1. Intro anomalous Hall effect2. Berry phase and Karplus-Luttinger theory3. Anomalous Nernst Effect in CuCr2Se4

4. Nernst effect from anomalous velocity

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

Supported by NSF

ISQM-Tokyo05

0.0 0.5 1.0 1.5 2.00

1

2

3

4

5

6

7

0.0 0.5 1.0 1.5 2.00.0

0.2

0.4

0.6

0.8

1.0

10

25

75

125

0H ( T )

300

200

150

100

50

300

250

xy (

m

)

250225

200

175

150

50

100

5 K

xy (

m

)

0H ( T )

105 K

25

75

275

125

175

225

x = 1.0x = 0.85

Anomalous Hall effect (AHE) in ferromagnet (CuCr2Se4: Br)

xyxy HR 0 MRsxy

J

xy

H

A brief History of the Anomalous Hall Effect

1954 Karplus Luttinger; transport theory on lattice Discovered anomalous velocity v = eE x .Earliest example of Berry-phase physics in solids.

1955 Smit introduced skew-scattering model (semi-classical). Expts confusing

1958-1964 Adams, Blount, Luttinger Elaborations of anomalous velocity in KL theory

1962 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)1973 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

1890? Observation of AHE in Ni by Erwin Hall1935 Pugh showed xy’ ~ M

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

e3 x dv = 0

Parallel transport

complex vectors 2/w)(vψ̂ i 221 /)e(en̂ i

angular rotatn is a phase ien̂ψ̂ n̂n̂ idd

constraint angle

v acquires geometric angle relative to local e1

Change Hamiltonian H(r,R) by evolving R(t)Constrain electron to remain in one state |n,R)

Electron wavefcn, constrained to surface |nR), acquires Berry phase

RRR nind

|n,R) defines surfacein Hilbert space

Parallel transport RR nin

ieRn

0 i

|n,R)

Berry phase and Geometry

Electrons on a Bravais Lattice 1

Berry vector potential

(r)(r) kr.k

ni

n uekBloch state

kk*

kkX nn uiuxdcell

3

Constraint! Confined to one band

k

k

(k)

AdamsBlountWannier

x.E(k) eH perturbation

Drift in k space, ket acquires phase ien k||

k||k nin

X(k).kd

Parallel transport n̂n̂ idd

X(k))((k) k iVH ext

X(k)k

k

E

k-space

E(k)v k e

Semiclassical eqn of motion

extVHH 0

Vext causes k to change slowly

kkk*X(k) nn uiurd 3

Motion in k-space sees an effective magnetic field

Equivalent semi-class. eqn of motion

x = R x = R + X(k) Gauge transf.

x fails to commute with itself!

X(k)x ki

X(k),],[

kijk

ji ixx

(X(k) = intracell coord.)

In a weak electric field,

E(k)x],[v

x.E

k eHi

eHH

0

(k) acts as a magnetic field in k-space,a quantum area ~ unit cell.

Karplus-Luttinger, Adams, Blount, Kohn, Luttinger, Wannier, …

R x

X(k)

k

kkkvJ gfe 02

Karplus Luttinger theory of AHE

Boltzmann eqn.

Anomalous velocity

Equilibrium FD distribution contributes!0

kf

Anomalous Hall current

1. Independent of lifetime involves f0k)

2. Requires sum over all k in Fermi Sea.but see Haldane (PRL 2004)

3. Berry curvature vanishes if time-reversal symm. validkΩ

kk

k vE

ef

g0

(B = 0)

Berry curvature

kkk ΩEv e

kk

k ΩEJ 02H 2 fe

2enxy '

• Luttinger’s anomalous velocity theory ’xy indpt of xy ~ 2

• Smit’s skew-scattering theory

’xy linear in xy ~

In general, xy = xy2

2enxy 'KL theory

Anderson, Phys. Rev. 115, 2 (1959).Kanamori, J. Phys. Chem. Solids 10, 87 (1959).Goodenough, J. Phys. Chem. Solids 30, 261 (1969)

Ferromagnetic Spinel CuCr2Se4

Cu

Goodenough-Kanamori rules

180o bonds: AF (superexch dominant)

90o bonds: ferromag.(direct exch domin.)

OCu

Se

Cr

0.0 0.2 0.4 0.6 0.8 1.00

50

100

150

200

250

300

350

400

450

CuCr2Se

4-xBr

x

TC (

K )

X

0 1 2 3 4 50.0

0.5

1.0

1.5

2.0

2.5

3.0

5 K

x = 1.0 x = 0.85 x = 0.5 x = 0.25 x = 0

CuCr2Se

4-xBr

x

M (

B /

Cr)

0H ( T )

• Tc decreases slightly as x increases.• At 5 K, Msat ~ 2.95 B /Cr for x = 1.0

• doping has little effect on ferromagnetism.

Effect of Br doping on magnetization

0 50 100 150 200 250 3000.01

0.1

1

10

0

0.1

0.25

0.5 (A,B)

0.6

0.85 (A)

0.85 (B)

1 (B)

x = 1 (A)

CuCr2Se

4-xBr

x (

m

cm )

T( K )

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

320102 cmnH

0.0 0.2 0.4 0.6 0.8 1.00

1

2

3

4

5

6

7

0.0

0.2

0.4

0.6

0.8

1.0

n H (

pe

r F

.U.

)

CuCr2Se

4-xBr

x

n H (

102

1 cm

-3 )

X

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

0.0 0.5 1.0 1.5 2.00.00

0.02

0.04

0.06

0.08

0.10

0.0 0.5 1.0 1.5 2.0-0.05

-0.04

-0.03

-0.02

-0.01

0.00

0.01

0H ( T )

300

300 K

200

150

10075

250

xy (

m

)

275

250

225

200

100-150 K

5-50

xy (

m

)

0H ( T )

125

225

175

5-50

x = 0.6x = 0.25

0.0 0.5 1.0 1.5 2.00

1

2

3

4

5

6

7

0.0 0.5 1.0 1.5 2.00.0

0.2

0.4

0.6

0.8

1.0

10

25

75

125

0H ( T )

300

200

150

100

50

300

250

xy (

m

)

250225

200

175150

50

100

5 K

xy (

m

)

0H ( T )

105 K

25

75

275

125

175

225

x = 1.0x = 0.85

• Large positive AHE, at 5K, , x = 1 . mxy 700

0.0 0.5 1.0 1.5 2.0-0.005

0.000

0.005

0.010

0.015

0.020

0.0 0.5 1.0 1.5 2.00.000

0.005

0.010

0.015

0.020

0.025

0H ( T )

300

350 K350 K

200

150100

50

300

250

xy (

m

)

275

250

225

200

175150

50

100

5

xy (

m

)

0H ( T )

5

x = 0.1x = 0

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

Wei Li Lee et al. Science (2004)

If ’xy ~ n,

then

’xy /n ~ 1/(n)2

~ 2

Observed A implies<>1/2 ~ 0.3 Angstrom

Fit to ’xy/n = A2

2enxy '

• impurity scattering regime

• 70-fold decrease in , from x = 0.1 to x = 0.85.

• xy/n is independent of

• Strongest evidence to date for the anomalous-velocity theory

An

An

Hxy

Hxy

/

08.095.1,/'

'

M J (per carrier)

JH (per carrier)

Brominedopantconc.

E

Doping has no effect on anomalous Hall current JH per hole

With increasing disorder, J decreases, but AHE JH is constant

Vy

HH

T

E

x

y

Ey/| |= Q0 B + QS MQS, isothermal anomalous Nernst coeff.

y

x

z

Tx

Tx

Anomalous Nernst Effect

Tx

)( T>Hx

yz

E>

Longitudinal and transverse charge currents in applied gradient

).(E.J T >>

xyxyyN TEe ||/

Total charge current

Nernst signal

Measure , eN, S and tanH to determine xy

Final constitutive eqnHxy Se tanN

0.0 0.5 1.0 1.5 2.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0x = 0.25

200

175150

125

100

75

50

25

105 K

Ey /

gra

d.T

(

V /

K )

0H ( T )

0.0 0.5 1.0 1.5 2.0

-2.0

-1.5

-1.0

-0.5

0.0 105 K

x= 0.6

150

125

100

75

50

25

175-200

Ey /

gra

d.T

(

V /

K )

0H ( T )

Wei Li Lee et al. PRL (04)

0.0 0.5 1.0 1.5 2.0

-2.0-1.8-1.6-1.4-1.2-1.0-0.8-0.6-0.4-0.20.00.2 x = 0.85

5 K10

25

200175

150

125

100

75

50

Ey /

gra

d.T

(

V /

K )

0H ( T ) 0.0 0.5 1.0 1.5 2.0

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2x = 1.0

350

300

250

225

200

175

150

125

1007550

2515

Ey /

gra

d.T

(

V /

K )

0H ( T )

)( TJ xyxy

kkk ΩEv e

lv)(

kk

k

f

Ts

>

zxxs

xy kvf

T

kk

k )(

Nernst effect current with Luttinger velocity

(KL velocity term)

Leading orderIn E and (-grad T)

1. Dissipationless (indpt of )2. Spontaneous (indpt of H)3. Prop. to angular-averaged

Peltiertensor

F

F

Bxy N

NTek

3

2

3

22

eN non-monotonic in x

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

Empirically, xy = gTNF

3D density of states

FB

xy NTek

A

2

A = 34 A2

Wei Li Lee et al. PRL (04)

F

F

Bxy N

NTek

3

2

3

22

Comp. with Luttinger result

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

’xy = n A 2

3. Confirms existence of dissipationless current Measured <>1/2 ~ 0.3 A.

4. Measured xy from Nernst, thermopower and Hall angle Found xy ~ TNF,

consistent with Luttinger velocity term

End

Parallel transport of a vector on a surface (Levi-Civita)

cone flattened on a plane

= 2(1-cos)

e transported without twisting about normal r

Parallel transport on C : e.de = 0

e acquires geometric angle

2(1-cos) on sphere

r

e

de(Holonomy)

de normal to tangent plane

0ˆ*ˆ Parallel transport

Local coord. frame (u,v)

n̂*n̂ i

n̂*n̂i

e.de = 0

Generalize to complex vectors

Geometric phase i) arises from rotation of local coordinate

frame,ii) is given by overlap between n and dn.

Local tangentplane

Nernst effect from Luttinger’s anomalous velocity

Area A is of the order of ~ xy ~ 1/3 unit cell section

xyB

xy e

Tk 2

In general,

Fxy N

FB

xy NTek

A

2

Since

we have

X(k),],[

kijk

ji ixx

kkk ΩEv e

Atom Electron on lattice

(R)A])[/( R VeiMH 221

R kslow variable

Berry gauge potential

(r)(r) kr.k

ni

n uekProduct wave fcn

R||RA R nin kkk*X(k) nn uiurd 3

“magnetic” fieldAB eff X(k)k

(r)(R)R)(r, RnN

inteN HHHH R)(r,(R)extVHH 0

Hamiltonian

effective H (k)X(k))( k iVH ext

fast variabler r in cell

Electrons on a Bravais Lattice 1

R x

X(k)Center of wave packet

Wannier coord. within unit cell

Berry vector potential

(r)(r) kr.k

ni

n uekBloch state

kk*

kkX nn uiuxdcell

3

kR ikXRx

Constraint! Confined to one band

k

k

(k)

AdamsBlountWannier

AB eff

R

Beff

(R)A])[/( R VeiMH 221

Berry phase in moving atom

Nuclear R(t) changes gradually but electron constrained to stay in state |n,R)

Electron wavefunction acquires Berry phase

Integrate over fast d.o.f.

Nucleus moves in an effective field

R A.R

dB

R||RA R nin

(Berry curvature)

(r)(R)R)(r, RnN product wave fcn

Bie

R)(r,(R) eN HHH

exp(i

Electron wavefcn acquires Berry phase

R A.R

dB

R||RA R nin

Nucleus moves in closed path R(t), butelectron is constrained to stay at eigen-level |n,R)

Constraint + parameter change Berry phase, fictitious Beff field on nucleus

connection

curvatureAB eff

• Boltzmann transport Eq. with anomalous velocity term.

,

,])([2

])([2

,)(

,ˆ,)(

,]][[2

2

,

02

30

2

0

30

expansionSommerfelduse

inlineartermkeep

useand

k

xkZ

k

kyx

kk

kZ

xy

xk

k

k

kk

dSd

f

Tm

e

kdTT

fkeJ

T

xk

ETT

f

kdfEek

eJ

k

k

g

g

TnCxy

e.temperaturisTandionconcentratcarrierisn,const.isCwhere

X(k)x ki

X(k),],[

kijk

ji ixx

In a weak electric field,

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

E(k)x],[v

x.E

k eHi

eHH

0

(k) is an effective magnetic field in k-space (Berry curvature)

Electrons on a lattice 3

kB~

gg mg ~,/1~ **

J. Phys. 34, 901 (1973)

Dissipationless, indept of

Nozieres-Lewiner theory

X(k)Rr

SEJ SO2

H 2 ne

•Anomalous Hall current JH

•Anomalous Hall effect in semiconductor with spin-orbit coupling

• Enhanced g factor and reduced effective mass

2SOSO 1where )/(,SkX(k) g

Electrons on a Lattice 2

BvEk ee AB

Predicts large Hall effect in lattice with broken time reversalKarplus Luttinger 1954, Luttinger 1958

Eqns. of motion?

kkv

k = 0 only ifTime-reversal symm.or parity is broken

X(k) a funcn. of k

E

kΩE e

Berry potential

Berry curvature

kkk X

0.0 0.5 1.0 1.5 2.0

-1.6

-1.4

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2 x = 0

225

125

75

200

175

150

100

5025

105 K

Ey /

grad

.T (

V

/ K

)

0H ( T )

0.0 0.5 1.0 1.5 2.0

-1.8

-1.6

-1.4

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

300275250

x = 0.1

225

125

75

200

175

150

100

5025

105 K

Ey /

grad

.T (

V

/ K

)

0H ( T )

Wei-Li Lee et al., PRL 2004

,S ,Sk- use

2

,][2

],[2

032

032

k

kH

kH

fkdEeJ

EfkdeJ

,

,]][[22 303

E

kdfEek

ekdfeJk

kkk

inlineartermkeep

kg

SEenJ H

22

0 50 100 150 200 250 300

-10

-5

0

5

10

15

20

25

0 50 100 150 200 250 3000

200

400

600

800

1000

1200

1400

T( K )

0.5 (B)

x = 0.6R

s (

10-8

m3 /C

)

0

0.1

0.25

0.5 (A)

R

s (

10-8

m3 /C

)

CuCr2Se

4-xBr

x

T( K )

1.0 (B)

0.85 (A)

0.85 (B)

x = 1.0 (A)

• 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 (T) .

0 50 100 150

0

-1

-2

-3

-4

-5

x = 1 x = 0.85 x = 0.6 x = 0.25 x = 0.1 x = 0

CuCr2Se

4-xBr

x

QS (

V

/K-T

)

T(K)

0 50 100 150

0.0

-0.2

-0.4

-0.6

xy (

V/K

--m

)

x = 1 x = 0.85 x = 0.6 x = 0.25 x = 0.1 x = 0

T(K)

• Qs same order for all x,• xy linear in T at low T.

Wei-Li Lee et al., PRL 2004