5.8: The temperature dependence of current in the OFF state

Both nanoscale and larger transistors have a small quantum capacitance in the OFF state, which is also known as subthreshold since \(V_{GS} < V_{T}\).

But even if the density of states is zero between \(\mu_{S} > E > \mu_{D}\), at higher temperatures, some electrons may be excited into empty states well above the Fermi Energy. If the density of states is very low at the Fermi Energy, but higher far from the Fermi level, then we can model the Fermi distribution by an exponential tail. Recall that this is known as a non-degenerate distribution; see Figure 5.8.1.

Equation (5.5.3) becomes

\[ N_{S} = \int^{\infty}_{-\infty} g(E-U)e^{-(E-\mu_{S})/kT}dE \nonumber \]

Now changing the variable of integration to \(E’=E-U\)

\[ N_{S} = \int^{\infty}_{-\infty} g(E’)e^{-(E’+U-\mu_{S})/kT}dE’ \nonumber \]

Simplifying

\[ N_{S} = e^{-U/kT}\int^{\infty}_{-\infty} g(E’)e^{-(E’-\mu_{S})/kT}dE’ \nonumber \]

Similarly,

\[ N_{D} = e^{-U/kT}\int^{\infty}_{-\infty} g(E’)e^{-(E’-\mu_{D})/kT}dE’ \nonumber \]

Thus, from Eq. (5.5.3) the current is

\[ I = \frac{q}{\tau}\text{exp}\left[\frac{qV_{GS}}{kT} \right] \cdot \int^{\infty}_{-\infty} g(E’)\left( e^{-(E’-\mu_{S})/kT} - e^{-(E’-\mu_{D})/kT}\right) \nonumber \]

Equation (5.8.5) holds in the limit that \(C_{G} \gg C_{S}, C_{D}\). In general, we find that the current in the subthreshold region is

\[ I = I_{0}\text{exp}\left[\frac{qV_{GS}}{kT} \frac{C_{G}}{C_{ES}}\right] \nonumber \]

Taking logarithm of both sides we find,

\[ \text{log}_{10}I=\frac{q}{KT}\frac{C_{G}}{C_{ES}}(\text{log}_{10}e)V_{GS}+\text{log}_{10}I_{0} \nonumber \]

The slope, S, of the subthreshold regime is usually expressed gate volts per decade of drain current. At room temperature, the optimum, when \(C_{G} \gg C_{S}, C_{D}\), is

\[ S=\frac{kT}{q}\frac{1}{\text{log}_{10}e} \approx 60 \text{ mV/decade} \nonumber \]

The slope becomes much sharper at low temperatures; see Figure 5.8.2.

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