$n$-type doping of $\mathrm{CuIn}{\mathrm{Se}}_2$ and $\mathrm{CuGa}{\mathrm{Se}}_2$

The efficiency of $\mathrm{CuIn}{\mathrm{Se}}_2$ based solar cell devices could improve significantly if $\mathrm{CuGa}{\mathrm{Se}}_2$, a wider band gap chalcopyrite semiconductor, could be added to the $\mathrm{CuIn}{\mathrm{Se}}_2$ absorber layer. This is, however, limited by the difficulty of doping $n$-type $\mathrm{CuGa}{\mathrm{Se}}_2$ and, hence, in its alloys with ${\mathrm{CuInSe}}_2$. Indeed, wider-gap members of semiconductor series are often more difficult to dope than lower-gap members of the same series. We find that in chalcopyrites, there are three critical values of the Fermi energy $E_F$ that control $n$-type doping: (i) $E_F^{n,\mathrm{pin}}$ is the value of $E_F$ where the energy to form Cu vacancies is zero. At this point, the spontaneously formed vacancies ($=$acceptors) kill all electrons. (ii) $E_F^{n,\mathrm{comp}}$ is the value of $E_F$ where the energy to form a Cu vacancy equals the energy to form an $n$-type dopant, e.g., ${\mathrm{Cd}}_{\mathrm{Cu}}$. (iii) $E_F^{n,\text{site}}$ is the value of $E_F$ where the formation of Cd-on-In is equal to the formation of Cd-on-Cu. For good $n$-type doping, $E_F^{n,\mathrm{pin}}$, $E_F^{n,\mathrm{comp}}$, and $E_F^{n,\text{site}}$ need to be as high as possible in the gap. We find that these quantities are higher in the gap in $\mathrm{CuIn}{\mathrm{Se}}_2$ than in $\mathrm{CuGa}{\mathrm{Se}}_2$, so the latter is difficult to dope $n$-type. In this work, we calculate all three critical Fermi energies and study theoretically the best growth condition for $n$-type $\mathrm{CuIn}{\mathrm{Se}}_2$ and $\mathrm{CuGa}{\mathrm{Se}}_2$ with possible cation and anion doping. We find that the intrinsic defects such as ${\mathrm{V}}_{\mathrm{Cu}}$ and ${\mathrm{In}}_{\mathrm{Cu}}$ or ${\mathrm{Ga}}_{\mathrm{Cu}}$ play significant roles in doping in both chalcopyrites. For group-II cation (Cd, Zn, or Mg) doping, the best $n$-type growth condition is $\mathrm{In}∕\mathrm{Ga}$-rich, and maximal Se-poor, which is also the optimal condition for stabilizing the intrinsic ${\mathrm{In}}_{\mathrm{Cu}}∕{\mathrm{Ga}}_{\mathrm{Cu}}$ donors. Bulk $\mathrm{CuIn}{\mathrm{Se}}_2$ can be doped at equilibrium $n$-type, but bulk $\mathrm{CuGa}{\mathrm{Se}}_2$ cannot be due to the low formation energy of intrinsic Cu-vacancy. For halogen anion doping, the best $n$-type materials growth is still under $\mathrm{In}∕\mathrm{Ga}$-rich, and maximal Se-poor conditions. These conditions are not best for halogen substitutional defects, but are optimal for intrinsic ${\mathrm{In}}_{\mathrm{Cu}}∕{\mathrm{Ga}}_{\mathrm{Cu}}$ donors. Again, $\mathrm{CuGa}{\mathrm{Se}}_2$ cannot be doped $n$-type by halogen doping, while $\mathrm{CuIn}{\mathrm{Se}}_2$ can.

Figure 1

$n$-type doping rules for CIS and CGS. The terms “anion” and “cation” refer to the host atoms, i.e., Cu, In, Ga, and Se.

Figure 2

The allowed chemical potential domains (shaded area) for (a) CIS and (b) CGS, shown in the $(\Delta {\mu }_{\mathrm{Cu}},\Delta {\mu }_{\mathrm{III}})$ plane. The white regions are areas which are excluded due to the formation of competing phases specified in the figures.

Figure 3

(Color online) Contour plot of the defect formation energies $\Delta H$ in CIS, illustrating the dependence on chemical potentials $\Delta {\mu }_{\mathrm{Cu}}$ and $\Delta {\mu }_{\mathrm{In}}$ (dependence on $\Delta {\mu }_{\mathrm{Se}}$ is implicit). Dark shading corresponds to low $\Delta H$ (high concentration), light shading to high $\Delta H$ (low concentration). The contour spacing corresponds to 0.1 eV. the ideal choices, according to the rules of Fig. 1 are: (a) Minimize $\Delta H\left({\mathrm{Cd}}_{\mathrm{Cu}}\right)$ for Cd-doping [Rule (1)]. (b) Minimize $\Delta H\left({\mathrm{Cl}}_{\mathrm{Se}}\right)$ for Cl-doping [Rule (1)]. (c) Maximize $\Delta H\left({\mathrm{V}}_{\mathrm{Cu}}\right)$ [Rule (3)]. (d) Minimize $\Delta H\left({\mathrm{In}}_{\mathrm{Cu}}\right)$ [Rule (4)]. (e) Maximize $\Delta H\left({\mathrm{Cd}}_{\mathrm{In}}\right)$ [Rule (5)].

Figure 4

${\mathrm{Cl}}_{\mathrm{Se}}$ and related defect formation energies under “point N” growth conditions. $E_F^{n,\text{comp}}$ indicates the Fermi energy where $\Delta H$ of the donor ${\mathrm{In}}_{\mathrm{Cu}}^{++}$ $\left({\mathrm{Ga}}_{\mathrm{Cu}}^{++}\right)$ and the acceptor $V_{\mathrm{Cu}}^{-}$ intersect. The vertical line indicates the self-consistently calculated equilibrium Fermi energy $E_F^{\mathrm{eq}}$ at $T=800\mathrm{K}$.

Figure 5

${\mathrm{Cd}}_{\mathrm{Cu}}$ and related defect formation energies under “point N” growth conditions. $E_F^{n,\text{site}}$ indicates the Fermi energy where $\Delta H$ of the donor ${\mathrm{Cd}}_{\mathrm{Cu}}^{+}$ and the acceptor ${\mathrm{Cd}}_{\mathrm{In}}^{-}$ (${\mathrm{Cd}}_{\mathrm{Ga}}^{-}$) intersect. The vertical line indicates the self-consistently calculated equilibrium Fermi energy $E_F^{\mathrm{eq}}$ at $T=800\mathrm{K}$.

Figure 6

Total DOS of (a) CGS:${\mathrm{Cd}}_{\mathrm{Cu}}$ and (b) CGS:${\mathrm{Cd}}_{\mathrm{Ga}}$, as well as the corresponding atom PDOS of the (c) ${\mathrm{Cd}}_{\mathrm{Cu}}$ and (d) ${\mathrm{Cd}}_{\mathrm{Ga}}$ dopant. The presented density-of-states includes a Gaussian broadening of 50 meV. Dotted lines indicate the VBM.

Figure 7

Schematic picture of the transition energies (in units of eV) of divalent donors $(+∕0)$ and acceptors $(0∕-)$ in CIS and CGS, referenced to the CBM and VBM, respectively. The transition energy values shown are without Makov-Payne correction.

Figure 8

The self-consistently calculated defect concentrations for Cd doping in CIS and CGS at $T=800\mathrm{K}$. The doping balance $\Delta c=c_{{\mathrm{Cd}}_{\mathrm{Cu}}}+2c_{{\mathrm{In}}_{\mathrm{Cu}}}-c_{{\mathrm{Cd}}_{\mathrm{In}}}-c_{{\mathrm{V}}_{\mathrm{Cu}}}$ is shown in black bars.

Figure 9

The defect concentration of halogen defects and intrinsic defects under “halogen favored” (point P) and “${\mathrm{In}}_{\mathrm{Cu}}$ favored” (point N) conditions in CIS. The growth temperature of 800 K is used. The doping balance $\Delta c=c_{{\mathrm{X}}_{\mathrm{Se}}}+2c_{{\mathrm{In}}_{\mathrm{Cu}}}-c_{{\mathrm{V}}_{\mathrm{Cu}}}$ (X represents halogen atoms) is shown in black bars.

Figure 10

Total energy difference $\Delta E$ per hole between the neutral semiconductor host and the positively charged $\left(q\right)$ host, accomodating a hole gas in the valence band. The calculation is done for the 8 atom elementary cell of CIS and CGS. $1∕q=1000$ corresponds to a hole concentration of about $5\times {10}^{18}$ ${\mathrm{cm}}^{-3}$.

Figure 11

Schematic figure showing the (a) band-filling correction to the (b) LDA finite supercell approach, and (c) the band-gap corrections. The shaded area show schematically the density-of-states of the donor electron $s-$ states.