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

Nernst effect in unconventional density waves Bal´ azs D´ ora, Kazumi Maki, Attila Virosztek Bojana Korin-Hamzi´ c, Mark Kartsovnik, Carmen Almasan Outline: • What are UDW? • General properties of α -(BEDT-TTF) 2 KHg(SCN) 4 • Phase diagram of CeCoIn 5 • Landau level formation • Angular dependent magnetoresistance • Thermoelectric power, Nernst coefficient

UDW Hamiltonian: ′ � � � ξ ( k )( a + k ,σ a k ,σ − a + k − Q ,σ a k − Q ,σ ) + ∆( k , σ ) a + k ,σ a k − Q ,σ + ∆( k , σ ) a + H = k − Q ,σ a k ,σ . k ,σ � a + k ,σ a k − Q ,σ � ∼ ∆( k , σ ): non-local interaction (on site and direct Coulomb, exchange, pair-hopping and bond-charge). The spectrum: �� ξ ( k ) − ξ ( k − Q ) � 2 E ± ( k , σ ) = ξ ( k ) + ξ ( k − Q ) + | ∆( k , σ ) | 2 ± 2 2 The general form of the gap in quasi-1D: ∆( l ) = ∆ 0 + ∆ 1 cos( l y b ) + ∆ 2 sin( l y b ) + ∆ 3 cos( l z c ) + ∆ 4 sin( l z c ) wavevector dependent=unconventional, � m ( Q ) � = 0, � n ( Q ) � = 0, “hidden-order”. ∆( σ ) = ∆( − σ ): UCDW, ∆( σ ) = − ∆( − σ ): USDW 1

3 2.5 2 E + ∆ 1.5 1 0.5 0 3 2 1 1 0.5 0 0 −1 −2 −0.5 bk y a ( k x − k F ) −3 −1 ∆( k ) = ∆ sin( bk y ), ε ( k ) = − 2 t a cos( ak x ) − 2 t b cos( bk y ), t a / ∆ = 2, t b / ∆ = 0 . 1 2

Order parameters: phase gap order parameter UCDW ∆ cos( bk y ) electric current density UCDW ∆ sin( bk y ) kinetic energy density USDW ∆ cos( bk y ) spin current density USDW ∆ sin( bk y ) 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. 3

The thermodynamic properties are identical to that of d-wave SC. Consequences: 3 ··· ∆ 0 C ∼ T 2 2.5 - - ∆ i � =0 χ ∼ T g ( ω ) /g 0 2 σ ( ω → 0) ∼ constant, ω 2 1.5 σ ( ω → 2∆) ∼ constant or diverges 1 Possible materials with UDW ground state: 0.5 • URu 2 Si 2 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 ω/ ∆ • α − (BEDT-TTF) 2 KHg(SCN) 4 • α − (BEDT-TTF) 2 I 3 • 2H-TaSe 2 • pseudogap: (TaSe 4 ) 2 I, HTSC, transition metal oxides (SrRuO 3 , BaRuO 3 ) 4

α − (BEDT-TTF) 2 MHg(SCN) 4 salt (M=K, Rb, Tl, NH 4 ) • quasi-one dimensional Fermi surface ⇒ DW instability • M=NH 4 : 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 c a Fermi surface B-T phase diagram ⇒ a kind of CDW 5

Properties of CeCoIn 5 Similarities with high T c superconductors: • quasi-2-dimensional structure (tetragonal) • d-wave SC UDW • proximity of AF T • presence of non-Fermi liquid phase Fermi liquid d-wave SC QCP B 6

Angular dependent magnetoresistance in CeCoIn 5 0.094 σ [H=4 T (red), 5 T (blue), 8 T (green), 10 T (black)] 0.093 0.092 0.091 0.09 0.089 0.088 0.087 0.086 0.085 0 20 40 60 80 100 120 140 160 180 θ 7

Effect of magnetic field 1. Landau levels, continuum model: n ∂ x L n ] + iv ⊥ ( − 1) n [ R + n =1 , 2 ( − iv [ R + n ∂ x R n − L + n ∂ y R n − L + n ∂ y L n ] − i ∆ b [exp( iϕ ) R + n ∂ y L n + exp( − iϕ ) L + � d r � H = n ∂ y R n ]) E Ψ = ( − iv a ∂ x ρ 3 + ∆ ceBx cos( θ ) ρ 1 )Ψ , � ⇒ E n = µ ± 2 nv a ∆ ce | B cos( θ ) | � In quasi 2D: E n = µ ± 2 n ∆ ceB | v a cos( θ ) − v ⊥ sin( θ ) | 3 2.5 2. Conductivity: E N E + 2 ∆ 1.5 1 σ n � 0.5 σ = cosh 2 ( βE n / 2), 0 3 2 n 1 1 0.5 0 −1 0 bk y a ( k x − k F ) −2 −0.5 y −3 −1 x 8

Angular dependent magnetoresistance in α -(BEDT-TTF) 2 KHg(SCN) 4 3500 3500 3000 3000 2500 2500 R ⊥ (15 T, θ ) (Ohm) R ⊥ (15 T, θ ) (Ohm) 2000 2000 1500 1500 1000 1000 500 500 0 0 −100 −80 −60 −40 −20 0 20 40 60 80 100 −100 −80 −60 −40 −20 0 20 40 60 80 100 θ ( ◦ ) θ ( ◦ ) experiment theory Current perpendicular to the a-c plane at T = 1 . 4K and B = 15 T 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 ◦ , . . . . 9

Angular dependent magnetoresistance in CeCoIn 5 0.094 σ (H=4 T (circle), 5 T (triangle), 8 T (square), 10 T (star) 0.093 0.092 0.091 0.09 0.089 0.088 0.087 0.086 0.085 0 20 40 60 80 100 120 140 160 180 θ 10

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: v D = ( E × B ) /B 2 . Heat current: J h = TS v D . S = g (0) e | B cos( θ ) | � ln(1 + exp( − βE n )) + βE n (1 + exp( βE n )) − 1 � � , m ∗ n for small T and large B : S = 2 g (0) e | B cos( θ ) | � � � �� � �� βE 1 βE 1 ln(2) + 2 ln 2 cosh − βE 1 tanh . m ∗ 2 2 α xy = − S | cos( θ ) | Bσ α xy = 1 � L 2D ��� � � � �� � βE 1 βE 1 1 + γ 2 B 2 − 2 e ln(2) + 2 ln 2 cosh − βE 1 tanh , 2 2 σ γ = eτ/m 11

Seebeck and Nernst coefficient in α -(BEDT-TTF) 2 KHg(SCN) 4 0 2.5 T = 1 . 4 K −1 2 T = 4 . 8 K T = 4 . 8 K −2 1.5 S xy ( µ V/K) S ( µ V/K) −3 1 T = 5 . 8 K −4 0.5 T = 6 . 9 K T = 1 . 4 K −5 0 −0.5 −6 5 10 15 20 6 8 10 12 14 16 18 20 22 B (T) B (T) 12

Seebeck and Nernst coefficient in CeCoIn 5 25 55 50 20 45 S xx ( µ V/K) 15 S xx ( µ V/K) 40 10 35 30 5 25 0 20 -5 15 0 2 4 6 8 10 12 0 2 4 6 8 10 12 B (T) B (T) 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) 0.5 0 0 -0.5 α xy ( µ V/K) α xy ( µ V/K) -0.5 -1 -1 -1.5 -1.5 -2 -2 -2.5 0 2 4 6 8 10 12 0 2 4 6 8 10 12 B (T) B (T) 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) 2 KHg(SCN) 4 is well described by Q1D UCDW • Q2D UDW is consistent with the pseudogap (non-Fermi liquid) phase of CeCoIn 5 14