Chapter 8
Numerical Results

In the previous chapters comparisons for the proposed methods have mostly been given for a   +  +
n  nn   -diode. The aim of this chapter is to show that the proposed methods blend well and lead to considerable performance improvements for realistic device geometries. First, spatially two-dimensional simulations are presented for a MOSFET-like device. Then, it is shown that the combination of the proposed extensions to the SHE method allow for the simulation of a fully three-dimensional FinFET on an average workstation.

The average workstation is considered to be a machine equipped with a Intel Core i7 960 CPU, 12 GB RAM and a NVIDIA GeForce GTX 580. For simplicity, Dirichlet boundary conditions are used at the contacts and a parabolic energy band model is employed. Self-consistency of the Poisson equation, the SHE equations for electrons, and the drift-diffusion equation for holes is obtained by means of a damped Gummel iteration. The energy spacing is set to 12.5  meV, and acoustical, optical and ionized impurity scattering mechanisms are considered. For simplicity, doping profiles are chosen constant in each segment of the device. Note that doping profiles in realistic devices are smeared out due to unavoidable diffusion, which is beneficial for the numerical stability. Since the more challenging case of constant doping profiles is considered here, the results in this chapter can be transferred to realistic doping profiles.

The meshes used in the next sections are generated with Netgen  [91]. Simulator output is written to VTK files  [59], which are then processed for visualization by the open-source software ParaView  [58].


The adaptive variable-order scheme presented in Chap. 6 is applied to the simulation of an n  -channel MOSFET in unipolar approximation with a channel length of 22  nm. Simulations are carried out using the unstructured triangular mesh depicted in Fig. 5.7. The doping concentration in the source and drain regions are set to 1020   cm−3   , and to 1016   cm−3   in the channel and in the bulk regions. A high-k hafnium oxide with a thickness of 3.5  nm is used. Doping profiles change abruptly between the individual regions for simplicity. Note that this is not realistic from a technological point of view, and is usually more challenging for the numerical point of view. The source and the bulk contacts are grounded, while the drain contact is biased at 0.9  Volts. The gate voltage is set to 0.7  Volts. Due to the unipolar approximation with rather high doping in the channel region, there is a spurious current flow between the drain and the body contact. Nevertheless, the characteristic behavior in the channel is observed and is used for an evaluation.


Figure 8.1: Comparison of the convergence of a drift-diffusion simulation and the SHE simulation. The iteration for the SHE method saturates because of the discretization error of the potential-dependent grid in the total energy variable.

To obtain self-consistency with the Poisson equation, a damped Gummel iteration of a first-order SHE with 100  iterations and a damping coefficient of 0.6  is carried out. The discrete  2
l   -norm of the potential update vector in each iteration is depicted in Fig. 8.1. Comparable convergence behavior is obtained for both the drift-diffusion and the SHE approach. Note that the SHE simulation starts with a lower, yet significant initial update, because the result of the drift-diffusion simulation is used as initial guess for the SHE simulation. As the iteration progresses, the SHE method saturates, which stems from the discretization error with respect to the potential-dependent band edge influencing the simulation domain in (x,H )  -space, cf. Sec. 2.3. In any case, the potential update is essentially homogeneous over the device, hence the infinity norm of the potential update is around   − 5
10   .

(a) Electron density (cm− 3  ).
(b) Density of electrons above 1  eV (cm−3  ).

Figure 8.2: Macroscopic quantities obtained from a SHE simulation of the MOSFET. Densities are computed from a fifth-order SHE, while self-consistency is ensured for a first-order SHE only.

Simulated carrier concentrations are depicted in Fig. 8.2. Densities have been computed from a uniform fifth-order SHE, while Gummel iterations for self-consistency have been carried out for a first-order SHE only. The inconsistencies near the source and drain contacts are due to the lack of full self-consistency of the fifth-order SHE with the Poisson equation, which is amplified by the use of Dirichlet boundary conditions. Consequently, it is concluded that it is insufficient to ensure self-consistency with a first-order SHE only.

Inside the channel a pinch-off can readily be observed. Furthermore, the plot of the carrier density above one electron Volt shows the exponential increase of carriers with high energy along the channel. It has to be emphasized that such an information cannot be obtained from macroscopic transport models such as the drift-diffusion model. For the unipolar simulation of a MOSFET considered here, a high population of heated electrons is obtained all over the drain region, which is about ten orders of magnitude higher than in the source region.

(a) Potential (V).
(b) Distribution Function (a.u.).

Figure 8.3: Plot of the electrostatic potential and the normalized isotropic part f0,0   of the distribution function.

Fig. 8.3 shows a plot of the electrostatic potential and the energy distribution function. Due to the small device dimensions, carriers are injected quasi-ballistically into the drain region. This leads to high values of the distribution function at kinetic energies around 0.9  eV, which is the cause of the high concentration of electrons above 1 eV shown in Fig. 8.2. Furthermore, the Dirichlet boundary conditions at the drain contact lead to a boundary layer of the distribution function, which is a purely numerical effect. Therefore, other boundary conditions such as those proposed by Hong et al.  [42] should be used for predictive device simulation.

(a) Error indicator computed for the first adaption step.
(b) Expansion orders after the first adaption step.

(c) Error indicator computed for the second adaption step.
(d) Expansion orders after the second adaption step.
Figure 8.4: Error indicator and expansion order distribution after the first and second adaption step. At the contacts, the error indicator is set to − 10  and a first-order expansion is preserved. Here, the bulk is kept at a fixed first order, since the contribution to transport is negligible. The vertical axis denotes total energy.

(a) Even Unknowns.
(b) Odd Unknowns.

Figure 8.5: Comparison of the number of unknowns of the linear system. The use of adaptive expansion orders leads to a reduction of unknowns by about a factor 1.5  for a third-order expansion, and a factor of 2  for a fifth-order expansion.

The decay-based strategy as discussed in Chap. 6 is applied for an adaptive variable-order SHE using the same parameters as for the simulation of the n+nn+   diode to the MOSFET. Clearly, the obtained results are not the best possible, yet they allow for judging the need for parameter adjustments over a wider range of devices. Fig. 8.4 shows the expansion order indicators obtained from a uniform first-order SHE and from the subsequent variable-order simulation. The expansion orders after the first and the second adaption step are particularly increased near the band-edge and around the channel. In the drain region, a hot energy tail of the distribution function leads to an increase of the expansion order above 0.8  eV only. At very high energies, a first-order expansion is preserved. It is worthwhile to note that almost all third-order expansion nodes become fifth-order expansion nodes after the second adaption step. Additional savings from a better parametrized error indicator are therefore expected. Nevertheless, the number of unknowns in Fig. 8.5 is reduced by a factor of up to two for fifth-order expansions, and execution times are reduced by a factor of almost five due to the sparser population of the system matrix at lower expansion orders, cf. Fig. 8.6. The high gain in execution time may also be due to better caching possibilities on the CPU, because data operations are more localized and thus less cache misses occur. Performance gains from a parallelization are comparable to the results in Sec. 7.4. It has to be mentioned that a uniform fifth-order SHE of the MOSFET leads to a linear system and a preconditioner which is too large in order to fit into GPU RAM. Therefore, additional fallback-mechanisms are required when using GPU acceleration with SHE.


Figure 8.6: Comparison of execution times for the preconditioner setup and the linear solver run using eight CPU threads. The sparser structure of the system matrix leads to a reduction of execution times by a factor of up to four.

(a) Average carrier energies (energy in eV, scale in nm).
(b) Comparison of the relative error of carrier energies.

Figure 8.7: Average carrier energy in the simulated MOSFET. A fifth-order uniform expansion is taken as reference for the comparison of the errors along a line 2  nm below the gate oxide, with x = 0  corresponding to the center of the channel. The convergence of the SHE method can readily be seen and adaptive expansions essentially agree with corresponding uniform expansions.

A comparison of the average carrier energy in Fig. 8.7 again confirms convergence of the SHE method and further shows that adaptive expansion orders lead to accuracy comparable to uniform expansions. The kinks in the error function are due to the interpolation of the solution quantities along a line passing through the mesh at a depth of 2  nm below the gate oxide. It is crucial to note that unlike in the case of the n+nn+   diode simulated in Sec. 6.3, a first-order SHE leads to a significant error in the average carrier energy, thus confirming that higher-order SHE is indeed required for modern scaled-down devices.

8.2 FinFET

While transistors have been fabricated as planar devices over decades, the small feature sizes of modern devices leads to the undesired effect that the gate loses control over the current flow. One of the possible remedies currently employed is the use of fully three-dimensional device layouts, such that the channel is fabricated as a fin, and the gate is wrapped around the fin in order to have better control over the current flow. A schematic view of such a so-called FinFET is given in Fig. 8.8 and investigated in the following. Since three faces of the channel are used to control current flow, the device is sometimes also referred to as trigate transistor.


Figure 8.8: Schematic view of the simulated FinFET geometry. The gate is shown in blue, the gate oxide in green, the fin in red and the substrate in yellow.

Due to symmetry considerations along the fin, it is sufficient to simulate only one half of the device. The simulated device has a channel length of 18  nm and the mesh used for the simulation is depicted in Fig. 5.8. A constant doping of   20
10   cm−3   is applied in the source and drain contacts, while a doping of   12
10   cm−3   is applied in the channel and in the bulk. Again, the device is simulated in an unipolar approximation. The source contact is grounded, the drain contact is biased by 0.3  Volt, a gate voltage of 0.8  Volt is applied, and the bulk contact is grounded. Due to the difference in the built-in potential, the Dirichlet boundary conditions for the potential at the bulk contact are set to − 0.5  Volt. The results presented in the following are obtained from a uniform first-order SHE, and at the end of this section an adaptive SHE up to third order is discussed.


Figure 8.9: Carrier density (in cm− 3   ) in the simulated FinFET. The color scale is truncated below 1015   cm−3   in order to have a higher resultion in the channel.

Fig. 8.9 shows the carrier density inside the device. The accumulation of carriers near the gate oxide is clearly visible, especially along the oxide on top. In the center of the channel, a region of lower carrier density is obtained.


Figure 8.10: Density of carriers with energy above 1  eV (in cm−3   ) in the simulated FinFET.

The density of carriers above 1  eV shown in Fig. 8.10 shows a high concentration of hot carriers in the source and drain region towards the channel. The peak concentration is obtained at the beginning of the gate oxide in the source region.


Figure 8.11: Potential in the simulated FinFET. The reference potential is the built-in potential in the source and drain region.

The potential distribution inside the device is depicted in Fig. 8.11, which also depicts the potential inside the oxide. Note that the oxide also extends towards the bulk, where three times the thickness around the fin is chosen. However, the oxide grown on the bulk does not contribute to carrier transport significantly.


Figure 8.12: Average carrier energy (in eV) in the simulated FinFET.

Average carrier energies are shown in Fig. 8.12. In contrast to the MOSFET with a high drain-source bias, the low drain-source bias for the FinFET leads to high average carrier energies mostly in the source close to the gate oxide. The elevated carrier energies deeper in the source are due to the high bulk bias.


Figure 8.13: Average expansion order after the first adaption step in the simulated FinFET.

While the uniform first-order SHE has lead to 443500  even-order unknowns leading to a memory consumption of 5  GB, a uniform third-order expansion results in 3976475  even-order unknowns and a memory consumption beyond 20  GB. Since this is beyond the memory provided by the workstation, only an adaptive SHE up to third-order is carried out. Using the same adaptive strategy based on the rate of decay of the expansion coefficients as for the MOSFET, 2242040  even-order unknowns are obtained, which is still too large for the workstation. Reducing the threshold value from − 6.0  to − 1.0  and clamping the bulk and the interior of the channel to first-order results in 860145  unknowns, which just fits into the 12  GB budget. The expansion order averaged over the full energy range is depicted in Fig. 8.13. The adaptive strategy readily resolves the crucial regions underneath the gate oxide, even though a rather high threshold is chosen.