Transcript Electrons on a triangular lattice in Na-doped Cobalt Oxide Yayu Wang, Maw Lin Foo, Lu Li, Nyrissa Rogado, S.

Electrons on a triangular lattice in Na-doped Cobalt Oxide
Yayu Wang, Maw Lin Foo, Lu Li, Nyrissa Rogado,
S. Watauchi, R. J. Cava, N.P.O.
Princeton University
1.
2.
3.
4.
5.
Frustration on triangular lattice
Large thermopower in NaxCoO2
ARPES
Hall effect
Phase diagram
Supported by NSF, ONR
Geometrical Frustration on triangular lattice
H = -J
Si Sj
(i,j)
Antiferromagnetic Ising model
?
Impossible to have
AF alignment
on all 3 bonds
Ground state is disordered and highly degenerate
Resonating valence bond model(s) 1971, 1987
Spin Ice in pyrochlores 1998
Frustrated magnetic states in spinels 1999
Nax CoO2
building block
tilt
Terasaki, Uchinokura 1997
Octahedra tilted to
form a layer
Co
Na
Na ions (dopants) sandwiched btw
layers of tilted CoO2 octahedra
Co ions define a triangular lattice
as grown
Co
Na
Resistivity of NaxCoO2 (x ~ 0.71)
Terasaki et al., PRB 1997
Wang et al. Nature ‘03
Susceptibility of insulators vs metals
c
Curie
~ 1/ T
Susceptibility
c = dM/dH
In metals, c small and indept of T
c
0
kT
Pauli
DOS
T
energy
AF spins
1/c
In Antiferromagnets
c= C/(T + q)
free spins
q = TN (Neel temp)
0
T
Susceptibility c has Curie-Weiss form
0.20
1/c (T.m ole/em u)
q ~ 55K
0.15
• AF Neel temperature
TN ~ 60-100 K
• Magnitude of c implies
Co4+ ions spin S = ½
Co3+ is diamagnetic (S = 0),
H || c
0.10
Co3+
H || ab
0.05
0.00
-50
0
50
100
T (K)
150
200
Co4+
Metallic resistivity but antiferromagnetic in spin response
(Curie-Weiss Metal)
0.20
1/c (T.m ole/em u)
q ~ 55K
0.15
H || c
0.10
H || ab
0.05
0.00
-50
0
50
100
T (K)
150
200
Thermopower and Peltier coef.
holes
J
E
JQ
Heat current density JQ
accompanies charge current density J
Ratio of currents JQ/J = P (Peltier coeff)
S = P / T = JQ/ JT
Large thermopower S of NaxCoO2
Terasaki et al Phys. Rev. B (1997)
100
Na 1.36 Co 2 O 4
x = 0.71
80
Q ( m V/K)
• Large thermopower
~10 times Sommerfeld
60
value at 300 K
40
20
Sommerfeld
0
0
50
100 150 200 250 300
T (K)
Thermopower
Classical gas
J = nev
JQ = n kBT v
kBT
e
Peltier coef.
P = JQ/J = kBT/e
Seebeck coef.
S = P / T = kB/e
Semiconductor
Natural unit of S
kB/e = 86 mV/K
JQ = n v D
S = (kB/e)(D/kBT)
m
D
Thermopower of conventional metals
“Excitation picture”
hole
particle
e(k)
Fermi level
m
Fermi Gas in E field
Charge currents add
mass currents cancel
Heat currents cancel
E
vacancies
hole
excitations
S = JQ/JT strongly suppressed
dk = c-k
S ~ (kB/e) (T/TF)
TF ~ 50,000 K
~ 86 x 10-2 mV/K
particle
excitations
E
S virtually indepndnt of H
Field dependence of S in NaxCoO2
Wang et al. Nature ‘03
1.4
H || - Ñ T
1.2
1.0
B
Q ( m V/K)
10K
-
T
0.8
0.6
8
0.4
7
3.3
6
• In-plane field
0.2
2.5K
5.3
H || - T
0.0
-0.2
• Strong field suppression
of Thermopower
4.4
0
2
4
6
8
10
m 0 H (T)
12
14
V
Spin contribution to thermopower (Chaikin Beni, 1976)
JQ
J
J = nev
Spin entropy per carrier = kBlog 2
JQ = nv kBT log 2
S = JQ/JT = (kB/e) log 2 ~ 60 mV/K
Not signif. in conv. metals
S(H,T) curve is a function of H/T only
100
Na 1.36 Co 2 O 4
Q (m V/K)
80
60
40
20
0
0
50
100 150 200 250 300
T (K)
Conclusion:
1. Spin entropy is the source
for enhanced thermopower
2. Key for new thermo-electric
materials -- Spin
Wang et al. Nature ‘03
In NaxCoO2, hole density nh = 1-x
Co 3d states
NaxCoO2
Multiple electronic phases vs. Na content
Superconductivity
Water intercalated superconductor
Takada et al., Nature (2003).
• pairing symmetry:
s, p or d-wave?
• Why is water essential?
• What is pairing mechanism:
e-ph or e-e or magnetic?
NaxCoO2·y H2O, x ~ 0.35, y ~ 1.30
Superconductor with Tc ~ 4.5K
T-linear Hall coefficient
Yayu Wang, 03
RH conv. metal
Why is RH T-linear?
Hopping Hall current in triangular lattice (Holstein, ‘61)
sH ~ t12 t23 t 31 ~ i
2
t3
exp(ia)
t12
Peierls phase a = 2p f/f0
f
1
t13
High-frequency RH* in tJ model (B.S. Shastry ‘93, ‘03)
sH ~ i(bt)3 exp(ia)
bt << 1 (b = 1/T)
s ~ (bt)2
R*H ~ sH/Hs2 ~ (bt) -1
T-linear
M2S-RIO conf. Rio de Janeiro, May 28th 2003 (N.P. Ong)
3
ARPES: Weak quasiparticle dispersion
Single-particle hopping :
Z. Hasan et al. (PRL ‘04)
Small bandwidth  Low degeneracy T
t < 0 and |t| ~ 10 meV (bandwidth < 100 meV)
Kinetic energy (eV)
Fermi Surface of Na0.71CoO2 measured by ARPES
Hasan et al.
Large hole-like FS
Hopping integral
t ~ 10 meV
Fermi velocity < 0.4 eV.A
Behavior of quasi-particles versus temperature
Integrated Intensity(arb. Units)
7
6
5
4
3
2
Resistivity is T-linear
below 100K
0
40
80
120
ARPES Quasiparticles are coherent
only below 150K
NaxCoO2
Insulating state
as grown
Multiple electronic phases vs. Na content
Foo et al.
PRL ‘04
Fine-tuning of Na content in NaxCoO2 single crystals
Foo et al., condmat/0312174 (2003), PRL ‘04
• Reduce the Na content by a series of chemical de-intercalation
• x = 0.75, as grown crystals of Floating zone or flux method
x = 0.68: NaClO3 in water
x = 0.50: I2 in Acetonitrile
Stronger oxidation agent
x = 0.31: Br2 in Acetonitrile
High-quality crystals with Na content 0.31 < x < 0.75
Calibration of the Na content vs. c-axis lattice parameter
Calibration procedure
11.3
c-axis (angstroms)
NaxCoO2
11.2
• treat powder and crystals
under same conditions
11.1
• powder x-ray diffraction to
get c-axis lattice constant
• ICP-AES to determine the
Na contents of powders
11.0
10.9
• x vs. c-axis calibration curve
powder
crystal
10.8
0.3
0.4
0.5
0.6
Na content
0.7
• from the c-axis of crystal,
extract the Na content
0.8
1.6
NaxCoO2
-3
c (10 emu/mole.Oe)
1.4
0.75
1.2
1.0
0.68
0.8
0.6
0.31
0.4
0.50
0.2
0.0
88K
53K
0
50
100
150
200
250
300
T (K)
•
•
x = 0.50 (1/2):
Two kinks at Tc1=88K and Tc2=53K in c
Resistivity shows insulating behavior below T=53K
An unexpected insulator at x = ½
0.5
25
2.0
NaxCoO2
NaxCoO2
x=0.50
20
88K
1.5
53K
1.0
 (m.cm)
 (m.cm)
53K
0.50
88K
d/dT
0.0
15
10
-0.5
0.5
0.31
0.71
0.0
0
50
100
150
T (K)
200
250
5
300
0
0
-1.0
20 40 60 80 100 120 140 160
T (K)
b
a
b*
a*
Na
Na vacancy
Electron diffraction at 300K shows the superlattice
formed by the Na ions, consistent with a zig-zag order
Zendbergen et al., condmat/0403206 (2004)
Thermal Conductivity
Hall coefficient
Foo et al., PRL ‘04
0
NaxCoO2
as grown
Multiple electronic phases vs. Na content
Spin
ordered
Foo et al.
PRL ‘04
Further enhancement of thermopower
x = 0.71
x=
Thermopower
of 0.88
NaxCoO2
250
S
S (mV/K)
200
Na0.88CoO2
150
Na0.71CoO2
100
50
Sommerfeld
0
0
50
100
150
200
T (K)
250
300
350
P = S2 s
250
2
Power Factor S s ( mW/cm . K )
Power factor of NaxCoO2
200
Na0.88CoO2
x ~ 0.85
2
150
100
50
x = 0.71 Na0.71CoO2
0
0
50
100
150
200
T (K)
250
300
350
Unusual electronic behavior in NaxCoO2
Strongly correlated s = ½ holes hopping on triangular lattice
•
Paramagnetic Metal (x ~ 1/3)
High conductivity, superconducting with H2O intercalatn.
•
Charge-ordered Insulator (x = ½)
Na ion ordering, hole ordering (stripes?),
giant thermal conductivity
•
Curie-Weiss metal (x ~ 2/3)
Curie-Weiss susceptibility, metallic cond., large thermopower
from spin entropy, T-linear Hall coef.
•
Spin Ordered Phase (x > ¾)
Even larger thermopower, field-induced metamagnetism
1.6
1.4
0.6
0.75
1.2
1.0
0.2
0.8
0.6
0.31
0.0
-0.2
0.4
-0.4
0.2
-0.6
0.0
x=0.31
T=5K
0.4
M (a.u.)
-3
c (10 emu/mole.Oe)
0.8
NaxCoO2
-0.8
0
50
100
150
T (K)
•
•
•
200
250
300
-5 -4 -3 -2 -1 0 1 2 3 4 5
m0H (T)
x = 0.31 (~ 1/3), parent compound of the SC
c is T-independent, not Curie-Weiss
M-H curves are linear at low T, no ferromagnetic order
Magnetic properties rather normal
3
100
NaxCoO2
NaxCoO2
0.71
2
RH (10 m /C)
3
60
1
-9
S (mV/K)
80
40
0.31
0.31
0
20
0.71
-1
0
0
50
100
150
T (K)
•
•
•
•
x = 0.31 (~ 1/3):
200
250
300
0
50 100 150 200 250 300 350
T (K)
Smaller high temperature thermopower
Smaller Hall coefficient, weaker T-dependent
larger hole concentration (~3x1022/cm3) and reduced correlation
Consistent with ARPES (MZ Hasan et al., and Hong Ding et al.)
x = 0.71 (~ 2/3)
•
c Curie-Weiss, AF interaction
•
 is T-linear at low T
•
S large, ~90 mV/K at 300K
•
RH strong T-linear
Curie-Weiss metal
Strong magnetic interaction
and electron correlation
x = 0.31 (~ 1/3)
•
c is T-independent, non
Curie-Weiss
•
 smaller, T 2 at low T
•
S small, ~34 mV/K at 300K
•
RH weaker T-dependence
Paramagnetic (T 2) metal
More like conventional metal
Sodium ion ordering versus x
Lynn, Cava et al.
0
120
-100
100
-200
80
0.71
S (mV/K)
S (mV/K)
100
-300
60
40
-400
0.31
NaxCoO2
x=0.50
-500
-600
0.50
0
50
100
150
200
250
300
20
0
0
100
200
300
T (K)
T (K)
S have giant negative values below Tc1
The number of holes are strongly reduced,
the residual charge carriers seem to be electron like
400
0
8
NaxCoO2
6
RH (10 m /C)
3
-400
0.50
4
-9
-9
3
RH(10 m /C)
-200
-600
-1000
0.31
NaxCoO2
x=0.50
-800
2
0
0.71
-2
0
50
100
150
T (K)
200
250
300
0
50 100 150 200 250 300 350
T (K)
RH becomes negative and the amplitude is 100 times larger
• charge density reduces by ~ 100 times
• particle-hole symmetry at low T
Possible charge ordering in NaxCoO2
electron
electron
hole
electron
hole
hole
x = 1/3
x = 2/3
3 a  3 a
3 a  3 a

electron
x = 1/2

hole

1/4 < x < 1/3
dome shape SC
< x < 1/4
Schaak et al., Nature0(2003)
3/4 < x < 1
CoO2: No results,
NaCoO2:
X = 1/2 Magnetic xordering?
~ 2/3 per Co site,
1 electron perDoped
Co site,
1 pair of electron
Mott Insulator?
insulator
x ~ 1/3Charge ordered
Motohashi
etinsulator.
al.,magnetic
PRB (2003)interaction
Strong
Mott insulator?
band
Sugiyama
et electron
al., condmatcorrelation
(2003)
More like conventional
metal
and
Maw-lin Foo et al., condmat
(2003)
Bayrakci et al., condmat (2003)
To appear in PRL (2004)
4
1.0
Na 1.36 Co 2 O 4
3
3
R H (10 cm /C)
0.6
2
-3
 (m  cm)
0.8
0.4
0
0.2
0.0
1
-1
0
100
200
300
T (K)
• Good conductor
•  is T-linear below 100 K
0
100
200
300
T (K)
Hall coefficient
n2D ~ 4× 1022 /cm2
400
1.0
NaxCoO2
0.71
60
0.8
NaxCoO2
0.6
40
0.31
0.4
 (m.cm)
 (m.cm)
50
30
20
30K
0.2
0.0
10
0
0
50
0
500 1000 1500 2000 2500
100 150 200 250 300
2
2
T (K)
T (K)
•
•
•
x = 0.31 (~ 1/3), parent compound of the SC
Better metal,  is smaller that x = 0.71
R ~ T2 below 30K,  ~ 10 mcm at 4K
More like a conventional Fermi liquid
Thermoelectric and Peltier effects
holes
J
J
JQ
Heat current density JQ
accompanies charge current density J
Ratio of currents JQ/J = P (Peltier coeff)
S = P / T = JQ/ JT
specimen
JQ
p
n
Thermoelectric cooler
Systematic change vs x except at x = ½
Susceptibility
Resistivity
Foo et al., PRL ‘04
•
Na
•
CoO2
Co
Na
CoO2
Transition metal oxide
tunable carrier density
Quasi-2D:
ρc/ρab~200 at 4K
•
Triangular Co lattice with
AF interaction—Frustrated
magnetic system
•
Enhanced thermopower
Co
I. Terasaki et al (1997)
Na
•
Superconductivity
K. Takada et al (2003)
Strong-Correlation System:
Kubo formula:
S ( 2 ) / S (1 )  m / e
Q=
,
T
,
m
 s 
=
E , V
T  N 
s spin : Spin Entropy
g s : Spin Degeneracy
Free Spin model:
Qspin
gmH
gmH


kB
gmH
gmH 
k BT
k BT
=
e
)
t anh(
)
ln(e
e 
k BT
k BT 
P.M.Chaikin et al (1976)
x=g m B B/k B T
0
1
2
3
4
0.6
Q ( m V/K)
T=2.5K, H||- Ñ T
raw data
0.4
fit
• Close fit using
free-spin model
0.2
0.0
• From fit:
Landé factor g ~ 2.2
0
2
4
6
8
m 0 H (T)
10
12
14
2.0
Th
phosphor
-bronze
- T Tc
LSCO-0.17
1.5
T0
V
Q ( m V/K)
1.0
10
0.5
7.5
0.0
• Stotal = S0 - Swire
• Phosphor bronze wire
is H-independent
• All obs. field dependence
from NaxCo2O4
5K
-0.5
-1.0
0
5
10
15
20
m 0 H (T)
25
30
Materials Constraint
semiconductor
metal


S ~ D/T
S ~ T/TF
1000 mV/K
T
T
Difficult to have ZT larger than 0.01 below 200 K.
5 mV/K
350
30
0.50
300
25
250
20
 (W/mK)
 (W/mK)
0.31
15
0.71
10
5
0
200
150
100
0
50
100
150
200
250
T (K)
Thermal conductivity:
mostly from phonons
300
50
0
0.71
0
0.31
50
100
150
200
T (K)
Much larger thermal conductivity: longer phonon mean free path
• Na ion ordering: reduced scattering by disordered Na ions in the Na layer
• charge ordering: steep increase below 88K, reduced electron-phonon
scattering in the CoO2 plane,
NaxCoO2
Several
electronic
phases vs. Na
content
Foo et al. PRL (’04)
Low-energy electronic structure of Na0.7CoO2
ARPES work by Hasan Group : cond-mat/0308438
Results on Na0.7CoO2 :
O
Weak quasiparticle dispersion : narrow bandwidth
O
Signatures of Strong Correlation (Large Hubbard U)
O
Fermisurface : Large rounded Hole-like, small vf (anisotropic)
O
Thermal behavior of QPs: coherent QP only below 150K
Strong Correlation (Large Hubbard U)
Single-particle hopping :
7
U ~ 5 eV
70
t < 0 and |t| ~ 10 meV (bandwidth < 100 meV)
Renormalization by a factor of 10
69
o Large hole-like Fermisurface
68
67
10
66
65
64
63
62
61
60
8
area(arb. Units)
Normalized Intensity(arb. Units)
Momentum
o
Temperature behavior of quasiparticles
BL 12 new results
h=
o
Fermi Surface of Na0.7CoO2
Integrated Intensity(arb. Units)
Weak quasiparticle dispersion
o Weak hexagonal anisotropy
6
4
o Fermi velocity < 0.4 eV.A
2
6
5
4
3
2
59
0
40
80
120
0
57
55
52
50
55
60
65
70
Photon Energy(eV)
Resistivity is T-linear
up to 100K
50
integrated intensity of 11eV peak v.s. photon energy
15 14 13 12 11 10
9
Binding Energy(eV)
Kinetic energy (eV)
Narrow band
Resonance profile of valence satellite
 a measure of Hubbard U ~ 5 eV
n(k) :  integration –100 meV to +25 meV
ARPES Quasiparticles are coherent
only below 150K
Strong Correlation (Large Hubbard U)
BL 12 new results
U ~ 5 eV
h=
70
69
68
67
10
66
63
62
61
60
59
8
area(arb. Units)
Normalized Intensity(arb. Units)
65
64
6
4
2
0
57
55
52
50
55
60
65
70
Photon Energy(eV)
50
integrated intensity of 11eV peak v.s. photon energy
15 14 13 12 11 10
9
Binding Energy(eV)
Narrow band
Resonance profile of valence satellite
 a measure of Hubbard U ~ 5 eV
NaxCoO2: Curie-Weiss Metal
(Curie Weiss)
resistivity
c
Pauli
TT
In NaxCoO2 (x = 0.70),
susceptibility implies spin-1/2 local moments,
instead of degenerate electron gas
0
0
40
40
-1
30
-2
10
25
H || ab
-3
15
20
-4
30
( H - 0)/ 0 (%)
(  H -  0)/ 0 (% )
35
-1
4.3K
20
-2
H || c
6
-3
-5
4.3
Na 1.36 Co 2 O 4
0
2
4
6
10
8
m 0 H (T)
10
12
14
-4
0
2
4
6
8
10
m 0 H (T)
Magneto-resistance also from Spin effect,
similar anisotropy between in-plane and c-axis.
12
14
ONR workshop 2004, Tampa
Two figures of merit for thermoelectrics
Max temp. difference
S = thermopower
 = resistivity
 = thermal conductivity
DTmax = ½ ZT2
1. Power factor
S2/
2. The ZT number
ZT = (S2/)(T/)
ZT ~ 1 in Bi2Te3 (at 300 K)
Maximize ZT and minimize resistivity
(physically conflicting demands)
0.060
Na0.88CoO2
ZT
0.040
0.020
Na0.71CoO2
0.000
0
50
100
150
T (K)
200
250
300