Noncollinear magnetic orders in PtFePt and PtMnPt stripes tailored by external electric fields: First-principles interactions and model energy landscapes

The stable and metastable magnetic orders in one-dimensional PtFePt and PtMnPt stripes, as well as the collective spin reorientations connecting them, are investigated by combining first-principles electronic theory with a detailed characterization of the associated energy landscapes. The relative stability of collinear and noncollinear magnetic orders is determined within density-functional theory by computing the frozen-magnon dispersion relations <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mrow> <a:msubsup> <a:mi>ɛ</a:mi> <a:mrow> <a:mi>δ</a:mi> <a:mi>γ</a:mi> </a:mrow> <a:mi>α</a:mi> </a:msubsup> <a:mrow> <a:mo>(</a:mo> <a:mi>q</a:mi> <a:mo>)</a:mo> </a:mrow> </a:mrow> </a:math> as functions of wave number <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"> <b:mi>q</b:mi> </b:math> , spin-polarization plane <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"> <c:mrow> <c:mi>δ</c:mi> <c:mi>γ</c:mi> </c:mrow> </c:math> , and chirality <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"> <d:mrow> <d:mi>α</d:mi> <d:mo>=</d:mo> <d:mo>±</d:mo> <d:mn>1</d:mn> </d:mrow> </d:math> . The effective interactions between the local magnetic moments <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"> <e:msub> <e:mrow> <e:mi mathvariant="bold-italic">μ</e:mi> </e:mrow> <e:mi>i</e:mi> </e:msub> </e:math> at the <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"> <g:mrow> <g:mn>3</g:mn> <g:mi>d</g:mi> </g:mrow> </g:math> transition metal atoms are derived, including the local magnetic anisotropy energies <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"> <h:msubsup> <h:mi>K</h:mi> <h:mi>i</h:mi> <h:mi>δ</h:mi> </h:msubsup> </h:math> , the symmetric exchange couplings <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"> <i:msubsup> <i:mi>J</i:mi> <i:mrow> <i:mi>i</i:mi> <i:mi>j</i:mi> </i:mrow> <i:mi>δ</i:mi> </i:msubsup> </i:math> , and the Dzyaloshinskii-Moriya (DM) vectors <j:math xmlns:j="http://www.w3.org/1998/Math/MathML"> <j:msub> <j:mi mathvariant="bold-italic">D</j:mi> <j:mrow> <j:mi>i</j:mi> <j:mi>j</j:mi> </j:mrow> </j:msub> </j:math> , where <l:math xmlns:l="http://www.w3.org/1998/Math/MathML"> <l:mrow> <l:mi>δ</l:mi> <l:mo>=</l:mo> <l:mi>x</l:mi> <l:mo>,</l:mo> <l:mi>y</l:mi> <l:mo>,</l:mo> <l:mi>z</l:mi> </l:mrow> </l:math> denotes the direction relative to the stripe geometry. The consequences of applying an external electric field (EF) on the magnetic couplings are quantified. Significant DM interactions are triggered by the EF, which breaks the inversion symmetry of the stripes. First-nearest-neighbor DM coupling dominates in ferromagnetic PtFePt, whereas second-nearest-neighbor DM coupling dominates in antiferromagnetic PtMnPt. The magnetic energy landscapes of the corresponding classical spin Hamiltonians <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>H</m:mi> <m:mo>[</m:mo> <m:msub> <m:mrow> <m:mi mathvariant="bold-italic">μ</m:mi> </m:mrow> <m:mi>i</m:mi> </m:msub> <m:mo>]</m:mo> </m:mrow> </m:math> are systematically explored without imposing any constraints on the orientations of <o:math xmlns:o="http://www.w3.org/1998/Math/MathML"> <o:msub> <o:mrow> <o:mi mathvariant="bold-italic">μ</o:mi> </o:mrow> <o:mi>i</o:mi> </o:msub> </o:math> . The metastable magnetic configurations involve domain walls (DWs) superimposed on the ferromagnetic (PtFePt) or antiferromagnetic (PtMnPt) backgrounds, whose width can be tuned by the applied EF. The morphology of the transition states and the minimum-energy paths connecting the local minima reveal the mechanisms for the creation, annihilation, and translation of magnetic domains (MDs). One observes that MDs with walls of opposite chirality, yielding a total winding number <q:math xmlns:q="http://www.w3.org/1998/Math/MathML"> <q:mrow> <q:msub> <q:mi>η</q:mi> <q:mi>z</q:mi> </q:msub> <q:mo>=</q:mo> <q:mn>0</q:mn> </q:mrow> </q:math> , identical to that of the ground state, are unstable, whereas domains with walls of the same chirality and <r:math xmlns:r="http://www.w3.org/1998/Math/MathML"> <r:mrow> <r:msub> <r:mi>η</r:mi> <r:mi>z</r:mi> </r:msub> <r:mo>=</r:mo> <r:mo>±</r:mo> <r:mn>1</r:mn> </r:mrow> </r:math> are stabilized by significant energy barriers. Extremely small upper bounds for the energy barriers associated with domain-wall translations along the stripes are found, implying remarkably high mobility. The roles of the microscopic interactions <s:math xmlns:s="http://www.w3.org/1998/Math/MathML"> <s:msubsup> <s:mi>K</s:mi> <s:mi>i</s:mi> <s:mi>δ</s:mi> </s:msubsup> <s:mo>,</s:mo> <s:mo> </s:mo> <s:msubsup> <s:mi>J</s:mi> <s:mrow> <s:mi>i</s:mi> <s:mi>j</s:mi> </s:mrow> <s:mi>δ</s:mi> </s:msubsup> </s:math> , and <t:math xmlns:t="http://www.w3.org/1998/Math/MathML"> <t:msub> <t:mi mathvariant="bold-italic">D</t:mi> <t:mrow> <t:mi>i</t:mi> <t:mi>j</t:mi> </t:mrow> </t:msub> </t:math> in the topological protection, energy barriers, and relative stability of magnetic configurations are disclosed. The present investigations may serve as a methodological blueprint for advancing our understanding of the local-environment dependence of interactions between magnetic moments, their role in collective responses and relaxation processes, and the energy landscapes that underlie the physics of magnetic low-dimensional systems and nanostructures.