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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: (b) level above CB bottom: (c) level below VB top: deep donor (d-like) autoionization irrelevant for semi- → hydrogenic donor (s-like) conducting properties CB 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) VB J exchange interaction from canonical (Schrieffer- Wolff) transformation antiferromagnetic, in agreement with experiment 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 → 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 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 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: G(k + q,) (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? first zero If typical impurity separation ¿ 1/2kF: 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: 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/0408561 etc. 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 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)

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