Introduction to the phenomenology of HiTc

superconductors.

Patrick Lee and T. Senthil

MIT

1. Basic physics: doped Mott insulator. (Early sections in Lee, Nagaosa and Wen, Rev Mod Phys,78,17(2006) and Lee, Reports of Progress in Physics, 71,012501(2008))

2. Introduction to experimental methods.Thermodynamic measurements: specific heat, spin susceptibility.

Transport: resistivity, Hall, magneto-resistance (angle dependence ADMR).

thermo-power, thermal conductivity.

quantum oscillations.

AC conductivity, optical, microwave and IR, time domain spectroscopy.

Neutron scattering

NMR

ARPES

Tunneling and STM.

3. Pseudo-gap physics.

Corner sharing octahedrals.

3d

eg

t2g dxy,dyz,dzx

dz2, dx2-y2

Octahedral

field splitting

X2-y2

z2

CuO plane: strongly-correlated electron system

Undoped CuOUndoped CuO22 plane: plane: Mott Insulator due toMott Insulator due to

ee-- - e - e-- interaction interactionVirtual hopping inducesVirtual hopping induces

AF exchange J=4tAF exchange J=4t22/U/U

CuOCuO22 plane with doped holes: plane with doped holes:

LaLa3+3+ Sr Sr2+2+: La: La2-x2-xSrSrxxCuOCuO44

tt

One hole per site: should be a metal according to band theory.

Mott insulator.

Ogata and Fukuyama, Rep. Progress in Physics, 71, 036501 (2008)

Charge transfer insulator.

Electron picture Hole picture

Mott insulator

Also from Raman scattering.

Largest J known among transition metal oxide, except for the Cu-O chain compound where J=220meV.

By fitting the spin wave dispersion measured by neutron scattering. (also needs a small ring exchange term.)

Spin flip breaks 6 bonds, costs 3J.

Doping a charge transfer insulator: The “Zhang-Rice singlet”

Symmetric orbital

centered on Cu.Anti-symmetric orbital

Due to AF exchange between Cu and O, the singlet symmetric orbital gains a large energy, of order 6 eV. This singlet orbital can hop with effective hopping t given by:

What is unique about the cuprates?

Pure CuO2 plane Single band Hubbard model, or its strong coupling limit, the t-J model.

Dopeholes t

J t 3 J1) low dimension

2) H = J Si · Sjnn

large J = 135 meV

Competition:

t favors delocalization of electrons

J favors ordering of localized spins3) quantum spin S =1/2 (NNN hopping t’ may explain asymmetry

Between electron and hole doping )

Fermi liquid theory in a nut-shell:

2. Luttinger Theorem: volume of Fermi surface is the same as free fermion, ie For n carriers it is n/2 mod 1 of the area of the BZ.

1. Well defined quasi-particles exist provided 1/<<E near the Fermi energy. Usually 1/ ~ T^2. The electron spectral function has the form

3. Physical quantities are given by free fermion expressions except for Landau parameters. Except for tunneling, z does not appear.

Low doping: AF order. Unit cell is doubled. We have small pockets of total area equal to x times the area of BZ.

Doping x holes in a Mott insulator.

Large doping: no unit cell doubling.

Total Fermi surface area is

Area in the reduced BZ is

1. Single hole.

2. Small doping

3. Superconducting state.

4. Fermi liquid.

5. Pseudo-gap.

How many ways does Nature have to deal with doping a Mott insulator?

Electron doped.

3 Dimension. Brinkman-Rice Fermi liquid.

AF with localized carriers.

Micro phase separation: stripes

Organic ET salts. Metal-insulator transition by tuning U/t.

Possibility of a “spin liquid”.

Doping yields a superconductor.

A second family of HiTc superconductors!

Electron doped side: AF persists to x=0.13 and the doped electrons are localized.

What is the origin of the p-h asymmetry? Hopping of electron on Cu (d10) is physically different from hopping of a Zhang-Rice singlet located on the oxygen. One possibility is polaron effect is stronger on the electron side.

J=31 meV

X<0.2 commensurate spin order, localized hole. (polaron effect?)

0.2<x diagonal stripe with 1 hole per Ni. (microscopic phase separation into Ni2+ and Ni3+).

Non-metallic until x=0.9

Now ½ hole per linear distance along the stripe (2 Cu sites) : mobile charge.

Smaller J means it is deeper in the Mott phase. Effective hopping is also small and polaron effects favor localized carriers.

Tokura et al, PRL 70, 2126 (1993).

X=0 is a band insulator, x=1 is a Mott insulator.

For x=1, Ti is d1 and has S=1/2. Very small optical gap (0.2eV). Surprisingly small TN=150K, (reduced due to orbital degeneracy).

3 dim perovskite structure.

Specific heat = T

This is an example of “Brinkman-Rice Fermi liquid”.

Diverging mass near the Mott insulator. m*/m=1/xh, z=xh.

e^2n/m* is proportional to xh , even though Fermi surface is “large” and has volume x=1-xh as inferred from the Hall effect.

Metal- insulator transition by tuning U/t.

U/t

x

AF Mott insulator

metal

Cuprate superconductor

Organic superconductor

Tc=100K, t=.4eV, Tc/t=1/40.

Tc=12K, t=.05eV, Tc/t=1/40.

X = Cu(NCS)2, Cu[N(CN)2]Br, Cu2(CN)3…..

Q2D organics -(ET)2X

anisotropic triangular lattice

dimer model

ET

X

t’ / t = 0.5 ~ 1.1

t’t t

Mott insulator

Q2D antiferromagnet -Cu[N(CN)2]Cl

t’/t=0.75

Is the Mott insulator necessarily an AF?

“Slater vs Mott”.

Until recently, the experimental answer is yes.

A digression on spin liquid.

Q2D spin liquid -Cu2(CN)3

Q2D antiferromagnet -Cu[N(CN)2]Cl

t’/t=1.06No AF order down to 35mK.J=250K.

t’/t=0.75

Magnetic susceptibility, Knight shift, and 1/T1T

• Finite susceptibility and 1/T1T at T~0K : abundant low energy spin excitation (spinon Fermi surface ?)

C nuclear

[A. Kawamoto et al. PRB 70, 060510 (04)]

Wilson ratio is approx. one at T=0.

is about 15 mJ/K^2moleSomething happens around 6K.

Partial gapping of spinon Fermi surface due to spinon pairing?

From S. Yamashita,.. K. Kanoda, Nature Physics, 4,459(2008)

More examples have recently been reported.

ET2Cu(NCS)2 9K sperconductor ET2Cu2(CN)3 Insulator spin liquid

M. Yamashita ...Matsuda ,Nature Physics 5,44(2009) Belin, Behnia, PRL81,4728(1998)

Thermal conductivity

Superconductivity in doped ET, (ET)4Hg2.89Br8, was first discovered Lyubovskaya et al in 1987. Pressure data form Taniguchi et al, J. Phys soc Japan, 76, 113709 (2007).

Doping of an organic Mott insulator.

Note the common feature of high Tc and organics:

• Proximity to Mott insulator.

• singlet and d-wave pairing.

Is it possible to have superconductivity in purely repulsive models, and if so, how do we understand it?

Note that in d-wave pairing, we avoid on-site repulsive energy. By making singlet pairs, we can gain exchange energy.

1. The one hole problem.

Theory for t-J model: self consistent Born approx. of hole scattered by AF magnon works very well. (Kane,Lee and Read, 1989 , Schmitt-Rink et al,….).

Main conclusions: the dispersion is given by an effective hopping of order J. The hole spectrum has a coherent part with relative spectral weight (J/t) and a broad incoherent part spreading over t.

ARPES data: review by X. J. Zhou et al, cond-mat0604284)

Not the whole story: line width very broad (300meV) and comparable to dispersion. To explain this, need to include strong electron phonon coupling (polaron). Line-shape is interpreted as Franck-Condon effect as in molecular H2. However, the peak of the spectral function is still given by the bare dispersion.

Message: one band t-J model works, but need strong e-phonon coupling.

Ideal for 2 dim. Assume parallel momentum is conserved. Measure ejected electron energy and infer the energy and momentum of the hole left behind.

Surface sensitive probe.

Resolution a few meV.

Recent Laser ARPES employs VUV lasers (about 7 eV). Energy resolution 0.26meV. Deeper penetration. Limited k space coverage. No tunability.

Light Source VUV Laser SynchrotronEnergy Resolution (meV) 0.26 5~15

Momentum Resolution (Å-1)

0.0036(6.994eV)

0.0091(21.1eV)

Photon Flux(Photons/s) 1014~1015 1012-1013

Electron Escape Depth (A)

30~100 5~10

Photon Energy Tunability Limited Tunablek-Space Coverage Small Large

Advantages and Disadvantages of VUV Laser ARPES

Laser and Synchrotron are complementary.

Bi2Sr2CaCu2O8+YBCOLSCO

BSCCO or Bi-2212

Simple, x is known, disorder. Low Tc. Cleanest. Doping by varying

oxygen conc. on chains.

Cleavage plane. Disorder.

.

Bi-2201

(Bi2Sr2CuO6+x)

Eisaki et al, PRB 69, 064512 (2004)

With further increase of layers, Tc does not go up further. The inner planes have less hole and may be AF ordered.

2. Small doping.

DC transport.

Boltzmann conductivity: =ne^2/m

Ando et al, PRL 87, 017001 (01)

Hall effect:

RH=1/nec

x=0.03 sample, from Padilla et al, PRB72,060511(2005)

Anomalous T dependence.

Optical conductivity Timusk and Statt,Rep Prog Phys 62,61 (99)

Drude formula for simple metal:

Extended Drude formula:

Padilla et al, PRB72,060511(05)

Include high frequency incoherent part.

From reflectivity or ellipsometry, deduce Re and Im parts of

Conclusion from transport measurements:

No divergent mass enhancement. m*/me~4.

Drude spectral wt (n/m*) is proportional to x with no T dependence. This wt becomes the superfluid density in the SC.

Scattering rate is roughly 2kT and becomes linear in at high frequencies.

Weight of delta function is the superfluid density and is proportional to x

Neutron scattering:

If there is long range AF order, Bragg peaks appear at G’s.

The direction of the ordered moment can be determined by rotating G.

In the absence of long range order, we can measure equal time correlation function by integrating over

Local moment picture works. Reduced from classical moment of unity due to quantum fluctuations of S=1/2.

NMR Local probe. Does not require large samples. Very important for the study of new materials.

1. Knight shift. Proportional to spin susceptibility, but free from impurity contributions. Line is often broadened by random distribution of local fields. Need good quality material. The shift and onset of line broadening can be used to measure spin order.

2. Spin relaxation rate. Measures the low energy spectrum of spin fluctuations.

3. NQR. Measure local electric field distribution.

S (T)+ VV +core+impurity(T)

K=KS (T) + KVV + Kcore

KS ~ S KVV ~ VV

Form factor F(q) peaks at different q for different sites.

For example, planar oxygen site does not see AF q=( but Cu site does.

For metals, Korringa relation:

One component vs two component system: validity of the one band Hubbard model.

Knight shift on different sites have identical T dependence.

Takigawa et al PRB43, 247 (91).

Theoretically, C. Varma believes that 3 band Hubbard model with interaction V between Cu and O is needed. He proposes the existence of orbital currents in the plane between Cu and O. These currents occur within the unit cell and does not change the unit cell.

Orbital currents have been observed by neutron scattering. The onset of these currents seem related to T*, the pseudo-gap scale.

However, the moments are about 45 degrees from the plane. Numerical studies find orbital currents between planar and apical oxygen. (Weber et al, ArXiv 0803.3983). Perhaps these effects do not affect the Fermi surface.

Li ..Bourges, Greven, Nature 455, 372 (2008).

There is also reports of T breaking (ferromagnetic like) by polar Kerr effect at slightly lower temperature. (Xia,…Kapitulnik,PRL 100,127002 (08))

3. Properties of the superconductor.

Pairing is d symmetry.

Phase sensitive measurements.

Tsuei and Kirtley Rev Mod Phys 2000.

1. tri-crystal experiment, IBM 1993. ½ flux vortex at the junction. Standard hc/2e votex everywhere else.

2. Corner SQUID. Wollman et al 1993.

ARPES. Node along diagonal.

Ding et al Nature 382, 51 1996.

Dirac cone characterized by vF and v

Importance of phase fluctuations.

Superfluid stiffness Ks is related to the Drude spectral wt..

It is measured by London penetration depth.

Thermal excitation of nodal qp gives linear T reduction.

Microwave cavity perturbation expt, or by muon precession relaxation rate which measures the magnetic field distribution near the vortex. Note very long several thousand angstrom) implies very small stiffness or superfluid density.

Uemura plot: linear relation between Tc and ns/m*.

From Boyce et al, Physica C 341,561 (00)

A distribution of magnetic field (eg caused by the overlapping fields of vortices) causes a damping of the oscillations.

Another set-up is zero field muSR, which is very sensitive to static (on the scale of the muon lifetime of 2 micro-sec) internal magnetic field (as low as a few gauss) due to magnetic ordering or spin glass freezing.

+ve Muons relax to certain (often unknown) sites.

In 2D phase fluctuations destroy SC order via the Kosterlitz-Thouless mechanism of proliferation of vortices and anti-vortices. They predict a universal relation:

The dynamics of phase fluctuations is probed by microwave conductivity by Corson et al Nature 398,221 (1999) in UD Bi-2212 Tc 74K.

(More about fluctuation SC via Nernst effect and diamagnetism later.)

For a SC:

Scaling function:

Then Tc is controlled by Ks, not by the energy gap as in BCS theory. Strong violation of BCS relation 2/kTc~4.

Isotope effect.

Summary:

Substantial isotope effect on Tc for underdoped, but little or no isotope effect for optimal and overdoped.

However, there is isotope effect on ns/m* for all doping. (unexplained: needs better understanding of e-phonon in strongly correlated materials.)

On the other hand there is no isotope effect on Fermi velocity by Laser ARPES, while there is shift in “kink” energy. (Iwasawa..Dessau,PRL101,157005(08)

m*/m=1+, but usually has no isotope effect.

Khasanov …Keller, PRB 73,214528 (06)

Qualitatively consistent with the idea that in UD, Tc is controlled by ns/m*.

If Tc~ns/m*, we expect Tc/Tc=-2line but datais closer to Tc/Tc=- (line B)

YBCO

However, in practice Tc has a more complicated dependence on ns/m*.

Other probes of nodal quasi-particles:

1. Quasi-particle dispersion shifted by electromagnetic gauge field A.

Volovik (1993) pointed out that near a vortex,

Set R to the average spacing between vortices.

Predicts a specific heat which goes as sqrt(B) and observed by K. Moler.

2. Universal ac conductivity and thermal conductivity. ( Lee, 93, Durst and Lee 2000)

Taillefer, PRL 79, 483 (97)

Use to measure vvF.

Raman scattering (electronic).

Devereaux and Hackl, Rev Mod Phys 79, 175(2007)

Non-resonant

resonant

Probe particle-hole charge excitation with a form factor.

Expand the polarization tensor in terms of irreducible representation of lattice point symmetry. For square lattice:

(k)=

(k)=

Expected contribution from quasi-particle, quasi-hole excitation.

The initial slope is proportional to .

The broad continuum comes from incoherent electronic excitations.

Summary:

The superconducting state is singlet d-wave pairing. The nodes dominate low temperature properties and are well characterized.

In the underdoped region, Tc is determined by phase fluctuation and not by the vanishing of the pairing gap. As a result, the energy gap is large even though Tc is small.

While unusual, a lot of the physical properties of the superconducting state at low temperature can be understood based on a conventional physical picture.

As we will see, questions remain as to what happens at higher temperature above Tc and in a high magnetic field which restores the resistive state. Furthermore, the precise behavior of the gap near the anti-node ( remains to be clarified.