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Journal of Material Sciences & Engineering
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Assessing Biaxial Stress and Strain in 3C-SiC/Si (001) by Raman Scattering Spectroscopy

Talwar DN1*, Wan L2, Tin CC3 and Feng ZC2

1Department of Physics, Indiana University of Pennsylvania, USA

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

3Department 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.

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Abstract

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.

Keywords

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

Introduction

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 (~1014-1019 cm-3). The ability of SiC to form native silicon dioxide (SiO2) 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 ~109 dyn/cm2. 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.

Experimental: V-CVD Growth of 3C-SiC/Si (001) Epifilms

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.

Raman Scattering Spectroscopy

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 Eqution for observing the lattice modes. Thus, for Si crystal a triply degenerate phonon ωo(Ґ) at the center of the Brillouin zone (i.e., Ґ-point) is allowed while in 3C-SiC material a doubly degenerate TO mode (ωTO(Ґ)) is forbidden and a non-degenerate LO phonon (ωLO(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 Eqution and Eqution 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 ωLO(Ґ), ωTO(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 ωLO(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 ωLO(Ґ), ωTO(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.

material-sciences-engineering-raman-spectra-geometry

Figure 1: Raman spectra in the near back-scattering geometry of 3C-SiC/Si (sample #113 of film thickness 12.8 μm). An Ar+ 488 nm line is used as the excitation source while keeping the laser power level adjusted to 200 mW (a) blue line on Si substrate and (b) redline without Si substrate.

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 ωo(Ґ) Si phonon line near ~520 cm-1 and ωLO(Ґ), ωTO(Ґ) modes of 3C-SiC near ~972 cm-1, ~796 cm-1, respectively. The relative peak intensities and lineshapes of ωo(Ґ) mode for Si substrate and the ωLO(Ґ), ωTO(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 ωo(Ґ) phonon line and a rise of the ωLO(Ґ) mode intensity (Figure 2) with respect to ωTO(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.

material-sciences-engineering-thickness-raman-spectra

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 ωo(Ґ) Si phonon line near ~520 cm-1 and ωLO(Ґ), ωTO(Ґ) modes of 3C-SiC near ~972 cm-1, ~796 cm-1, respectively.

Theoretical Background: Universal Axial Stress and Strain

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:

Eqution (1a)

and Eqution (1b)

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

Eqution (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 Equtionyy(=ε||) and Eqution. gain, one can separate the axial stress into a hydrostatic term X and a uniaxial term P along the z-axis:

Eqution (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 (Sij) and elastic stiffness constants (Cij), respectively as:

Eqution (4a)

Eqution (4b)

Eqution (4c)

Eqution (4d)

Both compliance and stiffness tensors are related [51] through S=C-1 for evaluating compliance coefficients (S11, S12, S44) from the known elastic (C11, C12, C44) 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, νH=-1 and for uniaxial stress along the z-axis Eqution.

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]:

Eqution (5)

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

Eqution (6)

provide two phonon modes given by:

Eqution (7)

one having a doublet

Eqution (8a)

and the other a singlet

Eqution (8b)

Stress induced modes: Hydrostatic case

In a hydrostatic case (P=0), with ao changing to a: Eqution

Eqution (9a)

Eqution (9b)

where Eqution is the bulk modulus.

The Grüneisen constant (hydrostatic stress) γo is:

Eqution (10)

where ω is the mode frequency and Vo crystal volume. By using Eqs. [8(a-b) and 10] one gets:

Eqution (11)

where Eqution

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

Eqution (12)

Eqution

where Eqution is the shear-deformation parameter.

Stress induced modes: Uniaxial case

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

Eqution (14)

one gets:

Eqution (15a)

Eqution (15b)

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

Eqution (15c)

Eqution (15d)

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

Eqution (16a)

Eqution (16b)

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

Eqution (17a)

Eqution (17b)

Eqution (17c)

where we defined Eqution

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

Eqution (18a)

 

Eqution (18b)

where the two stress coefficients

Eqution (19)

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

Eqution (20a)

Eqution (20b)

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

Eqution (21)

from the observed Raman mode frequencies Eqution

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

Eqution (22)

Eqution (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.

Results: Data Analysis

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 Cij 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 Table 2 we have listed the derived values of elastic compliances (S11, S12, S44), S11+2S12, S and Poisson ratio νS, from elastic constants (C11, C12, C44) reported [52-66] by various research groups.

3C-SiC
c11 c12 c44 s11 s12 s44 s11 + 2s12 ŝ νs Ref.
(x*1012 dyn/cm2) (x*10-13 cm2/dyn) (x*10-13 cm2/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 [ωLO(Ґ), ωTO(Ґ)] phonons in bulk 3C-SiC crystals by using RSS with applied X up to 22.5 GPa:

Eqution (24a)

Eqution (24b)

where the mode frequencies ωTO, ωLO are expressed in cm-1 while X is in GPa (1010 dyn/cm2). 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 X2 term in the right hand side of Eqs. (24a-b) is trivial and can be neglected. Therefore, one can determine the hydrostatic Grüneisen parameters Eqution from the linear relationship (Eq. (10)) involving Eqution. For 3C-SiC, the valuesEqution estimated by Olego et al. used an average bulk modulus data Bo (=321.9 GPa) of Si and C (Table 3) [36,37]. A correction was made to the Eqution values by exploiting the experimental bulk modulus Bo (=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 Eqution provided strong corroborations to the amended values. While Feng et al. [41] employed Eqution for assessing strains in 3C-SiC epifilms – one would expect improvement in the strain values if accurate results of Eqution are adopted.

material-sciences-engineering-the-long-wavelength-phonon

Figure 3: The long wavelength ωLO(Ґ), ωTO(Ґ) phonon mode frequencies of 3C-SiC vs. hydrostatic pressure (GPa). The symbols represent experimental data [42] whereas the solid lines correspond to the data fits and dashed lines represent the linear fit to the experimental results.

  ω0 (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

Table 3: Zone-center optical phonon frequencies [TO(Ґ), LO(Ґ)] and their hydrostatic pressure derivatives dω/dX for 3C-SiC. The phonon frequencies ωo are in cm-1 and the hydrostatic pressure X is in GPa. For comparison we have listed the related data for zone-center modes [ωo (cm-1)] of diamond, Si, and Ge crystals. The mode Grüneisen parameters γo, the bulk moduli B (in GPa) and their pressure derivatives B′ employed to calculate Eqution 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 ωLO, ωTO and Eqution in the near-backscattering geometry for several “as-grown” 3C-SiC/Si (001) and “free-standing” films. By using Eqs. [(21) and (22-23)] along with the Raman data of optical phonons and hydrostatic stress coefficients Eqution we obtained two values of X – took their average result and assigned an error bar (Table 4). From Table 4, the uncertainty of the calculated stress X is about 10% which agrees very well with the estimated value from the experimental error of 10% to 20% from the pressure dependent RSS mode frequencies. The results of in-plane and normal strains are evaluated for 3C-SiC films by exploiting Eqs. [4(a)-(b): setting P=-X] along with the elastic compliance data of S11=3.074x10-13 cm2/dyn and S12=-0.787x10-13 cm2/ dyn from Table 2 [54]. The calculated values reported in Table 4 of biaxial stress (X~0.45-0.87 GPa) and inplane (ε | ~0.1-0.2%) strains as well as normal (ε||~-0.07 to -0.14%) strains for several V-CVD grown samples of different thickness are found in good agreement with the results reported in the literature. The positive and negative signs of ε|| and ε indicate that in-plane strains are tensile while the normal strains are compressive, respectively for all 3C-SiC/Si (001) samples. In order to justify the use of elastic deformation theory for 3C-SiC films, we have studied the volume dependent shifts of optical phonons by using a well known Murnaghan’s equation of state with Bo (≡227 GPa) and B'o (≡4.1):

Eqution (25)

Sample # dSiC µm Equtioncm-1 Equtioncm-1 Equtioncm-1 Equtioncm-1 X(GPa) Eqution(%) Eqution(%) 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 S11=3.074 × 10-13 cm2/dyn and S12=-0.787 × 10-13 cm2/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 forEqution as compared to non-linear behavior of the hydrostatic pressure X>10 GPa dependent shifts for optical modes ωLO, ωLO (Figure 3). This means that for X>10 GPa, the deviation from a linear behavior of pressure dependent mode frequencies ωLO, ωTO (Figure 3) is caused by the nonlinear pressure – volume relationship. In other words if X2 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.

material-sciences-engineering-plot-strained-substrate

Figure 4: Plot ofEqution for the strained 3C-SiC grown on Si (001) substrate.

Summary and Conclusion

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.

Acknowledgements

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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