5. free e fermi gas

7/28/2019 5. Free e Fermi Gas

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Metallic structure

Crystals can be classified by binding:

ionic, covalent, Van der Waals, metallic

In case of metallic bond,the mean free path ~ 108 inter-atomic distance

Alkali metals: Li, Na, K, Cs, Rb have bcc structure

Noble metals: Lu, Ag, Au have fcc structure



Free e Fermi gas

Why condensed matter so transparent to conduction es?

(1) es are not scattered by cores arranged on periodiclattice because matter waves travel freely in a periodic lattice

(2) conduction e

s are scattered only infrequently by other conduction es Pauli exclusion principle

Free e Fermi gas = gas of free es subject to Pauli principle

This theory preceded quantum mechanics

+ + +

+ + +

Eg. Na metal

• Conduction es are 3s valence es

• Atomic core contains 10 es

• Cores occupy only 15% volume!

• Free es move throughout the bulk

volume



•In an alkali metal the atomic cores occupy a relatively small

part (-15 percent) of the total volume of the crystal, hut in a

noble metal (Cu, Ag, Au) the atomic cores are relatively larger

and may be in contact with each other.

•The common crystal structure at room temperature is bcc for

the alkali metals and fcc for the noble metals. •A monovalent crystal which contains N atoms will have N

conduction electrons and N positive ion cores.eg Na+ •The Fermi energy eF is defined as the energy of the topmost

filled level in the ground state of the N electron system



Energy levels in one dimension

L

e

Consider an e of mass m confined to L by

an infinite barrier

( ) of e is solution of the Schordinger eqnn x

H 0V 2, / 2 , / H p m p i d dx

22

2

2

nn n n

d H

m dx

Boundary conditions: (0) 0, ( ) 0n n L

2 1sin ;

2n n

n

x n L

cos ;nd n n A x

dx L L

sin ;n

n A x

L

22

2 sin

nd n n

A xdx L L

22

2n

n

m L



22

2n

n

m L

n m s

e occupancy

1 1

1

2 1

13 1

1

4 0

0

If there are 6 e s,

If = top most filled level F n

If = total number of e s N

2 ; F n N

22

2

F F

n

m L

22

2 2 F

N

m L



Free e gas in 3DFree-particle Schrodinger equation in 3D:

2 2 2 2

2 2 2 ( ) ( )2k k k r r m x y z

If e s are confined to cube of edge , the w.f. is a standing wave L

( ) sin( / )sin( / )sin( / )n x y z r A n x L n y L n z L

Periodic boundary conditions: ( , , ) ( , , )k k x L y z x y z

Wave fns. satisfying above conditions are that of travelling plane wave

( ) exp( )k r ik r 2 4

with 0, , ,... xk L L

& similarly for & y z k k

exp[ ( )] exp[ 2 ( ) / ] xik x L i n x L L exp( 2 / ) exp( 2 )i n x L i n

exp( ) xik x2 2

2 2

( ) ik r

k

r e x x

2

( )

ik r

xik e

2

( ) x k k r



2

2 2 2( ) ( )

2x y z k k k k k k r r

m

2 2 2 2

2 2 2( ) ( )

2k k k r r

m x y z

2

2

2 ( ) ( )k x k r k r x

22

2k k

m 2& k

( ) ( )k k p r i r ( )k k r

p mv k k

vm

22

2 F F

k m



3There exists one distinct triplet , , for (2 /L) of -space x y z k k k k

3In sphere of volume 4 / 3, total number of orbitals F k

3

3

4 / 32

(2 / )

F k N

L

3 3

23

F L k

3

23

F Vk

1/32

3 F N k V

2/32 2

3

2 F

N

m V

F

F

k v

m

1/32

3 N

m V

6 1~10 m

~1 to 15 eV

6~10 m/s



Density of states

( )dN

d

D

2/32 2

3

2

F

N

m V

3/2

2 2

2( ) 2

V mD

3/2

2 2

2{No of orbitals of energy

3

V m N

3The expression for can be written as ln ln constant

2 N N

3

2

dN d

N

3( )

2

dN N

d D

3( )

2

F

F

N D

3

2

B F

N

k T

ff f



Effect of temperature



H t it f



Heat capacity of e gas

Total electronic thermal energy B

F

NT E k T

T

= 2 el B

F

E T C Nk T T

At low ( ) we can use ( ) to get B F el T k T C D

0 0( ) ( ) ( )

F

E d f d D D

3We know that ( )

2 F

B F

N

k T D

2 21( )

3 el F BC k T D

21

2

el B

F

T C Nk

T

E i t l H t C it



C = gT + AT 3

At temperatures very much lower the T F , the heat

capacity may be written as the sum of electronic and

lattice contributions

where g and A are constants characteristic of the material

Experimental Heat Capacity

El t i l d ti it



Electrical conductivity

1

F e E v B

c{ 0 eE B

dv

F mdt

dk

dt

dk eE

dt / dk e Edt

( ) (0) / k t k eEt / k eE

Wh t ff t th ti f ?



What affects the motion of es? Phonons

◦ Quantization of lattice vibrations

◦ Analogous to photon with wave-particle duality

◦ Sound waves in crystals propagate through phonons

◦ Thermal vibrations thermally excited phonons

◦

A phonon of wave vector k interacts with other particles if it had a momentum k

Impurities

Lattice imperfections



e s collide with impurities, lattice imperfections and phonons

steady state in electric field can be achieved

e s drift due to application of electric field

incremental velocity

eE

vm

/ k eE

In constant , with e s of charge , per unit volume, E n q e

Electric current density j nq v2

Ohm's law

ne E

jm

Conductivity / j E

2

ne

m

2Electrical resistivity

m

ne



Mean Free Path

How do we determine the collision time?

Electrons have a mean free path that they travel before a

collision occurs = v F

At low temperatures, most of the mobile electrons are right at the

Fermi surface, so v = v F (Fermi velocity) At these temperature, one can have mean free paths of the

order of ~ 1 cm for very pure crystals (even upto 10 cm for some

extremely pure metals!)

Eg. Cu v F = 1.57×108cm/s

(300 K) 3 × 10-6 cm

(4 K) 0.3 cm

More collisions at high temperatures lead to shorter collision

times and hence tends to increase with temperature.



Experimental Resistivities

At room temperature (300 K), the electrical resistivity is

dominated by electron collisions with phonons

At low temperatures (~ 4 K), it is determined by collisions

with impurities

The rates of these collisions are dependent on one another

Matthiessen’s rule: = L + i

Phonon resistivity

(Related to

concentration of

phonons & T dependent)

Imperfection

resistivity (T

independent)



References



References

A. Beiser – “Concepts of Modern Physics”, 6 Ed., Tata

McGraw-Hill (New Delhi, 2003)

Charles Kittel – “Introduction to Solid State Physics”, 7

Ed., John Wiley and Sons (New York, 1996)

www.wikipedia.org

5. free e fermi gas

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