Picosecond x-ray strain rosette reveals direct laser excitation of coherent transverse acoustic phonons

Using a strain-rosette, we demonstrate the existence of transverse strain using time-resolved x-ray diffraction from multiple Bragg reflections in laser-excited bulk gallium arsenide. We find that anisotropic strain is responsible for a considerable fraction of the total lattice motion at early times before thermal equilibrium is achieved. Our measurements are described by a new model where the Poisson ratio drives transverse motion, resulting in the creation of shear waves without the need for an indirect process such as mode conversion at an interface. Using the same excitation geometry with the narrow-gap semiconductor indium antimonide, we detected coherent transverse acoustic oscillations at frequencies of several GHz.


INTRODUCTION
We present our numerical scheme for simulating time-resolved x-ray di↵raction in GaAs with laser-induced strain generation and propagation perpendicular (symmetric reflection) as well as non-perpendicular (non-symmetric reflection) to the surface orientation. We employ a model assuming electron-hole plasma and impulsive strain generation upon laser excitation followed by relaxation dynamics such as Auger recombination and free-carrier and thermal di↵usion. The Poisson e↵ect is used to couple longitudinal to uniaxial strain. The e↵ect of the transient strain field on the x-ray di↵raction curves for three non-collinear reflections [004], [113], and [202] are calculated using the method proposed by Wie et al. [1].

UNIAXIAL STRAIN MODEL
In order to simulate the strain dynamics, we begin with a 1D carrier-driven model for longitudinal strain in semiconductors consisting of impulsive strain propagation and lattice relaxation via carrier di↵usion and recombination described previously [2]. When a semiconductor crystal is illuminated by a laser beam with photon energy greater than the electronic energy bandgap, free carriers are generated within the optical penetration depth. With the availability of femtosecond laser pulses, an electron-hole plasma density of 10 14 to 10 19 cm 3 can be achieved, which is sufficient to induce impulsive strains due to the deformation potential and free-carrier di↵usion into the bulk crystal followed by relatively slow thermal di↵usion. In our strain model, two assumptions are made: (i) upon the femtosecond laser excitation on the crystal, the energy relaxation from the laser photons to creating the electron-hole plasma takes place instantaneously, and (ii) the electron-hole pairs relay their energy immediately to the lattice.
These are reasonable approximations because these processes occur on a sub-picosecond time scale, which is much shorter than the time-resolution of a synchrotron x-ray pulse ( 90 ps FWHM). A portion of the laser photon energy, E p , is used to promote electrons from the valence band to the conduction band while the excess energy E p E g is transferred to the lattice as heat, where E g is the electronic energy band gap. At a relatively low excitation level below the carrier saturation, the laser energy is deposited on the crystal surface within the 1/e absorption depth, ⇣. The initial free carrier density and temperature profiles are given by where n(z, t) is the free carrier density, T (z, t) is the lattice temperature and C l is the lattice heat capacity per unit volume. Using an absorbed fluence of F = 0.42 mJ/cm 2 and E p = 1.55 eV, we expect to have a carrier density of 2 ⇥ 10 19 cm 3 within the optical penetration depth.
Transient removal of the carrier population takes place via non-radiative Auger decay, radiative recombination and ambipolar di↵usion which are characterized by the constants A, B and D p respectively. This results in lattice heating and eventual thermal di↵usion, D t , characterized by @n @t = D p @ 2 n @z 2 An 3 Bn 2 @T @t = D t @ 2 T @z 2 + An 3 E g C l . ( The electronic strain is generated both from the free carriers via the deformation potential [3], ↵ p and temperature via the thermal expansion coe cient, ↵ t , ✏ e (z, t) = ↵ p n(z, t) + ↵ t T (z, t).
Rapid expansion of the lattice near the surface launches two counter propagating acoustic pulses along the surface normal direction. Consequently, we expect three longitudinal strain components consisting of the decaying electronic strain and two resulting traveling strain components ✏ + and ✏ , where a ⇡ phase shift exists upon reflection from the surface of the crystal, and the longitudinal strain is given by

3D STRAIN MODEL
Extending this model to describe transverse strain requires three assumptions. First, we presume that the electronic strain is isotropic, which is consistent with a picture of the deformation potential exerting uniform pressure in all directions in response to the sudden generation of free charge carriers. Second, we recognize that the 1D initial conditions also permit a shear wave along the surface normal direction [4]. Third, we propose that this transverse strain is driven by the Poisson ratio ⌫, which is an elastic response to the longitudinal lattice compression such that The material parameters listed in Table I were used to calculate the transient response of the lattice. First, Eq. 2 is solved numerically to give the electron density and thermal profiles at each time point. This is inserted into Eq. 3 to calculate the total generated strain, which is then allowed to propagate according to the elastic wave equation.

X-RAY DIFFRACTION CURVE CALCULATION
The strain rosette analysis shown in Fig. 2  Bragg angle for cubic crystals of arbitrary orientation, is used to calculate the time-resolved di↵raction curves.
For each reflection, the position and lineshape of the x-ray di↵raction curves are calculated numerically using the formulation derived by Wie et al. [1,5], in which depth-dependent strain fields are integrated into the solution of the Takagi-Taupin equations. In previous time-resolved x-ray simulations, where mostly symmetric reflection geometries have been investigated, the transverse strain term has been left out for simplicity. In our simulations, numerical depth step size is kept su ciently thin such that the maximum strain change within a single layer does not exceed 3⇥10 8 per time step. Table II shows the list of parameters used for the x-ray di↵raction calculations and Fig. 1 shows the results of the simulations for di↵erent di↵raction planes. Peak shifts for each reflection were determined by applying an asymmetric Gaussian fit to the x-ray di↵raction curves.