4
SERIES RESISTANCE, CHANNEL
LENGTH AND WIDTH, AND
THRESHOLD VOLTAGE
4.1
INTRODUCTION
Semiconductor device and circuit performance is generally degraded by series resistance
that depends on the series and shunt resistance, on the device, on the current flowing
through the device, and on a number of other parameters. The series resistance rs depends
on the semiconductor resistivity, on the contact resistance, and sometimes on geometrical factors. Series resistance may be very large before causing device degradation. For
example, in a reverse-biased photodiode with a photocurrent in the nano-amperes range,
series resistance is a minor consideration. However, series resistances of a few ohms are
detrimental for solar cells and power devices. The effect of rs on capacitance and carrier
concentration profiling measurements is discussed in Chapter 2. The aim of the device
designer should be a design in which series resistance is negligibly small for that device.
However, since rs cannot be zero, it is important to be able to measure it. The effective
channel length and width of a MOSFET are important device parameters because they
are required for modeling and they usually differ from the mask-defined and the physical
dimensions and the threshold voltage is one of the most important MOSFET parameters.
Methods to determine these are discussed.
4.2
4.2.1
PN JUNCTION DIODES
Current-Voltage
The current of a pn junction is often written as a function of the diode voltage Vd as
I = Io (eqV d /nkT − 1)
(4.1)
Semiconductor Material and Device Characterization, Third Edition, by Dieter K. Schroder
Copyright  2006 John Wiley & Sons, Inc.
185
186
SERIES RESISTANCE, CHANNEL LENGTH AND WIDTH, AND THRESHOLD VOLTAGE
rs
I
+
+
Vd
V
−
−
Fig. 4.1
Equivalent circuit of a diode.
where Io is the saturation current and n the diode ideality factor. The diode voltage Vd
is the voltage across the space-charge region and excludes any voltage drops across the
p and n quasi-neutral regions. If both Io and n are constant, then a plot of log(I ) versus
Vd yields a straight line for Vd > nkT /q.
A semiconductor diode can be represented by the equivalent circuit of Fig. 4.1, consisting of an ideal diode in series with resistance rs . When current flows through the
device, the diode terminal voltage V is
V = Vd + I R s
(4.2)
With series resistance Eq. (4.1) becomes
I = Io (eq(V −I rs )/nkT − 1)
(4.3)
The current in pn junction diodes is due to two components: space-charge region
(scr) recombination/generation and quasi-neutral region (qnr) recombination/generation,
leading to the I –V relationship
I = Io,scr (eq(V −I rs )/nkT − 1) + Io,qnr (eq(V −I rs )/nkT − 1)
(4.4)
Equation (4.4) is plotted in Fig. 4.2 for forward bias. There are four distinct regions in
the figure. For I rs ≪ V ≪ nkT /q, the current depends linearly on voltage (eqV /nkT − 1 ≈
qV /nkT ), giving a non-linear curve on the semilog plot. For V ≫ nkT /q, the current is
dominated by scr recombination at low current and by qnr recombination at higher current.
The breakpoint between the two current components occurs at V = 0.3 V in this example.
The I –V curve deviates from linearity at high current due to series resistance rs .
Extrapolating the two linear regions to V = 0 gives Io,scr and Io,qnr . The slope is
given by
d log I
(4.5)
m=
dV
Knowing the slope and sample temperature allows the ideality factor to be determined
from the relationship
q
q
=
(4.6)
n=
ln(10)mkT
2.3mkT
We will generally use the logarithm to base 10, written as “log”, instead of the logarithm
to base e, written as “ln”, because experimental data are usually plotted on “log”, not
“ln”, scales.
PN JUNCTION DIODES
187
100
Current (A)
10
∆V = Irs
−2
10−4
qnr
10−6
Slope m =
q/2.3nkT
10−8
scr
Io,scr
10−10
Io,qnr
10−12
Fig. 4.2
0
0.2
0.4
0.6
Voltage (V)
0.8
1
Current versus voltage for a diode with series resistance. Upper dashed line is for rs = 0.
The deviation of the log(I )–V curve from linearity at high currents is 1V = I rs ,
allowing rs to be determined according to
rs =
1V
I
(4.7)
Since the Schottky diode current-voltage behavior is similar to pn junctions, we will use
Fig. 3.38 for the rs extraction. Figure 4.3(a) gives that part of the I –V curve where rs is
negligible and n = 1.1 from the slope. Figure 4.3(b) shows the part of the rs -dominated
curve. The deviation from linearity, according to Eq. (4.7), gives rs = 0.8 Ä.
The resistance can also be obtained from the diode conductance gd = dI /dV . In the
region where rs is important, qnr recombination dominates and the current
I ≈ Io,qnr eq(V −Irs )/nkT
(4.8)
gives
gd =
qI (1 − rs gd )
nkT
(4.9)
We can write Eq. (4.9) as1
1
nkT
+ I rs
=
gd
q
(4.10)
suggesting a plot of I /gd versus I . Such a plot has an I = 0 intercept of nkT /q and slope
rs , as shown in Fig. 4.4(a).
Equation (4.9) can also be written as
q(1 − rs gd )
gd
=
I
nkT
(4.11)
Plotting gd /I versus gd , the gd = 0 intercept is q/nkT , the gd /I = 0 intercept is 1/rs
and the slope is qrs /nkT , as shown in Fig. 4.4(b). Careful measurements have revealed
the approach of Eq. (4.11) to give the most reliable results,2 although Fig. 4.4 shows the
scatter in (b) to be more severe than in (a) because both axes require a differentiation
of the data. Comparing Figs. 4.3 and 4.4 for rs extraction brings out an important point.
A slope method is generally more accurate than a single point method to determine an
188
SERIES RESISTANCE, CHANNEL LENGTH AND WIDTH, AND THRESHOLD VOLTAGE
10−2
Current (A)
10−3
Slope = 15.8
10−4
10−5
10−6
0
0.05
0.1
Voltage (V)
0.15
0.2
(a)
100
Current (A)
10−1
∆V = 0.12 V
I = 0.15 A
10−2
10−3
10−4
0.1
0.2
0.3
0.4
Voltage (V)
(b)
Fig. 4.3 Current versus voltage for the diode in Fig. 3.38. (a) low voltage where rs can be neglected,
(b) high-voltage where rs dominates.
unknown quantity. Since experimental data exhibit small errors, slope methods allow
smoothing of the data, whereas single point measurements incorporate any experimental
uncertainties in the parameter determination.
The diode conductance can be measured by superimposing a small ac voltage δV
on the dc voltage V and measuring the in-phase component δI with a lock-in amplifier
to obtain gd = δI /δV .3 Because of the exponential dependence of current on voltage,
δV should be kept as low as possible. Alternately, one can differentiate the I –V curve.
Again, because of the exponential nature of the curve, dc voltage steps should be less
than 1 mV. Using the semilog plot, where gd = Id [ln (I )]/dV , voltage steps as high as
10 mV are permissible.2
4.2.2
Open-Circuit Voltage Decay (OCVD)
Open-circuit voltage decay is a method to determine the minority carrier lifetime of pn
junctions as discussed in Chapter 7 and can also be used to determine the diode series
resistance, as illustrated in Fig. 4.5. The diode is forward biased. At t = 0 switch S is
opened, and the open-circuit diode voltage is monitored as a function of time. The lifetime
PN JUNCTION DIODES
189
0.15
I/gd (V)
0.1
Slope = rs
0.05
Intercept = nkT/q
0
0
0.05
0.1
Current (A)
(a)
0.15
50
Intercept = q/nkT
gd /I (V−1)
40
30
Intercept
= 1/rs
20
Slope = qrs/nkT
10
0
0
0.25
0.5
0.75
gd (S)
1
1.25
(b)
Fig. 4.4
(a) I /gd versus I, (b) gd /I versus gd for the device of Fig. 4.3.
Voc(t)
I
S
V
rs
Voc(0−)
∆V
Voc(0+)
Voc(t)
0
Fig. 4.5
t
Open-circuit voltage decay of a pn junction showing the voltage discontinuity at t = 0.
is determined from the slope of the Voc − t curve. The series resistance is obtained from
the voltage discontinuity 1V at t = 0.3
The voltage drop across the diode just before opening the switch Voc (0− ) consists of
the diode voltage Vd and the voltage drop across any device resistances
Voc (0− ) = Vd + I rs
(4.12)