Isovalent dopant-vacancy clustering in germanium
锗中的等价掺杂剂空位聚类

Germanium  锗
Tin  锡
Lead  铅
Carbon  碳
Isovalent dopant  等价掺杂剂
Vacancy  空缺
Radiation hardening  辐射硬化

The frantic rate of transistor development as described by Moore’s law has led to the incorporation of more advanced semiconductor and oxide materials in transistors as compared to the more conventional Si and its native oxide silicon dioxide $\left({\mathrm{SiO}}_2\right)$$\left({\mathrm{SiO}}_2\right)$(SiO_(2))\left(\mathrm{SiO}_{2}\right) [1-6]. This was realized by innovative processes such as chemical vapor deposition (CVD) that allowed the formation of numerous oxide materials on top of the semiconductor, therefore there was no need to select the semiconductor material based on its native oxide (for example ${\mathrm{SiO}}_2$${\mathrm{SiO}}_2$SiO_(2)\mathrm{SiO}_{2} for Si ) [7,8]. Semiconductor materials such as Ge and silicon germanium ( ${\mathrm{Si}}_{1-\mathrm{x}}{\mathrm{Ge}}_{\mathrm{x}}$${\mathrm{Si}}_{1-\mathrm{x}}{\mathrm{Ge}}_{\mathrm{x}}$Si_(1-x)Ge_(x)\mathrm{Si}_{1-\mathrm{x}} \mathrm{Ge}_{\mathrm{x}} ) are advantageous as compared to Si due to their higher carrier mobilities, and hence conductivities [9,10]. Although Ge is isostructural with Si its defective processes and energetics are considerably different and this should be considered for doping processes [11-13]. Of particular importance is the lower formation energy of vacancies in Ge as compared to Si as this will result in a significantly higher concentration of vacancies in Ge [11-13]. These vacancies in turn will interact with dopant atoms to form ${\mathrm{D}}_{\mathrm{n}}{\mathrm{V}}_{\mathrm{m}}$${\mathrm{D}}_{\mathrm{n}}{\mathrm{V}}_{\mathrm{m}}$D_(n)V_(m)\mathrm{D}_{\mathrm{n}} \mathrm{V}_{\mathrm{m}} clusters in Ge [14-16]. The drive for $n$$n$nn-type and $p$$p$pp-type dopants to form ${\mathrm{D}}_{\mathrm{n}}{\mathrm{V}}_{\mathrm{m}}$${\mathrm{D}}_{\mathrm{n}}{\mathrm{V}}_{\mathrm{m}}$D_(n)V_(m)\mathrm{D}_{\mathrm{n}} \mathrm{V}_{\mathrm{m}} clusters is the reduction of the total energy due to elastic and electrostatic interactions [14-16]. For isovalent dopants, the contribution to cluster stability is the reduction of the total energy due to relieve of stresses (i.e. elastic interaction) (see Table 7).
摩尔定律所描述的晶体管发展速度极快，导致晶体管中掺入了更先进的半导体和氧化物材料，而更传统的硅及其天然氧化物二氧化硅 $\left({\mathrm{SiO}}_2\right)$$\left({\mathrm{SiO}}_2\right)$(SiO_(2))\left(\mathrm{SiO}_{2}\right) [1-6]。这是通过化学气相沉积（CVD）等创新工艺实现的，该工艺允许在半导体顶部形成许多氧化物材料，因此无需根据其天然氧化物（例如 ${\mathrm{SiO}}_2$${\mathrm{SiO}}_2$SiO_(2)\mathrm{SiO}_{2} Si）来选择半导体材料[7,8]。与 Si 相比，锗和硅锗（ ${\mathrm{Si}}_{1-\mathrm{x}}{\mathrm{Ge}}_{\mathrm{x}}$${\mathrm{Si}}_{1-\mathrm{x}}{\mathrm{Ge}}_{\mathrm{x}}$Si_(1-x)Ge_(x)\mathrm{Si}_{1-\mathrm{x}} \mathrm{Ge}_{\mathrm{x}} ）等半导体材料具有更高的载流子迁移率，因此具有更高的导电率[9,10]。尽管锗与 Si 是同构的，但其缺陷过程和能量学却有很大不同，掺杂过程中应考虑这一点[11-13]。与 Si 相比，锗的空位形成能较低，因为这将导致锗的空位浓度显著升高[11-13]。这些空位又会与掺杂剂原子相互作用，在锗中形成 ${\mathrm{D}}_{\mathrm{n}}{\mathrm{V}}_{\mathrm{m}}$${\mathrm{D}}_{\mathrm{n}}{\mathrm{V}}_{\mathrm{m}}$D_(n)V_(m)\mathrm{D}_{\mathrm{n}} \mathrm{V}_{\mathrm{m}} 簇[14-16]。 $n$$n$nn 型和 $p$$p$pp 型掺杂剂形成 ${\mathrm{D}}_{\mathrm{n}}{\mathrm{V}}_{\mathrm{m}}$${\mathrm{D}}_{\mathrm{n}}{\mathrm{V}}_{\mathrm{m}}$D_(n)V_(m)\mathrm{D}_{\mathrm{n}} \mathrm{V}_{\mathrm{m}} 簇的驱动力是由于弹性和静电相互作用而降低总能量[14-16]。对于等价掺杂剂，对簇稳定性的贡献是由于应力的消除（即弹性相互作用）而减少总能量（见表 7）。

In Si isovalent dopant-vacancy clustering has been thoroughly studied as it suppresses the formation of thermal donors, restrains
在 Si 中，等价掺杂剂空位聚类已被彻底研究，因为它抑制了热供体的形成，抑制了
dislocation movement, increases mechanical properties, and enhances oxygen precipitation [17-24].
位错运动，增加机械性能，增强氧沉淀[17-24]。

For the present work, we employed spin-polarized density functional theory (DFT) calculations to predict the lowest energy structures of substitutional (D), substitutional-vacancy (DV), vacancy-substitutionalvacancy (VDV) and divacancy-substitutional dopant ( $\mathrm{D}=\mathrm{C},\mathrm{Si},\mathrm{Sn}$$\mathrm{D}=\mathrm{C},\mathrm{Si},\mathrm{Sn}$D=C,Si,Sn\mathrm{D}=\mathrm{C}, \mathrm{Si}, \mathrm{Sn} and Pb ) in Ge . The electronic structures, formation and binding energies of these defects were also calculated.
在目前的工作中，我们采用自旋极化密度泛函理论（DFT）计算来预测 Ge 中取代（D）、取代空位（DV）、空位-取代空位（VDV）和离位-取代掺杂剂（ $\mathrm{D}=\mathrm{C},\mathrm{Si},\mathrm{Sn}$$\mathrm{D}=\mathrm{C},\mathrm{Si},\mathrm{Sn}$D=C,Si,Sn\mathrm{D}=\mathrm{C}, \mathrm{Si}, \mathrm{Sn} 和 Pb）的最低能结构。还计算了这些缺陷的电子结构、形成和结合能。

The plane wave DFT code VASP (Vienna $Ab$$Ab$AbA b initio Simulation Package, version 5.3.5) was used throughout this study [25]. VASP plane wave basis sets and projected augmented wave (PAW) potentials [26]. Here for all the calculations, the plane wave basis set the cut-off energy is set to 500 eV . Here the valence electronic configurations of $\mathrm{C},\mathrm{Si},\mathrm{Ge}$$\mathrm{C},\mathrm{Si},\mathrm{Ge}$C,Si,Ge\mathrm{C}, \mathrm{Si}, \mathrm{Ge}, Sn and Pb were $2s^{2}2p^2,3s^{2}3p^2,4s^{2}4p^2,5s^{2}5p^2$$2s^{2}2p^2,3s^{2}3p^2,4s^{2}4p^2,5s^{2}5p^2$2s^(2)2p^(2),3s^(2)3p^(2),4s^(2)4p^(2),5s^(2)5p^(2)2 s^{2} 2 p^{2}, 3 s^{2} 3 p^{2}, 4 s^{2} 4 p^{2}, 5 s^{2} 5 p^{2} and $6s^{2}6p^2$$6s^{2}6p^2$6s^(2)6p^(2)6 s^{2} 6 p^{2}, respectively. Exchange correlation is described by the generalized gradient approximation (GGA) as parameterized by Perdew, Burke and Ernzerhof (PBE) [27]. A 250 Ge atomic site supercell is used for the defect calculations with the cell size being 19.3 A. For the Ge supercell, $4\times 4\times 4$$4\times 4\times 4$4xx4xx44 \times 4 \times 4 Monkhorst-Pack k-point meshes were used [28]. The conjugate gradient algorithm [29] to obtain energy minimum structures (full relaxation of atomic positions and lattice constants). The force tolerance was set to $0.001\mathrm{eV}/\text{Å}$$0.001\mathrm{eV}/\text{Å}$0.001eV//"Å"0.001 \mathrm{eV} / \AA, whereas the accuracy for electronic minimization to ${10}^{-4}$${10}^{-4}$10^(-4)10^{-4} eV .
本研究使用平面波 DFT 代码 VASP（Vienna $Ab$$Ab$AbA b initio Simulation Package，5.3.5 版）[25]。VASP 平面波基集和投影增强波（PAW）电位[26]。在这里，对于所有计算，平面波基设置为截止能量设置为 500 eV。这里，Sn 和 Pb 的 $\mathrm{C},\mathrm{Si},\mathrm{Ge}$$\mathrm{C},\mathrm{Si},\mathrm{Ge}$C,Si,Ge\mathrm{C}, \mathrm{Si}, \mathrm{Ge} 价电子构型分别是 $2s^{2}2p^2,3s^{2}3p^2,4s^{2}4p^2,5s^{2}5p^2$$2s^{2}2p^2,3s^{2}3p^2,4s^{2}4p^2,5s^{2}5p^2$2s^(2)2p^(2),3s^(2)3p^(2),4s^(2)4p^(2),5s^(2)5p^(2)2 s^{2} 2 p^{2}, 3 s^{2} 3 p^{2}, 4 s^{2} 4 p^{2}, 5 s^{2} 5 p^{2} 和 $6s^{2}6p^2$$6s^{2}6p^2$6s^(2)6p^(2)6 s^{2} 6 p^{2} 。交换相关性由 Perdew、Burke 和 Ernzerhof（PBE）参数化的广义梯度近似（GGA）来描述[27]。250 Ge 原子位点超级单胞用于缺陷计算，电池尺寸为 19.3 A。对于 Ge 超级单体， $4\times 4\times 4$$4\times 4\times 4$4xx4xx44 \times 4 \times 4 使用了 Monkhorst-Pack k 点网格[28]。共轭梯度算法[29]获得能量最小结构（原子位置和晶格常数的完全弛豫）。力公差设置为 $0.001\mathrm{eV}/\text{Å}$$0.001\mathrm{eV}/\text{Å}$0.001eV//"Å"0.001 \mathrm{eV} / \AA ，而电子最小化的精度为 ${10}^{-4}$${10}^{-4}$10^(-4)10^{-4} eV 。

Fig. 1. (a) Optimised structure of bulk Ge, (b) charge density plot showing the covalent bonding electron in the lattice and © total density of states (DOS) plot. Red vertical dot lines correspond to the Fermi energy level.
图 1.（a）体锗的优化结构，（b）电荷密度图显示晶格中的共价键电子和©总态密度（DOS）图。红色垂直点线对应于费米能级。

Table 1  表1
Calculated lattice parameters of elemental bulk phases of dopants considered in this study. The experimental values are provided in parentheses.
本研究考虑的掺杂剂元素体相的计算晶格参数。实验值在括号中提供。

Formation energies and substitution energies are crucial concepts in the study of doping in semiconductors. Doping involves introducing impurities into a semiconductor to modify its electrical properties. While formation energy determines the likelihood of the dopant incorporating into the host lattice, the substitution energy helps in understanding the stability of the dopant within the lattice, which impacts the efficiency and effectiveness of the doping process.
形成能和取代能是半导体掺杂研究中的关键概念。掺杂涉及将杂质引入半导体以改变其电性能。虽然形成能决定了掺杂剂掺入宿主晶格的可能性，但取代能有助于了解掺杂剂在晶格内的稳定性，这会影响掺杂过程的效率和有效性。

The substitutional doping energy for a single dopant atom (here $\mathrm{D}=$$\mathrm{D}=$D=\mathrm{D}= $\mathrm{C},\mathrm{Si},\mathrm{Sn}$$\mathrm{C},\mathrm{Si},\mathrm{Sn}$C,Si,Sn\mathrm{C}, \mathrm{Si}, \mathrm{Sn} and Pb ) in the Ge supercell was calculated using:
Ge 超级单胞中单个掺杂剂原子（此处和 $\mathrm{D}=$$\mathrm{D}=$D=\mathrm{D}= $\mathrm{C},\mathrm{Si},\mathrm{Sn}$$\mathrm{C},\mathrm{Si},\mathrm{Sn}$C,Si,Sn\mathrm{C}, \mathrm{Si}, \mathrm{Sn} Pb ）的取代掺杂能计算如下：
${\mathrm{E}}_{\text{sub}}={\mathrm{E}}_{\mathrm{D}\bullet \mathrm{Ge}\mathrm{\_}\text{supercell}}+{\mathrm{E}}_{\mathrm{Ge}}-{\mathrm{E}}_{\mathrm{Ge}}{}_{\text{-supercell}}-{\mathrm{E}}_{\mathrm{D}}$${\mathrm{E}}_{\text{sub }}={\mathrm{E}}_{\mathrm{D}\bullet \mathrm{Ge}\mathrm{\_}\text{supercell }}+{\mathrm{E}}_{\mathrm{Ge}}-{\mathrm{E}}_{\mathrm{Ge}}{}_{\text{-supercell }}-{\mathrm{E}}_{\mathrm{D}}$E_("sub ")=E_(D∙Ge_"supercell ")+E_(Ge)-E_(Ge)_("-supercell ")-E_(D)\mathrm{E}_{\text {sub }}=\mathrm{E}_{\mathrm{D} \bullet \mathrm{Ge} \_ \text {supercell }}+\mathrm{E}_{\mathrm{Ge}}-\mathrm{E}_{\mathrm{Ge}}{ }_{\text {-supercell }}-\mathrm{E}_{\mathrm{D}}
where ${\mathrm{E}}_{\mathrm{D}\bullet \mathrm{Ge}\mathrm{\_}\text{supercell}}$${\mathrm{E}}_{\mathrm{D}\bullet \mathrm{Ge}\mathrm{\_}\text{supercell }}$E_(D∙Ge_"supercell ")\mathrm{E}_{\mathrm{D} \bullet \mathrm{Ge} \_ \text {supercell }} is the total energy of a D atom occupying a Ge site in the Ge supercell, ${\mathrm{E}}_{\mathrm{Ge}}$${\mathrm{E}}_{\mathrm{Ge}}$E_(Ge)\mathrm{E}_{\mathrm{Ge}} is the total energy per atom in bulk Ge , where bulk refers to the crystalline form of the material, ${\mathrm{E}}_{{\mathrm{Ge}}_{-\text{supercell}}}$${\mathrm{E}}_{{\mathrm{Ge}}_{-\text{supercell }}}$E_(Ge_(-"supercell "))\mathrm{E}_{\mathrm{Ge}_{- \text {supercell }}} is the total energy of the Ge supercell and $E_D$$E_D$E_(D)E_{D} is the total energy of a bulk dopant per atom. To calculate the formation energy of a DV defect cluster in Ge we used:
式中 ${\mathrm{E}}_{\mathrm{D}\bullet \mathrm{Ge}\mathrm{\_}\text{supercell}}$${\mathrm{E}}_{\mathrm{D}\bullet \mathrm{Ge}\mathrm{\_}\text{supercell }}$E_(D∙Ge_"supercell ")\mathrm{E}_{\mathrm{D} \bullet \mathrm{Ge} \_ \text {supercell }} 是 在 Ge 超级单胞中占据 Ge 位点的 D 原子的总能量， ${\mathrm{E}}_{\mathrm{Ge}}$${\mathrm{E}}_{\mathrm{Ge}}$E_(Ge)\mathrm{E}_{\mathrm{Ge}} 是体状 Ge 中每个原子的总能量，其中体状是指材料的晶体形式， ${\mathrm{E}}_{{\mathrm{Ge}}_{-\text{supercell}}}$${\mathrm{E}}_{{\mathrm{Ge}}_{-\text{supercell }}}$E_(Ge_(-"supercell "))\mathrm{E}_{\mathrm{Ge}_{- \text {supercell }}} 是 Ge 超级单胞体的总能量， $E_D$$E_D$E_(D)E_{D} 是每个原子体掺杂剂的总能量。为了计算 Ge 中 DV 缺陷簇的形成能量，我们使用了：

where ${\mathrm{E}}_{\mathrm{DV}\mathrm{\_}\text{supercell}}$${\mathrm{E}}_{\mathrm{DV}\mathrm{\_}\text{supercell }}$E_(DV_"supercell ")\mathrm{E}_{\mathrm{DV} \_ \text {supercell }} is the total energy of a Ge supercell containing a DV defect cluster.
式中 ${\mathrm{E}}_{\mathrm{DV}\mathrm{\_}\text{supercell}}$${\mathrm{E}}_{\mathrm{DV}\mathrm{\_}\text{supercell }}$E_(DV_"supercell ")\mathrm{E}_{\mathrm{DV} \_ \text {supercell }} 是包含 DV 缺陷簇的 Ge 超级单胞的总能量。

To calculate the binding energy of a DV pair in Ge the following relation is used:
为了计算 DV 对的结合能，单位为 Ge，使用以下关系：
${\mathrm{E}}_{\mathrm{b}}(\mathrm{DV})={\mathrm{E}}_{{\mathrm{DV}}_{\text{\_supercell}}}+{\mathrm{E}}_{{\mathrm{Ge}}_{\text{-supercell}}}-{\mathrm{E}}_{\mathrm{D}\bullet \mathrm{Ge}\mathrm{\_}\text{supercell}}-{\mathrm{E}}_{{\mathrm{VGe}}_{\text{-supercell}}}$${\mathrm{E}}_{\mathrm{b}}(\mathrm{DV})={\mathrm{E}}_{{\mathrm{DV}}_{\text{\_supercell }}}+{\mathrm{E}}_{{\mathrm{Ge}}_{\text{-supercell }}}-{\mathrm{E}}_{\mathrm{D}\bullet \mathrm{Ge}\mathrm{\_}\text{supercell }}-{\mathrm{E}}_{{\mathrm{VGe}}_{\text{-supercell }}}$E_(b)(DV)=E_(DV_("_supercell "))+E_(Ge_("-supercell "))-E_(D∙Ge_"supercell ")-E_(VGe_("-supercell "))\mathrm{E}_{\mathrm{b}}(\mathrm{DV})=\mathrm{E}_{\mathrm{DV}_{\text {_supercell }}}+\mathrm{E}_{\mathrm{Ge}_{\text {-supercell }}}-\mathrm{E}_{\mathrm{D} \bullet \mathrm{Ge} \_ \text {supercell }}-\mathrm{E}_{\mathrm{VGe}_{\text {-supercell }}}
where ${\mathrm{E}}_{{\mathrm{VGe}}_{-\text{supercell}}}$${\mathrm{E}}_{{\mathrm{VGe}}_{-\text{supercell }}}$E_(VGe_(-"supercell "))\mathrm{E}_{\mathrm{VGe}_{- \text {supercell }}} is the total energy of a supercell that contains a Ge vacancy.
其中 ${\mathrm{E}}_{{\mathrm{VGe}}_{-\text{supercell}}}$${\mathrm{E}}_{{\mathrm{VGe}}_{-\text{supercell }}}$E_(VGe_(-"supercell "))\mathrm{E}_{\mathrm{VGe}_{- \text {supercell }}} 是包含 Ge 空位的超级单体的总能量。

Finally, the cohesive energies were calculated using:
最后，使用以下方法计算内聚能：
${\mathrm{E}}_{\text{coh}}(\mathrm{D}=\mathrm{C},\mathrm{Si},\mathrm{Sn}$${\mathrm{E}}_{\text{coh }}(\mathrm{D}=\mathrm{C},\mathrm{Si},\mathrm{Sn}$E_("coh ")(D=C,Si,Sn\mathrm{E}_{\text {coh }}(\mathrm{D}=\mathrm{C}, \mathrm{Si}, \mathrm{Sn} and Pb$)=E_D^{\text{isolated}}-E_D^{\text{bulk}}$$)=E_D^{\text{isolated }}-E_D^{\text{bulk }}$)=E_(D)^("isolated ")-E_(D)^("bulk "))=E_{D}^{\text {isolated }}-E_{D}^{\text {bulk }}
${\mathrm{E}}_{\text{coh}}(\mathrm{D}=\mathrm{C},\mathrm{Si},\mathrm{Sn}$${\mathrm{E}}_{\text{coh }}(\mathrm{D}=\mathrm{C},\mathrm{Si},\mathrm{Sn}$E_("coh ")(D=C,Si,Sn\mathrm{E}_{\text {coh }}(\mathrm{D}=\mathrm{C}, \mathrm{Si}, \mathrm{Sn} 和铅 $)=E_D^{\text{isolated}}-E_D^{\text{bulk}}$$)=E_D^{\text{isolated }}-E_D^{\text{bulk }}$)=E_(D)^("isolated ")-E_(D)^("bulk "))=E_{D}^{\text {isolated }}-E_{D}^{\text {bulk }}
where $E_D^{\text{isolated}}$$E_D^{\text{isolated }}$E_(D)^("isolated ")E_{D}^{\text {isolated }} and $E_D^{\text{bulk}}$$E_D^{\text{bulk }}$E_(D)^("bulk ")E_{D}^{\text {bulk }} are the total energies of an isolated gas phase M atom and the M atom in the bulk, respectively.
其中 $E_D^{\text{isolated}}$$E_D^{\text{isolated }}$E_(D)^("isolated ")E_{D}^{\text {isolated }} 和 $E_D^{\text{bulk}}$$E_D^{\text{bulk }}$E_(D)^("bulk ")E_{D}^{\text {bulk }} 分别是孤立气相 M 原子和体中 M 原子的总能量。

The Bader charge analysis [30] was employed to calculate the charges on the doped atoms and the nearest neighbor Ge atoms. Short-range dispersive forces were modeled using a semi-empirical method (DFT + D3) as implemented in the VASP code by Grimme et al. [31]. The GGA is a commonly used functional in DFT that often underestimates bandgaps in semiconductors and insulators. To address this issue, the GGA + U method adds a corrective term to better account for on-site Coulomb and exchange interactions in localized $p$$p$pp or $d$$d$dd or $f$$f$ff electron systems. The GGA + U method provides a robust framework for modeling materials with strongly correlated electrons, leading to reliable and experimentally consistent results [32]. The localized Ge $p$$p$pp states were described by including the orbital-dependent Coulomb potential (Hubbard U) and the exchange parameter J within the DFT + U calculations, as parameterized by Dudarev et al. [33]. The values of $\mathrm{U}=0\mathrm{eV}$$\mathrm{U}=0\mathrm{eV}$U=0eV\mathrm{U}=0 \mathrm{eV} and $\mathrm{J}=3.33\mathrm{eV}$$\mathrm{J}=3.33\mathrm{eV}$J=3.33eV\mathrm{J}=3.33 \mathrm{eV} were applied to describe the localized $p$$p$pp states of Ge as reported in previous studies [34,35].
采用 Bader 电荷分析[30]计算掺杂原子和最近邻锗原子的电荷。使用 Grimme 等[31]在 VASP 代码中实现的半经验方法（DFT + D3）对短程色散力进行建模。GGA 是 DFT 中常用的函数，通常低估半导体和绝缘体中的带隙。为了解决这个问题，GGA + U 方法添加了一个校正项，以更好地解释局部 $p$$p$pp 或 $d$$d$dd /或 $f$$f$ff 电子系统中的现场库仑和交换相互作用。GGA+U 方法为具有强相关电子的材料建模提供了一个稳健的框架，从而获得可靠且实验一致的结果[32]。根据 Dudarev 等[33]的参数化，在 DFT+U 计算中包括轨道相关库仑势（Hubbard U）和交换参数 J 来描述局部 Ge $p$$p$pp 态。将 和 $\mathrm{J}=3.33\mathrm{eV}$$\mathrm{J}=3.33\mathrm{eV}$J=3.33eV\mathrm{J}=3.33 \mathrm{eV} 的 $\mathrm{U}=0\mathrm{eV}$$\mathrm{U}=0\mathrm{eV}$U=0eV\mathrm{U}=0 \mathrm{eV} 值用于描述先前研究报道的 Ge 的局部 $p$$p$pp 状态 [34,35]。

Ge forms in the diamond crystal lattice (space group Fd $\bar{3}\mathrm{m}$$\overline{3}\mathrm{m}$bar(3)m\overline{3} \mathrm{~m}, No: 227), with lattice parameters $\mathrm{a}=\mathrm{b}=\mathrm{c}=5.66\text{Å}$$\mathrm{a}=\mathrm{b}=\mathrm{c}=5.66\text{Å}$a=b=c=5.66"Å"\mathrm{a}=\mathrm{b}=\mathrm{c}=5.66 \AA (refer to Fig. 1a). The calculated relaxed configuration has a lattice parameter of $\mathrm{a}=\mathrm{b}=\mathrm{c}=$$\mathrm{a}=\mathrm{b}=\mathrm{c}=$a=b=c=\mathrm{a}=\mathrm{b}=\mathrm{c}= $5.58\text{Å}$$5.58\text{Å}$5.58"Å"5.58 \AA in excellent agreement with the experimental results [36]. The charge density mainly localised on the strong Ge-Ge covalent bonds is shown in Fig. 1b. The predicted band gap ( 0.70 eV ) is in excellent agreement and only slightly underestimates the experimentally determined band gap $(0.74\mathrm{eV})$$(0.74\mathrm{eV})$(0.74eV)(0.74 \mathrm{eV}) [37,38] consistently with previous studies in this level of theory [34,35](see Fig. 1c).
Ge 在金刚石晶格（空间群 Fd $\bar{3}\mathrm{m}$$\overline{3}\mathrm{m}$bar(3)m\overline{3} \mathrm{~m} ，编号：227）中形成，具有晶格参数 $\mathrm{a}=\mathrm{b}=\mathrm{c}=5.66\text{Å}$$\mathrm{a}=\mathrm{b}=\mathrm{c}=5.66\text{Å}$a=b=c=5.66"Å"\mathrm{a}=\mathrm{b}=\mathrm{c}=5.66 \AA （参见图 1a）。计算出的松弛构型的晶格参数 $\mathrm{a}=\mathrm{b}=\mathrm{c}=$$\mathrm{a}=\mathrm{b}=\mathrm{c}=$a=b=c=\mathrm{a}=\mathrm{b}=\mathrm{c}= $5.58\text{Å}$$5.58\text{Å}$5.58"Å"5.58 \AA 与实验结果非常吻合[36]。电荷密度主要局限于强锗-锗共价键，如图 1b 所示。预测的带隙（0.70 eV）非常一致，仅略微低估了实验确定的带隙 $(0.74\mathrm{eV})$$(0.74\mathrm{eV})$(0.74eV)(0.74 \mathrm{eV}) [37,38]，与该理论水平的先前研究[34,35]一致（见图 1c）。

Calculated lattice parameters of elemental bulk phases of dopants together with their experimental values are provided in Table 1.
表1提供了掺杂剂元素体相的计算晶格参数及其实验值。

Here we calculated the formation energies of a single vacancy and a di-vacancy. Different divacancy configurations are shown in the ESI (see Fig. S1). As expected, the formation of a single vacancy is endoergic with a formation energy of 3.13 eV , which is in agreement with previous theoretical results [43]. In a recent DFT study, we calculated the vacancy formation energy in Si to be 3.63 eV [44] and this is consistent with the picture that the formation of vacancies in Ge is easier as compared to Si . The divacancy defect has a significant binding energy as it reduces the dangling bonds of two unbound vacancies from 8 to 6.
在这里，我们计算了单个空位和双空位的形成能量。ESI 中显示了不同的 divacancy 配置（见图 S1）。正如预期的那样，单个空位的形成是内能的，形成能量为 3.13 eV，这与之前的理论结果一致[43]。在最近的一项 DFT 研究中，我们计算出 Si 中的空位形成能量为 3.63 eV[44]，这与 Ge 中空位形成比 Si 更容易形成的情况一致。分离缺陷具有显着的结合能，因为它将两个未结合空位的悬空键从 8 减少到 6。

The minimum energy structures of the $\mathrm{D}(\mathrm{D}=\mathrm{C},\mathrm{Si},\mathrm{Sn}$$\mathrm{D}(\mathrm{D}=\mathrm{C},\mathrm{Si},\mathrm{Sn}$D(D=C,Si,Sn\mathrm{D}(\mathrm{D}=\mathrm{C}, \mathrm{Si}, \mathrm{Sn} and Pb$)$$)$)) substitutional atoms in Ge are given in Fig. 2(a-d). In the relaxed configurations, all four dopants form tetrahedral coordination with the nearest neighbor Ge atoms [see Fig. 2(e-h)]. The C-Ge bond distances in the ${\mathrm{CGe}}_4$${\mathrm{CGe}}_4$CGe_(4)\mathrm{CGe}_{4} tetrahedral unit were found to be shorter than the D-Ge bond distances in the ${\mathrm{DGe}}_4$${\mathrm{DGe}}_4$DGe_(4)\mathrm{DGe}_{4} tetrahedral units. This is because of the smaller atomic radius of C ( $1.90\text{Å}$$1.90\text{Å}$1.90"Å"1.90 \AA ) than that of Ge ( $2.34\text{Å}$$2.34\text{Å}$2.34"Å"2.34 \AA ). The shortest bond distance of $2.20\text{Å}$$2.20\text{Å}$2.20"Å"2.20 \AA arising from the strongest bond between the Ge and the C is reflected in the largest ionization energy difference between them (C: 11.26 and Ge: 7.89). The smaller ionization energy of Ge leads to the formation of positive Bader charges on the Ge atoms ( $\sim +1.00$$\sim +1.00$∼+1.00\sim+1.00 on each Ge ) in the ${\mathrm{CGe}}_4$${\mathrm{CGe}}_4$CGe_(4)\mathrm{CGe}_{4} tetrahedral unit. The electrons lost from four Ge atoms are gained by the $C$$C$CC substitutional to maintain the charge neutrality as evidenced by the negative Bader charge on the $\mathrm{C}(-3.91)$$\mathrm{C}(-3.91)$C(-3.91)\mathrm{C}(-3.91). An endothermic substitution energy of 1.42 eV indicates that this process is unfavorable. Doping of C introduces a smaller contraction in the volume (see Table 2) due to the formation of shorter C-Ge bonds ( $2.10\text{Å})$$2.10\text{Å})$2.10"Å")2.10 \AA) than $\mathrm{Ge}-\mathrm{Ge}$$\mathrm{Ge}-\mathrm{Ge}$Ge-Ge\mathrm{Ge}-\mathrm{Ge} bonds ( $2.41\text{Å}$$2.41\text{Å}$2.41"Å"2.41 \AA ) in the bulk Ge. A favorable (exothermic) substitution energy of -0.83 eV is noted for the dopant Si concerning bulk Si as a reference state. The preference for this dopant is due to the atomic radius of $\mathrm{Ge}(2.34\text{Å})$$\mathrm{Ge}(2.34\text{Å})$Ge(2.34"Å")\mathrm{Ge}(2.34 \AA) matching closely with that of $\mathrm{Si}(2.32\text{Å})$$\mathrm{Si}(2.32\text{Å})$Si(2.32"Å")\mathrm{Si}(2.32 \AA). Substitution becomes favorable concerning the gas phase atom as a reference state and its substitution energy is calculated to be -1.03 eV . There is a gradual increase in the substitution energy with increasing atomic radius. An unfavorable substitution for C arises from its larger cohesive energy. A smaller volume change for the dopant Si is reflected in the closer match between the Ge-Si and Ge-Ge bond distances (Si-Ge: $2.40\text{Å}$$2.40\text{Å}$2.40"Å"2.40 \AA and Ge-Ge: $2.41\text{Å}$$2.41\text{Å}$2.41"Å"2.41 \AA ). The Bader charges on the ${\mathrm{SiGe}}_4$${\mathrm{SiGe}}_4$SiGe_(4)\mathrm{SiGe}_{4} units are
Ge 中和 $\mathrm{D}(\mathrm{D}=\mathrm{C},\mathrm{Si},\mathrm{Sn}$$\mathrm{D}(\mathrm{D}=\mathrm{C},\mathrm{Si},\mathrm{Sn}$D(D=C,Si,Sn\mathrm{D}(\mathrm{D}=\mathrm{C}, \mathrm{Si}, \mathrm{Sn} Pb $)$$)$)) 取代原子的最小能结构如图 2（a-d）所示。在松弛构型中，所有四种掺杂剂都与最近邻的锗原子形成四面体配位[见图 2（e-h）]。发现 ${\mathrm{CGe}}_4$${\mathrm{CGe}}_4$CGe_(4)\mathrm{CGe}_{4} 四面体单元中的 C-Ge 键距离短于 ${\mathrm{DGe}}_4$${\mathrm{DGe}}_4$DGe_(4)\mathrm{DGe}_{4} 四面体单元中的 D-Ge 键距离。这是因为 C （ $1.90\text{Å}$$1.90\text{Å}$1.90"Å"1.90 \AA ） 的原子半径比 Ge （ $2.34\text{Å}$$2.34\text{Å}$2.34"Å"2.34 \AA ） 的原子半径小。Ge 和 C 之间最强键产生的最短键距 $2.20\text{Å}$$2.20\text{Å}$2.20"Å"2.20 \AA 反映在它们之间的最大电离能差（C：11.26 和 Ge：7.89）。锗较小的电离能导致 ${\mathrm{CGe}}_4$${\mathrm{CGe}}_4$CGe_(4)\mathrm{CGe}_{4} 四面体单元中的锗原子（每个锗 $\sim +1.00$$\sim +1.00$∼+1.00\sim+1.00 ）上形成正巴德电荷。从四个 Ge 原子中损失的电子通过 $C$$C$CC 取代获得以保持电荷中性，如 $\mathrm{C}(-3.91)$$\mathrm{C}(-3.91)$C(-3.91)\mathrm{C}(-3.91) 上的负 Bader 电荷所证明的那样。1.42 eV 的吸热取代能表明该过程不利。由于形成比本体 Ge 中的键 （ $2.10\text{Å})$$2.10\text{Å})$2.10"Å")2.10 \AA) $2.41\text{Å}$$2.41\text{Å}$2.41"Å"2.41 \AA ） 更 $\mathrm{Ge}-\mathrm{Ge}$$\mathrm{Ge}-\mathrm{Ge}$Ge-Ge\mathrm{Ge}-\mathrm{Ge} 短的 C-Ge 键，C 的掺杂导致体积收缩较小（见表 2）。对于作为参考态的掺杂剂 Si，注意到 -0.83 eV 的有利（放热）取代能。对这种掺杂剂的偏好是由于原子半径与 的 $\mathrm{Si}(2.32\text{Å})$$\mathrm{Si}(2.32\text{Å})$Si(2.32"Å")\mathrm{Si}(2.32 \AA) 原子半径紧密 $\mathrm{Ge}(2.34\text{Å})$$\mathrm{Ge}(2.34\text{Å})$Ge(2.34"Å")\mathrm{Ge}(2.34 \AA) 匹配。相原子作为参考态的取代变得有利，其取代能计算为 -1.03 eV。随着原子半径的增大，取代能逐渐增加。 C 的不利取代源于其较大的内聚能。掺杂剂 Si 的较小体积变化反映在 Ge-Si 和 Ge-Ge 键距离（Si-Ge： $2.40\text{Å}$$2.40\text{Å}$2.40"Å"2.40 \AA 和 Ge-Ge： $2.41\text{Å}$$2.41\text{Å}$2.41"Å"2.41 \AA ）之间的更紧密匹配中。 ${\mathrm{SiGe}}_4$${\mathrm{SiGe}}_4$SiGe_(4)\mathrm{SiGe}_{4} 单位的 Bader 费用为

Fig. 2. The minimum energy structures of a single (a) C , (b) Si , © Sn and (d) Pb substitutional atoms in Ge . The dopants form tetrahedral units (e-h) with nearest neighbor Ge atoms, with the numbers corresponding to the Bader charges.
图 2.Ge 中单个 （a） C 、 （b） Si 、 Sn 和 © （d） Pb 取代原子的最小能结构。掺杂剂与最近的邻 Ge 原子形成四面体单元 （e-h），其数字对应于巴德电荷。

Table 2  表2
The predicted cohesive energy, substitution energy, Bader charges on substitutionals, Ge-D bond distances and volume changes due to doping. Substitution energies calculated using gas phase atoms as reference states are provided in parentheses.
预测的内聚能、取代能、取代物上的巴德电荷、Ge-D 键距离和掺杂引起的体积变化。括号中提供了使用气相原子作为参考态计算的取代能。