# Device Model and Features of Operation

We consider a GBL-FET with the structure shown in Fig. 5.1. It is assumed that the back gate, which is positively biased by the pertinent voltage F_{b} > 0, provides the formation of the electron channel in the GBL between the Ohmic source and drain contacts. A relatively short top gate serves to control the source-drain current by forming the potential barrier (its height A_{m} depends on the top gate voltage V_{t} and other voltages) for the electrons propagating between the contacts.

Figure 5.1 Schematic view of the GBL-FET structure.

We shall assume that the GBL-FETs under the conditions when the electron systems in the source and drain sections are degenerate, i.e., % » *k _{B}T.* This implies that the back gate voltage is sufficiently high to induce necessary electron density in the source and drain sections.

In the GBL-FET the energy gap is electrically induced by the back gate voltage [32-34] (see also [18]). Thus in GBL-FETs, the back gate plays the dual role: it provides the formation of the electron channel and the energy gap. Since the electric field component directed perpendicular to the GBL plane in the channel section below the top gate (gated section) is determined by both k_{b} and *V _{t},* the energy gap can be different in different sections of the GBL channel:

*E*(source section),

_{gs }*E*(gated section), and

_{g}*E*(drain section) [17,18]. At sufficiently strong top-gate voltage (K

_{&d}_{t}< V

_{th}< 0, where 7

_{th}is the threshold voltage), the gated section becomes depleted. Since the energy gaps in GBLs are in reality not particularly wide, at further moderate increase in |k

_{t}|, the gated section of the channel becomes filled with holes (inversion of the charge in the gated section) if

*V*k

_{t}<_{in}

_{in}is the inversion voltage. As a result, the GBL-FETs with moderate energy gap are characterized by the threshold and inversion voltages: k

_{th}< 0 and

*V*0. The explicit formulas for к

_{in}<_{й1 }and

*V*shall be given in the following. The cases

_{m}*V*and

_{t}= V_{th}*V*= K

_{t}_{in }correspond to the alignment of the Fermi level in the source section of the channel with the conduction band bottom and the valence band top, respectively, in the gated section.

# Main Equations of the Model

Due to relatively high energy of optical phonons in graphene, the electron scattering in the GBL-FET channel is primarily due to disorder and acoustic phonons. Considering such quasielastic scattering, the quasiclassical Boltzmann kinetic equation governing the steady state electron distribution function/_{p} = /_{p}(x) in the gated section of the channel can be presented as

Here *e* = *e* is the electron charge, £_{p} = *p ^{2}/2m, m* is the electron effective mass in GBL,

**p**= (

*p*) is the electron momentum in the GBL plane (z = 0), w(q) is the probability of the electron scattering on disorder and acoustic phonons with the variation of the electron momentum by quantity

_{x},p_{y}**q**=

*(q*v

_{x},q_{y}),_{A}. =

*p*and axis

_{x},*x*is directed in this plane (from the source contact to the drain contact, i.e., in the direction of the current). For simplicity, we disregard the effect of "Mexican hat" (see, for instance, Ref. [34]) and a deviation of the real energy spectrum in the GBL from the parabolic one (the latter can be marked in the source and drain sections with relatively high Fermi energies). The effective mass in GBLs

*m*= Yi/2v

**where Yi = 0.35 - 0.43 eV is the inter-layer hopping integral and v**

_{w},_{w}- 10

^{s}cm/s is the characteristic velocity of electrons and holes in GLs [3, 34-36], so that

*m*= (0.03 - 0.04)m

_{0}, where

*m*is the bare electron mass. One of the potential advantages of GBL-FETs is the possibility of ballistic transport even if the top-gate length

_{0}*L*is not small. In such GBL- FETs, one can neglect the right-hand side term in Eq. (5.1).

_{t}As in Refs. [10,15, 37, 38], we use the following equation for the electric potential *tp= tp* (x) = y/(x, z)|_{z=0} in the GBLplane:

Here, L_ and X_{+} are the electron and hole sheet densities in the channel, respectively, *к* is the dielectric constant of the layers between the GBL and the gates and *W _{b}* and

*W*are the thicknesses of these layers. In the following, we put

_{t}*W*(except Section 5.5). Equation (5.2) is a consequence of the two-dimensional Poisson equation for the electric potential у/ (x, z) in the GBL-FET gated section

_{b}= W_{t}= W*[-LJ2 < x < LJ2*and

*-W*where L

_{b}< z < W_{b}_{t}is the length of the top gate) in the weak nonlocality approximation [37]. This equation provides the potential distributions, which can be obtained from the two-dimensional Poisson equation by expansion in powers of the parameter <5= (W

_{b}

^{3}+ W

_{t}

^{3})/15(W

_{b}+ W

_{t})/L

^{2}=

*W*where

^{2}/1SL^{2},*L*is the characteristic scale of the lateral inhomogeneities (in the x-direction) assuming that

The boundary conditions for Eqs. (5.1) and (5.2) are presented as

where /_{sp} and /_{dp} are the electron distribution functions in the source and drain sections of the channel. The functions /_{s p} and /_{dp }are the Fermi distribution functions with the Fermi energies £_{Fs} and e_{F},_{d} which are determined by the back gate and drain voltages, *V _{b}* and

*V*[17,18] (see also the Appendix):

_{d}where *b = a _{B}/QW, a_{B} = kh^{2}/me^{2}* is the Bohr radius, and

*h*is the reduced Planck constant. In the following, we shall assume that

*b*1, so that £

_{f},s -

*beV*and £

_{b}_{Fid}^

*be[V*- U

_{b}_{d}). In particular, if

*a*4 nm (GBL on Si0

_{B}=_{2}) and

*W*= 10 nm, one obtains

*b -*0.05. Due to a smallness of

*b,*we shall disregard a distinction between

*V*and

_{A}**V*because

_{d }*V*7

_{d}-_{d}* =

*bV*-c

_{A}*V*(as shown in the Appendix). Restricting ourselves by the consideration of GBL-FETs operation at not too high drain voltages, we also neglect the difference in the Fermi energies in the source and drain sections, i.e., put £

_{d}_{Fd}= £

_{Fs}= £

_{F}.

The source-drain dc current density (current per unit length in the direction perpendicular its flow) can be calculated using the following formulas:

In this case, Eq. (5.1) with boundary conditions (5.3) yields

where *T* is the temperature, *k _{B}* is the Boltzmann constant, and 0(£) is the unity step function. Using Eqs. (5.6) and (5.7), we obtain

Equation (5.8) can be presented in the following form:

Here (see, for instance, Ref. [23])

is the characteristic current density, and <5_{m} = (% - *A _{m})/k_{B}T,* and

*U*are the normalized voltage swing and drain voltage, respectively. At

_{A}= eV_{d}/k_{B}T*m*= 4 x 10"

^{29}g and

*T =*300 K,/

_{0}= 2.443 A/cm.

Figure 5.2 shows the dependences of the source-drain current / normalized by the value *J _{0}* as a function of the

*U*calculated using Eq. (5.9) for different values of <5

_{d}_{m}.

The GBL-FET transconductance *g* is defined as

Equations (5.9) and (5.11) yield

The obtained formulas for the source-drain current and transconductance can be simplified in the following limiting cases.

**Figure 5.2 **Normalized source-drain current *J/l _{0}*

**versus normalized drain voltage**

*U*

_{d}**at different values of normalized top-gate voltage swing <5**

_{m}.

## High Top-Gate Voltages

At high top-gate voltages, which correspond to the sub-threshold voltage range, the barrier height exceeds the Fermi energy (Д_{т}» £_{F}), so that 5_{m} » 1. In this case (the electron system in the gated section is nondegenerate), using Eqs. (5.9) and (5.12), we obtain

## Near Threshold Top-Gate Voltages

In this case, Д_{т} > £j:, i.e., |<5_{m}| < 1, Eqs. (5.9) and (5.12) yield

at low drain voltages *eV _{d} < k_{B}T* (

*U*1), and

_{d}<

at high drain voltages *eV _{d}» k_{B}T(U_{d}* » 1). Here, Co = [ ln[exp(-

*J* о

) + 1] a 0.678, Ci = P2) + 1] a 0.536, and C_{2} = P«f| *JO Jo*

exp(C ^{2})/[exp(^{2}- 0.337. In the limit £_{F} = A_{m}. Eqs. (5.13) and

- (5.14) provide the values
*J*and*g*close to those obtained from Eqs. - (5.17) and (5.18), which are rigorous in such a limit.

## Low Top-Gate Voltages

At low top-gate voltages, A_{m} < £_{F} from Eqs. (5.9)

Here,

withy_{0} = *I _{0} [k_{B}T)^{3}'^{2}.*

Using Eqs. (5.19) and (5.20), one obtains
at *eV _{d}* « £

_{F}— A

_{m}, and

at *eV _{d}* » £

_{f}- A

_{m}.

The dependences shown in Fig. 5.2 describe implicitly the dependences of / calculated using the universal Eq. (5.9) on the back-gate, top-gate, and drain voltages as well as on the geometrical parameters. Equations (5.13)-(5.25) provide these dependences in most interesting limits. However, to obtain the explicit formulas for *J* as well as for *g,* one needs to determine the dependences of the barrier height A_{m} on all voltages and geometrical parameters. Since the electron densities in the gated section in the limiting cases under consideration are different, the screening abilities of the electron system in this section and the potential distributions are also different. The latter leads to different vs *V _{t}* relations.