On the Casimir Effect in the High Tc Cuprates

1. On the Casimir Effect in the High Tc Cuprates Achim Kempf Canada Research Chair in the Physics of Information Departments of Applied Mathematics and Physics University of WaterlooPerimeter Institute for Theoretical Physics Waterloo, Ontario, Canada QFEXT07, Leipzig, Sep. 2007

2. High Tc superconductors Discovery: In mid 1980s Soon after: Creation of materials with Tc up to 120K Since then: Increase in Tc reached plateau at about 150K Today still open: Is plateau approaching a natural limit?Or could Tc go higher if we better understood the mechanism? Status of theory: Much is known about details of the microscopic mechanism (e.g. d-wave instead of s-wave). But: The phonon-phonon interaction is too weak to bind pairs at 100K. The key question: How can Cooper pairs be energetically stable at around 100 K and more ?

3. High Tc superconductors Generic properties: Above Tc: • Essentially insulating ceramic material, e.g. YBCO. Below Tc: • It becomes superconducting in parallel layers of Cu-O • Properties of the in-between layers: • essentially insulating • precise composition is, experimentally, of little significance ! Thus suggests:The layering itself may be important.

4. Casimir effect ? • Below Tc: • The parallel superconducting layers should imply a Casimir effect • Casimir energy is negative => expect lowering of the energy. • Question: • Could this energy lowering account for the condensation energy of the Cooper pairs? • The proposed scenario: • Electrons are energetically driven to use whatever microscopic mechanism there may be available to create superconductivity. • Conversely: Cooper pairs would be stable because their break-up would destroy the Casimir effect and this would have to raise the energy. • Thus, not only: Cooper pairs => superconductivity but also:Superconductivity => Cooper pairs

5. Assumptions for obtaining an order of magnitude estimate: • Above Tc: • Assume crystal is essentially homogeneous insulating ceramic. • Thus, assume Casimir effect negligible. (Note: Next better ansatz would be: Drude model for Cu-O layers in normal state) • Below Tc: • Use plasma sheet model for the Cu-O planes • Assume vacuum between the Cu-O layers • At transition from just above to just below Tc:Plasma sheets’ Casimir energy = Condensation energy

6. The Cu-O layers as plasma sheets In which regime is the system? • Thin sheets (individual Cu-O layers): • we will here model the Cu_O layer thickness as infinitesimal. • Small sheet separations: • Cu-O sheet separation is typically a ≈ 1nm • But Cu-O sheets are essentially transparent at 1nm scale wavelengths. • E.g., the London penetration depth is typically two to three orders of magnitude larger • Thus, contributions to Casimir effect: • Only from reflectivity changes at wavelengths that are several orders of magnitude larger than the layer separation. => Expectation: Casimir effect suppressed by several orders of magnitude.

7. How much is the Casimir effect suppressed? • Size of drop in Casimir energy at Tc?Recall: expression for plasma sheets with small separation, a, (Bordag 2006):Note: this energy expression is dominated by TM “surface” plasmons. • Amount of suppression, e.g., for layer separation a = 1nm ? For typical values, n = 1014 (cm)-2, q = 2e, and m = 10 me:

8. Tc for two parallel Cu-O planes ? • Ansatz: • Relate as usual to the density of states and the energy gap, where and • Resulting Tc:

9. Tc for two parallel Cu-O planes ? • Consider realistic values, such as a = 1nm, n = 1014(cm)-2, η = 1.76, m = 5 me, • Insert in equation above: • Obtain: => The order of magnitude could be right.

10. Casimir force among Cu-O layers? Example: • Use typical cuprate values as above, • Use typical elastic modulus: This yields a Casimir-induced contraction in c-direction: Measurable ?

11. Conclusions The model: • Above Tc: no Casimir effect • Below Tc: Casimir effect of thin plasma sheets separated by vacuum Findings: • Order of magnitude of Tc for cuprates around 100K explainable. • Tc generally increases with decreasing a, as is the case experimentally. If correct, generic predictions: • Layering is key to energetics of High Tc supercondutors • To increase Tc, must look for materials • with small spacings of potentially superconducting layers • large change difference in Casimir effect between normal and SC phase Example:Carbon nanotubes’ Tc should rise when layered - as is the case experimentally

12. Outlook • Must improve the model: • Normal phase: use Drude model • SC phase: Plasma model for Cu-O layers, dielectric in between. Even better: Try to use measured values for reflection properties. • From two to infinitely many layers (with multiple different spacings): • Real cuprates possess multiple layer spacings in unit cell and in between unit cells • Calculation nontrivial because nontrivial periodicity and high transparency, i.e. multiple refeltions/transmissions. • Compare with precision-measured Tc as a function of the spacings, a. • Example: measurements known for epitaxial superlattices of YBCO. • Try to increase Tc with design of new layered materials: • Need as small as possible spacings between potentially superconducting layers • Need as large as possible difference in Casimir effect between normal and SC phases