Characterization of Series Resistance and Mobility Degradation Parameter and Optimizing Choice of Oxide Thickness in Thin Oxide N-Channel MOSFET

Abstract

We present two methods to extract the series resistance and the mobility degradation parameter in short-channel MOSFETs. The principle of the first method is based on the comparison between the exponential model and the classical model of effective mobility and for the second method is based on directly calculating the two parameters by solving a system of two equations obtained by using two different points in strong inversion at small drain bias from the characteristic 𝐼_𝑑 (𝑉_𝑔). The results obtained by these techniques have shown a better agreement with data measurements and allowed in the same time to determine the surface roughness amplitude and its influence on the maximum drain current and give the optimal oxide thickness.

1. Introduction

Accurate model parameter extraction is crucial for modeling modern MOSFET devices. Extensive work abounds in the literature dedicated to this subject. Free-carrier mobility degradation and source-and-drain series resistance are two parameters of special importance for MOSFET characterization that are particularly cumbersome to extract independently from each other. Both of these parameters produce similar effects on the device’s transfer characteristics, 𝐼_𝑑 (𝑉_𝑔), a fact that complicates their accurate extraction.

Several ingenious procedures have been proposed to circumvent this difficulty [1–5]. Another method was proposed to extract these parameters from the drain current versus gate voltage characteristics in the saturation region using several devices of different mask channel lengths [6]. An alternative procedure was recently proposed to extract the source-and-drain series resistance independently of mobility degradation by using bias conditions under which the channel carrier mobility is kept constant [7]. Another recent works presented direct fitting and direct calculating methods in strong inversion and by exploiting the characteristic 𝐼_𝑑 (𝑉_𝑔) [8, 9].

In what follows we present a two procedures to be applied to the strong inversion 𝐼_𝑑 (𝑉_𝑔) characteristics of a single transistor, measured at a small drain bias. It is based on the exploiting of exponential model of effective mobility. The procedure has been validated using data from a single experimental short channel device.

The drain current, 𝐼_𝑑, at very small drain bias, can be expressed in terms of intrinsic voltages as𝐼_𝑑=𝑊𝐿⋅𝜇_eff⋅𝐶_ox⋅𝑉_𝐺−𝑉_𝑡⋅𝑉_𝐷, (1) where 𝑊 is the channel width, 𝐿 is the channel length, 𝐶_ox is the oxide capacitance, 𝜇_eff is the effective free-carrier mobility, 𝑉_𝐺 is the intrinsic gate voltage, 𝑉_𝐷 the intrinsic drain voltage, and 𝑉_𝑡 is the threshold voltage.

If the source-and-drain series resistance is significant, the device’s intrinsic gate and drain voltages are𝑉_𝐺=𝑉_𝑔−𝐼_𝑑⋅𝑅_sd2,𝑉_𝐷=𝑉_𝑑−𝐼_𝑑⋅𝑅_sd, (2) where 𝑉_𝑔 and 𝑉_𝑑 are the externally applied gate and drain voltages, respectively, and 𝑅_sd is the total source-and-drain series resistance.

The model of variation mobility with effective field considers that the attenuation of effective mobility is particularly due to Surface Roughness Scattering [10]. This work generalized all the classical models [9, 11] and give a physical meaning to the different used parameters.

The effective mobility is given by the exponential model in strong inversion:𝜇_eff=𝜇₀⋅exp−𝜃_𝑖⋅𝑉_𝐺−𝑉_𝑡, (3) where𝜃_𝑖=𝜂⋅𝛽⋅Δ⋅𝐶_ox𝜀_si. (4)𝜃_𝑖 is the intrinsic attenuation coefficient of mobility, η is constant parameter equal to 0.5 for electrons and 0.33 for holes, 𝛽=𝑞/𝑘𝑇 is the inverse of the thermal potential, 𝜀_si is the silicon permittivity, Δ is the surface roughness amplitude, and 𝜇₀ represents the low-field mobility.

Assuming that 𝑉_𝑔−𝑉_𝑡≫𝐼_𝑑⋅𝑅_sd/2, relation (1) becomes𝐼_𝑑𝑉=𝐾⋅^𝑔−𝑉_𝑡exp𝜃_𝑖⋅𝑉_𝑔−𝑉_𝑡+𝐾⋅𝑅_sd/𝑉_𝑑⋅𝑉_𝑔−𝑉_𝑡, (5) where𝑊𝐾=𝐿⋅𝜇₀⋅𝐶_ox⋅𝑉_𝑑. (6) Introducing the series resistance effect, the effective mobility becomes𝜇_eff=11/𝜇₀⋅exp−𝜃_𝑖⋅𝑉_𝑔−𝑉_𝑡+𝜃_𝑅sd𝑉_𝑔−𝑉_𝑡, (7) where𝜃_𝑅sd=𝐾⋅𝑅_sd𝑉_𝑑. (8) The transconductance is given by𝑔_𝑚=𝑑𝐼_𝑑𝑑𝑉_𝑔=𝐾⋅1−𝜃_𝑖⋅𝑉_𝑔−𝑉_𝑡exp𝜃_𝑖⋅𝑉_𝑔−𝑉_𝑡exp𝜃_𝑖⋅𝑉_𝑔−𝑉_𝑡+𝜃_𝑅sd⋅𝑉_𝑔−𝑉_𝑡². (9)

2. First Method

According to the classical model [10], and for low fields, the comparison of the first-order development of the exponential term in relation (3) gives the relation between the intrinsic coefficient attenuation mobility in exponential model and its value extrinsic 𝜃_𝑒 in the classical model: 𝜃_𝑒=𝜃_𝑖+𝜃_𝑅sd. (10) The values of 𝐾 and 𝜃_𝑒 are determined, respectively, by the slope in the plot of 𝑌₀(𝑉_𝑔)=𝐼_𝑑/√𝑔_𝑚 and 𝐻₀(𝑉_𝑔)=1/√𝑔_𝑚 which are straight line at relative low field.

For thin oxide and at high field, the characteristic 𝑌(𝑉_𝑔)=𝐼_𝑑/√𝑔_𝑚, which has the advantage to be independent of series resistance, presents a deviation with regard to 𝑌₀(𝑉_𝑔).

We note that 𝑔_𝑚 can reach the value zero at 𝑉_𝑔𝑚:𝜃_𝑖=1𝑉_𝑔𝑚−𝑉_𝑡. (11) Experimentally we use the device below this value, meanwhile we use the characteristic (𝑌(𝑉_𝑔)/𝑌₀(𝑉_𝑔)−1) versus 𝑉_𝑔. The value at ten per; cent of this characteristic is corresponding to the critical value (𝑉_𝑔𝑚−𝑉_𝑡)/2=1/2𝜃_𝑖; combining (11) with relations (3) and (10) we obtain the surface roughness amplitude Δ and series resistance 𝑅_sd, respectively.

3. Second Method

The procedure involves first extracting the expression exp(𝜃_𝑖(𝑉_𝑔−𝑉_𝑡)) from the drain current equation and using the function 𝑌=𝐼_𝑑/√𝑔_𝑚.

Setting 𝑥=𝑉_𝑔−𝑉_𝑡, and after some mathematical manipulation we obtain the following expression:𝑌⁻²⋅𝑥1−𝜃_𝑖=1⋅𝑥𝐼_𝑑−𝑅_sd𝑉_𝑑. (12) Secondly, we choose two points in the characteristic 𝐼_𝑑 (𝑉_𝑔) in strong inversion 𝑉_𝑔1 and 𝑉_𝑔2 (𝑥₁ and 𝑥₂), and we solve the following system:𝑌⁻²𝑉_𝑔1⋅𝑥₁1−𝜃_𝑖⋅𝑥₁=1𝐼_𝑑𝑉_𝑔1−𝑅_sd𝑉_𝑑,𝑌⁻²𝑉_𝑔2⋅𝑥₂1−𝜃_𝑖⋅𝑥₂=1𝐼_𝑑𝑉_𝑔2−𝑅_sd𝑉_𝑑. (13) Determining the parameter 𝜃 is done using the subtraction of the two system equations; we obtain a second order equation whose appropriate solution is:𝜃_𝑖=𝐵⋅𝑥₁+𝐶⋅𝑥₂+𝐶⋅𝑥₁−𝐴⋅𝑥₂2⋅𝐶⋅𝑥₁⋅𝑥₂+𝐵⋅𝑥₁+𝐶⋅𝑥₂+𝐶⋅𝑥₁−𝐴⋅𝑥₂²+4⋅𝑥₁⋅𝑥₂⋅(𝐴−𝐵−𝐶)2⋅𝐶⋅𝑥₁⋅𝑥₂, (14) where 𝐴=𝑌⁻²𝑉_𝑔1⋅𝑥₁,𝐵=𝑌⁻²𝑉_𝑔2⋅𝑥₂,1𝐶=𝐼_𝑑𝑉_𝑔1−1𝐼_𝑑𝑉_𝑔2. (15) The series resistance expression can be written using one of the two system equations:𝑅_sd=𝑉_𝑑⋅1𝐼_𝑑𝑉_𝑔2−𝑌⁻²𝑉_𝑔2⋅𝑥₂1−𝜃_𝑖⋅𝑥₂. (16) The choice of 𝑉_𝑔1 and 𝑉_𝑔2 is determined in mobility degradation effect zone.

4. Results and Discussion

In order to validate the previous procedures, the device used in this work has the parameters: channel width 𝑊=4 μm, gate oxide thickness 𝑡_ox=5 nm, channel length 𝐿=0.1 μm, and channel doping Na=10¹⁶ cm⁻³.

The transfer characteristics were measured at a small drain voltage: 𝑉_𝑑=20 mV. The application of our model with calculated parameters (Table 1) shows a good agreement with experimental data (Figure 1).

Table 1 

Values of the different extracted parameters.

Figure 1 

Transfer characteristics (symbols). Also shown is the corresponding model (continuous line) obtained from the drain current model using the calculated parameters (Table 1).

In relatively weak field we use model (1) as an approximation of our model; we plot this function 𝑌(𝑉_𝑔). The parameter 𝑉_𝑡 can be extracted from the 𝑉_𝑔 axis intercept of the observed straight lines of 𝑌 versus 𝑉_𝑔 plot (Figure 2).

Figure 2 

Characteristic 𝑌 versus gate voltage.

Figure 3 shows the characteristic (𝑌(𝑉_𝑔)/𝑌₀(𝑉_𝑔)−1) versus 𝑉_𝑔 which allows determining the intrinsic attenuation parameter.

Figure 3 

Characteristic 𝑌(𝑉𝑔) to 𝑌0(𝑉𝑔) ratio versus gate voltage.

5. Optimization of Drain Current

At 𝑉_𝑔=𝑉_𝑔𝑚, the drain current is maximal, its value is𝐼_𝑑max=𝑉_𝑑𝑅_chmin+𝑅_sd, (17) where 𝑅_chmin is the minimal resistance of channel: 𝑅_chmin=Δ𝛽𝜂𝐿𝑊𝜇₀𝜀_si⋅𝑒. (18) We note its independence of the oxide thickness and its proportionality of the surface roughness amplitude and of the channel length.

On the other hand, if we plot 𝐼_𝑑 versus the oxide thickness 𝑡_ox for a given gate voltage 𝑉_𝑔 (Figure 4), we can find the optimal oxide thickness. In fact, for very thin oxide the collisions on surface roughness predominate, and for thick oxide the inversion charge decreases, and consequently the drain current.

Figure 4 

Drain current versus oxide thickness for 𝑉𝑔=1.5 V.

6. Conclusion

We have presented two procedures that permit us to extract the mobility degradation factor, the series resistance, and the surface roughness thickness of MOSFET using the exponential model of mobility. The procedures are simple and accurate and based on calculating the model parameters using the experimental data measured in strong inversion at small drain bias. The exponential model of mobility allows also us to optimize the oxide thickness for the transistor.