Hindawi Publishing CorporationActive and Passive Electronic ComponentsVolume 2013 Article ID 153157 9 pageshttpdxdoiorg1011552013153157

Research ArticlePotential and Quantum Threshold Voltage Modeling ofGate-All-Around Nanowire MOSFETs

M Karthigai Pandian1 N B Balamurugan2 and A Pricilla3

1 Pandian Saraswathi Yadav Engineering College Sivagangai India2Thiagarajar Engineering College Madurai India3 St Michaelrsquos College of Engineering and Technology Sivagangai India

Correspondence should be addressed to M Karthigai Pandian karthickpandiangmailcom

Received 21 March 2013 Revised 15 August 2013 Accepted 15 August 2013

Academic Editor Gerard Ghibaudo

Copyright copy 2013 M Karthigai Pandian et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

An improved physics-based compact model for a symmetrically biased gate-all-around (GAA) silicon nanowire transistor isproposed Short channel effects and quantummechanical effects caused by the ultrathin silicon devices are considered inmodellingthe threshold voltage Device geometrics play a very important role in multigate devices and hence their impact on the thresholdvoltage is also analyzed by varying the height and width of silicon channelThe inversion charge and electrical potential distributionalong the channel are expressed in their closed forms The proposed model shows excellent accuracy with TCAD simulations ofthe device in the weak inversion regime

1 Introduction

Semiconductor nanowires are attractive components forfuture nanoelectronics since they can exhibit a wide rangeof device function and at the same time serve as bridgingwires that connect larger scale metallization The nanoscaleFETs based on silicon nanowires have notable attentionfor their potential applications in electronics industry In acontinuous effort to increase current drive and better controlover SCEs silicon-on-insulator (SOI) MOS transistors haveevolved from classical planar and single-gate devices into3D devices with a multigate structure (double- triple- orgate-all-around devices)Thesemultigate nanowire FETs thatprevent the electric field lines from originating at the drainfrom terminating under the channel region are now widelyrecognized as one of the most auspicious solutions formeeting the roadmap requirements in the decananometerscale Multigate device structures of nanowire transistorspave the way for better electrostatic control and as a resultintrinsic channels get higher mobility and current [1]

CMOS devices can be scaled down up to a channel lengthof 10 nm when the number of gates in the device is increasedIn such transistors the short channel effects are controlled

by the device geometry and hence an undoped or lightlydoped ultrathin body is used to sustain the channel Variousdevice structures such as double gate fully depleted SOItrigate and all around gate structures have been extensivelyinvestigated to restrict SCEswithin a limitwhile achieving theprimary advantages of scaling that is higher performancelower power and ever increasing integration density [2] Thescaling theory and the analytical SCEs model for nanowiretransistors based on the concept of natural length are success-ful to a certain extent To address the issue of 2D effects in thegate insulator a more generalized concept of scale length hasbeen proposed recently [3 4]

Modeling of quantum confinement and transport in ananowire transistor has been addressed in the literature [5ndash7] The undoped cylindrical body GAA field effect tran-sistor that has a great control over the corner effects andchannel is considered to be promising candidate for sub-45 nm regime [8 9] An analytical threshold voltage modelfor GAA nanoscale MOSFETs (Figure 1) considering thehot carrier induced interface charges has been proposed byGhoggali et al in 2008 [10] Quantum confinement and itseffect on threshold voltage variations in short channel GAAdevices have been studied in 2009 [11] A compact analytical

2 Active and Passive Electronic Components

W

H

Vg

Vds

Source Drain

TOX

Figure 1 Schematic diagram of a gate-all-around SiNWMOSFET

threshold voltage model proposed by Te-Kuang deals withthe interface trapped charges in a nanowire channel [12]A physically based classical model for body potential of acylindrical GAA nanowire transistor has been proposed byRay andMahapatra in 2008 [13] and a quasianalytical modelfor predicting the potential of a nanowire FET has beenproposed by De Michielis et al in 2010 [14]

In this paper quantum threshold voltage modeling of alightly doped gate-all-around silicon nanowire transistor isproposed In modeling the threshold voltage the quantumeffects are also taken into account as the quantization ofelectron energy in ultrathin devices can never be ignoredOne important consequence of the quantum mechanicalcarrier distribution in accordance with the device behavioroccurs when the device geometrics and the silicon thicknessare varied so a reliable compact model for the nanowiretransistors must also take into account quantum effectsresulting out of these variations The proposed physicallybased closed form quantum threshold voltage model holdsgood for ultrathin and ultrashort channel gate all arounddevices and does not discuss any unphysical fitting parameterThe compact threshold voltage model is obtained by solvingthe 3DPoisson equation and 2D Schrodinger equations in theweak inversion region These equations are then consistentlysolved to obtain the potential distribution and inversioncharge density

2 Threshold Voltage Modeling

Here we consider a lightly doped nanowire MOSFET in theweak inversion region where both fixed and mobile chargedensities in the channel are negligible We have assumeda flat potential on the plane perpendicular to the source-drain direction Poisson-Schrodinger equations should besolved consistently to obtain the potential and inversioncharge density But in the weak inversion regime we haveapproximated the Poisson equation as Laplace equation withthe inversion charge density neglected and thus the twoequations are decoupledThemidgapmetals are used for gateintended to suppress the silicon gate poly depletion inducedparasitic capacitances [15]The 3D Poisson equation is solvedto obtain the threshold voltage in the weak inversion regionincluding the parabolic band approximation The potentialdistribution in insulator and silicon regions can be expressedas

1205752120595 (119909 119910 119911)

1205751199092+1205752120595 (119909 119910 119911)

1205751199102+1205752120595 (119909 119910 119911)

1205751199112= 0 (1)

The potential 120595 in terms of 119909 (width) 119910 (height) and119911 (length) is to be determined The boundary conditionsdefined by the physics of the device are given by

120595(119909minus119867

2minus 119879OX 119911) = 119881119892

120595 (minus119882

2minus 119879OX 119910 119911) = 119881119892

1015840

120575120595

120575119909

10038161003816100381610038161003816100381610038161003816119909=0= 0

120575120595

120575119910

10038161003816100381610038161003816100381610038161003816119910=0

= 0

120595 (119909 119910 0) = 120595bi

120595 (119909 119910 119871) = 120595bi + 119881ds

(2)

For a gate-all-around device we have to find the insulatorpotential on all sides of the channel under considerationSo the height and width of the channel are also taken intoaccount The insulator potential is now expressed as

120595 (119909 119910 119911) =119881119892minus 120595bi minus Φms

119879OX(119909 minus

119882

2) + 120595bi

for 1198822lt 119909 lt

119882

2+ 119879OX 0 lt 119910 lt

119867

2

120595 (119909 119910 119911) =119881119892minus 120595bi minus Φms

119879OX(119910 minus

119867

2) + 120595bi

for 0 lt 119909 lt 119882

2

119867

2lt 119910 lt

119867

2+ 119879OX

(3)

where 119881119892is the gate voltage 120595bi is the built-in potential

119871 is the channel length and 119881ds is the drain to sourcevoltage which is negligible for low 119881ds Φms is the workfunction difference By applying the superposition principlethe electrostatic potential can be now written as

120595 (119909 119910 119911) = 119880119871(119909 119910 119911) + 119880

119877(119909 119910 119911) + 119881

119892(119909 119910) (4)

Here 119881119892(119909 119910) is the 1D solution of the Poisson equation that

satisfies the gate boundary conditions119880119871satisfies the source

boundary condition but it is bound to have a null value onthe gate and drain boundaries Similarly119880

119877satisfies the drain

boundary condition and it is bound to have a null valueon the gate and source boundaries On further evaluation

Active and Passive Electronic Components 3

the term 119881119892+ 119880119871is found to satisfy the potential equation

when119880119877is on null value and in an exact repetition the term

119881119892+ 119880119877satisfies the potential equation when 119880

119871is on null

value From (1)

120595119909119909+ 120595119910119910+ 120595119911119911= 0 (5)

By solving the above equation using LDE method we obtainthe value of 120595 Then the limits are applied on the equationusing the boundary conditions Now the potentials 119880

119871is

given by

1198801198711= sum

119899

sum

119898

119862 times 119870119860119899119898

sin(Λ119899(119909 minus

119882

2minus 119879OX))

times cos (119872119898119910) sinh(sum

119899119898

(119871 minus 119911))

for 1198822lt 119909 lt

119882

2+ 119879OX 0 lt 119910 lt

119867

2

(6)

1198801198712= sum

119899

sum

119898

119862 times 119870119861119899119898

cos (Λ119899119909)

times sin(119872119898(119910 minus

119867

2minus 119879OX))

times sinh(sum119899119898

(119871 minus 119911)) for 0 lt 119909 lt 119882

2

119867

2lt 119910 lt

119867

2+ 119879OX

(7)

1198801198713= sum

119899

sum

119898

119862 times 119870119862119899119898

cos (Λ119899119909) cos (119872

119898119910)

times sinh(sum119899119898

(119871 minus 119911))

for 0 lt 119909 lt 119882

2 0 lt 119910 lt

119867

2

(8)

Similarly the values of potential 119880119877are also derived as

follows

1198801198771= sum

119899

sum

119898

119862 times 119870119875119899119898

sin(Λ119899(119909 minus

119882

2minus 119879OX))

times cos (119872119898119910)

times sinh(sum119899119898

119911) for 1198822lt 119909 lt

119882

2+ 119879OX

0 lt 119910 lt119867

2

1198801198772= sum

119899

sum

119898

119862 times 119870119876119899119898

cos (Λ119899119909)

times sin(119872119898(119910 minus

119867

2minus 119879OX)) sinh(sum

119899119898

119911)

for 0 lt 119909 lt 119882

2

119867

2lt 119910 lt

119867

2+ 119879OX

1198801198773= sum

119899

sum

119898

119862 times 119870119877119899119898

cos (Λ119899119909) cos (119872

119898119910)

times sinh(sum119899119898

119911) for 0 lt 119909 lt 119882

2

0 lt 119910 lt119867

2

(9)

where 119888 = 1sum119899119898119871 and 119870

119860119899119898 119870119861119899119898

119870119862119899119898

119870119875119899119898

119870119902119899119898

and 119870

119877119899119898are constants From (6) and (8) 119880

119871and (120597119880

119871)120597119909

are found to be continuous in the 119909 direction (119909 = 1198822)The first derivative function (120597119880

1198711)120597119909 itself has 120576OX120576si

times discontinuities at the silicon insulator interfaces Thusapplying continuity in both equations we proceed to equate(6) and (8) as follows

minus119870119860119899119898

sin (Λ119899119879OX) = 119870119862119899119898 cos(

Λ119899119882

2) (10)

Differentiating (6) with respect to 119909

1205971198801198711

120597119909= sum

119899

sum

119898

119862 times 119870119860119899119898

(Λ119899) cos(Λ

119899(119909 minus

119882

2minus 119879OX))

times cos (119872119898119910) sinh(sum

119899119898

(119871 minus 119911))

(11)

Differentiating (8) with respect to 119909

1205971198801198713

120597119909= sum

119899

sum

119898

119862 times 119870119888119899119898

(Λ119899) (minus sin (Λ

119899119909))

times cos (119872119898119910) times sinh(sum

119899119898

(119871 minus 119911))

(12)

Equating (11) and (12)

119870119860119899119898

120576OX cos (Λ 119899119879OX) = minus119870119862119899119898120576si sin(Λ119899

2) (13)

Dividing (10) by (13)

120576si tan (Λ 119899119879OX) minus 120576OXcot(Λ119899119882

2) = 0 (14)

Likewise from (7) and (8) 119880119871and (120597119880

119871120597119910) are found to

be continuous in the 119910 direction (119910 = 1198672) The function(1205971198801198712120597119910) itself has discontinuities at the silicon insulator

interfaces which are proportional to the dielectric constant120576OX120576si Thus applying continuity in both equations andequating (7) and (8) we get

minus119870119861119899119898

sin (119872119898119879OX) = 119870119862119899119898 cos(

119872119898119867

2) (15)

4 Active and Passive Electronic Components

Differentiating (7) with respect to 119910

1205971198801198712

120597119910= sum

119899

sum

119898

119862 times 119870119861119899119898

cos (Λ119899119909) (119872

119898)

times cos(119872119898(119910 minus

119867

2minus 119879OX))

times sinh(sum119899119898

(119871 minus 119911))

(16)

Differentiating (8) with respect to 119910

1205971198801198713

120597119910= sum

119899

sum

119898

119862 times 119870119862119899119898

cos (Λ119899119909) (119872

119898)

times (minus sin (119872119898119910)) sinh(sum

119899119898

(119871 minus 119911))

(17)

Equating (16) and (17)

119870119861119899119898

120576OX cos (119872119898119879OX) = minus119870119862119899119898120576si sin(119872119898119867

2) (18)

Dividing (15) by (18)

120576si tan (119872119898119879OX) minus 120576OXcot(119872119898119867

2) = 0 (19)

This natural length is an easy guide for choosing deviceparameters and has simple physical meaning that a smallnatural length corresponds to superb short channel effectimmunity [4] The value of Λ

119899and 119872

119898depends on device

parameters The potential 119880119871can be modified as

1198801198711198991198981

= 120572119899119898

times 119862 times sin(Λ119899(119909 minus

119882

2minus 119879OX))

times cos (119872119898119910) times sinh(sum

119899119898

(119871 minus 119911))

for 1198822lt 119909 lt

119882

2+ 119879OX 0 lt 119910 lt

119867

2

1198801198711198991198982

= 120573119899119898

times 119862 times cos (Λ119899119909)

times sin(119872119898(119910 minus

119867

2minus 119879OX)) sinh(sum

119899119898

(119871 minus 119911))

for 0 lt 119909 lt 119882

2

119867

2lt 119910 lt

119867

2+ 119879OX

1198801198711198991198983

= 120574119899119898

times 119862 times cos (Λ119899119909) cos (119872

119898119910)

times sinhsum119899119898

(119871 minus 119911) for 0 lt 119909 lt 119882

2 0 lt 119910 lt

119867

2

(20)

Similarly the potential 119880119877can be modified as

1198801198771198991198981

= 120572119899119898

times 119862 times sin(Λ119899(119909 minus

119882

2minus 119879OX))

times cos (119872119898119910)

times sinh(sum119899119898

119911) for 1198822lt 119909 lt

119882

2+ 119879OX

0 lt 119910 lt119867

2

1198801198771198991198982

= 120573119899119898

times 119862 times cos (Λ119899119909)

times sin(119872119898(119910 minus

119867

2minus 119879OX)) times sinh(sum

119899119898

119911)

for 0 lt 119909 lt 119882

2

119867

2lt 119910 lt

119867

2+ 119879OX

1198801198771198991198983

= 120574119899119898

times cos (Λ119899119909) cos (119872

119898119910)

times sinh(sum119899119898

119911) for 0 lt 119909 lt 119882

2 0 lt 119910 lt

119867

2

(21)

Now the 119892119899119898

can be obtained from the potential equations(20) by using different multipliers in different regions

1198921198991198981

=120576OX sin (Λ 119899 (119909 minus1198822 minus 119879OX)) cos (119872119898119910)

2120576si sin (Λ 119899119879OX) cos (119872119898 (1198672))

for 1198822lt 119909 lt

119882

2+ 119879OX 0 lt 119910 lt

119867

2

1198921198991198982

=120576OX cos (Λ 119899119909) sin (119872119898 (119910 minus 1198672 minus 119879OX))

2120576si cos (Λ 119899 (1198822)) sin (119872119898119879OX)

for 0 lt 119909 lt 119882

2

119867

2lt 119910 lt

119867

2+ 119879OX

1198921198991198983

= minuscos (Λ

119899119909) cos (119872

119898119910)

cos (Λ119899(1198822)) cos (119872

119898(1198672))

for 0 lt 119909 lt 119882

2 0 lt 119910 lt

119882

2

(22)

Subsequently the constants 120572119899119898 120573119899119898 and 120574

119899119898are evaluated

suitably

120572119899119898

= cos(Λ119899

119882

2) sin (119872

119898119879OX)

120573119899119898

= cos(119872119898

119867

2) sin (Λ

119899119879OX)

120574119899119898

= sin (119872119898119879OX) sin (Λ 119899119879OX)

(23)

From (20) the potential 119880119871119899119898

can be rewritten as

1198801198711198991198981

= minus cos(Λ119899

119882

2) sin (119872

119898119879OX)

times sin (Λ119899119879OX) cos(119872119898

119867

2)

Active and Passive Electronic Components 5

1198801198711198991198982

= minus cos(119872119898

119867

2) sin (Λ

119899119879OX)

times cos(Λ119899

119882

2) sin (119872

119898119879OX)

1198801198711198991198983

= minus sin (119872119898119879OX) sin (Λ 119899119879OX)

times cos(Λ119899

119882

2) cos(119872

119898

119867

2)

(24)

By multiplying with the corresponding orthogonal conjugatefunctions and integrating coefficients of119880

119871can be obtained

The coefficients of 119880119877are also obtained in a similar method

119894119899119898

= (int

119867+119879OX

0

int

119882+119879OX

0

(120595 (119909 119910 0))

minus119881 (119909 119910) 119892119899119898(119909 119910) 119889119909 119889119910)

times (int

119867+119879OX

0

int

119882+119879OX

0

119880119871119899119898

(119909 119910 0) 119892119899119898(119909 119910) 119889119909 119889119910)

minus1

119895119899119898

= (int

119867+119879OX

0

int

119882+119879OX

0

(120595 (119909 119910 119871))

minus119881 (119909 119910) 119892119899119898(119909 119910) 119889119909 119889119910)

times (int

119867+119879OX

0

int

119882+119879OX

0

119880119871119899119898

(119909 119910 119871) 119892119899119898(119909 119910) 119889119909 119889119910)

minus1

(25)

The above integrals (25) are evaluated to obtain explicitexpressions for 119894

119899119898and 119895119899119898

as follows

119894119899119898

= (119881119892minus 120595bi) 120596119899119898

119895119899119898

= (119881119892minus 120595bi minus 119881ds) 120596119899119898

120596119899119898

=Ω119899119898

120585119899119898

(26)

where

Ω119899119898

=120576OXΛ 119899 tan ((Λ 119899119882) 2) + 120576OX119872119898 tan (1198721198981198672)

2120576si119879OXΛ2

1198991198722119898

minus2

Λ119899119872119898

tan(Λ119899119882

2) tan(

119872119898119867

2)

120585119899119898

=sin (Λ

119899119879OX)

16 cos (Λ1198991198822)

(119882 +sin (Λ

119899119882)

Λ119899

) 1199031

+sin (119872

119898119879OX)

16 cos (1198721198981198672)

(119867 +sin (119872

119898119867)

119872119898

) 1199032

(27)

And the values of 1199031and 1199032are given by

1199031=120576OX119879OX cos (1198721198981198672)

120576si sin (119872119898119879OX)+119867 sin (119872

119898119879OX)

2 cos (1198721198981198672)

1199032=120576OX119879OX cos (Λ 1198991198822)

120576si sin (Λ 119899119879OX)+119882 sin (Λ

119899119879OX)

2 cos (Λ1198991198822)

(28)

The potential equation is now rewritten as

120595 (119909 119910 119911) = 119881119892+ (119881119892minus 120595bi)

timessum

119899

sum

119898

120588119899119898

sinh(sum119899119898

(119871 minus 119911))

+ (119881119892minus 120595bi minus 119881ds)sum

119899

sum

119898

sinh(sum119899119898

119911)

(29)

where

120588119899119898(119909 119910) = minus120596

119899119898120574119899119898

cos (Λ119899119909) cos (119872

119898119910)

sinh (sum119899119898119871)

(30)

Once the potential distribution at every point of the cross-section of the channel is known we calculate the inversioncharge density by using surface integral over the surface areaof the channel When the integrated charge at virtual sourcebecomes equal to critical charge the gate voltage of a lightlydoped body device is nearly equal to the threshold voltage ofthe device Hence the inversion charge can be expressed as

119876 = int

1198672

minus1198672

int

1198822

minus1198822

119902119899119894119890(120595119880119879)119889119909 119889119910 (31)

where 119902 is the elementary charge 119880119879is thermal voltage and

119899119894is the intrinsic carrier concentrationThe charge equation can now be approximated as

119876 asymp 119882119867119902119899119894119890((120595((311988214)(311986714)119911119888))119880119879) (32)

Here 119911119888is the virtual source position which is half of the

channel length for low 119881ds Using the inversion charge wecan obtain the classical threshold model as expressed in thefollowing

119881TC = (119880119879 ln119876

119882119867119902119899119894

+ 2119881bi12058811 (3119882

143119867

14)

times sinh(sum11119871

2))

times (1 + 212058811(3119882

143119867

14) sinh(

sum11119871

2))

minus1

(33)

3 Quantum Threshold Voltage Modeling

AsMOSFETdevices are further scaled into the deep nanome-ter regime it has become necessary to include quantummechanical effects while modeling their device behaviorIn this paper we approximate the actual potential well as

6 Active and Passive Electronic Components

E11

Ec

Ec

998779Ec

Ei119909i119910

Eco

Figure 2 Band diagram perpendicular to the gate-square wellpotential of a GAA silicon nanowire transistor

the square well potential since it is difficult to solve theSchrodinger equation to obtain the potential expressed in(29) The square well potential of a gate-all-around nanowiretransistor is shown in Figure 2 The quantum charge of thedevice is expressed as

119876 = sum

119894119909

sum

119894119910

119902int

infin

119864119894119909119894119910

1198731D119891 (119864) 119889119864 (34)

where1198731D is the 1D density-of-states and 119891(119864) is the Fermi-

Dirac distribution function 119864 is the energy of the electronwave The terms 119894

119909and 119894119910are positive natural numbers

In silicon six energy valleys are found to be present inits band structure (two lower energy valleys two middleenergy valleys and two higher energy valleys) If the thinfilm of device has equal height and width the two lowerenergy valleys and two middle energy valleys are combinedtogether to produce four lower energy valleys and the othertwo higher energy valleys remain in their own stateThus thecharge is given by

119876 = 119902sum

119894119909

sum

119894119910

radic(119898119911

2120587ℎ2)int

infin

119864119894119909119894119910

(119864 minus 119864119894119909119894119910)minus12

1 + 119890 ((119864 minus 119864119865) 119896119879)

119889119864 (35)

where 119898119911is the mass of the valley which is perpendicular to

the direction of quantizationThe Fermi energy level is muchlower than the conduction band energy in weak inversionregion Hence the charge equation can be approximated as

119876 = 119902radic(119898119911

2120587ℎ2)sum

119894119909

sum

119894119910

int

infin

119864119894119909119894119910

119890((119864119865minus119864)119896119879)

(119864 minus 119864119894119909119894119910)12119889119864 (36)

Using the Schrodinger equation the value of 119864119894119909119864119894119910is deter-

mined by the following formulation [5]

119864119894119909119894119910

= 119864co +ℎ21205872

2[1

119898119909

(119894119909

119868119909

)

2

+1

119898119910

(119894119910

119868119910

)

2

] (37)

where the conduction band energy is given as

119864co =119864119892

2minus 119902120595 (0 0 119911

119888) (38)

Using (36) and (37) the integrated charge can be obtained as

119876 = 119902radic(2119896119879119898

119905

ℎ2)sum

119894119909

sum

119894119910

exp(minus119864co + 1198641 (119894119909 119894119910)

119896119879)

+ 119902radic(2119896119879119898

119905

ℎ2)sum

119894119909

sum

119894119910

exp(minus119864co + 1198642 (119894119909 119894119910)

119896119879)

+ 119902radic(2119896119879119898

119905

ℎ2)sum

119894119909

sum

119894119910

exp(minus119864co + 1198643 (119894119909 119894119910)

119896119879)

(39)

where

1198641(119894119909 119894119910) =

ℎ21205872

2[1

1198981

(119894119909

119882)

2

+1

119898119905

(119894119910

119867)

2

]

1198642(119894119909 119894119910) =

ℎ21205872

2[1

119898119905

(119894119909

119882)

2

+1

1198981

(119894119910

119867)

2

]

1198643(119894119909 119894119910) =

ℎ21205872

2[1

119898119905

(119894119909

119882)

2

+1

119898119905

(119894119910

119867)

2

]

(40)

Here the 119898119905and 119898

1are the transverse and longitudinal

effective masses of the energy valleys of silicon The lengths119894119909and 119894119910carry distinct values contingent on the direction of

quantization Finally the quantum threshold voltage modelbecomes

119881TQ = (119864119892

2119902+ (

119896119879

119902) In(119876119879

120591)

+2120595bi12058811 (0 0) sinh((sum11119871)

2))

times (1 + 212058811(0 0) sinh(

(sum11119871)

2))

minus1

(41)

where

120591 = 119902radic(2119896119879119898

119905

ℎ2) exp(minus1198641 (1 1)

119896119879)

+ 119902radic(2119896119879119898

119905

ℎ2) exp(minus1198642 (1 1)

119896119879)

+ 119902radic(2119896119879119898

119905

ℎ2) exp(minus

1198643(1 1)

119896119879)

(42)

Active and Passive Electronic Components 7

10 20 30 40 50 6002

022

024

026

028

03

032

034

036

038

04

Channel length (nm)

Pote

ntia

l (V

)

Proposed modelTCAD simulation

Figure 3 Constant electrostatic potential obtained from the analyt-ical solution of a gate-all-around silicon nanowire transistor is 03 Vfor different channel length with a height of119867 = 9 nm and channelwidth of 119882 = 9 nm TCAD simulation shows that the potential isconstant at 0296V

The impacts on the threshold voltage due to quantum effectsare acquired by using the following equation

119881TC = 119881TQ + Δ119881119879 (43)

Here Δ119881119879is the difference between the quantum threshold

voltage and the classical threshold voltage

4 Results and Discussion

Figure 3 shows the electrostatic potential of the proposed gateall around transistor and it is found to be constant value at03 V Continuously varying the 119899 and 119898 terms in (4) hasno impact on the potential as it remains constant along theinsulator boundaries This is totally in contrast to the resultsobtained in [15] where the potential is found to be linearlyvarying in the insulator boundaries The constant potentialhas to be deduced as the resultant of the gate voltage appliedsymmetrically across the four sides of the transistor TheTCAD simulation of the device shows that the electrostaticpotential is constant at 0296V The simulation results arefound to in acceptance with the TCAD results

Figure 4 represents the variation of total quantum inte-grated charge with the gate voltage Equation (39) is usedto obtain the integrated charge with only one energy leveland one series term It clearly shows that the decrease inthe film thickness leads to the increase in the quantumthreshold voltage which is actually due to the increase inenergy quantization of the transistor With the height andlength of the device being constant the width of the device isvaried and henceforth the variation of charge in accordancewith the gate voltage is illustrated in Figure 4

The variation of quantum threshold voltage with widthand height of the film at a channel length of 20 nm is

0 01 02 03 04 05 06 070

05

1

15

2

25

3

35

4times10minus15

Gate voltage (V)

Char

ge (c

mminus1)

W = 9e minus 9

W = 5e minus 9

W = 3e minus 9

TCAD simulation for W = 9e minus 9

TCAD simulation for W = 5e minus 9

TCAD simulation for W = 3e minus 9

Figure 4 Variation of quantum integrated charge at virtual sourcewith gate voltage for different film widths where the height andlength are119867 = 9 nm 119871 = 20 nm

2

8

times10minus9

times10minus9

Film height

2

44

66

8

10

1003

0305

031

0315

032

Film width

Qua

ntum

thre

shol

d vo

ltage

(V)

ProposedTCAD

Figure 5 Variation of quantum threshold voltage with film heightand width for channel length (119871 = 20 nm)

shown in Figure 5 The short channel effects tend to decreasealong with the energy quantisation and this can be furtherexplained as a result of increase in the effective band gap ofsilicon due to quantum effects The effect of confinementexpressed as the difference in the threshold voltage and itsvariation with the channel length 119871 is illustrated in Figure 6

8 Active and Passive Electronic Components

2 3 4 5 6 7 8 9 10

times10minus9

0014

0016

0018

002

0022

0024

0026

0028

003

0032

Film height

Proposed modelTCAD simulation

ΔVT

(mV

)

Figure 6 Variation of threshold voltagewith filmwidth for differentheight at 119871 = 20 nm

3 4 5 6 7 8

times10minus9

02

022

024

026

028

03

032

034

036

038

04

Film height

Thre

shol

d vo

ltage

(V)

ClassicalQuantum

TCAD simulation for classicalTCAD simulation for quantum

9

Figure 7 Variation of quantum and classical threshold voltage withfilm height Here 119871 = 20 nm and119882 = 9 nm

Themost important thing about this gate all around nanowiretransistor is that any change in one of the dimensions canbe nullified by proper tuning of other dimensions as thetransistor is symmetric about its height and width

Figure 7 shows the variation of the classical thresholdvoltage and quantum threshold voltage with the film height ata constant width of 9 nmThe value of the classical thresholdvoltage ranges from 027V to 029V for the correspondingchanges in the film height Similarly the quantum thresholdvoltage ranges from 03V to 031 V It shows that the devicehas a highly improved control over the threshold voltageTheTCAD results justify the simulation results

5 Conclusion

In this paper a quantum threshold voltage model for a GAAsilicon nanowire transistor is proposed by solving the 3DPoisson and Schrodinger equations Analytical expressionsfor potential and the inversion charge are expressed intheir closed forms The results show that the integratedcharge and the threshold voltage calculated in accordancewith the quantum effects of this proposed model are highlyimprovedThe future considerations include deriving the I-Vcharacteristics of the gate all around nanowire transistors andstudying the impact of scaling on various device parametersFinally to conclude this model provides an analytical anduseful way for the threshold voltage evaluations in gate allaround nanowire devices with a unified formalism employedin both classical and quantum mechanical approaches

References

[1] J Wang E Polizzi and M Lundstrom ldquoA computationalstudy of ballistic silicon nanowire transistorsrdquo in Proceedings ofthe IEEE International Electron Devices Meeting pp 695ndash698December 2003

[2] J P Colinge FINFETS and Other Multi-Gate TransistorsSpringer New York NY USA 2007

[3] D J Frank Y Taur andH-S PWong ldquoGeneralized scale lengthfor two-dimensional effects in MOSFETrsquosrdquo IEEE ElectronDevice Letters vol 19 no 10 pp 385ndash387 1998

[4] B Yu L Wang Y Yuan P M Asbeck and Y Taur ldquoScaling ofnanowire transistorsrdquo IEEE Transactions on Electron Devicesvol 55 no 11 pp 2846ndash2858 2008

[5] G D Sanders C J Stanton and Y C Chang ldquoTheory of trans-port in silicon quantum wiresrdquo Physical Review B vol 48 no15 pp 11067ndash11076 1993

[6] M-Y Shen and S-L Zhang ldquoBand gap of a silicon quantumwirerdquo Physics Letters A vol 176 no 3-4 pp 254ndash258 1993

[7] J P Colinge X Baie V Bayot and E Grivei ldquoQuantum-wireeffects in thin and narrow SOI MOSFETsrdquo in Proceedings of theIEEE International SOI Conference pp 66ndash67 October 1995

[8] C P Auth and J D Plummer ldquoScaling theory for cylindricalfully-depleted surrounding-gate MOSFETrsquosrdquo IEEE ElectronDevice Letters vol 18 no 2 pp 74ndash76 1997

[9] J-T Park and J-P Colinge ldquoMultiple-gate SOI MOSFETsdevice design guidelinesrdquo IEEE Transactions on ElectronDevices vol 49 no 12 pp 2222ndash2229 2002

[10] Z Ghoggali F DjeffalM A Abdi D Arar N NLakhdar and TTBendib ldquoAn analytical threshold voltagemodel for nanoscalerdquoin Proceedings of the 3rd International Design and TestWorkshop(IDT rsquo08) pp 93ndash97 December 2008

[11] Y-S Wu and P Su ldquoQuantum confinement effect in short-channel gate-all-around MOSFETs and its impact on the sensi-tivity of threshold voltage to process variationsrdquo in Proceedingsof the IEEE International SOI Conference October 2009

[12] C Te-Kuang ldquoA compact analytical threshold-voltage modelfor surrounding-gateMOSFETswith interface trapped chargesrdquoIEEE Electron Device Letters vol 31 no 8 pp 788ndash790 2010

[13] B Ray and SMahapatra ldquoModeling and analysis of body poten-tial of cylindrical gate-all-around nanowire transistorrdquo IEEETransactions on Electron Devices vol 55 no 9 pp 2409ndash24162008

Active and Passive Electronic Components 9

[14] L De Michielis L Selmi and A M Ionescu ldquoA quasi-analytical model for nanowire FETs with arbitrary polygonalcross sectionrdquo Solid-State Electronics vol 54 no 9 pp 929ndash9342010

[15] P R Kumar and S Mahapatra ldquoQuantum threshold voltagemodeling of short channel quad gate silicon nanowire transis-torrdquo IEEE Transactions onNanotechnology vol 10 no 1 pp 121ndash128 2011

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2 Active and Passive Electronic Components

W

H

Vg

Vds

Source Drain

TOX

Figure 1 Schematic diagram of a gate-all-around SiNWMOSFET

threshold voltage model proposed by Te-Kuang deals withthe interface trapped charges in a nanowire channel [12]A physically based classical model for body potential of acylindrical GAA nanowire transistor has been proposed byRay andMahapatra in 2008 [13] and a quasianalytical modelfor predicting the potential of a nanowire FET has beenproposed by De Michielis et al in 2010 [14]

In this paper quantum threshold voltage modeling of alightly doped gate-all-around silicon nanowire transistor isproposed In modeling the threshold voltage the quantumeffects are also taken into account as the quantization ofelectron energy in ultrathin devices can never be ignoredOne important consequence of the quantum mechanicalcarrier distribution in accordance with the device behavioroccurs when the device geometrics and the silicon thicknessare varied so a reliable compact model for the nanowiretransistors must also take into account quantum effectsresulting out of these variations The proposed physicallybased closed form quantum threshold voltage model holdsgood for ultrathin and ultrashort channel gate all arounddevices and does not discuss any unphysical fitting parameterThe compact threshold voltage model is obtained by solvingthe 3DPoisson equation and 2D Schrodinger equations in theweak inversion region These equations are then consistentlysolved to obtain the potential distribution and inversioncharge density

2 Threshold Voltage Modeling

Here we consider a lightly doped nanowire MOSFET in theweak inversion region where both fixed and mobile chargedensities in the channel are negligible We have assumeda flat potential on the plane perpendicular to the source-drain direction Poisson-Schrodinger equations should besolved consistently to obtain the potential and inversioncharge density But in the weak inversion regime we haveapproximated the Poisson equation as Laplace equation withthe inversion charge density neglected and thus the twoequations are decoupledThemidgapmetals are used for gateintended to suppress the silicon gate poly depletion inducedparasitic capacitances [15]The 3D Poisson equation is solvedto obtain the threshold voltage in the weak inversion regionincluding the parabolic band approximation The potentialdistribution in insulator and silicon regions can be expressedas

1205752120595 (119909 119910 119911)

1205751199092+1205752120595 (119909 119910 119911)

1205751199102+1205752120595 (119909 119910 119911)

1205751199112= 0 (1)

The potential 120595 in terms of 119909 (width) 119910 (height) and119911 (length) is to be determined The boundary conditionsdefined by the physics of the device are given by

120595(119909minus119867

2minus 119879OX 119911) = 119881119892

120595 (minus119882

2minus 119879OX 119910 119911) = 119881119892

1015840

120575120595

120575119909

10038161003816100381610038161003816100381610038161003816119909=0= 0

120575120595

120575119910

10038161003816100381610038161003816100381610038161003816119910=0

= 0

120595 (119909 119910 0) = 120595bi

120595 (119909 119910 119871) = 120595bi + 119881ds

(2)

For a gate-all-around device we have to find the insulatorpotential on all sides of the channel under considerationSo the height and width of the channel are also taken intoaccount The insulator potential is now expressed as

120595 (119909 119910 119911) =119881119892minus 120595bi minus Φms

119879OX(119909 minus

119882

2) + 120595bi

for 1198822lt 119909 lt

119882

2+ 119879OX 0 lt 119910 lt

119867

2

120595 (119909 119910 119911) =119881119892minus 120595bi minus Φms

119879OX(119910 minus

119867

2) + 120595bi

for 0 lt 119909 lt 119882

2

119867

2lt 119910 lt

119867

2+ 119879OX

(3)

where 119881119892is the gate voltage 120595bi is the built-in potential

119871 is the channel length and 119881ds is the drain to sourcevoltage which is negligible for low 119881ds Φms is the workfunction difference By applying the superposition principlethe electrostatic potential can be now written as

120595 (119909 119910 119911) = 119880119871(119909 119910 119911) + 119880

119877(119909 119910 119911) + 119881

119892(119909 119910) (4)

Here 119881119892(119909 119910) is the 1D solution of the Poisson equation that

satisfies the gate boundary conditions119880119871satisfies the source

boundary condition but it is bound to have a null value onthe gate and drain boundaries Similarly119880

119877satisfies the drain

boundary condition and it is bound to have a null valueon the gate and source boundaries On further evaluation

Active and Passive Electronic Components 3

the term 119881119892+ 119880119871is found to satisfy the potential equation

when119880119877is on null value and in an exact repetition the term

119881119892+ 119880119877satisfies the potential equation when 119880

119871is on null

value From (1)

120595119909119909+ 120595119910119910+ 120595119911119911= 0 (5)

By solving the above equation using LDE method we obtainthe value of 120595 Then the limits are applied on the equationusing the boundary conditions Now the potentials 119880

119871is

given by

1198801198711= sum

119899

sum

119898

119862 times 119870119860119899119898

sin(Λ119899(119909 minus

119882

2minus 119879OX))

times cos (119872119898119910) sinh(sum

119899119898

(119871 minus 119911))

for 1198822lt 119909 lt

119882

2+ 119879OX 0 lt 119910 lt

119867

2

(6)

1198801198712= sum

119899

sum

119898

119862 times 119870119861119899119898

cos (Λ119899119909)

times sin(119872119898(119910 minus

119867

2minus 119879OX))

times sinh(sum119899119898

(119871 minus 119911)) for 0 lt 119909 lt 119882

2

119867

2lt 119910 lt

119867

2+ 119879OX

(7)

1198801198713= sum

119899

sum

119898

119862 times 119870119862119899119898

cos (Λ119899119909) cos (119872

119898119910)

times sinh(sum119899119898

(119871 minus 119911))

for 0 lt 119909 lt 119882

2 0 lt 119910 lt

119867

2

(8)

Similarly the values of potential 119880119877are also derived as

follows

1198801198771= sum

119899

sum

119898

119862 times 119870119875119899119898

sin(Λ119899(119909 minus

119882

2minus 119879OX))

times cos (119872119898119910)

times sinh(sum119899119898

119911) for 1198822lt 119909 lt

119882

2+ 119879OX

0 lt 119910 lt119867

2

1198801198772= sum

119899

sum

119898

119862 times 119870119876119899119898

cos (Λ119899119909)

times sin(119872119898(119910 minus

119867

2minus 119879OX)) sinh(sum

119899119898

119911)

for 0 lt 119909 lt 119882

2

119867

2lt 119910 lt

119867

2+ 119879OX

1198801198773= sum

119899

sum

119898

119862 times 119870119877119899119898

cos (Λ119899119909) cos (119872

119898119910)

times sinh(sum119899119898

119911) for 0 lt 119909 lt 119882

2

0 lt 119910 lt119867

2

(9)

where 119888 = 1sum119899119898119871 and 119870

119860119899119898 119870119861119899119898

119870119862119899119898

119870119875119899119898

119870119902119899119898

and 119870

119877119899119898are constants From (6) and (8) 119880

119871and (120597119880

119871)120597119909

are found to be continuous in the 119909 direction (119909 = 1198822)The first derivative function (120597119880

1198711)120597119909 itself has 120576OX120576si

times discontinuities at the silicon insulator interfaces Thusapplying continuity in both equations we proceed to equate(6) and (8) as follows

minus119870119860119899119898

sin (Λ119899119879OX) = 119870119862119899119898 cos(

Λ119899119882

2) (10)

Differentiating (6) with respect to 119909

1205971198801198711

120597119909= sum

119899

sum

119898

119862 times 119870119860119899119898

(Λ119899) cos(Λ

119899(119909 minus

119882

2minus 119879OX))

times cos (119872119898119910) sinh(sum

119899119898

(119871 minus 119911))

(11)

Differentiating (8) with respect to 119909

1205971198801198713

120597119909= sum

119899

sum

119898

119862 times 119870119888119899119898

(Λ119899) (minus sin (Λ

119899119909))

times cos (119872119898119910) times sinh(sum

119899119898

(119871 minus 119911))

(12)

Equating (11) and (12)

119870119860119899119898

120576OX cos (Λ 119899119879OX) = minus119870119862119899119898120576si sin(Λ119899

2) (13)

Dividing (10) by (13)

120576si tan (Λ 119899119879OX) minus 120576OXcot(Λ119899119882

2) = 0 (14)

Likewise from (7) and (8) 119880119871and (120597119880

119871120597119910) are found to

be continuous in the 119910 direction (119910 = 1198672) The function(1205971198801198712120597119910) itself has discontinuities at the silicon insulator

interfaces which are proportional to the dielectric constant120576OX120576si Thus applying continuity in both equations andequating (7) and (8) we get

minus119870119861119899119898

sin (119872119898119879OX) = 119870119862119899119898 cos(

119872119898119867

2) (15)

4 Active and Passive Electronic Components

Differentiating (7) with respect to 119910

1205971198801198712

120597119910= sum

119899

sum

119898

119862 times 119870119861119899119898

cos (Λ119899119909) (119872

119898)

times cos(119872119898(119910 minus

119867

2minus 119879OX))

times sinh(sum119899119898

(119871 minus 119911))

(16)

Differentiating (8) with respect to 119910

1205971198801198713

120597119910= sum

119899

sum

119898

119862 times 119870119862119899119898

cos (Λ119899119909) (119872

119898)

times (minus sin (119872119898119910)) sinh(sum

119899119898

(119871 minus 119911))

(17)

Equating (16) and (17)

119870119861119899119898

120576OX cos (119872119898119879OX) = minus119870119862119899119898120576si sin(119872119898119867

2) (18)

Dividing (15) by (18)

120576si tan (119872119898119879OX) minus 120576OXcot(119872119898119867

2) = 0 (19)

This natural length is an easy guide for choosing deviceparameters and has simple physical meaning that a smallnatural length corresponds to superb short channel effectimmunity [4] The value of Λ

119899and 119872

119898depends on device

parameters The potential 119880119871can be modified as

1198801198711198991198981

= 120572119899119898

times 119862 times sin(Λ119899(119909 minus

119882

2minus 119879OX))

times cos (119872119898119910) times sinh(sum

119899119898

(119871 minus 119911))

for 1198822lt 119909 lt

119882

2+ 119879OX 0 lt 119910 lt

119867

2

1198801198711198991198982

= 120573119899119898

times 119862 times cos (Λ119899119909)

times sin(119872119898(119910 minus

119867

2minus 119879OX)) sinh(sum

119899119898

(119871 minus 119911))

for 0 lt 119909 lt 119882

2

119867

2lt 119910 lt

119867

2+ 119879OX

1198801198711198991198983

= 120574119899119898

times 119862 times cos (Λ119899119909) cos (119872

119898119910)

times sinhsum119899119898

(119871 minus 119911) for 0 lt 119909 lt 119882

2 0 lt 119910 lt

119867

2

(20)

Similarly the potential 119880119877can be modified as

1198801198771198991198981

= 120572119899119898

times 119862 times sin(Λ119899(119909 minus

119882

2minus 119879OX))

times cos (119872119898119910)

times sinh(sum119899119898

119911) for 1198822lt 119909 lt

119882

2+ 119879OX

0 lt 119910 lt119867

2

1198801198771198991198982

= 120573119899119898

times 119862 times cos (Λ119899119909)

times sin(119872119898(119910 minus

119867

2minus 119879OX)) times sinh(sum

119899119898

119911)

for 0 lt 119909 lt 119882

2

119867

2lt 119910 lt

119867

2+ 119879OX

1198801198771198991198983

= 120574119899119898

times cos (Λ119899119909) cos (119872

119898119910)

times sinh(sum119899119898

119911) for 0 lt 119909 lt 119882

2 0 lt 119910 lt

119867

2

(21)

Now the 119892119899119898

can be obtained from the potential equations(20) by using different multipliers in different regions

1198921198991198981

=120576OX sin (Λ 119899 (119909 minus1198822 minus 119879OX)) cos (119872119898119910)

2120576si sin (Λ 119899119879OX) cos (119872119898 (1198672))

for 1198822lt 119909 lt

119882

2+ 119879OX 0 lt 119910 lt

119867

2

1198921198991198982

=120576OX cos (Λ 119899119909) sin (119872119898 (119910 minus 1198672 minus 119879OX))

2120576si cos (Λ 119899 (1198822)) sin (119872119898119879OX)

for 0 lt 119909 lt 119882

2

119867

2lt 119910 lt

119867

2+ 119879OX

1198921198991198983

= minuscos (Λ

119899119909) cos (119872

119898119910)

cos (Λ119899(1198822)) cos (119872

119898(1198672))

for 0 lt 119909 lt 119882

2 0 lt 119910 lt

119882

2

(22)

Subsequently the constants 120572119899119898 120573119899119898 and 120574

119899119898are evaluated

suitably

120572119899119898

= cos(Λ119899

119882

2) sin (119872

119898119879OX)

120573119899119898

= cos(119872119898

119867

2) sin (Λ

119899119879OX)

120574119899119898

= sin (119872119898119879OX) sin (Λ 119899119879OX)

(23)

From (20) the potential 119880119871119899119898

can be rewritten as

1198801198711198991198981

= minus cos(Λ119899

119882

2) sin (119872

119898119879OX)

times sin (Λ119899119879OX) cos(119872119898

119867

2)

Active and Passive Electronic Components 5

1198801198711198991198982

= minus cos(119872119898

119867

2) sin (Λ

119899119879OX)

times cos(Λ119899

119882

2) sin (119872

119898119879OX)

1198801198711198991198983

= minus sin (119872119898119879OX) sin (Λ 119899119879OX)

times cos(Λ119899

119882

2) cos(119872

119898

119867

2)

(24)

By multiplying with the corresponding orthogonal conjugatefunctions and integrating coefficients of119880

119871can be obtained

The coefficients of 119880119877are also obtained in a similar method

119894119899119898

= (int

119867+119879OX

0

int

119882+119879OX

0

(120595 (119909 119910 0))

minus119881 (119909 119910) 119892119899119898(119909 119910) 119889119909 119889119910)

times (int

119867+119879OX

0

int

119882+119879OX

0

119880119871119899119898

(119909 119910 0) 119892119899119898(119909 119910) 119889119909 119889119910)

minus1

119895119899119898

= (int

119867+119879OX

0

int

119882+119879OX

0

(120595 (119909 119910 119871))

minus119881 (119909 119910) 119892119899119898(119909 119910) 119889119909 119889119910)

times (int

119867+119879OX

0

int

119882+119879OX

0

119880119871119899119898

(119909 119910 119871) 119892119899119898(119909 119910) 119889119909 119889119910)

minus1

(25)

The above integrals (25) are evaluated to obtain explicitexpressions for 119894

119899119898and 119895119899119898

as follows

119894119899119898

= (119881119892minus 120595bi) 120596119899119898

119895119899119898

= (119881119892minus 120595bi minus 119881ds) 120596119899119898

120596119899119898

=Ω119899119898

120585119899119898

(26)

where

Ω119899119898

=120576OXΛ 119899 tan ((Λ 119899119882) 2) + 120576OX119872119898 tan (1198721198981198672)

2120576si119879OXΛ2

1198991198722119898

minus2

Λ119899119872119898

tan(Λ119899119882

2) tan(

119872119898119867

2)

120585119899119898

=sin (Λ

119899119879OX)

16 cos (Λ1198991198822)

(119882 +sin (Λ

119899119882)

Λ119899

) 1199031

+sin (119872

119898119879OX)

16 cos (1198721198981198672)

(119867 +sin (119872

119898119867)

119872119898

) 1199032

(27)

And the values of 1199031and 1199032are given by

1199031=120576OX119879OX cos (1198721198981198672)

120576si sin (119872119898119879OX)+119867 sin (119872

119898119879OX)

2 cos (1198721198981198672)

1199032=120576OX119879OX cos (Λ 1198991198822)

120576si sin (Λ 119899119879OX)+119882 sin (Λ

119899119879OX)

2 cos (Λ1198991198822)

(28)

The potential equation is now rewritten as

120595 (119909 119910 119911) = 119881119892+ (119881119892minus 120595bi)

timessum

119899

sum

119898

120588119899119898

sinh(sum119899119898

(119871 minus 119911))

+ (119881119892minus 120595bi minus 119881ds)sum

119899

sum

119898

sinh(sum119899119898

119911)

(29)

where

120588119899119898(119909 119910) = minus120596

119899119898120574119899119898

cos (Λ119899119909) cos (119872

119898119910)

sinh (sum119899119898119871)

(30)

Once the potential distribution at every point of the cross-section of the channel is known we calculate the inversioncharge density by using surface integral over the surface areaof the channel When the integrated charge at virtual sourcebecomes equal to critical charge the gate voltage of a lightlydoped body device is nearly equal to the threshold voltage ofthe device Hence the inversion charge can be expressed as

119876 = int

1198672

minus1198672

int

1198822

minus1198822

119902119899119894119890(120595119880119879)119889119909 119889119910 (31)

where 119902 is the elementary charge 119880119879is thermal voltage and

119899119894is the intrinsic carrier concentrationThe charge equation can now be approximated as

119876 asymp 119882119867119902119899119894119890((120595((311988214)(311986714)119911119888))119880119879) (32)

Here 119911119888is the virtual source position which is half of the

channel length for low 119881ds Using the inversion charge wecan obtain the classical threshold model as expressed in thefollowing

119881TC = (119880119879 ln119876

119882119867119902119899119894

+ 2119881bi12058811 (3119882

143119867

14)

times sinh(sum11119871

2))

times (1 + 212058811(3119882

143119867

14) sinh(

sum11119871

2))

minus1

(33)

3 Quantum Threshold Voltage Modeling

AsMOSFETdevices are further scaled into the deep nanome-ter regime it has become necessary to include quantummechanical effects while modeling their device behaviorIn this paper we approximate the actual potential well as

6 Active and Passive Electronic Components

E11

Ec

Ec

998779Ec

Ei119909i119910

Eco

Figure 2 Band diagram perpendicular to the gate-square wellpotential of a GAA silicon nanowire transistor

the square well potential since it is difficult to solve theSchrodinger equation to obtain the potential expressed in(29) The square well potential of a gate-all-around nanowiretransistor is shown in Figure 2 The quantum charge of thedevice is expressed as

119876 = sum

119894119909

sum

119894119910

119902int

infin

119864119894119909119894119910

1198731D119891 (119864) 119889119864 (34)

where1198731D is the 1D density-of-states and 119891(119864) is the Fermi-

Dirac distribution function 119864 is the energy of the electronwave The terms 119894

119909and 119894119910are positive natural numbers

In silicon six energy valleys are found to be present inits band structure (two lower energy valleys two middleenergy valleys and two higher energy valleys) If the thinfilm of device has equal height and width the two lowerenergy valleys and two middle energy valleys are combinedtogether to produce four lower energy valleys and the othertwo higher energy valleys remain in their own stateThus thecharge is given by

119876 = 119902sum

119894119909

sum

119894119910

radic(119898119911

2120587ℎ2)int

infin

119864119894119909119894119910

(119864 minus 119864119894119909119894119910)minus12

1 + 119890 ((119864 minus 119864119865) 119896119879)

119889119864 (35)

where 119898119911is the mass of the valley which is perpendicular to

the direction of quantizationThe Fermi energy level is muchlower than the conduction band energy in weak inversionregion Hence the charge equation can be approximated as

119876 = 119902radic(119898119911

2120587ℎ2)sum

119894119909

sum

119894119910

int

infin

119864119894119909119894119910

119890((119864119865minus119864)119896119879)

(119864 minus 119864119894119909119894119910)12119889119864 (36)

Using the Schrodinger equation the value of 119864119894119909119864119894119910is deter-

mined by the following formulation [5]

119864119894119909119894119910

= 119864co +ℎ21205872

2[1

119898119909

(119894119909

119868119909

)

2

+1

119898119910

(119894119910

119868119910

)

2

] (37)

where the conduction band energy is given as

119864co =119864119892

2minus 119902120595 (0 0 119911

119888) (38)

Using (36) and (37) the integrated charge can be obtained as

119876 = 119902radic(2119896119879119898

119905

ℎ2)sum

119894119909

sum

119894119910

exp(minus119864co + 1198641 (119894119909 119894119910)

119896119879)

+ 119902radic(2119896119879119898

119905

ℎ2)sum

119894119909

sum

119894119910

exp(minus119864co + 1198642 (119894119909 119894119910)

119896119879)

+ 119902radic(2119896119879119898

119905

ℎ2)sum

119894119909

sum

119894119910

exp(minus119864co + 1198643 (119894119909 119894119910)

119896119879)

(39)

where

1198641(119894119909 119894119910) =

ℎ21205872

2[1

1198981

(119894119909

119882)

2

+1

119898119905

(119894119910

119867)

2

]

1198642(119894119909 119894119910) =

ℎ21205872

2[1

119898119905

(119894119909

119882)

2

+1

1198981

(119894119910

119867)

2

]

1198643(119894119909 119894119910) =

ℎ21205872

2[1

119898119905

(119894119909

119882)

2

+1

119898119905

(119894119910

119867)

2

]

(40)

Here the 119898119905and 119898

1are the transverse and longitudinal

effective masses of the energy valleys of silicon The lengths119894119909and 119894119910carry distinct values contingent on the direction of

quantization Finally the quantum threshold voltage modelbecomes

119881TQ = (119864119892

2119902+ (

119896119879

119902) In(119876119879

120591)

+2120595bi12058811 (0 0) sinh((sum11119871)

2))

times (1 + 212058811(0 0) sinh(

(sum11119871)

2))

minus1

(41)

where

120591 = 119902radic(2119896119879119898

119905

ℎ2) exp(minus1198641 (1 1)

119896119879)

+ 119902radic(2119896119879119898

119905

ℎ2) exp(minus1198642 (1 1)

119896119879)

+ 119902radic(2119896119879119898

119905

ℎ2) exp(minus

1198643(1 1)

119896119879)

(42)

Active and Passive Electronic Components 7

10 20 30 40 50 6002

022

024

026

028

03

032

034

036

038

04

Channel length (nm)

Pote

ntia

l (V

)

Proposed modelTCAD simulation

Figure 3 Constant electrostatic potential obtained from the analyt-ical solution of a gate-all-around silicon nanowire transistor is 03 Vfor different channel length with a height of119867 = 9 nm and channelwidth of 119882 = 9 nm TCAD simulation shows that the potential isconstant at 0296V

The impacts on the threshold voltage due to quantum effectsare acquired by using the following equation

119881TC = 119881TQ + Δ119881119879 (43)

Here Δ119881119879is the difference between the quantum threshold

voltage and the classical threshold voltage

4 Results and Discussion

Figure 3 shows the electrostatic potential of the proposed gateall around transistor and it is found to be constant value at03 V Continuously varying the 119899 and 119898 terms in (4) hasno impact on the potential as it remains constant along theinsulator boundaries This is totally in contrast to the resultsobtained in [15] where the potential is found to be linearlyvarying in the insulator boundaries The constant potentialhas to be deduced as the resultant of the gate voltage appliedsymmetrically across the four sides of the transistor TheTCAD simulation of the device shows that the electrostaticpotential is constant at 0296V The simulation results arefound to in acceptance with the TCAD results

Figure 4 represents the variation of total quantum inte-grated charge with the gate voltage Equation (39) is usedto obtain the integrated charge with only one energy leveland one series term It clearly shows that the decrease inthe film thickness leads to the increase in the quantumthreshold voltage which is actually due to the increase inenergy quantization of the transistor With the height andlength of the device being constant the width of the device isvaried and henceforth the variation of charge in accordancewith the gate voltage is illustrated in Figure 4

The variation of quantum threshold voltage with widthand height of the film at a channel length of 20 nm is

0 01 02 03 04 05 06 070

05

1

15

2

25

3

35

4times10minus15

Gate voltage (V)

Char

ge (c

mminus1)

W = 9e minus 9

W = 5e minus 9

W = 3e minus 9

TCAD simulation for W = 9e minus 9

TCAD simulation for W = 5e minus 9

TCAD simulation for W = 3e minus 9

Figure 4 Variation of quantum integrated charge at virtual sourcewith gate voltage for different film widths where the height andlength are119867 = 9 nm 119871 = 20 nm

2

8

times10minus9

times10minus9

Film height

2

44

66

8

10

1003

0305

031

0315

032

Film width

Qua

ntum

thre

shol

d vo

ltage

(V)

ProposedTCAD

Figure 5 Variation of quantum threshold voltage with film heightand width for channel length (119871 = 20 nm)

shown in Figure 5 The short channel effects tend to decreasealong with the energy quantisation and this can be furtherexplained as a result of increase in the effective band gap ofsilicon due to quantum effects The effect of confinementexpressed as the difference in the threshold voltage and itsvariation with the channel length 119871 is illustrated in Figure 6

8 Active and Passive Electronic Components

2 3 4 5 6 7 8 9 10

times10minus9

0014

0016

0018

002

0022

0024

0026

0028

003

0032

Film height

Proposed modelTCAD simulation

ΔVT

(mV

)

Figure 6 Variation of threshold voltagewith filmwidth for differentheight at 119871 = 20 nm

3 4 5 6 7 8

times10minus9

02

022

024

026

028

03

032

034

036

038

04

Film height

Thre

shol

d vo

ltage

(V)

ClassicalQuantum

TCAD simulation for classicalTCAD simulation for quantum

9

Figure 7 Variation of quantum and classical threshold voltage withfilm height Here 119871 = 20 nm and119882 = 9 nm

Themost important thing about this gate all around nanowiretransistor is that any change in one of the dimensions canbe nullified by proper tuning of other dimensions as thetransistor is symmetric about its height and width

Figure 7 shows the variation of the classical thresholdvoltage and quantum threshold voltage with the film height ata constant width of 9 nmThe value of the classical thresholdvoltage ranges from 027V to 029V for the correspondingchanges in the film height Similarly the quantum thresholdvoltage ranges from 03V to 031 V It shows that the devicehas a highly improved control over the threshold voltageTheTCAD results justify the simulation results

5 Conclusion

In this paper a quantum threshold voltage model for a GAAsilicon nanowire transistor is proposed by solving the 3DPoisson and Schrodinger equations Analytical expressionsfor potential and the inversion charge are expressed intheir closed forms The results show that the integratedcharge and the threshold voltage calculated in accordancewith the quantum effects of this proposed model are highlyimprovedThe future considerations include deriving the I-Vcharacteristics of the gate all around nanowire transistors andstudying the impact of scaling on various device parametersFinally to conclude this model provides an analytical anduseful way for the threshold voltage evaluations in gate allaround nanowire devices with a unified formalism employedin both classical and quantum mechanical approaches

References

[1] J Wang E Polizzi and M Lundstrom ldquoA computationalstudy of ballistic silicon nanowire transistorsrdquo in Proceedings ofthe IEEE International Electron Devices Meeting pp 695ndash698December 2003

[2] J P Colinge FINFETS and Other Multi-Gate TransistorsSpringer New York NY USA 2007

[3] D J Frank Y Taur andH-S PWong ldquoGeneralized scale lengthfor two-dimensional effects in MOSFETrsquosrdquo IEEE ElectronDevice Letters vol 19 no 10 pp 385ndash387 1998

[4] B Yu L Wang Y Yuan P M Asbeck and Y Taur ldquoScaling ofnanowire transistorsrdquo IEEE Transactions on Electron Devicesvol 55 no 11 pp 2846ndash2858 2008

[5] G D Sanders C J Stanton and Y C Chang ldquoTheory of trans-port in silicon quantum wiresrdquo Physical Review B vol 48 no15 pp 11067ndash11076 1993

[6] M-Y Shen and S-L Zhang ldquoBand gap of a silicon quantumwirerdquo Physics Letters A vol 176 no 3-4 pp 254ndash258 1993

[7] J P Colinge X Baie V Bayot and E Grivei ldquoQuantum-wireeffects in thin and narrow SOI MOSFETsrdquo in Proceedings of theIEEE International SOI Conference pp 66ndash67 October 1995

[8] C P Auth and J D Plummer ldquoScaling theory for cylindricalfully-depleted surrounding-gate MOSFETrsquosrdquo IEEE ElectronDevice Letters vol 18 no 2 pp 74ndash76 1997

[9] J-T Park and J-P Colinge ldquoMultiple-gate SOI MOSFETsdevice design guidelinesrdquo IEEE Transactions on ElectronDevices vol 49 no 12 pp 2222ndash2229 2002

[10] Z Ghoggali F DjeffalM A Abdi D Arar N NLakhdar and TTBendib ldquoAn analytical threshold voltagemodel for nanoscalerdquoin Proceedings of the 3rd International Design and TestWorkshop(IDT rsquo08) pp 93ndash97 December 2008

[11] Y-S Wu and P Su ldquoQuantum confinement effect in short-channel gate-all-around MOSFETs and its impact on the sensi-tivity of threshold voltage to process variationsrdquo in Proceedingsof the IEEE International SOI Conference October 2009

[12] C Te-Kuang ldquoA compact analytical threshold-voltage modelfor surrounding-gateMOSFETswith interface trapped chargesrdquoIEEE Electron Device Letters vol 31 no 8 pp 788ndash790 2010

[13] B Ray and SMahapatra ldquoModeling and analysis of body poten-tial of cylindrical gate-all-around nanowire transistorrdquo IEEETransactions on Electron Devices vol 55 no 9 pp 2409ndash24162008

Active and Passive Electronic Components 9

[14] L De Michielis L Selmi and A M Ionescu ldquoA quasi-analytical model for nanowire FETs with arbitrary polygonalcross sectionrdquo Solid-State Electronics vol 54 no 9 pp 929ndash9342010

[15] P R Kumar and S Mahapatra ldquoQuantum threshold voltagemodeling of short channel quad gate silicon nanowire transis-torrdquo IEEE Transactions onNanotechnology vol 10 no 1 pp 121ndash128 2011

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Active and Passive Electronic Components 3

the term 119881119892+ 119880119871is found to satisfy the potential equation

when119880119877is on null value and in an exact repetition the term

119881119892+ 119880119877satisfies the potential equation when 119880

119871is on null

value From (1)

120595119909119909+ 120595119910119910+ 120595119911119911= 0 (5)

By solving the above equation using LDE method we obtainthe value of 120595 Then the limits are applied on the equationusing the boundary conditions Now the potentials 119880

119871is

given by

1198801198711= sum

119899

sum

119898

119862 times 119870119860119899119898

sin(Λ119899(119909 minus

119882

2minus 119879OX))

times cos (119872119898119910) sinh(sum

119899119898

(119871 minus 119911))

for 1198822lt 119909 lt

119882

2+ 119879OX 0 lt 119910 lt

119867

2

(6)

1198801198712= sum

119899

sum

119898

119862 times 119870119861119899119898

cos (Λ119899119909)

times sin(119872119898(119910 minus

119867

2minus 119879OX))

times sinh(sum119899119898

(119871 minus 119911)) for 0 lt 119909 lt 119882

2

119867

2lt 119910 lt

119867

2+ 119879OX

(7)

1198801198713= sum

119899

sum

119898

119862 times 119870119862119899119898

cos (Λ119899119909) cos (119872

119898119910)

times sinh(sum119899119898

(119871 minus 119911))

for 0 lt 119909 lt 119882

2 0 lt 119910 lt

119867

2

(8)

Similarly the values of potential 119880119877are also derived as

follows

1198801198771= sum

119899

sum

119898

119862 times 119870119875119899119898

sin(Λ119899(119909 minus

119882

2minus 119879OX))

times cos (119872119898119910)

times sinh(sum119899119898

119911) for 1198822lt 119909 lt

119882

2+ 119879OX

0 lt 119910 lt119867

2

1198801198772= sum

119899

sum

119898

119862 times 119870119876119899119898

cos (Λ119899119909)

times sin(119872119898(119910 minus

119867

2minus 119879OX)) sinh(sum

119899119898

119911)

for 0 lt 119909 lt 119882

2

119867

2lt 119910 lt

119867

2+ 119879OX

1198801198773= sum

119899

sum

119898

119862 times 119870119877119899119898

cos (Λ119899119909) cos (119872

119898119910)

times sinh(sum119899119898

119911) for 0 lt 119909 lt 119882

2

0 lt 119910 lt119867

2

(9)

where 119888 = 1sum119899119898119871 and 119870

119860119899119898 119870119861119899119898

119870119862119899119898

119870119875119899119898

119870119902119899119898

and 119870

119877119899119898are constants From (6) and (8) 119880

119871and (120597119880

119871)120597119909

are found to be continuous in the 119909 direction (119909 = 1198822)The first derivative function (120597119880

1198711)120597119909 itself has 120576OX120576si

times discontinuities at the silicon insulator interfaces Thusapplying continuity in both equations we proceed to equate(6) and (8) as follows

minus119870119860119899119898

sin (Λ119899119879OX) = 119870119862119899119898 cos(

Λ119899119882

2) (10)

Differentiating (6) with respect to 119909

1205971198801198711

120597119909= sum

119899

sum

119898

119862 times 119870119860119899119898

(Λ119899) cos(Λ

119899(119909 minus

119882

2minus 119879OX))

times cos (119872119898119910) sinh(sum

119899119898

(119871 minus 119911))

(11)

Differentiating (8) with respect to 119909

1205971198801198713

120597119909= sum

119899

sum

119898

119862 times 119870119888119899119898

(Λ119899) (minus sin (Λ

119899119909))

times cos (119872119898119910) times sinh(sum

119899119898

(119871 minus 119911))

(12)

Equating (11) and (12)

119870119860119899119898

120576OX cos (Λ 119899119879OX) = minus119870119862119899119898120576si sin(Λ119899

2) (13)

Dividing (10) by (13)

120576si tan (Λ 119899119879OX) minus 120576OXcot(Λ119899119882

2) = 0 (14)

Likewise from (7) and (8) 119880119871and (120597119880

119871120597119910) are found to

be continuous in the 119910 direction (119910 = 1198672) The function(1205971198801198712120597119910) itself has discontinuities at the silicon insulator

interfaces which are proportional to the dielectric constant120576OX120576si Thus applying continuity in both equations andequating (7) and (8) we get

minus119870119861119899119898

sin (119872119898119879OX) = 119870119862119899119898 cos(

119872119898119867

2) (15)

4 Active and Passive Electronic Components

Differentiating (7) with respect to 119910

1205971198801198712

120597119910= sum

119899

sum

119898

119862 times 119870119861119899119898

cos (Λ119899119909) (119872

119898)

times cos(119872119898(119910 minus

119867

2minus 119879OX))

times sinh(sum119899119898

(119871 minus 119911))

(16)

Differentiating (8) with respect to 119910

1205971198801198713

120597119910= sum

119899

sum

119898

119862 times 119870119862119899119898

cos (Λ119899119909) (119872

119898)

times (minus sin (119872119898119910)) sinh(sum

119899119898

(119871 minus 119911))

(17)

Equating (16) and (17)

119870119861119899119898

120576OX cos (119872119898119879OX) = minus119870119862119899119898120576si sin(119872119898119867

2) (18)

Dividing (15) by (18)

120576si tan (119872119898119879OX) minus 120576OXcot(119872119898119867

2) = 0 (19)

This natural length is an easy guide for choosing deviceparameters and has simple physical meaning that a smallnatural length corresponds to superb short channel effectimmunity [4] The value of Λ

119899and 119872

119898depends on device

parameters The potential 119880119871can be modified as

1198801198711198991198981

= 120572119899119898

times 119862 times sin(Λ119899(119909 minus

119882

2minus 119879OX))

times cos (119872119898119910) times sinh(sum

119899119898

(119871 minus 119911))

for 1198822lt 119909 lt

119882

2+ 119879OX 0 lt 119910 lt

119867

2

1198801198711198991198982

= 120573119899119898

times 119862 times cos (Λ119899119909)

times sin(119872119898(119910 minus

119867

2minus 119879OX)) sinh(sum

119899119898

(119871 minus 119911))

for 0 lt 119909 lt 119882

2

119867

2lt 119910 lt

119867

2+ 119879OX

1198801198711198991198983

= 120574119899119898

times 119862 times cos (Λ119899119909) cos (119872

119898119910)

times sinhsum119899119898

(119871 minus 119911) for 0 lt 119909 lt 119882

2 0 lt 119910 lt

119867

2

(20)

Similarly the potential 119880119877can be modified as

1198801198771198991198981

= 120572119899119898

times 119862 times sin(Λ119899(119909 minus

119882

2minus 119879OX))

times cos (119872119898119910)

times sinh(sum119899119898

119911) for 1198822lt 119909 lt

119882

2+ 119879OX

0 lt 119910 lt119867

2

1198801198771198991198982

= 120573119899119898

times 119862 times cos (Λ119899119909)

times sin(119872119898(119910 minus

119867

2minus 119879OX)) times sinh(sum

119899119898

119911)

for 0 lt 119909 lt 119882

2

119867

2lt 119910 lt

119867

2+ 119879OX

1198801198771198991198983

= 120574119899119898

times cos (Λ119899119909) cos (119872

119898119910)

times sinh(sum119899119898

119911) for 0 lt 119909 lt 119882

2 0 lt 119910 lt

119867

2

(21)

Now the 119892119899119898

can be obtained from the potential equations(20) by using different multipliers in different regions

1198921198991198981

=120576OX sin (Λ 119899 (119909 minus1198822 minus 119879OX)) cos (119872119898119910)

2120576si sin (Λ 119899119879OX) cos (119872119898 (1198672))

for 1198822lt 119909 lt

119882

2+ 119879OX 0 lt 119910 lt

119867

2

1198921198991198982

=120576OX cos (Λ 119899119909) sin (119872119898 (119910 minus 1198672 minus 119879OX))

2120576si cos (Λ 119899 (1198822)) sin (119872119898119879OX)

for 0 lt 119909 lt 119882

2

119867

2lt 119910 lt

119867

2+ 119879OX

1198921198991198983

= minuscos (Λ

119899119909) cos (119872

119898119910)

cos (Λ119899(1198822)) cos (119872

119898(1198672))

for 0 lt 119909 lt 119882

2 0 lt 119910 lt

119882

2

(22)

Subsequently the constants 120572119899119898 120573119899119898 and 120574

119899119898are evaluated

suitably

120572119899119898

= cos(Λ119899

119882

2) sin (119872

119898119879OX)

120573119899119898

= cos(119872119898

119867

2) sin (Λ

119899119879OX)

120574119899119898

= sin (119872119898119879OX) sin (Λ 119899119879OX)

(23)

From (20) the potential 119880119871119899119898

can be rewritten as

1198801198711198991198981

= minus cos(Λ119899

119882

2) sin (119872

119898119879OX)

times sin (Λ119899119879OX) cos(119872119898

119867

2)

Active and Passive Electronic Components 5

1198801198711198991198982

= minus cos(119872119898

119867

2) sin (Λ

119899119879OX)

times cos(Λ119899

119882

2) sin (119872

119898119879OX)

1198801198711198991198983

= minus sin (119872119898119879OX) sin (Λ 119899119879OX)

times cos(Λ119899

119882

2) cos(119872

119898

119867

2)

(24)

By multiplying with the corresponding orthogonal conjugatefunctions and integrating coefficients of119880

119871can be obtained

The coefficients of 119880119877are also obtained in a similar method

119894119899119898

= (int

119867+119879OX

0

int

119882+119879OX

0

(120595 (119909 119910 0))

minus119881 (119909 119910) 119892119899119898(119909 119910) 119889119909 119889119910)

times (int

119867+119879OX

0

int

119882+119879OX

0

119880119871119899119898

(119909 119910 0) 119892119899119898(119909 119910) 119889119909 119889119910)

minus1

119895119899119898

= (int

119867+119879OX

0

int

119882+119879OX

0

(120595 (119909 119910 119871))

minus119881 (119909 119910) 119892119899119898(119909 119910) 119889119909 119889119910)

times (int

119867+119879OX

0

int

119882+119879OX

0

119880119871119899119898

(119909 119910 119871) 119892119899119898(119909 119910) 119889119909 119889119910)

minus1

(25)

The above integrals (25) are evaluated to obtain explicitexpressions for 119894

119899119898and 119895119899119898

as follows

119894119899119898

= (119881119892minus 120595bi) 120596119899119898

119895119899119898

= (119881119892minus 120595bi minus 119881ds) 120596119899119898

120596119899119898

=Ω119899119898

120585119899119898

(26)

where

Ω119899119898

=120576OXΛ 119899 tan ((Λ 119899119882) 2) + 120576OX119872119898 tan (1198721198981198672)

2120576si119879OXΛ2

1198991198722119898

minus2

Λ119899119872119898

tan(Λ119899119882

2) tan(

119872119898119867

2)

120585119899119898

=sin (Λ

119899119879OX)

16 cos (Λ1198991198822)

(119882 +sin (Λ

119899119882)

Λ119899

) 1199031

+sin (119872

119898119879OX)

16 cos (1198721198981198672)

(119867 +sin (119872

119898119867)

119872119898

) 1199032

(27)

And the values of 1199031and 1199032are given by

1199031=120576OX119879OX cos (1198721198981198672)

120576si sin (119872119898119879OX)+119867 sin (119872

119898119879OX)

2 cos (1198721198981198672)

1199032=120576OX119879OX cos (Λ 1198991198822)

120576si sin (Λ 119899119879OX)+119882 sin (Λ

119899119879OX)

2 cos (Λ1198991198822)

(28)

The potential equation is now rewritten as

120595 (119909 119910 119911) = 119881119892+ (119881119892minus 120595bi)

timessum

119899

sum

119898

120588119899119898

sinh(sum119899119898

(119871 minus 119911))

+ (119881119892minus 120595bi minus 119881ds)sum

119899

sum

119898

sinh(sum119899119898

119911)

(29)

where

120588119899119898(119909 119910) = minus120596

119899119898120574119899119898

cos (Λ119899119909) cos (119872

119898119910)

sinh (sum119899119898119871)

(30)

Once the potential distribution at every point of the cross-section of the channel is known we calculate the inversioncharge density by using surface integral over the surface areaof the channel When the integrated charge at virtual sourcebecomes equal to critical charge the gate voltage of a lightlydoped body device is nearly equal to the threshold voltage ofthe device Hence the inversion charge can be expressed as

119876 = int

1198672

minus1198672

int

1198822

minus1198822

119902119899119894119890(120595119880119879)119889119909 119889119910 (31)

where 119902 is the elementary charge 119880119879is thermal voltage and

119899119894is the intrinsic carrier concentrationThe charge equation can now be approximated as

119876 asymp 119882119867119902119899119894119890((120595((311988214)(311986714)119911119888))119880119879) (32)

Here 119911119888is the virtual source position which is half of the

channel length for low 119881ds Using the inversion charge wecan obtain the classical threshold model as expressed in thefollowing

119881TC = (119880119879 ln119876

119882119867119902119899119894

+ 2119881bi12058811 (3119882

143119867

14)

times sinh(sum11119871

2))

times (1 + 212058811(3119882

143119867

14) sinh(

sum11119871

2))

minus1

(33)

3 Quantum Threshold Voltage Modeling

AsMOSFETdevices are further scaled into the deep nanome-ter regime it has become necessary to include quantummechanical effects while modeling their device behaviorIn this paper we approximate the actual potential well as

6 Active and Passive Electronic Components

E11

Ec

Ec

998779Ec

Ei119909i119910

Eco

Figure 2 Band diagram perpendicular to the gate-square wellpotential of a GAA silicon nanowire transistor

the square well potential since it is difficult to solve theSchrodinger equation to obtain the potential expressed in(29) The square well potential of a gate-all-around nanowiretransistor is shown in Figure 2 The quantum charge of thedevice is expressed as

119876 = sum

119894119909

sum

119894119910

119902int

infin

119864119894119909119894119910

1198731D119891 (119864) 119889119864 (34)

where1198731D is the 1D density-of-states and 119891(119864) is the Fermi-

Dirac distribution function 119864 is the energy of the electronwave The terms 119894

119909and 119894119910are positive natural numbers

In silicon six energy valleys are found to be present inits band structure (two lower energy valleys two middleenergy valleys and two higher energy valleys) If the thinfilm of device has equal height and width the two lowerenergy valleys and two middle energy valleys are combinedtogether to produce four lower energy valleys and the othertwo higher energy valleys remain in their own stateThus thecharge is given by

119876 = 119902sum

119894119909

sum

119894119910

radic(119898119911

2120587ℎ2)int

infin

119864119894119909119894119910

(119864 minus 119864119894119909119894119910)minus12

1 + 119890 ((119864 minus 119864119865) 119896119879)

119889119864 (35)

where 119898119911is the mass of the valley which is perpendicular to

the direction of quantizationThe Fermi energy level is muchlower than the conduction band energy in weak inversionregion Hence the charge equation can be approximated as

119876 = 119902radic(119898119911

2120587ℎ2)sum

119894119909

sum

119894119910

int

infin

119864119894119909119894119910

119890((119864119865minus119864)119896119879)

(119864 minus 119864119894119909119894119910)12119889119864 (36)

Using the Schrodinger equation the value of 119864119894119909119864119894119910is deter-

mined by the following formulation [5]

119864119894119909119894119910

= 119864co +ℎ21205872

2[1

119898119909

(119894119909

119868119909

)

2

+1

119898119910

(119894119910

119868119910

)

2

] (37)

where the conduction band energy is given as

119864co =119864119892

2minus 119902120595 (0 0 119911

119888) (38)

Using (36) and (37) the integrated charge can be obtained as

119876 = 119902radic(2119896119879119898

119905

ℎ2)sum

119894119909

sum

119894119910

exp(minus119864co + 1198641 (119894119909 119894119910)

119896119879)

+ 119902radic(2119896119879119898

119905

ℎ2)sum

119894119909

sum

119894119910

exp(minus119864co + 1198642 (119894119909 119894119910)

119896119879)

+ 119902radic(2119896119879119898

119905

ℎ2)sum

119894119909

sum

119894119910

exp(minus119864co + 1198643 (119894119909 119894119910)

119896119879)

(39)

where

1198641(119894119909 119894119910) =

ℎ21205872

2[1

1198981

(119894119909

119882)

2

+1

119898119905

(119894119910

119867)

2

]

1198642(119894119909 119894119910) =

ℎ21205872

2[1

119898119905

(119894119909

119882)

2

+1

1198981

(119894119910

119867)

2

]

1198643(119894119909 119894119910) =

ℎ21205872

2[1

119898119905

(119894119909

119882)

2

+1

119898119905

(119894119910

119867)

2

]

(40)

Here the 119898119905and 119898

1are the transverse and longitudinal

effective masses of the energy valleys of silicon The lengths119894119909and 119894119910carry distinct values contingent on the direction of

quantization Finally the quantum threshold voltage modelbecomes

119881TQ = (119864119892

2119902+ (

119896119879

119902) In(119876119879

120591)

+2120595bi12058811 (0 0) sinh((sum11119871)

2))

times (1 + 212058811(0 0) sinh(

(sum11119871)

2))

minus1

(41)

where

120591 = 119902radic(2119896119879119898

119905

ℎ2) exp(minus1198641 (1 1)

119896119879)

+ 119902radic(2119896119879119898

119905

ℎ2) exp(minus1198642 (1 1)

119896119879)

+ 119902radic(2119896119879119898

119905

ℎ2) exp(minus

1198643(1 1)

119896119879)

(42)

Active and Passive Electronic Components 7

10 20 30 40 50 6002

022

024

026

028

03

032

034

036

038

04

Channel length (nm)

Pote

ntia

l (V

)

Proposed modelTCAD simulation

Figure 3 Constant electrostatic potential obtained from the analyt-ical solution of a gate-all-around silicon nanowire transistor is 03 Vfor different channel length with a height of119867 = 9 nm and channelwidth of 119882 = 9 nm TCAD simulation shows that the potential isconstant at 0296V

The impacts on the threshold voltage due to quantum effectsare acquired by using the following equation

119881TC = 119881TQ + Δ119881119879 (43)

Here Δ119881119879is the difference between the quantum threshold

voltage and the classical threshold voltage

4 Results and Discussion

Figure 3 shows the electrostatic potential of the proposed gateall around transistor and it is found to be constant value at03 V Continuously varying the 119899 and 119898 terms in (4) hasno impact on the potential as it remains constant along theinsulator boundaries This is totally in contrast to the resultsobtained in [15] where the potential is found to be linearlyvarying in the insulator boundaries The constant potentialhas to be deduced as the resultant of the gate voltage appliedsymmetrically across the four sides of the transistor TheTCAD simulation of the device shows that the electrostaticpotential is constant at 0296V The simulation results arefound to in acceptance with the TCAD results

Figure 4 represents the variation of total quantum inte-grated charge with the gate voltage Equation (39) is usedto obtain the integrated charge with only one energy leveland one series term It clearly shows that the decrease inthe film thickness leads to the increase in the quantumthreshold voltage which is actually due to the increase inenergy quantization of the transistor With the height andlength of the device being constant the width of the device isvaried and henceforth the variation of charge in accordancewith the gate voltage is illustrated in Figure 4

The variation of quantum threshold voltage with widthand height of the film at a channel length of 20 nm is

0 01 02 03 04 05 06 070

05

1

15

2

25

3

35

4times10minus15

Gate voltage (V)

Char

ge (c

mminus1)

W = 9e minus 9

W = 5e minus 9

W = 3e minus 9

TCAD simulation for W = 9e minus 9

TCAD simulation for W = 5e minus 9

TCAD simulation for W = 3e minus 9

Figure 4 Variation of quantum integrated charge at virtual sourcewith gate voltage for different film widths where the height andlength are119867 = 9 nm 119871 = 20 nm

2

8

times10minus9

times10minus9

Film height

2

44

66

8

10

1003

0305

031

0315

032

Film width

Qua

ntum

thre

shol

d vo

ltage

(V)

ProposedTCAD

Figure 5 Variation of quantum threshold voltage with film heightand width for channel length (119871 = 20 nm)

shown in Figure 5 The short channel effects tend to decreasealong with the energy quantisation and this can be furtherexplained as a result of increase in the effective band gap ofsilicon due to quantum effects The effect of confinementexpressed as the difference in the threshold voltage and itsvariation with the channel length 119871 is illustrated in Figure 6

8 Active and Passive Electronic Components

2 3 4 5 6 7 8 9 10

times10minus9

0014

0016

0018

002

0022

0024

0026

0028

003

0032

Film height

Proposed modelTCAD simulation

ΔVT

(mV

)

Figure 6 Variation of threshold voltagewith filmwidth for differentheight at 119871 = 20 nm

3 4 5 6 7 8

times10minus9

02

022

024

026

028

03

032

034

036

038

04

Film height

Thre

shol

d vo

ltage

(V)

ClassicalQuantum

TCAD simulation for classicalTCAD simulation for quantum

9

Figure 7 Variation of quantum and classical threshold voltage withfilm height Here 119871 = 20 nm and119882 = 9 nm

Themost important thing about this gate all around nanowiretransistor is that any change in one of the dimensions canbe nullified by proper tuning of other dimensions as thetransistor is symmetric about its height and width

Figure 7 shows the variation of the classical thresholdvoltage and quantum threshold voltage with the film height ata constant width of 9 nmThe value of the classical thresholdvoltage ranges from 027V to 029V for the correspondingchanges in the film height Similarly the quantum thresholdvoltage ranges from 03V to 031 V It shows that the devicehas a highly improved control over the threshold voltageTheTCAD results justify the simulation results

5 Conclusion

In this paper a quantum threshold voltage model for a GAAsilicon nanowire transistor is proposed by solving the 3DPoisson and Schrodinger equations Analytical expressionsfor potential and the inversion charge are expressed intheir closed forms The results show that the integratedcharge and the threshold voltage calculated in accordancewith the quantum effects of this proposed model are highlyimprovedThe future considerations include deriving the I-Vcharacteristics of the gate all around nanowire transistors andstudying the impact of scaling on various device parametersFinally to conclude this model provides an analytical anduseful way for the threshold voltage evaluations in gate allaround nanowire devices with a unified formalism employedin both classical and quantum mechanical approaches

References

[1] J Wang E Polizzi and M Lundstrom ldquoA computationalstudy of ballistic silicon nanowire transistorsrdquo in Proceedings ofthe IEEE International Electron Devices Meeting pp 695ndash698December 2003

[2] J P Colinge FINFETS and Other Multi-Gate TransistorsSpringer New York NY USA 2007

[3] D J Frank Y Taur andH-S PWong ldquoGeneralized scale lengthfor two-dimensional effects in MOSFETrsquosrdquo IEEE ElectronDevice Letters vol 19 no 10 pp 385ndash387 1998

[4] B Yu L Wang Y Yuan P M Asbeck and Y Taur ldquoScaling ofnanowire transistorsrdquo IEEE Transactions on Electron Devicesvol 55 no 11 pp 2846ndash2858 2008

[5] G D Sanders C J Stanton and Y C Chang ldquoTheory of trans-port in silicon quantum wiresrdquo Physical Review B vol 48 no15 pp 11067ndash11076 1993

[6] M-Y Shen and S-L Zhang ldquoBand gap of a silicon quantumwirerdquo Physics Letters A vol 176 no 3-4 pp 254ndash258 1993

[7] J P Colinge X Baie V Bayot and E Grivei ldquoQuantum-wireeffects in thin and narrow SOI MOSFETsrdquo in Proceedings of theIEEE International SOI Conference pp 66ndash67 October 1995

[8] C P Auth and J D Plummer ldquoScaling theory for cylindricalfully-depleted surrounding-gate MOSFETrsquosrdquo IEEE ElectronDevice Letters vol 18 no 2 pp 74ndash76 1997

[9] J-T Park and J-P Colinge ldquoMultiple-gate SOI MOSFETsdevice design guidelinesrdquo IEEE Transactions on ElectronDevices vol 49 no 12 pp 2222ndash2229 2002

[10] Z Ghoggali F DjeffalM A Abdi D Arar N NLakhdar and TTBendib ldquoAn analytical threshold voltagemodel for nanoscalerdquoin Proceedings of the 3rd International Design and TestWorkshop(IDT rsquo08) pp 93ndash97 December 2008

[11] Y-S Wu and P Su ldquoQuantum confinement effect in short-channel gate-all-around MOSFETs and its impact on the sensi-tivity of threshold voltage to process variationsrdquo in Proceedingsof the IEEE International SOI Conference October 2009

[12] C Te-Kuang ldquoA compact analytical threshold-voltage modelfor surrounding-gateMOSFETswith interface trapped chargesrdquoIEEE Electron Device Letters vol 31 no 8 pp 788ndash790 2010

[13] B Ray and SMahapatra ldquoModeling and analysis of body poten-tial of cylindrical gate-all-around nanowire transistorrdquo IEEETransactions on Electron Devices vol 55 no 9 pp 2409ndash24162008

Active and Passive Electronic Components 9

[14] L De Michielis L Selmi and A M Ionescu ldquoA quasi-analytical model for nanowire FETs with arbitrary polygonalcross sectionrdquo Solid-State Electronics vol 54 no 9 pp 929ndash9342010

[15] P R Kumar and S Mahapatra ldquoQuantum threshold voltagemodeling of short channel quad gate silicon nanowire transis-torrdquo IEEE Transactions onNanotechnology vol 10 no 1 pp 121ndash128 2011

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4 Active and Passive Electronic Components

Differentiating (7) with respect to 119910

1205971198801198712

120597119910= sum

119899

sum

119898

119862 times 119870119861119899119898

cos (Λ119899119909) (119872

119898)

times cos(119872119898(119910 minus

119867

2minus 119879OX))

times sinh(sum119899119898

(119871 minus 119911))

(16)

Differentiating (8) with respect to 119910

1205971198801198713

120597119910= sum

119899

sum

119898

119862 times 119870119862119899119898

cos (Λ119899119909) (119872

119898)

times (minus sin (119872119898119910)) sinh(sum

119899119898

(119871 minus 119911))

(17)

Equating (16) and (17)

119870119861119899119898

120576OX cos (119872119898119879OX) = minus119870119862119899119898120576si sin(119872119898119867

2) (18)

Dividing (15) by (18)

120576si tan (119872119898119879OX) minus 120576OXcot(119872119898119867

2) = 0 (19)

This natural length is an easy guide for choosing deviceparameters and has simple physical meaning that a smallnatural length corresponds to superb short channel effectimmunity [4] The value of Λ

119899and 119872

119898depends on device

parameters The potential 119880119871can be modified as

1198801198711198991198981

= 120572119899119898

times 119862 times sin(Λ119899(119909 minus

119882

2minus 119879OX))

times cos (119872119898119910) times sinh(sum

119899119898

(119871 minus 119911))

for 1198822lt 119909 lt

119882

2+ 119879OX 0 lt 119910 lt

119867

2

1198801198711198991198982

= 120573119899119898

times 119862 times cos (Λ119899119909)

times sin(119872119898(119910 minus

119867

2minus 119879OX)) sinh(sum

119899119898

(119871 minus 119911))

for 0 lt 119909 lt 119882

2

119867

2lt 119910 lt

119867

2+ 119879OX

1198801198711198991198983

= 120574119899119898

times 119862 times cos (Λ119899119909) cos (119872

119898119910)

times sinhsum119899119898

(119871 minus 119911) for 0 lt 119909 lt 119882

2 0 lt 119910 lt

119867

2

(20)

Similarly the potential 119880119877can be modified as

1198801198771198991198981

= 120572119899119898

times 119862 times sin(Λ119899(119909 minus

119882

2minus 119879OX))

times cos (119872119898119910)

times sinh(sum119899119898

119911) for 1198822lt 119909 lt

119882

2+ 119879OX

0 lt 119910 lt119867

2

1198801198771198991198982

= 120573119899119898

times 119862 times cos (Λ119899119909)

times sin(119872119898(119910 minus

119867

2minus 119879OX)) times sinh(sum

119899119898

119911)

for 0 lt 119909 lt 119882

2

119867

2lt 119910 lt

119867

2+ 119879OX

1198801198771198991198983

= 120574119899119898

times cos (Λ119899119909) cos (119872

119898119910)

times sinh(sum119899119898

119911) for 0 lt 119909 lt 119882

2 0 lt 119910 lt

119867

2

(21)

Now the 119892119899119898

can be obtained from the potential equations(20) by using different multipliers in different regions

1198921198991198981

=120576OX sin (Λ 119899 (119909 minus1198822 minus 119879OX)) cos (119872119898119910)

2120576si sin (Λ 119899119879OX) cos (119872119898 (1198672))

for 1198822lt 119909 lt

119882

2+ 119879OX 0 lt 119910 lt

119867

2

1198921198991198982

=120576OX cos (Λ 119899119909) sin (119872119898 (119910 minus 1198672 minus 119879OX))

2120576si cos (Λ 119899 (1198822)) sin (119872119898119879OX)

for 0 lt 119909 lt 119882

2

119867

2lt 119910 lt

119867

2+ 119879OX

1198921198991198983

= minuscos (Λ

119899119909) cos (119872

119898119910)

cos (Λ119899(1198822)) cos (119872

119898(1198672))

for 0 lt 119909 lt 119882

2 0 lt 119910 lt

119882

2

(22)

Subsequently the constants 120572119899119898 120573119899119898 and 120574

119899119898are evaluated

suitably

120572119899119898

= cos(Λ119899

119882

2) sin (119872

119898119879OX)

120573119899119898

= cos(119872119898

119867

2) sin (Λ

119899119879OX)

120574119899119898

= sin (119872119898119879OX) sin (Λ 119899119879OX)

(23)

From (20) the potential 119880119871119899119898

can be rewritten as

1198801198711198991198981

= minus cos(Λ119899

119882

2) sin (119872

119898119879OX)

times sin (Λ119899119879OX) cos(119872119898

119867

2)

Active and Passive Electronic Components 5

1198801198711198991198982

= minus cos(119872119898

119867

2) sin (Λ

119899119879OX)

times cos(Λ119899

119882

2) sin (119872

119898119879OX)

1198801198711198991198983

= minus sin (119872119898119879OX) sin (Λ 119899119879OX)

times cos(Λ119899

119882

2) cos(119872

119898

119867

2)

(24)

By multiplying with the corresponding orthogonal conjugatefunctions and integrating coefficients of119880

119871can be obtained

The coefficients of 119880119877are also obtained in a similar method

119894119899119898

= (int

119867+119879OX

0

int

119882+119879OX

0

(120595 (119909 119910 0))

minus119881 (119909 119910) 119892119899119898(119909 119910) 119889119909 119889119910)

times (int

119867+119879OX

0

int

119882+119879OX

0

119880119871119899119898

(119909 119910 0) 119892119899119898(119909 119910) 119889119909 119889119910)

minus1

119895119899119898

= (int

119867+119879OX

0

int

119882+119879OX

0

(120595 (119909 119910 119871))

minus119881 (119909 119910) 119892119899119898(119909 119910) 119889119909 119889119910)

times (int

119867+119879OX

0

int

119882+119879OX

0

119880119871119899119898

(119909 119910 119871) 119892119899119898(119909 119910) 119889119909 119889119910)

minus1

(25)

The above integrals (25) are evaluated to obtain explicitexpressions for 119894

119899119898and 119895119899119898

as follows

119894119899119898

= (119881119892minus 120595bi) 120596119899119898

119895119899119898

= (119881119892minus 120595bi minus 119881ds) 120596119899119898

120596119899119898

=Ω119899119898

120585119899119898

(26)

where

Ω119899119898

=120576OXΛ 119899 tan ((Λ 119899119882) 2) + 120576OX119872119898 tan (1198721198981198672)

2120576si119879OXΛ2

1198991198722119898

minus2

Λ119899119872119898

tan(Λ119899119882

2) tan(

119872119898119867

2)

120585119899119898

=sin (Λ

119899119879OX)

16 cos (Λ1198991198822)

(119882 +sin (Λ

119899119882)

Λ119899

) 1199031

+sin (119872

119898119879OX)

16 cos (1198721198981198672)

(119867 +sin (119872

119898119867)

119872119898

) 1199032

(27)

And the values of 1199031and 1199032are given by

1199031=120576OX119879OX cos (1198721198981198672)

120576si sin (119872119898119879OX)+119867 sin (119872

119898119879OX)

2 cos (1198721198981198672)

1199032=120576OX119879OX cos (Λ 1198991198822)

120576si sin (Λ 119899119879OX)+119882 sin (Λ

119899119879OX)

2 cos (Λ1198991198822)

(28)

The potential equation is now rewritten as

120595 (119909 119910 119911) = 119881119892+ (119881119892minus 120595bi)

timessum

119899

sum

119898

120588119899119898

sinh(sum119899119898

(119871 minus 119911))

+ (119881119892minus 120595bi minus 119881ds)sum

119899

sum

119898

sinh(sum119899119898

119911)

(29)

where

120588119899119898(119909 119910) = minus120596

119899119898120574119899119898

cos (Λ119899119909) cos (119872

119898119910)

sinh (sum119899119898119871)

(30)

Once the potential distribution at every point of the cross-section of the channel is known we calculate the inversioncharge density by using surface integral over the surface areaof the channel When the integrated charge at virtual sourcebecomes equal to critical charge the gate voltage of a lightlydoped body device is nearly equal to the threshold voltage ofthe device Hence the inversion charge can be expressed as

119876 = int

1198672

minus1198672

int

1198822

minus1198822

119902119899119894119890(120595119880119879)119889119909 119889119910 (31)

where 119902 is the elementary charge 119880119879is thermal voltage and

119899119894is the intrinsic carrier concentrationThe charge equation can now be approximated as

119876 asymp 119882119867119902119899119894119890((120595((311988214)(311986714)119911119888))119880119879) (32)

Here 119911119888is the virtual source position which is half of the

channel length for low 119881ds Using the inversion charge wecan obtain the classical threshold model as expressed in thefollowing

119881TC = (119880119879 ln119876

119882119867119902119899119894

+ 2119881bi12058811 (3119882

143119867

14)

times sinh(sum11119871

2))

times (1 + 212058811(3119882

143119867

14) sinh(

sum11119871

2))

minus1

(33)

3 Quantum Threshold Voltage Modeling

AsMOSFETdevices are further scaled into the deep nanome-ter regime it has become necessary to include quantummechanical effects while modeling their device behaviorIn this paper we approximate the actual potential well as

6 Active and Passive Electronic Components

E11

Ec

Ec

998779Ec

Ei119909i119910

Eco

Figure 2 Band diagram perpendicular to the gate-square wellpotential of a GAA silicon nanowire transistor

the square well potential since it is difficult to solve theSchrodinger equation to obtain the potential expressed in(29) The square well potential of a gate-all-around nanowiretransistor is shown in Figure 2 The quantum charge of thedevice is expressed as

119876 = sum

119894119909

sum

119894119910

119902int

infin

119864119894119909119894119910

1198731D119891 (119864) 119889119864 (34)

where1198731D is the 1D density-of-states and 119891(119864) is the Fermi-

Dirac distribution function 119864 is the energy of the electronwave The terms 119894

119909and 119894119910are positive natural numbers

In silicon six energy valleys are found to be present inits band structure (two lower energy valleys two middleenergy valleys and two higher energy valleys) If the thinfilm of device has equal height and width the two lowerenergy valleys and two middle energy valleys are combinedtogether to produce four lower energy valleys and the othertwo higher energy valleys remain in their own stateThus thecharge is given by

119876 = 119902sum

119894119909

sum

119894119910

radic(119898119911

2120587ℎ2)int

infin

119864119894119909119894119910

(119864 minus 119864119894119909119894119910)minus12

1 + 119890 ((119864 minus 119864119865) 119896119879)

119889119864 (35)

where 119898119911is the mass of the valley which is perpendicular to

the direction of quantizationThe Fermi energy level is muchlower than the conduction band energy in weak inversionregion Hence the charge equation can be approximated as

119876 = 119902radic(119898119911

2120587ℎ2)sum

119894119909

sum

119894119910

int

infin

119864119894119909119894119910

119890((119864119865minus119864)119896119879)

(119864 minus 119864119894119909119894119910)12119889119864 (36)

Using the Schrodinger equation the value of 119864119894119909119864119894119910is deter-

mined by the following formulation [5]

119864119894119909119894119910

= 119864co +ℎ21205872

2[1

119898119909

(119894119909

119868119909

)

2

+1

119898119910

(119894119910

119868119910

)

2

] (37)

where the conduction band energy is given as

119864co =119864119892

2minus 119902120595 (0 0 119911

119888) (38)

Using (36) and (37) the integrated charge can be obtained as

119876 = 119902radic(2119896119879119898

119905

ℎ2)sum

119894119909

sum

119894119910

exp(minus119864co + 1198641 (119894119909 119894119910)

119896119879)

+ 119902radic(2119896119879119898

119905

ℎ2)sum

119894119909

sum

119894119910

exp(minus119864co + 1198642 (119894119909 119894119910)

119896119879)

+ 119902radic(2119896119879119898

119905

ℎ2)sum

119894119909

sum

119894119910

exp(minus119864co + 1198643 (119894119909 119894119910)

119896119879)

(39)

where

1198641(119894119909 119894119910) =

ℎ21205872

2[1

1198981

(119894119909

119882)

2

+1

119898119905

(119894119910

119867)

2

]

1198642(119894119909 119894119910) =

ℎ21205872

2[1

119898119905

(119894119909

119882)

2

+1

1198981

(119894119910

119867)

2

]

1198643(119894119909 119894119910) =

ℎ21205872

2[1

119898119905

(119894119909

119882)

2

+1

119898119905

(119894119910

119867)

2

]

(40)

Here the 119898119905and 119898

1are the transverse and longitudinal

effective masses of the energy valleys of silicon The lengths119894119909and 119894119910carry distinct values contingent on the direction of

quantization Finally the quantum threshold voltage modelbecomes

119881TQ = (119864119892

2119902+ (

119896119879

119902) In(119876119879

120591)

+2120595bi12058811 (0 0) sinh((sum11119871)

2))

times (1 + 212058811(0 0) sinh(

(sum11119871)

2))

minus1

(41)

where

120591 = 119902radic(2119896119879119898

119905

ℎ2) exp(minus1198641 (1 1)

119896119879)

+ 119902radic(2119896119879119898

119905

ℎ2) exp(minus1198642 (1 1)

119896119879)

+ 119902radic(2119896119879119898

119905

ℎ2) exp(minus

1198643(1 1)

119896119879)

(42)

Active and Passive Electronic Components 7

10 20 30 40 50 6002

022

024

026

028

03

032

034

036

038

04

Channel length (nm)

Pote

ntia

l (V

)

Proposed modelTCAD simulation

Figure 3 Constant electrostatic potential obtained from the analyt-ical solution of a gate-all-around silicon nanowire transistor is 03 Vfor different channel length with a height of119867 = 9 nm and channelwidth of 119882 = 9 nm TCAD simulation shows that the potential isconstant at 0296V

The impacts on the threshold voltage due to quantum effectsare acquired by using the following equation

119881TC = 119881TQ + Δ119881119879 (43)

Here Δ119881119879is the difference between the quantum threshold

voltage and the classical threshold voltage

4 Results and Discussion

Figure 3 shows the electrostatic potential of the proposed gateall around transistor and it is found to be constant value at03 V Continuously varying the 119899 and 119898 terms in (4) hasno impact on the potential as it remains constant along theinsulator boundaries This is totally in contrast to the resultsobtained in [15] where the potential is found to be linearlyvarying in the insulator boundaries The constant potentialhas to be deduced as the resultant of the gate voltage appliedsymmetrically across the four sides of the transistor TheTCAD simulation of the device shows that the electrostaticpotential is constant at 0296V The simulation results arefound to in acceptance with the TCAD results

Figure 4 represents the variation of total quantum inte-grated charge with the gate voltage Equation (39) is usedto obtain the integrated charge with only one energy leveland one series term It clearly shows that the decrease inthe film thickness leads to the increase in the quantumthreshold voltage which is actually due to the increase inenergy quantization of the transistor With the height andlength of the device being constant the width of the device isvaried and henceforth the variation of charge in accordancewith the gate voltage is illustrated in Figure 4

The variation of quantum threshold voltage with widthand height of the film at a channel length of 20 nm is

0 01 02 03 04 05 06 070

05

1

15

2

25

3

35

4times10minus15

Gate voltage (V)

Char

ge (c

mminus1)

W = 9e minus 9

W = 5e minus 9

W = 3e minus 9

TCAD simulation for W = 9e minus 9

TCAD simulation for W = 5e minus 9

TCAD simulation for W = 3e minus 9

Figure 4 Variation of quantum integrated charge at virtual sourcewith gate voltage for different film widths where the height andlength are119867 = 9 nm 119871 = 20 nm

2

8

times10minus9

times10minus9

Film height

2

44

66

8

10

1003

0305

031

0315

032

Film width

Qua

ntum

thre

shol

d vo

ltage

(V)

ProposedTCAD

Figure 5 Variation of quantum threshold voltage with film heightand width for channel length (119871 = 20 nm)

shown in Figure 5 The short channel effects tend to decreasealong with the energy quantisation and this can be furtherexplained as a result of increase in the effective band gap ofsilicon due to quantum effects The effect of confinementexpressed as the difference in the threshold voltage and itsvariation with the channel length 119871 is illustrated in Figure 6

8 Active and Passive Electronic Components

2 3 4 5 6 7 8 9 10

times10minus9

0014

0016

0018

002

0022

0024

0026

0028

003

0032

Film height

Proposed modelTCAD simulation

ΔVT

(mV

)

Figure 6 Variation of threshold voltagewith filmwidth for differentheight at 119871 = 20 nm

3 4 5 6 7 8

times10minus9

02

022

024

026

028

03

032

034

036

038

04

Film height

Thre

shol

d vo

ltage

(V)

ClassicalQuantum

TCAD simulation for classicalTCAD simulation for quantum

9

Figure 7 Variation of quantum and classical threshold voltage withfilm height Here 119871 = 20 nm and119882 = 9 nm

Themost important thing about this gate all around nanowiretransistor is that any change in one of the dimensions canbe nullified by proper tuning of other dimensions as thetransistor is symmetric about its height and width

Figure 7 shows the variation of the classical thresholdvoltage and quantum threshold voltage with the film height ata constant width of 9 nmThe value of the classical thresholdvoltage ranges from 027V to 029V for the correspondingchanges in the film height Similarly the quantum thresholdvoltage ranges from 03V to 031 V It shows that the devicehas a highly improved control over the threshold voltageTheTCAD results justify the simulation results

5 Conclusion

In this paper a quantum threshold voltage model for a GAAsilicon nanowire transistor is proposed by solving the 3DPoisson and Schrodinger equations Analytical expressionsfor potential and the inversion charge are expressed intheir closed forms The results show that the integratedcharge and the threshold voltage calculated in accordancewith the quantum effects of this proposed model are highlyimprovedThe future considerations include deriving the I-Vcharacteristics of the gate all around nanowire transistors andstudying the impact of scaling on various device parametersFinally to conclude this model provides an analytical anduseful way for the threshold voltage evaluations in gate allaround nanowire devices with a unified formalism employedin both classical and quantum mechanical approaches

References

[1] J Wang E Polizzi and M Lundstrom ldquoA computationalstudy of ballistic silicon nanowire transistorsrdquo in Proceedings ofthe IEEE International Electron Devices Meeting pp 695ndash698December 2003

[2] J P Colinge FINFETS and Other Multi-Gate TransistorsSpringer New York NY USA 2007

[3] D J Frank Y Taur andH-S PWong ldquoGeneralized scale lengthfor two-dimensional effects in MOSFETrsquosrdquo IEEE ElectronDevice Letters vol 19 no 10 pp 385ndash387 1998

[4] B Yu L Wang Y Yuan P M Asbeck and Y Taur ldquoScaling ofnanowire transistorsrdquo IEEE Transactions on Electron Devicesvol 55 no 11 pp 2846ndash2858 2008

[5] G D Sanders C J Stanton and Y C Chang ldquoTheory of trans-port in silicon quantum wiresrdquo Physical Review B vol 48 no15 pp 11067ndash11076 1993

[6] M-Y Shen and S-L Zhang ldquoBand gap of a silicon quantumwirerdquo Physics Letters A vol 176 no 3-4 pp 254ndash258 1993

[7] J P Colinge X Baie V Bayot and E Grivei ldquoQuantum-wireeffects in thin and narrow SOI MOSFETsrdquo in Proceedings of theIEEE International SOI Conference pp 66ndash67 October 1995

[8] C P Auth and J D Plummer ldquoScaling theory for cylindricalfully-depleted surrounding-gate MOSFETrsquosrdquo IEEE ElectronDevice Letters vol 18 no 2 pp 74ndash76 1997

[9] J-T Park and J-P Colinge ldquoMultiple-gate SOI MOSFETsdevice design guidelinesrdquo IEEE Transactions on ElectronDevices vol 49 no 12 pp 2222ndash2229 2002

[10] Z Ghoggali F DjeffalM A Abdi D Arar N NLakhdar and TTBendib ldquoAn analytical threshold voltagemodel for nanoscalerdquoin Proceedings of the 3rd International Design and TestWorkshop(IDT rsquo08) pp 93ndash97 December 2008

[11] Y-S Wu and P Su ldquoQuantum confinement effect in short-channel gate-all-around MOSFETs and its impact on the sensi-tivity of threshold voltage to process variationsrdquo in Proceedingsof the IEEE International SOI Conference October 2009

[12] C Te-Kuang ldquoA compact analytical threshold-voltage modelfor surrounding-gateMOSFETswith interface trapped chargesrdquoIEEE Electron Device Letters vol 31 no 8 pp 788ndash790 2010

[13] B Ray and SMahapatra ldquoModeling and analysis of body poten-tial of cylindrical gate-all-around nanowire transistorrdquo IEEETransactions on Electron Devices vol 55 no 9 pp 2409ndash24162008

Active and Passive Electronic Components 9

[14] L De Michielis L Selmi and A M Ionescu ldquoA quasi-analytical model for nanowire FETs with arbitrary polygonalcross sectionrdquo Solid-State Electronics vol 54 no 9 pp 929ndash9342010

[15] P R Kumar and S Mahapatra ldquoQuantum threshold voltagemodeling of short channel quad gate silicon nanowire transis-torrdquo IEEE Transactions onNanotechnology vol 10 no 1 pp 121ndash128 2011

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Active and Passive Electronic Components 5

1198801198711198991198982

= minus cos(119872119898

119867

2) sin (Λ

119899119879OX)

times cos(Λ119899

119882

2) sin (119872

119898119879OX)

1198801198711198991198983

= minus sin (119872119898119879OX) sin (Λ 119899119879OX)

times cos(Λ119899

119882

2) cos(119872

119898

119867

2)

(24)

By multiplying with the corresponding orthogonal conjugatefunctions and integrating coefficients of119880

119871can be obtained

The coefficients of 119880119877are also obtained in a similar method

119894119899119898

= (int

119867+119879OX

0

int

119882+119879OX

0

(120595 (119909 119910 0))

minus119881 (119909 119910) 119892119899119898(119909 119910) 119889119909 119889119910)

times (int

119867+119879OX

0

int

119882+119879OX

0

119880119871119899119898

(119909 119910 0) 119892119899119898(119909 119910) 119889119909 119889119910)

minus1

119895119899119898

= (int

119867+119879OX

0

int

119882+119879OX

0

(120595 (119909 119910 119871))

minus119881 (119909 119910) 119892119899119898(119909 119910) 119889119909 119889119910)

times (int

119867+119879OX

0

int

119882+119879OX

0

119880119871119899119898

(119909 119910 119871) 119892119899119898(119909 119910) 119889119909 119889119910)

minus1

(25)

The above integrals (25) are evaluated to obtain explicitexpressions for 119894

119899119898and 119895119899119898

as follows

119894119899119898

= (119881119892minus 120595bi) 120596119899119898

119895119899119898

= (119881119892minus 120595bi minus 119881ds) 120596119899119898

120596119899119898

=Ω119899119898

120585119899119898

(26)

where

Ω119899119898

=120576OXΛ 119899 tan ((Λ 119899119882) 2) + 120576OX119872119898 tan (1198721198981198672)

2120576si119879OXΛ2

1198991198722119898

minus2

Λ119899119872119898

tan(Λ119899119882

2) tan(

119872119898119867

2)

120585119899119898

=sin (Λ

119899119879OX)

16 cos (Λ1198991198822)

(119882 +sin (Λ

119899119882)

Λ119899

) 1199031

+sin (119872

119898119879OX)

16 cos (1198721198981198672)

(119867 +sin (119872

119898119867)

119872119898

) 1199032

(27)

And the values of 1199031and 1199032are given by

1199031=120576OX119879OX cos (1198721198981198672)

120576si sin (119872119898119879OX)+119867 sin (119872

119898119879OX)

2 cos (1198721198981198672)

1199032=120576OX119879OX cos (Λ 1198991198822)

120576si sin (Λ 119899119879OX)+119882 sin (Λ

119899119879OX)

2 cos (Λ1198991198822)

(28)

The potential equation is now rewritten as

120595 (119909 119910 119911) = 119881119892+ (119881119892minus 120595bi)

timessum

119899

sum

119898

120588119899119898

sinh(sum119899119898

(119871 minus 119911))

+ (119881119892minus 120595bi minus 119881ds)sum

119899

sum

119898

sinh(sum119899119898

119911)

(29)

where

120588119899119898(119909 119910) = minus120596

119899119898120574119899119898

cos (Λ119899119909) cos (119872

119898119910)

sinh (sum119899119898119871)

(30)

Once the potential distribution at every point of the cross-section of the channel is known we calculate the inversioncharge density by using surface integral over the surface areaof the channel When the integrated charge at virtual sourcebecomes equal to critical charge the gate voltage of a lightlydoped body device is nearly equal to the threshold voltage ofthe device Hence the inversion charge can be expressed as

119876 = int

1198672

minus1198672

int

1198822

minus1198822

119902119899119894119890(120595119880119879)119889119909 119889119910 (31)

where 119902 is the elementary charge 119880119879is thermal voltage and

119899119894is the intrinsic carrier concentrationThe charge equation can now be approximated as

119876 asymp 119882119867119902119899119894119890((120595((311988214)(311986714)119911119888))119880119879) (32)

Here 119911119888is the virtual source position which is half of the

channel length for low 119881ds Using the inversion charge wecan obtain the classical threshold model as expressed in thefollowing

119881TC = (119880119879 ln119876

119882119867119902119899119894

+ 2119881bi12058811 (3119882

143119867

14)

times sinh(sum11119871

2))

times (1 + 212058811(3119882

143119867

14) sinh(

sum11119871

2))

minus1

(33)

3 Quantum Threshold Voltage Modeling

AsMOSFETdevices are further scaled into the deep nanome-ter regime it has become necessary to include quantummechanical effects while modeling their device behaviorIn this paper we approximate the actual potential well as

6 Active and Passive Electronic Components

E11

Ec

Ec

998779Ec

Ei119909i119910

Eco

Figure 2 Band diagram perpendicular to the gate-square wellpotential of a GAA silicon nanowire transistor

the square well potential since it is difficult to solve theSchrodinger equation to obtain the potential expressed in(29) The square well potential of a gate-all-around nanowiretransistor is shown in Figure 2 The quantum charge of thedevice is expressed as

119876 = sum

119894119909

sum

119894119910

119902int

infin

119864119894119909119894119910

1198731D119891 (119864) 119889119864 (34)

where1198731D is the 1D density-of-states and 119891(119864) is the Fermi-

Dirac distribution function 119864 is the energy of the electronwave The terms 119894

119909and 119894119910are positive natural numbers

In silicon six energy valleys are found to be present inits band structure (two lower energy valleys two middleenergy valleys and two higher energy valleys) If the thinfilm of device has equal height and width the two lowerenergy valleys and two middle energy valleys are combinedtogether to produce four lower energy valleys and the othertwo higher energy valleys remain in their own stateThus thecharge is given by

119876 = 119902sum

119894119909

sum

119894119910

radic(119898119911

2120587ℎ2)int

infin

119864119894119909119894119910

(119864 minus 119864119894119909119894119910)minus12

1 + 119890 ((119864 minus 119864119865) 119896119879)

119889119864 (35)

where 119898119911is the mass of the valley which is perpendicular to

the direction of quantizationThe Fermi energy level is muchlower than the conduction band energy in weak inversionregion Hence the charge equation can be approximated as

119876 = 119902radic(119898119911

2120587ℎ2)sum

119894119909

sum

119894119910

int

infin

119864119894119909119894119910

119890((119864119865minus119864)119896119879)

(119864 minus 119864119894119909119894119910)12119889119864 (36)

Using the Schrodinger equation the value of 119864119894119909119864119894119910is deter-

mined by the following formulation [5]

119864119894119909119894119910

= 119864co +ℎ21205872

2[1

119898119909

(119894119909

119868119909

)

2

+1

119898119910

(119894119910

119868119910

)

2

] (37)

where the conduction band energy is given as

119864co =119864119892

2minus 119902120595 (0 0 119911

119888) (38)

Using (36) and (37) the integrated charge can be obtained as

119876 = 119902radic(2119896119879119898

119905

ℎ2)sum

119894119909

sum

119894119910

exp(minus119864co + 1198641 (119894119909 119894119910)

119896119879)

+ 119902radic(2119896119879119898

119905

ℎ2)sum

119894119909

sum

119894119910

exp(minus119864co + 1198642 (119894119909 119894119910)

119896119879)

+ 119902radic(2119896119879119898

119905

ℎ2)sum

119894119909

sum

119894119910

exp(minus119864co + 1198643 (119894119909 119894119910)

119896119879)

(39)

where

1198641(119894119909 119894119910) =

ℎ21205872

2[1

1198981

(119894119909

119882)

2

+1

119898119905

(119894119910

119867)

2

]

1198642(119894119909 119894119910) =

ℎ21205872

2[1

119898119905

(119894119909

119882)

2

+1

1198981

(119894119910

119867)

2

]

1198643(119894119909 119894119910) =

ℎ21205872

2[1

119898119905

(119894119909

119882)

2

+1

119898119905

(119894119910

119867)

2

]

(40)

Here the 119898119905and 119898

1are the transverse and longitudinal

effective masses of the energy valleys of silicon The lengths119894119909and 119894119910carry distinct values contingent on the direction of

quantization Finally the quantum threshold voltage modelbecomes

119881TQ = (119864119892

2119902+ (

119896119879

119902) In(119876119879

120591)

+2120595bi12058811 (0 0) sinh((sum11119871)

2))

times (1 + 212058811(0 0) sinh(

(sum11119871)

2))

minus1

(41)

where

120591 = 119902radic(2119896119879119898

119905

ℎ2) exp(minus1198641 (1 1)

119896119879)

+ 119902radic(2119896119879119898

119905

ℎ2) exp(minus1198642 (1 1)

119896119879)

+ 119902radic(2119896119879119898

119905

ℎ2) exp(minus

1198643(1 1)

119896119879)

(42)

Active and Passive Electronic Components 7

10 20 30 40 50 6002

022

024

026

028

03

032

034

036

038

04

Channel length (nm)

Pote

ntia

l (V

)

Proposed modelTCAD simulation

Figure 3 Constant electrostatic potential obtained from the analyt-ical solution of a gate-all-around silicon nanowire transistor is 03 Vfor different channel length with a height of119867 = 9 nm and channelwidth of 119882 = 9 nm TCAD simulation shows that the potential isconstant at 0296V

The impacts on the threshold voltage due to quantum effectsare acquired by using the following equation

119881TC = 119881TQ + Δ119881119879 (43)

Here Δ119881119879is the difference between the quantum threshold

voltage and the classical threshold voltage

4 Results and Discussion

Figure 3 shows the electrostatic potential of the proposed gateall around transistor and it is found to be constant value at03 V Continuously varying the 119899 and 119898 terms in (4) hasno impact on the potential as it remains constant along theinsulator boundaries This is totally in contrast to the resultsobtained in [15] where the potential is found to be linearlyvarying in the insulator boundaries The constant potentialhas to be deduced as the resultant of the gate voltage appliedsymmetrically across the four sides of the transistor TheTCAD simulation of the device shows that the electrostaticpotential is constant at 0296V The simulation results arefound to in acceptance with the TCAD results

Figure 4 represents the variation of total quantum inte-grated charge with the gate voltage Equation (39) is usedto obtain the integrated charge with only one energy leveland one series term It clearly shows that the decrease inthe film thickness leads to the increase in the quantumthreshold voltage which is actually due to the increase inenergy quantization of the transistor With the height andlength of the device being constant the width of the device isvaried and henceforth the variation of charge in accordancewith the gate voltage is illustrated in Figure 4

The variation of quantum threshold voltage with widthand height of the film at a channel length of 20 nm is

0 01 02 03 04 05 06 070

05

1

15

2

25

3

35

4times10minus15

Gate voltage (V)

Char

ge (c

mminus1)

W = 9e minus 9

W = 5e minus 9

W = 3e minus 9

TCAD simulation for W = 9e minus 9

TCAD simulation for W = 5e minus 9

TCAD simulation for W = 3e minus 9

Figure 4 Variation of quantum integrated charge at virtual sourcewith gate voltage for different film widths where the height andlength are119867 = 9 nm 119871 = 20 nm

2

8

times10minus9

times10minus9

Film height

2

44

66

8

10

1003

0305

031

0315

032

Film width

Qua

ntum

thre

shol

d vo

ltage

(V)

ProposedTCAD

Figure 5 Variation of quantum threshold voltage with film heightand width for channel length (119871 = 20 nm)

shown in Figure 5 The short channel effects tend to decreasealong with the energy quantisation and this can be furtherexplained as a result of increase in the effective band gap ofsilicon due to quantum effects The effect of confinementexpressed as the difference in the threshold voltage and itsvariation with the channel length 119871 is illustrated in Figure 6

8 Active and Passive Electronic Components

2 3 4 5 6 7 8 9 10

times10minus9

0014

0016

0018

002

0022

0024

0026

0028

003

0032

Film height

Proposed modelTCAD simulation

ΔVT

(mV

)

Figure 6 Variation of threshold voltagewith filmwidth for differentheight at 119871 = 20 nm

3 4 5 6 7 8

times10minus9

02

022

024

026

028

03

032

034

036

038

04

Film height

Thre

shol

d vo

ltage

(V)

ClassicalQuantum

TCAD simulation for classicalTCAD simulation for quantum

9

Figure 7 Variation of quantum and classical threshold voltage withfilm height Here 119871 = 20 nm and119882 = 9 nm

Themost important thing about this gate all around nanowiretransistor is that any change in one of the dimensions canbe nullified by proper tuning of other dimensions as thetransistor is symmetric about its height and width

Figure 7 shows the variation of the classical thresholdvoltage and quantum threshold voltage with the film height ata constant width of 9 nmThe value of the classical thresholdvoltage ranges from 027V to 029V for the correspondingchanges in the film height Similarly the quantum thresholdvoltage ranges from 03V to 031 V It shows that the devicehas a highly improved control over the threshold voltageTheTCAD results justify the simulation results

5 Conclusion

In this paper a quantum threshold voltage model for a GAAsilicon nanowire transistor is proposed by solving the 3DPoisson and Schrodinger equations Analytical expressionsfor potential and the inversion charge are expressed intheir closed forms The results show that the integratedcharge and the threshold voltage calculated in accordancewith the quantum effects of this proposed model are highlyimprovedThe future considerations include deriving the I-Vcharacteristics of the gate all around nanowire transistors andstudying the impact of scaling on various device parametersFinally to conclude this model provides an analytical anduseful way for the threshold voltage evaluations in gate allaround nanowire devices with a unified formalism employedin both classical and quantum mechanical approaches

References

[1] J Wang E Polizzi and M Lundstrom ldquoA computationalstudy of ballistic silicon nanowire transistorsrdquo in Proceedings ofthe IEEE International Electron Devices Meeting pp 695ndash698December 2003

[2] J P Colinge FINFETS and Other Multi-Gate TransistorsSpringer New York NY USA 2007

[3] D J Frank Y Taur andH-S PWong ldquoGeneralized scale lengthfor two-dimensional effects in MOSFETrsquosrdquo IEEE ElectronDevice Letters vol 19 no 10 pp 385ndash387 1998

[4] B Yu L Wang Y Yuan P M Asbeck and Y Taur ldquoScaling ofnanowire transistorsrdquo IEEE Transactions on Electron Devicesvol 55 no 11 pp 2846ndash2858 2008

[5] G D Sanders C J Stanton and Y C Chang ldquoTheory of trans-port in silicon quantum wiresrdquo Physical Review B vol 48 no15 pp 11067ndash11076 1993

[6] M-Y Shen and S-L Zhang ldquoBand gap of a silicon quantumwirerdquo Physics Letters A vol 176 no 3-4 pp 254ndash258 1993

[7] J P Colinge X Baie V Bayot and E Grivei ldquoQuantum-wireeffects in thin and narrow SOI MOSFETsrdquo in Proceedings of theIEEE International SOI Conference pp 66ndash67 October 1995

[8] C P Auth and J D Plummer ldquoScaling theory for cylindricalfully-depleted surrounding-gate MOSFETrsquosrdquo IEEE ElectronDevice Letters vol 18 no 2 pp 74ndash76 1997

[9] J-T Park and J-P Colinge ldquoMultiple-gate SOI MOSFETsdevice design guidelinesrdquo IEEE Transactions on ElectronDevices vol 49 no 12 pp 2222ndash2229 2002

[10] Z Ghoggali F DjeffalM A Abdi D Arar N NLakhdar and TTBendib ldquoAn analytical threshold voltagemodel for nanoscalerdquoin Proceedings of the 3rd International Design and TestWorkshop(IDT rsquo08) pp 93ndash97 December 2008

[11] Y-S Wu and P Su ldquoQuantum confinement effect in short-channel gate-all-around MOSFETs and its impact on the sensi-tivity of threshold voltage to process variationsrdquo in Proceedingsof the IEEE International SOI Conference October 2009

[12] C Te-Kuang ldquoA compact analytical threshold-voltage modelfor surrounding-gateMOSFETswith interface trapped chargesrdquoIEEE Electron Device Letters vol 31 no 8 pp 788ndash790 2010

[13] B Ray and SMahapatra ldquoModeling and analysis of body poten-tial of cylindrical gate-all-around nanowire transistorrdquo IEEETransactions on Electron Devices vol 55 no 9 pp 2409ndash24162008

Active and Passive Electronic Components 9

[14] L De Michielis L Selmi and A M Ionescu ldquoA quasi-analytical model for nanowire FETs with arbitrary polygonalcross sectionrdquo Solid-State Electronics vol 54 no 9 pp 929ndash9342010

[15] P R Kumar and S Mahapatra ldquoQuantum threshold voltagemodeling of short channel quad gate silicon nanowire transis-torrdquo IEEE Transactions onNanotechnology vol 10 no 1 pp 121ndash128 2011

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

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VLSI Design

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Electrical and Computer Engineering

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Advances inOptoElectronics

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Chemical EngineeringInternational Journal of Antennas and

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International Journal of

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Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

6 Active and Passive Electronic Components

E11

Ec

Ec

998779Ec

Ei119909i119910

Eco

Figure 2 Band diagram perpendicular to the gate-square wellpotential of a GAA silicon nanowire transistor

the square well potential since it is difficult to solve theSchrodinger equation to obtain the potential expressed in(29) The square well potential of a gate-all-around nanowiretransistor is shown in Figure 2 The quantum charge of thedevice is expressed as

119876 = sum

119894119909

sum

119894119910

119902int

infin

119864119894119909119894119910

1198731D119891 (119864) 119889119864 (34)

where1198731D is the 1D density-of-states and 119891(119864) is the Fermi-

Dirac distribution function 119864 is the energy of the electronwave The terms 119894

119909and 119894119910are positive natural numbers

In silicon six energy valleys are found to be present inits band structure (two lower energy valleys two middleenergy valleys and two higher energy valleys) If the thinfilm of device has equal height and width the two lowerenergy valleys and two middle energy valleys are combinedtogether to produce four lower energy valleys and the othertwo higher energy valleys remain in their own stateThus thecharge is given by

119876 = 119902sum

119894119909

sum

119894119910

radic(119898119911

2120587ℎ2)int

infin

119864119894119909119894119910

(119864 minus 119864119894119909119894119910)minus12

1 + 119890 ((119864 minus 119864119865) 119896119879)

119889119864 (35)

where 119898119911is the mass of the valley which is perpendicular to

the direction of quantizationThe Fermi energy level is muchlower than the conduction band energy in weak inversionregion Hence the charge equation can be approximated as

119876 = 119902radic(119898119911

2120587ℎ2)sum

119894119909

sum

119894119910

int

infin

119864119894119909119894119910

119890((119864119865minus119864)119896119879)

(119864 minus 119864119894119909119894119910)12119889119864 (36)

Using the Schrodinger equation the value of 119864119894119909119864119894119910is deter-

mined by the following formulation [5]

119864119894119909119894119910

= 119864co +ℎ21205872

2[1

119898119909

(119894119909

119868119909

)

2

+1

119898119910

(119894119910

119868119910

)

2

] (37)

where the conduction band energy is given as

119864co =119864119892

2minus 119902120595 (0 0 119911

119888) (38)

Using (36) and (37) the integrated charge can be obtained as

119876 = 119902radic(2119896119879119898

119905

ℎ2)sum

119894119909

sum

119894119910

exp(minus119864co + 1198641 (119894119909 119894119910)

119896119879)

+ 119902radic(2119896119879119898

119905

ℎ2)sum

119894119909

sum

119894119910

exp(minus119864co + 1198642 (119894119909 119894119910)

119896119879)

+ 119902radic(2119896119879119898

119905

ℎ2)sum

119894119909

sum

119894119910

exp(minus119864co + 1198643 (119894119909 119894119910)

119896119879)

(39)

where

1198641(119894119909 119894119910) =

ℎ21205872

2[1

1198981

(119894119909

119882)

2

+1

119898119905

(119894119910

119867)

2

]

1198642(119894119909 119894119910) =

ℎ21205872

2[1

119898119905

(119894119909

119882)

2

+1

1198981

(119894119910

119867)

2

]

1198643(119894119909 119894119910) =

ℎ21205872

2[1

119898119905

(119894119909

119882)

2

+1

119898119905

(119894119910

119867)

2

]

(40)

Here the 119898119905and 119898

1are the transverse and longitudinal

effective masses of the energy valleys of silicon The lengths119894119909and 119894119910carry distinct values contingent on the direction of

quantization Finally the quantum threshold voltage modelbecomes

119881TQ = (119864119892

2119902+ (

119896119879

119902) In(119876119879

120591)

+2120595bi12058811 (0 0) sinh((sum11119871)

2))

times (1 + 212058811(0 0) sinh(

(sum11119871)

2))

minus1

(41)

where

120591 = 119902radic(2119896119879119898

119905

ℎ2) exp(minus1198641 (1 1)

119896119879)

+ 119902radic(2119896119879119898

119905

ℎ2) exp(minus1198642 (1 1)

119896119879)

+ 119902radic(2119896119879119898

119905

ℎ2) exp(minus

1198643(1 1)

119896119879)

(42)

Active and Passive Electronic Components 7

10 20 30 40 50 6002

022

024

026

028

03

032

034

036

038

04

Channel length (nm)

Pote

ntia

l (V

)

Proposed modelTCAD simulation

Figure 3 Constant electrostatic potential obtained from the analyt-ical solution of a gate-all-around silicon nanowire transistor is 03 Vfor different channel length with a height of119867 = 9 nm and channelwidth of 119882 = 9 nm TCAD simulation shows that the potential isconstant at 0296V

The impacts on the threshold voltage due to quantum effectsare acquired by using the following equation

119881TC = 119881TQ + Δ119881119879 (43)

Here Δ119881119879is the difference between the quantum threshold

voltage and the classical threshold voltage

4 Results and Discussion

Figure 3 shows the electrostatic potential of the proposed gateall around transistor and it is found to be constant value at03 V Continuously varying the 119899 and 119898 terms in (4) hasno impact on the potential as it remains constant along theinsulator boundaries This is totally in contrast to the resultsobtained in [15] where the potential is found to be linearlyvarying in the insulator boundaries The constant potentialhas to be deduced as the resultant of the gate voltage appliedsymmetrically across the four sides of the transistor TheTCAD simulation of the device shows that the electrostaticpotential is constant at 0296V The simulation results arefound to in acceptance with the TCAD results

Figure 4 represents the variation of total quantum inte-grated charge with the gate voltage Equation (39) is usedto obtain the integrated charge with only one energy leveland one series term It clearly shows that the decrease inthe film thickness leads to the increase in the quantumthreshold voltage which is actually due to the increase inenergy quantization of the transistor With the height andlength of the device being constant the width of the device isvaried and henceforth the variation of charge in accordancewith the gate voltage is illustrated in Figure 4

The variation of quantum threshold voltage with widthand height of the film at a channel length of 20 nm is

0 01 02 03 04 05 06 070

05

1

15

2

25

3

35

4times10minus15

Gate voltage (V)

Char

ge (c

mminus1)

W = 9e minus 9

W = 5e minus 9

W = 3e minus 9

TCAD simulation for W = 9e minus 9

TCAD simulation for W = 5e minus 9

TCAD simulation for W = 3e minus 9

Figure 4 Variation of quantum integrated charge at virtual sourcewith gate voltage for different film widths where the height andlength are119867 = 9 nm 119871 = 20 nm

2

8

times10minus9

times10minus9

Film height

2

44

66

8

10

1003

0305

031

0315

032

Film width

Qua

ntum

thre

shol

d vo

ltage

(V)

ProposedTCAD

Figure 5 Variation of quantum threshold voltage with film heightand width for channel length (119871 = 20 nm)

shown in Figure 5 The short channel effects tend to decreasealong with the energy quantisation and this can be furtherexplained as a result of increase in the effective band gap ofsilicon due to quantum effects The effect of confinementexpressed as the difference in the threshold voltage and itsvariation with the channel length 119871 is illustrated in Figure 6

8 Active and Passive Electronic Components

2 3 4 5 6 7 8 9 10

times10minus9

0014

0016

0018

002

0022

0024

0026

0028

003

0032

Film height

Proposed modelTCAD simulation

ΔVT

(mV

)

Figure 6 Variation of threshold voltagewith filmwidth for differentheight at 119871 = 20 nm

3 4 5 6 7 8

times10minus9

02

022

024

026

028

03

032

034

036

038

04

Film height

Thre

shol

d vo

ltage

(V)

ClassicalQuantum

TCAD simulation for classicalTCAD simulation for quantum

9

Figure 7 Variation of quantum and classical threshold voltage withfilm height Here 119871 = 20 nm and119882 = 9 nm

Themost important thing about this gate all around nanowiretransistor is that any change in one of the dimensions canbe nullified by proper tuning of other dimensions as thetransistor is symmetric about its height and width

Figure 7 shows the variation of the classical thresholdvoltage and quantum threshold voltage with the film height ata constant width of 9 nmThe value of the classical thresholdvoltage ranges from 027V to 029V for the correspondingchanges in the film height Similarly the quantum thresholdvoltage ranges from 03V to 031 V It shows that the devicehas a highly improved control over the threshold voltageTheTCAD results justify the simulation results

5 Conclusion

In this paper a quantum threshold voltage model for a GAAsilicon nanowire transistor is proposed by solving the 3DPoisson and Schrodinger equations Analytical expressionsfor potential and the inversion charge are expressed intheir closed forms The results show that the integratedcharge and the threshold voltage calculated in accordancewith the quantum effects of this proposed model are highlyimprovedThe future considerations include deriving the I-Vcharacteristics of the gate all around nanowire transistors andstudying the impact of scaling on various device parametersFinally to conclude this model provides an analytical anduseful way for the threshold voltage evaluations in gate allaround nanowire devices with a unified formalism employedin both classical and quantum mechanical approaches

References

[1] J Wang E Polizzi and M Lundstrom ldquoA computationalstudy of ballistic silicon nanowire transistorsrdquo in Proceedings ofthe IEEE International Electron Devices Meeting pp 695ndash698December 2003

[2] J P Colinge FINFETS and Other Multi-Gate TransistorsSpringer New York NY USA 2007

[3] D J Frank Y Taur andH-S PWong ldquoGeneralized scale lengthfor two-dimensional effects in MOSFETrsquosrdquo IEEE ElectronDevice Letters vol 19 no 10 pp 385ndash387 1998

[4] B Yu L Wang Y Yuan P M Asbeck and Y Taur ldquoScaling ofnanowire transistorsrdquo IEEE Transactions on Electron Devicesvol 55 no 11 pp 2846ndash2858 2008

[5] G D Sanders C J Stanton and Y C Chang ldquoTheory of trans-port in silicon quantum wiresrdquo Physical Review B vol 48 no15 pp 11067ndash11076 1993

[6] M-Y Shen and S-L Zhang ldquoBand gap of a silicon quantumwirerdquo Physics Letters A vol 176 no 3-4 pp 254ndash258 1993

[7] J P Colinge X Baie V Bayot and E Grivei ldquoQuantum-wireeffects in thin and narrow SOI MOSFETsrdquo in Proceedings of theIEEE International SOI Conference pp 66ndash67 October 1995

[8] C P Auth and J D Plummer ldquoScaling theory for cylindricalfully-depleted surrounding-gate MOSFETrsquosrdquo IEEE ElectronDevice Letters vol 18 no 2 pp 74ndash76 1997

[9] J-T Park and J-P Colinge ldquoMultiple-gate SOI MOSFETsdevice design guidelinesrdquo IEEE Transactions on ElectronDevices vol 49 no 12 pp 2222ndash2229 2002

[10] Z Ghoggali F DjeffalM A Abdi D Arar N NLakhdar and TTBendib ldquoAn analytical threshold voltagemodel for nanoscalerdquoin Proceedings of the 3rd International Design and TestWorkshop(IDT rsquo08) pp 93ndash97 December 2008

[11] Y-S Wu and P Su ldquoQuantum confinement effect in short-channel gate-all-around MOSFETs and its impact on the sensi-tivity of threshold voltage to process variationsrdquo in Proceedingsof the IEEE International SOI Conference October 2009

[12] C Te-Kuang ldquoA compact analytical threshold-voltage modelfor surrounding-gateMOSFETswith interface trapped chargesrdquoIEEE Electron Device Letters vol 31 no 8 pp 788ndash790 2010

[13] B Ray and SMahapatra ldquoModeling and analysis of body poten-tial of cylindrical gate-all-around nanowire transistorrdquo IEEETransactions on Electron Devices vol 55 no 9 pp 2409ndash24162008

Active and Passive Electronic Components 9

[14] L De Michielis L Selmi and A M Ionescu ldquoA quasi-analytical model for nanowire FETs with arbitrary polygonalcross sectionrdquo Solid-State Electronics vol 54 no 9 pp 929ndash9342010

[15] P R Kumar and S Mahapatra ldquoQuantum threshold voltagemodeling of short channel quad gate silicon nanowire transis-torrdquo IEEE Transactions onNanotechnology vol 10 no 1 pp 121ndash128 2011

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Active and Passive Electronic Components 7

10 20 30 40 50 6002

022

024

026

028

03

032

034

036

038

04

Channel length (nm)

Pote

ntia

l (V

)

Proposed modelTCAD simulation

Figure 3 Constant electrostatic potential obtained from the analyt-ical solution of a gate-all-around silicon nanowire transistor is 03 Vfor different channel length with a height of119867 = 9 nm and channelwidth of 119882 = 9 nm TCAD simulation shows that the potential isconstant at 0296V

The impacts on the threshold voltage due to quantum effectsare acquired by using the following equation

119881TC = 119881TQ + Δ119881119879 (43)

Here Δ119881119879is the difference between the quantum threshold

voltage and the classical threshold voltage

4 Results and Discussion

Figure 3 shows the electrostatic potential of the proposed gateall around transistor and it is found to be constant value at03 V Continuously varying the 119899 and 119898 terms in (4) hasno impact on the potential as it remains constant along theinsulator boundaries This is totally in contrast to the resultsobtained in [15] where the potential is found to be linearlyvarying in the insulator boundaries The constant potentialhas to be deduced as the resultant of the gate voltage appliedsymmetrically across the four sides of the transistor TheTCAD simulation of the device shows that the electrostaticpotential is constant at 0296V The simulation results arefound to in acceptance with the TCAD results

Figure 4 represents the variation of total quantum inte-grated charge with the gate voltage Equation (39) is usedto obtain the integrated charge with only one energy leveland one series term It clearly shows that the decrease inthe film thickness leads to the increase in the quantumthreshold voltage which is actually due to the increase inenergy quantization of the transistor With the height andlength of the device being constant the width of the device isvaried and henceforth the variation of charge in accordancewith the gate voltage is illustrated in Figure 4

The variation of quantum threshold voltage with widthand height of the film at a channel length of 20 nm is

0 01 02 03 04 05 06 070

05

1

15

2

25

3

35

4times10minus15

Gate voltage (V)

Char

ge (c

mminus1)

W = 9e minus 9

W = 5e minus 9

W = 3e minus 9

TCAD simulation for W = 9e minus 9

TCAD simulation for W = 5e minus 9

TCAD simulation for W = 3e minus 9

Figure 4 Variation of quantum integrated charge at virtual sourcewith gate voltage for different film widths where the height andlength are119867 = 9 nm 119871 = 20 nm

2

8

times10minus9

times10minus9

Film height

2

44

66

8

10

1003

0305

031

0315

032

Film width

Qua

ntum

thre

shol

d vo

ltage

(V)

ProposedTCAD

Figure 5 Variation of quantum threshold voltage with film heightand width for channel length (119871 = 20 nm)

shown in Figure 5 The short channel effects tend to decreasealong with the energy quantisation and this can be furtherexplained as a result of increase in the effective band gap ofsilicon due to quantum effects The effect of confinementexpressed as the difference in the threshold voltage and itsvariation with the channel length 119871 is illustrated in Figure 6

8 Active and Passive Electronic Components

2 3 4 5 6 7 8 9 10

times10minus9

0014

0016

0018

002

0022

0024

0026

0028

003

0032

Film height

Proposed modelTCAD simulation

ΔVT

(mV

)

Figure 6 Variation of threshold voltagewith filmwidth for differentheight at 119871 = 20 nm

3 4 5 6 7 8

times10minus9

02

022

024

026

028

03

032

034

036

038

04

Film height

Thre

shol

d vo

ltage

(V)

ClassicalQuantum

TCAD simulation for classicalTCAD simulation for quantum

9

Figure 7 Variation of quantum and classical threshold voltage withfilm height Here 119871 = 20 nm and119882 = 9 nm

Themost important thing about this gate all around nanowiretransistor is that any change in one of the dimensions canbe nullified by proper tuning of other dimensions as thetransistor is symmetric about its height and width

Figure 7 shows the variation of the classical thresholdvoltage and quantum threshold voltage with the film height ata constant width of 9 nmThe value of the classical thresholdvoltage ranges from 027V to 029V for the correspondingchanges in the film height Similarly the quantum thresholdvoltage ranges from 03V to 031 V It shows that the devicehas a highly improved control over the threshold voltageTheTCAD results justify the simulation results

5 Conclusion

In this paper a quantum threshold voltage model for a GAAsilicon nanowire transistor is proposed by solving the 3DPoisson and Schrodinger equations Analytical expressionsfor potential and the inversion charge are expressed intheir closed forms The results show that the integratedcharge and the threshold voltage calculated in accordancewith the quantum effects of this proposed model are highlyimprovedThe future considerations include deriving the I-Vcharacteristics of the gate all around nanowire transistors andstudying the impact of scaling on various device parametersFinally to conclude this model provides an analytical anduseful way for the threshold voltage evaluations in gate allaround nanowire devices with a unified formalism employedin both classical and quantum mechanical approaches

References

[1] J Wang E Polizzi and M Lundstrom ldquoA computationalstudy of ballistic silicon nanowire transistorsrdquo in Proceedings ofthe IEEE International Electron Devices Meeting pp 695ndash698December 2003

[2] J P Colinge FINFETS and Other Multi-Gate TransistorsSpringer New York NY USA 2007

[3] D J Frank Y Taur andH-S PWong ldquoGeneralized scale lengthfor two-dimensional effects in MOSFETrsquosrdquo IEEE ElectronDevice Letters vol 19 no 10 pp 385ndash387 1998

[4] B Yu L Wang Y Yuan P M Asbeck and Y Taur ldquoScaling ofnanowire transistorsrdquo IEEE Transactions on Electron Devicesvol 55 no 11 pp 2846ndash2858 2008

[5] G D Sanders C J Stanton and Y C Chang ldquoTheory of trans-port in silicon quantum wiresrdquo Physical Review B vol 48 no15 pp 11067ndash11076 1993

[6] M-Y Shen and S-L Zhang ldquoBand gap of a silicon quantumwirerdquo Physics Letters A vol 176 no 3-4 pp 254ndash258 1993

[7] J P Colinge X Baie V Bayot and E Grivei ldquoQuantum-wireeffects in thin and narrow SOI MOSFETsrdquo in Proceedings of theIEEE International SOI Conference pp 66ndash67 October 1995

[8] C P Auth and J D Plummer ldquoScaling theory for cylindricalfully-depleted surrounding-gate MOSFETrsquosrdquo IEEE ElectronDevice Letters vol 18 no 2 pp 74ndash76 1997

[9] J-T Park and J-P Colinge ldquoMultiple-gate SOI MOSFETsdevice design guidelinesrdquo IEEE Transactions on ElectronDevices vol 49 no 12 pp 2222ndash2229 2002

[10] Z Ghoggali F DjeffalM A Abdi D Arar N NLakhdar and TTBendib ldquoAn analytical threshold voltagemodel for nanoscalerdquoin Proceedings of the 3rd International Design and TestWorkshop(IDT rsquo08) pp 93ndash97 December 2008

[11] Y-S Wu and P Su ldquoQuantum confinement effect in short-channel gate-all-around MOSFETs and its impact on the sensi-tivity of threshold voltage to process variationsrdquo in Proceedingsof the IEEE International SOI Conference October 2009

[12] C Te-Kuang ldquoA compact analytical threshold-voltage modelfor surrounding-gateMOSFETswith interface trapped chargesrdquoIEEE Electron Device Letters vol 31 no 8 pp 788ndash790 2010

[13] B Ray and SMahapatra ldquoModeling and analysis of body poten-tial of cylindrical gate-all-around nanowire transistorrdquo IEEETransactions on Electron Devices vol 55 no 9 pp 2409ndash24162008

Active and Passive Electronic Components 9

[14] L De Michielis L Selmi and A M Ionescu ldquoA quasi-analytical model for nanowire FETs with arbitrary polygonalcross sectionrdquo Solid-State Electronics vol 54 no 9 pp 929ndash9342010

[15] P R Kumar and S Mahapatra ldquoQuantum threshold voltagemodeling of short channel quad gate silicon nanowire transis-torrdquo IEEE Transactions onNanotechnology vol 10 no 1 pp 121ndash128 2011

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

8 Active and Passive Electronic Components

2 3 4 5 6 7 8 9 10

times10minus9

0014

0016

0018

002

0022

0024

0026

0028

003

0032

Film height

Proposed modelTCAD simulation

ΔVT

(mV

)

Figure 6 Variation of threshold voltagewith filmwidth for differentheight at 119871 = 20 nm

3 4 5 6 7 8

times10minus9

02

022

024

026

028

03

032

034

036

038

04

Film height

Thre

shol

d vo

ltage

(V)

ClassicalQuantum

TCAD simulation for classicalTCAD simulation for quantum

9

Figure 7 Variation of quantum and classical threshold voltage withfilm height Here 119871 = 20 nm and119882 = 9 nm

Themost important thing about this gate all around nanowiretransistor is that any change in one of the dimensions canbe nullified by proper tuning of other dimensions as thetransistor is symmetric about its height and width

Figure 7 shows the variation of the classical thresholdvoltage and quantum threshold voltage with the film height ata constant width of 9 nmThe value of the classical thresholdvoltage ranges from 027V to 029V for the correspondingchanges in the film height Similarly the quantum thresholdvoltage ranges from 03V to 031 V It shows that the devicehas a highly improved control over the threshold voltageTheTCAD results justify the simulation results

5 Conclusion

In this paper a quantum threshold voltage model for a GAAsilicon nanowire transistor is proposed by solving the 3DPoisson and Schrodinger equations Analytical expressionsfor potential and the inversion charge are expressed intheir closed forms The results show that the integratedcharge and the threshold voltage calculated in accordancewith the quantum effects of this proposed model are highlyimprovedThe future considerations include deriving the I-Vcharacteristics of the gate all around nanowire transistors andstudying the impact of scaling on various device parametersFinally to conclude this model provides an analytical anduseful way for the threshold voltage evaluations in gate allaround nanowire devices with a unified formalism employedin both classical and quantum mechanical approaches

References

[1] J Wang E Polizzi and M Lundstrom ldquoA computationalstudy of ballistic silicon nanowire transistorsrdquo in Proceedings ofthe IEEE International Electron Devices Meeting pp 695ndash698December 2003

[2] J P Colinge FINFETS and Other Multi-Gate TransistorsSpringer New York NY USA 2007

[3] D J Frank Y Taur andH-S PWong ldquoGeneralized scale lengthfor two-dimensional effects in MOSFETrsquosrdquo IEEE ElectronDevice Letters vol 19 no 10 pp 385ndash387 1998

[4] B Yu L Wang Y Yuan P M Asbeck and Y Taur ldquoScaling ofnanowire transistorsrdquo IEEE Transactions on Electron Devicesvol 55 no 11 pp 2846ndash2858 2008

[5] G D Sanders C J Stanton and Y C Chang ldquoTheory of trans-port in silicon quantum wiresrdquo Physical Review B vol 48 no15 pp 11067ndash11076 1993

[6] M-Y Shen and S-L Zhang ldquoBand gap of a silicon quantumwirerdquo Physics Letters A vol 176 no 3-4 pp 254ndash258 1993

[7] J P Colinge X Baie V Bayot and E Grivei ldquoQuantum-wireeffects in thin and narrow SOI MOSFETsrdquo in Proceedings of theIEEE International SOI Conference pp 66ndash67 October 1995

[8] C P Auth and J D Plummer ldquoScaling theory for cylindricalfully-depleted surrounding-gate MOSFETrsquosrdquo IEEE ElectronDevice Letters vol 18 no 2 pp 74ndash76 1997

[9] J-T Park and J-P Colinge ldquoMultiple-gate SOI MOSFETsdevice design guidelinesrdquo IEEE Transactions on ElectronDevices vol 49 no 12 pp 2222ndash2229 2002

[10] Z Ghoggali F DjeffalM A Abdi D Arar N NLakhdar and TTBendib ldquoAn analytical threshold voltagemodel for nanoscalerdquoin Proceedings of the 3rd International Design and TestWorkshop(IDT rsquo08) pp 93ndash97 December 2008

[11] Y-S Wu and P Su ldquoQuantum confinement effect in short-channel gate-all-around MOSFETs and its impact on the sensi-tivity of threshold voltage to process variationsrdquo in Proceedingsof the IEEE International SOI Conference October 2009

[12] C Te-Kuang ldquoA compact analytical threshold-voltage modelfor surrounding-gateMOSFETswith interface trapped chargesrdquoIEEE Electron Device Letters vol 31 no 8 pp 788ndash790 2010

[13] B Ray and SMahapatra ldquoModeling and analysis of body poten-tial of cylindrical gate-all-around nanowire transistorrdquo IEEETransactions on Electron Devices vol 55 no 9 pp 2409ndash24162008

Active and Passive Electronic Components 9

[14] L De Michielis L Selmi and A M Ionescu ldquoA quasi-analytical model for nanowire FETs with arbitrary polygonalcross sectionrdquo Solid-State Electronics vol 54 no 9 pp 929ndash9342010

[15] P R Kumar and S Mahapatra ldquoQuantum threshold voltagemodeling of short channel quad gate silicon nanowire transis-torrdquo IEEE Transactions onNanotechnology vol 10 no 1 pp 121ndash128 2011

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Active and Passive Electronic Components 9

[14] L De Michielis L Selmi and A M Ionescu ldquoA quasi-analytical model for nanowire FETs with arbitrary polygonalcross sectionrdquo Solid-State Electronics vol 54 no 9 pp 929ndash9342010

[15] P R Kumar and S Mahapatra ldquoQuantum threshold voltagemodeling of short channel quad gate silicon nanowire transis-torrdquo IEEE Transactions onNanotechnology vol 10 no 1 pp 121ndash128 2011

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of