Institute of Technical Physics 1 conduction Hyperbolic heat conduction equation (HHCE) Outline 1....

Institute of Technical Physics 1

Hyperbolic heat conductionconduction equation (HHCE)

Outline

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


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?

UdivQ 0

t

Q T

Q t T t =

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.


Maxwell-Cattaneo equationt

Q TQ

Velocity: Dv

2 2x ~ t

2x ~ t

t : damped wave-like transport

t : diffusive energy transport

2ze e

2e e

2qe e

qe2e

T I

a tt

T T1 1A e I

a tz

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

0,0 0,5 1,0 1,5 2,0 2,50,3

0,4

0,5

0,6

0,7

0,8

0,9

1,0

1,1

ETTM: delay = 0fs

Phonon temperature divided by Tm

Tph /

Tm

t (ps)

Al

L = 30fs


0,0 0,5 1,0 1,5 2,0 2,50,3

0,4

0,5

0,6

0,7

0,8

0,9

1,0

1,1

ETTM: delay = 0fs ETTM: delay = 30fs


Tph /

Tm

t (ps)

Al

Schmidt, Husinsky and Betz– PRL 85 (2000) 3516


0,0 0,5 1,0 1,5 2,0 2,50,3

0,4

0,5

0,6

0,7

0,8

0,9

1,0

1,1

ETTM: delay = 0fs ETTM: delay = 30fs TTM: delay = 0fs


Tph

/ T

m

t (ps)

Al

Schmidt, Husinsky and Betz– PRL 85 (2000) 3516


David Funk et al. – HPLA 2004

0 0.5 1 1.5 2 2.5 3 3.5 4 4.50.7

0.75

0.8

0.85

0.9

0.95

1

1.05

t (ps)

Relative change of reflectivity

TTM

ETTM

AuL = 130fs


-1 0 1 2 3 4 50,0

0,2

0,4

0,6

0,8

1,0

experiment fit


Aud=20nm

I =12GW/cm2

L=100fs

R/R

(no

rmal

ized

t (ps)


-1 0 1 2 3 4 50,0

0,2

0,4

0,6

0,8

1,0

experiment fit electron temperature theory


Aud=20nm

I =12GW/cm2

L=100fs

R/R

(no

rmal

ized

t (ps)


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


Derivations of the MCE

1. Simple Taylor expansion: QQ t Q t

t

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)


2. Classical irreversible thermodynamics

Based on the assumption of local thermal equilibrium,

Onsager linear relations Ji = Lik·Xk

and positive entropy production

Fourier’s law q = - gradT parabolic diff. equation

local in space and time, no memory, close to equilibrium


3. Extended thermodynamics

Based on an extension of thermodynamical variables (S, T, p, V, fluxes)

qq T

t

Taking into account only the heat flux q one finds:

hyperbolic diff. equation

Temperature:

eq

1 1const.q q

T

nonlocal, with memory, far from equilibrium


Evolution of the classical entropy of an isolated system described by the HHCEand of the extended entropy


The physics behind the hyperbolic heat conduction

or what is the physical meaning of

Ec

Ev

Egap

E

Simplified scheme of a semiconductor

Assume:1. Initial density in Ec is zero2. Valence band is flat and thin

Both assumption are not essentialbut comfortable


fs laser pulse hits the target and excites a large number of electrons into

the

conduction band

Ec

Ephoton = L

Ev

Egap

E

Ec

Ephoton = L

Ev

Egap

EEel= L - Egap


2

exB

3h

k

Electrons thermalize very fast due to the large available phase

space

L gape

B

ET

k

an intensive quantity

Electron temperature starts to relax with characteristic time:

( )e eT Qe

ex

c T

h

Important point, electronic specific heat is an extensive quantity

Heat exchange coefficient 2

exB

3h

k

2

exB

3h

k


Electron density – Beer’s law

electron density

distance

n

Since ce ~ ne·Te follows T ~ ne·Te

That’s why, Te relaxes faster with

increasing distance leading to a build up of a temperature gradient


Relaxation time of electron system

( )e eT Qe

ex

c T

h

2D F

Qe 2th

v

3 v

Relation with the Drude scattering time


An example: Ti = 300K, ni =(0; 1016)cm-3 (!), Egap = 0.5eV, Lopt

= 20nm

EL = 1eV, L = 100fs, nf = 1018cm-3

0 10 20 30 40 50 60 70 800

1000

2000

3000

4000

5000

6000Electron temperature

z (nm)

T (

K)

dotted: ni = 0cm-3

solid: ni = 1016cm-3

Times:red: 50fsgreen: 100fsblue: 500fsblack: 1ps


Temperature

gradient

0 20 40 60 80 1000

200

400

600

800

1000

1200Thermal current

z (nm)q

Thermal current q = - 0(Te/T0)Te

Times:red: 50fs, green: 100fsblue: 500fs, black: 1ps

0 20 40 60 80 100160

140

120

100

80

60

40

20

0

20

Institute of Technical Physics 1 conduction Hyperbolic heat conduction equation (HHCE) Outline 1....

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