Modeling of Transition Metal Dichalcogenide Transistors for

SPICE Simulation

Chandan Yadav and Yogesh Singh Chauhan Department of Electrical Engineering Indian Institute of Technology Kanpur

Kanpur, India-208016

MOS-AK, Dec. 2016, Berkley, USA

Outline

Introduction TMD materials Model Model Validation Conclusion

2003 90nm

2005 65nm

2007 45nm

2009 32nm

Strained Silicon High-K metal gate

ox

depoxSi XTε

ελ =0

Scaling 1971, 10µm

Bulk Si MOSFETs

021λλ =DGFET03

1λλ =Trigate041λλ =QGFET

thlongthTh VVV ∇−= ,

C.-H. Jan et al. IEDM 2012 S. Natarajan et al. IEDM 2014 J. P. Colinge, SSE 2004

2nd-Generation FinFET@14nm node FinFET@22nm node

Source: Intel

22nm@2011 14nm@2014

Require better

λ0(natural length): measurement of source/drain influence in channel.

law

)exp(0λg

th

LV −∝∇

Introduction Planer Bulk MOSFET to 3D FinFETs

µξinvds WQI =

*

1m

∝µ

ββ

υξµ

µµ /1

0

0

1

+

=

sat

*

1m

inj ∝υ

Improve transport properties

Digital application NMOS PMOS

InAs, InGaAs, Ge, InSb

Ge, InSb

http://www.electronicsweekly.com/uncategorised/gan-on-si-power-transistors-french-lab-leti-2015-07/

High power and high speed Z. Geng et al. IEEE ESDERC 2016

2D material Transistors

J. A. Alamo, nature 2011

Introduction New Materials to improve transport properties

Outline

Introduction TMD materials Model Model Validation Conclusion

TMD Materials

A. K. Giem and K. S. Novoselov “The rise of graphene” nature materials, vol. 6, 2007

Graphene (2D) : Atomically thin film of graphite

Fullerene (0D) Nanotube (1D)

Graphite (3D)

The Nobel Prize in Physics 2010 was awarded jointly to Andre Geim and Konstantin Novoselov "for groundbreaking experiments regarding the two-dimensional material graphene“.

Fig.: One-atom-thick single crystals.

Birth of 2D materials

TMD Materials

http://www.novitas.eee.ntu.edu.sg/Research/ResearchHighlights/Pages/Nanotechnologies.aspx?print=1

2D materials Table: Schematic representation of the periodic table with highlighted transition metal (blue) and chalcogen (yellow) elements that form layered TX2 materials.

A. Kuc “Low-dimensional transition-metal dichalcogenides” Chem. Modell., 2014, 11, 1–29

Fig. Band edge position of some 2D layered semiconducting materials and work function of some representative metals.

Feng Wang et. al. Nanotechnology 26 (2015) 292001

TMD Materials: Properties Layered Growth Flexible

Tunable bandgap

Hetero-integration

D. Johnson, IEEE Spectrum , Nov. 2014. T. Roy, ACS Nano, 2014, 8 (6), pp. 6259–6264.

H. J. Kwon et. al. Appl. Phys. Lett. 106, 113111 (2015)

Variable permittivity and polarizibility

P. Kumar, Phys. Rev. B 93, 195428, 2016 Kuc et al., Phys. Rev. B 83, 245213 (2011), Liu et al., ACS Nano 8, 4033 (2014).

TMD Materials: Applications

D. Akinwande et. al. Nature Comm. 2014 TMD-Materials

K. H. Wang et. al. Nature Nanotech. 2012

K. H. Wang et. al. Nature Nanotech. 2012

Bio-Medical

K. K. Jadeh, et. al. Adv. Funct. Mater. 2015, 25, 5086–5099

Wensi Zhang et. al., Nanoscale, 2015

Recent Progress in TMD channel FETs

S. B. Desai et. al., vol. 354, Oct. 2016

Scaling of transistor upto 1nm gate length: MoS2 channel transistor

L. Yu et. al., Nano Lett. 2016, 16, 6349−6356

Working Circuits

Y. Liu et. al. Nano Lett., 2016, 16 (10), pp 6337–6342

Y. J. Lee et. al., vol. 4, no. 5, Sept. 2016

MoS2 Nanowire

Outline

Introduction TMD materials Model Model Validation Conclusion

Model

Fig.: Schematic under consideration for modeling.

From Gauss’s law, charge density (Qs) for ultra-thin TMDs [1] can be expressed as:

( ) ( )sbmsbgboxsmsfgoxs VCVCQ ϕϕ −∆Φ−+−∆Φ−=

( )sgeffoxs VCQ αϕ−=

(1)

Rearrangement of (1) leads to

where,

0tbgox

boxfggeff VV

CC

VV −+=ox

box

CC

+= 1α

bmsbox

oxmst C

CV ∆Φ+∆Φ=0

Note: Vt0 is used further as fitting parameter to tune threshold voltage.

(2)

Qs for symmetrical double gate: apply same voltage at front and back gate and use Cox=Cbox.

φs →surface potential

[1] W. Cao, J. Kang, W. Liu, and K. Banerjee, IEEE Trans. Electron Devices, vol. 61, no. 12, pp. 4282–4290, Dec. 2014.

Model From charge neutrality we can write

( )ADs NNnpqQ −+−=

Using Fermi-Dirac statistics (fFD) »mobile electron charge (n)

( ) ( )dEEfEqn FDE

eD

C

∫∞

= ,2ρ

»mobile hole charge (p)

( ) ( )( )dEEfEqp FD

E

hD

V

−= ∫∞−

1,2ρ

( ) 2

*

,2 2 πρ esv

eDmgg

E = ( ) 2

*

,2 2 πρ hsv

hDmggE =

is 2D density of states (DOS) of electron.

where, where,

is 2D density of states (DOS) of hole.

( )

−+

=

kTEE

Eff

FD

exp1

1and Ef → Fermi level energy

EC → Minima of conduction band EV → Maxima of valance band me

* → electron effective mass mh

* → hole effective mass

(3)

(4) (5)

Model From charge neutrality we can write

( )ADs NNnpqQ −+−=

Using Fermi-Dirac statistics (fFD) »mobile electron charge (n)

( ) ( )dEEfEqn FDE

eD

C

∫∞

= ,2ρ

»mobile hole charge (p)

( ) ( )( )dEEfEqp FD

E

hD

V

−= ∫∞−

1,2ρ

( ) 2

*

,2 2 πρ esv

eDmgg

E =( ) 2

*

,2 2 πρ hsv

hDmggE =

where, where,

( )

−+

=

kTEE

Eff

FD

exp1

1

and

ρ2D,e →Density of states (DOS) for electrons ρ2D,h →Density of states (DOS) for hole Ef → Fermi level energy EC → Minima of conduction band EV → Maxima of valance band me

* → electron effective mass mh

* → hole effective mass kB → Boltzmann constant

(3)

(4) (5)

−+=

th

seDB v

VTqkn

ϕρ exp1ln,2

(6) V is the quasi Fermi potential

Model Surface potential for n-type FETs

»From (2), (3), and (6)

−+−−=

th

s

ox

eDB

ox

impgeffs v

VCTqk

CqN

Vϕρ

αϕ exp1ln1 ,2

Implicit equation in φs and needs numerical solution→ undesirable for compact model.

Use Boltzmann statistics for initial guess of φs

−−−−=

th

oximpgeff

ox

eDth

ox

impgeffs v

CqNVVC

qWv

CqNV

αα

αρ

ααϕ exp,2

2

0

Lambert-W function Use Hally’s method to refine φs0

+−= 2'

''

' 21

rrr

rrδ

δϕϕ += 0ss

where, ( )

imp

th

seDBsgeffox

qN

vVTqkVCr

−

−+−−=

ϕραϕ exp1ln,20

(7)

(8)

(9)

Model Drain current (Ids) model for n-type FETs

»Using DD model and current continuity

Wg → device width and μeff → effective mobility ( )

( )dx

VdVd

dVQWdxdVQWI s

sseffgseffgds

−−

==ϕ

ϕµµ

assuming φs – V =u → d(φs – V)=du, we can write

,exp1ln,2

−+=

−−=

th

seDB

geffoxs v

VTqkVuV

CQ ϕρα

αdu

dQCdu

dV s

oxα11−−=

dxduQW

dxdQ

QCW

I sgeffs

sox

geffds µ

αµ

−−=

( ) ( )

duvu

LWTqk

uuL

NWqQQ

CLW

I

d

s

u

u thg

eDgeffB

dsg

impgeffddss

oxg

geffds

∫

+−

−+−=

exp1ln

2

,2

22

ρµ

µαµ

Involve polylogarithm→ not suitable for compact model

0==

Vssu ϕdsVVsdu

== ϕ

dsVVsdd QQ=

=0=

=Vsss QQ

(10)

(11)

Model Drain current (Ids) model for n-type FETs

»Using DD model and current continuity

Wg → device width and μeff → effective mobility ( )

( )dx

VdVd

dVQWdxdVQWI s

sseffgseffgds

−−

==ϕ

ϕµµ

assuming φs – V =u → d(φs – V)=du, we can write

,exp1ln,2

−+=

−−=

th

seDB

geffoxs v

VTqkVuV

CQ ϕρα

αdu

dQCdu

dV s

oxα11−−=

( ) ( ) duvu

LWTqk

uuL

NWqQQ

CLW

Id

s

u

u thg

eDgeffBds

g

impgeffddss

oxg

geffds ∫

+−−+−= exp1ln

2,222 ρµµ

αµ

Involve polylogarithm→ not suitable for compact model

0==

Vssu ϕdsVVsdu

== ϕ

dsVVsdd QQ=

=0=

=Vsss QQ

(10)

(11)

(12)

Model Drain current (Ids) model for n-type FETs

,0,

0,expexp1ln

>>

<<

≈

+

thth

thth

th

vufor

vu

vufor

vu

vu

,0,

2

0,expexpexp1ln

2

2

2

2

>>

−

<<

−

≈

+∫

thth

d

th

sth

thth

d

th

sthu

u th

vufor

vu

vuv

vufor

vu

vuv

duvud

s

−+

++

−

−+

++

≈

+∫

th

d

th

d

th

s

th

s

th

u

u th

vu

vu

vu

vu

vduvud

s exp1

2exp1ln1

exp1

2exp1ln1

exp1ln

22

ββ

»Regional Integration

»Unification of regional Integration with proposed smoothing function

Fig.: Plot of smoothing function

(13)

(14)

(15)

Model Drain current (Ids) model for n-type FETs

Velocity saturation effect

ʋsat → saturation velocity ξx = dφs/dx → electric field along channel length Ids,cor → drain current after velocity saturation effect

2,

1

+

=

xsat

eff

dscords

v

II

ξµ

g

sssdx L

ϕϕξ

−=

»Average electric field and resultant Ids,cor

( )2,

1

−+

=

sssdgsat

eff

dscords

Lv

II

ϕϕµ

(16)

(17)

Model Drain current (Ids) model for n-type FETs

Interface trap states effect

Changes subthreshold slope Vary threshold voltage leads to mobility degradation

»The threshold voltages changes charge, therefore, including Ntrap, we write

∑

+−+

=i

th

iits

itraptrap

vqEV

DN

,

,

exp1ϕ

Interface trap states (Ntrap) occupied by electrons are [1]

( ) trapsgeffoxs qNVCQ −−= αϕ

The further calculation of φs and Ids considering Ntrap is performed as presented in the previous slides.

(18)

(19)

[1] W. Cao, J. Kang, W. Liu, and K. Banerjee, IEEE Trans. Electron Devices, vol. 61, no. 12, pp. 4282–4290, Dec. 2014.

Outline

Introduction TMD materials Model Model Validation Conclusion

Model validation Surface potential, transfer and output characteristics of DGFET and XOIFET

DGFET DGFET DGFET

DGFET XOIFET XOIFET

Simulation data: W. Cao, J. Kang, W. Liu, and K. Banerjee, IEEE Trans. Electron Devices, vol. 61, no. 12, pp. 4282–4290, Dec. 2014.

Model validation Transfer and output characteristics of MoS2 and WSe2 NFETs

Experimental data: A. Sachid et. al., Adv. Mater., vol. 28, no. 13, pp. 2547–2554, Apr. 2016.

Experimental data: H. Fang et. al., Nano Lett., vol. 13, no. 5, pp. 1991–1995, 2013.

Model validation Transfer and output characteristics of MoTe2 NFET and WSe2 PFET NFETs

Experimental data: H. Fang et. al. Nano Lett., vol. 12, no. 7, pp. 3788–3792, 2012.

Experimental data: H. Xu, et. al. ACS Nano., vol. 9, no. 5, p. 4900–4910, Apr. 2015.

Model validation Capacitance: less than Cox is due to quantum capacitance and interface trap states. Inverter characteristics with experimental data. Gummel symmetry test of the developed model.

Experimental data: H. Wang et. al. Nano Lett., vol. 12, no. 9, pp. 4674–4680, 2012.

Gummel symmetry test

Conclusion

Developed explicit model of the surface potential and drain current.

Included Fermi-Dirac statistics in model development.

Included 2D DOS to correctly predict the DOS related quantity i.e. quantum capacitance effect.

Model shows excellent fitting with the simulation and experimental data.

Validated with measured data from inverter circuit.