Abstract
Film/substrate structures may undergo a localized thermal load, which can induce stresses, deformation and defects. In this paper, we present the solutions of temperature and stresses in a film/substrate structure under a local thermal load on the film surface. Then, the generalized Stoney formula, which connects the curvature of deformation and the stress field is obtained. The present solution takes into account the non-uniformity of the temperature field both in the width and thickness directions of the film. The thermo-mechanical solution is applied to the analysis of the temperature distribution, stresses, and damage of a GaN/sapphire system during the laser lift-off (LLO) process. It is shown that the laser with the Gaussian distribution of energy density causes much smaller tensile stresses at the edge of the heated area in the film than the laser with the uniform distribution of energy density, and thus can avoid damage to the GaN films separated from the substrate.
Keywords
Film/substrate; Thermal load; Stress; Laser lift-off; Damage
1. Introduction
Film/substrate structures are widely used in microelectronic and optoelectronic devices for decades and these structures usually undergo a local thermal load in applications (e.g., Bowden et al., 1998, Freund and Suresh, 2003 and Chui et al., 1998; Gotsmann et al., 2010). The mismatch of the coefficients of thermal expansion (CTE) between the film and the substrate causes deformation and stresses in the structure, and the stresses may in turn induce defects in the film, and influence the properties of the device. For example, the thermal atomic force microscopy (AFM) or scanning thermal microscopy (SThM) probe is used to heat poly ethylene terephthalate (PET) films to induce crystallization of the material (Zhou et al., 2008 and Duvigneau et al., 2010), and the SThM is also used in nanoscale lithography on molecular resist films to fabricate various microstructures (Mamin, 1996, King et al., 2006, Szoszkiewicz et al., 2007, Lee et al., 2008 and Pires et al., 2010). A similar situation arises in the laser lift-off technique (LLO; Elgawadi and Krasinski, 2008 and Sun et al., 2008) for separating GaN films from substrates. The working principle of the LLO technique is to use a laser beam to heat the interface between the film and the substrate. The GaN near the interface heated by the laser decomposes into metal Ga and nitrogen gas, so the film is separated from the substrate under a proper temperature. During the LLO process, the laser heating results in a non-uniform temperature distribution in the film both in the width and thickness directions. This non-uniform temperature distribution results in stresses, bending, and even damage in the film (Jain et al., 1996, Kaiser et al., 1998, Zheng et al., 2007 and Jang et al., 2010). Therefore, in these techniques, the details of the temperature distribution and stresses in the target films on the substrates are essential to an accurate control of the microstructures of the products.
The deformation of a film/substrate structure due to a thermal loading and/or surface stress has been paid much attention (e.g., Stoney, 1909, Freund and Suresh, 2003, Zhang and Dunn, 2004, Huang and Rosakis, 2005, Weissmüller and Duan, 2008 and Yi and Duan, 2009). The Stoney formula gives the relation between the film stress and the curvature (Stoney, 1909). However, the Stoney formula is only valid under six assumptions, the details of these assumptions can be found in the work of Huang and Rosakis (2005). With the developments of nanotechnology and biotechnology, the Stoney formula is widely used and generalized to different situations. Freund (2000) obtained the relation between the mismatch strain and the curvature in a film/substrate system in the nonlinear deformation range; Huang and Rosakis, 2005 and Huang and Rosakis, 2007 relaxed the sixth assumption, namely, all of the stress and curvature components are constant over the whole surface of the film, and derived the generalized Stoney formula for film/substrate structures under a temperature distribution, which is non-uniform in the width direction and uniform in the height direction of the structure. Feng et al., 2006 and Feng et al., 2008 applied the theory of Huang and Rosakis (2005) to a thin film/substrate system with different radii and the multi-layer film/substrate structures, and obtained the stress distribution under a non-uniform temperature distribution in the direction of the span (Feng et al., 2008).
Damage in film/substrate structures under various mechanical loading conditions has been extensively and intensively explored. Hutchinson and Suo (1992) summarized several common damage patterns in film/substrate structures, and gave the critical conditions for the defect emerging. Zhu et al. (2005) simulated the crack formation from pegs in thermal barrier systems, and estimated the critical conditions for the initiation and propagation of the cracks. Pramanik and Zhang (2011) investigated the residual stress and fracture in a silicon-on-sapphire system, and found that the fracture can be minimized by controlling the thickness of the buffer layer. Besides damage analysis of film/substrate structures in the field of solid mechanics, there exist some studies of the mechanical problems of LLO (Kozawa et al., 1995, Tavernier and Clarke, 2001 and Elgawadi and Krasinski, 2008). In these analyses, the temperature in the film induced by the laser heating is assumed to be uniform, but the temperature is found to be actually non-uniform both in the width and the height directions in experiments (Sun et al., 2008 and Sun, 2009). This non-uniformity of the temperature in the film during the LLO process is due to two factors. The first is that the energy density of the laser may be non-uniform; for example, the commonly used laser energy density satisfies the Gaussian distribution. The second is that the heat transfer during the LLO process can induce the non-uniformity of the temperature. The experiments of Sun (2009) show that cracks usually occur at the edge of the laser irradiated area, while almost no crack appears in the center of the irradiated area; and the GaN films separated by the laser with the Gaussian energy density have fewer defects than those by the laser with uniform energy density.
At present, to the authors’ knowledge, there is no thermo-mechanical solution to the temperature and stress distributions in film/substrate structures under a local thermal load on the film surface. In particular, a systematic analysis of the temperature distribution, stress fields, and damage of the GaN/sapphire structures in the LLO technique is lacking. In this paper, we present the solutions of temperature and stress distributions in a film/substrate structure under a local thermal load on the film surface. Then, the connection between the film stress and the curvature of deformation of the structure under the non-uniform temperature distribution is established. Finally, we apply the solutions to the LLO process of a GaN/sapphire structure, and analyze the stress field and damage mechanism of the separated GaN films.
2. Temperature distribution due to local thermal load
2.1. Temperature distribution in film/substrate structure
The schematic diagram of a film/substrate structure subjected to a local thermal load on the film surface is shown in Fig. 1. The thicknesses of the film and the substrate are hf and hs, respectively. The cylindrical coordinate (r,φ,z) is adopted, as shown in Fig. 1.
In the techniques of heat induced micro-cantilever sensors, the LLO of GaN films, and the thermal AFM probe heating, both the operation time and the conduction time of heat in the films are usually very short; therefore, the analysis is carried out under two assumptions: first, the interface between the film and substrate is adiabatic due to the large thermal resistance at the interface; second, the thermal interchange between the film surface and the environment is neglected. If the initial time, from which the film starts to be heated, is t=0, the temperature T(r) at a position r in the film satisfies (Bechtel, 1974),
equation(1)
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where and k are the density, thermal capacity and thermal conductivity of the film, respectively.
As shown in Fig. 1, the film is heated on the surface (z=0). For the interface between the film and substrate, when the application time of heat source on the film is very short and the film is thick, the heat cannot reach the interface. Moreover, for many kinds of film/substrate structures, the heat resistance at the interface is large. Therefore, the interface is assumed to be adiabatic (Wong, 1999). The boundary condition in Fig. 1 can be written as,
equation(2)
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where I(t,r,φ) is the energy density of the local thermal load.
By using the Green function method, we can get the solution to Eqs. (1) and (2), i.e.,
equation(3)
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where the Green functions G and Gz are given by ( Bechtel, 1974),
equation(4)
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equation(5)
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where K=k/(ρC),Ω denotes the heated area. t′,r′ and φ′ are variables. From Eq. (3), we can see that at any given time and position, the temperature distribution can be determined by the energy density and the application area of the local thermal load. Next, we will discuss two kinds of energy density.
2.1.1. Uniform energy density
If the local thermal load is uniformly distributed in a circular area with a radius of a (i.e., I(t,r,φ) is a constant I0), while the energy density is zero outside the circle, then the temperature distribution in the film is axially symmetric, and Eq. (3) becomes
equation(6)
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2.1.2. Gaussian energy density
The energy density of the local thermal load is non-uniform, but satisfies the Gaussian distribution, i.e.,
equation(7)
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where I0 is the energy density at the center of the heat source (i.e., the origin, c.f., Fig. 1), and λ is the parameter of the Gaussian distribution. Since the Gaussian distribution is axisymmetric, the temperature distribution is also axisymmetric. Then Eq. (3) can be simplified as
equation(8)
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There are two special cases of the Gaussian distribution: λ approaches infinity, and λ approaches zero.
(a)
λ→∞: Under this condition, the energy density of the heat source is uniform. The temperature distribution in the heated area of the film is uniform in the width direction, namely, it is a function of z and t , and independent of r . Then Eq. (3) becomes
equation(9)
′2+y-y′24K(t-t′)dx′dy′dt′
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=I0ρC∫0tGzdt′
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where ∞∞ in Eq. (9) denotes that the integration is carried out in the infinite area. In the derivation of Eq. (9), the following identity is used:
equation(10)
14Kπt-t′∬∞exp-x-x′2+y-y′24K(t-t′)dx′dy′=1
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When the film thickness is much smaller than the size of the heated area, we can use Eq. (9) to calculate the temperature distribution in the film.
(b)
λ→0λ→0: Under this condition, the heat source becomes a point source, and the energy distribution can be expressed by using a delta function, i.e.,
equation(11)
I=I0δ(r)I=I0δr
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Then Eq. (8) becomes
equation(12)
T=I0ρC∫0tGz∬Ωδr′G(r,0,t;r′,φ′,t′)r′dr′dφ′dt′=I0ρC∫0tGzG(r,0,t;0,0,t′)dt′=I0ρC∫0tGz4Kπ(t-t′)exp-r24K(t-t′)dt′
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When the size of the heated area is much smaller than the film thickness, we can use Eq. (12) to calculate the temperature distribution in the film.
2.2. Numerical results of temperature distribution
The film used in the calculation is a GaN film with hf=4μm, and the material parameters are: ρ=6.11gcm-3,k=2.1Wcm-1K-1,C=35.3Jmol-1K-1. We use the circular heat sources with different energy densities and operation times in the calculation, and the temperature is obtained by solving Eqs. (3), (4) and (5) using the Matlab. Fig. 2(a) and (b) show the temperature distribution (obtained by Eqs. (4), (5) and (6)) in a GaN film with a uniform energy density (the energy density is 600mJ/cm2, and the operation time is 38 ns). Fig. 2(a) shows the temperature distribution on the film surface, and we can see that the temperature is very high and remains a constant inside the heated area. However, the temperature decreases rapidly with a large gradient at the edge of the heat source. Fig. 2(a) shows the temperature distribution in the cross section (thickness direction) of the film, and it is found that the temperature gradient in the thickness direction is very large as well. From Fig. 2(a) and (b), we can find that there are large temperature gradients both in the thickness and the width directions of the film for the uniform energy density. Fig. 2(c) and (d) show the temperature distribution (obtained by Eqs. (4), (5) and (8) in a GaN film (hf=4μm). The energy density at the center of the heat source is 600mJ/cm2, and the operation time is 38ns. By comparing Fig. 2(c) and (d) with Fig. 2(c) and (d), it is found that the temperature distribution induced by the heat source of the Gaussian energy density has a smaller temperature gradient and diffusion length than those of the uniform energy density. Fig. 2(e) shows the temperature distribution in the thickness direction of the film. It is found that the temperature gradient in the thickness direction of the film increases with the increase of the film thickness under the same heating time, that is, the temperature gradient for the case of t=500ns,hf=400μm is larger than that for t=500ns,hf=40μm.
Fig. 2.
Temperature distribution in the film. (a) On the film surface (z=0) with a uniform energy density; (b) in the cross section of film with a uniform energy density (); (c) on the film surface (z=0) with the Gaussian energy density; (d) in the cross section of film with the Gaussian energy density (); (e) temperature distribution at r=0 along z direction with a uniform energy density. Tmax denotes the temperature at the center of the heat source.
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The temperature gradients both in the thickness and width directions (with the uniform energy density or Gaussian energy density) of the film are very large (c.f., Fig. 2). The large temperature gradients will induce a non-uniform stress distribution in the film/substrate structure. In the following, we will derive the stress and curvature of deformation of the film/substrate structure.
3. Generalized Stoney formula with a non-uniform temperature
3.1. Stress in film/substrate structure
As mentioned in the introduction, the classical Stoney formula is only valid when six assumptions are satisfied. The sixth assumption is that the stresses are uniform both in the thickness and the width directions, and there is no shear stress. However, according to the above calculation of temperature, large temperature gradients exist both in the thickness and the width directions of the film, and this will induce a non-uniform stress field as well as shear stresses in the film/substrate structure. In this case, the classical Stoney formula is not valid anymore. The previous studies of Huang and Rosakis, 2005 and Huang and Rosakis, 2007 and Feng et al. (2008) are for the cases where the temperature is uniform in the thickness direction and non-uniform in the width direction of the film. In this section, we will investigate the stresses and curvatures of the film/substrate system under the condition that the temperature is non-uniform both in the thickness and the width directions.
In order to solve this problem, we divide the film into a number of very thin layers. If the film is divided into infinitely thin layers, the temperature is uniform in the thickness direction in each of the thin layer. Therefore, the film/substrate structure with the non-uniform temperature distribution both in the thickness and the width directions can be regarded as a multi-layer film/substrate structure with a uniform temperature in the thickness direction in each layer and non-uniform temperature in the width direction (c.f., Fig. 3). The Young modulus, the Poisson ratio, and the thickness of the substrate are denoted by Es,νs and hs, respectively, and the Young modulus and the Poisson ratio of the film are Ef and νf, respectively. The thickness of the i th layer is hi. Assume that the total thickness of the film is much smaller than the substrate, namely, . According to the elastic theory, the stress components in the i th layer satisfies ( Feng et al., 2008)
equation(13)
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equation(14)
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equation(15)
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where εi=-αfTi(r), αf is the CTE of GaN, Ti(r) is the temperature in the ith layer, which can be derived from Eq. (3).
Fig. 3.
Schematic diagram of a multi-layer film/substrate structure.
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The in-plane displacement component (uf) in the width direction and the derivative of the displacement component (dw/dr) in the thickness direction at an arbitrary point of the film are (Feng et al., 2008)
equation(16)
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equation(17)
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where B1 and B2 are given by
equation(18)
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equation(19)
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is the area average value of εi,
equation(20)
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where A is the area of the film. Substituting Eq. (16) into Eqs. (13) and (14), we can get the stress in the i th layer
equation(21)
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equation(22)
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where η is an integration variable. Because we divide the film into thin layers with a infinitesimal thickness, we have
equation(23)
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Substituting Eq. (23) into Eqs. (15), (21) and (22), we get the stress components σrr, σφφ and τ at an arbitrary point in the film, i.e.,
equation(24)
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equation(25)
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equation(26)
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where ζ is an integration variable, ε∗ is the height average of ε, and is the area average of ε∗,
equation(27)
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The relation between the curvature and the displacements of the film satisfies
equation(28)
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Substituting Eqs. (17) and (23) into Eq. (28), we get
equation(29)
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equation(30)
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In the following, we will give the generalized Stoney formula of the film/substrate structure with a non-uniform temperature distribution both in the thickness and the width directions.
3.2. Generalized Stoney formula with non-uniform temperature
The total film thickness is assumed to be much smaller than the substrate thickness, such that the terms containing hf/hs in Eqs. (24) and (25) can be neglected. Then from Eqs. (24), (25), (29) and (30), we can get
equation(31)
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equation(32)
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equation(33)
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equation(34)
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Eq. (32) and Eq. (34) show that κrr-κφφ is proportional to σrr-σφφ,
equation(35)
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However, we cannot get the similar relation between κrr+κφφ and σrr+σφφ from Eqs. (31) and (33). To get the generalized Stoney formula, we need two additional relations between the stress and curvature. We define the area average of κrr+κφφ,
equation(36)
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Substituting Eq. (33) into Eq. (36) yields
equation(37)
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Define the thickness average of σrr+σφφ,
equation(38)
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From Eqs. (33), (37) and (38) we can get
equation(39)
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It is noted that there exists a relation between the average values of σrr+σφφ and κrr+κφφ (shown in Eq. (39)). (σrr+σφφ)∗ is related not only to the local curvature κrr+κφφ, but also to the non-local term . If the temperature distribution in the thickness direction is uniform, Eq. (39) reduces to that of Huang and Rosakis (2005), i.e.,
equation(40)
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Similarly, from Eqs. (26), (31) and (39) we can get the relation between the interface shear stress and the curvature,
equation(41)
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Eqs. (35), (39) and (41) form the generalized Stoney formula of the film/substrate structure with a non-uniform temperature both in the thickness and width directions. It is noted that there exists a shear stress in the film under the non-uniform temperature distribution in the width direction, whereas it is absent under the uniform temperature distribution in the width direction. We will show below (Section 4) that the interface shear stress has remarkable influence on the damages in the film.
3.3. Numerical results of curvature
Here, we calculate the curvature of the film/substrate structure under a thermal load on the surface. The film and substrate materials are GaN and sapphire, respectively. The thicknesses of the film and the substrate are and , respectively. The substrate thickness is much smaller than its size in the width direction. The material parameters of GaN used in the calculation are as follows (Liu and Edgar, 2002 and Williams and Moustakas, 2007): . The material parameters of sapphire are (Williams and Moustakas, 2007): and νs=0.25. We also consider two kinds of laser, namely, the circular heat source with a radius of and a uniform energy density , and an operation time , and the heat source with the Gaussian energy density (λ=1/2, , with the energy density at the center of the heat source ), and an operation time . The curvatures are calculated by Eqs. (29) and (30), and are shown in Fig. 4. Besides the results obtained by the present method, we calculate the curvatures using the theory of Huang and Rosakis (2005), in which we use the average temperature in the thickness direction, and the results are also shown in Fig. 4. We can see that the curvatures κrr and κφφ vary with positions. κrr is negative inside the heated area, and positive outside the heated area. However, κφφ is negative in the area and its absolute value reduces as the distance from the center of the heat source increases. The curvatures calculated by the present method have a remarkable difference from those given by the method of Huang and Rosakis (2007), especially, the curvature κφφ at the center (r=0). This means that if the temperature distribution in the thickness direction is non-uniform, we cannot simply use the average temperature to calculate the stresses and curvatures in the film.
Fig. 4.
Curvature calculated by the present method, and the theory of Huang and Rosakis (2005). “H&R” denotes the results obtained by the theory of Huang and Rosakis (2007), and “ED” denotes energy density.
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In following sections, we will apply the above solutions to the analysis of the stress field and damage mechanisms of the GaN/sapphire structure.
4. Stress analysis of GaN/sapphire during LLO
Fig. 5 shows the schematic diagram of the LLO separation process. During this process, a laser is used to irradiate the backside of the sapphire substrate. Because the bandgap of the sapphire is larger than the photon energy of the laser, the sapphire does not absorb the laser energy. In this case, the laser energy can get through the sapphire substrate, and reach the GaN/sapphire interface. As the bandgap of GaN is smaller than the photon energy, GaN would absorb the laser energy (Elgawadi and Krasinski, 2008 and Sun et al., 2008). Therefore, the laser energy will transfer into thermal energy, and make the GaN near the interface decompose into Ga and nitrogen gas. In this way, the film and substrate will be separated. A receptor wafer is bonded on the film surface, which can effectively avoid damage of the film (Fig. 5(b), Tavernier and Clarke, 2001).
Fig. 5.
Schematic diagram of LLO separation process. (a) GaN/sapphire structure before separation; (b) GaN/sapphire structure under separation (the red area denotes the GaN buffer layer that decomposes into metal Ga and nitrogen gas); (c) GaN/sapphire structure after separation. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
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We carried out experiments of LLO separation of GaN films on sapphire substrates. The GaN films in our experiments are grown on sapphire substrates by Metal–Organic Chemical Vapor Deposition (MOCVD) at the temperature around 1000 °C. The thicknesses of the film and the substrate are 4 μm and 430 μm, respectively. The radius of the GaN/sapphire structure is 2.5 cm. The silicon film with a thickness of about 300 μm is used as the receptor wafer. We used two kinds of laser. The first is the laser with a uniform energy density with the parameters as follows: laser radius ; wavelength ; period 25 ns; energy density 600 mJ/cm2. The second is the laser with the Gaussian energy density with λ=1, and its parameters are as follows: laser radius ; wavelength 248nm; period 38 ns; center energy density 600 mJ/cm2.
In the LLO process, the GaN film at the interface is heated to a temperature higher than 1000 °C, which can induce stresses in the film. The stress may produce cracks and other defects both in the film and the substrate. The GaN films transferred to the silicon films by these two kinds of laser are shown in Fig. 6(a) and (b). Fig. 6(a) shows the GaN film heated by the laser with the uniform energy density. Cracks appear at the edges of the laser irradiated areas. A similar phenomenon has been observed by Tavernier and Clarke (2001). However, Fig. 6(b) shows that there is no crack in the GaN when we use the laser with the Gaussian energy density. These phenomena cannot be explained by the existing theory. Thus, in the following, we will investigate the stress distribution, and the damage mechanism of the film during the LLO process by using our theory.
Fig. 6.
(a) Cracks on the separated surface of the GaN film at the edge of the laser irradiating area (Sun, 2009). (b) The separated surface of a GaN film, which was transferred to the silicon film by the LLO process (the Gaussian energy density) (Sun, 2009).
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4.1. Temperature distribution during LLO process
The laser acts on the interface between the film and the substrate in the LLO process (c.f., Fig. 5(b)). However, the local thermal load in the solutions of the temperature and stress distributions of the film/substrate structure derived in Sections 2 and 3 is applied on the outer surface of the film. Therefore, for the separation problem of the GaN/sapphire structure by LLO, instead of the boundary condition in Eq. (2), the boundary conditions of the LLO problem are
equation(42)
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Eqs. (1) and (42) are the basic equations for the temperature distribution in the film/substrate structure by LLO. The solution to Eqs. (1) and (42) has the similar form to Eq. (3), but Gz in Eq. (5) should be replaced by
equation(43)
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The decomposition temperature of the GaN film depends on the lattice quality. The better quality of the GaN film, the higher its decomposition temperature (Furtado and Jacob, 1983). During the deposition process, there are plenty of defects in the initial layer (several hundred nanometers in the film from the substrate) due to the lattice mismatch between the GaN and the sapphire. Generally, this layer is referred to as the buffer layer. Since most of the stresses generated by the mismatch between the film and the substrate have been released by the defects in the buffer layer, the layer grown above the buffer layer has a much better quality, and it is referred to as the epitaxial layer. The decomposition temperature of the buffer layer is about 750 °C (Furtado and Jacob, 1983), whereas it is about 1000 °C for the epitaxial layer (Morimoto, 1974). The objective of the LLO is to decompose the buffer layer, while keeping the epitaxial layer intact. Therefore, the temperature of the buffer layer and the epitaxial layer should be between 750 °C and 1000 °C.
From Eq. (3), we can see that the temperature distribution in the film depends on the properties of GaN (density, thermal capacity and thermal conductivity), and the laser energy distribution. Now we will obtain the appropriate laser energy distribution that can make the GaN buffer layer decompose. To compare the results with different energy distributions of the laser, the laser period is set to 38 ns and the energy density at the center is 600 mJ/cm2. The energy density distribution of the laser varies with the parameter λ, with λ=∞ denoting the uniform energy density distribution. In Fig. 7, the temperature distribution (obtained by Eqs. (4), (8) and (43) during the LLO process is shown. Fig. 7(a) shows the temperature on the interface (z=-hf) with the Gaussian distribution laser. From Fig. 7(a), we can see that the temperature at the laser center (r=0) is very high, and decreases with the distance away from the center. When λ⩽1/2, the temperature in 50% of the laser irradiated area is lower than 750 °C, such that the film cannot be separated completely from the substrate. However, when λ⩾1, the temperature in the total laser irradiated area is higher than 750 °C. Therefore, in order to separate the film from the substrate, one should choose the laser parameter λ⩾1 for the Gaussian distribution laser in the LLO technology.
Fig. 7.
Temperature in the GaN film during the LLO process. (a) Temperature in the width direction (at ; ) with the Gaussian energy density and (b) temperature along the thickness direction (at r=0) with a uniform energy density.
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Fig. 7(b) shows the temperature distribution at r=0 in the thickness direction with a uniform energy density. In Fig. 7(b), d denotes the distance from the interface. We can see that the diffusion length of the thermal energy in the duration of laser irradiation is about when . For (c.f. the black line), the point with 1000 °C corresponds to above the interface, and the point with 750 °C corresponds to above the interface. Therefore, when , the thickness range of the GaN film, whose temperature is between 750 °C and 1000 °C, is . Under this temperature distribution, the thickness of the decomposed GaN layer should be less than .
From the above analysis, we can conclude that the temperature distribution in the film is non-uniform (both in the thickness and width directions) during the LLO process, no matter what is the form of the laser energy density (uniform or the Gaussian distribution). It is noted that Tavernier and Clarke (2001) analyzed the separation mechanisms of the GaN/sapphire structure by LLO. They assumed that the temperature in the width direction of the whole film is uniform (the laser is actually applied only on a localized area), while the temperature in the thickness direction is non-uniform. Under the assumption of the uniform temperature in the width direction (Tavernier and Clarke, 2001), the stresses in the film are very different from the actual ones in the LLO process. In the following, we will analyze the stress distribution in the film under the non-uniform temperature distribution.
4.2. Stress distribution during LLO process
In the following, we will analyze the stress distribution in the GaN film. The normal stress and the shear stress at the interface between the film and the substrate are obtained from Eqs. (24) and (26). The temperature in these equations have been given in Section 4.1 both for the uniform and the Gaussian energy densities.
4.2.1. Stress with uniform energy density
We analyze the stress field under the non-uniform temperature distribution (both in the thickness and width directions) induced by the laser with a uniform energy density. The stresses can be calculated from Eqs. (24) and (26). In order to investigate the effect of the non-uniformity of temperature distribution, we compare the stresses induced by the uniform and nonuniform temperature fields in the width direction while the temperature in the thickness direction is nonuniform. Fig. 8(a) shows the normal stress at the center and edge of the laser (at r=0 and r=a) along the thickness direction. Case (A) denotes the stress calculated using the temperature distribution derived under the assumption of uniform temperature in the width direction, which is also considered in the work of Tavernier and Clarke (2001). Case (B) denotes the stress obtained from our solution. The normal stresses at r=0 for the two cases are the same, but those at the edge (r=a) are different.
Fig. 8.
(a) Normal stress along the thickness direction and (b) shear stress on the interface (z=-hf). The area I denotes the laser irradiated area, while II denotes the laser un-irradiated area.
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Fig. 8(b) shows the shear stress distribution on the interface from our solution. There is a large shear stress peak (0.8 GPa) at the edge of the laser irradiated area (). For the film in our experiment (shown in Fig. 6(a)), the shear stress is 0.64 GPa (). This means that the longer the irradiation time, the higher the interface shear stress. The non-uniformity of the temperature in the width direction has a large effect on the interface shear stress and the normal stress in the film (c.f. Fig. 8(a) and (b)). Therefore, the assumption of the uniform temperature in the width direction cannot describe the actual stress state in the LLO process.
4.2.2. Stress with Gaussian energy density
Fig. 9(a) shows the normal stress at the center and edge of the laser irradiated area with the uniform and the Gaussian energy densities. There is almost no difference between the normal stresses at the center for these two kinds of energy density. However, at the edge of the irradiated area, the normal stress is smaller for the laser with the Gaussian energy density than that with the uniform energy density. Fig. 9(b) shows the interface shear stresses for the two energy densities. The peak of the interface shear stress is reduced effectively (Fig. 9(b)) with the Gaussian energy distribution.
Fig. 9.
(a) Normal stresses at the center and edge of the laser irradiated area with the Gaussian energy density (λ=1) and the uniform energy density and (b) interface shear stresses with the two energy densities. “ED” denotes “energy density”.
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When we use the laser with the Gaussian energy density, the temperature (c.f. Fig. 7(a)) in the film varies with λ, so does the interface shear stress. Fig. 10(a) shows that the interface shear stress at the edge of the laser irradiated area decreases rapidly as λ becomes small. The maximum interface shear stress appears at the edge when λ>1/2. However, when λ<1/2, the maximum interface shear stress appears around the center. From the above analysis, we can see that the larger the parameter λ, the larger the interface temperature (c.f., Fig. 7(a)) and the interface shear stress (c.f., Fig. 10(a)). Therefore, a large λ is good to decompose the buffer layer, but it will induce large interface shear stress at the edge of the irradiated area. During the LLO process, the interface shear stress is a function of the laser radius a , the parameter λ, film thickness hf, etc. If the laser radius and film thickness are given, there is a critical value (λcr), which can minimize the maximum interface shear stress (c.f., Fig. 10(b)). For example, when λ=0.46, the maximum interface shear stress is reduced to about 0.1 GPa for , and it is 0.07 GPa for when λ=0.36.
Fig. 10.
(a) Interface shear stress (z=-hf) with the Gaussian distribution and (b) variation of the maximum interface shear stress with λ.
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4.3. Damage mechanism during LLO process
Cracks, which appear at the edges of the laser irradiated areas (c.f., Fig. 6(a)), are one of the most common defects during the LLO process. According to the analysis in Section 4.2, we know that the stress induced by the laser irradiation might be one of the factors to induce the cracks. Moreover, during LLO process, the laser energy makes the GaN near the interface decompose into Ga and nitrogen gas, and the nitrogen pressure is another factor to induce cracks. Therefore, the damage of GaN films is induced by the stress from irradiation and the nitrogen pressure simultaneously. In the following, we will analyze the effect of these stress and the nitrogen pressure on the damage of GaN films.
When the laser irradiated area of a film is separated from the substrate, while the surrounding area is still bonded to the substrate, the separated area can be regarded as an clamped plate. According to the theory of Timoshenko and Woinowsky-Krieger (1961), the normal stress in the GaN film induced by the nitrogen gas is
equation(44)
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where P is the nitrogen pressure, which can be determined from the temperature. In our experiments, the temperature of the nitrogen is about 1000 °C, and then according to the experimental results of Karpinski et al. (1984), P is about 10 MPa.
When the stress state induced by the laser irradiation and the nitrogen pressure is known, we can calculate the maximum tensile stress (σm) in the film by combing the results from Eqs. (24), (25) and (26) and (44), and the numerical results are shown in Fig. 11. For our experiments, at the edge of the laser irradiated area, for the laser with the uniform energy density, and for the laser with the Gaussian energy density. If a receptor wafer is used, the corresponding maximum tensile stresses at the edge of the laser irradiated area are 0.57 GPa and 0.1 GPa, respectively. The former is much larger than the latter. We do not find the value of the failure tensile stress of the GaN film under high temperature. However, it is noted that the yield shear stress of GaN at 900–1000 °C is about 0.15 GPa (Yonenaga, 2003). This indicates that the laser with the uniform energy may introduce cracks in the film (c.f. Fig. 6(a)), whereas that with the Gaussian distribution is unlikely in this case. Moreover, the maximum tensile stress emerges at the edge of the laser irradiated area. For the uniform energy density and Gaussian energy density, the angle θ of the direction of the maximum tensile stress relative to the interface is equal to 43° and 45°, respectively. As shown in Fig. 6(a) and (b), there are many cracks at the laser irradiated area when the laser with the uniform energy density is used; whereas there is no crack with the Gaussian energy density.
Fig. 11.
The maximum tensile stress in the GaN film. θ in the insert denotes the direction of the stress. “Uni” denotes “Uniform”, “Gs” denotes “Gaussian”, and “Rcp” denotes that the silicon receptor wafer is used.Figure options
5. Conclusions
We present the solutions of the temperature and stress fields induced by a local thermal load in a film/substrate structure. The localization of the heat generates a highly non-uniform temperature and thus a non-uniform stress field. We also derive the generalized Stoney formula that relates the curvature of deformation of the structure to the stress field taking into account the non-uniformity of the temperature field. The solutions can be used to analyze the temperature fields, deformation and damage mechanisms in film/substrate systems subjected to a local thermal load. As an application, we analyze the temperature and stress fields in a GaN/sapphire system during the LLO separation process. The distribution of the energy density (e.g., the uniform energy density and the Gaussian energy density) has a large effect on the stress distribution, especially, the interface shear stress. The laser with the uniform energy density induces a much larger maximum tensile stress in the GaN film than that with the Gaussian energy density. The large maximum tensile stress is one of the important factors that cause the damage of the separated GaN films. The thermo-mechanical solution can also be used to choose a laser with proper parameters which ensures efficient separation of the GaN films from the substrates while avoiding damaging the films.
Keywords:gan substrate, gan on sic substrate,gan metal substrate,gan oj gaas substrate ,gan bulk substrate,gan on zno substrate.
Source:Sciencedirect
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