Computational Materials Physics · Database: Calculated Stacking fault energy Green function...

title

narrated by

Hans L. Skriver

Computational Materials Physics

thanks toJ. KollarB. JohanssonG. JohannessonO. JepsenJ.-P. JanK. Jacobsen

L. VitosP. SöderlindS.I. SimakA. RubanN. RosengårdJ. Nørskov

A. NiklassonS. MirbtA. McMahanA.R. MackintoshW. LambrechtP.A. Korzhavyi

O. ErikssonN.E. ChristensenT. BligårdO.K. AndersenM. AldénI. Abrikosov

Center for Atomic-scale Materials Physics CAMP-DTU Lyngby

Eugene Wigner and Frederick Seitz (1955)

Wigner and SeitzIf one had a great calculating machine, one might apply it to the problem of solving the Schrödinger equationfor each metal and obtain thereby the interesting physical properties, such as the cohesive energy, the lattice constant, and similar parameters. It not clear, however, that a great deal would be gained by this. Presumably the results would agree with the experimentally determined quantities and nothing vastly new would be learned from the calculation.

It would be preferable instead to have a vivid picture of the behaviour of the wave functions, a simple description of the essence of the factors which determine cohesion and an understanding of the origins of variation in properties from metal to metal.

The task before us is not a purely scientific one;it is partly pedagogic.

wignerSeitz

Eriksen's Theorem

Physics is an experimentalscience












eriksen

coh3

Ca

Sc

TiV

Cr

Mn

Fe Co Ni

CuSr

Y

Zr

NbMo Tc Ru

Rh

Pd

AgBa

Lu La

Hf

TaW

Re Os

Ir

Pt

Au

0 2 4 6 8 100

100

200

300

400

500

600

700

800

900

1000

d-occupation

(kJ/

mol

e)

3d 4d 5d

C. Kittel: Introduction to Solid State physics, 7th edition (1996)

Measured Cohesive Energies

Cohesive EnergyEnergy difference betweenatom in the ground stateand the energy of the solidper atom at zero temperature

Ecoh = Ebond - ÇE atom

Ç Eatom= Epro + ÇE

E pro

Ç E : Calculated from local exchange integrals

: From experiment (Moore 1958)

coh2

E pro

ÇE

metal

atom

Ecoh

E bond (dn )*

(dn+1)*

(dn+1)

M.S. Brooks and B. Johansson, J. Phys. F13, L197 (1983)

M.S. Brooks and B. Johansson, J. Phys. F13, L197 (1983)

coh1

Cohesive Energy

The Friedel Model

E bond = -ÄEF

0D(E) ( E - Ea ) dE

= - ( E_

Ea- ) nd

ÄEF

0D(E) dEnd =

= ndW20 ( 10 - nd )

d-band occupation:

Bonding energy:

J. Friedel in The Physics of Metals (ed. J.M. Ziman, 1969)

friedelm

D(E)

E

W

0

EF

E_ Ea

10W

E bond

nd

2520

W

0 10

Morruzzi

Local Density Theory of MetallicCohesion

V.L. Morruzzi, A.R. Williams and J.F. Janak, Phys. Rev. B15, 2854 (1977)

Calculations by

using P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964)

W. Kohn and L. Sham, Phys. Rev. 140, A1133 (1965)

L. Hedin and B.I. Lundqvist, J. Phys. C 4, 2064 (1971)

U. von Barth and L. Hedin, J. Phys. C 5, 1629 (1972)

DFT

LDA

LSD

and fast KKR A.R. Williams, J.F. Janak and V.L. Morruzzi, Phys. Rev. B6, 4509 (1972)

Li

Na

K

Rb

Cs

Be

Mg

Ca

Sr

Ba

Sc Ti

ZrY

Lu Hf

V

Nb

Ta

Cr

Mo

W

Mn

Tc

Re

Ru

Os

Fe Co

Rh

Ir

Ni Cu

Pd

Pt

Ag

Au

fcc bcc hcp

Crystal Structures

cstruc

canon

O.K. Andersen, J. Madsen, U.K. Poulsen,O. Jepsen and J. Kollar, Physica 86-88B, 249 (1977)

Canonical Band Theory

struce-4d

0 2 4 6 8 10

-40

-30

-20

-10

0

10

20

30

40

50

Ag

Pd

Rh

Ru

Tc

MoNb

ZrYSr

4d metals

[mR

y]

d-occupation

bcc

fcchcp

Theory

"Exp."

fcchcp

Li

Na

K

Rb

Cs

Be

Mg

Ca

Sr

Ba

Sc Ti

ZrY

Lu Hf

V

Nb

Ta

Cr

Mo

W

Mn

Tc

Re

Ru

Os

Fe Co

Rh

Ir

Ni Cu

Pd

Pt

Ag

Au

fcc bcc hcp

Calculated Structural EnergyH.L. Skriver, Phys. Rev. B31, 1909 (1985)

A.R. Miedema and A.K. Niessen, CALPHAD 7, 27 (1983)

strDB

d-occupation [states/atom]

LMTO-ASA plus Madelung correction

Database:Calculated Structural energydifferences

H.L. Skriver, Phys. Rev. B31, 1909 (1985)

Andersen's force theorem

H.L. Skriver, Phys. Rev. Lett. 49, 1768 (1982)

Barth-Hedin XC

Li - Pa at experimental volume

stackDB

Instrinsic stacking fault

LMTO-ASA

Database:Calculated Stacking fault energy

Green function approach, semi-infinite geometry

Transition metals at experimental volume

N.M. Rosengaard, H.L. Skriver, Phys. Rev. B47, 12 865 (1993)

LDA: Perdew-Zunger,Ceperley-Alder

surfaceTension

J.N. Schmit: Ph.D. Thesis (1978), University of Liege; in Zangwill: Physics at Surfaces (1988)

Atomic number

Experimental Surface Tension

eotvos

1200 1250 1300 1350 1400 1450 1500 1550 16001.0

1.2

1.4

1.6

1.8

2.0

Surface tension of Cu

Temperature [K]

Sur

face

tens

ion

[J/m

2 ]

Tm ~ 20 %

Shaler and Wulff, Trans. AIME 185, 186 (1949) Belforti and Lopie, Trans. AIME 227, 20 (1963)

Eötvös Law

Ó Ò(T) Tm( - T)=Ó(Tm)+

Surface tension (energy) islinear in temperature.

Discontinuity at melting isapproximately 20 %.

Surface Energy

E coh = ndW20 ( 10 - nd )

Friedel:

Energy required to transforma bulk atom into a surface atomwith a corresponding increase in surface area

E surf = nd ( 10 - nd )WÔ20

Tight-Binding:

W = 1 - (8/12)Ô 1/2 ~ 0.19

surfaceEnergy

D(E)

E

W

0

EF

E_

10W

E surf

nd

2520

0 10

WÔ

D(E)

E

0

EF

E_

10W- WÔ

E surf

W- WÔ

surft-4d

36 38 40 42 44 46 480

1

2

3

RbSr

Y

Zr

NbMo

TcRu

Rh

Pd

Ag

Surface tensionat the melting temperature

4d metals

Atomic number

Sur

face

tens

ion

(J/m

2 )

: de Boer et al., Transition Metals Alloys in Cohesion and Structure, Vol 1 (North-Holland, 1988)

: Vitos et al., Surf. Sci. 411, 186 (1998); DFT-LMTO-FCD

Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd0

1

2

3

4

Calc. fcc111De Boer

4d metals

Sur

face

ener

gy(J

/m2 )

"Experimental" Surface Energy

surfe-4d

0.0 0.2 0.4 0.6 0.8 1.00.0

0.5

1.0

1.5

2.0

2.5

Mo

Ag

Pd

Rh

RuTc

NbZr

Y

Sr CalculatedFriedel theory

4d metals

d-band filling

Sur

face

ener

gy(e

V/a

tom

)

0 2 4 6 8 10 120

1

2

3

4

5

RbSr

Y

Zr

Nb

MoTc

Ru

Rh

Pd

Ag

Calculatedde Boer

4d metals

Number of valence electrons

Sur

face

ener

gy(J

/m2 )

: de Boer et al., Transition Metals Alloys in Cohesion and Structure, Vol 1 (North-Holland, 1988)

: Vitos et al., Surf. Sci. 411, 186 (1998); DFT-LMTO-FCD

Calculated Surface Energy

surfeDB

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

Re

Tc

Mn

Calculated surface energies for themost close-packed surfaces

W

Cr

Mo

3d4d5d

Transition metals

d-band filling

Sur

face

ener

gy(e

V/a

tom

)

0

1

2

3

4

-1

-2

-3

Su

rfac

e en

erg

y (

J/m

)2

Ste

p e

ner

gy

(10

J/

m)

-10

fcc (100)

Surface energy

Y Zr Nb Mo Tc Ru Rh Pd Ag

(100) x (111)

(100) x (010)

0

1

2

3

4

-1

-2

-3

LMTO-ASA plus full charge-density correction FCD

Database:Calculated Surface, step and kink energy

L. Vitos, A.R. Ruban, H.L. Skriver, J. Kollar, Surf. Sci. 411, 186 (1998)


Perdew, Burke, Ernzerhof GGA

Li - Pu at calculated volume

L. Vitos, H.L. Skriver, J. Kollar, Surf. Sci. 425, 212 (1999)

Esurf = E2DFCD E 3D

FCDN atom( N vac+ ) - N atom

segrth

cÛ0 1

EAsurf

EBsurf

Esegr

cÛA B

Surface Segregation EnergyBulk

Esurf = - EÛ'

A1-cB c

Surface A 1-c BcÛÛ

Friedel model

EAsurfEB

surfEB -> Asegr ~ -

dEbulk

dc c=0=

EB -> Asegr,Û =

dEsurf

dc c =0ÛÛ

ÌÛ'

( Ebulk ) -

Ü

Ü c Û

segrem

0,0 0,2 0,4 0,6 0,8 1,00,0

0,5

1,0

1 , 5

2.0

Wf(1-f)

Sur

face

Ene

rgy

(eV

/ato

m)

d-band filling

Friedel model

Solute

Hos

tAg

Y

Zr Zr

Nb

Nb

Nb

Mo

Mo

Mo

Tc

Tc

Tc

Ru

Ru

Ru

RhRh

Rh

Pd

Pd

Pd

Ag

Ag

Surface Segregation Energy

esegr

hcp

bcc

bcc

bcc

bcc

hcp

fcc

fcc

hcp

bcc

bcc

hcp

hcp

fcc

fcc

fcc

hcp

bcc

bcc

hcp

hcp

fcc

fcc

fcc

Zr Nb Mo Tc Ru Rh

Nb Mo Tc Ru RhZr Pd Ag

Zr

Nb

Mo

Tc

Ru

Rh

Pd

Ag

Friedel model

Esegr

-0.7 ... -0.3 eV-0.3 ... -0.05 eV-0.05 ... 0.05 eV0.05 ... 0.3 eV0.3 ... 0.7 eV> 0.7 eV

< - 0.7 eV

Ti V Cr Mn Fe Co Ni Hf Ta W Re Os Ir Pt

Ti

V

Cr

Mn

Fe

Co

Ni

Cu

Cu Au

Nb

Mo

Hf

Ta

W

Re

Os

Ir

Pt

Au

PdAg3d

4d5dHos

t[fc

c(11

1), b

cc(1

10),

hcp

(000

1)]

3d 4d 5dSolute

Zr

Tc

Ru

Rh

Pd

Ag

A.V. Ruban, H.L. Skriver, J.K. Nørskov, Phys. Rev. B59, 15990 (1999)

Calculated Segregation Energy

scls-th

Surface Core Level Shift

ÇSB = Ebulksol E

surfsol-

EsegrZ+1 -> Z~

B. Johansson and N. Mårtensson, Phys. Rev. B21, 4427 (1980)

Thermodynamical model of a fully screened core-hole

(Z*) (Z*)

Friedel model, d -band filling f

EsegrZ+1 -> Z Esurf

Z+1E

Zsurf-=

~ ~ W10

(1 - 2f)dEsurf

df

ZZ* ~ Z+1

initial state

final states

XPS surfa

ce Z

XPS bulk Z

scls-5d

Calculated Surface Core Level Shift

B. Johansson and N. Mårtensson, Phys. Rev. B21, 4427 (1980)

Thermodynamical model

EsegrZ+1 -> ZÇSB =

Friedel model, d -band filling f

EsegrZ+1 -> Z ~ W

10(1 - 2f)

A.V. Ruban and H.L. Skriver,Comp. Mat. Sci. 15, 119 (1999)

0.0 0.2 0.4 0.6 0.8 1.0

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

W

Re

Pt

5d metals

Esegrgr(Z+1)

Exp. SCLS

Friedel

Sur

face

cor

e-le

vel s

hift

(eV

)

d-band filling

IrOs

Ta

Hf

hcp bcc bcc hcp hcp fcc fcc

segrDB

1 2 3 4 5 60.00

0.25

0.50

0.75

1.00

0.00

0.25

0.50

0.75

1.00

cP

t

Calculated Experiment

0.00

0.25

0.50

0.75

1.00

Rh75

Pt25

(111)

(100)

(110)

Surface segregation profile

Surface layer

T = 1300 K

LMTO-ASA plus multipolecorrection

Database:Calculated Surface segregationenergy

A.V. Ruban, H.L. Skriver, J.K. Nørskov, Phys. Rev. B59, 15 990 (1999)


LDA: Perdew-Zunger Ceperley-Alder

Transition - transition metal atcalculated volume

A.V. Ruban, H.L. Skriver, Comp.Mat. Sci. 15, 119 (1999)

LSD: Vosko-Wilk-Nusair

Coherent Potential Approximation CPA

vacDB

Lu Hf Ta W Re Os Ir Pt Au

1

2

3

4

Y Zr Nb Mo Tc Ru Rh Pd Ag

1

2

3

4

Vac

ancy

form

atio

n en

thal

py (

eV)

Sc Ti V Cr Mn Fe Co Ni Cu

1

2

3

4 3d

4d

5d

CalculatedExperimentalRecommended values

Database:Calculated Vacancy formationenergy

P.A. Korzhavyi, I.A. Abrikosov, B. Johansson,A.V. Ruban, H.L. Skriver, Phys. Rev. B59, 11 693 (1999)

Supercell approach: Order-NLocally Self-consistent GreensFuntion LSGF method

Perdew, Burke, Ernzerhof GGA

Transition metals

s,p,d,f LMTO-ASA plus multipole correction

H1VF (P=0) = ~E(1,V) - E(0,V)N-1

N

vacstr

Y Zr Nb Mo Tc Ru Rh Pd Ag-3

-2

-1

0

1

2

Ene

rgy

diffe

renc

e (e

V)

bcc

fcc

bcc

P.A. Korzhavyi, I.A. Abrikosov, B. Johansson,A.V. Ruban, H.L. Skriver, Phys. Rev. B59, 11 693 (1999)

Calculated LSGF

E vacÑ

Vacancy Formation EnergyEmpirically

E 1VF ~ E coh

13

Structural contribution, i.e., beyond Friedel, to the vacancy formationenergy

E cohÑ E coh

Friedel= E str

Ñ+

E vacÑ

~ z z-1z[ ] 1/2

z-1z[ ]

1/2

( 1 - )

- z

E cohÑ

E strÑ

imp4dDB

Zr Nb Mo Tc Ru Rh Pd AgHost

-3

-2

-1

0

1

2

3

4

Esol (e

V)

Zr NbMo Tc RuRh Pd Ag

Database:Calculated impurity solution energy

LMTO-ASA

A.V. Ruban, H.L. Skriver, J.K. Nørskov, Phys. Rev. Lett. 80, 1240 (1998)

Perdew-Zunger Ceperley-Alder XC

4d transition metals at calculatedvolume

Coherent Potential Approximation

EsolÑ =

dEA1-cB c

dc c=0

Solution of B -> A

EAÑ EB

Ò+ -

impstr

1 2 3 4 5 6 7 8 9 10Nd

-1.0

-0.5

0.0

0.5

1.0

Est

r

bcc-

hcp (e

V)

ZrNbMoTcRu

Host:

Y

Zr

Nb

Mo

Tc

Ru

Rh

Pd

Ag

A.V. Ruban, H.L. Skriver, J.K. Nørskov, Phys. Rev. Lett. 80, 1240 (1998)

Ç

Impurity Solution EnergyCrystal structure contribution to thesolid solubility in transition metalalloys

NdA

ÇEsolÑ-Ò = ÇEstr

Ñ-Ò2

NdB

NdC

( ) ÇEstrÑ-Ò Nd

A( )-

NdC

= +12 ( )

Computational Materials Physics · Database: Calculated Stacking fault energy Green function...

Documents

Transcript of Computational Materials Physics · Database: Calculated Stacking fault energy Green function...