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

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

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

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

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

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

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

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

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)

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 theoryLuttinger & Kohn, PR 97, 869 (1955)

Without spin-orbit coupling (now for single hole):

Write wave function in Bloch form:

periodic part

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)

correctly gives heavy-hole, light-hole, split-off bands

respects point-group of crystal

only for region close to

Fermi surface, Dietl et al. (2000)

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

(2) Slater-Koster tight-binding theorySlater & Koster, PR 94, 1498 (1954), for GaAs: Chadi, PRB 16, 790 (1977)

tight-binding theory: consider atomic orbitals, express h1|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 |ki) diagramm:

q) =

G(k + q,)

G’(k,)

s’ s’

unperturbed Green function

for single parabolic band:

singularityat 2kF

Anomaly at 2kF from scattering between locally parallel portions of the Fermi surface

2kF

Hole magnetization in real space: Fourier transform

Oscillating and decaying magnetization around impurity spin, leads to:

Interaction:

FM

AFM

Friedel oscillations

with

Interaction oscillates on length scale 1/2kF = F/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

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:

k

EF

spin- hole density:

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

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

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.

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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 Mn- doped samples (S = 5/2) should collapse onto one curve – but don‘t

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

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 approximationSimilar 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)