РГЗ 7,8 лабы / Варианты РГЗ / #16 / Word / p8_news_2009_06

Contents

8

Band-Structure Calculation and the

Fundamentals of Strain Engineering

Latest Edition

Over the past three months, there are signs that the economy has hit bottom. However, the business outlook of the overall environment remains mixed. Despite the continued weakness in the economy, many semiconductor companies are persevering and remain committed to the research and development of new process technologies and products. Driven by the need to reduce product development costs, our customers continue to rely on TCAD tools to help achieve their product development targets withfewerengineeringwafersandinashorter period of time. In keeping with this mission, I am pleased to announce the release of TCAD Sentaurus Version C-2009.06, which includes significant enhancements to keep pace with the challenging semiconductor technology roadmap.

The demand for 3D simulation continues to increase as our customers tackle the tough challenges of designing new-generation devices and optimizing current-generation devices. Described in the main article of this editionofTCADNewsaretheenhancements in 3D geometry operations and meshing for handling complex device and cell geometries.Inaddition,newmodelscovering the latest advances in process and device technologies, as well as key improvements to the numeric solvers and new usability features, are introduced in this article. These enhancements enable exploring emerging devices such as tunneling FETs and phasechange memory, as well as studying the impact of physical effects such as random dopant fluctuation and device reliability. The second article highlights the recent development of a band-structure calculator, which is proving to be a very useful tool for investigating and understanding the effect of stress on band structure and electronic transport in silicon. The band-structure calculator is available in Sentaurus Device Monte Carlo Version C-2009.06.

While we await a rebound of the economy, I trust that you will enjoy reading about these exciting developments, compliments of our TCAD team, and using these new enhancements in TCAD Sentaurus Version C-2009.06. As always, I look forward to your comments and feedback.

With warm regards,

Terry Ma

VP Engineering, TCAD

Contact TCAD

For further information and inquiries: tcad_team@synopsys.com

effect of damage, the coimplantation model is disabled automatically when the PAI model is active.

~ I–V Profile Shift ~

Traditionally, interstitial and vacancy profiles after implantation are obtained by multiplying the as-implanted dopant profile by ifactor and vfactor, respectively. Yet this simple approach ignores the fact that, due to momentum transfer, the interstitial distribution is slightly deeper than the vacancy distribution. This release now allows users to shift pointdefect profiles with a shift vector (three component double arrays: frenkel.pair. offset, i.plus.offset, and v.plus. offset). These parameters can be applied to both Monte Carlo and analytic implants, and offer great flexibility in modeling the spatial separation of interstitial and vacancy profiles.

~ Oxygen Knock-on Effect in Monte Carlo Implantation ~

Monte Carlo implantation in Sentaurus Process provides a general model for recoil implantation, such as the oxygen knock-on effect. Recoil implantation is handled in the same way as in the cascade damage model, except that no vacancies are created at the displaced lattice sites and the recoil species are not recorded as interstitials when they stop. Instead, a new dataset is created for the recoil species. Oxygen is a predefined recoil species, and the oxygen knock-on effect can be simulated by simply specifying the parameter recoils in the implant command.

~ Dose Split Model in Monte Carlo Implantation ~

In conventional pseudoparticle Monte Carlo implantation, all particles have the same weight. In contrast, a new dose split model uses a “smart” weighting algorithm, which attributes a lower weight to earlier particles relative to later particles. This prevents the crystal from amorphizing prematurely, allowing more particles to undergo channeling. This model can drastically reduce the noise in the channeling tails. The dose split model is switched off by default and is activated by the command pdbSet MCImplant DoseSplit 1. The dose split model is especially effective for high-dose amorphizing implants, such as an arsenic implant with a dose of 8x1015 cm–2. For a typical run with the same number of particles, the dose split model is approximately 2 to 3 times slower compared to the conventional approach. However, the improvement in the statistics of the channeling tails is at least two orders of magnitude. To achieve the same statistical significance in the channeling tail, the conventional approach requires at least 100 times more particles, making the effective speedup when using the dose split model approximately 30 to 50 times.

~ Miscellaneous Improvements in Monte Carlo Implantation ~

Other improvements addressing Monte Carlo implantation in Version C-2009.06 include:

• Damage accumulation in polysilicon. The same damage accumulation model as in silicon is implemented in polysilicon. However, this model is switched off by default and is activated by the command pdbSet PolySilicon Damage 1.

•Ge effect on implantation in polysilicon. The impact of high Ge concentrations on subsequent implants is now considered.

•Point implant. An implantation can be

performed with all the particles entering a central location in the device surface. This fictional implant mode is useful for examining ion-channeling behaviors in crystalline materials.

•Randomize implant. This feature, which allows users to obtain different results

(due to different random seeds) without modifying the input file, is useful for statistical analysis.

New Features in Sentaurus Process Kinetic Monte Carlo

~ New Charged Clusters and Reactions ~

Starting with this release, impurity clusters are no longer restricted to be neutral. A new parameter, e0_Complex, accounts for the cluster charge, which is used in two different ways:

•First, the charges from clusters and substitutional dopants are both considered when calculating the Fermi level.

•Second, the cluster charge is used to include corrections to satisfy charge neutrality when the cluster traps or emits mobile species.

The limitation on impurity clusters emitting and accepting only neutral particles or pairs has been removed as well. For example, for a B2I2 cluster, previously it could only break as

B2I2 ↔ Bi0 + Bi0. The new paths Bi+ + Bi–, Bi+ + Bi0, and Bi0 + Bi– are now accessible.

Some possibilities, like Bi– + Bi–, are still forbidden given that they are electrostatically repulsive. Microscopic reversibility is very carefully maintained: If the Bi+ + Bi– reaction is allowed, B2I2 can break that way as well.

In cases where the reactions do not conserve thecharge,thesimulatorincludesautomatically a Fermi level–dependent term in all captures and emissions. For example, if B2I2 is defined as negative, the reaction B2I2– ↔ Bi+ + Bi– is not neutral. The simulator automatically takes care of the recombined e– on the righthand side of the reactions and also properly accounts for the potential energies of Bi– and Bi+. This introduces local dependencies on the reactions, in contrast to previous versions where, with the exception of stress and SiGe dependencies, the emission rates of impurity clusters for a given particle were global.

~ New Mixed Clusters ~

Several studies have involved the formation of mixed clusters, that is, more than one dopant grouped together with point defects. In particular, [5] uses AsPV and AsPI clusters, while [6] discusses FB clusters, and [7] suggests a contribution of the BCI cluster, which some studies [8] show as stable. Consequently, the impurity cluster model implemented in Sentaurus Process Kinetic Monte Carlo (Sentaurus Process KMC) and the model explained in the first section have been rewritten from the beginning to include generic mixed clusters.

In this way, users can perform atomistic simulationstofurtherelucidatethecontribution of these complex mechanisms. For example, as shown in Figure 7, the definition of BCI allows users to change the active boron by having the carbon dose fixed at 1x1015 ions/ cm2 and the BCI potential energy vary from –4.6 to –5.0 eV. The reduction in active boron implies an increase in the BCI concentration.

Figure 7. Different boron activations produced by varying the potential energy of a BCI cluster after carbon coimplant; carbon dose is fixed at 1x1015 ions/cm2.

A generic mixed cluster is any combination of dopants with or without point defects. In Sentaurus Process KMC, its definition is straightforward: To define PAs2V, both the As and P clusters must be switched on, and the activation energy, capture volume, charge and emission prefactors for PAs2V must be defined.

Upon specifying the reactions leading to the cluster (such as PV+As2, PAs+AsV, and PAs2+V), the model is ready to use.

This model also incorporates the features previously explained; the mixed clusters and the incoming and emitted particles do not need to be neutral. Finally, the development of this model allowed some reorganization of the impurity cluster parameters. In particular, the distinction between I and V impurity clusters has disappeared, together with the necessity (and potential mistakes) of repeating the pure dopant-cluster parameters (for example, potential energies of B2 or As4). Some cluster emissions, not previously considered due to these distinctions between I and V clusters, are now included, for example, As4 → Asi + As3V, B2 → Bi + BV, F3 → F3V + I.

~ Oxidation-enhanced Diffusion ~

An oxidation-enhanced diffusion (OED) model has been included in Sentaurus Process KMC, which changes the injection of point defects from the oxidizing interface, with the limitation that there is no boundary movement. This model is based on a similar continuum model [9]. In particular, the new model assumes that:

→ →

j · n = Ks([I ] − [I ]*) − Gox

whereKs=1/6νm(I)λ2/Lr, νm(I)isthemigration frequency for I, λ is the jumping distance, and Lr is the recombination length. The OED part is included in the term Gox, defined as:

where:

→

•  θ, Eθ, Vθ, Vscale, and Gpow are parameters.

•  V→ox is the oxidation speed given by the continuum solver.

•  → is the vector normal to the interface. n

•  Gscale is a term defined to account for Fermi level defects.

~ Extension of Allcharges Model ~

The model Allcharges, previously used for point defects only, has been extended to include all impurities. This model applies to the interaction and reemission of particles from interfaces.

In particular, Interface implements a three-phase segregation model allowing the emission and capture of neutral point defects and impurities only, which does not explicitly include Fermi-level dependencies; while the Allcharges model accepts all incoming particles regardless of their charge. In equilibrium, both models produce similar concentrations but Allcharges is typically faster in reaching equilibrium.

~ Parallel Charge Model ~

The computation of doping concentrations that is run periodically during KMC annealing has been parallelized. This doping is used to update the local Fermi level and the rates of several simulation events. This performance improvement can be switched on by setting math numThreadsKMC=<n>, where n is the number of required threads.

For example, n = 4 speeds up the charge model by a factor of 2 (in systems with four or more CPUs). However, since the charge model is only a small percentage of the whole KMC simulation, the improvement may be unnoticed except in simulations performing several charge updates, as is the case of simulations with temperature ramps.

~ Improved Support for Save and Reload ~

The command struct has been modified to include KMC restarting information, which minimizes effort when switching between KMC and continuum simulations, especially in Sentaurus Workbench projects running several splits.

Performance Improvements in Sentaurus Process

~ Numeric Methods ~

With Sentaurus Process Version C-2009.06, userscansignificantlyacceleratetheirprocess simulations, benefiting from improvements made in analytic 3D implantation, matrix and mechanics assembly, linear solvers, and parallelization.

A new modified Newton method for iterative linear solvers is available using the diffuse command. The full Newton method performs a matrix factorization at each nonlinear iteration, while the modified Newton tries to reuse one matrix factorization (or built preconditioner) for several nonlinear iterations. The full Newton method has a record of robustness, but the modified Newton method, based on the stable preconditioned iterative solver ILS, is fast and extremely robust.

Inert diffusion steps can also be accelerated using a newly developed ordering of mesh nodes and equations. These ordering algorithms improve the distribution of entries in the assembly matrices. For large structures (such as 3D meshes), they speed up the solution steps in serial and parallel modes. Table 1 shows a comparison with Version A-2008.09.

~ Multithreaded 3D Analytic Implantation ~

The major computational procedures of analytic implantation include retrieving the implant moments from implant tables, setting up the models (such as Gaussian, Pearson, or dualPearson)foreachmaterial,andcomputing the convolution integrals for all nodes in the device. Since generally the convolution integration is the most computationally intensive step, and the integration for each node is independent of each other, this is an ideal case for multithreaded parallelization.

In this release, 3D analytic implantation was parallelized using Intel® Threading Building Blocks, achieving good results. In certain cases, nearly ideal scaling with the number of cores or CPUs is achieved.

In typical situations, excluding the serial part, the scaling factor is approximately 2 times, 3.9 times, and 6.5 times for 2, 4, and 8 threads, respectively. For 8 to 16 threads, although scaling degrades from the ideal case, it is still not saturated. On a 16-core machine running 16 threads, a 35% speedup over 8 threads is achieved.

Multithreaded parallelization in 3D analytic implantation is enabled by specifying the command math numThreadsImp3d=<n>, where n is the number of threads and generally should be equal to the number of cores or CPUs available on a machine.

~ Performance Improvement of 3D Analytic Implantation in Serial Mode ~

The performance of 3D analytic implantation in serial mode has also been improved, ranging

Table 1. Performance improvement in 3D strained-silicon NMOS process simulation (RTP step, 3D mesh with 133K nodes, assembly matrices of size 732K with 14.9 million nonzero entries, time in seconds).

from 10% to 20% when compared to Version A-2008.09.

~ Performance Improvement of Monte Carlo Implantation in Serial Mode ~

In Version A-2008.09, multithreaded parallelization in Monte Carlo implantation achieved a significant performance improvement, with a speedup factor of approximately 9 to 12 times for 16 threads running on a 16-core machine. This release includes further performance improvements in serial mode without sacrificing scalability. Compared to Version A-2008.09, the speedup of Monte Carlo implantation in serial mode ranges from 25% for implantation into amorphous material to 45% for implantation into crystalline silicon. This model is now generally faster than Crystal-TRIM (which is also available in Sentaurus Process). For example, with identical implant conditions and comparable resulting dopant profiles, the Monte Carlo implantation model is approximately 5% to 10% faster for implants into crystalline silicon and is almost twice as fast as Crystal-TRIM for implants into amorphous materials.

Creating Meshes for Sentaurus Device within Sentaurus Process

The mesh engine Sentaurus Mesh is now fully integrated into Sentaurus Process, allowing the creation within Sentaurus Process of meshes optimized for device simulation upon completion of the process flow. Mesh refinement requirements are different for process and device simulations. For example, when using adaptive meshing during process simulations, typically the gradients of all dopants, clusters, and point defects are considered. However, for device simulations, only the gradients of the net active doping concentration are relevant. In process simulations, most interface boundaries must be resolved adequately to account for segregation effects and for pointdefect recombination. For MOSFET device simulations, on the other hand, mainly the gate oxide interfaces must be resolved very finely to capture the inversion layer as well as poly depletion effects. All other interfaces are less critical. Therefore, when creating a mesh optimized for device simulations, all previously defined lines and refinement boxes are deleted and a new device simulation–oriented meshing strategy is defined.

All other operations required to prepare a structure for device simulation can be performed in Sentaurus Process. These steps include:

• Adjusting the doping concentration in the polysilicon gate. This is a simple way to match poly depletion effects to C–V measurement data.

•Clipping the substrate. A deep substrate is needed in process simulations to account for the fast point-defect diffusion but, typically, it is not necessary in device simulations.

• Reflecting the structure. The symmetry of MOS devices can speed up process simulations by considering only half of the structure, but obviously the full structure is needed for device simulations.

•Assigning contacts. Electrical contacts can be defined directly in Sentaurus Process.

•Setting region names explicitly. This can be useful to activate some physical models only in certain regions of the device structure during device simulations.

To use this last capability, the region name must be known. However, Sentaurus Process automatically assigns unique region names as needed during etching and deposition operations. Therefore, the final name of a region is not always known a priori. To avoid manually checking the name of a given region, it is now possible to explicitly set region names in Sentaurus Process.

With Sentaurus Mesh integrated within it, Sentaurus Process can now export meshed structures directly to Sentaurus Device, avoiding the intermediate remeshing step fulfilled by Sentaurus Structure Editor in past versions and simplifying the simulation flow. An example illustrating this capability is distributed with Version C-2009.06 of TCAD Sentaurus.

Sentaurus Device

High-k Metal Gates

Sentaurus Device Version A-2007.12 included a mobility model with high-k insulator degradation terms to account for remote Coulombscattering(RCS)andremotephonon scattering (RPS). In this release, the model is enhanced to include numeric damping factors that ensure the mobility degradation is active only close to the relevant interfaces. These damping factors are functions of the distance to the silicon–oxide interface. Previously, a user-defined fixed offset parameter allowed the adjustment of the distance parameter to the actual distance to the high-k insulator– SiO2 interface inducing the remote scattering. However, this approach is limited to planar systems. The new model makes the damping factors dependent on the actual distance to the nearest high-k insulator–SiO2 interface. This makes the high-k insulator degradation model more flexible and easier to use.

Besides the mobility reduction due to RCS and RPS, the metal/high-k insulator–SiO2 gate stack alters the electrostatics in the channel. For example, Tatsumura et al. [10] showed that the formation of a dipole layer at the high-k–SiO2 interface can lead to a threshold voltage shift even without additional mobility-degrading scattering.

The task of simulating the effects of the dipole layer at the high-k–SiO2 interface can be divided into two independent parts:

•Given the material system and processing conditions, what is the resulting dipole layer (charge value and distribution as well as layer thickness)?

•Given a dipole layer, what are the electrostatic effects on the electrical characteristics of the device?

The physical origin of the formation of the electrical dipole layer at the high-k–SiO2 interface is the subject of active research (see, for example, [11][12]) and an accepted model suitable for TCAD simulation has not yet emerged.

However, the second simulation task can be fully addressed with TCAD Sentaurus. An efficient way to simulate the dipole layer is to introduce the dipole layer as an explicit region at the high-k–SiO2 interface and to apply opposing charges to the interfaces of this region. Figure 8 shows the TCAD Sentaurus

Dipole Layer Thickness [Å]

0.4

0.2

V [V] t

0

Dipole Surface Charge [cm−2]

Figure 8. Dipole layer–induced threshold voltage shift. Green circles: Threshold voltage shift simulated with Sentaurus Device as function of dipole surface charge for a dipole layer thickness of 5 Å. Purple circles: Simulated threshold voltage shift as function of dipole layer thickness for a dipole surface charge of 1013 cm–2. Red and blue lines correspond to results of the analytic model given in the literature [10].

dopant locations back into a doping profile that is usable by Sentaurus Device. This atomization/doping-assignment technique is very efficient and can be used to create an unlimited number of randomized structures.

The atomization process is accomplished by first obtaining the average number of dopants at each mesh node in the initial structure by multiplying the dopant density by the nodal volume. This average number of dopants is treated as the expectation value for a Poisson distribution random-number generator, which returns a new number of dopants to associate with each node.

These dopants are then distributed randomly within each nodal volume. The atomization process described here is statistically equivalent to the method used by Frank et al. [16] for creating discrete dopants from a continuous profile.

Sentaurus Mesh offers three dopingassignment methods to accommodate different approaches found in the literature for treating discrete dopants in device simulation programs:

•The Sano method [17] assigns a doping function to each discrete dopant that represents the long-range portion of the Coulombic potential of the particle.

•The nearest grid point (NGP) method assigns the doping of a particle to the nearest mesh node.

•The cloud-in-cell (CIC) method distributes the doping of a particle to the vertex nodes of the element in which the particle is located.

When the NGP method or CIC method is selected, device simulations typically include density-gradient quantum corrections to avoid nonphysical charge trapping caused by the sharply resolved Coulomb potentials associated with the discrete dopants [18].

For example, Sentaurus Mesh was used to create 100 randomizations of the L = 20 nm, W = 50 nm n-channel MOSFET shown in Figure 11.

The doping-assignment method for this example was CIC. Sentaurus Device was then used to simulate the Id–Vg characteristics for the randomized structures. For these simulations, density-gradient corrections for both electrons and holes were included. The results are shown in Figure 12.

Figure 13 shows the distribution of threshold voltages extracted from these results.

Tunneling FET Simulations

Band-to-band tunneling is one of the most important leakage-generation mechanisms in MOSFETs as aggressive MOSFET downscaling has continually increased the electric field inside transistors. Band-to-band tunneling is important especially in alternative channel materials such as Ge, InAs, and InGaAs that exhibit a small band gap and effective mass. Recently, tunneling FETs (TFETs) [19] have gained significant interest because of their potential for realizing a subthreshold slope less than 60 mV/decade. The device structure of TFETs is similar to that of NMOSFETs but a p+ instead of an n+ source is introduced to make a reverse-biased

DopingConcentration [cm−3]

4.4e+20

2.8e+17 1.8e+14

-2.0e+12

-3.3e+15

-5.2e+18

Figure 11. Initial continuous doping n-channel MOSFET: L = 20 nm, W = 50 nm.

Figure 12. Id–Vg characteristics for 100 randomized devices.

Threshold Voltage [V]

Figure 13. Threshold voltage distribution for 100 randomized structures.

p+–i–n+ diode whose tunneling barrier can be modulated by the gate electrode. The major problem of TFETs is relatively small oncurrents, and TFETs using different materials and heterostructures have been proposed to improve the current drive. To explore the device performance of TFETs, it is important that band-to-band tunneling models in device simulators account for the nonlocal tunneling process.

Sentaurus Device now supports a physicsbased nonlocal band-to-band tunneling model that accounts for both the coherent and phonon-assisted tunneling processes. Contrary to the tunneling at interfaces or contacts where the tunneling direction is unchanged during the simulation, the tunneling direction in band-to-band tunneling changes with bias conditions and must be defined dynamically. The model searches for the tunneling path and calculates the tunneling barrier with the following assumptions:

•The tunneling path is a straight line with its direction opposite to the gradient of the valence band at the start position.

•The tunneling energy is equal to the valence band energy at the start position and equal to the conduction band energy plus band offset at the end position.

•When the tunneling path encounters Neumann boundaries or semiconductor– insulator interfaces, it undergoes specular reflection.

Therefore, the model does not require user specification of the nonlocal mesh. The band- to-band tunneling generation rate across the tunneling barrier is obtained from the path integration, which extends Kane’s coherent band-to-band tunneling and Keldysh’s phonon-assisted band-to-band tunneling models [20] to arbitrary tunneling barriers. Up to three different tunneling processes can be considered simultaneously. In addition, mole fraction–dependent tunneling parameters can be defined in mole fraction–dependent materials.

As an illustration, a simulated double-gate (DG) TFET device is shown in Figure 14 and the simulated I–V characteristics are shown in Figure 15. Figure 16 shows the hole and electron generation rates inside the Ge source TFET.

Hydrogen Transport Effects of Device Degradation

Oxide degradation caused by bias and temperature stress is believed to be related to the silicon–hydrogen bond depassivation and the subsequent hydrogen transport that

Figure 14. Symmetric DG TFET structure

(LG = 30 nm, Lch = 20 nm, tox = 1 nm, and tSi = 10 nm).

Figure 15. Calculated Id–Vg characteristics of DG TFETs based on nonlocal band-to-band tunneling model. Introducing Ge at the source extension increases the drain current without enhancing the off-state leakage current.

Figure 16. Calculated hole and electron generation rates inside Ge source TFET when Vd = Vg = 2 V.

may involve chemical reactions [21][22][23]. Sentaurus Device now provides a general framework for hydrogen transport–related degradation models. This framework accounts for multidimensional drift-diffusion equations for hydrogen atoms (X1), hydrogen molecules (X2), and hydrogen ions (X3):

+ Rnet + ri ([Xi ] − [Xi ]0 ) = 0

where Di is the diffusion coefficient, Edi is the diffusion activation energy, KiQ is the number of charges, Rnet is the net recombination rate due to chemical reactions, ri is the explicit recombination rate, [Xi] is the volume density, and [Xi]0 is the reference density for element Xi. In addition, users can specify arbitrary numbers of interface and bulk reactions among mobile elements (hydrogen atoms, hydrogen molecules, hydrogen ions), electrons (X4), and holes (X5) defined by the following reaction equation:

where the nonnegative integers αi and βi are the particle numbers of element Xi to be removed and created by the forward reaction. Finally, the multistate configuration can be used to model reactions between the mobile hydrogen elements and localized hydrogen states such as silicon–hydrogen bonds at the silicon–oxide interface.

As an example, consider the degradation of a MOSFET due to negative bias stress using the generalized reaction-diffusion model.

In the model, the hole capture process breaks the silicon–hydrogen bond at the silicon–oxide interface into a hydrogen atom and positively charged silicon dangling bond. As the generated hydrogen atoms diffuse into the gate oxide, some of them change into hydrogen molecules, which can also diffuse into the oxide. As a boundary condition, the polysilicon–oxide interface is modeled as a hydrogen sink.

Figure 17 shows the calculated density of hydrogen atoms and hydrogen molecules in the oxide region and the density of generated silicon dangling bonds at the silicon–oxide interface as a function of stress time.

TCAD News June 2009

Enhanced Raytracer Features for Solar-Cell Simulation

Solar technology is emerging as an important growth sector in the semiconductor industry. Various enhancements have been added to Sentaurus Device to facilitate the design of modern solar cells.

Simulation of solar cells revolves around optics and transport physics. The optics is handled primarily with raytracing, and a compact memory model for the raytracer has been introduced. The compact memory model does not store the raytree as it is being created, so it relieves the use of significant memory. A reduced structure is used, and ingenious ways of extracting information from this reduced structure have been implemented without disturbing the multithreading feature of raytracing. Therefore, raytracing of 3D devices with many starting rays can be performed without requiring vast amounts of memory.

The raytracer has also been included in the unified interface for optical generation. Ramping of various excitation variables has been simplified, together with the feature of addingbackgroundopticalgenerationthrough file input. The excitation variables that can be ramped include intensity, wavelength, wave direction, and polarization. A new feature of enabling the input of a set of user-defined starting rays is available. This feature adds flexibility to input excitation, for example, users can now utilize a scattered set of starting rays from all directions.

One approach to model textured layer stacks, as they are used in modern thin-film solar-cell designs, is to use the transfer matrix method (TMM) inside the raytracer to calculate reflection and transmission coefficients at particular interfaces. However, some layers in the stack can be electrically active and can contribute to optical carrier generation (see Figure 21).Toaccountforsuchaphenomenon, the TMM contact in the raytracer has been modified to collect optical generations as rays traverse the TMM contact. The collected optical generations can then be distributed into specific regions of the electrical grid. A quantum efficiency parameter can also be set to fine-tune the absorption effectiveness of the thin-film layers.

Optical Generation [cm−3 s−1] 1.0e+22

1.0e+21

1.0e+20

1.0e+19

1.0e+18

1.0e+17

Figure 21. Modern solar cells contain absorbing thin-film material that generates electron–hole pairs. The optical generation profiles at different wavelengths are shown: (left) 0.3 µm, (middle) 0.5 µm, and (right) 0.7 µm.

The revised refractive index and extinction coefficient of silicon are plotted as functions of wavelength in Figure 22. These values are updated for the wavelength range of 0.84 µm to 10 µm to reflect the latest published measurement data.

All these enhancements to raytracing and silicon data offer greater versatility and accuracy for solar-cell simulations using Sentaurus Device.

Extended Precision

Simulating wide-bandgap semiconductor devices can be challenging due to extremely small intrinsic carrier densities. Frequently, artificial carrier generation is introduced to increase the leakage current level and to improve the numeric convergence. Starting

transport through the oxide layer. The hot carriers were simply sent to the closest vertex across the oxide layer. For small-memory devices, with increased electric fields and multiple charge-storage regions in the form of nanocrystals, the closest-vertex algorithm solution becomes inaccurate as it is a purely geometric approach.

Version C-2009.06 allows a more realistic approach for accurately tracing HCI currents. The oxide layer is replaced by a wide-bandgap semiconductor with insulator properties. The carrier transport equations are solved within the wide-bandgap semiconductor, and the injected hot carriers are converted into a current boundary condition at the interface. The hot carriers then follow the path towards the charge-storage regions (nanocrystals) as governed by the transport equations. This approach computes the full solution for the carrier distribution in the emulated oxide layer.

Inaddition,semiconductorregionswithcharge boundary conditions are now allowed inside wide-bandgap semiconductor regions. Such regions with charge boundary conditions can be used to model the nanocrystals as floating gates. Instead of solving the full transport equations inside the nanocrystals, only the simpler charge equilibrium equation is solved. These two new features can be used together for an accurate and robust simulation of hotcarrier tracing in memory devices.

~ Improvements in Box Method ~

A new algorithm to obtain box method measures and coefficients has been implemented in Version C-2009.06. The CVPL_Average algorithmisextremelystable and more accurate, especially for complex 3D meshes with a large number of flat elements per vertex. In a 3D simulation, the Voronoï box is a convex polyhedron. The new algorithm describes such a convex polyhedron as the intersection of multiple halfspaces. This approach leads to a more accurate and stable algorithm. The algorithm is available in both Sentaurus Device and Sentaurus Process.

~ Parallel Assembly ~

For Version C-2009.06, Synopsys has continued its efforts to provide incremental improvements in the parallelization of the matrix assembly. The domain decomposition approach previously introduced has been further refined to reduce overhead and to increase scalability. Furthermore, the Sentaurus Device code base has been reorganized to enable further improvements in the future.

~ FDTD for Oblique Incident Waves on Periodic Structures ~

Sentaurus Device Electromagnetic Wave Solver (EMW) has new enhancements to handle periodic simulations of CMOS image sensors and solar cells with oblique plane wave excitations. A sine–cosine algorithm has been implemented with an updated FDTD computation engine to simulate oblique periodic harmonic optical-generation results. Mur, Hidgon, and convolutional perfectly matched layer (CPML) boundary conditions are available with this release.

Usability Improvements in Sentaurus Device

~ Simplified Nonlocal Mesh Construction ~

Sentaurus Device offers a very versatile nonlocal tunneling model. This model relies on 1D meshes constructed internally by Sentaurus Device. Apart from the physical models, users can control the construction of these 1D meshes, which allows them to fine-tune the behavior and performance of the model. The price for this flexibility is that the specification of the 1D mesh puts a significant burden on users.

To relieve this burden, in this release, Sentaurus Device simplifies the specification of the 1D mesh for the typical application:

tunneling through an insulating barrier. In the new specification mode, users simply list the regions that belong to the tunneling barrier; they do not need to worry about interfaces, contacts, or the maximum length of 1D mesh lines. The physical models are specified as for the full 1D mesh specification mode. Therefore, users have the choice between the ease of use of the new specification mode and the flexibility of the full specification mode.

~ Continuation Method ~

The adaptive continuation method was introduced to accurately model complicated device phenomena such as breakdown in bipolar devices or latch-up in MOSFETs. The simulation of these phenomena usually involves tracking a multivalued curve in the I–V plane with abrupt changes and turning points. Adaptive continuation enables automated simulation along the curve, a capability not available with other methods.

In the adaptive continuation method, the simulation advances to the next operating point if a solution has converged and the angle between the last segment and the previous segment on the I–V curve is small. The step size along the I–V curve is adjusted based on the magnitude of this angle. Previous versions of Sentaurus Device computed the angle in the global I–V plane defined by the simulation window.Oneofthelimitationsofthisapproach was the wide distribution of the angle values across the simulation window due to the large range span along the I-axis. In this context, it was difficult to control the curve smoothness by global angle constraints, which in general are only optimal for a part of the simulation window. The current version allows the angle to be measured in a local scaled I–V plane instead of the global I–V plane. In this case, angle values become more uniform across the simulation window and relatively independent of the large span of the window along the I-axis.

Consequently,morepreciseusercontrolofthe step size in the simulation window using curve smoothness criteria is possible. In previous versions, users could control the maximum arc length variation allowed along the traced I–V curve by setting the MaxStep parameter. A smaller value for MaxStep would create a denser I–V curve. It was relatively difficult for users to guess a good MaxStep for arc length variations along the I–V curve based on variations of voltage or current at the adaptive continuation contact.

Two more practical parameters for limiting the maximum arc length variation are introduced in this release: MaxVstep and MaxIstep (or its variants in logarithmic scale). MaxVstep limits the allowed step (arc length variation) so that its projection on the V-axis does not exceed MaxVstep. A similar criterion is valid for MaxIstep.

The adaptive continuation method can also be used in a mixed-mode environment. All the device contacts, except the adaptive continuation contact, are allowed to be connected to a circuit.

~ Flexible Break Criteria ~

In addition to global break criteria in the Math section, Sentaurus Device supports sweepspecific break criteria. Instead of terminating the entire simulation, only the current sweep (Quasistationary or Transient command) is terminated, and Sentaurus DeviceproceedstothenextSolve command. In addition, the following new break criteria are available:

•Device power: This option is attractive for power devices, where the device power is more relevant than electrode currents.

•Mixed mode: This option is used to monitor node voltages and currents into a node, and to terminate a Solve command if predefined limits are exceeded.

~ Parameter Extraction ~

The extraction of MOSFET model parameters, such as threshold voltage and saturation current, is a frequent objective in device simulation. Until now, it was only possible to extract these parameters by launching an Inspect script after the completion of a Sentaurus Device simulation.

Starting with Version C-2009.06, all the Tcl extraction commands in Inspect are now also available directly in Sentaurus Device. To extract the threshold voltage, for example, users would first ramp the gate voltage to produce an Id–Vg plot. Within the Tcl command file of Sentaurus Device, the Inspect command f_VT can be invoked to obtain the threshold voltage.

Afterwards,thedevicesimulationcancontinue to extract additional MOSFET parameters as required. This new feature helps to simplify the complexity of Sentaurus Workbench projects. It can be used to extract many parameters within a single-device simulation, or the value of an extracted parameter can be used to guide subsequent Solve commands.

New Features in Sentaurus Band Structure

Sentaurus Band Structure now supports direct output of band-structure data in the file formats expected by Sentaurus MOCA. This includes writing of combined band energy and group velocity files, of lookup files for final-state selection in scattering processes (energy-orderedfiles),andofdensity-of-states (DOS) files. A new method computeDOS has been added to both the EPM::Crystal class and the AnalyticBandSolver class; in addition to the optional file output, DOS data is presented to the Tcl interpreter as a “list” object for further processing.

There is a new command for constructing k- vector lists that uniformly cover the irreducible Brillouin zone wedge of a diamond or zincblende crystal with strain-induced symmetry reduction.

EPM::AtomicSpecies has been extended to allow greater flexibility in specifying the local pseudopotential form factor. In addition to specifying sample values for a cubic spline or a1, …, a6 parameters for the Friedel interpolation formula, it is now possible to supply a user-defined Tcl procedure for the evaluationoftheformfactor.Formfactorvalues are cached internally to avoid performance degradation resulting from frequent calls to this procedure during Hamiltonian assembly.

Sentaurus Workbench

Usability Improvements

Sentaurus Workbench Version C-2009.06 introduces advanced view options to help users manage large projects.

Now, any part of the simulation flow can be hidden or displayed, for example, a complete tool step or a few parameters or variables. A new dialog box is provided to give full control of the project view.

Multiple options in View > Tree Options and View > Table Options display or hide the tool row, the tool comments row, the parameter row, the experiment numbers, and other parts of the project view.

There is also a choice between the traditional horizontalorientationofthesimulationflowand the new vertical flow representation, where the rows become columns (see Figure 25). Depending on the size and layout of a project, it may be more convenient to work with one or the other.

In addition, users can switch the project view from the traditional full mode to the compact mode, where only varying parameterization parts of the simulation flow with extracted variables are displayed, with all other parts of the flow hidden (see Figure 26).

The compact mode can be used to focus on the active parameterization part of the

project and can be useful for large design-of- experiments (DoE) projects.

Finally, users can utilize mouse operations to customize the width of all columns and the height of all rows in the project table by manually changing their size.

in Silicon by Carbon Implantation and Characterization of MOSFET’s with Carbon-Implanted Channels,”

IEEE Transactions on Electron Devices, vol. 44, no. 9, pp. 1544–1551, 1997.

Figure 25. Different parts of project view in vertical orientation.

Introduction

The periodicity of crystalline solids leads to a peculiar phenomenon: the formation of energy bands. Between bands, there may be energy ranges that do not contain any states (band gaps). A band gap between the highest occupied and the lowest unoccupied state is characteristic of semiconductor materials or (if the band gap is very large) insulators.

The dispersion relation of a band describes

the relationship between crystal momentum

≡ –

pcrys h k and energy E. It is the generalization of the ordinary relationship between kinetic energy and momentum E = ||p||2/2m to a particle moving in the periodic lattice potential. Group velocity v and reciprocal effective mass 1/m* (now a tensor quantity, not a scalar) of a particle in a crystal, respectively, can be obtained by taking first and second derivatives of the dispersion relation with regard to crystal momentum.

An external force F acting on a particle in a crystal changes its crystal momentum pcrys instead of its true momentum p:

p˙crys ≡ h−k˙ = F

Together, these relationships describe the propagation of a particle in a band subject to an external force in the absence of collisions (and quantum effects). Scattering processes limit the time over which the particle can be accelerated. This leads to the simplest possible model of carrier mobility μ, in which the drift velocity is determined by the effective mass m*, the particle charge q, the relaxation time τ, and the electric field E:

More detailed analysis will lead to more refined transport models, up to full-band device Monte Carlo simulations that use all the information contained in the band structure.

Empirical Pseudopotential Method

Sentaurus Band Structure is part of Sentaurus Device Monte Carlo. It implements bandstructure calculation for crystalline solids using the empirical pseudopotential method (EPM) [1][2][3] as well as analytic methods for valence bands (six-band k·p method [4]) and silicon-like conduction bands (ellipsoidal model; two-band k·p method [5]).

In the pseudopotential method, band energies and derived quantities are calculated directly from the shape of the unit cell of the crystal, and the positions and species of the atoms contained in it. The atoms, in turn, are characterized by pseudopotential

form factors. The pseudopotential operator combines the effect of the atomic nucleus and inner (“core”) electrons. Using it instead of the normal electrostatic potential in Schrödinger’s equation leads to correct valence energies but ignores uninteresting core states.

parameters are enough to obtain a reasonable approximation to the band structure of silicon [1]. For improved accuracy, Sentaurus Band Structure adds nonlocal correction and spin-orbit coupling terms [2][3]. Strain effects are accommodated by extending the local pseudopotential form factor from a set of isolated points to a function of crystal momentum transfer.

Band structures of alloys such as Si1–xGex are accessible using the virtual crystal approximation (VCA): Each lattice position is occupied by a statistical mixture of the constituent atomic species (atomic form factors are interpolated linearly in the mole fraction).

Sentaurus Band Structure also supports the calculation of complex band structures, which contain information about the evanescent states involved in quantum-mechanical tunneling.

Band Structure under <110> Uniaxial Stress

Recently, renewed interest in band-structure calculation was driven by the advent of strain engineering in semiconductor devices [6]. Previously,theemergenceofmechanicalstress in semiconductor processes was considered an undesirable source of reliability and yield issues. In contrast, strain engineering involves the intentional creation of large mechanical stress in order to improve transport properties. The effect of strain-enhanced mobility can be traced to the modification of the band structure under mechanical stress: Stress deforms an elastic material. At the microscale, this changes the shape of the unit cell of the crystal and, potentially, the arrangement of atoms within the unit cell [7]. The altered unit-cell layout translates to a changed band structure.

Forexample,Figure1showsenergyisosurfaces of the first valence band for both relaxed silicon and silicon under 2 GPa of compressive stress along <110>. In unstrained silicon, the energy surfaces have highly symmetric star shapes. In strained silicon, the symmetry