New Concept of Differential Effective Mobility in MOS Transistors

The effective mobility μeff 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 Id in linear regime by the inversion charge Qi as μeff = L W. Id Qi.Vd . (1)


Introduction
The effective mobility  eff 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 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 gc (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.

Differential Mobility Concept
For a given DC bias, if the gate voltage of V g is increased, the drain current of I d will accordingly augment and the inversion charge of 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 Therefore, (2) can be expressed in terms of transconductance, g m = I d /V g , and gate-to-channel capacitance, C gc = Q i /V g , as It should be noted that  diff 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 = I d /V b , and, C gc by the body (or back gate)-to-channel capacitance, C bc = Q i /V b .
In all the cases, given the definitions of g m and C gc (or C bc ), it is easy to show that the differential effective mobility  diff and the effective mobility  eff are related to each other as 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 dc , associated with the total inversion charge Q i , and (ii) the AC centroid, X ac , related to the incremental inversion charge Q i .In the case of a front gate modulation, X ac can be obtained from the capacitance as [3,4] where C ox is the front gate oxide capacitance and  si the silicon permittivity.X ac and X dc are related by the following differential equation [3]: It can be shown by integration of (6) that X dc can be calculated from X ac as where Q ith is a specific value of the inversion charge near threshold.One can show from simulation that X dc and X ac merge at threshold where (7) tends to the limit X dc = X ac (Q ith ).

Results and Discussion
C gc (V g ) and I d (V g ) measurements have been performed on FD-SOI and bulk devices.Here the  diff 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 si = 10 nm.
Figure 1 shows typical I d (V g ) and C gc (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  diff and  diffb refer to the font gate and back gate differential mobilities, respectively.As is usual  eff is found to be significantly attenuated at high inversion, mainly due to surface roughness (SR) scattering.Note that  diff is degrading faster than  eff with Q i , whereas  diffb is slightly decreasing before reaching a plateau of higher value.
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 si ) of the total inversion charge, X dc , and of incremental inversion charges for front gate and back gate modulation, X ac and X acb (see Figure 3(a)).As expected, X dc and X ac 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 acb , is almost constant with Q i and remains around the middle of the silicon film (≈0.5 t si ).This allows us now to understand why  diffb was found nearly constant with Q i and with a higher value.Indeed,  diffb refers to the effective mobility of carriers residing nearly in the middle of the film.In contrast, X ac corresponds to carriers with a decreasing mobility as they are approaching the front interface, subjected to enhanced SR scattering.
Semiclassical TCAD simulations have been performed in such FD-SOI structures by considering two mobility approaches, i.e., either a local  eff model or a global one.In the local approach,  eff is a spatial function of the local electric field E y like  eff =  0 /(1 + E y /E c ) [5] and E c is a critical field.In the global approach,  eff is calculated for the whole channel, using the effective electric field   = ∫   0   () () / ∫   0 ()  (n being the carrier density) [6,7], as  eff =  0 /(1 + E eff /E c ).The simulation results shown in Figures 2(b  local mobility model provides an overall good description of the experimental mobility data (Figure 2(a)).Indeed, in the global approach,  diffb is strongly degraded at strong inversion due to the E eff increase with V g , whereas, in the local model,  diffb 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

Figure 1 :
Figure 1: Typical I d (V g ) (a) and C gc (V g ) (b) characteristics obtained on p type FD-SOI MOSFETs for two substrate voltages V b (W = 10 m, L = 10 m).
) and 2(c) clearly indicate that only the Q i (10 12 q/cm 2

Figure 2 :Figure 3 :
Figure 2: Variations of  eff ,  diff , and  diffb with inversion charge Q i as obtained from experiment (a) and from simulation in the local (b) or global (c) approaches ( 0 = 220 cm 2 /Vs, E c = 3 × 10 5 V/cm).