SURFACE SEGREGATIONS IN RANDOM ALLOYS FROM FIRST-PRINCIPLES THEORY

Igor A. Abrikosov

Department of Physics and Measurements Technology,

Linköping University

ACKNOWLEDGEMENTS

• A. Ponomareva• N. Skorodumova• A. V. Ruban • S. I. Simak • L. Pourovski• S. Shallcross

CONTENTS :

• Introduction: first principles calculations at T=0 and at T>0

• Ordering effects in the bulk: multicomponent alloys

• Surface segregations in NiPt and NiPd.• Segregation energies in Fe-Cr system.• Cr-rich part of the diagram: importance of

correlation effects.

Density Functional Theory(DFT)

∑=

++=

+∇−=

−=

occr

vvvreffv

reffvm

effH

relonerH

XCEXTH

ψψσρ

ρρρσ

σ

σψεσψ

*),...RR,R,,(

)()()(),...RR,R,,(

),...RR,R,,(2

),...RR,R,,(),...RR,R,,(

M21

M21

M212

2

M21M21

h

)(),...RR,R,,( M21 rjj

jCr ϕσψ ∑=

)()R,...,R,R,,,,(* M21 rrrrGH ′−=′ δσσ

F F

C

A

BC

D

Structures:

A

B

CD

}{ is σσ =r

⎩⎨⎧

=otherwise

atomby occupied is site if

1-

1 Aiiσ

1 1 -1 1 1

1 -1 1 1 -1

1 1 -1 1 1

-1 1 1 1 -1

,... ,s

kji

s

ji σσσσσ

∑ ⎟⎠

⎞⎜⎝

⎛−=−=s

sB kT

EZTkF exp Zln

...),3(),2(

)1()0(

++

++=

∑∑s

kjis

sji

s

tot

VV

VVE

σσσσσ

σ

Calculations of effective interatomic potentials

The Connolly-Williams method

1. Choose structures fcc L12 L10 DO22

2. Calculate Etot: E(fcc) E(L12) E(L10) E(DO22)

with predefined correlation functions [ ],... , ,

)1,3()2,2()1,2(

kjijiji σσσσσσσ

...),2()1()0( ∑++=s

jis

tot VVVE σσσ

min

2

...}{ :L.S.Mby found are thenIf =∑⎥⎥

⎦

⎤

⎢⎢

⎣

⎡∑−≤

m f fVf

mEVNN strpot

The Monte Carlo method

Calculations of averages at temperature T:Z

Tk

EA

A s B

ss∑ ⎟⎟

⎠

⎞⎜⎜⎝

⎛−

=exp

Create the Marcov chain of configurations: ⎟⎟⎠

⎞⎜⎜⎝

⎛−=

Tk

E

ZP

B

ss exp

1

Balance at the equilibrium state: ⎟⎟⎠

⎞⎜⎜⎝

⎛−→=⎟⎟

⎠

⎞⎜⎜⎝

⎛−→

Tk

EssW

Tk

EssW

B

s

B

s 'exp)'(exp)'(

EΔ )10( exp 0

0

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

≤≤>⎥⎦

⎤⎢⎣

⎡ Δ−>Δ

≤Δ

rrTkE

E

E

B

Atoms exchanged

ΔE

Example: ordered phases in Cu2NiZn• 21 concentration and volume dependent effective

cluster interactions

• Electronic structure calculations using O(N) LSGF method

• 32 different atomic distributions at fixed concentration (144 atom supercell)

• VNi-Zn(nn)=12.8mRy > VCu-Zn(nn)=5.0 mRy >

VCu-Ni(nn)= 2.5 mRy

• VNi-Zn(4nn)= -2.5mRy, all other ECI are small

• Cluster expansion represents total energy calculations with average accuracy better than 0.015 mRy, and with the maximal error 0.2 mRy (or 4% of the ordering energy)

Example: ordered phases in Cu2NiZn• 21 concentration and volume dependent effective

cluster interactions

• Electronic structure calculations using O(N) LSGF method

• 32 different atomic distributions at fixed concentration (144 atom supercell)

• VNi-Zn(nn)=12.8mRy > VCu-Zn(nn)=5.0 mRy >

VCu-Ni(nn)= 2.5 mRy

• VNi-Zn(4nn)= -2.5mRy, all other ECI are small

• Cluster expansion represents total energy calculations with average accuracy better than 0.015 mRy, and with the maximal error 0.2 mRy (or 4% of the ordering energy)

Calculations of effective interatomic potentials

The generalized perturbation method

1. Calculate electronic structure of a random alloy (for example, use the CPA):

)( ,~ BAtg

...2

1)(

)'(

''

RR

jiRR

RRoneone VcEE σσ∑+= -determine a perturbationof the band energy due to small varioations of the correlation functions

2.

where the effective interatomic interactionsare given by an analytical formula:

{ }∫−= ARRR

BRRR

ARRR tttdEV ''''

1γγ

π

Example: bulk ordering in NiPt

Method Δrnd(UR)

(mRy/atom)

Δrnd(R)

(mRy/atom)

Δ´L10(UR)

(mRy/atom)

CPA-GPM

1.57 -1.03 -6.06

FP-CWM

Lu et al.1.72 -2.23 -5.71

The new surface Monte Carlo methodIn order to represent the bulk chemical potential, the obtained by bulk MCfixed reservoir of atoms is used:

Bulk reservoirSurface sample

Vacuum

)()( ABEBAEE surfbulk →Δ+→Δ=Δ

)10( exp 0

0

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

≤≤>⎥⎦

⎤⎢⎣

⎡ Δ−>Δ

≤Δ

rrTkE

E

E

B

The new surface Monte Carlo method

Only one fixed bulk configuration of the reservoir is used.

How do the results depend on the size of the reservoir?

The dependence of the surface layer energy on size of reservoir in NiPd(100)

Configuration of the (111) surface of the Ni49Pt51 substoichometric ordered alloy

Surface segregations in the NiPt and NiPd alloys

1. A segregation reversal phenomenon has been observed at the surfaces of

NiPt random alloys: Pt segregates towards the (100) and (111) surfaces, Ni

segregates towards the (110) surface.

2. No such effect has been found for the isoelectronic NiPd alloys. The strong

Pd segregations have been observed on all low-indexed surfaces.

There are bulk ordered phases NiPt(L10) and Ni3Pt(L12)

in the Ni-Pt system

No bulk ordering occur in NiPd down to T=400K

SGPM surface potentials for Ni50Pt50(Ni50Pd50) in K (σPt(Pd)=1)

Layer 1 2 3 4 5

V(1)V(1)

bulk167(-1613) -883(-648) -265(-126) 163(35) 0(0)

(110)V(2,1)

483(224) 555(262) 571(279) 586(280) 556(279)

V(2,1)

2433(1017) 2286(1097) 2379(1132) 2225(1116) 2225(1116)

V(2,1)

781(261) 686(296) 556(279) 556(279) 556(279)

V(1)V(1)

bulk-705(-1118) 231(2) -219(-60) 0 -

(111) V(2,1)

1373(820) 1852(854) 1979(871) 1668(837) -

V(2,1)

1927(921) 1857(861) 1668(861) 1668(837) -

Example: bulk ordering in NiPtTransition to L10 in Ni50Pt50: Tc

exp=917 K, Tcteor=925 K

Segregation profiles in the Ni50Pt50 and Ni50Pd50 random alloys

Ni-Pt (111)

Ni-Pt (111)

Configuration of the (111) surface of the NiPt stoichometric and substoichometric ordered alloys

Ni50Pt50 ordered alloyNi49Pt51 ordered alloy

(111) surface

[ ]kind" wrong" of neighboursnearest theofnumber ≈antiE

}{ is σσ =r

⎩⎨⎧

=otherwise

atomby occupied is site if

1-

1 Aiiσ

1 1 -1 1 1

1 -1 1 1 -1

1 1 -1 1 1

-1 1 1 1 -1

,... ,s

kji

s

ji σσσσσ

∑ ⎟⎠

⎞⎜⎝

⎛−=−=s

sB kT

EZTkF exp Zln

...),3(),2(

)1()0(

++

++=

∑∑s

kjis

sji

s

tot

VV

VVE

σσσσσ

σ

EMTO vs Full-potential: c/a ratio in ordered alloys

CONCLUSIONS :

• There are problems.

• We are here to solve them!