of Investigation mechanical stability of possible structures of PtN using first-principles computations
S. K. R. Patil,1 S. V. Khare,2* B. R. Tuttle,3 J. K. Bording,4 and S. Kodambaka5
1
Department of Mechanical Engineering, The University of Toledo, Toledo, Ohio, USA. Department of Physics and Astronomy, The University of Toledo, Toledo, Ohio, USA. School of Science , Behrend College, Penn State University at Erie, PA, 16513. Center for Functional Nanomaterials, Brookhaven National Lab, Upton, NY 11973, USA.
2
3
4
5
IBM Research Division, T.J. Watson Research Center, Yorktown Heights, New York
10598, USA
ABSTRACT We report an ab initio study of the mechanical stability of platinum nitride (PtN), in four different crystal structures, the rock salt (rs-PtN), zinc blende (zb-PtN), cooperite and a face centered orthorhombic phase. Only the rs-PtN phase is found to be stable and has the highest bulk modulus B = 284 GPa. Its electronic density of states shows no band gap making it metallic. The zb-PtN phase does not stabilize or harden by the nitrogen vacancies investigated in this study. Therefore, the experimental observation of super hardness in PtN remains a puzzle. I. INTRODUCTION Metal and semi-conductor nitrides are an important class of materials having properties of fundamental interest as well as those used in a variety of applications.1,2,3,4
*
Corresponding author email: [e-mail]
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Despite the wide interest in making ever better nitrides for applications, the noble metal nitrides have evaded discovery till recently. Despite the wide interest in metal nitrides, noble metal nitrides have evaded discovery until the recent synthesis of gold 5 and platinum nitrides. In 2004 Gregoryanz et al.6 reported the synthesis of platinum nitride, PtN. This compound was formed using laser-heated diamond anvil-cell techniques at pressures greater than 45 GPa and temperatures exceeding 2000 K. The compound was then recovered completely at room temperature and pressure and analyzed by electron microprobe techniques. Compositional profiles showed that the Pt to N ratio was close to 1:1 with a little variation given by the formula PtN1 x where x < 0.05. The synchrotron xray diffraction experiment revealed PtN to be face-centered cubic but was unable to distinguish between zinc-blende (zb-PtN) and rocksalt (rs-PtN) structures due to a much stronger Pt signal caused by the large difference between masses of Pt and N. But PtN had a first order Raman spectrum and hence rocksalt structure was ruled out.6 As the two first order bands obtained6 seemed to correspond to Raman active peaks of a zincblende structure, the PtN synthesized was concluded to be of this form. The bulk modulus (B) of this zinc-blende PtN was determined to be 372 5 GPa. This B is comparable to 382 GPa of super hard cubic zinc-blende structure BN7. Thus PtN with a zinc-blende structure is the first noble metal nitride experimentally identified to be a super hard material. However, theoretical calculations fail to confirm the high bulk modulus extracted from experiment. Recent theoretical investigations have applied different density functional methods to calculate the lattice constants and bulk moduli of various forms of PtN. These studies conclude that the lattice constant of the zb-PtN is consistent with
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experiments. However, these calculations draw contradictory conclusions about the high bulk modulus of PtN found in experiment. The bulk modulus reported in Ref. [8] concurs with the experimental value but the value reported by Ref. [9] contradicts both the experiment and Ref. [8], since it is smaller by a factor of one half. Another investigation, in Ref. [10] gives a variety of different values for B depending on the method used. These explorations motivated us to study theoretically different crystal structures of PtN as possible candidates for super-hardness. We restricted our study to compounds with 1:1 stoichiometric ratio of Pt:N (except for PtN2, cf. section VIII). Two of these, the zb- and rs- PtN phases were motivated by results of X ray measurements6. The zb-PtN was found to be mechanically unstable and transformed to a lower energy fco-PtN phase for strain type 1 given in Table I. Thus fco-PtN was studied as a potential phase for super hard PtN. PtS exists in cooperite phase and hence cooperite PtN (co-PtN) was also studied as a possible phase for super hard PtN. The main results of our investigations are as follows. The zb-PtN was found to have a lattice constant close to experiment; however, it was also found to be mechanically unstable and transformed to a lower energy fco-PtN. The bulk modulus of zb-PtN was found from two different methods to be almost half the experimentally derived value consistent with the value reported in Ref. [9]. Our bulk modulus results are therefore consistent with Ref. [9]. The higher values reported in Refs. [8] and [10] are higher by a factor of two due to errors in calculation. Only the rs-PtN phase was found to be mechanically stable i.e. its elastic constants obey the conditions
(C11 C12 ) > 0, (C11 + 2C12 ) > 0,
C11 > 0 , and C 44 > 0 . It has the highest bulk
modulus (B = 284 GPa) greater than the value of 230 GPa for the zb-PtN phase. This lower value of B in the zb-PtN phase (compared to the experimental value of 372 GPa6 )
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and also its instability led us to investigate the effect of N vacancies on these properties. We found that for nitrogen vacancy concentrations of 3.7% and 12.5% that bracketed the value of the maximum 5% reported in experiment, the zb-PtN phase remained unstable and its B hardly changed. Our study indicates that further experimental investigation needs to be carried out to find the cause of the stability and high B of the zb-PtN. The rest of the paper is organized as follows. Section II gives the details of the ab initio method used. Section III describes the structure of four different phases of PtN studied. Section IV illustrates the method of calculation of elastic constants. In section V we investigate the stability of the four phases. Section VI reports the band structure and density of states of rs-PtN. Section VII gives results involving introduction of N vacancies. Section VIII gives the results and discussion on bulk modulus calculated for different phases of PtN studied.
II AB INITIO METHOD We performed first-principles total energy calculations within the local density approximation (LDA) and also generalized gradient approximation (GGA) to the density functional theory 11 (DFT) using the suit of codes VASP. 12 , 13 , 14 , 15 Core electrons are implicitly treated by ultra soft Vanderbilt type pseudopotentials 16 as supplied by G. Kresse and J. Hafner17. For each calculation, irreducible k-points are generated according to the Monkhorst-Pack scheme. 18 Convergence is achieved with 408 k- points in the irreducible part of Brillouin zone for zb-PtN and rs-PtN structures and with 512 and 864 k- points for cooperite and orthorhombic structures respectively. The single-particle wave functions have been expanded in a plane-wave basis using a 224 eV kinetic energy cutoff.
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All atoms are allowed to relax until a force tolerance of 0.03 eV/ is reached for each atom. Tests using a higher plane-wave cutoff and a larger k-point sampling indicate that a numerical convergence better than 1.0 meV is achieved for relative energies.
III. CRYSTAL STRUCTURE We investigated four different phases of PtN, the (i) zb-PtN structure (space group F43m ) 19 , (ii) rs-PtN structure (space group Fm3m )19, (iii) face-centered orthorhombic structure (fco) (space group Fddd )19 and (iv) cooperite (PtS) structure (space group P 4 2 / mmc )19. The unit cells for the first three phases are shown Figs. 1, 2, and 3 respectively. They consist of three lattice constants of the conventional unit cell a, b, and c with lattice vectors a1 =
1 1 1 (0, b, c), a 2 = (a, 0, c) and a 3 = (a, b, 0). The 2 2 2
1 (a1 + a 2 + a 3 ) . The first two
basis consists of a Pt atom at (0, 0, 0) and an N atom at
phases have c = b = a, giving them a cubic symmetry. The first phase has = 4, while the second and third phases have = 2. Fig. 4. shows the unit cell of co-PtN having lattice constants a and c. The lattice vectors are a1 = (a, 0, 0), a 2 = (0, a, 0) and a 3 = (0, 0, c). Two Pt atoms at (0, 0, 0),
3 1 a 2 + a 3 make up the basis. 4 2 The equilibrium lattice constants a, b, and c were varied independently (when they were different) to obtain the absolute minimum in total energy for each structure. All basis atoms were allowed to relax fully. Table V summarizes the equilibrium lattice
1 (a1 + a 2 + a 3 ) and two N atoms at 1 a 2 + 1 a 3 , 4 2 2
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constants of the different PtN structures. The structure with the lowest total energy per formula unit of PtN was co-PtN. Hence the formation energies, E f r t , of the other phases are reported with respect to tetragonal structure in Table V.
IV. ELASTIC CONSTANTS Elastic constants are the measure of the resistance of a crystal to an externally applied stress. For small strains Hookes law is valid and the crystal energy E is a quadratic function of strain.20 Thus, to obtain the total minimum energy for calculating the elastic constants to second order, a crystal is strained and all the internal parameters relaxed. Consider a symmetric 3 3 strain tensor which has matrix elements ij (i, j = 1, 2, and 3) defined by Eq. (1)
ij I ij e i + 1 1 I ij e 9 i j . 2
(
)
(1)
Such a strain transforms the three lattice vectors defining the unstrained Bravais lattice { a k , k = 1, 2, and 3} to the strained vectors21 { a k , k = 1, 2, and 3} as given by Eq. (2)
a = (I + ). a k , k
(2)
where I is defined by its elements, I ij = 1 for i = j and 0 for i j . Each lattice vector a k or
a k is a 3 1 matrix. The change in total energy due to above strain (1) is
E E({ei }) E 0 V 1 6 6 P(V0 ) + Cijei e j + O e3 , = 1 i V0 V0 V0 2 i =1 j =1
({ })
(3)
where V0 is the volume of the unstrained lattice, E 0 is the total minimum energy at this unstrained volume of the crystal, P( V0 ) is the pressure of the unstrained lattice, and V is the new volume of the lattice due to strain in Eq. (1). In Eq. (3), C ij = C ji due to crystal
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symmetry.20 This reduces the elastic stiffness constants C ij , from 36 to 21 independent
elastic constants in Eq. (3). Further crystal symmetry20 reduces the number to 9 ( C11 ,
C12 , C13 , C 23 , C 22 , C 33 , C 44 , C 55 , C 66 ) for orthorhombic crystals, 6 ( C11 , C12 , C13 ,
C 33 , C 44 , C 66 ) for tetragonal crystals and 3 ( C11 , C12 , C 44 ) for cubic crystals. A proper choice of the set of strains { e i , i = 1, 2, .... 6 }, in Eq. (3) leads to a parabolic relationship between E / V0 ( E E E 0 ) and the chosen strain. Such choices for the set { e i } and the corresponding form for E are shown in Table I22 for cubic, Table II23
for tetragonal and Table III24 for orthorhombic lattices. For each lattice structure of PtN studied, we strained the lattice by 0%, 1%, and 2% to obtain the total minimum
energies E(V) at these strains. These energies and strains were fit with the corresponding parabolic equations of E / V0 as given in Tables I, II and III to yield the required second order elastic constants. While computing these energies all atoms are allowed to relax with the cell shape and volume fixed by the choice of strains { e i }.
V. MECHANICAL STABILITY
1 The strain energy C ij ei e j of a given crystal in Eq. (3) must always be positive for all 2 possible values of the set { e i }; otherwise the crystal would be mechanically unstable. 1 This means that the quadratic form C ij ei e j must be positive definite for all real 2 values of strains unless all the strains are zero. This imposes further restrictions on the
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elastic constants C ij depending on the crystal structure. These stability conditions can be found out by standard algebraic methods.25
a. zb-PtN
For cubic crystal structures such as those of zb-PtN or rs-PtN, the necessary conditions for mechanical stability are given by26
(C11 C12 ) > 0 , (C11 + 2C12 ) > 0, C11 > 0 ,
C 44 > 0 .
(4)
The elastic constants are determined by applying the strains listed in Table I. C11 C12 is obtained by using the strain combination on the first row of Table I. Table IV shows the numerical values of our computation of all the elastic constants of zb-PtN. These values satisfy all the stability conditions of Eq. (4) except the condition that (C11 C12 ) > 0. Thus we have concluded that the zb-PtN is mechanically unstable which contradicts the experimental data.6 We now turn our attention to the strain used on the second row of Table I which is an isotropic strain and it yields, B
C11 + 2 C12 .We obtained a B value of 230 GPa, 3
which is lower than the experimentally reported value of 372 GPa by 38%.6 The fit of these isotropically strained volumes and corresponding total minimum energies to Murnaghan equation of state27 yielded B = 231 GPa. Thus our theoretical calculations suggest that the experimentally observed structure is unstable and with a B that is far larger than the theoretically expected value. However, the experiment found that the precise stoichiometry for their PtN sample was given by PtN 1 x with 0 < x < 0.05. To investigate the effect of N vacancies on the stability of zb-PtN and value of B, we did
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further calculations which are described in section VII. Based on our results we conclude that N vacancies only soften the material and do not explain the large experimental value for B (372 GPa). These disagreements between our theoretically computed properties and the experimental results for zb-PtN motivated us to explore other possible structures of PtN which could potentially yield very large values of the elastic constants and hence superhardness. Since Pt has a large value of B = 298 GPa28 it would seem plausible to have such an expectation.6 The X-ray diffraction part of the experimental measurements could not distinguish6, 8 between the zb-PtN and rs-PtN structural types since they both had face centered cubic (fcc) symmetry. This is because of the much weaker signal of N atoms than that of Pt atoms due to a large difference in their atomic numbers. Also many other mono transition metal nitrides, such as CrN, NbN, VN, and ZrN exist in the NaCl phase. So we explored this phase next.
b. rs-PtN
As rs-PtN is cubic, it has to satisfy all the conditions in Eq. (4) to be mechanically stable. These conditions are indeed satisfied as seen from the calculated elastic constants in Table IV, making it mechanically stable. The calculated B with the parabolic fit of strain 2 in Table I was found to be 284 GPa. As zb-PtN is unstable and rs-PtN is stable and because of the fcc structure reported by the X ray analysis6, one would be tempted to conclude that PtN is a rock salt structure like many other monotransition metal nitrides. But the calculated lattice constant of rock salt PtN, 0.45036 nm (by GGA) and 0.44071 nm (by LDA) varies substantially from 0.4801 nm measured experimentally.6 The value
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of B we obtained was 284 GPa, still off from the experimental value of 372 GPa. It is thus not clear whether the observed PtN is in NaCl structure. We nonetheless, explored the electronic properties of this stable phase which are described in section VI. With an interest toward finding a super-hard form of PtN we explored two other structures of PtN having 1:1 stoichiometry. The first of these, the tetragonal (cooperite) structure was motivated by the existence in this form of PtO which could exist as a contaminant in the experimental sample.29
c. co-PtN
The stability criteria for a tetragonal crystal26 are:
(C11 C12 ) > 0 , (C11 + C 33 2c13 ) > 0, (2C11 + C 33 + 2C12 + 4C13 ) > 0,
C11 > 0, C 33 > 0, C 44 > 0, C 66 > 0 .
(5)
The elastic constants of tetragonal PtN are shown in Table IV. The calculated elastic constant C 44 is negative violating Eq. (5) and so is labeled unstable in Table IV. For the strain types 1, 2, 4, 5 and 6 in Table II the total minimum energy for strained lattice was less than that of the unstrained lattice indicating the transformation of the tetragonal structure to either monoclinic or triclinic structures. Hence, the elastic constants
C11 , C12 , C13 , C 66 , and C 33 are labeled unstable in Table IV, making the tetragonal cell
mechanically unstable. Thus the formation of stable PtN in cooperite phase is ruled out. The fourth structure we investigated was discovered by noticing that C11 C12 < 0 in Table IV for the zb-PtN. Under the strain corresponding to C11 C12 the ZnS structure transforms to a face centered orthorhombic (fco) structure.
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d. fco-PtN
The mechanical stability criteria for face centered orthorhombic24 PtN are:
(C 22 + C33 2C23 ) > 0, (C11 + C22 + C33 + 2C12 + 2C13 + 2C 23 ) > 0,
C11 > 0, C 22 > 0, C33 > 0, C 44 > 0, C55 > 0, C 66 > 0. (6)
The calculated elastic constants are shown in Table IV. All the elastic constants obey the mechanical stability criteria given in Eq. (6) except for C 44 . Hence the possibility of PtN crystallizing in face centered orthorhombic phase is eliminated. For strain type 4 in Table III the fco PtN transforms to a triclinic phase, which we did not investigate. We now conclude that of the four forms of PtN we studied only the rs-PtN is mechanically stable. We describe its electronic structure next.
VI. ELECTRONIC STRUCUTRE OF rs-PtN
The band structure of this phase along a high symmetry direction is shown in Fig. 5. The calculated density of states (DOS) is shown in Figure 6. There is no band gap in the at DOS the fermi level (EF) and hence rs-PtN is metallic. The bands near the fermi level are mainly contributed by platinum d-orbitals while the lowest band is mainly the nitrogen sorbital. The electronic density of states is calculated using 408 irreducible k-points and a 0.2 eV smearing of the energy levels to provide a smooth DOS plot. The DOS between 5 eV and +1 eV is dominated by the Pt metal states and compares well to the photoemission spectra of platinum 30 . Figure 7. shows the projected density of states (PDOS) of Pt and N atoms in s, p and d orbitals. As seen from PDOS, the d electrons of Pt contribute to the majority of the DOS near the Fermi level.
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VII. NITROGEN VACANCIES
The experimental specimen of PtN 1 x was sub-stoichiometric with 0 < x < 0.05. This sub-stoichiometry may be the reason for both its stability and high value of B. To check for such a trend, we performed a calculation with a larger supercell31 to create Pt27N26 (i.e. x = 0.0370) which is comparable to the experimental specimen. The equilibrium lattice constant for this sub stoichiometric zb-PtN was found to be 0.46019 nm and B was found to be 221 GPa. Thus the bulk modulus is reduced slightly at this nitrogen vacancy concentration. The other sub-stoichiometry computed was a supercell32 of Pt8N7 (i.e. x = 0.125). The lattice constant and bulk modulus for this compound were 0.45556 nm and 238 GPa respectively. Thus this composition hardly raises the value of B to the value reported in experiment. The strain type 1 of Table I for which the zb-PtN was unstable was applied to the above mentioned two super cells. These calculations showed that the stability criterion (C11 C12 ) > 0 was not satisfied for these sub-stoichiometric forms either. Values of x in our calculations for sub-stoichiometric cases range from 0 (no vacancies), 0.037 and 0.125. These values cover the experimental range for x from 0 to 0.05 and beyond. It seems unlikely then that PtN 1 x ( 0 < x < 0.05 ), can be stabilized by the presence of vacancies alone. However, stabilizing and hardening effects due to other types of defects or impurities induced by the high pressure and high temperature production technique used in the experiment cannot be ruled out.
VIII. BULK MODULUS RESULTS AND DISCUSSION
Table V lists our values of B for zb-PtN and rs-PtN calculated using VASP with LDA and also the generalized gradient approximation, GGA. As a test of our method we also
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calculated the bulk modulus of zinc blende structure of BN and rock salt OsN with LDA. Our B = 382 GPa for BN is in very good agreement with the experimental value of 382 GPa7 and previous LDA value of 403 GPa.33 Also our calculated bulk modulus value of 380 GPa for OsN is in good agreement with 372 GPa obtained by a previous ab initio calculation.34 However, our results using VASP with GGA for PtN differ significantly from those of Ref. [8]38 as seen in Table V. Our value of B = 192 GPa with GGA for zbPtN is about 48% smaller than their value of 371 GPa which is almost identical to the experimental value6. However, our value matches quite well with the value of 194 GPa found in Ref. [9]. Note that an all electron method WIEN2K35 was used in both Refs. [8] and [9], while our VASP method is based on pseudo-potentials. For a proper comparison we also computed these bulk moduli with WIEN2K. We used FLAPW calculations with LDA and GGA as done in Refs. [8] and [9] using WIEN2K. Non over lapping muffin-tin sphere radii of 0.0100 nm and 0.0085 nm were used for Pt and N atoms respectively. Using this method we obtained B = 178 GPa for zb-PtN as seen in Table V, which is in good agreement with our value of 192 GPa obtained using VASP. A similar situation arises for rs-PtN with GGA where our WIEN2K result is B = 233 GPa in agreement with VASP value of 226 GPa but different from 431 GPa of Ref. [8]38. The LDA and GGA bulk moduli for zb-PtN obtained in Ref. [9], which also uses WIEN2K are given in Table V. They are in good agreement with our present work. For zb-PtN, the shear modulus,
C = C11 C12 = -17 GPa obtained in Ref. [9] is in good agreement with -15.5 GPa 2
obtained by us. R. Yu et al.9, have suggested that the experimental sample may contain excess N atoms, and hence may well be the PtN2 flourite phase in the Fm 3 m space group. As a
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check, we also calculated the bulk modulus and elastic constants of PtN2 using VASP. The bulk modulus was found to be 300 GPa. The computed elastic constants C11, C12 and C44 were 495 GPa, 193 GPa and 109 GPa respectively, satisfying the mechanical stability conditions for cubic lattices as given in Eq. (4). These values for fluorite PtN2 are in good agreement with those obtained in Ref. [9]. It is interesting to note that though the value for B clearly disagrees with the experimental value it is the only phase for which C44 is of extremely high value 495 GPa comparable to that of super hard BN36. It is also interesting to note that values for B in Ref. [8]38 are approximately WIEN2K and those of Ref. [9]. 1 of our values using 2
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SUMMARY
Using first principles calculations we have computed properties of PtN, a recently synthesized noble metal nitride6. Using our ab initio calculations the experimental zincblende structure of PtN reported6 was found to be mechanically unstable and its bulk modulus was found to be 38% lower than in experiment. Upon introduction of N vacancies in this zb-PtN structure we found that it remained unstable and the bulk modulus did not change substantially. The role of other types of impurities or defects causing the stability and super-hardness cannot be ruled out. Further experimental investigation is needed to understand the underlying causes behind the stability and super-hardness, which are not explained by our ab initio calculations. To find super-hardness in other forms of PtN we also investigated its rock-salt, cooperite, and face centered orthorhombic phases. Of these only rs-PtN was found to be stable. X-ray diffraction measurements6 have identified this form as a possible structural candidate showing fcc symmetry. However, our calculated lattice constant 0.45036 nm differs from the experimental value of 0.48010 nm by 6.2%. We also find no evidence for super-hardness in this form. The electronic band structure and total density of states of this stable form were studied. This form shows no band gap and is metallic consistent with experimental observation. All our computations and those of others8,9,10 reveal that more experiments need to be performed to ascertain the true nature of the newly discovered PtN material.
This work was partially supported by the Division of Materials Sciences, Office of Basic Energy Science, US Department of Energy. S. V. Khare acknowledges support from the
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University of Toledo, Office of Research, URAF 2005 award. We thank Ohio Supercomputer Center for providing computing resources.
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27
28
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30
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31
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32
A 2 2 2 array of unit cells with 2 atoms per unit cell in fcc (1 Pt and 1 N), results in 16 atoms (8 Pt and 8 N atoms). We removed 1 N atom to get Pt8N7.
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Just before submission we rec...