from __future__ import annotations
import pydantic.v1 as pd
from tidy3d.components.base import Tidy3dBaseModel
[docs]
class ConstantMobilityModel(Tidy3dBaseModel):
"""Constant mobility model
Example
-------
>>> import tidy3d as td
>>> mobility_model = td.ConstantMobilityModel(mu=1500)
"""
mu: pd.NonNegativeFloat = pd.Field(
..., title="Mobility", description="Mobility", units="cm²/V-s"
)
[docs]
class CaugheyThomasMobility(Tidy3dBaseModel):
"""The Caughey-Thomas temperature-dependent carrier mobility model.
Notes
-----
The general form of the Caughey-Thomas mobility model [1]_ is of the form:
.. math::
\\mu_0 = \\frac{\\mu_{max} - \\mu_{min}}{1 + \\left(N/N_{ref}\\right)^z} + \\mu_{min}
where :math:`\\mu_0` represents the low-field mobility and :math:`N` is the total doping (acceptors + donors).
:math:`\\mu_{max}`, :math:`\\mu_{min}`, :math:`z`, and :math:`N_{ref}` are temperature dependent,
the dependence being of the form
.. math::
\\phi = \\phi_{ref} \\left( \\frac{T}{T_{ref}}\\right)^\\alpha
and :math:`T_{ref}` is taken to be 300K.
The complete form (with temperature effects) for the low-field mobility can be written as
.. math::
\\mu_0 = \\frac{\\mu_{max}(\\frac{T}{T_{ref}})^{\\alpha_2} - \\mu_{min}(\\frac{T}{T_{ref}})^{\\alpha_1}}{1 + \\left(N/N_{ref}(\\frac{T}{T_{ref}})^{\\alpha_3}\\right)^{\\alpha_N(\\frac{T}{T_{ref}})^{\\alpha_4}}} + \\mu_{min}(\\frac{T}{T_{ref}})^{\\alpha_1}
The following table maps the symbols used in the equations above with the names used in the code:
.. list-table::
:widths: 25 25 75
:header-rows: 1
* - Symbol
- Parameter Name
- Description
* - :math:`\\mu_{min}`
- ``mu_min``
- Minimum low-field mobility for :math:`n` and :math:`p`
* - :math:`\\mu_{max}`
- ``mu_n``
- Maximum low-field mobility for :math:`n` and :math:`p`
* - :math:`\\alpha_1`
- ``exp_1``
- Exponent for temperature dependence of the minimum mobility coefficient
* - :math:`\\alpha_2`
- ``exp_2``
- Exponent for temperature dependence of the maximum mobility coefficient
* - :math:`\\alpha_N`
- ``exp_N``
- Exponent for doping dependence.
* - :math:`\\alpha_4`
- ``exp_4``
- Exponent for the temperature dependence of the exponent :math:`\\alpha_N`
* - :math:`N_{ref}`
- ``ref_N``,
- Reference doping parameter
.. [1] M. Caughey and R.E. Thomas. Carrier mobilities in silicon empirically related to doping
and field. Proceedings of the IEEE, 55(12):2192–2193, December 1967
Example
-------
>>> import tidy3d as td
>>> mobility_Si_n = td.CaugheyThomasMobility(
... mu_min=52.2,
... mu=1471.0,
... ref_N=9.68e16,
... exp_N=0.68,
... exp_1=-0.57,
... exp_2=-2.33,
... exp_3=2.4,
... exp_4=-0.146,
... )
>>> mobility_Si_p = td.CaugheyThomasMobility(
... mu_min=44.9,
... mu=470.5,
... ref_N=2.23e17,
... exp_N=0.719,
... exp_1=-0.57,
... exp_2=-2.33,
... exp_3=2.4,
... exp_4=-0.146,
... )
Warning
-------
There are some current limitations of this model:
- High electric field effects not yet supported.
"""
# mobilities
mu_min: pd.PositiveFloat = pd.Field(
...,
title=r"$\mu_{min}$ Minimum electron mobility",
description="Minimum electron mobility at reference temperature (300K) in cm^2/V-s. ",
)
mu: pd.PositiveFloat = pd.Field(
...,
title="Reference mobility",
description="Reference mobility at reference temperature (300K) in cm^2/V-s",
)
# thermal exponent for reference mobility
exp_2: float = pd.Field(
..., title="Exponent for temperature dependent behavior of reference mobility"
)
# doping exponent
exp_N: pd.PositiveFloat = pd.Field(
...,
title="Exponent for doping dependence of mobility.",
description="Exponent for doping dependence of mobility at reference temperature (300K).",
)
# reference doping
ref_N: pd.PositiveFloat = pd.Field(
...,
title="Reference doping",
description="Reference doping at reference temperature (300K) in #/cm^3.",
)
# temperature exponent
exp_1: float = pd.Field(
...,
title="Exponent of thermal dependence of minimum mobility.",
description="Exponent of thermal dependence of minimum mobility.",
)
exp_3: float = pd.Field(
...,
title="Exponent of thermal dependence of reference doping.",
description="Exponent of thermal dependence of reference doping.",
)
exp_4: float = pd.Field(
...,
title="Exponent of thermal dependence of the doping exponent effect.",
description="Exponent of thermal dependence of the doping exponent effect.",
)
