EE 130/230M
Spring 2013
UNIVERSITY OF CALIFORNIA
College of Engineering
Department of Electrical Engineering and Computer Sciences
Prof. Liu & Dr. Xu
Homework Assignment #14 Solution
Problem 1: BJT small-signal model
Using the simplified hybrid-pi circuit model:
Assume the small signal emitter current is iE
KCL : iE   g m vBE 
vBE
dv
 C BE
r
dt
Re call : g m 
qI C

, r 
kT
gm
Substituting :
dvBE  g m
1


dt
 C r C

iE
0
 vBE 
C



i r
Boundary conditions : vBE ( t  0)  vBE 0 , vBE (t  )    E 
 1  g m r

 (at steady state)


i r 
i r
Solving : vBE   vBE 0  E   e  t / E  E 
1  g m r 
1  g m r

1
rC
where, E 
  
gm
1
g m r  1

C r C
kT
 C
kT
kT
kT
qI C
Re writing :  E 
 C
  C j ,BE  Cd ,BE 
 (C j ,BE  g m F )
 1
qI C
qI C
qI C
E F 
kT C j ,BE
q IC
Problem 2: Base transit time and BJT transient response
Given an ideal Si npn BJT maintained at 300K with uniform base doping NB = 1017 cm-3, quasineutral region width WB = 0.25m, minority-carrier lifetime in the base B = 10 ns, operating at
the edge of saturation (VBC = 0V), at a collector current IC of 1 mA:
(a) The base transit time is
W2
WB 2
(0.25  104 cm)2
t  B 

 15 ps
2 DB 2 kT 
2  0.026  800
B
q
(b) The excess minority-carrier charge stored in the base
QB = ICt t =1´10-3 ´15´10-12 =15´10-15 C
where the value for the base transit time, τt, is determined in part (a). If VBC is increased in
magnitude (keeping in mind that the sign of VBC is negative when the npn BJT is biased in
the forward-active mode, with VBE held constant), QB will decrease and IC will increase. This
is a consequence of the base width modulation effect: the width of the quasi-neutral base
region decreases as the B-C depletion region width increases with increasing reverse bias on
the B-C junction.
(c) If the BJT is operating in active mode and the base current is suddenly doubled at time t = 0:
Given that I C  1mA and  B  0.1 s :
We know,  
0.1 s
 6666.67
t 
15 ps
1mA
 0.15 A
 6666.67
If I B is increased to 2 I Bo  0.3 A
 I Bo 
IC
B

From the charge control equation (for t  0) :
dQB
Q
 2 I Bo  B
dt
B
The general solution is QB (t )  2 I Bo B  Ae
 t
B
Initial condition : QB (t  0 )  I Bo B  A  0
 A   I Bo B
 QB (t )  I Bo B (2  e
 t
B
)
Finally ,sin ce :
 t
Q (t ) I 
iC (t )  B  Bo B (2  e  B )
t
 iC (t )  1(2  e
t
t
0.1 s
)mA
(d) For a fixed collector current iC ( VCC/RL in a standard circuit where the BJT is used as a
digital switch – ref. Lecture 27 Slide 9) in the ON state, a BJT will switch off more
quickly from active mode operation because there is less stored excess minority-carrier
charge in the quasi-neutral base region. Note: When a BJT is operating in saturation mode,
iC < iB so iB > iC/ hence QB is larger than in active mode.
Problem 3: Advanced MOSFET structures
(a) In a thin-body MOSFET, OFF-state leakage current between the source and drain regions is
significantly reduced due to the elimination of sub-surface leakage paths. The source/drain
junction depth is also effectively reduced, so that the source/drain regions support a
smaller fraction of the depletion charge underneath the gate electrode and hence the
short-channel effect is reduced. This facilitates scaling to shorter gate length.
Due to reduced pn junction area, there is less capacitive coupling between the drain and the
channel region; therefore, the capacitive coupling between the gate and the channel region is
more dominant, so the sub-threshold swing (S) is improved. For a given OFF-state leakage
current (maximum) specification, then, the threshold voltage (VT) can be lower and hence
the supply voltage (VDD) required to achieve the (minimum) ON-state drive current
specification can be lower.
(b) In a thin-body MOSFET structure (TSi < 10 nm), there is no need to use a high level of body
doping to suppress the short-channel effect and drain-induced barrier lowering (DIBL);
therefore, the thin Si body region is usually lightly doped (below 1018 cm-3). Since the
maximum depletion width (WT) is > 30 nm for a body dopant concentration = 1018 cm-3, TSi <
WT so that the body is fully depleted.
(c) The primary advantage of a multiple-gate (double-gate or tri-gate) thin-body MOSFET
structure over a single-gate thin-body MOSFET structure is that the body thinness
requirement is relaxed, i.e. the body does not need to be as thin and hence undesirable
quantum-confinement effects – which make VT very sensitive to variations in TSi – can be
avoided for very short gate length (LG < 20 nm). Other benefits include enhanced field-effect
mobility (due to lower average transverse electric field strength in the inversion layers) for
higher ON-state drive current, and possibly better layout area efficiency (i.e. current per unit
area on the surface of a Si wafer) if the Si fin of a double-gate FinFET is very tall, for more
compact circuit implementation.