Strain Shift Coefficient Research Articles

Raman strain–shift measurements and prediction from first-principles in highly strained silicon

This work presents how first-principles simulations validated through experimental measurements lead to a new accurate prediction of the expected Raman shift as a function of strain in silicon (Si). Structural relaxation of a strained primitive cell is first performed to tackle the relative displacement of the Si atoms for each strain level. Density functional perturbation theory is then used to compute the energy of the optical phonon modes in highly strained Si and retrieve the strain–shift trend. The simulations are validated by experimental characterization, using scanning electron microscopy (SEM) coupled with backscattering Raman spectroscopy, of silicon microbeams fabricated using a top-down approach. The beams are strained up to 2% thanks to the internal tensile stress of silicon nitride (Si3N4) actuators, allowing a validation of the perturbation theory in high-strain conditions. The results are compared with the phonon deformation potentials theory, and the uncertainty caused by the various parameters found in the literature is discussed. The simulated strain–shift coefficients of $$-175.77\,{\text {cm}}^{-1}$$ (resp. $$-400.85\,{\text {cm}}^{-1}$$ ) and the experimental one of $$-160.99\,{\text {cm}}^{-1}$$ (resp. $$-414.97\,{\text {cm}}^{-1}$$ ) are found for the longitudinal optical LO (resp. transverse optical TO $$_{1}$$ ) mode, showing good agreement.

The effects of an activation-dependent increase in titin stiffness on whole muscle properties using finite element modeling

Active state titin’s effects have been studied predominantly in sarcomere or muscle fiber segment level and an understanding of its functional effects in the context of a whole muscle, and the mechanism of those is lacking. By representing experimentally observed calcium induced stiffening and actin-titin interaction induced reduced free spring length effects of active state titin in our linked fiber-matrix mesh finite element model, our aim was to study the mechanism of effects and particularly to determine the functionally more effective active state titin model. Isolated EDL muscle of the rat was modeled and three cases were studied: passive state titin (no change in titin constitutive equation in the active state), active state titin-I (constitutive equation involves a higher stiffness in the active state) and active state titin-II (constitutive equation also involves a strain shift coefficient accounting for titin’s reduced free spring length). Isometric muscle lengthening was imposed (initial to long length, lm = 28.7 mm to 32.7 mm). Compared to passive state titin, (i) active state titin-I and II elevates muscle total (lm = 32.7 mm: 14% and 29%, respectively) and active (lm = 32.7 mm: 37.5% and 77.4%, respectively) forces, (ii) active state titin-II also shifts muscle’s optimum length to a longer length (lm = 29.6 mm), (iii) active state titin-I and II limits sarcomere shortening (lm = 32.7 mm: up to 10% and 20%, respectively). Such shorter sarcomere effect characterizes active state titin’s mechanism of effects. These effects become more pronounced and functionally more effective if not only calcium induced stiffening but also a reduced free spring length of titin is accounted for.

Evaluation of Strain-Shift Coefficients for SiSn by Raman Spectroscopy

1. Introduction Silicon tin (SiSn) is a new promising candidate material for next-generation optical devices since its energy band-gap is theoretically expected to a direct band-gap as Sn fraction increases [1, 2]. The lattice constant difference (strain states) between the substrate and the SiSn thin film directly affects a band-gap, a carrier mobility, and device design. Therefore, it is important to evaluate strain states in SiSn thin film and accurate strain evaluation techniques are indispensable.Raman spectroscopy is one of the powerful strain evaluation techniques because it is a high spatial resolution and nondestructive measurement. However, the strain shift coefficient of SiSn has not been experimentally investigated in contrast to silicon germanium, germanium tin, and highly accurate strain measurement cannot be performed by Raman spectroscopy. In this study, Raman measurement was performed to extract the strain-shift coefficient of SiSn thin films, accurately. 2. Experimental method SiSn thin films (Sn fraction: 0.50, 0.90, 1.8, 2.0, and 6.0%) were grown on (001)-Si substrate by molecular beam epitaxy. Their thickness was set to 50 nm. Sn fraction and in-plane biaxial strain were confirmed by X-ray diffraction (XRD).The Si1-x Sn x thin films were evaluated by Raman spectroscopy. Excitation source of ultraviolet laser (λ = 355 nm) was used and focal length of the Raman spectrometer was 2,000 mm. Wavenumber resolution is approximately 0.1 cm-1. The Raman spectra of the Si1-x Sn x thin films for Si-Si vibration mode were measured to determine the strain-shift coefficient. 3. Results and Discussion Figure 1 shows Raman spectra of Si1-x Sn x with x of 0.06 and bulk Si (Si-Si mode) for comparison. As a result, asymmetrical broadening on the low-wavenumber side with slight peak shift in the Raman spectrum for the SiSn thin film were observed. In the Raman scattering, this broadening is often explained using the so-called spatial correlation model, which reproduces the relaxation of momentum conservation due to random atom positions.Figure 2 shows the relationship between Raman peak shift of SiSn for Si-Si mode (Δω LO) and in-plane strain estimated by reciprocal space mapping using XRD. In this study, D’Costa model (see below equation) was used to obtain strain-free Raman shift for Si-Si mode [3]: ω 0 = 520 – 141.8x. (1)It was revealed that the relationship between Raman peak shift of SiSn for Si-Si mode and in-plane strain has clear linear dependence. From the slope of the straight line which calculated by linear approximation as shown in Fig. 2, the value of the strain-shift coefficient b was found to be -732 cm-1. This value shows good agreement with that of strained Si [4] and SiGe [5]. On the basis of the result, it is suggested that the strain-shift coefficient b is not affected by random atom positions (alloy state), and the value of b may not change by composition and strain state.In conclusion, these insights play an important role in the strain evaluation for the next-generation SiSn device by Raman spectroscopy and in the understanding the relationship between phonon behavior and strain. Acknowledgements This work was supported partly the Japan Society for the Promotion of Science (JSPS) (19K21971 and 17J08240).

Evaluation of Strain-Shift Coefficients for SiSn by Raman Spectroscopy

We report on the strain-shift coefficient of silicon tin (SiSn) alloys obtained by ultraviolet (UV) Raman spectroscopy. The values of the Raman peak shift of the Si-Si mode for the SiSn (Sn fraction: 0.5, 0.9, 1.8, 2.2, and 6.0%) thin films were found to increase with Sn fraction and it indicates that compressive strain is induced in the SiSn films. The strain-shift coefficient of the Si-Si mode obtained from linear fits to the measurement data (UV Raman spectroscopy and X-ray diffraction reciprocal space mapping) has a good agreement with the values of the strain-shift coefficients about strained Si and strained silicon germanium, and commercial high-purity Si reported by previous studies. The strain-shift coefficient determined by in this work contribute to realize strain measurements for the next-generation SiSn devices such as near-infrared optical and thermoelectric devices.

Determination of phonon deformation potentials and strain-shift coefficients in Ge-rich Si1−xGex using bulk Ge-rich Si1−xGex crystals and oil-immersion Raman spectroscopy

The strain-free Raman shift of the Ge–Ge mode, , of Ge-rich Si1−xGex (x: Ge fraction) was determined accurately from the bulk Ge-rich Si1−xGex samples fabricated by the Czochralski (Cz) method. Using the obtained , the phonon deformation potentials (PDPs), p and q, and the strain-shift coefficient bLO of isotropic biaxial strained Ge-rich Si1−xGex thin films were extracted by oil-immersion Raman spectroscopy using Raman peak shifts of longitudinal and transverse optical (LO and TO) phonon modes. As a result, it was confirmed that these parameters are almost constant with small variations and that the strain-shift coefficient bLO is in good agreement with ab initio calculations. The parameters determined in this work are essential to realize accurate strain measurements using Raman spectroscopy for Ge-rich Si1−xGex devices.

Determination of Phonon Deformation Potentials in Carbon-doped Silicon

Background and purpose Carbon-doped silicon (Si:C) is used as a stressor to the channel of n-type MOSFET [1]. However, there are few report directly evaluated the strain states of Si:C. The strain-Raman coefficient that relate Raman shift to in-plane strain has not been derived yet. In this study, we determined the strain-Raman coefficient for Si:C using water-immersion Raman spectroscopy in conjunction with X-ray diffraction (XRD) measurement. Experimental method Si:C films were grown on (001) Si substrate by molecular beam epitaxy. The film thickness were 43, 50 and 37 nm, with C concentration of 0.600%, 0.826% and 1.14% respectively. The film thickness and the C concentration were confirmed by cross section TEM and SIMS measurement, respectively. Raman peak shift was measured by water immersion Raman spectroscopy. For the Raman evaluation, the numerical aperture of the water immersion lens and the refractive index of the medium were 1.2 and 1.3, respectively. The wavelength of the excitation light source and the focal length of the spectrometer were 355 nm and 2000 mm. Here, the penetration depth is approximately 5 nm. All the Raman spectroscopy were performed under the LO-active conditions. The in-plane and out-of-plane strain were measured by XRD. Part of the XRD measurement was performed at SPring-8 using synchrotron radiation as X-ray source. Results and Discussion Figure 1 shows the Raman peak shift obtained from Si:C with various C concentration. The spectra were calibrated with the strain-free single crystalline-Si (c-Si) at 520 cm-1. From Fig. 1, the Raman shift is decreased as the C concentration increases. This reflect the tensile strain is applied to Si:C. Table 1 shows the results of the XRD, and the in-plane and out-of-plane strain in Si:C. Here, ε || and ε ⊥ are the in-plane and out-of-plane strain, respectively. The in-plane and out-of-plane lattice spacing of c-Si is 1.357525 Å. From Table 1, it is clear that the tensile strain is applied to Si:C. This is consistent with the Raman spectroscopy result. There should be a relationship such as the following equation between Raman shift and in-plane strain. ω Si:C = ω Si + ax + bε || (1) Here, ω Si:C and ω Si are the Raman shift of Si:C and Si, x is C concentration, and a and b are the alloy-disorder coefficient and strain-shift coefficient, respectively. In this study, we used the value of a coefficient to be 210 cm-1 to determine b coefficient [2]. From Eq. (1), the in-plane strain ε || can be obtained by the following equation using LO phonon Raman wavenumber shift from the strain-free Raman shift ω 0 Δω LO = ω Si:C – ω 0 = bε ||, (2) ω 0 = ω Si + ax = 520 + 210x. (3) Figure 3 shows Raman wavenumber shift Δω LO as a function of the in-plane strain ε || obtained from the XRD. The dotted line is the linear fit to the experimental data. As shown in Fig. 3, although the C concentration x range is very narrow, we confirmed that the Δω LO shows a linear dependence on the in-plane strain ε ||. Finally, we have succeeded to determine the value of strain-Raman (b) coefficient: b = -539.5 cm-1. Acknowledgements The XRD measurement of 0.600 and 1.06% Si:C was performed at SPring-8 (JASRI) with proposal number of 2015A1971, 2015B1925.

Determining the directional strain shift coefficients for tensile Ge: a combined x-ray diffraction and Raman spectroscopy study

In this work the calibration of the directional Raman strain shift coefficient for tensile strained Ge microstructures is reported. The strain shift coefficient is retrieved from micro-Raman spectroscopy measurements in combination with absolute strain measurements from x-ray diffraction using focused synchrotron radiation. The results are used to fit the phonon deformation potentials. A linear dependence of the phonon deformation potentials p and q is revealed. The method can be extended to provide strain calibration of Raman experiments also in other material system.

In-Plane Biaxial Strain Evaluation Induced in Ge1-X Sn x Films Using Oil-Immersion Raman Spectroscopy

We performed evaluation of anisotropic strain relaxation in the patterned Ge1-x Sn x mesa by Raman spectroscopy. For strain evaluation, several coefficients, e.g., strain shift coefficient b and phonon deformation potentials (PDPs: p and q), are indispensable. In this study, using thin Ge1-x Sn x films which were epitaxially grown on Ge substrate by MOCVD, the b coefficient and PDPs were derived. From results of Raman spectroscopy and XRD, the b coefficient was -399.3±5.6 cm-1. And, the p and q show only small variation in the Sn concentration range less than 3.2%. These derived coefficients were used to evaluate the biaxial strain in the patterned Ge1-x Sn x mesa by oil-immersion Raman spectroscopy. As a result, we confirmed uniaxial strain relaxation in the Ge1-x Sn x stripe pattern.

Examination of phonon deformation potentials for accurate strain measurements in silicon–germanium alloys with the whole composition range by Raman spectroscopy

The phonon deformation potentials (PDPs), p and q, of Si1−xGex with the whole range of the Ge concentration x were examined in detail in pursuit of accurate strain measurements by Raman spectroscopy. An oil-immersion Raman technique was adopted to extract the PDPs of Si1−xGex, in which a complex sample preparation process or a stress-introduction device is not necessary. The strain-shift coefficients bLO and bTO, which can be calculated using the obtained PDPs, were compared with the values in the literature, and we suggested which values were best for application to accurate strain measurements. Ab initio calculation was also performed to understand the behavior of the PDPs throughout the whole range of x in Si1−xGex.

Raman measurements of uniaxial strain in silicon nanostructures

The strain-shift coefficient used to convert Raman shifts to strain depends on multiple factors including phonon deformation potentials (PDPs). PDPs have been reported for silicon, which differ by 30%. This leads to varying strain-shift-coefficients. Using the wrong strain-shift coefficient affects the strain determined. The discrepancies in the reported PDPs were previously ascribed to surface stress relaxation and the opacity of the material to the laser radiation. This paper shows that surface orientation and scattering geometry are major factors behind the PDPs discrepancies. The work further demonstrates that different PDPs are required to accurately characterize transverse optical and longitudinal optical Raman modes. The effects of sample geometry and stress have also been studied. State of the art devices use nanowires and thin films under large values of uniaxial stress; however, previous experiments to determine PDPs and strain-shift coefficients in silicon have been limited to bulk material and stress only in the range 0–2 GPa. In this work, the strain-shift coefficient of silicon nanostructures is determined for a large range of geometries and applied stress values (0–4.5 GPa). Strain in the samples has been measured using three independent techniques: analytical calculations, finite element simulations, and by direct visual inspection of the samples elongation using scanning electron microscopy. Raman shifts have been measured using 458 nm and 364 nm laser radiations. The combination of these techniques and the large number of samples (up to 85) has allowed the accurate determination of the strain-shift coefficient for the technologically important (100) silicon surface.

Effects of composition, strain, and atomic disorder on optical phonon frequencies in Si1−xGex

Optical phonon shift coefficients are studied over the entire composition range of Si1−xGex, taking into account the effects of composition, atomistic disorder and strain. Ab initio calculations complement various experimental observations found in the literature by revealing that strain effects on the optical phonons—both hydrostatic and biaxial—vary nonlinearly with composition. Additionally, the pure mode frequencies of the highest optical phonon modes in unstrained Si1−xGex are found to shift linearly with composition, while the mixed-mode frequency shifts quadratically with composition around x=0.5. While atomic randomness affects the composition dependence of the optical phonons, the strain-shift coefficients are approximately insensitive to the local atomic configuration. Short range atomic ordering increases the linear and nonlinear effects.

Strain in epitaxial Si/SiGe graded buffer structures grown on Si(100), Si(110), and Si(111) optically evaluated by polarized Raman spectroscopy and imaging

We report on the characterization, thanks to Raman spectroscopy and imaging of tensely strained Si films pseudomorphically grown on (001), (110), and (111) SiGe virtual substrates. The samples studied here are those described in the work of Destefanis et al. [J. Appl. Phys 106, 043508 (2009)]. They consist in 17-nm-thick strained Si layers grown at 650 °C with SiH4 as a gaseous precursor on top of polished SiGe virtual substrates of various surface orientations. We first derived the exact component array of the strain/stress field along the different growth directions. Because the relation between strain or stress and the Raman frequencies are complex, we also derive the strain-shift coefficients for the different substrate orientations considered in this work and the polarization selection rules. Visible and near-UV Raman spectroscopies were used to extract the in-plane lattice parameter of the SiGe virtual substrates and the tensile strain in the thin Si epitaxial layers on top. We have notably investigated thanks to Raman imaging the in-plane distribution of strain in Si layer/SiGe buffer stacks grown on (110) and (111) Si substrates. Original surface arrays have been highlighted for each surface orientation. Promising results have been obtained for (110) SiGe virtual substrates in terms of strain and layer quality while the technological usefulness of the (111) ones is more questionable.

Measurement of phonon pressure coefficients for a precise determination of deformation potentials in SiGe alloys

AbstractFor an effective use of Raman scattering as strain characterization tool in SiGe nanostructures a precise knowledge of the phonon deformation potentials (DPs) is strictly necessary. The optical phonon DPs can be determined by means of Raman scattering measurements from the cleaved edge of a biaxially strained SiGe alloy layer grown pseudomorphically on silicon and subsequently capped. Due to uncertainties in the literature values of the unstrained phonon frequencies it turns out that the desired degree of accuracy is only attained by complementing the Raman measurements from the edge with that of the hydrostatic pressure coefficient of the optical phonons. For that purpose we have grown by molecular beam epitaxy up to seven partially strained Si1–x Gex alloys on Si spanning the entire compositional range and measured the dependence on hydrostatic pressure of the frequency of the Ge‐like, Si‐like and mixed Si–Ge optical modes. After correcting for the pressure dependent biaxial stress induced by the Si substrate on the alloy layer and taking into account the dependence on alloy composition of the bulk modulus we obtain a fairly constant value of the Grüneisen parameter around 1.0 for all three optical modes in the whole range of Ge contents. We also determined the strain shift coefficients for the three modes, which are essentially independent of Ge content between 0.4 and 1. Our results are in very good agreement with recentcalculations of the SiGe phonon deformation potentials using a modified Keating model, which settles the longstanding issue about the large discrepancies between results from different experiments. (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Strain in Epitaxial Si/SiGe Graded Buffer Structures Grown on Si (100), Si (110) and Si(111): a Raman Spectroscopy Study

We report preliminary results on the Raman characterization of strained-Si films pseudomorphically grown on (001), (110) and (111) SiGe virtual substrates. Because the relation between strain or stress and the Raman frequencies are complex, we first derive the strain-shift coefficients for the different substrate orientations considered in this work. Then, visible and near-UV Raman spectroscopies were used to extract the in-plane lattice parameter of the virtual substrate and the strain in the thin silicon epitaxial layers grown on top. Finally, we investigated the in-plane strain distribution in strained Si/SiGe buffer layers grown on (110) and (111) silicon substrates by Raman imaging. In-plane strain fluctuations are observed both in the epilayer and the virtual substrate for all substrate orientations.

Phonon strain shift coefficients in Si1−xGex alloys

A comprehensive study of the biaxial strain-induced shift of the Si1−xGex Raman active phonon modes is presented. High-resolution Raman measurements of Si1−xGex/Si heterostructures have been compared to x-ray diffraction data. Our approach, unlike previous works, is effective to decouple and quantify separately the effect of strain and composition on the phonon frequencies, yielding an accurate determination of the phonon strain shift coefficients in the entire composition range. Our results show that the strain shift coefficients are independent of the composition, a result which is in good agreement with theoretical calculations, performed within the framework of valence force-field theory.

Characterization of electrical and structural properties of strained-Si-on-insulator layers

The electrical and structural properties of strained-Si-on-insulator (sSOI) wafers were investigated. The strain, calculated from two-dimensional reciprocal space mapping, was found to be 0.78%, which is comparable to that of fully relaxed Si1−xGex film with Ge concentration of 20.6at.%. Based on the Raman peak shift combined with measured value of strain, the strain shift coefficient is extracted to be −736cm−1. The pseudo-metal-oxide-semiconductor field-effect transistor measurements, employed to characterize the electrical properties of sSOI wafers, showed that both electron and hole mobilities are enhanced by strain. The enhancement factor of electron mobility is larger than that of hole mobility.

Composition dependence of the phonon strain shift coefficients of SiGe alloys revisited

By combining Raman scattering from the cleaved edge and under hydrostatic pressure, we have accurately determined the tetragonal phonon deformation potentials of strained Si1−xGex alloys in the entire compositional range for the Ge-like, Si-like, and mixed Si–Ge optical modes. A known biaxial strain is induced on thin alloy layers by pseudomorphic epitaxial growth on silicon and subsequent capping. We also determine the strain shift coefficient of the three modes, which are essentially independent of Ge content between 0.4 and 1. This is key information for an effective use of Raman scattering as strain-characterization tool in SiGe nanostructures.

Strain Behaviors of Si[sub 1−x]Ge[sub x] Grown on Oxidized and Etched Si[sub 1−x]Ge[sub x

The strain of Si 1-x Ge x (x = 0.15) was controlled by oxidation temperature, time, and initial Ge fraction in underlying Si 1-x Ge x . Followed by oxidation and HF cleaning of Si 1-x Ge x , epi-Si 1-x Ge x (x = 0.15) films were deposited on the Ge pileup layer, which is formed by the ultrahigh-vacuum chemical vapor deposition technique at 550°C. Through the X-ray rocking curve measurements and Raman scattering measurements, the strain-shift coefficient [b(x)] was evaluated. By using oxidized and etched Si 1-x Ge x for the substrate, the strain of deposited Si 1-x Ge x was changed from the partially relaxed to the tensile-strained state, a method for growing relaxed Si 1-x Ge x .

Modelling of the phonon strain shift coefficients in Si1-xGex alloys

In the Raman spectrum of epitaxial Si1-xGex alloys, the composition effect, that determines the three mode peak position, is superimposed to the phonon shift caused by strain. Therefore, the experimental spectra interpretation remains not straightforward, unless vibrational properties calculations of such systems are available. In this paper, reticular dynamic calculations within a modified Keating model, the Anharmonic Keating, are discussed. A new set of model parameters is introduced, providing simple and predictive supercell calculations to investigate accurately the three Raman-active phonon modes in Si1-xGex alloys. The recalculated AK model is successfully employed to model the zone centre Raman-active phonon modes of relaxed and hydrostatically stressed Si1-xGex alloys. The results of the calculations are in quantitative agreement with experimental data reported in literature.

Strain-induced shift of phonon modes in [formula omitted] alloys

A theoretical approach to the lattice dynamics of Si 1 - x Ge x alloys within a valence force field framework is discussed. A modified Keating model, the anharmonic Keating, was employed to perform supercell calculations in order to investigate accurately the three Raman active optical phonon modes. Theoretical results on both relaxed and strained SiGe samples are shown. A detailed study of the biaxial strain-induced shift of phonon modes of epitaxial SiGe alloys has been performed in all the composition range. Furthermore, a comparison of experimental data with the model results on phonon strain shift coefficients will be also discussed.