LECTURE 5
Electron dynamics
1. The action of the external force and the conduction effective mass of an electron.
We have previously shown (L. 4) that in the vicinity of the energy bands maxima or
minima, the dispersion relation E(k) can conveniently be described by the electron effective
mass. Now we will show that the somehow similar notion of the conduction effective mass
can be successfully used for the description of the dynamic properties of electron in a crystal.
In order to describe the movement of an electron in a complex crystal potential, we can
consider it as a moving wave packet. The electron moving in the periodic potential has some
average velocity <v>, which we can associate with the group velocity of the wave packet.
e
k
v grad >= < (1.5)
Taking into account the relationship E = e, where e is the wave packet frequency, we can
rewrite this equation to the form:
E grad
k
v
1
>= < (2.5)
This relationship has a quite general character and does not depend on the form of the
dependence E(k).
If an external force F acts on an electron in the crystal, then the classical equation of motion is
still valid for averaged values:
F v
k
> =<
dt
E d ) (
(3.5)
provided that the inequality aF s E
g
is fulfilled, E
g
is the energy gap between given band and
its closest neighbour, and a is the crystal lattice parameter. The meaning of this condition is
that the force is sufficiently weak in comparison to the atomic forces, in order not to excite the
electrons to or from the neighbouring bands.
Taking into account that
dE
dt
E
k
dk
dt
d
dt
grad E
i
i
i
( )
( )
k k
k
k
= =
c
c
(4.5)
and using (3.5) we obtain
F v
k
v < >= < >
d
dt
(5.5)
Comparing both sides of this equation we see that the pseudo-momentum k plays the role of
an averaged momentum <p>:
F
p k
=
< >
=
d
dt
d
dt
( )
(6.5)
Thus, the wave packet cantered on the wave vector k has an average momentum
<p>=k (7.5)
The average acceleration of the electron obtained from (2.5) is
d v
dt
d
dt
E
k
E
k k
dk
dt
i
i i m
m
m
< >
=
|
\
|
.
| =
|
\
|
.
|
|
1 1
2
c
c
c
c
(8.5)
Defining the tensor of the reciprocal of the effective mass as
1 1
2
2
~
m
E
k k
ik i m
=
c
c
(9.5)
1
we can rewrite this equation in the form:
d v
dt m
d k
dt m
F
i
ik
k
k
ik
k
k
< >
= =
1 1
~ ~
(10.5)
The difference between this tensor and the tensor of the reciprocal of the effective mass of
density of states defined previously by equation (6.4) is that this tensor is defined for a general
point of Brillouin zone, not only for an extremum point, and in the result, away from the
extremum point it can depend on k. Only in the vicinity of an extremum point, both the
tensors are equal.
Reducing (10.5) to the principal tensor axes we obtain:
d v
dt m
F
i
i
i
< >
=
1
~
(11.5)
For convenience, in the following we will drop the notation of averaging < >, still assuming
that we will deal with the averaged (in the classic sense) values of momentum and velocity.
Like in the previous case (L. 4), after the diagonalisation of the tensor of the reciprocal of the
effective mass, we can introduce the tensor of the effective mass
~
m
i
as a reciprocal of the
diagonalised tensor of the reciprocal of the effective mass. We can than write
~
m
i
dv
dt
F
i
i
=
(12.5)
In the case of the scalar effective mass
-
m
we have
m
d
dt
*
v
F =
(13.5)
Here we see that the effective mass tensor plays the role of the classical dynamic mass in the
classical equation of motion.
We have seen that in some of the semiconductors with cubic structure, the conduction band
isoenergetic surfaces have the form of ellipsoids of revolution. In such a case we deal with
two components of the tensor of the reciprocal of the effective mass only: the longitudinal (
l
m
) and the lateral (
t
m
) one. The longitudinal is in the direction of the longer axis, and the lateral
is in the direction of the shorter axis of the ellipsoid. In such a case it is possible to describe
their electrical transport phenomena with the help of an averaged effective mass:
|
|
.
|
\
|
+ =
-
t l
c
m m
m
~
2
~
1
3
1 1
, (14.5)
which is a scalar quantity.
The effective mass describes the motion of electrons in crystals in the sense that it contains
the effect of the system of huge internal forces. So we can consider that the effect of an
external force is such as if it acted on a free particle with this effective mass.
Thus, if
0 <
-
m
, the force and the acceleration have opposite directions, and if
0 >
-
m
, they
have the same directions.
It results from (9.5) that for isotropic dispersion relation E(k) the effective mass given by the
following relationship:
2
2
2
1 1
dk
E d
m
=
-
(15.5)
It should, however, be pointed out that it result from the general theory of the electron
transport that in the case of spherical dispersion relation, the effective mass that describe the
electron gas is defined in the following way:
2
dk
dE
k m
2
1 1
=
-
(16.5)
This effective mass is called the momentum effective mass or the transport effective mass. If
the dispersion relation is both spherical and parabolic, the definitions (15. 5) and (16.5) are
obviously equivalent. However, in the case of non-parabolicity the definitions give different
effective masses. The non-parabolicity can occur at some distance from the extremum point
only, and is important in the case of the narrow-gap semiconductors such as InSb or InAs.
It must be kept in mind that the effective masses have nothing to do with the inertial mass.
The inertial mass of an electron in a crystal remains the same as the inertial mass of an
electron in free space.
2. The effective mass and the density of states of electrons
For the evaluation of some physical parameters of a semiconductor, which are
characterised by its assembly of electrons as a whole, e.g. the electron concentration at a given
temperature, we have to know the number of the electron states falling into a unit energy
range in the crystal. This quantity is named the electron density of states distribution and is
denoted as g(E). The number of states in the energy range between E and E+dE is
}
+ s s
|
.
|
\
|
=
dE E E E
k d N
a
dE E g
) (
3
3
2
2 ) (
k
t
(17.5)
where
z y x
dk dk dk k d
3
The right hand side of (17.5) is the volume of an energy shell located between E and E+dE in
the k space divided by the volume
3
) / 2 ( a t /N occupied by a single energy state and
multiplied by the factor of 2 corresponding to the two allowed spin projections. N is the
number of atoms in the crystal, so that the crystal volume
3
Na V =
For the elementary volume in the k space we can take
= dSdk k d
3
(18.5)
where dS is an elementary isoenergetic surface element, as shown in Fig. 1.5, and
dk is the
component of k perpendicular to the surface element.
d S
k
x
k
y
k
z
3
Fig. 1.5. The surface element dS of the isoenergetic surface S (given by the condition E(k)
=const.). This particular surface is the effect of superposition of two ellipsoids of revolution.
The group velocity <v> of an electron moving in a given band depends on crystallographic
direction and is given by
= V >= <
k
E
E v
c
c
1 1
k
(19.5)
hence }
> <
=
S
v
dS V
E g
3
) 2 (
2
) (
t
(20.5)
Assume now that the isoenergetic surface is a sphere, i.e. the band is an isotropic one. Then
the integration gives the result
2
4 k S t =
, and for the velocity we have
k
E
v
c
c
1
>= <
(21.5)
Introducing it into (20.5) and assuming that our crystal has a unit volume, 1 = V we obtain the
density of states of a unit volume crystal:
E
k k
E g
c
c
t
2
) (
|
.
|
\
|
= (22.5)
In particular, for the quadratic dispersion relation:
E
k
m
=
-
2 2
2
(23.5)
we obtain
2
1
2
3
2
1
3 2
2
) ( E
m
E g
t
-
=
(24.5)
This equation explains why the quantity
-
m
is called the density of states effective mass.
In the case of isotropic bands, the assumption of the quadratic dispersion is reasonable,
because the expansion in series around an extremum point
0
k gives for the first term a
constant and for the second one a term quadratic in k.
The valence bands of the IV group (of the periodic table) semiconductors Si and Ge and also
the valence bands of the group III-V and II-VI semiconductors are isotropic at k~0, but for
large ks the isoenergetic surfaces are warped. We also know already that there exist
semiconducting materials with anisotropic bands. The conduction bands of Si and Ge and also
the valence and the conduction bands of the IV-VI semiconducting compounds belong to this
group. The isoenergetic surfaces of these materials are ellipsoids of revolution. In the vicinity
of the extremum points of each of these ellipsoids the dispersion law has the form:
|
|
.
|
\
|
+ + =
l
z
t
y
t
x
m
k
m
k
m
k
E
2
2
2 2
2
) (
k
(25.5)
where
t
m and
l
m are the transversal and longitudinal effective masses respectively.
It may be shown that for this dispersion law, the density of states can also be written in the
form of (24.5) if the effective mass
-
m
is substituted with:
( )
3 / 1
3 2 1
3 / 2
m m m L m
m ef
= (26.5)
where L
m
is the number of the effective minima in the first Brillouin zone. Therefore, for Ge
we have 4 =
m
L , and for Si we have 6 =
m
L .
In a similar way we can introduce a scalar effective density of states mass for the mentioned
warped surfaces. In general, we can introduce an effective scalar density of states mass for
4
isoenergetic surfaces of arbitrary form. The smaller the effective mass, the wider the electron
band in question, and smaller the density of states.
3. The holes
We will now consider the motion of electrons in the Brillouin zone. It results directly
from equation (4.4) of previous lecture and (2.5) that for arbitrary k
) ( ) ( k v k v =
(27.5)
This is because the derivative of an even function E(k) is an odd function.
If no external force is acting on electrons, then the total current from the electrons having the
wave vectors k in the first Brillouin zone is equal to the sum of all currents generated by the
individual electrons
k
j . We must also take into account that for a given k two alignments of
spin ( 2 / 1 = s ) are allowed. Therefore we have
0 ) ( 2
,
= = =
k k
k
k v j j e
s
tot
(28.5)
The total current vanishes because for each electron with the velocity v we can find in the
zone a corresponding electron with the velocity v.
This situation will not change if an external force is applied to a crystal with entirely filled
energy bands with electrons. In such a crystal, in accord with the Pauli exclusion principle, the
electrons can only exchange their momenta, which does not change the total momentum, and
thereby the total current.
This situation does change, however, when we apply the external force to a band filled only
partly with electrons. In such a case the total current is
=
s
n tot
s e
,
) , ( ) , (
k
k v s k j o
(29.5)
where o
n
( , ) k s is equal to one, when the state (k,s) is filled with electron, and is equal to zero,
when the state is not filled by an electron. If the state is not occupied with an electron, we say
that it is occupied by a hole.
If by analogy to the electron, we denote by
) , ( s k
p
o
a state occupied by a hole, then obviously
the following relationship must be fulfilled:
) , ( 1 ) , ( s k s k
n p
o o =
In such a case we can rewrite (27.5) into the form:
= =
s
p
s
p tot
s e s e
, ,
) , ( ) , ( ) , ( )] , ( 1 [
k k
k v s k k v s k j o o
(30.5)
An important conclusion from this equation is that an electric current in a conduction band not
entirely filled with electrons can be represented by a current of positively charged holes in
amount of
s
p
,
) , (
k
s k o
and moving with the velocities v(k).
Since the current j flows in the direction of the electric field E, it follows from the
equivalence of (29.5) and (30.5) that the action of the field is to increase the number of
electrons with the velocities v(k) directed against the field, or equivalently, that the effect of
the field is to increase the number of holes moving in the field direction.
Assume that there are no external potential gradients, which could affect the electrons. In this
case the total energy flow carried by the electrons from the entire Brillouin zone is in each
point of the crystal equal to zero:
0 ) ( ) ( ) , (
,
= =
k v k s k w
k s
n
E o
(31.5)
5
This is because ) , ( ) , ( s s
n n
k k = o o , ) ( ) ( k k = E E and ) ( ) ( k v k v = and as a result we
have
) ( ) , ( k s k E
n
o
v(k)=-
) ( ) , ( k s k E
n
o
v(-k) (32.5)
which means that in the sum (31.5) there always appear two components that cancel each
other.
Now assume that an electrical potential appears. In such a case each electron gains the
energy -e, and the total energy current is now:
) ( ] ) ( )[ , (
) ( ] ) ( )][ , ( 1 [ ) ( ] ) ( )[ , (
,
, ,
k v k s k
k v k s k k v k s k w
k
k k
+ =
= = =
s
p
s
p
s
n
e E
e E e E
o
o o
(33.5)
Thus, the energy current carried out by electrons is equivalent to the energy current carried out
by positively charged holes with negative energies. The direction of the energy axis for
electrons is opposite to the energy axis for holes.
If an energy band is almost entirely filled with electrons, then the electron effective mass at
the top of the band has a negative value. To such a band we can apply the effective mass
approximation, writing it in the form:
=
s
s
s
m
p
E E
2
) ( ) (
2
0
k k (34.5)
In this equation we can treat the mass (-m
s
)>0 as the mass of a hole. In this interpretation
(32.5) describes the dispersion law E(k) for holes.
Now we can also, as we did it for electrons, introduce for holes the definitions of various
effective masses, e.g. the transport or the density of states effective masses and so on.
The holes should be considered as positively charged particles in the conduction processes,
which may take place in the presence of magnetic, or other fields. It may occur that the
conduction processes take place simultaneously in two bands, of which one is a hole band and
the other is an electron band.
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