Mechanical Strength and Reliability of a Novel Thin Monocrystalline Silicon Solar Cell

Dewei Xu* and Paul S. Ho Microelectronics Research Center The University of Texas at Austin

Austin, TX 78758 dwxu@mail.utexas.edu

Rajesh A. Rao, Leo Mathew, Scott Smith, Sayan Saha, Dabraj Sarkar, Curt Vass and Dharmesh Jawarani AstroWatt Inc.

Austin, TX 78758

Abstract— Thin crystalline silicon solar cells, on the order of a few to tens of μm thick, are of interest due to significant material cost reduction and potentially high conversion efficiency. These thin silicon films impose stringent mechanical strength and handling requirements during wafer transfer, cell processing and module integration. Quantitative mechanical and fracture analyses to address reliability issues become necessary. Based on a bi-material foil composed of thin monocrystalline silicon and a supporting substrate fabricated from a novel SOM® (Semiconductor on Metal) kerf-less exfoliation process, closed-form mechanical analyses are introduced and developed to evaluate their strength and fracture behaviors. These analyses include the thermal stress field in the device silicon layer and supporting substrate, the fracture behavior and effects of pyramid structures from surface texturing and the energy release rate at the silicon-substrate interface. It is shown that the introduction of the intrinsic compressive residual strain in the SOM® substrate expands the processing temperature spectrum. The developed analysis and methodology can be readily extended to other thin film solar cell structures with various configurations of device layers and supporting substrates.

Keywords- Thin crystalline silicon solar cells; mechanical strength; reliability; thermal misfit strain; stress intensity factor; energy release rate; cell process temperature limit

I. INTRODUCTION Thin crystalline silicon solar cells are of much interest due

to their potentially high efficiency and low material cost. Simulations have shown that 10~50 um is the optimal thickness for maximum efficiency with appropriate backside passivation or a back-surface-field (BSF) and light trapping schemes [1-3]. In addition, thin crystalline silicon solar cells are more apt to be used in bifacial cell architectures since the backside efficiency increases with decreasing cell thickness [3, 4]. The low material cost of thin crystalline silicon solar cells is self-evident. If crystalline silicon solar cells consume silicon with thickness on the order of tens of microns resulting from a kerf-less process, the silicon usage per watt-peak (WP) can be

reduced in an order of magnitude and such thin crystalline silicon solar cells would obscure the margin of other thin-film solar cells in cost reduction over silicon PVs yet with high efficiency and long-term robustness [2].

The benefits of thin crystalline silicon solar cells are exciting with respect to the material cost reduction and potentially higher converting efficiency. However, the challenges to commercialize this effort involved with the mechanical handling and reliability issues are not trivial. Thicker substrate themselves can behave as mechanical support. It is necessary to add a supporting substrate for thin crystalline films to improve handling and yield capability during wafer transfer, cell processing and module integration. For example, assuming a vacuum chuck used during wafer transfer, the resulting maximum mechanical stress in the silicon wafer 2/p hσ ∝ , where p is the pressure difference between the atmosphere and the vacuum and h represents the wafer thickness. When wafer thickness reduces to a fraction of current cell thickness (180~200 μm), the corresponding stress may increase by more than one order of magnitude and hence a supporting substrate becomes necessary. Since silicon solar cells are fabricated at high temperatures, the resulting thermal stress due to the difference of the coefficients of thermal expansion (CTE) between silicon and the supporting substrate material determines the allowable temperature limit for cell processing and module integration. The temperature endurance of the silicon-substrate bilayer structure should be cautiously evaluated to choose appropriate cell fabrication recipes. Note that the resulting bow and fracture issues from the thermal residual stress after firing due to the CTE mismatch between front silicon solar cells and rear aluminum paste layer is already a problem for solar cells at current thickness of ~200 μm [5, 6]. Furthermore, the state-of-art silicon solar cells are textured on the front surface with pyramid structures to improve the light-trapping in these devices [7, 8]. These pyramids can behave as initial crack sources under the thermal stresses and lead to the breakage of thin crystalline silicon films at high temperature. Such fracture-related reliability issues are more sensitive during cell processing and module integration of thin crystalline silicon soar cells.

978-1-4577-1680-5/12/$26.00 ©2012 IEEE 4A.3.1

So far there is lack of quantitative analyses to predict the mechanical deformation, stress, and fracture properties of thin crystalline solar cells in terms of surface texture structures, residual stress originating from fabrication processes and required temperature excursions during cell processing and module integration. Such studies are essential to quantify and evaluate the capabilities of the handling and reliability and yield loss issues of thin crystalline silicon solar cells. In this paper, on the basis of a monocrystalline thin solar cell developed using a patented exfoliation process with a typical bilayer cell structure of silicon device and rear contact metal layer, quantitative mechanical and fracture analysis to address reliability issues are introduced and developed. Based on these analyses, we discuss several critical reliability issues for the SOM® thin silicon solar cells, such as, the determination of the intrinsic compressive residual strain and the temperature limit for the cell processing and module integration.

II. SOM® MONOCRYSTALLINE THIN SILICON SOLAR CELLS AstroWatt Inc. has developed a kerfless exfoliation

process to produce 25μm thin exfoliated monocrystalline silicon foils from a parent wafer and, due to the significantly less silicon consumed in the solar cell, the production cost can be as low as $0.26/Wp [9, 10]. This approach involves forming a flexible metal foil over a silicon substrate using an electrochemical deposition process. Subsequently an annealing process is performed during which internal stresses are created in the silicon substrate due to the compressive plastic residual strain developed in the metal foil during annealing [11]. The exfoliation is aided and controlled by a proprietary tool which applies mechanical force at a predetermined location on the wafer, leading to fracture along a sub-surface plane of the substrate. The thickness of the exfoliated silicon layer is controlled by varying the electroplated metal thickness, the annealing process, and the mechanical tool parameters. The flexible metal foil behaves as a mechanical support for 25 μm monocrystalline silicon and also serves as rear contact layer in the cell structure. Fig. 1 shows a picture of 125 mm pseudo-square composite foil with 25 μm exfoliated monocrystalline silicon layer. It can be seen that this composite foil is bowed due to the presence of residual compressive strain from annealing process.

Figure 1 Picture of exfoliated 125 mm pseudo-square composite foil with ~25 µm monocrystalline Si (100).

III. THEORETICAL FORMULATIONS In this section, a stress analysis is first presented for the bi-

material composite foil in terms of the compressive plastic

residual strain and processing temperature. Based on this stress analysis, fracture solutions considering cracks from textured pyramids and interfacial debonding are introduced.

A. Stress analysis Fig. 2a schematically shows the bi-material foil composed

of a silicon layer and a supporting substrate (metal for SOM® foils) with a compressive residual strain 0ε and a curvature κ (concave downward). At room temperature, the residual strain

0ε is the misfit strain between two layers, as illustrated in Fig. 2b after cutting along the interface. It is noted that the curvature of and the stress distributions in the composite foil are determined by the misfit strain which varies with the change of temperature. In this following, the misfit strain in terms of 0ε and temperature T , the corresponding curvature, and the resulting stress distributions are analyzed.

Assuming that the thicknesses of the silicon layer and the substrate are 1h and 2h , respectively; Young’s modulus 1E and Poisson’s ratio 1v for silicon and 2E and 2v for substrate material, respectively; the coefficients of thermal expansion for silicon and substrate material are 1α and 2α , respectively. The assumptions made in the analysis are as follows: (i) all materials are linear elastic; (ii) the coefficients of thermal expansion remain constant; (iii) the bi-material foil develops to a cylindrical surface and plain strain conditions are assumed along its axis; (iv) the interface between silicon and substrate is perfectly bonded.

Figure 2 (a) Schematics of the bi-material foil composed of silicon and supporting substrate with a compressive residual strain 0ε . The misfit strain

0ε at room temperature (b), a thermal misfit strain TεΔ (c) and a total misfit strain εΔ (d).

At temperature T , the misfit strain (Fig. 2c) between the silicon film and the substrate due to the difference between the coefficients of the thermal expansion is

2 1 0 2 1( )( ) ( )T T T Tε α α α αΔ = − − = − Δ , (1)

(c)

TεΔ(b)

y

x

0ε

0Tε ε εΔ = Δ −

(d)

(a)

Si Substrate

1 / κ

4A.3.2

(a) (b)

Si Metal Si

where the TΔ is the temperature differential with respect to room temperature 0T (in the following, T and TΔ are used interchangeably). Therefore, the total misfit strain (Fig. 2d) in terms of TεΔ and the residual compressive strain 0ε is

0 2 1 0( )T Tε ε ε α α εΔ = Δ − = − Δ − . (2)

The resulting curvature of the bi-material foil and the stress field due to the misfit strain εΔ can be derived from engineering composite beam theories with the x-y coordinate system whose origin is set at the interface (see Appendix for detailed derivation). The results are similar to the study of bi-metal thermostats [12].

The curvature of the bi-material foil of silicon and substrate is

( )1 2 1 2 1 2 2 1 02 4 3 2 2 3 2 41 1 1 2 1 2 1 2 1 2 1 2 1 2 2 2

6 ( ) ( )4 6 4E E h h h h T

E h E E h h E E h h E E h h E hα α ε

κ+ − Δ −

=+ + + +

, (3)

where, 21 1 1/ (1 )E E v= − and 2

2 2 2/ (1 )E E v= − . The curvature is defined as positive when the resulting cylindrical surface is concave upward. For example, at room temperature ( 0TΔ = ), the resulting surface of the composite foil is concave downward and the corresponding curvature is negative, as shown in Fig. 1.

The stress distributions in the silicon and substrate layers are given as

( ) 2 21 2 2 2 1 0 2 2 1 1

1 11 1 2 2 1 1 2 2

( )( ) ( )

2( )E E h T E h E hy E y

E h E h E h E hα α ε

σ κ− Δ − −= − ++ +

for 10 y h< ≤ , (4)

and ( ) 2 2

1 2 1 2 1 0 2 2 1 12 2

1 1 2 2 1 1 2 2

( )( ) ( )

2( )E E h T E h E hy E y

E h E h E h E hα α ε

σ κ− Δ − −= − − ++ +

for 2 0h y− ≤ ≤ , (5)

respectively. In Eqs. (4) or (5), the first term represents the axial stress and the second term is the associated bending stress.

B. Fracture and Debonding Analysis State-of-art silicon solar cells are textured on front surface

to reduce reflectance and improve light trapping and the pyramid structure obtained by wet etching is the most common in monocrystalline silicon PV industry [7, 8]. Fig. 3a shows pyramid structures of a textured surface on a 25 μm SOM® monocrystalline silicon film [13]. Such textured structures are beneficial to the solar cell performance and however, their existence is detrimental to the mechanical handling, reliability and yield of solar cells, especially when the silicon or device layer becomes thinner and thinner. These pyramid structures behave as initial cracks which propagate under a critical level of tensile stress and finally lead to the breakage of the silicon film. Fig. 4a shows such a crack resulting from a temperature field which runs through a SOM® silicon film. The corresponding view was on the interface side of the silicon film (the metal substrate was etched away) due to the low

reflectivity of the textured surface. A succinct fracture analysis is given in the following.

Assuming these pyramids with the same height of a , they are simplified as uniformly distributed edge cracks or periodic

Figure 3 (a) Pyramid structures of textured surface on a 25 μm monocrystalline silicon film from SOM® technology [13], and mechanical mode of these pyramids as periodic edge cracks (b) and a single edge crack (c).

Figure 4 Optical images show the failure modes of the SOM® silicon film resulting from temperature field: (a) a crack run through the silicon film and (b) a narrow strip of silicon film debonded from the metal substrate.

edge cracks on the silicon front surface, schematically shown in Fig. 3b. The Mode I (opening mode) stress intensity factor for such periodic cracks on a semi-infinite plane can be represented as [14]

1.12I pK F aσ π= , (6)

where PF is the coefficient of the stress intensity factor under periodic cracks on a semi-infinite plane; σ is the tensile stress at silicon top surface and 1 1( )hσ σ≈ from Eq. (4). The coefficient of PF ranges from 0 to 1 and its determination is not trivial. It is determined by the size and orientation of these edge cracks (or pyramids) and their spacing [15-17]. As a rule of thumb, the resulting stress intensity factor from periodic edges cracks is smaller than the one corresponding to a single edge crack under the same stress field as shown in Fig. 3c where 1PF = . This rule holds for non-uniformly distributed edge cracks if this single edge crack represents the longest one. In this study, a conservative estimation of 1PF ≈ is applied.

The coefficient 1.12 in Eq. (6) is valid for an edge crack on a semi-infinite plane or 1/ 0a h ∝ . However, when silicon

(b)

a

a

(c)

(a)

4A.3.3

wafers become thinner ( 1h decreases), the influence of silicon layers with finite thickness should be considered. A geometrical dimensionless coefficient is given by considering an edge crack on a finite plane [14]

( ) ( )( )31 11 1 1

1

2 ( / 2 )/ 0.752 2.02 / 0.37 1 ( / 2 )

( / 2 )Tan a h hf a h a h Sin a h

Cos a h aπ

ππ π

= + + − . (7)

The asymptotic of the geometrical coefficient ( )1/ 1.12f a h = as 1/ 0a h ∝ (i.e., the pyramid height a is small compared to the silicon thickness). In summary,

1 1 1( / ) ( )IK f a h h aσ π≈ (8)

is used in the following to evaluate the Mode I stress intensity factor corresponding to pyramid structures.

The other failure mode, other than the cracking or breakage of silicon film, is the interfacial debonding between the silicon film and the supporting substrate. Fig. 4b shows an example of such interfacial deboding: a strip of silicon initiated from the edge of the SOM® foil and debonded from the metal substrate. From the stress analysis above, it is known that the stress field is discontinuous at the silicon and substrate interface where debonding can occur under a temperature field. The energy release rate is applied to quantify and evaluate this debonding behavior at the interface. It is the elastic strain energy stored in the silicon film and substrate per unit length since the stresses in both silicon and supporting substrate are released after debonding. Therefore, the energy release rate at the interface can be represented as

1

2

2 201 2

01 2

( ) ( )12

h

h

y yG dy dyE E

σ σ−

⎛ ⎞= +⎜ ⎟

⎝ ⎠∫ ∫ . (9)

Substitute Eqs. (4) and (5) into Eq. (9), the equation becomes

( )( )

23 31 2 1 2 1 1 2 2 2 1 0

2 4 3 2 2 3 2 41 1 1 2 1 2 1 2 1 2 1 2 1 2 2 2

( ) ( )2 4 6 4

E E h h E h E h TG

E h E E h h E E h h E E h h E hα α ε+ − Δ −

=+ + + +

. (10)

Eq. (10) is consistent with the energy release rate derived from stress intensify factors [18].

A special case is noted here. This case is to predict the energy release rate in the situation where the SOM® bi-material composite foil is held circumferentially and keep flat during cell processing (various types of chemical and physical vapor depositions or CVD/PVD) in order to obtain uniform and high quality solar cells, i.e., the energy release rate as 0κ = . Substituting Eqs. (4) and (5) into Eq. (9) and considering

0κ = , the corresponding energy release rate

( )( ) ( )211 2 1 2 2 1 0 1 1 2 22 ( ) /G E E h h T E h E hα α ε= − Δ − + . (11)

It can be seen that for the same misfit strain εΔ or 2 1 0( ) Tα α ε− Δ − , the energy release rate from Eq. (11)

( 0κ = ) is always larger than the one from Eq. (10) as the composite foil can deform without constraint. This observation can be explained from the fact that the stresses (Eqs. (4) and

(5)) in the composite foil are alleviated due to the resulting bending moment as the foil is not constrained and therefore, there is less stored strain energy corresponding to the total misfit εΔ to drive the debonding at the silicon-substrate interface.

IV. EXPERIMENTAL, RESULTS AND DISCUSSIONS In this section, we are going to discuss several critical

issues for the handling and reliability of thin monocrystalline silicon solar cells based on the SOM® substrates: the determination of the residual strain 0ε and the process temperature limit for cell fabrication and module integration.

The related material properties are used in the following discussions: Young’s modulus 1 170 GPaE = , Poisson’s ratio

1 0.22v = and the coefficient of thermal expansion 6 o

1 2.6 10 / Cα −= × for silicon [19]; measured Young’s modulus 2 205 GPaE = , Poisson’s ratio 2 0.31v = and the coefficient of thermal expansion 6 o

2 13.4 10 / Cα −= × for the metal supporting substrate.

A. Determination of residual strain 0ε

The compressive residual strain 0ε resulting from the annealing process is intrinsic in the SOM® process and plays a critical role in the exfoliation process. Due to this residual strain, the in-plane stress (the first term in Eq. (4)) in the silicon layer of exfoliated composite foil is compressive at room temperature ( 0TΔ = ), which provides additional advantage to the already sturdy structure of SOM® bi-material composite foil and improves reliability and robustness during handling and transferring. The existence of this residual strain also increases the temperature limit during cell process and module integration. An accurate determination of the residual strain 0ε is the basis of all discussions in this study.

The exfoliated bi-material composite foil is bowed due to the existence of 0ε (Fig. 1). Eq. (3) shows the variation of the resulting curvature with TΔ and 0ε and can be simplified as

( )2 1 0( )c Tκ α α ε= − Δ − , (12)

where, 1 2 1 2 1 22 4 3 2 2 3 2 4

1 1 1 2 1 2 1 2 1 2 1 2 1 2 2 2

6 ( )4 6 4

E E h h h hcE h E E h h E E h h E E h h E h

+=+ + + +

.

From Eq. (12), it can be seen that

0 2 1 0( ) Tε α α= − Δ , (13)

where 0TΔ is the temperature differential with respect to room temperature at zero curvature ( 0κ = ). Once the temperature differential 0TΔ is known, so is 0ε . 0TΔ can be approximately determined by placing the SOM® composite foil on a hot plate and recording the temperature as the foil becomes flat. However, a curve-fit of the variation of curvature with TΔ (Eq. (12)) is used to accurately determine 0TΔ and the associated implementation is introduced in the following.

4A.3.4

The bowed composite foil was placed on a hot plate and the corresponding curvature can be obtained through measuring the height H of the bow (neglecting second-order higher terms)

( )2 22 / / 4H H Lκ = + , (14) where L is the side length of the composite foil. H was measured using a digimatic indicator (Mitutoyo Corp., Japan) which was attached to an extension arm and placed above the composite foil. By varying the temperature of the hot plate, the height of the bow at each temperature was recorded and the according curvature was evaluated using Eq. (14) with

125 mmL = .

Fig. 5a show the pictures of the composite foil at room temperature 22 °C on a hot plate and Fig. 5b shows the corresponding picture when the foil became approximately flat (~85 °C). Fig. 5c shows the variation of measured curvature with the temperature differential TΔ and corresponding linear curve-fit. From the curve-fit, the temperature differential at zero curvature 0 64.7 T CΔ = (the intercept of the TΔ -axis) and 4

0 6.98 10ε −= × . The corresponding slope 0.144 m-1 from curve-fitting is close to the prediction from Eq. (12) with actual dimensions of the foil within 6% deviation. Furthermore, since the residual strain 0ε in the metal layer is generated during annealing the bilayer structure of metal foil and parent wafer and remains after the exfoliation process, a bending beam test was conducted to determine 0ε after annealing yet before the exfoliation. The values of the residual strain 0ε determined from these two measurements are quite consistent.

Figure 5 Pictures of the composite foil on a hot plate at room temperature 22 °C (a) and ~85 °C (b). (c) The variation of measured curvature with the temperature differential TΔ and corresponding linear curve-fit. Each data is the average of three measurements.

B. Temperature limit during cell process The handling and transferring capability of SOM® thin

monocrstalline silicon films are implemented by the metal supporting substrate with a compressive residual strain 0ε . For the thin crystalline silicon backed by a metal layer, due to the

CTE mismatch between silicon and metal, thermal tensile stresses at high temperature can develop. Under the resulting tensile stress, initial cracks can propagate and run through the entire silicon layer, especially when the silicon surface is textured. Furthermore, debonding at the interface between the silicon layer and metal substrate can also occur at high temperatures. The high temperature limit that the SOM® bi-material composite foil can endure without reliability issues, such as breakage in silicon film or debonding at the interface, has to be determined in order to choose appropriate cell processing strategy and recipes. These issues are studied in the following on the basis of the analysis in Section 3.

Note that during cell manufacturing, the SOM® composite foil is usually clamped circumferentially by a specially designed holder and kept flat and therefore, the corresponding curvature 0κ = . Substituting Eq. (4) ( 0κ = ) into Eq. (8), the Mode I stress intensity in the silicon layer

( )1 2 2 2 1 01

1 1 2 2

( )( / )I

E E h TK f a h a

E h E hα α ε

π− Δ −

=+

, (15)

where 1( / )f a h is the geometrical coefficient as shown in Eq. (7).

The energy release rate at the interface can be evaluated using Eq. (11)

( )( )

21 2 1 2 2 1 0

1 1 2 2

( )2

E E h h TG

E h E hα α ε− Δ −

=+

.

There are some general observations for the fracture behavior (Eqs. (15) and (11)) for SOM® bi-material composite foils. First, for a constant silicon thickness 1h , the thicker the metal substrate, the higher the stress level in the silicon layer and hence the larger the IK and G . Second, the stress intensity factor IK is a linear function of the temperature field the energy release rate G is quadratic. The higher the cell processing temperature ( 0T> Δ ), the higher the IK and G . Thirdly, the compressive residual strain 0ε reduces the total misfit strain 2 1 0( ) Tε α α εΔ = − Δ − by offsetting the thermal misfit strain 2 1( )T Tε α αΔ = − Δ and consequently, decreases the values of both IK and G . Finally, from Eq. (15), the height a of pyramids of textured structure has a significant impact on IK especially for a thin monocrystalline Si film. The second and third terms in Eq. (15) depend on the pyramid height a and its relative thickness ratio to silicon thickness 1h . For example, IK corresponding to the standard practice

5 μma ≈ and 1 200 μmh ≈ in current PV industry [7, 8] is 2.3 times the one corresponding to 1 μma ≈ . For the SOM® composite foil, the targeted silicon thickness 1 25 μmh = and the effect of textured pyramids on the stress intensity factor IK is more significant. Therefore, smaller pyramids for the SOM®

solar cell texturing are desirable and 2 μma ≈ (Fig. 3a) is used [13]. Detailed discussions are elaborated in the following.

-10

-8

-6

-4

-2

0

2

4

-20 0 20 40 60 80

y = -9.3451 + 0.14447x R= 0.99594

κ (m

-1)

ΔT (°C)

(c)

(a) (b)

4A.3.5

Figure 6 The variations of IK and G with the temperature differential TΔ .

ICK and CG are plotted as boundaries for IK and G curves, respectively.

In order to avoid the crack development or interfacial debonding, both IK and G have to be less than the fracture toughness ICK and the critical energy release rate CG , respectively, i.e.

I ICK K≤ and CG G≤ . (16)

For a (100) monocystalline Si film, 1 MPaICK m≈ [20] and the critical energy release rate 2 2

1/ 5.6 J/mC ICG K E= ≈ for a perfectly bonded interface. The conditions in Eq. (16) are used to determine the processing temperature limit for a SOM®

bi-material composite foil with silicon film thickness 1 25 μmh = , metal substrate thickness 2 50 μmh = and

pyramid height 2 μma = .

Fig. 6 shows both the variations of IK and G with the temperature differential TΔ with respect to room temperature 22 °C. It can be seen that when TΔ is below zero curvature temperature 0TΔ as determined in Section IV.A, the silicon film is in compression and therefore IK vanishes. As the temperature increases further, IK increases linearly. At the same time, G varies with temperature parabolically and also varnishes at zero curvature temperature. ICK and CG are plotted as boundaries for IK and G curves, respectively. The intersection point of IK and ICK curves o310 CKTΔ = is the processing temperature limit to avoid crack propagation and breakage in silicon film. Similarly, the intersection point of G and CG curves o240 CGTΔ = sets up the temperature limit to prevent interfacial debonding. The maximum allowable process temperature differential maxTΔ is the lesser of KTΔ and GTΔ , i.e., max min( , ) 240K GT T TΔ = Δ Δ = °C or the highest process temperature maxT = 262 °C at room temperature 22 °C . It can be also seen that from Fig. 6 that GTΔ is less than KTΔ for a pyramid height 2 μma ≈ , which is consistent with our experimental observations. In practice, the chipped-off silicon flakes at edges or corners of the composite foils resulting from interfacial debonding were observed at high temperature during cell fabrications.

The advantage of the compressive residual strain 0ε to the processing temperature tolerance is further emphasized. Without 0ε , the entire IK and G curves would translate to the left by the value of zero curvature temperature 0TΔ and both vanish at the origin in Fig. 6. Consequently, the maximum processing temperature maxT would be reduced by the amount of 0 65TΔ ≈ °C and drop to below 200 °C. It may be difficult to find processes to fabricate junctions for silicon solar cells at lower than 200 °C. Furthermore, the existence of 0ε may be beneficial to the conversion efficiency of the silicon solar cells due to the increased silicon bandgap [6, 21] since the silicon bandgap intends to increase under compressive stress [22]. In summary, the compressive residual strain 0ε plays critical roles not only in the exfoliation process but also in cell processing by expanding the cell processing temperature spectrum.

It is concluded that the temperature limit of cell processing for the SOM® bi-material composite foil is around 200~300 °C. In a conventional silicon solar cell process, a high temperature diffusion step up to 1000°C is usually practiced and this diffusion process cannot be used to process the SOM® bi-material foil into solar cells. A heterojunction cell architecture with a process temperature of ~ 200 °C is appropriate for solar cells based on the SOM® process [23]. Fig. 7 shows the I-V curve of a 9.1mm x 9.1mm area of the cell with heterojunction cell structure processed at 200 °C made from the SOM®

composite foil with ~25 μm silicon and ~ 50 μm metal substrate. The cell characteristics are shown in the table beside the figure and the efficiency of this unoptimized device (without intrinsic a-Si passivation) is about 15% [13].

Figure 7 I-V characteristic of a 9.1mm x 9.1mm solar cell formed on a 25μm SOM® monocrystalline silicon film.

V. CONCLUSIONS Quantitative mechanical and fracture analyses were

introduced and developed to address the mechanical strength and reliability issues of thin crystalline silicon solar cells fabricated from a patented exfoliation technology. These analyses include the thermal stress field in the crystalline silicon layer and supporting substrate, the fracture behavior of pyramid structures and the energy release rate at the silicon-substrate interface. On the basis of these analyses, the intrinsic compressive residual strain from the fabrication of SOM®

monocrystalline silicon films was determined experimentally; the temperature limit for cell processing and module integration was determined and a heterojunction solar cell structure was

Voc 580 mV Jsc 33.6 mA/cm2

FF 76.7% Efficiency 14.92%

0

0.2

0.4

0.6

0.8

1

1.2

0

5

10

15

0 40 80 120 160 200 240 280 320

KI

G

G (J m

-1)

ΔT (°C)

KI (M

Pa

m-1

/2)

KIC

GC

ΔTK

ΔTG

ΔT0

4A.3.6

chosen. In this way, a robust and reliable SOM® thin crystalline solar cells can be achieved.

The developed solutions and methodology can be readily applied to study other thin film solar cell structures with various configurations of device layers and supporting substrates.

ACKNOWLEDGMENT The authors would like to thank Moses Ainom, Ricardo

Garcia, Ariam Gurmu, and Rachel Stout for their invaluable technical support and preparation of samples.

REFERENCES [1] M.J. Kerr, et al., Limiting efficiency of crystalline silicon solar cells

enhanced to coulomb-enhanced auger recombination, Progress in Photovoltaics, 11 (2003) 97.

[2] C.A. Wolden, et al., Photovoltaic manufacturing: Present status, future prospects, and research needs, Journal of Vacuum Science & Technology A, 29 (2011) 030801.

[3] K.A. Munzer, et al., Thin monocrystalline silicon solar cells, IEEE Transactions on Electron Devices, 46 (1999) 2055.

[4] A.C. Pan, et al., Effect of thickness on bifacial silicon solar cells, 2007. [5] T. van Amstel, et al., Towards a better understanding of the thermo-

mechanical behavior of h-pattern cells during metallization, in: Pvsc: 2008 33rd ieee photovoltaic specialists conference, vols 1-4, Ieee, New York, 2008, pp. 1371.

[6] A. Luque, S. Hegedus, Handbook of photovoltaic science and engineering, 2nd ed., Wiley, Chichester, West Sussex, U.K., 2011.

[7] Z.Q. Xi, et al., Investigation of texturization for monocrystalline silicon solar cells with different kinds of alkaline, Renewable Energy, 29 (2004) 2101.

[8] E. Vazsonyi, et al., Improved anisotropic etching process for industrial texturing of silicon solar cells, Sol. Energy Mater. Sol. Cells, 57 (1999) 179.

[9] R.A. Rao, et al., A novel low cost 25μm thin exfoliated monocrystalline si solar cell technology, in: 37th ieee photovoltaic specialists conference, June 2011.

[10] L. Mathew, D. Jawarani, Us patent 7749884, in, July 2010. [11] D. Jawarani, et al., A low cost kerfless exfoliation technology for 25 um

thin monocrystalline solar cells, in: the Fifth International Workshop on Crystalline Silicon Solar Cells, Boston, November, 2011.

[12] S. Timoshenko, Analysis of bi-metal thermostats, Journal of the Optical Society of America, 11 (1925) 233.

[13] S. Saha, et al., Light-trapping in exfoliated ~25µm si foil for solar cells, Submitted, (2011).

[14] H.P.P.C. Tada, I.G. Rankin, The stress analysis of cracks handbook, 3rd ed., ASME Press, New York, 2000.

[15] O.L. Bowie, Solution of plane crack problems by mapping techniques, in: G.C. Sih (Ed.) Method of analysis and solution of crack problems: , Noordhoff International Publishing 1973, pp. 1.

[16] X.Q. Jin, L.M. Keer, Solution of multiple edge cracks in an elastic half plane, International Journal of Fracture, 137 (2006) 121.

[17] C.E. Freese, Periodic edge cracks of unequal length in a semi-infinite tensile sheet, International Journal of Fracture, 12 (1976) 125.

[18] Z.G. Suo, J.W. Hutchinson, Interface crack between 2 elastic layers, International Journal of Fracture, 43 (1990) 1.

[19] G. Simmons, H. Wang, Single crystal elastic constants and calculated aggregate properties: A handbook, 2d ed., Mass., M.I.T. Press, Cambridge, 1971.

[20] Y.L. Tsai, J.J. Mecholsky, Fractal fracture of single-crystal silicon, J. Mater. Res., 6 (1991) 1248.

[21] P. Baruch, et al., On some thermodynamic aspects of photovoltaic solar-energy conversion, Sol. Energy Mater. Sol. Cells, 36 (1995) 201.

[22] H. Unlu, A thermodynamic model for determining pressure and temperature effects on the bandgap energies and other properties of some semiconductors, Solid-State Electronics, 35 (1992) 1343.

[23] M. Taguchi, et al., Hit (tm) cells - high-efficiency crystalline si cells with novel structure, Progress in Photovoltaics, 8 (2000) 503.

APPENDIX A stress analysis for a bi-material composite beam with a

misfit strain εΔ is given in the following, schematically shown in Fig. A1. Two equal and opposite forces ( P and P− ) and a moment M per unit length are introduced to remove

εΔ .

From the equilibrium equation

1 2( ) / 2M P h h= + . (A1)

The compatibility equation requires

1 1 2 2/ /P h E P h EεΔ = + . (A2)

Therefore

1 2 1 2 1 1 2 2/ ( )P E E h h E h E hε= Δ + . (A3)

For the cross section of the bi-material composite beam, the distance between the neutral axis (XX´) and the x-axis or the interface

2 212 2 1 1 1 1 2 22 ( ) / ( )E h E h E h E hδ = − + (A4)

and the bending stiffness 2 4 3 2 2 3 2 41 1 1 2 1 2 1 2 1 2 1 2 1 2 2 2

1 1 2 2

4 6 412( )

E h E E h h E E h h E E h h E hE h E h

+ + + +Ω =+

. (A5)

The resulting curvature of the bi-material composite beam

1 2 1 2 1 22 4 3 2 2 3 2 41 1 1 2 1 2 1 2 1 2 1 2 1 2 2 2

6 ( )4 6 4

M E E h h h hE h E E h h E E h h E E h h E h

εκ + Δ= =Ω + + + +

. (A6)

The stress in the composite beam is the sum of the in-plane and bending stress resulting from P and M , respectively. The stresses in the up-layer and bottom-layer are respectively given as

1 2 21 1 1

1 1 1 2 2

( ) ( ) ( )P E E hy E y E yh E h E h

εσ κ δ κ δΔ= − + = − ++

for 10 y h< ≤ , (A7)

and 1 2 1

2 2 22 1 1 2 2

( ) ( ) ( )P E E hy E y E yh E h E h

εσ κ δ κ δΔ= − − + = − − ++

for 2 0h y− ≤ ≤ . (A8)

Figure A1 Schematic of a bi-material composite beam with a misfit strain εΔ between two layers (a) and the equivalent load set to remove the misfit strain (b).

(a)

P− P−

PP

M M

(b)

εΔ 1 1,E h

2 2,E h

y

xX’X

δ

4A.3.7