MEMS Electrical Fundamentals

Transcription

1 ROCHESTER INSTITUTE OF TECHNOLOGY MICROELECTRONIC ENGINEERING MEMS Electrical Fundamentals Dr. Lynn Fuller webpage: Electrical and Microelectronic Engineering Rochester Institute of Technology 82 Lomb Memorial Drive Rochester, NY microe program webpage: MEMSOutline.ppt Page 1

2 OUTLINE Introduction Resistors for temperature and light Resistor as Heaters Uniform Doped pn Junction Photodiodes and LEDs Diode Temperature Sensors Capacitors Capacitors as Sensors Chemicapacitor Diaphragm Pressure Sensor Condenser Microphone Capacitors as Electrostatic Actuators Signal Conditioning Page 2

3 INTRODUCTION Resistor heaters are used in many MEMS applications including ink jet print heads, actuators, bio-mems, chemical detectors and gas flow sensors. Resistors are used as temperature sensors, strain sensors and light sensors. Diodes are used for sensing temperature and light. Capacitors are used for sensing displacement in accelerometers and gyroscopes. Capacitors are also used in chemical sensors, liquid level sensing and microphones. This module will discuss the electrical fundamentals for these applications. Page 3

4 OHM S LAW I R = V/I = 1/slope + V I R V - Resistor a two terminal device that exhibits a linear I-V characteristic that goes through the origin. The inverse slope is the value of the resistance. Page 4

5 THE SEMICONDUCTOR RESISTOR Resistance = R = L/Area = s L/w ohms Resistivity = Rho = 1/( qµ n n + qµ p p) ohm-cm Sheet Resistance = Rhos s = 1/ ( q µ(n) N(x) dx) ~ 1/( qµ Dose) ohms/square s = / t w t Rho is the bulk resistivity of the material (ohm-cm) Rhos is the sheet resistance (ohm/sq) = Rho / t L R Area Note: sheet resistance is convenient to use when the resistors are made of thin sheet of material, like in integrated circuits. R = Rho L / Area = Rhos L/W Page 5

6 Mobility (cm 2 / V sec) MEMS Electrical Fundamentals MOBILITY ^13 10^14 holes 10^15 10^16 electrons 10^17 10^18 Total Impurity Concentration (cm -3 ) 10^19 10^20 Electron Arsenic and hole mobilities in silicon Boron at 300 K as functions Phosphorus of the total dopant concentration (N). The values plotted are the results of the curve fitting measurements from several sources. The mobility curves can be generated using the equation below with the parameters shown: µ(n) = µ mi + (µ max-µ min ) {1 + (N/N ref ) } From Muller and Kamins, 3 rd Ed., pg 33 Parameter Arsenic Phosphorous Boron µ min µ max N ref 9.68X10^ X10^ X10^ Page 6

7 TEMPERATURE EFFECTS ON MOBILITY Derived empirically for silicon for T in K between 250 and 500 K and for N (total dopant concentration) up to 1 E20 cm-3 µn (T,N) = 88 Tn Tn [ N / (1.26E17 Tn 2.4 )] ^0.88 Tn µp (T,N) = 54.3 Tn Tn [ N / (2.35E17 Tn 2.4 )]^ 0.88 Tn Where Tn = T/300 From Muller and Kamins, 3 rd Ed., pg 33 Page 7

8 MOBILITY CALCULATIONS This spreadsheet calculates the mobility from the equations given by Kamins, Muller and Chan, shown on the previous two pages. Page 8

9 TEMPERATURE COEFFICIENT OF RESISTANCE R/ T for semiconductor resistors R = Rhos L/W = Rho/t L/W assume W, L, t do not change with T Rho = 1/(qµn + qµp) where µ is the mobility which is a function of temperature, n and p are the carrier concentrations which can be a function of temperature (in lightly doped semiconductors) as T increases, µ decreases, n or p may increase and the result is that R usually increases unless the decrease in µ is cancelled by the increase in n or p Page 9

10 DIFFUSED RESISTOR Aluminum contacts Silicon dioxide n-wafer Diffused p-type region The n-type wafer is always biased positive with respect to the p-type diffused region. This ensures that the pn junction that is formed is in reverse bias, and there is no current leaking to the substrate. Current will flow through the diffused resistor from one contact to the other. The I-V characteristic follows Ohm s Law: I = V/R Sheet Resistance s ~ 1/( qµ Dose) ohms/square Ro = s L/W ohms Page 10

11 Desired resistor network RESISTOR NETWORK 500 W W Layout if Rhos = 100 ohms/square L W Page 11

12 R AND C IN AN INTEGRATED CIRCUIT 741 OpAmp R C Estimate the sheet resistance of the 4000 ohm resistor shown. Page 12

13 DIFFUSION FROM A CONSTANT SOURCE N(x,t) = No erfc (x/2 D p t p ) Solid Solubility Limit, No N(x,t) p-type erfc function Wafer Background Concentration, NBC n-type x Xj into wafer for erfc predeposit Q A (tp) = Q A (tp)/area = 2 No (Dptp) / Dose Where Dp is the diffusion constant at the predeposit temperature and tp is the predeposit time Page 13

14 DIFFUSION FROM A LIMITED SOURCE N(x,t) = Q A (tp) Exp (- x 2 /4Dt) Dt Gaussian function for erfc predeposit Q A (tp) = Q A (tp)/area = 2 No (Dptp) / Dose Where D is the diffusion constant at the drive in temperature and t is the drive in diffusion time, Dp is the diffusion constant at the predeposit temperature and tp is the predeposit time Page 14

15 DIFFUSION CONSTANTS AND SOLID SOLUBILITY DIFFUSION CONSTANTS BORON PHOSPHOROUS BORON PHOSPHOROUS TEMP PRE or DRIVE-IN PRE DRIVE-IN SOLID SOLID SOLUBILITY SOLUBILITY Dp or D Dp D NOB NOP 900 C 1.07E-15 cm2/s 2.09e-14 cm2/s 7.49E-16 cm2/s 4.75E20 cm E20 cm E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E21 Page 15

16 ION IMPLANTED RESISTOR Aluminum contacts Silicon dioxide n-wafer Ion Implanted p-type region Like the diffused resistor but more accurate control over the sheet resistance. The dose is a machine parameter that is set by the user. Sheet Resistance s ~ 1/( qµ Dose) ohms/square Also the dose can be lower than in a diffused resistor resulting in higher sheet resistance than possible with the diffused resistor. Page 16

17 THIN FILM RESISTOR Aluminum contacts Silicon dioxide n-wafer Thin Layer of Poly Silicon Or Metal For polysilicon thin films the Dose = film thickness,t, x Solid Solubility No if doped by diffusion, or Dose ~ 1/2 ion implanter dose setting if implanted The Sheet Resistance Rhos ~ 1/( qµ Dose) + Rgrain boundaries ohms/square For metal the Sheet Resistance is ~ the given (table value) of bulk resistivity, Rho, divided by the film thickness,t. Rhos = Rho / t ohms/square Page 17

18 POLY SILICON Grain boundary take up some of the implanted dose. They also add resistance to the resistor that is less sensitive to temperature and doping concentration. We assume grain size ~ equal to ½ the film thickness (t) and the number of grains equals the path length (L) divided by grain size (t/2). Each grain boundary adds a fixed resistance which is found empirically. (example 0.9 ohms) Page 18

19 CALCULATION OF RESISTANCE Dose/t = Concentration Page 19

20 POLY RESISTOR Use t=0.6, L=390u, W=10u Dose =1e16cm-2, N-type poly, and 0.9 ohms per boundary Poly R = 1/SLOPE = 1/0.681m = 1468 ohm Rhos = 1468/39 =37.6 ohm/sq Page 20

21 RESISTOR I-V CHARACTERISTICS Use t=1.5u, L=500u, W=100u Dose =0.5e15cm -2 p-type single crystal silicon Gives R=683 ohms R= 1/1.44e-3 = 694 ohms Page 21

22 CLOSE UP OF RESISTORS AND THERMOCOUPLE Aluminum N+ Poly Thermocouple Green P+ Diffused Resistor 200 um wide x 180 um long Use t=1.5, L=180, w=200 dose =2e14, p-type crystalline, Rmeas = 207 Red N+ Polysilicon Resistor 60 um x 20 um + 30 to contact so L/W ~ 6 Use t=0.5, L=120, w=20 dose =1e16, n-type poly, and 0.9 ohms per boundary Rmeas = 448 Page 22

23 RESISTOR TEMPERATURE RESPONSE I Cold Hot R = L/(W xj) ohms 1/( qµ n n + qµ p p) V L,W,xj do not change with light, µn and µp does not change with light but can change with temperature, n and p does not change much in heavy doped semiconductors (that is, n and p is determined by doping) Page 23

24 RESISTOR LIGHT RESPONSE No light Full light R = L/(W xj) ohms 1/( qµ n n + qµ p p) L,W,xj do not change with light, µn and µp does not change with light but can change with temperature, n and p does not change much in heavy doped semiconductors (that is, n and p is determined by doping) Page 24

25 HEATERS Final steady state temperature depends on power density in watts/cm2 and + V I R = s L/W the thermal resistance from heater to ambient - P= IV = I 2 R watts Page 25

26 THERMAL CONDUCTIVITY Power input Temp above ambient Rth = 1/C L/Area where C=thermal conductivity L= thickness of layer between heater and ambient Area = cross sectional area of the path to ambient Thermal Resistance, Rth to ambient Temp ambient Page 26

27 THERMAL PROPERTIES OF SOME MATERIALS MP Coefficient Thermal Specific C of Thermal Conductivity Heat Expansion ppm/ C w/cmk cal/gm C Diamond Single Crystal Silicon Poly Silicon Silicon Dioxide Silicon Nitride Aluminum Nickel Chrome Copper Gold Tungsten Titanium Tantalum Air Water watt = cal/sec Page 27

28 HEATER EXAMPLE Example: Poly heater 100x100µm has sheet resistance of 25 ohms/sq and 9 volts is applied. What temperature will it reach if built on 1 µm thick oxide? Power R thermal = V 2 /R = 81/25 = 3.24 watt = 1/C L/Area = (1/0.014 watt/cm C)(1e-4cm/(100e-4cm x100e-4cm)) = 71.4 C/watt Temperature = T ambient + (3.24) (71.4) = T ambient C Page 28

29 TEMPERATURE SENSOR EXAMPLE Example: A poly heater is used to heat a sample. The temperature is measured with a diffused silicon resistor. For the dimensions given what will the resistance be at 90 C and 65 C xj=t=1µm Nd= n =1E18cm -3 (n-type) R(T,N) = 1 q µ n (T,N) n L Wt 50µm 50,000 10µm q µ n (T=273+90=363,N=1e18) n = 1.6e-19 (165) 1e18 = 26.4 q µ n (T=273+65=338,N=1e18) n = 1.6e-19 (196) 1e18 = 31.4 R=1894 ohms R=1592 ohms Page 29

30 TEMPERATURE SENSOR EXAMPLE Dose/t = Concentration Page 30

31 DIODES AND HEATERS Poly Heater on top of Diodes VDD I ~ (VDD-0.6)/R R Vo ~ mV/ C I T2 T1 Integrated n-well series resistor. V T2>T1 Page 31

32 UNIFORMLY DOPED PN JUNCTION P+ - Phosphrous donor atom and electron Space Charge Layer Ionized Immobile Phosphrous donor atom p = N A n = N P+ D B- Ionized Immobile Boron acceptor atom B- + B- + B- + B- P+ P+ P+ - P+ - P+ - Boron acceptor atom and hole p-type B- B- P+ P+ P+ - P+ - qn A W 1 =qn D W 2 charge density, +qn D -W 1 W2 n-type +V R x -qn A Electric Field, Potential, +V R Page 32

33 UNIFORMLY DOPED pn JUNCTION From Physical Fundamentals: Potential Barrier - Carrier Concentration: = KT/q ln (N A N D /ni 2 ) From Electric and Magnetic Fields : Gauss s Law, Maxwells 1st eqn: = Relationship between electric flux D and electric field : D = Poisson s Equation: 2 = - / D Definition of Electric Field: = - V Page 33

34 EXAMPLE CALCULATIONS Width of space charge layer depends on the doping on both sides and the applied reverse bias voltage and temperature. Page 34

35 V RB = reverse breakdown voltage CURRENTS IN PN JUNCTIONS Is Id Forward Bias V D Reverse Bias Id p n + V D - Vbi = turn on voltage ~ 0.7 volts for Si Id = Is [EXP (q VD/KT) -1] Ideal diode equation Page 35

36 INTEGRATED DIODES p-wafer n+ p+ n-well p+ means heavily doped p-type n+ means heavily doped n-type n-well is an n-region at slightly higher doping than the p-wafer Note: there are actually two pn junctions, the well-wafer pn junction should always be reverse biased Page 36

37 REAL DIODES Series Resistance =1/4.82m=207 Junction Capacitance ~ 2 pf Is = 3.02E-9 amps BV = > 100 volts Size 80µ x 160µ P N Page 37

38 DIODE SPICE MODEL MEMS Diode Model Parameter Default Value Extracted Value Is reverse saturation current 1e-14 A 3.02E-9 A N emission coefficient 1 1 RS series resistance ohms VJ built-in voltage 1 V 0.6 CJ0 zero bias junction capacitance 0 2pF M grading coefficient BV Breakdown voltage infinite 400 IBV Reverse current at breakdown 1E-10 A - DXXX N(anode) N(cathode) Modelname.model Modelname D Is=1e-14 Cjo=.1pF Rs=.1.model RITMEMS D IS=3.02E-9 N=1 RS=207 +VJ=0.6 CJ0=2e-12 M-0.5 BV=400 Page 38

39 DIODE TEMPERATURE DEPENDENCE Id = I S [EXP (q VD/KT) -1] Neglect the 1 in forward bias, Solve for VD VD = (KT/q) ln (Id/I S ) = (KT/q) (ln(id) ln(is)) Take dvd/dt: note Id is not a function of T but Is is eq 1 dvd/dt = (KT/q) (dln(id)/dt dln(is)/dt) + K/q (ln(id) ln(is)) zero VD/T from eq 1 Rewritten dvd/dt = VD/T - (KT/q) ((1/Is) dis/dt ) Now evaluate the second term, recall Is = qa (Dp/(LpNd) +Dn/(LnNa))ni 2 eq 2 Note: Dn and Dp are proportional to 1/T Page 39

40 DIODE TEMPERATURE DEPENDENCE and ni 2 (T) = A T 3 e - qeg/kt This gives the temperature dependence of Is Is = C T 2 e - qeg/kt eq 3 Now take the natural log ln Is = ln (C T 2 e - qeg/kt) Take derivative with respect to T (1/Is) d (Is)/dT = d [ln (C T 2 e -qeg/kt) ]/dt = (1/Is) d (CT 2 e -qeg/kt )dt = (1/Is) [CT 2 e -qeg/kt (qeg/kt 2 ) + (Ce -qeg/kt )2T] = (1/Is) [Is(qEg/KT 2 ) + (2Is/T)] Back to eq 2 dvd/dt = VD/T - (KT/q) [(qeg/kt2 ) + (2/T)]) dvd/dt = VD/T - Eg/T - 2K/q) Page 40

41 EXAMPLE: DIODE TEMPERATURE DEPENDENCE dvd/dt = VD/T - Eg/T - 2K/q Silicon with Eg ~ 1.2 ev, VD = 0.6 volts, T=300 K dvd/dt =.6/ /300) - (2(1.38E-23)/1.6E-19 = -2.2 mv/ I T2 T1 V T1<T2 Page 41

42 DIODE AS A TEMPERATURE SENSOR P+ N+ Poly Heater Buried pn Diode, N+ Poly to Aluminum Thermocouple Compare with theoretical -2.2mV/ C Page 42

43 SPICE FOR DIODE TEMPERATURE SENSOR Page 43

44 space charge layer PHOTODIODE B - + B- + B- p-type B - I P+ P+ B - B - B - P+ B - B - - P+ - + P+ P+ + P+ electron and hole pair - P+ Phosphrous donor atom and electron P+ Ionized Immobile Phosphrous donor atom B - Ionized Immobile Boron acceptor atom + B- - P+ - P+ - P+ n-type - P+ Boron acceptor atom and hole Page 44

45 CHARGE GENERATION IN SEMICONDUCTORS E = h = hc / What wavelengths will not generate e-h pairs in silicon. Thus silicon is transparent or light of this wavelength or longer is not adsorbed? From: Micromachined Transducers, Gregory T.A. Kovacs Material Bandgap 300 K max (um) GaN ZnO SiC CdS GaP CdSe GaAs InP Si Ge PbS PbTe InSb Page 45

46 CHARGE GENERATION vs WAVELENGTH E = h = hc / h = e-34 j/s = (6.625 e-34/1.6e-19) ev/s E = 1.55 ev (red) + E = 2.50 ev (green) B - B- E = 4.14 ev (blue) P+ B - B - n-type P+ - P B- p-type B - P+ B - B P P+ P+ + P+ - P+ - P+ B - - P+ I Page 46

47 Adsorption Coefficient, (1/cm) ADSORPTION VERSUS DISTANCE V I 1.00E+06 Adsorption Coefficient vs Wavelength For Silicon n p I 1.00E E E E E E E-01 + V - I No Light More Light Most Light V 1.00E E E E E Wavelength (nm) f(x) = f(0) exp - x Find % adsorbed for Green light at x=5 µm and Red light at 5 µm Page 47

48 PN JUNCTION DESIGN FOR PHOTO DIODE 100% % 0µm 1µm 2µm 3µm 4µm Space Charge Layer Page 48

49 SINGLE AND DUAL PHOTO CELL Isc = ua Pmax=0.32V 0.909uA =0.29uW Isc = ua Page 49

50 CAPACITORS Capacitor - a two terminal device whose current is proportional to the time rate of change of the applied voltage; I = C dv/dt a capacitor C is constructed of any two conductors separated by an insulator. The capacitance of such a structure is: C I + V - C = o r Area/d d where o is the permitivitty of free space Area r is the relative permitivitty Area is the overlap area of the two conductor separated by distance d o = 8.85E-14 F/cm r air = 1 r SiO 2 = 3.9 Page 50

51 OTHER CAPACITOR CONFIGURATIONS Two Dielectric Materials between Parallel Plates C1 C2 C Total = C1C2/(C1+C2) Example: A condenser microphone is made from a polysilicon plate 100 µm square with 1000Å silicon nitride on it and a second plate of aluminum with a 1µm air gap. Calculate C and C if the aluminum plate moves 0.1 µm. Page 51

52 DIELECTRIC CONSTANT OF SELECTED MATERIALS Vacuum 1 Air Acetone 20 Barium strontium titanate 500 Benzene Conjugated Polymers 6 to 100,000 Ethanol 24.3 Glycerin 42.5 Glass 5-10 Methanol 30 Photoresist 3 Plexiglass 3.4 Polyimide 2.8 Rubber 3 Silicon 11.7 Silicon dioxide 3.9 Silicon Nitride 7.5 Teflon 2.1 Water Page 52

53 CALCULATIONS Page 53

54 OTHER CAPACITOR CONFIGURATIONS Interdigitated Fingers with Thickness > Space between Fingers C = (N-1) o r L h /s h = height of fingers s = space between fingers N = number of fingers L = length of finger overlap Example: Page 54

55 OTHER CAPACITOR CONFIGURATIONS Interdigitated Fingers with Thickness << Space between Fingers C = LN 4 o r 0 0 n=1 1 2n-1 Jo2 (2n-1) s 2(s+w) Jo = zero order Bessel function w = width of fingers s = space between fingers N = number of fingers L = length of finger overlap Reference: Lvovich, Liu and Smiechowski, Page 55

56 OTHER CAPACITOR CONFIGURATIONS Two Long Parallel Wires Surrounded by Dielectric Material Capacitance per unit length C/L C/L = 12.1 r / (log [(h/r) + ((h/r) 2-1) 1/2 ] h = half center to center space r = conductor radius (same units as h) Reference: Kraus and Carver Example: Calculate the capacitance of a meter long connection of parallel wires. Solution: let, h = 1 mm, r = 0.5mm, plastic er = 3 the equation above gives C/L = 63.5 pf/m C = 63.5 pf Page 56

57 OTHER CAPACITOR CONFIGURATIONS Coaxial Cable Capacitance per unit length C/L C/L = 2 r / ln(b/a) b = inside radius of outside conductor a = radius of inside conductor Reference: Kraus and Carver Example: Calculate the capacitance of a meter long coaxial cable. Solution: let b = 5 mm, a = 0.2mm, plastic er = 3 the equation above gives C/L = 51.8 pf/m C = 51.8 pf Page 57

58 CAPACITORS AS SENSORS d C C1 C2 C d One plate moves relative to other changing gap (d) Center plate moves relative to the two fixed plates d One plate moves relative to other changing overlap area (A) C1 C2 Page 58

59 CAPACITORS AS SENSORS Change in Space Between Plates (d) Change in Area (A) microphone Change in Dielectric Constant (er) gyroscope position sensor Page 59

60 Two conductors separated by a material that changes its dielectric constant as it selectively absorbs one or more chemicals. Some humidity sensors are made using a polymer layer as a dielectric material. CHEMICAL SENSOR Change in Dielectric Constant (er) chemical sensor Page 60

61 REFERENCES 1. Device Electronics for Integrated Circuits, Richard S. Muller, Theodore I. Kamins, John Wiley & Sons., 3 rd edition, Micromachined Transducers, Gregory T.A. Kovacs, McGraw- Hill, Microsystem Design, Stephen D. Senturia, Kluwer Academic Press, Optimization and fabrication of planar interdigitated impedance sensors for highly resistive non-aqueous industrial fluids, Lvovich, liu and Smiechowski, Sensors and Actuators B:Chemical, Volume 119, Issue 2, 7 Dec. 2006, pgs Solar Cells, Martin A. Green, Prentice-Hall Page 61