Influence of pressure on the properties of the multigap type-I superconductor BeAu

We report on studies of the superconducting and normal-state properties of the noncentrosymmetric superconductor BeAu under hydrostatic pressure conditions. The room-temperature equation of state (EOS) reveals the values of the bulk modulus (<a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:msub><a:mi>B</a:mi><a:mn>0</a:mn></a:msub></a:math>) and its first derivative (<b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:msubsup><b:mi>B</b:mi><b:mn>0</b:mn><b:mo>′</b:mo></b:msubsup></b:math>) at ambient pressure to be <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mrow><c:msub><c:mi>B</c:mi><c:mn>0</c:mn></c:msub><c:mo>≃</c:mo><c:mn>133</c:mn><c:mspace width="0.28em"/><c:mi>GPa</c:mi></c:mrow></c:math> and <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:mrow><e:msubsup><e:mi>B</e:mi><e:mn>0</e:mn><e:mo>′</e:mo></e:msubsup><e:mo>≃</e:mo><e:mn>30</e:mn></e:mrow></e:math>, respectively. Up to the highest pressures studied (<f:math xmlns:f="http://www.w3.org/1998/Math/MathML"><f:mrow><f:mi>p</f:mi><f:mo>≃</f:mo><f:mn>2.2</f:mn><f:mspace width="0.28em"/><f:mi>GPa</f:mi></f:mrow></f:math>), BeAu remains a multigap type-I superconductor. The analysis of <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"><h:mrow><h:msub><h:mi>B</h:mi><h:mi mathvariant="normal">c</h:mi></h:msub><h:mrow><h:mo>(</h:mo><h:mi>T</h:mi><h:mo>,</h:mo><h:mi>p</h:mi><h:mo>)</h:mo></h:mrow></h:mrow></h:math> data within the self-consistent two-gap approach suggests the presence of two superconducting energy gaps, with the gap-to-<j:math xmlns:j="http://www.w3.org/1998/Math/MathML"><j:msub><j:mi>T</j:mi><j:mi mathvariant="normal">c</j:mi></j:msub></j:math> ratios <l:math xmlns:l="http://www.w3.org/1998/Math/MathML"><l:mrow><l:msub><l:mi mathvariant="normal">Δ</l:mi><l:mn>1</l:mn></l:msub><l:mo>/</l:mo><l:msub><l:mi>k</l:mi><l:mi mathvariant="normal">B</l:mi></l:msub><l:msub><l:mi>T</l:mi><l:mi mathvariant="normal">c</l:mi></l:msub><l:mo>∼</l:mo><l:mn>2.3</l:mn></l:mrow></l:math> and <p:math xmlns:p="http://www.w3.org/1998/Math/MathML"><p:mrow><p:msub><p:mi mathvariant="normal">Δ</p:mi><p:mn>2</p:mn></p:msub><p:mo>/</p:mo><p:msub><p:mi>k</p:mi><p:mi mathvariant="normal">B</p:mi></p:msub><p:msub><p:mi>T</p:mi><p:mi mathvariant="normal">c</p:mi></p:msub><p:mo>∼</p:mo><p:mn>1.1</p:mn></p:mrow></p:math> for the larger and smaller gaps, respectively [<t:math xmlns:t="http://www.w3.org/1998/Math/MathML"><t:mrow><t:mi mathvariant="normal">Δ</t:mi><t:mo>=</t:mo><t:mi mathvariant="normal">Δ</t:mi><t:mo>(</t:mo><t:mn>0</t:mn><t:mo>)</t:mo></t:mrow></t:math> is the zero-temperature value of the gap and <w:math xmlns:w="http://www.w3.org/1998/Math/MathML"><w:msub><w:mi>k</w:mi><w:mi mathvariant="normal">B</w:mi></w:msub></w:math> is the Boltzmann constant]. With increasing pressure, <y:math xmlns:y="http://www.w3.org/1998/Math/MathML"><y:mrow><y:msub><y:mi mathvariant="normal">Δ</y:mi><y:mn>1</y:mn></y:msub><y:mo>/</y:mo><y:msub><y:mi>k</y:mi><y:mi mathvariant="normal">B</y:mi></y:msub><y:msub><y:mi>T</y:mi><y:mi mathvariant="normal">c</y:mi></y:msub></y:mrow></y:math> increases while <cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML"><cb:mrow><cb:msub><cb:mi mathvariant="normal">Δ</cb:mi><cb:mn>2</cb:mn></cb:msub><cb:mo>/</cb:mo><cb:msub><cb:mi>k</cb:mi><cb:mi mathvariant="normal">B</cb:mi></cb:msub><cb:msub><cb:mi>T</cb:mi><cb:mi mathvariant="normal">c</cb:mi></cb:msub></cb:mrow></cb:math> decreases, suggesting that pressure enhances (weakens) the coupling strength between the superconducting carriers within the bands where the larger (smaller) superconducting energy gap has opened. The superconducting transition temperature <gb:math xmlns:gb="http://www.w3.org/1998/Math/MathML"><gb:msub><gb:mi>T</gb:mi><gb:mi mathvariant="normal">c</gb:mi></gb:msub></gb:math>, the zero-temperature values of the superconducting gaps <ib:math xmlns:ib="http://www.w3.org/1998/Math/MathML"><ib:msub><ib:mi mathvariant="normal">Δ</ib:mi><ib:mn>1</ib:mn></ib:msub></ib:math> and <kb:math xmlns:kb="http://www.w3.org/1998/Math/MathML"><kb:msub><kb:mi mathvariant="normal">Δ</kb:mi><kb:mn>2</kb:mn></kb:msub></kb:math>, and the zero-temperature value of the thermodynamic critical field <mb:math xmlns:mb="http://www.w3.org/1998/Math/MathML"><mb:mrow><mb:msub><mb:mi>B</mb:mi><mb:mi mathvariant="normal">c</mb:mi></mb:msub><mb:mrow><mb:mo>(</mb:mo><mb:mn>0</mb:mn><mb:mo>)</mb:mo></mb:mrow></mb:mrow></mb:math> decrease with increasing pressure, with the rates of <ob:math xmlns:ob="http://www.w3.org/1998/Math/MathML"><ob:mrow><ob:mi mathvariant="normal">d</ob:mi><ob:msub><ob:mi>T</ob:mi><ob:mi mathvariant="normal">c</ob:mi></ob:msub><ob:mo>/</ob:mo><ob:mi mathvariant="normal">d</ob:mi><ob:mi>p</ob:mi><ob:mo>≃</ob:mo><ob:mo>−</ob:mo><ob:mn>0.197</ob:mn><ob:mspace width="0.28em"/><ob:mi mathvariant="normal">K</ob:mi><ob:mo>/</ob:mo><ob:mi>GPa</ob:mi></ob:mrow><ob:mo>,</ob:mo><ob:mo> </ob:mo><ob:mrow><ob:mi mathvariant="normal">d</ob:mi><ob:msub><ob:mi mathvariant="normal">Δ</ob:mi><ob:mn>1</ob:mn></ob:msub><ob:mo>/</ob:mo><ob:mi mathvariant="normal">d</ob:mi><ob:mi>p</ob:mi><ob:mo>≃</ob:mo><ob:mo>−</ob:mo><ob:mn>0.034</ob:mn><ob:mspace width="0.28em"/><ob:mi>meV</ob:mi><ob:mo>/</ob:mo><ob:mi>GPa</ob:mi></ob:mrow><ob:mo>,</ob:mo><ob:mo> </ob:mo><ob:mrow><ob:mi mathvariant="normal">d</ob:mi><ob:msub><ob:mi mathvariant="normal">Δ</ob:mi><ob:mn>2</ob:mn></ob:msub><ob:mo>/</ob:mo><ob:mi mathvariant="normal">d</ob:mi><ob:mi>p</ob:mi><ob:mo>≃</ob:mo><ob:mo>−</ob:mo><ob:mn>0.029</ob:mn><ob:mspace width="0.28em"/><ob:mi>meV</ob:mi><ob:mo>/</ob:mo><ob:mi>GPa</ob:mi></ob:mrow></ob:math>, and <cc:math xmlns:cc="http://www.w3.org/1998/Math/MathML"><cc:mrow><cc:mi mathvariant="normal">d</cc:mi><cc:msub><cc:mi>B</cc:mi><cc:mi mathvariant="normal">c</cc:mi></cc:msub><cc:mrow><cc:mo>(</cc:mo><cc:mn>0</cc:mn><cc:mo>)</cc:mo></cc:mrow><cc:mo>/</cc:mo><cc:mi mathvariant="normal">d</cc:mi><cc:mi>p</cc:mi><cc:mo>≃</cc:mo><cc:mo>−</cc:mo><cc:mn>2.65</cc:mn><cc:mspace width="0.28em"/><cc:mi>mT</cc:mi><cc:mo>/</cc:mo><cc:mi>GPa</cc:mi></cc:mrow></cc:math>, respectively. The measured <hc:math xmlns:hc="http://www.w3.org/1998/Math/MathML"><hc:mrow><hc:msub><hc:mi>B</hc:mi><hc:mi mathvariant="normal">c</hc:mi></hc:msub><hc:mrow><hc:mo>(</hc:mo><hc:mn>0</hc:mn><hc:mo>)</hc:mo></hc:mrow></hc:mrow></hc:math> values plotted as a function of <jc:math xmlns:jc="http://www.w3.org/1998/Math/MathML"><jc:msub><jc:mi>T</jc:mi><jc:mi mathvariant="normal">c</jc:mi></jc:msub></jc:math> follow an empirical scaling relation established for conventional type-I superconductors.