# Problem 9.20 Threshold bias for an n-channel MOSFET: In the text we used a criterion that the inversion of the MOSFET channel occurs when V s = ;2 F w

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1 Prof. Jasprit Singh Fall 2001 EECS 320 Homework 11 The nals for this course are set for Friday December 14, 6:30 8:30 pm and Friday Dec. 21, 10:30 am 12:30 pm. Please choose one of these times and inform me of your choice if you haven't done so. This homework is not due back to me. However, please do the homework and check against the solutions that are also given. On Dec. 11 we will use the class time to review the various topics covered in this course. Problem 1: Problem 9.18 of the text. Problem 2: Problem 9.20 of the text. Problem 3: Problem 10.8 of the text. SOLUTIONS Problem 9.18 ANMOSwithV T of 1.5 V is operated at V GS = 5 V and I DS =100A. Determine if the device is in linear or saturation regime. k = ZC ox L =20A=V 2 Solution Let us calculate the saturaton current for the biasing conditions I D (sat) = k 2 (V GS ; V T ) 2 = 10(3:5) 2 =122:5 A The current given (100 A) is smaller than the saturation current so the device is in the linear regime. Using the linear expression I DS = k (V GS ; V T ) V DS we get V DS =1:43 V 1

2 Problem 9.20 Threshold bias for an n-channel MOSFET: In the text we used a criterion that the inversion of the MOSFET channel occurs when V s = ;2 F where e F = (E F ; E Fi ). Consider another criterion in which we say that inversion occurs when the electron density at the Si/SiO 2 interface becomes cm ;3. Calculate the gate threshold voltage needed for an MOS device with the following parameters for the two dierent criteria: d ox = 500 A ms = 1:0 V N a = cm ;3 Solution For the transistor under consideration the value of F is given by F = k BT ; `n N a e n i = ;(0:026) `n = ;0:17 volt 1: The surface band bending required for inversion according to the 1 st criterion is V s = ;2 F =0:34 volt In the second criteria, the surface bending should be such that the surface electron concentration is cm ;3. The bulk electron concentration is n o = n2 i p o = 2: cm ; cm ;3 =2: cm ;3 The surface bending needed to generate a surface density of10 16 cm ;3 is V s (2) = k B T `n n(surface) =0:026 `n n(bulk) 2: = 0:52 volt Thus, according to the second criteria, the surface bending is considerably greater. The gate voltage needed for inversion is According to the 1 st criterion V GS = V fb + V s + sf s C ox V GS (1) = V fb + V s (1) + (4e sn a j F j) 1=2 C ox 2

3 According to the 2 nd criterion, V GS (2) = V fb + V s (2) + (4e sn a V s (2)) 1=2 C ox C ox = ox = 3:9 8:84 10;14 F=cm =6:9 d ox ( ;8 10 ;8 F=cm 2 cm) The 1 st criterion gives V GS (1) = 1:0+0:34 + (4 1:6 10;19 11:9 8:84 10 ;14 0:17) 1=2 (6:9 10 ;8 ) = 1:355 volt V GS (2) = 1:54 volt Problem 10.8 Consider an n-type 1.0 m FET made from GaAs and Si. The device cuto frequency is limited by transit time eects. Using the velocity- eld relations for electrons in GaAs and Si, estimate the cuto frequency for the devices when (a) V D =0:5 V, i.e., for low-power applications and (b) V D =10V, i.e., for high-power applications. Assume that the average eld in the channel is V D =L. Solution i) The average eld in the channel is F = V D L = 0:5 V 1:0 10 ;4 cm =5 103 V=cm The velocity in GaAs and Si is from Appendix B v(gaas) = 1: cm=s The transit time for the GaAs device is The cuto frequency is The transit time for the Si device is v(si) = cm=s t tr = 1:0 10;4 cm 1: cm=s =6:67 10;12 s f T = 1 2 tr =23:86 GHz t tr = 1:0 10;4 cm =1: ;11 cm=s f T = 11:1 GHz 3

4 ii) The average eld is F = 10 V 1:0 10 ;4 cm =105 V=cm In this case, the electron velocity in GaAs and Si is about 10 7 cm/s. The cuto frequency is about 16 GHz in both devices. The example shows the benets of GaAs and Si over low power applications. 4

5 ADVANCED COURSES YOU CAN TAKE IF YOU ARE IN- TERESTED IN SEMICONDUCTOR DEVICES AND PHYSICS EECS 421: This course covers more details of semiconductor devices. It is an advanced version of EECS 320. EECS 423 and 425: These are good courses on semiconductor fabrication issues. EECS 420: This course would be useful for students who want to have a better understanding of quantum mechanics and its applications in technology. 5

6 A TRIAL FINAL EXAM WITH SOLUTIONS Problem 1:(10 points) ASip ; n diode is reverse biased at 5.0 V. Electrons from the p-side enter the depletion region and are swept away by the depletion region eld to the n-side. Calculate and plot the velocity versus distance as the electrons go from the p-side depletion region edge to the n-side. N d = N a =10 16 cm ;3 The electron velocity-eld relation is v = F 1+ F v s with = 1000 cm 2 =V:s and v s =10 7 cm=s. In this problem we have to use the discussion in Section 5.2 to obtain the eld prole in the depletion region of a diode. The eld increases linearly from the p-side depletion region, ;W p, to the p-n junction point and then decreases towards zero at the n-side depletion region edge, W n. On the p-side the depletion region eld is F (x) =; en ax ; en aw p We can see that the built in voltage of the diode is 0.7 V. The depletion region width on the p-side and n-side can be calculated to be (total voltage is 5:7 V) The maximum eld at x =0is W p =6:1 10 ;5 cm = W n F m = ; en aw p We see that the velocity values are: =9: V=cm x = ;W p = ;0:61 m : F =0 v =0 x = ;0:31 m : F =4: V=cm v =710 6 cm=s x = 0: F =9: V=cm v =9: cm=s etc: The velocity distance curve is symmetric around x = 0. 6

7 Problem 2:(10 points)inasinpnbjtacurrentgain,, of 100 is required. Also the transit time of electrons across the base is to be 10 ps. N de = cm ;3 N dc = 1: cm ;3 D b = 20:0 cm 2 =s L b = 15:0 m D e = 10:0 cm 2 =s L e = 5:0 m Emitter dimensions = 100m 100m (i) Calculate the base width needed for the device. (ii) Calculate the base doping needed for this device. (iii) In 3 4 sentences describe the problems with this device and how these problems can be solved by using a heterojunction bipolar transistor. You may make the following approximations : The device is in the forward active mode and the reverse bias collector current is zero. W b is much smaller than L b. To nd the base width needed we use the relation for the transit time across the base (remember there is no eld across the base and the charge ows by diusion, not drift: see Example 3.17 or Eqn. 7.99) This gives for the base width needed t tr = W 2 b 2D b W b = p 2(20cm 2 =s)(20 10 ;12 s)=0:2 m Using the desired value of = 100 we now nd that the base doping is N ab = p b0 =4: cm ;3 In this device the base is quite thin and the device will therefore suer from Early voltage problems and punch through problems. In the HBT the base can be doped very heavily and the gain can still be large. Thus we canhave ahigh speed and high gain device. Problems 3:(10 points) An n-mesfet is made from GaAs and the maximum acceptable gate current density is1.0a=cm 2. Apower supply is available that can provide a V GS value of 5:0 V. 7

8 The device parameters are: e b = 0:8 V h = 0:15 m N d = cm ;3 L = 2:0 m Z = 20:0 m n = 6000cm 2 =V:s = 13 0 (i) What is the range of V GS that is allowable for the device. (ii) Calculate the maximum transconductance, g m, of the device in the saturation region. We use the Schottky diode equation to nd the prefactor, J s of the current that would ow in the gate-semiconductor junction. J s = R T 2 exp Using R = 8 Acm ;2 K ;2 for GaAs we get ; e b k B T J s =3:12 10 ;8 A=cm 2 The maximum forward bias that we can have before the gate-semiconductor current density reaches 1 A/cm 2 is then given by 10 ;6 A=cm 2 =3:12 10 ;8 A=cm 2 exp(ev=k B T ) This gives V =0:09 V. This is the upper limit on the gate bias V GS. For the device we can see that V bi = 0:78 V and V p = 3:13 V. The lower limit on the gate bias is then V GS = ;(3:13 ; 0:78) = ;2:4 V. The transconductance in saturation is given by This gives (using V GS =0:09 V) g m (sat) =g 0 1 ; Vbi ; V GS V p 1=2! g m (sat) =1:5 10 ;2 S Problem 4: (10 points) Consider an n-mosfet at room temperature. The device is to be used as a load resistor in a circuit and is biased so that V GS = V DS. The resistance of the MOSFET (i.e V DS =I D ) is to be 100 k when V DS =3:0 V. 8

9 (i) Calculate the width of the gate, Z, needed to accomplish this. (ii) Note that the MOSFET resistance changes with V GS or V DS value. Calculate the MOSFET resistance at V DS =3:5 V. The other parameters for the device are the following: n = 600 cm 2 V ;1 s ;1 Gate Length = 2:0 m d ox = 500 A V T = 1:0 V In this problem we can see that since we always have V DS > (V GS ; V T ), the device is in saturation. At abiasofv DS =3:0 V the current is We then have This gives (C ox =6:89 10 ;4 F/m 2 ) I D = 3:0V 10 5 =3 10;5 A 3 10 ;5 A = ZC ox 2L (V GS ; V T ) 2 Z =0:72 m Using the value of Z and noting that the device is in saturation we ndthat at a bias of V DS =3:5 V, The resistance is then 75.2 k. I D =4:65 10 ;5 A 9

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