Crystal field analysis of Dy and Tm implanted silicon for photonic and quantum technologies

We report the lattice site and symmetry of optically active Dy and Tm implanted Si. Local symmetry was determined by fitting crystal field parameters (CFPs), corresponding to various common symmetries, to the ground state splitting determined by photoluminescence measurements. These CFP values were then used to calculate the splitting of every J manifold. We find that both Dy and Tm ions are in a Si substitution site with local tetragonal symmetry. Knowledge of rare-earth ion symmetry is important in maximising the number of optically active centres and for quantum technology applications where local symmetry can be used to control decoherence. ©2014 Optical Society of America OCIS codes: (160.2540) Fluorescent and luminescent materials; (160.6000) Semiconductor materials. References and links 1. W. L. Ng, M. A. Lourenco, R. M. Gwilliam, S. Ledain, G. Shao, and K. P. Homewood, "An efficient roomtemperature silicon-based light-emitting diode," Nature 410, 192-194 (2001). 2. M. A. 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Introduction
Silicon is unprecedented in its technological applications, however, its indirect electronic band gap makes it a poor light emitter due to fast non-radiative decay dominating over slower radiative routes, which limits its photonic applications. Dislocation engineering by boron implantation can be used to inhibit non-radiative decay paths and allow band edge photoluminescence (PL) and electroluminescence (EL) from Si [1]. The same dislocation engineering technique, accompanied by rare-earth (RE) implantation can be used to obtain PL and EL at specific wavelengths determined by the RE optical transitions [2,3]. As a consequence, RE implanted Si is a promising platform for future photonic applications. However, comparatively little is known about the local environment of optically active RE ions implanted into silicon, which limits the optimisation of this technology. The atomic shielding of the f-shell electrons in REs means that the effect of the host crystal field on their energy levels is weaker when compared with transition metal ions [4,5] or bismuth [6,7]. Nevertheless, the complex f-shell inter-electron (coulomb) and spin-orbit interactions, along with the simultaneous influence of the crystal field, give rise to intricate energy level splitting that is highly dependent on the local crystal field environment of the RE and can reveal detailed structural information. Because of the low number of optically active ions in implanted samples compared to bulk doped samples, measuring the crystal field splitting from absorption measurements in implanted samples is extremely difficult. PL spectra can often be detected, but are inherently weak because above band gap excitation is required, and this only penetrates the first few hundred nm from the sample's surface. Therefore, various peaks in the crystal field split spectra can be undetected or unresolved.
Er implanted Si has been investigated extensively in comparison to other REs, largely because of the telecoms applications of its 1.5 µm luminescence. Extended X-ray absorption fine structure (EXAFS) measurements of Er-implanted, high O impurity Czochralski-grown Si, which exhibit strong PL at 1.54 µm, show Er is six-fold coordinated to oxygen [8]. Strong PL can also be observed from Er and O coimplanted Si [9]. By contrast, in low O impurity FZ Si, no PL was observed and Er was coordinated with 12 Si atoms [8]. Contradictory Rutherford backscattering (RBS) measurements have indicated that Er implanted in Si is located [10], and is not located [11], at an interstitial site. Other RBS and PL measurements indicate that when annealed at over 800°C, 80% of Er occupies substitutional sites with tetrahedral symmetry [12]. Electron paramagnetic resonance (EPR) spectra of Er and O coimplanted Si indicate two Er 3+ centers having monoclinic C 1h symmetry [13] and trigonal symmetry [14]. The emission channelling technique indicates that Er in FZ Si occupies tetrahedral interstitial sites [15], and first-principles calculations indicate that the lowest energy configuration of Er in Si is Er 3+ at a tetrahedral interstitial site [16]. These contradictory experimental findings on the local structure of Er in Si indicate that processing and annealing conditions have a strong influence in the local environment of Er implanted into Si. It is therefore highly likely that processing conditions can influence the local environment of other REs implanted into Si.
RE ions are also ideal systems for quantum technologies because the shielding of their f-shell electrons offers an atomic scale barrier to decoherence. Long decoherence times, up to 4 ms, for internal RE transitions have been demonstrated in RE doped optical crystals [17]. Major sources of decoherence arise from the interaction with the nuclear spin or from the thermal bath of electrons of the host. Silicon has an inherently low nuclear spin, and isotopically pure silicon has zero nuclear spin. Recently, a room temperature decoherence time of 39 min has been demonstrated for phosphorous impurites in isotopically pure 28 Si [18]. Combining the shielding of RE f-electrons with the low nuclear spin and processing pedigree of silicon offers a novel system in which to implement quantum technologies. Since the decoherence of RE quantum states is strongly influenced by the crystal field and nuclear spin of its local environment, knowledge and control of this environment is critical for the implementation of quantum technologies based on RE implanted Si.
The symmetry of the RE ion local environment is particularly important for RE quantum states, for example the magnetic moment of single neutral Ho atoms on a platinum substrate has been shown to have lifetimes of the order of minutes, which could be useful for the storage of qubits [19]. This long lifetime was explained because in the particular case of total angular momentum ,J, = 8 for Ho and the C 3v symmetry of the Pt <111> substrate, the eigenstates of the crystal field Hamiltonian form three distinctive groups which can be used to show that the matrix elements describing a substrate electron-induced transition between degenerate m J magnetic sublevels vanish. This means the Ho atoms are in effect invisible to the electrons moving through the Pt. Any RE atom with J = 3n + 1 or 3n +2, for any interger n ≥ 0, should also show long lifetimes; this leaves only Pr and Pm, the later having no stable isotopes [19]. C 3v symmetry can occur for substitutional impurities in Si if there is distortion in the <111> direction, or if a pair of inequivalent substitutional atoms are bonded [20]. In other work relating the symmetry of REs to their applications in quantum technologies, it has been shown that increasing the ground state tunnelling gap of RE complexes can minimise decoherence from nuclear spins [21]. The design of RE systems with large tunnelling gaps requires an m J ground state doublet that can mix with off-diagonal crystal field parameters allowed by the below cubic symmetry of the RE; such symmetries include D 4d [21]. Since the implantation and post implant annealing conditions for Er implanted Si can give rise to different symmetries for Er, it may be possible to engineer a RE implanted Si system with a symmetry that stabilizes the magnetic moment, for example Pr or Ho with C 3v symmetry, or decreases decoherence from nuclear spins, for example Tb or Ho with D 4d symmetry, by optimising the processing conditions. This would require a method to rapidly determine the symmetry of the sample, such as analysing the PL from the sample, which would allow the development of processing conditions which produce a RE symmetry favorable for quantum technology applications.
The usefulness of RE doped materials for quantum technologies can be determined by measuring the spin decoherence time using spin echo [22], or the optical decoherence time using photon echo [23]. Tm and Dy doped materials have received little attention for quantum technology applications compared to other REs. Photon echo measurements of Tm 3+ doped LaF 3 and YAG revealed zero field decoherence times of 7 and 75µs, respectively [24]. The magnetic field dependence and nonexponential character of the echo decays indicated that decoherence was due to nuclear spin fluctuations of the host [24]. Since silicon, and in particular isotopically pure 28 Si, have low nuclear spin we expect the decoherence time to be significantly improved in Tm implanted Si. In this work we report for the first time crystal field analysis of the PL from Dy and Tm ions implanted into Si which is used to determine symmetry and structural parameters. Our analysis method could be used as a quick technique to screen the processing conditions of optically active RE implanted into Si, which would be useful for the development of quantum and photonic technologies based on RE implanted Si.

Experimental
Dy and Tm implanted Si samples were fabricated by room temperature Dy or Tm implantation (10 13 ion/cm 2 at 400 keV) into n-type Si <100> substrates (2-7 Ωcm) previously implanted with B at an energy of 30 keV and a dose of 10 15 ion/cm 2 . The samples were then annealed in N 2 ambient at 850°C for 5 min. Photoluminescence (PL) measurements were obtained by placing the samples in a continuous flow liquid nitrogen cryostat at 80 K.
Samples were excited by a 532 nm diode-pumped laser line at a power density of ∼0.3 W⁄cm 2 , optical emission was dispersed with a monochromator and detected with an extended InGaAs detector using standard phase sensitive detection.

Photoluminescence measurements
We show the PL spectra of Dy and Tm implanted Si in Figs. 1 and 2, respectively. In each case we overlapped PL spectra corresponding to transitions from multiple J manifolds to the ground state to reduce uncertainty in the position of energy levels in the ground state. For Dy 3+ (4f 9 ) the quantum numbers of the ground state are S = 5/2, L = 5 and J = 15/2, for Tm 3+ (4f 12 ) they are S = 1, L = 5 and J = 6. The PL spectra for Dy implanted Si in the 7000 to 7800 cm -1 region contains a mixture of 6 H 9/2 → 6 H 15/2 and 6 F 11/2 → 6 H 15/2 transitions [3], which is caused by overlapping of the 6 H 9/2 and 6 F 11/2 manifolds [25]. This makes determination of the ground state splitting difficult, however, the 5300 to 5900 cm -1 region contains only the 6  allows the ground state splitting to be discerned. Figure 1(a) shows the 6 H 11/2 → 6 H 15/2 transition of Dy 3+ ; we were able to identify eight peaks in the spectra. Group theory arguments predict J+1/2 doublets (eight peaks) for hexagonal, tetragonal and lower symmetry, whereas cubic symmetry would be expected to give rise to two doublets and three quartets (five peaks) [26]. To verify the identity of the transitions we aligned the 6 H 11/2 → 6 H 15/2 transition in Fig.  1(a) with the 6 H 9/2 → 6 H 15/2 and 6 F 11/2 → 6 H 15/2 mixture of transitions in Fig. 1(b) by means of offsetting the transition energy scale by 1675 cm -1 . An additional alignment is found by offsetting Fig. 1(b) by a further 216 cm -1 , this should correspond to the separation of the manifolds in the 6 H 9/2 / 6 F 11/2 mixed state, although we are unable to ascertain which of these manifolds the alignments in Fig. 1(b) and 1(c) belong to. All the peaks in Fig. 1(a) are aligned with features in the spectra in Figs. 1(b) and 1(c), apart from those at 5733 and 5887 cm -1 , the most likely explanation being that these peaks are too weak to be detected. Many extra peaks occur in the 6 H 9/2 / 6 F 11/2 → 6 H 15/2 transition, which could be due to transition from thermally populated higher energy levels, so-called "hot-peaks", in the 6 H 9/2 / 6 F 11/2 mixed manifold.  Fig. 2(b) corresponds to a small peak in Fig. 2(a) at 5111 cm -1 .

Crystal field parameter fitting
We analysed the splitting of a single J multiplet (the ground state), which was determined from PL spectra, to fit CFPs. The Hamiltonian (H) of a RE in a crystal field can be thought of as consisting of a free ion component (H F ) and a crystal field perturbation (V CF ) H F includes Coulomb and spin-orbit interactions, V CF can be thought of as a perturbation generated by the same mechanisms that produce bonding. The multipole expansion of V CF is defined as the linear combination of a set of spherical tensors, ( ) k q C , and k q B which are structural factors referred to as the crystal field parameters which represent the symmetry of the environment [30] ( ) As such, crystal field contributions can be built from a superposition of contributions from the individual ligands. Using the triangular relationship of the so-called 3jm symbols which describe the coupling of angular momenta, the possible values for rank k crystal field operators are restricted to 0, 1, 2, 3, 4, 5 and 6. Hermiticy and time-reversal invariance allow only even rank k operators. Since rank k = 0 does not produce any crystal field splitting, only rank k = 2, 4 and 6 are considered for CFP analysis [31]. To fit CFPs to our experimental data requires the calculation of energy levels from an initial estimate of CFPs. To construct the energy matrix for this calculation, the 3jm symbols were combined with the reduced matrix elements This determines the diagonal components of the ( ) k q C spherical tensor operator matrix which are combined with the crystal field parameters k q B and summed over the k values to produce the energy matrix. Diagonalizing this energy matrix returns the energy levels of the J multiplet. In order to fit CFPs we determined CFPs that minimise the sum of squares difference between the calculated and experimental energy levels. To avoid local minima we systematically searched the parameter space with starting values between previously reported maximum and minimum values. Each site symmetry of the RE has its own particular set of non-vanishing CFPs. In order to determine the site symmetry of our REs, we fitted sets of CFPs corresponding to commonly occurring symmetries. The results are shown in Table 1 which shows that the best fit occurs for tetragonal symmetry for both the REs. For cubic symmetry the constraints:  [33]. In some respects the comparison with cubic symmetry is superfluous because we observed more PL lines in all the REs than would be expected for cubic symmetry. However, the fitted energy levels shown in Figs. 3(a) and 3(b) allowed us to reconfirm that cubic symmetry is unlikely even if there is uncertainty in the position of the PL lines. Orthorhombic and lower symmetries cannot be reliably fitted to the Dy data because there are more crystal field parameters than energy levels. An important test for the validity of fitted crystal field parameters is that they change gradually when progressing through the lanthanide series. This is the case for our fitted tetragonal parameters for Dy and Tm. Table 1 also shows that when comparing CFPs for symmetries other than tetragonal for Dy and Tm, there is no apparent relationship, indicating that these symmetries are not those of the REs. Figures 3(a) and 3(b) show the experimentally determined energy levels (E exp ) along with those determined by fitting sets of CFPs (E fit ) corresponding to various symmetries. Comparing these reconfirms that tetragonal symmetry is the best fit for all the REs. We used the fitted crystal field parameters for tetragonal symmetry to calculate the entire energy level structure of Dy and Tm in Si using Lanthanide [34] software, which is shown up to 10000 cm -1 in Fig. 3 (c). The largely host independent free ion parameters were taken from fitted absorption spectra of REs in LaF 3 [35]. The calculated energy levels were in good agreement with the absolute energies of the PL spectra we observed. The splitting of the excited states may be confirmed in the future if and when it becomes possible to measure by absorption or other means.

Interpretation of crystal field parameters
The CFPs obtained by fitting spectroscopic data are referred to as experimental CFPs and they measure the strength of the interaction between the f-shell electrons of lanthanides and their surrounding crystalline environment. However, these parameters do not in themselves give a direct insight into the mechanisms of the crystal field splitting [36]. The rank 4 CFPs are dominated by covalency and closely related overlap contributions [31]. The crystal structure of silicon is the same as that of diamond with tetrahedrally coordinated Si atoms with cubic symmetry and O h point group and Fd3m space group with a unit cell dimension of 5.431 Ǻ and an equilibrium distance between Si atoms of 2.352 Ǻ. The substitutional site in Si has four nearest neighbours with tetrahedral coordination, the symmetry is nominally cubic, but our CFP fitting shows that the RE has tetragonal symmetry. A tetrahedally coordinated interstitial site also exists in Si, with the same relative position to its four nearest neighbours as the substitutional site. However, the substitutional site has twelve second nearest neighbours with different relative positions to the six second nearest neighbours in the interstitial site. These substitutional and interstitial sites are the only likely sites for the RE in Si, in order to distinguish between these sites and determine their geometry we tested the two electrostatic models illustrated in Figs. 4(a) and 4(b) using a modified version of SIMPRE [37] software. The SIMPRE software uses a point charge electrostatic model in which N ligands are represented by their point charges (Z). The CFPs in Stevens normalisation ( q k A ) are calculated with the following formalism [21,38] ( ) where R i , θ i , and φ i are the effective polar coordinates of the point charge and Z i is the effective point charge, associated to the i-th ligand with the RE at the origin; e is the electron, p kq are the prefactors of the spherical harmonics and Z kq are the tesseral harmonics expressed in terms of the polar coordinates for the i-th donor atom R i , θ i , and φ i . The Stevens normalisation CFPs ( q k A ) used in Eq. 4(a)-4(c) are related to the Wybourne normalisation CFPs ( k q B ) that we use with the appropriate coefficients [31,39]. The SIMPRE software does not take into account screening or covalency. However, here we use it to distinguish between a substitutional and an interstitial site, rather than to obtain meaningful structural parameters. We fitted CFPs calculated by SIMPRE to our experimentally determined CFPs by varying the dimensional parameters, a and b, of the tetragonal primitive cell and the effective charge (Z) of Si atoms on the RE; if a ≠ b, the RE has D 2d symmetry. The results of the fitting are shown in Table 2  parameters. The electrostatic model is known to underestimate rank 6 CFPs [40], but this could also be because the point charge electrostatic model is over-simplified for the covalent Si system we are investigating. Therefore, structural parameters calculated from fitting the point charge electrostatic model to our experimental CFPs may be shifted from their true values to mimic the charge distribution in covalent Si [41], but should still be related to the true values. In this respect our fitted structural parameters can be viewed as estimates. The 2 0 B parameter belongs only to the tetragonal component of the crystal field [42], it is therefore dependent on the degree of axial asymmetry i.e. the difference between a and b, whereas the other parameters are more dependent on the absolute values of a and b and Z, therefore we can have more confidence in the degree of axial asymmetry than absolute values determined by the point charge electrostatic model. In contrast with the substitutional electrostatic model, the interstitial electrostatic model gives the incorrect sign for 4 0 B in Dy and the incorrect sign for 4 0 B and 4 4 B in Tm, along with a significantly poorer fit, indicated by the σ value. The large difference in Z for Dy and Tm in the interstitial model may also be unphysical. Although the point charge model does not take into account screening or covalency, it does suggest that Dy and Tm exist in the tetrahedrally coordinated substitutional site illustrated in Fig. 4(a). Both Dy and Tm have tetragonal D 2d symmetry, however the primitive cell for Dy has significantly higher axial asymmetry than Tm. Our effective charges are comparable with those previously reported: the effective charge a a b L 1 in III-V compounds is 2-2.5 [43], the Mulliken charges of Si atoms in zeolites was found to be ~2.5 [44]. The axial asymmetry could be caused by our dislocation engineering technique which introduces a uniaxial strain field caused by dislocation loops aligned to the four nonparallel <111> lattice planes [1,45,46]. Another possibility is the formation of a complex, with axial asymmetry, by charge compensation of the RE 3+ by other defects such as vacancies or impurities. The D 2d symmetry that we have determined for Dy 3+ and Tm 3+ implanted Si does not match the symmetries that we identified as being important for quantum technology applications.
These symmetries were C 3v symmetry required for the stabilization of the magnetic moment Ho atoms or the D 4d symmetry required for increased ground state tunneling gaps. Nevertheless, based on previous work on Er implanted Si, different processing conditions should vary the local environment and symmetry of Dy 3+ and Tm 3+ implanted Si, which can we quickly and easily determined by our analysis method. Also, all the RE and symmetry combinations that result in a stabilization of the magnetic moment, as in Ho atoms on a C 3v symmetry substrate [19], have not yet been calculated. So there may be different symmetries that result in stabilization of the magnetic moment.

Conclusions
We report PL measurement and analysis from multiple J manifolds in Dy and Tm implanted Si. Comparing these manifolds allowed us to identify the ground state splitting with a higher degree of certainty than has been achieved previously. We used this data on the ground state splitting to fit sets of CFPs that correspond to various common symmetries. This allowed us to determine that the symmetry of the Dy and Tm ions in the silicon lattice is tetragonal for both ions. We used our fitted CFPs along with previously published free ion parameters to calculate the splitting of every J manifold of Dy and Tm implanted Si, which we report graphically up to just above the band gap of Si. We used a point charge electrostatic model to fit CFPs to our experimentally determined CFPs using structural models for an interstitial and a substitutional site in Si. This allowed us to determine that both Dy and Tm exist in a tetrahedrally coordinated substitution site with tetragonal symmetry. The degree of tetragonal distortion was significantly higher in Dy compared to Tm. We propose that this distortion is produced by a strain field caused by our dislocation engineering technique. This analysis, which reveals detailed symmetry information on RE implanted Si, could be important for designing ion implantation parameters that lead to systems with long decoherence times which would be valuable for quantum technologies based on RE implanted Si.