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

Disorder and transport in DMS high metallic insulating/localized low 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 CB VB 2+ 1–

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

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

“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 = F e–it the transition rate is k space element

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)

N(0): density of states at Fermi energy elastic: energy does not change explicitly symmetric in k and k´ (not so in higher orders!) For  scatterers, , we get and thus Here W is a constant, apart from the energy-conservation factor For later convenience we write this constant as (defining 1/) N(0): density of states at Fermi energy

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

since Similarly:

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

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

DMS: Additional spin scattering with impurity spins distribution of local magnetic quantum 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)

rscr: electronic screening length 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)As see potential of charged defects substitutional MnGa: Ql = –1 antisites AsGa: Ql = +2 Mn-interstitials: Ql = +2 rscr: electronic screening length

PR distinguishes between extended and localized states 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 extended states (conducting): localized states (insulating): PR distinguishes between extended and localized states

PR/L3 PR » L3, large PR » L0, small  metallic down to T ! 0 insulating/localized: only activated hopping  PR/L3 PR » L3, large PR » L0, small extended localized larger size L mobility edge 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!

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

defect diffusion during growth and annealing incorporation in correlated positions during growth lead to correlated defect positions Monte Carlo simulations to find low-energy configurations C.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

Effect on VB holes PR vs. 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?

p (holes per Mn) small, increases with x p large, decreases with x Ohno, JMMM 200, 110 (1999) Edmonds et al., APL 81, 3010 (2002) p (holes per Mn) small, increases with x shift of EF → MIT around x ~ 0.01 OK p large, decreases with x always metallic OK requires clustering random defects: always insulating

Lightly doped DMS: Percolation picture For low concentrations x of magnetic impurities in III-V DMS Kaminski & 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 activated hopping 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

→ Spin-orbit coupling 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 feel the spin magnetization? → Spin-orbit coupling Three mechanisms: skew scattering side-jump scattering intrinsic Berry-phase effect (no scattering)

(1) Skew scattering Smit (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 kin t ! –1 kout t ! 1 scattering region 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

kout kin t ! 1 t ! –1 (2) Side-jump scattering Berger (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

H(k) commutes with (projection of j onto k) → simultaneous eigenstates (3) Intrinsic k space Berry-phase Karplus & 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 H(k) commutes with (projection of j onto k) → simultaneous eigenstates eigenvalues of : j = –3/2, –1/2, 1/2, 3/2 light holes heavy holes heavy holes

 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 Schrödinger equation for wave packet (Sundaram & Niu) gives  omit scattering

 acts like an inhomogeneous magnetic field in k space 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

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)

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

Define hole and impurity magnetizations (z components) in 4-band subspace 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

(non-equilibrium magnetization in z direction) Anisotropic spin diffusion from spin-orbit coupling in VB fastest along axis of local magnetization From hydrodynamic equations obtain magnetic 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

U: applied voltage, Li: Hall-bar dimensions, : detector band width 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