free electron theoryfree electron theory

free electron theory

1. Free Electron Theory
Many solids conduct electricity.
There are electrons that are not bound to atoms but are able to move through the
whole crystal.
Conductingsolidsfallintotwomainclasses;metalsandsemiconductors.
andincreasesbytheadditionofsmall
amounts of impurity. The resistivity normally decreases monotonically with
decreasing temperature.
andcanbereducedbytheadditionof
small amounts of impurity.
Semiconductors tend to become insulators at low T.68
( ) ;10 10
metals
RT m

  ( ) ( )
pure semiconductor metal
RT RT


2. Why mobile electrons appear in some
solids and others?
Whentheinteractionsbetweenelectronsareconsideredthis
becomesaverydifficultquestiontoanswer.
Thecommonphysicalpropertiesofmetals;
•Greatphysicalstrength
•Highdensity
•Goodelectricalandthermalconductivity,etc.
Thischapterwillcalculatethesecommonpropertiesofmetals
usingtheassumptionthatconductionelectronsexistandconsist
ofallvalenceelectronsfromallthemetals;thusmetallicNa,Mg
andAlwillbeassumedtohave1,2and3mobileelectronsper
atomrespectively.
Asimpletheoryof‘freeelectronmodel’whichworks
remarkablywellwillbedescribedtoexplainthesepropertiesof
metals.

3. Why mobile electrons appear in some
solids and not others?
Accordingtofreeelectronmodel(FEM),the
valanceelectronsareresponsibleforthe
conductionofelectricity,andforthisreasonthese
electronsaretermedconductionelectrons.
Na
11
￫1s
2
2s
2
2p
6
3s
1
Thisvalanceelectron,whichoccupiesthethirdatomicshell,
istheelectronwhichisresponsiblechemicalpropertiesof
Na.
Valance electron (loosely bound)
Core electrons

4. WhenwebringNaatomstogethertoformaNa
metal,
NahasaBCCstructureandthedistancebetween
nearestneighboursis3.7A˚
The radius of the third shell in Na is 1.9 A˚
SolidstateofNaatomsoverlapslightly.Fromthis
observationitfollowsthatavalanceelectronisno
longerattachedtoaparticularion,butbelongsto
bothneighbouringionsatthesametime.
Na metal

5. Theremovalofthevalanceelectronsleaves
apositivelychargedion.
Thechargedensityassociatedthepositive
ioncoresisspreaduniformlythroughoutthe
metalsothattheelectronsmoveina
constantelectrostaticpotential.Allthe
detailsofthecrystalstructureislostwhen
thisassunptionismade.
+
+
+
+
+ +
Avalanceelectronreallybelongstothewhole
crystal,sinceitcanmovereadilyfromoneionto
itsneighbour,andthentheneighbour’s
neighbour,andsoon.
Thismobileelectronbecomes aconduction
electroninasolid.
AccordingtoFEMthispotentialistakenaszero
andtherepulsiveforcebetweenconduction
electronsarealsoignored.

6. Therefore,theseconductionelectronscanbe
consideredasmovingindependentlyinasquare
welloffinitedepthandtheedgesofwell
correspondstotheedgesofthesample.
Considerametalwithashapeofcubewithedge
lengthofL,
ΨandEcanbefoundbysolvingSchrödingerequation
0
L/2
V
L/22
2
2
E
m
   0V
Since,( , , ) ( , , )x L y L z L x y z   
•By means of periodic boundary conditions Ψ’s are running waves.

7. The solutions of Schrödinger equations are plane waves,
where V is the volume of the cube, V=L
3
So the wave vector must satisfy
where p, q, r taking any integer values; +ve, -ve or zero.()11
( , , )
x y z
i k x k y k zi k r
x y z e e
VV



Normalization constantNa p 2
,where k





 2
Na p
k

 22
k p p
Na L

 2
x
kp
L

 2
y
kq
L

 2
z
kr
L


; ;

8. The wave function Ψ(x,y,z) corresponds to an
energy of
the momentum of
Energy is completely kinetic22
2
k
E
m
 2
2 2 2
()
2
x y z
E k k k
m
   ( , , )
x y z
p k k k 22
21
22
k
mv
m
 2 2 2 2
m v k pk

9. Weknowthatthenumberofallowedkvalues
insideasphericalshellofk-spaceofradiuskof2
2
( ) ,
2
Vk
g k dk dk


whereg(k)isthe
densityofstatesper
unitmagnitudeofk.

10. The number of allowed states
per unit energy range?
Eachkstaterepresentstwopossibleelectron
states,oneforspinup,theotherisspindown.( ) 2 ( )g E dE g k dk ( ) 2 ( )
dk
g E g k
dE
 22
2
k
E
m
 2
dE k
dk m
 2
2mE
k ()gE 2 ( )gk dk
dE 2
2
2
V
 k k 2
2mE 2
m
k 3/ 2 1/ 2
23
(2 )
2
()
V
mEgE



11. Ground state of the free electron
gas
Electronsarefermions(s=±1/2)andobey
Pauliexclusionprinciple;eachstatecan
accommodateonlyoneelectron.
Thelowest-energystateofNfree
electronsisthereforeobtainedbyfilling
theNstatesoflowestenergy.

12. ThusallstatesarefilleduptoanenergyE
F,
known asFermi energy,obtainedby
integratingdensityofstatesbetween0andE
F,
shouldequalN.Hence
Remember
Solve for E
F(Fermi energy);2 / 3
22
3
2
F
N
E
mV


 3/ 2 1/ 2
23
(2 )
2
()
V
mEgE

 3/ 2 1/ 2 3/ 2
2 3 2 3
00
( ) (2 ) (2 )
23
FF
EE
F
VV
N g E dE m E dE mE

  

13. The occupied states are inside the Fermi sphere in k-space
shown below; radius is Fermi wave number k
F.22
2
F
F
e
k
E
m

k
z
k
y
k
x
Fermi surface
E=E
F
k
F2 / 3
22
3
2
F
N
E
mV


 From thesetwo equation k
F
can be found as,1/ 3
2
3
F
N
k
V



The surface of the Fermi sphere represent the
boundary between occupied and unoccupied k
states at absolute zero for the free electron gas.

14. Typicalvaluesmaybeobtainedbyusing
monovalentpotassiummetalasanexample;for
potassiumtheatomicdensityandhencethe
valanceelectrondensityN/Vis1.402x10
28
m
-3
so
that
Fermi (degeneracy) Temperature T
F by19
3.40 10 2.12
F
E J eV

   1
0.746
F
kA

 F B F
E k T 4
2.46 10
F
F
B
E
TK
k
  

15. Itisonlyatatemperatureofthisorderthatthe
particlesinaclassicalgascanattain(gain)
kineticenergiesashighasE
F.
OnlyattemperaturesaboveT
Fwillthefree
electrongasbehavelikeaclassicalgas.
Fermimomentum
Thesearethemomentum andthevelocityvalues
oftheelectronsatthestatesontheFermi
surfaceoftheFermisphere.
So,FermiSphereplaysimportantroleonthe
behaviourofmetals.FF
Pk F e F
P m V 61
0.86 10
F
F
e
P
V ms
m

  

16. 2 / 3
22
3
2.12
2
F
N
E eV
mV


 1/ 3
2
13
0.746
F
N
kA
V


  
 61
0.86 10
F
F
e
P
V ms
m

   4
2.46 10
F
F
B
E
TK
k
   Typical values of monovalent potassium metal;

18. The free electron gas at finite temperature
AtatemperatureTtheprobabilityofoccupation
ofanelectronstateofenergyEisgivenbythe
Fermidistributionfunction
Fermidistributionfunctiondeterminesthe
probabilityoffindinganelectronattheenergy
E.( ) /
1
1
FB
FD E E k T
f
e




19. E
FE<E
F E>E
F
0.5
f
FD(E,T)
E( ) /
1
1
FB
FD E E k T
f
e



Fermi Function at T=0
and at a finite temperature
f
FD=? At 0°K
i.E<E
F
ii.E>E
F( ) /
1
1
1
FB
FD E E k T
f
e


 ( ) /
1
0
1
FB
FD E E k T
f
e




20. Fermi-Dirac distribution function at
various temperatures,

21. T>0
T=0
n(E,T)
E
g(E)
E
F
n(E,T)numberoffree
electronsperunitenergy
rangeisjustthearea
undern(E,T)graph.( , ) ( ) ( , )
FD
n E T g E f E T
Number ofelectronsperunitenergyrange
accordingtothefreeelectronmodel?
Theshadedareashowsthechangeindistribution
betweenabsolutezeroandafinitetemperature.

22. Fermi-Diracdistributionfunctionisa
symmetric function; at finite
temperatures,thesamenumberoflevels
belowE
Fisemptiedandsamenumberof
levelsaboveE
Farefilledbyelectrons.
T>0
T=0
n(E,T)
E
g(E)
E
F

23. Heat capacity of the free electron
gas
Fromthediagramofn(E,T)thechangeinthe
distributionofelectronscanberesembledinto
trianglesofheight1/2g(E
F)andabaseof2k
BTso
1/2g(E
F)k
BTelectronsincreasedtheirenergyby
k
BT.
T>0
T=0
n(E,T)
E
g(E)
E
F
Thedifferenceinthermal
energyfromthevalueat
T=0°K21
( ) (0) ( )( )
2
FB
E T E g E k T

24. Differentiating with respect to T gives the
heat capacity at constant volume,2
()
v F B
E
C g E k T
T


 2
()
3
33
()
22
FF
F
F B F
N E g E
NN
gE
E k T

 22 3
()
2
v F B B
BF
N
C g E k T k T
kT
 3
2
vB
F
T
C Nk
T

 

Heat capacity of
Free electron gas

25. Transport Properties of Conduction Electrons
Fermi-Diracdistributionfunctiondescribesthe
behaviourofelectronsonlyatequilibrium.
Ifthereisanappliedfield(EorB)ora
temperaturegradientthetransportcoefficientof
thermalandelectricalconductivitiesmustbe
considered.
Transport coefficients
σ,Electrical
conductivity
K,Thermal
conductivity

26. Total heat capacity at low temperatures
where γ and βare constants and they can
be found drawing C
v/T as a function of T
23
C T T
Electronic
Heat capacity
Lattice Heat
Capacity

27. Equationofmotionofanelectronwithanapplied
electricandmagneticfield.
ThisisjustNewton’slawforparticlesofmassm
e
andcharge(-e).
Theuseoftheclassicalequationofmotionofa
particletodescribethebehaviourofelectronsin
planewavestates,whichextendthroughoutthe
crystal.Aparticle-likeentitycanbeobtainedby
superposingtheplanewavestatestoforma
wavepacket.e
dv
m eE ev B
dt
   

28. The velocity of the wavepacket is the group
velocity of the waves.Thus
So one can use equation of mdv/dt1
ee
d dE k p
v
mmdk dk

    22
2
e
k
E
m
pk

 e
dv v
m eE ev B
dt

    
 
= mean free time between collisions. An electron
loses all its energy in time
(*)

29. In the absence of a magnetic field, the applied E
results a constant acceleration but this will not
cause a continuous increase in current. Since
electrons suffer collisions with
phonons
electrons
The additional term cause the velocity v to
decay exponentially with a time constant when
the applied E is removed.e
v
m



 

30. The Electrical Conductivty
In the presence of DC field only,eq.(*) has the
steady state solution
Mobility determines how fast the charge carriers
move with an E.e
e
vE
m


a constant of
proportionality
(mobility)e
e
e
m


Mobility for
electron

31. Electrical current density,J
Where n is the electron density and v is drift
velocity.Hence()J n e v N
n
V
 2
e
ne
JE
m

 JE 2
e
ne
m

 e
e
vE
m


Electrical conductivity
Ohm’s law1


 L
R
A

 Electrical Resistivity and Resistance

32. Collisions
Inaperfectcrystal;thecollisionsofelectronsare
withthermallyexcitedlatticevibrations
(scatteringofanelectronbyaphonon).
Thiselectron-phonon scatteringgivesa
temperaturedependent collisiontime
whichtendstoinfinityasT0.
Inrealmetal,theelectronsalsocollidewith
impurity atoms, vacancies and other
imperfections,thisresultinafinitescattering
time evenatT=0.()
ph
T 0


33. The total scattering rate for a slightly imperfect
crystalat finite temperature;
So the total resistivity ρ,
This is known as Mattheisen’s rule and illustrated in
following figure for sodium specimen of different
purity.0
1 1 1
()
ph
T  

Due to phonon
Due to imperfections02 2 2
0
()
()
e e e
I
ph
m m m
T
ne ne T ne
  
  
    
Ideal resistivity
Residual resistivity

34. Residual resistance ratio
Residual resistance ratio = room temp. resistivity/ residual resistivity
and it can be as high as for highly purified single crystals.6
10
Temperature
pure
impure

35. Collision time   10 1
5.3 10 ( )
pureNa
residual
xm

  71
( ) 2.0 10 ( )
sodium
RT x m

  28 3
2.7 10n x m

 e
mm 14
2
2.6 10
m
xs
ne



 11
7.0 10xs
 6
1.1 10 /
F
v x m s ( ) 29l RT nm ( 0) 77l T m
can be found by taking
at RT
at T=0F
lv
Taking ; and
Thesemeanfreepathsaremuchlongerthantheinteratomic
distances,confirmingthatthefreeelectronsdonotcollidewiththe
atomsthemselves.

36. Thermal conductivity, Kmetals non metals
KK
 1
3
VF
K C v l V
C
Due to the heat tranport by the conduction electrons
Electronscomingfromahotterregionofthemetalcarry
morethermalenergythanthosefromacoolerregion,resultingina
netflowofheat.Thethermalconductivityl F
v B
kT F
 F
lv 21
2
F e F
mv
where is the specific heat per unit volume
is the mean free path; and Fermi energy
isthemeanspeedofelectronsresponsibleforthermalconductivity
sinceonlyelectronstateswithinabout of changetheir
occupationasthetemperaturevaries.2 2 2
21 1 2
()
3 3 2 3
B
V F B F
F e e
N T nk T
K C v k
V T m m
  
     2
2
vB
F
T
C Nk
T
 
 

where

37. Wiedemann-Franz law2
e
ne
m

 22
3
B
e
nk T
K
m

 2
2
82
2.45 10
3
Kk
x W K
Te



  


B
The ratio of the electrical and thermal conductivities is independent of the
electron gas parameters;82
2.23 10
K
L x W K
T

  
Lorentz
number
For copper at 0 C