We use the first principles methods to study the electronic structure and optical properties of G-type anti-ferromagnetic hexagonal-YMnO3. Ground state properties of this material were calculated within density functional theory (DFT) using the DFT + U formalism. We calculated the quasiparticle band structure of this material using many body perturbation theory within the GW approximation. In order to understand the optical response of this material, we solved the Bethe–Salpeter equation and calculated the absorption spectrum. Our calculated optical band gap of 1.45 eV agrees well with the experimental value of 1.55 eV. We find an exciton binding energy of 0.21 eV for this material.

Multiferroic materials are increasingly becoming materials of interest for a wide variety of applications. This is primarily owing to the fascinating coexistence and interplay between orbital, charge, and spin degrees of freedom in these materials.1 Applications of these materials include data storage, capacitors, transducers, actuators,2,3 etc. Although most existing technologically important ferroelectrics are oxides with perovskite structure, recent studies have shown coexistence of these degrees of freedoms in hexagonal manganites. These manganites have found applications as non-volatile memory materials4 and gate ferroelectrics in field-effect transistors.5 Furthermore, it has been shown that ferroelectric polarization driven carrier separation can lead to increase in solar cell efficiency in these materials6 as well.

One of the manganites, hexagonal-YMnO3 (h-YMnO3), has received a lot of attention in the recent years.2 h-YMnO3 is a multiferroic, which has been found to be a ferroelectric below 914 K and also has anti-ferromagnetic order with a Neel temperature of 80 K.7 The transition from paraelectric to ferroelectric distorts the crystal structure of h-YMnO3 by tilting the oxygen bi-pyramids surrounding Mn ions and also bending the intermediate Y layers.8 As a result the symmetry reduces from centrosymmetric P63/mmc to a non-centrosymmetric P63cm space group and the primitive cell contains six formula units of YMnO3 instead of two in case of the paraelectric phase.

The electronic structure of this h-YMnO3 and its dependence on magnetic ordering have been studied extensively in the past using both first principles density functional theory (DFT) methodology9,10 and different experimental techniques.11,12 However, as has been pointed out in Ref. 11, while the theoretical studies have been successful in explaining ferroelectricity, they have failed to reproduce the band gaps obtained from optical absorption or photoluminescence experiments.12 The reason for this discrepancy is the well known “band gap problem” in DFT.13,14 In previous calculations based on DFT, quasiparticle energies are assumed to be the Kohn–Sham eigenvalues obtained from solving the Kohn–Sham DFT equations self-consistently. But as is well-known, Kohn–Sham DFT eigenvalues do not have the physical meaning of electron addition and removal energies.13 Quasiparticle (QP) energies can be calculated within the framework of many body perturbation theory. In practice, they can be calculated by including the self-energy correction to the Kohn–Sham eigenvalues. Similarly, to obtain the optical spectra, it is important to include electron-hole interaction and other excitonic effects in the calculation. The GW approximation13,15 is one such state-of-the-art methodology which has been very successful in reproducing quasiparticle and optical properties of various systems—semiconductors and insulators,13,16–18 metals,19,20 and nano-structured materials.21,22

In this article, we report results from GW-BSE calculation done on G-type anti-ferromagnetically ordered phase of h-YMnO3. Furthermore, we also constructed maximally localized Wannier functions (MLWF)23,24 to understand the electronic structure as well as the metal-ligand hybridization in this transition metal oxide.

We performed the first principles calculations to understand the properties of h-YMnO3. Our DFT calculations were performed using plane waves and pseudopotentials as implemented in the Quantum Espresso25 package. We used the Perdew–Burke–Ernzerhof (PBE)26 approximation to the exchange-correlation functional. In the case of h-YMnO3, the PBE starting point leads to incorrect relative ordering of the orbitals.10 As a result, we used the rotationally invariant formulation of PBE + U27 in our calculations with a U of 10.1 eV and J of 0.88 eV for the Mn d orbitals. The choice of U and J is based on constrained DFT calculation within local density approximation (constrained-LDA) done on similar materials, LaMnO3 and CaMnO328 as there are no constrained LDA calculations on YMnO3. The calculated values of U and J in this constrained LDA study were found to be U = 10.1 eV and J = 0.88 eV for LaMnO3 and U = 10.0 eV and J = 0.88 eV for CaMnO3. Our choice of U and J can be a posteriori justified based on the comparison of our PBE + U density of states (DOS) with experiment.29 Norm-conserving pseudopotentials,30 which include semi-core states (4s, 4p in case of Y and 3s, 3p in case of Mn), have been employed in all the calculations to capture the strong exchange-interaction present from those states. The unit cell of h-YMnO3 has 30 atoms containing six formula units of YMnO3. The Brillouin zone was sampled using a 4 × 4 × 2 k-grid. The wavefunctions were expanded using plane waves up to a cutoff energy of 250 Ry.

Once we have solved the Kohn–Sham equations and obtained the PBE + U wavefunctions, we constructed MLWFs23,24 using the Wannier9031 package. One can rewrite the Hamiltonian of the system in MLWF basis. Such a Hamiltonian can be written in terms of four blocks—one objective orbital block (Ho(k)), some other block of states (Ho(k)) that hybridizes with objective orbital block, and V(k) (V(k)) which accounts for the amount of hybridization between those two groups of states.32,33 Schematically the Hamiltonian in the MLWF basis can be written as

H(k)=[Ho(k)V(k)V(k)Ho(k)].
(1)

If one chooses the orbitals of interest as the objective block, makes the rest of the blocks zero, and diagonalizes the resulting Hamiltonian, the eigenvalues will give us energy levels of the objective orbitals without the effects of hybridization with other states. We used this method to find energies of the unhybridized orbitals of interest.

In order to calculate the quasiparticle energies within the GW framework, we used the PBE + U mean-field starting point. In the GW formalism13,15 one can obtain the quasiparticle energies as a correction to Kohn–Sham eigenvalues using the following equation:

EnkQP=ϵnkDFT+nk|Σ(EnkQP)VxcDFT|nk,
(2)

where EnkQP is the GW quasiparticle energy, ϵnkDFT is the Kohn–Sham eigenvalue obtained from the starting mean field calculation, Σ(EnkQP) is the electron self energy operator, which is calculated within GW approximation, VxcDFT is the exchange correlation potential (including the U correction), and |nk is the mean field wavefunction for band n with wavevector k. The self energy operator is constructed using the mean field G and W, called the G0W0 approximation.

To obtain the optical response function, we solve the Bethe–Salpeter equation (BSE)34–36 

(EckQPEvkQP)AvckS+vckvck|Keh|vck=ΩSAvckS,
(3)

where AvckS is the exciton wavefunction (in the quasiparticle state representation), ΩS is the excitation energy, Keh is the electron-hole interaction kernel, and v (c) indicates valence (conduction) states. We can compute the imaginary part of the dielectric function from the solution to the BSE using the expression

ϵ2(ω)=16π2e2ω2S|e·0|v|S|2δ(ωΩS),
(4)

where

0|v|S=vckAvckSvk|v|ck,
(5)

where v is the velocity operator, e is the direction of polarization of incident light, and e is the charge of an electron.

We have used the BerkeleyGW37 package to perform one-shot G0W0 and BSE calculations. While there are several other GW schemes available, such as partial self-consistent GW and self consistent GW,38–42 owing to computational expense we restrict ourselves to the one-shot G0W0 scheme. The static dielectric matrix was calculated with a 40 Ry energy cutoff in a plane-wave basis and extended to finite frequencies within the generalized plasmon pole model.13 In both dielectric matrix and self-energy operator (Σ) calculations we have included 2400 bands. In order to ensure convergence of our calculation with respect to the number of bands, we added a static remainder43 term to the self energy (see supplementary material for details). For the BSE calculation, the kernel was interpolated from 4 × 4 × 2 coarse k-grid to a 10 × 10 × 5 fine k-grid. To ensure that the 10 × 10 × 5 fine grid is sufficient, we increased the fine grid to 12 × 12 × 6 and found no significant difference in the absorption spectrum. We have included 20 valence and 3 conduction bands in this part of the calculation to construct the electron-hole interaction kernel. The number of bands included in the calculation was sufficient to ensure the convergence of the absorption spectrum up to ∼1.5 eV from the absorption-edge.

Previous first principles calculations44,45 have shown that several different collinear and non-collinear magnetic orderings are possible in h-YMnO3. According to Ref. 44 the triangular frustrated anti ferromagnetic (TAFM) phase is the ground state of this material, whereas the G-type anti ferromagnetic (GAFM) phase has slightly higher energy than that of TAFM phase. Ref. 45 has reported several other phases involving non-collinear magnetic ordering beside TAFM phase having total energies within 5 meV (per formula unit). However, we are interested in studying the room temperature properties of h-YMnO3. At room temperature, the magnetic ordering in h-YMnO3 is expected to be paramagnetic. Nevertheless, we have chosen the GAFM phase to perform our calculations due to the computational difficulty in calculating the paramagnetic phase or any of the phases involving non-collinear magnetic moments. While the GAFM is not paramagnetic, this magnetic ordering is a reasonable representation of the actual material at room temperature. This magnetic ordering would be better than a non-magnetic phase owing to the presence of local magnetic moments. In Fig. 1 the crystal structure of h-YMnO3 has been shown with the GAFM magnetic ordering.

FIG. 1.

The crystal structure of h-YMnO3 and magnetic ordering in GAFM phase. Silver and golden spheres represent Y and O atoms, respectively. Blue and red spheres are Mn atoms colored differently to differentiate up and down magnetic moments. The oxygen bi-pyramids surrounding each Mn have been shaded with the same color convention corresponding to the magnetic moment of the Mn atom at the center.

FIG. 1.

The crystal structure of h-YMnO3 and magnetic ordering in GAFM phase. Silver and golden spheres represent Y and O atoms, respectively. Blue and red spheres are Mn atoms colored differently to differentiate up and down magnetic moments. The oxygen bi-pyramids surrounding each Mn have been shaded with the same color convention corresponding to the magnetic moment of the Mn atom at the center.

Close modal

In our calculation, we have used the lattice parameter values (a = b = 6.13 Å and c = 11.34 Å) obtained from first principles calculation44 done on h-YMnO3. The values are in excellent agreement with experimental values (a = 6.13 Å and c = 11.4 Å).46 We have then relaxed the atomic coordinates such that forces are less than 10−3 Ry/au. In Table I we have listed the Wyckoff notations and fractional atomic coordinates obtained from our calculation and compared them with the values obtained from high-resolution neutron powder diffraction patterns47 at room temperature (300 K). We find that our structure is in excellent agreement with the experimental structure.

TABLE I.

Comparison between atomic positions from our calculations and experimental data obtained from high-resolution neutron powder diffraction pattern.

ExperimentalaTheoretical
AtomWyckoffxyzxyz
Y(1) 2(a) 0.0000 0.0000 0.2727 0.0000 0.0000 0.2775 
Y(2) 4(b) 0.3333 0.6667 0.2320 0.3335 0.6666 0.2317 
Mn 6(c) 0.3330 0.0000 0.0000 0.3338 0.0000 0.0010 
O(1) 6(c) 0.3076 0.0000 0.1625 0.3057 0.0000 0.1649 
O(2) 6(c) 0.6414 0.0000 0.3360 0.6400 0.0003 0.3373 
O(3) 2(a) 0.0000 0.0000 0.4754 0.0006 0.0006 0.4794 
O(4) 4(b) 0.3333 0.6667 0.0163 0.3343 0.6667 0.0193 
ExperimentalaTheoretical
AtomWyckoffxyzxyz
Y(1) 2(a) 0.0000 0.0000 0.2727 0.0000 0.0000 0.2775 
Y(2) 4(b) 0.3333 0.6667 0.2320 0.3335 0.6666 0.2317 
Mn 6(c) 0.3330 0.0000 0.0000 0.3338 0.0000 0.0010 
O(1) 6(c) 0.3076 0.0000 0.1625 0.3057 0.0000 0.1649 
O(2) 6(c) 0.6414 0.0000 0.3360 0.6400 0.0003 0.3373 
O(3) 2(a) 0.0000 0.0000 0.4754 0.0006 0.0006 0.4794 
O(4) 4(b) 0.3333 0.6667 0.0163 0.3343 0.6667 0.0193 
a

Reference 47.

Fig. 2(a) shows the electronic band structure of h-YMnO3 obtained from PBE + U calculations. As can be seen from the figure, the bands are almost flat along the A-Γ direction while they disperse significantly along the other directions. This is because of the fact that apical O atoms are bonded with three Y atoms and one Mn atom, whereas the planar O atoms are bonded with three Mn atoms and one Y atom. It is evident from orbital resolved density of states (DOS) figure (Fig. 2(b)) that for energies close to the valence band maxima (VBM) Mn(3d) states hybridize more strongly with O(2p) states than Y(4d) states do. As a result, hopping in a-b plane is significantly greater than that along c direction. This causes the bands near VBM, which originate mostly from hybridization between Mn(3d) and O(2p) states, to have a significantly smaller dispersion along the A-Γ direction than the Γ-K direction.

FIG. 2.

(a) PBE + U band-structure (blue solid) along with the GW band-structure (red dashed) of h-YMnO3. Valence band maxima (VBM) are set to be zero for energy axis. The special k-points used for plotting the band structures are Γ (0, 0, 0), K (13,13, 0), M (12, 0, 0), L (12,0,12), and A (0, 0, 12). (b) Orbital resolved DOS from PBE + U calculation. In case of Mn(3d) states solid (black) and dashed (blue) color have been used to differentiate between states coming from Mn atoms hosting up and down magnetic moments.

FIG. 2.

(a) PBE + U band-structure (blue solid) along with the GW band-structure (red dashed) of h-YMnO3. Valence band maxima (VBM) are set to be zero for energy axis. The special k-points used for plotting the band structures are Γ (0, 0, 0), K (13,13, 0), M (12, 0, 0), L (12,0,12), and A (0, 0, 12). (b) Orbital resolved DOS from PBE + U calculation. In case of Mn(3d) states solid (black) and dashed (blue) color have been used to differentiate between states coming from Mn atoms hosting up and down magnetic moments.

Close modal

In h-YMnO3 the O(2p) bands are almost completely filled and located within ∼6 eV below VBM. Without any onsite Hubbard interaction (U) the Mn(3d) bands lie above the O(2p) levels.10 However, in the presence of a strong U, the majority Mn(3d) bands go below O(2p) (4–8 eV below VBM), whereas the minority Mn(3d) bands go to 4–8 eV above VBM. The application of U significantly changes the character of the top of the valence band from Mn(3d) to O(2p). Most of the Y(4d) bands are unoccupied lying in the range 4–8 eV above VBM.

In Figs. 3(a)–3(h) we have shown the MLWFs resembling Mn(3d) and O(2p) states. Fig. 3(i) shows that, in case of Y atom, capped trigonal prismatic crystal field splits 4d levels such that dyz/dzx are energetically most stable. The dz2 and dxy/dx2y2 are higher in energy by 0.59 eV and 1.79 eV, respectively.

FIG. 3.

Mn(3d)-like Wannier functions (a) dz2, (b) dzx, (c) dyz, (d) dxy, and (e) dx2y2, and O(2p) like Wannier functions (f) pz, (g) px, and (h) py. Red and blue colors are for isosurfaces of identical absolute values but opposite signs. (i) Crystal field splitting of Y(4d), Mn(3d), and O(2p) levels obtained by diagonalizing the respective subspace of the Hamiltonian in Wannier function basis. For Mn(3d) the spin up and down orbitals are indicated. For O(2p) orbitals contributions from apical (ap) and planar (pl) atoms are shown separately. (j) Positions of the bands originating from the hybridization between different 3d orbitals of Mn and 2p orbitals of O from diagonalizing the respective subspace of the Hamiltonian in Wannier function basis. (k) Orbital resolved DOS for Mn(3d) and O(2p) levels.

FIG. 3.

Mn(3d)-like Wannier functions (a) dz2, (b) dzx, (c) dyz, (d) dxy, and (e) dx2y2, and O(2p) like Wannier functions (f) pz, (g) px, and (h) py. Red and blue colors are for isosurfaces of identical absolute values but opposite signs. (i) Crystal field splitting of Y(4d), Mn(3d), and O(2p) levels obtained by diagonalizing the respective subspace of the Hamiltonian in Wannier function basis. For Mn(3d) the spin up and down orbitals are indicated. For O(2p) orbitals contributions from apical (ap) and planar (pl) atoms are shown separately. (j) Positions of the bands originating from the hybridization between different 3d orbitals of Mn and 2p orbitals of O from diagonalizing the respective subspace of the Hamiltonian in Wannier function basis. (k) Orbital resolved DOS for Mn(3d) and O(2p) levels.

Close modal

In h-YMnO3,48 MnO5 ions form a trigonal bi-pyramid structure. The crystal field in such environment splits the Mn(3d) levels into two low lying doubly degenerate states e2g and e1g (dxz/dyz and dxy/dx2y2) and one singly degenerate state a1g (dz2). Our calculation clearly shows that (Fig. 3(i)) the doubly degenerate e2g is the lowest in energy among the occupied Mn(3d) levels with e1g and a1g levels at energies 0.84 eV and 4.41 eV higher than e2g. In the presence of onsite Hubbard interaction (U) each of the Mn(3d) states split as well. On an average this state splits by ∼11.5 eV, which is consistent with the value of U we chose (10.1 eV). Fig. 3(j) shows the positions of the bands originating from the hybridization between Mn (3d) and O (2p) orbitals from diagonalization of the respective subspace of the Wannier Hamiltonian.

In Fig. 3(k) we have shown orbital resolved DOS of Mn(3d) and O(2p) orbitals. The orbital resolved DOS results are from a PBE + U calculation. Owing to the tilting of the MnO5 trigonal bi-pyramids, the local z-axis (along apical O–Mn bond) does not align with the crystallographic c-axis. In order to obtain the orbital resolved DOS, we rotated the crystal such that the local z-axis of each Mn atom aligns with the Cartesian z-axis. The figure shows that the first 3 conduction bands at 0.5–2.0 eV above VBM and the 3 valence bands at 5–6 eV below VBM have mostly Mn(dz2) and O(2p) character. The e1g valence bands of majority spin are more dispersive than a1g. These e1g bands lie 5–8 eV below VBM, whereas e2g valence bands lie 6.5–8 eV below VBM. If we take a closer look at Fig. 3(k) we can see that the top of the valence band mostly consists of O(2p) states strongly hybridized with e2g and e1g states of Mn(3d). The bottom of the conduction band on the other hand arises predominantly from the hybridization of a1g states of Mn(3d) with O(2p) states. To confirm these characterizations explicitly, we diagonalize the Hamiltonian of MnO5 written in Wannier function basis in the subspace of these orbitals. We consider the hybridization of O(2p) orbitals with a1g, e1g + e2g and all the Mn(3d) orbitals. We include both planar and apical O atoms in the calculation because they both hybridize significantly with Mn(3d) states. We find that the resulting bands shown in Fig. 3(j) lie in the same energy windows as seen from the orbital resolved DOS (Fig. 3(k)).

Several first principles calculations based on DFT and DFT + U have been done previously on h-YMnO3. Zhong et al.44 have reported results for ferromagnetic (FM), A-type anti ferromagnetic (AAFM), GAFM, and TAFM phases. In this study, based on PBE, the authors found the band gap of GAFM phase to be 0.38 eV. Qian et al.10 used the LMTO-ASA method (with U = 10.1 eV and J = 0.88 eV) and reported a band gap of 1.1 eV for AAFM phase of h-YMnO3. Novák et al.49 did LDA + U calculation for FM phase of the same material. The authors in this work found that the band gap varies from 0.45 eV to 0.9 eV when the value of U is varied from 9 eV to 12 eV. Qian et al.50 also studied the optical properties of h-YMnO3 within LDA + U. In this study, the authors have considered GAFM ordering of Mn moments with a smaller value of U and J (U = 8 eV and J = 0.88 eV) and found a band gap of 0.48 eV. Our PBE + U band structure as well as DOS agrees qualitatively with Ref. 50.

It is evident from the band structure (Fig. 2(a)) that h-YMnO3 is a direct band gap semiconductor. Fig. 4 shows the quasiparticle shifts of first 50 conduction and 6 valence bands of h-YMnO3 calculated within G0W0. Quasiparticle corrections shift the valence bands towards more negative energies and raise the conduction bands slightly. As a result the gap opens up from 0.59 eV within PBE + U to 1.66 eV within GW. In recent years it has been pointed out51,52 that exclusion of semicore electrons from the charge density while using it in generalized plasmon pole model is important in order to obtain the correct band gap. We have also calculated the band gap by excluding the semicore electrons from the charge density. In h-YMnO3 this changes the band gap by less than 30 meV. Fig. 4 also shows that the quasiparticle shifts do not produce a rigid shift of the band structure. As can be seen in Fig. 2(a), the dispersion of the second valence band between Γ and K increases significantly after quasiparticle corrections. Fig. 4 shows that this trend continues throughout the valence band. The quasiparticle shifts change the dispersion of the bands as well as their energies relative to each other. Fig. 2(a) shows the G0W0 band-structure of h-YMnO3 obtained from our calculation. Furthermore, the valence bands dispersion changes significantly between PBE + U and GW.

FIG. 4.

Quasiparticle corrections to the PBE + U eigenvalues for first 50 valence and 6 conduction bands at high-symmetry points Γ (green triangle), M (red square), L (blue star), and A (black circle) in the first BZ. Dashed lines indicate the valence-band maximum (VBM) and conduction band minimum (CBM).

FIG. 4.

Quasiparticle corrections to the PBE + U eigenvalues for first 50 valence and 6 conduction bands at high-symmetry points Γ (green triangle), M (red square), L (blue star), and A (black circle) in the first BZ. Dashed lines indicate the valence-band maximum (VBM) and conduction band minimum (CBM).

Close modal

Photoelectron spectroscopy of YMnO329 has found that the a1g, e2g, and e1g levels originating from Mn(3d) states lie at 4.1, 5.9, and 7.1 eV below VBM, respectively. Our orbital resolved DOS calculation within PBE + U suggests position of those states at 5.3, 6.7, and 7.4 eV below VBM (Fig. 3(k)). The ordering of the Mn (3d) levels agrees with the experimental result, which is a consequence of the choice of U. It indicates that the value of U chosen for our mean-field PBE + U calculation is appropriate. However, after quasiparticle correction within G0W0, these states shift away from VBM by 1.6–2.1 eV making the agreement with experiment worse. It is well known that the GW approximation is the more appropriate theory (compared with PBE + U) for describing the photoemission experiments.13,15 This mismatch between GW quasiparticle levels and experimental data can be due to two reasons. As the states we are trying to calculate are highly localized d-states they are much harder to converge within the GW framework. We have chosen parameters of our GW calculation such that the quasiparticle gap is converged but that may not be sufficient to accurately calculate the quasiparticle energies of Mn(3d) states. Furthermore, it has been pointed out in the literature that the d-band positions are sometimes not well described within the G0W0 approximation.54,55 One has to go beyond the G0W0 approximation to either the GW0 or self-consistent GW to describe these d-band energy positions well.

To compare our results with the optical absorption experiments we solve the Bethe–Salpeter equation and compute the optical absorption spectrum. Fig. 5(a) shows the imaginary part of the macroscopic dielectric function calculated by solving BSE. The polarization of light is assumed to be parallel (top panel) and perpendicular (bottom panel) to the c-axis. The theoretical curves have been broadened using Gaussian broadening of width 100 meV. In the figure, we show the optical response function both with and without the presence of electron-hole interaction. We find that upon including the electron-hole interaction, the first absorption peak is at 1.45 eV. This calculated optical gap is in excellent agreement with the optical spectroscopy data available for this material (1.55 eV).56 We obtain the exciton binding energy by subtracting the absorption onset from the quasiparticle gap. Our calculated exciton binding energy in this material is 0.21 eV.

FIG. 5.

(a) Imaginary part of the macroscopic dielectric function when light polarization is along the c-axis (top panel) and perpendicular to c-axis (bottom panel). The blue solid and red dashed curves show the calculation with or without electron-hole interaction, respectively. The black dotted curve shows the experimental absorption.53 (b) Real-space exciton wavefunction corresponding to lowest energy exciton (1.45 eV). Grey, green, and cyan spheres denote Y, Mn, and O atoms, respectively. The hole position is shown by the black sphere.

FIG. 5.

(a) Imaginary part of the macroscopic dielectric function when light polarization is along the c-axis (top panel) and perpendicular to c-axis (bottom panel). The blue solid and red dashed curves show the calculation with or without electron-hole interaction, respectively. The black dotted curve shows the experimental absorption.53 (b) Real-space exciton wavefunction corresponding to lowest energy exciton (1.45 eV). Grey, green, and cyan spheres denote Y, Mn, and O atoms, respectively. The hole position is shown by the black sphere.

Close modal

In Fig. 5(b) we show the exciton wavefunction corresponding to the lowest energy exciton (1.45 eV). The plots are isosurface of modulus squared of the real-space exciton wavefunction. The isosurface of the exciton wavefunction was plotted by choosing 2% of the maximum value. We fix the hole position to be near the Mn atom along the bond between Mn and planar O atom. As we can see from Fig. 5(b), the exciton wavefunction is localized within a unit cell in c direction but has a much larger spread in a-b plane. The reason behind this strong localization in the c direction is that the lowest energy transitions mostly arise from the transition between almost flat bands in A-Γ direction. These bands have almost no contribution from Y(4d) states. As a result, the exciton wavefunction is localized in plane of the Mn–O polygons with no amplitude on the Y atoms. This type of bi-dimensional excitons have also been observed in other transition metal oxides such as TiO2.57 

For energies below 2 eV, in case of light polarization along c-axis our calculated absorption spectra do not show the narrow peak at 1.69 eV (1.8 eV (Ref. 11)), when compared with the experimental results.53 In our calculation we find some excitations in the energy range 1.45 eV–1.75 eV. However, they have very small oscillator strengths and are dark, whereas in case of light polarization perpendicular to c-axis (parallel to a-b plane) we see a much more intense peak at 1.54 eV as compared with the experimental data. The experimental data has a peak at 1.61 eV.11,53 These results show that while we correctly predict the peak positions in absorption spectra as well as the anisotropy in absorption along different directions, the oscillator strengths of these peaks are not captured properly within the GW-BSE formulation. In our calculations, we have ignored the effects due to phonons, ionic screening, and defects, which can lead to such discrepancies.57 In any real material at finite temperature phonons (or polarons) and defects (such as oxygen vacancies) will certainly affect optical properties substantially. Furthermore, as the BSE calculation has been performed within the Tamm–Dancoff approximation and using static screening, this could also lead to discrepancies in oscillator strengths.58 

We have used the state-of-the-art GW-BSE methodology to understand the electronic structure of G-type anti-ferromagnetic phase of h-YMnO3 starting from PBE + U mean field. Our calculation shows that the valence and conduction bands in this material arise due to Mn(3d) and O(2p) orbitals. Our calculated GW band structure shows a band gap of 1.66 eV. Furthermore, our optical band gap is in very good agreement with the experimental value of 1.55 eV. The exciton binding energy in this material is quite large (0.21 eV) for it to be used in photovoltaic applications. In addition, the GW band structure of YMnO3 indicates that transport along the c direction is also expected to be difficult. This along with the anisotropic absorption below 2 eV will make this material difficult to use for photovoltaic applications.

See supplementary material for the convergence of the self energy with respect to the number of bands.

The authors would like to thank Professor S. Raghavan, Professor S. Avasthi, Professor R. Ranjan and Professor P. Ramamurthy for useful discussions. T.B. thanks S. Sinha Roy for helpful discussions. The authors would also like to thank the anonymous referees for their insightful comments. This work is supported under the US-India Partnership to Advance Clean Energy-Research (PACE-R) for the Solar Energy Research Institute for India and the United States (SERIIUS), funded jointly by the U.S. Department of Energy (Office of Science, Office of Basic Energy Sciences, and Energy Efficiency and Renewable Energy, Solar Energy Technology Program, under Subcontract No. DE-AC36-08GO28308 to the National Renewable Energy Laboratory, Golden, Colorado) and the Government of India, through the Department of Science and Technology under Subcontract No. IUSSTF/JCERDC-SERIIUS/2012 dated 22nd Nov. 2012. We thank Super Computer Research and Education Centre (SERC) at IISc for the computational facilities.

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