Charge transport in correlated metals – I
I.a The basics
I.b Charge transport in the cuprates – the SCES poster-child
Nigel Hussey High Field Magnet Laboratory Radboud University, Nijmegen
ICTP workshop, Trieste, 12th July 2015
Charge transport in correlated metals – II
II.a Bad metals
II.b Kadowaki-Woods ratio The link between transport and thermodynamics
Nigel Hussey High Field Magnet Laboratory Radboud University, Nijmegen
ICTP workshop, Trieste, 13th July 2015
Outline
• Introduction
• Boltzmann transport theory
• Introduction to the cuprates
• Charge transport in the cuprates
ICTP workshop, Trieste, 12th July 2015
Introduction
• Often the first thing to be measured, but the last to be understood…
• “What scatters may also pair” Hence, electrical resistivity is a powerful, albeit coarse, probe of superconductivity
What makes dc transport measurements such an important probe of correlated electron systems?
Gunnarsson et al., RMP 75 1085 (03)
Drude model
Drude assumed that electrons were scattered by random collisions with the immobile ion cores. He assumed a mean free time τ between collisions so that after a time t, the number of electrons having survived without collisions was If the electric field E has been present for this time t, then an unscattered electron will have achieved a drift velocity of v = (-eEt/m) and have travelled a distance This gives for the total electronic transport in the direction of the applied field which is equivalent to n0 electrons having mean drift velocity for time τ. Finally, for a metal containing n electrons / m3, the current density with the corresponding electrical conductivity
( ) ( )τtntn −= exp0
( )mteEx 22=
( )
−=−
−=
∫∫
∞∞
meEnttt
meEnt
tnx
20
0
20
0
dexp2
ddd ττ
τ
2τ3 ( )meEv τ−=
( )m
EnenevEj τσ2
=−==
mne τσ
2
=
Despite the crude approximations and wildly incorrect assumptions, the Drude expression serves as an excellent, practical way to form simple pictures and rough estimates of properties whose deeper comprehension may require analysis of real complexity. How come? Well, some of it is fortuitous. Drude then estimated the velocity from the kinetic equation From this he extracted a mean-free-path of ~1 nm, i.e. the approximate interatomic spacing! Of course, while this seemed reasonable to Drude, it was way off the mark... The workability of the Drude model reflects the fact that two of the fundamental assumptions (the action due to the Lorentz force and the exponential decay in n(t)) are also found to be equally applicable to Bloch waves and fermionic quasiparticles.
( ) sec10 1cm2300 -14×≈⇒Ω≈ τµρ
/sm105232
21 ≈⇒= vTkmv B
Drude model
( ) ( ) ( ) ( ) [ ]( )BvEffpp×+−=+−= ettt
tt
τdd( ) ( ) ( ) ( ) ( )( )2dd d1d tOtttttt ++−=+ fpp τ
Drude model
Hall effect ac conductivity In steady state, current is independent of t Seek steady state solution of the form px and py thus satisfy Thus where ωc = eB/m (B//z) Thus Current density Hall field is determined by condition jy = 0 where Hence, Hall coefficient
τω
τω
yxcy
xycx
ppeE
ppeE
−+−=
−−−=
0
0
×+−−= BpEpp
me
t τdd
yxcy
xycx
jjE
jjE
+−=
+=
τωστωσ
0
0
xxc
y jneBjE
−=
−=⇒
0στω
neBjE
Rx
yH
1−==
( )tet
Epp−−=
τdd
( ) ( ) ( ) tit ωω −= expRe pp
( )ωτ
σωσi−
=1
0
( ) ( ) ( ) ( )( ) ( ) ( )ωωσ
ωτωωω 0
02
1E
Epj =−
=−
=i
mnem
ne
( ) ( ) ( )ωτωωω 0Epp ei −−=−
( ) ( ) ( ) tit ωω −= expRe 0EE
( )tfk ,r
(i) Carriers move in and out of the region r
(ii) The k-vector will be changed by external fields
(iii) Carriers are scattered
kκk v
rr fftt
f k ∇⋅−=∂∂
⋅∂∂
−=
∂∂
diff
kk
kkk
∂∂
⋅−=∂∂
⋅∂∂
−=
∂∂ kk ff
ttf
field
[ ]( )BvEk k ×+=
ewhere
( ) ( ) ( )τ
0kk
kkkkk kkk ff,Qfffftf −
−=′′−−−=
∂∂
∫ ′′ d11coll
Local concentration of carriers “occupancy” in the state k in the neighbourhood of the point r in space and time t
relaxation time approximation
Distribution functions
Boltzmann equation – 3 key points
0collfielddiff
=
∂∂
+
∂∂
+
∂∂
tf
tf
tf kkk
Note that this is steady state, not equilibrium state fk
0 For electrons We are most concerned with small departures from fk
0
( ) 1/exp1
+−=
Tkf
Bµεk
0k
∂∂
∝−=ε
0k0
kkkfffg
Fermi distribution
∂∂
∝−=ε
0k0
kkkfffg
f k0 (ε)
µ or εF
- (dfk 0/dε)
ε
1
0
0collfielddiff
=
∂∂
+
∂∂
+
∂∂
tf
tf
tf kkk
⇒−= 0kkk ffg
( ) ( )
k
0k
0k
k
0kk
0k
k
0k εµε
µεµε ∂
∂−=
∂∂
∂∂−
=∂∂
⇒+−
=fff
TTf
Tkf
B
and 1/exp
1
[ ]( )
[ ]( )k
BvEr
v
kBvEv
kk
kκ
k
kk
kkκ
∂∂
⋅×++∂
∂⋅+
∂∂
−=
∂∂
⋅×+−
∇∂∂
+∇∂∂
⋅−
gegtf
fefTTf
coll
000
µµ
[ ]( )coll
∂∂
−=∂∂
⋅×+−
∇∂∂
+∇∂∂
⋅−⇒tffefT
Tf kk
kkk
κ kBvEv
µ
µ
[ ]( )coll
∂∂
−=∂∂
⋅×+−∇⋅−⇒tffef kk
kkκ kBvEv
Boltzmann equation – all aboard the Chain Rule!
Assume “infinite homogeneous medium”, constant temperature and zero magnetic field; and for simplicity, let us make the phenomenological relaxation time approximation Hence
( )τ
kkkkk gt
gt
fgtf
−=∂
∂=
∂+∂
=
∂∂ 0
coll
( ) ( ) τtegtg −= 0kk
Evκk
k ⋅
∂∂
−=⇒ τε
efg k0
coll
0
∂∂
−=⋅
∂∂
−⇒tfefk k
κk
Evε
Linearized Boltzmann equation
( ) [ ]k
Bvr
vEv kκkk
κk ∂
∂⋅×+
∂∂
⋅+
∂∂
−=
∇−+∇
−⋅
∂∂
− kkk gegtf
eeT
Tf
coll
0 1 µµεε X X X X
( ) teefg ttt
k ′⋅
∂∂
−= ′−−
∞−∫ d
0τ
εEvκ
kk
Boltzmann vs. Drude
( ) ( ) τtegtg −= 0kk
0collfielddiff
=
∂∂
+
∂∂
+
∂∂
tf
tf
tf kkk
( ) ( ) [ ]( )BvEpp×+−−= et
tt
τdd
fielddiff
∂∂
+∂
−=
∂∂
−tfg
tf kkk
τ
[ ]( )k
BE kk
kk
∂∂
⋅×+−∂
−=
∂∂
−fveg
tf
τdiff
c.f. Drude ( ) τtentn −= 0
Thus, despite their extreme starting points, both models consider a steady state solution involving forces acting on charged particles or wave packets with a distribution of velocities.
Current response
A n electrons/unit vol.
v
v
( ) jijii kgekfn EvJ kk σππ
==⇒= ∫∫ 33
33 d
41d
22
( )0d 30 =∫ kfei kv
vvxJ neAneAAI === ˆ
Similarly for thermal current ( ) jijii Tkg ∇−=−= ∫ κµεπ
33 d
41
kvJ
jijii kge EvJ k σπ
== ∫ 33 d
41
∫
∂∂
−⋅=⇒ kfe jiij3
02
3 d 4
1ε
τπ
σ vv
and Eκk ⋅
∂∂
−= vefg τε
0
Consider spherical Fermi surface
d3k
kF
dϕ dkr
dθ kF dθ
∫∫∫∫∫∫
∂∂
−=
∂∂
−=
∂∂
−ππ
ϕϕθεεε 0
2
0
003
0
d sindd dd d FFrFr kkkfSkfkf
dc electrical conductivity
dc electrical conductivity
Recall where cosγ is the angle between vF and kF = 1 for isotropic 3D FS And for most metals, behaves as a δ-function. Hence, Note that this is the same surface integral that appears in the expression for the electronic density of states
rF kdcosd1 γεε vv kkkk
=⇒∇=
( )( ) ∫∫ ∫∫ ∫ =−=
∂∂
−γ
εεδε cos
dd d d d 0
F
FFrFnFr
SSkSkfv
k
∂∂
−ε
0f
( )( )∫ ∇
=kn
Fn
Sgεπ
ε 14d
3
dc electrical conductivity
Hence,
θϕϕτπ
σπ π
d d sin
4
2
0 0
23
23D ∫ ∫
⋅= F
F
jiij k
ve vv
ϕθθϕτπ
ϕθϕτπ
σπ ππ π
∫ ∫∫ ∫ ==2
0 0
2323
22
0 0
23
2
d dcossin4
d d sin 4 FFF
F
xxxx vkek
vvve
22
2
0
32
0
23
22
3
d sind cos4
Fxx
FFxx
ke
vke
πσ
ϕϕθθπ
τσππ
=⇒
=⇒ ∫∫π 4/3
c.f. Drude result: ( )
22
22
2
323
3
2
33*34
22
* FF
FFF kekvek
mek
mne
πτ
πτπ
πτσ =
=
== QED
( )θϕ cossinFx vv =
Assume, isotropic, cylindrical Fermi surface
1cos =⇒ γ
FFF
xx kc
ec
vkeπ
ϕϕπ
τσπ
2d cos
2
22
0
22
2
==⇒ ∫
c.f. Drude result: ( )
F
F
FFF k
ce
kve
ck
mek
cmne
πτ
πτππ
πτσ
22*2
22
*
22222
3
2
=
=
== QED
π
Fxx kc
eπ
σ2
2
=
cπ2
vF || kF
|kF |
∫∫⋅
=−
ππ
π
ϕγ
τπ
σ2
03
22D d
cosd
4 FF
jic
czij k
vke vv
In-plane conductivity for quasi-2D metal
ϕγ
ϕτππ
ϕγ
τπ
σπππ
π
d cos
cos24
d cos
d4
2
0
2
3
22
03
2
∫∫∫
==
−
FFF
F
xxc
czxx
vkc
ekv
vvke
Conductivity in a magnetic field
In metals, the effect of a magnetic field is usually to deviate the trajectory of the carriers from their electric-field induced path. The two major corresponding changes to the electrical conductivity tensor are: (i) the well-known Hall effect emerges as a result of cross terms (σxy etc...) (ii) the longitudinal (diagonal) conductivity also changes - magnetoresistance The Hall effect can be of either sign, depending on the majority or most mobile carrier type. The magnetoresistance, on the other hand, is almost always positive. Why? A definition of magnetoresistance The increase in the longitudinal resistance caused by the additional scattering all mobile carriers experience per unit length in the direction of the applied electric field in the presence of an applied magnetic field.
-e
B
L
X X X X X
Conductivity in a magnetic field
Within the relaxation time approximation; Continuous series Hence e.g. To calculate σxy
(1), simply work through with the applied magnetic field H//z and the electric field E//y and calculate the response Jx with the first-order Jones-Zener equation.
[ ]τε
kkk ggefe −=∂∂
⋅×+
∂∂
⋅⇒k
BvvE kk
κ
0
( ) [ ]
∂∂
−⋅
∂∂
⋅×−=k
κk vEk
Bvε
ττ 0k
nn
kfeeg
( ) [ ] kfeveev kj
n
in
ij3
0
3 d 4
1∫
∂∂
−
∂∂
⋅×−=k
k kBv
εττ
πσ
( ) [ ]k
Bvr
vEv kκkk
κk ∂
∂⋅×+
∂∂
⋅+
∂∂
−=
∇−+∇
−⋅
∂∂
− kkk gegtf
eeT
Tf
coll
0 1 µµεε X X X
Interlude: Taking into account the shape of quasi-2D FS
γ ϕ−γ
ϕ
dϕ
dkF
kF dϕ
γ γ
( )( )[ ]
∂∂
=⇒
= − ϕ
ϕγ
ϕγ F
F
F kk
k lntand
dtan 1
The relation between vF and kF
For B//c: where and Finally, NB (i) σxy
(1) probes the variation of the mean-free-path around the Fermi surface. NB (ii) σxy
(1) in a quasi-2D metal does NOT depend on the carrier density
( ) [ ] kfeveev kyxxy
30
31 d
41
∫
∂∂
−
∂∂
⋅×−=k
k kBv
εττ
πσ
B vk
[ ]//k
BvF ∂∂
=∂∂
⋅×k
Bvk
ϕγ
∂∂
=∂
∂
Fkkcos
//
∫∫∫−
=
∂∂
−ππ
π
ϕγε
2
0
30
dcos1dd F
F
c
cz kkkf
v
( ) ( ) ( )[ ]∫∫ −∂∂
−−
=∂∂−
=ππ
ϕγϕτϕ
γϕτπ
ϕϕ
ππ
σ2
022
32
023
31 dsincos
2d2
4 FFy
xxy vvc
Bec
Be
Hall conductivity in a quasi-2D conductor
//k∂∂
Thus σxx(2) as expected, is negative and scales with B2
Consider isotropic cylinder:
( ) [ ] [ ]
( ) ( )[ ]∫
∫
−∂∂
∂∂
−=
∂∂
−
∂∂
⋅×−
∂∂
⋅×−=
π
ϕγϕϕ
γϕ
γϕπ
ετττ
πσ
2
032
24
30
32
dcoscoscos2
d 4
1
F
kxxxx
kcBe
kfeveeevk
kk kBv
kBv
( )
( ) ( )222
222
23
324
0
2 22
τωππσ
σc
FFFxx
xx
kBe
kec
ckBe
−=−
=⋅−
=⇒
In-plane magnetoresistance in a quasi-2D metal
=
=
F
F
c
keBvmeB
*ω
( ) ( )FF
xx ckBe
ckBe
3
3242
02
2
32
3242
2dcoscos
2
πϕ
ϕϕϕ
πσ
π
−=∂
∂= ∫
In-plane magnetoresistance in a quasi-2D metal
Magnetoresistance MR is zero in strictly isotropic system even though magnetoconductance is always finite
2
0
)1(
)0(
)2(
0
−−=
∆
xx
xy
xx
xx
xx
xx
σσ
σσ
ρρ
( )
( )
( ) ( ) 0
22
22
22
2
22
232
0
=−+=
−+=
⋅−+=
∆
τωτω
τω
ππ
τωρρ
cc
Fc
Fc
xx
xx
keB
kec
cBe
( ) ( )c
Bec
Bec
Bexy 2
232
0
222
232
022
231
2dcos
2dsincos
2
πϕϕ
πϕ
ϕϕϕ
πσ
ππ
==∂
∂−= ∫∫ π
Fxx kc
eπ
σ2
2
=
Kohler’s rule
If the only effect of a change of temperature or of a change of purity of the metal is to alter τtr(k) to λτtr(k), where λ is not a function of k, then ∆ρ/ρ is unchanged if B is changed to B/λ. Since , , the product ∆ρ.ρ (= ∆ρ/ρ . ρ2) is independent of τtr and a plot of ∆ρ/ρ versus (B/ρ)2 is expected to fall on a straight line with a slope that is independent of T (provided the carrier concentration remains constant). Kohler’s rule is obeyed in a large number of standard metals, including those with two types of carriers, provided that changes in temperature or purity simply alter τtr(k) by the same factor. Deviations from Kohler’s rule imply a change in carrier content, possibly a change in dimensionality, or most likely, a change in the variation of τtr(k) around the Fermi surface.
Narduzzo et al., PRL 98, 146601 (07)
( )2τωρρ c∝∆
• Their transition temperatures are anomalously high
• Their superconducting order parameter is unconventional (d-wave)
• The superconductivity emerges out of a highly correlated insulating state
• Their normal metallic state is unlike anything that has been seen before
Tab 10~ ααρ +
Martin et al., Phys. Rev. B (1990)
Bi2+xSr2-xCuO6+δ
Ando et al., Phys. Rev. B (1999)
Bi2+xSr2-xLaxCuO6
Why are high temperature superconductors interesting?
Zaanen Senthil Schmalian Vojta Randeria
“The metallic state at optimal doping embodies the enlightenment. Rather than being complicated, this ‘bad’ metal shows a sacred simplicity – symbolized, for example, by its linear resistivity…”
“Most mysterious is the strange optimally doped metal. There simply do not seem to be (m)any good ideas on how to think about it. Yet it is empircally characterized by simply stated laws.”
“The biggest mystery is the linear resistivity at optimal doping.”
“The difficulties lie with the normal state, featuring phenomena like the apparent linear-in-temperature resistivity at optimal doping.”
“However, many questions about the precise nature of the transition from the superconductor to the Mott insulator, the possibilities of various competing/coexisting ordered states at low doping and, most particularly, the description of the non-Fermi-liquid normal states remain unclear at the present time.”
High temperature superconductivity and Nobel Laureates
J. G. Bednorz (1987) K. A. Muller (1987)
A. A. Abrikosov (2003) P. W. Anderson (1977) P. G. de Gennes (1991)
A. W. Geim (2010) V. L. Ginzburg (2003) A. J. Heeger (2000) H. Kroemer (2000)
R. W. Laughlin (1998) A. J. Leggett (2003) N. F. Mott (1977)
J. R. Schrieffer (1972) F. Wilczek (2004)
The key structural element, the copper-oxide plaquette, appears either in a square planar, or an octahedral arrangement. Hence, the Jahn-Teller effect is not necessarily important in all cuprates.
Crystal and electronic structure
Tl2Ba2CuO6+δ La2-xSrxCuO4
copper
oxygen
apical oxygen
La/Sr
Electronic configuration of Cu 1s2 2s2 2p6 3s2 3p6 3d9 4s2 Electronic configuration of Cu2+ 1s2 2s2 2p6 3s2 3p6 3d9 = 1 carrier/unit cell Electronic configuration of Cu(2-x)+ 1s2 2s2 2p6 3s2 3p6 3d9-x = Hole-doped CuO2 plane
Crystal and electronic structure
(La3+)2Cu2+(O2-)4
(La3+)2-x(Sr2+)xCu(2-x)+(O2-)4
t2g orbitals eg orbitals
At half filling
Mott insulator 3d9 3d9 3d8 3d10 Charge transfer insulator 3d9 3d10 L
+U
O Cu
+U
+∆pd
~ 7-8 eV
+∆pd
Beyond half filling…
Doping away from half-filling strongly frustrates the spin background – leads to a strong suppression of the AFM ordering temperature. Frustration is significantly greater for spin exchange along the coordination axis than diagonally to it. Inherent in-plane anisotropy within the CuO2 plane.
“Nodal-anti-nodal dichotomy” O Cu
+t’
Phase diagram
AFM suppressed with the addition of only a few percent of holes. Superconductivity emerges at a hole concentration p ~ 0.05. Maximum Tc occurs at an optimal doping of p = 0.16. Superconductivity vanishes again around p > 0.30. Dip in Tc also seen around p = 0.125 – so-called 1/8-anomaly. Normal state phase diagram dominated by strange metallic phase and the pseudogap.
Phase diagram
AFM suppressed with the addition of only a few percent of holes. Superconductivity emerges at a hole concentration p ~ 0.05. Maximum Tc occurs at an optimal doping of p = 0.16. Superconductivity vanishes again around p > 0.30. Dip in Tc also seen around p = 0.125 – so-called 1/8-anomaly. Normal state phase diagram dominated by strange metallic phase and the pseudogap.
In underdoped cuprates, possibly an additional temperature scale Tf due to extended region of phase fluctuating superconductivity
Phase diagram – evolution of in-plane resistivity
At low doping, ρab(T) is non-metallic at low T. Superconductivity develops out of insulating ground state.
Note almost exponential decrease in ρab(300) with increased doping, showing how the mobility of doped holes improves with doping.
Optimally doped cuprates
Fxx kc
eπ
σ2
2=
Yoshida et al., PRB 61 R15035 (99)
YBa2Cu3O7 Tl2Ba2CuO6+δ
Tyler + Mackenzie, PhysicaC 282 1185 (1997)
Negative intercepts imply that T-linear resistivity cannot continue to absolute zero – there has to be a crossover to a
higher exponent of T at the lowest temperature
1) T-linear resistivity is observed only in a narrow composition range near optimal doping; the sharp crossover to supralinear resistivity on the OD side being more suggestive of electron correlation effects than phonons.
2) The absence of resistivity saturation in OP La2-xSrxCuO4 up to 1000K argues
against a dominant e-ph mechanism.
3) The frequency dependence of 1/τtr, extracted from extended Drude analysis of the in-plane optical conductivity, is inconsistent with an electron-boson scattering response due to phonons. In particular, Γ(ω) does not saturate at frequencies corresponding to typical phonon energies in HTC.
4) It has proved extremely problematic to explain the quadratic T-dependence of the inverse Hall angle cotθH(T) in a scenario based solely on e-ph scattering.
The problem(s) with phonons
,
Strong T-dependence over a very wide temperature and doping range. What does it signify?
Hall effect in hole-doped cuprates
,
Hwang et al., PRL 72 2636 (94)
Marked, almost 1/T increase in RH(T) could indicate a reduction in carrier number with decreasing temperature (Recall that RH = 1/ne) However, this increase in RH(T) is occurring in range above pseudogap temperature.
Strong T-dependence over a very wide temperature and doping range. What does it signify?
,
Hall effect in hole-doped cuprates
Hwang et al., PRL 72 2636 (94)
Moreover, in the underdoped regime, below T = T*, RH(T) begins to decrease with T in the pseudogap regime. This suggests that RH(T) does not in any way reflect the variation of n(T). Cuprates are also largely single-band.
,
Ando + Murayama, PRB 60 R6991 (94)
Hall effect in hole-doped cuprates
Variation in ρab(T) and 1/RH(T) with Zn-doping in Y123 does not appear to be correlated with one another…
However when data plotted as cotθH(T), it shows a very simple T2 dependence.
The inverse Hall angle
,
Chien et al., PRL 67 2088 (91)
Variation in ρab(T) and 1/RH(T) with Zn-doping in Y123 does not appear to be correlated with one another…
However when data plotted as cotθH(T), it shows a very simple T2 dependence.
The inverse Hall angle
,
2Hxx
xyxy
BBR
σσρ
==
xyσ
xxσHθ
ab
xxxy
xx
xx
xy
BR
BR
ρθ
σσσθ
σσ
θ
HH
HH
H
cot
cot
tan
=⇒
==⇒
=
This so-called separation of lifetimes is most apparent at optimal doping, where ρab(T) ~ T and cotθH(T) ~ T2. Such remarkable transport behaviour is a hallmark of the cuprates that has engaged some of the greatest condensed matter thinkers.
The separation of lifetimes
,
Tyler + Mackenzie, Physica C 282 1185 (97)
Ono + Ando, PRB 75 024515 (07)
The quasi-2D heavy-fermion compound CeCoIn5 shows a similar phenomenon of separation of lifetimes between Tc and Tcoh ~ 20 K.
We are not alone…
,
Nakajima et al., JPSJ 75 023705 (06)
1) The two-lifetime model of Anderson and co-workers
Transport theories – the “Big Three”
[ ]
( )
( )
( ) ( )22
2H
2
)0(
)1(
)0(
)2(
2
HH)1(H
2
H
30H3
3)(
1
1cot1
11
d4
BTAT
BTATT
TTT
kfvk
Bvve
xx
xy
xx
xx
ab
ab
tr
tr
xy
xx
trab
tr
trj
n
kin
ij
+∝∝
−−=
∆⇒
+∝∝∝=⇒∝
∝∝⇒∝
∂∂
−
∂∂
×−= ∫
τσσ
σσ
ρρ
ττττ
σσθ
τ
τρ
τ
εττ
πσ
Kohler’s rule violated all the way up to room temperature
Violation of Kohler’s rule
Harris et al., PRL 75 1391 (95)
T-dependence of in-plane MR extremely well described by Anderson’s two-lifetime model
The non-integer inverse Hall angle
,
Konstantinovic et al., PRB 62 R11989 (00)
Ando + Murayama, PRB 60 R6991 (99)
This non-integer exponent in the inverse Hall angle and its doping-dependence are very hard to capture within this model
2) The marginal Fermi-liquid model of Varma and co-workers
The anisotropic T-independent elastic scattering term γ(k) was argued to originate from small-angle scattering off impurities located out of the plane. ⇒ scattering rate anisotropy reflects that of DOS, i.e. ⇒ Violation of isotropic-l approximation and strong variation in RH(T) and Unfortunately, this model struggles to account for both the in-plane MR and the evolution of the exponents in OD cuprates.
( )kγλτ
+∝ Ttr
1
Cu O La, Sr
( )2
H1cot
∝
tr
Tτ
θ
( ) ( )ϕϕτ Fv
11∝
Carter+Schofield, PRB 66 241102(R) (02)
Transport theories – the “Big Three”
3) Anisotropic single lifetime models
3a) Nearly AFM FL model 3b) Cold spots model 3c) Anisotropic scattering rate saturation model
TTfc
∝∝
1;1 2\
( ) ( )Fτ
ϕϕ 12sin20 +Γ∝Γ
Carrington et al., PRL 69 2855 (92) Stojkovic+Pines, PRL 76 811 (96)
Ioffe+Millis, PRB 58 11631(98)
Hussey, EPJB 31 495 (03) Hussey, JPCM 20 123201 (08)
( ) ( ) ( )
maxidealeff
22
210ideal
1112sin
Γ+
Γ=
Γ
Γ+Γ+Γ=Γ TTϕϕϕ
Transport theories – the “Big Three”
Altogether now!
( ) ( ) ( )
maxidealeff
22
210ideal
1112sin
Γ+
Γ=
Γ
Γ+Γ+Γ=Γ TTϕϕϕ
( )kγλτ
+∝ Ttr
1MFL Two-lifetime
( ) ( )Fτ
ϕϕ 12sin20 +Γ∝Γ
Cold Spots model NAFL
2
H
1;1 TTtr
∝∝ττ
21;1 TTcf
∝∝