Luigi Paolasini paolasini@esrf - European Synchrotron ... · Stephen Blundell: “Magnetism in...

Luigi Paolasini [email protected]

L. Paolasini - LECTURES ON MAGNETISM- LECT.8

LECTURE 8: “COMPETING INTERACTIONS”

-  Frustrated magnetism. - Low dimensional magnetism - Confined magnetic systems

References: Moessner and A. Ramirez, Physics Today 59, 24 (2006) P.W Anderson, Physical Rev. 102, 1008 (1956) J. Villain, J. Phys. C: Solid State Phys. 10 1717 (1977) Stephen Blundell: “Magnetism in Condensed Matter”, Oxford Master series.


The strong competition between interactions leads to emergent behaviors and highly degenerate ground states with unusual properties

Examples: -  Frustrated magnetism -  Heavy fermion behaviour -  Fermi liquid -  Unconventional superconductivity -  Giant magneto-resistance -  Vortex matter

The ground states can be tuned by slight modifications of the internal or external parameters, e.g., lattice deformations, chemical substitution, or the application of a magnetic field.


Linus Pauling

L. Pauling, JACS 1935

Ice possesses a finite configurational entropy at zero temperature (Boltzmann’s relation):

S= kB ln(Ngs)=0.81 cal/(K•mol)

Configurational disorder intrinsic to the protons in ice Ice rule: each oxygen have 2 long and 2 short bonds

for N=2 Oxigens and 2N=4 hydrogens 22N=16 configurations only 6 are energetically favourables Number of ground states: Ngs<22N(6/16)N

First recognition of geometrical frustration


Triangular planar Ising magnet: Wannier (1950)

Pyroclore Ising magnets: “spin ice”: Anderson (1956)

The geometry of the lattice precludes the simultaneous minimization of all the interactions

Si±1, J>0

Magnetic analogues of liquids and ice The lattice geometry inhibits the formation of a simple, ordered spin configuration. Leads to a degenerate manifold of GS rather than a single stable GS configuration. Emergence of unconventional phase transitions and unusual magnetic excitations

Cooperative paramagnetic regime Villain (1977)

Frustration is caused by competing interactions due to site disorder

Degeneracy is hallmark of frustration!


Ground constraint for each triangle:

Ising nearest neighbour exchange interaction:

These systems have a finite entropy in the ground state Wannier (1950) Ω=exp(S/kB) S≈0.34 N kB

In a triangular lattice we have esponentially many ground states

1 frustrated bond per triangle


Flippable spins experience vanishing exchange field


Cluster of q mutually interconnected Heisenberg spins Antiferromagnetic interaction between nearest neighbours Continuous S=(SxSySz) and |S|=1

All the states in which the total sum of the spin is zero are ground state

Constraint counting: D=F-K F = total number of degree of freedom K = number of ground state constraints D = ground state degeneracy

The frustration is characterized by the number of possible ground states

F=2q Each spin Si has 2 degree of freedom (ω and φ) K =6 ΣSi=0 and -3 d.o.f. due to global rotations D=2q-6

Classical Heisenberg models

D=0

D=2


2D- Kargome Pyroclore

Vertex-sharing triangles Vertex-sharing tetrahedra

Triangular

Vertex-sharing triangles Each site belong 6 triangles Each site belong 2 triangles

In order to maximize the ground state degeneracy for a fixed number of spins, we need to minimize the number of constraints

Each site belong 2 tetrahedra

We can extend this method of counting to treat lattices rather than clusters.

S~0.5 N kB S~0.203 N kB S~0.34 N kB


Definition: regime at low temperature T ≪ J which is continuously connected to high-temperature paramagnetic phase

Phase transitions occur much below the Curie-Weiss temperature: Tc ≪ ΘCW

Curie-Weiss susceptibility

Frustration/fluctuation parameter

Short-range spin correlations: space and time spin fluctuations which persist down to T=0


Ramirez, 1994


A frustrated magnet even at T=0 can be closer to a paramagnet

A local arrangement of spins is possible at zero energy cost preserving the cluster constraint ΣiSi=0.

At high temperature the spins are free Sf=NkBln2

At low temperature correlations between spins develop and the entropy drop.

For a frustrated magnet the entropy is finite at T=0 because the large degeneracy of the ground state:

S0= ½ NkBln(3/2)

The same evaluated by Pauling for ice.

Entropy in Spin Ice:

Entropy of spin-ice Dy2Ti2O7

Ramirez (1999)

Unfrustrated magnet


Site randomness: random substitution of magnetic atoms in a matrix of non-magnetic atoms without short range order.

Competing interactions: The random distances between magnetic atoms lead to RKKY-type interactions of different sign.

Cooperative freezing temperature: Locally correlated clusters units develop progressively at low temperature until the system freeze at Tf into one of the multidegenerate ground state.

Low temperature spin-glass state: - absence of long range order - metastability (field cooling behaviour) - slow relaxation behaviour

Competing interaction on diluted magnetic metal alloys (Ex.: Cu1-xMnx alloys)

x=0.0108

x=0.0202

Cu1-xMnx

Tf Tf

Field cooling

In general the spin-glass behaviour can be induced by Site or bond randomness.


Competing interaction in small ferromagnetic particles: Demagnetizing energy ! domain wall energy

Critical size for a single domain:

K magnetocristalline anisotropy constant (Ksin2θ)

Néel relaxation time

Superparamagnetic materials : single domain ferromagnetic nanoparticles in a non magnetic matrix. The magnetization randomly flip direction under the influence of temperature

τ0≈10-9 sec.

The state of the nanoparticle (superparamagnetic or blocked) depends on the measurement time.


D=3 Isotropic

D=2 Planar anisotropy

D=1 Uniaxial anisotropy

The dimesional order parameter D (=1, 2, 3) depends on the number of components of Sx, Sy, Sz of the spin operator S.

One dimensional spin chain - No long range order expected for nearest neighbours interactions in a Ising-like spin chain (q=1, D=1) -  Interchain interactions lead to a long range order at low temperature below a characteristic crossover temperature. -  Above the crossover region complex magnetic excitations develop with a characteristic low dimensional signature.


Heisenberg isotropic excitations = magnons -  spin-flip excitation propagating with a dispersion relation -  magnons are bosons (S=1) -  gapless excitations (massless Goldstone modes):

Ising ferromagnetic spin chain excitations -  domain wall creation which moves freely along the chain -  no dispersion relations -  the departure from the Ising-like behaviour can cause a dispersion modulation

Heisenberg AF spin chain excitations = Spinons -  spin-½ excitations in an antiferromagnet -  spinons are fermions -  Dispersion relation for an AF:

J>0 antiferromagnetic exchange

-  Neutron scattering involve the change of S=1 spin, and so it create a pair of spinons, giving rise to a continuum of excitation below the dispersion curve:


Continuum


Peierls instability: -  Competition between elastic energy and energy saving by opening a gap in the energy spectrum close to the Fermi surface -  Nesting between Fermi surfaces by a wavevector q=2kF -  Spontaneaous dimerization of a one-dimensional chain -  Metal insulator transition

Example: Cromium develops a spin density wave below TN=310K with q=0.96 (2π/a) with a resistivity increase.

Peierls excitations: -  The Peierls dimerization produces a singlet state (spin-Peierls state) and an alternating modulation of exchange interactions -  Energy excitations between the singlet (J1) and triplet (J2)

Gapped energy spectrum, proportional to the dimerization


Haldane chains: -  chains with integer spin S=1,2,3,… (triplet ground state) -  non-linear quantum fluctuations -  the excitation spectrum have a gap (Haldane gap)

Spin Ladders - Coupling between two or more spin chains -  The intrachain coupling is of the same order of the interchain coupling.

Three-legs spin-½ ladders (Ex. Sr2Cu3O5): -  Formation of singlet states leaves an isolated spin -  the system is analogue to a ½-spin chain -  Excitation spectrum is gapless

Two-legs spin-½ ladders (Ex. SrCu2O3): -  For strong intrachain coupling J >>J -  Formation of singlet states -  Energy cost J to break a singlet into a triplet -  Excitation spectrum have an energy gap

Doped spin ladders (Ex. Sr14-xCaxCu24O21): -  Hole doping tends to break the singlet state -  Engineering superconductivity in two leg spin ladders


Metallic thin films -  Atoms at surface have a reduced number of nearest neighbours -  Reduced bandwidth => increase of density of states at EF -  Stoner Criterion can be satisfied => enhanced magnetic properties

Ex.: Ferromagnetic Palladium or Vanadium thin films

Engineering metastable phases - Lattice mismatch between substrate and the magnetic film -  The forces at interface drive the magnetic thin film into different phase

Ex.: Co (hexagonal) => bcc Co on AsGa substrate of fcc Co on fcc Cu substrate

Surface anisotropy - The magnetic anisotropy and the easy axis of magnetization can be altered at the surface:

Surface anisotropy t=thickness 2=two interfaces Dominates in thin films Induces perpendicular magnetization

Volume anisotropy -  Lattice strain -  uniaxial crystal // surface

Shape anisotropy Dipolar contribution Dominate in thick films Reduced for rough surfaces Induces

Ean.= K sin2θ K=2Ks/t + Kν + µ0M2


Change of the resistance of a material under the application of a magnetic field:

Electronic properties of Nickel At T>Tc - Narrow d band => Stoner criterion satisfied and md* >> me -  Because s bands are mostly free, ms* ≈ me => Conductivity σ dominated by 4s electrons - and transition mainly s-> d

At T<TC -  All unoccupied d states are antiparallel -  only half s electrons can make transitions -  Reduced scattering at low temperatures -  A decrease of resistivity is expected at low temperature

unoccupied d states

At T<TC and H="≠0

The resistivity R(H) decrease because the spin polarization increase, allowing more s->d transitions!


Large negative magnetoresistance discovered in Fe/Cr/Fr multilayers Competition between spin up and spin down electron currents

RKKY interaction between spins of ferromagnetic film separated by non-magnetic metal interfaces. Modulation of the interlayer coupling tuned by the Cr thickness tCr:

Two current model: Electron spin transport is affected by the F of AF interlayer coupling.

H>Hs => F coupling:

H=0 => AF coupling:

Fert and Grünberg (1988)


Phase transitions between different quantum phases, which can only be accessed by varying a physical parameter like pressure, magnetic field and doping level. A quantum critical point is a special class of continuous phase transition that takes place at the absolute zero

In a classical phase transition (T-driven) the correlation length and correlation time diverges approaching Tc. (=>ω->0)

In the QPT the critical fluctuations are quantum mechanical in nature, exhibiting scale invariance in both space and in time and affect a wide range of temperatures above the QCP

Examples: -  Heavy fermion superconductors -  Non-Fermi liquids -  Organic superconductors -  HTc superconductivity

Luigi Paolasini paolasini@esrf - European Synchrotron ... · Stephen Blundell: “Magnetism in...

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