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The thermal-mechanical properties of transition metal carbonitrides can be affected by the concentration and ordering of vacancies besides the C/N atomic ratio. However, there are few reports on the vacancy ordered structure of ternary transition metal carbonitrides. In the present paper, the first-principles method is used to study the vacancy ordered structures, mechanical properties, electronic properties and the effect of vacancies on the ternary Hf-C-N system. Firstly, the crystal structures of Hf-C-N system is examined by the first-principles and evolutionary algorithms implemented in USPEX under ambient pressure, and eight thermodynamical stable vacancy ordered structures are found, each of which has a rock-salt structure, and is also dynamical and mechanical stable, which are verified by the calculations of their phonon dispersion curves and elastic constants. The vacancies are occupied at the [Hf 6] octahedral interstices, which replace the positions of non-metal atoms. Their crystallographic data such as space group, lattice constants are also predicted. To the best of our knowledge, there is no report on the Hf-C-N vacancy ordered structures and these structures investigated here in this work are all found for the first time. Then their mechanical properties are calculated. The Hf-C-N vacancy ordered structures all have very high bulk, shear and elastic modulus and hardness. It is found that except for C∶N = 1∶4, for the Hf-C-N system with the same C/N ratio the moduli, Vickers hardness values, and Pugh’s ratios decrease with the increase of the concentration of vacancy. However, the Vickers hardness of Hf 6CN 4(the concentration of vacancy is equal to 1/6) is higher than that of Hf 5CN 4(no vacancy), that is so-called vacancy hardening. Finally, the electronic density of states and the crystal orbital Hamilton populations are calculated. The chemical bonding of Hf-C-N vacancy ordered structure is analyzed, which is a mixture of covalence and metallic and is similar to that of binary transition metal carbides and nitrides. With the increase of the concentration of vacancy, the total bond strength decreases, and then the modulus decreases for Hf-C-N compound.
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Compound Space group Lattice constants/Å ΔH/(eV·atom–1) CN CV Hf6C4N $C2 $/m a= 5.679,b= 9.799,c= 5.671,β= 70.6o –0.0899 5 1/6 Hf6C3N $C2 $ a= 5.658,b= 9.763,c= 9.262,β= 144.8o –0.0980 4 1/3 Hf6C3N2 $C2 $m a= 5.660,b= 9.783,c= 5.619,β= 109.6o –0.1038 5 1/6 Hf3CN $C2 $ a= 5.632,b= 9.705,c= 5.625,β= 109.8o –0.1107 4 1/3 Hf6C2N3 $C2 $ a= 5.624,b= 9.725,c= 5.602,β= 109.6o –0.1047 5 1/6 Hf4CN2 Cmmm a= 6.427,b= 9.147,c= 3.235 –0.1082 4/5 1/4 Hf6CN3 $C2 $/m a= 5.592,b= 9.658,c= 6.455,β= 125.3o –0.0894 4 1/3 Hf6CN4 $C2 $/m a= 5.580,b= 9.681,c= 5.587,β= 70.3o –0.0815 5 1/6 
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Compounds C11 C22 C33 C44 C55 C66 C12 C13 C23 Hf6C4N-$C2 $/m 414.3 406.6 415.6 158.0 170.6 148.7 94.1 116.1 104.5 Hf6C3N-$C2 $ 358.5 362.8 352.2 100.0 114.3 132.3 87.5 98.3 91.6 Hf6C3N2-$C2 $/m 414.6 417.4 407.6 152.2 157.6 147.8 111.9 115.0 116.3 Hf3CN-$C2 $ 354.5 363.5 348.7 90.6 103.6 128.7 102.1 109.5 101.3 Hf6C2N3-$C2 $ 409.7 418.7 418.1 149.6 160.2 148.9 123.4 122.9 126.5 Hf4CN2-Cmmm 373.4 368.8 406.8 142.2 133.1 135.8 146.4 112.0 124.4 Hf6CN3-$C2 $/m 361.1 358.4 351.7 84.9 99.8 124.9 108.1 121.7 114.2 Hf6CN4-$C2 $/m 401.2 414.1 403.5 146.5 157.2 139.8 134.0 139.9 147.8 DownLoad:
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Compound B/GPa G/GPa E/GPa μ G/B HV/GPa Hf6C4N 229.0 140.8 350.6 0.2449 0.6148 17.5 Hf5C4N[19] 260.6 201.3 480.3 0.1928 0.7727 29.9 Hf6C3N 180.9 121.5 297.9 0.2256 0.6717 17.8 Hf4C3N[19] 262.2 202.1 482.4 0.1934 0.7707 29.9 Hf6C3N2 214.0 151.1 366.9 0.2143 0.7059 22.1 Hf3CN 188.0 113.4 283.3 0.2489 0.6031 14.6 Hf2CN[19] 268.1 198.5 477.6 0.2031 0.7403 28.1 Hf6C2N3 221.3 149.7 366.6 0.2239 0.6766 20.7 Hf4CN2 212.7 132.8 329.7 0.2417 0.6242 17.1 Hf3CN2[19] 272.8 185.1 452.8 0.2233 0.6786 23.9 Hf6CN3 195.4 108.9 275.6 0.2650 0.5574 12.7 Hf4CN3[19] 276.2 179.6 442.8 0.2328 0.6504 22.2 Hf6CN4 207.2 156.1 374.4 0.1989 0.7535 24.6 Hf5CN4[19] 279.0 171.5 427.0 0.2449 0.6147 20.0 DownLoad:
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Compound –ICOHP Compound –ICOHP Hf—C Hf—N Hf—Hf Hf—C Hf—N Hf—Hf Hf6C4N 3.373 3.567 0.529 Hf6C3N2 3.181 2.990 0.571 Hf5C4N 3.373 3.033 0.459 Hf4CN2 3.474 3.111 0.650 Hf6C3N 3.350 3.067 0.718 Hf3CN2 3.551 3.091 0.541 Hf4C3N 3.319 3.029 0.454 Hf6CN3 3.408 3.211 0.570 Hf6C3N2 3.607 3.103 0.530 Hf4CN3 3.321 3.159 0.520 Hf3CN 3.277 3.211 0.737 Hf6CN4 3.675 3.179 0.591 Hf2CN 3.483 2.802 0.490 Hf5CN4 3.319 3.017 0.500 DownLoad:
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