Hyperbolic heat conduction equation
(HHCE)
Bernd Hüttner CPhysFInstP, Stuttgart
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
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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?
U
 divQ  0
t
Q  T
 = const.
Q  t   T  t 
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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Maxwell-Cattaneo equation
Q

 Q  T
t
2
2

D

T

T

Te
1
1
I
qe
Velocity: e v 
 z 
e
 A  e I  qe 
 
2
2
ae  t ae  t
e
t
z

t   :
x2 ~ t2
damped wave-like transport
t   :
x2 ~ t
diffusive energy transport
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Phonon temperature divided by T m
1,1
1,0
0,9
Al
Tph / Tm
0,8
0,7
ETTM: delay = 0fs
0,6
0,5
L = 30fs
0,4
0,3
0,0
0,5
1,0
1,5
2,0
2,5
t (ps)
Schmidt, Husinsky and Betz– PRL 85 (2000) 3516
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Phonon temperature divided by T m
1,1
1,0
0,9
Al
Tph / Tm
0,8
0,7
ETTM: delay = 0fs
ETTM: delay = 30fs
0,6
0,5
0,4
0,3
0,0
0,5
1,0
1,5
2,0
2,5
t (ps)
Schmidt, Husinsky and Betz– PRL 85 (2000) 3516
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Phonon temperature divided by T m
1,1
1,0
0,9
Al
Tph / Tm
0,8
0,7
ETTM: delay = 0fs
ETTM: delay = 30fs
TTM: delay = 0fs
0,6
0,5
0,4
0,3
0,0
0,5
1,0
1,5
2,0
2,5
t (ps)
Schmidt, Husinsky and Betz– PRL 85 (2000) 3516
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Relative change of reflectivity
1.05
1
Au
L = 130fs
0.95
TTM
0.9
0.85
0.8
ETTM
0.75
0.7
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
t (p s)
David Funk et al. – HPLA 2004
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Relative change of reflectivity
Au
d=20nm
2
I =12GW/cm
L=100fs
1,0
R/R (normalized
0,8
0,6
experiment
fit
0,4
0,2
0,0
-1
0
1
2
3
4
5
t (ps)
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Relative change of reflectivity
Au
d=20nm
2
I =12GW/cm
L=100fs
1,0
R/R (normalized
0,8
0,6
experiment
fit
electron temperature
theory
0,4
0,2
0,0
-1
0
1
2
3
4
5
t (ps)
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The physical defects of hyperbolic heat conduction equation
Körner and Bergmann - Appl. Phys. A 67 (1998) 397
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.
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Derivations of the MCE
(0. Maxwell (1867) has suppressed the term because he assumed
that the time is too short for a measurable effect)
1. Simple Taylor expansion:
2. From the Boltzmann equation
Q  t    Q  t   
Q
t
Hüttner – J. Phys.: Condens. Matter 11 (1999) 6757
3. In the frame of the Extended Irreversible Thermodynamics
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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 = - l gradT
parabolic diff. equation
local in space and time, no memory, close to equilibrium
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3. Extended thermodynamics
Based on an extension of thermodynamical variables (S, T, p, V, fluxes)
Temperature:
1
1

 const.q  q
 Teq
Taking into account only the heat flux q one finds:

q
 q  T
t
hyperbolic diff. equation
nonlocal, with memory, far from equilibrium
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Evolution of the classical entropy of an isolated system described by the HHCE
and of the extended entropy
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The physics behind the hyperbolic heat conduction
or what is the physical meaning of 
Simplified scheme of a semiconductor
E
Ec
Egap
Assume:
1. Initial density in Ec is zero
2. Valence band is flat and thin
Both assumption are not essential
but comfortable
Ev
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fs laser pulse hits the target and excites a large number of electrons into the
conduction band
E
E
Eel= L - Egap
Ec
Egap
Egap
Ephoton = L
Ev
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Ec
Ephoton = L
Ev
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Electrons thermalize very fast due to the large available phase space
Te 
L  E gap
an intensive quantity
kB
Electron temperature starts to relax with characteristic time:
Heat exchange coefficient
ce (Te )
T 
 Qe
h ex
3 3
222
 
h ex
h ex
kBk B
Important point, electronic specific heat is an extensive quantity
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Electron density – Beer’s law
electron density
Since ce ~ ne·Te follows T ~ ne·Te
n
That’s why, Te relaxes faster with
increasing distance leading to a
build up of a temperature gradient
distance
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Relaxation time of electron system
ce (Te )
T 
 Qe
h ex
Relation with the Drude scattering time
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Qe
D v 2F

3 v2th
19
An example: Ti = 300K, ni =(0; 1016)cm-3 (!), Egap = 0.5eV, Lopt = 20nm
EL = 1eV, L = 100fs, nf = 1018cm-3
Electron temperature
6000
dotted: ni = 0cm-3
solid: ni = 1016cm-3
5000
T (K)
4000
Times:
red: 50fs
green: 100fs
blue: 500fs
black: 1ps
3000
2000
1000
0
0
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20
30
40
z (nm)
50
60
70
80
20
Thermal current q = - 0(Te/T0)Te
Temperature gradient
20
Thermal current
1200
0
1000
20
40
800
q
60
600
80
400
100
200
120
140
0
160
0
20
40
60
80
0
100
20
40
60
80
100
z (nm)
Times:
red: 50fs, green: 100fs
blue: 500fs, black: 1ps
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