Pressure-induced phase transition in the electronic structure of palladium nitride

We present a combined theoretical and experimental study of the electronic structure and equation of state (EOS) of crystalline ${\text{PdN}}_2$. The compound forms above 58 GPa in the pyrite structure and is metastable down to 11 GPa. We show that the EOS cannot be accurately described within either the local density or generalized gradient approximations. The Heyd-Scuseria-Ernzerhof exchange-correlation functional (HSE06), however, provides very good agreement with experimental data. We explain the strong pressure dependence of the Raman intensities in terms of a similar dependence of the calculated band gap, which closes just below 11 GPa. At this pressure, the HSE06 functional predicts a first-order isostructural transition accompanied by a pronounced elastic instability of the longitudinal-acoustic branches that provides the mechanism for the experimentally observed decomposition. Using an extensive Wannier function analysis, we show that the structural transformation is driven by a phase transition of the electronic structure, which is manifested by a discontinuous change in the hybridization between $\text{Pd}d$ and $\text{N}p$ electrons as well as a conversion from single to triple bonded nitrogen dimers. We argue for the possible existence of a critical point for the isostructural transition, at which massive fluctuations in both the electronic as well as the structural degrees of freedom are expected.

Figure 1

(Color online) Measured Raman spectra obtained after reaction between Pd and ${\text{N}}_2$. Spectrum obtained in experimental Run 1 at $\sim 57\text{GPa}$. The inset shows in more detail the frequency range from 750 to $1200{\text{cm}}^{-1}$. Peaks are indexed assuming pyrite-${\text{PdN}}_2$. The low-frequency peaks are due to nitrogen.

Figure 2

(Color online) Raman spectra for a range of pressures. Left panel: experimental unprocessed data from Run 2. The lowest spectrum is obtained from the recovered sample after increasing pressure to 36 GPa from a minimum of 11 GPa. Right panel: calculated spectra (LDA). The spectra are normalized to maximum peak height.

Figure 3

(Color online) Experimental Raman intensities of the ${\text{A}}_{\text{g}}$ (squares) and ${\text{E}}_{\text{g}}$ (circles) modes as a function of pressure. The solid lines have been fitted to the filled data points only.[36] The inset shows the calculated band gaps of pyrite-${\text{PdN}}_2$, pyrite-${\text{PtN}}_2$, and baddelyite-${\text{IrN}}_2$. The band gaps of ${\text{PtN}}_2$ and ${\text{IrN}}_2$ are underestimated since they were calculated using a GGA exchange-correlation functional.

Figure 4

(Color online) Equation of state as obtained from theory (HSE06, red solid, PBE, green dashed, and LDA, blue dotted) and experiment (Ref. 3) (yellow dashed-dotted) state. The inset shows the nitrogen bond length as a function of atomic volume and the gray areas indicates the two-phase region.

Figure 5

(Color online) Experimental and theoretical HSE06 Raman frequencies as a function of volume. The corresponding pressures as obtained from the equation of state are shown at the top of the figure. The gray area indicates the two-phase region (see Fig. 4).

Figure 6

(Color online) Bulk modulus as obtained from theory (HSE06, red solid and PBE, green dashed). The gray area indicates the two-phase region.

Figure 7

(Color online) Orbital-projected band structures of ${\text{PdN}}_2$ at three representative pressures. The line color indicates the relative admixture of $\text{Pd}d$ (red) and $\text{N}p$ (green) states. Note the band crossing at the R point in the middle panel.

Figure 8

(Color online) Partial density of states (arbitrary units) for $\text{Pd}d$ states as a function of volume and energy. The vertical bars indicate the two-phase region.

Figure 9

(Color online) (a) Spread of occupied MLWFs. The red solid line indicates the sum of all occupied MLWFs. The letters in the legend correspond to the MLWFs depicted in Fig. 10. (b) Total spread of occupied the MLWFs as a function of volume and nitrogen dimer bond length. The dashed line separates the nitrogen single and triple bonded regions.

Figure 10

(Color online) (Center) Pyrite-${\text{PdN}}_2$ conventional cell. [(a)–(j)] Isosurfaces of MLWFs orbitals, (a)–(c) and (e)–(i) correspond to occupied MLWFs in the high- and low-pressure regions, respectively. (d) and (j) correspond to the lowest unoccupied MLWFs. The green and orange surfaces denote positive and negative values, respectively. (a) $t_{2\text{g}}$-like states (three per Pd), (b) N-N single bond (one per dimer), (c) Pd-N bond (three MLWFs per nitrogen), (d) $\text{Pd}{e}_{\text{g}}$-like states (e) $\text{Pd}{t}_{2\text{g}}$-like states (three per Pd), [(f) and (g)] $\text{Pd}{e}_{\text{g}}$-like states (one each per Pd), (h) $\text{N}s$ derived state (one per nitrogen), (i) N-N triple bond (three per dimer, each rotated $120°$ about the N-N axis), and (j) ${\text{N}}_{2}2p{\pi }^{\ast }$-like states.

Figure 11

(Color online) Band structures for the occupied states in the high- (left panel) and low-pressure cases (right panel), respectively. The line color indicates the projection of the eigenstates onto the MLWFs.

Figure 12

(Color online) Schematic illustration of the relation between pressure and volume that gives rise to a first-order transition at low temperatures and becomes second order as temperature is increased. The inset exemplifies a possible corresponding phase diagram.