Medical, Pharma, Engineering, Science, Technology and Business

^{1}Department of Physics, Indiana University of Pennsylvania, USA

^{2}Laboratory of Optoelectronic Materials and Detection Technology, Guangxi Key Laboratory for Relativistic Astrophysics, School of Physical Science & Technology, Guangxi
University, China

^{3}Department of Physics, Auburn University, Auburn, USA

- *Corresponding Author:
- Talwar DN

Department of Physics

Indiana University of Pennsylvania

975 Oakland Avenue, 56 Weyandt Hall

Indiana, Pennsylvania 15705- 1087, USA

**Tel:**7247627719

**E-mail:**[email protected]

**Received Date:** January 28, 2017; **Accepted Date:** February 16, 2017; **Published Date:** February 26, 2017

**Citation: **Talwar DN, Wan L, Tin CC, Feng ZC (2017) Assessing Biaxial Stress and
Strain in 3C-SiC/Si (001) by Raman Scattering Spectroscopy. J Material Sci Eng 6:324. doi: 10.4172/2169-0022.1000324

**Copyright:** © 2017 Talwar DN, et al. This is an open-access article distributed
under the terms of the Creative Commons Attribution License, which permits
unrestricted use, distribution, and reproduction in any medium, provided the
original author and source are credited.

**Visit for more related articles at** Journal of Material Sciences & Engineering

Highly strained 3C-SiC/Si (001) epilayers of different thicknesses (0.1 μm-12.4 μm) prepared in a vertical reactor configuration by chemical vapor deposition (V-CVD) method were examined using Raman scattering spectroscopy (RSS). In the near backscattering geometry, our RSS results for “as-grown” epilayers revealed TO- and LO-phonon bands shifting towards lower frequencies by approximately ~2 cm-1 with respect to the “free-standing” films. Raman scattering data of optical phonons are carefully analyzed by using an elastic deformation theory with inputs of hydrostatic-stress coefficients from a realistic lattice dynamical approach that helped assess biaxial stress, inplane tensile- and normal compressive-strain, respectively. In each sample, the estimated value of strain is found at least two order of magnitude smaller than the one expected from lattice mismatch between the epilayer and substrate. This result has provided a strong corroboration to our recent average-t-matrix Green’s function theory of impurity vibrational modes – indicating that the high density of intrinsic defects at the 3C-SiC/Si interface are possily responsible for releasing the misfit stresses and strains. Unlike others, our RSS study in “as-grown” 3C-SiC/Si (001) has reiterated the fact that for ultrathin epilayers (d<0.4 μm) the optical modes of 3C-SiC are markedly indistinctive. The mechanism responsible for this behavior is identified and discussed. PACS: 78.20.-e 63.20.Pw 63.20.D.

3C-SiC/Si (001); Raman scattering; Stress and strain; Elastic theory

Silicon carbide (SiC) is one of the very few IV-IV compound
semiconductors – exhibiting exceptional mechanical, electrical and
chemical properties [1-14]. The novel characteristics of SiC have made
it suitable for fabrication of many important modern [1-10] devices
for microelectronic, optoelectronic and sensor application needs.
The scientific interest in SiC is stimulated by a strong chemical bond
between Si and C atoms which provides the material a wider-bandgap,
extreme hardness, high thermal stability, chemical inertness, higher
thermal conductivity, high melting temperature, large bulk modulus,
high critical (breakdown) electric field strength, and low dielectric
constant. Among other wide-bandgap semiconductors, SiC is rather
distinctive for controlling both n- and p-type dopants [11-18] across
the broader concentration range (~10^{14}-10^{19} cm^{-3}). The ability of SiC to
form native silicon dioxide (SiO_{2}) is an advantage [19-21] leading to its
use in device fabrications. SiC has also been considered as a substitute
for Si to make Schottky diodes and metal–oxide–semiconductor fieldeffect
transistors (MOSFETs) for high-power, high-temperature, and
high-frequency applications. Both crystalline and polycrystalline SiC
have become attractive in recent years to design micro- and nanoelectro-
mechanical systems (MEMS, NEMS) [1-10]. While silicon
carbide is currently being used to fabricate green, blue and ultraviolet
light-emitting diodes (LEDs) – the emerging market [22,23] of utilizing
heteroepitaxy/homoepitaxy SiC films is in high-power switches and
microwave devices.

SiC occurs in more than 200 different crystalline structures [9-14] called polytypes. While every polytype is perceived by its own stacking sequence of Si-C bilayers – each structure displays its explicit set of distinct electrical and vibrational properties. The customary polytypes that are being developed for commercial needs include the cubic (3C-SiC), hexagonal (4H-SiC, 6H-SiC), and rhombohedral (15R-SiC, 21R-SiC) structures. The original work by Nishino et al [24] proposed a multistep chemical vapor deposition (CVD) method to prepare 3C-SiC on Si. Many attempts have been made in recent years to improve the growth [25-30] mechanisms of 3C-SiC/Si (001) epifilms. The prospect of attaining large area 3C-SiC/Si (001) epilayers by CVD and molecular beam epitaxy (MBE) appears to be very encouraging. Despite the successful growth of 3C-SiC/Si (001) the quality of epilayers is still lacking thus impeding their use in the fabrication of electronic devices. On the other hand, a positive trade-off is its low cost advantage and there is a greater prospect of scalability in the fabrication of devices using Si as compared to 4H-SiC that sustained the current interest in the growth of 3C-SiC/Si for future applications.

There exists a considerable difference in the lattice constants (19.8%) and thermal expansion coefficients (8%) between 3C-SiC and Si. This leads to biaxial strain in 3C-SiC/Si (001) epilayers which might instigate modifying their physical and chemical properties. Whether one will be able to make practical use of such highly strained structures in electronic devices is still an open question. It is quite possible, however, that a large lattice mismatch in 3C-SiC/Si (001) leads to breaking of atomic bonds in epilayers which generates high density of dislocations or intrinsic defects [31]. It is likely that these defects stimulate releasing interfacial strains in epilayers. Therefore, it is imperative to evaluate biaxial stress in 3C-SiC/Si (001) epilayers for further progress in device engineering. Apart from the x-ray diffraction study (which is insensitive to ultrathin layers) [32,33] the most frequently used technique for stress estimation is the Raman scattering spectroscopy (RSS) [34-46]. Earlier, the method has been used successfully to appraise microstrains in bulk semiconductors under hydrostatic and uniaxial stress [36,37]. As the RSS approach is precise, sensitive, convenient and non-destructive it can be employed for studying the biaxial strain in thin epilayers grown on thick substrates including 3C-SiC/Si (001) [41-45].

The purpose of this work is to explore both theoretically and
experimentally the problem of assessing residual stress and strain in
3C-SiC/Si (001) epilayers with large lattice mismatch. By using RSS,
we will study the optical phonon shifts in a number of 3C-SiC/Si (001)
samples grown by CVD method in the vertical reactor configuration
(V-CVD) [30]. A T64000 Jobin Yvon triple advanced research Raman
spectrometer, equipped with an electrically cooled charge coupled
device (CCD) detector, is employed to measure the optical phonon
frequencies in the near backscattering geometry. The observed phonon
shifts will be assimilated in a conventional elastic deformation [40-46]
theory to appraise the stresses and strains inside the 3C-SiC films. Our
calculated biaxial stress in the V-CVD grown 3C-SiC/Si (001) epifilms
of varied thickness fall well within the range of 0.45-0.94 GPa, i.e., an
order of ~10^{9} dyn/cm^{2}. Theoretical results are compared and discussed
with the existing experimental and other simulated data. Despite a
considerable difference (~19.8%) in the lattice constants [30] between
the bulk Si and 3C-SiC materials – the study has offered significantly
lower (two-order of magnitudes) values of inplane strains (i.e., ~0.1-
0.2%) as well as normal (i.e., ~-0.07 to -0.14%) strains within the
3C-SiC films grown on Si substrate. While the simulated results are
quite surprising – the outcome has undoubtedly offered support to our
earlier speculations of high density dislocations and/or intrinsic defects
near the 3C-SiC/Si interface [31] which are likely to be responsible for
releasing misfit stresses and strains in 3C-SiC films.

3C-SiC epifilms used in the present RSS study are grown on
(001) Si substrates under normal atmospheric pressure environment
using CVD method in a vertical reactor configuration. Although, we
employed 1 in diameter Si wafers as substrates – our reactor is capable
of scaling up to handle 3 in diameter Si substrates for growing 3C-SiC
epifilms. The V-CVD system utilizes a rotating SiC-coated susceptor
heated by a radio-frequency (RF) induction power supply and is capable
of operating at atmospheric and low pressure modes. The vertical
configuration has several advantages including substrate rotation to
give a large-area thickness uniformity, convenient in-situ monitoring
of substrate parameters, and easy implementation of various growth
enhancement procedures. The method used here to grow 3C-SiC/Si
(001) epilayers consisted of three main steps described in details [30]
elsewhere. It is to be noted that 3C-SiC/Si (001) epilayers are prepared
at 1 atm and 1360°C with source ratio of Si/C (of ~0.33) using growth
time τ between 2 min and 4 h at a rate of 3.2 ± 0.1 μm h^{-1} achieving the
film thicknesses *d*, between 0.1 μm and 12.8 μm (**Table 1**). The set of
single-crystalline 3C-SiC films used in the RSS study show uniformly
smooth and mirror-like surfaces without macro-cracks – even for the
thinnest film.

Sample # | Growth time | Epifilm thickness d µm |
---|---|---|

125 D | 2 min | 0.1 |

125 C | 15 min | 0.8 |

125 B | 30 min | 1.6 |

125 A | 45 min | 2.4 |

119 A | 1 hr | 3.2 |

119 B | 3 hr | 9.6 |

113 | 4 hr | 12.8 |

**Table 1:** Properties of V-CVD grown 3C-SiC/Si (001) samples at 1 atm and 1360°C.
The source ratio of Si/C was set at approximately ≅0.33 with different growth times
of 2 min, 15 min, 30 min, 45 min, and 1 hr, 3 hr, and 4 hr, respectively.

RSS is a powerful and non-destructive technique to provide
valuable information on the vibrational characteristics of materials for assessing the epilayer thickness, strain, disorder, and site selectivity
of defects [46,47]. The method is particularly suited for probing local
atomic- and/or nanoscale structural changes in SiC materials while
making careful analysis of its subtle spectral variations. Since RSS
efficiency depends upon the polarizability of electron cloud – the
process is quite sensitive to light elements involved in producing
covalent bonds including SiC. The strong Si-C bonding in 3C-SiC with
large bandgap stimulates higher Raman efficiency – requiring incident
laser light of visible spectral range with reduced intensity to prevent
significant heating of the material samples. We have performed room
temperature RSS measurements in the near backscattering x(y',y') x
geometry on several 3C-SiC epifilms of diverse thickness (~0.1 and
12.8 μm) grown on thicker (~200 and 400 μm) Si-substrates. A T64000
Jobin Yvon triple advanced research Raman spectrometer equipped
with electrically cooled CCD detector is employed with an Ar^{+} 488 nm
line as an excitation source, while keeping the power level adjusted to
200 mW.

Assessing the lattice phonons in an ideal backscattering geometry for
perfect diamond/zb materials requires strict wavevector conservation
[47,48] and polarization selection rules. In the first-order RSS, this
constraint limits the phonon wavevector to for observing the
lattice modes. Thus, for Si crystal a triply degenerate phonon ω_{o(Ґ)} at the center of the Brillouin zone (i.e., Ґ-point) is allowed while in
3C-SiC material a doubly degenerate TO mode (ω_{TO(Ґ)}) is forbidden
and a non-degenerate LO phonon (ω_{LO(Ґ)}) is permitted. By applying the
hydrostatic pressure X up to 22.5 GPa in bulk 3C-SiC crystals, Raman
scattering spectroscopy [37] was used earlier to measure the changes in
the long wavelength optical phonon frequencies that helped evaluate
the mode Grüneisen parameters and of the optical modes.

**Near backscattering Raman spectra to assess stress and strain
in 3C-SiC/Si (001)**

One must note that the V-CVD grown 3C-SiC/Si (001) epilayers
are perceived with biaxial stress in 3C-SiC films due to differences in
Si and 3C-SiC lattice constants and thermal expansion coefficients
[30]. An elastic deformation theory developed here can be applied to
this system for assessing stressess and strains. We used RSS method
to study the optical phonon shifts in (i) ″as grown″ 3C-SiC epilayers
prepared on Si (001) with no processing done after V-CVD growth,
and (ii) self-supported ″free-standing″ 3C-SiC films in which the Si
substrate is removed with KOH etching solution. As an example, we
have displayed (**Figure 1a and 1b**) our results of the RSS measurements
recorded in the near back-scattering geometry for a 12.8 μm thick (a)
“as-grown” and (b) “free-standing” film. Clearly, in the “as-grown”
sample, the observed ω_{LO(Ґ)}, ω_{TO(Ґ)} modes near ~972 cm^{-1}, ~796 cm^{-1} are seen shifting towards lower frequencies by approximately ~2 cm^{-1} when Si substrate is etched away. In **Figure 1a** we have also noticed a
weak phonon feature appearing between ~938–950 cm^{-1} almost ~20-35 cm^{-1} lower than the ω_{LO(Ґ)} phonon line. No such trait emerged, however,
in the “free standing” film. By examining several “as-grown” 3C-SiC/
Si (001) samples we have realized the weak feature in only a few cases
and certainly not in the “free-standing” films. We strongly believe
attributing this characteristic to interface states between 3C-SiC and Si.
Again, the weak trait has neither affected the ω_{LO(Ґ)}, ω_{TO(Ґ)} phonon lines
nor it changed the stress calculations. By using a conventional elastic
deformation theory we have evaluated the bi-axial strains in 3C-SiC
epilayers by incorporating the observed Raman optical phonon shifts
in the “as-grown” and “free-standing” films.

**Thickness dependent Raman spectra of 3C-SiC/Si (001)**

In **Figure 2** we have reported RSS measurements in the near backscattering
geometry for seven of the V-CVD grown 3C-SiC/Si (001)
samples (**Table 1**) in which the film thickness d is varied between 100
Å to 12.8 μm. Except for the 100 Å thick sample, we observed in all
other “as-grown” materials (**Figure 2**) the ω_{o(Ґ)} Si phonon line near
~520 cm^{-1} and ω_{LO(Ґ)}, ω_{TO(Ґ)} modes of 3C-SiC near ~972 cm^{-1}, ~796 cm^{-1},
respectively. The relative peak intensities and lineshapes of ω_{o(Ґ)} mode
for Si substrate and the ω_{LO(Ґ)}, ω_{TO(Ґ)} phonons of 3C-SiC epifilms showed
variations with growth time τ or film thickness *d*. In thicker “as-grown” samples, one expects less penetration of the laser light through 3C-SiC
into Si substrate which inflicts a decrease in the intensity of the ω_{o(Ґ)} phonon line and a rise of the ω_{LO(Ґ)} mode intensity (**Figure 2**) with respect
to ω_{TO(Ґ)} phonon. This observation clearly indicates improvement in
the crystalline quality of the thicker epilayers prepared with increased
growth time τ. Unlike others [44] our RSS study has reiterated (**Figure
2**) the fact that in “as-grown” samples with thin epilayers (*d*<0.4 μm) the
3C-SiC optical modes are markedly indistinctive. We will incorporate
the observed optical phonon shifts in “as-grown” and “free-standing”
films to our elastic deformation theory to estimate both “in-plane” and
“normal” strains in epilayers of different thickness.

**Figure 2:** Thickness dependent Raman spectra in the near back-scattering
geometry for seven V-CVD “as-grown” 3C-SiC/Si (001) samples, where d is
varied between 0.1 μm to 12.8 μm. Except for 0.1 μm thick epifilm, in all other
samples we observed ω_{o(Ґ)} Si phonon line near ~520 cm^{-1} and ω_{LO(Ґ)}, ω_{TO(Ґ)} modes of 3C-SiC near ~972 cm^{-1}, ~796 cm^{-1}, respectively.

To comprehend the ″pressure-dependent″ vibrational properties in semiconductors one can: (i) apply the ″hydrostatic pressure″ in bulk samples using diamond anvil cell, (ii) perform ″uniaxial-stress″ on large size specimens, and (iii) examine ″biaxial-stress″ in thin films prepared on mis-matched substrates. In each case the ″pressure″ is considerd as a perturbation. By adopting a conventional elastic deformation theory, one can derive expressions for the three cases in terms of Raman-stress coefficients and optical mode frequencies to empathize experimental data.

By using an elasticity theory [49,50] for a continuous media – the
strain (ε) and stress (σ) tensors can be linked to the elastic-stiffness *C* and elastic-compliance *S* tensors via:

(1a)

and (1b)

For uniaxial stress, the off-diagonal elements of stress and strain are zero. In a generalized form, the axial stress and strain tensors are symmetric about the z axis [41]:

(2)

In Eq. (2), if *X* or *Z*<0 the stress is compressive and it is tensile
when *X* or *Z*>0. The elements of strain tensor can be rewritten as =ε_{yy}(=ε_{||}) and . gain, one can separate the axial stress into a
hydrostatic term *X* and a uniaxial term *P* along the *z*-axis:

(3)

with *Z*=*X* + *P*. By using Eqs. (1 and 2) it is straight forward to link
the strain and stress elements to the elastic compliance (S_{ij}) and elastic
stiffness constants (C_{ij}), respectively as:

(4a)

(4b)

(4c)

(4d)

Both compliance and stiffness tensors are related [51] through *S*=*C*^{-1} for evaluating compliance coefficients (S_{11}, S_{12}, S_{44}) from the
known elastic (C_{11}, C_{12}, C_{44}) constants. Again, the negative ratio of
strains perpendicular and parallel to the stress axis is known as the Poisson ratio ν. For the hydrostatic pressure, ν_{H}=-1 and for uniaxial
stress along the z-axis .

If the zb crystal is deformed either internally (residual stress) or externally (applied stress) – the three optical phonon frequencies can be obtained by solving the secular equation [40]:

(5)

where *p*, *q*, *r* are the symmetry allowed anharmonic parameters known
as phonon deformation potentials. Here, the term represents the shift of perturbed (strained: ) optical phonons from the unstrained modes , and *u _{i}* are the
components of eigen vectors. For axial stress with strains (Eq. 2) the
non-trivial solutions of Eq. (5):

(6)

provide two phonon modes given by:

(7)

one having a doublet

(8a)

and the other a singlet

(8b)

**Stress induced modes: Hydrostatic case**

In a hydrostatic case (*P*=0), with *a*_{o} changing to a:

(9a)

(9b)

where is the bulk modulus.

The Grüneisen constant (hydrostatic stress) *γ*_{o} is:

(10)

where *ω* is the mode frequency and *V*_{o} crystal volume. By using Eqs.
[8(a-b) and 10] one gets:

(11)

where

If a cubic cell of lattice constant *a*_{o} is distorted to a tetragonal cell of lattice constants *a* and *c*, respectively, then . Consequently, one can rewrite Eqs.
[8(a)-(b)] as:

(12)

where is the shear-deformation parameter.

**Stress induced modes: Uniaxial case**

If a uniaxial stress is applied in the z-direction:

(14)

one gets:

(15a)

(15b)

Solving the above two Eqs. [15(a)-(b)], it can be shown that:

(15c)

(15d)

Substituting Eqs. [15(c-d)] into Eqs. [(12) and (13)] one can obtain

(16a)

(16b)

Subtracting Eqs. [16(a)-(b)], we will have

(17a)

(17b)

(17c)

where we defined

**Stress induced modes: Biaxial case**

The 3C-SiC/Si(001) system is regarded as one with biaxial stress (i.e., *P*=-*X* : cf. Eq. 3 ) in the film due to difference in the substrate lattice
constants and thermal expansion coefficients. With Eq. [4 (d)], one can
rewrite Eqs. [8(a)-(b)] as

(18a)

(18b)

where the two stress coefficients

(19)

can be determined by RS experiments. Again it is customary to add superscripts TO, LO on for classifying the doublet and singlet (cf. Eqs. 18(a-b)) modes i.e.,

(20a)

(20b)

Subtracting Eq. (20 a) from Eq. (20 b) one can obtain

(21)

from the observed Raman mode frequencies

Using Eq. (21) with hydrostatic stress coefficients known from the pressure dependent RS experiments, we can get two values of X from Eqs. [20(a-b)]:

(22)

(23)

If we take an average of the two *X* values and assign error bar, we
can appraise both in-plane and normal strains in 3C-SiC films by using
Eqs. [4(a)-(b)] and setting *P*=-*X*. It is to be noted that the classical
elastic deformation theory assumes a linear relationship between stress
and strain (Eqs. (10 and 15-17)) – which may not be very realistic at
higher *X*>10 GPa. In our analyses from RSS data of optical phonons
the estimated values of hydrostatic pressure components are seen to
fall well within *X*<1 GPa in all V-CVD grown 3C-SiC/Si (001) samples.
It is, therefore, strongly believed that in the elastic deformation theory
one would anticipate nearly <5% error in assessing strains.

**Elastic constants**

Accurate knowledge of elastic constants for 3C-SiC is crucial for
engineering MEMS and/or NEMS devices and evaluating stress and
strains in 3C-SiC/Si (001) epilayers (Eqs. (4a-b)). The published data
on elastic constants by exploiting diverse experimental and theoretical
methods is, however, rather conflicting [52-66]. The complete result on
lattice dynamics of 3C-SiC has been reported earlier[67] by inelastic
x-ray scattering (IXS) method. The pressure dependent optical phonon
shifts in bulk 3C-SiC crystals are investigated with applied hydrostatic
pressure (X) up to 22.5 GPa [37]. Recently, we have adopted a realistic
rigid-ion model (RIM) to simulate phonon dispersions of 3C-SiC
at 1 atm [31] and 22.5 GPa [67]. In the RIM scheme, the short- and
long-range Coulomb interactions are optimized by least-square fitting
procedures using lattice constants, critical point phonon energies
as input while elastic constants *C _{ij}* and their pressure derivatives
are deliberated as constraints to match the IXS [67] and pressuredependent
phonon data. For 3C-SiC, the best fit values of phonon
dispersions find recent experimental [54] and local density functional
data of elastic constants more reliable than the results available from
earlier measurements. In

3C-SiC | |||||||||
---|---|---|---|---|---|---|---|---|---|

c_{11} |
c_{12} |
c_{44} |
s_{11} |
s_{12} |
s_{44} |
s_{11} + 2s_{12} |
ŝ | ν_{s} |
Ref. |

(x*10^{12} dyn/cm^{2}) |
(x*10^{-13} cm^{2}/dyn) |
(x*10^{-13} cm^{2}/dyn) |
|||||||

5.4 | 1.8 | 2.5 | 2.222 | -0.556 | 4 | 1.111 | 2.5 | 0.25 | [52] Exp. |

3.9 | 1.42 | 2.56 | 3.183 | -0.85 | 3.906 | 1.484 | 2.718 | 0.267 | [53] Exp. |

3.95 | 1.36 | 2.36 | 3.074 | -0.787 | 4.237 | 1.499 | 2.575 | 0.256 | [54] Exp. |

3.523 | 1.404 | 2.329 | 3.673 | -1.047 | 4.294 | 1.58 | 2.988 | 0.285 | [55] Cal. |

3.489 | 1.384 | 2.082 | 3.7 | -1.051 | 4.803 | 1.598 | 2.972 | 0.284 | [56] Cal. |

3.71 | 1.69 | 1.76 | 3.77 | -1.18 | 5.682 | 1.41 | 3.51 | 0.313 | [57] Cal. |

2.89 | 2.34 | 0.554 | 12.56 | -5.62 | 18.05 | 1.321 | 13.764 | 0.447 | [58] Cal. |

3.9 | 1.426 | 1.911 | 3.188 | -0.854 | 5.233 | 1.481 | 2.729 | 0.268 | [59] Cal. |

4.2 | 1.26 | 2.87 | 2.764 | -0.638 | 3.484 | 1.488 | 2.286 | 0.231 | [60] Cal. |

3.9 | 1.34 | 2.53 | 3.111 | -0.795 | 3.953 | 1.52 | 2.57 | 0.256 | [61] Cal. |

4.05 | 1.35 | 2.54 | 2.963 | -0.741 | 3.937 | 1.481 | 2.5 | 0.25 | [62] Cal. |

4.2 | 1.2 | 2.6 | 2.727 | -0.606 | 3.846 | 1.515 | 2.2 | 0.222 | [63] Cal. |

3.84 | 1.32 | 2.41 | 3.16 | -0.808 | 4.149 | 1.543 | 2.571 | 0.256 | [64] Cal. |

4.151 | 1.319 | 2.654 | 2.845 | -0.686 | 3.768 | 1.473 | 2.397 | 0.241 | [65] Cal. |

3.63 | 1.54 | 1.49 | 3.687 | -1.098 | 6.711 | 1.49 | 3.211 | 0.298 | [66] Cal. |

**Table 2:** Calculated elastic compliance and other parameters of 3C-SiC obtained from the existing sets of elastic constants available in the literature.

**Grüneisen parameter**

Earlier, Olego et al. [36,37] described the hydrostatic pressure
dependent optical [ω_{LO(Ґ)}, ω_{TO(Ґ)}] phonons in bulk 3C-SiC crystals by
using RSS with applied X up to 22.5 GPa:

(24a)

(24b)

where the mode frequencies ω^{TO}, ω^{LO} are expressed in cm^{-1} while X
is in GPa (10^{10} dyn/cm^{2}). For 3C-SiC, the optical phonon shifts at low
temperature (6 K) have also been reported with X up to 15 GPa [42]. In **Figure 3**, a linear fit to the phonon data by dotted lines has revealed that
for *X*<10 GPa the *X*^{2} term in the right hand side of Eqs. (24a-b) is trivial
and can be neglected. Therefore, one can determine the hydrostatic
Grüneisen parameters from the linear relationship (Eq. (10)) involving . For 3C-SiC, the values estimated by Olego et al. used an average bulk modulus data *B*_{o} (=321.9 GPa)
of Si and C (**Table 3**) [36,37]. A correction was made to the values by exploiting the experimental bulk modulus *B*_{o} (=227 GPa) of
3C-SiC. While the experimental results of hydrostatic mode Grüneisen
parameters for 3C-SiC fall within <3% – our RIM calculations [67] of provided strong corroborations to the amended
values. While Feng et al. [41] employed for assessing strains in 3C-SiC epifilms – one would expect improvement in the
strain values if accurate results of are adopted.

ω_{0} (cm-1) |
dω/dX | γo | B (GPa) | B | |
---|---|---|---|---|---|

3C-SiCa | TO(Ґ) 797.7 | 3.88 | 1.56,b 1.10c | 227 | 4.1 |

LO(Ґ) 973.6 | 4.59 | 1.55,b 1.09c | |||

LO-TO 175.9 | 0.654 | ||||

C (dia)a | 1330 | 2.9 | 0.96 | 442 | 4.09 |

Si | 523.9 | 5.1 | 0.96 | 99 | 4.24 |

Ge | 304.6 | 4.02 | 1 | 75.8 | 4.55 |

^{a}Ref. [42].

^{b}Ref. [36-37].

^{c}Our

**Table 3:** Zone-center optical phonon frequencies [TO(Ґ), LO(Ґ)] and their hydrostatic pressure derivatives dω/dX for 3C-SiC. The phonon frequencies ω_{o} are in cm^{-1} and the
hydrostatic pressure *X* is in GPa. For comparison we have listed the related data for zone-center modes [ω_{o} (cm^{-1})] of diamond, Si, and Ge crystals. The mode Grüneisen
parameters γ_{o}, the bulk moduli B (in GPa) and their pressure derivatives B′ employed to calculate for 3C-SiC are also listed.

**Stress and strain in V-CVD 3C-SiC/Si (001) films**

In **Table 4** we have displayed the RSS results of optical phonons *ω ^{LO}*,

(25)

Sample # | d_{SiC} µm |
cm^{-1} |
cm^{-1} |
cm^{-1} |
cm^{-1} |
X(GPa) | (%) | (%) | Ref. |
---|---|---|---|---|---|---|---|---|---|

125 A | 2.4 | 795.5 | 972.8 | 794.2 | 971.6 | 0.446 ± 0.045 | 0.102 | -0.072 | [our] |

119 A | 3.2 | 795.7 | 972.9 | 793.9 | 971.8 | 0.552 ± 0.056 | 0.126 | -0.087 | [our] |

119 B | 9.6 | 795.8 | 973.2 | 793.5 | 971.5 | 0.739 ± 0.075 | 0.169 | -0.116 | [our] |

113 | 12.8 | 796 | 973.4 | 793.3 | 971.4 | 0.868 ± 0.088 | 0.199 | -0.137 | [our] |

487* | 4 | 795.5 | 972.4 | 794.3 | 969.9 | 0.575 ± 0.058 | 0.132 | -0.091 | [41] |

475 B | 4.5 | 795.1 | 972.4 | 793.1 | 971.1 | 0.622 ± 0.063 | 0.142 | -0.098 | [41] |

462 | 6 | 796.3 | 973.6 | 793.1 | 970.3 | 1.138 ± 0.115 | 0.26 | -0.179 | [41] |

475 | 7 | 796.3 | 972.8 | 794.3 | 969.9 | 0.810 ± 0.082 | 0.185 | -0.127 | [41] |

**Table 4:** Comparison of the Raman scattering data on optical phonons (cm^{-1}) used for assessing the stresses and strains in V-CVD grown 3C-SiC/Si (001). We used elastic
compliance values of S_{11}=3.074 × 10^{-13} cm^{2}/dyn and S_{12}=-0.787 × 10^{-13} cm^{2}/dyn (Table 3) from the experimental data of elastic constants [54].

The results displayed in **Figure 4** for small *X* clearly reveal a linear dependence of the relative change for as compared to non-linear behavior of the hydrostatic pressure *X*>10
GPa dependent shifts for optical modes *ω*^{LO}, *ω*^{LO} (**Figure 3**). This means
that for *X*>10 GPa, the deviation from a linear behavior of pressure
dependent mode frequencies *ω*^{LO}, *ω*^{TO} (**Figure 3**) is caused by the nonlinear
pressure – volume relationship. In other words if *X*^{2} terms from
the right-hand side of Eqs. (24a-b)) are neglected – one would expect
nearly ~5% error in the optical mode frequencies for *X*>10 GPa and less
than 0.5% for *X*<1 GPa. This result is quite significant as the evaluated
hydrostatic pressure component X in all V-CVD grown 3C-SiC/Si (001)
samples are found to be <1 GPa – validateing our elastic deformation
theory of assessing strains in epifilms grown on mismatched substrates.

In summary, we have carried out extensive RSS measurements on
several highly mismatched thin epitaxially grown 3C-SiC films on thick
Si (001) substrates using V-CVD method. The 3C-SiC/Si (001) material
system is perceived as one with a biaxial stress due to differences in
the lattice constants and thermal expansion coefficients. In the V-CVD
approach, while keeping Si/C ratio at ~0.33 we have prepared 3C-SiC
films on Si by varying the growth time between 2 min to 4 h. A
conventional elastic deformation theory is used to derive the necessary
expressions involving stress coefficients – correlating them with
Raman phonon shifts and hydrostatic, uniaxial and biaxial stresses.
For bulk 3C-SiC, we used the existing pressure dependent phonon
[37,42] measurements to estimate the hydrostatic-stress coefficients.
The experimental data of long wavelength optical phonons [LO(Ґ) and
TO(Ґ)] in the near-backscattering geometry is compared for several
“as-grown” 3C-SiC/Si (001) and “free-standing” samples having film thickness ranging between 2.4 μm to 12.8 μm. The analysis of Raman
scattering phonon data has not only helped us appraise the crystalline
quality of films but also facilitated assessing the stresses and strains in
several V-CVD grown 3C-SiC/Si (001) samples of different thickness.
Our estimated results of biaxial stresses (X~0.45-0.87 GPa) as well as
inplane (ε | ~0.1–0.2%) and normal (ε_{ ⊥} ~-0.07 to -0.14%) strains using
elastic deformation theory with elastic constants from Philippe Djemia
provided values in good agreement with data reported in the literature
by different research groups [41,44]. In 3C-SiC/Si epifilms, while the
appraised average value of the biaxial stress (~0.651 GPa) is an order
of magnitude smaller – the strain estimates are found two-order of
magnitudes smaller than the lattice misfits between the bulk 3C-SiC
and Si. Although this result is quite interesting – it provides strong
corroboration to our recent study of impurity vibrational modes based
on average-*t*-matrix Green’s function theory implying that there exists
a high density of intrinsic defects at the 3C-SiC/Si interface which is
possibly responsible for releasing misfit stresses and strains.

The author (DNT) wishes to thank Dr. Deanne Snavely, Dean College of Natural Science and Mathematics at Indiana University of Pennsylvania for the travel support and the Innovation Grant that he received from the School of Graduate Studies making this collaborative research possible. The work at Guangxi University is supported by National Natural Science Foundation of China (NO. 61367004) and Guangxi Key Laboratory for the Relativistic Astrophysics- Guangxi Natural Science Creative Team funding (No. 2013GXNSFFA019001).

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