Happy Birthday Eugene!
Wishing you great beach-soccer games with good friends till 120 !
Field-induced Quantum Critical Route to a Fermi Liquid in High-Tc Superconductors
CUNYMarch 13, 2009
L. Krusin-ElbaumIBM T.J. Watson Research Center, Yorktown Heights, New York
Strange metal
Antecedent states of matter that become unstable in favor of high Tc – commonly referred as the `normal state’
Key to the origins of high Tc
Phase diagram
Pseudogap phenomenon: friend or foe?
In the `foeIn the `foe’’ views views QCPQCP
• d-density wave (staggered flux state: Laughlin, Chakravarty, Lee, etc.)
• Loop-current order (broken time-reversal symmetry: C. Varma)
• Mixed order parameters (dx2-y2+idxy, dx2-y2+is, etc.)
• RVB: Strange metal beyond the pseudogap energy scale no need for QCP (P.W. Anderson)
Strange metalAF
PG
`Normal’ state Many states, many transitions
CUNYMarch 13, 2009
Latest: Some evidence for the time-reversal symmetry breaking in the PG phase
Novel magnetic order in the pseudogap phase of YBa2Cu3O6+x and HgBa2CuO4+δBourges et al., PRL 96, 197001 (2006); condmat/arXiv:0805.2959 (2008), also Mook et al, 2008.
neutrons
Polar Kerr effect
Spontaneous Kerr rotation in zero field near PG in YBa2Cu3O6+x Kapitulnik et al., PRL 100, 127002 (2008) CUNY
March 13, 2009
•Quantum vortex liquid in La2-xSrxCuO4L. Li et al., Nature Phys. 3, 311 (2007).
The Debate: A Friend, Perhaps?
Vortices and pseudogap•Nernst effect in La2-xSrxCuO4
Z. A. Xu et al., Nature 406, 486 (2000).
•THz conductivity in Bi2Sr2CaCu2O8+y J. Corson et al., Nature 398, 221 (1999).
Vortex-like excitations (superconducting fluctuations) exist above Tc.
Pseudogap - an ultimate upper limit to the vortex state?
@ 60 T and above?
CUNYMarch 13, 2009
Phase diagram deduced from ARPES & transportBi-2212
A. Kaminski, S. Rosenkranz, H. M. Fretwell, Z. Z. Li, H. Raffy, M. Randeria, M. R. Norman, and J. C. Campuzano, Phys. Rev. Lett. 90, 207003 (2003)
coherent metal phase @ low-T & high hole doping p two well defined spectral peaks in ARPES (due to
coherent bilayer splitting, superlinear ρincoherent metal phase @ high-T & low p linear ρ, single broad feature in ARPES
crossover @ TxCUNYMarch 13, 2009
Ultrahigh magnetic fields to kill superconductivity and to examine ‘normal’ state far on the overdoped side of the dome
We have our big hammer:
Searching for: Transformation into a conventional metal Quantum (or not) phase transition(s) between `normal’ states of matter antecedent to high-Tc
CUNYMarch 13, 2009
Pseudogap Closed by Zeeman Splitting
The right-hand-side translates onto the Zeeman energy scale on the left-hand-side as (gµB/kB)H.
Hpg and T* obtained separately in the same crystals in the overdoped regime, give a scaling gµBHpg = kBT* with g = 2.0 (inset).
T. Shibauchi, L. Krusin-Elbaum, M. Li, M.P. Maley, and P.H. Kes, Phys. Rev. Lett. 86, 5763 (2001)
200
150
100
50
0
µ 0H
pg (
T)
2001000T* (K)
gµBHpg = kBT*
700
600
500
400
300
200
100
0
T o
r (g
µ B/k
B)H
(K
)
0.250.200.150.10p
500
400
300
200
100
0
µ0 H
(T)
Hpg
Tc
T*Hsc
•Pseudogap closing field Hpgdecreases with doping
•Zeeman scaling gµBHpg = kBT*holds → suggesting spin-singletcorrelations in forming the pseudogap
•Peak field scales with Tc(p)
CUNYMarch 13, 2009
Tc vs doping `dome’ Tc/Tc
max = 1-82.6 (p-0.16)2
c-axis resistivity ρc: a powerful probe of the pseudogap•Intrinsic tunneling junctions along the c axis (layered structure with large anisotropy)
• ρc probes the low-energy DOS in the bulk
•Recovery of the DOS by magnetic field →
negative interlayer magnetoresistance (MR)
•Pseudogap closing field Hpg = H* can be evaluated
T. Shibauchi et al., Phys. Rev. Lett. 86, 5763 (2001); Phys. Rev. B 67, 064514 (2003).
L. Krusin-Elbaum, T. Shibauchi, C. H. Mielke, Phys. Rev. Lett. 92, 097005 (2004).
T. Watanabe et al., Phys. Rev. Lett. 84, 5848 (2000).
Bi2Sr2CaCu2O8+y
T* from the deviation from the T-linear metallic dependence consistent with the tunneling spectra and the static susceptibility.
T = 110 K
ρc sensitive to the (π,0) points (`hot spots’) of the Fermi surface, where the pseudogap first opens up
CUNYMarch 13, 2009
2.5
2.0
1.5
1.0
0.5
0.0
ρ c (
Ω c
m)
6050403020100
µ0H (T)
2.5 K
20 K
3.5 K
50 K
40 K
30 K
10 K7.5 K
5 K
4.2 K
H0ρ
Hsc
2.0
1.5
1.0
0.5
0.0
0.1 1 10 100
40 K
ρcn
Hpg H0ρ
Hsc
1.34
1.32
1.30
1.28
1.26
1.24
1.22
1.20
ρ c (
Ω c
m)
120110100908070
T (K)
T*T*T*
0 T30 T
58.5 T
OD (Tc = 67 K)
0.4
0.3
0.2
0.1
0.0
∆ρc
(Ω c
m)
100806040200µ0H (T)
T (K)62.270.481.895.4
Hpg
b
4
3
2
1
0
ρ c (
Ω c
m)
25020015010050
T (K)
-10
-8
-6
-4
-2
0
MR
@ 31.2 T
(%)T*
OD (Tc = 78 K)
0 T
∆ρc a
T. Shibauchi et al., Phys. Rev. B 67, 064514 (2003).
ρc(H,T) in Bi2Sr2CaCu2O8+y crystals
T. Shibauchi, L. Krusin-Elbaum, M. Li, M.P. Maley, and P.H. Kes, PRL 86, 5763 (2001).
CUNYMarch 13, 2009
Field anisotropy of pseudogap closing field
L. Krusin-Elbaum, T. Shibauchi, C. H. Mielke, Phys. Rev. Lett. 92, 097005 (2004).
Hpgab / Hpg
c = 1.35 ± 0.1Anisotropy of g-factor [T. Watanabe et al., Phys. Rev. Lett. 84, 5848 (2000)]
gc /gab = 1.3
(χc(T) ~ 1.6 χab(T))
Zeeman scaling
gcµBHpgc = gabµBHpg
ab ~ kBT*
Triplet excitation @high H overcomes spin-singlet correlations responsible for the gap in the spin spectrum and orbital contribution is very small.
χc/χab=(gc /gab)2
CUNYMarch 13, 2009
M. R. Norman et al., Adv. Phys. 54, 715 (2005).
Phase diagram
CUNYMarch 13, 2009
Y. Kubo et al., Phys. Rev. B 43, 7875 (1991).
D. N. Basov and T. Timusk, Rev. Mod. Phys. 77, 721 (2005).
ρ(T)=ρ(0)+AT2
Fermi liquid metal
ρ(T) ~ Tn
1n2
Non-Fermi liquid
polycrystals
Phase diagram
Heavily overdoped Tl2Ba2CuO6+x
-How n-FL state transforms into FL state ?-Magnetic field effect?
M. R. Norman et al., Adv. Phys. 54, 715 (2005).
CUNYMarch 13, 2009
Can We Get to a Coventional Fermi Liquid by Applying Magnetic Field?
0.040
0.035
0.030
0.025
0.020
0.015
ρ c (Ω
cm
)
6004002000
T1.3
(K1.3
)
µ0H = 0 T
0.04
0.03
0.02
0.01
0.00
ρ c (Ω
cm
)
100806040200T (K)
µ0H = 0 T
Tl2Ba2CuO6 (Tc~15 K)
ρc = ρc0 + A0T
2 +CT
At zero field,ρc(T)-ρc0 ~ Tn n=1.3
orρc(T)= ρc0 + A0T2+CT
Non-Fermi liquid(strange metal)
c-axis longitudinal magneto-transport(less afflicted with orbital contributions)
M. Abdel-Jawad et al., Nat. Phys. 2, 821 (2006).
ρab(T) = ρab0 + AT2+CT
CUNYMarch 13, 2009
0.040
0.035
0.030
0.025
0.020
0.015
ρ c (Ω
cm
)
6004002000
T1.3
(K1.3
)
µ0H = 0 T
0.04
0.03
0.02
0.01
0.00
ρ c (Ω
cm
)
100806040200T (K)
µ0H = 0 T
Tl2Ba2CuO6 (Tc~15 K)
ρc = ρc0 + A0T
2 +CT
M. Abdel-Jawad et al., Nat. Phys. 2, 821 (2006).
ρab(T) = ρab0 + AT2+CT
~T2 ~T
isotropic anisotropic
1/τ
At zero field,c-axis longitudinal magneto-transport(less afflicted with orbital contributions)
1/ζcτ
T & momentum dependence of transport scattering rate τ
2 channels: conventional ( T2)anisotropic ( T) , same symmetry as d-gapCUNY
March 13, 2009
0.04
0.03
0.02
0.01
0.00
ρ c (Ω
cm
)
100806040200T (K)
µ0H = 0 T
Tl2Ba2CuO6 (Tc~15 K)
µ0H = 45 T
ρc = ρc(0) + A(45 T)T
2
ρc = ρc0 + A0T 2
+CT
0.040
0.035
0.030
0.025
0.020
0.015
ρ c (Ω
cm
)
6004002000
T1.3
(K1.3
)
µ0H = 0 T
Field-induced transformation from the n-FL to FL state
c-axis longitudinal magneto-transport(less afflicted with orbital contributions)
Measured in a 45-T hybrid magnet in Tallahassee (NHMFL)
At zero field,ρc(T)-ρc0 ~ Tn n=1.3
orρc(T)= ρc0 + A0T2+CT
Non-Fermi liquid(strange metal)
At 45 T (H//c),ρc(T)= ρc(0)+A(45 T)T2
Fermi liquid
CUNYMarch 13, 2009
40
35
30
25
20
15
ρ c (m
Ω c
m)
1000080006000400020000
T 2
(K2)
45 T 40 T 35 T 30 T 25 T 20 T 15 T 11.5 T
TFL
-6
-4
-2
0
2
ρ c -
ρ c(0)
-A
(H)T
2 (
µΩ c
m)
1000080006000400020000T
2 (K
2)
TFL
above TFL
~Tn (n<2)
remnants of the non-Fermi liquid behaviorbelow TFL
A(H)T2
Fermi liquid behavior with field dependent A(H)
CUNYMarch 13, 2009
40
35
30
25
20
15
ρ c (m
Ω c
m)
1000080006000400020000
T 2
(K2)
25 T
-6
-4
-2
0
2
ρ c -
ρ c(0)
-A
(H)T
2 (
µΩ c
m)
1000080006000400020000T
2 (K
2)
TFLabove TFL
~Tn (n<2)
remnants of the non-Fermi liquid behaviorbelow TFL
A(H)T2
Fermi liquid behavior with field dependent A(H)
CUNYMarch 13, 2009
40
35
30
25
20
15
ρ c (m
Ω c
m)
1000080006000400020000
T 2
(K2)
45 T 40 T 35 T 30 T 25 T 20 T 15 T 11.5 T
TFL
-6
-4
-2
0
2
ρ c -
ρ c(0)
-A
(H)T
2 (
µΩ c
m)
1000080006000400020000T
2 (K
2)
TFL
above TFL
~Tn (n<2)
remnants of the non-Fermi liquid behaviorbelow TFL
A(H)T2
Fermi liquid behavior with field dependent A(H)
CUNYMarch 13, 2009
100
80
60
40
20
0
T (
K)
403020100
µ0H (T)
SC
FL
n-FL
HQCP
6
5
4
3
2
1
0
A (
µΩcm
/K2 )
50403020100µ0H (T)
A0
HQCP
TFL
HFL
Quantum critical point @ HQCP ~ Hc2
in the T 0 limit normal state above Hc2 is aFermi liquid- consistent with recent observation of Wiedemann-Franz law C. Proust et al., PRL 89, 147003 (2002)
H-T diagram of the normal state in OD Tl-2201deviation from T2
CUNYMarch 13, 2009
0.04
0.03
0.02
0.01
0.00
ρ c (Ω
cm
)
403020100µ0H (T)
100 K 70 K 50 K 30 K 20 K 15 K 5.0 K 1.5 K0.56 K
Hsc
Hirr
H // c3
2
1
0
δρc
(mΩ
cm
)40302010
µ0H (T)
70 K
HFL
1.5 K
5.0 K
10 K
20 K
30 K
50 K
(b)(a)
Field dependence of ρcδρc: deviation from linear magnetoresistance
Above Hsc~8 T, superconductivity is destroyed (normal state).Above HFL, longitudinal magnetoresistance is linear in H.
0.1
1
10
100
µ 0 H (
T)
60 50 40 30 20 10 0 T (K)
H 0 ρ
H sc
H pg
H sp
vortex liquid
2D vortex solid
3D vortex lattice
T. Shibauchi et al., Phys. Rev. B 67, 064514 (2003).
Overdoped Bi2Sr2CaCu2O8+y
~Hc2(0)
Hc2(0)
CUNYMarch 13, 2009
100
80
60
40
20
0
T (
K)
403020100
µ0H (T)
SC
FL
n-FL
HQCP
6
5
4
3
2
1
0
A (
µΩcm
/K2 )
50403020100µ0H (T)
A0
HQCP TFL
HFL
ρc(T)= ρc(0)+ A(H)T2
Divergent behavior of A(H)near Hc2(0)
A(H) = D (H-HQCP)-α
α=0.62, HQCP=7.4 T
Kadowaki-Woods relation(Aγ2) suggests enhanced
mass near QCP.
Field-induced QCP (2nd order transition at T = 0 K)
Crossover line at finite temperatures
deviation from T2
deviation from H-linear
CUNYMarch 13, 2009
Kohler plot of normal-state MR against µ0H/ρcn(0) ζcτ
Temperature dependent violation of the Kohler’s rule
ρcn(0) is the normal-state, zero-field ρc(T)
Exp.
isotropicζcτsimulations
4-fold basal-plane anisotropicζcτConventional FL scaling of all curves in the Kohler plot
At high T , Kohler rule can be violated by other mechanisms(e.g., orbital effects)
we observe consistent deviation at low T below HFL intrinsic effect, not an artifact of ζcτ CUNY
March 13, 2009
Linear c-axis magnetoresistance: a quantum phenomenon in the
Fermi liquidLinear high-field transverse MR:
old storyMR in Bi up to 32 T, Kapitza, 1928
“the linear Kapitza law”
Quantum Magnetoresistance QMR:a consequence of only one Landau band filled
MR in AgTe chalcogenides,R. Xu, A. Husmann, T.F. Rosenbaum, M.-L. Saboungi, J.E. Enderby, and P.B. Lttlewood, Nature 390, 57 (1997)
QMR in layered semimetals: condition Larmour frequency ħeH/mc >>Tdistance between lowest Landau bands >T and > EF in the lowest band; small hopping between layers, small charge densities n, large hopping mass M, mass in the layers m,
n (M/m)1/2(eH/ħc)3/2
Semimetals, high-Tc cupratesA.A. Abrikosov, Phys. Rev. B 60, 4231 (1999); Eur. Phys. Lett. 49, 789 (2000)
Linear high-field c-axis QMRnew story MR in layered cuprates
for ; ni doping
coherent & incoherent resonant tunnelingthrough two centers, (d/α)1/2 = coher. dist. quantum interferenceLandau level split with H;m
c-axis
@ high H
one can not expect orbital quantization effects, but this happens when only lowest Landau level contributes to tunneling
A.A. Abrikosov, Phys. Rev. B 61, 5928 (2000)
CUNYMarch 13, 2009
Summary so far• Overdoped Tl2Ba2CuO6+x still shows
some remnants of non-Fermi liquidbehavior. ρc(T)-ρc0 ~ T1.3 or = A0T
2+CT
• At high fields, standard Fermi liquidbehavior recovers at low temperatures.ρc(T)= ρc(0)+A(H)T2
• Fermi liquid coefficient A(H) shows a divergent behavior near Hc2(0), suggesting field-induced quantum criticality.
• It bears a striking resemblance to that in heavy-fermion superconductorCeCoIn5, suggesting a common underlying physics in these strongly correlated electron systems.
100
80
60
40
20
0
T (
K)
403020100
µ0H (T)
SC
FL
n-FL
HQCP
6
5
4
3
2
1
0
A (
µΩcm
/K2 )
50403020100µ0H (T)
A0
HQCP
T. Shibauchi, L. Krusin-Elbaum, M. Hasegawa, Y. Kasahara, R.Okazaki, and Y. Matsuda, Proc. Natl. Acad. Sci., 105, 7120 (2008)
Heavy fermionsCeMIn5 (M = Rh, Co)
High-Tc cuprates
CUNYMarch 13, 2009
Summary so far• Overdoped Tl2Ba2CuO6+x still shows
some remnants of non-Fermi liquidbehavior. ρc(T)-ρc0 ~ T1.3 or = A0T
2+CT
• At high fields, standard Fermi liquidbehavior recovers at low temperatures.ρc(T)= ρc(0)+A(H)T2
• Fermi liquid coefficient A(H) shows a divergent behavior near Hc2(0), suggesting field-induced quantum criticality.
• It bears a striking resemblance to that in heavy-fermion superconductorCeCoIn5, suggesting a common underlying physics in these strongly correlated electron systems.
Heavy fermionsCeMIn5 (M = Rh, Co)
High-Tc cuprates
T. Shibauchi, L. Krusin-Elbaum, M. Hasegawa, Y. Kasahara, R.Okazaki, and Y. Matsuda, Proc. Natl. Acad. Sci., 105, 7120 (2008).
CUNYMarch 13, 2009
So How About the Doping Dependence ?
n-FL to Fl transition: Doping dependence
CUNYMarch 13, 2009
10 100
0.01
0.1
1
n =
2
[ρab
- ρ
0] (10
-3Ω
cm
)
T (K)
ABCDE
Tl-2201 n = 1
100 75 50 25 0
1.0
1.5
2.0 n
Tc (K)
ρ = ρ0 + AT n
Y. Kubo et al., Phys. Rev. B 43, 7875 (1991)
L. Krusin-Elbaum et al., subm. Nature (2009)
n-FL to Fl transition: Scaling
CUNYMarch 13, 2009
• vertical phase boundary @ T = 0
• from Clausius-Clapeyron relation it requires∆S ( H/T) = 0 across the transition
• consistent with 2nd order quantum phase transition
• magnetic susceptibility ξ decreases in the PG state
• ground state E on the PG state [@ H = 0, in simple MFT]
E N(0)(T*)2
• then the field that kills the PG is given by
½ ξ (H*)2 N(0)(T*)2, where ξ µB2N(0).
• the relation µB H*(p) T*(p) followed down to T*(pc0) 0 defines QCP @ H = 0.
L. Krusin-Elbaum et al., subm. Nature (2009)
n-FL to Fl transition: Field-Doping dependence
CUNYMarch 13, 2009
• Fermi liquid regime becomes dominant with heavier hole (over)doping
0.24 0.26 0.280.0
0.5
1.0
1.5
2.0
2.5
3.0
TFL
(H)
slope
dT/d
H | F
L (
K/T
)doping p
L. Krusin-Elbaum et al., subm. Nature (2009)
n-FL to Fl transition: Critical fluctuation regime
CUNYMarch 13, 2009
•Tx nearly vertical n-FL regime is not ‘wedge’-like
• px(H) is field-linear, consistent with scaling of H* & T*
scaling
• dome shrinks with increasing H• n-FL wedge shifts to the left• QCP shifts to the left• at fixed doping TFl increases with increasing H• at fixed T HFL increases with doping• consistency with our diagram ?
T
doping
increasing H
n-FL
FL
• two possibilities: wedge or s-shape
V. Aji & C.M. Varma, PRL 99, 067003 (2007)
T
doping
n-FL
FL
QCP & field dependence of n-FL
Tx (~ζx) ~ τc-1 exp(-b/p-pc)
CUNYMarch 13, 2009
Where Are We Now?• By killing superconductivity with strong magnetic in heavily doped cuprate
superconductors we can map both onsets of the pseudogap T* and Fermi liquid TFL in a previously unexplored regime.
• T* and TFL converge in the T 0 limit @ a critical doping pc0 QCP
• Scaling properties of T*(p,H) and TFL(p,H) demonstrate that `strange metal’state in between T* and TFL is governed by a QCP.
• In magnetic field the QCP shifts toward lower doping with the concurrent suppression of Tc quantum critical fluctuations & superconductivity are intimately linked.
• Normal state competing order to high Tc.
• As far as theory the devil is in the detail !!!!
CUNYMarch 13, 2009
Thanks!
CollaboratorsTakasada Shibauchi (Dept. of Physics, Kyoto University, Japan)
Yuichi Kasahara, Ryuji Okazaki (students)
Yuji Matsuda________Masashi Hasegawa (Dept. of Mater. Sci. & Eng., Nagoya University, Japan)________
Chuck Mielke, Ross McDonald (NHMFL Los Alamos, USA) – 60-65 T pulsed magnets
Bruce Brandt (NHMFL Tallahassee, USA) – 45 T hybrid dc magnet
Sudip Chakravarty (UCLA), Chandra Varma (UC Riverside), Mike Norman (Argonne), Manfred Sigrist (ETH-Zurich), Hiroshi Kontani (Nagoya U.)
acknowledgements:
CUNYMarch 13, 2009
J. Paglione et al., Phys. Rev. Lett. 91, 246405 (2003).
A. Bianchi et al., Phys. Rev. Lett. 91, 257001 (2003).
Field-induced QCP in quasi-2D heavy fermion superconductor CeCoIn5
M. A. Tanatar et al., Science 316, 1320 (2007).
100
80
60
40
20
0
T (
K)
403020100
µ0H (T)
SC
FL
n-FL
HQCP
6
5
4
3
2
1
0
A (
µΩcm
/K2 )
50403020100µ0H (T)
A0
HQCP
overdoped Tl2Ba2CuO6+x
The similarity suggests a common underlying physics (likely of magnetic origin)in these strongly correlated electron systems.
CUNYMarch 13, 2009
Field-induced antiferromagnetic order in cupratesLa2-xSrxCuO4 (x = 0.10)
underdopedLa2-xSrxCuO4 (x = 0.163)
Tl2Ba2CuO6+δ (Tc = 85 K)slightly overdoped
B. Lake et al., Nature 415, 299 (2001).
B. Lake et al., Science 219, 1759 (2001).
K. Kauyunagi et al., PRL 90, 197003 (2003).
spatially resolved NMR
AF vortex cores
Field induces `striped' AF order.
Vortex state -- inhomogeneous mixture of a SC spin fluid and a material containing a nearly ordered AF
7.5 T
0 T
neutrons
optimally doped
CUNYMarch 13, 2009
Phase diagram deduced from STEMBi-2212
A. Gomes et al., Nature 447, 569-572 (2007). all still far from the end of the dome !!!
Gap evolution with doping Gap map – sort of
CUNYMarch 13, 2009
0
100
200
300
400
0.05 0.10 0.15 0.20 0.25 0.30 0.350
100
200
300
400
500
doping
Tem
pera
ture
Tc
Hsc
Hpg
Bi-2212Tl-2201
µ 0H (
T)
hole concentration p
T*
strange metal
Fermiliquid
pseudogapped metal
T (
K)
Bi-2212Tl-2201
a
Close look at the phase diagram in the OD regime
Hsc ~ 1.4 Tc
but
Zeeman scale
gµBHpg = kBT*
0 100 2000
50
100
150
Bi-2212 Tl-2201
2µBH
pg = k
BT*
µ 0Hpg
(T
)
T* (K)
H // c
CUNYMarch 13, 2009
T = 0.6 Tc
Doping dependence in the heavily OD regime
CUNYMarch 13, 2009
H-T diagram of the pseudogap state in BSCCO
1
10
100
µ 0H
sc (
T)
1.00.80.60.40.20.0
T / Tc
UD (Tc = 68 K)UD (Tc = 90 K)OD (Tc = 78 K)OD (Tc = 67 K)
120
100
80
60
40
20
0
µ 0H
(T
)
120100806040200
T (K)
Hsc
Hpg
OD (Tc = 67 K)
Hirr
(a) (b)
`flat’ temperature dependence of Hpg
exponential T dependence of peak field Hsc
H-T diagram showing the pseudogap closing field Hpg , and two characteristic fields of the superconducting state: the field Hsc [close, but below Hc2, at which quasiparticle tunneling overtakes Josephson (Cooper pair) tunneling] and the irreversibility field Hirr .
CUNYMarch 13, 2009
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