Universal chemical formula dependence of ab initio low-energy effective Hamiltonian in single-layer carrier-doped cuprate superconductors: Study using a hierarchical dependence extraction algorithm

We explore the possibility to control the superconducting transition temperature at optimal hole doping <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:msubsup><a:mi>T</a:mi><a:mrow><a:mi>c</a:mi></a:mrow><a:mi>opt</a:mi></a:msubsup></a:math> in cuprates by tuning the chemical formula (CF). <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:msubsup><b:mi>T</b:mi><b:mrow><b:mi>c</b:mi></b:mrow><b:mi>opt</b:mi></b:msubsup></b:math> can be theoretically predicted from the parameters of the low-energy effective Hamiltonian with one antibonding (AB) <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mrow><c:mi>Cu</c:mi><c:mn>3</c:mn><c:msub><c:mi>d</c:mi><c:mrow><c:msup><c:mi>x</c:mi><c:mn>2</c:mn></c:msup><c:mo>−</c:mo><c:msup><c:mi>y</c:mi><c:mn>2</c:mn></c:msup></c:mrow></c:msub><c:mo>/</c:mo><c:mi mathvariant="normal">O</c:mi><c:mn>2</c:mn><c:msub><c:mi>p</c:mi><c:mi>σ</c:mi></c:msub></c:mrow></c:math> orbital per Cu atom in the <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:msub><e:mi>CuO</e:mi><e:mn>2</e:mn></e:msub></e:math> plane, notably the nearest-neighbor hopping amplitude <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"><f:mrow><f:mrow><f:mo>|</f:mo></f:mrow><f:msub><f:mi>t</f:mi><f:mn>1</f:mn></f:msub><f:mrow><f:mo>|</f:mo></f:mrow></f:mrow></f:math> and the ratio <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"><g:mrow><g:mrow><g:mi>u</g:mi><g:mo>=</g:mo><g:mi>U</g:mi><g:mo>/</g:mo><g:mo>|</g:mo></g:mrow><g:msub><g:mi>t</g:mi><g:mn>1</g:mn></g:msub><g:mrow><g:mo>|</g:mo></g:mrow></g:mrow></g:math>, where <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"><h:mi>U</h:mi></h:math> is the onsite effective Coulomb repulsion. However, the CF dependence of <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"><i:mrow><i:mrow><i:mo>|</i:mo></i:mrow><i:msub><i:mi>t</i:mi><i:mn>1</i:mn></i:msub><i:mrow><i:mo>|</i:mo></i:mrow></i:mrow></i:math> and <j:math xmlns:j="http://www.w3.org/1998/Math/MathML"><j:mi>u</j:mi></j:math> is a highly nontrivial question. In this paper, we propose the universal dependence of <k:math xmlns:k="http://www.w3.org/1998/Math/MathML"><k:mrow><k:mrow><k:mo>|</k:mo></k:mrow><k:msub><k:mi>t</k:mi><k:mn>1</k:mn></k:msub><k:mrow><k:mo>|</k:mo></k:mrow></k:mrow></k:math> and <l:math xmlns:l="http://www.w3.org/1998/Math/MathML"><l:mi>u</l:mi></l:math> on the CF and structural features in hole doped cuprates with a single <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:msub><m:mi>CuO</m:mi><m:mn>2</m:mn></m:msub></m:math> layer sandwiched between block layers. To do so, we perform extensive calculations of <n:math xmlns:n="http://www.w3.org/1998/Math/MathML"><n:mrow><n:mrow><n:mo>|</n:mo></n:mrow><n:msub><n:mi>t</n:mi><n:mn>1</n:mn></n:msub><n:mrow><n:mo>|</n:mo></n:mrow></n:mrow></n:math> and <o:math xmlns:o="http://www.w3.org/1998/Math/MathML"><o:mi>u</o:mi></o:math> and analyze the results by employing a machine-learning method called hierarchical dependence extraction (HDE). The main results are (a) <p:math xmlns:p="http://www.w3.org/1998/Math/MathML"><p:mrow><p:mrow><p:mo>|</p:mo></p:mrow><p:msub><p:mi>t</p:mi><p:mn>1</p:mn></p:msub><p:mrow><p:mo>|</p:mo></p:mrow></p:mrow></p:math> has a main-order dependence on the radii <q:math xmlns:q="http://www.w3.org/1998/Math/MathML"><q:msub><q:mi>R</q:mi><q:mi mathvariant="normal">X</q:mi></q:msub></q:math> and <s:math xmlns:s="http://www.w3.org/1998/Math/MathML"><s:msub><s:mi>R</s:mi><s:mi mathvariant="normal">A</s:mi></s:msub></s:math> of the apical anion X and cation A in the block layer. (<u:math xmlns:u="http://www.w3.org/1998/Math/MathML"><u:mrow><u:mrow><u:mo>|</u:mo></u:mrow><u:msub><u:mi>t</u:mi><u:mn>1</u:mn></u:msub><u:mrow><u:mo>|</u:mo></u:mrow></u:mrow></u:math> increases when <v:math xmlns:v="http://www.w3.org/1998/Math/MathML"><v:msub><v:mi>R</v:mi><v:mi mathvariant="normal">X</v:mi></v:msub></v:math> or <x:math xmlns:x="http://www.w3.org/1998/Math/MathML"><x:msub><x:mi>R</x:mi><x:mi mathvariant="normal">A</x:mi></x:msub></x:math> decreases.) (b) <z:math xmlns:z="http://www.w3.org/1998/Math/MathML"><z:mi>u</z:mi></z:math> has a main-order dependence on the ionic charge <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML"><ab:msub><ab:mi>Z</ab:mi><ab:mi mathvariant="normal">X</ab:mi></ab:msub></ab:math> of X and the hole doping <cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML"><cb:mi>δ</cb:mi></cb:math> of the AB orbital. (<db:math xmlns:db="http://www.w3.org/1998/Math/MathML"><db:mi>u</db:mi></db:math> decreases when <eb:math xmlns:eb="http://www.w3.org/1998/Math/MathML"><eb:mrow><eb:mrow><eb:mo>|</eb:mo></eb:mrow><eb:msub><eb:mi>Z</eb:mi><eb:mi mathvariant="normal">X</eb:mi></eb:msub><eb:mrow><eb:mo>|</eb:mo></eb:mrow></eb:mrow></eb:math> increases or <gb:math xmlns:gb="http://www.w3.org/1998/Math/MathML"><gb:mi>δ</gb:mi></gb:math> increases.) We elucidate and discuss the microscopic mechanism of items (a) and (b). We demonstrate the predictive power of the HDE by showing the consistency between items (a) and (b) and results from previous works. The present results provide a basis for optimizing superconducting properties in cuprates and possibly akin materials. Also, the HDE method offers a general platform to identify dependencies between physical quantities.