Migration of isolated point defects in CuNb

Migration of Isolated point defects at a model CuNb interface

Kedarnath Kolluri, and M. J. Demkowicz

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

Acknowledgments: R. G. Hoagland, J. P. Hirth, B. Uberuaga, A. Kashinath, A. Vattré, X.-Y. Liu, A. Misra, and A. Caro

Interface contains arrays of misfit dislocations separating coherent regions

General features of semicoherent fcc-bcc interfaces

〈110〉Cu〈111〉Nb

〈11

2〉C

u〈

112〉

Nb

Cu-Nb

〈110〉Cu〈111〉Nb

〈112〉 Cu〈112〉 Nb

Cu-Nb

M. J. Demkowicz et al., Dislocations in Solids Vol. 14 (2007)

one set only

Two sets of misfit dislocations with Burgers vectors in interface plane

Structure of interfaces: Misfit dislocations

An coherent state (where there are no dislocations) is necessary for this analyses

Structure of CuNb KS interface

Interfacial Cu atomsCu atoms Nb atoms

〈112〉〈111〉

〈110〉

〈111〉〈112〉

〈110〉

Cu interfacial plane

〈110〉Cu

〈11

2〉C

u

1 nmMDI

Cu-Nb KS

Structure of interfaces: Misfit dislocations

K. Kolluri, and M. J. Demkowicz, unpublished

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Cu-Nb KS Cu-Fe NW Cu-V KS

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Form

atio

n en

ergy

(eV

)A

ngle

with

-ve

x ax

is

coherent

• A general method to identify dislocation line and Burgers vectors

• Assumption: A coherent patch exists at the interface

• Advantage: Reference structure not required

• Limitations: Dislocation core thickness cannot be determined (yet)

Misfit dislocation intersections (MDIs) are point defect traps

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Form

atio

n en

ergy

(eV

)A

ngle

with

-ve

x ax

is

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Form

atio

n en

ergy

(eV

)A

ngle

with

-ve

x ax

is

structure vacancy formation energies

Cu-Nb KS interface

b1

!1

Set 2

Set 1

a

La2

a1

I1

b1

!1

Set 1

Set 2

a1

a2

L

Lb1

!1

Set 1

Set 2

b

3L

(a) (b) (c)

Misfit dislocation intersections (MDIs) are point defect traps

M. J. Demkowicz, R. G. Hoagland, J. P. Hirth, PRL 100, 136102 (2008)

Vacancy

Point defects delocalize at MDI to form kink-jog pairs

Interstitial

MDIs are point defect traps

Point defects delocalize at MDI to form kink-jog pairs

M. J. Demkowicz, R. G. Hoagland, J. P. Hirth, PRL 100, 136102 (2008)

Vacancy Interstitial

Structure of isolated point defects in Cu-Nb

• Defect at these interfaces “delocalize”

• knowledge of transport in bulk can not be ported

• Migration is along set of dislocation that is predominantly screw

• In the intermediate step, the point defect is delocalized on two MDI

Vacancy

Interstitial

Point defects migrate from one MDI to another in CuNb

b1

!1

Set 2

Set 1

L

a2 a1 Set 1

Set 2

a1

a2

L

Lb1

!1

b1

!1

Set 1

Set 2

3L

• Thermal kink pairs nucleating at adjacent MDI mediate the migration

• Migration barriers 1/3rd that of migration barriers in bulk

KJ1

KJ3´KJ4

Cu

〈112〉

〈110〉Cu

KJ2´

KJ4

KJ3

KJ2KJ1

Cu

〈112〉

〈110〉Cu

a bIVacancy

Step 1

! (reaction coordinate)

t

ca

I

t t

b

" E

(eV

)

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t

I

t

b

t

"Ea-b = 0.06 - 0.12 eV"Ea-I = 0.25 - 0.35 eV"Ea-t = 0.35 - 0.45 eV

VacancyInterstitial

Isolated point defects in CuNb migrate from one MDI to another



b1

!1

Set 2

Set 1

L

a2 a1 Set 1

Set 2

a1

a2

L

Lb1

!1

b1

!1

Set 1

Set 2

3L

KJ1

KJ3´KJ4

Cu

〈112〉

〈110〉Cu

KJ2´

KJ4

KJ3

KJ2KJ1

Cu

〈112〉

〈110〉Cu

a bIVacancy

Step 1


t

ca

I

t t

b

" E

(eV

)

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t

I

t

b

t


VacancyInterstitial




Set 2

b1

!1

a1

a2

Set 1

L

L

b1!1

Set 1

Set 2

b1

!1

Set 1

Set 2

3L

KJ1

KJ3´KJ4

KJ2´

Cu

〈112〉

〈110〉Cu

cb IVacancy

Step 2


t

ca

I

t t

b

" E

(eV

)

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t

I

t

b

t


VacancyInterstitial

Thermal kink pairs aid the migration process



The width of the nucleating thermal kink pairs determines the barrier

ΔEact = 0.35 - 0.45 eV ΔEact = 0.60 - 0.67 eV

(d) (e) (f)

Vacancy

(a) (b) (c)

Interstitial

1nm

Thermal kink pairs aid the migration process

Multiple migration paths and detours

Migration paths (CI-NEB)

• Not all intermediate states need to be visited in every migration

• The underlying physical phenomenon, however, remains unchanged


t

ca

I

t t

b "

E (

eV

)

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t

I

t

b

t


VacancyInterstitial

Entire migration path can be predicted

Key inputs to the dislocation model

• Interface misfit dislocation distribution

• Structure of the accommodated point defects

Analysis of the interface structure may help predict quantitatively

point-defect behavior at other semicoherent interfaces

Δ E

(eV

)

s s

0

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a 0

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b

IDislocation model

Atomistics

K. Kolluri and M. J. Demkowicz, Phys Rev B, 82, 193404 (2010)

KJ1

KJ3´KJ4

Cu

〈112〉

〈110〉Cu

KJ2´

KJ4

KJ3

KJ2KJ1

Point defect migration rates from simulations

FORMATION, MIGRATION, AND CLUSTERING OF . . . PHYSICAL REVIEW B 85, 205416 (2012)

FIG. 15. (Color online) (a) Total energy change (filled squares),kink-jog core energy (filled triangles), and the energy from the dislo-cation model (continuous curve) for the direct migration mechanism.Filled circles show the kink-jog core volume. The arrow on the leftshows the range of formation energies computed for a aCu

4 !112" jogon a screw dislocation in fcc Cu and the arrow on the right shows thecorresponding formation volumes. (b) Plan view of the interface Cu(gold) and Nb (gray) atoms with a point defect in extended state B.Arrows mark the location of kink-jogs, the numbers are values of S,and red lines mark the nominal locations of set 2 misfit dislocationcores.

The above discrepancy arises because the core energy of thejog, which is assumed constant for all states in our dislocationmodel [and therefore does not appear in Eq. (1)], actually variesalong the direct migration path. To estimate the core energyof the kink-jog, we summed differences in atomic energiesbetween the core atoms and corresponding atoms in a defect-free interface. The kink-jog core is taken to consist of 19 atoms:the 5-atom ring in the Cu terminal plane and the 7 neighboringCu and Nb atoms from each of the two planes adjacent to the Cuterminal plane. Core volumes were computed in an analogousway. The core energies of the migrating jog are plotted asfilled triangles in Fig. 15(a) and are in good semiquantitativeagreement with the overall energy changes occurring alongthe direct migration path. Core volumes are plotted as filledcircles.

Figure 15(b) shows the Cu and Nb interface planes witha point defect in the extended state B. Arrows mark thelocations of the two kink-jogs and red lines mark the nominallocations of set 2 misfit dislocation cores. The numbers are

values of the displacement parameter S. At all values of Sexcept S # {3,4}, the kink-jog resides in the vicinity of a set2 misfit dislocation, which affects its structure. The atomicconfigurations of the kink-jog at S # {3,4} were compared tothat of a constricted l = aCu

4 !112" jog on a screw dislocationin fcc Cu.61,62 Depending on the choice of reference energiesand volumes, the core energy and volume of the l = aCu

4 !112"jog were found to be $0.8%1.1 eV and $0.4!o%0.6!o,respectively, where !o = 13.339 A3 is the atomic volume offcc Cu. These values compare very well with those obtainedfor the jog at S # {3,4}, which are also the states where thekink-jog core energy is largest.

Thus, the true energy barrier for the direct migrationmechanism is roughly equal to the difference in the formationenergies of the jog at the MDI and that of an isolated jog on ascrew dislocation. The dislocation model may be modified toaccount for such a behavior by allowing the core radius " tochange with the distance between the jogs (S). Although irrel-evant in the Cu-Nb interface, the direct migration mechanismmay occur in other interfaces where the activation energy forthermal-kink-pair nucleation is comparable to the differencein kink-jog core energies described above. Furthermore, weexpect that the direct migration mechanism, when active,would be highly pressure sensitive on account of its highactivation volume.

D. Temperature dependence of point defect migration

As described in Sec. III C, delocalized interface pointdefects jump between MDIs through multiple steps. KineticMonte Carlo (kMC) simulations63 may be used to determinethe temperature dependence of the effective migration ratedue to these numerous transitions. Since the vacancy andinterstitial migration is along set 1 misfit dislocations, weconsider migration only in one dimension. The transitions wetake into account along with their activation energies are listedin Table I. In each transition listed in Table I, the start and endstates are connected through just one path, but there may bemore than one end state accessible for a given starting state.For example, a defect at its initial state A has two I states,

TABLE I. Transitions occurring during migration of individualpoint defects that were considered in kMC simulations, theircorresponding activation energy barriers, and number of distinct endstates for a given start state.

Transition Activation energy Number oftype (eV) distinct end states

A & I 0.40 2A & B 0.40 2I (near A) & B 0.15 1I (near A) & A 0.15 1B & A 0.35 1B & I 0.35 2B & I ' 0.20 1B & C 0.35 1I (near C) & C 0.15 1I (near C) & B 0.15 1I ' & B 0.15 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

Jum

p ra

te (n

s-1) 1

� = �0e� 0.4eV

kBT

1e-05

0.0001

0.001

0.01

0.1

1300 1000 800 700 600 500

Inverse of Temperature (K-1)

• Migration rates are reduced because there are multiple paths

• Transition state theory may be revised to explain reduced migration rates

Migration is temperature dependent

K. Kolluri and M. J. Demkowicz,Phys Rev B, 85, 205416 (2012)

Jum

p ra

te (n

s-1)

1e-05

0.0001

0.001

0.01

0.1

1300 1000 800 700 600 500



1� = �0e

� 0.4eVkBT




Jum

p ra

te (n

s-1)

1� = �⇥0

�1

kBTe�Eacte�

kBT

1� = �0e

� 0.4eVkBT

1e-05

0.0001

0.001

0.01

0.1

1300 1000 800 700 600 500



FORMATION, MIGRATION, AND CLUSTERING OF . . . PHYSICAL REVIEW B 85, 205416 (2012)

migration process with their multiple minima may be thoughtof as a single “flat” degree of freedom at the saddle point.65

Taking initial and final states corresponding to a defectresiding at neighboring MDIs, we attempt to represent all ofthe intermediate states with a single “effective” saddle pointof this kind. To account for the multiple states that comprisethe effective saddle point, we consider the saddle point tobe a hypersurface with one translational mode. Therefore,the saddle point has N ! 2 vibrational degrees of freedominstead of N ! 1 degrees of freedom as in the case of migrationinvolving a single jump through a unique saddle point. Hence,Eqs. (5) and (6) become

! =!

kBT

2"

A0"N!2

j=11# "j

#kBT2"

"Nj=1

1#j

#kBT2"

e! E(S)

kB T

e! E(A)

kB T

, (7)

! =

$2"

kBT

A0"N!2

j=11# "j"N

j=11#j

e! E(S)!E(A)

kB T = # "0

1#kBT

e! Eact

kB T , (8)

where A0 is a constant contributed by the translational mode65

to the configurational partition function. The attempt fre-quency predicted by this expression is temperature dependent.

The expression in Eq. (8) is identical to those obtainedfor models of an overdamped elastic spring on a nonlinearpotential surface. Such a spring also has a translational modeand has been used as a model for nucleation and motion of kinkpairs on overdamped solitons in spatially one-dimensionalsystems65–69 and for nucleation of kink pairs on dislocationsin two-dimensional Frankel-Kontorova models.70 Our kMCresults fit very well to Eq. (8) with an effective activationbarrier Eact

eff = 0.398 ± 0.002 eV, as illustrated by the blackline in Fig. 16(a).

The numerical value of the # "0 may be determined directly

from MD simulations by counting the number of times a defectjumps from one MDI to another in a fixed time interval.We assumed that point defect migration follows a Poissonprocess71 in which the probability that exactly s events occurin a time interval t is given by

p(t/$,s) = (t/$ )se!t/$

s!. (9)

Here, $ = 1!

is the average waiting time for a defectto migrate to an adjacent MDI. We performed N0 = 64independent MD runs of a vacancy at an MDI in the Cu-Nb

interface. These runs were repeated at three different temper-atures: T $ {600,700,800} K. The duration of each run was8.11 ns (ttot). In each run, migration events were identified bydirect inspection of atomic configurations recorded at intervalsof 40.5 ps. From the investigation described in Sec. III C1,we know that the typical duration of a complete migrationevent at T = 800 K is 32.5 ps. Thus, the selected timeinterval between consecutive recordings minimizes the totalnumber of configurations that must be saved and analyzedwhile ensuring that no more than one migration event occursbetween recordings. While no migration events were observedin some runs, as many as three distinct ones were observedin others. For a given temperature, we identify the probabilityp(ttot/$,s) = n(s)

N0that the point defect migrated to an adjacent

MDI exactly s times, where n(s) is the number of runs in whichexactly s migration events occurred and plotted in Fig. 17 ashistograms. We use these probabilities to determine $ from aleast-squares fit in s to

ln[(s!)p(t/$,s)] = s ln(t/$ ) ! t/$. (10)

Good fits are obtained for all three temperatures, confirmingour assumption that point defect migration follows a Poissonprocess (Fig. 17). The jump rates for each temperature,obtained by fitting, are plotted in Fig. 16(b) as filled graycircles with uncertainties corresponding to the error in theleast-squares fit. The gray line is the least-squares fit of Eq. (8)to the rates obtained from MD. The activation energy obtainedfrom our kMC model (Eact

eff = 0.398 ± 0.002 eV) is well withinthe uncertainty of the activation energy found by fitting the MDdata, namely, Eact

eff = 0.374 ± 0.045 eV.The effective attempt frequency for defect migration ob-

tained by fitting the MD data is # "0 = 6.658 % 109 ± 2.7 %

106 s!1. This value is several orders of magnitude lower thantypical attempt frequencies for point defect migration in fccCu, namely, 1012!1014 s!1.72–74 A mechanistic interpretationfor such a low migration attempt frequency is not immediatelyforthcoming. One possible explanation is that it arises fromthe large number of atoms participating in the migrationprocess. The attempt frequency for migration of compact pointdefects might be expected to be on the order of the Einsteinfrequency because it involves the motion of only one atom.However, the migration mechanism discussed here involvescollective motion of many atoms. Their collective oscillationin a vibrational mode that leads up to the saddle point fordefect migration may have a considerably lower frequency

FIG. 17. Comparison between the MD data (histograms) to the fits obtained by assuming that point defect migration follows a Poissonprocess (continuous curves and data points).

205416-11




MDkMC

0.001

0.01

0.1

1

1300 1000 800 700 600 500

Jum

p ra

te (n

s-1)



• Modified rate expression is fit to MD statistics to obtain attempt frequency

• Attempt frequency is much lower than is normally observed for point defects

model(Eact

eff

= 0.398±0.002 eV) is well within the uncertainty of the activation energy found

by fitting the MD data, namely E

act

eff

= 0.374± 0.045 eV.

⌫

00 = 6.658⇥ 109 ± 2.7⇥ 106s�1 The effective attempt frequency for defect migration ob-

tained by fitting the MD data is ⌫ 00 = 6.658⇥ 109 ± 2.7⇥ 106s�1.This value is several orders

of magnitude lower than typical attempt frequencies for point defect migration in fcc Cu,

namely 1012�1014 s�1 69–71. A mechanistic interpretation for such a low migration attempt

frequency is not immediately forthcoming. One possible explanation is that it arises from

the large number of atoms participating in the migration process. The attempt frequency

for migration of compact point defects might be expected to be on the order of the Einstein

frequency because it involves the motion of only one atom. However, the migration mech-

anism discussed here involves collective motion of many atoms. Their collective oscillation

in a vibrational mode that leads up to the saddle point for defect migration may have a

considerably lower frequency than an Einstein oscillator. This interpretation, however, is

at odds with other collective processes, such as the spontaneous transformation of small

voids to stacking fault tetrahedra, whose effective attempt frequency was several orders of

magnitude higher than the Einstein frequency72.

Delocalized point defect migration from one MDI to another may also involve passage

through several intermediate metastable states that do not assist migration: the I0 states

described in section III C 1. Therefore, the defect is likely to spend more time between

the initial and final states than it would had there been only one saddle point, lowering

the effective attempt frequency. If this were to completely account for the lowering of the

attempt frequency, however, then nucleation of I0 states would have to occur several orders

of magnitude more frequently than the completion of a migration step, which is not what we

observe. Finally, conventional transition state theory overestimates attempt frequencies by

assuming that every time a point defect crosses the saddle point, it reaches the final state.

In reality, however, a saddle point may be recrossed several times before reaching the final

state73, reducing the value of the pre-factor as derived by transition state theory61. Further

work is needed to determine which, if any, of these explanations is the correct one.

28

model(Eact

eff

= 0.398±0.002 eV) is well within the uncertainty of the activation energy found

by fitting the MD data, namely E

act

eff

= 0.374± 0.045 eV.

E

act

eff

= 0.374± 0.045 eV ⌫

00 = 6.658⇥ 109 ± 2.7⇥ 106s�1 The effective attempt frequency

for defect migration obtained by fitting the MD data is ⌫ 00 = 6.658⇥ 109± 2.7⇥ 106s�1.This

value is several orders of magnitude lower than typical attempt frequencies for point defect

migration in fcc Cu, namely 1012�1014 s�1 69–71. A mechanistic interpretation for such a

low migration attempt frequency is not immediately forthcoming. One possible explanation

is that it arises from the large number of atoms participating in the migration process. The

attempt frequency for migration of compact point defects might be expected to be on the or-

der of the Einstein frequency because it involves the motion of only one atom. However, the

migration mechanism discussed here involves collective motion of many atoms. Their collec-

tive oscillation in a vibrational mode that leads up to the saddle point for defect migration

may have a considerably lower frequency than an Einstein oscillator. This interpretation,

however, is at odds with other collective processes, such as the spontaneous transformation

of small voids to stacking fault tetrahedra, whose effective attempt frequency was several

orders of magnitude higher than the Einstein frequency72.

Delocalized point defect migration from one MDI to another may also involve passage

through several intermediate metastable states that do not assist migration: the I0 states

described in section III C 1. Therefore, the defect is likely to spend more time between

the initial and final states than it would had there been only one saddle point, lowering

the effective attempt frequency. If this were to completely account for the lowering of the

attempt frequency, however, then nucleation of I0 states would have to occur several orders

of magnitude more frequently than the completion of a migration step, which is not what we

observe. Finally, conventional transition state theory overestimates attempt frequencies by

assuming that every time a point defect crosses the saddle point, it reaches the final state.

In reality, however, a saddle point may be recrossed several times before reaching the final

state73, reducing the value of the pre-factor as derived by transition state theory61. Further

work is needed to determine which, if any, of these explanations is the correct one.

28

K. Kolluri and M. J. Demkowicz, Phys Rev B, 85, 205416 (2012)

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

Technology

Migration of isolated point defects in CuNb