磁性激发研究：关联电子系统的中子散射探索

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"Lucy Helme的博士论文‘Neutron Scattering Studies of Correlated Electron Systems’主要探讨了通过中子散射研究相关电子系统的主题，特别是关注于系统中的磁激发。研究涉及三种不同的竞争相互作用，包括电荷序、磁序和晶格自由度。实验数据通过与理论模型的对比进行分析，以深入了解材料的物理特性。" 在这篇论文中，作者首先详细介绍了中子散射技术在研究相关电子系统中的应用，这是一种非破坏性的探测手段，能够揭示物质内部的微观结构和动力学行为。中子因其独特的性质，可以与物质内的原子核和磁矩相互作用，因此特别适用于探索材料的磁性和晶格结构。在第一章中，论文可能涵盖了中子散射的基本原理和在凝聚态物理中的重要性，讨论了如何利用这一技术来研究电子的相关性以及这些相关性如何影响材料的宏观性质。第二章可能概述了所研究的三个相关电子系统的基本特性和背景，每个系统都展示了不同的竞争相互作用。这些系统的选择是为了提供一个广泛的物理现象样本，包括但不限于超导、磁性相变和电荷密度波等。第三章是论文的核心部分，集中研究了稀土氧化物PrO2中的Jahn-Teller效应。Jahn-Teller效应是一种由于电子关联引起的结构畸变现象，此处通过无弹性中子散射研究了晶体场跃迁。作者比较了实验数据与点电荷模型的预测，发现晶体场能级随温度的变化可归因于该效应。后续章节可能进一步分析了实验结果，讨论了磁激发的特征、动力学行为以及它们如何与电荷和晶格有序相互作用。此外，论文可能还探讨了这些发现对理解高温超导性、磁性材料的磁有序和量子临界现象等基础物理问题的贡献。通过这样的研究，作者不仅深化了对相关电子系统基本物理现象的理解，也为开发新型功能材料和优化现有技术提供了理论支持。这篇论文对于从事凝聚态物理、材料科学和中子散射技术研究的学者具有重要的参考价值。

Chapter 1. Introduction to Correlated Electron Systems 6

By assuming that the spin parts of the wavefunction are normalized we can write the

diﬀerence in energies between the singlet and triplet state as

J = E

− E

∗

HΨ

−

∗

HΨ

(1.5)

= −2

∗

)ψ

∗

) H ψ

)ψ

) dr

and this deﬁnes the exchange constant (or exchange integral), J. It is straightforward

to show that we can write an eﬀective spin Hamiltonian [11]

H = (E

− E

· S

= JS

· S

, (1.6)

so J is a measure of the interaction between the two spins. If J < 0, E

< E

and

the triplet state S = 1 is favoured; if J > 0, E

> E

and the singlet state S = 0 is

favoured.

Generalizing these ideas to a many-body system is a complicated problem, but

it was recognised early on by Heisenberg that equation 1.6 could be generalized to

provide a go od description of the interactions between magnetic moments on all

neighbouring ions in a compound. The result is known as the Heisenberg Hamiltonian:

H =

i,j

· S

, (1.7)

where the sum is over all pairs of spins i, j (each pair is counted only once), and J

is the exchange constant describing the interaction between i and j. In this notation

J > 0 favours antiparallel alignment of moments (antiferromagnetic alignment) and

J < 0 favours parallel alignment of moments (ferromagnetic alignment).

When electrons on neighbouring ions interact directly, with no intermediary, the

exchange interaction is known as direct exchange. The strength of this interaction

is strongly dependent on the overlap of the electron shells of the two ions. There

is often insuﬃcient overlap for a strong direct exchange interaction, either because

the wavefunctions lie very close to the nucleus (e.g. in rare earth 4f electrons),

or because the ions are too far apart (e.g. in transition metal oxides where

the magnetic ions are separated by oxygen). In these cases an indirect exchange

interaction may b e important. In ionic solids the most important interaction is known

as superexchange, an indirect exchange interaction between two non-neighbouring

magnetic ions which is mediated by a non-magnetic ion between them. Figure 1.3

represents the superexchange interaction in a magnetic oxide between two magnetic

ions M mediated by an oxygen ion. The M ions are represented by one type of

d orbital, and vertical arrows represent the spins of electrons distributed over the

bond in the antiferromagnetic and ferromagnetic interactions. The antiferromagnetic

arrangement is more favourable to electron hopping, and therefore allows the electrons

to become delocalized over the structure, lowering the kinetic energy. Superexchange

therefore generally favours an antiferromagnetic alignment, with J > 0.

Ferromagnetic superexchange is possible, see ref. [11], but the interaction is weaker and less

common.

Chapter 1. Introduction to Correlated Electron Systems 8

Figure 1.5: The eﬀect of an external applied ﬁeld on an antiferromagnet. (a)

No applied ﬁeld. (b) Field B perpendicular to ordered moments.

parallel to ordered moments (Spin-ﬂop).

In general an antiferromagnet can usually be considered as two interlocking

sublattices of spins: one of ‘up’ spins and one of ‘down’ spins. In zero ﬁeld the

magnetic moments of the two sublattices cancel so the net magnetization is zero. To

make a measurement of the magnetization a small magnetic ﬁeld is applied along

a particular direction. If the ﬁeld is applied perpendicular to the ordered moment

direction (see ﬁg. 1.5b) then b oth ‘up’ and ‘down’ spins tilt slightly towards the

ﬁeld and a component of magnetization is measured along the ﬁeld direction. If the

small ﬁeld is applied parallel to the ordered moment direction of one sublattice, and

antiparallel to the other (see ﬁg. 1.5c), the net eﬀect on the two sublattices is zero

and no magnetization component is measured. This gives rise to large anisotropies

in magnetization measurements of antiferromagnets in the ordered phase.

If a strong magnetic ﬁeld is applied perpendicular to the moment direction the

spins tilt in the same way, see ﬁg. 1.5b. As the ﬁeld increases φ decreases until

the spins are ﬁnally parallel. However, if a strong magnetic ﬁeld is applied parallel

to the ordered moments the eﬀect is more interesting. As the ﬁeld is increased the

spins remain as shown in ﬁg. 1.5c until at a critical ﬁeld they suddenly ﬂip into the

conﬁguration shown in ﬁg. 1.5d. This is known as a spin-ﬂop transition.

1.1.3 Charge Order

In a doped material, the replacement of one ion by another ion of diﬀerent valence

introduces excess holes (or electrons) into the material. At high temperatures the

excess holes are generally randomly distributed, but on cooling they may form a

periodic arrangement due to the repulsive Coulomb interaction between them. This

ordered state of localized charge is known as charge order (CO).

Charge ordering occurs in slightly diﬀerent ways in diﬀerent layered materials,

but the pattern of ordering is highly dependent on doping level. As an example we

Chapter 1. Introduction to Correlated Electron Systems 10

ordering pattern. This is demonstrated in the lower row of ﬁg. 1.6 which shows

antiferromagnetic order on a square lattice with three diﬀerent charge orders (red

dashed boxes show the magnetic unit cells). Although these diagrams illustrate charge

and magnetic order that is commensurate with the crystal lattice, incommensurate

charge order is also observed. The incommensurability can be due to charge

density waves, wave-like modulations in the charge which can have a periodicity

incommensurate with the lattice, or due to discommensurations in the charge order.

1.1.4 Orbital Order and Lattice Eﬀects

Magnetic ions in strongly correlated electron systems are often transition metals

or rare earths with valence electrons in the 3d or 4f shells. These orbitals are not

spherically symmetric so the symmetry of the local environment of the magnetic ion

can determine which orbitals are favoured as the ground-state, and can aﬀect the

orientation of the orbitals within the crystal structure.

As an example, ﬁgure 1.7 shows a magnetic 3d ion in an octahedral environment

of surrounding oxygen ions. We consider the two d-orbitals with four-fold angular

dependence in the xy plane: d

and d

, shown in ﬁg. 1.7(b, c). It is clear that

the d

orbital has a much lower overlap with the p orbitals of the neighbouring

oxygen ions than the d

orbital, and hence will have lower energy. This leads to

a splitting between the levels known as t

(which include d

, d

and d

), and the

levels (d

and d

Eﬀects like these can lead to the orbitals aligning into a periodic pattern in the

crystal lattice in order to minimize the energy of the conﬁguration: this is known

as orbital ordering. One example is the manganate compound La

1−x

1+x

MnO

which is isostructural to the high-T

cuprates discussed earlier. The phase diagram

of La

1−x

1+x

MnO

is shown in ﬁg. 1.8a [13]. At half doping (x = 0.5) the

compound exhibits charge, magnetic and orbital ordering within the layers, as shown

Figure 1.7: (a) A magnetic ion surrounded by oxygen ions forming an

octahedron. (b) d

orbital of the magnetic ion surrounded by

the p orbitals of the oxygens. The d

orbital is lower in energy

than the d

orbital (c). (d) The resulting splitting in energy

levels.

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磁性激发研究：关联电子系统的中子散射探索

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