Fitzgerald-1999 Light is always quantized: Photoelectric effect Einstein • Photoelectric effect shows that E=h ν even outside the box I,E, λ e-metal block Maximum electron energy,
Trang 13.225 15
• Compton, Planck, Einstein
– light (xrays) can be ‘particle-like’
• DeBroglie
– matter can act like it has a ‘wave-nature’
• Schrodinger, Born
– Unification of wave-particle duality, Schrodinger
Equation
Wave-particle Duality: Electrons are not just particles
© E Fitzgerald-1999
Light is always quantized: Photoelectric effect (Einstein)
• Photoelectric effect shows that E=h ν even outside the box
I,E, λ
e-metal block
Maximum electron energy,
Emax
ν
ν
Emax =h(ν-νc)
!
Trang 2DeBroglie: Matter is Wave
N λ =2dsin θ
For small θ, θ~λ/d, so λmust be on order of
d in order to measure easily
© E Fitzgerald-1999
extremes)
Unification: Wave-particle Duality
(kx t
i
=
Ψ
k and p known exactly
i n
ne n n
∑
= Ψ
∞
=
generalized
i n
ne n n
∑
= Ψ
© E Fitzgerald-1999
)
Trang 33.225 19
Quantum Mechanics - Wave Equation
Classical Hamiltonian
QM Operators
E z y x
V
m
2
2
∆
=
i
p h
t i
E
∂
∂
−
= h
3 Average or expectation value of variable
∫
=
v
op ψ dV
α ψ
t i z
y x
V
∂
−
−
= +
∆
2
2
2
© H.L Tuller-2001
Assume ψ (x,y,z,t) separable into ψ (x,y,z) and φ (t)
Applying separation of variables:
ε
φ φ
ψ
∂
∂
−
= +
∇
−
t i
V
m
1
2
2
2
h
Time-Dependent Equation:
( ) t Ae i ( ε t Ae ωt
φ = − h = −
ω
ε h =
Time-Independent Equation:
⇒
)
Trang 43.225 21
Free Particle
• One dimensional V = O
ψ ψ
ε
2
2
k
m
dx
d
−
=
−
=
h
ikx Ae
=
ψ
) (
) , ( x t Aei kx ωt
• Momentum
k
dx x i
p x = ∫ ψ * h ∂ ∂ ψ = h
m
p
m
k
2
2
2 2
2
=
= h
ε
k
© H.L Tuller-2001
Particle in Box
2
2 2
;
h
ε
ψ = Ae ikx +Be− ikx k = m
Boundary Conditions:
0 ) ( )
0
( = ψ d =
ψ
0 )
0
( = A+B=
0 ) (
)
( d = A e ikd −e − ikd =
d
n
= n =1,2,3
=
d
x
n
ψ 2 sin n =1,2,3
V 0
=
x x = d
∞
•
•
•
∞
Trang 53.225 23
Particle in Box
z k y k x k A z
y
x
n ( , , ) = sin 1 ⋅ sin 2 ⋅ sin 3
ψ
2
3
2
2
2
1
i = π = 1,2,3
i
n
m
k n
n n md
n
2 ) (
8
2
2
2
3
2
2
2 1
2
2
h
=
ε
n = Quantum numbers
• Degeneracy
First excited state 112, 211, 121
© H.L Tuller-2001
Consequence of Electrons as Waves on Free Electron Model
Standing wave picture Traveling wave picture
0
0
L
L
n
k
e
e
e
L x
x ikx
L ik ikx
π
2
1
) ( ) (
) (
=
=
=
+ Ψ
=
Ψ
+
L
Trang 6Representation of E,k for 1-D Material
m
p m
k E
2 2
2 2 2
=
= h
E
k
∆k=2π/L
m
k E m
k dk dE
∆
=
∆
= 2 2 h h
states
electrons
En
En-1
En+1 m=+1/2,-1/2
All e- in box accounted for
EF
Total number of electrons=N=2*2kF*L/2 π
© E Fitzgerald-1999
Trang 73.225 1
Representation of E,k for 1-D Material
2 1 2
2 2
1 )
(
2
−
=
=
=
=
E m k
m L dE
dk
dk
dN
E
g
L
k
h
h π π
π
g(E)=density of states=number of electron states per energy per length
length is determined by the crystal structure and valence
occupied electron state at T=0
2
or 2 2
π π
n k mE k
L
N
h
m
k dk dE
mE k m
k E
2
2
2 ; 2 2
h
h h
=
=
=
© E Fitzgerald-1999
Representation of E,k for 2-D Material
E(kx,ky)
kx
ky
m
k
k
2
)
( 2 2
= h
© E Fitzgerald-1999
Trang 83.225 3
Representation of E,k for 3-D Material
kx
ky
kz
Ε(kx,ky,kz)
2π/L
m k k k
2 ) ( 2 2 2
2 + +
= h
Fermi Surface or Fermi Sphere
kF
m
k
v F
F = h
m
k
E F
F 2
2 2
h
=
B
F
F k
E
T =
3
2
2
)
(
h
mE
m
E
g
π
=
1
2
3 n
k F = π
© E Fitzgerald-1999
)
So how have material properties changed?
• The Fermi velocity is much higher than
kT even at T=0! Pauli Exclusion raises
the energy of the electrons since only 2
e- allowed in each level
• Only electrons near Fermi surface can
interact, i.e absorb energy and
contribute to properties
TF~104K (Troom~102K),
EF ~100Eclass, vF 2~100vclass 2
Trang 93.225 5
Effect of Temperature (T>0): Coupled electronic-thermal properties in conductors
properties
each energy state/level
Originates from:
N possible configurations
EF
1
1
) (
+
T k E E
b F e f
If E-EF/kbT is large (i.e far from
E E
b
F
e f
) ( −
−
=
© E Fitzgerald-1999
Fermi-Dirac Distribution: the Fermi Surface when T>0
µ ~EF
f(E)
1
T=0
T>0 0.5
kbT
kbT
E
All these e- not perturbed by T
fBoltz
Boltzmann-like tail, for
v
v T
U
∂
∂
=
Heat capacity of metal (which is ~ heat capacity of free e- in a metal):
(
~
~
U ∆ ⋅∆ b ⋅ F ⋅ b F ⋅ b U=total energy of
electrons in system
T k E g T
U
v v
2 ) (
2⋅ ⋅
=
∂
∂
= Right dependence, very close to exact derivation
© E Fitzgerald-1999
Trang 103.225 7
Electrons in a Periodic Potential
• Rigorous path: H Ψ =E Ψ
• We already know effect: DeBroglie and electron diffraction
• Unit cells in crystal lattice are 10-8 cm in size
• Electron waves in solid are λ =h/p~10-8 cm in size
• Certain wavelengths of valence electrons will diffract!
© E Fitzgerald-1999
Diffraction Picture of the Origin of Band Gaps
• Start with 1-D crystal again
λ ~a
a
1-D
θ
a
n
k
k
a
n
π
λ
π
λ
=
=
=
2
2
Take lowest order, n=1, and consider an incident valence electron moving to the right
x
a
i o
o
x
a
i i
i
e
a
k
e
a
k
π
π ψ
π ψ
π
−
=
−
=
=
=
;
;
Reflected wave to left:
Total wave for electrons with diffracted wavelengths:
x
a
i
x
a
o i
a
o i
s
o
i
π ψ ψ
ψ
π ψ ψ
ψ
ψ ψ
ψ
sin
2
cos
2
=
−
=
= +
=
±
=
a k k
k i o
π
2
=
−
=
∆