Chapter 3 Introduction to Quantum Theory of...

W.K. Chen Electrophysics, NCTU 1

Chapter 3 Introduction to Quantum Theory of Solids


Outline

Allowed and forbidden energy bands

Electrical conduction in solid

Extension to three dimension

Density of state

Statistical mechanics


3.1 Allowed and forbidden energy bands

222

4

2)4( n

emE

o

on

hπε−

=

In one-electron atom

Only discrete values of energyare allowed

The probability function shows that the electron is not localized at a given radius

We can extrapolate these single-atoms results to a crystal and quantitatively derive the concepts of allowed and forbidden energy bands


3.1.1 Formation of energy bands

Fig. (a) shows the radial probability density function for the lowest electron energy state of the single, non-interacting hydrogen atom Fig. (b) shows the radial probability curves for two atoms that are in close proximity to each other

Due to Pauli exclusion principle, no two electrons may occupy the same quantum state, the interaction of two electrons give each discrete quantized energy level splitting into two discrete energy levels


By pushing the numbers of hydrogen type atoms together with a regular periodic arrangement, the initial quantized energy level will split into a band of discrete energy levels

The Pauli exclusion principle states that the joining of atoms to form a system (crystal) does not alter the total number of quantum states

The discrete energy must split into a band of energies in order that each electron can occupy a distinct quantum state

For a system with 1019 one-electron atoms with a width of allowed energy band of 1 eV, if the discrete energy states are equidistant, then the energylevels are separated by 10-19 eV, which is extremely small


Example 3.1 Small change of kinetic energy

21

22 2

1

2

1 υυ mmE −=Δ

Consider an electron traveling at a velocity of 107 cm/s. Assume the velocity increases by a value of 1 cm/s. Find the increase in kinetic energy

υυυυ

υυυυυυυ

υυυ

Δ=Δ≈Δ⇒

Δ+Δ+=Δ+=⇒

Δ+=

11

21

21

21

22

12

)2(2

1

)(2)(

Let

mmE

eV 107.5106.1

1011.9

J 1011.9)01.0)(10)(1011.9(

919

28

285311

−−

−

−−

×=××

=Δ

×=×=Δ≈Δ

E

mE υυ

eV 1085.2106.1

1056.4J 1056.4)10)(1011.9(

2

1

2

1 219

212125312 −

−

−−− ×=

××

=×=×== υmE


222

4

2)4( n

emE

o

on

hπε−

=

Discrete energy levels for a hydrogen atom

eVE

E

eVE

E

eVE

E

eVE

85.04

51.13

4.32

6.13

21

4

21

3

21

2

1

−==

−==

−==

−=

1E

2E

4E

3E

eV 2.10=ΔE

eV 89.1=ΔEeV 66.0=ΔE


For a atom contains electrons up through the n=3 energy level, when we bought these atoms close together from initially far apart

The n=3 energy shell will begin to interact initially, followed by n=2 energy shell interaction and finally the innermost n=1 shell interaction

If ro is the equilibrium interatomic distance, then at this distance we have Bands of allowed energies (electrons may occupy)

Bands of forbidden energies

allowed band

forbidden band


For silicon atoms, having 14 electrons,

2 electrons occupy n=1 energy level

8 electrons occupy n=2 energy level

4 remaining valence electrons occupying n=3 energy level are involved in chemical reactions

Isolated silicon atom

silicon crystal

For silicon crystal3s states corresponds to n=3 and l=0 ( containing 2 quantum states)

(n=3, l=0, s=-1/2, +1/2)

3p states corresponds to n=3 and l=1 ( containing 6 quantum states)

n=3, l=+1, m=1, s=-1/2, +1/2

m=0, s=-1/2, +1/2

m=-1, s=-1/2, +1/2


The upper band is assigned as conduction band

the lower band is call valence band

Since each Si atom has only 4 valence electrons

At 0 K, electrons are in the lowest wnergy states

all states in the valence band will be full, and all states in the conduction band will be empty

As the interatomic distance decreases, the 3s and 3p states and gradually merge together to form broad band

At the equilibrium interatomic distance, the broad band has again split with 4 quantum states per atom in the lower band and 4 quantum states per atom in the upper band


3.1.2 The Kronig-Penney ModelFig. (a) shows potential function of a single, noninteracting, one-electron atom

Fig. (b) shows potential function for the case when several atoms are in close proximity arranged, one-dimensionalarray

Fig. (c) is the net potential function in Fig. (b)

One-dimensional square periodic potential function is used in K.P model


Kronig-Penny Model

jkxexux )()( =ψ wavetraveling :

)(periodith function w periodic:)(

Qjkxe

baxu +

Block Theorem

The theorem states that all one-electron wave functions involving periodically varying potential energy functions must be of the form

tjjkx eexu

txtxω

φψ−⋅=

=Ψ

)(

)()(),(

The general time-independent wave function

)( ωω hh

== EE

⎩⎨⎧

<<−<<

=0

0 0)(

xbV

axxV

o


0)())((2)(

22

2

=−+∂

∂xxVE

m

x

x ψψh

Time-independent Schrodinger wave equation

jkxexux )()( =ψ

)0( 0)()(2)(

:IIRegion

)0( 0)(2)(

:IRegion

22

2

22

2

≤≤−=−

+∂

∂

≤≤=+∂

∂

xbxVEm

x

x

axxmE

x

x

o ψψ

ψψ

h

h


0)()()(

2)(

2222

22

2

=−−+ xukdx

xdujk

dx

xud β γβ jVEm o =

−=

2

)(2

h

Region II :

u2(x): the amplitude of the wave function in region II

0)()()(

2)(

1221

21

2

=−−+ xukdx

xdujk

dx

xud α 22 2

h

mE=α

Region I:

u1(x): the amplitude of the wave function in region I

222 2

homV

−=αβ

0for )(

0for )(

)()(2

)()(1

<<−+=

<<+=

+−−

+−−

xbBeCexu

axBeAexu

xkjxkj

xkjxkj

ββ

αα

if Vo > E

if E >Vo


0for )(

0for )(

)()(2

)()(1

<<−+=

<<+=

+−−

+−−

xbBeCexu

axBeAexu

xkjxkj

xkjxkj

ββ

αα

4 unknowns, 4 BCs

The wave function ψ(x) and its derivative ∂ψ/∂x must be continuous⇒u(x) and its derivative ∂u/∂x must be continuous

jkxjkx

jkx

exjkuex

xu

x

x

exux

)()()(

)()(

+∂

∂=

∂∂

=ψ

ψ

⇒

+∂

∂=+

∂∂

⇒

∂∂

=∂

∂

⇒=⇒=

++

+

−−

−

+−

+−

+−

+−+−

jkxjkx

x

jkxjkx

x

xx

jkxjkx

exjkuex

xuexjkue

x

xu

x

x

x

x

exuexuxx

)()(

)()(

)()(

)()( )()(

ψψψψ )()( −− = xuxu

+− ∂∂

=∂

∂

xx x

xu

x

xu )()(


⇒= +− )0( )0( :B.C.1 12 uu

⇒∂∂

=∂∂

+−

:B.C.20

2

0

2

x

u

x

u

0for )(

0for )(

)()(2

)()(1

<<−+=

<<+=

+−−

+−−

xbBeCexu

axBeAexu

xkjxkj

xkjxkj

ββ

αα

bxax x

u

x

u

−== ∂∂

=∂∂ 21 :B.C.4

)( )( :B.C.3 21 buau −= )()()()( bkjbkjakjakj BeCeBeAe ++−−+−− +=+ ββαα

)()(

)()()()(

)()(

bkjbkj

akjakj

ekek

ekek++−−

+−−

+−−=

++−ββ

αα

ββ

αα

0)()()()( =++−−+−− DkCkBkAk ββαα

DCBA +=+


From the above 4 eq, derived from 4 BCs, we obtain the parameter k in relation of the total energy E (through the parameter α)and the potential function Vo (through the parameter β)

)(cos))(cos(cos))(sin(sin2

)( 22

bakbaba +=++− βαβααβ

βα

Kronig-Penny Model

⎩⎨⎧

<<−<<

=0

0 0)(

xbV

axxV

o


3.1.3 The k-space diagram

Case 1: Free particle:bVo is finite value, andb=0Vo=0

2

2

h

mE=α

2

2

h

mE==αβ


)( 22


βα

0 , )(2

2>

−== γγβ

h

EVmjj o

V(x)

E



)(2

22

bakbaba +=++− ααααα

αα

particle) (free coscos kaa =α

0 1

k=α

kpmmmmE=====

hhhh

υυα

2

221

2

)(22

p: particle momentumk: wave number

⎪⎩

⎪⎨⎧

==

=⇒

m

k

m

pE

kp

22

222 h

h

p: momentum of wave motion

⇒

kmE

=⇒2

2

h

For the case of free particle, parabolic relation is valid between energy E and momentum p.


Case 2: bound electrons in crystal:bVo is finite value, butb→0Vo→∞ ⇒ E<<Vo

Periodic δ function potential barrier

V(x)

Periodic δ function potential barrier


2

2

h

mE=α 0 ,

)(2

2>

−==⇒<< γγβ

hQ

EVmjjVE o

o

)(cos))(cosh(cos))(sinh(sin2

)( 22

bakbaba +=+− γαγααγαγ

The above eq does not lend itself to an analytical solution, but must be solved using numerical or graphical techniques to obtain the relation between k, Eand Vo.


)( 22


βα

kaaa

abamVo cos)(cossin

2=+ α

αα

h

akaa

aP cos)(cos

sin' equation k -E =+ ααα 2

'h

bamVP o=

energy term momentum term

decaying wave at barrier region

b→0 ⇒sinhγb≈ γb & coshγb→1 ][2

1cos ][

2

1sin θθθθ θθ jjjj eeee

j−− +=−=Q

2

2

h

mE=α


2cosh

2sinh

xxxx eex

eex

−− +=

−=

x

2

sinhxx ee

x−−

=

2cosh

xx eex

−+=

][2

1cos ][

2

1sin θθθθ θθ jjjj eeee

j−− +=−=Q


)(cossin

')( aa

aPaf α

ααα +=

1cos)(1 +≤=≤− kaaf α

2

2

h

mE=αThe parameter α is related to the total energy E of the particle

a is fixed, k varies



E-k diagram

The right figure shows the energy E as a function of the wave number k

The plot shows the concept of allowed energy bands for particle propagating in the crystal lattice

Since the energy E is discontinuities, we also have the concept of forbidden energy for particles in the crystal


Solution

628.2=aα

For ka=0

kaaa

aPaf cos)(cos

sin')( =+= αααα

Example 3.2 bandwidth

Using numerical technique (trial and error), we find

1)(cossin

' +=+ aa

aP α

αα

eV 053.1J 1068.1)105)(1011.9(2

)10054.1()628.2(

2

)628.2(

628.22

2

1921031

2342

2

22

22

=×=×××

==

==⇒=

−−−

−

maE

amE

amE

lower

upper

h

hhαα

The lowest allowed band occurs as ka changes from 0 to π

Coefficient P’ =10, potential width a=5Å


For ka=π, as seen form the figure it happens αa=π

eV 50.1J 10407.2

)105)(1011.9(2

)10054.1(

2

2

19

21031

2342

2

22

2

=×=

×××

==

==

−

−−

−ππ

πα

maE

amE

a

upper

upper

h

h

eV 447.0053.150.1 =−=−=Δ lowerupper EEE


)2cos()2cos(cos ππ nkankaka −=+=kaaa

aP cos)(cos

sin' =+ ααα

energy term momentum term

Displacement of portions of the curve in left figure by 2π still satisfy the E-kequation


The entire E versus k plot is contained within -π/a<k<π/a

For free electron, the particle momentum and the wave number k are related by

Given the similarity between the free electron solution and the results of the single crystal,

The parameter ħk in a single crystal is referred to as the crystal momentum, which is not the actual momentum of the electron in the crystal , but is a constant of motion that includes the crystal interaction

electron) (free momentum k p h=

k h:momentum crystal


3.2 Electrical conduction in solids

For covalent bonding of silicon crystal, each silicon atom is surrounded by 8 valence electrons. 4 from itself, 4 from the 4 nearest Si neighbor

For N silicon atoms in the crystal, there are 4 N energy states in lower valence band and 4N energy states in higher conduction band. Each energy state allow only one electron to reside

At T=0K, all valence electrons are lying in their lowest energy states, valence band, i.e. the valence band is fully occupied and the conduction band is completely empty


At T>0K, a few valence band (VB) electrons may gain enough thermal energy to break the covalent band and jump into conduction band (CB)

VB: covalent bonding position

CB: elsewhere

Since the pure semiconductor is neutrally charged, as the negatively charged electron breaks away from its covalent bonding position, a positively charged “empty state” is created in the original covalent bonding position in the VB

VB

CB

Covalent bond breaking


At T=0K, the energy states in the VB in E-k diagram are completely full, and the states in CB are empty (Fig. (a))

At T>0K, a few valence band (VB) electrons may gain enough thermal energy to break the covalent band and jump into CB (Fig (b))


3.2.3 Drift current

)cmsec

#( densityFlux

)sec

#(

)(Flux

2⋅=

Φ=

=ΔΔ

=Δ

=Δ

=Φ

υφ

υυ

nA

nAt

tnA

t

nA

t

N

l

l l

A

N: total number of flow chargen: volume density of flow chargeA: cross-sectional areaυ: average drift velocityl: traveling length of carrier per Δt

)cm

A( density current Drift

2dqnqJ υφ == Ampere

∑=

=n

iiqJ

1

υ υi: the velocity of the ith charged carrier


E-k diagram under external biasUnder no external force, the electron distribution in CB is an even function of k, since the net momentum is zero under thermal equilibrium.

Under applied force, electrons in CB can gain energy and a net momentum

dtFFdxdE υ==

TheoremEenergy -Work


3.2.3 Electron effective massThe movement of an electron in a lattice is different from that of an electron in free space

maFFF exttotal =+= int

Fint: due to the interaction between the moving electron and other charged particles, such as ions, protons and else electrons in the lattice

amFext*=

Free space crystal

maFext =r

extFr

m*: effective massWhich takes into account of the particle mass and the effects of internal forces

amFext*=

r

extFr


E-k diagram and particle mass in free space

electron) (free 1

:velocityk

E

∂∂

=h

υ

( )

)(

curve) (parabolic 222

1 222

kp

m

k

m

pmE

ooo

h

h

=

=== υ

2

2

2

⇒

===∂∂

⇒o

o

oo m

m

m

p

m

k

k

E υhhh

⇒

=∂∂

=∂∂

oo mm

k

kk

E 22

2

2

)2

( hh

11

:mass2

2

2 k

E

mo ∂∂

=h

The first derivative of E with k is related to the velocity of free particle

The second derivative of E versus k is inversely proportional to the mass of free particle


Conduction bandThe energy near the bottom of CB may be approximated by a parabola, just as that of free particle

0 , 12

1 >=− CkCEE c

electron) (free

11 :mass

2

2

2 k

E

mo ∂∂

=h

electron)(Block

11

:mass effective2

2

2* k

E

mn ∂∂

=h

Electron effective mass & E-k diagram in crystal


Effective mass parameter is used to relate quantum mechanics andclassical mechanics

The larger the bowing of E-k diagram, the small the effective mass is

21

2

2

2*

12

2

11

2C

11

2 ),0( 2

hh=

∂∂

=⇒

=∂∂

>=∂∂

k

E

m

Ck

ECkC

k

E

n

amF next*=

If we apply an electric field, the acceleration is

**

E

nn

ext

m

e

m

Fa

−==

Thus acceleration of Block electron with external force is

EE: electric fieldE: total energy

Fext

a


3.2.4 Concept of holeThe empty state is created when a covalent bond is broken

The movement of a valence electron into the empty state is equivalent to the movement of a positively charged empty state itself

The above positive charge carrier is called “hole”

The crystal now has a second equally important charge carrier, hole, that can give rise to a current


The drift current in VB

∑−=)(

)(filledi

ieJ υ

Under external biased voltage, it is the electrons that moving in the VB, not the positively charged ions

The drift current due to electrons in VB

ion for

electronfor , 1

++=

−== ∑=

eq

eqqJn

iiυQ

∑∑ ++−=)()(

)()(emptyi

ifulli

i eeJ υυ

emptyfilled filled

= +


band) valence(full ?)()(

=− ∑fulli

ie υ

The velocity of individual electron is given

1

)(k

EEe ∂

∂=h

υ

The valence band is symmetric in k and every state in VB is occupied so that every electron with a velocity lυ l, there is a corresponding electron with a velocity -lυ l

Since the valence band is full, the distribution of electrons with respect to k has no chance to be changed by an external applied. The net drift current density generated from a completely full band electrons, then, is zero

0)()(

=− ∑fulli

ie υ


The net drift current due to electrons in VB is entirely equivalent to placing a positively charged particle in the empty state (assigned as hole) and assuming all other states in the band are neutrally charged

The associated hole velocity is exactly the velocity of the electron who occupy the empty state

The net drift current density due to electrons in VB

∑∑∑ +=++−=)()()(

)()()(emptyi

iemptyi

ifulli

i eeeJ υυυ0

hole neutralneutral


Valence bandThe energy near the bottom of VB may be approximated by a parabola, just as that of free particle

22kCEE =−υ

02C

11

2 ),0( 2

22

2

2

2*

22

2

22

<−

=∂∂

=⇒

−=∂∂

>−=∂∂

hh k

E

m

Ck

ECkC

k

E

e

Effective mass & E-k diagram in crystal

An electron moving near the top of valence band behaves as if it has a negative mass


Thus acceleration of Block electron near the top of VB with external force is

EeamF eext )(* −==

**

**0

pe

e

ext

e

ext

m

eE

m

eE

m

F

m

Fa

+=

−=

>==

emptyfilled filled

EFext

a

me: negative quantity

We may rewrite

hole neutralneutral

EFext

a

2

2

2**

111

k

E

mm ep ∂∂

−=−=h

**ep mm −=

1

)()(k

EEE eh ∂

∂−=−=h

υυ


Now we can model VB as having particles with a positive electronic charge and a positive effective mass, so it will move in the same direction as the

direction of applied field

The new particle is the hole

The density of hole is the same as the density of empty states in VB


3.2.5 Metals, insulators and semiconductors

Insulator:The allowed bands are either completely empty or completely full

Since empty band and full band contribute no current at all, the resistivity of an insulator is very large

The bandgap of insulator is usually on the order of 3.5 to 6 eV or larger


Semiconductor:The allowed bands are at conditions either almost empty or almost full

Both the electrons in CB and holes in VB can contribute the current

The bandgap of semiconductor is on the order of 1 eV

The resistivity of semiconductor can be controlled and varied over many orders of magnitude


Metal:The band diagram for a metal may be in one of two forms

partially full band

overlapped conduction and valence bands

In the case of partially full band, many electrons are available for conduction, so that the material can exhibit a large electrical conductivity

For overlapped conduction and valence bands , there are large numbers of electrons as well as large numbers of holes can move, so this material can also exhibit a very high electrical conductivity


3.3 Extension to three dimensionsThe basic concept of allowed and forbidden energy bands comes from the electrons moving in periodic potential arrangement in crystal lattice

For three dimensional crystal, the distance between atoms varies as the direction through the crystal changes


The E-k diagram is symmetric in k so that no new information is obtained by displaying the negative axis

In following plot of E-k diagrams, E-k diagrams at two directions are plotted. [100] portion of the diagram with +k to the right and [111] portion of the diagram with +k to the left are plotted


Direct bandgap

the minimum conduction band is at k=0 so that the transitions for electron between CB & VB can take place with no change in crystal momentum

Indirect bandgap

the minimum conduction band occurs not at k=0 so that the transitions for electron between CB & VB includes an interaction with the crystal


Effective mass concept in 3-dim crystal


3.4.1 Density of states for free electronTo determine the density of allowed quantum states as a function of energy, we need to consider an appropriate mathematical model

Consider a free electron confined to a three-dimensional cubic with length awith infinite potential well

elsewhere ),,(

0for

0for

0for 0),,(

∞=<<<<<<=

zyxV

az

ay

axzyxV

a

a

a

Using the separation of variables technique in 3 dimensional Schrodinger’swave equation, we can have

22222

2zyx kkkk

mE++==

h xkr

ykr

zkrk

r


, ,a

nka

nka

nk zzyyxx

πππ===

Two-dimensional plot of allowed quantum states in k space

nx, ny, nz: positive integers

Positive and negative values of kx,ky or kz have the same energy and represent the same quantum state

The volume Vk of a single quantum state in k-space is

3)(a

Vk

π=


The density of quantum state in k-space for whole cubic box is

kT V

dkkdkkg

24)

8

1(2)(

π=

2: due to two spin states allowed for each quantum state1/8: due to the positive values of kx,ky, and kz

32

2

)( adkk

dkkgT ⋅=π


Density of quantum states versus energy

dEEmh

aadE

E

mmEdEEg

dEEgadkk

dkkgk

⋅⋅=⋅⋅⋅=

=⋅=

2/33

33

22

32

2

)2(4

2

112)(

)()(

ππ

π

hh

dEE

mdE

Emdk

mEkmE

k

2

11

2

12

1

21

,2

22

hh

hhQ

=⋅⋅=

==

The number of energy states for whole cubic box between E and E+dE, i.e., between k and k+dk is given by

dEEh

mdEE ga

dkkdkkgk 3

2/33

2

2 )2(4)( )(

ππ

=⇒⋅=


3

)()()(

a

dEEg

V

dEEgdEEg TT ==

The density of quantum states per unit energy per unit volume of the crystal

dEEh

mdEEg

3

2/3)2(4)(

π=

The density of states is proportional to the energy E. As the energy of free particle becomes large, the number of available quantum states increase

The density of states is dependent on the mass of the free particle

g(E)

E


Example 3.3 Density of statesCalculate the total density of states for free electron per unit volume with energies between 0 and 1 eV

321

3272/319334

2/331

2/33

2/3

1

03

2/31

0

states/cm 105.4

m105.4)106.1(3

2

)10625.6(

)]1011.9(2[4

3

2)2(4

)2(4)(

−

−−−

−

×=

×=×⋅⋅××⋅

=

⋅⋅=

== ∫∫

N

N

Eh

mN

dEEh

mdEEgN

eVeV

π

π

π

The density of states is typically a large number ranging from 1018 to 1022

cm-3

The density of quantum states in semiconductor is usually less than the density of atoms in semiconductor crystal


3.4.2 Density of states for semiconductorThe parabolic relationship between energy and momentum of a free electron is

m

k

m

pE

222

22 h==

The E-k curve near k=0 at conduction band can be approximated as a parabola, so we have

m

k

m

pEE c 22

222 h

==−

The density of states in CB is

dEEEh

mdEEg c

nc −=

3

2/3* )2(4)(

π


Similarly, the E-k parabolic relationship for a free hole in VB

*2

2

*

2

22 pp m

k

m

pEE

h==−υ

The density of states for holes (electrons) in VB is

dEEEh

mdEEg p −= υυ

π3

2/3* )2(4)(


3.5 Statistical mechanicsIn dealing with large numbers of particles, we are interested only in the statistical behavior of the group as a whole rather than in the behavior of each individual particle

There are three distribution laws determining the distribution of particles among available energy states


Maxwell-Boltzmann distributionParticles are distinguishable

No limit to the number of particles allowed in each energy state

The exemplary case is low-pressure gas molecules in a container

Bose-Einstein distributionParticles are indistinguishable

No limit to the number of particles allowed in each energy state

The exemplary case is photon

Fermi-Dirac distributionParticles are indistinguishable

Only one particle is permitted in each energy state

The exemplary case is electrons in crystal


3.5.2 The Fermi-Dirac Probability FunctionFor ith energy level with gi quantum statesOnly one particle is allowed in each state

First particle:gi ways of choosing to place the particle

Second particle:(gi -1) ways of choosing to place the particle

Ni particle:[gi –(Ni-1)] ways of choosing to place the particle

)!(

!))1(()1)((

ii

iiiii Ng

gNggg

−=−−− L

Then the total number of ways of arranging Ni particles in the ith energy level is


Since the particles are indistinguishable, the interchange of any two electrons does not produce a new arrangement

The actual number of independent ways to distribute the Ni particles in the ithlevel is

)!(!

!

iii

ii NgN

gW

−=


))()((

exp1

1)(

)(

)(EgEN

kT

EEEf

dEEg

dEEN

fE <

⎟⎟⎠

⎞⎜⎜⎝

⎛ −+

==

N(E)dE (number density): Number of particles per unit volume per unit energy (E to E+dE)

g(E)dE: number of quantum states per unit volume per unit energy


Meaning of Fermi energy

At T=0K, The probability of a quantum state being occupied is unity for E<Ef

The probability of a quantum state being occupied is zero for E>Ef

exp1

1)(

⎟⎟⎠

⎞⎜⎜⎝

⎛ −+

=

kT

EEEf

fE

for 1)(

for 0)(

fE

fE

EEEf

EEEf

<=

>=


3.5.3 Features of Fermi-Dirac FunctionFor T>0K

The probability of a state being occupied at E=Ef is always ½

( ) 2

1

0exp1

1)( =

+=EfE


Example 3.6 3kT above the Fermi-energy

%74.420.091

1

3exp1

1

exp1

1)( =

+=

⎟⎠⎞

⎜⎝⎛+

=

⎟⎟⎠

⎞⎜⎜⎝

⎛ −+

=

kTkT

kT

EEEf

fE

At energy above Ef, the probability of a state being occupied by an electron become significantly less than unity


Example 3.7 99% probability

K756 eV06529.0

30.0exp1

1101.0

exp1

11)(1

=⇒=⇒

⎟⎠⎞

⎜⎝⎛ −+

−=

⎟⎟⎠

⎞⎜⎜⎝

⎛ −+

−=−

TkT

kT

kT

EEEf

fE

Assume Fermi energy is 6.25 eV, Calculate the temperature at which there is 1 % probability that a state 0.30 below the Fermi energy level will not contain an electron

eV/K1062.8

J/K1038.15

23

−

−

×=

×=k


Maxwell-Boltzmann Approximation

exp1

1)(

⎟⎟⎠

⎞⎜⎜⎝

⎛ −+

=

kT

EEEf

fE

kTEEkT

EEEf f

fE 3for exp)( >−⎥

⎦

⎤⎢⎣

⎡ −−≈

When E-Ef>>kT, the exponential term in the denominator is much greater than the unity

300KTat eV 026.0 ==kT

The Fermi-Dirac probability for E-Ef> 3kT can be approximated by Boltzmann distribution

For E-Ef>3kT, the Boltzmann approximation is slightly larger than the Fermi-Dirac curve with inaccuracy less than 5%


Example 3.8 3kTCalculate the energy at which the difference between Boltzmannapproximation and the Fermi-Dirac function is 5%

kTkTEEkT

EE

kT

EE

kT

EE

kT

EE

kT

EEkT

EE

ff

ff

f

f

f

3)05.0

1ln( 05.0exp

05.01exp1exp

05.0

exp1

1

exp1

1exp

≈=−⇒=⎟⎟⎠

⎞⎜⎜⎝

⎛ −−

=−⎭⎬⎫

⎩⎨⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛ −+⋅⎟⎟

⎠

⎞⎜⎜⎝

⎛ −−

=

⎟⎟⎠

⎞⎜⎜⎝

⎛ −+

⎟⎟⎠

⎞⎜⎜⎝

⎛ −+

−⎟⎟⎠

⎞⎜⎜⎝

⎛ −−

Chapter 3 Introduction to Quantum Theory of...

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