Prototypes of Nonrelativistic Spin Splitting and Polarization in Symmetry Broken Antiferromagnets

Antiferromagnets that break both space-time reversal and translation-spin-rotation symmetries were recently predicted [L.-D. Yuan, Z. Wang, J.-W. Luo, E. I. Rashba, and A. Zunger, ] to possess splitting between the otherwise spin-degenerate energy bands even without the relativistic spin-orbit coupling (SOC). Here, we point out that such nonrelativistic spin splitting (NRSS)—in particular, “spin splitting type 4” (SST-4) symmetry-broken antiferromagnets—can be divided into subgroups having distinct patterns of spin splitting and spin textures, depending on additional auxiliary symmetries of spin interconversion and polarity. These SST-4 subgroups include the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>α</a:mi> </a:math> -type (no spin-interconverting symmetry) having spin splitting at the Brillouin zone center, as well as the <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:mi>β</c:mi> </c:math> subgroup in which a rotation symmetry is applied and determines the alternating spin texture and the <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:mi>γ</e:mi> </e:math> subgroup having exclusively reflection spin-interconverting symmetry. Unlike ferrimagnets, the <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:mi>α</g:mi> </g:math> -type compounds are shown to have tiny net magnetization at finite temperature and thus avoid the adverse effect of the stray field. The <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:mi>α</i:mi> </i:math> and <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"> <k:mi>β</k:mi> </k:math> subgroups can be either polar or nonpolar, whereas the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"> <m:mi>γ</m:mi> </m:math> subgroup is polar only, providing a basis for possible switching by external fields. The combination of NRSS-enabling and auxiliary symmetries is used here as a filter for identifying previously synthesized compounds as specific prototypes. Their characteristic splitting and spin polarization are calculated by density functional theory to the benefit of potential future experimental testing. Interesting results are as follows: (i) SOC-independent NRSS can exceed the magnitude of the SOC-induced Rashba and Dresselhaus spin splitting in semiconductors. (ii) Examples of predicted <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"> <o:mi>α</o:mi> </o:math> -type insulating compounds include <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"> <q:mrow> <q:msub> <q:mi>BiCrO</q:mi> <q:mn>3</q:mn> </q:msub> </q:mrow> </q:math> (nonpolar) and <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"> <s:mrow> <s:msub> <s:mi>Mn</s:mi> <s:mn>2</s:mn> </s:msub> <s:msub> <s:mi>ScSbO</s:mi> <s:mn>6</s:mn> </s:msub> </s:mrow> </s:math> (polar), the latter having spin splitting of 158 meV and 160 meV in the valence and conduction bands, respectively. (iii) The <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"> <u:mi>β</u:mi> </u:math> -type ( <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" display="inline"> <w:mrow> <w:msub> <w:mrow> <w:mtext>Cu</w:mtext> </w:mrow> <w:mrow> <w:mn>2</w:mn> </w:mrow> </w:msub> <w:msub> <w:mrow> <w:mtext>Y</w:mtext> </w:mrow> <w:mrow> <w:mn>2</w:mn> </w:mrow> </w:msub> <w:msub> <w:mrow> <w:mtext>O</w:mtext> </w:mrow> <w:mrow> <w:mn>5</w:mn> </w:mrow> </w:msub> </w:mrow> </w:math> and <y:math xmlns:y="http://www.w3.org/1998/Math/MathML" display="inline"> <y:mrow> <y:msub> <y:mrow> <y:mtext>FeF</y:mtext> </y:mrow> <y:mrow> <y:mn>2</y:mn> </y:mrow> </y:msub> </y:mrow> </y:math> ) and <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" display="inline"> <ab:mi>γ</ab:mi> </ab:math> -type compounds ( <cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML" display="inline"> <cb:mrow> <cb:mrow> <cb:msub> <cb:mrow> <cb:mtext>Mn</cb:mtext> </cb:mrow> <cb:mrow> <cb:mn>4</cb:mn> </cb:mrow> </cb:msub> <cb:msub> <cb:mrow> <cb:mtext>Nb</cb:mtext> </cb:mrow> <cb:mrow> <cb:mn>2</cb:mn> </cb:mrow> </cb:msub> <cb:msub> <cb:mrow> <cb:mtext>O</cb:mtext> </cb:mrow> <cb:mrow> <cb:mn>9</cb:mn> </cb:mrow> </cb:msub> </cb:mrow> </cb:mrow> </cb:math> and <eb:math xmlns:eb="http://www.w3.org/1998/Math/MathML" display="inline"> <eb:mrow> <eb:msub> <eb:mrow> <eb:mtext>FeScO</eb:mtext> </eb:mrow> <eb:mrow> <eb:mn>3</eb:mn> </eb:mrow> </eb:msub> </eb:mrow> </eb:math> ) are distinguished both by their auxiliary symmetries and polarity. The spin textures of <gb:math xmlns:gb="http://www.w3.org/1998/Math/MathML" display="inline"> <gb:mi>γ</gb:mi> </gb:math> -type compounds are mirror reflected with spin degeneracy of the wave vectors on that mirror. These observations will likely broaden the experimental playing field of NRSS physics significantly.