TERCERA SESIÓN

In this section, the Gabor-Morlet wavelet transform is first presented as a straight- forward method to analyse the time evolution of the frequency-dependent amplitude of CR-waveforms, and allows the identification of the effective contribution of one or more CRs at different magnetic fields. This is further demonstrated in Section 4.2.2, where the model used to obtain fits in the time- and frequency-domains is outlined.

4.2.1 The Gabor-Morlet wavelet waveform

The Gabor-Morlet wavelet transformG(ω, t0, σ) is defined as: G(ω, t0, σ) = Z ∞ −∞ Eind,B(t)·e−t 02_/2σ2 ·eiωtdt. (4.9)

where the second product term within the integral represents a Gaussian centred at timet0 and with width defined byσ. A judicious choice ofσ in the range 0.5< σ < 2.5 ps permits the amplitude of the CR (at a certain frequencyω) to be examined at timet0, at the expense of a poorer frequency resolution. The time-averaged (Fourier) transform is recovered in the limitσ→ ∞.

In Figure 4.5(a) the Gabor-Morlet transform is first applied, with width σ = 1.5 ps, to a simulated CR-waveform consisting of a single CR - this has been obtained from Equation 4.14 in the next section, usingE0(t) as the measured trans-

mitted THz-pulse at B = 0, cyclotron frequency fc = 0.5 THz and τ = 2 ps. The amplitude |G(ω, t0, σ)| is also reported for the experimental data of SiGe1 at dif- ferent magnetic fields in Figure 4.5 (b,c,d), with σ = 1.5 ps. Two temporal regimes can be distinguished. At early times (below 3 ps)|G|exhibits a broadband response corresponding to the driving electric field E0(t), green waveform. At t0 > 3.5 ps, |G| evolves into a more narrow-band cyclotron resonance and represents the free- induction decay of the inter-LL transitions. While at B = 5.5 T, Figure 4.5(d), multiple resonances can be clearly resolved, as indicated by the red and blue dashed horizontal lines, at B = 3.0 T and 5.0 T there is no clear splitting either in |G|

[Figure 4.5(b,c)] or in the transmission in the frequency-domain [Figure 4.4]. As in the case of the substrate’s response in Figure 4.2, this can be related to the fact that the splitting in the frequency-domain is comparable to the linewidth of the CR transitions (Section 2.6.1).

To examine whether a broad resonance consists of multiple narrower reso- nances, time slices of the Gabor-Morlet amplitude |G| - at fixed frequencies - are

Figure 4.5: Gabor-Morlet wavelet transform of: (a) a single CR-waveform simulated using Equation 4.14 [Section 4.2.2]; (b,c,d) the experimental CR-waveforms for the SiGe1 sample at different B (black lines). (e) Time slice at 0.5 THz [horizontal red dashed line in (a)] showing an exponential decay of |G|. (f,g,h) Deviation from the exponential decay as a result of multiple interfering CRs.

plotted with a logarithmic y axis in Figure 4.5(e-h). Since the CR-waveform in Figure 4.5(a) consists of a single CR, the time slice at f = 0.5 THz exhibits an exponential decay with lifetime τ = 2 ps [Figure 4.5(e)]. In Figure 4.5(f-h), the de- viation from an exponential decay (dashed lines) indicates that there are multiple, interfering CRs even at B = 3 T and 5 T. These CRs present different lifetime τ as a consequence of the different occupancy of the initial and final LL states, see Section 2.6.1. This will be analysed in detail, later in Section 5.8, by calculating the joint density of states. Furthermore, the high frequency resonance at 5.5 T, red lines in Figure 4.5(h), is characterised by a shorter decay time as a result of the broader linewidth, Γ =~/τ, in Figure 4.5(d). In conclusion, the Gabor-Morlet

waveform provides a sensitive analysis which permits beats between spin-split CRs to be observed.

4.2.2 Modelled CRs in the time- and frequency-domains

This section outlines the fitting methods in the time- and frequency-domain used to estimate frequencies, carrier densities and lifetimes of thei-th CR at different B. Modelling the transmission in both domains assures precision in evaluating these parameters. The origin of the induced electric field in the time-domain, given by Equation (4.3), is first explained using both the quantum mechanical and classical approach. Secondly, the parallel agreement of the simulations in both domains is deeply analysed in order to clarify the presence of more CRs at differentB.

Quantum mechanics view. In the quantum mechanical view every CR can be seen as a two-level system where holes are excited from the LL state|1i to|2i with an energy differenceE2−E1defining the cyclotron frequencyωc= (E2−E1)/~. The

density matrix formalism describes the dynamics of such a system by solving the Optical Bloch Equations (OBEs), as described for instance in Arikawaet al. [101]. For short pulses, the analytic solutions of the OBEs are provided by a perturbation theory which, in the first order, leads to the homogeneous free decay of the diagonal elementρ12 of the density matrix as a result of the interaction with an electric field E0(t): ρ12(t) =e−iωcti Z t −∞ e−(t−t0)/τE0(t0)eiωct 0 dt0. (4.10) The two-level system defines a macroscopic polarizationP = 2Nsysµ12Re[ρ12], where µ12 is the dipole matrix element between the considered states andNsysis the num-

ber of two-level systems. The induced electric Eind,B(t) ∝dP/dt has frequency ωc and decay rate 1/τ.

Classical view. In the classical (Drude) picture, the cyclotron motion of holes within the quantum well in thei-th state, with a density pi and effective mass m∗i, is described by the dynamical conductivity [98]:

σi(t) = [2πpie2/mi∗]e−t/τicosωc,it . (4.11) The interaction of the THz field with holes defines an induced current:

jind,i(t) = Z t −∞ E0(t0)σi(t−t0)dt0 = = [2πpie2/m∗i] Z t −∞ E0(t0)e−(t−t 0_)/τ i_{cos [ω} c,i(t−t0)]dt0, (4.12)

which has a structure similar to Equation (4.10). In this picture, the reemitted THz field due to thei-th transition,Eind,i∼djind,i/dt, can be written as [98]:

Eind,i(t) = [2πpie2/m∗i] q ω_c,i2 + 1/τ_i2× Z t −∞ E0(t0)e−(t−t 0_)/τ i_cos[ω c,i(t−t0)]dt0. (4.13) The normalised experimental CR-waveforms at different B were fitted by taking into account the sum of thei-th CR arising from transitions between different LLs:

Eind(t) = X i Ai Z t −∞ E0(t0)e−(t−t 0_)/τ i_cos[ω c,i(t−t0)]dt0, (4.14)

owing to the determination of the frequency, effective mass and lifetime of the CRs contributing toEind(t).

Considering different lifetimes for the CRs is justified by the different occu- pation of initial and final states - as in the transition scheme of Figure 2.22 - as well as by the Gabor-Morlet analysis in Section 4.2.1.

Modelled transmission. As Equation (4.14) was used to fit the normalised CR- waveforms, the factorsAi were varied between 0 and 1. The determination ofpi was achieved by usingωc,i, m∗i and τi, obtained from the CR-waveform fits, to simulate the transmission in the frequency-domain by means of the stack model outlined in Section 4.1.2. Thei-th contribution to TB(ω) was taken into account by modelling the complex refractive index for the Ge-QWs, as given by Equation (4.5). The depth of the transmission spectra at the cyclotron frequency fc,i is related to pi through the conductivity σi(ω) defined in Equation (4.6) or to the tensor element σxx in Equation (2.65).

To clarify the adopted model, it is useful to first consider the CR-waveform and the transmission spectrum atB = 5.5 T for the SiGe1 sample reported in Figure 4.6(c,f), respectively. The modelled CR-waveform, black line, agrees well using three CRs (as clearly resolved in the transmission spectrum) whose waveforms are plotted with different colours and present different phase, frequency, decay time and amplitude. The strongest contribution (purple) with A1 = 0.57 corresponds to the

highest cyclotron frequency,fc,1. Once the density p1 was determined such that the

Figure 4.6: Examples of modelled CR-waveforms and transmission spectra for the sample SiGe1. (a,b,c) Comparison of the agreeement of fits with data (thick red line) in the time- domain which considered the contribution of three (black line) or two (blue) or one CR (cyan). The contribution of the different CRs, in the case of the 3CRs-fit, are plotted in the bottom part of the figure (purple, green and yellow lines). (d,e,f) Comparison of the agreement in the frequency-domain with parameters obtained from the time-domain fits above.

the hole densitypi, which participates in thei-th transition, was determined using:

pi =p1 Ai A1 s ω_c,2₁+ 1/τ₁2 ω_c,i2 + 1/τ_i2. (4.15) Here the other parameters are as obtained from the time-domain fits. The modelled conductivityσ =P

iσi results in a simulated transmission (black line) which well agrees with the experiment (red line).

For B ≥ 4 T, three CRs were used for the SiGe1 sample. For instance, at B = 4 T - where two CRs are resolved in the transmission - the best fit was obtained in both domains by considering three CRs [Figure 4.6(a,d)]. At B = 5 T in Figure 4.6(b), fitting with three CRs (black line), instead of two (blue line), does not effectively improve the agreement between fit and data but, as shown in Figure

Figure 4.7: Comparison between time-domain fits obtained using one or two CRs forB ≤ 3 T. (e,f,g) SiGe1 data CR-waveforms (red lines) and fits which consider one CR (blue line) and two CRs (black line). Values ofχ2indicate the different agreement. The deviation from a single exponential is evident atB= 1.2 (a) and 3.0 T (d). (e,f,g) Clear deviation from a single CR-waveform (red line) and the two CRs fit (black) for SiGe2.

4.6(e), the latter case does not agree as well as in the three CRs case. Conversely, the comparable agreement with data for the one (cyan line) and three CRs fits in the frequency-domain transmission of Figure 4.6(e) does not give rise to the same agreement with the CR-waveform in Figure 4.6(b)

In conclusion, as the relative depth of the different features is well repro- duced for the two samples, the agreement given by transferring the time-domain parameters to the frequency-domain is clearly visible in Figure 4.4.