EXPERIMENTAL METHODS AND SAMPLES Depending on the experimental configuration (1–3), the pump and probe experiments were performed on two different fs setups

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MASTER of PHYSICS

We consider the following properties of two metals (copper and titanium) given in the table.

(Table from Lejman et al JOSA B 2014)

We consider that a pulse laser of 100 fs interacts with the metal. This laser energy pulse is 1 nJ. The laser excites a surface of the metal of 0.1mm * 0.1 mm.

1) Give the value of the electronic temperature in both metal just after the laser light absorption.

For a quantum electron gas we have :

!E =C_e!T_e ="eT_e!T_e

After an integration we obtain :

phase transitions [26]. Hot carriers can also play a significant role in femtochemistry processes [27].

2. EXPERIMENTAL METHODS AND SAMPLES

Depending on the experimental configuration (1–3), the pump and probe experiments were performed on two different fs setups. A first setup employs a mode-locked Ti:Sapphire cavity-dumped fs laser of 500 kHz repetition rate. The laser beam output at 800 nm is split into a pump and a probe beam by a polarizing beam splitter. The probe beam passes through a second-harmonic generation (SHG) BBO crystal to obtain a probe wavelength of 400 nm. The pump is chopped by an acousto-optic modulator at around 100 kHz. Both beams are focused by ×10 microscope objectives. The spot size of the laser beams were around 20 μm in diameter, measured with the knife method. This spot size is much larger than the sample thickness, so that the 1D approximation is appro- priate for the description of transport properties of coherent acoustic phonon or electron wave packets (i.e., plane-wave approximation). The transient optical signal recorded with a balanced photodiode is processed with a lock-in amplifier at the pump modulation frequency. The second setup em- ployed a Ti:Sapphire laser of tunable wavelength. The laser beam output is split into a pump beam and a probe beam, which pumps an optical parametric oscillator (OPO) having an intra-cavity SHG doubling crystal. The OPO allows the probe wavelength to be tunable from 520 to 600 nm with a step of around 1 nm. The pump beam is chopped with an electro-optic modulator at around 100 kHz, and the transient optical signal is recorded with a balanced photodiode and processed with a lock-in amplifier at the pump modulation frequency. The pump fluence range is20–60 μJ∕cm²and that of the probe fluence is around 5 times smaller. In all experiments, the linearity of the optical response has been checked by performing different experiments with variable probe fluence.

We have chosen to study the superdiffusive transport of electronic energy in copper for different reasons. First, electrons in copper are known to have large mean free paths of the order of hundreds of nanometers. Copper has a small electron–phonon coupling constant, with g≈20–60×10¹⁵ W · m⁻³· K⁻¹ [4,6,28] (see Table1). Second, copper has a d band optical transition centered at 574 nm [29], which lies within the optical range of our OPO (520–600 nm). Consequently, the OPO output has a probe wavelength suitable for the inves- tigation of the hot electrons dynamics around the Fermi energy, and also to probe with better efficiency the coherent acoustic phonons. It has been shown in the past that, indeed, the detection of coherent acoustic phonons in copper at 800 nm is difficult [16,33]. High purity (99.9999%) copper film has been deposited onto a 40 nm thick titanium film lying on a glass substrate (both sides polished quartz glass) of 1 mm thickness (see Fig. 1). Contrary to copper, titanium exhibits a much larger electron–phonon coupling constant (g∼200–500×10¹⁵ W · m⁻³· K⁻¹) [4] (see Table 1) that per- mits the generation of quite high frequency coherent acoustic phonons [21], and can, therefore, be considered as an efficient thermoelastic transducer. To protect the copper free surface from oxidation, a∼40nm thin layer of cerium oxide has been deposited.

3. RESULTS AND DISCUSSION

A. Generation of Coherent Acoustic Phonons by Superdiffusive Hot Electrons

Following the classical picosecond acoustic scheme, we performed experiments in front–back configuration 1 [see Fig.2(a)]. In this case, the laser pump excites the titanium film and, through a thermoelastic process, leads to the generation of picosecond acoustic pulses that propagate within the copper film, and are detected at the front surface. The transient optical reflectivity signal exhibits different features. We can easily observe some periodic events at time aroundt≈45, 145, and 230 ps. Two of these successive events are separated by a time interval very close to2τ_Cu!2H∕V_Cu. This time ex- actly corresponds to a round trip time of the longitudinal acoustic phonons in copper. Consequently, we can attribute these events to the acoustic pulse photo-generated in the titanium film, and traveling back and forth within the copper film at the longitudinal acoustic speed V_Cu!4730 m∕s [34]. The short longitudinal acoustic pulses in metals and detection of successive acoustic echoes have been already ex- tensively described in picosecond acoustics and is not of concern here [15,35–37]. However, to clearly explain the detection process in the presence of the thin cerium oxide pro- tecting layer (conf. 1), a complete numerical modeling will be given in Subsection3.C. Furthermore, for the signal of conf. 1, we can note that a sharp offset of the transient reflectivity signal appears before the first acoustic pulse detection. This sharp peak, of about 3 ps duration, appears prior to the arrival of the acoustic phonons coming from the titanium film, within a delay of τ_Cu!H∕V_Cu. As a consequence, this sharp peak corresponds to the so-called zero time, which defines the onset of pump excitation. Such a peak has already been observed in similar time-of-flight experiments in gold [3] and silver [10], and is attributed to the arrival of superdiffusive laser-excited electrons. We will discuss the optical signature of these hot electrons ultrafast transport in Subsection 3.B.

In the second front–back configuration, the pump beam excites the copper film and the probe beam detects the

Table 1. Some Physical Properties of Copper and Titanium

Copper Titanium Optical Properties: skin depth (nm)

[29,30], Wavelength (nm)

400 ∼14.5 ∼14.4

572 ∼14.5 ∼16.8

780 ∼12.4 ∼18.8

800 ∼12 ∼19.3

Mechanical Properties

Sound velocityV_S"m∕s# 4730 5090

Densityρ"kg∕m³# 8960 4500

Thermal and Transport Properties

Thermal expansion (K⁻¹) [31] 16×10⁻⁶ 8.5×10⁻⁶ Lattice heat capacity (at 300 K)

(J · mol⁻¹· K⁻¹)

24.4 25.06

Electronic heat capacity (J · m⁻³· K⁻¹), T_e is the electronic temperature

96.6T_e [31] 313 T_e [31,32]

Electron–phonon coupling constantg (10¹⁵ W · m⁻³· K⁻¹) [4,6,28]

20–60 200–500 Heat conductivityκ (W · m⁻¹· K⁻¹) [31] 400 80

284 J. Opt. Soc. Am. B / Vol. 31, No. 2 / February 2014 Lejmanet al.

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!E = 1

2!_e(T_{e, f}² "T_e,i²)

with Te,f and Te,I the final and initial electrons temperature.

The final temperature is then :

T_e,_f = T_e,i² + 2!E

!_e

The parameter λe is given in the table above. We will consider that the initial temperature is 300K. We have then to calculate the energy par unit of volume deposited by the laser. We will consider that the energy absorbed is 1nJ. That energy is deposited in a volume determined by the area of the irradiated surface and the penetration of the light (skin depth ζ) . The volume where the energy is deposited is then :

V = Ae^!x/^! dx

0

"

# ⁼^$^%^!^A^!^e^!^x/^!^&^'0

"

= A!

IN the case of copper for example, we can take a skin depth ζ=12nm. So V becomes V=1.2.10^-‐15 m^-‐3. So the energy per volume is ΔE=1nJ/V =0.83.10⁶ J/m³.

So the final temperature is :

T_e,_f = 300² + 2!0.83!10⁶ 96.6

2) Considering the electron-‐phonon coupling parameters given in the table, can you estimate the characteristic time after which the electron subsystem has given its energy back to the lattice subsystem.

!_e!ph " C_e / g

So for example for copper, that time, for an electronic temperature Te =500K, is

!_e!_ph " 96.6 * 500 / 40.10¹⁵ =1.2ps

3) Give the final value of the lattice temperature.