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Band Theory of Solids

Dr. Anurag Srivastava Web address: http://tiiciiitm.com/profanurag Email: [email protected] Visit me: Room-110, Block-E, IIITM Campus

Crystal Physics ABV- IIITM-Gwalior (MP) India Energy band structures of solids

Crystal Physics ABV- IIITM-Gwalior (MP) India Formation of Energy Bands

From , we know that the energy of the bound of the hydrogen atom is quantized with associated radial probability density functions. The wave function for the lowest electron energy state

When two hydrogen atoms are brought in close proximity, their wave functions will overlap , which means the two will interact. This interaction results in the quantized energy level splitting into two discrete energy levels.

Crystal Physics ABV- IIITM-Gwalior (MP) India Formation of Energy Bands

Similarly, when a number of hydrogen-type atoms that are arranged in a periodic lattice and initially very far apart are pushed together, the initial energy level will split into a band of discrete energy levels.

According to the Pauli exclusion principle, the total number of quantum states will remain the same after the joining of atoms to form a system (crystal).

There will be many energy levels within the allowed band in order to accommodate all of the electrons in a crystal. As an example, suppose that we have a system of 1019 one-electron atoms and the width of the energy band at the equilibrium inter-atomic distance is 1 eV. If the spacing between neighboring energy levels is the same, the difference in neighboring energy levels will be 10−19 eV, which is extremely small so that we have a quasi-continuous energy distribution through the

Crystal Physics ABV- IIITM-Gwalior (MP) India Energy bands in solid  different conductivity

It is to clearly show two overlapping energy bands, not filling of electron states in real space.

K-Ch.8 Fig.1 Real space coordinates

Crystal Physics ABV- IIITM-Gwalior (MP) India Energy Band

 When 2 Si atoms are brought together:

- Linear combinations of atomic orbitals (LCAO) for two-electron wave functions (1, 2) of atoms leads to 2 distinct “normal” modes: a higher energy anti-bonding (anti- symmetric) orbital, and a lower energy bonding (symmetric) orbital (Pauli‟s exclusion principle) - For bonding state: an electron in the region between the two nuclei is attracted by two nuclei  V(r) is lowered in this region  electron probability density is higher in this region than for anti-bonding state  It is the lowering of E of bonding state that causes cohesion of crystal

Crystal Physics ABV- IIITM-Gwalior (MP) India distribution of an isolated silicon atom

Crystal Physics ABV- IIITM-Gwalior (MP) India Example: consider an electron traveling at a velocity of 107 cm/sec. if the velocity increases by 1 cm/sec, calculate the change in its kinetic energy.

Solution:

Comment: the kinetic energy change is orders of magnitude larger than the energy spacing in the allowed energy band, which suggests that the discrete energies within an allowed energy band can be treated as a quasi- continuous distribution. Crystal Physics ABV- IIITM-Gwalior (MP) India Allowed and Forbidden Energy Bands

Consider again a periodic arrangement of atoms. Each atom contains electrons up to n = 3 energy level. If these atoms are brought together, the outermost electrons in the n = 3 energy shell will begin to interact and split into a band of allowed energies. As the atoms move closer, the electrons in the n = 2 shell, and finally the innermost electrons in the n = 1 shell, will also form two bands of allowed energies.

Crystal Physics ABV- IIITM-Gwalior (MP) India 1.12 eV (Si)

Crystal Physics ABV- IIITM-Gwalior (MP) India Crystal Physics ABV- IIITM-Gwalior (MP) India Crystal Physics ABV- IIITM-Gwalior (MP) India Crystal Physics ABV- IIITM-Gwalior (MP) India Energy band structures of Siand GaAs. Circles (º) indicate holes in the valence bands and dots (•) indicate electrons in the conduction bands

Crystal Physics ABV- IIITM-Gwalior (MP) India Crystal Physics ABV- IIITM-Gwalior (MP) India Crystal Physics ABV- IIITM-Gwalior (MP) India e Ee

Ei

Ep

Intrinsic semiconductor. (a) Schematic band diagram. (b) Density of states. (c) Fermi distribution function. (d) Carrier concentration.

Crystal Physics ABV- IIITM-Gwalior (MP) India Intrinsic carrier densities in Siand GaAs as a function of the reciprocal of

Crystal Physics ABV- IIITM-Gwalior (MP) India Crystal Physics ABV- IIITM-Gwalior (MP) India Crystal Physics ABV- IIITM-Gwalior (MP) India for Siand GaAsas a function of temperature and impurity concentration. The dependence of the bandgap on temperature

Crystal Physics ABV- IIITM-Gwalior (MP) India Crystal Physics ABV- IIITM-Gwalior (MP) India Crystal Physics ABV- IIITM-Gwalior (MP) India Crystal Physics ABV- IIITM-Gwalior (MP) India Crystal Physics ABV- IIITM-Gwalior (MP) India Crystal Physics ABV- IIITM-Gwalior (MP) India Crystal Physics ABV- IIITM-Gwalior (MP) India Crystal Physics ABV- IIITM-Gwalior (MP) India Crystal Physics ABV- IIITM-Gwalior (MP) India Crystal Physics ABV- IIITM-Gwalior (MP) India Simplified schematic drawing of the Czochralskipuller. Clockwise (CW), counterclockwise (CCW).

Crystal Physics ABV- IIITM-Gwalior (MP) India Crystal Physics ABV- IIITM-Gwalior (MP) India Crystal Physics ABV- IIITM-Gwalior (MP) India Fermi Energy (EF) and Fermi-Dirac Distribution Function f(E)

Fermi Energy (EF) Fermi Energy is the energy of the state at which the probability of electron occupation is ½ at any temperature above 0 K.

It is also the maximum kinetic energy that a free electron can have at 0 K.

The energy of the highest occupied level at temperature is called the Fermi Energy or Fermi Level.

Crystal Physics ABV- IIITM-Gwalior (MP) India The Fermi energy at 0 K for metals is given by

N - number of possible quantum states 3N  2 / 3  h 2  E    V - volume F     8m  m - mass of electron     h - planck's constant

When temperature increases, the Fermi level or Fermi energy also slightly decreases.

The Fermi energy at non–zero ,

 2   2  k T  E  E 1    F F0  12  E     F0   Here the subscript „0‟ refers to the quantities at zero kelvin.

Crystal Physics ABV- IIITM-Gwalior (MP) India Fermi-Dirac Distribution Function f(E)

In quantum statistics, a branch of physics, Fermi–Dirac statistics describe a distribution of particles over energy states in systems consisting of many that obey the "Pauli exclusion principle". It is named after Enrico Fermi and Paul Dirac, each of whom discovered the method independently (although Fermi defined the statistics earlier than Dirac).

Fermion: is a particle that follows Fermi–Dirac statistics. These particles obey the Pauli exclusion principle. include all quarks and leptons, as well as all composite particles made of an odd number of these, such as all baryons and many atoms and nuclei. Fermions differ from bosons, which obey Bose–Einstein statistics.

Crystal Physics ABV- IIITM-Gwalior (MP) India Fermi-Dirac Distribution Function f(E)

We can approximate the average energy level at which an electron is present is with the Fermi-Dirac distribution: where E is the energy level, k is the Boltzmann constant, T is the (absolute)

temperature, and EF is the Fermi level. The Fermi level is defined as the of electrons, as well as the (hypothetical) energy level where the probability of an electron being present is 50%.

Crystal Physics ABV- IIITM-Gwalior (MP) India The significance of the Fermi energy is most clearly seen by setting T=0. At absolute zero, the probability is =1 for energies less than the Fermi energy and zero for energies greater than the Fermi energy. We picture all the levels up to the Fermi energy as filled, but no particle has a greater energy. This is entirely consistent with the Pauli exclusion principle where each quantum state can have one but only one particle.

Crystal Physics ABV- IIITM-Gwalior (MP) India Important Definitions:

Crystal Physics ABV- IIITM-Gwalior (MP) India Crystal Physics ABV- IIITM-Gwalior (MP) India Effect of Temperature on f(E)

Crystal Physics ABV- IIITM-Gwalior (MP) India Boltzmann Approximation

Probability that a state is empty (i.e. occupied by a hole):

Crystal Physics ABV- IIITM-Gwalior (MP) India cnx.org/content/m13458/latest Equilibrium Distribution of Carriers

 Obtain n(E) by multiplying gc(E) and f(E)

Energy band Density of Probability of Carrier diagram States, gc(E) × occupancy, f(E) = distribution, n(E)

Crystal Physics ABV- IIITM-Gwalior (MP) India cnx.org/content/m13458/latest  Obtain p(E) by multiplying gv(E) and 1-f(E)

Energy band Density of Probability of Carrier diagram States, gv(E) ×occupancy, 1-f(E) = distribution, p(E)

Crystal Physics ABV- IIITM-Gwalior (MP) India Crystal Physics ABV- IIITM-Gwalior (MP) India