Ab initio DFT study of ideal shear strength of polytypes of silicon carbide
Ab initio density functional calculations are performed to investigate the ideal shear deformation of SiC poly types (3C, 2H, 4H, and 6H). The deformation of the cubic and the hexagonal poly types in small strain region can be well represented by the elastic properties of component Si4C-tetrahedrons...
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Інститут проблем міцності ім. Г.С. Писаренко НАН України
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irk-123456789-482382013-08-20T22:04:53Z Ab initio DFT study of ideal shear strength of polytypes of silicon carbide Umeno, Y. Kinoshita, Y. Kitamura, T. Научно-технический раздел Ab initio density functional calculations are performed to investigate the ideal shear deformation of SiC poly types (3C, 2H, 4H, and 6H). The deformation of the cubic and the hexagonal poly types in small strain region can be well represented by the elastic properties of component Si4C-tetrahedrons. The stacking pattern in the polytypes affects strain localization, which is correlated with the generalized stacking fault (GSF) energy profile of each shuffle-set plane, and the ideal shear strength. Compressive hydrostatic stress decreases the ideal shear strength, which is in contrast with the behavior of metals. Выполнены функциональные расчеты ab initio плотности с целью изучения идеальной сдвиговой деформации политипов SiC (3С, 2Н, 4Н, 6Н). Деформирование кубических и гексагональных политапов в области малых деформаций характеризуется упругими свойствами составляющих тетраэдров, Si4C. Характер укладки в политипах оказывает воздействие на локализацию деформаций (что коррелирует с профилем энергии обобщенного дефекта укладки каждой перемещенной плоскости) и идеальную прочность на сдвиг. Сжимающее гидростатическое напряжение снижает идеальную прочность на сдвиг, что отличает поведение этих материалов от металлов. 2008 Article Ab initio DFT study of ideal shear strength of polytypes of silicon carbide / Y. Umeno, Y. Kinoshita, T. Kitamura // Проблемы прочности. — 2008. — № 1. — С. 8-13. — Бібліогр.: 18 назв. — англ. 0556-171X http://dspace.nbuv.gov.ua/handle/123456789/48238 539. 4 en Проблемы прочности Інститут проблем міцності ім. Г.С. Писаренко НАН України |
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Научно-технический раздел Научно-технический раздел |
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Научно-технический раздел Научно-технический раздел Umeno, Y. Kinoshita, Y. Kitamura, T. Ab initio DFT study of ideal shear strength of polytypes of silicon carbide Проблемы прочности |
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Ab initio density functional calculations are performed to investigate the ideal shear deformation of SiC poly types (3C, 2H, 4H, and 6H). The deformation of the cubic and the hexagonal poly types in small strain region can be well represented by the elastic properties of component Si4C-tetrahedrons. The stacking pattern in the polytypes affects strain localization, which is correlated with the generalized stacking fault (GSF) energy profile of each shuffle-set plane, and the ideal shear strength. Compressive hydrostatic stress decreases the ideal shear strength, which is in contrast with the behavior of metals. |
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Article |
| author |
Umeno, Y. Kinoshita, Y. Kitamura, T. |
| author_facet |
Umeno, Y. Kinoshita, Y. Kitamura, T. |
| author_sort |
Umeno, Y. |
| title |
Ab initio DFT study of ideal shear strength of polytypes of silicon carbide |
| title_short |
Ab initio DFT study of ideal shear strength of polytypes of silicon carbide |
| title_full |
Ab initio DFT study of ideal shear strength of polytypes of silicon carbide |
| title_fullStr |
Ab initio DFT study of ideal shear strength of polytypes of silicon carbide |
| title_full_unstemmed |
Ab initio DFT study of ideal shear strength of polytypes of silicon carbide |
| title_sort |
ab initio dft study of ideal shear strength of polytypes of silicon carbide |
| publisher |
Інститут проблем міцності ім. Г.С. Писаренко НАН України |
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2008 |
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Научно-технический раздел |
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http://dspace.nbuv.gov.ua/handle/123456789/48238 |
| citation_txt |
Ab initio DFT study of ideal shear strength of polytypes of silicon carbide / Y. Umeno, Y. Kinoshita, T. Kitamura // Проблемы прочности. — 2008. — № 1. — С. 8-13. — Бібліогр.: 18 назв. — англ. |
| series |
Проблемы прочности |
| work_keys_str_mv |
AT umenoy abinitiodftstudyofidealshearstrengthofpolytypesofsiliconcarbide AT kinoshitay abinitiodftstudyofidealshearstrengthofpolytypesofsiliconcarbide AT kitamurat abinitiodftstudyofidealshearstrengthofpolytypesofsiliconcarbide |
| first_indexed |
2025-07-04T08:32:44Z |
| last_indexed |
2025-07-04T08:32:44Z |
| _version_ |
1836704559649521664 |
| fulltext |
Scientif ic and Technical
Section
UDC 539. 4
A b I n it io D F T S tu d y o f I d e a l S h e a r S tr e n g th o f P o ly ty p e s o f S il ic o n C a r b id e
Y . U m en o ,1a Y . K in osh ita ,2 b and T . K itam u ra2,c
1 Institute o f Industrial Science, The University of Tokyo, Tokyo, Japan
2 Graduate School o f Engineering, Kyoto University, Kyoto, Japan
a umeno@iis.u-tokyo.ac.jp, b yusuke-kinoshita@t02 .mbox.media.kyoto-u.ac.jp,
c kitamura@me.kyoto-u.ac.jp
Ab initio density functional calculations are performed to investigate the ideal shear deformation o f
SiC poly types (3C, 2H, 4H, and 6H). The deformation o f the cubic and the hexagonal poly types in
small strain region can be well represented by the elastic properties o f component Si4C-
tetrahedrons. The stacking pattern in the polytypes affects strain localization, which is correlated
with the generalized stacking fault (GSF) energy profile o f each shuffle-set plane, and the ideal
shear strength. Compressive hydrostatic stress decreases the ideal shear strength, which is in
contrast with the behavior o f metals.
K e y w o rd s : ideal strength, shear deform ation, ab initio sim ulation, silicon carbide.
In trod u ction . S ilicon carbide (SiC) p ossesses prominent properties such as high
m echanical strength, chem ical stability and large band gap energy, and has been w idely
used as thermal and m echanical functional material, electrom agnetic functional material,
etc. D etailed investigations in atom istic and electronic level are required for SiC crystals
because they have a variety o f polytype structures characterized by stacking sequence [ 1],
w hich contributes to their interesting m echanical properties. Thus, w ith the aim to
elucidate its m echanical deform ation behavior, not on ly experim ental studies but also
theoretical approach such as atom istic m odeling have been carried out [2]. A b initio (first
principles) calculations have also been performed [3 -5 ] to g ive reliable theoretical
insights to the m echanical properties o f SiC around the equilibrium state. H ow ever, the
investigations o f the m echanical properties around h igh ly strained conditions are
indispensable for understanding o f deform ation behavior o f crystals. A lthough ab initio
investigations o f the tensile properties o f 3C (/3)-SiC by L i and W ang [6 ] and o f the shear
by Ogata et al. [7] have brought som e interesting results, this issue deserves further
studies. In particular, it is important to theoretically evaluate the ultimate strength under
ideal shear deform ation o f polytypes, w hich is relevant to the critical shear stress at the
onset o f d islocation nucleation from a pristine crystal, to understand the p lasticity in the
atom istic scale. M oreover, the response o f the ideal shear strength to com pressive stresses
is worth investigating because local lattice configurations m ay receive shear deform ation
in com bination w ith normal stresses in experim ents, nam ely nanoindentation.
Since the m echanical behavior at atom ic scale is strongly correlated w ith the
electronic nature and it is difficult for empirical interatomic potentials to correctly
represent various properties aw ay from the equilibrium state, it is important to study the
m echanical deform ation by atom istic and electronic m odeling, nam ely the ab initio
m ethodology.
© Y. U M EN O , Y. K IN O SH ITA , T. K ITA M U R A , 2008
8 ISSN 0556-171X. Проблемы прочности, 2008, № 1
mailto:umeno@iis.u-tokyo.ac.jp
mailto:yusuke-kinoshita@t02.mbox.media.kyoto-u.ac.jp
mailto:kitamura@me.kyoto-u.ac.jp
Ab Initio DFT Study o f Ideal Shear Strength
In this study, w e perform ab initio calculations based on the density functional
theory (D FT) to investigate the ideal shear deform ation o f SiC polytypes (3C, 2H, 4H,
and 6H) w ith the aim to provide fundamental know ledge about the m echanical properties
o f the crystals including their behavior under h igh ly sheared strain conditions and the
ideal strength, focusing on the effect o f the intrinsic polytype structure on the strain
localization and the ideal strength. We further explore the effect o f hydrostatic pressure on
the ideal strength.
S tru ctu re o f S iC P olytyp es. SiC consists o f tetrahedrons w here vertices are
occupied by silicon atom s w ith carbons located in the center o f gravity. The crystal
p ossesses various structures (SiC polytypes) w ith different stacking sequence, w hich are
denoted in R am sdel’s notation as nX , w here n is the number o f layers along the c-axis
per periodic cycle and X is the identifier o f crystal structure (C: Cubic and H:
H exagonal). Figure 1 depicts the structures o f 3C, 2H, 4H, and 6H polytypes. In this
study, shear deform ation on the c-plane, w hich is ( 111) in cubic structure and (0001 ) in
hexagonal, is studied because it is associated w ith an important slip system o f SiC. A s is
schem atically delineated in Fig. 2, cubic (3C) and hexagonal (2H , 4H, 6H, ...) crystals
have different symmetry in shear deformation due to the stacking structure [8 ]. Concerning
shear on the c-plane, 3 C -S iC has three-fold sym m etry resulting in different geom etrical
configurations betw een shear deform ations in and its opposite direction([121]). On the
other hand, hexagonal polytypes have six-fo ld sym m etry in shear deform ation on (0001 )
plane because their stacking consists o f Si4C-tetrahedrons facing opposite directions. The
shear deform ations in a direction and its opposite (e.g ., [0 1 1 0] and [0 1 10]) are therefore
identical in hexagonal polytypes.
[ 101]
[in]
[2110]
[0001]
Fig. 1. Schematics o f stacking sequence o f SiC polytypes.
S im u lation P rocedure. We performed ab initio DFT calculations based on the
projector augm ented w ave (PAW) m ethod w ithin the framework o f generalized gradient
approxim ation (G G A ) using the V ienna A b Initio Sim ulation Package VASP [9, 10]. The
plane-w ave cu to ff energy w as set to 500 eV and the PW 91-G G A functional [11] was
adopted.
In the setup the x, y , and z axes are in [ 0 1 1 0]([ 1 2 1 ]), [2 1 1 0 ]( 101), and
[0001]([111]), respectively. Shear deform ation under zero and nonzero hydrostatic stress
is sim ulated as follow s: A fter finding equilibrium lattice parameters o f undeform ed
crystals by energy minimization under the hydrostatic stress o h , shear deformation y zx is
applied to each sim ulation ce ll w here atom ic configuration are relaxed until all the forces
are b elow 0.005 eV /A and normal strains o f the ce ll are adjusted so that normal stress
com ponents are w ithin ± 1 0 0 M Pa from predeterm ined o h .
ISSN 0556-171X. npodxeMbi npounocmu, 2008, N 1 9
Y. Umeno, Y. Kinoshita, and T. Kitamura
[1010] [1100]
3C Hexagonal
Fig. 2. Schematics o f symmetry in shear deformation o f 3C and hexagonal SiC crystals.
R esu lts and D iscu ssion . Stress-S tra in R elationship. Figure 3 show s stress-strain
relationships o f the cubic and hexagonal polytypes (3C, 2H, 4H, and 6H) obtained by the
ab initio calculations. The curve o f 3C differs from those o f the hexagonal polytypes
because o f the difference in the stacking structure, w hich is explained in m ore detail in
[8 ]. A lthough the stress-strain relations up to y = 0.2 are alm ost identical betw een the
hexagonal polytypes, the polytype structure affects the deform ation behavior at higher
strains and thus the ideal strength; the m axim um stress o f 2H is the highest and o f 6H the
low est. We find nontrivial effect o f the structure o f polytypes on the ideal strength r is;
i.e ., r is o f 6H (29.83 GPa) is about 10% low er than that o f 2H (32 .97 GPa). This is
ob viously due to the stacking pattern (structure) affecting the m echanical properties,
w hich w ill be d iscussed later on. The ideal strength o f 3C -S iC obtained here, 30.3 GPa,
com pares w ell w ith the value evaluated by the loca l density approxim ation (L D A ) by
Ogata (29.5 GPa [12]).
35
0 0.1 0.2 0.3 0.4 0.5
Shear strain
Fig. 3. Stress-strain curves o f SiC polytypes.
The ideal (theoretical) shear strength can be correlated w ith the critical shear
strength o f d islocation nucleation in a pure crystal. For exam ple, Bahr et al. [13]
demonstrated in their study o f nanoindentation o f tungsten and iron single crystals that the
m axim um shear stress required for d islocation nucleation show s an excellent agreement
w ith the theoretical shear strength. Ohta et al. [14] devised a sophisticated experimental
procedure to evaluate the critical shear stress for dislocation nucleation in silicon, w hich
also com pares w ell w ith the theoretical strength [15]. To the best or our know ledge, there
has been no experim ental w ork extracting the critical shear stress for dislocation
nucleation in SiC , but w e believe that the value w e obtained in this study m ust be a good
prediction. Experimental evaluation o f the critical shear stress for d islocation nucleation
in S iC is h igh ly desirable although it can be dem anding due to the requirement o f special
techniques such as preparation o f specim ens w ith an ideal shape.
10 ISSN 0556-171X. npodxeMbi npounocmu, 2008, N 1
Ab Initio DFT Study o f Ideal Shear Strength
N orm al Strains an d Volume. C hanges in normal strains and volum e o f the SiC
polytypes during shear deform ation are presented in Fig. 4. The hexagonal polytypes
show nearly the sam e evolution o f normal strains w ith increasing shear strain. In SiC(3C)
the evolution o f normal strains is different and changes in £ xx and £ yy are more obvious
than in the hexagonal polytypes; the former decreases and the latter increases as the shear
strain grow s. R elative volum e, V/V0 = (1 + £ xx )(1 + £ yy )(1 + £ ^ ), how ever, changes
sim ilarly both in 3C and the hexagonal polytypes; i.e ., the volum e decreases w ith
increasing shear strain.
0.1
0.05
0 ,
■3
E -0.05
- 0.1
-0.15
by
S im c ;
SiC{2K
SiC(4H)
SiC(6H)
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
Shear strain y7I
1.1
1 . 0 5
§
ci
E
IT 0.95
0.9
S i C ( 3 C ) +
S i C ( 2 H j X
S i C ( 4 H ) 0
. S i C ( 6 H ) A
0.1 0.2 0.3 0.4 0.5
Shear strain y71[
b
Fig. 4. Changes in normal strains (a) and volume during shear o f SiC polytypes (b).
Strain L ocaliza tion . To investigate the deform ation o f each SUC -tetrahedron lattice,
w e now show in Fig. 5a the “bond shear strain,” y b , representing deform ation o f each
atom ic bond as in the schem atic. In the hexagonal polytypes y b differs depending on the
layer, signifying that bonds o f specific layers deform more than the others. This is
analogous to non-uniform deform ation or strain localization in inhom ogeneous materials
(structure). U nlike 3C, inhom ogeneous stacking structure intrinsically existing in the
hexagonal polytypes causes the strain localization, w hich affects the ideal strength.
The strain localization show s deviation from the intuitive picture o f the deform ation
that lattices A, B, and C are ‘softer’ and A ', B ', and C ' are ‘stiffer’. In 6H -S iC , w hile
lattices A and B accom m odate large and alm ost identical bond shear strain, lattice C show s
a sm all deform ation and its bond shear strain is even sm aller than that o f C '. This im plies
that the deform ability o f the bond is affected b y the stacking discontinuity betw een
lattices C and B ' (C ' and A); i.e ., lattice C is stiffened by the overlaying lattice B ' (and
similarly, C ' is softened by A). This effect is seen in the other hexagonal polytypes as w ell.
The difference in deform ation among the layers can be explained by the generalized
stacking fault (G SF) energy o f the shuffle-set layers show n in Fig. 5b. Here, the lattice
over a shuffle-set plane is rigidly shifted along the x direction w ithout atomic relaxation
w hile the lattice b elow is fixed, and the energy increase as a function o f the rigid shift is
evaluated (see the schem atic in the figure).
The profile o f GSF energy depends on the layer, m eaning that the bond in each layer
has different “deform ability.” The layer show ing low er peak in 0 < x s < 1 has a higher
deformability, w hich corresponds to strain localization found in Fig. 5a. The GSF energy
profile supports the above-m entioned hypothesis o f the m echanism that the deform ability
o f the bond in question is affected by the stacking discontinuity. The GSF energy
landscape can be a profile representing the deform ability o f each bond subject to shear
although it does not incorporate the effect o f atom istic relaxation.
ISSN 0556-171X. npodxeMbi npounocmu, 2008, N 1 11
Y. Umeno, Y. Kinoshita, and T. Kitamura
Normalized shift displacement x:;
b
a
Fig. 5. (a) Evolution o f bond shear strain, y b. A, B, C and those with a prime denote the
tetrahedrons as depicted in Fig. 1. (b) GSF energy landscape o f 2H and 4H with a shuffle set being
rigidly shifted along the x direction. The abscissa is the shift displacement normalized with respect
to the lattice width, xs = xdj X .
E ffect o f P ressure. Figure 6 compares the ideal shear strengths o f the polytypes
under zero and nonzero hydrostatic com pression. In the figure, both the abscissa and the
ordinate are norm alized b y r 0s , the ideal shear strength under no com pression.
Hydrostatic com pression significantly decreases the ideal strength in all the hexagonal
polytypes studied here. The response o f the ideal shear strength to com pression can be
explained b y the volum e change during shear; i.e ., the system s contract as the shear strain
grow s and the com pressive normal stress helps the shear deform ation. This phenom enon
is in contrast w ith the properties o f m etals, where, in general, the ideal shear strength
increases under com pressive pressure [16, 17].
1
0.S8ojl
^ 0-06
w
0.94
■ffi
gg 0.92
CD
| 0.9
CO0
1 0.88o
ffi 0.86
N
| 0.84
z 0.82
0,8
-1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0
Normalized hydrostatic stress oh/T?
Fig. 6 . Ideal shear strength as a function hydrostatic stress. Both abscissa and ordinate are
normalized by r is (the ideal shear strength at Oh = 0).
12 ISSN 0556-171X. npo6neMbi npoHuocmu, 2008, № 1
Ab Initio DFT Study o f Ideal Shear Strength
It w as demonstrated by Krenn et al. [18] that the effect o f com pressive stress to the
shear strength is important in the interpretation o f the critical stress for plastic deform ation
found in nanoindentation experim ents, because the contribution o f normal stress
com ponents can change the ideal strength in shear. Therefore, our finding here is crucial
as it show s that the effect o f com pressive stress to the shear strength o f covalent system
can be different even qualitatively from that o f m etals. A s it has been pointed out, the
relation betw een shear and normal stresses exhibits a strong anisotropic character [16] and
dependence on atom species [17]. Further extensive studies for various crystals and stress
conditions w ill be necessary to elucidate its m echanism .
C on clusions. We have investigated the ideal shear deform ation o f SiC polytypes
(3C, 2H, 4H and 6H) by m eans o f ab initio DFT calculations based on the generalized
gradient approximation. The variety o f the stacking pattern in the polytypes causes strain
localization, w hich is correlated w ith the GSF energy profile o f each shuffle-set plane, and
difference in the ideal shear strength. We also exam ined the effect o f hydrostatic
com pression to the shear deform ation to reveal that the com pressive stress decreases the
ideal shear strength in all the polytypes studied here, w hich is in contrast to m etals, where
in general the ideal shear strength is increased by com pression. More extensive studies
w ill be required to elucidate the m echanism o f the effect o f the normal stress because it
can be h ighly anisotropic and susceptible to the interatomic bonds o f the atom species
Acknowledgments. One o f the authors (Y.U.) acknowledges financial support from the Grant-in-Aid
for Scientific Research of Japan Society o f the Promotion of Science (JSPS, No. 1876008).
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Received 28. 06. 2007
ISSN 0556-171X. npo6neMbi npouHocmu, 2008, № 1 13
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