Journal of Alloys and Compounds 592 (2014) 42–47

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Journal of Alloys and Compounds

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Theoretical investigation on the structural phase transition, elasticproperties and hardness of RhSi under high pressure

0925-8388/$ - see front matter � 2014 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.jallcom.2014.01.012

⇑ Corresponding authors.E-mail addresses: scu_kuang@163.com (X.-Y. Kuang), lucheng@nynu.edu.cn (C.

Lu).

Jing-Jing Wang a, Xiao-Yu Kuang a,⇑, Yuan-Yuan Jin a, Cheng Lu b,⇑, Xiao-Fen Huang c

a Institute of Atomic and Molecular Physics, Sichuan University, Chengdu 610065, Chinab Department of Physics, Nanyang Normal University, Nanyang 473061, Chinac Department of Physics, Sichuan Normal University, Chengdu 610068, China

a r t i c l e i n f o

Article history:Received 24 September 2013Received in revised form 3 January 2014Accepted 3 January 2014Available online 10 January 2014

Keywords:RhSiStructural phase transitionHardnessHigh pressure

a b s t r a c t

We have carried out a systematic study on the structural phase transition, elastic properties and hardnessof B31 and B20-phase RhSi under high pressure by first-principles calculations using the pseudopotentialplane-wave method. The calculated values at zero pressure are in good agreement with experiments andsimilar theoretical calculations. Among them, the mechanism of the high pressure induced structuralphase transition of RhSi has been studied in a detailed fashion. Under compression, the theoretical calcu-lations show that RhSi will undergo a structural phase transition from B31-phase to B20-phase at5.59 GPa. Based on the semiempirical method of the evaluation of covalent crystal, the hardness Hm ofRhSi is estimated under high pressure. Remarkably, the theoretical results suggest that at zero pressure,the hardness value of B31-RhSi is 14.8 GPa and that of B20-RhSi is 19.0 GPa.

� 2014 Elsevier B.V. All rights reserved.

1. Introduction

Transition metal silicides have been the subject of considerableinterest for several decades, first because they provide a model sys-tem for the detailed study of silicon-based microelectronics and sec-ond because the numerous technological applications. They havewide applications in many fields, which include the use as hightemperature structural materials for the rocket- and jet-engineparts, for furnace construction, i.e. Super Kanthal, as temperaturesensors for turbine engines, in manufacturing ceramic materialsand in the production of high temperature coatings [1–4]. The lowelectrical resistance allows the application in the semiconductorindustry, such as the design of low resistance gate, ohmic contactsand integrated circuits [5]. Among them, particularly attention hasbeen focused on the 4d transition metal silicides RhSi, and a numberof experimental and theoretical works have been reported.

Using Rutherford backscattering spectroscopy (RBS), Seemann-Bohlin X-ray diffraction (XRD), cross-sectional transmission elec-tron microscopy (TEM), sheet resistivity via four-point probe, andschottky barrier height techniques, Psaras et al. performed a com-parative study on the structure and growth kinetics of RhSi on sin-gle crystal, polycrystalline, and amorphous silicon substrates. Theysuggested that all three silicon substrates form an identical

rhodium silicide compound, RhSi, indicating that the crystallinityof the substrate has no effect on the resulting rhodium silicide[6]. Petersson et al. presented the XRD measurements on the reac-tions of platinum metal films with silicon substrates and suggestedthat RhSi formed by diffusion-controlled kinetics at temperaturesof about 350 �C with the activation energy of 1.9 eV [7]. In orderto gain a deeper insight into the structural properties of the rho-dium-silicon system, Schellenberg et al. investigated a number ofthe intermetallic compounds of rhodium–silicon alloys by meansof XRD, differential thermal analysis, metallography and electricalresistivity measurements and reported an accurate phase diagramof the Rh–Si system [8].

Some theoretical calculations have been performed on rhodiumsilicide. Most recently, Altintas reported calculations for the elec-tronic and structural properties of RhSi crystals in different phases[9]. Using the density functional theory with the generalized gradi-ent correction, Imai and Watanabe calculated the electronic struc-ture of RhSi along the high symmetry directions of the Brillouinzone near the Fermi level [10]. The theoretical results indicatedthat RhSi is metallic. In spite of the great present interest in re-search on RhSi, many of physical properties are only just beginningto be studied. In particular, there have been practically no system-atic studies of the mechanical stability and hardness of RhSi,knowledge of which is essential for their practical utilization. Theaim of present work is twofold. On the one hand, it is to give a com-prehensive investigation on the high pressure structural phasetransition of RhSi from B31-phase to B20-phase. On the other hand,we are motivated by the hope that such an investigation might

0 1 2 3 4 5

0.975

0.980

0.985

0.990

0.995

1.000

a/a0

b/b0

c/c0

V/V0

Stru

ctur

al p

aram

eter

rat

ios

Pressure (GPa)

Fig. 1. The structural parameter ratios a/a0, b/b0, c/c0, and V/V0 as functions ofpressure for B31phase RhSi. Where a0, b0, c0, and V0 represent the lattice parametersand cell volume determined at the zero pressure, respectively.

J.-J. Wang et al. / Journal of Alloys and Compounds 592 (2014) 42–47 43

contribute to further understanding on the structural properties ofother transition metal silicides under high pressure.

2. Method

The structural optimization and properties calculations are performed using thepseudopotential plane-wave method within the framework of the density func-tional theory and implemented through the Cambridge Serial Total Energy Package(CASTEP) [11–13]. The ultrasoft Vanderbilt potentials are used and the exchange–correlation terms are considered by the Perdew–Berke–Ernzerhof (PBE) form ofthe generalized gradient approximation (GGA) [14]. Pseudo-atomic calculationsare performed for Rh 4d85s1 and Si 3s23p2. In all the high-accuracy computations,the cutoff energies of the plane-wave basis are chosen as 700 eV and 550 eV forB31 and B20-phase RhSi, respectively [15]. For the Brillouin-zone sampling, weuse the 6 � 10 � 5 and 8 � 8 � 8 Monkhorst–Pack mesh for B31 and B20-phaseRhSi, respectively [16]. The tolerances for geometry optimization are set as the dif-ference in total energy being within 10�5 eV/atom. The accuracy of the total ener-gies obtained within the framework of density functional theory is in many casessufficient to predict which structure at a given pressure has lowest free energy.The formation enthalpies are calculated from the formula DH = E(solidRhSi) � E(metal Rh) � E(solid Si) for RhSi. The carbite-type structure is used for so-lid Si [17].

3. Results and discussion

3.1. Structural properties

The structural properties are so important to understand the so-lid properties from the microscopic viewpoint [18,19]. Based onthe first-principles calculations, we optimize the conventional unitcells of the two phases RhSi. The calculated lattice parameters,primitive cell volume V0, formation enthapies DH are listed in Ta-ble 1, together with the available experimental and theoreticaldata for comparison. We find that the calculated lattice parametersof the B31 and B20-phase RhSi are in good agreement with exper-imental data and theoretical values. The obtained negative forma-tion enthalpies of the B31 and B20-phase RhSi indicate that theyare thermodynamic stable and can be synthesized easily, whichis supported by the previous experimental report [25]. In orderto investigate the pressure-dependence behaviors of the structuralproperty for RhSi, we concentrate our investigation on the pressureranging from 0 to 35 GPa. In Fig. 1, we present the lattice parame-ter ratios a/a0, b/b0, c/c0, and the volume ratios V/V0 as the func-tions of pressure for B31-RhSi. From Fig. 1, we can see that withthe increasing pressure, the cell volume ratios changes from 1 to0.975. All the lattice parameters are decreasing, and specially, thevalues of a/a0 and c/c0 change are lower than b/b0, which meansthat the b axis is easier to compress than a and c axes.

3.2. Structural phase transition under high pressure

In order to shed more light on the phase stability origin of RhSi,the Gibbs free energy G for the two involved phases has been

Table 1The calculated and experimental lattice constants (Å), primitive cell volume V0 (Å3), formaderivative of the bulk modulus B00 for RhSi in B31 and B20 phases at zero pressure.

Structure a b c

B31-RhSi Present work 5.606 3.105 6Expt. [20] 5.532 3.063 6Expt. [21] 5.552 3.069 6Theor. [9] 5.475 3.036 6Theor. [22] 5.586 3.104 6

B20-RhSi Present work 4.734Expt. [21] 4.686Expt. [23] 4.626Expt. [24] 4.675Theor. [9] 4.630Theor. [22] 4.716

calculated. In general, the Gibbs free energy G can be expressedas [26,27]

G ¼ Eþ PV � TS ð1Þ

where E is the internal energy, S is the vibrational entropy, P is thepressure and V is the volume. At T = 0 K, the Gibbs free energy G isequal to the enthalpy H

H ¼ Eþ PV ð2Þ

The transition pressure Pt for the structural phase transitions isdetermined by the usual condition of equal enthalpies in the bothphases, i.e., HB31(Pt) = HB20(Pt). The variations of the enthalpy as thefunction of the pressure for B31 and B20-phase RhSi are presentedin Fig. 2. The calculations show that with the increase of pressure,RhSi will undergo a structural phase transition from B31-RhSi toB20-RhSi at 5.59 GPa.

3.3. Elastic properties and mechanical stability

Elastic properties of a solid provide important informationabout various fundamental solid-state phenomena. Most impor-tantly, knowledge of elastic stiffness coefficients is essential formany practical applications related to the mechanical propertiesof a solid, such as internal strain, thermo-elastic stress, and loaddeflection [28].

tion enthalpy per formula unit DH (eV), bulk modulus B0 (GPa), and the first pressure

V0 DH B0 B00

.455 112.399 �0.916 208.7 4.82

.362 107.801

.374 108.607

.312 104.919

.410 111.143 �0.88 208.8

106.155 �0.890 230.4 4.79102.89898.996102.17599.253104.887 �0.86 225.7

0 2 4 6 8 10-0.03

-0.02

-0.01

0.00

0.01

0.02

5.59GPa

B31-RhSi

B20-RhSi

B20-RhSiB31-RhSi

H-H

B20

( eV

/cel

l)

Pressure (GPa)

Fig. 2. The enthalpies of RhSi in both B31 and B20 structures. The phase transitionappears at 5.59 GPa.

44 J.-J. Wang et al. / Journal of Alloys and Compounds 592 (2014) 42–47

To gain further insight into the mechanical stability and elasticproperties of RhSi, we have calculated the elastic constants usingthe ab initio stress–strain relations. Based on the Hooke’s law, theelastic stiffness tensor cijkl can be expressed as [29]

cijkl ¼@rijðxÞ@ekl

� �X

ð3Þ

where rij and ekl are the applied stress and Eulerian strain tensors, Xand x are the coordinates before and after the deformation. Underthe hydrostatic pressure P, we have [29,30]

cijkl ¼1

VðxÞ@2EðxÞ@eij@ekl

!X

þ P2

2dijdkl � dikdjl � dildjk� �

ð4Þ

where d is the finite strain variable. The theoretical elastic constantsare given in Table 2. It is worth pointing out that our theoreticalcalculations of the elastic constants for both B31 and B20-phaseRhSi are coincidence well with the theoretical report [22].

Table 2Calculation elastic constants Cij (GPa), bulk modulus B (GPa), shear modulus G (GPa), Younpercent compressibility of shear moduli factors (AB and AG), and Debye temperature HD (

B31-RhSi

Present work Theor.

0/GPa 1/GPa 2/GPa 3/GPa 4/GPa 5/GPa 0/GPa

C11 416.5 422.9 430.4 438.2 438.2 451.3 417.3C22 227.6 233.5 240.1 243.1 243.1 255.6 230.1C33 351.5 363.8 371.4 374.5 374.5 390.6 359.9C44 100.7 103.9 105.8 106.3 106.3 108.0 116.2C55 114.0 115.8 118.7 121.0 121 124.9 114.8C66 86.4 88.8 90.9 91.8 91.8 95.5 95.4C12 110.7 114.1 118.7 122.6 122.6 129.7 130.9C13 139.3 144.5 150.2 154.1 154.1 162.0 145.9C23 162.1 164.4 168.8 174.6 174.6 182.5 165.1BV 202.2 207.4 213.0 217.6 221.1 227.3 210.1BR 188.6 193.3 198.8 203.0 206.9 212.8 196.4GV 99.1 101.5 103.4 104.1 105.1 107.3 102.9GR 90.9 93.8 95.7 95.8 96.9 99.2 94.3B 195.4 200.3 205.9 210.3 214.0 220.1 203.3G 95.0 97.6 99.5 100.0 101.0 103.2 98.6E 245.3 251.9 257.1 258.9 261.9 267.8 254.7m 0.29 0.29 0.29 0.29 0.3 0.3 0.29A1 0.82 0.84 0.84 0.84 0.84 0.83 0.96A2 1.78 1.72 1.73 1.80 1.80 1.78 1.77A3 0.82 0.83 0.84 0.84 0.84 0.85 0.99AB 3.48 3.52 3.45 3.47 3.47 3.29 3.36AG 4.30 3.96 3.86 4.16 4.16 3.90 4.37HD 482.3 488.4 492.8 493.7 496 501.0 490.5

Generally, the elastic constants of a given crystal should satisfythe generalized mechanical stability criteria. For the orthorhombicphase B31-RhSi, the criteria for mechanical stability are given by[31]

C11 > 0;C22 > 0;C33 > 0;C44 > 0;C55 > 0;C66 > 0;

½C11 þ C22 þ C33 þ 2ðC12 þ C13 þ C23Þ� > 0; ð5Þ

ðC11 þ C22 � 2C12Þ > 0; ðC11 þ C33 � 2C13Þ > 0;ðC22 þ C33 � 2C23Þ > 0:

For the cubic phase B20-RhSi, the criteria for mechanicalstability are given by [31]

C11 > 0; C44 > 0; C11 > jC12j; ðC11 þ 2C12Þ > 0: ð6Þ

From the theoretically obtained elastic constants, we do findthat the two phases RhSi satisfy the mechanical stability criteria,which implies that both B31 and B20-phase RhSi are elasticallystable. The Voigt bulk modulus BV, Reuss bulk modulus BR, Voigtshear modulus GV, Reuss shear modulus GR, bulk modulus BH, shearmodulus GH, Young’s modulus E, Poisson’s ratio m, shear elasticanisotropic factors (A1, A2 and A3), and the percent compressibilityof shear moduli factors (AB and AG) are also shown in Table 2. Theformulae of elastic moduli and elastic anisotropy factors are shownin Supporting Information. It is found that our calculated resultsare consistent well with previous theoretical calculations [22].

To comprehend the pressure dependence behavior of RhSi, wehave calculated elastic constants depending on pressure, and the re-sults are shown in Table 2 and plotted in Fig. 3. We can observe thatall the elastic constants increase monotonically with pressure. Espe-cially, under the effect of pressure, C11 varies largely compared withvariations of others. To investigate the pressure dependence behav-iors of the elastic modulus, we concentrate our investigation on thepressure ranging from 0 to 35 GPa. From Table 2 and Figs. A.1–A.3 (inSupporting Information), we can observe that the pressure has animportant influence on the bulk modulus (B), shear modulus (G)and Young’s modulus (E). We also can find that as the pressureincreases, all these elastic modulus for B31 and B20-phase RhSiincrease monotonically, respectively. The elastic anisotropy at high

g’s modulus E (GPa), Poisson’s ratio m, shear elastic anisotropic factors (A1, A2 and A3),K) for RhSi in B31 and B20 phases under zero pressure and high pressure.

B20-RhSi

[22] Present work Theor. [22]

0/GPa 10/GPa 20/GPa 30/GPa 35/GPa 0/GPa

325.5 399.4 468 535.1 568.5 323.8

96.9 109.4 120.7 132.1 138.2 99.4

178.2 212.0 246.5 279.1 295.8 177.9

227.3 274.5 320.6 364.4 386.7 226.5227.3 274.5 320.6 364.4 386.7 226.587.6 103.1 116.9 130.5 137.5 88.886.0 102.5 116.7 130.4 137.5 86.8227.3 274.5 320.6 364.4 386.7 226.586.8 102.8 116.8 130.4 137.5 87.8230.1 274.3 314.2 349.6 368.7 233.20.33 0.33 0.34 0.34 0.34 0.321.32 1.17 1.09 1.03 1.01 1.361.32 1.17 1.09 1.03 1.01 1.361.32 1.17 1.09 1.03 1.01 1.360 0 0 0 0 00.91 0.29 0.08 0.01 0 1.14458.9 496.4 526.3 553.7 567.2 460.5

0 10 20 30 40

100

200

300

400

500

B20-RhSiC11

C22

C33

C44

C55

C66

C12

C13

C23

Ela

stic

con

stan

ts (

GP

a)

Pressure (GPa)

B31-RhSi

Fig. 3. Pressure dependence of elastic constants in both B31 and B20-phase RhSiconsider the phase transition.

0 10 20 30

480

500

520

540

560B20-RhSi

Deb

ye te

mpe

ratu

re (

GP

a)

Pressure (GPa)

B31-RhSi

Fig. 5. Pressure dependence of the Debye temperature HD in both B31 and B20-phase RhSi consider phase transition.

0

3

6

9

-15 -10 -5 0 5 10

0

3

6

9

Rh-dSi-sSi-ptotal

0 GPaB31-RhSi

(a)

Rh-dSi-sSi-ptotal

5 GPa

Energy(eV)

)Ve/snortcele(

setat sfo

ytisneD

Rh-d(b)B20-RhSi

J.-J. Wang et al. / Journal of Alloys and Compounds 592 (2014) 42–47 45

pressure is important for understanding the evolution of bonding inthe system [18]. Fig. 4 shows the percent shear anisotropy (AG) inB31-RhSi and B20-RhSi under high pressure. We can see that withthe increasing pressure, AG decrease monotonically for B31-RhSi,but vary not monotonically for B20-RhSi. Moreover, the shear elasticanisotropic factors (A1, A2, A3) under different pressure are calcu-lated (see Table 2 and Figs. A.4–A.6 in Supporting Information).We can find that all elastic anisotropy factors decrease monotoni-cally for B31-RhSi, but vary not monotonically for B20-RhSi. Thevariations of the elastic anisotropy factors at different pressuresshow that B31-RhSi is anisotropic. But for B20-RhSi, when the pres-sure increases to 35 GPa, the value of elastic anisotropy factors(which equal to 1.01) are close to 1, which indicate that B20-RhSiis isotropic.

3.4. Debye temperature

The Debye temperature, which is related to the average soundvelocity mm, is obtained by the equation [19]:

HD ¼hkB

34pVa

� �1=3

mm ð7Þ

0 10 20 30

0.00

0.01

0.02

0.03

0.04

AG

(%)

Pressure (GPa)

B31-RhSi B20-RhSi

Fig. 4. Pressure dependence of the percent compressibility of anisotropy factor (AG)in both B31 and B20-phase RhSi consider phase transition.

0

3

6

9

-15 -10 -5 0 5 10 15

0

3

6

9

0 GPa Si-sSi-ptotal

Den

sity

of

stat

es (

elec

tron

s/eV

)

20 GPa Rh-dSi-sSi-ptotal

Energy (eV)

Fig. 6. Calculated PDOSs and TDOSs for B31 and B20-phase RhSi under zero andhigh pressure.

where h, KB, Va and mm are Planck’s constant, Boltzmann’s constant,atomic volume and sound velocity, respectively. The sound velocityis given by [19]

Table 3Calculated bond number n, bond length dl, Mulliken overlap population Pl, bond volume ml

b ; the metallic population P0 and Vickers hardness (GPa) for B31 and B20-phase RhSi atzero and high pressure.

Bond n dl (Å) Pl P0 mlb (Å3) Hl

m Hm

B31-RhSi (0/GPa) Si–Rh 8 2.436 0.315 0.0009 4.265 20.7 14.8Si–Rh 4 2.440 0.350 0.0009 4.286 22.8Si–Si 8 2.527 0.220 0.0009 4.760 12.0Si–Si 4 2.694 0.260 0.0009 5.764 10.3

B31-RhSi (5/GPa) Si–Rh 8 2.423 0.300 0.0003 4.193 20.3 15.3Si–Rh 4 2.426 0.350 0.0003 4.207 23.6Si–Si 8 2.506 0.215 0.0003 4.638 12.3Si–Si 4 2.658 0.270 0.0003 5.536 11.5

B20-RhSi (0/GPa) Si–Rh 12 2.507 0.480 0.0023 3.403 45.9 19.0Si–Si 12 2.932 0.130 0.0023 5.444 12.2

B20-RhSi (20/GPa) Si–Rh 12 2.455 0.460 0.0028 3.185 49.1 13.9Si–Si 12 2.860 0.150 0.0028 5.032 7.4

46 J.-J. Wang et al. / Journal of Alloys and Compounds 592 (2014) 42–47

mm ¼13

2m3

tþ 1

m3l

� �� ��1=3

ð8Þ

where mt and ml are the transverse and the longitudinal velocities,respectively, which can be obtained from Navier’s equation [32]:

mt ¼

ffiffiffiffiGq

s; ml ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3Bþ 4G

3q

sð9Þ

The calculated values of Debye temperature of RhSi under both zeroand high pressure are also listed in Table 2 and plotted in Fig. 5. At0 GPa, The Debye temperature (HD = 482.3 K) for B31-RhSi and(HD = 458.9 K) for B20-RhSi are in excellent agreement with theprevious theoretical results (HD = 490.5 K for B31-RhSi andHD = 460.5 K for B20-RhSi) [22].

3.5. Density of states

The total density of states (TDOSs) and partial density of states(PDOSs) for B31 and B20- phase RhSi under both zero and highpressure are shown in Fig. 6. Zero point energy represents Fermi le-vel. We can see that the two phases RhSi are both metallic due tothe finite pseudogap at the Fermi level, which is consistent withother similar theoretical reports [10,22]. The existence of finitepseudogap induces that not all the bonding states are filled andsome extra electrons are required to reach maximum stability inthis compound. It also appears that, for both B31 and B20-phaseRhSi, the bonding states are primarily composed of Rh 4d orbitalsand Si 3s orbitals while Rh 4d orbitals and Si 3p orbitals constitutethe antibonding states. From Fig. 6, we can also see that the state-resolved DOSs for the Rh and Si atoms under zero and high pres-sure reveal a strong hybridization between the Rh states and Sistates. Interestingly, the peaks of the DOSs move away from theFermi level with the increasing pressure. The valence bands tendto broaden and shift toward the lower-energy region, whereasthe conduction band levels shift toward the higher-energy regionobviously.

3.6. Hardness

Hardness is understood as the extent to which a given solid re-sists both elastic and plastic deformation. In general, the hardnessof covalent crystals is intrinsic and equivalent to the sum of resis-tance of each bond per unit area to indenter. This resistant force ofbond can be characterized by energy gap, and the number of bondper unit area is determined by valence electron density. Based onthis assumption, we represent a semiempirical method for theevaluation of hardness in covalent crystals [33]. Thus, the hardnesscan be calculated using the following relations [33–35]:

Hm ¼ ½Ylð740ðPl � P0Þml�5=3

b Þnl

�1=P

nl

¼ ½Ylðhl

m Þnl

�1=P

nl

ð10Þ

where

P0 ¼ nfree

Vð11Þ

nfree ¼Z EF

EP

NðEÞdE ð12Þ

vlb ¼ ðd

lÞ3=X

v½ðdmÞ3Nv

b � ð13Þ

Pl is the Mulliken overlap population, mlb is the volume, nl is the

number of l type bond, dl is the bond length, and Nlb is the l type

bond number per unit volume.P0 is the metallic population, nfree is the number of free electron

in a cell, V is the volume of the cell, and EF is the Fermi energy. InTable 3, we present the bond parameters and hardness for the twophases RhSi at zero and high pressure. It is found that, at zero pres-sure, the hardness of B20-RhSi is 19.0 GPa, which is larger thanB31-RhSi with 14.8 GPa. The comparisons of the hardness for RhSiat zero and high pressure show that the compressive stress may in-crease the hardness of B31-RhSi, but decrease the hardness of B20-RhSi. Unfortunately, as far as we know, there are no experimentaland theoretical data available related to the hardness of RhSi athigh pressure in the literature for our comparison. Consequently,our work is a first attempt in this direction and our results couldinspire further research interests for the transition metal silicidesin the near future.

4. Conclusions

In the present paper, we have undertaken a study on the struc-tural phase transition, elastic properties and hardness of RhSi un-der high pressure using the pseudopotential plane-wave methodbased on the density functional theory. Our theoretical resultsshow that the structural phase transition of RhSi from B31-phaseto B20-phase occur at 5.59 GPa. We have also performed an accu-rate calculation of the pressure-dependent density of states of B31and B20-phase RhSi. The results reveal that both of the two phaseshave metallic property. Based on the Mulliken overlap populationanalysis, it is found that RhSi is hard material. The origin of thehardness of the two phases can be traced to the large Mullikenoverlap population Pl of Rh–Si bonds.

J.-J. Wang et al. / Journal of Alloys and Compounds 592 (2014) 42–47 47

Acknowledgements

This work was supported by the National Natural Science Foun-dation of China (nos. 11104190, 11274235, and 11304167) and theDoctoral Education Fund of Education Ministry of China (nos.20100181110086 and 20111223070653).

Appendix A. Supplementary material

Supplementary data associated with this article can be found, inthe online version, at http://dx.doi.org/10.1016/j.jallcom.2014.01.012.

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