Semiconductor physics

CNS182 Caltech, Liu&Delbruck 1/10/2007
MOS transistors – the dominant technology
p+ p+ n+ n+
np-
We need to understand enough about semiconductors and
junctions to understand how MOS transistors work
•Insulators, conductors, semiconductors
•Crystal structure of silicon
•Band structure (valence, conduction, and forbidden bands)
•Holes and electrons
•Mechanisms of charge transport (diffusion & drift)
•Doping with donors and acceptors
•Fermi-Dirac distribution
•Law of mass action (np=n i 2 )
•p-n junction
•Reverse biased junction and its capacitance
The Diamond Structure of Silion
Each atom is
covalently
bonded to 4
neighbors
Si dominates because it has a clean oxide interface: SiO 2
Energy bands arise from periodic strcture of
crystal
Wavefunctions of
electron in a box
Energy levels
A crystal is like a periodic box
•Only wavefunctions with discrete
nonzero energies act like free
particles
•Wavefunctions at forbidden
energies die off exponentially
E c
Conductors, Semiconductors and Insulators
Energy bands
Electrons and holes
Mead, 1988
Grove, 1969
• Electrons in the conduction band act like free particles
• Holes are bubbles in the valence band and also act like
free particles
• The electrons move, but it is easier to talk about the
vacancy (the hole) moving, just like it is easier to talk
about a bubble moving than about the water around it
moving
• Quantum-mechanically, a hole acts like an electron,
except that holes have positive charge and the
effective mass in silicon is 2.5 times larger for a hole
than for an electron
Lecture 2, Semiconductor physics 1

CNS182 Caltech, Liu&Delbruck 1/10/2007
Energy
relative
to Fermi
level in
kT units
The meaning of energy in the band
diagram
E g=1.2eV=50kT
The Fermi-Dirac distribution of exclusionary
state occupation at thermal equilibrium
States above Fermi level are occupied with Boltzman distribution
−( E−Ef)/ kT
p(occupied) ≈ e
States at the Fermi level are ½ occupied
States below Fermi
level are unoccupied
with Boltzman
distribution
−( E f −E)/
kT
p(unoccupied) ≈ e
Probability of occupation of a state
The thermal energy
• Each degree of freedom of a system in
thermal equilibrium has average energy kT/2
• The thermal voltage kT/q is the voltage a
single charge falls through to pick up the
thermal energy kT
• kT/q=25mV=1/40V at room temperature
• kT/q is the natural scale of voltage for
electronic systems in thermal equilibrium
• The band gap of silicon (1.2eV) is about 50
times kT
An intrinsic semiconductor (no dopants)
Band
diagram
Density
of states
Fermi-Dirac
distribution
Carrier
concentrations
Donors and Acceptors in the periodic table A donor atom donates a free electron
Binding energy of electron is
reduced from free atom
binding energy (~0.5 eV) by
silicon dielectric constant
ε Si ≈12ε 0
0.
5eV
Ebinding ≈ ≈ 0.
05eV
≈ 2kT
12
At 300K, nearly all donor
electrons are free – because
there are many more ways to
be free than bound
Lecture 2, Semiconductor physics 2

CNS182 Caltech, Liu&Delbruck 1/10/2007
Doping levels
• Concentration of Si is about 10 23 /cm 3
• Doping can vary from about 10 15 /cm 3 to
10 19 /cm 3
• These doping levels still represent only
a tiny fraction of the total atoms, from
10 -8 to 10 -4
The law of mass action: np=n i 2
• In equilibrium, more holes means less
electrons, and vice-versa.
• np=n i 2
• n i is the intrinsic carrier density
• n i increases with temperature
• At room temperature n i is 10 10 /cm 3 , or
about 1/10 13 Si atoms.
Mobility is a function of electric field
Grove 1969
An n-type semiconductor
A p-type semiconductor
An electric field causes carriers to drift
J = μ(
Eqn )
Lecture 2, Semiconductor physics 3
Current
Flux
Charge density
Electric field
Mobility
J ≈ μqnE
for E that causes velocities
much less than the thermal velocity
A density gradient causes carriers to
diffuse
Diffusion current
J = −Dq∇n Current
Flux
Diffusion
constant
Charge
density

CNS182 Caltech, Liu&Delbruck 1/10/2007
Drift and diffusion are related by
The Einstein relation
J =−Dq∇n J ≈ μqnE
diffusion
kT
D = μ
q
drift
This relation got Einstein his Nobel prize
A P-N junction
N-type P-type
Electric field
Diffusion of holes from p region
Diffusion of electrons from n region
In equilibrium, Drift = Diffusion for electrons and holes
Electrostatic
Potential: potential
energy of positive
charge
Potential energy of
negatively charged
electron
Electrostatic potentials in a PN
junction
N-type P-type
φ
φ e
E
A P-N junction
Mobile majority carriers
N-type P-type
Fixed ions
“Depletion region”
“Space-charge region”
Charges, fields, and potentials in a PN
junction
N-type P-type
Lecture 2, Semiconductor physics 4
Charge
density
Electric
field
N D
Electrostatic
potential
Q
φ
E
N A
kT N N
ϕ T = log( )
q n
A D
2
i
Typically, the built-in voltage, ϕ T , is about 0.75V
Carrier densities in a PN junction
log scale
n
n i
N-type P-type
depletion region—both n and p are far below doping values
p
np=n i 2

CNS182 Caltech, Liu&Delbruck 1/10/2007
Carrier densities in a PN junction
N-type P-type
linear scale
n
ni depletion region—both n and p are far below doping values
p
np=n i 2
MOS transistors – the dominant
technology
p+ p+ n+ n+
n
Next week:
Understanding how MOS transistors work in weak
and strong inversion
p
What was covered
– Insulators, conductors, semiconductors
– Crystal structure of silicon
– Band structure (valence, conduction, and
forbidden bands)
– Holes and electrons
– Mechanisms of charge transport (diffusion &
drift)
– Doping with donors and acceptors
– Fermi-Dirac distribution
– Law of mass action (np=n i 2 )
– p-n junction
Lecture 2, Semiconductor physics 5