Lecture #8 Non-linear phononics

Transcription

1 Lecture #8 Non-linear phononics Dr. Ari Salmi

2 Last lecture High pressure phononics can give insight into phase transitions in materials SASER can be used to generate high fluences of phonons at a narrow frequency band

3 Phononics at high temperatures and pressures Anisotropic measurements Measure the arriving nanoacoustic wavefronts on the surface full elasticity tensor (silicon)

4 The SASER phonon laser Maryam et al., Nature Communications 2013 Combine these into a phononic laser f = 325 GHz

5 Nonlinear phononics Matemaattis-luonnontieteellinen tiedekunta / Henkilön nimi / Esityksen nimi

6 Propagation in lattices Nonlinear phenomena Higher harmonics Shock waves Solitons (Practical uses - Nanoacoustics I)

7 Propagation in lattices Generalized Hooke s law Where C is the stiffness tensor and ρ the crystalline density If one writes the elastic energy as a function of strain Murnaghan, Am. J. Math. 1937

8 Propagation in lattices This can be written in a familiar form if some assumptions are made Initial stress = 0 alpha = 0 Larger order terms are small

9 Propagation in lattices However, if the strain is not small, the larger order terms are not zero nonlinear terms in the elasticity This leads to a stiffness tensor of Van Capel, Ultrasonics 2015

10 Propagation in lattices This has four terms Geometry Anharmonicity of the lattice The elastic constants are also frequency dependent D a coefficient for shear waves and F the dispersion coefficient, usually positive (normal dispersion)

11 Propagation in lattices After adding attenuation, one can write the general non-linear equation for ultrahigh frequency waves Linear part Nonlinear elasticity Shear frequency dependency General frequency dependency (dispersion) In practice, this is never solved Attenuation

12 Propagation in lattices Nonlinear elasticity In practice, ultrafast ultrasonics is usually 1-D (why?) One can write the 1-D equation for a propagating non-linear wave Here this is written in a co-moving frame General frequency dependency (dispersion) Attenuation Linear part

13 Propagation in lattices Some more simplifications for numerical analysis So, the equation can be written in a form of

14 Propagation in lattices Attenuation So, the equation can be written in a form of Linear part Nonlinear elasticity Where the T s are times different mechanisms take to affect the signal shape General frequency dependency (dispersion)

15 Non-linear equations When Tprop/Tatt 0 (Very low attenuation in the medium) Kadomtsev Petviashvili equation Equation for long wavelength water waves Line solitons!

16 Non-linear equations When Tprop/Tatt 0 (Very low attenuation in the medium) and Tprop/Tdiff 0 (Very low diffraction in the medium) Korteveg-de Vries equation Solitons

17 Non-linear equations When Tprop/Tdisp 0 (Very low dispersion in the medium) Khokhlov-Zabolotskaya-Kouznetsov equation Non-linear acoustic beams in dispersive media

18 Non-linear equations When Tprop/Tdisp 0 (Very low dispersion in the medium) and Tprop/Tdiff 0 (Very low diffraction in the medium) Burgers equation Shock waves

19 Numerical simulations Increasing power non-linear effects

20 Numerical simulations Increasing power non-linear effects

21 Measurements in practice This increases the frequency spectrum of the generated phonons

22 Measurements in practice Bojahr et al., PRL 2015 Ultrasound-like generation of second harmonic of phonons

23 Nonlinearity in room temperature High frequencies high attenuation low dispersion Low focusing (large beam focal spot) long diffraction time Simplifies the equation Only two parts remaining nonlinearity and attenuation To get nonlinear effects, nonlinearity must dominate 0 0

24 Nonlinearity in room temperature Depends on the initial Reynolds number R (for a Gaussian pulse) Nonlinear effects when R >~ 1 Certain distance for a shock wave to form

25 Nonlinearity in room temperature Shock waves at a high enough fluence!

26 Nonlinearity with lower dispersion Low focusing (large beam focal spot) long diffraction time Low attenuation Simplifies the equation Korteweg-de Vries equation 0 0

27 Nonlinearity with lower dispersion This has a solution of the form Which has an interesting property Only one parameter - k S Amplitude Speed of the propagation Width

28 Nonlinearity with lower dispersion k S is also the spectral width of the generated nonlinear pulse Fourier transform of the solution

29 Nonlinearity with lower dispersion k S must be numerically determined from a transcendental equation

30 Nonlinearity with lower dispersion This forms a soliton at high enough amplitudes

31 Soliton trains If there is significant dispersion, soliton trains form Dispersive KdV equation

32 Phonon shock waves and solitons Reed et al., PRL 2008 Simulation prediction of shock-wave induced phonon soliton trains in GaN Predicted that they could be used to generate THz EM radiation from a AlN/GaN crystal

33 Phonon shock waves and solitons Armstrong et al., Nature Physics 2009 Measured the emitted THz radiation

34 Current SoA and outlooks Currently, only longitudinal nonlinear waves have been studied Shear?

35 Current SoA and outlooks Acoustic ablation?

36 Current SoA and outlooks Femtosecond (i.e. THz) solitons?

37 Take-home Three types of non-linear phenomena for phonons Second (third) harmonic generation Shock waves Solitons