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```					Chapter 7. Vacancies

What you have to learn from this chapter?
 State of Matter;
 Internal Energy, Entropy, Gibbs Free energy;
 Statistical Mechanical Definition of Entropy;
 Equilibrium Vacancy Concentration;
 Vacancies in Semiconductor Materials;
 Vacancy Motion
 States of Matter: strongly depends on temperature;
three approaches
(1) thermodynamics: the macroscopic laws to predict
the final states of matter => a lot of experimental
observation => equation of state
(2) statistical mechanics: linking the distribution of
atomic behaviors to the overall observed
macroscopic behaviors in equilibrium
(3) kinetics: (a) linking the state parameters using
mechanics of atom or molecular; (b) the transient
behavior of matter from state  to state , could
involve physics of individual process;
 Internal Energy: U; all energy in a system (kinetics
energy, potential energy, zero point energy, cohesive
energy, energy of phonons in a lattice). * A state
property (independent of reaction path)!
 From first law of thermodynamics: dU=dQ+dW;
(conservation of energy! The increase of the internal
energy of a system equals to the energy [heat] added
to the system minus the energy [work] spent by the
system.)
* Fixed dQ, i.e. Q constant => work done by a
heat-isolated gas piston => volume expands
=> -PdV;
* Fixed dW , i.e. W constant or no work is done to
the system => entropy of the system  (ways to
divided energy among particles )
 dU  TdS  PdV
 Entropy: definition of S in thermodynamics
dQ 
B
S  SB  S A               Reaction path independent
A T 
 rev
dQ            dQ
dS         ; dS 
T rev         T irrev
S = Q/T (entropy); if T=1o, then the entropy is Q/1o, equal to the
heat delivered to a 1o reservoir.
When a parameter is path independent => a state property
 Spontaneous Reaction: typically happened in a
far-from-equilibrium (non-equilibrium) conditions
=> irreversible reaction; e.g. water-ice transformation
at temperature < 0oC!
 Gibbs Free Energy (G) :
G  U  PV  TS  H  TS
U: internal energy; P: pressure; V: volume;
T: absolute temperature; S:entropy; H: enthalpy
dH  dU  PdV  VdP  dH  TdS  PdV  PdV  VdP
 dH  TdS  VdP
At constant pressure  dH  TdS  dQ
H: easier to deal with for a constant pressure system
dG  dH  TdS  SdT  TdS  VdP  TdS  SdT  VdP  SdT
 The Gibbs free energy is zero for any reversible
reaction that takes place at constant temperature and
constant pressure. dG=0!
G  H  TS ; for constant temperature
Use  to represent conditions not necessarily in a
reversible case!
H  Q     ; at constant pressure   G  Q  TS
=> For spontaneous reaction or irreversible reaction
G <0 !
 Statistical Mechanical Definition of Entropy:
 Entropy (S) is a state properties. It is a measure of the
disorder of the system. The entropy difference
between two states (I and II) is SI-SII => related to
energy difference.
 Partition between two idea gases (A & B) at constant
temperature and pressure. No work
A       B
done and no heat transferred!             T nA nB T
G  Q  TS  TS                     P   VA VB P State 1
The mixing of gases is a                  Mixture of A & B
spontaneous, or irreversible                     T
V        P
reaction! => G < 0 => S > 0.                          State 2
Entropy of mixing (S)! System becomes more
disorder.
Disorder of a system could be characterized by the
probability ( ) of finding a system in a specific
configuration.
 The combination of probability 1 and  2 is  1·  2!
=> Imply S is related to  by ln ! Since TS equals
to energy => S has a unit of kB! => S= kB ln =>
S = S2- S1 = kB ln 2/ 1 !
 Find the relative probability of state 1 (without
separation) vs state 2.
If one A atom is introduced into the undivided box,
the probability of finding it in VA is VA/V. Continue
this process for nA A atom => (VA/V)nA. Similar
process for B atoms => (VB/V)nB.
 The probability of finding all nA A atoms in VA and
all nB B atoms in VB (i.e. state 1) is (VA/V)nA(VB/V)nB.
Similarly, the probability of finding all nA A atoms
in V and all nB B atoms in V (i.e. state 2) is
(V/V)nA(V/V)nB = 1
=> Probability ratio (2/1): (VA/V) -nA(VB/V) -nB
=> S = S2- S1 = kB ln [(VA/V) -nA(VB/V) -nB] !
 VA n A  VB nB           VA          VB 
S  k ln       knA ln    knB ln  
 V   V  
                            V           V 
n A VA             nB VB
n  n A  nB ;              XA ;           X B  (1  X A )
n V                n V
S  knX A ln X A  kn(1  X A ) ln 1  X A 
  RX A ln X A  (1  X A ) ln 1  X A 
 Effect of temperature on the distribution of particles in
different energy states:
 The  in the following cases:
(a) system with distinguishable particles
(b) Indistinguishable particles + Pauli exclusion
principles (Fermions, e.g. electrons)
(c) Indistinguishable particles (no Pauli exclusion
principles; Bosons, e.g. phonon)
 System with discrete energy levels:
E1, E2, E3, …., Ei
n1, n2, n3, …., ni number of particles in the
corresponding E
n1!, n2!, n3!, …., ni! number of ways to put
indistinguishable particles
n1+ n2 + n3 +… + ni = N N!: disregard energy levels
* Case (a): the number of ways to distribute all
distinguishable particles into all energy states
N !   ni!  Wd
i            W: probability of exchanging particles in
N!              different levels; only make sense for
Wd 
 ni !
i
distinguishable particles!

* An example: there are two energy levels E1 and E2,
2 (n1) particles in E1 and 3 (n2) particles in E2=> N=5
E1               E2             1   2    3   4   5

Indistinguishable sites:
=> 5!
Any three in E2 is the same regardless of the sequence, e.g.
Assume 123 is in E2, there are 6 (3!) ways of arrangement
(123, 213, 321, 132, 231, 312)
Similar for E1 => 2!.
5! = 2!3!Wd => Wd=10
10 ways to distribute 5 particles into two groups with 2 and 3
particles, respectively. (12, 345; 13, 245; 14, 235; 15, 234;
23, 145; 24, 135; 25, 134; 34, 125; 35, 124; 45, 123)

If every energy levels Ei have gi states =>
Γ  Wd   gi    gi 
ni                ni    N!
 N!
gi  i
n

i                   i               ni !
i
i  ni !

* Case (b): Fermion
ni particles in the gi states available at the
ith energy level Ei;
Pauli exclusion principle => ni  gi;
gi !
ni ! gi  ni !
       gi !    
Γ   n ! g  n ! 

i  i     i    i 

* Case (c): Bosons
ni particles in the gi states available at the
ith energy level Ei;

Using gi wall to separated ni particles
( gi  1  ni )!
Γ 
i   ni ! gi  1!
* (a-1):
S  k ln Γ      Γ  N !
gi n
i

ni !
S           g   n1        i
 ln N !     i
 ln N !  ni ln gi  ln ni !
k         i ni !           i

 N ln N  N   ni ln gi  ni ln ni  ni (NlnN is a constant)
S      Si                             Si
    ni ln gi  ni ln ni  ni ;      ni ln gi  ni ln ni  ni
k   i k      i                        k
Si 1  Si 
   n ni  ln gi  ln ni ni
k i
k          
S      Si
         ln gi  ln ni
k   i   k      i
Thermodynamics: G =-TS - N + E + PV
E:internal energy; : average energy change per
particle (chemical potential)
At equilibrium and constant volume; G=V=0
=>-TS - N + E =0
N  ni ; E   Eini
i               i
=> 0    kT (ln gi  ln ni )ni   ni   Eini
                       
i                             i        i
gi
=>     ni   kT (ln n )    Ei   0
                       
i               i           
gi               ni       Ei    
=>  kT (ln )    Ei  0   exp            
ni               gi          kT 
Maxwell Boltzmann distribution
* (b-1):
S
  ln g i ! ln ni ! ln g i  ni !
k   i

  g i ln g i  ni ln ni  g i  ni  ln g i  ni 
i
Si
 ln( gi  ni )  ln ni
ni
        gi  ni            
G=0 => ni  kT ln                        Ei   0
i                 ni              
gi  ni Ei            gi         Ei   
ln                       exp                1
ni         kT         ni         kT 
gi
ni                           Fermi-Dirac
 Ei   
exp 
 kT      1      distribution
* (c-1): similar process
gi  ni Ei         g        E  
ln                    i  exp  i       1
ni         kT     ni        kT 
gi
ni                        Bose-Einstein
 Ei   
exp 
 kT     1     distribution

Typical energy level are on the order of  =>
continuous approximation
Number of states at the ith level gi => replaced by
density of states D(Ei)!
=> The particle number in energy E
D( E )
n( E )                     f ( E )  D( E )   Bose-Einstein
E  
exp 
 kT     1                       distribution
D( E )                           Fermi-Dirac
n( E )                     f ( E )  D( E )
E  
 1
exp                                   distribution
 kT 
M-B       T1
n                                 Fermi-Dirac statistics 
B-E                                 Maxwell Boltamann
T2          E    statistics at E >> F
F
 Vacancies:
 Assume a crystal containing N lattice sites, includes
n0 atoms and nv vacant site (vacancies).
i.e. N = n0 + nv; motion of vacancy and creation
of vacancy: see Fig. 7.1 & 7.2 (Schottky defect).
 Work required to form a vacancy: w; => to form nv
vacancies => nvw
Gv  H v  TSv At constant pressure enthalpy
Gv  nv w  TSv   = work: HV = nvw
 At least two contributions to SV : vibrational entropy
(less important) will be neglected at the following
discussion) and configurational entropy.
N!       Number of ways to arrange
S  k ln Γ   Γ
n0! nv !   Vacancies in the lattice
Stirling Approximat ion : ln N!  N ln N  N
S  k ln N! ln n0! ln nv !
 k N ln N  N  n0 ln n0  n0  nv ln nv  nv 
 k (n0  nv ) ln N  n0 ln n0  nv ln nv 
Gv  nv w  kT (n0  nv ) ln( n0  nv )  n0 ln n0  nv ln nv 
Minimize Gibb free energy with respect to nv.
dGv                              1                          1        
 0  w  kT (n0  nv )             ln( n0  nv )  nv  ln nv 
dnv                          (n0  nv )                     nv       
nv
0  w  kT ln
(n0  nv )
=> nv  (n0  nv ) exp( w / kT )  N exp( w / kT )
The changes of the free energy of the lattice (with N
site) due to the presence of the vacancies (nv site):
* Including the vibrational entropy into the discussion
Svf : vibration al entropy ;
SC  k (n0  nv ) ln N  n0 ln n0  nv ln nv 
G  nv w  T Sc  nv Svf 
In equilibrium,
dG           dG                 dS
 0 =>         w  TSv  T
f
0
dnv           dnv                dnv
nv                   Svf        w
=> w  TSv  T ln  0 => nv  N exp              exp   
f
 k         
N                                kT 
For metals, w is between 0.5 to 1 eV (48-96 kJ/mol)
Typical rule: w 10kTm!
Cu: w=83.7 kJ/mol => at 300K => nv/N  4.4510-15.
=> at 1350K => nv/N  6.110-4.
Si: w=2.4 eV; Sfv = 1.1k
=> at 1200K => nv/N  2.610-10.
Svf            Svf   
Order of magnitude            1  exp 
 k        order of 3

k                     
The equilibrium concentration of neutral vacancies in
Si is independent of Fermi level. N = 5.0x1022 cm-3
=> [Vx] = 1.5x1023exp(1.1k/k)exp(-2.4/kT)
= 3.9x1013 cm-3 at 1200 K.
GV   nvw             GV     nvw
n1e                     n2e
nv                            nv
low T
High T
-T1S
-T2S
 Vacancies in simple semiconductor materials:
Charge states of vacancy become important!

VSi0, VSi+
VSi-, VSi=
VSi0          VSi+           VSi-
VSix      Donor state    Acceptor state
Energy level for charged vacancies in Si:
VSi+
1.2                                    EC 0.6                    EC
Energy (eV)

0.25
1.0
Eg 1.05                      T=0 K        Eg 0.5        T=1400 K
0.6                               0.06
0                                     EV   0                    EV
VSi   =         -
VSi VSi        +              VSi= VSi-
 The concentration of charged vacancies is governed
by the Fermi-Dirac statistics and is given by
[VT ]         where EF: Fermi level
[V ] 
-

1  e E  E F / kT
                  E-:the energy level for
acceptor state vacancy
[VT]: total vacancy concentration.                 8
[VT]=[Vx]+[V-]+[V=]+[V+]
            
V+       V-

[V ]  [VT ] exp E F  E / kT
-                                                                     V=

log10[V]
            
0
=>                                                       Vx
[V ]  [VT ] exp E   E F / kT
[V-] is significant for n-type Si         -8
[V+] is significant for p-type Si                     0        0.4    0.8        1.2
EF (eV)
Similar results could be used                p-type Si              n-type Si
for other types of charged
defects; e.g. interstitials.
 Chemical vacancies: some intermetallic compounds
are characterized by a particular electron
concentration; e.g. NiAl is stable phase when the
e/a =3/2; Al: 3S23P1 (1 valence electron);
Ni: 3d84S2 (2 valence electron); for some reason,
the compound is Ni deficient, i.e. Ni1-xAl;
introducing vacancies into the compound could
balance the e/a => stabilize the structure!
 Chemical vacancies are stable and belong to part of
phase.
 Vacancies could be produced by irradiation the
materials with high energy particles; these vacancies
belong to microstructure of the material =>
annealing could remove these vacancies.
 Vacancy Motion (Diffusion of vacancy):
 The probability of an

Energy
rv
atom with vibrational                qm
energy greater than              A      B                  x
qm is proportional to                AB
qm
p  exp(  )    Probability of a successful jump
kT
rv: Number of atoms jumps per second into a vacancy
rv  A exp  m      qm: migration
k 0   0 exp  m     
-q                                    -q
                                    
    kT      energy                      kT 

ra: Number of jumps per second made by an atom
ra 
nv        - qm      exp  - q f    
A exp  m
-q    
A exp                                      kT 
n0             kT               kT               
Equilibrium vacancy concentration = exp(-qf/kT);
qf: formation energy
 - q f  qm  
ra  A exp 
              kT 

 Interstitial Atoms and Divacancies:
 Interstitial atoms are atoms that occupied sites that
are not the lattice sites of a structure. There are some
specific sites (open region) suitable for interstitials in
a structure. e.g. Fig. 7.7, 7.8.
The migration energies for interstitials vary a lot.
The description of the textbook about Fig. 7.9 is