296 CHAPTER 6. BIPOLAR JUNCTION TRANSISTORS
where the current and the excess charge are evaluated at the boundary. Using the
expression for minority current in terms of the diffusion coefficient and the charge density
gradient, we have
Dp
d(δp)
dx
|boundary=vrecomδp|boundary
Consider the case where an excess hole density ofδp(x 1 )is injected across a depletion
region into ann-side. The boundary of the contact is at a positionx 2. The distance
(x 2 −x 1 )Lpso that the hole concentration can be assumed to decrease linearly.
Expressδp(x 2 )in terms of the surface recombination velocity,δp(x 1 ), and the diffusion
coefficientDpand (x 2 −x 1 ).
Problem 6.31Consider an npn bipolar transistor where the base is graded from
Al 0. 04 Ga 0. 96 As at the emitter end to GaAs at the collector end. The emitter material is
Al 0. 22 Ga 0. 78 AsanditisgradedtoAl 0. 04 Ga 0. 96 As at the base.
(a) Sketch the detailed band diagram of the HBT in equilibrium, and under forward active
bias. What is the quasi-electric field,EB, in the base?
(b) Solve the drift-diffusion equation to obtain an expression for the base minority carrier
concentration, n(x), in terms of the total current JC,EB,andWB.Assumeμ=
1000 cm^2 /Vs.
Problem 6.32An npn HBT has a collector consisting of slabs of two different materials,
A and B. The velocity in material A isvswhile that in material B isv 2 s. The thickness of
these slabs is equal, ie.WA=WB=W 2 C,whereWCis the total collector width. Assume
that the doping in the collector isND, and that the voltageVCBdepletes the entire
collector region.
(a) Consider the case when slab A is adjacent to the base. Draw the charge, electric field
and potential profiles in the collector when the Kirk threshold current is reached. What is
the Kirk threshold of this transistor? How does it compare with a transistor whose
collector is made up of material A throughout?
(b) Repeat the above but now with slab B adjacent to the base. Compare the two cases and
explain the result.
(c) Assume that the breakdown field in material A is less than in material B. Which
material should be placed adjacent to the base to maximize the breakdown voltage under
high injection?
Problem 6.33For some unknown reason, possibly dislocated assisted diffusion, the base
doping in the bipolar transistor diffuses back into the collector forming a base profile
shown in figure 6.26.
(a) Calculate the transit time across the depletion region of the base-collector junction for
this transistor.
(b) Assuming that the collector remains fully depleted, and that the current density is
measured by the collector current divided by the collector area, how is the Kirk current
threshold for the transistor affected? Comment on your answer. Assumevs=10^7 cm/s
andμ= 1000cm^2 /V.sfor electrons. Apply both the impulse function and charge control
