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
1.
2.
3.
4.
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
Anomalous Hall effect (AHE) in ferromagnet (CuCr2Se4: Br)
1.0
xy (  m )
x
7
10
5K
25
50
75
100
125
150
0.8
J
y
x = 0.85
0.6
25
50
75
100
125
150
175
4
3
225
0.0
0.0
10
5
200
0.2
5K
6
175
0.4
x = 1.0
xy (  m )
H
2
200
250
1
275
300
0.5
1.0
1.5
2.0
0
0.0
0.5
0H ( T )
0H ( T )
 xy  R0 H   xy
1.0
 xy  Rs M
225
250
300
1.5
2.0
A brief History of the Anomalous Hall Effect
1890? Observation of AHE in Ni by Erwin Hall
1935 Pugh showed xy’ ~ M
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
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
e3 x dv = 0
complex vectors
angular rotatn is a phase
ψ̂  (v  i w) / 2
ψ̂  n̂ ei
v acquires geometric
angle  relative to local e1
n̂  (e1  i e 2 ) / 2
d   n̂ id n̂
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
|n,R)
Parallel transport
 i   0
  n R ei
  n R i n R
Electron wavefcn, constrained to surface |nR), acquires Berry phase 
   d R  n R i n R
e(k)
Electrons on a Bravais Lattice 1
Constraint!
Bloch state
Confined to one band
 nk (r)  ei k .r un k (r)
H  e (k)  e E . x
k
perturbation
k
|  | n k ei
Drift in k space, ket acquires phase
Parallel transport
Adams
Blount
Wannier
  n k | i | n k
d   n̂ id n̂
   d k . X(k)
X k   d x u n k i k u n k
3
cell
*
Berry vector potential
Semiclassical eqn of motion
E
H  H 0  Vext
k

k-space
Vext causes k to change slowly
X(k)   d 3 r u * n k i k un k
x=R
x = R + X(k)
Gauge transf.
H  e (k)  Vext (ik  X(k))
Motion in k-space sees an effective magnetic field 
Equivalent semi-class. eqn of motion

   k  X(k)

 v   k e (k)  e E 
x fails to commute with itself!
X(k)
x  ik  X(k)
[ x , x ]  ie ijk ,
i
R
x
Karplus-Luttinger, Adams, Blount,
Kohn, Luttinger, Wannier, …
j
k
(X(k) = intracell coord.)

    X(k)
In a weak electric field,
H  H0  e E . x

 v  i[ H , x]  k e (k)  e E 
(k) acts as a magnetic field in k-space,
a quantum area ~ unit cell.
Karplus Luttinger theory of AHE
Boltzmann eqn.

J  2e v k f  gk
0
k

 f k0 
e E v t k
g k   
 e 
k
Anomalous velocity
vk  e k  e E Ωk
Equilibrium FD distribution
Anomalous Hall current
f k0
(B = 0)
contributes!
J H  2e E   f Ωk
2
Berry curvature
0
k
k
1. Independent of lifetime t
(involves f0k)
2. Requires sum over all k in Fermi Sea.
but see Haldane (PRL 2004)
3. Berry curvature
Ωk
vanishes if time-reversal symm. valid
e2
 xy '  n


In general,
xy = xy2
• Luttinger’s anomalous velocity theory
’xy indpt of t a xy ~ 2
• Smit’s skew-scattering theory
’xy linear in t
KL theory
a xy ~ 
e2
 xy '  n


Ferromagnetic Spinel CuCr2Se4
Cu
O
180o bonds: AF
(superexch dominant)
Se
Cu
Cr
90o
bonds: ferromag.
(direct exch domin.)
Goodenough-Kanamori rules
Anderson, Phys. Rev. 115, 2 (1959).
Kanamori, J. Phys. Chem. Solids 10, 87 (1959).
Goodenough, J. Phys. Chem. Solids 30, 261 (1969)
Effect of Br doping on magnetization
450
3.0
350
2.5
300
2.0
M ( B / Cr)
TC ( K )
400
250
200
150
100
50
5K
CuCr2Se4-xBrx
1.5
x = 1.0
x = 0.85
x = 0.5
x = 0.25
x=0
1.0
0.5
CuCr2Se4-xBrx
0
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0
X
• Tc decreases slightly as x increases.
• At 5 K, Msat ~ 2.95 B /Cr for x = 1.0
• doping has little effect on ferromagnetism.
1
2
3
4
0H ( T )
5
CuCr2Se4-xBrx
1.0
7
x = 1 (A)
10
CuCr2Se4-xBrx
6
1 (B)
0.8
5
-3
nH ( 10 cm )
0.6
1
21
 ( m cm )
0.85 (B)
0.5 (A,B)
0.25
0.1
0.6
4
3
0.4
2
0.1
0
0.2
1
0.01
0
50
100
150
200
T( K )
250
300
0
0.0
0.2
0.4
0.6
X
• At 5 K,  increases over 3 orders as x goes from 0 to 1.0.
• nH decreases linearly with x. nH  2  1020 cm3 , for x =1.0.
0.8
0.0
1.0
nH ( per F.U. )
0.85 (A)
0.01 x = 0.25
300 K
0.00
250
225
0.10
0.08
xy (  m )
xy (  m )
175
150
-0.02
125
-0.03
100
75
-0.04
100-150 K
5-50
200
-0.01
x = 0.6
200
0.06
225
0.04
250
275
0.02
5-50
-0.05
0.0
0.5
1.0
1.5
300
2.0
0H ( T )
• x = 0.25, negative AHE at 5K.
• x = 0.6 , positive AHE at 5K.
0.00
0.0
0.5
1.0
0H ( T )
1.5
2.0
x = 0.85
25
50
75
100
125
150
0.8
xy (  m )
7
10
5K
0.6
25
50
75
100
125
150
175
4
3
225
0.0
0.0
10
5
200
0.2
5K
6
175
0.4
x = 1.0
xy (  m )
1.0
2
200
250
0.5
1.0
1.5
225
250
300
1.5
2.0
1
275
300
2.0
0H ( T )
• Large positive AHE, at 5K,
0
0.0
0.5
1.0
0H ( T )
 xy  700   m , x = 1
.
x=0
0.025
0.020
x = 0.1
350 K
350 K
300
300
xy (  m )
0.020
250
0.015
5
50
xy (  m )
0.015
250
0.010
0.005
200
0.010
275
225
50
5
200
0.000
0.005
100
0.000
0.0
0.5
150
1.0
1.5
-0.005
0.0
2.0
0H ( T )
• x=0 , AHE unresolved below 100K.
• x=0.1, non-vanishing negative AHE at 5 K.
100 150
0.5
175
1.0
0H ( T )
1.5
2.0
Wei Li Lee et al. Science (2004)
e2
 xy '  n


If ’xy ~ n,
then
’xy /n ~ 1/(nt)2
~ 2
Fit to ’xy/n = A2
Observed A implies
<>1/2 ~ 0.3 Angstrom
• impurity scattering
regime
 xy' / nH  A  ,   1.95  0.08
  xy' / nH  A
• 70-fold decrease in t,
from
x = 0.1 to x =
0.85.
• xy/n is independent of
t
• Strongest evidence to
date for the anomalousvelocity theory
Doping has no effect on anomalous Hall current JH per hole
E
JH (per carrier)
M
J (per carrier)
Bromine
dopant
conc.
With increasing disorder,
J decreases, but AHE JH is constant
Anomalous Nernst Effect
Ey/|
 xT |
= Q0 B + QS 0M
QS, isothermal anomalous Nernst coeff.
z
x
Vy
y
Ey
  xT
 xT
H
H
I0
Longitudinal and transverse charge currents in applied gradient


J   . E  .( T )
Total charge current
eN  E y / | T |  xy  xy
 xy  eN  S tan H
Nernst signal
Final constitutive eqn
Measure , eN, S and tanH to determine xy

  xT
z
x
 
E

y

H
 ( T )
x= 0.6
10
Ey / grad.T (  V / K )
5K
Ey / grad.T (  V / K )
0.0
x = 0.25
25
-0.5
-1.0
75
125
0.0
1.0
75
100
125
150
200
0.5
50
-1.5
150
175
-2.5
10
25
-1.0
100
-2.0
5K
-0.5
50
-1.5
0.0
1.5
0H ( T )
2.0
-2.0
0.0
175-200
0.5
1.0
1.5
0H ( T )
2.0
Wei Li Lee et al. PRL (04)
0.2 x = 0.85
10
0.0
5K
-0.2
25
-0.4
50
-0.6
75
-0.8
100
-1.0
125
-1.2
-1.4
150
-1.6
175
200
-1.8
-2.0
0.0
0.5
1.0
1.5
0H ( T )
x = 1.0
0.0
Ey / grad.T (  V / K )
Ey / grad.T (  V / K )
0.2
-0.2
-0.4
-0.6
-0.8
-1.0
15
350
25
50
75
300
100
125
250
150
175
225
-1.2
2.0
0.0
200
0.5
1.0
0H ( T )
1.5
2.0
Nernst effect current with Luttinger velocity
J y   yx (  xT )

 
ks
 vk  e k  e E Ωk
Leading order
In E and (-grad T)
(e k   )  f 
 
 v l
T  e k 
(KL velocity term)
 xy  
ks
(e k   )  f 
 
vx k x  z
T  e k 
2
 2 ek B T 2  e  N 

 
 xy 

3  3  e  F
1.
2.
3.
Dissipationless (indpt of t)
Spontaneous (indpt of H)
Prop. to angular-averaged 
 NF
Peltier
tensor
eN non-monotonic in x
xy decreases monotonically with x
Wei Li Lee et al. PRL (04)
3D density of states
Empirically,
xy = gTNF
 ek B 2T 
NF
 xy  A




A = 34 A2
Comp. with Luttinger result
2
 2 ek B T 2  e  N 

 
 xy 

3  3  e  F
 NF
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
’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)
e transported without twisting about normal r
 = 2(1-cos)
cone flattened on a plane
Parallel transport on C :
e.de = 0
de normal to tangent plane
r
e acquires geometric angle
  2(1-cos) on sphere
(Holonomy)
e
de
Generalize to complex vectors
Local tangent
plane
Local coord.
frame (u,v)
e.de = 0
Parallel transport
ˆ * ˆ  0
  i n̂*  n̂
   i n̂*  n̂
Geometric phase 
i)
arises from rotation of local coordinate frame,
ii)
is given by overlap between n and dn.
Nernst effect from Luttinger’s anomalous velocity

i
j
k
[ x , x ]  ie ijk ,     X(k)
 vk  e k  e E Ωk
In general,
Since
we have
 k B 2T   xy

 xy  
 e
e


 xy
e
 NF
 ek B 2T 
NF
 xy  A




Area A is of the order of  ~ DxDy ~ 1/3 unit cell section
Atom
Electron on lattice
H  H N (R)  H e (r, R)  H int
(r, R)  N (R) n R (r)
R
r
A  n R | iR | n R
B eff    A
Hamiltonian
Product wave fcn
slow variable
fast variable
Berry gauge potential
“magnetic” field
H  (1 / 2M )[ i R  e A]2  V (R)
effective H
H  H 0  Vext
 nk (r)  ei k .r un k (r)
k
r in cell
X(k)   d 3 r u * n k i k un k

   k  X(k)
H  Vext (ik  X(k))  e (k)
e(k)
Electrons on a Bravais Lattice 1
Constraint!
Bloch state
Adams
Blount
Wannier
Confined to one band
 nk (r)  e
i k .r
k
un k (r)
k
Center of wave packet
Wannier coord.
X(k)
R  ik
x  R  Xk
within unit cell
X k   d 3 x u n k i k u n k
R
x
*
cell
Berry vector potential
Berry phase in moving atom
product wave fcn
H  H N (R)  H e (r, R)
(r, R)  N (R) n R (r)
Nuclear R(t) changes gradually but electron constrained to stay in state |n,R)
G
Electron wavefunction acquires Berry phase
   e i B
R
B   d R . A
G
A  n R | iR | n R
Integrate over fast d.o.f.
H  (1 / 2M )[ i R  e A]2  V (R)
G
Beff
B eff    A
(Berry curvature)
R
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
 gexp(iB)
B   d R . A
G
Constraint + parameter change
A  n R | iR | n R
connection
B eff    A
curvature
Berry phase, fictitious Beff field on nucleus
• Boltzmann transport Eq. with anomalous velocity term.

 

e k
J  2  e[   eE  ][ f k0  g k ] d 3 k ,
k

 f k0 
t    e   (T ) , and use E  k x xˆ ,
 g k   

 e   k  T
t

k 

keep term linear in (T ) ,

 f k0   e  

3


(T ) ] d k
 
 Z [t 
J y   2 e
 e   k  T
t

k 

2
kx
 f k0 
[ Z 2 (e   ) ] de dS ,

  yx
  e   k , x
 k e
k 

use Sommerfeld expansion ,
2e 2

mT
  xy  C n T
where C is const. , n is carrier concentrat ion and T is temperatur e.
Electrons on a lattice 3
x  ik  X(k)
[ x , x ]  ie ijk ,
i
j
k

    X(k)
(
 ~
  Bk
)
1. (k) -- a “Quantum area” -- measures uncertainty in x; (k)~ DxDy.
In a weak electric field,
H  H0  e E . x

 v  i[ H , x]  k e (k)  e E 
2. (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
g * ~ 1 / e g , m* ~ e g
r  R  X(k)
X(k)  SO k S, where SO  (1 / e g )2
•Anomalous Hall current JH
J H  2ne 2 SO E S
Dissipationless, indept of t
Electrons on a Lattice 2
Eqns. of motion?
 k  e E e v B
B   A

k  k  Xk
 vk  e k  e E Ωk
Berry potential
Berry curvature
X(k) a funcn. of k
E
k = 0 only if
Time-reversal symm.
or parity is broken
Predicts large Hall effect in lattice with broken time reversal
Karplus Luttinger 1954, Luttinger 1958
-0.2
50
25
75
5 K 10
100
-0.4
125
-0.6
150
-0.8
-1.0
175
-1.2
200
-1.4
225
-1.6
0.0
0.5
1.0
0.2
x = 0.1
5 K 10
0.0
25
-0.2
50
-0.4
75
-0.6
100
125
-0.8
150
-1.0
-1.2
175
-1.4
200
225
-1.6
250
-1.8
275
Ey / grad.T (  V / K )
Ey / grad.T (  V / K )
0.2 x = 0
0.0
300
1.5
0H ( T )
2.0
0.0
0.5
1.0
1.5
0H ( T )
2.0
Wei-Li Lee et al., PRL 2004


 
3
e k
0
3




J  2  e k f k d k  2  e[   eE  ][ f k  g k ] d k ,
k

keep term linear in E ,
 0  
 J H  2e  d k f k [ E  ],


3
2
0
 J H  2e E  [  d k f k  ] ,

 



use   -  k  S,     k    2S,
2
3
 
 JH  2ne  E  S
2
1400
800
3
-8
3
x = 1.0 (A)
15
10
1.0 (B)
600
0.5 (A)
0.5 (B)
5
0
0
400
0.85 (B)
200
0
x = 0.6
20
-8
1000
Rs ( 10 m /C )
1200
Rs ( 10 m /C )
25
CuCr2Se4-xBrx
0.85 (A)
0
50 100 150 200 250 300
0.1
-5
0.25
-10
0
50 100 150 200 250 300
T( K )
T( K )
• 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) .
-0.6
x=1
x = 0.85
x = 0.6
x = 0.25
x = 0.1
x=0
-4
QS (  V/K-T )
x=1
x = 0.85
x = 0.6
x = 0.25
x = 0.1
x=0
CuCr2Se4-xBrx
-3
-0.4
xy ( V/K--m )
-5
-2
-0.2
-1
0.0
0
0
50
100
T(K)
• Qs same order for all x,
• xy linear in T at low T.
150
0
50
100
T(K)
Wei-Li Lee et al., PRL 2004
150