Lecture #5
OUTLINE
• Intrinsic Fermi level
• Determination of EF
• Degenerately doped semiconductor
• Carrier properties
• Carrier drift
Read: Sections 2.5, 3.1
Intrinsic Fermi Level, Ei
• To find EF for an intrinsic semiconductor, use the fact
that n = p:
Nce
 ( Ec  E F ) / kT
 Nve
 ( E F  Ev ) / kT
Ec  Ev kT  N v 
  Ei
EF 

ln 
2
2  Nc 
*

m
Ec  Ev 3kT  p  Ec  Ev
Ei 

ln * 
m 
2
4
2
 n
Spring 2007
EE130 Lecture 5, Slide 2
n(ni, Ei) and p(ni, Ei)
• In an intrinsic semiconductor, n = p = ni and EF = Ei :
n  ni  N c e  ( Ec  Ei ) / kT
 N c  ni e
n  ni e
Spring 2007
( Ec  Ei ) / kT
( E F  Ei ) / kT
p  ni  N v e  ( Ei  Ev ) / kT
 N v  ni e
p  ni e
EE130 Lecture 5, Slide 3
( Ei  Ev ) / kT
( Ei  E F ) / kT
Example: Energy-band diagram
Question: Where is EF for n = 1017 cm-3 ?
Spring 2007
EE130 Lecture 5, Slide 4
Dopant Ionization
Consider a phosphorus-doped Si sample at 300K with
ND = 1017 cm-3. What fraction of the donors are not ionized?
Answer: Suppose all of the donor atoms are ionized.
 Nc 
Then EF  Ec  kT ln 
  Ec  150meV
 n 
Probability of non-ionization 
1
1  e ( ED  EF ) / kT
1

 0.017
(150meV  45meV ) / 26 meV
1 e
Spring 2007
EE130 Lecture 5, Slide 5
Nondegenerately Doped Semiconductor
• Recall that the expressions for n and p were derived using
the Boltzmann approximation, i.e. we assumed
Ev  3kT  EF  Ec  3kT
3kT
Ec
EF in this range
3kT
Ev
The semiconductor is said to be nondegenerately doped in this case.
Spring 2007
EE130 Lecture 5, Slide 6
Degenerately Doped Semiconductor
• If a semiconductor is very heavily doped, the Boltzmann
approximation is not valid.
In Si at T=300K: Ec-EF < 3kT if ND > 1.6x1018 cm-3
EF-Ev < 3kT if NA > 9.1x1017 cm-3
The semiconductor is said to be degenerately doped in this case.
• Terminology:
“n+”  degenerately n-type doped. EF  Ec
“p+”  degenerately p-type doped. EF  Ev
Spring 2007
EE130 Lecture 5, Slide 7
Band Gap Narrowing
• If the dopant concentration is a significant fraction of
the silicon atomic density, the energy-band structure
is perturbed  the band gap is reduced by DEG :
8
DEG  3.5 10 N
1/ 3
300
T
N = 1018 cm-3: DEG = 35 meV
N = 1019 cm-3: DEG = 75 meV
Spring 2007
EE130 Lecture 5, Slide 8
Mobile Charge Carriers in Semiconductors
• Three primary types of carrier action occur
inside a semiconductor:
– Drift: charged particle motion under the influence
of an electric field.
– Diffusion: particle motion due to concentration
gradient or temperature gradient.
– Recombination-generation (R-G)
Spring 2007
EE130 Lecture 5, Slide 9
Electrons as Moving Particles
In vacuum
F = (-q)E = moa
In semiconductor
F = (-q)E = mn*a
where
mn* is the electron effective mass
Spring 2007
EE130 Lecture 5, Slide 10
Carrier Effective Mass
In an electric field, E, an electron or a hole accelerates:
 q
a
mn*
q
a *
mp

electrons
holes
Electron and hole conductivity effective masses:
m*n /m 0
m*p /m 0
Spring 2007
Si
0.26
0.39
Ge
0.12
0.30
EE130 Lecture 5, Slide 11
GaAs
0.068
0.50
Thermal Velocity
Average electron kinetic energy 
vth 
3kT

*
mn
3
1
kT  mn*vth2
2
2
3  0.026eV  (1.6 10 19 J/eV)
0.26  9.110 31 kg
 2.3 105 m/s  2.3 107 cm/s
Spring 2007
EE130 Lecture 5, Slide 12
Carrier Scattering
• Mobile electrons and atoms in the Si lattice are
always in random thermal motion.
– Electrons make frequent collisions with the vibrating atoms
• “lattice scattering” or “phonon scattering”
– increases with increasing temperature
– Average velocity of thermal motion for electrons: ~107 cm/s @ 300K
• Other scattering mechanisms:
– deflection by ionized impurity atoms
– deflection due to Coulombic force between carriers
• “carrier-carrier scattering”
• only significant at high carrier concentrations
• The net current in any direction is zero, if no electric
2
3
field is applied.
1
electron
4
Spring 2007
EE130 Lecture 5, Slide 13
5
Carrier Drift
• When an electric field (e.g. due to an externally applied
voltage) is applied to a semiconductor, mobile chargecarriers will be accelerated by the electrostatic force. This
force superimposes on the random motion of electrons:
3
2
1
electron
4
5
E
• Electrons drift in the direction opposite to the electric field
 current flows
 Because of scattering, electrons in a semiconductor do not achieve
constant acceleration. However, they can be viewed as quasi-classical
particles moving at a constant average drift velocity vd
Spring 2007
EE130 Lecture 5, Slide 14
Electron Momentum
• With every collision, the electron loses momentum
mn* vd
• Between collisions, the electron gains momentum
(-q)Etmn
tmn is the average time between electron scattering events
Spring 2007
EE130 Lecture 5, Slide 15
Carrier Mobility
mn*vd = (-q)Etmn
|vd| = qEtmn / mn* = n E
• n  [qtmn / mn*] is the electron mobility
Similarly, for holes: |vd| = qEtmp / mp*  p E
• p  [qtmp / mp*] is the hole mobility
Spring 2007
EE130 Lecture 5, Slide 16
Electron and Hole Mobilities
 has the dimensions of v/E :
 cm/s cm 2 



V/cm
V

s


Electron and hole mobilities of selected
intrinsic semiconductors (T=300K)
 n (cm2/V∙s)
 p (cm2/V∙s)
Spring 2007
Si
1400
Ge
3900
GaAs
8500
InAs
30000
470
1900
400
500
EE130 Lecture 5, Slide 17
Example: Drift Velocity Calculation
a) Find the hole drift velocity in an intrinsic Si sample for E = 103
V/cm.
b) What is the average hole scattering time?
Solution:
a)
b)
vd = n E
p 
qt mp
*
p
m
Spring 2007
 t mp 
m*p  p
q
EE130 Lecture 5, Slide 18
Mean Free Path
• Average distance traveled between collisions
l  vtht mp
Spring 2007
EE130 Lecture 5, Slide 19
Summary
• The intrinsic Fermi level, Ei, is located near midgap
– Carrier concentrations can be expressed as
functions of Ei and intrinsic carrier concentration, ni :
n  ni e
( E F  Ei ) / kT
p  ni e
( Ei  E F ) / kT
• In a degenerately doped semiconductor, EF is
located very near to the band edge
• Electrons and holes can be considered as quasiclassical particles with effective mass m*


– In the presence of an electric field , carriers move
with average drift velocity vd   , where  is the
carrier mobility
Spring 2007
EE130 Lecture 5, Slide 20