small–signal source–drain conductance gdsm by using –parameters measurements is described first. Then, both the MOSFET model adopted by this work, and
igure 6–1 shows the small–signal equivalent circuit of a MOSFET [107], where Rg, Rs and Rd
onnection parasitics. Besides, the parasitics of th
capacitance are used to model the intrinsic device. As mentioned in [104] and [105], the small–signal source–drain conductance determined from the real part of the (without pads and
wher o the Z–parameters after de–em edding the influences of pad parasitics and interconnections.
In this section, the extraction of S
the parameters extraction method will be presented.
A. Extraction of gdsm from S–parameters measurements F
are series resistances; Lg, Ls and Ld represent the interc
e pads are modeled by the capacitors Cpg and Cpd in series with the resistors Rpg and Rpd, respectively. With the small–signal equivalent circuit shown in Fig. 6–1, gdsm, as a function of gate bias, can be derived directly from the Z–parameter data. In this work, the Z–parameters of test MOSFETs are obtained from S–parameters measurements, after de–embedding the influence of pad parasitics and interconnections, performed at various gate voltages while keeping the drain bias to be zero. Fig. 6–2 shows the equivalent circuit, which is valid for a gate voltage above pinchoff and at zero drain voltage [104–106], where a distributed channel resistance R and a distributed gate ch'
'
d
interconnections parasitics) at low frequency is regarded as the measured data g Cg
Z22
dsm and expressed as:
(6–1)
[ ]
sd gds
1 1
frequency low dsm d
R Z R
g = = + +
Re 22 1
e the superscript d refers t b
B. Descriptions Of MOSFET Model and Extraction Method
Figure 6–3 shows the schematic circuit model of a MOS transistor, where g, d and s denote the DC drain current and the small–signal sour
(6–2)
(6–3)
x is the gate apacitance per unit area, µeff is the effective inversion layer mobility, βeff is the gain factor, RT is the
chan
(6–4)
conductance between source and drain terminals consists of the source/drain series resistances and the
external nodes, while d’ and s’ denote the internal nodes. The
ce–drain conductance gdsm (including source/drain series resistance) of a MOS transistor in the linear region can be expressed as follows [89]:
( ) ( )
where Weff and Leff are the effective channel width and length, respectively, Co
c
sum of source and drain series resistance, VT is the device threshold voltage, Vds and Vgs are external drain–source voltage and gate–source voltage, respectively (Vds = Vd’s’ + RTIds, Vgs = Vgs’ + RTIds/2).
In the following extraction procedure, in order to avoid the drain–bias effects mentioned in Section 6–1, zero drain bias is set in the S–parameters measurement. This makes the intrinsic
nel conductance gds expressed as:
In actual small–signal conductance gdsm measurement, the inverse of the measured
inverse of intrinsic channel conductance. Hence, the measured small–signal source–drain
(
gs T)
eff(
gs T)
.conductance gdsm can be expressed as
(6–5)
er ob xpressed as follows [102], [103]:
(6–6)
wher ters [102], [103].
Conventional extraction methods were developed in the past for the case θ2 = 0 [94]. The
(6–7)
where
irst and second rder derivatives, which can be expressed as:
In SPICE–based model, the gate–voltage dependent curve of inversion lay m ility µeff is
( )
1 . 1 1
1
T gs eff T ds
T R V V
R g
+ −
=
+
β
( ) ( )
gdsm =
e
.
1 1 2 2
0
T gs T
gs V V V
V − + −
= +
θ
eff
θ µ µ
e µ0 is the low field mobility, θ1 and θ2 are the mobility degradation parame
approximation θ2 = 0 implies that the current is always an increasing function of gate voltage.
However, in a situation that for a MOSFET with thin oxide thickness, this assumption is incorrect, and the current can decrease at high gate voltages resulting in a negative transconductance [107], [108]. The reason for this effect is attributed to the strong dependence of the carrier mobility on the oxide surface roughness scattering. This mechanism is taken into account by the quadratic mobility dependence of gate voltage via the coefficient θ2.
By introducing equation (6–6) into equation (6–5), we have the following expression:
( )
( ) (
0 gs) (
T)
2 .dsm
V g V
1+ 1 +RT 0 Vgs −VT + 2 Vgs −VT
= −
θ β
θ
β
0
.
0
eff ox effC W
L
β = µ
Then, inverting equation (6–7) and differentiating once and twice results in the f o
(6–8)
echnology, the relat From equation (6–9), the function F2 (Vgs) can be defined as follows:
From equation (6–10), the fact that the function F2 (Vgs) is a linear function of which the th
meter β0. For a given device, by the use of a simple straight–line fit to the numerically derived experimental quantity (β0/2)1/3(Vgs–VT) in the plot of function F2 (Vgs) versus Vgs, the threshold voltage VT and parameter β0 can be obtained from the x–axis intercept and slope, respectively.
According to equation (6–8), the threshold voltage VT and parameter β0 determined above serves to generate a plot of the function F1 (Vgs) versus 1/(Vgs–VT)2 which can be used to extra
ility degradation parameter θ2. The parameter θ2 can be determined from the y–axis intercept of a best curve–fit of experimental data F1 (Vgs).
After having threshold voltage VT, parameters β0 and θ2, the two remaining parameters RT and θ1 can be determined by optimization to fit equ
By performing the above extraction procedure for devices with several different geometries (i.e., drawn gate width Wdrawn and drawn gate length Ldrawn) fabricated by the same t
ionship between parameters β0 and Wdrawn, and the relationship between parameters 1/β0 and Ldrawn can be also obtained [109]. With the standard relationship between drawn gate length and effective gate length: Leff = (Ldrawn–∆L), and the one between drawn gate width and effective gate width: Weff = (Wdrawn–∆W) [90], where ∆L and ∆W are taken as constant in the same technology, the
parameter β0 can be expressed as:
(6–11)
0 v sus
e used to determine the effective values of W and L by linear extrapolation, respectively. From the
d Discussion
plied to determine the parameters of the test devices sing S–parameters measurement. S–parameters are measured in the common source–substrate conf
ce–drain conductance gdsm is determined from the real part of p
( )
( )
00
µ
β
⋅ ⋅∆
−
∆
= − ox
drawn
drawn C
L L
W W
From equation (6–11), the plots of parameter β0 versus Wdrawn and parameter 1/β er Ldrawn can b
plot of parameter β0 versus Wdrawn, the parameters ∆W can be obtained (with known value of Cox) from the x–axis intercept of the best–fit linear line for the experimental data. With the determined parameter ∆W, the parameters ∆L and µ0 can be obtained (with known value of Cox) from the x–axis intercept and slope of the best–fit linear line for the experimental data, respectively, from the plot of parameter 1/β0 versus Ldrawn.