Micro and Nanoelectronics
ELL732
Lecture 6
Dhiman Mallick
Department of Electrical Engineering, IIT Delhi
1
Semester I – 2024 - 2025
The Effective Mass
❖ The movement of an electron in a lattice will be different from that of an electron in free space.
❖ In addition to an externally applied force, there are internal forces in the crystal due to positively
charged ions or protons and negatively charged electrons, which will influence the motion of electrons
in the lattice.
❖ We can write
where Ftotal , Fext , and Fint are the total force, the externally applied force, and the internal forces,
respectively, acting on a particle in a crystal. The parameter a is the acceleration and m is the rest mass of
the particle.
❖ Since it is difficult to take into account all of the internal forces, we can write
where the acceleration a is now directly related to the external force.
❖ The parameter m*, called the effective mass, considers the particle mass and also the effect of the
internal forces.
The Effective Mass
❖ The effective mass of an electron in a crystal can be related to the E versus k curves.
❖ The second derivative of E with respect to k is inversely proportional to the mass of the particle.
❖ For the case of a free electron, the mass is a constant (nonrelativistic effect), so the second derivative
function is a constant.
❖ The mass of the electron is also a positive quantity.
❖ If we apply an electric field to the free electron and use Newton’s classical equation of motion,
❖ where a is the acceleration, E is the applied electric field, and e is the magnitude of the electronic
charge.
❖ The motion of the free electron is in the opposite direction to the applied electric field because of the
negative charge.
The Effective Mass
The energy near the bottom of conduction band may be The energy near the top of valence band may be
approximated by a parabola approximated by a parabola
where m* is called the effective mass. Since C1>0, m*>0 where m* is called the effective mass. Since C1>0, m*>0
If an electric field is applied to the electron in the If an electric field is applied to the electron in the
bottom of the conduction band, acceleration is bottom of the conduction band, acceleration is
where mn* is the effective mass of the electron. where mn* is the effective mass of the electron.
Problem 1: Two possible conduction bands are shown in the E versus k diagram given in the figure. State
which band will result in the heavier electron effective mass; state why.
Problem 2: Two possible valence bands are shown in the E versus k diagram given in the figure. State which
band will result in the heavier hole; state why.
Problem 3: The E versus k diagram for a particular allowed energy band is shown in the figure. Determine (a)
the sign of the effective mass and (b) the direction of velocity for a particle at each of the four positions
shown.
In three-dimensional space
One problem encountered in extending the
potential function to a three-dimensional crystal
is that the distance between atoms varies as the
direction through the crystal changes.
Direct and Indirect Semiconductors
Energy-band structures of (a) GaAs and (b) Si.
Direct and Indirect Semiconductors
❖ In GaAs, the minimum conduction band energy and maximum valence band energy occur at the same k
value. A semiconductor with this property is said to be a direct bandgap semiconductor.
❖ Transitions between the two allowed bands can take place with no change in crystal momentum.
❖ Significant effect in optical properties - GaAs and other direct bandgap materials are ideally suited for
use in semiconductor lasers and other optical devices – all compound semiconductors are not direct.
❖ A semiconductor whose maximum valence band energy and minimum conduction band energy do not
occur at the same k value is called an indirect bandgap semiconductor.
❖ A transition in an indirect bandgap material must necessarily include an interaction with the crystal so
that crystal momentum is conserved.
❖ Example – Si, Ge.
Density of States
❖ The number of carriers that can contribute to the conduction process is a function of the number of
available energy or quantum states by the Pauli exclusion principle.
❖ Density of allowable energy states - the number of available energy or quantum states per unit crystal
volume.
❖ We must determine the density of these allowed energy states as a function of energy in order to
calculate the electron and hole concentrations.
The density of quantum states is a function of energy E .
As the energy of this free electron becomes small, the number of available quantum states decreases.
Density of States
The density of allowed electronic energy states in the
conduction band
for E≥Ec. As the energy of the electron in the conduction
band decreases, the number of available quantum states
also decreases.
The density of allowed electronic energy states in the
conduction band
for E≤Ev.
Fermi-Dirac Distribution
In dealing with large numbers of particles, we are interested only in the statistical behavior of the group as
a whole rather than in the behavior of each individual particle.
The number density N(E) is the number of particles per unit volume per unit energy.
The function g(E) is the number of quantum states per unit volume per unit energy.
The function fF(E) is called the Fermi–Dirac distribution or probability function - gives the probability that a
quantum state at the energy E will be occupied by an electron. The energy EF is called the Fermi energy .
fF(E) is the ratio of filled to total quantum states at any energy E.
Fermi-Dirac Distribution
The Fermi probability function versus energy for different
The Fermi probability function versus energy for T= 0K.
temperatures.
Fermi-Dirac Distribution
The probability of a state being occupied, fF(E), and the probability of a state being empty,
1 - fF(E).
References
• Semiconductor Physics and Devices- Basic Principles by
Donald A. Neamen
• Solid State Electronic Devices by Ben G. Streetman and
Sanjay Kumar Banerjee
• Physics of Semiconductor Devices by S.M. Sze and Kwok K.
Ng
• State-of-the-art Research Papers
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