The effect of sharp-corner emendation of irregular FinFETs on electrothermal characteristics

Abstract

The present study is an attempt to investigate the impacts of channel modification and the capabilities of amended sharp-corner FinFETs from thermal and electrical points of view. It also provides a new definition for gate-oxide and channel capacitance of irregular fin shape by replacing the contribution of each area in the total gate capacitance expression of the square FinFET. This definition determines the subthreshold reliability of the amended sharp-corner FinFETs. As a function of gate insulator capacitance, channel capacitance, depletion charge per unit length, and fin area, mobile electron concentrations are derived for amended sharp-corner FETs by assuming an arbitrary channel potential profile to simplify the formulation. The comparison results demonstrate that an amended FinFET with a partial cylindrical shape at the top region of fin (PC-FinFET) by higher gate controllability adjusts the hot carrier effects, reduces DIBL, improves the subthreshold characteristics as well as short-channel effects, while the amended-channel FinFET with extended round-bottom region reduces the self-heating effects, attenuates the thermal resistance, and moderates the thermal dependence of electrical characteristics. Therefore, it is deduced that modified-channel FinFET (MC-FinFET), with both cylindrical top and extended bottom regions, has improved thermal and electrical stabilities in both subthreshold and saturation modes in comparison with a conventional thin-film FinFET. The superiority of the MC-FinFET, which was evaluated with three-dimensional simulations, demonstrates the ability of this structure as a high-performance device over the other eliminated sharp-corner FinFETs.

Appendix

The first step of the universal core approach [32] for a double gate FinFET is utilizing the arbitrary potential assumption, instead of differential form of Poisson’s equation as expressed in Eq. (19a), [(a) for rectangular and (b) for cylindrical cross sections].

$$\begin{aligned} \varphi \left( x \right)= & {} \left( {2\left| x \right| /W_{\mathrm{Si}} } \right) \times \left( {\varphi _s -\varphi _0 } \right) +\varphi _0 \end{aligned}$$

(19a)

$$\begin{aligned} \varphi \left( r \right)= & {} \left( {r/R} \right) \times \left( {\varphi _s -\varphi _0 } \right) +\varphi _0 \end{aligned}$$

(19b)

where ${\varphi }_{s }$ and ${\varphi }_{0 }$ are the surface and center potentials. Then, Using Gauss’s law and the boundary conditions at each interface, the relation between the surface potential, center potential and gate voltage is determined by Eq. (20):

$$\begin{aligned} \frac{\varepsilon _\mathrm{ox} }{t_\mathrm{ox} }\left( {V_\mathrm{G} -V_\mathrm{FB} -\varphi _s } \right) =\varepsilon _\mathrm{si} \frac{d\varphi }{dx}|_{{{x}}={W}_{{\mathrm{si}}/2} } =\frac{2\varepsilon _\mathrm{si} }{W_\mathrm{si} } \Delta \varphi =-\frac{Q_\mathrm{t} }{2} \end{aligned}$$

(20)

where $t_\mathrm{ox}$ is the oxide thickness, $V_\mathrm{FB}$ is the flat-band voltage, and $Q_{\mathrm{t}}$ is the total charge density in channel. Then, by utilizing Boltzmann’s statistics for the carriers by Eq. (21) and using the channel potential, we can obtain the charge density and the total charge per unit area in the channel by Eq. (22):

$$\begin{aligned} \rho \left( x \right)= & {} -qN_\mathrm{A} -q\frac{n_i^2 }{N_\mathrm{A} }\hbox {e}^{\left( {\varphi \left( x \right) -V} \right) /v_T } \end{aligned}$$

(21)

$$\begin{aligned} Q_\mathrm{total}= & {} Q_d +Q_\mathrm{in} =-qN_\mathrm{A} W_{\mathrm{Si}} \nonumber \\&-\,q\frac{n_i^2 }{N_\mathrm{A} }\mathop \int \limits _{-W_{\mathrm{Si}/2} }^{W_{\mathrm{Si}/2} } \hbox {e}^{\left( {\varphi -V} \right) /v_T }\hbox {d}x\nonumber \\= & {} -qN_\mathrm{A} W_{\mathrm{Si}} -q\frac{n_i^2 W_{\mathrm{Si}} }{N_\mathrm{A} }\hbox {e}^{\frac{\varphi _0 -V}{v_T }}\left( {\hbox {e}^{\frac{\Delta \varphi }{v_T }}-1} \right) \end{aligned}$$

(22)

where V is the electron quasi-Fermi potential,$Q_{\mathrm{e}}$ and $Q_{\mathrm{d} }$ are the total mobile electron and depletion charges per unit area. With the help of the Gauss’s law and charge density, surface and center potentials can be determined. Finally by replacing ${\Delta } \varphi$ by ${- (W_\mathrm{si} / 4 \varepsilon _\mathrm{si}) \times Q}_{ t}$ from (20) in Eq (22)${, \varphi }_{\mathrm{s}}$ as a function of $Q_{\mathrm{e}}$ and $Q_{\mathrm{d}}$ is determined. When ${ \varphi }_{\mathrm{s}}$ is obtained from the total charge equation and replaced in Gauss’s law, a relation between total mobile electron charge per unit area $(Q_{\mathrm{e}})$ in the channel and gate voltage is determined by Eq. (23)

$$\begin{aligned}&V_{\mathrm{G}} -V_\mathrm{FB} +\frac{\hbox {t}_{\mathrm{ox}} }{2{\varepsilon }_{\mathrm{ox}} }{Q}_{\mathrm{d}} -V\nonumber \\&\quad =-\frac{\hbox {t}_{\mathrm{ox}} }{2{\varepsilon }_{\mathrm{ox}} }{Q}_{\mathrm{e}} \nonumber \\&\qquad +v_{\mathrm{T}} \ln \left( {\frac{-\frac{{Q}_{\mathrm{e}} \left( {{Q}_{\mathrm{e}} +{Q}_{\mathrm{d}} } \right) }{2v_{\mathrm{T}} {\varepsilon }_{{\mathrm{si}}} /W_{\mathrm{Si}} }}{q\frac{n_i^2 }{N_\mathrm{A} }W_{\mathrm{Si}} \left[ {1-\hbox {exp}\frac{W_{\mathrm{Si}} }{2v_{\mathrm{T}} {\varepsilon }_{{\mathrm{si}}} }({Q}_{\mathrm{e}} +{Q}_{\mathrm{d}} )} \right] }} \right) \end{aligned}$$

(23)

Using this approach for cylindrical arbitrary potential by Eq. (19b) in the channel of the cylindrical cross section, to obtain the cylindrical cross section’s charge model, then analyzing and Comparing the results of different regular geometrical cross sections (e.g., DG and Cy-GAA-FETs) mobile charge model can be generalized. Therefore, a single expression that relates the mobile electron charge density per unit length $(Q_{\mathrm{e}})$ with the applied gate voltage is written in the following form:

$$\begin{aligned}&V_\mathrm{G} -V_\mathrm{FB} +\frac{Q_{\mathrm{d},n} }{C_{\mathrm{g},n} }-V\nonumber \\&\quad =-\frac{Q_{\mathrm{e},n} }{C_{\mathrm{g},n} }\nonumber \\&\qquad +v_T \ln \left( {\frac{-Q_{\mathrm{e},n} }{q\frac{n_i^2 }{N_\mathrm{A} }A_{\mathrm{ch},n} }\frac{-\left( {Q_{\mathrm{e},n} +Q_{\mathrm{d},n} } \right) /v_T C_{\mathrm{ch},n} }{1-exp\frac{Q_{\mathrm{e},n} +Q_{\mathrm{d},n} }{v_T C_{\mathrm{ch},n} }}} \right) \nonumber \\ \end{aligned}$$

(24)