Download Appendix B: Boltzmann Transport Theory

Modern Physics for Engineers
Jasprit Singh
Copyright Q 2004 WILEY-VCH Verlag GmhH & Co. KGaA
APPENDIX
B
BOLTZMANN
TRANSPORT THEORY
Transport of electrons in solids is the basis of many modern technologies. The Boltzmann transport theory allows us to develop a microscopic model for macroscopic
quantities such as mobility, diffusion coefficient, and conductivity. This theory has
been used in Chapter 8 to study transport of electrons and holes in materials. In
this appendix we will present a derivation of this theory.
B.l
BOLTZMANN TRANSPORT EQUATION
In order to describe the transport properties of an electron gas, we need t o know
the distribution function of the electron gas. The distribution would tell us how
electrons are distributed in momentum space or k-space (and energy-space) and
from this information all of the transport properties can be evaluated. We know
that at equilibrium the distribution function is simply the Fermi-Dirac function
1
f ( E ) = exp
(%tE>+
1
P.1)
This distribution function describes the equilibrium electron gas and is independent of any collisions that may be present. While the collisions will continuously
remove electrons from one h-state t o another, the net distribution of electrons is
always given by the Fermi-Dirac function as long as there are no external influences
to disturb the equilibrium.
To describe the distribution function in the presence of external forces,
we develop the Boltzmann transport equation. Let us denote by f k ( ~ the
)
local
concentration of the electrons in state Ic in the neighborhood of T . The Boltzmann
approach begins with an attempt to determine how f k ( ~ changes
)
with time. Three
possible reasons account for the change in the electron distribution in k-space and
r-space:
353
BOLTZMANN TRANSPORT THEORY
354
Figure B.1: At time 1 = 0 particles at position T -6stVk reach the position r at a later time
6t. This simple concept is important in establishing the Boltzmann transport equation.
1. Due to the motion of the electrons (diffusion), carriers will be moving into
and out of any volume element around T .
2. Due to the influence of external forces, electrons will be changing their momentum (or &-value) according to h d k / d t = F e z t .
3. Due t o scattering processes, electrons will move from one k-state to another.
We will now calculate these three individual changes by evaluating the
partial time derivative of the function fk(T) due to each source.
B.l.l
Diffusion-Induced Evolution of
fk(T)
If uk is the velocity of a carrier in the state k, in a time interval t , the electron
moves a distance t u k . Thus the number of electrons in the neighborhood of r at
time 6t is equal t o the number of carriers in the neighborhood of T - 6t Vk at time
0, as shown in Fig. B . l
We can thus define the following equality due to the diffusion
fk(T,
6 t ) = fk(T - 6t
bk, 0)
(B.2)
or
B.1.2 External Field-Induced Evolution of f k ( T )
The crystal momentum k of the electron evolves under the action of external forces
according t o Newton’s equation of motion. For an electric and magnetic field ( E
and B ) , the rate of change of Ic is given by
B.1. BOLTZMANNTRANSPORTEQUATION
355
In analogy t o the diffusion-induced changes, we can argue that particles at
time t = 0 with momentum k - k 6t will have momentum k at time 6 t and
which leads t o the equation
= - k’- a f k
dk
v x B
h
B.1.3
dk
Scattering-Induced Evolution of
fk(T)
We will assume that the scattering processes are local and instantaneous and change
the state of the electron from k to k ’ . Let W ( k , k ’ )define the rate of scattering
from the state k to k’ if the state k is occupied and 12‘ is empty. The rate of change
of the distribution function fk(v) due to scattering is
The ( 2 ~ in) the
~ denominator comes from the number of states allowed in a k-space
volume d3k‘. The first term in the integral represents the rate at which electrons
are coming from an occupied k’ state (hence the factor f k’ ) to an unoccupied kstate (hence the factor (1 - f k ) ) . The second term represents the loss term.
Under steady-state conditions, there will be no net change in the distribution function and the total sum of the partial derivative terms calculated above will
be :
Let us define
9k = f k
- f:
03.9)
where f: is the equilibrium distribution.
We will attempt to calculate gk, which represents the deviation of the distribution function from the equilibrium case.
Substituting for the partial time derivatives due to diffusion and external
fields we get
Substituting
fk
= f: + gk
(B.ll)
356
BOLTZMANN TRANSPORT THEORY
We note that the magnetic force term on the left-hand side of Eqn. B . l l is proportional t o
e
b k ' TI (Vk x B )
and is thus zero. We remind ourselves that (the reader should be careful not to
confuse E k , the particle energy and E , the electric field)
vk
1
1 dEk
--
(B.12)
h dlc
and (in semiconductor physics, we often denote p by E F )
f: =
[w]
+
1
exp
1
(B.13)
Thus
T
(B.14)
Also
(B.15)
Substituting these terms and retaining terms only to second-order in electric
field (i.e., ignoring terms involving products g k . E ) , we get, from Eqn. €3.11,
The equation derived above is the Boltzmann transport equation.
We will now apply the Boltzmann equation to derive some simple expressions for conductivity, mobility, etc., in semiconductors. We will attempt to relate
the microscopic scattering events to the measurable macroscopic transport properties. Let us consider the case where we have a uniform electric field E in an infinite
system maintained at a uniform temperature.
B.l. BOLTZMANNTRANSPORTEQUATION
k=O
357
k-
Figure B.2: This figure shows that at time 1 = 0, the distribution function is distorted
by some external means. If the external force is removed, the electrons recover to the
equilibrium distribution by collisions.
T h e Boltzmann equation becomes
(B.17)
Note t h a t only the deviation gk from the equilibrium distribution function above
contributes to t h e scattering integral.
As mentioned earlier, this equation, although it looks simple, is a very
complex equation which can only be solved analytically under fairly simplifying
assumptions, We make a n assumption t h a t the scattering induced change in the
distribution function is given by
(B.18)
We have introduced a time constant r whose physical interpretation can
be understood when we consider what happens when the external forces have been
removed. In this case the perturbation in t h e distribution function will decay according to t h e equation
(B.19)
T h e time r thus represents t h e time constant for relaxation of the perturbation as shown schematically in Fig. B.2. T h e approximation which allows us to
write such a simple relation is called the relaxation time approximation (RTA).
According t o this approximation
(B.20)
BOLTZMANN TRANSPORT THEORY
358
Figure B.3: T h e dispIaced distribution function shows t h e effect of a n appIied eIectric
field.
Note that we have not defined how 7- is to be calculated. We have merely
introduced a simpler unknown that still needs to be determined. The lc-space distribution function may be written as
Using the relation
We have
=
;(h - -e;E)
(B.21)
This is a very useful result which allows us to calculate the non-equilibrium
function fk in terms of the equilibrium function f o . The recipe is very simpleshift the original distribution function for h values parallel to the electric field by
e r E / h . If the field is along the z-direction, only the distribution for k, will shift.
This is shown schematically in Fig. B.3. Note that for the equilibrium distribution
function, there is an exact cancellation between positive velocities and negative
velocities. When the field is applied, there is a net shift in the electron momenta
and velocities given by
6p = h6k = -erE
erE
= -m’
(B.22)
B. 1, BOLTZMA N N TRANSPORT EQUATION
This gives, for the mobility,
p=-
359
er
m’
If the electron concentration is n, the current density is
(B.23)
J = neSv
- ne2rE
m*
or the conductivity of the system is
u=-
ne2 r
m’
(B.24)
This equation relates a microscopic quantity r to a macroscopic quantity 6.
So far we have introduced the relaxation time T , but not described how it
is to be calculated. We will now relate it to the scattering rate W ( k ,k ’ ) , which can
be calculat#ed by using the Fermi golden rule. We have, for the scattering integral,
Let u s examine some simple cases where the integral on the right-hand side becomes
simplified.
Elastic Collisions
Elastic collisions represent scattering events in which the energy of the electrons
remains unchanged after the collision. Impurity scattering and alloy scattering discussed in Chapter 8 fall into this category. In the case of elastic scattering the
principle of microscopic reversibility ensures that
W ( k lk ’ ) = W ( k ’ ,k )
(B.25)
,
i.e., the scattering rate from an initial state k to a final state k is the same as that
for the reverse process. The collision integral is now simplified as
(B.26)
The simple form of the Bollzmann equation is (from Eqn. B.17)
-
2)
at
scattering
(B.27)
BOLTZMANN TRANSPORT THEORY
360
JX
Figure B.4: Coordinate system illustrating a scattering event.
The relaxation time was defined through
-3)
-
at
.T
scattering
(B.28)
Substituting this value in the integral on the right-hand side, we get
or
(Wk
- V k ( ) w(k,k’)d3k’ . E
(B.30)
and
(B.31)
In general, this is a rather complex integral to solve. However, it becomes
considerably simplified for certain simple cases. Consider, for example, the case of
isotropic parabolic bands and elastic scattering. In Fig. B.4 we show a geometry for
the scattering process. We choose a coordinate axis where the initial momentum is
along the z-axis and the applied electric field is in the y-z plane. The wavevector
after scattering is given by lc represented by the angles cr and 4. Assuming that
the energy bands of the material is isotropic, Ivk[ = (vk’1. We thus get
I
(B.32)
B.1. B O L T Z M A N N T R A N S P O R T EQUATION
36 1
We can easily see from Fig. B.4 that
cos 8’ = sin B sin a sin
or
+ cos 0 cos a
-= t a n B s i n a s i n 4 + c o s a
cos 8’
cos 0
When this term is integrated over C#J to evaluate T , the term involving s i n 4 will
integrate to zero for isotropic bands since W ( k k, ) does not have a #J dependence,
only an a dependence. Thus
I
1
-7-= / W ( k , k ’ ) ( I - c o s a ) d 3 k ’
(B.33)
This weighting factor (1- cos a ) confirms the intuitively apparent fact that
large-angle scatterings are much more important in determining transport properties than small-angle scatterings. Forward-angle scatterings ( a = 0), in particular,
have no detrimental effect on (T or p for the case of elastic scattering.
Inelastic Collisions
In the case of inelastic scattering processes, we cannot assume that W ( k , k ’ ) =
W ( k ’ ,k ) . As a result, the collision integral cannot be simplified to give an analytic
result for the relaxation time. If, however, the system is non-degenerate, i.e., f ( E )
is small, we can ignore second-order terms in f and we have
Under equilibrium we have
or
W ( k ’k)
, = &(k,
J
k’
k)
(€3.36)
Assuming that this relation holds for scattering rates in the presence of the applied
field, we have
(B.37)
The relaxation time then becomes
(B.38)
The Boltzmann is usually solved iteratively using numerical techniques.
362
BOLTZMANN TRANSPORT THEORY
B.2 AVERAGING PROCEDURES
We have so far assumed that the incident electron is on a well-defined state. In a
realistic system the electron gas will have an energy distribution and r , in general,
will depend upon the energy of the electron. Thus it is important to address the
appropriate averaging procedure for r . We will now do so under the assumptions
that the drift velocity due to the electric field is much smaller than the average
thermal speeds so that the energy of the electron gas is still given by 3 k ~ s T / 2 .
Let us evaluate the average current in the system.
(B.39)
The perturbation in the distribution function is
(B.40)
If we consider a field in the z-direction, the average current in the zdirection is from Eqns. B.39 and B.40
(B.41)
The assumption made on the drift velocity ensures that v; = v 2 / 3 , where v is the
total velocity of the electron. Thus we get
(B.42)
Now we note that
1
-m*(v2)=
2
+kBT =
3
-kBT
2
rn'(v2)/3
also
(B.43)
Substituting in the right-hand side of Eqn. B.42, we get (using 3kBT = m (v2))
(B.44)
B.2. AVERAGING PROCEDURES
.
.
363
Thus, for the purpose of transport, the proper averaging for the relaxation
time is
(B.45)
Here the double brackets represent an averaging with respect to the perturbed distribution function while the single brackets represent averaging with the equilibrium
distribution function.
For calculations of low-field transport where the condition vz = v2/3 is
valid, one has to use the averaging procedure given by Eqn. B.45 to calculate mobility or conductivity of the semiconductors. For most scattering processes, one finds
that it is possible t o express the energy dependence of the relaxation time in the
form
T(E) = To(E/kBT)’
(B.46)
where ro is a constant and s is an exponent which is characteristic of the scattering
process. We will be calculating this energy dependence for various scattering processes in the next two chapters. When this form is used in the averaging of Eqn.
B.45, we get, using a Boltzmann distribution for fo(k)
where p = hk is the momentum of the electron.
Substituting y = p2/(2m*kgT), we get
(B.48)
To evaluate this integral, we use I?-functions which have the properties
(B.49)
and have the integral value
qa)=
1
co
ya-le-ydy
(B.50)
In terms of the I?-functions we can then write
(B.51)
If a number of different scattering processes are participating in transport,
the following approximate rule (Mathiesen’s rule) may be used to calculate mobility:
1
1
(B.52)
-=c,
Tzot
2
1
where the sum is over all different scattering processes.
(B.53)