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Institute of Technical Physics 1 conduction Hyperbolic heat conduction equation (HHCE) Outline 1. Maxwell – Cattaneo versus Fourier 2. Some properties of the HHCE 3. Objections against the HHCE – a misunderstanding 4. A physical explanation of the relaxation time Bernd Hüttner CPhysFInstP, Stuttgart 1. Maxwell – Cattaneo versus Fourier

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Institute of Technical Physics 2 1. What is wrong with the parabolic heat conduction equation? It predicts an infinite propagation velocity for a finite thermal pulse ! How can this happens? = const. The cause and effect in this case occur at the same instant of time, implying that its position is interchangeable, and that the difference between cause and effect has no physical significance.

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Institute of Technical Physics 3 Maxwell-Cattaneo equation Velocity: damped wave-like transport diffusive energy transport

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Institute of Technical Physics 4 Schmidt, Husinsky and Betz– PRL 85 (2000) 3516 L = 30fs

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Institute of Technical Physics 5 Schmidt, Husinsky and Betz– PRL 85 (2000) 3516

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Institute of Technical Physics 6 Schmidt, Husinsky and Betz– PRL 85 (2000) 3516

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Institute of Technical Physics 7 David Funk et al. – HPLA 2004 Au L = 130fs

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Institute of Technical Physics 8

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10 In this paper the HHCE is inspected on a microscopic level from a physical point of view. Starting from the Boltzmann transport equation we study the underlying approximations. We find that the hyperbolic approach to the heat current density violates the fundamental law of energy conservation. As a consequence, the HHCE predicts physically impossible solutions with a negative local heat content. The physical defects of hyperbolic heat conduction equation Körner and Bergmann - Appl. Phys. A 67 (1998) 397

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Institute of Technical Physics 11 Derivations of the MCE 1. Simple Taylor expansion : 2. From the Boltzmann equation Hüttner – J. Phys.: Condens. Matter 11 (1999) 6757 3. In the frame of the Extended Irreversible Thermodynamics ( 0. Maxwell (1867) has suppressed the term because he assumed that the time is too short for a measurable effect)

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Institute of Technical Physics 12 2. Classical irreversible thermodynamics Based on the assumption of local thermal equilibrium, Onsager linear relations J i = L ik ·X k and positive entropy production Fourier’s law q = - l gradTparabolic diff. equation local in space and time, no memory, close to equilibrium

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Institute of Technical Physics 13 3. Extended thermodynamics Based on an extension of thermodynamical variables (S, T, p, V, fluxes) Taking into account only the heat flux q one finds: hyperbolic diff. equation Temperature: nonlocal, with memory, far from equilibrium

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Institute of Technical Physics 14 Evolution of the classical entropy of an isolated system described by the HHCE and of the extended entropy

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Institute of Technical Physics 15 The physics behind the hyperbolic heat conduction or what is the physical meaning of EcEc EvEv E gap E Simplified scheme of a semiconductor Assume: 1.Initial density in E c is zero 2.Valence band is flat and thin Both assumption are not essential but comfortable

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Institute of Technical Physics 16 fs laser pulse hits the target and excites a large number of electrons into the conduction band EcEc E photon = L EvEv E gap E EcEc E photon = L EvEv E gap E E el = L - E gap

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Institute of Technical Physics 17 Electrons thermalize very fast due to the large available phase space an intensive quantity Electron temperature starts to relax with characteristic time: Important point, electronic specific heat is an extensive quantity Heat exchange coefficient

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Institute of Technical Physics 18 Electron density – Beer’s law Since c e ~ n e ·T e follows T ~ n e ·T e That’s why, T e relaxes faster with increasing distance leading to a build up of a temperature gradient

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Institute of Technical Physics 19 Relaxation time of electron system Relation with the Drude scattering time

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Institute of Technical Physics 20 An example: T i = 300K, n i =(0; 10 16 )cm -3 (!), E gap = 0.5eV, L opt = 20nm E L = 1eV, L = 100fs, n f = 10 18 cm -3 dotted: n i = 0cm -3 solid: n i = 10 16 cm -3 Times: red: 50fs green: 100fs blue: 500fs black: 1ps

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Institute of Technical Physics 21 Temperature gradient Thermal current q = - 0 (T e /T 0 ) T e Times: red: 50fs, green: 100fs blue: 500fs, black: 1ps

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Institute of Technical Physics 22

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Hyperbolic Heat Equation 1-D BTE assume Integrate over velocity space assuming that is an averaged relaxation time (4.63a) (4.64) or.

Hyperbolic Heat Equation 1-D BTE assume Integrate over velocity space assuming that is an averaged relaxation time (4.63a) (4.64) or.

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