J Comput Electron (2014) 13:180–185DOI 10.1007/s10825-013-0497-0

Carrier scattering and impact ionization in bilayer graphene

M. Saeidmanesh · M.H. Ghadiry · M. Khaledian ·M.J. Kiani · Razali Ismail

Published online: 12 September 2013© Springer Science+Business Media New York 2013

Abstract Transport properties of carriers in bilayer graph-ene (BLG) were studied. Several analytical models were de-veloped for drift velocity, scattering rate and ionization coef-ficient of BLG for the first time. Then, the joint effect of tem-perature and potential difference of layers were addressed onthe modeled parameters. The accuracy of the proposed mod-els for drift velocity and scattering rate was verified by thesimulation results of published works. In addition, the an-alytical results of ionization coefficient of BLG were com-pared with those of silicon.

Keywords Bilayer graphene · Drift velocity · Scatteringrate · Lucky drift · Ionization

1 Introduction

Although scaling of FETs has been the key to reduce powerconsumption and propagation delay of integrated circuits forthe last decade, there is a concern on the limit of this pro-cess in the near future [1]. Hence, ensuring the performanceimprovement of integrated circuits demands introduction ofnew alternative materials. The possibility of using BLG asthe channel material in FETs to overcome the silicon basedFETs’ problems, such as leakage current [2], has been ad-dressed in some literatures [3, 4].

The stack of two Graphene layers—the so-called BilayerGraphene (BLG)—attracts significant attention due to its in-

M. Saeidmanesh · M. Khaledian · M.J. Kiani · R. Ismail (B)Faculty of Electrical and Electronics, Universiti TechnologiMalaysia, Skudai, Malaysiae-mail: razali@fke.utm.my

M.H. GhadiryDepartment of Computer Engineering, Arak Branch, IAU, Arak,Iran

teresting electronic properties [5, 6]. The coupling of twographene layers in the AB-stacking rises the hyperbolic en-ergy dispersions for valance and conductance bands that areseparated by the interlayer interaction. Although, BLG is agapless material in its pristine form, it is predicted throughtheoretical studies [7, 8] that the band gap can be induced bya potential difference between graphene layers. The poten-tial difference can be resulted either by changing the dopingor by an external bias. Having the controllable band-gap isone of the most exceptional properties of BLG that makes ita promising material for post silicon nanoelectronic devices.

A cross section view of a BLG field effect transistor(FET) and a close-up view of its channel are shown inFig. 1. Nanoscale dimension, on the other hand, results theincrement of hot-electron generation and high electric field,whereby the latter causes impact ionization to occur [9].Impact-ionisation coefficient (α) is defined as the proba-bility of reaching the threshold energy (Et ) per unit dis-tance along the direction of the electric field (F ) whereimpact ionization happens. Impact ionization can cause un-desirable effects such as excess gate current, high electronmobility and substrate current in MOSFETs; the formercauses power dissipation and the latter causes bipolar ac-tions (latch-up) [9]. Therefore it is necessary to study theimpact ionization in BLG as a material for future generationnanoelectronic.

Although there are some analytical expressions to de-scribe ionization coefficient for Silicon, there has been noattempt to do this for BLG. Maes et al. [9] have reviewedthe previous ionization modeling for silicon. The theory ofimpact ionization for silicon has been fundamentally intro-duced by Wolff [10], Shockley [11] and Ridly [12].

The energy gained by a carrier between two succes-sive collisions (E = qFλm) is assumed to be significantlylower than the energy loss in the time between two col-

J Comput Electron (2014) 13:180–185 181

Fig. 1 (a) BLG transistor witha top gate and (b) BilayerGraphene (BLG)

lisions in the Lucky Ballistic (LB) theory of Shockley,where λm is the mean free path. In this theory the ioniza-tion coefficient is given by α = exp(−Et/qFλm)/l wherel = Et/qF is the distance travelled ballistically by a car-rier, starting at zero energy, to gain the threshold energy (Et )

and exp(−Et/qFλm) is the probability of avoiding the col-lision [9].

The opposite assumption is made in Wolff theory of ion-ization i.e. the energy gained by carriers is assumed to bemuch greater than the lost energy. In this model the elec-tron’s energy increases gradually as a result of numerouscollisions. The ionization coefficient in this case is given byα = (1/l) exp(−Et/F

2).The Lucky Drift (LD) theory proposed by Ridley is based

on the difference between energy relaxation (τE) and mo-mentum relaxation (τm) rates. In this theory an electron canreach Et through different modes, (1) in LB mode by start-ing from zero energy and avoiding collision, (2) by expe-riencing some momentum relaxation and not energy relax-ation collisions while drifting in the electric field and (3) bycombining these two modes i.e. starting from zero energyand gaining E through ballistic motion and thence reachingto Et through drift motion. The ionisation coefficient basedon LD model is given by

α = 1

λm

r(exp( −lλE

) − r exp( −lλm

))

1 − exp( lλE

) − r2(1 − exp( −lλm

))(1)

where r = λm/λE .

2 Preliminaries

At high electric field, where the impact ionization occurs,the drift current is the dominant part of drain current. Hence,in this section the drift velocity as well as momentum relax-ation time are analytically modelled for BLG.

2.1 Drift velocity

According to the definition, the average velocity of carrierscan be given by:

〈v〉 =∫ ∞

0 |v|DoS(E)f (E)dE

n(2)

where f (E) = 1/(1+ e(E−EF )/KBT ) is the Fermi-Dirac dis-tribution function which gives the probability of occupationof a state at any energy level and n is the carrier concentra-tion of BLG. In this function EF is the Fermi Energy, KB

is the Boltzmann constant, T is temperature and DoS is thedensity of states, which is written as:

DoS(E) = 1

2π�2

[

m∗ +V (

V 2+2t2⊥V 2+t2⊥

)1/2

2υF [2m∗(E − Ec)]1/2

]

(3)

where t⊥ = 0.35 eV is the inter-layer hopping parameter,Ec = Eg/2 + (V1 + V2)/2 is the conduction band edge,υF ≈ 1 × 106 m s−1 is the Fermi Velocity, � is the reducedPlanck’s constant, V = V1 − V2 is the potential differencebetween graphene layers and m∗ is electron effective massof electron in BLG which is given by:

m∗ = t⊥(V 2 + t2⊥)3/2

2V (V 2 + 2t2⊥)

1

υ2F

(4)

By using the Fermi-Dirac Integral approximations andmaking the substitutions of x = (E − Ec)/KBT and ηF =(EF − Ec)/KBT , (2) can be rewritten as:

〈v〉 = 1

nπ�2

[

KBT 2f (1) + KBT Ecf (0)

+ �√

KBT

υf

√8m∗ f

(−1

2

)]

(5)

where � = V (V 2+2t2⊥V 2+t2⊥

)1/2, f (−12 ), f (0) and f (1) are

Fermi-Dirac integrals of order (−1/2), 0 and 1 respectively.The drift velocity with respect to electric field is illustratedin Fig. 2. To verify the accuracy of the model, analyticalresults were compared against experimental data extractedfrom the work of Li et al. [13]. It can be seen that for highelectric field regions the drift velocity will be saturated.Hence, in our model, drift velocity can be treated as an in-variable parameter with respect to electric field.

2.2 Relaxation time

The carrier motion in the channel can be affected by twokinds of collision, firstly the collision with impurities and

182 J Comput Electron (2014) 13:180–185

Fig. 2 Drift velocity of BLG for a wide range of electric field. Simu-lation data are extracted from the work of Li et al. [13]

secondly with other carriers. To evaluate the momentum re-laxation rate each collision is weighted with the fractionalchange as:

1

τm(E)= Σk′(1 − cos θ)S

(k, k′) (6)

where S(k, k′) is the transition rate from state k to state k′and is given by [14]:

S(k, k′) = 2π

�

∣∣ψ∗(−→k ).ψ

(−→k′ )d−→r ∣

∣2

×(

D2A�ωβ

ρmAv2s

)

Nβδ(E′ − E

)(7)

where ρm = 7.6e(−8) g/cm2 is the mass density of graph-ene, DA = 16.5 eV is the acoustic deformation potential, vs

is the sound velocity, ωβ = vsβ (with β = |k − k′|) is theacoustic phonon energy, � is the reduced Planck’s constant,Nβ = 1/(exp(βωβ) − 1) is the phonon occupation numberand

E(k) = Eg

2+ �

2

2m∗(|k| − kg

)2 + V1 + V2

2(8)

is the energy dispersion of BLG [15] near the finite wavevector of kg which is given by

kg = V

2υF�

√V 2 + 2t2⊥V 2 + t2⊥

(9)

where vf ≈ 1 × 106 m s−1 is the Fermi Velocity. The small-est band gap of BLG which takes place at kg is expressedas:

Eg = V t⊥√V 2 + t2⊥

(10)

Potential difference of layers induces the electric field ofF = V/td , where td = 0.35 nm is the interlayer distance inBLG. Electron effective mass and energy bandgap of BLGwith respect to high electric field ranges are depicted inFig. 3. The density of symbols indicates the saturation be-haviour of these parameters.

Fig. 3 Electron effective mass (a), and energy bandgap (b) of BLGversus electric field

Fig. 4 Scattering rate BLG for different values of temperature as afunction of carrier energy. The simulation data of monolayer grapheneare extracted from Fang et al. [17]

Using the wave function of BLG [16]

ψ(−→k ) = 1√

2

(1

eiθ

)

e(i−→k −→r ) (11)

the overlap probability of (7), |ψ∗(−→k ).ψ(−→k′ )d−→r |2, can be

simplified to [ 12 (1 + cos θ)]. Then, substituting (7) in (6),

momentum relaxation time can be obtained by applying∫δ(x −x′)f (x′)dx′ = f (x) and converting summation into

the integral (for more details, see the Appendix):

1

τm(E)= D2

AωβNβ

2ρmv2s

(2m∗

�+

√2m∗

E − Ec

)

(12)

Figure 4 shows the scattering rate of BLG as a function ofelectron energy. The scattering rate is found to increase withincreasing temperature for all ranges of electron energy. AtT = 300 the analytical results of BLG model are comparedwith the simulation data of monolayer graphene extractedfrom the work of Fang et al. [17]. The temperature depen-dence of scattering rate reported by Simpson et al. [18] con-firms our model. Although the rate for BLG is higher thanthat of monolayer graphene, the trend is the same for bothof them.

Considering the high-energy phonon as the dominantscattering mechanism, the obtained momentum relaxation

J Comput Electron (2014) 13:180–185 183

Fig. 5 Temperature dependence of momentum relaxation mean freepath. The analytical results of model are shown by solid line and thesimulation data with symbols

time can be used to derive the other scattering parametersas:

τE(E) = τm

�ωop

E

λE(E) = υF τE, λm = υF τm

(13)

where ωop = 0.15 eV/� is optical phonon frequency andE = (1/2)m∗

eυd2 is the electron’s energy. In Fig. 5 the mo-

mentum relaxation mean free path is plotted as a functionof temperature. The analytical results are compared withthe simulation data extracted from the study done by Bhat-tacharya and Mahapatra [19].

3 Impact ionization model

The proposed model is on the basis of the fact that the ion-ization coefficient has the overall form of (α = Ptotal/ l)

where Ptotal is the total probability of reaching the thresh-old energy and l is the distance travelled by a carrier prior toimpact ionization. In the proposed model an electron is as-sumed to reach the threshold energy through either ballisticmotion or drift motion.

3.1 Ballistic motion

According to Fang et al. [20], the dominant scattering mech-anism at low energies is the elastic acoustic phonon scatter-ing. If PBM(F, t) is the probability that an electron avoidssignificant momentum relaxation and energy relaxation col-lisions during travel time t , it is true that

PBM(F, t) = exp

(

−∫ t

0

dt

τm

)

(14)

When an electron travels in an electric field with strengthF , it gains the energy of E = qFx, where x is the distance

Fig. 6 Electron Effective mass (a), Threshold Energy and EnergyBandgap (b) of BLG versus Interlayer potential

travelled by the electron. Taking x = λE = τEυd , the en-ergy gain rate can be obtained as dE/dt = qFυd . There-fore, PBM(F,E) can be given by

PBM(F,E) = exp

(

−∫ Et

0

dE

τmυdqF

)

(15)

By plugging (12) in (15) and performing the integral, onecan rewrite the total probability of ballistic motion in theform of

PBM(F,Et ) = exp

( −D2AωβNβ

2ρmv2s VdqF

[2m∗

�Et

+ √8m∗(Et − Ec) − √

2m∗Ec

])

(16)

where Et is the threshold energy of ionization. For semicon-ductors with parabolic and direct band structure [21], Et isgiven by

Et = Eg

2m∗ + m∗h

m∗ + m∗h

(17)

where m∗h = 1.25m∗, is the hole effective mass. As illus-

trated in Fig. 6, the effective mass, threshold energy andenergy bandgap of BLG enter the saturation region as in-terlayer potential increases.

3.2 Drift motion

Without experiencing inelastic collisions, a lucky electronundergoes two steps of gaining energy while reaching thethreshold energy. In the first step, the electron moves towardthe field direction and reaches the energy level E while expe-riencing neither elastic nor inelastic collisions. In this casethe probability of reaching level E of energy is given by

PS1(F, t) = PBM(F, t)dt

τm

(18)

where dt/τm is the probability of first collision. Convertingthe integral variable to energy, (18) can be rewritten as

PS1(F,E) = PBM(F,Et )

τmυdqFdE (19)

184 J Comput Electron (2014) 13:180–185

Fig. 7 Ionisation coefficient of BLG versus low range electric field forthree different interlayer potential energies. The simulation results ofsilicon are extracted from Overstraeten and Man [22]

Then, the second step starts after the first collision when theelectron energy increases from E to Et with the probabilityof

PS2(F,E) = exp

(

−∫ Et

E

dE

τE(E)υdqF

)

= exp

(−�wopD2AωβNβ

ρmv2s m

∗V 3d qF

[2m∗(Et − E)

�

+ 2m∗(√

Et − Ec − √E − Ec)

])

(20)

Accordingly the total probability of drift motion is given by

PDM(F,E) =∫ Et

0PS1(F,E)PS2(F,E) (21)

where the Integral variable dE is included in PS1. One cansolve the integral numerically to obtain PDM(E,F ) BLG.Hence, the total probability of reaching the threshold energyis given by

Ptotal(F,E) = PBM(F,Et ) + PDM(F,Et ) (22)

The ionization coefficient values for BLG with respectto low electric field for three different values of interlayerpotential energies is depicted in Fig. 7. The analytical re-sults of the proposed model for BLG are compared againstthe simulation data of silicon extracted from the investiga-tion reported by Overstraeten and Man [22]. Although, thevalues for BLG are more than those of silicon at the wholerange of the electric field, it can be seen that the trend is thesame for both themes. In addition, as Fig. 6 shows, thresh-old energy increases with potential energy and consequentlyionization coefficient decreases. In Fig. 8, ionization coeffi-cient of BLG is illustrated as a function of electric field. Inthis figure the effects of high electric field and temperatureare taken into account as well. It can be seen that as eitherpotential energy of layers or temperature increases the ion-ization coefficient decreases. This implies that perpendicularelectric field applied to the layers in BLG can be used to tunethe ionization coefficient.

Fig. 8 Ionisation coefficient of BLG versus electric field (includinghigh ranges) for two values of interlayer potential energies and tem-peratures

4 Conclusion

The drift velocity, scattering rate and ionization coefficientof BLG have been analytically modeled in this paper. Thesimulation results of published works were used to verifythe accuracy of the proposed models. The ionization modelwas on the basis of lucky drift theory of ionization which hasbeen developed in previous works. The effect of temperatureand interlayer potential has been studied on the modeled pa-rameters. It was found that interlayer potential of BLG canbe used as a tuning parameter for ionization coefficient. Inaddition, the obtained results revealed that the ionization co-efficient of BLG is higher than that of silicon.

Acknowledgements The authors would like to thank the ResearchManagement Centre (RMC) of Universiti Teknologi Malaysia (UTM)for providing excellent research environment in which to complete thiswork.

Appendix

Plugging (7) in (6) and converting summation into integral:

1

τm(E)= 2

�π

(D2

A�ωβ

ρmAv2s

)

Nβ

×∫ ∞

0δ

(

Ec + �

2m∗(|k| − kg

)2 − E

)

kdk

× 1

2

∫ 2π

0(1 − cos θ)(1 + cos θ)dθ (23)

According to the definition of Dirac delta function:∫

δ(x − x′)f

(x′)dx′ = f (x) (24)

the (23) can be rewritten as:

1

τm(E)= A

2�

(D2

A�ωβ

ρmAv2s

)

Nβ

√2m∗

E − Ec

×∫ ∞

0δ(s − √

E − Ec)

(√2m∗�

s + kg

)

ds (25)

J Comput Electron (2014) 13:180–185 185

where

s = �√2m∗

(|k| − kg

)(26)

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