## From local degrees of freedom to correlated states in anisotropic 3d transition metal compounds

##### Abstract

Anisotropy plays a crucial role in a wide variety of magnetic systems. In
low-dimensional materials it can stabilise magnetic structures, overcoming the
tendency for thermal
fluctuations to disorder the magnetic correlations. It can
also give rise to exotic dynamics such as nonlinear excitations and amplitude fluctuations that are not present in purely isotropic magnets. The origin of
magnetic anisotropy lies in the physics of the individual magnetic ion and
the crystallographic environment in which it nds itself. The nature of the
magnetocrystalline anisotropy is therefore highly dependent on both the crystal
structure and the species of magnetic ion. This dependence on the particulars of
the system gives rise to starkly different phenomena in different compounds.
In this Thesis, the physics of a number of anisotropic 3d transition metal
compounds will be investigated, with a particular focus on the interplay between
the single-ion physics and correlated phenomena. The Thesis begins with a
discussion of the nature of magnetic interactions in the solid state, focusing on the
quantum mechanical nature of the spin and orbital degrees of freedom that give
rise to magnetism. Chapter 2 then provides an overview of neutron scattering
{ the principle experimental technique employed in this Thesis. This chapter
concentrates on the instrumentation and neutron scattering theory required to
interpret the results detailed in the later chapters and includes sections on both
time-of-flight and triple-axis spectroscopy.
Following the two introductory chapters, Chapter 3 explores the low energy
dynamics of quasi-one-dimensional, large-S quantum antiferromagnets with easyaxis
anisotropy. Such a situation is present in some 3d transition metal
compounds based on ions such as Fe³⁺ or Mn²⁺. A description of these
systems is developed using a semiclassical nonlinear δ model. The saddle point
approximation leads to a sine-Gordon equation which supports soliton solutions.
These correspond to the movement of spatially extended domain walls. Long-range magnetic order in spin chain compounds is typically a consequence of a weak
inter-chain coupling. Below the ordering temperature, the coupling to nearby
chains leads to an energy cost associated with the separation of two domain
walls. From the kink-antikink two-soliton solution, an effective confi nement
potential is computed. At distances that are large compared to the size of the
solitons the potential is linear, as expected for point-like domain walls. At small
distances the gradual annihilation of the solitons weakens the effective attraction
and renders the potential quadratic. By numerically solving the effective onedimensional
Schrödinger equation with this nonlinear con finement potential,
the soliton bound state spectrum is computed. The theory is then applied to
CaFe₂O₄, an anisotropic magnet based upon an antiferromagnetic zig-zag network
of 3d⁵ Fe³⁺ ions with S = 5/2 and L = 0. Neutron scattering measurements are
able to resolve seven discrete energy levels for spectra recorded slightly below the
Néel temperature TN ≈ 200 K. These modes are well described by the nonlinear
confi nement model in the regime of large spatially extended solitons.
Chapter 4 concerns a jeff = 1/2 magnet α-CoV₂O₆, where spin-orbit coupling
is much larger than the inter-ion coupling and hence the jeff = 1/2 manifold is
well separated from spin-orbital levels. Here, the anisotropy originates from a
small crystallographic distortion which can be treated as a small perturbation
motivating an effective S = 1/2 Hamiltonian with an Ising/uniaxial symmetry.
Low temperature magnetisation data show the existence of magnetisation
plateaux, yet these are not accompanied by Bragg peaks in neutron diffraction
data and hence are not indicative of transitions to new phases of long-range
magnetic order. By application of the Lieb-Schultz-Mattis theorem, the existence
of these magnetisation plateaux is reconciled with the absence of corresponding
Bragg peaks in α-CoV₂O₆. This analysis relates the underlying symmetries of the
ground state to the magnetisation. The presence of uniaxial anisotropy is shown
to stabilise metastable short-range magnetic order at different fi eld strengths and
temperatures, constructed from antiphase boundaries.
Remaining on the theme of metastable antiphase boundary order, Chapter 5
returns to the S = 5=2 antiferromagnet CaFe₂O₄ which exhibits two magnetic
orders that show regions of coexistence at some temperatures. By applying
neutron scattering and a Green's function formalism, the spin wave excitations
in this material are characterised, elucidating the relevant terms in the spin
Hamiltonian. In doing so, it is suggested that the low temperature A phase
order (↑↑🡣🡣) finds its origins in the freezing of antiphase boundaries created by
thermal
fluctuations in a parent B phase order (↑↑🡣🡣). The low temperature
magnetic order observed in CaFe₂O₄ is thus the result of a competition between
the exchange coupling along c, which favors the B phase, and the single-ion
anisotropy, which stabilises thermally-generated antiphase boundaries, leading to
static metastable A phase order at low temperatures.
In Chapter 6, an iron-rich sample of the two-dimensional van der Waals itinerant
ferromagnet Fe₃GeTe₂ is investigated using neutron scattering. The excitations
are shown to be predominantly two-dimensional in nature and broadened, as
expected for an itinerant magnet. The anisotropy strength is shown to be greater
in magnitude than has been reported in a recent study of iron- deficient samples,
hinting at a crucial role of the iron concentration in the single-ion properties of
Fe3GeTe2. A model of domain walls is developed and the extracted exchange
parameters from the neutron scattering results are used to calculate the expected
domain wall width, based on bulk exchange parameters. This is then compared
with scanning tunnelling microscopy (STM) data which are reflective of the
surface physics. Strong agreement is found with the STM data suggesting that
the surface properties are similar to that of the bulk.
Chapter 7 concerns another two-dimensional van der Waals ferromagnet, VI₃.
Unlike the 3d⁵ transition metal compound discussed in the preceding chapter,
VI₃ is formed from a honeycomb of 3d² V³⁺ ions which carry an orbital degree
of freedom. Here the Green's function formalism is extended to treat systems
with an orbital degree of freedom, treating the spin-orbit coupling and crystal
distortions explicitly. Neutron scattering is used to understand the nature of the
low energy spin dynamics in VI₃, demonstrating the existence of two qualitatively
different low energy modes. The neutron data are then modelled using the Green's
function formalism, allowing a connection to be made between the spectrum
and the crystallographic structure and indicating the presence of two differently
distorted domains. It is shown that the anisotropy arising due to the cooperative
effect of spin-orbit coupling and crystal distortions allows for the stable two-dimensional
magnetism at finite temperature.
Finally, in Chapter 8 the Green's function formalism is extended to treat
noncollinear structures. This formalism is then applied to the noncollinear chargeordered
antiferromagnet RbFe²⁺Fe³⁺F₆ - a system of mixed valance formed by
two coupled networks of Fe²⁺ (3d⁶) and Fe³⁺ (3d⁵) chains. The spin-orbit coupling
and effect of crystal distortions on the Fe²⁺ion are considered explicitly and the
neutron scattering response is calculated using the noncollinear Green's function
formalism. In addition to spin-orbit excitons, it is shown that the low symmetry
of the Fe²⁺ coordination may give rise to low energy amplitude
fluctuations that
are not captured by linear spin wave theory. It is suggested that noncollinear
magnets with low local symmetry may provide candidate systems for stable low
energy amplitude modes in condensed matter systems.