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Hard superconducting nitrides

Contributed by Russell J. Hemley, January 7, 2005
Abstract
Detailed study of the equation of state, elasticity, and hardness of selected superconducting transitionmetal nitrides reveals interesting correlations among their physical properties. Both the bulk modulus and Vickers hardness are found to decrease with increasing zeropressure volume in NbN, HfN, and ZrN. The computed elastic constants from first principles satisfy c _{11} > c _{12} > c _{44} for NbN, but c _{11} > c _{44} > c _{12} for HfN and ZrN, which are in good agreement with the neutron scattering data. The cubic δNbN superconducting phase possesses a bulk modulus of 348 GPa, comparable to that of cubic boron nitride, and a Vickers hardness of 20 GPa, which is close to sapphire. Theoretical calculations for NbN show that all elastic moduli increase monotonically with increasing pressure. These results suggest technological applications of such materials in extreme environments.
Hard superconducting materials are of considerable interest for specific electronic applications. Superconductivity has been discovered in diamond, generally believed to be the hardest material having very high shear and bulk moduli (1, 2), with a superconducting transition temperature (T _{c}) near 4 K when doped with boron (3). However, the transitionmetal compounds having the sodium chloride (B1) structure (e.g., NbN, NbC, ZrN, or HfN) are also hard superconductors but with relatively higher T _{c}s. The transition temperatures of solid solutions of NbN and NbC can reach a maximum value of 17.8 K, which is close to those found for the cubic A15type compounds such as Nb_{3}Sn and V_{3}Si (4). The refractory characteristics of these transitionmetal nitrides and carbides have been applied as coatings to increase the wear resistance, for instance, in cutting tools as well as for magnetic storage devices. The unusual hardness enhancement in these materials has been theoretically shown to originate from a particular σband of bonding states between the nonmetal p orbitals and the metal d orbitals that strongly resists shearing strains (5). At the moment, there is a need to investigate elastic and mechanical properties of these superconductors under simulated extreme working conditions.
Here, we report both experimental and theoretical studies of the equation of state, elasticity, and hardness of selected superconducting transitionmetal nitrides. We find that the cubic δNbN superconducting phase possesses a bulk modulus of 348 GPa, comparable to that of cubic boron nitride, and a Vickers hardness of 20 GPa, which is close to sapphire (Al_{2}O_{3}) (6). The results indicate that these nitrides are good candidates for engineering hard superconducting materials.
Experimental and Theoretical Details
Equations of state studies were based on angledispersive synchrotron powder xray diffractometry with a diamond anvil cell. The diffraction experiments were carried out at the synchrotron beam line 16IDB of the Advanced Photon Source High Pressure Collaborative Access Team. A 500 × 500μm^{2} monochromatic beam of wavelength λ = 0.4219 Å was focused by using a pair of bimorph mirrors to a 10 × 12μm^{2} beam. The diffraction data were recorded on a MAR345 imaging plate and integrated by using fit2d software (7). The geometric parameters and the wavelength were calibrated by using a silicon standard (NIST 640c standard). Powder samples were loaded into a hole in stainlesssteel gaskets with a nominal diameter of 100 μm and 10 to 15μm thickness between two 400μmdiameter flat diamond culets. A ruby chip served as an optical pressure sensor in the diamond anvil cell (8). LiF was chosen as a pressure medium for NbN, and silicon oil was used as a pressure medium in the HfN and ZrN samples. The powder samples with grain sizes <2 μm were obtained by crushing the single crystals of NbN, HfN, and ZrN. Single crystals grown by the zoneannealing technique have been detailed (9–11). Hardness measurements were performed on the nitride single crystals by means of a Vickers indentation method with a pyramidal diamond indenter.
The thermodynamical stability of these transitionmetal nitrides are examined by firstprinciples calculations. All calculations are based on the density functional theory within the local density approximation. We used a planewave basis and pseudopotentials with the abinit (version 4.3.2) package (12) to optimize structures and calculate total energies and electronic structures. The opium program (13) was used to generate pseudopotentials for N, Nb, Zr, and Hf. Planewave basis sets with a cutoff of 50 Ha were tested and found to be highly converged. A dense 16 × 16 × 16 kpoint mesh was used over the Brillouin zone.
The elastic moduli of these transitionmetal nitrides were calculated from stress (σ) changes caused by very small strain (ε) through the relation c _{ij} = σ_{i}/ε_{j} (14). Cubic crystals have only three independent elastic constants, namely c _{11}, c _{12}, and c _{44}, for the B1 structure. The bulk modulus K _{0} was calculated from fitting energy and volume data to the Vinet equation of state (15).
Results and Discussion
Xray diffraction patterns of NbN were taken up to 51.2 GPa. Typical patterns taken in the course of increasing pressure are shown in Fig. 1. The ambient pressure phase for NbN is fcc, as evidenced by the observations of (111), (200), (220), (311), and (222) peaks, and the measured lattice parameter is a _{0} = 4.379(2) Å. The (111) peak of LiF shifts much faster than the (200) peak of NbN under pressure, almost overlapping at 30 GPa but separating again at 51.2 GPa. At high pressures, the same diffraction peaks still remain, indicating that there is no phase transformation up to 51.2 GPa. The measured lattice parameter at 51.2 GPa is 4.213(3) Å, which corresponds to a volume compression V/V _{0} of 0.89.
Fig. 2 shows the measured pressurevolume curve or equation of state for NbN, HfN, and ZrN samples to 51.2 GPa at room temperature. The experimental data were fitted to the Vinet equation of state (16) with x = (V/V _{0})^{1/3}, where V _{0} and V are the zeropressure and compressed volumes and K _{0} and K _{0}′ represent the bulk modulus at zero pressure and its pressure derivative. Metals with simple B1 NaCl structure always have a characteristic K _{0}′ of ∼4.0 (17). Taking K _{0}′ of 4.0 for our samples, we obtained the bulk moduli at ambient condition of 348, 260, and 248 GPa for NbN, HfN, and ZrN, respectively. The bulk modulus was found to decrease with increasing zeropressure volume V _{0} from 83.98(11) Å^{3} for NbN through 93.14(14) Å^{3} for HfN to 95.69(9) Å^{3} for ZrN. Our calculated values of K _{0} are 354, 306, and 285 GPa for NbN, HfN, and ZrN, respectively. The K _{0} obtained from the fits to the static data are close to our calculated results and previous theoretical predictions (18, 19). The calculated zeropressure volumes are 82.26, 87.94, and 92.28 Å^{3} for NbN, HfN, and ZrN, respectively. These values agree with our experimental results at ambient pressure. Note that the bulk modulus for NbN is comparable to that of the superhard material cubic boron nitride (6).
The elastic moduli were obtained from a linear fit by the stressstrain relation. For cubic lattices, the Poisson's ratio ν can be expressed as ν = c _{12}/(c _{11} + c _{12}) (20). The computed results of these parameters for cubic NbN, HfN, and ZrN at ambient condition are listed in Table 1. We also estimated the elastic moduli from the initial slopes of the dispersion curves obtained from neutron scattering data (9–11). The results also are presented in Table 1 for comparison. As can be seen, the firstprinciples calculations and neutron scattering measurements give a qualitatively similar tendency of the three elastic constants in these materials. That is, c _{11} > c _{12} > c _{44} for NbN, but c _{11} > c _{44} > c _{12} for HfN and ZrN. The obtained elastic moduli in HfN are in excellent agreement for both methods. However, the value of c _{11} obtained from firstprinciples calculations is larger than that from neutron scattering measurements in NbN and ZrN. Both methods give an almost constant Poisson's ratio of ∼0.16 for these nitrides. It is worth noting that the calculated values of K′_{0} are consistent with the value of 4.0 characteristic of many materials.
The Young's and shear moduli (E and G) are two important quantities for technological and engineering applications and provide fundamental description of a material's mechanical behavior. The former is defined as the ratio between stress and strain. The latter, which is related to bond bending, depends on the nature of the bond and decreases as a function of ionicity. For the compound to have a high shear modulus and high hardness, directional bonding and a rigid structural topology are typically required. The shear modulus G can be calculated from the Viogt approximation [G = (c _{11}  c _{12} + 3c _{44})/5]. Notice that in the cubic system the Young's modulus E is not isotropic (20). The variation with direction depends on with l _{i} being the unit vector. The quantity is zero for the directions of the cube axes (001) and has its maximum value of 1/3 in the (111) direction. In the cubic nitrides, the Young's modulus is a minimum for the (111) and a maximum for (001) directions. In Table 1, we also summarized these parameters obtained from the resulting elastic constants. Both the firstprinciples calculations and neutron scattering measurements provide a consistent estimation of the Young's and shear moduli for these nitrides.
We investigated theoretically the highpressure behavior of the elastic moduli for cubic δNbN. Fig. 3a shows the pressure dependence of the elastic constants. All constants increase monotonically with increasing pressure. These elastic constants satisfy the generalized elastic stability criteria for cubic crystals under hydrostatic pressure (21), c _{11} + c _{12} > 0, c _{44} > 0, and c _{11}  c _{12} > 0. The first criterion is closely related to the bulk modulus, which is obviously necessary for stability. The latter two are thought to play a more subtle role since c_{44} involves shearing the metalN bonds and c _{11}  c _{12} involves stretching (and compression) of the metalN bonds with a combination of bending and stretching of metal–metal bonds. We note that the inequality c _{11} > c _{12} > c _{44} is satisfied over a wide pressure range (i.e., to 50 GPa).
The elastic anisotropy at high pressure is important for understanding the evolution of bonding in the system. A measure of the anisotropy of the elasticity is A = 2c _{44}/(c _{11}  c _{12}) for cubic crystals. For isotropic elasticity, the two shear moduli (c _{11}  c _{12})/2 and c _{44} are equal and the anisotropy becomes unity. In Fig. 3b we plotted the pressure dependence of A for cubic δNbN. A shows a gradual decrease with increasing pressure. The rapid increase of c _{11} with pressure is expected because of enhanced nearestneighbor interaction, which leads to a stretching of the metalN bonds (c _{11}  c _{12}). On the other hand, pressure induces sp → d electron transfer from N to Nb atoms (22), giving rise to the occupation of σbonding states derived from d–d interactions. This band is lowered under shear strain (5), which gives a modest negative contribution to c _{44}. As a result, the pressureinduced decrease of A is dominated by stretching of the metalN bonds rather than the gradual shearing of the structure.
Aggregate values for the bulk K _{0} and shear G modulus were obtained from the individual elastic constants (Fig. 3c ). The bulk modulus increases with pressure and reaches 557 GPa at pressure of 50 GPa. At 20 GPa, the bulk modulus of δNbN reaches the zeropressure value of the bulk modulus of diamond (1). The shear modulus also increases across the whole pressure range. The behavior of the moduli suggests an increasing hardness in this material with pressure.
Using conventional microhardness testing techniques, the Vickers hardness H _{V} was measured under a loading force from 0.49 to 9.8 N. The loading time was fixed at 15 s. Fig. 4 shows the measured H _{V} for the studied transitionmetal nitrides under various loads. From the results, it can be seen that the hardness decreases with increasing lattice parameter. The average hardness values are 20.0, 19.5, and 17.4 GPa under a load of 0.49 N for NbN, HfN, and ZrN, respectively. For comparison, our measured H _{V} of 20 GPa for cubic δNbN is slightly lower than the reported ≈23 GPa in a δNbN_{0.93} film (23). The measured H _{V} of 15.5 GPa in HfN under a loading force of 9.8 N is very close to the reported of 14.5 GPa in HfN ceramics (24). Another observation of our results is that the Vickers hardness decreases with the increase in load. Similar behavior has been reported for cubic boron nitride and Al_{2}O_{3} (25). Note that the cubic δNbN superconducting phase has a the bulk modulus of 348 GPa, comparable to that of cubic boron nitride, and a Vickers hardness of 20 GPa, the same as sapphire. Recent studies in films (23) revealed that the hardness of the hexagonal δ′NbN phase with high nitrogen concentration can reach to 40 GPa. The relatively large bulk modulus and high hardness in the superconducting cubic δNbN phase are favorable for potential harddevice applications.
NbN has the highest T _{c} among the transitionmetal nitrides. Currently, NbN films are being examined in applications of radio frequency superconducting accelerator cavities (26), superconducting quantum interference devices (27), superconducting hotelectron photodetectors (28), and IR sensors (29). Our results reveal that this material is highly incompressible compared with HfN and ZrN because of its high values of the bulk modulus and Vickers hardness. Application of pressure increases both the bulk and shear moduli. The enhancement of hardness is therefore expected under high pressure. Meanwhile, T _{c} was found to increase with pressure in this compound (30). These combined features make NbN a good candidate along with hard superconductors, such as borondoped diamond, for possible applications under extreme conditions.
Acknowledgments
Work at the Carnegie Institution was supported by U.S. Department of Energy Grant DEFG0202ER4595, Office of Naval Research Grant N000140210506, the Department of Energy/Office of Basic Energy Sciences, Department of Energy/National Nuclear Security Administration, Carnegie/Department of Energy Alliance Center Grant DEFC0303N00144, the National Aeronautics and Space Administration, the National Science Foundation, and the W. M. Keck Foundation. Work at the Los Alamos National Laboratory was performed under the auspices of the Department of Energy. The HighPressure Collaborative Access Team is a collaboration involving the Carnegie Institution of Washington, Lawrence Livermore National Laboratory, the University of Hawaii, the University of Nevada Las Vegas, and the Carnegie/Department of Energy Alliance Center at the Argonne National Laboratory.
Footnotes

↵ ∥ To whom correspondence should be addressed. Email: r.hemley{at}gl.ciw.edu.
 Copyright © 2005, The National Academy of Sciences
References
 ↵
 ↵
 ↵
 ↵
 ↵

↵
Haines, J., Léger, J. M. & Bocquillon, G. (2001) Annu. Rev. Mater. Res. 31 , 123.

↵
Hammersley, A. P., Svensson, S. O., Hanfland, M., Fitch, A. N. & Häusermann, D. (1996) High Press. Res. 14 , 235248.

↵
Mao, H. K., Xu, J. A. & Bell, P. M. (1986) J. Geophys. Res. 91 , 46734676.
 ↵
 ↵
 ↵
 ↵
 ↵

↵
Cohen, R. E., Gülseren, O. & Hemley, R. J. (2000) Am. Mineral. 85 , 338344.

↵
Vinet, P., Ferrante, J., Rose, J. H. & Smith, J. R. (1987) J. Geophys. Res. 92 , 93199325.

↵
Bridgman, P. W. (1945) Proc. Am. Acad. Arts Sci. 76 , 17.
 ↵
 ↵

↵
Nye, J. F. (1957) Physical Properties of Crystals (Clarendon, Oxford).
 ↵

↵
Palanivel, B., Kalpana, G. & Rajagopalan, M. (1993) Phys. Stat. Sol. B 176 , 195202.
 ↵
 ↵
 ↵

↵
Benvenuti, C., Chiggiato, P., Parrini, L. & Russo, R. (1993) Nuclear Instru. Methods Phys. Res. A 336 , 1622.

↵
Faucher, M., Fournier, T., Pannetier, B., Thirion, C., Wernsdorfer, W., Villegier, J. C. & Bouchiat, V. (2002) Physica C 368 , 211217.
 ↵
 ↵

↵
Chen, X. J., Struzhkin, V. V., Kung, S., Mao, H. K., Hemley, R. J. & Christensen, A. N. (2004) Phys. Rev. B 70, 014501, 16.
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