Magnetic exchange and dipolar interactions in GdRu2Si2 : Three-dimensional magnetism in a layered magnet

<a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mrow><a:msub><a:mi>GdRu</a:mi><a:mn>2</a:mn></a:msub><a:msub><a:mi>Si</a:mi><a:mn>2</a:mn></a:msub></a:mrow></a:math> has recently drawn significant attention as a centrosymmetric magnet capable of hosting a short-period skyrmion square lattice (SkL) in the absence of Dzyaloshinskii-Moriya interaction (DMI). In this system, Gd atoms are arranged on a square lattice, forming 2D layers separated by the Ru-Si network in the out-of-plane direction. RKKY-type exchange between the Gd moments results in an exchange frustration, which is the main source of non-collinearity in the spiral phases of the system. So far, the literature has emphasized the importance of in-plane <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mover accent="true"><b:mi>Q</b:mi><b:mo>⃗</b:mo></b:mover></b:math> vectors in understanding the magnetic phases of the system as they appear and have been observed in the 2D Gd layers. In this work, we calculate the Gd-Gd magnetic exchange interactions (<d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:msub><d:mi>J</d:mi><d:mrow><d:mi>i</d:mi><d:mi>j</d:mi></d:mrow></d:msub></d:math>) and perform atomic spin dynamics (ASD) simulations, providing new insights about <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:mrow><e:msub><e:mi>GdRu</e:mi><e:mn>2</e:mn></e:msub><e:msub><e:mi>Si</e:mi><e:mn>2</e:mn></e:msub></e:mrow></e:math>. Our calculated <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"><f:msub><f:mi>J</f:mi><f:mrow><f:mi>i</f:mi><f:mi>j</f:mi></f:mrow></f:msub></f:math> shows that the strongest magnetic interaction occurs between Gd atoms along the [111] body-diagonal direction of the unit cell. This, along with the body-centered tetragonal structure of the Gd sublattice, points to the presence of a hitherto ignored modulation vector, <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"><g:msub><g:mover accent="true"><g:mi>Q</g:mi><g:mo>⃗</g:mo></g:mover><g:mrow><g:mo>[</g:mo><g:mn>111</g:mn><g:mo>]</g:mo></g:mrow></g:msub></g:math>, along the [111] direction in the spiral phases of the system. We confirm this with ASD simulations and establish that the magnetic phases in <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"><i:mrow><i:msub><i:mi>GdRu</i:mi><i:mn>2</i:mn></i:msub><i:msub><i:mi>Si</i:mi><i:mn>2</i:mn></i:msub></i:mrow></i:math> are far more complex than they appear to be in the 2D Gd layers. This interlayer modulation vector <j:math xmlns:j="http://www.w3.org/1998/Math/MathML"><j:msub><j:mover accent="true"><j:mi>Q</j:mi><j:mo>⃗</j:mo></j:mover><j:mrow><j:mo>[</j:mo><j:mn>111</j:mn><j:mo>]</j:mo></j:mrow></j:msub></j:math>, along with the intralayer <l:math xmlns:l="http://www.w3.org/1998/Math/MathML"><l:msub><l:mover accent="true"><l:mi>Q</l:mi><l:mo>⃗</l:mo></l:mover><l:mrow><l:mo>[</l:mo><l:mn>100</l:mn><l:mo>]</l:mo></l:mrow></l:msub></l:math> and <n:math xmlns:n="http://www.w3.org/1998/Math/MathML"><n:msub><n:mover accent="true"><n:mi>Q</n:mi><n:mo>⃗</n:mo></n:mover><n:mrow><n:mo>[</n:mo><n:mn>010</n:mn><n:mo>]</n:mo></n:mrow></n:msub></n:math>, determines the total magnetic ordering of the system. Despite its layered appearance, both structurally and magnetically, it is in reality a strong 3D magnet, requiring additional considerations for its proper theoretical modeling. Taking these into account, our ASD simulations provide an excellent description of the experimentally observed magnetic phase transitions. Furthermore, our work reveals the critical influence of dipolar interactions on the ground-state magnetic properties of <p:math xmlns:p="http://www.w3.org/1998/Math/MathML"><p:mrow><p:msub><p:mi>GdRu</p:mi><p:mn>2</p:mn></p:msub><p:msub><p:mi>Si</p:mi><p:mn>2</p:mn></p:msub></p:mrow></p:math>, a direct consequence of the large magnetic moment of Gd. We observed a competition between dipolar interactions and uniaxial anisotropy, with each favoring a different magnetic ordering. However, the dominance of the dipolar interaction is evident in the observed ground state. This suggests a weak uniaxial anisotropy in the system, which aligns with the spherical symmetry of the Gd <q:math xmlns:q="http://www.w3.org/1998/Math/MathML"><q:mrow><q:mn>4</q:mn><q:mi>f</q:mi></q:mrow></q:math> states. Our findings offer a deeper understanding of <r:math xmlns:r="http://www.w3.org/1998/Math/MathML"><r:mrow><r:msub><r:mi>GdRu</r:mi><r:mn>2</r:mn></r:msub><r:msub><r:mi>Si</r:mi><r:mn>2</r:mn></r:msub></r:mrow></r:math>'s complex magnetism and highlight the potential for similar important interlayer effects in other layered magnetic systems.