Diluted Magnetic SemiconductorsDiluted Magnetic SemiconductorsProf. Bernhard Heß-Vorlesung 2005Prof. Bernhard Heß-Vorlesung 2005

Carsten TimmCarsten TimmFreie Universität BerlinFreie Universität Berlin

Overview

1. Introduction; important concepts from the theory of magnetism

2. Magnetic semiconductors: classes of materials, basic properties, central questions

3. Theoretical picture: magnetic impurities, Zener model, mean-field theory

4. Disorder and transport in DMS, anomalous Hall effect, noise

5. 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 semiconductors:

H. Ohno, J. Magn. Magn. Mat. 200, 110 (1999)

S.A. Wolf et al., Science 294, 1488 (2001)

J. König 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 general 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 Magnetism (World Scientific, Singapore, 2000)

1. Introduction; important concepts from the theory of magnetism

Motivation: Why magnetic semiconductors?

Theory of magnetism:

• Single ions

• Ions in crystals

• Magnetic interactions

• Magnetic order

Why magnetic semiconductors?

(1) Possible applications

Nearly incompatible technologies in present-day computers:

semiconductors: processing ferromagnets: data storage

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 magnetoresistance effect: Spin transistor (spin-orbit coupling)Datta & Das, APL 56, 665 (1990)

Review on spintronics:Žutić et al., RMP 76, 323 (2004)

Possible advantages of spintronics:

spin interaction is small compared to Coulomb interaction → less interference

spin current can flow essentially without dissipation J. König et al., PRL 87, 187202 (2001); S. Murakami, N. Nagaosa, and S.-C. Zhang, Science 301, 1348 (2003) → less heating

spin can be changed by polarized light, charge cannot

spin is a nontrivial quantum degree of freedom, charge is not

higher miniaturization

Quantum computerClassical bits (0 or 1) replaced by quantum bits (qubits) that can be in a superposition of states.

Here use spin ½ 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 positions and interactions of moments

Vision:

new effects due to competition of old effects

Theory of magnetism: Single ions

Magnetism of free electrons:

Electron in circular orbit has a magnetic momentl

l

re

ve

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 intrinsic 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:

Electron spin: with (Stern-Gerlach experiment!)

→ 2 states ↑,↓ , 2-dimensional spin Hilbert space

→ operators are 2£2 matrices

Commutation relations: [xi,pj] = i~ij 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, …: principall = 0, …, n – 1: angular momentumm = –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 from nucleus and averaged charge density of all other electrons

assume 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 earths (Gd, Ce)5f shell: actinides (U, Pu)

2sp shell: organic radicals (TTTA, N@C60)

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

Degeneracy 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.

Empirical: Hund’s rulesHund’s 1st rule: S ! Max has lowest energyHund’s 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 many-particle states: 2S+1Lwhere L is given as a letter: L 0 1 2 3 4 5 6 ...

S P D F G H I ...

Spin-orbit (LS) coupling

(2L+1)(2S+1) -fold degenaracy partially lifted by relativistic effects

r

v –eZe in rest frame

of electron: r

–v

–e

Zemagnetic field at electron position (Biot-Savart):

energy of electron spin in field B:

?

Coupling of the si and li: Spin-orbit coupling

Ground state for one partially filled shell:

less than half filled, n < 2l+1: si = S/n = S/2S (Hund 1)

more 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 Jackson’s book)

over occupied 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 = |L–S|, …, L+S, energy depends on quantum numbers L, S, J. Remaining degeneracy is 2J+1 (from Jz)

n < 2l+1 ) > 0 ) J = Min = |L–S| has lowest energy

n > 2l+1 ) < 0 ) J = Max = L+S has lowest energyHund’s 3rd rule

Notation: 2S+1LJ Example: Ce3+ with 4f1 configurationS = 1/2, L = 3 (Hund 2), J = |L–S| = 5/2 (Hund 3)

gives 2F5/2

The different g-factors of L and S lead to a complication:

With g ¼ 2 we naively obtain the magnetic moment

But M is not a constant of motion! (J is but S is not.) Since [H,J] = 0 andJ = L+S, L and S precess about the fixed J axis:

L

SS

2S+L = J+S

J

J+S||

Only the time-averaged moment can be measured

Landé g-factor

?

Theory of magnetism: Ions in crystalsCrystal-field effects:Ions behave differently in a crystal lattice than in vacuum

Comparison of 3d (4d, 5d) and 4f (5f) ions:Both typically 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+)

1s2sp3sp

3d

1s2sp

3spd4spd 4f5sp

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 (5f)

strong overlap with d orbitals strong crystal-field effects …stronger than spin-orbit coupling treat crystal field first, spin-orbit coupling as small perturbation (single-ion picture not applicable)

weak overlap with f orbitals weak crystal-field effects …weaker than spin-orbit coupling treat spin-orbit coupling first, crystal field partially lifts 2J+1 fold degeneracy

de

t2

vacuum cubic tetragonal

Single-electron states, orbital part: Many-electron states:

multiplet with fixed L, S, J

2J + 1 states

vacuum crystal

Total spin:

if Hund’s 1st rule coupling > crystal-field splitting: high spin (example Fe2+: S = 2)

if Hund’s 1st rule coupling < 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:

Consider 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 momentumorbital effect in 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

Lz

E

0

is real for any state since all eigenvalues are real

Theory of magnetism: Magnetic interactions

The phenomena of magnetic order require interactions between moments

Ionic crystals: Dipole interaction of two ions is weak, cannot explain magnetic order

Direct exchange interaction

Origin: Coulomb interaction

without proof: expansion into Wannier functions and spinors

yields

electron creation operator

with…

with and

exchanged

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 Hubbard model

2nd order perturbation theory for small hopping, t ¿ U:

local Coloumb interaction

Hubbardmodel

exchanged

Prefactor positive (J < 0) → antiferromagnetic interaction

Origin: reduction of kinetic energy

allowed forbidden

Kinetic exchange through intervening nonmagnetic ions: Superexchange, e.g. FeO, 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 terms

• biquadratic exchange:

• ring exchange (square):

Hopping between partially filled d-shells & Hund‘s first rule: Double exchange, e.g. manganites, possibly Fe, Co, Ni

Hund

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 rotational symmetry in spin and real space

Parmenter (1973)

tt´

Idea: Canonical transformationSchrieffer & Wolff (1966), Chao et al., PRB 18, 3453 (1978)

unitary transformation (with Hermitian operator T) → same physics formally expand in choose T such that first-order term (hopping) vanishes neglect third and higher orders (only approximation) set = 1obtain 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)

EF

Theory of magnetism: Magnetic order

We now restrict ourselves to pure spin momenta, denoted by Si.

For negligible 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 directionare also ground states → degeneracy

H is invariant under spin rotation, specific ground states are not→ 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

Assuming classical spins: Si are vectors of fixed length S

The ground state can be shown to have the form

with

generalhelical 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 Heisenberg 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 interactionwith an effective field (molecular field)

write (so far exact):

thermal average ofexpectation values

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 :

S BS

0

1

Non-trivial solutions appear 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 interactionhas maximum at q = 0, thus for z neighbors

Full solution of mean-fieldequation: numerical

(analytical results inlimiting 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 small field (linear response!)

results in

For a density n of magnetic ions:

Curie-Weiß law

T0: “paramagnetic Curie temperature”

Ferromagnet:(critical temperature,Curie temperature)

diverges at Tc like (T–Tc)–1

1/

0

General helical magnet:

grows for T ! Tc but does not diverge(divergence at T0 preemptedby magnetic ordering)

1/

0 c

possible T0

(can be negative!)

Mean-field theory can also treat much more complicated cases, e.g.,with magnetic anisotropy, in strong magnetic field etc.