Ro is shown above. The incoming beam is accounted by a plane wave Eoexp (-. iKoRo), where EO is the electric field vector and KO the wave vector. The angle.
A STUDY ON MOTT INSULATOR: SYNTHESIS AND CHARACTERISATION OF Sr2IrO4
JADAVPUR UNIVERSITY KOLKATA-700032
Master’s Dissertation in the Department of Physics, Jadavpur University June, 2017
By Nirman Chakraborty PG-II (Day) Exam roll: MPD1722003
Supervisor Dr. Sugata Ray Professor Department of Material Science Indian Association for the Cultivation of Science Jadavpur, Kolkata-700032
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CERTIFICATION This is to certify that the Master’s dissertation entitled “A COMPREHENSIVE STUDY ON MOTT INSULATOR: SYNTHESIS AND CHARACTERISATION OF Sr2IrO4” submitted by Mr Nirman Chakraborty in the Department of Physics, Jadavpur University in partial fulfilment of the requirements for the degree of Master of Science in Physics from Jadavpur University, Kolkata is entirely based on his own work in my laboratory at the IACS. He has completed this whole project with sheer dedication, enthusiasm, and punctuality. Neither this report nor any part of it has been submitted for any diploma or academic award anywhere.
Sugata Ray
Prof. Argha Deb Head Department of Physics Jadavpur University
Prof. Sukhen Das Project Co-ordinator Department of Physics Jadavpur University
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Acknowledgement It gives me extreme pleasure to express deep gratitude to my guide Dr. S Ray for accepting me as a project student and allowing working in his lab. I am highly indebted to Dr. S K Neogi for guiding me throughout the whole process. Without him, I could not have had such a wonderful experience. I would thank Mr Rafikul Ali Saha for guiding me, helping with all possible means and bearing with my consistent persuasions, including questions, suggestions, materials for the work, etc. Finally, I am extremely thankful to the Department of Physics, Jadavpur University for giving me this precious opportunity to nurture my passion in such a constructive way and supporting me, both in the days of happiness and crisis.
Nirman Chakraborty PG-II (Day) Department of Physics Jadavpur University Kolkata-700032
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Contents 1. Introduction 2. Understanding Mott transition a) Formation of band structure in solids b) Band structure study of normal insulators c) Introduction to Mott physics d) Band structure in Mott insulators 3. Experimental study using Sr2IrO4 as an illustration a) Sample preparation b) Structural characterisation: XRD analysis i) Introduction to XRD as an efficient tool for structure analysis ii) XRD of sample prepared iii) XRD of sample normalised with respect to the standard figure c) Study of its lattice structure and classification on the basis of Group Theory 4. Resistivity measurement of the sample: Basic principles and experimental results 5. Magnetic measurements 6. An important application of Mott insulators in technology 7. Future scope 8. Conclusion 9. Bibliography
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1. Introduction Mott insulators are a class of materials that should conduct electricity under conventional band theories, but are insulators when measured (particularly at low temperature). This is due to electron-electron interactions, which are not considered in conventional band theories. The band gap in a Mott insulator exists between bands of like characters, such as 3d character, whereas the band gap in charge transfer insulators exist between cation and anion states, say O 2p and Ni 3d bands in NiO. Although the band theory in solids had been very successful in describing various electrical properties of materials, in 1937 Jan Hendrik de Boer and Evert Johannes Willem Verwey pointed out that a variety of transition metal oxides predicted to be conductors by band theory (because they have an odd no of electrons per unit cell) are insulators. Neville Mott and Rudolf Peirels then predicted that this anomaly could be explained by including interaction between electrons. In 1949 Mott proposed a model for NiO as an insulator, whose conduction is based on formula :(Ni2+O2-)2→Ni3+O2-+Ni1+O2In this situation, formation of a band gap preventing conduction can be understood as competition between Coulomb potential U between 3d electrons and the transfer integral t of 3d electrons between neighbouring atoms (the transfer integral is a part of the tight binding approximation). The total energy gap is then E=U-2zt, where z is the no of nearest neighbours. In general Mott insulators occur when the repulsive Coulomb potential U is large enough to create an energy gap. One of the simplest theories of Mott insulators is the 1963 Hubbard model. Mottism denotes the additional ingredient, aside from ferromagnetic ordering, which is necessary to describe a Mott insulator. Mott insulators are of growing interest in advanced Physics research, having application in thin film magnetic heterostructures and high temperature superconductivity. This kind of insulator can become a conductor by changing some parameters like composition, pressure, strain, voltage or magnetic field. This effect is known as Mott transition and can be used to build smaller FETs, switches and memory devices than possible with conventional materials. 5|Page
1. a) Mott physics The transition between metallic and insulator phases can be interpreted in some cases using the usual band picture. Consider a simple metal with an even no. of electrons per atom. Since the number of states in any band is twice the number of elementary cells, materials possessing an even no. of electrons per atom and which crystallize in a structure with a monatomic unit cell shall have completely filled or empty bands , unless the bands overlap in energy. It is precisely due to the overlap of the bands that the divalent elements of the second group of the periodic table are metals. When the distance between the atoms is increased, the overlap of the wave functions of neighbouring atoms decreases and the band narrows. The atomic energy levels are recovered in the large distance limit, when the atoms become independent. The narrowing of the bands is displayed below:
i.
Variation of bandwidth with distance between the atoms1
The overlap between the bands vanishes at a critical value of the interatomic distance. If the lower band is completely filled and upper band is completely empty, the system becomes insulating. For smaller lattice parameters, the material is metallic. This kind of metal-nonmetal transition occurring as the lattice parameter is changed (it can be achieved e.g. by applying pressure) is known as the band overlap or Wilson transition. The insulating state is a band insulator or a Bloch Wilson insulator. The interaction with the periodic one particle potential plays the dominant role in the formation of bands and this transition can be interpreted in the one dimensional picture without taking into account the electron-electron interaction.
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b) Band structure of Mott insulators In order to form a physical picture for Mott transition, consider a crystal built of monovalent atoms. The electrons outside the closed shell form a half filled band and the system is metallic even if the distance between the atoms is so large that the overlap of wave functions is exponentially small and the probability for an electron to hop to neighbouring sites is almost negligible. This is because electron- electron interactions are neglected in the band structure calculations. If the wave function of the half-filled band described in terms of extended Bloch functions is expressed in terms of localised Wannier states, configurations with two electrons on some atoms and no electrons on others occur with high probability. These configurations have much higher energy than the configurations with one electron sitting on each atom when the intra-atomic coulomb repulsion is taken into account. Therefore when the band is sufficiently narrow, the relatively high energy charge fluctuations are expected to be blocked and the electrons become localized. To make estimation as to where it happens, MOTT suggested that the dominant factor determining whether an electron system is metallic or non-metallic is the competition between kinetic energy and coulomb energy that tends to bind electrons to atoms. A simple estimate can be given by assuming that the system is metallic if the Fermi energy €F, which characterises the kinetic motion of electrons, is larger than the Coulomb energy. For electrons with an effective mass m*moving in a medium with dielectric constant €r, the coulomb energy can be estimated by
𝑒2
4𝜋𝜖0 ∈𝑟 𝑎0∗
, where 𝑎0∗ =𝑎0 ∈rme/m* is the effective Bohr radius of
the medium with a0 as the true Bohr radius. This energy can also be written as ℏ2 ℰ𝑟 𝑚𝑒 𝑎0 𝑎0∗
ℰ𝑓 =3𝜋 2
=
ℏ2 𝑚∗ 𝑎0∗
2/3 ℏ2 2𝑚
2
. Expressing the Fermi energy in terms of electron density
𝑛𝑒 2/3 . Comparison of the two expressions above gives
1/3
𝑎𝑜∗ 𝑛𝑒 >C……*, as the condition for metallic phase with C of order unity. This system is insulating for smaller densities. A somewhat different consideration leads to the same result. When the atoms are far apart, the electrons are bound to the positive ions by the coulomb 7|Page
potential. The radius of the lowest energy state is the Bohr radius, the electrons are localized and the system is insulating. When the atoms are closer and the density of the mobile electrons is ne, a detached electron is attracted by the positive ion left behind via a potential screened by the other electrons. The screening length can be estimated as the inverse of the Thomas Fermi wave number 2 𝑞𝑇𝐹 = 4𝜋𝜌(𝜖𝑓 )𝑒 2 =(3/𝜋)1/3
4 𝑎0
1/3
𝑛𝑒
When the screening length is less than the Bohr radius, the screened potential cannot bind the electron any more, the electrons are free to move and the system becomes metallic. Comparing 1/qTF with ao leads to condition (*) for the existence of metallic state. Although these considerations do not say anything about this, MOTT emphasized that the localized electrons will have localized magnetic moments in the insulating state, and these moments will persist in the correlated metallic state. The insulating nature of the state is not due to magnetism, but due to the onsite coulomb repulsion, which operates in the paramagnetic state as well. The moments disappear at higher electron densities. J.HUBBARD proposed a model which incorporates electron-electron interactions and accommodates the eventual magnetic properties of the system. In its simplest form the model assumes that there is no degenerate Wannier state at each lattice site with energy €o. Electrons can feely hop to neighbouring sites if that site is empty. However when two electrons with same spin share the same site, they repel each other. The repulsion is characterized by the Hubbard potential U given by ᵻ ℋ = −𝑡∑𝜎 𝑐𝑗,𝜎 𝑐𝑙,𝜎 + U∑𝑗 𝑛𝑗↑ 𝑛𝑗↓ - µ ∑𝑗 (𝑛𝑗↑ + 𝑛𝑗↓ ) ᵻ Here 𝑐𝑗,𝜎 and 𝑐𝑗,𝜎 are the creation and annihilation operator for electrons of spin 𝜎 at j th site, the first term being the kinetic energy term, t is hopping amplitude. It describes the destruction of an electron of spin 𝜎 at 𝑙 and its creation at j. indicates that hopping is allowed only between the adjacent sites. The second term is the interaction term; it goes throughout all sites and adds energy U if the site is doubly occupied. The third term stands for the contribution of ᵻ chemical potential µ. And 𝑛𝑗,𝜎 =𝑐𝑗,𝜎 𝑐𝑗,𝜎 is the number operator. To understand the Physics of the model we consider first the case where the number of electrons is equal to the number of lattice sites and treat the interaction between electrons in the mean-field approximation, an
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approximation which may be applicable for weak couplings, when U is smaller than the bandwidth determined by the hopping amplitude. The electrons fill half of the band in one-particle picture and the system is metallic. On the other hand, in the opposite limit, when U>>t, the half-filled Hubbard model is equivalent to an antiferromagnetic Heisenberg model with exchange coupling J= -2t2/U. The spins can be exchanged but the electrons do not carry current; the system is insulating. If an electron hops to a neighbouring site, double occupancy lasts only for a very short time due to the strong on-site repulsion and one of the electrons has to hop back to the temporarily empty site. The electron energies are around €0 and €0+U, around meaning that the atomic energy levels are broadened into bands of width W=2zt owing to the finite hopping probability, z being the no of nearest neighbours. The number of allowed states is equal to the number of sites in both bands, even if the spin quantum number is taken into account. The bands are so well separated when W is small compared to U. They are so called lower and upper Hubbard bands. For one electron per site, which would give a half filled band in the usual electronic band structure, the lower Hubbard band is completely filled while the upper one is empty. The system behaves as an insulator. Thus as a consequence of competition between the Coulomb repulsion and the hopping to the neighbouring sites, the system behaves differently in the limits Ut. We may expect that the separated bands exist above a certain critical value of the dimensionless coupling U/t. The gap between them disappears at (U/t)c and the two bands merge into one. The expected density of states is depicted below:
ii.
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The expected density of states in the Hubbard model in the limits tu, and at the critical value of coupling.1
Whether the system is metallic or insulating is determined by the ratio of the energy scales U/t. We know however that the exchange coupling between the localized moments introduces another energy scale into the problem, which is typically smaller than the band splitting and is related to the formation of localized moments. Mott insulators, where the moments are ordered antiferromagnetically by this exchange coupling are sometimes known as Mott-Heisenberg insulators, restricting the name Mott Hubbard insulator to materials in which localized magnetic moments do not display long range order.
3 a) Sample preparation A polycrystalline sample of Sr2IrO4 was prepared by solid state reaction from a stoichiometric mixture of predried (5000c for 12h) SrCO3 (Aldrich, 99.99%) and IrO2 (Aithica, 99.99%). The finely mixed powders were heated in air as follows: 24 hours at 6000c, 12 hours at 8000c, 24 hours at 10000c and 24 hours at 10500c. After each step, the sample was cooled to room temperature in furnace over a period of several hours. The reaction used is cited below:
2SrCO3 + IrO2 = Sr2IrO4 + 2CO2
b)Structural characterisation and XRD analysis i) XRD as an efficient tool Diffraction effects are observed when electromagnetic radiation impinges on periodic structures with geometrical variations on the length scale of the wavelength of radiation. The interatomic distances in crystals and molecules amount to 0.15-0.4 nm which corresponds to the electromagnetic spectrum with the wavelength of x rays having photon energy between 3 and 8 keV. Accordingly, phenomenon like constructive and destructive interference should become observable when molecular and crystalline structures are exposed to x rays. Now the phenomenon of elastic scattering of x rays by the electrons, also 10 | P a g e
known as Thomson scattering, where the electron oscillates as a Hertz dipole at the frequency of the incident radiation. The wavelength of the x rays is hence conserved, contrary to photoelectric and Compton effects.
iii.
Scattering of x rays by a single electron
Elastic scattering for a single free electron of charge e, mass m and at position Ro is shown above. The incoming beam is accounted by a plane wave Eoexp (iKoRo), where EO is the electric field vector and KO the wave vector. The angle between K and Ko is the scattering angle denoted by 2Ɵ. We can define
2Ɵ=arccos
𝐾𝐾𝑂
. Hence the amplitude of the scattered wave at R can be
written as E(R) =EO
1
𝑒2
4𝜋𝜀𝑂 𝑅 𝑚𝑐 2
SinW could lead to a typical spin s=1/2 Mott insulator (b.). However a reasonable U cannot lead to an insulating state as seen from the fact that Sr 2RhO4 is a normal metal. As the SO coupling is taken into account, the t2g states effectively correspond to the orbital angular momentum L=1 states. In the strong SO coupling limit, the t2g band splits into effective total angular momentum Jeff=1/2 doublet and Jeff=3/2 quartet bands(c.). Note that Jeff=1/2 is energetically higher than Jeff=3/2, seemingly against Hund’s rule, since the Jeff=1/2 is branched off from J5/2(5d5/2) manifold due to large crystal field as in (e.). As a result, with the filled J eff=3/2 band and one remaining electron in the Jeff =1/2 band, the system is effectively reduced to a half filled Jeff=1/2 single band system(c.). The Jeff=1/2 spin-orbit integrated states form a narrow band so that even small U opens a Mott Gap, making it a Jeff =1/2 Mott insulator(d.). The narrow band width is due to reduced hopping elements of the Jeff=1/2 states with isotropic orbital and mixed spin behaviour.
5) Magnetic measurements
ZFC: Zero Field Cooling FC: Field Cooling
xi.
Magnetic moment vs. Temp. Curve
The magnetic measurements of the sample were done in a SQUID (MPMS) (Superconducting Quantum Interference Device). The sample was tested for 17 | P a g e
magnetisation, as the response function, when subjected to an external field of 100 Oe. The curve for cooling the sample when subjected to the field shows that the magnetic moment shows a steady increase from 300k and near 2 k shows a spike, clearly indicating that the sample attains a more ordered nature at low temperature, thus dipoles get aligned in preferential manner, resulting in a large magnetisation. The pattern is different in zero field case, where the magnetisation is obviously lower than the former case, showing a hump at 236 k. Then it attains a negative value near 150 k and finally a positive spike near 2 k. Sr2IrO4 where Ir ion is in +4 state shows a weak ferromagnetism8 at relatively high transition temperature of 236 k. The slight negative value in M at low temperature suggests diamagnetism, which can be explained as: the Para film in which the sample is placed for measurement is diamagnetic in nature. When the amount of sample to be tested is small, this diamagnetic behaviour dominates at low temperatures in ZFC case.
6) Application Korean researchers12 have developed an innovative power interruption technology Mott metal-Insulator transition (Mott MIT) device. The device is said to reduce the size and enhance the performance of traditional electromagnetic switches and circuit breakers. The Mott MIT signifies the phenomenon that a Mott insulator can be abruptly converted into a metal or vice versa, without the structural phase transition. A Mott MIT Critical Temperature Switch (CTS) has been developed which generates a current control (or signal) at a critical temperature between 67oc and 85oc as the unique characteristic of Vanadium Oxide. After that, MIT switches have been applied to types of electromagnetic switches that interrupt an electric current in case of over current. A traditional electromagnetic switch –which takes the role of interrupting electricity through the mechanical switching when it conducts an over current-is composed of both an electromagnet called magnetic contactor, which connects or disconnects signals of main power and the thermal overload relay, with an on-off switching function controlled by temperature. The overload relay is composed of both an expensive and delicate mechanical switch with a large size 18 | P a g e
and a bimetal that is made of two metals with different thermal expansion coefficients joined together. The metal has the property of bending in any direction when heat is applied. The bending force of the bimetal controls the mechanical switch inducing the on-off switching. However the bimetal undergoes a change in its bending property during long time usage, therefore the accuracy of the overload relay drops, resulting into drop in performance. In order to solve the problem, the MIT CTS can be used as an on-off sensor instead of the bimetal. In this case, mechanical switch is replaced by a simple electric circuit controlling the electromagnet, which means that mechanical switching is converted to electrical. Therefore the MIT overload relay becomes small in size by removing the large mechanical switch and is also independent of external conditions. Hence the MIT electromagnetic switch has a reliable and accurate electronic switching mechanism.
7) Further scope The measurements done above could well be repeated with a Sr2IrO4 sample doped with an electron so that some new and interesting properties could be studied. (Observation of a d-wave gap in electron-doped Sr2IrO4: Nature Physics 12, 37-41(2016)). The Mott insulating stage is a manifestation of strong electron interactions in nominally metallic systems. It has been reported15 that unconventional superconductivity is induced by carrier doping in the J eff =1/2 Mott insulator. The superconducting state has been found to be stable only by electron doping and not by hole doping.
8) Conclusion Mott Physics and study of Mott insulators introduces us to a new scheme of insulators which can be studied diligently using the Hubbard model. One striking point of the Mott insulator is that drastic electronic state change emerges associated with the insulator-metal transition. In the vicinity of the Mott transition, a wide variety of unprecedented phenomenon like hightemperature superconductivity, colossal magneto resistance and large thermoelectric effect arises from interplay among charge, spin and orbital 19 | P a g e
degrees of freedom. While their functional effects should form the significant basis of future oxide electronics, the detailed mechanisms are still under debate. The Mott insulating state has been first found11 and studied in a set of simple 3d transition-metal oxides like Ti2O3, V2O3, Cr2O3, MnO, FeO, CoO and CuO. Among a number of ternary and multinary compounds, the most prototypical target material is perovskite oxides such as R1-xAxMO3 and R2-xAxMO4 (R, A and M being rare earth metals, alkaline earth metals and 3d transition metal elements).
10) Bibliography 1. Fundamentals of the Physics of solids, Vol-3, Jeno Solyom, Springer- Verlag. 2. Introduction to the Physics of electrons in solids, Henri Alloul, SpringerVerlag . 3. The physical principles of magnetism, Allan H. Morish, IEEE press. 4. Introduction to solid state Physics, Charles Kittel, Wiley. 5. Solid state Physics, Ashcroft Mermin, Cengage publishers. 6. Elements of group theory for Physicists, A.W. Joshi, New Age International Publishers. 7. Elements of X ray diffraction, B.D.Cullity, Addison-Wesley. 8. N.S Kini, A M Strydom, H S Jeevan and S ramakrishnan, J. Phys.:Condens. Matter 18(2006) 8205-8216. 9. Ben Ranjbar, Brendan J. Kennedy, Journal of solid state chemistry 232 (2015). 10. B.L. Chamberland, A R Philpotts, Journal of Alloys and Compounds, 182(1992). 11. M Uchida, Spectroscopic study on charge-spin-orbital coupled phenomenon in Mott Transition oxides, Springer Theses. 12. Hyun Tak Kim, New power interruption technology based on Mott device, Electronics online. 20 | P a g e
13. Vikram V. Deshpande, Bhupesh Chandra, Robert Caldwell, Science, vol. 323, Issue 5910. 14. B. J. Kim, Hosub Jin, J.Y. Kim, B. G. Park, C.S.Leem, Jaejun Wu,T.W. Noh, C. Kim, S. J.Oh, J. H. Park, V. Durairaj, G. Cao and E. Rotenberg. Physical Review Letters 101, 076402 (2008). 15. Hiroshi Watanabe, Tomonori Shirakawa and Seiji Yunoki, Physical Review Letters 110, 027002 (2013).
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