Optically modulated magnetic resonance of erbium implanted silicon

Er implanted Si is a candidate for quantum and photonic applications; however, several different Er centres are generated, and their symmetry, energy level structure, magnetic and optical properties, and mutual interactions have been poorly understood, which has been a major barrier to the development of these applications. Optically modulated magnetic resonance (OMMR) gives a spectrum of the modulation of an electron paramagnetic resonance (EPR) signal by a tuneable optical field. Our OMMR spectrum of Er implanted Si agrees with three independent measurements, showing that we have made the first measurement of the crystal field splitting of the 4I13/2 manifold of Er implanted Si, and allows us to revise the crystal field splitting of the 4I15/2 manifold. This splitting originates from a photoluminescence (PL) active O coordinated Er centre with orthorhombic C2v symmetry, which neighbours an EPR active O coordinated Er centre with monoclinic C1h symmetry. This pair of centres could form the basis of a controlled NOT (CNOT) gate.


Introduction
Optical characterisation of Er implanted Si presents a number of difficulties. The low absorption and emission cross-section of Er, low overall number of implanted ions, and low depth of implants (typically no more than ~2 μm), makes direct optical measurements, particularly absorption extremely challenging. PL can be readily obtained by indirect excitation, where above band gap radiation generates excitons which are trapped by, and transfer their energy to, Er 3+ centres from which radiative relaxation gives rise to characteristic emission at 1.5 μm (ref. 1). The crystal field splitting of the 4 I15/2 ground state of Er implanted Si can be determined from PL measurements; however, the splitting of the 4 I13/2 excited state remains elusive, with only the first two or three levels determined from so called "hot lines", i.e. PL transitions from thermally populated crystal field states in the excited state manifold 4 . The difficulty in distinguishing hot lines from each other, and other PL lines, along with the requirement of cryogenic temperature for PL from Er implanted Si makes identification of higher lying 4 I13/2 crystal field states from hot lines unlikely.
When co-implanted with O, Er generates a variety of centres that are detectable by PL and EPR. In addition, molecular beam epitaxy (MBE) grown Er doped Si generates PL lines similar to those observed in Er implanted Si and Zeeman measurement of this PL have been reported 5,6 . EPR measurements have identified a set of monoclinic and trigonal O coordinated Er centres 7,8,9,10 . PL measurements identify a cubic Si coordinated Er centre and at least one additional O coordinated Er centre with lower than cubic symmetry 4, 11 , although the symmetry and energy level structure of this centre has not been fully identified. Zeeman measurements of MBE grown Er doped Si have identified an orthorhombic centre with proposed O and Si coordination, but this centre was not observed in EPR measurements of Er implanted Si. The properties of these centres are summarised in Table 1; however, the relationship between these centres is not currently understood. Further understanding of these centres will assist researchers in developing processing strategies to favour one particular type of centre. This would benefit both photonic and QT applications of Er implanted Si.
The OMMR technique we use in this study involves the modulation of the EPR signal from Er implanted Si by a tuneable laser resonant with an Er centre's electron dipole transition. The OMMR technique can be thought of as the inverse of traditional optically detected magnetic resonance Hughes et al. Sci. Rep. 9, 19031 (2019) 4 (ODMR) techniques, which involve the modulation of optical luminescence or absorption signals, by a microwave field. Whereas the OMMR technique involves the modulation of a microwave absorption signals by an optical field. Absorption based ODMR could, in principle, be used to probe the excited state of Er implanted Si. However, since absorption measurements are not feasible, neither would ODMR based on absorption.
In this work, we use OMMR to measure, for the first time, the crystal field splitting of the first excited state of Er implanted Si. We determine the energy level structure of a PL active O coordinated Er centre and identify it as having orthorhombic C2v symmetry. We also report previously unreported PL lines from Er implanted Si. We show that the monoclinic EPR centre and orthorhombic PL centre are distinct, but highly localised centres. This pair of centres could be exploited for CNOT gates in an Er implanted Si based quantum computer architecture. occurs under laser irradiation, centred at ~970 G with a peak-to-peak width of ~320 G. Figure 1 shows a contour plot constructed from many OMMR spectra taken at magnetic fields between 480 and 2300 G. The strong optically generated EPR resonance appears as two bands at 850 and 1170 G, since the second lock-in gives the absolute value of the modulated EPR signal. The sample alignment used in the OMMR measurement is shown in Figure 1 (c). The strongest OMMR spectrum, at 1173 G, is shown in Figure 2(a). Four broad bands can be observed in laser energy centred at 6390, 6520,  When the sample is rotated in the (11 ̅ 0) plane, both the narrow EPR and broad optically generated resonances shift in magnetic field. By making small angle adjustments we were able to approximately align the zero-crossing point, at 961 G, of the broad optically generated resonance with the monoclinic EPR line at 950 G. At 961 G the OMMR spectrum switches from the broad spectrum to one with seven narrow peaks at 6360, 6389, 6415, 6506, 6548, 6579 and 6592 cm -1 , with FWHM of ~15 cm -1 , and reduces in intensity by an order of magnitude, as shown in Figure 2 (b). This weak narrow spectrum cannot be resolved in Figure 1 (b). At different sample orientations, when the zerocrossing point of the broad optically generated resonance does not correspond with a narrow EPR resonance, the OMMR spectrum is largely featureless, see supplementary Figure S1, demonstrating that the 961 G OMMR spectrum is related to the monoclinic EPR line at 950 G. G. Arrows indicate the identified peaks. The temperature was 10 K and the microwave frequency was 9.37 GHz.

Photoluminescence measurements
PL measurements at 60 K of the Er implanted Si sample are shown in Figure 3. We identify ~17 peaks in this spectrum, all with FWHM ~20 cm -1 . The PL peaks associated with a Si coordinated cubic Er centre (Er-C) have previously been unambiguously determined from PL measurements of Er implanted Si with low O concentrations 11,12 . We observe the same peaks in Figure 3, identified with We propose that the peaks observed in the OMMR spectra in Figure 2(a) and (b) are an indirect measurement of the crystal field splitting of the 4 I13/2 manifold, and that there are two distinctive types of OMMR spectra: a strong broad spectrum, as in Figure 2 (a), and a weak narrow spectrum, as in Figure 2 (b), which represent two different Er centres. To test this hypothesis, we performed crystal field analysis on the known energy splitting of the 4 I15/2 manifold determined by PL to see if they predict the splitting observed in our OMMR spectra.

Crystal field analysis
Since all the reported narrow EPR resonances belong to low symmetry centres, we assume that the broad optically generated EPR resonance shown in Figure 1(a), which has an OMMR spectrum shown in Figure 2 (a), originates from the Er-C cubic centre. To confirm this, we fitted a set of cubic crystal field parameter (CFPs) to the cubic PL lines identified in Figure 3, which, as shown in Figure 4  similar to those previously reported for the Er-C centre 4 . Using these fitted CFPs we calculated the splitting of the 4 I13/2 manifold and compared it to the OMMR at 1173 G; however, there is no match (RMSD = 153.0 cm -1 ). We can therefore discount OMMR at 1173 G as originating from the Er-C centre.
The 961 G OMMR spectrum could originate from the Er-O1 centre, which has undetermined, lower than cubic symmetry 4 . It has the predicted seven lines for the splitting of the 4 I13/2 manifold with lower than cubic symmetry. This could be the monoclinic symmetry of the OEr-1' centre 8 , which has a resonance at around the same magnetic field, or the orthorhombic symmetry of the Er-1 centre identified by Zeeman measurements 5,6 . Applying the same procedure used for Er-C presented a problem. There are seven energy levels reported in the Er-O1 centre and five corresponding peaks that could initially be identified in our PL spectrum. This is not enough peaks to fit either the nine CFPs for orthorhombic symmetry, or fourteen CFPs for monoclinic symmetry. Analysis of g tensors indicate the monoclinic OEr-1' centre can be approximated with a tetragonal field 14 . We attempted fitting a set of tetragonal D2d CFPS to the seven Er-O1 lines; however, the fit was poor, and the calculated splitting of the 4 I13/2 manifold did not match the 961 G OMMR spectrum. We then tried fitting a set of tetragonal CFPs to the 961 G OMMR, and, as shown in Figure 4 (b), there was a good fit (RMSD = 11.6 cm -1 ). Using these tetragonal CFPs, we calculated the splitting of the 4 I15/2 manifold, but these didn't match our low symmetry PL peaks or the Er-O1 energy levels. However, if we assume that peak 4 in Figure 3 is the ground state, there is a good match to our non-cubic PL if we also assume peaks 1 and 2 contain two closely spaced peaks, as shown in Figure 4 (b). We also note that the splitting of the first two levels of the 4 I13/2 manifold in Er-O1, measured by PL hot lines, matches the splitting of the first two levels of the 961 G OMMR.
Given the information from the tetragonal fit in Figure 4 (b), we propose that peak 4 is the crystal field ground state and that if peaks 1 and 2 contain two levels we can revise our non-cubic PL energy level structure to that shown in Figure 4 (c). Along with PL hot lines from ref. 4, this gives a total of ten energy levels, which allows us to fit orthorhombic CFPs. There is an excellent fit (RMSD = 3.5 cm -1 ), and using these CFPs we can calculate the rest of the 4 I13/2 splitting, which gives a very good match to the 961 G OMMR (RMSD = 9.0 cm -1 ). This accurate prediction of the OMMR spectrum from independent measurements shows conclusively that the 961 G OMMR spectrum is a measurement of the splitting of the 4 I13/2 manifold of the non-cubic PL centre. Using all the non-cubic PL and 961 G OMMR lines allows us to refine the orthorhombic fit, as shown in Figure 4

OMMR mechanism
The mechanism for optical excitation of Er implanted Si is well established and involves the generation of carrier pairs by above band-gap irradiation. These excitons are then trapped by an Er- To understand the mechanism that gives rise to the OMMR signal, we examined possible ways that absorption into the 4 I13/2 manifold could increase the EPR signal. We can assume that at 10 K the EPR signal in our OMMR measurement arises from X-band microwave absorption in the Zeeman levels of the crystal field ground state of the 4 I15/2 manifold, as shown in Figure 5. Given a long enough spin-lattice relaxation time, T1, which is reasonable at 10 K, this Zeeman transition will readily saturate, so a modulation of the EPR signal can arise from repopulation of the Zeeman ground state. Since the OMMR spectrum shows the splitting of the 4 I13/2 manifold, absorption into each 4 I13/2 crystal field level will relax back to the 4 I15/2 Zeeman ground state and increase the EPR signal. We know from the offset energy, see Methods, that this absorption must come from a level 227 cm -1 above the crystal field ground state. In rare earth doped semiconductors, the 4f ground state manifold was previously assumed to lie deep within the valence band 18,19,20 , based partly on the separation of the 4f-levels of isolated rare earth ions from the vacuum level. We have recently made the first observation of direct optical transitions from the silicon conduction band to internal 4f-levels of implanted Ce, Eu, and Yb, which gave a significant enhancement of emission 21 . We also showed that their 4f ground state manifolds lie ~1000 cm -1 above the valence band. This precedent of band state to 4f-level transitions indicates that transitions from the valence band to the 4 I13/2 excited state of Er implanted Si are feasible, and may be a significant enhancement on intra 4f transitions. We therefore propose a possible mechanism for the OMMR process, illustrated in Figure 5, which involves optically induced transitions from the valence band to the 4 I13/2 manifold, which subsequently relax to the 4 I15/2 Zeeman ground state and enhance the EPR signal, with an optical spectral dependence matching the crystal field splitting of the 4 I13/2 manifold. The offset energy Eoff, indicates that the 4 I15/2 manifold is partially buried in the valence band.
At 10 K, relaxation from 4 I13/2 to 4 I15/2 should be almost entirely radiative, with a radiative relaxation time, τr ~ms. However, coupling to defect sates could cause non-radiative decay. We have measured the spin-lattice relaxation time of the monoclinic EPR centre, T1Mo, in a sample with a 3×10 17 cm -3 Er concentration to be ~ms, see supplementary Figure S3, and we were able to measure EPR resonances from this sample. Since EPR from the orthorhombic Er-O1R centre can't be measured, even from a sample with 1×10 19 cm -3 Er, its spin-lattice relaxation time, T1Or, should be >>ms, in order for the Zeeman ground state to become saturated. Therefore, we expect τr <<T1Or, which would allow pumping of the Zeeman ground state by transitions from the 4 I13/2 to manifold. The actual crystal field splitting is shown, the Zeeman splitting is only shown for the crystal field ground state for clarity.

g tensor calculation
The set of CFPs we have obtained from tetragonal, orthorhombic and monoclinic fitting can be used to calculate the expected EPR g tensors, see Methods. We would expect the calculated g tensors to match those of the OEr-1' centre identified by Carey et al 8 , since the OMMR measurement can only be observed at the magnetic field of a resonance corresponding to this centre. However, Table 2 shows the g tensors of all the Er centres identified from EPR measurements of Er implanted Si, those obtained from Zeeman measurements of MBE grown Er doped Si and our calculated g tensors. The EPR g tensors correspond only to the crystal field ground state of the 4 I15/2 manifold, but Zeeman  shows that the Er-O1R and Er-1 centres are the same centre and this centre is a separate centre to any of the identified EPR centres. The symmetry of the Er-O1R centre should therefore also be same orthorhombic C2v symmetry as the Er-1 centre. However, there must be a link between the Er-O1R centre and the OEr-1' centre since the OMMR spectrum of the Er-O1R centre is only observed when at the EPR resonance of the OEr-1' centre, this implies that microwave absorption, and hence population of the spin-up state of the OEr-1' centre allows the OMMR mechanism of the Er-O1R centre to proceed. This indicates that the OEr-1' and Er-O1R centres exist in close proximity, possibly as a dimer. This has important implications for QT applications of Er implanted Si, since single molecular magnets (SMMs) containing two weakly coupled rare earths with different coordination environments have been proposed as the basis of a two qubit CNOT gate 22 , and SMMs containing two Tb 3+ ions have been shown to meet all the conditions needed for a universal CNOT gate 23 . The OEr-1' and Er-O1R centres appear to be the solid-state analogue of these SMM systems but have all the device fabrication advantages of being based in Si. In addition, the strong axial anisotropy displayed in the g tensors of the OEr-1' centre and, in particular, the Er-O1R centre are a requirement for the realisation of CNOT gates 22 . There are also implications for the photonic applications of Er implanted Si, where the ability to switch a 1.5 µm optical transition on with microwave or magnetic pulses could have applications in signal processing. We propose that the reason the Er-O1R centre is not EPR active is because the 4 I15/2 crystal field ground state has a long T1, which is consistent with our model for the OMMR mechanism. This long T1 could make the EPR signal significantly weaker than centres with a shorter T1 24 . The dramatic difference in the g tensor of the OEr-1' and Er-O1R centres is in line with previous findings that subtle differences in the structure of Dy SMMs can have dramatic effects on their magnetic properties 25 .

Conclusions
The OMMR spectrum of Er implanted Si at 961 G is accurately predicted by crystal field analysis of PL measurements, accurately predicts g tensors from Zeeman measurements, and agrees with PL hotline measurements. This agreement with three sets of independent measurements shows beyond any reasonable doubt that the 961 G OMMR spectrum is a measurement of the splitting of the 4 I13/2 manifold, which represents the first measurement of the crystal field splitting of the 4 I13/2 manifold of Er implanted Si. The OMMR spectrum originates from the Er-O1R centre and is only observed at a magnetic field which is at an EPR resonance of the OEr-1'centre, which indicates that these two centres are in close proximity. The Er-O1R centre is not observed in EPR measurements, which we propose is due to a longer T1 than the OEr-1' and other EPR active centres.
An energy offset in the OMMR spectrum, compared to what would be expected from a direct absorption measurement, indicates that the OMMR signal originates from transitions from the top of the valence band, and that the 4 I15/2 manifold is partially buried in the valence band. Because the OMMR mechanism involves transitions from the valence band, it may be restricted to rare-earth doped semiconductors, it may also requires a spin T1 long enough to allow saturation of the Zeeman transition. However, the OMMR technique could be used by other researchers investigating rare-earth doped semiconductors.

Ion implantation and annealing
The sample was prepared by implanting Er and O into P-doped <100> 500 µm thick Si wafer supplied by Topsil at 77 K. The unimplanted wafer had a measured resistivity of 8000 ± 500 Ωcm, corresponding to a P concentration of 5.5 ± 0.3 × 10 11 cm -3 . A range of implant energies was used to give a flat ion concentration profile down to a depth of around 1.5 µm, as illustrated in supplemetary Figure S4. Doses were chosen to give Er and O concentrations of 10 19 and 10 20 cm -3 , respectively.
Isotope specific implantation was used so that only the zero nuclear spin 166 Er was implanted. After implantation the sample was annealed at 450°C for 30 min to smooth the crystalline-amorphous interface, then at 620°C for 180 min to recrystallize the amorphized region then at 850°C for 30 s to activate the Er for EPR measurments. It was found that annealing at 850°C significantly increased the EPR signal strength. This is in contrast to previous work where the same smoothing and recrystallization anneal was used, but the activation anneal was 900°C for 30 s. 8,9,10 This is probably due to inconstancies between different annealing furnaces. The relative EPR signal intensities for different activation annealing conditions are shown in supplemetary Figure S5.

EPR and OMMR measurements
EPR measurements were taken on a Brucker EMX EPR spectrometer, incorporating a super high-Q resonator with optical access. The field modulation was 100 kHz, and the microwave frequency was

PL and electrical measurements
PL spectra were obtained by placing the sample in a cold finger LN2 cryostat at 60 K, dispersing the fluorescence generated by a 462 nm 50 mW laser diode in a Bentham TMc300 monochromator, with a resolution of 3 nm, and detecting with an IR PMT coupled with standard phase sensitive detection.
All spectra were corrected for the system response. Thermopower and conductivity measurements were carried out using a method described previously 26 .

Crystal field analysis
The Hamiltonian (H) of the Er 3+ in our OMMR measurement can be described as accounts for the interactions that occur in a free Er ion. There are many interactions thought to occur in the free ion, these can be broken down as follows 27 .
Where 0 is a constant representing the kinetic energy of the f electrons and their coulomb interactions with the nucleus and electrons in filled shells. and represent electron-electron intra-shell coulomb and spin-orbit interactions, respectively. ∑ ( )

=1
are a set of four corrective terms which include two and three body operators. Together, these give 20 parameters to represent . Each rare earth has its own set of parameters, and these vary little between hosts.
We used those given by Carnall et al. for Er:LaF3 28 . describes a perturbation generated by ligands of the host crystal lattice surrounding the Er 3+ ion.
The multipole expansion of is defined as the linear combination of a set of spherical tensors, ( ) , and structural factors, , which are referred to as crystal field parameters and represent the symmetry of the environment 29 .
Details of the construction of are given elsewhere 30 . Each site symmetry of Er 3+ has its own particular set of non-vanishing CFPs.
The Zeeman interaction, , is given by = . (4) Where is the Landé factor, is the Bohr magneton, J is the angular momentum operator, and H is the magnetic field strength 31 . was around two orders of magnitude smaller than the crystal field widths. This meant Zeeman splitting was not observed on the OMMR spectra, and was therefore not considered during the fitting procedure. To fit OMMR and PL lines, differences between the eigenvalues of H and experimental energy levels were minimised with a least squares fitting algorithm.

Offset energy
At cryogenic temperatures, and if direct intra-manifold transitions are being considered, the highest energy PL transition between two manifolds in a rare earth ion represents the energy separation of the crystal field ground states of the manifolds. Similarly, the lowest energy absorption transition between the same two manifolds represents the same energy separation. The absolute energies of the OMMR and PL measurements are therefore highly significant for our model. From Figure 3, the highest energy PL peak of the Er-O1R center is at 6587 cm -1 , whereas the lowest energy transitions for the represents the splitting of the first excited state manifold, it must originate from a state that is an offset energy, Eoff, above the crystal field ground state of the 4 I13/2 ground state manifold, where Eoff = 6587-6360 = 227 cm -1 . This is also important for our crystal field analysis since the inter-manifold separation calculated by is only applicable to direct inter-manifold experimentally observed transitions. Therefore, for our crystal field analysis, Eoff was added to the peak energies of the 961 G OMMR spectrum in Figure 2 (b).

g tensor calculation
Each crystal field doublet has two sets of eigenvectors: | +⟩ and | −⟩. One for each pair of degenerate eigenvalues. The diagonal components of the g tensor, gx, gy, gz, are calculated using the first order perturbation expressions 31 .

Data Availability
The datasets generated during the current study are available in the Mendely Data repository, http://dx.doi.org/10.17632/5g67t8gsbc.1