! Master USTH Thermal expansion and thermoelasticity

Master USTH

1

Thermal expansion and thermoelasticity

I-1/ The general picture of a lattice energy U(r) is given in Fig. 1. In the case we consider it as harmonic (red curve in Fig. 1) with the simplified equation given by :

U(r)=U0+a. (r-r0)²

can you explain, without equation, simplify by symmetry principles, why there is no thermal expansion ?

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Figure 1 : representation of the lattice energy U(r) versus the interatomic distance r. Uo is the binding energy.

For a purely harmonic oscillator, whatever the vibratinal energy, the average position will be always in the middle of the segment. So that when the temperature increases, the number of phonons increases too but the average poistion <r> remains the same.

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I-2/ The average force that undergoes a solid is <F> with <F>=<-dU(r)/dr>. (the averge holds on time and space)

If the solid is isolated, what is the average force ? The average force is then <F>=2a<r-r0>.

Master USTH

2

What is then the average interatomic position <r-r0> ?

At equilibrium, and for an isolted solid, the average force that undergoes the solid is <F>=0, so that for this harmonic potential the equilibrium position is <r-r0> =0.

Deduce the value of the linear thermal expansion coefficient (α=d<r>/dT) ?

The thermal expansion is α=d<r-r0>/dT =0 since the average postion is also always 0.

I-3/ Do the same calculation for an anharmonic potential : U(r)=U0+a. (r-r0)² + b. (r-r0) ³

Deduce <F>,<r-r0> and α.

The average force in that case is <F>=2a<r-r0>+3b<(r-r0)²>

For an isolated system, the average force that undergoes the system is <F>=0 so that the mean position becomes :

<r-r0>=(-3b/2a) <(r-r0)²>

This indicates that the mean position is connected to the quadratic displacement.

In that case, because we take an aharmonic potential, we can see that the mean position is no more always 0.

We know in thermodynamic (equipartition energy law, see Maxwell-Boltzmann statistics) that the elastic energy is related to ½ C(r-r0)²=kT (where C, k and T are the elastic constant of an oscillation, the Boltzmann constant and the temperature). So, as a first order, we can say that the mean posiion <r-r0> will be proportionnal to the temperature (a linear thermal expansion is the common situation).

The thermal expansion will be :

α=d<r>/dT =d<r-r0>/dT=d ((-3b/2a) <(r-r0)²>)/dT=d( (-3b/2a)*(2k/C)*T) /dT

=-(3bk/aC)

Master USTH

3

We can sketch this situation on the following figure where, in the anharmonic potential, the mean position deviates from ro as soon as the vibrational energy increases.

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the relaxation. !In the limit of g ", T_e!T_i at all times, and Eqs. #3$ and #4$ yield the standard equation of Fourier heat diffusion, with heat capacity C!C_e"C_i .] After this relax- ation, Fourier thermal diffusion with thermal diffusivity D

!%/(C_e"C_i)&%/C_i is the dominant diffusion process. The large ratio D_e/D&C_i /C_e'100 implies that nonequilibrium electron diffusion is the dominant transport mechanism for times #(_e. For brevity we shall henceforth refer to nonequi- librium electron diffusion as simply electron diffusion in this paper.

With the above linear approximation, Eqs. #3$ and #4$ can be solved analytically in the frequency domain and, by in- corporating the strain generation according to Eq. #2$, the frequency spectrum of the acoustic strain pulse can be obtained:¹⁶

)˜#z,*$! 3B+˜S#*$*² exp#i*z/v$ ,₀%v⁴#1$i*C_i /g$!1"#-*/v$².

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This equation describes the acoustic pulse shape when it has left the near-surface region. Here */2/ is the acoustic fre- quency, ˜S(*) is the Fourier transform of S(t), and p is de- fined by

p!

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Equation #5$ is fairly complex, but it represents a rela- tively straightforward process. Before comparing theory and experiment it is useful to make some rough estimates of the length scales involved in electron and thermal diffusion. Af- ter the pump pulse arrival and during a time of the order of the energy relaxation time of the electrons (_e!C_e/g the nonequilibrium electrons diffuse and transport their energy to a depth 'z_e!(D_e(_e)^1/2!(%/g)^1/2.^14,16 This results in a spa- tially distributed collection of acoustic sources, giving acous- tic pulse broadening #to a temporal duration of 'z_e /v for z_e&-).¹⁴ The relative values of z_e and - therefore determine the importance of electron diffusion on the picosecond acoustic phonon generation. For both Ni and Cr z_e'- '15 nm, as shown in Table I, and so electron diffusion sig- nificantly affects the generation of acoustic phonon pulses in these metals. Once the electrons have relaxed, the lattice temperature T_i continues to vary because of thermal diffu- sion. Provided that the thermal diffusivity is reasonably small,⁴⁰ the broadening effect of thermal diffusion on the acoustic pulses is roughly determined by thermal diffusion during a time (_ac'-/v, equal to the acoustic propagation time across the optical absorption depth. The thermal diffusion length corresponding to this time is given by z_i!(D(_ac)^1/2!(D-/v)^1/2.⁴¹ For both Ni and Cr, D/-v '0.3, and z_i'8 nm '-/2. This distance is smaller than the corresponding length z_e for electron diffusion.

Quantitative calculations of the strain pulse shape are now given. We show the calculated strain pulse and inward sur- face displacement !$0z!$1₀^")(z,t)dz. variations for Ni in Figs. 3#a$ and 3#b$. Similar strain and displacement curves were also found for Cr, and are not shown here. The three strain profiles are evaluated numerically by an inverse Fou- rier transform of Eq. #5$. We have used the relevant physical constants from the literature #Table I$. In particular, bulk values for the thermal conductivity are used because the grain size of the films is large compared to the diffusion lengths z_e and z_i . The solid curves correspond to the strain pulses including both thermal and electron diffusion, the dashed-dotted curves are the profiles taking into account only thermal diffusion (g "),⁴² and the dotted curves are

FIG. 3. Calculated profiles in the time domain for #a$ the strain pulse ) and #b$ the inward surface displacement $0z for Ni at a position far from the surface in the absence of ultrasonic attenuation

#plotted relative to the center of the strain pulse$. #c$ The strain

pulse spectrum for Ni. The solid curve is the strain profile including both thermal and electron diffusion. The dashed-dotted curve is the strain profile including only the effect of thermal diffusion #g "$.

The dotted curve is the strain profile in the absence of diffusion

(g ", % 0.$ The effect of the finite pump and probe pulse du-

rations are included in these curves. The scales are normalized with respect to the heights for no diffusion processes and in the limit of short optical pulse durations.

PICOSECOND ACOUSTIC PHONON PULSE GENERATION . . . PHYSICAL REVIEW B 67, 205421 #2003$

205421-5

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Figure 7: (color online) (a) Left part: schematic view of the intraband relaxation process (red arrows) which leads to incoherent phonon emission which contributes to the local increase of the phonon pres- sure. The interband process is sketched by the green arrow. Right part : because of anharmonicty, the equilibrium position increases when the vibrational energy of the lattice increase. (b) CAP emitted by ther- moelastic process in strong electron-phonon coupling metal Ni [5].

The lattice and electron heat di⇥usion has a weak influence. Copy- right (2003) by The American Physical Society.

increase of temperature) only if the interatomic poten- tial is anharmonic. This consideration is general and independent on the process of lattice heating (electronic energy release by intraband electron-phonon coupling, by non-radiative interband recombination, ). The varia- tion of the internal pressure is connected to lattice ther- mal expansion as :

= 1

3 B ( ⌦ P

⌦ T )

(14)

The integration of this equation provides the variation of the external stress according to :

⌃

= 3 B T

(15)

From the microscopic point of view, the lattice heat- ing comes from the energy transfer from the initially ex- cited electrons towards the phonons sublattice (see Fig / 3(c) for the TTM model). The hot carriers interact in- deed with phonon to relax towards lower energy levels.

When the relaxation processes take place some acous- tic phonons are emitted (depending on the solid, either optical or / and acoustic phonon are emitted and some optical phonons can also decay into acoustic phonons).

These emitted phonons have no phase relation between each other, so that they contribute to the increase of the population of thermal phonons. The calculation of the pressure induced by a modification of the phonon pop- ulation has been derived in classical books of solid state physics [51] or physical acoustics [70] and we have a general expression :

⌃

= ⇥

k

⇤ n

(k) ~ ⌥

⇥

= ⇥

C

⇤ T

(z , t) (16) where ⇥

is an averaged Gr¨uneisen coe ⇤ cient among all k-dependent Gr¨uneisen factors ⇥

= (V /⌥

)( ⇤⌥

/⇤V ) = (1 /⌥

)( ⇤⌥

/⇤⌅ ). C

⇤T

cor- responds to the change of lattice vibrational energy

⇤n

(k) ~ ⌥

.

In metals, once the electrons and phonons have ther- malized (see TTM model in Fig. 3(b)), the lattice tem- perature can be estimated simply by assuming a com- plete transfer of energy from the electron to the phonon subsystem with ⇤T

= Nh⇧/C

, where N(m

) is the photoexcited carriers concentration, C

(J.m

.K

) the lattice heat capacity and h ⇧ the pump quanta energy. In the case of semiconductors, ultrafast lattice heating is connected to the intraband relaxation process of elec- trons (holes) in the conduction (valence) band and is :

⇤T

= N (h⇧ E

) /C

, where E

is the band gap. This calculation does not account for possible interband re- laxation processes, so that only a part of the absorbed energy is converted into heating. The photoexcited car- riers accumulated at the bottom and top of the conduc- tion and valence band contribute to the deformation po- tential stress as discussed in previous Part 3.1 (see Eq.

11). If the carriers undergo non-radiative interband re- laxation process (non-radiative recombination), then we have to take into account an additional source of lattice heating such as ⇤T

= NE

/C

, where E

is the band gap.

3.2.2. Experimental observations of photogeneration of CAP by TE process

Metals : While deformation potential is usually more easy to evidence in semiconductors, thermoelastic pro- 10

I-4/ A laser interacts with a solid and gives rise to a lattice thermal heating. The energy provided by the laser is ΔE and the lattice heat capacity is Cv. What is the lattice temperature increase ?

ΔT =ΔE/Cv

I-5/ What is the consequence of this lattice heating on a the thermal expansion of a harmonic and anharmonic lattice ?

If we consider a harmonic lattice, the heating will increase the population of phonons but no thermal expansion (α=0) wil occur. As a consequence, no phonon pressure appears (no thermoelastic effect). We remind that the linear thermal expansion is α =− 1

3B

∂P

∂T

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V

.

Only when the lattice is anharmonic, some thermal expansion can occur as discussed in the first questions of this exercice. so for as soon as the lattice temperature increases, the internal pressure increases too (the pressure is here negative because the systems expands – thermal expansion) with :

∂P =−3Bα∂T