Bo E' Sernelius, Dept' of Physics and Measurement Technology, Linkping University, SE581 83 Linkping

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Bo E. Sernelius, Dept. of Physics and Measurement
Technology, Linköping University, SE-581 83
Linköping, Sweden
Full inclusion of spatial dispersion resolves the
controversy regarding the finite-temperature
Casimir-force between two metal plates.

• QFEXT05

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Outline of the talk

• Neglect of spatial dispersion,
• Derivation of the forces
• Results
• Nernst heat theorem
• Inclusion of spatial dispersion
• Derivation of the forces
• Results
• Nernst heat theorem
• Conclusion

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Spatial dispersion

• To find the electromagnetic normal-modes of a
  system it is absolutely necessary to take the
  frequency dependence of the dielectric function
  into account.
• In some situations it is also necessary to
  include the momentum-, or wave-vector-dependence,
  i.e., to take spatial dispersion into account.

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Neglect of Spatial Dispersion

• Then one may use the experimentally measured
  dielectric properties of the objects to find the
  normal modes of the system.
• From these one finds the interaction energy and
  forces between the objects.
• At zero temperature the energy is just the zero
  point energy of the normal modes.
• At finite temperature also the occupation numbers
  affects the results.

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How are the modes obtained?

• One solves Maxwells equations in all regions and
  use the standard boundary conditions at all
  interfaces.
• The parallel components of the E and H fields are
  continuous across the interfaces.
• The normal components of the B and D fields are
  continuous across the interfaces.
• The standard Fresnel coefficients may be used.

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Condition for modes between two half spaces
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Modes contributing to van der Waals and Casimir
forces
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Modes between metal plates
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Interaction energy
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Interaction Potential between two Gold plates
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Interesting region expanded
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Energy correction factor in different
approximations
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TM and TE integrands
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Larger separation
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Even larger separation
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Temperature dependence of Entropy for modes
between two gold plates. The Nernst heat theorem
is obeyed!
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Summands or integrands
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The Nernst heat theorem, the third law of
thermodynamics.

• The Nernst heat theorem is obeyed as long as
  there are any defects in the crystal.
• This is of course the case in all real systems
• For a perfect crystal without impurities and
  other lattice defects the Nernst heat theorem is
  not obeyed.
• Why this is so is an academic question.
• For Nernsts heat theorem to be fulfilled the
  Integrands have to be continuous functions of
  frequency at the small frequency end. The
  defect-less metal-crystal case has a TE integrand
  that is a step-function at zero frequency

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Defect-less metal crystal
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Spatial Dispersion

• Taking spatial dispersion into account makes the
  calculations more involved.
• The Fresnel coefficients can no longer be used
• The experimentally obtained dielectric functions
  can no longer be used
• One has to resort to theoretical dielectric
  functions
• Both transverse and longitudinal versions are
  needed.
• We use the RPA (random phase approximation) or
  Lindhard dielectric functions

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Basics assumptions

• The dielectric properties on each side of the
  interface is assumed to be valid all the way up
  to the interface.
• We assume ideal interfaces, sharp with no
  defects.
• This means that the in-plane momentum, k, is
  conserved. Thus k is a good quantum number for
  the modes.
• For isotropic materials there is no preferred
  direction in the xy-plane
• We arbitrarily choose the propagation of the
  mode to be in the x-direction.

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• The modes at an interface are characterized by
  the 2D (two-dimensional) wave vector, k, in the
  plane of the interface.
• We let a general wave vector be denoted by q, and
  k is then the in-plane component.

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Charge and current densities

• The modes are associated with induced volume
  charge and current densities inside the objects
• Induced two dimensional charge and current
  densities at the interfaces
• The charge and current densities are not
  independent
• They are coupled through the equation of
  continuity

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Potentials in coulomb gauge
Fields from potentials
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Assume a surface charge density at an interface

• A time varying charge density is associated with
  a current density via the equation of continuity

• Thus the charge density couples to a current
  density in the x- direction only

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• The electric fields associated with this mode are
  in the plane of incidence (xz-plane) and the
  magnetic fields are perpendicular to this plane.
• The fields are p-polarized or TM-fields
  (Transverse Magnetic)
• One could be led to believe that there are also
  modes associated with a Jy-component and with no
  accompanying induced charge density.
• However, there are no such solutions to the
  Maxwell equations whose resulting fields obey the
  boundary conditions.

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• in the geometry with two parallel interfaces
  there are modes generated by coupled
  Jy-components at the two interfaces.
• These modes are TE-modes with s-polarized fields.

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Let us now return to our mode

• The charge density gives rise to a scalar
  potential and
• the current density to a vector potential.
• Since we are working in Coulomb gauge, only the
  transverse part of the current contributes to the
  vector potential.
• To get the transverse part we subtract the
  longitudinal part from the current density

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Study a single interface
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The B-field
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In real space we have
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g-functions
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g-functions in vacuum
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Gamma-functions
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Conditions for modes at a flat interface

• The modes at a single flat interface are
  TM-modes.
• The magnetic field has only y-components while
  the electric field has both x- and z-components.
• The condition for modes is expressed in terms of
  these g-functions.

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Two parallel interfaces

• At a single interface there is also a component
  of the current pointing in the y-direction, but
  this does not couple to the charge density and do
  not help in the formation of self-sustained
  fields.
• Now if we have two interfaces these current
  components at the two interfaces may couple and
  give rise to a mode this is the TE-mode.
• The scalar potential does not contribute to the
  fields in this case.

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• The TE fields are genuinely transverse in
  character and only the transverse dielectric
  function enters the relations.
• The TM-modes on the other hand have both
  longitudinal and transverse character and both
  types of dielectric function enter the formalism.

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Condition for modes between two interfaces
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Interaction energy
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Tilde-g-functions

• The tilde over a g-function means the g-function
  calculated inside the medium minus the
  corresponding function calculated in vacuum.
• The tilde versions of the g-functions were
  introduced since they represent faster converging
  integrals. To benefit from this the two integrals
  in each tilde version of a g-function should be
  combined into one.

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TM and TE Integrands
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Same figure but with linear scales
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Expanded energy scale
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• We find that the drop in the TE-integrand is so
  close to the axis that the zero-temperature
  result is not affected.
• For room-temperature the drop affects only the
  TE, n 0 term.
• The integrands are continuous so the Nernst heat
  theorem is fulfilled even in absence of
  dissipation.

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Results for different approximations
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Why is the TE modes behaving as they do?

• Let us look at GTE

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Integrand for the gb(k,w) function for w 0.
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Integrand for the gb(k,w) function for k 0.01
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Summary

• I have presented the results for the force
  between two metal plates taking spatial
  dispersion into account.
• One then needs theoretical dielectric functions.
• I used the RPA or Lindhard dielectric functions,
  both transverse and longitudinal versions.
• No fiddling around with asymptotic expansions.
• In the results presented here no dissipation was
  assumed.
• The same striking results as we obtained earlier
  when spatial dispersion was neglected but
  dissipation was included
• The academic question involving the Nernst heat
  theorem for a system without defects was
  resolved.

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References

• The present work on spatial dispersion, Bo E.
  Sernelius, Phys. Rev. B 71, 235114 (2005).
• Our previous work on dissipation M. Boström and
  Bo E. Sernelius, Thermal effects on the Casimir
  force in the 0.1-5 mm range, Phys. Rev. Lett. 84,
  4757 (2000) Bo E. Sernelius and M. Boström, 
  Dispersion forces between real metal plates at
  finite temperature and the Nernst heat theorem ,
  in Quantum Field Theory Under the Influence of
  External Conditions, Ed. K. A. Milton (Rinton
  Press, USA, 2004) M. Boström and Bo E.
  Sernelius,  Entropy of the Casimir effect between
  real metal plates, Physica A 339, 53 (2004).
• Experimental paper by Lamoreaux S.K. Lamoreaux,
  Phys. Rev. Lett.78, 5 (1997).
• If you want to read more about the modes Bo E.
  Sernelius, Surface Modes in Physics, Wiley-VCH
  2001.