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Geometric limits of coherent III-V core/shell nanowires
O. Salehzadeh, K. L. Kavanagh, and S. P. Watkins
Citation: J. Appl. Phys. 114, 054301 (2013); doi: 10.1063/1.4816460
View online: http://dx.doi.org/10.1063/1.4816460
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Geometric limits of coherent III-V core/shell nanowires
O. Salehzadeh, K. L. Kavanagh, and S. P. Watkins
Department of Physics, Simon Fraser University, Burnaby, British Columbia, V5A 1S6, Canada
(Received 26 May 2013; accepted 8 July 2013; published online 1 August 2013)
We demonstrate the application of a simple equilibrium model based on elasticity theory to estimate
the geometric limits of dislocation-free core/shell nanowires (NWs). According to these calculations,
in a coherent core/shell structure, tangential strain is the dominant component in the shell region and
it decreases quickly away from the heterointerface, while axial strain is the dominant component in
the core and is independent of the radial position. These strain distributions energetically favour the
initial relief of axial strain in agreement with the experimental appearance of only edge dislocations
with line directions perpendicular to the NW growth axis at the core/shell interfaces. Such
dislocations were observed for wurtzite InAs/InP and zincblende GaAs/GaP core/shell NWs with
dimensions above the coherency limits predicted by the model. Good agreement of the model was
also found for experimental results previously reported for GaAs/InAs and GaAs/GaSb core/shell
NWs. V
C2013 AIP Publishing LLC.[http://dx.doi.org/10.1063/1.4816460]
INTRODUCTION
The nanowire (NW) geometry is expected to facilitate
the growth of dislocation free axial and radial heterostruc-
tures with dimensions above the critical thicknesses known
for thin films. This characteristic of the semiconductor
NWs has opened a new window to design and fabricate
core/shell NW based-devices.
1,2
The rational design of
these devices requires the knowledge of the coherency lim-
its in core/shell NWs. Otherwise, dislocation formation
will degrade the device performance.
3
Several groups have
attempted to determine the coherency limit and/or strain
distribution in core/shell NWs using elasticity theory
via finite-element analysis,
4
variational methods,
5
or an an-
alytical approach.
6,7
Using the variational approach,
Raychaudhuri and Yu
5
determined the total strain energy
of a core/shell structure numerically with the assumption
that the strain components are position-independent in the
core and shell regions. Then, they estimated the critical
dimensions by numerically determining the shell thickness,
for a given core radius, for which it became energetically
favorable to include a dislocation. Haapamaki et al. deter-
mined the total strain energy analytically considering the
strain components to be position-dependent.
6
Then they
obtained the critical dimension of InAs/Al
x
In
1x
As core/
shell NWs by determining the geometric limits at which
the total strain energy exceeded the dislocation energy.
Both of these models predicted an increase in the critical
shell thickness with decreasing core radius and the pres-
ence of a critical core radius under which the critical shell
thickness tended to infinity.
5–7
However, comparison of
predicted results with experimental data for core/shell
NWs is rather limited to a specific material system and a
limited geometric range.
6,8
In this work, we have grown zincblende (ZB) and wurt-
zite (WZ) III-V core/shell NWs and calculated the expected
strain distribution and the geometric limits of the coherent
radial heterostructure NWs. We considered the strain compo-
nents to be position-dependent, similar to Ref. 6, and
determined the total strain energy and strain components
numerically, similar to Ref. 5. Our numerical results are in
agreement with our experimental results found for wurtzite
InAs/InP and zincblende GaAs/GaP core/shell NWs and
results reported in the literature.
EXPERIMENT
The NWs were grown via the vapor-liquid-solid (VLS)
mechanism. A 0.2 nm Au layer was deposited on a (111)B
Si-doped GaAs substrate by thermal evaporation. The Au-
coated wafer was then annealed for 10 min (at 460 C under
H
2
(3 standard liters per minute) and tertiarybutylarsine
(TBAs) overpressure) in a vertical metalorganic vapor phase
epitaxy (MOVPE) reactor operating at a pressure of 50 Torr.
This resulted in Au nanoparticles with sizes in the range of
15 nm–110 nm. Trimethylindium (TMIn) (flow rate
9.9 lmol/min) and TBAs (flow rate 66 lmol/min) were used
as the group III and V precursors to grow the InAs core NWs
for 400 s. The growth of the InP shell was achieved by
switching off the TBAs and switching on the tertiarybutyl-
phosphine (TBP) (flow rate 960 lmol/min) for 250 s. The
V/III ratio was 6.6 and 97 for the growth of the InAs core
and InP shell, respectively. The sample was then cooled
under TBP/H
2
. Both InAs core and InP shell materials were
grown at 460 C. To grow GaAs/GaP core/shell NWs, the
GaAs core was grown using trimethylgallium (TMGa, flow
rate of 21.4 lmol/min) and TBAs (flow rate 164 lmol/min).
The growth of the GaP shell was achieved by switching off
the TMGa and TBAs and switching on the TEGa (flow rate
15.1 lmol/min) and TBP (flow rate 960 lmol/min) for
200–400 s. The V/III ratio was 7.6 and 63.7 for the growth of
the GaAs core and GaP shell, respectively. The sample was
then cooled under TBP/H
2
. Both the GaAs core and GaP
shell materials were grown at 410 C.
Field-emission scanning electron microscopy (SEM)
and scanning transmission electron microscopy (STEM)
(operated at 200 kV) were used for structural and energy dis-
persive spectroscopy (EDS) analyses.
0021-8979/2013/114(5)/054301/8/$30.00 V
C2013 AIP Publishing LLC114, 054301-1
JOURNAL OF APPLIED PHYSICS 114, 054301 (2013)
MODEL
Wurtzite core/shell NWs
In this model, we ignored the faceting of the NW and
considered two coaxial cylinders with a core radius of r
c
and
a shell thickness of tas shown in Fig. 1. A cylindrical coordi-
nate system was defined at the core/shell interface (defined
by unit vectors e
r
,e
h
,e
z
). The magnitude and distribution of
the strain components (e
i
k
(r), where istands for either core
(c) or shell (s) and k¼r, h,z) in the core and shell regions
are mainly determined by the constraints that (1) coherency
has to be maintained at the core/shell interface and (2) the
total strain energy in the system has to be a minimum. The
interfacial strain components at the hetero-interface are
defined as follows:
ec
zðr¼rcÞ¼fc
z¼azac
z
ac
z
;(1a)
ec
hðr¼rcÞ¼fc
h¼ahac
h
ac
h
;(1b)
es
zðr¼rcÞ¼fs
z¼azas
z
as
z
;(1c)
es
hðr¼rcÞ¼fs
h¼ahas
h
as
h
;(1d)
where a
k
and a
i
k
are, respectively, the strained and relaxed
lattice constants in the k¼z, hdirections and i¼cor sfor
core or shell, respectively.
In general, strain components can be determined from
the components of the displacement vector (~
uðr;zÞ¼urðrÞer
þuzðzÞez) as follows:
ei
r¼@ui
rðrÞ
@r;(2a)
ei
h¼ui
rðrÞ
r;(2b)
ei
z¼@ui
zðzÞ
@z;(2c)
where u
r
and u
z
are the components of the displacement vec-
tor along the e
r
and e
z
directions, respectively. We should
note that the NW symmetry in the azimuthal direction results
in u
h
(h)¼0(u
h
(0)¼u
h
(2p)¼0). The displacement compo-
nents can be determined using the equilibrium equations of
elasticity
9
@ri
r
@rþri
rri
h
r¼0;(3a)
@ri
z
@z¼0;(3b)
where r
i
k
is the stress along the kth direction and normal to a
plane whose outward normal is along the kth direction
(i¼c,s stands for core and shell). The values of r
i
k
are
related to the stiffness constants (c
nm
)by
ri
r¼c11ei
rþc12ei
hþc13ei
z;(4a)
ri
h¼c12ei
rþc11ei
hþc13ei
z;(4b)
ri
z¼c13ei
rþc13ei
hþc33ei
z:(4c)
Substituting Eqs. (4a)–(4c) in Eqs. (3a) and (3b) leads to the
following partial differential equations:
@2ui
r
@r2þ@ui
r
r@rui
r
r¼0;(5a)
@2ui
z
@z2¼0:(5b)
Equations (5a) and (5b) have the following general
solutions:
ui
r¼airþbi
r;(6a)
ui
z¼cizþui:(6b)
The coefficients in Eqs. (6a) and (6b) can be determined by
imposing the following boundary conditions:
1. u
c
r
is finite as rapproaches zero. Therefore, b
c
¼0.
2. u
i
z
(z ¼0) ¼0 which results in u
i
¼0.
3. ec
zðr¼rcÞ¼fc
z.
4. ec
hðr¼rcÞ¼fc
h.
5. Coherency at the core/shell interface results in
FIG. 1. Schematic of the core/shell NW geometry with a core radius of r
c
,
shell thickness of t, and length of L. The dashed loop represents an edge dis-
location loop with Burger’s vector along the NW growth direction.
054301-2 Salehzadeh, Kavanagh, and Watkins J. Appl. Phys. 114, 054301 (2013)
es
zðr¼rcÞ¼ac
z
as
zðfc
zþ1Þ1;(7a)
es
hðr¼rcÞ¼ac
h
as
hðfc
hþ1Þ1:(7b)
6. Stress normal to the free surface is zero
(rs
rðrcþtÞ¼0).
Using these boundary conditions, we determined the coeffi-
cients in Eqs. (6a) and (6b) as a function of c
i
nm
,f
c
z
,f
ch
,r
c
,
and t. For a given r
c
and t, the values of f
c
z
,f
ch
were deter-
mined numerically by minimizing the total strain energy in
the core/shell structure which is given by
Ustrainðrc;t;fc
z;fc
hÞ
L¼1
2ðrc
0ð2p
0
rc
kec
krdrdh
þ1
2ðrcþt
rcð2p
0
rs
kes
krdrdh;(8)
where Lis the length of the NW and sums over the repeated
indices were assumed.
Partial relaxation of the heterostructure can happen by
insertion of a dislocation at the interface. The actual misfit
dislocation formation mechanism and configuration for core-
shell NWs may be complicated and, in general, depends on
the NW dimensions and the misfit between core and shell
material.
10,11
Here, we only consider the case of a pure edge
dislocation loop of radius r
c
with a Burger’s vector of magni-
tude balong the NW growth direction (Fig. 1). Similar to
other reports, we found only dislocations with such an edge
component experimentally and their formation mechanism
whether by glide or climb processes
8,10–12
remain to be con-
firmed. In wurtzite semiconductors, a dislocation loop
around the core has energy equal to
13
Udis
L¼1
2Kb2rcln 32rc
b
2
;(9)
where
K¼ð
c13 þc13Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
c44ð
c13 c13Þ
c11ð
c13 þc13 þ2c44Þ
s;
c13 ¼ffiffiffiffiffiffiffiffiffiffiffiffi
c11c33
p:
The critical dimensions can be estimated for a given core radius
by numerically determining the critical shell thickness at which
U
strain
(Eq. (8)) exceeds U
dis
(Eq. (9)), under the assumption
that there is no energy barrier for dislocation nucleation.
Zincblende core/shell NWs
The above approach could be modified for zincblende
core/shell NWs. Equations (4a)–(4c) change to
ri
r¼c11ei
rþc12ei
hþc12ei
z;(10a)
ri
h¼c12ei
rþc11ei
hþc12ei
z;(10b)
ri
z¼c12ei
rþc12ei
hþc11ei
z:(10c)
Also, the dislocation loop energy changes to
14
Udis
L¼b2l
2ð1Þrcln 32rc
b
2
;(11)
where land are the shear modulus and Poisson’s ratio,
respectively. The values of l,, and c
ij
are summarized in
Table I. To determine the stiffness constants for the wurtzite
structure, Martin’s transformations were employed.
15
RESULTS
Figure 2shows the strain components as a function of
radial distance from the center of a wurtzite InAs/InP core/
shell NW (solid lines) with core radius of 20 nm and shell
thickness of 10 nm. This graph was generated numerically
by minimizing the total strain energy in Eq. (8) and using
Eqs. (2a)–(2c) and (7a) and (7b). The InAs core, which has a
larger lattice constant than the InP shell, is under compres-
sive strain (negative value) in all directions and the strain
components are position independent. Also, the radial and
tangential strains are equal in the core regions, similar to the
case of thin film heterostructures, as expected due to symme-
try considerations. The shell region is under tensile strain in
the e
z
and e
h
directions, while it is compressed in the radial
direction. The radial compression in the shell is a direct
result of the 6th boundary condition listed above. In the shell
region, the strain components in the e
h
and e
r
directions are
position dependent, while the strain is position independent
in the e
z
direction. In the case of a wurtzite InP/InAs core/
shell NW (dashed lines), the signs of the strain components
are opposite to the ones in the InAs/InP core/shell NW. The
strain accommodation in the InP core (InAs shell) of the InP/
InAs NW is slightly lower (higher) than the strain in the
InAs core (InP shell) of the InAs/InP NW. Qualitatively,
similar results were obtained for zincblende NWs. These
results are in qualitative agreement with the results reported
in Ref. 7for Si/Ge NWs; however, there is a clear discrep-
ancy between our results and those of Ref. 6where it was
reported that the core with the larger lattice constant is under
tensile strain, and the shell with the smaller lattice constant
is under compressive strain, in the e
z
and e
h
directions.
Figure 3(a) shows the calculated interfacial axial and
tangential strain components (at r¼r
c
) of an InAs core with
a radius of 20 nm and InP shell with thickness varying in the
TABLE I. Summary of reported values of the shear modulus, l; the
Poisson’s ratio, ; and the stiffness constants, c
ij
for selected compound
semiconductors with ZB structure and calculated c
ij
for WZ structure.
C
11
(GPa) C
12
(GPa) C
44
(GPa) l(GPa)
GaAs (ZB)
16
119 53.8 59.5 32.8 0.31
GaP (ZB)
16
140.5 62.1 70.3 39.2 0.31
GaSb (ZB)
17
88.3 40.2 43.2 24.0 0.31
InAs (ZB)
16
83.3 45.3 39.6 19.0 0.35
InP (ZB)
16
101.1 56.1 45.6 22.5 0.36
InSb (ZB)
18
66.0 38.0 30.0 15.1 0.35
C
11
C
12
C
13
C
33
C
44
GaAs (WZ) 142.0 48.7 35.9 154.9 38.4
InAs (WZ) 100.3 42.1 31.6 110.8 23.0
InP (WZ) 120.3 52.3 40.7 131.9 27.1
054301-3 Salehzadeh, Kavanagh, and Watkins J. Appl. Phys. 114, 054301 (2013)
range of 1 nm–50 nm. With increasing shell thickness, the
interfacial axial strain in the InP shell decreases continuously
from 0.029 to 0.002, while the interfacial axial strain in the
InAs core increases from 0.004 to 0.029. A similar trend
with much smaller gradient was observed in the case of the tan-
gential strains in the core/shell interface. However, it is clear
that the interfacial tangential strain in the shell remains sub-
stantially larger than that in the core region. These numerical
results indicate that, for thick shells, the interfacial tangential
strain concentrates in the shell region, while the axial strain
concentrates in the core. The tangential strain in the shell is
position-dependent and drops quickly away from the core/shell
interface, while it is position-independent in the core.
Therefore, energetically, it is more favourable for the shell to
accommodate the tangential strain to minimize the total strain
energy. On the other hand, the axial strain is position-
independent in both core and shell regions. For thin shells,
axial strain can be accommodated by the shell, while for
thicker shells, axial strain must be accommodated by the core
in order to lower the total strain energy. This partitioning of the
strain fields in core/shell NWs should therefore result in critical
shell thicknesses above the known values for thin films.
The total strain energy of the core/shell NW and the dis-
location energy per unit length are plotted as a function of
shell thickness in Fig. 3(b). Strain energy increases with
increasing shell thickness and exceeds the dislocation energy
for shell thicknesses larger than 45 nm. Therefore, the critical
shell thickness for a NW with radius of 20 nm is 45 nm. The
saturation of strain energy for thick shells is due to the fact
that tangential strain in the shell drops away from the inter-
face (see Fig. 2) and the axial strain in the core reaches its
maximum limit (see Fig. 3). In the core region, axial strain is
the dominant component and therefore the formation of dis-
location loops with line direction perpendicular to the NW
axis can most effectively relieve the axial strain, significantly
lowering the total strain energy.
Figure 4(a) shows the change in the axial and tangential
strain components at the interface of a core/shell NW with a
fixed shell thickness of 14 nm and varying core radius in the
range of 5 nm–85 nm. The axial strain in the core decreases
with increasing core radius, while it increases in the shell
region. Weaker effects were observed for tangential strains.
These results confirm that a thinner core can accommodate a
higher degree of strain compared with a thicker one and
therefore the critical shell thickness should be larger for a
thinner core. The strain and dislocation energies are plotted
in Fig. 4(b) as a function of shell thickness which intersect at
r
c
¼63 nm, meaning that the critical shell thickness of an
InAs/InP core/shell NW with radius of 63 nm is 14 nm.
FIG. 2. Plot of the strain components
as a function of radial distance from
the center of the core. Solid lines and
dashed lines correspond to wurtzite
InAs/InP and InP/InAs core/shell
NWs, respectively, with core radius of
20 nm and shell thickness of 10 nm.
The vertical dotted line represents the
core/shell interface.
FIG. 3. (a) Plot of the change in the interfacial strain components (at r¼r
c
)
and (b) elastic strain and dislocation energies per unit length as a function of
shell thickness of wurtzite InAs/InP core/shell NWs with a fixed core radius
of 20 nm.
054301-4 Salehzadeh, Kavanagh, and Watkins J. Appl. Phys. 114, 054301 (2013)
Figure 5shows the dependence of critical shell thickness
on the core radius of wurtzite InAs/InP and zincblende
GaAs/GaP core/shell NWs. Consistent with previous
reports,
5–7
the critical shell thickness tends to infinity for
NWs with core radii smaller than a critical core radius. For
NWs with core radii larger than the critical core radius, the
shell thickness must be below a certain value to maintain
coherency at the hetero-interface. The square and circle data
points are experimental results which will be discussed later.
For NWs with core radii larger than 100 nm, the critical shell
thickness is independent of the core radius. The plot of the
calculated critical core radius (r
c0
) and critical shell thick-
ness for zincblende core/shell NWs with a particular core ra-
dius of 100 nm (t
c0
) as function of lattice mismatch between
core and shell materials (f
0
) is shown in Fig. 6.r
c0
is 21 nm
for zincblende InAs/InP core/shell NWs (mismatch of 3.2%)
and drops to 2.9 nm for zincblende InAs/GaAs core/shell
NWs (mismatch of 7.2%). Similarly, t
c0
drops from 15.1 nm
to 2.4 nm by increasing the mismatch from 3.2% to 7.2%.
The graphs in Fig. 6are fairly linear indicating the power
law dependence of r
c0
and t
c0
on f
0
. The fits to the points
were obtained by r
c0
¼310 f
02.5
and t
c0
¼130 f
02.0
.We
should note that r
c0
and t
c0
of a wurtzite InAs/InP core/shell
NWs are 18.2 nm and 13.4 nm, respectively, which are
smaller than the values obtained for zincblende (r
c0
¼21 nm
and t
c0
¼15.2 nm) InAs/InP core/shell NWs. Similarly, in
the case of wurtzite InAs/GaAs NWs, r
c0
and t
c0
are 2.3 nm
and 2.2 nm, respectively, which are smaller than the corre-
sponding values for zincblende structures.
Experimental results: Comparison with model
Figure 7(a) shows a bright field (BF) TEM image of an
InAs/InP core/shell NW with a core radius of 17 nm and
shell thickness of 20 nm. Energy dispersive X-ray spectros-
copy analyses were carried out on these NWs and the shell
formation and its thickness were verified. The shell thickness
is uniform along the NW growth direction, except for the
tapered neck region which consists of InP formed during the
shell growth via the VLS mechanism. Dark field (DF) TEM
images taken using (0002) and (2110) diffracted spots are
shown in Figs. 7(b) and 7(c), respectively. These images
clearly indicate that the NW is free of stacking faults and dis-
locations. The corresponding selected area diffraction (SAD)
pattern, along the [0110] direction, from the middle part of
the core/shell structure is shown in Fig. 7(d). The observed
single set of spots indicate that the structure is coherent. A
high resolution (HR) TEM image of the InAs/InP interface is
shown in Fig. 7(e) confirming that the structure is free of dis-
locations. The wurtzite structure of this NW can be inferred
from the ABAB, stacking sequence along the [0001] growth
direction. EDS analyses were performed (not shown here)
FIG. 4. (a) Plot of the change in the interfacial strain components (at r¼r
c
)
and (b) elastic strain and dislocation energies per unit length as a function of
shell thickness of InAs/InP core/shell NWs with fixed shell thickness of
14 nm.
FIG. 5. Plot of the calculated critical shell thickness as a function of core ra-
dius for wurtzite InAs/InP (solid line) and zincblende GaAs/GaP (dashed
line) core/shell NWs. The open and solid data points are the experimental
results for NWs with and without dislocations detected, respectively. The
squares are for InAs/InP core/shell NWs, while the circles are for GaAs/GaP
NWs. The coherency is maintained for thicknesses below the curves.
FIG. 6. Plot of the critical core radius as a function of lattice mismatch
between core and shell materials with zincblende structure (left). The critical
shell thicknesses of core/shell NWs with a particular core radius of 100 nm
are plotted on the right.
054301-5 Salehzadeh, Kavanagh, and Watkins J. Appl. Phys. 114, 054301 (2013)
and the formation of an InP shell around InAs core NWs was
confirmed.
Figures 8(a) and 8(b) show BF and (0002) DF TEM
images of an InAs/InP core/shell NW with a core radius of
30 nm and shell thickness of 25 nm. The shell thickness of
25 nm is larger than the predicted critical shell thickness of
19 nm for a NW with this core radius (see Fig. 5). The corre-
sponding SAD pattern, indicating a (0110) sample orienta-
tion, is shown in Fig. 8(c). In all TEM images shown in Figs.
7and 8, the incident electron beam is perpendicular to the
NW growth axis. The observed contrast in Figs. 8(a) and
8(b) corresponds to the presence of edge dislocations at the
core/shell interface perpendicular to the NW growth direc-
tion. Another example of a dislocation is shown in Fig. 8(d)
for a NW with core radius of 37 nm and shell thickness of
22 nm and the corresponding SAD pattern is shown in Fig.
8(e). A HRTEM image of the region indicated by a white
square in Fig. 8(d) is shown in Fig. 8(f) indicating the inser-
tion of an extra plane in the InP shell. The dislocations have
Burger’s vectors along the NW growth direction. The con-
trast observed between misfit dislocation pairs in these NWs
may be related to a complete dislocation loop or other defect,
such as a stacking fault from partial dislocations. The
observed contrast in Figs. 8(a),8(b) and 8(d) should not be
confused with pre-existing basal plane {0001} stacking
faults. In the case of a core/shell NW with stacking faults,
the stacking faults propagate from the core into the shell. An
example of a core/shell NW with stacking faults (core radius
of 15 nm and shell thickness of 22 nm) is shown in Fig. 8(g).
The strong parallel contrast is due to stacking faults that do
not stop at the core/shell interface but propagate into the
shell to the outer surface.
The observed dislocations in Fig. 8have an average
spacing (D
e
)of4067 nm. These edge dislocations relax the
axial strain with respect to the lattice mismatch strain, 0.032,
by a percentage given by 1
0:032
b
De, where bis the size of the
Burger’s vector equal to a/2, or 0.35 nm, equivalent to
(28 64)%. This magnitude of axial strain relaxation, 0.009,
is too small to be detectable in SAD patterns. No evidence of
relaxation in other directions was detected either from
images or SAD patterns.
The solid and open square data points plotted in Fig. 5
correspond to NWs free of dislocations and with disloca-
tions, respectively, as a function of their shell thickness and
core radius. These experimental results are consistent with
our numerical predictions for the critical geometries (solid
and dotted lines). Our TEM results indicate that an InP shell
with a thickness of 35 nm could be grown coherently on an
InAs NW with a radius of 10 nm. Recently reported coherent
InAs/InP core/shell NWs with core radius of 20 nm and shell
thickness of 20 nm are consistent with our numerical
predictions.
19
Figures 9(a) and 9(b) show a BF TEM image and corre-
sponding SAD pattern (near a h112isample orientation) of a
zincblende GaAs/GaP core/shell NW with a core radius of
25 nm and a shell thickness of 13 nm. The shell thickness of
FIG. 7. (a) Bright field and dark field TEM images of an InAs/InP core/shell
NW taken by (b) (0002) and (c) (2110) diffracted spots, (d) selected area dif-
fraction pattern of the middle of the core/shell NW, and (e) a high resolution
TEM image of the InAs/InP interface. The arrow in (e) shows the InAs/InP
interface.
FIG. 8. Examples of TEM investiga-
tions of strain relaxed wurtzite InAs/
InP core/shell NWs (a) BF and (b)
(0002) DF TEM images for a NW with
a core radius of 30 nm and shell thick-
ness of 25 nm with a corresponding
SAD pattern in (c). (d) (2110) DF
TEM image of another NW with a core
radius of 37 nm and shell thickness of
22 nm with a corresponding SAD pat-
tern in (e) and HRTEM in (f). (g)
(0002) DF TEM image of a NW with a
core radius of 15 nm and a shell thick-
ness of 22 nm. The observed contrasts
in (a), (b), and (d) are perhaps due to
the formation of loop dislocations,
while the contrast in (g) is due to the
formation of stacking faults.
054301-6 Salehzadeh, Kavanagh, and Watkins J. Appl. Phys. 114, 054301 (2013)
13 nm is larger than the predicted critical shell thickness
(10 nm) for a NW with this core radius (see Fig. 5). The
HRTEM image shown in Fig. 9(c) clearly indicates the inser-
tion of an extra plane in the GaP shell. The observed disloca-
tions have Burger’s vectors along the NW growth direction
and line directions perpendicular to the [111] growth direc-
tion. The average spacing of the observed edge dislocations
is 45 64 nm giving an average relaxation of 23 62%.
The solid and open circle data points plotted in Fig. 5
correspond to NWs free of dislocations and with disloca-
tions, respectively, as a function of their shell thickness and
core radius. According to the TEM analysis, the GaAs/GaP
core/shell NWs with core radii in the range of 25 nm–45 nm
and shell thicknesses larger than 10 nm were relaxed. The
thinner shells of 5–6 nm were found to grow coherently on
GaAs NWs with radii of 22–25 nm. These experimental
results are consistent with our numerical predictions. The
previously reported GaAs/GaP core/shell NWs with core ra-
dius of 25 nm and shell thickness of 25 nm
20
should have
relaxed according to our numerical predictions. Even though
detailed TEM investigations were not carried out on the
reported GaAs/GaP core/shell NWs, the presence of Moir
e
fringes in their BF TEM image
20
clearly indicates that the
structure was partially relaxed.
In a previous work, we found zincblende GaAs/GaSb
core/shell NWs with core radii larger than 10 nm and shell
thicknesses larger than 4 nm to relax via the formation of
periodic edge dislocations (Burger’s vectors along [111]
direction), while the shells with thicknesses below 2 nm
grew coherently on the GaAs core.
21
These results, summar-
ized in Fig. 10, are consistent with our model. In another
work, wurtzite and zincblende InAs/GaAs core/shell NWs
with core radii larger than 10 nm and shell thicknesses larger
than 2.5 nm were reported to relax
3
in agreement with our
numerical calculations (data also shown in Fig. 10).
Finally, the level of agreement between theory and
experiment is remarkable considering that we have assumed
no energy barrier for dislocation nucleation and motion via
glide or climb processes. In addition, we have neglected any
effect of facets on the strain distribution. The effects of these
factors appear to be too small to be detectable. However,
they may play an important role in the dislocation formation
mechanism.
22
We should note that, dislocation pairs or loops
were not observed in zincblende GaAs/GaP and GaAs/GaSb
core/shell NWs
21
where dislocation formation by glide proc-
esses on oblique {111} planes have been reported in Si/Ge
NWs with dimensions above the coherency limits.
8
In conclusion, a model to estimate the critical dimen-
sions of core/shell NWs based on elasticity theory was pre-
sented. The numerical calculations were carried out for
various III-V core/shell NWs. The theory was found to be
consistent with experimental results previously reported for
GaAs/GaSb and InAs/GaAs core/shell NWs and to the TEM
results found in this work for wurtzite InAs/InP and zinc-
blende GaAs/GaP core/shell NWs. All core/shell NWs stud-
ied here with dimensions above the coherency limits,
predicted by the model, relax axially via the formation of
FIG. 9. (a) BF TEM image and (b) cor-
responding SAD pattern of a GaAs/
GaP core/shell NW with a core radius
of 25 nm and shell thickness of 13 nm.
(c) HRTEM image of the core/shell
interface, indicating the presence of an
edge dislocation inside the white
circle.
FIG. 10. Plot of the calculated critical shell thickness as a function of core
radius for zincblende GaAs/GaSb (dashed-dotted curve), zincblende InAs/
GaAs (dotted curve), and wurtzite InAs/GaAs (dashed curve) core/shell
NWs. The open and solid data points are the experimental results for NWs
with and without dislocations detected, respectively. The experimental
results for InAs/GaAs core/shell NWs are from Ref. 3.
054301-7 Salehzadeh, Kavanagh, and Watkins J. Appl. Phys. 114, 054301 (2013)
edge dislocations at the core/shell interface with line direc-
tion perpendicular to the NW growth direction. Numerical
results indicate that a uniform axial strain is the dominant
component in the core region, while tangential strain that
decreases quickly away from the heterointerface is the domi-
nant component in the shell. This distribution of strain in a
cylindrical geometry favours relaxation via edge dislocations
(line directions perpendicular to the growth direction) for
NWs of large core radius, relieving axial strain first.
1
X. Jiang, Q. Xiong, S. Nam, F. Qian, Y. Li, and C. M. Lieber, Nano Lett.
7, 3214 (2007).
2
S. Manna, S. Das, S. P. Mondal, R. Singha, and S. K. Ray, J. Phys. Chem.
C116, 7126 (2012).
3
K. L. Kavanagh, J. Salfi, I. Savelyev, M. Blumin, and H. E. Ruda, Appl.
Phys. Lett. 98, 152103 (2011).
4
J. Gronqvist, N. Sondergaard, F. Boxberg, T. Guhr, S. Aberg, and H. Q.
Xu, J. Appl. Phys. 106, 053508 (2009).
5
S. Raychaudhuri and E. T. Yu, J. Vac. Sci. Technol. B 24, 2053 (2006).
6
C. H. Haapamaki, J. Baugh, and R. R. LaPierre, J. Appl. Phys. 112,
124305 (2012).
7
T. E. Trammell, X. Zhang, Y. Li, L.-Q. Chen, and E. Dickey, J. Cryst.
Growth 310, 3084 (2008).
8
S. A. Dayeh, W. Tang, F. Boioli, K. L. Kavanagh, H. Zheng, J. Wang, N.
H. Mack, G. Swadener, J. Y. Huang, L. Miglio, K.-N. Tu, and T. Picraux,
Nano Lett. 13, 1869 (2013).
9
J. P. Hirth and J. Lothe, Theory of Dislocations, 2nd ed. (Wiley, New
York), p. 31.
10
M. Y. Gutkin, I. A. Ovid’ko, and A. G. Sheinerman J. Phys. Condens.
Matter 15, 3539 (2003).
11
I. A. Ovid’ko and A. G. Sheinerman, Adv. Phys. 55, 627 (2006).
12
I. A. Goldthrope, A. F. Marshall, and P. C. McIntyre, Nano Lett. 8, 4081
(2008).
13
Y. Chou and J. Eshelby, J. Mech. Phys. Solids 10, 27 (1962).
14
P. M. Anderson and J. R. Rice, J. Mech. Phys. Solids 35, 743 (1987).
15
R. M. Martin, Phys. Rev. B 6, 4546 (1972).
16
S. C. Jain, M. Willander, and H. Maes, Semicond. Sci. Technol. 11, 641
(1996).
17
W. F. Boyle and R. J. Sladek, Phys. Rev. B 11, 2933 (1975).
18
L. H. DeVaux and F. A. Pizzarello, Phys. Rev. 102, 85 (1956).
19
S. G. Ghalamestani, M. Heurlin, L.-E. Wernersson, S. Lehmann, and K. A.
Dick, Nanotechnology 23, 285601 (2012).
20
M. Montazeri, M. Fickenscher, L. M. Smith, H. E. Jackson, J. M.
Yarrison-Rice, J. H. Kang, Q. Gao, H. H. Tan, C. Jagadish, Y. Guo, J.
Zou, M. Pistol, and C. E. Pryor, Nano Lett. 10, 880 (2010).
21
O. Salehzadeh, K. L. Kavanagh, and S. P. Watkins, J. Appl. Phys. 113,
134309 (2013).
22
G. Perillart-Merceroz, R. Thierry, P.-H. Jouneau, P. Ferret, and G.
Feuillet, Appl. Phys. Lett. 100, 173102 (2012).
054301-8 Salehzadeh, Kavanagh, and Watkins J. Appl. Phys. 114, 054301 (2013)
