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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 6 — Mar. 25, 2013
  • pp: 7162–7170
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Strain induced bandgap and refractive index variation of silicon

Jingnan Cai, Yasuhiko Ishikawa, and Kazumi Wada  »View Author Affiliations


Optics Express, Vol. 21, Issue 6, pp. 7162-7170 (2013)
http://dx.doi.org/10.1364/OE.21.007162


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Abstract

We present a study of the influence of high strain on the bandgap and the refractive index of silicon. The results of photoluminescence show that with the strain applied, the silicon bandgap can be adjusted to 0.84 eV and the refractive index of silicon increases significantly. 1.4% change of refractive index of silicon was observed. The strain-induced bandgap shrinkage and absorption coefficient change of silicon are considered as the main cause of the significant refractive index change. The present work indicates that the application of strain is promising to control the refractive index of silicon in devices so that applications such as compensation of thermal effect in optical devices can be achieved.

© 2013 OSA

1. Introduction

2. Design and calculation

To characterize the strain effect on the refractive index of silicon, a simple silicon cantilever beam structure is designed. In order to apply the strain to the silicon beam, a probe is preferred to push down the cantilever beam to generate the strain. Generally, the way to use a probe is just to push down the beam. However, the maximum pushing depth is usually limited, e.g. 3μm, and the thickness of the top silicon layer is only 200-300 nm. That is to say, the strain might not be large enough to incur a remarkable change of refractive index. To generate high strain on the same structure, instead of simply pushing down the beam, we try to use the probe with an included angle of 30-45 degree to the substrate and push the beam down and forward slightly so that more bending of the beam can be achieved. Figure 1
Fig. 1 (a) The shape of the designed Si cantilever beam. (b), (c) the schematic cross-section image showing how the beam bends by the operation of probe. The undercut region was generated while removing the BOX by wet etching. (b): Bending Method One, directly pushing down the beam; (c): Bending Method Two, pushing the beam down and forward so that the beam is more bent.
is the schematic of the designed shape of the silicon cantilever beam and the two methods to push the beam.

Defining that a SOI wafer with 250 nm of top silicon layer and 3μm of buried oxides is used for patterning and a cantilever beam size of 2.5μm in width and 15 μm in length is prepared. Based on this configuration, the strain field in the structure was numerically solved by the finite-element method (FEM) using a commercial software package (Abaqus CAE). Figure 2 (a)
Fig. 2 (a) The modeled structure of the silicon cantilever beam for simulation. (b) Typical strain distribution obtained by the simulator using Method Two.
shows the modeled structure for the simulation by Abaqus CAE. To simulate the different bending configurations, different forces in the scale of 10−5 Newton were added onto the silicon cantilever beam in the simulation. The properties of materials such as the Young’s Modulus and the Poisson ratio can be input for the simulation. The corresponding results of strain distribution can be obtained by running the Abaqus CAE software. For the Method One, the beam-end was vertically pressed down to the bottom by a certain force. For the Method Two, a vertical downward force F1 followed by a forward force F2 were applied on to the beam, as shown in Fig. 2 (b). By modifying the values of the forces, the moving distance of the beam-end along the F2 direction can be adjusted, thus the different bending configurations can be achieved. Figure 2 (b) is the typical simulation result of the bending configuration using Method Two. The strain distribution and the value of strain in a designated location can be obtained from this simulation result. The simulation results show that if the beam is pushed downward as Method One shown in Fig. 1(b), the max strain in the beam will be 0.2%. If Method Two is implemented as shown in Fig. 1(c) or Fig. 2(b), the max strain, which distributes on top of the convex region on the beam, can reach to 2.5% or even higher. Typically, the application of strain induces photoelastic effect and this effect contributes to the variation of refractive index. Based on the theory [19

19. C. G. Van de Walle, “Band lineups and deformation potentials in the model-solid theory,” Phys. Rev. B Condens. Matter 39(3), 1871–1883 (1989). [CrossRef] [PubMed]

,20

20. M. Huang, “Stress effects on the performance of optical waveguides,” Int. J. Solids Struct. 40(7), 1615–1632 (2003). [CrossRef]

], it can be calculated that 0.5%, 1.5% and 2.5% of strains can lead to 0.067%, 0.201% and 0.335% variation of refractive index, respectively.

3. Experiments

The silicon cantilever beam structure as mentioned above was patterned by an ADVANTEST F5112 electron beam writer. The beam patterns were formed via reactive ion etching, using Cl2/O2 as the etchant. HF solution was used to etch the buried silicon dioxide to form the final beam structure. The micro photoluminescence (PL) measurement was carried out to characterize the strained silicon cantilever beam. The information of PL the resonance peaks allows the examination of the important parameters of the beam structure, such as the effective refractive index. A blue laser source with 457 nm in wavelength is used to excite the sample. During the measurement, the signal of luminescence was accumulated for 20 times. A tungsten probe with spherical tip radius of about 2 micron was used to generate the strain in the silicon beam. In order to avoid the thermal effect, the tungsten probe was operated by slightly touching the free silicon beam. This operation can help create the similar thermal condition to the case of bending beam.

4. Results and discussions

Figure 3
Fig. 3 Typical PL spectra of Si cantilever beam under different strain conditions. The photos in the right side were taken during each operation. The numbers are the TM mode numbers for the Febry-Perot resonator.
provides the typical PL spectra of the relaxed and strained silicon cantilever beam under different bending conditions. It should be noted here that all the measurements were operated using the same silicon cantilever beam. We name the cantilever beam using the pushing condition of Method One (in Fig. 1) as the “bent” beam, and name the ones using Method Two bending configurations as the “bent1” beam with a slight bending, the “bent2” beam with a moderate bending, and the “bent3” beam with a large bending. An obvious change of bandgap can be observed if the beam is bent using Method Two, which indicates a larger strain in the silicon beam. In general, by pushing the beam forward, the strain increases. To find the bandgaps of the silicon under various strain conditions, Lorentzian fittings were performed to fit the broad PL peaks. The Febry-Perot peaks in the PL spectra were not counted when the fittings were performed. Those fitted PL peak positions represent the bandgaps of the silicon under different strain conditions. The results of fittings show that the peak positions of the broad PL spectra are 1122.0 ± 0.8 nm for the “relaxed” beam, 1123.2 ± 0.8 nm for the “bent” beam, 1194.6 ± 0.8 nm for the “bent1” beam, 1309.5 ± 0.5 nm for the “bent2” beam, 1471.5 ± 2.5 nm for the “bent3” beam, and 1122.6 ± 0.8 nm for the “restored” beam, respectively. It can be derived that the bandgap of silicon is 1.11 eV for the “relaxed” beam, 1.10 eV for the“bent” beam, 1.04 eV for the “bent1” beam, 0.95 eV for the “bent2” beam, and 0.84 eV for the “bent3” beam, respectively. These results demonstrate that the application of strain is effective to adjust the bandgap of silicon. The model-solid theory using the deformation potentials [19

19. C. G. Van de Walle, “Band lineups and deformation potentials in the model-solid theory,” Phys. Rev. B Condens. Matter 39(3), 1871–1883 (1989). [CrossRef] [PubMed]

] is a well-developed method to calculate the correlations between the strain and the bandgap for semiconductors. In this work, the strain as a function of silicon bandgap was calculated based on this theory. Since the bandgaps of the strained silicon can be easily estimated from the PL spectra, the strain value of each bending configuration can be obtained by checking the calculated strain-bandgap relations. By means of this method, the calculated strain in the “bent1” beam is 0.61%, the strain in the “bent2” beam is 1.32% and the strain in the “bent3” beam is 2.15%. In addition, it should be mentioned that when the beam is restored, the peaks return to the original positions, indicating that the deformation of the beam is elastic rather than plastic. Therefore, the shift of broad PL peak should be attributed to the change of silicon bandgap, rather than the defects created by plastic deformation. Another study [17

17. J. Munguía, G. Bremond, J. M. Bluet, J. M. Hartmann, and M. Mermoux, “Strain dependence of indirect band gap for strained silicon on insulator wafers,” Appl. Phys. Lett. 93(10), 102101 (2008). [CrossRef]

] on the strain induced bandgap change of silicon on insulator reported that 0.87%, 1.22% and 1.54% of strain can induce 0.13 eV, 0.184 and 0.239 eV bandgap change of the silicon, respectively. Under the same strain, the silicon beam we reported yields slightly smaller change (about 0.05 eV) of bandgap than the one reported in reference [17

17. J. Munguía, G. Bremond, J. M. Bluet, J. M. Hartmann, and M. Mermoux, “Strain dependence of indirect band gap for strained silicon on insulator wafers,” Appl. Phys. Lett. 93(10), 102101 (2008). [CrossRef]

]. Noticing that the thickness of the silicon in this reference is only 9 nm, which is much thinner than the 250 nm silicon layer in our work, the dimension effect might contribute to the difference between those two results.

According to the results, there exists obvious resonance for each spectrum, which is due to the Febry-Perot resonance in the vertical direction along the silicon beam. In order to identify those Febry-Perot resonance peaks, the effective refractive index should be calculated. In this work, the mode solving analysis based on finite element method is carried out by using the commercial software Apollo Photonic Solutions Suite. The effective refractive indices can be calculated using this mode solving. For a Febry-Perot resonator, the on-resonance condition can be shown as,
mλ=2nL.
(1)
where λ is the on-resonance wavelength, n is the effective refractive index, L is the length of the optical path and m is the mode number. Based on Eq. (1), the resonance peak from each mode can be identified by matching the calculated effective refractive index to the measured resonance peak. Some of the matched modes, TM-12, TM-13, and TM-14 are shown in Fig. 3. Setting the typical resonance peak position of “relaxed” beam as the reference position, the shifts of peaks for TM-14, TM-13 and TM-12 modes can be calculated and listed in Table 1

Table 1. Typical peak shifts of free and strained silicon beams

table-icon
View This Table
. The presented data show that the shifts increase with the strain. For the “bent2” beam, the shifts of resonance peaks are about 6.5-10.2 nm. For the beam with larger strain, the shifts of peaks can reach up to 18 nm. Normally, the resonance peak shift is derived from the change of refractive index.

In order to find out the possible reasons that cause such a significant variation of refractive index, we try to consider the factors as follows.

  • (1) Estimation by the photoelastic effects.

    According to the simulation results, when the beam is bent, there exists the tensile strain on the top part of the beam. A zero strain line locates in the center region and compressive strain distributes on the other side, as illustrated in Fig. 4
    Fig. 4 Cross sectional view of the strain distribution and the corresponding changes of refractive indices of different locations in the bent silicon cantilever beam that has 1.535% tensile strain at eh top surface.
    . Basically, the change of refractive index Δn is proportional to the stress-optic constant [20

    20. M. Huang, “Stress effects on the performance of optical waveguides,” Int. J. Solids Struct. 40(7), 1615–1632 (2003). [CrossRef]

    ]. Due to the special strain distributions inside the silicon beam, different locations inside the beam yield different variations of refractive index. Taking the beam with 1.535% tensile strain at the top surface as the example, the changes of refractive indices in different locations can be calculated based on photoelastic theory, and the values of those changes inside the silicon beam are indicated and shown in Fig. 4. It is obvious that the tensile part and compressive part lead to -Δn and + Δn effect. Since the cross section of the beam layer can be considered containing the layers with different refractive indices, as shown in Fig. 4, the waveguide model consisting seven stacking layers with different refractive indices are created for the calculation of effective refractive index. Based on this model, the Apollo Photonic Solution Suit mode solving result indicates that for the 250 nm slab, only a less than 10−3 of effective refractive index variation compared with the index for non-strain condition. For the cases of the other strain conditions, similar results can be obtained. Due to the opposite (positive and negative on two sides) distributions of strains in the silicon beam, the effective refractive index of this thin 250nm-thick silicon layer is neutralized. Thus, the calculation results of photoelastic effects cannot match the experimental values observed and other factors should be counted in.

    Fig. 4Cross sectional view of the strain distribution and the corresponding changes of refractive indices of different locations in the bent silicon cantilever beam that has 1.535% tensile strain at eh top surface.
  • (2) The shape change of the silicon beam under strain.

    This could cause extra effect on the shift of resonance. However, the mode solving results indicate that the shape change of the resonator causes less than 10−2 change of refractive index, which is relatively smaller than the as measured refractive index change. Thus the change of the resonator shape is not the dominant factor that affects the resonance.

  • (3) The thermal effect.

    In general, the fluctuation of temperature can be incurred by the exciting laser. The bandgap of silicon thereby shrinks when the temperature rises. It is reported that the 1Kelvin temperature change can lead to 6 × 10−5 of silicon Δn/n [21

    21. J. A. McCaulley, V. M. Donnelly, M. Vernon, and I. Taha, “Temperature dependence of the near-infrared refractive index of silicon, gallium arsenide, and indium phosphide,” Phys. Rev. B Condens. Matter 49(11), 7408–7417 (1994). [CrossRef] [PubMed]

    ], which means 0.1% of refractive index change requires more than 20K raise of temperature. However, the value of peak shift is determined by the relative difference between two peaks rather than the absolute peak positions. Keeping the identical thermal environments, i.e. keeping the cantilever beam at identical temperature, can remove the noise of thermal effect. During the measurement, the same output of laser power was used. Owing to the identical thermal environments in the measurements, the peak shift generated by the thermal effects is expected to be small. In fact, compared with the “relaxed” and the “bent1” spectra, no obvious resonance peak shift is observed, which also provides the evidence of the identical thermal environment during the measurements. Therefore, we can claim that the thermal effect is not the main factor to shift the resonance peak.

  • (4) The strain-induced bandgap shift and thus the refractive index variation.

    Fig. 5The estimation of refractive index change (dashed line) according to the calculation from the fitting results shown in inset, and the refractive index change calculated from experimental data (scattered points). Inset: the estimation of silicon refractive index change by considering the effect of bandgap shift. The black dotted data points are obtained from reference [25].

Therefore, based on the discussion above, we can get that the large change of silicon refractive index could mainly be caused by the change of absorption coefficient induced by the high strain in silicon.

This result provides the hint that the application of strain could be a promising way to adjust the bandgap as well as the refractive index of silicon. Because the strain can be generated and released freely and quickly, and the index change can reach to 1% or even more, this application of stain can be operated in many fields such as compensation for thermal-induced fluctuation in optical filters, modulators. Moreover, the significant bandgap shift to ~0.8 eV provides the possibility of making silicon-based photodetectors even for wavelength of 1.3μm and 1.55 μm.

5. Summary

Utilizing a silicon micro cantilever beam, the study of high strain on refractive index of silicon was carried out in this work. The photoluminescence results showed that significant reduction of bandgap and increase of refractive index can be realized if high strain is applied. The bandgap of silicon can be reduced to around 0.8 eV under strain. The refractive index varies with the increase of strain and 1.4% of refractive index variation can be achieved. It is indicated that the large change of refractive index is attributed to the strain-induced bandgap shift and absorption coefficient change of silicon. This work predicts that the application of strain on silicon could be a way for many applications based on the adjustment of refractive index.

Acknowledgments

This research is partly supported by the Japan Society for the Promotion of Science (JSPS) through its “Funding Program for World-Leading Innovative R&D on Science and Technology (FIRST Program)”. The samples were fabricated using an EB writer F5112 + VD01 in VLSI Design and Education Center (VDEC), the University of Tokyo, donated by ADVANTEST Corporation with the collaboration with Cadence Corporation.

References and links

1.

L. Pavesi and D. Lockwood, Silicon Photonics (Springer-Verlag, 2004).

2.

H. F. Hamann, A. Weger, J. A. Lacey, Z. Hu, P. Bose, E. Cohen, and J. Wakil, “Hotspot-limited microprocessors: direct temperature and power distribution measurements,” IEEE J. Solid-State Circuits 42(1), 56–65 (2007). [CrossRef]

3.

F. Xia, M. Rooks, L. Sekaric, and Y. Vlasov, “Ultra-compact high order ring resonator filters using submicron silicon photonic wires for on-chip optical interconnects,” Opt. Express 15(19), 11934–11941 (2007). [CrossRef] [PubMed]

4.

A. Liu, L. Liao, D. Rubin, H. Nguyen, B. Ciftcioglu, Y. Chetrit, N. Izhaky, and M. Paniccia, “High-speed optical modulation based on carrier depletion in a silicon waveguide,” Opt. Express 15(2), 660–668 (2007). [CrossRef] [PubMed]

5.

N. A. Yebo, P. Lommens, Z. Hens, and R. Baets, “An integrated optic ethanol vapor sensor based on a silicon-on-insulator microring resonator coated with a porous ZnO film,” Opt. Express 18(11), 11859–11866 (2010). [CrossRef] [PubMed]

6.

J. Welser, J. L. Hoyt, and J. F. Gibbons, “Electron mobility enhancement in strained-Si n-type metal-oxide- semiconductor field-effect transistors,” IEEE Electron Device Lett. 15(3), 100–102 (1994). [CrossRef]

7.

Y. Ishikawa, K. Wada, D. D. Cannon, J. Liu, H.-C. Luan, and L. C. Kimerling, “Strain-induced band gap shrinkage in Ge grown on Si substrate,” Appl. Phys. Lett. 82(13), 2044–2046 (2003). [CrossRef]

8.

P. H. Lim, S. Park, Y. Ishikawa, and K. Wada, “Enhanced direct bandgap emission in germanium by micromechanical strain engineering,” Opt. Express 17(18), 16358–16365 (2009). [CrossRef] [PubMed]

9.

R. S. Jacobsen, K. N. Andersen, P. I. Borel, J. Fage-Pedersen, L. H. Frandsen, O. Hansen, M. Kristensen, A. V. Lavrinenko, G. Moulin, H. Ou, C. Peucheret, B. Zsigri, and A. Bjarklev, “Strained silicon as a new electro-optic material,” Nature 441(7090), 199–202 (2006). [CrossRef] [PubMed]

10.

C. Schriever, C. Bohley, J. Schilling, and R. B. Wehrspohn, “Strained silicon photonics,” Materials 5(12), 889–908 (2012).

11.

M. Cazzanelli, F. Bianco, E. Borga, G. Pucker, M. Ghulinyan, E. Degoli, E. Luppi, V. Véniard, S. Ossicini, D. Modotto, S. Wabnitz, R. Pierobon, and L. Pavesi, “Second-harmonic generation in silicon waveguides strained by silicon nitride,” Nat. Mater. 11(2), 148–154 (2011). [CrossRef] [PubMed]

12.

Y. Amemiya, Y. Tanushi, T. Tokunaga, and S. Yokoyama, “Photoelastic effect in silicon ring resonators,” Jpn. J. Appl. Phys. 47(4), 2910–2914 (2008). [CrossRef]

13.

C. Schriever, C. Bohley, and R. B. Wehrspohn, “Strain dependence of second-harmonic generation in silicon,” Opt. Lett. 35(3), 273–275 (2010). [CrossRef] [PubMed]

14.

K.-H. Hong, J. Kim, S.-H. Lee, and J. K. Shin, “Strain-driven electronic band structure modulation of Si nanowires,” Nano Lett. 8(5), 1335–1340 (2008). [CrossRef] [PubMed]

15.

H. Nakamura, “Strain effects on the band structure for Si nanowires,” in Proceedings of IEEE Conference on Nanotechnology (IEEE NANO Organizers, 2009), pp. 555–558.

16.

Y. Ishikawa, K. Wada, D. D. Cannon, J. Liu, H.-C. Luan, and L. C. Kimerling, “Strain-induced band gap shrinkage in Ge grown on Si substrate,” Appl. Phys. Lett. 82(13), 2044–2046 (2003). [CrossRef]

17.

J. Munguía, G. Bremond, J. M. Bluet, J. M. Hartmann, and M. Mermoux, “Strain dependence of indirect band gap for strained silicon on insulator wafers,” Appl. Phys. Lett. 93(10), 102101 (2008). [CrossRef]

18.

K. Yoshimoto, R. Suzuki, Y. Ishikawa, and K. Wada, “Bandgap control using strained beam structures for Si photonic devices,” Opt. Express 18(25), 26492–26498 (2010). [CrossRef] [PubMed]

19.

C. G. Van de Walle, “Band lineups and deformation potentials in the model-solid theory,” Phys. Rev. B Condens. Matter 39(3), 1871–1883 (1989). [CrossRef] [PubMed]

20.

M. Huang, “Stress effects on the performance of optical waveguides,” Int. J. Solids Struct. 40(7), 1615–1632 (2003). [CrossRef]

21.

J. A. McCaulley, V. M. Donnelly, M. Vernon, and I. Taha, “Temperature dependence of the near-infrared refractive index of silicon, gallium arsenide, and indium phosphide,” Phys. Rev. B Condens. Matter 49(11), 7408–7417 (1994). [CrossRef] [PubMed]

22.

K. Bücher, J. Bruns, and H. G. Wagemann, “Absorption coefficient of silicon: An assessment of measurements and the simulation of temperature variation,” J. Appl. Phys. 75(2), 1127–1132 (1994). [CrossRef]

23.

A. A. Patrin and M. I. Tarasik, “Optical-absorption spectrum of silicon containing internal elastic stresses,” J. Appl. Spectrosc. 65(4), 598–603 (1998). [CrossRef]

24.

V. Lucarini, J. J. Saarinen, K.-E. Peiponen, and E. M. Vartiainen, Kramers–Kronig Relations in Optical Materials Research (Springer-Verlag, 2005).

25.

E. D. Palik, Handbook of Optical Constants of Solids (Academic, 1985).

OCIS Codes
(040.6040) Detectors : Silicon
(130.0250) Integrated optics : Optoelectronics
(160.4760) Materials : Optical properties
(250.5230) Optoelectronics : Photoluminescence

ToC Category:
Integrated Optics

History
Original Manuscript: January 17, 2013
Revised Manuscript: March 5, 2013
Manuscript Accepted: March 5, 2013
Published: March 14, 2013

Citation
Jingnan Cai, Yasuhiko Ishikawa, and Kazumi Wada, "Strain induced bandgap and refractive index variation of silicon," Opt. Express 21, 7162-7170 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-6-7162


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References

  1. L. Pavesi and D. Lockwood, Silicon Photonics (Springer-Verlag, 2004).
  2. H. F. Hamann, A. Weger, J. A. Lacey, Z. Hu, P. Bose, E. Cohen, and J. Wakil, “Hotspot-limited microprocessors: direct temperature and power distribution measurements,” IEEE J. Solid-State Circuits42(1), 56–65 (2007). [CrossRef]
  3. F. Xia, M. Rooks, L. Sekaric, and Y. Vlasov, “Ultra-compact high order ring resonator filters using submicron silicon photonic wires for on-chip optical interconnects,” Opt. Express15(19), 11934–11941 (2007). [CrossRef] [PubMed]
  4. A. Liu, L. Liao, D. Rubin, H. Nguyen, B. Ciftcioglu, Y. Chetrit, N. Izhaky, and M. Paniccia, “High-speed optical modulation based on carrier depletion in a silicon waveguide,” Opt. Express15(2), 660–668 (2007). [CrossRef] [PubMed]
  5. N. A. Yebo, P. Lommens, Z. Hens, and R. Baets, “An integrated optic ethanol vapor sensor based on a silicon-on-insulator microring resonator coated with a porous ZnO film,” Opt. Express18(11), 11859–11866 (2010). [CrossRef] [PubMed]
  6. J. Welser, J. L. Hoyt, and J. F. Gibbons, “Electron mobility enhancement in strained-Si n-type metal-oxide- semiconductor field-effect transistors,” IEEE Electron Device Lett.15(3), 100–102 (1994). [CrossRef]
  7. Y. Ishikawa, K. Wada, D. D. Cannon, J. Liu, H.-C. Luan, and L. C. Kimerling, “Strain-induced band gap shrinkage in Ge grown on Si substrate,” Appl. Phys. Lett.82(13), 2044–2046 (2003). [CrossRef]
  8. P. H. Lim, S. Park, Y. Ishikawa, and K. Wada, “Enhanced direct bandgap emission in germanium by micromechanical strain engineering,” Opt. Express17(18), 16358–16365 (2009). [CrossRef] [PubMed]
  9. R. S. Jacobsen, K. N. Andersen, P. I. Borel, J. Fage-Pedersen, L. H. Frandsen, O. Hansen, M. Kristensen, A. V. Lavrinenko, G. Moulin, H. Ou, C. Peucheret, B. Zsigri, and A. Bjarklev, “Strained silicon as a new electro-optic material,” Nature441(7090), 199–202 (2006). [CrossRef] [PubMed]
  10. C. Schriever, C. Bohley, J. Schilling, and R. B. Wehrspohn, “Strained silicon photonics,” Materials5(12), 889–908 (2012).
  11. M. Cazzanelli, F. Bianco, E. Borga, G. Pucker, M. Ghulinyan, E. Degoli, E. Luppi, V. Véniard, S. Ossicini, D. Modotto, S. Wabnitz, R. Pierobon, and L. Pavesi, “Second-harmonic generation in silicon waveguides strained by silicon nitride,” Nat. Mater.11(2), 148–154 (2011). [CrossRef] [PubMed]
  12. Y. Amemiya, Y. Tanushi, T. Tokunaga, and S. Yokoyama, “Photoelastic effect in silicon ring resonators,” Jpn. J. Appl. Phys.47(4), 2910–2914 (2008). [CrossRef]
  13. C. Schriever, C. Bohley, and R. B. Wehrspohn, “Strain dependence of second-harmonic generation in silicon,” Opt. Lett.35(3), 273–275 (2010). [CrossRef] [PubMed]
  14. K.-H. Hong, J. Kim, S.-H. Lee, and J. K. Shin, “Strain-driven electronic band structure modulation of Si nanowires,” Nano Lett.8(5), 1335–1340 (2008). [CrossRef] [PubMed]
  15. H. Nakamura, “Strain effects on the band structure for Si nanowires,” in Proceedings of IEEE Conference on Nanotechnology (IEEE NANO Organizers, 2009), pp. 555–558.
  16. Y. Ishikawa, K. Wada, D. D. Cannon, J. Liu, H.-C. Luan, and L. C. Kimerling, “Strain-induced band gap shrinkage in Ge grown on Si substrate,” Appl. Phys. Lett.82(13), 2044–2046 (2003). [CrossRef]
  17. J. Munguía, G. Bremond, J. M. Bluet, J. M. Hartmann, and M. Mermoux, “Strain dependence of indirect band gap for strained silicon on insulator wafers,” Appl. Phys. Lett.93(10), 102101 (2008). [CrossRef]
  18. K. Yoshimoto, R. Suzuki, Y. Ishikawa, and K. Wada, “Bandgap control using strained beam structures for Si photonic devices,” Opt. Express18(25), 26492–26498 (2010). [CrossRef] [PubMed]
  19. C. G. Van de Walle, “Band lineups and deformation potentials in the model-solid theory,” Phys. Rev. B Condens. Matter39(3), 1871–1883 (1989). [CrossRef] [PubMed]
  20. M. Huang, “Stress effects on the performance of optical waveguides,” Int. J. Solids Struct.40(7), 1615–1632 (2003). [CrossRef]
  21. J. A. McCaulley, V. M. Donnelly, M. Vernon, and I. Taha, “Temperature dependence of the near-infrared refractive index of silicon, gallium arsenide, and indium phosphide,” Phys. Rev. B Condens. Matter49(11), 7408–7417 (1994). [CrossRef] [PubMed]
  22. K. Bücher, J. Bruns, and H. G. Wagemann, “Absorption coefficient of silicon: An assessment of measurements and the simulation of temperature variation,” J. Appl. Phys.75(2), 1127–1132 (1994). [CrossRef]
  23. A. A. Patrin and M. I. Tarasik, “Optical-absorption spectrum of silicon containing internal elastic stresses,” J. Appl. Spectrosc.65(4), 598–603 (1998). [CrossRef]
  24. V. Lucarini, J. J. Saarinen, K.-E. Peiponen, and E. M. Vartiainen, Kramers–Kronig Relations in Optical Materials Research (Springer-Verlag, 2005).
  25. E. D. Palik, Handbook of Optical Constants of Solids (Academic, 1985).

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