Download 1-7 If the lattice constant of silicon is 5

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 V( x )  0
for 0  x  a , 0  y  a , 0  z  a , and V( x )  
elsewhere. Start with Schrödinger’s wave equation, use the separation of variables
technique, and show that the energy is quantized and is given by
E nx n y nz
 2 2 2

(n x  n y2  n z2 )
2
2ma
Where n x = 1,2,3…, n y = 1,2,3…., n z = 1,2,3…
(a) Estimate the tunneling probability of particle with an effective mass of 0.067 mo
(an electron in gallium arsenide), where mo is the mass of an electron, tunneling
through a rectangular potential barrier of height V0  0.8eV and width 15Å. The
particle kinetic energy is 0.2eV . (b) Repeat part (a) if the effective mass of the
particle is 1.08mo (an electron in silicon).
5. Show that the probability of an energy state being occupied by E above the
Fermi energy is the same as the probability of a state being empty E below the
Fermi level.
6. Assume the Fermi energy level is exactly in the center of the bandgap energy of a
semiconductor at T  300K . (a) Calculate the probability that energy state at
E  Ec  kT 2 is occupied by an electron for Si, Ge, and GaAs. (b) Calculate the
probability that an energy state at E  Ev  kT 2 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).
2m p kT 3 / 2
2mn*kT 3 / 2
) , N v  2(
) )
(given N c  2(
2
h
h2
*
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 p o if E Fi  E F  0.325eV .
(b)Assuming that p o from part (a) remains constant, determine the value of
E Fi  E F when T  400K . (c)Find the value of n o in both parts (a) and (b).
10. For the Boltzmann approximation to be valid for a semiconductor, the Fermi level
must be at least 3kT below the donor level in an n-type material and at least
3kT above the acceptor level in a p-type material. If T  300K , 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 T  450K is doped with boron at a concentration of
1.5  1015 cm 3 and with arsenic at a concentration of 8  1014 cm 3 . (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 T  300K varies linearly from x  0 to
x  0.01cm . The hole diffusion current density is J dif  20 A cm 2 , and the hole
concentration at x  0 is p  4  1017 cm 3 . Determine the hole concentration at
x  0.01cm .
14. A silicon semiconductor at
T  300K
is homogenously doped with
N d  5  1015 cm 3 and N a  0 . (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 N d  0 and N a  5  1016 cm 3 .
15. A silicon crystal having a cross-sectional area of 0.001cm 2 and a length of
10 -3 cm is connected at its ends to a 10-volt battery. At T  300K , we want a current
of 100mA in the silicon. Calculate: (a) the required resistance R , (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 N d  1015 cm 3 .
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 m. 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 m.
17. As show in the following figure, for a semiconductor in thermal equilibrium with
a nonuniform donor impurity concentration N d (x) . Please prove the induced
 kT  1 dN d ( x)
electric field  ( x)   
.
 e  N d ( x) dx
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 () is as shown in fig. (b), and derive the peak
e
( N d xn2  N a x 2p ) , and (d) prove
value of electric field, (c) prove Vbi 
2 s
 2  V N  Nd 
W   s bi ( a
)
Na Nd 
 e
Fig.
1/ 2
(a)
.
Fig.
(b)
19. A silicon Hall device at T  300K . A Hall Effect device is fabricated with the
following geometry: d  10 3 cm , W  10 2 cm and L  10 1 cm . The following
parameters are measured:
BZ  1000
I x  0.75mA , V x  15V , VH  5.8mV
and
gauss  10 -1 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 (W L)  10 and
t ox  425 Å, determine VT and  n .
4. Chap 6, problems 6.46
Consider an ideal n-channel MOSFET with a width-to-length ratio of
2
(W L)  10 , an electron mobility of  n  400 cm V - s , an oxide thickness
of  t ox  475 Å, and a threshold voltage of VT  0.65V . (a) Determine
the maximum value of source resistance so that the saturation
transconductance g ms is reduced by no more than 20 percent from its
ideal value when VGS = 5V. (b) Using the value of rs calculated in part
(a), how much is g ms 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
16
3
Consider an n-channel MOSFET with N a  10 cm and t ox  450 Å, If
rj  0.3m
and L  1m , 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
N a  1017 cm 3
submicron
MOSFET
with
L  0.25m ,
Z  5m ,
2
7
2
,  n  500 cm V - s , C O  3.45  10 F cm , 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
N a  1017 cm 3 , Q f q  1011 cm 2 , and an n  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 n  -polysilicon gate having
N a  5  1017 cm 3 , d  4nm , and d Si  30nm , calculate the threshold voltage.
Assume that Q f , Qot , and Qm are all zero.
MOSC
1. (a) Calculate the maximum space charge width x dT and maximum
'
(max) in p-type Silicon, gallium arsenide,
space charge density QSD
and germanium semiconductors of an MOS structure. Let T=300K and
assume N a  1016 cm 3 .
(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
'
QSD
(max)  7.5  10 9 C cm 2 .
(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
N a  5  1015 cm 3 . 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 N d  5  1015 cm 3 . Please plot VT versus t ox over the
range 20Å  t ox  500Å .
5. Consider an aluminum gated NMOS capacitor with substrate doping
concentration of N d  5  1015 cm 3 . Assume the t ox  500Å . Please plot
VT versus temperature over the range 100K  t ox  450K .
6. Consider a MOS capacitor structure on n-type Si substrate with
t ox  100Å by using Aluminum gate with work function of 4.1eV. (a)
Please plot the band diagram at accumulation, inversion and depletion.
(b) Assuming the N d  1016 cm 3 . Please calculate the flat band voltage
and the threshold voltage.
7. Consider an aluminum gate-silicon dioxide-p-type silicon MOS
structure with t ox  450Å . the silicon doping is N a  2  1016 cm 3 and
the flat band voltage is VFB=-1.0V. Determine the fixed oxide charge
'
Q SS
.
8. An ideal MOS capacitor with an aluminum gate has a silicon dioxide
thickness of t ox  400Å on a p-type silicon substrate doped with an
acceptor concentration of N a  1016 cm 3 . Determine the capacitances
'
'
'
, C min
and C inv
at (a) f = 1 Hz and (b) f = 1MHz. (c)
C ox , C FB
Determine the VFB and VT .
9. Determine the metal-semiconductor work function difference  MS 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
N a  6  1015 cm 3 . (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 n   p  n  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 1.10m . The transistor dopings are
N E  1017 cm 3 , N B  1016 cm 3 , and N C  1015 cm 3 . (a) For T  300K ,
calculate the B-E voltage at which the minority-carrier electron
concentration at x  0 is 10 percent of the majority-carrier hole
concentration. (b) At this bias, determine the minority-carrier hole
concentration at x '  0 . (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 T  300K with
doping of N E  5  1017 cm 3 , N B  1016 cm 3 , and N C  5  1014 cm 3 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 T  300K has an area of 10 -3 cm 2 , neutral
base width of 1.m , and doping concentrations of N E  1018 cm 3 ,
N B  1017 cm 3 , and N C  1016 cm 3 . Other semiconductor parameters
are DB  20 cm 2 s ,  E 0   B 0  10 7 s , and  C 0  10 6 s . Assuming the
transistor is biased in the active region and the recombination factor
is unity, calculate the collector current for (a) VBE  0.5V , (b)
I E  1.5mA , and (c) I B  2A .
28. Consider an npn silicon bipolar transistor at T  300K with the
following parameters:
DB  20 cm 2 s
DE  10 cm 2 s
 B 0  10 7 s
 E 0  5  10 8 s
N B  1016 cm 3
xE  0.5m
The recombination factor,  , has been determined to be   0.998 .
We need a common-emitter current gain of   120 . Assuming that
 T   , determine the maximum base width, x B , and the minimum
emitter doping , N E , to achieve this specification.
34. A silicon pnp bipolar transistor at T  300K has uniform dopings of
N E  1018 cm 3 , N B  1016 cm 3 , and N C  1015 cm 3 . The metallurgical
base width is 1.2m . Let DB  10 cm 2 s and  B 0  5  10 7 s . Assume
that the minority-carrier hole concentration in the base can be
approximated by a linear distribution. Let VEB  0.625V . (a)
Determine the hole diffusion current density in the base for VBC  5V ,
VBC  10V , and VBC  15V . (b) Estimate the Early voltage.
49. Assume the base transit time of a BJT is 100ps and carries cross the
1.2m B-C space charge region at a speed of 1017 cm 3 . The
emitter-base junction charging time is 25ps and the collector
capacitance and resistance are 0.1Pf and 10 , respectively.
Determine the cutoff frequency.
49. 考慮葉柏仕-摩爾模型，並且令基極端點為開路，因此 IB = 0。證
明當外加
一個集極-射極電壓時，我們得到
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[2]、Semiconductor Devices physics and technology, S.M. Sze
ex1 For an ideal p  n  p transistor, the current components are given
by
I Ep  3mA ,
I En  0.01mA ,
I Cp  2.99mA , and
I Cn  0.001mA .
Determine (a) the emitter efficiency  , (b) the base transport factor
 T , (c) the common-base current gain  0 , (d) I CBO , (e) the
common-emitter current gain  0 , (f) I CEO .
ex2 An ideal p   n  p transistor has impurity concentrations of 1019 ,
1017 , and 5  1015 cm 3 in the emitter, base, and collector regions,
respectively; the corresponding lifetimes are 10 -8 , 10 -7 , and 10 -6 s .
Assume that an effective cross section area A is 0.05 mm 2 and the
emitter-base
junction
is
forward-biased
to
0.6V.
Find
the
common-base current gain of the transistor. Note that the other
device parameters are DE  1cm 2 s , D p  10 cm2 s , Dc  2 cm 2 s , and
W  0.5m .
ex4 A HBT has a bandgap of 1.62eV for the emitter, and a bandgap of
1.42eV for the base. A BJT has a bandgap of 1.42eV for both the
emitter and base materials; it has an emitter doping of 1018 cm 3 and a
base doping 1015 cm 3 . (a) If the HBT has the same doping and the
same  0 . (b) If the HBT has the same emitter doping and the same
 0 as the BJT, how much can we increase the base doping of the
HBT? Assume that all other device parameters are the same.
ex5 Consider a thyristor in which the leakage currents I 1 and I 2 are 0.4
and 0.6mA, respectively. Explain the forward-backing characteristics
when ( 1   2 ) is 0.01 and 0.9999.
1. An n  p  n transistor has a base transport factor  T of 0.998, an
emitter efficiency of 0.997, and an I Cp of 10 nA. (a) Calculate  0
and  0 for the device. (b) If I B  0 , what is the emitter current?
3. A silicon p  n  p transistor has impurity concentrations of 5  1018 ,
2  1017 , and 1016 cm 3 in the emitter , base, and collector, respectively.
The base width is 1.0m , and the device cross-sectional area is
0.2 mm 2 . When the emitter-base junction is forward biased to 0.5V
and the base-collector junction is reverse biased to 5V, calculate (a)
the neutral base width and (b) the minority carrier concentration at
the emitter-base junction.
10. Show that the base transport factor  T can be simplified to
1  (W 2 L2p ) .
2. Given that an ideal transistor has an emitter efficiency of 0.999 and
the collector-base leakage current is 10A , calculate the active
region emitter current due to holes if I B  0 .
23. A Si transistor has D p of 10 cm 2 s and W of 0.5m . Find the
cutoff frequencies for the transistor with a common-base current gain
 0 of 0.998. Neglect the emitter and collector delays.
25. Consider a Si1-x Gex Si HBT with x  10% in the base region (and
0% in emitter and collector region). The bandgap of the base region
is 9.8% smaller than that of Si. If the base current is due to emitter
injection efficiency only, what is the expected change in the
common-emitter current gain between 0  and 100  C ?