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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
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?
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.
Institute of Technical Physics 3
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
Institute of Technical Physics 5
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
Phonon temperature divided by Tm
Tph /
Tm
t (ps)
Al
Schmidt, Husinsky and Betz– PRL 85 (2000) 3516
Institute of Technical Physics 6
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
Phonon temperature divided by Tm
Tph
/ T
m
t (ps)
Al
Schmidt, Husinsky and Betz– PRL 85 (2000) 3516
Institute of Technical Physics 7
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
Institute of Technical Physics 8
-1 0 1 2 3 4 50,0
0,2
0,4
0,6
0,8
1,0
experiment fit
Relative change of reflectivity
Aud=20nm
I =12GW/cm2
L=100fs
R/R
(no
rmal
ized
t (ps)
Institute of Technical Physics 9
-1 0 1 2 3 4 50,0
0,2
0,4
0,6
0,8
1,0
experiment fit electron temperature theory
Relative change of reflectivity
Aud=20nm
I =12GW/cm2
L=100fs
R/R
(no
rmal
ized
t (ps)
Institute of Technical Physics 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
Institute of Technical Physics 11
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)
Institute of Technical Physics 12
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
Institute of Technical Physics 13
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
Institute of Technical Physics 14
Evolution of the classical entropy of an isolated system described by the HHCEand of the extended entropy
Institute of Technical Physics 15
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
Institute of Technical Physics 16
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
Institute of Technical Physics 17
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
Institute of Technical Physics 18
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
Institute of Technical Physics 19
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
Institute of Technical Physics 20
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
Institute of Technical Physics 21
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 22