Intro anomalous Hall effect Berry phase and Karplus-Luttinger theory

1. Geometry and the intrinsic Anomalous Hall and Nernst effects Wei-Li Lee, Satoshi Watauchi, Virginia L. Miller, R. J. Cava, and N. P. O. Princeton University • Intro anomalous Hall effect • Berry phase and Karplus-Luttinger theory • Anomalous Nernst Effect in CuCr2Se4 • Nernst effect from anomalous velocity Supported by NSF ISQM-Tokyo05

2. H J y x Anomalous Hall effect (AHE) in ferromagnet (CuCr2Se4: Br)

3. A brief History of the Anomalous Hall Effect 1890? Observation of AHE in Ni by Erwin Hall 1935 Pugh showed rxy’ ~ M • Karplus Luttinger; transport theory on lattice • Discovered anomalous velocity v = eE x W. • Earliest example of Berry-phase physics in solids. • Smitintroduced skew-scattering model (semi-classical). Expts confusing 1958-1964 Adams, Blount, Luttinger Elaborations of anomalous velocity in KL theory • Kondo, MarazanaApplied skew-scattering model to • rare-earth magnets (s-f model) but RH off by many orders of magnitude. • 1970’s Berger Side-jump model (extrinsic effect) • Nozieres LewinerAHE in semiconductor. Recover Yafet result (CESR) 1975-85 Expt. support for skew-scattering in dilute Kondo systems (param. host). Luttinger theory recedes. 1983 Berry phase theorem. Topological theories of Hall effect 1999-2003 Berry phase derivation of Luttinger velocity (Onoda, Nagaosa, Niu, Jungwirth, MacDonald, Murakami, Zhang, Haldane)

4. constraint angle Parallel transport of vector v on curved surface Constrain v in local tangent plane; no rotation about e3 Parallel transport v acquires geometric angle a relative to local e1 e3 x dv = 0 complex vectors angular rotatn is a phase

5. Parallel transport Berry phaseand Geometry Change Hamiltonian H(r,R) by evolving R(t) Constrain electron to remain in one state |n,R) |n,R) defines surface in Hilbert space Electron wavefcn, constrained to surface |nR), acquires Berry phase a

6. e(k) k Electrons on a Bravais Lattice 1 Adams Blount Wannier Constraint!Confined to one band Bloch state k perturbation Drift in k space, ket acquires phase Parallel transport Berry vector potential

7. k Semiclassical eqn of motion E Vext causes k to change slowly W Gauge transf. k-space x = R x = R + X(k) Motion in k-space sees an effective magnetic field W Equivalent semi-class. eqn of motion

8. X(k) R x Karplus-Luttinger, Adams, Blount, Kohn, Luttinger, Wannier, … x fails to commute with itself! (X(k)= intracell coord.) In a weak electric field, W(k)acts as a magnetic field in k-space, a quantum area ~ unit cell.

9. Karplus Luttinger theory of AHE Boltzmann eqn. Anomalous velocity (B = 0) Equilibrium FD distribution contributes! Berry curvature Anomalous Hall current • 1. Independent of lifetime t (involves f0k) • 2. Requires sum over allk in Fermi Sea. • but see Haldane (PRL 2004) • 3. Berry curvature vanishes if time-reversal symm. valid

10. In general, rxy = sxyr2 • Luttinger’s anomalous velocity theory • s’xy indpt of ta rxy ~ r2 • Smit’s skew-scattering theory • s’xy linear in t a rxy ~ r KL theory

11. Cu Ferromagnetic Spinel CuCr2Se4 Cu O 180o bonds: AF (superexch dominant) Se Cr Anderson, Phys. Rev. 115, 2 (1959). Kanamori, J. Phys. Chem. Solids 10, 87 (1959). Goodenough, J. Phys. Chem. Solids 30, 261 (1969) 90o bonds: ferromag. (direct exch domin.) Goodenough-Kanamori rules

12. Effect of Br doping on magnetization • Tc decreases slightly as x increases. • At 5 K, Msat ~ 2.95 mB /Cr for x = 1.0 • doping has little effect on ferromagnetism.

13. At 5 K, increases over 3 orders as x goes from 0 to 1.0. • nH decreases linearly with x. , for x =1.0.

14. x = 0.25, negative AHE at 5K. • x = 0.6 , positive AHE at 5K.

15. Large positive AHE, at 5K, , x = 1 .

16. x=0 , AHE unresolved below 100K. • x=0.1, non-vanishing negative AHE at 5 K.

17. Wei Li Lee et al. Science (2004) If s’xy ~ n, then r’xy /n ~ 1/(nt)2 ~ r2 Fit to r’xy/n = Ar2 Observed A implies <W>1/2 ~ 0.3 Angstrom

18. impurity scattering regime • 70-fold decrease in t, from x = 0.1 to x = 0.85. • sxy/n is independent of t • Strongest evidence to date for the anomalous-velocity theory

19. E JH (per carrier) M J (per carrier) Bromine dopant conc. Doping has no effect on anomalous Hall current JH per hole With increasing disorder, J decreases, but AHE JH is constant

20. z x Anomalous Nernst Effect Ey/| |= Q0 B + QSm0M QS, isothermal anomalous Nernst coeff. Vy y H H I = 0

21. Longitudinal and transverse charge currents in applied gradient Total charge current Nernst signal Final constitutive eqn Measure r, eN, S and tanqH to determine axy z y H x

22. Wei Li Lee et al. PRL (04)

24. Nernst effect current with Luttinger velocity Peltier tensor (KL velocity term) Leading order In E and (-grad T) • Dissipationless (indpt of t) • Spontaneous (indpt of H) • Prop. to angular-averaged W

25. eN non-monotonic in x

26. axy decreases monotonically with x Wei Li Lee et al. PRL (04)

27. 3D density of states Empirically, axy = gTNF A = 34 A2 Comp. with Luttinger result Wei Li Lee et al. PRL (04)

28. Summary 1. Test of KL theory vs skew scattering in ferromagnetic spinel CuCr2Se4-xBrx. 2. Br doping x = 0 to 1 changes r by 1000 at 5 K r’xy = n Ar2 3. Confirms existence of dissipationless current Measured <W>1/2 ~ 0.3 A. 4. Measured axy from Nernst, thermopower and Hall angle Found axy ~ TNF, consistent with Luttinger velocity term

29. End

30. e de Parallel transport of a vector on a surface (Levi-Civita) e transported without twisting about normal r a = 2p(1-cosq) cone flattened on a plane Parallel transport on C : e.de = 0 e acquires geometric angle a = 2p(1-cosq) on sphere de normal to tangent plane r (Holonomy)

31. Local coord. frame (u,v) e.de = 0 Generalize to complex vectors Local tangent plane Parallel transport • Geometric phase a • arises from rotation of local coordinate frame, • is given by overlap between n and dn.

32. Nernst effect from Luttinger’s anomalous velocity In general, Since we have Area A is of the order of W ~ DxDy ~ 1/3 unit cell section

33. Atom Electron on lattice Hamiltonian Product wave fcn R k slow variable r r in cell fast variable Berry gauge potential “magnetic” field effective H

34. e(k) Wannier coord. within unit cell k Electrons on a Bravais Lattice 1 Adams Blount Wannier Constraint!Confined to one band Bloch state k Center of wave packet X(k) x R Berry vector potential

35. Beff R G R Berry phase in moving atom product wave fcn Nuclear R(t) changes gradually but electron constrained to stay in state |n,R) G Electron wavefunction acquires Berry phase Integrate over fast d.o.f. (Berry curvature) Nucleus moves in an effective field

36. R Constraint + parameter change Berry phase, fictitious Beff field on nucleus Nucleus moves in closed path R(t), but electron is constrainedto stay at eigen-level |n,R) G Electron wavefcn acquires Berry phase Y gYexp(icB) connection curvature

37. Boltzmann transport Eq. with anomalous velocity term.

38. Electrons on a lattice 3 1.W(k) -- a “Quantum area” -- measures uncertainty in x; W(k)~ DxDy. In a weak electric field, 2. W(k)is an effective magnetic field in k-space (Berry curvature)

39. Nozieres-Lewiner theory J. Phys. 34, 901 (1973) • Anomalous Hall effect in semiconductor with spin-orbit coupling • Enhanced g factor and reduced effective mass • Anomalous Hall current JH Dissipationless, indept of t

40. Berry potential Berry curvature X(k) a funcn. of k Wk = 0 only if Time-reversal symm. or parity is broken E Electrons on a Lattice 2 Eqns. of motion? Predicts large Hall effect in lattice with broken time reversal Karplus Luttinger 1954, Luttinger 1958

41. Wei-Li Lee et al., PRL 2004

43. Rs chanes sign when x >0.5. • |Rs| increases by over 4 orders when varying x. • Rs(T) is not simple function or power of r(T) .

44. Qs same order for all x, • axy linear in T at low T. Wei-Li Lee et al., PRL 2004