Relativistic quasiparticle self-consistent electronic structure of hybrid halide perovskite photovoltaic absorbers

Solar cells based on a light absorbing layer of the organometal halide perovskite ${\mathrm{CH}}_3{\mathrm{NH}}_3$PbI${}_3$ have recently surpassed $15\%$ conversion efficiency, though how these materials work remains largely unknown. We analyze the electronic structure and optical properties within the quasiparticle self-consistent GW approximation. While this compound bears some similarity to conventional $sp$ semiconductors, it also displays unique features. Quasiparticle self-consistency is essential for an accurate description of the band structure: Band gaps are much larger than what is predicted by the local-density approximation (LDA) or GW based on the LDA. Valence band dispersions are modified in a very unusual manner. In addition, spin-orbit coupling strongly modifies the band structure and gives rise to unconventional dispersion relations and a Dresselhaus splitting at the band edges. The average hole mass is small, which partially accounts for the long diffusion lengths observed. The surface ionization potential (work function) is calculated to be 5.7 eV with respect to the vacuum level, explaining efficient carrier transfer to TiO${}_2$ and Au electrical contacts.

Figure 1

QSGW band structure for ${\mathrm{CH}}_3{\mathrm{NH}}_3$PbI${}_3$ (left) and NH${}_4$PbI${}_3$ (right). Zero denotes valence band maximum. Bands are colored according to their orbital character: green depicts I 5$p$, red depicts Pb 6$p$, and blue depicts Pb 6$s$. Points denoted M and R are zone-boundary points close to ($\frac{1}{2}$,$\frac{1}{2}$,0) and ($\frac{1}{2}$,$\frac{1}{2}$,$\frac{1}{2}$), respectively. The valence band maximum and conduction band minimum are shifted slightly from R as a consequence of the $\mathbf{L}·\mathbf{S}$ coupling. Valence bands near $-2$ eV (conduction bands near +3 eV) are almost purely green (red) showing that they consist largely of I $5p$ (Pb $6p$) character. Bands nearer the gap are darker as a result of intermixing with other states. Light-gray dashed lines show corresponding bands in the LDA. The dispersionless state near $-$5 eV corresponds to a molecular level of methylammonium. In QSGW this state is pushed down to $-$7.7 eV. The dispersion of the highest valence bands is very poorly described by the LDA, as described in the text.

Figure 2

Left: constant energy contours in the ($1\overline{1}0$) plane for the upper valence band of ${\mathrm{CH}}_3{\mathrm{NH}}_3$PbI${}_3$. The origin corresponds to the R point and [111] and [$11\overline{2}$] are the horizontal and vertical axes of $k$. Energy contours are in increments of 2.5 meV, so that the outermost contour corresponds approximately to RT. Corresponding contours in the (111) plane (not shown) appear similar. Valence bands have two maximal points near $R\pm 0.005$ [$11\overline{2}$]. At low temperature and low doping, ($E_F<5$ meV) the two extrema act as independent centers with approximately spherical effective masses. At high doping ($E_F>20$ meV) or high temperature, holes effectively see a single band maximum with roughly elliptical constant energy surfaces. Panels (b) and (c) show energy $E_v\left(k\right)\equiv E\left(R\right)-E\left(k\right)$ for the lower valence band. This band has a single maximum at $R$, with approximately spherical dispersion. Panel (b) shows $E_v\left(k\right)$, on a log-log scale in the [$1\overline{1}0$], [111], and [$11\overline{2}$] directions as red solid, green dashed, and blue dotted lines, respectively. For comparison a parabolic band with effective mass 0.1$m$ is shown as a gray dot-dashed line. Panel (c) plots $h^2{k}^2/\left(2m{E}_v\right)$ against $E_v$, which may be taken as a definition of the effective mass (see text). $E_v$ is in eV; $k$ is in units $2\pi /a$.

Figure 3

Optical absorption spectrum calculated within the RPA from the QSGW potential, for ${\mathrm{CH}}_3{\mathrm{NH}}_3$PbI${}_3$ and NH${}_4$PbI${}_3$. The absorption is smaller than, but comparable to that of GaAs, shown for comparison. Note that similar measurements have been reported in Ref. [31].