Post on 31-Jan-2016
description
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