DOCSLIB.ORG

Ferroelectricity Induced by Interatomic Magnetic Exchange Interaction

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Ferroelectricity Induced by Interatomic Magnetic Exchange Interaction Ferroelectricity induced by interatomic magnetic exchange interaction Xiangang Wan,1† Hang-Chen Ding,2 Sergey Y. Savrasov3 and Chun-Gang Duan2,4‡ 1Department of Physics and National Laboratory of Solid State Microstructures, Nanjing University, Nanjing 210093, China 2Key Laboratory of Polar Materials and Devices, Ministry of Education, East China Normal University, Shanghai 200062, China 3Department of Physics, University of California, Davis, One Shields Avenue, Davis, CA 95616, USA 4National Laboratory for Infrared Physics, Chinese Academy of Sciences, Shanghai 200083, China † Electronic address: [email protected] ‡ Electronic address: [email protected] or [email protected] Multiferroics, where two or more ferroic order parameters coexist, is one of the hottest fields in condensed matter physics and materials science1-9. However, the coexistence of magnetism and conventional ferroelectricity is physically unfavoured10. Recently several remedies have been proposed, e.g., improper ferroelectricity induced by specific magnetic6 or charge orders2. Guiding by these theories, currently most research is focused on frustrated magnets, which usually have complicated magnetic structure and low magnetic ordering temperature, consequently far from the practical application. Simple collinear magnets, which can have high magnetic transition temperature, have never been considered seriously as the candidates for multiferroics. Here, we argue that actually simple interatomic magnetic exchange interaction already contains a driving force for ferroelectricity, thus providing a new microscopic mechanism for the coexistence and strong coupling between ferroelectricity and magnetism. We demonstrate this mechanism by showing that even the simplest antiferromagnetic (AFM) insulator MnO, can display a magnetically induced ferroelectricity under a biaxial strain. The combination of different ferroic properties, proposed to explain the ferroelectricity in magnetic spiral especially ferroelectricity and (anti)ferromagnetism, structures. Electric polarization can also be induced by provides additional degree of freedom to control collinear spin order in the frustrated magnet with several magnetic and dielectric properties of the material. Such species of magnetic ions6,15-17. There are also proposals to functionality is of great potential in applications to realize multiferroic state in composite systems or next-generation fast, portable and low-energy materials with nanoscale inhomogeneity18,19. consumption data storage and processing devices. Nevertheless, currently all of the known magnetically Unfortunately, magnetism and ferroelectricity tend to be driven single-phase multiferroics require either mutually exclusive, as conventional ferroelectric Dzyaloshinskii-Moriya interaction (DMI)20,21, which is perovskite oxides usually require transition metal (TM) small in strength, or competing exchange interactions in ions with a formal configuration d0, whereas magnetism, real space. Therefore, they generally have complex in contrast, needs TM ions with partially filled d shells10. magnetic order, low transition temperature (below As a consequence, simultaneous occurrence of several ten K) and small electric polarization (generally ferromagnetism and ferroelectricity is hard to be two to three orders of magnitude smaller than typical achieved, especially at room temperature. There are ferroelectrics), making them still far away from practical indeed some exceptions in Bismuth related magnetic applications. Therefore searching new mechanism for oxides, e.g., BiFeO3 and BiMnO3. In these compounds, multiferroicity is of both fundamental and technological however, the magnetism and ferroelectricity are widely interest. believed to have different origins, rendering that the In this Letter, we demonstrate that regardless the magnetoelectric coupling rather weak9. spin-orbit coupling, simple interatomic magnetic To gain a strong magnetoelectric coupling, vast exchange interaction already provides a strong driving efforts have been devoted to search the improper force to break the inversion symmetry of the system, ferroelectricity where electric dipoles are induced by which is necessary for the occurrence of ferroelectricity. magnetism6. Phenomenological theory suggests that It is therefore a new microscopic mechanism for the spatial variation of magnetization is essential for the coexistence and strong coupling between ferroelectricity magnetically induced electric polarization11. Several and magnetism, and all of the magnetic insulators with microscopic mechanisms12-14, all emphasizing the simple magnetic structure need to be revisited. Using importance of spin-orbital coupling, have also been band structure calculations, we numerically confirm this 1 new mechanism by illustrating that even simplest for M1/M2 and O site. t1/t2 are the hopping integrals antiferromagnets, i.e. MnO, can display a magnetically between M1/M2 and O. Based on the standard induced ferroelectricity under biaxial strain. Schrieffer-Wolff transformation22, we can eliminate the O orbital and obtain the effective hopping integral between M1 and M2 site: tt 12 teff ∝ (2) εεdo− It is well known that the hopping integral is inversely − y − y proportional to the bond-length: tr11∼ , tr22∼ , where r1(2) is the bond length of M1(2)-O, y is a positive value and strongly depends on both the bond type and the Figure 1 | Illustration of ferroelectricity induced by participated orbital23. A longitudinal displacement of O indirect magnetic exchange interaction between ion u will then change the effective hopping to anion-mediated magnetic cations. Here M and M are 11 1 1 2 t ∝ =C (3) magnetic cations, O is oxygen anion. When O atom, eff yy yy εεd −+0 ()()ru ru − ()() ru + ru − which originally sits in the middle of M1 and M2 ions, is where C is a parameter and not sensitive to u, r is the shifted along the M1(2)-O bond direction (x) by a small distance between magnetic ion and center O-site. It is displacement u, the magnetic exchange interactions of interesting to notice that regardless the parameter C and the system will increase and may support the O off-center the superscript y in Eq. (3), a displacement u which movement. An electric dipole is then formed, as shown breaks inversion symmetry always increases the effective by an arrow in the picture. hopping between the magnetic ions. To study the interatomic magnetic exchange To see the effect of u on the magnetic ordering interaction and the associated magnetic ordering energy, energy, we take the superexchange, which is one of the we consider a three atoms case, where a diamagnetic ion most common mechanisms in magnetic insulators, as an such as oxygen ion sits between two transition metal example. It is well known that for superexchange the 24 ions, as shown in Fig.1. As revealed by Sergienlo and interatomic exchange interaction can be written as : Dagotto13, DMI provides a driving force for the oxygen 4t 2 J =− eff (4) ion to shift perpendicularly to the spin chain. Due to the U small strength, however, the energy gain from DMI is where U is the Coulomb interaction for the magnetic usually less than the ordinary elastic energy, orbital. Thus, an off-center displacement of O ion can consequently only a few compounds show magnetically enhance the interatomic exchange interaction. Usually 6 induced ferroelectricity . Here, we consider the effect of the magnetic ordering energy has the form of longitudinal displacement of diamagnetic ions (shown by − J SS⋅ , thus for the AFM case, the energy gain up ∑ij ij i j the dot line in Fig.1) on the magnetic ordering energy. As is well known, in the above case, the distance between to the second order of the longitudinal displacement of O magnetic ions usually is much larger than the radii of d/f ions u is 112y orbital which carry magnetic moments, thus the direct Δ∝−Eu + ≈− 2 (5) exchange is negligible, and the hybridization between ()()ru+−22442yyyy ru r r+ magnetic and diamagnetic ions is essential for the Thus we prove that regardless the exact formula for the indirect magnetic exchange coupling. Therefore, we first hopping dependence on distance, an off-center distortion discuss the hopping integrals in this system. For can definitely lower the total energy by an amount ∼ u 2 . simplicity, here we only show the case of one orbital per Above we have discussed the superexchange in ion, and it is easy to generalize our results to multi-orbital one-band case. For multi-band superexchange and even cases. The kinetic energy then can be written as: double exchange mechanism25, the exchange coupling J Hcccccc=++εεε+++ is also proportional to the effective hopping, just the kin∑∑∑ d d11σσ d o o σσ o d d 2 σ d 2 σ σσσ relationship between J and teff in these mechanisms may ++++tcchctcchc(++ . .) ( . .) (1) not be as simple as shown in Eq. (4). Keeping in mind 12∑∑do12 d o σσ that the effective hopping is a function of 1(ru22− ), an where the operator cc+ (), cc+ (), cc+ () dd11σ σ dd22σ σ ooσ σ off-center distortion will therefore always lower the magnetic energy. Moreover, regardless the specific form creates (annihilates) a spin σ electron at M1, M2 and O site, respectively. ε d and ε o are the on-site energies 2 Figure 2 | Energy difference as a function of O offcenter displacement. a, Calculated energy differences between zero-temperature AFM and FM state of MnO for a series of O off-center displacement at different biaxial strains: no strain (red square), -3% (blue triangle), -5% (green circle).
Load more

Recommended publications
  • Effect of Exchange Interaction on Spin Dephasing in a Double Quantum Dot
    PHYSICAL REVIEW LETTERS week ending PRL 97, 056801 (2006) 4 AUGUST 2006 Effect of Exchange Interaction on Spin Dephasing in a Double Quantum Dot E. A. Laird,1 J. R. Petta,1 A. C. Johnson,1 C. M. Marcus,1 A. Yacoby,2 M. P. Hanson,3 and A. C. Gossard3 1Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA 2Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 76100, Israel 3Materials Department, University of California at Santa Barbara, Santa Barbara, California 93106, USA (Received 3 December 2005; published 31 July 2006) We measure singlet-triplet dephasing in a two-electron double quantum dot in the presence of an exchange interaction which can be electrically tuned from much smaller to much larger than the hyperfine energy. Saturation of dephasing and damped oscillations of the spin correlator as a function of time are observed when the two interaction strengths are comparable. Both features of the data are compared with predictions from a quasistatic model of the hyperfine field. DOI: 10.1103/PhysRevLett.97.056801 PACS numbers: 73.21.La, 71.70.Gm, 71.70.Jp Implementing quantum information processing in solid- Enuc. When J Enuc, we find that PS decays on a time @ state circuitry is an enticing experimental goal, offering the scale T2 =Enuc 14 ns. In the opposite limit where possibility of tunable device parameters and straightfor- exchange dominates, J Enuc, we find that singlet corre- ward scaling. However, realization will require control lations are substantially preserved over hundreds of nano- over the strong environmental decoherence typical of seconds.
    [Show full text]
  • PART 1 Identical Particles, Fermions and Bosons. Pauli Exclusion
    O. P. Sushkov School of Physics, The University of New South Wales, Sydney, NSW 2052, Australia (Dated: March 1, 2016) PART 1 Identical particles, fermions and bosons. Pauli exclusion principle. Slater determinant. Variational method. He atom. Multielectron atoms, effective potential. Exchange interaction. 2 Identical particles and quantum statistics. Consider two identical particles. 2 electrons 2 protons 2 12C nuclei 2 protons .... The wave function of the pair reads ψ = ψ(~r1, ~s1,~t1...; ~r2, ~s2,~t2...) where r1,s1, t1, ... are variables of the 1st particle. r2,s2, t2, ... are variables of the 2nd particle. r - spatial coordinate s - spin t - isospin . - other internal quantum numbers, if any. Omit isospin and other internal quantum numbers for now. The particles are identical hence the quantum state is not changed under the permutation : ψ (r2,s2; r1,s1) = Rψ(r1,s1; r2,s2) , where R is a coefficient. Double permutation returns the wave function back, R2 = 1, hence R = 1. ± The spin-statistics theorem claims: * Particles with integer spin have R =1, they are called bosons (Bose - Einstein statistics). * Particles with half-integer spins have R = 1, they are called fermions (Fermi − - Dirac statistics). 3 The spin-statistics theorem can be proven in relativistic quantum mechanics. Technically the theorem is based on the fact that due to the structure of the Lorentz transformation the wave equation for a particle with spin 1/2 is of the first order in time derivative (see discussion of the Dirac equation later in the course). At the same time the wave equation for a particle with integer spin is of the second order in time derivative.
    [Show full text]
  • Comment on Photon Exchange Interactions
    Comment on Photon Exchange Interactions J.D. Franson Johns Hopkins University, Applied Physics Laboratory, Laurel, MD 20723 Abstract: We previously suggested that photon exchange interactions could be used to produce nonlinear effects at the two-photon level, and similar effects have been experimentally observed by Resch et al. (quant-ph/0306198). Here we note that photon exchange interactions are not useful for quantum information processing because they require the presence of substantial photon loss. This dependence on loss is somewhat analogous to the postselection required in the linear optics approach to quantum computing suggested by Knill, Laflamme, and Milburn [Nature 409, 46 (2001)]. 2 Some time ago, we suggested that photon exchange interactions could be used to produce nonlinear phase shifts at the two-photon level [1, 2]. Resch et al. have recently demonstrated somewhat similar effects in a beam-splitter experiment [3]. Because there has been some renewed discussion of this topic, we felt that it would be appropriate to briefly summarize the situation regarding photon exchange interactions. In particular, we note that photon exchange interactions are not useful for quantum information processing because the nonlinear phase shifts that they produce are dependent on the presence of significant photon loss in the form of absorption or scattering. This dependence on photon loss is somewhat analogous to the postselection process inherent in the linear optics approach to quantum computing suggested by Knill, Laflamme, and Milburn (KLM) [4]. Our interest in the use of photon exchange interactions was motivated in part by the fact that there can be a large coupling between a pair of incident photons and the collective modes of a medium containing a large number N of atoms.
    [Show full text]
  • Heisenberg and Ferromagnetism
    GENERAL I ARTICLE Heisenberg and Ferromagnetism S Chatterjee Heisenberg's contributions to the understanding of ferromagnetism are reviewed. The special fea­ tures of ferrolnagnetism, vis-a-vis dia and para magnetism, are introduced and the necessity of a Weiss molecular field is explained. It is shown how Heisenberg identified the quantum mechan­ ical exchange interaction, which first appeared in the context of chemical bonding, to be the es­ sential agency, contributing to the co-operative ordering process in ferromagnetism. Sabyasachi Chatterjee obtained his PhD in Though lTIagnetism was known to the Chinese way back condensed matter physics in 2500 BC, and to the Greeks since 600 BC, it was only from the Indian Institute in the 19th century that systematic quantitative stud­ of Science, Bangalore. He ies on magnetism were undertaken, notably by Faraday, now works at the Indian Institute of Astrophysics. Guoy and Pierre Curie. Magnetic materials were then His current interests are classified as dia, para and ferromagnetics. Theoretical in the areas of galactic understanding of these phenomena required inputs from physics and optics. He is two sources, (1) electromagnetism and (2) atomic the­ also active in the teaching ory. Electromagnetism, that unified electricity and mag­ and popularising of science. netiSlTI, showed that moving charges produce magnetic fields and moving magnets produce emfs in conductors. The fact that electrons (discovered in 1897) reside in atoms directed attention to the question whether mag­ netic properties of materials had a relation with the mo­ tiOll of electrons in atoms. The theory of diamagnetism was successfully explained to be due to the Lorentz force F = e[E + (l/c)v x H], 011 an orbiting electron when a magnetic field H is ap­ plied.
    [Show full text]
  • Short-Range Versus Long-Range Electron-Hole Exchange Interactions in Semiconductor Quantum Dots
    RAPID COMMUNICATIONS PHYSICAL REVIEW B VOLUME 58, NUMBER 20 15 NOVEMBER 1998-II Short-range versus long-range electron-hole exchange interactions in semiconductor quantum dots A. Franceschetti, L. W. Wang, H. Fu, and A. Zunger National Renewable Energy Laboratory, Golden, Colorado 80401 ~Received 25 August 1998! Using a many-body approach based on atomistic pseudopotential wave functions we show that the electron- hole exchange interaction in semiconductor quantum dots is characterized by a large, previously neglected long-range component, originating from monopolar interactions of the transition density between different unit cells. The calculated electron-hole exchange splitting of CdSe and InP nanocrystals is in good agreement with recent experimental measurements. @S0163-1829~98!51144-7# One of the most intriguing features of the spectroscopy of clear, however, to what extent the exchange interaction in semiconductor quantum dots is the energy shift between the quantum dots is affected by dielectric screening. Recent di- zero-phonon absorption and emission peaks observed in size- rect calculations8,15 for InP and CdSe nanocrystals revealed selective spectroscopies of monodispersed samples.1–7 The that the unscreened exciton splitting is significantly larger redshift of the emission peak, of the order of 10 meV, has than the measured redshift, suggesting that the screening of been measured in Si,1 CdSe,2–5 InP,6 and InAs ~Ref. 7! the LR interaction may be important. nanocrystals grown with different techniques. Although sev- ~iv! How to calculate the exchange splitting. In standard eral models have been proposed to explain the redshifted approaches based on the effective-mass approximation emission, recent measurements and calculations for CdSe ~EMA! the e-h exchange interaction is described by a short- ~Refs.
    [Show full text]
  • ± Μbb 2Μbb Ef
    Paramagnetic Susceptibility ms probability A) Bound Electrons in Atoms x B +½ p ½e s=½ Curie Law: 1/T +x +B ½ p+ ½e With increasing temperature T the alignment of the magnetic moments in a B field is less effective, due to thermal spin flips. Calculation for an electron (spin ½, negative charge): Energy of an electron in a magnetic field: U = msgBB = BB (ms= ½, g=2) The probabilities p are proportional to the Boltzmann factor: x p exp(U/kBT) = exp(BB / kBT) = e 1 x with x = BB/kBT << 1 x +x x The normalization condition p+ + p = 1 gives p= e / (e + e ) ½ ½ x The magnetization M is obtained by separating the electron density N into opposite spins with magnetic moments B and probabilities p : +x x +x x M = N (p+B pB) = N B (e e ) / (e + e ) NBx = NB(BB/kBT) 2 = 0 M/B 0 N B /kBT The 1/T law originates from 1/kBT in the Boltzmann factor exp(U/kBT). B) Free Electrons in a Metal Pauli Paramagnetism: independent of T , D(EF) , analogous to: cV D(EF) The cross-hatched area corresponds to EF 2 B the excess spins, B i.e. to the induced magnetization M. D(E) BB D(E) 1 Energy Bands of Ferromagnets Like the bands of a Pauli paramagnet in B-field, the band structure E(k) of a ferromagnet can be decomposed into two sets of bands, one for majority spin ( , “spin up”, || B ), the other for minority spin ( , “spin down”, || B ). They are separated by the magnetic exchange splitting Eex , which ranges from 0.3 eV in Ni to 2 eV in Fe.
    [Show full text]
  • Paolasini Magnetism Lecture4.Ppsx
    Luigi Paolasini [email protected] LECTURE 4: “MAGNETIC INTERACTIONS” - Dipole vs exchange magnetic interactions. - Direct and indirect exchange interactions. - Anisotropic exchange interactions. - Interplay between orbital and magnetic order. L. Paolasini - LECTURES ON MAGNETISM- LECT.4 - Follows directly from Maxwell equations - Direct interaction between two magnetic moments - depends on their relative orientation The dipolar interaction is very weak: µ1µ2 r ~ 1Å, U => U ~ 10-23J = 1K 3 4π r µ1=µ2=1µB Long range interaction, responsible for demagnetizing field and ferro- magnetic domains. Dipolar interaction are important only when the exchange interactions are small. L. Paolasini - LECTURES ON MAGNETISM- LECT.4 - Magnetism is fully quantum mechanical Competition between the kinetic energy of the electrons squeezed in a small box and the Coulomb repulsion The size of atoms is given by the balance of these two terms: ħ2 π2 Kinetic energy ~ ≈ eV 2m L2 Kinetic energy ~ ≈e 2eV Coulomb energy ~ ≈ eV 4πε0 L Spin-orbit ≈ meV Magneto-crystalline anisotropy ≈ µeV L. Paolasini - LECTURES ON MAGNETISM- LECT.4 Light atoms: Heavy atoms: Z small Z huge Electrons drawn tightly Electrons far apart: together by nucleus: Weakly interacting Strongly interacting and prone to correlated behavior L. Paolasini - LECTURES ON MAGNETISM- LECT.4 Quantum-mechanical description of identical particle - Indistinguishable particles: - Bosons: share the same quantum state photons, gluons, phonons, helium-4 atoms. - Fermions: follows the Pauli exclusion principle Electrons, neutrinos, quarks, protons, neutrons, helium-3 atoms - Symmetry of quantum states - Symmetrical (bosons) and anti-symmetrical (fermions) overall wavefunctions - Statistical properties of identical particles system: - Fermions: Fermi-Dirac statistic - Bosons: Bose-Einstein statistic L. Paolasini - LECTURES ON MAGNETISM- LECT.4 Pauli spin matrices Electron Wavefunction => SPINOR: representation of the spin part of electron wavefunction ψ(x,+1/2) => ψ(x,-1/2) => L.
    [Show full text]
  • 5. Exchange Interaction
    5. Exchange interaction The dipole interaction between the magnetic moments of the electrons is much too weak for supporting magnetic order of materials at high temper- atures. For explaining the observed magnetism, we need to find a strong interaction between the electrons. We might think that an interaction de- pending explicitly on the spin (the magnetic moments) of the electrons is needed; no such interaction is known though. In 1928, Werner Heisenberg realized that the responsible interaction is the Coulomb repulsion between electrons; this is a strong interaction but does not explicitly depend on the spin. Spin selectivity is due to quantum mechanics, in particular the Pauli principle: Two electrons with parallel or antiparallel spin behave dif- ferently even though the fundamental interaction is the same; the spatial * * wave function ψ(r1, r2) has to be antisymmetric and symmetric, respec- tively. One consequence is that two electrons with parallel spin cannot be in the same place. In order to discuss how the Coulomb interaction term leads to an exchange interaction of the spins, we first write this interaction in second quantization; we will introduce this very useful representation first. 5.1 Occupation number representation for fermions So far, we have discussed the Hamiltonian in so-called first quantization: * Ne Ne 2 Ne Pi * * * H = H0 + H1 , H0 = hi = + V(ri) , H1 = u(ri − rj) 2m Xi=1 Xi=1 Xi=1 Xi<j (5.1) * * where u(ri − rj) is either the bare Coulomb potential 2 * * e u(ri − rj) = * * (5.2) |ri − rj| or an effective, screened interaction. First quantization implies that the an- tisymmetry of the wave function has to be taken into account by working 44 with Slater determinants which is rather cumbersome.
    [Show full text]
  • Exchange and Ordering in Magnetic Materials
    Exchange and ordering in magnetic materials Claudine Lacroix, Institut Néel, Grenoble 1-Origin of exchange 2- Exchange in insulators: superexchange and Goodenough- Kanamori rules 3- Exchange in metals: RKKY, double exchange, band magnetism 4- Magnetic ordering: different types of orderings, role of dimensionality, classical vs quantum spins Cargese, 27/02/2013 Various types of ordered magnetic structures: Type of magnetic order depends on the interactions Various microscopic mecanisms for exchange interactions in solids: -Localized / itinerant spin systems -Short / long range -Ferro or antiferro 2 Exchange and ordering in magnetic materials 1-Origin of exchange 2- Exchange in insulators:superexchange and Goodenough-Kanamori rules 3- Exchange in metals: RKKY, double exchange, band magnetism 4- Magnetic ordering: different types of orderings, role of dimensionality, classical vs quantum spins Cargese, 27/02/2013 Origin of exchange interactions: - electrostatic interactions - Pauli principle Interatomic exchange: Hydrogen molecule Exchange interactions are due to Coulomb repulsion of electrons Hamiltonian of 2 H nuclei (A, B) + 2 electrons (1,2): H = H0(r1-Ra) +H0(r2-RB) + Hint •A •1 •B H = p2/2m + U(r) 0 •2 Hint: Coulomb interaction 2 possibilities for the total electronic spin: S=0 or S=1 4 Wave function of the 2 electrons: Ψ(1,2) = φ(r1,r2)χ(σ1,σ2) φ(r1,r2) : orbital part χ(σ1,σ2) : spin part Pauli principle: wave function Ψ(1,2) should be antisymmetric Ψ(1,2) = - Ψ(2,1) ⇒ either φ symmetric, χ antisymmetric or φ antisymmetric, χ symmetric
    [Show full text]
  • 9 Indistinguishable Particles and Exchange
    9 Indistinguishable Particles and Exchange Quantum mechanics allows us to predict the results of experiments. If we conduct an experiment with indistinguishable particles a correct quantum description cannot allow anything which distin- guishes between them. For example, if the wavefunctions of two particles overlap, and we detect a particle, which one is it? The answer to this is not only that we don’t know, but that we can’t know. Quantum mechanics can only tell us the probability of finding a particle in a given region. The wavefunction must therefore describe both particles. The Schroedinger equation is then: " # h¯2 − (∇2 + ∇2) + V (r ) + V (r ) Φ(r , r ) = EΦ(r , r ) 2m 1 2 1 2 1 2 1 2 where the subscripts label each particle, and there are six coordinates, three for each particle. Φ is a wave in six dimensions which contains the information we can measure: the probability of finding particles at r1 and r2, but not what we can’t measure: which particle is which. What basis states would be appropriate for Φ? An approximation is to use a product such as Φ(r1, r2) = |a(r1)b(r2)i where a(r1) and b(r2) are one-particle wavefunctions of atoms 1 and 2. This allows us to separate the two particle equation into two one particle equations: −h¯2 −h¯2 [ ∇2 + V (r )]|a(r )i = E |a(r )i;[ ∇2 + V (r )]|b(r )i = E |b(r )i 2m 1 1 1 1 1 2m 2 2 2 2 2 2 provided that the particles do not interact (n.b ∇1 does not act on b(r2)).
    [Show full text]
  • Introduction to Heisenberg Model
    Introduction to Heisenberg model Javier Junquera Most important reference followed in this lecture Magnetism in Condensed Matter Physics Stephen Blundell Oxford Master Series in Condensed Matter Physics Exchange interactions If relativistic effects are not considered, then the electric interaction between particles does not depend on their spins In the absence of a magnetic field, the Hamiltonian of a system of particles in electric interaction does not contain spin operators A look to the hamiltonian: a difficult interacting many-body system. Kinetic energy operator for the electrons Potential acting on the electrons due to the nuclei Electron-electron interaction Kinetic energy operator for the nuclei Nucleus-nucleus interaction Exchange interactions If relativistic effects are not considered, then the electric interaction between particles does not depend on their spins In the absence of a magnetic field, the Hamiltonian of a system of particles in electric interaction does not contain spin operators When it is applied to a wavefunction, it does not affect the spinoidal variables The wave function of a system of particles can be written as a product It depends only on the spin It depends only on the variables spatial coordinates of the particles The Schrödinger equation determines only , leaving arbitrary The indistinguisibility of the particles leads to symmetry conditions on the wave function Despite the fact that the electric interaction is spin-independent, there is a dependendency of the energy of the system with respect the total
    [Show full text]
  • Chapter 5 Ferromagnetism
    Chapter 5 Ferromagnetism 1. Mean field theory 2. Exchange interactions 3. Band magnetism 4. Beyond mean-field theory 5. Anisotropy 6. Ferromagnetic phenomena Comments and corrections please: [email protected] Dublin February 2007 1 1. Mean field theory The characteristic feature of ferromagnetic order is spontaneous M magnetisation s due to spontaneous alignment of atomic magnetic moments, which disappears on heating above a critical temperature known as the Curie point. The magnetization tends to lie along certain easy directions determined by crystal structure (magnetocrystalline anisotropy) or sample shape. 1.1 Molecular field theory Weiss (1907) supposed that in addition to any externally applied field H, there is an internal ‘molecular’ field in a ferromagnet proportional to its magnetization. Hi M = nW s Hi must be immense in a ferromagnet like iron to be able to induce a ! significant fraction of saturation at room temperature; nW 100. The origin of these huge fields remained a mystery until Heisenberg introduced the idea of the exchange interaction in 1928. Dublin February 2007 2 m mB m i Magnetization is given by the Brillouin function, < > = J(x) where x = µ0 H /k BT. m m The spontaneous magnetization at nonzero temperature Ms = N< > and M0= N . In zero external field, we have B Ms/M0 = J(x) (1) But also by eliminating Hi from the expressions for Hi and x, 2 Ms/M0 = (NkBT/µ0M0 nW)x 2 2 which can be rewritten in terms of the Curie constant C = µ0Ng µB J(J+1)/3kB. Ms/M0 = [T(J+1)/3CJn W]x (2) The simultaneous solution of (1) and (2) is found graphically , or numerically.
    [Show full text]