1、Diluted Magnetic Semiconductors Prof. Bernhard He-Vorlesung 2005 Carsten Timm Freie Universitt Berlin,Overview Introduction; important concepts from the theory of magnetism Magnetic semiconductors: classes of materials, basic properties, central questions Theoretical picture: magnetic impurities, Ze
2、ner model, mean-field theory Disorder and transport in DMS, anomalous Hall effect, noise Magnetic properties and disorder; recent developments; questions for the future,http:/www.physik.fu-berlin.de/timm/Hess.html,These slides can be found at:,Literature,Review articles on spintronics and magnetic s
3、emiconductors: H. Ohno, J. Magn. Magn. Mat. 200, 110 (1999) S.A. Wolf et al., Science 294, 1488 (2001) J. Knig et al., cond-mat/0111314 T. Dietl, Semicond. Sci. Technol. 17, 377 (2002) C.Timm, J. Phys.: Cond. Mat. 15, R1865 (2003) A.H. MacDonald et al., Nature Materials 4, 195 (2005),Books on genera
4、l solid-state theory and magnetism: H. Haken and H.C. Wolf, Atom- und Quantenphysik (Springer, Berlin, 1987) N.W. Ashcroft and N.D. Mermin, Solid State Physics (Saunders College Publishing, Philadelphia, 1988) K. Yosida, Theory of Magnetism (Springer, Berlin, 1998) N. Majlis, The Quantum Theory of M
5、agnetism (World Scientific, Singapore, 2000),1. Introduction; important concepts from the theory of magnetismMotivation: Why magnetic semiconductors?Theory of magnetism: Single ions Ions in crystals Magnetic interactions Magnetic order,Why magnetic semiconductors?,(1) Possible applications,Nearly in
6、compatible technologies in present-day computers:,ferromagnetic semiconductors: integration on a single chip? single-chip computers for embedded applications: cell phones, intelligent appliances, security,More general: Spintronics,Idea: Employ electron spin in electronic devices,Giant magnetoresista
7、nce effect:,Spin transistor (spin-orbit coupling) Datta & Das, APL 56, 665 (1990),Review on spintronics: uti et al., RMP 76, 323 (2004),Possible advantages of spintronics:spin interaction is small compared to Coulomb interaction less interferencespin current can flow essentially without dissipation
8、J. Knig et al., PRL 87, 187202 (2001); S. Murakami, N. Nagaosa, and S.-C. Zhang, Science 301, 1348 (2003) less heatingspin can be changed by polarized light, charge cannotspin is a nontrivial quantum degree of freedom, charge is not,higher miniaturization,Quantum computer Classical bits (0 or 1) rep
9、laced by quantum bits (qubits) that can be in a superposition of states. Here use spin as a qubit.,new functionality,(2) Magnetic semiconductors: Physics interest,Universal “physics construction set”,Control over magnetism by gate voltage, Ohno et al., Nature 408, 944 (2000)Vision:,control over posi
10、tions and interactions of moments,Theory of magnetism: Single ions,Magnetism of free electrons: Electron in circular orbit has a magnetic moment,with the Bohr magneton,l is the angular momentum in units of ,The electron also has a magnetic moment unrelated to its orbital motion. Attributed to an int
11、rinsic angular momentum of the electron, its spin s.,In analogy to orbital part:g-factor In relativistic Dirac quantum theory one calculates Interaction of electron with its electromagnetic field leads to a small correction (“anomalous magnetic moment”). Can be calculated very precisely in QED:,Elec
12、tron spin: with (Stern-Gerlach experiment!) 2 states , , 2-dimensional spin Hilbert space operators are 22 matrices,Commutation relations: xi,pj = iij leads to sx,sy = isz etc. cyclic. Can be realized by the choice si i/2 with the Pauli matrices,quantum numbers: n = 1, 2, : principal l = 0, , n 1: a
13、ngular momentum m = l, , l: magnetic (z-component) in Hartree approximation: energy nl depends only on n, l with 2(2l+1)-fold degeneracy,Magnetism of isolated ions (including atoms):Electrons & nucleus: many-particle problem!Hartree approximation: single-particle picture, one electron sees potential
14、 from nucleus and averaged charge density of all other electronsassume spherically symmetric potential eigenfunctions:,angular part; same for any spherically symmetric potential Ylm: spherical harmonics,Totally filled shells have and thus,nd shell: transition metals (Fe, Co, Ni) 4f shell: rare earth
15、s (Gd, Ce) 5f shell: actinides (U, Pu) 2sp shell: organic radicals (TTTA, NC60),Magnetic ions require partially filled shells,Many-particle states: Assume that partially filled shell contains n electrons, then there are,possible distributions over 2(2l+1) orbitals degeneracy of many-particle state,D
16、egeneracy partially lifted by Coulomb interaction beyond Hartree:,commutes with total orbital angular momentum and total spin, L and S are conserved, spectrum splits into multiplets with fixed quantum numbers L, S and remaining degeneracy (2L+1)(2S+1). Typical energy splitting Coulomb energies 10 eV
17、. Empirical: Hunds rules Hunds 1st rule: S ! Max has lowest energy Hunds 2nd rule: if S maximum, L ! Max has lowest energy,Arguments: (1) same spin & Pauli principle electrons further apart lower Coulomb repulsion (2) large L electrons “move in same direction” lower Coulomb repulsion,Notation for ma
18、ny-particle states: 2S+1L,where L is given as a letter:,Spin-orbit (LS) coupling (2L+1)(2S+1) -fold degenaracy partially lifted by relativistic effects,in rest frame of electron:,r,v,e,Ze,magnetic field at electron position (Biot-Savart):,energy of electron spin in field B:,Coupling of the si and li
19、: Spin-orbit coupling Ground state for one partially filled shell:less than half filled, n 2l+1: si = S/2S (filled shell has zero spin),This is not quite correct: rest frame of electron is not an inertial frame. With correct relativistic calculation: Thomas correction (see Jacksons book),over occupi
20、ed orbitals,unoccupied orbitals,Electron-electron interaction can be treated similarly. In Hartree approximation: Z ! Zeff Z in ,L2 and S2 (but not L, S!) and J L + S (no square!) commute with Hso and H:,J assumes the values J = |LS|, , L+S, energy depends on quantum numbers L, S, J. Remaining degen
21、eracy is 2J+1 (from Jz),Notation: 2S+1LJ,Example: Ce3+ with 4f1 configuration S = 1/2, L = 3 (Hund 2), J = |LS| = 5/2 (Hund 3) gives 2F5/2,The different g-factors of L and S lead to a complication: With g 2 we naively obtain the magnetic moment,But M is not a constant of motion! (J is but S is not.)
22、 Since H,J = 0 and J = L+S, L and S precess about the fixed J axis:,Only the time-averaged moment can be measured,Land g-factor,Theory of magnetism: Ions in crystals,Crystal-field effects: Ions behave differently in a crystal lattice than in vacuum Comparison of 3d (4d, 5d) and 4f (5f) ions: Both ty
23、pically loose the outermost s2 electrons and sometimes some of the electrons of outermost d or f shell,3d (e.g., Fe2+),4f (e.g., Gd3+),partially filled shell on outside of ion strong crystal-field effects,partially filled shell inside of 5s, 5p shell weaker effects,partially filled,3d (4d, 5d),4f (5
24、f),strong overlap with d orbitalsstrong crystal-field effectsstronger than spin-orbit couplingtreat crystal field first, spin-orbit coupling as small perturbation (single-ion picture not applicable),weak overlap with f orbitalsweak crystal-field effectsweaker than spin-orbit couplingtreat spin-orbit
25、 coupling first, crystal field partially lifts 2J+1 fold degeneracy,Single-electron states, orbital part:,Many-electron states:,multiplet with fixed L, S, J,2J + 1 states,vacuum,crystal,Total spin:if Hunds 1st rule coupling crystal-field splitting: high spin (example Fe2+: S = 2)if Hunds 1st rule co
26、upling crystal-field splitting: low spin (example Fe2+: S = 0),If low and high spin are close in energy spin-crossover effects (interesting generalized spin models) Remaining degeneracy of many-particle ground state often lifted by terms of lower symmetry (e.g., tetragonal),Total angular momentum: C
27、onsider only eigenstates without spin degeneracy. Proposition:,for energy eigenstates,Proof: Orbital Hamiltonian is real: thus eigenfunctions of H can be chosen real. Angular momentum operator is imaginary:,is imaginary,On the other hand, L is hermitian,Quenching of orbital momentum orbital effect i
28、n transition metals is small (only through spin-orbit coupling),With degeneracy can construct eigenstates of H by superposition that are complex functions and have nonzero hLi,is real for any state since all eigenvalues are real,Theory of magnetism: Magnetic interactions,The phenomena of magnetic or
29、der require interactions between moments Ionic crystals:Dipole interaction of two ions is weak, cannot explain magnetic orderDirect exchange interaction,Origin: Coulomb interaction,without proof: expansion into Wannier functions and spinors ,yields,electron creation operator,with,with,and,exchanged,
30、Positive J favors parallel spins ferromagnetic interaction Origin: Coulomb interaction between electrons in different orbitals (different or same sites),Kinetic exchange interaction,Neglect Coulomb interaction between different orbitals ( direct exchange), assume one orbital per ion: one-band Hubbar
31、d model,2nd order perturbation theory for small hopping, t U:,local Coloumb interaction,Hubbard model,exchanged,Prefactor positive (J 0) antiferromagnetic interaction Origin: reduction of kinetic energy,allowed,forbidden,Kinetic exchange through intervening nonmagnetic ions: Superexchange, e.g. FeO,
32、 CoF2, cuprates,Higher orders in perturbation theory (and dipolar interaction) result in magnetic anisotropies:on-site anisotropy: (uniaxial), (cubic)exchange anisotropy: (uniaxial)dipolar:Dzyaloshinskii-Moriya: as well as further higher-order termsbiquadratic exchange:ring exchange (square):,Hoppin
33、g between partially filled d-shells & Hunds first rule: Double exchange, e.g. manganites, possibly Fe, Co, Ni,Magnetic ion interacting with free carriers:Direct exchange interaction (from Coulomb interaction)Kinetic exchange interaction,with,tight-binding model (with spin-orbit),Hd has correct rotat
34、ional symmetry in spin and real space,Parmenter (1973),Idea: Canonical transformation Schrieffer & Wolff (1966), Chao et al., PRB 18, 3453 (1978),unitary transformation (with Hermitian operator T) same physicsformally expand in choose T such that first-order term (hopping) vanishesneglect third and
35、higher orders (only approximation)set = 1 obtain model in terms of Hband and a pure local spin S:,Jij can be ferro- or antiferromagnetic but does not depend on , (isotropic in spin space),Theory of magnetism: Magnetic order,We now restrict ourselves to pure spin momenta, denoted by Si. For negligibl
36、e anisotropy a simple model is,Heisenberg model,For purely ferromagnetic interaction (J 0) one exact ground state is,(all spins aligned in the z direction). But fully aligned states in any direction are also ground states degeneracy H is invariant under spin rotation, specific ground states are not
37、spontaneous symmetry breaking,For antiferromagnetic interactions the ground state is not fully aligned! Proof for nearest-neighbor antiferromagnetic interaction on bipartite lattice:,tentative ground state: but (for i odd, j even),does not lead back to not even eigenstate! This is a quantum effect,A
38、ssuming classical spins: Si are vectors of fixed length S The ground state can be shown to have the form,with,general helical order,usually Q is not a special point incommensurate order,Q = 0: ferromagnetic,arbitrary and the maximum of J(q) is at q = Q,Exact solutions for all states of quantum Heise
39、nberg model only known for one-dimensional case (Bethe ansatz) Need approximations Mean-field theory (molecular field theory) Idea: Replace interaction of a given spin with all other spins by interaction with an effective field (molecular field),write (so far exact):,thermal average of expectation v
40、alues,fluctuations,only affects energy,use to determine hhSiii selfconsistently,Assume helical structure:,then,Spin direction: parallel to Beff Selfconsistent spin length in field Beff in equilibrium:,Brillouin function:,Thus one has to solve the mean-field equation for :,Non-trivial solutions appea
41、r if LHS and RHS have same derivative at 0:,This is the condition for the critical temperature (Curie temperature if Q=0),Coming from high T, magnetic order first sets in for maximal J(Q) (at lower T first-order transitions to other Q are possible),Example: ferromagnetic nearest-neighbor interaction
42、 has maximum at q = 0, thus for z neighbors,Full solution of mean-field equation: numerical (analytical results in limiting cases),fluctuations (spin waves) lead to,Susceptibility (paramagnetic phase, T Tc): hhSiii = hhSii = B,(enhancement/suppression by homogeneous component of Beff for any Q) For
43、small field (linear response!),results in,For a density n of magnetic ions:,Curie-Wei law,T0: “paramagnetic Curie temperature”,Ferromagnet: (critical temperature, Curie temperature) diverges at Tc like (TTc)1,General helical magnet: grows for T ! Tc but does not diverge (divergence at T0 preempted by magnetic ordering),Mean-field theory can also treat much more complicated cases, e.g., with magnetic anisotropy, in strong magnetic field etc.,
