The document summarizes the theoretical picture of dilute magnetic semiconductors (DMS). It describes the Zener model where magnetic impurities interact with charge carriers via exchange interaction. It then discusses the mean field approximation used to calculate the Curie temperature. For higher doping concentrations, a virtual crystal approximation is used to replace impurity spins with a smooth spin density. The model explains several experimental observations but cannot explain some properties like the shape of magnetization curves. At very low doping, a bound magnetic polaron model applies where carriers hop between localized acceptor levels aligned with impurity spins.

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. We follow T. Dietl, Ferromagnetic semiconductors, Semicond. Sci.
Technol. 17, 377 (2002) and J. König et al., cond-mat/0111314
DMS: Basic theoretical picture
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. Transition-metal ions in II-VI and III-V DMS
(a) level in the gap:
deep donor (d-like)
(b) level above CB bottom:
autoionization
→ hydrogenic donor (s-like)
Three cases (here for donors):
CBCB
(c) level below VB top:
irrelevant for semi-
conducting properties
VB
III-V
M3+
! M2+
M3+
! M4+
II-VI
M2+
! M1+
M2+
! M3+
acceptor
donor

4. 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)
Mn in II-VI semiconductors: no levels in gap, stable Mn2+
(half filled)
→ only introduces spin 5/2, no carriers
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. 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
Wierzbowska et al., PRB 70,
235209 (2004)
 Mn d-orbital weight at EF, VB top,
CB bottom: not seen in photoemission
LDA+U: phenomenological incorporation
of Hubbard U in d orbitals
d orbitals
 Mn d-orbital weight shifted away from
EF, better agreement
 similar results from other methods
going beyond LSDA: GGA, SIC-LDA

7. 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
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
MBE growth also introduces compensating donors:
antisites AsGa and interstitials Mni

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, 195211 (2004)
Consider the high-concentration case first
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

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:
(1) Kohn-Luttinger k ¢ p theory
(2) 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):
Write wave function in Bloch form:
periodic part

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 ukα 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 Γ (k = 0)

12. With spin-orbit coupling: treat
similarly. Obtain 6-band Hamiltonian:
components are bilinear in ki
Abolfath et al., PRB 63, 054418
(2001)
 correctly gives heavy-hole, light-hole, split-off bands
 respects point-group of crystal
 only for region close to Γ
Fermi surface, Dietl et al. (2000)
Γ

13. Spherical approximation for p-type semiconductors
(G. Zaránd, A.H. MacDonald etc.)
For light and heavy holes only: 4-band approximation
average over all angles:
hole total angular momentum
for heavy (–) and light (+) holes
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 hψ1|H|ψ2i, 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. 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) = χ(q) B(q)
 χ(q) from perturbation theory of 1st order for eigenstates (complicated
integral over k vector of states |ψkσi)
diagramm:
χ(q) =
Gσ(k + q,ω)
Gσ’(k,ω)
sσσ’ sσ’σ
unperturbed Green function

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

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

18. Interaction oscillates on length scale 1/2kF = λF/2
What do we expect?
 If typical impurity separation ¿ 1/2kF:
Many neighboring impurity spins within first ferromagnetic maximum,
weaker alternating interaction at larger distances → ferromagnetism
 If typical separation > 1/2kF:
r
E
first zero
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. 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:
k
EF
spin-σ hole density:

21. 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 » ni
4/3
Compare expriment:
Ohno, JMMM 200, 110 (1999)
bad
sample

23. 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
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

24. 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
Dietl, cond-mat/0408561 etc.
☻
☻
☻
☻
☻
☻
☻
experimentallyconfirmed
?
?
Diamond:
Mn replaces C2,
low spin, deep level
→ no DMS?
Erwin et al. (2003)

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. Dietl et al., PRB 63, 195205 (2001)
Magnetization: numerical solution of equations for |hSi| and |hsi|
For k ¢ p Hamiltonian:
Curves become more
Brillouin-function-like
for increasing nh

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. 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, 107203 (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, 029701
(2003)