Disorder and Transport in Diluted Magnetic Semiconductors: An In-depth Analysis

1. 4. Disorder and transport in DMS, anomalous Hall effect, noise • Disorder and transport in DMS •Weak potential disorder scattering: Semiclassical transport theory •Strong potential disorder scattering •Interlude: Spatial defect correlations •Lightly doped DMS: Percolation picture •Anomalous Hall effect

2. insulating/localized low CB 2+ metallic high 1– 1– VB Disorder and transport in DMS • Why should disorder be important? • Direct observations: • some samples are Anderson insulators • metallic samples have high residual resistivity (kFl is not large) • metal-insulator transition Physical reasoning: • many dopands (acceptors/donors) in random positions • compensation → low carrier density→weak electronic screening • compensation →charged defects

3. What type of disorder?Example: (Ga,Mn)As • Single Mn acceptor:Binding energy is ~3/4 due to Coulomb attraction, ~1/4 to exchange→Coulomb potential disorder gives larger contribution • Coulomb disorder is strongly enhanced by presence of compensating defects • antisites AsGa (double donors) • interstitials Mni (double donors) Fully aligned impurity spins & “large” wave functions (not strongly localized)→ each carrier sees many aligned spins→ mean-field limit, weak exchange disorder MacDonald et al., Nature Materials 4, 195 (2005) Moral: Neglecting Coulomb but keeping J is questionable

4. k r Weak potential disorder scattering: Semiclassical transport theory Boltzmann equation: without scattering the phase-space density does not change in the comoving frame Thus vk: velocity, F: force Disorder scattering described by scattering integral: with

5. “in” “out” transition rate from |k´i to |ki due to disorder derivation from full equation of motion of density operator : see Kohn & Luttinger, PR 108, 590 (1957) How can we calculate W?Assume the disorder potential to be a small perturbation V: →Leading order perturbation theory(see, e.g., Landau/Lifschitz, vol. 3) For a periodic perturbation V = Fe–it the transition rate is k space element

6. static perturbation: let ! 0 and F!V • potential consists of random short-range scatterers: this gives shift r (ll´ terms drop out under averaging over impurity configurations) integration over 3k: leads to (ni = Nimp/Volume: density of impurities)

7. For scatterers, , we getand thus • elastic: energy does not change • explicitly symmetric in k and k´ (not so in higher orders!) Here W is a constant, apart from the energy-conservation factor For later convenience we write this constant as (defining1/) N(0): density of states at Fermi energy

8. Semiclassical equations of motion:Consider a wave packet narrow in r and k space Equations of motion for average position and momentum (short notation): integration by parts

9. since Similarly:

10. Writing and dropping h…i we obtain Resistivity: In steady state , then the current density is then (parabolic band) In homogeneous electromagnetic field: Thus (c = 1) for parabolic band (~ = 1) Drude conductivityD

11. Normal Hall effect: Assume , E component perpendicular to j follows from Hall electric field Thus the Hall conductivity is and the Hall coefficient

12. DMS: Additional spin scattering with impurity spins distribution of local magneticquantum numbers, m = –S,…,S paramagnetic phase (T > Tc):Drude resistivity with total scattering rate ferromagnetic phase (T < Tc):complicated; different density of states for ", # etc. spin-orbit effects can be included: C.T. et al., PRB 69, 115202 (2004)

13. Strong potential disorder scattering • Strong disorder goes beyond the previous description. Approaches: • diagrammatic disorder perturbation theory (not discussed here) • numerical diagonalization Example:VB holes in (Ga,Mn)Assee potential of charged defects • substitutional MnGa: Ql= –1 • antisites AsGa: Ql = +2 • Mn-interstitials: Ql = +2 rscr: electronic screening length

14. extended states (conducting): localized states (insulating): • Method:C.T., Schäfer & von Oppen, PRL 89, 137201 (2002) • write H in suitable basis (here: Hband is parabolic, choose plane waves) • numerical diagonalization→ spectrum, eigenfunctions • calculate participation ratios for all eigenstates with normalization PR distinguishes between extended and localized states

15. PR/L3 insulating/localized:only activated hopping PR » L3, large larger size L extended localized mobility edge PR » L0, small metallic down to T! 0  • are states at Fermi energy extended? →conductor (“metallic”) are they localized? → (Anderson) insulator • Problem:Following this approach all (Ga,Mn)As samples should be insulating • Solution: Must consider detailed spatial distribution of defects!

16. Interlude: Spatial defect correlations Why not fully random defects? many defects of charges –1 and +2,compensation (few holes),weak screening of Coulomb interaction large Coulomb energy of defects random defects cost high energy

17. defect diffusion during growth and annealing • incorporation in correlatedpositions during growth lead to correlated defect positions Monte Carlo simulationsto find low-energy configurationsC.T. et al., PRL 89, 137201 (2002);C.T., J. Phys.: C. M. 15, R1865 (2003) Hamiltonian of defects: MC: at least 20£20£20 fcc unit cells(cartoons are 10£10£10) Snapshot:formation of clusters

18. Effect on VB holes • PRvs. electron energy for various numbers of MC steps (“annealing times”) 0 MC steps: fully random • random defects →no gap – contradicts experiments • …and tendency towards localization • correlated defects → weak smearing of VB edge • …and extended states (except close to VB edge) Where is the Fermi energy?

19. requires clustering random defects: always insulating

20. Lightly doped DMS: Percolation picture • For low concentrations x of magnetic impurities in III-V DMSKaminski & Das Sarma, PRB 68, 235210 (2003) following Berciu & Bhatt (2001), Erwin & Petukhov (2002), Fiete et al. (2003) etc. • hole in hydrogenic impurity state, spin antiparallel to impurity moment:Bound magnetic polaron (BMP) • low concentration: transport by thermally activatedhopping from BMP to empty impurity site → conductivity vanishes for T! 0 • higher concentration →percolation→ conducting • ferromagnetism if aligned clusters percolate, transport/magnetic percolation governed by different energies (Lecture 5) • no structure in resistivity at Tc since only sparse percolating cluster orders

21. Anomalous Hall effect (AHE) In conducting ferromagnets one generically observes a Hall voltage in the absence of an applied magnetic field (M: magnetization) or Compare normal Hall effect: How can the orbital motion feelthe spin magnetization? →Spin-orbit coupling • Three mechanisms: • skew scattering • side-jump scattering • intrinsic Berry-phase effect (no scattering)

22. kout t!1 kin t!–1 scattering region • (1) Skew scatteringSmit (1958), Kondo (1962) etc. • pure potential scattering • band structure with spin-orbit coupling: • in bulk crystals, e.g., k¢p • in assymmetric quantum well:Rashba term Scattering theory (second-order Born approximation) gives contribution to Hall resistivity Note: opposite situation, spin-orbit scattering of carriers in bands without spin-orbit coupling is sometimes also called skew scattering

23. (2) Side-jump scatteringBerger (1970) etc. • pure potential scattering • band structure with spin-orbit coupling: • in bulk crystals, e.g., k¢p • in assymmetric quantum well:Rashba term kout kin t!1 t!–1 scattering region Scattering theory gives contribution to Hall resistivity for random alloys (high resistivity) dominates over skew scattering Note: opposite situation is again also called side-jump scattering

24. H(k) commutes with (projection of j onto k) → simultaneous eigenstates eigenvalues of : j = –3/2,–1/2, 1/2, 3/2 • (3) Intrinsic k space Berry-phaseKarplus & Luttinger, PR 95, 1154 (1954)Jungwirth, Niu & MacDonald, PRL 88, 207208 (2002) • no scattering necessary (intrinsic effect) • band structure with spin-orbit coupling Example: p-type DMS, 4-band spherical approximation light holes heavy holes heavy holes

25.  Idea:Sundaram & Niu, PRB 59, 14915 (1999) Consider narrow wave packet in slowly varrying scalar and vector potential, , A • packet center describes orbit r(t) in real space • …accompanied by orbit k(t) in k space • spin of packet has to follow k-dependent quantization axis • closed orbit: additional quantum (Berry) phase proportional to solid angle enclosed by spin path on sphere Detailed, more general derivation from Schrödingerequation for wave packet (Sundaram & Niu) gives omit scattering

26. with the Berry curvature where |ui is the periodic part of the Bloch function– essentially the spin part In the absence of an appied magnetic field acts like an inhomogeneous magnetic field in k space For heavy holes: Now obtain the current density

27. Fermi function For E along x direction and M along z () along M by symmetry):Jungwirth et al., PRL 88, 207208 (2002) • independent of scattering term in Boltzmann equation (intrinsic) • contribution to Hall conductivity,not resistivity • AH is indeed proportional to magnetization (if not too large), in agreement with experiments • correct order of magnitude (6-band model)

28. Anomalous Hall effect above TcC.T., von Oppen & Höfling, PRB 69, 115202 (2004) In the paramagnetic phase • Does the AHE play any role here? • Idea: the temporal correlation functiondoes not vanish • gives nonzero Hall voltage noise • related to dynamical susceptibility: Start from Boltzmann equations potential scattering holes: spin-flip scattering impurity spins: j, m: magnetic quantum numbers of holes / impurities

29. in 4-band subspace Define hole and impurity magnetizations (z components) scattering integrals also contain overlap integrals of spin states Derive hydrodynamic equations for the magnetizations, e.g., for holes: Bh, Bi are effective fields containing coupling to i, h, respectively Note anisotropic diffusion term

30. Anisotropic spin diffusion • from spin-orbit coupling in VB • fastest along axis of local magnetization From hydrodynamic equations obtainmagnetic susceptibilities of holes / impurities (non-equilibrium magnetization in z direction) t = (T–Tc)/Tc Dynamics of collective spin-wave modes is purely diffusive and anisotropic

31. From impurity susceptibility obtain anomalous Hall voltage noise • assume intrinsic Berry-phase origin → contribution to Hall conductivity • relate correlations of Hall voltage to correlations of impurity spins • integrate over time to obtain noise U: applied voltage, Li: Hall-bar dimensions, : detector band width Noise is critically enhanced for T!Tc