brief introduction to superconductivity in the cuprates · 2010. 5. 11. · introduction to...
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Introduction to superconductivity in the
Jules CarbotteMcMaster and CIFAR
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A Famous Discovery!
• 1986J.G. Bednorz and K.A. Müller
• Nobel Prize 1987 – Fastest one ever!
La2-xBaxCuO4
Tc ~36 K
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Breaking the Liquid Nitrogen Barrier!
• 1987
Paul Chu and co-workers
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Based on idea of cooper pairsequal and opposite momentumand spinsPairs overlap in r-space so many
body condensate,all pairs in same wave function
Macromolecule,quantum mechanics at macroscopic level
Coherence length much larger then free electron spacing
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John Bardeen Leon N. Cooper John R. Schrieffer
BCS theory 1957 physics nobel prize 1972
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Aoki cond-mat 0811.1656
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YBaCuO 123
CuO chain
CuO2 plane
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Cu oxide plane is modeled with atoms on square lattice “ a “witheach site filled which corresponds to half filling of BZMeasure doping from half filling as reference. Hole doping. MOTT insulatorin band theory would be a metal
“a”
e- e- e- e-
e- e- e- e-
Send to other none CuO2Plane= hole doping
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“a”
e- e- e-
e- e- e- e-
Hole
Electron cannot hop to occupied site because of Hubbard U [largerepulsive energy] . Can only hop to empty siteso at half filling MOTT insulator
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“a”
e- e- e-
e- e- e- e-
e-
Antiferromagnet has twice the unit cell and half the BZ
e-
e-
e-
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LAFB
Z
Copper –oxygen B Z
At half filling a metal in band theory
Mott insulator because of large Uno double occupancy
Leads to pseudogap
Gutzwiller factors narrow bands and account for reduced coherence Large incoherent background
U-AFBZ
L-AFBZ
[pi/a ,pi/a]
On AFMBZ
New energyscale PG
Superconddome
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Phase Diagram of High TC
T
DOPING
AFPseudogap
Superconducting
Underdoped Region
Presence of both Pseudogap and Superconducting gap.
More doping means more holes
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0.16 0.270.05
Doping level, p
The superconducting domeTe
mpe
ratu
re
TcTc
max
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LSCO: Tcmax = 40K
Y123Bi-2212: Tc
max = 91KTl-2201
0.16 0.270.05
Doping level, p
The superconducting domeTe
mpe
ratu
re
TcTc
max
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LSCO: Tcmax = 40K
Y123Bi-2212: Tc
max = 91KTl-2201
0.16 0.270.05
Doping level, p
The superconducting domeTe
mpe
ratu
re
TcTc
max
Optimally doped (OPT)
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LSCO: Tcmax = 40K
Y123Bi-2212: Tc
max = 91KTl-2201
0.16 0.270.05
Doping level, p
The superconducting domeTe
mpe
ratu
re
TcTc
max
Optimally doped (OPT)
Underdoped (UD)
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LSCO: Tcmax = 40K
Y123Bi-2212: Tc
max = 91KTl-2201
0.16 0.270.05
Doping level, p
The superconducting domeTe
mpe
ratu
re
TcTc
max
Optimally doped (OPT)
Underdoped (UD) Overdoped (OD)
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Phase diagram of the cuprates
Basov and Timusk, Rev. Mod. Phys 77, 721 (2005)
Doping level, p
Tem
pera
ture
Non-FLPseudogap
AFMFL
d-SC Tc
T*
0.05 0.270.160.0
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Phase diagram of the cuprates
Basov and Timusk, Rev. Mod. Phys 77, 721 (2005)
Doping level, p
Tem
pera
ture
Non-FLPseudogap
AFMFL
d-SC
LSCO
Bi-2212Tl-2201
Y123
Tc
T*
0.05 0.270.160.0
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At zero temperature, no absorption till 2Δ, one Δ to pull an electron out of condensate and one more when it is placed back in.This process blocks states that can no longer be used to form condensate
condensate
Takes energy gap to pull an electron out of condensate or to put one in
Macromolecule, all electrons bound together
photon
Creates 2 excitations
Process requires twice gap
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Specific Heat of Al
Phillips, Phys. Rev. 114, 67 (1959)
Superconducting State
Normal State
Classic BCS with s-wave gap
Note the exponential drop at low temperature and a jump at Tc
~T
Tc
Second order phase transition
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Because of gap, takes energy delta Δ to release an electron from condensate and make an excitation [quasiparticle].
Specific heat is exponentially activated at low temperature.
Note 1/T dependence, still exponential dominates at low T.
Exponential activation
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Kirill Samokhin,Brock University
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Density of electronic states in s- and d-wave superconductor
In d-wave distribution of gaps from 0to maximum gap
Just depression ,no real gap
cos[2θ]
s
d
Inverse square root singularity
Weaker log singularity
+
-
--
+
linear
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...
.
.
.
..
..
..
..... .
......
.
S-wave few excited electrons
D-wave more excited electronsonly around nodes
Temperature creates excitationsout of ground state
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Specific heat for many dopings
0.13 to 0.21
0.08 to 0.23
Increasing doping
Loram et. al. PRB 69,060502 [2004]
Specific heat gammaIs C{T}/T
Linear in T
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Cuprates are near half filling for the CuO2 Brillouin ZoneIn band theory this would be a metalBecause of MOTT physics its an insulator at half fillingMott physics
e -
e -
e -
e - e -
Hopping to empty siteis okHopping to filled siteis energetically not favorable because of Hubbard Ubig on site repulsionNO double occupancy
Empty state, hole doping
Lattice parameter a
e-
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Use the model of Yang, Rice and Zhang [YRZ] PRB73 ,174501 [2006] based on RVB resonating valence bond, spin liquid, has a quantum critical point [QCP]at doping x=0.2 where a pseudogap develops in the electronicstructure as MOTT insulator is approached
Illes et.al. PRB 79 ,100505 [2009]
Pseudo gap
Superconducting gap
Pseudo gap modifies electronic structureFermi surface reconstruction
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Use the model of Yang, Rice and Zhang [YRZ] PRB73 ,174501 [2006] based on RVB resonating valence bond, spin liquid, has a quantum critical point [QCP]at doping x=0.2 where a pseudogap develops in the electronicstructure as MOTT insulator is approached
Illes et.al. PRB 79 ,100505 [2009]
Pseudo gap
Superconducting gap
Pseudo gap modifies electronic structureFermi surface reconstruction
QCP x=0.2
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Usual large Fermi surface of Fermi liquid theoryfor tight binding bands near half filling
Top right corner of Two-D Cu-O2B.Z.
Reconstructed Fermi surface due to pseudogapand approach to Mott Insulator ; metallicityis reduced
Luttinger hole pocket,small fermi surfacefront is weighted order 1, back little weightFewer zro energy excitations
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For x=0.19 can have holes and electron pockets [near B Z boundary]
Electron pocketNegative energy
Hole pocketPositive energy
Gaped connecting contour
Strongly aware of AFBZ
Weakly perturbedBy pseudogap
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Second energy scale associated with Mott transition to insulating state
Density of states N[w]doping
Look here,newEnergy scale
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Pseudogap does not changeLow temperature law or its slope
Linear ,NO change
Gama is specific heatover temperature
Of course ,the gutzwiller coherence factor will come in additionally
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Dirac point isonly active spotat low temperature
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Optical Properties in BCS
photon
Metal surfaceIncident on metalsurface
Can get:• reflection• absorption• transmission
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In conventional superconductors, tunneling has been method of choice to get information on gap and phonons
Optics has been hard, good metals reflectance near 1
In poor metals such as oxides, optics has been great!
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Reflectance is an experimentally measured quantity
From it can get optical the conductivity as a function of energy
Has real and imaginary part
Real part is absorptive part
Interested in conductivity in energy range of gap and phonon energies: far infrared
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DRUDE model
DC value
electron mass
velocity
elastic scattering time
electron charge
electric field
n: electron density
Drude Conductivity
No damping term no absorption
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Real [left] and imaginary [right] part of DRUDE conductivity
Width at half maximum is optical scattering rate 1/τ − here it is 1.0
Plasma frequency ωp
Absorptive part
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At zero temperature, need one Δ to pull an electron out of condensate and one more to place it back in.This process blocks states that can no longer be used to form condensate
condensate
Takes energy gap to pull an electron out of condensate or to put one in
Macromolecule, all electrons bound together
photon
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Real part of conductivitys-wave superconductor
Missing area goes into a delta function at origin Optical spectral weight conserved
Fairly dirty case
Nam, Phys. Rev. 156, 487 (1967)
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Comparison of real part of conductivity in s- and d- wave BCS at zero temp.
E. Schachinger and J.P. Carbotte, Models in Methods of High-Tc Superconductivity, Vol 2, Edited by J K Srivastava and S M Rao, pp73-169
No real gap
Real gap
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0ptical conductivity has real and imaginary part Real part is absorptive part
In superconducting state, imaginary part is related tothe penetration depth
Free space superconductor
Magnetic field decays on lengthscale of penetration depth
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Low temperature behaviour of superfluid density in s- and d-waves-wave is exponentially activatedd-wave is linear in temperature
s-wave
d-wave
Inverse square of London penetration depth is proportional to superfluid density
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Comparison of London penetration depth for s- and d-wave symmetry in BCS
Experimental data in YBCO: D. A. Bonn et al, PRB 50, 4051 (1994)
Penetration depth is distance an external magnetic field can penetrate into a superconductor [screening supercurrents are set up]
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Highly underdoped orthoII YBCOPure d-wavelinear in Tat low temp
Dirty d-waveCrossover to T**2due to scattering
Crossover from linear to quadraticHuttema et.al. PRB 80,104509 [2009]
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1/λ2(T) For Various Dopings
x = 1.2xopt
x = xopt
x = 0.9xopt
x = 0.8xopt
x = 0.7xopt
x = 0.6xopt
x = 0.5xopt
This trend is seen in experimentsee Anukool et al. (Cambridge)PRB 80, 024516 (2009)
Fisher et.al. G-McM-group
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Raman in d-wave superconductor
Depends on polarization of the light , nodal, antinodal are different
Photon in
Photon out
Electron hole- particle pair created
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Raman scattering
Different polarization of light ,have different sampling factors [images different parts of k-space]
B1g samples most antinodal and B2g nodal direction
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Le Tacon el.al. Nature Physics 2 ,537[2006]
Peak energy scaleup
Peak energy scaledown
Less doping ,more MOTT
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YRZ theory of underdoped cupratesLeblanc et.al. PRB 81,064504 [2010]
G-McM-group
Both scales are partof YRZ modelNo pseudogap in nodaldirectionCan dominate anti-nodal direction
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Hufner et. Al. Rep. Prog. Phys. 71, 062501 [2008]
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There are two gaps . Superconducting gap
and a normal state gap associated with loss of
metalicity as Mott transition to insulating state
is approached
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Hard to escape there are two gaps in underdoped cupratesone superconducting gap ,the other a pseudogap associated with Mott physics
Mott physics
e -
e -
e -
e - e -
Hopping to empty siteis okHopping to filled siteis energetically not favorable because of Hubbard Ubig on site repulsionNO double occupancy
Empty state, hole doping
Lattice parametera
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Angular resolved photo emission ARPES
Photon in, electron oute-
photon
Measures electron dispersion curve
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Luttinger contour no gap
Extended contour: gaped part
A F B Z
Fermi contourof nearestapproachARPES
Measure along red contour and front part of luttinger fermi surface ,back has little weight
Lightly weighted side
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Kondo et. al. Nature, 457,296 [2009]
ARPES
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YRZ theory applied to ARPES ,Leblanc et.al. Phys. Rev. B 81, 064504 [2010]
Total is square root of sum of squares of pseudogap [na ]and superconducting gap
superconducting gap
pseudogap
combination
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Chatterjee et.al. Nature Physics 6,99 [2010]
Highly underdoped Bi2212 NO sign of second gap scale!or arcs around 45 degree from luttingr pockets
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END
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For x=0.19 can have holes and electron pockets [near B Z boundary]
Electron pocketNegative energy
Hole pocketPositive energy
Gaped connecting contour
Strongly aware of AFBZ
Weakly perturbedBy pseudogap
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Hot spot
Contour nearest approach
ARPES measures dispersion curves for occupied states. Can “see” if there are states of zero energies [real Fermi surface] .
gap
No gap
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x = 0.20
x = 0.16
Dirac point is only active spot at low temperature
Dirac point
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Kamerlingh-Onnes1911
Discovery of superconductivity 1911
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Temperature excites electrons out of fermi sea,create particle hole excitations
Number is N[0] *T*TInternal energy U changegoes like above and
Specific heat like T
In s-wave superconducting state there is a gap and so exponential activation
In a d-wave superconductor have distribution of gaps and DOS N[w] is linearin w so U goes like T**3 and specific heat like T**2
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I NIS
Vcondensate
Normal NSuperconducting S
Process requires only one gap
Compare with tunneling
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Hard to miss second gap ,perhaps seen best in c-axis opticsIt is there in normal state above Tc
Flat incoherent behavior
D Basov-group