M J Edwards
email: m.j.edwards2@bath.ac.uk

C R Bowen

D W E Allsopp

A C E Dent

Modelling Wafer Bow in Silicon-Polycrystalline CVD Diamond Substrates for GaN-based Devices

Keywords: Finite-element, CVD diamond, GaN, Silicon, Modelling PACS: 62.20.-x:Mechanical properties of solids, 62.20.D-: Elasticity, 68.55.jd: Thickness, 68.65.Ac: Multilayers

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Introduction

Gallium Nitride (GaN) is a robust material that is stable at temperatures up to 1000°C, has a wide bandgap of 3.4eV and a Young's modulus (E <0001> ) of ~150 GPa at room temperature [START_REF] Reeber | High Temperature Elastic Constant Prediction of Some Group III-Nitrides[END_REF][START_REF] Kim | Nanopatterned Surface Effect on the Epitaxial growth of InGaN/GaN Multi-quantum Well Light Emitting Diode Structure[END_REF]. These properties make it suitable for use in harsh environments. The wide band gap and high breakdown field of GaN make it suitable for power electronic devices, from which heat needs to be extracted for optimum performance. In order to create a viable GaN sensor, high power diode or transistor, a suitable substrate is required to support the device in harsh operating conditions. Silicon-polycrystalline diamond substrates are potentially advantageous, since both materials have high melting points, good mechanical and chemical stability. Polycrystalline diamond is also an excellent thermal conductor, with a conductivity of 2270 J/Kms at 20°C, and has a Young's modulus of 1050GPa at the same temperature.

Crystalline silicon provides a substrate for the growth of the polycrystalline diamond and, if thinned to a few microns, provides a crystalline base from which epitaxial GaN can be grown. The (111) orientated silicon is the preferred substrate orientation for epitaxial GaN growth, although growth on Si(100) and Si(110) is also possible [START_REF] Dadgar | Epitaxy of GaN on silicon-impact of symmetry and surface reconstruction[END_REF][START_REF] Joblot | Growth of wurtzite-GaN on silicon (100) substrate by molecular beam epitaxy[END_REF]. However, silicon-polycrystalline wafers are observed to bow when the substrate is cooled from the typical CVD diamond growth temperature (~700-800°C) [START_REF] Zimmer | Wide area polycrystalline diamond coating and stress control by sp(3) hot filament CVD reactor[END_REF][START_REF] Zimmer | Aspects of scaling CVD diamond reactors[END_REF] to room temperature (25°C). One cause of the bowing is a mismatch in the coefficients of thermal expansion (CTE), , between silicon and polycrystalline diamond, which are shown in Figure 1 along with CTE data for GaN. At low temperatures the CTE of diamond is lower than silicon while at higher temperatures the diamond CTE is higher than silicon. The aim this work is to use finite-element simulations to model the mechanical behaviour of silicon-polycrystalline diamond wafers, and to predict their bowing behaviour as a function of temperature and geometry. This work shows that the bow of a 100 mm diameter wafer is predicted to exceed 1mm; which will render processing of devices via photolithography very difficult. For example, the requirements for modern semiconductor products are a wafer bow of <40 µm for wafers of 300mm diameter [START_REF] Zimmer | Aspects of scaling CVD diamond reactors[END_REF].

This paper will highlight the modelling techniques explored in this work, an analytical model based on thermal stresses in bi-layers, and a finite element (FE) model. The temperature dependent coefficients of thermal expansion (CTE), , Poisson's ratios, , and Young's modulii, E, of both silicon and polycrystalline diamond were used in the model. Initially, a linear average of the temperature dependent material properties was calculated from 25 to 800°C, since the analytical method assumed that , and E were independent of temperature. The temperature independent properties were also used in the FE model to validate the boundary conditions and geometry used, through comparison with the analytical results. Once the FE model was validated, it was extended to include the influence of temperature dependent material properties. Issues relating to (i) the presence of a columnar diamond structure leading and anisotropy of the film and (ii) changes in silicon orientation from (111) to (100) are discussed.

Material Properties of Polycrystalline CVD Diamond and Silicon

Materials properties were collected to determine the key material properties of silicon and polycrystalline diamond. Both materials feature a cubic crystal structure and hence require three stiffness coefficients, (c 11 , c 12 , c 44 ) to describe the elastic response. Polycrystalline diamond consists of randomly orientated crystalline grains, making it approximately isotropic at the macroscopic level [START_REF] Wilks | Properties and Applications of Diamond[END_REF][START_REF] Field | The Properties of natural and synthetic diamond[END_REF][START_REF] Szuecs | Temperature dependence of Young's modulus and degradation of chemical vapor deposited diamond[END_REF], although a columnar structure at larger film thicknesses could also be present [START_REF] May | Diamond thin films: a 21st-century material[END_REF]. At this stage both materials were assumed isotropic for simplicity; texturing of the diamond film will be discussed later in the paper. Consequently, only the Young's modulus, E, and x-y Poisson's ratio, , are required for both materials, rather than the full stiffness matrix. This assumption aided the development of a simple analytical model, discussed in section 3, where all materials in the bi-layered wafer system are treated as isotropic. The CTE, , of both silicon and diamond is isotropic owing to its crystallographic structure, such that x = y = z . [START_REF] Pickrell | Thermal expansion of polycrystalline diamond produced by chemical vapor deposition[END_REF][START_REF] Watanabe | Linear thermal expansion coefficient of silicon from 293 to 1000 K[END_REF]

Material Properties of Polycrystalline CVD Diamond

For polycrystalline diamond, without columnar grains [START_REF] May | Diamond thin films: a 21st-century material[END_REF], the material consists of grains of randomly orientated diamond material with little variation in the elastic modulus along any direction. Sources [START_REF] Wilks | Properties and Applications of Diamond[END_REF][START_REF] Field | The Properties of natural and synthetic diamond[END_REF] report relatively low elastic coefficients for polycrystalline diamond due to imperfections and impurities. Furthermore, the experimental data of Szuecs et al [START_REF] Szuecs | Temperature dependence of Young's modulus and degradation of chemical vapor deposited diamond[END_REF] shows that the Young's modulus of CVD polycrystalline diamond is not affected by the grain size if the defect density is constant. Modern growth techniques now used to grow polycrystalline diamond produce significantly reduced defect densities and its Young's modulus can approach that of optical grade diamonds [START_REF] Szuecs | Temperature dependence of Young's modulus and degradation of chemical vapor deposited diamond[END_REF]. The values for the elastic modulii of diamond at room temperature were taken from McSkimin and Andreatch [START_REF] Mcskimin | Elastic moduli of diamond as a function of pressure and temperature[END_REF] a commonly cited source and are quoted in table 1, although the potential influence of the variation in the elastic modulus is also discussed later in the paper.

578

The Young's modulus and Poisson's ratio at room temperature (25°C) for diamond were calculated using the elastic modulii values in table 1 and standard formulae [START_REF] Mcskimin | Elastic moduli of diamond as a function of pressure and temperature[END_REF] to be E RT = 1050 GPa and = 0.1. However, it has been found that the Young's modulus of CVD diamond degrades faster with respect to temperature than predicted in McSkimin and Andreatch [START_REF] Mcskimin | Elastic moduli of diamond as a function of pressure and temperature[END_REF] due to the presence of contaminants. It has been shown experimentally by Szuecs et al [START_REF] Szuecs | Temperature dependence of Young's modulus and degradation of chemical vapor deposited diamond[END_REF] that the variation of Young's modulus (E) with temperature, T, of CVD diamond undergoes a linear regression of the form [START_REF] Reeber | High Temperature Elastic Constant Prediction of Some Group III-Nitrides[END_REF] where the temperature coefficient, c T , has an average value of -1.027 x 10 -4 K -1 . Using a room temperature Young's modulus, E RT [START_REF] Mcskimin | Elastic moduli of diamond as a function of pressure and temperature[END_REF], equation (1), and the average value for c T from [START_REF] Szuecs | Temperature dependence of Young's modulus and degradation of chemical vapor deposited diamond[END_REF], the temperature dependent Young's modulii for polycrystalline CVD diamond were calculated and are shown in table 2. The Poisson's ratio is assumed constant between -100 and 1000°C. Table 2. Elastic and thermal properties of CVD diamond used in the model [START_REF] Xu | Theory of the thermal expansion of Si and diamond[END_REF][START_REF] Wort | Thermal Properties of bulk polycrystalline CVD diamond[END_REF][START_REF] Szuecs | Temperature dependence of Young's modulus and degradation of chemical vapor deposited diamond[END_REF][START_REF] Wort | Diamond as an electronic material[END_REF]. Temp (°C) E (GPa) [START_REF] Mcskimin | Elastic moduli of diamond as a function of pressure and temperature[END_REF] (10 -6 K -1 ) [7,8] -

( ) [ ] 25 1 - + = T c E E T RT

Material Properties of (111) Silicon

While (100) orientated silicon can be used for diamond on silicon [START_REF] Rudder | Direct deposition of polycrystalline diamond films on Si(l00) without surface pre-treatment[END_REF] this work initially considers (111) orientation silicon since it is for c-plane GaN epitaxial growth [START_REF] Liu | GaN nano-pendeo-epitaxy on Si (111) substrates[END_REF]. The elastic modulii of isotropic silicon and their temperature dependence are determined using the approach of Varshni [START_REF] Varshni | Temperature dependence of the elastic constants[END_REF], which is derived from the Einstein model for a solid and are given by

) 1 /( / 0 - - = T t ij ij e u c c (2)
where u and t are fitting parameters and c ij 0 is the value for the elastic constant at 0K. The 0K elastic modulii values are based on the work in McSkimin [START_REF] Mcskimin | Measurement of Elastic Constants at Low Temperatures by Means of Ultrasonic Waves -Data for Silicon and Germanium Single Crystals, and for Fused Silica[END_REF], who measured the elastic modulii between 78 and 300K using ultrasonic wave techniques. The elastic modulii 0K values and fitting constants used in this work are shown in table 3. As the silicon is (111) orientated in the model, the compliance coefficients, s ij , need to be calculated from the elastic modulii from the standard formulae in [START_REF] Cho | Characterization of Young's modulus of silicon versus temperature using a ''beam deflection" method with a four-point bending fixture[END_REF]. Using the compliance coefficients calculated from [START_REF] Wilks | Properties and Applications of Diamond[END_REF], the Young's modulus, E (111), and Poisson's ratio, (111), of silicon along the (111

)
+ - + = υ (4)
The derivation of equations ( 3) and (4) can be found in Cho [START_REF] Cho | Characterization of Young's modulus of silicon versus temperature using a ''beam deflection" method with a four-point bending fixture[END_REF]. Equations ( 3) and (4) can also be derived using the compliance coefficients for rotated axes in cubic crystals from Wortman and Evans [START_REF] Wortman | Young's modulus, shear modulus, and Poisson's ratio in silicon and germanium[END_REF]. Although (111) silicon has an independent shear modulus, it has been treated as an isotropic material [START_REF] Cho | Characterization of Young's modulus of silicon versus temperature using a ''beam deflection" method with a four-point bending fixture[END_REF]. Initial exploration of this problem has indicated that the error caused by assuming that (111) silicon is isotropic is less than 1% and will have little effect on the model results shown in this paper. Therefore, it has been assumed that (111) silicon has isotropic mechanical properties. The temperature dependent Young's modulii and Poisson's ratios, calculated using equations (3) and (4), are presented in table 4 for (111) silicon alongside its temperature dependent CTEs [START_REF] Brozel | Thermal Properties[END_REF]. Table 4. Elastic and thermal properties of silicon used in the model [START_REF] Xu | Theory of the thermal expansion of Si and diamond[END_REF][START_REF] Brozel | Thermal Properties[END_REF][START_REF] Varshni | Temperature dependence of the elastic constants[END_REF][START_REF] Mcskimin | Measurement of Elastic Constants at Low Temperatures by Means of Ultrasonic Waves -Data for Silicon and Germanium Single Crystals, and for Fused Silica[END_REF][START_REF] Cho | Characterization of Young's modulus of silicon versus temperature using a ''beam deflection" method with a four-point bending fixture[END_REF][START_REF] Wortman | Young's modulus, shear modulus, and Poisson's ratio in silicon and germanium[END_REF]. The data in table 4 shows that (111) silicon has a high degree of thermal stability, with only a small decrease in both E and as the temperature increases.

Temp (°C) E (111) (GPa) (111) (10 -6 K -1 ) [

Analytical Model

The lack of detailed bowing measurements of the bowing of polycrystalline diamond -silicon composite wafers means that a method of confirming the prediction of the finite element model is required. A comparison is made with the predictions of an analytical model based on the theory of thermal stresses in elastic multilayer structures described by Hsueh [START_REF] Hsueh | Thermal stresses in elastic multilayer systems[END_REF] and figure 2.

Analytical model with temperature independent properties.

In the analytical method, it is assumed that the silicon-polycrystalline diamond wafer is 100 mm in diameter, perfectly elastic and cooled to room temperature (25°C) from a growth temperature of 800°C ( T = 775°C); this is similar to the deposition temperatures used in Zimmer et al [START_REF] Zimmer | Wide area polycrystalline diamond coating and stress control by sp(3) hot filament CVD reactor[END_REF]. Due to a mismatch in CTEs the film and substrate contract by differing amounts (figure 2b) and since the strain in the wafer is constrained it bows due to asymmetric stresses (figure 2c). If the silicon substrate (with a higher CTE) contracts to a greater degree than the diamond, the silicon is under tension while the diamond film is under compression. The stress-strain distribution in such a bi-layered system is dependent on three parameters. These parameters are the uniform strain component, c, the bending axis, t b , and the radius of curvature, r. The stress-strain parameters can be determined through the total strain in the system, , the normalized biaxial stresses, Sb and Fb , and three sequential boundary conditions discussed in [START_REF] Hsueh | Thermal stresses in elastic multilayer systems[END_REF]. Firstly, the model assumes that both silicon and polycrystalline diamond have isotropic properties to simplify the derivation. Secondly, as the wafer undergoes biaxial strain it has planar geometry and the Young's modulii of the substrate (silicon) and film (diamond) are adjusted according to [START_REF] Hsueh | Thermal stresses in elastic multilayer systems[END_REF] such that

s S Sb E E ν - = 1 (5) F F Fb E E ν - = 1 (6)
where E S and E F are the Young's modulii of the film and substrate respectively, with E Sb and E Fb being the adjusted Young's modulii that take the biaxial strain of the system into account. The Poisson's ratios of the film and substrate are F and S respectively. The normalized biaxial stresses on film and substrate materials and the total strain in the system are

r t z c b - + = ε (7) ) ( T E S Sb Sb ∆ - = α ε σ (8) ) ( T E F Fb Fb ∆ - = α ε σ (9)
Using boundary conditions derived from equations (7) -( 9) [START_REF] Hsueh | Thermal stresses in elastic multilayer systems[END_REF], it is possible to derive the universal strain component, c, the bending axis, t b , and the radius of curvature, r, for an elastic bi-layered strip.

F Fb S Sb F F Fb S S Sb t E t E T t E t E c + ∆ + = ) ( α α (10) ) ( 2 2 2 F Fb S Sb S Sb F Fb b t E t E t E t E t - = (11)
[ ]

) ( ) ( 3 ) 3 2 ( ) 3 2 ( 2 2 2 2 T c t E T c t E t t t E t t t E r F F Fb S S Sb b F F Fb b S S Sb ∆ - - ∆ - - + + = α α (12)
With r derived it is possible to calculate the bow, b, of the wafer using geometry.

Determining Wafer Bow from the Radius of Curvature.

It is possible using simple geometry to determine the bow, b, of the silicon-polycrystalline diamond wafer from its radius of curvature, r, and radius of the wafer, R W . Wafer bow is defined as the change in displacement in the z direction, shown in figure 3. It is assumed in the model that the wafer is a perfect circle and therefore has rotational symmetry; this means that using a two-dimensional derivation as shown in figure 2 is acceptable. Figure 3 shows the geometry used to derive the resultant equation for determining wafer bow and is:

- ± - = r R cos r R cos r R cos R r r b W W W W π π π 4 2 4 3 2 2 2 2 (13)
The sign of the "±" is dependent on the sign of the radius of curvature, with a minus sign used for a negative radius of curvature and vice versa.

It is possible to simplify equation (13) further when 0, when b becomes

2 2 W R r r b - - - = (14)
This small angles approximation holds when the bow is below 3mm and the radius of curvature is negative.

Finite Element (FE) Model

The FE model was developed using commercial software (ANSYS 11.0), chosen for its multi-physics capacity that allows the temperature dependent behaviour of the structural properties to be considered. The model assumes that the wafer obeys simple linear elasticity with no plastic deformation of the substrate or the presence of intrinsic stress within the diamond film. Before a FE model can be used it needs to be tested against either measured data or a rigorous analytical method. There is limited experimental data on the bow of silicon-polycrystalline diamond wafers [START_REF] Zimmer | Wide area polycrystalline diamond coating and stress control by sp(3) hot filament CVD reactor[END_REF][START_REF] Zimmer | Aspects of scaling CVD diamond reactors[END_REF], therefore the only reliable method of validating the model was to compare its results with those of the above analytical approach in which the wafer is assumed to act as an elastic bi-layered strip under biaxial strain. This is a reasonable assumption because the wafer has rotational symmetry, and can be assumed elastic over a wide range of stresses. Zimmer et al. [START_REF] Zimmer | Wide area polycrystalline diamond coating and stress control by sp(3) hot filament CVD reactor[END_REF][START_REF] Zimmer | Aspects of scaling CVD diamond reactors[END_REF] indicated that a non-uniform temperature distribution across the siliconpolycrystalline diamond wafer during processing leads to asymmetric stress distributions that can distort the wafer to a saddle shape. However, using hot filament CVD technology Zimmer et al. produced a thermally uniform wafer [START_REF] Zimmer | Wide area polycrystalline diamond coating and stress control by sp(3) hot filament CVD reactor[END_REF], with a radial stress profile, and more symmetric wafer bow. One of the disadvantages of the analytical method is that it assumes that the stiffness and CTEs do not change with respect to temperature. Figure 4(a) shows schematically the structure modelled using finite element analysis; it consists of a top film layer of polycrystalline CVD diamond and a bottom substrate layer of (111) silicon and is modelled in ANSYS using PLANE13 elements [26]. For efficient computation, axisymmetric boundary conditions were used, as shown in figure 4(a). The PLANE13 element type was chosen because it is fully axisymmetric and its degrees of freedom include the two required spatial dimensions and temperature. The wafer was clamped at the bottom edge to provide a simple measure of wafer bow.

The bow of the wafer was calculated for different thicknesses of the silicon substrate, t s , between 1000 and 3µm. The underlying interest in composite polycrystalline diamond -silicon wafers is their potential as substrates with superior heat sinking for high power GaN electronic and optoelectronic devices, exploiting the superior thermal conduction of polycrystalline diamond. As such, a thick silicon layer is undesirable as it would impair the performance of any potential device. Thus, the primary purpose of the FE model is to model the bowing behaviour of the wafer as it is cooled and subsequently mechanically thinned down to the required silicon thickness. Referring to figures 4(a) and 4(b), the bow is defined as the change in displacement in the z-direction from the bottom right-hand corner of the wafer with respect to the origin.

Comparison of FE Model and the Analytical Method

The main limitation of the analytical model is that it assumes that the Young's modulii, Poisson's ratios and CTEs of both polycrystalline CVD diamond and silicon do not vary with temperature, which could lead to errors in predicted bow. However, it is possible to include the temperature dependence of properties using a linear average of the temperature dependent material properties from table 2 and table 4. The temperature independent averages of the Young's modulii, Poisson's ratios and CTEs of silicon and polycrystalline diamond between 25 and 800°C used in the analytical and FE model are in table 5. The higher linear average CTE of silicon, compared to diamond, leads to the wafer bowing as in figure 2c with the silicon under tension and diamond in compression. In the example used in this work the bowing behaviour of a 100mm diameter composite siliconpolycrystalline CVD diamond wafer, with a polycrystalline CVD diamond thickness t F = 100µm and temperature independent material properties from table 5, have been calculated using the analytical and FE method. The thickness of the silicon substrate, t S , was varied between 3µm and 1000µm and the wafer bow was measured after cooling the composite substrate from 800 to 25°C ( T=775°C). Figure 5 shows a comparison of the predicted wafer bow found from the FE model and the analytical model. The bowing values calculated from the analytical model are shown by the continuous line, whilst those found from the temperature independent FE model are shown by the square data points ( ).

Excellent agreement is observed between the analytical and temperature independent FE models, indicating mutual consistency and implies that the boundary conditions and mesh densities employed in the FE approach are appropriate. Good agreement is expected because both models use identical material properties and have identical geometries. It is of interest to note that both models predict that the bowing is a maximum when the thickness of the silicon almost equals that of the polycrystalline diamond film (t S ~tF ), with the bow then falling as the silicon thickness decreases to zero. The bowing is also reduced, but with a monotonically decreasing rate as the silicon substrate thickness is increased to 1000µm. This behaviour is readily understood at the two extremes, the composite wafer bow is governed by the stiffness of the silicon (t S >>t F ) or the stiffness of the polycrystalline diamond (t S <<t F ) at these conditions.

The results from figure 5 expose a problem likely to arise if a thick silicon substrate is to be thinned by etching or polishing after growth of the CVD polycrystalline diamond to enable optimum heat extraction of any GaN semiconductor device grown on the free silicon surface. The increase in bow as the silicon is thinned will increase the likelihood of wafer breakage owing to the build up of tensile stress in the silicon. One of the advantages of the FE model is that it can be readily extended to include the temperature dependent material properties from table 2 and table 4 as shown by open triangular data points ( ) in figure 5. The most significant result from the temperature dependent FE model is that it predicts a lower degree of wafer bowing in comparison to the results from the analytical and temperature independent FE models. In fact at low temperatures the CTE of diamond is lower than silicon while at higher temperatures the CTE of diamond is higher than silicon (Figure 1). This is significant because this suggests that the temperature independent models overestimate the degree of bowing. Understanding these effects is vital because it is important to engineer the degree of bow to be as small as possible. One of the ways the bowing effects have been explored using the temperature dependent FE model is by varying the thickness of the polycrystalline diamond film, as shown in figure 6.

Figure 6 shows the predicted bow of five different wafers with diamond film thicknesses between 25 and 200 µm. The same process of simulating the thinning of the silicon layer from 1000 to 10 µm was applied to each wafer. Figure 6 reveals that the maximum bow is smaller as the polycrystalline diamond layer thickens. The temperature dependent FE model predicts that increasing the thickness of the polycrystalline diamond from 100 to 200 µm approximately halves the wafer bow. The maximum bow is reduced when the polycrystalline diamond thickness is increased due to the diamond having a stiffening effect on the wafer. When the thickness of the polycrystalline diamond layer approaches 200 µm, the 100 mm diameter wafer will have a predicted maximum bow of ~1 mm from centre to edge. The breadth of this maximum means that the wafer will not see the dramatic increases in bow associated with thinning the silicon layer of wafers with a polycrystalline diamond thickness of ≤100 µm, indicating a potentially viable design. Refering to Figure 1, the CTE of GaN is greater than both diamond and silicon. If GaN is epitaxially grown on the free silicon surface, i.e. the underside of the silicon in Figures 2c and4b, then the higher CTE of GaN could increase bow; although the GaN device layers are expected to be relatively thin (a few microns) [START_REF] Zimmer | Enhancing semiconductors with diamond[END_REF].

A final point is that the analytical and FE models explored in this paper assume that the wafer is perfectly elastic with no plastic deformation of the silicon substrate. Lattice dislocations can partially relax the wafer, reducing the overall amount of bow; however experimental measurements by Zimmer et al [START_REF] Zimmer | Wide area polycrystalline diamond coating and stress control by sp(3) hot filament CVD reactor[END_REF] show that on removal of the diamond by oxidation there was no observable plastic deformation of the silicon.

Comparison of FE Model with Experimental Data

The only available experimental data found in the literature for bow of silicon -polycrystalline diamond wafers is in Zimmer et al [START_REF] Zimmer | Wide area polycrystalline diamond coating and stress control by sp(3) hot filament CVD reactor[END_REF]. The thickness of the silicon layer was 525 µm [START_REF] Chandler | sp3 Diamond: Wafer thickness[END_REF] and the wafer diameter was 100 mm [START_REF] Zimmer | Wide area polycrystalline diamond coating and stress control by sp(3) hot filament CVD reactor[END_REF]. The deposition layer of polycrystalline diamond was increased from 2 to 20 µm at 2 µm intervals. FE models of diamond-silicon wafer were generated at three different temperatures (715, 755 and 800°C) to compare with the experimental observations; where 715 and 755 °C are the growth temperatures used in [START_REF] Zimmer | Wide area polycrystalline diamond coating and stress control by sp(3) hot filament CVD reactor[END_REF]. The temperature dependent elastic and CTE data of silicon and diamond were used in the model and figure 7 indicates the model predictions.

Zimmer et al highlighted that a cooler processing temperature (e.g. 715°C) leads to a higher compressive stress in the diamond films and a higher degree of bow compared to 755°C. It was stated that this observation did not correlate with published CTE data for diamond and silicon. However the modelling results in figure 7, where bow is only generated by CTE mismatch, clearly demonstrate that higher processing temperatures can lead to reduced bow due the CTE of diamond being larger than silicon at higher temperatures (Figure 1). It was also experimentally observed that the rate of change of stress, and bow, decreased as the diamond film becomes thicker. This is also observed in figure 7 and is related to the overall stiffness of the diamond-silicon wafer increasing with increasing diamond thickness. The absolute value of bow in the model is greater than the experimental measurements by a factor of three to four. This could be related to plastic deformation of the silicon, the diamond film exhibiting a lower elastic modulus compared to the model values or intrinsic stress within the diamond layer. For example, papers [START_REF] Wilks | Properties and Applications of Diamond[END_REF][START_REF] Field | The Properties of natural and synthetic diamond[END_REF] report relatively low elastic coefficients for polycrystalline diamond due to imperfections and impurities. To examine the influence of changes in the elastic modulus of diamond on bow, the experimental data of Zimmer et al. was compared with model data for diamond grown at 715°C at different thicknesses on a silicon wafer 525 µm thick and 100 mm diameter [START_REF] Zimmer | Wide area polycrystalline diamond coating and stress control by sp(3) hot filament CVD reactor[END_REF], figure 8. As observed in figure 7, the rate of change of bow decreases as the diamond film becomes thicker as observed with the experimental data in Figure 8 ( ). The best agreement corresponds to a low diamond elastic modulus of 300GPa. Chiu et al. [START_REF] Chiu | Elastic modulus of and residual stresses in diamond films[END_REF] and Kim et al. [START_REF] Kim | Measurement of Residual Stress in Diamond Films Obtained Using Chemical Vapor Deposition[END_REF] demonstrated that the elastic modulus of diamond CVD films decreased considerably (from 1000 to 300GPa) with increasing CH 4 /H 2 . Windischmann et al [START_REF] Windischmann | Intrinsic stress in diamond films prepared by microwave plasma CVD[END_REF] indicated that an intrinsic and residual tensile stress can be present in diamond films grown on silicon which could also influence bow and the level of stress varies with process temperatures and CH 4 /H 2 ratio [START_REF] Chiu | Elastic modulus of and residual stresses in diamond films[END_REF]. Zimmer et al. reported a decrease in bow with increased diamond thickness as a result of intrinsic stress [START_REF] Zimmer | Enhancing semiconductors with diamond[END_REF]; the level of the intrinsic stress can depend on quality, growth conditions, microstructure, texture and surface state (roughness) along with non-uniform heating.

Orientation of silicon and diamond

As the diamond layer grows thicker the diamond can develop an orientation in the growth direction and grains become columnar [START_REF] May | Diamond thin films: a 21st-century material[END_REF]. For diamond textured along [111] the modulii 1/S 11 = 1168GPa and 1/S 33 = 1206GPa [32] at room temperature, which are both higher than the 1050GPa for polycrystalline material used in the model (see both [32] and Table 2). Finite element models were undertaken with a 100mm diameter (111) silicon substrate and a 100µm diamond layer. By replacing the polycrystalline diamond with orientated diamond the maximum bow reduced from 2733µm to 2673µm; the higher stiffness of the diamond restricting the wafer bow to a small extent. While the (111) orientated silicon is the preferred substrate orientation for epitaxial GaN growth, the (100) silicon surface is also used [START_REF] Rudder | Direct deposition of polycrystalline diamond films on Si(l00) without surface pre-treatment[END_REF]. The elastic modulus (E) and Poisson ratio (ν) exhibit an isotropic character on the (111) silicon plane [START_REF] Cho | Characterization of Young's modulus of silicon versus temperature using a ''beam deflection" method with a four-point bending fixture[END_REF] with E=169GPa and ν=0.26 at room temperature, see [START_REF] Cho | Characterization of Young's modulus of silicon versus temperature using a ''beam deflection" method with a four-point bending fixture[END_REF] and Table 4. For (100) silicon, E varies from 169GPa to 130GPa along the (100) plane and Poisson ratio varies from 0.278 to 0.062. Based on finite element models run for (100) silicon with polycrystalline diamond the maximum bow (where the thickness of the diamond and silicon was similar) decreased from 2733µm to 2543µm, which may be due to lower effective silicon stiffness limiting deformation of the diamond film (ignoring Poisson effects). For both of these cases the changes in bow are relatively small compared to issues related to precise diamond elastic properties used due to individual growth conditions and the presence of intrinsic stress; although it seems wafer bow is more sensitive to changes in the elastic properties of the more compliant silicon substrate compared to the diamond. This may warrant a more detailed study to examine the influence of silicon and diamond orientation [32] and wafer geometry on bow.

Conclusions

A FE model and analytical method have been developed to determine the bow of silicon-polycrystalline CVD diamond composite wafers. The model found that the maximum bow and resulting peeling moment of a particular wafer structure occurred at a point where the thicknesses of both the silicon and polycrystalline diamond layers were nearly identical, in close agreement with an analytical model. The finite element approach enabled the variation in silicon and diamond properties with temperature to be modelled, which led to a smaller predicted wafer bow when compared to models using a single linear average and allows for prediction of bow at different growth conditions. The results demonstrate that the growth of diamond on a thick silicon substrate can lead to lower levels of bow. As the silicon substrate is subsequently thinned to reduce its effective thermal resistance the bow increases considerably, especially when the thicknesses of the two materials are similar. The model also predicts experimental observations that lower processing temperatures can lead to larger degree of bow and the rate of change of bow decreases with increasing diamond thickness. Differences between model and experimental data are likely to be due to differences in the experimental conditions and materials properties/quality, such as elastic properties and the presence of intrinsic stress in the diamond layer. The modelling approach could also be used to evaluate the influence of diamond thinning on wafer bow [START_REF] May | The New Diamond Age?[END_REF]. Realising the desirable combination of high thermal conductivity and large Young's modulus of polycrystalline CVD diamond and the high crystal quality of (111) silicon as a platform for epitaxial growth of GaN for sensors and high temperature electronics will require close optimisation of the growth temperature and thicknesses of the two layers forming the composite substrates.

Figure 1 .Figure 2 .

 12 Figure 1. Thermal expansion coefficients of silicon[9], diamond [7,8] and GaN [10],

Figure 3 .

 3 Figure 3. Geometry used to determine the wafer bow.

Figure 4 .Figure 5 .Figure 6 .Figure 7 .

 4567 Figure 4. (a) Schematic of the wafer model. (b) FE model calculation of wafer bow, which is the displacement in the -z direction.

Figure 8 .

 8 Figure 8. Bow as function of diamond stiffness for 525 µm thick Si at 715°C

Table 1 .

 1 Elastic

		modulii of
	diamond at room temperature
	[17].
	Modulus	(GPa)
	c 11	1079
	c 12	124

Table 3 .

 3 0K Elastic modulii and fitting constants of silicon[START_REF] Varshni | Temperature dependence of the elastic constants[END_REF].

	c ij	c ij 0 (GPa) u (GPa)	t (K)
	c 11	167.5	5.33	407.0
	c 12	65.0	2.31	334.8
	c 44	80.0	2.25	497.4

Table 5 .

 5 Temperature independent (linearly averaged) material properties of silicon and polycrystalline CVD diamond between the temperatures of 25 and 800°C.

	Material	E (25 -800°C) /	(25 -800°C)	(25 -800°C) /
		GPa		10 -6 K -1
	Silicon	165.5	0.26	3.45
	Polycrystalline Diamond	1014.0	0.1	2.8

Table 2

 2

	Data
	Zimmer et al. [4]
	900 GPa
	800 GPa
	700 GPa
	600 GPa
	500 GPa
	400 GPa
	300 GPa
	200 GPa

Acknowledgements

The research leading to these results has received funding from the European Community's Seventh Framework Programme FP7/2007-2011 under grant agreement n°214610, project MORGaN. This publication reflects only the author's views and that the Community is not liable for any use that may be made of the information contained therein.


M Carbucicchio
email: massimo.carbucicchio@fis.unipr.it

R Ciprian

L Nasi

Effects of deposition temperature and elemental layer thickness on the properties of Fe/Co multilayers grown by molecular beam epitaxy

Fe/Co multilayers were grown at different temperatures up to 200°C onto MgO-(100) monocrystalline substrates, varying both the Co and Fe elemental layer thickness. All samples show a multilayered structure where the elemental layers are separated by a thin interfacial region which, for high-temperature growths, is constituted by an equiatomic FeCo compound. The multilayers grown at room temperature show an in-plane uniaxial magnetic anisotropy. The increase of the growing temperature determines an improvement of the sample smoothness and the appearance of two preferred in-plane orientations of the easy magnetization axis having different strength. By increasing the Co layer thickness, the in-plane magnetic behaviour becomes isotropic, while by increasing the Fe layer thickness, an out-of-plane contribution to the magnetization vector establishes which is responsible for the appearance of a well defined magnetic morphology constituted by stripe-like domains.

Introduction

The great interest addressed to Fe/Co multilayers is due to their unique magnetic properties which found applications in the magnetic recording and the development of giant and tunneling magnetoresistance systems. These properties are strongly influenced by the microstructure and can be finely tailored adjusting the growing parameters. In particular both coercivity and anisotropy can be tailored by changing the film thickness, the order deposition of the elemental layers [1] and the degree of interface roughness [2] and crystallinity [3][4][5]. These last characteristics in turn, are largely depending on the nature and crystallinity of the substrates [1,6] and on the deposition temperature, which can determine the establishing of peculiar crystallographic textures [6][7][8][9]. While many works have been carried out on Fe/Co films grown at room temperature, low attention has been addressed to the effects that the growing temperature plays on the microstructural and magnetic properties of Fe/Co multilayers.

In the present work, thin Fe/Co multilayers were deposited by molecular beam epitaxy (MBE) at different temperatures onto MgO-(100) monocrystalline substrates, changing the Fe and Co layer thickness. The magnetic and microstructural properties were studied as a function of both the growing temperature and the elemental layer thickness.

The increase of the growing temperature leaves almost unchanged the coercivity of the multilayers while appreciably influences the in-plane uniaxial anisotropy and the Fe/Co interface intermixing. On the other hand, the elemental layer thickness influences the anisotropy dispersion along the easy magnetization axis of the multilayers grown at room temperature. For higher growing temperatures, the increase of the elemental layer thickness appreciably modifies the magnetic properties giving rise to an isotropic in-plane behaviour and to a perpendicular contribution to the magnetization vector.

Experimental

Co/Fe multilayers, 60 nm thick, were grown in ultra-high vacuum onto MgO-(100) monocrystalline substrates by means of a MBE system equipped with electron guns, at different temperatures ranging from room temperature (RT) up to 200°C. The deposition rates were for both elements ~0.6 nm/min, and the layer thickness was controlled "in-situ" by means of a quartz microbalance. All multilayers were grown starting from Co and capped with an overlayer of Co 3 nm thick. To study the effects of the Co and Fe layer thickness on the magnetic properties of the multilayers, two series of samples were grown. In the first series the Fe layer thickness was fixed to 9 nm while the Co layer thickness (t Co ) was changed in the range 3 ÷ 11 nm. In the second series the Co layer thickness was 3 nm and the Fe thickness (t Fe ) was changed in the range 9 ÷ 17 nm. In the following the two series of samples were labelled as T-N[Co(t Co )/Fe(t Fe )] where T is the growing temperature in °C and N the Co/Fe bilayer repetition number.

Conversion Electron Mössbauer Spectroscopy (CEMS) was performed by means of a 50 mCi 57 Co(Rh) source, with the -rays at an incident angle of 15°. The isomer shifts are referred to -Fe. The spectra were fitted with a superposition of Lorentzian lines by means of a leastsquares minimization routine with a combination of linear and non-linear regressions.

The X-ray diffraction (XRD) measurements were performed in -2 geometry by means of a Philips X'Pert diffractometer using the Cu(K ) radiation ( = 0.154 nm).

The magnetic measurements were carried out with a magneto-optical Kerr effect (MOKE) magnetometer using a s-polarized 633 nm He-Ne laser light and the modulation-polarization technique. The angular dependence of the normalized remanence, ( ) / r s M M α , was studied applying the magnetic field parallel to the film plane at different angles, . r M is the remanence recorded for zero applied field and s M is the saturation magnetization. The magnetic interactions among ferromagnetic grains and layers were studied by means of the M-plot, where ( )

M H ∆ is defined as: ( ) ( ) ( ) 1 2 IRM DCD M H M H M H ∆ = -- .
( )

IRM M H and
( ) DCD M H are respectively the isothermal and dc demagnetized remanence curves normalized to the saturation magnetization, and are obtained by plotting the remanence as a function of an increasing positive/negative magnetic field applied to a previously demagnetized/saturated samples. For a non-interacting assembly of particles M ∆ is zero for any values of H. When exchange-coupling or demagnetizing interactions establish among the grains, M ∆ shows positive or negative values respectively. Conventional and high resolution transmission electron microscopy (HRTEM), high angle annular dark field (HAADF) and energy dispersive X-ray spectroscopy (EDS) in scanning mode (STEM) were performed on the 200-3[Co(11)/Fe (9)] and 200-3[Co(3)/Fe(17)] multilayers using a JEOL 2200FS microscope working at 200 keV. The STEM-HAADF and selective area EDS analyses provide atomic number (Z)-dependent contrast images, thus revealing the overall quality of the multilayers. Cross-section specimens were prepared by mechanical grinding and dimpling, followed by Ar ion milling in a Gatan PIPS apparatus.

Atomic and Magnetic Force Microscopy (AFM, MFM) were performed in ultra-high vacuum in tapping mode. In particular, MFM images were obtained with a CoCr-coated tip probe, in lift mode, at a distance of 40 nm from the sample surface. The values of the surface roughness were calculated from AFM images as the average deviation of the roughness profile from a mean line over the evaluation length. The sensitivity of AFM allows measuring the surface roughness over a very wide dynamic range from 0.03 nm to 500 nm.

Microstructural Properties

The CEMS spectrum measured for the RT-5[Co(3)/Fe(9)] multilayer, figure 1(a), can be fitted as the superposition of three contributions: a main sextet (α) due to α-Fe, and two series of sextets (ζ and χ), whose hyperfine magnetic field (H hf ) distributions are reported beside the spectrum. These contributions can be attributed respectively to a FeCo alloy having an almost equiatomic composition, and to a solid solution of Fe into Co, Co(Fe) ss [6]. The corresponding XRD pattern reported in figure 2(a 9)] multilayer a decrease of the amount of the Co(Fe) ss can be deduced, followed by a slight increase of the α-Fe amount. For the film grown at 200°C, the Mössbauer spectrum, figure 1(b), shows the presence of α-Fe and CoFe, while the contribution from Co(Fe) ss has completely disappeared. This fact can be explained considering the higher atomic mobility which favours an elemental "layer-bylayer" growth, this way lowering the interface roughness and defectiveness, and therefore the presence of low density zones where atom interdiffusion can preferentially occur [6]. From the relative areas of the contributions to the Mössbauer spectrum, the thickness of the CoFe interfacial layer was evaluated to be ~1.4 nm, a value that remains practically unaffected by the increase of the thickness of the elemental layers.

The XRD spectrum of the 100-5[Co(3)/Fe( 9)] shows beside the reflection due to the substrate, the presence of a small (200) reflection which corresponds to the bcc Fe(200) and bcc FeCo(200) reflections [5,8]. By increasing the growing temperature up to 200°C, figure 2(b), the full width at half maximum of the (200) reflection slightly decreases while its intensity increases, suggesting an improvement of the sample crystallinity and an increase of the grain size. No other diffraction peaks were observed in the spectrum indicating the absence of any additional phases and crystal orientations, and therefore testifying the presence of a preferred [100] texture of the multilayer [5,8]. The satellite peaks often observed around the main reflection in multilayered structures are absent probably due to the very low thickness of the multilayers which can hinder the establishing of a good structural coherence among the layers in particular on going toward the sample surface.

Magnetic Properties

The angular dependence of the normalized remanence / r s M M and the M-plot for the Fe/Co multilayers are reported in figure 7. The magnetic behaviour of the RT-5[Co(3)/Fe (9)] is characterized by a sine-like polar plot indicating the presence of an in-plane uniaxial anisotropy, figure 7(a). The corresponding M ∆ curve reported in figure 7(b), except for a small positive contribution, is almost equal to zero for any applied field, indicating that the multilayer behaves as a non-interacting assembly of particles.

For the multilayers grown at room temperature, the increase of the Co layer thickness determines only a slight decrease of the anisotropy dispersion and therefore a slight increase of the in-plane anisotropy. On the contrary the increase of the Fe layer thickness induces an increase of the anisotropy dispersion allowing the establishing of a more isotropic magnetic behaviour. By increasing the growing temperature, the coercivity remains practically unchanged while the normalized remanence evaluated along the easy axis monotonically increases (figure 8 for the 5[Co(3)/Fe(9)] multilayers). On the contrary along the hard direction the normalized remanence remains unchanged up to growing temperatures of ~60°C, upon which it slightly increases indicating the establishing of a small perpendicular contribution to the magnetization vector. Moreover, a second preferred in-plane orientation of the easy magnetization axis establishes, and strengthens, giving rise, for the multilayer grown at 200°C, to a magnetic behaviour having a four-fold symmetry [figure 7(a)]. Along the hard direction, i.e. at 45°with respect to both the easy directions, the in-plane hysteresis loop shows a "transcritical shape", figure 9(a), which is typical of systems showing an appreciable out-of-plane component of the magnetization vector. The loop is due to the superposition of two contributions: (i) an abrupt transition at low applied fields due to wall displacement and (ii) an almost linear approach to the saturation for high applied fields, due to the progressive rotation of the magnetic moments [11]. The positive values of the M ∆ curve in figure 7(b) indicate that the high-temperature growth favours the establishing among the layers of a significant exchange-coupling interaction. By increasing the Co layer thickness up to 6 nm, 200-4[Co(6)/Fe(9)], the two in-plane preferred orientations of the easy magnetization axis become comparable in strength, giving rise to a more in-plane isotropic behaviour as shown in figure 7(c). This may well be linked to the establishing of the [011] crystallographic texture, figure 2(c). The M ∆ values become negative indicating the presence of demagnetizing interactions, figure 7(d). This change of the M ∆ sign cannot be ascribed to a lower interlayer exchange-coupling interaction due for example to the presence of second phases into the multilayers or to the interfacial FeCo regions, since:

(i) no appreciable difference can be detected between the Mössbauer spectrum of the 200-4[Co(6)/Fe (9)] and that of the 200-5[Co(3)/Fe (9)] having a lower Co layer thickness, and (ii) the shape of the MOKE loops suggests a behaviour typical of a single-layer magnet.

On the contrary, it seems reasonable to assume that, among the small grains that form in the uppermost part of the multilayer [figures 3(c) and 4(a)], magnetostatic interactions can establish, in this way reducing the effectiveness of the interlayer exchange-coupling interactions.

A further increase of the Co layer thickness up to 11 nm leaves unchanged the magnetic behaviour, but determines the establishing of exchange-coupling interactions among both grains and layers, figure 7(d), reasonably because of a higher grain-packing fraction that gives raise to a smoother sample surface, figure 3(d).

The coercivity for the Fe/Co multilayers grown at 200°C, figure 10 The coercivity as a function of the Fe layer thickness, reported in figure 10, initially decreases reaching a minimum for t Fe = 12 nm and then slightly increases. This trend can be explained considering that the Fe/Co multilayers can be treated as exchange-coupled spring magnets, i.e. systems constituted by soft (Fe) and hard (Co) magnetic phases exchange coupled at the interfaces. The increase of the soft Fe layer thickness gives rise to the initial softening of the magnetic behaviour of the multilayer. When the thickness of the soft layer becomes much higher than that of the hard one the structural disorder increases favouring an increase of the preferential sites for the domain wall pinning [12].

Conclusions

Fe/Co multilayers were grown at different temperatures onto MgO-(100) monocrystalline substrates, changing the Co and Fe layer thickness.

By growing the multilayers at room temperature, a thin interfacial region forms between the elemental layers constituted by FeCo and Co(Fe) ss . The magnetic behaviour is typical of a system showing an in-plane uniaxial magnetocrystalline anisotropy. The increase of the elemental layer thickness induces a slight decrease (in the case of Co) or increase (in the case of Fe) of the anisotropy dispersion along the easy magnetization direction.

The increase of the growing temperatures determines a lower diffusion at the interfaces with the gradual disappearance of the Co(Fe) ss and the formation of the only FeCo compound in the multilayers grown at 200°C. For this temperature, the samples are characterized by (i) regularly modulated layers of Fe and Co, which start to rough at a depth of ~30 nm from the surface, (ii) a [100] crystallographic texture, and (iii) two in-plane preferred orientations of the easy magnetization axis.

The increase of the Co layer thickness determines the establishing of an isotropic in-plane magnetic behaviour and a preferred [011] crystallographic texture with a structural relationship The increase of the Fe layer thickness up to 12 nm, for growing temperatures of 200°C, determines (i) an improvement of the [100] texture of the Fe/Co multilayers and (ii) an appreciable out-of-plane component of the magnetization vector which gives rise to a magnetic morphology constituted by stripe-like domains. For Fe layer thickness higher than 12 nm, the magnetization vector preferentially lies into the film plane and the structural relationship [011]Fe bcc ||[100]Co fcc ||[100]MgO establishes as in the case of the multilayers with the Co layer thickness higher than 3 nm.

  ) shows only the (200) reflection due to the MgO substrate.

Figure 1

 1 Figure 1 CEMS spectra for the (a) RT-5[Co(3)/Fe(9)] and (b) 200-5[Co(3)/Fe(9)] multilayers. The hyperfine field distributions for the χ and ζ contributions are reported beside the spectra. The increase of the growing temperature up to 200°C, contrary to what normally expected, determines a lower intermixing at the interfaces. From the analysis of the relative areas of the contributions to the Mössbauer spectrum performed for the 100-5[Co(3)/Fe(9)] multilayer a decrease of the amount of the Co(Fe) ss can be deduced, followed by a slight increase of the α-Fe amount. For the film grown at 200°C, the Mössbauer spectrum, figure1(b), shows the presence of α-Fe and CoFe, while the contribution from Co(Fe) ss has completely disappeared. This fact can be explained considering the higher atomic mobility which favours an elemental "layer-bylayer" growth, this way lowering the interface roughness and defectiveness, and therefore the presence of low density zones where atom interdiffusion can preferentially occur[6]. From the relative areas of the contributions to the Mössbauer spectrum, the thickness of the CoFe interfacial layer was evaluated to be ~1.4 nm, a value that remains practically unaffected by the increase of the thickness of the elemental layers.

Figure 2

 2 Figure 2 XRD pattern for the Fe/Co multilayers.

Figure 3

 3 Figure 3 AFM images (2 × 2 µm 2 ) for the Fe/Co multilayers.

Figure 4

 4 Figure 4 (a) HAADF cross section image of the 200-3[Co(11)/Fe(9)] sample; (b) EDS map for Mg (red), Co (blu) and Fe (green) taken in the same region.

Figure 5

 5 Figure 5 (a) (010) cross-sectional HREM of sample 200-3[Co(11)/Fe(9)] with the FFT pattern of the MgO substrate in the inset; (b) FFT pattern of the Co/MgO interface region and (c) portion of the FFT pattern taken at the first Fe/Co interface.

Figure 6 (

 6 Figure 6 (010) cross-sectional HREM of the multilayer surface showing three different bcc Fe(Co) grains separated by grain boundaries. The FFT patterns of the differently oriented grains are shown in the insets.

Figure 7

 7 Figure 7 The

Figure 8

 8 Figure 8 Coercivity and normalized remanence evaluated along the easy and hard axis directions as a function of growing temperature for the 5[Co(3)/Fe(9)] multilayers.

Figure 9

 9 Figure 9 Longitudinal hysteresis loops for (a) 200-5[Co(3)/Fe(9)] measured with the field applied along the in-plane hard direction and (b) 200-4[Co(3)/Fe(12)] multilayer.

  , increases as a function of the Co layer thickness and reaches a value of ~80 Oe for the 200-3[Co(11)/Fe(9)] multilayer.

Figure 10

 10 Figure 10 Behaviour of the coercivity as a function of the Co and Fe thickness for the Fe/Co multilayers grown at 200°C.

Figure 11

 11 Figure 11 MFM images (2 × 2 µm 2 ) for (a) 200-4[Co(3)/Fe(12)] and (b) 200-3[Co(3)/Fe(17)].

  [011]Fe bcc ||[100]Co fcc ||[100]MgO. Depending on the Co layer thickness, the intergrain interactions favour alternatively the magnetized or demagnetized state of the multilayer.


Jian Chen

Gerard A Bell

Hanshan Dong

James F Smith

Ben D Beake

A study of low temperature mechanical properties and creep behavior of polypropylene using a new sub-ambient temperature nanoindentation test platform

Keywords: nanoindentation, sub-ambient temperature, polypropylene, creep

Mechanical properties and creep behaviour of an atactic-polypropylene (aPP) have been studied in the vicinity of its glass transition temperature (-18ºC) via a nanoindentation platform integrated with a sub-ambient temperature capability. All low temperature tests were validated by measurements on a fused silica reference sample from 25ºC to -30ºC. The fused silica results showed virtually invariant elastic modulus with temperature over this range consistent with literature measurements by sonic resonance. Hardness and elastic modulus of aPP increased as the test temperature decreased and the amorphous regions went through the glass transition. The creep behaviour was analysed using two approaches: (i) a logarithmic method, and (ii) the Boltzmann integral method. The results showed that the creep extent decreased as the temperature was reduced, and for the time constants obtained there were upper-limit values at -10ºC, about 8ºC above the quoted glass transition temperature. The strain rate sensitivity obtained by the logarithmic method also showed a maximum at -10ºC.

Introduction

Nanoindentation enables local probing of spatial variations in mechanical properties, time-dependent behaviour, and strain rate dependence. Such knowledge of materials on a small scale is often key to their optimization for practical applications [1][2][3][4][START_REF] Beake | Nanomechanical Testing under Nonambient Conditions[END_REF]. Normally, these tests are carried out under ambient conditions that may not be close to the actual working conditions for many applications. For example, temperature can play an important role in materials properties; indeed, many mechanical properties are strongly temperature-dependent, for instance a system can operate safely at one temperature and fail when the temperature is changed [6][7][START_REF] Fox-Rabinovich | Surface &amp[END_REF][9]. To address this problem, Confidential: not for distribution. Submitted to IOP Publishing for peer review 15 September 2010 several approaches have been used to probe the surface properties of materials under their actual service conditions. For example, nanoindentation has been utilized to test polymers, metals and hard coatings at elevated temperatures [9][10][11][12][13], and the results have been correlated with the performance of coatings in extreme applications such as high speed machining.

Similarly, various engineering activities are carried out at sub-ambient temperatures, including aerospace, marine and cryo-machining applications [7,[14][15][16][17]. Materials in-service may be subjected to mechanical loading at low temperatures, for instance. For optimization of wear resistance, in particular, it is highly desirable to determine the mechanical properties under true in-service conditions. Although there are macro-scale sub-ambient temperature commercial mechanical testers [7], up until now there have been no reports of low temperature nanoindentation capability.

A popular engineering polymer, polypropylene, is in the rubbery state at room temperature. Its mechanical properties are expected to vary strongly with reducing temperature as it transforms to the glassy state [14,[START_REF] Zrida | Polymer Testing[END_REF][START_REF] Cowie | [END_REF][20], i.e. when the material passes through its glass transition temperature. Moreover, polypropylene shows temperature-dependent viscoelastic behaviour under loading. Thus its time-dependence around the glass-rubber transition temperature is also of interest in this study.

In light of the above comments, a newly-developed sub-ambient temperature capability based on the commercial NanoTest platform TM (Micro Materials Ltd. Wrexham, UK) was therefore utilized to investigate the mechanical properties and creep behaviour of polypropylene around its glass-rubber transition temperature. There are several approaches to analyzing nanoindentation creep data. For example, Bower et al. [21] employed a similitude analysis for the stress and strain field during indentation. Applying a transformation method, the stress exponent of the creep response can be obtained from the displacement-time data during the hold period. Likewise, other researchers [22][23][24][25] have analysed creep data assuming that the steady-state creep strain rate follows a power law. Of particular significance in this study, two different approaches were adopted to analyse the temperature-dependent creep behaviour of the polypropylene.

Experimental

The sub-ambient temperature measurement capability was developed and integrated with the NanoTest instrument in a manner which preserved normal sample motion and operation routines. This nanoindentation instrument employs horizontal loading as described in detail previously [26]. For sub-ambient measurements three key design changes were made: 1) environmental control, 2) cooling on the sample side, 3) cooling on the indenter side. This capability provided cooling of both the sample and the diamond indenter down to -30ºC without water vapour condensation and thermal drift problems. A reference sample (fused silica) was tested to validate this equipment. A schematic of the new system is shown in Figure 1(a). The design used the same methodology -separate thermal control of both indenter and sample to minimise/eliminate heat flow on contact -used in high-temperature nanoindentation [11,13], which has been successful in performing tests to 750ºC. The sample and diamond indenter were attached to thermoelectric coolers (Peltier elements) as shown in Figure 1(b). To eliminate condensation at sub-ambient temperatures, the whole pendulum system was enclosed in a sealed chamber, which was purged with argon gas. This reduced the water vapour concentration to 300ppm (V). The detailed system design is documented elsewhere [START_REF] Bell | ENSINGER, TECAFINE PP Data Sheet[END_REF].

The test materials were fused silica and atactic-polypropylene (aPP). The fused silica was obtained from Starna Scientific Ltd. Essex UK. Fused silica is commonly used as a nanoindentation reference material, with a Young's modulus of 72 GPa. The aPP was obtained from ENSINGER Ltd, Mid Glamorgan, UK [28] with a quoted glass transition temperature of approximately -18ºC. A Berkovich indenter was used for all tests. Indentations were load-controlled to different peak loads at various loading rates as shown in Table 1. The loading history included a hold at peak load before unloading in order to minimize the influence of viscoelastic deformation during subsequent unloading. For aPP, this hold period was set to 20s, and displacement data at constant load was collected during this period to analyse the creep response of the aPP. The Oliver and Pharr method [1] was used to extract hardness and elastic modulus values from the unloading data. Unloading rates were set equal to the loading rates. The separation distance between indentations was 35µm and each test was repeated 5 times. To study the time-dependent behaviour of aPP at different temperatures, the creep data were analysed and fitted using two approaches: (i) the logarithmic method discussed in Refs. 23 and 24, and (ii) the Boltzmann integral method discussed in Refs. [25][26][START_REF] Bell | ENSINGER, TECAFINE PP Data Sheet[END_REF] The logarithmic equation can be expressed as

+ = ∆ 1 ln L h t A d τ (1)
where ∆d and t h are the increase in depth and hold time during hold period, A and L τ are termed the extent parameter and the time constant, respectively. Equation (1) can also be expressed as the creep strain ( ) ( )

+ = = = ∆ 1 ln 0 0 L h h h t t d A t d d τ (2)
where d(t h =0) is the initial penetration depth at the beginning of the hold (creep) period, ∆d/d(t h =0) is the creep strain, A/d(t h =0) (the dimensionless fractional increase in depth during creep ) is known as the strain rate sensitivity parameter [START_REF] Chudoba | Surface &amp[END_REF].

Although only two variables -measures of extent and rate -are used to describe the visco-deformation behaviour, in practice the creep curves normally show a quasi-logarithmic form [30], thus producing an excellent fit to the raw data.. This method can be used not only for linear viscoelastic materials, but also non-linear viscoelastic materials, and with the quality of the fit it is possible to predict the creep response over a relatively long time. In the past, this method has been used for a range of polymer systems to identify changes in creep behaviour with density of cross-linking [26].

The major limitations of this method are that the equation is empirical and does not explicitly take into account differences in the time taken to reach the peak load, though the fitting parameters reflect this. It is not possible to deconvolute the particular contributions of elasticity, plasticity, viscoelasticity and visco-plasticity by this approach.

In other work [31][32][33][34][35], assuming the materials is linear viscoelastic, different combinations of springs and dashpots have been used to simulate the instantaneous elasticity, instantaneous plasticity, viscoelasticity and visco-plasticity. The creep compliance function is then deduced. The penetration depth can be calculated using the Boltzmann integral operator. This methodology has been successfully adapted by Oyen [4,[34][35][36] to simulate the whole loading and holding curve. However, a better fit required more variables, which can degrade the accuracy of the solution [36]. Compared to the logarithmic method, this needs more time and a careful selection of the initial value to make the iteration converge. In the present study, a free spring in series with a Kelvin parallel spring and dashpot element was used as the temperature effect can be studied using the time constant from the only dashpot. The background of this method is briefly described as follows.

For a conical-pyramidal indenter, the indentation depth can be expressed as

( ) ( ) - ⋅ = t du du dP u t J m t h 0 2 (3)
where h(t) is the penetration depth, m is constant, P is the load, t is the time, u is a dummy variable of integration for time and J(t) is the material's creep function. The creep function J(t) can be expressed as

( ) - - = B t c c t J τ exp 1 0 (4)
where c 0 and c 1 are the creep function coefficients, τ B is the creep time. The solved equation (3) using the creep function of equation ( 4) gives

2 1 1 ' 1 ' 0 1 exp exp ) ( - - - = B R B R t t k c kt c t h τ τ τ (5)
where t R is the loading time,

Results

Fused silica was used as a reference sample and tested from room temperature (25ºC) down to -30ºC. To ensure isothermal contact conditions, prior to indentation the diamond was held at the specimen surface for 5 to 30s under minimal load. The recorded loading-unloading curves showed good repeatability. The elastic modulus (E) was found to be (72 ± 2) GPa over the temperature range +25 to -30ºC which agrees with the data in Ref [START_REF] Munro | Elastic Moduli Data for Polycrystalline Ceramics[END_REF] .

The hardness (H) and elastic modulus (E) results of aPP were measured at different loading rates and temperatures. Indentation tests with the highest loading rate 25mN/s (corresponding to only 2s loading time) resulted in approximately 10% higher modulus for all temperatures studied. There was no clear difference in H and E between the three slowest loading rates, 10mN/s, 2.5mN/s and 1mN/s (loading time 5s, 20s and 50s, respectively). The average values of H and E of aPP were therefore determined from these and are shown in Figure 2. It is clear that the hardness and elastic modulus increased as the testing temperature decreased. The scattering of data can be attributed to the relatively high surface roughness of aPP (Ra ~0.67 µm). To minimize the effect of the roughness, a high testing load of 50mN was used. The corresponding maximum depth was about 4000 nm, significantly higher than the Ra value.

Figure 3 illustrates a typical quasi-logarithmic creep curve. The curves obtained at all test temperatures were of this conventional, continuous form, with no discrete relaxation events such as those reported by Ngan on PE [START_REF] Li | [END_REF].

Discussion

Hardness and elastic modulus on aPP

The elastic modulus values for the fused silica reference sample were essentially independent of the test temperature over the range studied. This result is in agreement with studies using sonic resonance reported by NIST [30], which also show virtually invariant properties of the fused silica over this range. These results provided validation of the performance of the nanoindentation test capability to -30ºC.

Figure 2 shows that the hardness and elastic modulus of aPP exhibited the expected strong dependence on the testing temperature. The increase in H and E can be attributed to the transition of the amorphous regions in the semi-crystalline aPP from a 'rubbery' state at room temperature to a 'glassy' state as the temperature was decreased to its glass transition temperature (-18ºC). The aPP has appreciable crystallinity, and for that reason the change in the slope of elastic modulus going through the glass transition is less pronounced than it would have been for an amorphous polypropylene. In previous work by one of the authors on PET around glass transition temperature [11] it was found that the changes in elastic modulus against temperature in elevated temperature nanoindentation depended on the sample crystallinity; for the highest crystallinity PET sample at the change was smooth, whilst for lower crystallinity samples the slope changes dramatically. We attribute the lack of strong slope dependence going through glass transition temperature to the high crystallinity of this aPP sample.

The scatter in the measured H and E values of aPP (Figure 2) can be attributed to the high level of surface roughness. This effect was alleviated by employing relatively high loads (and hence penetration depths). At 50mN the maximum depth at peak load decreased from ~4800nm at 25ºC to ~3000nm at -30ºC. Tweedie et al. have recently reported that contact loading may increase the glass transition temperature, so that polymers below their bulk glass transition temperature may exhibit significantly higher than expected stiffness, particularly when the penetration depth is under 200nm [START_REF] Munro | Elastic Moduli Data for Polycrystalline Ceramics[END_REF], consistent with other reports of increasing elastic modulus on polymeric materials as the indentation depth is reduced. Brostow et al. [39,40] also found densification inside the scratching groove for polypropylene, resulting in increased hardness. Tweedie et al. suggest that the contact loading creates an interfacial region of confined molecular mobility, adjacent to the probe, either via intermolecular interactions or via stretching or alignment of macromolecular chains. Nevertheless, the contribution of this modified region to the overall mechanical response of the bulk polymer decreases with increasing contact depth. To exclude this possibility, a series of comparative indentations at the testing load 25mN was also carried out; the resultant values of E were essentially identical to those at 50mN within the experimental error. The measured increase in E at sub-ambient temperatures does not appear therefore to be overestimated due to the decreasing indentation depths at 50mN.

For a time-dependent material, it is important to compensate the creep effects on the measured H and E -such effects can be strong and have been found to give very misleading results in certain circumstances [2,[START_REF] Chudoba | Surface &amp[END_REF]. It has been suggested elsewhere that the combination of a slow loading rate, long hold period and fast unload can minimize artefacts in modulus determination on viscoelastic materials. The hold period (20s) was sufficiently long to avoid the presence of any 'nose' shape -as observed by Ngan and others for shorter holding periods [2, 24, 25, 41] -even with the slowest unloading rate of 1mN/s.

In contrast, at the highest load rate of 25mN/s, the peak load was reached in only 2 s and the resultant modulus values were about 10% higher than at the other loading rates for all temperatures. The influence of visco-deformation on the measurement of the unloading contact stiffness has been discussed in [2,[START_REF] Chudoba | Surface &amp[END_REF] and the applicability of the Feng & Ngan compliance correction equation (equation ( 6)) [2] has been investigated.

• • + = P D S S h u 1 1 ( 6
)
where S is the contact compliance, S u is the elastic contact stiffness at the onset of unloading,

• h D is the indenter displacement recorded at the end of the hold period, and

• P is the unloading rate at the onset of unloading.

However, after application of this equation the calculated modulus (Ec) showed a much stronger dependence on the loading/unloading rate with a minimum in Ec at 10mN/s -about 40% less than at 1mN/s. It may be attributed to the different loading history up to the unload which resulted in the different microstructure of the time-dependent material at the onset of unload, and so the tip-sample contact stiffness S should be different [25]. The constant hardness and modulus observed on aPP for indentations at 1, 2.5 and 10mN/s suggests that the presence of the 20s hold period was effective in minimizing the influence of viscoelasticity on the measured hardness and modulus for these slower loading rates.

The higher E at 25 mN/s is a result of the smaller depth at the end of loading period (D max ). At room temperature, the maximum depth for the loading rate of 25mN/s (~4200 nm) was much less than for the other loading rates of 1 to 10mN/s (4480-4750 nm). The depth after creep for the 25 mN/s tests was extrapolated to 70s using the logarithmic equation to investigate whether the apparently higher E was simply a result of a shorter total loading time before unloading. The corresponding maximum penetration depth was about 4350 nm -considerably less than those from the tests using the slower loading rates. It is possible therefore that the indentation behaviour of polypropylene shows a more complex dependence on loading rate. It could be explained by an increase in yield strength of polypropylene with the strain rate [START_REF] Zrida | Polymer Testing[END_REF]. Thus extra energy could be required for plastic deformation during indentation. Similar behaviour has been observed in uniaxial PET [42].

Creep analysis

As described in the Experimental section, the creep behaviour was studied using the logarithmic method and Boltzmann integral method. All hold periods were fitted well using the logarithmic equation as shown in Figure 3.

The variation in the creep extent parameter (A) with temperature from indentations with a loading time of 20s is shown in Figure 5. The dimensionless parameter (A/d(t h =0)) has been considered as a measure of stain rate sensitivity [36], which enables meaningful comparisons of different testing conditions. Our results are shown in Figure 6.a. It can be seen that the value of A/d(t h =0) is between 0.028 and 0.032 over the entire temperature range, with the highest A/d(t h =0) at -10ºC . The creep time constant (τ L ) from the loading time 20s has been plotted against the testing temperatures in Figure 6.b. It is clear that there is also a maximum τ L value at -10ºC.

It has been shown previously that the actual time to reach the maximum load influences the creep kinetics dramatically [30]. When a constant loading rate was considered for each particular temperature, it was found that τ L increased with increased loading rate, exhibiting a power law relationship as shown in Figure 7.

Clearly, the correlation coefficient (and Figure 3) indicates that the experimental creep data is extremely well described by the logarithmic equation. The fitted creep extent parameter, A, decreased with decreasing the test temperature. The dimensionless creep strain sensitivity, A/d(t h =0), and the creep time constant, τ L , showed upper-limit values appearing at -10ºC. At the same time, the creep time constant, τ L , showed a power law relationship with the loading time.

The creep data has also been fitted using the Boltzmann integral method. The adjusted square of the correlation coefficient (R 2 ) is shown in Figure 8. We can see that the quality of fit is improved upon decreasing the loading rate.

As shown in Figure 9.a, the time constant (τ B ) also reached a maximum at -10ºC. This indicated that both methods are effective for revealing the effect of glass transition on creep. The time constant also strongly depended on the loading rates. Figure 9.b shows that the time constant (τ B ) at -10ºC exhibited a power-law relationship with the loading rates, which agrees with the founding using the logarithmic method (Figure 7).

The fitted variables c 0 ' and c 1 ' are shown in Figures 10.a and 10.b. We can see that c 0 ' decreases with the temperature. As c 0 ' is proportional to the instantaneous elasticity, this is consistent with the measured E (Figure 2) increasing as the temperature decreases. The creep extent (c 1 ') showed strong dependence on temperature and loading rate, consistent with the fitted creep extent (A) in Figure 5 obtained using the logarithmic equation. More creep occurred for higher loading rates and temperature.

The creep mechanism in polymers is explained by the Eyring theory as proposed by Berthoud et al. and Tweedie et al. [43,44]. For a creep event to occur, the polymer chains must overcome a potential barrier by thermal activation at a particular reaction rate. At higher temperatures, the thermal barrier is easier to be overcome and thus the movement of the polymer chains, such as translation and rotation becomes easier than at lower temperatures. Therefore, the creep extent parameter (c 1 ) would be expected to decrease with a decrease in testing temperature. As the variable (c 1 ') is proportional to c 1 , it decreases with decreasing temperature as shown in Figure 10.b.

We assume that the creep extent (c 1 ) obeys an Arrhenius relationship:

- = RT Q k extent creep exp _ ( 7
)
where k is constant, R is the gas constant, T is the Kelvin temperature and Q is the activation energy. After taking logarithms, the activation energy Q B (Boltzmann integral method) can be obtained from the slope (-Q/R) of the ln(c') vs. 1/T curve as shown in Figure 11. The calculated Q B is 11.0±2.9kJ/mol.

As shown in Figures 6 and9.a, in the temperature range considered there were maximum values of A/d(t h =0) and τ L using the logarithmic method, and τ B for the Boltzmann integral method, at -10ºC. In reference [43], the creep stain sensitivity (A/d(t h =0)) for a range of amorphous and semi-crystalline polymers was related to the difference between their glass transition (Tg) and testing temperatures. Similarly for the present work, these parameters indicate a property change around Tg.

Although we have only performed nanoindentation measurements at 10ºC intervals and cannot resolve the exact temperature at which peak values of A/d(t h =0), τ L and τ B occur, it is notable that the temperature at which we have determined peak values of these parameters, -10ºC, is 8ºC above the supplier quoted T g (-18ºC) of the studied aPP. As described in reference [START_REF] Cowie | [END_REF], reported glass transition temperatures for polypropylene are influenced by a wide range of factors, such as sample tacticity, sample crystallinity and chain length, as well as the measurement technique used. In Dynamic Mechanical Analysis measurements (DMA), the peak in tan delta is typically offset to a few degrees higher than the T g determined from the inflexion point in the storage modulus versus temperature graph [6,30]. It has been suggested previously that, based on previous ambient and elevated temperature measurements on semi-crystalline and amorphous polymers, changes in A/d(t h =0) are correlated with changes in the tan delta peak [12]. It is notable that the offset in A/d(t h =0) (and the time constants) from the glass transition appears the same (typically ~8-10ºC) for elevated temperature, ambient temperature and sub-ambient nanoindentation measurements for a wide range of polymers.

The peak value of creep parameters at glass transition temperature indicated different creep mechanisms which have been proposed in references [12,45] correlating the creep rate behaviour with the difference between T g and the testing temperature (y = Tg-T). When the testing temperature is higher than T g , the free volume mechanism of creep was implicated as there is more free volume available for molecular motions [6,45,46]. At lower temperatures (e.g. for y>50ºC) brittleness and crack propagation are likely to be the dominant mechanisms of creep [43]. It was suggested that around the T g , these two creep mechanisms could be weak resulting in the increase in the creep time. Whatever the mechanism, it is clear that the indentation creep kineticsboth extent and rate -subtly vary in the vicinity of the glass transition. The measured hardness from unloading curve analysis is slightly lower than expected at -10ºC (Figure 2) and this appears to be consistent with the comparatively enhanced creep at this temperature.

The peak values of A/d(t h =0) and τ L for the logarithmic method at -10ºC are about 9% and 16% higher, respectively, than the average of the others. The greater increase in τ L is consistent with the data in [9] that the parameter τ L was more sensitive than A/d(t h =0) to changes in the glass transition region. As mentioned above, the average value of τ L from all the tested temperatures can be fitted well to a power law relationship vs. loading time. This agrees with the finding [12] that the power law fit for aPP is a better fit than a linear relationship and it is not possible to determine a measurable creep cut-off time from the indentation measurements.

Conclusions

The hardness and elastic modulus of atactic-polyprolythene increased as the testing temperature decreased through the glass transition range and the polymer changed from rubbery to glassy. A temperature reduction of -55ºC from room temperature led to a change of about 100% and 150% increments for hardness and elastic modulus, respectively.

The indentation creep behaviour of atactic-polyprolythene can be fitted well using both the logarithmic method and the Boltzmann integral method.

The creep extent (A and c 1 ') decreased with decreasing test temperature. The time constants (τ L and τ B ) increased with the loading time with a power-law relationship.

A/d(t h =0), τ L and τ B indicated a creep rate change at -10ºC, which is around the glass transition temperature.

The fit for the Boltzmann integral method was better at lower loading rates.

Figures

Figure 4 .

 4 Figure 4.a shows typical creep curves at different temperatures after constant loading at a rate of 25mN/s. It can be seen that the initial depth produced before the hold period was reduced upon decreasing the temperature. The creep depth measured at different conditions (temperature and loading rate) is shown in Figure 4.b. It can be seen that the total creep depth during the hold period decreased with decreasing temperature and loading rate.

Figure 1 (Figure 3 Figure 8

 138 Figure 1 (a) NanoTest pendulum assembly and specimen stage with environmental enclosure; (b) Peltier cooling stages on sample and indenter Figure 2 Averaged hardness and elastic modulus of aPP tested at different temperatures Figure 3 Superimposed experimental creep data and logarithmic expression. Testing conditions: (i) load 50mN, (ii) loading rate 25mN/s, (iii) temperature -10ºC; Fitting parameters: A=97.47nm, B=5.46s -1 ; d(t h =0)=3088.1nm Figure 4 (a) Illustrative penetration depth during hold period for the loading rate=25mN/s; (b) Increase of penetration depth during hold period at different testing temperatures Figure 5 Creep extent parameter (A) vs. test temperature fitted using logarithmic equation

Figure 1 (Figure 2 Figure 11 (

 1211 Figure 1 (Figure1(a)_(b).tif)

Table 1 Nanoindentation test conditions

 1

	No. Sample	Peak Load (mN)	Loading rate (mN/s)	Loading time (s)	Hold period (s)	Temperatures (ºC)
	1	FS	100	5	20	5
	2 3 4	aPP aPP aPP	50 50 50	25 10 2.5	2 5 20	20 20 20	RT (25), 0, -10, -20, -30
	5	aPP	50	1	50	20

Acknowledgement:

The authors would like to thank the late Professor Tom Bell for his contribution to the first part of the project. The authors also wish to express their appreciations to Prof. Witold Brostow at University of North Texas for helpful discussions. Finally, the financial support from EPSRC, UK (EP/C535061/1) is gratefully acknowledged.