transport, scattering generation/recombinationsoktyabr/nnse508/nnse508...total electron current:...
TRANSCRIPT
NNSE508 / NENG452 Lecture #13
1
Lecture contents
• Transport, scattering
• Generation/recombination
Band-to-band
recombination Trap-assisted (SRH)
recombination
and generation
Auger
recombination
Ec
Et
Ev
E
NNSE508 / NENG452 Lecture #13
2
Electron transport: General considerations
Motion in real space = thermal motion
+ drift + scattering
Mean free path:
Motion in wavevector space: inside a
band valley (unless intervalley
scattering is involved)
Current density is proportional to drift
velocity of carriers
2
A
cmdJ env
How “free” carriers react on external electric field ?
th mv
NNSE508 / NENG452 Lecture #13
3 Electron transport: phenomenological approach
Kinetic energy per electron in 3D (from
kinetic theory of gases)
Thermal velocity
Force on a “free” electron includes
electric and “friction” with
momentum relaxation time m :
In the steady state drift velocity is
proportional to the field (m –drift
mobility):
And current density (s –conductivity)
gives Ohm’s law:
For semiconductor containing both
electron and holes:
sm enenvJ
Tkmv
B2
3
2
2
m
de
de
vme
dt
dvm
**
m
*
e
md
m
ev
2
*
cm
Vs
m
e
e
m
m
nm
een
e
m
*
2ms
he pne mms
1 2
7310
*
Bth d
k T cmv v
m s
-1
-1
3D:(Ω cm)
2D:(Ω/sq.)s
electric field)(
NNSE508 / NENG452 Lecture #13
4
Scattering mechanisms
In low electric fields
• ionized impurities
• acoustic phonons
– Deformation potential
– Piezoelectric
In high electric fields
• optical phonons
• intervalley scattering
At high concentrations
• carrier-carrier scattering
To understand how these mechanisms affect mobility one needs to
consider dependence (E)
Mathiessen’s rule for relaxation time:
...
11111
niiiopacm
NNSE508 / NENG452 Lecture #13
5 Conductivity effective mass
• For non-degenerate conduction band minimum (G-
point):
• Indirect conduction band minimum: effective mass
will depend on direction!
– For example: Si along [100] direction:
– Conductivity effective mass of Si [100] electrons::
• Valence band maximum: need to add up
contributions of light and heavy holes:
• Conductivity effective mass of holes
tlc mmm
21
3
11*
G mme
*
2 2
2 2 26 6
m mn ex ey ez
l t
e en nJ J J J
m m
mhhh
lh
lhhhlhp e
m
p
m
pJJJ 2
2323
2121
*
11
hhlh
hhlh
hh
hh
lh
lh
v mm
mm
m
p
m
p
pm
Effective masses and low field mobilities at room temperature:
NNSE508 / NENG452 Lecture #13
6 Temperature dependence of mobility
Mobility in n-Si
From Yu and Cordona, 2003,
and Shur, 2003
Mobility in p-Si
Mobility in n-GaAs Relaxation time m depends on energy, and
therefore mobility depends on temperature
LO phonon scattering
Intravalley acoustic phonon
scattering + (2TA+LO)
intervaley phonon scattering
NNSE508 / NENG452 Lecture #13
7
High field electron transport
Hot electrons transfer energy into thermal
vibrations of the crystal lattice (phonons).
Such vibrations can be modeled as harmonic
oscillations with a certain frequency, wph .
The energy levels of a harmonic oscillator
are equidistant with the energy difference
between the levels equal to
Two types of phonons scatter electrons
differently: acoustical (slow) and optical (fast):
Hence, process for a hot electron can be
represented as follows. The electron
accelerates in the electric field until it
gains enough energy to excite optical phonon:
phphE w
meVE optopt 40 w
kTE acac w
NNSE508 / NENG452 Lecture #13
8
Intervalley scattering in high electric field
GaAs has an L valley just 0.29 eV higher than
G -valley
At high fields electrons can be transferred
into L-valleys. The conductivity will
depend on concentrations and mobilities
of electrons in both valleys:
Model for dependence of drift velocity on
electric field in GaAs
From Shur, 2003
LLNNe mms GG
NNSE508 / NENG452 Lecture #13
9 Diffusion and drift of carriers
• If the carrier concentration is changing from point-to point, the carriers
will redistribute to equalize the concentration = Diffusion
• Diffusion is described by the first Fick’s law:
Electron diffusion current
Hole diffusion current
• If electric field (weak) is applied:
Total electron current:
Total hole current:
plays the role of an effective field
• Diffusion coefficient and diffusion current have sense only if
concentration change is small on the mean free path:
• Substituting the gradient with the effective field:
neDJ nn
peDJ pp
(Flux of particles)= -(Diffusion coef.) x (gradient of concentration)
neDneJ nnn m
peDpeJ pop m
D n
n
n
e
TkB
nn
1Tk
e
B
NNSE508 / NENG452 Lecture #13
10 Equilibrium of drift-diffusion: Einstein relation
At equilibrium condition (no total current)
diffusion current is equal to drift current:
For nondegenerate semiconductor:
Carrier concentration is changing in the
electric field, following the potential j(r):
Gradient of electron concentration is :
And substituting, obtain Einstein relation:
0 neDne nnm
Tk
EENn
B
CFC exp
Tk
xenxn
B
)(exp)( 0
j
jTk
enx
Tk
enn
BB
)(
EF
EC
Drift flux
Diffusion flux
me
TkD B *
e
m
m
em
NNSE508 / NENG452 Lecture #13
The rate of change of the carriers between x
and x + dx is determined by fluxes and
generation/recombination of carriers:
Continuity equation for electrons:
And similar for holes
Continuity equation
CE
x x dx
x
( )nJ x ( )nJ x dx
VE
nGnR
( )1 nJ xdx
q x
( ) ( )( , )
( , ) ( , )n nn n
J x J x dxn x tAdx A G x t R x t Adx
t q q
( , ) 1( , ) ( , ) ( , )n n n
n x tdivJ x t G x t R x t
t q
( , ) 1( , ) ( , ) ( , )p p p
p x tdivJ x t G x t R x t
t q
The continuity equation allows us to
calculate the carrier distribution in the
presence of generation and recombination
3
1,G R
cm s
NNSE508 / NENG452 Lecture #13
0
0( )
t t
p t p Ge
12 Generation/Recombination
Band-to-band
recombination Trap-assisted (SHR)
recombination
and generation
Auger
recombination
Ec
Et
Ev
E
Recombination mechanisms • In the simplest model net recombination
rate (intrinsic = response of the
semiconductor) is proportional to the
EXCESS carrier density (n0, p0 –
equilibrium concentrations, ’s - minority
carrier lifetimes):
• Un might be negative – net generation
• At thermodynamic equilibrium, the
generation rate and the recombination
rate are equal:
• After generation, excess concentration
decays exponentially decays:
Ec
Ev
E
Ionization due to
energetic
particle/photon
Generation mechanisms
E-h pairs generation
by light
Ephoton=hn>Eg
0p p p
p
p pU R G
P
0n pU U
0n n n
n
n nU R G
3
1
cm s
Extrinsic generation
rate, not material
response
NNSE508 / NENG452 Lecture #13
13 Band-to-band transitions in direct-bandgap
and indirect-bandgap semiconductors
• Low emission probability due to 2nd
order (phonon-assisted) process
(momentum conservation)
• Long radiative lifetime low radiative
quantum efficiency
• Si, Ge, diamond, GaP, AlAs…
• High emission probability
• Short radiative lifetime high radiative
quantum efficiency, efficient LED,
lasers…
• GaAs, InGaAs, GaN, InP, InGaSb…
NNSE508 / NENG452 Lecture #13
14 Optical absorption
From Colinge, 2005
• Beer-Lambert absorption law
• Significant absorption at
• Indirect bandgap – low near
edge absorption, gradual
spectral slope (momentum
conservation)
• Direct bandgap – sharp edge
• Absorption spectrum depends
on density of sates in the bands
and transition probabilities.
Ephoton = hn > Eg
I(x) =I0 exp(-ax)
NNSE508 / NENG452 Lecture #13
15
Optical absorption and non-equilibrium carriers
• Intensity [W/cm2=J/s-cm2] of light is exponentially
decreasing due to absorption
• Absorbed intensity in a layer with a thickness d:
• In case of uniform generation (low absorption) ad <<<1,
can be always applied to a thin slice
• Number of generated e-h pairs is equal to the number of
photon absorbed. When generation is uniform:
• Density of both electrons and holes increase under band-tp-
band optical excitation by the same amount
• Higher energy photons generate “hot” carries that relax
towards the band extrema
• G/R is typically slower (10-9 - 10-5 s) than energy
relaxation due to scattering (10-14 - 10-12 s)
Allows to treat electrons and holes
statistically independent i.e.
introduce separate quasi-Fermi
levels for electrons and holes.
0absI I da
I(x) =I0 exp(-ax)
0 0
d
absI I I e a
opt opt
n p
I IdG G
d h h
aa
n n
Dn = Dp
Extrinsic generation rate
NNSE508 / NENG452 Lecture #13
16
Radiative recombination rate Rr and radiative life time r:
Radiative recombination rate Rr (B- radiative or
Bi-molecular recombination coefficient)
consists of equilibrium and non-equilibrium parts:
Two cases:
High injection rate (bimolecular recombination):
Low injection:
Radiative recombination
DD ))(( 00 ppnnBBnpRr
dt
dpR
nn
dt
dnr
r0
00 , pnn D
2nBRr D
0nn D
nBr
D
1
00
1pnB
r
in a doped material one of the
concentrations dominates
200 iBnnpnnB DD
Considering optical excitation: Dn = Dp
Might depend
on n
New non-equilibrium Rr (=Ur)
and
NNSE508 / NENG452 Lecture #13
17
Radiative recombination
From Yu and Cordona, 2003
Radiative lifetime for low injection
• Minority carrier lifetime
For example for p-material lifetime of
injected electrons will be 0
1Bp
r
0nn D
NNSE508 / NENG452 Lecture #13
for centers where the recombination rate is highest (i.e. for )
18
Non-radiative recombination through
recombination centers Nt: Shockley-Hall-Reed
(SHR) processes
Consider electron G/R: nt - centers occupied
with electrons ( f –Fermi distribution function)
lifetime:
At equilibrium, electron emission
:
Shockley-Hall-Reed (SHR) Recombination
2
exp expSHR
i
t i i tp i n i
np nU
E E E En n p n
kT kT
2
0
i
SHR
n n
nn
n npU
t iE E
n n nU R G
p p pU R G
_
11n n th n t t t
n
R v n N n f Es
n n t tG e N f E
_
1n
n th n tv N
s
_ exp i tn n th n i
E Ee v n
kTs
Combining
electrons and
holes
2
iSHR
n i p i
np nU
p n n n
For minority carriers (e.g.
electrons in p-type) 0p p n
NNSE508 / NENG452 Lecture #13
19
Surface recombination in quasi-neutral p-type region
With surface recombination velocity sn [cm/s]
Auger recombination contribute at high carrier
concentrations:
In general in case of external generation G :
where lifetime is a result of various processes: non-
radiative through recombination centers, radiative,
Auger recombination:
Lifetime at high excitation conditions:
Other recombination mechanisms
, 0surf n nU s n n
n
nnG
dt
dn
0
Arnn
1111
0
322 CnnpCpnCR pnA
, ip n p n
21CnBnA
n
Uniform (photo)generation:
( ) ( ) 11
p
x
Ln n nn
n n n
s Ln x p x G e
s L
D D
x
n
n0 Ln
nG
NNSE508 / NENG452 Lecture #13
Lecture recap
20
• Ohmic behavior results from scattering
• Drift-diffusion
• Recombination rate
• Continuity equation (1D)
neDneJ nnn m
1 nn n
JnG R
t q x
0n n n
n
n nU R G
Minority carrier lifetime