Download - 4. Carrier Transport
4. Carrier transport 2102385
• Why do we want to know about carrier transport in semiconductors?
• Transport mechanisms– Drift (under external electric field)– Diffusion (under concentration gradient)
– Drift-diffusion inter-relationship
1
4. Carrier Transport
Information (I, V)
Information transfer =carrier transport ( I )
Information read-out (I, V)
want large bandwidth must understand transport mechanisms
amout of infotime Mbps
external force
(battery, -field)
Contents
4.1 Drift
4.2 Diffusion
4.3 Current densities
4.4 Einstein relation
4.5 Continuity equations
v.2018.AUG
all about motion ofCB electrons & VB holes
4. Carrier transport 2102385 2
Transport: in a semiconductor
( ) ( ) ( )
x
t > 0, x = 0 t > 0, x > 0 t = 0
Drift Diffusion Current densities Einstein relation Continuity equations
Individual: yes
Overall: no
Individual: yes
Overall: yes
Carriers motion due to thermal energy (equilibrium condition) does not lead to net
displacement of “carriers”. No net charge transport.
External electric fields cause the carriers to drift apart. Net charge transport results.
4.1 Drift
Drift is the underlying mechanism of conduction in MOSFET (C7).
(ข)
(ค)
4. Carrier transport 2102385 3
Transport: in a circuit (semiconductor + metal + battery)
Drift Diffusion Current densities Einstein relation Continuity equations
electrons holes
generation recombination(sam
e as
C3
, sl
ide
#6
)
Metal-semiconductor junctions serve as
fast EHP generation-recombination center
All electronic circuits consist of wires (metals, electrons only) and chips
(semiconductors, electrons and holes). How can electrons/holes complete a circuit?
Under electric fields, both types of carriers drift.
Metal-semiconductor junctions serve as
low-resistance contacts
4. Carrier transport 2102385
note 1: p = Momentum, hole concentrationnote 2: t = mean free Time (of equilibrium carriers),
recombination lifetime (of escess carriers)reminder: Force = rate of change of Momentum
4
Acceleration (by -field)
• for single e-: net force = - ex
• for n electrons/cm3:
Deceleration (by collisions)
• assume truly random collisions
• initial momentum lost every t (mean free time,
average time between scattering events)
x
field
x nedt
dp
tx
collisions
x p
dt
dp
0x
•
steady-state current flow• average momentum per e-:• average velocity per e-:
• Current density = # e- crossing unit area per unit time n<vx> × e
• Drude model of conduction (1900): Ohm’s law (1827)
• If carriers include both electrons (n) and holes (p):
00 xx
collisions
x
field
x nep
dt
dp
dt
dp
t
x
nn
x
x
xx
x
m
e
m
pv
en
pp
t
t
**
23*
2
cm
A
s
cm
cm
Coulomb
m
nevenJ x
n
xx
electron
electron unit
t
t
nem
ne
n
*
2
pn pene
xxv
xxJ
*
nm
et
drift velocity
Drift Diffusion Current densities Einstein relation Continuity equations
4.1.1 Mobility (…) r high
rr
r
JA
I
L
V
A
LIIRV
;mobility
conductivity
current density
4. Carrier transport 2102385
102
103
104
p
n
Ge, 300 K
102
103
104
Si, 300 KMo
bil
ity
(cm
2/V
-s)
n
p
1014
1015
1016
1017
1018
1019
102
103
104
GaAs, 300 K
Doping Concentration (cm-3)
n
p
5
(ND, NA) [ND+, Na
-] [ionized] impurity scattering t
(N)22
2*
* ;
dkEdm
m
e
t
)(
)(
Ap
Dn
N
N
(…)
Impurity scattering
The higher the doping conc, the lower the mobility
t = mean free timerecombination lifetime
minority carrier lifetime
4. Carrier transport 2102385 6
carrier typescattering mechanisms
ionized impurity lattice carrier-carrier
majority
minority
1014
1015
1016
1017
1018
1019
1020
10
100
1,000
Si, 300 K
Mo
bil
ity
(cm
2/V
-s)
Doping Concentration (cm-3
)
n,
n,
p,
p,
(…)
)(
)(
Ap
Dn
N
N
)(
)(
Dp
An
N
N
(N,type): majority minority
4. Carrier transport 2102385 7
• Matthieson’s rule: or
mechanism which causes lower mobility dominates
21
111
ttt
21
111
low temperatures limit
(ionized) impurity scattering
high temperatures limit
phonon/lattice scattering
As+
rc
KE > |PE|e
KE = 1/2mev
2
KE |PE|
KE < |PE|
Scattering of electrons by an ionized impurity
h+
e
Thermal vibrations of atoms can break bonds and thereby create electron-hole pairs.
2 10 100
105
106
E
lect
ron
Mo
bil
ity
n (
cm2/V
-s)
Temperature (K)
ND = 41013
cm-3
combined
piezoelectric
neutral
impurity
deformation potential
ionized
impurity polar
optical phonon
GaAs
(…) (T)
1t
2t
t = mean free time (s)1/t = scattering rate (Hz)
4. Carrier transport 2102385 8
Example: Si
n(ND,T)
100 200 400 600 800 1000
102
103
104
E
lect
ron M
obil
ity
n (
cm2/V
-s)
Temperature (K)
Si
ND = 1019
cm-3
1016
1017
1018
1014
T3/2
T3/2
impurity
scattering
lattice
scattering
(…) (N,T)
4. Carrier transport 2102385
log(n)
INTRINSIC
EXTRINSIC
IONIZATION
log( )
log( )
T-3/2
T3/2
Lattice
scattering
Impurity
scattering
1/TLow TemperatureHigh Temperature
T
Metal
Semiconductor
LO
GA
RIT
HM
ICS
CA
LE
Res
isti
vit
y
Temperature dependence of electrical conductivity for a doped (n-
type) semiconductor.9
note:
log() vs 1/T
log(r) vs T
&
)()()( TeTnT
ne
r
4.1.2 Resistivity & Conductivity
Drift Diffusion Current densities Einstein relation Continuity equations
(…) r high
)/1(
)(
T
T
r
Reminder: = C3, #20
= C4, #7
4. Carrier transport 2102385 10
1014
1015
1016
1017
1018
1019
1020
1021
10-4
10-3
10-2
10-1
100
101
102
p-GaAs
n-GaAs
p-Si
n-Si
300 K
Res
isti
vit
y (
-cm
)
Doping Concentration ND, N
A (cm
-3)
)(Nr
1% alloy, solid solubility limits
4. Carrier transport 2102385 11
• (low-field) mobility – default• at high fields: Ohm’s law is not true
• at high fields, vd ~ vth (107 cm/s) carriers are hot added energy transferred to lattice, not to increase vdrift (scattering limits velocity)
• mobility is meaningless• (saturation) current density:
v
J
IRV
Tkvm Bth 2/2*• velocities– drift velocity (vdrift) = (see graph)
• at low fields• vs at high fields
– vth does not contribute to current (random motion)
– vdrift does not contribute to current (directed by )
driftthtotal vvv
velocity saturation (vs) | impact ionization (II)
snevJ
4.1.3 High (electric) field effects
102
103
104
105
106
105
106
107
108
Si (holes)
Si (electrons)
300 K
Dri
ft V
elo
city
(cm
/s)
Electric Field (V/cm)
GaAs (electrons)
GaAs (holes)
Drift Diffusion Current densities Einstein relation Continuity equations
(…) r high
vs is the cause of saturation current Id-sat in nanoscale CMOS
! no
Si b
reakd
ow
n (II)
Ebr
4. Carrier transport 2102385 12
A
A
C
b
c
d
x
D
B
EC
EV
A/B
c
velocity saturation (vs) | impact ionization (II)
Drift Diffusion Current densities Einstein relation Continuity equations
(…) r high
II is the cause of breakdown of many devices (diodes, transistors), effective mostly while devices are under reverse bias
nMP
nPPnn
1
1)1( 2
cause: energetic carriers break bond (ionize), generating 2 additional carriers upon each collision (impact)
effect: carrier multiplication—by a factor M related to ionization probability P according to
4. Carrier transport 2102385 13
t0
x = 0
x
t1
t2 t
y = 0
x
t0
t1
t2
t
y =
0
y
4.2 DiffusionDrift Diffusion Current densities Einstein relation Continuity equations
Diffusion is a natural process occuring in many branches of science and engineering.
Carrier diffusion results from non-uniform creation of excess carriers.
Diffusion is the underlying mechanism of conduction in diodes (C5) & BJTs (C6).
4. Carrier transport 2102385
Diffusion process:
• Spatial variation (gradient) in n and p diffusion (net motion of carriers from regions of high to low concentration)
• equal prob. of moving in both directions• rate of e- flow in +x-direction per unit area (flux density)
n(x) p(x)
n(l)
n(-l) n(0)
x
n p
In
Ip
0
14
Important parameters
: flux (#/area-time)
l: mean free path
t: mean free time
D: diffusion coefficient
I: current
Units
: /cm2-s or (cm-2-s-1)
l: cm
t: s
D: cm2/s
I: A (or C/s)
)()(2
1
2
)(
2
)(0
lnlnlllnlln
xn
ttt
dx
xdnD
dx
xdnl
dx
xdnln
dx
xdnln
lx
n
n
)()(
)()0(
)()0(
2
1)(
2
t
t
How is I related to ?
“first-order approximation”(Taylor series)
4. Carrier transport 2102385 15
• electrons flux:
• holes flux:
• electron current density:
• hole current density:
dx
xdnDx nn
)()(
dx
xdpDx pp
)()(
dx
xdnqD
dx
xdnDqdiffJ nnn
)()()(.)(
dx
xdpqD
dx
xdpDqdiffJ ppp
)()()(.)(
Diffusion current:
Diffusion flux: (#/cm2-s)
(A/cm2)
q = |electronic charge| = 1.6*10-19 C
4. Carrier transport 2102385 16
• Drift and diffusion components of e- and h+:
dx
xdpqDxxpqxJ
dx
xdnqDxxnqxJ
ppp
nnn
)()()()(
)()()()(
Ex
n-Type Semiconductor
x
Hole Drift
Hole Diffusion
Electron Drift
Electron Diffusion
Semitransparent electrode
Light
drifts
diffusions
Total current:)()( xJxJ pn
Agree with the
+/- signs?
4.3 Current densitiesDrift Diffusion Current densities Einstein relation Continuity equations
In actual devices, current will be dominated by one of the four terms, i.e.drift
diffusion NFET, PFET npn, pnp
4. Carrier transport 2102385
EC
EV
EF
x
E n(x)
x
E
0
x = 0
Ei
17
• Device: assumed optical excitation from the left (as in previous slide), or n-Si doped by surface diffusion to attain the profile
• Mechanism: diffusion causes -field which causes drift (oppose diffusion).
• Open circuit:set Jp(x) = 0
dx
dE
qdx
xdVx
dxdE
i
F
1)()(.2
0/.1
q
kTD
dx
xdp
xp
Dx
p
p
...
)(
)(
1)(
diffusiondrift
Conditions:
4.4 Einstein relationDrift Diffusion Current densities Einstein relation Continuity equations
Si, 300 K
1014
1015
1016
1017
1018
1019
1020
0
200
400
600
800
1,000
1,200
1,400
Doping Concentration (cm-3
)
(c
m2/V
-s)
0
5
10
15
20
25
30
35
D (cm
2/s)
(see p.162)
Equilibrium requirement:
(Streetman #3.5, #5.1.1)0/ dxdEF
• Open circuit:set Jp(x) = 0
4. Carrier transport 2102385 18
4.5 Continuity equationsDrift Diffusion Current densities Einstein relation Continuity equations
A dx
G R J (x) J (x+dx) I
Changes in carrier concentrations due to
- generation G and recombination R were considered in C3
- drift Jdrift and diffusion Jdiff (combined J(x)) were considered in 4.1, 4.2 (4.3)
- the 4 mechanisms are now considered together, forming basic equation that can
determine changes in carrier concentration (in space x and time t) of all devices
pp
p
nnn
RGx
J
qt
p
RGx
J
qt
n
1
1
The rates of increase of electrons and holes
in the shaded volume:
… (detailed derivation pp.165-6)
p
ppp
n
nnn
pG
x
p
dx
pdD
t
p
nG
x
n
dx
ndD
t
n
t
t
2
2
2
2
Continuity equations for electrons & holes
… C3, #37,30
… C4, #16
a continuity equation: a partial differential equation which
gives a relation between the amount of the quantity and the
"transport" of that quantity. It states that the amount of the
conserved quantity at a point or within a volume can only
change by the amount of the quantity which flows in or out of
the volume. "Quantity" which can flow or move, such as mass,
energy, electric charge, momentum, number of molecules, etc.
(electrodynamics, hydro-, fluid dynamics...) (source: Wiki)
4. Carrier transport 2102385 19
Solutions: 1. t | 2. x
Drift Diffusion Current densities Einstein relation Continuity equations
4.5.1
0 2 4 6 80.0
0.2
0.4
0.6
0.8
1.0
n
(G
opt n
)
t/tn
0 2 4 6 8 10 12 14 16
t/tn
( ) ( )
0.1
0.9
t = t0
nt
nop eGtntt /
1)(A.
ntt
nop eGtntt /)( 0)(B.
A B
A & B (material parameters) can be used to predict rise & fall times (device/circuit parameters).
n
nnn
nG
x
n
dx
ndD
t
n
t
2
2
Case 1. uniform illumination of a semiconductor (no bias): on @ t = 0, off @ t = t0
Determine n(t), p(t)
@ A Gop
@ B 0“uniform”
(in space, x)
4. Carrier transport 2102385 20
Solutions: 1. t | 2. x
Drift Diffusion Current densities Einstein relation Continuity equations
4.5.2
The equation can be used to design layer thickness for optical absorption (solar cells, photocells...)
0 1 2 3 40.0
0.2
0.4
0.6
0.8
1.0
t/tn
p
/p
x/Lp
0 1 2 3 40.0
0.2
0.4
0.6
0.8
1.0x = 0
t/tn
p
/p
( ) ( ) = 2
0.25 = 1
x
x = 0
22
2
22
2
ppp
nnn
L
p
D
p
dx
pd
L
n
D
n
dx
nd
t
t
pp
nn
LxLx
LxLx
eCeCxp
eCeCxn/
2
/
1
/
2
/
1
)(
)(
nnn DL t ppp DL t
pLxpexp
/)(
Diffusion equations
Diffusion length:
Lp = (for the whole) average distance the number of holes reduced to 1/e
Lp = (for a hole) average distance a hole diffuses before recombining
general solutions
Case 2.1 steady illumination of a semiconductor (no bias) at one surface
Case 2.2 steady injection of carriers in biased p-n junctions
Determine carrier distribution: n(x), p(x)
n
nnn
nG
x
n
dx
ndD
t
n
t
2
2
“steady”(in time, t)
no bias- complete absorption @ surface- no carriers generated inside bulk
specific solution
(C1 = 0)
4. Carrier transport 2102385 21
Drift Diffusion Current densities Einstein relation Continuity equations
(…) r high 1. t | 2. x
Conclusions
1. Drift of carriers under electric field
• Mobility is a function of doping concentration N and type (min or maj carrier), T
• Resistivity r and conductivity also functions of N, T
• At high fields, velocity saturates at vs and at very high fields material breaks down by impact ionization (II)
2. Diffusion of carriers under concentration gradient (due to doping or uneven excitation)
3. Current density
4. Einstein relation:
5. Continuity equations:
• t-dependent solution (rise/fall time – pulse excitation)
• x-dependent solution (diffusion length – uniform, steady state excitation)
he pene
driftJ
dx
xdpqD
dx
xdnqDqJ pndiff
)(,
)(
diffdrifttotal JJJ
q
kTD
tDL