2.3.3. Interpolation Schemes

Introduction

As our software addresses simulations for a broad range of semiconductor materials, these based on binary compounds (like GaAs) and single elements (like Si), a unified naming of the alloys becomes problematic if one tries to follow standards in the literature. For example, SixGe1-xis a binary alloy (two elements), while GaxIn1-xAs is a ternary alloy (three elements), even though their parameters are interpolated using exactly the same schemes.

On the other hand side, in both cases there are only two component materials (pure materials) involved in formation of the alloy, Si and Ge in the first case, and GaAs and InAs in the second case. Therefore, in this documentation, we will refer to all pure materials (which parameters are typically tabulated in literature, like: GaAs, InN, ZnO, Si, etc.) as component materials and naming of interpolation schemes will be based on the number of these components materials involved in them. With such formalism, both SixGe1-x and GaxIn1-xAs are two-component alloys.

Attention

Syntax of the database is consistent with standard naming for III-V and II-VI material systems. Therefore, regardless of the number of elements forming component materials, they are referred to as binaries; and the simplest available alloys are ternary alloys.

Two-component alloys

Two-component alloys are typically called binary alloys when group-IV are mixed (IV-IV) and ternary alloys in the case of III-V or II-VI binary compounds (III-V-V, III-III-V, II-VI-VI, and II-II-VI). Examples of such alloys are: SixGe1-x, GaxIn1-xN, and GaAsxSb1-x.

Material parameters of two-component alloys are interpolated based on material parameters of two components and a proper bowing parameter \(b_\mathrm{AB}\) for the alloy, if defined. Three interpolation schemes are available in nextnano++ for this type of alloys: Linear, Quadratic, and Cubic.

Linear - no bowing

If only parameters of the component materials are defined then a linear interpolation is used to evaluate values of the parameters for the alloy.

IV-IV

For alloys of type AxB1-x, the scheme reads

\[P_\mathrm{AB}\left(x\right) = x\cdot P_\mathrm{A} + \left[1-x\right]\cdot P_\mathrm{B},\]

where \(P_\mathrm{AB}\left(x\right)\) is an interpolated material parameter of a two-component alloy AxB1-x based on parameters \(P_\mathrm{A}\) and \(P_\mathrm{B}\) describing pure components A and B, respectively.

III-III-V and II-II-VI

For alloys of type AxB1-xC, the scheme reads

\[P_\mathrm{ABC}\left(x\right) = x\cdot P_\mathrm{AC} + \left[1-x\right]\cdot P_\mathrm{BC},\]

where \(P_\mathrm{ABC}\left(x\right)\) is an interpolated material parameter of a two-component alloy AxB1-xC based on parameters \(P_\mathrm{AC}\) and \(P_\mathrm{BC}\) describing pure components AC and BC, respectively.

III-V-V and II-VI-VI

For alloys of type ABxC1-x, the scheme reads

\[P_\mathrm{ABC}\left(x\right) = x\cdot P_\mathrm{AB} + \left[1-x\right]\cdot P_\mathrm{AC},\]

where \(P_\mathrm{ABC}\left(x\right)\) is an interpolated material parameter of a two-component alloy ABxC1-x based on parameters \(P_\mathrm{AB}\) and \(P_\mathrm{AC}\) describing pure components AB and AC, respectively.

Syntax

Let’s consider an alloy GaInAs with AC being GaAs and BC being InAs. All the parameters of GaAs and InAs needs to be defined within binary_zb{} or binary_wz{}. To recognize the alloy and relate names of component materials, one needs to also define ternary_zb{} or ternary_wz{}, but no bowing parameters needs to be defined there, zeroes are assumed.

binary_zb{
    name    = GaAs
    valence = III_V

    # All the parameters of GaAs here (P_A)
}

binary_zb{
    name    = InAs
    valence = III_V

    # All the parameters of InAs here (P_B)
}

ternary_zb{
    name       = "Ga(x)In(1-x)As"
    valence    = III_V
    binary_x   = GaAs
    binary_1_x = InAs

    # No bowing parameters specified here
}

Quadratic - constant bowing

If bowing parameters are specified in the database using keywords If linear interpolation is not sufficient, quadratic interpolation with a bowing parametr can be used instead.

IV-IV

For alloys of type AxB1-x, the scheme reads

\[P_\mathrm{AB}\left(x\right) = x\cdot P_\mathrm{A} + \left[1-x\right]\cdot P_\mathrm{B} - x\left[1-x\right]\cdot b_\mathrm{AB},\]

where \(P_\mathrm{AB}\left(x\right)\) is an interpolated material parameter of a two-component alloy AxB1-x based on parameters \(P_\mathrm{A}\) and \(P_\mathrm{B}\) describing pure components A and B, respectively, and \(b_\mathrm{AB}\) is a bowing parameter for the alloy.

III-III-V and II-II-VI

For alloys of type AxB1-xC, the scheme reads

\[P_\mathrm{ABC}\left(x\right) = x\cdot P_\mathrm{AC} + \left[1-x\right]\cdot P_\mathrm{BC} - x\left[1-x\right]\cdot b_\mathrm{ABC},\]

where \(P_\mathrm{ABC}\left(x\right)\) is an interpolated material parameter of a two-component alloy AxB1-xC based on parameters \(P_\mathrm{AC}\) and \(P_\mathrm{BC}\) describing pure components AC and BC, respectively, and \(b_\mathrm{ABC}\) is a bowing parameter for the alloy.

III-V-V and II-VI-VI

For alloys of type ABxC1-x, the scheme reads

\[P_\mathrm{ABC}\left(x\right) = x\cdot P_\mathrm{AB} + \left[1-x\right]\cdot P_\mathrm{AC} - x\left[1-x\right]\cdot b_\mathrm{ABC},\]

where \(P_\mathrm{ABC}\left(x\right)\) is an interpolated material parameter of a two-component alloy ABxC1-x based on parameters \(P_\mathrm{AB}\) and \(P_\mathrm{AC}\) describing pure components AB and AC, respectively, and \(b_\mathrm{ABC}\) is a bowing parameter for the alloy.

Syntax

For quadratic interpolation of a certain material paramter, one has to specify a bowing paramter \(b_{AB}\) inside the groups ternary_zb{} or ternary_wz{}.

binary_zb{
    name    = GaAs
    valence = III_V

    # All the parameters of GaAs here (P_A)
}

binary_zb{
    name    = InAs
    valence = III_V

    # All the parameters of InAs here (P_B)
}

ternary_zb{
    name       = "Ga(x)In(1-x)As"
    valence    = III_V
    binary_x   = GaAs
    binary_1_x = InAs

    # Some bowing parameters  (b_AB)
}

Cubic - composition-dependent bowing

If a constant bowing paramter \(b_{AB}\) is not sufficient for interpolation of the parameters, like for highly-mismatched alloys or dilute nitrides, one can use a scheme where the bowing parameter is assumed to be linearly dependent on the mole fraction x, \(b_\mathrm{AB}\left(x\right)\).

IV-IV

For alloys of type AxB1-x, the scheme reads

\[P_\mathrm{AB}\left(x\right) = x\cdot P_\mathrm{A} + \left[1-x\right]\cdot P_\mathrm{B} - x \left[1-x\right]\cdot b_\mathrm{AB}\left(x\right)\]
\[b_\mathrm{AB}\left(x\right) = x \cdot b_\mathrm{AB\rightarrow A} + \left[1-x\right] \cdot b_\mathrm{AB\rightarrow B},\]

where \(P_\mathrm{AB}\left(x\right)\) is an interpolated material parameter of a two-component alloy AxB1-x based on parameters \(P_\mathrm{A}\) and \(P_\mathrm{B}\) describing pure components A and B, respectively. The \(b_\mathrm{AB\rightarrow A} = b_\mathrm{AB}\left(1\right)\) is a bowing parameter for nearly pure A, while the \(b_\mathrm{AB\rightarrow B} = b_\mathrm{AB}\left(0\right)\) is a bowing parameter for nearly pure B.

III-III-V and II-II-VI

For alloys of type AxB1-xC, the scheme reads

\[P_\mathrm{ABC}\left(x\right) = x\cdot P_\mathrm{AC} + \left[1-x\right]\cdot P_\mathrm{BC} - x\left[1-x\right]\cdot b_\mathrm{ABC},\]
\[b_\mathrm{ABC}\left(x\right) = x \cdot b_\mathrm{ABC\rightarrow AC} + \left[1-x\right] \cdot b_\mathrm{ABC\rightarrow BC},\]

where \(P_\mathrm{ABC}\left(x\right)\) is an interpolated material parameter of a two-component alloy AxB1-xC based on parameters \(P_\mathrm{AC}\) and \(P_\mathrm{BC}\) describing pure components AC and BC, respectively. The \(b_\mathrm{ABC\rightarrow AC} = b_\mathrm{ABC}\left(1\right)\) is a bowing parameter for nearly pure AC, while the \(b_\mathrm{ABC\rightarrow BC} = b_\mathrm{ABC}\left(0\right)\) is a bowing parameter for nearly pure BC.

III-V-V and II-VI-VI

For alloys of type ABxC1-x, the scheme reads

\[P_\mathrm{ABC}\left(x\right) = x\cdot P_\mathrm{AB} + \left[1-x\right]\cdot P_\mathrm{AC} - x\left[1-x\right]\cdot b_\mathrm{ABC},\]
\[b_\mathrm{ABC}\left(x\right) = x \cdot b_\mathrm{ABC\rightarrow AB} + \left[1-x\right] \cdot b_\mathrm{ABC\rightarrow AC}\]

where \(P_\mathrm{ABC}\left(x\right)\) is an interpolated material parameter of a two-component alloy ABxC1-x based on parameters \(P_\mathrm{AB}\) and \(P_\mathrm{AC}\) describing pure components AB and AC, respectively. The \(b_\mathrm{ABC\rightarrow AB} = b_\mathrm{ABC}\left(1\right)\) is a bowing parameter for nearly pure AB, while the \(b_\mathrm{ABC\rightarrow AC} = b_\mathrm{ABC}\left(0\right)\) is a bowing parameter for nearly pure AC.

Example and Syntax

Let’s consider the bowing paramters of energy gaps in \(Al_xGa_{1-x}As\) based on the Table XII. in [vurgaftmanjap2001]. The direct gap has a bowing parameter given by the formula

\[b_\mathrm{AlGaAs}\left(x\right) = -0.127 + 1.310 \cdot x\]

while indirect gaps to the points L and X have bowing parameters 0 and 0.055, respectively. Therefore, two bowing parameters needs to be included in the database, the one at mole fraction x=0 to describe the interpolation for small amounts of Al, near GaAs:

\[b_\mathrm{AlGaAs \rightarrow GaAs} = b_\mathrm{AlGaAs}\left(0\right) = -0.127 + 1.310 \cdot 0 = -0.127,\]

and at x=1 to describe the interpolation for small amounts of Ga, near AlAs:

\[b_\mathrm{AlGaAs \rightarrow AlAs} = b_\mathrm{AlGaAs}\left(1\right) = -0.127 + 1.310 \cdot 1 = 1.183.\]

Finally, the fraction-dependent bowing parameter is given by

\[b_\mathrm{AlGaAs}\left(x\right) = x \cdot b_\mathrm{AlGaAs \rightarrow AlAs} + \left[1-x\right] \cdot b_\mathrm{AlGaAs \rightarrow GaAs}\]

To use this of interpolation, one should not use ternary_zb{} or ternary_wz{} groups to define bowing parameters. Instead, groups bowing_zb{} or bowing_wz{} should be used to define valued of the bowing for extrememal concentrations, x=0 and x=1. The groups ternary2_zb{} and ternary2_wz{} should be used to relate all the bowing parameters and component materials for the alloy.

binary_zb{
    name    = AlAs
    valence = III_V

    # All the parameters of GaAs here (P_A)
}

binary_zb{
    name    = GaAs
    valence = III_V

    # All the parameters of InAs here (P_B)
}

# Al(x)Ga(1-x)As: (x=1)
bowing_zb{
    name       = "AlGaAs_Bowing_AlAs"
    valence    = III_V

    conduction_bands{
        Gamma{ bandgap = -0.127 + 1.310 * 1 }  # b_AB(x=1)
        X    { bandgap = 0.055              }  # b_AB(x=1)
    }
}

# Al(x)Ga(1-x)As: (x=0)
bowing_zb{
    name       = "AlGaAs_Bowing_GaAs"
    valence    = III_V

    conduction_bands{
        Gamma{ bandgap = -0.127 + 1.310 * 0 }  # b_AB(x=0)
        X    { bandgap = 0.055              }  # b_AB(x=0)
    }
}

ternary2_zb{
    name       = "Al(x)Ga(1-x)As"
    valence    = III_V

    binary_x   = AlAs
    binary_1_x = GaAs
    bowing_x   = AlGaAs_Bowing_AlAs   # b_AB(x=1)
    bowing_1_x = AlGaAs_Bowing_GaAs   # b_AB(x=0)
}

Note, that there is no bowing parameter specified for the indirect band gap to the L valley, which is equivalent to using linear interpolation (the bowing equal zero).

Hint

An alternative approach can be to use analytical formulas to define the bowing parameter with the mole fraction as a variable.

Three-component alloys

Three-component alloys are typically called ternary alloys when group-IV are mixed (IV-IV-IV) and quaternary alloys in the case of III-V or II-VI binary compounds (III-V-V-V, III-III-III-V, II-VI-VI-VI, and II-II-II-VI).

Examples of such alloys are: SixGey Sn1-x-y, AlxGayIn1-x-yN, and GaPxAsySb1-x-y.

IV-IV-IV

For alloys of type AxByC1-x-y, having \(w = 1-x-y\), the scheme reads

\[\begin{split}\begin{aligned} P_\mathrm{ABC}\left(x,y\right) &= P'_{AB}\left(x,y\right) + P'_{AC}\left(x,w\right) + P'_{BC}\left(y,w\right)\\ &- xy\cdot b'_\mathrm{AB}\left(x,y\right) - xw\cdot b'_\mathrm{AC}\left(x,w\right)- yw\cdot b'_\mathrm{BC}\left(y,w\right) \\ &- xyw\cdot b_\mathrm{ABC}. \end{aligned}\end{split}\]

The \(P'_{AB}\left(x,y\right)\), \(P'_{AC}\left(x,w\right)\), and \(P'_{BC}\left(y,w\right)\) are linear combinations of parameters \(P_\mathrm{A}\), \(P_\mathrm{B}\), and \(P_\mathrm{C}\).

\[\begin{split}\begin{aligned} P'_{AB}\left(x,y\right) & = x\cdot P_\mathrm{A} + y\cdot P_\mathrm{B} \\ P'_{AC}\left(x,w\right) & = x\cdot P_\mathrm{A} + w\cdot P_\mathrm{C} \\ P'_{BC}\left(y,w\right) & = y\cdot P_\mathrm{B} + w\cdot P_\mathrm{C} \\ \end{aligned}\end{split}\]

The \(b'_\mathrm{AB}\left(x,y\right)\), \(b'_\mathrm{AC}\left(x,w\right)\), and \(b'_\mathrm{BC}\left(y,w\right)\) are two-component bowing parameters. They can be equal zero, constant or dependent on mole fraction as:

\[\begin{split}\begin{aligned} b'_\mathrm{AB}\left(x,y\right) & = \frac{x\cdot b_\mathrm{AB\rightarrow A}+y\cdot b_\mathrm{AB\rightarrow B}}{x + y} \\ b'_\mathrm{AC}\left(x,w\right) & = \frac{x\cdot b_\mathrm{AC\rightarrow A}+w\cdot b_\mathrm{AC\rightarrow C}}{x + w} \\ b'_\mathrm{BC}\left(y,w\right) & = \frac{y\cdot b_\mathrm{BC\rightarrow B}+w\cdot b_\mathrm{BC\rightarrow C}}{y + w}. \end{aligned}\end{split}\]

The \(b_\mathrm{ABC}\) is a three-component bowing parameter.

III-III-III-V and II-II-II-VI

For alloys of type AxByC1-x-yD, having \(w = 1-x-y\), the scheme reads

\[\begin{split}\begin{aligned} P_\mathrm{ABCD}\left(x,y\right) &= P'_{ABD}\left(x,y\right) + P'_{ACD}\left(x,w\right) + P'_{BCD}\left(y,w\right)\\ &- xy\cdot b_\mathrm{ABD}\left(x\right) - xw\cdot b_\mathrm{ACD}\left(x\right)- yw\cdot b_\mathrm{BCD}\left(x\right) \\ &- xyw\cdot b_\mathrm{ABCD}, \end{aligned}\end{split}\]

The \(P'_{ABD}\left(x,y\right)\), \(P'_{ACD}\left(x,w\right)\), and \(P'_{BCD}\left(y,w\right)\) are linear combinations of parameters \(P_\mathrm{AD}\), \(P_\mathrm{BD}\), and \(P_\mathrm{CD}\).

\[\begin{split}\begin{aligned} P'_{ABD}\left(x,y\right) & = x\cdot P_\mathrm{AD} + y\cdot P_\mathrm{BD} \\ P'_{ACD}\left(x,w\right) & = x\cdot P_\mathrm{AD} + w\cdot P_\mathrm{CD} \\ P'_{BCD}\left(y,w\right) & = y\cdot P_\mathrm{BD} + w\cdot P_\mathrm{CD} \\ \end{aligned}\end{split}\]

The \(b'_\mathrm{ABD}\left(x,y\right)\), \(b'_\mathrm{ACD}\left(x,w\right)\), and \(b'_\mathrm{BCD}\left(y,w\right)\) are two-component bowing parameters. They can be equal zero, constant or dependent on mole fraction as:

\[\begin{split}\begin{aligned} b'_\mathrm{ABD}\left(x,y\right) & = \frac{x\cdot b_\mathrm{ABD\rightarrow AD}+y\cdot b_\mathrm{ABD\rightarrow BD}}{x + y} \\ b'_\mathrm{ACD}\left(x,w\right) & = \frac{x\cdot b_\mathrm{ACD\rightarrow AD}+w\cdot b_\mathrm{ACD\rightarrow CD}}{x + w} \\ b'_\mathrm{BCD}\left(y,w\right) & = \frac{y\cdot b_\mathrm{BCD\rightarrow BD}+w\cdot b_\mathrm{BCD\rightarrow CD}}{y + w}. \end{aligned}\end{split}\]

The \(b_\mathrm{ABCD}\) is a three-component bowing parameter.

III-V-V-V and II-VI-VI-VI

For alloys of type ABxCyD1-x-y, having \(w = 1-x-y\), the scheme reads

\[\begin{split}\begin{aligned} P_\mathrm{ABCD}\left(x,y\right) &= P'_{ABC}\left(x,y\right) + P'_{ABD}\left(x,w\right) + P'_{ACD}\left(y,w\right)\\ &- xy\cdot b_\mathrm{ABC}\left(x\right) - xw\cdot b_\mathrm{ABD}\left(x\right)- yw\cdot b_\mathrm{ACD}\left(x\right) \\ &- xyw\cdot b_\mathrm{ABCD}, \end{aligned}\end{split}\]

The \(P'_{ABC}\left(x,y\right)\), \(P'_{ABD}\left(x,w\right)\), and \(P'_{ACD}\left(y,w\right)\) are linear combinations of parameters \(P_\mathrm{AB}\), \(P_\mathrm{AC}\), and \(P_\mathrm{AD}\).

\[\begin{split}\begin{aligned} P'_{ABC}\left(x,y\right) & = x\cdot P_\mathrm{AB} + y\cdot P_\mathrm{AC} \\ P'_{ABD}\left(x,w\right) & = x\cdot P_\mathrm{AB} + w\cdot P_\mathrm{AD} \\ P'_{ACD}\left(y,w\right) & = y\cdot P_\mathrm{AC} + w\cdot P_\mathrm{AD} \\ \end{aligned}\end{split}\]

where \(b_\mathrm{ABC}\), \(b_\mathrm{ABD}\), and \(b_\mathrm{ACD}\) are two-component bowing parameters and \(b_\mathrm{ABCD}\) is a three-component bowing parameter.

The \(b'_\mathrm{ABC}\left(x,y\right)\), \(b'_\mathrm{ABD}\left(x,w\right)\), and \(b'_\mathrm{ACD}\left(y,w\right)\) are two-component bowing parameters. They can be equal zero, constant or dependent on mole fraction as:

\[\begin{split}\begin{aligned} b'_\mathrm{ABC}\left(x,y\right) & = \frac{x\cdot b_\mathrm{ABC\rightarrow AB}+y\cdot b_\mathrm{ABC\rightarrow AC}}{x + y} \\ b'_\mathrm{ABD}\left(x,w\right) & = \frac{x\cdot b_\mathrm{ABD\rightarrow AB}+w\cdot b_\mathrm{ABD\rightarrow AD}}{x + w} \\ b'_\mathrm{ACD}\left(y,w\right) & = \frac{y\cdot b_\mathrm{ACD\rightarrow AC}+w\cdot b_\mathrm{ACD\rightarrow AD}}{y + w}. \end{aligned}\end{split}\]

The \(b_\mathrm{ABCD}\) is a three-component bowing parameter.

Syntax

As the two-component bowing parameters can be linearly dependent on composition, constant, or equal zero, one needs to begin with defining the parameters for all material components with the bowing parameters for two-component alloys, following the syntax described before in sections Linear, Quadratic, and Cubic.

The three-component bowing parameter can be specified in the groups quaternary_zb{} or quaternary_wz{}. The role of these groups is to associate all component-materials, two-component bowing parameters with a name of the three-component alloy and to define the three-component bowing parameters if some of them are non-zero.

Let’consider SixGey Sn1-x-y. The parameters for three material components, Si, Ge, and, Sn need to be defined, as well as up to three sets of constant bowing parameters (or up to six sets of composition dependent bowing parameters), for SiGe, GeSn, and SiSn. The structure of database for this alloy with constant bowing parameters can be as follows.

binary_zb{
    name    = Si
    valence = IV_IV

    # All the parameters of Si here (P_A)
}

binary_zb{
    name    = Ge
    valence = IV_IV

    # All the parameters of Ge here (P_B)
}

binary_zb{
    name    = Sn
    valence = IV_IV

    # All the parameters of Sn here (P_C)
}

ternary_zb{
    name       = "Si(x)Ge(1-x)"
    valence    = IV_IV
    binary_x   = Si
    binary_1_x = Ge

    # Optional bowing parameters  (b_AB)
}

ternary_zb{
    name       = "Si(x)Sn(1-x)"
    valence    = IV_IV
    binary_x   = Si
    binary_1_x = Sn

    # Optional bowing parameters  (b_AC)
}

ternary_zb{
    name       = "Ge(x)Sn(1-x)"
    valence    = IV_IV
    binary_x   = Ge
    binary_1_x = Sn

    # Optional bowing parameters  (b_BC)
}

quaternary_zb {
    name       = "Si(x)Ge(y)Sn(1-x-y)"
    valence    = IV_IV
    binary1    = Si
    binary2    = Ge
    binary3    = Sn
    ternary12  = "Si(x)Ge(1-x)"
    ternary13  = "Si(x)Sn(1-x)"
    ternary23  = "Ge(x)Sn(1-x)"

    # Optional bowing parameters  (b_ABC)
}

Attention

The following sections are not finished.

Four-component alloys

Four-component alloys with a stoichiometry AxB1-xCyD1-yare typically used only for III-V and II-VI material systems (III-III-V-V and II-II-VI-VI). They are typically called quaternary alloys.

An exemplary alloy is GaxIn1-xPxAs1-y.

III-III-V-V and II-II-VI-VI

For alloys of type AxB1-xCyD1-y, having \(u=1-x\) and \(v=1-y\), the scheme reads

\[\begin{split}\begin{aligned} P_\mathrm{ABCD}\left(x,y\right) & = xy\cdot P_\mathrm{AC} + uy\cdot P_\mathrm{BC} + xv\cdot P_\mathrm{AD} + uv\cdot P_\mathrm{BD}\\ & - xuy \cdot b'_\mathrm{ABC}\left(x,u\right) - xuv \cdot b'_\mathrm{ABD}\left(x,u\right) - xyv \cdot b'_\mathrm{ACD}\left(y,v\right) - uyv \cdot b'_\mathrm{BCD}\left(y,v\right) \\ & - xuyv \cdot b_\mathrm{ABCD} \end{aligned}\end{split}\]

Groups required like for three-component alloys but instead of using quaternary_zb{} one should use quaternary4_zb{}.

### Indium Aluminum Arsenide Antimonide (InAlAsSb) ###

quaternary4_zb {
    name      = "In(x)Al(1-x)As(y)Sb(1-y)"
    valence   = III_V
    binary1   = InAs
    binary2   = AlAs
    binary3   = AlSb
    binary4   = InSb

    ternary12 = "In(x)Al(1-x)As"   # Note: In(x)Al(1-x)As and In(1-x)Al(x)As are equivalent
    ternary23 = "AlAs(x)Sb(1-x)"   #      as can be seen in the above equation.
    ternary34 = "Al(x)In(1-x)Sb"   #      So one has to use the name that is already defined in the database.
    ternary14 = "InAs(x)Sb(1-x)"
}
IV-IV-IV-IV

For alloys of type AxByCzD1-x-y-z, having \(w=1-x-y-z\), the scheme reads

\[\begin{split}\begin{aligned} P_\mathrm{ABCD}\left(x,y\right) & = x\cdot P_\mathrm{A} + y\cdot P_\mathrm{B} + z\cdot P_\mathrm{C} + w\cdot P_\mathrm{D}\\ & - xy \cdot b'_\mathrm{AB}\left(x,y\right) - xz \cdot b'_\mathrm{AC}\left(x,z\right) - xw \cdot b'_\mathrm{AD}\left(x,w\right) \\ & - yz \cdot b'_\mathrm{BC}\left(y,z\right) - yw \cdot b'_\mathrm{BD}\left(y,w\right) - zw \cdot b'_\mathrm{CD}\left(z,w\right) \\ & - xyz \cdot b'_\mathrm{ABC} - xyw \cdot b'_\mathrm{ABD} - xzw \cdot b'_\mathrm{ACD} - yzw \cdot b'_\mathrm{BCD} \\ & - xyzw \cdot b_\mathrm{ABCD} \end{aligned}\end{split}\]
quinternary_zb : _alloy_zb{ TYPE=group OPT=1

    binary_a{ TYPE=string }              #  A
    binary_b{ TYPE=string }              #  B
    binary_c{ TYPE=string }              #  C
    binary_d{ TYPE=string }              #  D

    ternary_ab{ TYPE=string }            #  A(x)B(1-x)
    ternary_ac{ TYPE=string }            #  A(x)C(1-x)
    ternary_ad{ TYPE=string }            #  A(x)D(1-x)
    ternary_bc{ TYPE=string }            #  B(x)C(1-x)
    ternary_bd{ TYPE=string }            #  B(x)D(1-x)
    ternary_cd{ TYPE=string }            #  C(x)D(1-x)

    quaternary_abc{ TYPE=string }        #  A(x)B(y)C(1-x-y)
    quaternary_abd{ TYPE=string }        #  A(x)B(y)D(1-x-y)
    quaternary_acd{ TYPE=string }        #  A(x)C(y)D(1-x-y)
    quaternary_bcd{ TYPE=string }        #  B(x)C(y)D(1-x-y)

    # from base group, optional quinternary bowing parameters
}

Six-component alloys

III-III-III-V-V and II-II-II-VI-VI

For alloys of type AxByC1-x-yDzE1-z, having \(u=1-x-y\) and \(w=1-z\), the scheme reads

\[\begin{split}\begin{aligned} P_\mathrm{ABCDE}\left(x,y,z\right) & = xz\cdot P_\mathrm{AD} + yz\cdot P_\mathrm{BD} + uz\cdot P_\mathrm{CD}\\ & + xw\cdot P_\mathrm{AE} + yw\cdot P_\mathrm{BE} + uw\cdot P_\mathrm{CE}\\ & - xyz \cdot b'_\mathrm{ABD}\left(x,y\right) - xuz \cdot b'_\mathrm{ACD}\left(x,u\right) - yuz \cdot b'_\mathrm{BCD}\left(y,u\right) \\ & - xyw \cdot b'_\mathrm{ABE}\left(x,y\right) - xuw \cdot b'_\mathrm{ACE}\left(x,u\right) - yuw \cdot b'_\mathrm{BCE}\left(y,u\right) \\ & - xzw \cdot b'_\mathrm{ADE}\left(z,w\right) - yzw \cdot b'_\mathrm{BDE}\left(z,w\right) - uzw \cdot b'_\mathrm{CDE}\left(z,w\right) \\ & - xyzw \cdot b'_\mathrm{ABDE} - xuzw \cdot b'_\mathrm{ACDE} - yuzw \cdot b'_\mathrm{BCDE} \\ & - xyuz \cdot b'_\mathrm{ABCD} - xyuw \cdot b'_\mathrm{ABCE} \\ & - xyuzw \cdot b_\mathrm{ABCDE} \end{aligned}\end{split}\]
quinternary6_zb : _alloy_zb{ TYPE=group OPT=1

    binary_a_d{ TYPE=string }            #  AD
    binary_b_d{ TYPE=string }            #  BD
    binary_c_d{ TYPE=string }            #  CD
    binary_a_e{ TYPE=string }            #  AE
    binary_b_e{ TYPE=string }            #  BE
    binary_c_e{ TYPE=string }            #  CE

    ternary_ab_d{ TYPE=string }          #  A(x)B(1-x)D
    ternary_ac_d{ TYPE=string }          #  A(x)C(1-x)D
    ternary_bc_d{ TYPE=string }          #  B(x)C(1-x)D
    ternary_ab_e{ TYPE=string }          #  A(x)B(1-x)E
    ternary_ac_e{ TYPE=string }          #  A(x)C(1-x)E
    ternary_bc_e{ TYPE=string }          #  B(x)C(1-x)E
    ternary_a_de{ TYPE=string }          #  AD(x)E(1-x)
    ternary_b_de{ TYPE=string }          #  BD(x)E(1-x)
    ternary_c_de{ TYPE=string }          #  CD(x)E(1-x)

    quaternary_abc_d{ TYPE=string }      #  A(x)B(y)C(1-x-y)D
    quaternary_abc_e{ TYPE=string }      #  A(x)B(y)C(1-x-y)E
    quaternary_ab_de{ TYPE=string }      #  A(x)B(1-x)D(y)E(1-y)
    quaternary_ac_de{ TYPE=string }      #  A(x)C(1-x)D(y)E(1-y)
    quaternary_bc_de{ TYPE=string }      #  B(x)C(1-x)D(y)E(1-y)

    # from base group, optional quinternary bowing parameters
}

Eight-component alloys

\(A_xB_{1-x}C_{y}D_{1-y}E_{z}F_{1-z}\)

quaternary8_zb : _alloy_zb{ TYPE=group OPT=1

    binary_a_c_e{ TYPE=string }          #  ACE
    binary_b_c_e{ TYPE=string }          #  BCE
    binary_a_d_e{ TYPE=string }          #  ADE
    binary_b_d_e{ TYPE=string }          #  BDE
    binary_a_c_f{ TYPE=string }          #  ACF
    binary_b_c_f{ TYPE=string }          #  BCF
    binary_a_d_f{ TYPE=string }          #  ADF
    binary_b_d_f{ TYPE=string }          #  BDF

    ternary_ab_c_e{ TYPE=string }        #  A(x)B(1-x)CE
    ternary_ab_d_e{ TYPE=string }        #  A(x)B(1-x)DE
    ternary_ab_c_f{ TYPE=string }        #  A(x)B(1-x)CF
    ternary_ab_d_f{ TYPE=string }        #  A(x)B(1-x)DF
    ternary_a_cd_e{ TYPE=string }        #  AC(x)D(1-x)E
    ternary_b_cd_e{ TYPE=string }        #  BC(x)D(1-x)E
    ternary_a_cd_f{ TYPE=string }        #  AC(x)D(1-x)F
    ternary_b_cd_f{ TYPE=string }        #  BC(x)D(1-x)F
    ternary_a_c_ef{ TYPE=string }        #  ACE(x)F(1-x)
    ternary_b_c_ef{ TYPE=string }        #  BCE(x)F(1-x)
    ternary_a_d_ef{ TYPE=string }        #  ADE(x)F(1-x)
    ternary_b_d_ef{ TYPE=string }        #  BDE(x)F(1-x)

    quarternary_ab_cd_e{ TYPE=string }   #  A(x)B(1-x)C(y)D(1-y)E
    quarternary_ab_cd_f{ TYPE=string }   #  A(x)B(1-x)C(y)D(1-y)F
    quarternary_ab_c_ef{ TYPE=string }   #  A(x)B(1-x)CE(y)F(1-y)
    quarternary_ab_d_ef{ TYPE=string }   #  A(x)B(1-x)DE(y)F(1-y)
    quarternary_a_cd_ef{ TYPE=string }   #  AC(x)D(1-x)E(y)F(1-y)
    quarternary_b_cd_ef{ TYPE=string }   #  BC(x)D(1-x)E(y)F(1-y)

    # from base group, optional quinternary bowing parameters
}

Note

If you need other interpolation schemes for your research, raise a support ticket attaching formulas of your interest, related references, and explanation why it’s valuable.

A brief introduction to quaternaries is shown in this Powerpoint presentation (Quaternaries.pptx , Quaternaries.pdf).