Professur für Anorganische Chemie II MRC07 Dr. Anna Isaeva Basics

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1 Professur für Anorganische Chemie II MRC07 Dr. Anna Isaeva Basics Dr. Anna Isaeva Dresden, 8th October 2018

2 Recommended literature Dr. Martin Valldor (IFW Dresden) Lecture series Spins Do ( , Neubau Chemie, Bergstraße 66, Seminar room 398)

3 A sneak peek into my own research in topological insulators The page is in English

4 Semiconductor s surface Possible metallic surface states Degenerated spin states Rashba effect Spin-resolved surface states Local spin currents Spin-resolved surface states Resilient against backscattering

5 Topological insulator A bulk semiconductor (insulator) with spin-resolved (metallic) surface states that are robust against backscattering (TR symmetry preserved) and exhibit spin-momentum locking 3D weak TI Spin-resolved edge states 3D strong TI (Balents, Moore, 2007) Spin-resolved surface states D. Kong, Yi Cui. Nature Chem. 3, 845 (2011).

6 HgTe/CdTe quantum wells: 2D TI O.A. Pankratov et al., Solid State Commun. 61, 93 (1987). B.A. Bernevig et al., Science 314, 1757 (2006). M. König et al. Science 318, 766 (2007).

7 Tetradymite-type 3D strong TIs Bi 2 Te 3 Helical surface states (Dirac cone) n-doped Bulk insulator Te Bi Nature Phys. 5, 438 (2009), Science 325, 178 (2009), PRL 103, (2009).

8 AC II, TUD: Structure variety of new 3D WTIs (W:weak) Bi 2 TeI GaGeTe Dual TI Bi 14 Rh 3 I 9 I. Rusinov et al. Sci. Rep. 6, (2016). A. Zeugner et al. Chem. Mater. 29, 1321 (2017). N. Avraham et al. arxiv.org: A. Zeugner et al. Chem. Mater. 30, 5272 (2018). B. Rasche et al. Nature Mater. 12, 422 (2013). B. Rasche et al. Chem. Mater. 25, 2359 (2013). B. Rasche et al. Sci. Rep. 6, (2016). C. Pauly et al. Nature Phys. 11, 338 (2015).

9 Optimization of crystal growth (mostly by Chemical Transport Reactions) Bi 2 TeI β-bi 4 I 4 Bi 14 Rh 3 I 9 Slide 15

10 Pursuit of magnetic topological insulators MnBi 2 Te 4 First AFM TI arxiv.org: Cr-doped (Bi,Sb) 2 Te 3 below 30 mk: C.-Z. Chang et al. Science. 340, 167 (2013). c Mol [emu/mol] 0,22 0,20 0,18 0,16 0,14 0,12 0,10 0,08 0,06 0,04 0,02 T N =24,1(5)K T N =24,2(1)K H ^ ab: m 0 H = 1 T, zfc H ^ ab: m 0 H = 1 T, fcw H ab: m 0 H = 1 T, zfc H ab: m 0 H = 1 T, fcw 0, T [K] Slide 32

11 Spintronics GMR effect (1988, A. Fert and P. A. Grüneberg; Nobel Prize in Physics in 2007) Quantum mechanical effect Electron scattering is dependent on the spin orientation M NM M antipatallel magnetisation Rel. electrical resistance, % M NM M Magnetic field, H C. Chappert et al., Nature Mater. 6, 813 (2007)

12 Historical overview Observation and application (from the ancient times on) Fe 3 O 4 = (Fe 3+ ) 2 Fe 2+ (O 2 ) 4 A mixed valence compound Structure type: (inverse) spinel Magnisia, Greece General formula for spinels: A 3+ 2B 2+ O 4 Normal spinel: cations A in octahedral sites, B in tetrahedral Inverse spinel: ½ of A in tetrahedral sites, ½ of A and B in octahedral 5μ B Lodestone or Magnetite Fe 3 O 4 5μ B + 4μ B μ net ~ 4μ B ferrimagnetic ordering

13 Outline 1.1. Macroscopic level Classification of magnetic materials based on their susceptibility 1.2. Atomic level Spin-orbit coupling. Magnetocrystalline anisotropy. Coupling schemes in many-electron systems Diamagnetism of valence (core) electrons and itinerant electrons Paramagnetism of valence (core) electrons and itinerant electrons 1.3. Interacting (magnetic) momenta Exchange interactions (spin-spin) or correlation effects. Direct exchange Superexchange, double exchange RKKY-interactions Anisotropic interactions (for instance, Dzyaloshinski-Moriya) 2.4. Cooperative magnetism Magnetically ordered states Magnetic frustration, spin glass, spin ice Band magnetism Metal-insulator transitions

14 1.1. Macroscopic level Magnetism as interaction with an external magnetic field Macroscopic picture We are studying the response of a macroscopic sample introduced into an external magnetic field

15 1.1. Macroscopic level Magnetism as interaction with an external magnetic field Macroscopic picture We are studying the response of a macroscopic sample introduced into an external magnetic field (a usual approach to explore the magnetic properties of a (unknown) sample) Experimental parameters/variables: magnetic field strength H (A/m) magnetic flux density B (T = Nm/A) magnetization M (A/m) susceptibility χ (dimensionless) permeability μ (μ 0 μ r ) or relative permeability μ r permeability of vacuum (permeability constant) μ 0 = 4p 10 7 V s/a m

16 Characteristic values Magnetism as interaction with an external magnetic field Cause (H) Mediation (B) Response (M) or electric current Maxwell equations Lorentz force magnetic dipols what the sample sees B = μ 0 H (free space) B = μ 0 μ r H (in the solid) B = μ 0 (H + M) M = χ H μ r = 1 + χ

17 Classification of magnetic materials Based on susceptibility or permeability. Susceptibility is discontinuous μ r = 1 + χ 0 µ r < 1 diamagnetic material µ r = 1 vacuum µ r > 1 paramagnetic material µ r 1 magnetically ordered material (non-linear dependence cannot be assessed based on the absolute values of permeability or susceptibility ) μ = μ r μ 0 χ < 0 χ = 0 χ > 0 diamagnetic material vacuum paramagnetic material χ = M / H χ 0 magnetically ordered materials (non-linear dependence, e. g. hysteresis) any other complex dependence

18 A note on paramagnetism Paramagnetism Magnetization is induced and persists until the external magnetic field is applied. In contrast to that, ferromagnetism (or any other longrange magnetic ordering) is stable without an applied field (spontaneous). Large positive experimental χ values do not suffice for a sample to be characterized as ferro(ferri)magnetic. Paramagnetic materials achieve saturation in high magnetic fields.

19 Magnetic properties are not a permanent characteristic of a material at hand. They are being studied within a parameter space (magnetic fields, temperature, physical or chemical pressures are variables). Magnetic phase transitions are often observed, in particular along the temperature scale. Methods of synthesis/crystal growth and the associated physical shape of the sample (crystal, powder, nanoscaled product, thin film, etc.) can (strongly) influence the magnetic characteristics of a given material. Also due to confinement effects: spatial confinement: nanostructures; thin films structure confinement: 2D, 1D and 0D structure fragments, various types of chemical bonding between the fragments electronic confinement: 2DEG at interfaces, surface; heterostructures Phase diagrams, incl. magnetic ones (H, T, p, etc.). The range of available magnetic fields: from 50 µt on the Earth surface to Saturation magnetization of iron Outmost weak magnetic field in a lab Magnetic properties in a parameter space ca. 2 T 10 9 T Brain waves (Physikalisch-Technischen Bundesanstalt, Berlin) Outmost strong (stable) magnetic field in a lab 45 T (Tallahassee, USA)

20 1.2. Atomic level Magnetism as a phenomenon dealing with interactions (orientation) of magnetic moments (mostly spins) Microscopic picture We are looking at interactions and mutual adjustments of elemental magnets or magnetic centres. Cooperative, collective (electron-based) phenomena. Electron correlations and competing ground states.

21 1.2. Atomic level Magnetism as a phenomenon dealing with interactions (orientation) of magnetic moments (mostly spins) Microscopic picture We are looking at interactions and mutual adjustments of elemental magnets or magnetic centres. Cooperative, collective (electron-based) phenomena. Electron correlations and competing ground states. Experimental parameters/variables: Quantum numbers Angular (orbital) momentum L, (total) spin momentum (S), total angular momentum J Bohr magneton μ B = A m 2

22 Characteristic values Magnetism as a phenomenon dealing with interactions (orientation) of magnetic moments (mostly spins) Spin angular momentum (S) = eigenrotation of an electron (core electrons and itinerant electrons). A metaphor of planetary rotation is deceiving. Experimental confirmation: Stern-Gerlach experiment (1922) Ag [Kr]4d 10 5s 1 atoms in an inhomogeneous magnetic field Orbital angular momentum (L) = electron is an electric charge on a circular orbit, loop current (core electrons) Experimental confirmation: Einstein-de Haas effect (1915) A rod made of a ferromagnetic material is suspended on a string. It starts to rotate around its axis when it is magnetized along its length. Reason: conservation of angular momentum. Orbital angular momentum is a quantum-mechanical analog of the classic angular momentum.

23 Einstein-de Haas experiment Demonstrates a connection between the classic angular momentum and macroscopic magnetization of a macroscopic body and the angular momentum of an electron. Relates the observed magnetic moment to its angular quantum number. Rotation is a consequence of the conservation of angular momentum.

24 Einstein-de Haas experiment Demonstrates a connection between the classic angular momentum and macroscopic magnetization of a macroscopic body and the angular momentum of an electron. Relates the observed magnetic moment to its angular quantum number. Rotation is a consequence of the conservation of angular momentum. Magnetic moment m is related to: Current loops (orbital motion of electric charge) Spin magnetic moments of elementary particles m I m = I ds, S area Units Am 2 Current loop is an orbital motion of charge and orbital motion of a particle mass (angular momentum L) m and L are proportional: m = γ L, γ g-factor, gyromagnetic ratio This experiment proves that the angular momentum is real.

25 Magnetic moment m I B E pot = m B Potential energy is minimal when the moment is parallel to the external field. The relation is written by analogy with an electric dipole. The notion of magnetic dipole. Are there magnetic monopoles? In non-uniform magnetic field, there will be a magnetic force proportional to the magnetic field gradient, acting on a magnetic dipole: F loop = (m B), F z = m z B z z TU Dresden, Magnetismus Folie 25

26 Magnetic moment m I B E pot = m B Potential energy is minimal when the moment is parallel to the external field. The relation is written by analogy with an electric dipole. The notion of magnetic dipole. Are there magnetic monopoles? In homogenous magnetic field, there is no force but there is torque: T = m B Angular momentum and torque lead to magnetic precession with Larmor frequency γ B

27 Bohr magneton m B = eħ 2m = J T = ev T = emu One Bohr magneton corresponds to the moment of the electron in a hydrogen atom with the Bohr radius of Å and angular quantum number l = 1. In quantum mechanics, γ = m B, so that ħ m = m B L, m = gm B ԦS, g Landé-factor, g = 2 (free electron) For electrons in periodic solids, both core and itinerant electrons, the Landéfactor is slightly different from 2. m = m B (L + g ԦS) total magnetic moment

28 Stern-Gerlach experiment Ag: [Kr]4d 10 5s 1 V B = -m z B F z = -(dv B /dz) = m z (db/dz) If the spin eigenvalues were similar to the orbital ones (the l = 1 state), we would expect to see 3 deflected beams. If the space quantization were due to the magnetic quantum number m l, m l states were always odd (2l +1) and should produce an odd number of lines. m s (secondary spin quantum number) ranges from s to +s and generates (2s + 1) values. Fermions (incl. electrons) take up half-integer s values. TU Dresden, Magnetismus Folie 28

29 A History of Spin Zeeman effect (1896) Sodium yellow doublet (splitting of the emission lines without external magnetic field: a direct consequence of spin-orbit coupling) Na atom: the 3p level splits into 2 states with total angular momentum (J) J = 3/2 and J = 1/2 in the presence of the internal magnetic field caused by orbital motion and spin. In the reference system of a core electron, the nucleus with a charge +Ze is rotating around the electron and generates a magnetic field. In this field, the electron has an additional potential energy: U m = μ s B = μ z B z = g s μ B S z B z = μ B B z TU Dresden, Magnetismus Folie 29

30 Spin-orbit coupling I Spin angular momentum (S) = eigenrotation of an electron (core and itinerant electrons) Core electrons = loop currents angular momentum (L) Spin-orbit coupling (in a rotating reference system of an electron, a positively charged nucleus (+Ze) is rotating on a circular orbit and generating a magnetic field): j 2 = j = l + s l 2 + s 2 + 2l s Energy of SOC: E SO = μ s B orb = A l s A SOC constant, A~ Z4 n 6, Z atomic number, n principal quantum number E SO lies in the range between 10 and 100 mev R. Gross, A. Marx. Festkörperphysik. 2. Auflage TU Dresden, Magnetismus Folie 30

31 A History of Spin Zeeman effect (1896) Sodium yellow doublet (splitting of the emission lines without external magnetic field: a direct consequence of spin-orbit coupling) Bohr-Sommerfeld atomic model (1913) Paschen-Back effect (1921) Stern-Gerlach experiment (1922) S. Goudsmith, G. Uhlenbeck (1925, Universität Leiden): electron has its own orbital momentum W. Pauli (1927): has determined the complete set (incl. spin) of commuting variables (CSCO), or the complete set of quantum numbers describing a quantum system, Pauli exclusion principle TU Dresden, Magnetismus Folie 31

32 Zeeman effects Zeeman effect(s): Splitting of energy states in external magnetic field Sodium atom in external field (all z-component of total angular momentum J: Term: 2S+1 L J, selection rules

33 Splitting of spectral lines Angular momentum quantum number is taken into account, Total spin = 0 Magnetic quantum number and spin qantum number are decoupled TU Dresden, Magnetismus Folie 33

34 Momenta coupling in many-electron systems How do angular l i and spin s i moments of individual electrons couple in case of a manyelectron system (e.g. a valence shell)? j i = l i + s i jj-coupling Russell-Saunders Russell-Saunders scheme (light elements, e.g. 3d-transition metals with L = 2, but also rare-earth elements of the 4f row L = 3): all single orbital angular momenta couple to the total angular momentum L = σ i l i, all single spin momenta couple to the total spin moment S = σ i s i, then L and S couple to the total total angular momentum J = L + S. jj-scheme (for large atomic numbers Z): first pairs of individual l i and s i moments couple to j i, then these couple to the total angular momentum J. This is a case of strong spin-orbit coupling. TU Dresden, Magnetismus Folie 34

35 Spin-orbit coupling II Energy of spin-orbit coupling: E SO = μ s B orb = A l s E SO corresponds to the energy that is gained when spin aligns parallel to l from the initial perpendicular position to l. Since orbital angular momentum l often has a preferred (energetically favorable) crystallographic orientation (thanks to crystal-field effects (electrostatic potential of some symmetry), degeneracy of 3d-electons), the spin momentum s will also orient parallel to this direction via SOC effect. In this spirit, spin-orbit coupling strengthens magnetocrystalline anisotropy. Magnetocrystalline anisotropy leads to the effect of easy magnetization along particular crystallographic axis. In other words, the lattice (atomic arrangement in general) can influence the spin orientation via spin-orbit effects. Other types of magnetic anisotropy: Shape anisotropy: easy magnetization axis in the longest direction of measurement Magnetoelastic anisotropy (induced anisotropy). TU Dresden, Magnetismus Folie 35

36 Further considerations Quenching of orbital angular moment ( spin-only magnetism ) in compounds of transition metals, band magnetism Strong SOC may unquench the orbital momenta affected by electrostatic potential Chemical bonding (differing orbital overlap between the adjacent atoms). Electron-electon interactions (Pauli exclusion principle, Coulomb repulsion, wave function symmetry). Solid state chimera TU Dresden, Magnetismus Folie 36

37 Characteristic values Magnetism as a phenomenon dealing with interactions (orientation) of magnetic moments (mostly spins) Total magnetic moment is a sum of all contributions of all electrons in a solid. Both core and itinerant electrons cause dia- and paramagnetic contributions. TU Dresden, Magnetismus Folie 37

38 Magnetic response of core and itinerant electrons Localized electrons Core electrons Delocalized electrons Itinerant electrons R. Gross, A. Marx. Festkörperphysik. 2. Auflage TU Dresden, Magnetismus Folie 38

39 Magnetic response of core and itinerant electrons χ van Vleck χ Pauli χ Larmor χ Langevin Localized electrons Itinerant (quasi-free) electrons Paramagnetism Langevinparamagnetism, χ = C/T Contributions of spin and orbital angular momenta of electrons on partially occupied shells Van Vleckparamagnetism. Spin and orbital momenta of closed shells. Pauli-paramagnetism Contributed by the spin momenta Diamagnetism Atomic or Larmordiamagnetism Contributed by the orbital angular momenta Landau-diamagnetism Contributed by the orbital angular momenta TU Dresden, Magnetismus Folie 39

40 Diamagnetism Induced magnetic moments are antiparallel to the field creating them. Induced currents weaken the external field in accordance with the Lenz s law. Note: Bohr-van Leeuwen theorem (1911 and 1919): The net magnetization of an electron ensemble in thermal equilibrium is equal to zero at finite temperatures and finite electric or thermal fields. Consequently, electron interactions are impossible in the classic, nonrelativistic concept. Both in closed shells (Larmor-D.) and itinerant electrons (Landau-D.): χ ~ Universal effect Generally temperature-independent magnetismus/kapitel-2.pdf Examples: water, bismuth, nitrogen

41 Diamagnetism Diamagnetic levitation (strong fields up to 15 T) High Field Magnet Laboratory, University of Nijmegen, 1997 Andre Geim: Ig-Nobel Prize (2000) Nobel Prize (2010) Literature: A.Geim, Physics Today, 9 (1998), gnetism.pdf M.V. Berry, A.K.Geim, European Journal of Physics, 18 (1997), TU Dresden, Magnetismus Folie 41

42 Paramagnetism Existing magnetic moments of electrons align parallel to the applied magnetic field. Temperature-dependent effect (Langevin-P.) in atoms with partially filled shells: χ ~ Temperature-independent, weak effect (Pauli-P.) for itinerant electrons: χ ~ No interaction between the moment, the effect vanisches when the applied field is cancelled. Potential energy of a magnetic moment m in an external field B: U = m B U is minimized when m is parallel to B (Zeeman energy) Examples: Na (3s-electrons) alkali metals (closed shells and competion between Pauli-P. and diamagnetism) Elemental copper, Cu 2+ salts magnetismus/kapitel-2.pdf TU Dresden, Magnetismus Folie 42

43 Langevin paramagnetism (Curie law) Paramagnetism of core electrons (only for partially filled valence shells) The magnetic moment of one localized magnetic centre (without strong spin-orbit coupling) : μ 2 = g J2 J(J+1)μ B2, g J = (J(J+1) + S(S+1) L(L+1)) / (2J(J+1)) + 1 g J Landé factor (characterizes coupling between orbital and spin moments) Langevin paramagnetism follows the Curie law: χ mol = C T mit C = μ 0 N A µ 2 3k B material-specific Curie constant Gd 2 (SO 4 ) 3 8H 2 O TU Dresden, Magnetismus Folie 43

44 Curie-Weiss law Very often is accompanied by magnetic ordering (cooperative magnetism) below a critical Curie temperature (T c ) Above the T c the Curie-Weiss law is applied: C χ mol = T Θ p ferro para ferro para The paramagnetic Curie temperature of Weiss constant Θ p reflects the sum of all interactions between the microscopic magnetic moments. TU Dresden, Magnetismus Folie 44

45 Cooperative phenomena Antiferromagnets: MnF 2, MnO, YBa 2 Cu 3 O 6, LaMnO 3 Ferrimagnets: Fe 3 O 4 (Magnetit), Y 3 Fe 5 O 12 (yttrium iron garnet) Helikale magnets: Tb, Dy, Ho Spinglass: Cu 1 x Mn x, Mn x Ge 1 x TU Dresden, Magnetismus Folie 45

46 Pauli paramagnetism Free electrons (metals) Why is there no temperature dependence in contrast to Langevin P. (c = C/T)? Explanation: Quasi-free electrons obey the Fermi statistics. The Fermi temperature (T F ) of the free electron gas is much higher than the room temperature. Consequently, only a small portion of electrons can flip their spin in the narrow energy interval near the Fermi edge (Pauli principle). The amount of these electrons is k BT E F that = T T F, so c Pauli = C T T T F = C T F (no temperature dependency) R. Gross, A. Marx. Festkörperphysik. 2. Auflage Temperature-independent, χ ~ 10 5 Landau-D. of itinerant electrons: c Landau = c Pauli /3 Deviations from the exact ratio due to interactions with the lattice potential (dependent from the effective mass of itinerant electrons m*) c Pauli can increase thanks to electronelectron interactions (e.g. palladium metal) TU Dresden, Magnetismus Folie 46

47 Van-Vleck paramagnetism Found in atoms and ions with electron configurations that lack one electron for the half-filled state (e.g. d 4, f 6 ) Temperature dependent, but only notable at low temperatures Quantum effect: Contribution is connected to field-induced electronic transition between an excited and the (non-magnetic, non-degenerate) ground state χ ~ 10 4 Examples: molecular crystals, Eu- and Sm-containing compounds Eu 2 O 3, Eu 3+ [Xe]4f 6 TU Dresden, Magnetismus Folie 47

48 Paramagnetism Cu [Ar]3d 10 4s 1 Resistivity: 16,8 nωm cm 3 mol -1 Na [Ne]3s 1 Resistivity: 47 nωm cm 3 mol -1 CuO: magnetic semiconductor CuS: weak Pauli paramagnetics, metallic CuS = (Cu + ) 3 (S 2- )(S 2 ) 2- [p] Or (Cu + ) 2 Cu 2+ S 2 2- S 2- TU Dresden, Magnetismus Folie 48

49 Magnetic susceptibility For localized moments: J = 0, S = 0, L = 0: only Larmor dia J = 0, S = L 0: (Larmor dia) + van Vleck paramagnetism J 0: (Larmor-Dia + van Vleck para) + Langevin paramagnetism Empirical rules to find the ground state of a multi-electron system (Hund s rules): 1. The total spin momentum S is maximal. 2. The total angular momentum L is maximal. 3. J = L S for less than half-filled shell, J = L + S for more than halffilled shell. Applies when the Russell-Saunders coupling is justified, when SOC is much weaker than Coulomb repulsion.