Proc. R. Soc. A (2005) 461, 2521–2543
doi:10.1098/rspa.2005.1470
Published online 24 June 2005
Observations of nanoindents via
cross-sectional transmission
electron microscopy: a survey of
deformation mechanisms
B Y S. J. L LOYD 1 A. C ASTELLERO 2 , F. G IULIANI 1 , Y. L ONG 1 ,
K. K. M C L AUGHLIN 1 , J. M. M OLINA -A LDAREGUIA 3 ,
N. A. S TELMASHENKO 1 , L. J. V ANDEPERRE 1 AND W. J. C LEGG 1
1
Department of Materials Science and Metallurgy, University of Cambridge,
Pembroke Street, Cambridge CB2 3QZ, UK
([email protected])
2
Department of Materials, Swiss Federal Institute of Technology Zürich,
ETH-Hönggerberg, Wolfgang-Pauli-Strasse 10, 8093 Zürich, Switzerland
3
CEIT, P. Manuel Lardizabal 15, 20018 San Sebastian, Spain
Examination of cross-sections of nanoindents with the transmission electron microscope
has recently become feasible owing to the development of focused ion beam milling of
site-specific electron transparent foils. Here, we discuss the development of this technique
for the examination of nanoindents and survey the deformation behaviour in a range of
single crystal materials with differing resistances to dislocation flow. The principal
deformation modes we discuss, in addition to dislocation flow, are phase transformation
(silicon and germanium), twinning (gallium arsenide and germanium at 400 8C), lattice
rotations (spinel), shear (spinel), lattice rotations (copper) and lattice rotations and
densification (TiN/NbN multilayers). The magnitude of the lattice rotation, about the
normal to the foil, was measured at different positions under the indents. Indents in a
partially recrystallized metallic glass Mg66Ni20Nd14 were also examined. In this case a
low-density porous region was formed at the indent tip and evidence of shear bands was
also found. Further understanding of indentation deformation will be possible with threedimensional characterization coupled with modelling which takes account of the variety
of competing deformation mechanisms that can occur in addition to dislocation glide.
Mapping the lattice rotations will be a particularly useful way to evaluate models of the
deformation process.
Keywords: transmission electron microscope; focused ion beam; nanoindentation;
Peierls stress; shear bands; lattice rotations
1. Introduction
Indentation has been used to measure hardness for over a century yet the
relationship between hardness and macroscopic material properties such as the
yield stress is still not fully understood (Cheng & Cheng 2000). Nanoindenters
Received 1 October 2004
Accepted 18 February 2005
2521
q 2005 The Royal Society
2522
S. J. Lloyd and others
that record the indenter depth as a function of load are now widely used and
allow the characterization of samples that would be difficult or impossible with
conventional mechanical testing owing to the small size of the material in one or
more dimensions, for example, a thin film less than a few microns thick. In such a
case, the ability reliably to determine material properties using indentation is
particularly important. In addition, owing to the decreasing size of components
in semiconductor and microelectromechanical systems technologies (Spearing
2000) there is increasing interest in how the mechanical properties at submicron
length-scales may vary from those in the bulk (Gao & Huang 2003; Lawn 2004).
For example, cracking may be avoided in normally brittle ceramics if the stressed
volume is small enough and, hence, room temperature plastic flow may be
investigated by nanoindentation (Lloyd et al. 2002). In each case, in order to
understand the mechanical property data obtained it is important to observe the
defects associated with the deformation as directly as possible. In this article, we
present observations of nanoindents in a range of materials observed in crosssection using transmission electron microscopy (TEM). While TEM has been
used to investigate nanometer scale deformation for nearly 50 years (Whelan
2002), it is only recently that improvements in sample preparation have made it
feasible to examine the subsurface localized deformation associated with
nanoindentation.
Subsurface indent deformation has been investigated with other techniques for
many years. For example, the slip lines under indents in MgO have been
examined by cleaving and etching (Keh 1960), as have crack distributions under
indents in sodalime glass (Hagan & Swain 1978). Indented semiconductor
multilayers have been cleaved and imaged in the scanning electron microscope
(Castell et al. 1993). Indents in metals (Chaudhri 1993) and metallic glasses
(Donovan 1989) have also been sectioned and examined in the optical microscope
and by scanning electron microscopy. More recently, focused ion beam (FIB)
milling has been used to section and image indents in, for example, multilayer
thin films (Tsui et al. 1999). However, all these observations have been made
under relatively large indents and none of these samples could be examined with
the TEM.
One of the usual limitations of TEM—that only a small region of a few
micrometres across is thin enough to be imaged—is not a problem in the
examination of nanoindentation since the deformation is itself highly localized and
contained within the typical area of thin foil that can be made. This localization is
particularly useful for cases where the deformation would otherwise be hard to
locate, such as the shear bands in metallic glasses in a uniformly strained sample.
However, conventional methods to prepare electron transparent foils are not
sufficiently site-specific to section an indent reliably, although cross-sectional
TEM samples have been made of scratches or wedge-shaped indents (Hultman
et al. 2000; Kramer et al. 2004). The only way to create a TEM cross-section of a
specific indent is to use a FIB which can quickly and reliably create thin foils with
submicron precision (Langford & Petford-Long 2001a). Indent deformation in a
plane parallel to the indented surface can be examined in the TEM using foils made
by conventional methods (Hockey 1971; Hockey & Lawn 1975; Page et al. 1992;
Stelmashenko & Brown 1996). In this case an array of indents is back-thinned to
the indented surface.
Proc. R. Soc. A (2005)
2523
Deformation mechanisms of nanoindents
10–1
theoretical shear strength
GaAs
sapphire
Si,Ge
spinel TiN
t /G
MgO
NaCl
KC1
Mo
W
Cu
Peierls stress
10–3
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
d/b
Figure 1. The ratio of shear flow stress (t) and shear modulus (G) for a range of materials with
different ratios of atom spacing parallel to the slip plane with that normal to it (d/b). The
experimental flow stresses were estimated from hardness measurements, assuming hardness is
approximately six times the shear flow stress (Cheng & Cheng 2000). The data for MgO is plotted
using d/b for the hard slip system (Lloyd et al. 2002). Lines are also shown for the theoretical shear
strength and the Peierls stress as a function of d/b, as explained further in the text.
The first TEM examination of an indent cross-section prepared in a FIB was
by Saka & Nagaya (1995). This and much of the earliest work concentrated on
silicon and other semiconductor materials (Le Bourhis & Patriarche 2000; Lloyd
et al. 2001a; Bradby et al. 2002a). Nanoscale multilayers were first examined by
this method by Lloyd et al. (1999). Indents in metals have also been
investigated, for example by Ando et al. (1999).
This type of TEM analysis is a ‘post-mortem’ examination in that it is the final
structure after the indenter has been removed that is observed. Thus, the
sequence of phase transformations occurring under loading and unloading in, for
example, silicon (Kailer et al. 1997) cannot be observed directly. In situ
indentation in the TEM allows real-time imaging of the deformation process but
often with insufficient time resolution to record the motion of individual
dislocations (Minor et al. 2004). In addition, the limitation of using a thin foil
means the deformation can differ markedly to that seen in the bulk. For example,
a phase transformation is not observed on indenting a silicon thin foil (Stach
et al. 2001).
Dislocation flow becomes more difficult as the ratio of the atom spacing
parallel to the slip plane with that normal to it, d/b, is increased. However, at
lower values of d/b the stress required to move a dislocation is almost equal to
the theoretical shear strength (see figure 1). This shows the change in both the
theoretical shear strength and the Peierls stress with d/b. The latter is a
modification of the original analysis (Peierls 1940; Cottrell 1965) that shows good
agreement with the experimental data (Clegg et al. in pressa). The line
representing the theoretical shear strength (tt) is given by the expression (Kelly
1966), tt/GZ(1/6p)(d/b)K1, where G is the shear modulus. At low values of d/b,
Proc. R. Soc. A (2005)
2524
S. J. Lloyd and others
electron beam
in TEM
(a)
small
indents
50 µm
marker indents
Pt strip covering
indent
Ga ions
in FIB
(b)
1 µm
cuts to relieve stress
thin foil
indent
Pt protective layer
Figure 2. (a) Schematic illustrating how the indents were sectioned in the FIB to allow
examination with a TEM. (b) Secondary electron FIB image of the final stages of preparation of a
thin foil through a 50 mN indent in silicon. The residual stresses associated with the phase
transformations at the indent caused bending of the foil that was not alleviated by cutting the
edges of the foil.
where the Peierls stress and the theoretical shear strength become more nearly
equal, one might expect that other deformation mechanisms, such as twinning or
phase changes, might be more common. In this paper we present a preliminary
survey of the indentation behaviour in a number of materials with a range of
Peierls stresses to illustrate the power of this type of TEM analysis in elucidating
the range of deformation mechanisms available.
2. Experimental
Single crystal samples were indented in their (001) surface using a Nanotest 600
(Micromaterials, UK) indenter with a diamond Berkovich tip. Loading rates
were typically approximately 1 mN sK1.
TEM foils were prepared using a FEI FIB200 workstation as described in more
detail elsewhere (Lloyd et al. 2002). The specimen geometry is shown in figure 2a.
Typically, the indents were sectioned though the tip of the indent with the thin
foil normal approximately parallel to either the [110] or [100] zone axis, although
the foil plane was not always exactly parallel to one of the indent ridges. The
position of the thin foil could be controlled to within approximately 200 nm. The
‘trench’ approach used here proved to be more reliable than the ‘lift-out’ method
and also allowed further thinning of the specimens (Langford & Petford-Long
2001a). Often the foils bent in the final stages of milling owing to the relaxation
of elastic strains in the sample and the bending was not always fully alleviated by
making cuts at the sides of the foil. For example, bending was localized at the site
of the indent in silicon because of the volume changes associated with the phase
transformations that occur (figure 2b).
Proc. R. Soc. A (2005)
Deformation mechanisms of nanoindents
2525
TEM observations were made on a Philips CM30 TEM operating at 300 kV.
Rotations about an axis perpendicular to the plane of the foil were measured by
recording diffraction patterns using a small selected area aperture (diameter ca
30 nm) and measuring the relative rotation of the pattern to within 0.58. Local
misorientations were also present that tilted the crystal out of the plane of the
foil. However, for the sections made close to the indent tip, examined in this
paper, these misorientations were typically less than 58 and are the subject of a
separate study (McLaughlin et al. 2004). It should be noted that foil bending due
to residual elastic stresses in materials with a high yield stress will hamper the
measurement of these misorientations. However, the lattice rotations discussed
in this paper will be largely unaffected by foil bending.
3. Results and discussion
(a ) Silicon and related semiconductors
Figure 3a,b shows indents in (001) silicon at loads of 30 and 60 mN, respectively.
In both cases, a transformed region is evident immediately under the area of the
indenter tip with dislocation activity seen outside the transformed region, which
in the case of the 60 mN indent is located in lobes spreading out from the base of
the transformed zone. A crack running perpendicular to the surface and
originating at the base of the transformed region can be seen in the 60 mN indent.
This is likely to have formed on loading (Lawn & Wilshaw 1975) and would
account for the observation of material having been squeezed into it (figure 3b).
A similar crack is not observed in the 30 mN indent but the moiré contrast below
the transformed zone is probably due to the presence of a ‘half-penny’ crack in
the plane of the foil (Saka et al. 2002). The transformed zone is the same shape at
both loads, consistent with the self-similar deformation predicted for a Berkovich
indenter and the width of the transformed zone scales as expected with the
square root of the load.
The measured hardness from this range of loads was 10.8G0.8 GPa, similar to
the pressure required to cause a transformation to the b-Sn tetragonal structure
in a diamond anvil cell. If the hardness is governed by the phase transformation
pressure, it is surprising that dislocation activity is observed outside the
transformed region and suggests that stresses are high enough to cause
dislocation flow. However, recent calculations (Vandeperre et al. 2004) have
shown that dislocation flow will occur in addition to the phase transformation
when the yield strength is lower than 1.5 times the transformation pressure.
Further, if the transformed material has a low resistance to plastic flow (as
suggested by the observation that the transformed material has extruded into
the crack under the 60 mN indent; figure 3b) the pressure gradient between the
indenter and the edge of the plastic zone is reduced substantially. Thus,
the pressure on the indent, that is, the hardness, becomes equal to the pressure
needed for the phase transformation and is lower than it would have been in the
absence of a phase transformation. The hardness of silicon is virtually
temperature independent up to approximately 500 8C because the pressure
needed for complete accommodation of the deformation by dislocation flow is
higher than the transformation pressure. Germanium behaves in a similar
manner under indentation (Lloyd et al. 2002) but the hardness is reduced to
Proc. R. Soc. A (2005)
2526
S. J. Lloyd and others
(a)
200 nm
(b)
200 nm
Figure 3. Bright field images of indents in silicon formed with loads of (a) 30 mN and (b) 60 mN.
The structure is observed here after the load has been removed, allowing the high pressure phase to
transform back to a mixture of other structures (Lloyd et al. 2002).
4G1 GPa at 400 8C (Vandeperre et al. 2004) compared with 9.9G0.6 GPa at
room temperature. At 400 8C the phase transformation is avoided because the
pressure needed for accommodation of the indenter by twinning and dislocation
flow is below the phase transformation pressure. A finely balanced competition
between these deformation processes, which are probably also sensitive to strain
rate, may be the reason that others have found twinning rather than a phase
transformation under indents in germanium (Bradby et al. 2002b).
Competition between deformation processes as a function of temperature has
also been observed in gallium arsenide. The plastic deformation mechanisms in
gallium arsenide at room temperature are twinning and dislocation flow, with the
former accommodating the majority of the material displacement (Lloyd et al.
2001a). (Despite having the same lattice as silicon and germanium, a phase
transformation is not observed as the transformation pressure is much higher
Proc. R. Soc. A (2005)
Deformation mechanisms of nanoindents
2527
than the room temperature hardness.) At 300 8C the plastic zone becomes much
broader and twinning, and while present, ceases to be the dominant mechanism
(Giuliani et al. 2003). The reduction in twinning at elevated temperatures may
be understood on the basis that twins nucleate at a critical surface shear. For
example, spherical indentation of InxGa(1Kx)As superlattices resulted in twins
only forming towards the edge of the indentation where the surface shear was
greatest (Lloyd et al. 2003). If the yield stress is too low, there is insufficient
elastic displacement to cause twinning before the stress is relaxed by dislocation
flow. The critical surface shear was estimated to be approximately 78 (Lloyd et al.
in press).
In summary, for silicon and related semiconductors there is competition
between deformation processes, including phase transformation, twinning and
dislocation flow as a function of temperature, with the one that occurs at the
lowest stress controlling the hardness. The Peierls stress (figure 1) is sufficiently
high in silicon and germanium at room temperature that phase transformation is
the easier deformation mechanism.
(b ) Spinel (MgAl2O4)
Preliminary observations of indents in sapphire, spinel and magnesia have
previously been presented (Lloyd et al. 2002). The deformation in spinel is
examined in more detail in this article. Spinel was chosen because of its relatively
high hardness (measured here to be 16 GPa), cubic symmetry and its {111} h110i
primary slip provides five independent slip systems that can accommodate an
arbitrary shape change. Despite its high Peierls stress (figure 1), similar to the
semiconductors discussed above, twinning is not observed in spinel (Mitchell
1999).
The observations here were made on a series of indents at loads of 30–100 mN
in the (001) surface of spinel with one of the indent ridges aligned approximately
along h110i. A bright field image of an 80 mN indent with a diffraction pattern
taken from the deformed zone is shown in figure 4a. The shear bands are shown
more clearly for a 60 mN indentation at higher magnification in figure 4b.
Figure 4c schematically illustrates the principal features of the deformation
pattern. Figure 5 shows the variation in the shear band spacing as a function of
distance from the indent tip (i.e. along the black lines on figure 4b) for both sides
of the indent (‘steep’ and ‘shallow’ as defined in figure 4b) and two indenter
loads.
Several deformation modes can be discerned from figure 4: a cone of sheared
{111} planes under the indent impression surrounded by a region of tangled
dislocation lines and rotation of the crystal about an axis perpendicular to the
plane of the foil. In addition, some pile-up is evident at the sides of the indent and
residual compressive elastic strain of around 0.5% is detectable on the streaked
portion of the 440 reflections in the diffraction pattern in figure 4a. Elastic strain
in the 002 direction is more likely to be relieved on removal of the load. Cracks
are formed (figure 4a) perpendicular to the surface which nucleate at the
intersection of slip bands, consistent with the model of Cottrell (1958). At lower
loads smaller cracks were formed in similar positions. No cracks were evident for
indents formed with loads less than 50 mN.
Proc. R. Soc. A (2005)
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S. J. Lloyd and others
(a)
200 nm
(b)
'shallow'
'steep'
100 nm
(c)
width
high
dislocation
density
depth
shear bands
crack
Figure 4. (a) Bright field image of an 80 mN indent in spinel with an inset diffraction pattern.
(b) Bright field image of a 60 mN indent in spinel showing the region of shear bands under the
indent more clearly. (The vertical line is an artefact of montaging images from two negatives.) The
white lines mark the change in angle of the shear bands with respect to the surface on the steep side
of the indent. The shear band spacing, plotted in figure 5, is measured from the indent tip along the
black lines. (c) Schematic illustrating the principle components of the deformation pattern in
spinel. The width and depth of the plastic zone, plotted in figure 7, are defined here.
The shear bands and crystal rotations are now considered in more detail.
While there is a large statistical scatter there is a trend to increase the shear
band spacing with increasing distance from the indent tip, the effect being more
marked on the shallow side of the indent (figure 5). The pattern of shear band
spacing is very similar for the two loads shown, although the spacing on the steep
side of the indent is a little smaller for the larger load. The minimum shear band
width is around 10 to 20 nm. The increase of the shear band spacing with
distance from the indenter tip implies there is a limit to the amount of
Proc. R. Soc. A (2005)
2529
Deformation mechanisms of nanoindents
width (nm)
150
100
80 mN, shallow
50 mN, shallow
80 mN, steep
50 mN, steep
50
0
500
1000
1500
distance (nm)
2000
2500
Figure 5. Slip band spacing in spinel as a function of distance from the indenter tip (see figure 4b)
for the 50 and 80 mN indents. The error in measurement of the shear band spacing is
approximately 10 nm.
displacement occurring through any one shear band, possibly due to strain
hardening. Hence, a high concentration of shear bands is found close to the
indenter tip where the vertical displacements are largest, whereas away from the
tip a lower density is sufficient to accommodate the relatively small vertical
displacements. This is similar to the observation by Mendelson (1972) that the
slip band spacing in silicon single crystals at elevated temperatures decreased
with increased plastic strain.
The discrete, ‘stepped’ nature of the deformation may be understood in the
following way. The material initially deforms elastically until the elastic stress
reaches a critical value at which shear is initiated, relieving most or all of the
elastic stresses which then build up again under further loading until a new shear
band forms, and so on. This type of process has been proposed by a number of
others, including Yoffe (1982), Bull et al. (1989) and Lawn (2004). The fact that
the shear band spacing is relatively independent of load is consistent with a
model of this type.
Lattice rotations about an axis perpendicular to the [110] zone axis are shown
in figure 6a at different points of the cross-section of a 50 mN indent. At each
point the crystal structure is undistorted from the bulk (see the inset diffraction
patterns) but the local orientation of the crystal varies so that a diffraction
pattern taken with a larger selected area aperture, that includes the majority of
the deformed zone such as that shown in figure 4a, shows streaked spots.
It is clear from figure 6a that the rotations are only found in the region
immediately below the indent impression. The rotations are greatest near the
indent tip and are similar on the steep and shallow side of the indent, although
the magnitude of the rotation decreases with distance from the indent tip along
the surface on the shallow side. The magnitude of the rotations below the indent
decreases rapidly to zero within a depth of about twice the residual indent depth
below the original surface. Dark field images, taken from the highest rotation
angle streaks on opposite sides of the 004 reflection, are shown in figure 6b,c.
These also indicated that the magnitude of the rotations decreases with distance
from the crystal surface. The direction of rotation on either side of the indent
Proc. R. Soc. A (2005)
2530
S. J. Lloyd and others
(a)
200 nm
(b)
100 nm
(c)
100 nm
Figure 6. (a) Bright field image of a 50 mN indent in spinel with lattice rotations (in degrees) at
selected positions indicated. Lattice rotations were measured relative to the region marked ‘A’
from selected area diffraction patterns. A clockwise rotation is taken as positive. The inset
diffraction patterns on the left and right are from the region closest to the indent tip on the steep
and shallow sides of the indent, respectively. (b) and (c) are dark field images of the surface of the
same indent taken using streaks from either side of the 004 reflection.
shown in figure 6 is clockwise and anticlockwise for the steep and shallow sides of
the indent, respectively. The magnitude of the crystal rotation is less than the
angle of the surfaces in the residual indent (approximately 20 and 108 for the
steep and shallow sides of the indent, respectively), indicating that over half of
the total residual displacement must be accommodated by the shear bands on the
steep side of the indent. A combination of shear and rotation may account for
why the shear bands on the steep side of the indent appear to be rotated over 208
relative to the undistorted crystal (see the white lines marked on figure 4b), even
though the measured crystal rotation on the diffraction patterns is less than half
this value.
The observations here suggest that the rotations do not appear to be a direct
consequence of the shear band formation. Firstly, while misorientation is
sometimes observed either side of a shear band, there is often no misorientation
present (e.g. see the identical diffraction contrast on either side of the shear
bands in the vicinity of ‘A’ in figure 6a). Secondly, the magnitude of the rotation
rapidly falls off with distance from the indented surface over a region in which
the shear band density remains approximately constant. Hence, we conclude that
the lattice rotation is a separate deformation process, which most likely occurs
first since it can happen continuously without any nucleation as soon as the
indenter contacts the surface, initially as an elastic process. This is supported by
the ideas of Page et al. (1992), who show that a critical load is required to induce
plastic deformation in defect-free surfaces. If the distortion of the lattice at the
surface is modelled as an elastic shear, the stress required for a shear of 88 (for
GZ108 GPa) is 15 GPa—the same as the hardness. However, this agreement is
Proc. R. Soc. A (2005)
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Deformation mechanisms of nanoindents
5
width
depth
distance (mm)
4
3
2
1
0
4
5
6
7
8
load1/2 (mN1/2)
9
10
11
Figure 7. Plastic zone size in spinel as a function of the square root of indenter load. The width and
depth are defined in figure 4c. The error in the measurement of the plastic zone size is
approximately G100 nm.
probably fortuitous since to model the rotation process accurately would require
a much more sophisticated calculation. Furthermore, the 88 lattice rotation
measured is not necessarily entirely produced elastically. Once the crystal shear
bands operate, the crystal is probably fixed in its rotated orientation so that even
any elastic component cannot fully relax on unloading. The net effect is similar to
the reorientation of the lattice found in twins (i.e. a martensitic transformation)
except that the shear is not so localized and varies in magnitude across the
indented area.
The size of the deformed zone (with dimensions defined in figure 4c) plotted as
a function of the square root of maximum load is shown in figure 7. For the small
range of loads examined, equally good linear
fits can be made plotting the plastic
pffiffiffiffiffiffiffiffiffi
zone dimensions as a function of load and load but the latter is consistent with
theory (Johnson 1985). Extrapolating these data to zero distance predicts a finite
load of approximately 1 mN to initiate yield, consistent with the observations of
Page et al. (1992). The width to depth ratio has a constant value of
approximately 2.15 over this load range. A constant width to depth ratio
would be expected for the self-similar indenter geometry used here and a value
close to 2 is consistent with the common assumption that indent deformation can
be modelled as an expanding hemispherical cavity (Johnson 1985), although the
detailed pattern of deformation is clearly strongly affected by the
crystallography.
While a full investigation of size effects is outside the scope of this paper we
note that the presence of at least two deformation processes naturally leads to
the possibility of a size effect, if the relative proportions of each vary with load. If
the volume of rotated material varies linearly with indent depth then a larger
proportion of the deformation will be accommodated by rotation at low loads.
Given that elastic recovery is more likely for the rotated material than the
sheared material, the lower loads would be expected to show a reduced residual
area and, hence, a higher hardness, in a similar way to the model proposed by
Bull et al. (1989).
Proc. R. Soc. A (2005)
2532
S. J. Lloyd and others
500 nm
Figure 8. Bright field image of a 5 mN indent in the (001) surface of copper single crystal with a
diffraction pattern (inset) from the indented region. This image was taken with the crystal tilted so
the 002 systematic row was diffracting strongly. Lattice rotations relative to position ‘A’ are
marked in degrees, as for figure 6a.
(c ) Copper
Peierls stresses are so low in f.c.c. metals such as copper (see figure 1) that the
macroscopic yield stress is determined by the barrier crystal defects provide to
dislocation glide. The yield stress in a single, relatively defect-free crystal (as
examined here) will be low, so that almost all the indent deformation will be
plastic in contrast to the high elastic strains in the ceramics discussed above.
Nevertheless, the rapid hardening caused by indentation means high stresses can
be achieved, particularly at very low loads (McElhaney et al. 1998).
A bright field image of a 5 mN indent in a mechanically polished (001) surface
of a single copper crystal is shown in figure 8 with a diffraction pattern from the
indented region inset. Streaked spots in the diffraction pattern arise from local
rotations of the crystal perpendicular to the [100] zone axis. The rotations are
marked on figure 8, which was obtained with the crystal tilted to the 002
systematic row so that the undeformed crystal around the edge of the figure was
strongly diffracting. ‘Speckled’ contrast arising from damage associated with the
focused ion milling is most apparent in the strongly diffracting regions (Ando
et al. 1999).
The distribution of crystal rotations are a little different to those found in
spinel. In copper the rotation is greater on the steeper side of the indent and for
both sides of the indent the rotation is smaller near the indent tip than at the
edge of the indent impression. Some rotation is also present near the surface
outside the indent impression. The rotations reduced to zero within a distance of
twice the residual indent depth from the original surface. Interestingly, a
‘cellular’ pattern of misoriented regions, often observed in metals deformed to
high strains (Stobbs et al. 1976; Hughes 2001), is not apparent here. Instead, the
constraint of the indentation geometry creates a region where the rotation
decreases continuously from the surface.
The maximum value of the rotations is larger than that found in spinel and is also
large compared with typical lattice rotations associated with local deformation
Proc. R. Soc. A (2005)
Deformation mechanisms of nanoindents
2533
around hard particles in homogeneously strained samples. For example, rotations
of up to 28 were observed locally around silica particles in strained, dispersionstrengthened copper single crystals (Chapman & Stobbs 1969). Often the
misorientations found in metals are explained in terms of geometrically necessary
dislocations (GND; Ashby 1970; Hughes et al. 2003), and the indentation size effect
has been modelled using GND (Stelmashenko et al. 1993).
The density of GND (rG) in a bent crystal can be estimated from the equation rGZK/b (Ashby 1970) where K is the curvature of the crystal in mK1. In
figure 8 the maximum crystal curvature is from the indent tip (assumed to have a
rotation of 0) to the point on the steep side of the indent with a rotation of 118,
a distance of approximately 300 nm. Thus, the curvature is 6.3!105 mK1 giving
rGZ2.5!1015 mK2, corresponding to a dislocation spacing of approximately
20 nm. Although this is a minimum dislocation density for this region (given that
statistically stored dislocations will also be present) it is not physically
unreasonable. Initial attempts to measure the dislocation density in this region
have been hampered by the relatively high foil thickness and dislocation loops
introduced as damage from the FIB milling (Ando et al. 1999). The stress (tf)
required to move a dislocation
this ‘forest’ of GND can be estimated
pffiffiffiffiffithrough
ffi
from the equation, tf Z ðGb rG Þ=3 (Haasen 1986), to give tfZ180 MPa for
GZ42 GPa. Assuming a factor of 6 relating tf and H (see figure 1) this would give
a hardness increase of approximately 1.1 GPa.
While the structure of the final rotated crystal may be described in terms of
GND it is not necessarily formed from the coordinated motion of an array of
dislocations, just as deformation twins are not necessarily formed by partial
dislocations gliding along parallel slip planes—even though their structure is
geometrically equivalent to such a process having occurred (Christian &
Mahajan 1995). It is possible that the small region of crystal in immediate
contact with the indenter is rotated through cooperative small movements of
atoms more akin to a martensitic transformation. For a small region of material
the stress required for such a process may not be dissimilar to that required to
move dislocations through the GND forest. Unique behaviour can occur under
very localized loading, as discussed by Tadmor et al. (1999). For example, in
their simulations of indentation in aluminium they show that twins can form,
even though it is usually thought that twinning does not occur in aluminium.
Whatever the mechanism causing the crystal rotation it is only accommodating
part of the indent displacement since the measured lattice rotation is much less
than the angle of the indented surface, as was also observed in spinel. Dislocations
must also be moving material away from the indent impression. If the rotations
comprise a greater proportion of the deformed volume at small loads they may
account for the marked size effect seen in copper. Interestingly, the hardening
predicted above from the GND is of a similar magnitude to hardening observed
for small indents in a copper single crystal (McElhaney et al. 1998). Work is in
progress to investigate these ideas further (McLaughlin et al. 2004).
(d ) TiN/NbN multilayers
The hardness of multilayers with nanometer scale periods (L) has been studied
extensively because of the period-dependent properties they sometimes display
with hardnesses greatly exceeding their monolithic components, particularly for
Proc. R. Soc. A (2005)
2534
S. J. Lloyd and others
100 nm
Figure 9. Bright field image of a 30 mN indent in a TiN/NbN multilayer (LZ14 nm) grown on
(001) MgO.
L in the range 1–10 nm (Chung & Sproul 2003). Thus, there is the attractive
possibility of tuning the hardness of films by tailoring the design of the multilayer
to control the hardness independently of the material properties (and Peierls
stresses) of the component layers (Lloyd & Molina-Aldareguia 2003). In some
cases, where a hard material such as an oxide or nitride is layered with a soft
metal, hardness enhancements beyond the rule of mixtures are not surprising—
exactly the same behaviour is found in dispersion-strengthened alloys (Haasen
1986). However, the enhancements found when two similar materials are layered
and the way the hardness varies with the period are still not fully understood
(Clegg et al. in pressb).
In addition to their interest as systems with variable and enhanced hardness,
multilayers are also useful in understanding the deformation processes occurring
because they provide internal markers that show how the material deformed at a
smaller and more controllable scale than is possible with naturally layered
materials such as pearlite (Porter et al. 1978). Here, we discuss the deformation
of a TiN/NbN multilayer with a period (L) of 14 nm and sputter deposited on
MgO (001) with a 50 V substrate bias (see Molina-Aldareguia et al. (2002) for
details of the deposition process). Its hardness was 24G2 GPa, similar to the
hardness of the monolithic components grown under the same conditions.
Figures 9 and 10 are bright field images of a 30 mN indent in the multilayer
illustrating the principle deformation patterns. In figure 9 the layers are seen to
be bent to follow the indenter face. Shear bands are also evident to the side of the
indent running along the {011} planes, which are the favoured slip planes for this
structure. The multilayer has a columnar structure (some column boundaries are
indicated on figure 9) and the column boundaries can act as weak points where
shear occurs (Molina-Aldareguia et al. 2002). Figure 10 is a bright field image
taken with the undeformed part of the multilayer tilted close to the [100] zone
axis, with the lattice rotations about this axis at different positions under the
indent indicated. The pattern of rotations is different to that seen in both spinel
and copper. They are confined to a cone of material defined by the indent
impression and a depth below the indent tip that is approximately 0.8 of the
Proc. R. Soc. A (2005)
Deformation mechanisms of nanoindents
2535
100 nm
Figure 10. Bright field image of the same indent as in figure 9. Lattice rotations relative to position
‘A’ are marked in degrees, as for figure 6a. The white lines mark the approximate position of the
compressed cone of material.
width of the indent impression (as approximately indicated by the white lines on
figure 10). The rotations are greatest around the indent tip (like spinel) but
decay much less rapidly with distance from the indented surface. Unlike spinel
and copper the lattice rotation is similar to the angle of the indent impression at
the surface of the indentation. However, the crystal rotation is much greater than
the inclination of the layers below the surface so the layers are virtually
horizontal at approximately 400 nm below the surface, but there is still a crystal
rotation of 108. Measurements of L under the indentation show that the cone
marked in figure 10 also defines a region where L is reduced by up to 10%, while
it maintains the undeformed value outside this cone. A similar pattern was found
under a 10 mN indentation.
Since the multilayer is continuous across the top of the indented surface it
must have been stretched (by approximately 4%) in this region and hence
reduced in thickness to maintain constant volume. However, such an effect
cannot account for the reduction in L further from the surface where the layers
are virtually horizontal. Given that there is no evidence of material being
squeezed out sideways either here or in similar longer period multilayers (Lloyd
et al. 1999), the most likely explanation for the reduction in L is that the material
has been compressed into both voids present in the structure and atomic scale
vacancies. The structure of NbN grown under similar conditions is known to be
highly defective on an atomic scale (see Lloyd et al. 2001b) and a high density of
voids can be seen in these multilayers. Figure 11a shows an area of film with the
compressed, indented region on the right and the uncompressed multilayer on
the left at a defocus of C2 mm to increase the visibility of the voids by Fresnel
contrast around their edges. A marked decrease in porosity in the compressed
region is evident, indicating that the multilayer has densified under the indent.
An estimate for the degree of densification that can be achieved in these
multilayers can be made from figure 11b, which shows a multilayer grown under
Proc. R. Soc. A (2005)
2536
(a)
S. J. Lloyd and others
outside indent
steep side
of indent
(b)
50 nm
(c)
growth
direction
20 nm
Figure 11. (a) Bright field image of the TiN/NbN multilayer shown in figure 9; taken in weakly
diffracting conditions at a defocus of C2 mm. A relatively pore-free region on the steep side of the
indent impression can be seen with the adjacent uncompressed region showing a high pore
concentration. The pores can be identified by the dark Fresnel contrast in their centre. The
position of some pores is indicated by the white arrows. (b) Bright field image, at a defocus of
K2 mm, of a TiN/NbN multilayer grown under similar conditions to the multilayer shown in
(a) but about twice as slowly so the base of the multilayer (bottom of image) was at the growth
temperature of approximately 750 8C for approximately 5 h. L is reduced and no pores are
observed close to the substrate but pores further from the substrate (top of image) can be identified
by the bright Fresnel contrast at their centre. The position of some pores is indicated by the white
arrows. (c) Schematic diagram illustrating how the multilayer may be deforming to give crystal
rotations of approximately 108 (indicated by the tilted squares) yet maintaining untilted layer
interfaces (indicated by the horizontal black lines).
similar conditions but about twice as slowly so the base of the multilayer was
held at 750 8C for approximately 5 h. As a result, the composition amplitude was
reduced at the base of the multilayer (Molina-Aldareguia 2002), less porosity is
evident and L is reduced by 15% (figure 11b). Another TiN/NbN multilayer with
a longer L (50 nm) and grown under different conditions on MgO (111) in two
twin variants had a much higher porosity. Indents in this structure exhibited
reductions in L of over 20% immediately under the indent with no indication of
any material laterally flowing away from the indent (Lloyd et al. 1999).
The residual depth of the indent in figure 9 is approximately 12% of the total
multilayer thickness. Given that the compression observed in the ‘cone’ (which
nearly extends to the base of the multilayer) under the indent is around 10%, it
appears that the vast majority of the surface displacement is accounted for by
densification. The ‘cone’ shape therefore reflects the displacements of the surface
in the indent impression, with the largest displacement accommodated by
densification of the greatest depth of multilayer.
The observation that the crystal rotation can be greater than the rotation of
the layers is at first sight surprising, and illustrates the danger of solely
examining the position of the layer interfaces to infer the deformation pattern.
Proc. R. Soc. A (2005)
Deformation mechanisms of nanoindents
2537
Phenomenologically it could be accounted for by deformation that both rotated
the crystal lattice and also periodically sheared the lattice to maintain interfaces
that are, on average, unrotated, as illustrated schematically in figure 11c.
Densification would still be required to account for the reduction in L, and the
rearrangement of atoms under compressive stress in what is already a highly
defective structure (Lloyd et al. 2001b) may produce a deformation pattern
similar to that shown in figure 11c.
(e ) Metallic glass Mg66Ni20Nd14
In glasses, the Peierls stress is effectively infinite, because no dislocations can
exist in a material with no crystal lattice. Hence, plastic flow must occur by
other mechanisms. Recently, there has been a great increase in interest in the
mechanical properties of metallic glasses owing to the ability to make bulk
glasses, and also the observations of steps in load-depth plots from
nanoindentation thought to be associated with shear band formation (Schuh &
Nieh 2004). Shear bands cannot ordinarily be observed in post-mortem TEM
sections of a deformed region of glass since the dilated structure in the shear band
relaxes back to the bulk material once the stress is released. Only in places where
the structure cannot relax, such as the foil surface, can shear bands be observed
(Donovan & Stobbs 1981). Here, we examined a partially recrystalized glass of
composition Mg66Ni20Nd14 (for more details see Madge 2003) so that the small
crystals (hexagonal Mg2Ni) could be used as markers to indicate the path of
any shear bands. The hardness at loads of 5 and 15 mN was measured to be
5G0.1 GPa.
Figure 12a is a bright field image at a defocus of C20 mm from a 15 mN
indent showing that a low density, porous region has formed approximately
400 nm in diameter at the indent tip. As a result, the profile of the indenter tip
is steeper at the low density region (see the white lines in figure 12a),
presumably because it provided less elastic recovery on removal of the load.
Given that the surface is being compressed, it is surprising to find low-density
material here, but it may have formed from a high concentration of dilated
material in shear bands emanating from the indent tip whose free-volume
relaxes to form voids on removal of the load, as is found for isolated shear bands
at the surface (Donovan & Stobbs 1981). Evidence for shear bands under the
indent can be seen in figure 12b, predominantly found at angles approximately
458 to the surface. The high volume fraction of crystals may have inhibited
shear band propagation so that the steps in the load depth plots observed in the
fully amorphous alloy are not resolved for the partially recrystallized sample
examined here (Greer et al. 2004).
4. Conclusions and future directions
This preliminary survey of deformation under nanoindents has shown the wide
variety of processes that can occur, including phase transformation, twinning,
dislocation flow, lattice rotation, shearing, densification and void formation,
some of which are unique to this localized high pressure loading and not found
under homogeneous deformation of bulk samples. Modelling of the indentation
process must take account of the competing processes available in order reliably
Proc. R. Soc. A (2005)
2538
S. J. Lloyd and others
(a)
indent
tip
low-density
region
100 nm
(b)
indent tip
lowdensity region
shear
bands
100 nm
Figure 12. Indents in partially crystallized metallic glass Mg66Ni20Nd14 with indented loads of
(a) 15 mN and (b) 5 mN. Defocus of (a) is C20 mm. Possible location of some of the shear bands is
indicated by the white arrows in (b).
to relate hardness measurements to macroscopic properties such as the yield
stress. Clearly, dislocation glide is not the only important mechanism that must
be considered.
Despite the wide variety of behaviour observed, there are some similarities
between the different deformation modes. Large crystal rotations were seen in all
the crystals examined except silicon, and these were usually combined with
shear. Even the crystal rearrangement required for the phase transformation in
silicon is partly achieved through shear, so the transformation pressure is
reduced if shear stresses are present (Hu et al. 1986). Similarly, twinning is a
special case of a crystal rotation that results in a plastic shear of the crystal. The
stresses required for these processes can approach the theoretical strength of the
crystal, and thus, may only be supported where the Peierls stress is very high or
dislocation motion is difficult due to a high density of defects. Even in copper
high stresses can be supported. The peak hardness observed for small loads in
single crystals is 2.7 GPa (McElhaney et al. 1998), one-fifteenth of the shear
modulus.
The lattice rotations may be best understood as similar to a martensitic
transformation, i.e. the motion of many atoms moving small distances in a
coordinated way due to the high degree of material constraint under indentation.
It is not clear to what extent the rotations are elastically recoverable,
Proc. R. Soc. A (2005)
Deformation mechanisms of nanoindents
2539
particularly in spinel. Experiments on indents in MgO have shown that asterism
in diffraction patterns due to crystal rotations surrounding the indent impression
was eliminated after a high temperature anneal, implying that these were due to
elastic stresses (Brown et al. 1988). However, here we have examined lattice
rotations under the indent. No rotation was found to the side of the indent
(figure 6a), and previous work has shown that the deformation pattern in MgO
differs markedly to that in spinel (Lloyd et al. 2002).
Metallic glasses behave in some respects more like spinel than a crystalline
metal because there is no dislocation flow so the yield stresses are high, close to
the theoretical strength. (The hardness of the Mg66Ni20Nd14 discussed above was
approximately one-fifth of its shear modulus.) The serrated load-depth plots
obtained in some metallic glasses are consistent with a model of progressive
increments in elastic strain being relieved by yielding through shear band
formation, similar to that discussed above for spinel (Schuh & Nieh 2004). Even
the increase in width between the shear bands with distance from the indent tip
in spinel is analogous to the increasing elastic strain between the displacement
steps with increasing load found in the metallic glasses (Greer et al. 2004). The
shear process itself, however, is different in the metallic glass and spinel. In the
metallic glass the shear band is thought to consist of 10–20 nm wide region of
material dilated by approximately 10% (Donovan & Stobbs 1981), more akin to
the volume changes associated with reconstructive phase transitions, such as
those observed in silicon.
While the TiN/NbN multilayer exhibited shear, lattice rotation and a ‘cone’
under the indent in which the deformation was concentrated like spinel, the
densification process was unique. The multilayer did not show any superhardening effect, but this is not likely to be because the films were porous, since
even multilayers that do show hardening can also contain voids (Shinn et al.
1992). Superhardening is usually explained in terms of different mechanisms that
inhibit dislocation flow in some way, building on the initial theories of Koehler
(1970), who proposed strengthening in laminates with different elastic properties
due to the repulsion of dislocations from layers with the higher modulus.
However, these models still lack quantitative agreement with experiment, and
they also assume that the yield processes occurring in nanoindentation are the
same, albeit more localized, as those occurring under homogeneous loading
(Clegg et al. in pressb). Furthermore, in the case of nitride systems in which some
of the highest hardnesses are achieved it is not clear why additional barriers for
dislocation motion should be so effective when the intrinsic lattice resistance of
the monolithic components is so high, and thus other deformation mechanisms
are likely to be favoured. Recent work has shown that factors such as coherency
strain may actually reduce hardness where the Peierls stress is high (Lloyd et al.
in press). Rather, the hardness changes found in nitride films may be explained
by differences in the compressive stress which may lead to the growth of
structures with differing ability to densify. Clearly, theories of multilayer
hardening need to be evaluated in the light of the actual deformation observed in
particular samples.
The preliminary analyses presented here are limited to one two-dimensional
section, usually including the indent tip. A full characterization should examine
sections at different positions through the indent impression to give a threedimensional picture, and also at different loads and relative orientations of the
Proc. R. Soc. A (2005)
2540
S. J. Lloyd and others
indent (with threefold symmetry) and the indented surface (which in the
examples here had fourfold symmetry). In addition, the lattice rotations must
be measured in three dimensions and not just about an axis perpendicular to
the plane of the foil as here. Work is in progress to do this in copper by
determining the local crystal orientation from Kikuchi patterns (McLaughlin
et al. 2004). Greater resolution is possible than in the case of the recently
developed X-ray analysis of the subsurface rotations around indents in copper
(Yang et al. 2004). In other materials where more of the indent deformation is
stored elastically, bending of the foil must be taken into account when measuring
the local plastic misorientations in three dimensions. In addition, measurement
of dislocation densities in indented metals will be necessary to test models
involving GND, but to do this may require the surface damage introduced by the
FIB to be removed (Langford & Petford-Long 2001b).
The ability to examine such localized deformation in the TEM in a controlled
way is a very recent development and is thus a new technique to assist our
understanding of nanoindentation. In particular, a map of the lattice rotations
under indents may provide a ‘fingerprint’ of deformation that can be readily
compared with simulations of the deformation process. Ultimately, it should be
possible to create a deformation ‘map’ on the basis of the material bonding and
structure, which predicts the different types of deformation behaviour most
probable under varying indentation conditions, and then to design material
structures to display a specified hardness.
We are grateful for financial support from The Royal Society, the Basque government, the
Department of Trade and Industry, the Natural Science and Engineering Research Council of
Canada, the Overseas Research Students Award Scheme and the Cambridge Commonwealth
Trust. We also thank Dr S. Madge for provision of the metallic glass sample, H. M. Lloyd for the
micrograph shown in figure 11b and Dr E. Yoffe for useful discussions.
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