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 Waterloo Perimeter 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: Question: The proposed scenario:
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: Findings: If correct, generic predictions:
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