Elucidating magnetic structure with optical dopants: erbium-doped Gd$_2$SiO$_5$

Abstract

The narrowness of the optical transitions of rare-earth-ion dopants makes them highly sensitive probes of their environment. We measured the optical transitions Er$^{3+}$ dopants to determine the previously unknown magnetic ordering of Gd$_{2}$SiO$_{5}$ -- a promising host for quantum applications of rare-earth dopants. By measuring the transitions' magnetic-field dependence we determined an antiferromagnetic ordering with spins oriented along or slightly canted from the crystal's $a^*$ axis. The optical transitions are narrower than the coupling to gadolinium spins revealing information about the coupling strengths. We further optically measured a Néel temperature of $1.86\pm0.01_\mathrm{stat.}\pm0.07_\mathrm{syst.}$ K, and assembled a phase diagram in applied field and temperature showcasing a triple point where two gadolinium sites order semi-independently from each other. At high applied field the erbium dopants show long optical coherence times up to 0.4 ms at 3 T; at low fields these are probably limited by three low-frequency magnon modes below 10 GHz, observed directly. This study can be used to benchmark a method of magnetic structure determination.

Figures (11)

• Figure 1: (a) Monoclinic unit cell of a GSO crystal dramicaninLuminescenceStructuralProperties2006. Gadolinium ions are labelled with their oxygen coordination number, CN7 (purple) or CN9 (dark green). CN7 ions lie approximately on (100) sheets and have approximate $C_s$ ligand symmetry with mirror plane shown for one of the ions in pink. CN9 ions have approximate $C_{3v}$ ligand symmetry. The $C_3$ axes are shown with arrows and one of the mirror planes for a CN9 ion is shown in lime green. Erbium ions can substitute any site, and within the unit cell there are four subsites with differently oriented crystal fields. (b) A photograph of the sample inside a bulk loop-gap cavity with the front plate removed. The sample is annotated as well as a loop in the cavity giving an inductance $L$, and a gap giving a capacitance $C$, which makes the cavity an analog to a $LC$-resonator. Below: diagram of crystal axes in this sample. The $a^*$ axis is within 0.5° of the normal of the top and bottom faces, whereas the other axes are only approximate. The sign of the $b$ axis is unknown. (c) Closed photograph of the cavity without microwave cables.
• Figure 1: Comparison of synthetic spectra from the effective spin-½ model to real spectra for site 2 at the base temperature of the dilution refrigerator. Parameters given in main text. The assumed misalignment was 0.5°. (a,d) Synthetic spectra. (b,e) Experimental data. (c,f) Experimental data with calculated transitions from the ground state of the model overlaid. Spin-up (spin-down) sublattices at zero field are shown as purple (red) lines. Dotted lines are the sublattices with higher $|B_\mathrm{mf}|$. The difference between (a--c) and (d--f) is just the colour scale.
• Figure 2: (a) Optical transmission spectra of Er:GSO at $\approx\qty{60}{\milli\K}$. Site 1 is seen around 196, whereas site 2 is around 196.1. Blue and red arrows next to strong absorption features label like-to-like transitions of antiferromagnetic sublattices and the erbium spin state of that sublattice at that field. Blue (red) arrows label the ions that are spin up (down) at zero applied field. Top inset: weak transitions from site 1 observed behind site 2 lines. Bottom inset: A pair of weak transitions of site 2 seen around 0.4. Weak lines at other fields are possibly satellite lines or due to cooperative excitation of neighbouring gadolinium spins. (b) Energy levels of site 2. Red and blue are magnetic sublattices that order antiferromagnetically, pointing along the $a^*$ axis. In the $Z_1$ ground doublet, the bold lines are the ground states and their spin orientation is annotated. Dotted lines are the unpopulated state. The strong transitions are like-to-like transitions that preserve spin direction, hence the excited $Y_1$ state that is observed in those transitions is bold, whereas the other is dotted. At about 0.52, the ground state of one of the subsites changes spin direction as the Zeeman splitting goes through zero and therefore we measure a different excited state. The $g$-factors are found from the strong transitions as well as the weak lines shown in the bottom inset of (a). (c) Approximate model for site 1 energy levels. The $g$-tensor does not have a principal axis along $a^*$ and/or the spins are canted from $a^*$, hence like-to-unlike transitions are stronger and energy levels do not cross.
• Figure 2: Comparison of synthetic spectra from the effective spin-½ model to real spectra for site 2 at 830±10. (a,d) Synthetic spectra. (b,e) Experimental data. (c,f) Experimental data with calculated transitions from the ground state of the model overlaid. Spin-up (spin-down) sublattices at zero field are shown as purple (red) lines. Solid (dashed) lines are transitions from the ground (first excited) state. The difference between (a--c) and (d--f) is just the colour scale. The differences to the cold spectra in Extended Data Fig. \ref{['fig:toysite2']} are that the assumed temperature is higher, the exchange splitting is reduced to 93 of the cold value to account for reduction in mean field, and a Lorentzian FWHM of 2 is used.
• Figure 3: (a) A phase diagram assembled through optical measurements near a triple point. Light green lines in the background show the extent of our measurements. Phase transition fields were determined as sudden global changes on optical transmission spectra. Where the optical spectra show features from both phases, the range of coexistence in applied field is shown by error bars. Points with larger temperature uncertainty were measured by sweeping the sample's temperature instead. For this graph the systematic error from the temperature sensor is not included as it is expected to be consistent. At 2.95 and 1.011, the optical spectra had features from all three spectra, and this point is marked as the triple point. Dashed lines and background colour are guides to the eye. FM: ferromagnetic; AFM: antiferromagnetic; PM: paragmagnetic; SC: spin canting. (b) Optical spectra at 950±7 showing the phase $1\leftrightarrow2$ transition. The four main transitions of site 1 approach two degenerate pairs. At that field, a phase transition occurs and at higher applied field only one main transition is observed. We therefore believe that site 1 becomes ferromagnetically ordered. Site 2 goes from four transitions to two, so we believe that it is still ordered antiferromagnetically, but that the former two$\rightarrow$four splitting was due to nonequivalent site 1 neighbours. (c) Phase-transition between phases $1\leftrightarrow3$ at 1.29±0.01. (d) Optical transmission through the sample at site 2 as the sample is slowly cooled through and beyond the Néel temperature at zero applied field. (e) The spectra from (d) are fit to a sum of two Lorentzian line shapes. The splitting and full widths at half maximum (FWHM) are shown and are used to determine the Néel temperature where their temperature dependence changes slope suddenly. Solid black lines show the fitted parameters in the range over which they are fitted, and the dotted black line shows the extracted Néel temperature.