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全エネルギーの使い道白井光雲
大学院講義「先端物質設計論」
大阪大学産業科学研究所ナノテクセンター
2016年
www.cmp.sanken.osaka-u.ac.jp/~koun/Lecs/Material_Design16.pdf
第1節 第一原理計算における全エネルギー
なぜ全エネルギーか?1. Electronic energy
1–1
• 全電子からの見方
• 一電子からの見方バンド計算
DFT計算
1つ1つの電子エネルギーは分かる
相互作用をしている多粒子系では限界
1つ1つのエネルギーを足しても全エネルギーにならない
相互作用をしている多粒子系の実験は、全エネルギーの差
結合エネルギー電子分光、光分光も全エネルギーの差
全エネルギーは全てを決める1. Electronic energy
1–2
物質の結合
その結合エネルギーをどう計算するか?
電荷密度ρ(r)が分かれば求まる
(イオン,金属,共有性,ファン・デル・ワールス結合)
ρ(r) Ε
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
3. Chemical bond
1–3
3. Chemical bond
1–3
Variational Method
Potentialex) H2 molecule
final charge densityevolution of charge density
Total energy
Etot[ρ] = T + Uion[ρ] + UH[ρ] + Uxc[ρ]
kinetic energyelectron-ioninteraction
electron-electroninteraction
Ψ −12m
∇ j2 Ψ
j∑
ρ(r)Vion (r)dr∫
Vion (r) = −Ze2
| r − R |R∑
UH[ρ] = ρ(r)VH(r)dr∫Uxc[ρ] = ρ(r)Vxc (r)dr∫
VH(r) = e dr ' ρ(r ')r − r '∫
approximate Uxc
(LDA)
1. Electronic energy
1–4
Binding energy
Cohesive energy
Formation energy
1. Electronic energy
1–5
Immediate applications of Etot
Eb[A-B] = E[A] + E[B] – E[AB]
Ecoh[A(sol)] = E[A(gas)] – E[A(sol)]
Eform[AmBn] = E[AmBn] – (mE[A]+ nE[B])
1. Electronic energy
1–6
Cohesive and formation energies
1. Electronic energy
1–7
Etot Ry/cell eV/atom
E(B.C.) eV/atom
B12C3 -102.2094 -92.709 -92.709
alternate -102.3036 -92.795 -92.795 B12 -68.0843 -77.195 -92.686 diamond -22.7329 -154.650
B (atom) -5.1709 -70.354 -85.535 C (atom) -10.7498 -146.260 Ecoh(B) 6.841 Ecoh (C) 8.390 Ecoh (B12C3) 7.260 Eform(B12C3) 0.023
alternate 0.109
E(B.C.) = 154E(B) + E(C)[ ]
Exp.
5.777.37
0.146
Formation energy of boron carbide
D. M. Bylander, L. KleinmanPRB 42 1394 (1990)PRB 42 1316 (1990)
mixed gas
mixed solid
1. Electronic energy
1–8
Boron Carbide
rh
in
rh
ci
1
2
cc
c
ci
x y
z
3
4
c
0.023 eV 0.109 eV∆H = 0.146 eV (exp.)
全ての変化1. Electronic energy
1–9
A B
Q=∆H反応熱
反応の方向性
>0
<0
exothermic
endothermic
∆G=∆H–T∆S>0
<0
inhibit
proceed
=0 equilibrium
第2節 全エネルギーと固有値
orbital energy
ionization energy
Meaning of KS levels
...
2. One-electron level
2–1
εi
kXΓ
εi
Ii = E(!,ni ,!) − E(!,ni −1,!)
I (1) = E(N ) − E(N −1),I (2) = E(N −1) − E(N − 2),
Etot = I (i )i=1
N
∑
If it were
then
2. One-electron level
2–2
Etot = εii=1
N
∑
I (i ) = εN +1− i
Ii = E(!,ni ,!) − E(!,ni −1,!) = εi
Etot = εii=1
N
∑ −12
ρ(r)VH (r)dr∫ − ρ(r) Vxc (r) − εxc (r)[ ]dr∫
Actually,
In the two-electron picture, a single-electron energy is not defined.
2. One-electron level
2–3
One-electron model Two-electron model
Ground state
Excited state
Statistics of impurity levels in a gap in semiconductors
2. One-electron level
2–4
nd =1
12exp β εd − µ( ){ }+1
1exp β εd − µ( ){ }+1
nd =1+ exp −β εd − µ +U( ){ }
12exp β εd − µ( ){ }+1+ 12 exp −β εd − µ +U( ){ }
FD distribution
U → ∞
U → 0
Ionization energies and eigenvalues are different things.
2. One-electron level
2–5
He atom
Relaxation of wave functions by removing an electron.
1s -1.8359
He+
1s -4.0
total energy-5.7234
C. C. Roothaan et al.Rev. Mod. Phys. 32 (1960) 186.
(Ry units)
Ionization energy1.7234
(-1.1404)
(-5.6685)
LDA
1 11.26
2 24.383
3 47.887
4 64.492
5 392.077
6 489.981
sumi I(i) 1030.08
Ionization potentials of carbon atom
(eV)Carbon
LSD
2p↑ 3.725
↓ 5.903
2s↑ 11.465
↓ 13.838
1s↑ 258.937
↓ 259.813
Etot 1019.501
orbital energy
CRC Handbook of Chemistry and Physics, 67th ed.
2. One-electron level
2–6
The relaxation effect of wave function becomes insignificant when N → ∞.
2. One-electron level
2–7
Koopmans’ theorem
E(!,ni ,!) − E(!,ni −1,!) = εi
HF approach
DFT
transition state
Perdew, et al.
Koopmans’ theorem
Janak theorem
E(!,nN ) − E(!,nN −1) = εN
E(!,ni ,!) − E(!,ni −1,!) ≈ εi (!,ni − 0.5,!)
2. One-electron level
2–8
∂E(!ni!)∂ni
= εi
E(!,ni ,!) − E(!,ni −1,!) = εi
Significance of eigenvalues
the highest occupied state
2. One-electron level
2–9
Energy gap problem
k
N electrons
EDFT
k
N+1 electrons
EDFT
µ(N)
µ(N+1)Eg
Eg,DFT
∆
Ec = Etot(N +1) − Etot
(N )
Ev = Etot(N ) − Etot
(N −1)
Eg = µ (N +1) − µ (N )
= εN +1(N) − εN
(N)( ) + εN +1(N+1) − εN +1
(N)
= εN +1(N+1) − εN
(N)
Δ
第3節 半導体中の 不純物
Why does impurity exist?
ni = Ns exp −GF
kT⎛⎝⎜
⎞⎠⎟
GF = HF − TSF
formation enthalpy
formation entropy
3–1-1
introduction of n defects
equilibrium condition
G = G0 + n(HF − TSF ) − TSd
∂G∂n
= 0
entropy of disorder
Sd = k lnW W =
N(N −1)!(N − n +1)n!
=N !
(N − n)!n!
→ k N(lnN −1) − (N − n)[ln(N − n) −1]− n(lnn −1){ }
GF = T ∂Sd∂n
→ k ln Nn
nN
= exp −GF
kT⎡⎣⎢
⎤⎦⎥= exp SF
k⎡⎣⎢
⎤⎦⎥exp −
HF
kT⎡⎣⎢
⎤⎦⎥
3. Application
3–1-2
3. Application
3–2-1
donor and acceptor
“Positively charged states of an impurity are defined as donor states, and negatively charged states are defined as acceptor states.”
S. T. Pantelides, Rev. Mod. Phys. 50 (1978) 797.
Which species are donor, and which ones are acceptor?
VO Donor→
0
10En
ergy
\e
V\
M Σ Γ X∆ ∆S R T M Z X Γ
SrTiO3
O2p
Ti3d
1T1u
1T1g
2Eg3Eg
2T1u1T2u
3 O2–
6 eTi4+
Sr2+
4 e
2 e
3. Impurity physics
(i) Oxygen vacancy in SrTiO3
3–2-2
W. Luo, et al., Phys. Rev. B 70, 124109 (2004)
3. Impurity physics
3–2-3
HBC
HT(i)
C. G. van de Walle, et al., Phys. Rev. B 39, 10791 (1989)
BC
I
+
p-like bonding
sp anti-bondingp
HSi
anti-bonding
3. Impurity physics
(ii) Hydrogen in Si
3–2-3
3. Impurity physics
(iii) Vacancies in GaN
S. Limpijumnog & C. G. van de Walle
#31,32
#33 0.0442 = 0.60eV
0.2056 = 2.80eV
VGa
VN
Acceptor→
Donor→
J. Neugebauer & C. G. van de Walle Phys. Rev. B 50, 8067 (1994)
#29
#30
=3.36eV0.2476
3–2-4
3. Application
Charge states of impurity
conduction band
valence band
Ec
EvEA
ED Ed
Ea
donor and acceptor
3–3-1
3. Application
donor and accepter levels
EA = Etot(N +1)(A) − Etot
(N ) (A)
ED = Etot(N ) (D) − Etot
(N −1)(D)
donor
acceptor
3–3-2
3. Application
Presentation of impurity states
donor level
F(q) = E(q) + qµformation energy
F(+) = E(+) + µ
acceptor level F(−) = E(−) − µ
1
EcEDEv
F(0)
Ed
2
34
F(+)=E(+)+µ
1
EcEAEv
F(0)
Ea
2
34
F( )=E( ) µ
3–3-3
How effective is this doping?
problem of self-compensation
special topics
3–4-1
0.0 0.2 0.4 0.6 0.8Fermi level (eV)
Form
atio
n en
ergy
(eV)
-1
0
1
2
3
SiGa
AsGa
SiASVGa
VAs
Ga-poor Si-doped GaAs
3–4-2
500 600 700 800 900 1000 1100 1200
T (K)
0.0
0.2
0.4
0.6
0.8
1.0
eV)
n (
cm-3
)
1016
1017
1018
1019
1020
n (
cm-3
)
1016
1017
1018
1019
1020
Si concentation (cm-3)1016 1018 1020 1022 1024
T=1200K
T=1200K
NSi = N0 exp −βΔHSi( )
n = Nc* exp −β(Ec − µ){ }
ΔHSi = 0.57
ND+ − NA
− = n − p
NSi+ = NSi
ND0 = ND
12exp β εd − µ( ){ }+1
3–4-3
Def
ect c
once
ntra
tion
(cm
-3)
1016
1018
1020
1022
Si concentration (cm-3)1016 1018 1020 1022 1024
1014
1014
SiGa+
VGa-3
SiAs-
AsGa0
第4節 熱力学的エネルギー 諸量との関係
internal energy
kinetic energy of atoms
U = Ekin +Φ({Rl})Etot
!"# $#
Rl = Rl0 + ul
Φ({Rl}) = Φ(0) ({R0l}) +Φ(1) +Φ(2) +Φ(3) + ...
Φ(2) =12
∂2Φ∂Rl
2l∑ ul
2
U = Φ(0) ({R0l})Etot
0! "# $# + Ekin +Φ(2)
Uharm! "# $# +Φ(3) + ...
4. Thermodynamics
4–1
ΔU = ΔQ − pΔV
phonon contributionelastic contribution
– Harmonic approximation –
• bulk modulus • force constant
V and u are independent variables.
but ...
B = −V ∂2U∂V 2 f = ∂2U
∂u2
∆L = Nu
Uph (u) =Uph
0 +12fiui
2
i∑ =Uph
0 + !ωqnqq∑Φel (V ) = Φel (V0 ) +
12BV0ε
2
4. Thermodynamics
4–2
Uharm = Φel (V ) +Uph (u)
phonon contribution to thermodynamic quantities
free energy
specific heat
Z0 = Tr{e−βH0 } = sinh(xq / 2)⎡⎣ ⎤⎦
−1,
q∏ xq = β!ωq
F0 = −kT lnZ0 =12!ωq
q∑
zero-point energy"#$ %$
+ kT ln(1− e−β!ωq )q∑
U = −∂∂βlnZ
Cv = −T ∂2F∂T 2
4. Thermodynamics
4–3
4. Thermodynamics
4–4
ω 2 u1u2
⎡
⎣⎢⎢
⎤
⎦⎥⎥=
k11 k12k21 k22
⎡
⎣⎢⎢
⎤
⎦⎥⎥
u1u2
⎡
⎣⎢⎢
⎤
⎦⎥⎥
u
1 2f = k u
f2 = - u2k12 = - u1 u2
2E
Dynamic matrix
Phonon frequency
Contour map of free energy4. Thermodynamics
4–5