Homework answers / question archive / EHB 442E HW#1 Due: 29 September 2016 @9:30 am — No late homework will be accepted

EHB 442E HW#1 Due: 29 September 2016 @9:30 am — No late homework will be accepted

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EHB 442E HW#1

Due: 29 September 2016 @9:30 am — No late homework will be accepted.

1) Consider an electron with a normalized wave function defined as p(x) = MaVaxe~™ for x >

OandOfor x < 0.

a) Show that M =2.

b) For what value of x does P(x) = |g(x)|* is at peak.

c) Calculate <x >.

d) What is the probability of finding the particle between x = Oand x = 1/a?

2) In a photoelectric experiment, the threshold wavelength for the ejection of photoelectrons

from zinc is 310 nm. Calculate the work function for zinc. Also, calculate the velocity for the

photoelectrons by light of wavelength 2000 A other than threshold.

EHB 442E HW#1

Due: 21 September 2017 @9:30 am — No late homework will be accepted.

 

1) An electron is described by a plane wave function as g(x, t) = Aexp|[j(10x — 7t)]. Calculate

the expectation value of the x — component of momentum, the y— component of momentum

and the energy of the electron.

 

2) The uncertainty in position is 12 A for a particle of mass 5 X 1072? kg. Determine the minimum

uncertainty in

 

a. The momentum of the particle.

b. The kinetic energy of the particle.

EHB 442E HW#2

Due: 13 October 2016 @9:30 am — No late homework will be accepted.

 

1) A semiconductor has a conduction band structure described approximately by E = Ep, —

Acos ak, — B(cos Bk, + cos Bk;). Assuming that the band minimum occurs near ky = ky =

k, = 0 and the effective mass for electrons (m,,) is given by the geometric mean of effective

masses in all k directions, find an expression for mp.

 

2) Derive an expression for the intrinsic Fermi level in terms of the conduction band energy (E,),

valence band energy (£,), and electron and hole effective masses (m;, and m,).

 

3) Calculate the thermal equilibrium electron and hole concentration in Si at T = 300 K for the case

when the Fermi energy level is 0.22 eV below the conduction band energy (E,). For Si, the band

gap energy (E,) at 300 K is 1.11 eV. Density of states effective mass values for Si, m;, and my,

are given as 1.08 mp and 0.56 mp, respectively, where mpg is the free electron rest mass.

 

4) A doped Si sample of thickness 3 mm shows a Hall voltage of V, = 3 mV for current a density

of J =300A/m*, under a magnetic field of B, =1Wb/m?. Find the type of the

semiconductor and the doping concentration.

EHB 442E HW#2

Due: 5 October 2017 @9:30 am — No late homework will be accepted.

 

1) Find the approximate binding energy for GaN having the following specifications €, = 9.7

andm, = 0.13mp.

 

2) Asemiconductor has an electron band structure E(k) = (4k* +5) eV where k has the units

of A. Calculate the effective mass of the electrons.

 

3) Derive an expression for the intrinsic Fermi level in terms of the conduction band energy (E-),

valence band energy (£,), and electron and hole effective masses (m;, and m,).

 

4) A hypothetical semiconductor has an intrinsic concentration of 1 x 10!4cm™? at 300 K; it has

conduction and valence band effective densities of states Ne and Ny both equal tol x

107°cm7?.

 

a. What is the bandgap E,°

 

b. If the semiconductor is doped with Np = 1x 10?’donors/cm?, what are the

equilibrium electron and hole concentrations at 300K?

 

c. Ifthe same piece of semiconductor, already having Np = 1 X 10/’donors/cm?, is also

doped with N, = 2 x 10’’donors/cm?, what are the new equilibrium electron and

hole concentrations at 300 K?

 

d. Consistent with the answer to part (c), what is the Fermi level position with respect to

the intrinsic Fermi level E; — E; ?

 

5) Consider a Si bar with width of 0.02 cm, thickness of 8 um, and length of 0.6 cm. In a Hall Effect

measurement, magnetic field generated at z-direction is B, = 10 Wb/cm? and the current

flowing is 0.8 MA. If Vig = 1 mV and VA, = 50 mV, then find the type, concentration, and

mobility of the majority carriers.

EHB 442E HW#3

Due: 27 October 2016 @9:30 am — No late homework will be accepted.

1) A semiconductor material with the intrinsic charge carrier concentration n; is linearly doped

across the sample, which is of thickness t. Donor concentration Ng varies from 0 at x = 0 to

Nao at x =f.

a. Write equations for total electron and hole concentrations as a function of distance x.

b. Determine electron and hole diffusion current densities given that the diffusion

coefficients are Dy and Dy.

c. Find the expression for Fermi Level (E;) as a function of x and the intrinsic level E;.

2) Assume that a photoconductor in the shape of a bar of length L and area A has a constant

voltage V applied, and it is illuminated such that gon EHP/cm?-s are generated uniformly

throughout. If Uy >> My, we can assume the optically induced change in current Al is

dominated by the mobility u“, and the lifetime T,, for electrons. Show that AI = qgALQoptn/Tt

for this photoconductor, where T; is the transit time of electrons drifting down the length of

the bar via two approaches:

a. By using the relation between the electron mobility u“, and the electron velocity v,,

across the bar.

b. By considering the change in optically induced charge in the bar over the transit time T;.

3) Boron is diffused into an intrinsic Si sample, giving the acceptor distribution shown below.

Sketch the equilibrium band diagram and show the direction of the resulting electric field,

for N,(x) > n; . Repeat for phosphorous, with Np(x) > n; .

KO

X

4) Boron is implanted into an n-type Si sample (Np = 101®cm73), forming an abrupt junction of

square cross section with area = 2 X 1072cm?. Assume that the acceptor concentration in the

p-type region is N, = 4x 10*®cm™®. Calculate Vo, Xno, Xpo, Q4, and Eo for this junction at

equilibrium (300 K). Sketch the electric field and the charge density.

EHB 442E HW#3

Due: 19 October 2017 @9:30 am — No late homework will be accepted.

 

1) Asemiconductor material with the intrinsic charge carrier concentration n,; is doped across the

sample in a semi-parabolic fashion, which is of thickness t. Donor concentration Ng varies from

Ngo atx =OtoOat x=t.

 

a. Write equations for total electron and hole concentrations as a function of distance x.

 

b. Determine electron and hole diffusion current densities given that the diffusion

coefficients are Dy and Dy.

 

c. Find the expression for Fermi Level (E;) as a function of x and the intrinsic level E;.

 

2) Assume that a photoconductor in the shape of a bar of length L and area A has a constant

voltage V applied, and it is illuminated such that gon EHP/cm?-s are generated uniformly

throughout. If Uy >> My, we can assume the optically induced change in current Al is

dominated by the mobility wu, and the lifetime T,, for electrons. Show that AI = qeALQoptn/Tt

for this photoconductor, where T; is the transit time of electrons drifting down the length of

the bar via two approaches:

 

a. By using the relation between the electron mobility u, and the electron velocity v,,

across the bar.

b. By considering the change in optically induced charge in the bar over the transit time T;.

 

3) Boron is diffused into an intrinsic Si sample, giving the acceptor distribution shown below.

Sketch the equilibrium band diagram and show the direction of the resulting electric field,

for N,(x) > n; . Repeat for phosphorous, with Np(x) > n; .

 

/

X

 

4) Boron is implanted into an n-type Si sample (Np = 10+°cm 3), forming an abrupt junction of

Square cross section with area = 3 X 1072cm?. Assume that the acceptor concentration in the

p-type region is N4 = 4 x 10*?cm7?. Calculate Vo, xno, Xpo, Q4, and Ep for this junction at

equilibrium (400 K). Sketch the electric field and the charge density.

EHB 442E HW#4

Due: 24 November 2016 @9:30 am — No late homework will be accepted.

 

1) Calculate the electric field in the n-side neutral region of a silicon diode. Assume that the diode

has a reverse saturation current, />, and is biased with a forward current of Ve at

temperature Ty. Assume that the n-side has a doping concentration of Np, the electron

mobility is “, and the cross section area of the diode is A.

 

2) Assume holes are injected from p*-n junction into a short n region of length l. If 6,,(x,) varies

linearly from Ap, at x, = 0 to zero at the ohmic contact (x, = lL), find the steady state charge

in the excess hole distribution Q,, and the current I.

 

3) A Schottky barrier is formed between a metal having a work function of 4.3 eV and p-type Si

(electron affinity = 4 eV). The acceptor doping in the Si is 107” cm?.

 

a Draw the equilibrium band diagram, showing a numerical value for qeVp.

b Draw the band diagram with 0.3 V forward bias. Repeat for 2 V reverse bias.

 

4) Metal with a work function of 4.6 eV is deposited on Si (electron affinity = 4 eV) and acceptor

doping is 107° cm®. Draw the equilibrium band diagram and mark off the Fermi level, the band

edges, and the vacuum level. Is this a Schottky or ohmic contact, and why? By how much should

the metal work function be altered to change the type of the contact? Explain with reference

to the band diagram.

EHB 442E HW#4

Due: 23 November 2017 @9:30 am — No late homework will be accepted.

 

1) Calculate the electric field in the p-side neutral region of a silicon diode. Assume that the diode

has a reverse saturation current, Is, and is biased with a forward current of Vy at

temperature Ty. Assume that the p-side has a doping concentration of Ny, the hole mobility

iS [ly and the cross section area of the diode is A.

 

2) Assume electrons are injected from n*-p junction into a short p region of length J. If bn(Xp)

varies linearly from Any at X» = 0 to zero at the ohmic contact (x, = !), find the steady state

charge in the excess hole distribution Q, and the current I.

 

3) A Schottky barrier is formed between a metal having a work function of 4.5 eV and p-type Si

(electron affinity = 3.8 eV). The acceptor doping in the Si is 107° cm”.

 

a Draw the equilibrium band diagram, showing a numerical value for qeVp.

b Draw the band diagram with 0.2 V forward bias. Repeat for 2.2 V reverse bias.

 

4) Metal with a work function of 4.4 eV is deposited on Si (electron affinity = 3.9 eV) and acceptor

doping is 107” cm®. Draw the equilibrium band diagram and mark off the Fermi level, the band

edges, and the vacuum level. Is this a Schottky or ohmic contact, and why? By how much should

the metal work function be altered to change the type of the contact? Explain with reference

to the band diagram.

EHB 442E HW#5

Due: 8 December 2016 @9:30 am — No late homework will be accepted.

 

1) A Si p-n-p transistor has the following properties at room temperature:

 

Ty =Ty = 0.1 us Nz = 101? cm7? = Emitter concentration

Dy, = Dy = 10 cm*/s Nz = 101° cm? = Base concentration

Nc = 10!? cm7? = Collector concentration

 

We = 3 um = Emitter width

 

W =1.5 um = Metallurgical base width, i.e. the distance between base-emitter junction and

base-collector junction

 

A=10-° cm? = Cross-sectional area

 

If Vep = OV and Vep = 0.6 V, calculate the following:

 

a) Neutral base width (Wp)

 

b) Base transport factor

 

c) Emitter injection efficiency

 

d) a,B andy.

 

e) Ic, Ip and Ig.

 

f) Base Gummel number.

 

2) Draw the corresponding equivalent circuit for the Ebers-Moll model of the p-n-p transistor

above.

 

3) A Si p-n-p BJT at T=300K has uniform dopings of Ne = 101° cm73, Nz = 1018 cm=3, Nz =

101° cm7~3. The metallurgical base width is 1.2 um.

 

a. Calculate the peak electric field at the CB junction and the CB depletion capacitance per

unit area for normal biasing, with a CB bias of 30 V.

 

b. Estimate the neutral base width narrowing at this voltage, ignoring the EB depletion

region.

 

4) Show that the expression for the excess minority hole concentration by (x) ina uniformly doped

p-n-p bipolar transistor in forward active region having the base width Wg, the hole diffusion

length L,, the emitter-base voltage Veg, the base doping Ng and the intrinsic carrier

concentration n; can be found by solving the diffusion equation as:

 

2 —

Ti Jexp (dete) — 1] sinh (“@ =~) — sinh (

Np kpT Ly Ly

5, (x) =-2>9>-/S— Se _ So

P . We

sinh (=)

p

EHB 442E HW#5

Due: 7 December 2017 @9:30 am — No late homework will be accepted.

 

1) A Si p-n-p transistor has the following properties at room temperature:

 

Ty =Ty = 0.1 us Nz = 102° cm~? = Emitter concentration

Dy, = Dy = 10 cm*/s Nz = 101° cm~3 = Base concentration

Nc = 10!? cm7? = Collector concentration

 

We = 3 um = Emitter width

 

W =1.5 um = Metallurgical base width, i.e. the distance between base-emitter junction and

base-collector junction

 

A=10-° cm? = Cross-sectional area

 

If Vep = OV and Vep = 0.6 V, calculate the following:

 

a) Neutral base width (W,)

 

b) Base transport factor

 

c) Emitter injection efficiency

 

d) a,B andy.

 

e) Ic, Ip and Ig.

 

2) Draw the corresponding equivalent circuit for the Ebers-Moll model of the p-n-p transistor

above.

 

3) Show that the expression for the excess minority hole concentration by (x) ina uniformly doped

p-n-p bipolar transistor in forward active region having the base width Wg, the hole diffusion

length L,, the emitter-base voltage Veg, the base doping Ng and the intrinsic carrier

concentration n; can be found by solving the diffusion equation as:

 

2 _

Ne [exp (“¢-42) — 1] sinh (“5 —) — sinh (zt

6,(x) = TTT

sinh (=)

p

 

4) ASisolar cell 2 cm x 2 cm with J,, = 42 nA has an optical generation rate of 101? EHP /cm?s

within L, = L, = 2 .5um of the junction. If the depletion width is 1.5 um, calculate the short-

circuit current and the open circuit voltage for this cell.