Raman-strain relations in highly strained Ge: Uniaxial #100#, #110# and biaxial (001) stress A. Gassenq, S. Tardif, K. Guilloy, I. Duchemin, N. Pauc, J. M. Hartmann, D. Rouchon, J. Widiez, Y. M. Niquet, L. Milord, T. Zabel, H. Sigg, J. Faist, A. Chelnokov, F. Rieutord, V. Reboud, and V. Calvo
Citation: Journal of Applied Physics 121, 055702 (2017); doi: 10.1063/1.4974202 View online: http://dx.doi.org/10.1063/1.4974202 View Table of Contents: http://aip.scitation.org/toc/jap/121/5 Published by the American Institute of Physics
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JOURNAL OF APPLIED PHYSICS 121, 055702 (2017)
Raman-strain relations in highly strained Ge: Uniaxial h100i, h110i and biaxial (001) stress A. Gassenq,1,a) S. Tardif,1 K. Guilloy,1 I. Duchemin,1 N. Pauc,1 J. M. Hartmann,2 D. Rouchon,2 J. Widiez,2 Y. M. Niquet,1 L. Milord,2 T. Zabel,3 H. Sigg,3 J. Faist,4 A. Chelnokov,2 F. Rieutord,1 V. Reboud,2 and V. Calvo1 1
Univ. Grenoble Alpes, CEA-INAC, 17 rue des Martyrs, 38000, Grenoble, France Univ. Grenoble Alpes, CEA-LETI, Minatec Campus, 17 rue des Martyrs, 38054, Grenoble, France 3 Laboratory for Micro- and Nanotechnology, Paul Scherrer Institute, 5232, Villigen, Switzerland 4 Institute for Quantum Electronics, ETH Zurich, 8093, Z€ urich, Switzerland 2
(Received 13 October 2016; accepted 30 December 2016; published online 1 February 2017) The application of high values of strain to Ge considerably improves its light emission properties and can even turn it into a direct band gap semiconductor. Raman spectroscopy is routinely used for strain measurements. Typical Raman-strain relationships that are used for Ge were defined up to 1% strain using phonon deformation potential theory. In this work, we have studied this relationship at higher strain levels by calculating and measuring the Raman spectral shift-strain relations in several different strain configurations. Since differences were shown between the usual phonon deformation potential theory and ab-initio calculations, we highlight the need for experimental calibrations. We have then measured the strain in highly strained Ge micro-bridges and micro-crosses using Raman spectroscopy performed in tandem with synchrotron based microdiffraction. High values of strain are reported, which enable the calibration of the Raman-strain relations up to 1.8% of in plane strain for the (001) biaxial stress, 4.8% strain along h100i, and 3.8% strain along h110i. For Ge micro-bridges, oriented along h100i, the nonlinearity of the Raman shift-strain relation is confirmed. For the h110i orientation, we have shown that an unexpected non-linearity in the Raman-strain relationship has also to be taken into account for high stress induction. This work demonstrates an unprecedented level of strain measurement for the h110i uniaxial stress and gives a better understanding of the Raman-strain relations in Ge. Published by AIP Publishing. [http://dx.doi.org/10.1063/1.4974202]
I. INTRODUCTION
Silicon and germanium are the dominant materials for electronic integrated circuits but the indirect bandgap of these group IV elements prevents them from being used as efficient light-emitting components. Two approaches have been proposed to enhance the light emission of Ge by doping or/and straining the material. Lasers have been made using heavily n-doped Ge (in order to fill the indirect L-valley).1 Even if the physical interpretation is controversial,2 the experiment has been reproduced3 confirming the high intrinsic lasing threshold of the doping approach. Alternatively, it has been predicted that a large tensile strain in Ge can transform it into a direct band gap material.4 Several methods are currently being explored to apply strain in Ge including the use of deposited stressor layers,5–9 applied external mechanical stress in Ge membranes,10,11 strain redistribution,12–14 or by growing thin Ge layers on InGaAs buffers.15,16 The strain that is required to achieve a direct bandgap Ge crystal is theoretically around 4.5% strain for uniaxial tensile stress,17–20 and around 1.8% strain for biaxial tensile stress.18,21,22 Today, these targeted strain amplitudes have been experimentally reached for both stress configurations,9,23–25 paving way towards mid-infrared lasers fully compatible with a)
Author to whom correspondence should be addressed. Electronic mail:
[email protected]
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Complementary Metal Oxide Semiconductor technologies. However, accurate measurements of the strain amplitude at the micro-scale are now essential for comparison with theoretical calculations, which predict the indirect to direct transition at a certain strain level. X-ray diffraction (XRD) is the method of choice to determine strain in crystalline materials. Using synchrotron sources, these measurements can be performed down to the submicrometer scale.6,26–29 However, synchrotrons are not easily accessible and strain measurements in Ge devices are mainly performed in laboratory using Raman spectroscopy. Raman spectroscopy is a commonly used method for strain evaluation that has the advantage of being well spatially resolved (i.e., down to the micron scale), fast and readily available in laboratories. Since mechanical stress influences the phonon frequency, the strain can be evaluated by measuring the Raman spectral shift. For hydrostatic strain, the Gr€uneisen formalism was developed.30,31 For non-hydrostatic strain, the Ramanstrain relationships are highly dependent on both the material and the stress configuration.32–34 Recently, we have calibrated the Raman spectral shift strain relation in Ge micro-bridges under h100i uniaxial stress using X-ray micro-diffraction at the European Synchrotron Radiation Facility (ESRF) synchrotron and Raman spectroscopy.23 Nevertheless, the h110i stress configuration in Ge is also of interest for hole mobility enhancement35 and photonics applications.8 While the Raman
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shift-strain conversion rules for the h110i orientation have been measured only up to 2.2 cm1,33 spectral shifts up to 10 cm1 have been recently observed experimentally.8 A determination of the Raman shift-strain relation for large h110i stress is thus needed. In this work, we have studied the Raman spectral shiftstrain relationship for different stress configurations, both theoretically and experimentally. We have first compared the Raman-strain relations, calculated using the usual semiempirical model based on the phonon deformation potentials33,34,36 to ab initio calculations. Since some discrepancies were shown, we have highlighted the need for the experimental calibrations. We have therefore performed synchrotron-based Laue micro-diffraction in tandem with backscattering Raman spectroscopy measurements in Ge micro-crosses and micro-bridges under biaxial and uniaxial stress, respectively. For the Ge micro-crosses, the Raman shift-strain relation was measured up to 1.8% strain. Our results are in good agreement with previous results (where the Raman shift-strain relationship had already been calibrated up to 2.6% strain in thin layers37–39) showing that the membrane approach combined with micro-XRD is relevant to measure Raman-strain relationships in thicker layers. For the Ge micro-bridges oriented along h100i, the nonlinearity of the Raman shift-strain relationships was confirmed.23 For the micro-bridges along h110i, the Raman shift strain relationship was measured up to an unprecedented strain amplitude of 3.8% and an unexpected nonlinearity was also observed. For the latter, we thus provide an updated Raman shift-strain relationship (i.e., much larger than the previous calibration limited to around 1% (Ref. 33)) by fitting our data. II. THEORY
In this part, we first provide the equations which will be compared to our experimental data in Section IV (more details about those equations are given in the Appendix). The ratios between the strain values along different orientations (Equations (1)–(3)), hereafter called “strain ratios,” depend on the nature of the stress (e.g., biaxial, uniaxial…). They are equivalent to anisotropic Poisson ratios. Equation (1) corresponds to h100i uniaxial stress, Equation (2) to h110i uniaxial stress, and Equation (3) to (001) bi-axial stress. The strain components ei correspond to the longitudinal strain along axis i A: B:
e010 e001 S12 ¼ ¼ ; e100 e100 S11
e001 2 S12 ; ¼ e110 S11 þ S12 þ S44 =2
C:
(1)
e110 S11 þ S12 S44 =2 ; ¼ e110 S11 þ S12 þ S44 =2 (2)
e010 e100 S11 þ S12 ¼ ¼ : e001 e001 2 S12
(3)
Additionally, the three possible Raman frequencies expressed as functions of the phonon deformation potentials12,33,34,38 (p, q, and r) and of the elastic coefficients (S11,
S12, and S44) are given in Equations (4)–(6). They were defined with the semi-empirical linear model based on the Raman secular equation33,34,36 typically used for strainedGe.8,12,14,17,40,41 Equation (4) corresponds to h100i uniaxial stress, Equation (5) to h110i uniaxial stress, Equation (6) to (001) biaxial stress and Equation (7) to h111i uniaxial stress. pS11 þ 2qS12 e100 ; 2S11 x0 pS12 þ qðS11 þ S12 Þ ¼ e100 ; 2S11 x0
A : Dx1 ¼ Dx2 ¼ DxTO ¼ Dx3 ¼ DxLO
(4)
pðS11 þ S12 Þ þ qðS11 þ 3S12 Þ6rS44 e110 ; 2x0 ðS11 þ S12 þ S44 =2Þ pS12 þ qðS11 þ S12 Þ e110 ; ð5Þ Dx3 ¼ DxLO ¼ x0 ðS11 þ S12 þ S44 =2Þ
B : Dx1;2 ¼ DxTO ¼
1 S11 þ 3S12 e100 ; pþq C : Dx1 ¼ Dx2 ¼ DxTO ¼ S11 þ S12 2x0 pS12 þ qðS11 þ S12 Þ ð6Þ e100 ; Dx3 ¼ DxLO ¼ x0 ðS11 þ S12 Þ ð p þ 2qÞðS11 þ 2S12 Þ þ 2rS44 e111 ; 2x0 ðS11 þ 2S12 þ S44 Þ ð p þ 2qÞðS11 þ 2S12 Þ rS44 Dx3 ¼ DxLO ¼ ð7Þ e111 ; 2x0 ðS11 þ 2S12 þ S44 Þ
D : Dx1 ¼ Dx2 ¼ DxTO ¼
Dx ¼ k e:
(8)
In this formalism, the Raman shift-strain relationships are linear, as generally expressed in Equation (8). These are plotted as straight lines in Figure 1 for the different stress configurations, using the following constant phonon deformation potentials: p/x02 ¼ 1.47, q/x02 ¼ 1.93, and r/x02 ¼ 1.11 from Refs. 42 and 43 elastic coefficients: S11 ¼ 9.6 103, S12 ¼ 2.6 103, and S44 ¼ 1.49 102 GPa1 from Refs. 44 and 45 and the unstrained phonon frequency x0 of Ge at 301 cm1.37,38,43 The corresponding linear coefficients k are shown in Figure 1 as well. For the h110i stress, there are 3 non-degenerate solutions (Figure 1(b)), as shown in Equation (5) and demonstrated in Refs. 33, 46, and 47. Longitudinal Optical (LO) phonons are usually accessed using retro-diffusion experiments. We have compared our calculated LO k coefficients to calculated21,25,48,49 and measured37,50,51 values from the literature in Table I. The theoretical values21,25,48,49 were estimated with phonon deformation potentials measured up to 1% strain.33 Our LO coefficients calculated in the linear model presented above are in very good agreement with the reported strain shift coefficients from the literature. The Raman shift-strain relationship was also calculated using ab initio calculations for the three different stress configurations measured hereafter and compared with the linear Raman-strain relations. The dependence of the phonon frequency on strain was determined from local-density approximation exchange correlation functions using the Abinit Density Functional Theory (DFT) package.53–55 In order to circumvent metallicity problems arising from the DFT
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biaxial stress. In the h110i uniaxial case, stress configurations were solved up to 3% strain. In this last configuration, the shear component gives rise to numerical instabilities at higher strain that could not be circumvented within the present numerical scheme, even with very dense k-point sampling and energy cut-off. Since some differences are found between the linear-model and ab initio calculations, the measurements of the Raman-strain relations are needed. III. EXPERIMENTS A. Samples
FIG. 1. Theoretical strain-Raman wavenumber shifts from 0% to 5% strain in Ge for (a) uniaxial stress along h100i, (b) uniaxial stress along h110i, (c) biaxial stress along (001), and (d) uniaxial stress along h111i.
calculation in crystalline Ge, we enforced a complete occupation of the valence band structure over the whole Brillouin zone, corresponding effectively to a 0 K electronic temperature. The relaxation of the strained crystalline structure along the axis perpendicular to the applied stress has been obtained for the 8 atom orthorhombic supercell. Response function phonon calculations were performed on the resulting relaxed 2 atom unit cell. Phonon band structures were thus simulated by ab-initio modeling for different strain states (dashed lines in Figure 1). The Raman shift-strain relation has been computed up to 5% strain for h100i uniaxial stress and (001)
TABLE I. Summary of the different Raman-strain linear coefficients reported in the literature.
Stress configuration
e (%)
Uni-axial h100i
e100
Uni-axial h110i
e110
Bi-axial (001)
Uni-axial h111i
e100 ¼ e010
e111
kLO (cm1)
References
1.53 1.54 1.52 2.04 2.25 2.02 4.19 4.24 4.15 4.08 3.90 3.84 4.42 4.34
This work 12 and 52 17 This work 8 35 This work 25 37 51 21 50 This work 48 and 49
To measure the Raman-strain relationships for different stress configurations, we have used Micro-crosses25 and micro-bridges12,13,23 fabricated from optical Ge-On-Insulator (GeOI) wafers.56–58 Figures 2(a)–2(c) present the process flow used. GeOI films are 350 nm thick with a built-in thermal strain evaluated to be around 0.16% (Figure 2(a)). Layer patterning was performed using e-beam lithography and dry etching (Figure 2(b)). The Ge membranes were then released using SiO2 under-etching by anhydrous HF vapors and alcohol vapors (Figure 2(c)). After the under-etching of the buried oxide, tensile strain is concentrated in the central, i.e., the narrowest part of the suspended membranes. Figures 2(d) and 2(f) present the design parameters used to tune the strain level in Ge membranes in two cases: uniaxial stress12 (Figure 2(d)) and biaxial stress25 (Figure 2(f)). Figures 2(e) and 2(g) show Scanning Electron Microscopy images of some of the fabricated membranes. Figure 3 presents the three different designs. Such simulations have been performed to evaluate the strain homogeneity compared to the measurement probe size (i.e., the Raman laser or X-ray focal spot). We have calculated inplane strains in typical Ge membranes using COMSOL Multiphysics. The modeling details are given in Refs. 25, 59, and 60. Figures 3(a) and 3(b) show the in-plane strain e// along the stress orientation corresponding to the h100i and h110i crystallographic directions, respectively, while Figure 3(c) displays e// ¼ (e100 þ e010)/2 for (001) biaxial stress. In every case, the strain homogeneity allows us to measure the strain state in the center of the membrane with a 1 lm diameter probe as experimentally confirmed in Ref. 59. B. Raman spectroscopy measurements
Raman spectroscopy was performed with a 785 nm wavelength incident laser and a 1 lm diameter spot. The light was focused onto the sample surface using a 100 short working distance objective with a 0.9 numerical aperture. The laser intensity (9 lW (Ref. 23)) was low enough to avoid heating effects.61 The Raman spectral shifts were measured by fitting the retro-diffused light spectra with Lorentzian functions. A bulk Ge substrate was used as a reference to determine the Raman wavenumber in the relaxed material (for 0% strain). Figure 4 shows the measured spectra for several membranes with an under-etching of 30 lm (Refs. 25 and 60) and different design parameters (defined in Figures 2(d) and 2(f)). Figure 4(a) shows spectra for micro-bridges stressed along the h100i direction. A maximum Raman shift
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shift of 14 cm1 is reported for the TO2 mode. Note that such large Raman spectral shifts are the highest reported value up to date for the h110i stress induction in Ge (compared to 10 cm1)8. Figure 4(c) shows spectra of (001) bi-axially stressed micro-crosses. A maximum Raman shift of 8 cm1 is detected, which is also very close to the maximum reported values for biaxial stress using strain redistribution.23 C. Micro-diffraction measurement
FIG. 2. Process flow for membrane fabrication starting from (a) Ge-on insulator stack followed by (b) Ge patterning and (c) membrane under-etching; design parameters used to tune the tensile (d) uniaxial and (f) biaxial stress; tilted SEM images of Ge membranes for (e) uniaxial and (g) biaxial stress induction.
of 9.5 cm1 is measured, which is very close to the maximum reported values in the literature.23 Figure 4(b) shows spectra of micro-bridges oriented along the h110i direction. In that case, two phonon modes are detected, and both LO and TO2 modes are accessed due to the large numerical aperture of our Raman spectrometer.33 A maximum Raman
In order to establish relationships between the measured Raman wavenumber shift and strain, all samples measured by Raman spectroscopy were also measured by micro-XRD. The strain state in the Ge micro-bridges was measured using X-ray Laue and rainbow-filtered Laue micro-diffraction at beam-line BM32 of the ESRF in Grenoble.59 The synchrotron X-ray beam was focused to a 0.5 lm 0.5 lm spot size on the central part of the micrometer size membranes. The Ge Laue diffraction patterns were recorded using a 2D camera. Details concerning the measurement method are given in Refs. 23, 59, 62, and 63. Since Laue diffraction can be used to access the full strain tensor in Ge micro-bridges,59 we have measured the strains along all directions in the center of our micro-bridges and micro-crosses. The measured strains along the stress directions are provided in red in Figure 4, close to each corresponding Raman spectra. Figure 5 shows the different strain ratios (defined in Eqs. (1)–(3)) for all the investigated micro-bridges and micro-crosses; the horizontal lines indicate the theoretical values (Equations (1)–(3) with S11 ¼ 9.6 103, S12 ¼ 2.6 103, and S44 ¼ 1.49 102 GPa1).44,45 The indicated uncertainties come from the strain uncertainties, which have been
FIG. 3. Calculated longitudinal strain in Ge membranes under (a) h100i uniaxial stress; (b) h110i uniaxial stress and (c) (001) biaxial stress.
FIG. 4. Raman spectra of different Ge micro-bridges for (a) uniaxial stress induction along h100i and (b) uniaxial stress induction along h110i; (c) Raman spectra of different Ge micro-crosses used for (001) biaxial strain induction.
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FIG. 5. Theoretical and measured strain ratios using Laue micro-diffraction performed on Ge membranes for uniaxial and biaxial stress. For uniaxial stresses, ex is the strain along the stress direction. For the biaxial stress, ex is the strains along the h100i direction.
evaluated to be around 0.1%.23,59 A very good agreement is found between the calculated values and the measured data, which is a clear confirmation that the strain redistribution method allows the application of tunable pure uniaxial or biaxial stresses in Ge crystals. IV. DISCUSSION
Figure 6 presents the measured strain as a function of the Raman spectral shift for the three measured stress configurations. Experimental data are plotted as dots, the usual linear models33 are plotted as straight lines, ab initio calculations are plotted as dashed lines, and empirical data fitting as dotted lines. For the biaxial stress, we found a good agreement between the experimental, semi-empirical, and ab initio Raman shift-strain relations. Indeed, the Raman linear models have already been calibrated for such levels of strain using thin Ge layers grown on lattice mismatched buffers.37–39 We confirm here that it can be extended to a thicker layer as well. For the uniaxial stress along h100i, a deviation of the linear coefficient is shown, as previously reported.23 For the uniaxial stress along h110i, we also report an unexpected non-linearity of the Raman-strain relationships for both measured LO and TO2 phonon modes. Indeed, for the maximum reported Raman spectral shift, the expected strain value by the usual linear model should be 4.2%, while only 3.8% is measured. As a consequence, the previously reported highest strain value extracted from Raman spectroscopy for uniaxial stress along h110i in Ge is overestimated.8 In order to provide the new Raman shift-strain relationship, we have fitted our experimental data with polynomial equations (Equation (9)). The resulting empirical coefficients a and b are compared in Table II to the inverse of the linear coefficients k resulting from Equation (8). a and b coefficients have 5% and 10% standard errors, respectively, coming from the fit deviation.23 Given those uncertainties, we found a good agreement between 1/k and a, because the
FIG. 6. Measured strain by micro-XRD (dots) and calculated strain by ab initio calculations (dashed lines) or linear coefficients (straight lines) versus Raman shift for the uniaxial stresses along h100i, h110i and bi-axial stress along (001).
linear coefficient 1/k has been defined only for low levels of strain (i.e., where the b coefficient does not significantly impact the Raman shift-strain relationship). However, for the uniaxial stresses, the data fitting gives a deviation from the linear model, which is higher than the strain measurement uncertainties for over 3% strain. In that case, the usual phonon deformation potentials are not sufficient to describe the Raman shift-strain relations e ¼ a Dx þ b Dx2 :
(9)
V. CONCLUSION
In conclusion, we have studied the Raman shift-strain relationship for different stress configurations. Since some variations were found between the linear-model and ab initio calculations, we have experimentally measured the Ramanstrain relations. In the case of (001) biaxial stress, we confirm that the Raman-strain relation defined using thin strained layers (10 nm) is also valid for much thicker Ge layers. For the h100i and h110i uniaxial stresses, we have shown that the TABLE II. Raman-strain relationships using linear empirical models (Table I) compared to polynomial empirical models resulting from Laue microdiffraction performed in tandem with Raman spectroscopy on Ge membranes. 2
Stress configuration
e (%)
1/k (cm)
a (cm)
b (cm )
Bi-axial (001) LO Uni-axial h110i LO Uni-axial h110i TO2 Uni-axial h100i LO
e100 ¼ e010 e110 e110 e100
0.23 0.48 0.32 0.65
0.23 0.48 0.33 0.68
0 0.0045 0.0035 0.0190
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project “Straintronics,” as well as the Swiss National Science foundation SNF.
Raman shift-strain relations are not linear. Our work gives a better understanding in the Raman shift relationship in Ge and report the measurement of a record breaking 3.8% strain in germanium for the h110i uniaxial stresses.
APPENDIX: PHONON DEFORMATION POTENTIAL THEORY
ACKNOWLEDGMENTS
In this appendix, we give details about the Raman shiftstrain relationship using the usual phonon-deformation potential theory12,32–34,38 for the four main stress configurations studied in pure Ge.8,12,17,21,25,35,37,48–52 Such a model is based on the Raman dynamical secular relationship (Equation (A1)),32,36 which yields three different eigenvalues ki (i ¼ 1, 2, 3) corresponding to the three possible phonon gamma modes non-degenerated with strain
The authors would like to thank the Platforme de Technologie Amont and the 41 platform in Grenoble for the clean room facilities and the beamline BM32 at ESRF for synchrotron-based measurement as well as David Cooper and Dhruv Singhal for their help correcting the English. This work was supported by the CEA DRF-DRT Phare projects “Photonics” and “Operando,” the CEA-Enhanced Eurotalent
pexx þ qðeyy þ ezz Þ ki 2rexy 2rexz
2rexy peyy þ qðexx þ ezz Þ ki 2reyz
The relationships between the different oriented strains were expressed from the generalized Hooke law (Equation (A2)) in the crystallographic base (Equation (A3)) 0 1 0 10 1 rxx exx S11 S12 S12 B eyy C B S12 S11 S12 C B ryy C 0 B C B CB C B ezz C B S12 S12 S11 C B rzz C B C¼B C :B C; B eyz C B B C S44 =2 0 0 C B C B C B ryz C @ exz A @ 0 0 S44 =2 0 A @ rxz A exy rxy 0 0 S44 =2 (A2) exx ¼ e100 ;
eyy ¼ e010 ;
ezz ¼ e001 :
2reyz ¼ 0: pezz þ qðexx þ eyy Þ ki 2rexz
(A1)
The various relationships linking the 6 strain components ei,j (i, j ¼ x, y or z) are obtained from the Hooke law and the stress tensor r (h100i uniaxial stress: Eq. (A4), h110i uniaxial stress: Eq. (A5), (001) bi-axial stress: Eq. (A6) and h111i uniaxial stress: Eq. (A7)) 0 1 r0 B 0 C B C B 0 C B C; eyy ¼ ezz ¼ S12 exx ; exy ¼ exz ¼ eyz ¼ 0; rA ¼ B C 0 S11 B C @ 0 A 0
(A3)
(A4)
0
1 r0 =2 B C B r0 =2 C B C B 0 C B C rB ¼ B C; B 0 C B C B 0 C @ A r0 =2 B : exy ¼
exx ¼
S11 þ S12 e110 ; S11 þ S12 þ S44 =2
S44 =2 e110 ; S11 þ S12 þ S44 =2
ezz ¼
2 S12 e110 ; S11 þ S12 þ S44 =2
e110 ¼
exz ¼ eyz ¼ 0; 0
B B B rC ¼ B B B @
r0 r0 0 0 0 0
S11 þ S12 S44 =2 e110 ; S11 þ S12 þ S44 =2
(A5)
1 C C C C; C C A
eyy ¼ exx ¼
S11 þ S12 ezz ; 2 S12
exz ¼ exy ¼ eyz ¼ 0;
(A6)
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0
1 r0 =3 B C B r0 =3 C B C B r =3 C B 0 C rD ¼ B C; B r0 =3 C B C B r =3 C @ 0 A r0 =3
20
(A7)
S12 þ 2 S11 e111 ; S11 þ 2 S12 þ S44 S44 =2 e111 : exy ¼ exz ¼ eyz ¼ S11 þ 2 S12 þ S44 exx ¼ eyy ¼ ezz ¼
By substituting the relationships between the different strains (Equations (A4)–(A7)) in the Raman secular equations (Equation (A1)), the three possible Raman frequencies can be expressed as functions of the phonon deformation potentials (p, q, and r) and the elastic coefficients (S11, S12, and S44). Equation (4) concerns uniaxial stress along h100i, Equation (5) uniaxial stress along h110i, Equation (6) (001) biaxial stress and Equation (7) uniaxial stress along h111i. 1
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