1. Diluted magnetic semiconductors aim to integrate semiconductor processing and ferromagnetic data storage on a single chip. Magnetic semiconductors are classes of materials that exhibit both semiconducting and magnetic properties. 2. The lecture discusses the theoretical picture of magnetic impurities in semiconductors based on the Zener model and mean-field theory. It also covers disorder, transport properties, and the anomalous Hall effect in diluted magnetic semiconductors. 3. The final sections discuss magnetic properties in the presence of disorder and recent developments, as well as open questions for the future of magnetic semiconductors.

1. Diluted Magnetic SemiconductorsDiluted Magnetic Semiconductors
Prof. Bernhard Heß-Vorlesung 2005Prof. Bernhard Heß-Vorlesung 2005
Carsten TimmCarsten Timm
Freie Universität BerlinFreie Universität Berlin

2. 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:

3. 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)

4. 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

5. 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

6. 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)

7. 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 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
functionality

8. (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

9. Theory of magnetism: Single ions
Magnetism of free electrons:
Electron in circular orbit has a magnetic moment
l
µl
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.

10. 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~δij leads to [sx
,sy
] = isz
etc. cyclic.
Can be realized by the choice si
´ σi
/2 with the Pauli matrices

11. quantum numbers:
n = 1, 2, …: principal
l = 0, …, n – 1: angular momentum
m = –l, …, l: magnetic (z-component)
in Hartree approximation:
energy εnl 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
Yl
m
: spherical harmonics

12. 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

13. 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 energy
Hund’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

14. Notation for many-particle states: 2S+1
L
where 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 –e
Ze in rest frame
of electron:
r
–v
–e
Ze
magnetic field at electron position (Biot-Savart):
energy of electron spin in field B:
?

15. 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

16. Electron-electron interaction can be treated similarly.
In Hartree approximation: Z ! Zeff < Z in λ
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 ) λ > 0 ) J = Min = |L–S| has lowest energy
 n > 2l+1 ) λ < 0 ) J = Max = L+S has lowest energy
Hund’s 3rd rule
Notation: 2S+1
LJ Example: Ce3+
with 4f1
configuration
S = 1/2, L = 3 (Hund 2), J = |L–S| = 5/2 (Hund 3)
gives 2
F5/2

17. 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 and
J = L+S, L and S precess about the fixed J axis:
L
S
S
2S+L = J+S
J
J+S||
Only the time-averaged moment can
be measured
Landé g-factor
?

18. 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 s2
electrons and sometimes some of
the electrons of outermost d or f shell
3d (e.g., Fe2+
) 4f (e.g., Gd3+
)
1s
2sp
3sp
3d
1s
2sp
3spd
4spd
4f
5sp
partially filled shell on outside of
ion → strong crystal-field effects
partially filled shell inside of
5s, 5p shell → weaker effects
partially filled

19. 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
d
e
t2
vacuum cubic tetragonal
Single-electron states, orbital part: Many-electron states:
multiplet with fixed L, S, J
2J + 1 states
vacuum crystal

20. 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

21. 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
Lz
E
0
is real for any state since
all eigenvalues are real

22. 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 φ and spinors χ
yields
electron creation operator
with…

23. 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

24. 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, CoF2, cuprates…

25. 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

26. 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)
t
t´

27. Idea: Canonical transformation
Schrieffer & Wolff (1966), Chao et al., PRB 18, 3453 (1978)
unitary transformation (with Hermitian operator T) → same physics
 formally expand in ε
 choose T such that first-order term (hopping) vanishes
 neglect third and higher orders (only approximation)
 set ε = 1
obtain model in terms of Hband and a pure local spin S:
Jij can be ferro- or antiferromagnetic but does not depend on σ, σ´
(isotropic in spin space)
EF

28. 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 direction
are also ground states → degeneracy
H is invariant under spin rotation, specific ground states are not
→ spontaneous symmetry breaking

29. 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

30. Assuming 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,

31. 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 interaction
with an effective field (molecular field)
write (so far exact):
thermal average of
expectation values
fluctuations
only affects energy use to determine hhSiii selfconsistently

32. Assume helical structure:
then
Spin direction: parallel to Beff
Selfconsistent spin length in field Beff in equilibrium:
Brillouin function:

33. Thus one has to solve the mean-field equation for σ :
σ
S BS
0
1
σ
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)

34. Example:
ferromagnetic nearest-neighbor interaction
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

35. Susceptibility (paramagnetic phase, T > Tc): hhSiii = hhSii = χ 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”

36. Ferromagnet:
(critical temperature,
Curie temperature)
χ diverges at Tc like (T–Tc)–1
Τ
1/χ
0 Τ0
General helical magnet:
χ grows for T ! Tc but does
not diverge
(divergence at T0 preempted
by magnetic ordering)
Τ
1/χ
0 Τc
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.