Potential Distributions, Source-Drain Current, and Transconductance

At the point x = x₀ corresponding to the condition eY=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

In the cases V_b + V_t = 0 and -(K_b + V,) > V_b » %/e, Eq. (5.45) yields x₀ = -LJ2 + Л In[4bVJ{fb + 2b)(V_b + V_t)] andx₀ = -LJ2 + 2A₀£_F/ [-e[V_b + V,)] - -LJ2 + 2ЬЛ₀ VJ[-{V_b + 7_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 x₀ coincides with the position of the barrier maximum x_m. Considering that at x₀ = 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 - q₀^_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 + 2b)V_B. In particular, Eqs. (5.48) and (5.49) at V_t = U_th, yield J_th - J₀C₀. For a GBL-FET with L_t = 40 nm, W = 10 nm, at T = 300 K, one obtains/_th ^ 1.656 А/cm andg_th^ 2167 mS/mm.

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 < A₀ -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_Q. 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

when V_dt- 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_t). As follows from Eqs. (5.33) and

(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_b = 5 (% = 0.25 eV), we obtaing_on -1467 mS/mm. Comparing Eqs. (5.49) and (5.54), we

findg₀Jg_th ~ b^z/2 1^еУъ/^кв^т - 0.158 and g/g_th on/g_th - 0.68.

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_b = 5 V, e_F ^ 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 V_t -v_tht-v_th = 0, and V_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_t - V_th. 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 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 - _th-