Transcript Document

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
DMS: Basic theoretical picture
We follow T. Dietl, Ferromagnetic semiconductors, Semicond. Sci.
Technol. 17, 377 (2002) and J. König et al., cond-mat/0111314
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
Transition-metal ions in II-VI and III-V DMS
M2+ ! M1+
acceptor
M3+ ! M2+
M3+ ! M4+
M2+ ! M3+
donor
II-VI
III-V
Three cases (here for donors):
(a) level in the gap:
deep donor (d-like)
CB
(b) level above CB bottom:
autoionization
→ hydrogenic donor (s-like)
(c) level below VB top:
irrelevant for semiconducting properties
CB
VB
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 (SchriefferWolff) transformation
antiferromagnetic, in agreement with experiment
VB
J
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)
(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,
235209 (2004)
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
→
intermediate x →
high x
→
holes bound to acceptors, hopping
…overlap to form impurity band
…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
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)
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
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 spin 1/2
impurity spin 5/2
hole position
impurity position
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)
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):
periodic part
Write wave function in Bloch form:
 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)
With spin-orbit coupling: treat
similarly. Obtain 6-band Hamiltonian:
components are bilinear in ki
Abolfath et al., PRB 63, 054418
(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 

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
(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
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:
(q) =
G(k + q,)
s’
s ’
G’(k,)
unperturbed Green function
 for single parabolic band:
singularity
at 2kF
2kF
Anomaly at 2kF from scattering between locally
parallel portions of the Fermi surface
Hole magnetization in real space: Fourier transform
FM
with
Friedel oscillations
AFM
Oscillating and decaying magnetization around impurity spin, leads to:
Interaction:
Interaction oscillates on length scale 1/2kF = F/2
What do we expect?
 If typical impurity separation ¿ 1/2kF:
first zero
E
r
Many neighboring impurity spins within first ferromagnetic maximum,
weaker alternating interaction at larger distances → ferromagnetism
 If typical separation > 1/2kF:
E
r
ferro-, antiferromagnetic interactions equally common → no long-range order
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)
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:
EF
k
Assuming weak effective field: EZ ¿ EF
Obtain Tc: linearize Brillouin function
insert
= 1 at Curie temperature
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
Compare expriment:
Ohno, JMMM 200, 110 (1999)
bad
sample
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
Results for group-IV, III-V, and II-VI host semiconductors:
experimentally confirmed
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.
Diamond:
?
Mn replaces C2,
low spin, deep level
→ no DMS?
?
Erwin et al. (2003)
Magnetization: Numerical solution of equations for |hSi| and |hsi|,
parabolic band
Note that system
parameters only
enter through S
and Tc
All curves for Mndoped samples (S
= 5/2) should
collapse onto one
curve – but don‘t
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, 195205 (2001)
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
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)
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)