Cation Vacancy in Wide Bandgap III‐Nitrides as Single‐Photon Emitter: A First‐Principles Investigation

Abstract Single‐photon sources based on solid‐state material are desirable in quantum technologies. However, suitable platforms for single‐photon emission are currently limited. Herein, a theoretical approach to design a single‐photon emitter based on defects in solid‐state material is proposed. Through group theory analysis and hybrid density functional theory calculation, the charge‐neutral cation vacancy in III‐V compounds is found to satisfy a unique 5‐electron‐8‐orbital electronic configuration with Td symmetry, which is possible for single‐photon emission. Furthermore, it is confirmed that this type of single‐photon emitter only exists in wide bandgap III‐nitrides among all the III‐V compounds. The corresponding photon energy in GaN, AlN, and AlGaN lies within the optimal range for transfer in optical fiber, thereby render the charge‐neutral cation vacancy in wide‐bandgap III‐nitrides as a promising single‐photon emitter for quantum information applications.


Group theory and many-electron effect analysis of the vacancy defect
We first use an analytic model to understand the defect energy level and the corresponding many-electron configuration. For a vacancy with T d symmetry, the Hamiltonian H that projected on sp 3 dangling bond orbitals ψ i (i = 1, 2, 3, 4) can be written as The dangling bond orbital basis is not orthogonal and there is an overlap integral S = ⟨ψ i |ψ j ⟩(i ̸ = j). The calculated eigenvalues are where A 1 is a nondegenerate state, the wavefunction can be expressed as and T 2 is a triple degenerate state, the corresponding wavefunctions are For the case that S ≪ 1, the energy splitting between A 1 and T 2 is given as ∆, the relative position of A 1 and T 2 depends on the sign of ∆, i.e. the sign of ⟨ψ i |H|ψ j ⟩.

S1
To assess the many-electron effect on the energy level of the defect center with different electronic configuration, we build the Slater determinant wavefunction | . . . mn . . .⟩, where the basis |m⟩, |n⟩ are composed by A 1 (v) and T 2 (t x , t y , t z ). The total energy is expressed as ⟨. . . mn . . . |H| . . . mn . . .⟩. The electron Hamiltonian can be separated into two parts Under the tight-bonding approximation that neglecting all the overlap integrals, the one-electron integrations can be expressed as The two-electron integrals can be expressed as where J and J ′ are the two-electron integrals of the dangling bond orbital basis ψ i

Electronic structure calculation
The atomic positions were optimized until the maximum force on each atom was less than 0.01 eV/Å with PBE functional, and the electronic structures are calculated with the Heyd-Scuseria-Ernzerhof (HSE06) [6] hybrid functional by using the PWmat [7,8] package. The mixing parameter of the explicit Hartree-Fock exchange energy is set to be 0.32 for AlN, GaN, and GaAs, 0.25 for InN, AlP, GaP, InP, AlAs, and InAs. According to the previous theoretical study, the hybrid functional is accurate for the transitions between internal defect levels. [9] The ONCV-PWM pseudo potential [10] with an energy cutoff of 50 Ry is adopted. For all the calculations, the Al (3s 2 3p 1 ), Ga (4s 2 4p 1 ), In (5s 2 5p 1 ), N (2s 2 2p 3 ), P(3s 2 3p 3 ), As (4s 2 4p 3 ), and Mg (2s 2 2p 6 3s 2 ) are treated as valence electrons, and the spin polarization effect is considered. The calculated bandgap for wurtzite InN, GaN, and AlN are 0.69 eV, 3.59 eV, and 5.95 eV; the calculated bandgap for zinc blende InP, GaP, and AlP are 1.27 eV, 2.39 eV, and 2.37 eV; the calculated bandgap for zinc blende InAs, GaAs, and AlAs are 0.21 eV, 1.40 eV, and 2.11 eV, which are consistent with experimental results. [11,12] The calculated band structures and the corresponding orbital contributions for AlGaN with different Al ratios are shown in Figure S2. For wurtzite GaN and AlN, the conduction band minimum is contributed by the s state of the N atom. The valence band maximum states are more complex. In the absence of spin-orbit interaction, the top of the valance states at the Γ point is split into a nondegenerate crystalfield split hole (CH) state (contributed by N pz orbital) and a twofold-degenerate state (contributed by N px and N py orbitals) due to the non-cubic crystal-field splitting ∆ cr . With the presence of spin-orbit interaction ∆ so , the twofold-degenerate state is further split into the heavy hole (HH) and light hole (LH) states. Theoretical and experimental investigations indicate that the order of these states is different between GaN and AlN [13,14,12]. Here, the calculated crystal-field splitting energy ∆ cr for GaN and AlN are 29 meV and -222 meV, respectively. Previous theoretical calculations of ∆ cr for GaN and AlN are 42 meV and -217 meV [14], this indicates that the HSE06 functional can correctly describe the band structure of GaN and AlN.

Absorption spectra and radiative lifetime
The absorption spectrum was calculated with the random phase approximation (RPA) method, which is based on direct Fermi's Golden rule, here, ∂H/∂k is the momentum operator, E i is the eigenvalue of state |ψ i ⟩.
The radiative lifetime is calculated by Fermi's Golden rule, [15] the concrete form of the radiative rate is given by where ω is the frequency of emission photon, n is the index of refraction, we use the value n of 2.16, 2.38, and 2.59 for AlN, GaN, and InN, a linear combination of n is used for the alloy, µ ij is the transition dipole moment, ϵ 0 is the vacuum permittivity,h is the reduced Planck constant, c is the vacuum speed of light. According to the relationship of ij me , the radiative rate W rad (ω) and radiative lifetime τ rad = 1/W rad (ω) can be obtained.

Thermodynamic stability calculation for point defect
The formation energies of defects are defined as here α is the defect type, q is its charge, E(α, q) is the supercell energy with the defect α with charge q, E(host) is the supercell energy without defect, n i is the number of element i changed from defect α, µ i is the corresponding chemical potential, E V BM (host) is the host valence band maximum (VBM) eigenenergy, E F is the Fermi energy relative to E V BM (host). ∆V = V (α, q, R) − V (host, R), where R is a place far away from the defect. [16] This formation energy can be used to estimate the defect concentration under the equilibrium condition. Based on Eq. S13, the formation energy of neutral defects does not change with E F , while for the charged defects, the formation energy varies linearly with E F and the slope is equal to the charge state q. The calculated chemical potentials for binary and elemental phases are µ (

Atomic defect structure for band structure calculation
The atomic structures of V cation in III-V compounds are shown in Figure S3. For V cation in III-phosphide and III-arsenide, a zinc-blende type supercell containing 215 atoms is used with a (1×1×1) K-point sampling; for V cation in III-nitride, a wurtzite type supercell containing 191 atoms is used with a (2×2×1) K-point sampling.
GaAs InN InP InAs Figure S3: Supercell structures of III-V compounds with V cation for band structure calculation, the V cation is highlighted by the black circle. The blue, green, pink, gray, purple, and orange balls represent Al, Ga, In, N, P, and As atoms, respectively.
The atomic structures of V cation in InGaN alloy are shown in Figure S4, a (1×2×1) mesh is chosen for K-point sampling for In 0.125 Ga 0.875 N 127-atom supercell, a (1×2×2) mesh is chosen for K-point sampling for In 0.25 Ga 0.75 N 215-atom supercell.

Band structure of V cation
The band structure of V cation in InGaN calculated with HSE06 functional is shown in Figure S6, the supercell structures are shown in Figure S4. The band structure of V cation +Mg cation /Mg − cation in GaN/AlN calculated with HSE06 functional is shown in Figure S7, the supercell structures are shown in Figure S5.

Defect energy level and absorption spectrum for V cation
The defect energy level and absorption spectrum of V cation in In 0.125 Ga 0.875 N calculated with HSE06 functional are shown in Figure S8, a 255-atom supercell is used with a Γ point only K mesh sampling.  The defect energy level and absorption spectrum of V cation in strained GaN/AlN calculated with HSE06 functional are shown in Figure S9, a 399-atom supercell is used with a Γ point only K mesh sampling for both GaN and AlN.

2.317
(a 1 )  Figure S9: (a 1 )-(d 1 ) are the atomic supercell structures for V Ga in 95%strain-GaN, V Ga in 105%strain-GaN, V Al in 95%strain-AlN, and V Al in 105%strain-AlN, the corresponding defect energy levels in the spin-down channel (the VBM is set to zero) and absorption spectrum calculated with HSE06 functional are shown in (a 2 )-(d 2 ) and (a 3 )-(d 3 ), respectively. S10

Defect energy level and absorption spectrum for V cation
The defect energy level and absorption spectrum of V Ga in AlGaN alloy calculated with HSE06 functional are shown in Figure S10. A 399-atom supercell is used for Al 0.25 Ga 0.75 N, a 359-atom supercell is used for Al 0.5 Ga 0.5 N, a 359-atom supercell is used for Al 0.8 Ga 0.2 N, all the calculation use a Γ point only K mesh sampling.

Ground and excited state geometries of V cation
The interatomic distance (in Å) between the four N atoms (index 1 to 4) around V catoion at ground (excited) states are depicted in Table S2, index 1 represents the N atom along the (0001) direction, index 2-4 represent the remaining N atoms.

Moment matrix for V cation
The calculated moment matrices |⟨ψ i |p|ψ j ⟩| 2 (in atomic unit) of V cation in III-nitrides (corresponding to Table 1) among VBM, CBM, and all the defect levels in the spin-down channel are shown below.