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Page 6 of 17 Liu et al. J Mater Inf 2023;3:17 https://dx.doi.org/10.20517/jmi.2023.19

Table 1. The calculated properties of HECs in the present work at ground state (0 K), including equilibrium atomic energy (E ),
0
equilibrium atomic volume (V ), equilibrium bulk modulus (B ), the derivative of B (B’), and lattice parameter (a)
0
0
0
Composition E (eV/atom) V (Å /atom) B (GPa) B’ a (Å)
3
0
0
0
(TaZrU)C -9.7059 13.3873 235.14 4.11 9.5097
(YZrU)C -8.8862 15.1241 173.71 4.45 9.9043
TaC -11.1001 11.2312 324.88 4.23 4.48
a
b
b
-11.1 , -11.1 b 322.15 , 365.3 j 4.22 b 4.479 , 4.42 j
ZrC -9.7315 13.0724 222.11 3.97 4.71
b
a
b
-9.74 , -9.73 b 219.3 , 220.1 e 4.06 b 4.711 , 4.71 e
UC -10.3533 14.9923 186.77 3.66 4.93
f
f
158 , 191 g 4.96 , 4.97 g
YC -7.8109 16.4216 123.98 4.06 5.08
j
j
128 , 124.3 e 4.451 d 5.08 , 5.09 e
a
; Jiang et al., first-
The present first-principles calculations utilize PAW-GGA-PBE. Ye et al., first-principles calculations with PAW-GGA-PBE [81] b
[82] d [83] e
principles calculations with PAW-GGA- PBE ; Maibam et al., first-principles calculations with GGA- PBE ; Korir et al., first-
[84] f [85] g
principles calculations with PAW-GGA- PBE ; Shi et al., first-principles calculations with PAW-GGA- PBE ; Mei et al., first-principles
calculations with PAW-GGA-PBE [86] j ; Isaev et al., first-principles calculations with GGA-PBE [87] .

Figure 2. The calculated Energy-Volume (E-V) curves of (TaZrU)C and (YZrU)C.

binary carbides. Furthermore, the bulk modulus B of (TaZrU)C and (YZrU)C are also comparable to the
0
average value of the three binary carbides. Similar to most HECs, the equilibrium properties of MECs,
(TaZrU)C and (YZrU)C, follow the mixture rule , as seen in their equilibrium atomic energy, volume,
[88]
lattice constant, and bulk modulus.

Mechanical properties are critical to the industrial applications of materials. Interestingly, (TaZrU)C has a
greater bulk modulus B than (YZrU)C, indicating that (TaZrU)C has a higher strength than (YZrU)C and
0
will be validated by the following bonding charge density analysis. However, the first derivative of bulk
modulus with respect to pressure B ’ of (TaZrU)C is smaller than that of (YZrU)C. Despite the absence of
0
experimental data for comparison, the current findings can serve as valuable references for future
experimental and theoretical research.