5.1 Mobility

Mobility describes the relation between drift velocity of electrons or holes and an applied electric field in a solid.

$${\ensuremath{\mathitbf{v}}}_\mathrm{d} = \hat{\mu} {\ensuremath{\mathitbf{E}}}$$ (5.1)

Here, $v_\mathrm{d}$ is the drift velocity, ${\ensuremath{\mathitbf{E}}}$ is the electric field, and $\hat{\mu}$ denotes the mobility tensor.

In a semiconductor the mobility of electrons is different from that of holes. The reason is the different band structure and scattering mechanisms of these two carrier types. When one charge carrier is dominant the conductivity of a semiconductor is directly proportional to the mobility of the dominant carrier. In Figure 5.1a and 5.1b the electron mobility of Si is plotted. The mobility in Si is a strong function of temperature and impurity concentration.

Figure 5.1: Experimental electron mobility data for Si [Jacoboni77,Tosic81].

[a]

[b]

The device characteristics of MOSFETs is strongly influenced by transport in the inversion layer. Thus, the lattice mobility, representing a bulk quantity, cannot be directly used as a model parameter. In fully depleted silicon-on-insulator (FDSOI) and ultra-thin-body (UTB) MOSFETs all charge carriers reside in the inversion layer, thus quantum confinement and surface roughness scattering have to be taken into account when modeling the mobility.

The low-field mobility in inversion layers, when analyzed as a function of the confining electric field, is a function of doping, gate-voltage, back-bias voltage, and gate oxide thickness. Sabnis and Clemens found that the mobility data shows a universal behavior [Sabnis79], if it is plotted as a function of the effective field

$$E_\mathrm{eff}=\frac{e}{\ensuremath{\kappa_\mathrm{si}}{}} (N_\mathrm{depl} + \eta N_\mathrm{s})\ .$$ (5.2)

Here, $\eta \sim 1/2$ for electrons, $N_\mathrm{depl}$ and $N_\mathrm{s}$ are the depletion and inversion charge densities, respectively, and $\ensuremath{\kappa_\mathrm{si}}$ is the dielectric constant of Si. This led to the conclusion that the MOSFET inversion layer mobility is a reproducible property associated with the Si-SiO$_2$ system, and not a parameter sensitive to nominal process variations.

On the basis of this work, investigations for different substrate orientations were performed. It was found that a similar universal behavior is achieved when the value of $\eta$ [Takagi94] is properly adapted.

The first mobility studies were performed on MOSFETs with a spatially uniform doping profile in the channel. Later on, the validity of the universality of the effective mobility in channels with arbitrary doping profiles was investigated. It was discovered that the doping dependence of the effective mobility can be eliminated if plotted as a function of $E_\mathrm{eff} \propto (\eta N_\mathrm{s} + b N_\mathrm{depl})$, where the coefficient $b$ is sensitive upon the shape of the doping profile [Vasileska97].

With the advent of SOI technologies, the physical basis and limitation of the universal nature of the effective mobility was examined for fully depleted SOI inversion layers. It was found that the universal mobility behavior does not hold if the electron density distribution reaches the lower surface of the Si layer. For such SOI structures with extremely thin Si film thickness $T_\mathrm{si}$ it has been predicted that there exists another kind of universal mobility behavior as a function of the inversion electron density $N_\mathrm{s}$, independent of the impurity concentration $N_\mathrm{sub}$ and the buried oxide layer thickness $T_\mathrm{box}$ [Shoji97].

The effective mobility $\mu_\mathrm{eff}$ in Si inversion layers can be experimentally determined from the drain conductance $g_\mathrm{d}$ in the linear region

$$\ensuremath{\mu_\mathrm{eff}}= \frac{L}{W} \frac{ g_\mathrm{d}(V_\mathrm{g})}{N_\mathrm{s}(V_\mathrm{g})}\ ,$$ (5.3)

where $L$ and $W$ are the transistor length and width, respectively, $N_\mathrm{s}$ is the surface carrier concentration, and $V_\mathrm{g}$ is the gate voltage. The value of $E_\mathrm{eff}$ can be calculated from (5.2) using

$$N_\mathrm{depl} = \sqrt{4 \ensuremath{\kappa_\mathrm{si}}\phi_\ma... ...athrm{B} = \frac{\ensuremath {{\mathrm{k_B}}}T}{e} \ln(N_\mathrm{sub} / n_i)\ .$$ (5.4)

Here, $N_\mathrm{sub}$ is the substrate impurity concentration, $E_i - e \phi_\mathrm{B}$ is the bulk Fermi energy, and $n_i$ is the intrinsic carrier concentration of Si ( $n_i = 1.4e10$ cm$^{-3}$ at 300 K).

Figure 5.2 shows that the mobility characteristics of MOSFET inversion layers can be split into three distinctive regions. At low inversion charge densities (low vertical fields), mobility is limited by scattering with doping atoms and charges at the Si-SiO$_2$ interface (Coulomb scattering). Going to higher inversion densities, phonon scattering gains importance and dominates over Coulomb scattering. At large $N_\mathrm{s}$ scattering with surface roughness limits the total mobility.

Figure: Universal mobility in Si inversion layer at $T=300 K$.

To understand how the different scattering mechanisms affect the performance of a device on a circuit level, the device switching trajectory in the $V_\mathrm{DS}$- $V_\mathrm{GS}$ space has to be considered. It has been found that for the delay of a ring oscillator the mobility at low and intermediate is important, because there the device spends comparatively more time than in the high gate field region [Mujtaba95].

E. Ungersboeck: Advanced Modelling Aspects of Modern Strained CMOS Technology