東北大学機関リポジトリTOUR

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Device model for graphene bilayer field-effect

transistor

著者

尾辻泰一

journal or

publication title

Journal of Applied Physics

volume

105 number

10 page range

104510-1-104510-9

year

2009

URL

http://hdl.handle.net/10097/47802

doi: 10.1063/1.3131686

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Device model for graphene bilayer field-effect transistor

V. Ryzhii,1,2,a兲M. Ryzhii,1,2A. Satou,1,2T. Otsuji,2,3and N. Kirova4 1

Computational Nanoelectronics Laboratory, University of Aizu, Aizu-Wakamatsu 965-8580, Japan 2

Japan Science and Technology Agency, CREST, Tokyo 107-0075, Japan 3

Research Institute for Electrical Communication, Tohoku University, Sendai 980-8577, Japan 4

Laboratoire de Physique des Solides, Univ. Paris-Sud, CNRS, UMR 8502, F-91405 Orsay Cedex, France 共Received 19 December 2008; accepted 14 April 2009; published online 26 May 2009兲

We present an analytical device model for a graphene bilayer field-effect transistor共GBL-FET兲 with a graphene bilayer as a channel and with back and top gates. The model accounts for the dependences of the electron and hole Fermi energies as well as energy gap in different sections of the channel on the bias back-gate and top-gate voltages. Using this model, we calculate the dc and ac source-drain currents and the transconductance of GBL-FETs with both ballistic and collision dominated electron transport as functions of structural parameters, the bias back-gate and top-gate voltages, and the signal frequency. It is shown that there are two threshold voltages, Vth,1and Vth,2,

so that the dc current versus the top-gate voltage relation markedly changes depending on whether the section of the channel beneath the top gate共gated section兲 is filled with electrons, depleted, or filled with holes. The electron scattering leads to a decrease in the dc and ac currents and transconductances, whereas it weakly affects the threshold frequency. As demonstrated, the transient recharging of the gated section by holes can pronouncedly influence the ac transconductance resulting in its nonmonotonic frequency dependence with a maximum at fairly high frequencies. © 2009 American Institute of Physics.关DOI:10.1063/1.3131686兴

I. INTRODUCTION

The features of the electron and hole energy spectra in graphene provide the exceptional properties of graphene-based heterostructures and devices.1–6 However, due to the gapless energy spectrum, the interband tunneling7 can sub-stantially deteriorate the performance of graphene field-effect transistors 共G-FETs兲 with realistic device structures.8–11 To avoid drawbacks of the characteristics of G-FETs based on graphene monolayer with zero energy gap, the patterned graphene 共with an array of graphene nanoribbons兲 and the graphene bilayers can be used in graphene nanoribbon FETs 共GNR-FETs兲 and in graphene bilayer FETs 共GBL-FETs兲, re-spectively. The source-drain current in GNR-FETs and GBL-FETs, as in the standard GBL-FETs, depends on the gate voltages. The positively biased back gate provides the formation of the electron channels, whereas the negative bias voltage between the top gate and the channels results in forming a potential barrier for electrons which controls the current. By properly choosing the width of the nanoribbons, one can fabricate graphene structures with a relatively wide band gap12 共see also Refs.13–16兲. Recently, the device dc and ac character-istics of GNR-FETs were assessed using both numerical14 and analytical17–19 models. The effect of the transverse elec-tric field 共to the GBL plane兲 on the energy spectrum of GBLs20–22 can also be used to manipulate and optimize the GBL-FET characteristics. A significant feature of GBL-FETs is that under the effect of the transverse electric field not only the density of the two-dimensional electron gas in the GBL varies, but the energy gap between the GBL valence and conduction bands appears. This effect can markedly

influ-ence the FET characteristics. The structure of a GBL-FET is shown in Fig. 1. In this paper, we present a simple analytical device model for a GBL-FET, obtain the device dc and ac characteristics, and compare these characteristics with those of GNR-FETs.

The paper is organized as follows. In Sec. II, we con-sider the GBL-FET band diagrams at different bias voltages and estimate the energy gaps and the Fermi energy in differ-ent sections of the device. Section III deals with the Boltz-mann kinetic equation, which governs the electron transport at dc and ac voltages and the solutions of this equation. The cases of the ballistic and collision dominated electron trans-port are considered. In Secs. IV and V, the dc transconduc-tance and the ac frequency-dependent transconductransconduc-tance are calculated using the results of Sec. III. Section VI deals with the demonstration and analysis of the main obtained results, numerical estimates, and comparison of the GBL-FET prop-erties with those of GNR-FETs. In Sec. VII, we draw the main conclusions. In the Appendix, some intermediate calcu-lations related to the dynamic recharging of the gated section by holes due to the interband tunneling are singled out.

a兲_{Electronic mail: v-ryzhii}_{共at兲u-aizu.ac.jp.}

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II. GBL-FET ENERGY BAND DIAGRAMS

We assume that the bias back-gate voltage Vb⬎0, while

the bias top-gate voltage Vt⬍0. The electric potential of the

channel at the source and drain contacts are ␸= 0 and ␸ = Vd, respectively, where Vd is the bias drain voltage. The

former results in the formation of a 2DEG in the GBL. The distribution of the electron density⌺ along the GBL is gen-erally nonuniform due to the negatively biased top gate forming the barrier region beneath this gate. Simultaneously, the energy gap Eg is also a function of the coordinate x共its

axis is directed in the GBL plane from the source contact to the drain contact兲 being different in the source, top-gate, and drain sections of the channel共see Fig.2兲. Since the net top-gate voltage apart from the bias component Vtcomprises the

ac signal component␦V共t兲, the height of the barrier for elec-trons entering the section of the channel under the top gate 共gated section兲 from the source side can be presented as

⌬共t兲 = ⌬0+␦⌬共t兲. 共1兲

Depending on the Fermi energy in the extreme sections of the channel, in particular, on its value ␧F in the source

section and on the height of the barrier in this section ⌬₀, there are three situations. The pertinent the GBL-FET energy band diagrams are demonstrated in Fig.2. The spatial distri-butions of electrons and holes in the GBL channel are differ-ent depending on the relationship between the top-gate volt-age Vt and two threshold voltages, Vth,1 and Vth,2. These

threshold voltages are determined in the following.

When Vth,2⬍Vth,1⬍Vt, the top of the conduction band in

the gated section is below the Fermi level关Fig.2共a兲兴. In this case, an n+_-n-n+_{structure is formed in the GBL channel. At}

Vth,2⬍Vt⬍Vth,1, the Fermi level is between the top of the

conduction band and the bottom of the valence band in this section 关Fig.2共b兲兴. This top-gate voltage range corresponds to the formation of an n+_-i-n+ _{structure. If V}

t⬍Vth,2⬍Vth,1,

both band edges are above the Fermi level关Fig.2共c兲兴, so that

n+_{-p and p-n}+_{junctions are formed beneath the edges of the}

top gate. In the first and third ranges of the top gate voltage 共“a” and “b” ranges兲, the electron and hole populations of the gated section are essential. In the second range共range b兲, the gated section is depleted. In the voltage range a, the source-drain current is associated with a hydrodynamical electron flow共due to effective electron-electron scattering兲 in the gated section. In this case, the source-drain current and GBL-FET characteristics are determined by the conductivity of the gated section, which, in turn, is determined by the electron density and scattering mechanisms including the electron-electron scattering mechanism, and by the self-consistent electric field directed in the channel plane. In such a situation, different hydrodynamical models of the electron transport共including the drift-diffusion model兲 can be applied 共see, for instance, Refs.23–27兲.

If Vth,2⬍Vt⬍Vth,1, considering the potential distribution

in the direction perpendicular to the GBL plane invoking the gradual channel approximation28,29 and assuming for sim-plicity that the thicknesses of the gate layers separating the channel and the pertinent gates, Wband Wt, are equal to each

other Wb= Wt= W, we obtain ⌬0= − e 共Vb+ Vt兲 2 , ␦⌬共t兲 = − 1 2e␦V共t兲, 共2兲

where e =兩e兩 is the electron charge. In the voltage range in question, the electron system in the gated section is not de-generate. This voltage range, as well as the range Vt⬍Vth,2

⬍Vth,1, corresponds to the GBL-FET “off-state.” Similar

for-mulas take place for the barrier height from the drain side 共with the replacement of ⌬0by⌬0+ eVd兲.

In the cases when Vth,2⬍Vth,1⬍Vt or Vt⬍Vth,2⬍Vth,1,

⌬0= − e 共Vb+ Vt兲 2 ⫾ 2␲eW ␬ ⌺0⫿, ␦⌬共t兲 = −1 2e␦V共t兲 ⫾ 2␲eW ␬ ␦⌺⫿共t兲. 共3兲

Here ⌺₀⫿+␦⌺⫿共t兲 are the electron and hole densities in the gated section and ␬ is the dielectric constant of the gate layers. In the most interesting case when the electron densi-ties in the source and drain sections are sufficiently large, so that the electron systems in these sections are degenerate. Considering this, the height of the barrier⌬₀ is given by

⌬0= − e Vt共aB/8W兲 关1 + 共aB/4W兲兴 ⯝ − eVt

冉

aB 8W

冊

, 共4兲 ⌬0= − e Vt共aB/8W兲 + 共Vt− Vb兲共d/2W兲 关1 + 共aB/4W兲兴 , 共5兲

when Vth,2⬍Vth,1⬍Vt and Vt⬍Vth,2⬍Vth,1, respectively.

Here aB=␬ប2/me2is the Bohr radius, d is the effective

spac-ing between the graphene layers in the GBL which accounts for the screening of the electric field between these layers.20,21 This quantity is somewhat smaller than the real spacing between the graphene layers in the GBL d0

⯝0.36 nm. The Bohr radius aB can be rather different in

different materials of the gate layers. In the cases of Si02and

=

_∆(t)

(c)

(d)

<

E

g,s

_E

_g,d

_E

_g,s

ε

F

eV

d

FIG. 2. Band diagrams at different top gate bias voltages共Vb⬎0,Vd= 0兲: 共a兲

Vth,2⬍Vth,1⬍Vt, 共b兲 Vth,2⬍Vt⬍Vth,1 共depleted gated section兲, and 共c兲 Vt

⬍Vth,2⬍Vth,1 共gated section filled with holes兲. Panel 共d兲 corresponds to

Vth,2⬍Vt⬍Vth,1but with Vd⬎0.

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Hf02 共with k⯝20 30,31兲 gate layers, aB⯝4 nm and aB

⯝20 nm, respectively. In deriving Eqs.共4兲and共5兲, we have taken into account that in real GBL-FETs,共aB/8W兲Ⰶ1.

For the normal GBL-FET operation, the electron densi-ties, ⌺s− and ⌺d−, induced by the back-gate voltage in the

source and drain sections, respectively, should be sufficiently high markedly exceeding their thermal value ⌺T

= 2 ln 2mkBT/␲ប2, where kBis the Boltzmann constant and T

is the temperature. This occurs when Vb⬎ln 2共8W/aB兲

⫻共kBT/e兲+VT. In such a case, i.e., at sufficiently high

back-gate voltages, the Fermi energy in the source section is given by ␧F= kBT 关1 + 共aB/8W兲兴 ln

冋

exp

冉

aB 8W eVb kBT

冊

− 1

册

. 共6兲 Here we considered that the electron density in the source section⌺s−=␬Vb/4␲eW共the electron density in the drain

sec-tion of the channel is approximately equal to ⌺d−=␬共Vb

− Vd兲/4␲eW兲. If aB= 4 – 20 nm at T = 300 K, VT

⯝0.035–0.173 V. As follows from Eq. 共6兲, the Fermi en-ergy in the source section at VbⲏVTis a linear function of

Vb, practically independent of the temperature, and it can be

presented as ␧F⯝ eVb 共aB/8W兲 关1 + 共aB/8W兲兴 ⯝ eVb

冉

aB 8W

冊

Ⰷ kBT. 共7兲 Setting W = 5 nm, aB= 4 – 20 nm, and Vb= 1 V, we obtain

␧F= 48– 91 meV. This case corresponds to the electron

den-sity in the source section is⌺s

−_⯝4⫻1012_{− 2⫻10}13 _cm−2_{. At}

VdⱗVb, the electron density in the drain section is somewhat

smaller but of the same order of magnitude.

Comparing Eqs.共2兲, 共4兲, and 共5兲, one can see that the height of the barrier ⌬₀ increases with increasing absolute value of the top-gate voltage rather slow in the voltage ranges a and “c” in contrast with its steep increase in the voltage range b. Since the energy gaps in GBLs E_g,s, Eg, and

Eg,ddepend on the local transverse electric field,20–22they are

different in different sections of the channel depending on the bias voltages,

E_g,s=edVb 2W , Eg= ed₀共Vb− Vt兲 2W , Eg,d= ed共Vb− Vd兲 2W . 共8兲 One can see that at Vt⬍0 and Vd⬎0, one obtains Eg⬎Eg,s

ⱖEg,d. Since dⰆaB, the energy gap in the source section is

much smaller than the Fermi energy in this section. Indeed, assuming d⯝d0= 0.36 nm and W = 5 nm for Vb= 1 V from

Eq. 共8兲, we obtain E_g,s⯝36 meV. However, the energy gap in the gated section at sufficiently high top-gate voltages can be relatively large共see below兲. Naturally, an increase in the top-gate voltage leads to an increase in the Fermi energy in the source共drain兲 section as well to an increase in the energy gaps in all sections.

The threshold voltages V_th,1and V_th,2 are determined by the conditions⌬0=␧Fand⌬0=␧F+ Eg, respectively. The

lat-ter implies that the Fermi energy of holes in the gated section ␧F

共hole兲₌_⌬

0−␧F− Eg= 0. As a result, the threshold voltages are

given by Vth,1⯝ − Vb

冉

1 + aB 4W

冊

, Vth,2⯝ − Vb

冉

1 + aB 4W+ d0 W

冊

. 共9兲 Since one can assume that dⰆW, the threshold voltages are close to each other, 兩V_th,1兩ⱗ兩V_th,2兩 with 兩V_th,2− V_th,1兩 ⯝共2d0/W兲Vbⲏ4Eg,s/e. The values of the energy gap in the

gated section at the threshold top gate voltages are given by Eg兩V_t=Vth,1ⱗ Eg兩Vt=Vth,2⯝

ed₀Vb

W ⯝ 2Eg,s

冉

d₀

d

冊

. 共10兲 Using the same parameters as in the above estimate of the energy gap in the source section, for the energy gap in the gated section at Vt⯝Vth,1⯝−1 V, we obtain Eg⯝72 meV.

In the following we restrict our consideration by the situa-tions when the height of the barrier for electrons in the gated section is sufficiently large 共so that ⌬₀⬎␧F兲, which

corre-sponds to the band diagrams shown in Figs.2共b兲 and2共c兲.

III. BOLTZMANN KINETIC EQUATION AND ITS SOLUTIONS

The quasiclassical Boltzmann kinetic equation govern-ing the electron distribution function fp= fp共x,t兲 in the

sec-tion of the channel covered by the top gate 共gated section兲 can be presented as ⳵fp ⳵t +vx ⳵fp ⳵x =

冕

d 2_qw_共q兲共f p+q− fp兲␦共␧p+q−␧p兲. 共11兲

Here, taking into account that the electron共and hole兲 disper-sion relation at the energies close to the bottom of the con-duction band is virtually parabolic with the effective mass m 共m⬃0.04m0, where m0 is the bare electron mass兲, for the

energy of electron with momentum p =共px, py兲 we put ␧p

= p2_{/2m=␧, v}

x= px/m=p cos␪/m, where cos␪= px/p 共the

x-axis and the y-axis are directed in the GBL plane兲 and w共q兲 is the probability of the electron scattering on disorder and acoustic phonons with the variation of the electron momen-tum by quantity q. The density of the electron 共thermionic兲 current, J = J共x,t兲, in the gated section of the channel 共per unit length in the y-direction兲 can be calculated using the following formula:

J = 4e 共2␲ប兲2

冕

d

2_p_v

xfp, 共12兲

where ប is the reduced Planck constant. Disregarding the electron-electron collisions in the gated section of the chan-nel共due a low electron density in this section in contrast with the source and drain sections where the electron-electron col-lisions are essential兲, we consider two limiting cases: ballis-tic transport of electrons across the gated section and strongly collisional electron transport.

A. Ballistic electron transport

If␦V共t兲=␦V_␻e−i␻t, where ␦V␻Ⰶ兩Vt兩 and ␻ are the

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dis-tribution function can be searched as fp= F0+␦F␻共x兲e−i␻tand

⌬=⌬0+␦⌬␻e−i␻t. Assuming that eVdⰇkBT and solving Eq.

共11兲with the boundary conditions, fp兩p_xⱖ0,x=0= exp

冋

␧F−⌬共t兲 − ␧

⌰共px兲. 共15兲

Here, ⌰共px兲 is the unity step function. The first boundary

condition given by Eq.共14兲corresponds to quasiequilibrium electron distribution in the source section of the channel and the injection of electrons with the kinetic energy exceeding the barrier height ⌬共t兲 from the source section to the gated section 共at x=0兲. The injection of electrons from the drain source to the gated section 共at x=Lt兲 is neglected due to

eVdⰇkBT; this inequality leads to rather high barrier near the

drain edge of the gated section. The presence of the unity step function⌰共px兲 in Eqs.共16兲and共17兲reflects the fact that

there are no electrons propagating backward due to the ab-sence of the electron scattering in the gated section.

Using Eqs.共12兲,共14兲, and共15兲, we arrive at the follow-ing formulas for the dc and ac components, J0 and␦J␻, of

the current at the drain edge of the gated section 共i.e., at x = Lt兲, J₀= e

冑

2m共kBT兲 3/2 ␲3/2_ប2 exp

冉

␧F−⌬0 kBT

冊

= J₀B, 共16兲 ␦J_␻ J0 =

冉

−␦⌬␻ kBT

⫻

冋

C

冉

冑

␻␶ 2

冑

␰

冊

+ iS

冉

冑

␻␶ 2

冑

␰

冊

册

,

where ␶= Lt

冑

m/2kBT is the effective ballistic transit time

across the gated section of electrons with the thermal veloc-ityvT=

冑

2kBT/m and C共x兲 and S共x兲 are Fresnel’s cosine and

In the case of strongly collisional electron transport, the distribution function in the gated section is close to isotropic and it can be searched in the form

fp= F + g cos␪. 共18兲

Here F = F共␧,x,t兲 is the symmetrical part of the electron dis-tribution function 共which is generally not the equilibrium function兲. The second term in Eq.共18兲presents the asymmet-ric part of the distribution function with g = g共␧,x,t兲. A simi-lar approach was used for the calculation of characteristics of heterojunction bipolar transistors32 共see also Refs. 19 and 33兲. As a result, after the averaging of Eq.共11兲over the angle ␪, one can arrive at the following coupled equations:

⳵F ⳵t = −

冑

␧ 2m ⳵g ⳵x, ⳵gF ⳵t +␯g = −

冑

2␧ m ⳵F ⳵x. 共19兲

Here in the case of w共q兲=w=const, which corresponds to the scattering of electrons on short-range defects, ␯= mw/2. Equation共19兲is reduced to the following equation for func-tion F: ⳵ ⳵t

冉

⳵F_␧ ⳵t +␯F

冊

= ␧ m ⳵2_F ⳵x2. 共20兲

In the most interesting case when eVdⰇkBT, the boundary

conditions for Eq. 共20兲at x = 0 and x = Lt can be adopted in

the following form: F兩x=0= exp

冋

␧F−⌬共t兲 − ␧

kBT

册

, F兩x=L_t⯝ 0. 共21兲

The boundary condition under consideration imply that at x = 0 there is the electron injection from the source section of the channel, whereas at x = Lt an effective extraction of the

electrons into the drain section occurs due to a strong pulling dc electric field. Due to a strong electron scattering a signifi-cant portion of the injected electrons returns back to the source section.

Setting as above ␦V共t兲=␦V␻e−i␻t and, hence, F = F₀ +␦F_␻e−i␻tand g = F₀+␦g_␻e−i␻t, we obtain

d2F0 dx2 = 0, 共22兲 d2␦F_␻ dx2 − m␻共␻+ i␯p兲 ␧ ␦F␻= 0, 共23兲 and arrive at F0= exp

冋

␧F−⌬0−␧ kBT

2␧ mexp

冋

␧F−⌬0−␧ kBT

册

, 共26兲

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␦g_␻= − i 共␻+ i␯兲

冑

2␧ m ⳵␦F_␻ ⳵x = − i exp

冋

␧F−⌬0−␧ kBT

册

冑

2␻ 共␻+ i␯兲 ⫻cosh关␣␻共x − Lt兲兴 sinh共␣␻Lt兲

冉

␦⌬_␻ kBT

冊

. 共27兲

After that, using Eqs.共12兲,共26兲, and 共27兲, we arrive at the following formulas for J0 _and_␦_J␻_{共at x=L}

t兲: J0= e 2共kBT兲2 ␲ប2_L t␯ exp

冉

␧F−⌬0 kBT

冊

= J₀C, 共28兲 ␦J_␻ J0 =

冉

−␦⌬␻ kBT

冊

冕

0 ⬁ d␰

冑

␰e−␰H_␻共␰兲. 共29兲 Here H␻共␰兲 = i␶␯ sinh

冑

关2␻共␻+ i␯兲␶2/␰兴

冑

2␻ ␻+ i␯.

According to Eq. 共28兲, J₀C⬀1/Lt␯. One needs to stress that

the collisional case under consideration corresponds actually to ␯␶Ⰷ1. In the frequency range ␻Ⰶ␯/␶2_, _H

␻共␰兲⯝

冑

␰. At ␻Ⰷ␯/␶2_{, one obtains}

冉

␧F−⌬0 kBT

冊

R0= eJ₀C 2kBT R0, 共34兲 when Vt⬍Vth,2⬍Vth,1.

As follows from the comparison of Eq. 共30兲 with Eq. 共31兲and Eq. 共33兲with Eq.共34兲, the GBL-FET dc transcon-ductance in the top gate voltage range V_th,2⬍Vt⬍Vth,1 共in

the range b兲 might be much larger than that when Vt⬍Vth,2

⬍Vth,1 共in the range c兲 since aBⰆ8W. This is due to

rela-tively slow increase in⌬₀with increasing兩Vt兩 when the hole

density in the gated section becomes essential.

The voltage dependences of the dc transconductance can be obtained using Eqs. 共30兲,共31兲,共33兲, and 共34兲and invok-ing Eqs. 共2兲, 共4兲, and 共6兲. In particular, in a rather narrow voltage range Vth,2⬍Vt⬍Vth,1, one obtains

G₀C=

冑

␲ ␯␶G0 B_{⬀ exp}

冋

eVb kBT

冉

aB 8W

冊

⫻exp

冋

e共Vt− Vth,2兲 kBT

冉

aB 8W

冊

册

. 共36兲 V. GBL-FET AC TRANSCONDUCTANCE

According to the Shockley–Ramo theorem,34,35 the source-drain ac current is equal to the ac current induced in the highly conducting quasineutral portion of the drain sec-tion of the channel and in the drain contact by the electrons injected from the gated section. This current is determined by the injected ac current given by Eq.共16兲or Eq.共28兲, as well as the electron transit-time effects in the depleted portion of the drain section. However, if ␻␶dⰆ1, where the ␶d is the

electron transit time in depleted region in question, the in-duced ac current is very close to the injected ac current.19,36,37 Since, at moderate drain voltages, the length of depleted portion of the drain section Ld can usually be

shorter than the top gate length Lt, in the most practical range

of the signal frequencies ␻ⱗ␶−1_{, one can assume that} _␶

d

0 ⬁ d␰

冑

␰e−␰H_␻共␰兲, 共38兲 respectively.

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In the range of gate voltages Vth,2⬍Vt⬍Vth,1共range b兲, Eq.共2兲 yields

冏

⳵␦⌬_␻ ⳵␦V␻

冏

V_d = − e 2. 共39兲

In this case, Eqs.共37兲and共38兲result in G_␻B= eJ0 B 2kBT

冕

0 ⬁ d␰

冑

␰e−␰F_␻共␰兲, 共40兲 G_␻C= eJ0 C 2kBT

冕

0 ⬁ d␰

冑

␰e−␰H_␻共␰兲. 共41兲 As follows from Eqs. 共40兲 and 共41兲, the characteristic frequencies of the ac transconductance rolloff are 1/␶ and ␯/␶2_{in the case of the ballistic and collisional electron}

trans-port, respectively, i.e., the inverse times of the ballistic and diffusive transit across the gated section of the channel. In-deed, the quantity␯/␶2can be presented as D/Lt2, where D is

the electron diffusion coefficient.

The situation becomes more complex in the range of the top gate bias voltages Vt⬍Vth,2⬍Vth,1 共range c兲. As follows

from Eq.共4兲in this voltage range, the quantity␦⌬_␻is deter-mined not only by the ac voltage ␦V_␻ but also by the ac component of the hole density in the gated section ␦⌺_␻+. Moreover, at sufficiently high signal frequencies, the hole system in the gated section cannot manage to follow the variation of the ac voltage. Taking into account the dynamic response of the hole system 共see Appendix兲, instead of Eq. 共39兲one can obtain

冏

⳵␦⌬␻ ⳵␦V_␻

冏

V_d

= −e

2R␻. 共42兲

Here共see Appendix兲 R␻=

冉

_4WaB

冊

再

冉

_4WaB

冊

+ 1 1 − i␻␶r

冎

−1 =R0

冉

1 − i␻␶r 1 − i␻␶rR0

冊

, 共43兲 where␶r is the time of the gated section recharging

associ-ated with changing of the hole density due to the tunneling or/and generation-recombination processes. Generally,␶r

de-pends on the top gate length Lt.

Accounting for Eq.共42兲, we arrive at the following for-mulas for the GBL-FET ac transconductance when Vt

⬍Vth,2⬍Vth,1: G_␻B= eJ0 B 2kBT R␻

冕

0 ⬁ d␰

冑

␰e−␰F_␻共␰兲, 共44兲 G_␻C= eJ0 C 2kBT R␻

冕

0 ⬁ d␰

冑

␰e−␰H_␻共␰兲. 共45兲 If␻Ⰷ␶r −1

, one obtainsR_␻⯝1 and the ac transconductances in both b and c ranges of the top gate voltage are close to each other 关compare Eqs. 共40兲and共41兲 with Eqs. 共42兲 and 共44兲兴. However, at low signal frequencies 共␻Ⰷ␶r

−1_{兲, the ac}

transconductance given by Eqs.共44兲and共45兲for the voltage

range c are markedly smaller than those given by Eqs.共40兲 and共41兲valid in the voltage range b.

VI. ANALYSIS OF THE RESULTS AND DISCUSSION

Comparing G₀B and G₀Cgiven by Eqs.共30兲and共35兲, we obtain G₀C/G₀B=

冑

␲/␯␶. This implies that the above ratio markedly decreases with increasing collision frequency共with decreasing electron mobility兲 and the top gate length, i.e., with the departure from the ballistic transport.

As shown above, the dc current steeply drops in a nar-row top-gate voltage range Vth,2⬍Vt⬍Vth,1. Indeed, the ratio

J₀B兩V_t=V_th,2/J0 B_兩 V_t=V_th,1⯝exp关−共ed0Vb/WkBT兲兴. Setting W = 5 nm, T = 300 K, and Vb= 1 – 2 V, we find J₀B兩V_t=V_th,2/J0 B_兩

V_t=V_th,1⯝3⫻10−3– 6⫻10−2. The estimate of the

dc current at Vt= Vth,2⯝−Vb 共which might be interesting for

the GBL-FET applications in digital large scale circuits兲 with W = 5 nm, T = 300 K, and Vb= 1 – 2 V yields J0

B_⯝1

⫻10−3_{– 2}_⫻10−2 _A_{/cm. At T=77 K and V}

b= 1 V, one

ob-tains J₀B⯝7⫻10−7 _A_{/cm. In the case of a GBL-FET with}

the width H = 1 ␮m, the latter corresponds to the character-istic value of the “off-current” J₀BH⯝70 pA. Similar values can be obtained at T = 300 K when −Vt= Vb⯝4.6 V.

As follows from Eq.共35兲, the GBL dc transconductance in the range of the top gate voltages Vth,2⬍Vt⬍Vth,1is

par-ticularly large when VtⱗVth,1. This is due to a sharp

voltage-sensitivity of the dc current and its relatively high values at such voltages. Indeed, using Eq. 共28兲 at T = 300 K and Vt

ⱗVth,1, we obtain G0

B_{ⱗ2500 mS/mm. In GBL-FETs with}

WtⰆWb, the dc transconductance can be even larger.

The pre-exponential factor in the right-hand side of Eq. 共36兲is proportional to a small parameter共aB/8W兲. The

argu-ment of the exponential function in this equation comprises small parameters 共aB/8W兲 and 共d0/2W兲. This implies that

the dc transconductance in the voltage range Vt⬍Vth,2

⬍Vth,1 described by Eq. 共36兲 is relatively small and is a

fairly weak function of the top-gate voltage. As follows from Eqs. 共35兲 and共36兲, the ratio of the dc transconductance at VtⱗVth,2 to that at VtⱗVth,1 is equal approximately to the

following small value: 共aB/4W兲exp关ed0共Vth,2− Vb兲/2WkBT兴

=共aB/4W兲exp共−2eEg,sd0/dkBT兲; it is smaller then the ratio of

the dc currents by parameter 共aB/4W兲.

Figure3shows the ac transconductance兩G_␻兩 normalized by the dc transconductance at the ballistic transport G₀Bas a function of the normalized signal frequency ␻␶ calculated for GBL-FETs with both ballistic and collisional electron transports. It is assumed that the top gate voltage is in the range Vth,2⬍Vt⬍Vth,1. The inset in Fig.3 shows the

depen-dence of the normalized threshold frequency ␻t␶ on ␯␶

⬀␯Lt. The threshold frequency is defined as that at which

兩G_␻兩/G0= 1/

冑

2. One can see that 兩G␻兩 pronouncedly

de-creases with increasing collision frequency. However, as seen from the inset in Fig.3, the decrease in␻twith

increas-ing ␯ is markedly slower: the ratio of␻t at␯␶= 2 and␻t at

␯␶= 6共i.e., three times larger兲 is approximately equal to 1.42. Setting Lt= 100– 500 nm and T = 300 K for the threshold

frequency ft=␻t/2␲ at the ballistic transport, we obtain ft B

⯝0.485−0.97 THz. To realize the near ballistic regime of the electron transport 共␯␶Ⰶ1兲 in GBL-FETs with such gate

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lengths, the electron mobility ␮⬎共1−5兲⫻104 cm2/V s is required. The possibility of the latter mobilities at room tem-peratures was discussed recently共see, for instance, Ref. 6兲. At a shorter top gate, Lt= 75 nm, one obtains ft

B

⯝1.29 THz. In the case of a GBL-FET with relatively long top gate and moderate mobility 共Lt= 500 nm and ␮= 2

⫻104 _cm2_{/V s兲 when the effect of scattering is strong 共}_␯

⯝2⫻1012 _s−1_and_␯␶_{⯝2.7兲, we obtain f}

t

C_{⯝94 GHz.}

In Fig.4, the similar dependences calculated for a GBL-FET with collisional electron transport at V_th,2⬍Vt⬍Vth,1

and at Vt⬍Vth,2⬍Vth,1 are demonstrated. Since in the latter

top-gate voltage range the electron scattering on holes accu-mulated in the gated section can be strong, so that the real-ization of the ballistic transport at such top-gate voltages might be problematic, only the dependences corresponding to the collisional electron transport are shown. As follows from Eqs.共44兲and共45兲and seen from Fig. 4, the ac trans-conductance at the top gate voltages Vt⬍Vth,2⬍Vth,1is fairly

small at low frequencies␻ⱗ␶r−1 being close to the dc

trans-conductance共due to a smallness of parameter R⯝aB/4W兲,

whereas it becomes much larger in the intermediate fre-quency range␶r−1Ⰶ␻ⱗ␶−1共or␶r−1Ⰶ␻ⱗ␯␶−2兲. This is due to

the effect of holes in the gated section. Owing to this effect, the low frequency noises can be effectively suppressed.

Using the above results for GBL-FETs and those

ob-tained previously19 for GNR-FETs, one can compare the GBL and GNR characteristics. In particular, considering the expressions for J₀Bfound for GBL-FETs and GNR-FETs, we obtain J₀B/J₀B,GNR= 1/2. For the case of collisional transport, one obtains J₀C/J₀C,GNR⬃1. As a result, the GBL-FET and GNR-FET dc transconductances are close to each other. The ratio of the GBL-FET and GNR-FET ac transconductances at high signal frequencies is G_␻B/G_␻B,GNR⬀1/

冑

␻␶, i.e., the GBL-FET ac transconductance falls more steeply with in-creasing␻than the GNR-FET ac transconductance.

The GBL-FET dc and ac characteristics obtained are valid if the interband tunneling source-drain current through the n+_{-p and p-n}+_{junctions beneath the edges of the top gate}

are small in comparison with the thermionic current created by the electrons overcoming the potential barrier in the gated section. Such a tunneling current can be essential in the volt-age range c 共Vt⬍Vth,2⬍Vth,1兲 depending on the energy gap

near the top gates and the length of the n+_{-p and p-n}+

junc-tions in question. This implies that there is a limitation when the top-gate voltage is not too high Vtin comparison with the

threshold voltage Vth,2 共i.e., when VtⱗVth,2兲, so that the

cal-culated characteristics correspond to the most interesting voltage range where the ac transconductance can be rather large.

VII. CONCLUSIONS

We presented an analytical device model for a GBL-FET. Using this model, we calculated the GBL-FET dc and ac characteristics and shown that:

共1兲 The dependence of the dc current on the top gate voltage is characterized by the existence of three voltage ranges, corresponding to共a兲 the population of the gated section by electrons,共b兲 the depletion of this section, and 共c兲 its essential filling with holes, and determined by the top-gate threshold voltages Vth,1and Vth,2.

共2兲 The ac current is most sensitive to the top-gate voltage Vtand the dc and ac transconductances are large when

Vth,2⬍VtⱗVth,2.

共3兲 The electron scattering in the gated section results in a marked reduction in the dc and ac transconductances. However, the threshold frequency corresponding to 兩G␻兩/G0= 1/

冑

2 decreases with increasing collision

fre-quency relatively smoothly.

共4兲 The transient recharging of the gated section by holes 共at Vt⬍Vth,2兲 leads to a nonmonotonic frequency

depen-dence of the ac transconductance with a pronounced maxima in the range of fairly high frequencies. This effect might be used for the optimization of GBL-FETs with reduced sensitivity to low frequency noises. 共5兲 The fabrication of GBLs with high electron mobility at

elevated temperatures opens up the prospects of realiza-tion of terahertz GBL-FETs with ballistic electron trans-port operating at room temperatures and surpassing FETs based on A₃B₅compounds.

ACKNOWLEDGMENTS

The authors are grateful to Professor S. Brazovskii 共Uni-versity Paris-Sud兲 and Professor E. Sano 共Hokkaido

Univer-0 2 4 6 8 10 ωτ 0 0.2 0.4 0.6 0.8 1 |Gω |/G 0 B 0 2 4 6 ντ 0 0.5 1 1.5 ωt τ ν = 0 ντ = 2 ντ = 6 ντ = 4

FIG. 3. The ac transconductance共normalized by G₀B_{兲 vs}_␻␶_{calculated for}

GBL-FETs with ballistic共␯= 0兲 and collisional electron transport 共␯␶= 2 , 4, and 6兲 at the top gate voltage in range “b,” i.e., Vth,2⬍Vt⬍Vth,1. For Lt

= 100 nm and T = 300 K,␻␶= 10 corresponds to f =␻/2␲⯝3.3 THz. The inset shows the normalized threshold frequency␻t␶as a function of

param-eter␯␶. 10-2 10-1 100 101 ωτ 0 0.2 0.4 0.6 0.8 1 |Gω |/G 0 B V_th,2< V_t< V_th,1 ντ = 2 V_t< V_th,2 τ_r/τ = 50

FIG. 4. The ac transconductance共normalized by G₀B_{兲 vs}_␻␶_{calculated for}

GBL-FETs with collisional electron transport共␯␶= 2兲 at the top voltage in range c共Vt⬍Vth,2⬍Vth,1兲 at␶r/␶= 50.

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sity兲 for fruitful discussions and valuable information. The work was supported by the Japan Science and Technology Agency, CREST, Japan. One of the authors共N.K.兲 acknowl-edges the support by the INTAS Grant No. 7972 and by the ANR program in France共Project No. BLAN07-3-192276兲.

APPENDIX: DYNAMIC RESPONSE OF THE HOLE SYSTEM IN THE GATED SECTION

At sufficiently large values兩Vt兩, the gated section is

es-sentially populated with the holes. As follows from Eq.共4兲, ␦⌬␻= −1

2e␦V␻− 2␲e2W

␬ ␦⌺␻+. 共A1兲

The ac component of the hole density␦⌺_␻+ obeys the conti-nuity equation, which can be presented in the following form: − i␻␦⌺_␻+=␦G_␻+␦J␻ T_兩 x=0−␦J␻T兩x=L_t eLt . 共A2兲

Here␦G_␻ is the variation of the generation pf holes in the gated section 共associated, say, with the generation of holes by the thermal radiation38兲 and␦J_␻T兩_x=0and␦J_␻T兩_x=L

tare

the interband tunneling ac currents near the source and drain edges of the top gate, respectively. For normal operation of GBL-FETs, these tunneling current should relatively small. This is achieved in GNL-FETs by proper choice of the en-ergy gap in the different sections of the channel, Eg,s, Eg, and

Eg,d, which should be not too small. The terms in the

right-hand side of Eq.共A2兲can be presented as

␦G_␻= Kg共␦⌬␻−␦␧F+兲, 共A3兲 ␦J_␻T兩x=0−␦J␻T兩x=L_t= 2共␦⌬␻−␦␧F +_兲 eRt . 共A4兲

Here ␦␧F+⯝共␲ប2/2m兲␦⌺+␻, Kg= 2m/␲ប2␶g, where ␶g is the

characteristic time of the generation-recombination processes in the gated section, Rtis the tunneling resistance of the p-n

junctions induced by the negative top gate voltage near the edges of the top gate; ln Rt⬀Eg3/2/E储⬀共Vb− Vt兲3/2/Vt. From

−1 , 共A5兲

where␶RC= RtCt is the time of the gated section recharging

by the tunneling currents, Rtis the tunneling resistance of the

p-n junctions induced by the negative top gate voltage near the edges of the top gate, the quantity ␶r=␶RC␶g/关␶RC

+共aB/4W兲␶g兴 is the characteristic time of the gated section

recharging by holes, and Ct=␬Lt/2␲W is the capacitance of

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東北大学機関リポジトリTOUR

Device model for graphene bilayer field-effect

transistor

著者

尾辻 泰一

journal or

publication title

Journal of Applied Physics

volume

105

number

10

page range

104510-1-104510-9

year

2009

URL

http://hdl.handle.net/10097/47802

Device model for graphene bilayer field-effect transistor

冉

冊

=

E

E

E

E

ε

(a)

(b)

∆(t)

(c)

(d)

<

E

E

E

ε

eV

冋

冉

冊

册

冉

冊

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冕

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冉

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冉

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尾辻泰一

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