Electron spin resonance

High-field, high-frequency Electron spin resonance (ESR) spectra of powdered sam- ples were recorded on a home-built spectrometer at the EMR facility, National High Magnetic Field Laboratory, Tallahassee, Florida, USA. Microwave frequencies in the range 52 ≤ ν ≤ 626 GHz at temperatures ranging from ≈ 3 K to 80 K were used in the measurement. The instrument is a transmission-type device and uses no resonance cavity. Powdered samples are loaded into thin teflon vessels and low- ered into the cryostat. The microwaves were generated by a phase-locked Virginia Diodes source, generating frequency of 13±1 GHz, and equipped with a cascade of frequency multipliers to generate higher harmonic frequencies. The resultant signal was detected using a cold bolometer. A superconducting magnet capable of reaching fields up to 15 T was employed.

ESR is an experimental technique that is highly suited to determining the difference between ground-state and excited energy levels in condensed matter [27]. Using a combination of frequency dependent and temperature dependent studies, the sign and magnitude of single-ion anisotropy (see section 1.2.3) parameters can be obtained [21]. In ESR measurements, microwave radiation (109−1011 GHz) is supplied to a sample in a magnetic field. The magnetic field splits the energy levels via the Zeeman mechanism [115]. If the injected photon has the same energy (ν) as the difference between the energy levels (∆E), then the photon will be absorbed by the spin. This induces a transition of the spin to a higher energy state:

∆E =hν=gµ_bµ0H∆ms, (2.9)

where ∆ms=±1 is the allowed change in the spin state. This causes a peak in the ESR spectrum. At low fields, energy-level mixing occurs. This allows transitions between spin states with ∆ms=±2. This transition is called the half-field transition as it occurs at approximately half the mean-field of the ∆ms=±1 transitions. As both ν and ∆E can be changed (the latter tuned using different µ0H), there are two different methods to obtaining spectra; by scanning the frequency or the field. Because of difficulty in scanning microwave frequencies and the common use of

Figure 2.7: Resonances observed in ESR spectra of powdered samples with(a)the Hamiltonian in Eq. 2.10 (D 6= 0 and E 6= 0) and (b) uniaxial symmetry (D 6= 0 andE = 0). Adapted from Ref. [12].

resonant cavities for signal detection, most ESR spectrometers keep the frequency constant and change the field.

In this thesis, ESR measurements have been used to determine the single-ion parameters ofS= 1 powdered compounds. Spectra of these samples can exhibit up to nine resonances, three for each crystal axis that field can be applied along. The position of each of these resonances can be calculated by exact diagonalisation of theS= 1 Hamiltonian: ˆ H=DX i ˆ Sz2 i +E X i ˆ Sx2 i −Sˆ y2 i +µ_bµ0 X i g·H·Sˆi. (2.10)

This yields, for fields parallel (||) to thex,yandz axes, the expected resonance field for a particular frequency ν and unknown single-ion anisotropy parameters D and

E [12]: ∆ms=±1 transitions µ0H || x−axis : µ0H = _gxµ1 b q hν±3 2 1 3D−E 2 −1 4(D+E) 2_. µ0H || y−axis : µ0H = _gyµ1 b q hν±3 2 1 3D+E 2 −1 4(D−E) 2_. µ0H || z−axis : µ0H = _gzµ1 b q (hν±D)2−E2_. ∆ms=±2 transitions µ0H || x−axis : µ0H = _2gxµ1 b q (hν)2−(D+E)2. µ0H || y−axis : µ0H = _2gyµ1 b q (hν)2−(D−E)2. µ0H || z−axis : µ0H = _2gz1_µ b q (hν)2−4E2_. (2.11)

Representative first derivative of the transmission spectra for anS = 1 pow- dered sample are presented in Fig. 2.7 showing the expected shapes of the resonance peaks with components of the spin state with ∆ms = ±1. The first derivative is commonly used for convenience as the position of the resonances can be observed more easily [116]. For low symmetry, the y and z transitions (field applied parallel to the y and z axis) are peak shapes, whereas the x transition are peak derivative shapes. For uniaxial symmetry (E = 0), the z transition is a peak shape and the xy transition is a peak derivative shape. Due to the large amount of resonances that may occur during a measurement, it is necessary to label them appropriately. The labels for this thesis are presented in Fig. 2.7 and Fig. 2.8. For a field applied parallel to the z axis, the half-field transition is labelled ζ, the low-field is Ψ and high-field is. For xy transitions, the half-field is labelledα, the low-field isβ and the high-field is γ. If there is a finite E term, then the subscripts x and y will be used to distinguish between the two directions for the α, β and γ transitions. In an ESR measurement of a powdered sample, ESR spectra at different frequencies are made. All the observed transitions (with known field and frequency values) were then fitted to the simultaneous equations in Eq. 2.11 to obtain the single-ion

Figure 2.8: Ground state energy level diagram of a D-only S = 1 magnetic ion with easy-plane anisotropy which shows the labelling of the transitions in ESR measurements in this project. The field is applied parallel to(a)thez-axis and (b)

xy plane. In the presence of an E term in the Hamiltonian, the xy energy-levels split andα,β and γ resonances will be labelled withx and y subscripts depending on the transition they refer to.

anisotropy parameters and intrinsic g-tensor. Further details of the fitting process can be found in Ref. [30].

It is not possible to obtain the sign of D for powdered S = 1 compounds from fitting frequency-dependent ESR data in this manner [12, 116]. This is because the positions of ESR transitions do not change between easy-plane and easy-axis anisotropy. However it is possible to determine the sign of D from observing how the intensities of the resonance peaks changes as the temperature is varied. At very low temperatures, most of the spins occupy the ground state. A transition from this state to an excited state will have a very high intensity. A transition between two excited states will have a very low intensity due to the low occupancy of these energy levels. As the temperature increases, the transition from the ground state will decrease in intensity as spins increasingly occupy the higher energy levels. Tran- sitions between the excited levels will therefore have larger intensities. This can be used to determine the sign ofDas the ground state will be different for positive and negativeD. For example, the low-field xy (β) transition in Fig. 2.8 is between the ground state ms = −1 and excited ms = 0 level for D > 0. At low temperatures it will have a much higher intensity than the high-field xy (γ) transition which is between two excited states. For easy-axis anisotropy, these change. Theγresonance is due to the transition between the ground statems =−1 and excitedms = 0 level whilst theβ transition is now between the excitedms= 0 andms= +1 levels. The γ resonance will have a much higher intensity than the β transition at low temper- atures. Hence, there are two methods to determine the sign ofD. The first is the difference in intensity between low and high-field resonances at a given temperature due to transitions with the field aligned along one crystal axes. However, due to the

large magnitudes of D in the compounds in this project, the high-field transitions are often out of the field/frequency range of most spectrometers. In this case the second method can be used: by measuring ESR spectra with the same frequency at different temperature the sign of D can be determined by whether the intensity of one particular resonance increases or decreases.