tidy3d.SemiconductorMedium

class SemiconductorMedium

Bases: AbstractChargeMedium

This class is used to define semiconductors.

Parameters:

• attrs (dict = {}) – Dictionary storing arbitrary metadata for a Tidy3D object. This dictionary can be freely used by the user for storing data without affecting the operation of Tidy3D as it is not used internally. Note that, unlike regular Tidy3D fields, attrs are mutable. For example, the following is allowed for setting an attr obj.attrs['foo'] = bar. Also note that Tidy3D` will raise a TypeError if attrs contain objects that can not be serialized. One can check if attrs are serializable by calling obj.json().

• name (Optional[str] = None) – Optional unique name for medium.

• frequency_range (Optional[Tuple[float, float]] = None) – [units = (Hz, Hz)]. Optional range of validity for the medium.

• allow_gain (bool = False) – Allow the medium to be active. Caution: simulations with a gain medium are unstable, and are likely to diverge.Simulations where ‘allow_gain’ is set to ‘True’ will still be charged even if diverged. Monitor data up to the divergence point will still be returned and can be useful in some cases.

• nonlinear_spec (Union[NonlinearSpec, NonlinearSusceptibility] = None) – Nonlinear spec applied on top of the base medium properties.

• modulation_spec (Optional[ModulationSpec] = None) – Modulation spec applied on top of the base medium properties.

• viz_spec (Optional[VisualizationSpec] = None) – Plotting specification for visualizing medium.

• heat_spec (Union[FluidSpec, SolidSpec, SolidMedium, FluidMedium, NoneType] = None) – DEPRECATED: Use td.MultiPhysicsMedium. Specification of the medium heat properties. They are used for solving the heat equation via the HeatSimulation interface. Such simulations can beused for investigating the influence of heat propagation on the properties of optical systems. Once the temperature distribution in the system is found using HeatSimulation object, Simulation.perturbed_mediums_copy() can be used to convert mediums with perturbation models defined into spatially dependent custom mediums. Otherwise, the heat_spec does not directly affect the running of an optical Simulation.

• permittivity (ConstrainedFloatValue = 1.0) – [units = None (relative permittivity)]. Relative permittivity.

• N_c (PositiveFloat) – [units = cm^(-3)]. $N_c$ Effective density of states in the conduction band.

• N_v (PositiveFloat) – [units = cm^(-3)]. $N_v$ Effective density of states in the valence band.

• E_g (PositiveFloat) – [units = eV]. Band-gap energy

• mobility_n (Union[CaugheyThomasMobility, ConstantMobilityModel]) – Mobility model for electrons

• mobility_p (Union[CaugheyThomasMobility, ConstantMobilityModel]) – Mobility model for holes

• R (Tuple[Union[AugerRecombination, RadiativeRecombination, ShockleyReedHallRecombination], ...] = []) – Array containing the R models to be applied to the material.

• delta_E_g (Optional[SlotboomBandGapNarrowing] = None) – Bandgap narrowing model.

• N_a (Union[NonNegativeFloat, SpatialDataArray, Tuple[Union[tidy3d.components.tcad.doping.ConstantDoping, tidy3d.components.tcad.doping.GaussianDoping], ...]] = 0) – [units = 1/cm^3]. Units of 1/cm^3

• N_d (Union[NonNegativeFloat, SpatialDataArray, Tuple[Union[tidy3d.components.tcad.doping.ConstantDoping, tidy3d.components.tcad.doping.GaussianDoping], ...]] = 0) – [units = 1/cm^3]. Units of 1/cm^3

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

\[\begin{equation} - \nabla \cdot \left( \varepsilon_0 \varepsilon_r \nabla \psi \right) = q \left( p - n + N_d^+ - N_a^- \right) \end{equation}\]

\[\begin{equation} q \frac{\partial n}{\partial t} = \nabla \cdot \mathbf{J_n} - qR \end{equation}\]

\[\begin{equation} q \frac{\partial p}{\partial t} = -\nabla \cdot \mathbf{J_p} - qR \end{equation}\]

As well as iso-thermal, the system is considered to be at \(T=300\). This restriction will be removed in future releases.

The above system requires the definition of the flux functions (free carrier current density), \(\mathbf{J_n}\) and \(\mathbf{J_p}\). We consider the usual form

\[\begin{equation} \mathbf{J_n} = q \mu_n \mathbf{F_{n}} + q D_n \nabla n \end{equation}\]

\[\begin{equation} \mathbf{J_p} = q \mu_p \mathbf{F_{p}} - q D_p \nabla p \end{equation}\]

where we simplify the effective field defined in [1] to

\[\begin{equation} \mathbf{F_{n,p}} = \nabla \psi \end{equation}\]

i.e., we are not considering the effect of band-gab narrowing and degeneracy on the effective electric field \(\mathbf{F_{n,p}}\). This is a good approximation for non-degenerate semiconductors.

Let’s explore how material properties are defined as class parameters or other classes.

Symbol	Parameter Name	Description
\(N_a\)	N_a	Ionized acceptors density
\(N_d\)	N_d	Ionized donors density
\(N_c\)	N_c	Effective density of states in the conduction band.
\(N_v\)	N_v	Effective density of states in valence band.
\(R\)	R	Generation-Recombination term.
\(E_g\)	E_g	Bandgap Energy.
\(\Delta E_g\)	delta_E_g	Bandgap Narrowing.
\(\sigma\)	conductivity	Electrical conductivity.
\(\varepsilon_r\)	permittivity	Relative permittivity.
\(q\)	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 \(T=300K\)

• Steady state only

• Dopants are considered to be fully ionized

Note

• Both \(N_a\) and \(N_d\) 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.

Attributes

attrs

Methods

Inherited Common Usage

N_c

N_v

E_g

mobility_n

mobility_p

R

delta_E_g

N_a

N_d

__hash__()

Hash method.