# New Concept of Differential Effective Mobility in MOS Transistors

Authors: K. Bennamane; G. Ghibaudo
Journal: Active and Passive Electronic Components
 (2019)

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## Abstract

A new concept of differential effective mobility is proposed. It characterizes the effective mobility of an increment of drain current resulting from a small increase of inversion charge in MOSFET channel. It allows us to show that the effective mobility can be described by a local electric field approach and not entirely by an effective electric field model.


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## Body

## 1. Introduction

The effective mobilityμ
e
f
f is one of the most important device parameters characterizing the transport in MOS transistors. The effective mobility in a MOSFET is intimately related to the average mobility of the carriers forming the inversion channel. From an experimental point of view, the effective mobility can be obtained by normalizing the drain current I
d in linear regime by the inversion charge Q
i as(1)
μ
e
f
f
=
L
W
.
I
d
Q
i
.
V
d
.where V
d is the drain voltage, L is the gate length, and W is the gate width. In general, the inversion charge is obtained by integration of the gate-to-channel capacitance C
g
c
(
V
g
) in the so-called split C-V technique [1, 2].In this work, we propose a new concept for the mobility, namely, the differential effective mobility, which characterizes the effective mobility of an increment of drain current resulting from a small increase of inversion charge.


## 2. Differential Mobility Concept

For a given DC bias, if the gate voltage ofδ
V
g is increased, the drain current of δ
I
d will accordingly augment and the inversion charge of δ
Q
i. So, in analogy to (1), a differential effective mobility associated with the mobility of the small amount of carriers induced in the inversion layer by the gate voltage increase can be defined by(2)
μ
d
i
f
f
=
L
W
.
δ
I
d
δ
Q
i
.
V
d
.Therefore, (2) can be expressed in terms of transconductance, g
m
=
δ
I
d
/
δ
V
g, and gate-to-channel capacitance, C
g
c
=
δ
Q
i
/
δ
V
g, as(3)
μ
d
i
f
f
=
L
W
.
g
m
C
g
c
.
V
d
.It should be noted thatμ
d
i
f
f can be evaluated not only for the normal (or front) gate voltage but also for the back gate voltage V
b, i.e., the body bias for a bulk device, the substrate voltage for FD-SOI transistors, or the back gate voltage for double gate MOSFETs. In this case, in (3), g
m should be replaced by the body (or back gate) transconductance, g
b
=
δ
I
d
/
δ
V
b, and, C
g
c by the body (or back gate)-to-channel capacitance, C
b
c
=
δ
Q
i
/
δ
V
b.In all the cases, given the definitions ofg
m and C
g
c (or C
b
c), it is easy to show that the differential effective mobility μ
d
i
f
f and the effective mobility μ
e
f
f are related to each other as(4)
μ
d
i
f
f
=
μ
e
f
f
+
Q
i
.
δ
μ
e
f
f
δ
Q
i
.As will be shown below, it is interesting to discuss the notion of differential mobility in relation to the centroid of the inversion charge. Two charge centroids can similarly be defined [2]: (i) the DC centroid, X
d
c, associated with the total inversion charge Q
i, and (ii) the AC centroid, X
a
c, related to the incremental inversion charge δ
Q
i. In the case of a front gate modulation, X
a
c can be obtained from the capacitance as [3, 4](5)
X
a
c
=
ε
s
i
1
C
g
c
-
1
C
o
x
.where C
o
x is the front gate oxide capacitance and ε
s
i the silicon permittivity.X
a
c and X
d
c are related by the following differential equation [3]:(6)
X
a
c
=
X
d
c
+
Q
i
.
δ
X
d
c
δ
Q
i
.It can be shown by integration of (6) that X
d
c can be calculated from X
a
c as(7)
X
d
c
=
1
Q
i
-
Q
i
t
h
∫
Q
i
t
h
Q
i
X
a
c
u
d
uwhere Q
i
t
h is a specific value of the inversion charge near threshold. One can show from simulation that X
d
c and X
a
c merge at threshold where (7) tends to the limit X
d
c
=
X
a
c
(
Q
i
t
h
).


## 3. Results and Discussion

C
g
c
(
V
g
) and I
d
(
V
g
) measurements have been performed on FD-SOI and bulk devices. Here the μ
d
i
f
f concept is illustrated with data taken on FD-SOI p type transistors, but similar results have been obtained on n and p type bulk structures. The FD-SOI devices feature a 2.2 nm gate oxide, a 145 nm bottom oxide, and an undoped silicon channel of thickness t
s
i
=
10 nm.Figure1 shows typical I
d
(
V
g
) and C
g
c
(
V
g
) characteristics for two substrate biases V
b. These curves have been used to calculate the corresponding g
m
(
V
g
), g
b
(
V
g
), and Q
i
(
V
g
) characteristics. The effective mobility and differential effective mobility have then been evaluated using (1) and (3). Their variations with inversion charge are shown in Figure 2(a), where μ
d
i
f
f and μ
d
i
f
f
b refer to the font gate and back gate differential mobilities, respectively. As is usual μ
e
f
f is found to be significantly attenuated at high inversion, mainly due to surface roughness (SR) scattering. Note that μ
d
i
f
f is degrading faster than μ
e
f
f with Q
i, whereas μ
d
i
f
f
b is slightly decreasing before reaching a plateau of higher value.Figure 1
TypicalI
d
(
V
g
) (a) and C
g
c
(
V
g
) (b) characteristics obtained on p type FD-SOI MOSFETs for two substrate voltages V
b (W = 10 μm, L = 10 μm).
(a)
(b)Figure 2
Variations ofμ
e
f
f, μ
d
i
f
f, and μ
d
i
f
f
b with inversion charge Q
i as obtained from experiment (a) and from simulation in the local (b) or global (c) approaches (μ
0
=
220 cm2/Vs, E
c
=
3
×
10
5 V/cm).
(a)
(b)
(c)In order to better interpret these mobility data, we have extracted using (5)-(7) the variations with Q
i of the normalized centroids (
X
/
t
s
i
) of the total inversion charge, X
d
c, and of incremental inversion charges for front gate and back gate modulation, X
a
c and X
a
c
b (see Figure 3(a)). As expected, X
d
c and X
a
c are getting closer to the front channel interface (zero on y-axis of Figure 3) as the transistor is pushed into stronger inversion [3, 4]. In contrast, the centroid of the incremental inversion charge induced by the back gate modulation, X
a
c
b, is almost constant with Q
i and remains around the middle of the silicon film (≈0.5 t
s
i). This allows us now to understand why μ
d
i
f
f
b was found nearly constant with Q
i and with a higher value. Indeed, μ
d
i
f
f
b refers to the effective mobility of carriers residing nearly in the middle of the film. In contrast, X
a
c corresponds to carriers with a decreasing mobility as they are approaching the front interface, subjected to enhanced SR scattering.Figure 3
Variations ofX
d
c, X
a
c, and X
a
c
b with inversion charge Q
i as obtained from experiment (a) and simulation (b).
(a)
(b)Semiclassical TCAD simulations have been performed in such FD-SOI structures by considering two mobility approaches, i.e., either a localμ
e
f
f model or a global one. In the local approach, μ
e
f
f is a spatial function of the local electric field E
y like μ
e
f
f
=
μ
0
/
(
1
+
E
y
/
E
c
) [5] and E
c is a critical field. In the global approach, μ
e
f
f is calculated for the whole channel, using the effective electric field E
e
f
f
=
∫
0
t
s
i
E
y
(
y
)
n
(
y
)
d
y
/
∫
0
t
s
i
n
(
y
)
d
y (n being the carrier density) [6, 7], as μ
e
f
f
=
μ
0
/
(
1
+
E
e
f
f
/
E
c
). The simulation results shown in Figures 2(b) and 2(c) clearly indicate that only the local mobility model provides an overall good description of the experimental mobility data (Figure 2(a)). Indeed, in the global approach, μ
d
i
f
f
b is strongly degraded at strong inversion due to the E
e
f
f increase with V
g, whereas, in the local model, μ
d
i
f
f
b is almost constant, as in the experiment, since E
y cancels around midchannel. Note also from Figure 3 that the simulated variations of the centroids with Q
i well agree with the experimental ones, which emphasizes the analysis consistency. Finally, in Figure 4, in order to get a better physical insight, we have plotted the variations of the various mobilities μ
e
f
f, μ
d
i
f
f, and μ
d
i
f
f
b as a function of their associated centroids X
d
c, X
a
c, and X
a
c
b. Several features are worth noticing from these plots: (i) μ
e
f
f
(
X
d
c
) and μ
d
i
f
f(X
a
c) nearly fall on the same graph, (ii) μ
d
i
f
f
b(X
a
c
b) prolongates the μ
d
i
f
f(X
a
c) trends to smaller centroid values, and (iii) regarding simulation results, only the local mobility model provides again the good trend.Figure 4
Variations ofμ
e
f
f, μ
d
i
f
f, and μ
d
i
f
f
b with associated centroid X
d
c, X
a
c, and X
a
c
b as obtained from experiment (a) and from local (b) and global (c) approaches.
(a)
(b)
(c)


## 4. Conclusions

The concept of differential effective mobility has been demonstrated for the first time. It allowed us to show that the effective mobility can be described by a local electric field approach rather than an effective electric field one. This means that one cannot fully model the carrier mobility in MOSFET without a local model, especially in TCAD simulation. However, the existence of aμ
e
f
f(E
e
f
f) universal curve and its use for experimental data analysis might remain more simple and appropriate for transport near the interface in single gate operation mode.



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*Source: 5716230-2019-03-05.xml*