Scattering processes

An irreversible phase loss of coherent superpositions of states |b and |a leads to homoge-neous broadening. Now we want to discuss various possible scattering mechanisms contribut-ing to homogeneous broadencontribut-ing and their relevance for the transport processes observed in our femtosecond experiment.

In general one can distinguish between two different kinds of scattering processes, namely intersubband and intrasubband scattering as schematically shown in Fig. 6.32 (a). Intra-subband scattering corresponds to transitions of electrons between two states within the same subband [A in Fig. 6.32 (a)] whereas intersubband scattering represents a transition between different subbands [B in Fig. 6.32 (a)]. There are a number of different scattering mechanisms relevant in quantum cascade structures. The most important ones are electron-electron, electron-LO phonon, and electron-impurity scattering.

Electron-impurity scattering The ionized donors in the injector region of the quantum cascade structure provide Coulomb potentials which can act as scattering centers for the electrons. Since the impurities are immobile such scattering events are elastic, i.e., there is no energy exchange. Nevertheless, those lead to momentum relaxation as depicted in Fig. 6.32 (b). Because of the small wavefunction overlap, intersubband electron-impurity scattering is very weak. Hence, this scattering mechanism leads mainly to intrasubband momentum relaxation.

Figure 6.32: (a) Schematics of intersubband (A) and intrasubband (B) scattering. (b) Electron-impurity scattering is an elastic scattering mechanism, i.e., there is only momentum but no energy relaxation. (c) Electron-LO phonon scattering leads to intraband transitions (A and C) or intersubband transitions (B and D). (d) Various Coulomb scattering processes between two electrons.

Electron-LO phonon scattering An important scattering mechanism is the scattering of electrons with LO phonons. Here, energy and momentum are exchanged. Due to the small energy dispersion, LO phonons have a nearly k-independent energy of ¯h ω_LO = 36 meV in GaAs. Scattering between a LO phonon and an electron leads to the exchange of the energy

¯

h ω_LO. Both inter- and intrasubband scattering are possible as shown in Fig. 6.32 (c).

Two different scattering processes can be distinguished, LO phonon emission [A and B in Fig. 6.32(c)] and LO phonon absorption [C and D in Fig. 6.32 (c)]. The scattering rate for LO phonon emission is proportional to 1 +n_LO(T_L), where

n_LO(T_L) = 1 exp^{¯h ω}_k ^LO

BTL

−1

is the thermal LO phonon population determined by the Bose distribution. The LO phonon absorption rate is proportional to n_LO(T_L). Electrons being at least ¯h ω_LO above the re-spective subband minimum can spontaneously emit a LO phonon with a rate of 10¹³ s⁻¹ [148, 81, 4].

LO phonon scattering is the dominant energy exchange process between the electron gas and the lattice.

Electron-electron scattering The Coulomb interaction causes scattering among the elec-trons. It is strongly modified by screening in the presence of other mobile charges in the electron gas. Scattering events among the electrons are accompanied by momentum and en-ergy exchange. Electron-electron scattering is the dominant mechanism for thermalization within a single subband and between different subbands. Since the Coulomb interaction is a long-range interaction the electron gas represents a complicated, coupled many-body system.

In the simplest approximation, electron-electron scattering is described in terms of indepen-dent two-electron interactions. In this approximation the possible inter- and intrasubband processes are depicted in Fig. 6.32 (d) [149]. (A) There are pure intrasubband processes where the interactions and the transitions occur within a single subband. This is the dom-inant process for intrasubband thermalization. (B) Interaction between two electrons, each in a different subband, can exchange energy and in-plane momentum while staying within the respective subbands after the interaction. (C) There are scattering processes where two electrons are initially in different subbands, and then exchange subbands. Processes B and C lead to thermalization among two subbands. (D) Finally, there are interactions where two electrons are initially in the same subband, but both move to another subband after the interaction.

However, this simple picture of electron-electron scattering via two-electron interaction is an approximation and an exact theoretical modelling requires a full quantum kinetic description including full dynamic screening [150,151,152]. This is still an unsolved problem for quasi-two dimensional electron plasmas.

Scattering in quantum cascade structures After this general consideration we will now discuss the relevance of the various scattering processes in quantum cascade structures as found in literature and compare these results with our experiment.

In Ref. [124] Harrison has carried out detailed calculations for electron-electron and electron-LO phonon scattering rates in quantum cascade structures using an approach based on Fermi’s golden rule. According to these calculations intersubband scattering rates via

LO phonon emission are on the order of 10¹¹ s⁻¹ rising to 3×10¹²s⁻¹ between the strongly coupled subbands 2 and 1. At the same time the electron-electron intersubband rates are on the order of 10⁹ −10¹⁰ s⁻¹. From this he claims that in quantum cascade structures intersubband scattering is dominated by LO phonon scattering. Intrasubband scattering is dominated by electron-electron scattering, for which rates greater than 5×10¹³ s⁻¹ were calculated. He concludes that these high scattering rates drive the system towards a ther-malized electron distribution in each subband having the same electron temperature for all subbands. In these calculations the author has assumed a priori that the electron temper-ature is the same as the lattice tempertemper-ature which was assumed to be 77 K. We want to emphasize that the temperature is a critical parameter for the various scattering mechanisms and that its determination is an important task.

The question of the relation between the lattice and carrier temperature was addressed by Iotti and Rossi [126]. The authors carried out Monte Carlo simulations taking into account electron-LO phonon and electron-electron scattering via two-electron interaction in a semiclassical approach. They confirmed the result of Harrison [124] that there is a thermalized electron distribution in each of the subbands and that the temperature is the same for all subbands. Assuming a lattice temperature of T_L = 77 K they calculated an electron temperature ofT_C = 600−700 K.

However, a simple estimation suggests that this carrier temperature is too high: We as-sume a Boltzmann distribution for the electrons and an intrasubband electron-LO phonon scattering rate of RLO = 10¹³ s⁻¹ [4]. Taking into account LO phonon emission and ab-sorption according to the LO phonon population n_LO(T_L) the carrier temperature T_C as a function of the current densityI and lattice temperatureT_L reads

TC(I, TL) = −h ω¯ LO

where N is the total 2-dimensional carrier density in the structure (N = 4×10¹¹ cm⁻²). In Fig. 6.33the carrier temperature is plotted as a function of current density for various lattice temperatures. The carrier temperature increases more or less linearly with the current. For a lattice temperature of T_L= 10 K the carrier temperature is T_C = 180 K at I = 7 kA/cm² [Fig. 6.33, dashed line]. The carrier temperature is not essentially higher forTL = 100 K [Fig.

6.33, dotted line]. This estimation demonstrates that we can assume a carrier temperature on the order of T_C = 200 K at low lattice temperatures. It increases linearly with higher lattice temperatures. The solid line in Fig. 6.33 indicates the carrier temperature for a lattice temperature of T_L = 300 K. At I = 7 kA/cm² we observe a carrier temperature on the order of T_C = 400 K. With a simple estimation we can also calculate the temperature increase after pump depletion. In the time-resolved experiment, the pump pulse promotes electrons into the lower subbands of the active region. These subbands are energetically around ∆E_2g = 120 meV above the subbands a and b [cf. Fig. 6.18 (b)]. Due to the strong carrier-carrier scattering with rates > 10¹³ s⁻¹ we can assume that the electron gas is thermalized within t_D = 400 fs. Using the estimation carried out in section 6.6.1 the electron density in the upper laser subband 3 is around 10 % of the total carrier densityn_tot. We have seen that a pump pulse with an energy of 7.5 nJ saturates the gain completely.

Thus, the number of de-excited electrons is n_ex = 0.1n_tot. Assuming no energy relaxation from the electron gas to the lattice, the additional energy input due to these electrons reads E_add = ∆E_2gn_ex. With these consideration we can estimate the carrier temperature increase

Figure 6.33: Carrier temperature T_C as a function of current density for various lattice temperatures T_L. The solid line indicates a lattice temperature of T_L = 300 K, the dotted line indicates T_L = 100 K, and the dashed line T_L= 10 K.

after pump depletion. ForE_pump = 7.5 nJ and T_L= 10 K the carrier temperature increases fromT_C = 180 K to T_C = 300 K.

Now we discuss the relevance of the various scattering mechanisms for the electron trans-port. In quantum Monte Carlo simulations taking into account electron-LO phonon scat-tering but neglecting electron-electron scatscat-tering Iotti and Rossi studied the dynamics of electron wavepacket dynamics in quantum cascade laser structures [128]. At t = 0 they prepared the system fully localized in the lowest injector subband. As time evolves, they observed wavepacket oscillations between the injector and the upper laser subband decaying on a time scale of several picoseconds. This dephasing time is an order of magnitude larger than the decay time observed in our experiment. Obviously, the reason for this is the neglect of electron-electron scattering in the calculations. The same authors also presented simula-tions including electron-electron scattering but using a semi-classical Monte Carlo approach.

From these calculations they concluded that energy-relaxation and dephasing processes are strong enough to destroy any phase-coherence effect on a sub-picosecond time scale [128].

The rather high electron-electron scattering rate of>5×10¹³s⁻¹ obtained from those the-oretical calculations are much larger than the homogeneous dephasing rate 1/T₂ <5×10¹²s⁻¹ determined from our experiment. Obviously, only a fraction of scattering events leads to de-phasing of the coherent superposition of states|a and|b. This is in contrast to predictions of Fermi’s Golden Rule, which assumes that particles are scattered into eigenstates of the unperturbed Hamiltonian. On short time scales, however, quantum mechanics allows also coherent superpositions of such final states, a concept known as scattering-induced coherence [153, 154].