Download Carrier Transport

Carrier Transport
The topic of carrier transport will now be analyzed. First, carrier transport as a result of
drift will be studied followed by an analysis of carrier transport caused by diffusion.
Important equations of state will be derived. This topic will lead us directly into the
analysis of the PN junction in the next section.
Drift
Pierret defines drift as the charged-particle motion in response to an applied electric field.
From our earlier analysis of charge carriers in semiconductors, properties of the various
electron states near the bottom of the conduction band and near the top of the valence
band and the introduction to the concept of holes, we can make the following
generalization:
When an electric field ( E ) is applied across a semiconductor, negatively charged
electrons will accelerate in a direction opposite E and positively charged holes will
accelerated in a direction parallel to E.
The carriers will not accelerate indefinitely because of scattering from various sources
such as impurity atoms (both ionized and neutral), phonon scattering, carrier-carrier
scattering and other scattering mechanisms. Averaged over time, the carriers will tend to
have a certain time averaged drift velocity vd that, for electric fields that are not
excessively large, is linearly proportional to the applied field E. This relation can be
written as:
v d  E
where  is called the mobility and will have a different value for conduction band
electrons and valence band holes denoted by nand p respectively. The mobilities are
also very dependent on the semiconductor material. For instance, GaAs has a
significantly higher electron mobility but comparable hole mobility relative to Si. For
very high fields, vd is no longer linearly proportional to E and tends to acquire a constant
value denoted by vsaturation.
The electron and hole drift current density can easily be shown to have the following
forms:
J p drift  e p pE
J n drift  e n nE
The mobility of a perfect crystal without any defects and at very low temperature should
approach infinity. However, because of a variety of defects and scattering processes, the
mobility in reality is finite. A few of the scattering processes are:
1.
2.
3.
4.
5.
Phonon scattering
Ionized impurity scattering
Scattering by neutral impurity atoms and defects
Carrier-carrier scattering
Piezoelectric scattering
These scattering processes limit the mobility and is accounted for by assigning
component mobilities to each process and using Matthiessen’s rule (similar to adding up
resistances in an electrical circuit) to obtain the total mobility:
1
n
1
p


1
 Ln
1
 Lp


1
 In
1
 Ip
 ...
 ...
where n is the total electron mobility, Ln is the phonon scattering mobility component,
In is the ionized impurity scattering mobility component. Likewise for second equation
for hole mobility. Typically phonon scattering and ionized impurity scattering are the
dominant factors limiting mobilities. These scattering processes have the following
dependencies:
L  T
3
2
3
T 2
I 
NI
where N I  N A  N D .
Resistivity
The resistivity () is an important electrical property of materials. It is defined as the
proportionality constant relating current density and electric field:
J
1

E  E
where  is called the conductivity. The resistivity and conductivity can be determined
from previous equations we have studied:
J  J N Drift  J P Drift  q n n   p pE
Thus, it is easily seen that:

1

 q n n   p p 
Hall Effect
The hall effect is used to determine information about the carriers in semiconductor
materials. Using the Hall effect, it is possible to determine whether electrons or holes are
primarily responsible charge transport.
The Hall coefficient is defined as:
RH 
Ey

J x Bz
VH w
BI
To get a handle on what RH is, let us derive from first principles the equation for RH. The
lorentz force on a moving positively charged hole in a magnetic field is:
F  qv  B  qE  0
Hence, we have that vd 
Ey
Bz
. Using the relation J x  qpvd for holes, we have:
J x  qp
Ey
Bz
Finally, we see that for and p-type semiconductor with the primary charge carriers as
holes, RH is a positive quantity with a value of:
RH 
1
qp
Similarly for n-type materials with the primary charge carriers as electrons, RH is a
negative quantity with a value of:
RH  
1
qn
Diffusion
Diffusion is a physical phenomenon generally thought of as residing in the
thermodynamics field of physics and plays a crucial role in most physical systems you
can envision. Hence, it should come as no surprise that it is of central importance in the
electronic behavior of semiconductors. If you have not studied thermodynamics, don’t
fret, for we have been introducing concepts from different areas of physics throughout
this course. We have brought together concepts from different areas of physics to
produce a synthesis upon which we have developed a powerful foundation to predict the
physical, electronic and optical properties of semiconductors. First it was the very
geometrical field of crystallography joined with electromagnetism to predict x-ray
diffraction, then it was crystallography and electromagnetism joined with quantum
mechanics to describe energy bands and electron and hole states, finally it will be
crystallography, electromagnetism and quantum mechanics joined with thermodynamics
to predict carrier transport and electronic devices. Onwards we go...
Pierret defines diffusion to be “a process whereby particles tend to spread out or
redistribute as a result of their random thermal motion, migrating on a macroscopic scale
from regions of high particle concentration into regions of low particle concentration”. If
there is no external applied force or chemical potential (i.e., the system is uniform
throughout), then diffusion leads to a uniform distribution of particles. Figure 6.12 in
Pierret gives a good visual description of diffusion and the resulting electrical current.
Pierrent gives a simplified proof of the equations for the diffusion current resulting in the
common sense answer that the diffusion current is proportional to the gradient of the
carrier concentration. The result is:
J P diff  qDP p
J N diff  qDP n
where DP and DN are the hole and electron diffusion coefficients respectively.
Einstein Relationships
The Einstein relationship is a relationship between the diffusion coefficients (DP and DN)
and the mobilities of the carriers (N and P). Pierret then mentions three facts to which I
will add an additional one:
1. Under equilibrium conditions, the Fermi level inside a material (or inside a group
of materials in intimate contact) is invariant as a function of position; that is
dE F
 0 . This fact is obvious once the concept of the Fermi level is understood
dx
as the level where electron states with energies below this level are predominantly
filled and electron states with energies above this level are mostly empty
(analogous to a lake of water where the top of the water is the “Fermi level”, most
of the water is below the surface “Fermi level” of the lake in its low energy
available “states”).
2. Fact number 2 states what we found in chapter 4 that as one dopes a
semiconductor more and more n-type, the Fermi level departs from the intrinsic
Fermi level and approaches the conduction band. Similarly, as one dopes a
semiconductor more and more p-type, the Fermi level moves towards the valence
band.
3. One third fact I will insert is fact that an applied electric field  causes a bending
in the valence and conduction band and hence the intrinsic Fermi level according
to the following equation:

1 dE c 1 dEV 1 dEi


q dx
q dx
q dx
See chapter 4.3.2 for a detailed explanation of this but it is fairly obvious. From
electromagnetism, the electric field is equal to the negative gradient of the
potential and the potential is proportional to the potential energy. Hence:
  V 
1 d Ec  Eref  1 d EV  Eref 

q
dx
q
dx
Figure 6.14 of Pierret is a good example of band bending as a result of a
nonuniformly doped semiconductor:
These facts being stated, we can continue. Under equilibrium conditions, the current
density is zero. Hence:
J N  J N drift  J N diff  q n n  qD N
dn
0
dx
Using information from fact #3 above,  
1 dE c
and for nondegenerate semiconductors
q dx
n  N C e  EF  EC  KT we have:
dn
n dEC
nq
 1 dEC 
 N C e  EF  EC  KT  



dx
KT dx
KT
 KT dx 
Using this result in the equation for JN, we have:
DN 
KT
N
q
DP 
KT
P
q
Similarly for JP, we find:
These two relations between the diffusion coefficients and the mobility for electrons and
holes are called Einstein relationships. These will prove useful.
Equations of State
Collecting the results from the first two sections, we have that the total current is a sum of
the conduction band electrons and valence band holes: J = JN + JP where:
J N  q N n  qDN n
J P  q P p  qDN p
These last two equations can be cast into a simpler form by the introduction of the
concept of quasi-Fermi levels FN and FP for electrons and holes respectively. Pierret
states that “these energy levels are by definition related to the nonequilibrium carrier
concentration in the same way EF is related to the equilibrium carrier concentration”.
Stated mathematically, this is:
n  ni e  FN  Ei  KT
FN  Ei  KT ln n ni 
p  ni e  Ei  FN  KT
FP  Ei  KT ln  p ni 
In general FN and FP will be two distinct values that will tend back to EF as the
semiconductor goes back to equilibrium. It can be shown by differentiating p and n
above to get p and n , substituting this in for JP and JN and using the Einstein
relationships that:
J P   P pFP
J N   N nFN
These equations are extremely useful for analysis of energy bands and current transport
in electronic devices.
Continuity Equations
The charge continuity equations will now be studied, first in general and then applied to
minority carriers under low level injection and low electric fields. Continuity equations
are used throughout physics and engineering and relate the time dependence of some
concentration to other functional relationships of the concentration. For instance,
concentration of mass can be written as the time dependence of mass density which can
be shown to be:
d
 J m
dt
where J m  v is the “mass current” where v is the velocity of the mass particles. A
good description of this concept, the derivation and proof is given in Vector Calculus by
Jerrold E. Marsden pg. 544-545:
The same type of continuity equation can be applied to charge carriers with the
realization that charge concentration can increase or decrease because of other
mechanisms besides carrier transport, namely recombination and generation. Hence we
can write:
dn 1
 J N  rN  g N
dt q
dp
1
  J P  rP  g P
dt
q
where the divergence of the electron and hole current account for the drift and diffusion
current contribution to rate of change in time of the charge concentration, the terms rN
and rP are the electron and hole recombination rates respectively and the terms gN and gP
are other electron and hole generation processes respectively (such as photoexcitation,
....). These are the general rate equations that will be used in quite often in device
analysis. Let us apply these equation to a simplified situation of minority carrier
diffusion.
Minority Carrier Diffusion Equations
Let us make the following assumptions:
1.
2.
3.
4.
5.
6.
The system is one-dimensional.
The analysis is restricted to only minority carriers.
The electric field is negligible.
The equilibrium carrier concentrations are not a function of position.
Low level injection conditions are held throughout the system.
No other recombination or generation processes except for “photoexcitation”.
Under these conditions, we have:
dn 

d  qD N

1
1 dJ N 1 
d 2n
dx 
J N 

 DN 2
q
q dx
q
dx
dx
dn dno dn dn



dx
dx
dx
dx
rN 
n
n
dn dno dn dn



dt
dt
dt
dt
gN = GL
where GL is the number of electron-hole pairs generated per sec-cm3 by the absorption of
externally introduced photons. If the semiconductor is not subjected to illumination, then
GL = 0. Using these simplifications, we obtain the minority carrier diffusion equations:
dn p
dt
 DN
d 2 n p
dx
2

n p
n
 GL
dp n
d 2 p n p n
 DP

 GL
dt
p
dx 2
where the subscripts have been added as Pierret does to denote the fact that these are
minority carrier concentrations.