Nernst effect in unconventional density waves Bal azs D ora, - - PowerPoint PPT Presentation

Nernst effect in unconventional density waves

Bal´ azs D´

• ra, Kazumi Maki, Attila Virosztek

Bojana Korin-Hamzi´ c, Mark Kartsovnik, Carmen Almasan Outline: • What are UDW?

• General properties of α-(BEDT-TTF)2KHg(SCN)4
• Phase diagram of CeCoIn5
• Landau level formation
• Angular dependent magnetoresistance
• Thermoelectric power, Nernst coefficient

UDW Hamiltonian:

H =

′

• k,σ
• ξ(k)(a+

k,σak,σ − a+ k−Q,σak−Q,σ) + ∆(k, σ)a+ k,σak−Q,σ + ∆(k, σ)a+ k−Q,σak,σ

• .

a+

k,σak−Q,σ ∼ ∆(k, σ): non-local interaction (on site and direct Coulomb, exchange, pair-hopping and

bond-charge). The spectrum:

E±(k, σ) = ξ(k) + ξ(k − Q) 2 ± ξ(k) − ξ(k − Q) 2 2 + |∆(k, σ)|2

The general form of the gap in quasi-1D:

∆(l) = ∆0 + ∆1 cos(lyb) + ∆2 sin(lyb) + ∆3 cos(lzc) + ∆4 sin(lzc)

wavevector dependent=unconventional, m(Q) = 0, n(Q) = 0, “hidden-order”. ∆(σ) = ∆(−σ): UCDW, ∆(σ) = −∆(−σ): USDW

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−1 −0.5 0.5 1 −3 −2 −1 1 2 3 0.5 1 1.5 2 2.5 3

bky a(kx − kF)

E+ ∆

∆(k) = ∆ sin(bky), ε(k) = −2ta cos(akx) − 2tb cos(bky), ta/∆ = 2, tb/∆ = 0.1

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Order parameters:

phase gap

• rder parameter

UCDW ∆ cos(bky) electric current density UCDW ∆ sin(bky) kinetic energy density USDW ∆ cos(bky) spin current density USDW ∆ sin(bky) spin kinetic energy density

These phases are known as: orbital antiferromagnet, staggered flux phase, d-density wave, bond-order wave, spin nematic state and spin bond-order wave.

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The thermodynamic properties are identical to that of d-wave SC. Consequences: C ∼ T 2 χ ∼ T σ(ω → 0) ∼constant, ω2 σ(ω → 2∆) ∼ constant or diverges Possible materials with UDW ground state:

• URu2Si2
• α−(BEDT-TTF)2KHg(SCN)4
• α−(BEDT-TTF)2I3
• 2H-TaSe2
• pseudogap: (TaSe4)2I, HTSC, transition metal oxides (SrRuO3, BaRuO3)

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.5 1 1.5 2 2.5 3

ω/∆ g(ω)/g0

···∆0

• - ∆i=0

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α−(BEDT-TTF)2MHg(SCN)4 salt (M=K, Rb, Tl, NH4)

• quasi-one dimensional Fermi surface ⇒ DW instability
• M=NH4: superconductor at T = 1 K
• M=K, Rb, Tl: phase transition at T = 8 − 12 K
• no X-ray (CDW) or spin signal (SDW) ⇒ hidden-order
• threshold electric field consistent with UDW

a c Fermi surface B-T phase diagram⇒ a kind of CDW

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Properties of CeCoIn5

Similarities with high Tc superconductors:

• quasi-2-dimensional structure (tetragonal)
• d-wave SC
• proximity of AF
• presence of non-Fermi liquid phase

B UDW Fermi liquid d-wave SC T QCP

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Angular dependent magnetoresistance in CeCoIn5

20 40 60 80 100 120 140 160 180 0.085 0.086 0.087 0.088 0.089 0.09 0.091 0.092 0.093 0.094

θ σ [H=4 T (red), 5 T (blue), 8 T (green), 10 T (black)]

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Effect of magnetic field

• 1. Landau levels, continuum model:

H =

• dr

n=1,2 (−iv [R+ n ∂xRn − L+ n ∂xLn] + iv⊥(−1)n [R+ n ∂yRn − L+ n ∂yLn] − i∆b [exp(iϕ)R+ n ∂yLn + exp(−iϕ)L+ n∂yRn])

EΨ = (−iva∂xρ3 + ∆ceBx cos(θ)ρ1)Ψ, ⇒ En = µ ±

• 2nva∆ce|B cos(θ)|

In quasi 2D: En = µ ±

• 2n∆ceB|va cos(θ) − v⊥ sin(θ)|

x y EN

−1 −0.5 0.5 1 −3 −2 −1 1 2 3 0.5 1 1.5 2 2.5 3

bky a(kx − kF)

E+ ∆

• 2. Conductivity:

σ =

• n

σn cosh2(βEn/2),

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Angular dependent magnetoresistance in α-(BEDT-TTF)2KHg(SCN)4

−100 −80 −60 −40 −20 20 40 60 80 100 500 1000 1500 2000 2500 3000 3500

θ (◦) R⊥(15T, θ) (Ohm)

−100 −80 −60 −40 −20 20 40 60 80 100 500 1000 1500 2000 2500 3000 3500

θ (◦) R⊥(15T, θ) (Ohm)

experiment theory

Current perpendicular to the a-c plane at T = 1.4K and B = 15T for φ = −77◦, −70◦, −62.5◦, −55◦, −47◦, −39◦, −30.5◦, −22◦, −14◦, −6◦, 2◦, 10◦ 23◦, 33◦, 41◦, 48.5◦, 56◦, 61◦, 64◦, 67◦, 73◦, 80◦, 88.5◦, 92◦ and 96◦ from bottom to top. The curves are shifted from their original position along the vertical axis by n × 100Ohm, n = 0 for φ = −77◦, n = 1 for φ = −70◦, . . . .

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Angular dependent magnetoresistance in CeCoIn5

20 40 60 80 100 120 140 160 180 0.085 0.086 0.087 0.088 0.089 0.09 0.091 0.092 0.093 0.094

θ σ (H=4 T (circle), 5 T (triangle), 8 T (square), 10 T (star)

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Thermoelectric coefficients

Seebeck coefficient: thermally excited quasiparticles, carrying energy, formulated similarly to

• resistivity. First three Landau levels are used.

Nernst coefficient: in an applied electric and magnetic field, the quasiparticle orbits drift as: vD = (E × B)/B2. Heat current: Jh = TSvD. S = g(0)e|B cos(θ)| m∗

• n
• ln(1 + exp(−βEn)) + βEn(1 + exp(βEn))−1

, for small T and large B: S = 2g(0)e|B cos(θ)| m∗

• ln(2) + 2 ln
• 2 cosh
• βE1

2

• − βE1 tanh
• βE1

2

• .

αxy = −S| cos(θ)| Bσ αxy = 1 σ

• L2D

1 + γ2B2 − 2e

• ln(2) + 2 ln
• 2 cosh
• βE1

2

• − βE1 tanh
• βE1

2

• ,

γ = eτ/m

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Seebeck and Nernst coefficient in α-(BEDT-TTF)2KHg(SCN)4

5 10 15 20 −0.5 0.5 1 1.5 2 2.5

B(T) S(µV/K)

T = 1.4 K T = 4.8 K T = 5.8 K T = 6.9 K

6 8 10 12 14 16 18 20 22 −6 −5 −4 −3 −2 −1

B(T) Sxy(µV/K)

T = 1.4 K T = 4.8 K 12

Seebeck and Nernst coefficient in CeCoIn5

2 4 6 8 10 12

• 5

5 10 15 20 25

B(T) Sxx(µV/K)

2 4 6 8 10 12 15 20 25 30 35 40 45 50 55

B(T) Sxx(µV/K)

T = 1.3 K, 1.65 K, 2.5 K, 3.5 K, 4.8 K 7.3 K, 10.5 K 15 K (from bottom to top)

2 4 6 8 10 12

• 2
• 1.5
• 1
• 0.5

0.5

B(T) αxy(µV/K)

2 4 6 8 10 12

• 2.5
• 2
• 1.5
• 1
• 0.5

B(T) αxy(µV/K) 13

Conclusions

• non-local interactions
• transition is metal to metal instead of metal to insulator
• gapless excitations around the zeros of the gap
• In magnetic field: Landau levels, particles living around nodes dominate the low-T high-H be-

haviour, gapped excitations

• The low-temperature phase of α-(BEDT-TTF)2KHg(SCN)4 is well described by Q1D UCDW
• Q2D UDW is consistent with the pseudogap (non-Fermi liquid) phase of CeCoIn5

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