Two –Temperature Model (Chap 7.1.3)

Two –Temperature Model (Chap 7.1.3)

6 Dec 2011Injoo Hwang

Thermal Engineering & Manufacturing Process Laboratory

Contents

1. Two – Temperature Model1

2. Conclustion2


Two-Temperature Model

Two – Temperature Model A pair of coupled nonlinear equations governing the effective temperatures of electrons and phonons (by S.I. Anisimov et al. 1974)

Assumed in the Two – Temperature Model

Electron System Phonon System

Equilibrium Equilibrium

Mutually

Non-Equilibrium



aseee

e qΤΤGΤtΤC

ses

s ΤΤGtΤC

: Electron System

: Phonon System

: the volumetric heat capacityepe cC )(

G

aq

e

sps cC )(

: the electron – phonon coupling constant

: the source term ( during the laser pulse )

: the thermal conductivity ( heat conduction by phonons is neglected )∵



Examining parameters

esee

epeeepe TTkncmncC

F

2B

2

, 2)(

- The thermal conductivity

eevc F,31

e

e

mTkn

F

F

2B

2 223

1

es

eqe

e

e TT

Tmkn

3

2B

2

F

2B

2

, 2 Tknc e

ev e

e m F2

FF2

- The coupling constant- The volumetric heat capacity

s

aee

TvnmG

6

22

eq

Bae

e

eeq

aee kvn

knmvnmG

18

)(36

24

2B

2

22

or

se

eeq T

knm

2

B2

3

G is independent of temperature and proportional to the square of the speed of sound in the metal


Two-Temperature Model3/1

DB

34

aa nhkv

0

2D

2B

0

2D

2B

3

2

518.0412

s

e

s

e

Tkn

TknG

goldfor K)W/(m109.2 316

The associated electron and phonon temperatures near the surface

Ultrafast thermoreflectance experiments

tpt

eT

sTaq

Specimen

Probe beam

Pump beam

Thermoreflec-tance signal

The electron temperature rises quickly during the pulse and begins to decrease afterward

The lattice temperature gradually increases until the electron and lattice systems reach a thermal equilibrium

Measuring the effective electron temperature by no contact thermometer which performs the femtosecond or picosecond thermoreflectrance technique



aseee

e qTTGTtTC se

ss TTGtTC

as

see

e qtTCT

tTC

t

TCC

CqT

CtT e

s

e

s

ae

s

s

2

aseee

e qTTGTtTC asee

ee q

tTT

tGT

ttT

tC

tq

tTG

tTGT

ttTC ase

ee

e

2

2

2

tq

GtTT

tGtT

GC

tT ae

eees

11 2

2

2

tq

GtTT

tGtT

GC

tT

CC

CqT

Cae

eeee

s

e

s

ae

s

11 2

2

22

2

222

tT

GCC

tTCC

tq

GCqT

tGCT eseeseasa

es

e

2

2qT2

T2 1

tT

tT

tqqT

tT eeaa

ee

qT

q

T

1

see

se

se

s

CCCGCC

CCGC

Differential equations for the electron temperature



2

2q2

T2 1

tT

tTqT

tT ssa

ss

Differential equations for the phonon temperature

se CC / GCs /T TTq // GCCCC esee

- Identical to the lagging heat equations

- Following the general trends

- τq : Thermalization time

(thermal time constant for the electron system to reach

an equilibrium with the phonon system)



- For noble metals at room temperature -

• Relaxation time : τ ~ 30 to 40 fs• Thermalization time : τq ~ 0.5 to 0.8 ps• Retatdation time : τT ~ 60 to 90 ps

The two-temperature model cannot be applied to t < τ ( the limitation of Fourier’s law )

- Difficult issues -

The processes below 20 fs (relaxation time of Cr is about 3 fs)

• Electron – electron inelastic scattering• Thermionic emission, ionization, phase transformation, chemical reaction• The reduced pulse width include widened frequency spectrum• Increased pulse intensity, decreased pulse energy


Conclusion

Two – Temperature Model (their own local equilibrium not in mutual equilibrium)

Examine parameters (the volumertic heat capacity, the thermal conductivity, the coupling constant)

The behavior of the electron and phonon temperatures

Non contact thermometer ( thermoreflctance technique known as pump-and-probe method)

Derive PDE for the electron or phonon temperature

Documents

Two –Temperature Model (Chap 7.1.3)