The role of magnetoplasmons in Casimir force calculations
In this paper we review the role of magneto plasmon polaritons in the Casimir force calculations. By applying an external constant magnetic field a strong optical anisotropy is induced on two parallel slabs reducing the reflectivity and thus the Casimir force. As the external magnetic field increases, the Casimir force decreases. Thus, with an an external magnetic field the Casimir force can be controlled.The calculations are done in the Voigt configuration where the magnetic field is parallel to the slabs. In this configuration the reflection coefficients for TE and TM modes do not show mode conversion.
The optics of surfaces plays an important role in the calculation of the Lifshitz-Casimir formula lifshitz (). Indeed the Lifshitz formula can be obtained from the sum of surface-polariton modes between two metallic slabs sernelius (); roman1 (); roman2 (); intravaia (); lambrecht (). In metallic and semiconducting surfaces the effect of an externally applied magnetic field leds to the excitation of magnetoplasmon modes. This external magnetic field changes significantly the behavior of the plasma modes and induces an optical anisotropy that is magnetic field dependent coco (). This has the effect of reducing the Casimir force significantly. This reduction on the force has been applied to the problem of pull-in dynamics in micro and nano electrodynamical systems (mems and nems), and as has been shown to increase the detachment length in cantilever mems and nems esquivel09 ().
To illustrate this point, consider the Drude model in the presence of the external magnetic field. The equation of motion for the electrons in the material is
where is the effective mass of the electron, the charge and is the relaxation time. Assuming an harmonic electric field the current can be found and thus the conductivity can be calculated palik ()
where , is the cyclotron frequency, and is the Levi-Civita symbol. The dielectric tensor is obtained from
Clearly if we recover the results for the isotropic case.
Ii Voigt and Faraday Configurations
For an arbitrary direction of the magnetic filed, the calculation of the dispersion relation of the surface magneto plasmons and of the optical reflectivity is difficult. To simplify the problem, specific directions of the magnetic field have to be chosen manvir (). In the so called Faraday configuration, the magnetic field is perpendicular to the slab. In this case, there is mode conversion upon reflection from the slab. This is, if a TE wave is incident, the reflected wave will consists of a TE and TM modes, similar for an incident TM mode.
The second configuration, that will be used in this paper, is the Voigt geometry where the magnetic field is parallel to the slabs. In this case there is no mode conversion upon reflection. Consider a slab parallel to the plane. In the Voigt geometry the external magnetic field points along the axis. In this case, the components of the dielectric tensor are given by manvir (); brion ()
and and . The other components are equal to zero. In these equations is the background dielectric function, the plasma frequency, In the absence of the magnetic field, and the plates become isotropic. For simplicity we have not included the Drude damping parameter. In the rest of the paper we will use the dimensionless variable , that gives the relative importance of the external magnetic field. In Figure (1) we have plotted the dielectric function components as given by Eq. (4) after rotation of the frequency to the complex plane for a value of , showing the strong anisotropy of the system.
In the material slab, the dispersion relation is obtained from the wave equation
where is the dielectric tensor and . Upon replacement of its components (Eq.(4)) into Eq.(5), the nontrivial solution of the resulting equation is obtained if
Outside the slab we assume there is vacuum and we have the usual dispersion
The reflection coefficient can now be calculated taking into account that outside the slab, for a TM polarized wave the field is of the form
and within the slab
where and are given by Eqs.(6) and (7) respectively. From the corresponding electric fields and by applying the boundary conditions, the reflection coefficients can be found. The detailed procedure for the Voigt configuration can be found in Ref.(manvir ()).
Iii Reduction of the Casimir force with an external magnetic field
To study the effect of the external magnetic field on the Casimir force we use Lifhitz formula
where is the Matsubara frequencies and the reflectivities for or polarized modes (TM and TE, respectively). This expression for the Casimir force can be used only for the Voigt configuration since there is no mode conversion. The Faraday configuration, that will be presented elsewhere, requires a more general expression for the Casimir force, as the one discussed by Bruno bruno (). The reflectivities and are replaced in Eq. (10) by a reflectivity matrix whose components are , , , , where the first subindex represents the polarization of the incident wave and the second sub-index the polarization of the reflected wave.
In Figure (2) we plot the Casimir force normalized to the ideal case , for several values of the reduced frequency . All frequencies are normalized to the plasma frequency and the distances are normalized to the plasma wavelength . The value for the background dielectric function is for as reported by Palik palik2 (). In general, semiconductors (e.g. ) can be used, since they exhibit a strong magnetoplasmon response. The important feature of Fig. (2), is that as the magnetic field increases the Casimir force decreases. For high magnetic fields, there is a drop in as a function of separation, this drop is more significant with increasing magnetic field.
In this paper we have reviewed briefly the principles of magnetoplasmons in semiconductors and its effect on the calculations of the Casimir force between parallel slabs. In particular we consider the Voigt configuration, where the magnetic field is parallel to the surface of the slabs. This external magnetic field induces a strong optical anisotropy that reduces the reflectivity. This has the effect of reducing the Casimir force as the external magnetic field increases. In a future work, the effect of the external magnetic field on the Casimir torque will be considered.
Acknowledgements: The authors acknowledge partial support from DGAPA-UNAM grant no.project No. IN-113208 and CONACyT project No. 82474, VIEP-BUAP and SEP-BUAP-CA 191.
- (1) E. M. Lifshitz, Sov. Phys. JETP 2, 73 (1956).
- (2) B. Sernelius, Surface Modes in Physics, (Wiley-VCH, Berlin, 2001).
- (3) C. Noguez, C.E. Romám-Velázquez, R. Esquivel-Sirvent and C. Villarreal, Europhys. Lett. 67, 191 (2004).
- (4) C. Noguez, C.E. Romám-Velázquez, C. Villarreal and R. Esquivel-Sirvent, Phys. Rev. A 69, 042109 (2004).
- (5) F. Intravaia and A. Lambrecht, Phys. Rev. Lett. 94, 110404 (2005)
- (6) F. Intravaia, C. Henkel, and A. Lambrecht Phys. Rev. A 76, 033820 (2007).
- (7) R. Garcia-Serrano, G. Martinez, P. Hernández, M. Palomino-Ovado, and G. H. Cocoletzi, Phys. Status Solidi B 246 , 1357 (2009).
- (8) R. Esquivel-Sirvent, M. Palomino-Ovado, and G. H. Cocoletz, Appl. Phys. Lett. 95 051909 (2009).
- (9) E.D.Palik and J.K. Furdyna, Rep. Prog. Phys. 33, 1193 (1970).
- (10) M. S. Kushwaha, Surf. Sci. Rep. 41, 1 (2001).
- (11) J.J. Brion, R. F. Wallis, A. Hartstein, E. Burstein, Phys. Rev. Lett 28, 1455 (1972).
- (12) P. Bruno, Phys. Rev. Lett. 88 , 240401 (2002).
- (13) E.D. Palik, R. Kaplan, R.W. Gammon, H. Kaplan, R. F. Wallis and J.J. Quinn, Phys. Rev. B 13, 2497 (1976).