Nonlinear and Ultrafast Optical Investigations of Correlated Materials Citation Chu, Hao (2017) Nonlinear and Ultrafast Optical Investigations of Correlated Materials. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/Z9VD6WHV. https://resolver.caltech.edu/CaltechTHESIS:06092017-141136995 Abstract This thesis comprises studies of 3 d -5 d transition metal oxides with various degrees of electronic correlation using nonlinear harmonic generation rotational anisotropy as well as time-resolved optical reflectivity methods. Specifically, we explored photo-induced phase transition in Ca 2 RuO 4 and Sr 2 IrO 4 , discovered novel electronic phases in doped Sr 2 IrO 4 and Sr 3 Ir 2 O 7 , and investigated different types of antiferromagnetic orders in transition metal trichalcogenides MPX 3 . Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: Correlated Materials, Nonlinear Optics, Second Harmonic Generation, Ultrafast, Photo-induced Phase Transition, Cuprate, Iridate, 2D Materials, Trichalcogenide Degree Grantor: California Institute of Technology Division: Engineering and Applied Science Major Option: Applied Physics Thesis Availability: Public (worldwide access) Research Advisor(s): Hsieh, David Group: Institute for Quantum Information and Matter Thesis Committee: Hsieh, David (chair) Eisenstein, James P. Yeh, Nai-Chang Refael, Gil Defense Date: 11 May 2017 Record Number: CaltechTHESIS:06092017-141136995 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:06092017-141136995 DOI: 10.7907/Z9VD6WHV Related URLs: URL URL Type Description http://dx.doi.org/10.1038/nmat4836 Related Document UNSPECIFIED http://dx.doi.org/0.1038/nphys3517 Related Document UNSPECIFIED https://doi.org/10.1103/PhysRevLett.114.096404 Related Document UNSPECIFIED http://doi.org/10.1063/1.4891417 Related Document UNSPECIFIED Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 10334 Collection: CaltechTHESIS Deposited By: Hao Chu Deposited On: 13 Jun 2017 20:56 Last Modified: 02 Jun 2020 21:47 Thesis Files Preview PDF - Final Version See Usage Policy. 7MB Repository Staff Only: item control page

Nonlinear and Ultrafast Optical Investigations of Correlated Materials Thesis by Hao Chu In Partial Fulfillment of the Requirements for the degree of Doctor of Philosophy in Applied Physics California Institute of Technology Pasadena, California 2017 Defended May 11th 2017 ii © 2017 Hao Chu All Rights Reserved iii Acknowledgements Looking back at the journey I have taken during graduate school, there are many people whom I would like to thank for having shaped, inspired or accompanied me. First and foremost I would like to thank my advisor David Hsieh, a man of immense vision and ideas. Never stopping, David pushes the realm of our scientific inquiry to newer and newer frontiers. For this, and for all other guidance I have received from him, I owe him deep gratitude. I would also like to thank colleague Darius Torchinsky and Wes Erbsen, Daisy Raymondson, Mike Walls from KMLabs for teaching me ultrafast optics from scratch. Without their guidance, many of the experiments we did would be difficult if not impossible to start. Alberto de la Torre taught me generously on the scientific part. His knowledge of theory and experiment on correlated materials is impressive. I have never failed to deepen my understanding of correlated physics every time I discuss with him. I have also learnt a lot about 2D materials from Alon Ron, another colleague of mine. I would also like to thank colleagues Tejas Deshpande, Liuyan Zhao, John Harter, Junyi Shan, Honglie Ning, with whom I interacted everyday and collaborated on a few different projects. From all these people I learnt a great deal about condensed matter physics, and more broadly how to be a scientist. On this subject of growing to be a scientist, I am also grateful to Prof. Brent Fultz, Prof. Sandra Troian and Prof. Nai-Chang Yeh for bringing me to Caltech and introducing me to the world of condensed matter physics. Frank Rice, for whom I was the teaching assistant for quite a few terms, showed me a great example of an experimental physicist and the kind of teacher I would like to be. I would also like to thank the many precious friends I made at Caltech, who shared my passions and always supported me. I want to thank Ding Ding and Bilin Zhuang, whom I came to know in my first year and with whom I shared many late nights working on problem sets and discussing science and life. Many of my thanksgiving eves were spent with them and to me they are like my family. I would also like to thank Chan U Lei, who was my apartment-mate for a long time and who shared the same passions and encouraged me in times of frustration. Yiyang Liu, Daodi Lu, Lei Fu are the people with whom I had many intellectual discussions and learnt a lot about interesting things outside physics: history, politics, etc. I would also like to thank Yejun Feng and Yishu Wang from Rosenbaum’s group for teaching me Xray scattering and sharing scientific ideas with me. Much more than these people, iv I would also like to thank Wei-Hsun Lin, Yingrui Chang, Xilin Lv, Junle Jiang, Xuan Zhang, Sijia Dong, Fan Liu, Jingbo Huang, Xiaoying Tan, Menghua Zhuang, and many others whom I cannot name all at once here. They made up the colorful spectrum of my life through graduate school and I always relish the memories I had with them. I would also like to thank my undergraduate friend Leizhi Sun, a physicist too, who is always in touch to share his latest aspirations. And also to my friends and teachers from Jin Ling Middle School, I owe them deep thanks for nurturing a free and independent soul in me and preparing me for all the adventures I would embark on in my life. These are the friendships, no matter how distant in time and space, I will always carry in my heart. To those who brought me onto the journey of physics, I owe them deep thanks and appreciations. These include Fritjof Capra with his the Tao of Physics, which opened the mind-boggling door of physics to me. My high school friend Leiyu Xie inspired me to take interest in arts and humanities and offered great help to me when I was applying for college. My undergraduate advisor Lyman Page is a physicist whose devotion to research and students has deeply moved me and inspired me to become a physicist and a teacher like him from day one. I have also been tremendously inspired, at different stages of my life, by undergraduate friend Jingyuan Zhang and primary school classmate Mingming Yang, who are physicists themselves too, to pursue my passion to the fullest and to the best of my capability. Last but not least, I would also like to thank Mengran Yang, who encouraged me and lightened me up when I was frustrated. Thanks to her, I am upbeat and always ready for new challenges in research and in life. To my parents, Jin Chu and Qian Gu, who cultivated a scientific curiosity and gift in me since my childhood, I owe them my deepest gratitude. I thank them for giving me a carefree environment to explore and pursue anything that interests me. I still remember that one evening during my 5th grade in elementary school, my dad mentioned that Nanjing University, a leading research university in my hometown, developed state-of-art laboratories for condensed matter physics research. Those magical words of condensed matter physics had since taken root in me and led me to where I am this day. For these and for the unyielding love and support they have given to me, I thank my parents with all my heart. v Abstract This thesis comprises studies of 3d-5d transition metal oxides with various degrees of electronic correlation using nonlinear harmonic generation rotational anisotropy as well as time-resolved optical reflectivity methods. Specifically, we explored photo-induced phase transition in Ca2 RuO4 and Sr2 IrO4 , discovered novel electronic phases in doped Sr2 IrO4 and Sr3 Ir2 O7 , and investigated different types of antiferromagnetic orders in transition metal trichalcogenides MPX3 . Ca2 RuO4 is an antiferromagnetic insulator in which the delicate balance between microscopic competing interactions can be tipped over by a gentle external influence. In particular, heating Ca2 RuO4 above 350 K induces a first-oder insulatorto-metal transition together with a structural phase transition. Using time-resolved pump probe optical reflectivity, we explored the possibility of photo-induced insulatorto-metal transition in Ca2 RuO4 . While transient optical reflectivity shows a metallike response above a critical pump fluence, a metallic signature is absent in the transient terahertz reflectivity. Our results suggest a finite spectral weight being transferred to the Mott gap, although a metastable photo-induced metallic phase is absent. This is consistent with the lack of photo-induced structural phase transition which would otherwise stabilize the metallic phase. The layered perovskite iridate Srn+1 Irn O3n+1 is iso-structural to the cuprate high temperature superconductors and realize an interesting analogue to the cuprate family. Electron-doped Sr2 IrO4 has recently been shown to exhibit the pseudogap phase as well as possibly d-wave superconductivity. We first used time-resolved optical reflectivity to investigate the temperature evolution of the electronic structure in electron- and hole-doped Sr2 IrO4 . Then, using second harmonic generation rotational anisotropy, we refined the crystallographic structure of Sr2 IrO4 and identified an inversion breaking hidden order phase in both the parent and hole-doped Sr2 IrO4 at low temperatures. We find that the loop-current model, which was proposed to explain the pseudogap phase of cuprates, well explains this microscopic hidden order in Sr2 IrO4 . The bilayer Sr3 Ir2 O7 is a weak spin-orbit coupled antiferromagnetic insulator. Contrary to its single layer cousin, no interesting electronic phases have been reported in electron or hole-doped Sr3 Ir2 O7 . Using time-resolved optical reflectivity, we found evidence for a closing of the insulating gap in Sr3 Ir2 O7 above TN . In addition, in vi electron-doped metallic Sr3 Ir2 O7 , we discovered a charge-density-wave-like instability that onsets at a previously reported structural phase transition temperature. However, this electronic instability is of a peculiar nature in that it might be shortranged or fluctuating in time, which is similar to the fluctuating charge density wave recently observed in underdoped cuprate. Transition metal trichalcogenides MPX3 , where M = {Mn, Fe, Ni}, X = {S, Se}, are layered van der Waals materials that undergo magnetic transitions at low temperatures. Different types of magnetic order, including the Néel order and the antiferromagnetic stripe order, realize depending on the ratios between J1 , J2 and J3 . We demonstrate that second harmonic generation (SHG) is sensitive to the microscopic magnetic structures. In particular, the SHG process couples to Néel order parameter in MnPS3 but not the antiferromagnetic stripe order parameter in FePS3 . Our experiment proves that SHG can be used as a sensitive and non-invasive tools for identifying the different magnetic structures in 2D materials. vii Published Content 1. Chu, H. et al. A charge density wave-like instability in a doped spin orbit assisted weak Mott insulator. Nature Materials 16, 200–203 (2017). DOI: 10.1038/nmat4836 Contribution: Designed and conducted experiment. Analyzed data. Wrote paper. 2. Zhao, L., Torchinsky, D. H., Chu, H., et al. Evidence of an odd-parity hidden order in a spin- orbit coupled correlated iridate. Nature Physics 12, 32 (2016). DOI: 10.1038/nphys3517 Contribution: Conducted experiment. 3. Torchinsky, D. H., Chu, H., et al. Structural Distortion-Induced Magnetoelastic Locking in Sr2IrO4 Revealed through Nonlinear Optical Harmonic Generation. Phys. Rev. Lett. 114, 096404 (2015). DOI: 10.1103/ PhysRevLett.114.096404 Contribution: Designed and conducted experiment. Analyzed data. 4. Torchinsky, D. H., Chu, H., et al. A low temperature nonlinear optical rotational anisotropy spectrometer for the determination of crystallographic and electronic symmetries. Rev. Sci. Instrum. 85, 083102 (2014). DOI: 10.1063/1.4891417 Contribution: Conducted experiment. Analyzed data. vii Table of Contents Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter I: Introduction to correlated materials . . . . . . . . . . . . . . . . . 1.1 Hubbard model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Crystal field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Spin-orbit interaction . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Phases of cuprate high-Tc supercondutors . . . . . . . . . . . . . . . Chapter II: Experimental techniques . . . . . . . . . . . . . . . . . . . . . . 2.1 Second harmonic generation rotational anisotropy . . . . . . . . . . 2.2 Time-resolved optical reflectivity . . . . . . . . . . . . . . . . . . . Chapter III: Adsorbate effect in optical pump probe experiments . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Time-dependent reflectivity from adsorbate-affected Sr2 IrO4 . . . . . Chapter IV: Optical pump probe investigation of electronic and structural instabilities in Ca2 RuO4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Experiment and results . . . . . . . . . . . . . . . . . . . . . . . . . Chapter V: Optical pump probe investigations of the electronic structures of electron- and hole-doped Sr2 IrO4 . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Experimental . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter VI: Hidden order phase in Sr2 IrO4 revealed by second harmonic generation rotational anisotropy . . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Experiment and results . . . . . . . . . . . . . . . . . . . . . . . . . Chapter VII: Charge-density-wave-like instability in (Sr1−x Lax )3 Ir2 O7 revealed by time-resolved optical reflectivity . . . . . . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Experiment and results . . . . . . . . . . . . . . . . . . . . . . . . . Chapter VIII: Second harmonic generation investigations of antiferromagnetic orders in MPX3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Experiment and results . . . . . . . . . . . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii v vii 1 1 3 5 6 11 11 13 16 16 16 20 20 20 29 29 29 30 37 37 38 48 48 48 56 56 58 63 1 Chapter 1 Introduction to correlated materials The band theory of solids laid down the theoretical foundation for quantitative investigations of the electronic properties of solids. It was highly successful in understanding the transport properties of elemental metals and semiconductors, with verdict that a crystal with an odd number of electrons per unit cell is a metal. However, the discovery of a few insulating transition metal oxides posed serious challenge to the band theory. CoO, MnO and NiO all have an odd number of electrons per unit cell but they turned out to be electrically insulating. Nevill Mott realized that it is the strong Coulomb interaction between the electrons that is responsible for the insulating nature of these transition metal oxides. Charge correlation splits the otherwise half-filled conduction band into a fully filled valence band and an empty conduction band with a gap in between. The importance of these so-called Mott insulators was not realized until the discovery of high temperature superconductors half a century later. The high temperature superconductors are doped Mott insulators and exhibit a plethora of interesing electronic phases. Since then, it was realized from extensive researches on high temperature superconductivity that in addition to charge, the lattice, spin and orbital degrees of freedom could all play important roles in interaction and lead to the rich phase diagrams of correlated materials. 1.1 Hubbard model Hubbard introduced a model Hamiltonian to account for the competition between kinetic hopping (t) and Coulomb correlation (U), which favors banding and localization respectively: X X † H= −t i j ciσ c jσ + U ni↑ ni↓ (1.1) hi jiσ i † † where niσ = ciσ ciσ is the number operator, ciσ and ciσ are the creation and annihilation operators for an electron with spin σ at site i. t 0 s are intra- and inter-site (nearest neighbour) hopping energies: t ii describes the mean energy of the band while t i j describes the bandwidth. † In the U = 0 limit, ciσ and ciσ can be expanded in terms of the creation and annihi- 2 lation operators of the Bloch states: 1 X ik·Ri † † = √ ciσ e ckσ , N k The first term in (1.1) becomes X † Ekσ ckσ ckσ , 1 X −ik·Ri ciσ = √ e ckσ . N k Ekσ = X t i j eik·(Ri −R j ) (1.2) (1.3) hi ji kσ Since t i j has the same periodicity as the lattice, we see that the eigenstates of the non-interacting Hamiltonian become the free traveling Bloch states. On the other hand if t i j = 0, the Hamiltonian becomes diagonal and X E= t ii (ni↑ + ni↓ ) + Uni↑ ni↓ (1.4) ı At half filling, E is minimized if there is exactly one electron on each site i, i.e. ni↑ ni↓ = 0. The kinetic energy gain t i j for hopping is 0 while the penalty U for double occupancy as a result of hopping is heavy, hence electrons are localized to their sites and an insulating ground state results. We see that the two limits of the Hubbard model correspond to a band metal and a Mott insulator. For half filling and the limit of large U, it can be shown that the effective Hamiltonian for (1.1) becomes the antiferromagnetic Heisenberg model: X H = −J Si · S j (1.5) hi ji where J = −4t i2j /U. This can be understood from the virtual exchange process envisioned below. If the electronic spins on neighbouring sites are parallel, hopping is forbidden by the Pauli principle. However, if they are antiparallel, then one electron can hop onto its neighbouring site, creating double occupancy and hence an energy cost U. The electron can then hop back to its original site, removing the energy cost U. This process is dubbed virtual hopping or virtual exchange and acts as a perturbation to the antiferromagnetic ground state. By second order perturbation theory, the energy of the perturbed state is lowered from the original state by an amount proportional to −t i2j /U. This explains why Mott insulators are usually antiferromagnetic. According to this effective Hamiltonian, the low energy excitations of this half-filled large-U Hubbard model are spin excitations, corresponding to the configuration where neighbouring spins are flipped during the virtual exchange process. 3 Although antiferromagnetic order is usually associated with Mott insulators, it is not a priori. In fact, one needs to distinguish between a Mott insulator and a Slater insulator. In the latter, antiferromagnetic order doubles the size of the unit cell and opens up a gap at the folded Brillouin zone boundary in the momentum space. In a Mott insulator, the fundamental reason for opening up a gap at the Fermi energy is electron correlation. Whether TN = TI MT defines a practical way for distinguishing between a Mott insulator and a Slater insulator. In Chapter 7, we will discuss this issue again for the 5d spin-orbit assisted Mott insulator Sr3 Ir2 O7 . 1.2 Crystal field In addition to Coulomb repulsion, lattice, or crystal field, also has important effects on the electronic ground state of correlated materials. A free atom in a spherical potential has orbital eigenstates described by spherical harmonics: ψnlm (r) = Rnl (r)Ylm (θ, φ) (1.6) The orbital angular momentum of such a state is given by the azimuthal quantum number l, with projection m along the z-axis. For a partially filled shell, Hund’s rule states that the lowest energy configuration is the state with (1) maximum total P spin angular momentum (S = |m s | = | si |), (2) maximum total orbital angular moP mentum (L = |ml | = | mi |), (3) minimum J for less than half-filled shell (J = L −S) and maximum J for more than half-filled shell (J = L + S). Both spin angular momentum and orbital angular momentum of the electrons contribute to the magnetic moment of the ion, which is then given by µ=g p J (J + 1) µ B (1.7) where g is the Landé g-factor defined as g =1+ J (J + 1) + S(S + 1) − L(L + 1) 2J (J + 1) (1.8) For transition metals with d electrons on the outmost shell, the orbital angular momentum component is often found missing from the measured magnetic moment. This happens because in a crystalline environment, the ionic potential is no longer spherical, instead it is reduced to discrete rotational symmetries in addition to other symmetries. In a q-fold rotational symmetric crystalline environment, the ionic potential can be expanded in terms of the spherical harmonics and the only nonvanishing terms are Ylm with m as an integer multiple of q. As a result, this potential 4 only couples orbital momentum states with quantum numbers m1 and m2 such that m1 − m2 = N q where N is an integer. In an octahedral environment, Y2+2 and Y2−2 are coupled together to give rise to the new d xy and d x 2 −y2 eigenstates:  1  +2 −2 d x y = √ Y2 − Y2 2i  1  d x 2 −y2 = √ Y2+2 + Y2−2 2 A similar transformation gives the d xz , d yz , d z2 states:  1  −1 +1 d xz = √ Y2 − Y2 2  i  −1 +1 d yz = √ Y2 + Y2 2 d z2 = Y20 P Since Y2+m and Y2−m contribute equally to these states, L z = mi = 0, i.e. these new eigenstates do not possess orbital angular momentum. Moreover, they split the degeneracy of the spherical harmonic states of the spherical potential. In an octahedral environment, the energy of {d xy , d xz , d yz } is lowered while the energy of {d x 2 −y2 , d z2 } is raised. They are designated as the t 2g and eg states respectively and their energy difference is often referred to as 10Dq or simply crystal field splitting. It is instructive to look at the expectation value of L z in the basis of |ψi > = {d xz , d yz , d x y , d x 2 −y2 , d z2 }, 0 i 0 0 0 + *. .. −i 0 0 0 0 /// D E (1.9) ψi L z ψ j = ... 0 0 0 2i 0 /// // .. . 0 0 −2i 0 0 / , 0 0 0 0 0 This means that while an eigenstate of a crystal field by itself does not possess orbital angular momentum, transition among different eigenstates may generate an orbital angular momentum. Even so, the orbital angular momentum among the eg states is completely quenched (bottom right 2 × 2 matrix). Transition among the t 2g states, on the other hand, can generate a finite orbital angular momentum, which is isomorphic to that of the p-orbitals up to a minus sign: D ψi L z ψ j E *. 0 −i 0 +/ = .. i 0 0 // , , 0 0 0 - with |ψi i = {px , py , pz } (1.10) 5 Transition between the t 2g and eg states is thwarted by their energy difference. Similar arguments can be given to the other components of the orbital angular momentum operator L. Therefore, under crystal field splitting, the orbital angular momentum becomes L = 1 for the t 2g states and vanishes for the eg states. 1.3 Spin-orbit interaction While crystal field quenches orbital angular momentum, spin-orbit interaction, on the other hand, prefers a different linear combination of the spherical harmonics and ’unquenches’ the orbital angular momentum. In 3d transition metal spin-orbit effect is weak, but in 5d transition metal spin-orbit interaction may have energy comparable to Coulomb interaction and crystal field splitting. In such systems, the crystal field orbital states are no longer the eigenstates of the spin-orbit coupled Hamiltonian. Instead, we need to consider the eigenstates of the total angular momentum operator J = L + S. For L = 1 and S = 1/2, i.e. t 2g states, the Jeff states can be found as linear combinations of the crystal field orbital states using the Clebsch-Gordan table: X Jeff , m Jeff = α ml m s |l, ml i |s, m s i ml ,m s which gives + + 1 1 3 3 , ± = |1, ±1i , ± 2 2 2 2 + r + r + 1 1 1 1 3 1 1 2 ,± = |1, ±1i , ∓ + |1, 0i , ± 2 2 3 2 2 3 2 2 r + + r + 1 1 2 1 1 1 1 1 ,± = ± |1, ±1i , ∓ ∓ |1, 0i , ± 2 2 3 2 2 3 2 2
√ where on the right hand side |1, ±1i = ∓d xz −id yz / 2 and |1, 0i = d x y . Plugging them in, we find √ √ √ 1/2i −1/ 3 1/ 3 −i/ 3 d x y↑ + |1/2, *. +/ *. √ √ √ +/ *. .. |3/2, 1/2i // = .. −2/ 6 −1/ 6 i/ 6 // .. d yz↓ /// (1.11) √ √ 0 1/ 2 i/ 2 - , d xz↓ , |3/2, −3/2i - , etc. These Jeff states are the eigenstates of the spin-orbit coupled Hamiltonian. They describe the 5d electron states in Srn+1 Irn O3n+1 as we will see in Chapter 5 through Chapter 7. 6 1.4 Phases of cuprate high-Tc supercondutors As discussed in previous sections, correlated materials exhibit lattice, charge, spin, and orbital degrees of freedom. Under appropriate experimental conditions, they exhibit macroscopic orders known as phases. Furthermore, these microscopic degrees of freedom often compete with one another and result in a kaleidoscope of phases in close proximity of one another. In such cases, small changes in temperature, pressure, electromagnetic field, or carrier doping, can tip the balance between the competing interactions and drive the system from one phase to another. A good example of this is Ca2 RuO4 , in which either a moderate amount of heat (∼ 360 K), or pressure (∼ 0.5 GPa), or electric field (∼ 40 V/cm) is enough to drive the system from an insulator into a metal. In Chapter 4, we exploit such instability and investigate the possibility of inducing an insulator-to-metal transition in Ca2 RuO4 using light. On the other hand, the multitude of macroscopic phases that can be realized in correlated materials is best exemplified by the cuprate high-Tc superconductor, which exhibits the Mott insulating phase, high temperature superconductivity (HTSC), pseudogap, charge-density-wave (CDW), nematic order at different temperature and doping levels [Fig. 1.1]. Apart from the unsolved mystery of HTSC, the pseudogap and the charge-density-wave phase have attracted a lot of attention recently regarding their roles in the formation of HTSC. Below we introduce the basic phenomenology of these two phases based on studies on cuprate high-Tc superconductors. In Chapter 6, using second harmonic generation rotational anisotropy we show that Sr2 IrO4 also manifests a hiddren phase consistent with theoretical models of the pseudogap phase. In Chapter 7, we provide evidence for a charge-density-wave-like instability in electron-doped Sr3 Ir2 O7 using time-resolved optical reflectivity. Superconductivity is characterized by the formation of an energy gap at the Fermi level below which Cooper pairs form. The Cooper pairs form a superfluid that flows with zero electrical resistance. Breaking of the Cooper pairs by thermal excitation or phonon scattering is thwarted by the presence of the energy gap. As with an ordinary electronic wavefunction, the superconducting wavefunction is composed of the orbital part and the spin part. The orbital wavefunction takes angular momentum values of L = 0, 1, 2, etc, and inherits the same symmetries prescribed for the corresponding atomic wavefunctions. The spin component takes on corresponding values S = 0, 1, such that the total wavefunction is antisymmetric under the exchange of two electrons. Conventional superconductors are characterized by L = 0 and S = 0, which gives rise to a uniform superconducting gap in the Brillouin 7 past decade, research in high RH(0) < 0 RH(0) > 0 RH(0) > 0 (small) nH(0) ~ p nH(0) ~ 1+ p UD OP OD B=0 B>0 e debate T* icting, with Temperature (a.u.) Strange metal Pseudogap T0 TI AFI th the discovery of (quantum) TN SC 0 of the charge density within CO 0 0.05 0.1 0.15 0.2 0.25 Hole doping, p Fermi TC liquid 0.3 Figure 1.1: Phase diagram of La2−x Srx CuO4 [1]. The pseudogap temperature T∗ extrapolates to 0 at near p ∼ 0.2 doped holes per CuO2 plaquette. In the underdoped regime, a short-ranged charge order (CO) onsets below T0 and competes with superconductivity, causing a dip in Tc . Tφ denotes the temperature below which superconducting phase fluctuations are observed. zone. High-Tc superconductors, in most cases, are characterized by L = 2 and S = 0, and are called d-wave superconductors. A d-wave superconducting gap exhibits nodes along the (π, π) and equivalent directions in the Brillouin zone, where the phase φ(k) of superconducting gap function ∆(k)eiφ(k) switches sign [Fig. 1.2]. However, even at temperatures above Tc and below a characteristic T∗ (pseudogap tempreature), part of the Fermi surface is already gapped with a finite density of states residing along the ’Fermi arcs’ around the position of the nodes. As temperature cools towards Tc , these Fermi arcs shrink towards the position of the nodes, 8 the process of which is depicted in Fig. 1.2. Such Fermiology is unique to HTSC and the phase between Tc and T∗ is termed the pseudogap phase. One interpretation of the microscopic origin of the pseudogap phase is that Cooper pairs form in the pseudogap phase without phase coherence. This is based on the observation of Bogoliubov quasiparticle interference, a signature of delocalized Cooper pairs, at energy < T∗ in the pseudogap phase. However, excitation at energy ∼ T∗ in the pseudogap phase breaks local C4 symmetry of the underlying cuprate lattice and is distinct from the homogeneous low energy excitations. Understanding the holistic picture of the pseudogap phase is still underway and is crucial for understanding the mechanism of HTSC. Figure 1.2: Illustration of the temperature evolution of Fermi surface in underdoped cuprate. From left to right: (one quadrant of) Fermi surface of an underdoped cuprate at T < Tc , Tc < T < T∗ and T > T∗ . The pseudogap phase (Tc < T < T∗ ) is characterized by the Fermi arcs extending from the position of the d-wave gap nodes with other parts of the Fermi surface gapped out. Another prominent feature of the cuprate phase diagram is the charge order in underdoped samples [Fig. 1.1], which is also referred to as the charge density wave. Neutron diffraction provided the first evidence of charge order in hole-doped cuprates. In underdoped La1.6-x Nd0.4 Srx CuO4 , the static charge order manifests as a set of satellite peaks around Bragg peaks at wave vectors Qcharge = (H ± qcharge , K ± qcharge , L) [Fig. 1.3(b)(c)]. In addition, neutron diffraction also found modulation of the antiferromagnetic order at twice the wavelength of the charge modulation. It manifests as a set of magnetic peaks around the AF wave vector (1/2, 1/2, L) at QAF = (1/2 ± qAF , 1/2 ± qAF , L) with qAF = 1/2 · qcharge [Fig. 1.3(b)-(d)]. For x = 0.125, qcharge and qAF are determined to be 0.25 and 0.125 respectively. Therefore, the real space charge and spin order can be visualized as Fig. 1.3(a), where the doped holes arrange themselves into one dimensional ’charge rivers’ separating two Intensity (counts/10 min) a Intensity (counts/2 min) Access provided by California Institute of Technology on 05/04/17. For personal use only. 9 CuO2: nh = 0.125 1,500 c (0, 2 + q, 0) b (020) 1,400 (110) 1,300 1,200 1,100 –0.35 400 350 –0.25 q d –0.15 (0.5, 0.5 + q, 0) 300 250 200 150 100 –0.2 –0.1 0.0 q 0.1 0.2 Figure 1 Figure discovery 1.3: Charge orderorder in Lain1.6-x Nd0.4 Srx CuO neutron diffrac4 revealed Nd0.4 Srby Neutron scattering of stripe 12%-doped La1.6−x x Cu2 O4 . (a) Schematic tion a. Schematic the charge and the spinCu order hole-doped LNSCO. representation of [2]. the stripe pattern: of Circles represent sitesinin12.5% the CuO 2 plane, with arrows denoting Circle denote Cu sites in the CuO plane. Arrows denote Cu spins, with black 2 the Cu spins and gray circles indicating the location of the doped holes. (b) (HK0) projectionand of the white colors denoting anti-phase relation. Gray circles denote the location of doped reciprocal space showing the location of the measured charge and spin satellites (black circles) along the (010) holes. b.respectively; (HK0) projection of the represent reciprocalBragg space reflections. showing the(c,d location charge scattering and (110) directions, white circles ) Elasticofneutron and magnetic peaks (black) and Bragg peaks (white). c. Charge order at (0, 1.75, measurement of the (c) charge-order peak at (0, 1.75, 0) and (d ) magnetic peaks at (0.5, 0.375, 0) and 0) with revealed elastic scattering. d. Magnetic order at Reference (0.5, 0.375,300)with andpermission. (0.5, 0.625, 0), scansby along theneutron arrows marked in panel b. Adapted from (0.5, 0.625, 0). was determined to be located at wave vectors QAF = (±1/2, ±1/2, L).1 Around the same time, the first experimental hint of stripedomains. order was revealed by the independent neutron opposite antiferromagnetic Direct visualizations of the charge order are scattering measurements of Yoshizawa et tunneling al. (17) and Birgeneau et al.In(18), whoSTM reported provided by scanning microscopy (STM). Bi2212, revealsincommensurate periSr0.08 CuO4 inside (8% hole doping) La1.89 Sr0.11 CuO4 (11% AF modulations in underdoped La1.92 conductance odic modulations in tunneling vortices where and superconductivity hole doping), respectively, suggesting the presence an[Fig electronic phase characterized by the is suppressed by externally applied magnetic of field 1.4]. This argues for the quasi-static ordering of the doped holes. A subsequent study by Thurston et al. (19) revealed that competition scenario between HSTC and charge density wave. Additional evidence the incommensurate AF order could only be found in the superconducting (SC) phase of LSCO for this interpretation comes from the temperature-doping phase diagram, where Tc [it was later shown to be present also in the insulating phase, albeit with a different modulation is suppressed or even fully eliminated (in the case of LBCO) by the presence of vector (20)]. The full momentum structure and doping dependence of incommensurate spin charge density wave [Fig. 1.1]. In addition to the static charge order observed by order in LSCO was later uncovered by Cheong et al. (21) and shown to manifest itself as a set diffraction and STM, charge density wave may also exist dynamically. This has of four magnetic peaks at QAF−IC = (1/2 ± δIC , 1/2, L) and QAF−IC = (1/2, 1/2 ± δIC , L) (see demonstrated [3] where signatures of the amplitudeincommensurability. and phase modes representing the doping-dependent diagram in been Figure 1b), with δinICref The experimental observations from neutron scattering spurred an intense theoretical activity, aimed at addressing two primary phenomenological findings: (a) the momentum structure of the incommensurate AF order and its relationship to the ordering of the doped holes, and (b) the emergence of incommensurate AF order only within the SC phase, which pointed at an A <0,π > c a <π, π> 10(0, ¼) Q of a fluctuating charge density wave in underdoped LSCO are captured by timeS resolved optical techniques. Understanding the mechanism that gives rise to the prevalent charge order phenomenon in high-Tc superconductors is also important for understanding HTSC and isÅbeing intensely pursued. 100 150 Fourier amplitude (pS) b 100 Å pA 2 Q A 15 mV 100 0 mV 50 –15 mV 0 0 d 0 0.2 0.4 0.6 0.8 1.0 k (2π/a0) Figure 1.4: Periodic modulation of tunneling conductance in the vortices of suFigure 2 perconducting Bi2 Sr2 CaCu2 O8+δ [2]. The vortices are formed with an externally Scanning tunneling microscopy (STM) explorations of charge order in Bi-based cuprates. (a) L applied magnetic field. Inside the vortices where superconductivity is suppressed, temperature topographic map of a cleaved surface of slightly overdoped (T c = 89 K) Bi2 Sr2 Ca the tunneling conductance shows modulation with a periodicity of 4 lattice sites. (Bi2212). (b) Differential tunneling conductance (dI/dV ) map (integrated between 1 and 12 m an applied magnetic field of 5 and 0 T, showing the appearance of period-4 density modulation magnetic vortex cores, and over the same spatial region of panel a. Adapted from Reference 35 permission. (c) Momentum space representation of the periodic structures observed using STM including the Fourier peaks from the atomic lattice (A) and the structural supermodulation (S), charge-order peaks (Q) at Q ∼ (±0.21, 0) and Q ∼ (0, ±0.21), in reciprocal lattice units (r.l.u.) resolved Fourier-transformed conductance map along the (10) direction, showing no dependen charge-order peak position (Q) on energy, thereby demonstrating the independence of this phe from quasiparticle interference effects. Adapted from Reference 41 with permission. (e–g) (Pane next page) Conductance maps in Na-doped Ca2 CuO2 Cl2 at different bias voltages, and (h–j) cor 2D Fourier transforms whose line cuts (k–m) along the Cu-O bond directions show the chargefrequencies and the lack of energy dispersion in the momentum-resolved structures. Adapted fr Reference 42 with permission. 374 Comin · Damascelli 11 Chapter 2 Experimental techniques The development of ultrafast femtosecond laser over the past decades made possible nonlinear optics as well as time-resolved measurements of electronic and structural dynamics of solids. We developed and employed second harmonic generation rotational anisotropy as well as time-resolved optical reflectivity techniques for investigating correlated materials. The experiments described below are enabled by a KMLabs Wyvern 1000 ultrafast amplified laser system. This system outputs 1 mJ pulses at 10 kHz repetition rate. We use leakage beams through dielectric mirrors, which have less than 1 µJ pulse energy, for our experiments. These pulses typically have a temporal width of 70 fs and a wavelength around 795 nm (∼ 1.5 eV) with a FWHM of 50 nm. 2.1 Second harmonic generation rotational anisotropy Electromagnetic waves travelling through a medium induces an oscillating electric dipole P, magnetic dipole M, and electric quadrupole Q̂ in the medium at frequencies ωi of the incident waves as well as linear combinations of these frequencies: P(ω1 +ω2 +...), etc. Mathematically, this is expressed as the expansion of multipole moments in terms of the electric and magnetic fields of the incident light [4]: P ∝ χ ee : E + χ em : H + χ eee : EE + χ eem : EH + χ emm : HH + O(E,H) 3 M ∝ χ me : E + χ mm : H + χ mee : EE + χ mem : EH + χ mmm : HH + O(E,H) 3 Q̂ ∝ χ qe : E + χ qm : H + χ qee : EE + χ qem : EH + χ qmm : HH + O(E,H) 3 They radiate according to   ∂ 2 Q̂  ∂M  ∂2P − µ0 ∇ 2 , S = µ0 2 + µ0 ∇ × ∂t ∂t ∂t where the electric dipole contribution (∝ P) is usually the strongest contribution. In the case of a monochromatic incident light, as is the case for a narrow-band laser, the radiated field consists of harmonics at ω, 2ω, 3ω, etc. Second harmonic generation refers to the O(E,H)2 processes such as P(2ω) = χ eee E(ω)E(ω), M(2ω) = χ mee E(ω)E(ω) or Q̂(2ω) = χ qee E(ω)E(ω). Since the process involves two copies 12 of the incident light field, the intensity of SHG scales as the square of the intensity of the input light. It is particularly enhanced by ultrafast pulsed laser with large peak electric field. The specific form (i.e. non-zero independent elements) of the multipole tensor χ is constrained by the symmetry of the media as well as the nature of the tensor itself. For example, if C is a symmetry operation of a crystal, then the second order electric eee = χ eee ≡ C 0 C 0 C 0 χ eee . susceptibility tensor χ eee for the crystal has to obey χijk i i j j k k ijk i0j0k0 Additionally, if χ is an axial tensor, then the right hand side of the equation has to be multiplied by the determinant of the symmetry operation; under time reversal operation, a polar tensor is invariant while an axial tensor changes sign. Applying these rules to a centrosymmetric medium, we find that χ eee vanishes but χ mee and χ qee are allowed. If a phase transition breaks inversion, then χ eee becomes allowed. Therefore, SHG can be used as a sensitive probe for inversion breaking, and more generally symmetry breaking, during phase transitions. In such experiments, symmetry breaking would either manifest as a discontinuous increase in SHG intensity, or a reduction in discrete symmetry. In fact, second harmonic generation rotational anisotropy (SHG-RA) has been demonstrated in NiO [5], Bi-YIG/GGG [6] with normally incident beam, and in GaFeO3 [7] with obliquely incident beam where the polarizations of the beams are rotated while the scattering plane remains fixed. The limitation of a normally incident experiment is that tensor elements involving the z component are not probed. On the other hand, in the latter oblique incident experiment where the scattering plane remains fixed, the SHG signal at each angle is a different linear combination of tensor elements. Extracting individual tensor elements from such data could be challenging. Therefore, it is desirable to conduct SHG-RA experiment with oblique incidence on a rotating sample. The challenge is that it is difficult to rotate a sample inside an optical cryostat and keep the incident beam from walking on the sample. This difficulty is overcome by an ingenious design where the sample is fixed and the rotation of the scattering plane is accomplished by the use of a rotating transmissive phase grating [8]. A drawing of the rotating phase grating scheme is shown in Fig. 2.1. A normally incident laser beam (red) is diffracted by the phase grating into two constructive first orders with opposite wavevectors in the xy plane (normal to the optical axis). The phase grating is placed at the focal point of a convex lens which brings the two diverging beams to parallel to each other. Both beams then get focused onto the sample by a reflective objective. We make sure that the sample 13 Figure 2.1: Drawing of a rotating scattering plane defined by a rotating phase grating. is placed at the focal point of the reflective objective by checking that both beams are overlapped on the sample for all orientations of the phase grating. After this, we block one of the diffracted beam and reflect the other beam into a photomultiplier tube (PMT). A linear polarizer and a set of color filters are placed in front of the PMT to select the desired polarization and wavelength of the SHG. The phase grating, the PMT, as well as a 1/2-waveplate (not shown in Fig. 2.1) in front of the phase grating are rotated simultaneously to simulate the rotation of the lab frame. The layout of the entire SHG-RA experiment is shown in Fig. 2.2. 2.2 Time-resolved optical reflectivity Another way ultrafast laser can be useful for material research is the time resolution it enables. After shining a pulse of light (pump) onto a sample and perturbing its ground state, we can monitor the subsequent change in the optical properties of the sample using another weak pulse of light (probe). A straightforward property to measure is the optical reflectivity, but other properties like optical transmissiv- 14 Figure 2.2: Layout of the SHG-RA experiment. Regen: regeneratively amplified ultrafast laser. OPA: optical parametric amplifier. P: polarizer. WP: waveplate. L1(2): convex lens 1(2). PM: phase mask (phase grating). LPF: long pass filter. DM: dichroic mirror. RO: reflective objective. Cryo: cryostat. A: analyzer. SPF: short pass filter. IF: interference (band pass) filter. PMT: photomultiplier tube. ity, terahertz reflectivity, Kerr-rotation angle, second harmonic generation rotational anisotropy, etc. can also be measured transiently. Figure 2.3 shows the layout of a time-resolved optical transmissivity experiment. Here, optical parametric amplifier (OPA) allows one to selectively generate light of different wavelength that can be used to pump and probe specific transitions in a material. The pump beam is usually mechanically chopped or optically modulated in time (not shown) so that the change in transient optical properties is also periodically modulated. This enables detection by lock-in amplifier. The delay between the pump and the probe is set by a mechanical delay stage. For transient optical reflectivity, we employ a balanced detection scheme which compensates for variations in laser pulse-to-pulse energy by subtracting a reference beam from the signal. Additionally, for degenerate pump probe reflectivity experiment, we set the pump and probe polarizations to perpendicular to each other so that pump scatter noise is minimized. By employing all these schemes, we are able to achieve a signal-to-noise ratio on the order of 10−5 for degenerate pump probe reflectivity experiments. In addition to the ability to resolve electronic and structural dynamics on ultrafast timescales, pump probe experiment also realizes the possibility of driving ultrafast phase transition. To name a few examples: photo-induced insulator-to-metal transition has been demonstrated in VO2 [9], photo-induced structural phase transition has been demonstrated also in VO2 and La0 .42Ca0 .58MnO3 [10]. More remark- 15 Figure 2.3: Schematic of a pump probe reflectivity experiment. ably, evidences for photo-induced and photo-enhanced superconductivity have been reported in K3 C60 [11] and YBa2 Cu3 O6.48 [12] respectively. These examples comprise only a small fraction of all the photo-induced phase transitions that have been demonstrated. In some of these experiments, optical excitation in the visible range is enough to induce phase transitions while in others, specific phonon modes need to be activated by mid-IR pulses before phase transitions can take place. 16 Chapter 3 Adsorbate effect in optical pump probe experiments 3.1 Introduction Optical measurements are sensitive to the surface properties of the investigated samples. This is particularly true for measurements in cryogenic environment. In our optical pump probe reflectivity experiments on Rh-doped Sr2 IrO4 , we find that residual gases in the cryostat condenses onto the sample surface at low temperature and leads to oscillations in reflectivity. To ensure an adsorbate-free environment for these experiments, the use of a cover slip above the sample is necessary, in addition to a vacuum better than 5 × 10−7 mbar. Detailed descriptions of the phenomenology of surface adsorption in optical pump probe reflectivity studies are presented below. 3.2 Time-dependent reflectivity from adsorbate-affected Sr2 IrO4 Single crystal Sr2 Ir1−x Rhx O4 (x = 0.05 & x = 0.11) samples are cleaved in air and quickly transferred to an optical cryostat (Janis ST-500), which is pumped to a vacuum of ∼ 1 × 10−6 mbar at room temperature by a turbo pump station (Pfeiffer HiCube). Liquid nitrogen is then used to cool the sample down. At 180 K, the vacuum inside the cryostat reaches 2 × 10−7 mbar. Under such vacuum condition, optical pump probe reflectivity data are taken during the cooling process with degenerate 800 nm beams. Figure 3.1 shows the reflectivity transients from the x = 0.11 sample as a function of temperature. At or above 180 K, the reflectivity transients show small changes as a function of temperature. Below 180 K, the reflectivity transient starts to change wildly: it first flips sign and reaches maximum negative magnitude at 140 K, then it decreases in magnitude while developing a positive component (120 K - 80 K). Such temperature dependence below 180 K is not reproducible upon additional measurements. Moreover, even at a fixed temperature below 180 K, the reflectivity transient evolves on a timescale of tens of minutes. Figure 3.2 shows the time dependence of the reflectivity transient from the same sample at 140 K. The change in the reflectivity transient on an hourly timescale is obvious, and the pump-induced reflectivity change at t = 0 oscillates between positive and negative on a timescale of ∼ 20 hrs. This is similar to the trend manifested in Fig. 3.1, where ∆R/R(t = 0) shows oscillating behavior in time too. Similar oscillation in reflectivity transients below 180 K is also observed on the x 17 = 0.05 sample in similar vacuum condition. 10 "R/R (10-4 ) 5 0 240 K 220 K 200 K 180 K 160 K 140 K 120 K 100 K 80 K -5 -10 -15 0 1 2 Time (ps) Figure 3.1: Temperature dependent pump probe reflectivity transients from Sr2 Ir1−x Rhx O4 (x = 0.11). At or above 180 K, reflectivity transient shows small change as a function of temperature. Below 180 K, reflectivity transient shows non-monotonic change as a function of temperature. Moreover, the temperature dependent reflectivity transients below 180 K shown here are not reproducible upon additional measurements. To gain further insight on the time dependence of the reflectivity transient below 180 K, we monitor the bare reflectivity of the x = 0.05 sample below 180 K as a function of time. Figure 3.3 shows the time dependence of reflectivity at 80 K and 140 K respectively. Reflectivity oscillates with a period of ∼ 40 hr and ∼ 20 hr respectively at the two temperatures. Moreover, the minimum of the pump probe transient at t = 0 also varies periodically with reflectivity, consistent with our observations in Fig. 3.1 and Fig. 3.2. Such periodicity disappears when temperature is increased to above 170 K. From these observations, we deduce that the oscillation in reflectivity is caused by interference of the reflected beams from the sample surface and the surface of gaseous molecules condensed onto the sample surface when temperature is below their freezing point. Because these gaseous molecules are continuously condensed onto the sample surface, the interference between the two reflections continuously evolves between constructive and destructive, leading to oscillations in reflected intensity. The period of oscillation at 80 K is understandably longer than that at 140 K because the gaseous molecules have less kinetic energy at lower 18 1 hr 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 "R/R (10-3 ) 2 1 0 -1 0 1 2 Time (ps) Figure 3.2: Time dependent pump probe reflectivity transients from Sr2 Ir1−x Rhx O4 (x = 0.11) at 140 K. Reflectivity transients are taken at 1 hr intervals and show periodic variation between positive and negative pump-induced reflectivity change. The period of oscillation inferred from the data plotted here is about 20 hr. temperature, thereby slowing down the adsorption process. Since a vacuum of 2 × 10−7 mbar is approaching the operational limit of a turbo pump in combination with an optical cryostat, we search for ways for preventing the absorption process from happening other than improving vacuum condition. One solution is to employ a cover slip onto the sample holder rim above the sample using vacuum grease. At low temperatures, the cover slip acts as a cold surface to condense the gaseous molecules and prevents them from reaching the sample surface. After the installation of a cover slip, we found that the reflectivity of the sample measured on tens of hours timescale becomes stable and dominated by laser noise [Fig. 3.4]. Subsequent pump probe reflectivity measurements on other samples using a cover slip in combination with a vacuum of ∼ 10−7 mbar has always shown a stable reflectivity at cryogenic temperature. Therefore, the gaseous adsorption problem can be completely prevented by the cover slip method. 19 1.0 ∆R/R(t=0) R R (a.u.) ∆R/R(t=0) (10 -3 ) 8 4 80 K 140 K 0 0 0 20 40 Time (hr) 60 0 10 20 30 Time (hr) 40 50 Figure 3.3: Time dependence of reflectivity and ∆R/R(t = 0) on Sr2 Ir1−x Rhx O4 (x = 0.05). Reflectivity and the minimum of the reflectivity transient at t = 0 is shown as a function of time at 80 K and 140 K. Reflectivity oscillates with a period of 15 - 40 hr, while the reflectivity transient minimum also varies with the same periodicity. The oscillatory behavior is due to continuous condensation of gaseous molecules onto the sample surface over time. 2 1 1 0 0 R (a.u.) ∆R/R(t=0) (10 -4) ∆R/R(t=0) R -1 140 K -2 10 15 Time (hr) 20 Figure 3.4: Time dependence of reflectivity and ∆R/R(t = 0) on Sr2 Ir1−x Rhx O4 (x = 0.11) with cover slip. With a cover slip in thermal contact with the sample holder above the sample, oscillations in the sample reflectivity and the reflectivity transient both disappear. 20 Chapter 4 Optical pump probe investigation of electronic and structural instabilities in Ca2 RuO4 4.1 Introduction The layered perovskite Ca2 RuO4 realizes an orbital-ordered antiferromagnetic insulator under ambient conditions [13–19]. This phase is stabilized by a delicate balance between competing microscopic interactions which makes it extremely fragile against external perturbations [20–24]. Heating Ca2 RuO4 above 360 K induces a first-order insulator-to-metal transition (IMT) concomitant with a structural phase transition where the c-axis of the lattice is elongated [13, 25]. The insulator-tometal transition can also be induced by isovalent substitution [13, 26] or carrier doping [26–29], a small electric field of ∼ 40 V/cm [30, 31], or a modest pressure of ∼ 0.5 GPa [32–34]. The multitude of IMT pathways in Ca2 RuO4 testifies the delicate balance between the microscope degrees of freedom in it. Since photoexcitation is capable of inducing phase transitions in 3d transition metal oxides [9, 35–40], it would be interesting to explore how it affects the electronic instability in Ca2 RuO4 . 4.2 Experiment and results Pristine Ca2 RuO4 crystals are known to undergo irreversible structural damage when heated above TI MT [29]. In our study, we use 8% Fe-doped Ca2 RuO4 crystals that are structurally robust through thermal cycles across TI MT [41]. The insulatorto-metal transition is well captured in optical reflectivity where an 8% drop is observed at 800 nm across TI MT = 369 K, a temperature slightly higher than the undoped parent compound. This is consistent with our ellipsometry measurement shown in Fig. 4.1, where reflectivity at 800nm (1.55 eV) in the metallic phase is slightly lower than the insulating phase due to spectral weight transfer to low energies across the IMT. In our time-resolved optical reflectivity experiment, we focus cross-polarized 800 nm pump and probe light onto sample at near normal incidence. A sensitivity of 10−5 in ∆R/R is achieved by using a balanced detection scheme. Figure 4.2 shows the transient response of Ca2 Ru0.92 Fe0.08 O4 to optical excitation across TI MT . In the metallic phase (T > TI MT ), impulsive optical excitation induces a sharp resolution-limited rise in reflectivity at time t = 0 followed by a fast decay E. Photon energy dependence of optical conductivity and reflectivity 21 2 Reflectivity (b) 400 K 1.5 300 K 0.3 1 0.2 0 1 2 3 Energy (eV) 1 2 3 σ (103Ω −1cm−1) (a) 0.4 0.5 0 4 Energy (eV) Figure 4.1: (a) Reflectivity and (b) the real part of optical conductivity of FIG.Ca S5. Photon energy dependence of the (T= (a) optical and (b) O4 in the insulating 300 K)reflectivity and metallic (T optical = 400 conductivity K) phase of 2 Ru 0.92 Fe0.08 a function energy. 8 %as Fe-doped Ca2of RuO 4 below (T = 300 K) and above (T = 400 K) TIMT . within the first 100 fs [Fig. 4.2(a)]. A more gradual equilibration process ensues, which is modulated by a coherent Ag optical phonon [42] on shorter time scales and experiments measured by known ellipsometry, whichscattering were used[43] to on determine the scales optical rea coherentwere acoustic phonon as Brillouin longer time flectivity and optical as is a cooled functioninto of photon [Fig.phase, S5]. The behavior [Fig. 4.2(a) inset].conductivity As the sample the Mottenergy insulating the sign of the opticalreverses conductivity TIMT is A highly consistent that reported of ∆R/R [Fig. across 4.2(b)]. The phononwith mode softens nearinTundoped g optical I MT Ca2and RuOundergoes an abrupt blue-shift as the lattice structure changes from L-Pbca to 4 [3, 4]. S-Pbca across MT . To estimate theTIconversion from absorbed pump fluence (F ) to number of photons of The values of the optical constants of the 8 % Fe-doped Ca2 RuO4 crystals used in our N /Aξ energy hν absorbed Ru site (n), used the formula = NRuph/Vu.c.optical whereexcitation, Nph = F A/hν In order to betterper characterize thewe metallic phase undernimpulsive FWHM Fe we number performoffluence probepulse, studies O4 area above is the photonsdependent absorbed pump per pump A =onπ Ca × 2(Ru )2 is of the 0.08the 20.92 TI MT . Figure 4.3(a) shows transients depth, at a fewNRu different flupump beam on the crystal, ξ isthe thereflectivity optical penetration is the pump number of Ru ences T =cell 385and K. VUp to the highest fluence (40 mJ/cm ) sampled, the pump inatoms peratunit u.c. is the volume per unit cell [5]. Given a fluence threshold of 13 2 duced 2 reflectivity change at t = 0 increases linearly with pump fluence [Fig. 4.3(a) mJ/cm , a pump beam spot size of 64 µm at FWHM and an optical penetration depth of inset]. However, the relaxation dynamics of photo-excited carriers at short time scales (< 100 fs) changes in a nonlinear way with pump fluence, which is visualof light (800 nm) and k is the imaginary part of the complex refractive index measured by ized by a comparison of the reflectivity transients normalized to their maxima for ellipsometry, we estimate the value of n corresponding to Fth to be ∼ 0.6 per Ru site. different pump fluences [Fig. 4.3(b)]. The carrier relaxation rate becomes slower as pump fluence increases. Optical pump probe reflectivity studies on gold and silver 6 nano structures have established that carrier relaxation time in metals increases with fluence [44, 45]. This can be understood from the two temperature model describing the heat exchange between the electronic subsystem and the lattice subsystem, 95 nm as determined from the Beer-Lambert law ξ = λ/(4πk), where λ is the wavelength 22 (a) Metal 395K 2 385K ∆R/R (10−3) 1 2 0 20 40 1 0 0 0 Mott Insulator 350K 320K 0.5 300K 1 0 −0.5 −1 1.5 (b) 0 1 0 20 2 Time (ps) 40 3 Figure 4.2: Temperature dependent time-resolved optical reflectivity traces. Data are taken (a) above and (b) below TI MT with a pump fluence of 3.8 mJ/cm2 . Traces are vertically offset for clarity. Insets show traces on a longer timescale for T = 395 K and 300 K. and the assumption that ∆R/R measures n, the photo-excited carrier density [46], through which the electronic temperature is defined after electron thermalization. The two temperature model is given by Ce (Te )dTe /dt = −g(Te − Tp ), Cp (Tp )dTp /dt = g(Te − Tp ), 23 0 4 t=0 20 40 Fluence (mJ/cm2) 2 0 40 mJ/cm 19 8.7 3.8 2.0 2 (b) 1 ∆R/R (Norm) ∆R/R (10−2) 5 (a) 0 −0.1 0 0.1 −0.1 Time (ps) 0 0.1 Figure 4.3: Fluence dependence of transient reflectivity at T = 385 K (a) Pump induced reflectivity change at t = 0. Inset shows the linear dependence of ∆R/R(t=0) on fluence. (b) The same set of reflectivity transients normalized to ∆R/R(t=0) to emphasize the change in relaxation dynamics at different pump fluences. (Te - Teq)/(Tinitial - Teq) 1 100 K 150 K 200 K 250 K 0.8 0.6 0.4 0.2 0 0 1 2 3 Time 4 5 Figure 4.4: Simulation of the variation of electronic temperature via heat exchange with lattice. Te : electronic temperature, Tinitial : initial electronic temperature, Teq : equilibrium temperature reached after the electron-lattice thermalization. Legend shows the different values of Tinitial . 24 where g is the electron-phonon coupling constant, Te and Tp (Ce and Cp ) are the electron and lattice temperature (specific heat) respectively. For metals, Ce (Te ) = γTe , where γ = π 2 k B2 g( )/3, k B is the Boltzmann constant and g( ) is the density of states at the Fermi energy. Figure 4.4 shows the time evolution of Te computed from the two temperature model using realistic values of Ce and Cp for different pump fluences. Indeed, a slower carrier relaxation time is expected for a higher pump fluence, which qualitatively agrees with both our results and ref[44, 45]. Therefore, a fluence dependent carrier relaxation can be regarded as a generic feature of metallic system in optical pump probe reflectivity studies. To examine the possibility of a photo-induced insulator-to-metal transition in the low temperature insulating phase, we perform fluence dependent pump probe studies on Ca2 Ru0.92 Fe0.08 O4 at ambient temperature. Figure 4.5 shows that the reflectivity transient scales linearly with pump fluence up to Fth ∼ 13 mJ/cm2 . Beyond that, ∆R/R(t = 0) saturates and starts to decrease just before the damage threshold around 40 mJ/cm2 [Fig. 4.5(b)]. In addition, carrier relaxation rate does not change below Fth but slows down above Fth [Fig. 4.5(c)(d)], which consistent with the response of a metallic system. We can eliminate photo-heating as the origin of the relaxation rate slowing-down because for the same pump fluence the relaxation dynamics remains unchanged from 295 K to 350 K [Fig. 4.6], and a pump fluence at Fth is not able to significantly heat the sample from 295 K. Since 1.55 eV is close to the isosbestic point in the optical conductivity spectrum of Ca2 Ru0.92 Fe0.08 O4 , reflectivity change at this wavelength is expected to be small between the insulating phase and the metallic phase. A more dramatic change is expected in the terahertz (THz) spectrum. Figure 4.7 shows the THz reflectivity transient at a few difference pump fluences across Fth , where only a small change (∆E/E < 1%) is observed after optical excitation. In addition, the THz reflectivity transient shows a long-lived plateau from 1 ps onwards, which persists to hundreds of picoseconds. Although the peak of THz reflectivity transient also saturates above Fth , the magnitude of the plateau does not show any critical behavior with pump fluence. Instead, it scales linearly with pump fluence [Fig. 4.7(b)]. The lack of dramatic increase in ∆E/E above Fth is inconsistent with a photo-induced metastable metallic phase. Instead, the observed change in ∆E/E is consistent with heating. The linear fluence dependence of the plateau suggests that the material continues to absorb photon even beyond the apparent saturation in ∆R/R and ∆E/E. Therefore, although a metastable metallic phase is unlikely to be present above Fth , a funda- 25 0 0 −2 ∆R/R (10 ) t = 0 ps Fth ~ 13mJ/cm2 −3 (a) −0.1 ∆R/R (Norm) 0 −1 −1 −3 (b) 2 01.2 mJ/cm 0.1 0.2 0.3 2.8 5.5 10 13 16 21 26 31 0 t = 0.2 ps −0.3 −0.4 −0.5 (c) −0.1 −2 0.1 0.2 0.3 Time (ps) −0.6 (d) 10 20 30 2 Fluence (mJ/cm ) Figure 4.5: Fluence dependence of transient reflectivity at T = 295 K.(a) Fluence dependent reflectivity transients show (b) a linear increase in ∆R/R(t=0) with fluence up to 13 mJ/cm2 , which is marked as Fth and the red dotted line. (c) The same set of reflectivity transients normalized to ∆R/R(t=0) to contrast their relaxation dynamics. (d) To directly visualize the change in relaxation rate, the value of ∆R/R(t=0.2 ps) is extracted from each reflectivity transient and plotted as a function of fluence. A discontinuity is also observed near Fth ∼ 13 mJ/cm2 . mental change in the electronic structure must have taken place to account for the saturation as well as the change in the carrier relaxation. A most likely scenario is a partial spectral weight transfer into the Mott gap by photo-excitation above Fth , either due to photo-doping effect or reduced electron correlation. A finite spectral weight at the Fermi energy would give rise to a slower carrier relaxation with increasing pump fluence, as predicted by the two tempera- 26 ∆ R/R (Norm) 0 295 K 310 320 330 340 350 -1 -0.1 0 0.1 0.2 0.3 Time (ps) Figure 4.6: Temperature dependence of transient reflectivity. The fluence used is 3.8 mJ/cm2 . The reflectivity transient shows little dependence on temperature. ∆E/E (10 -3 ) 12 12 4 mJ/cm2 9 18 38 8 8 4 4 0 ∆E/E(t = 0) ∆E/E(t > 10 ps) -2 0 2 Time (ps) 4 0 10 20 30 0 Fluence (mJ/cm2) Figure 4.7: Fluence dependence of transient terahertz reflectivity at T = 295 K. (a) THz reflectivity transient for a few different pump fluences across Fth ∼ 13 mJ/cm2 . A long-lived plateau in ∆E/E is observed from 1 ps onwards, which persists out of the window to hundreds of picoseconds. (b) Fluence dependence of the ∆E/E peak (blue) and plateau (green). ture model. However, this state is distinct from a truly metallic state characterized by the collapse of the Mott gap and a high THz reflectivity. In fact, lattice dynamics reveals that Ca2 Ru0.92 Fe0.08 O4 retains the low temperature insulating structure 27 even above Fth , which does not support a metallic electronic state [23]. Figure 4.8(a) shows the coherent optical phonon oscillations extracted from the optical reflectivity transient at different fluences. At low fluences, the coherent oscillation is centered at 3.3 THz and can be attributed to an Ag optical phonon mode [42]. As fluence increases towards Fth , damping becomes stronger. This is most likely due to better screening by more photoexcited electrons. Simultaneously, a second mode becomes visible near Fth . This mode is heavily damped and is only resolvable within the first 1 ps. Fourier transform shows that the Ag mode does not show any softening across Fth [Fig. 4.8(b)] while in thermally induced IMT it completely softens at TI MT [Fig. 4.8(c)]. Since the Ag mode is robust through Fth , we conclude that Ca2 Ru0.92 Fe0.08 O4 retains the insulating S-Pbca lattice structure above Fth . The electronic properties of Ca2 RuO4 have been shown to be highly sensitive to the occupancy of the d xy and d xz/yz orbitals within the 2/3 - filled t 2g manifold. Above TI MT where the L-Pbca structure is realized, the d x y and d xz/yz orbitals are approximately equally occupied and the system realizes an electronically isotropic metal. Below TI MT , on the other hand, cluster dynamical mean-field theory calculations [23] show that the L-Pbca to S-Pbca structural distortion shifts the orbital occupancies by altering their crystal-field splitting and relative bandwidths, creating a completely filled d x y band and half-filled d xz/yz bands. This, in turn, generates a Mott instability in the d xz and d yz bands that drives the system into a d xy ferro-orbital ordered insulating phase. Our results suggest that optical excitation within the t 2g complex using 1.55 eV photons induces spectral weight transfer into the Mott gap without collapsing the Mott gap. As a result, the material gains some metallicity but is far from being a real metal. Moreover, the lattice structure does not undergo a first-order transition as in the thermally-induced IMT case, therefore any transient metallicity would not be supported by the low temperature S-Pbca structure. As a result of all these, optical excitation at 1.55 eV energy does not induce a metastable metallic phase in Ca2 RuO4 . This is in stark contrast with VO2 which has a similar thermally induced IMT concomitant with a monoclinic-to-rutile structural phase transition, and strong photoexcitation in VO2 induces an IMT together with the structural phase transition [9, 37]. The metallic phase in VO2 is metastable and persists for over 100 ps. Such contrasting lifetimes may be due to the fact that the there exist metastable structural phases in VO2 close in energy to the high temperature rutile phase [47, 48], which can be accessed via optical excitation and which are commensurate with an itinerant or- 28 (b) 9.6 mJ/cm2 (c) 395K 8.0 385K 6.6 ∆R/R (Norm) 5.1 TIM 360K 350K Fc 4.1 340K 3.3 330K 2.5 320K 1.7 FFT Intensity (a.u.) (a) 310K 0.9 300K 0.4 295K 1 1.5 Time (ps) 2 2 3 4 2 3 4 Frequency (THz) Figure 4.8: Fluence dependence of phonon dynamics in Ca2RuO4 at T = 295 K. (a) Background-subtracted and normalized (by fluence) reflectivity transients showing coherent phonon oscillations. At low fluence, the coherent phonon is attributed to an Ag mode. Above the threshold fluence, Fth , an additional mode, which is quickly damped within one or two periods, becomes noticeable. (b)(c) Fourier spectrum of the Ag mode at (b) different fluences and (c) different temperatures. bital occupancy. On the other hand, no competing metastable structural phases have been observed in Ca2 RuO4 , making photo-induced IMT difficult to realize. It will also be interesting to search for photo-induced phase transitions in wider classes of ruthenates such as PbRuO3 [49] and even 5d transition metal oxides where the effects of strong spin-orbit coupling give rise to additional competing microscopic interactions. 29 Chapter 5 Optical pump probe investigations of the electronic structures of electron- and hole-doped Sr2 IrO4 5.1 Introduction The Ruddlesden-Popper series of iridium oxides, Srn+1 Irn O3n+1 , realize a unique correlated electronic system somewhat analogous to the cuprate high temperature superconductor family. Both materials crystalize in tetragonal lattice structure, exhibit antiferromagnetic insulating ground state [50], develop similar fermiology upon doping [51–54]. In particular, the single layer Sr2 IrO4 has been shown to host pseudogap [51] and possibly d-wave superconductivity [52] upon electron doping. Interestingly, unlike the undoped cuprate which is a Mott insulator, Sr2 IrO4 shows characteristics of a Slater insulator [55, 56]. Both experiment and theory have suggested that magnetic order is indispensable for the insulating ground state of Sr2 IrO4 [55–57]. On the other hand, La- and Rh-substitution which dopes electron and hole respectively into Sr2 IrO4 are shown to modify the electronic ground state and magnetic order of Sr2 IrO4 . In particular, in Sr2 Ir1−x Rhx O4 , TN decreases continuously with x up to x = 0.17, where magnetic ordering is completely suppressed and Sr2 Ir1−x Rhx O4 becomes a metal [54, 58, 59]. A similar suppression of magnetic ordering and an insulator-to-metal transition is induced in (Sr1−x Lax )2 IrO4 near x = 0.05 [53, 54, 58]. While the electronic ground state of the doped Sr2 IrO4 have been measured at cryogenic temperature [51, 53], a detailed study of their temperature evolution across TN has not been done to our knowledge. This is in part due to issues with energy resolution at elevated temperatures for probes like angle-resolved photoemission spectroscopy (ARPES). In this study, we use time-resolved optical reflectivity, which is suitable for probing electronic structure at elevated temperatures, to study their evolution across TN . In addition, we show that by photo-heating or photo-doping these samples, we close the insulating gap in Sr2 IrO4 . 5.2 Experimental For our time-resolved optical reflectivity study, we employ a regeneratively amplified Ti:Sapphire laser that outputs 80 fs pulses centered around 795 nm (1.5 eV) at 10 kHz repetition rate. The pump beam passes through a delay stage and a mechanic chopper chopping at half of the laser repetition rate. The pump and probe 30 beams are cross-polarized and focused at near normal incidence onto the sample with a spotsize of 60 µm FWHM. The reflected probe beam from the sample is detected by a photodiode connected to a lock-in amplifier. All time-resolved reflectivity measurements are carried out in vacuum better than 2 × 10−7 mbar. A balanced detection geometry involving a second reference probe beam, together with polarization filtering, allows a signal-to-noise ratio better than 10−5 in ∆R/R. Our samples include electron-doped (Sr1−x Lax )2 IrO4 with x = 0.045, hole-doped Sr2 Ir1−x Rhx O4 with x = 0.03, 0.11, 0.15, as well as the insulating parent. These single crystals were grown using the self flux technique. The La and Rh concentration was determined by energy-dispersive X-ray spectroscopy (EDX). Details of the growth procedure can be found in ref [60]. 5.3 Results The insulating ground state of Sr2 IrO4 is due to strong spin-orbit coupling of the 5d electrons which splits the d 5 electron configuration of Ir4+ into a fully filled Jeff = 3/2 band and a narrow half-filled Jeff = 1/2 band. Moderate Coulomb interaction is then able to split the Jeff = 1/2 band into an empty upper Hubbard band (UHB) and a fully-occupied lower Hubbard band (LHB). Optical conductivity measurements have shown that the Jeff = 3/2 valence band and the Jeff = 1/2 LHB are separated from the Jeff = 1/2 UHB approximately by 1 eV and 0.5 eV respectively [61]. Therefore, the 1.5 eV excitation pulse promotes electrons from the bottom of the Jeff = 3/2 valence band to the Jeff = 1/2 UHB and the degenerate probe beam monitors the subsequent relaxation of the photo-excited electrons. Earlier time-resolved optical reflectivity experiment on Sr2 IrO4 has established a bi-exponential relaxation dynamics for the photo-excited electrons for excitation fluence less than 20 µJ/cm2 [56]. In this model ∆R/R = Ae−t/τ1 + Be−t/τ2 + C, τ1 was interpreted as the time constant for electron thermalization via electron-phonon scattering, τ2 was understood as the time constant for bimolecular recombination of electron-hole pairs across the Mott gap. By a few picosecond delay, these dynamical processes are completed and the probed region of the sample reaches a quasi-thermal equilibrium. This part of the reflectivity transient is described by the constant C representing the change in reflectivity due to the increased temperature of the sample after photo-excitation. C decays to zero on a much longer timescale limited by phonon-phonon scattering time. Figure 5.1 shows the reflectivity transients from Sr2 IrO4 at a few different temperatures across TN ∼ 240 K. To visualize 31 the slow component of the bi-exponential relaxation, we plot the normalized reflectivity transients on log scale in Fig. 5.1(b). By comparing these reflectivity transients to an exponential function e−t/τ with τ = 1 ps, it can be seen that the τ2 is on the order of 1 ps. Similar exponent is also observed in the relaxation of ∆R/R in La- and Rh-doped samples and is the main focus of this study. #10 -3 10 0 0 10 "R/R -2 -1 -3 77 K 120 K 160 K 200 K 240 K 280 K 77 K 120 K 160 K 200 K 240 K 280 K -4 -5 10 -2 -6 0 2 Time (ps) 4 60 1 2 3 4 "R/R(Norm) -1 10 -3 Time (ps) Figure 5.1: . (a) Reflectivity transients from Sr2 IrO4 across TN ∼ 240 K using 20 µ/cm2 excitation fluence. (b) The same reflectivity transients in (a) with their constant offset C subtracted and then normalized to the peak, plotted on log scale to emphasize the slow component of the bi-exponential relaxation. The dotted line is e−t/τ with τ = 1 ps. Figure 5.2 shows the normalized reflectivity transients from Sr2 IrO4 as well as the La- and Rh-doped Sr2 IrO4 from 77 K to 280 K. The Néel temperature for these three samples are 240 K, 210K and negative respectively. In Sr2 IrO4 , the relaxation rate first slows down on heating up towards TN . Once above TN , it increases again. This is corroborated by the fit results shown in Fig. 5.2(d), where τ2 peaks near TN . In comparison, the La- and Rh- doped samples show monotonic increase in relaxation rate in the same temperature range. Previous time-resolved optical reflectivity experiment has found evidence for Mott gap closing above TN in Sr2 IrO4 [56]. This is corroborated by the peaking of τ2 near TN in our data. The divergence of electron relaxation time near gap-closing is both predicted by theory and experimentally observed in a number of systems [56, 62, 63]. In the presence of a gap ∆ at the Fermi energy, non-radiative recombination of an electron-hole pair creates a boson, typically in the form of a phonon, of energy ∆. This boson can un- 32 dergo the reverse process to self-annihilate and create a new pair of electron-hole, or decay anharmonically into bosons of smaller energy. The rate of the two processes are determined by the number density of the electron-hole pairs and the low energy bosons, which is dominated by thermal population. Near the gap-closing temperature, ∆ is small and thermally occupied bosons with energy greater than ∆ are abundant. Therefore, electron-hole pair recombination is thwarted, leading a divergence in their relaxation time. Detailed mathematical formulation of this model can be found in ref [63]. Here, it suffices to note that τ2 diverges near TN in Sr2 IrO4 but monotonically decreases in La- and Rh-doped Sr2 IrO4 . Together with the lack of bimolecular recombination signature in the latter two samples, this indicates that the gaps in the two systems are already at least partially filled with a finite density of states down to 77 K. Further understanding of metallic response in time-resolved optical reflectivity can be obtained from the fluence dependence of relaxation rate. When a gap is present at the Fermi energy, as discussed in [56] the rate of electron-hole recombination increases with excitation fluence due to the greater probability for an electron to find a hole. However, when there is a finite density of states (DOS) at the Fermi energy, the number of excited electrons, n, after electrons thermalize among themselves is directly related to the electronic temperature by an approximate linear relation: dn ∝ dTe . Since ∆R/R measures n, it is also a measure of the electronic temperature. Under this interpretation, τ2 , the slow component of reflectivity relaxation, describes the heat exchange between the electron and phonon subsystems that brings the entire system to quasi-thermal equilibrium via electron-phonon scattering. Using a two temperature model, we can describe the heat exchange between the electrons and the lattice by the coupled equations Ce (Te )dTe /dt = −g(Te − Tp ) (5.1) Cp (Tp )dTp /dt = g(Te − Tp ) (5.2) where g is the electron-phonon coupling constant, Te and Tp are the electron and phonon temperatures respectively. For metals, the electronic specific heat Cp (Te ) = γTe , where γ = π 2 k B2 g( )/3, k B is the Boltzmann constant and g( ) the DOS at the Fermi energy. Using experimentally determined values of γ for Rh-doped Sr2 IrO4 and Cp (T), we simulate the heat exchange process and find that photo carrier relaxation becomes slower with higher excitation fluence, as shown in Fig. 5.3(d). Indeed, fluence dependent reflectivity transients on La- and Rh-doped Sr2 IrO4 at 33 0 1 2 3 4 5 0 1 2 3 4 0 0.1 0 -0.1 -0.1 -0.2 "R/RNorm 5 4.5% La Sr2IrO4 -0.2 -0.3 -0.3 -0.4 -0.4 77 K 120 K 160 K 200 K 240 K 280 K -0.5 -0.6 -0.7 -0.8 77 K 120 K 160 K 200 K 240 K 280 K -0.5 -0.6 "R/RNorm 0.1 -0.7 -0.8 -0.9 -0.9 -1 0.1 -1 1 0 0.95 -0.1 3% Rh 0.9 -0.3 -0.4 0.85 77 K 120 K 160 K 200 K 240 K 280 K -0.5 -0.6 -0.7 -0.8 0.8 Sr2IrO 4 4.5% La 3% Rh -0.9 -1 0 1 2 3 Time (ps) 4 =2 (ps) "R/RNorm -0.2 5 100 150 200 250 0.75 0.7 Temperature (K) Figure 5.2: Reflectivity transients from (a) Sr2 IrO4 , (b) (Sr1−x Lax )2 IrO4 with x = 0.045, (c) Sr2 Ir1−x Rhx O4 with x = 0.03, for a range of different temperature using 200 µ/cm2 excitation fluence. The reflectivity transients have their offset C subtracted and then normalized to the peak in order to better visualize the relaxation rate. (d) The time constant of the slow exponential, τ2 , from fit is plotted against temperature for the three samples. 77 K exhibit a relaxation rate that slows down with increasing pump fluence: Fig. 5.3(b)(c). On the other hand, Fig. 5.3(a) shows that at 77 K, the relaxation rate in Sr2 IrO4 increases with fluence initially (< 1 mJ/cm2 ), consistent with the bimolecular recombination of electron-hole pairs, but then it slows down with higher excitation fluence (> 1 mJ/cm2 ), signifying a filling of DOS at the Fermi energy. One plausible explanation for this photo-induced gap filling is that the electron temperature is heated above TN by photo-excitation, reproducing the Mott gap closing above TN in equilibrium measurements. The electron temperature could be esti- 34 0 1 2 3 4 5 0 1 2 3 4 5 0 -0.1 -0.1 -0.2 "R/RNorm 0.1 0 -0.2 Sr2IrO4 -0.3 4.5% La -0.3 -0.4 -0.4 -0.5 -0.5 -0.6 -0.6 -0.7 -0.7 -0.8 -0.8 -0.9 -0.9 -1 0.1 -1 0 1.2 1 1 0.8 0.8 0.6 "R/RNorm 0.1 "R/RNorm -0.2 -0.3 3% Rh -0.4 -0.5 0.09 mJ/cm 2 0.2 mJ/cm 2 1.0 mJ/cm 2 2.3 mJ/cm 2 3.8 mJ/cm 2 6 mJ/cm 2 -0.6 -0.7 -0.8 -0.9 0.6 0 2 4 0.4 6 "Te /Te Norm =2 (ps) -0.1 2 Fluence (mJ/cm ) 0.2 -1 0 0 1 2 3 Time (ps) 4 0 5 1 2 3 4 5 Time Figure 5.3: Reflectivity transients from (a) Sr2 IrO4 , (b) (Sr1−x Lax )2 IrO4 with x = 0.045, (c) Sr2 Ir1−x Rhx O4 with x = 0.03 at 77 K for a range of different excitation fluences. The reflectivity transients are normalized in the same way as in Fig. 5.2. (d) Simulation of the decay of electron temperature over time for different initial temperatures. Each decay is normalized to the initial temperature change. Inset shows the time constant of the slow exponential, τ2 , extracted from fit against fluence for the three samples. mated using Q e = 1/2γTe2 , and the amount of pulse energy absorbed is ∆Q e = (1 - e−1 )FVm /ξ, where F is the pump fluence, Vm is the molar volume, and ξ is the optical penetration depth of Sr2 IrO4 at 795 nm. ξ is estimated similarly to [56]. Using this, a fluence of 14 µJ/cm2 is found to be necessary to heat the electrons instantaneously to TN = 240 K, way much smaller than the threshold fluence of 1 mJ/cm2 . However, if we stipulate that the entire electron-lattice system be heated above TN after thermal equilibration, then the two temperature model above gives 35 a critical fluence of about 3 mJ/cm2 , which is close to our experimentally observed threshold fluence. Another explanation for the photo-induced gap filling is photo-doping. Since each absorbed photon creates an electron-hole pair, photo-excitation can be thought as producing similar effects as chemical doping. The density of electron-hole pairs created by the pump pulse can be estimated by n = (1 - e−1 )N ·N A/(A·ξ), where N is the number of photons per pump pulse, N A Avogadro’s number, A the area of the pump beam, ξ the optical penetration depth. Since La- and Rh-doping on the order of a few percent is enough to create a finite DOS at the Fermi energy, an electron-hole pair density on the order of n ∼ 0.01 is sufficient for inducing an insulator-to-metal transition. This translates into a fluence on the order of a few 100 µJ/cm2 . The inset of Fig. 5.3(d) shows that τ2 increases with fluence more steeply for Sr2 IrO4 (> 1 mJ/cm2 ) than for La- and Rh-doped samples. This is expected because γ is proportional to the DOS at the Fermi energy. Experimentally γ has been measured to be ∼ 1.8 mJ/mol·K2 for Sr2 IrO4 and increases to ∼ 7 mJ/mol·K2 for 11% Rh-doped Sr2 IrO4 . A larger γ reduces the initial electron temperature, which according to the two temperature model [Fig. 5.3(d)], causes τ2 to be smaller. Fig. 5.4(a)(b) shows the reflectivity transients from 11% Rh-doped Sr2 IrO4 as a function of temperature and fluence. Interestingly, a negative component reminiscent of the lower Rh-doped samples is present in reflectivity transient at low temperature and low fluence. It disappears with both temperature and fluence, leaving behind a purely positive response similar to the 15% Rh-doped sample [Fig. 5.4(d)]. For the 15% Rh-doped sample, however, ∆R/R is entirely positive and is linear with fluence. The insensitivity of the relaxation rate to fluence in 15% Rh-doped Sr2 IrO4 might be due to a much larger value of γ in the sample. Overall, our data suggest that the high temperature or the high fluence limit of the 11% Rh-doped Sr2 IrO4 approaches that of 15% Rh-doped Sr2 IrO4 in the optical pump probe reflectivity study, suggesting similar effects between temperature increase and photo-excitation in Rh-doped Sr2 IrO4 . In conclusion, our pump probe reflectivity study of differently doped Sr2 IrO4 shows that a finite density of states is present in the Mott gap above TN in parent Sr2 IrO4 and above 77 K in Rh- and La-doped Sr2 IrO4 . Moreover, photoexcitation induces a similar filling of the Mott gap in parent Sr2 IrO4 . All these observations point to a similar role between heating and thermal excitation in Sr2 IrO4 . 36 0 1 2 3 5 0 1 2 3 77 K 140 K 200 K 260 K 6 Δ R/R(10-4) 4 5 5 12 0.09 mJ/cm 2 0.2 mJ/cm 2 1.0 mJ/cm 2 2.3 mJ/cm 2 6.0 mJ/cm 2 8.5 mJ/cm 2 4 3 4 10 8 6 4 11% Rh 2 2 0 11% Rh 1 1 0.09 mJ/cm 2 0.2 mJ/cm 2 1.0 mJ/cm 2 2.3 mJ/cm 2 6.0 mJ/cm 2 8.5 mJ/cm 2 11% Rh 0.6 -4 0.5 Δ R/RNorm 0.2 mJ/cm 2 2.3 mJ/cm 2 8.5 mJ/cm 2 0.8 Δ R/RNorm* -2 77 K 0 1 Δ R/RNorm* 7 0.4 0 0.2 270 K 0 0 1 2 3 Time (ps) 4 77 K 15% Rh 5 0 1 2 3 Time (ps) 4 5 -0.5 Figure 5.4: (a) Reflectivity transients from Sr2 Ir1−x Rhx O4 with x = 0.11 for different temperatures with an excitation fluence of 200 µJ/cm2 . (b) Reflectivity transients from x = 0.11 sample at 77 K with different excitation fluences, normalized by the fluence. (c) Reflectivity transients from x = 0.11 sample at 270 K with different excitation fluences, normalized by the peak. (d) Reflectivity transients from x = 0.15 sample at 77 K with different fluences, normalized in the same way as in Fig. 5.2. 37 Chapter 6 Hidden order phase in Sr2 IrO4 revealed by second harmonic generation rotational anisotropy 6.1 Introduction In an archetypical Mott insulator, strong Coulomb repulsion (U) forbids intersite hopping and splits an otherwise half-filled conduction band into a fully filled lower Hubbard band (LHB) and an empty upper Hubbard band (UHB) below and above the Fermi energy. This is exemplified by 3d transition metal oxides (TMOs), the first discovered Mott insulators. In 4d and 5d TMOs, the delocalization of 4d and 5d electrons reduces U and places many TMOs near the border of insulator-tometal transition. The fragility of the insulating ground states in these compounds is illustrated in Ca2 RuO4 , which is a 4d TMO. 5d TMOs, with even weaker U, are therefore expected to be metallic. Despite having an odd number of electrons per formula unit and a very small U, 5d TMO Sr2 IrO4 has an insulating gap of about 0.4 eV as determined from optical conductivity measurement [61]. The insulating nature of Sr2 IrO4 in the presence of delocalized 5d electrons arises from strong spin-orbit coupling (SOC) interaction of the 5d electrons. Strong SOC splits the five t 2g orbital states in Ir4+ into lowerlying spin-orbit coupled Jeff = 3/2 states and higher-lying Jeff = 1/2 states. The Jeff = 3/2 states are fully filled by four electrons while the Jeff = 1/2 states are partially occupied by one electron. With the reduced bandwidth of the Jeff = 1/2 states, small Coulomb interaction is now able to open up an insulating gap in Sr2 IrO4 [50]. As shown in Fig. 6.1(a), Sr2 IrO4 crystalizes in the tetragonal K2 NiF4 structure as La2 CuO4 [50]. Despite strong SOC, the magnetic interaction in Sr2 IrO4 is Heisenberg-type with three-dimensional long range order below TN ∼ 230 K. The Jeff = 1/2 moments arrange in a canted antiferromagnetic order which produces a weak in-plane moment [Fig. 6.1(b)]. Due to the similarity in the lattice, electronic and magnetic structure between Sr2 IrO4 and cuprate high-Tc superconductor (HTSC) parent compound, it is anticipated that electronic phases similar to those in cuprate HTSC might also realize in doped Sr2 IrO4 . Almost concurrent with our experiments, pseudogap and possibly d-wave superconductivity are reported in electron-doped Sr2 IrO4 [51, 52]. In this study we use SHG-RA to search for differences between the initial and intermediate the denominator factors involving energies and states (ℏ)wim and the lifetime broadening energy lifetimes of the intermediate states in Eq. 2 can G. The interference between various scattering drop out. A careful analysis of the scattering paths is directly reflected in the scattering inten- intensity can show that the wave function given sities of the photon, and in this way the valence by Eq. 1 represents the ground 38 state in Sr2IrO4 electronic states can be detected with phase sen(4). symmetry-broken phases in Sr IrO that might be similar to those in cuprate HTSC. 2 A Sr2IrO4 4 B H=0 C H>HC b due to t than SO state wil become other lim be equal T=10 K a z=5/8 z=3/8 c b a z=1/8 Fig. 3. pattern ofstructure Sr2IrO4. (A) Sr2IrO Figure 6.1:Magnetic Latticeordering and magnetic of Layered Sr2 IrO4crystal . A. structure Each unitof cell of4, of a tetragonal unit cell (space group I41/acd) with lattice parameters a ≈ 5.5Å and Sr2consisting IrO4 consists of four layers of IrO6 octahedra, which undergo a staggered correc ≈ 26Å (4). The blue,◦ red, and purple circles represent Ir, O, and Sr atoms, respectively. (B) lated rotation of ∼ 12 about the c-axis. 1/2 moments a canted eff =moments Canted antiferromagnetic ordering patternB. of The Jeff =J1/2 (arrows)order withininIrO 2 planes antiferromagnetic fashion below T , which gives rise to a weak in-plane moment. and their stacking pattern along the cNaxis in zero field and in the weakly ferromagnetic state, Figure taken from [64]. determined from the x-ray data shown in (C) to (E) (4). (C and D) L-scan profile of magnetic x-ray diffraction (l = 1.1Å) along the (1 0 L) and (0 1 L) direction (C) and the (0 0 L) direction (D) at 10 K in zero field. The huge fundamental Bragg peak at (0 0 16) and its background removed in (D). r.l.u., reciprocal lattice unit. (E) L-scan of magnetic x-ray diffraction (l = 6.2wereExperiment and results 1.1Å) along the (1 0 L) direction at 10 K in zero field and in the in-plane magnetic field of The SHG-RA areThe performed using 800 nmof (1.5 eV) beams ≈0.3 T parallelexperiments to the plane. (F) temperature dependence the intensity of thefocused magnetic (1 ∼ 0 19) peak (red circles) field H ≈ 0.3 T. The onto 50 µm-diameter spotinonthe thein-plane sample.magnetic The pulse energy for the 800temperaturenm beam dependent magnetization in the in-plane field of 0.5 T is shown by the solid line. is ∼100 nJ and the pulse duration is ∼ 100 fs. The Sr2 IrO4 samples are cleaved in air and then quickly transferred to the optical cryostat and pumped down to vacuum. The reported crystalline point group of Sr2 IrO4 is the centrosymmetric www.sciencemag.org SCIENCE 4/mmm VOL 323 (space group I41 /acd) [64–66]. Figure 6.2 shows the SHG-RA patterns collected from Sr2 IrO4 at room temperature. A centrosymmetric lattice forbids bulk electric dipole radiation χiEj D k E j (ω)E k (ω), instead SHG has to come from bulk electric quadrupole radiation χiQD E (ω)∇ k El (ω) or surface electric dipole radiation j kl j (where inversion symmetry is broken and χ E D does not vanish). Assuming the bulk electric quadrupole radiation, the SHG-RA data can be nicely fit assuming a centrosymmetric 4/m point group. The reported 4/mmm point group requires mirror planes in the ac and bc planes, which are broken due to the rotation of the SHG-RA peaks from the a, b axes. Moreover, the form of χQD 4/mmm constrains the Pin -Sout (PS) and Sin -Sout (SS) polarization combinations to be eightfold rotational 18 D T=10 K 13 E 19 14 1 T=10 K 16 F (1 0 17) Intenstiy net moment Intensity (arb. units) z=7/8 0 100 6 MARCH 2009 P H Y S I C A L R E V I E W L E39 TTERS 0.1 (f) b a 0.0 0.6 (c) b a 0.1 a 0.0 0.15 a 0.15 295 K Data (g) b b 0.6 a a 0.0 0.6 (h) 0.2 0.0 0.2 (d) b 0.00 IPS(3ω)/IPP (3ω) 0.05 0.6 (b) b IPP(3ω)/IPP (3ω) IPS(2ω)/IPP (2ω) a a 0.00 (c) ISP(2ω)/IPP (2ω) (a) b 0.05 (b) (d) (e) b ISS(3ω)/IPP (3ω) ISS(2ω)/IPP (2ω) (a) ISP(3ω)/IPP (3ω) PRL 114, 096404 (2015) 1.0 0.5 0.0 0.5 1.0 180 K Data FIG. 3 (color online). THG-RA pattern (b) PS, (c) SP, and (d) PP geometries. T using 1200 nm incident and 400 nm r T ¼ 295 K (black circles) and T ¼ 1 The magnitudes of all patterns are norm Red lines are best fits to the 295 K data us induced THG-RA calculated using the point group. IPP(2ω)/IPP (2ω) shows THG-RA patterns with best fits induced THG from a 4=m point group 1.0 the SHG data (Fig. 2), the peaks and va 0.5 patterns are rotated away from the high of the crystal, which cannot be repr a 0.0 a 4=mmm point group [33]. Moreover, th 0.5 the lobe amplitudes in the PS and SP g only be described using a 4=m point gro 1.0 expression for the PS and SP patte 4=mmm point group, which is propor Data 4/m 4/mmm cannot account for these features [33]. The SHG-RA and THG-RA pattern m 222 422 mm2 Sr2 IrO4 exhibits a globally reduced metry that is best described by the I41 FIG. 2 (color online). SHG-RA patterns (open circles) taken Figure 6.2: SHG-RA patterns collected from Sr2 IrO4 atTheroom temperature. implies that the c- and d-glide planes p under (a) SS, (b) PS, (c) SP, and (d) PP geometries. data (a)(b)(c)(d) SHG-RA patterns four polarization combinations. in-I41 =acd description are ac were collected usingin 800 nm different incident and 400 nm reflected light existThe in the T ¼ 295are K. normalized The magnitudes allpeak patterns to pattern. tensities of all atpatterns toofthe in are thenormalized Pin -Pout (PP) can onlyRed occur if the tetragonal disto the PP trace. Red and cyan lines are best fits to bulk electric octahedra on the two sublattices are in (green) solid line is fit to the 4/m (4/mmm) point group assuming bulk electric quadrupole induced SHG-RA calculated using centrosymmetric and 4(b)] [33]. Owing to the str quadrupole radiation. to other point groups 4=m and (e)(f)(g)(h) 4=mmm pointFits groups, respectively. (e)–(h) assuming Best fit bulk electric coupling in Sr2 IrO4 [36–38], this results to bulk electric dipole induced SHG calculated using the quadrupole radiation. distortion, which is present in our d three proposed noncentrosymmetric point groups mm2, 222, and below the Néel temperature T N ¼ 240 422 as well as the monoclinic point group m for comparison. Responses that are absent in the plots are symmetry forbidden. likely influence how the magnetic mo octahedral rotations. The most widely used model [2] f measurements are established bulk sensitive probes of relationship between the moment can centrosymmetric crystals [34,35] because the bulk electric octahedral rotation angle α [Fig. 4(a dipole contribution χ ED ijkl (even rank) is allowed. Figure 3 b b 40 symmetric in intensity. This clearly contradicts the PS and SS data. Together, under the assumption of bulk electric quadrupole radiation, these evidences suggest that the correct crystalline point group of Sr2 IrO4 is 4/m instead of 4/mmm. In order to rule out the possibility that a non-negligible surface electric dipole radiation is present and interferes with the bulk electric quadrupole radiation to give rise to the observed SHG-RA patterns, we carry out third harmonic generation rotational anisotropy (THG-RA) experiment. THG is bulk-sensitive because the χiEj D kl is allowed in centrosymmetric media. Therefore, compared to the bulk electric dipole contribution to THG, surface electric dipole contribution to THG is negligible. The experiment is performed using 1.2 µm incoming beam produced from an optical parametric amplifier. Figure 6.3 shows the THG-RA patterns collected from Sr2 IrO4 at room temperature. Similar to SHG-RA patterns, the peaks and valleys in THG-RA patterns are also rotated from the a, b axes. Additionally, for THG-RA, ED χ4/mmm also constrains PS and SP patterns to be eightfold rotational symmetric in intensity, which is again contradicted by the data. Instead, the bulk electric dipole contribution from the 4/m point group describes the THG-RA data remarkably well. Therefore, we conclude that the crystalline symmetry of Sr2 IrO4 is indeed described by the point group 4/m (space group I41 /a). The reduction of crystalline symmetry from 4/mmm to 4/m necessitates the removal of the c- and d-glide planes in the I41 /acd description. This can only occur if the tetragonal distortions of the iridium-oxygen octahedra on the two sublattices are inequivalent [Fig. 6.4(b)]. This has an important consequence on the relationship between the Jeff = 1/2 moment canting and the IrO6 octahedra rotation in Sr2 IrO4 . A widely studied model for understanding the relationship between the moment canting angle φ and the octahedral rotation angle α [Fig. 6.4(a)] is developed by Jacklie and Khaliullin [67] for a I41 /acd space group with uniform octahedral distortion. It was found that φ/α depends only on one parameter θ defined through tan(2θ) = √ 2 2λ/(λ-2∆), where λ and ∆ are the strength of spin-orbit coupling and tetragonal crystal field splitting. A perfect magnetoelastic locking (φ/α = 1) is predicted in the cubic limit (∆ = 0) [Fig. 6.4(c)]. Using commonly accepted values [50, 67–70] of λ (∼ 400 meV) and ∆ (∼ 140 meV), the JK model predicts φ/α ∼ 0.7, which contradicts recent neutron and resonant x-ray diffraction studies that report values of φ = 13(1)◦ [71] and 12.2(8)◦ [72] and α = 11.8(1)◦ [71], which indicate a nearly perfect magnetoelastic locking. If the model is modified to allow a staggered tetragonal distortion between the two week ending 6 MARCH 2015 PHYSICAL REVIEW LETTERS (f) 0.6 (c) b a a a 0.0 0.15 a 0.15 295 K Data (g) b a (h) b 0.2 b a 0.0 0.2 (d) b 0.00 IPS(3ω)/IPP (3ω) 0.6 (b) b IPP(3ω)/IPP (3ω) a a ISS(3ω)/IPP (3ω) (a) b ISP(3ω)/IPP (3ω) (e) 41 1.0 b 0.5 a 0.0 0.5 1.0 180 K Data 295 K Fit 3 (color online). THG-RA patterns taken under (a) SS, Figure 6.3:FIG. THG-RA patterns collected from Sr IrO . (a)(b)(c)(d) THG-RA (b) PS, (c) SP, and (d) PP geometries. The2 data4 were collected patterns in four different polarization combinations normalized to the peak in the using 1200 nm incident and 400 nm reflected light at both Pin -Pout (PP) pattern. red solid line and is fit Tto¼the180 4/m group assuming bulk T¼ 295 KThe (black circles) K point (green circles). electric The radiation. magnitudes of all patterns are normalized to the PP trace. a dipole Red lines are best fits to the 295 K data using bulk electric dipole induced THG-RA calculated using the centrosymmetric 4=m sublattices point (∆1 = −∆2 ≡ ∆) [68, 73], which is consistent with the I41 /a space group. group, we found that φ/α becomes very insensitive to both λ and ∆ [Fig. 6.4(d)]. Therefore, shows we conclude that patterns a staggered tetragonal distortion explains the reduction THG-RA with best fits to bulk electric dipole induced THGfrom from4/mmm a 4=m point group Similar to of crystalline symmetry (space groupoverlayed. I41 /acd) to 4/m (space group the SHG data (Fig. 2), the peaks and valleys of the THG-RA I41 /a) as well as almost perfect magnetoelastic locking in Sr2 IrO4 . patterns are rotated away from the high symmetry directions of thecrystalline crystal, which cannot by to fitsearch to a for possiWitha the correct symmetry of Srbe 2 IrOreproduced 4 , we are ready a 4=mmm point group [33]. Moreover, there is a modulation of the SHGble symmetry-broken phases in it at low temperature. In Fig. 6.5, we plot the lobe amplitudes in the PS and SP geometry data that can RA patterns in the PP and PS polarization channels at 295 K and 170 K respectively. only be described using a 4=m point group. The mathematical The rotational symmetry the SHG-RA patterns is clearly reduced from expression forof the PS and SP patterns derived using a C4 to C1 2 at 170K. We can rulepoint out structural and antiferromagnetic as the origin 4=mmm group, which is proportional order to j sinð4ψÞj , of this /m 4/mmm cannot account for neutron these features [33]. X-ray diffraction studies have symmetry breaking: extensive and resonant The SHG-RA and THG-RA together show that 422 mm2 found no evidence for any structural changepatterns in Sr2 IrO 4 below room temperature Sr2 IrO4 exhibits a globally reduced bulk structural sym[64, 71, 72,metry 74] while antiferromagnetic order below TN group. ∼ 230 This K is described that isthebest described by the I41 =a space patterns (open circles) taken 0 bydata the centrosymmetric orthorhombic magnetic group thought 2/m1 which implies that the c- and d-glide planespoint previously to retains d (d) PP geometries. The dent and 400 nm reflected light exist in the I41 =acd description are actually absent, which all patterns are normalized to can only occur if the tetragonal distortions of the oxygen s are best fits to bulk electric octahedra on the two sublattices are inequivalent [Figs. 4(a) culated using centrosymmetric and 4(b)] [33]. Owing to the strong magnetoelastic respectively. (e)–(h) Best fit coupling in Sr2 IrO4 [36–38], this staggered tetragonal uced SHG calculated using the PRL 114, 096404 (2015) P H Y S I C A L R E V I E42W L E T T E R S point of our model is a m that includes both SOC Its ground state is a K spin composition is de of SOC and tetragonal a Jeff ¼ 1=2 descriptio tetragonal splitting. By splitting (Δ1 and Δ2 ) on each sublattice will in orbital compositions. W pseudospin-1=2 degrees following superexchang H ¼ Σn;n0 JSn Sn0 þ δJ z Szn Szn0 where the isotropic ex δJz and δJ xy , and Dzya functions of the mic Coulomb interaction, a FIG.Inequivalent 4 (color online). (a) distortions Illustration of of IrO an 6IrO axes point along the a 2 plane in Figure 6.4: tetragonal octahedra and spinSr IrO . The oxygen octahedra rotate about the c axis by %α 4 rn;n0 is the unit vecto octahedron2 locking. (a) Illustration of an IrO2 plane in Sr2 IrO4 . The oxygen creating a two sublattice structure. The magnetic moments couple minimization of the octahedra rotate about the c-axis by ±α creating a two sublattice structure. The to the lattice and exhibit canting angles %ϕ. (b) An unequal magnetic moments couple to the lattice and exhibit canting angles ±φ. (b) An employed unto calculate tetragonal distortion Δ2 )theontwo thesublattices two sublattices as by the 1 and equal tetragonal distortion (∆1 (Δ and ∆2 ) on as required methods are fully descr required space group. (c) The λratio as a for the 1 =aφ/α I41 /a space group. by (c) the The I4 ratio as a function of both and ∆ϕ=α calculated In the case of a function of both λ and Δ calculated for the case of uniform and case of uniform and (d) staggered (∆1 = -∆2 ) tetragonal distortion (Δ1 ¼ Δ2 ≡ Δ) as requ (d) staggered (Δ1 ¼ −Δ2 ) tetragonal distortion using Eq. (1), we find that for a fixed assuming a Coulomb energy U 1 ¼ 2.4 eV, a Hund’s coupling from 1 as a function o J H ¼ 0.3 eV,orhopping t ¼ 0.13 order eV, and ¼ 11.5°. C2 symmetry. Ferroantiferroelectric canαalso be ruled out owing to the [Fig. 4(c)]. However, if absence of any anomaly in the temperature dependence of the dielectric constant is staggered between sub and Khaliullin (JK) assuming an I41 =acd space of Sr2 IrOJackeli for this symmetry4 above 170 K [75]. The most plausible explanation consistent with an I41 = group, which onlyof allows a uniform broken phase is the ordering a higher for multipolar degreetetragonal of freedom disthat coexists remarkably insensitive tortion. In the JK model, the ratio ϕ=α depends only on a p ffiffi shows that the magnitu with the Jeff = 1/2 moments in the IrO6 octahedra. parameter θ defined by tanð2θÞ ¼ 2 2λ=ðλ − 2ΔÞ, where λ enced by the spatially and Δtheare the strengths of SOCorder, and we uniform tetragonal To investigate symmetry of this multipolar note that 20/m and m10 distortion are rather than splitting,subgroups respectively. perfect magnetoelastic locking the only CEF two magnetic of theAcrystallographic point group 4/m that do oxygen not octahedron, wh ¼ 1)rotation is predicted cubic limit (Δ ¼ Both 0) where 0 include 2(ϕ=α (two-fold along c)inasthe a symmetry operation. 20/m and m1 local tetragonal distortio the local magnetic degrees of freedom are derived from a of perfect magne break global inversion symmetry and allow a bulk electric dipole contributionvation to Jeff ¼ 1=2 Kramers doublet. Any mixing between the Jeff ¼ SHG on 1=2 top ofand the existing bulk states electricintroduced quadrupole contribution. A coherent sumAlthough we curren Jeff ¼ 3=2 through tetragonal measure of Δ1 and Δ2 distortion causes ϕ=α to be either smaller or larger than 1 the sign of tetragonal C depending on whether the oxygen octahedra are elongated observations of perfect (Δ > 0) or compressed (Δ < 0). Using commonly accepted ence of noncubic struc [2,10,22,39,40] values of λ (∼400 meV) and Δ (∼140 meV) sures of Δ1 and Δ2 usin LETTERS OI: 10.1038/NPHYS3517 43 c b Pin−Pout T = 295 K ϕ 1.0 Pin−Sout 0.06 0.04 0.5 a O 0.02 Ir Sr d 2ω Pin Sout c d Pout Pin−Pout Pin−Sout 0.06 1.0 0.04 0.5 0.02 ϕ T = 170 K Sr2 IrO 4 (001) Figure 6.5: SHG-RA patterns collected from Sr2 IrO4 at 295 K and 170 K. All patterns to layer the peak in4 the Pin -Pout pattern at 295 K.shows Orange solid en order in Sr2 IrO4 . a, Crystal structureare of a normalized single perovskite in Sr2 IrO (tetragonal point group 4/m). Inset to-electric loop-current order, withisthe in he direction of currentbulk flow.electric b, Schematic of the RA-SHG line fit arrows to the pointing 4/m point group assuming quadrupole radiation. Purple larizations of the obliquely incident fundamental (in) and outgoing beam (out) can be independently selected solid line is fit tobeam the coherent sum ofSHG bulk electric quadrupole radiation from a 4/m ndicular (S) to the light scattering plane (shaded). RA-SHG data are acquired by measuring the SHG intensity I(2!) 0 point group and bulk electric dipole radiation from a 2 /m magnetic point group. of Sr2 IrO4 as a function of the angle ' between the scattering plane and crystal a–c-plane. c, RA-SHG data collected under n geometries using = 800 nm incident light at T = 295 K. d, Analogous data to c collected at T = 170 K. All data sets are ale, normalized to a value of 1, which corresponds to ⇠20 fW. The high-temperature data are fitted to time-reversal of The the low-temperature crystallographic quadrupole from the 4/m point group and le-induced SHG (orange lines). dataelectric can only be fitted to the radiation coherent sum of time-reversal invariant 0 d and time-reversal broken bulk electric-dipole-induced SHG (purple lines), as described in the text. the hidden-order-induced electric dipole radiation based on either the 2 /m or m10 magnetic group fits the data extremely [Fig.antiferromagnetic 6.5]. This indicates c 4/m crystallographic point group point of two subgroups (20 /m and m10 )well of the pointthat the 0 0 y well (Fig. 1c). group. Both 2 /m parity and m1 thecannot global arise inversion multipolar hidden order breaks andbreak hence fromsymmetry the parity-even al symmetry of the RA-SHG patterns of the crystal and thus allow a bulk ED contribution to SHG magnetic multipolar orders recently proposed for Sr IrO4 below TN [76,with 77]. re (T = 170 K) is reduced from C4 to on top of the existing EQ contribution,2 which is consistent sed by a structural distortion because the large changes observed in the 0SHG amplitude. Moreover, A microscopic model that satisfies the 20/mpattern or m1 calculated magnetic using point agroup symmetry nant X-ray diffraction studies show no expressions for the RA-SHG coherent 2–5 ymmetry below room temperature . sum of the crystallographic andisthe hidden-order-induced is the magneto-electric loop-current order, EQ which predicted to exist in the pseuby the antiferromagnetic structure that ED contributions (Supplementary Section 2), with the elements dogap region of theEQcuprates [78–80] but can in principle persist even ED at half-filling because that has a centrosymmetric of ijkl derived from a 4/m point group and those of ijk derived [81]. In this model, pair aof20counter-circulating current in each nt group (2/m10 ) with C from aeither /m or m10 point group, fit theloops data forms extremely well CuO2 2 symmetry erro- or antiferroelectric order can also (Fig. 1d). However this indicates that an ordering of higher-rank square plaquette and can be described by a toroidal pseudovector order parameter ence of any anomaly in the temperature parity-even P magnetic multipoles, which preserves global inversion [82,T 83] defined as Ω= rcannot where r is the location of the orbital magnetic moc constant of Sr2 IrO4 above = 170 K symmetry, the origin of our RA-SHG results. i ×mi , be ing of a higher multipolar degree of the other hand, our results can be explained are by an ordering ment mi inside theOnplaquette. Four degenerate configurations possible because the Jeff = 1/2 moment in each IrO6 of higher-rank parity-odd magnetic multipoles that preserves the ost plausible explanation. An ordering translational symmetry of either the antiferromagnetic structure or magnetic multipoles with the same the crystallographic structure. A microscopic model that satisfies he antiferromagnetic structure has, in this condition as well as the 20 /m or m10 point group symmetries is in Sr2 IrO4 below TN (refs 22,23). Such the magneto-electric loop-current order (Supplementary Section 4), e difficult to detect using diffraction- which is predicted to exist in the pseudogap region of the 14–18,20 24 NATURE PHYSICS DOI: 10.1038/NPHYS3517 44 b 295 K a Domain I Ω Domain II 100 µm 175 K Domain III Domain IV 3 | Degenerate ground-state of the hidden Figure 6.6:Figure Degenerate ground states ofconfigurations the loop-current modelorder. The loop-current Four di�erent types of RA-SHG patterns found by performing local order Ω can exhibit four distinct configurations, which are all captured by SHG-RA within all of the domains mapped in Fig. 2b. Red lines are on differentmeasurements regions of the Sr2 IrO4 sample. fits to the expressions described in the text. The large lobes are shaded pink to emphasize the orientation of each pattern. Schematics of the four degenerate magneto-electric loop-current order configurations are shown the two intra-unit-cell current loops can lie along either of theThe tworeddiagonals in the below each pattern to illustrate the possible correspondence. arrows denote thehave direction of the ⌦ in eachconfigurations, plaquette. square plaquette and can either of toroidal the twomoment time-reversed which correspond to four 90◦ -rotated directions of the pseudovector [Fig. 6.6]. Therefore, which suggests that they are not pinned to structural defects in 100 µm four types the of domains arexamine expected to manifest in the loop current phase. crystal. To whether the different patches observed in Fig. 2b correspond to domains with different ⌦ orientation, we To search for these domains in the hidden-order Sr2 IrO perform wide4 , we performed local RA-SHG measurementsphase withinofeach of the patches. exhaustive the entire crystal areawhich in Fig.sees 2b the reveals a change field SHG An imaging in thestudy Pin -Sover channel largest out polarization total of only four types of patches, characterized by the four distinct among all RA-SHG polarization channels. SHG images are collected from a ∼ 500 µm × 500 patterns (Supplementary Section 6) shown in Fig. 3, which µm regionare of the ab surface room temperature andcopies at 175 [Fig.shown 6.7]. At room exactly 0 -, 90 -,at180 - and 270 -rotated of K those rder domains. a,b, Wide-field in Fig. 1. These results are consistent with the hidden phase in a single temperature, SHG image has a uniform intensity that is consistent with he cleaved (001) plane of Sr2 IrO4 Sr2 IrO4 being a magneto-electric loop-current order—but other crystallographic domain. On that cooling = 175 regions of certainly higher and lower eometry at ' = 78 at T = 295 K (a) microscopic models obey to theTsame set K, of symmetries ht and dark regions present in the be ruled out. intensities cannot with lateral dimension on the order of ∼ 100 µm become visible. Upon mains of the hidden order. The dark The ordering temperature T of the hidden phase in Sr2 IrO4 was thermal cycling, the distribution and of these domains of arethe rearranged, sugcrometre-sized spots near the determined by measuring theshapes temperature dependence SHG he high- and low-temperature thepinned Pin –Souttopolarization geometry where the hiddengesting thatintensity they areinnot structural defects in the crystal. Figure 6.6 shows ave either of two time-reversed to four 90 -rotated directions of l, one therefore expects domains order-induced changes are largest. The onset of change is clearly observed at T ⇠ 232 K, as shown in Fig. 4a, which is close to but slightly higher than the value of TN ⇠ 230 K determined by d.c. magnetic susceptibility measurements, and evolves with an order-parameter-like behaviour on further cooling (Supplementary Section 7). In a Landau free energy expansion (Supplementary LETTERS NATURE P 45 a b Pin−Sout 295 K a Domain I Ω 100 µm b Pin−Sout 175 K Domain III Figure 3 | Degenerate ground-state Four di�erent types of RA-SHG patte measurements within all of the dom fits to the expressions described in t to emphasize the orientation of each degenerate magneto-electric loop-c below each pattern to illustrate the p arrows denote the direction of the to which suggests that they are n the crystal. To examine whethe Fig. 2b correspond to domains performed local RA-SHG measu An exhaustive study over the en 0 1 total of only four types of patche I(2ω ) (a.u.) RA-SHG patterns (Supplementa are exactly Figure 6.7:Figure Domains of the hidden order phase revealed by SHG images. (a) 0 -, 90 -, 180 - and 2 2 | Spatial mapping of hidden order domains. a,b, Wide-field in Fig. 1. These results are con SHG imagereflection at 295 SHG K shows uniform intensity across(001) the plane sample consistent microscopy images of the cleaved of Srsurface 2 IrO4 Sr2 IrO4 being a magneto-electr with a single crystalline domain. (b) SHG image at 175 K reveals domains higher measured under Pin –Sout polarization geometry at ' = 78 at T = 295 K (a) ofmicroscopic models that obey th and at T = 175corresponding K (b). A patchwork light and configurations dark regions present theloop current and lower intensities to ofdifferent of inthe cannot be ruled out. low-temperature image arises from domains of the hidden order. The dark pseudovector. The ordering temperature T curves at the sample edge and dark micrometre-sized spots near the determined by measuring the te sample centre that are visible in both the high- and low-temperature intensity in the Pin –Sout polariza images are structural defects. order-induced changes are larg observed at T ⇠ 232 K, as sho in the square plaquette and can have either of two time-reversed but slightly higher than the va configurations, which correspond to four 90 -rotated directions of d.c. magnetic susceptibility me the pseudovector. In a real material, one therefore expects domains order-parameter-like behaviour of all four types to be populated. Section 7). In a Landau free e To search for domains of the hidden order in Sr2 IrO4 we Section 8), a bilinear coupling b 100 µm b x=0 1.0 TN TΩ TN 0.5 x = 0.11 1.0 TΩ ∆IPin−Sout (2ω) normalized ∆IPin−Sout (2ω) normalized 1.0 c x = 0.04 ∆IPin−Sout (2ω) normalized a TN 46 0.5 0.5 four types of domains from an extensive survey of SHG-RA patterns in different regions of the sample surface. These SHG-RA patterns are 0◦ , 90◦ , 180◦ and 270◦ rotated copies of those in Fig. 6.5, and represent the four 90◦ -rotated directions of the loop current pseudovector. Although other microscopic models that obey the 0.0 0.0 0.0 same set of symmetries cannot be ruled out, our observation of the four types of symmetry-broken domains is consistent order being the magneto100 200 100with the hidden 200 T (K) T (K) electric loop-current order. d TΩ 100 200 T (K) 250 Sr2Ir1−xRhxO4 200 SHG (TΩ) 150 d.c. magnetic susceptibility (TN) Refs 8,27 and Supplementary Informati T (K) Hidden order Resonant X-ray di�raction (TN) Ref. 28 100 SRO Néel order ARPES (pseudogap present) Ref. 8 50 LRO 0 0.00 0.05 0.10 0.15 Rh concentration, x Figure 4 | Temperature and6.8: hole-doping dependence the hidden order. a-c, Change in SHG from Srorder. Figure Temperature andof hole-doping phase diagram ofintensity the hidden 2 Ir1 x Rhx O4 at x = 0 (a), x = 0.04 x = 0.11 (c) measured relative to their room-temperature values as a function of temperature. Data were taken under Pin –Sout polarization geometr Red open circle denotes the hidden order onset temperature, TΩ , determined from ' = 78 . The error bars represent the standard deviation over 60 independent measurements. No di�erence was observed SHG experiment with error bar indicated. Black filled diamond denotes TN from between curves measu cooling and heating. Lines are guides to the eye. The dashed red lines mark the transition temperature T of the hidden order phase deduced from d.c. magnetic susceptibility measurement. Green filled triangle denotes TN from data and the dashed black lines mark the Néel temperature TN determined from d.c. magnetic susceptibility measurements. d, Temperature versu resonant X-ray diffraction [59]. Orange filled square denotes pseudogap region phase diagram of Sr2 Ir1 x Rhx O4 showing the boundaries of the hidden order and the long-range (LRO) and short-range (SRO) Néel ordered region ARPES [60]. where a pseudogapfrom is present are also marked, although a pseudogap phase boundary is not yet experimentally known. on hole-doped Sr2 Ir1 x Rhx O4 crystals (Supplementary Section 9) through higher doping may be a route to achieving high-Tc Additional insight for the nature of this hidden order comes from a temperatureto track the evolution of T as a function of Rh concentration or parity-odd29,30 superconductivity in the iridates. Although 28 doping 27phase diagram X-ray of thediffraction hidden order phase. theoretical To do so, studies we determine the or(x). Bulk magnetization and resonant studies are required to establish the microscopi have shown thatdering Néel ordering persists for x . 0.17 and that T is of the hidden phase, the fact that it bears the symmetr N a series of hole-doped Sr Ir temperature TΩ of the hidden phase for 2 1-x Rhx O4 monotonically suppressed with x. But no evidence of any broken magneto-electric loop-current order already suggests sever crystals, which long-range antiferromagnetic order macroscopic is increasingly suppressed symmetry phases beyondinthe Néelthephase has been reported in esting responses, including a linear magneto 18 Sr2 Ir1 x Rhx O4 sowith far. doping. Remarkably, SHG measurements and TΩ isour determined by the onsetshow of the effect increase in non-reciprocal SHG intensityoptical in the rotation17,18 . that the hidden phase transition observed in the parent compound symmetry-broken direction of Pin -Salthough channel. In parent Sr2 IrO4 , TΩ out polarization also persists on hole doping (Fig. 4b,c) and that, T Methods is found be 232 K, close to slightly higher Methods than TN and ∼ 230 as determined is suppressed with x, theto splitting between Tbut and TN grows anyKassociated references are available in the monotonically from approximately 2 K to 75 K between x = 0 and version of the paper. by d.c. magnetic susceptibility measurements. In Sr2 Ir1-x Rhx O4 , TΩ and TN bex ⇠ 0.11. This provides strong evidence that the Néel and hidden orders are not trivially tied, but are independent and distinct Received 5 August 2015; accepted 15 September 2015; electronic phases. published online 26 October 2015 Our finding of a hidden broken symmetry phase in proximity to an antiferromagnetic Mott insulator reveals a striking parallel References between the cuprate and Sr IrO phase diagrams (Fig. 4d), which 47 come increasingly separated as x increases [Fig. 6.8]. This indicates that the Néel order and the hidden order are not trivially tied, but are independent and distinct electronic phases. In fact, in a Landau free energy expansion, a bilinear coupling between the antiferromagnetic Néel and hidden order parameters is forbidden by symmetry, so there is no apriori reason that the two orders should couple or coincide. The close proximity of TΩ and TN observed in parent Sr2 IrO4 suggests some microscopic mechanism by which one order can induce the other through a biquadratic coupling: for example, an enhancement of the superexchange coupling between Jeff = 1/2 moments due to the presence of hidden order. As long range antiferromagnetism is suppressed by hole doping, fluctuations in the hidden order is less likely to affect the ordering of the Jeff = 1/2 moments, hence the increasing separation between TΩ and TN at higher hole dopings. Our finding of a hidden symmetry-broken phase in proximity to an antiferromagnetic Mott insulator reveals a striking parallel between the cuprate and Sr2 IrO4 phase diagrams, which is further strengthened by recent observations of a pseudogap region in Sr2 Ir1-x Rhx O4 using angle-resolved photoemission spectroscopy [60]. Driving the hidden phase to a quantum critical point [84] through higher doping may be a route to achieving high-Tc [85] or parity-odd [86, 87] superconductivity in the iridates. Although further theoretical studies are required to establish the microscopic origin of the hidden phase, the fact that it bears the symmetries of a magneto-electric loop-current order already suggests several interesting macroscopic responses, including a linear magneto-electric effect [79] and non-reciprocal optical rotation [79, 88] . 48 Chapter 7 Charge-density-wave-like instability in (Sr1−x Lax )3 Ir2 O7 revealed by time-resolved optical reflectivity 7.1 Introduction Similar to the single-layer Sr2 IrO4 , Sr3 Ir2 O7 is also a spin-orbit-assisted Mott insulator. Its ground state is an antiferromagnetic insulator with a charge gap ∆ of order 100 meV, as determined by optical conductivity [89], scanning tunnelling microscopy (STM) [90] and angle-resolved photoemission spectroscopy (ARPES) [91] measurements. However, Sr3 Ir2 O7 does not adhere to the strict description of a Mott insulator, where the insulating gap is opened by Coulomb interaction, a case distinct from a Slater insulator where the insulating gap is opened by the antiferromagnetic order. There is evidence from transport [92], optical conductivity [89] and ARPES [93] studies that the electronic density of states (DOS) near the Fermi level changes in the vicinity of the antiferromagnetic transition (TN ∼ 280 K) in Sr3 Ir2 O7 . Moreover, resonant inelastic X-ray scattering experiments [94] [95] show that the energy scale of magnetic exchange is comparable to ∆, raising questions about the insulating nature of Sr3 Ir2 O7 . In addition, unlike the rich electronic phases that have been observed for electron and hole-doped Sr2 IrO4 , no electronic instability similar to those in cuprate high-Tc superconductors has been observed in doped Sr3 Ir2 O7 . Electron-doped (Sr1−x Lax )3 Ir2 O7 induces an insulator-to-metal transition with simultaneous suppression of long-range antiferromagnetism [96]. In addition, a structural phase transition at low temperature with both La- and Ca-substitutions is also observed. Other than these, there has been no report of collective electronic phenomena in electrondoped Sr3 Ir2 O7 . In order to better understand the insulating nature of Sr3 Ir2 O7 and scrutinize electron-doped Sr3 Ir2 O7 for electronic instabilities, we put it to the test of time-resolved optical reflectivity, which is sensitive to small change of electronic structure near Fermi energy even at elevated temperatures. 7.2 Experiment and results In time-resolved optical reflectivity experiment, a pump pulse promotes electrons from the valence band to the conduction band, and a probe pulse monitors the subsequent recombination of the electron-hole pairs. In the case of an insulator, 49 small changes of the electronic structure across the gap can dramatically affect the electron-hole (e-h) recombination, which in turn manifests in time-resolved optical reflectivity. We apply this technique to Sr3 Ir2 O7 first to identify the relationship between the insulating gap and the antiferromagnetic order. Figure 7.1(a) shows the reflectivity transients of Sr3 Ir2 O7 measured with a pump and probe photon energy of 1.55 eV across the Néel temperature (TN ). At all temperatures, pump excitation causes an abrupt drop in reflectivity followed by a slower recovery on the picosecond timescale. Around TN , both the magnitude of the drop and the recovery time become strongly temperature dependent. To understand the temperature dependence in greater detail, the individual reflectivity transients are fitted to an exponential decay function ∆R/R = Aet/τ + C, where A is the amplitude of the reflectivity drop at t = 0, τ is the characteristic photo-carrier relaxation time and C describes much slower equilibration processes such as heat diffusion. Figure 7.1(b) shows that upon heating towards TN , A declines sharply in magnitude while τ exhibits a divergent behaviour, both of which are consistent with the insulating gap closing at TN . Microscopically, this can be understood using the phenomenological Rothwarf-Taylor model [63], where A is treated as being representative of the fractional change in the number of e-h pairs introduced by the pump pulse and τ is the characteristic time for the population of such pairs to decay. Recombination of an e-h pair across the gap is accompanied by the emission of a phonon with energy greater than or equal to ∆. Since these phonons can in turn excite additional e-h pairs, τ is ultimately determined by the time required for these phonons to anharmonically decay into phonons of energy less than ∆ or to escape from the probed region. As the gap closes upon approaching TN , more thermally populated low-energy phonons become available for e-h pair production, impeding the recombination of photo-excited e-h pairs and reducing their ratio to thermally excited e-h pairs, which respectively cause τ to diverge and A to drop precipitously. In the limit where the ratio of photo-excited to thermally excited e-h pairs is small, analytical expressions can be derived [62] for the temperature dependence of both A and τ, assuming a Bardeen-Cooper-Schrieffer (BCS)-like temperature dependence of the gap below TN , which fit the data remarkably well, as shown in Fig. 7.1(b). These results confirm that the insulating state of Sr3 Ir2 O7 is only marginally stable and that the insulating gap closes concurrently with the loss of antiferromagnetic long-range order, which differs from the strongly correlated Mott state realized in cuprates [97]. The fact that no drastic changes in band structure are observed across TN by ARPES [93] and optical conductivity [89] measurements suggests that the LETTERS a 4 6 s) y (p a l e D 2 NATURE 50 220 0 240 T (K) 260 280 300 10 2 5 ∆R/R (10−4) ∆R/R (10−4) 0 −5 −10 b 1.2 1.0 −5 x=0 T < TN 0.0 1.3 0 Dela −10 T > TN CB εF VB 0 xc ∼ 2 0 M −2 Je� = 3/2 −4 τ (ps) A (a.u.) Figure 2 | Doping-depende Three-dimensional surface −6 of La doping (x) and delay. 0.6 x = 0, 1.3, 2.7, 3.6, 4.7 and 5. at 120 K using a pump fluen −8 0.4 magnified view of the delay main figure, showing the hi −10 the low-frequency electron 0.2 180 200 220 240 260 280 300 samples, respectively. The e T (K) insulator-to-metal transitio space cut as determined by FigureFigure 7.1: 1Temperature-dependent reflectivitytransients transients Ir72 O7 (x = the 0). data. | Temperature-dependent reflectivity of of Sr3Sr Ir23O a, Three-dimensional surface plot of the transient reflectivity as a function (x = 0). a, Three-dimensional surface plot of the transient reflectivity as a of tem−2 perature and delay measuredand using a pump fluence µJcm . Black lines are function of temperature delay measured usingofa 80 pump fluence of superposed atop the exp 2 raw traces at select temperatures. Pronounced changes are observed around 80 µJ cm . Black lines are raw traces at select temperatures. Pronounced TN ∼ from coherent oscillatio 280 K.changes b, Temperature dependence the K. amplitude A (open orange circles) and are observed around TN of ⇠ 280 b, Temperature dependence of decaythe time τ (filledAblue circles) the reflectivity extracted from fitsalso to been observed by Ra amplitude (open orangeofcircles) and decaytransients time ⌧ (filled blue circles) x > 2% on the other han the single decay function. The superposed and blue curves are of theexponential reflectivity transients extracted from fits to the orange single exponential modulation is present (F best fits to the Rothwarf-Taylor model described in the main text. inset decay function. Error bars epresent the standard deviation of theThe best fit depicts reported Raman active p the transfer of spectral into the insulating gap above TNeach while the Jeff = 1/2 parameters acrossweight 20 independent measurements made at which diminishes in am valence band (VB)and andcan conduction (CB)toremain intact. fluctuations. temperature largely be band attributed laser intensity (probably due to screeni The superposed orange and blue curves are best fits to the Rothwarf–Taylor in the metallic regime, su model described in the main text. The inset depicts the transfer of spectral To understand the weight into the insulating gap above TN while the Je� = 1/2 valence band studied its detailed temp (VB) and conduction band (CB) remain intact. 0.8 spectral weight into the gap (Fig. 1b inset) and is concentrated at momentum values near the X points of the Brillouin zone, as shown temperature evolution o the most metallic sample both the low-frequency clearly resolved in both 51 gap closure in Sr3 Ir2 O7 occurs primarily through the transfer of high-energy spectral weight into the gap (Fig. 7.1(b) inset) and is concentrated at momentum values NATURE MATERIALS DOI: 10.1038/NMAT4836 near the X points of the Brillouin zone, as shown by ARPES [93]. T (K) 260 280 1 5.8 4.7 300 4 x = 2.7% 3.6 5 2.7 ∆R/R (10−4) 5 0 −5 x=0 x = 1.3 % xc ∼ 2 0.0 ,x( 0 3 Delay ( p 6 s) −10 −8 −10 280 300 A (a.u.) −6 X %) x ∼ 2% εF εF M M M ng opi d x = 5.8% La- εF 0 −4 0 1.3 x=0 −2 10 ∆R/R (10−5) 10 Delay (ps ) 2 3 X X M M M FigureFigure 7.2:2 | Doping-dependent Doping-dependent reflectivity transients (Sr La ) Ir O . reflectivity transients of (Sr1 ofx La x )1-x 3 Ir2 O7x. 3 2 7 Three-dimensional surface plotplot of of thethetransient asa afunction function of La Three-dimensional surface transientreflectivity reflectivity as dopingof(x) and delay. Black lines are the raw traces of the x = 0, 1.3, 2.7, 3.6, 4.7 La doping (x) and delay. Black lines are the raw traces of the and 5.8% doped samples. All 5.8% tracesdoped were samples. measuredAllattraces 120 Kwere using a pump fluence x = 0, 1.3, 2.7, 3.6, 4.7 and measured −2 2 of 140atµJcm . Thea pump top inset shows a magnified viewtop of inset the delay bounded 120 K using fluence of 140 µJ cm . The showsrange a by themagnified coloured view triangles in the main figure, showing the high-frequency coherent of the delay range bounded by the coloured triangles in the A1g phonon modeshowing and the the low-frequency electronic observed in the main figure, high-frequency coherentmode A1g phonon mode andx = 1.3% and x the = 2.7% samples, respectively. The evolution of the structure across the low-frequency electronic mode observed in the x = band 1.3% and x = 2.7% insulator-to-metal transition (x ∼ 2%) along the M-X-M momentum space cut as samples, respectively. The cevolution of the band structure across the determined by ARPES [98, 99] is(xillustrated schematically the data. insulator-to-metal transition the M–X–Mbelow momentum c ⇠ 2%) along space cut as determined by ARPES13,14 is illustrated schematically below the data. ransients of Sr3 Ir2 O7 e transient reflectivity as It a has recently been shown that electron doping Sr3 Ir2 O7 by substituting Sr with ng a pump fluence of La can also suppress antiferromagnetism and force the insulating gap to close [96, superposed atop the exponential decay (Fig. 2 inset), which arises emperatures. Pronounced 98–100]. However, in the thermally driven (Fig. mode 7.1), ARPES from coherentunlike oscillations of a 4.4 THz A1gcase phonon that hasstudies mperature dependence of have shown that observed doping across the critical value in of Sr x 3inIr2(Sr Lax28). )3 Ir2 For O7 , which is also been by Raman scattering O71-x(ref. all y time ⌧ (filled blue circles) x > 2% on the other hand, an additional low-frequency (⇠1 THz) o the single exponential on the level of a few percent, causes a discontinuous shift of the band edges and modulation is present (Fig. 2 inset) that does not correspond to any d deviation of the best fitthe formation of electron pockets around the M points of the Brillouin zone [98, reported Raman active phonon mode. Unlike the A1g phonon mode, nts made at each which diminishes in amplitude upon entering the metallic regime er intensity fluctuations. (probably due to screening by free carriers), this mode appears only fits to the Rothwarf–Taylor in the metallic regime, suggesting an electronic origin. cts the transfer of spectral To understand the origin of this low-frequency mode, we e Je� = 1/2 valence band 52 99]. The contrast between these two mechanisms of gap closing is highly apparent from ultrafast optical reflectivity data. As shown in Fig. 7.2, the doping-induced gap closing across x c ∼ 0.02 (2%) is manifested as a change in sign of ∆R/R, which is consistent with an insulator-to-metal transition that is realized by a collective electronic rearrangement [101]. Interestingly, across x c ∼ 2%, an abrupt change is also observed in the collective excitations of the system. For x < 2%, we observe a high-frequency modulation superposed on top of the exponential decay (Fig. 7.2 inset), which arises from coherent oscillations of a 4.4 THz A1g phonon mode that has also been observed by Raman scattering in Sr3 Ir2 O7 [102]. For all x > 2% on the other hand, an additional low-frequency (∼ 1 THz) modulation is present (Fig. 7.2 inset) that does not correspond to any reported Raman active phonon mode. Unlike the A1g phonon mode, which diminishes in amplitude upon entering the metallic regime (probably due to screening by free carriers), this mode appears only in the metallic regime, suggesting an electronic origin. To understand the origin of this low-frequency mode, we studied its detailed temperature dependence. Figure 7.3(a) shows the temperature evolution of the reflectivity transients obtained from the most metallic sample (x ∼ 5.8%) available. At low temperatures, both the low-frequency mode and the A1g phonon mode are clearly resolved in both the raw data and its Fourier transform (Fig. 7.3(a) inset). As the temperature increases, the frequency and the amplitude of the low-frequency mode continuously decrease until the mode is no longer resolvable near 200 K. To extract quantitative information from this data, we fit the oscillatory component of the reflectivity transients to a damped sinusoid of the form ADW e−Γt sin(2πt/τDW + φ), where ADW , Γ and τDW are the amplitude, damping rate and period of the low-frequency mode, respectively, and φ is a phase offset. Figure 7.3(b) shows that ADW exhibits an order-parameter-like onset below a critical temperature TDW ∼ 210 K and Fig. 7.3(c) shows that both Γ and τDW diverge upon approaching TDW , indicating a softening of the mode to zero energy. These behaviours are consistent with coherent amplitude oscillations of an electronic order parameter that develops via a continuous phase transition below TDW . In particular, they are highly reminiscent of the coherent charge density wave amplitudons that have previously been observed by time-resolved optical reflectivity in systems such as K0.3 MoO3 [103] or the underdoped cuprates [3, 104]. A recent neutron diffraction experiment found evidence of a weak intra-unit-cell structural distortion in metallic (Sr1-x Lax )3 Ir2 O7 crystals below a critical tempera- LETTERS 53 NATURE MATERIALS DOI: 10.1038/NMAT4836 180 K 0 1 2 3 4 5 Frequency (THz) 5 10 K 4 c 300 15 5 200 0 100 T (K) 200 τDW (ps) 2 4 Delay (ps) 2 1.6 Density wave metal 50 TDW 1 0 0 3 Γ −1 (ps) 2.0 230 K AFM insulator 100 4 1 150 5 2.4 2 PM metal 250 10 0 3 d 20 T (K) FT amp (a.u.) 6 ∆R/R (10−3) b La (x = 5.8%) 7 ADW (a.u.) a 1.2 6 0 100 T (K) 200 0 0 TS 0 2 4 6 La-doping, x (%) Figure 3 | Coherent oscillations of an electronic order parameter. a, Reflectivity transients of 5.8% La doped Sr3 Ir2 O7 plotted for a series of temperatures separated by 20 K. The traces are vertically o�set for clarity. A higher pump fluence of 400 µJ cm 2 was used to better visualize the weak temporal modulations but they are present even at lower fluences (see Fig. 2). Two of the modulation peaks are marked with triangles to guide the eye. The Fourier 3 2 at 180 7 K is shown in the inset. The red line is a best fit to the sum of two Lorentzian transform of the background-subtracted reflectivity transient measured functions and a constant background. b,c, Temperature dependence of the oscillation amplitude (ADW ), and period (⌧DW ) and damping time ( 1 ), respectively, extracted from −2 fits to a damped sinusoidal function described in the main text. Error bars represent the 95% confidence intervals of the fitted values. d, Temperature versus doping phase diagram of (Sr1 x Lax )3 Ir2 O7 . The paramagnetic (PM) to antiferromagnetic (AFM) phase boundary and the line of structural phase transitions (Ts ) are based on data from ref. 12, whereas the line of density wave phase transitions (TDW ) is based on time-resolved optical reflectivity data. Error bars denote the uncertainty in TDW estimated by extrapolating to the temperature at which ADW goes to zero. Figure 7.3: Coherent oscillations of an electronic order parameter. a, Reflectivity transients of 5.8% La doped Sr Ir O plotted for a series of temperatures separated by 20 K. The traces are vertically offset for clarity. A higher pump fluence of 400 µJcm was used to better visualize the weak temporal modulations but they are present even at lower fluences (see Fig. 7.2). Two of the modulation peaks are marked with triangles to guide the eye. The Fourier transform of the backgroundCa (x ∼ 7%) The red line highlysubtracted reminiscent of the coherent charge density wavemeasured amplitudons at 180 K is shown in the reflectivity transient inset. that have previously been observed by time-resolved optical 5K is a inbest fit to sum of (ref. two29)Lorentzian and a constant background. b,c, reflectivity systems suchthe as K MoO or the under- functions 2 dopedTemperature cuprates . 40 K dependence of the oscillation amplitude (ADW ), and period (τDW ) and A recent neutron diffraction experiment found evidence −1 ), respectively, 80 K of a damping weak intra-unit-cell distortion in extracted metallic time (Γstructural from fits to a damped sinusoidal func1 (Sr La ) Ir O crystals below a critical temperature T that is also K (∼T ) tion thespatial main text. Error bars represent the 95% confidence120intervals of of order 200described K (ref. 12). Noin new periodicity was reported belowthe T , fitted which isvalues. consistentd, with the absence of additional Temperature versus doping phase diagram of (Sr1-x160La K x )3 Ir2 O7 . superlattice modulations and band folding in low-temperature 0 The paramagnetic (PM) to antiferromagnetic (AFM) phase boundary and the line STM and ARPES data on (Sr La ) Ir O , respectively. On the basis of the observations that the structural distortion of structural phase transitions (TS ) arewith based on data from ref [96], whereas the line 5K emerges upon very light La doping, appears to compete −1 antiferromagnetic ordering and is suppressed in temperature by of density wave phase transitions (TDW ) is based on time-resolved optical reflectivisovalent substitution of Sr + by Ca + , it was suggested that there is a data.contribution Error bars the uncertainty strongity electronic to thedenote lattice distortion that enhances in TDW estimated by extrapolating to the T (ref. 12). By carrying out transient reflectivity measurements temperature at which ADW goes to zero. −2 0.3 10,11 2 7 s s 12 13,14 1 x 2 x 3 2 7 2 s FT amp (a.u.) x 3 ∆R/R (10−3) 1 x 3 s analogous to Fig. 3a–c for a series of different La doping levels (see 0 1 2 3 4 5 6 Supplementary Section 5 for detailed comparison), we identified Frequency (THz) the region of the temperature versus doping phase diagram over which our observed electronic order exists (Fig. 3d). Remarkably, its −1 0 1 2 3 4 5 6 7 TS coincides that iswell also 200 K [96]. No new spatial periodicity phase ture boundary with of that order of the structural transition, Delay (ps) was reported reaffirming a cooperative interaction between the structural and Figure 4 | of Reflectivity transients superlattice of insulating (Sr1 x Ca below additional modulaelectronic order T parameters. x )3 Ir2 O7 . Transient S , which is consistent with the absence To confirm that the low-frequency oscillations observed in reflectivity of ⇠7% Ca substituted Sr3 Ir2 O7 plotted over a range of spanning structural transition Ts ⇠on 120 K (see and bandfrom folding in low-temperature STM [96] andits ARPES [98,temperature 99] data (Sr1 x tions Lax )3 Ir2 O a Fermi surface instability and not temperatures 7 originate simply from the structural distortion below Ts , we performed time- Supplementary Section 6). The traces are vertically o�set for clarity. The A1g phonon are clearly resolved both the raw data and Fourier (Sroptical On the basis of theoscillations observations thatin the structural resolved reflectivity measurements on isovalent substituted 1-x La x )3 Ir2 O 7 , respectively. (Sr1 x Cax )3 Ir2 O7 , which remains insulating like Sr3 Ir2 O7 but transform (inset), but the electronic mode is absent. The red line in the inset is a best fitappears to the sum of ato Lorentzian function with and a constant background. distortion emerges upon very light La doping, compete antiferundergoes the same structural distortion as that observed in metallic (Sr1 x Lax )3 Ir2 O7 , albeit at considerably lower temperature (see romagnetic ordering andtheisreflectivity suppressed by isovalent substitution of Supplementary Section 6). Figure 4 shows transientsin temperature 4.4 THz is again observed and does not undergo any measurable of (Sr0.93 Ca0.07 )3 Ir2 O7 taken both above and below its structural shift across Ts . However, the low-frequency mode that was observed 2+ 2+ Sr temperature by Ca (T,s it was thatmode there a strong contribution tothethe transition ⇠ 120 K).suggested The A1g phonon at is in metallic (Sr1 electronic absent both in raw data x Lax )3 Ir2 O7 is clearly lattice distortion that enhances T [96]. By carrying out transient reflectivity mea© 2017 Macmillan Publishers Limited, part of Springer Nature. All rights reserved. surements analogous to Fig. 7.3(a)-(c) for a series of different La doping levels, we identified the region of the temperature versus doping phase diagram over which NATURE MATERIALS | ADVANCE ONLINE PUBLICATION | www.nature.com/naturematerials S 3 nic order parameter. a, Reflectivity transients of 5.8% La doped Sr3 Ir2 O7 plotted for a series of temperatures �set for clarity. A higher pump fluence of 400 µJ cm 2 was used to better visualize the weak temporal er fluences (see Fig. 2). Two of the modulation peaks are marked with triangles to guide the eye. The Fourier ctivity transient measured at 180 K is shown in the inset. The red line is a best fit to the sum of two Lorentzian mperature dependence of the oscillation amplitude (ADW ), and period (⌧DW ) and damping time ( 1 ), 54 sinusoidal function described in the main text. Error bars represent the 95% confidence intervals of the fitted observed electronic order exists [Fig. (AFM) 7.3(d)].phase Remarkably, its the phase agram of (Sr1 x Lax )3 Ir2 Oour paramagnetic (PM) to antiferromagnetic boundary and line bound7 . The on data from ref. 12, whereas line of density wavethat phase (TDW )transition, is based onreaffirming time-resolved ary the coincides well with oftransitions the structural a cooperative uncertainty in TDW estimated by extrapolating to the temperature at which ADW goes to zero. interaction between the structural and electronic order parameters. density wave amplitudons by time-resolved optical O3 (ref. 29) or the under- Ca (x ∼ 7%) 5K 2 40 K FT amp (a.u.) ∆R/R (10−3) eriment found evidence 80 K distortion in metallic 1 temperature Ts that is also 120 K (∼Ts) l periodicity was reported he absence of additional 160 K lding in low-temperature 0 x Lax )3 Ir2 O7 , respectively. the structural distortion 5K appears to compete with −1 pressed in temperature by was suggested that there is a ce distortion that enhances reflectivity measurements −2 erent La doping levels (see 0 1 2 3 4 5 6 omparison), we identified Frequency (THz) oping phase diagram over s (Fig. 3d). Remarkably, its −1 0 1 2 3 4 5 6 7 of the structural transition, Delay (ps) etween the structural and 4 | Reflectivity transientsof ofinsulating insulating (Sr O27O . Transient 1 1-x x Ca x )x3)Ir32Ir Figure Figure 7.4: Reflectivity transients (Sr La 7 . Transient rereflectivity of ⇠ 7% Ca substituted Sr Ir O plotted over a range oscillations observedflectivity in 7 of ∼ 7% Ca substituted Sr3 Ir23O72 plotted over a rangeofof temperatures spanning its structural transition ⇠ 120 are K (see surface instability and spanning not temperatures its structural transition temperature TS ∼temperature 120 K. TheTstraces vertically Supplementary Section 6). The traces are vertically o�set for clarity. The ow Ts , we performed timeoffset for clarity. The A1g phonon oscillations are clearly resolved in both the raw phonon oscillations are clearly resolved in both the raw data and Fourier ts on isovalent substituted data andA1g Fourier transform (inset), but the electronic mode is absent. The red line in transform but the electronic mode is absent. The red line in the inset lating like Sr3 Ir2 O7 the butinset is a best(inset), fit to the sum of a Lorentzian function and a constant background. as that observed in metallic is a best fit to the sum of a Lorentzian function and a constant background. y lower temperature (see s the reflectivity transients 4.4 THz observed and does not undergo measurable To confirm that is theagain low-frequency oscillations observed in any (Sr1-x Lax )3 Ir2 O7 origiand below its structural shift across Ts . However, the low-frequency mode that was observed nate from a Fermi surface instability and not simply from the structural distortion The A1g phonon mode at in metallic (Sr1 x Lax )3 Ir2 O7 is clearly absent both in the raw data below TS , we performed time-resolved optical reflectivity measurements on isovaTION | www.nature.com/naturematerials lent substituted (Sr1-x Cax )3 Ir2 O7 , which remains insulating like Sr3 Ir2 O73but un2017 Macmillan Publishers Limited, part ofthe Springer All rightsdistortion reserved. as that observed in metallic (Sr1-x Lax )3 Ir2 O7 , dergoes sameNature. structural albeit at considerably lower temperature. Figure 7.4 shows the reflectivity transients of (Sr0.93 Ca0.07 )3 Ir2 O7 taken both above and below its structural transition temperature (TS ∼ 120 K). The A1g phonon mode at 4.4 THz is again observed and does not undergo any measurable shift across TS . However, the low-frequency mode that 55 was observed in metallic (Sr1-x Lax )3 Ir2 O7 is clearly absent both in the raw data and its Fourier transform (Fig. 7.4 inset), thus precluding a purely structural origin of the mode. No static long-range magnetic order has been observed by neutron or resonant Xray diffraction, or magnetic susceptibility measurements in metallic (Sr1-x Lax )3 Ir2 O7 crystals [96]. In addition, we have performed resonant inelastic X-ray scattering measurements [105] that show both quasi two-dimensional magnon modes and exciton-like spin-orbital entangled modes in (Sr1-x Lax )3 Ir2 O7 to exist only at frequencies far above 1 THz. These observations rule out any spin-related instabilities at the 1 THz energy scale and instead point towards a charge density wave-like instability as being responsible for the low-energy mode. However, this interpretation must be reconciled with the absence of any signatures of a new spatial periodicity below TDW in other experiments. One possibility is that the charge density wave order is very short-ranged like in the case of under-doped cuprates [3, 104], or even temporally fluctuating, which would naturally explain the strong damping of its collective modes [Fig. 7.3(c)] in comparison to those in static long-range ordered charge density wave systems such as K0.3 MoO3 [103]. Although a detailed microscopic understanding of this density wave instability in (Sr1-x Lax )3 Ir2 O7 will require more extensive theory and experiments, our work shows generally that iridates in the weak Mott limit can also support interesting electronic instabilities, possibly realizing strongly spin-orbit coupled analogues of those found in their lighter transition metal oxide counterparts. 56 Chapter 8 Second harmonic generation investigations of antiferromagnetic orders in MPX3 8.1 Introduction Since the first successful exfoliation of graphene [106, 107], which is a semimetal, the family of 2D van der Waals (vdW) materials has quickly expanded to include spin-valley coupled semiconductors [108, 109], Ising superconductor [110, 111], and charge-density-wave material [112]. Adding to the list is the quest for materials that exhibit long range magnetic order in the 2D limit, which is thought to be highly suppressed by thermal fluctuations [113]. MPX3 , with transition metal M and chalcogen X = {S, Se, Te}, are honeycomb-structured layered vdW materials that exhibit magnetism. They are exfoliable materials to test the Mermin-Wagner theorem for magnetism in the 2D limit. Recently, long ranger ferromagnetism was verified by magneto-optical Kerr microscopy in monolayer CrI3 [114] and bilayer CrGeTe3 [115]. Similarly, long range antiferromagnetic stripe order has also been verified by scanning Raman spectroscopy in monolayer FePS3 [116], where the stripe order doubles the unit cell and leads to the appearance of additional Raman peaks below TN . Unlike the antiferromagnetic stripe order in NiPS3 and FePS3 , Néel-type antiferromagnetic order exists in MnPS3 [117–120]. Magnetic susceptibility measurements and spin wave model calculations found that the exchange interaction in MnPS3 is Heisenberg-like [120, 121], whereas NiPS3 is better described by the XY model [120] and FePS3 by the Ising model [120, 122]. Whether the Heisenberg antiferromagnet survives to the monolayer limit is a question to be answered by experiments. This also has implication for a recent proposal of spin-valley coupling in monolayer antiferromagnetic MnPS3 which can lead to circular dichroic hall effect and topological edge states at the antiferromagnetic domain walls [123]. However, magneto-optical Kerr effect is not sensitive to collinear antiferromagnetism. Moreover, Néel order preserves the unit cell of MnPS3 , therefore it does not fold the Brillouin zone and introduce new Ramon mode to a Raman scattering experiment. A microscopic probe that is sensitive to the Néel transition on a honeycomb lattice therefore has to be found. Fortunately, due to the inequivalence of the two 57 Figure 8.1: Lattice structure of MPX3 and SHG-RA experimental geometry. Lattice structure of monoclinic MPX3 looking down along (a) c-axis and (b) baxis. The magnetic M = Mn, Fe, Ni ions are arranged in a 2D hexagonal lattice in the ab plane. Each unit cell consists of two layers of MPX3 displaced along the a-axis. (c) Geometry of the SHG-RA experiment. The scattering plane is defined normal to the ab plane of MPX3 . sublattices of a honeycomb structure, the Néel order in MnPS3 necessarily breaks inversion symmetry. Second harmonic generation (SHG), which is extremely sensitive to inversion-breaking [4], and which can be easily adapted to microscopy, is therefore the perfect tool for studying the Néel order in MnPS3 . We apply this this technique to MnPS3 , FePS3 , and NiPS3 , and demonstrate that SHG very well distinguishes between the Néel order and the antiferromagnetic stripe order in these compounds. A microscopic model is offered at the end to explain how the SHG process couples to Néel order parameter but not the antiferromagnetic stripe order 58 parameter. 8.2 Experiment and results (a) FePS3 (b) NiPS3 50 K 130 K (c) MnPS3 1 60 K 200 K (d) MnPS3 2.5 100 K fit (2/m) 30 K fit (2'/m) Figure 8.2: SHG-RA patterns for MPX3 above and below TAF . The Sin -Sout data from (a) FePS3 (b) NiPS3 above and below their respective TAF show negligible change across TAF . In contrast, in MnPS3 the intensity of SHG (d) below TN is much stronger compared to (c) above TN . Moreover, the symmetry of the Sin Sout pattern also changes across TN . In (c)(d) the solid lines are best fit to the bulk electric quadrupole radiation from 2/m point group and bulk electric dipole radiation from 20/m magnetic point group respectively. The data in (c) is normalized to the maximum intensity in (d). The crystalline point group of MPX3 is 2/m [124], which consists of a 2-fold rotation along the crystalline b-axis and a mirror plane perpendicular to it. Figure 8.1(a)(b) shows the crystal structure of MPX3 . The transition metal M are arranged on the vertices of a hexagon, with chalcogens on top and below forming two oppositely pointed equilateral triangles related by inversion. Different layers of MPX3 are displaced along the crystalline a-axis. However, coupling between 59 the layers is weak and the structure can thought of as quasi-2D. In particular, this 2D character is manifested in second harmonic generation rotational anisotropy (SHG-RA) patterns. Figure 8.2 shows the SHG-RA patterns from MnPS3 , FePS3 and NiPS3 from the Sin -Sout polarization channel, which all exhibit close to sixfold rotational symmetry in intensity above TAF . Since 2/m is centrosymmetric, bulk electric dipole contribution to SHG is forbidden by symmetry. Instead, the Sin -Sout patterns can be explained in terms of either surface electric dipole (ED) contribution Pi (2ω) = χiEj D k E j (ω)E k (ω) or bulk electric quadrupole (QD) contriQD bution Pi (2ω) = χi j kl E j (ω)∇ k El (ω). In both cases, the sign of Pi (2ω) alternates between adjacent peaks. Hence the Sin -Sout intensity data, which is proportional to |Pi (2ω)| 2 , in fact implies three-fold rotational symmetry. This is the correct rotational symmetry of a single MPX3 plane. The magnetic interactions in MPX3 are frustrated by competing first, second and third nearest neighbor exchange interactions {J1 , J2 , J3 }. The magnetic ground state of MPX3 ranges from in-plane ferromagnetic to Néel, and to antiferromagnetic stripe order, with ferro- or antiferromagnetic inter-plane couplings. Due to the high level of frustration, it was proposed that both mechanical stress and carrier doping can switch the magnetic order from one to another [125, 126]. Figure 8.3 shows the magnetic structures of the three materials in our study. MnPS3 undergoes an inplane Néel transition at 78 K. FePS3 exhibits an in-plane antiferromagnetic stripe order that is antiferromagnetically coupled between the planes below 123 K. NiPS3 also an exhibits in-plane antiferromagnetic stripe order which is however ferromagnetically coupled between the planes below 155 K. While the in-plane Néel order breaks inversion in MnPS3 , both forms of stripe order in FePS3 and NiPS3 preserve inversion: inversion plus a translation by lattice constants reproduces the same stripe order [Fig. 8.3]. Since bulk ED radiation, the strongest contribution to SHG, is only allowed in non-centrosymmetric media, we expect SHG to be greatly enhanced below TN in MnPS3 . Figure 8.2 shows that while the change in the SHG-RA patterns for FePS3 and NiPS3 across TAF is small, in MnPS3 there is indeed a dramatic increase in SHG intensity below TN , signifying bulk ED contribution. In addition, the three-fold rotational symmetry inferred from the Sin -Sout pattern above TN is reduced to onefold below TN . A recent neutron polarimetry study on MnPS3 found that in the Néel phase, the magnetic moments are oriented at a finite angle ∼ 8◦ from the c∗ axis (ĉ∗ ≡ â × b̂) [124]. This projects a finite moment in the ab plane, which breaks 60 Figure 8.3: Magnetic structures of MPX3 . (a) Below 78 K, MnPS3 develops inplane Néel order which breaks inversion symmetry due to the inequivalence of the two sublattices of a honeycomb lattice. (b) Below 123 K, FePS3 develops in-plane antiferromagnetic stripe order. Between the planes the coupling is antiferromagnetic. Dashed arrow denotes the symmetry operation (inversion through the center *) which maps the magnetic structure onto itself. (c) Below 155 K, NiPS3 also develops in-plane antiferromagnetic stripe order, with ferromagnetic coupling across the planes. Inversion (1) plus lattice translation (2) maps the magnetic structure onto itself. the three-fold rotational symmetry of the paramagnetic phase and gives rise to the observed one-fold rotational symmetry. The low temperature SHG-RA pattern can be fitted to the bulk ED contribution from the 20/m magnetic point group, to which the antiferromagnetic structure of MnPS3 belongs to. To investigate the Néel order in MnPS3 more systematically, we measure the intensity of peaks in the Sin -Sout pattern as a function of temperature. Figure 8.4(a) shows the temperature dependence of the SHG peak intensity. In MnPS3 , the SHG intensity has a power dependence on temperature from which we can extract χ(T ) ∝ (TN − T) β with β = 0.2 because SHG intensity is proportional to | χ| 2 . In contrast, in FePS3 , χ shows negligible temperature dependence. While a macroscopic consideration based on symmetry alone can explain the change in SHG intensity, below we propose a microscopic theory based on a model developed for Cr2 O3 [127] to explain why SHG couples to the Néel order parameter in MnPS3 but not the antiferromagnetic stripe order parameter in FePS3 . SHG involves a two-photon absorption process and a single-photon emission process. Based on optical absorption studies on MnPS3 , the two-photon absorption process in our SHG experiment using 1.5 eV incoming light corresponds to the 6 A (6 S) → 4 T (4 D) intra-ionic d-d transition in Mn2+ [128, 129]. Although the 1g 2g TN 4 3 2 1 0 20 40 60 T (K) 80 16 SHG Intensity (a.u.) SHG Intensity (a.u.) 61 TAF 12 8 4 0 80 100 120 T (K) 140 Figure 8.4: Temperature dependence of SHG intensity for MnPS3 and FePS3 . The temperature dependence of SHG intensity for (a) MnPS3 and (b) FePS3 across their respective TAF = 78 K and 127 K. The orange and blue curves correspond to the temperature dependence of different peaks marked in the insets. The solid lines in (a) are best simultaneous fit of the two sets of data to I (T ) = A(TN − T ) 2 β plus a constant offset, with β = 0.2. absorption edge in MnPS3 , which corresponds to P4+ , S2− → Mn2+ charge transfer, is also close to 3 eV, it moves to higher energy at lower temperature [128, 130]. As such, any two-photon absorption due to this transition is suppressed at low temperature, impeding the SHG process. In contrast, the 6 A1g (6 S) → 4 T2g (4 D) transition is insensitive to temperature change, and hence more likely to account for the absorption process of SHG. In addition to this assumption, we assume moderate spin-orbit coupling as well as a small local polar distortion of the Mn2+ ionic potential, which has recently been suggested to occur in MnPS3 [131] to explain its magnetoelectric effect [124]. First of all, spin-orbit coupling, which is weak although finite in 3d transtion metal, mixes the t 2g and eg states of Mn. For the toy model, we consider its effect on the d x y and d x 2 −y2 states: D E d xy , m s L · S d x 2 −y2 , m s = 2i hSz i where hSz i ≡ hm s |Sz |m s i. On the other hand, the inversion symmetry of the Mn2+ ionic potential is broken by its nearest Mn2+ neigbhours as well as a trigonal distortion of the sulphur octahedron around it. This mixes the 3d and 4p states [132] of Mn2+ . Treating both spin-orbit interaction and local polar distortion as small 62 perturbations, the new ionic eigenstates for Mn2+ can be written as E E E d˜xy = d x y , m s + λ hSz i d x 2 −y2 , m s + η |p, m s i E E E d˜x 2 −y2 = d x 2 −y2 , m s − λ hSz i d x y , m s + η 0 p0, m s where λ denotes the spin-orbit coupling strength, η, η 0 are proportional to the local parity-breaking molecular field, and |p, m s i, |p0, m s i are Mn 4p orbitals. E E Assuming d xy and d x 2 −y2 are the initial and final states of the two-photon absorption process, the SHG process in the ED channel can be written as XD E ED ED d˜xy r d˜x 2 −y2 d˜x 2 −y2 r α α r d˜x y α where |αi denotes intermediate state. The leading contribution in the expansion is proportional to λ 0 η 0 . It comprises terms involving only the unperturbed 3d states: ED ED E D d xy r d x 2 −y2 d x 2 −y2 r α α r d xy . However, since d orbitals are parity-even, the integrand is odd and the integral vanishes. The next leading contribution is proportional to either λ 1 η 0 or λ 0 η 1 . The contribution proportional to λ 1 η 0 vanishes too because it comprises terms similar to those in λ 0 η 0 , except for a coefficient of λhSz i in front. The contribution proportional to λ 0 η 1 also vanishes when summing over the unit cell because neighbouring Mn are related by inversion and hence the local polar distortion η switches sign between them [Fig. 8.3]. The leading contribution to the ED radiation is then proportional to λ 1 η 1 . In other words, it is proportional to   X λη i hSz ii = λη hSz i1 − hSz i2 + hSz i3 − hSz i4 + hSz i5 − hSz i6 i where the term in parenthesis defines the order parameter of the Néel order in  MnPS3 . Since the antiferromagnetic stripe order parameter is hSz i1 + hSz i2 +  hSz i3 − hSz i4 − hSz i5 − hSz i6 , it does not couple to the ED process of SHG. In conclusion, we have demonstrated that second harmonic generation is a sensitive probe for the Néel type antiferromagnetic order in MnPS3 and distinguishes it from the antiferromagnetic stripe order in FePS3 and NiPS3 . In addition, we offered a microscopic theory to explain why SHG couples to the Néel order parameter but not the antiferromagnetic stripe order parameter in MPX3 . Our experiment paves the way towards verifying 2D long range antiferromagnetic order in MnPS3 . 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