Compound Semiconductors

The constitutive equation for compound semiconductors is somewhat different than elemental semiconductors because the reference reservoir is taken as a reservoir of the stoichiometric compound. [8,9,10] The chemical potentials of the components of the compound can vary according to environmental considerations. [11] Taking GaAs as an example, the chemical potentials of the components are fixed according to

$$\mu_{GaAs}=\mu_{Ga}+\mu_{As}$$ (3)

To calculate the energy of formation of a vacancy on a site in GaAs, the constitutive relation is

$$E_{form}=E_{total}-n^{Ga}\mu_{Ga}-n^{As}\mu_{As}$$ (4)

Substituting equation 3 yields

$$E_{form}=E_{total}-n^{Ga}\mu_{Ga}-n^{As}(\mu_{GaAs}-\mu_{Ga})$$ (5)

which can be rearranged to

$$E_{form}=E_{total}-n^{As}\mu_{GaAs}-(n^{Ga}-n^{As})\mu_{Ga}$$ (6)

The energy of formation is given relative to a perfect crystal of GaAs, with $n^{As}$ GaAs molecules. As mentioned above, the value of $\mu_{Ga}$ can vary according to environmental conditions and the value of $\mu_{As}$ will change according to equation 3.

The range of allowed values for $\mu_{Ga}$ and $\mu_{As}$ is bracketed by formation of bulk Ga or As metals. When $\mu_{Ga\:(bulk)} < \mu_{Ga}$ or $\mu_{As\:(bulk)} < \mu_{As}$ then the precipitation of elemental solids is preferred to the formation of GaAs. Because the GaAs of the system is in equilibrium with bulk GaAs, equation 3 can be rewritten.

$$\mu_{Ga}+\mu_{As}=\mu_{GaAs}=\mu_{GaAs\:(bulk)}=\mu_{Ga\:(bulk)}+\mu_{As\:(bulk)}+H_{f\:GaAs}$$ (7)

This relation² can be used to give the range for $\mu_{Ga}$ to satisfy equation 6. One limit is given simply by $\mu_{Ga}=\mu_{Ga\:(bulk)}$. For the other limit the value $\mu_{As}=\mu_{As\:(bulk)}$ is substituted into the left hand side of 7 and results in $\mu_{Ga}=\mu_{Ga}+H_{f\:GaAs}$. [10] The allowed range for $\mu_{Ga}$ is $(\mu_{Ga\:(bulk)}+H_{f\:GaAs})<\mu_{Ga}<\mu_{Ga\:(bulk)}$.