Influence of temperature, electric field and spin Zeeman field on the quantum capacitance of the germanene monolayer material

Abstract In this work, we investigate the quantum capacitance (C Q ) of monolayer germanene under the influences of temperature, the electric field, and the Zeeman effect. The results show that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi>C</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>Q</mml:mi> </mml:mrow> </mml:msub> </mml:math> reaches a minimum value at the Dirac point ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi>E</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>F</mml:mi> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:math> ), which is a typical characteristic of Dirac materials. At low temperatures ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>T</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mn>0</mml:mn> <mml:mspace width="0.25em"/> <mml:mi mathvariant="normal">K</mml:mi> <mml:mo>,</mml:mo> <mml:mspace width="0.25em"/> <mml:mn>20</mml:mn> <mml:mspace width="0.25em"/> <mml:mi mathvariant="normal">K</mml:mi> </mml:math> ), <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi>C</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>Q</mml:mi> </mml:mrow> </mml:msub> </mml:math> is nearly zero at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi>E</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>F</mml:mi> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:math> , reflecting a significant reduction in the available density of states. As the temperature increases, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi>C</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>Q</mml:mi> </mml:mrow> </mml:msub> </mml:math> also increases, particularly at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi>E</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>F</mml:mi> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:math> , due to the thermal excitation of electronic states. The presence of an electric field and the Zeeman effect do not break the symmetry of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi>C</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>Q</mml:mi> </mml:mrow> </mml:msub> </mml:math> around the Dirac point but affect its minimum value at low temperatures. Notably, our results indicate that variations in electric field energy do not significantly alter the value or overall trend of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi>C</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>Q</mml:mi> </mml:mrow> </mml:msub> </mml:math> . These findings highlight influence of temperature, electric field and spin Zeeman field on the quantum capacitance of the germanene, which is crucial for quantum electronic applications. The ability to fine-tune <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi>C</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>Q</mml:mi> </mml:mrow> </mml:msub> </mml:math> while maintain structural stability under room temperature conditions makes germanene a promising candidate for high-performance electronic devices.