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\begin_body

\begin_layout Chapter
Introduction to two-dimensional molecular magnetism
\begin_inset CommandInset label
LatexCommand label
name "chap:mmintro"

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
begin{sloppypar}
\end_layout

\end_inset

In the following two chapters (
\begin_inset CommandInset ref
LatexCommand nameref
reference "chap:cuhf2pyz2x"

\end_inset

 and 
\begin_inset CommandInset ref
LatexCommand nameref
reference "chap:other2dmm"

\end_inset

), I will discuss a series of molecular materials which act as model systems
 for exploring fundamental magnetism.
 In particular, the structures of the compounds studied here give rise to
 magnetic exchange interactions which are significantly larger within planes
 in the materials than between those planes.
 These materials therefore behave approximately as collections of isolated
 two-dimensional magnetic layers, with a small inter-layer coupling which
 becomes significant at low enough temperatures.
 This chapter provides some introductory material relevant to both chapters,
 first giving a brief overview of the physics of the model being probed,
 the 
\begin_inset Formula $S=\frac{1}{2}$
\end_inset

 two-dimensional square-lattice quantum Heisenberg antiferromagnet, in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "sec:2DSLQHA"

\end_inset

; then describing molecular magnetism, concentrating on the structures examined
 in this thesis in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "sec:molecular-magnetism"

\end_inset

; and finally, an overview is given of the methods by which we can parametrise
 the low dimensionality of materials in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "sec:exchange-anisotropy"

\end_inset

.
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
end{sloppypar}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
The work in these chapters has been published, as indicated in Ref.
\begin_inset space ~
\end_inset


\begin_inset CommandInset citation
LatexCommand citealp
key "Steele2011-2d-molmag-muons"

\end_inset

.
\end_layout

\begin_layout Section
The 
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
texorpdfstring{
\end_layout

\end_inset


\begin_inset Formula $S=\frac{1}{2}$
\end_inset


\begin_inset ERT
status open

\begin_layout Plain Layout

}{S = 1/2}
\end_layout

\end_inset

 2D square-lattice quantum Heisenberg antiferromagnet
\begin_inset CommandInset label
LatexCommand label
name "sec:2DSLQHA"

\end_inset


\end_layout

\begin_layout Standard
The 
\begin_inset Formula $S=\frac{1}{2}$
\end_inset

 two-dimensional square-lattice quantum Heisenberg antiferromagnet (2DSLQHA)
 continues to be one of the most important theoretical models in condensed
 matter physics
\begin_inset CommandInset citation
LatexCommand cite
key "Manousakis1991-2DSLHAFM"

\end_inset

.
 Experimental realisations of the 2DSLQHA in crystals also contain an interactio
n between planes, so that the relevant model describing the coupling of
 electronic spins 
\begin_inset Formula $\bm{S}_{i}$
\end_inset

 gives rise to the Hamiltonian
\begin_inset Foot
status collapsed

\begin_layout Plain Layout
This Hamiltonian employs the single-
\begin_inset Formula $J$
\end_inset

 convention such that, in the case of just two 
\begin_inset Formula $S=\frac{1}{2}$
\end_inset

 interacting spins, the separation between the singlet and triplet energies
 is equal to the interaction strength 
\begin_inset Formula $J$
\end_inset

.
\end_layout

\end_inset

 
\begin_inset Formula 
\begin{equation}
\mathcal{H}=J\sum_{\langle i,j\rangle_{xy}}\bm{S}_{i}\cdot\bm{S}_{j}+J_{\perp}\sum_{\langle i,j\rangle_{z}}\bm{S}_{i}\cdot\bm{S}_{j},\label{eq:2D-Hamiltonian}
\end{equation}

\end_inset

where 
\begin_inset Formula $J$
\end_inset

 (
\begin_inset Formula $J_{\perp}$
\end_inset

) is the strength of the in- (inter-) plane coupling and the first (second)
 summation is over neighbours parallel (perpendicular) to the 2D 
\begin_inset Formula $xy$
\end_inset

-plane.
 Any 2D model (
\begin_inset Formula $J_{\perp}=0$
\end_inset

) with continuous symmetry will not show long-range magnetic order (LRO)
 for 
\begin_inset Formula $T>0$
\end_inset

 due to a divergence of infrared fluctuations
\begin_inset CommandInset citation
LatexCommand citep
key "Mermin1966-and-Wagner-no-1-2D-order,Berezinskii1971-Mermin-and-Wagner"

\end_inset

.
 However, layered systems approximating 2D models (
\begin_inset Formula $J_{\perp}\neq0$
\end_inset

) will inevitably enjoy some degree of interlayer coupling and this will
 lead to magnetic order, albeit at a reduced temperature due to the influence
 of quantum fluctuations.
 Quantum fluctuations are also predicted to reduce the value of the magnetic
 moment in the ground state of the 2DSLQHA to around 60% of its classical
 value
\begin_inset CommandInset citation
LatexCommand cite
key "Manousakis1991-2DSLHAFM"

\end_inset

, and this reduction is often seen in the ordered moments of real materials.
 In layered materials that approximate the 2DSLQHA, the measurement of the
 antiferromagnetic ordering temperature 
\begin_inset Formula $T_{\mathrm{N}}$
\end_inset

 is often problematic due not only to this reduction of the magnetic moment,
 but also to short-range correlations that build up in the quasi-2D layers
 above 
\begin_inset Formula $T_{\mathrm{N}}$
\end_inset

.
 These correlations lead to a reduction in the size of the entropy change
 that accompanies the phase transition, reducing the size of the anomaly
 in the measured specific heat
\begin_inset CommandInset citation
LatexCommand cite
key "sengupta2003-bulkmeasuresbad"

\end_inset

.
 It has been shown in a number of previous cases that 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
muSR
\backslash

\end_layout

\end_inset

 measurements do not suffer from these effects and therefore represent an
 effective method for detecting magnetic order in complex anisotropic systems
\begin_inset CommandInset citation
LatexCommand citep
key "Blundell2007-muSR-lowD-magnets-review,goddard2008-2DHMexchange,Lancaster2006-CuPzN,Lancaster2007-Cupyz2ClO42"

\end_inset

.
\end_layout

\begin_layout Section
Molecular magnetism
\begin_inset CommandInset label
LatexCommand label
name "sec:molecular-magnetism"

\end_inset


\end_layout

\begin_layout Standard
The rich chemistry of molecular materials allows for the design and synthesis
 of a wide variety of highly-tunable magnetic model systems
\begin_inset CommandInset citation
LatexCommand cite
key "Blundell2004-molmagreview"

\end_inset

.
 Magnetic centres, exchange paths and the surrounding molecular groups can
 all be systematically modified, allowing investigation of their effects
 on magnetic behaviour.
 In particular, the existence of different exchange paths along different
 spatial directions can result in quasi--low-dimensional magnetic behaviour
 (i.e.
\begin_inset space \space{}
\end_inset

systems with magnetic interactions constrained to act in a two-dimensional
 plane or along a one-dimensional chain).
 Such systems have the potential to better approximate low-dimensional models
 than many traditional inorganic materials.
 In addition, these molecular materials can have exchange energy scales
 of order 
\begin_inset Formula $J/k_{\mathrm{B}}\sim\SI{10}{\kelvin}$
\end_inset

 which are accessible with typical laboratory magnetic fields
\begin_inset CommandInset citation
LatexCommand cite
key "goddard2008-2DHMexchange"

\end_inset

 of 
\begin_inset Formula $B\sim\SI{10}{\tesla}$
\end_inset

 allowing an additional avenue for their experimental study.
 This contrasts with typical inorganic low-dimensional systems where the
 exchange is found to be 
\begin_inset Formula $J/k_{\mathrm{B}}\sim\SI{1000}{\kelvin}$
\end_inset

 and fields of 
\begin_inset Formula $B\sim\SI{1000}{\tesla}$
\end_inset

 would be needed to significantly perturb the spin system.
\end_layout

\begin_layout Standard
It has also been shown
\begin_inset CommandInset citation
LatexCommand cite
key "Xiao2009-XY-q2DHA,Sengupta2009-nonmonotonic-TN-B-CuHF2pyz2BF4"

\end_inset

 that a small 
\begin_inset Formula $XY$
\end_inset

-like anisotropy exists in some molecular materials.
 Although 
\begin_inset Formula $J_{\perp}$
\end_inset

 is the decisive energy scale for the magnetic ordering, this anisotropy
 has been shown to have an influence on the ordering temperature
\begin_inset CommandInset citation
LatexCommand cite
key "Xiao2009-XY-q2DHA"

\end_inset

 and determines the shape of the low-field 
\begin_inset Formula $B$
\end_inset

--
\begin_inset Formula $T$
\end_inset

 phase diagram of these systems
\begin_inset CommandInset citation
LatexCommand cite
key "Sengupta2009-nonmonotonic-TN-B-CuHF2pyz2BF4"

\end_inset

.
\end_layout

\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status open

\begin_layout Plain Layout
\align center
\begin_inset Graphics
	filename figures/mmintro/ligands/ligands.eps

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption

\begin_layout Plain Layout
Bridging ligands used in the compounds described in this thesis: (a) pyrazine
 (N
\begin_inset Formula $_{\text{2}}$
\end_inset

C
\begin_inset Formula $_{\text{4}}$
\end_inset

H
\begin_inset Formula $_{\text{4}}$
\end_inset

, abbreviated pyz); and (b) (ii) pyridine-
\emph on
N
\emph default
-oxide (C
\begin_inset Formula $_{\text{5}}$
\end_inset

H
\begin_inset Formula $_{\text{5}}$
\end_inset

NO, abbreviated pyo).
 Dashed lines indicate where the ligands bond to other parts of molecular
 structures.
\begin_inset CommandInset label
LatexCommand label
name "fig:ligands"

\end_inset


\end_layout

\end_inset


\end_layout

\end_inset

Several classes of molecular magnetic material closely approximate the 2DSLQHA
 model, and the following two chapters report the results of 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
muSR
\backslash

\end_layout

\end_inset

 measurements performed on several such materials.
 These systems are self-assembled coordination polymers, based around paramagnet
ic ions such as Cu
\begin_inset Formula $^{\text{2+}}$
\end_inset

, linked by neutral bridging ligands and coordinating anion molecules.
 The materials are based on combinations of three different ligands: (i)
\begin_inset space ~
\end_inset

pyrazine (N
\begin_inset Formula $_{\text{2}}$
\end_inset

C
\begin_inset Formula $_{\text{4}}$
\end_inset

H
\begin_inset Formula $_{\text{4}}$
\end_inset

, abbreviated pyz) and (ii)
\begin_inset space ~
\end_inset

pyridine-
\emph on
N
\emph default
-oxide (C
\begin_inset Formula $_{\text{5}}$
\end_inset

H
\begin_inset Formula $_{\text{5}}$
\end_inset

NO, abbreviated pyo), both of which are planar rings; and (iii)
\begin_inset space ~
\end_inset

the linear bifluoride ion [(HF
\begin_inset Formula $_{\text{2}}$
\end_inset

)
\begin_inset Formula $^{-}$
\end_inset

], which is bound by strong hydrogen bonds F
\begin_inset Formula $\cdots$
\end_inset

H
\begin_inset Formula $\cdots$
\end_inset

F.
 The pyz and pyo ligands are shown in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "fig:ligands"

\end_inset

.
 
\end_layout

\begin_layout Standard
Specifically, 
\begin_inset CommandInset ref
LatexCommand formatted
reference "chap:cuhf2pyz2x"

\end_inset

 discusses the molecular system [
\emph on
M
\emph default
(HF
\begin_inset Formula $_{\text{2}}$
\end_inset

)(pyz)
\begin_inset Formula $_{\text{2}}$
\end_inset

]
\emph on
X
\emph default
, where 
\emph on
M
\emph default

\begin_inset Formula $^{\text{2+}}$
\end_inset


\begin_inset space ~
\end_inset

= Cu
\begin_inset Formula $^{\text{2+}}$
\end_inset

 is the transition metal cation and 
\emph on
X
\emph default

\begin_inset Formula $^{-}$
\end_inset

 is one of various anions (e.g.
\begin_inset space ~
\end_inset

BF
\begin_inset Formula $_{\text{4}}^{-}$
\end_inset

, ClO
\begin_inset Formula $_{\text{4}}^{-}$
\end_inset

, PF
\begin_inset Formula $_{\text{6}}^{-}$
\end_inset

 
\emph on
etc.
\emph default
).
 
\begin_inset CommandInset ref
LatexCommand formatted
reference "chap:other2dmm"

\end_inset

 then reports the results of measurements on other quasi-2D systems.
 First, [Cu(pyz)
\begin_inset Formula $_{\text{2}}$
\end_inset

(pyo)
\begin_inset Formula $_{\text{2}}$
\end_inset

]
\emph on
Y
\emph default

\begin_inset Formula $_{\text{2}}$
\end_inset

, with 
\emph on
Y
\emph default

\begin_inset Formula $^{-}$
\end_inset


\begin_inset space ~
\end_inset

= BF
\begin_inset Formula $_{\text{4}}^{-}$
\end_inset

 or PF
\begin_inset Formula $_{\text{6}}^{-}$
\end_inset

, in which pyo ligands bridge Cu(pyz)
\begin_inset Formula $_{\text{2}}$
\end_inset

 planes.
 Then, the quasi-2D non-polymeric compounds [Cu(pyo)
\begin_inset Formula $_{\text{6}}$
\end_inset

]
\emph on
Z
\emph default

\begin_inset Formula $_{\text{2}}$
\end_inset

, where 
\emph on
Z
\emph default

\begin_inset Formula $^{-}$
\end_inset

=BF
\begin_inset Formula $_{\text{4}}^{-}$
\end_inset

, ClO
\begin_inset Formula $_{\text{3}}^{-}$
\end_inset

 or PF
\begin_inset Formula $_{\text{6}}^{-}$
\end_inset

 are examined.
 Materials in which either Ni
\begin_inset Formula $^{\text{2+}}$
\end_inset

 (
\begin_inset Formula $S=1$
\end_inset

) or Ag
\begin_inset Formula $^{\text{2+}}$
\end_inset

 (
\begin_inset Formula $S=\frac{1}{2}$
\end_inset

) form the magnetic species in the quasi-2D planes rather than Cu
\begin_inset Formula $^{\text{2+}}$
\end_inset

 are also investigated.
\end_layout

\begin_layout Section
Parametrising exchange anisotropy
\begin_inset CommandInset label
LatexCommand label
name "sec:exchange-anisotropy"

\end_inset


\end_layout

\begin_layout Standard
In order to assess the extent to which these systems approximate the 2DSLQHA,
 we can quantify their dimensionality by comparing the transition temperature
 
\begin_inset Formula $T_{\mathrm{N}}$
\end_inset

 to the exchange parameter 
\begin_inset Formula $J$
\end_inset

.
 The temperature 
\begin_inset Formula $T_{\mathrm{N}}$
\end_inset

 can be extracted using 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
muSR 
\end_layout

\end_inset

, whilst 
\begin_inset Formula $J$
\end_inset

 can be obtained reliably from pulsed-field magnetisation measurements
\begin_inset CommandInset citation
LatexCommand cite
key "goddard2008-2DHMexchange"

\end_inset

, heat capacity or magnetic susceptibility.
 Further, quantum Monte Carlo simulations allow this to be related to the
 interlayer exchange 
\begin_inset Formula $J_{\perp}$
\end_inset

, and the correlation length of quantum fluctuations, 
\begin_inset Formula $\xi$
\end_inset

.
 
\end_layout

\begin_layout Standard
Mean-field theory predicts a simple relationship for the ratio of the transition
 temperature 
\begin_inset Formula $T_{\mathrm{N}}$
\end_inset

 and the exchange 
\begin_inset Formula $J$
\end_inset

 given by
\begin_inset CommandInset citation
LatexCommand cite
key "blundell-magnetism-in-condensed-matter-book"

\end_inset

 
\begin_inset Formula 
\begin{equation}
\frac{k_{\mathrm{B}}T_{\mathrm{N}}}{J}=\frac{2}{3}zS(S+1),\label{eq:mean-field-kBTNoverJ}
\end{equation}

\end_inset

where 
\begin_inset Formula $k_{\mathrm{B}}$
\end_inset

 is Boltzmann's constant, 
\begin_inset Formula $z$
\end_inset

 is the number of nearest neighbours and 
\begin_inset Formula $S$
\end_inset

 is the spin of the magnetic ions.
 In the pseudocubic [Cu(HF
\begin_inset Formula $_{\text{2}}$
\end_inset

)(pyz)
\begin_inset Formula $_{\text{2}}$
\end_inset

]
\emph on
X
\emph default
 systems, 
\begin_inset Formula $S=\frac{1}{2}$
\end_inset

 and 
\begin_inset Formula $z=6$
\end_inset

, and 
\begin_inset CommandInset ref
LatexCommand formatted
reference "eq:mean-field-kBTNoverJ"

\end_inset

 yields 
\begin_inset Formula $k_{\mathrm{B}}T_{\mathrm{N}}/J=3$
\end_inset

.
 However, the reduced dimensionality increases the prevalence of quantum
 fluctuations, depressing the transition temperature, and in [Cu(HF
\begin_inset Formula $_{\text{2}}$
\end_inset

)(pyz)
\begin_inset Formula $_{\text{2}}$
\end_inset

]BF
\begin_inset Formula $_{\text{4}}$
\end_inset

 we find 
\begin_inset Formula $k_{\mathrm{B}}T_{\mathrm{N}}/J\approx0.25$
\end_inset

, which is indicative of large exchange anisotropy.
\end_layout

\begin_layout Standard
Combining the experimental measures of 
\begin_inset Formula $T_{\mathrm{N}}$
\end_inset

 and 
\begin_inset Formula $J$
\end_inset

 with the results of quantum Monte Carlo (QMC) simulations allows us to
 deduce the exchange anisotropy 
\begin_inset Formula $J_{\perp}/J$
\end_inset

 in the system
\begin_inset CommandInset citation
LatexCommand cite
key "goddard2008-2DHMexchange"

\end_inset

.
 Specifically, QMC simulations
\begin_inset CommandInset citation
LatexCommand cite
key "yasuda2005-2DHA-JvsTN"

\end_inset

 for 2DSLQHA where 
\begin_inset Formula $10^{-3}\leq J_{\perp}/J\leq1$
\end_inset

 are well described by the expression 
\begin_inset Formula 
\begin{equation}
\frac{J_{\perp}}{J}=\mathrm{e}^{b-4\uppi\rho_{\mathrm{s}}/T_{\mathrm{N}}},\label{eq:Yasuda-J-Jprime}
\end{equation}

\end_inset

where 
\begin_inset Formula $\rho_{\mathrm{s}}$
\end_inset

 is the spin stiffness and 
\begin_inset Formula $b$
\end_inset

 is a numerical constant.
 For 
\begin_inset Formula $S=\frac{1}{2}$
\end_inset

, the appropriate parameters are 
\begin_inset Formula $\rho_{\mathrm{s}}/J=0.183$
\end_inset

 and 
\begin_inset Formula $b=2.43$
\end_inset

.
 This expression allows a better estimate of 
\begin_inset Formula $k_{\mathrm{B}}T_{\mathrm{N}}/J$
\end_inset

 in a 3D magnet: evaluating for 
\begin_inset Formula $J_{\perp}/J=1$
\end_inset

 yields 
\begin_inset Formula $k_{\mathrm{B}}T_{\mathrm{N}}/J=0.95$
\end_inset

.
 This is lower than the crude mean-field estimate because mean-field theory
 takes no account of fluctuations.
 Estimates of 
\begin_inset Formula $J$
\end_inset

 for our materials, from pulsed magnetic field studies except where noted,
 along with calculated 
\begin_inset Formula $J_{\perp}/J$
\end_inset

 ratios, are shown in the summary tables throughout these chapters.
\end_layout

\begin_layout Standard
Another method of parametrising the exchange anisotropy is to consider the
 predicted correlation length of two-dimensional correlations in the layers
 at the temperature at which we observe the onset of LRO.
 The larger this length, the better isolated the layers can be supposed
 to be.
 This can be estimated by combining an analytic expression for the correlation
 length in a pure 2DSLQHA
\begin_inset CommandInset citation
LatexCommand cite
key "Hasenfratz1991-2DQHA-correlation-length"

\end_inset

, 
\begin_inset Formula $\xi_{\mathrm{2D}}$
\end_inset

, with quantum Monte Carlo simulations to obtain an expression
\begin_inset CommandInset citation
LatexCommand cite
key "Beard1998-2DSLQHA-large-correlation-lengths,Kastner1998-cuprates-review"

\end_inset

 appropriate for 
\begin_inset Formula $1\leq\xi_{\mathrm{2D}}/a\leq350,000$
\end_inset

, 
\begin_inset Formula 
\begin{equation}
\frac{\xi_{\mathrm{2D}}}{a}=0.498\mathrm{e}^{1.131J/k_{\mathrm{B}}T}\left[1-0.44\left(\frac{k_{\mathrm{B}}T}{J}\right)+\mathcal{O}\left(\frac{k_{\mathrm{B}}T}{J}\right)^{2}\right],\label{eq:Beard-correlation-length}
\end{equation}

\end_inset

where 
\begin_inset Formula $a$
\end_inset

 is the square lattice constant, and 
\begin_inset Formula $T$
\end_inset

 is the temperature.
 This formula yields 
\begin_inset Formula $\xi_{\mathrm{2D}}(T_{\mathrm{N}})\approx0.5a$
\end_inset

 for the mean-field model (
\begin_inset Formula $k_{\mathrm{B}}T_{\mathrm{N}}/J=3$
\end_inset

), and 
\begin_inset Formula $\xi_{\mathrm{2D}}(T_{\mathrm{N}})\approx a$
\end_inset

 for 
\begin_inset Formula $k_{\mathrm{B}}T_{\mathrm{N}}/J=0.95$
\end_inset

 from quantum Monte Carlo simulations [i.e.
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand formatted
reference "eq:Yasuda-J-Jprime"

\end_inset

 with 
\begin_inset Formula $J_{\perp}/J=1$
\end_inset

].
 By comparison, in [Cu(HF
\begin_inset Formula $_{\text{2}}$
\end_inset

)(pyz)
\begin_inset Formula $_{\text{2}}$
\end_inset

]BF
\begin_inset Formula $_{\text{4}}$
\end_inset

 
\begin_inset CommandInset ref
LatexCommand formatted
reference "eq:Beard-correlation-length"

\end_inset

 gives 
\begin_inset Formula $\xi_{\mathrm{2D}}(T_{\mathrm{N}})\approx50a$
\end_inset

, showing a dramatic increase in the size of correlated regions which build
 up in the quasi-2D layers before the onset of LRO.
\end_layout

\end_body
\end_document
