<i>Ab initio</i> study of the vanadium-based half Heusler alloys <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>VRh</mml:mi> <mml:mi>Z</mml:mi> </mml:mrow> </mml:math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/Mat

We embrace a high-throughput investigation strategy beginning from the electronic structure properties to examine three vanadium-based half Heusler alloys with 18 valence electrons, <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mrow> <a:mi>VRh</a:mi> <a:mi>Z</a:mi> </a:mrow> </a:math> <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"> <b:mrow> <b:mo>(</b:mo> <b:mi>Z</b:mi> <b:mo>=</b:mo> <b:mi>Ge</b:mi> <b:mo>,</b:mo> <b:mi>Si</b:mi> <b:mo>,</b:mo> <b:mo> </b:mo> <b:mtext>and</b:mtext> <b:mo> </b:mo> <b:mi>Sn</b:mi> <b:mo>)</b:mo> </b:mrow> </b:math> , for thermoelectric application. This work is inspired by expected contrasting electrical transport essentials, particularly high electrical conductivity <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"> <c:mi>σ</c:mi> </c:math> and high Seebeck coefficient <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"> <d:mrow> <d:mi>S</d:mi> </d:mrow> </d:math> for attaining enhanced figure of merit <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"> <e:mrow> <e:mo>(</e:mo> <e:mrow> <e:mi>Z</e:mi> <e:mi>T</e:mi> </e:mrow> <e:mo>)</e:mo> </e:mrow> </e:math> performance. We employ the electronic fitness function <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"> <f:mrow> <f:mo>(</f:mo> <f:mtext>EFF</f:mtext> <f:mo>)</f:mo> </f:mrow> </f:math> , which evaluates the magnitude of decoupling <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"> <g:mi>σ</g:mi> </g:math> and <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"> <h:mi>S</h:mi> </h:math> for high <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"> <i:mrow> <i:mi>Z</i:mi> <i:mi>T</i:mi> </i:mrow> </i:math> by a distinct band structure. The impact of varying the Fermi level on the power factor, electrical conductivity, and Seebeck coefficient is explored. The effects of exchange correlation are reported by use of the Perdew-Burke-Ernzerhof (PBE) functional in line with the framework of the generalized gradient approximation (GGA), as implemented in the density functional theory. To rigorously assess the reliability of the GGA-PBE functional, subsequent electronic structure calculations are carried out on the three compounds using the <j:math xmlns:j="http://www.w3.org/1998/Math/MathML"> <j:mrow> <j:mi mathvariant="italic">GW</j:mi> </j:mrow> </j:math> approximation. We obtain lattice constants of <l:math xmlns:l="http://www.w3.org/1998/Math/MathML"> <l:mrow> <l:msub> <l:mi>a</l:mi> <l:mn>0</l:mn> </l:msub> <l:mo>=</l:mo> <l:mn>5.79</l:mn> <l:mspace width="0.16em"/> <l:mi>Å</l:mi> <l:mo>,</l:mo> <l:mspace width="0.16em"/> <l:msub> <l:mi>a</l:mi> <l:mn>0</l:mn> </l:msub> <l:mo>=</l:mo> <l:mn>5.76</l:mn> <l:mspace width="0.16em"/> <l:mi>Å</l:mi> </l:mrow> </l:math> , and <p:math xmlns:p="http://www.w3.org/1998/Math/MathML"> <p:mrow> <p:msub> <p:mi>a</p:mi> <p:mn>0</p:mn> </p:msub> <p:mo>=</p:mo> <p:mn>6.09</p:mn> <p:mspace width="0.16em"/> <p:mi>Å</p:mi> </p:mrow> </p:math> for VRhGe, VRhSi, and VRhSn, respectively. All materials exhibit ductility as indicated by their values of Cauchy pressure, as well as Poisson's ratio and Pugh's ratio values. The VRhSn machinability index is the highest at 2.2456. The highest hardness is realized in VRhSi, which is at 7.9957 GPa. Based on the Kleinman parameter, bond stretching is negligible in all <r:math xmlns:r="http://www.w3.org/1998/Math/MathML"> <r:mrow> <r:mi>VRh</r:mi> <r:mi>Z</r:mi> </r:mrow> </r:math> materials. <s:math xmlns:s="http://www.w3.org/1998/Math/MathML"> <s:mrow> <s:mi>VRh</s:mi> <s:mi>Z</s:mi> </s:mrow> </s:math> spatial dependence plots indicate that all investigated compounds are anisotropic. In this study, dynamic stability is observed in all compounds due to the nonexistence of phonon modes that correspond to imaginary frequencies. The derived electronic energy band structure reveals that all three alloys exhibit semiconductor properties with band gap energies of 0.50, 0.32, and 0.42 eV in VRhGe, VRhSi, and VRhSn, respectively, corresponding to indirect transitions. The maximum values realized for the EFF when <t:math xmlns:t="http://www.w3.org/1998/Math/MathML"> <t:mrow> <t:mi>T</t:mi> <t:mo>=</t:mo> <t:mn>1000</t:mn> <t:mspace width="0.16em"/> <t:mtext>K</t:mtext> </t:mrow> </t:math> using <v:math xmlns:v="http://www.w3.org/1998/Math/MathML"> <v:mrow> <v:mi mathvariant="italic">GW</v:mi> </v:mrow> </v:math> are <x:math xmlns:x="http://www.w3.org/1998/Math/MathML"> <x:mrow> <x:mn>0.69</x:mn> <x:mo>×</x:mo> <x:msup> <x:mrow> <x:mn>10</x:mn> </x:mrow> <x:mrow> <x:mo>−</x:mo> <x:mn>19</x:mn> </x:mrow> </x:msup> <x:msup> <x:mrow> <x:mi>W</x:mi> </x:mrow> <x:mfrac> <x:mn>5</x:mn> <x:mn>3</x:mn> </x:mfrac> </x:msup> <x:mtext>m</x:mtext> <x:msup> <x:mrow> <x:mtext>s</x:mtext> </x:mrow> <x:mfrac> <x:mrow> <x:mo>−</x:mo> <x:mn>1</x:mn> </x:mrow> <x:mn>3</x:mn> </x:mfrac> </x:msup> <x:msup> <x:mrow> <x:mtext>K</x:mtext> </x:mrow> <x:mrow> <x:mo>−</x:mo> <x:mn>2</x:mn> </x:mrow> </x:msup> </x:mrow> <x:mo>,</x:mo> <x:mo> </x:mo> <x:mrow> <x:mn>0.66</x:mn> <x:mo>×</x:mo> <x:msup> <x:mrow> <x:mn>10</x:mn> </x:mrow> <x:mrow> <x:mo>−</x:mo> <x:mn>19</x:mn> </x:mrow> </x:msup> <x:msup> <x:mrow> <x:mi>W</x:mi> </x:mrow> <x:mfrac>