On Integration-based Methods for MOSFET Model

On Integration-based Methods for MOSFET Model Parameter Extraction Adelmo Ortiz-Conde, Francisco J. García Sánchez, and Ramón Salazar Solid State Electronics Laboratory, Universidad Simón Bolívar, Apartado Postal 89000, Caracas 1080-A, Venezuela * Email: {ortizc, fgarcia, rsalazar }@ieee.org

2. Extraction methods using the linear region In order to critically assess and compare the different extraction methods based on the linear region, we will apply them all to extract the value of the threshold voltage from the measured transfer characteristics of an n-channel bulk MOSFET [3] with a 5µm mask channel width, a 0.18µm mask channel length, and a 3.2nm gate oxide thickness. 2.1 Integral method The first integration-based method for MOSFET was developed in 1997 [4] to extract the threshold voltage and it was generalized in 1998 [5] to extract the effective channel length. The following integral difference function was proposed to eliminate the effects of the parasitic drain and source series resistances in the extraction of the MOSFET parameters, [1,4]:

Rm

Vgb

0

0

∫ Vgb dRm −

∫ Rm dVgb

, (1)

where Rm = Vds/ID is the total measured resistance from drain to source, and Vgb = Vmax - Vg is a variable related to the gate-source voltage, Vg, and the maximum gate voltage under consideration is Vmax. After some algebraic manipulations, the following function may be obtained: D(V gb , R mV gb ) =

2V gb K

+

V gb K (V max − V gb − VT )

V gb ⎤ 2(V max − VT ) ⎡ + ln ⎢1 − ⎥ ( − ) K V V max T ⎦ ⎣

, (2)

where VT is the threshold voltage, and K is the parameter associated with the device make-up: W K= Co µ 0 , (3) Lm − ∆L Here, Co is the oxide capacitance per unit area, W is the channel width, and µ0 is the low-field mobility in the channel. The mobility model used here has the form:

µ eff =

µ0

, (4) 1 + θ (Vg − VT ) where µeff is the effective mobility in the channel, and the parameter θ accounts for mobility degradation. 1010 108

Experimental Results

106

K = 9.95 mA/V VT = 0.59 V

(V)

1. Introduction The main motivation for using integration-based parameter extraction methods [1-3] is primarily to eliminate parasitic series resistance effects, and secondly to alleviate the impact of experimental data noise on the extracted parameter values. This article will review and scrutinize the following existing integration-based methods for extracting threshold voltage in MOSFETs, biased in the linear region: (1) integral method [4,5]; (2) transition method [6]; and (3) Normalized mutual integral difference operator method [7,8]. We will also review the following two integration-based methods for extracting threshold voltage in the saturation region: (1) H function method [9], and (2) G1 function method [10]. The H function method, originally developed for amorphous MOSFETs, is applied successfully to short-channel single crystal MOSFETs. This paper will also review a method [11] to evaluate the location of a maximum value of a given function with high level of noise by using integration. Finally, we will briefly mention how distortion may be evaluated by the use of this type of integration methods [12-15].

D(Vgb , Rm ) =

104

D1

Abstract This article reviews integration-based modelparameter extraction methods for MOSFETs. It comprises three different methods that use the transfer characteristics measured under linear regime operation conditions. Additionally two other methods are included for extraction under saturation conditions. An integration-based method to evaluate the location of a maximum value of a given function is also included. Finally, the possibility of evaluating distortion is briefly introduced.

102

Extracted fit with: 2

100 Vd =10 mV

10-2

Integral method

10-4 0.0

0.5

1.0 Vgb (A)

1.5

Fig. 1. The Integral method implemented using the plot of function D1 versus Vgb of the test n-channel bulk MOSFET.

978-1-4244-2186-2/08/$25.00 ©2008 IEEE

0.6

0

.

ID

(5)

ID

Vg

0

0

∫ V g dI D − ∫ I D dV g . (6)

Vg I D

Transition method

0.5 VT = 0.49 V

0.4 0.2

VT = 0.45 V Vd =10 mV 0.5

1.0 Vg (V)

G1

0.3 0.2 0.1 Vd =10 mV 0.0 10-11 10-10 10-9 10-8 10-7 10-6 10-5 10-4 ID (A)

Fig. 2. Plot of function G1 versus ID for the transition method [3]. This method evaluates the threshold voltage from the maximum value of G1.

1.5

Fig. 3. Plot of function P versus Vg for the normalized mutual integral difference operator method. This method evaluates the threshold voltage from the value of Vg at the maximum value of P.

Using integration by parts we find that this new function P is essentially the normalized previous function G1: G (V , I ) . (7) P (VGS , I D ) = 1 GS D VGS According to this method, VT is the value of Vg at the maximum value of P, which is 0.49V in Fig. 3. It is important to point out that this method and the transition method yield slightly different results because of the two different assumptions. 3. Extraction methods from the saturation region 3.1 H function method Although the H function method was originally developed for non-crystalline MOSFETs [9], it may be also applied to short-channel crystalline MOSFETs. The extraction of VT in these devices introduces the following additional difficulties: First, the saturation drain current in strong inversion is usually modeled by an equation of the form [9,16] K , (8) I Dsat = V g − VT m 2 where K is a conductance parameter with units of AV-m and m an empirical parameter which can be different from 2, the value used in conventional MOSFET models. Second, the value of parameter m cannot be easily extracted from a simple plot of log(IDsat) versus log(Vg) because practical operation values of Vg are usually not large enough to validate the approximation: (Vg - VT) ≈ Vg . Third, it is not clear at what point the IDsat versus Vg plot could be linearly extrapolated, since the curve does not present an inflexion point because the mobility of these devices raises as Vg is increased. A method to extract the threshold voltage that circumvents some of these difficulties is based on the following function which can be numerically computed from the measured IDsat (Vg) characteristics:

(

0.4 (V)

0.6

0.0

∫ I D (V g )dV g

A plot of G1 versus ln(ID) should be a straight line below threshold where the current is dominated by diffusion and consequently it is predominantly exponential. As soon as Vg=VT, function G1 drops abruptly. This is observed in Fig. 2 for the test bulk device. The maximum value of G1, which is 0.49 V, corresponds to the threshold voltage of the device. 2.3 Normalized mutual integral difference operator method This method is based on a normalized version of the integral difference function, as follows[7,8]:

P (V g , I D ) =

0.8

Normalized mutual integral difference operator method

0.0

Vg

G1 (V g , I D ) = V g − 2

1.0

P

The procedure for extracting VT and K, is as follows: First, the value of D is calculated from (1) using the measured current-voltage data. Next, the value of K and VT are determined by fitting (2) to the previously calculated D as illustrated in Fig. 1. Finally, ∆L is extracted from (3) by the linear extrapolation of K-1 versus Lm , as is shown in [5]. Although this method produces accurate results, it is cumbersome to implement. 2.2 Transition method This method [6] uses the sub-threshold-to-strong inversion transition region of MOSFETs to extract the threshold voltage. It uses an auxiliary operator that involves integration of the drain current as a function of gate voltage. The following function G1 is numerically calculated from the measured data.

)

0.6

1.0

H function method 0.5

Experimental Results Extracted straight line with:

0.8

m = 1.14 VT = 0.59 V

( IDsat / IDsat-max )(1/m)

H (V)

0.4

Experimental results for various values of m

0.3 0.2 0.1

m=1 1.14 1.5 2

0.6 0.4 0.2

Vd = Vg

Vd = Vg

0.0

0.0 0.0

0.5

1.0 Vg ( V )

0.0

1.5

Fig. 4. Function H for the experimental data of the device described in section 2. The slope of the straight line for strong inversion implies m=1.14. The intercept of the straight line to the gate bias axis is VT =0.59 V. Vg

( )

H Vg =

∫ I Dsat (V g ) dV g 0

.

I Dsat

(9)

After substitution of (8) into (9), and assuming that the variation of K with respect to Vg is insignificant, and that the subthreshold contribution is negligible, we obtain:

( ) (Vmg −+V1T )

H Vg =

,

(10)

which means that H(Vg) behaves linearly in the strong inversion region. Therefore, a plot of function H versus Vg has a slope that defines the value of m and a Vg axis intercept which gives the sought after value of VT . After having found m and VT, the remaining parameter in (8), K , may be easily evaluated from (8). Figure 4 shows a plot of the numerical calculation of H(Vg) according to (10) for the device described in section 2. For strong inversion the curve is seen to behave approximately as a straight line with m=1.14 and VT=0.59 V. It is important to point out that m