Half-Excitonic Insulator: A Single-Spin Bose-Einstein Condensate

First-principles calculations reveal an unusual electronic state (dubbed as half excitonic insulator) in monolayer $1T\text{−}M{X}_2$ ($M=\mathrm{Co}$, Ni and $X=\mathrm{Cl}$, Br). Its one spin channel has a many-body ground state due to excitonic instability, while the other is characterized by a conventional band insulator gap. This disparity arises from a competition between the band gap and exciton binding energy, which exhibits a spin dependence due to different orbital occupations. Such a state can be identified by optical absorption measurements and angle-resolved photoemission spectroscopy. Our theory not only provides new insights for the study of exciton condensation in magnetic materials but also suggests that strongly correlated materials could be fertile candidates for excitonic insulators.

Figure 1

Schematic diagrams that classify magnetic insulators in terms of the exciton $E_t$: (a) an ordinary magnetic insulator when $E_t>0$ (both spins), (b) a magnetic EI when $E_t<0$ (both spins), and (c) a magnetic HEI when $E_t<0$ (single spin). “$\uparrow$” and “$\downarrow$” denote electron spin. Thick black dashed lines denote the energy zero, and the thin colored dashed lines denote the energetically most favorable excitons in each spin. Note that these magnetic systems can still be ferromagnetic or antiferromagnetic depending on their total magnetic moment.

Figure 2

(a) Atomic structure (left) and Ni-centered local octahedron (right) of monolayer $1T\text{−}{\mathrm{NiCl}}_2$ with Ni (Cl) atoms colored in brown (green). (b) Magnetic phases: FM (top), SAFM (middle), and ZAFM (bottom). Open and filled circles are lattice sites on the Ni plane with opposite local moments. Red dashed rectangles are the supercells used for energy calculations. (c) Total magnetic moment vs temperature, showing a Curie temperature of 118 K. (d) Phonon spectrum of the FM (ground) state.

Figure 3

Spin-resolved PBE (a) band structure and (b) projected density of states, and (c) the corresponding $G_0{W}_0$ band structure. In (a) and (c), the minority spin gaps are indicated. In (b), arb. units is the abbreviation for arbitrary units. Energy zero is at the top of valence band.

Figure 4

(a) Imaginary part of the dielectric function. The black solid line is obtained from the BSE with contributions from individual spins color coded with red and blue, vertical lines denoting the corresponding $E_g$. The gray solid line is obtained from the independent-particle (IP) approximation, i.e., ignoring electron-hole interactions. (b) Exciton energies (vertical lines), superimposed on the imaginary part of the BSE dielectric function in (a) in the low-energy region. $X_i$ and $D_i$ denote dark excitons and the peak positions of ${ϵ}_2$, which are indicative of bright excitons. (c) Reciprocal-space (left) and real-space (right) exciton wave functions modulus. The real-space wave functions are highly localized on Ni atoms. To show the fine details, we truncate the wave functions modulus at 20% of their maximum values. In the plots, the hole (the green dot) is fixed at central Ni.

Figure 5

A schematic illustration of the formation of $X_1$ exciton. It corresponds to a $|d^8{d}^8⟩\to |d^7{d}^9⟩$ excitation. “$\uparrow$” and “$\downarrow$” denote occupied $d$ electrons with up and down spin. An $X_1$, as can be seen in the encircled area, is formed by elevating a down-spin electron (red “$\downarrow$”) to the nearest neighbor Ni, leaving a hole (red filled circle), which makes two adjacent Ni atoms differ in their valency ($+3$ and $+1$).