Impact ionization

CHANGELOG.md

@@ -17,6 +17,7 @@ and this project adheres to [Semantic Versioning](https://semver.org/spec/v2.0.0
 - Non-isothermal Charge simulations are now available. One can now run this type of simulations by using the `SteadyChargeDCAnalysis` as the `analysis_spec` of a `HeatChargeSimulation`. This type of simulations couple the heat equation with the drift-diffusion equations which allow to account for self heating behavior.
 - Because non-isothermal Charge simulations are now supported, new models for the effective density of states and bandgap energy have been introduced. These models are the following: `ConstantEffectiveDOS`, `IsotropicEffectiveDOS`, `MultiValleyEffectiveDOS`, `DualValleyEffectiveDOS`.
 - Added the Hurkx model for direct band-to-band tunneling `HurkxDirectBandToBandTunneling`.
+- Added Selberherr's model for impact ionization `SelberherrImpactIonization`.
 
 ### Changed
 - `LayerRefinementSpec` defaults to assuming structures made of different materials are interior-disjoint for more efficient mesh generation.

tests/test_components/test_heat_charge.py

@@ -2382,3 +2382,13 @@ def test_generation_recombination():
         E_0=1,
         sigma=2,
     )
+
+    # make sure we can build a SelberherrImpactIonization
+    _ = td.SelberherrImpactIonization(
+        alpha_n_inf=7.03e5,
+        alpha_p_inf=1.582e6,
+        E_n_crit=1.23e6,
+        E_p_crit=2.03e6,
+        beta_n=1,
+        beta_p=1,
+    )

tidy3d/__init__.py

@@ -88,6 +88,7 @@
     IsotropicEffectiveDOS,
     MultiValleyEffectiveDOS,
     RadiativeRecombination,
+    SelberherrImpactIonization,
     ShockleyReedHallRecombination,
     SlotboomBandGapNarrowing,
     TemperatureBC,
@@ -685,6 +686,7 @@ def set_logging_level(level: str) -> None:
     "ScalarModeFieldCylindricalDataArray",
     "ScalarModeFieldDataArray",
     "Scene",
+    "SelberherrImpactIonization",
     "Sellmeier",
     "SemiconductorMedium",
     "ShockleyReedHallRecombination",

tidy3d/components/tcad/generation_recombination.py

@@ -282,3 +282,81 @@ class HurkxDirectBandToBandTunneling(Tidy3dBaseModel):
         "semiconductors sigma is typically 2.0, while for indirect "
         "semiconductors sigma is typically 2.5.",
     )
+
+
+class SelberherrImpactIonization(Tidy3dBaseModel):
+    """
+    This class defines the parameters for the Selberherr impact ionization model. Two formulations are available that
+    depend on the driving field, as described in [1]_ (:math:`\\| E \\|`) and [2]_ (:math:`E \\cdot J_{\\nu} / \\| E \\|` for :math:`\\nu = n,p`).
+
+    Notes
+    -----
+
+        The impact ionization rate ``\\alpha_{\\nu}`` (for :math:`\\nu = p` (holes) and :math:`\\nu = n` (electrons)) is defined by:
+
+        .. math::
+
+            \\alpha_{\\nu} = \\alpha_{\\nu}^\\infty \\cdot \\exp \\left( - \\left( \\frac{E_{\\nu}^{\\text{crit}} \\cdot |\\mathbf{J}_{\\nu}|}{E \\cdot \\mathbf{J}_{\\nu}} \\right)^{\\beta_{\\nu}} \\right)
+
+        where :math:`\\alpha_{\\nu}^\\infty`, :math:`E_{\\nu}^{\\text{crit}}`, and :math:`\\beta_{\\nu}` are material-dependent parameters.
+
+    Example
+    -------
+        >>> import tidy3d as td
+        >>> default_Si = td.SelberherrImpactIonization(
+        ...   alpha_n_inf=7.03e5,
+        ...   alpha_p_inf=1.582e6,
+        ...   E_n_crit=1.23e6,
+        ...   E_p_crit=2.03e6,
+        ...   beta_n=1,
+        ...   beta_p=1,
+        ...   formulation='PQ'
+        ... )
+
+    References
+    ----------
+        .. [1] Selberherr, Siegfried. Analysis and simulation of semiconductor devices. Springer Science & Business Media, 1984.
+        .. [2] Vassil Palankovski and Rüdiger Quay. Analysis and simulation of heterostructure devices. Springer Science & Business Media, 2004.
+    """
+
+    alpha_n_inf: pd.PositiveFloat = pd.Field(
+        ...,
+        title="Electron ionization coefficient at infinite field",
+        description="Electron ionization coefficient at infinite field.",
+        units="1/cm",
+    )
+    alpha_p_inf: pd.PositiveFloat = pd.Field(
+        ...,
+        title="Hole ionization coefficient at infinite field",
+        description="Hole ionization coefficient at infinite field.",
+        units="1/cm",
+    )
+    E_n_crit: pd.PositiveFloat = pd.Field(
+        ...,
+        title="Critical electric field for electrons",
+        description="Critical electric field for electrons.",
+        units="V/cm",
+    )
+    E_p_crit: pd.PositiveFloat = pd.Field(
+        ...,
+        title="Critical electric field for holes",
+        description="Critical electric field for holes.",
+        units="V/cm",
+    )
+    beta_n: pd.PositiveFloat = pd.Field(
+        ...,
+        title="Exponent for electrons",
+        description="Exponent for electrons.",
+    )
+    beta_p: pd.PositiveFloat = pd.Field(
+        ...,
+        title="Exponent for holes",
+        description="Exponent for holes.",
+    )
+
+    formulation: str = pd.Field(
+        "PQ",
+        title="Formulation",
+        description="Formulation used for impact ionization. Options are 'Selberherr' "
+        "or 'PQ' for Selberherr and Palankovski and Quay formulations, respectively.",
+    )

tidy3d/components/tcad/types.py

@@ -26,6 +26,7 @@
     DistributedGeneration,
     HurkxDirectBandToBandTunneling,
     RadiativeRecombination,
+    SelberherrImpactIonization,
     ShockleyReedHallRecombination,
 )
 from tidy3d.components.tcad.mobility import CaugheyThomasMobility, ConstantMobilityModel
@@ -52,6 +53,7 @@
     RadiativeRecombination,
     ShockleyReedHallRecombination,
     HurkxDirectBandToBandTunneling,
+    SelberherrImpactIonization,
 ]
 BandGapNarrowingModelType = Union[SlotboomBandGapNarrowing]