# Potential Distributions, Source-Drain Current, and Transconductance

To obtain the explicit dependences of the source-drain current and the transconductance on the gate voltages V_{b} and *V _{t}* as well as on the drain voltage

*V*one needs to find the relationship between the barrier height Д

_{d},_{т}and these voltages. This necessitates the calculations of the potential distribution in the channel. The latter can be found from Eq. (5.2) in an analytical form in the following limiting cases.

## High Top-Gate Voltages: Sub-threshold Voltage Range (Δm >> εF)

When the barrier height ^ exceeds the Fermi energy f_{F}, the electron density is low in the gated section and, hence, one can disregard the contribution of the electron charge in this section. In such a limit, we arrive at the following equation for the potential:

where Л_{0} = ^2/3*W* and *F _{0} =* -(K

_{b}

*+*V

_{t})/2. Solving Eq. (5.26) considering boundary conditions (5.4), for the case of high top-gate voltages we obtain

Limiting our consideration by the GBL-FETs with not too short top gate (L_{t}» *W),* Eq. (5.27) can be presented as

Equation (5.28) yields

where q_{0} = exp(L_{t}/2A_{0})/2. Simultaneously, for the position of the barrier top one obtains

The terms in the right-hand side of Eq. (5.29) containing parameter q_{0} reflect the effect of the top-gate geometry (finiteness of its length). This effect is weakened with increasing top barrier length L_{t}. The effect of drain-induced barrier lowering in the case under consideration is described by the last term in the right-hand side of Eq. (5.29).

Equation (5.29) yields (ЭД_{т}/Э1/,) = -(e/2)(l - qo’^{1})- Invoking Eqs. (5.13) and (5.14), we obtain

Here, F_{th} = -[1 + 2b/(l - q_{0}^{_1} )]кь - ^{_}(1 ^{+} 2b) *V _{b}.* The rightmost factors in the right-hand sides of Eqs. (5.31) and (5.32), associated with the effect of drain-induced barrier lowering, lead to an increase in

*g*with increasing F

_{d}not only at

*eV*~

_{d}*k*at

_{B}Tbut*eV*»

_{d}*k*« exp

_{B}T:g*{eV*One can see that in the range of the top-gate voltages under consideration, the GBL-FET transconductance exponentially decreases with increasing | V

_{d}/2ri_{0}k_{B}T)._{t}+ K

_{b}| and

where at *T* = 300 К the characteristic value of the transconductance - 4330 mS/mm.

## Near Threshold Top-Gate Voltages (Δm ≥ εF)

At *eV _{d} < k_{B}T* -с

*£р,*taking into account that the electron distribution is characterized by the equilibrium Fermi distribution function, the electron density in the gated section can be presented in the following form:

Considering this, we reduce Eq. (5.2) to

Here,

so that Л/Л_{0} = *yfb **<* 1. The solution of Eq. (5.35) with boundary condition (5.4) is given by

From Eq. (5.36) we obtain

where q = exp *{LJ* 2Л)/2, and the position of the barrier top is

Since Л < Л_{0}, one obtains q » q_{0}, and the terms in Eq. (5.37) containing parameter q can be disregarded. This implies that the effects of top-gate geometry and drain-induced barrier lowering are much weaker (negligible) in the case of the top-gate voltages in question in comparison with the case of high top-gate voltages.

Substituting from Eq. (5.37) into Eq. (5.16), for low drain voltages we arrive at

At relatively high drain voltages *(eV _{d}* > £

_{F}»

*k*the electron charge in the source portion of the gated section (x < x

_{B}T),_{m}, where x

_{m}is the coordinate of the barrier top) is primarily determined by the electrons injected from the source. The electron injection from the drain at high drain voltages is insignificant. Hence, the electron charge in the drain portion of the gated section can be disregarded. In this case, Eq. (5.2) can be presented as

at *-LJ2* < x < x_{0}, and

atx_{0}

At the point x = x_{0} corresponding to the condition e

Y=Xo + £_{F} = 0, the solutions of Eqs. (5.40) and (5.41) should be matched:

Solving Eqs. (5.40) and (5.41) with conditions (5.4) and (5.42), we obtain the following formulas for the potential

*LJ*2 _{0} < x< *LJ2* as well as an equation for x_{0}:

Figure 5.3 Barrier profile Д = *-e*

at different top-gate voltage l/_{t} and drain voltage l/_{d} for GBL-FETs with different top-gate length *L _{t}.* Upper and lower pairs of curves correspond to Ц-Ц„ = -1.5 V and

*V*- V,

_{t}_{h}^ 0, respectively;

*W =*10 nm,

*b =*0.05, and

*V*5.0 V.

_{b}=

In the cases V_{b} + *V _{t}* = 0 and -(K

_{b}+ V,) > V

_{b}» %/e, Eq. (5.45) yields x

_{0}=

*-LJ2*+ Л In[4

*bVJ{fb*+

*2b)(V*V

_{b}+_{t})] andx

_{0}=

*-LJ2*+ 2A

_{0}£

_{F}/

*[-e[V*+ V,)] -

_{b}*-LJ2*+ 2ЬЛ

_{0}

*VJ[-{V*+ 7

_{b}_{t})], respectively. When -(V

_{b}+ l^t) —> +0, the matching point shifts toward the channel center. [f-(K

_{b }+ V

_{t}) increases, the matching point tends to the source edge of the channel. In this case, the role of the electron charge in the vicinity of the source edge diminishes, and the potential distribution tends to that given by Eq. (5.27).

At the threshold, the matching point x_{0} coincides with the position of the barrier maximum x_{m}. Considering that at x_{0} = x_{m}, both the left-hand and right-hand sides of Eq. (5.45) are equal to zero, for the barrier top height near the threshold at relatively high drain voltages we obtain the following:

Both Eqs. (5.29) and (5.46) correspond to the situations when the electron density in a significant portion of the channel is fairly low. However there is a distinction in the dependence of Д_{т} on *V _{d}* (the pertinent coefficients differ by factor of two). This is because in the first case the barrier top is located near the channel center, whereas in the second case it is shifted to the vicinity of the source edge [compare Eqs. (5.30) and (5.47)].

Using Eqs. (5.19) and (5.50), we obtain (Э Д_{т}/Э V_{t}) = -(e/2)(l - q_{0}^{_1}) and arrive at the following formula for the transconductance near the threshold, i.e., when *V _{t} - Vtb*

Here, as above, *V _{tb} *--(1 + 2

*b)V*In particular, Eqs. (5.48) and (5.49) at

_{B}.*V*U

_{t}=_{th}, yield

*J*For a GBL-FET with L

_{th}- J_{0}C_{0}._{t}= 40 nm,

*W =*10 nm, at

*T =*300 K, one obtains/

_{th}^ 1.656 А/cm and

*g*2167 mS/mm.

_{th}^## Low Top-Gate Voltages (Δm < εF)

At relatively low top-gate voltages when Д_{т} < e_{F}, the electron system is degenerate not only in the source and drain sections but in the gate section as well. In this top-gate voltage range, the spatial variation of the potential is characterized by A ^ *W 2b/3.* As a result, for Д_{т }one obtains an equation similar to Eq. (5.37). Since A < A_{0} -c *L _{v}* the parameter determining the effect of the top-gate geometry and the effect of drain-induced barrier lowering is

*r*= exp(L

_{t}/2A)/2 »

*r*As a consequence, one can neglect the effects in question in the top-gate voltage range under consideration. As a result, one can arrive at

_{Q}.

when *V _{d}t-* V

_{th}),

when *V _{d}> b(V_{t}* - K

_{th}), and

Considering Eq. (5.51), at low top-gate voltages we obtain
when *eV _{d} » e_{F}- ^ be(V_{b} + VJ,* and

when *eV _{d} « e_{F} - ^ be(V_{b}* +

*V*As follows from Eqs. (5.33) and

_{t}).(5.34), the transconductance is proportional to a small parameter *b ^{z/1}.* This is because the effect of the top-gate potential is weakened due to a strong screening by the degenerate electron system in the gated section. As a result, the transconductance at low top-gate voltages is smaller than that at the top-gate voltage corresponding to the threshold. Assuming that

*b*= 0.05 and

*V*= 5 (% = 0.25 eV), we obtain

_{b}*g*-1467 mS/mm. Comparing Eqs. (5.49) and (5.54), we

_{on}find*g _{0}Jg_{th}* ~

*b*- 0.158 and

^{z/2}1^{еУ}ъ/^{к}в^{т}*g/g*

_{th}*- 0.68.*on/g

_{t}h

Since the source-drain current at high top-gate voltages decreases exponentially when *-V _{t}* increases, the transconductance decreases as well.

Figure 5.3 shows the barrier (conduction band) profile Д = *-e(p *in the GBL-FETs calculated using Eqs. (5.27), (5.36), and (5.44) for different applied voltages and top-gate lengths. As demonstrated, the barrier height naturally decreases with increasing *V _{t} - V_{dv}* At the threshold (V

_{t}= F

_{th}), the barrier height is equal to the Fermi energy (at

*b =*0.05 and

*V*5 V,

_{b}=*e*^ 0.25 eV). One can see that shortening of the top-gate leads to a marked decrease in the barrier height (the short-gate effect). The source-drain current as a function of the drain voltage for different top-gate voltage swings is demonstrated in Fig. 5.4. The dependences corresponding to

_{F}*V*t-v 0, and V

_{t}-v_{th}_{th}=

_{t}- F

_{th}> 0 were calculated using formulas from subsections A, B, and C, respectively. Figure 5.5 shows that the transconductance as a function of the top-gate voltage swing exhibits a pronounced maximum at

*V*This maximum is attributed to the following. At high top-gate voltages, the effect of screening is insignificant due to low electron density in the channel. As a result, the height of the barrier top is rather sensitive to the top-gate voltage variations. The source-drain current in this case is exponentially small, so that the transconductance is small. In contrast, at low top-gate voltages, the screening by the electrons in the channel is effective, leading to a much weaker control of the barrier height by the top voltage [pay attention to parameter

_{t}- V_{th}.*b*« 1 in Eqs. (5.50)—(5.54)]. Despite, a strong source-drain current provides a moderate values of the transconductance. However, in the near threshold voltage range, both the sensitivity of the barrier height and the source-drain current to the top-gate voltage are fairly large.

**Figure 5.4 **Source-drain current./ versus drain voltage *V _{d}* at different values of the top-gate voltage swing

*Vt-V**

Figure 5.5 Transconductance *g* versus top-gate voltage swing l/_{t} - _{t}h-