Excerpt from the Proceedings of the COMSOL Conference 2008 Hannover

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1 Excerpt from the Proceedins of the COMSO Conference 8 Hannover Modelin and Characterization of Superconductin MEMS for Microwave Applications in Radioastronomy N. Al Cheikh *,1, P. Xavier 1, J.-M. Duchamp 1, K. Schuster 1 Institute of Microelectronics, Electromanetism and Photonics (IMEP-AHC) Grenoble (France), Institute of Millimetrics Radio Astronomy (IRAM) Grenoble (France) * IMEP-AHC - Minatec - INPG 3 Parvis ouis Néel - BP 57, 3816 Grenoble Cedex 1, alcheikn@minatec.inp.fr Abstract: Superconductin GHz electronics circuits are frequently used in Radio Astronomy instrumentation. The features of these instrumentations can be sinificantly improved by usin tuneable capacitances, which can be realized by electrically actuated, micromechanical brides (MEMS) made of superconductin Niobium (Nb). In order to analyse the electromechanical behaviour of such devices and the intrinsic stress radient induced durin the fabrication process, COMSO simulations of the brides have been performed. The simulations are in areement with the theoretical approach and are experimentally validated. Keywords: tuneable capacitances, microwave, superconductor, MEMS, radiostronomy. 1. Introduction Superconductin electronic components are widely used in the field of radio astronomy, because of their low noise and losses performances. Furthermore, Superconductor- Insulator-Superconductor (SIS) mixers are implemented in much radio astronomy instrumentation due to their hih sensitivity. Nevertheless, some transmitter s sinals such as radars and satellites can perturb broadband astronomical sinal and saturate SIS mixers because of their hih amplitude [1]. A solution to this problem is based on a narrow frequency band detection swept on the entire observed band (between 7GHz and 115GHz at the IRAM). For this purpose, variable tunin elements are of increasin interest. Tuneable capacitances, so-called varactors, represent one important class of tuneable circuit elements. They are widely applied in wireless communication or radars, where they act as basic elements for adjustin the operation rane of variable filters, phase shifters or transceivers. New devices manufactured with superconductin materials meet these demands and are well established in this field []. These variable devices based on capacitive MEMS were fabricated by the IRAM which is interested in usin these components for aile impedance matchin networks and filters [3]-[4]. The brides are actuated by applyin a suitable voltae between the bride and the round electrode underneath (fiure 1). Fiure 1. Schematic picture of an air bride with electrode underneath for electrostatic actuation a) without and b) with applied voltae. This paper presents how to describe a D MEMS takin into account coupled electrostatics and mechanics problems. This study is only the first part of a reional project SupraMEMS. The final oal of the project is to propose a multiphysics model (electromechanical, microwave and superconductin) of these tunable capacitances. In the first section, the MEMS fabrication process is described. In the second one, measurements of capacitance value versus the voltae characteristic C(V) are compared with a simple capacitance model. The assumption of a bendin effect of the bride layer is proposed. A more accurate model is proposed in the third section to extract the new capacitance value versus the voltae characteristic C(V) and the new pull in voltae V PI. Then this model is confirmed by COMSO simulations in section four. In this section, the slope ede variation is also taken into account. Finally this paper draws out conclusions on the bendin effect of the air bride on the MEMS capacitance feature.

2 . Fabrication process For decade now, many superconductin electronic circuits are based on Niobium (Nb) with a critical temperature of 9 K. The fabrication process of IRAM MEMS is described on fiure a. A Nb transmission line is realized in the first step. Then a photoresist polymer AR-4/8 (ARESIST GmbH) is used as a sacrificial layer. This layer has a thickness about 5 µm. Smooth rims of the sacrificial resist layer are necessary to avoid breakin of the Nb film at side ede of the MEMS. To obtain sufficiently smooth edes, the AR-4/8 was baked in a convection oven before exposure and after development. The Nb layer was sputtered, in the third step, by DC-manetron sputterin in steps of 9 s deposition and 5 min break (fiure b). The break delays are needed to reduce surface heatin and thus deformation or polymerization of AR-4/8 layer. The widths of the brides are defined in this step by a neative tone photoresist mask. For a ood coverae of the structures, the same thick resist AR 4/8 with a reversal process is used. Throuh this resist mask, which covers the surface of the Nb brides, the no-covered parts were etched by RIE (Reactive Ion Etchin) as shown in fiure c. The final step wash away the sacrificial layer in hot acetone at 7 C (fiure d). 3. First experimental observation Two sizes of MEMS are desined. The first one has 5µm hih, 1 µm lon, 1 µm wide, but the capacitive part has only 9 µm lon. This MEMS capacitance topoloy is closed to the plane capacitance one (fiure 3). So a first approximation of the capacitance can be obtained with the simple equation (1) ε. W. C( ) =, (1) is the heiht of the bride, W the width and the lenth. ε is the air permittivity equal to 8, F/m. For the first size; MEMS should have a capacitance value C() about 16 ff. The second MEMS size is 4 µm hih, 6 µm lon, 1 µm wide. In this case C() is near 13.3 ff. Fiure 3. Picture of an air bride with electrode underneath for electrostatic actuation. The inset shows the bendin of the bride layer and the reduced heiht due to stress in the Nb film. Capacitance-voltae measurements are shown on fiure 4. We can noticed that C() values are between 3.4 ff and 33.4 ff, two times reater than the expected values. Fiure. Schematic of the fabrication process for sputtered Nb air brides. Fiure 4. Measured capacitance-voltae for various MEMS.

3 Of course, there are some uncertainties on the MEMS lenth and width durin the process and ede effect are not takin into account in equation (1) but these errors can not justify a 1 % error on C value. We assume that the only parameter able to explain this difference is the heiht of the air bride. It should be smaller than expected at the middle of the MEMS lenth. It is probably due to the bendin of the bride layer after removin the sacrificial layer and releasin the MEMS (fiure 3 insert). This effect could have two causes. First; a lenthenin can be caused by a tensile stress and shrinkin of the sacrificial layer durin deposition and coolin time. Second; a variation of MEMS side slope could occur when the bride is released. 4. Theoretical beam modellin In this section, we present a new electromechanical model of these devices. This model takes into account technoloical imperfections such as the bendin of the bride. This theoretical study of the air bride shape ives new values for C(V) and the pull in voltae V PI. 4.1 New C() equation We assume that the bendin bride is a Fixed-Fixed beam with a lenth equal to +. The heiht of the bride alon the lenth of the MEMS can be described by a cosine equation. ymax πx y( x) = 1 cos () y max is the maximal difference between the expected heiht and the heiht with bendin in the middle of the MEMS. The new capacitance equation becomes C ε W (3) y x ( ) = dx ( ) If these capacities can be described as the sum of elementary plane capacitance, interation result is. ε W. C( ) = (4) y max 1 With this new expression for C(), the larest of the bendin of the bride, obtained for 33.4 ff (fiure 4), must be between 71% to 75% of the heiht. The ap variation is between 3.6 µm and 3.8 µm. 4. Eneretical approach In this sub-section, we describe the variation of the voltae V(y max ) when we took into account an eneretical approach where the equations of the device electrostatic enery and of the mechanical enery are equalled. The expression of mechanical enery is iven by the theorem of Clapeyron [7] : U m 1 = εdv σ v (5) where σ =π E (ymax' ) 4 is an estimated value of the induced axial stress [8], dv is the volume variation, y max is the maximal bucklin in the middle after application of the voltae and E is the youn modulus. ε = ymax' ymax) π ( 4 is the strain of the beam [6]. Thus, in Equation (3), the expression of the mechanical enery becomes: π 4 EW.. h y ( max ' ymax ' ymax ) U m = (6) 3 where h is the thickness of the beam. The expression of the electrostatic enery is iven by : 1 U e = C( V ) V (7) The expression of the capacitance C(V) is the same as C() when y max is replaced by y max because the profile is still assumed to be of cosine shape before and durin the application of the voltae. So the expression of C(y max ) is: ( ) C V = W. ε (8) y 1 ' max ( V ) Finally, the equation (6) is balanced with equation (8) to find the expression of the voltae variation V(y max ): V = y max' 4 1 y y y π Eh max ' ( max ' max ) (9) 16 4 ε W

4 The pull-in voltae V PI can be derived when the variation of the voltae V(y max ) is plotted. The maximum of the curve is the value of V PI ; for example, on the curve havin C()=33fF, the deduced value of V PI is 88.4 V (fiure 5). These curves are very close to the ones experimentally obtained. These results validate the COMSO modelin of the superconductin MEMS. 5. COMSO Multiphysics beam simulations To confirm these previous theoretical results of the bendin beam, simulations with COMSO have been done. This section describes the different steps necessary to simulate correctly the IRAM MEMS in COMSO Multiphysics. Fiure 5. V(y max ) variation,. Besides the calculated curves of C(V) follow easily usin the values of y max which has come from V(y max ) in the expression of C(y max ) (fiure 6). These curves are calculated takin the parameters in the Tab 1. Fiure 6. Theoretical Voltae dependence of the capacitance for different air brides. Symb. (µm) y max(µm) h(nm) (µm) O X * Tab1: the heiht of the MEMS is iven by and in the centre by ( -ymax '),where ( -ymax) is the heiht when V = (initial deflection). 5.1 Simulations and results of C() value In the first step, to find the values of the capacitance C(), the method consists in applyin a force on the free boundary (fiure 7) of a no-bucklin beam in order to create a bucklin. Then the schema of the simulated bucklin beam will be extracted (fiure 8). In this new schema, the electrostatical domain is applied to find C(). In this context, a D simulation is used in the Model Naviator ; from the COMSO MEMS Module, the Structural Mechanics is used for Plane stress and from The COMSO Multiphysics Module the Deformed Mesh, where the COMSO Multiphysics translates the application-mode equations between the fixed and movin frames. The proram keeps pictures of the movements usin the movin Mesh (AE). After that, for the Modelin eometry, we draw a Fixed-Fixed beam soft bucklin and soft board. In this way, we can put the necessary properties for plane stress and for Boundary Conditions. The first in the Subdomain settin concerned the values of youn s modulus (E), Poisson s ratio (υ) and the thickness; the second lead to choice between the fixed beam and the free beam. When we can put a force on the Free beam, after every application of the force, we can find the value of y max to know C(y max ). The second step consists in extrudin the schema of the bucklin beam after the application of a force (fiure 8); we can used for that in COMSO Mesh the Create Geometry From Mesh. The third step consists in usin the electrostatic in the Mems Module, use the Electrostatics application mode which solves the electric potential (see fiure 9).

5 Fiure 7. schematic of the application of mechanical Force on fixed-fixed beam. Fiure 9. Electric Field manitude for Fixed Fixed beam. Fiure 8. Extrude the schema of fiure 7 where the created Geometry from Mesh is used Now C() can be calculated from the stored electric enery in the capacitance, U e (7) and the voltae across the capacitor: U e C( ) = (1) V Same steps are done for the beam with the supports where we put the condition fixed boundary on the board and the Force is only applied on the Free beam (Fiure 1). Fiure 1. Electric Field manitude for beam with supports. 5. Simulations and results of C(V) and V PI To calculate C(V) a couplin between mechanical and electrostatical is necessary. The first step consists in takin into account the extrude schema of MEMS when the plain stress, Movin Mesh and Electrostatics are applied in the COMSO simulation. Here electrostatical forces named Fes_nTx_emes and Fes_nTx_emes are automatically enerated variables from the Electrostatics application mode (_emes) and define the x- and y- components of the electrostatic force which are used in place of the mechanical force for the couplin Electrostatics Mechanics and for the boundary condition for electrostatics (fiure 11) when we put Vin in the V Boundary.

6 Fiure 13. Comparison between measures, theory and simulations of the Mems with the support Fiure 11. Boundary condition of the MEMS by the couplin Electrostatics-Mechanics used with Mesh(AE) In this couplin, the values of the capacitances cannot be calculated; only the variation of the place of the free beam (Fiure 1) and the voltae V PI. 6. Conclusions In this paper, the necessary steps to simulate particular MEMS used in radioastronomy applications on COMSO Multiphysics are described. The obtained C(V) results are validated from experimental capacitance measurements and theoretical calculations. 7. References Fiure 1. Surface Electric potential, the white color represents the displacement y after the couplin Electrostatics-Mechanics In fact, then durin the simulation an error messae ( Failed to find a solution for all parameters, even when usin the minimum parameter step. ) appears, this indicates that the pull-in voltae has been reached. To find C(V), the schema of fiure 1 is extruded to a new schema after every application of the voltae because it is needed to apply the same steps as in the pararaph 5.1 to calculate the capacitance of Fiure 13. A ood areement between measures, theory and simulations is observed. 1. K.H. Gundlach and M. Schicke, Supercond. Sci. Technol, 13, R171,. J. Zmuidzinas and P.. Richards, Proc, IEEE, 9(1), 1597, 4 3. J. Yao, J. Micromech Microen, 1, R9, 4. R. Borwick, P. Stupar, J.F. DeNatale, R. Anderson, R. Erlandson, IEEE Trans Microwave Theory and Techniques, 51(1), 315, 3 5. H. Guckel,T. Randazzo, D.W. Burns, A Simple Technique for the determination of mechanical Strain in Thin Films with Applications to Polysilicon, 15/11/ W.Fan and J.A.Wickert, Post Bucklin of Micromechined beams, J.Micromech Microen, , 4 Auust F.Fey, Analyse des structures et milieu continus, Mecanique des structures, Pae S. Pamidihantam, R. Puers, K. Baert and H.C. Tilmans, J.Micromech Microen, , June. 8. Acknowledements This work has been founded from the French Rhône-Alpes Reion, in the frame of the SUPRAMEMS project (MicroNano cluster) and in cooperation with the partners IN yon, Ecole des Mines de St Etienne and Infiniscale.