17 empty lattice approximation - binghamton...
TRANSCRIPT
![Page 1: 17 Empty lattice approximation - Binghamton Universitybingweb.binghamton.edu/~suzuki/SolidStatePhysics/17...2018/03/21 · 11. Energy band of real systems: comparison with the empty](https://reader035.vdocuments.net/reader035/viewer/2022081604/6112bb65d9cc8e0b00433c1b/html5/thumbnails/1.jpg)
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.
![Page 2: 17 Empty lattice approximation - Binghamton Universitybingweb.binghamton.edu/~suzuki/SolidStatePhysics/17...2018/03/21 · 11. Energy band of real systems: comparison with the empty](https://reader035.vdocuments.net/reader035/viewer/2022081604/6112bb65d9cc8e0b00433c1b/html5/thumbnails/2.jpg)
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
![Page 3: 17 Empty lattice approximation - Binghamton Universitybingweb.binghamton.edu/~suzuki/SolidStatePhysics/17...2018/03/21 · 11. Energy band of real systems: comparison with the empty](https://reader035.vdocuments.net/reader035/viewer/2022081604/6112bb65d9cc8e0b00433c1b/html5/thumbnails/3.jpg)
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
![Page 4: 17 Empty lattice approximation - Binghamton Universitybingweb.binghamton.edu/~suzuki/SolidStatePhysics/17...2018/03/21 · 11. Energy band of real systems: comparison with the empty](https://reader035.vdocuments.net/reader035/viewer/2022081604/6112bb65d9cc8e0b00433c1b/html5/thumbnails/4.jpg)
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)
![Page 5: 17 Empty lattice approximation - Binghamton Universitybingweb.binghamton.edu/~suzuki/SolidStatePhysics/17...2018/03/21 · 11. Energy band of real systems: comparison with the empty](https://reader035.vdocuments.net/reader035/viewer/2022081604/6112bb65d9cc8e0b00433c1b/html5/thumbnails/5.jpg)
5
EêE0
G MM XX G0.5 1.0 1.5
1
2
3
4
5
![Page 6: 17 Empty lattice approximation - Binghamton Universitybingweb.binghamton.edu/~suzuki/SolidStatePhysics/17...2018/03/21 · 11. Energy band of real systems: comparison with the empty](https://reader035.vdocuments.net/reader035/viewer/2022081604/6112bb65d9cc8e0b00433c1b/html5/thumbnails/6.jpg)
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
![Page 7: 17 Empty lattice approximation - Binghamton Universitybingweb.binghamton.edu/~suzuki/SolidStatePhysics/17...2018/03/21 · 11. Energy band of real systems: comparison with the empty](https://reader035.vdocuments.net/reader035/viewer/2022081604/6112bb65d9cc8e0b00433c1b/html5/thumbnails/7.jpg)
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 .
![Page 8: 17 Empty lattice approximation - Binghamton Universitybingweb.binghamton.edu/~suzuki/SolidStatePhysics/17...2018/03/21 · 11. Energy band of real systems: comparison with the empty](https://reader035.vdocuments.net/reader035/viewer/2022081604/6112bb65d9cc8e0b00433c1b/html5/thumbnails/8.jpg)
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)
![Page 9: 17 Empty lattice approximation - Binghamton Universitybingweb.binghamton.edu/~suzuki/SolidStatePhysics/17...2018/03/21 · 11. Energy band of real systems: comparison with the empty](https://reader035.vdocuments.net/reader035/viewer/2022081604/6112bb65d9cc8e0b00433c1b/html5/thumbnails/9.jpg)
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)
![Page 10: 17 Empty lattice approximation - Binghamton Universitybingweb.binghamton.edu/~suzuki/SolidStatePhysics/17...2018/03/21 · 11. Energy band of real systems: comparison with the empty](https://reader035.vdocuments.net/reader035/viewer/2022081604/6112bb65d9cc8e0b00433c1b/html5/thumbnails/10.jpg)
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
![Page 11: 17 Empty lattice approximation - Binghamton Universitybingweb.binghamton.edu/~suzuki/SolidStatePhysics/17...2018/03/21 · 11. Energy band of real systems: comparison with the empty](https://reader035.vdocuments.net/reader035/viewer/2022081604/6112bb65d9cc8e0b00433c1b/html5/thumbnails/11.jpg)
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
![Page 12: 17 Empty lattice approximation - Binghamton Universitybingweb.binghamton.edu/~suzuki/SolidStatePhysics/17...2018/03/21 · 11. Energy band of real systems: comparison with the empty](https://reader035.vdocuments.net/reader035/viewer/2022081604/6112bb65d9cc8e0b00433c1b/html5/thumbnails/12.jpg)
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
![Page 13: 17 Empty lattice approximation - Binghamton Universitybingweb.binghamton.edu/~suzuki/SolidStatePhysics/17...2018/03/21 · 11. Energy band of real systems: comparison with the empty](https://reader035.vdocuments.net/reader035/viewer/2022081604/6112bb65d9cc8e0b00433c1b/html5/thumbnails/13.jpg)
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
![Page 14: 17 Empty lattice approximation - Binghamton Universitybingweb.binghamton.edu/~suzuki/SolidStatePhysics/17...2018/03/21 · 11. Energy band of real systems: comparison with the empty](https://reader035.vdocuments.net/reader035/viewer/2022081604/6112bb65d9cc8e0b00433c1b/html5/thumbnails/14.jpg)
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
![Page 15: 17 Empty lattice approximation - Binghamton Universitybingweb.binghamton.edu/~suzuki/SolidStatePhysics/17...2018/03/21 · 11. Energy band of real systems: comparison with the empty](https://reader035.vdocuments.net/reader035/viewer/2022081604/6112bb65d9cc8e0b00433c1b/html5/thumbnails/15.jpg)
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
![Page 16: 17 Empty lattice approximation - Binghamton Universitybingweb.binghamton.edu/~suzuki/SolidStatePhysics/17...2018/03/21 · 11. Energy band of real systems: comparison with the empty](https://reader035.vdocuments.net/reader035/viewer/2022081604/6112bb65d9cc8e0b00433c1b/html5/thumbnails/16.jpg)
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
![Page 17: 17 Empty lattice approximation - Binghamton Universitybingweb.binghamton.edu/~suzuki/SolidStatePhysics/17...2018/03/21 · 11. Energy band of real systems: comparison with the empty](https://reader035.vdocuments.net/reader035/viewer/2022081604/6112bb65d9cc8e0b00433c1b/html5/thumbnails/17.jpg)
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
![Page 18: 17 Empty lattice approximation - Binghamton Universitybingweb.binghamton.edu/~suzuki/SolidStatePhysics/17...2018/03/21 · 11. Energy band of real systems: comparison with the empty](https://reader035.vdocuments.net/reader035/viewer/2022081604/6112bb65d9cc8e0b00433c1b/html5/thumbnails/18.jpg)
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
![Page 19: 17 Empty lattice approximation - Binghamton Universitybingweb.binghamton.edu/~suzuki/SolidStatePhysics/17...2018/03/21 · 11. Energy band of real systems: comparison with the empty](https://reader035.vdocuments.net/reader035/viewer/2022081604/6112bb65d9cc8e0b00433c1b/html5/thumbnails/19.jpg)
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
![Page 20: 17 Empty lattice approximation - Binghamton Universitybingweb.binghamton.edu/~suzuki/SolidStatePhysics/17...2018/03/21 · 11. Energy band of real systems: comparison with the empty](https://reader035.vdocuments.net/reader035/viewer/2022081604/6112bb65d9cc8e0b00433c1b/html5/thumbnails/20.jpg)
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
![Page 21: 17 Empty lattice approximation - Binghamton Universitybingweb.binghamton.edu/~suzuki/SolidStatePhysics/17...2018/03/21 · 11. Energy band of real systems: comparison with the empty](https://reader035.vdocuments.net/reader035/viewer/2022081604/6112bb65d9cc8e0b00433c1b/html5/thumbnails/21.jpg)
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
![Page 22: 17 Empty lattice approximation - Binghamton Universitybingweb.binghamton.edu/~suzuki/SolidStatePhysics/17...2018/03/21 · 11. Energy band of real systems: comparison with the empty](https://reader035.vdocuments.net/reader035/viewer/2022081604/6112bb65d9cc8e0b00433c1b/html5/thumbnails/22.jpg)
22
Fig. Energy dispersion (the empty lattice approximation) for thr bcc metal. J. Yamashita, The
Theory of Electrons in Metals (Asakura, 1973, in Japanese)
![Page 23: 17 Empty lattice approximation - Binghamton Universitybingweb.binghamton.edu/~suzuki/SolidStatePhysics/17...2018/03/21 · 11. Energy band of real systems: comparison with the empty](https://reader035.vdocuments.net/reader035/viewer/2022081604/6112bb65d9cc8e0b00433c1b/html5/thumbnails/23.jpg)
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
![Page 24: 17 Empty lattice approximation - Binghamton Universitybingweb.binghamton.edu/~suzuki/SolidStatePhysics/17...2018/03/21 · 11. Energy band of real systems: comparison with the empty](https://reader035.vdocuments.net/reader035/viewer/2022081604/6112bb65d9cc8e0b00433c1b/html5/thumbnails/24.jpg)
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).
![Page 25: 17 Empty lattice approximation - Binghamton Universitybingweb.binghamton.edu/~suzuki/SolidStatePhysics/17...2018/03/21 · 11. Energy band of real systems: comparison with the empty](https://reader035.vdocuments.net/reader035/viewer/2022081604/6112bb65d9cc8e0b00433c1b/html5/thumbnails/25.jpg)
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.
![Page 26: 17 Empty lattice approximation - Binghamton Universitybingweb.binghamton.edu/~suzuki/SolidStatePhysics/17...2018/03/21 · 11. Energy band of real systems: comparison with the empty](https://reader035.vdocuments.net/reader035/viewer/2022081604/6112bb65d9cc8e0b00433c1b/html5/thumbnails/26.jpg)
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
![Page 27: 17 Empty lattice approximation - Binghamton Universitybingweb.binghamton.edu/~suzuki/SolidStatePhysics/17...2018/03/21 · 11. Energy band of real systems: comparison with the empty](https://reader035.vdocuments.net/reader035/viewer/2022081604/6112bb65d9cc8e0b00433c1b/html5/thumbnails/27.jpg)
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
![Page 28: 17 Empty lattice approximation - Binghamton Universitybingweb.binghamton.edu/~suzuki/SolidStatePhysics/17...2018/03/21 · 11. Energy band of real systems: comparison with the empty](https://reader035.vdocuments.net/reader035/viewer/2022081604/6112bb65d9cc8e0b00433c1b/html5/thumbnails/28.jpg)
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
![Page 29: 17 Empty lattice approximation - Binghamton Universitybingweb.binghamton.edu/~suzuki/SolidStatePhysics/17...2018/03/21 · 11. Energy band of real systems: comparison with the empty](https://reader035.vdocuments.net/reader035/viewer/2022081604/6112bb65d9cc8e0b00433c1b/html5/thumbnails/29.jpg)
29
E
0.6 0.4 0.2 0.0 0.2 0.4 0.6
2
4
6
8