Theory of carrier dynamics and time resolved reflectivity in ${\mathrm{In}}_x{\mathrm{Mn}}_{1-x}\mathrm{As}∕\mathrm{Ga}\mathrm{Sb}$ heterostructures

We present detailed theoretical calculations of two color, time-resolved pump-probe differential reflectivity measurements. The experiments modeled were performed on ${\mathrm{In}}_x{\mathrm{Mn}}_{1-x}\mathrm{As}∕\mathrm{Ga}\mathrm{Sb}$ heterostructures and have shown pronounced oscillations in the differential reflectivity as well as a time-dependent background signal. Previously, we showed that the oscillations resulted from a generation of coherent acoustic phonon wave packets in the epilayer and were not associated with the ferromagnetism. Now we take into account not only the oscillations, but also the background signal which arises from photoexcited carrier effects. The two color pump-probe reflectivity experiments are modeled using a Boltzmann equation formalism. We include photogeneration of hot carriers in the ${\mathrm{In}}_x{\mathrm{Mn}}_{1-x}\mathrm{As}$ quantum well by a pump laser and their subsequent cooling and relaxation by emission of confined LO phonons. Recombination of electron-hole pairs via the Schockley-Read carrier trapping mechanism is included in a simple relaxation time approximation. The time-resolved differential reflectivity in the heterostructure is obtained by solving Maxwell’s equations and by comparing the experiments. Phase space filling, carrier capture and trapping, band-gap renormalization, and induced absorption are all shown to influence the spectra.

Figure 1

Schematic diagram of the heterostructure consisting of a $25\mathrm{nm}$ ${\mathrm{In}}_{0.91}{\mathrm{Mn}}_{0.09}\mathrm{As}$ quantum well and a $820\mathrm{nm}$ GaSb barrier grown on a GaAs substrate. The band gap as a function of position in each of the layers is also shown.

Figure 2

(Color online) Background model dielectric function at $T=0\mathrm{K}$ as a function of photon energy used in computing the dielectric function in the ${\mathrm{In}}_x{\mathrm{Mn}}_{1-x}\mathrm{As}$ quantum well.

Figure 3

(Color online) Model dielectric functions at $T=0\mathrm{K}$ as a function of photon energy for (a) GaSb and (b) GaAs used in calculating the dielectric functions in the GaSb barrier and GaAs substrate.

Figure 4

(Color online) Band structure of a $25\mathrm{nm}$ ferromagnetic ${\mathrm{In}}_{1-x}{\mathrm{Mn}}_x\mathrm{As}$ quantum well with infinite barriers and $x=9\%$ at (a) $T=300\mathrm{K}$ and (b) $T=20\mathrm{K}$. The Curie temperature is taken to be $T_c=55\mathrm{K}$. The Fermi energies $E_f$ for a hole density of $p={10}^{19}{\mathrm{cm}}^{-3}$ are indicated by the horizontal lines.

Figure 5

Coherent phonon strain field as a function of position for delay times ranging from $0\text{to}140\mathrm{ps}$. The curves have been offset for clarity.

Figure 6

(Color online) Derivative of the complex GaSb dielectric function with respect to strain as a function of the probe photon energy. The solid line is the real part and the dashed line is the imaginary part.

Figure 7

(Color online) Theoretical and experimental differential reflectivity for probe wavelengths of (a) 650 and (b) $775\mathrm{nm}$ due to variations in the time- and space-dependent dielectric functions.

Figure 8

Experimental (a) and theoretical (b) coherent phonon differential reflectivity oscillations for probe wavelengths of 650, 775, and $850\mathrm{nm}$.

Figure 9

(Color online) Coherent phonon differential reflectivity oscillation period vs probe wavelength. The experimental data are shown as solid circles, the dashed line shows the oscillation period calculated from the theory described in the text, and the solid line shows the oscillation period estimated from Eq. (62).

Figure 10

(Color online) Derivative of the complex GaSb dielectric function with respect to strain as a function of the probe wavelength in the experimentally relevant wavelength range. The solid line is the real part and the dashed line is the imaginary part.