Institute of Solid State Physics
Technische Universität Graz
Extrinsic semiconductors
The introduction of impurity atoms that can add electrons or holes
is called doping.
n-type : donor atoms contribute electrons to the conduction band.
Examples: P, As in Si.
p-type : acceptor atoms contribute holes to the valence band.
Examples: B, Ga, Al in Si.
Institute of Solid State Physics
Technische Universität Graz
n and p
The electron density and hole density are:
 EF − Ec 
n = N c exp 

 k BT 
 Ev − EF 
p = N v exp 

k
T
 B

The law of mass action:
 Eg 
np
= n= N v N c exp  −

k
T
 B 
2
i
Temperature dependence
k BT ≈ E D
200 K
1000 K
n ≈ ND
ni
 Eg 
N v N c exp  −

 k BT 
n-type
n-type ND > NA, p ~ 0
 EF − Ec 
n N=
N c exp 
=

D
k
T
 B

 Nc 
E=
Ec − k BT ln 

F
N
 D
For n-type, n ~ density of donors,
p = ni2 /n
p-type
p-type NA > ND, n ~ 0
 Ev − EF 
=
p N=
N v exp 

A
 k BT 
 Nv 
E=
Ev + k BT ln 

F
N
 A
For p-type, p ~ density of acceptors,
n = ni2/p
Intrinsic / Extrinsic
Intrinsic: n = p
Conductivity strongly temperature dependent near room
temperature
Extrinsic: n ≠ p
Conductivity almost temperature independent at room
temperature
Intrinsic semiconductors
Extrinsic semiconductors
intrinsic
extrinsic
log10(ni) cm-3
Ge
Si
freeze-out
GaAs
1/T
ni
300 K
 Eg 
N v N c exp  −

2
k
T

B 
At high temperatures, extrinsic
semiconductors have the same
temperature dependence as intrinsic
semiconductors.
Why dope with donors AND acceptors?
Bipolar transistor
collector
n
base
p
emitter
n+
lightly doped p substrate
MOSFET
Bipolar Junction Transistor
Ionized donors and acceptors
For Ev + 3kBT < EF < Ec- 3kBT
N D+ =
ND
 EF − ED 
1 + 2 exp 

k
T


B
N A− =
Boltzmann approximation
NA
 E A − EF 
1 + 4 exp 

k
T


B
4 for materials with light
holes and heavy holes (Si)
2 otherwise
ND = donor density cm-3
NA = donor density cm-3
ND+ = ionized donor density cm-3
NA- = ionized donor density cm-3
Mostly, ND+ = ND and NA- = NA
Charge neutrality
Carrier concentration vs. Fermi energy
n + NA- = p + ND+
Calculating EF(T) numerically
http://lamp.tu-graz.ac.at/~hadley/psd/L4/eftplot.html
degenerately doped semiconductor
Degenerate semiconductor
Heavily doped semiconductors are called degenerately doped
ND > 0.1 Nc -> EF in the conduction band
NA > 0.1 Nv -> EF in the valence band
Heavy doping narrows the band gap
The Boltzmann approximation is not valid
Degenerate semiconductors = metal
Technische Universität Graz
Institute of Solid State Physics
Carrier Transport
Ballistic transport
Drift
Diffusion
Generation and recombination
The continuity equation
High field effects
Ballistic transport



dv

F=
ma =
−eE =
m
dt

 −eEt 
v
=
+ v0
m
2
 −eEt


x
=
+ v0t + x0
2m
Electrons moving in an electric field follow parabolic
trajectories like a ball in a gravitational field.
electron velocity
The electrons that participate in conduction are moving fast.
Classical equipartition:
1
2
mvth2 = 32 k BT
At 300 K, vth = 107 cm/s.
Time between scattering events tsc
mean free path:  = vthτsc ~ 10 nm ~ 200 atoms
Drift (diffusive transport)



*
* dv
F=
−eE =
ma=
m
dt

  eE
v =−
v0
(t − t0 )
*
m
<v0> = 0
t0
t+t0
v0
v
<t - t0> = τsc < average time between scattering events


 −eEτ sc −eE
=
vd =
m*
m*v
drift velocity:
time between two collisions
v - v0
t0 τsc t
vd ,n =
− µn E
vd , p =
µpE
Drift
drift velocity:
vd ,n =
− µn E
vd , p =
µpE





j=
−nevd ,n + pevd , p =
σE
( neµn + peµ p ) E =
=
µ
for Si:
−eτ sc −e
=
m*
m*v
µn = 1500 cm2/Vs
µp = 450 cm2/Vs
For E = 1000 V/cm
vd = 106 cm/s
Drift
Solid state electronic devices, Streetman and Banerjee
 


vd ,n =
− µn E vd , p =
µpE





j=
−nevd ,n + pevd , p =
ne
µ
+
pe
µ
E
=
σ
E
( n
p)
Matthiessen's rule
1
=
τ sc
1
τ sc ,lattice
+
1
τ sc ,impurity
mostly temperature independent
phonons, temperature dependent
1
=
µ
1
µlattice
1
σ= = neµn + peµ p
ρ
+
1
µimpurity
doping increases the conductivity
by increasing the carrier density
but decreases the mobility
Mobility
calculator
http://www.pvlighthouse.com.au/calculators/mobility%20calculator/mobility%20calculator.aspx
Resistivity
calculator
http://www.pvlighthouse.com.au/calculators/Resistivity%20calculator/Resistivity%20calculator.aspx
Crossed E and B fields
B
E
Ballistic transport

  

F =ma =−e E + v × B
(
)
<v>
E
Diffusive transport
Hall angle:
<v>
θH
 eBzτ sc 
=
θ H tan  − * 
m 

−1
Magnetic field (diffusive transport)



vd

−eE =
F=
ma =
e
µ




v


F=
ma =
−e E + vd × B =
e d
(
)
µ
If B is in the z-direction, the three components of the force are
− µ ( Ex + vdy Bz ) =
vdx
− µ ( E y − vdx Bz ) =
vdy
− µ Ez =
vdz
Magnetic field
vd , x =
− µ Ex − µ Bz vd , y
− µ E y + µ Bz vd , x
vd , y =
vd , z = − µ Ez
If Ey = 0,
vd , y = − µ Bz vd , x
tan θ H = − µ Bz
The Hall Effect (diffusive regime)
− µ Ex − µ Bz vd , y
vd , x =
vd , y =
− µ E y + µ Bz vd , x
vd , z = − µ Ez
If vd,y = 0,
Ey = vxBz = VH/W = RH jxBz
VH = Hall voltage, RH = Hall Constant
The Hall Effect (diffusive regime)

 
F q v×B
=
(
Ey = vxBz = VH/W = RH jxBz
)
VH = Hall voltage, RH = Hall Constant
jx = I /A
vx = - jx/ne
for n-type
vx = jx/pe for p-type
RH = - 1/ne
for n-type
RH = 1/pe
for p-type