Intrinsic defects and dopability of zinc phosphide

Zinc phosphide (Zn${}_3$P${}_2$) could be the basis for cheap and highly efficient solar cells. Its use in this regard is limited by the difficulty in $n$-type doping of the material. In an effort to understand the mechanism behind this, the energetics and electronic structure of intrinsic point defects in zinc phosphide are studied using generalized Kohn-Sham theory and utilizing the Heyd, Scuseria, and Ernzerhof (HSE) hybrid functional for exchange and correlation. Novel “perturbation extrapolation” is utilized to extend the use of the computationally expensive HSE functional to this large-scale defect system. According to calculations, the formation energy of charged phosphorus interstitial defects are very low in $n$-type Zn${}_3$P${}_2$ and act as “electron sinks,” nullifying the desired doping and lowering the Fermi-level back toward the $p$-type regime. This is consistent with experimental observations of both the tendency of conductivity to rise with phosphorus partial pressure, and with current partial successes in $n$-type doping in very zinc-rich growth conditions.

Figure 1

Zn${}_3$P${}_2$ allowed chemical potentials. The red and green lines are the stability limits for Zn${}_3$P${}_2$ and ZnP${}_2$, respectively. The dark purple region corresponds to the range of chemical potentials, where we are assured of forming zinc phosphide crystals.

Figure 2

Tetragonal Zn${}_3$P${}_2$ has a 40-atom unit cell with $P$42/nmc symmetry.

Figure 3

DFT band structure—calculated with HSE functional. Partial density of states shows makeup of VBM mainly of phosphorus $p$ character, while the CBM is mixed $p$ and $s$ character from phosphorus and zinc states.

Figure 4

GGA to HSE band alignment. Band gaps for GGA (left) and HSE (right) are labeled at the $\gamma$ point. Calculated offsets for the VBM and CBM are shown.

Figure 5

$V_{\text{Zn}}$ electronic structure: partial density of states of the neutral defect for the four nearest neighboring phosphorus atoms. Defect states highlighted in yellow show mainly $p$ character. Bond lengths are given relative to distance to defect site in perfect bulk cell.

Figure 6

$V_{\text{Zn}}$ formation energy. Acceptor-type defect, plotted for the 0, $-$1, $-$2 charged states as the Fermi level varies from the top of the VBM ($p$ type) to the bottom of the CBM ($n$ type). Favorable growth condition on the left.

Figure 7

$V_{\mathrm{P}}$ electronic structure: partial density of states of the $+$2 charged defect for the nearest neighboring zinc atoms. Defect states highlighted in yellow show mainly Zn-$p$ and Zn-$d$ character. Bond lengths are given relative to distance to defect site in perfect bulk cell.

Figure 8

$V_{\mathrm{P}}$ formation energy. Donor-type defect, plotted for the 0, $+$1, $+$2 charged states as the Fermi level varies from the top of the VBM ($p$ type) to the bottom of the CBM ($n$ type). Favorable growth condition on the left.

Figure 9

${\text{Zn}}_i$ electronic structure: partial density of states of the $-$2 charged defect for the interstitial Zn site. Defect states highlighted in yellow show mainly Zn-$s$ character. Bond lengths are given relative to distance to defect site in perfect bulk cell.

Figure 10

${\text{Zn}}_i$ formation energy. Donor-type defect, plotted for the 0, $+$1, $+$2 charged states as the Fermi level varies from the top of the VBM ($p$ type) to the bottom of the CBM ($n$ type). Favorable growth condition on the left.

Figure 11

${\text{P}}_i$ electronic structure: partial density of states of the neutral defect for the interstitial phosphorus site. Defect states highlighted in yellow show mainly P-$p$ character. Bond lengths are given relative to distance to defect site in perfect bulk cell.

Figure 12

${\text{P}}_i$ formation energy. Acceptor-type defect, plotted for the 0, $-$1, $-$2 charged states as the Fermi level varies from the top of the VBM ($p$ type) to the bottom of the CBM ($n$ type). Favorable growth condition on the left.

Figure 13

Potential alignment correction. Differences in electrostatic potential between perfect and defect cells far from defect area are used to correct charged cell energies.

Figure 14

Perturbation extrapolation workflow: assuming that the perfect unit cell GGA forms a complete basis for the supercell defect wave functions, we project the GGA defect states onto the GGA perfect states and then use the offsets between the GGA and HSE unit cells to extrapolate HSE supercell behavior.

Figure 15

Slab geometry for band alignment determination. Red regions show areas where the potential was integrated in the bulk and vacuum regions.