Chapter5 carrier transport phenomena

solid states physics

1. Microelectronics I
Chapter 5: Carrier Transport
Phenomena
Transport; the process by which charged particles (electrons
and holes) move

2. Microelectronics I
Understanding of electrical properties ( I-V characteristics)
Basic current equation;
En eI
⋅
⋅
⋅
∝
μ
e; electronic charged (constant, 1.6 x 10-19 C)
u; mobility ( figure of merit that reflect the spee d)
n; carrier concentration
E; Electric field
MCarrier concentration (electron, n
oand hole, p
o)
iChapter 4
MCarrier transport (current)
iThis chapter
E; Electric field

3. Microelectronics I
Carrier Transport
“Drift”
The movement of carrier due
to electric field (E)
“Diffusion”
The flow of carrier due to density
gradients (dn/dx)
divider
electron
V
+
-
E
electron
divider
electron

4. Microelectronics I
5.1 Carrier Drift
Drift current density
Consider a positively charged hole,
MWhen electric field, E, is applied, the hole accele rates
eE am F
p
= =
*
m*p; effective mass of hole, a; acceleration, e; el ectronic charge
MHowever, hole collides with ionized impurity atoms and with thermally vibrating
lattice atom
hole
Lattice atom
Ionized impurity atom
E

5. Microelectronics I
hole
Lattice atom
Ionized impurity atom
E
Holes accelerates
due to E Involves in collision
(“Scattering Process”)
cLoses most of energy
Gain average drift velocity, v
dp
E v
p dp
μ
=
µ
p; Hole mobility (unit; cm
2
/Vs)
Describes how well a carrier move due to E

6. Microelectronics I
Drift current density, J
drf(unit; A/cm
2
) due to hole
dp drf p
epv J
=
|
pE e J
p drf p
μ
=
|
Drift current density due to electron
nE e J
n drf n
μ
=
|
Total drift current;
Ep n e J
p n drf
) (
μ
μ
+
=
The sum of the individual electron and hole drift c urrent densities

7. Microelectronics I
Mobility effects
*
p
cp
p
m
e
τ
μ
=
Mobility is important parameter to determine the co nductivity of material
*
n
cn
n
m
e
τ
μ
=
τ; mean time between collisions If τ=10
-15
s, in average, every 10
-15
s, carrier involves in collision @ scattering
Two dominant scattering mechanism
1. Phonon or lattice scattering
2. Ionized scattering

8. Microelectronics I
1. Lattice scattering or phonon scattering
At temperature, T > 0 K, atoms randomly vibrate. Th is thermal vibrations cause a
disruption of the periodic potential function. This resulting in an interaction
between carrier and the vibrating lattice atoms.
Mobility due to lattice scattering, µ
L
2
/
3
−
∝
T
μ
2
/
3
−
∝
T
L
μ
As temperature decreases, the probability of a scat tering event decreases. Thus,
mobility increases

9. Microelectronics I
electron
hole

10. Microelectronics I
2. Ionized Ion scattering
Coulomb interaction between carriers and ionized im purities produces scattering
or collusion. This alter the velocity characteristi cs of the carriers.
Mobility due to ionized ion scattering, µ
I
L
N
T
2/3
∝
μ
Total ionized impurity concentration
I
N
Total ionized impurity concentration
MIf temperature increases, the random thermal veloci ty of a carrier increases,
reducing the time the carrier spends in the vicinit y of the ionized impurity center.
This causes the scattering effect decreases and mob ility increases.
MIf the number of ionized impurity centers increase s, then the probability of a
carrier encountering an ionized impurity centers in creases, thus reducing
mobility

11. Microelectronics I

12. Microelectronics I
The net mobility is given by
I L
μ μ μ1 1 1
+ =
Due to phonon scatteringDue to ionized ion scattering

13. Microelectronics I
Conductivity
E Ep n e J
p n drf
σ
μ
μ
=
+
=
) (
Drift current σ; conductivity [(i.cm)
-1
]
) (p n e
p n
μ
μ
σ
+
=
electron
hole
Function of electron and hole concentrations and mo bolities
Ρ; resistivity [i.cm
]
) (
1 1
p n e
p n
μ μ σ
ρ
+
= =

14. Microelectronics I

15. Microelectronics I
L
+
-V
I
Area, A
Bar of semiconductor
I
V
Current density,
AI
J=
Electric field,
LV
E=
IR I
A
L
I
A
L
V
L
V
A
I
E
J
=





=





=
=
=
ρ
σ
σ
σ
Resistance, R is a function of resistivity, or
conductivity, as well as the geometry of the
semiconductor

16. Microelectronics I
Consider p-type semiconductor with an acceptor doping N
a(N
d=0) in which N
a>>n
i
p e p n e
n p n
μ
μ
μ
σ
≈
+
=
) (
Assume complete ionization
ρ
μ σ1
≈ ≈
a n
Ne
Function of the majority carrier

17. Microelectronics I
ex.; Consider compensated n-type Silicon at T=300 K with a co nductivity of σ=16
(ccm)
-1
and an acceptor doping concentration of 10
17
cm
-3
. Determine the donor
concentration and the electron mobility.
Solution;
At T=300 K, we can assume complete ionization. (N
d-N
a>>n
i)
) 10 () 10 6.1( 16
) (
17 19
− × =
− = ≈
−
d n
a d n n
N
N N ene
μ
μ μ σ
To determine µn and Nd, we can use figure mobility vs. impurity concentration with
trial and error
) 10 ( 10
17 20
− =
d n
N
μ

18. Microelectronics I
rIf N
d=2 x 10
17
cm
-3
, impurity
concentration, N
I= N
d
+
+N
a
-
=3 x 10
17
cm
-3
. from the figure, µn= 510
cm
2
/Vs. so σ=8.16 (ccm)
-1
.
rIf N
d=5 x 10
17
cm
-3
, impurity
concentration, N
I= N
d
+
+N
a
-
=6x 10
17
cm
-3
. from the figure, µn= 325
cm
2
/Vs. so
σ
=20.8 (
c
cm)
-
1
.
cm
2
/Vs. so
σ
=20.8 (
c
cm)
-
1
.
N
dshould be between 2 x 10
17
and 5 x
10
17
cm
-3
. after trial and error.
Nd= 3.5 x 10
17
cm
-3
µn=400 cm
2
/Vs
σ= 16 (ccm)
-1

19. Microelectronics I
Ex 2.; designing a semiconductor resistor with a specified resistance to handle a given current density A Si semiconductor at T=300 K is initially doped with d onors at a concentration of
N
d=5 x 10
15
cm
-3
. Acceptors are to be added to form a compensated p-type
material. The resistor is to have a resistance of 10 k cand handle a current
density of 50 A/cm
2
when 5 V is applied.
Solution;
When 5 V is applied to 10 kcresistor, the current, I
mA
RV
I5.0
10
5
4
= = =
If the current density, J is limited to 50 A/cm
2
, the cross-sectional area, A is
2 5
3
10
50
10 5.0
cm
JI
A
−
−
=
×
= =

20. Microelectronics I
Consider that electric field, E is limited to 100 V/cm. Then the length of the
resistor, L is
The conductivity, σof the semiconductor is
cm
EV
L
2
10 5
100
5
−
×= = =
1
5 4
2
) (5.0
10
10
10 5
−
−
−
Ω =
×
×
= =cm
RA
L
σ
The conductivity of the compensated p-type semiconductor is
) (
d a p p
N N e p e
−
=
≈
μ
μ
σ
Here, the mobility is function of total ionized impuri ty concentration Na+Nd

21. Microelectronics I
Using trial and error, if Na=1.25x10
16
cm
-3
, then Na+Nd=1.75x10
16
cm
-3
, and the
hole mobility, from figure mobility versus impurity con centration, is approximately
µp=410 cm
2
/Vs. The conductivity is then,
492.0 ) 10 )5 5. 12(( 410 10 6.1 ) (
15 19
= × − × × × = − =
−
d a p
N N e
μ σ
This is very close to the value we need. From the calculati on
L=5x10
-2
cm
A=10
-5
cm
2
Na=1.25x10
16
cm
-3

22. Microelectronics I
Velocity Saturation
E v
d
μ
=
Drift velocity increase linearly with applied electric fi eld.
rAt low electric field,
vd increase linearly
with applied E. with applied E. slope=mobility
rAt high electric field,
vd saturates
cConstant value

23. Microelectronics I
Carrier diffusion
Diffusion; process whereby particles from a region of high concentrat ion toward
a region of low concentration.
divider
Carrier
Electron concentration, n
Position x
Electron diffusion
current density
Electron flux
dxdn
eD J
dx
dn
De J
n dif nx
n dif nx
=






− −=
|
|
)(
D
n; electron diffusion coefficient

24. Microelectronics I
Hole centration, p
Hole diffusion current density
Hole flux
dxdp
eD J
dx
dp
eD J
p dif px
p dif px
−=






− =
|
|
Hole centration, p
Position x
current density
D
p; hole diffusion coefficient
Diffusion coefficient; indicates how well carrier move as a result of
density gradient.

25. Microelectronics I
Total Current Density
Total Current
Density
Electron drift
current
hole drift
current
Electron diffusion
current
hole diffusion
current
dif px drf p dif nx drf n
J J J J J
| | | |
+
+
+
=
dx
dp
eD
dx
dn
eD E ep E en J
p n x p x n
− + + =
μ μ
1-D
3-D
p eD n eD E ep E en J
p n p n
∇
−
∇
+
+
=
μ
μ

26. Microelectronics I
Mobility,µ; indicates how well carrier moves due to electrical field
Diffusion coefficient, D; how well carrier moves due to density gradient
Here, we derive
relationship between mobility and diffusion
coefficient
using nonuniformly doped semiconductor model
“Einstein relation”
Graded impurity distribution
nonuniformly doped semiconductor
electron
x
E
C
E
F
E
v
x
Energy-band diagram

27. Microelectronics I
E
C
E
F
E
v
x
MDoping concentration decreases as x increases
MElectron diffuse in +x direction
MThe flow of electron leaves behind positively
charged donor
Induce electrical field, E
x, given by
x
dN
kT
E
d
)
(
1


−
=
…eq.1
dx
x
dN
x N e
kT
E
d
d
x
)
(
)(
1


−
=
Since there are no electrical connections, there is no curr ent(J=0)
0
)(
)(= + =
dx
x dN
eD Ex Ne J
d
n x d n
μ
…eq.1
…eq.2 Electron current

28. Microelectronics I
From eq.1 and 2,
e
kT D
n
n
=
μ
Hole current must also be zero. We can show that
e
kT D
p
=
μ
e
p
μ
e
kT D D
p
p
n
n
= =
μ μ
Diffusion coefficient and mobility are not independent parameters.
The relationship between this 2 parameterc“Einstein relation”

29. Microelectronics I
The Hall effect Using the effect, we can determine
MThe type of semiconductor
MCarrier concentration
Mmobility
Magnetic field
Applied electrical field
Force on charged particle
in magnetic field (“Lorentz
force”)
B qv F
×
=

30. Microelectronics I
Mthe Lorentz force on electron
and hole is in –y direction
MThere will be buildup of negative
charge (n-type) or positive charge
(p-type) at y=0
MAs a results, an electrical field
called
“Hall field, E
H
”
is induced.
called
“Hall field, E
H
”
is induced.
Hall field produces
“Hall voltage,
V
H”
In y-direction,
Lorentz force will be balanced by force due to Hall field
z x H
H
z x
WBv V
W
V
q B qv
=
= ×
(p-type)
Polarity of V
His used to determine the type of semiconductor

31. Microelectronics I
For p-type
) )( (Wd ep
I
v
x
x
=
d
eV
BI
p
epd
BI
V
H
z x
z x
H
=
=
Can calculate the hole concentration in p-type
d
eV
H
For n-type
d eV
BI
n
end
BI
V
H
z x
z x
H
−=
−=
Note that V
His negative for n-type

32. Microelectronics I
When we know the carrier concentration, we can calculate ca rrier mobility
x p x
E ep J
μ
=
Wd epV
LI
L
E ep
Wd
I
x
x
p
x p x
=
=
μ
μ
Similar with n-type, mobility is determined from
Wd enV
LI
x
x
n
=
μ