A meta-GGA perspective on the altermagnetism of RuO 2

The metallic oxide <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:msub> <a:mi>RuO</a:mi> <a:mn>2</a:mn> </a:msub> </a:math> hosts a fascinating edge case of magnetism: while nonmagnetic in ideal bulk material, density functional theory (DFT) predicts an altermagnetic ground state within the <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"> <b:mrow> <b:mi>DFT</b:mi> <b:mo>+</b:mo> <b:mi>U</b:mi> </b:mrow> </b:math> method. The magnetic state of strained or doped thin films remains controversial, but evidence for a nontrivial magnetic state is ample. Here, I study the altermagnetic ground state of <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"> <c:msub> <c:mi>RuO</c:mi> <c:mn>2</c:mn> </c:msub> </c:math> on a higher rung of Jacob's ladder of density functional approximations, the meta-GGA level including the kinetic energy density and the density Laplacian. While the workhorse functional of solid-state physics is a generalized gradient approximation (GGA), the modern <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"> <d:mrow> <d:msup> <d:mrow> <d:mi mathvariant="normal">r</d:mi> </d:mrow> <d:mn>2</d:mn> </d:msup> <d:mtext>SCAN-L</d:mtext> </d:mrow> </d:math> functional has been established as a general-purpose functional which can replace GGA, while systematically improving solid-state properties without introducing spurious errors like erroneous magnetic ground states. Comparison of local spin-density approximation <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"> <f:mrow> <f:mo>(</f:mo> <f:mi>LSDA</f:mi> <f:mo>)</f:mo> <f:mo>+</f:mo> <f:mi>U</f:mi> </f:mrow> <f:mo>,</f:mo> <f:mo> </f:mo> <f:mrow> <f:mi>GGA</f:mi> <f:mo>+</f:mo> <f:mi>U</f:mi> </f:mrow> </f:math> , and meta- <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"> <g:mrow> <g:mi>GGA</g:mi> <g:mo>+</g:mo> <g:mi>U</g:mi> </g:mrow> </g:math> results on <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"> <h:msub> <h:mi>RuO</h:mi> <h:mn>2</h:mn> </h:msub> </h:math> shows systematic enhancement of the exchange interaction, leading to a reduction of the onset value of the Hubbard <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"> <i:mi>U</i:mi> </i:math> parameter at different levels of density functional approximation. However, the magnetic ground state, studied at the experimental lattice constants, remains nonmagnetic with <j:math xmlns:j="http://www.w3.org/1998/Math/MathML"> <j:mrow> <j:msup> <j:mrow> <j:mi mathvariant="normal">r</j:mi> </j:mrow> <j:mn>2</j:mn> </j:msup> <j:mtext>SCAN-L</j:mtext> </j:mrow> </j:math> . I demonstrate that altermagnetism is easily formed upon lattice expansion, hole doping, and uniaxial strain on the <l:math xmlns:l="http://www.w3.org/1998/Math/MathML"> <l:mi>c</l:mi> </l:math> axis. The <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mrow> <m:mi mathvariant="normal">r</m:mi> </m:mrow> <m:mn>2</m:mn> </m:msup> <m:mtext>SCAN-L</m:mtext> </m:mrow> </m:math> calculations set conservative thresholds for distortions and doping levels for the onset of altermagnetism in a parameter-free framework.