Finite Element IDE supports simulation with alloy materials, which combine two base materials (either unary, binary compound, or ternary alloy) to create new alloy. Physically, this is typically accomplished by depositing a material where a fraction of its constituent atoms are substituted with those of another species. An example is the alloy of aluminium arsenide (AlAs) and gallium arsenide (GaAs): replacing some of the gallium atoms in the crystal lattice with aluminium atoms creates the alloy Al_{x}Ga_{1-x}As, where x is the mole fraction of aluminium.

In this page, we will describe the interpolation procedure used to derive the properties of the alloy material from its base materials. In addition, important details about how to apply and interpret the material properties are provided.

## Binary/Ternary alloys (Electrical/Thermal parameters)

### Base materials

Base materials that the alloy consists of. This can be any unary (e.g. Si, Ge) or binary compound (e.g. GaAs).

### Interpolation formulas

The properties P(A_{1-x}B_{x}) of binary alloys, such as SiGe, are interpolated from the corresponding properties of the base materials (P(A) and P(B)) according to the formula

$$ P\left(A_{1-x} B_x\right)=\left(1-x\right)P\left(A\right)+xP\left(B\right)+x\left(1-x\right)C, $$

The properties P(A_{1-x}B_{x}D) of ternary alloys, such as AlGaAs, are interpolated from the corresponding properties of the base materials (P(AD) and P(BD)) according to the formula

$$ P\left(A_{1-x} B_xD\right)=\left(1-x\right)P\left(AD\right)+xP\left(BD\right)+x\left(1-x\right)C, $$

where x is the mole fraction and C is the bowing parameter (quadratic coefficient). If the bowing parameters are not known, they can be set to 0 in the first approximation.

There are two interpolation schemes used for creating the binary or ternary alloy from the two specified semiconductors:

**1. Single valley interpolation: **

In this scheme, the lowest valley from semiconductor A and the lowest valley from semiconductor B are considered and by interpolating between the two, the new valley for the alloy is calculated.

**2. Multi valley interpolation:**

In this scheme, all valleys from both semiconductors are considered. For example, in the diagram below, we interpolate first between the two L valleys of semiconductor A and B and then we interpolate between the two gamma valleys of semiconductor A and B and then we take the lower valley of the two interpolation results. This is a more accurate interpolation method. If we choose to use the multi-valley method, we should have available data for all valleys of both semiconductors in the material database.

The actual interpolation uses Vegard's law [1] to determine the resulting properties for each valley depending on the mole fraction x of each semiconductor in the alloy. For example for band gap energy of Al(x)Ga(1-x)As, we have:

$$E_{g,AlGaAs}=(1-x)E_{g,GaAs}+xE_{g,AlAs}+bx(1-x)$$

Where, b is the bowing parameter in the same units as the interpolated parameter, in this case eV. Often the bowing parameter is zero, but may be modified in the alloy material property tabs.

The only exception is mobility and thermal conductivity. Mobility and thermal conductivity are interpolated by harmonic mean, in which case the reciprocal equation holds:

$$\frac{1}{\mu_{AlGaAs}}=\frac{1-x}{\mu_{GaAs}}+\frac{x}{\mu_{AlAs}}+\frac{x(1-x)}{b}$$

When interpolating by harmonic mean, the term with 1/b is ignored if the bowing parameter is set to zero.

As highlighted in the overview, many of the semiconductor properties are derived from the characteristics of the conduction band minima, with the lowest-energy minima determining the material behavior. As an example, GaAs is a direct band gap semiconductor, but has higher energy conduction band minima near the X and L symmetry points. Conversely, silicon is an indirect band gap semiconductor, with the lowest energy conduction band minima located at L.

Comparison of the band gap energy of the silicon germanium alloy |

In the alloy of two different materials, the symmetry points of the lowest energy conduction band minima for the base semiconductors may be the same for both (e.g. InGaAs) or different (e.g. SiGe). There are two choices to interpolate the properties of the base semiconductors

- Multi-valley: interpolate the properties each pair of conduction band minima independently, then choose those from the lowest-energy minima
- Single-valley: interpolate the properties of the lowest energy conduction band minima from each base material, ignoring differences in the location (nearest point of symmetry).

Typically, the first choice (multi-valley) is the most physically realistic; however, single-valley interpolation is useful for strained materials (e.g. SiGe) and materials where information about the higher energy conduction band valleys is not available.

**Electrons vs. Holes**

It is important to note that the conduction band minima define the properties for the electrons (effective mass, mobility, etc.). The properties for holes are defined by the valence band maxima, which are typically symmetric around the Γ-point. In the material database, the properties for holes are repeated for each band to make them more accessible, however only the properties defined for the active (lowest energy) conduction band are used in the simulation.

**Abrupt Transitions**

For certain combinations of base semiconductors using the multi-valley selection model, properties or model parameters may be unavailable for the higher-energy bands. This can be indicated in the material properties by setting the parameter to zero.

In the example screenshot below, the electron mobility for the higher-energy Γ conduction band valley for AlSb is set to zero. Alloys with AlSb that mix the Γ valley properties will not interpolate this parameter – instead, the electron mobility for the other base semiconductor will be used for all mole fractions. Electron mobility set to zero disables interpolation with that parameter.

A second case relates to the transitions between direct and indirect band-gap materials, when multi-valley interpolation is selected. In this scenario, the coefficients of the recombination rate models may be expected to change abruptly. An example of this behavior is the radiative recombination rate, which will be much larger for a direct band gap material than an indirect material. In this scenario, it is more accurate to use two different values for the recombination rate coefficient, depending on the direct or indirect nature of the band gap.

To account for this behaviour, the interpolation for recombination rate properties can be disabled by choosing “abrupt” interpolation. With that selection, the coefficients are chosen directly from the base semiconductor whose active (lowest energy) conduction band is the active valley for the alloy. An example of the electron lifetime for AlxGa1-xAs is compared to the band gap in the figure below. When the alloy transitions from the direct (Γ-valley) band gap of GaAs to the indirect (X-valley) band gap of AlAs at x=0.42, the carrier lifetime is adjusted to match that of the corresponding material.

### Electronic

For everything in this tab, we are setting the bowing parameter for the corresponding property in the same units as the property. Each alloy can take four sets of bowing parameters (one for single-valley interpolation, and one each for Ec valley gamma, L, or X when using multi-valley interpolation).

### Recombination

For everything in this tab, we have the choice to either set the bowing parameter for the corresponding property or use an abrupt method, which just means that instead of a smooth interpolation between the values, there will be an abrupt change from the value in one semiconductor to the other.

### KdotP

For everything in this tab, we are setting the bowing parameter for the corresponding property in the same units as the property.

### Thermal

For everything in this tab, we are setting the bowing parameter for the corresponding property in the same units as the property.

## Quaternary alloy (Electrical/Thermal parameters)

### Interpolation type

There are two quaternary types supported:

- A
_{x}B_{1-x}C_{y}D_{1-y}(two group III and two group V elements, such as In_{x}Ga_{1−x}As_{y}P_{1−y}) - A
_{x}B_{y}C_{1-x-y}D (three group III elements and one group V element, such as Al_{x}Ga_{y}In_{1-x-y}As)

### Base materials

Base materials that the alloy consists of. These must be ternary alloys (e.g. AlGaAs).

### Interpolation formulas

Quaternary alloys of type A_{x}B_{1-x}C_{y}D_{1-y} (two group III and two group V elements) are composed from the interpolation of ternary alloy constituents [2]:

$$ P\left(A_xB_{1-x}C_yD_{1-y}\right)=\frac{x\left(1-x\right)\left[\left(1-y\right)P\left(A_xB_{1-x}D\right)+yP\left(A_xB_{1-x}C\right)\right]+y\left(1-y\right)\left[xP\left(AC_yD_{1-y}\right)+\left(1-x\right)P\left(BC_yD_{1-y}\right)\right]}{x\left(1-x\right)+y\left(1-y\right)}, $$

for composition fractions x and y. For example, a combination of the properties of In_{x}Ga_{1−x}P, In_{x}Ga_{1−x}As, InAs_{y}P_{1−y}, and GaAs_{y}P_{1−y} is used to define the properties of In_{x}Ga_{1−x}As_{y}P_{1−y}.

Quaternary alloys of type A_{x}B_{y}C_{1-x-y}D (three group III elements and one group V element) are composed from the interpolation of ternary alloy constituents [2]:

$$ P\left(A_xB_yC_{1-x-y}D\right)=\frac{xyP\left(A_{1-u}B_uD\right)+y(1-x-y)P\left(B_{1-v}C_{v}D\right)+x(1-x-y)P\left(A_{1-w}C_{w}D\right)}{xy+y(1-x-y)+x(1-x-y)}, $$

for composition fractions x and y and u = (1-x+y)/2, v = (2-x-2y)/2, w = (2-2x-y)/2. For example, a combination of the properties of Al_{1-u}Ga_{u}As, Ga_{1-v}In_{v}As, and Al_{1-w}In_{w}As, is used to define the properties of Al_{x}Ga_{y}In_{1-x-y}As.

### Custom interpolation

For some properties, such as band gap at room temperature Eg, and some quaternary materials, such as InGaAsP and AlGaInAs, a custom interpolation formula can be set if available. These formulas provide better accuracy compared to default interpolation.

NOTE: Surface recombination properties of alloys are not interpolated from the semiconductor materials but rather are defined at the boundary conditions for the alloy itself in a manner similar to semiconductor materials. |

### References

[1] Vegard, L. (1921). "Die Konstitution der Mischkristalle und die Raumfüllung der Atome". Zeitschrift für Physik 5 (1): 17–26.

[2] Vurgaftman et al., J. Appl. Phys., 89, 5815 (2001)