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Diluted Magnetic Semiconductors Prof. Bernhard Heß-Vorlesung 2005 Carsten Timm Freie Universität Berlin

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Diluted Magnetic Semiconductors

Prof. Bernhard Heß-Vorlesung 2005

Carsten Timm

Freie Universität Berlin

- Introduction; important concepts from the theory of magnetism
- Magnetic semiconductors: classes of materials, basic properties, central questions
- Theoretical picture: magnetic impurities, Zener model, mean-field theory
- Disorder and transport in DMS, anomalous Hall effect, noise
- Magnetic properties and disorder; recent developments; questions for the future

These slides can be found at:

http://www.physik.fu-berlin.de/~timm/Hess.html

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)

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)

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

(1) Possible applications

Nearly incompatible technologies in present-day computers:

ferromagnetic semiconductors: integration on a single chip?

single-chip computers for embedded applications:cell phones, intelligent appliances, security

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 dissipationJ. 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

new functionality

Quantum computer

Classical 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 effects due to competition of old effects

(2) Magnetic semiconductors: Physics interest

Control over magnetism

by gate voltage, Ohno et al., Nature 408, 944 (2000)

Vision:

Universal “physics construction set”

control over positions and interactions of moments

ve

re

l

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 intrinsic angular momentum of the electron, its spins.

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

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

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

Totally filled shells have and thus

Magnetic ions require partially filled shells

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)

Many-particle states:

Assume that partially filled shell contains n electrons, thenthere 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 rules

Hund’s 1st rule:S! Max has lowest energyHund’s 2nd rule: if Smaximum, 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

–e

Ze

r

?

Notation for many-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:

–e

r

Ze

–v

magnetic field at electron position (Biot-Savart):

energy of electron spin in field B:

This is not quite correct: rest frame of electron is not an inertial frame. With correct relativistic calculation: Thomas correction (see Jackson’s book)

- 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)

over occupied orbitals

unoccupied orbitals

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

- n > 2l+1 ) < 0 )J = Max = L+S has lowest energy

Hund’s 3rd rule

Electron-electron interaction can be treated similarly.In Hartree approximation: Z!Zeff < Z in

L2 and S2 (but notL, 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)

Notation: 2S+1LJ

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

gives 2F5/2

S

L

J

J+S||

2S+L = J+S

?

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

With g¼ 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 Sprecess about the fixed J axis:

Only the time-averaged moment can be measured

Landé g-factor

4f

3d

4spd

3sp

3spd

2sp

2sp

1s

1s

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 typically loose the outermost s2electrons and sometimes some of the electrons of outermost d or f shell

3d (e.g., Fe2+)

4f (e.g., Gd3+)

partially filled

partially filled shell inside of 5s, 5p shell →weaker effects

partially filled shell on outside of ion →strong crystal-field effects

d

t2

vacuum

tetragonal

cubic

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

Single-electron states, orbital part:

Many-electron states:

multiplet with fixed L, S, J

2J + 1 states

vacuum

crystal

- 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

is real for any state since all eigenvalues are real

E

Lz

0

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 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

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

electron creation operator

yields

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

local Coloumb interaction

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

exchanged

allowed

forbidden

Prefactor positive (J < 0) → antiferromagnetic interaction

Origin: reduction of kinetic energy

- Kinetic exchange through intervening nonmagnetic ions:Superexchange, e.g. FeO, CoF2, cuprates…

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

- 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):

t

- Magnetic ion interacting with free carriers:
- Direct exchange interaction (from Coulomb interaction)
- Kinetic exchange interaction

tight-binding model (with spin-orbit)

with

Parmenter (1973)

Hd has correct rotational symmetry in spin and real space

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 = 1

obtain model in terms ofHband 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 negligible anisotropy a simple model is

Heisenberg model

For purely ferromagnetic interaction (J > 0) oneexact 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

arbitrary and the maximum of J(q) is at q = Q,

Assuming classical spins:Si are vectors of fixed length S

The ground state can be shown to have the form

generalhelical order

Q = 0: ferromagnetic

with

usually Q is not a special point →incommensurate order

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):

fluctuations

thermal average ofexpectation values

only affects energy

use to determine hhSiii selfconsistently

then

Spin direction: parallel to Beff

Selfconsistent spin length in field Beff in equilibrium:

Brillouin function:

1

S BS

0

Thus one has to solve the mean-field equation for :

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)

ferromagnetic nearest-neighbor interaction

has 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”

0

T0

T

1/

0

Tc

T

possible T0(can be negative!)

Ferromagnet:(critical temperature,Curie temperature)

diverges at Tc like (T–Tc)–1

General helical magnet:

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

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

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