Microelectronics Reliability xxx (2013) xxx–xxx

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Microelectronics Reliability

journal homepage: www.elsevier .com/locate /microrel

Analytical model for threshold voltage of double gate bilayer graphenefield effect transistors

0026-2714/$ - see front matter � 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.microrel.2013.08.003

⇑ Corresponding author. Tel.: +60 7 553222.E-mail address: razali@fke.utm.my (R. Ismail).

Please cite this article in press as: Saeidmanesh M et al. Analytical model for threshold voltage of double gate bilayer graphene field effect tranMicroelectron Reliab (2013), http://dx.doi.org/10.1016/j.microrel.2013.08.003

M. Saeidmanesh a, M. Rahmani a, H. Karimi b, M. Khaledian a, Razali Ismail a,⇑a Faculty of Electrical Engineering, Universiti Teknologi Malaysia – UTM, Johor Bahru, Johor 81310, Malaysiab Malaysia Japan International Inst. of Technology, Universiti Teknologi Malaysia – UTM, Johor 81310, Malaysia

a r t i c l e i n f o a b s t r a c t

Article history:Received 7 August 2012Received in revised form 1 August 2013Accepted 5 August 2013Available online xxxx

A new model for threshold voltage of double-gate Bilayer Graphene Field Effect Transistors (BLG-FETs) ispresented in this paper. The modeling starts with deriving surface potential and the threshold voltagewas modeled by calculating the minimum surface potential along the channel. The effect of quantumcapacitance was taken into account in the potential distribution model. For the purpose of verification,FlexPDE 3D Poisson solver was employed. Comparison of theoretical and simulation results shows a goodagreement. Using the proposed model, the effect of several structural parameters i.e. oxide thickness,quantum capacitance, drain voltage, channel length and doping concentration on the threshold voltageand surface potential was comprehensively studied.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

As predicted by Moore, the semiconductor industry has beenfacing an exponential growth of the number of transistors per chipduring the last three decades. It is also predicted by ITRS (Interna-tional Technology Roadmap for Semiconductors) that the gatelength would scale down to 4.5 nm by 2023 [1]. However, main-taining this trend is a major challenge for both the industry andscientific community due to arising short channel effects. As a re-sult new device structures including FinFETs, nanowire FETs, andrecently carbon nanotube field-effect transistors (CNTFETs) andgraphene nanoribbon FETs have been proposed. Among themgraphene based devices (either single layer graphene or bilayergraphene) have attracted the attention of scientific communitydue to their fascinating electronic properties such as quantum halleffect, high carrier mobility and their ability to be scaled down [2–5].

On the other hand, the gapless nature of single layer graphenewhich is considered as the main obstacle on its application ingraphene based electronics [6], causes the gate voltage to lose itscontrol on switching off the device [7]. To overcome this drawback,bilayer graphene can be used where the band-gap is induced byintroducing a potential difference between two layers as a resultof an external perpendicular electric field [8,9]. Moreover, the po-tential difference can be realized with an applied gate field whichmeans the band-gap can be controlled by gate bias [10]. Recently,the feasibility of using bilayer graphene as channel material is ad-dressed in some analytical device models [11–14].

The surface potential is a fundamental variable in the derivationof various short channel effects. Thus, it is highly desirable for bi-layer graphene to model the surface potential analytically withthe detailed device physics for developing the threshold voltagemodel.

The paper is organized as follows. In Section 2 the potential dis-tribution along the channel is modeled for the proposed structure,the quantum capacitance is also modeled and subsequently in-cluded in the potential model. In Section 3 the threshold voltageis modeled based on the potential model. Section 4 deals withthe analysis of obtained results and illustrations. In Section 5 themain conclusions are drawn.

2. Theoretical model for potential distribution

A schematic cross section of a double gate BLGFET with the def-inition of the geometrical characteristics are shown in Fig. 1 wheretch, tg, tox are bilayer graphene, single layer graphene and oxidethicknesses respectively and L is the channel length. The first andsecond graphene layers are arranged in AB-stacking [15] as shownin Fig. 2.

Using the common Poisson’s equation the potential distribu-tion, U(x,y), for any point (x,y) of BLG channel is given by [16]:

@2Uðx; yÞ@x2 þ @

2Uðx; yÞ@y2 ¼ qðND þ niÞ

eg

0 6 x 6 tch;0 6 y 6 Lð1Þ

where eg is the dielectric constant of graphene; q is the electroncharge; ND [in cm�3] is the doping concentration and ni = bn/tchc[in cm�3] is the intrinsic carrier concentration where n is the two

sistors.

Fig. 2. A typical AB-stacked bilayer graphene [4].

2 M. Saeidmanesh et al. / Microelectronics Reliability xxx (2013) xxx–xxx

dimensional carrier concentration of bilayer graphene which is gi-ven by [17]:

n ¼Z 1

0DOSðEÞ½f ðE� EFSÞ þ f ðE� EFDÞ�dE ð2Þ

where f ðE� EFiÞ ¼ 1=ð1þ eðE�EFiÞ=KBTÞ; EFsðEFdÞ is the Fermi energy ofsource (drain) and DOS is the density of state

DOSðEÞ ¼ m�

2p�h2 1þ �hkg

2m�ðE� EcÞ½ �1=2

" #ð3Þ

where Ec is the conduction band edge, m⁄ and E are effective massand energy of electron in BLG respectively, �h is the reduced Planck’sconstant, t\ = 0.35 eV is the interlayer hopping parameter and kg isthe wave vector in which the smallest gap is observed which isgiven by [18]

kg ¼V

2tF�h

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiV2 þ 2t2

?

V2 þ t2?

sð4Þ

where V = V1 � V2 is interlayer potential, tF � 1 � 106 m s�1 is FermiVelocity [10].

In nanoscale devices where tox is small, quantum capacitancewhich is connected in series with oxide capacitance, should be ta-ken into account in overall gate capacitance [19]. To include theeffect of quantum capacitance into the potential distribution ofEq. (1), the surface charge density is used

Q ¼Z L

0

Z tch

0qðND þ niÞdxdy ð5Þ

As a result, q(ND + ni) = Q/(L tch). In addition, the surface charge den-sity using the Gauss theorem can be written as

Q ¼ tch CfgðUch;f þ Vfb � VfgÞ þ CbgðUch;b þ Vfb � VbgÞ� �

ð6Þ

where Cfg(Cbg) is the front (back) gate oxide capacitance, Uch,f(Uch,b)is the self consistent potential in the central region of the front(back) channel and Vfb is the flat band voltage, the voltage at whichthere is no band bending in the semiconductor, and is given by[20,21]:

Vfb ¼ /m �vg

qþ Eg

2þ KBT

qln

ND

ni

� �� �ð7Þ

where /m is the metal work function, vg is the electron affinity, T isthe temperature and KB is the Boltzmann constant. For symmetricstructures it is assumed that Cg = Cfg = Cbg = eox/tox, Uch = Uch,f = Uch,b

where eox is the oxide dielectric. To gain a better insight in devicecapacitances, the electrostatics of device is shown in Fig. 3

from which the differential capacitance seen by each gate isgiven by

Cd;i ¼ Cg 1� @Uch;i

V ig

� �ð8Þ

where (i = f,b) indicates front and back gates. According to Fig. 3 onecan rewrite Cd,i as

Oxide

Source Drain

Top Gatetox

OxideBack Gate

x

y

tch

L

Second Layer

First Layer

Fig. 1. Cross view of bilayer graphene double gate transistor.

Please cite this article in press as: Saeidmanesh M et al. Analytical model forMicroelectron Reliab (2013), http://dx.doi.org/10.1016/j.microrel.2013.08.003

Cd;i ¼CgðCg þ CqÞ

2Cg þ Cqð9Þ

from Eqs. (8) and (9) we have

Uch;i ¼CgVig

2Cg þ Cqð10Þ

consequently Eq. (6) can be obtained as

Q ¼ tchCgCg

2Cg þ Cq� 1

� �ðVfg þ VbgÞ þ 2Vfb

� �ð11Þ

In addition, the quantum capacitance is given by

Cq ¼ q2 @ni

@Eð12Þ

where E is energy. Substituting ni in Eq. (12), the quantum capaci-tance is written as

Cq ¼m�

2p�h2tch

DðEÞXi¼S;D

f ðE� EFiÞ ð13Þ

where DðEÞ ¼ 1þ �hkg

2m�ðE�EcÞ½ �1=2

h i. The Mexican-hat structure of the

band in BLG provides a large DOS and makes quantum capacitancecomparable to Cg. Now the effect of quantum capacitance can be in-cluded into the potential distribution of Eq. (1)

@2Uðx; yÞ@x2 þ @

2Uðx; yÞ@y2 ¼ 1

eg

QLtg

� �ð14Þ

Inasmuch as in the strong inversion region the charge controls thechannel potential along the y-direction [23,22], Eq. (1) is valid forweak inversion region where the potential can be approximatedby a simple parabolic function along the (x) [24,25]:

Uðx; yÞ ¼ P0ðyÞ þ P1ðyÞxþ P2ðyÞx2 ð15Þ

where coefficients P0, P1 and P2 are functions of y only and aresolved with the boundary conditions of:

Fig. 3. Equivalent circuit of device electrostatics.

threshold voltage of double gate bilayer graphene field effect transistors.

M. Saeidmanesh et al. / Microelectronics Reliability xxx (2013) xxx–xxx 3

Uð0; yÞ ¼ Uf ðyÞUðtch; yÞ ¼ UbðyÞdUðx; yÞ

dx

x¼0¼ eox

eg

Uf ðyÞ � V 0fgtox

dUðx; yÞdx

x¼tch

¼ eox

eg

V 0bg �UbðyÞtox

ð16Þ

where V 0fg ¼ Vfg � Vfb;V0bg ¼ Vbg � Vfb;Uf ðyÞðUbðyÞÞ is the potential

on front (back) channel surface. The coefficients Pi(i = 0, 1, 2) canbe determined by applying the boundary conditions of Eq. (16) inEq. (15)

P0ðyÞ ¼ Uf ðyÞ ð17aÞ

P1ðyÞ ¼eox

eg

Uf ðyÞ � V 0fgtox

ð17bÞ

P2ðyÞ ¼ð1þ Cg=CchÞV 0fg þ V 0bg � ð2þ Cg=CchÞUf ðyÞ

t2chð1þ 2Cch=CgÞ

ð17cÞ

where Cch = eg/tch. Substitution Eq. (17) in Eq. (15) one can obtainthe potential distribution in every point (x,y) of channel

Uðx; yÞ ¼ Uf ðyÞ þeox

eg

Uf ðyÞ � V 0fgtox

x

þð1þ Cg=CchÞV 0fg þ V 0bg � ð2þ Cg=CchÞUf ðyÞ

t2chð1þ 2Cch=CgÞ

x2 ð18Þ

The front surface potential, (x = 0), can be obtained by substitutingEq. (18) in Eq. (1)

@2Uf ðyÞ@2y

� aUf ðyÞ ¼ b ð19Þ

where

a ¼ 2ð2þ Cg=CchÞt2

chð1þ 2Cch=CgÞð20aÞ

b ¼ðQtch=LÞð1þ 2Cch=CgÞ � 2ð1þ Cg=CchÞV 0fg � 2V 0bg

t2chð1þ 2Cch=CgÞ

ð20bÞ

solving Eq. (19) with boundary conditions of Uf(0) = Vbi andUf(L) = Vbi + Vds, where Vbi = VTln (ND/ni) is the built-in potential ofthe channel-drain and channel-source junctions with VT = KBT/q asthermal voltage, the front gate surface potential along the channelcan be given by

Uf ðyÞ ¼ C1expðffiffiffiap

yÞ þ C2expð�ffiffiffiap

yÞ � ba

ð21Þ

where

0 5 100.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

L (nm)

Surfa

ce P

oten

tial (

V)

0 10.1

0.2

0.3

0.4

0.5

0.6

L (

Cq=3µF/cm2

Cq=5 µF/cm2

Cq=3

Cq=

L= 10 nm L= 20 n

Fig. 4. Surface potential along channel distance for different channel lengths and quantuthe calculated results from the analytical model are represented by the solid lines.

Please cite this article in press as: Saeidmanesh M et al. Analytical model forMicroelectron Reliab (2013), http://dx.doi.org/10.1016/j.microrel.2013.08.003

C1 ¼#S �#Sexpð

ffiffiffiap

LÞ � #D

expðffiffiffiap

LÞ � expð�ffiffiffiap

LÞð22aÞ

C2 ¼#Sexpð

ffiffiffiap

LÞ � #D

expðffiffiffiap

LÞ � expð�ffiffiffiap

LÞð22bÞ

#S ¼baþ Vbi ð22cÞ

#D ¼baþ Vbi þ VDS ð22dÞ

To verify the accuracy of potential distribution model the Flex-PDE program was utilized which has solved the Poisson’s equationnumerically within the defined boundary conditions. The followingparameters were used in the simulation: tox = 1 nm, tch = 1.2 nm,ND = 1 � 1018 cm�3, ni = 5 � 1016 cm�3, Vfg = 0.3 V, Vbg = 0.1 V,Vds = 0.1 V and Vfb = 0.2 V. The simulation and modeling resultsare illustrated in Fig. 4 for L = 10, 20, 30 nm and Cq = 3, 5 lF/cm2.It can be seen that channel length affects the position of minimumpotential and hence the threshold voltage value. There is a goodagreement between the simulation and modeling results.

3. Theoretical model for threshold voltage

According to the definition, threshold voltage is defined as thegate voltage at which the minimum surface potential is twice ofthe Fermi potential [26], i.e.

Uminð0; yÞ ¼ 2/f ð23Þ

where /f = KT/qln(ND/ni) is the Fermi potential. The position of min-imum potential along the channel which is called virtual cathode[27] can be calculated from Eq. (21). Finally, by plugging the ob-tained Umin in Eq. (23) and solving the relevant equations for Vfg,the threshold voltage is given by:

Vth ¼DðB� DVds þ EUf þ VbiÞ

AðE=4� DÞ ð24Þ

where

D ¼ expðffiffiffiap

LÞ þ expðffiffiffiffiffiffiffi�ap

LÞ � 2

E ¼ expðffiffiffiffiffiffi2ap

LÞ þ expðffiffiffiffiffiffiffiffiffiffi�2ap

LÞ

A ¼t2

chCgð1þ 2 CchCgÞð Cg

2CgþCq� 1Þ � 2ð1þ Cg

CchÞ

h i2Lð2þ Cg

CchÞ

B ¼tchCg ð Cg

2CgþCqÞVbg þ 2Vfb

h itchL ð1þ 2 Cch

CgÞ þ 2ð1þ Cg

CchÞVfb � 2V 0bg

h i2ð2þ Cg

CchÞ

ð25Þ

0 20nm)

0 10 20 300.1

0.2

0.3

0.4

0.5

0.6

L (nm)

µF/cm2 Cq=3µF/cm2

5 µF/cm2Cq=5 µF/cm2

m L= 30 nm

m capacitances. The simulation results of FlexPDE are represented by symbols and

threshold voltage of double gate bilayer graphene field effect transistors.

0 5 10 15 200.2

0.3

0.4

0.5

0.6

0.7

Position in the Channel (nm)

Surfa

ce P

oten

tial (

V)

Vds=0 VVds=0.1 VVds=0.2 V

Fig. 5. Surface potential along channel distance for different values of drain–sourcevoltages with tox = 1 nm, ND = 1 � 1018 cm�3 and ni = 5 � 1016 cm�3.

5 10 15 200.15

0.2

0.25

0.3

0.35

L (nm)

Thre

shol

d Vo

ltage

(V)

tox=1 nm

tox=2 nm

tox=3 nm

Fig. 6. Threshold voltage versus channel length for different oxide thickness’s withVds = 0.1 V, ND = 1 � 1017 cm�3 and ni = 5 � 1016 cm�3.

5 10 15 200

0.1

0.2

0.3

0.4

L (nm)

Thre

shol

d Vo

ltage

(V)

ND=1e+017ND=5e+017ND=9e+017

Fig. 7. Threshold voltage versus channel length for different doping concentrationwith Vds = 0.1 V, tox = 1 nm and ni = 5 � 1016 cm�3.

5 10 15 200.2

0.25

0.3

0.35

0.4

L (nm)

Thre

shol

d Vo

ltage

(V)

Vds=0.05 VVds=0.1 VVds=0.15 V

Fig. 8. Threshold voltage versus channel length for different drain source voltageswith tox = 1 nm, ND = 1 � 1017 cm�3 and ni = 5 � 1016 cm�3.

4 M. Saeidmanesh et al. / Microelectronics Reliability xxx (2013) xxx–xxx

4. Results and discussion

We have analytically calculated potential distribution andthreshold voltage of double gate BLG-FETs for different doping con-centrations, drain–source voltages and structural dimensions. Aschannel length becomes shorter the depletion region increaseswhich produces a huge surface potential that decreases the barrierheight. This phenomena is depicted in Fig. 5 where surface poten-tial is plotted versus channel length with Vds as parameter. It can beseen that the minimum surface potential value and its location aredependent on the drain voltage which is a sign of drain inducedbarrier lowering (DIBL).

In Fig. 6 the variation of threshold voltage as a function of chan-nel length is illustrated for different gate-oxide thickness (tox). Itcan be seen that threshold voltage increases as tox increases. Thisis because of the fact that gate-oxide electric field increases as tox

decreases. Even for an invariant gate voltage this electric fieldincrement induces more charge near the interface by decrementof threshold voltage.

In Fig. 7 the variation of threshold voltage versus channel lengthis shown for different doping concentrations (ND). It is illustratedthat as ND increases the threshold voltage also increases. This isdue to the fact that source-channel barrier increases with ND incre-ment which results in threshold voltage increment. Also, it is ob-served that threshold voltage decreases as source–drain distance(channel length) reduces that is because of the increment of elec-tric field effect on depletion regions of source and drain junctions.

Please cite this article in press as: Saeidmanesh M et al. Analytical model forMicroelectron Reliab (2013), http://dx.doi.org/10.1016/j.microrel.2013.08.003

Finally, Fig. 8 shows the effect of drain–source voltage onthreshold voltage along the channel. It can be seen that the thresh-old voltage reduces as drain–source voltage increases, and the ef-fect of drain–source voltage is also more significant for thesource side of the channel which represents the DIBL.

5. Conclusions

An analytical expression has been presented for potential distri-bution and threshold voltage of double gate BLG-FETs taking intoaccount the effect of quantum capacitance in potential model.For sake of verification, the analytical results were compared withthe simulation results of FlexPDE and a good agreement has beendemonstrated. The results show that channel length, doping con-centration and drain voltage could be used to adjust the thresholdvoltage. However, it has been proven that channel length of lessthan 10 nm and doping concentration in the order of 1017 cm�3

are effective. In addition, drain–source voltage is effective for chan-nels with L < 10 nm. As doping in the order of 1017 cm�3 is difficultand device with dimensions of less than 10 nm is still hard tofabricate, it is concluded that adjusting the threshold voltage inBLG-FET using L, Vds and ND is not a flexible approach.

Acknowledgements

The authors thank the Research Management Centre (RMC) ofUniversiti Teknologi Malaysia (UTM) for providing excellent re-search environment in completing this work.

threshold voltage of double gate bilayer graphene field effect transistors.

M. Saeidmanesh et al. / Microelectronics Reliability xxx (2013) xxx–xxx 5

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threshold voltage of double gate bilayer graphene field effect transistors.