全エネルギーの使い道koun/lecs/material_design...全エネルギーは全てを決める...

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


1–5

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])


1–6

Cohesive and formation energies


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


1–8

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


2–2

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.


2–3

One-electron model Two-electron model

Ground state

Excited state

Statistics of impurity levels in a gap in semiconductors


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.


2–5

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.


2–6

The relaxation effect of wave function becomes insignificant when N → ∞.


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,!)


2–8

∂E(!ni!)∂ni

= εi

E(!,ni ,!) − E(!,ni −1,!) = εi

Significance of eigenvalues

the highest occupied state


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

全エネルギーの 使い道koun/lecs/material_design...全エネルギーは全てを決める...

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全エネルギーの使い道koun/lecs/material_design...全エネルギーは全てを決める...