Semiconductor Crystals - November 8, 2005
After studying this section, you should be able to explain:
What are semiconductor crystals? How do I calculate the band gap in these crystals?
What causes Excitations to form electron-hole pairs? What is Intrinsic Conduction:
Electron-hole pair formation? What is Extrinsic Conduction: Doping? What is
Recombination and Carrier Lifetime? What are Compound Semiconductors?
__________________________________________________________________________
What are Semiconductor Crystals?
Semiconductor crystals are characterized lower concentration of charge
carriers (either electrons or holes) than metals.
At room temperature,
their electrical conductivity is ~10-5 to 105 S/m.
They have band gaps much
narrower than insulators (0.2-6.5eV).
Sources of Excitations of electron-hole pairs
1. Thermal Excitations – Thermal energy (heat) causes electrons to cross
the band-gap.
2. Optical Excitations – Photons having energy, h, large enough to excite
electrons across the band-gap.
3. Impurities – Doping pure semiconductors with atoms having more or fewer
electrons causes energy states associated with dopant to donate to the
conduction band or accept from the valence band.
Heat Input
(T)
Electrons excited
from Valence to
Conduction Band
{D(), f(), n}
Conduction
Hole Left behind
in the Valence
Band (D(h), f(h),
p)
Under applied
field, , e and h
move creating
current (e, h)
Intrinsic Semiconductors (No impurities states)
Intrinsic conduction arises when there is a low impurity concentration in
the semiconductor and so increase in temperature determines the electrical
properties.
Semiconductors showing intrinsic conductivity have small band
gaps so that thermal excitations are enough to cause electrons to move from
the valence band to the conduction band.
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Semiconductor Crystals
When the band gap is large
1
relative to temperature, there will be few intrinsic carriers and
conductivity will drop.
The band gap energy (Eg) is the energy difference between the lowest
unoccupied band and the highest valence band.
Band Gap = Conduction Band Edge – Valence Band Edge
Conductivity arises from electron mobility in the conduction band and hole
(vacant sites left behind by electrons) mobility in the valence band.
Band Gap in Semiconductor Crystals
Semiconductor materials are characterized by a strong temperature
dependence of conductivity.
Typically, these materials have a lower
concentration of charge carriers (either electrons or holes) than metals.
At room temperature, their electrical resistivity is ~10-2 to 109 ohm-cm.
Semiconductor Type
III-V (trivalent-
Al, Ga, In
N, As, P
Zn, Cd, Hg
S, Se, Te, O
C, Si, Ge, Sn
C, Si, Ge, Sn
pentavalent)
II-VI (divalenthexavalent)
IV-IV (Diamond Type)
Measurement of Intrinsic Conductivity
Band gap is determined by Optical Absorption measurements of two types:
1. Direct Optical Absorption Measurements: Photons are absorbed at
frequencies
g reveal the crystal to have a band gap of E g   g .
Such
measurements are made by scanning the range of photon frequencies until
absorption occurs.
2. Indirect Optical Absorption Measurements: Indirect absorption are
made when the band gap is too large for a single photon to be absorbed,
then indirect absorption occurs.
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Semiconductor Crystals
In this case, a phonon is emitted in
2
the excitation process, so that,  g  E g   
frequency.
 is the phonon
Typically, E g is greater than  
3. Temperature dependence of Electrical Conductivity: The intrinsic
conductivity and carrier concentration is controlled by the ratio of
energy gap to temperature since the carrier concentration (n or p) is
proportional to kbT / Eg.
The exact equation is obtained by integrating
the product of conduction band Density of States (called D()) with the
Fermi-Dirac distribution (called f()).

n or p  D ( ) f ( )d
The range on the integration is Ec to  for n-type (electron carriers)
and -  to Ev for p-type (hole carriers)
The bulk carrier concentration determined using Hall measurement
(http://www.eeel.nist.gov/812/meas.htm).
Temperature dependence of the
carrier concentration can be used to calculate the band gap.
Calculation of Intrinsic Electron Charge Carrier Concentration
We must use the density of states and the Fermi-Dirac function to calculate
the intrinsic charge carriers for Electrons.
Fermi Dirac Function, f():
The Fermi-Dirac Distribution gives the probability that an energy level, ,
will be occupied by an electron:
f ( ) 
1
.
exp[(   ) / k BT ]  1
In the case of semiconductors, we wish to calculate the number of electrons
excited to the conduction band at temperature, T.
In this case, we call 
the Fermi Level (the highest occupied level of the valence band).
In most
cases, >>kBT, resulting in the Fermi-Dirac distribution taking the form
of the Boltzman Distrubution,
f ( )  exp     / k B T 
Density of States Function, D():
The Density of States is defined as the number of orbitals per unit energy
(or dN/d).
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Semiconductor Crystals
3
dN
V
D( ) 

d 2 2
 2m e * 


 2 


3/ 2
( k  E c )1 / 2
Intrinsic electron charge carrier concentration:
The concentration of electrons in the conduction band then becomes, n,
where

n
 D( ) f ( )d
Ec
*
1  2me 
n
2 2   2 
*
1  2me 
n
2 2   2 
3/ 2
3/ 2 
 (  E
Ec
c)
1/ 2
     
exp 
 d
 k BT 

  
  
 (  E c )1 / 2 exp
d
exp
 k B T  Ec
 k BT 

Integrating yields
 m *k T 
n  2 e B2 
 2 


3/ 2
   Ec  

exp
 k BT 
Here, the density of states function uses the effective hole mass (which is
different than the effective electron mass).
In fact, the effective hole
mass is larger due to the following:
1. Band curvature in E(k) near top and bottom of band
2. Band curvature in E(k) leads to higher correction to electron/hole
mass (always fractional)
3. Higher correction leads to lower effective mass.
Density of occupied electron states vs. Density of electron states
Conduction Band
Eg
0
Energy Gap
Valence Band
Density of states D() for semiconductors in the
vicinity of the energy gap is parabolic at the bottom of
the conduction band and the top of the valence band.
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Semiconductor Crystals
4
Density of electron and hole states, D() assumes a parabolic shape near
the energy gap given by the parabolic function (i.e. there are more energy
bands as we move away from the energy gap.
(However, fermi distribution
actually causes a reduction of density of occupied electron states near top
of conduction band.)
We have examined the density of electron D() states.
be occupied by electrons.
Not all states will
In order to determine the density of occupied
electron states {D()f()}, we have employed the Fermi-Dirac Function!!!
Now, let’s Calculate the density of occupied electron states, Ne
Conduction Band
Energy Gap
Eg
Valence Band
0
Density of occupied states is D()f() decreases exponentially to
approach zero assymptotically in the conduction band away from the
energy gap.
Calculation of Intrinsic Hole Charge Carrier Concentration
We will again use the density of states and the Boltzmann function to
calculate the intrinsic charge carriers for Holes.
Fermi Dirac Function, f():
For holes, the distribution becomes 1-f()electrons.
f ( )  1  exp     / k B T 
 exp     / k B T 
Density of States Function, D():
The equation for the density of states is
dN
V
D( ) 

d 2 2
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Semiconductor Crystals
 2me 
 2 
  
3/ 2
( E v   k )1 / 2
5
Intrinsic electron charge carrier concentration:
The concentration of holes in the conduction band then becomes, p, where

p   D( ) f ( )d
Ec
Resulting in
 m *k T 
p  2 h B2 
 2 


3/ 2
 E    

exp v
 k BT 
Calculating the Energy Gap from Electron and Hole carrier concentration
Measurements
Multiplying n and p, we can have an equation which does not depend on the
fermi energy (it cancels).
The product only depends on the magnitude of
Eg.
 m *k T 
if p  2 h B2 
 2 


3/ 2
 E    

exp v
 k BT 
 m *k T
n  2 e B2
 2





3/ 2
   E c  

exp
 k BT 
 E  
    
   
  Ec  
 exp
 exp

then np  N c N v exp v  exp
 k BT 
 k BT 
 k BT 
 k BT 


  Eg 

np  N c N v exp

 k BT 
with
 m *k T
Nv = 2 h B2
 2





3/ 2
and
 m *k T
Nc = 2 e B2
 2

Furthermore, for an intrinsic semiconductor, n=p.




3/ 2
Thus,
n  p  N c  N v  exp[ ( E g ) / 2kbT ]
Given the value of the Energy Gap, we can now calculate the Fermi Energy
The fermi energy is located within the energy gap, however, we are not
certain where.
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Semiconductor Crystals
6
We can determine the location of the fermi level for intrinsic
semiconductors by the following:
In intrinsic conduction, one electron excitation to the conduction level
produces one hole in the valence level, therefore, n = p and
   Ec  

n  N c exp
k
T
B


And
 E    

p  N v exp v
 k BT 
   Ec  
 E    
  N v exp v

N c exp
 k BT 
 k BT 
   E c  

exp
k B T 
Nv


Nc
 E    

exp V
k
T
B


   Ec  EV    
Nv

 exp
Nc
k BT


 2  E g  
 2  Ec  EV  
Nv
Nv

 or
 exp
 exp

Nc
Nc
k BT


 k BT 
N v  mh* 


Since
N c  me* 

 2  E g
Nv
 exp
Nc
 k BT
3/ 2
    m




*
h
me*




3/ 2
Taking the ln of both sides, we have,

 2  E g

 k T
B

   3 ln  m


2


*
h
me*




Simplification yields the Fermi energy,

Eg
2

 m* 
3
k B T ln  h* 
4
 me 
However, Since the energy gap (e.g. Si = 1.1 eV) is much larger than
kbT=0.025eV (near room temperature).
We see that for Intrinsic
Semiconductors, the Fermi level lies at the center of the Band Gap:  
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Semiconductor Crystals
Eg
2
7
!
Intrinsic Carrier Mobility, e
h
The intrinsic carrier mobility is defined as the drift velocity per unit
electric field.
The units of carrier mobility is [m/s/J] or [s/kg].
=|v|/E
in units of [s/kg]
The electrical conductivity, , is defined as =ne2/m.
In terms of carrier mobility, this conductivity can be written as:
=(e + h)
=(nee + peh)
where n and p are the electron and hole concentrations, respectively.
Hence, the conductivity can be thought of as the sum of the contribution
due to electrons and the contribution due to holes.
By comparison of the electrical conductivity based upon the drift velocity
of electrons within the Fermi sphere, we see that the intrinsic carrier
mobility becomes:
e=ee/me and h=e h/mh
where e and h are the time between collisions with ion cores.
Temperature Dependence of mobility during Intrinsic conduction
There is a temperature dependence of carrier mobility.
This temperature
dependence is thought to be small compared with the temperature dependence
of carrier concentration dependence of exp(-Eg/2kbT).
In other words, the
temperature dependence of intrinsic conduction is thought to be controlled
by the formation of electron-hole pairs rather than the mobility of
electrons and holes.
Impurity atoms enter the structure by either substitution of the lattice
atoms or by fitting into interstitial sites (e.g. tetrahedral, octahedral,
or cubic interstices).
In general, atoms which have similar size to the
host structure atoms will occupy substitutional sites rather than
interstitial sites.
We will confine our discussion of dopants in
semiconductors to those which occupy substitutional sites.
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Semiconductor Crystals
8
Types of Impurities
Since Si and Ge (Group 4 elements) are tetrahedrally coordinated and there
are 4 valence electrons on each atom (Figure 1).
If an impurity atom from
Group 5 (e.g. As, P, Sb) is added as a dopant element, there will be an
extra valence electron left unbonded and free to participate in conduction
(Figure 2).
Alternatively, if an impurity element from Group 3 (e.g. B,
Al, Ga, In) is added as a dopant element, there will be an extra hole left
over to participate in conduction (Figure 3).
Si
Si
Si
Si
Si
Si
________________________
Figure 1. Tetrahedrally
Semiconductor Crystals
Coordinated Silicon
9
Si
Si
+
As
excess
electron
Si
Si
Si
Figure 2. Tetrahedrally Coordinated Silicon doped with As (Group 5 element).
A conduction electron is left over. The result is an n-type semiconductor.
Conduction eSi
Si
Si
+
Conduction
hole
B
Si
Si
Figure 3. Tetrahedrally Coordinated Silicon doped with B (Group 3 element).
A conduction hole is left over. The result is a p-type semiconductor.
Donors are substitutional atoms which ‘give up’ an electron to the crystal
(e.g. Group 5 elements).
Acceptors are substitutional atoms which consume
an electron (leaving behind a hole).
Donor State Electron Mobility
The donor electron is attracted by coulomb potential to the impurity ion.
This potential is defined by
E=e/r
where  is the static dielectric constant of the crystal after doping, r is
the distance traversed by the electron in the presence of the electric
field, and e is the electronic charge.
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Semiconductor Crystals
10
The donor ionization energy is given by, Ed=(me4)/(82h2)
Materials having high dielectric constants will inhibit electron mobility
in a given electric field.
Therefore, insulating materials are known for
having a high dielectric constant.
Materials such as SiO2, ZrO2, and
Al2O3 have high dielectric constants and are sometimes called dielectric
materials.
(An interesting sidenote: as the semiconductor industry moves
toward smaller and smaller devices—Moore’s Law, the SiO2 gate dielectrics
used to insulate silicon are becoming too thin to inhibit electron
tunneling (e.g. current across the dielectric occurs)—thereby leading to
the study of materials having higher dielectric constants than SiO2 to
substitute as gate dielectrics.)
A donor level (dashed line) is near the conduction band.
Since electrons
conduct, these are known as n-type conductors.
As we increase our donor dopant concentration, we have an increase in the
fermi energy.
From equations which show the exponential dependence of
concentration on fermi level, we know that this is a logarithmic increase:
Ef
Log Nd or Na
An acceptor level is near to the valence band.
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Semiconductor Crystals
11
Ionization Energy and Bohr Radius of Donor States (pp 222)
Before the electron can participate in conduction, it must first be
liberated from the donor (e.g. As in Figure 2).
The energy required to
ionize the donor impurity can be estimated as
Ed 
e 4 me
2 
2

2
13.6 me
2 m
in eV.
Notice that this is equivalent to the ionization energy of atomic hydrogen
upon 2.
(Side note on effective mass, me pp 209-214.
We note that effective mass
is mass after we accounting for effects due to the energy band curvature.
We may assume that me=0.2m in Si and me=0.1m in Ge).
The Bohr radius of the donor atom is :
ad 
 2
me e
2

 0.53
in Angstroms
*
me / m
Significance of the Bohr radius arises when we attempt to understand
conduction in doped semiconductors.
The Bohr radius must be large enough
to result in overlapping of impurity bands (~80Ang for very small effective
electron mass—as in Si me*=0.2m and Ge me*=0.1m).
Electron conduction,
thus, occurs by movement of electrons from one donor impurity to the next.
One final note is that electrons transport from one unoccupied donor site
(e.g. As+) to another unoccupied donor site.
If an electron already exists
at the donor site (e.g. As0), then transport of the next electron will not
occur.
Acceptor State Electron Mobility (pp 224)
Acceptors will receive electrons from the neighboring atoms, leaving behind
holes in the host lattice structure.
Ionization energies of acceptors
requires the input of energy (because the host lattice must form a hole).
Relative to Intrinsic Conduction
Intrinsic conduction occurs in pure, undoped semiconductors.
In this case,
the number of electrons and holes are equal.
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Semiconductor Crystals
12
Thermal Ionization of Donors and Acceptors
Donor and acceptors may be ionized by increasing temperature.
The equation
governing this temperature effect is the same as the equation for thermal
ionization of hydrogen,
  Ed 

n  no N d exp 
 2k B T 
where
 me k B T 


2
 2 
where Nd is the donor concentration.
no  2
3
We may write ,
Nd + (Si/Ge) Nd+ + e- + (Si/Ge) for the donor reaction with the host
lattice
Na + (Si/Ge)  Na- + h+ + (Si/Ge)
for the acceptor reaction with the host
lattice
As such, the law of chemical equilibrium suggests that the reaction
equilibrium constant, k is
k=[ Nd+] [e-]/[ Nd ] for the donor reaction
k=[ Na-][h+] /[Nd ] for the acceptor reaction.
We may call the electron carrier concentration, n=[e-].
Therefore, n=[ Nd+]
satisfies charge neutrality.
Likewise, we may call the hole carrier concentration, p= [h+] = [ Na-]
Recombination Lifetime
By law of mass action, product of n and p yields an expression which, at a
given temperature – T, is proportional to the rate at which an electronhole pair will recombine in the reaction e+h=photon (called the
recombination reaction).
The emitted photon will have an energy equal to
that of the band gap,
Eg = h
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Semiconductor Crystals
13
This quantity is useful if we consider that an incoming photon will
generate excess electron-hole pairs above the concentration formed via
thermal excitations and doping.
Excess Charge carrier produced by optical excitations:
Extrinsic: Without radiation, we have a balance between thermal excitation
and recombination.
In the presence of radiation, we increase the number of
electron-hole pairs.
With optical production, a much greater change is
produced in the minority carrier concentration (10^4 holes increase to
10^14 holes increase by a factor of million while 10^16 electrons increase
by 10^10
factor of 1 part in 1 million.
Intrinsic: Without radiation, equal number of electron-hole pairs.
Increase in the number of electron hole pairs after optical excitation.
In extrinsic cases, after the light is removed, the decrease in carrier
concentration depends on excess minority charge carrier (since it was most
effected) in an equation of the form: exp(-t/
minority carrier lifetime.
 is the excess
The minority carrier lifetime depends on dopant
concentration and other defects which may effect the recombination
reaction.
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Semiconductor Crystals
14