PROBLEMS- CH1 SEMICONDUCTOR FUNDAMENTALS
SC or S/C: semiconductor
Q1) What is the probability of finding an electron at energy 0.18 eV above Ec at room temperature (300 °K) if EF is 0.1
eV below Ec? (Ans: f(E)=2.1x10 5)
Q2) A Si sample is doped B acceptors at a level of 10 14 cm-3, find n and p:
a) at room temperature (Ans: p = 1014 cm-3, n = 106 cm-3)
b) at 470 K (Ans: p =1.28 10 14 cm-3, n = 2.8x1013 cm-3)
(Let ni=1010 at 300 K and me= 1.18 mo, mh= 0.81mo, Eg= 1.1 for Si)
Q3) For a GaAs sample is doped with In acceptors at a level of 5x1017 cm-3, find p and Ef relative to Ev at room
temperature. (mh= 0.52mo), (Ans: p=5x10 17 cm-3, Ef-Ev=0.076 Ev)
Q4) Calculate the thermal equilibrium concentration of electrons and holes for a given Fermi energy. Consider Si at
T=300 K so that Nc=2.8x1019 cm-3 and Nv=1.04x10 19 cm-3. Assume that the Fermi energy is 0.25 eV below the conduction
band. Bandgap energy of Si is 1.12 eV. (Ans: n=1.8x10 15 cm-3, p= 2.7x10 4 cm-3)
Q5) Determine the thermal equilibrium electron and hole concentrations for a given doping concentration. Consider an ntype Si SC at T=300 K in which Nd=1016 cm-3 and Na=0. The intrinsic carrier concentration is assumed to be ni=1.5x10 10
cm-3.. (Ans: n=1x1016 cm-3, p= 2.25x10 4 cm-3)
Q6) Calculate the thermal equilibrium electron and hole concentrations in a germanium (Ge) sample for a given doping
density. Consider a Ge sample at T=300 K in which Nd=5x1013 cm-3 and Na=0. Assumed that ni=2.4x1013 cm-3. (Ans:
n=5.97x10 13 cm-3, p= 9.65x1012 cm-3)
Q7) Calculate the thermal equilibrium electron and hole concentrations in a compensated p-type semiconductor (SC).
Consider a Si SC at T=300 K in which Nd=3x1015 cm-3 and Na=1016 cm-3. Assume ni=1.5x1010 cm-3. (Ans: n=3.21x104
cm-3, p= 7x1015 cm-3)
Q8) Determine the required impurity concentration in a SC material. A Si device with n type material is to be operated at
T=550 K. At this temperature, the intrinsic carrier concentration must contribute no more than 5 percent of the total
electron concentration. Determine the minimum donor concentration required to meet this specification. (Ans:
Nd=1.39x1015 cm-3)
Q9) Determine the required donor impurity concentration to obtain a specified Fermi energy. Si at T=300 K contains an
acceptor impurity concentration of Na=10 16 cm-3. Determine the concentration of donor impurity atoms that must be added
so that the silicon is n type and the Fermi energy is 0.20 eV below the conduction band edge. (Ans: Nd=2.24x1016 cm-3)
Q10) Determine the Fermi-energy position and the maximum doping at which the Boltzmann approximation is still valid.
Consider p- type Si, at T=300 K, doped with boron. We may assume that the limit of the Boltzmann approximation occurs
Ef-Ea=3kT. Ionization energy is Ea-E v=0.045 eV for boron silicon. (Ans: Na=3.2x1017 cm-3)
PROBLEMS- CH2 CARRIER TRANSPORT AND RECOMBINATION
SC or S/C: semiconductor
Q1) Calculate the mean free time of an electron having mobility of 1000 cm2/V-s at 300 K; also calculate the mean free
path. Assume me*=0.26mo, thermal velocity of electrons is 10 7cm/s.
(Ans: mean free time, τ: 0.148ps; mean free path=14.8 nm)
Q2) Find the room-temperature resistivity of an n-type silicon doped with 10 16 phosphorus atoms/cm3. µn=1300 cm2/Vs. (Ans: resistivity, ρ= 0.48 Ω-cm)
Q3) A sample of Si is doped with 10 16 phosphorus atoms/cm3. Find the Hall coefficient and Hall voltage in a sample
-4
with w=500 µm, A=2.5 cm2, I= 1 mA, and Bz=10 wb/cm2. (Ans: RH= ─ 625 cm3/C, VH= ─1.25 mV)
Q4) Assume that, in n-type SC at T=300 K, the electron concentration varies linearly from 1x1018 to 7x1017 cm-3 over a
distance of 0.1 cm. Calculate the diffusion current density if the electron diffusion coefficient is D n=22.5 cm2/s. (Ans:
J n=10.8 A/cm2)
Q5) Minority carriers (holes) are injected into a homogeneous n-type SC sample at one point. An electric field of 50
V/cm is applied the sample, and the field moves these minority carriers a distance of 1cm in 100 µs. Find the drift
velocity and the diffusivity of the minority carriers. (Ans: vd=10 4 cm/s, Dp=5.18 cm2/s)
Q6) A uniformly doped SC resistor (doping level Nd=1016 cm-3 and cross sectional-area A=20 µm2) conducts 2 mA
of current. The resistor is connected by copper wires with cross section of 0.1 area mm2. Determine and
compare the drift velocities of the electrons in the SC and copper regions. The concentration of electrons in
copper is ncu=8.1x1022 cm-3. (Ans: vd(SC)=62.5 km/s; vd(cu)=1.54 µm/s; As seen the results there are many
more electrons in the copper wires that can move 4x1010 times slower and still supply the necessary current to
the resistor.)
Q7) The mobility of electrons in pure GaAs at 300 K is 8500 cm2/V-s. Calculate the relaxation time (i.e mean free
time). If the GaAs sample is doped at Nd=1017 cm-3, the mobility decreases to 5000 cm2/V-s. Calculate the mobility and
relaxation time due to ionized impurity scattering. Assume me*=0.067mo (Ans: τL=3.24x10 -13 s µ I=1.2X104 cm2/V-s,
τ I=4.6x10-13 s)
Q8) Consider two SC samples, one Si and one GaAs. Both materials are doped n-type at Nd= 1017 cm-3. Assume 50
percent of the donors are ionized at 300 K. Calculate the conductivity of the samples. Compare this conductivity to the
conductivity of undoped samples. Write a comment about your results.
µn (Si)=1000 cm2/V-s, µp(Si)=350 cm2/V-s, ni(Si)=1.5x1010 cm-3
µn (GaAs)=8000 cm2/V-s, µp(GaAs)=400 cm2/V-s, ni(GaAs)=1.84x106 cm-3
(Ans: Si: σ doped=8 (Ω-cm)-1, σundoped=3.24x10-6 (Ω-cm)-1)
(Ans: GaAs : σdoped=64 (Ω-cm)-1, σundoped=2.47x10-9 (Ω-cm)-1)
Q9) Two scattering mechanism are present in an n-type silicon sample. With both mechanism presents, the overall
effective electron mobility in the sample is 563 cm2/V-s. If doping this sample causes its conductivity to increase by a
factor of 105 over the intrinsic conductivity, calculate the change in Fermi energy caused by this doping at room
temperature. Assume the intrinsic electron and hole mobilities are 1350 cm2/V-s and 480 cm2/V.s, respectively. (Ans:
Ef─Ei=0.33 eV)
Q10) Using Boltzmann statistics, calculate the position of the electron and hole quasi-Fermi levels when electron-hole
density of 10 17 cm-3 is injected into pure (undoped) silicon at 300 K. Repeat if 10 15 cm-3 electron and holes are injected.
E g(Si)=1.1 Ev. (Ans: Efn─Efp=0.833 eV for 10 17 cm-3; Efn─Efp=0.59 eV for 10 15 cm-3)
Q11) The absorption coefficients near the bandedges of GaAs and Si are ~104 cm-1 and 10 3 cm-1, respectively. What is
the minimum thickness of a sample in each case that can absorb 90% of the incident light?
(Ans: thickness (GaAs) =2.3 µm, thickness (Si) =23 µm, as seen the results Si requires a very thick active
absorption layer in order to function)
OPTICAL DETECTOR: In optical detector light is absorbed to create electron and holes. These are collected and
generate a signal.
Q12) In a p-type GaAs sample, electrons are injected from a contact. If the minority carrier mobility is 4000 cm2/V.s at
300 K, calculate the diffusion length for the electrons. The recombination time is 0.6 ns. (Ans: Ln=2.5 µm)
Q13) Consider a Si sample of length L. The diffusion coefficient for electrons is 25cm2/s and electron lifetime is 0.01
µs. An excess electron concentration is maintained at x=0 and x=L. The excess concentrations are: Δn(x=0)=2x10 18
cm-3; Δn(x=L)= ─1x1014 cm-3.
Calculate and plot the excess electron distribution from x=0 to x= L. Do the calculations for the following values: L=
10 µm and 0.5 µm.
(Ans: Constant values are -3.7x10 16 cm-3 and 2.03x1018 cm-3 for 10 µm
-9.03x1018 cm-3 and 1.1x1019 cm-3 for 0.5 µm)
If we plot excess carrier concentration with respect to length we see that for the shorter length, the carrier
density falls off linearly.
PROBLEMS-CH3 pn JUNCTION ELECTROSTATICS
Q1) A silicon diode is fabricated by starting with an n-type (Nd=10 16 cm-3) substrate, into which indium is diffused to
form as a p-type region doped at 10 18 cm-3. Assuming that an abrupt p+- n junction is formed by the diffusion process,
(a) calculate the fermi level positions in the p- and n- regions, (b) determine the contact potential in the diode, (c)
calculate the depletion widths on the p- and n- side.
(Ans: Ec─Efn=0.206 eV, Efp─Ev=0.06 eV, ψo=0.838 V, xd=0.32 µm)
Q2) Consider a 20 µm diameter p-n diode fabricated in silicon. The donor density is 1016 cm-3 and the acceptor density
is 10 18 cm-3. Calculate the following in this diode at 300 K: (a) the depletion widths and electric field profile under
reverse biases of 0, 5, and 10 V, and under a forward bias of 0.5 V, (b) what are the charges in the depletion region
for these biases? Ans: Vr=0 V, x p=32 Å, xn=0.32 µm, Ɛ= ─4.95x10 4 V/cm, Q=1.61x10-13 C
V r=5 V, x p=84.5 Å, x n=0.845 µm, Ɛ= ─1.3x105 V/cm, Q=4.24x10 -13 C
V r=10 V, xp=115.1 Å, xn=1.151 µm, Ɛ= ─1.78x105 V/cm, Q=5.78x10 -13 C
Vf=0.5 V, xp=20 Å, xn=0.20 µm, Ɛ= ─3.14x10 4 V/cm, Q=1x10-13 C
(We can see that at a reverse bias of ~10 V, the peak field is beginning to approach the breakdown field for Si, which is
around 3x105 V/cm)
Q3) For a silicon one-sided abrupt junction with N a=1019 cm-3 and Nd=1016 cm-3, calculate the depletion layer width
and the maximum field zero bias (T=300 K). (Ans: xd=0.343 µm, |Ɛ m|= 5.2x10 4 V/cm)
Q4) For a silicon one-sided abrupt junction with Na=2x10 19 cm-3 and Nd=8x10 15 cm-3, calculate the junction
capacitance at zero bias and reverse bias of 4 V (T=300 K).
(Ans: Cj (0 V) = 2.728x10-8 F /cm2, Cj (-4 V) = 1.172x10-8 F /cm2)
PROBLEMS- CH4 pn JUNCTION UNDER FORWARD BIAS
Q1) Consider an ideal diode model for a silicon p-n diode with Nd=1016 cm-3 and N a=1018 cm-3. The diode area is 10 -3
cm-2. Transport properties of the diode are given by the following values at 300 K:
n-side:
µp=300 cm2 V-1 s-1
µn=1300 cm2 V -1 s-1
Dp=7.8 cm2 s -1
Dn=33 cm2 s-1
p-side:
µp=100 cm2 V-1 s-1
µn=280 cm2 V-1 s-1
Dp=2.6 cm2 s -1
Dn=7.3 cm2 s-1
(Note that the mobility is a lot lower in the heavily doped p-side because of the increased ionized impurity scattering.)
Assume that τp= τn=1µs. Calculate the diode current. (Ans: Is=1x10-14 A)
Q2) We will define a p-n diode to be “turned on” when the current density reaches 103 A /cm2 (this is an approximate
criterion). Calculate the turn-on or cut-in voltage for a Si and GaAs p-n diode with following parameters (same for both
diodes) N d=N a=1017 cm-3, τp= τn=10 -8 s. (Dp=12.5 cm2 s-1, Dn=35 cm2 s -1 for Si and Dp=10 cm2 s-1, D n=220 cm2 s-1 for
GaAs) (Ans: 0.8 V for Si, 1.26 V for GaAs)
Q3) An important use of a forward-biased p-n diode is as an emitter in a bipolar transistor. In the emitter it is desirable
that the current be injected via only one kind of charge. The diode efficiency is thus defined as (Jn is the current density
carried by electron injection into the p-side)
=
J
J
Consider a GaAs p-n diode with the following parameters:
Electron diffusion coefficient,
Dn=30 cm2 s-1
Hole diffusion coefficient
Dp=15 cm2 s-1
p-side doping
Na=5x10 16 cm-3
n-side doping
Nd=5x10 17 cm-3
Electron minority carrier time
τn=10-8 s
Hole minority carrier time
τp=10-7 s
Calculate the diode injection efficiency (this is called the emitter efficiency in a bipolar transistor). (Ans: γinj =0.98)
Q4) A forward biased pn+ GaAs diode is used as an LED at 300 K. Assume that all the electrons are injected into the
top p region combine with holes to generate photons. Calculate the photon generation rate and optical power coming
from the LED if the device has the parameters of the diode in question 3 (Q3), is forward biased at 1 V. The diode area
is 0.1 mm2. Energy bandgap of GaAs is 1.43 eV. (Ans: photon generation rate=1.9x10 14 s-1, p=4.3x10-5 W)
Q5) An important application of p-n (or p-i-n) diodes is their use as detectors of optical radiation. The optical signal
creates electron-hole pairs that are collected as a current if the pairs are in the depletion region and regions within the
diffusion lengths of the depletion region consider the following parameters of a silicon p-n diode at 300 K:
Diode area
A=10 4 µm2
Electron diffusion coefficient
Dn=20 cm2 s-1
Hole diffusion coefficient
Dp=12 cm2 s-1
p-side doping
Na=2x10 16 cm-3
n-side doping
Nd=1x10 17 cm-3
Electron minority carrier time
τn=10-8 s
Hole minority carrier time
τp=10-8 s
Applied reverse voltage
Vr =15 V
Calculate the photocurrent of the device if the electron-hole (e-h) pairs are generated from an optical signal at a rate
GL=1022 cm-3s-1 and the photocurrent, IL, is “q A G L (x d+ Ln+ Lp)”.
( xd is the depletion width and Ln and Lp are the diffusion lengths for electron and hole) (Ans: IL=0.155 mA)
Q6) Consider the Si p-n junction diode with a cross sectional area of 2x10-4 cm2. Calculate the generation current
density for a reverse bias of 4 V. the parameters of the diode are Na=5x1016 cm-3, N d=1016 cm-3, ni=9.65x109 cm-3,
Dn=21 cm2 s-1, Dp=10 cm2 s-1 and assume τg =τn= τp=5x10-7 s. (Ans: Jgen=2.66x10-7 A/cm2)
Q7) For an ideal abrupt silicon p+-n junction with Nd=8x1015 cm-3, calculate the stored minority carriers per unit area in
the neutral n-region when forward bias of 1V is applied. The diffusion length of the holes is 5 µm.
(Ans: Qp =4.69x10 -2C/cm2)
Q8) Consider a long p+-n diode that is biased to carry a forward current of 1 mA. The junction capacitance is 100 pF. If
the minority carrier lifetime is τp=1µs, what is the admittance of the diode at 300 K for a 1 MHz signal.
(Cdiff= Kf*(q/kT)* I* τp, where Kf is a factor which is 1/2 for long base diodes and 2/3 for narrow base devices.)
(Ans: y=0.038 +j 0.12 A/ V)
PROBLEMS- CH5 BIPOLAR JUNCTION TRANSISTOR (BJT)
(For following questions emitter thickness is much greater than minority carrier diffusion length)
Q1) A silicon pnp transistor at 300 K has a device area of 10 -3 cm2, base width of 1 µm, and minority carrier diffusion
length (for both electrons and holes) of 10 µm. The doping profiles are NAE=2x1018 cm-3, NDB=1017 cm-3, and NAC=10 16
cm-3. Calculate the collector current for the forward active mode cases where (a) Vbe=0.6 V, (b) IB=2 µA. Assume that
Db=6 cm2s-1 and De=20 cm2s-1. (Ans: (a) Ic=0.2268 mA, (b) Vbe= 0.576 V)
Q2) In a bipolar transistor, the base width plays a key role in the design consideration. If the base width is too small,
the device can suffer punchthrough and may have too large base resistance (which is inversely proportional to the base
width). If the base width is too large, the base transport factor becomes small.
Consider an npn Si bipolar transistor with base doping of 1016 cm-3. If the electron diffusion coefficient in the base is
10 cm2s-1 and the electron lifetime is 10 -6 s, calculate the base width required to have transport factor of 0.997. Assume
that the neutral base width is equal to the actual basewidth. (Ans: xb=2.5 µm)
Q3) An npn silicon transistor has the following parameters at 300K: αN= α F=0.99, αI= αR=0.25. Calculate the saturation
voltage VCE when (a) IC= 1 mA, IB=20 µA, (b) IC= 5 mA, IB=75 µA.
(Ans: (a) VCE(sat) =0.149V, (b) VCE(sat) =0.167V ) We see that as IB increases the value of VCE(sat) increases, but the
change is rather small.
Q4) Consider an npn Si bipolar transistor that is to be designed with an emitter injection efficiency of γ=0.995. To
maintain a reasonable base resistance, the base is doped p-type with N A=1016 cm-3. Calculate the n-type doping needed
in the emitter. Assume that Le=10*x b and Dn~Dp. (Ans: ND=2x1017 cm-3)
Q5) A silicon pnp transistor at 300 K has a device area of 10 -3 cm2, base width of 1 µm, and minority carrier diffusion
length (for both electrons and holes) of 10 µm. The doping profiles are NAE=2x1018 cm-3, NDB=1017 cm-3, and
NAC=1016 cm-3. Ic=0.2268 mA and Vbe=0.6 V. Assume that Db=6 cm2s-1 and De=20 cm2s-1. Calculate the current gains
and the transconductance of the device.
(Ans: β=46, α=0.978, g m=8.72 mS=mA/V)
Q6) This example will examine the dependence of current gain on the minority carrier recombination in the base.
Consider a silicon npn transistor with the following parameters at 300K:
Emitter doping,
Nde= 5x1017 cm-3
Base doping,
Nab= 1017 cm-3
Base width,
xb= 1 µm
Diffusion coefficient, Db=De=20 cm2/s
Calculate the current gain for the two cases: (a) minority carrier lifetime for the electrons and holes=10 -6 s; (b) minority
carrier lifetime=10-8 s. Such a reduction in lifetime can be obtained in silicon by introducing defects.
(Ans: (a) α=0.9953, β=210.7, (b) α=0.931, β=13.6)
PROBLEMS
CH6 METAL OXIDE SEMICONDUCTOR FIELD EFFECT TRANSOSTORS- MOSFET
ɛox=3.9ɛo, ɛs=11.7ɛo, where ɛo= 8.854x10-14 F/cm
Q1) Consider an NMOS having Vt =2 V. What is the region of operation (triode, saturation, or cutoff) if
(a) Vgs=1 V and Vds= 5 V
(b) Vgs=3 V and Vds= 0.5 V
(c) Vgs=3 V and Vds= 6 V
(d) Vgs=5 V and Vds= 6 V
Ans: (a) cutoff (b) triode (c) saturation (d) saturation
Q2) A certain enhancement mode NMOS transistor has W= 160 µm, L= 2 µm, kn’=50 µm/V2, and Vt= 2 V. Plot
the drain characteristic curves to scale for Vgs= 0, 1, 2, 3, 4, and 5 V.
Q3) Consider a process technology for which Lmin= 0.4 µm, tox= 8 nm, µn=450 cm2/V.s, and Vt= 2 V.
’
(a) Find Cox, and kn
(b) For MOSFET with W/L= 8 µm/0.8 µm, calculate the values of Vgs, Vdsmin needed to operate the transistor in the
saturation region with a dc current Id=100 µA.
(c) For the device in (b), find the value of Vgs required to cause the device to operate as a 1000-Ω resistor for very
small Vds.
’
2
Ans: (a) Cox=4.32 fF/ µm2, kn = 194 µm/V ; (b) Vgs=1.02 V, Vdsmin=0.32 V; (c) Vgs=1.22 V
Q4) An enhancement type NMOS transistor with Vt= 0.7 V has its source terminal grounded and a 1.5 V dc applied
to gate. In what region does the device operate for (a) Vd= 0.5 V, (b) Vd= 0.9 V, (c) Vd= 3 V.
Ans: (a) triode (b) saturation (c) saturation
Q5) An enhancement type NMOS transistor with Vt= 0.7 V has its source terminal grounded and a 1.5 V dc applied
2
to gate. The device has µnCox= 100 µm/V , W= 10 µm, and L=1 µm, find the value of drain current that for (a)
Vd= 0.5 V, (b) Vd= 0.9 V, (c) Vd= 3 V.
Ans: (a) 275 µA (b) 320 µA (c) 320 µA
Q6) An enhancement type NMOS transistor with Vt= 0.7 V conducts a current Id= 100 µA when Vgs=Vds=1.2 V.
Find the value of Id Vgs=1.5 V and Vds= 3 V. Also, calculate the value of drain to source resistances rds for small Vds
and Vgs= 3.2 V.
Ans: (a) 256 µA (b) 500 Ω
Q7) Calculate the on-resistance of an NMOS transistor for Vgs=2 V and 5 V if Vt=1 V and kn=125 µA/V2. (Note
that: on resistance means drain to source resistance, rds)
Ans: (a) 4 KΩ (b) 1KΩ
Q8) What is the region of operation and drain current of an NMOS transistor having Vt=1 V, kn=0.5 mA/V2, and
λ=0.02 V. (a) Vgs=1 V, Vds= 5 V; (b) Vgs=2 V, Vds= 0.5 V; (c) Vgs=2 V, Vds= 2 V
Ans: (a) cutoff, 0A (b) triode, 375 µA (c) saturation, 520 µA
Q9) Identify the source, drain, gate and the bulk terminals, and find the current I in the following figures. Assume
Vt=0.7 V, W/L=10, and k’=100 µA/V2.
─ 0.2 V
0.2 V
I
I
5V
5V
(a)
(b)
Ans: (a) 840 µA (b) ─880 µA
Q10) (a) What is the W/L ratio required for an NMOS transistor to have an on-resistance of 500 Ω when Vgs=5 V
’
2
and Vsb=0 V? Assume Vt=0.75 and k =100 µA/V .
(b) Repeat for Vgs=3.3 V.
Ans: (a) 4.71/1
(b) 7.84/1
’
Q11) Calculate kn for NMOS transistor with µn = 500 cm2 /V.s for a oxide thickness of (a) 50 nm, (b) 20 nm, (c) 10
nm and (d) 5 nm.
2
Ans: (a) 34.5 µA/V
2
(b) 86.3 µA/V
2
(c) 173 µA/V
2
(d) 345 µA/V
Q12) What is the charge density (C/cm2) in the channel if the oxide thickness is 25 nm and the oxide voltage exceeds
the threshold voltage by 1 V?
Ans: 1.38x10-7 C/cm2
Q13) What is the electron velocity in the channel if µn = 500 cm2 /Vs and electric field is 2000 V./s.
Ans: ─1x10-6 cm/s