1. 3. Theoretical picture: magnetic impurities, Zener model, mean-field theory
DMS: Basic theoretical picture
• Transition-metal ions in II-VI and III-V DMS
• Higher concentrations of Mn in II-VI and III-V DMS
• The “Standard Model” of DMS
• DMS in weak doping limit

2. DMS: Basic theoretical picture
We follow T. Dietl, Ferromagnetic semiconductors, Semicond. Sci. Technol. 17, 377 (2002) and J. König et al., cond-mat/
Consider the (by now) standard system: Mn-doped III-V DMS (excluding wide-gap), e.g., (Ga,Mn)As, (In,Mn)As, (In,Mn)Sb
Goals:
understand mechanism of ferromagnetic ordering
learn where to look for desired properties:
• high Tc
• high mobility
• strong coupling between carriers and spins

3. Three cases (here for donors):
Transition-metal ions in II-VI and III-V DMS
II-VI
M2+ ! M1+
M2+ ! M3+
acceptor
donor
III-V
M3+ ! M2+
M3+ ! M4+
Three cases (here for donors):
(a) level in the gap: deep donor (d-like)
(b) level above CB bottom: autoionization → hydrogenic donor (s-like)
(c) level below VB top: irrelevant for semi- conducting properties CB CB VB

4. Controversial in III-N and III-P, may be deep acceptor
Mn in II-VI semiconductors: no levels in gap, stable Mn2+ (half filled) → only introduces spin 5/2, no carriers
Mn in III-V semiconductors: acceptor level below VB top (hole picture!) → hydrogenic acceptor level
Mn3+ becomes Mn2+ (spin 5/2) + weakly bound hole (experimental binding energy: 112 meV)
Controversial in III-N and III-P, may be deep acceptor
Interaction between Mn2+ and holes consists of
Coulomb attraction (accounts for ~ 86 meV)
exchange interaction from canonical (Schrieffer- Wolff) transformation
antiferromagnetic, in agreement with experiment VB J

5. E[n] is not known → approximations
Ab-initio calculations for Mn in DMS:
Density functional theory starts from Hohenberg-Kohn (1964) theorem:
For given electron-electron interaction (Coulomb) the potential V (due to nuclei etc.) and thus the Hamiltonian and all properties of the system are determined by the ground-state electronic density n0(r) alone.
Now write the energy E[n(r)] as a functional of density n(r) for given V. Can show that E is minimized by n = n0.
E[n] is not known → approximations
Local density approximation (LDA): Unknown (exchange-correlation) term in E[n] is written as
partially neglects correlations between electrons
Local spin density approximation (LSDA): keep full spin density s(r)

6. (Ga,Mn)As with 3.125% Mn: typical results
d orbitals
Mn d-orbital weight at EF, VB top, CB bottom: not seen in photoemission
LDA+U: phenomenological incorporation of Hubbard U in d orbitals
Mn d-orbital weight shifted away from EF, better agreement
similar results from other methods going beyond LSDA: GGA, SIC-LDA
Wierzbowska et al., PRB 70, (2004)

7. What is “low”, “intermediate”, and “high” for (Ga,Mn)As?
Higher concentrations of Mn in II-VI and III-V DMS
no carriers (II-VI): short-range antiferromagnetic superexchange → paramagnetic at low Mn concentration x, spin-glass at higher x
with holes (not fully compensated III-V):
low x → holes bound to acceptors, hopping intermediate x → …overlap to form impurity band high x → …merges with valence band
MBE growth also introduces compensating donors: antisites AsGa and interstitials Mni
Big question:
What is “low”, “intermediate”, and “high” for (Ga,Mn)As?
Governed by Mn separation nMn–1/3 vs. acceptor effective Bohr radius aB

8. Experimental evidence for holes with VB character in III-As and III-Sb:
metallic conduction at low T, not thermally activated hopping
high-field Hall effect
Photoemission: anion p-orbital character
Raman scattering
very-high-field (500 T) cyclotron resonance of VB holes, not d-like Matsuda et al., PRB 70, (2004)
But does not fully rule out a separate impurity band of hydrogenic states
Experimental evidence that VB holes couple to impurity spins:
large anomalous Hall effect
spin-split VB, leading to large magnetoresistance effects
Consider the high-concentration case first

9. The “Standard Model” of DMS (T. Dietl, A.H. MacDonald et al.)
Step 1: Zener model [Zener, Phys. Rev. 83, 299 (1951)]
In terms of VB holes and impurity spins – here for single parabolic band:
hole position
impurity position
hole spin 1/2
impurity spin 5/2
Notes:
canonical transformation really gives scattering form
…and is not local
no potential scattering – disorder only from exchange term
(unrealistic band structure – can be improved)

10. The first (band) term can be improved to get a realistic band structure
Two main approaches:
Kohn-Luttinger k ¢ p theory
Slater-Koster tight-binding theory
(1) Kohn-Luttinger k ¢ p theory Luttinger & Kohn, PR 97, 869 (1955)
Without spin-orbit coupling (now for single hole):
periodic part
Write wave function in Bloch form:

11. treat k ¢ p term as small perturbation (valid if only small k are relevant)
degenerate perturbation theory up to second order: if ground state is N-fold degenerate the Hamiltonian is, to 2nd order,
6-band Kohn-Luttinger Hamiltonian for VB top (still no spin-orbit): 3 periodic functions uk with p-orbital symmetry (one nodal plane per site)
Cannot calculate A, B, C precisely due to electron-electron interaction → treat as fitting parameter to actual band structure close to  (k = 0)

12. With spin-orbit coupling: treat
similarly. Obtain 6-band Hamiltonian:
components are bilinear in ki Abolfath et al., PRB 63, (2001)
Fermi surface, Dietl et al. (2000) 
correctly gives heavy-hole, light-hole, split-off bands
respects point-group of crystal
only for region close to 

13. Spherical approximation for p-type semiconductors (G. Zaránd, A. H
Spherical approximation for p-type semiconductors (G. Zaránd, A.H. MacDonald etc.)
For light and heavy holes only: 4-band approximation
for heavy (–) and light (+) holes
average over all angles:
hole total angular momentum
Spherical approximation
heavy holes: light holes:
Reasonable at small doping for some quantities

14. (2) Slater-Koster tight-binding theory Slater & Koster, PR 94, 1498 (1954), for GaAs: Chadi, PRB 16, 790 (1977)
tight-binding theory: consider atomic orbitals, express h1|H|2i, i.e. hopping matrix elements t, by 2- and 3-center integrals
these integrals are not correct – no electron-electron interaction
thus view them as fitting parameters: choose to fit the resulting band structure to known energies, usually at high-symmetry points in k space
respects full symmetry (space group)
Chadi (1977): with only NN hopping (few parameters) quite good description of VB, including spin-orbit coupling

15. Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction
Motivation for following steps: RKKY interaction
Idea: In the Zener model, impurity spins polarize the carriers by means of the exchange interaction. Other impurity spins are aligned by this polarization → interaction between impurity spins
Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction
localized impurity spin S → acts like magnetic field B(q) ~ S
induces hole magnetization m(q) = (q) B(q)
(q) from perturbation theory of 1st order for eigenstates (complicated integral over k vector of states |ki)
diagramm:
G(k + q,)
G’(k,)
s’
s’
(q) =
unperturbed Green function

16. 2kF for single parabolic band: singularity at 2kF
Anomaly at 2kF from scattering between locally parallel portions of the Fermi surface

17. Hole magnetization in real space: Fourier transformFM
AFM
Friedel oscillations
with
Oscillating and decaying magnetization around impurity spin, leads to:
Interaction:

18. E r E r Interaction oscillates on length scale 1/2kF = F/2
What do we expect?
If typical impurity separation ¿ 1/2kF:
first zero r E
Many neighboring impurity spins within first ferromagnetic maximum, weaker alternating interaction at larger distances → ferromagnetism
If typical separation > 1/2kF: r E
ferro-, antiferromagnetic interactions equally common → no long-range order

19. Step 2: Virtual crystal approximation
Replace impurity spins by smooth spin density
Ignores all disorder
valid in stongly metallic regime (high x)
…but not for all quantities (e.g., not for resistivity)
requires impurity separation < 1/2kF (see RKKY interaction)

20. EF k Step 3: Mean-field approximation
Hole spins only see averaged impurity spins and vice versa. In homogeneous system: M(ri) = ni S
Selfconsistent solution: Impurity spins:
Hole spins, assuming a parabolic band:
spin- hole density: k EF

21. = 1 at Curie temperature Assuming weak effective field: EZ ¿ EF
Obtain Tc: linearize Brillouin function
insert
= 1 at Curie temperature

22. Gives mean-field Curie temperature
where N(0) is the density of states at the Fermi energy (one spin direction)
For weak compensation nh ¼ ni, then Tc » ni4/3
bad sample
Compare expriment: Ohno, JMMM 200, 110 (1999)

23. Beyond simple parabolic band: result for Tc remains valid
enhancement of Tc by ferromagnetic (Stoner) interactions of VB holes: Fermi liquid factor AF » 1.2 (from LSDA)
reduction of Tc by short-range antiferromagnetic superexchange: correction term –TAFM (very small in III-V DMS, but not in II-VI)
ni is the concentration of active magnetic impurities (not interstitials etc.)
Dietl et al., PRB 55, R3347 (1997); Science 287, 1019 (2000) etc. but in our notation
Dietl et al. (1997) showed that this theory is equivalent to writing down a Heisenberg-type model with interactions calculated from RKKY theory and applying a mean-field approximation to that

24. experimentally confirmed
Results for group-IV, III-V, and II-VI host semiconductors:
5% of cations replaced by Mn (2.5% of atoms for group-IV) hole concentration nh = 3.5 £ 1020 cm-3 ☻
experimentally confirmed ?
Diamond:
Mn replaces C2, low spin, deep level → no DMS?
Erwin et al. (2003)
Dietl, cond-mat/ etc.

25. Magnetization: Numerical solution of equations for |hSi| and |hsi|, parabolic band
Note that system parameters only enter through S and Tc
All curves for Mn- doped samples (S = 5/2) should collapse onto one curve – but don‘t

26. Magnetization: numerical solution of equations for |hSi| and |hsi|
For k ¢ p Hamiltonian:
Curves become more Brillouin-function-like for increasing nh
Dietl et al., PRB 63, (2001)

27. Experiments well explained within k ¢ p/Zener/VCA/MF theory
order of magnitude of Tc
optical conductivity
photoemission (partly)
X-ray magnetic circular dichroism
magnetic anisotropy & strain
anomalous Hall effect – perhaps not for (In,Mn)Sb
Experiments that cannot be explained
(change of) shape of magnetization curves → Lecture 5
weak localization & metal-insulator transition → Lecture 4
critical behavior of resistivity → Lecture 5
photoemission: appearance of flat band
giant magnetic moments in (Ga,Gd)N → Lecture 5

28. Jij also decays exponentially on scale aB
DMS in weak-doping limit (R. Bhatt et al.)
Step 1: Zener model for hopping between localized acceptor levels, hole spin aligned (in antiparallel, Jpd<0) to impurity spin (bound magnetic polaron)
Valid if acceptor Bohr radius aB is small compared to typical separation
Bhatt, PRB 24, 3630 (1981):
Jij also decays exponentially on scale aB
Step 2: Mean-field approximation Similar to band model but with position-dependent effective field
Step 3: Impurity average (or large system)

29. Advantage: takes disorder into account
Problems:
mean-field Tc determined by strongest coupling, real Tc determined by weak couping between clusters (percolation)
only for very small concentrations x ¿ 1% [applied incorrectly by Berciu and Bhatt, PRL 87, (2001)]
Upper limit for impurity concentration: Width of impurity band must be small compared to acceptor binding energy (band does not overlap VB)
For x ~ few percent: exceedingly broad “IB”, merged with VB (and CB!)
C.T. et al., PRL 90, (2003)