Chapter 2.
Basic Semiconductor Physics
9/12/2008
1
2.1 Introduction
• Si, GaAs, InP: the published experimental and
theoretical values of certain parameters differ
considerably from reference to reference.
• Even worse for less matured materials (e.g., SiC
or GaN), alloys and strained layers.
• The 11 basic semiconductors: the cubic crystal
type (Ge, Si, GaAs, AlAs, InAs, InP, GaP), the
hexagonal (wurtzite) crystal type (SiC (4H & 6H),
GaN, AlN).
• A linear interpolation scheme used for
approximating material parameters unavailable.
2
• If the material parameters are known for 2 basic binary
compounds AC & BC, the composition-dependent material
parameter TAxB1-xC(x) of the ternary compound (AxB1-xC)
can be estimated by
TAx B1xC ( x)  xBAC  (1  x) BBC
• For a quaternary alloy of composition AxB1-xCyD1-y,
QAx B1xCy D1y ( x, y)  xyBAC  x(1  y)BAD  (1  x) yBBC  (1  x)(1  y)BBD
• This method is valid only when reliable data for the alloy
of interest are not available.
• Although numerical device simulation accounts for more
physical effects and is often more accurate, analytical
approaches nonetheless enjoy great popularity.
3
2.2 Free-Carrier Densities
2.2.1 Band diagrams and Band structure
• Carrier density=the number of carriers/cm3
• Under the thermal equilibrium condition, heat is
only energy applied to the semiconductor.
• If other types of energy are applied, the
equilibrium may be perturbed.  nonequilibrium
• EG = EC – EV (Energy state ≈ Energy level)
4
5
• Generation: the electron is lifted to the conduction
band, leaving behind an unoccupied state in the
valence band.
• Recombination: a free electron falls down from
the conduction band to the valence band, thereby
releasing energy (≥ the bandgap) and filling the
hole.
• At T=0 K, all allowed states below EF are filled
with electrons and all states above EF are empty.
• The density of electrons and holes can also be
increased by intentionally incorporating impurities
(dopants) into the semiconductor.
6
• Energy band structure
7
2.2.2 Carrier Statistics
• The density of states (the density of allowed
energy states) in the conduction band, N(E):
3/ 2
 2mn,ds 
N ( E )  4  2  ( E  Ec )1/ 2
 h 
mn,ds: the density-of-states effective electron mass.
• The effective density of states in the conduction
band, NC, defined as
 2mn,ds k BT 

N C  2
2
h


• Therefore,
3/ 2
1/ 2
 E  EC 
2

N (E) 
N C 
  k BT 
1
k BT
8
• The probability that E is occupied by an electron
is governed by the Fermi-Dirac distribution
function f(E): f ( E )  1
(2-6)
1 e
 E  EF

 k BT



At E = EF, f(E) is ½.
• The density of electrons at a certain energy: the
density of states × the occupation probability =

(2-7)
n0   N ( E ) f ( E )dE
EC
1/ 2
2 N C  E  EF 
1





EC
k
T

 B  1  exp  E  EF

 k BT

1
dE
 k BT



9
E  EC
• Changing the variable from E to x with x 
k BT

n0 
2 NC



0
x

E  EC
1  exp  x  F
k BT




dx
(2-10)
Fermi integral of the order ½, which
unfortunately cannot be solved analytically.
• Similarly, the hole density in valence band:
1/ 2
 EV  E 
2
p( E ) 
NV 

k
T

 B

 2 mh ,ds k BT 
NV  2 

2
h


1
k BT
3/ 2
10
• The probability that the energy state is empty, i.e.,
not occupied by and electron:
fh (E)  1  f (E)  1 
1
E  EF
1  exp
k BT
• After some algebraic manipulations,
p0 
2 NV



0
x

EV  EF
1  exp  x 
k BT

EV  E
dx where x 
k BT



(2-14)
11
2.2.3 Approximations for the Carrier Densities
• Boltzmann Statistics
– For E-EF>3kBT, the exponential term in (2-6) is >>1,
 E  EF 
(2-16)
f ( E )  exp  

k
T
B


– Similarly, for EF-E>3kBT
 E E
f h ( E )  exp   F

k
T
B


– f(E) and fh(E) are called the Boltzmann distribution
functions. (2-5)&(2-16)(2-7):
 E E 
n0  N C exp   C F 
k BT 

(2-18)
 E E 
p0  NV exp   F V 
k BT 

12
– Semiconductors with above conditions are
nondegenerate.
– For degenerate semiconductors (n-type with E-EF< 3kBT
and p-type with EF-E<3kBT), the Fermi-Dirac statistics
should be used.
– Nondegenerate and degenerate semiconductors
correspond to lightly and heavily doped materials,
respectively.
– The electron density = the hole density in an intrinsic
semiconductor
 E E 
n0  ni  NC exp   C F 
k BT 

 E E 
p0  ni  NV exp   F V 
k BT 

13
14
– Solving the above equations for NC and NV, inserting
the results into (2-18),
E E 
n0  ni exp  F i 
 k BT 
E E 
p0  ni exp  i F 
 k BT 
– Under thermal equilibrium  mass action law
 E 
n0 p0  ni2  NC NV exp   G 
 kBT 
– The charge neutrality is maintained in any
semiconductor device regardless of the doping.
(Electrons and holes are mobile charges, whereas
ionized dopants are fixed.)
p  N D  n  N A
15
• Complete Dopant Ionization-Approximation 1
– Assume all donors are ionized in an n-type material and
the dopant concentration>>ni  n0 ≈ ND,
ni2
p0 
– Similarly, for a p-type, p0 ≈ NA,
ND
ni2
n0 
NA
• Complete Dopant Ionization-Approximation 2
– Low doping or a large intrinsic carrier concentration
– In an n-type material,
ni2
n0  N D  p0  N D 
n0
n0 
ND 
N D2  4ni2
, p0 
2
(2-30)
N D2  4ni2  N D
2
16
– Similarly, for a p-type,
N A  N A2  4ni2
p0 
, n0 
2
N A2  4ni2  N A
2
• Incomplete Dopant Ionization
– The dopant atoms are not 100% ionized.
– ND>ND+ in n-type, NA>NA- in p-type.
– This occurs either at low temperature or when the
dopant’s activation energy is high.
– At room temperature, complete ionization can be
assumed for most semiconductors, e.g., Si, GaAs, and
InGaAs.
– In SiC, EDA and EAA are larger than those in the abovementioned materials, and at room temperature only a
portion of the dopants are ionized.
17
– For n-type, the density of ionized donors with EDA and
ED is described using the Fermi-like distribution
function:
1
fD 
 ED  EF 
1
1  exp 

g
k
T
B


ground state degeneracy factor g is added.
– The density of ionized donors (i.e., the density of donor
states not occupied by an electron) can be:




1

N D  N D (1  f D )  N D 1 

 ED  EF  
1
 1  exp 
 
g
 k BT  

(2-37)
18
– Assuming the electron density ≈ the density of ionized
1
donors, (2-37) n  N
(2-38)
0
D
– Rearranging (2-18)
 E  ED 
1  g exp  F

k
T
B


EF  k BT ln
n0
 EC
NC
– Insert this into (2-38) and solving a quadratic equation
 E  ED
N
n0  C exp   C
2g
k BT



EC  ED
ND
exp
 1
  1  4 g

NC
k BT


– The density of ionized acceptors: N A  N A f A
– The probability fA of an acceptor state occupied by an
electron can be:
1
fA 
 E  EF 
1  g exp  A

k
T
B


19
– Finally, using (2-19)
 E  EV
N
p0  V exp   A
2g
k BT



E A  EV
NA
exp
 1
  1  4 g

NV
k BT


• Joyce-Dixon Approximation
 n
EF  EC
 ln  0
k BT
 NC

 n0

a
 1

 NC
2
3
4
2
3
4

 n0 
 n0 
 n0 

a

a

a




2
3
4
N
N
N

 C
 C
 C
 p0 
 p0 
 p0 
 p0 
 p0 
EV  EF
 ln 
  a1 
  a2 
  a3 
  a4 

k BT
N
N
N
N
N
 V 
 V 
 V 
 V 
 V 
20
21
2.3 Carrier Transport
2.3.1 Introduction
• 2 driving forces for carrier transport: electric field and
spatial variation of the carrier concentration.
• Both driving forces lead to a directional motion of carriers
superimposed on the random thermal motion.
• To calculate the directional carrier motion and the currents
in a semiconductor, classical & nonclassical models can be
used.
• The classical models assume that variation of E-field in
time is sufficiently slow so that the transport properties of
carriers (mobility or diffusivity) can follow the changes of
the field immediately.
22
• If carriers are exposed to a fast-varying field, they
may not be able to adjust their transport properties
instantaneously to variations of the field, and
carrier mobility and diffusivity may be different
from their steady-state values  nonstationary
• Nonstationary carrier transport can occur in
electron devices under both dc and ac bias
conditions.
• Whether the field acting on the carrier is varying
in time or nor is important.
• In III-V transistors, nonstationary transport plays
an important role, but is much less important in Si
transistors. In SiC transistors, it is entirely
neglected.
23
2.3.2 Classical Description of Carrier Transport
A. Carrier Drift.
• Assume thermal equilibrium for a semiconductor having a
spatially homogeneous carrier concentration with no
applied E-field. No driving force for directional carrier
motion. The carriers not in standstill condition but in
continuous motion due to kinetic energy. For electron in
the conduction band, E  32 k T  m2 v where vth is the thermal
velocity, mn* is the conductivity effective electron mass.
• The average time btw 2 scattering events is the mean free
time and the average distance a carrier travels btw
collisions is the mean free path.  Fig. 2.5 (a)
• Applying V, the E-fields adds a directional component to
the random motion of the electron.  Fig. 2.5 (b)
*
n
kin
B
2
th
24
• The mean electron velocity: vn= -μnE
• The directed unilateral motion of carriers caused by E-field
is drift velocity.
• Similarly, vp = μpE
• A change in E-field instantaneously results in a change of
the drift velocity.
25
• Band diagram: qV=ΔEC= ΔEV= ΔEi= ΔEF
26
B. Low-Field Carrier Drift.
• The drift velocity is linearly dependent on the field.
• The low-field mobilities μ0n and μ0p depend on both the
doping concentration and on the temperature.
• The dependence of the low-field mobility is modeled
empirically      max   min
0
min
 N
1  
 N ref




27
• The mobility of minority
carriers can be
considerably higher than
that of majority carriers.
• The electron low-field
mobilities for Si and GaAs
in Fig 2.7
• The low-field electron and
hole mobilities of different
undoped semiconductors in
Fig 2.8
28
29
• Temperature-dependent low-field mobility
 T 
 0 (T )   0 (300 K )  

300
K


n
30
C. High-Field Carrier Drift
• v-E characteristics for Si and for semiconductors
with similar v-E characteristics can be modeled by
the empirical expression
1/ 






1
v(E )  0 E 
 



E
0
 1

  vsat  
 
 
• The saturation velocity vsat is temperaturedependent and decreases with increasing
temperature. For electrons and holes in Si,
vsat
2.4 107

cm / s
 T 
1  0.8 exp 

 600 K 
31
32
• In GaAs or InP, not only can a sublinear slope of the v-E
characteristics be observed, but the velocity actually
decreases after reaching a peak value at a certain critical
field and approaches asymptotically to a saturation value.
• Fig 2.11 illustrates the electron population in the lower and
upper valleys and the stationary v-E characteristics for
GaAs.
33
34
D. Drift Current Densities
• The drift current density JDr is described by the product of
the density, the charge, and the drift velocity of the drifting
carriers.
J Dr,n  qnvn  qnn (E )E
J Dr, p  qnv p  qn p (E )E
• Under low-field conditions where v = μ0E holds,
J Dr,n  qn0 nE
J Dr, p  qn0 pE
• The total drift current density JDr = JDr,n +JDr,p.
35
E. Carrier Diffusion and Diffusion Current Densities
• In a nonuniform carrier distribution, carriers tend to move
from higher carrier concentration to lower carrier
concentration.  diffusion
• At x = x0, the probability that an electron moves to the left
is the same as the probability that it moves to the right.
• The average rate P1 of electrons flowing from x1 toward x0
1
and crossing the plane x = x0 is
P1  n1vth
2
36
• Approximating the concentration at x1, P  1 v n  dn x 
1
th 0
dx
2
• Analogously, the average rate of electrons flowing from x-1
toward x0 is P  1 v n  dn x 
1
2
th
0
dx
• The net rate P of electrons crossing the plane at x0 in +x
direction is the difference btw P-1 and P1,

P  vth  dn
dx x
• Introducing the electron diffusion coefficient Dn=vthΔx,
P   Dn
dn
dx
• The electron diffusion current density JDi,n can be obtained
from the net rate of electron diffusion by multiplying P
with the charge of an electron, i.e., by –q, as
J Di,n  qDn
dn
dx
37
• Similarly,
J Di, p  qD p
dp
dx
• JDi = JDi,n + JDi,p
• Driving force for a diffusion current is the gradient of the
carrier concentration, whereas the driving force for a drift
current is the electric field.
• The main difference btw these driving forces is that the
field acts directly on the carriers and give rise to a
directional motion of every carrier in the sample, whereas
an individual carrier does not feel any force from a carrier
concentration gradient.
• A relationship btw mobility and diffusion coefficient 
Einstein relation:
(2-70)
k BT
k BT
Dn 
n , Dp 
p
q
q
38
F. Alternative Expressions for the Current Density
J  J Dr,n  J Di,n  J Dr, p  J Di, p
J n  J Dr,n  J Di,n  qnnE  qDn
• Since
dn
dx
d
E   , EF  E  q
dx
• Similarly,
J p  p p
dEF
dx
,
J n  qnnE  qDn
dn
dx
 E  E 
d 
 d 
 qnn     qDn  ni exp  F i  
dx 
 dx 
 k BT  
 n n
dEF
dx
• Therefore, the general driving force for a current is the
gradient of the Fermi level, and it is impossible to identify
from whether this driving force is caused by a field or a
gradient of the carrier concentration.
39
2.3.3 Nonclassical Description of Carrier Transport
• The basic assumption of the classical description of carrier
drift  a sudden change of the field results in an
immediate change of carrier drift velocity: Not correct!!
• According to classical mechanics, a particle with a certain
mass cannot change its velocity instantaneously because of
inertia, even if the driving force for the motion does.
• Carriers possess a certain mass (the effective mass) and
need a certain transition time to change their kinetic energy
and their velocity after a sudden variation of the field.
• The behavior of carriers during the transition period is
called carrier dynamics or nonstationary carrier transport.
• Such a transport was first investigated using Monte Carlo
Simulation for Si and GaAs.
40
• It can also be described easily by the relaxation time
approximation (RTA). The heart of the RTA is the energy
and momentum balance equations, for a homogeneous
semiconductor: dE
E  E0
d (m * v )
m*v
 qEv 
,
 qE 
dt
E
dt
p
where
τE and τp: the energy-dependent effective relaxation times
for carrier energy and momentum,
E0: the carrier energy at thermal equilibrium (E = 0),
m*: the conductivity effective electron mass (the effective
mass related to transport phenomena),
m*×v : momentum P of the carrier.
• The balance btw the amount of energy and momentum that
carriers gained from the applied field, and the energy and
momentum loss caused by scattering. The effects of all
different scattering events are combined in τE and τp
41
• Stationary v-E, E-E, and m*-E are needed to simulate
nonstationary transport.
• Let us consider an n-type GaAs sample, at time t, the
electron possess E(t), P(t) and m*(t) and travel with v(t).
Now assume that during the small time interval Δt after t
the applied field changes from E(t) to E(t+Δt). The
evolution of the electron velocity:
– Calculate τE(t+Δt) and τP(t+Δt) by assuming stationary conditions
(d/dt=0)
Est (t )  E0
mst* (t )vst
 E (t  t ) 
, p (t  t ) 
qE st (t )vst (t )
qE st (t )
– Calculate the electron energy and momentum at t+Δt. For this, the
balance equations are discretized.

E (t )  E0 

E (t  t )  E (t )  t  qE (t  t )v(t ) 
 E (t  t ) 


P(t ) 

P(t  t )  P(t )  t  qE (t  t ) 

(
t


t
)
P


42
43
– Set Est(t+Δt) = E(t+Δt) and deduce the new stationary
values Est(t+Δt), vst(t+Δt) and mst*(t+Δt) from Fig 2.15
P(t  t )
– Calculate the electron velocity v(t+Δt) from v(t  t )  m* (t  t )
st
– Increase the time by another Δt and proceed with step 1.
44
• Velocity overshoot: when a large field is applied, the
electron velocity reaches a much higher peak than that
predicted by the stationary v-E characteristics.
45
2.4 PN Junctions
46
• In Fig.2.18, the field is in –x direction and is the driving
force for a drift current. This drift tendency is in the
opposite direction of the diffusion tendency. With no bias
applied to the pn junction, the diffusion and drift
tendencies compensate each other, and no net carrier flow
occurs.
• The electron affinity χ is the energy difference btw the
vacuum level and the conduction band edge.
• The energy barrier btw the p- and n-type regions is the
same for the conduction and valence bands  built-in
voltage Vbi of the pn junction.
• Derivation of Vbi: When no external voltage is applied, no
net current flows across the junction.
(2-84)
dn
J  J  J  qEn  qD
0
n
Dr, n
Di, n
n
n
dx
47
• Rearranging (2-84) and applying Einstein relation (2-70)
 qEnn dx  qDn dn  0  qEnn dx  k BT n dn  0
d  Edx 
k BT dn
q n
• Integrating across the space-charge region (-xp ~ xn),
xn
k BT dn k BT n( xn )

ln
 xp q
n
q
n(  x p )
Vbi   d  
 xp
xn
k BT
ND
k BT N D N A

ln 2

ln 2
q
ni / N A
q
ni
• To find the thickness of the space-charge region, using
Poisson equation, where ε = ε0εr
d 2
dE
q

  ( p  n  ND  N A )
2
dx
dx

48
• In the space-charge region on the p-type side, simplified to
d 2 q
 NA
dx 2 
• Applying the boundary condition, i.e., E-field = 0 at x=-xp,
q
(2-89)
E   N (x  x )

A
p
• Similarly, on the n-type side,
q
E  N A ( x  xn )

(2-90)
49
• Integrating (2-89) & (2-90) gives the potential distribution,
using B.C. (φ=0 at x=-xp, φ=Vbi at x=xn),
q
N A ( x  x p ) 2 on the p-type side,
2
q
 ( x)  Vbi  N D ( x  xn ) 2 on the n-type side
2
 ( x) 
• Because the potential in a pn junction is continuous,
 (0)   (0),  (0)   (0)
2Vbi
2Vbi
ND
NA
xp 
, xn 
q N A (ND  N A )
q ND (ND  N A )
d sc  x p  xn 
2Vbi N D  N A
q ND N A
 the thickness of the space-charge region under the
thermal equilibrium (no voltage is applied)
50
• For the case of a voltage applied,
xp 
2 (Vbi  Vpn )
q
2 (Vbi  Vpn )
ND
NA
, xn 
N A (ND  N A )
q
ND (ND  N A )
2 (Vbi  Vpn ) N D  N A
d sc  x p  xn 
q
ND N A
• We next discuss the minority carrier concentrations at the
edges of the space-charge region.
• If no external voltage is applied, n(-xp) = n0p, p(xn) = p0n.
• Under forward-bias condition (Vpn>0), the drift and
diffusion currents no longer compensate each other.
• The extrinsic field caused by the applied voltage is in the
opposite direction compared to the intrinsic field originated
from the uncompensated acceptor and donor ions in s-c
region.
51
• Thus, the resulting field is smaller compared to that of the
equilibrium case, the drift current is decreased, and the
diffusion current becomes the dominant tendency. 
injection of electrons from n-type to p-type, and holes are
injected into n-type region.  minority carrier
concentrations in the s-c region and at the edges of s-c
region are increased.
• Fermi level is no longer a straight line across the pn
junction.  no longer single Fermi level, Eqs. (2-22)&(223) are not valid.  quasi-Fermi levels are introduced. 
split into 2 separate quasi-Fermi levels for Efn and Efp.
n  ni exp
p  ni exp
EFn  Ei
k BT
Ei  EFp
(2-99)
(2-100)
k BT
52
• Multiplying above 2 eqs,
np  n exp
2
i
EFn  EFp
k BT
, p  n exp
2
i
qV pn
kBT
 n exp
2
i
V pn
(2-101)
VT
53
• For a forward bias applied to the pn junction, Eqs.
(2-99)-(2-101) result in increased minority carrier
densities in the entire space-charge region.
• In the p-type bulk region, electrically neutral 
Δn= Δp  even true at the edge of s-c region.
p( x p )  p0 p  n( x p )  n0 p
• In most cases, n0p is much smaller compared to the
other charge components.
p0 p 
4n0 p
Vpn 
n(  x p ) 
exp
 1 since ni2  n0 p p0 p
 1

2 
p0 p
VT

54
• Similarly, for the minority carrier density at x=xn,
Vpn 
n0 n 
4 p0 n
p( xn ) 
exp
 1
 1

2 
n0 n
VT

• For low-level injection, i.e., for small Vpn ((1+x)n≈1+nx)
and low injected minority carrier densities,
V pn
ni2
n( x p )  n0 p exp

exp
VT N A
VT
V pn
V pn
ni2
p( xn )  p0 n exp

exp
VT N D
VT
V pn
• For high-level injection, (large Vpn)
n( x p )  ni exp
p( xn )  ni exp
V pn
2VT
V pn
2VT
55
• At x=-xp, E=0, assuming no recombination taking place,
J  J Di,n ( x p )  J Di, p ( x p )
 J Di,n ( x p )  J Di, p ( xn )
dn
dp
 qDn
 qDp
dx  x p
dx xn
(2-111)
56
• The characteristic lengths Ln and Lp are minority carrier
diffusion lengths for electrons and holes.  This is the
average distance an excess minority carriers can move by
diffusion in a sea of majority carriers before they disappear
by recombination.
• Defined as Ln  Dn n , Lp  Dp p
where Dn and Dp: the minority electron and hole diffusion
coefficients, τn and τp : the minority electron and hole
lifetimes.
• To determine (2-111), assume a 1st order approximation.
J  qDn
n( x p )  n0 p
Ln
 qDp
p0 n  p( xn )
Ln
57
• Assuming complete ionization, (2-27),(2-29), (2-105) and
(2-106),

ni2
ni2  Vpn 
J  q  Dn
 Dp
 1
 exp
N D Ln  
VT 
 N A Ln
 Valid for the long-base junction
• Junctions with quasineutral regions shorter than the
minority carrier diffusion lengths are short-base junctions.


Vpn 
Vpn 

ni2
ni2
J  q  Dn
 Dp
 1  J s  exp
 1
  exp
 N A ( xBp  x p )
N D ( xBn  xn )  
VT
VT




58
2.5 Schottky Junctions
• A rectifying junction formed by a metal and a semiconductor.
• For microwave transistors, the most important Schottky
junction is the junction btw a metal and an n-type
semiconductor serving as a gate electrode in MESFETs and
HEMTs.
59
• In real Schottky junctions, there are charged interface
states located at the metal-semiconductor interface, and the
effect of the interface states on the behavior of the
Schottky junction is typically more significant compared to
that of the work function difference itself.
• Although many theories for the Schottky junction have
been proposed, commonly the Vbi is not calculated using an
expression derived from a physical mode, but rather is
taken from experimental data.
• A quantity closely related to Vbi is the Schottky barrier
height φb. qVbi  b  ( EC  EF )
60
61
• Thickness of the space-charge region: using
Poisson equation, d 2 ( x) qN D
dx
2


– First, consider the zero applied voltage using the
boundary conditions that at x = dsc, E =0, φ = Vbi.
qN D
 ( x)  Vbi 
( x  d sc ) 2
2
– Since φ = 0 at x = 0,
2Vbi
d sc 
qN D
– With Vapp,
d sc 
2 (Vbi  Vapp )
qN D
62
2.6 Impact Ionization
• Carriers in a semiconductor are accelerated and pick up
energy when E-field is applied.
• During the scattering process, they transfer part of the
energy to the lattice.
• If the field is sufficiently high, a carrier can gain so much
energy that during a collision with the lattice it can break a
bond and create a free electron. Due to the energy
transfer, a valence electron is moved to the conduction
band, thereby creating a free electron and a free hole.
• As a result, an electron-hole pair is generated and we have
3 carriers after the scattering event compared to only 1
carrier before the process. Impact ionization
63
• The sum of the energies of the 3 carriers after
scattering = the energy of the original carrier
before scattering.
• The same holds for the momentum.
• After impact ionization, all 3 carriers are
accelerated by the field. If the field remains high
enough, they can create further electron-hole pairs,
which are again accelerated and create more
electron-hole pairs.  An enormous increase in
the total number of carriers is the avalanche
multiplication.
• Avalanche can lead to device breakdown, and, if
the currents are not limited, may destroy the
device.
64
• The voltage at which avalanche becomes noticeable is the
breakdown voltage BV.
• Obviously the minimum kinetic energy a carrier must gain
to cause impact ionization is slightly larger than the
bandgap energy.
• The increase in carrier concentration by impact ionization
is described by the electron-hole pair generation rate GII
defined by
GII   n nvn   p pv p
where αn and αp: the electron and hole ionization rates =
the number of electron-hole pairs generated by an electron
(hole) per unit distance traveled.
65
• Local E-field model proposed by Selbeherr
  bp   p 
  bn n 
 n  an exp     ,  p  a p exp    
  E  
  E  
66
67
68
• 2 conclusions
– The ionization coefficients increase considerably with increasing
field.
– Semiconductors with larger bandgaps show lower ionization
coefficients for a given field.
• Not the E-field but the kinetic energy of traveling electrons
and holes is the main reason for impact ionization. 
Impact ionization model should not be a function of the
local field but rather than the carrier energy.
• It is extremely difficult to obtain realistic field and carrier
energy distributions by analytical means.
• The largest E-fields and the highest probabilities for
impact ionization in microwave transistors occur in the
space-charge regions of pn junction.
69
• In bipolar transistors, the critical region is the c-b
space-charge region, in MOSFETS it is the drainbulk pn junction, in MESFETs and HEMTs, it is
the Schottky junction of the gate near the drain
region. Concerning breakdown, all these junctions
behave like a one-sided abrupt junction.
70
• The breakdown voltage BV can be estimated
 E C2 1
BV 
2q N
where N is the ionized background concentration
in the lightly doped side of the pn junction or in
the semiconductor side of the Schottky junction.
71
• The probability of impact ionization is related to
the bandgap. From the values of the critical fields
in Fig.2.26 and the bandgaps in Table 2.1, a
polynomial fit of the form EC=f(EG) can be made.
• Using fitting, one can calculate the approximate
critical fields for a semiconductor not given in
Fig.2.26.
• The approximated critical field and Eq.(2-126)
then lead to the breakdown voltage for any
semiconductor device of interest. In general, a
large bandgap and a large breakdown voltage are
desirable for microwave transistors.
72
2.7 Self Heating
• When a voltage is applied to a semiconductor device and
thereby a current passes through it, electric power is
dissipated into heat, or in other words, electric energy is
transformed into thermal energy. In the case of dc
operation, the power dissipated, Pth is given by Pth = VI
where V is the dc voltage drop across the device and I is
the dc current flowing through it. The heat spreads
throughout the semiconductor and finally leaves the
semiconductor chip, i.e., it is transferred to the
surroundings. Furthermore, the heat leads to an increase in
the temperature inside the device.  self heating
• Because all the material parameters of semiconductors are
temperature dependent, the knowledge of the temperature
inside the device is critical for the accurate modeling of
device behavior.
73
• This is especially true for power transistors, in
which a large amount of heat is generated. For the
analysis of self-heating and for the thermal design
of power transistors, the temperature rise in the
transistor, the temperature distribution, and the
thermal resistance of the transistor are of primary
concern.
• 3 mechanisms by which heat can leave the device
and the chip: convection, radiation, conduction
74
75
HW2
• Derive the Einstein relation between mobility and
diffusion coefficient.
• Derive the energy and the momentum balance
equations.
• Explain the following
– Brillouin zone
– Plasma oscillation
– Plasma frequency
76