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MASTER of PHYSICS
Examination P208 “Ultrafast Phenonema” (duration 2 hours, no document) Lecturer : Prof. P. Ruello, Univ Maine, FRANCE.
I-‐ ELECTRON-‐PHONON COUPLING
We consider the following properties of two metals (copper and titanium) given in the table (Table from Lejman et al JOSA B 2014).
A pulse laser of 100 fs interacts with metals at room temperature (T
i=300K). The laser excites the surface S of the metal with S= 0.1mm * 0.1 mm. The laser energy per pulse that is absorbed by the metal (Ti and Cu) is 1 nJ. The wavelength of the exciting laser is 800 nm and the corresponding penetration depth is given in the table.
1) Give the value of the electronic temperature just after the laser light absorption for both Cu and Ti. We will consider here that only electrons have absorbed the light energy.
2) Considering the Sommerfled model for the electronic subsystem, we can show that the electronic pressure P
eat thermodynamic equilibrium is :
P
e= 2 3
E V
-‐-‐-‐-‐-‐-‐-‐ Laser action à
!P
e= 2 3
!E V
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 cav- ity-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 μ m 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 fre- quency. The pump fluence range is 20 – 60 μ J ∕ cm
2and that of the probe fluence is around 5 times smaller. In all experi- ments, 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, elec- trons in copper are known to have large mean free paths of the order of hundreds of nanometers. Copper has a small elec- tron – phonon coupling constant, with g ≈ 20 – 60 × 10
15W · m
−3· K
−1[4,6,28] (see Table 1). 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 en- ergy, 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 × 10
15W · 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 ∼ 40 nm 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 tran- sient optical reflectivity signal exhibits different features. We can easily observe some periodic events at time around t ≈ 45, 145, and 230 ps. Two of these successive events are separated by a time interval very close to 2τ
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 ti- tanium 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 de- tection 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 de- tection process in the presence of the thin cerium oxide pro- tecting layer (conf. 1), a complete numerical modeling will be given in Subsection 3.C. Furthermore, for the signal of conf. 1, we can note that a sharp offset of the transient reflectivity sig- nal 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 ∼14.4572
∼14.5 ∼16.8780
∼12.4 ∼18.8800
∼12 ∼19.3Mechanical Properties
Sound velocity
VS"m
∕s
#4730 5090
Density
ρ"kg
∕m
3#8960 4500
Thermal and Transport Properties
Thermal expansion (K
−1) [31]
16×10−6 8.5×10−6Lattice heat capacity (at 300 K)
(J · mol
−1· K
−1)
24.4 25.06
Electronic heat capacity (J · m
−3· K
−1), T
eis the electronic temperature
96.6
T
e[31] 313 T
e[31,32]
Electron
–phonon coupling constant
g(10
15W · m
−3· K
−1) [4,6,28]
20
–60 200
–500 Heat conductivity
κ(W · m
−1· K
−1) [31] 400 80
284 J. Opt. Soc. Am. B / Vol. 31, No. 2 / February 2014 Lejman
et al.where E is the total kinetic energy of electrons included in the volume V. What is the increase of the electronic pressure Δ P
e(we consider the volume V as fixed during that fast process of electron excitation):
3) What is the expression, as a function of electronic heat capacity and the electron-‐
phonon coupling constant g, of the characteristic time after which the electron subsystem has given its energy back to the lattice subsystem ? Give the value for Cu and Ti. Is there any relationship between these values obtained for Cu and Ti and the strong difference reported for the heat conductivity of both metals ?
4) After the interaction of the hot electrons with phonons (electron-‐phonon thermalization process), the lattice reaches a final temperature T
f. What is the value ? (titanium molar mass = 47.9 g.mol
-‐1, titanium density = 4.51 g.cm
-‐3).
5) Considering the definition of the thermal expansion α:
! = ! 1 3B
"P
"T
#
$ %
V
