Kondo-semimetal to Fermi-liquid phase crossover in black phosphorus to pressure-induced orbital-nematic gray phosphorus

We perform a comparative study of the electronic structures and electrical resistivity properties of black $\left(A17\right)$ and of pressurized gray $\left(A7\right)$ phosphorus, showing band-selective Kondoesque electronic reconstruction in these layered $p$-band phosphorus allotropes. Based on density functional dynamical mean-field theory calculations, we show that gray phosphorus can host a three-dimensional Kondo semimetal to a Fermi liquid phase crossover under anisotropic compression, which lifts in-layer orbital degeneracy. Therein, the $3p$ spectrum is almost unaffected both in the semiconducting and semimetallic Kondo phases, however in the Fermi liquid regime strong electronic reconstruction is predicted to exist in the orbital nematic phase of compressed gray phosphorus. These findings contribute to the microscopic understanding of the role played by dynamical multiorbital electronic interactions in the low energy spectrum of correlated semiconductors and topological semimetals.

Figure 1

Layered structure of black (left panel) and gray (right panel) phosphorus. See Ref. [16] for details on the $A17$ (black) to $A7$ (gray) structural transformation of pressurized phosphorus.

Figure 2

Comparison between the LDA orbital resolved density of states (DOS) for bulk BP (dashed line) and GP (solid line), showing sizable one-particle band narrowing of the conduction band states of bulk GP. Notice as well the (narrow gap) insulating and semimetallic nature of BP and GP, respectively.

Figure 3

LDA (dashed line) and $\mathrm{LDA}+\mathrm{DMFT}$ (solid line) orbital resolved DOS of bulk GP, showing spectral weight redistribution over large energy scales. The $\mathrm{LDA}+\mathrm{DMFT}$ spectral functions of BP are shown for comparison. The stars indicate the energy position of the incipient lower- and upper-Hubbard bands in the valence and conduction band states of bulk BP.

Figure 4

Imaginary part of the orbital-resolved self-energies $\left[Im{\mathrm{\Sigma }}_a\left(\omega \right)\right]$ within the Kondo insulating phases of bulk BP and GP. Notice the substantial reduction of the near zero constant value of $\left[Im{\mathrm{\Sigma }}_a\left(\omega \right)\right]$ near $E_F$ in GP, implying stronger Kondo correlations effects in GP compared to BP.

Figure 5

$T$ dependence of electrical resistivities for the Kondo insulating state and semimetallic states of bulk BP and GP, respectively, showing qualitative good accord with transport data [14, 15] for bulk BP. Particularly interesting is the insulating behavior at low temperatures intrinsic to narrow band semiconductors and the minimum value around 40 K in BP, which is pushed to lowers temperatures in semimetallic GP. Noteworthy is the region, where $\rho \left(T\right)$ is linear in $ln\left(T\right)$ for bulk GP (as seen in the right inset), the fingerprint of Kondo resistivity. Left inset displays the resistivity curves of GP for different $U$ values, showing the changes in the $ln\left(T\right)$ corrections at low temperatures due to increased metallicity in the bulk.

Figure 6

Orbital resolved $\mathrm{LDA}+\mathrm{DMFT}\left(U=10.0\mathrm{eV},J_H=0.5\mathrm{eV}\right)$ spectral functions in the orbital nematic phase of bulk GP, showing their evolution with increasing the orbital field $\mathrm{\Delta }$. The anisotropic energy gap suppression in the valence and conduction bands of the $p_x$ and $p_y$ orbitals on increasing $\mathrm{\Delta }$ goes hand-in-hand with large scale spectral weight transfer. Particularly interesting are the crossing points at different energies in the spectral functions.

Figure 7

Orbital-resolved self-energies (imaginary part, $\left[Im{\mathrm{\Sigma }}_a\left(\omega \right)\right])$ across the Kondo insulating to Fermi liquid crossover of orbital nematic GP. Notice the substantial reduction of the near zero constant value of $\left[Im{\mathrm{\Sigma }}_a\left(\omega \right)\right]$ near $E_F$ in all channels upon increasing the orbital field $\mathrm{\Delta }$ and the smooth crossover to a $-{\omega }^2$ dependence characteristic of correlated Fermi liquid metals.

Figure 8

$T$ dependence of electrical resistivities for the Kondo semimetallic and metallic state of orbital nematic GP. Particularly interesting is the suppression of the $ln\left(T\right)$ behavior and the resistivity upturn with increasing $\mathrm{\Delta }$. The inset displays resistivity curves of GP for the different $\mathrm{\Delta }$ values on a linear scale, showing the changes in $\rho \left(T\right)$ towards a $T^2$ dependence characteristic of Fermi liquid metals.