Spin-lattice couplings and effect of displacements on magnetic interactions of a skyrmion system PdFe/Ir(111)

PdFe/Ir(111) has attracted tremendous attention for next-generation spintronics devices due to existence of magnetic skyrmions with the external magnetic field. Our density functional theoretical calculations in combination with spin dynamics simulation suggest that the spin spiral phase in fcc stacked PdFe/Ir(111) flips into the skyrmion lattice phase around B _{ext} \sim <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:msub> <mml:mi/> <mml:mrow> <mml:mi>e</mml:mi> <mml:mi>x</mml:mi> <mml:mi>t</mml:mi> </mml:mrow> </mml:msub> <mml:mo>∼</mml:mo> </mml:mrow> </mml:math> 6 T. This leads to the microscopic understanding of the thermodynamic and kinetic behaviours affected by the intrinsic spin-lattice couplings (SLCs) in this skyrmion material for magneto-mechanical properties. Here we calculate fully relativistic SLC parameters from first principle computations and investigate the effect of SLC on dynamical magnetic interactions in skyrmion multilayers PdFe/Ir(111). The exchange interactions arising from next nearest-neighbors (NN) in this material are highly frustrated and responsible for enhancing skyrmion stability. We report the larger spin-lattice effect on both dynamical Heisenberg exchanges and Dzyaloshinskii-Moriya interactions for next NN compared to NN which is in contrast with recently observed spin-lattice effect in bulk bcc Fe and CrI _3 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mi/> <mml:mn>3</mml:mn> </mml:msub> </mml:math> monolayer. Based on our analysis, we find that the effective measures of SLCs in fcc (hcp) stacking of PdFe/Ir(111) are \sim 2.71 ( \sim 2.36) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mo>∼</mml:mo> <mml:mn>2.71</mml:mn> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mo>∼</mml:mo> <mml:mn>2.36</mml:mn> <mml:mo stretchy="false" form="postfix">)</mml:mo> </mml:mrow> </mml:math> and \sim 14.71 ( \sim21.89) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mo>∼</mml:mo> <mml:mn>14.71</mml:mn> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mo>∼</mml:mo> <mml:mn>21.89</mml:mn> <mml:mo stretchy="false" form="postfix">)</mml:mo> </mml:mrow> </mml:math> times stronger for NN and next NN respectively, compared to bcc Fe. The linear regime of displacement for SLC parameters is ≤ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mo>≤</mml:mo> </mml:math> 0.02 Å which is 0.72% of the lattice constant for PdFe/Ir(111). The microscopic understanding of SLCs provided by our current study could help in designing spintronic devices based on thermodynamic properties of skyrmion multilayers.