3. Theoretical picture: magnetic impurities, Zener model, mean-field theory

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/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

3. M2+! M1+ acceptor M3+! M2+ M3+! M4+ M2+! M3+ donor II-VI III-V CB CB VB Transition-metal ions in II-VI and III-V DMS 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

4. 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 (Schrieffer- Wolff) transformation antiferromagnetic, in agreement with experiment 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 inE[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, 235209 (2004)

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 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–1/3vs. 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-likeMatsuda 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

9. hole spin 1/2 impurity spin 5/2 hole position impurity position 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: • 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-Luttingerk¢ptheoryLuttinger & 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 kiAbolfath 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 

13. Spherical approximation heavy holes: light holes: Reasonable at small doping for some quantities 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 averageover all angles: hole total angular momentum

14. (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 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. G(k + q,) s’ s’ G’(k,) 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) = unperturbed Green function

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

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

18. first zero E r E r • 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: 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) = niS Selfconsistent solution:Impurity spins: Hole spins, assuming a parabolic band: spin- hole density:

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

22. bad sample Gives mean-field Curie temperature where N(0) is the density of states at the Fermi energy(one spin direction) For weak compensationnh¼ni, then Tc»ni4/3 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 Diamond: Mn replaces C2, low spin, deep level→no DMS? Erwin et al. (2003) Dietl, cond-mat/0408561 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¢pHamiltonian: Curves become more Brillouin-function-like for increasing nh Dietl et al., PRB 63, 195205 (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. 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)

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 concentrationsx¿ 1% [applied incorrectly by Berciu and Bhatt, PRL 87, 107203 (2001)] Upper limit for impurity concentration:Widthof impurity band must besmall 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)