lecture+15+mak +free+electron+gas2

7/30/2019 Lecture+15+MAK +Free+Electron+Gas2

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PHY 3201 FIZIK KEADAAN PEPEJAL

Loosely bounded valence electrons areattracted towards nucleus of other atoms

Electrons spread out among atomsforming electron clouds.The removal of the valance electronsleaves a positively charged ion.The charge density associated thepositive ion cores is spread uniformlythroughout the metal so that theelectrons move in a constantelectrostatic potential.This potential is taken as zero and therepulsive force between conductionelectrons are also ignored.


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V

xL 2222

2/22

2

Ln

mn Lmn

02

22

2

nnn m

x

The boundary conditions are = 0 at x = 0 = 0 at x = L At x = L,


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EF EE

F

0.5

f FD(E,T)

E

( ) /

11 F B FD E E k T

f e

f FD=? At 0K

i. EEF

( ) /1 1

1 F B FD E E k T f

e

( ) /

10

1 F B FD E E k T f

e

The distribution of electrons among the levels is usually described by thedistribution function, f ( ), which is defined as the probability that the level is occupied by an electron.


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FREE ELECTRON GASIN THREE DIMENSION

The free particle Schrodinger equation in threedimension is

)()(r

k

r z y xm k 2

2

2

2

2

22

2

If the electrons are confined to a cube of edge L, thesolution is the standing wave

z k yk xk Ar z y xn sinsinsin)(

Ln

k L

nk

Ln

k z z y

y x

x

,,where


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Periodic Boundary ConditionsTo apply the boundary conditions to the free electronswave function in a solid, it is more convenient to write the

solution to the Schrodinger equation as:

For a cubic solid of dimension L, the wave function must

satisfy the relation


7/14PHY 3201 FIZIK KEADAAN PEPEJAL

If we impose the above conditon in the x direction

To satisfy the boundary condition

,..4,2,0 L L

k x

and similarly for k y and k z.



The energy of the orbital with wavevector k

2222

22

22 z y xk k k k

mk

m

In the ground state of a system of N free electrons, theoccupied orbitals may be represented as points inside asphere in k space. Therefore all the occupied states lieinside the sphere of radius k F . The energy at the surfaceof this sphere is the Fermi energy. The magnitude of the

wavevector k F and the Fermi energy are related by thefollowing equation:2

2

2 F F k

m



The Fermi energy and the Fermi wavevector (momentum) are determined by the number of valenceelectrons in the system. In order to find the relationshipbetween N and k F , we need to count the total number of orbitals in a sphere of radius k F which should be equal to

N . There are two available spin states for a given set of k x , k y , and k z . The volume in the k space which isoccupies by this state is equal to (2 / L)3. Thus in thesphere of 4k F3/3 the total number of states is

N k V

Lk

F F 3

23

3

32342

//.



3123 /

V

N k F

Then

which depends only on the particle concentration.

Therefore, the Fermi energy is3222 3

2

/

V N

m F

The particle velocity in the orbital k is given byThe electron velocity v F at the Fermi surface is

m/

3123/

V N

mmk

v F F



Meaning of the Fermi TemperatureThe Fermi temperature is not the temperature of theelectron gas!

It is a measure of where the Fermi energy is at(typically on the order of ~ 10000 K)

So, for most metals say at room temperature, notmany electrons are excited above the Fermi energy.



In phase space a surface of constant energy is a sphere, isas schematically shown in thepicture.

Any "state", i.e. solution of the

Schroedinger equation with aspecific k , occupies thevolume given by one of thelittle cubes in phase space.

The number of cubes fittinginside the sphere at energy E thus is the number of all energy levels up to E

Density of States


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The density of states, D() is defined as the number of

orbitals per unit energy range,

With to obtain the total number of

orbitals of energy

d dN

D )(23

22

23

/

mV N

232

221

23

22 N mV D

/

/

)(

lecture+15+mak +free+electron+gas2

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