Abstract:
A micro electro mechanical system device includes a fixed electrode that includes a first electrode group, and a movable electrode that moves with respect to the fixed electrode as voltage is applied and includes a second electrode group that opposes the first electrode group, wherein electrodes of at least one among the first electrode group and the second electrode group are connected via a resistor.

Description:
This application is a continuation of International Application No. PCT/JP2007/052977, filed Feb. 19, 2007, the disclosure of which is incorporated herein in its entirety by reference. 
    
    
     FIELD 
     The embodiments discussed herein are related to a MEMS device and an optical switch. 
     BACKGROUND 
     A micro electro mechanical system (MEMS) device is made up of small parts created by fine processing technology for semiconductors. An optical switch that switches optical paths in an optical communication system uses a MEMS device (see, for example, Japanese Laid-Open Patent Application Publication No. 2004-70053). 
     The MEMS device applies high voltage to electrodes to gather electrical charge at each electrode and yields a mechanical displacement by electrostatic force. Since the extent of the mechanical displacement can be adjusted by the voltage applied, the MEMS device is used for many purposes in addition to optical switches. 
       FIG. 15  is a perspective view of electrodes of a conventional MEMS device. As depicted in  FIG. 15 , the MEMS device includes a fixed electrode  1510  and a movable electrode  1520 . On the fixed electrode  1510  and the movable electrode  1520 , electrodes  1511 - 1514  and  1521 - 1524  are arranged in a comb-like configuration. 
     When voltage is applied to the fixed electrode  1510  and the movable electrode  1520 , the electrodes  1511 - 1514  take positive charge and the electrodes  1521 - 1524  take negative charge. In this way, electrostatic force works between the electrodes  1511 - 1514  and the electrodes  1521 - 1524 , whereby the movable electrode  1520  moves in the direction indicated by an arrow  1530  with respect to the fixed electrode  1510 . 
     When the MEMS device is applied to an optical switch, a mirror is fixed to the movable electrode  1520 . Voltage is applied to the fixed electrode  1510  and the movable electrode  1520  to displace the movable electrode  1520  and change the angle at which the mirror reflects light. As a result, the optical path is switched by controlling the voltage for the fixed electrode  1510  and the movable electrode  1520 . 
     However, if the voltage applied is increased in order to increase the amount of displacement of the movable electrode  1520  and improve performance of the MEMS device, occasionally spark discharge occurs between the fixed electrode  1510  and the movable electrode  1520 . The spark discharge results in unexpected movement of the movable electrode  1520  or damage to the fixed electrode  1510  and the movable electrode  1520 . 
       FIG. 16  is an explanatory diagram for the principle of spark discharge. As depicted in  FIG. 16 , increasing the voltage applied between electrodes  1601  and  1602  accelerates an electron  1612  and a gas molecule  1611  colliding with the electron  1612  is ionized. As reference numeral  1620  indicates, a positively charged ion  1621  generated by the ionization collides with the electrode  1602 , a negative electrode. A large, instantaneous electric current due to secondary electron emission caused by the collision is called spark discharge. 
       FIG. 17  is an explanatory diagram illustrating spark discharge occurring in the conventional MEMS device. As indicated by reference numeral  1701 , when spark discharge occurs between an electrode  1512  and an electrode  1521 , charges on electrodes near the electrodes  1512  and  1521  (or on all electrodes) disappear by recombination. 
     Thus, electrostatic attractive force displacing the movable electrode  1520  drops instantaneously. An instantaneous drop in electrostatic attractive force causes unexpected behavior of the movable electrode  1520 . In particular, when the MEMS device is used as an optical switch in an optical network, the spark discharge causes unfavorable displacement of the movable electrode  1520  and disturbs the optical path, resulting in a communications breakdown on a network. 
     In addition, once the spark discharge occurs between the electrodes  1512  and  1521 , it takes time to supply the electrodes  1512  and  1521  with charges from other electrodes, a power source, or ground, and to reduce a potential difference between the electrodes  1512  and  1521 . As a result, the spark discharge occurs for a long time and consequently, it takes time for the movable electrode  1520  to return to an original position. Furthermore, since the spark discharge occurs for a long time, current due to the discharge increases, whereby the displacement of the movable electrode  1520  becomes larger. 
       FIG. 18  is a circuit diagram that is equivalent to the MEMS device depicted in  FIG. 17 . The fixed electrode  1510  and the movable electrode  1520  of the MEMS device are equivalent to multiple condensers connected in parallel. As depicted in  FIG. 18 , the condensers connected in parallel can be thought of as one large condenser  1801 . 
     The spark discharge between the fixed electrode  1510  and the movable electrode  1520  is equivalent to a state where a switch  1802 , which is connected to both ends of the condenser  1801 , is instantaneously turned on. As the switch  1802  is turned on, charges on the condenser run through the switch  1802  and disappear. 
       FIG. 19  is a graph depicting the displacement of the mirror of the conventional MEMS device due to spark discharge. In  FIG. 19 , the horizontal axis is time [sec]. The vertical axis is an angle [°] of rotation of a mirror fixed on the movable electrode  1520 . Reference numeral  1901  indicates a point at which the spark discharge occurred. As depicted in  FIG. 19 , the maximum displacement of the mirror fixed on the movable electrode  1520  occurs when the spark discharge occurs, and the displacement decreases over time. 
     One way to deal with this problem is to cover the entire surface of the fixed electrode  1510  and the movable electrode  1520  with insulating film, preventing secondary electrons from being generated at the collision of positive ions against electrodes. One example is a vacuum deposition method where coating material is heated to a high temperature and applied to the MEMS device by exposing a surface of the MEMS device to the vapors under vacuum conditions. 
     However, when the shape of the fixed electrode  1510  and the movable electrode  1520  is complicated as depicted in  FIG. 15 , it is difficult to coat the electrodes uniformly. In addition, since high voltage is applied to the fixed electrode  1510  and the movable electrode  1520 , insulating film coating the electrodes may cause residual polarization; even after the voltage is changed, polarization before the change of voltage remains. In this case, even if the voltage is controlled, actual displacement and planned displacement do not coincide. 
     SUMMARY 
     According to an aspect of the invention, a micro electro mechanical system device includes a fixed electrode that includes a first electrode group; and a movable electrode that moves with respect to the fixed electrode as voltage is applied and includes a second electrode group that opposes the first electrode group. Further, electrodes of at least one among the first electrode group and the second electrode group are connected via a resistor. 
     The object and advantages of the invention will be realized and attained by means of the elements and combinations particularly pointed out in the claims. 
     It is to be understood that both the foregoing general description and the following detailed description are exemplary and explanatory and are not restrictive of the invention, as claimed. 
    
    
     
       BRIEF DESCRIPTION OF DRAWINGS 
         FIG. 1  is a perspective view illustrating a MEMS device according to a first embodiment; 
         FIG. 2  is a perspective view of electrodes of the MEMS device according to the first embodiment; 
         FIG. 3  is a circuit diagram equivalent to the MEMS device depicted in  FIG. 2 ; 
         FIG. 4  is a perspective view of the MEMS device illustrating operation of the MEMS device according to the first embodiment; 
         FIG. 5  is a table illustrating an effect of the MEMS device according to the first embodiment; 
         FIG. 6  is a graph depicting displacement due to spark discharge in the MEMS device according to the first embodiment; 
         FIG. 7  is a perspective view of one example of fabrication of a resistor of the MEMS device according to the first embodiment; 
         FIG. 8  is a graph illustrating a relationship between impurity density and resistance; 
         FIG. 9  is a circuit diagram equivalent to an example where electrodes of the MEMS device according to the first embodiment are modified; 
         FIG. 10  is a graph depicting the displacement of a mirror of the MEMS device due to spark discharge; 
         FIG. 11  is a perspective view of electrodes of a MEMS device according to a second embodiment; 
         FIG. 12  is a perspective view of electrodes of a MEMS device according to a third embodiment; 
         FIG. 13  is a perspective view of an example of modification of the electrodes of the MEMS device according to the third embodiment; 
         FIG. 14  is a plan view of an optical switch according to a fourth embodiment; 
         FIG. 15  is a perspective view of electrodes of a conventional MEMS device; 
         FIG. 16  is an explanatory diagram for the principle of spark discharge; 
         FIG. 17  is an explanatory diagram illustrating spark discharge occurring in the conventional MEMS device; 
         FIG. 18  is a circuit diagram that is equivalent to the MEMS device depicted in  FIG. 17 ; and 
         FIG. 19  is a graph depicting the displacement of the mirror of the conventional MEMS device due to spark discharge. 
     
    
    
     DESCRIPTION OF EMBODIMENTS 
     Preferred embodiments of the present invention will be explained with reference to the accompanying drawings. 
       FIG. 1  is a perspective view illustrating a MEMS device according to a first embodiment. As depicted in  FIG. 1 , a MEMS device  100  includes a substrate  200 , a mirror  300 , an actuator  400 , and an actuator  500 . The MEMS device  100  is an optical switch that changes the angle of the mirror  300  a small amount and switches paths of light reflected at the mirror  300 . 
     The mirror  300  is rotatable about the x-axis and the y-axis. The actuator  400  rotates the mirror  300  about x-axis. The actuator  400  includes a fixed electrode  410 , a movable electrode  420 , and a voltage applying unit  430 . The fixed electrode  410  is fixed to the substrate  200 . 
     The movable electrode  420  is rotatable about an axis  201  of the substrate  200 . The movable electrode  420  rotates about the axis  201  and changes a relative position with respect to the fixed electrode  410 . The mirror  300  is fixed to the movable electrode  420 . The voltage applying unit  430  applies voltage to the fixed electrode  410  and the movable electrode  420 . A voltage control unit (not depicted) is connected to the voltage applying unit  430  and controls the voltage applied to the fixed electrode  410  and the movable electrode  420  by the voltage applying unit  430 . 
     The voltage control unit controls the applied voltage, for example, at tens to hundreds of volts. In this way, the voltage applied to the fixed electrode  410  and the movable electrode  420  causes electrostatic attractive force between the fixed electrode  410  and the movable electrode  420  and rotates the movable electrode  420  about the axis  201 . As the movable electrode  420  rotates, the mirror  300  fixed to the movable electrode  420  also rotates about the axis  201 . 
     As a result, the angle at which the mirror  300  reflects light changes. When voltage applied to the fixed electrode  410  and the movable electrode  420  is changed, the degree of rotation of the movable electrode  420  changes; whereby the angle at which the mirror  300  reflects light can be controlled. When voltage applied to the fixed electrode  410  and the movable electrode  420  is changed, optical paths can be switched. 
     The actuator  500  rotates the mirror  300  about y-axis. The structure of the actuator  500  is similar to that of the actuator  400  and thus the explanation thereof is omitted. The actuator  400  rotates the mirror  300  about the x-axis and the actuator  500  rotates the mirror  300  about the y-axis; whereby the angle at which the mirror  300  reflects light can be changed two-dimensionally. 
       FIG. 2  is a perspective view of electrodes of the MEMS device according to the first embodiment. In  FIG. 2 , parts identical to those depicted in  FIG.1  are indicated by the same reference numerals used in  FIG. 1  and explanations therefor are omitted. As depicted in  FIG. 2 , the fixed electrode  410  includes a fixed electrode unit  410   a  that is fixed to the substrate  200  and to which voltage is applied, and multiple electrodes  411 - 414  (first electrode group) that are arranged in a comb-like configuration on the fixed electrode  410   a.    
     The movable electrode  420  includes a movable electrode unit  420   a  to which voltage is applied, and multiple electrodes  421 - 424  (second electrode group) that are formed on the movable electrode unit  420   a . The electrodes  421 - 424  of the movable electrode  420  face the electrodes  411 - 414  of the fixed electrode  410  and are arranged in a comb-like configuration. 
     In this explanation, the electrodes  411 - 414  and the electrodes  421 - 424  take a comb-like configuration where flat plates are arranged at regular intervals. There is sufficient space between the electrodes  411 - 414  and the electrodes  421 - 424  for the movable electrode  420  to move with respect to the fixed electrode  410  when voltage is applied to the fixed electrode  410  and the movable electrode  420 . 
     At the base of each electrode  411 - 414  of the fixed electrode  410 , a resistor  411   a - 414   a  is disposed. The base of the respective electrodes  411 - 414  indicates an end part of the respective electrodes  411 - 414  with respect to the fixed electrode  410 , or a part near the end part. At the base of each electrode  421 - 424  of the movable electrode  420 , a resistor  421   a - 424   a  is disposed. For each of the resistors  411   a - 414   a  and the resistors  421   a - 424   a , a resistor of 10 kΩ may be used. 
     In this way, the electrodes  411 - 414  of the fixed electrode  410  are connected via the resistors  411   a - 414   a  so that the respective, electrodes  411 - 414  are electrically isolated. The electrodes  421 - 424  of the movable electrode  420  are also connected via the resistors  421   a - 424   a  so that the respective electrodes  421 - 424  are electrically isolated. 
       FIG. 3  is a circuit diagram equivalent to the MEMS device depicted in  FIG. 2 . As depicted in  FIG. 3 , the electrodes  411 - 414  and the electrodes  421 - 424  facing the electrodes  411 - 414  are equivalent to multiple condensers  311 - 314  connected in parallel. 
     Since the condensers  311 - 314  are electrically isolated by resistors  331 - 334  and resistors  341 - 344 , the condensers  311 - 314  are multiple independent condensers and cannot be considered as one large condenser. When the electrodes  411 - 414  and the electrodes  421 - 424  are taken as the condensers  311 - 314 , the shape and position of the respective electrodes  411 - 414  and  421 - 424  are designed so that capacity of the respective condensers  311 - 314  becomes 1.4 pF. 
     The voltage applying unit  430  that applies voltage to the fixed electrode  410  and the movable electrode  420  is equivalent to a power source  320 . The resistors  411   a - 414   a  at the bases of the electrodes  411 - 414  are equivalent to the resistors  331 - 334  at the ends of the condensers  311 - 314 . The resistors  421   a - 424   a  are equivalent to the resistors  341 - 344  at the ends of the condensers  341 - 344 . 
     The spark discharge is equivalent to a state where a switch  350  connecting both ends of one of the condensers  311 - 314  (in this explanation, the condenser  314 ) is instantaneously turned on. After the switch  350  is instantaneously turned on, charges on the condenser  314  disappear through the switch  350 . 
     The condensers  311 - 313  to which the switch  350  is not connected are electrically isolated from the condenser  314  by the resistors  331 - 334  and the resistors  341 - 344 . Thus, charges on the condensers  311 - 313  remain even after the switch  350  is instantaneously turned on. 
       FIG. 4  is a perspective view of the MEMS device illustrating operation of the MEMS device according to the first embodiment. As depicted in  FIG. 4 , positive charges gather on the fixed electrode  410  and negative charges gather on the movable electrode  420  to create electrostatic attractive force. When the spark discharge occurs between the electrode  412  and the electrode  421 , as expressed by reference numeral  401 , charges on the electrodes  412  and  421  disappear. 
     The electrode  412  is electrically isolated from the electrodes  411 ,  413 , and  414  by the resistor  412   a . Therefore, flow of charges from the electrodes  411 ,  413 , and  414  to the electrode  412  is suppressed. The electrode  421  is also electrically isolated from the electrodes  422 - 424  by the resistor  421   a . Thus, flow of charges from the electrodes  422 - 424  to the electrode  421  is suppressed. 
     In this way, electrodes that lose charges are limited to the electrodes  412  and  421  between which the spark discharge occurred; whereby decrease of electrostatic attractive force between the fixed electrode  410  and the movable electrode  420  is suppressed. For instance, when the fixed electrode  410  and the movable electrode  420  each has  20  electrodes arranged in a comb-like configuration, even if spark discharge occurs between one pair of electrodes, the electrostatic attractive force is maintained between the other 19 pairs of electrodes. Therefore, decrease in the electrostatic attractive force between the fixed electrode  410  and the movable electrode  420  is reduced to one twentieth. 
     Since electrical current due to the spark discharge is converted to voltage at the resistors  412   a  and  421   a , potential of the electrodes  412  and  421  fluctuates with small electrical current. Thus, large electrical current flows at the spark discharge, and the potential difference between the electrodes  412  and  421  changes instantaneously. As a result, a period of the spark discharge shortens and the current due to the spark discharge decreases. 
       FIG. 5  is a table illustrating an effect of the MEMS device according to the first embodiment. In the table of  FIG. 5 , the first column depicts a resistance [Ω] of the resistors  411   a - 414   a  and  421   a - 424   a  of the fixed electrode  410  and the movable electrode  420 . The second column depicts a voltage drop [%] between the fixed electrode  410  and the movable electrode  420  when the spark discharge occurs between the fixed electrode  410  and the movable electrode  420 . 
     The third column depicts a variation [°] of a rotation angle of the mirror  300  immediately after the spark discharge due to the spark discharge and the voltage drop between the fixed electrode  410  and the movable electrode  420 . The fourth column depicts a RC time constant [sec] of the equivalent circuit depicted in  FIG. 3 . 
     As depicted in the second row of the table, the voltage drop of the conventional MEM device without the resistors  411   a - 414   a  and  421   a - 424   a  (the resistance is set to 1Ω) is 100% and the comparison is made with cases below. The variation of the rotation angle of the mirror  300  is 2.55°. The RC time constant is 0.1 psec. 
     As depicted in the third row of the table, when the resistance of the resistors  411   a - 414   a  and  421   a - 424   a  is set to 1 kΩ, even if the spark discharge occurs between a pair of the electrodes  411 - 414  and  421 - 424 , the voltage drop is 45.3%. The variation of the rotation angle of the mirror  300  is 1.04°. The RC time constant of the equivalent circuit is 100 psec. 
     As depicted in the fourth row of the table, when the resistance of the resistors  411   a - 414   a  and  421   a - 424   a  is set to 10 kΩ, even if the spark discharge occurs between a pair of the electrodes  411 - 414  and  421 - 424 , the voltage drop is 7.7%. The variation of the rotation angle of the mirror  300  is 0.19°. The RC time constant of the equivalent circuit is 1 nsec. 
     As explained, resistors of high resistance are disposed at the bases of the electrodes  411 - 414  and  421 - 424  so that the variation of the rotation angle of the mirror  300  at the spark discharge can be greatly reduced. In particular, by the insertion of the resistors  411   a - 414   a  and  421   a - 424   a  of 10 kΩ at the bases of the electrodes  411 - 414  and  421 - 424 , the variation of the rotation angle of the mirror  300  is reduced to the extent that a communications system is not affected. The resistors  411   a - 414   a  and  421   a - 424   a  may have a resistance in the range of 1 kΩ to 20 kΩ with consideration of manufacturing limitations. 
     The resistors  411   a - 414   a  and  421   a - 424   a  at the bases of the electrodes  411 - 414  and  421 - 424  forms a RC filter in the equivalent circuit (see  FIG. 3 ). The RC time constant of the RC filter increases up to about 1 nsec. However, since the rotation of the mirror  300  is a mechanical movement, variation of the rotation angle of the mirror  300  is sufficiently slow compared with the RC time constant. 
     Specifically, a response speed of the rotation of the movable electrode  420  and the mirror  300  is about 1 msec, calculated from resonance frequency. This response speed is about one million times slower than the RC filter&#39;s RC time constant of 1 nsec. Thus, the RC time constant of this equivalent circuit is absorbed in the response speed of the movable electrode  420  and the mirror  300  and can be ignored. 
       FIG. 6  is a graph depicting displacement due to spark discharge in the MEMS device according to the first embodiment. The horizontal axis of  FIG. 6  is time [sec]. The vertical axis is the rotation angle [°] of the mirror  300  of the MEMS device  100 . Reference numeral  601  indicates the time at which the spark discharge occurred. As indicated in  FIG. 6 , the variation of the rotation angle of the mirror  300  at the time when the spark discharge occurs is greatly reduced compared with the conventional MEMS device (see  FIG. 19 ). In addition, the recovery time from the occurrence of the spark discharge to convergence to the original angle of the mirror  300  shortens. 
       FIG. 7  is a perspective view of one example of fabrication of a resistor of the MEMS device according to the first embodiment. Often a semiconductor of a single crystal has high resistivity and thus impurities are purposely introduced. Electrons or holes are donated by the doped impurity ions due to the thermal energy of the base material and the donated electrons or holes act as carriers. The resistivity of a region into which impurities are introduced lowers in proportion to the amount of impurities and thus electrical current flows more easily. 
     On the other hand, when impurities whose ion valence is opposite the ion valence of the base material are introduced to the base material, the resistivity of the region in which the impurities are introduced increases. For instance, when trivalent ions have been added to tetravalent silicon and pentavalent ions are further added to the silicon, holes originating from the trivalent ions and electrons originating from the pentavalent ions are recombined; whereby carriers at the conduction band and the valence band decreases. 
     In this case, the resistivity of the region into which pentavalent ions are added can be increased. When the number of trivalent ions is equal to the number of tetravalent ions, the resistivity increases up to a value nearly equal to the resistivity of an intrinsic semiconductor, into which no impurity has been introduced. Ions for increasing the resistivity of the base station are called compensation ions. 
     Here, as depicted in  FIG. 7 , n-type S i    702  and n-type S i    703  are formed to sandwich S i O 2    701 , and the patterning of a resist layer  704  is performed on a surface of the n-type S i    702 . The compensation ions  705  are introduced from above the resist layer  704 , and the resistivity of a region  706  that is not covered by the resist layer  704  is increased. A resist layer may be formed on the n-type S i    703  and the compensation ions  705  may be introduced (not depicted in  FIG. 7 ); whereby the resistivity of part of the n-type S i    703  is increased. 
     After the S i O 2    701  and the resist layer  704  are removed from the n-type S i    702  and the n-type S i    703 , the n-type S i    702  and the n-type S i    703  with resistivity increased in part are obtained. Cutting of the n-type S i    702  and the n-type S i    703  makes the comb-like fixed electrode  410  and movable electrode  420  or various electrodes which will be described later. When the compensation ions  705  are accelerated in the electric field, the compensation ions  705  can be introduced not only to a surface of but also deeper into the n-type S i    702  and the n-type S i    703 . 
       FIG. 8  is a graph illustrating a relationship between impurity density and resistance. In  FIG. 8 , the horizontal axis is the density of an impurity [1/m3] that is introduced into a to-be-a-resistor part of the fixed electrode  410  and the movable electrode  420 . The vertical axis is the resistance [Ω] formed in the fixed electrode  410  and the movable electrode  420 . As indicated by reference numeral  801 , in general, the density of impurity is about 1023/m3 and the resistance is about 1Ω. 
     On the other hand, according to the MEMS device  100  of the first embodiment, as indicated by reference numeral  802 , a resistor of about 10 kΩ can be formed after the compensation ions  705  are introduced into the to-be-a-resistor part  706  and the density of impurity is reduced to about 1019/m3. Here, one example of a process of forming a resistor has been introduced but the process is not limited to this example, and various methods may be used. 
       FIG. 9  is a circuit diagram equivalent to an example where electrodes of the MEMS device according to the first embodiment are modified. In  FIG. 9 , parts identical to those depicted in  FIG. 3  are indicated by the same reference numerals used in  FIG. 3  and explanations therefor are omitted. In the first embodiment above, the fixed electrode  410  and the movable electrode  420  include the resistors  411   a - 414   a  and the resistors  421   a - 424   a  respectively but only one of the fixed electrode  410  and the movable electrode  420  may have such resistors. 
       FIG. 10  is a graph depicting the displacement of a mirror of the MEMS device due to spark discharge. With respect to  FIG. 10 , only the electrodes  411 - 414  of the fixed electrode  410  include resistors of 20 kΩ respectively. Reference numeral  1001  indicates the point of time at which the spark discharge has occurred. As indicated by reference numeral  1001 , the variation of the rotation angle of the mirror  300  at the spark discharge is greatly reduced compared with the conventional MEMS device (see  FIG. 19 ). 
     As explained, even if only one among the fixed electrode  410  and the movable electrode  420  includes resistors, an effect of the embodiments can be obtained. Further, when resistors are used of a larger resistance compared with the resistors used when included in both the fixed electrode  410  and the movable electrode  420 , an advantage can be obtained in a similar manner to a case where the resistors are included in both the fixed electrode  410  and the movable electrode  420 . 
     Though the resistance of the resistors  411   a - 414   a  and  421   a - 424   a  is about 10 kΩ or 20 kΩ in the above explanation, resistance of the resistors is determined based on the shape of the fixed electrode  410  and the movable electrode  420 , and on the voltage applied to the fixed electrode  410  and the movable electrode  420  by the voltage applying unit  430  (this is also the case in the subsequent embodiments). 
     The larger voltage applied to the fixed electrode  410  and the movable electrode  420  by the voltage applying unit  430  is, the higher the resistance of the resistors is set. The shape of the electrode indicates an interval between electrodes  411 - 414  and an interval between electrodes  421 - 424 , an interval between opposing electrodes  411 - 414  and  421 - 424 , the electrode area opposing another electrode, and so on. 
     The resistance of the resistors is set such that the movement of charges between the electrodes  411 - 414  and the electrodes  421 - 424  is suppressed. In this way, electrodes that lose charges are limited to those at which a spark discharge has occurred and the decrease of the electrostatic attractive force between the fixed electrode  410  and the movable electrode  420  is prevented. 
       FIG. 11  is a perspective view of electrodes of a MEMS device according to a second embodiment. According to the MEMS device  100  of the first embodiment, the electrodes  411 - 414  and the electrodes  421 - 424  are formed in a comb-like configuration on the fixed electrode  410  and the movable electrode  420 ; however, configuration of the fixed electrode  410  and the movable electrode  420  is not limited to the first embodiment; namely, multiple electrodes may be arranged in array-like configuration on the fixed electrode  410  and the movable electrode  420 . 
     A MEMS device  100  according to the second embodiment includes, for example, as depicted in  FIG. 11 , a movable electrode  1110 , a fixed electrode  1120 , and a fixed electrode  1130 . The movable electrode  1110  is rotatable about an axis  1140 . The movable electrode  1110  includes a flat plate unit  1110   a  and a flat plate unit  1110   b.    
     The flat plate unit  1110   b  is placed on a back side and at a central part of the flat plate unit  1110   a , the flat plate unit  1110   b  being perpendicular to the flat plate unit  1110   a . Therefore, a vertical sectional view of the movable electrode  1110  has a T-shape. On a surface of the flat plate unit  1110   a , a mirror (not depicted) is fixed. In two spaces separated by the flat plate unit  1110   b , multiple parallel electrodes  1111  (second electrode group) and multiple parallel electrodes  1112  (second electrode group) are disposed. 
     On the fixed electrode  1120 , multiple parallel electrodes  1121  (first electrode group) opposing the parallel electrodes  1111  on the movable electrode  1110  are disposed. On the fixed electrode  1130 , multiple parallel electrodes  1131  (first electrode group) opposing the parallel electrodes  1112  on the movable electrode  1110  are disposed. 
     When voltage is applied to the movable electrode  1110  and the fixed electrodes  1120  and  1130 , electrostatic attractive force arises between the movable electrode  1110  and the fixed electrodes  1120  and  1130 . According to the voltage applied, the parallel electrodes  1111  are attracted to the parallel electrodes  1121  or the parallel electrodes  1112  are attracted to the parallel electrodes  1131 . 
     In this way, the movable electrode  1110  rotates about the axis  1140 . As the movable electrode  1110  rotates, the mirror fixed on the movable electrode  1110  also rotates about the axis  1140 ; whereby an angle at which the mirror reflects light changes. Controlling the voltage applied to the movable electrode  1110  and the fixed electrodes  1120  and  1130  changes the degree of rotation and the direction of rotation of the movable electrode  1110 ; whereby an angle at which the mirror reflects light can be adjusted. 
     At the base of each parallel electrode  1111 , a resistor  1111   a  is disposed. At the base of each parallel electrode  1112 , a resistor  1112   a  is disposed. At the base of each parallel electrode  1121 , a resistor  1121   a  is disposed. At the base of each parallel electrodes  1131 , a resistor  1131   a  is disposed. 
     In this way, the parallel electrodes  1111  are connected to each other via the resistors  1111   a  and are electrically isolated. The parallel electrodes  1112  are connected to each other via the resistors  1112   a  and are electrically isolated. 
     The parallel electrodes  1121  are connected to each other via the resistors  1121   a  and are electrically isolated. The parallel electrodes  1131  are connected to each other via the resistors  1131   a  and are electrically isolated. An equivalent electric circuit for this example is similar to the equivalent circuit depicted in  FIG. 3 . 
     As a result, even if spark discharge occurs between a pair of electrodes  1111 ,  1112 ,  1121 , and  1131 , electrodes that lose charges are limited to the pair of electrodes between which the spark discharge has occurred; whereby the electrostatic attractive force between the other parallel electrodes is not lowered. Thus, the variation of the rotation angle of the mirror fixed on the movable electrode  1110  at the spark discharge can be reduced. 
     Further, in this structure, an aspect ratio of each of resistors  1111   a ,  1112   a ,  1121   a ,  1131   a  to the movable electrode  1110 , the fixed electrode  1120 , the fixed electrode  1130  is small. Thus, processes for forming the resistors  1111   a ,  1112   a ,  1121   a ,  1131   a  on the movable electrode  1110 , the fixed electrode  1120  and the fixed electrode  1130  are facilitated. 
       FIG. 12  is a perspective view of electrodes of a MEMS device according to a third embodiment.  FIG. 13  is a perspective view of an example of modification of the electrodes of the MEMS device according to the third embodiment. In  FIGS. 12 and 13 , parts identical to those depicted in  FIG. 2  are indicated by the same reference numerals used in  FIG. 2  and explanations therefor are omitted. 
     In the explanation of the MEMS device  100  according to the first and second embodiments, a resistor (for example resistors  411   a - 414   a ) is disposed at the base of each electrode (for example electrodes  411 - 414 ) of the fixed electrode  410  and the movable electrode  420 ; however, disposition of the resistor is not limited to the embodiments, i.e., it suffices that electrodes are connected to each other via a resistor and are electrically isolated. 
     For instance, as depicted in  FIG. 12 , multiple electrodes may be electrically isolated by one resistor. Specifically, at the fixed electrode  410 , a resistor  1201  may be formed on a fixed electrode unit  410   a  on which the electrodes  411 - 414  are provided. As a result, the electrodes  411 - 414  are connected to each other via the resistor  1201  and are electrically isolated. 
     At the movable electrode  420 , a resistor  1202  may be formed on the movable electrode unit  420   a  on which the electrodes  421 - 424  are provided. As a result, the electrodes  421 - 424  are connected to each other via the resistor  1202  and are electrically isolated. An equivalent electric circuit in this case is similar to the equivalent circuit depicted in  FIG. 3 . 
     Further, as depicted in  FIG. 13 , for example, resistors  1311 - 1313  may be disposed in the fixed electrode unit  410   a  such that the resistors  1311 - 1313  separate the electrodes  411 - 414 . As a result, the electrodes  411 - 414  are connected to each other via the resistors  1311 - 1313  and are electrically isolated. 
     Resistors  1321 - 1323  may be provided in the movable electrode unit  420   a  such that the resistors  1321 - 1323  separate the electrodes  421 - 424 . As a result, the electrodes  421 - 424  are connected to each other via the resistors  1321 - 1323  and are electrically isolated. An equivalent electric circuit in this case is similar to the equivalent circuit depicted in  FIG. 3 . 
     The MEMS device  100  and the modified MEMS device  100  according to the third embodiment may be applied to the MEMS device  100  of the first or second embodiment (see  FIGS. 2 ,  9 ,  11 ). 
       FIG. 14  is a plan view of an optical switch according to a fourth embodiment. As depicted in  FIG. 14 , the MEMS devices  100  of the above embodiments are arranged on a circuit board  1400  to form a packaged optical switch. The MEMS devices  100  are connected to the circuit board  1400  by wire bonding. 
     The circuit board  1400  includes a voltage control unit (not depicted) that controls the voltage to be applied to the fixed electrode  410  and the movable electrode  420  by the voltage applying unit  430 . The voltage control unit controls the voltage applied to each of the MEMS devices  100  arranged on the circuit board  1400  and adjusts the angle at which the mirrors  300  of the MEMS devices reflect light. 
     As explained, according to a MEMS device and an optical switch of the embodiments, the reduction of electrostatic attractive force between a fixed electrode and a movable electrode upon spark discharge can be prevented. Therefore, according to a MEMS device and an optical switch of the embodiments, unexpected variation of the movable electrode upon spark discharge is suppressed; whereby stable behavior of the MEMS device and the optical switch can be achieved. Further, the time period that the spark discharge persists is shortened and the recovery time of the movable electrode to return to an original position is also shortened. 
     In the explanation of the embodiments above, a MEMS device is applied to an optical switch; however, application of the MEMS device according to the embodiments is not limited to an optical switch. 
     According to the embodiments, even if spark discharge occurs, unexpected variation of a movable electrode can be prevented. In addition, the entire surface of an electrode may be covered with an insulating film. In this way, the spark discharge is curbed and unexpected variation of the movable electrode is suppressed. As a result, a MEMS device such as an optical switch can operate more stably. 
     All examples and conditional language recited herein are intended for pedagogical purposes to aid the reader in understanding the invention and the concepts contributed by the inventor to furthering the art, and are to be construed as being without limitation to such specifically recited examples and conditions, nor does the organization of such examples in the specification relate to a showing of the superiority and inferiority of the invention. Although the embodiments of the present invention have been described in detail, it should be understood that the various changes, substitutions, and alterations could be made hereto without departing from the spirit and scope of the invention.