Éçª w m - osaka universitykoun/lecs/material_design13.pdf · 5 392.077 6 489.981 sum i i(i)...
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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
2013年
12013年 11月 26日 火曜日
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第1節 第一原理計算における全エネルギー
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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
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1. Electronic energy
1–2
Etot =
−12m
ϕ i ∇2 ϕ i
i∑ + ρ(r)Vion (r)dr∫ + ρ(r)VH(r)dr∫ + ρ(r)εxc (r)dr∫
εii=1
N
∑ −12
ρ(r)VH (r)dr∫ − ρ(r) Vxc (r) − εxc (r)[ ]dr∫
=
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Binding energy
Cohesive energy
Formation energy
1. Electronic energy
1–3
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))
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1. Electronic energy
1–4
Cohesive and formation energies
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1. Electronic energy
1–5
Etot
Ry/cell eV/atom
E(B.C.)
eV/atomB12C3 -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
-92.686
B (atom) -5.1709 -70.354 -85.535 C (atom) -10.7498 -146.260
-85.535
Ecoh(B) 6.841 Ecoh (C) 8.390 Ecoh (B.C.) 7.260 Eform(B.C.) 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
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1. Electronic energy
1–6
• structure
Structure of Boron Carbide
rh
in
rh
ci
1
2
cc
c
ci
x y
z
3
4
c
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第2節 全エネルギーと固有値
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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
∑
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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,
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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
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Ionization energies and eigenvalues are different things.
2. One-electron level
2–4
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
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1 11.260
2 24.383
3 47.887
4 64.492
5 392.077
6 489.981
sumi I(i) 1030.080
Ionization potentials of carbon atom
(eV)Carbon
LSD
2p↑ 3.725
2p↓ 5.903
2s↑ 11.465
2s↓ 13.838
1s↑ 258.937
1s↓ 259.813
Etot 1019.501
orbital energy
CRC Handbook of Chemistry and Physics, 67th ed.
2. One-electron level
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The relaxation effect of wave function becomes insignificant when N → ∞.
2. One-electron level
2–6
Koopmans’ theorem
E(,ni ,) − E(,ni −1,) = εi
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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
2–7
∂E(ni)∂ni
= εi
E(,ni ,) − E(,ni −1,) = εi
Significance of eigenvalues
the highest occupied state
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2. One-electron level
2–10
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
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第3節 半導体中の不純物
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3. Application
3–7
Impurity states
conduction band
valence band
Ec
EvEA
ED Ed
Ea
donor and acceptor
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3. Application
3–9
Which species are donor, and which ones are
acceptor?
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3. Application
3–8
“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.
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How does impurity exists?
ni = Ns exp −GF
kT⎛⎝⎜
⎞⎠⎟
GF = HF − TSF
formation enthalpy
formation entropy
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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
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chemical bonds of impurity
H in Siex)
3–1
BC
AB
T C
H
T
BC site = stable site
BC site
T site
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H impurity levels
3–2
BC site p-like bonding state
+
s-like bonding
anti-bonding
sp bonding s
HSi
+
p-like bonding
sp anti-bondingp
HSi
anti-bonding
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3. Application
3–7
Charge states of impurity
conduction band
valence band
Ec
EvEA
ED Ed
Ea
donor and acceptor
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3. Application
3–9
donor and accepter levels
EA = Etot(N +1)(A) − Etot
(N ) (A)
ED = Etot(N ) (D) − Etot
(N −1)(D)
donor
acceptor
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3. Application
3–10
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
23
4
F(+)=E(+)+µ
1
EcEAEv
F(0)Ea
2
34
F( )=E( ) µ
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3. Application
3–11
shallow levels
Ec
µ
F(+)
EvED
F(0)
donor
Ec
µ
F(–)
Ev EA
F(0)
acceptor
“Donor and acceptor states refer to the change in the charge states”
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3. Application
3–12
Ecµ
F(+)
Ev
F(0)
E(0/+)
F(+)
F(0)
donor
n type
F(+)
F(0)
acceptor
p type
deep level I
Both donor and acceptor are possible.
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3. Application
3–13
deep level II
Ec
F(+)
Ev
F(0)
E(0/+)F(–)
E(0/–)
two-charge states amphoteric
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3. Application
3–14
Ex)
PL peak
FZ + anneal
S.D. Brotherton, et. al., J. Appl. Phys. 65, 1826 (1987)
Cu in Si
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3. Application
3–15
interpretation
D AAA
Ev Ec
DLTS
Nt
hole emission
hole capture
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Vacancy in Si
special topics
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EPR spectra of vacancy in Si
by Watkins
T=4.2 K
T=20.4 K
H || <100>
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Watkins, Inst. Phys. Conf. Ser. 23 (1975) p. 1.
Hole emission from V+
light illumination → V+
after turn off: V+ → V + h+
ESR signal
decay of ESR signal
B doped
Ev
V+
Ea = 0.045
B-
0.006
0.039
excited
In doped
Ev
V+
Ea = 0.16
In-
0.057
T = 20.4 K
EF
V+ = 0.05
time
cons
tant
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Watkins, et al., Inst. Phys. Conf. Ser. 46 (1979) p. 16.
DLTS signal
carrier injection → V++
Watkins, et al., Phys. Rev. Lett. 44 (1980) 593.
Ev
V
0.05
Ec
0.130+
+2+
V++ = 0.13
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Ishisada, Master Thesis (2009)
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Interstitial in Si
special topics
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EF (eV)
0.0 1.0
10
5
6
7
8
9
H f(eV) THB
(0)
(++)
(+)
3. Application
3–16
Formation enthalpy
R. Car, et al., PRL 52, 1814 (1984)
F(q) = E(q) + qµ
Interstitial Siex)
Later, Hf has been lowered by about 1 eV.
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3. Application
3–17
Si interstitial in n-Si donor level
G. D. Watkins, MRS Symp. Proc. 469, 139 (1997)
ED = E(0) − E(+)
Ec
Ev
0.4
0
(+)
(+)
(++)0.4
1.2
XTBTHT
Sii+ + e-Sii2+ + 2 e-
Sii0
8
7
6
5
4Form
atio
n en
thal
phy
[eV ]
1.2
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Example
self-interstitial in crystal Si
3. Application
3–5
Cs = 5x10 22 cm-3
Ci = Cs exp −EF − TSFkT
⎛⎝⎜
⎞⎠⎟
<110> splitting interstitialcy
Ef = 3.3 eV
Sf configurational entropy
vibrational entropy
Sf,c/k = ln 6 = 1.8
Sf,v/k = 3.9
CI [cm-3] = 2.0x10 25 exp[–Ef/kT]
1
2
3
4
5
6
7
8
65
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concentration
3. Application
3–6
CI [cm-3] = 2.0x10 25 exp[–Ef/kT]
self-interstitial in crystal Si
D =CI
Cs
dI +Cv
Cs
dv + Dx
dI = 1x10 -4 [cm2/s]
P.E. Blöchl et al., Phys. Rev. Lett. 70, 2435 (1993)
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第4節 熱力学的エネルギー、諸量との関係
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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
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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)
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phonon contribution to thermodynamic quantities
free energy
specific heat
Z0 = Tr{e−βH0 } = sinh(xq / 2)⎡⎣ ⎤⎦
−1,
q∏ xq = βωq
F0 = −kT lnZ0 =12ωq
q∑
zero-point energy
+ kT ln(1− e−βωq )q∑
U = −∂∂βlnZ
Cv = −T ∂2F∂T 2
4. Thermodynamics
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V or p ?
(b) elastic contribution
~ V dependence
isotropic case
anisotropic case
need of fully tensors
stress
strain
p = −∂U∂V
Uharm = Φel (V ) +Uph (u)
εijσ ij
4. Thermodynamics
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(c) anharmonic interactions
~ cross term between T and V
4. Thermodynamics
4–5
F(ε, ϑ ) = F0 +12Fεεε
2 + Fϑϑ +12Fϑϑϑ
2 + Fεϑεϑ
−Vσ =∂F∂ε
= Fεεε + Fεϑϑ = 0 ε = −FεϑFεε
ϑ
F(ϑ ) = F0 − S0ϑ +12
Fϑϑ −Fεϑ
2
Fεε
⎛⎝⎜
⎞⎠⎟ϑ 2
stress-free condition
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Example
stability of α and β borons
4. Thermodynamics
4–6
-78.5
-78.0
-77.5
-77.0
-76.5
-20 -10 0 10 20Pressure (GPa)
-boron-boronα
β
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0 500 1000 1500 2000 2500Temperature (K)
∆H = Hβ – Hα
A. Masago, KS
∆F = Fβ – Fα
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phase diagram boron
4. Thermodynamics
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