STUDY OF THE EXCITED STATES OF THE QUANTUM ANTIFERROMAGNETS

Transcription

1 STUDY OF THE EXCITED STATES OF THE QUANTUM ANTIFERROMAGNETS A THESIS SUBMITTED TO THE UNIVERSITY OF MANCHESTER FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN THE FACULTY OF ENGINEERING AND PHYSICAL SCIENCES 013 By Mohammad Ghanim Merdan School of Physics and Astronomy

2 Contents Abstract 8 Declaration 10 Copyright 11 Acknowledgements 1 Publications 13 1 Magnetism and Spin Lattice Systems Introduction Classification of Magnetic Materials Diamagnetism Paramagnetism Ferromagnetism Antiferromagnetism and Ferrimagnetism Origin of Exchange Interaction The Spin Hamiltonian Holstein-Primakoff Transformation Bogoliubov Transformation Spin-Wave Excitation via Classical Dynamics Spin-Wave Theory Bipartite Lattice Antiferromagnets Triangular Lattice Antiferromagnets Quasi-One-Dimensional Hexagonal Antiferromagnets The Coupled Cluster Method for Quantum Antiferromagnetism 5.1 Introduction

3 . The Formalism of CCM The CCM Applications to spin- 1 XXZ Model The LSUB Approximation Scheme The SUB Approximation Scheme The SUB+LSUB4 Approximation Scheme The Excitations Spectra Longitudinal Excitations in Quantum Antiferromagnets Introduction The Longitudinal Mode Excitation Spectra The Longitudinal Modes in Quantum Antiferromagnetic Models Square Lattice Antiferromagnets Triangular Lattice Antiferromagnets Quasi-1D Hexagonal Antiferromagnets Discussion and Conclusions Ground and Excited States of the CCM The Longitudinal Excitation Future Work A The Ground Bra-State in the SUB+LSUB4 Scheme 100 Word count:

4 List of Tables 1.1 Ground states of Nickel and Manganese ions of 3d n electron systems The ground state energy per spin for the 1D spin-1/ XXZ model for the full SUB, LSUB4 and SUB+LSUB4 for some values of The ground-state energy per spin for the D spin-1/ XXZ model in the SUB+LSUB4 scheme for some values of, together with that of the full SUB, SUB+g4 a, and LSUB4 schemes [4]

5 List of Figures 1.1 The main categorization of magnetic properties of materials with localized magnetic moments [] Distance in the hydrogen molecule which consist of two protons and two electrons Magnon dispersion for ferromagnets and antiferromagnets. For small q, the dispersions are quadratic and linear for ferromagnets and antiferromagnets respectively Representation of the spin motion in antiferromagnets. (a) Spin wave with plan view of an array. (b) Two spin waves propagating along a line of the spins with the same frequency Representation of the spin motion in ferromagnets. (a) Spin wave with plan view of an array. (b) The spins viewed in prospective Antiparallel spin structure of square antiferromagnetic lattice D plot of the linear spin-wave energy spectrum of Eq. (1.98), for a square-lattice Heisenberg model at = Ground state energy per spin for 1D, D and 3D spin- 1 Heisenberg model as a function of The sublattice magnetization for 1D, D and 3D as a function of the anisotropy The 10 spin structure of a triangular lattice antiferromagnet (a) The intensity of the linear spin-wave energy spectrum (dimensionless) ω q of Eq. (1.1) for a triangular-lattice Heisenberg model at = 1. (b) 3D plot of the linear spin-wave energy spectrum E(q) for a triangular-lattice Heisenberg model at = The classical spin structure of the quasi-1d hexagonal antiferromagnets: (a) on the ab-plane, and (b) the three-dimensional structure The rotation of up pointing spins coordinates by 180 o around y axis

6 1.14 The three spin-wave excitation spectra (in colors) for CsNiCl 3 with J = 0.345, J = and D = 0 THz, along the symmetry direction (0, 0, π + πη), (4πη, 0, π) and ( 4π, 0, π + πη). Also included is the 3 gapped y-mode (black, denoted as y ) with D = using the anisotropy term of Eq. (1.130). The solid and dash with the blue color on the lines indicate the zx + -mode and zx -mode respectively The sublattice magnetization for 1D spin-1/ XXZ model as a function of for the LSUB (dash line), the full SUB (solid line) [4] with exact result of Ref [84]. (Ref [4], adapted.) The ground state energy per spin for 1D spin-1/ XXZ model as a function of for the LSUB, full SUB and SUB+LSUB4 approximation schemes [83], together with the exact result of Refs. [81, 8]. (Ref [83], adapted.) The graphical representation of the ten local configurations in Eqs. (.93)- (.96) for the short-range part of the SUB+LSUB4 scheme. The flipped spins with respect to the Néel state are indicated by the crosses The structure of the computer programme of the SUB+LSUB4 scheme The ground state-energy per spin as a function of for spin-1/ XXZ model in the LSUB, full SUB, SUB+g4 a and SUB+LSUB4 schemes. The critical terminating points for each scheme are also indicated The staggered magnetization for the D spin-1/ XXZ model for the LSUB, full SUB, SUB+g4 a and SUB+LSUB4 schemes The excitation energy gap E(0) for the D spin-1/ XXZ Heisenberg model as a function of, for the full SUB, SUB+g4 a and SUB+LSUB4 schemes. The two gap values at = 1 are given by the LSUB4 scheme ( ) and LSUB8 scheme ( ) of Ref. [76] where the high-order excitation correlations are included as discussed in the text The spin-wave excitation spectra for the D spin-1/ XXZ Heisenberg model at c for the CCM (SUB and SUB+LSUB4) results, and at = 1 for the linear spin-wave theory (LSWT), the series expansion (SE) [89], and quantum Monte Carlo calculations [90]. The energy spectra in (a) are for q x = q y and those in (b) are for q y = The excitation spectrum E(q) of the longitudinal mode together with spin-wave excitation spectrum E(q) of the square lattice with an anisotropy = [110]

7 3. The hexagonal first Brillouin zone of a triangular lattice in reciprocal space. The coordinates of the labeled points are, Γ = (0, 0), P = (π/3, 0), L = (π, 0), Q = (4π/3, 0), M = (π, π/ 3), K = (π/3, π/ 3) and O = (0, π/ 3) The excitation spectrum E(q) of the longitudinal mode together with spin-wave excitation spectrum E(q) along (LMΓKP QMO) of the BZ of the Fig. 3. with an anisotropy = The gap value at K and Q points is 0.03zsJ for the longitudinal mode, and zsJ for the transverse spin-wave modes The longitudinal modes L ± as derived from Eq. (3.3) together with the spin-wave y- and zx ± modes as derived from Eq. (1.138) for CsNiCl 3 along the symmetry direction (0, 0, π +πη), (4πη, 0, π) and ( 4π 3, 0, π + πη). The longitudinal modes L ± calculated from the first order approximation and after including the second term in Eq. (3.17) are indicated by the dash and solid lines respectively (a) The first Brillouin zone of a quasi-1d hexagonal antiferromagnets. The points (0, 0), (π/3, π/ 3), (π/3, 0), (4π/3, 0), (π, π/ 3), and (0, π/ 3) all at q z = π are denoted as Q, K, P, Q, L, O respectively, and the similar points but at q z = 0 are denoted as Γ, K, P, Q, L, O respectively. (b) The hexagonal Brillouin zone at q z = π with some symmetry points in conventional notations for the quasi-1d systems The longitudinal mode L along the path Q K P QL O of the hexagonal Brillouin zone of Fig. 3.5(b) together with the spin-wave y and zx modes for CsNiCl 3. The longitudinal modes L calculated from the first order approximation and after including the second term in Eq. (3.17) are indicated by the dash and solid lines respectively

8 Abstract THE UNIVERSITY OF MANCHESTER Doctor of Philosophy Study of the excited states of the quantum antiferromagnets by Mohammad Ghanim Merdan, 013 We investigate the quantum dynamics of the spins on different Heisenberg antiferromagnetic spin lattice systems. Firstly, we applied the coupled-cluster method to the spin-1/ antiferromagnetic XXZ model on a square lattice by employing an approximation which contains two-body long-range correlations and high-order four-body local correlations. Improvement is found for the ground-state energy, sublattice magnetization, and the critical anisotropy when comparing with the approximation including the two-body correlations alone. We also obtain the full excitation spectrum which is in good agreement with the quantum Monte Carlo results and the high-order spin-wave theory. Secondly, we study the longitudinal excitations of quantum antiferromagnets on a triangular lattice by a recently proposed microscopic many-body approach based on magnon-density waves. We calculate the full longitudinal excitation spectra of the antiferromagnetic Heisenberg model for a general spin quantum number in the isotropic limit. Similar to the square lattice model, we find that, at the center of the first hexagonal Brillouin zone Γ(q = 0) and at the magnetic ordering wavevectors ±[Q = (4π/3, 0)], the excitation spectra become gapless in the thermodynamic limit, due to the slow, logarithmic divergence of the structure factor. However, these longitudinal modes on two-dimensional models may be considered as quasi-gapped, as any finite-size effect or small anisotropy will induce a large energy gap, when compared with the counterpart of the transverse spin-wave excitations. We have also investigated the excited states of the quasi-one-dimensional quantum antiferromagnets on hexagonal lattices, including the longitudinal modes based on the magnon-density waves. A model Hamiltonian with a uniaxial single-ion anisotropy is first studied by 8

9 a spin-wave theory based on the one-boson method; the ground state thus obtained is employed for the study of the longitudinal modes. The full energy spectra of both the transverse modes (i.e., magnons) and the longitudinal modes are obtained as functions of the nearest-neighbor coupling and the anisotropy constants. We have found two longitudinal modes due to the non-collinear nature of the triangular antiferromagnetic order, similar to that of the phenomenological field theory approach by Affleck. The excitation energy gaps due to the anisotropy and the energy gaps of the longitudinal modes without anisotropy are then investigated. We then compares our results for the longitudinal energy gaps at the magnetic wavevectors with the experimental results for several antiferromagnetic compounds with both integer and non-integer spin quantum numbers, and we find good agreements after the higher-order contributions are included in our calculations. 9

10 Declaration No portion of the work referred to in this thesis has been submitted in support of an application for another degree or qualification of this or any other university or other institute of learning. 10

11 Copyright (i) The author of this thesis (including any appendices and/or schedules to this thesis) owns certain copyright or related rights in it (the Copyright ) and he has given The University of Manchester certain rights to use such Copyright, including for administrative purposes. (ii) Copies of this thesis, either in full or in extracts and whether in hard or electronic copy, may be made only in accordance with the Copyright, Designs and Patents Act 1988 (as amended) and regulations issued under it or, where appropriate, in accordance with licensing agreements which the University has from time to time. This page must form part of any such copies made. (iii) The ownership of certain Copyright, patents, designs, trade marks and other intellectual property (the Intellectual Property ) and any reproductions of copyright works in the thesis, for example graphs and tables ( Reproductions ), which may be described in this thesis, may not be owned by the author and may be owned by third parties. Such Intellectual Property and Reproductions cannot and must not be made available for use without the prior written permission of the owner(s) of the relevant Intellectual Property and/or Reproductions. (iv) Further information on the conditions under which disclosure, publication and commercialisation of this thesis, the Copyright and any Intellectual Property and/or Reproductions described in it may take place is available in the University IP Policy (see policies/intellectual-property.pdf), in any relevant Thesis restriction declarations deposited in the University Library, The University Library s regulations (see regulations) and in The University s policy on presentation of Theses. 11

12 Acknowledgements I am grateful to my supervisor Yang Xian for his support, help and efforts during my study. I am also grateful to my colleagues C. Fullerton, T. Brett, R. Morris, A. Bladon, A. Black and J. Challenger, T. Biancalani, P. Ashcroft, J. Sanders, G. Constable for their assistance and cooperation. 1

13 Publications M. Merdan and Y. Xian, Coupled-cluster calculations for the ground and excited states of the spin-half X X Z model, J.Phys: Condens. Matter 3, (011). M. Merdan and Y. Xian, Longitudinal excitations in triangular Lattice antiferromagnets, J. of Low Temp. Phys. 171, 797 (01). M. Merdan and Y. Xian, Excited states of quasi-one-dimensional hexagonal quantum antiferromagnets, Phys. Rev. B 87, (013). 13

14 Chapter 1 Magnetism and Spin Lattice Systems 1.1 Introduction The magnetic property of a system arises from electrons (charge e, mass m) which are known to have an intrinsic spin magnetic moment. The z-component of the magnetic moment of atoms, molecules and solid is associated with electron spins µ z = gµ B m s, (1.1) where g is Landé s factor which is approximately equal to, µ B is the Bohr magneton (e /m = A.m ), and m s = s, s 1,, (s 1), s and s is the spin quantum number [1]. The classification of materials is dependant upon their magnetic properties. While some atoms or ions have permanent magnetic moment, others do not. For example, ferromagnetism is displayed in substances which have permanent moments and spontaneous alignments. When a magnetic field is applied, the dipolar magnetic fields of these moments interact and the interaction increases the spontaneous alignment. On the other hand, in antiferromagnetic materials the coupling between moments is such that the neighbourings moment tend to align along opposite directions. The long range order is described as two opposite ferromagnetic sublattices and the net magnetization should be equal. However, when the temperature is increased above zero, the thermal fluctuation reduces the alignment in a substance which has long range magnetic order. Increasing the temperature will make fluctuations larger and larger until the magnetic order is completely destroyed. The corresponding transition temperature is known as the Curie temperature for ferromagnets 14

15 1.1. INTRODUCTION 15 and Néel temperature for antiferromagnets []. However, the elementary excitations for the ferromagnets and antiferromagnets occur when the spin of one atom is reversed and the actual states can be linear combinations of many of these states. This excitation is the well known spin-wave excitation, i.e. quasiparticle excitation. The quasiparticle excitation corresponds to Anderson s spin-wave excitations [3] which are referred to as magnon, as described in Sec 1.8. An important method of quantum many body theory known as the coupled cluster method (CCM) has also been applied to spin systems to investigate the antiferromagnetic spin-wave excitations [4]. In this method the bra and ket wavefunctions are not Hermitian to one another because a linear approximation is involved in the bra wavefunction. A general variational theory for the ground and excited states has been proposed by extending the CCM to a variational formalism in which the bra and ket wavefunctions are Hermitian to one another [5, 6]. The CCM is different from the other techniques for quantum many-body system by dealing with infinite systems from the outset. On the other hand, in the application of CCM to spin systems, we need to make approximations within the cluster expansion terms to make this method solvable in practice. These approximations can be done by making selections of some included terms in the cluster expansions of the correlations operator that is used to parameterize the many-body wave functions, as we will describe more fully in Chapter. Another type of excitation known as the longitudinal excitation has been investigated theoretically and confirmed experimentally in various antiferromagnetic compounds. This kind of excitation correspond to the oscillation in the magnitude of the magnetic ordering parameter. The longitudinal excitation has been confirmed in various antiferromagnetic structures with Néel-like long range order at low temperature such as KCuF 3 [7] with s = 1/ and the hexagonal ABX 3 -type antiferromagnets such as CsNiCl 3 and RbNiCl 3 [8, 9] with both s = 1. Recently a microscopic many-body theory has been proposed [10] for the longitudinal excitations of spin-s antiferromagnets. In quantum antiferromagnets with a Néel-like order the longitudinal excitation has been identified as the collective mode of the magnon density waves, as we will describe in detail in Chapter 3. The thesis is organised into four chapters. In chapter one, we provide a brief explanation for the classification of the magnetic materials. We discuss the exchange interactions which describe how the spins interact. The interaction between spins is well explained by the Heisenberg Hamiltonian that we use to obtain the classical and quantum spin-wave excitations for antiferromagnets. The applications of the linear

16 16 CHAPTER 1. MAGNETISM AND SPIN LATTICE SYSTEMS spin-wave theory to different antiferromagnetic lattice models such as the bipartite lattice, the triangular lattice and quasi-one-dimensional hexagonal systems have also been presented. For the quasi-1d model, we obtain the low-energy spin wave spectra as a function of the uniaxial single-ion anisotropy. To our knowledge, this anisotropy dependence of the spin wave spectra has not been published before. In Chapter, we consider the CCM to investigate the ground state and excited states of a square lattice antiferromagnets. The CCM formalism is described in detail and some approximation schemes within the CCM are used to calculate the expectation values of the ground and excitation energies. We will also present our numerical results for the square lattice model using the so called SUB+LSUB4 approximation scheme of the CCM. In Chapter 3, we investigate the longitudinal excitation in quantum antiferromagnets using a microscopic many body approach. The longitudinal excitation will be described in detail and we will present numerical results for some antiferromagnetic models such as the square lattice model, the triangular lattice model and quasi-one dimensional hexagonal systems. In Chapter 4, we will discuss our numerical results and outline ideas for future work. 1. Classification of Magnetic Materials The substances with magnetic order can be classified according to the particular alignment pattern which the moments display. The main classification of magnetic materials is shown in Fig. 1.1 []. The magnetic properties of materials can be investigated with reference to magnetic susceptibility, which is defined as χ = M B, (1.) where M is the magnetization, and B is the magnetic field intensity. A susceptibility is frequently related to unit mass or to a mole of the substance. The molar susceptibility is written as χ M. Materials with a negative susceptibility are known as diamagnetic and materials with a positive susceptibility are known as paramagnetic. In this section, we briefly describe the classification of magnetic materials Diamagnetism Substances whose atoms or ions have no permanent magnetic moments are called diamagnetic if the magnetization induced by an applied field is in the opposite direction

17 1.. CLASSIFICATION OF MAGNETIC MATERIALS 17 Magnetic properties of substances Magnetic moment? Not Permanent Diamagnetism Permanent Long range order? No Paramagnetism Yes Parallel orientations? Yes Ferromagnetism No Antiparallel moments Unequal magnitude Ferrimagnetism Equal magnitude Antiferromagnetism Figure 1.1: The main categorization of magnetic properties of materials with localized FIG. magnetic 1. The moments main []. categorization of magnetic properties of materials. to the applied field. The microscopic application of Lenz s law describes the response of diamagnetic materials to an external applied magnetic field. Since the induced moments oppose the external magnetic field, the diamagnetic materials are characterized by a negative scalar susceptibility.

18 18 CHAPTER 1. MAGNETISM AND SPIN LATTICE SYSTEMS 1.. Paramagnetism Substances that consist of permanent magnetic moments and no spontaneous long range order are known as paramagnetic. Without an applied magnetic field, the orientations of magnetic moments have random orientation in thermal equilibrium and there is no net magnetic moment is displayed. When an external field is applied to these materials in high temperature, it will produce a partial alignment of the moments making a net magnetic moment. The alignment of magnetic moments tend to be parallel to an applied magnetic field, so the susceptibility is positive for isotropic paramagnets Ferromagnetism Materials that contain permanent moments that align spontaneously are called ferromagnetic. The interaction of moments via their dipolar magnetic fields give an increase to spontaneous alignment. In the absence of an applied field, a state of uniform magnetization is typically not observed in ferromagnets, because this state has a large magnetostatic energy. Instead, the magnetization remains uniform only within small regions known as domains, and it has different orientations in neighboring domains. In this way, the volume-averaged magnetization and the magnetostatic energy of the sample are both greatly reduced. When an external magnetic field is applied the alignment of domains will be with magnetic field giving rise to the net magnetization. Hence the isotropic ferromagnet, has positive scalar susceptibility Antiferromagnetism and Ferrimagnetism The quantum mechanical coupling in some substances between moments is such that the nearest neighbors moments are pointing in opposite directions. The long range order can be represented in the term of two opposing ferromagnetic sublattices. When the net magnetizations of the two sublattices are equal, the substances are called antiferromagnetic, whereas, if the net magnetizations are unequal, the material is ferrimagnetic. However, antiferromagnets behave like anisotropic paramagnets. When the external magnetic field is absent, the magnetizations of the two sublattices cancel, giving zero net magnetic moment. The susceptibility in the direction of the moments is very small at low temperature, because the field applies no net torque to the aligned moments. In contrast, the transverse susceptibility is non-zero even for T = 0, because the moments are free to turn in the direction of the applied field.

19 1.3. ORIGIN OF EXCHANGE INTERACTION 19 Other kind of magnetic ordering may also be found. For example, in many rare earth metals, the alignment of spins of neighboring magnetic moments can be in a spiral or helical pattern, and the materials are called helimagnets [11]. There are also magnetic materials with mixed interactions called spin glasses which show a random, yet cooperative, freezing of the spins at a well defined temperature T f (freezing temperature) [1]. Spin glasses display a metastable frozen state without the usual magnetic long range-ordering below the freezing temperature. Another state that can be achieved in a system with interacting spins is called a spin liquid in which the constituent localized moments are highly correlated but still fluctuate strongly down to absolute zero temperature [13]. However, in the previous sections, we described the magnetic materials with localized magnetic moments. In some cases, the magnetic moment in metals are associated with the conduction electrons which are delocalized and can wander freely through the sample; they are called itinerant electrons. Diamagnetism, paramagnetism and ferromagnetism may all occur in the case of delocalized magnetic moments. The low-energy ordered states of the metals are called spin-density waves (SDW). The ground state is described by a periodic modulation of the spin density whose period is related to the Fermi wave vector [1]. The SDW arises in highly anisotropic, low dimensional metals and it is different from the spin-wave excitation of ferromagnetism and antiferromagnetism described in the next sections. 1.3 Origin of Exchange Interaction We discuss in this section the origin of the Heisenberg Hamiltonian, and we can see later how the spin Hamiltonian operator depends upon the interaction between two spin operators. We expect the first interaction to play a role which is the dipole-dipole interaction [1]. If the distance between two magnetic moments µ 1 and µ is r, then the energy is equal to E = µ o 4πr 3 [µ 1 µ 3 r (µ 1 r)(µ r)], (1.3) where µ = 4π 10 7 N/A is the permeability of free space. The order of magnitude of this effect can be easily estimated for two moments, and each moment is µ 1µ B separated by r 1 Å to be approximately µ /4πr J. This energy is equivalent to nearly 1K in temperature; however, this interaction is too weak to account for magnetic ordering. To understand the origin of the Heisenberg Hamiltonian,

20 0 CHAPTER 1. MAGNETISM AND SPIN LATTICE SYSTEMS the hydrogen molecule, which is represented in Fig. 1., is used as the prototype for exchange interaction. This molecule consists of two atoms whose spins interact []. The total energy of the system can be written as the summation of the kinetic energy of electrons plus the potential energy of charges. The potential energy is attractive between opposite charges and repulsive between like charges. Figure 1.: Distance in the hydrogen molecule which consist of two protons and two electrons. H = p 1 m + p m e ( ), (1.4) 4πε 0 r 1a r b r 1b r a r 1 R ab where m is the mass of the electron, p i are the electron momenta, r ij is the distance between particles i, j and R ab is the distance between protons. These protons are considered fixed such that the kinetic energy associated with the motion of protons can be neglected because their mass compared with the electron s mass is large. The magnetic interaction between the electron spins is also neglected because the electrostatic interaction is much stronger than the magnetic interaction. With considering ψ the wave function of the system, the expectation value of the energy of the system can be written as E = ψ Hψd 3 r 1 d 3 r ψ ψd 3 r 1 d 3 r. (1.5) If the atomic diameter is small compared to the distance between atoms, then it is possible to use the wave function of the unperturbed hydrogen atom. The hydrogen

21 1.3. ORIGIN OF EXCHANGE INTERACTION 1 atom wave functions ϕ a and ϕ b satisfy ( p 1 m ( p m e 4πε 0 r 1a e 4πε 0 r b ) ϕ a (r 1 ) = E 0 ϕ a (r 1 ), (1.6) ) ϕ b (r ) = E 0 ϕ b (r ). (1.7) The symmetric and antisymmetric wavefunctions, ψ + and ψ, can be constructed using the atomic wavefunctions as basis functions ψ + = 1 [ϕ a (r 1 )ϕ b (r ) + ϕ a (r )ϕ b (r 1 )], (1.8) The above equations can be written as ψ = 1 [ϕ a (r 1 )ϕ b (r ) ϕ a (r )ϕ b (r 1 )]. (1.9) ψ ± = 1 [ϕ a (r 1 )ϕ b (r ) ± ϕ a (r )ϕ b (r 1 )]. (1.10) The normalization constant can be found as ψ ±(r 1, r )ψ ± (r 1, r )d 3 r 1 d 3 r = 1 ± α, (1.11) where α = ϕ a(r)ϕ b (r)d 3 r, is the overlap integral, and the normalization integral for each basis function is equal to unity ϕ aϕ a d 3 r = ϕ bϕ b d 3 r = 1. (1.1) Using Eqs. (1.4) and (1.10), we obtain ψ ±(r 1, r )H(r 1, r )ψ ± (r 1, r )d 3 r 1 d 3 r = E 0 (1 ± α ) + V ± U, (1.13) where V and U are the Coulomb integral and exchange integral respectively, defined as V = ϕ a(r 1 )ϕ b(r )H i (r 1, r )ϕ a (r 1 )ϕ b (r )d 3 r 1 d 3 r, (1.14)

22 CHAPTER 1. MAGNETISM AND SPIN LATTICE SYSTEMS U = ϕ a(r )ϕ b(r 1 )H i (r 1, r )ϕ a (r 1 )ϕ b (r )d 3 r 1 d 3 r, (1.15) where H i is the interaction part of the Hamiltonian H i = The symmetric and antisymmetric energies are given by e ( ). (1.16) 4πε 0 r 1 R ab r 1b r a E ± = E 0 + V ± U (1 ± α ), (1.17) the difference between symmetric and antisymmetric energies is E + E = V α U 1 α 4. (1.18) The difference between the symmetric and antisymmetric states depend on the spin orientation through the Pauli exclusion principle, therefore the wave function must be the product of the space and the spin parts Ψ = ψ ± s m s. (1.19) In a two-electron system, we have two possibilities s = 0 and s = 1. So, the antisymmetric singlet state with s = 0 can be constructed as 00 = 1 ( ), (1.0) and the symmetric triplet state with s = 1 is given by 11 =, 10 = 1 ( + ), 1 1 =. (1.1) It is quite useful to find an operator related to spin orientation giving us the energy of the state, this can be achieved by using the properties of total spin operator S ψ + 00 = s(s + 1) ψ + 00 = 0, (1.)

23 1.4. THE SPIN HAMILTONIAN 3 S ψ 1 m s = s(s + 1) ψ 1 m s = ψ 1 m s. (1.3) The spin operator would be H spin = E [ E E + ] S. (1.4) We will here refer to singlet and triplet states rather than symmetric and antisymmetric respectively. The total operator can be expanding as S = (S 1 + S ) (S 1 + S ) and the spin Hamiltonian of Eq. (1.4) can be rewritten as H spin =E [ ][ ] E E + S 1 + S + S 1 S (1.5) =E [ ][ ] E E + s(s + 1) + S 1 S (1.6) = 1 [ ] 1 [ ] E+ + 3E + E E 4 + S1 S (1.7) where s = 1/ for the electron. The low lying excitation above the ground state is more important than the zero of energy. Ignoring the constant first term of Eq. (1.7) and using Eq. (1.18) the Hamiltonian in Heisenberg form is given as H spin = J S 1 S, (1.8) where J = 1 (E + E ) = U V α 1 α 4. (1.9) 1.4 The Spin Hamiltonian The Heisenberg Hamiltonian of the two-electron system of Eq. (1.8) can be extended to N-spin systems by summing all individual two-spin interacting terms of spins on i and j with exchange magnetic parameter J ij. The general Heisenberg Hamiltonian can be written as [14] H = J ij S i S j. (1.30) ij The Hamiltonian may be written in terms of spin operators (S x, S y, S z ) as H = ij J ij [ σ(s x i S x j + S y i Sy j ) + Sz i S z j ]. (1.31)

24 4 CHAPTER 1. MAGNETISM AND SPIN LATTICE SYSTEMS The spin operators satisfy the commutation relations [S x i, S y j ] = i Sz i δ ij, [S y i, Sz j ] = i S x i δ ij, (1.3) [Si z, Sj x ] = i S y i δ ij. The Hamiltonian can be classified into three models depending on the values of parameters σ and. when σ = = 1, H is the Heisenberg model, when σ = 0 and = 1, is the Ising model. This model was developed by the physicist Wilhelm Lenz, who gave this problem to his student E. Ising. Weiss theory was the only theory of magnetism available around 190, which showed the phase transition independence of lattice dimensionality between paramagnetism and ferromagnetism. A phase transition has been calculated mathematically for two lattice dimensions by Onsager [15] in However, in 195 Ising found out the one dimension model has no phase transition and he could not find a solution in two and three dimensions. The Hamiltonian equation in this model was given by H = ij J ij S i S j µ B B 0 S i, (1.33) i the second term in the above equation refers to external magnetic field B 0, namely the Zeeman term [14]. For N lattice points, each point on 1, and 3 dimension is described by spin variable S i and can take just ±1 S i = ±1 ; i = 1,,..., N. (1.34) This model has many applications in statistical mechanics including binary alloys, fluids, magnetic insulator, etc. When σ = 1 and = 0, in Eq. (1.31), H is referred to as the XY model, in the classical XY-model, the unit vectors S i lie in two dimensions on each lattice site. The Hamiltonian of classical XY model is given by the following, H = ij J ij cos(θ i θ j ), (1.35)

25 1.5. HOLSTEIN-PRIMAKOFF TRANSFORMATION 5 where θ is the angle of spin with arbitrary axis in the xy-plane. The XY Hamiltonian operator in the Heisenberg model can be written as H = ij J ij (Si x Sj x + S y i Sy j ). (1.36) For the case of antiferromagnetism J > 0, this model is completely soluble in one dimension for the model with nearest neighbor interaction [16]. 1.5 Holstein-Primakoff Transformation Holstein and Primakoff [17] expressed the angular momentum operators in terms of boson creation and annihilation operators. This transformation has been applied to antiferromagnetism by Kubo [18] to investigate thermodynamic properties of antiferromagnets. Oguchi [19] applied this transformation to antiferromagnetic model and found the correction to Anderson s spin wave theory rising from the interaction between spin waves. The Holstein-Primakoff transformation is S + = s ˆf a, S = sa ˆf, S z = s a a, (1.37) where S ± are known as the raising and lowering operators and defined as S = S x is y. (1.38) These operators obey the commutation relations of Eq.(1.3) [S z i, S j ] = S i δ ij, [S + i, S j ] = Sz i δ ij. (1.39) ˆf is the non-linear operator ˆf = 1 a a s, (1.40)

26 6 CHAPTER 1. MAGNETISM AND SPIN LATTICE SYSTEMS with the boson operators obeying the commutation relations [a i, a j ] = δ ij, [a i, a j ] = [a i, a j ] = 0. (1.41) 1.6 Bogoliubov Transformation Bogoliubov [0] introduced a linear canonical transformation for the boson operators when joining with the theory of superfluidity and superconductivity. This transformation is known as Bogoliubov transformation. For each value of the wave vector q the linear transformation is as follows α q = u q a q v q a q ; α q = u q a q v q a q, (1.4) where coefficients (u q, v q ) have two conditions: (i) the new boson operators should be linearly independent for all values of q [α q, α q ] = 0 ; [α q, α q] = 0, (1.43) (ii) these operators also should satisfy the canonical commutation relation [α q, α q] = 1. (1.44) By using u q vq = 1, it is possible to parameterize u q and v q in the form of hyperbolic functions u q = cosh(θ q ) and v q = sinh(θ q ), (1.45) the inverse of Eqs. (1.4) can be given as a q = u q α q + v q α q ; a q = u q α q + v q α q. (1.46) 1.7 Spin-Wave Excitation via Classical Dynamics When the temperature is at absolute zero T = 0, the crystal s magnetic moments are essentially completely ordered. The magnetic moments decrease with increasing the temperature. In this case, enough thermal energy is gained by the elementary moments

27 1.7. SPIN-WAVE EXCITATION VIA CLASSICAL DYNAMICS 7 to turn versus the internal field. This deviation from perfect order can be described classically in terms of spin waves, or quantum mechanically in terms of magnons, which are quantized spin waves [1]. We take the 1D Heisenberg model as an example to demonstrate the classical spinwave excitation. The classical ground state of a simple antiferromagnet has antiparallel spin for the sublattices as we have mentioned before. Consider N spins of the value s on a line; the Heisenberg interaction for this case is N H = J S n S n+1, (1.47) n=1 with J > 0. If we consider S n as classical vector, the dot product of the spins vector in the ground state is s, with the ground state energy given by E 0 = JNs. (1.48) Classically, the dispersion relation can be obtained, and the terms in Eq. (1.47) which contain the nth spin can be written as JS n (S n 1 + S n+1 ), (1.49) the magnetic moment is given by µ n = gµ B S n, (1.50) then Eq. (1.49) become µ n [(J/gµ B )(S n 1 + S n+1 )]. (1.51) The energy associated with the magnetic moment is given as U n = µ n B n, (1.5) where, B n is the magnetic field, from Eq. (1.51) we can write B n = (J/gµ B )(S n 1 + S n+1 ). (1.53)

28 8 CHAPTER 1. MAGNETISM AND SPIN LATTICE SYSTEMS The rate of change of the angular momentum is equal to the torque µ n B n ds n dt = (gµ B / )S n B n = (J/ )S n (S n 1 + S n+1 ). (1.54) In Cartesian components ds x n dt = (J/ )[S y n(s z n 1 + S z n+1) S z n(s y n 1 + S y n+1)], (1.55) ds y n dt = (J/ )[S z n(s x n 1 + S x n+1) S x n(s z n 1 + S z n+1)], (1.56) ds z n dt = (J/ )[S x n(s y n 1 + S y n+1) S y n(s x n 1 + S x n+1)]. (1.57) When the amplitude of the excitation is low, we can find an approximation for the linear equations by approximating all the spin up by (S z = s) with even indices n compose the sublattice a, and spin down (S z = s) with odd indices n + 1 compose the sublattice b. The terms of the product of S x and S y are also neglected in the equation for ds z /dt. The linearized equations are ds x n dt = (Js/ )(S y n + S y n 1 + S y n+1), (1.58) ds y n dt = (Js/ )(S x n + S x n 1 + S x n+1). (1.59) We can write the equations for a spin on the sublattice b as ds x n+1 dt = (Js/ )(S y n+1 + S y n + S y n+), (1.60) ds y n+1 dt = (Js/ )(S x n+1 + S x n + S x n+), (1.61) Using S + = S x + is y we get ds z n dt = dsz n+1 dt = 0. (1.6) ds + n dt = (ijs/ )(S + n + S + n 1 + S + n+1), (1.63)

29 1.7. SPIN-WAVE EXCITATION VIA CLASSICAL DYNAMICS 9 ds + n+1 dt The solution should have the following form = (ijs/ )(S + n+1 + S + n + S + n+). (1.64) S + n = u exp[i(n)qa iet] ; S + n+1 = υ exp[i(n + 1)qa iet]. (1.65) Using the above expression, with Eqs. (1.63) and (1.64) we obtain Eu = 1 E ex(u + υe iqa + υe iqa ), (1.66) Eυ = 1 E ex(υ + ue iqa + ue iqa ), (1.67) where E ex = Js/, the solution of Eqs. (1.66) and (1.67) is given by E q = E ex[1 cos qa], (1.68) or E AF q = E AF ex sin qa. (1.69) This antiferromagnetic dispersion relation is quite different from that for a ferromagnet with energy spectrum given by E F = E F ex(1 cos qa). (1.70) The antiferromagnetic and ferromagnetic spectra are plotted in Fig For qa 1, Eq. (1.70) can be written as Eq F = ( 1 E exa ) q. (1.71) This energy spectrum is proportional to q whereas for an antiferromagnet, the energy spectrum is linear in q (see Fig. 1.3). This behavior of antiferromagnetic spectrum is similar to that of the frequency of a phonon of the lattice vibration in the same limit. The excitation of a spin system is represented in a circular motion for each spin as shown in Figs. 1.4 and 1.5.

30 30 CHAPTER 1. MAGNETISM AND SPIN LATTICE SYSTEMS E F q Eq/Eex E AF q π π/ hω= 4JS 0 sin( ka). π/ π (117) q For k = 1, the energy spectrum changes linear versus k and we have hω= 4JS sin( ka). (117) Figurediscussed 1.3: Magnon some experimental dispersion forresults ferromagnets of some antiferromagnet and antiferromagnets. in section For (4). small q, For k = 1, the energy spectrum changes linear versus k and we have the dispersions are quadratic and linear for ferromagnets and antiferromagnets respectively. discussed some experimental results of some antiferromagnet in section (4). (a) (a) (a) (b) (b) Figure 1.4: Representation of the spin motion in antiferromagnets. (a) Spin wave with plan view (b) FIG. of 4. an array. Spin wave (b) Two representation spin waves of propagating antiferromagnet. along (a) a line plan of of the an spins array with the same frequency. containing a spin wave. (b) spin wave propagating along a line of spin. FIG. 4. Spin wave representation of antiferromagnet. (a) plan of an array containing a spin wave. (b) spin wave propagating along a line of spin. The spin wave dispersion relation is different for ferromagnet, since the two sublattices are in the same directions. The energy spectrum of ferromagnet, is The spin wave dispersion relation is different for ferromagnet, since the two given as, sublattices are in the same directions. The energy spectrum of ferromagnet, is h ω= 4 JS(1 cos ka), (118) given as, for long wavelengths ka = a, so that (1 cos ka) ( ka) / and h ω= 4 JS(1 cos ka), (118)

31 It is clear that the frequency is proportional to k which is different from It antiferromagnet is clear that the relation. frequency The is physical proportional picture to of k ferromagnetic which is different spin wave from are antiferromagnet shown in Fig. 5. relation. The physical picture of ferromagnetic spin wave are 1.8. SPIN-WAVE THEORY 31 shown in Fig. 5. (a) (a) (a) (b) (b) (b) Figure 1.5: Representation of the spin motion in ferromagnets. (a) Spin wave with plan view FIG. of 5. an The array. spin (b) motion The spins ferromagnet. viewed in prospective. (a) The spin viewed in perspective. FIG. 5. The spin (b) motion Plan viewed in ferromagnet. of an array (a) with The a spin spin viewed wave. in perspective. 1.8 Spin-Wave Theory (b) Plan viewed of an array with a spin wave. Bloch The first quantum [] introduced physical a picture spin wave is essentially theory in different the Heisenberg from the model classical of ferromagnetism. The since picture The quantum we quantum have physical spin of energy up picture and of is down a spin essentially so, wave the is different possible called afrom magnon. excitation the classical At is a to low picture flip temperature, since sublattice the he found we have a spin temperature spin the up opposite and dependency down direction so, ofthe the i.e. possible magnetization. from excitation S z to In Sz 1931, is as to shown flip Bethe the in [3] introduced sublattice Fig. 6. the In wave spin the in function case the of opposite antiferromagnet, of the Heisenberg direction to i.e. antiferromagnetic express from S the z to spin Swave zlinear as excitations, shown chain model in with Fig. nearest the 6. spin-flip neighbor In the operators case interaction. of antiferromagnet, can be The introduced ground to state express as energy a linear the spin was combination, obtained wave excitations, by of Hulthen spin [4] the in operators, spin-flip Anderson operators [3] gave can be a good introduced approximation as a linear to determine combination, the ground of spin state operators, X xs i i ; X% energy and excitation spectrum in the same style of Klein xs % + i i, and Smith [5] at absolute (10) i i zero temperature X xs i i ; X% with coefficients in two chosen dimensions as, D and three dimensions xs % + i i, 3D. In the next(10) sections i i the ground state energy and the spontaneous with coefficients chosen as, magnetization ik. r of the Heisenberg Hamiltonian of different antiferromagneticnsystems will be discussed. N xi = xi( k) = e ; i ik. ri x% i = x% i( k) = e. (11) ik. r xi = xi( k) = e ; i ik. ri x% i = x% i( k) = e. (11) N 37 N Bipartite Lattice Antiferromagnets The crystal may be divided into two sublattices a for spin-up and b for spin-down; the nearest neighbors of an atom on a lie on b and vice versa. The angle between spins of the sublattice a and b is 180 as shown in Fig. 1.6 for a square lattice. Eq. (1.30) represents the general expression for the antiferromagnetic Hamiltonian and can be

32 3 CHAPTER 1. MAGNETISM AND SPIN LATTICE SYSTEMS Figure 1.6: Antiparallel spin structure of square antiferromagnetic lattice. written in terms of the raising and lowering operators H = J ij [1 (S i S+ j + S + i S j ) + ] Sz i Sj z, (1.7) where the nearest-neighbor exchange constant is J > 0 for an antiferromagnet, the sum on i, j runs over all the nearest neighbor pairs of the bipartite lattice once and is the magnetic anisotropy. Magnetic substances have a magnetic anisotropy when the internal energy depends upon the magnetization direction with respect to crystallographic axes [6]. The Heisenberg model corresponds to the isotropic case i.e. = 1, while the limit of will give the Ising model and the XY model is given by = 0. We now make the Holstein-Primakoff transformation for the a-sublattice and for the b-sublattice. The lowering and raising operators can be written as S + ai = s 1 a i a i s a i ; S ai = sa i 1 a i a i s, (1.73) S + bj = sb j 1 b j b j s ; S bj = s where index i for a-sublattice, index j for b-sublattice. However 1 b j b j s b j, (1.74) S a iz = s a i a i ; S b jz = s + b j b j. (1.75)

33 1.8. SPIN-WAVE THEORY 33 By using the binomial expansion for the operator ˆf = 1 ˆn/s in Hilbert space ˆf(s) = 1 ˆn 4s 1 3 ˆn, (1.76) s with ˆn = a a for a-sublattice and ˆn = b b for b-sublattice. The operator is called the spin deviation operator. If the correction terms in the binomial expansion are neglected by setting ˆf = 1, and using the Bloch s theorem to Fourier transform [7], the spin wave variables are introduced as a q = (/N) 1/ i b q = (/N) 1/ j e iqr i a i ; a q = (/N) 1/ i e iqr j b j ; b q = (/N) 1/ j e iqr i a i, e iqr j b j. (1.77) The sum is over N/ atoms for sublattices a and b. In the expansion of Eqs. (1.73), (1.74) and (1.75) the leading terms are S + ai = (s/n)1/ ( q e iqr i a q ) ; S ai = (s/n)1/ ( q e iqr i a q), (1.78) S + bj = (s/n)1/ ( q e iqr j b q) ; S bj = (s/n)1/ ( q e iqr j b q ), (1.79) S a iz = s N qq e i(q q ).ri a qa q ; S b jz = s + N qq e i(q q ).rj b qb q. (1.80) The summation over q is executed on the first Brillion zone (BZ) of the sublattice; the first Brillion zone has half the volume of the BZ of the total lattice. Therefore, the number of points N of these sublattices is half the number of antiferromagnetic atoms in the crystal. The wave number extends only over N/ value from π to π. For example, the linear chain of length N By applying the following equation q = πn N, n = 1 N +,..., 0,...1 N. (1.81) j e i(q q )r j = 1 Nδ qq, (1.8)

34 34 CHAPTER 1. MAGNETISM AND SPIN LATTICE SYSTEMS the total Hamiltonian can be transformed to magnon variables as, after the linear approximation of Eqs. (1.76), (1.78), (1.79) and (1.80) H = 1 NzJs + H 0, (1.83) where z is the coordination number, and H 0 is the linear term in magnon variables H 0 = Jzs q [γ q (a qb q + a q b q ) + (a qa q + b qb q )], (1.84) with γ q = z 1 ϱ e iq.ϱ, (1.85) where the vector ϱ connects the atom i with the nearest neighbors on the Bravais lattice. If a lattice has an inversion symmetry, thus γ q is symmetric γ( q) = γ(q). (1.86) Using the Bogoliubov transformations for the two bosonic operators a q = u q α q + v q β q ; a q = u q α q + v q β q, (1.87) b q = u q β q + v q α q ; b q = u q β q + v q α q, (1.88) with Eq. (1.84), the linear Hamiltonian becomes H 0 = Jzs q { [γ q (u q + v q) + u q v q ](α qβ q + α q β q ) + [γ q u q v q + (u q + v q)](β qβ q + α qα q ) + γ q u q v q + v q}. (1.89) This Hamiltonian can be diagonalized by setting the factor on a non diagonal term equal to zero γ q (u q + v q) + u q v q = 0, (1.90) by using a simple algebra and using hyperbolic identities to determine γ q γ q = tanh θ, (1.91)

35 1.8. SPIN-WAVE THEORY 35 the constant term in Eq. (1.89) is given as c q = γ q u q v q + v q, (1.9) or we have thus c q = [cosh θ q (1 γ q / ) 1], (1.93) cosh θ q = 1, (1.94) (1 γq / ) 1/ c q = [(1 γ q / ) 1/ 1]. (1.95) The total Hamiltonian of Eq. (1.83) is given as H = E 0 + q E q (α qα q + β qβ q ), (1.96) where E 0 is the ground state energy E 0 = 1 NzJs + Jzs q [(1 γ q / ) 1/ 1], (1.97) and, E q is the energy spectrum of spin waves E q = J zs(1 γ q / ) 1/. (1.98) The energy spectrum for a square lattice for instance is represented in the surface plot of Fig.1.7 with two zero modes, one at q = 0, and the another at the antiferromagnetic ordering wave vector Q = (π, π). These modes are the direct result of having broken the continuous rotational symmetry and are known as Goldstone modes [8]. The sum over q is replaced by an integral, as there are N/ values of q [1 (1 q γ q / ) 1/ ] = 1 NI D, (1.99) I D = 1 (π) D π π [1 (1 γ q / ) 1 / ] dq 1 dq...dq D, (1.100)

36 36 CHAPTER 1. MAGNETISM AND SPIN LATTICE SYSTEMS Eq/Jzs π π/ 0 q y π/ π π 0 π/ q x π π/ Figure 1.7: 3D plot of the linear spin-wave energy spectrum of Eq. (1.98), for a squarelattice Heisenberg model at = 1. where, D is the number of dimensions. This integral can be calculated by expanding the square root in Eq. (1.98) using the binomial theorem and then to keep only the first term. However, this approximation is not useful because of the slow convergence of the series if just the first two terms are taken. Therefore, it is appropriate to choose numerical integration, which give the possibility to avoid the approximation for multidimensional cases. The numerical integrals have been calculated by Anderson and Kubo for one, two and three dimensions. In one dimension, γ q is given by, with z 1 = and D = 1 γ q D=1 = e iqa + e iqa = cos qa, (1.101) for two dimensions, with z = 4 and D = γ q D= = 1 (cos q xa + cos q y a), (1.10) for three dimensions, with z 3 = 6 and D = 3 (simple cubic) γ q D=3 = 1 3 (cos q xa + cos q y a + cos q z a). (1.103)

37 1.8. SPIN-WAVE THEORY 37 The ground state energy is given by E 0 D = 1 zj Ns (1 + I D /s). (1.104) The numerical results at = 1 are I 1 = 0.363, I = and I 3 = The ground state energy E 0 for one, two and three dimensions as a function of anisotropy is plotted in Fig Δ D 0.6 E0/JN D D Figure 1.8: Ground state energy per spin for 1D, D and 3D spin- 1 Heisenberg model as a function of. Magnetization is defined as the amount of magnetic moments per unit volume V M = N m V µ s, (1.105) where, N m is the number of magnetic moments. The diagonal method of the Bogoliubov transformation made the expression for the sublattice magnetization easy to calculate by using the z-component of the up-sublattice M = s N a qa q, (1.106) q