A New Approach for the Extraction of Threshold Voltage for MOSFET’s

Jen Shuang Wong, Jian Guo Ma, Kiat Seng Yeo, Manh Anh Do

School of Electrical and Electronic Engineering,Nanyang Technological University, Nanyang Avenue,

Singapore 639798, jswong@pmail.ntu.edu.sg

ABSTRACT

A new approach for the extraction of threshold voltage(Vth) is proposed, namely, the “Third Derivative of Drain-Source Current” method or simply “TD” method. Thismethod extracts the Vth by finding the Vgs where the thirdderivative of Ids is maximal. It meets the threshold conditionrequirement, which is the onset of the inversion channelcreation. The method has been tested on transistors down to0.25µm channel length and is found to be fast, accurate andsimple to implement using standard measurementequipment.

Keywords: Semiconductor, MOSFET’s, Threshold Voltage,Extraction Method

1 INTRODUCTION

The voltage between the gate and source, Vgs, of a MOSdevice determines the concentration of the carriers in thechannel; the gate voltage for which the channel currentbecomes significant is called the threshold voltage, Vth.Since the threshold voltage defines the start of MOStransistor’s operation, an accurate extraction of thisparameter is crucial.

There are a number of threshold voltage extractionmethods, however, the extracted values for the Vth varyfrom methods to methods [1-5]. Hence, a universal, easy toimplement and accurate Vth extraction method is necessary.The Constant Current (CC) method defines the thresholdvoltage as the Vgs when ( )effeffdods L/WII ×= [1].

However, it has an ambiguous definition on the criticaldrain current, Ido. Although the Linear Extrapolation (LE)method [2] is widely used, the extracted Vth value maychange significantly because of variations in the extrinsicresistance. The Second-Derivative-of-the-Logarithm-of-drain-current (SDL) method and Tranconductance Change(TC) method [3,4] define the threshold voltage as where thedrift current contribution in Ids is equal to the diffusioncurrent contribution. However, it is not the thresholdcondition. The Match-Point (MP) method defines thethreshold voltage as the gate voltage for which theexponential extrapolation of sub-threshold current deviatesby 5% from the measured current [5]. Its extracted valuesmeet the threshold condition, i.e. the onset of the inversion

channel formation. Nonetheless, more time is consumed asextrapolation of the exponential component need to be donefirst.

A new approach for the extraction of Vth is proposed,namely, the “Third Derivative of Drain-Source Current”method or simply “TD” method. This method extracts theVth by finding the Vgs where the third derivative of Ids ismaximal. This method meets the threshold conditionrequirement, which is the onset of the inversion channelcreation.

2 THE NEW METHOD, THEORY

To illustrate the new method, understanding of theMOSFET’s operation is necessary. In this section, we willfirst concentrate on the study of the transistor’s operationusing a nMOSFET’s as an example. Later of this section,we will then focus on the new method.

2.1 MOSFET’s Operation

Fig. 1 shows the MOSFET’s channel inversion. Withthe drain and source grounded, the gate controls the chargein the channel. When a small positive biased voltage isapplied to the gate of a nMOSFET, the conditions withinthe channel will change. For the condition of Vgs > 0, freeholes that are present in the p-type silicon are repelled withthe accumulation of positive charge at the gate, thusforming a depletion region. The depth of the depletionregion from the silicon surface, WD is given by

( )A

SSiD qN

Wφε

=2

(1)

where εSi is the permitivity of silicon, φS is the surfacepotential and NA is the substrate doping concentration. Thisdepletion region is formed over both the length and widthdirections. Increasing the positive gate voltage further willeventually lead to the saturation of the depletion depth at,

( )A

BSimaxD qN

Wφε

=22

(2)

where φB is the bulk potential. Once the saturation of thedepletion region is reached, additional gate bias will attractnegative mobile electrons to the surface. When enoughelectrons have accumulated in the channel area, the surface

Source DrainGate

N+ N+

Inversion layerL

Vg

=Vth

Vd

=0Vs

P-type substrate

VB=0

WD

Fig.1 MOSFET’s channel inversion

of the silicon changes from hole-dominated to electron-dominated material and is said to have inverted. Under thiscondition, a conducting n-channel or inversion layer isformed under the gate between the two n+-diffusion,namely, source and drain, as shown in Fig. 1. Furtherincrease in gate voltage will only increase the surfacepotential slowly beyond 2φB, whereby the increased gatevoltage drops across the gate oxide. The minimum gatevoltage, which is required to form the conducting channelor an inverted layer, is called the threshold voltage Vth. Wecan classify the above four stated operation regions asdepletion region, weak inversion, moderate inversion, andstrong inversion.

Fig. 2 shows the magnitude of inversion layer chargeper unit area versus the Vgs. The onsets of weak, moderateand strong inversion, VLO, VMO, VTO can be expressedrespectively as:

bsFFFBLO VVV −φγ+φ+= (3)

bsFFFBMO V22VV −φγ+φ+= (4)

bsSSFBTO VVV −φγ+φ+= (5)

where, VFB is the flat-band voltage, φF is the Fermipotential, φS is the surface potential, and γ is the bodyfactor. The theoretical definition of Vth is based on the“strong-inversion” condition at which the φS is twice of theφF. Therefore, from eqns. (4) and (5), It is clear that VTO

will be equal to VMO if the transistor is biased in the "stronginversion" condition.

2.2 The New Method

It is clear that the VTO and VMO can be easily determinedfrom Fig. 2. VTO can be determined by finding the Vgs

where the second derivative of the drain-source current ismaximum which is the TC method. Whereas, VMO is equalto the Vgs where the third derivative of Ids is maximumwhich is the proposed method. The graphical representationof the proposed method is illustrated in Fig. 3. Thetransconductance (Gm) curve is obtained by differentiatingthe drain current (Ids) with respect to the gate voltage (Vgs)and once again to give the curve Gm' whereas the Gm”curve can be obtained by differentiating the Gm’ curveagain. The position of the peak of the Gm” curve on the Vgs

axis is taken as the threshold voltage.Fig. 4 shows that the extracted Vth using TD method

coincides with the extracted Vth using MP. Since thisapproach is physically defined, it is valid for all kinds oftechnology variations and it is independent of tolerances inchannel dimensions and extrinsic resistance.

3 RESULTS AND DISCUSSION

The new approach is verified with MOSFET’s datadown to 0.25µm. Fig. 5 compares the extracted results ofdifferent methods with Vds biasing at (a) 0.5V and (b) 2.5V.As we can see, the extracted Vth values using TD methodare very close to the Vth values extracted using MP method.While TC method over estimates the value of Vth. Thereason for this over-estimation is that, conventionally, wewill treat the surface potential equal to 2φF, neglecting aadditional term ∆φ which has value of several thermalvoltage.

Vgs

|QI|

VLO VMO VHOVFB

Slope = Cox

Depletion Weak

inversion

Moderate

inversion

Strong

inversion

Fig.2 Magnitude of inversion layer charge perunit area vs. Vgs

VTO

4 CONCLUSION

A new approach (TD method) for the extraction of Vth isproposed. It has been desmostrated with MOSFET’s datawith channel length down to 0.25µm. This method isphysically reliable and is found to be fast, accurate andsimple to implement.

ACKNOWLEDGMENT

The authors are grateful to Chartered SemiconductorManufacturing Ltd (CSM) for fabricating our teststructures.

REFERENCES

[1] L. A. Akers and J. J. Sanchez, “Threshold voltagemodels of short, narrow and small geometry MOSFETs:Subthreshold characteristics,” Jpn. J. Appl. Phys., vol.29,pp. L2279-L2282, 1990.

[2] L. A. Akers, “The inverse-narrow-width effect,”IEEE Electron Device Lett., vol. EDL-7, pp. 419-21, July1986.

0 0.5 1 1.5 2 2.5 3-6

-5

-4

-3

-2

-1

0

1

2

3

4x 10

-3

TD

MP

Vgs

Ids

log(Ids)/5000

Vth

Fig.4 Comparison of extractionapproaches between TD and MPmethods

Fig.5 Extracted Vth by different methodsversus channel lengths from 0.25 to 1µm at(a) Vds=0.5V and (b) Vds=2.5V

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.3

0.4

0.5

0.6

0.7

0.8

0.9

1

ChannelLength(a) Vds=2.5V

Vth

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

ChannelLength

(a) Vds=0.5V

Vth

Fig.3 Threshold voltage extraction by TDmethod and TC method.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9-6

-4

-2

0

2

4

6x 10

-5

Gm

Gm'

Gm''

Vgs

Vth

Vth extracted by TC

CCSDLTC

MP

TD

Gm, Gm’, Gm”

[3] K. Aoyama, “A method for extracting the thresholdvoltage of MOSFET’s based on current components’, inSimulation of Semiconductor Devices and Processes,Vienna, Austria: Springer-Verlag, vol.6, pp.118-121, 1995.

[4] H. S. Wong, M. H. White, T. J. Krutsick, and R. V.Booth, “Modeling of transconductance degradation andextraction of threshold voltage in thin oxide MOSFET’s,”Solid State Electron., vol. 30, pp. 953-968, 1987.

[5] B. El-Kareh, W. R. Tonti, and S. L. Titcomb, “Asubmicron MOSFET parameter extraction technique,” IBMJ. Res. Develop., vol. 34, pp. 243-249, 1990.