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

!"

!#

!$

(110)

" %" &" '" ("

"

%"

&"

'"

200 GPaPlotRange->{0., 0.025}

0 GPa

" %" &" '" (" )"

"

%"

&"

'"

" %" &" '" (" )"

"

%"

&"

'"

PlotRange->{0., 0.005}

Transition of metal to insulator

metal Na

Y. Ma, et al., Nature 458, 12 (2009)

calculated charge density

BCC HCP

1. Introduction