Subsections

• 4.5.1 Universal Mobility
• 4.5.2 Semi-Empirical Modeling

4.5 Inversion Layer Mobility

The transport of carriers in the inversion layer is different from that in the bulk. Carriers in the channel region experience the irregularities at the Si/SiO$_2$ interface. The resulting surface mobility is lower than the bulk mobility, since the carriers in the channel undergo surface roughness scattering in addition to the bulk scattering mechanisms.

The electric field component normal to the Si/SiO$_2$ interface causes the formation of a potential well which confines carriers to a region close to the interface to form a two dimensional electron gas (2DEG) or hole gas (2DHG). The carrier motion is thus quantized in the direction normal to the interface thereby leading to the formation of sub-bands within the conduction or valence bands as shown in Fig. 4.5a.

For a (100) surface two sets of sub-bands, the primed and the unprimed subband ladders, are formed due to the different quantization masses. Within the triangular well approximation, the energy level of $i^{th}$ subband is inversely proportional to the quantization mass. Therefore, the subbands in the $\Delta _2$ valley are less separated in energy, and in absolute values also are lower in energy than the subbands of $\Delta_4$ valleys. The mobility can then be calculated from the subband structure using numerical methods by taking into account the scattering rates between the subbands. Such an approach is, however, quite time intensive and therefore should be replaced by a more effective analytical modeling approach. Table 4.5 lists a number of semi-empirical inversion layer mobility models employed in today's device simulators.

Simulator	Surface mobility models
ATLAS	Darwish, Lombardi, Shirahata, Shin, Watt,
MEDICI	Mujtaba, Shirahata, Shin, Watt, Darwish
DESSIS	Lombardi/Darwish, Reggiani,
MINIMOS-NT	Selberherr, Lombardi

Table 4.2: List of inversion layer mobility models used in general purpose device simulators. The models are due to: Darwish [Darwish97], Lombardi [Lombardi88], Mujtaba [Mujtaba94], Selberherr [Selberherr90], Shin [Shin89], Shirahata [Shirahata92], Watt [Watt87], Reggiani [Reggiani99]

Figure 4.5: Energy lineups of the conduction subbands in (a) unstrained Si and (b) Strained Si. Strain changes the lineup of the energies and thus is able to remove the degeneracy of subband ladders.

4.5.1 Universal Mobility

Experimental evidences indicate that the inversion layer mobility, when investigated as a function of the electric field component normal to the Si/SiO$_2$ interface, is a function of the doping concentration, the gate and substrate bias and the oxide thickness. The effective mobility, is a spatial average of the mobility profile in the inversion layer. Sabnis and Clemens [Sabnis79] demonstrated that that if the effective mobility of electrons is plotted as a function of the effective transverse electric field in the inversion layer, the universal mobility curve is obtained.

The effective field for the electrons in the inversion layer is defined as the average of the normal electric field $E_y(y)$ experienced by the electrons weighted by the electron concentration $n_y(y)$.

$$E_{\text{\text{eff}}} = \frac{\displaystyle\int_{0}^{y_i}E_y(y) n(y) dy} {\displaystyle\int_{0}^{y_i}n(y) dy}$$ (4.89)

The integration in (4.89) is performed over the depth of the inversion layer, $y_i$. In terms of the field at the top ( $E_{\text{top}}$) and bottom ( $E_{\text{bottom}}$) of the inversion layer, the effective field becomes

$$E_{\text{eff}} = \frac{E_{\text{top}} + E_{\text{bottom}} }{2} = \frac{1}{\epsilon _{si}}\left( \frac{Q_{\text{inv}}}{2} + Q_{\text{depl}} \right).$$ (4.90)

To arrive at (4.90), the relations

$$E_{\text{top}} = \frac{ Q_{\text{inv}} + Q_{\text{depl}} } {\epsilon _{si}},$$ (4.91)

$$E_{\text{bottom}} = \frac{ Q_{\text{depl}} } {\epsilon _{si}},$$ (4.92)

derived using Gauss's law are used. $Q_{\text{inv}}$ and $Q_{\text{depl}}$ denote the inversion and depletion charge densities, respectively, and can be computed as

$$Q_{\text{inv}} = -q \int_{0}^{y_i} n(y) dy$$ (4.93)

$$Q_\mathrm{depl} = \sqrt{4 \ensuremath{\kappa_\mathrm{si}}\phi_\mathrm{B} N_\mathrm{sub}/q}% ,\quad\mathrm{where}\quad$$ (4.94)

The definition of $E_{\text{eff}}$ in (4.90) is valid for electrons on (100) oriented surfaces and can be generalized to

$$E_{\text{eff}} = \frac{1}{\epsilon } \left( \eta Q_{\text{inv}} + Q_{\text{depl}} \right).$$ (4.95)

The parameter $\eta$ is dependent on the orientation of the crystal surface and can assume values different from $1/2$ due to valley repopulation effects [Lee91]. For estimating the electron mobilities on (110) and (111) oriented surfaces, $\eta \approx 1/3$ can be assumed.

Figure 4.6: Universal mobility in the Si , inversion layer.

Similar to (4.89) the effective mobility, $\mu_{\text{eff}}$, in Si inversion layers can be defined as

$$\mu_{\text{\text{eff}}} = \frac{\displaystyle\int_{0}^{y_i} \mu(y) n(y) dy} {\displaystyle\int_{0}^{y_i}n(y) dy}.$$ (4.96)

Experimentally, the effective mobility can be determined from the drain current relation (4.24) through

$$\mu_{\mathrm{eff}}= \frac{L}{W} \frac{ g_\mathrm{d}(V_\mathrm{g})}{Q_\mathrm{inv}(V_\mathrm{g})} ,$$ (4.97)

where ${g_d = \frac{\partial I_d}{\partial V_{ds}} }$ denotes the drain conductance. The different regimes of the inversion layer mobility are shown in Fig. (4.6). At low effective fields, carrier mobility is dominated by Coulomb scattering, which is more effectively screened at higher effective fields. At moderate effective fields, phonon scattering determines the mobility. Finally, in the high-effective field regime, surface roughness scattering limits the carrier mobility. The universal nature of the carrier mobility is attributed to the phonon and surface roughness mobilities.

4.5.2 Semi-Empirical Modeling

The effect of strain on the inversion layer mobility can be understood in terms of the modification of the subband structure. In the presence of strain, each subband ladder experiences and additional energy shift as shown in Fig. 4.5b. Since this shift depends on the valley orientation, it may be different for each valley and consequently the degeneracy between the subband ladders can be lifted. Although significant effort has been put by several groups on the calculation of the mobilities using subband Monte Carlo techniques [Gamiz02,Roldan96,Fischetti02,Ungersboeck06,Rashed95,Jungemann03b,Fan04], little work has been done in the area of developing simplified analytical inversion layer mobility models suitable for device simulations. This is primarily because of the increased complexity of the physical effects involved in the inversion layer in the presence of strain.

A semi-empirical modeling approach based on Darwish's mobility model [Darwish97] for epitaxially grown Si on relaxed SiGe was suggested by [Roldan03]. The effect of strain was incorporated in the model through two functions $\alpha(x)$ and $\beta(x)$, where x denotes the mole fraction of Ge in the underlying SiGe substrate. The model was compared with several experimental data and demonstrated good agreement to one data set.

Figure 4.7: Effective mobility versus field for strained Sireported from various groups. MIT [Currie01], Stanford [Welser94b], IBM [Huang01,Rim01,Rim02], Toshiba [Mizuno00]

Fig. (4.7) shows the effective mobility versus field for biaxially strained Si grown on relaxed SiGe, as obtained by several experimental groups. The figure reveals a large amount of scatter in the effective mobility values for the same Ge content, reported by various groups. Possible reasons for this scatter are the different processing conditions adopted. It has also been speculated that electrons in the inversion layer experience a reduced surface roughness scattering [Fischetti02]. The ambiguity in the physical explanation of reduced surface roughness together with the process variations makes the physical modeling of effective mobility a daunting task.

In this work, the surface mobility is thus modeled semi-empirically. A formalism similar to that proposed by Darwish [Darwish97] is used.

$$\frac{1}{\mu_0} = R\cdot\left(\frac{1}{\mu_{b}} + \frac{1}{\mu_{ac}} + \frac{1}{\mu_{sr}^{-1}} \right).$$ (4.98)

Here the terms $\mu_b$, $\mu_{ac}$ and $\mu_{sr}$ denote the bulk, acoustic phonon, and surface roughness limited mobilities, respectively. The bulk mobility $\mu_b$ is obtained by projecting the mobility tensor (4.45) along the direction of the driving force vector. In the high effective field regime, the inversion layer mobility is affected by surface acoustic phonons and the mobility can be expressed as [Darwish97]

$$\hspace*{-0cm} \mu_\ensuremath{{\mathrm{ac}}} = B\left(\frac{E_\e... ...t)^2\right] \left(\frac{E_\ensuremath{{\mathrm{ref}}}} {{E_\perp}}\right)^{1/3}$$ (4.99)

The formulation in (4.99) is based on the considerations of the classical and quantum mechanical thickness of the inversion layer, as suggested in [Schwarz83]. $E_\perp$ denotes the electric field component perpendicular to the current direction.

For very high effective fields, the mobility is limited by surface roughness scattering and has been modeled as

$$\mu_\ensuremath{{\mathrm{sr}}} = \left[D_1 + D_2 \cdot \left(\fra... ...\right] \left(\frac{S_\ensuremath{{\mathrm{ref}}}} {{E_\perp}}\right)^{\gamma},$$ (4.100)

$$\gamma = A + 0.5\left(\frac{n(r) + 10^{15} }{N_I}\right)^{S_\ensuremath{{\mathrm{exp}}}}.$$ (4.101)

The exponent $\gamma$ is a function of the local inversion charge. The increase of the exponent with increasing inversion charge has been attributed to an increase of inter-subband scattering at higher effective fields [Mori79]. In (4.99) and (4.101), a doping dependence is necessary in order to achieve an agreement between the measured and calculated $\mu_\ensuremath{{\mathrm{eff}}}$.

The parameter $R$ in (4.98) is a function of the enhancement factor $f$ and is given by

$$R(f) = \mu^L_{\text{str}}(f) / \mu_{\text{uns}}$$ (4.102)

Here $\mu^L_{\text{str}}(f)$ denotes the pure lattice mobility (excluding the doping dependence) of strained Si and can be obtained by projecting the mobility (4.45) along the direction of the driving force vector. The experimental data in Fig. 4.7 reveal that the mobility enhancement is higher for lower effective fields and vice-versa. To capture this effect into the model, the enhancement factor has been modeled as

$$f = \left\{ \begin{array}{ll} \displaystyle f_1, \hspace*{1cm} \h... ...displaystyle f_2,\hspace*{1cm} \hspace*{2mm} E_\perp > E_2. \end{array} \right.$$ (4.103)