The tensile test in transition metal disilicides with C11b structure is simulated by ab initio electronic structure calculations using full potential linearized augmented plane wave method (FLAPW). Full relaxation of both external and internal parameters is performed. The theoretical tensile strength of MoSi2 and WSi2 for [001] loading is determined and compared with those of other materials

Friák, M.

Vitek, Vaclav

Šob, M.

English

ScholarlyCommons@Penn

University of PennsylvaniaScholarlyCommonsDepartmental Papers (MSE) Department of Materials Science & EngineeringNovember 2000Ab initio simulation of a tensile test in MoSi2 andWSi2M. FriákAcademy of Sciences of the Czech RepublicM. ŠobMasaryk UniversityVaclav VitekUniversity of Pennsylvania, vitek@lrsm.upenn.eduFollow this and additional works at: http://repository.upenn.edu/mse_papersCopyright Materials Research Society. Reprinted from MRS Proceedings Volume 646.2000 Fall Meeting Symposium NHigh-Temperature Ordered Intermetallic Alloys IXPublisher URL: http://www.mrs.org/members/proceedings/fall2000/n/N4_8.pdfThis paper is posted at ScholarlyCommons. http://repository.upenn.edu/mse_papers/10For more information, please contact libraryrepository@pobox.upenn.edu.Recommended CitationFriák, M., Šob, M., & Vitek, V. (2000). Ab initio simulation of a tensile test in MoSi2 and WSi2. Retrieved fromhttp://repository.upenn.edu/mse_papers/10Ab initio simulation of a tensile test in MoSi2 and WSi2AbstractThe tensile test in transition metal disilicides with C11b structure is simulated by ab initio electronic structurecalculations using full potential linearized augmented plane wave method (FLAPW). Full relaxation of bothexternal and internal parameters is performed. The theoretical tensile strength of MoSi2 and WSi2 for [001]loading is determined and compared with those of other materials.CommentsCopyright Materials Research Society. Reprinted from MRS Proceedings Volume 646.2000 Fall Meeting Symposium NHigh-Temperature Ordered Intermetallic Alloys IXPublisher URL: http://www.mrs.org/members/proceedings/fall2000/n/N4_8.pdfThis conference paper is available at ScholarlyCommons: http://repository.upenn.edu/mse_papers/101Ab initio simulation of a tensile test in MoSi2and WSi2M. Friaky, M.Sob, and V. VitekzInstitute of Physics of Materials, Academy of Sciences of the Czech Republic,Zizkova 22,CZ-616 62 Brno, Czech Republic, mafri@ipm.czyDepartment of Solid State Physics, Faculty of Science, Masaryk University, Kotlarska 2,CZ-611 37 Brno, Czech RepubliczDepartment of Materials Science and Engineering, University of Pennsylvania,3231 Walnut St., Philadelphia, PA 19104-6272, U. S. A.ABSTRACTThe tensile test in transition metal disilicides with C11bstructure is simulated by abinitio electronic structure calculations using full potential linearized augmented plane wavemethod (FLAPW). Full relaxation of both external and internal parameters is performed.The theoretical tensile strength of MoSi2and WSi2for [001] loading is determined andcompared with those of other materials.INTRODUCTIONTransition metal (TM) silicides are considered as a very promising basis for a newgeneration of high-temperature structural materials that can signicantly improve thethermal eÆciency of energy conversion systems and advanced engines. The reason is that athigh temperature they combine the ductility and thermal conductivity of metals with highstrength and corrosion resistance of ceramics. Intrinsic oxidation resistance is due to theformation of silicon oxide lms at surfaces and high creep strength is related to low diusioncoeÆcients. The melting temperature is much higher than that of Ni-based superalloys orNi and Ti-based aluminides and is comparable to that of silicon-based ceramics. The largestimpediment is low ductility and/or toughness at ambient temperatures.The purpose of this paper is to investigate, from rst principles, the electronic structureand ground state of TM-disilicides, MoSi2and WSi2, with C11bstructure, to simulate atensile test for ideal crystal without defects, including full relaxation of both external andinternal parameters, and to determine thus the theoretical tensile strength for the [001]loading.TENSILE TEST SIMULATIONThe tensile strength of materials is usually limited by presence of internal defects, mostlydislocations. In a defect-free crystal, the tensile strength is several order of magnitude higherand is comparable with elastic moduli. Most of the calculations of theoretical (ideal) strengthis based on empirical potentials with the parameters adjusted to experimental data. However,most of these experimental data correspond to the equilibrium ground state. Therefore, theMat. Res. Soc. Symp. Vol. 646 © 2001 Materials Research SocietyN4.8.12semiempirical approaches adapted to the equilibrium state may not be valid for materialsloaded close to their theoretical strength limits.In the rst-principles (ab initio) electronic structure calculations, we start from thefundamental quantum theory. The only input is atomic numbers of the constituent atomsand, usually, some structural information. This approach is reliable even for highly non-equilibrium states.To simulate the tensile test, we rst calculate the total energy of the material in theground state. Then, in the second step, we apply some elongation of the crystal along theloading axis (in the [001] direction in the present case; the loading axis is denoted as 3) bya xed amount "3that is equivalent to the application of a tensile stress 3. Subsequently,we fully relax both the stresses 1and 2in the directions perpendicular to the axis 3 aswell as the internal degrees of freedom. In this way, we nd the contractions "1and "2whichcorrespond to zero tensile stresses 1and 2and the new values of internal parameters. TheC11bstructure is tetragonal and keeps its tetragonal symmetry during the tensile test alongthe [001] axis. Therefore, 1= 2and we minimize the total energy as a function of latticeparameter a and internal parameter  (dened e.g. in [1, 2], see also Fig. 2).The tensile stress 3is then given by3=2c0@E@"31a2;where E is the total energy per basis (i.e. one TM atom and two Si atoms) and c0is thelattice parameter in the direction of loading in the ground state. The inexion point in thetotal energy dependence yields the maximum of the tensile stress; if some other instabilitydoes not occur before reaching the inexion point, it also corresponds to the theoreticaltensile strength.DETAILS OF THE CALCULATIONSIn order to obtain reliable ab initio total energies of materials during tensile testsimulations, the methods using a shape approximation of the crystal potential (for examplespheroidization, as in the LMTO-ASA method or in standard KKR and APW approaches)are not adequate [3, 4]. Instead, full-potential treatments must be employed. In this studywe utilized the full-potential linearized augmented plane waves (FLAPW) code described indetail in [5]. The electronic structure calculations were performed self-consistently withinthe local density approximation (LDA).When simulating the tensile test, crystal lattices are severely distorted and some atomsmay move very close together. Therefore, the muÆn-tin radii must be suÆciently small toguarantee non-overlapping of the muÆn-tin spheres at every stage of the test. We use themuÆn-tin radii equal to 2.3 a.u. for transition metal atoms and 2.1 a.u. for silicon. Theseare kept constant in all calculations presented here. The product of muÆn-tin radius andthe maximum reciprocal space vector, RMTkmax, is equal to 10, the maximum l value forthe waves inside the atomic sphere, lmax, and the largest reciprocal vector G in the Fourierexpansion of the charge, Gmax, are set to 12 and 15, respectively, and the number of k-pointsin the rst Brillouin zone is equal to 2000.N4.8.23RESULTS AND DISCUSSIONFirst we determined the ground state properties for MoSi2and WSi2. The values oflattice parameters, a and c, internal parameter , and ratio of calculated ground-statevolume over the experimental one, V=Vexp(Ref. [2]), are summarized and compared withother calculations and experiments in Table 1. Our results are in very good agreement withboth experimental and previous theoretical data.parameters of MoSi2a c  V=Vexp[2]FLAPW-this work 6.004 14.689 0.0021 0.974exp. Ref. [2] 6.051 14.836 0.0019 1.000exp. Ref. [6] 6.059 14.830 0.0020 1.002exp. Ref. [7] 6.047 14.855 { 0.999FLAPW Ref. [1] 6.089 14.897 0.0022 1.017LMTO-FP Ref. [8] 6.021 14.740 { 0.984LMTO-ASA Ref. [9] 6.026 14.780 { 0.988parameters of WSi2FLAPW-this work 6.020 14.721 0.0013 0.975exp. Ref. [2] 6.068 14.870 0.0014 1.000exp. Ref. [7] 6.070 14.891 { 1.010FLAPW Ref. [1] 6.104 14.866 0.0018 1.020Table 1: Ground-state lattice parameters of MoSi2and WSi2.The theoretical tensile strengths are summarized in Table 2. They are comparable withthe tensile strengths predicted for, e.g., NiAl [10] or W [11]. Unfortunately, to the bestof our knowledge there are no measurements of ideal tensile strength of MoSi2and WSi2(whiskers, nanoindentation) and, therefore, we were not able to compare our results withthe experimental ones.material MoSi2WSi2NiAl [10] W [11]structure C11bC11bB2 A2th(GPa) 37 38 46 29"30.18 0.18 0.21 0.12Table 2: Theoretical tensile strengths thof MoSi2and WSi2and the corresponding "3compared with those of other materials.In Figure 1, we display the dependences of total energy E, internal parameter , tensilestress 3and the lattice parameter a on "3in MoSi2. The total energy has a parabolic shapearound the minimum; it becomes almost at in the neighbourhood of the inexion pointcorresponding to the maximum of tensile stress. During the deformation the value of latticeconstant perpendicular to the loading axis decreases nearly monotonously. On the otherhand, the internal parameter increases with increasing "3except for small neighbourhood ofN4.8.34the ground state where it is almost constant. Let us note that the tendency of  to increasewith "3is in agreement with the interpretation of recent experimental data [12].00.010.020 0.1 0.2"3eeeeeeeeeeeeeee5.75.85.96.06.10 0.1 0.2a(a.u.)"3eeeeeeeeeeeeeee020400 0.1 0.23(GPa)"3eeeeeeeeeeeeeerr00.51.01.50 0.1 0.2E E0(eV=basis)"3eeeeeeeeeeeeeeerrFigure 1: Variations of total energy E per basis (one Mo and two Si atoms), internalparameter , tensile stress 3and lattice parameter a during the simulation of the tensiletest in MoSi2. Here E0is the ground-state energy and "3is the strain in the [001] direction.The position of the inexion point in the energy dependence and the maximum of the tensilestress are denoted by a dashed line.In Figure 2 we remind the denition of the internal parameter  [1, 2] and show fourdierent types of bonds between constituent atoms of the repeat cell in MoSi2. It wasproposed by Tanaka et al. [2] that the Mo-Si bonds along the [001] direction (thin dash-dotted lines in Fig. 2) are weaker than the Mo-Si bonds in the other directions (thick dashedlines in Fig. 2) and, on the contrary, Si-Si bonds along the [001] direction (thick full linesin Fig. 2) are stronger than those in the other directions (thin dashed lines in Fig. 2).Our calculations conrm this suggestion regarding the Mo-Si bonds. In the C11bstructure,N4.8.45four Mo atoms in the (001) plane form a square of side a which constitutes the basis of abipyramid completed by two Si atoms above and below the center of this square. It seemsthat the atoms forming these bipyramids try to keep together during the tensile test in the[001] direction. Namely, the edges of the bipyramid are the Mo-Si bonds the length of whichis nearly constant during the test even when exceeding the theoretical tensile strength (fullcircles in the left-hand part of Fig. 2). In agreement with Ref. [2], they can be denotedas "strong" bonds. The Si-Si bonds in the [001] direction, constituting the height of thebipyramid, are elongated under tension (open triangles in Fig. 2); this is related to thedecrease of the lattice parameter a (Fig. 1). However, the Si-Si bonds in the other directionsare extended approximately in the same way (open circles in Fig. 2).The behavior of the Si-Si bonds is somewhat dierent under compression. The lengthof the non-[001] Si-Si bonds is nearly constant whereas the length of the [001] Si-Si bondschanges signicantly, similarly as the length of the [001] Mo-Si bonds. Therefore, we candistinguish the "strong" and "weak" Mo-Si bonds, but it is not possible to introduce the"strong" and "weak" Si-Si bonds. The situation in WSi2is similar.Let us note here that the relaxation of the internal parameter  during the tensile test iscrucial. If  were kept constant the [001] Si-Si and Mo-Si bonds would behave very similarly.The same would be true for non-[001] Si-Si and Mo-Si bonds.4.64.855.25.45.65.866.26.46.6-0.05 0 0.05 0.1 0.15 0.2 0.25 0.3l(a.u.)"3||44444444444444444{ { {eeeeeeeeeeeeeeeeee{ { {uuuuuuuuuu uuuuuuuu{  {??????????????????SiMo1/3+Figure 2: Variations of the length of atomic bonds during the tensile test simulation inMoSi2. The right-hand side of the gure displays the (110) plane in MoSi2with the C11bstructure; the Si atoms are represented by small circles and Mo atoms by large circles. Bondsbetween Mo-Si and Si-Si atoms are shown as thick dashed and thin dash-dotted lines (the"strong" and "weak" Mo-Si bonds) and thick full and thin dashed lines (the [001] and theother Si-Si bonds). The internal parameter  is dened as deviation from the ideal value of1/3 [1, 2].N4.8.56CONCLUSIONSWe simulated the tensile test in ideal MoSi2and WSi2loaded along the [001] axis usingthe rst-principles full-potential electronic structure calculations and determined theoreticaltensile strength of those materials. The analysis of bond lengths variation under an uniaxialstress shows that, in accordance with Ref. [2], it is possible to distinguish "strong" and"weak" TM-Si bonds but the behavior of Si-Si bonds is more complex and exhibits a tension-compression asymmetry.ACKNOWLEDGEMENTSThis research was supported by the Ministry of Education of the Czech Republic (ProjectNo. ME-264), by the National Science Foundation{International Programs (Grant No. INT-96-05232), by the Grant Agency of the Academy of Sciences of the Czech Republic (ProjectNo. A1010817), by the Grant Agency of the Czech Republic (Project No. 106/99/1178),and by the U.S. Department of Energy, Basic Energy Sciences (Grant No. DE-FG02-98ER45702). A part of this study has been performed in the framework of the COSTAction P3 (Project No. OC P3.10). The use of the computer facility at the MetaCenterof the Masaryk University, Brno, and at the Boston University Scientic Computing andVisualization Center is acknowledged.REFERENCES1. L. F. Mattheiss, Phys. Rev. B, 45, 3252 (1992).2. K. Tanaka, K. Nawata, K. Yamamoto, H. Inui, M. Yamaguchi and M. Koiwa, in Proc.of the U.S.-Japan Workshop on Very High Temperature Structural Materials (1999), p.67.3. O.K. Andersen, M. Methfessel, C.O. Rodriguez, P. Blochl, and H.M. Polatoglou, inAtomistic Simulations of Materials: Beyond Pair Potentials, edited by V. Vitek andD.J. Srolovitz (Plenum, New York-London, 1989), p. 1.4. M.Sob, L.G. Wang, and V. Vitek, Comp. Mat. Sci., 8, 100 (1997).5. P. Blaha, K. Schwarz, and J. Luitz, WIEN97, Technical University of Vienna 1997(improved and updated Unix version of the original copyrighted WIEN-code, whichwas published by P. Blaha, K. Schwarz, P. Sorantin, and S.B. Trickey, Comput. Phys.Commun., 59, 399 (1990).6. Y. Harada, M. Morinaga, D. Saso, M. Takata and M. Sakata, Intermetallics, 6, 523(1998).7. M.-A. Nicolet and S.S. Lau, in VLSI Electronics: Microstructure Science, ed. N.G.Einspruch and G.B. Larrabee (Academic, New York, 1983), Vol. 6, p. 329.8. M. Alouani, R. C. Albers, and M. Methfessel, Phys. Rev. B, 43, 6500 (1991).9. S. Tang, K. Zhang, and X. Xie, J. Phys. C, 21, L777 (1988).10. M.Sob, L.G. Wang, and V. Vitek, Phil. Mag. B, 78, 653 (1998).11. M.Sob, L.G. Wang, and V. Vitek, Mat. Sci. Eng. A, 234-236, 1075 (1997).12. K. Tanaka et al., this Proceedings.N4.8.6

Ab initio simulation of a tensile test in MoSi\u3csub\u3e2\u3c/sub\u3e and WSi\u3csub\u3e2\u3c/sub\u3e

Kosmopolis

University of Pennsylvania
ScholarlyCommons
Departmental Papers (MSE) Department of Materials Science & Engineering
November 2000
Ab initio simulation of a tensile test in MoSi2 and
WSi2
M. Friák
Academy of Sciences of the Czech Republic
M. Šob
Masaryk University
Vaclav Vitek
University of Pennsylvania, vitek@lrsm.upenn.edu
Follow this and additional works at: http://repository.upenn.edu/mse_papers
Copyright Materials Research Society. Reprinted from MRS Proceedings Volume 646.
2000 Fall Meeting Symposium N
High-Temperature Ordered Intermetallic Alloys IX
Publisher URL: http://www.mrs.org/members/proceedings/fall2000/n/N4_8.pdf
This paper is posted at ScholarlyCommons. http://repository.upenn.edu/mse_papers/10
For more information, please contact libraryrepository@pobox.upenn.edu.
Recommended Citation
Friák, M., Šob, M., & Vitek, V. (2000). Ab initio simulation of a tensile test in MoSi2 and WSi2. Retrieved from
http://repository.upenn.edu/mse_papers/10
Ab initio simulation of a tensile test in MoSi2 and WSi2
Abstract
The tensile test in transition metal disilicides with C11b structure is simulated by ab initio electronic structure
calculations using full potential linearized augmented plane wave method (FLAPW). Full relaxation of both
external and internal parameters is performed. The theoretical tensile strength of MoSi2 and WSi2 for [001]
loading is determined and compared with those of other materials.
Comments
Copyright Materials Research Society. Reprinted from MRS Proceedings Volume 646.
2000 Fall Meeting Symposium N
High-Temperature Ordered Intermetallic Alloys IX
Publisher URL: http://www.mrs.org/members/proceedings/fall2000/n/N4_8.pdf
This conference paper is available at ScholarlyCommons: http://repository.upenn.edu/mse_papers/10
1
Ab initio simulation of a tensile test in MoSi2 and WSi2
M. Friaky, M. Sob, and V. Vitekz
Institute of Physics of Materials, Academy of Sciences of the Czech Republic, Zizkova 22,
CZ-616 62 Brno, Czech Republic, mafri@ipm.cz
yDepartment of Solid State Physics, Faculty of Science, Masaryk University, Kotlarska 2,
CZ-611 37 Brno, Czech Republic
zDepartment of Materials Science and Engineering, University of Pennsylvania,
3231 Walnut St., Philadelphia, PA 19104-6272, U. S. A.
ABSTRACT
The tensile test in transition metal disilicides with C11b structure is simulated by ab
initio electronic structure calculations using full potential linearized augmented plane wave
method (FLAPW). Full relaxation of both external and internal parameters is performed.
The theoretical tensile strength of MoSi2 and WSi2 for [001] loading is determined and
compared with those of other materials.
INTRODUCTION
Transition metal (TM) silicides are considered as a very promising basis for a new
generation of high-temperature structural materials that can signicantly improve the
thermal eÆciency of energy conversion systems and advanced engines. The reason is that at
high temperature they combine the ductility and thermal conductivity of metals with high
strength and corrosion resistance of ceramics. Intrinsic oxidation resistance is due to the
formation of silicon oxide lms at surfaces and high creep strength is related to low diusion
coeÆcients. The melting temperature is much higher than that of Ni-based superalloys or
Ni and Ti-based aluminides and is comparable to that of silicon-based ceramics. The largest
impediment is low ductility and/or toughness at ambient temperatures.
The purpose of this paper is to investigate, from rst principles, the electronic structure
and ground state of TM-disilicides, MoSi2 and WSi2, with C11b structure, to simulate a
tensile test for ideal crystal without defects, including full relaxation of both external and
internal parameters, and to determine thus the theoretical tensile strength for the [001]
loading.
TENSILE TEST SIMULATION
The tensile strength of materials is usually limited by presence of internal defects, mostly
dislocations. In a defect-free crystal, the tensile strength is several order of magnitude higher
and is comparable with elastic moduli. Most of the calculations of theoretical (ideal) strength
is based on empirical potentials with the parameters adjusted to experimental data. However,
most of these experimental data correspond to the equilibrium ground state. Therefore, the
Mat. Res. Soc. Symp. Vol. 646 © 2001 Materials Research Society
N4.8.1
2
semiempirical approaches adapted to the equilibrium state may not be valid for materials
loaded close to their theoretical strength limits.
In the rst-principles (ab initio) electronic structure calculations, we start from the
fundamental quantum theory. The only input is atomic numbers of the constituent atoms
and, usually, some structural information. This approach is reliable even for highly non-
equilibrium states.
To simulate the tensile test, we rst calculate the total energy of the material in the
ground state. Then, in the second step, we apply some elongation of the crystal along the
loading axis (in the [001] direction in the present case; the loading axis is denoted as 3) by
a xed amount "3 that is equivalent to the application of a tensile stress 3. Subsequently,
we fully relax both the stresses 1 and 2 in the directions perpendicular to the axis 3 as
well as the internal degrees of freedom. In this way, we nd the contractions "1 and "2 which
correspond to zero tensile stresses 1 and 2 and the new values of internal parameters. The
C11b structure is tetragonal and keeps its tetragonal symmetry during the tensile test along
the [001] axis. Therefore, 1 = 2 and we minimize the total energy as a function of lattice
parameter a and internal parameter  (dened e.g. in [1, 2], see also Fig. 2).
The tensile stress 3 is then given by
3 =
2
c0
@E
@"3
1
a2
;
where E is the total energy per basis (i.e. one TM atom and two Si atoms) and c0 is the
lattice parameter in the direction of loading in the ground state. The inexion point in the
total energy dependence yields the maximum of the tensile stress; if some other instability
does not occur before reaching the inexion point, it also corresponds to the theoretical
tensile strength.
DETAILS OF THE CALCULATIONS
In order to obtain reliable ab initio total energies of materials during tensile test
simulations, the methods using a shape approximation of the crystal potential (for example
spheroidization, as in the LMTO-ASA method or in standard KKR and APW approaches)
are not adequate [3, 4]. Instead, full-potential treatments must be employed. In this study
we utilized the full-potential linearized augmented plane waves (FLAPW) code described in
detail in [5]. The electronic structure calculations were performed self-consistently within
the local density approximation (LDA).
When simulating the tensile test, crystal lattices are severely distorted and some atoms
may move very close together. Therefore, the muÆn-tin radii must be suÆciently small to
guarantee non-overlapping of the muÆn-tin spheres at every stage of the test. We use the
muÆn-tin radii equal to 2.3 a.u. for transition metal atoms and 2.1 a.u. for silicon. These
are kept constant in all calculations presented here. The product of muÆn-tin radius and
the maximum reciprocal space vector, RMTkmax, is equal to 10, the maximum l value for
the waves inside the atomic sphere, lmax, and the largest reciprocal vector G in the Fourier
expansion of the charge, Gmax, are set to 12 and 15, respectively, and the number of k-points
in the rst Brillouin zone is equal to 2000.
N4.8.2
3
RESULTS AND DISCUSSION
First we determined the ground state properties for MoSi2 and WSi2. The values of
lattice parameters, a and c, internal parameter , and ratio of calculated ground-state
volume over the experimental one, V=Vexp (Ref. [2]), are summarized and compared with
other calculations and experiments in Table 1. Our results are in very good agreement with
both experimental and previous theoretical data.
parameters of MoSi2
a c  V=Vexp [2]
FLAPW-this work 6.004 14.689 0.0021 0.974
exp. Ref. [2] 6.051 14.836 0.0019 1.000
exp. Ref. [6] 6.059 14.830 0.0020 1.002
exp. Ref. [7] 6.047 14.855 { 0.999
FLAPW Ref. [1] 6.089 14.897 0.0022 1.017
LMTO-FP Ref. [8] 6.021 14.740 { 0.984
LMTO-ASA Ref. [9] 6.026 14.780 { 0.988
parameters of WSi2
FLAPW-this work 6.020 14.721 0.0013 0.975
exp. Ref. [2] 6.068 14.870 0.0014 1.000
exp. Ref. [7] 6.070 14.891 { 1.010
FLAPW Ref. [1] 6.104 14.866 0.0018 1.020
Table 1: Ground-state lattice parameters of MoSi2 and WSi2.
The theoretical tensile strengths are summarized in Table 2. They are comparable with
the tensile strengths predicted for, e.g., NiAl [10] or W [11]. Unfortunately, to the best
of our knowledge there are no measurements of ideal tensile strength of MoSi2 and WSi2
(whiskers, nanoindentation) and, therefore, we were not able to compare our results with
the experimental ones.
material MoSi2 WSi2 NiAl [10] W [11]
structure C11b C11b B2 A2
th (GPa) 37 38 46 29
"3 0.18 0.18 0.21 0.12
Table 2: Theoretical tensile strengths th of MoSi2 and WSi2 and the corresponding "3
compared with those of other materials.
In Figure 1, we display the dependences of total energy E, internal parameter , tensile
stress 3 and the lattice parameter a on "3 in MoSi2. The total energy has a parabolic shape
around the minimum; it becomes almost at in the neighbourhood of the inexion point
corresponding to the maximum of tensile stress. During the deformation the value of lattice
constant perpendicular to the loading axis decreases nearly monotonously. On the other
hand, the internal parameter increases with increasing "3 except for small neighbourhood of
N4.8.3
4
the ground state where it is almost constant. Let us note that the tendency of  to increase
with "3 is in agreement with the interpretation of recent experimental data [12].
0
0.01
0.02
0 0.1 0.2

"3
e e ee
ee
e
e
e
e
e
e
e
e
e
5.7
5.8
5.9
6.0
6.1
0 0.1 0.2
a
(a
.u
.)
"3
e
e
e
ee
e
e
e e
e
e
e
e
e e0
20
40
0 0.1 0.2

3
(G
P
a
)
"3
e
e
e
e
e
e
e
e
e
e e
e e
e
r
r
0
0.5
1.0
1.5
0 0.1 0.2
E
 
E
0
(e
V
=b
a
si
s)
"3
e
e eee
e
e
e
e
e
e
e
e
e
e
r
r
Figure 1: Variations of total energy E per basis (one Mo and two Si atoms), internal
parameter , tensile stress 3 and lattice parameter a during the simulation of the tensile
test in MoSi2. Here E0 is the ground-state energy and "3 is the strain in the [001] direction.
The position of the inexion point in the energy dependence and the maximum of the tensile
stress are denoted by a dashed line.
In Figure 2 we remind the denition of the internal parameter  [1, 2] and show four
dierent types of bonds between constituent atoms of the repeat cell in MoSi2. It was
proposed by Tanaka et al. [2] that the Mo-Si bonds along the [001] direction (thin dash-
dotted lines in Fig. 2) are weaker than the Mo-Si bonds in the other directions (thick dashed
lines in Fig. 2) and, on the contrary, Si-Si bonds along the [001] direction (thick full lines
in Fig. 2) are stronger than those in the other directions (thin dashed lines in Fig. 2).
Our calculations conrm this suggestion regarding the Mo-Si bonds. In the C11b structure,
N4.8.4
5
four Mo atoms in the (001) plane form a square of side a which constitutes the basis of a
bipyramid completed by two Si atoms above and below the center of this square. It seems
that the atoms forming these bipyramids try to keep together during the tensile test in the
[001] direction. Namely, the edges of the bipyramid are the Mo-Si bonds the length of which
is nearly constant during the test even when exceeding the theoretical tensile strength (full
circles in the left-hand part of Fig. 2). In agreement with Ref. [2], they can be denoted
as "strong" bonds. The Si-Si bonds in the [001] direction, constituting the height of the
bipyramid, are elongated under tension (open triangles in Fig. 2); this is related to the
decrease of the lattice parameter a (Fig. 1). However, the Si-Si bonds in the other directions
are extended approximately in the same way (open circles in Fig. 2).
The behavior of the Si-Si bonds is somewhat dierent under compression. The length
of the non-[001] Si-Si bonds is nearly constant whereas the length of the [001] Si-Si bonds
changes signicantly, similarly as the length of the [001] Mo-Si bonds. Therefore, we can
distinguish the "strong" and "weak" Mo-Si bonds, but it is not possible to introduce the
"strong" and "weak" Si-Si bonds. The situation in WSi2 is similar.
Let us note here that the relaxation of the internal parameter  during the tensile test is
crucial. If  were kept constant the [001] Si-Si and Mo-Si bonds would behave very similarly.
The same would be true for non-[001] Si-Si and Mo-Si bonds.
4.6
4.8
5
5.2
5.4
5.6
5.8
6
6.2
6.4
6.6
-0.05 0 0.05 0.1 0.15 0.2 0.25 0.3
l
(a
.u
.)
"3
||
4
4
4
4
4
4
4
4
44
444
4
44
4
{ { {
e e ee
ee e
e
e
e
e
e
e
e
e e
e
e
{ { {
u u uuuu u
u u u u u u u u u u
u
{  {
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
Si
Mo
1
/3
+

Figure 2: Variations of the length of atomic bonds during the tensile test simulation in
MoSi2. The right-hand side of the gure displays the (110) plane in MoSi2 with the C11b
structure; the Si atoms are represented by small circles and Mo atoms by large circles. Bonds
between Mo-Si and Si-Si atoms are shown as thick dashed and thin dash-dotted lines (the
"strong" and "weak" Mo-Si bonds) and thick full and thin dashed lines (the [001] and the
other Si-Si bonds). The internal parameter  is dened as deviation from the ideal value of
1/3 [1, 2].
N4.8.5
6
CONCLUSIONS
We simulated the tensile test in ideal MoSi2 and WSi2 loaded along the [001] axis using
the rst-principles full-potential electronic structure calculations and determined theoretical
tensile strength of those materials. The analysis of bond lengths variation under an uniaxial
stress shows that, in accordance with Ref. [2], it is possible to distinguish "strong" and
"weak" TM-Si bonds but the behavior of Si-Si bonds is more complex and exhibits a tension-
compression asymmetry.
ACKNOWLEDGEMENTS
This research was supported by the Ministry of Education of the Czech Republic (Project
No. ME-264), by the National Science Foundation{International Programs (Grant No. INT-
96-05232), by the Grant Agency of the Academy of Sciences of the Czech Republic (Project
No. A1010817), by the Grant Agency of the Czech Republic (Project No. 106/99/1178),
and by the U.S. Department of Energy, Basic Energy Sciences (Grant No. DE-FG02-
98ER45702). A part of this study has been performed in the framework of the COST
Action P3 (Project No. OC P3.10). The use of the computer facility at the MetaCenter
of the Masaryk University, Brno, and at the Boston University Scientic Computing and
Visualization Center is acknowledged.
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N4.8.6


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Ab initio simulation of a tensile test in MoSi\u3csub\u3e2\u3c/sub\u3e and WSi\u3csub\u3e2\u3c/sub\u3e

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