The Low-Temperature Expansion of the Casimir-Polder Free Energy of an Atom with Graphene

• Main Authors: Nail Khusnutdinov, Natalia Emelianova
• Format: Article
• Language: English
• Published: MDPI AG 2021-03-01
• Series: Universe
• Subjects: Casimir-Polder force; graphene; dispersion forces
• Online Access: https://www.mdpi.com/2218-1997/7/3/70

We consider the low-temperature expansion of the Casimir-Polder free energy for an atom and graphene by using the Poisson representation of the free energy. We extend our previous analysis on the different relations between chemical potential <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>μ</mi></semantics></math></inline-formula> and mass gap parameter <i>m</i>. The key role plays the dependence of graphene conductivities on the <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>μ</mi></semantics></math></inline-formula> and <i>m</i>. For simplicity, we made the manifest calculations for zero values of the Fermi velocity. For <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>μ</mi><mo>></mo><mi>m</mi></mrow></semantics></math></inline-formula>, the thermal correction <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∼</mo><msup><mi>T</mi><mn>2</mn></msup></mrow></semantics></math></inline-formula>, and for <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>μ</mi><mo><</mo><mi>m</mi></mrow></semantics></math></inline-formula>, we confirm the recent result of Klimchitskaya and Mostepanenko, that the thermal correction <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∼</mo><msup><mi>T</mi><mn>5</mn></msup></mrow></semantics></math></inline-formula>. In the case of exact equality <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>μ</mi><mo>=</mo><mi>m</mi></mrow></semantics></math></inline-formula>, the correction <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∼</mo><mi>T</mi></mrow></semantics></math></inline-formula>. This point is unstable, and the system falls to the regime with <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>μ</mi><mo>></mo><mi>m</mi></mrow></semantics></math></inline-formula> or <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>μ</mi><mo><</mo><mi>m</mi></mrow></semantics></math></inline-formula>. The analytical calculations are illustrated by numerical evaluations for the Hydrogen atom/graphene system.