This section describes the material models available in the electrical/thermal material database. Different models are used to realize two different types of material properties, electrical and thermal, each belonging to a different solver, CHARGE and HEAT, respectively.

## Electrical Properties

The material models used to define the electrical properties of materials fall into one of five general classes: conductors, insulators, semiconductors, alloys, and fluids.

### Conductor

Conductors are treated as perfect electrical conductors in a CHARGE simulation. Therefore, the electrostatic potential is constant over the entire domain of the conductor. Conductors are used to specify electrostatic boundary conditions, and must be associated with a contact in an electrical simulation.

- WORK FUNCTION: The defining characteristic of the conductor is its work function, which describes the energy cost of removing an electron from the material.

### Insulator

Materials that are defined as insulators are treated as ideal electrical insulators with a constant isotropic DC permittivity. Insulators contain no free charge, and specify flux boundary conditions in an electrical simulation.

- RELATIVE DIELECTRIC PERMITTIVITY: The relative permittivity (or dielectric constant) of the material is equal to the square of the refractive index, and is assumed to be the DC (zero frequency) value.

### Semiconductor

Semiconductors, like insulators, are band gap materials. The band gap of a semiconductor is typically small enough to allow a significant fraction of electrons to be thermally excited from the valence band to the conduction band at room temperature (300K). The band gap for a semiconductor typically ranges from 0.5-1.5eV. When energetically excited to a conduction band state, electrons leave behind a positively charged mobile vacancy, known as a hole, which behaves much like a free electron in the semiconductor. The mobility of the electrons and holes and the rates at which they are generated and recombine are determined by the models described in this section.

Semiconductors are described by a comprehensive set of parametric models that account for the behavior of free charge in an ideal crystal lattice and corrections to that behavior due to interactions with impurities, interactions with other charges, and temperature effects. See the Semiconductor page for details of its Electronic Properties and Recombination.

### Binary Alloy

Binary Alloys consist of two semiconductors. The bandgap and other properties for alloys are determined from the properties of the semiconductors it consists of.

As semiconductors are described by a comprehensive set of parametric models, so are alloys. See the Alloys page for details.

### Fluid

Liquid and gaseous materials are defined as fluids in the material database. In the CHARGE solver, fluids are treated as insulators. They contain no free charge, and specify flux boundary conditions in an electrical simulation.

- RELATIVE DIELECTRIC PERMITTIVITY: The relative permittivity (or dielectric constant) of the material is equal to the square of the refractive index, and is assumed to be the DC (zero frequency) value.

## Thermal Properties

The material models used to define the thermal properties of materials fall into one of two general classes: non-fluids (which includes conductors, insulators, semiconductors, and alloys) and fluids.

### Solid

This includes semiconductors, conductors, alloys, and insulators. All these material types are defined using similar models describing their density, specific heat, thermal conductivity, and electrical conductivity.

In the HEAT solver, solid materials are defined by their thermal properties. The properties are listed as follows:

**Heat Transport Properties**:

- DENSITY (kg/m3): The density of the material in SI unit.
- SPECIFIC HEAT (J/kg/K): The specific heat of the material in SI unit. Can be defined as a constant value or using a temperature dependent model. The temperature dependence of specific heat of solids is modeled using the following equation,

$$ c_L(T)= c_{L,300}+c_1 \frac{(\frac{T}{300})^{\beta}-1}{(\frac{T}{300})^{\beta} + \frac{c_1}{c_{L,300}}}$$

where, cL(T) is the specific heat at temperature T, cL,300 is the specific heat at 300 K, and c1 is a constant. - THERMAL CONDUCTIVITY (W/m/K): The thermal conductivity of the material in SI unit. Can be defined as a constant value or using a temperature dependent model. The temperature dependence of thermal conductivity of solids is modeled using the following equation,

$$ K_L(T) = K_{300}\Big(\frac{T}{300}\Big)^{\eta} $$

where, KL(T) is the thermal conductivity at temperature T, K300 is the thermal conductivity at 300 K, and η is a constant.

**Electrical Conduction Properties**:

- ELECTRICAL CONDUCTIVITY (S/m): The electrical conductivity of the material in SI unit. Can be defined as a constant value or using a temperature dependent model. The temperature dependence of electrical conductivity of solids is modeled using the following equation,

$$\sigma(T) = \Big[ \frac{1}{\sigma_0}\big[ 1 + \alpha(T-T_0) \big] \Big]^{-1} $$

where, σ(T) is the electrical conductivity at temperature T, σ0 is the electrical conductivity at temperature T0, and α is a constant.

The doping dependency can also be enabled in the model, which is based on deriving the mobility using the Caughey-Thomas model:

$$ \mu_{n,p} = \mu^{min,300} _{n,p}\overline{T}^{\eta_1}+ \frac{ \mu^L _{n,p}-\mu^{min,300} _{n,p}\overline{T}^{\eta_1}}{1+(\frac{N^+_D+N^-_A}{N_{300} \overline{T}^{\eta_2}})^{P_{300} \overline{T}^{\eta_3}}}$$

Each coefficient in the formula has temperature dependence (A = A_300 * (T/300)^η), T with an over-bar is the temperature scaled by 300K, and the lattice mobility µL is determined from the material model. After this, electrical conductivity is obtained by:

$$\sigma =n q \mu $$

### Fluid

**Heat Transport Properties**

- DENSITY (kg/m3): The mass density of the fluid in SI unit. Can be defined as a constant value or using a temperature dependent model. The temperature dependent model is applicable for gases only and the mass density is calculated using the ideal gas law,

$$ \rho(T) = P \Big( \frac{R_{\text{specific}}}{T} \Big) $$

where, ρ(T) is the mass density at temperature T, P is the pressure, and Rspecific is the specific gas constant. - SPECIFIC HEAT (J/kg/K): The specific heat of the fluid in SI unit. Can be defined as a constant value or using a temperature dependent model. The temperature dependence of specific heat of fluids is modeled using the following equation,

$$ c_L(T)= c_{L,300}+c_1 \frac{\Big(\frac{T}{300}\Big)^{\beta}-1}{\Big(\frac{T}{300}\Big)^{\beta} + \frac{c_1}{c_{L,300}}}$$

where, CL(T) is the specific heat at temperature T, CL,300 is the specific heat at 300 K, and c1 is a constant. - THERMAL CONDUCTIVITY (W/m/K): The thermal conductivity of the fluid in SI unit. Can be defined as a constant value or using a temperature dependent model. The temperature dependence of thermal conductivity of fluids is modeled using the following equation,

$$ K_L(T) = K_{300}\Big(\frac{T}{300}\Big)^{\eta} $$

where, KL(T) is the thermal conductivity at temperature T, K300 is the thermal conductivity at 300 K, and η is a constant.

- DYNAMIC VISCOSITY (Pa.s): The dynamic viscosity of the fluid in SI unit. Can be defined as a constant value or using a temperature dependent model. The temperature dependence of dynamic viscosity of fluids is modeled using the Sutherland's formula,

$$ \mu(T) = C_1 \frac{T^{3/2}}{T + S} $$

where, μ(T) is the dynamic viscosity at temperature T, S is the Sutherland's constant for the particular fluid, and C1 is given by,

$$ C_1 = \mu_{300}(300 + S) $$

where, μ300 is the dynamic viscosity of the fluid at 300 K. - THERMAL EXPANSIVITY (1/K): The thermal expansivity of the fluid in SI unit. Can be defined as a constant value or using a temperature dependent model. The temperature dependence of thermal expansivity of fluids is calculated using the ideal gas law that gives,

$$ \beta = \alpha / T $$

where, β is the thermal expansivity at temperature T and α is a constant whose value is 1 for ideal gases.

### Solid Alloy

Binary Alloys consist of two materials. The density and other thermal properties for alloys are determined from the properties of the materials it consists of.

Thermal alloys use the same interpolation method as electrical alloys to obtain material properties from their base materials. See the Alloys page for details.