Anisotropic Quantum Corrections for 3-D Finite-Element Monte Carlo Simulations of Nanoscale Multigate Transistors

Anisotropic 2-D Schrödinger equation-based quantum corrections dependent on valley orientation are incorporated into a 3-D finite-element Monte Carlo simulation toolbox. The new toolbox is then applied to simulate nanoscale Si Siliconon-Insulator FinFETs with a gate length of 8.1 nm to study the contributions of conduction valleys to the drive current in various FinFET architectures and channel orientations. The 8.1 nm gate length FinFETs are studied for two cross sections: rectangular-like and triangular-like, and for two channel orientations: 〈100〉 and 〈110〉. We have found that quantum anisotropy effects play the strongest role in the triangular-like 〈100〉 channel device increasing the drain current by ~13% and slightly decreasing the current by 2% in the rectangular-like 〈100〉 channel device. The quantum anisotropy has a negligible effect in any device with the 〈110〉 channel orientation.

, [2]. These nanoscale device structures possess a very complex 3-D geometry created by the fabrication process flow [3]. The resulting irregular transistor shapes can be precisely described by the 3-D finiteelement (FE) method that is essential to determine quantum Manuscript  confinement, which will play a crucial role in carrier density distribution and carrier transport along the device channel [4].
In this paper, we report on anisotropic FE Schrödinger equation-based quantum corrections (QCs) incorporated into in-house 3-D FE Monte Carlo (MC) device toolbox [5], [6]. The MC transport engine has already included anisotropic bandstructure [6]- [8] using k-vector transformations [9], but the Schrödinger equation QCs were approximated by isotropic electron effective mass tensor (EMT) [5]. Here, we extend the calibration-free Schrödinger equation QCs into three separate valleys (see Fig. 1) using longitudinal and transverse electron effective masses. The QC approach in the 3-D MC device simulations is more efficient than the multisubband MC [10], [11], especially in 3-D simulations of multigate devices with cross sections in the range of 5-20 nm and at large applied biases when carriers undergo frequent intersubband transitions in addition to intrasubband ones while still delivering the expected predictive power [6], [12]. The developed 3-D FE MC toolbox with anisotropic FE Schrödinger QCs is applied to nanoscale n-channel Si Silicon-on-Insulator (SOI) FinFETs with a gate length of 8.1 nm designed following the International Technology Roadmap for Semiconductors (ITRS) specifications [13]. In order to fully exploit the capability of the anisotropic QCs, we consider two FinFETs with different cross sections: 1) rectangular-like [ Fig. 2 two different channel orientations: 1) the (110) top surface and (100) sidewalls referred to as the 100 channel and 2) the (100) top surface and (110) sidewalls as the 110 channel. While the impact of anisotropy in multigate transistors was investigated in the past [15], this new FE toolbox allows to determine how much improvement in performance can be related to the particular cross section/channel orientation and which cross-sectional dimension is critical to predict the accurate quantitative contribution to the electron transport.

II. 3-D MONTE CARLO SIMULATION TOOLBOX
The 3-D FE ensemble MC device simulation toolbox uses anisotropic nonparabolic bandstructure for transport [7] with all Si-related electron scattering mechanisms, including interface roughness and ionized impurity scatterings. More details on the 3-D MC transport model can be found in [5], [6], and [16]. The QCs, essential in nanoscale MOSFETs, are incorporated using the solutions of the 2-D FE Schrödinger equation [5] that need no calibration [17]. Initially, the 3-D FE MC simulation toolbox with the 2-D Schrödinger equation-based QCs [5] used an isotropic (scalar) effective mass. This implies that the same quantum potential is seen by all the particles independently of the orientation of valleys neglecting, thus the confinement-induced valley splitting [5], [12].
Here, we extend the 2-D Schrödinger-based QCs incorporating anisotropy by separating contributions according to the valley orientation [18]. The separate QCs for each valley accurately account for quantum confinement in nanoscale nonplanar Si channels, and have diverse effects in various device cross sections and channel orientations.
We start with a brief description of the simulation process shown in the flowchart in Fig. 3. At the beginning, we solve Schrödinger-Poisson equations at equilibrium to obtain initial distribution of particles depending on the valley population. The injection of particles in the source/drain is adjusted using a velocity-weighted Maxwellian distribution proportional to the valley population as well as considering the respective effective mass in the transport direction. The EMT is constructed depending on the valley orientation. In the 100 orientation, the ellipsoid principal axes of valleys  coincide with a device coordinate system. In the 110 orientation, the ellipsoid principal axes are not aligned with the device coordinate system, and therefore, a transformation of coordinates is performed [19]. When the bandgap is large enough (E g > 1 eV), the effective mass approximation accurately represents the complex bandstructure for energies close to the conduction band and, thus, holds well in the case of Si [20]. Table I shows the EMT as well as effective transport mass for the 100 and 110 channel orientations.
The 3-D FE mesh of the simulated device contains predefined 2-D planes perpendicular to the transport direction, where E is the energy,h is the reduced Planck's constant, is the 2-D wave function, and U (y, z) = −[qV (y, z) + χ(y, z)] is the 2-D potential energy with χ(y, z) being the electron affinity and q the electron charge [5]. The wave function is assumed to be zero at the outer boundary of the 2-D slice so it penetrates into surrounding oxide. The Schrödinger equation (1) is solved separately for each of the three valleys marked 1, 2, and 3, as shown in Fig. 1. The resulting wave function (Fig. 4) is then used to calculate the 2-D quantum density for each of the three valleys as where m * Tr is the electron effective transport mass, k B is the Boltzmann constant, T is the temperature, and E F n is the quasi-Fermi level. These separate calculations for each valley imply that the effect of valleys splitting with a distinctive population in each of the valleys is considered. The 2-D density is interpolated onto the 3-D simulation domain to obtain a separate QC potential for each valley as [5], [17] where n ien (r) is the effective intrinsic carrier concentration of electrons and holes and φ n is the quasi-Fermi potential for electrons. Finally, particles are moved in the quantumcorrected potential of the respective valley according to where t denotes time and k is the wave vector of the particle.

III. APPLICATION TO NANOSCALED Si SOI FinFETs
The developed 3-D FE MC device simulation toolbox with anisotropic 2-D Schrödinger-based QCs is then employed to study the performance of sub-10 nm gate length Si SOI FinFETs designed according to ITRS specifications [13] with two cross sections (see Fig. 2) and two channel orientations. Thanks to the FE simulation domains with 21 slices distributed along the transport direction for the 2-D Schrödinger-based QCs, the triangular-like and rectangularlike device geometries accurately follow realistic transistor shapes, including rounded corners due to etching processes used in their fabrication. These 8.1-nm gate length multigate FinFETs have the same channel perimeter of 26.5 nm giving an area of 49.5/29.6 nm 2 for the rectangular-like/triangular-like cross section, a high-K dielectric gate-stack with an equivalent oxide thickness of 0.55 nm, and a Gaussian n-type doping in the source/drain using a standard deviation (σ x ) of 2.61 nm [6]. To study the subthreshold slope (SS) (see Table II), we use a 3-D FE drift-diffusion transport model [6], because the ensemble MC is too noisy to accurately calculate very small currents and the source-to-drain tunneling is relatively small in these devices [12].
Figs. 5 and 6 show the I D -V G characteristics on logarithmic and linear scales at low and high drain biases of 0.05 and 0.6 V for the 8.1-nm gate length triangularlike and rectangular-like FinFETs, respectively, with the 100 and 110 channel orientations. The drain current is normalized to the gate perimeter (see Fig. 2). The drain current at V G = 0 V is very small and becomes visibly affected by statistical noise and numerical errors inherent to the MC technique [21]. Both triangular-like and rectangularlike devices deliver a higher current for the 100 channel orientation than for the 110 one at both low and high drain biases due to higher electron mobility in the 100 crystallographic orientation in the bulk Si despite that confinement changes the carrier transport to low-dimensional diminishing the difference. An important comparison is the difference of 20% and 33% for the rectangular and triangular FinFET, respectively, between the drive current of 100 and 110 orientations for the 8.1-nm gate length multigate transistors.  This difference is only between 10% and 12% for the devices with the gate lengths of 12.8 and 10.7 nm, respectively [12]. Note that for manufacturing, the 110 orientation is preferred, because it is largely beneficial to the hole mobility in p-type transistors assuming that the difference between the 100 and 110 orientations drive current is small. Table II shows electrostatic integrity characteristics for rectangular-like and triangular-like FinFETs illustrating the effect of anisotropic QC against isotropic QC. The threshold voltage (V T ) for each device variant is very close to 0.25 V. The triangular-like shapes exhibit a better SS and draininduced barrier lowering than rectangular-like ones thanks to a better confinement of the channel density (see Fig. 11). The drive currents for both devices are a bit reduced with anisotropic QCs, except for the triangular-like device in the 100 channel orientation. Fig. 5(a) and (b) shows the I D -V G characteristics for the triangular-like FinFET at low and high drain biases of 0.05 and 0.6 V, respectively, for both orientations comparing the anisotropic QCs against the isotropic one. In the 100 orientation, the ON-current is increased by 13% and 9% at low and high drain biases, respectively, while in the 110 orientation, the ON-current is very similar with isotropic and anisotropic QC at both biases. In addition, the anisotropic simulations give better SS as indicated by the blue dashed line segment of 60-mV/decade slope. The same comparison of I D -V G characteristics but for the rectangular-like FinFET is shown in Fig. 6(a) and (b). The current is reduced only by 2% when using more realistic anisotropic QCs at both low and high drain biases in contrast to the triangular-like shape transistors, which have a stronger confinement and a better gate control over the channel.  The acceleration is less pronounced under the gate due to enhanced phonon scattering at large kinetic energy. The velocity starts to saturate at the beginning of the gate due to an enhanced interface roughness and phonon scattering, especially, in the triangular device. Finally, it declines at the heavily doped drain due to a strong ionized impurity scattering coupled with phonon emission [7].
When using the anisotropic QC, the overall (sum of contributions from the three valleys) average electron velocity becomes slightly lower in both devices compared with the average velocity obtained from the simulations with isotropic QC, except for the triangular-like device in the 100 channel. In the 100 channel, the 1 velocity is the smallest in both device shapes, because the heaviest mass lies in the transport direction and a lighter mass along the confinement direction. The 2 and 3 velocities are equal in the rectangular-like device because of the shape symmetry. In the triangular-like 100 channel, the 2 velocity is larger than the 3 velocity, because the quantum confinement along 2 is the strongest. In the 110 channel, the situation is opposite with the 3 velocity being the largest in both the device shapes, because it has the lightest effective transport  mass. The 1 and 2 velocities are equal, because they have equal effective transport masses (Table I).
Figs. 9 and 10 compare the average valley population along the 100 and 110 channel orientations at V G = 0.8 V and V D = 0.6 V for the triangular and rectangular FinFETs, respectively, using the anisotropic QC. In the 100 orientation, the 1 valley is the most populated in the rectangular-like FinFET [ Fig. 10 (left)]. Although the 2 and 3 valleys have the same effective transport mass, the same population occurs only for the rectangular FinFET. In the triangular FinFET [ Fig. 9 (left)], the 2 valley is more populated than the 1, because it lies in a strongly confined y-direction that gives, along with the velocity profile, more current in the 100 triangular device. In the 110 orientation [Figs. 9 and 10 (right)], the 3 valley with the smallest effective transport mass has the smallest population in both devices, while the 1 and 2 valleys are equally populated. Fig. 11(a) and (b) shows the average electron density cross sections in the middle of the channel for both rectangular and triangular devices, respectively, at V G = 0.8 V and V D = 0.6 V. Note that a different scale for each device is used to show the contrast between high and low areas of density. In both devices, there is volume inversion. In the rectangular device [ Fig. 11(a)], the electron density is distributed mostly at the top and the bottom. The density in the triangular device [ Fig. 11(b)] is much larger and distributed toward the narrow top due to a stronger confinement.

IV. CONCLUSION
A new anisotropic QC using the solutions of 2-D FE Schrödinger equation on the slices along the channel of multigate transistors has been incorporated into the 3-D FE MC device toolbox [6]. The new 3-D FE MC device toolbox with the anisotropic QCs accounting for valley orientation has been then applied for a study of quantum anisotropy effects in the 8.1-nm gate length SOI FinFETs with the 100 and 110 channel orientations comparing rectangularlike and triangular-like cross sections. These two channel orientations and two cross sections allowed us to explore the significance of quantum confinement anisotropy in nanoscale multigate transistors. These findings elevate the triangular-like shape multigate transistors for sub-10-nm technology over the rectangular-like ones and point out loss of more than 30% of the drive current in the 110 channel transistors preferred for the integration into CMOS.