Relaxation of Hot Quasiparticles in a d-WaveSuperconductor

A. Rosch (Cologne), P. Howell (Karlsruhe)

P. Hirschfeld (Florida)

• pump-probe experiments in high-temperature superconductors

• relaxation of high-energy quasi-particles

• momentum conservation and diffusion in momentum space

Phase Diagram of the Cuprates

Fermi liquid?

T

AF

M in

sulato

r

pseudogap

strange metal

hole concentration

superconductor

best understood (??): physics within superconducting phase

Scenario 1: Quasiparticles in superconducting phase

Microwave conductivity σ(ω) in

YBa2Cu3O6−δ Hosseini et al. (1999)

well fitted by thermal QPs

Agreement with λ−2(T).

τ−1tr decreases fast below Tc

Umklapp scattering of QPs

Walker & Smith, PRB 61, 11285 (2000)

Scenario 2: Quantum Critical Scaling

σ(ω) in Bi2Sr2CaCu2O8 requires more complicated model

→ ~τ−1qp ≃ 0.8kBT

This agrees with ARPES:

Nodal QPs of energy ω show ‘quan-

tum critical scaling’:

~τ−1qp ≃ max[~ω, kBT ]

above and below Tc

(marginal Fermi liquid ??)

♦ = 48 K, △ = 90 K, ◦ = 300 KValla et al., Science 285, 2110 (1999)

Pump–Probe Experiments in Superconductors

Pump with laser pulse — probe optical response

cascade of pair−breaking down to gap

phonons equilibrated with hot QP transport energy out of system

Step 0: excitation of QP (1 eV)

Step 1 ( < 1ps):

Step 2 (1−10ps):

Step 3 (ns − s):

This talk !

µ

Rothwarf, Taylor (1967)

Pump–Probe Reflectivity on YBa2Cu3O6.5

time-resolved reflectivity δR ∝ ∫

dω ω2δσ(ω) after pumping

Segre, Gedik, Orenstein, Bonn, Liang, Hardy et al., PRL 88, 137001 (2002)

QPs relax very fast down ∼∆0, then slowly with τ−1 ∝ T3

Mystery:QP energy ∼∆0 ≫ T so expect τ independent of T.

More experiments...

in general: very different time-scales measured in pump-probe experi-ments in HTc depending on doping, sample quality, frequency, inten-sity, temperature

some recent experiments:

• Diffusion of nonequilibrium quasi-particles in a cuprate superconductor, (Gedik et

al. Science 2003):writing a interference pattern on crystal to obtain diffusion

• Dynamics of Cooper pair formation in Bi2Sr2CaCu2O8−δ (Kaindl et al.,Science 2003,

PRB, 2005):time-dependence of σ(4meV < ω < 11meV )

• Nonequilibrium quasiparticle relaxation in the vortex state

of La2−xSrxCuO4, (Bianchi et al. PRL 2005)

• Abrupt transition in quasiparticle dynamics at op-

timal doping in a cuprate superconductor system

(Gedik, et al. PRL 2005): different sign and differ-ent density dependence for under- and overdopedsamples!

underdoping vs. overdoping (Gedik, et al. PRL 2005)

need other hot QP to free trapped excitations for underdoped system?

Fermi Liquid Interpretation

Photoinduced QPs create cascade of QPs, i.e. excite pairs out of

condensate.

Some QPs end up near gap antinodes with energy ∆0.

How do these ‘hot’ QPs relax?

E and k conservation ⇒bottleneck for pair-breaking

processes.

Intermediate T:

Umklapp scattering from thermal

QPs with activation gap . ∆0.

Low T:

Slow relaxation by scattering from

thermal QPs at nodes.

antinode

thermal QPs

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NB: Clean crystals ⇒ negligible impurity scattering

Bottleneck for Pair-Breaking

Scattering process d†−p−qd

†pd

†k+qdk

Idea: antinodal velocity < nodal velocity ⇒antinodal QP cannot lose enough energy to generate nodal QP-pairs

EANk = ∆0 cos 2θ +

2k2E2F

∆0 cos 2θ

ENp = 2

√

E2Fp2

⊥ + ∆20p

2‖

Energy lost by antinodal QP:

EANk − EAN

k+q = 2q∆0 sin 2θ

Energy needed to create pair:

/4

p

p

q

θα

−p−q

k

ENp + EN

−p−q

= 2

√

E2Fq2 cos2 α + ∆2

0q2 sin2 α ≃ 2qEF cos(π

4 + θ)

⇒ Pair-breaking only possible if θ ≃ π4

No scattering for |k − kF| .1

√2

∆20

E2F

cos2 2θ

How else can Quasiparticles Relax?

Also bottleneck for phonons and spin waves.

What about other QP scattering processes?

Hint =∑

1,2,3,4

[

V2−3(u1v2 + v1u2)(u3v4 + v3u4)d†4↑d

†3↓d

†2↑d

†1↓

+ V2+1(u1u2 − v1v2)(u3v4 + v3u4)d†4↑d

†3↓d

†2σd1σ

+ V2+1(u1v2 + v1u2)(u3v4 + v3u4)d†4σd

†3σ̄d2σ̄d1σ

+ V2−3(u1u4 − v1v4)(u2u3 − v2v3)d†4σd

†3σ′d2σ′d1σ

+ h.c.]

Consider QP–QP scattering terms:

Low density of hot QPs, so scatter off thermal QPs.

Umklapp Scattering

QP–QP scattering: d†k+qd

†p−q+Gdpdk

Large momentum transfer possible, q ∼ kF, but thermal QP can’t bearbitrarily close to node

⇒ τ−1U ∼ e−Eact/T with activation energy Eact . ∆0

(−π,−π)

(−π,π)

(π,−π)

(π,π)

"hot" QP

QPthermal

gap

G=(2 ,2 )

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π π exponentially suppressed

at low T!

Quasi-Elastic Scattering

Scattering from nodal QPs: d†k+qd

†p−qdpdk

Discrepancy between nodal and

antinodal velocities

⇒ q = (q‖,q⊥) ∼ ( T∆0

, TEF

) ≪ kF

p

p

/4π

k

p

q

−q

small transferred momentum ⇒ diffusion in momentum space

hot QP reaches quasi-equilibrium with nodal QPs: |k−kF | ∼ √T∆0/EF

• for T ≪ ∆30

E2F

⇒ |k − kF| ≪∆2

0E2

F

⇒ hot QP inside bottleneck

• for T ≫ ∆30

E2F

⇒ |k − kF| ≫∆2

0E2

F

⇒ hot QP outside bottleneck

Boltzmann Equation at T ≪ ∆30/E2

F (1)

Calculate evolution of antinodal distribution gk(t).

∂gk

∂t= −

∫

dq[

gkGqq(q, Ek − Ek+q) − gk−qGqq(q, Ek−q − Ek)]

,

Gqq(q, ε) = |Vqq|2∫

dp fp(1 − fp−q) δ(Ep − Ep−q + ε)

Hot QPs ‘thermalise’ in radial direction → transformation:

gk(t) =Ck(t)

hθe−βEk , hθ =

∫

(kF + k)dk e−βEk

Quasi-equilibrium in radial direction

⇒ Ck only weakly dependent on |k|.

Ck = C0θ + (k − 〈k〉)C1

θ + · · ·

Boltzmann Equation at T ≪ ∆30/E2

F (2)

Project out C0θ and do gradient expansion since q ≪ kth.

∂

∂t

[

C0θ(t)

hθ

]

=

[

Fθ∂

∂θ+ D

∂2

∂θ2

]

C0θ(t)

hθ(1)

where F = 4β∆0D =7π5

80

|Vqq|2T4

∆30E

2F

Recall gk(t) =Ck(t)

hθe−βEk Equilibrium? C0

θ(t) = hθ.√

Particle conservation? N =∫

dkgk =∫

dθ C0θ

Re-write (1) as

∂

∂tC0

θ = −F∂

∂θ

(

θC0θ

)

+ D∂2

∂θ2C0

θ

Right-hand side has only gradient terms.√

⇒ Structure of equation + ratio F/D fixed by symmetries!

Diffusion Equation

∂

∂tC0

θ = −F∂

∂θ

(

θC0θ

)

︸ ︷︷ ︸

force due to energygradient from antinode

+D∂2

∂θ2C0

θ︸ ︷︷ ︸

diffusion

Dominant contribution from φ ≃ π4

⇒ Zig-zig diffusion in momentum space thermal

QPs

Cθ(t) =1

σ(t)√

2πexp

[

− θ2

2σ2(t)

]

, σ2(t) = (D/F + a2/2)e2Ft − D/F

Hot QP reaches node after time:

τθ ∼ 1

Fln min

[√

F

D,1

a

]∼

∆30E

2F

|Vqq|2T4

Check

What about next terms in Ck = C0θ + (k − 〈k〉)C1

θ + · · · ? Project out

equation for C1θ:

C1θ (t) = C1

θ (0) exp

[

− tE2F

τθ∆20

]

⇒ assumption of radial equilibrium justified.

Intermediate Temperatures (T ≫ ∆30/E2

F)

|k − kF| ∼√

T∆0

EF≫ ∆2

0

E2F

so outside pair-breaking bottleneck

But

• detailed balance satisfied (both pair-breaking and recombination)

• QPs are created near nodes ⇒ number of antinodal QPs conserved

• Small momentum transfer, q ≪ kF

→ same form of diffusion equation,with different D

τθ ∼E5

F

|V|2√

T 5∆30

with equal contributions from pair-breaking and QP–QP scattering.

Comparison with Experiment

Latest data (Gedik et al. 2004) consistent

with activated behaviour with

Eact ≃ 8 meV

Why does activated behaviour

dominate at such low T?

Compare quantitatively:

(using ∆0EF

≃ 18 and ∆0 ≃ 20 meV for Y123):

Umklapp scattering: τ−1U ∼ |V|2

EFe−Eact/T

Quasielastic scattering: τ−1θ ∼ |V|2

EF

(

∆0

EF

)4(T

∆0

)5/2

(for T ≫ ∆30/E2

F ≃ 3 K)

Only comparable at T ≃ 4 K

⇒ need to measure at lower T to see diffusive process

Conclusions

• Pump-probe experiments:

New information about strongly correlated systems!

How do QP interact? How does SC form dynamically? ...

• Within Fermi liquid picture: bottleneck for pair-breaking.

• Hot QPs relax slowly by scattering from thermal QPs.

• Analytic solution of Boltzmann equation: diffusion in momentum

space and power-law behaviour at low temperature

• Many open questions and challenges from experiments

Role of phonons? (main player in conv. SC)

Main discrepancy: density dependence quantitatively inconsistent

(2 orders of magnitude too weak)

P. C. Howell, A. Rosch, PRL 92, 037003 (2004)