Fluctuations of the Fluctuations of the superconducting order - PowerPoint PPT Presentation

Fluctuations of the Fluctuations of the superconducting order parameter superconducting order parameter as an origin of the Nernst Nernst effect effect as an origin of the Karen Michaeli and Alexander M. Finkel’stein

Nernst Effect- - High High Tc Tc Materials Materials Nernst Effect The Nernst signal E ν = y −∇ ⋅ T B x . Y. Wang, et al 2005 Disappearance of phase Usually explained by Pairing must coherence at T C although the the existence of survive above T C gap is still finite vortices. . Anderson, 2007 Raghu, et al, 2008 Mukerjee and Huse 2004

Nernst Effect – – Conventional Conventional Nernst Effect Superconductors Superconductors The Nernst signal The strong Nernst signal above Tc can not be explained by the vortex-like fluctuations. E ν = y −∇ ⋅ T B x Nb 0 Si . 15 0 . 85 A. Pourret, et al 2007 It has been suggested that the fluctuations of the order parameter cause the effect.

Nernst Effect - - Metals Metals Nernst Effect The electric current in response to temperature gradient in a system with two species of particles (electrons and holes): ( ) ( ) ( ) The Boltzamann equation for the δ ∂ ε δ ε f k f e v f = ⋅ ∇ × ⋅ / 0 / m e h k k e h k v T B distribution function: τ ∂ ∂ k T c k k k d d ∫ ∫ = − δ + δ j e v f e v f The electric current : ( ) ( ) π k π k e d e d h 2 2 ( ) ⎡ ⎤ ∂ ε ε 2 τ ε 2 τ ∇ = ∫ k d f v v T − = 0 0 x k k k k k x j e ⎢ ⎥ The longitudinal electric current: ( ) ∂ ε e π d 2 ⎣ ⎦ d d T k The transverse electric current: ( ) ∂ ε ε τ ∇ 2 = ∫ [ ] eB k ( ) d f v T ω C = ω τ − − ω τ ≠ 0 0 y k k k x j e ( ) ∂ ε π e C C d 2 mc d T k

Particle-hole symmetry does constrain the magnitude of the Nernst effect. Under the approximation of a constant density of states: ( ) τ ∇ ε ∂ ε 2 ( ) v T d f ∫ = ω τ ν ε = 2 0 2 0 y F x k k j e ( ) 0 ∂ ε π e C k d 2 d T k For the collective modes the effective density of states is far from being a constant. The charged modes such as The neutral modes are fluctuations of superconducting not deflected by the order parameter generate the Lorentz force. Nernst effect.

Particle-hole symmetry does constrain the magnitude of the Nernst effect. Under the approximation of a constant density of states: ( ) τ ∇ ε ∂ ε 2 ( ) v T d f ∫ = ω τ ν ε = 2 0 2 0 y F x k k j e ( ) 0 ∂ ε π e C k d 2 d T k The charged modes such as The neutral modes are fluctuations of superconducting not deflected by the order parameter contribute to the Lorentz force. Nernst effect.

σ α ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ j E ⎜ ⎟ = ⎜ ⎟ ⎜ ⎟ The Nernst Nernst Coefficient Coefficient The ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ~ α κ ∇ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ j T E α α σ − α σ ≈ E xy e = = y xy xx xx xy σ N e −∇ σ + σ 2 2 N xx T x xy xx α xx vanishes under the approximation of a constant density of states A. Pourret, et al 2007

The Nernst Nernst Coefficient Coefficient The ∇ The vertex of the temperature gradient is: ( ) T ε ε v T Integrating over the frequency ε terms that in the electric conductivity are real becomes imaginary. For example - the Aslamazov-Larkin term at B → 0 lnT/Tc<<1 : ∇ ω ω ∂ [ ] T d q d n ( ) ( ) ∫ ∝ η ω − ω 2 , , AL P j e L q L q ( ) + π ∂ ω π 1 e 4 R A 2 d T T L is the propagator of the fluctuations of the superconducting order parameter: − 1 ⎡ ⎤ πω 1 T i = + η 2 + χω ln m ⎢ ⎥ L q , ν R A 8 ⎣ ⎦ T T C χ – is zero under the constant density of states approximation

Nernst Effect- - Kubo Formula Kubo Formula Nernst Effect Luttinger approach – Luttinger approach – J. M. Luttinger 1964. Thermal conductivity: Introducing a gravitational field that is coupled to the Hamiltonian density: � ( ) ( ) ( ) ∫ ∫ − = + γ st H h r dr e r h r dr 0 0 Deriving the Kubo formula for the linear response to the field: � The E.O.M for the density matrix: ( ) ρ & + ∇ = [ ] d t ( ) 0 h j = ρ , i H t E dt ( ) − β = ∇ γ H j r e 0 j j E E E Luttinger connected between the response to the gravitational field and the temperature gradient .

Magnetization Magnetization There has been a long discussion about the contribution of magnetization to the thermoelectric transport currents. For example: Obraztsov Sov. Phys. Solid State 1965 Smrcka and Streda J. Phys. C 1977 Cooper, Halperin and Ruzin PRB 1997 The heat current that describes the change in the entropy. In the presence of magnetic field the thermodynamic expression for the heat contains the magnetization term: μ dQ = TdS = dE - dN + MdB. The Kubo formula is not enough the contribution from the magnetization must be added.

Nernst Effect- - Quantum Kinetic Equation Quantum Kinetic Equation Nernst Effect ⎛ < ⎞ ⎛ ⎞ R K G G G G ( ) ˆ ⎜ ⎟ ⎜ ⎟ = ⇔ , ; ' ' The Keldysh Green function: T G r t r t ⎜ ⎟ ⎜ ⎟ > A G G ⎝ ⎠ ⎝ ⎠ G ~ T The current response to temperature gradient : [ ] ( ) ( ) ( ) ( ) < ˆ ˆ = ∇ ∇ + ∇ ∇ ˆ ˆ 2 j i e v T G T e v T L T Δ e e The quasi-particles excitations The fluctuations of the order parameter ) [ ] ( − 1 ˆ ˆ = λ − Π , ; ' , ' L r t r t 1 1 v e and v Δ are the renormalized velocities ( ) ( ) ˆ ˆ ∇ and are the solution of the quantum kinetic equation. ∇ ; , ; ' , ' ; , ; ' , ' G T r t r t L T r t r t 1 1 1 1

Nernst Effect- - Magnetization Magnetization Nernst Effect + ⎛ ⎞ ' r r ˆ ∇ = − ε ⎜ ; , ' ; ⎟ G T R r r 2 ⎝ ⎠ ( ) ˆ ∇ T r − ε ; ' ; G r ( ) ˆ ⋅ ∇ ∂ − ε ' ; R T G r r − ε 0 ∂ ε T Translation invariant part Local equilibrium part ( ) ( ) ˆ ∇ = − Π ∇ L T L T L 0 0 − 1 ⎡ ⎤ ε ∇ ⎢ ⎥ d T ( ) → ∫ ∂ = < − ε − j y lim ' ; ⎢ ⎥ ln 99999 ie MG r r ∂ 0 B π e 2 ⎢ ⎥ ' T r r ⎢ ⎥ ⎣ ⎦ According to the third law of The Peltier coefficient is thermodynamics related to the flow of entropy α → → 0 0 when T

The Peltier Peltier Coefficient Coefficient The The contributing diagrams: and the magnetization: ( ) ⎧ ⎫ ⎡ ⎤ ⎛ ω + Ω + ⎞ ∂ ∞ ⎪ 1 / 2 ⎪ 1 ⎛ 1 ⎞ 4 N eB T ∑∑ eDH α = − − ν − ψ ⎜ + ⎟ − ψ ln ln ⎜ ⎟ Ω = mag ⎨ ⎢ m C ⎥ ⎬ ⎜ ⎟ ∂ π π xy 2 4 2 ⎪ ⎝ ⎠ ⎪ ⎝ ⎠ c B ⎣ T T ⎦ ⎩ ⎭ = ω c 0 N C m ~ ~ ~ Ω ~ Ω = C π C 4 ~ Ω C / T

X The Peltier Peltier Coefficient Coefficient The T << ~ ln 1 Ω C << Ω C / T T T C Classical fluctuations – coincide with Ω the Phenomenological and e α ≈ C microscopic result of Ussishkin et al, ( ( ) ) xy 192 ln / T T T B 2002 and Ussishkin 2003 C Experimental data from A. Pourret, et al 2007 film of thickness 35 Nb Si nm 0 . 15 0 . 85 = 380 and T C mK 2 = D = 385 cm T MF 0 . 187 mK sec C

X The Peltier Peltier Coefficient Coefficient The ~ Ω / C T C T Ω C << >> ln 1 T T C Ω e α ≈ C ( ) π 2 xy 24 ln / T T T C

X The Peltier Peltier Coefficient Coefficient The ~ Ω / C T C T Ω C << >> ln 1 T T C 1 < ω < Quantum fluctuations – T τ ⎛ ⎞ ⎛ ⎞ 1 T ⎜ ⎟ − ln ⎜ ln ⎟ ln ln The diagrams yield contributions of the order: ⎜ ⎟ τ ⎝ ⎠ ⎝ ⎠ T T C The logarithmically divergent terms are canceled out by the magnetization Trace of the third law of thermodynamics

X The Peltier Peltier Coefficient Coefficient – – The High Magnetic field High Magnetic field X ~ Ω / C T C B >> ln 1 Ω C >> T B C Ω C The diagrams include contributions proportional to . T These terms are canceled out by the magnetization. The Nernst signal goes to zero at T → 0. 2 eT α ≈ ( ) Ω xy 3 ln / B B Consistent with the C C third law of thermodynamics.

The Peltier Peltier Coefficient as a Function of the Coefficient as a Function of the The Magnetic Field Magnetic Field X ~ Ω / C T C α xy

The Peltier Peltier Coefficient as a Function of the Coefficient as a Function of the The Magnetic Field Magnetic Field X ~ Ω / C T C Ω e α ≈ C ( ( ) ) xy 192 ln / T T T B C α xy