Subsections

• 3.1 Subband Monte Carlo Model
• 3.2 Surface Roughness Scattering
• 3.3 Influence of SRS
• 3.4 Comparison with Bulk Simulations

3. Homogeneous Transport in Inversion Layers

WITH CONTINUOUSLY downscaling of the device geometry, the influence of quantum mechanical effects on the device characteristics starts to increase. Transport parameters are fundamentally affected by surface roughness scattering [146,147] and quantization [148]. In [149], surface roughness scattering has been approximated using the semi-empirical Matthiesen rule. However, the influence of inversion layer effects on higher-order transport parameters has not been studied satisfactorily yet. To point out the impact of these effects on the transport parameters, a self-consistent SMC simulator has been developed. The object of investigations is a homogeneous bulk UTB SOI MOSFET, where the impact of inversion layer effects on the carrier transport is very high.

3.1 Subband Monte Carlo Model

Investigations of inversion layer effects using the SMC simulations, have already been given in numerous publications [150,151,152]. However, a description concerning higher-order transport parameters in the inversion layer, which is important for the characterization of macroscopic transport in realistic devices, has not been done satisfactorily yet.

In order to study the influence of quantization and surface roughness scattering on higher-order transport parameters within high-fields, a self-consistent SMC simulator has been developed [50,153,154] (see sketch in Fig. 3.1).

Figure 3.1: Principle data flow of the parameter extraction for higher-order transport models. While transport is treated in the SMC code, the influence of the confinement perpendicular to the oxide interface is carried out by the Schrödinger-Poisson solver.

Subband energy levels and wavefunctions are initially determined self-consistently with the Poisson equation assuming Fermi-Dirac statistics. Based on the subband structure, Monte Carlo calculations are performed taking into account phonon induced, impurity and surface roughness scattering. The non-parabolicity of the band structure is treated by Kane's model [70]. The scattering rates are strongly affected by high driving fields, which results in a shift of the wavefunctions. The consequence of this shift is a change in the overlap integral of the scattering operator and therefore a change in the subband occupations, which leads to a modified carrier concentration. This concentration can be obtained as

$$\hspace*{-0.7cm} n_{\mathrm{tot}}(z) =  2 \ensuremath{\sum_{\ensu... ..._{\ensuremath{n^{'}}}}^2(z)\ensuremath{{\;}\mathrm{d}}\ensuremath{\mathcal{E}}}$$ (3.1)

$$+2 \ensuremath{\sum_{\ensuremath{n^{''}}}}\ensuremath{\int\limits... ...uremath{n^{''}}}}^2(z)\ensuremath{{\;}\mathrm{d}}\ensuremath{\mathcal{E}}}{\;}.$$

The subband occupation number is represented by the Fermi-Dirac distribution function $\ensuremath{f}$ . After convergence is reached, which is achieved by an exchange of energy eigenvalues, wavefunctions, and subband occupations, the transport parameters can be extracted.

In Fig. 3.2, a result of the self-consistent loop between the Schrödinger-Poisson solver and the MC simulator is presented. Several velocities for different inversion layer concentrations  $\ensuremath {N}_\mathrm {inv}$ as a function of the lateral field for a UTB SOI MOSFET device with a channel thickness of  ${\mathrm{4}}{\;}{\mathrm{nm}}$ and a substrate doping of  ${\mathrm{2\times10^{16}}}{\;}{\mathrm{cm^{-3}}}$ have been extracted.

Figure 3.2: Carrier velocity as a function of the driving field for different inversion layer concentrations. Due to surface roughness scattering, the carrier velocity decreases for increasing inversion layer concentrations. The maximum velocity is below the saturation velocity of bulk Si.

3.2 Surface Roughness Scattering

Surface roughness scattering (SRS) is the main scattering process in high inversion layer concentrations, which has a strong impact on the transport parameters (see Fig. 3.3).

Figure 3.3: The effective mobility as a function of the effective field [7]. For increasing bulk doping and for low inversion layer concentrations Coulomb scattering is the main scattering process, while for increasing  $\ensuremath {N}_\mathrm {inv}$ phonon scattering becomes more important.

Surface roughness can be seen as a barrier at the interface, whose position has a small and slowly varying displacement  $\ensuremath{\Delta_\ensuremath{\ensuremath{\mathitbf{r}}} }$ . Here, $\ensuremath{\ensuremath{\mathitbf{r}}}$ is the two-dimensional vector in the plane of the interface [47].

$\ensuremath{\Delta_\ensuremath{\ensuremath{\mathitbf{r}}} }$ can be expressed by its Fourier components

$$\ensuremath{\Delta_\ensuremath{\ensuremath{\mathitbf{r}}} }=\ensu... ...h{\ensuremath{\mathitbf{q}}}\ensuremath{\ensuremath{\mathitbf{r}}} \right){\;},$$ (3.2)

whereas the power spectrum  $\ensuremath{\langle \mathrm{\vert}\ensuremath{\Delta_\ensuremath{\ensuremath{\mathitbf{q}}}}\mathrm{\vert}^2 \rangle}$ is usually modeled with a Gaussian form as [155]

$$\ensuremath{\langle \mathrm{\vert}\ensuremath{\Delta_\ensuremath{... ...bda^2\exp\left(-\ensuremath{\ensuremath{\mathitbf{q}}}^2\Lambda^2/4\right){\;},$$ (3.3)

or with an exponential shape as described in [156,157]. $\ensuremath{\ensuremath{\mathitbf{q}}}$ is the difference between the incoming wavevector  $\ensuremath{\ensuremath{\mathitbf{k}}}$ and the wavevector $\ensuremath{\ensuremath{\mathitbf{k}}}^{'}$ after the scattering event. $\Lambda$ is the correlation length of the oxide thickness fluctuations and should be understood as the minimum distance between two points at which thicknesses are considered independent. The SRS matrix elements  $\ensuremath{V^{\mathrm{SRS}}_{\mathrm{\ensuremath{\ensuremath{\mathitbf{k}}}\ensuremath{\ensuremath{\mathitbf{k}}}^{'}}}}$ have been defined and discussed in [158] as

$$\ensuremath{V^{\mathrm{SRS}}_{\mathrm{\ensuremath{\ensuremath{\ma... ...nsuremath{n}^{'}}}}}{\mathrm{d}\ensuremath{x}}\Big\vert_{\ensuremath{x}=0}{\;}.$$ (3.4)

Therefore, the derivative of the wave function at the interface plays an important role. The quantization direction in equation (3.4) is in  $\ensuremath{x}$ direction. In [47] it has been shown that the scattering potential depends linearly on the inversion layer concentration  $\ensuremath {N}_\mathrm {inv}$ defined as [47,1]

$$\ensuremath{N}_\mathrm{inv}=\ensuremath{g_v}\left(\frac{\sqrt{\en... ...{i}}{k_\ensuremath{\mathrm{B}}\ensuremath{T_\ensuremath{n}}}\right)\right){\;},$$ (3.5)

and on the depletion concentration  $\ensuremath{N}_\mathrm{dep}$ which can be expressed as [7]

$$\ensuremath{N}_\mathrm{dep}=\left(4 N\ensuremath{\epsilon}_{\math... ...2}\mathrm{ln}\left(\frac{N}{\ensuremath{n_\mathrm{i}}}\right)\right)^{1/2}{\;}.$$ (3.6)

$N$ and $\ensuremath{n_\mathrm{i}}$ denote the substrate and the intrinsic concentration, respectively. The influence of this dependence on higher-order transport parameters will be shown in the sequel. Inserting the scattering matrix into Fermi's golden rule (see equation (1.137)) and integrating over the whole space, the SRS time  $\ensuremath{\tau_{\mathrm{SR}}}$ can be written as [47]

$$\frac{1}{\ensuremath{\tau_{\mathrm{SR}}}}=\frac{2\pi}{\hbar }\ens... ...left(\ensuremath{\ensuremath{\ensuremath{\mathitbf{k}}}^{'}}\right)\right){\;}.$$ (3.7)

With a large correlation length $\Lambda$ , the interface is locally flat and therefore the term  $\ensuremath{\langle \mathrm{\vert}\ensuremath{\Delta_{\ensuremath{\ensuremath{... ...suremath{\ensuremath{\ensuremath{\mathitbf{k}}}^{'}}}}^2\mathrm{\vert} \rangle}$ tends to zero, which means that the surface roughness scattering is not effective. For small $\Lambda$ the relaxation time is determined by the product of $\Delta$ and $\Lambda$ (see equation (3.3)). In fact, for relatively small correlation lengths, a considerable scattering rate is observed, if the amplitude of thickness variations $\Delta$ is high enough. The energy in the denominator of the expression  $\ensuremath{V^{\mathrm{SRS}}_{\mathrm{\ensuremath{\ensuremath{\mathitbf{k}}}\e... ...h{\mathitbf{k}}}-\ensuremath{\ensuremath{\ensuremath{\mathitbf{k}}}^{'}}\right)$ reflects the fact that scattering with small wave vectors predominates, $\left(1-\mathrm{cos}\left(\ensuremath{\theta}_{\ensuremath{\ensuremath{\mathitbf{k}}}\ensuremath{\ensuremath{\ensuremath{\mathitbf{k}}}^{'}}}\right)\right)$ is the essential component entering the relaxation time [1], while the delta function is related to the elastic nature of the process.

3.3 Influence of Surface Roughness Scattering on Higher-Order Transport Parameters

The main differences between bulk transport and transport in an inversion layer is the occurrence of quantum mechanical effects such as quantum confinement, subbands and SRS. Due to quantum confinement, carriers cannot move in the direction perpendicular to the interface. Thus, carriers have to be treated as a two-dimensional gas, which has a considerable impact on the transport properties.

In Fig. 3.4, higher-order mobilities as a function of the inversion layer concentration  $\ensuremath {N}_\mathrm {inv}$ are shown. In a bulk MOSFET within low fields, the carrier mobility  $\ensuremath {\ensuremath {\mu }_{\mathrm {0}}}$ fits the measurement data of Takagi [7,159] quite well, while a significant reduction is observed in the quantized  ${\mathrm{4}}{\;}{\mathrm{nm}}$ channel region of a UTB SOI MOSFET [160].

Figure 3.4: Higher-order mobilities as a function of the inversion layer concentration  $\ensuremath {N}_\mathrm {inv}$ for different lateral fields. For high fields the difference of the mobilities decreases. For low fields in a bulk MOSFET the carrier mobility is equal to the measurement data of Takagi.

A considerable deviation of higher-order mobilities for low $\ensuremath {N}_\mathrm {inv}$ and low fields can be observed, while for high fields this deviation disappears. All mobilities are constant for high fields and are not affected anymore by SRS.

Figure 3.5: Conduction band edge as a function of the position for different inversion layer concentrations. For high  $\ensuremath {N}_\mathrm {inv}$ , the carriers are closer to the interface and hence they are more affected by surface roughness scattering than for low  $\ensuremath {N}_\mathrm {inv}$ .

Figure 3.6: Conduction band and wavefunctions of a UTB SOI MOSFET for different electric fields. Both the wavefunctions and subbands are shifted with increasing lateral electric fields. The conduction band edge is affected by the change in the subband occupations.

The reason for the constant mobilities for high fields can be explained with Fig. 3.5 and Fig. 3.6.

In Fig. 3.5 the conduction band edge for several  $\ensuremath {N}_\mathrm {inv}$ is presented. Due to the band bending for increasing $\ensuremath {N}_\mathrm {inv}$ , the carriers move closer to the interface, and therefore the influence of SRS becomes stronger, which results in a lowering of the mobilities. For high fields, the distance of the carriers to the interface increases, which reduces the influence of SRS on the mobilities. This is demonstrated in Fig. 3.6. Here, the behavior of the conduction band edge together with the first two subbands and their wavefunctions for different driving fields of ${\mathrm{30}}{\;}{\mathrm{kV/cm}}$ and ${\mathrm{100}}{\;}{\mathrm{kV/cm}}$ is presented. As pointed out before, there is a shift of the conduction band edge and subbands to lower values for increasing driving fields. The result of this band edge shift is that the wavefunction is shifted away from the interface, which means that the probability of finding a carrier near the interface decreases. As a consequence, the influence of surface characteristics on the transport parameters decreases as well. Therefore, the effect of SRS on the mobilities is drastically reduced for high fields.

Figure 3.7: Influence of surface roughness scattering on $\ensuremath {\ensuremath {\mu }_{\mathrm {0}}}$ , $\ensuremath {\ensuremath {\mu }_{\mathrm {1}}}$ , and $\ensuremath {\ensuremath {\mu }_{\mathrm {2}}}$ as a function of the lateral field for different $\ensuremath {N}_\mathrm {inv}$ . For low fields, surface roughness scattering has a strong impact, while for high fields the mobilities are unaffected by SRS.

Figure 3.8: Ratio between the mobilities neglecting and considering SRS for high and low  $\ensuremath {N}_\mathrm {inv}$ values as a function of the lateral field. For low  $\ensuremath {N}_\mathrm {inv}$ , the mobilities are unaffected. The carrier mobility is more affected by SRS than the higher-order mobilities.

This is visible in Fig. 3.7. Here, higher-order mobilities as a function of the electric field for different inversion layer concentrations have been calculated both with and without surface roughness scattering. For fields above  ${\mathrm{120}}{\;}{\mathrm{kV/cm}}$ , the effect of surface roughness scattering on the carrier mobility can be neglected, while for the energy flux mobility and the second-order energy flux mobility even at  ${\mathrm{100}}{\;}{\mathrm{kV/cm}}$ the scattering process has only a minor impact. Thus, higher-order mobilities are not so much affected by SRS as the carrier mobility.

This is demonstrated in Fig. 3.8, where the ratios of higher-order mobilities with and without SRS for high and low inversion layer concentrations of  $\mathrm{1.2}\times{\mathrm{10^{13}}}{\;}{\mathrm{cm^{-2}}}$ and  $\mathrm{2.8}\times{\mathrm{10^{12}}}{\;}{\mathrm{cm^{-2}}}$ , respectively, are plotted. The difference in the low  $\ensuremath {N}_\mathrm {inv}$ regime between the ratio curves of  $\ensuremath {\ensuremath {\mu }_{\mathrm {0}}}$ and the higher-order mobilities is not as high as in the strong inversion regime. This can be explained as follow: Surface roughness scattering changes the momentum relaxation time  $\ensuremath {\tau _\mathrm {0}}$ and therefore the velocity, which is proportional to  $\ensuremath {\tau _\mathrm {0}}$ , is shifted to lower values for increasing  $\ensuremath {N}_\mathrm {inv}$  (see Fig. 3.10). Therefore, the antisymmetric part of the distribution function is reduced for high inversion layer concentrations following that the distribution function becomes more isotropic. The effect is that the impact of SRS on the energy flux is not as high as on the carrier flux. This is pointed out in Fig. 3.9. where the ratio of the momentum relaxation time and the energy-flux relaxation time $\ensuremath {\tau _\mathrm {3}}$ is presented. Due to the strong increase for high $\ensuremath {N}_\mathrm {inv}$ of $\ensuremath {\tau _\mathrm {0}}$ compared to $\ensuremath {\tau _\mathrm {3}}$ , the higher-order mobilities, which are proportional to the relaxation times, are not as effected by SRS as $\ensuremath {\ensuremath {\mu }_{\mathrm {0}}}$ .

Figure 3.9: Ratio of the momentum relaxation time  $\ensuremath {\tau _\mathrm {0}}$ and energy flux relaxation time  $\ensuremath {\tau _\mathrm {3}}$ for different  $\ensuremath {N}_\mathrm {inv}$ as a function of the driving field. Surface roughness scattering influences  $\ensuremath {\tau _\mathrm {0}}$ more than  $\ensuremath {\tau _\mathrm {3}}$ , especially for high inversion layer concentrations.

Figure 3.10: Influence of surface roughness scattering on the electron velocities for different  $\ensuremath {N}_\mathrm {inv}$ . For decreasing  $\ensuremath {N}_\mathrm {inv}$ , SRS loses the influence on the carrier velocity. Due to non-parabolic bands and quantization, the velocity is below the saturation velocity of the bulk.

The dependence of  $\ensuremath {\ensuremath {\mu }_{\mathrm {0}}}$ on  $\ensuremath {N}_\mathrm {inv}$ has an impact on the carrier velocity as pointed out in Fig. 3.10. There, the velocity for high inversion layer concentration and low inversion layer concentration is shown. The difference in the high field case between the carrier velocities considering once SRS and neglecting SRS for the high inversion layer concentration case is about ${\mathrm{25}}{\;}{\mathrm{\%}}$ , while the velocities in the high and low $\ensuremath {N}_\mathrm {inv}$ neglecting SRS respectively, yield the same result.

Figure 3.11: Influence of surface roughness scattering on  $\ensuremath {\tau _\mathrm {1}}$ and $\ensuremath {\tau _\mathrm {2}}$ as a function of the kinetic energy of the carriers for different  $\ensuremath {N}_\mathrm {inv}$ . Due to the elastic scattering nature of SRS, the relaxation times are not affected by SRS.

Figure 3.12: Energy relaxation time (left side) and the second-order relaxation time (right side) as a function of the inversion layer concentration for different lateral electric fields. For a field of $\mathrm {50 kV/cm}$ and high $\ensuremath {N}_\mathrm {inv}$ , $\ensuremath {\tau _\mathrm {1}}$ and $\ensuremath {\tau _\mathrm {2}}$ increase in contrast to high-fields, where the energy relaxation time remain constant.

Fig. 3.11 shows the energy relaxation time and the second-order relaxation time as a function of the kinetic energy of the carriers for low and high inversion layer concentrations considering and neglecting SRS. Due to the elastic nature of SRS, there is no change in the energy relaxation time and in the second-order relaxation time.

Figure 3.13: First subband occupation of the unprimed, primed, and double primed valleys as a function of the inversion layer concentration for fields of  $\mathrm {50 kV/cm}$ and  $\mathrm {100 kV/cm}$ . Due to the light mass of the unprimed valley in transport direction, the subband occupation number is higher than in the other valleys.

In Fig. 3.12, $\ensuremath {\tau _\mathrm {1}}$ and $\ensuremath {\tau _\mathrm {2}}$ as a function of $\ensuremath {N}_\mathrm {inv}$ for different lateral electric fields are presented. As can be observed, the relaxation times for high fields are constant except for a lateral field of  ${\mathrm{50}}{\;}{\mathrm{kV/cm}}$ . There is a significant change in the second-order relaxation time. This can be explained with Fig. 3.13, where the first subband occupation as a function of  $\ensuremath {N}_\mathrm {inv}$ of the unprimed, primed, and double primed valleys for lateral fields of ${\mathrm{50}}{\;}{\mathrm{kV/cm}}$ , and ${\mathrm{200}}{\;}{\mathrm{kV/cm}}$ is shown. Due to the fast increase of the occupation number of the first subband in the unprimed valley at  ${\mathrm{50}}{\;}{\mathrm{kV/cm}}$ compared to the high-field case, where the occupation is constant, the change in the second-order relaxation times increases as well for high  $\ensuremath {N}_\mathrm {inv}$ .

Fig. 3.14 demonstrates the mobilities in each valley and the total average mobility as a function of the channel thickness. The mobilities are indirectly proportional to the effective masses. As has been pointed out in Fig. 1.10, the velocity of the unprimed valley is high compared to the primed and double primed valleys. This is due to the light conduction mass of the unprimed valley and therefore the mobility is also high compared to the other valleys. The maximum peak in the total average mobility is due to a high occupation of the unprimed ladder for a thickness of ${\mathrm{2}}{\;}{\mathrm{nm}}$ as demonstrated in Fig. 3.15. By increasing the channel thickness to ${\mathrm{3}}{\;}{\mathrm{nm}}$ , the occupation of the unprimed valley decreases and the occupation of the primed valley with the heavy conduction mass increases. The total mobility curve has a minimum at about ${\mathrm{4}}{\;}{\mathrm{nm}}$ bulk thickness. At this value, the occupation of the primed valley has its maximum. After the occupation of the double primed valley starts to increase, the total mobility increases too, until the occupation numbers of each valley reach a saturation value. Furthermore, it has been reported in [155] that for ultra thin body devices the carrier mobility is proportional to the device thickness to the power of six, which is in good agreement with the results.

Figure 3.14: Mobilities of the unprimed, primed, and double primed valleys, and total average mobility as a function of the channel thickness. At about $\mathrm {3 nm}$ , there is a maximum in the total mobility. This is the point where the mobility of the unprimed valley is already high while the carrier mobility of the remaining valleys is low.

Figure 3.15: Populations of the unprimed, primed, and double primed valleys as functions of the channel thickness.

3.4 Comparison with Bulk Simulations

To point out just the influence of quantization on higher-order transport parameters, the subband results have been compared to three-dimensional bulk data. Surface roughness scattering has not been considered in these subband simulations.

In Fig. 3.16, $\ensuremath {\ensuremath {\mu }_{\mathrm {0}}}$ , $\ensuremath {\ensuremath {\mu }_{\mathrm {1}}}$ , and $\ensuremath {\ensuremath {\mu }_{\mathrm {2}}}$ are compared to bulk simulations with a doping of  ${\mathrm{10^{16}}}{\;}{\mathrm{cm^{-3}}}$ . As can be observed, the higher-order mobilities of the 2D electron gas are below the mobilities of the 3D bulk simulations, especially in the low field regime.

Figure 3.16: Carrier, energy-flux, and second-order energy flux mobility of SMC and bulk MC simulations. The mobilities obtained by bulk simulations are higher than in subband simulations. For high fields, the mobilities from subband simulations yield the same value as from bulk simulations.

Figure 3.17: Comparison of the energy relaxation time and second-order energy relaxation time using 2D SMC data and 3D bulk MC data. For high energies both simulations converge to the same value.

This behavior can be explained as follows: Due to Heisenberg's uncertainty principle, there is a wider distribution of momentum in the quantization area, because of a higher localization of the particles than in the bulk. Hence, there are more bulk phonons available that can assist the transition between electronic states. This will lead to an increase of the phonon rates and a decrease of the mobilities [161].

Due to the higher probability of scattering with phonons in the subband case, the energy relaxation time and the second-order energy relaxation time are lower than in the bulk as demonstrated in Fig. 3.17. This is also the case in the carrier temperatures as visualizes in Fig. 3.18.

Here, the subband carrier temperature is below the bulk temperature due to the fact, that an increase of the phonon scattering probability decreases the carrier temperature. However, for high energies, the relaxation times and the temperatures of the subband simulations converge to the bulk results due to the occupation of higher subbands.

Figure 3.18: Comparison of the carrier temperature using 2D SMC data and 3D bulk MC data. Due to higher phonon scattering in the 2D case the temperature is lower than in the 3D case.

Figure 3.19: Subband occupations as functions of the subband ladders in the unprimed, primed, and double primed valleys for electric fields of  $\mathrm {1 kV/cm}$ and  $\mathrm {100 kV/cm}$ . For high fields, the subband ladders in the primed and double primed valleys are occupied.

Fig. 3.19 shows the subband occupation as a function of the subband ladders for a low field and a high field. As pointed out, for an electric field of  $\ensuremath{E}={\mathrm{1}}{\;}{\mathrm{kV/cm}}$ , the first subband in the unprimed valley is highly occupied, while the ladders in the primed and double primed valleys are more or less unoccupied. The situation changes for a field of  $\ensuremath{E}={\mathrm{100}}{\;}{\mathrm{kV/cm}}$ . The carriers gain more energy, which results in the occupation of higher subbands. The occupation values of the first two ladders in the primed and double primed valleys are higher than in the unprimed valley.

A study of the influence of important inversion layer effects on higher-order transport parameters using the SMC method has been given. The investigations made in this chapter are based on a homogeneous bulk subband system, where all spatial gradients of the macroscopic transport models are negliable. The next chapter is devoted to higher-order transport models in real devices.