# Thermal stress modelling of diamond on GaN/III-Nitride membranes

Jerome A. Cuenca<EMAIL_ADDRESS>Matthew D. Smith  Daniel E. Field
Fabien C-P. Massabuau  Soumen Mandal  James Pomeroy  David J. Wallis  Rachel
A. Oliver  Iain Thayne  Martin Kuball  Oliver A. Williams  School of Physics
and Astronomy, Cardiff University Cardiff, Wales, CF24 3AA, UK  School of
Engineering, University of Glasgow, G12 8LT, UK  Centre for Device
Thermography and Reliability, University of Bristol, BS8 1TL, UK  Cambridge
Centre for Gallium Nitride, Department of Materials Science and Metallurgy,
Cambridge, CB3 0FS, UK  Department of Physics, SUPA, University of
Strathclyde, Glasgow, G4 0NG, UK  School of Engineering, Cardiff University
Cardiff, Wales, CF24 3AA, UK

###### Abstract

Diamond heat-spreaders for gallium nitride (GaN) devices currently depend upon
a robust wafer bonding process. Bonding-free membrane methods demonstrate
potential, however, chemical vapour deposition (CVD) of diamond directly onto
a III-nitride (III-N) heterostructure membrane induces significant thermal
stresses. In this work, these thermal stresses are investigated using an
analytical approach, a numerical model and experimental validation. The
thermal stresses are caused by the mismatch in the coefficient of thermal
expansion (CTE) between the GaN/III-N stack, silicon (Si) and the diamond from
room temperature to CVD growth temperatures. Simplified analytical wafer bow
models underestimate the membrane bow for small sizes while numerical models
replicate the stresses and bows with increased accuracy using temperature
gradients. The largest tensile stress measured using Raman spectroscopy at
room temperature was approximately 1.0 $\pm 0.2$ GPa while surface
profilometry shows membrane bows as large as
$58\text{\,}\mathrm{\SIUnitSymbolMicro m}$. This large bow is caused by
additional stresses from the Si frame in the initial heating phase which are
held in place by the diamond and highlights challenges for any device
fabrication using contact lithography. However, the bow can be reduced if the
membrane is pre-stressed to become flat at CVD temperatures. In this way, a
sufficient platform to grow diamond on GaN/III-N structures without wafer
bonding can be realised.

###### keywords:

cvd diamond, gallium nitride, membranes, thermal stress, finite element
modelling

††journal: Journal of LaTeX Templates

##  1 Introduction

Increasing demand for high power microwave devices has driven research into
gallium nitride (GaN), in particular, high electron mobility transistors
(HEMTs)[1]. Current device technologies either have impressive high power
handling (e.g. Si) or high frequency capability (e.g. GaAs and InP), though
not both. GaN is a possible candidate although, despite significant progress,
its power handling capability is still limited by self-heating[1]. This is due
to the limited thermal conductivity of GaN with values between 160 and 210
W/mK[2]. Additionally, some substrates for GaN also have a low thermal
conductivity such as sapphire (Al2O3) and silicon (Si) at around 30 and 150
W/mK, respectively[2, 3, 4]. This can be overcome with sufficient heat
extraction by integration with substrates of much higher thermal conductivity
such as silicon carbide (SiC) at 400 W/mK, which is commonly used for
electronic applications. Diamond is another candidate which in its single
crystal form is capable of achieving thermal conductivities in excess of 2,000
W/mK[5, 6]. Several studies have shown reduced operating temperatures of GaN
devices when using integrated diamond heat spreading technology[7, 8, 9].

GaN-on-diamond is not trivial to fabricate, but can be realised either by
surface activated wafer bonding[10, 11, 12, 13, 14] or heteroepitaxial growth.
Growth of GaN onto diamond[15] or diamond onto GaN can be achieved using
interlayers such as silicon nitride (SiN) or group-III-Nitride (III-N)
structures. Examples of III-N structures include aluminium nitride (AlN) and
aluminium gallium nitride (AlGaN)[16, 17, 6, 18, 19]. In the wafer bonding
approach, it is imperative to achieve a sub-nanometre surface roughness in
order for Van der Waals bonding to occur. However, it has been calculated that
a large thermal barrier resistance of Van der Waals bonded diamond on GaN is
expected, reducing the efficacy of diamond heat spreading technology[20]. For
heteroepitaxial growth of diamond onto III-N structures, this can only be
achieved at present with polycrystalline diamond (PCD) using chemical vapour
deposition (CVD) at lower than atmospheric pressure and at high temperatures
(0.1 to 0.2 atm at temperatures from 600 to 800 ∘C[8]). Additionally in
contrast to single crystal diamond (SCD), PCD can be grown over large areas,
making this approach more commercially viable. A number of CVD methods exist
for depositing PCD. Microwave plasma assisted chemical vapour deposition
(MPCVD) offers diamond growth over small areas (<100 mm diameter wafers) and
can be optimised for very fast growth rates (50 to
$150\text{\,}\mathrm{\SIUnitSymbolMicro m}$ h-1)[21]. Larger area coverage
(>100 mm diameter wafers) can be achieved with hot filament (HFCVD), although
repeatability is challenging and growth rates are generally much lower (1 to
$4\text{\,}\mathrm{\SIUnitSymbolMicro m}$ h-1) [22, 23, 24].

Wafer Bonding MethodMembrane MethodGaNAlN/AlGaNSiSi
TempOxide/NitrideGaNAlN/AlGaNSiAlN/AlGaNGaNOxide/NitrideSi
TempDiamondAlN/AlGaNGaNOxide/NitrideSi
TempDiamondAlN/AlGaNGaN$\circlearrowleft$$\circlearrowleft$AlN/AlGaNSiGaNGaNSiSiAlN
AlGaNGaNSiSiDiamondAlN AlGaN

Figure 1: Approaches to diamond on GaN/III-N integration using wafer bonding
steps prior to CVD diamond growth (top) and membrane fabrication (bottom). ‘Si
Temp’ is defined as a temporary Si carrier wafer.

Direct diamond growth on GaN is challenging owing to a number of factors. The
first is that a stable carbide bond is difficult to achieve on either side of
the GaN faces which can be overcome using interlayers that form carbides. The
second issue is that GaN is widely deposited using metal organic chemical
vapour deposition (MOCVD) where the Ga-polar HEMT layer is typically at the
top of the stack. While growing diamond on the Ga-polar side has been
demonstrated for thin films at less than $1\text{\,}\mathrm{\SIUnitSymbolMicro
m}$[7], this creates additional challenges to embed the diamond layer during
device fabrication. Also, it has been shown that the thermal conductivity of
diamond can be very low at these thicknesses[8]. Therefore, the diamond must
ideally be grown on the underside or the N-polar side of the GaN/III-N stack.
The N-polar side is chemically very different to the Ga-polar surface and
thick film growth requires significant investigation[19]. This requires wafer
bonding to a temporary carrier, flipping, etching and then growing the diamond
on the exposed III-N stack as is shown in Fig. 1. Finally, a prominent issue
is the thermal stress and strain (bow) from the mismatch in the coefficient of
thermal expansion (CTE) between the layers in the stack[25]. High temperature
CVD diamond growth generates large stresses upon cooling which bows the wafer
and may mechanically fracture the GaN. Additionally, these high temperatures
may also result in complete destruction of the GaN/III-N stack and so ensuring
the growth temperature is as low as possible mitigates the risk of damage or
fracturing upon cooling.

A potential method which enables fast access to the correct polarity face of
the GaN/III-N stack for diamond growth, without the need for costly and time
consuming wafer bonding, is to use a GaN/III-N membrane, as is shown in Fig.
1. This can be achieved using an inductively coupled plasma (ICP)[17]. This
membrane technique has been realised for other GaN applications including
light emitting diodes, planar photonic devices[26, 27, 28] and impedance based
sensors[29, 30, 31]. The details of the integration of diamond with GaN
membranes are seldom reported, specifically owing to the fragile nature of the
membrane and the high temperatures required to grow diamond. Previous reports
have shown diamond on GaN/III-N membranes from 0.5 to 5 mm in diameter under
stresses of around 0.7 GPa due to the thermal cycling [17]. The stress is
lower than the experimentally reported mechanical strength of GaN (compressive
strength of 10 to 15 GPa and tensile strength of 4 to 7.5 GPa)[32, 33, 34] and
far less than density functional theory (DFT) estimates of the tensile
strength (30 to 40 GPa)[35, 36]. The stresses are also equivalent to those
measured in a typical diamond on GaN/III-N sample produced through the wafer
bonding method (0.67 to 1 GPa)[37], demonstrating a potentially viable
approach for circumventing wafer bonding. However, the membrane deformations
and bow are not reported which may significantly impact device fabrication
using contact lithography. Since the membrane structure is far more complex
when compared to a cylindrical wafer stack, an understanding of the parameters
which cause these deformations and induced thermal stresses is required before
considering this approach for any device fabrication.

In this work, the thermal stress and strains (membrane bow) associated with
the growth of CVD diamond on GaN/III-N membranes have been investigated using
analytical modelling, numerical modelling and experimental work. First, a
simplified Stoney analytical model is briefly described for stress in a two
layer cylindrical stack. Following this a finite element model (FEM) is
presented which calculates the anticipated thermal stresses in circular
membranes and investigates the effect of varying the geometry, materials and
temperature. Next, experimental data is presented on the growth and
characterisation of diamond on GaN/III-N membranes to validate the model.
Finally, comparisons and discussions are drawn, particularly on the
limitations of the CVD diamond on GaN/III-N membrane approach.

##  2 Analytical Model

###  2.1 Coefficient of Thermal Expansion (CTE)

Thin film deposition at elevated temperatures requires careful consideration
of stress. At room temperature, the net stresses are due to a combination of
intrinsic stresses and extrinsic stresses. Intrinsic stresses arise mainly due
to the imperfections in the microstructure of the deposited film such as
crystal grain boundaries, voids, impurities and defects caused by the lattice
mismatch between the film and the substrate. For example, the lattice mismatch
between GaN and Si results in residual intrinsic stresses after MOCVD[38].
Additionally, it is well-known that intrinsic stresses are also present in CVD
diamond depending on the experimental conditions[39, 40]. Extrinsic stresses
are due to externally applied forces such as mechanical loading or thermal
stress, the latter of which is a well-known challenge for thin film deposition
and is the subject of this study.

Figure 2: Temperature dependent CTE of diamond, Si, AlN and GaN[25, 41, 42].

Thermal stresses arise due to the thermomechanical properties of the layers in
the stack. In isolation, each material preferentially expands/contracts to
different volumes at different temperatures. However, for a three dimensional
(3D) sample with all layers bound axially in the $z$ direction, the
deformations in the $xy$ plane are constrained. The CTE of diamond and the
layers in a typical GaN-on-Si wafer are shown in Fig. 2. AlN and GaN have very
similar values at typical CVD growth temperatures of 600 to 1000 ∘C and a
small difference at room temperature of around 1.5 $\times 10^{-6}$
$\mathrm{K}^{-1}$. Diamond and Si have lower values than the GaN/III-N layers
across the temperature range. At room temperature Si intersects with AlN while
diamond is much lower than all of the materials. It is therefore expected that
after cooling to room temperature, the CVD diamond on the membrane will be
held in compression, whilst the GaN and III-N layers in tension. Thus, careful
consideration of the tensile strength of the GaN/III-N stack must be made to
ensure that the membrane does not fracture through upon cooling. The resultant
deformations caused by thermal stress can be determined for a simplified
cylindrical wafer stack using the following analytical model.

###  2.2 Analytical Model

In a cylindrical coordinate system ($\rho,\theta,z$), consider an axially
symmetric wafer stack consisting of two unbound layers of differing
thicknesses ($t_{\textrm{film}}$ and $t_{\textrm{sub}}$) with differing CTE
($\alpha_{\textrm{film}}$ and $\alpha_{\textrm{sub}}$). The layer thicknesses
in the $z$ direction are much smaller than the radius of the cylindrical
stack. When the temperature is lowered, depending on the CTE, the volumes of
the layers shrink with a negligible change in thickness compared to a much
larger radial change. Using the linear thermal expansion model, the resultant
radial 1D strains are:

$\varepsilon_{\textrm{film}}=\frac{r_{\textrm{film}}(T)}{r_{\textrm{0,film}}}=1+\alpha_{\textrm{film}}(T)\left[T-T_{0}\right]$
(1)
$\varepsilon_{\textrm{sub}}=\frac{r_{\textrm{sub}}(T)}{r_{\textrm{0,sub}}}=1+\alpha_{\textrm{sub}}(T)\left[T-T_{0}\right]$
(2)

where $T$ is the temperature in K, $r(T)$ is the temperature dependent radius
of the layer in m, $r_{0}$ denotes the initial radius in m at a reference
temperature of $T_{0}$ and $\alpha(T)$ is the temperature dependent CTE of the
layer in K-1. If the layers are bound, the radial strain at the interface must
be equal, with one layer being held in tension while the other is held in
compression. Therefore, an additional strain is required to keep the materials
bound at the interface, equal to the difference between the two unbound
strains:

$\Delta\varepsilon=\varepsilon_{\textrm{sub}}-\varepsilon_{\textrm{film}}=\left[T-T_{0}\right]\left[\alpha_{\textrm{sub}}(T)-\alpha_{\textrm{film}}(T)\right]=\Delta
T\Delta\alpha$ (3)

The calculated stress associated with this strain is related through the
Young’s modulus:

$\sigma=E\cdot\Delta T\Delta\alpha$ (4)

where $\sigma$ is the induced thermal stress in Pa and $E$ is the Young’s
modulus of the layer in Pa. Considering 2D axial symmetry however, this
generated 1D radial stress is not the equilibrium state as the two layer
system is not restricted along the $z$ axis. The induced internal stress thus
results in a finite strain in the $z$ direction. For an axially symmetric
sample, this results in a radius of curvature $R$ through a relation with the
biaxial modulus, otherwise known as the Stoney formula[39, 43, 44]:

$R=\frac{E_{\textrm{sub}}}{\left(1-\nu_{\textrm{sub}}\right)}\frac{t_{\textrm{sub}}^{2}}{t_{\textrm{film}}}\frac{1}{6}\frac{1}{\sigma}$
(5)

where $R$ is the radius of curvature of the substrate,
$E_{\textrm{sub}}/\left(1-\nu_{\textrm{sub}}\right)$ is the biaxial modulus of
the substrate and $\nu_{\textrm{sub}}$ is the Poisson’s ratio. With the radius
of curvature known, the wafer bow can be calculated for a given wafer
diameter. This is simply related through arclength[25]:

$w=R\left(1-\textrm{cos}\left[\frac{r_{\textrm{sub}}}{R}\right]\right)$ (6)

where $w$ is the displacement of the central point in the $z$ direction of the
wafer in m or bow. Knowing the diameter of the wafer ( $2r_{\textrm{sub}}$),
one can predict the radius of curvature of a cylindrical stack for a given
thermal stress.

GaN on Si Heat Diamond on GaN Cool (a) (b) Figure 3: Analytical model of bow
using Eq. (5) for two scenarios: (a) the heating of a cylindrical GaN on Si
stack to CVD growth temperatures, resulting in an upward bow and (b) the
cooling of a cylindrical diamond on GaN stack from CVD growth temperatures to
room temperature, also resulting in an upward bow.

The stresses in a cylindrical stack that are associated with the CVD diamond
growth process at various deposition temperatures can be approximated using
typical thicknesses, diameters and values of $E$, $\nu$ and $\alpha(T)$ for
GaN and diamond as given in Fig. 2 and Table 1. Fig. 3 shows models of GaN-on-
Si wafers heated to CVD growth temperatures and diamond on GaN wafers cooled
to room temperature. Here, the GaN/AlGaN and AlN nucleation layers are grouped
and modelled as one thin GaN layer ($t_{\textrm{film}}$=
$5\text{\,}\mathrm{\SIUnitSymbolMicro m}$) while the substrate is modelled as
Si upon heating ($t_{\textrm{sub}}$= $75\text{\,}\mathrm{\SIUnitSymbolMicro
m}$) or diamond on cooling ($t_{\textrm{sub}}$=
$50\text{\,}\mathrm{\SIUnitSymbolMicro m}$). With a constant radius of
curvature, it is clear that the bow increases with diameter. At temperatures
between 600 and 800 ∘C, the initial heating bow is associated with a stress of
around 0.3 to 0.41 GPa while the cooling bow is associated with a stress
around of 0.43 to 0.53 GPa. An important note is that the analytical model
implies that membranes of 0.5 mm in diameter have a bow of less than
$1\text{\,}\mathrm{\SIUnitSymbolMicro m}$. Low temperature growths (<800 ∘C)
are chosen in this study since most sucessful growths on GaN occur at lower
temepratures to minimise thermal stress. This analytical model predicts a net
stress in the GaN as high as 0.94 GPa. This value is lower than the reported
tensile strength of GaN (4 to 7.5 GPa at room temperature) and similar to
those reported in previous studies[17, 34].

Table 1: Material properties used in analytical and numerical models. Material | Young’s Modulus | Poisson’s | Density | Thermal | Specific | Ref.
---|---|---|---|---|---|---
|  $E$ (GPa) | Ratio |  $\rho$ (g/m3) | Conductivity | Heat |
|  | $\nu$ |  |  $\kappa$ (W/mK) |  $C_{p}$ (J/kg$\cdot$K) |
Diamond | 1000 | 0.2 | 3.51 | 2270 | 0.52 | [45, 46]
GaN | 295 | 0.25 | 6.16 | 253 | 0.42 | [33]
AlN | $310.2-0.02T\textrm{e}^{-533T}$ | 0.23 | 2.92 | 321 | 0.78 | [47]
Si | $172.3-0.01T$ | 0.26 | 2.33 | 130 | 0.70 | [25, 41]

A prominent limitation of this model is its simplicity; the membrane structure
in reality is not a millimetre sized GaN-on-Si wafer but a millimetre sized
GaN membrane that is suspended by a much larger Si border. The stresses
induced by the Si cannot be taken into account since it is bound to both the
edges of the membrane and the rest of the GaN film. It is also assumed that
the diamond thickness is constant on the top of the membrane, which is not
true since lateral growth will occur on the edges of the Si. Such model
complexities can be readily implemented using a numerical model.

##  3 Numerical Model

Diamond \- 1 to $50\text{\,}\mathrm{\SIUnitSymbolMicro m}$Si \- 75 to
$500\text{\,}\mathrm{\SIUnitSymbolMicro m}$III-N \- 1 to
$5\text{\,}\mathrm{\SIUnitSymbolMicro m}$GaN \-
$2\text{\,}\mathrm{\SIUnitSymbolMicro m}$15 mm15 mm$z$$x$$y$

$S$$S$..$f_{2}$$f_{1}$.$f_{0}$Boundary
Conditions$f_{0}\rightarrow\vv{\bm{u}}(0,0,w)$
$f_{1}\rightarrow\vv{\bm{u}}(0,v,0)$ $f_{2}\rightarrow\vv{\bm{u}}(u,0,0)$
$S\rightarrow$ Symmetry
$T_{0}=25{}^{\circ}$C$S$$S$..$f_{2}$$f_{1}$.$f_{0}$$T_{0}=720{}^{\circ}$C

Figure 4: Cross-sectional diagram of the base numrical membrane model, not to
scale. Boundary conditions show symmetry planes to reduce the model size and
fixed constraints which restrict the velocity field ($\vv{\bm{u}}$) at
specific points ($f$) .

###  3.1 Model Description

The wafer stack has been modelled in the COMSOL Multiphysics® FEM package
using the Thermal Stress module which incorporates both Solid Mechanics and
Heat Transfer. A cross section schematic of the model stack is shown in Fig. 4
and consists of a GaN device layer on the bottom, a lumped III-N layer, a Si
frame which defines the membrane pattern and a CVD diamond layer. The lumped
III-N layers represent a combination of the AlN nucleation layer and AlGaN
strain relief layers, conservatively approximated as AlN. The base model
thicknesses are given as $t_{\textrm{GaN}}=$
$2\text{\,}\mathrm{\SIUnitSymbolMicro m}$, $t_{\textrm{III-N}}=$
$3\text{\,}\mathrm{\SIUnitSymbolMicro m}$, $t_{\textrm{Si}}=$
$75\text{\,}\mathrm{\SIUnitSymbolMicro m}$ and $t_{\textrm{Dia}}=$
$50\text{\,}\mathrm{\SIUnitSymbolMicro m}$. The dimensions have been chosen to
represent a typical MOCVD sample which has been thinned from the Si side and
cleaved to dimensions of 15 mm by 15 mm for MPCVD. This sample size has been
chosen since successful diamond depositions have been achieved on AlN on Si in
previous work at this size in a high power density plasma[17, 48]. Also,
smaller samples minimise any diamond thickness variation over the sample owing
to the small plasma size at these conditions. The boundary conditions of the
structure are also shown in Fig. 4 whereby the centre point of the GaN is
fixed in $xy$ space and free to displace in the $z$ direction. Two symmetry
planes are also defined to reduce the size of the model. The growth
temperature of the base structure is homogenous with no gradients and chosen
to match the experimental pyrometer data in the centre of the sample during
diamond growth for a low temperature recipe (720 ∘C)[17]. The material
parameters are the same as those used in the analytical model with additional
thermomechanical properties, shown in Table 1.[32, 41, 42, 46, 47, 49, 50]

|  |  |  |  |
---|---|---|---|---|---
0\. Initial | 1\. Membrane | 2\. Diamond | 3\. Diamond | 4\. Combined cooled |
GaN-on-Si | heated to CVD | deformed to fit | stress-relieved | diamond and |
membrane | temperatures | onto structure | and cooled | initial membrane |
Figure 5: Steps taken in the FEM with colour plot showing the von Mises stress
in GPa and the resultant deformation field, $\times 10$ $z$ scale for
visibility.

The model is run in an automated stepped process as is shown in Fig. 5. Step 0
shows the initial geometry of the model. Step 1 is the heating model, where
the structure is heated to growth temperatures to obtain the initial thermal
stresses in the absence of any diamond. Convergence of this step is crucial as
it defines the initial shape of the deposited diamond. This step typically
converged to two solutions, whereby the GaN-on-Si membrane either had a
positive bow (GaN displaces upwards into the Si) or a negative bow (GaN
displaces away from the Si), although with the same magnitude. The presented
results show the positive bow solutions since this was what was found in later
experiments. Step 2 uses this displacement field to deform the diamond layer
to fit on top of the stressed membrane geometry. Step 3 isolates the diamond
layers and relieves the previously computed stress in this deformed
configuration as it is assumed that the diamond is deposited stress-free. The
diamond layer is then cooled in isolation to obtain the deformation at room
temperature. The reason why the diamond is separately cooled is for stability
during model convergence. This can only be assumed if there is no hysteresis
in the thermal cycling. Step 4 integrates the cooled diamond by stretching the
initial room temperature membrane structure over the cooled diamond to give
the final deformation and stresses associated with this state. To validate
this stepped process, the absence of diamond in the final step shows no
deformation.

###  3.2 Numerical Model Results

The modelled membrane results are shown in Fig. 6, 7 and 8 where the dotted
lines show the heated sample and the solid lines show the final state where
the cooled diamond is integrated with the structure. A positive $z$
displacement refers to a membrane bow into the Si while a negative $z$
displacement refers to away from the Si, as per the defined coordinate system
in Fig. 4. The von Mises stress denote the residual stress associated with the
displacement fields.

|
---|---
(a) | (b)
Figure 6: Simulated results for varying the CVD growth temperature are given in (a) and (b) for the membrane bow and the average residual stress in the GaN membrane, relative to step 0, respectively. Base model layer thicknesses have been used: $t_{\textrm{GaN}}=$ $2\text{\,}\mathrm{\SIUnitSymbolMicro m}$, $t_{\textrm{III-N}}=$ $3\text{\,}\mathrm{\SIUnitSymbolMicro m}$, $t_{\textrm{Si}}=$ $75\text{\,}\mathrm{\SIUnitSymbolMicro m}$ and $t_{\textrm{Dia}}=$ $50\text{\,}\mathrm{\SIUnitSymbolMicro m}$.  |
---|---
(a) | (b)
|
(c) | (d)
Figure 7: Simulated results for varying $t_{\textrm{Dia}}$ are given in (a) and (b) for the membrane bow and the average residual stress in the GaN membrane relative to step 0, respectively. Results for varying $E_{\textrm{Dia}}$ are given in (c) and (d) for the membrane bow and stress, respectively. The traces marked ‘All’ refer to the heating phase of all the model cases since the diamond is introduced afterwards. The base model layer thicknesses have been used for other layers: $t_{\textrm{GaN}}=$ $2\text{\,}\mathrm{\SIUnitSymbolMicro m}$, $t_{\textrm{III-N}}=$ $3\text{\,}\mathrm{\SIUnitSymbolMicro m}$ and $t_{\textrm{Si}}=$ $75\text{\,}\mathrm{\SIUnitSymbolMicro m}$. A deposition temperature of $T_{g}=720$ ∘C was used as to simulate low temperature growth recipes for GaN/III-N membranes[17]. The base value of $E_{\textrm{Dia}}=1$ TPa. |
---|---
(a) | (b)
|
(c) | (d)
Figure 8: Simulated results for varying $t_{\textrm{III-N}}$ are given in (a)
and (b) for the membrane bow and the average residual stress in the GaN
membrane, relative to step 0, respectively. Results for varying
$t_{\textrm{Si}}$ are given in (c) and (d) for the membrane bow and stress,
respectively. The base model layer thicknesses have been used for the other
layers: $t_{\textrm{GaN}}=$ $2\text{\,}\mathrm{\SIUnitSymbolMicro m}$,
$t_{\textrm{III-N}}=$ $3\text{\,}\mathrm{\SIUnitSymbolMicro m}$,
$t_{\textrm{Dia}}=$ $50\text{\,}\mathrm{\SIUnitSymbolMicro m}$ and
$t_{\textrm{Si}}=$ $75\text{\,}\mathrm{\SIUnitSymbolMicro m}$. A deposition
temperature of $T_{g}=720$ ∘C was used as to simulate low temperature growth
recipes for GaN/III-N membranes[17]

Fig. 6 shows the base model displacement as a function of membrane diameter
and at varying growth temperatures. It is clear from Fig. 6.a that the bow
increases with membrane diameter. It is also shown that, as expected, the bow
increases with deposition temperature, although to a lesser degree, resulting
in an increase in stress. The heated stresses in Fig. 6.b are approximately
0.2 GPa for larger membranes and increase to 0.4 GPa for smaller membranes.
These values are in the same range as the heated stresses for the analytical
model. However, the heated bow is much larger than the analytical model; for
example in Fig. 6 upon changing the temperature from 600 to 800 ∘C, the 5 mm
membrane bows, respectively, from 80 to $90\text{\,}\mathrm{\SIUnitSymbolMicro
m}$ in the numerical model while in Fig. 3, the calculated bow for the same
membrane is 22 to $30\text{\,}\mathrm{\SIUnitSymbolMicro m}$ in the analytical
model. This large difference is attributed to the additional thermal stress
contribution from the Si border which is not taken into account in the
analytical approach, demonstrating a significant limitation in using Eq. (4)
with (5) to predict the bow in membrane structures. After cooling with the
diamond introduced, the overall bow is relatively unchanged, implying that
when the diamond is deposited, it attempts to hold the deformed membrane into
place upon cooling owing to its extreme hardness and low CTE across the
temperature range. This results in a large increase in the residual stresses,
as the initial membrane model in Step 0 should be flat at room temperature but
is held in the bowed state. These residual stresses also decrease as the
growth temperature is reduced. As shown in Fig. 6.b, GaN/III-N membranes with
diameters between 1 and 5 mm will be under a stress between 1.0 to 1.7 GPa
which is still less than the tensile strength (4 to 7.5 GPa at room
temperature)[34].

Fig. 7 shows the models for varying the diamond thickness, where the traces
marked ‘All’ stipulate the result for all cases for the heating models; the
initial geometry is not varying only the diamond and its properties. A lower
limit of $1\text{\,}\mathrm{\SIUnitSymbolMicro m}$ has been chosen since it
has been shown that thinner diamond films result in incredibly low thermal
conductivities[8]. The diamond thickness appears to have minimal effect on the
observed bow but a significant effect on the residual stress in the membrane.
This result also demonstrates that even a
$1\text{\,}\mathrm{\SIUnitSymbolMicro m}$ thin diamond layer is enough to hold
the membranes into the heated bowed state by several tens of micrometres owing
to the extreme stiffness of diamond. To further demonstrate this, Fig. 7.c and
7.d show the results for varying the diamond Young’s Modulus
($E_{\textrm{Dia}}$) for a $1\text{\,}\mathrm{\SIUnitSymbolMicro m}$ thick
diamond layer. Decreasing the stiffness decreases the final membrane bow and
stress. At very low values of around 200 GPa, the diamond is no longer rigid
enough to fully sustain the deformation thereby reducing the final bow and
stress. However, such low values of $E_{\textrm{Dia}}$ are not expected for
CVD diamond, even for ultra-nano-crystalline (UNCD) films[46, 51].

Fig. 8.a and 8.b show the effect of varying the III-N stack thickness,
revealing minimal differences in both bow and cooled stress. The heated bow
and stress, however, increases with thickness which is not favourable at high
temperatures since GaN weakens towards 1000 ∘C[52]. A similar case is found
for increasing the Si thickness, although a much larger bow is seen in Fig.8
across all membrane sizes. The average stress in the membrane is also
marginally relieved upon cooling as the Si is thickened, shown in Fig. 8.d.
Since the Si thickness can be varied by a much larger range at tens of microns
compared to the III-N layers at a few microns, changes to the Si are expected
to have a more profound effect on the bow and stress than the III-N layers.
This implies a potential route to minimising the bow through Si thickness,
however it would need to be much less than
$75\text{\,}\mathrm{\SIUnitSymbolMicro m}$.

In the presented models, it is apparent that after CVD diamond deposition, the
final stress in the membranes is within the mechanical limits of GaN, however,
the anticipated bow of large membranes is significant, imposing challenges for
device manufacturing through contact photo-lithography. The models imply that
only small membranes are a potentially viable route. To verify these findings,
circular GaN/III-N membrane samples of varying diameter were fabricated with
similar dimensions to the models. The effect of CVD diamond deposition on the
mechanical properties was investigated through measurement.

##  4 Experiment

There were 3 samples in total, fabricated using commercially obtained wafers,
diced to square dimensions of 15 mm by 15 mm. The fabrication of the membranes
and details can be found in previous work[17]. In brief, the as received
GaN/III-N on Si wafer was etched from the Si side using a high power ICP at
900 W, followed by membrane patterning using photolithography and finally a
lower power ICP etch at 600 W to completely remove the Si and expose an AlN
layer on the GaN/III-N membranes. The thickness of the remaining Si handle
border was was measured using a micrometer. Following this, CVD diamond growth
was achieved using a Carat systems CTS6U MPCVD reactor. The temperature of the
sample was measured using a Williamson pyrometer (Model DWF-24-36C) which uses
dual wavelengths at around $2\text{\,}\mathrm{\SIUnitSymbolMicro m}$ to
minimise emissivity errors. Prior to growth, a non-ultrasonic seeding method
was used owing to the fragility of the membranes. Briefly, the exposed
GaN/III-N membranes were first pre-treated in a N2/H2 microwave plasma with a
forward power of 1.5 kW at 20 Torr. This step allows some control over the
seeding density by decreasing the streaming potential of the AlN. This brief
plasma pre-treatment stage has been shown to increase the oxygen surface
content on AlN by partially etching and subsequent adsorption of oxygen onto
the surface once the sample is brought to atmosphere[48]. Following this, a
nano-diamond colloid solution was pipetted onto the exposed GaN/III-N membrane
side, covering both the membrane and the Si handle border. The samples were
rinsed carefully in de-ionised water and dried on a hotplate for 10 minutes
set to 115∘C. Diamond growth was achieved in a CH4/H2 microwave plasma with a
forward power of 5.5 kW at 110 to 120 Torr and a CH4 concentration of 3% at a
total flow rate of 500 sccm. At these conditions, the pyrometer measured a
temperature of around 720 to 750 ∘C. With a typical growth rate at these
conditions of approximately $2$ to $3\text{\,}\mathrm{\SIUnitSymbolMicro
m}$/hour, growth runs were conducted for 19 hours to achieve a diamond
thickness of around 38 to $57\text{\,}\mathrm{\SIUnitSymbolMicro m}$.

Before |  |
---|---|---
|  |

After |  |
---|---|---
|  |
5 mm | 2 mm | 0.5 mm

Initial Heating |
---|---
|

Figure 9: Photos of the 3 GaN/III-N membrane samples before (top row), after
MPCVD diamond growth (middle row) and at the beginning of MPCVD growth (bottom
row) with numerical model of the displacement field in micrometres ($z$ scale
$\times$10 for visibility). All samples are grown one by one, positioned
centrally on the sample holder as shown in the bottom photo.

Following growth, surface profilometry was used to determine the membrane bow
with a profiler tip of diameter of $12\text{\,}\mathrm{\SIUnitSymbolMicro m}$,
a force of 1 mg and a maximum travel time of 60 seconds across the membrane.
Raman spectroscopy was used to determine the structural signatures of the
GaN/III-N and CVD diamond and infer the stress in the membranes, achieved
using a Renishaw inVia Raman spectrometer with a 514 nm laser. To ensure that
there is an adhered interface between the CVD diamond and the N-polar side of
the III-N stack, cross-sectional observation of the samples was conducted by
high angle annular dark field scanning transmission electron microscopy
(HAADF-STEM) and energy dispersive X-ray spectroscopy (EDX) in an FEI Tecnai
Osiris microscope operated at 200 kV. The TEM foil was prepared by focused ion
beam (FIB) from the GaN/III-N side of the membrane.

The fabricated membranes are shown in Fig. 9, before, after and at the start
of CVD diamond growth. After deposition, the 0.5 and 2 mm diameter membranes
have high optical transparency, implicit of a successful growth, however, the
5 mm diameter membrane shows a much darker film implying that the GaN membrane
is significantly damaged. This is likely due to the membrane bowing upwards
towards the CH4/H2 microwave plasma and melting. Referring to the image of the
sample inside the MPCVD chamber during the first hour of growth in Fig. 9, it
is clear that the centre of the membrane is at elevated temperatures. Also,
the sample corners are at much higher temperatures than the rest of the
sample. This is corroborated by the displacement fields of the numerical model
that shows that these regions are significantly elevated during growth.

Figure 10: Cross-sectional HAADF-STEM microgaph and EDX data of CVD diamond
grown on a 0.5 mm GaN/III-N membrane.

###  4.1 Scanning Transmission Electron Microscopy

A Cross sectional STEM micrograph of the CVD diamond grown on the GaN/III-N
membrane stack (0.5 mm diameter) is shown in Fig. 10. The EDX analysis shows
the carbon rich layer bound to an aluminium and nitrogen rich layer, where the
former is the deposited CVD diamond layer and the latter corresponds to an AlN
nucleation layer ($\sim$$130\text{\,}\mathrm{nm}$). Underneath the AlN
nucleation layer, the AlGaN strain relief layers can be clearly seen
($\sim$$3.6\text{\,}\mathrm{\SIUnitSymbolMicro m}$) which is then followed by
the GaN device layer ($\sim$$1.7\text{\,}\mathrm{\SIUnitSymbolMicro m}$) and
AlGaN barrier layers which forms the device 2DEG. In between the diamond and
the AlN nucleation layer shows a very thin layer of silicon and oxygen, most
likely SiO2 based on the atomic ratios. This is a curious finding owing to the
absence of any oxygen in the gas precursors. There are two possible origins
for this SiO2 layer. The first is from leftover Si after the ICP etch which
then forms an oxide when brought to atmosphere the MPCVD stages. The second is
that the Si border is etched in the MPCVD plasma pre-treating stage before
seeding, resulting in redeposition of the Si on the membrane and subsequent
oxide formation when the sample is brought to atmosphere; more discussion on
this is given later.

###  4.2 Raman Spectroscopy

Figure 11: Raman spectra of the 0.5 mm GaN membrane sample at stages during the fabrication process. Spectra were taken at different spot locations for (a) free-standing diamond reference, (b) as received GaN-On-Si wafer, (c) and (d) GaN side after membrane fabrication, (e) and (f) GaN side after CVD diamond growth and (g) and (h) diamond side after CVD diamond growth. Background subtraction has been achieved using a linear fit. |
---|---
(a) | (b)
|
(c) | (d)
Figure 12: Magnified view of the Raman spectra of the GaN/III-N membranes. (a)
and (b) show the spectra at the Si peak and GaN $E_{2}$(high) peak while (c)
and (d) show the spectra at the diamond peak for membrane diameters of 0.5 mm
and 2 mm, respectively. Color schemes are the same as Fig. 11. Numerical
values are the peak average with standard deviation of 3 different
measurements of a similar region. Spectra obtained with a laser wavelength of
514 nm and background subtraction using a linear fit while Lorentzian peak
fits are obtained using least squares curve fitting.

The Raman spectra of the 0.5 mm membrane sample is shown in Fig. 11 at
different stages of the fabrication process. Fig. 11.a shows a free-standing
diamond sample produced at similar growth conditions with the Si removed. This
sample shows a sharp peak at 1333 cm-1 associated with diamond and minimal
signs of non-diamond carbon impurities. Fig. 11.b shows the as-received GaN-
on-Si wafer with the Si substrate contribution at 520 cm-1 and the GaN/III-N
layers at around 569 cm-1 and 734 cm-1, corresponding to the $E_{2}$(high) and
the $A_{1}$ longitudinal optical modes of GaN, respectively[53]. After
photolithography and ICP etching to create the membrane, Fig. 11.c shows the
spectra on the membrane and demonstrates that no Si mode could be detected,
implicit of a successful etch, while Fig. 11.d shows that on the Si border the
GaN/III-N features are relatively unchanged. Fig. 11.e to Fig. 11.h show the
spectra after diamond growth. Fig. 11.e shows that after growth the GaN/III-N
on the Si border is minimally changed, however, on the membrane in Fig. 11.f,
there is also a sharp diamond peak at around 1333 cm-1. This is due to the
$\sim$$50\text{\,}\mathrm{\SIUnitSymbolMicro m}$ thick diamond layer on the
underside of the GaN/III-N membrane. There also exists very small and broad D
and G bands at 1350 cm-1 and 1580 cm-1, attributed to a very low concentration
of non-diamond (graphitic like) carbon that is introduced during MPCVD growth.
Fig. 11.g shows the diamond grown on the Si. Finally, Fig. 11.h shows the
Raman spectra on the GaN membrane from the diamond side which has a similar
spectra to the membrane side in Fig 11.f.

Since the diamond layer is much thicker than the GaN/III-N layers, the high
intensity diamond peak dominates the spectra on the membrane, masking the
GaN/III-N features. Magnified views of the GaN and diamond regions are given
in Fig. 12. For both the 0.5 mm and 2 mm samples, the GaN $E_{2}$(high) peaks
are clearly shown, however, for the 5 mm sample this peak was not consistently
found at multiple regions on the membrane. This is likely caused by damage
owing to the elevated regions as is shown in Fig. 9. For the 0.5 mm and 2 mm
samples, the GaN $E_{2}$(high) peaks are shifted to lower frequencies of
$-2.8\pm 0.7$ cm-1 and $-3.0\pm 0.4$ cm-1, respectively. This implies that the
GaN/III-N layers are held in tension when the diamond is deposited which is
anticipated from the stress models. The estimated magnitude of the residual
biaxial stress can be calculated using a linear frequency shift relation of
2.9 cm-1 GPa-1[53]. Using the $E_{2}$(high) peak from the as-received
GaN/III-N on Si wafer as the reference peak, an average tensile stress over
various regions of the 0.5 and 2 mm membranes gave approximately 1.0 $\pm 0.2$
GPa and 1.0 $\pm 0.1$ GPa, respectively. The magnified diamond peak region in
Fig. 12.b shows minimal shift between the free-standing diamond film reference
and at various regions on the grown sample. The free-standing film shows a
marginal compressive shift to the value of 1332.8 $\pm 0.02$ cm-1 from the
typically observed value of 1332.5 cm-1[45, 54]. In contrast to the GaN/III-N
spectra which all show tensile shifts from the reference GaN-on-Si wafer, the
diamond peaks in both samples show varying global shifts from the reference
film; compressive shifts for the 0.5 mm sample and tensile shifts for the 2 mm
sample. These global shifts here show very small differences (around 0.2 GPa
using a linear shift relation of 2.87 cm-1 GPa-1[55]) in the intrinsic
stresses in the deposited diamond.

|
---|---
(a) | (b)
|
(c) | (d)
Figure 13: Surface profilometry of GaN/III-N membranes before and after CVD
diamond growth. (a) to (c) show two traces taken orthogonally across the
membrane. The bow ($\Delta z$) is the average of the displacement over the two
orthogonal directions, with the largest standard deviation of
$2\text{\,}\mathrm{\SIUnitSymbolMicro m}$. The stress is estimated from Eq.
(5) using the curvature for the initial bow to obtain the initial stress
($\sigma_{0}$) of a pre-stressed GaN film on Si and the curvature for the
final bow to obtain the final stress ($\sigma_{1}$) for a GaN film on diamond.
The stress is estimated using thicknesses: $t_{\textrm{GaN/III-N}}=$
$5.3\text{\,}\mathrm{\SIUnitSymbolMicro m}$, $t_{\textrm{Dia}}=$
$50\text{\,}\mathrm{\SIUnitSymbolMicro m}$, $t_{\textrm{Si}}=$
$75\text{\,}\mathrm{\SIUnitSymbolMicro m}$. (d) shows $\Delta z$ and
$\sigma_{1}$ as a function of membrane diameter.

###  4.3 Surface Profilometry

The surface profilometry of the GaN side of the membranes before and after CVD
growth is shown in Fig. 13. In addition to the prediction of thermal stresses,
the analytical model can also be used to infer the stresses using Eq. (5) and
the defined radii of curvature. An indication as to whether a simplified
analytical model can be used accurately or not can be determined by
corroborating these values with the Raman spectroscopy measurements. A clear
trend of increasing bow with diameter is found, however, large discrepancies
are noted for the implied stress data. Starting with the smallest 0.5 mm
diameter membrane, the initial bow was minimal at less than
$1\text{\,}\mathrm{\SIUnitSymbolMicro m}$. After diamond growth, the bow was
around $7\text{\,}\mathrm{\SIUnitSymbolMicro m}$. The calculated stress using
Eq. (5) suggests an initial value of around 0.8 GPa and increases to a very
large value of around 23.6 GPa after growth , a value much higher than the
experimentally measured tensile strength[34]. The 2 mm membrane showed a
larger initial bow of $10\text{\,}\mathrm{\SIUnitSymbolMicro m}$ and increased
by $17\text{\,}\mathrm{\SIUnitSymbolMicro m}$ after CVD diamond growth. The
initial stress is similar at 0.8 GPa with a stress after growth of 5.4 GPa,
much lower than the 0.5 mm membrane. Finally, the 5 mm membrane shows a much
larger initial strain in the opposite direction. While the initial strain is
different, the stresses can be compared with the other samples based on the
net stress. Also, after diamond growth, a considerable difference is found
between the orthogonal traces, with a large region warped in the centre of the
membrane. This is due to the fact that the large membrane has bowed
significantly away from the heat sunk sample holder and pushed further into
the plasma resulting in thermal runaway and damage to the GaN/III-N film.
Using the initial curvature from the edges of the membrane, the maximum
estimated net bow is considerably larger at around
$58\text{\,}\mathrm{\SIUnitSymbolMicro m}$.

##  5 Discussion

There are several key findings of this thermal stress study that should be
considered before designing and fabricating diamond on GaN/III-N membranes:
(1) analytical and numerical modelling accuracy, (2) damage to larger
membranes, (3) etching and redeposition of the Si border and finally (4)
approaches to reducing membrane bow.

###  5.1 Modelling accuracy

The presented analytical model does not give an accurate reproduction of the
experimental results; larger bow for large membranes, overall smaller stresses
than those found in experiment and most pertinent is the almost a non-existent
bow for much smaller membranes. This is indeed due to the additional stresses
caused by the Si border that are not analytically modelled. The numerical
model produces a better representation of the experimental data by exhibiting
a measurable bow at smaller sizes, however, there still exists a discrepancy
in larger sizes as with the analytical model. Fig. 7, show bows of almost
twice the experimental values, implying that the induced thermal stress is
much higher in the model.

One important factor that has not been considered in either of the models is
the temperature gradient across the sample. In Fig. 9, it is clear that the
corners of the sample are at much higher temperatures, owing to the bowing
upwards of the edges into the plasma. The base thermal stress model assumes a
uniform temperature across the whole sample. A simple linear spatial
temperature gradient can be imposed to replicate this phenomenon that is
scaled to the sample size through the following:

Figure 14: Comparison of membrane bow from experimental measurements,
analytical model and numerical model with additional temperature gradients
introduced. The numerical model parameters used were: $t_{\textrm{GaN}}=$
$1.7\text{\,}\mathrm{\SIUnitSymbolMicro m}$, $t_{\textrm{III-N}}=$
$3.6\text{\,}\mathrm{\SIUnitSymbolMicro m}$, $t_{\textrm{Dia}}=$
$50\text{\,}\mathrm{\SIUnitSymbolMicro m}$, $t_{\textrm{Si}}=$
$75\text{\,}\mathrm{\SIUnitSymbolMicro m}$ and a deposition temperature of
$T_{g}=720$ ∘C and variation induced using Eq. 7. The analytical model uses
the same constant deposition temperature and thicknesses with a combined
GaN/III-N layer and the result showing the combined heating and cooling
stresses.

$T(x,y)=T_{g}\left(1+\beta\frac{\sqrt{2(x^{2}+y^{2})}}{W_{\textrm{sub}}}\right)$
(7)

where $T$ is the spatially dependent temperature in ∘C, $T_{g}$ is the growth
temperature in the centre of the membrane, $\beta$ is a temperature variation
parameter, $x$ and $y$ are the initial positions in Cartesian coordinates and
$W_{\textrm{sub}}$ is the width of whole sample. Fig. 14 demonstrates the use
of a temperature gradient, which drastically reduces the bow in the centre to
align with experimental values while increasing the stress by a small amount.
For example, at a $\beta$ value of 0.5, the temperature linearly increases
from 720 ∘C at the centre to 1,080 ∘C at the corner tip of the sample which in
turn creates additional stresses which counteract the central bow. This
demonstrates the importance of using numerical models for examining thermal
stresses in GaN/III-N membrane structures.

###  5.2 Heat sinking during MPCVD

The damage to the largest membrane marks an interesting issue in the CVD
diamond on GaN/III-N membrane approach. This comes down to a matter of bow and
thermal runaway. Note that while the net bow of the 5 mm diameter membrane is
$58\text{\,}\mathrm{\SIUnitSymbolMicro m}$ its vertical displacement above the
zero point ($\sim$$30\text{\,}\mathrm{\SIUnitSymbolMicro m}$) is not that much
larger than the 2 mm diameter membrane
($\sim$$24\text{\,}\mathrm{\SIUnitSymbolMicro m}$) which was not damaged. A
plausible reason as to why the larger membrane sustained damage is that the
heat sinking capability is less effective over larger areas owing to the poor
thermal conductivity of GaN and AlGaN. The temperature at the membrane surface
was likely much higher than for the other samples. This ultimately limits any
large area CVD diamond growth over GaN/III-N membranes unless appropriate heat
sinking is employed.

###  5.3 Si etching and deposition

The thin layer of SiO2 at the interface between the CVD diamond and the
N-polar side of the AlN is a potentially serendipitous finding. This SiO2
layer facilitates a stable carbide bond for the diamond to the III-N stack; a
SiN bond with the N-polar side of the AlN and a SiC bond with the diamond.
There are two cases where this layer may be deposited. The first is at the ICP
membrane etch stage where the SiO2 layer is simply due to an incomplete etch
through the Si and the remaining few nanometres oxidise from exposure to air
after the sample is brought up to atmosphere. A second origin is during the
MPCVD N2/H2 plasma pre-treating stage before the seeding process where the Si
border is etched and redeposited on the membrane. There are numerous studies
on Si in H2 microwave plasmas which demonstrate the production of the volatile
silane (SiH4) gas that diffuses to and reacts with a target substrate to
redeposit Si[56, 57]. Pure SiH4 as a source gas can also be used to co-deposit
diamond onto substrates[58]. Once the sample is brought up to atmosphere after
the N2/H2 plasma pre-treatment, the deposited silicon oxidises to form a SiO2
layer and the diamond seeds are introduced. It’s also shown that the N-Polar
face of AlN has been shown to have a high concentration of surface oxide
groups[59] which may contribute to the formation of SiO2. From a stress
perspective, the adhesion strength needs investigation as high stresses in
this layer will result in delamination. This layer may be incredibly important
for creating an interface with N-polar faces and CVD diamond, subject of
another study[60].

###  5.4 Reducing membrane bow

Finally, the micron large membrane bows found for the CVD diamond on GaN/III-N
membranes presents the biggest challenge for device manufacturing using
contact lithography. Such large membrane bows cannot be dealt with as is,
however, it is important to note that the results in this study demonstrate
the net membrane bows. This means that if the membrane is sufficiently pre-
stressed before growth in the opposite direction, the bow upon heating will be
counteracted and the CVD diamond would be deposited on a flat GaN/III-N
membrane. As per the model findings, the extreme stiffness of the diamond
should then hold the membrane flat upon cooling, resulting in far less bow.
This would also solve the heat sinking issues for larger membranes in point
(2) as the membrane would not bow towards the plasma and become damaged. As a
final processing step, the membranes can then be laser cut to reveal a free-
standing GaN/III-N/diamond stack, ready for device processing. This idea of
course requires significant GaN-on-Si wafer development to produce pre-
stressed wafers, out of scope of this study.

##  6 Conclusion

The thermal stresses of the membrane approach for the integration of chemical
vapour deposited (CVD) diamond with gallium nitride (GaN) on group-III-N
(III-N) layers have been investigated using analytical and numerical modelling
and experimental results. In utilising the Stoney analytical model in the
presented way, the anticipated thermal stresses in GaN/III-N membrane
structures are not accurately reproduced to the values found in experiment.
While this is covered using numerical modelling, the result still requires
careful consideration of the spatial temperature distribution over the sample
whereby simple linear gradients appear to suffice. The dependence of various
parameters including layer thicknesses, temperature and diamond stiffness on
the membrane stress and bow has been modelled with the overall conclusion that
the membrane deformation when it is heated defines the expected structure
after cooling to room temperature. A CVD diamond layer as thin as
$1\text{\,}\mathrm{\SIUnitSymbolMicro m}$ is enough to lock this deformation
in place, meaning that this initial deformation is the key to reducing bow.
Experimental measurements using Raman spectroscopy corroborate the numerical
models and additionally, simply using the radius of curvature from surface
profilometry data to infer the stress is invalid at sizes lower than several
millimetres. The use of a Si border in this membrane method also introduces
SiO2 at the interface between the diamond and the GaN/III-N stack. The
membrane bow is the biggest issue that must be addressed as large membranes
bow towards the microwave plasma and become damaged, however, if the membranes
can be pre-stressed before growth, the final bow is expected to reduce.

##  7 Acknowledgements

This project has been supported by Engineering and Physical Sciences Research
Council (EPSRC) under program Grant GaN-DaME (EP/P00945X/1). J. A. Cuenca is
an EPSRC Postdoctoral researcher. DJ Wallis acknowledges support of EPSRC
fellowship (EP/N01202X/2).

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