Vật lý điện tử GS Le Tuan

C H A P T E R 2
FUNDAMENTAL QUANTUM CONCEPTS
1.1 The Bohr model
The results of emission spectra experiments led Niels Bohr (1913) to construct a model for the
hydrogen atom, based on the mathematics of planetary systems. If the electron in the hydrogen
atom has a series of planetary-type orbits available to it, it can be excited to an outer orbit and then
can fall to any one of the inner orbits, giving off energy corresponding to one of the lines of Fig.
2.1.To develop the model, Bohr made several postulates
è Electrons exist in certain stable, circular orbits about the nucleus. This assumption implies that the
orbiting electron does not give off radiation as classical electromagnetic theory would normally
require of a charge experiencing angular acceleration; otherwise, the electron would not be stable
in the orbit but would spiral into the nucleus as it lost energy by radiation
è The electron may shift to an orbit of higher or lower energy, thereby gaining or losing energy
equal to the difference in the energy levels (by absorption or emission of a photon of energy h
w).
è The angular momentum p
q
of the electron in an orbit is always an integral multiple of Planck’s
constant divided by 2p (h/2p is often abbreviated Ñ for convenience), p
q
= n Ñ, (n = 1, 2, 3, 4, ).
Although Bohr proposed this ad hoc relationship simply to explain the data, one can see that this
is equivalent to having an integer number of de Broglie wavelengths fit within the circumference
of the electron orbit (Prob. 2.2). These were called pilot waves, guiding the motion of the
electrons around the nucleus. The de Broglie wave concept provided the inspiration for the
Schrödinger wave equation in quantum mechanics discussed later.
Fig. 2.1. Electron orbits and transitions in the Bohr model of the hydrogen atom. Orbit spacing is not drawn to scale.
Bohr supposed that the electron in a stable orbit of radius r about the proton of the hydrogen
atom, we can equate the electrostatic force between the charges to the centripetal force:
(1.1)
-

1
4 pe
0
q
2
r
2
=-
mv
2
r
where m is the mass of the electron, and v is its velocity. From assumption 3 we have
(1.2)
p
q
= mvr = nÑ
Since n takes on integral values, r should be denoted by r
n
to indicate the nth orbit. Then Eq.
(2-1) can be written
(1.3)
m
2
v
2
=
n
2
Ñ
2

r
n
2
Substituting Eq. (2.3) into Eq. (2.1), we find that the n-th radius is
(1.4)
r
n
=
1
4 pe
0
n
2
Ñ
2
mq
2
= n
2
r
1
, n = 1, 2, 3, 4, 
therefore r
1
= a
0
=
Ñ
2
mq

2
º 0.529 Þ has the name as the first Bohr radius. We noted that the allowed
radii are r
n
= r
1
, 4 r
1
, 9 r
1
, 16 r
1
,
The electron velocity in each of its orbits can be estimated from (2.2) - (2.4)
(1.5)
v
n
= n
Ñ
mr
n
=
1
n
v
1
with v
1
=
1

4 pe
0

q
2
Ñ
º 2.2 ä 10
6
m/s.
From (2.3) we have the total energy of the electron as the summ of the kinetic and potential
ones, as following,
(1.6)
E
n
= E
d
n
+ E
t
n
=
mv
n
2
2
+
-
1
4 pe
0

q
2
r
n
=-
mq
4
4 pe
0

2
2 Ñ
2
1
n
2
=-
1
n
2
E
1
with the ionization energy for hydrogen atom E
1
º - 13.6 eV.
The transitions in hydrogen emission spectra have been explained (see Fig, 2.1), for example,
between the orbits n
1
and n
2

(1.7)
h n
21
= Ñ w
21
= 
mq
4
4 pe
0

2
2 Ñ
2
hc

1
n
1
2
-
1
n
2
2
=
E
1
hc
1

n
1
2
-
1
n
2
2
= R
1
n
1
2
-
1
n
2
2
with R is the Rydberg constant, which is frequently serving as an unity of energy in atomic physics.
Although the Bohr model was immensely successful in explaining the hydrogen spectra, numer-
ous attempts to extend the "semi-classical" Bohr analysis to more complex atoms such as hclium
proved to be futile. Success along these lines had to await further development of the quantum
mechanical formalism. Nevertheless, the Bohr analysis reinforced the concept of energy quantiza-
tion and the attendant failure of classical mechanics in dealing with systems on an atomic scale.
Moreover, the quantization of angular momentum in the Bohr model clearly extended the quantum
concept, seemingly suggesting a general quantization of atomic-scale observables.
2 SSED Le Tuan Chapters 2.nb
1.2 Black body radiaton
In this part we will very briefly recall some important moments in hystory of physics at the end
of the XIX century. We know that heat transfer takes place through the processes of conduction,

convection and radiation. The first two processes can take place only in a medium, while the last
does not need a medium .Robert Kirchhoff studied (1860) the problem of radiation and the proper-
ties of perfect absorb materials. He pointed out that when such a material is heated, it will emit
radiations of all wavelengths i.e. it will be a perfect emitter. Such a body is referred to as a black
body.For instance, a hollow body whose walls are at the same temperature behaves like a black
body. They pointed out that it will be a perfect emitter emitting through a tiny hole in its surface,
radiations of all wavelengths. Further, any radiation entering such a cavity will undergo infinitely
many reflections inside and will lose energy at every reflection.
Thus no incident radiation will emerge out of the tiny hole. In other words, it is also a perfect
absorber. Thus such a cavity, which can be easily constructed , is almost a perfect black body.
In 1879, the Austrian physicist Josef Stefan reassessed the work of Dulong and Petit. He made
the corrections and calculated the pure radiation component of heat transfer and the went to the law
for the total emission power of black body
(1.8)
E =sT
4
with the Stefan constant s = 5.70 ä 10
-8
W m
-2
K
-4
. Then, Wien (1893) calculated the energy
u(
l
, T) emitted per unit volume per sec in unit wavelength interval.
(1.9)
ul, T =
f
l,

T

l
5
ïl
p
T = b
where b = 2.898 ä 10
-3
m.K is the Wien constant anf the nature of the function f(
l
, T) still is
unknown. Based on Newtonian physics, by very adequate calculation of the emission within the
black body cavity to be made up of a series of standing waves, Ragleigh and Jeans came to the
famous formula:
(1.10
)
ul, T =
8 p
k

T
l
4
Fig. 2.2. The black body emission spectra. The outstanding violet catastrophe with Rayleigh-Jeans formula led Planck to
wor
k
out the concept of quantization of energy transfer.
However, it completely fails near the low wavelength end where it diverges to an infinite value
for u(

l
, T). This is often referred to as the Ultra-violet Catastrophe.
SSED Le Tuan Chapters 2.nb 3
The important step was taken by Max Planck around 1900. Planck suggested an interpolation
formula by taking for Wien's function `f(l, T)' the form:
(1.11
)
f l, T =
8 p
k
b
Exp
b
l T
- 1
where k as the Boltzmann constant and b as an adjustable parameter.
At that time, the physicists thought that the atoms of a material which absorbed or emitted
radiation as harmonic oscillators. Hence, oscillators were in equilibrium with radiation exchanging
energy with it. Planck knew that radiations are electromagnetic waves. Hence the cavity was filled
with these waves. These waves were absorbed and emitted by the oscillators that behaved like
classical `pendulums'. Planck believed in the electromagnetic nature of radiation. But the oscillator
mechanism of absorption and emission of electromagnetic waves was more a theoretical model. He
arriveed at his own formula for the black-body spectrum, suggesting that a harmonic oscillator can
not have any energy but only in integral multiples of a quantum of energy e
0
= h
n
, where n is the
natural frequency of the oscillator and h a constant to be determined. In other words, Planck quan-
tized the permitted energies of an oscillator. Thus, they were strictly non-classical in nature with

energies e = n h
n
, n being an integer. These oscillators are in thermal equilibrium at any tempera-
ture. Therefore, Planck invoked the Boltzmann distribution to describe them. Accordingly, the
average oscillator energy e of the system of oscillators becomes:
(1.12
)
e=
h
n

Exp
h n
kT
- 1
with the Planck constant h = 6.626 ä 10
-34
J.s.
Then, the total energy of radiation per unit volum e at l per unit wavelength interval is:
(1.13
)
ul, T =
8 p hc
Exp
hc
l kT
- 1l
5

It is worth to say that at the limit T Ø ¶ we can return again to the well-known Ragleigh -

Jeans formula.
4 SSED Le Tuan Chapters 2.nb
1.3 Basic Formalism
1.3.1 The five postulates of Quantum Mechanism
The formulation of quantum mechanics, also called wave mechanics focuses on the wave
function, Y(x,y,z,t), which depends on the spatial coordinates x, y, z, and the time t. In the following
sections we shall restrict ourselves to one spatial dimension x, so that the wave function depends
solely on x. An extension to three spatial dimensions can be done easily. The wave function Y(x,t)
and its complex conjugate
Y
*
(x,t) are the focal point of quantum mechanics, because they provide a
concrete meaning in the macroscopic physical world: The product
Y
*
(x, t)dx is the probability to
find a particle, for example an electron, within the interval x and x + dx. The particle is described
quantum mechanically by the wave function Y(x,t). The product Y*(x,t)Y(x,t) is therefore called the
window of quantum mechanics to the real world.
Quantum mechanics further differs from classical mechanics by the employment of operators
rather than the use of dynamical variables. Dynamical variables are used in classical mechanics,
and they are variables such as position, momentum, or energy. Dynamical variables are contrasted
with static variables such as the mass of a particle. Static variables do not change during typical
physical processes considered here. In quantum mechanics, dynamical variables are replaced by
operators which act on the wave function. Mathematical operators are mathematical expressions
that act on an operand. For example, (d / dx) is the differential operator. In the expression (d / dx)
Y(x,t), the differential operator acts on the wave function, Y(x,t), which is the operand. Such
operands will be used to deduce the quantum mechanical wave equation or Schrödinger equation.
The postulates of quantum mechanics cannot be proven or deduced. The postulates are hypothe-
ses, and, if no violation with nature (experiments) is found, they are called axioms, i. e. non-

provable, true statements.
Postulate 1
The wave function (x,y,z,t) describes the temporal and spatial evolution of a quantummechani-
cal particle. The wave function (x,t) describes a particle with one degree of freedom of motion.
Postulate 2
The product Y*(x,t)Y(x,t) is the probability density function of a quantum-mechanical particle.
Y*(x,t)Y(x,t)dx is the probability to find the particle in the interval between x and x + dx. Therefore,
(1.14
)

-¶
+¶
Y*x, tY x, t„ x = 1
If a wave function (x, t) fulfills Eq. (2.14), then (x, t) is called a normalized wave function.
Equation (2.14) is the normalization condition and implies the fact that the particle must be
located somewhere on the x axis.
Postulate 3
The wave function (x, t) and its derivative (/ x) (x, t) are continuous in an isotropic
medium.
(1.15
)
Limit

x, t, x Ø x
0
=

x
0
, t

(1.16
)
Limit
∑
∑x
 x, t, x Ø x
0
=
∑
∑x
 x, t
x=x
0
SSED Le Tuan Chapters 2.nb 5
In other words, (x, t) is a continuous and continuously differentiable function throughout
isotropic media. Furthermore, the wave function has to be finite and single valued throughout
position space (for the one-dimensional case, this applies to all values of x).
Postulate 4
Operators are substituted for dynamical variables. The operators act an the wave function (x,
t). In classical mechanics, variables such as the position, momentum, or energy are called dynami-
cal variables. In quantum mechanics operators rather than dynamical variables are employed.
Table 2.1 shows common dynamical variables and their corresponding quantummechanical
operators
Table 2.1. Common physical dynamic variables and quantummechanic operatotors
Dynamic variables Operators
in positionspace in momentumspace
Position, x x 
—
i


p
x
 i—

p
x
Function of position, f x
poteial energy, U x
f x;Ux f 
—
i

p
x
  f i—

p
x

Momentum, p
x
i—

x

—
i

x
p

x
Function of momentum, f p
x
 f  i—

x
  f 
—
i

x
 f p
x

Kinetic energy,
1
2
mv
2

—
2
2m

2
x
2
p
x
2

2m
Total energy, E
—
i

t
—
i

t
Total energy, E 
—
2
2m

2
x
2
U x
p
x
2
2m
 U 
—
i

p
x


Postulate 5
The expectation value, , of any dynamical variable , is calculated from the wave function
according to
(1.17
)
=

-¶
+¶x
Y
*
x, tx
`
Y x, t„ x =<Y
*
Ï x
`
ÏY>
6 SSED Le Tuan Chapters 2.nb
1.3.2 Some properties of quantummechanical operators
Eigenfunctions and eigenvalues
Any mathematical rule which changes one function into some other function is called an opera-
tion. Such an operation requires an operator, which provides the mathematical rule for the opera-
tion, and an operand which is the initial function that will be changed under the operation. Quan-
tum mechanical operators act on the wave function Y(x, t). Thus, the wave function Y(x, t) is the
operand. Examples for operators are the differential operator (d / dx) or the integral operator 
…dx. In the following sections we shall use the symbol x
`
for an operator and the symbol f(x) for
an operand. The definition of the eigenfunction and the eigenvalue of an operator is as follows: If

the effect of an operator
x
`
operating on a function f(x) is that the function f(x) is modified only by
the multiplication with a scalar, then the function f(x) is called the eigenfunction of the operator x
`
,
that is
(1.18
)
x
`
Yx, t =l
s
Yx, t
where l
s
is a scalar (constant). l
s
is called the eigenvalue of the eigenfunction. For example, the
eigenfunctions of the differential operator are exponential functions, because
(1.19
)
„
„ x
e
l
s
x
=l

s
e
l
s
x
where l
s
is the eigenvalue of the exponential function and the differential operator.
Linear operators and commutation law
Virtually all operators in quantum mechanics are linear operators. An operator is a linear
operator if
(1.20
)
x
`
c Yx, t = c x
`
Yx, t
where c is a constant. For example d / dx is a linear operator, since the constant c can be
exchanged with the operator d / dx. On the other hand, the logarithmic operator (log) is not a linear
operator, as can be easily verified. In classical mechanics, dynamical variables obey the commuta-
tion law. For example, the product of the two variables position and momentum commutes, that is
(1.21
)
xp
x
= p
x
x
However, in quantum mechanics the two linear operators, which correspond to x and p, do not

commute, as can be easily shown. One obtains
(1.22
)
x
`
p
`
Yx, t = x 
Ñ
i

∑
∑x
Yx, t
and alternatively
(1.23
)
p
`
x
`
Yx, t =
Ñ
i
∑
∑
x
x Yx, t  =
Ñ
i

Yx, t+
Ñ
i
x
∑
∑
x
Yx, t
Linear operators do not commute, since the result of Eqs. (2.22) and (2.23) are different.
SSED Le Tuan Chapters 2.nb 7
Hermitian operators
In addition to linearity, most of the operators in quantum mechanics possess a property which is
known as hermiticity. Such operators are hermitian operators, which will be defined in this section.
The expectation value of a dynamical variable is given by the 5th Postulate according to (2.17).
The expectation value is now assumed to be a physically observable quantity such as posi-
tion or momentum. Thus, the dynamical variable is real, and is identical to its complex
conjugate.
(1.24
)
=

-¶
+¶x
Y
*
x, tx
`
Y x, t„ x =<Y
*
Ï x

`
ÏY>
<YÏ x
`
*
ÏY
*
>=

-¶
+¶x
Y x, t x
`
*
Y
*
x, t„ x = 
*

Operators x
`
and x
`
*
, which satisfy Eq. (2.24) are called hermitian operators.
The definition of an hermitian operator is in fact more general than given above. In general,
hermitian operators satisfy the condition
(1.25
)


-¶
+¶x
Y
1
*
x, tx
`
Y
2
x, t„ x =

-¶
+¶x
Y
2
x, t x
`
*
Y
1
*
x, t„ x
where y
1
(x) and y
2
(x) may be different functions. If y
1
1(x) and y
2

(x) are identical, Eq. (2.23)
simplifies into Eq. (2.24)
There are a number of consequences and implications resulting from the hermiticity of an
operator. Two more properties of hermitian operators will explicitly mentioned. First, eigenvalues
of hermitian operators are real. To prove this, suppose x
`
is an hermitian operator with eigenfunc-
tion y(x) and eigenvalue l. Then
and also due to hermiticity of the operator
Since Eqs. (4.17) and (4.19) are identical, therefore l = l*, which is only true if l is real. Thus,
eigenvalues of Hermitian operators are real.
Second, eigenfunctions corresponding to two unequal eigenvalues of an hermitian operator are
orthogonal to each other. This is, if x
`
is an hermitian operator and y
1
(x) and y
2
(x) are eigenfunc-
tions of this operator and l
1
and l
2
are eigenvalues of this operator then
Time-Independent Formulation
If the particle in the system under analysis has a fixed total energy E, the quantum mechanical
formulation of the problem is significantly simplified. Consider the general expression for the
energy expectation value in the total volume space ý as deduced from Table 2.1 and Postulate 5:
(1.26
)

< E >=

ý
Y
*
r

, t -
Ñ
i
∑Yr

, t
∑t
„ý
By the normalization requirements, one can find that <E> = E = constant, so
(1.27
)
-
Ñ
i
∑Yr

, t
∑t
= E Yr

, t
Indeed, if we make a direct substitution of the last expression into the above, the desired result
will be obtained

8 SSED Le Tuan Chapters 2.nb
(1.28
)
< E >=

ý
Y
*
-
Ñ
i
∑Y
∑t
„ý = E

ý
Y
*
Y„ý = E = constant
In the result, we can have a general solusion of the form:
(1.29
)
Yr

, t =Yx, y, z, t =yx, y, ze
-i
Et
Ñ
for so-called , reminiscing about the main Newtonian equation in classical mechanics:
(1.30

)
p
2
2 m
+ Ur

 = E
total
ó H
`
Yr

, t =
t
ó
-
Ñ
2
2 m
“+Ur

 Yr

, t =-
Ñ
i
∑Yr

, t
∑t

where by definition
(1.31
)
“=
∑
2
∑ x
2
+
∑
2
∑ y
2
+
∑
2
∑z
2
After substitution the general solution into the time-dependent , we can be dealed now with the
time-independent Schrödiger equation for a steady processes:
(1.32
)
-
Ñ
2
2 m
“+Ur

 yr


 = E yr


where eigen value E is total energy and eigen function y(r

) for the patrticle in the steady state.
The function y(r

) must be finite, continuous, and single-valued for all values of x, y, z and has the
similar statistical meaning like Yr

, t does.
1.4 Simple Quantum Mechanics Problem Solutions
1.4.1 The infinite square-shaped quantum well
The infinite square-shaped well potential is the simplest of all possible potential wells. The one-
dimension infinite square well potential is illustrated in Fig. 2.3(a) and is defined as
Fig. 2.3. a) Schematic illustration of the 1-D infinite square quantum well (QW). The solutions of this QW are shown in the
terms of b) eigen-functions Y
n
xand eigen-state energies E
n
, and c) probability densities Y
n
*
xY
n
x.
SSED Le Tuan Chapters 2.nb 9
To find the stationary solutions for
y

n
(x) and
E
n,
we must find functions for
y
n
(x), which
satisfy the Schrödinger equation. The time-independent Schrödinger equation contains only the
differential operator d / dx, whose eigenfunctions are exponential or sinusoidal functions. Since the
Schrödinger equation has the form of an eigenvalue equation, it is reasonable to try only eigenfunc-
tions of the differential operator. Furthermore, we assume that
y
n
(x) = 0 for | x | > L / 2, because the
potential energy is infinitely high in the barrier regions. Since the 3rd Postulate requires that the
wave function be continuous, the wave function must have zero amplitude at the two potential
discontinuities, that is y
n
(x = ±L / 2) = 0. We therefore employ sinusoidal functions and differenti-
ate between states of even and odd symmetry in well region, like that
(1.33
)
y
n
x=
A cos
n+1p
L
, n = 0, 2, 4, and x §

L
2
A sin
n+1p
L
, n = 1, 3, 5, and x §
L
2
and vanishes in the outside
(1.34
)
y
n
x= 0,n = 0, 1, 2, and x >
L
2
The condition of continuity of wave functions yields the value of the constant A
(1.35
)
A =
2

L
By inspection of the Schrödinger equation with the determined above wave functions, one can
deduce the eigen values - energy levels for the QW:
(1.36
)
E
n
=

Ñ
2
2 m

n + 1p
L

2
, n = 0, 1, 2,
The lowest value has the name as ground state energy
(1.37
)
E
0
=
Ñ
2
p
2
2 mL
2
when other levels are n-th excited state energies with n = 1, 2, 3, , respectively. The spacing
between two adjacent energy levels, that is E
n
– E
n-1
, is proportional to n. Thus, the energetic
spacing between states increases with energy. The eigenstate energies are, as already mentioned,
expectation values of the total energy of the respective state. Its easily to show, based on Postulate
5, that the energy of a particle in an infinite square well is purely kinetic. The particle has no

potential energy.
1.4.2 The 1-D asymmetric and symmetric finite square-shaped quantum well
In contrast to the infinite square well, the finite square well has barriers of finite height. The
potential of a finite square well is shown in Fig. 2.4. The two barriers of the well have a different
height and therefore, the structure is denoted asymmetric square well. The potential energy is
constant within the three regions I, II, and III, as shown in Fig. 2.4. In order to obtain the
solutions to the Schrödinger equation for the square well potential, the solutions in a constant
p
otential will be considered first.
10 SSED Le Tuan Chapters 2.nb
Fig. 2. 4. 1-D asymmetric finite quantum well with the width of L and barrier heights U
I
and U
II
I
.
Assume that a particle with energy E is in a constant potential U. Then two cases can be distin-
guished, namely E > U and E < U. In the first case (E > U) the general solution to the timeindepen-
dent one-dimensional Schrödinger equation is given by
(1.38
)
yx = A coskx + B sinkx
where A and B are constants and
(1.39
)
k =
2 mE
Ñ
2
Insertion of the solution into the Schrödinger equation proves that it is indeed a correct solu-

tion. Thus the wave function is an oscillatory sinusoidal function in a constant potential with E >
U.
In the second case (E < U), the solution of the time-independent one-dimensional Schrödinger
equation is given by
(1.40
)
yx = Ce
k
x
+ De
-
k
x
where C and D are constants and
(1.41
)
k=
2 m U - E
Ñ
2
=
2 mU
Ñ
2
- k
2
Thus the wave function is an exponentially growing or decaying function in a constant potential
with E < U. Next, the solutions of an asymmetric and symmetric square well will be calculated.
The potential energy of the well is piecewise constant, as shown in Fig. 2.4. Having shown that the
wave functions in a constant potential are either sinusoidal or exponential, the wave functions in

the three regions I (x 0), II (0 < x < L), and III (x L), can be written as
(1.42
)
y
I
x = Ae
k
I
x
(1.43
)
y
II
x = A coskx + Bsinkx
(1.44
)
y
III
x = A coskL + BsinkLe
-k
III
x-L
When applying the boundary conditions for continuity of wave functions and their first differen-
tials, we have the first pair of them, i.e.
y
I
0 = y
II
0 and y
II

L = y
III
L, are already satisfied. The
second pair of equations,i.e.
y
I
'
0 = y
II
'
0 and y
II
'
L = y
III
'
L , gives
SSED Le Tuan Chapters 2.nb 11
(1.45
)
A k
I
– Bk= 0
(1.46
)
A k
I
I
I
cos kL– k sin kL+ B k

I
I
I
sin kL+ k cos kL = 0
This homogeneous system of equations has solutions, only if the determinant of the system
vanishes, so one obtains
(1.47
)
tankL =
k
L

k
I
L
+
k
III
L

k
2
L
2
-k
I
L k
I
I
I

L
which is the eigenvalue equation of the finite asymmetric square well.
For the finite symmetric square well, which is of great practical relevance, the eigenvalue
equation is given by
(1.48
)
tankL =
2
k
L
k
L
k
2
L
2
-k
2
L
2
where k = 
I
= 
I
II
. If is expressed as a function of k, then solving the eigenvalue equation
yields the eigenvalues of k and, the allowed energies E and decay constants , respectively. The
allowed energies are also called the eigenstate energies of the potential. We must give up a trivial
solution kL = 0 (and thus E = 0) which possesses no practical relevance. Non-trivial solutions of
the eigenvalue equation can be obtained by a graphical method. Fig. 2.5 shows the graph of the left-

hand and right-hand side of the eigenvalue equation. The dashed curve represents the right-hand
side of the eigenvalue equation. The intersections of the dashed curve with the periodic tangent
function are the solutions of the eigenvalue equation. The quantum state with the lowest non-trivial
solution is called the ground state of the well. States of higher energy are referred to as excited
states.
Fig. 2. 5. Graphical solution of the eigen-value equation for a symmetric QW. The function 2 k L
k
L/(k
2
L
2
-k
2
L
2
) is
expressed by the dashed curve. The crossing points of the tangent function and the dashed curve are solutions for the
eigen-value equation. The solution k L = 0 is a trivial solution having no practical relevance.
The dashed curve shown in Fig. 2.5 has two significant points, namely a pole and an end point.
The dashed curve has a pole when the denominator of the right-hand side of the eigenvalue equa-
tion vanishes, i. e., when kL = L
12 SSED Le Tuan Chapters 2.nb
(1.49
)
kL
Pole
=
mU
Ñ
2

L
and end point of the dashed curve, ensuring that k is still staying as a real value,
(1.50
)
kL
End point
=
2 mU
Ñ
2
L
There are no further bound state solutions to the eigenvalue equation beyond the end point.
Now once the eigenvalues of k and are known, one is allowed for the determination of the
constants A and B and the wave functions. It is possible to show that all states with even quantum
numbers (n = 0, 1, 2 … are of even symmetry with respect to the center of the well, i. e. (x) = (–
x
). All states with odd quantum numbers (n = 1, 3, 5 … are of odd symmetry with respect to the
center of the well, i. e. (x) = (–x). The even and odd state wave functions in the well are thus
of the form
(1.51
)
y
II
x =
A
II
cosk
II
x -
L

2
 , n = 0, 2, 4,
A
II
sink
II
x -
L
2
 , n = 1, 3, 5,
After normalization, one can determine the constant
A
I
I
. Then, by inspection of the Schrödinger
equation, the eigen value - energies - can be fixed.
We can also find the number of bound states, n
max
, in symmetric finite QW. Since the maxi-
mum energy of a bound state equals V
0
one finds:
(1.52
)
V
0
= E
max
= E
0

n
max
- 1
2
Then,
(1.53
)
n
max
= Int 1+
V
0
E
0
Also from the special case where n
max
= 2, one finds that there is only one bound state if V
0
<
E
0
.
1.4.3 1-D triangular quantum well
Consider a triangular well with constant electric field and an infinite barrier at x = 0, as shown
in Figure 2. 6.
Fig. 2. 6. 1-D triangular quantum well with electric field ¶.
SSED Le Tuan Chapters 2.nb 13
Schrödinger equation for this potential becomes:
(1.54
)

-
Ñ
2
2 m
d
2
yx
dx
2
+ q¶ x yx = E
n
yx , x > 0
Since the Airy function is a solution to y'' - x y = 0 and approaches zero as x approaches infin-
ity, one can write the solution to the Schrödinger equation as:
(1.55
)
yx = A Ai
2 m
Ñ
2
q
2
¶
2
1

3
q ¶ x - E
n


where A is a proportionality constant which can be determined by normalization. Since y(x = 0)
has to be zero at the infinite barrier the eigenvalues, E
n
, are obtained from
(1.56
)
E
n
=-
Ñ
2
q
2
¶
2
2 m
1

3
a
n
where a
n
is the n-th zero of the Airy function. The Airy function is shown in Figure 2. 7.
Fig. 2. 7. Airy function normalized to Ai(0) = 1.
Its zeros are approximately given by:
(1.57
)
a
n

=-
3 p
2
n -
1
4

2

3
, n = 1, 2,
and the corresponding energy values are:
(1.58
)
E
n
=
Ñ
2
2 m
1

3

3 p q ¶
2
n -
1
4


2

3
, n = 1, 2,
Of particular interest is the first energy level, obtained for n = 1, resulting in:
(1.59
)
E
1
=
Ñ
2
2 m
1

3
9 p q ¶
8
2

3
The description of the quantum states for particle in the one-dimension triangular quantum well
by the Airy function is mathematically adequate. Unfortunately, the practical calculation of quan-
tum states dealing with Airy functions is extremely tedious. That's why various approximation
approaches are being involved. The most frequently used one with success among them is standard
Fang - Howard wave function:
(1.60
)
z
x ==

k
3
2
1
2
xe
-
k
2
x
14 SSED Le Tuan Chapters 2.nb
Problems
2.1 Point A is at an eletrostatic potential of + 1 V relative to point B in vacuum. An electron
initially at rest at B moves to A. What energy (expressed in J and eV) does the electron have at A?
What's its velocity in m/s?
2. 2 Sketch an experimental setup in which a silver electrode (work function 4.73 eV) is sealed
in vacuum envelope with a second electrode and a voltage is applied between them to measure the
photoelectric effect. If light of wavelength 2164 Å is used, what voltage is required to reduce
photoelectric current between plates to zero? With the voltage turn off, what is the maximum
velocity of electrons (m/s) emitted into vacuum from the Ag surface?
2. 3 Calculate the Bohr radius (Å) and energy (eV) for n = 1, 2, and 3.
2. 4 Show that the constant in the equation n
21
= 
mq
4
2 Ñ
2
h 4 p ¶
0


2

1
n
1
2
-
1
n
2
2
 is the Rydberg con-
stant times the speed of light.
2. 5 Calculate l for the Lyman series to n = 5, the Balmer series to n = 7, and the Pasen series
to n = 10. Plot the result and find the wavelength limit for each of the three series?
2. 6 The de Broglie wavelength of a particle l = h/m V describes the wave-particle duality for
small particles such as electrons. What is the de Broglie wavelength (in Å) of an electron at 150
eV?The same question for electrons at 10 keV, which is typical of electron microscope? What are
the advantage of electron microscopes compairing with the one working witj visible light.
2. 7 Rewrite the position–momentum form of the Heisenberg uncertainty relation into one of
energy-time form.
2. 8 Separate the variables in time-dependent Schrödinger equation in order to obtain the time-
independent one and write out proerties of their solutions. Assuming Y(x) has a particular time-
independent value y
n
x, show that the corresponding energy equals the separation constant E
n
.
2. 9 Calculate the first three energy levels for an electron in an infinite quantum well with the

width of 10 Å.
is wavefunctions.
2. 10 Show that the choise of the magnetic quantum number m = , -3, -2, -1, 0, 1, 2, 3, is
proper, basing on the angular Schrödinger equation for electron in hydrogen atom.
2. 11 What do Li, Na, and K have in common? What do F, Cl, and Br have in common? What
are the electron configuratiosn for the ionized Na, Cl?
2. 12 The exciton is a hydrogen atom-like entity encountered in advanced semiconductor work.
It consists of an electron bound to a +q charged particle (a hole) of approximately equal mass. Bohr
atom results can be used in computing the allowed energy states of the exciton provided the
reduced mass, m
r
= m
+
m
-
/(m
+
+ m_) ª m
0
/2, replaces the electron mass in the Bohr atom formula
-
tion. In addition, the distance between the components of the exciton is always such that there are
intervening semiconductor atoms. Thus ¶
0
in the Bohr formulation must also be replaced by K
s
,
where K
s
is the semiconductor dielectric constant. Using K

s
= 11.8, determine the ground state (n =
1) energy of an exciton in Si.
2.13 Reflection High Energy Electron Diffraction (RHEED) has become a commonplace
technique for probing the atomic surface structures of materials. Under vacuum conditions an
electron beam is made to strike the surface of the sample under test at a glancing angle (J < 100).
The beam reflects off the surface of the material and subsequently strikes a phosphorescent screen.
Because of the wave-like nature of the electrons, a diffraction pattern characteristic of the first few
atomic layers is observed on the screen if the surface is flat and the material is crystalline. With a
distance between atomic planes of d = 5 Å, a glancing angle of 1
o
, and an operating de Broglie
wavelength for the electrons of 2dsinU, compute the electron energy employed in the technique.
SSED Le Tuan Chapters 2.nb 15
2.14 (a) Confirm, as pointed out in the text, that p
x
 = 0 for all energy states of a particle in a I-
D box.
(b) Verify that the normalization factor for wavefunctions describing a particle in a I-D box is
An =
2

a .
(c) Determine (x) for all energy states of a particle in a I-D box.
2.15 In examining the finite potential well solution, suppose we restrict our interest to energies
where x = E/Va § 0.01 and permit a to become very large such that a
Ñ
a x
max
>> p. Present an

argument that concludes the energy states of interest will be very closely approximated by those of
the infinitely deep potential well.
2.16 The symmetry of a problem sometimes allows one to simplify the mathematics leading to
a solution. If, for example, the x = 0 point in the finite potential well problem is moved to the
middle of the well, it becomes obvious that the wavefunction solution must be symmetric about x =
0; i.e., y(- x) = ± y(x).
16 SSED Le Tuan Chapters 2.nb
C H A P T E R 4
DENSITY OF STATES AND EQUILIBRIUM CARRIER
CONCENTRATION
1.1 Density of states in bulk semiconductors (3D)
Carriers occupy either localized impurity states or delocalized continuum states in the conduc-
tion band or valence band. In the simplest case, each impurity has a single, nondegenerate state.
Thus, the density of impurity states equals the concentration of impurities. The energy of the
impurity states is the same for all impurities (of the same species) as long as the impurities are
sufficiently far apart and do not couple. The density of continuum states is more complicated and
will be calculated in the following sections. Several cases will be considered including (i) a spheri-
cal, single-valley band, (ii) an anisotropic band, (iii) a band with multiple valleys, and (iv) the
density of states in a semiconductor with reduced degrees of freedom such as quantum wells,
quantum wires, and quantum boxes. Finally the effective density of states will be calculated.
1.1.1 Single-valley, spherical, and parabolic band
The simplest band structure of a semiconductor consists of a single valley with an isotropic (i.
e. spherical), parabolic dispersion relation. This situation is closely approximated by, for example,
the conduction band of GaAs. The electronic density of states is defined as the number of electron
states per unit volume and per unit energy. The finiteness of the density of states is a result of the
P
aul
i
principle, which states that only two electrons of opposite spin can occupy one volume
element in phase space. The phase space is defined as a six-dimensional space composed of real

space and momentum space. We now define a ‘volume’ element in phase space to consist of a
range of positions and momenta of a particle, such that the position and momentum of the particle
are distinguishable from the positions and momenta of other particles. In order to be distinguish-
able, the range of positions and momenta must be equal or exceed the range given by the uncer-
tainty relation. The volume element in phase space is then given by
(1.1)
Dx Dy Dz Dp
x
Dp
y
Dp
z
= 2 p Ñ
3
where the "volume" element in phase space is 2 p Ñ
3
. When we go to one-dimensional case, we
will have agian the well-known Heisenberd's principle
Dx Dp
x
= 2 p Ñ. Using the de Broglie relation
(p = Ñ k) the ‘volume’ of phase space can be written as
(1.2)
Dx Dy Dz Dk
x
Dk
y
Dk
z
= 2 p 

3
The density of states per unit energy and per unit volume, which is denoted by r
DOS
(E), allows
us to determine the total number of states per unit volume in an energy band with energies E
1
(bottom of band) and E
2
(top of band) according to
(1.3)
N =

E
1
E
2
r
DOS
E„ E
Note that N is the total number of states per unit volume, and r
DOS
(E) is the density of states
per unit energy per unit volume. To obtain the density of states per unit energy dE, we have to
determine how much unit-volumes of k-space is contained in the energy interval E and E + dE,
since we already know that one unit volume of k-space can contain two electrons of opposite spin.
In order to obtain the volume of k-space included between two energies, the dispersion relation
E
= E (k) will be employed. For a given dE one can easily determine the corresponding length in k-
space. The k-space length associated with an energy interval dE in one-dimensional case is simply
given by the slope of the dispersion relation. While the one-dimensional dispersion relations can be

illustrated easily, the three-dimensional dispersion relation cannot be illustrated in three-dimen-
sional space. To circumvent this difficulty, surfaces of constant energy in k-space are frequently
used to illustrate a three-dimensional dispersion relation. As an example, the constant energy
surface in k-space is illustrated in Fig. 4.1 for a spherical, single-valley band. A large separation of
the constant energy surfaces, i. e. a large Dk for a given DE, indicates a weakly curved dispersion
and a large effective mass.
Fig. 4. 1 Constant enerfy surface for a single-valley, isotropic band.
In order to obtain the volume of k-space enclosed between two constant energy surfaces, which
correspond to energies E and E + dE, we (first) determine dk associated with E and (second)
integrate over the entire constant energy surface. The ‘volume’ of k-space enclosed between the
two constant energy surfaces shown in Fig. 4. 2 is thus given by
Fig. 4. 2 Constant enerfy surfaces with energy E and E + dE used to calculate volume in k-space enclosed between the
two surfaces.
(1.4)
W
k-space
E = E

Surface
∑
k
∑Ek
„s
where ds is an area element of the constant energy surface. In a three-dimensional k-space we
use gradk = (∑ / ∑k
x
, ∑ / ∑k
y
, ∑ / ∑k
z

) and obtain
(1.5)
W
k-space
E = E

Surface
1
“
k
Ek
„s
Since an electron requires a volume of
2 p 
3
/2 = 4p
3
in phase space, the number of states per
unit volume is given by
(1.6)
NE =
1
4 p
3
E

Surface
1
“
k

Ek
„s
So finally, we obtain the density of states per unit energy and unit volume according to
2 SSED Le Tuan Chapter 4.nb
(1.7)
r
DOS
E =
1
4 p
3

Surface
1
“
k
Ek
„s
In this equation, the surface element ds is always perpendicular to the vector grad
k
E (k). Note
that the surface element ds is in k-space and that ds has the dimension m
-2
.
Next we apply the expression for the density of states to isotropic parabolic dispersion rela-
tions of a three-dimensional semiconductor. In this case the surface of constant energy is a sphere
of area 4p k
2
and the parabolic dispersion is E = Ñ
2

k
2
/ (2m*) + E
pot
where k is the wave vector.
Insertion of the dispersion in the last equation yields the density of states in a semiconductor with a
single-valley, isotropic, and parabolic band
(1.8)
r
DOS
3 D
E =
1
2 p
2
2 m
*
Ñ
2
3

2
E - E
pot
where E
pot
is a potential energy such as the conduction band edge or the valence band edge
energy, E
c
or E

v
, respectively.
1.1.2 Single-valley, anisotropic, parabolic band
In an anisotropic single-valley band, the dispersion relation depends on the spatial direction.
Such an anisotropic dispersion is found in III–V semiconductors in which the L- or X- point of the
Brillouin zone is the lowest minimum, for example in GaP or AlAs. The surface of constant energy
is then no longer a sphere, but an ellipsoid, as shown in Fig. 4.3. The three main axes of the ellip-
soid may have different lengths, and thus the three dispersion relations are curved differently. If the
main axes of the ellipsoid align with a cartesian coordinate system, the dispersion relation is
(1.9)
E =
Ñ
2
k
x
2
2 m
x
*
+
Ñ
2
k
y
2
2 m
y
*
+
Ñ

2
k
z
2
2 m
z
*
Fig. 4. 3 Ellipsoidal constant energy surface with a weakly curved dispersion relation along the k
x
axis and strongly curved
dispersion relation along the k
y
and k
z
axis.
The vector grad
k
E is given by grad
k
E = (Ñ
2
k
x
/m
x
*
, Ñ
2
k
y

/m
y
*
, Ñ
2
k
z
/m
z
*
). Since the vector grad
k
E is
perpendicular on the surface element, the absolute values of ds and grad
k
E can be taken for the
integration. Integration of Eq. (4.7) with the dispersion relation of Eq. (4.9) yields the density of
states in an anisotropic semiconductor with parabolic dispersion relations, i. e.
(1.10
)
r
DOS
E =
2
p
2
Ñ
3
m
x

*
m
y
*
m
z
*
E - E
pot
SSED Le Tuan Chapter 4.nb 3
If the main axes of the constant-energy ellipsoid do not align with the
k
x
,
k
y
, and
k
z
axes of the
coordinate system then m
x
*
, m
y
*
, and m
z
*
can be formally replaced by m

1
*
, m
2
*
, and m
3
*
.
Frequently, the constant energy surfaces are rotational ellipsoids, that is, two of the main axes
of the ellipsoid are identical. The axes are then denoted as the transversal and the longitudinal axes
for the short and long axes, respectively. Such a rotational ellipsoid is schematically shown in Fig.
4.3. A relatively light mass is associated with the (short) transversal axis, while a relatively heavy
mass is associated with the (long) longitudinal axis. If the masses are denoted as m
t
*
and m
l
*
for the
transversal and the longitudinal mass, respectively, Eq. (4.10) can be modified according to
(1.11
)
r
DOS
E =
2
p
2
Ñ

3
m
l
*
m
t
*2
E - E
pot
The anisotropic masses m
x
*
, m
y
*
, m
z
*
, m
t
*
, and m
l
*
are frequently used to define a density-ofstates
effective mass. This mass is given by
(1.12
)
m
DOS

*
= m
x
*
m
y
*
m
z
*

1

3
(1.13
)
m
DOS
*
= m
l
*
m
t
*2

1

3
The density of states is then given by

(1.14
)
r
DOS
E =
2
p
2
Ñ
3
m
DOS
*

3

2
E - E
pot
Note that for isotropic semiconductors the effective mass coincides with the density-of-states
effective mass.
1.1.3 Multiple valleys
At several points of the Brillouin zone, several equivalent minima occur. For example, eight
equivalent minima occur at the L-point as schematically shown in Fig. 4.4. Each of the valleys can
accommodate carriers, since the minima occur at different k
x
, k
y
, and k
z

values, i. e. the Pauli
principle is not violated. The density of states is thus obtained by multiplication with the number of
equivalent minima, that is
(1.15
)
r
DOS
E =
M
c
2
p
2
Ñ
3
m
1
*
m
2
*
m
3
*
E - E
pot
where M
c
is the number of equivalent minima and m
1

*
, m
2
*
, and m
3
*
are the effective masses for
motion along the three main axes of the ellipsoid.
Fig. 4. 4 Constant energy surfaces for the L-point of the Brillouin zone. The band structure consists of eight equivalent
rotational ellipsoids.
4 SSED Le Tuan Chapter 4.nb
Density of states in semiconductors with reduced
dimensionality (2D, 1D, 0D)
Semiconductor heterostructure allows one to change the band energies in a controlled
way and confine charge carriers to two (2D), one (1D), or zero (0D) spatial dimensions.
Due to the confinement of carriers, the dispersion relation along the confinement direc-
tion is changed. The change in dispersion relation results in a change in the density of
states.
Confinement of a carrier in one spatial dimension, e. g. the z-direction results in the
formation of quantum states for motion along this direction. Consider the ground state in
a quantum well of width L
z
with infinitely high walls. The ground-state energy is
obtained from the solution of Schrödinger’s equation and is given by
(1.16
)
E
0
=

Ñ
2
2 m
*
p
L
z
2
The particle in the quantum well can assume a range of momenta in the z-direction;
the range is given by the uncertainty principle, i. e.
(1.17
)
Dk
z
=
Dp
z
Ñ
=
2 p
L
z
The dispersion relation for motion along the confinement (z-) direction is thus given
by
(1.18
)
E = E
0
for entire range of k
z

The dispersion is flat, i. e. constant for all values of k
z
. The z-component of the vector
grad
k
E (see Eq. 4.9) is therefore zero and need not be considered.
We next consider the x- and y- direction and recall that the Schrödinger equation is
separable for the three spatial dimensions. Thus, the kinetic energy in the x y - plane is
given by
SSED Le Tuan Chapter 4.nb 5
(1.19
)
E =
Ñ
2
2 m
x
*
k
x
2
+ k
y
2

for a parabolic dispersion.
Fig. 4. 5 Constant energy surfaces of a a) 3-dimensional, b) 2-dimensional, and c) 1-dimensional systems. The
surfaces are a sphere, a circle, a point for 3D, 2D, and 1D systems, respectively.
The surface of constant energy for the dispersion relation given by Eq. (4.18) is
shown in Fig. 4.5, and is a circle around k

x
= k
y
= 0. The density of states of such a 2D
electron system is obtained by similar considerations as for the 3D case. The reduced
p
hase space now consists only of the x y -plane and the k
x
and k
y
coordinates. Corre-
spondingly, the two-dimensional density of states is the number of states per unit-area
and unit-energy. The volume of k-space between the circles of constant energy is given
by Eq. (4.5). The equation is evaluated most conveniently in polar coordinates in which
k
r
= k
x
2
+ k
y
2

1

2
is the radial component of the k vector. The surface integral reduces to
a line integral and the total length of the circular line is 2p k
r
. The volume of k-space

then obtained is
(1.20
)
W
k-space
2 D
E = dE

Surface
1
“
k
Ek
„s =
2 p m
*
Ñ
2
Since two electrons of opposite spin require a volume element of 2 p
2
in phase
space, the density of states of a 2D electron system is given by
(1.21
)
r
DOS
2 D
E =
m
*

p Ñ
2
, E ¥ E
0
6 SSED Le Tuan Chapter 4.nb
where
E
0
is the ground state of the quantum well system. For energies E ¥
E
0
, the 2D
density of states is a constant and does not depend on energy. If the 2D semiconductor
has more than one quantum state, each quantum state has a state density of Eq. (4.20).
The total density of states can be written as
(1.22
)
r
DOS
2 D
E =
m
*
p Ñ
2

n
sE - E
n


where E
n
are the energies of quantized states and s(E –E
n
) is the step function.
We next consider a one-dimensional (1D) system, the quantum wire, in which only
one direction of motion is allowed, e. g. along the x-direction. The dispersion relation is
then given by E = Ñ
2
k
x
2
/ (2m*). The ‘volume’ (i. e. length-unit) in k-space is obtained in
analogy to the
(1.23
)
W
k-space
1 D
E =

Surface
sk
x
- k
x
0

“
k

Ek
x

„s =
m
*
2 Ñ
2
E - E
0

, E ¥ E
0
The volume in phase space of two electrons with opposite spin is given by 2p and
thus the 1D density of states is given by
(1.24
)
r
DOS
1 D
E =
1
p Ñ
m
*
2 E - E
0

, E ¥ E
0

Note that the density of states in a 3-, 2- and 1-dimensional system has a functional
dependence on energy according to E
1

2
, E
0
, and E
- 1

2
, respectively. For more than
one quantized state, the 1D density of states is given by
(1.25
)
r
DOS
1 D
E =
1
p Ñ

n
m
*
2 E - E
n

sE - E
n


where E
n
are the energies of the quantized states of the wire.
SSED Le Tuan Chapter 4.nb 7
Fig. 4. 6 Electronic density of states of semiconductor with 3, 2, 1 and 0 degrees of fredoom for electron propaga-
tion. Systems with 2, 1 and 0 degrees of fredoom are reffered to as quantum wells, quantum wires, and quantum
dot, respectively.
Finally, we consider the density of states in a zero-dimensional (0D) system, the
quantum box. No free motion is possible in such a quantum box, since the electron is
confined in all three spatial dimensions. Consequently, there is no k-space available
which could be filled up with electrons. Each quantum state of a 0D system can therefore
be occupied by only two electrons. The density of states is therefore described by a d-
function.
(1.26
)
r
DOS
0
D
E = 2 sE - E
0

For more than one quantum state, the density of states is given by
(1.27
)
r
DOS
0 D
E =

1
p Ñ

n
2 sE - E
n

The densities of states for one quantized level for a 3D, 2D, 1D, and 0D electron
system are schematically illustrated in Fig. 4.6.
8 SSED Le Tuan Chapter 4.nb
1.3 Effective density of states in 3D, 2D, 1D, and 0D
semiconductors
The effective density of states is introduced in order to simplify the calculation of the popula-
tion of the conduction and valence band. The basic simplification made is that all band states are
assumed to be located directly at the band edge. This situation is illustrated in Fig. 4.7 for the
conduction band. The 3D density of states has square-root dependence on energy. The effective
density of states is d-function-like and occurs at the bottom of the conduction band.
Fig. 4. 7 Energy dependent density od states, r
DOS
3 D
, and effective density of states, N
c
3 D
, at the bottom of the conduction band.
An electronic state can be either occupied by an electron or unoccupied. Quantum mechanics
allows us to attribute to the state a probability of occupation. The total electron concentration in a
band is then obtained by integration over the product of state density and the probability that the
state is occupied, that is
(1.28
)

n =

E
bottom
E
top
r
DOS
E f E„ E
where f(E) is the (dimensionless) probability that a state of energy E is populated. The limits of the
integration are the bottom and the top energy of the band, since the electron concentration in the
entire band is of interest.
As will be shown in a subsequent section, the probability of occupation, f(E), is given by the
Maxwell–Boltzmann distribution, also frequently referred to as the Boltzmann distribution, is
given by
(1.29
)
f
B
E = exp -
E
-
E
F
k
B
T
where E
F
is the Fermi energy. Using Eq. (4.26), the electron concentration can be determined by

evaluating the integral.
The effective density of states at the bottom of the conduction band is now defined as the
density of states which yields, with the Boltzmann distribution, the same electron concentration as
the true density of states, that is
(1.30
)
n =

E
bottom
E
top
r
DOS
E f
B
E„ E = N
c
f
B
E = E
c

where N
c
is the effective density of states at the bottom of the conduction band and E
c
is the
energy of the bottom of this band. Strictly speaking, the effective density of states has no physical
meaning but is simply a mathematical tool to facilitate calculations. For completeness, Eqs. (4.26)

and (4.28) are now given explicitly using the Boltzmann distribution and the density of states of an
isotropic three-dimensional semiconductor:
SSED Le Tuan Chapter 4.nb 9