band theory of solids -...

Crystal Physics ABV- IIITM-Gwalior (MP) India

Band Theory of Solids

Dr. Anurag Srivastava

Web address: http://tiiciiitm.com/profanurag

Email: [email protected]

Visit me: Room-110, Block-E, IIITM Campus

http://tiiciiitm.com/profanurag

mailto:[email protected]

ABV- IIITM-Gwalior (MP) IndiaCrystal Physics

Energy band structures of solids


Formation of Energy Bands

From quantum mechanics, we know that the energy of the bound electron of

the hydrogen atom is quantized with associated radial probability density

functions.

When two hydrogen atoms are brought

in close proximity, their wave functions

will overlap , which means the two

electrons will interact.

This interaction results in the quantized

energy level splitting into two discrete

energy levels.

The wave function

for the lowest

electron energy state


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

Formation of Energy Bands


Energy bands in solid different conductivity

K-Ch.8 Fig.1Real space

coordinates

It is to clearly show two overlapping

energy bands, not filling of electron states

in real space.


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

Energy Band


Quantum state distribution of an isolated silicon atom


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.


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.


1.12 eV (Si)


Energy band structures of Siand GaAs. Circles (º) indicate holes in the

valence bands and dots (•) indicate electrons in the conduction bands


Intrinsic semiconductor. (a) Schematic band diagram. (b) Density of states. (c)

Fermi distribution function. (d) Carrier concentration.

eEe

Ep

Ei


Intrinsic carrier densities in Siand GaAs as a function of the

reciprocal of temperature


Fermi level for Siand GaAsas a function of temperature and impurity

concentration. The dependence of the bandgap on temperature


Simplified schematic

drawing

of the Czochralskipuller.

Clockwise (CW),

counterclockwise (CCW).


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 absolute

zero temperature is called the Fermi Energy or Fermi Level.

Fermi Energy (EF) and Fermi-Dirac

Distribution Function f(E)


The Fermi energy at 0 K for metals is given by

m

hNEF

8

3 23/2

When temperature increases, the Fermi level or Fermi

energy also slightly decreases.

The Fermi energy at non–zero temperatures,

22

0

0 121

F

FFE

TkEE

Here the subscript „0‟ refers to the quantities at zero kelvin.

N - number of possible quantum states

V - volume

m - mass of electron

h - planck's constant


In quantum statistics, a branch of physics, Fermi–Dirac statistics describe

a distribution of particles over energy states in systems consisting of

many identical particles 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).

Fermi-Dirac Distribution Function f(E)

Fermion: is a particle that

follows Fermi–Dirac statistics.

These particles obey the Pauli

exclusion principle.

Fermions 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.


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

chemical potential of electrons, as well as the (hypothetical) energy level where

the probability of an electron being present is 50%.

Fermi-Dirac Distribution Function f(E)


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.


Important

Definitions:


Effect of Temperature on f(E)


Boltzmann Approximation

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


Equilibrium Distribution of Carriers

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

Energy band

diagram

Density of

States, gc(E)

Probability of

occupancy, f(E)Carrier

distribution, n(E)× =

cnx.org/content/m13458/latest


Obtain p(E) by multiplying gv(E) and 1-f(E)

Energy band

diagram

Density of

States, gv(E)

Probability of

occupancy, 1-f(E)Carrier

distribution, p(E)× =

cnx.org/content/m13458/latest

band theory of solids -...

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