"""Defines heat material specifications"""
from __future__ import annotations
import pydantic.v1 as pd
from tidy3d.components.data.data_array import SpatialDataArray
from tidy3d.components.medium import AbstractMedium
from tidy3d.components.tcad.doping import DopingBoxType
from tidy3d.components.tcad.types import (
BandGapNarrowingModelType,
MobilityModelType,
RecombinationModelType,
)
from tidy3d.components.types import Union
from tidy3d.constants import (
CONDUCTIVITY,
ELECTRON_VOLT,
PERMITTIVITY,
)
class AbstractChargeMedium(AbstractMedium):
"""Abstract class for Charge specifications
Currently, permittivity is treated as a constant."""
permittivity: float = pd.Field(
1.0, ge=1.0, title="Permittivity", description="Relative permittivity.", units=PERMITTIVITY
)
@property
def charge(self):
"""
This means that a charge medium has been defined inherently within this solver medium.
This provides interconnection with the `MultiPhysicsMedium` higher-dimensional classes.
"""
return self
def eps_model(self, frequency: float) -> complex:
return self.permittivity
def n_cfl(self):
return None
[docs]
class ChargeInsulatorMedium(AbstractChargeMedium):
"""
Insulating medium. Conduction simulations will not solve for electric
potential in a structure that has a medium with this 'charge'.
Example
-------
>>> import tidy3d as td
>>> solid = td.ChargeInsulatorMedium()
>>> solid2 = td.ChargeInsulatorMedium(permittivity=1.1)
Note
----
A relative permittivity :math:`\\varepsilon` will be assumed 1 if no value is specified.
"""
[docs]
class ChargeConductorMedium(AbstractChargeMedium):
"""Conductor medium for conduction simulations.
Example
-------
>>> import tidy3d as td
>>> solid = td.ChargeConductorMedium(conductivity=3)
Note
----
A relative permittivity will be assumed 1 if no value is specified.
"""
conductivity: pd.PositiveFloat = pd.Field(
...,
title="Electric conductivity",
description=f"Electric conductivity of material in units of {CONDUCTIVITY}.",
units=CONDUCTIVITY,
)
[docs]
class SemiconductorMedium(AbstractChargeMedium):
"""
This class is used to define semiconductors.
Notes
-----
Semiconductors are associated with ``Charge`` simulations. During these simulations
the Drift-Diffusion (DD) equations will be solved in semiconductors. In what follows, a
description of the assumptions taken and its limitations is put forward.
The iso-thermal DD equations are summarized here
.. math::
\\begin{equation}
- \\nabla \\cdot \\left( \\varepsilon_0 \\varepsilon_r \\nabla \\psi \\right) = q
\\left( p - n + N_d^+ - N_a^- \\right)
\\end{equation}
.. math::
\\begin{equation}
q \\frac{\\partial n}{\\partial t} = \\nabla \\cdot \\mathbf{J_n} - qR
\\end{equation}
.. math::
\\begin{equation}
q \\frac{\\partial p}{\\partial t} = -\\nabla \\cdot \\mathbf{J_p} - qR
\\end{equation}
As well as iso-thermal, the system is considered to be at :math:`T=300`. This restriction will
be removed in future releases.
The above system requires the definition of the flux functions (free carrier current density), :math:`\\mathbf{J_n}` and
:math:`\\mathbf{J_p}`. We consider the usual form
.. math::
\\begin{equation}
\\mathbf{J_n} = q \\mu_n \\mathbf{F_{n}} + q D_n \\nabla n
\\end{equation}
.. math::
\\begin{equation}
\\mathbf{J_p} = q \\mu_p \\mathbf{F_{p}} - q D_p \\nabla p
\\end{equation}
where we simplify the effective field defined in [1]_ to
.. math::
\\begin{equation}
\\mathbf{F_{n,p}} = \\nabla \\psi
\\end{equation}
i.e., we are not considering the effect of band-gap narrowing and degeneracy on the effective
electric field :math:`\\mathbf{F_{n,p}}`. This is a good approximation for non-degenerate semiconductors.
Let's explore how material properties are defined as class parameters or other classes.
.. list-table::
:widths: 25 25 75
:header-rows: 1
* - Symbol
- Parameter Name
- Description
* - :math:`N_a`
- ``N_a``
- Ionized acceptors density
* - :math:`N_d`
- ``N_d``
- Ionized donors density
* - :math:`N_c`
- ``N_c``
- Effective density of states in the conduction band.
* - :math:`N_v`
- ``N_v``
- Effective density of states in valence band.
* - :math:`R`
- ``R``
- Generation-Recombination term.
* - :math:`E_g`
- ``E_g``
- Bandgap Energy.
* - :math:`\\Delta E_g`
- ``delta_E_g``
- Bandgap Narrowing.
* - :math:`\\sigma`
- ``conductivity``
- Electrical conductivity.
* - :math:`\\varepsilon_r`
- ``permittivity``
- Relative permittivity.
* - :math:`q`
- ``tidy3d.constants.Q_e``
- Fundamental electron charge.
Example
-------
>>> import tidy3d as td
>>> default_Si = td.SemiconductorMedium(
... N_c=2.86e19,
... N_v=3.1e19,
... E_g=1.11,
... mobility_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_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,
... ),
... R=([
... td.ShockleyReedHallRecombination(
... tau_n=3.3e-6,
... tau_p=4e-6
... ),
... td.RadiativeRecombination(
... r_const=1.6e-14
... ),
... td.AugerRecombination(
... c_n=2.8e-31,
... c_p=9.9e-32
... ),
... ]),
... delta_E_g=td.SlotboomBandGapNarrowing(
... v1=6.92 * 1e-3,
... n2=1.3e17,
... c2=0.5,
... min_N=1e15,
... ),
... N_a=0,
... N_d=0
... )
Warning
-------
Current limitations of the formulation include:
- Boltzmann statistics are supported
- Iso-thermal equations with :math:`T=300K`
- Steady state only
- Dopants are considered to be fully ionized
Note
----
- Both :math:`N_a` and :math:`N_d` can be either a positive number or an ``xarray.DataArray``.
- Default values for parameters and models are those appropriate for Silicon.
- The current implementation is a good approximation for non-degenerate semiconductors.
.. [1] Schroeder, D., T. Ostermann, and O. Kalz. "Comparison of transport models far the simulation of degenerate semiconductors." Semiconductor science and technology 9.4 (1994): 364.
"""
N_c: pd.PositiveFloat = pd.Field(
...,
title="Effective density of electron states",
description=r"$N_c$ Effective density of states in the conduction band.",
units="cm^(-3)",
)
N_v: pd.PositiveFloat = pd.Field(
...,
title="Effective density of hole states",
description=r"$N_v$ Effective density of states in the valence band.",
units="cm^(-3)",
)
E_g: pd.PositiveFloat = pd.Field(
...,
title="Band-gap energy",
description="Band-gap energy",
units=ELECTRON_VOLT,
)
mobility_n: MobilityModelType = pd.Field(
...,
title="Mobility model for electrons",
description="Mobility model for electrons",
)
mobility_p: MobilityModelType = pd.Field(
...,
title="Mobility model for holes",
description="Mobility model for holes",
)
R: tuple[RecombinationModelType, ...] = pd.Field(
[],
title="Generation-Recombination models",
description="Array containing the R models to be applied to the material.",
)
delta_E_g: BandGapNarrowingModelType = pd.Field(
None,
title=r"$\Delta E_g$ Bandgap narrowing model.",
description="Bandgap narrowing model.",
)
N_a: Union[pd.NonNegativeFloat, SpatialDataArray, tuple[DopingBoxType, ...]] = pd.Field(
0,
title="Doping: Acceptor concentration",
description="Units of 1/cm^3",
units="1/cm^3",
)
N_d: Union[pd.NonNegativeFloat, SpatialDataArray, tuple[DopingBoxType, ...]] = pd.Field(
0,
title="Doping: Donor concentration",
description="Units of 1/cm^3",
units="1/cm^3",
)
