Abstract
The product Jahn–Teller effect may occur for such coupled electron–phonon systems in solids where single electrons occupy double degenerate orbitals. We propose that the excited state of the neutral XV splitvacancy complex in diamond, where X and V labels a groupIV impurity atom of X = Si, Ge, Sn, Pb and the vacancy, respectively, is such a system with e_{g} and e_{u} double degenerate orbitals and E_{g} quasilocalized phonons. We develop and apply ab initio theory to quantify the strength of electron–phonon coupling for neutral XV complexes in diamond, and find a significant impact on the corresponding optical properties of these centers. Our results show good agreement with recent experimental data on the prospective SiV(0) quantum bit, and reveals the complex nature of the excited states of neutral XV color centers in diamond.
Introduction
Fluorescent, paramagnetic point defects in diamond may realize quantum bits for quantum technology. Splitvacancy complexes of groupIV impurity atom (X = Si, Ge, Sn, Pb) and vacancy, that is, XV defects with D_{3d} symmetry are in the focus of intense research. The negatively charged XV, that is, XV(−) defects have S = 1/2 spin state and fluoresce mostly in the visible.^{1,2,3,4,5,6,7,8,9,10,11} The inversion symmetry of the centers assumes virtually no Stark shift in the optical signals, which is a prerequisite for realization of indistinguishable single photon sources. Among these color centers, SiV(−) is the most studied,^{2,4,5,12,13,14,15,16,17,18,19,20,21,22} and stands out with a large Debye–Waller (DW) factor of 0.7, and the demonstration of quantum communication and sensor applications.^{12,13} On the other hand, the SiV(−) exhibits short spin coherence times due to phonon dephasing caused by the dynamic Jahn–Teller effect on the orbital doublet,^{18} thus cooling to the millikelvin regime is required for quantum bit operations.^{23,24} It is predicted that PbV(−) might have much longer spin coherence times because of the enlarged gap of the orbital doublet caused by spin–orbit interaction, but with the expense of smaller Debye–Waller factor than that of SiV(−).^{25}
Alternatively, by removing an electron from XV(−) centers, an orbital singlet with S = 1 ground state appears^{2,26,27,28,29} that should have intrinsically long coherence times. Recently, it has been demonstrated that SiV(0) exhibits spin coherence time almost up to a second and relaxation time nearly a minute^{30} at 20 K together with a nearinfrared fluorescence signal, and has been proposed for quantum communication applications.^{30} This observation naturally shifts the focus towards XV(0) color centers in diamond. However, the nature of the excited and shelving states and levels are far from being understood for SiV(0).^{31} In particular, the 946nm zerophononline (ZPL) optical transition of SiV(0) (see refs. ^{30,32,33,34,35,36}) was originally assigned to an ^{3}A_{2g} ↔ ^{3}A_{1u} electronic excitation from the ground state to the excited state;^{5,34} however, it has been very recently revealed that the excited state should be a ^{3}E_{u} state deduced from stress measurements.^{36} Furthermore, a dark ^{3}A_{2u} state below the ^{3}E_{u} by 6.7 meV was activated in the luminescence spectrum by exerting uniaxial stress on the diamond sample.^{36} The optical signals of other XV(0) centers have not yet been identified at all. First principles methods are major tools to explore the complex physics of point defects that can strongly contribute to understanding SiV(0) color center and identifying the other XV(0) color centers.
In this article, we present first principles results on the optical properties of XV(0) color centers in diamond. We show that the electrons and phonons are strongly coupled in the electronic excited states, and they constitute of a (e_{g} ⊗ e_{u}) ⊗ E_{g} product Jahn–Teller (pJT) system, where e_{g} and e_{u} refers to the corresponding electronic orbitals that are simultaneously coupled to quasilocalized E_{g} symmetry breaking local vibrational mode. This pJT effect is responsible for the anomalous optical spectrum of SiV(0). We briefly discuss our results in the context of quantum technology applications. We provide the theoretical optical signatures of the other XV(0) color centers too.
XV(0) defect has six carbon dangling bonds and the impurity atoms sits in the inversion center of diamond (see Fig. 2c) and exhibits D_{3d} symmetry. These six dangling bonds introduce a_{1g} ⊕ a_{2u} ⊕ e_{u} ⊕ e_{g} orbitals.^{2,5,14,29,34} The a_{1g} and a_{2u} levels fall in the valence band (VB) of the diamond.^{8} The e_{u} level is fully occupied by four electrons and it either resonant with the VB or pops up in the gap by increasing the X atomic number.^{25} The e_{g} level is occupied by two electrons in the band gap of diamond in the ground state. By promoting an electron from the e_{u} orbital to the e_{g} orbital (or forming a single hole on both orbitals), the lowest energy optically active and inactive excited states are formed. The 16 electronic configurations from these orbitals are
where we focus on the 12dimensional triplet subspace.
The excited triplet twohole wavefunctions and the ^{3}A_{2g}〉 ground state can be expressed by the following equations in the hole representation,
where we introduce the antisymmetrization operator \({\cal A}ab\rangle = (ab\rangle  ba\rangle )/\sqrt 2\) and symmetrization operator \({\cal S}ab\rangle = (ab\rangle + ba\rangle )/\sqrt 2\), and spinup (spindown) holes are labeled by ↑ (↓). It is worth to note that two singlet levels (^{1}A_{1g}〉, ^{1}E_{g}〉) for the \(e_g^2\) electronic configuration appear above the ^{3}A_{2g}〉 ground state and below the (e_{g}e_{u}) excited triplet levels of Eq. 1. Additionally, the (e_{g}e_{u}) excited singlet levels are expected to lie above the (e_{g}e_{u}) excited triplets due to the Coulomb repulsion.^{31,36} We show the single determinant states of the triplets in Eq. 2 in Fig. 1 in their respective S_{z} = +1 spin substate (see also Supplementary Note 1). We calculate these five \({\cal A}e_{ux}e_{gy}\rangle \otimes  \uparrow \uparrow \rangle\) wavefunctions by means of Kohn–Sham density functional theory (DFT) (see Methods).
Results
Formulation of the pJT Hamiltonian
Our DFT calculations indicate a strong Jahn–Teller distortion in the lowest energy triplet excited state of XV(0), going from the high D_{3d} symmetry to the low C_{2h} symmetry. This can be understood by considering the fact that e_{g} and e_{u} orbitals are occupied by a single hole in the excited state, thus they are both Jahn–Teller unstable. By applying the E ⊗ e Jahn–Teller theory on both orbitals in the strongly coupled limit and constructing an antisymmetric product \(\left( {{\cal A}e_{ux}^\varphi e_{gx}^\varphi \rangle } \right)\) of the two particles with adding the spin degrees of freedom, one will arrive to the following (see Eqs. S23–S30),
which corresponds to walking on the yellow circle of the adiabatic potential energy surface (APES) in Fig. 2a. An accurate solution can be found by solving the following Hamiltonian,
where \(\hat H_{{\mathrm{osc}}}\) is the twodimensional harmonic oscillator spectrum of the E_{g} phonon mode, \(\hat H_{{\mathrm{pJT}}}\) is the the pJT Hamiltonian, and \(\hat W\) is the electron correlation Hamiltonian between the triplet states in Eq. 2. We define \(\hat H_{{\mathrm{osc}}}\) by means of ladder operators (\(a^{\,}_{X,Y} , \, a_{X,Y}^\dagger\)) as \(\hat H_{{\mathrm{osc}}} = \hbar \omega _{\mathrm{E}}\mathop {\sum}\nolimits_\alpha ^{X,Y} \left( {a_\alpha ^\dagger a_\alpha + \frac{1}{2}} \right)\), where ℏω_{E} is the effective phonon energy.
The usual e ⊗ E linear Jahn–Teller Hamiltonian^{37,38,39,40} is modified to
where u/g labels the e_{u}/e_{g} orbital, and σ_{z} and σ_{x} are the standard Pauli matrices in Eq. 5. \(\hat \sigma _0\) is the twodimensional unit matrix that is introduced for the individual electron–phonon coupling strength F_{u} and F_{g}.^{41} \((\hat X,\hat Y) = (a_{(X,Y)}^\dagger + a_{(X,Y)})/\sqrt 2\) defines the twodimensional configuration space spanned by the E_{g} vibration mode through the harmonic oscillator ladder operators. See Supplementary Note 3 for details about the derivation of Eq. 5. Finally, we define the position of electronic levels by Λ and Ξ caused by static electronic correlation with the following expression (see Supplementary Note 2):
The overall Hamiltonian of the system in a 4 × 4 matrix notation is the following by combining Eqs. 2–6:
where we label the individual single determinant electronic wavefunctions at \(\hat H_{{\mathrm{pJT}}}\) for clarity (see also Supplementary Note 3). The diagonal part of the pJT matrix is selfexplanatory. If the geometry is distorted towards +X, the e_{ux}e_{gx}〉 wavefunction would lower its energy, by a joint F_{u} + F_{g} coupling strength. In the central part of diagonal Hamiltonian, the two Jahn–Teller effects are destructive, and the joint product Jahn–Teller strength is F_{u} − F_{g}.
Next, the parameters in Eq. 7 are determined by first principles DFT calculations, and the coupled electron–phonon Hamiltonian is solved (see Supplementary Note 3 for details).
Parameters from ab initio calculations
We show the ab initio parametrization of the full Hamiltonian for SiV(0), and discuss the results in detail. The key results are also summarized for the other XV(0) color centers in Table 1 as obtained by the same procedure. Our DFT ΔSCF method yields the total energy of the four, \({\cal A}e_{uy}e_{gy}\rangle\), \({\cal A}e_{ux}e_{gy}\rangle\), \({\cal A}e_{uy}e_{gx}\rangle\), and \({\cal A}e_{ux}e_{gx}\rangle\), electronic configurations (see also Supplementary Note 3 and Fig. S2). The APES of these states is depicted in Fig. 3a. This is our starting point to determine the parameters in Eq. 7.
In D_{3d} symmetry, \(E_{{\mathrm{tot}}}[{\cal A}e_{ux}e_{gy}\rangle ] = E_{{\mathrm{tot}}}[{\cal A}e_{uy}e_{gx}\rangle ]\) and \(E_{{\mathrm{tot}}}[{\cal A}e_{ux}e_{gx}\rangle ] = E_{{\mathrm{tot}}}[{\cal A}e_{uy}e_{gy}\rangle ]\), where E_{tot} is the DFT total energy (see Supplementary Note 2). Finally, the calculated energy separation is Λ = 78.3 meV for SiV(0).
We determine the position of ^{3}E_{u}〉 by a \(e_{u \pm }\rangle = (e_{ux}\rangle \pm ie_{uy}\rangle )/\sqrt 2\) transformation on the defect orbitals, thus twohole wavefunction will be a single determinant in this basis as \(\,{}^3E_u\rangle = {\cal A}e_{u \pm }e_{u \pm }\rangle\), which can be directly approximated by means of DFT (see Supplementary Note 2). Finally, Ξ = 44.9 meV is obtained by this procedure for SiV(0).
The effective vibration energy ℏω_{E} can be found by fitting to the parabola of the lowest APES curvature in Fig. 3a that results in 75.9 meV for SiV(0).
The electron–phonon coupling F_{g} and F_{u} parameters can be derived by reading out the characteristic Jahn–Teller energies \(E_{{\mathrm{JT}}}^{(1)}\) and \(E_{{\mathrm{JT}}}^{(2)}\) in the APES (see Fig. 3a) as follows:
In SiV(0), \(E_{{\mathrm{JT}}}^{(1)} = 258\,{\mathrm{meV}}\), that is very significant and seriously affect the calculated ZPL energy. On the other hand, \(E_{{\mathrm{JT}}}^{(1)} = 0.47\,{\mathrm{meV}}\), which is small, and results nearly identical to F_{g} and F_{u}. We note here that we neglect the quadratic Jahn–Teller terms in Eq. 5, that is, the APES in Fig. 3 is axially symmetric. However, the \({\cal A}e_{uy}e_{gy}\rangle\) has a bit smaller Jahn–Teller energy than that of \({\cal A}e_{ux}e_{gx}\rangle\) by 43, 46, 46, and 48 meV for SiV, GeV, SnV, and PbV, respectively, that would cause a quadratic Jahn–Teller effect. However, these energies are an order of magnitude smaller than that of \(E_{{\mathrm{JT}}}^{(1)}\), and would only lead to minor correction to the results from linear Jahn–Teller approximation. We explicitly proved this for XV(−) color centers in our previous study.^{25} Therefore, we still apply the linear Jahn–Teller approximation for the sake of simplicity.
Finally, all the parameters could be derived or read out from the calculated APES (see Table 1), thus one can setup the full Hamiltonian in Eq. 7. It is intriguing to use X as a continuous variable at Y = 0 in Eq. 7, and plot the solution in Fig. 3b. The contribution of the dark ^{3}A_{2u} state is shown by black balls, while the contribution of the optically active ^{3}A_{1u} state (z polarization) and ^{3}E_{u} state [(x,y) polarization] is depicted as red and teal balls, respectively. The lowest energy solution will apparently involve the dark ^{3}A_{2u} state. For the full quantum mechanical solution (where X and Y are operators), we use the following wavefunction ansatz:^{25,42,43}
where n, m〉 is the representation of E_{g} vibration that we consider up to 15quanta limit (n + m ≤ 15) for the low energy spectrum and up to 50quanta limit for high energies in Fig. 3c as explained in Supplementary Note 3. The vibronic spectrum shows up two deep levels that are separated by δ = 6.8 meV, where the deepest level belongs to the vibronic \(\,{}^3\tilde A_{2u}\rangle\) and the second level is associated with the vibronic \(\,{}^3\tilde E_u\rangle\) in SiV(0).
We also show the derived parameters and results for GeV, SnV, and PbV systems in Table 1. There is a clear trend that the ZPL energies increase with heavier impurity atom.
On the other hand, the spin–orbit coupling (SOC) will be significant for heavy impurity atoms, and rapidly increase with the atomic number of the impurity atom. Based on our previous calculations for XV(−) defects,^{25} the SOC on e_{u}〉 orbitals, λ_{u} is 7, 33, 100, and 250 meV for SiV, GeV, SnV, and PbV, respectively. This can be neglected for SiV(0) but can be significant for the other XV(0) defects, that might alter the ZPL energies SnV(0) and PbV(0). The simultaneous solution for pJT and SOC is out of the scope of this paper but might be required for the ultimately accurate description of the excited states of SnV(0) and PbV(0).
Discussion
Experimental data are only available for SiV(0), thus we can directly compare our results only to them. A recent stress measurement on the photoluminescence spectrum of SiV(0) revealed a dark state where the corresponding level was below the ZPL energy by 6.8 meV.^{36} Our calculations explain this feature by the pJT effect of the three triplet excited states. The lowest energy branch of the excited state triplets yield ≈50% ^{3}A_{2u}〉 and ≈50% ^{3}E_{u}〉 electronic character that can be anticipated from Eq. 3. The lowest energy vibronic state is the dark \(\,{}^3\tilde A_{2u}\rangle\) (951 nm), and the next vibronic level above it by 6.7 meV belongs to the optically allowed \(\,{}^3\tilde E_u\rangle\) (ZPL of 946 nm). We note that in ordinary e ⊗ E JT systems, a degenerate E level is the lowest vibronic state quickly followed by a nondegenrate A vibronic level by tunneling splitting energy.^{37,38,44,45,46} In our present pJT case, the order of these states are reversed, which is a clear signature of the manifestation of the pJT effect.
There are numerous consequences of this finding: (i) The optical polarization of the emitted photons at the ZPL (946 nm) is perpendicular to the symmetry of the axis. (ii) The emission will be strain dependent as the symmetry breaking strain can activate the 951nm ZPL transition, again with photon polarization perpendicular to the symmetry axis of the defect. (iii) The 946nm ZPL intensity will be temperature dependent, as it depends on the thermal occupation of the \(\,{}^3\tilde E_u\rangle\) over the lowest energy \(\,{}^3\tilde A_{2u}\rangle\). These properties were indeed observed in previous experiments.^{34,36}
We developed a theory for the excited state of XV(0) quantum bits in diamond, which revealed a pJT effect, that is, strong coupling of a localized vibration mode to multiple triplet electronic states. We showed that our theory can explain numerous experimental features of SiV(0) color center. We predicted the basic optical properties of the other XV(0) color centers too. Our results provide tools to experimentally test pJT systems in solid state. Our findings can be useful to guide experiments on these color centers for quantum bit applications.
Methods
Numerical solution of the Jahn–Teller Hamiltonian
We determined the energy levels of the product Jahn–Teller system (Eq. 7) by a numerical code implemented in GNU octave that we describe in the Supplementary Information.
DFT calculations
We characterized XV(0) color centers by plane wave supercell calculations within spinpolarized DFT as implemented in the vasp 5.4.1 code.^{47} We determined the electronic structure within the Born–Oppenheimer approximation where the ions are treated as classical particles and where the minimum energy is found by moving the atoms until the quantum mechanical forces acting on the ions fall below 10^{−3} eV/Å. We embedded the XV(0) defects in a 512atom diamond supercell. The Brillouin zone is sampled at the Γpoint. We applied an energy cutoff at 370 eV for expressing the plane wave basis set within the applied projector augmentation wave method.^{48,49} We calculated the excited states with the constrainedoccupation DFT method (ΔSCF method).^{50} We used the HSE06 hybrid functional^{51,52} that reproduces the experimental band gap and the charge transition levels in GroupIV semiconductors within 0.1 eV accuracy.^{53} For the electron–phonon coupling calculations of XV(0) defects, we applied the same machinery that could well reproduce the ZPL energies of XV(−) exhibiting the dynamic Jahn–Teller effect;^{25} thus, we expect similar performance for XV(0) too.
Data availability
The data from DFT calculations that support the findings of this study are available from Gergő Thiering (email: thiering.gergo@wigner.mta.hu) upon reasonable request.
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Acknowledgements
Support from ÚNKP173III New National Excellence Program of the Ministry of Human Capacities of Hungary, the National Research Development and Innovation Office of Hungary within the Quantum Technology National Excellence Program (Project Contract No. 20171.2.1NKP201700001), and the European Commission of H2020 ASTERIQS project (Grant No. 820394) is acknowledged. We thank the National Information Infrastructure Development Program for the highperformance computing resources in Hungary.
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G.T. performed the simulations and analyzed the data. A.G. conceived the research and wrote the manuscript with G.T. All authors discussed and commented on the manuscript.
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Thiering, G., Gali, A. The (e_{g} ⊗ e_{u}) ⊗ E_{g} product Jahn–Teller effect in the neutral groupIV vacancy quantum bits in diamond. npj Comput Mater 5, 18 (2019). https://doi.org/10.1038/s4152401901583
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