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 Dissipationless means independent of scattering rate. Supported by NSF ISQM-Tokyo05

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

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. Smit introduced skew-scattering model (semi-classical). Expts confusing 1958-1964 Adams, Blount, Luttinger Elaborations of anomalous velocity in KL theory Kondo, Marazana Applied 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 Lewiner AHE 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)

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

Berry phase and 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 Parallel transport Electron wavefcn, constrained to surface |nR), acquires Berry phase a

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

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

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

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 all k in Fermi Sea. but see Haldane (PRL 2004) 3. Berry curvature vanishes if time-reversal symm. valid

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

Ferromagnetic Spinel CuCr2Se4 180o bonds: AF (superexch dominant) Se Cr Spinel is famous for being a ferrite insulator(CuFe2O4,ex). But in CuCr2Se4 the Cu is nonmagnetic. Goodenough model – Cu ion is non-magnetic. 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

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.

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

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

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

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

If s’xy ~ n, then r’xy /n ~ 1/(nt)2 ~ r2 Fit to r’xy/n = Ar2 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

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

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

Anomalous Nernst Effect Ey/| | = Q0 B + QS m0M QS, isothermal anomalous Nernst coeff. Vy x z y H H I = 0

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

Wei Li Lee et al. PRL (04)

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

eN non-monotonic in x

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

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

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 A r2 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

End

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 e de (Holonomy)

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

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

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

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

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 R Integrate over fast d.o.f. R G Beff (Berry curvature) Nucleus moves in an effective field

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

Boltzmann transport Eq. with anomalous velocity term.

1. W(k) -- a “Quantum area” -- measures uncertainty in x; W(k)~ DxDy. 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)

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 Kohm-Luttinger Representation : H = K^2/2/m_0+g_0  S.B + k. =g*c/e/E2 , E2=g(g +)/(2 g + ) JH= (polarization current) + (current due to spin-orbit correction to the electrostatic driving force and to scattering potential) Dissipationless, indept of t

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

Wei-Li Lee et al., PRL 2004

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) .

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