Migration of isolated point defects in CuNb

1. Migration of Isolated point defects
at a model CuNb interface
Kedarnath Kolluri, and M. J. Demkowicz
Acknowledgments:
R. G. Hoagland, J. P. Hirth, B. Uberuaga, A. Kashinath, A. Vattré, X.-Y. Liu, A. Misra, and A.
Caro
Financial Support:
Center for Materials at Irradiation and Mechanical Extremes (CMIME) at LANL,
an Energy Frontier Research Center (EFRC) funded by
U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences

2. 〈112〉 〈112〉
Cu
Nb
General features of semicoherent fcc-bcc interfaces
Cu-Nb
〈110〉 〈111〉
Cu
Nb
Interface contains arrays of misfit dislocations separating coherent
regions

3. Structure of interfaces: Misfit dislocations
〈112〉 〈112〉
Cu
Nb
Cu-Nb
〈110〉 〈111〉
Cu
Nb
one set only
M. J. Demkowicz et al., Dislocations in Solids Vol. 14 (2007)
Two sets of misfit dislocations with Burgers vectors in interface plane
An coherent state (where there are no dislocations) is necessary for
this analyses

4. Structure of CuNb KS interface
〈112〉
〈111〉
Cu interfacial plane
〈110〉
〈111〉
〈112〉
Cu atoms
Nb atoms
Interfacial Cu atoms
MDI
1 nm
〈112〉
Cu
〈110〉
〈110〉
Cu

5. Structure of interfaces: Misfit dislocations
150
0.2
0.45
0
0.2
0.4
100
0.4
Cu-Nb KS
0.6
50
coherent
0.4
0.2
0.6
0.4
0.8
0.6
1
0.8
0.3
0.4
0.6
0.25
0
0.8
0.8
0.2
1 0.15
1
0
u-Fe NW
1
0
0.2
0.2
0.4
0.4
0.6 0.6
0.8
0.8
Cu-V KS
0
0.2
0.4
0.6
0.8
1
11 Kolluri, and M. J. Demkowicz,
0
0.2
K.
unpublished
A general method to identify dislocation line and Burgers vectors
•
Assumption: A coherent patch exists at the interface
0
•
Advantage: Reference structure not required
1.4
•
0.2
Limitations: Dislocation core thickness cannot be determined (yet)
1.2
1.4
1.2
gy (eV)
•
0.35
0.28
0.26
150
0.24
0.22
0.2
100
0.18
0.16
50
0.14
0.12
0.1
0
0.08
0.06
Angle with -ve x axis
0 0.5

6. 1
0 0.5
150
0.2
0.45
0.8
1
0.6
50
0.35
0.3
0
1.4
0.2
0.2
1.2
0.4
0.4
1
0.6
0.6
0.25
0
2
0.6
nm
0.8
0.2
1 0.15
1
0
0.4
0.8
0.8
1
0
0.2
0.8
0.6
0.2
0.4
1
0.4 1 0.6
0.6
0
0.8
0.8
0.2
11
0.4
Cu-V KS
1 nm
0.6
0.4
0.28
0.26
150
0.24
0.22
0.2
100
0.18
0.16
50
0.14
0.12
0.1
0
0.08
0.06
0.6
0.8
0.6
1 0.15
1
0
1
1.2
1.4
0.2
0.8
0.8
1.4
0.2
0.8
0.4
1.2
0.6
1
0.8
0.8
0.6
1
0.4
0.6
0.8
0
1
0
0.2
Cu-Nb KS interface
1.2
1
0.6
0.4
0.6 0.6
0.4
0.2
0.6
0.4
0.8
0.6
0.8
1
11
0.06
0.55
0.5
0.451.4
0.4
0.351.2
0.3
1
0.25
0.2
0.8
0.15
0.1
0.6
0.05
1
vacancy formation energies
0.4
1
0.2
0.4
1.4 nm
0
0.2
0
0.2
0
1
structure
1.4
1
0.8
0
tion energy (eV)
1
0.8
0.8
0.4
0.2
0
0.4
100
0.4
0.2
0
Angle with -ve x axis
0.4
0
Formation energy (eV)
1
Misfit dislocation intersections (MDIs) are
Cu-Nb KS
Cu-Fe NW
Cu-V KS
point defect traps
.2
0
0.6

7. Misfit dislocation intersections (MDIs) are
point defect traps
Vacancy
a
M. J. Demkowicz, R. G. Hoagland, J. P. Hirth, PRL 100, 136102 (2008)
a2
a1
L
a2
Set 2
!1
Se
t1
Point defects delocalize at MDI to form kink-jog pairs
L
b1
!1
a1

8. MDIs are point defect traps
Interstitial
M. J. Demkowicz, R. G. Hoagland, J. P. Hirth, PRL 100, 136102 (2008)
Point defects delocalize at MDI to form kink-jog pairs

9. Structure of isolated point defects in Cu-Nb
Vacancy
•
Interstitial
Defect at these interfaces “delocalize”
•
knowledge of transport in bulk can not be ported

10. Point defects migrate from one MDI to another in CuNb
Vacancy
Interstitial
•
Migration is along set of dislocation that is predominantly screw
•
In the intermediate step, the point defect is delocalized on two MDI

11. 0.45
0.4
Vacancy
I
KJ1t
t
Se
KJ3
!1
b1
•
t
KJ3´
a2
0.15
0.1
a1 L
!1
0.05
b1
b
t1
L
t1
L
0.2
t
t
I
I
0 a
•
KJ4
〈110〉
Cu
KJ4
0.25
Set 2
〈110〉
Cu
Se
t1
Step 1
t
Se
" E (eV)
a2 a1
KJ1
KJ2´
0.35
0.3
b
KJ2
〈112〉
Cu
a
〈112〉
Cu
Isolated point defects in CuNb migrate from
one MDI to another
Set 2
3L
Set 2
!1
b1
b
Interstitial
Vacancy
"Ea-b = 0.06 - 0.12 eV
"Ea-I = 0.25 - 0.35 eV
"Ea-t = 0.35 - 0.45 eV
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Thermal kink pairs nucleating at adjacent MDI mediate the migration 0.
Migration barriers
1/3rd
! (reaction coordinate)
that of migration barriers in bulk

12. Isolated point defects in CuNb migrate from
one MDI to another

13. Isolated point defects in CuNb migrate from
one MDI to another

14. 0.45
0.4
Vacancy
I
KJ1t
t
Se
KJ3
!1
b1
•
t
KJ3´
a2
0.15
0.1
a1 L
!1
0.05
b1
b
t1
L
t1
L
0.2
t
t
I
I
0 a
•
KJ4
〈110〉
Cu
KJ4
0.25
Set 2
〈110〉
Cu
Se
t1
Step 1
t
Se
" E (eV)
a2 a1
KJ1
KJ2´
0.35
0.3
b
KJ2
〈112〉
Cu
a
〈112〉
Cu
Isolated point defects in CuNb migrate from
one MDI to another
Set 2
3L
Set 2
!1
b1
b
Interstitial
Vacancy
"Ea-b = 0.06 - 0.12 eV
"Ea-I = 0.25 - 0.35 eV
"Ea-t = 0.35 - 0.45 eV
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Thermal kink pairs nucleating at adjacent MDI mediate the migration 0.
Migration barriers
1/3rd
! (reaction coordinate)
that of migration barriers in bulk

15. Thermal kink pairs aid the migration process
b
0.4
Vacancy
I
KJ1
KJ2´
KJ4
I
3L
0.15
t1
a2
Se
L
t1
0.2
t
t
KJ3´
0.25
a1 L
!1
b1
t
〈110〉
Cu
I
2
Set 0.1
•
t
t
b
Se
" E (eV)
Se
t1
Step 2
•
c
0.35
0.3
!1
t
〈112〉
Cu
0.45
0.05
b1
0 a
Set 2
Set 2
!1
b1
b
Interstitial
Vacancy
"Ea-b = 0.06 - 0.12 eV
"Ea-I = 0.25 - 0.35 eV
"Ea-t = 0.35 - 0.45 eV
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Thermal kink pairs nucleating at adjacent MDI mediate the migration 0.
Migration barriers
2/3rd
! (reaction coordinate)
that of migration barriers in bulk

16. Thermal kink pairs aid the migration process
(a)
(b)
(c)
(d)
(e)
(f)
Vacancy
Interstitial
ΔEact = 0.35 - 0.45 eV
ΔEact = 0.60 - 0.67 eV
1nm
The width of the nucleating thermal kink pairs determines the barrier

17. 0.45
t
Multiple migration paths and detours
0.4
t
t
t
0.35
t
" E (eV)
0.3
I
I
0.25
t
0.2
b
Migration paths
(CI-NEB)
Interstitial
0.15
Vacancy
0.1
0.05
0 a
0
b
"Ea-b = 0.06 - 0.12 eV
"Ea-I = 0.25 - 0.35 eV
"Ea-t = 0.35 - 0.45 eV
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.
! (reaction coordinate)
•
Not all intermediate states need to be visited in every migration
•
The underlying physical phenomenon, however, remains unchanged

18. Entire migration path can be predicted
0.5
0.5
0.45
0.45
0.4
0.4
0.35
0.35
0.3
0.3
0.25
0.25
I
0.2
0.15
0.2
I
0.15
0.1
0.1
0.05
0.05
0
a
0
Dislocation model
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
KJ1
1
KJ1
0 KJ2
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
s
Key inputs to the dislocation model
b
Atomistics
0
〈112〉
Cu
Δ E (eV)
0.55
KJ2´
〈110〉
Cu
KJ3
KJ4
•
Interface misfit dislocation distribution
•
KJ4
1
s
KJ3´
K. Kolluri and M. J. Demkowicz,
Phys Rev B, 82, 193404 (2010)
Structure of the accommodated point defects
Analysis of the interface structure may help predict quantitatively
point-defect behavior at other semicoherent interfaces

19. jog, which is assumed constant for all states in our dislocation
model [and therefore does not appear in Eq. (1)], actually varies
along the direct migration path. To estimate the core energy
of the kink-jog, we summed differences in atomic energies
between the core atoms and corresponding atoms in a defectfree interface. The kink-jog core is taken to consist of 19 atoms:
the 5-atom ring in the Cu terminal plane and the 7 neighboring
Cu and Nb atoms from each of the two planes adjacent to the Cu
terminal plane. Core volumes were computed in an analogous
way. The core energies of the migrating jog are plotted as
filled triangles in Fig. 15(a) and are in good semiquantitative
agreement with the overall energy changes occurring along
the direct migration path. Core volumes are plotted as filled
circles.
Figure 15(b) shows the Cu and Nb interface planes with
a point defect in the extended state B. Arrows mark the
locations of the two kink-jogs and red lines mark the nominal
locations of set 2 misfit dislocation cores. The numbers are
TABLE I. Transitions occurring during migration of individual
point defects that were considered in kMC simulations, their
corresponding activation energy barriers, and number of distinct end
states for a given start state.
Point defect migration rates from simulations
Transition
type
A→I
A→B
I (near A) → B
I (near A) → A
B→A
B→I
B→I
B→C
I (near C) → C
I (near C) → B
I →B
Activation energy
(eV)
Number of
distinct end states
0.40
0.40
0.15
0.15
0.35
0.35
0.20
0.35
0.15
0.15
0.15
2
2
1
1
1
2
1
1
1
1
1
205416-9
•
Hypothesis:
•
•
•
transition state theory is valid and
Rate-limiting step will determine the migration rate ≥ 0.4 eV
Validation:
•
kinetic Monte Carlo (since the migration path is not trivial)
•
Statistics from molecular dynamics

20. Migration is temperature dependent
Jump rate (ns-1)
0.1
1
=
0.01
0e
0.4eV
kB T
0.001
0.0001
1e-05
1300
1000
800
700
600
500
Inverse of Temperature (K-1)
K. Kolluri and M. J. Demkowicz,
Phys Rev B, 85, 205416 (2012)
•
Migration rates are reduced because there are multiple paths
•
Transition state theory may be revised to explain reduced migration rates

21. Migration is temperature dependent
Jump rate (ns-1)
0.1
1
=
0.01
0e
0.4eV
kB T
0.001
0.0001
1e-05
1300
1000
800
700
600
500
Inverse of Temperature (K-1)
K. Kolluri and M. J. Demkowicz,
Phys Rev B, 85, 205416 (2012)
•
Migration rates are reduced because there are multiple paths
•
Transition state theory may be revised to explain reduced migration rates

22. ln[(s!)p(t/τ,s)] = s ln(t/τ ) − t/τ.
(10)
Migration is temperature dependent
s are obtained for all three temperatures, confirming
0.1
tion that point defect1migration follows a Poisson
⇥
= 0 k 1T e
ig. 17). The jump rates for each temperature,
0.01
1
= 0e
y fitting, are plotted in Fig. 16(b) as filled gray
h uncertainties corresponding to the error in the
0.001
es fit. The gray line is the least-squares fit of Eq. (8)
obtained from MD. The activation energy obtained
0.0001
act
MC model (Eeff = 0.398 ± 0.002 eV) is well within
nty of 1e-05 activation energy found by fitting the MD
the
500
act
y, Eeff =1300 1000 0.045700 600
0.374 ± 800 eV.
Inverse of Temperature (K )
ctive attempt frequency for defect-1migration obfittingKolluri and M. J. Demkowicz, = 6.658 × 109 ± 2.7 ×
K. the MD data is ν0
Phys Rev several orders of
is value is B, 85, 205416 (2012) magnitude lower than
mpt frequencies for point defect migration in fcc
• Migration −1 .72–74 A mechanistic interpretation paths
12
14 rates are reduced because there are multiple
, 10 −10 s
ow migration attempt frequency is not immediately
• Transition state theory may be revised to explain reduced migration rates
g. One possible explanation is that it arises from
number of atoms participating in the migration
Jump rate (ns-1)
Eact
e
kB T
B
0.4eV
kB T

23. act
model(Eef f = 0.398 ± 0.002 eV) is w
Migration is temperature dependent
a
by fitting the MD data, namely Ee
Jump rate (ns-1)
1
MD
kMC
0.1
act
model(Eef f = 0.398 ± 0.002 eV) is well wi
act
by fitting the MD data, namely Eef f = 0
0
⌫0 = 6.658 ⇥ 109 ± 2.7 ⇥ 106 s
0.01
1
0
act
tained by0.374 ± 0.045 MD 0data is ⌫0
fitting the eV ⌫ 0 = 6.658 ⇥
Eef f =
0.001
1300
for defect migration obtained typical t
of magnitude lower than by fittingat
1000
800
700
600
500
69–71
value is 1012 1014
namelyseveral orderssof1 magnitude mec
. A lowe
Inverse of Temperature (K-1) migration in fcc Cu, namely 1012 1014
frequency is not immediately forth
K. Kolluri and M. J. Demkowicz, Phys Rev B, 85, 205416 (2012)
low migration attempt frequency is not im
•
•
the large number of atoms particip
Modified rate expression is fit to MD statistics to obtain attempt frequency
is that it arises from the large number of
Attempt frequency is much lower than for migration of for migration of compa
is normally observedcompact point defe
attempt frequency for point defects
frequency because it involves the m
der of the Einstein frequency because it i

24. Summary
•
Interface has defect trapping sites
– density of these sites depends on interface structure
•
Point defects migrate from trap to trap
– migration is multi-step and involves concerted motion of atoms
– migration can be analytically represented