E
F

/ k T 
1
 d

N a
  p a

N d
2E d
+
D
,

Chapter 1. Fundamentals
1.1 Bonds and bands
1.2 Impurities and defects
1.3 Statistical distribution of charge carriers
1.4 Charge transport

Charge Transport
• Ohm’s law
V

IR
• Equipartition of energy
E

1
2
* 2 m v th

3
2 k T
B
• Room temperature (300K) k T
B RT
~ 30meV, m c e
2 
0.5MeV, m * / m e

0.07 (GaAs) v th
/ c
 k T m c
B
2
3 / * ~ 10

3 v th
7
~ 10 cm/s

The complete notes for Chapter 1 are expected to be available after
Sept. 20, 2006.