Third-order and fifth-order nonlinear spin-current generation in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>g</mml:mi> </mml:math> -wave and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>i</mml:mi> </mml:math> -wave altermagnets and perfectly nonreciprocal spin current in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>f</mml:mi> </mml:math> -wave magnets
Motohiko Ezawa
Abstract
A prominent feature of <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mi>d</a:mi> </a:math> -wave altermagnets is the pure spin current generated in the absence of spin-orbit interactions. In the context of symmetry, there are <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"> <b:mi>s</b:mi> </b:math> -wave, <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"> <c:mi>p</c:mi> </c:math> -wave, <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"> <d:mi>d</d:mi> </d:math> -wave, <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"> <e:mi>f</e:mi> </e:math> -wave, <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"> <f:mi>g</f:mi> </f:math> -wave, and <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"> <g:mi>i</g:mi> </g:math> -wave magnets. In this paper, making an analytic study of two-band Hamiltonian systems coupled with electrons, we demonstrate unexpectedly that only the <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"> <h:mrow> <h:mi>ℓ</h:mi> <h:mi>th</h:mi> </h:mrow> </h:math> order nonlinear transverse spin current proportional to <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"> <i:msup> <i:mi>E</i:mi> <i:mi>ℓ</i:mi> </i:msup> </i:math> is generated in higher-wave symmetric magnets when the number of nodes is <j:math xmlns:j="http://www.w3.org/1998/Math/MathML"> <j:mrow> <j:mi>ℓ</j:mi> <j:mo>+</j:mo> <j:mn>1</j:mn> </j:mrow> </j:math> . Here <k:math xmlns:k="http://www.w3.org/1998/Math/MathML"> <k:mi>E</k:mi> </k:math> is applied electric field. The nonlinear spin current is essential provided the linear spin current is absent. Indeed, only the third-order nonlinear spin current is generated in <l:math xmlns:l="http://www.w3.org/1998/Math/MathML"> <l:mi>g</l:mi> </l:math> -wave altermagnets, while only the fifth-order spin current is generated in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>i</m:mi> </m:math> -wave altermagnets. In particular, only the second-order nonlinear spin current is generated in <n:math xmlns:n="http://www.w3.org/1998/Math/MathML"> <n:mi>f</n:mi> </n:math> -wave magnets, which leads to a perfect nonreciprocal spin current. On the other hand, there is no spin-current generation in <o:math xmlns:o="http://www.w3.org/1998/Math/MathML"> <o:mi>p</o:mi> </o:math> -wave magnets.