Zeeman spectroscopy and crystal-field analysis of low symmetry centres in Nd3+ doped Y2SiO5

We report on infrared to visible Zeeman absorption spectroscopy and parameterised crystal-field modelling of Nd3+ centres in Y2SiO5 through the use of experimentally inferred crystal-field energy levels and Zeeman directional electronic g values. We demonstrate that good agreement between the calculated and experimental crystal-field energy levels as well as directional Zeeman g values along all three crystallographic axes can be obtained. Further, we demonstrate that the addition of correlation crystal field effects successfully account for discrepancies that arise between the calculated and experimental values relevant to the 2H 11/2 (2) multiplet in a one-electron crystal field model.


Introduction
Rare-earth (RE) ions in various host crystals have interested the quantum information community for their potential use in quantum information storage and manipulation. This is due to their optically addressable transitions, long coherence times, and narrow homogeneous linewidths. Recently, lanthanide-doped Y 2 SiO 5 has come to the fore in quantum information storage or manipulation applications due to the * Author to whom any correspondence should be addressed.
Original Content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. low nuclear spin of its constituent ions that allows for the observation of long optical and spin coherence lifetimes [1]. Recently, a nuclear spin coherence time of over 6 h has been measured for Eu 3+ :Y 2 SiO 5 [2] using ZEFOZ (zero first order Zeeman) points [3]. The same technique has lead to ms range electron spin T 2 in 167 Er 3+ :Y 2 SiO 5 [4] and 171 Yb 3+ :Y 2 SiO 5 [5][6][7]. Since the ZEFOZ technique utilises avoided crossings of the complex hyperfine structures in lanthanide-doped crystals they have proven difficult to be found experimentally therefore an ability to calculate such ZEFOZ points would be a huge advantage. These points can be computationally predicted through the use of spin Hamiltonian. However, a parameterised crystal-field model the electronic and magnetic-hyperfine structure of an entire 4f N configuration provides a degree of transferability from ion to ion.
The coherence properties of Nd 3+ -doped Y 2 SiO 5 and the contributions from different decoherence mechanisms to these properties have been studied over the past decade with the aim of implementing quantum storage using this material [7][8][9][10][11][12][13][14]. The large magnetic moment of Kramers ions (e.g. Nd 3+ and Er 3+ ) results in strong dipole-dipole interactions and faster spin-lattice relaxation rates compared with non-Kramers ions (e.g. Pr 3+ and Eu 3+ ) where both spin-spin interaction and spin-lattice relaxation rates are low. As a result, efficient optical pumping for state initialisation in optical quantum storage can be challenging, which in turn leads to low storage efficiency [9]. However, in terms of applications, Kramers ions feature large Zeeman and hyperfine splittings allowing the realisation of large bandwidth quantum memories and could be interfaced with superconducting qubits working in the 1-10 GHz regime in hybrid quantum systems [15,16]. As for other Kramers ions such as Er 3+ and Yb 3+ in Y 2 SiO 5 , employing the ZEFOZ technique in Nd 3+ in Y 2 SiO 5 may increase the spin coherence time and the use of parameterised crystalfield modelling can be used to predict the required avoided crossings. Furthermore, Nd 3+ is neighbouring ion to Pr 3+ in the lanthanide series, so it may be possible to use of Nd 3+ crystal-field parameters to predict the ZEFOZ points (or 'clock transitions') of Pr 3+ in Y 2 SiO 5 . A coherence time of up to one minute has been demonstrated for Pr 3+ in Y 2 SiO 5 [17].
Recently, crystal-field analyses have been performed for several different RE ions in the C 1 centres of Y 2 SiO 5 , including Ce 3+ , Sm 3+ , Ho 3+ , Er 3+ , and Yb 3+ [18][19][20][21][22][23]. In this work, we report on Zeeman absorption spectroscopy and parameterised crystal-field modelling of both Nd 3+ centres in Y 2 SiO 5 through the use of experimentally inferred crystal-field energy levels and Zeeman directional electronic g values as well as utilising the electron paramagnetic resonance (EPR) measured ground-state g tensor. Experimental electronic levels determined from laser site-selective spectroscopy were reported in [24] whilst the EPR measurements are from the literature [15,25]. We demonstrate very good agreement between the calculated and experimental crystal-field energy levels and directional Zeeman g values.

Experimental
Y 2 SiO 5 (YSO) is a monoclinic silicate crystal having space group C 6 2 h . Trivalent RE ions substitute for the Y 3+ ion. The lattice constants of Y 2 SiO 5 are a = 10.4103 Å, b = 6.7212 Å, c = 12.4905 Å, and β = 102 • 39'. Here the crystallographic b axis corresponds to the C 2 rotation axis and the crystallographic a and c axes are located in the mirror plane which is perpendicular to the crystallographic b axis. Following the convention of Li et al we define the optical extinction axes as D 1 and D 2 which are located in the a-c mirror plane and are perpendicular to each other in addition to the b axis [26]. Each molecule of Y 2 SiO 5 contains two substitutional Y 3+ sites, site 1 (Y 1 ) and site 2 (Y 2 ), each with C 1 symmetry and are distinguished by their co-ordination numbers of six and seven, respectively [27]. Additionally, Nd 3+ in each of the crystallographic sites of YSO has four magnetic subclasses related by rotation around the C 2 symmetry axis of the crystal and inversion [28]. These four subclasses with different orientations can be classified into two groups. One group that are related through inversion and respond identically to the applied magnetic field. Another group that are related through rotation around the C 2 axis interact differently with the applied magnetic field. Therefore, there are two magnetically inequivalent subclasses for each crystallographic site in a magnetic field with arbitrary direction.
The sample employed in this study was an X2 phase, Y 2 SiO 5 crystal with 0.02 molar % of 145 Nd 3+ grown using the Czochralski process. The crystal had dimensions of 5.6 mm, 4.6 mm, and 4 mm along the D 1 , D 2 , dielectric, and b crystallographic axes, respectively. The Zeeman absorption spectra were collected using a Bruker Vertex 80 Fourier transform infrared (FTIR) spectrometer having a maximum spectral resolution of 0.075 cm −1 . In addition, these measurements made use of a 4 T superconducting magnet. The sample was attached to a copper mount which was then screwed into the bore of the magnet's solenoid and was cooled down to the liquid helium temperature at 4.2 K.

Results and discussion
The ground configuration of trivalent neodymium is 4f 3 , which consists of 182 two-fold degenerate Kramers states. The standard notation of a letter plus a numerical subscript is used here for labelling the crystal-field levels of the LSJ multiplets. Thus the ground state is denoted as 4 I 9/2 Z 1 , the first excited state Z 2 and so on. These follow the convention utilised in the standard 'Dieke diagram' to label the Stark levels of the crystal field splitting of the Nd 3+ manifolds. The experimental crystal field energy levels used in the current analysis were collected through laser site-selective and temperature-dependent absorption spectroscopy [24]; these energy levels are given in tables 1 and 2.

Zeeman absorption spectroscopy
In order to achieve a global minimum in a crystal-field fit for a low symmetry system we require directional information in addition to the crystal-field energy levels. The required information can be obtained by performing high-resolution Zeeman absorption spectroscopy along all three crystallographic axes, covering Y 2 SiO 5 :Nd 3+ absorption lines from the near infrared to the visible spectral regions.
A set of representative Zeeman absorption spectra obtained at 4.2 K are shown in figures 1-4 for the 4 I 9/2 (Z 1 ) −→ 4 F 3/2 (R 1 ) and 4 I 9/2 (Z 1 ) −→ 4 G 7/2 (E 1 ) transitions from sites 1 and 2. The displayed Zeeman absorption spectra were collected at a magnetic field strength of 4 T along the D 1 , D 2 and b crystal axes. The right panels show the experimental Zeeman split components (black dots) between 0 and 4 T with increments of 0.5 T for the corresponding magnetic field direction, with the calculated splittings (solid red lines) having zero field energies shifted as appropriate to overlay with the experimental points. It can be seen that the calculations are a good approximation to the experimental data. The experimental Zeeman components in these figures correspond to the peak positions obtained via line shape fits to the experimental absorption data. The linewidths obtained in these fits are in the range 0.2-0.3 cm −1 which correspond to the instrumental line broadening associated with the FTIR spectrometer. Figure 5 illustrates the splittings of the lowest crystal field levels of the ground state 4 I 9/2 and the excited state 4 I 11/2 of Nd 3+ . The upper and lower levels of the ground (excited) state are indicated by g u (e u ) and g l (e l ), respectively. The absorption transitions between the Zeeman split components are labelled a-d. This labelling is also used in figures 1-4. The g-values can then be extracted from the transition energies. For the ground state, the g-value can be calculated using and the excited state g-value can be calculated utilising where B is the applied magnetic field strength and µ B is the Bohr magneton. Provided that the ground state g-value is known for a direction of the applied field the excited state gvalue can be determined from equation (2). The Zeeman splitting of each crystal field doublet is characterised by an anisotropic g tensor. The ground state g tensors for Nd 3+ in Y 2 SiO 5 for both sites are known from EPR measurements. The EPR g tensors for sites 1 and 2 are provided in table 3. The g tensor given in [15] has been rotated in order to match the directional g-values, g x , g y , and g z , in the axis system with X, Y, and Z parallel to D 1 , D 2 , and b, respectively, in a right-handed coordinate system. These ground state directional g-values are given in tables 1 and 2 for each site along with the g-values corresponding to higher crystal field energy levels.
A total of 61 experimentally observed energy levels and 28 g values have been measured for site 1, with 64 experimentally observed energy levels and 51 g-values measured for site 2.

Parameterised crystal-field analysis
The Hamiltonian appropriate for modelling the 4f 3 configuration is [29], where H FI and H CF are the free-ion and crystal field Hamiltonians. H Z , and H z are the electronic and nuclear Zeeman effects in the presence of an external magnetic field. H HF and H Q are referred to as the nuclear magnetic dipole hyperfine and nuclear quadrupole hyperfine Hamiltonians, respectively. The complete free-ion Hamiltonian includes the configuration barycentre parameterised by E AVG , the aspherical Coulomb repulsion represented by the Slater parameters F k , and the spin-orbit interaction given by ζ. In addition, a set of parameterised corrective terms is also added to account for two and three-body interactions as well as higher order effects. Following Wybourne's normalisation the parameterised crystal field Hamiltonian is given [30]   where the summation is also over all 4 f electrons. The B k q parameters are referred to as the crystal-field parameters and the C (k) q (i) are spherical tensor operators. In equation (4), k = 2, 4, 6 and q = −k, . . . , k. All parameters with the exception of the axial (q = 0) parameters are complex, leading to a total of 27 independent values for the C 1 point group symmetry of yttrium sites in Y 2 SiO 5 .
In addition to the crystal-field transitions and directional Zeeman g values the fits incorporate ground-state magnetic splittings calculated from the EPR ground state g tensors at various magnetic field directions. The form of the applied Table 1. Theoretical and experimental electronic energy levels and g values for site 1 of Y 2 SiO 5 :Nd 3+ with the theoretical values determined from the calculated crystal field Hamiltonian. The theoretical and experimental values of g X , g Y , and g Z are given with the applied magnetic field being along a crystal axis, B ∥ D 1 , B ∥ D 2 , and B ∥ b, respectively. '-' indicates an unassigned energy level. All energy values are given in cm −1 . The numbers in parentheses are the g value errors.
Energies magnetic field follows a spiral spatial dependence given in equation (5).
where B 0 is the magnitude of the applied field, here chosen to be 0.05 T. Using this approach, fits to both sites incorporated 12 'synthetically generated' magnetic splittings. This method has proved to be particularly useful when the number of experimental energy levels and g-values included in the fit is less than the free parameters, since the additional data points calculated from the experimental g tensor increase the number of observables. This approach was employed in [19] for YSO:Ce 3+ .

Correlation crystal-field model for 2 H 11/2 (2)
The standard one-electron crystal field parameterisation cannot account for the crystal-field splittings observed for certain multiplets of the 4f 3 (Nd 3+ ), 4f 7 (Gd 3+ ), and 4f 10 (Ho 3+ ) configurations. These are 2 H 11/2 (2), 6 I 17/2 , and 3 K 8/2 for the above configurations, respectively. These discrepancies have been consistently observed in several crystals for these ions [33]. In particular the difficulty of fitting the crystal-field levels of the 2 H 11/2 multiplets of Nd 3+ using the one-electron crystal field model Hamiltonian has been discussed in [34]. There are two 2 H 11/2 multiplets in the 4f 3 configuration which are labelled 2 H 11/2 (1) and 2 H 11/2 (2) [35]. The two multiplets are mixed by the Coulomb and spin-orbit interactions. To approach this problem a model Hamiltonian is introduced that includes additional correlated two-electron crystal-field interactions represented by H CCF . The correlation crystal field Hamiltonian is parameterised in terms of orthogonal operators. According to Judd [36] and Reid [37], the orthogonal correlation crystal field is given by where G K iq are the correlation crystal field parameters and g are the corresponding orthogonal operators. For f electrons K runs from 0 to 12 accepting the even values only and q is determined by symmetry. g (K) i with K = 0 correspond to Coulomb interactions and with i = 1 to one-electron operators g (K) 1 = U (K) that can be absorbed in the standard one-electron term of the model Hamiltonian. Therefore, the Hamiltonian contains a large number of parameters, however, only a few of them are required to obtain an agreement between the experimentally observed and calculated crystal field energy levels of 2 H 11/2 (2) through fitting. Reid and co-workers showed in [33] that the required correlation crystal field operators are g (4) 2 , g (4) 10A , and g (4) 10B for the YAG:Nd 3+ system. The relevance to the 4th rank operators can be understood when we note that the reduced matrix element of the C (4) operator is unusually small for the above multiplet [38]. To reduce the number of additional parameters, we assume that the q dependence of G 4 iq (with i = 2, 10A, and 10B) correlation crystal field parameters follows that of the 4th rank one-electron crystal field parameter B 4 q . Thus, we have and only three independent correlation crystal field parameters are included in the fits. The analyses of Ng and Newman [39] and Li and Reid [35] suggest that that the major contribution to these parameters is the excitations from the lanthanide core states, e.g. 5p to 4 f. The results of the fits to the energy level data, Zeeman g-values, and ground state magnetic splittings are given in tables 1 and 2 for sites 1 and 2, respectively. The free-ion parameters that were held fixed in all stages of the calculations are listed in table 4. Those free-ion parameters that were allowed to vary included the Slater parameters F 2 , F 4 , F 6 , and the spin-orbit parameter ζ and are summarised in table 5. The number of fitted parameters for site 1 was 33 in total, including 27 one-electron crystal field parameters, 1 correlation crystal field parameter, and 5 free-ion parameters. The total number of parameters for site 2 was 35 including 27 one-electron crystal field parameters, 3 correlation crystal field parameters and Table 2. Theoretical and experimental electronic energy levels and g values for site 2 of Y 2 SiO 5 :Nd 3+ with the theoretical values determined from the calculated crystal field Hamiltonian. The theoretical and experimental values of g X , g Y , and g Z are given with the applied magnetic field being along a crystal axis, B ∥ D 1 , B ∥ D 2 , and B ∥ b, respectively. All energy values are given in cm −1 . The numbers in parentheses are the g value errors.
Energies   [23,31] utilised as the crystal field parameters fit initial values have been also given in this table for comparison. Since the addition of the spin Hamiltonian data and the fits to the data from the experimental spin Hamiltonian tensors adds several local minima to the solution space a basin hopping algorithm, which attempts a random step followed by a local minimisation was employed as a global optimisation routine [40,41]. The algorithm used for the local minimisation was the bound optimisation by quadratic approximation algorithm from the nonlinear-optimisation package [42]. The experimental and calculated ground state g tensors for both sites are given in table 3. It is possible to determine the g tensors from the wavefunctions of the Kramers doublets. However, to avoid ambiguities, we followed the method described in [43], which involves calculating the magnetic splittings in nine directions, and solving for the square of the g tensor. Taking the square root provides an unambiguous g tensor. We note that the same procedure was also applied to the experimental g tensors to enable direct comparison of the experimental and calculated g tensors.
To facilitate the comparison of the two crystal-field parameters set, the rotationally invariant crystal-field strength parameters, S k (k = 2,4,6 for f electrons) and S are introduced [44]. and These parameters are given in table 5 for YSO:Nd 3+ alongside the corresponding parameters for YSO:Er 3+ for both of their sites. The comparison reveals that the crystal field strength parameters of YSO:Nd 3+ are larger in size in the 4th and 6th rank crystal-field parameters. This agrees with the effect of lanthanide contraction across the series on the parameters, discussed by Carnall et al [32] for trivalent lanthanides in LaF 3 . The experimental and calculated crystal field energy levels of 2 H 11/2 (2) of sites 1 and 2 of YSO:Nd 3+ are given in table 6. Results from both one-electron and correlation crystal field Table 5. Calculated free-ion, single-electron, and correlation crystal-field parameters for sites 1 and 2 of Y 2 SiO 5 :Nd 3+ . The Y 2 SiO 5 :Er 3+ crystal-field parameters provided in [23,31] were used as initial values for the fits and are given for comparison. The initial values for the free-ion parameters were adopted from [32]. All values are in cm −1 . calculations are included and show a dramatic improvement in the fit utilising the correlation crystal field Hamiltonian with the more accurate calculation associated with site 1. This table also reveals that the crystal field splittings predicted by the standard one-electron crystal field calculation are noticeably smaller than their corresponding experimental values, a problem that is addressed by the inclusion of the correlation crystal field terms discussed earlier. In the analysis performed by Li and Reid [35] and also in [33] it is shown that the ratio of G 4 10A /B 4 0 always possesses a negative sign and is less than 1. In the current calculation, this parameter retains the same property and the ratio is determined to be −0.49 for site 1 and −0.08 for site 2. The best correlation crystal field fits for site 1 were obtained including just G 4 10A parameter whereas all three parameters were required to be included for the best fits obtained for site 2. Correlation crystal-field fits with only G 4 10A parameter for Nd 3+ in various hosts have been previously reported [35], and this parameter was shown to have the largest effect on the calculated energy levels of 2 H 11/2 (2). The fits including all three correlation crystal field parameters for site 1 yielded G 4 10A = −151 cm −1 , G 4 10B = 38 cm −1 , and G 4 2 = 964 cm −1 .

Conclusion
Using crystal field energy levels and Zeeman g-value data acquired for Y 2 SiO 5 :Nd 3+ , crystal field fits were performed for both C 1 symmetry centres. The directional Zeeman splitting data provides orientation information to the fits, allowing the determination of a unique set of crystal field parameters. For more accurate fits, and to account for the unknown phase of the Kramers doublet eigenvectors, the energy-level and Zeeman data were supplemented by synthetically generated ground state splittings for multiple directions of the applied magnetic field derived from the ground-state g tensors. Furthermore, the addition of correlation crystal field parameters successfully accounted for discrepancies between the theoretical and experimental values for the 2 H 11/2 (2) multiplet when only the one-electron crystal field model was used.

Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files).