INTERBAND TRANSITIONS
In a semiconductor at low frequencies, the principal electronic
Conduction mechanism is associated with free carriers
In this Chapter we wish to study the relationship between the electronic band structure and the optical
properties in crystals. We shall confine ourselves here to the case of inter-band transitions. Emphasis will
be given to the discussion of that part of the structure in the optical constants which may be understood on
the basis of symmetry properties of the crystal. The optical measurements have been accumulated for all
types of solids and have been interpreted on the basis of quantum theory of band-to-band transitions.
When the photon energy increases and becomes comparable to the energy gap, a new conduction process
can occur. A photon can excite an electron from an occupied state in the valence band to an unoccupied
state in the conduction band. This is called an inter-band transition and is represented schematically by
the picture in Fig. 1. In this process the photon is absorbed, an excited electronic state is formed and a
hole is left behind. This process is a quantum mechanical in nature. We now discuss the factors that are
important in these transitions.
1. THRESHOLD ENERGY:
The threshold energy in inter-band transitions is always equal to the energy band gap. We
can plot the real part of the conductivity 𝜎1 (𝜔) vs incident photon energy. When the
conductivity goes to zero, we get threshold value as shown in Fig. 2.
Figure 1: Schematic diagram of an allowed interband Figure 2: Real part of the conductivity for an allowed optical
𝝎
transition. transition. We note that 𝝈𝟏 (𝝎) = 𝜺𝟐 (𝝎).
𝟒𝝅
2. NATURE OF MATERIAL:
Materials are
i. If the relative position of the conduction band minimum and the valence band
maximum both lies at the same k, it is called direct band gap material.
Conservation of the crystal momentum yields
𝑘𝐶𝐵 = 𝑘𝑉𝐵
ii. If the relative position of the conduction band minimum and the valence band
maximum both do not lie at the same k, it is called indirect band gap material.
There involve phonon to conserve the crystal momentum. Hence the conservation
of the crystal momentum yields
𝑘𝐶𝐵 = 𝑘𝑉𝐵 ∓ 𝑞
Where, q is the phonon wave vector. Because of the phonon contribution in
indirect band gap transition, we do not use indirect band gap materials for optical
devices.
TRANSITION RATE OF DIRECT BANDGAP MATERIALS
If the relative positions of the conduction band minimum and the valence band maximum both
lies at the same k, it is called direct band gap material. The
Conservation of the crystal momentum yields must be hold in direct band gap materials.
𝑘𝐶𝐵 = 𝑘𝑉𝐵
Figure 4: Optical transitions from a discrete level to a continuum
of states.
To find the transition rate that is the transition probability per unit time for direct band gap
materials, we use Fermi-Golden rule, which is defined as:
2𝜋
𝑊𝑖−𝑓 = |𝑀|2 𝑔(ℏ𝜔)…………(1)
ℏ
Where, 𝑀 =< 𝑓|𝐻 ′ |𝑖 > is the matrix element that couples initial and final states of an electron.
It describes the size of the given perturbation, and 𝑔(ℏ𝜔) is the density of states (DOS).
It is used for the transitions between discrete levels such as those found in individual atoms to
continuous bands of levels. This situation is the usual one in solids, and is shown in Fig. 4.
Where 𝑔(ℏ𝜔)𝑑𝐸 is the number of states that fall in the energy range E to E+dE. Transitions are
possible to any of the states that fall within this energy range. The density of states factor in Eq.
(1) is very important in solid state problems. In the discussion below, we consider the matrix
element first, and we consider DOS afterwards.
Matrix Element
The matrix element describes the effect of the external perturbation caused by the light wave on
the electrons which are initially present in the valence band. It is given by
𝑀 =< 𝑓|𝐻 ′ |𝑖 >
= ∫ 𝜓𝑓∗ (𝒓) 𝐻 ′ (𝒓)𝜓𝑖 (𝒓)𝑑 3 𝒓…………………..(2)
Where, 𝐻 ′ is the perturbation associated with the light wave i.e. it represents interaction between
the atom and the radiation field. r is the position vector of the electron, 𝜓𝑖 (𝒓) and 𝜓𝑓 (𝒓) are the
wave functions of initial | 𝑖 > and final |𝑓 > states respectively. In order to find this matrix
element, we shall use the semi-classical approach in which the electrons are treated quantum
mechanically while the photons are described by electromagnetic waves (classically ).
To evaluate Matrix element in Eq. (2), we need to know the wave functions of the states, and
also the form of the perturbation due to the light wave. The perturbation due to the light can be
evaluated by calculating the effect of the electromagnetic field on the electron in the atom. From
classical electromagnetism we know that the field changes the momentum of a charged particle
from p to (p - qA), where q is the charge and A is the vector potential defined by
𝑩= 𝛁×𝑨
The Hamiltonian for an electron with q = -e in an electromagnetic field is therefore:
1
𝐻= (𝐩 + e𝐀)2 + 𝑉(𝒓)
2𝑚0
𝐩𝟐 𝑒 𝑒 2 𝐴2
= + 𝑉(𝒓) + (𝑷 ∙ 𝑨 + 𝑨 ∙ 𝑷) + = 𝑯𝟎 + 𝑯′
2𝑚0 2𝑚0 2𝑚0
𝐩𝟐 𝑒 𝑒 2 𝐴2
Where, 𝑯𝟎 = 2𝑚 + 𝑉(𝒓) and 𝑯′ = 2𝑚 (𝑷 ∙ 𝑨 + 𝑨 ∙ 𝑷) + are the Hamiltonians of an
0 0 2𝑚0
electron when there is no field and when the field is applied.
Let we try to simplify the perturbed Hamiltonian as,
𝑒 𝑒 2 𝐴2
𝑯′ = 2𝑚 (𝑷 ∙ 𝑨 + 𝑨 ∙ 𝑷) + …………….(3)
0 2𝑚0
We make the following approximations to simplify it,
1. For low intensity light, A will be small; therefore, the term involving 𝐴2 can be
neglected.
2. A and P commute i.e., 𝑷 ∙ 𝑨 − 𝑨 ∙ 𝑷 = 𝟎.
Using the above approximations, Eq. (3) can be written as:
𝑒
𝑯′ = 𝑚 𝑷 ∙ 𝑨………………………..(4)
0
Since E and B both vary in time and space of an electromagnetic wave, so A must also do so,
Therefore,
𝑨 = 𝑨𝟎 𝑒 𝑖(𝒌.𝒓−𝜔𝑡)
= 𝑨𝟎 𝑒 𝑖(𝒌.𝒓) 𝑒 −𝑖(𝜔𝑡)
By applying the Taylor’s series, we get
(𝒊𝒌.𝒓)2
= 𝑨𝟎 [𝟏 + (𝒊𝒌. 𝒓) + + ⋯ ]𝑒 −𝑖(𝜔𝑡)
2
At optical frequencies,
𝜆~1 𝜇𝑚
𝑟~10−10 𝑚 (size of an atom)
2𝜋|𝒓|
⇒ |𝒌. 𝒓| = ~10−3
𝜆
Therefore, 𝒌. 𝒓 and its higher powers can be neglected in the expression of A and it can be
written as
𝑨 ≅ 𝑨𝟎 𝑒 −𝑖(𝜔𝑡)
For 𝑡 → 0, we get
𝑨~𝑨𝟎
Therefore, Eq. (4) can be written as
𝑒
𝑯′ = 𝑷 ∙ 𝑨𝟎
𝑚0
So the matrix element in Eq. (2) will becomes,
𝑒
𝑀 = 𝑚 < 𝑓|𝑷 ∙ 𝑨𝟎 |𝑖 >…………………………(5)
0
The equation of motion for time dependent operator O is given by,
𝑑𝐎 𝑖
= [𝑯 , 𝑶]
𝑑𝑡 ℏ 𝟎
𝑖
= ℏ (𝑯𝟎 𝑶 − 𝑶𝑯𝟎 )
Similarly, we can write
𝑑𝐫 𝑖
= [𝑯 , 𝒓]
𝑑𝑡 ℏ 𝟎
𝑖
= ℏ (𝑯𝟎 𝒓 − 𝒓𝑯𝟎 )
𝑑𝐫
∴ < 𝑓|𝑷|𝑖 >= 𝑚0 < 𝑓 � � 𝑖 >
𝑑𝑡
𝑖
= ℏ 𝑚0 < 𝑓|(𝑯𝟎 𝒓 − 𝒓𝑯𝟎 )|𝑖 >
𝑖
= ℏ 𝑚0 ( 𝐸𝑓 − 𝐸𝑖 ) < 𝑓|𝒓|𝑖 >
(We use 𝐻𝜓 = 𝐸𝜓)
𝐴𝑠 𝐸𝑓 − 𝐸𝑖 = ℏ𝜔
∴ < 𝑓|𝑷|𝑖 >= 𝑖𝑚0 𝜔 < 𝑓|𝒓|𝑖 >………………………..(6)
Therefore, Eq. (5) becomes
𝑀 = 𝑖𝑒𝜔 < 𝑓|𝒓 ∙ 𝑨𝟎 |𝑖 >
=< 𝑓|𝑒𝒓 ∙ 𝑖𝜔𝑨𝟎 |𝑖 >
𝛿𝑨
As 𝑨~𝑨𝟎 and 𝑬 = − 𝛿𝑡 = 𝑖𝜔𝑨𝟎 , also 𝒑 = −𝑒𝒓, is the dipole moment.
∴ 𝑀 =< 𝑓|−𝒑 ∙ 𝑬|𝑖 >……………………(7)
By coparing (2) and (7), we get
𝐻 ′ = −𝒑 ∙ 𝑬………………………………..(8)
Where, 𝒑 = −𝑒𝒓, is the electric dipole moment. This is in fact exactly equal to the interaction
energy experienced by a dipole in an electric field. That’s why the interband transitions are also
known as electric dipole transitions.
The wave functions of electron
