ELEC 3908, Physical Electronics, Lecture 7
Generation, Recombination
and Diffusion
Lecture Outline
• Have described structure and processing of diode as well
as important doping and area related effects
• Now want to derive the basic static (dc) model for diode
current flow in terms of applied potential
• First need to look at two physical processes which play
critical roles in determining the bias-dependence of diodes
and bipolar devices
– Generation and recombination refer to the transitions of electrons
between the valence and conduction bands
– Diffusion is the movement of a particle under the influence of a
concentration gradient
ELEC 3908, Physical Electronics:
Generation, Recombination and Diffusion
Page 7-2
Generation and Recombination (GR)
•
•
•
Have found expressions for the electron and hole densities n and p for
material in thermal equilibrium
To characterize material under bias, require model for the nature of
electron and hole densities with time
Movement of an electron between the conduction and valence bands is
characterized by two processes:
–
Generation is the movement of an electron from the valence to
conduction bands, and therefore the creation of a hole
– Recombination is the movement of an electron from the conduction to
valence bands, and therefore the “destruction” of a hole - note that
destruction of the hole is just the filling of the valence orbital
ELEC 3908, Physical Electronics:
Generation, Recombination and Diffusion
Page 7-3
Generation and Recombination in Pure Material
•
•
•
In intrinsic (pure) silicon, holes
created by a full band to band
transition of an electron
Eg is relatively large, making
this transition unlikely at room
temperature
Other mechanisms are therefore
important
ELEC 3908, Physical Electronics:
Generation, Recombination and Diffusion
Page 7-4
Trap-Aided Generation and Recombination
•
•
•
•
In practice, impurities will
always be present in material
Impurities generate extra
energy levels in the crystal
structure called traps
Most important impurities are
those which introduce traps
near midgap (e.g. Au), since
this gives possibility of two
steps in energy of Eg/2
Even with great care in
processing, trap-aided GR
dominates full band to band GR
in silicon at room temperature
ELEC 3908, Physical Electronics:
Generation, Recombination and Diffusion
Page 7-5
Shockley-Read-Hall GR Model
• Complete analysis in the Shockley-Read-Hall model takes
into account the chances of an electron existing at the trap
energy, and trap position in the gap
• Expression for the net recombination rate U in a simplified
expression (equal capture release prob., midgap traps) is
n( x ) p( x ) − ni2
U=
τ o ( n( x ) + p( x ) + 2 ni )
• το is the average minority lifetime in seconds
• U is positive when net recombination is occurring,
negative when net generation is occurring
ELEC 3908, Physical Electronics:
Generation, Recombination and Diffusion
Page 7-6
GR - Equilibrium
• At equilibrium, np = ni2 (note that n does not have to be,
and in general will not be for a doped material, equal to p)
• Generation and recombination occur, but exactly balance
• The net recombination rate U is therefore 0, and no net
change in n or p will occur with time
Ec
np − ni2
U=
=0
τ o ( n + p + 2ni ) np =n2
i
np = ni2
G
R
Ev
ELEC 3908, Physical Electronics:
Generation, Recombination and Diffusion
Page 7-7
GR – Net Recombination
• When np is greater than ni2, excess carriers are present, and
recombination will dominate generation, and act to lower
the concentrations with time
• The net recombination rate U is therefore > 0
Ec
np − ni2
U=
>0
τ o ( n + p + 2ni ) np>n2
i
np > ni2
G
R
Ev
ELEC 3908, Physical Electronics:
Generation, Recombination and Diffusion
Page 7-8
GR – Net Generation
• When np is less than ni2, carrier concentrations are lower
than at equilibrium, and generation will dominate
recombination, acting to raise the concentrations with time
• The net recombination rate U is therefore < 0
Ec
np − ni2
U=
<0
τ o ( n + p + 2ni ) np<n2
i
np < ni2
G
R
Ev
ELEC 3908, Physical Electronics:
Generation, Recombination and Diffusion
Page 7-9
Summary - Physical Property of GR
• Looking at magnitude of n,p vs ni gives a valuable insight
into the physical property of generation and recombination
– np > ni2, U > 0 : If the np product is greater than ni2, U is positive,
indicating net recombination. Since recombination will lower n
and p, the np product will decrease with time (if possible)
– np = ni2, U = 0 : If the np product is equal to ni2, U is zero,
indicating no net generation or recombination, the equilibrium
condition. No net change in n or p will occur with time
– np < ni2, U < 0 : If the np product is less than ni2, U is negative,
indicating net generation. Since generation will increase n and p,
the np product will increase with time (if possible)
• GR therefore acts to restore equilibrium, and is the
counterbalancing force against a bias-induced disturbance
ELEC 3908, Physical Electronics:
Generation, Recombination and Diffusion
Page 7-10
Low Level Injection
• The SRH expression can be simplified for the case of an
increase in both n and p which is large compared to the
equilib. minority density but small compared to the equilib.
majority density - termed low level injection
• Example: disturbance in p-type material which raises np by
an amount which is large compared to npo but small
compared to the acceptor doping NA
• For the case of low level injection,
U
p − type
n
(
≈
p
− n po
τo
) ≡ Δn
τo
p
U
n − type
p
(
≈
n
− pno )
τo
≡
Δ pn
τo
• Note that U depends on the excess minority density only
ELEC 3908, Physical Electronics:
Generation, Recombination and Diffusion
Page 7-11
Example 7.1: Low Level Injection
The concentrations of electrons and holes in a p-type sample
of silicon with NA=2x1016 /cm3 are raised by 109 /cm3 by a
disturbance. Calculate the prediction of the net
recombination rate U using the full SRH model and the
approximate low level injection expressions assuming a
minority lifetime of 1 μsec.
ELEC 3908, Physical Electronics:
Generation, Recombination and Diffusion
Page 7-12
Example 7.1: Solution
• The equilibrium majority and minority densities are
16
po = N A = 2 × 10 / cm
3
n po
ni2
=
= 10 4 / cm 3
NA
• The new values of the densities after the disturbance are
p = p o + 10 9 ≈ 2 × 1016 / cm 3
n p = n po + 10 9 ≈ 10 9 / cm 3
• The full SRH and approximate models give
10 ⋅ 2 × 10 − (145
. × 10 )
3
15
10
/
cm
=
⋅ sec
−6
9
16
10
10 (10 + 2 × 10 + 2 ⋅ 145
. × 10 )
9
U=
U≈
Δn p
τo
16
10 2
109
= − 6 = 1015 / cm 3 ⋅ sec
10
ELEC 3908, Physical Electronics:
Generation, Recombination and Diffusion
Page 7-13
Diffusion
•
•
The physical process of diffusion is the movement of particles under
the influence of a concentration gradient.
Particles move so as to reduce the concentration gradient, i.e. from the
region of higher concentration to the region of lower concentration
ELEC 3908, Physical Electronics:
Generation, Recombination and Diffusion
Page 7-14
Diffusion Flux
•
The flux, or flow of particles,
under the influence of diffusion
is given by
dc( x )
Θ = −D
dx
•
•
•
Where Θ is the particle flux per
unit area (/cm2sec), D is the
diffusion coefficient (cm2/sec)
and c(x) is the per unit volume
concentration (/cm3)
Negative sign gives movement
in direction of decreasing conc.
D is a measure of ease of
movement of particles
ELEC 3908, Physical Electronics:
Generation, Recombination and Diffusion
Page 7-15
Electron and Hole Diffusion
•
Diffusion of electrons and holes
gives rise to current density
components
dn( x )
J n ,diff = qDn
dx
dp( x )
J p ,diff = − qD p
dx
•
Diffusion coefficients Dn and
Dp are given in terms of
mobilities μn and μp as
kT
Dn =
μ
q n
kT
Dp =
μ
q p
ELEC 3908, Physical Electronics:
Generation, Recombination and Diffusion
Page 7-16
Example 7.2: Diffusion Coefficients
Calculate the diffusion coefficients for silicon at 300K if the
electron and hole mobilities are μn=1350 cm2/Vsec and
μp=480 cm2/Vsec.
ELEC 3908, Physical Electronics:
Generation, Recombination and Diffusion
Page 7-17
Example 7.2: Solution
• Using the values of mobilities given and the value for kT/q
at 300K
kT
Dn =
μn = 0.02586 ⋅1350 = 34.9 cm 2 / sec
q
kT
Dp =
μ p = 0.02586 ⋅ 480 = 12.4 cm 2 / sec
q
ELEC 3908, Physical Electronics:
Generation, Recombination and Diffusion
Page 7-18
Diffusion Length
•
•
•
•
GR can influence electron/hole
behavior during diffusion
Example shows region where
both gen. and recomb.
occurring
Certain probability that carrier
will recombine while diffusing,
or that carriers will be gen.
Define diffusion lengths Lp and
Ln (cm) as the average distance
traveled before recombining, or
the length over which most of
the generation is occurring
Ln =
Dnτ o
Lp =
D pτ o
ELEC 3908, Physical Electronics:
Generation, Recombination and Diffusion
Page 7-19
Example 7.3: Diffusion Length
What are the diffusion lengths in silicon at 300K? Assume the
minority lifetime is 0.5 μsec.
ELEC 3908, Physical Electronics:
Generation, Recombination and Diffusion
Page 7-20
Example 7.3: Solution
• Using the values of diffusion length calculated earlier,
Ln = 34.9 ⋅ 0.5 ×10 −6 = 4.18 ×10 −3 cm = 41.8μm
L p = 12.4 ⋅ 0.5 × 10 −6 = 2.49 × 10 − 3 cm = 24.9μm
ELEC 3908, Physical Electronics:
Generation, Recombination and Diffusion
Page 7-21
Lecture Summary
• Electron transition to the conduction band (gaining energy)
creates an electron-hole pair - generation
• Electron transition to the valence band (losing energy)
destroys an electron-hole pair – recombination
• GR is characterised by the net recombination rate U
– np < ni2, U < 0, net generation
– np = ni2, U = 0, equilibrium
– np > ni2, U > 0, net recombination
• Diffusion is the movement of particles down a
concentration gradient, characterized (1D) by a diffusion
coefficient and the derivative of the particle density
• These concepts will be used in derivation of the ideal diode
equation
ELEC 3908, Physical Electronics:
Generation, Recombination and Diffusion
Page 7-22