Abstract:
A method of forming a photovoltaic device includes a plurality of quantum wells and a plurality of barriers. The quantum wells and barriers are disposed on an underlying layer. The barriers alternate with the quantum wells. One of the plurality of quantum wells and the plurality of barriers is comprised of tensile strained layers and the other of the plurality of quantum wells and the plurality of barriers is comprised of compressively strained layers. The tensile and compressively strained layers have elastic properties. The method includes selecting compositions and thicknesses of the barriers and quantum wells taking into account the elastic properties such that each period of one tensile strained layer and one compressively strained layer exerts substantially no shear force on a neighboring structure; providing the underlying layer; and forming the quantum sells and barriers on the underlying layer according to the derived compositions and thicknesses.

Description:
CROSS-REFERENCE TO RELATED PATENT APPLICATIONS 
     The present application is a Continuation of U.S. patent application Ser. No. 09/955,297 filed Sep. 19, 2001, which claims priority to Great Britain Patent Application 0118150.2 filed Jul. 25, 2001. The entire disclosures of U.S. patent application Ser. No. 09/955,297 and Great Britain Patent Application 0118150.2 are incorporated by reference herein, including the claims, specifications, abstracts, and drawings thereof. 
    
    
     BACKGROUND 
     This invention relates to an improved photovoltaic device/cell for the conversion of heat radiation into electricity. 
     Thermophotovoltaics (TPV) is the use of photovoltaic (PV) cells to convert heat radiation, e.g. from the combustion of fossil fuels or biomass, into electricity. The energy spectrum is often reshaped using selective emitters which absorb the heat and re-emit in a narrow band. The re-emitted radiation may be efficiently converted to electric power using a PV cell of appropriate low band-gap. Higher PV cell efficiencies can be achieved by introducing multi-quantum-wells (MQW) into the intrinsic region of a p-i-n diode if the gain in short-circuit current exceeds the loss in open-circuit voltage [K. W. J. Bartham and G. Duggan, J. Appl. Phys. 67, 3490 (1990). K. Barnham et al., Applied Surface Science 113/114, 722 (1997). K. Barnham, International Published Patent Application WO-A-93/08606 and U.S. Pat. No. 5,496,415 (1993)]. A Quantum Well Cell (QWC) in the quaternary system InGaAsP lattice-matched to InP substrates is a promising candidate for TPV applications as the effective band-gap can be tuned, out to about 1.65 μm (In 0.53 Ga 0.47 As), without introducing strain, by varying the well depth and width, to match a given spectrum. The enhancement in output voltage of a QWC is a major advantage for TPV applications [P. Griffin et al., Solar Energy Materials and Solar Cells 50, 213 (1998). C. Rohr et al., in Thermophotovoltaic Generation of Electricity: Fourth NREL Conf., Vol. 460 of AIP Conf. Proc. (American Institute of Physics, Woodbury, N.Y., 1999), pp. 83-92]. 
     There is considerable interest in extending the absorption to longer wavelengths for higher overall system efficiencies with lower temperature sources; and lower temperature fossil sources have also lower levels of pollution. Appropriate and inexpensive substrates of the required lattice constant and band-gap are not available, so the lower band-gap material is often strained to the substrate, introducing dislocations which increase non-radiative recombination. Freundlich et al. have proposed strained quantum well devices [U.S. Pat. No. 5,851,310 (1998), U.S. Pat. No. 6,150,604 (2000)], but these can only incorporate a restricted number of wells without creating dislocations. Freundlich proposes limiting the number of wells to a maximum of 20, which will not produce sufficient absorption for efficient generation however. In a MQW system, these dislocations can be reduced by strain-balancing the layers; alternating barriers and wells have bigger and smaller lattice-constants, but on average are lattice-matched to the substrate [N. J. Ekins-Daukes et al., Appl. Phys. Lett. 75, 4195 (1999)]. 
     SUMMARY 
     An exemplary embodiment relates to a method of forming a photovoltaic device that includes a plurality of quantum wells and a plurality of barriers. The quantum wells and barriers are disposed on an underlying layer. The barriers alternate with the quantum wells. One of the plurality of quantum wells and the plurality of barriers is comprised of tensile strained layers and the other of the plurality of quantum wells and the plurality of barriers is comprised of compressively strained layers. The tensile and compressively strained layers have elastic properties. The method includes selecting compositions and thicknesses of the barriers and quantum wells taking into account the elastic properties such that each period of one tensile strained layer and one compressively strained layer exerts substantially no shear force on a neighboring structure; providing the underlying layer; and forming the quantum sells and barriers on the underlying layer according to the derived compositions and thicknesses. 
     Another exemplary embodiment relates to a photovoltaic device that includes an underlying layer and a multiple quantum well portion formed of a plurality of quantum wells and a plurality of barriers disposed on the underlying layer such that the barriers alternate with the quantum wells. One of the plurality of quantum wells and the plurality of barriers is comprised of tensile strained layers and the other of the plurality of quantum wells and the plurality of barriers is comprised of compressively strained layers. The tensile and compressively strained layers have elastic properties. The tensile strained layers and the compressively strained layers have compositions and thicknesses that are selected taking into account the elastic properties such that each period of one tensile strained layers and one compressively strained layers exerts substantially no shear force on a neighboring structure. 
     Another exemplary embodiment relates to a photovoltaic device having a multiple well quantum portion formed upon a virtual substrate having a virtual substrate lattice constant that is different than a substrate lattice constant of an underlying substrate. The virtual substrate is InP 1-x As x  where 0&lt;x&lt;1 and the substrate is InP. 
     Another exemplary embodiment relates to a photovoltaic device having a multiple quantum well portion formed of strained alternating quantum well layers of In x Ga 1-x As, where x&gt;0.53, and barrier layers of Ga y In 1-y P, where y&gt;0. 
     Another exemplary embodiment relates to a photovoltaic device that includes a plurality of quantum wells and a plurality of barriers, the barriers alternating with the quantum wells. One of the plurality of quantum wells and the plurality of barriers is comprised of tensile strained layers and the other of the plurality of quantum wells and the plurality of barriers is comprised of compressively strained layers. The tensile strained layers and the compressively strained layers have compositions such that a period of one tensile strained layer and one compressively strained layer exerts substantially no shear force on a neighboring structure. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a bandgap diagram of a strain-balanced quantum well cell. The p- and n-regions are made of material that is lattice-matched to the InP substrate, e.g. In 0.53 Ga 0.47 As or InP. The quantum wells are made of In x Ga 1-x As with x&gt;0.53, and the barrier of In x Ga 1-x As with x&lt;0.53, GaInP or InGaAsP. 
         FIG. 2  is a schematic drawing of a strain-compensated quantum well cell where the width indicates the lattice parameter of the material when unstrained. 
         FIG. 3  is a graph of dark current densities of a strain-balanced quantum well cell (as depicted in  FIG. 2  but with 30 quantum wells) compared with bulk GaSb of similar effective bandgap (see  FIG. 4 ) and lattice-matched bulk InGaAs. 
         FIG. 4  is a graph of modeled internal quantum efficiency (with back-surface reflector) of a strain-balanced quantum well cell (as depicted in  FIG. 2  but with 30 quantum wells) compared with bulk GaSb and lattice-matched bulk InGaAs. 
         FIG. 5  is a graph of modeled internal quantum efficiency (with back-surface reflector) of a strain-balanced quantum well cell optimized for a Holmia emitter (not to scale). 
         FIG. 6  is a graph of the dark current of an AlGaAs/GaAs quantum well cell, where the data (dots) is fitted (black line). The modeled dark current density for a QWC with a higher band-gap barrier (grey line) is reduced. 
         FIG. 7  shows Lattice constant versus Bandgap of the material system In x Ga 1-x As 1-y P y . 
     
    
    
     DETAILED DESCRIPTION 
     According to an exemplary embodiment, a photovoltaic device has a multiple quantum well portion with alternating tensile strained layers and compressively strained layers, said tensile strained layers and said compressively strained layers having compositions such that a period of one tensile strained layer and one compressively strained layer exerts substantially no shear force on a neighboring structure. 
     The inventors have recognized that rather than seeking to provide an average lattice constant that matches the substrate, what is truly important is to balance the forces in the tensile and compressively strained layer to provide an average or effective zero stress system. A device providing an average lattice constant matching the substrate may still allow a significant build up of stress that will result in undesirable dislocations. 
     With this concept one can extend the absorption threshold to longer wavelength without introducing dislocations. 
     With a strain-balanced multi-quantum-well stack in the intrinsic region of a two-terminal photovoltaic device the absorption threshold can be extended to longer wavelengths. In particular, with high bandgap barriers the dark current can be reduced at the same time, and hence the conversion efficiency is increased significantly. 
     What is also helpful to achieve higher conversion efficiencies is an improved voltage performance, due to a lower dark current. This is provided by the higher barriers which may also be provided when balancing the strain. 
     According to another exemplary embodiment, a photovoltaic device has a multiple well quantum portion formed upon a virtual substrate having a virtual substrate lattice constant different than a substrate lattice constant of an underlying substrate, wherein said virtual substrate is InP 1-x As .x , where 0&lt;x&lt;1 and said substrate is InP. 
     Using an InP 1-x As x , based virtual substrate allows lattice matching to a quantum well system having a relatively large lattice constant, and typically desirable low bandgap. 
     According to another exemplary embodiment, a photovoltaic device has a multiple quantum well portion formed of strained alternating quantum well layers of In x Ga 1-x As, where x&gt;0.53, and barrier layers of Ga y In 1-y P, where y&gt;0. 
     This combination of layers allows provision of an advantageously high barrier energy within the multiple quantum well system which reduces the dark current. Furthermore, this composition is well suited to stress balancing and use with the above mentioned virtual substrate. 
     A photovoltaic cell to convert low energy photons is described, consisting of a p-i-n diode with a strain-balanced multi-quantum-well system incorporated in the intrinsic region. The bandgap of the quantum wells is lower than that of the lattice-matched material, while the barriers have a much higher bandgap. The high band-gap barriers reduce the dark current. Hence the absorption can be extended to longer wavelengths, while maintaining a low dark current. This leads to greatly improved conversion efficiencies, particularly for low energy photons from low temperature sources. This can be achieved by strain-balancing the quantum wells and barriers, where each individual layer is below the critical thickness and the strain is compensated by quantum wells and barriers being strained in opposite directions. The strain is compensated by choosing the material compositions and thicknesses of the layers in such a way that the average stress is zero, taking into account the elastic properties of the materials. Thereby the creation of misfit dislocations, which are detrimental to the dark current and hence to the cell conversion efficiency, can be avoided. The number of quantum wells that can be incorporated is therefore not limited by the build-up of strain, but only by the size of the i-region, and is typically 30-60 [This is an important advantage over Freundlich&#39;s strained QWs with a maximum number of about 20 (see U.S. Pat. No. 5,851,310 and U.S. Pat. No. 6,150,604)]. The width of the i-region is limited by the electric field that needs to be maintained across it. 
     The absorption can be further extended to longer wavelengths by introducing a strain-relaxed layer (virtual substrate) between the substrate and the active cell. The device is then grown on this virtual substrate and the layers are strain-balanced with respect to the new lattice constant. This allows one to effectively move to a specific lattice constant which is associated with a desired band gap for the lattice matched and strain-balanced materials. This is of particular interest for thermophotovoltaic applications with lower temperature sources, as one can extend the absorption towards the required long wavelengths. 
     As an example for a strain-compensated QWC, we consider a 30 well In 0.62 Ga 0.38 As/In 0.47 Ga 0.53 As (InP) QWC, grown by MOVPE, whose sample description is given in Table I. 
     
       
         
               
             
               
               
               
               
               
               
             
               
               
               
               
               
               
             
           
               
                 TABLE I 
               
             
             
               
                   
               
               
                 Sample description of a strain-compensated quantum well cell. 
               
             
          
           
               
                   
                 Thickness 
                   
                   
                   
                 Conc. 
               
               
                 Layers 
                 (Å) 
                 Material 
                 Function 
                 Doping 
                 (cm −3 ) 
               
               
                   
               
             
          
           
               
                 1 
                 1000 
                 In 0.53 Ga 0.47 As 
                 Cap 
                 p 
                 1E+19 
               
               
                 1 
                 7000 
                 InP 
                 Emitter 
                 p 
                 2E+18 
               
               
                 30 
                 120 
                 In 0.45 Ga 0.55 As 
                 Barrier 
                 i 
               
               
                 30 
                 120 
                 In 0.62 Ga 0.38 As 
                 Well 
                 i 
               
               
                 1 
                 120 
                 In 0.47 Ga 0.53 As 
                 Barrier 
                 i 
               
               
                 1 
                 5000 
                 InP 
                 Base 
                 n 
                 1E+18 
               
               
                   
                   
                 InP 
                 Substrate 
                 n 
               
               
                   
               
             
          
         
       
     
     In  FIG. 2  the strain-balancing conditions of one example are shown, where the average lattice-constant of wells and barriers is roughly the same as the InP substrate.  FIG. 1  shows a schematic diagram of the energy band-gaps of this kind of structure. This specific sample was not designed for TPV applications; the p-region, for example, is far too thick. It does not quite fulfill the ideal strain-balanced conditions, but is close enough to avoid strain relaxation, as is evident by the low dark current of the device (see  FIG. 3 ). In fact, the dark current density is even lower than in a very good lattice-matched bulk InGaAs/InP cell [N. S. Fatemi et al., in Proc. 26th IEEE PV specialists conf. (IEEE, USA, 1997), pp. 799-804] as shown in  FIG. 3 . In  FIG. 4  we show the spectral response (SR) (=external quantum efficiency) data of the strain-balanced QWC at zero bias. The effective band-gap, resulting from the material composition and the confinement, is about 1.77 μm, which is well beyond the band-edge of lattice-matched InGaAs. Hence the strain-balanced approach has enabled the absorption threshold to be extended out to 1.77 μm while retaining a dark current more appropriate to a cell with a band-edge of less than 1.65 μm. The band-edge of the strain-balanced QWC is similar to that of a GaSb cell, but it has a lower dark current (see  FIG. 3 ). Strain-balanced QWCs in InGaP/InGaAs on GaAs have demonstrated dark currents comparable to homogenous GaAs cells [N. J. Ekins-Daukes et al., Appl. Phys. Lett. 75, 4195 (1999)]. We have shown (see  FIG. 3 ) that, if anything, In x Ga 1-x As/In z Ga 1-z As (InP) cells with absorption edges out to 1.77 μm have lower dark currents than bulk InGaAs cells. To obtain even lower dark currents, we need a higher band-gap in the barriers. We can achieve that by using a different material for the barrier, such as In x Ga 1-x As 1-y P y  with y&gt;0 or GaInP as indicated in  FIG. 1 , and an example for such a device is given in Table II. 
     
       
         
               
             
               
               
               
               
               
               
             
               
               
               
               
               
               
             
           
               
                 TABLE II 
               
             
             
               
                   
               
               
                 Sample description of a strain-balanced quantum 
               
               
                 well cell with high bandgap barriers. 
               
             
          
           
               
                   
                 Thickness 
                   
                   
                   
                 Conc. 
               
               
                 Layers 
                 (Å) 
                 Material 
                 Function 
                 Doping 
                 (cm −3 ) 
               
               
                   
               
             
          
           
               
                 1 
                 1000 
                 In 0.53 Ga 0.47 As 
                 Cap 
                 p 
                 1E+19 
               
               
                 1 
                 1500 
                 InP 
                 Emitter 
                 p 
                 5E+18 
               
               
                 1 
                 50 
                 Ga 0.18 In 0.82 P 
                 Barrier 
                 i 
               
               
                 49 
                 100 
                 Ga 0.18 In 0.82 P 
                 Barrier 
                 i 
               
               
                 50 
                 100 
                 In 0.72 Ga 0.28 As 
                 Well 
                 i 
               
               
                 1 
                 50 
                 Ga 0.18 In 0.82 P 
                 Barrier 
                 i 
               
               
                 1 
                 5000 
                 InP 
                 Base 
                 n 
                 1E+18 
               
               
                   
                   
                 InP 
                 Substrate 
                 n 
               
               
                   
               
             
          
         
       
     
     We have developed a model which calculates the SR of multi-layer In x Ga 1-x As 1-y P y  devices, lattice-matched to InP (x=0.47 y) [M. Paxman et al., J. Appl. Phys. 74, 614 (1993), C. Rohr et al., in Thermophotovoltaic Generation of Electricity: Fourth NREL Conf., Vol. 460 of AIP Conf. Proc. (American Institute of Physics, Woodbury, N.Y., 1999), pp. 83-92], which has been extended to estimate the SR of strain-balanced In x Ga 1-x As/In z Ga 1-z As on InP [C. Rohr et al., in Proc. 26th International Symposium on Compound Semiconductors No. 166 in Institute of Physics Conference Series (Institute of Physics Publishing, Bristol and Philadelphia, 2000), pp. 423-426]. The cell efficiency can be determined given the measured dark current data of the cell, assuming superposition of dark and light current. For photovoltaic applications the p-region of a device would typically be as thin as 1500 Å (instead of 7000 Å) in order to increase the light level that reaches the active i-region where carrier separation is most efficient and to reduce free carrier absorption. A mirror on the back of a semi-insulating (i.e. charge neutral) substrate is particularly useful for QWCs as it enhances the well contribution significantly. The effect of such a mirror is simulated by doubling the light pass through the wells. The strain-balanced QWC is modeled with these modifications and, for the purpose of comparison, the reflectivity is removed to show the internal quantum efficiency in  FIG. 4 . 
     We compare our strain-balanced QWC as well as our lattice-matched InGaAsP QWCs with lattice-matched InGaAs monolithic interconnected modules (MIMs) [N. S. Fatemi et al., in Proc. 26th IEEE PV specialists conf. (IEEE, USA, 1997), pp. 799-804], one of the best lattice-matched bulk InGaAs/InP TPV cells, and with bulk GaSb [A. W. Bett et al., in Thermophotovoltaic Generation of Electricity: Third NREL Conf., Vol. 401 of AIP Conf. Proc. (American Institute of Physics, Woodbury, N.Y., 1997), pp. 41-53], currently the only material which is being used commercially for TPV applications. To compare efficiencies we assume “typical” TPV conditions of 100 kW/m 2  normalized power density, grid shading of 5%, and internal quantum efficiencies for all cells. A back surface reflector is an integral part of MIM technology and particularly useful for QWCs as it enhances the well contribution significantly. It also increases TPV system efficiency because longer wavelength radiation, that is not absorbed by the cell, is reflected back to the source. The efficiency projections for various illuminating spectra are calculated from data presented in  FIGS. 3 and 4  and are summarized in Table III. The relative efficiencies are rather more reliable than the absolute values. 
     
       
         
               
             
               
               
               
               
               
             
               
               
               
               
               
             
           
               
                 TABLE III 
               
             
             
               
                   
               
               
                 Comparison of predicted efficiencies (in %) of bulk InGaAs 
               
               
                 MIM, GaSb, lattice-matched and strain-balanced quantum well 
               
               
                 cells with back-mirror using internal quantum efficiencies, 
               
               
                 under various spectra at 100 kW/m 2 , and 5% grid shading: 
               
             
          
           
               
                   
                 Bulk InGaAs 
                 Bulk 
                 InGaAsP 
                 Strain-bal. 
               
               
                 Spectrum 
                 MIM 
                 GaSb 
                 QWC 
                 QWC 
               
               
                   
               
             
          
           
               
                 Solar × 100 
                 16 
                 16 
                 20 
                 19 
               
               
                 3200K blackbody 
                 18 
                 18 
                 22 
                 27 
               
               
                 2000K blackbody 
                 11 
                 11 
                 12 
                 22 
               
               
                 1500K blackbody 
                 5.5 
                 5.6 
                 4.8 
                 14 
               
               
                 MgO 
                 13 
                 15 
                 16 
                 41 
               
               
                 Ytterbia 
                 26 
                 25 
                 42 
                 32 
               
               
                 Erbia 
                 37 
                 37 
                 46 
                 43 
               
               
                 Holmia 
                 4.5 
                 5.4 
                 4.1 
                 39 
               
               
                   
               
             
          
         
       
     
     The lower dark current of the QWCs (see  FIG. 3 ) is the main reason for their higher efficiencies in Table III. The lattice-matched InGaAsP QWC shows higher efficiencies than the InGaAs MIM and GaSb in all cases except for black-body temperatures below about 2000 K. Higher black-body temperatures, for example 3200 K and the solar spectrum AM1.5 (approximating 5800 K) at 100 times concentration, are favourable for the lattice-matched InGaAsP QWC. At black-body temperatures around 2000 K and below, the strain-balanced QWC outperforms the others. Particularly with the MgO emitter, which was designed for a GaSb cell [L. Ferguson and L. Fraas, in Thermophotovoltaic Generation of Electricity: Third NREL Conference Vol. 401 of AIP Conf. Proc. (American Institute of Physics, Woodbury, N.Y., 1997), pp. 169-179], the strain-balanced QWC is significantly better and shows an efficiency which is about 50% higher than that of a GaSb cell (see Table III). 
     Based on these results it should be possible to use this concept of strain-balanced QWCs to extend the absorption threshold even further, beyond 2 μm, optimized for TPV applications with a Holmia emitter (see  FIG. 5 ). The efficiency for such a strain-balanced QWC with a Holmia emitter [M. F. Rose et al., Journal of Propulsion and Power 12, 83 (1996)] is predicted to reach 39% under the same conditions as discussed above. The more the band-edge of a PV cell is extended towards longer wavelengths, the more suitable it becomes for lower temperature sources. 
     The conversion efficiency can be further substantially increased by reducing the dark current. In strain-balanced devices, this can be achieved if higher band-gap material is used for the barriers as indicated in  FIG. 1  and Table II. 
     A model for the dark current behavior of QWCs is used in  FIG. 6 . In  FIG. 6 , a dark current density of an AlGaAs/GaAs quantum well cell is fitted, and it shows that the modeled dark current density for a QWC with a higher band-gap barrier is reduced and hence the efficiency will be increased. 
     In order to be lattice-matched to an InP substrate, the material composition of In x Ga 1-x As 1-y P y  must be chosen to lie on the vertical line in  FIG. 7  going through InP, which corresponds to x≈0.53+0.47 y. That means, the lowest bandgap one can achieve with lattice-matched material is with In 0.53 Ga 0.47 As, a bandgap of E g  ≈0.74 eV. Strain-compensation in a multi-layer system allows one to achieve lower effective band-gaps. The quantum wells are compressively strain, going down the branch from In 0.53 Ga 0.47 As towards InAs (i.e. x&gt;0.53), and to compensate the barriers have tensile strain going up the branch from In 0.53 Ga 0.47 As towards GaAs (i.e. x&lt;0.53). To improve the dark current with higher bandgap barriers one can use material compositions with y&gt;0 and the same lattice constant as before, i.e. going up on a vertical line in  FIG. 7 . To achieve high bandgap barriers, these may be formed of Ga y In 1-y P, where y&gt;0. In  FIG. 7  this composition follows the upper limit between InP and GaP. 
     By introducing a virtual substrate, still lower bandgaps can be reached as the lattice constant is increased by relaxed buffer layers. This shifts the base or reference line for strain-compensation towards the right in  FIG. 7 . This virtual substrate can be made of InAsP (upper branch in  FIG. 7 ) [Wilt et al., 28th IEEE PVSC (2000), p. 1024] instead of InGaAs. Such an InAsP buffer is better in confining the dislocations in the virtual substrate, which is crucial for successfully growing a strain-compensated multi-quantum well (MQW) structure on top of it. 
     The conditions for zero-stress strain-balance may be determined from the following considerations: 
     The strain ε for each layer i is defined as 
             ɛ   =         a   0     -     a   i         a   i             
where α 0  is the lattice constant of the substrate (or virtual substrate), and α i  is the natural unstrained lattice constant of layer i.
 
     A strain-balanced structure should be designed such that a single period composed of one tensile and one compressively strained layer, exerts no shear force on its neighbouring layers or substrate. To achieve such a zero stress situation, one needs to taken into account the differences in elastic properties of the layers. Applying linear elastic theory one can derive the following conditions 
     
       
         
           
             
               
                 
                   
                     
                       
                         ɛ 
                         1 
                       
                       ⁢ 
                       
                         t 
                         1 
                       
                       ⁢ 
                       
                         A 
                         1 
                       
                       ⁢ 
                       
                         a 
                         2 
                       
                     
                     + 
                     
                       
                         ɛ 
                         2 
                       
                       ⁢ 
                       
                         t 
                         2 
                       
                       ⁢ 
                       
                         A 
                         2 
                       
                       ⁢ 
                       
                         a 
                         1 
                       
                     
                   
                   = 
                   0 
                 
               
               
                 
                   ( 
                   
                     zero 
                     ⁢ 
                     
                       - 
                     
                     ⁢ 
                     stress 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     condition 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     a 
                     0 
                   
                   = 
                   
                     
                       
                         
                           t 
                           1 
                         
                         ⁢ 
                         
                           A 
                           1 
                         
                         ⁢ 
                         
                           a 
                           1 
                         
                         ⁢ 
                         
                           a 
                           2 
                           2 
                         
                       
                       + 
                       
                         
                           t 
                           2 
                         
                         ⁢ 
                         
                           A 
                           2 
                         
                         ⁢ 
                         
                           a 
                           2 
                         
                         ⁢ 
                         
                           a 
                           1 
                           2 
                         
                       
                     
                     
                       
                         
                           t 
                           1 
                         
                         ⁢ 
                         
                           A 
                           1 
                         
                         ⁢ 
                         
                           a 
                           2 
                           2 
                         
                       
                       + 
                       
                         
                           t 
                           2 
                         
                         ⁢ 
                         
                           A 
                           2 
                         
                         ⁢ 
                         
                           a 
                           1 
                           2 
                         
                       
                     
                   
                 
               
               
                 
                   5 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ( 
                     
                       Match 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       substrate 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       lattice 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       constant 
                     
                     ) 
                   
                 
               
             
             
               
                 
                   A 
                   = 
                   
                     
                       C 
                       11 
                     
                     + 
                     
                       C 
                       12 
                     
                     - 
                     
                       
                         2 
                         ⁢ 
                         
                           C 
                           12 
                           2 
                         
                       
                       
                         C 
                         11 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Layer 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     stiffness 
                   
                   ) 
                 
               
             
           
         
       
     
     where t 1  and t 2  are the thicknesses of layers  1  and  2 , and C 11  and C 12  are the elastic stiffness coefficients. 
     Although illustrative embodiments of the invention have been described in detail herein with reference to the accompanying drawings, it is to be understood that the invention is not limited to those precise embodiments, and that various changes and modifications can be effected therein by one skilled in the art without departing from the scope and spirit of the invention as defined by the appended claims.