Transcript Lecture #5 OUTLINE • Intrinsic Fermi level • Determination of EF • Degenerately doped semiconductor • Carrier properties • Carrier drift Read: Sections 2.5, 3.1

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:
N ce
 ( E c  E F ) / kT
EF 
Ec  Ev
2
 N ve
 ( E F  E v ) / kT
 Nv 
  Ei

ln 

2
N
 c
kT
*

 E E
m
E c  E v 3 kT
p
v
Ei 

ln  *   c
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
 N c  nie
n  ni e
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 ( E c  E i ) / kT
( E c  E i ) / kT
( E F  E i ) / kT
p  ni  N v e
 N v  ni e
p  ni e
EE130 Lecture 5, Slide 3
 ( E i  E v ) / kT
( E i  E v ) / kT
( E i  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.
Then E F
 Nc 
 E c  kT ln 
  E c  150 meV
 n 
Probability of non-ionization 

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1
1 e
( E D  E F ) / kT
1
1 e
(150 meV  45 meV ) / 26 meV
EE130 Lecture 5, Slide 5
 0 . 017
Nondegenerately Doped Semiconductor
• Recall that the expressions for n and p were derived using
the Boltzmann approximation, i.e. we assumed
E v  3 kT  E F  E c  3 kT
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 :
D E G  3 . 5  10
8
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:
a
a
 q
m
*
n
m
*
p
q
electrons
holes
Electron and hole conductivity effective masses:
m*n /m 0
m*p /m 0
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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
v th 
3 kT
m
*
n


3
kT 
2
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2
3  0 . 026 eV  (1 . 6  10
0 . 26  9 . 1  10
 2 . 3  10 m/s  2 . 3  10 cm/s
5
1
7
EE130 Lecture 5, Slide 12
 31
*
2
m n v th
 19
kg
J/eV)
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
*
m nvd
• 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 :
2
 cm/s
cm 



V/cm
V

s


Electron and hole mobilities of selected
intrinsic semiconductors (T=300K)
 n (cm2/V∙s)
 p (cm2/V∙s)
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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)
vd = n E
b)  p 
q t mp
m
*
p
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m p p
*
 t mp 
q
EE130 Lecture 5, Slide 18
Mean Free Path
• Average distance traveled between collisions
l  v tht 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  E i ) / kT
p  ni e
( E i  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 v d   , where  is the
carrier mobility

Spring 2007
EE130 Lecture 5, Slide 20