# GBL-FET Energy Band Diagrams

We assume that the bias back-gate voltage *V _{b} >* 0, while the bias top- gate voltage

*V*< 0. The electric potential of the channel at the source and drain contacts are

_{t}*tp =*0 and

*tp = V*respectively, where

_{d},*V*is the bias drain voltage. The former results in the formation of a 2DEG in the GBL. The distribution of the electron density E along the GBL is generally nonuniform due to the negatively biased top gate forming the barrier region beneath this gate. Simultaneously, the energy gap

_{d}*E*is also a function of the coordinate

_{g}*x*(its axis is directed in the GBL plane from the source contact to the drain contact) being different in the source, top-gate, and drain sections of the channel (see Fig. 2.2). Since the net top-gate voltage apart from the bias component

*V*comprises the ac signal component 5F(t), the height of the barrier for electrons entering the section of the channel under the top gate (gated section) from the source side can be presented as

_{t}

Depending on the Fermi energy in the extreme sections of the channel, in particular, on its value, e_{F} in the source section and on the height of the barrier in this section Д_{0}, there are three situations. The pertinent the GBL-FET energy band diagrams are demonstrated in Fig. 2.2. The spatial distributions of electrons and holes in the GBL channel are different depending on the relationship between the top-gate voltage *V _{t}* and two threshold voltages, and Vth,2- These threshold voltages are determined in the following.

When V_{thi2} < *V _{th l} < V_{t},* the top of the conduction band in the gated section is below the Fermi level (Fig. 2.2a). In this case, an n

^{+}-n-n

^{+ }structure is formed in the GBL channel. At V

_{th2}<

*V*F

_{t}<_{th l}, the Fermi level is between the top of the conduction band and the bottom of the valence band in this section (Fig. 2.2b). This top-gate voltage range corresponds to the formation of an

*n*structure. If

^{+}-i-n**V*

_{t}<*V*< Vi. both band edges are above the Fermi level (Fig. 2.2c), so that

_{th},2*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 (a and

^{+}*b*ranges), the electron and hole populations of the gated section are essential. In the second range (range

*b*), the gated section is depleted. In the voltage range

*a,*the source-drain current is associated with a hydrodynamical electron flow (due to effective electron-electron scattering) 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 electronelectron 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 (including the drift-diffusion model) can be applied (see, for instance, Refs. [23-27]).

Figure 2.2 Band diagrams at different top gate bias voltages (Уь > 0, *V _{d}* = 0): (a)

*V*< Vi < (b)

_{th 2}*V*th,i (depleted gated section), and (c) V,< V

_{th2}_{thj2}

< Vth,i (gated section filled with holes), Panel (d) corresponds to l/_{th},2 < *V _{t}* < Цьд but with l/d>0.

If fth,2 < *V _{t} < V_{thl},* considering the potential distribution in the direction perpendicular to the GBL plane invoking the gradual channel approximation [28, 29] and assuming for simplicity that the thicknesses of the gate layers separating the channel and the pertinent gates, W

_{b}and W

_{t}, are equal to each other

*W*we obtain

_{b}=W_{t}= W,

where *e* = |e| is the electron charge. In the voltage range in question, the electron system in the gated section is not degenerate. This voltage range, as well as the range *V _{t} < V*

_{tht2}*<*kth,i- correspond to the GBL-FET "off-state." Similar formulas take place for the barrier height from the drain side (with the replacement of Д

_{0}by Д

_{0}+

*eV*

_{d}).In the cases when V_{t}h,2<^th,i_{2}

Here Sq +5S^{+}(t) are the electron and hole densities in the gated section and к is the dielectric constant of the gate layers. In the most interesting case when the electron densities 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 Д_{0} is given by

when *V** _{th2}* < Vth,i < k

_{t}and

*V*V

_{t}<_{th2}< Цьд, respectively. Here

*a*is the Bohr radius,

_{B}= Kh^{2}/me^{2}*d*is the effective spacing between the graphene layers in the GBL which accounts for the screening of the electric field between these layers [20, 21]. This quantity is smaller than the real spacing between the graphene layers in the GBL

*d*

_{0 }*^*0.36 nm. The Bohr radius o

_{B}can be rather different in different materials of the gate layers. In the cases ofSi0

_{2}and Hf0

_{2}(with

*к ^*20 [30, 31]) gate gate layers, o

_{B}4 nm and

*a*20 nm, respectively. In deriving Eqs. (2.4) and (2.5), we have taken into account that in real GBL-FETs,

_{B}^*[a*

_{B}/QW]For the normal GBL-FET operation, the electron densities, and , induced by the back-gate voltage in the source and drain sections, respectively, should be sufficiently high markedly exceeding their thermal value Lp = 2 In *2тк _{в}Т/лЬ^{2},* where

*k*is the Boltzmann constant and

_{B}*T*is the temperature. This occurs when F

_{b}> In 2(8

*W/*a

_{B}) x

*[k*+

_{B}T/e)*V*In such a case, i.e., at sufficiently high back-gate voltages, the Fermi energy in the source section is given by

_{T}.Here we considered that the electron density in the source section S” = *KV _{b}/4neW* (the electron density in the drain section of the channel is approximately equal to Ej = /с(^ь

^{-}

*V*If

_{d})/4nelV).*a*4 - 20 nm at

_{B}=*T*= 300 K,

*V*0.035 - 0.173 V. As follows from Eq. (2.6), the Fermi energy in the source section at

_{T}-*V*is a linear function of

_{b}> V_{T}*V*practically independent of the temperature, and it can be presented as

_{b},Setting *W* = 5 nm, a_{B} = 4 - 20 nm, and K_{b} = 1 V, we obtain £_{F} = 48 - 91 meV. This case corresponds to the electron density in the source section is Z” - 4 x 10^{12} -2 x 10^{13} cm"^{2}. At *V _{d} < V_{b},* the electron density in the drain section is somewhat smaller but of the same order of magnitude.

Comparing Eqs. (2.2), (2.4), and (2.5), one can see that the height of the barrier Д_{0} increases with increasing absolute value of the top- gate voltage rather slow in the voltage ranges *a* and V’ in contrast with its steep increase in the voltage range *b.* Since the energy gaps in GBLs *E _{&s},* Eg, and £g.d depend on the local transverse electric field [20-22], they are different in different sections of the channel depending on the bias voltages,

One can see that at *V _{t} <* 0 and

*V*> 0, one obtains

_{d}^{>}£g,s - ^g.d- Since of Og, the energy gap in the source section is much smaller than the Fermi energy in this section. Indeed, assuming

*d*^ cf

_{0}= 0.36 nm and

*W =*5 nm for V

_{b}= 1 V from Eq. (2.8), we obtain

*E*36 meV. However, the energy gap in the gated section at sufficiently high top-gate voltages can be relatively large (see below). Naturally, an increase in the top-gate voltage leads to an increase in the Fermi energy in the source (drain) section as well to an increase in the energy gaps in all sections.

_{&s}The threshold voltages and are determined by the conditions Aq = % and Д_{0} = £_{F} + *E _{%},* respectively. The latter implies that the Fermi energy of holes in the gated section e{-

^{hole)}= Д

_{0}- e

_{F}- E

_{g}= 0. As a result, the threshold voltages are given by

Since one can assume that *d* « *W,* the threshold voltages are close to each other, |7_{thjl}| * th2* with |V

_{th},

_{2}- V

_{tM}| -

*(2d*

_{0}*/W)V*The values of the energy gap in the gated section at the threshold top gate voltages are given by

_{b}>4E_{&s}/e.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 *V _{t}* - V

_{tM}^ -1 V, we obtain

*E*meV. In the following we restrict our consideration by the situations when the height of the barrier for electrons in the gated section is sufficiently large (so that Д

_{g}^72_{0}> %), which corresponds to the band diagrams shown in Figs. 2.2b and c.