Transcript Document

Diluted Magnetic Semiconductors
Prof. Bernhard Heß-Vorlesung 2005
Carsten Timm
Freie 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
These slides can be found at:
http://www.physik.fu-berlin.de/~timm/Hess.html
Literature
Books on general solid-state
theory and magnetism:
Review articles on spintronics and
magnetic semiconductors:
H. Haken and H.C. Wolf, Atomund Quantenphysik (Springer,
Berlin, 1987)
H. Ohno, J. Magn. Magn. Mat. 200, 110
(1999)
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)
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
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
higher
miniaturization
 spin can be changed by polarized light, charge cannot
 spin is a nontrivial quantum degree of freedom,
charge is not
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
(2) Magnetic semiconductors: Physics interest
Control over magnetism
Universal “physics construction set”
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 moment
l
ve
re
l
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
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:
quantum numbers:
n = 1, 2, …: principal
l = 0, …, n – 1: angular momentum
m = –l, …, l: magnetic (z-component)
angular part; same for any
spherically symmetric potential
Ylm: spherical harmonics
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, 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 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
Notation for many-particle states: 2S+1L
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
v
Ze
–e
r
in rest frame
of electron:
r
Ze
magnetic field at electron position (Biot-Savart):
energy of electron spin in field B:
?
–v
–e
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)
over occupied orbitals
 more than half filled, n > 2l+1: si = –S/2S (filled shell has zero spin)
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 energy
Notation: 2S+1LJ
Hund’s 3rd rule
Example: Ce3+ with 4f1 configuration
S = 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 and
J = L+S, L and S precess about the fixed J axis:
Only the time-averaged moment can
be measured
S
S
2S+L = J+S
J
L
J+S||
Landé g-factor
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+)
3d
3sp
2sp
1s
4f (e.g., Gd3+)
partially filled
partially filled shell on outside of
ion → strong crystal-field effects
5sp
4spd
3spd
2sp
1s
4f
partially filled shell inside of
5s, 5p shell → weaker effects
3d (4d, 5d)
4f (5f)
 strong overlap with d orbitals
 weak overlap with f orbitals
 strong crystal-field effects
 weak crystal-field effects
 …stronger than spin-orbit
coupling
 …weaker than spin-orbit
coupling
 treat crystal field first, spin-orbit
coupling as small perturbation
(single-ion picture not applicable)
 treat spin-orbit coupling first,
crystal field partially lifts 2J+1
fold degeneracy
Single-electron states, orbital part:
d
Many-electron states:
multiplet with fixed L, S, J
e
t2
2J + 1 states
vacuum
cubic
tetragonal
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
is real for any state
since
all eigenvalues are real
On the other hand, L is hermitian
Quenching of orbital momentum
E
orbital effect in transition metals is small
(only through spin-orbit coupling)
0
Lz
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…
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
Hubbard
model
local Coloumb interaction
2nd order perturbation theory for small hopping, t ¿ U:
exchanged
allowed
Prefactor positive (J < 0) → antiferromagnetic interaction
Origin: reduction of kinetic energy
 Kinetic exchange through intervening nonmagnetic ions:
Superexchange, e.g. FeO, CoF2, cuprates…
forbidden
 Hopping between partially filled d-shells & Hund‘s first rule:
Double exchange, e.g. manganites, possibly Fe, Co, Ni
Hund
Higher orders in perturbation theory (and dipolar interaction) result
in magnetic anisotropies:
• on-site anisotropy:
• exchange anisotropy:
(uniaxial),
(cubic)
(uniaxial)
• dipolar:
• Dzyaloshinskii-Moriya:
as well as further higher-order terms
• biquadratic exchange:
• ring exchange (square):
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
t´
t
Idea: Canonical transformation
Schrieffer & 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
EF
obtain 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)
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
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
general
helical order
with
Q = 0: ferromagnetic
arbitrary and the maximum of J(q) is at q = Q,
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 interaction
with an effective field (molecular field)
write (so far exact):
thermal average of
expectation values
only affects energy
fluctuations
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  :
1

S
BS
Non-trivial solutions appear if LHS
and RHS have same derivative at 0:
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 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
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)
1/
 diverges at Tc like (T–Tc)–1
0
General helical magnet:
T0
T
Tc
T
1/
 grows for T ! Tc but does not
diverge
(divergence at T0 preempted
by magnetic ordering)
0
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.