Part 1: Semiconductor Physics
1. If the lattice constant of silicon is 5.43Å, calculate (a) the distance from the center of one silicon atom to the center of its nearest neighbor, (b) the number density of silicon atoms (#/cm3), and (c) the mass density (grams/cm3) of silicon.
2. Assume that each atom is a hard sphere with the surface of each atom in contact with the surface of its nearest neighbor. Determine the percentage of total unit cell volume that is occupied in (a) a simple cubic lattice, (b) a face-centered cubic lattice, (c) a body-centered cubic lattice, and (d) a diamond lattice.
3. A one-dimensional infinite potential well with a width of 12 Å contains an electron. (a)Calculate the first two energy levels that the electron may occupy. (b)If an electron drops from the second energy level to the first, what is the wavelength of a photon that might be emitted?
4. Consider a three-dimensional infinite potential well. The potential function is given by for , , , and elsewhere. Start with Schrödinger’s wave equation, use the separation of variables technique, and show that the energy is quantized and is given by
Where = 1,2,3…, = 1,2,3…., = 1,2,3…
(a) Estimate the tunneling probability of particle with an effective mass of (an electron in gallium arsenide), where is the mass of an electron, tunneling through a rectangular potential barrier of height and width 15Å. The particle kinetic energy is . (b) Repeat part (a) if the effective mass of the particle is (an electron in silicon).
5. Show that the probability of an energy state being occupied by above the Fermi energy is the same as the probability of a state being empty below the Fermi level.
6. Assume the Fermi energy level is exactly in the center of the bandgap energy of a semiconductor at . (a) Calculate the probability that energy state at is occupied by an electron for Si, Ge, and GaAs. (b) Calculate the probability that an energy state at is empty for Si, Ge, and GaAs.
7. As show in the following figure, what type of this semiconductor? (a) Relative Ec and Ev, write the equations of n0 and p0. (b) Relative EFi, write the equations of n0 and p0. Write the equations of (c) the mass action law, (d) the charge neutrality condition, and (e) the position of intrinsic Fermi-energy (EFi).
(given )
8. If the density of states function in the conduction band of a particular semiconductor is a constant equals to K, derive the expression for the thermal-equilibrium concentration of electrons in the conduction band, assuming Fermi-Dirac statistics and assuming the Boltzmann approximation is valid.
9. (a) Consider silicon at T = 300K. Determine if . (b)Assuming that from part (a) remains constant, determine the value of when . (c)Find the value of in both parts (a) and (b).
10. For the Boltzmann approximation to be valid for a semiconductor, the Fermi level must be at least below the donor level in an n-type material and at least above the acceptor level in a p-type material. If , determine the maximum electron concentration in a n-type semiconductor and maximum hole concentration in p-type semiconductor for the Boltzmann approximation to be valid in (a) silicon and (b) gallium arsenide.
11. A sample of silicon at is doped with boron at a concentration of and with arsenic at a concentration of . (a) Is the material n or p type? (b) Determine the electron and hole concentrations. (c) Calculate the total ionized impurity concentration.
12. As show in the following figure, explain why the intrinsic Fermi-energy is located closely to the midgap. If the electron effective mass is higher than that of hole, the intrinsic Fermi-energy will shift to above or below the midgap, explain your reason.
13. Determine the carrier density gradient to produce a given diffusion current density. The hole concentration in silicon at varies linearly from to . The hole diffusion current density is , and the hole concentration at is . Determine the hole concentration at .
14. A silicon semiconductor at is homogenously doped with and . (a) Determine the thermal equilibrium concentration of free electrons and free holes. (b) Calculate the drift current density for an applied ε-field of 30 V/cm. (c) Repeat parts (a) and (b) for and .
15. A silicon crystal having a cross-sectional area of and a length of is connected at its ends to a 10-volt battery. At , we want a current of 100mA in the silicon. Calculate: (a) the required resistance , (b) the required conductivity, (c)the density of donor atoms to be added to achieve this conductivity, and (d) the concentration of acceptor atoms to be added to form a compensated p-type material with the conductivity given from part(b) if the initial concentration of donor atom is .
16. Consider a sample of p-type silicon at T = 300 K. Assume the hole concentration varies linearly dropped from x = 0 to x = 50 mm. The hole concentration at x = 0 is p(0) = 1015 cm-3. The diffusion current density is found to be Jp = 0.25 A/cm2. If the hole diffusion coefficient is Dp = 10 cm2/s, find the hole concentration at x = 50 mm.
17. As show in the following figure, for a semiconductor in thermal equilibrium with a nonuniform donor impurity concentration. Please prove the induced electric field .
18. If a p-n junction has the charge density as shown in fig. (a), please explain (a) the reason that the built-in electric field (e) is as shown in fig. (b), and derive the peak value of electric field, (c) prove , and (d) prove .
Fig. (a) Fig. (b)
19. A silicon Hall device at . A Hall Effect device is fabricated with the following geometry: , and . The following parameters are measured: , , and tesla. Determine (a) the conductivity type (b) the majority carrier concentration, and (c) the majority carrier mobility.
20. As show in the following figure, a p-type semiconductor is contact with an n-type semiconductor to form a p-n junction, please draw the space charge region and explain (a) what is the meaning of space charge, and (b) where is the direction of internal built-in electric field. (c) Where is the maximum electric field?
Part 2: PN junction
Sze, Chap 4, problem 1,3, 4,5, 7, 17
example 5
Neamen, Chap 5, problem 2,4, 5, 14, 18,21, 37,40
Chap 9, problem 1,8, 9, 17, 28,31,49, 52
Part 3: MOS capacitor and MOSFET
[1]、An Introduction to Semiconductor Devices, Donald Neamen
1. Chap 6, review questions 2.
A. Sketch the energy band diagrams in an MOS capacitor with an n-type substrate in accumulation, depletion, and inversion modes.
B. Describe what is meant by an inversion layer of charge. Describe how an inversion layer of change can be formed in an MOS capacitor with a p-type substrate.
C. Why does the space charge region in the semiconductor of an MOS capacitor essentially reach a maximum width once the inversion layer is formed?
2. Chap 6, review questions 7.
A. What is a channel stop in an NMOS transistor?
B. What is meant by self-aligned source and drain contacts?
C. Sketch the p-well configuration in the CMOS structure.
3. Chap 6, problems 6.37
The experimental characteristics of an ideal n-channel MOSFET biased in the saturation region are show in Figure P6.37. IF and Å, determine and .
4. Chap 6, problems 6.46
Consider an ideal n-channel MOSFET with a width-to-length ratio of , an electron mobility of , an oxide thickness of Å, and a threshold voltage of . (a) Determine the maximum value of source resistance so that the saturation transconductance is reduced by no more than 20 percent from its ideal value when VGS = 5V. (b) Using the value of calculated in part (a), how much is reduced from its ideal when VGS = 3V ?
5. Chap 7, review questions 3
A. Discuss the effect of charge sharing on the threshold voltage as the channel length decreases.
B. Discuss the effect of charge sharing on the threshold voltage as the channel width decreases.
6. Chap 7, problems 7.14
Consider an n-channel MOSFET with and Å, If and , determine the threshold shift due to the short channel effect.
[2]、Semiconductor Devices physics and technology, S.M. Sze
1. Chap 6, problems 14
Derive the I-V characteristics of a MOSFET with the drain and gate connected together and the source and substrate grounded. Can one obtain the threshold voltage from these characteristics?
2. Chap 6, problems 17
For the devices stated in Prob. 16, find the transconductance.
Consider a submicron MOSFET with , , , , , and VT = 0.5 V.
Find the channel conductance for VG=1V and VD=0.1V
3. Chap 6, problems 23
A field transistor with a structure similar to Fig.21 in the text has , , and an polysilicon local interconnect as the gate electrode. If the requirement for sufficient isolation between device and well is VT > 20 V, calculate the minimum field oxide thickness.
4. Chap 6, problems 30
For an n-channel SOI device with -polysilicon gate having , , and , calculate the threshold voltage. Assume that , , and are all zero.
MOSC
1. (a) Calculate the maximum space charge width and maximum space charge density in p-type Silicon, gallium arsenide, and germanium semiconductors of an MOS structure. Let T=300K and assume .
(b) Repeat part (a) if T=200K
2. (a) Consider n-type silicon in an MOS structure. Let T=300K. Determine the semiconductor doping so that .
(b) Determine the surface potential that results in the maximum space charge
width.
3. A 400Å oxide is grown on p-type silicon substrate with . The flat-band voltage is -0.9V. Calculate the surface potential at the threshold inversion point as well as the threshold voltage assuming negligible oxide charge. Also find the maximum space charge width for the device.
4. Consider an aluminum gated NMOS capacitor with substrate doping concentration of . Please plot VT versus over the range .
5. Consider an aluminum gated NMOS capacitor with substrate doping concentration of . Assume the . Please plot VT versus temperature over the range .
6. Consider a MOS capacitor structure on n-type Si substrate with by using Aluminum gate with work function of 4.1eV. (a) Please plot the band diagram at accumulation, inversion and depletion. (b) Assuming the . Please calculate the flat band voltage and the threshold voltage.
7. Consider an aluminum gate-silicon dioxide-p-type silicon MOS structure with . the silicon doping is and the flat band voltage is VFB=-1.0V. Determine the fixed oxide charge .
8. An ideal MOS capacitor with an aluminum gate has a silicon dioxide thickness of on a p-type silicon substrate doped with an acceptor concentration of . Determine the capacitances , , and at (a) = 1 Hz and (b) = 1MHz. (c) Determine the and .
9. Determine the metal-semiconductor work function difference in an MOS structure with p-type silicon for the case when the gate is (a) aluminum, (b) n+ polysilicon, and (c) p+ polysilicon. Let . (6.4)
10. An ideal MOS capacitor is fabricated by using intrinsic silicon and n+ polysilicon gate. (a) Sketch the energy band diagram through the MOS structure under flat-band conditions. (b) Sketch the low-frequency C-V characteristics from negative to positive gate voltage. (6.24)
Part 4: Bipolar Transistor and Related Devices
[1]、An Introduction to Semiconductor Devices, Donald Neamen
1. Find β for a bipolar junction transistor with a nondegenerate emitter. Assume that emitter, base, and collector are noncompensated and that
NE=2x1018cm-3,
NB=1016cm-3
NC=1015 cm-3
WE=0.2um,
WB=0.1um
2. Fromthe equation, α=γαTM,, Where M is the carrier multiplication factor in the base-collector junction. For small base-collector voltage, M=1 and α=γαT and β=α/(1-α). Show that for avalanche breakdown, M=1+1/β.
3. For a uniformly doped bipolar transistor in thermal equilibrium, (a) sketch the energy-band diagram, (b) sketch the electric field through the device, and (c) repeat parts (a) and (b) for the transistor biased in the forward-active region.
8. A uniformly doped silicon npn transistor is to be biased in the forward-active region with the B-C junction reverse biased by 3V. The metallurgical base width is . The transistor dopings are , , and . (a) For , calculate the B-E voltage at which the minority-carrier electron concentration at is 10 percent of the majority-carrier hole concentration. (b) At this bias, determine the minority-carrier hole concentration at . (c) Determine the neutral base width for this bias.
13. Derive the expression for the excess minority-carrier hole concentration in the bias region of a uniformly doped pnp bipolar transistor operating in the forward-active region.
18. A uniformly doped silicon pnp bipolar transistor at with doping of , , and is based in the inverse-active region. What is the maximum B-C voltage so that the low-injection condition applies?
21. A silicon npn transistor at has an area of , neutral base width of , and doping concentrations of , , and . Other semiconductor parameters are , , and . Assuming the transistor is biased in the active region and the recombination factor is unity, calculate the collector current for (a) , (b) , and (c) .