PHYSICAL REVIEW B 70, 012102 (2004)
Properties of dislocation half loops in Au„100…: Structure, formation energy,
and diffusion barrier
F. El Gabaly,* R. Miranda, and J. de la Figuera
Departamento de Física de la Materia Condensada, Universidad Autónoma de Madrid, 28049 Spain
(Received 20 February 2004; published 21 July 2004)
Dislocation half-loops intersecting the surface have been detected by Scanning Tunneling Microscopy after
mild sputtering on Au共100兲. We determine their structure and formation energy by means of atomistic simulations. The Peierls barrier is also estimated from the simulations and found to be a few meV. This represents
a very efficient way of moving mass parallel to the surface under applied stress, where a net movement of
hundreds of atoms can take place with barriers smaller than that for the diffusion of a single adatom. Thermal
diffusion is not observed.
DOI: 10.1103/PhysRevB.70.012102
PACS number(s): 61.72.Ff, 61.72.Bb, 68.37.Ef, 07.05.Tp
Dislocations located close to the surface of a crystalline
solid are known to affect the mechanical properties of the
surface, but dislocations can also give rise to enhanced reactivity at points where they emerge from the solid,1 and influence strongly epitaxial growth.2 Ordered dislocation networks have been used as templates for the growth of arrays
of nanostructures.3 Certain dislocation configurations have
also been proposed to act as a pathway for the net transport
of atoms parallel to the surface with extremely low activation
barriers.4,5
Dislocations are usually treated within elastic theory introducing a cutoff length for atomic positions that are too far
away from equilibrium positions. Such an approach has been
extremely successful.6 Nevertheless there is an increasing
number of experimental results related to the behavior of
dislocations at the nanometer scale that require a fully atomistic modeling. Problems such as the estimate of the barriers
for dislocation motion, the formation of compact dislocation
networks in ultra-thin films,2 or the first steps of the nucleation of dislocations in nanoindentation experiments require
atomic scale total energy calculations that nowadays can be
based on semi-empiric potentials and simulations with thousands to millions of atoms. We present in this work an atomistic study of some of the properties of dissociated dislocation half-loops in Au共001兲7 with a comparison to
experimental observations of the half-loops obtained by both
ion-beam irradiation and nanoindentation.8 The results will
also be compared to studies based on simple dislocation geometries within the framework of dislocation theory in an
isotropic elastic continuum.9 The simplest expected subsurface structure will be obtained, and possible complications in
the experimental images will be addressed. Estimates for the
formation energies of half-loops of different sizes will be
presented, as will be the energy barrier for the glide of the
dislocation structures.
The experiments were performed in a ultra-high-vacuum
(UHV) Scanning Tunneling Microscope (STM) with a base
pressure of better than 2 ⫻ 10−10 Torr. The Au sample was
cleaned by cycles of Ar+ sputtering at 600 eV followed by
annealing. Annealing to 900 K produced flat terraces with
the well known 5 ⫻ 20 “hex” reconstruction of Au共100兲.10
This reconstruction appears in STM images as fringes ori0163-1829/2004/70(1)/012102(4)/$22.50
ented along 具110典 directions. These fringes arise from a
moiré-like effect between the topmost layer with an hexagonal arrangement and the lower bulk-like layer with square
symmetry.11
Annealing the Au共100兲 crystal at 600 K after sputtering
produces structures on the surface as shown in Fig. 1,
marked with black circles. These structures will be called
mesas from now on. They are probably present right after the
sputtering process, but the surface roughness at that stage
makes it impossible to locate them. Annealing at 900 K
makes them disappear from the surface. They are mostly
rectangular protrusions 0.6 Å high with lateral sizes in the
range of nanometers to tens of nanometers. The sides of the
protrusions are aligned with the compact 具110典 directions.
Two opposite edges of the mesas present an abrupt change in
height when crossing them, and their separation will be
called the “span” s of the structure. The other two decay
smoothly, and their separation will called the “width” w of
the structure from now on [see Fig. 1(b)]. The interpretation
of the mesas in terms of the dissociation of perfect edge
dislocation half-loops has been described in Refs. 7 and 8
and further discussed within the realm of dislocation theory
of an isotropic elastic continuum.9
The interstitial/vacancy character of those half-loops
which have the span aligned with the reconstruction fringes
(not shown) can be tested by observing whether on top of the
mesa there is a missing/extra reconstruction fringe with re-
FIG. 1. (a) STM image of the Au共100兲 surface after sputtering
and gentle annealing, 116 nm⫻ 160 nm in size. Several mesas
marked with black circles can be observed. The image has been
differentiated to enhance the contrast on the terraces. (b) STM image of a single mesa. The width w and the span s as defined in the
text are marked. The image size is 30.3 nm⫻ 30.3 nm.
70 012102-1
©2004 The American Physical Society
PHYSICAL REVIEW B 70, 012102 (2004)
BRIEF REPORTS
FIG. 2. (Color online) Atomistic simulation of the relaxation of
a triangle of interstitials. (a) Initial configuration, as seen from below the surface, showing only the top layer (gray) and the interstitial atoms (yellow). (b) Partially relaxed configuration, showing a
perfect edge-dislocation half-loop. (c) Completely relaxed configuration, giving the mesa structure. (d) Top view of the surface layer
of the EAM simulation.
spect to the unperturbed, reconstructed surface. In effect,
given the larger interatomic spacing in the bulk of the Au
crystal relative to the reconstructed uppermost layer, a missing fringe in the moiré pattern corresponds to an extra row of
interstitial atoms in the crystal with respect to the surface.12
All the half-loops where the test could be applied, both generated by sputtering and by nanoindentation,8 have an interstitial character and correspond to the insertion of an extra
plane of atoms.
To simulate the mesa structures we use atomistic simulations. The simulations are based on the embedded atom
method potential13,14 (EAM) with the DYNAMO code.15 The
cell dimensions are 29 atomic layers deep with 40 atoms per
side (92800 atoms). The top surface is traction free, while the
bottom is held fixed with periodic boundary conditions applied to the side surfaces. The Au EAM potential16 gives a
stacking fault energy of 30.7 mJ/ m2 in good agreement with
experimental values. The hexagonal top layer responsible for
the surface reconstruction in the experimental images has not
been included in the simulations, as it has been shown to
behave as a floating layer on top of the unreconstructed
lower layer.7 When the top view of the last layer is shown,
the color scale is keyed to the height of the atoms within the
layer. In the perspective views the centrosymmetry
parameter17 is employed to show only the atoms that have a
distorted environment: the top and bottom layers (grey), and
atoms located at dislocations (dark blue) and stacking faults
(yellow).
The starting configuration to simulate a mesa is a layer of
interstitials introduced in a 共111兲 plane [see Fig. 2(a)]. The
layer of extra atoms has a triangular shape and edges along
compact directions. The top row of interstitials is located just
above the surface plane. The precise initial location of the
atoms relative to the reference bulk atoms is slightly off ontop positions (the resulting relaxed structures are independent of that position within a range of 0.3 Å in the vertical
direction). On the other hand, if the starting location of the
interstitials corresponds to either extrinsic stacking fault positions or exact on-top positions they do not relax to a regular
dislocation half-loop. Instead, some of the extra atoms are
ejected to the surface. Likewise, vacancies in a triangular
plane give rise to vacancies in the surface layer. When relaxing statically the interstitial plane in the previously described
position, a perfect edge dislocation half-loop is first generated [see Fig. 2(b)]. Each perfect edge dislocation continues
the relaxation by dissociating within a 共111兲 plane into a
stacking fault bounded by Shockley partial dislocations6 [as
shown in Fig. 2(c)]. The precise shape of the stacking fault
region and the length of the stair-rod dislocation depends on
the depth of the dislocation half-loop. The top layer of the
EAM simulation is very similar in shape to the experimentally observed mesa, reproducing its 0.6 Å height [compare
Fig. 2(d) to Fig. 1(b)]. The length of the topside of the original triangle of interstitials is the final span of the mesa.
The precise structure of the dislocations underneath the
surface cannot be determined from the STM images alone as
these only show the uppermost atomic layer. The simplest
structure proposed corresponds to the V-shaped half-loop
shown in Fig. 2. A test of such a model is the experimentally
observed width-to-span relationship for mesas reported in
Ref. 9. For small spans, the width and span are proportional.
At larger spans, there is a saturation in the width with a
typical value of 7 nm. In the V-shaped model, the depth of
the structure and the span are proportional because the stacking faults are located on intersecting {111} planes. The saturation of the width at large spans (which means also deep
half-loops) is due to the equilibrium width of a long dislocation dissociated into a stacking fault ribbon.6 The EAM
simulations of simple V-shaped half-loops, as shown in Fig.
3(a), fall in the proportional part of the width-to-span curve.
The results are in excellent agreement with the experiments
reported for nanoindentation mesas.9 The smaller half-loops
present stacking faults of a triangular shape [see Fig. 3(b)],
with the Shockley partial dislocations aligned to closepacked directions. The larger half-loop of Fig. 3(c) shows a
tendency to have parallel Shockley partial dislocations close
to the surface.
So far it has been assumed that the dislocation structure
underneath the surface is the result of the dissociation of
V-shaped dislocation half-loops. Such structures are clearly
inadequate to explain experimental observations of nonsymmetric mesas as shown in Fig. 4(a). We propose that the
subsurface structure in these cases is an irregular, W-shaped
dislocation half-loop [see Figs. 4(b) and 4(c)]. A regular
W-shaped structure could originate symmetric mesas, but
would give rise to half-loops with a different width-to-span
relationship relative to the simple V-shaped half-loops [see
Fig. 3(a)]. The EAM simulations prove that such structures
are higher in total energy than the simple V-shaped halfloops. We calculate that a W-shaped symmetric structure
(similar to the one shown in Fig. 4(c) with both sides of
equal length) is 1.5 eV higher in energy than a simple
V-shaped half-loop made of a similar number of interstitials.
The dislocation half-loops should move easily if there is a
common glide direction for all the dislocations that comprise
the structure. The Shockley partials that form the edges of
the structure can move in each of the {111} planes in which
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PHYSICAL REVIEW B 70, 012102 (2004)
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FIG. 5. Peierls barrier for the displacement of V-shaped halfloops of different sizes, as estimated by the NEB method.
FIG. 3. (Color online) (a) Width-to-span relationship for
V-shaped simulated half-loops (empty diamonds) and for experimentally generated mesas (filled circles, from Ref. 9). The star
marks the width and the span of a W-shaped simulated half-loop.
Also shown are the perspective view of (b) a 7 atoms wide halfloop and (c) a 23 atoms wide half-loop.
each stacking fault resides. The intersection of both planes
defines a line along the axis of the stair-rod dislocation. Such
a direction is the common glide direction for the half-loop.
The movement of the half-loop in this direction will be hampered by the Peierls barrier6 of the dislocations involved. The
energy barrier for glide of the dislocation half-loops was calculated on a smaller slab of 13600 atoms using a modified
version of the DYNAMO code which implements the Nudged
Elastic Band Method18 (NEB). The results are presented in
Fig. 5. All the barriers obtained are extremely small. Even
for the 13 atoms-wide half-loop, corresponding to the net
movement of 91 atoms, the barrier for displacing the halfloop by one lattice distance is only 10.9 meV. Experimentally, the mesas have been observed to move under applied
stress.11 They have been assumed to move in the nanoindentation experiments as they are detected far away from the
point of nanoindentation. In fact they have been observed to
glide along the stair-rod axis in molecular dynamics simulations of nanoindentation9 under applied stress. Nevertheless,
we have not detected any thermal motion of dislocation halfloops a few times larger than those simulated here. This is
FIG. 4. (Color online) (a) STM image of a nonrectangular mesa.
The image size is 30 nm⫻ 30 nm. (b)–(c) EAM simulation of a
possible dislocation structure that can give rise to such a top view
image. (b) Top view; (c) perspective view.
the case for both the sputtering experiments (up to a temperature of 400 K) and the nanoindentation experiments. The
most likely explanation lies in the diffusion prefactor: we
expect the diffusion prefactor to be very low for large halfloops given the concerted motion involved in moving the full
structure. The number of atoms in this systems makes the
direct estimate of the diffusion prefactor from the simulations quite difficult. The experimental data provides a rough
estimate: given that no movement is detected at 300 K in
times of the order of hours and assuming a diffusion barrier
of the order of 25 meV for these slightly larger half-loops, a
prefactor of the order of 10−2 s−1 is obtained. This has to be
compared to the typical prefactor for adatom diffusion of
1013 s−1.
The prefactor is expected to depend strongly on the number of atoms of the half-loop. Although the (large) observed
half-loops do not move, the smaller ones could potentially
have an important role in the thermal mass-transport on the
surface. Unusual mechanisms, such as a single interstitial in
the surface layer of strained Cu共100兲 (a surface crowd ion)19
or subsurface diffusion below surfactant layers20 have been
recently proposed as efficient surface diffusion pathways.
The physical reason for the easy diffusion in the former case
is analogous to the one that facilitates glide of the dislocation
FIG. 6. Calculated formation energy of dissociated V-shaped
half-loops as a function of span.
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PHYSICAL REVIEW B 70, 012102 (2004)
BRIEF REPORTS
half-loops: the spread of the distortion caused by the defect
in the moving direction. In our case, the key process would
be the formation of dislocation half-loops. In order to estimate whether this surface diffusion pathway is relevant the
formation energies of half-loops of different sizes were calculated and they are shown in Fig. 6. Even the smallest loop,
three atoms deep, has a formation energy of 3.08 eV. For
bigger half-loops, the formation energy increases with the
size of the mesa. The formation energy per interstitial atom
decreases from 0.51 eV (3 atoms wide half-loop) to
0.174 eV (23 atoms wide half-loop). Based on this formation
energies, we rule out a significant contribution of half-loops
to thermal mass transport at surfaces.
In summary, dislocation half-loops have been observed by
STM after mild sputtering and annealing of a Au共100兲 crystal. The behavior of dissociated half-loops has been studied
by atomistic simulations employing EAM potentials. The
starting configuration to simulate a half-loop is a flat triangular plane of interstitials. Relaxing statically the configuration goes through the expected dislocation reactions giving
rise to a dissociated dislocation half-loop. The formation en-
ergies for half-loops of different sizes has been estimated,
ruling out a significant contribution to thermal mass transport
at the surface. The diffusion barrier for the half-loops has
been estimated by the NEB method, giving an extremely low
value of 5 – 12 meV. This small barrier explains how the
half-loops could be observed in nanoindentation experiments
quite far from the nanoindentation point, and provides an
efficient nonthermal mass-transport mechanism under applied stress along compact directions.
*Electronic address: [email protected];
10 D.
http://hobbes.fmc.uam.es
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