High-temperature superconductivity - Chalmersdelsing/Superconductivity/Lectures... · Tsuei&Kirtley, Rev. Mod. Phys., Vol. 72, No. 4, October 2000 D-wave symmetry Superconductivity

High-temperature

superconductivity

Superconductivity and Low temperature physics, FMI036

Alexey Kalabukhov Quantum Device Physics Laboratory, MC2

Electronic phase diagram

Superconductivity and Low temperature physics, FMI036 2

Superconductivity exists only in narrow doping range. Non-doped material is AF

insulator (”parent compund”). At high doping level cuprates are ”normal” metals.

Pseudogap region in low doping levels


Flux quantization

Low-Tc SQUID measures variation of magnetic flux in HTS ring:

Low-TC SQUID High-TC ring

T = 4 K

Vout

Mi

ΦR

SQUID voltage: Flux quantization: 𝑉𝑜𝑢𝑡~∆Φ𝑆~𝑀𝑖∆Φ𝑅 ∆Φ𝑅 = 𝑛Φ0 = 𝑛ℎ

𝑄


Flux quantization

C.E. Cough at el., Nature, 326, 855 (1987)

Measurements of flux jumps as a function of time in a HTS superconducting ring by LTS SQUID: Q = 2e

0 = h/2e Gough, C. E et al., 1987, Nature (London) 326, 855.


Important superconducting parameters

•Very short coherence lengths ~ interatomic distance

•Anisotropy in (a,b) and c axis

• Type II with high Hc2

“High-Temperature Superconducting Materials Science and Engineering” ed. Donglu Shi, 1995

Role of thermal fluctualtions


Ginznurg-Landau model: variation of the free energy:

Fluctuations assumed to be small if L << ξ:

This condition is not valid in some temperature region close to Tc:

GL fluctuation parameter

Change of the free energy:

Fluctuations are small if:

𝐹𝑠0 = 𝐹𝑛 + 𝛼 Ψ2 +

𝛽

2Ψ 4

𝛿Ψ∗(𝑟 )𝛿Ψ(0) 𝑟~𝜉 ≪ Ψ

𝐹𝑠𝑛 = 𝐹𝑠0 − 𝐹𝑛 =4𝜋𝛼2

2𝛽= 𝐻𝐶

2(𝑇) 𝜉3 𝑇 = 𝐻𝐶2(0) 𝜉3 0 𝑇 − 𝑇𝐶

1/2

𝑘𝐵𝑇𝐶 ≪ 𝐹𝑠𝑛

Role of thermal fluctualtions


In low-Tc superconductors: Fluctuations are negligible!

In high-Tc superconductors: 𝑘𝐹𝜉 0 < 10, ∆𝑇𝑓𝑙 ~ 1 𝐾 !

Fluctuation effects in cuprates are much stronger!!

Paraconductivity (Aslamasov-Larkin effect)


𝜎𝐴𝐿2𝐷 =

𝑒2

ℏ

𝑇𝐶𝑇 − 𝑇𝐶

𝜎𝐴𝐿3𝐷~

𝑇𝐶𝑇 − 𝑇𝐶

1/2

2D: Thin superconducting film, d < ξ

3D: Bulk superconductor

W. Lang et al., PRB 51, 9180 (1995)

Masked by inhomogeneous Tc

Can be used to analyze

coupling between CuO2-planes

Flux creep


Broadening of the resistive transition in HTS materials

Temperature fluctuations of flux result in “glassy” behavior –

depends on history of the state

ρ,

Ω∙c

m


Thermally activated flux flow (TAFF)

LTS: HTS (YBCO): HTS (BSCCO):

Vortex liquid: flux pinning is ineffective


Vortices in HTS: pancakes formation

Josephson effect


GHz/mV 483or

2sin

2

2

sin 12

eVtII

tteV

eV

t

II

JoJc

oJo

c

Josephson effect:

coupling of two

superconductors

through a ”weak link”

(S-I-S, S-N-S…)

DC and AC Josephson effects:

- dc supercurrent if I<IC

- Oscillating ac current if I>IC

Tunneling S-I-S JJs:


Barrier thickness: d ~ ξ ~ 10 Å in cuprates

How to make a Josephson Junction from HTS?

Grain boundaries in HTS


I

V

JCGB << JC BULK

- Weak link!

H. Hilgenkamp and J. Mannhart, Rev Mod Phys 74, p 485, APRIL 2002

high-Tc

epitaxial film

Bicrystal

substrate

H. Hilgenkamp and J. Mannhart, Rev Mod Phys 74, p 485, APRIL 2002

Grain boundaries in cuprates

Quality of the grain boundary depends

on the misorientation angle


Artificial grain boundaries


5 mm GB

GB

YBCO

Bicrystal technology and epitaxial

thin film deposition:

Bicrystal Bi-epitaxial

Bi-epitaxial Step-edge


30 40 0 10 20

misorientation angle (degree)

Jcgb

/ J

cg

0

0.2

0.4

0.6

0.8 [001] tilt

[100] tilt

[100] twist

T = 4.2 K

Artificial grain boundaries

Defects in grain boundaries


Grain boundary junctions are not S-I-S – there is a normal component (”shunt”)

Non-superconducting region,

due to non-stoichiometry

(oxygen) – S-N-S

Insulating region + localized

states, S-I-N-I-S


Intrinsic Josephson junctions

Tunneling between superconducting

CuO2-planes through insulating layers in

c-axis direction

Ar ion milling

Carving out a single junction


A. Yurgens, M. Torstesson, L. You APL 88, 222501 2006

Intrinsic Josephson junctions: fabrication

Symmetry of the order parameter


Symmetry of the Fermi surface of cuprates in CuO2-planes:

How superconducting order parameter will change in this case?

ARPES of fermi surface in

cuprates: 4-fold symmetry

Tsuei&Kirtley, Rev. Mod. Phys., Vol. 72, No. 4, October 2000

D-wave symmetry


𝑉𝑘,𝑘′ = −𝑉 BCS, cubic (spherical) symmetry.

∆𝛼,𝛽(𝑘)~ 𝑐𝛼,𝑘 , 𝑐𝛽,−𝑘 Pair wave function in general case:

Spin operators

(↑↓) Annihilation

operator

𝛼 = −𝛽

𝛼 = 𝛽

S = 0 L = 0 singlet, S-wave

S = 0 L = 2 singlet, D-wave

S = 1 L = 1 triplet, P-wave

Must be symmetric for

electron

commutations!

Pairing potential:


D-wave symmetry


∆𝑠(𝑘)~ ∆𝑠0 + ∆𝑠1 cos 𝑘𝑥 + cos 𝑘𝑦

∆𝑔(𝑘)~ ∆𝑔0 sin 2𝑘𝑥 sin 𝑘𝑦 − sin 2𝑘𝑦 sin 𝑘𝑥

∆𝑥2−𝑦2(𝑘)~ ∆𝑥2−𝑦20cos 𝑘𝑥 − cos 𝑘𝑦

∆𝑥𝑦(𝑘)~ ∆𝑥𝑦0sin 𝑘𝑥 sin 𝑘𝑦

In the limit to 2D case (x-y CuO2 planes, no dispersion

in z-direction):

s wave

dx2-y2 wave

g wave

dxy wave

Which configuraion is realized in HTS?



Symmetry of the order parameter

Symmetry of the order parameter in k-space:

D-wave symmetry: nodes and lobes, ns = 0 in nodes!


Phase sensitive experiments


Ψ𝑘 𝑟 , 𝑡 = Ψ𝑘 𝑒𝑖𝜑

𝑘~∆𝑥2−𝑦20cos 𝑘𝑥 − cos 𝑘𝑦 𝑒𝑖𝜑𝑘, 𝑘 = 𝐿, 𝑅

Supercurrent across grain boundary:

𝐼𝑆~ Ψ∗𝛻Ψ−Ψ𝛻Ψ∗ ~𝐼0 cos 2𝜃𝐿 cos 2𝜃𝑅 sin𝜑

Sigrist-Rice formula (clean limit)


𝜃𝐿 = 0, 𝜃𝑅 = 45° 𝐼𝑆 = 0 Equivalent to phase shift by π! -> Pi-junctions

Phase sensitive experiments



Pi-junction in a superconducting ring: flux quantization (neglecting self-inductance):

ℏ𝛻𝜑 = 2𝑒𝐴

ℏ 𝛻𝜑𝑑𝑙 = 2𝑒 𝐴 𝑑𝑙

ℏ 𝜋 + 2𝜋𝑛 = 2𝑒Φ

Φ = ℏ

2𝑒2𝜋 𝑛 +

1

2= Φ0 𝑛 +

1

2

𝜋

Half-integer flux quantum effect


Pairing symmetry: tri-crystal experiment

Pure dx2-y2 order parameter in tetragonal Tl2Ba2CuO6+d

C.C. Tsuei et al., Nature 387,481(1997).

LTS SQUID-microscope

Detecting spantaneous half flux quantum in tri-crystal junctions:

Models of HTS

• Challenges:

– High-TC: does not fit BCS

– Non-metallic ground state, quazi-2D

– Anti-ferromagnetic ordering

– Dependence of TC on doping

– D-wave symmetry

– Strong e-e correlation effects (insulating at

low doping)

– Pseudogap


Problem of high-Tc


𝜔𝐷 = 𝑐𝑆 6𝜋2𝑛 1/3 𝑇𝐷 =

ℏ𝜔𝐷𝑘𝐵

𝑇𝐶 = 1.14𝑇𝐷𝑒𝑥𝑝 −1

𝑁𝐹(0)𝑉

𝜆 ≡ 𝑁𝐹(0)𝑉

Al : TD = 428 K, λ ~ 0.18, TC = 1.6 K (experiment: 1.6 K)

YBCO : TD = 100 K, λ ~ 0.5, TC = 13 K (experiment: 92.5 K)

Critical temperature in

BCS model

Coupling constant, typically λ < 0.2

Debye temperature:

corresponds to a highest frequency mode of lattice vibrations (ωD)

Simple estimations:


The isotope effect

• Oxygen Isotope Effect: Replacement of O16 -> O18

• Very small change of TC in optimally doped regime

K.A. Muller, J.Phys.Cond.Matter, 19, 251002 (2007)

𝜔𝐷~1

𝑀=> 𝑇𝐶~

1

𝑀

𝛼 ≡ −𝑀 ∆𝑀

𝑇𝐶 ∆𝑇𝐶 ~0.5

Electron-phonon coupling is not

main paring mechanism in

cuprates?

Pseudogap


• Gap-like features in all underdoped cuprates above Tc

• Manifests in various experiments

• Preformation of cooper pairs OR non-superconducting phases?

Models of HTS

• Non-Fermi liquid models:

– Resonance valence band (RVB, t-J model): AFM

ground state, e-e pairing through magnetic

interactions (spin-density waves, SDW)

• Fermi-liquid models:

– Hartri-Fock calculations from Fermi liquid,

approximation of interacting electrons (BCS-like)


None of the models correctly predicts high-Tc in cuprates!

t-J model

• Doped Mott insulator

• No Fermi liquid – quasiparticle approach does not work

• Ground state: AFM insulator

• Underdoping: 1D state (”stripes”) hole delocalization

• Optimal doping: 2D state (Josephson tunneling between stripes),

superconducting coherence

• Overdoping: 3D state, loss of coherence

• Pairing is due to magnetic interactions

• Referred to RVB-model (Anderson 1987)


“Concepts in High Temperature Superconductivity”, E. W. Carlson, V. J. Emery, S. A. Kivelson, D.

Orgad, http://arxiv.org/abs/cond-mat/0206217v1

http://arxiv.org/abs/cond-mat/0206217v1




Experimental observation of stripes



Fermi-liquid models


- Hartree-Fock computations, Fermi-liquid hamiltonian. No initial

assumptions about the electronic state of the HTS

- Existence of Fermi surface from recent SdH experiments

- Pseudogap: transformation of Fermi surface into pockets (spin-

waves state)

- Conventional fermionic quasiparticles exist,but pairing due to

antiferomagnetic spin fluctuations

- Completely rules out Mott insulators (observed experimentally!)

R. B. Laughlin, PHYSICAL REVIEW B 89, 035134 (2014)

Fermi-liquid models


R. B. Laughlin, PHYSICAL REVIEW B 89, 035134 (2014)

This model seems to predict correctly Tc in cuprates:

Heavy-fermion superconductors


Large values of the electronic specific heat ->

“Heavy electrons”

CeCu2Si2 discovered by Steglich in 1979


Heavy-fermion superconductors

Phase diagram similar to cuprates

Co-existence of superconducting and magnetic phases

Organic superconductors


1963: First prediction by W.A. Little (Stanford Univ.), metal chains in organic

molecules

1980: Discovery of TMTSF, TC ~ 0.9 K at 12 kBar

1988: Other various organic superconductors, TC ~ 11.2 K at ambient pressure

TM2X, quasi-1D: ET2X, quasi-2D:



From: Paul Chaikin, NYU (http://www.physics.nyu.edu/~pc86/)

http://www.physics.nyu.edu/~pc86/

http://www.physics.nyu.edu/~pc86/



Highly anisotropic: 105

Phase boundary between superconductivity and anti-ferromagnetic order

Possible p-wave superconductors

2006: Oxypnictides


Hiroki Takahashi, Tokyo:

(Fe,As) LaO:

TC ~ 43 K

Yoichi Kamihara, Tokyo:

(Fe,P) LaO:

TC ~ 4 K

45

Compound (powder & single

crystals)

Tc Reference

LaOFeP ~5 K Y. Kamihara et al., J. Am. Chem.

Soc.128, 10012 (2006)

LaNiOP ~3 K T. Watanabe et al., Inorg. Chem. 46,

7719 (2007)

La[O1-xF-x]FeAs

La[O1-xCa2+x]FeAs

26 K (x=0.05-0.12)

0 K

Y. Kamihara et al., J. Am. Chem.

Soc.130, 3296 (2008)

La[O1-xFx]NiAs 3.8 K (x=0.1)

2.75 K (x=0)

Z. Li et al., arXiv:0803.2572

(La1−xSrx)ONiAs 3.7 K (x=0.1-0.2)

2.75 K (x=0)

L. Fang et al., arXiv:0803.3978

(La1−xSrx)OFeAs 25 K (x=0.13) H.-H. Wen et al., EPL 82, 17009 (2008)

Ce[O1−xFx]FeAs 41 K (x=0.2) G.F. Chen et al., arXiv:0803.3790

Pr[O1-xFx]FeAs

Nd[O1-xFx]FeAs

52 K (x=0.11) Z.-A. Ren et al., arXiv:0803.4283; Z.-A.

Ren et al., arXiv:0803.4234

Gd[O1−xFx]FeAs 36 K (x=0.17) P. Cheng et al., arXiv:0804.0835

Sm[O1− xFx]FeAs 55 K (x=0.1-0.2) Z.-A. Ren et al., arXiv:0804.2053;

R.H. Liu et al., arXiv:0804.2105

(Eu,Tm)[O1− xFx]FeAs no stable ZrCuSiAs structure G. F. Chen et al., arXiv:0803.4384

From: I. Eremin, Entanglement in Spin and Orbital Systems, Cracow 18-22 June 2008

Superconducting properties


Phase diagram:


J. Zhao et al., arXiv:0806.2528

• Parent compound: AFM normal metal • Layered structure • No pseudogap! • Electron and hole doped • Doping also possible by replacing Fe by Co • S-type symmetry of the order parameter

2001: MgB2


”High” critical temperature: TC ~ 39 K

Discovered by J. Akimitsu, Aoyama Nature, Vol. 410 No. 6824 (2001) pp.63-64.

Discovery of MgB2

• Discovered by J. Akimitsu, (2001)

• Metallic graphite-like structure

• ”High” critical temperature TC ~ 39 K

• Very difficult to make thin films


Nature 410, 63-64 (2001)


Cristina Buzea et al 2001

Supercond. Sci. Technol. 14

R115-R146

Superconducting properties of MgB2

• Coupling in B planes is stronger

• Double-gap model: Δπ ~ 2.0 meV, Δσ ~ 6.5 meV

• Explains TC and specific heat capacity


Two gaps in MgB2

P. Szabo et al., PRL 87 135002 (2001)

High-temperature superconductivity - Chalmersdelsing/Superconductivity/Lectures... · Tsuei&Kirtley, Rev. Mod. Phys., Vol. 72, No. 4, October 2000 D-wave symmetry Superconductivity

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