Density Functional Theory Study on Formation Energy, Formation Entropy, and Thermal Equilibrium Concentration of Tin and Vacancy in Germanium Layer
Authors: Sueoka Koji , Noda Yusuke 
Published: December 2024

Germanium tin (Ge1-x
Sn
x
)-related semiconductors are attractive for electronic and optoelectronic devices. One of the main problems with Ge1-x
Sn
x
layer growth is Sn precipitation since the thermal equilibrium concentration of Sn in the Ge matrix is as low as 1 at.% below 500oC [1]. There have been many experimental challenges to doping higher concentrations of Sn in Ge [2]. The purpose of this study is to derive the thermal equilibrium concentration of substitutional Sn (Sns) and interstitial Sn (Sni) in the Ge layer through theoretical calculations. The impact of the incorporation of vacancy (V) by Sn doping is another research topic in this study.

We carried out density functional theory calculations within the generalized gradient approximation (GGA) and sx-LDA [3] hybrid functional for electron exchange and correlation, as implemented in the CASTEP code. Two types of calculation models; two bulk cubic models and one (001) layer model, were prepared. For the bulk models, 64 and 512 Ge-atom supercells were prepared to calculate the formation energy (Ef
) of Sns, Sni at <110> dumbbell (D)-site, at tetrahedral (T)-site, at hexagonal (H)-site, V, and Sns
V in Ge crystals. The degeneracy number (w) of each defect was investigated and obtained as w = 1 (Sns), 12 (Sni at D-site), 1 (Sni at T-site), 3 (Sni at H-site), 3 (V), and 12 (V at nearest of Sns). The 64-atom model was also used for the formation (vibration) entropy (Sf
) calculations with the linear-response method [4]. For a layer model, 16 Ge atoms are included in each of the 32 layers. The depth dependence of Ef
of Sns, Sni, V, and SnV was obtained.

Table 1 summarizes the calculated Ef
and Sf
of each defect in bulk Ge crystal. It is clear that the Ef
of Sns from GGA is close to that from sx-LDA, while the Ef
of the other defects from GGA is far smaller than that from sx-LDA. This is due to the prediction of Ge as metallic from GGA. The defect levels of Sni, V, and SnV that should be formed in the bandgap (Eg
) are mixed with conduction band. By using the Ef
of Sns obtained from GGA (512 atoms) and the other defects from sx-LDA, the thermal equilibrium concentration Ceq
= w×4.42×1022 exp(Sf/k)exp(-Ef/kT) (cm-3) was determined. For Sns, Ceq
(Sns) = 4.42×1022 exp(1.80) exp(-0.28 eV/kT) cm-3 and Ceq
(Sns) = 3.71×1021 cm-3 at 500 oC, which is about 8.4 at.% of Ge concentration. This value is comparable to the experimental value of 1 at.% below 500oC [1]. The Ceq
(Sni) is negligible in comparison with the Ceq
(Sns) as the Ef
of Sni is larger than 4 eV. This result indicates that Sni atoms are not incorporated into the Ge matrix during layer growth.

Figure 1 (a) shows the arrangement of Sns atoms in the (001) layer model. The Ef
for Sns and Sni obtained from GGA are summarized in Table 2. The Ef
at the surface (1st layer) decreases dramatically to about 0 eV for Sns, and to 0.51 eV for Sni at T-site, as shown in Fig. 1 (b). By ignoring the difference in the number of sites and formation entropy of Sns and Sni on the surface, the ratio W of Sn present on the surface being distributed to interstitial and substitutional positions was obtained: W = Ceq
(Sni)/Ceq
(Sns) = exp(-0.51 eV/kT)/exp(0) = 0.05% at 500oC. Since Sni atoms are not incorporated into the Ge bulk, they will precipitate on the surface of the Ge layer.

Finally, Fig. 2 shows the charge-state dependence of the Ef
of V in Ge determined from sx-LDA. These results agree with the previous study using the HSE06 hybrid functional [5]. When the Fermi level is near the center of Eg
, there are three charge states of V: V
0, V
-1, and V
-2. The Ceq
(V) at the melting point is 2.9×1014 cm-3, which is one order smaller than the experimental value (2.9-3.9×1015 cm-3) [6]. It was also confirmed that the binding energy of Sn and V
0 is 0.54 eV, and Sn enters first then takes V from the surface.

Acknowledgement

This work was supported by JST, CREST Grant Number JPMJCR21C2, Japan.

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Figure 1