17 empty lattice approximation - binghamton...
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1
Empty lattice approximation
Masatsugu Sei Suzuki
Department of Physics, SUNY at Binghamton
(Date: March 21, 2018)
1. Empty lattice approximation
Actual band structures are usually exhibits as a plot of energy vs wavevector in the first
Brillouin zone.
When band energies are approximated fairly well by electron energies
2
2
2k
m k
ℏ
It is advsable to start a calculation by carrying the free electron energies back into the first zone.
We look for a reciprocal lattice vector G such that
' k k G
where k' is unrestricted and is the true wavevector in the empty lattice,
2
2
22
22 2 2
( ) '2
( )2
[( ) ( ) ( ) ]2
G
x x y y z z
m
m
k G k G k Gm
k k
k G
ℏ
ℏ
ℏ
.
with k in the first Brillouin zone and G allowed to run over the appropriate reciprocal lattice
points.
2
Fig. The Brillouin zone for the 2D square lattice. k' = k + G.
2. SC (simple cubic)
2( , , )
a
k , (Cartesian coordinate)
with , , and, are between -1/2 and 1/2, and
1 2 3
2 2( , , ) ( , , )h k l n n n
a a
G (Cartesian coordinate)
with h = n1, k = n2, and, l = n3 are integers.
2
2
222 2 2
1 2 3
( ) ( )2
2[( ) ( ) ( ) ]
2
Gm
n n nm a
k k Gℏ
ℏ
1 2
3
3
4
4
4
5
56
7
1
23
3
4
4
4
5
5
6
7
12
3
3
4
4
4
5
56
7
1
23
3
4
4
4
5
5
6
7
k'k
G
3
Fig. First Brillouin zone for the simple cubic lattice.
(a) M
= , and = 0. 2
10 .
2
2
222 2 2
1 2 3
( ) ( )2
2[( ) ( ) ]
2
Gm
n n nm a
k k Gℏ
ℏ
4
with
22
0
2
2
am
Eℏ
.
(b) MX
= 1/2, = 0, and changes from 1/2 to 0.
2
2
222 2 2
1 2 3
( ) ( )2
2 1[( ) ( ) ]
2 2
Gm
n n nm a
k k Gℏ
ℏ
(c) X
= = 0, and changes from 1/2 to 0.
2
2
222 2 2
1 2 3
( ) ( )2
2[( ) ]
2
Gm
n n nm a
k k Gℏ
ℏ
4. Numerical calculation (sc)
5
EêE0
G MM XX G0.5 1.0 1.5
1
2
3
4
5
6
Fig. Energy band of the empty sc lattice along the M, MX, and X line of the first Brillouin
zone .
5. fcc (face-centered cubic) lattice
2( , , )
a
k , (Cartesian coordinate)
with , , and, are between -1/2 and 1/2, and
1 2 3
2( , , )
2( , , )
h k l h k l h k la
n n na
G
(Cartesian coordinate)
with h, k, and, l are integers.
2
2
222 2 2
1 2 3
( ) ( )2
2[( ) ( ) ( ) ]
2
Gm
n n nm a
k k Gℏ
ℏ
7
Fig. First Brillouin zone for the fcc lattice. X = (0,0,1), W=(1/2, 0, 1), L = (1/2, 1/2, 1/2),
K = (1/4, 1/4, 1) or K = (3/4, 3/4, 0) in the units of 2/a. bx, by, bz are the reciprocal
lattice vectors of the conventional unit cell. . bx = (2/a) (1, 0, 0). by = (2/a) (0, 1,
0). bz= (2/a) (0, 0, 1).
(a) (0,0,0) →X(0,0,1)
2( , , )
a
k ,
where
= 0 and = 0. 10 .
8
2
2
2 2 2
0 1 2 3
( ) ( )2
[( ) ]
Gm
E n n n
k k Gℏ
with
22
0
2
2
am
Eℏ
.
(b) X(0,0,1)→W(1/2,0,1)
2( , , )
a
k ,
where
= 0 and = 1. 2/10 .
2
2
2 2 2
0 1 2 3
( ) ( )2
[( ) (1 ) ]
Gm
E n n n
k k Gℏ
(c) W(1/2,0,1)→L(1/2,1/2,1/2)
2( , , )
a
k ,
where
= 1 - and = 1/2. 2/10 .
2
2
2 2 2
0 1 2 3
( ) ( )2
[(1/ 2 ) ( ) (1 ) ]
Gm
E n n n
k k Gℏ
(d) (0,0,0)→L(1/2,1/2,1/2)
9
2( , , )
a
k ,
where
= = . 2/10 .
2
2
2 2 2
0 1 2 3
( ) ( )2
[(1/ 2 ) ( ) (1 ) ]
Gm
E n n n
k k Gℏ
(e) (0,0,0)→K(3/4,3/4,0)
2( , , )
a
k ,
where
= , =0. 4/30 .
2
2
2 2 2
0 1 2 3
( ) ( )2
[( ) ( ) ( ) ]
Gm
E n n n
k k Gℏ
(f) K(1/4,1/4,1)→X(0,0,1)
2( , , )
a
k ,
where
, =1, = 1/4 to 0.
2
2
2 2 2
0 1 2 3
( ) ( )2
[( ) ( ) (1 ) ]
Gm
E n n n
k k Gℏ
6. Numerical calculation (fcc)
10
G XX WW LL GG KK X
fcc
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.50
1
2
3
4
5
11
__________________________________________________________________________
7. bcc (body-centered cubic) lattice
EêE0
G XX WW LL GG KK X
fcc
0.5 1.0 1.5 2.0 2.5 3.0 3.5
2
4
6
8
10
12
2( , , )
a
k , (Cartesian coordinate)
with , , and, are between -1/2 and 1/2, and
1 2 3
2( , , )
2( , , )
k l l h h ka
n n na
G
(Cartesian coordinate)
with h, k, and, l are integers.
2
2
222 2 2
1 2 3
( ) ( )2
2[( ) ( ) ( ) ]
2
Gm
n n nm a
k k Gℏ
ℏ
13
Fig. First Brillouin zone for the bcc lattice. = (0,0,0), H = (1,0,0), N = (1/2, 1/2, 0), P = (1/2,
1/2, 1/2), in the units of 2/a. bx, by, bz are the reciprocal lattice vectors of the conventional
cubic unit cell. bx = (2/a) (1, 0, 0). by = (2/a) (0, 1, 0). bz= (2/a) (0, 0, 1).
(a) (0,0,0) →H(1,0,0)
2( , , )
a
k ,
where
14
= 0 and = 0. 10 .
2
2
2 2 2
0 1 2 3
( ) ( )2
[( ) ]
Gm
E n n n
k k Gℏ
with
22
0
2
2
am
Eℏ
.
(b) H(1,0,0)→P(1/2,1/2,1/2)
2( , , )
a
k ,
where
= = . 2/10 .
2
2
2 2 2
0 1 2 3
( ) ( )2
[(1 ) ( ) ( ) ]
Gm
E n n n
k k Gℏ
(c) P(1/2,1/2,1/2)→ (0,0,0)
2( , , )
a
k ,
where
= = , where changes from 1/2 to 0
2
2
2 2 2
0 1 2 3
( ) ( )2
[( ) ( ) ( ) ]
Gm
E n n n
k k Gℏ
15
(d) (0,0,0)→N(1/2,1/2,0)
2( , , )
a
k ,
where
= , = 2/10 .
2
2
2 2 2
0 1 2 3
( ) ( )2
[( ) ( ) ( ) ]
Gm
E n n n
k k Gℏ
(e) N (1/2,1/2,0)→ H (1,0,0)
2( , , )
a
k ,
where
+ = 1, =0. 12/1 .
2
2
2 2 2
0 1 2 3
( ) ( )2
[( ) (1 ) ( ) ]
Gm
E n n n
k k Gℏ
8. Numerical calculation for bcc
16
EêE0 bcc
G HH PP GG NN H0.0 0.5 1.0 1.5 2.0 2.5 3.00
1
2
3
4
5
17
EêE0 bcc
G HH PP GG NN H0.5 1.0 1.5 2.0 2.5 3.0
2
4
6
8
10
18
9. Radius of Fermi sphere for the fcc lattice
Fermi wavenumber kF:
nV
Vk
V
c
F 3
3 3
4
)2(2
,
where
n: electrons per primitive cell (fcc), or number of electrons per atom.
Vc: volume of primitive cell (fcc)
3
4
1aVc .
Then we get
na
kF 3
3
3
4
3
4
)2(
12
or
na
kF 3
23 12
or
3/1
2
32
n
akF
The Fermi energy is
3/2
0
3/2222
2
2
3
2
32
22
nE
n
amk
mFF
ℏℏ
with
19
22
0
2
2
am
Eℏ
.
In summary,
3/2
0 2
3
n
E
F
Table
10. Radius of Fermi sphere for the bcc lattice
Fermi wavenumber kF:
nV
Vk
V
c
F 3
3 3
4
)2(2
,
where
n: electrons per primitive cell (bcc), or number of electrons per atom.
Vc: volume of primitive cell (fcc)
3
2
1aVc .
n H 3 n
2 πL2ê3
1 0.610887
2 0.969723
3 1.2707
4 1.53934
5 1.78624
6 2.0171
7 2.23542
8 2.44355
9 2.64315
10 2.83549
11 3.0215
12 3.20195
13 3.37746
14 3.54851
15 3.71554
20
Then we get
na
kF 3
3
3
2
3
4
)2(
12
or
na
kF 3
23 6
or
3/1
4
32
n
akF
The Fermi energy is
3/2
0
3/2222
2
4
3
4
32
22
nE
n
amk
mFF
ℏℏ
with
22
0
2
2
am
Eℏ
.
In summary,
3/2
0 4
3
n
E
F
Table
21
11. Energy band of real systems: comparison with the empty lattice approximation
(i) Na (bcc)
Fig. Energy dispersion of Na (bcc). J. Yamashita, The theory of Electrons in Metals (Asakura,
1973, in Japanese)
n H 3 n
4 πL2ê3
1 0.384835
2 0.610887
3 0.800488
4 0.969723
5 1.12526
6 1.2707
7 1.40823
8 1.53934
9 1.66508
10 1.78624
11 1.90343
12 2.0171
13 2.12766
14 2.23542
15 2.34064
22
Fig. Energy dispersion (the empty lattice approximation) for thr bcc metal. J. Yamashita, The
Theory of Electrons in Metals (Asakura, 1973, in Japanese)
23
Fig. Energy dispersion of bcc from the empty lattice approximation (the present result).
(ii) Al (fcc)
EêE0 bcc
G HH PP GG NN H0.0 0.5 1.0 1.5 2.0 2.5 3.00
1
2
3
4
5
24
Fig. Energy dispersion of Al (fcc, full lines) compared with the free electron fcc metal (broken
lines) [After Segal (1961), Phys. Rev.124, 1797]. R.G. Chambers, Electrons in Metals and
Semiconductors (Chapman and Hall, 1990).
25
Fig. Energy dispersion of free electron fcc metal (empty lattice approximation) [Herman (1967),
in An Atomic Approach to the Nature and Properties of Materials (ed. Pask), Wiley, New
York]. R.G. Chambers, Electrons in Metals and Semiconductors, Chapman and Hall, 1990.
26
Fig. Energy dispersion of fcc from the empty lattice approximation (the present result).
APPENDIX
2D square lattice
2
2 22 2[( ) ( ) ]
2x yk h k k
m a a
ℏ
Here we put
2 2, ,x yk k
a a
G XX WW LL GG KK X
fcc
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.50
1
2
3
4
5
27
Thus we have the resulting energy dispersion as
2 2
22,
1( , ) [( ) ( ) ]
2
2
h k
E h k
m a
ℏ
where 0, 1, 2,....h , and 0, 1, 2,....k
(a) Energy dispersion along the ( ,0)xk direction (denoted by red arrow)
We make a plot of ( ,0)E as s function of for 1
2
28
(b) Energy dispersion along the ( , )x xk k direction (denoted by blue arrow)
We make a plot of ( , )E as s function of for 2
2
E
0.4 0.2 0.0 0.2 0.4
2
4
6
8
29
E
0.6 0.4 0.2 0.0 0.2 0.4 0.6
2
4
6
8
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