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Tips & Tricks
4. Device Simulation of Stress Effects in Front-End-of-Line Structures

4.1 Loading Stress
4.2 Coordinate System for Stress
4.3 Impact of Stress on Band Structure
4.4 Impact of Stress on Quantization Models
4.5 Stress-Dependent Mobility Models
4.6 Impact of Stress on Saturation Velocity
4.7 Discretization Options for Tensor Mobility Models
4.8 Plotting Stress-Dependent Mobility and Factors

Objectives

To address common questions and issues related to modeling the impact of stress in device simulations.

4.1 Loading Stress

In a typical TCAD flow, the stress tensor throughout a device is computed during the process simulation and is saved to a TDR file along with the device mesh and other quantities such as the doping. Then, Sentaurus Device loads the mesh and doping profile by specifying this TDR file for the Grid file within the File section, while the stress tensor is loaded by specifying the Piezo file within the File section:

File {
  Grid = device.tdr
  Piezo = device.tdr
}

If the Piezo file is not specified, no stress is loaded. As shown in this example, the Grid file and the Piezo file are often the same.

4.2 Coordinate System for Stress

The stress tensor is a 3x3 symmetric tensor and is defined by default within the simulation coordinate system. In a typical TCAD flow, the stress tensor is computed during the process simulation for a device with a particular orientation with respect to the crystallographic coordinate system. In Sentaurus Device, this simulation coordinate system must be specified using the LatticeParameters section of the parameter file. Typically, this section specifies the directions of the x- and y-axes of the simulation coordinate system relative to the crystallographic axes. Then, the direction of the z-axis is computed automatically. Some common examples are:

FinFET with a (110) sidewall and <110> channel in which the x-axis is along the channel direction, the y-axis is perpendicular to the sidewalls, and the z-axis is along the height of the fin:
```
LatticeParameters {
  X = ( 1,1,0)   #[1]
  Y = (-1,1,0)   #[1]
}
```
FinFET with a (100) sidewall and <100> channel in which the x-axis is along the channel direction, the y-axis is perpendicular to the sidewalls, and the z-axis is along the height of the fin:
```
LatticeParameters {
  X = (1,0,0)   #[1]
  Y = (0,1,0)   #[1]
}
```

4.3 Impact of Stress on Band Structure

Stress modifies the band structure of silicon by splitting the edges of the valleys and bands that make up the conduction and valence bands. Stress also can modify the shape, or dispersion, of the bands resulting in a change in the density-of-states (DOS). Together, these effects modify the threshold voltage in a MOSFET.

Options are available for considering the impact of stress on band structure. The most physical of these options treats the impact of stress on both the band edges and the dispersion, and is selected using the following syntax in the Physics section:

Physics {
  Piezo (
    Model (
      DeformationPotential(ekp hkp minimum)
      DOS(eMass hMass)
    )
  )
}

This set of models computes the level shifts to the conduction valleys and valence bands using the k·p models, taking the bottom-most conduction valley and the topmost valence band to define the band gap. The stress dependency of the DOS mass is considered as well.

4.4 Impact of Stress on Quantization Models

Quantization models, such as density gradient, consider the impact of stress primarily through changes in the band edges as specified by the DeformationPotential model. Some models have options for considering additional impacts of stress on quantization. Specifically for the density gradient model, there is an option to use the strained DOS mass in the density-gradient equation. However, it is recommended to use the default setting for this option in which only the relaxed DOS mass is used. Comparisons to detailed solutions of the Schrödinger equation show this setting produces better agreement for the threshold voltage.

4.5 Stress-Dependent Mobility Models

The mobility gain due to stress in a MOSFET channel depends strongly on the surface and channel orientations. In a FinFET device with multiple surface orientations for the top and sidewalls, it is convenient to use models that treat this orientation dependency automatically. This section provides comments and recommendations for these types of mobility model.

4.5.1 Subband Stress-Mobility Models for Electrons and Holes

The subband stress-mobility models work as corrections to the relaxed low-field mobility, including automatically accounting for different surface orientations. In particular, if in a FinFET, the Lombardi or IAL mobility model is selected to compute the relaxed low-field mobility with the AutoOrientation option, the subband stress-mobility models will, by default, automatically compute the stress-induced mobility gain for the different surface orientations in the fin.

For some surface orientations, most notably (110), the relaxed mobility is anisotropic and depends on the channel direction. It is necessary, therefore, to indicate to the subband models which channel direction to use. To select the required channel direction, specify either RelChDir110 for a <110> channel, or -RelChDir110 for a <100> channel, in the (e|h)Subband option list. RelChDir110 is selected by default.

To enable the subband stress-mobility models to work correctly for a MOSFET channel (not only for the bulk), the statement MultiValley(MLDA kpDOS) must be specified in the Physics section. This specification also activates the multivalley MLDA quantization model. If you want to use density gradient as the quantization model, the keyword -Density also must be specified to limit the use of the multivalley MLDA model to only the calculation of mobility.

Below are recommended models and options (which are verified and calibrated for multiple orientations and stress cases) to specify in the Physics section to activate the electron and hole subband stress-mobility models with density gradient as the quantization model and for a <110> channel direction:

Physics {
  MultiValley(MLDA kpDOS -Density)
  Piezo (
    Model (
      Mobility (
        eSubband(Fermi EffectiveMass Scattering)
        hSubband(Doping EffectiveMass Scattering(MLDA))
      )
    )
  )
}

The calculation of the hSubband model can be time consuming because it treats the strained valence band structure in great detail for each mesh vertex. To speed up the model, it is recommended to use the Doping option (compared to eSubband), which still gives good accuracy.

In addition, the hSubband model can require a long initial precomputation step before the start of any bias ramp. In this precomputation step, all the needed energy-dependent band- and mobility-related data is computed for each mesh vertex. Fortunately, this precomputation step lends itself well to parallelization and can be accelerated using a large number of threads.

The cost of this precalculation step can be spread over multiple runs of Sentaurus Device on the same device structure using the MVMLDAcontrols save/load feature. In this approach, the precomputation step is performed in a separate Sentaurus Device run with a large number of threads by doing only one Poisson solution and then saving the computed band-related data to a file using:

Math { MVMLDAcontrols(save= "file_name") }

In subsequent Sentaurus Device runs, the initial hSubband precomputation step can be avoided by loading this file using:

Math { MVMLDAcontrols(load= "file_name") }

4.5.2 Piezoresistance Model for Electrons and Holes

The piezoresistance model is a straightforward model in which the mobility gain is computed as either a linear or quadratic function of the stress, depending on whether the first-order or second-order option is selected. The appropriate piezoresistance coefficients are specified in the crystallographic coordinate system and are transformed by the tool to the simulation coordinate system.

When this model is applied to a FinFET channel, the surface orientation dependency of the stress-induced mobility gain and, therefore, the piezoresistance coefficients, should be considered. These coefficients can be specified for different surface orientations and used in conjunction with the AutoOrientation framework. When using the piezoresistance model in a FinFET channel, you must consider the following as well:

Unlike in a highly doped source/drain region in which the stress-induced mobility gain is suppressed due to high doping, the mobility gain (or reduction) in a MOSFET channel can become relatively large. At large stress, this gain typically saturates at a value that depends on the carrier type and surface/channel orientation. To accurately model this behavior, the MinStressFactor and MaxStressFactor parameters in the Piezoresistance section of the parameter file must be used.
The piezoresistance model is based on a bulk model, which for silicon means the resulting mobility reflects the cubic symmetry of relaxed silicon, that is, some surface/channel orientations in a MOSFET cannot be completely characterized by this model. In particular, for a (100)/<100> channel, you cannot model the mobility gain due to both transverse stress and vertical stress with this model.

4.5.3 MCmob Model for Electrons and Holes

MCmob is a nonlinear piezoresistance model that considers the impact of the main types of uniaxial stress on mobility: longitudinal, transverse, and vertical stress. This model is calibrated against reference data generated either by Sentaurus Device Monte Carlo or by the mobility calculation feature of Sentaurus Band Structure for a select set of surface/channel orientations. This model is compatible with the AutoOrientation framework.

For details and recommendations for the MCmob model, refer to the Advanced Calibration for Device Simulation User Guide.

4.6 Impact of Stress on Saturation Velocity

In general, stress can impact both low-field and high-field mobility. The impact of stress on high-field mobility is primarily through the effective saturation velocity. Detailed device Monte Carlo studies of bulk silicon have shown that the stress dependency of the saturation velocity is rather weak. Therefore, for long-channel devices, it is recommended to suppress any change in saturation velocity due to stress. This can be achieved by setting the SaturationFactor parameters in the Piezo statement of the Physics section to 0.0 (by default, they are set to 1.0):

Piezo (
  Model (
    Mobility (
      eSaturationFactor=0.0
      hSaturationFactor=0.0
    )
  )
)

For short-channel devices, quasiballistic transport is expected to play a larger role in device behavior. Device Monte Carlo studies have shown that quasiballistic transport is affected by stress. While drift-diffusion simulations cannot physically model quasiballistic transport, the effect of quasiballistic transport on device behavior can be approximated by changing the saturation velocity.

The effect of stress on quasiballistic transport can, therefore, be modeled as a stress dependency of the saturation velocity. For strained short-channel devices, the SaturationFactor parameter then becomes a fitting parameter, and a value larger than 0 might be more appropriate. Alternatively, the saturation velocity itself can be tuned to account for the combined impact of gate length and stress on quasiballistic transport.

4.7 Discretization Options for Tensor Mobility Models

Some of the stress-dependent mobility models produce a tensor mobility. There are several options for selecting how such models should be treated numerically and, in particular, how they are discretized onto the device mesh. For 3D devices, it is recommended to use the TensorFactor option in the Math section to improve stability and speed. This can be activated using:

Math { StressMobilityDependence=TensorFactor }

4.8 Plotting Stress-Dependent Mobility and Factors

With StressMobilityDependence=TensorFactor specified in the Math section, several field quantities related to stress-dependent mobility can be saved in the plot file as follows.

Plot name	Description
`(e\|h)MobilityStressFactorXX` and so on	Low-field mobility gain due to stress.
`(e\|h)TensorMobilityFactorXX` and so on	High-field mobility gain due to stress.
`(e\|h)TensorMobilityXX` and so on	High-field mobility due to stress.

When StressMobilityDependence=TensorFactor is specified in the Math section, only the diagonal components of these field quantities are relevant.