38  1
5.73
Lecture
#38
updated September 19, 20032:22 PM
LAST TIME
:
Infinite 1D Lattice II
H
2
+
localization
↔
tunneling:
overlap
bonding and antibonding orbitals
R vs.
distance below top
of barrier
a n
0
2
1D
∞
lattice:
1 state per ion
tunneling only between nearest neighbors
∞
H
matrix
0 = c
q
(E
0
– E) – A(c
q–1
+ c
q+1
)
∞
# of coupled equations
Usually solve for {c
q
} by setting determinant of coefficients = 0
and solving for E.
Can’t do this because determinant is
∞
.
TRICK:
expect equal probability of finding e
–
on each lattice site by analogy to
plane wave e
ikx
, where probability density is uniform at all sites along x,
try
c
q
= e
ikq
x
q
= q
integer
distance between lattice sites
H
H
=
−
−
=
=
=−∞
∞
∑
E
A
A
E
E
c
q
q
q
q
q
0
0
0
0
0
0
0
0
O
O
O
O
O O
ν
ϕ
ν
ϕ
ϕ
ν
TIGHTBINDING (KronigPenney) Model (see Baym pp. 116122)
Notice that this is similar to free particle e
ikx
, which seems rather strange because
particle is never really free in “tightbinding” model.
Variational wavefunction.
Minimize E.
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38  2
5.73
Lecture
#38
updated September 19, 20032:22 PM
c
q
2
1
=
divide through by
e
E
E
A
k
ikq
0
2
0
=
−
(
)
−
cos
E k
E
A
k
( )
=
−
0
2
cos
E(k)
k
E
0
—
−π
+ π
0
E varies continuously over an interval 4A, where A is the adjacent site interaction strength
or the “tunneling integral”
What happens when we look at
k
outside 
π
/
≤
k
<
π
/
“1st Brillouin Zone”
c
e
k
k
c
e
e
e
e
k
ikq
k
i k
q
ikq
i
q
ikq
=
′ =
+
π
=
=
=
′
+
π
π
2
2
2
(one additional
wavelength per
lattice spacing )
wavefunction is unchanged!
So if
k
goes outside 1st Brillouin Zone, get same
ψ
, so get same E
nothing new!
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 Spring '04
 RobertField
 free particle, meff, cos kl

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