1 Introduction

Recent developments in photonics have shed a new light on the vast, still unexplored nature of magnetism. For example, magneto-optical effects are widely employed to visualize the spatial distributions of magnetic domains, which underpin their well known applications in magnetic recording. Femtosecond laser pulses have recently revealed the dynamics of magnetism on a timescale even shorter than that of a single precession cycle. In addition, it has been demonstrated that light can even access a single spin, the smallest unit of magnetism; this is expected to work as a key element in quantum information and computation technologies. With the aid of optics, magnetism helps us to navigate our life in this era of modern informatics, and potentially of future quantum informatics - as magnetic compasses led the world to the Grand Navigations.

Among various magnets, we focus on two-sublattice antiferromagnets, a class of magnetic materials. The spins in their two types of sublattices are aligned opposite directions; they exhibit net no magnetic moment. This results in many attracting features. To name a few, antiferromagnetic domain boundaries are usually very thin, the typical domain size to sustain the magnetic order can be even a nanometer scale, and the spin dynamics in antiferromagnets are fast and have long coherence time. However, at the same time, the absence of magnetic moment has left them untamed, because their manipulation through an external field is quite formidable.

In this context, we propose that optical techniques pave a new way to manipulate antiferromagnetism. We demonstrate optical methods to control magnetic orders in MnF2 crystals and to manipulate magnetization dynamics in NiO; both these materials are representative antiferromagnets. The key concepts for their realization are to fully employ space-time symmetry of the crystal and to tune the polarization state of the control lasers precisely.

2 Space-Time Symmetry of Optical Susceptibilities

This Chapter overviews symmetric relations of electromagnetics and crystals, mainly as a reference for our experimental and theoretical studies presented in the following Chapters. Neumann's principle suggests that every macroscopic property of a crystal possesses the symmetry described by the point group of the crystal. Another important rule is Onsager's reciprocity relations, which is a direct consequence of medium symmetry under time reversal. This explains Faraday rotation, where light propagating in a nonmagnetic medium under an external magnetic field experiences rotation of its polarization azimuth.

However, when the time reversal does not act as a symmetry operation of the crystal (in other words, if it is not gray), careful treatments of Onsager's relations are required. We show two approaches (1) employing a combined operation of a spacial operation and the time reversal (2) an explicit treatment of time-odd variables. In particular, the former is useful when the magnetic point group of the crystal is black-and-white. In these cases, unique magneto-optical effects, such as dichroism between two orthogonal linear polarizations under magnetic field, are allowed [1]. Note that MnF2 is an example of such materials.

3 Physical Properties of MnF2

This Chapter describes a comprehensive survey of the structural, magnetic and optical properties ofMnF2. Main features in the optical spectrum of MnF2 in the visible region are dominated by d-d transitions inMn2+ ions. The ground states of Mn2+ ions take high-spin d5 configurations; this makes the transitions spin-forbidden. Weak magnetic dipole transitions that directly create single ionic excitations are allowed by virtue of the spin-orbit coupling. In addition to these weak magnetic dipole transitions, much stronger electric dipole transitions were observed. This cannot be explained by single-ionic transitions, because electric dipole processes are parityforbidden. In this Chapter, a mechanism proposed by Tanabe et al. that considers pair creations of excitons and magnons are reviewed. Optical absorption bands corresponding to this process are called magnon sidebands.

4 Selection Rules for Magneto-Optical Effects in MnF2

When an external magnetic field is applied along the c axis of a MnF2 crystal, it is known that dichroism between two linear polarizations is induced, as discussed in Chapter 2. This dichroism is odd with respect to the external magnetic field and the antiferromagnetism vector; it is called magneto-linear dichroism (MLD). Two types of antiferromagnetic domains in MnF2 [Fig. 1(a)] can be distinguished by this MLD. At the same time, magneto-circular dichroism (MCD) with a considerable strength was observed. A notifying feature of these MLD and MCD is that they have specifically different spectral shapes around the 6A1g ! (4A1g;4 Eg) absorption line of Mn2+. As introduced in Chapter 1, one of our goals is to observe and manipulate antiferromagnetic domains in MnF2. Although these magneto optical phenomena are particularly important for this purpose, lack of systematic measurements of the dichroism prevailed comprehensive understandings of their origins.

We measured systematically the magnetic field dependence of theseMLDandMCDof the 6A1g ! (4A1g;4 Eg) lines [Fig. 1(b)] at 6 K. They are separated into two main features [Fig. 1(c)]: both MLD andMCDthat originates from the magnetic dipole transition (Mπ), and the presence of a strong MLD and the absence of MCD at the edges of the magnon sideband (E(high) 2σ and E(low) 2σ ). We clarified that additional conditions restrict the dichroism when the cubic crystal field is much stronger than the spin-orbit coupling. This explains the reason why only significant MLD was observed for the magnon sideband. These experimental observations matched well with the calculated spectra of MLD and MCD considering dispersions of magnons and excitons [Figs. 1(e), (f), and (g)].

5 Optical Manipulation of Antiferromagnetic Domains

Just below the N´eel temperature (67 K) of MnF2, this crystal shows MLD, whose sign depends on the sign of the antiferromagnetic order parameter [Fig. 2(a)]. We observed spatial distributions of domains by means of this MLD. This MLD can also introduce an imbalance between the forming energies of two different antiferromagnetic domains when the crystal is optically pumped by an intense linearly polarized laser under a magnetic field. We employed this imbalance to control boundaries of antiferromagnetic domains in MnF2, as shown in Fig. 2(b). The position of the boundary depended on the polarization azimuth of the pump beam, which clearly indicates that the controlling mechanism is polarization-dependent optical pumping [Fig. 2(c)]. Wide varieties of applications are expected, such as formation of artificially frustrated antiferromagnets via optical patterning.

6 Dynamics of Magnetism Excited by Optical Pulses

Ultrafast control of magnetization via impulsive stimulated Raman scattering (ISRS) has recently attracted considerable interest. In Chapter 6, we describe a phenomenological theory for excitation of magnons in NiO by ISRS. This is formulated by the modulation of the optical susceptibility tensor under presence of magnons. Since NiO has a it gray magnetic point group, optical susceptibility should be even functions of the antiferromagnetism vector; this is important to determine the form of its susceptibility tensor. As a result, both circular and linear polarizations are found to contribute magnon excitation. There exist clear selection rules between the polarization states, and the modes and phases of the excited magnon modes. Methods to observe the induced magnons are also discussed.

7. Optomagnetism along a Threefold Axis

When the crystal has a uniaxial optical axis, the selection rules for ISRS be simplified by means of conservation laws of angular momentum [Fig. 3(a)]. Chapter 7 describes this simplified polarization selection rule for a light induced magnetization dynamics in NiO. Here, the role of discrete rotational symmetry of the crystal is clarified. When a fundamental light beam propagates along a continuous rotational axis, magnetization can only be induced along that axis [Fig. 3(b)], while generation of magnetization in the perpendicular plane is prohibited. One method to control such in-plane magnetization is to employ a linearly polarized laser pulse propagating along a threefold rotational axis of a crystal. With such geometry, the rotational analogue of the umklapp process allows a change of the angular momentum of the light field and induced magnetization by pm 3hbar [2]. This activates the otherwise forbidden path to access in-plane magnetization by ISRS and subsequent difference-frequency generation (DFG) [Figs. 3(c) and (d)]. As a demonstration, we observed radiation (~ 1 THz) from the magnetic oscillations induced by linearly polarized laser pulses along a [111] axis of a single crystal of antiferromagnetic NiO with micro multi-domain structure [Figs. 3(e), (f), and (g)]. Clear polarization dependence on the incident and the radiated photons was obtained [Fig. 3(h)], which agreed excellently with the theoretical prediction.

8. Envelope Helicity and the Vectorial Manipulation of Magnetization

In this Chapter, we discuss a general approach for controlling low-energy rotational excitations by ISRS [4]. ISRS is free from thermalization, because the scattered light carries away the excess energy [Fig. 4(a)]. A femtosecond pulse is employed because it has a broadband spectrum that covers the needed frequencies in a single pulse [Fig. 4(b)]. However, a design for a continuous spectrum in nonlinear process usually imposes a complex analysis of the electromagnetic and material parameters. This is because ISRS is intrinsically a two-photon process, and one photon can contribute to multiple scattering processes. Therefore, independent optimization of polarization states of each frequency component is no more a good strategy. Here, we theoretically show that the temporal trajectory of the envelope of the electromagnetic-field vector [Fig. 4(c)] enables us to intuitively interpret ISRS experiments and optimize the controlling laser pulse. To describe the envelope of a light pulse, instantaneous Stokes parameters (ISP) are defined under slowly varying envelope approximation. Envelope helicity is introduced to the Fourier component of the ISPs, which determines the change of angular momentum via ISRS. As an application, we employ this concept to interpret the vectorial manipulation of magnetization in an antiferromagnetic NiO [5].

This opens new routes to design experiments to manipulate low-energy rotational excitations in ultrafast timescales; not by simple heating and demagnetization effect, but rather by much more interesting phenomena based on coherent nonlinear optical transitions and material symmetries.

9. Concluding remarks

In summary, we demonstrated novel methods to manipulate antiferromagnets by employing space-time symmetry of the crystals and tuning azimuthal angles of linearly polarized lasers. This study covers interactions between light and magnetism in specific materials, i.~e., MnF_2 and NiO. It is worth unifying these two cases, under a single framework. Roughly speaking, the key concept unifying our results is the similarity of antiferromagnets and linearly polarized light in that angular momentum is canceled between their two degenerate constituents. It is widely believed that circularly polarized light is favorable for optical control of magnetism because such light carries spin angular momentum. On the other hand, a linearly polarized light contains same populations of left and right circularly polarized components, and their spin angular momenta are canceled. This is the reason why its abilities to control magnetism have been usually put aside. However, we showed that the relative phase between the two circular components, i.~e., the polarization azimuth, plays a key role to manipulate antiferromagnets through selection rules based on the space-time symmetry of the crystal.

Fig. 1 (a) Crystal and spin structure of two types of antiferromagnetic domains in MnF_2. (b) Absorption coefficient of MnF_2 for light propagating along its c axis. Experimental spectra of (c) MLD between x and y polarizations and (d) MCD. Calculated spectra of (e) optical absorption (f) MLD, and (g) MCD.

Fig. 2 (a) MLD spectra of a single domain MnF_2 around its N'eel temperature. Inset shows the temperature dependence of MLD for an optical wavelength of 396.25 nm. This wavelength was employed for both observation and manipulation of antiferromagnetic domains. (b) Line profile of the domain distribution. Different curves show the distributions under pumping by lasers with different azimuthal angles. (c) Change of the domain population at the pump position as a function of the pump laser azimuth.

Fig. 3 (a) Changes in angular momenta of the electromagnetic fields by creation or annihilation of photons propagating along the z-axis. The lower panels show schematics of the changes of the angular momentum in collinear scattering processes of photons to a mode with (b) the same helicity and (c) the opposite, and (d) a collinear DFG process allowed along a threefold axis involving two photons with opposite helicity. (e) Schematics of the experimental setup, (f) temporal waveform of the terahertz waves radiated from NiO and (g) their Fourier transformed spectrum. (g) Ellipticity and azimuth θ of the radiated waves as functions of the pump azimuth ψ, together with a theoretical line θ = -2ψ [3].

Figure 4: (a) Schematics of ISRS. A photon ( E(ω1)) is scattered to a different mode( E*(ω2)). The electronic system is transfered from the initial state (|i>) to the final state (|f>) via virtual intermediate states (|m>). (b)Spectrum of a femtosecond pulse that covers pairs of frequencies needed for ISRS. (c) An example of a light pulse that selects angular momentum transfer via ISRS. Top panel plots electric field, while bottom panel shows two components (S1 and S2) of the ISPs representing polarization azimuth.

極超短パルスレーザー技術の進歩によって、光による能動的な物質制御とそのダイナミクスの観測が可能となった。特に物質の磁性を光で制御する研究が活発化している。強磁性体の光制御は磁気記録などの応用上の観点からも興味がもたれ、光パルスによる高速磁化制御について多くの研究が行われている。一方近年、多彩なスピン構造をもつ物質が開拓され、磁性構造に起因する特異な物性が見いだされている。それらを光で制御することはあらたな光機能発現や物性制御という観点で興味が深い。本研究はその第一歩として反強磁性体に着目し、光による制御の原理と方法を調べたものである。反強磁性は、高温超伝導との関わりなど、強相関電子系の物性にとって重要な役割を果たす。また、強い交換磁場により、反磁性共鳴がテラヘルツ領域に現れ、マグノンの励起によってコヒーレントな磁気放射が生じるなど興味深い現象が見いだされている。しかし、反強磁性体は2つの副格子がもつ磁気モーメントがキャンセルしていることから、外場による制御が難しいという課題がある。本研究は、反強磁性体を対象とし、線形および非線形の光学応答について磁性結晶の群論的考察をもとに、光の偏光自由度を活用し、光による反強磁性体の制御と反強磁性体による光の制御について探索した。

本論文は英文によって執筆され、以下の9章からなる。以下に各章の内容を要約する。

第1章では、本論文の序論として、本研究の背景として、光と磁化の相互作用についてこれまでの研究について概説し、本研究目的と本論文の構成について述べている。

第2章では本研究の基礎となる、結晶の対称性と電磁応答の関係について概説している。磁気対称群を用いて、磁気光学効果・光磁気効果とOnsagerの定理について述べている。また、結晶の離散回転対称性と光の角運動量保存の関係について議論している。

第3章では、本研究の主題の一つである、MnF2結晶についてその光物性について概説している。特に、本論文で研究を行ったd-d電子遷移に起因する光学吸収について、弱い磁気双極子遷移と電気双極子許容のマグノンと励起子の協働励起について紹介している。

第4章では、MnF2結晶において磁気光学効果の実験を行い、そのスペクトル解析について述べている。まず、MnF2がその白黒磁気点群対称性に起因して、磁気直線二色性を示すことに着目し、円二色性との競合やスペクトル構造の特異性について調べている。印加磁場を系統的に変化させた実験を行い、直線二色性と円二色性を分離して観測することに成功した。その結果、磁気双極子遷移共鳴では、円・直線が共存し、6A1g→4A1g,4Egなど特定のd-d遷移に伴う励起子のマグノンサイドバンドでは直線二色性のみが観測されることを見いだしている。対称性を考慮した上で、マグノンと励起子の分散を取り入れた理論計算を行い、実験結果が再現されることを示している。これらの考察から、この電気双極子遷移許容の励起子マグノンサイドバンドはより高い対称性での電子構造によって規定されるものであることを示している。

第5章では、前章で議論した磁気直線二色性が結晶のドメイン観察に利用できることに着目し、光照射によるドメンイン制御について述べている。光励起下で、ネール点をまたいで冷却した際に光励起の効果によって、ドメイン界面が変化することが見いだされ、ドメイン制御法として利用できることを示している。前章で議論した磁気直線二色性に呼応した励起偏光依存性が観測され、本結晶の磁気的な対称性を反映していることが確認されている。

第6章では、瞬時的誘導ラマン散乱(ISRS Impulsive Stimulated Raman Scattering)によって光パルスによる反強磁性共鳴モードのマグノン励起について議論を行っている。NiO結晶を例として、磁気点群による考察を行い、マグノン励起の条件について考察している。反強磁性が単一ドメインの場合、励起パルスは直線偏光でも円偏光でもマグノンを励起できること、偏光によってマグノンモードや位相を選択励起できること導いている。またマグノンの観測方法についても述べている。

第7章では、結晶が3回回転対称軸をもつ場合、それにそって進む光ではISRSによって赤外活性なモードを励起できることを示している。この応用例としてマルチドメインのNiO結晶において[111]軸に沿って光を照射すると、磁気放射可能なマグノンを励起できることを示している。対称性の議論により、観測された磁気放射によるテラヘルツ電磁波の偏光特性が説明できることを示している。テラヘルツ波の波長に比べ十分小さなドメインがランダムに生じることにより、結晶が実効的に3回回転対称性を獲得したことによるものであることを示している。

前章では、偏光が決まったパルスによる励起を議論していた。この場合、どの様な偏光状態を用意しても、磁気振動は直線偏光となる。第8章では、光パルスの包絡線のねじれを用いることで、ISRSによってより自由に磁気振動の制御が可能となることを示している。NiOの例のように励起するマグノンの周波数が孤立している場合には、偏光角の異なる直線偏光のダブルパルスを用いることで、磁気励起をベクトル的に制御できる。ここではこの議論をさらに一般化し、周波数帯域、回転方向について選択的な回転励起の制御を行うことための光パルス波形の設計指針について議論している。

第9章では本研究のまとめを行い、今後の展望について述べている。

以上のように本研究は、反強磁性体結晶を舞台として、磁性結晶の対称性について群論的な考察をもとに、線形および非線形の光学過程について議論を行い、特異な磁気光学効果について実験理論両面から検討を行った。MnF2では、磁気光学効果としてd-d電子遷移に伴い、直線二色性と円二色性があらわれることとそれが分離可能であることを示し、結晶の対称性の議論からその起源を明らかにした。さらに光励起によって反強磁性ドメインが制御できることを示した。NiO結晶では、誘導ラマン過程を用いたパルス光による磁気的励起の制御について詳細に検討を行った。結晶の対称性を考慮して光の偏光の自由度を巧みに制御することで、磁気励起を自在に制御できることを示した。またこれを一般化し、より一般的にベクトル的に包絡関数が制御された光パルスを用いることで、誘導ラマン過程を介してより自由度の高い回転励起の方法を提案した。

これらの研究は、光と磁気の関わりについて、理論的および実験的検討により、新たな側面を見いだしたものであり、本研究の成果は今後の物理工学の発展に大きく寄与することが期待される。

よって、本論文は博士(工学)の学位論文として合格と認める。