Lecture+14+MAK +Free+Electron+Gas1

7/30/2019 Lecture+14+MAK +Free+Electron+Gas1

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PHY 3201 FIZIK KEADAAN PEPEJAL

Free Electron Fermi Gas

The properties of materials depends to a large extent

on the type of bonding and the energy distribution of

the bonding electrons. The bonding electrons in the

solid find themselves in the potential of their parent

nuclei, other nuclei and other electrons. Because ofthe intricate nature of the potential, the theory of

solids is quite complex.

However, it may be simplified by assuming simple

models for the potential appropriate to different typesof solids


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Many solids conduct electricity.

There are electrons that are not bound to atoms but are able tomove through the whole crystal.

Conducting solids fall into two main classes; metals and

semiconductors.

Metal - Resistivity increases by the addition of small amounts

of impurity. The resistivity normally decreases monotonically

with decreasing temperature.

Semiconductor Resistivity can be reduced by the addition ofsmall amounts of impurity.

Semiconductors tend to become insulators at low T.


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Why mobile electrons appear in some

solids and not others?

According to free electron model (FEM), the valance

electrons are responsible for the conduction of

electricity, and for this reason these electrons are

termed conduction electrons.

Na11 1s2 2s2 2p6 3s1

This valance electron, which occupies the third atomic

shell, is the electron which is responsible chemical

properties of Na.

Valance electron (loosely bound)

Core electrons


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The removal of the valance electrons leaves a

positively charged ion. The charge density associated the positive ion

cores is spread uniformly throughout the metal sothat the electrons move in a constant electrostaticpotential. All the details of the crystal structure islost when this assunption is made.

+

+ + +

+ +

A valance electron really belongs to the whole crystal,since it can move readily from one ion to its neighbour,and then the neighbours neighbour, and so on.

This mobile electron becomes a conduction electron in asolid.

According to Free Electron Fermi Gas this potential istaken as zero and the repulsive force betweenconduction electrons are also ignored.


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Free Electron Fermi Gas

The main assumptions for metals is as follows:

The atomic nuclei are fixed on the crystal lattice atregular positions.

The electrons in the inner shells of each atom are

tightly bound to the atom; the atomic nucleus ownsthem.

Electrons in the outer shells are not bound to anyparticular atom. They are fairly free to move around inthe crystal lattice, but they are still bound to the solid.

All atoms together provide a collective potential inwhich all the outer electrons are moving. The electronsin turn, by floating around from one atom to the other,provide bonds that keep all the atoms together.


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A free electron model is the simplest way to represent theelectronic structure of metals. According to this model, thevalence electrons of the constituent atoms of the crystalbecome conduction electrons and travel freely throughout

the crystal. The only restriction being that they cannotescape from the metal. Within this model we also neglectthe interaction of conduction electrons with ions of thelattice and the interaction between the conductionelectrons. Hence the conduction electrons in a metal are

modeled by a free electron gas.

This theory is called the free electron theory of metals


7/20PHY 3201 FIZIK KEADAAN PEPEJAL

The Time-IndependentSchrodinger Equation

The Schrodinger Equation is a mathematical description

of an electron as a wave as suggested by Schrodinger in

1926. The time independent Schrodinger Equation is

used when the properties of atomic systems have to be

calculated in stationary conditions, i.e. when the propertyof the surroundings of the electron does not change with

time. If the potential energy V, depends only on location

(and not, in addition, on time), :



The time independent Schrodinger equation has the

following form:

nnH

H is the Hamiltonian = Kinetic Energy +

Potential Energy

nis the energy of the electron in orbital n

n is the wavefunction for the orbital n

If we use the assumptions above, the Schrodinger equation for a

metal is simply that of a particle in a box and since the potential is

constant we might as well set it to zero. The Hamiltonian will then

consist of only the kinetic term

2

222

2))((

2

1

2 dx

d

mdx

di

dx

di

mm

pH



nnnndx

d

mH

2

22

2

Thus, the Schrodinger eqn. reduces to



Free electron

For one dimension the Schrodinger equation becomes0

222

2

nn

n m

x

This is the differential equation for an undampedvibration and have the following solution

kxAx sin)(

2

2 2

mk

2

2

2k

m

where

Let

02

2

2

nn

n kx



Since no boundary condition had to be considered for

the calculation of the free electrons, all values of theenergy are allowed i.e. one obtains an energy

continuum.


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Electon in a potential

well (bound electron)

Consider a dimensional potential box in which the

electron can move freely between two infinitely high

potential barriers. The potential barriers do not allow

the electron to escape from the potential well, which

means that = 0 forx0 andxL The potential energyinside the well is zero. Therefore

02

22

2

nn

n m

x

V

xL

xAn

n

2sin


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The boundary conditions are= 0 atx = 0

= 0 atx = L

Atx = L,

0

2

sin

LA nn

Which can only be satisfied if

Thus,

nL

n

2

or2

nnL

x

L

nAn

sin


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These solutions correspond to standing waves with a

different number of nodes within the potential well asis shown in Fig.1.

n

nmm

kmp

2

222

2222

2222

2/2

2

2

L

n

mnLmn


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Now we need to accommodate Nvalence electrons in

these quantum states. According to the Pauli exclusion

principle no two electrons can have their quantum

number identical.

The electronic state in a 1D solid is characterized bytwo quantum numbers that are n and ms, where n

describes the orbital n and msdescribes the magnetic

quantum number. Therefore, each orbital labeled by the

quantum numbern can accommodate two electrons,

one with spin up and one with spin down orientation.


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Let nFdenote the highest filled energy level, where we

start filling the levels from the bottom (n = 1) andcontinue filling higher levels with electrons until all N

electrons are accommodated. It is convenient to

suppose that Nis an even number.

The condition 2nF= Ndetermines nF, the value ofn forthe uppermost filled level.

The energy of the highest occupied level is called the

Fermi energy, F. For the one dimensional system ofN

electrons we find2222

222

L

N

mL

n

m

FF


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The ground state of the Nelectron system is illustrated

in Fig.2a: All the electronic levels are filled up to theFermi energy. All the levels above are empty.


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The distribution of electrons among the levels is usually

described by the distribution function, f(), which isdefined as the probability that the level is occupied by

an electron. Thus if the level is certainly empty, then, f()

= 0, while if it is certainly full, then f() = 1. In general, f()

has a value between zero and unity.It follows from the preceding discussion that the

distribution function for electrons at T = 0K has the form

F0

1

Ff )(


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The distribution function at non-zero temperature is

given by the Fermi distribution function. The Fermidistribution function determines the probability that an

orbital of energy is occupied at thermal equilibrium.

The quantityis called the chemical potential.

11

Tkf

B/exp)(

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