W.K. Chen Electrophysics, NCTU 1
Chapter 3 Introduction to Quantum Theory of Solids
W.K. Chen Electrophysics, NCTU 2
Outline
Allowed and forbidden energy bands
Electrical conduction in solid
Extension to three dimension
Density of state
Statistical mechanics
W.K. Chen Electrophysics, NCTU 3
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
W.K. Chen Electrophysics, NCTU 4
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
W.K. Chen Electrophysics, NCTU 5
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
W.K. Chen Electrophysics, NCTU 6
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
W.K. Chen Electrophysics, NCTU 7
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
W.K. Chen Electrophysics, NCTU 8
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
W.K. Chen Electrophysics, NCTU 9
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
W.K. Chen Electrophysics, NCTU 10
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
W.K. Chen Electrophysics, NCTU 11
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
W.K. Chen Electrophysics, NCTU 12
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
W.K. Chen Electrophysics, NCTU 13
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
W.K. Chen Electrophysics, NCTU 14
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
W.K. Chen Electrophysics, NCTU 15
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 )()(
W.K. Chen Electrophysics, NCTU 16
⇒= +− )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 +=+
W.K. Chen Electrophysics, NCTU 17
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
W.K. Chen Electrophysics, NCTU 18
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
)(cos))(cos(cos))(sin(sin2
)( 22
bakbaba +=++− βαβααβ
βα
0 , )(2
2>
−== γγβ
h
EVmjj o
V(x)
E
W.K. Chen Electrophysics, NCTU 19
)(cos))(cos(cos))(sin(sin2
)(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.
W.K. Chen Electrophysics, NCTU 20
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
W.K. Chen Electrophysics, NCTU 21
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.
)(cos))(cos(cos))(sin(sin2
)( 22
bakbaba +=++− βαβααβ
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
W.K. Chen Electrophysics, NCTU 22
2cosh
2sinh
xxxx eex
eex
−− +=
−=
x
2
sinhxx ee
x−−
=
2cosh
xx eex
−+=
][2
1cos ][
2
1sin θθθθ θθ jjjj eeee
j−− +=−=Q
W.K. Chen Electrophysics, NCTU 23
)(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
W.K. Chen Electrophysics, NCTU 24
W.K. Chen Electrophysics, NCTU 25
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
W.K. Chen Electrophysics, NCTU 26
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
W.K. Chen Electrophysics, NCTU 27
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
W.K. Chen Electrophysics, NCTU 28
)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
W.K. Chen Electrophysics, NCTU 29
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
W.K. Chen Electrophysics, NCTU 30
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
W.K. Chen Electrophysics, NCTU 31
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
W.K. Chen Electrophysics, NCTU 32
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))
W.K. Chen Electrophysics, NCTU 33
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
W.K. Chen Electrophysics, NCTU 34
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
W.K. Chen Electrophysics, NCTU 35
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
W.K. Chen Electrophysics, NCTU 36
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
W.K. Chen Electrophysics, NCTU 37
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
W.K. Chen Electrophysics, NCTU 38
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
W.K. Chen Electrophysics, NCTU 39
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
W.K. Chen Electrophysics, NCTU 40
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
= +
W.K. Chen Electrophysics, NCTU 41
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
W.K. Chen Electrophysics, NCTU 42
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
W.K. Chen Electrophysics, NCTU 43
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
W.K. Chen Electrophysics, NCTU 44
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
W.K. Chen Electrophysics, NCTU 45
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
W.K. Chen Electrophysics, NCTU 46
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
W.K. Chen Electrophysics, NCTU 47
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
W.K. Chen Electrophysics, NCTU 48
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
W.K. Chen Electrophysics, NCTU 49
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
W.K. Chen Electrophysics, NCTU 50
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
W.K. Chen Electrophysics, NCTU 51
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
W.K. Chen Electrophysics, NCTU 52
Effective mass concept in 3-dim crystal
W.K. Chen Electrophysics, NCTU 53
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
W.K. Chen Electrophysics, NCTU 54
, ,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=
W.K. Chen Electrophysics, NCTU 55
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
W.K. Chen Electrophysics, NCTU 56
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
=⇒⋅=
W.K. Chen Electrophysics, NCTU 57
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
W.K. Chen Electrophysics, NCTU 58
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
W.K. Chen Electrophysics, NCTU 59
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
W.K. Chen Electrophysics, NCTU 60
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)(
W.K. Chen Electrophysics, NCTU 61
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
W.K. Chen Electrophysics, NCTU 62
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
W.K. Chen Electrophysics, NCTU 63
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
W.K. Chen Electrophysics, NCTU 64
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
−=
W.K. Chen Electrophysics, NCTU 65
))()((
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
W.K. Chen Electrophysics, NCTU 66
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
<=
>=
W.K. Chen Electrophysics, NCTU 67
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
W.K. Chen Electrophysics, NCTU 68
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
W.K. Chen Electrophysics, NCTU 69
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
W.K. Chen Electrophysics, NCTU 70
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%
W.K. Chen Electrophysics, NCTU 71
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
≈=−⇒=⎟⎟⎠
⎞⎜⎜⎝
⎛ −−
=−⎭⎬⎫
⎩⎨⎧
⎟⎟⎠
⎞⎜⎜⎝
⎛ −+⋅⎟⎟
⎠
⎞⎜⎜⎝
⎛ −−
=
⎟⎟⎠
⎞⎜⎜⎝
⎛ −+
⎟⎟⎠
⎞⎜⎜⎝
⎛ −+
−⎟⎟⎠
⎞⎜⎜⎝
⎛ −−