Database

Quaternaries

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First of all there are two types of quaternaries:

• A_xB_1-xC_yD_1-y
  consisting of four binaries AC, AD, BC and BD and containing 2 group III and 2 group V elements.
• AB_xC_yD_1-x-y or B_xC_yD_1-x-yA
  consisting of three binaries AB, AC and AD and containing 1 group III and 3 group V elements or
  consisting of three binaries BA, CA and DA and containing 3 group III and 1 group V elements.

As a first approximation one could try to use simple linear interpolation from the binary parameters (Bs).

• Q(x,y) = xyB_AC + x(1-y)B_AD + (1-x)yB_BC + (1-x)(1-y)B_BD                (1)
• Q(x,y) = xB_AB + yB_AC + (1-x-y)B_AD                                              (2)

If relationships for the ternary parameters are available one should use the weighted sum of the related ternary values (Ts):

• Q(x,y) = { x(1-x) [ yT_ABC(x) + (1-y)T_ABD(x) ] + y(1-y) [ xT_ACD(y) + (1-x)T_BCD(y) ]  } /  [ x(1-x) + y(1-y) ]    (3)
• Q(x,y) = [ xyT_ABC(u) + y(1-x-y)T_ACD(v) + (1-x-y)xT_ABD(w) ]  /  [ xy + y(1-x-y) + (1-x-y)x ]                      (4)
  with   u = (1-x-y) / 2
           v = (2-x-2y) / 2
          w = (2-2x-y) / 2

The lattice constant is known to obey Vegard's law well, i.e. to vary linearly with composition.

The band gaps in many ternary alloys (A_xB_1-xC or CA_xB_1-x) can be approximated in the form of the usual quadratic function

T_ABC = xB_AC + (1-x) B_BC - x(1-x)C_ABC

where C_ABC is the bowing parameter.

Lattice matched quaternaries

If one considers lattice matched quaternaries (lattice matched to the common substrate materials GaAs, InP, InAs or GaSb) one can express Q(x,y) as Q(x,y(x)). As we only have one free parameter in this case, our simulator will simply use the same scheme as in the case of ternary alloys that also have only one free parameter. The quaternary will be represented as a combination of two lattice-matched constituents, one of which must be a ternary while the other may be either a binary or a ternary. The treatment is considerably simplified by the usual absence of any strong bowing of the band parameters for such an alloy, which is expected on theoretical grounds because the two constituents have identical lattice constants.

In the following we will assume two lattice matched binary or ternary end points, E and F, which are combined with arbitrarily composition z to form the lattice matched quaternary alloy E_zF_1-z. we then employ the expression

Q_EF(z) = zT_E + (1-z)T_F - z(1-z)C_EF                                                          (5)

where T_E and T_F are the values at the end poits and C_EF is the additional bowing associated with combining the two end point materials to form a quaternary. For lattice matched quateraries, using equation (5) with the available experimental evidence to determine C_EF for each property should lead to a better representation then either the procedure of (3) and (4) or simple linear interpolation (1, 2).

Example:

• AlGaInP lattice matched to GaAs
  
  (Al_zGa_1-z)_0.51In_0.49P       or, more accurately     (Al_0.52In_0.48P)_z(Ga_0.51In_0.49P)_1-z
  
  will be represented in our database as
  
  $ternary-zb-default
 ternary-type                = (Al(x)Ga(1-x))0.51In0.49P-zb-default
 binary(x)                   = Al0.52In0.48P-zb-default
 binary(1-x)                 = Ga0.51In0.49P-zb-default
 
 bow-conduction-band-masses  = 0d0     0d0     0d0
 ...
$end_ternary-zb-default

For this procedure, unfortunately, the materials Al0.52In0.48P-zb-default and Ga0.51In0.49P-zb-default have to be defined explicitly in the database.

(This text is taken party from Vurgaftman et al., JAP 89, 5815 (2001))