Subsections

• 3.3.1.1 Temperature dependence of the bandgap
• 3.3.1.2 Semiconductor Alloys

3.3.1 Bandgap Energy

The bandgap (or forbidden energy zone) is one of the most important semiconductor parameters. Various models define the temperature dependence of the bandgap energy in semiconductors (e.g. [112]). For an alloy $\mathrm {A}_{1-x}\mathrm {B}_x$, the temperature-dependent bandgaps of the constituents (A and B) are calculated first. The bandgap and the energy offset are then calculated depending on the material composition. This is important to assure consistency between the values for alloy materials at $x=0$ and $x=1$ and the values for the respective basic materials. For materials where the bandgap changes between direct and indirect the multiple valley conduction bands are considered.

3.3.1.1 Temperature dependence of the bandgap

In MINIMOS-NT the model of Varshni [112] is used for basic materials. The temperature dependence is calculated by (3.60), where $E_{\mathrm{g,0}}$ is the bandgap at 0 K.

$$E_{\mathrm{g}}= E_{\mathrm{g,0}}-\frac{\alpha\cdot T_{{\mathrm{L}}}^{2}}{\beta +T_{{\mathrm{L}}}}$$ (3.60)

The parameter values are summarized in Table 3.8. Note, for these materials always the lowest conduction band valley minimum is taken into account. In addition, the resulting bandgaps at 300 K, $E_{\mathrm{g,300}}$, are included in Table 3.9.

Table 3.8: Parameter values for modeling the bandgap energies

Material	Minimum	$E_{\mathrm{g,0}}$ [eV]	$\alpha$ [eV/K]	$\beta$ [K]	Reported $E_{\mathrm{g,0}}$ [eV]	References
Si	X	1.1695	4.73 $\cdot {10}^{-4}$	636	1.17	[90,86]
Ge	L	0.7437	4.774 $\cdot {10}^{-4}$	235	0.74	[86]
GaAs	G	1.521	5.58 $\cdot {10}^{-4}$	220	1.51-1.55	[113,114,115]
AlAs	X	2.239	6.0 $\cdot {10}^{-4}$	408	2.22-2.239	[94,90,108]
InAs	G	0.420	2.5 $\cdot {10}^{-4}$	75	0.414-0.43	[92,113]
InP	G	1.421	3.63 $\cdot {10}^{-4}$	162	1.42-1.432	[108,86,114]
GaP	X	2.338	5.771 $\cdot {10}^{-4}$	372	2.338-2.346	[108,86,90]

Table 3.9: Bandgap energies at room temperature compared to reported data

Material	Minimum	$E_{\mathrm{g,300}}$ [eV]	Reported $E_{\mathrm{g,300}}$ [eV]	References
Si	X	1.124	1.12-1.1242	[85,86,90]
Ge	L	0.663	0.66-0.67	[90,85,86]
GaAs	G	1.424	1.42-1.43	[116,95,104,85]
AlAs	X	2.163	2.14-2.168	[94,116,104,90]
InAs	G	0.360	0.354-0.37	[93,92,115,117]
InP	G	1.350	1.34-1.351	[90,115,92]
GaP	X	2.261	2.26-2.272	[85,92,90]

For $\mathrm{Si}$ two additional models can be chosen which are based on polynomial fits of second and third order. The first one after Gaensslen [118,119] was also used in MINIMOS 6. The second model is based on data from Green [120].

$$E_{\mathrm{g}}= E_0 + E_1\cdot\left(\frac{T_{\mathrm{L}}}{\mathrm... ...300 K}}\right)^2+E_3\cdot\left(\frac{T_{\mathrm{L}}}{\mathrm{300 K}}\right)^3$$ (3.61)

The parameter values for these models are summarized in Table 3.10.

Table 3.10: Parameter values for modeling the bandgap energies

Model	$E_0$ [eV]	$E_1$ [eV]	$E_2$ [eV]	$E_3$ [eV]	$E_{\mathrm{g,300}}$ [eV]
Gaensslen	1.1785	-0.02708	-0.02745	0.0	1.124
Green	1.17	0.00572	-0.06948	0.018	1.124

In Fig. 3.8 the results obtained with the three different models for Si are compared to data from [120]. In Fig. 3.9 and Fig. 3.10 the temperature dependence of the direct gap in GaAs and InP, respectively, are compared to other models.

3.3.1.2 Semiconductor Alloys

In the case of alloy materials the temperature-dependent bandgaps of the constituents, $E_{\mathrm{g}}^\mathrm {A}$ and $E_{\mathrm{g}}^\mathrm {B}$, are calculated by (3.60). However, for materials where the bandgap changes between direct and indirect the multiple valley conduction bands are considered. For that purpose, additional model parameters are needed for the higher energy valleys in the respective III-V binary materials (Table 3.11). In addition, the resulting bandgaps at 300 K, $E_{\mathrm{g,300}}$, are included in Table 3.12.

Table 3.11: Parameter values for modeling the bandgap energies

Material	Minimum	$E_{\mathrm{g,0}}$ [eV]	$\alpha$ [eV/K]	$\beta$ [K]	Reported $E_{\mathrm{g,0}}$ [eV]	References
GaAs	X	1.981	4.6 $\cdot {10}^{-4}$	204	1.9-1.91	[116,104]
AlAs	G	2.891	8.78 $\cdot {10}^{-4}$	332	2.907-3.02	[59,95,94,104]
InAs	X	2.278	5.78 $\cdot {10}^{-4}$	83
InP	X	2.32	7.66 $\cdot {10}^{-4}$	327	2.32-2.38	[109,84,121]
GaP	G	2.88	8.0 $\cdot {10}^{-4}$	300	2.869-2.895	[91,84,122]

Table 3.12: Bandgap energies at room temperature compared to reported data

Material	Minimum	$E_{\mathrm{g,300}}$ [eV]	Reported $E_{\mathrm{g,300}}$ [eV]	References
GaAs	X	1.899	1.9-1.91	[116,104]
AlAs	G	2.766	2.671-2.766	[93,59]
InAs	X	2.142	1.37-2.14	[116,108,123]
InP	X	2.21	2.21-2.30	[116,123,121]
GaP	G	2.76	2.73-2.85	[84]

The bandgap and the energy offset of an alloy $\mathrm {A}_{1-x}\mathrm {B}_x$ are calculated by

$$E_{\mathrm{g,X}}^\mathrm {AB} = E_{\mathrm{g,X}}^\mathrm {A}\cdot\left(1-x\right) + E_{\mathrm{g,X}}^\mathrm {B} \cdot x+C_\mathrm {g,X}\cdot(1-x)\cdot x$$ (3.62)

$$E_{\mathrm{g,\Gamma}}^\mathrm {AB} = E_{\mathrm{g,\Gamma}}^\mathrm {A}\cdot\left(1-x\right) + E_{\mathrm{g,\Gamma}}^\mathrm {B} \cdot x+C_\mathrm {g,\Gamma}\cdot(1-x)\cdot x$$ (3.63)

$$E_{\mathrm{g}}^\mathrm {AB} = \mathrm {min}\left(E_{\mathrm{g,X}}^\mathrm {AB};E_{\mathrm{g,\Gamma}}^\mathrm {AB}\right)$$ (3.64)

The bowing parameters $C_\mathrm {g,X}$ and $C_\mathrm {g,\Gamma}$ are summarized in Table 3.13. Additional bowing parameters $C_\mathrm {g}$ are given as a reference for the case when a one-valley bandgap fit is used.

$$E_{\mathrm{g}}^\mathrm {AB} = E_{\mathrm{g}}^\mathrm {A}\cdot\left(1-x\right) + E_{\mathrm{g}}^\mathrm {B} \cdot x+C_\mathrm {g}\cdot(1-x)\cdot x$$ (3.65)

Table 3.13: Parameter values for the bandgap of alloy materials

Material	$C_\mathrm {g,\Gamma}$ [eV]	$C_\mathrm {g,X}$ [eV]	$C_{\mathrm {g}}$ [eV]	Reported	References
SiGe			-0.4	-0.4	[124]
AlGaAs	0.0	-0.143	0.7	0$^\Gamma$, -0.143$^\mathrm {X}$	[93,125]
InGaAs			-0.475	-0.4, -0.475, -0.555	[108,90,71]
InAlAs	-0.3	-0.713	1.2	-0.689$^\Gamma$,-0.24$^\Gamma$	[108,90]
InAsP			-0.32	-0.101, -0.32	[108,90]
GaAsP	-0.21	-0.21	0.5	-0.176$^\Gamma$,-0.23$^\Gamma$	[108,126]
InGaP	-0.67	-0.17	0.6	-0.786$^\Gamma$,-0.6$^\Gamma$,-0.18$^\mathrm {X}$	[108,90,109]

For example, such a one-valley bandgap fit is used in the case of the technologically important strained Si$_{1-x}$Ge$_x$ grown on Si (see Fig. 3.11). In certain cases for $x>0.6$ the bandgap can become smaller than the one of pure Ge [127] depending on the strain. In the unstrained case, however, an $\mathrm {X}$-to-${\mathrm{L}}$ gap transition is observed at about $x=0.85$.

The material composition dependence of the $\Gamma$, ${\mathrm{L}}$, and $\mathrm {X}$-band gaps in Al$_x$Ga$_{1-x}$As at 300 K is shown in Fig. 3.12. A direct-to-indirect gap transition is observed at about $x=0.45$. The one-valley bandgap fit which is included for comparison gives a good agreement only for $x<0.28$. In Fig. 3.13 the temperature dependence of the bandgap in Al$_x$Ga$_{1-x}$As with Al content as a parameter is shown. In Fig. 3.14 the material composition dependence of the $\Gamma$, ${\mathrm{L}}$, and $\mathrm {X}$-band gaps in In$_x$Ga$_{1-x}$As at 300 K is shown. As can be seen this material has a direct bandgap for the entire composition range. Therefore, only the energy of the $\Gamma$ valley is taken into account. However, in the case of strain the bandgap can significantly differ [128]. For example, a good fit to the strained bandgap values of In$_x$Ga$_{1-x}$As grown on GaAs can be achieved by changing the $E_{\mathrm{g,0}}$ of InAs from 0.42 eV to 0.58 eV. In Fig. 3.15 the temperature dependence of the bandgap in In$_x$Ga$_{1-x}$As with In content as a parameter is shown. In the case of In$_x$Al$_{1-x}$As there is direct-to-indirect gap transition at about $x=0.3$ and the use of one valley fit is nearly impossible. It gives comparatively accurate values for the lattice matched case of $x=0.52$ and above (see Fig. 3.16). InAs$_x$P$_{1-x}$ has a direct gap for the complete material composition range so only the $\Gamma$ valley energy is calculated. The bowing parameter value suggested in [108] is used. GaAs$_{1-x}$P$_x$ has a direct-to-indirect gap transition at about $x=0.5$. As can be seen in Fig. 3.17 a one valley model can be successfully used for this material. In Fig. 3.18 the material composition dependence of the $\Gamma$, ${\mathrm{L}}$, and $\mathrm {X}$-bandgaps in Ga$_x$In$_{1-x}$P at 300 K is shown. The direct-to-indirect gap transition is at about $x=0.7$. Only at the technologically important value of $x=0.51$, when Ga$_x$In$_{1-x}$P lattice matches the one of GaAs the one-valley model gives a good fit for the bandgap. The temperature dependence of the bandgap in Ga$_x$In$_{1-x}$P with Ga content as a parameter is shown in Fig. 3.19.

Figure 3.8: Comparison of different models for the temperature dependence of the bandgap in Si

Figure 3.9: Comparison of different models for the temperature dependence of the bandgap in GaAs

Figure 3.10: Comparison of different models for the temperature dependence of the bandgap in InP

Figure 3.11: Material composition dependence of the $\Gamma$, ${\mathrm{L}}$, and $\mathrm {X}$-bandgaps in Si$_{1-x}$Ge$_x$ at 300 K

Figure 3.12: Material composition dependence of the $\Gamma$, ${\mathrm{L}}$, and $\mathrm {X}$-bandgaps in Al$_x$Ga$_{1-x}$As at 300 K

Figure 3.13: Temperature dependence of the bandgap in Al$_x$Ga$_{1-x}$As with Al content as a parameter

Figure 3.14: Material composition dependence of the $\Gamma$, ${\mathrm{L}}$, and $\mathrm {X}$-bandgaps in In$_x$Ga$_{1-x}$As at 300 K

Figure 3.15: Temperature dependence of the bandgap in In$_x$Ga$_{1-x}$As with In content as a parameter

Figure 3.16: Material composition dependence of the $\Gamma$, ${\mathrm{L}}$, and $\mathrm {X}$-bandgaps in In$_x$Al$_{1-x}$As at 300 K

Figure 3.17: Material composition dependence of the $\Gamma$, ${\mathrm{L}}$, and $\mathrm {X}$-bandgaps in GaAs$_{1-x}$P$_x$ at 300 K

Figure 3.18: Material composition dependence of the $\Gamma$, ${\mathrm{L}}$, and $\mathrm {X}$-bandgaps in Ga$_x$In$_{1-x}$P at 300 K

Figure 3.19: Temperature dependence of the bandgap in Ga$_x$In$_{1-x}$P with Ga content as a parameter