3.15.2.2 Diffusion Resistance Model for Variability Modeling

If $RGEOMOD = 1$, a diffusion resistance model for variability modeling will be invoked. The physically derived model captures the complex dependences of resistance on the geometry of FinFETs.

$RGEOMOD = 1$ is derived based on the FinFET structure (single-fin or multi-fin with merged source/drain). Figure 1 shows the cross section of a double-gate FinFET with raised source/drain ($RSD$) along the source-drain direction. $L_g$ (gate length) and $TOXP$ (physical oxide thickness, not shown in Fig. 1) are calculated in Section 3.1. A hard mask with thickness $TMASK$ often exists on top of the fin. If $TMASK = 0$, the model will assume there is no hard mask and the dielectric thickness on top of the fin is $TOXP$ (triple-gate FinFET). In the figure, $LSP$ is the spacer thickness, $LRSD$ is the length of the raised source/drain, $HFIN$ is the fin height, $TGATE$ is the gate height, and $HEPI$ is the height of the epitaxial silicon above the fin. These parameters are specified by the user.

Figure 1: Cross section of a raised source/drain double-gate FinFET and symbol definition

The resistivity of the raised source/drain can be specified with the parameter $RHORSD$. If $RHORSD$ is not given the resistivity is calculated using the following expressions [11]:

$\mu_{MAX} = \begin{cases} 1417 &\text{for NMOS} \\ 470.5 &\text{for PMOS} \end{cases} \qquad (3.461)$

$\mu_{RSD} = \begin{cases} 52.2 + \dfrac{\mu_{MAX} - 52.2}{1 + \Big( \dfrac{NSD}{9.68 \times 10^{22}} \Big)^{0.680}} - \dfrac{43.4}{1 + \Big( \dfrac{3.41 \times 10^{26} }{NSD} \Big)^{2.0}} &\text{for NMOS} \\ \\ 44.9 + \dfrac{\mu_{MAX} - 44.9}{1 + \Big( \dfrac{NSD}{2.23 \times 10^{23}} \Big)^{0.719}} - \dfrac{29.0}{1 + \Big( \dfrac{6.10 \times 10^{26} }{NSD} \Big)^{2.0}} &\text{for PMOS} \end{cases} \qquad (3.462)$

$\rho_{RSD} = \dfrac{1}{q \cdot NSD \cdot \mu_{RSD}} \qquad (3.463)$

where $NSD$ is the active doping concentration of the raised source/drain.

The diffusion resistance includes two components: the spreading resistance due to current spreading from the extension region into the raised source/drain ($R_{sp}$) and the resistance of the raised source/drain region ($R_{con}$).

The spreading resistance, $R_{sp}$ is derived by assuming the current spreads at a constant angle $\theta_{RSP}$ in the raised source/drain. Comparison with numerical simulation shows that $\theta_{RSP}$ is around 55 degrees. The spreading resistance is given as a function of the cross sectional area of the raised source/drain ($A_{rsd}$) and the effective fin area ($A_{fin}$):

$R_{sp} = \dfrac{\rho_{RSD} \cdot cot(\theta_{rsp})}{\sqrt{\pi} \cdot NFIN} \cdot \Bigg[ \dfrac{1}{\sqrt{A_{fin}}} - \dfrac{2}{\sqrt{A_{rsd}}} + \sqrt{\dfrac{A_{fin}}{A_{rsd}^2}} \Bigg] \qquad (3.464)$

$A_{fin}$ is given by

$A_{fin} = \begin{cases} HFIN \times TFIN &\text{for } HEPI \ge 0 \\ (HFIN + HEPI) \times TFIN &\text{for } HEPI < 0 \end{cases} \qquad (3.465)$

Here $HEPI < 0$ is the case where silicidation removes part of the silicon, forming a recessed source/drain (Fig. 2).

Figure 2: Lithography-defined FinFET with a smaller source/drain height compared to the fin height (silicide not shown).

The raised source drain cross sectional area ($A_{rsd}$) is given by

$A_{rsd} = \begin{cases} FPITCH \cdot HFIN + \Big[ TFIN + \\ \qquad (FPITCH - TFIN) \cdot CRATIO \Big] \cdot HEPI &\text{for } HEPI \ge 0 \\ FPITCH \cdot (HFIN + HEPI) &\text{for } HEPI < 0 \end{cases} \qquad (3.466)$

In the above formula, we have assumed a rectangular geometry for negative $HEPI$ (Fig. 2) and the cross sectional area is simply the fin pitch times the final height of the source/drain. For a positive $HEPI$, we have considered a $RSD$ formed by selective epitaxial growth, in which case the $RSD$ may not be rectangular (e.g. Fig. 3). In calculating the cross sectional area, we take into account the non-rectangular corner through the parameter $CRATIO$. $CRATIO$ is defined as the ratio of corner area filled with silicon to the total corner area. In the example given in Fig. 4, $CRATIO$ is 0.5.

Figure 3: FinFET with non-rectangular epi and top silicide

Figure 4: 2-D cross section of a FinFET with non-rectangular epi and top silicide

The calculation of the contact resistance ($R_{con}$) is based on the transmission line model [12]. $R_{con}$ is expressed as a function of the total area ($A_{rsd,total}$) and the total perimeter ($P_{rsd,total}$):

$R_{rsd,TML} = \dfrac{\rho_{RSD} \cdot l_t}{A_{rsd,total}} \cdot \dfrac{cosh(\alpha) + \eta \cdot sinh(\alpha)}{sinh(\alpha) + \eta \cdot cosh(\alpha)} \qquad (3.467)$

$\alpha = \dfrac{LRSD}{l_t} \qquad (3.468)$

$l_t = \sqrt{\dfrac{RHOC \cdot A_{rsd,total}}{\rho_{RSD} \cdot P_{rsd,total}}} \qquad (3.469)$

where $RHOC$ is the contact resistivity at the silicide/silicon interface. The total area and perimeter are given by

$A_{rsd,total} = A_{rsd} \times NFIN + ARSDEND \qquad (3.470)$

$P_{rsd,total} = (FPITCH + DELTAPRSD) \times NFIN + PRSDEND \qquad (3.471)$

$DELTAPRSD$ is the per-fin increase in perimeter due to non-rectangular raised source/drains. $ARSDEND$ and $PRSDEND$ are introduced to model the additional cross-sectional area and the additional perimeter, respectively, at the two ends of a multi-fin FinFET.

Figure 5: FinFET with a non-rectangular epi and silicide on top and two ends.

$SDTERM = 1$ indicates the source/drain are terminated with silicide (Fig. 5), while $SDTERM = 0$ indicates they are not. $\eta$ is given by

$\eta = \begin{cases} \dfrac{\rho_{RSD} \cdot l_t}{RHOC} &\text{for } SDTERM = 1 \\ \\ 0.0 &\text{for } SDTERM = 0 \end{cases} \qquad (3.472)$

In the case of the recessed source/drain, a side component of the contact resistance must be modeled as well. It is given by

$R_{rsd,side} = \dfrac{RHOC}{NFIN \cdot (-HEPI) \cdot TFIN} \qquad (3.473)$

Finally, the total diffusion resistance is given by

$\begin{aligned} R_{s,geo} &= R_{d,geo} = \dfrac{R_{rsd}}{NF} \cdot \Big[ RGEOA + RGEOB \times TFIN + \\ &RGEOC \times FPITCH + RGEOD \times LRSD + RGEOE \times HEPI \Big] \end{aligned}$

$(3.474)$

where

$R_{rsd} = \begin{cases} R_{rsd,TML} + R_{sp} &\text{for } HEPI \ge 0 \\ \\ \dfrac{(R_{rsd,TML} + R_{sp}) \times R_{rsd,side}}{(R_{rsd,TML} + R_{sp}) + R_{rsd,side}} &\text{for } HEPI < 0 \end{cases} \qquad (3.475)$

Fitting parameters $RGEOA$, $RGEOB$, $RGEOC$, $RGEOD$ and $RGEOE$ are introduced for fitting flexibility.

References

[11] G. Masetti, M. Severi, and S. Solmi, "Modeling of Carrier Mobility Against Carrier Concentration in Arsenic-, Phosphorus-, and Boron-Doped Silicon," IEEE Transaction on Electron Devices, vol. 30, no. 7, pp. 764-769, July 1983.

[12] H. H. Berger, "Model for contacts to planar devices," Solid-State Electronics, vol. 15, pp. 145-158, 1972.