5. Magnetic properties and disorder; recent developments; questions for the future

1. 5. Magnetic properties and disorder; recent developments; questions for the future • Magnetic properties and disorder • Anisotropic RKKY interaction •Effect of weak disorder on the magnetization •Disorder and magnetization: Strong disorder approach •Lightly doped, strongly disordered DMS: Percolation picture •Stability of collinear magnetic state, magnetic fluctuations •Spin-charge coupling: possible stripe order • •Magnetic order and transport: Resistive anomaly • Recent experiments – new puzzles • Magnetic semiconductors – what next?

2. Magnetic properties and disorder Complicated system of coupled carriers and spins defects defects impurity spins(random positions) impurity spins(random positions) Coulomb disorder Coulomb disorder carriers carriers carrier-carrierinteraction carrier-carrierinteraction spin-spin interaction spin-spin interaction carrier scattering carrier scattering magnetic ordermagnetic fluctuations magnetic ordermagnetic fluctuations

3. Experiments: Potashnik et al. (2001) Park et al., PRB 68, 085210 (2003) Mn-implanted GaAs:C (Ga,Mn)As • more disorder → straight/convex magnetization curves • annealing seems to reduce disorder (curves more mean-field-like) • magnetization for T! 0 is significantly less than saturation

4. Anisotropic RKKY interaction Reminder: local moment polarizes carrier band, other local moments see oscillating magnetization (“integrate out the carriers”) RKKY interaction for local carrier-impurity exchangeJpd and parabolic band(Lecture 3) is isotropic in real and spin space isotropic in real space isotropic in spin space • Zaránd & Jankó, PRL 89, 047201 (2002): local Jpd and spherical (4-band) approximation with spin-orbit interaction→HRKKY isotropic in real space, highly anisotropic in spin space→ frustration • Brey & Gómez-Santos: PRB 68, 115206 (2003): non-local Jpd with Gaussian form, realistic 6-band k¢p Hamiltonian→weakly anisotropic in real and spin space

5. Attempt at more realistic description:C.T. & MacDonald, PRB71, 155206 (2005) (a) Start from Slater-Koster tight binding theoryfor GaAs with spin-orbit coupling[Chadi (1977)] 16 bands (b) Incorporation of Mn d-orbitals hybridization with GaAs sp-orbitals:photoemission (Okabayashi), ab-initio(Sanvito) interactions: HubbardU and Hund‘s 1st ruleJH to preserve spherical symmetry of d-shell in real and spin space:Parmenter, PRB 8, 1273 (1973) – 5-orbital Anderson model does not

6. unitary transformationdoes not change the physics EF (c) Canonical perturbation theory for strong U,JH for single Mn Method: Chao et al., PRB 18, 3453 (1978),related to Schrieffer-Wolff transformation in (+), out (–) • expand in  • choose T (hermitian) to make linear order vanish • approximation: truncate after 2nd order • set = 1 • approximation: project onto ground state withN = 5, S = 5/2 projected energies:

7. with d5! d4 d5! d6 Inserting one finds spin scattering k dependent↔ nonlocal in real space • For small k, k´ only  = ´ = px,y,z • antiferromagnetic • correct order of magnitude

8. G J J S2 S1 G (d) RKKY interaction • two Mn impurities at 0 and R→ canonical transformation • integrate out the carriers • oscillating • ferromagnetic at small R • highly anisotropic in real space (from bands) • anisotropic in spin space for larger R(spin-orbit)

9. Can the spin anisotropy lead to non-collinear magnetization and reduction of average magnetization at T = 0(Zaránd & Jankó)? For magnetization in z direction typical effective field in transverse (x) direction is given by With our results for Jij this is small even at x = 0.01→no strong non-collinearity or reduction due to this mechanism Complementary ab-initio approach: Do LDA (+U)– usually without spin-orbit coupling – for supercell with 2 Mn impurities in various positions, extract J(R) from total energy → highly anisotropic in real space, isotropic in spin space (no spin-orbit!) Typically overestimates J(R) and thus Tc (while we underestimate it)

10. Effect of weak disorder on the magnetization Results for magnetization and Tc in Lecture 3 did not include disorder Inclusion of disorder: Coherent potential approximation (CPA)Takahashi & Kubo, PRB 66, 153202 (2002); also Bouzerar, Brey etc. impurity sites • only local Coulomb disorder potential (or zero in most other works) • (local J, simple semicircular DOS assumed) CPA: Approximate true scattering by multiple scattering at a single impurity embedded in an effective medium – good for low impurity concentration Cannot describe localization, difficult to treat extended scatterers (Coulomb)

11. increased magnetization Typical for CPA and DMFT calculations with point-like impurities:Tc decreases for high hole concentrations (weak compensation) x = 0.05 Origin: Impurity band broadens for higher spin polarization (since it is mostly due to large J) → spin polarization unfavorable for band filling & 1/2 1 hole per Mn (Ga,Mn)As: Takahashi & Kubo Problem: Always requires unphysically large J to obtain reasonable Tc, since no long-range Coulomb potentials→incorrect impurity-band physics

12. Disorder and magnetization: Strong disorder approach annealing C.T. et al., PRL 89, 137201 (2002) Zener model with potential disorder: Method:mean-field theory with disorder convex → concave for less disorder

13. weak link J´ Lightly doped, strongly disordered DMS: Percolation picture For low concentrations x of magnetic impurities in III-V DMSKaminski & Das Sarma, PRL 88, 247202 (2002); PRB 68, 235210 (2003),Alvarez et al., PRL 89, 277202 (2002), etc. • at T = 0: ferromagnetism if aligned clusters percolate • at T > 0: two clusters align if the weak coupling J´ between them is &kBT→ express by T-dependent BMP radius For lower T the aligned region growths and eventually percolates at Tc (Kaminski & Das Sarma) Exponentially small for small hole concentration, but not zero (since no quantum fluctuations) – compare VCA/MF: Tc/nh1/3 ni

14. holes only total Global magnetization is carried by sparse clusterat T.Tc: small Percolation theory:Kaminski & Das Sarma (2002) Monte Carlo simulations: Mayr et al., PRB 65, 241202(R) (2002) • Highly convex magnetization curves (upwards curvature) • MC: magnetization at T! 0 strongly reduced compared to saturation For clustered defects: large ordered clusters exist for T&Tc, percolate at Tc→ more rapid (Brillouin-function-like) increase of bulk magnetization

15. Stability of collinear magnetic state, magnetic fluctuations Question: Is the collinear state found by approximate (mean-field) methods stable against magnetic fluctuations? In particular with disorder? Expand energy around mean-field solution→density of states of magnetic excitations • Schliemann & MacDonald, PRL 88, 137302 (2002) etc.: spin disorder, no Coulomb disorder • DOS at negative energies: state not stable • …collective spin excitations involving many spins (high participation ratio) Schliemann, PRB 67, 045202 (2003): 6-band k¢ p model • solution not even stationary Goldstone mode

16. C.T., J. Phys.: Cond. Mat. 15, R1865 (2003): Spin and Coulomb disorder, parabolic band • DOS at negative energies: state not stable • clustered defects: DOS shifts away from zero →stiffer magnetic order Collinear state unstable due to anisotropic magnetic interactions – but argument from RKKY interaction suggests that the deviation is small

17. with Spin-charge coupling: possible stripe orderC.T., cond-mat/0509653 Carrier-mediated ferromagnetism→strong dependence of magnetism on hole concentration in (In,Mn)As, (Ga,Mn)As Landau theory for magnetization m and excess carrier concentration n: Ohno et al. (2000) magnetization charge Introduce electrostatic potential:for p-type DMS

18. Spin- and charge-density waves (stripes) periodic, anharmonic magnetization and carrier density for lowest energy typical solutions

19. Tc Tc Magnetic order and transport:Resistive anomaly in dirty itinerant ferromagnets Zumsteg & Parks, PRL 24, 520 (1970) Potashnik et al., APL 79, 1495 (2001) Ni (Ga,Mn)As R(T) ρ(T) dR/dT

20. Approach equivalent to: • perturbation theory, similar to inverse quasiparticle lifetime, but transport rate involves factor • anomaly from small momentum transfersq Theories for paramagnetic regime, T > Tc: (1)de Gennes & Friedel, J. Phys. Chem. Solids 4, 71 (1958) • scattering from magnetic fluctuations • close to Tc: critical slowing down →static, elastic Ornstein/Zernicke (sharp maximum)

21. (2) Fisher & Langer, PRL 20, 665 (1968) • disorder damping for large length scales ↔small q: electronicGreen function decays exponentially on scale l (mean free path) • no de Gennes-Friedel singularity from small q • weak singularity from large q ¼ 2kF, have to go beyond Ornstein/Zernicke/Landau theory: α: small anomalousspecific-heat exponent Equivalent to Boltzmann equation approach (Lecture 4), disorder and magnetic scattering treated on equal footing Problem: fails for magnetic correlation length (T)À l (mean free path), magnetization variations are explored by diffusive carriers

22. (phase coherence length) Beyond the Boltzmann approachC.T., Raikh & von Oppen, PRL 94, 036602 (2005) (a) Description of transport on large length scales • magnetization ~ constant in cells • conductivity of network: 3D resistor network • spatial average h…i • large system: equivalent to average over • quenched disorder •magnetization (thermal)

23. UCF UCF Correlation function spin ↑,↓ carriers have different Fermi energies but see same disorder (b) Two spin subbands: ↑,↓ universal conductance fluctuations (UCF)

24. is a scaling function H(y) of y = eff. Zeeman energy £ spin-orbit time,increases by factor of 2 in strong effective Zeeman field is a scaling function of x =eff. Zeeman energy £ diffusion time (Stone 1985, Altshuler 1985, Lee and Stone 1985) Correlations decrease with increasingZeeman energy (c) With spin-orbit coupling: realistic case (d) Typical magnetization assuming Gaussian fluctuations: long-wavelength modes

25. with scaling function maximum at Tc For Gaussian fluctuations: Beyond Gaussian fluctuations: Stronger singularity than in Fisher/Langer and de Gennes/Friedel theories Condition:Transport disorder-dominated at Tc (low Tc, strong disorder) – (In,Mn)Sb?

26. field-cooled sweep zero-field-cooled Recent experiments – new puzzles • Ferromagnetism in superdilute magnetic semiconductorsDhar et al., PRL 94, 037205 (2005); Sagepa et al., cond-mat/0509198 • GaN:Gd with Gd concentrations from 7£1015 to 2£1019 cm-3(x = 8£10-8 to 2£10-4) • wurtzite structure • formal valence Gd3+: isovalent, configuration 4d7, local spin S = 7/2 • high concentration of native donors (N vacancies) expected Observations: • room-temperature ferromagnetism (Tc» 360K for x = 8£10-8) • highly insulating

27. m absolute moment per Gd • giant magnetic moment per Gdfor low x • effective field acting on VB is reversed photoluminescence Magnetization must be carried by “something else” – native defects? How does very little Gd induce magnetic ordering?

28. Sn1-xXxO2 • “d0 ferromagnetism” in oxidesCoey et al., Nature Mat. 4, 173 (2005) • ferromagnetism in HfO2(no partially filled shells?) • magnetization extraplotates to nonzero value for strongly diluted DMS • Two species of substitutional Mn in (Ga,Mn)As: XAS, XMCD resultsKronast et al. (BESSY II Collaboration), submitted • Mn2+ with ~ 3d5 configuration, large moment, orders magnetically • Mn3+ with ~ 3d4 configuration, strong valence fluctuations, large moment, does not order Questions: What mechanism lifts the d5! d4 transition by several eV?Why does high-spin Mn3+ not participate in the ordering?

29. Magnetic semiconductors – what next? • Goals for DMS experiments: • control of growth dependence, reproducability • unconventional DMS (oxides etc.), concentrated magnetic semiconductors • other magnetic probes: NMR/NQR/SR and neutron scattering • dynamics: optical pump-probe and noise • crossover to antiferromagnetism, superconductivity, QHE… • …and theory: • study of crossover between weak doping (BMP‘s) and band picture • unconventional DMS (oxides etc.): different mechanisms? • better ab-initio methods to get hydrogenic impurity level ofMn in GaAs • detailed simulation of DMS growth to find defect distribution • selfconsistent theory of scattering and carrier-mediated magnetism

30. DMS/nonmagnetic semiconductor heterostructures Transport, disorder & magnetism delta-doped layers,single layer vs. superlatticemetallic or insulating?magnetic properties? FNF structure:RKKY coupling between layers,control by gate voltage– unlike metal structures DMS quantum dots • many local moments • few local moments DMS/nonmagnetic interfaces,spin injection MC simulation of interdiffusion cf. experiment:Kawakami et al.,APL 77, 2379 (2000)

31. Electronic correlations & quantum critical points • electronic correlations: (Ga,Gd)N? • at least two quantum critical points:• ferromagnetic end point• metal-insulator transition • …with overlapping critical regions • Griffiths-McCoy singularities: rare regions relevant for popertiesGalitski et al., PRL 92, 177203 (2004) Vision:DMS are ideal materials to study the interplay of disorder and electronic correlations: both are important and can be tuned Possible parallels to cuprates: indications that dopand-induced disorder is important in cuprates, McElroy et al., Science 309, 1048 (2005)

32. Diluted Magnetic Semiconductors Prof. Bernhard Heß-Vorlesung 2005 I am grateful for discussions and collaborations with G. Alvarez, W.A. Atkinson, M. Berciu, L. Borda, G. Bouzerar, L. Brey, H. Buhmann, K.S. Burch, S. Dhar, T. Dietl, H. Dürr, S.C. Erwin, G.A. Fiete, E.M. Hankiewicz, F. Höfling, P.J. Jensen, T. Jungwirth, P. Kacman, J. König, J. Kudrnovský, A.H. MacDonald, L.W. Molenkamp, W. Nolting, H. Ohno, F. von Oppen, C. Paproth, M.E. Raikh, F. Schäfer, J. Schliemann, M.B. Silva Neto, J. Sinova, C. Strunk, G. Zaránd and others