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Migration of isolated point defects in CuNb
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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K. Kolluri, and M. J. Demkowicz, unpublished
0
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Cu-Nb KS Cu-Fe NW Cu-V KS
1 nm
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1 nm 1.4 nm
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
0
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0.15
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0.35
0.4
0.45
0.5
0.55
Cu-Nb KS Cu-Fe NW Cu-V KS
1 nm
0
0
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0.8
1
0 0.2 0.4 0.6 0.8 1
0
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100
150
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1
0 0.2 0.4 0.6 0.8 1
0.6
0.8
1
1.2
1.4
1 nm 1.4 nm
Form
atio
n en
ergy
(eV
)A
ngle
with
-ve
x ax
is
0
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
0
50
100
150
0
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0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
0
50
100
150
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
Cu-Nb KS Cu-Fe NW Cu-V KS
1 nm
0
0
0.2
0.4
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1
0 0.2 0.4 0.6 0.8 1
0
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100
150
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0.8
1
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0
0.2
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0.8
1
0 0.2 0.4 0.6 0.8 1
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1
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0
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1
0 0.2 0.4 0.6 0.8 1
0.6
0.8
1
1.2
1.4
1 nm 1.4 nm
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
)
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
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
Isolated point defects in CuNb migrate from one MDI to another
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
! (reaction coordinate)
t
ca
I
t t
b
" E
(eV
)
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
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
• Thermal kink pairs nucleating at adjacent MDI mediate the migration
• Migration barriers 1/3rd that of migration barriers in bulk
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
! (reaction coordinate)
t
ca
I
t t
b
" E
(eV
)
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
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
Thermal kink pairs aid the migration process
• Thermal kink pairs nucleating at adjacent MDI mediate the migration
• Migration barriers 2/3rd that of migration barriers in bulk
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
! (reaction coordinate)
t
ca
I
t t
b "
E (
eV
)
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
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
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
0.05
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0.35
0.4
0.45
0.5
0.55
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
I
a 0
0.05
0.1
0.15
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0.25
0.3
0.35
0.4
0.45
0.5
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
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
Inverse of Temperature (K-1)
Migration is temperature dependent
1� = �0e
� 0.4eVkBT
• Migration rates are reduced because there are multiple paths
• Transition state theory may be revised to explain reduced migration rates
K. Kolluri and M. J. Demkowicz,Phys Rev B, 85, 205416 (2012)
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
Inverse of Temperature (K-1)
Migration is temperature dependent
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
• Migration rates are reduced because there are multiple paths
• Transition state theory may be revised to explain reduced migration rates
K. Kolluri and M. J. Demkowicz,Phys Rev B, 85, 205416 (2012)
MDkMC
0.001
0.01
0.1
1
1300 1000 800 700 600 500
Jum
p ra
te (n
s-1)
Inverse of Temperature (K-1)
Migration is temperature dependent
• 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