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

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

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.
Smit introduced skew-scattering model (semi-classical). Expts confusing
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)
Expt. support for skew-scattering in dilute
Kondo systems (param. host). Luttinger theory recedes.
1983 Berry phase theorem. Topological theories of Hall effect
Berry phase derivation of Luttinger velocity
(Onoda, Nagaosa, Niu, Jungwirth, MacDonald, Murakami, Zhang, Haldane)

4. 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

5. 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

6. Electrons on a Bravais Lattice 1k
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

7. 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

8. 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.

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

10. 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

11. 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

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. 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

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) 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

20. Anomalous Nernst Effect
Ey/| | = Q0 B + QS m0M
QS, isothermal anomalous Nernst coeff. Vy x z 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. Empirically, axy = gTNF A = 34 A2 3D density of states
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 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

29. End

30. 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)

31. 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.

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 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

34. Electrons on a Bravais Lattice 1k
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

35. 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

36. 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

37. Boltzmann transport Eq. with anomalous velocity term.

38. 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)

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
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

40. 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

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