Fitzgerald-1999 • Assume electrons with wave vectors k’s far from diffraction condition are still free and look like traveling waves and see ion potential, U, as a weak background poten
Trang 13.225 9
Diffraction Picture of the Origin of Band Gaps
Probability Density=probability/volume of finding electron=| ψ |2
x
a
x
a
s
a
π
ψ
π
ψ
2
2
2
2
cos
4
sin
4
=
=
a
a
• Only two solutions for a diffracted wave
• Electron density on atoms
• Electron density off atoms
• No other solutions possible at this wavelength: no free traveling wave
© E Fitzgerald-1999
• Assume electrons with wave vectors (k’s) far from diffraction condition are still free and look like traveling waves and see ion potential, U, as a weak background potential
• Electrons near diffraction condition have only two possible solutions
– electron densities between ions, E=Efree-U
– electron densities on ions, E= Efree+U
• Exact solution using H Ψ =E Ψ shows that E near diffraction conditions is also parabolic in k, E~k2
Nearly-Free Electron Model
© E Fitzgerald-1999
Trang 23.225 11
Nearly-Free Electron Model (still 1-D crystal)
m
p
m
k
E
2
2
2
2
2
=
k
∆k=2π/L
Quasi-continuous
k m
k E m
k dk dE
∆
=
∆
=
2 2
h h
states
π /a
π/a
Away from k=nπ/a, free electron curve
∆k=2π/a=G=reciprocal lattice vector
Near k=n π /a, band gaps form, strong interaction of e- with
U on ions
© E Fitzgerald-1999
Electron Wave Functions in Periodic Lattice
• Often called ‘Bloch Electrons’ or ‘Bloch Wavefunctions’
E
k
π /a 0
Away from Bragg condition, ~free electron
m
k E e m
U m
2
2 2 2
2 2
2
h h
−
Near Bragg condition, ~standing wave electron
( )x Gx Gx u( )x E U ( )x U
U m
H=− ∇ + o≈ o ; ≈cos or sin = ; = o
2
2 2
ψ h
Since both are solutions to the S.E., general wave is
( ) x u
eikx lattice free =
=
termed Bloch functions
Trang 33.225 13
Block Theorem
• If the potential on the lattice is U(r) (and therefore
U(r+R)=U(r)), then the wave solutions to the S.E are a
plane wave with a periodic part u(r) that has the periodicity
of the lattice
( ) ( r u r R )
u
r u e
r ik r
+
=
=
Note the probability density spatial info is in u(r):
* 2
*Ψ = Ψo u r ⋅ u r
Ψ
An equivalent way of writing the Bloch theorem in terms of Ψ :
e
r e
R r u e R r
R ik
r ik R r ik R
r ik
Ψ
= + Ψ
Ψ
= +
= + Ψ
⋅
⋅ + +
© E Fitzgerald-1999
Reduced-Zone Scheme
• Only show k=+- π /a since all solutions represented there
π/a
−π/a
© E Fitzgerald-1999
)
Trang 43.225 15
Real Band Structures
• GaAs: Very close to what we have derived in the nearly free electron model
• Conduction band minimum at k=0: Direct Band Gap
© E Fitzgerald-1999
Review of H atom
( ) ( ) ( )
ψ
ψ
φ θ ψ
E
H
r
R
=
Φ Θ
=
Do separation of variables; each variable gives a separation constant
φ separation yields ml
θgives
r gives nl
After solving, the energy E is a function of n
4
2 13.6 2
eV n
e Z E
o
−
=
−
=
h πε µ
mland Φand Θgive Ψthe shape (i.e orbital shape)l
The relationship between the separation constants (and therefore the quantum numbers are:)
n=1,2,3,…
=0,1,2,…,n-1
ml=- , - +1,…,0,…, ,
(ms=+ or - 1/2)
l
0
-13.6eV
U(r)
)
in
-1
Trang 53.225 17
Relationship between Quantum Numbers
Origin of the periodic table
© E Fitzgerald-1999
Bonding and Hybridization
• Energy level spacing decreases as atoms are added
• Energy is lowered as bonding distance decreases
• All levels have E vs R curves: as bonding distance decreases, ion core
repulsion eventually increases E
E
R
s p
Debye-Huckel
hybridization
NFE picture,
semiconductors
© E Fitzgerald-1999
Trang 63.225 1
Properties of non-free electrons
• Electrons near the diffraction condition are not
approximated as free
• Their properties can still be viewed as free e- if an
‘effective mass’ m* is used
π/a
−π/a
2 2
2
*
* 2 2
2
k E m m
k E
ec ec
∂
∂
=
= h h
2 2
2
*
* 2 2
2
k E m m
k E
ev ev
∂
∂
=
= h
h
Note: These electrons have negative mass!
m
k E
2
2 2
h
=
© E Fitzgerald-1999
Band Gap Energy Trends
Note Trends: 1 As descend column, MP decreases as does Eg while aoincreases
2 As move from IV to III-V to II-VI compounds become more ionic,
MP and Eg increase while aotends to decrease
6 / 10 3.56 / 3.16
1685 / 1770 1.1 / 3 5.42 / 5.46
1231 / 1510 / ? 0.72 / 1.35/ ? 5.66 / 5.65 / ?
508 / 798 / ? 0.08 /0.18 / 1.45 6.45 / 6.09 / ?
IV / III-V / II-VI*
∗Fill in as many of the question marks as you can C
Trang 73.225 3
Trends in III-V and II-VI Compounds
Band
Gap
Gap
(eV) )
Lattice Constant (A)
Lattice Constant (A)
SiGe
SiGe
Alloys
Alloys
Larger atoms, weaker bonds, smaller U, smaller Eg, higher µ , more costly!
© E Fitzgerald-1999
Energy Gap and Mobility Trends
Material
GaN AlAs GaP GaAs InP InAs InSb
Eg(eV)°K 3.39 2.3 2.4 1.53 1.41 0.43 0.23
µn(cm2/V·s)
150
180 2,100 16,000 44,000 120,000 1,000,000
Remember that:
*
m
e τ
2
2 2
*
1
1
k
E h
∂
=
© H.L Tuller-2001
Trang 83.225 5
Metals and Insulators
• E F in mid-band area: free e-, metallic
• E F near band edge
• E F in very large gap, insulator
© E Fitzgerald-1999
Semiconductors
• Intermediate magnitude band gap enables
free carrier generation by three mechanisms
– photon absorption
– thermal
– impurity (i.e doping)
• Carriers that make it to the next band are
free carrier- like with mass, m*
Trang 93.225 7
Semiconductors: Photon Absorption
• When Elight=h ν >Eg, an electron can be promoted
from the valence band to the conduction band
Ecnear band gap
Evnear band gap
E
k
Creates a ‘hole’ in the valence band
© E Fitzgerald-1999
Holes and Electrons
• Instead of tracking electrons in valence band, more convenient to track missing
electrons, or ‘holes’
• Also removes problem with negative electron mass: since hole energy increases as holes
‘sink’, the mass of the hole is positive as long as it has a positive charge
Decreasing electron energy
Decreasing electron energy
Decreasing hole energy
© E Fitzgerald-1999
Trang 103.225 9
Conductivity of Semiconductors
• Need to include both electrons and holes in the conductivity expression
* 2
* 2
h h e
e h
e
m
pe m
ne pe
µ µ
p is analogous to n for holes, and so are τ and m*
Note that in both photon stimulated promotion as well as thermal
promotion, an equal number of holes and electrons are produced, i.e n=p
© E Fitzgerald-1999
Thermal Promotion of Carriers
• We have already developed how electrons are promoted in energy with T: Fermi-Dirac distribution
• Just need to fold this into picture with a band-gap
EF
g(E)
gc(E)~E1/2 in 3-D
gv(E) Despite gap, at non-zero
temperatures, there is some possibility of carriers getting into the conduction band (and creating holes in the valence band)
Eg +