1. 4. Disorder and transport in DMS, anomalous Hall effect, noise
• Weak potential disorder scattering: Semiclassical transport theory
• Strong potential disorder scattering
• Interlude: Spatial defect correlations
• Lightly doped DMS: Percolation picture
• Anomalous Hall effect

2. Disorder and transport in DMS
high
metallic
insulating/localized
low
Why should disorder be important?
Direct observations:
some samples are Anderson insulators
metallic samples have high residual resistivity (kFl is not large)
metal-insulator transition
Physical reasoning:
many dopands (acceptors/donors) in random positions
compensation → low carrier density → weak electronic screening
compensation → charged defects CB VB 2+ 1–

3. What type of disorder? Example: (Ga,Mn)As
Single Mn acceptor: Binding energy is ~3/4 due to Coulomb attraction, ~1/4 to exchange → Coulomb potential disorder gives larger contribution
Coulomb disorder is strongly enhanced by presence of compensating defects
antisites AsGa (double donors)
interstitials Mni (double donors)
Fully aligned impurity spins & “large” wave functions (not strongly localized) → each carrier sees many aligned spins → mean-field limit, weak exchange disorder
MacDonald et al., Nature Materials 4, 195 (2005)
Moral: Neglecting Coulomb but keeping J is questionable

4. k r vk: velocity, F: force
Weak potential disorder scattering: Semiclassical transport theory
Boltzmann equation: without scattering the phase-space density does not change in the comoving frame r k
Thus
vk: velocity, F: force
Disorder scattering described by scattering integral:
with

5. “in”
“out”
transition rate from |k´i to |ki due to disorder
derivation from full equation of motion of density operator : see Kohn & Luttinger, PR 108, 590 (1957)
How can we calculate W? Assume the disorder potential to be a small perturbation V:
→ Leading order perturbation theory (see, e.g., Landau/Lifschitz, vol. 3)
For a periodic perturbation V = F e–it the transition rate is
k space element

6. static perturbation: let  ! 0 and F ! V
potential consists of random short-range scatterers:
this gives
shift r
(l  l´ terms drop out under averaging over impurity configurations)
integration over 3k:
leads to
(ni = Nimp/Volume: density of impurities)

7. N(0): density of states at Fermi energy
elastic: energy does not change
explicitly symmetric in k and k´ (not so in higher orders!)
For  scatterers, , we get and thus
Here W is a constant, apart from the energy-conservation factor
For later convenience we write this constant as (defining 1/)
N(0): density of states at Fermi energy

8. Semiclassical equations of motion: Consider a wave packet narrow in r and k space
Equations of motion for average position and momentum (short notation):
integration by parts

9. since
Similarly:

10. (c = 1) In homogeneous electromagnetic field: Thus
Writing and dropping h…i we obtain
for parabolic band (~ = 1)
Resistivity: In steady state , then
the current density is then (parabolic band)
Drude conductivity D

11. Normal Hall effect:
Assume , E component perpendicular to j follows from
Hall electric field
Thus the Hall conductivity is
and the Hall coefficient

12. DMS: Additional spin scattering with impurity spins
distribution of local magnetic quantum numbers, m = –S,…,S
paramagnetic phase (T > Tc): Drude resistivity with total scattering rate
ferromagnetic phase (T < Tc): complicated; different density of states for ", # etc.
spin-orbit effects can be included: C.T. et al., PRB 69, (2004)

13. rscr: electronic screening length
Strong potential disorder scattering
Strong disorder goes beyond the previous description. Approaches:
diagrammatic disorder perturbation theory (not discussed here)
numerical diagonalization
Example: VB holes in (Ga,Mn)As see potential of charged defects
substitutional MnGa: Ql = –1
antisites AsGa: Ql = +2
Mn-interstitials: Ql = +2
rscr: electronic screening length

14. PR distinguishes between extended and localized states
Method: C.T., Schäfer & von Oppen, PRL 89, (2002)
write H in suitable basis (here: Hband is parabolic, choose plane waves)
numerical diagonalization → spectrum, eigenfunctions
calculate participation ratios for all eigenstates
with normalization
extended states (conducting):
localized states (insulating):
PR distinguishes between extended and localized states

15. PR/L3 PR » L3, large PR » L0, small 
metallic down to T ! 0
insulating/localized: only activated hopping 
PR/L3
PR » L3, large
PR » L0, small
extended
localized
larger size L
mobility edge
are states at Fermi energy extended? → conductor (“metallic”) are they localized? → (Anderson) insulator
Problem: Following this approach all (Ga,Mn)As samples should be insulating
Solution: Must consider detailed spatial distribution of defects!

16. Interlude: Spatial defect correlations
Why not fully random defects?
many defects of charges –1 and +2, compensation (few holes), weak screening of Coulomb interaction
large Coulomb energy of defects
random defects cost high energy

17. defect diffusion during growth and annealing
incorporation in correlated positions during growth
lead to correlated defect positions
Monte Carlo simulations to find low-energy configurations C.T. et al., PRL 89, (2002); C.T., J. Phys.: C. M. 15, R1865 (2003)
Hamiltonian of defects:
MC: at least 20£20£20 fcc unit cells (cartoons are 10£10£10)
Snapshot: formation of clusters

18. Effect on VB holes
PR vs. electron energy for various numbers of MC steps (“annealing times”)
0 MC steps: fully random
random defects → no gap – contradicts experiments
…and tendency towards localization
correlated defects → weak smearing of VB edge
…and extended states (except close to VB edge)
Where is the Fermi energy?

19. p (holes per Mn) small, increases with x p large, decreases with x
Ohno, JMMM 200, 110 (1999)
Edmonds et al., APL 81, 3010 (2002)
p (holes per Mn) small, increases with x
shift of EF → MIT around x ~ OK
p large, decreases with x
always metallic OK
requires clustering
random defects:
always insulating

20. Lightly doped DMS: Percolation picture
For low concentrations x of magnetic impurities in III-V DMS Kaminski & Das Sarma, PRB 68, (2003) following Berciu & Bhatt (2001), Erwin & Petukhov (2002), Fiete et al. (2003) etc.
hole in hydrogenic impurity state, spin antiparallel to impurity moment: Bound magnetic polaron (BMP)
low concentration: transport by thermally activated hopping from BMP to empty impurity site → conductivity vanishes for T ! 0
higher concentration → percolation → conducting
ferromagnetism if aligned clusters percolate, transport/magnetic percolation governed by different energies (Lecture 5)
no structure in resistivity at Tc since only sparse percolating cluster orders

21. → Spin-orbit coupling Anomalous Hall effect (AHE)
In conducting ferromagnets one generically observes a Hall voltage in the absence of an applied magnetic field
(M: magnetization) or
Compare normal Hall effect:
How can the orbital motion feel the spin magnetization?
→ Spin-orbit coupling
Three mechanisms:
skew scattering
side-jump scattering
intrinsic Berry-phase effect (no scattering)

22. (1) Skew scattering Smit (1958), Kondo (1962) etc.
pure potential scattering
band structure with spin-orbit coupling:
• in bulk crystals, e.g., k ¢ p
• in assymmetric quantum well: Rashba term
kin
t ! –1
kout
t ! 1
scattering region
Scattering theory (second-order Born approximation) gives contribution to Hall resistivity
Note: opposite situation, spin-orbit scattering of carriers in bands without spin-orbit coupling is sometimes also called skew scattering

23. kout kin t ! 1 t ! –1 (2) Side-jump scattering Berger (1970) etc.
pure potential scattering
band structure with spin-orbit coupling:
• in bulk crystals, e.g., k ¢ p
• in assymmetric quantum well: Rashba term
kout
kin
t ! 1
t ! –1
scattering region
Scattering theory gives contribution to Hall resistivity
for random alloys (high resistivity) dominates over skew scattering
Note: opposite situation is again also called side-jump scattering

24. H(k) commutes with (projection of j onto k) → simultaneous eigenstates
(3) Intrinsic k space Berry-phase Karplus & Luttinger, PR 95, 1154 (1954) Jungwirth, Niu & MacDonald, PRL 88, (2002)
no scattering necessary (intrinsic effect)
band structure with spin-orbit coupling
Example: p-type DMS, 4-band spherical approximation
H(k) commutes with (projection of j onto k) → simultaneous eigenstates
eigenvalues of : j = –3/2, –1/2, 1/2, 3/2
light holes
heavy holes
heavy holes

25.  Idea: Sundaram & Niu, PRB 59, 14915 (1999)
Consider narrow wave packet in slowly varrying scalar and vector potential, , A
packet center describes orbit r(t) in real space
…accompanied by orbit k(t) in k space
spin of packet has to follow k-dependent quantization axis
closed orbit: additional quantum (Berry) phase proportional to solid angle enclosed by spin path on sphere
Detailed, more general derivation from Schrödinger equation for wave packet (Sundaram & Niu) gives 
omit scattering

26.  acts like an inhomogeneous magnetic field in k space
with the Berry curvature
where |ui is the periodic part of the Bloch function – essentially the spin part
In the absence of an appied magnetic field
 acts like an inhomogeneous magnetic field in k space
For heavy holes:
Now obtain the current density

27. Fermi function
For E along x direction and M along z ()  along M by symmetry): Jungwirth et al., PRL 88, (2002)
independent of scattering term in Boltzmann equation (intrinsic)
contribution to Hall conductivity, not resistivity
AH is indeed proportional to magnetization (if not too large), in agreement with experiments
correct order of magnitude (6-band model)

28. Anomalous Hall effect above Tc C. T
Anomalous Hall effect above Tc C.T., von Oppen & Höfling, PRB 69, (2004)
In the paramagnetic phase
Does the AHE play any role here?
Idea: the temporal correlation function does not vanish
gives nonzero Hall voltage noise
related to dynamical susceptibility:
Start from Boltzmann equations
potential scattering
holes:
impurity spins:
spin-flip scattering
j, m: magnetic quantum numbers of holes / impurities

29. Define hole and impurity magnetizations (z components)
in 4-band subspace
scattering integrals also contain overlap integrals of spin states
Derive hydrodynamic equations for the magnetizations, e.g., for holes:
Bh, Bi are effective fields containing coupling to i, h, respectively
Note anisotropic diffusion term

30. (non-equilibrium magnetization in z direction)
Anisotropic spin diffusion
from spin-orbit coupling in VB
fastest along axis of local magnetization
From hydrodynamic equations obtain magnetic susceptibilities of holes / impurities
(non-equilibrium magnetization in z direction)
t = (T–Tc)/Tc
Dynamics of collective spin-wave modes is purely diffusive and anisotropic

31. U: applied voltage, Li: Hall-bar dimensions, : detector band width
From impurity susceptibility obtain anomalous Hall voltage noise
assume intrinsic Berry-phase origin → contribution to Hall conductivity
relate correlations of Hall voltage to correlations of impurity spins
integrate over time to obtain noise
U: applied voltage, Li: Hall-bar dimensions, : detector band width
Noise is critically enhanced for T ! Tc