2019年、大学院講義
「半導体物性」
白井光雲、産業科学研究所
多電子理論からのアプローチ
✤ 1,半導体物理序論
✤ 2,密度汎関数理論
✤ 3,ドナー・アクセプターの電子論
✤ 4,深い準位の物理
✤ 5,伝導機構
✤ 6,光学特性と伝導特性の統一的解釈
✤ 7,エキシトン
講義内容
経験則から理論へ
ヤーン・テラー効果
有効質量近似
電子ー格子相互作用
光ー電子相互作用
全電子理論
1. Introduction
P.Y. Yu and M. Cardona, Fundamentals of Semiconductors, 4th, (Springer, 2010) Yu & Cardona, FSGeneral reference:
1. Introduction
1.1 Characteristics of covalent semiconductors
I. Stoichiometric composition: (8-n) rule
III. Strong orientational nature of covalent bond
II. Small number of coordination number: Nc
Nc = 4
Nc > 6 metal
Compounds, not alloys
brittle ductile脆性 延性
1. Introduction
1.1.1 Crystal Structure
Diamond
1
2
3
4
1
2
3
4
Graphite
Wurtzite
1
2
Zincblende
1
2
12
β-Tin
Td2
Oh7 D6h4Fd3m
T43m-
P6 /mmc3
P6 mc3
C6v4 m: a mirror plane with normal (100)c: glide plane in the c-axis with normal (120)
63: a six-fold screw along the c-axis
I4 /amd1
D4h19
When c/a=√(2), this is equivalent to diamond structure
1. Introduction
Si
GaAs
ZnSe
8-N rule
Bi2Te3
B2O3
1.1.2 Stoichiometric composition
Yu & Cardona, FS, p.15
IV
III-V
II-VI
V2-VI3
III2-VI3
1. Introduction
Se1-x-yAsxGeyAs2Se3
Topological connectivity
R. Zallen, The Physics of Amorphous Solids, Wiley (1983), ch. 3.
Topological model of glass J.C.フィリップス、「ガラスの物理」日経エレクトロニクス、1982.8.2, p. 163.
1. Introduction
0 5 10 15 200
2.5
5
7.5
10
12.5
15
0 0.1[el/Bohr^3]
Max= 0.0828092 at {4, 3}Min= 0.00336242 at {7, 14}
(110)
(110)
0 5 10 15 20 25 300
5
10
15
20
25
0 0.0045
[el/Bohr^3]Max= 0.00434729 at {18, 5}Min= 0.00174657 at {1, 1}
0 20 40 60 800
20
40
60
80
100
120
140
0 0.05[el/Bohr^3]
0 20 40 60 80 100 1200
50
100
150
200
0 0.35[el/Bohr^3]
Max= 0.337882 at {12, 152}Min= 0.00374256 at {44, 7}
Si
0 2.5 5 7.5 10 12.5 150
2.5
5
7.5
10
12.5
15
0 0.22[el/Bohr^3]Max= 0.212156 at {1, 8}
Min= 0.000791678 at {1, 1}
(100)NaCl
Graphite
Na
covalent bonding
metallic bonding
vdW bonding
ionic bonding
1–6
1.1.3 Covalent bonding
Highly oriented bonding
1. Introduction
1. Crystal growth: layer-by-layer growth
M.E. エバハート、「日経サイエンス」2000年2月号、44
2. Mechanical properties: brittle ductile脆性 延性
1. Introduction
Metal/nonmetal transition at high pressure
Insulator Metal
Si Diamond HCP
Nc 4 6
G. J. Ackland, in High-Pressure Surface Science and Engineering, eds. Y. Gogotsi and C. Domnich, IOP (2004), p.120.
1. Introduction
Diamond
1
2
Oh7 Fd3m
3 C2
1.1.3 Symmetry
(R|0) (R|τ)
6 S4
6 σd
8 C3
1 E
3 σh
6 C4
6 C2
8 S6
1 I
1. IntroductionX-ray Diffraction
S(hkl) = f j exp −2π i x jh+ y jk + z jl( )⎡⎣
⎤⎦
j∑
= fκ exp −2π i x jh+ y jk + z jl( )⎡⎣
⎤⎦
j∑
κ∑
Structural factor
κ: chemical speciesof symmetry equivalent
キッテル「固体物理学入門」(上)、第5版、丸善、(1976), p.66
FCC
Sκ (hkl)
Sκ (hkl)
奇数偶数混合のとき0
Diamond
1+ exp iπ (h+ k)⎡⎣ ⎤⎦ + exp iπ (k + l)⎡⎣ ⎤⎦ + exp iπ (l + h)⎡⎣ ⎤⎦
1+ exp iπ (h+ k)⎡⎣ ⎤⎦ + exp iπ (k + l)⎡⎣ ⎤⎦ + exp iπ (l + h)⎡⎣ ⎤⎦( ) 1+ exp iπ2 (k + l)⎡
⎣⎢
⎤
⎦⎥
⎛⎝⎜
⎞⎠⎟
FCCの場合の消滅則に加え
hklが全て偶数で、かつ
h+k+l ≠ 4n
1.2 Band theory1. Introduction
Direct-gapIndirect-gap
Brillouin zone
L
Γ
K W XZ
US
g1g2
kxky
kz
g3
1. Introduction
Wigner-Seitz cell in the reciprocal space
1. Introduction
1.2.1 Nearly-free-electron model
plane-wave expansionϕk (r) = ck+GG∑ e− i(k+G)⋅r
− !2
2mk +G( )2 ck+G + U (k +G ')ck+G '
G '∑ = εkck+G U (G) = 1
ΩU (r)eiG⋅r dr∫
Yu & Cardona, FS, p.53
(3) (3)
(2)
(2)
(3)
(2)
(2) (2)
(2) (2)
ΔΛ00
1
2
4
6
8
Nel
1. Introduction
Yu & Cardona, FS, p.53
Comparison of free-electron band to a real band
1. Introduction
Energy band with spin freedom
Ge GaAs
Yu & Cardona, FS, p.64
1. Introduction
Chemical bonding picture
1. Introduction
1.2.2 Tight-bindig model
1. Introduction
Chemical bonding picture
1. Introduction
s
3p
sp3 hybrid
A1
hybrid orbitals
atomic orbitals bands
valence band
conduction band
T2
1. Introduction
Correspondence to the band structure
1. Introduction
s band
p band
k
E
XΓ
1. Introduction
1.2.3 Properties of band gap
Bragg reflection and band gap
U (G) = 1Ω
Ua (r −Ra )eiG⋅r dr∫
a∑ = 1
ΩeiG⋅R j Uκ (r)e
iG⋅r dr∫κ j∑
= 1Ω
Sκ (G)Uκ (G)κ∑ Sκ(G): Structural factor
0 G k
O(U){
0 G/2 k
2|U |G
1. Introduction
Bragg reflection and band gap
Why not splitting?
Uκ(G) = 0
for diamond structure
but not for Zincblende
U (G) = 1ΩU (r)eiG⋅r dr∫
for G = X 2πa1,0,0( )
Yu & Cardona, FS, p.53
1. Introduction
1) Difference between bonding and antibonding states:
2) Width of bands: W
Eg
Vcb
Dia. > Si > Ge > Sn
at p=0
Vcb
W
As p increases,
W increases
Eg
Eg decreases
Magnitude of band gap
1. Introduction
Deformation potential
1. Introduction
Band closing
a = dEd lnV
B = dpd lnV
ap =dEdp
= aB
~10 eV
~100 GPa
~0.1 eV/GPa
Band closing is predicted at 10 GPa for Si.
However, …
1. Introduction
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200 GPaPlotRange->{0., 0.025}
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Transition of metal to insulator
metal Na
Y. Ma, et al., Nature 458, 12 (2009)
calculated charge density
BCC HCP
1. Introduction