# Chapter 3. Introduction to the Quantum Theory of Solids

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```					Chapter 3. Introduction to the
Quantum Theory of Solids
Allowed and Forbidden Energy Bands
Electrical Conduction in Solids
Extension to Three Dimensions
Density of States Function
Statistical Mechanics

Young-Hwan Lee
http://cafe.daum.net/lyh201circuit
E-mail : lyh201@hanyang.ac.kr
Mobile : 010-7178-1884
3.1 Allowed and Forbidden Energy Bands
※ Electronic energy states occur in bands of allowed states:
separated by forbidden energy bands
Pauli exclusion principle : only one electron is allowed to occupy any
given quantum state.
3.1.1 Formation of Energy Bands
Regular periodic arrangement of atoms →energy level will split into a
band of discrete energy levels by the Pauli exclusion principle

(a) Probability density function of an isolated hydrogen atom (b) Overlapping probability
density functions of two adjacent hydrogen atoms (c) The splitting of the n = 1 state.

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3.1 Allowed and Forbidden Energy Bands
 Example 3.1
Calculate the change in the kinetic energy of an electron when the
velocity(107 cm/s) changes by a small value(increases 1 cm/s).
1 2 1 2
E  mv2  mv1
2      2
v2  v1  v   v2  (v1  v) 2  v12  2v1v  (v) 2
2

v  v1
1
E  m(2v1v)  mv1v
2
E  (9.111031 )(105 )(0.01)  9.111028 [J]
9.1110 28
E               5.7 109 [eV]
1.6 1019
Hanyang University         fall – 2007                              3
3.1 Allowed and Forbidden Energy Bands

As atoms are brought together from infinity, the atomic orbitals overlap
and give rise to bands. Outer orbitals overlap first. The 3s orbitals give
rise to the 3s band, 2p orbitals to the 2p band, and so on.

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3.1 Allowed and Forbidden Energy Bands
Energy band theory of single crystal materials: energy band splitting
and the formation of allowed and forbidden bands
repulsion    attraction

8N states
4N electrons
(a) Schematic of an isolated silicon atom.

(b) The splitting of the 3s and 3p states of silicon
into the allows and forbidden energy bands.

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3.1 Allowed and Forbidden Energy Bands
3.1.2 The Kronig-Penney Model
3.1.3 The k-Space Diagram
The relationship of the energy and momentum for V0 = 0, P  = 0

Energy of free particle
p 2 k 2 2
E    
2m 2m
where k : wave number
p : momentum

sin a
f (a)  P           cos a
a
The shaded areas show the allowed
values of (αa) corresponding to real
values of k.

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3.1 Allowed and Forbidden Energy Bands
The energy E has discontinuities.
The concept of allowed energy bands for the particle propagation
in the crystal lattice and the concept of forbidden energies for the
particle in the crystal.

The E versus k diagram                 The E versus k diagram in the
reduced-zone representation

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3.2 Electrical Conduction in Solids
Consider electrical conduction in solids related to the band theory
3.2.1 The Energy Band and the Bond Model
 At T = 0 K, valence band는valence electrons에 의해 full filled band,
conduction band는empty band (filled band내의electrons은electrical
conduction에 기여하지 못함)
 At T > 0 K, valence band내 a few valence electrons이 covalent bond를
깨뜨리기에 충분한thermal energy를 얻어서 conduction band로
transition.

Covalent bonding in a    Breaking of a covalent   Generation of a charge
semi. at T = 0 K         bonding
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3.2 Electrical Conduction in Solids

(a) T = 0 K                 (b) T > 0 K
 temperature 가 증가함에 따라 transition되는 electrons의 수 증가.
→C.B.내에conduction electron & V.B.내empty state(hole)이 발생:
carriers (이carriers에 의해electrical conduction현상이 일어남.)

Hanyang University      fall – 2007                     9
3.2 Electrical Conduction in Solids
3.2.2 Drift Current
The drift current density due to the motion of positively charged ions
J  qNvd [A/cm 2 ]
where N : volume density [cm–3]
vd : average drift velocity [cm/s]
The drift current density due to the individual ion velocities
N
J  q  vi
i 1
If a force is applied to a particle and the particle moves, it must gain
energy.
dE  Fdx  Fvdt
The drift current density due to the motion of electrons
N
J  e vi
i 1

Hanyang University             fall – 2007                             10
3.2 Electrical Conduction in Solids
3.2.3 Electron Effective Mass
Effective mass: Taking into account the particle mass and the effect of
the internal forces.
→crystal내에서 결합에 구속되어 있지 않은 electron은 free space내
의 electron과 유사하게 결정내를 비교적 자유롭게 움직이나 atomic
core에 의한 periodic potential의 영향 때문에 conduction electron의
mass는 free space내의 electron의 mass와는 다르게 된다.
Ftotal  Fext  Fint  ma
where a : the acceleration
m : the rest mass of the particle
It is difficult to take into account all of the internal forces,
Fext  m*a
m* : the effective mass

Hanyang University           fall – 2007                           11
3.2 Electrical Conduction in Solids
The relationship between energy and momentum
p2  2k 2
E   
2m 2m
The relationship between momentum and wave number
p  k
The first derivative of E with respect to k
dE  2 k p                      1 dE p
                                v
dk   m    m                       dk m
The second derivative of E with respect to k
2     2                    2
d E                   1 d E 1
2
                  2   2

dk     m                dk     m
Hanyang University          fall – 2007           12
3.2 Electrical Conduction in Solids
Newton’s classical equation of motion
 eE
F  ma  eE                a
m
Apply the results to the electron in the bottom of an allowed
energy band. (fig. 3.16)
E  Ec  C1 (k ) 2
Taking the second derivative of E with respect to k,
2
d E2                    1 d E 2C1
 2C1               2
 2  2
dk 2                     dk    
If we apply an electric field to the electron in the bottom of the
allowed energy band, then the acceleration
 eE
a *
mn    mn* : effective mass of the electron
Hanyang University            fall – 2007                            13
3.2 Electrical Conduction in Solids
3.2.4 Concept of the Hole
The hole has a positive effective mass and positive electronic charge.

The movement of a hole in a semiconductor.

(a) Valence band with conventional electron-filled state and empty states.
(b) Concept of positive charges occupying the original empty states.
Hanyang University                fall – 2007                                 14
3.2 Electrical Conduction in Solids
3.2.5 Metals, Insulators, and Semiconductors
Completely filled                   Partially filled
Completely empty

Insulator              Semiconductor                    Band gap :
overlap

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3.3 Extension to Three Dimensions
3.3.1 The k-Space Diagram of Si and GaAs
Si: indirect band gap semiconductor
GaAs: direct band gap semiconductor

Energy band structure (a) GaAs and (b) Si

3.3.2 Additional Effective Mass Concepts
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3.4 Density of States Function
3.4.1 Mathematical Derivation
the density of quantum states        4 (2m)3 2
g (E) 
per unit volume of the crystal
E
h3
3.4.2 Extension to Semiconductors
 The density of allowed electronic energy
states in the conduction band
4 (2mn )3 2
*
gc ( E)                     E  Ec E  Ec
h3
 The density of allowed electronic energy
states in the valence band
4 (2m* )3 2
gv (E)                      Ev  E E  E v
p
3
h
 The density of allowed electronic energy
states within the forbidden energy gap
g ( E )  0 for         Ev  E  Ec           The density of energy states

Hanyang University               fall – 2007                                  17
3.4 Density of States Function
 Example 3.3
Calculate the density of states per unit volume with energies
between 0 and 1 eV.
1 eV       4 (2m)3 / 2 1 eV
N   g ( E )dE          3
    E dE
0                  h         0

4 (2m)3 / 2 2 3 / 2
or N        3
 E
h        3
4 (2  9.111031)3 / 2 2
N                           (1.6 1019 )3 / 2
(6.625 1034 )3      3
 4.5 10 27 m 3
or N  4.5 1021 states/cm3

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3.5 Statistical Mechanics
The electrical characteristics will be determined by the statistical
behavior of a large number of electrons
3.5.1 Statistical Laws
Maxwell-Boltzmann probability function
{    Bose-Einstein function
Fermi-Dirac probability function
3.5.2 Fermi-Dirac Probability Function
 Maxwell-Boltzmann probability function
 The particles are considered to be distinguishable by being numbered,
with no limit to the number of particles allowed in each energy state.
 gas molecules (noninteracting potential free particles
→ continuous energy)                                  g (E)
N (E) 
 Ei 
Boltzmann constant
exp    exp 
k T 

kB :1.380662×10–23 [J/K]                                       B 
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3.5 Statistical Mechanics
N(E) : the number of particles per unit volume per unit energy
g(E) : the number of quantum states per unit volume per unit energy

N (E)                     ( E  EF )             EF   
 f F ( E )  exp                   if  
               

g (E)                        k BT                k BT   

energy E인 quantum state가 particle에 의해 occupation probability
 Bose-Einstein function
 The particles are indistinguishable and, there is no limit to the
number of particles permitted in each quantum state.
→ Pauli’s exclusion principle에 지배되지 않는 입자.(photons)

f F (E) 
N (E)

1                       EF 
 E            
            
g (E)
exp( )  exp      1
k T                k BT 
 B 
energy E인 quantum state가 Bose particle에 의해 occupation probability
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3.5 Statistical Mechanics
 Fremi-Dirac Probability Function
 The particles are distinguishable, but now only one particle is
permitted in each quantum state.
 free electron in metals
N (E)                        1
 f F (E) 
g (E)                        E  EF    
1  exp 
 k T       

 B         
 energy E인 quantum state가 Fermi 입자에 의해 점유될 확률(또는
Fermi 입자가 존재할 확률)
 each quantum state 마다 1개의 입자가 존재할 수 있으므로 f(E)≤1
 어떤critical energy( E=EF; Fermi energy)에서 f(E)는 급속히 감소
→Pauli's exclusion principle의 효과를 반영한 것.

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3.5 Statistical Mechanics
3.5.3 The Distribution Function and the Fermi Energy
"Fermi-Dirac energy distribution(or probability) function”
energy E인quantum state가 Fermi 입자에 의해 점유될(또는, Fermi
입자가 존재할) 확률.
1
f F (E) 
 E  EF   
1  exp 
 k T      

 B        
at T = 0 K, when E < EF, fF (E – EF) = 1
when E > EF, fF (E – EF) = 0
at T > 0 K, when E > EF, fF (E = EF) = 1/2
All electrons have energies below the Fermi energy at T = 0 K.

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3.5 Statistical Mechanics
The probability of an energy
above EF being occupied
increase as the temperature
increases and the probability
of a state below EF being
empty increases as the
temperature increases.
The Fermi probability function versus
energy for different temperatures.

EF : Fermi energy →electron의 존재(점유) 확률이 ½인 energy
level
1                 ( E  EF ) 
f F (E)                      exp               
 E  EF             k BT     
1  exp 
  k BT 
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3.5 Statistical Mechanics
 Example 3.6
Calculate the probability that an energy state above EF is occupied
by an electron. T = 300 K.
At T = 300 K, E – EF = 3kT

1                1
f F (E)                     
 E  EF           3kT 
1  exp          1  exp      
 kT               kT 

1       1
                   0.0474
1  e 1  20.09
3

 4.74 %

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3.5 Statistical Mechanics

The probability of a state being occupied,            The Fermi-Dirac probability function and
and the probability of a state being empty .          the Maxwell-Boltzmann approximation.

The actual Boltzmann approximation is valid when exp[(E – EF)/kT] >>1.
  ( E  EF ) 
f F ( E )  exp               
      kT      
It is common practice to use the E – EF >>kT notation when applying
the Boltzmann approximation.

Hanyang University                  fall – 2007                                         25

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