Pull-in control due to Casimir forces using external magnetic fields
aa r X i v : . [ qu a n t - ph ] J u l Pull-in control due to Casimir forces using external magneticfields.
R. Esquivel-Sirvent, ∗ M. A. Palomino-Ovando, and G. H. Cocoletzi Instituto de F´ısica, Universidad Nacional Aut´onoma de M´exico,Apartado Postal 20-364, D.F. 01000, M´exico Facultad de Ciencias Fisico-Matem´aticas, Universidad Aut´onoma de Puebla,Apartado Postal 5214, Puebla 72000, M´exico Instituto de F´ısica, Universidad Aut´onoma de Puebla,Apartado Postal J-48, Puebla 72570, M´exico (Dated: August 30, 2018)
Abstract
We present a theoretical calculation of the pull-in control in capacitive micro switches actuatedby Casimir forces, using external magnetic fields. The external magnetic fields induces an opti-cal anisotropy due to the excitation of magneto plasmons, that reduces the Casimir force. Thecalculations are performed in the Voigt configuration, and the results show that as the magneticfield increases the system becomes more stable. The detachment length for a cantilever is alsocalculated for a cantilever, showing that it increases with increasing magnetic field. At the pull-inseparation, the stiffness of the system decreases with increasing magnetic field. ∗ Corresponding author. Email:raul@fisica.unam.mx etal. [7].The role of the Casimir force in MEMS was first demonstrated theoretically by Serryand Maclay [8, 9] and experimentally by Buks [10, 11], that showed that this force inMEMS/NEMS devices can cause the movable parts to pull-in and adhere to each other. Thepull-in or snap down in MEMS and NEMS occurs when the magnitude of the Casimir force(or for that matter any attractive force) between the plates overcomes an elastic restitutionforce. The pull-in in the presence of Casimir and Van der Waals forces has been reviewedby several authors[12, 13, 14, 15, 16, 17, 18]. Given that the pull-in is an undesirable effect,its suppression in the electrostatic case and in the presence of dispersive forces has also beenreported (see for example [19, 20]).In this paper we present a theoretical calculation of the effect of an external magneticfield on the stability of a capacitive micro/nano switch, when the only force of attractionpresent is the Casimir force. This external magnetic field can be used to control the pull-in dynamics since the Casimir force decreases with increasing magnetic field due to theexcitation of magnetoplasmons [21].To test the idea consider the simple capacitive switch depicted in Figure 1. It consists oftwo parallel plates, one is fixed and the other plate is attached to a linear spring of elasticconstant κ and is allowed to move along the z − axis . The plates are on the x − y plane.Given the frequency dependent dielectric function of the plates, the Casimir force F betweentwo parallel plates separated a distance L can be calculated using the Lifshitz formula. Forthe purpose of this paper we use the reduction factor η = F/F where F = − ~ cπ/ L isthe force between perfect conductors. The reduction factor is given by η = 120 L cπ Z ∞ QdQ Z ∞ dωk ( G s + G p ) , (1)where G s = ( r − s r − s exp ( − kL ) − − and G p = ( r − s r − s exp ( − kL ) − − . In theseexpressions, the factors r p,s are the reflection amplitudes for either p or s polarized light2 Q is the wavevector component along the plates, q = ω/c and k = p q + Q . Theabove expressions are evaluated along the imaginary frequency axis iω , a usual procedurein Casimir force calculations [7].To be able to decrease the Casimir force via an external magnetic field, we have to changethe reflectivities r p and r s . This can be done using anisotropic media in the so-called Voigtconfiguration. In this configuration the magnetic field is parallel to the plates along the x − axis .Besides being experimentally more feasible, in the Voigt configuration the Lifshitz formulaEq. (1) can be used since there is no mode conversion of the reflected waves. In generalfor anisotropic media, an incident p wave will reflect an s and a p wave and the same modemixing happens for incident s polarized waves. In this case the Lifshitz formula has to begeneralized using tensorial Green’s functions and the reflectivities take a matrix form [22].Let us consider that the slabs are made of a semiconductor such as InSb , whose opticalproperties are well known [23]. In the presence of an external magnetic field, the componentsof the dielectric tensor are [24] ǫ xx = ǫ L (cid:20) − ω p ω ( ω + iγ ) (cid:21) ,ǫ yy = ǫ L (cid:20) − ( ω + iγ ) ω p ω (( ω + iγ ) − ω c ) (cid:21) ,ǫ yz = ǫ L (cid:20) iω c ω p ω (( ω + iγ ) − ω c ) (cid:21) , (2)and ǫ zz = ǫ yy and ǫ zy = − ǫ yz . The other components are equal to zero. In these equations ǫ L is the background dielectric function, ω p the plasma frequency, γ the damping parameterand ω c = eB /m ∗ c is the cyclotron frequency for carriers of charge e and effective mass m ∗ . In the absence of the magnetic field, ω c = 0 and the plates become isotropic. Thisanisotropy is induced by the excitation of magneto-plasmons.The reflectivities for s and p polarized waves in the Voigt configuration can be calculatedusing the surface impedance approach, as recently shown by [21]. Using Eq.(1) the Casimirforce between the plates for different values of ω c was calculated by , showing a reduction ofthe Casimir force as a function of magnetic field. [21].3he analysis of the stability of a capacitive switch, is well known (see for example[12, 25,26] ); Given the total potential energy of the system is U = U elastic + U Casimir , the criticalpoint when the upper plate will jump to contact is obtained from the conditions dUdz = 0 and d Udz = 0. The first of these equations is simply the equilibrium of forces. For example, for aforce proportional to ∼ /r n , the pull-in distance is given by z in = ( n/n + 1) L , where L isthe initial separation of the plates. In the case of the Casimir force where n = 4 the pull-indistance is z in = 0 . L .In our case, using Eq.(1) the equilibrium condition is κ ( L − z ) = π ~ cA z η ( z, ω c ) , (3)where A is the area of the plates, and κ is Hooke’s constant.Introducing the dimensionless quantities ¯ z = z/L and Ω c = ω c /ω p , the previous equation,Eq.(3), is written as (1 − ¯ z )¯ z η (¯ z, Ω c ) = λ, (4)where the parameter λ is given by λ = F ( L ) κL . (5)The plot of the parameter λ as a function of plate separation ¯ z (bifurcation diagram) fordifferent values of ω c is shown in Figure (3). As a reference we have plotted the bifurcationdiagram for the Casimir force between perfect conductors. In this figure, as the magneticfield increases the maximum value of λ also increases. For each value of the magnetic field,the upper part of the curve before the fold, corresponds to stable solutions. In all cases, themaximum value of λ occurs at ¯ z = 0 .
8. The inset shows the reduction factor for differentvalues of the magnetic field at the pull-in separation. This calculation is based on thecalculations of ref. [21].A system that can be described by a simple-lumped system as that of Fig. (1) is acantilever switch. Consider a cantilever of rectangular cross section, of length l , width w ,thickness t and Young modulus E . In this case the spring constant or stiffness is given by κ = Ewt / l [27]. From the definition of the bifurcation parameter, we see that the pull-instiffness can be calculated. Similarly we can calculate the detachment length [26, 28, 29]of the cantilever. This is the maximum length of the cantilever that will not adhere to4 substrate and is also determined from the values of λ at pull-in. In Figure (3) we plotthe stiffness and detachment length for different values of the cyclotron frequency. As themagnetic field increases, the detachment length increases up to 60% with respect to thedetachment length at zero magnetic field and the stiffness shows a 20% decrease in its zeromagnetic field value.The reduction of the Casimir force due to the excitation of magnetoplasmons is used tocontrol the pull-in or jump to contact in capacitive switches actuated by Casimir forces.As the external magnetic field increases the force decreases allowing an increase of thebifurcation parameter λ at pull-in. From this value of λ in the pull-in stiffness and detachmentlength are calculated with increasing magnetic fields. In this paper we used dimensionlessquantities, to present the results in general. Once a particular system and material is chosen,the magnetic fields needed to reach a value of Ω c can be calculated. The additional parameterthat allows us to change Ω c in a given material is the plasma frequency, that can vary insome orders of magnitude depending on the carrier concentration [24]. Acknowledgments
Partial support provided by DGAPA-UNAM project No. IN-113208 and CONACyTproject No. 82474, VIEP-BUAP and SEP-BUAP-CA 191. [1] Lamoreaux S K (1997)
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FIG. 2: Bifurcation diagram as a function of magnetic field. From right to left the curves correspondto the cyclotron frequencies Ω c = 6 , , , ,
0. As a reference the bifurcation diagram for perfectconductors is also shown. The inset shows the reduction factor as a function of Ω c . The reductionfactor decreases with increasing magnetic field. This curve was calculated at a fixed separationbetween the plates z = 0 . L . Ω c DETACHMENT LENGTHSTIFFNESS
FIG. 3: The stiffness at the pull-in separation κ/κ and the detachment length l/l are plottedas function of the cyclotron frequency. The values of the stiffness and detachment length at zeromagnetic field are κ and l . The detachment length can be increased up to a 60% as the magneticfield, and hence the cyclotron frequency, increases.. The detachment length can be increased up to a 60% as the magneticfield, and hence the cyclotron frequency, increases.