Transcript Electronic Theory of Materials Properties

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)
H (r , , R1 , R 2 ,...RM )  
 (r , , R1 , R 2 ,...RM )
oneel
2

eff
eff
H

 2  v (r , , R1 , R 2 ,...RM )
2m
eff
v (r , , R1 , R 2 ,...RM )  vH (  )  vEXT (  )  v XC (  )
 (r , , R1 , R 2 ,...RM ) 
*


occ
 (r , , R1 , R 2 ,...RM )   C j j (r )
j
H * G(r , , r, , R1 , R 2 ,...,R M )   (r  r)
F
Structures:
F
A
D
A
C
B
B
C
D
C
 Es 
Z   exp 

 kT 
s
F  k BT ln Z

 s  { i }
s
1 if sitei is occupiedby atom A
i  
-1 otherwise
1
s
 i j ,  i j k ,...
1 -1 1
1 -1 1
1
1 -1
1 -1 1
-1 1
1
1
1
1 -1
Etot  V ( 0 )  V (1)  
V
s
( 2, s )
 i j  V
s
( 3, s )
 i j k  ...
Calculations of effective interatomic potentials
The Connolly-Williams method
1. Choose structures
with predefined
correlation functions
2. Calculate Etot:
fcc
 
i
L12
( 2,1)
j
E(fcc)
L10
,  i j
E(L12)
( 2, 2 )
DO22
,  i j k
E(L10)
( 3,1)

,...
E(DO22)
Etot  V ( 0)  V (1)   V ( 2, s )  i j ...
s
If N pot  N str then {V } are found by L.S.M:
2


f

E

...
V   min
 m 
f

m
f

The Monte Carlo method
Calculations of averages at temperature T:
Create the Marcov chain of configurations:
Balance at the equilibrium state:

E
E
 Es 
A
exp
s s   k T 
 B 
A 
Z
 Es 
1

Ps  exp  
Z
 k BT 
 Es 
 Es ' 



W ( s  s' ) exp 
 W ( s'  s) exp 

 k BT 
 k BT 
E  0






E
E  0 exp 


r
(
0

r

1
)




k
T
 B 


Atoms exchanged
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):
2.
Eone
1
 Eone (c)  VRR'  i j
2 RR'
( RR ')
...
where the effective interatomic interactions
are given by an analytical formula:
VRR '  
1


A
B
A
dE
t

t

t
R RR ' R ' R ' R R


~, t A( B)
g
-determine a perturbation
of the band energy due to
small varioations of the
correlation functions
Example: bulk ordering in NiPt
Method
rnd(UR)
rnd(R)
´L10(UR)
(mRy/atom) (mRy/atom) (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 method
In order to represent the bulk chemical potential, the obtained by bulk MC
fixed reservoir of atoms is used:
Bulk reservoir
Surface sample
Vacuum
E  Ebulk ( A  B)  Esurf (B  A)
E  0






E
E  0 exp 
 r (0  r  1)




 k BT 


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 (Pt(Pd)=1)
Layer l
1
2
3
4
5
167(-1613)
-883(-648)
-265(-126)
163(35)
0(0)
483(224)
555(262)
571(279)
586(280)
556(279)
V(2,1)ll1
2433(1017)
2286(1097)
V(2,1)ll2
781(261)
686(296)
556(279)
556(279)
-705(-1118)
231(2)
-219(-60)
0
-
V(2,1)ll
1373(820)
1852(854)
1979(871)
1668(837)
-
V(2,1)ll1
1927(921)
1857(861)
1668(861)
1668(837)
-
V(1)lV(1)bulk
(110)
V(2,1)ll
V(1)lV(1)bulk
(111)
2379(1132) 2225(1116) 2225(1116)
556(279)
Example: bulk ordering in NiPt
Transition to L10 in Ni50Pt50: Tcexp=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
(111) surface
Ni49
50Pt51
50 ordered alloy
Eanti  number of the nearest neighbours of " wrong " kind
 Es 
Z   exp 

 kT 
s
F  k BT ln Z

 s  { i }
s
1 if sitei is occupiedby atom A
i  
-1 otherwise
1
s
 i j ,  i j k ,...
1 -1 1
1 -1 1
1
1 -1
1 -1 1
-1 1
1
1
1
1 -1
Etot  V ( 0 )  V (1)  
V
s
( 2, s )
 i j  V
s
( 3, s )
 i j k  ...
EMTO vs Full-potential:
c/a ratio in ordered alloys
CONCLUSIONS :
• There are problems.
• We are here to solve them!