Magnetic compounds

Reminder of basics: - inter-ion exchange on the electronic level

leads via Hund’s rules to the direct-exchange (Heisenberg) Hamiltonian of the ionic crystal

- In Mott insulator, the effective antferromagnetic-exchange Hamiltonian is , where

- Effective (exchange) Hamiltonian

Superexchange interactions

Example: orbital hybridization of: into 3d10L (L denotes a hole in the O 2p shell) in CuO2 planes.

An effective (super)exchange interaction between the spin moment of Cu ions in (Cu-O-plane-containing) SrCuO2Cl2leads to the Mott-insulator state of s=1/2 spins of Cu ions (of a high Neel temperature TN=256K).

The overdoped Bi2Sr2CaCuO2O8+x (x is of the order of 0.1) is a high-Tc superconductor, thus, loosing the insulating property(t’/t ratio increases).

Antiferromagnetic superexchange(e.g. Mott insulators: VO2,NbO2,MnF2,NiF2,FeF2,NiO,MnO,CoO,FeO)

Ferromagnetic superexchange:two half-filled orbitals of the cationsoverlap with two orbitals of the same intervening anion

Goodenough-Kanamori rules

Example of a mixture of antiferromagnetic and ferromagnetic superexchange: ferrites of the spinel structure A2+B3+

2O2-4 ; e.g. Fe3O4, NiFe2O4, CoFe2O4, ZnFe2O4

The A-O-A and A-O-B superexchange interactions are antiferromagnetic,while (weaker) B-O-B superexchange is ferromagnetic. The result is a ferrimagnetic ordering.

Example of a superexchange-assisted antiferromagnetic spin-½ chain KCuF3

The (super)exchange interactions are anisotropic (spins of Cu ions interact along c-axis only) due to specific orientations of the d-orbitals of Cu

KCuF3 is a propotype of 1D Ising antiferromagnet due to a strong crystal-field-type anisotropy

Example: the structure of antiferromagneticNiO,CoO,FeO,MnO.

The magnetic sublattices form consecutive paralel layers since p-orbitals

Superexchange in diluted magnetic semiconductors (DMS)

In IIA1-xMnxVIB systems (e.d. Cd1-xMnxSe, Hg1-xMnxTe), the dominating mechanism of the exchange

is the antiferromagnetic superexchange Mn-B-Mn, (upon the dilution, Mn2+ ions substituteCd2+ ions while B2- ions are distorted).

For x<0.6, the exchange energy is not sufficient to order antiferromagnetically the system while magnetic-frustrationand/or random-anisotropy effects cause the spin-glass ordering (see next transparency).

Exchange frustration – a property of the antiferromagnetic exchange interactions.Consider a single triangle of a trigonal lattice or a triangle of a random lattice. The exchange energiesEexch=J(S1*S2+S1*S3+S2*S3) of the configurations A and B are the same while, for J>0, they are lower thanthe energy of the ferromagnetic ordering; we call the system „exchange frustrated” or „geometrically frustarted”.

Random anisotropy – follows from a random form of the crystal field (and coexists with exchange interactions)

A B

Mechanisms of magnetic disorder (spin-glass formation) in systems with superexchange

Zener (double-exchange) interaction

Note: the double-exchange is ferromagnetic interaction always

Double-exchange interaction is version of superexchange where p-orbital connects two different d-orbitals

Example is a „self-doped Mott insulator” CrO2. In spite of typical for Mott insulators lattice structure and high U, it is ferromagnetic due to the double exchange. The d-orbitals of Cr4+ ions spread over the lattice in different degreesbecause of the lattice distorsions. In the result CrO2 is a metallic ferromagnet.

Example: the Vervey transition in Fe3O4. Below TV, the system becomes insulating (a kind ofcharge-transfer insulator), the lattice symmetry is monoclinic. The ferromagnetic superexchange tends into double-exchange interaction, resulting in a jump in the magnetization

The double-exchange interaction in diluted systems

Example: the system considered by Zener (1951) was (La1-xAx)MnO3 , where A=Ca, Sr, Ba. The system is of the Perovskite structure (XIIA2+VIB4+X2-

3). Single ion of dopant gains one electron of d-orbital of Mn. In the doping range xϵ[0.2;0.4],the system is ferromagnetic due to the double exchange between Mn3+ and Mn4+ ions intermediated by O2- ions.

Tc of LSMO is about 300KFerromagnetism mechanism: wholes appear in a narrow TM-band, placed between the widevalence band and the empty conduction band, upon dopping with the acceptors „A”. Those wholescorrespond to the partial de-ionzation of Mn and they are responsible for the appearanceof the condutivity.

For x<0.2, LSMO becomes a Mott insulator, (then, the d-band is drastically narowing compared to the ferromagnetic phases).

Biquadratic exchange and noncolinear magnetism (in Mott insulators)

One diagonalizes the Hamiltonian written in basis of four-vertex states

After diagonalization, one writes down the effective Hamiltonian

Thus, the dominating terms of the exchange hamiltonian are of the order of while the biquadratic terms are of the order of

Canonical model of biquadratic exchange – a square (rectangle) plaquette

Example: non-collinear antiferromagnet (a Mott insulator with biquadratic exchange): UO2

Ground-state ion configurations:

hybridize into . Here denotes a hole in O 2p shell. Note that ab-initio calculations those include the temperature or lattice deformation, require the inclusion of 7s levels.

Macroscopic theory of the ordering in UO2 (Dzyaloshinskii, Man’ko 1964):averaging the spins over four sublattices, defining magnetization i 3 antiferromagnetic vectors and writting the free energy φ’with the Hamiltonian HBB or (in a more general way) on the basis of the cubic symmetry;

one finds the energy minimum for m=(0,0,0), l1=(1/2,0,1/2), l2=(1/2,1/2,0), l3=(0,1/2,1/2)

RKKY (Ruterman-Kittel-Kasuya-Yoshida) interaction („s-d model”)

Staring point: hyperfine interaction between the electron and a nucleus

The dipole-dipole interaction diverges for s-shell electron (the p,d,f,… orbitals are of the property of vanishing at r=0, independent of the radial part of the wavefunction). For the s-shell electrons, we apply the following trick. Consider the magnetic field around the nucleus

and apply the identity to obtain

Since , and , and none direction in space is distinguished,

integration over any sphere centered at the origin leads to

and, finally,

Note that for r≠0.

Hence, the energy of the hyperfine electron-nucleus interaction reads

Consider a paramagnetic dopant in a nonmagnetic environment, for instance Mn ions in a Cu lattice The conduction electrons can overlap with the magnetic moment of the Mn atom (localized moment of the electronic d-shell of Mn), thus, they correspond to the s-shell electrons in the hyperfine-interaction model. The quantized interaction energy reads

,where Sn denotes the spin of the Mn ions, while J(k-k’) is an exchange factor due to the Coulomb interaction of theion and electron

(the electrical charges and dielectric constant are omitted).

We evaluate the perturbation corrections of the first order to the wavefunction and Fermi energy of the spin-- (spin-+) electrondue to the magnetic dopants

and the corrections to the electronic densities

One derives the first term - derivation in next slayd

and the second termdenoting

Derivation of the density of states for spin-± electrons

The Fourier transform

leads us, for , to

Notice that ϱ± is divergent at r=Rn , thus, it is not relevant to the closest vicinity of the lattice site. Nevertheless, the spin polarization increases with closing to the site in an oscillatory way.

Including q=0 into the sum, one writes

The Hamiltonian that includes the anisotropy due to the spin-orbital effects of the Cu ions (in the first term), takes the form

Its static (zero-order) solution is

The first-order perturbation correction to the energy

and the second-order perturbation correction

are calculated in order to write effective Hamiltonian for the localized spin momenta

Example: in IIIA1-xMnxVB systems (e.g. As1-xMnxGa), the RKKY („Zener s-d”) exchange causes the ferromagnetic ordering.

Doping with Mn provides both the localized magnetic moments and the holes of enough concentration.

Ferromagnetism mechanism: for metals (heavily-p-doped semiconductors), the narrow TM band (of the magnetic dopant)is positioned well below the bottom of the conduction band (top of the valence band), thus, the d-electrons do not contribute to the conductivity. While in metals the RKKY interaction is mediated by s-electrons of the conduction band,in p-doped semiconductors, it is mediated by holes of the valence band.

Note: the ferromagnetic elements are s-d like magnets; they are not purely itinerant-electron systems

Dzyaloshinskii-Moriya interaction

Example: hematite; α-Fe2O3, (and Cr2O3) contain magnetic ions ferromagnetically-coupled in the planeswhile planes are antiferromagnetically-coupled. Resulting ordering is spiral (helical or conical)

with a (Morin) helical-conical transition at 260K in hematite. The conical state is calledweakly ferromagnetic

Dzyaloshinskii observation (1957): the helical and conical structures of hematite and MnCO3 can be describedmacroscopically with a classical Hamiltonian of lowered (relative to the exchangeHamiltonian) symmetry

Example: weakly ferromagnetic BiFeO3 – the only room-temperature bulk multiferroic

In a simplified model:

In a more detailed one:

Admixture of extremely strong spin-orbit interaction of Bi and large magnetic momentum of Fe leads to strong DM coupling

=>

Magnetoelectric effect:

Example: KMnO3

Consider two-sublattice magnet of the perovskite structure (Pbnm or D2h symmetry)In fig. 1,3 and 2,4 represent Mn ions of of sublattices 1 and 2, respectively, and the fluorine (orthorombic) distortion is indicated.

Rewriting the magnetic Hamiltonian, one finds the phases (ii), (iii)To be weakly ferromagetic (ferrimagnetic) due to the DM interaction

The Moriya’s idea (1960): include into the Hubbard Hamiltonian of the system the spin-orbit interaction due to the electrostatic field of the magnetic ions.

The p-d-hybridization mechanism of the electronic orbits (relevant to the superexchange)explains the influence of the spin-orbit coupling on the hopping amplitudes.

The resulting one-partice part of the Hamiltonian reads

Here bn’n denotes the usual hopping amplitudes, while Cn’nz and Cn’n

± denote the spin-orbit contributions to the hopping amplitudes

For p-d hybridized systems, V(r) originates from the magnetic ions usually

The second perturbation order for the energy relates to the Hamiltonian diagonalization for the Mott-Hubbard localized state

The last term is usually neglected since |C|<<|b|.

Alternative theory of DM interaction (relevant to localized-spin systems, Yoshida 1968): the second perturbation order with respect to the Coulomb and LS couplings leads us to the energy correction

Here n,n’ relate to the ground states of the coupled ions at R,R’ ,respectively, while m,m’ correspond to the excited states.

One sees that D is non-zero provided the inversion symmetry of the system is broken.

The following rules allow for determining the direction of „Dzyaloshinskii-vector” D:

Example: itinerant cubic helimagnet without inversion symmetry MnSi, (FeGe, Fe1-xCoxSi are similar).

A-phase of MnSi is a skyrmion lattice (Muhlbauer 2009)

Counterexample: in V2-yO3 one observes metal-insulator transition at y≈0.05 that is accompanied by changingthe Mott antiferromagnet into an incommensurate-spiral SDW. It results from the competition between the direct exchange (A,B in the picture) and the superexchange (C, D in the picture).

In doped structures, left-hand and right-hand chiralitiesare present simultanously in the samples (local SDW order).

Similar metal-insulator/SDW-Mott transition is observed in high-Tc superconductorLa2-xSrxCuO4

Note: earlier theories of cuprates assumed the presence of DM interaction

Spin glasses

Exchange frustration

Random single-ion anisotropy

Sign-changing exchange constant – RKKY exchange: J(RA-RB)~cos[kF(RA-RB)]/ (RA-RB)3

(the competition of distance-dependent ferromagnetic and antiferromagnetic interactions)

Dipole-dipole interaction --

~1/(RA-RB)3

Dzyaloshinskii-Moriya interaction with sign-changing constant – hyperfine interaction by Rutermann and Kittel is mixedwith spin-orbit-assisted electron jumping (Levy-Fert 1980). It is especially efficient in the canonical spin glassesdoped with Pd, Fe, Co, Pt

Mechanisms of magnetic disorder (spin-glass formation) in systems with RKKY exchange(classical SGs are noble metals Au, Ag, Cu, Pt weakly-diluted with transition metals Mn, Fe)

The Curie-Weiss temperature depends on the probabilities for the magnetic-moment-pairdistribution Px(R)

The cusp at the SG-paramagnet transition is typical.Additionally, field-cooled and zero-field-cooled statesare different in terms of non-zero order parameters

Ni79Mn21 ZFC

Complex hysteresis loop is due to local ferromagnetic order

In crystalline isolating; EuxSr1-xSe, EuxSr1-xTe, the SG phase results from the competion of different-signs exchangeinteractions within the pairs of magnetic (Eu) ions; nearest neighbours and next-nearest neighbours (JNNN/JNN≈-1/2).

In metallic; Al1-xGdx, (FexNi1-x)75P16B6Al3, (FexMn1-x)95P16B6Al3, and insulating; MnO*Al2O3*SiO2, CoO* Al2O3*SiO2, the SG phase is due to the amorphous structure.

Magnetic phase diagrams of SGs with ferromagnetic ions

Ising-SG and Heisenberg-SG theory

Heisenberg-SG (Gabay and Thoulouse 1981)

Ising-SG (Sherrington and Kirkpatrick 1975)

Edwards and Anderson (1975) have introduced the order parameter and proposed a mean-fieldtheory of the SG ordering, where the averaging over the replicas of the system is performed with the probabilty distribution