Variable capacitors are essential for building tunable RF systems. We present here the design and simulation of novel zipping variable capacitors with a high permittivity dielectric layer. Two modelling techniques are presented: finite element simulation and variational analysis. A capacitance ratio greater than 40 can be obtained for a 100µm x 25µm device which has a high permittivity dielectric layer (εr = 200). By shaping either the top electrode beam or the bottom electrode, continuously variable capacitance is achieved at low bias voltages

Holmes, A.S.

Pu, S.H

Yeatman, E.M.

English

Southampton (e-Prints Soton)

DESIGN AND SIMULATION OF ZIPPING VARIABLE CAPACITORS* S. H. Pu, A. S. Holmes, E. M. Yeatman Optical and Semiconductor Devices Group, Department of Electrical & Electronic Engineering Imperial College London, Exhibition Road, SW7 2AZ, UK suanhui.pu@imperial.ac.ukAbstract    Variable  capacitors  are  essential  for  building tunable  RF  systems.  We  present  here  the  design  and simulation of novel zipping variable capacitors with a high permittivity  dielectric  layer.  Two  modelling  techniques  are presented: finite element simulation and variational analysis. A capacitance ratio greater than 40 can be obtained for a 100m x 25m device which has a high permittivity dielectric layer (r = 200). By shaping either the top electrode beam or the  bottom  electrode,  continuously  variable  capacitance  is achieved at low bias voltages. Key  Words:  zipper  tunable  capacitor,  stable  zipping,  RF MEMS, high-k dielectric I  INTRODUCTION Variable  capacitors  are  key  components  in  the implementation  of  tunable  filters,  phase  shifters and  voltage-controlled  oscillators  (VCO).  In  the last  ten  years,  MEMS  variable  capacitors  have emerged  as  viable  alternatives  to  semiconductor varactor  diodes  mainly  due  to  the  higher  quality (Q) factors and extended tuning ranges that can be achieved [1]. Despite the lower tuning speeds of MEMS  variable  capacitors  relative  to  varactors, typical  operation  speeds  (on  the  order  of microseconds)  are  sufficient  for  many  RF applications [2].  MEMS variable capacitor designs can be broadly categorised into gap-tuning devices, where the gap between  the  capacitor  electrodes  is  changed  to achieve tuning, and area-tuning devices, where the electrode overlap area is adjusted [1]. The majority of these designs are electrostatically actuated due to  the  low  power  consumption  and  ease  of implementation.  Electrostatic  zipping  structures  have  been employed  in  MEMS  actuators  [3,4]  but  have remained  relatively  unexplored  for  variable capacitor applications. Hung et al have previously fabricated  zipping  variable  capacitors  with  a straight cantilever design [5]. In addition, Lincoln Lab has also produced a zipping capacitive switch [6]. This paper presents the design and modelling of a new type of zipping variable capacitor with a high permittivity dielectric layer.   II  CAPACITOR DESIGN II.1 ZIPPING VARIABLE CAPACITOR Figure 1 below shows a schematic cross-section of a zipping variable capacitor. By applying a control voltage Vb, a curved top electrode is pulled down onto  a  dielectric  layer,  hence  increasing capacitance. In the absence of a DC bias voltage, the capacitance (C0) of the device is dominated by the air gap underneath the top electrode, resulting in low capacitance. Once the top electrode is pulled down,  the  presence  of  the  high  permittivity dielectric allows a large capacitance ratio (Cmax/C0) to be obtained for a given device size.    Figure 1. Schematic of a zipping capacitor. The curved top electrode can be fabricated using a bi-layered structure with a thin top layer which is in biaxial tension prior to release, and a relatively stress-free  bottom  layer.  This  creates  a  bending moment when the structure is released, and thus an upward curvature. Figure 2 shows the cross-section of the bi-layered top electrode where the subscripts 1  and  2  refer  to  the  bottom  and  top  layers respectively; h is the thickness, b is the width and d denotes the distance from the surface to the neutral plane.  Figure 2. Cross-section of bi-layered top electrode. h2 h1 d2 d1  1 2 z y Neutral Plane x Vb Dielectric  layer Bottom electrode Glass substrate Tensile metal layer   Unstressed metal layer b * Presented at MicroMechanics Europe 2007, Guimarães, Portugal, 16 – 18 September 2007. Using plate theory, the released shape of the top electrode may be determined by  2222 22 2 1 122 ) ' E ' E ( 2) ( xhd hI Ix sr     (1) where 2 is the is the biaxial (in x and z directions) tension in the top layer, Ē is the biaxial modulus defined as Ē = E/(1-), E is the Young’s modulus,  is Poisson’s ratio and I’ is the moment of inertia per  unit  width  about  the  neutral  axis.  This expression is valid for small deflections where the radius  of  curvature  is  much  greater  than  the electrode thickness. II.2 PULL-IN INSTABILITY The curved top electrode is susceptible to the well-known  electrostatic  pull-in  instability  that  affects gap-tuning parallel plate capacitors. As the control voltage  is  increased  from  0,  the  capacitance increases from C0 until the critical pull-in voltage is  reached  when  the  entire  curved  beam  is straightened out onto the dielectric. This results in a  large  sudden  increase  in  capacitance  and essentially  the  device  functions  as  a  capacitive switch,  alternating  between  a  low  nominal  value and  a  high  switched  value.  Figure  3  shows  an example of the simulated C-V characteristic of one such device with a dielectric relative permittivity of 4 (SiO2). The closure of the air gap at pull-in results in a highly non-linear capacitance variation and  this  effect  will  be  magnified  when  a  high permittivity dielectric is used. 0.00.51.01.52.02.53.03.54.00 1 2 3 4 5 6 7 8 9 10 11Voltage [V]C [pF] Figure 3. C-V characteristic of zipping capacitor with constant electrode widths. In  order  to  achieve  continuously  variable capacitance, we propose two different strategies for allowing stable zipping of the top electrode as the bias  voltage  is  increased.  The  first  method  is  to vary the stiffness of the beam, increasing its width from the fixed end to the free end. Hence, as the top  electrode  is  pulled  onto  the  dielectric increasingly larger voltages are required to zip the electrode further. The second method for achieving controlled zipping is by varying the width of the bottom  electrode  [5]  and  hence  controlling  the electrostatic force per unit length along the device. The width is made largest at the end nearer to the fixed anchor and smallest at the free end. Figure 4(a)  shows  a  zipping  variable  capacitor  with  a shaped  top  electrode  and  figure  4(b)  shows  one with a shaped bottom electrode.  Figure 4. Achieving controlled zipping with (a) shaped top electrode (b) shaped bottom electrode. III  MODELLING AND RESULTS Two different modelling techniques were applied in the design of the zipping variable capacitors. An electromechanical  finite element model was  used to  evaluate  designs  with  variable  top  electrode width.  In  addition,  a  semi-analytical  variational approach  [7] was  used  to  evaluate both  types  of designs. III.1  ANSYS FINITE ELEMENT MODEL Due to the geometry of the device, 3D modelling is required.  However,  the  computation  times  are extremely long and hence a simplified 2D model was  created which approximates the beam  width function  using  an  equivalent  elastic  modulus function. Treating the bi-layered top electrode as a beam  when loaded  electrostatically, the moment-curvature relationship is given by  ) ' ' )( () () (12 2 1 1 I E I E x bx Mx Rx    (2) where b(x) is the width function, M is the bending moment and Rx is the radius of curvature. Next, we (a) (b) “Design and Simulation of Zipping Variable Capacitors”, presented at MicroMechanics Europe 2007, Guimarães, Portugal, 16 – 18 September 2007.  consider a simplified 2D (unit beam width) model consisting of a single layered beam in which the actual beam width function b(x) is modelled using an  equivalent  Young’s  modulus  function  Em(x). The  moment-curvature  relationship  of  this  2D model is then e m m x b h x Ex Mx R3 ) () ( 12) (1   (3) where the subscript  m  denotes the parameters  of the model and be is the actual width of the bottom electrode. By equating (2) and (3) and imposing M and Rx to be equal we can obtain an expression for Em e mm b hI E I E x bx E 32 2 1 1 ) ' ' )( ( 12) (   (4) The Young’s modulus of each element along the 2D beam is then specified using equation (4). III.2  VARIATIONAL ANALYSIS A  second,  semi-analytical  method  was  used  to solve  for  the  C-V  characteristic  of  the  zipping capacitors.  The  released  curvature  of  the  top electrode may be expressed as 2 ) ( x x sr     (5) where  is a function of the stress in the tensile layer,  the  thickness  of  the  two  layers  and  their elastic  properties.  When  the  top  electrode  is deflected and zipping, its shape is approximated by  0 ) (  x sd   a x   0  2 ) ( ) ( a x c x sd     L x a      (6) where  a  and  c  are  variational  shape  parameters. Figure 5 shows the released and deflected shapes of a zipping capacitor.  Figure 5. Released and deflected top electrode shapes. The top electrode is modelled as a single layered beam  and  its  mechanical  strain  energy  can  be expressed as      Lad rarm dxdxs s d EIdxdxs d EIU2220222 ) (2 2  (7) where I is a function of x for the case where the top electrode  width  b  is  a  function  of  x.  The electrostatic energy can be expressed as        Lad r deadd ee dxs tV bdxtV bU 20022121   (8) where  td  is  the  dielectric  thickness,  0  is  the permittivity  of  free  space  and  the  dielectric permittivity d = r0 (be is a function of x for the case  where  the  bottom  electrode  is  shaped). According  to  the  principle  of  virtual  work,  the equilibrium  shape  of  the  top  electrode  can  be determined  by  minimising  an  appropriate  energy function. Assuming the capacitor is actuated by a variable voltage source, the required function is  e m U U c a    ) , (   (9) Differentiating  (9)  with  respect  to  a  and  c,  and setting the derivatives to zero yields two equations in a and c. Since no closed-form solution has been found, the equations are solved numerically giving the deformed shape, sd. The capacitance can then be calculated from the shape. III.3  RESULTS FOR TOP ELECTRODE SHAPING Figure 6 shows the simulated C-V characteristic of a  zipping  capacitor  with  a  linear  beam  width function, b(x)=0.18x+20 (m) for 0  x  400. The bottom electrode width is 20m and the dielectric has a thickness of 0.1m and a relative permittivity of 4. 0.00.51.01.52.02.53.00 2 4 6 8 10 12 14 16 18Voltage [V]C [pF]Ansys 2D Simplified ModelSemi-Analytical ModelAnsys 3D Figure 6. C-V characteristic of a zipping capacitor with linear beam shape. The results from the different simulation methods show similar characteristics qualitatively but differ  2 22 2 1 1222 2 2' E ' E 22) ( x xI Ihd hx sr     2 ) ( ) ( a x c x sd    L - a  a x “Design and Simulation of Zipping Variable Capacitors”, presented at MicroMechanics Europe 2007, Guimarães, Portugal, 16 – 18 September 2007.  in actual values of zipping voltages. Nevertheless, the results all indicate that controlled zipping can be achieved by shaping the top electrode and the increase in capacitance with voltage is now much more gradual relative to that in figure 3. For this design, the achieved capacitance ratio exceeds 20.  Figure 7 shows the C-V characteristic of a zipping capacitor with a high permittivity dielectric (r = 200).  The  dielectric  thickness  is  0.1m  and  the bottom electrode width is 5m. The width function is b(x)=0.18x+5 (m) for 0  x  100. The high permittivity of the dielectric allows a much larger capacitance ratio (>40) to be obtained for a smaller device size. 0123456789100.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0Voltage [V]C [pF] Figure 7. C-V characteristic of a zipping capacitor with a linear beam shape and a high permittivity dielectric. III.4  RESULTS FOR BOTTOM ELECTRODE SHAPING Results  from  the  semi-analytical  model  for capacitors  with  shaped  bottom  electrodes  are shown in figure 8. Three different bottom electrode shapes  have  been  modelled  for  a  capacitor  of length  L  =  480m.  The  top  electrode  width  is 70m  and  the  dielectric  is  0.3m  thick  with  a relative permittivity of 4. As expected, the results indicate  that  controlled  zipping  with  gradual increase  in  capacitance  is  possible.  Capacitance ratios larger than 9 were obtained based on these designs. IV  CONCLUSIONS Zipping  MEMS  variable  capacitors  with  large tuning ranges have been presented. The use of a high  permittivity  dielectric  layer  gives  a  large capacitance ratio for a compact device size. Stable x  zipping, and hence gradual increase in capacitance is  achieved  by  shaping  either  the  top  electrode beam  or  the  bottom  electrode.  Two  different modelling  methods  have  been  discussed  which allow  the  C-V  characteristics  of  various  design options to be simulated quickly. 0.00.51.01.52.02.53.00 2 4 6 8 10 12 14 16 18Voltage [V]C [pF]Linear, be(x) = -0.12x + 70Quadratic 1, be(x) = 0.00025(x-L)^2+12.4Quadratic 2, be(x) = 70 - 0.00025x^2 Figure 8. C-V characteristics of capacitors with shaped bottom electrodes. Further  work  is  needed  to  develop  fabrication processes for making the zipping capacitors. As a first step, devices with a SiO2 dielectric layer will be  fabricated.  Subsequently,  high  permittivity dielectric materials such as lead zirconate titanate or barium strontium titanate will be integrated into high tuning range devices. REFERENCES [1]  J.  J.  Yao,  "RF  MEMS  from  a  device  perspective," Journal of micromechanics and microengineering, vol. 10, 2000, pp. R9-R38. [2]  G. M. Rebeiz, "RF MEMS switches and switch circuits," IEEE microwave magazine, vol. 2, 2001, pp. 59-71. [3]  K. Sato and M. Shikida, "Electrostatic film actuator with a large vertical displacement," presented at Micro Electro Mechanical  Systems,  1992,  MEMS  '92,  Proceedings, Travemunde, Germany, 1992. [4]  R.  Legtenberg,  J.  Gilbert,  S.  D.  Senturia,  and  M. Elwenspoek,  "Electrostatic  curved  electrode  actuators," Microelectromechanical Systems, Journal of, vol. 6, 1997, pp. 257-265. [5]  E. S. Hung and S. D. Senturia, "Tunable capacitors with programmable  capacitance-voltage  characteristic," presented  at  Technical  Digest,  Solid-State  Sensor  and Actuator Workshop 1998, Hilton Head Island, SC, USA, 1998. [6]  S. Duffy, C. Bozler, S. Rabe, J. Knecht, L. Travis, P. Wyatt, C. Keast, and M. Gouker, "MEMS microswitches for reconfigurable microwave circuitry," IEEE microwave and wireless components letters, vol. 11, 2001, pp. 106-8. [7]  S. D. Senturia, Microsystem design: Kluwer Academic, 2000. “Design and Simulation of Zipping Variable Capacitors”, presented at MicroMechanics Europe 2007, Guimarães, Portugal, 16 – 18 September 2007. 

Electrostatic curved electrode actuators,&quot;

Electrostatic film actuator with a large vertical displacement,&quot; presented at Micro Electro Mechanical Systems,

MEMS microswitches for reconfigurable microwave circuitry,&quot;

Microsystem design: Kluwer Academic,

RF MEMS from a device perspective,&quot;

RF MEMS switches and switch circuits,&quot;

Tunable capacitors with programmable capacitance-voltage characteristic,&quot; presented at Technical Digest, Solid-State Sensor and Actuator Workshop

Design and simulation of zipping variable capacitors

Design and simulation of zipping variable capacitors

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