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

Authors: Noureddine Maouhoub; Khalid Rais
Journal: Active and Passive Electronic Components
 (2011)

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## 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 characteristicId (Vg). 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.


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## Body

## 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,Id (Vg), 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 Id (Vg) [8, 9].In what follows we present a two procedures to be applied to the strong inversionId (Vg) 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,Id, at very small drain bias, can be expressed in terms of intrinsic voltages as(1)Id=WL⋅μeff⋅Cox⋅(VG-Vt)⋅VD,
where W is the channel width, L is the channel length, Cox is the oxide capacitance, μeff is the effective free-carrier mobility, VG is the intrinsic gate voltage, VD the intrinsic drain voltage, and Vt is the threshold voltage.If the source-and-drain series resistance is significant, the device’s intrinsic gate and drain voltages are(2)VG=Vg-Id⋅Rsd⁡2,VD=Vd-Id⋅Rsd⁡,
where Vg and Vd are the externally applied gate and drain voltages, respectively, and Rsd 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:(3)μeff=μ0⋅exp⁡(-θi⋅(VG-Vt)),
where(4)θi=η⋅β⋅Δ⋅Coxεsi.θi 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, εsi is the silicon permittivity, Δ is the surface roughness amplitude, and μ0 represents the low-field mobility.Assuming thatVg-Vt≫Id·Rsd/2, relation (1) becomes(5)Id=K⋅(Vg-Vt)exp⁡(θi⋅(Vg-Vt))+((K⋅Rsd⁡)/Vd)⋅(Vg-Vt),
where(6)K=WL⋅μ0⋅Cox⋅Vd.
Introducing the series resistance effect, the effective mobility becomes(7)μeff=1(1/(μ0⋅exp⁡[-θi⋅(Vg-Vt)]))+θRsd⁡(Vg-Vt),
where(8)θRsd⁡=K⋅Rsd⁡Vd.
The transconductance is given by(9)gm=dIddVg=K⋅(1-θi⋅(Vg-Vt))exp⁡(θi⋅(Vg-Vt))[exp⁡(θi⋅(Vg-Vt))+θRsd⁡⋅(Vg-Vt)]2.


## 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 θe in the classical model: (10)θe=θi+θRsd⁡.
The values of K and θe are determined, respectively, by the slope in the plot of Y0(Vg)=Id/gm and H0(Vg)=1/gm which are straight line at relative low field.For thin oxide and at high field, the characteristicY(Vg)=Id/gm, which has the advantage to be independent of series resistance, presents a deviation with regard to Y0(Vg).We note thatgm can reach the value zero at Vgm:(11)θi=1(Vgm-Vt).
Experimentally we use the device below this value, meanwhile we use the characteristic (Y(Vg)/Y0(Vg)-1) versus Vg. The value at ten per; cent of this characteristic is corresponding to the critical value (Vgm-Vt)/2=1/2θi; combining (11) with relations (3) and (10) we obtain the surface roughness amplitude Δ and series resistance Rsd, respectively.


## 3. Second Method

The procedure involves first extracting the expressionexp(θi(Vg-Vt)) from the drain current equation and using the function Y=Id/gm.Settingx=Vg-Vt, and after some mathematical manipulation we obtain the following expression:(12)Y-2⋅x1-θi⋅x=1Id-Rsd⁡Vd.
Secondly, we choose two points in the characteristic Id (Vg) in strong inversion Vg1 and Vg2 (x1 and x2), and we solve the following system:(13)Y-2(Vg1)⋅x11-θi⋅x1=1Id(Vg1)-Rsd⁡Vd,Y-2(Vg2)⋅x21-θi⋅x2=1Id(Vg2)-Rsd⁡Vd.
Determining the parameter θ is done using the subtraction of the two system equations; we obtain a second order equation whose appropriate solution is:(14)θi=(B⋅x1+C⋅x2+C⋅x1-A⋅x2)2⋅C⋅x1⋅x2+(B⋅x1+C⋅x2+C⋅x1-A⋅x2)2+4⋅x1⋅x2⋅(A-B-C)2⋅C⋅x1⋅x2,
where (15)A=Y-2(Vg1)⋅x1,B=Y-2(Vg2)⋅x2,C=1Id(Vg1)-1Id(Vg2).
The series resistance expression can be written using one of the two system equations:(16)Rsd⁡=Vd⋅(1Id(Vg2)-Y-2(Vg2)⋅x21-θi⋅x2).
The choice of Vg1 and Vg2 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 widthW=4 μm, gate oxide thickness tox=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:Vd=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.
Parameterθi (1/V)Rsd (Ω)Δ (nm)First method0.363107.530.31Second method0.36108.80.28Figure 1
Transfer characteristics (symbols). Also shown is the corresponding model (continuous line) obtained from the drain current model using the calculated parameters (Table1).In relatively weak field we use model (1) as an approximation of our model; we plot this function Y(Vg). The parameter Vt can be extracted from the Vg axis intercept of the observed straight lines of Y versus Vg plot (Figure 2).Figure 2
CharacteristicY versus gate voltage.Figure3 shows the characteristic (Y(Vg)/Y0(Vg)-1) versus Vg which allows determining the intrinsic attenuation parameter.Figure 3
CharacteristicY(Vg) to Y0(Vg) ratio versus gate voltage.


## 5. Optimization of Drain Current

AtVg=Vgm, the drain current is maximal, its value is(17)Idmax⁡=VdRchmin⁡+Rsd⁡,
where Rchmin is the minimal resistance of channel:
(18)Rchmin⁡=ΔβηLWμ0εsi⋅e.
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 plotId versus the oxide thickness tox for a given gate voltage Vg (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 forVg=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.



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*Source: 713129-2011-10-13.xml*