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PROC. 29th INTERNATIONAL CONFERENCE ON MICROELECTRONICS (MIEL 2014), BELGRADE, SERBIA, 12-14 MAY, 2014

Modeling Electrostatics of Double Gated Monolayer MoS2 Channel Field-Effect Transistors

Galina S. Sheredeko, Kirill S. Zemtsov, Gennady I. Zebrev

Abstract - Analytical modeling of double gated field-

effect transistor with conductive channel based on monolayer molybdenite (MoS2) is presented.

I. INTRODUCTION

Recently isolated monolayer material MoS2 has

properties which seem to be a very promising for future digital electronics [1,[2

]]. It combines several advantages of monolayer technologies and devoid in some extent of its shortages. Particularly, the most attractive characteristic of the MoS2 is that it has a rather wide intrinsic bandgap (in contrast to most widely studied two-dimensional material – graphene) and at the same time at least ostensibly ensures high enough carrier mobility. Another advantage of the single-layer MoS2 are the absence of parasitic depletion layers typical for the Si-MOSFETs and reduced geometric short-channel effects. Aiming to develop a convenient tool for simulation of the prospect MoS2 digital circuits the electrostatics of double-gate structures and current-voltage characteristics in linear mode of the single-layer MoS2 based FETs will be described in this work.

II. MONOLAYER FIELD-EFFECT TRANSISTOR

A. Double gate configuration of monolayer FET

A schematic view of the structure and the energy band diagram of monolayer double gate field-effect transistor are shown in Figs. 1-2.

Fig. 1. Double-gate structure energy band diagram monolayer FETs ors.

Monolayer conductor

ζ eV2

eV1

d1 d2

gate 1 gate 2

eVox2 eVox1

EG

Fig. 2. Double-gate structure energy band diagram

Equivalent circuit for double-gate monolayer FET is shown in Fig. 3.

C1

εF

CQ Cit

C2

V1 V2

Fig. 3. Equivalent circuit of double-gate monolayer structure

B. Carrier density and quantum capacitance as function of chemical potential

The total charge density per unit area nS in the

monolayer channel is determined by temperature T and chemical potential ζ

2

/ 2ln 1 exp v B G

SB

g mk T En

k Tζ

π −

= +

, (1)

where ħ is the Plank constant, kB is the Boltzmann constant, m is the carrier effective mass, vg is the valley degeneracy,

Galina S. Sheredeko, Kirill S. Zemtsov and Gennady I. Zebrev are with the Department of micro- and nanoelectronics of National Research Nuclear University MEPHI, Kashirskoe sh., 31, Moscow, Russia, E-mail: gizebrev@mephi.ru

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chemical potential ζ is assumed to be reckoned from the middle of the bandgap GE (~1.8 eV for MoS2 monolayer) so as positive (negative) values of ζ correspond to predominance of electrons (holes).

Quantum capacitance is defined as 2Q SC e dn dζ=

and negligible in all wide bandgap semiconductors for non-degenerate case when the Fermi level lies deeply in the bandgap. This fact is contrasted with the gapless graphene case where quantum capacitance is significant at all electric regimes including the charge neutrality point [3

Using Eqs.1 one obtains ].

02 ,

/ 21 exp

QSQ

G

B

CdnC e

d Ek T

ζ ζ= =

−+ −

(2)

where

20 2

vQ

g mC e

π= ≅

200-400 fF/µm2.

For degenerate case when / 2GEζ > the carrier density is expressed as follows

( )20 / 2 S Q Ge n C Eζ≅ × − . (4)

For non-degenerate 2D electron gas its density is exponentially dependent on the Fermi level position in the forbidden gap

0/ 2

exp GS Q T

B

Een C

k Tζ

ϕ −

≅

(5)

These relationships are determined by general statistics of 2D electron gas with a finite effective mass. In contrast, a value of the chemical potential (or, equivalently, position of the Fermi level) in the monolayer channel depends on electrostatics of the whole double gate structure and on many significant details, including interface trap density and electric connection configuration.

C. Electrostatics of monolayer double gate FET

Both gates are assumed to be positively biased with respect of the grounded source. Assuming for the sake of simplicity a uniform spectrum of interface trap density Dit, the charge neutrality condition is given by the equation

( )1 21 0 2 0

1 2S it

eV eVen C

d dζ ζ

ε ε ε ε ζ ζ− −

+ = + , (6)

or, equivalently

( )2 21 2

1 1 1

1 Sit enC C CV VC C C

ζζ

++ = + +

, (7)

where VG1(2) is the top (back) gate voltage with reference to the source,

( ) ( ) ( )1 2 1 2 1 2G FBV V V= − , ( ) ( )1 2 1 2/MS oxFBV eN Cϕ= − are the charge neutrality biases, ζ is the chemical potential in the monolayer, C1(2) is the sheet capacitance of top (back) oxide, 2

it itC e D= is the interface traps capacitance. The degeneration offset voltage VO up to an additive

term can be defined for the gate 1 at the grounded gate 2 (V2 = 0) as

( ) 2 01 1

1 1

/ 2 1 ,2

it G SO G

C C E enV V E

C eСζ

+= = = + +

(8)

where the density at the offset voltage can be estimated at

/ 2GEζ = using Eq.1

( ) 00 / 2 ln 2Q T

S S G

Cn n E

eϕ

ζ= = = >1012 cm2. (9)

Please do not confuse the degeneration offset voltage with the threshold voltage defined traditionally as extrapolation of the linear portion of the transfer characteristics.

1 1 0 1T O SV V en C≅ − . Using (7) and (8) one reads

( )( )02

1 11 1

1 ,2

S Sit GO

e n nC C EV V

C eСζ

ζ− + − = + − +

(10)

Expanding Eq.1 into the Taylor series in the vicinity of

the degeneration point one gets

( )2

00 0

1 .2 2 8 2

QG GS S Q

T

CE Een en Ce e

ζ ζ ζϕ

− ≅ × − + × −

This expansion is valid approximately only within the

range

2 .2

GT

Ee

ζ ϕ− <

in contrast to the high electron density region

2 2G TE eζ ϕ> + , where the electron density is approximated by Eq. 4.

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D. “Classical” high electron density region

High electron density correspond to relatively small de Broglie wavelength and, thereby, to the “classical” case. This circumstance is expressed through independence of electron density on quantum capacitance as it takes place in the silicon MOSFETs. Actually,

( )1 1 1S Ten C V V≅ − (11)

because of 0 2Q Q itC C C C≅ >> + , as a rule, in this region.

The inversion layer (“quantum”) capacitance plays rather minor role in the silicon FETs since it is very low in the subthreshold operation mode and extremely high in above threshold strong inversion regime.

E. Subthreshold electron density region

Fr the subthreshold region of the FET operation with very low charge density in the channel one can neglect electron density in Eq. 8 and one obtains

( )1 1 0 1

2 12 1G O S

it

E V V en Ce C C C

ζ− +

− ≅+ +

. (12)

Then

1 1 0 10

1

ln 1 exp O SS Q T

T

V V en Cen C

mϕ

ϕ − +

≅ + , (13)

where the ideality factor for gate 1 was introduced

( )1 2 11 itm C C C= + + .

F. Linear I-V characteristics The dependence of carrier density on gate voltage

determines the low-field and linear region of I-V characteristics

0D S DSWI e n VL

µ= , (14)

where W/L is the width to length ratio of the channel, 0µ is microscopic mobility, VDS is the drain bias.

We found an approximate set of relationships for description of I-V characteristics

1 10

2

1

ln 1 exp1

OS Q T

it QT

V Ven C

C C CC

ϕϕ

− ≅ + + + +

, (15)

0

1 1

1

1 exp

QQ

O

T

CC

V Vmϕ

≅ −

+ −

. (16)

Figure 4 shows comparison of the experimental I-V characteristics taken from [1] and simulation in the linear regime provided with Eqs. 14-16.

Fig. 4. Comparison of experimental data (points, [1]) and simulation (solid lines). Parameters W/L = 4/1.5 µm, dox1= 270 nm, dox2= 30 nm, , VDS = 0.01V are taken from [1], εox2 = 16, εox1 = 3.9, fitted parameters µ0 = 28 cm2/V×s, Cit = 6.0×10-7 fF/µm2, VO1 = 5 V, effective mass m = 0.37 m0 Dashed lines correspond to the two quasi-linear dependencies with field effect mobility µFE ≅ 28 cm2/V×s (VT ≅ - 27.5 V) and µFE ≅ 170 cm2/V×s (VT ≅ - 3 V). Unfortunately a restricted character of experimental data presented in [1] (dependence on the thick back gate voltage) does not allow to provide parameter extraction in a unique way. Particularly, the quantitative results in low the density region (VBG < VO1=5 V) depend significantly on the choice of effective mass and the grounded top gate capacitance value. This circumstance has caused recently controversy about the mobility extraction procedures [4

] in MoS2 field-effect transistors.

REFERENCES

[1] B. Radisavljevic, A. Radenovic, J. Brivio, V. Giacometti, A.

Kis, “Single-layer MoS2 transistors,”, Nature Nanotechnology, 2011, pp. 147-150.

[2] Youngki Yoon, Kartic Ganapathi, Sayeef Salahuddin, “How good can monolayer MoS2 transistors be”, Nano Letters, 2011, nl2018178

[3] G.I. Zebrev et al. “Using Capacitance Methods For Interface Trap Level Density Extraction In Graphene Field-Effect Devices,” MIEL 2012 Proceed.

[4] M. Fuhrer, J. Hone, “Measurements of mobility in dual-gated MoS2 transistors,” Nature Nanotechnology, 146-148, Vol.8, March 2013.