Model calculations

h = 58^◦ In the Bloch picture, we get the degree of excitation:

n_ex = 1 2

1−cos(θ)= 23.5%

This value is in good agreement with the one calculated above.

We have demonstrated that a strong nonlinear saturation (>20 %) of the 1–2 intersub-band excitation can be induced with incident pulses of only 1 pJ pulse energy. In the Bloch sphere picture, the strongest observed saturation of the free induction decay corresponds to a partial Rabi flop of up to 1/6 (60^◦) of the full cycle (360^◦) [Fig. 4.11 (b)].

4.5.2.2 Model calculations

Finally, we analyze the experimental data by model calculations using the Maxwell-Bloch equations for a homogeneously broadened two-level system [Eqs. (2.4)-(2.6)]. Our theoretical description is similar to that describing self-induced transparency experiments [131,132]. In general the full spatial and time dependence in the Maxwell-Bloch equations have to be taken into account. In this case, Eqs. (2.4)-(2.6) have to be solved numerically. In the following, we will first treat the case of linear excitation. Finally, we consider the case of nonlinear excitation.

We make some approximations which simplify the problem considerably:

• In the prism geometry applied in our measurements, the electric field experienced by the electrons is nearly identical for all individual quantum wells, as is evident from the calculated intensity pattern in Fig. 4.3 (b). This fact allows for a simplified theoretical description with a position independent polarization P_Res(z) ≡ P_Res. Using the new

coordinate system x =x, t =t−x/v_gr (we will skip the primes in the following) Eq.

(2.4) can be integrated along the interaction length L Eout=Ein+i ΩL

2₀c nPRes (4.3)

• In the linear case n_ex is very small so that Eq. (2.6) can be neglected.

• Further, we assume that the amplitude of the re-emitted free induction decay is much smaller than the incident field amplitude, i.e.,

|i ΩL

2₀c nP_Res| E_in (4.4)

The experimental data indicate that the free induction decay [Fig. 4.5 (c)] has an amplitude which is approximately a factor of three times smaller than the amplitude of the incident field [Fig. 4.5(a)]. Hence, this relation is not strictly satisfied. A direct comparison with the exact result for the linear case, however, justifies its application as will be shown below. With approximation (4.4), Eq. (2.5) can be treated independently from Eq. (2.4) considerably facilitating the solution of the Maxwell-Bloch equations.

The solution of Eq. (2.5) with an excitation field tuned to resonance (∆ω= 0) is P_Res =i₀c n A_abs In this calculation, the only free parameter is the incident field Ein. The absorption Aabs = ln(10)·a is taken from the linear absorption measurement [Fig. 4.1 (b)] and the dephasing time T₂ = 320 fs is taken from four wave mixing experiments [69]. The second term of Eq.

(4.6) is the re-emitted electric field, i.e., the free induction decay. It is emitted with a 180^◦ phase shift so that it destructively interferes with the incident fieldE_in as mentioned above.

The transmission coefficient [cf. Eq. (4.2)] is calculated by Fourier transforming Eq. (4.6) t(ω) = E_out(ω)

E_in(ω) = 1− 0.5A_abs

1−i(ω−ω₀)·T₂ (4.7) Linear single pulse experiment To model the data, in a first step we fit a Gaussian to the reference electric fieldERef [Fig. 4.5 (a)]. This yields the incident electric fieldEin [Fig.

4.5 (d)]. Then, using Eq. (4.6) we calculate the field transient after transmission through the quantum well sample E_out [Fig. 4.5 (e)]. The free induction decay E_{F ID}=E_out−E_in is plotted in Fig. 4.5 (f). The experimental data [Fig. 4.5 (b)] agree well with the calculated transient.

Now, we come back to the question of the validity of the relation (4.4). We compare calculations with and without Eq. (4.4). Without this approximation, an analytic solution for the linear response can be found

E_out =E_in−A_abs

0.0 0.5 1.0 1.5 0.0

0.5

-0.5 0.0 0.5 1.0 1.5 1E-5

1E-4 1E-3 0.01 0.1

Electric field envelope (arb. u.)

Delay time (ps)

Envelope (arb. u.)

Delay time (ps)

Figure 4.13: Calculated envelopes of electric field transients through the quantum well sam-ple. The solid line indicates the full Maxwell-Bloch calculation using Bessel functions. The dotted line corresponds to a calculation where the approximation (4.4) was made. Inset:

Field transient envelopes plotted on a logarithmic scale.

where J₁ is the first order Bessel function. In Fig. 4.13 the envelopes of the field transients calculated with and without approximation (4.4) are shown. A difference can only be ob-served on a logarithmic scale (inset of Fig. 4.13). This difference is much smaller than the noise in the measurement justifying a posteriori the application of Eq. (4.4) to model the experimental data.

Linear two pulse experiment In Figs. 4.6 (b) and 4.6 (d), the results of these model calculations in the case of two phase-locked incident pulses are shown. Such calculations are in excellent agreement with the data of Figs. 4.6 (a) and 4.6 (c).

Coherent nonlinear propagation In order to model the data shown in Fig. 4.7 it is important to note that the degree of excitation in (b) is rather high (n_ex ≈ 20 %). Hence, in a model calculation using the Maxwell-Bloch equations the last Eq. (2.6) can not be neglected any more. However, as in the model for the linear case we can still assume that the amplitude of the re-emitted free induction decay is small compared to the amplitude of the incident field [Eq. (4.4)] so that the Bloch equations (2.5) and (2.6) can be treated independently from Eq. (2.4). For a given E^{M IR}(t) we can calculate numerically P_Res(t) using Eqs. (2.5) and (2.6). With Eq. (4.3) we then determine the electric field transient transmitted through the quantum well sample.

-0.5 0.0 0.5 -2

0 2 4 6 8 10 12

Theory Experiment

Electric Field (arb. u.)

Delay Time (ps)

-0.5 0.0 0.5

Figure 4.14: Measured and calculated electric field transients for high excitation (I^pulse= 1.0 pJ) through a reference sample (upper traces), through the quantum well sample (middle traces), and the free-induction decay (lower traces) of the intersubband excitation in response to a single pulse.

In Fig. 4.14the calculated field transients for an incident pulse with energyI^pulse= 1.0 pJ are shown (right traces). It agrees well with the measured curves (left traces) [cf. Fig. 4.7 (b)]. The solid line in Fig. 4.11 (b) is the calculated ratio between the amplitude of the free-induction decay and the amplitude of the incident pulse E_{F ID}/E_in. This curve fits also quite well the experimentally determined amplitude ratio [Fig. 4.11 (b), symbols].

In conclusion, we have demonstrated coherent control of linear intersubband polarizations with subpicosecond dephasing times by weak ultrafast electric field transients in the mid-infrared. A nonlinear response, i.e., a saturation of the polarization amplitude by up to 20 %, can be induced with pulses of only 1 pJ energy. This nonlinear response follows exactly the prediction of an ideal homogeneously broadened two-level system. This fact may facilitate the realization of ultrafast switching devices based on coherent intersubband polarizations.

Direct measurement of electrically induced optical transmission changes in quantum cascade lasers

In this chapter, we present a direct study of optical transmission changes due to gain in an electrically driven quantum cascade structure. The normalized transmission change, i.e.,

T(I, λ)−T(0, λ)/T(0, λ) is measured as a function of both applied current I and wave-length λ. The main motivation for this experiment is to develop a novel measurement technique which combines ultrafast spectroscopy with high-current electrical pumping of semiconductor device structures. This technique gives experimental access to fundamental questions of semiconductor physics which could not be obtained by other means. One im-portant example, the physics of electron transport in semiconductor nanostructures in the presence of a dense electron plasma, will be discussed in the subsequent chapter.

The development of this technique implies considerable effort in the optical and electronic setup. Two things are very important. First, the light pulses have to penetrate highly doped, thick contact layers of the sample, which are necessary for electric pumping. Secondly, to prevent heating of the sample, the driving current has to be applied in pulsed mode. Thus, the combination of optical and synchronized electric pulses requires an elaborate electronic setup. For the experiments described in this chapter we use the cavity-dumped Ti:sapphire laser system presented in section 2.2.1 as it represents an easy tunable mid-infrared source.

Similar to the experiments discussed in the previous chapter we use a beam propagation geometry in which the optical beam is incident under an angle of 60^◦. This is an appropriate way to ensure a large coupling with the intersubband dipoles. To limit heat dissipation in the sample from the applied current, the quantum cascade structure is processed in circular mesas with a small surface. We have to take into account, however, that the mid-infrared light focus is rather large demanding a big mesa size for a suitable overlap. As a compromise, for the experiments discussed in this chapter we will use a sample which was originally developed for electroluminescence measurements. Here, the mesa has around half the size of the mid-infrared focus.

In the following we first introduce the quantum cascade sample. After the discussion of the effect of heating due to the electrical current, we will give a detailed explanation of the experimental realization to carry out a transmission change experiment. Then we present linear transmission change measurements to derive the gain coefficient. We analyze the data with respect to the effects resulting from the chosen beam propagation geometry. Taking

these effects into account we will determine the gain coefficient of the quantum cascade structure and compare our results with work of other groups published in literature.

5.1 Sample properties

The sample used for the transmission change experiment (sampleA₁) is a GaAs/Al_xGa_1−xAs quantum cascade structure (aluminum content: x= 33 %) grown at Thomson-CSF in Paris.

It contains 10 periods of the active layer structure similar to the one in Ref. [99] and repre-sents the layer structure of the first quantum cascade laser based on the GaAs/Al_xGa_1−xAs material system (aluminum content: x = 33 %). At this aluminum content AlGaAs is a direct gap semiconductor. The conduction band offset between AlGaAs and GaAs is

∆E_c = 295 meV [133].

The conduction band diagram of the active region and the relevant wavefunctions are shown in Fig. 5.1. The active region consists of three coupled GaAs quantum wells. The injector is a graded superlattice containing five coupled GaAs quantum wells. The injector region and the active region are separated by a 6.2 nm thick AlGaAs injection barrier and a 3.4 nm thick AlGaAs exit barrier. The inset of Fig. 5.1 is a schematics of the injector and the active region under bias.

To reduce the lifetime of subband 2 (lower laser states), the 1–2 subband spacing cor-responds to the energy of an LO phonon (¯h ω_LO = 36 meV). The calculated lifetimes of

Figure 5.1: Conduction band diagram of sample A₁ for an applied electric field of 48 kV/cm.

Probability densities |Ψ(x)|² are shown for the relevant wavefunctions (|g: ground state in the injector, |3: upper laser state, |2: lower laser state). The inset is a schematics of the injector and the active region divided by the injection barrier.

Table 5.1: Layer structure of sample A₁. material doping (Si) thickness (nm)

GaAs 2×10¹⁸ cm⁻³ 600 top contact

GaAs 20

10 repetitions

GaAs 3.2

AlGaAs 2.0

GaAs 2.8

AlGaAs 4×10¹⁷ cm⁻³ 2.3

GaAs 4×10¹⁷ cm⁻³ 2.3 injector

AlGaAs 4×10¹⁷ cm⁻³ 2.5 GaAs 4×10¹⁷ cm⁻³ 2.3

AlGaAs 2.5

GaAs 2.1

AlGaAs 6.2

GaAs 1.5

AlGaAs 2.0

GaAs 4.9 active

AlGaAs 1.7 region

GaAs 4.0

AlGaAs 3.4

GaAs 3.2

AlGaAs 2.0

GaAs 2.8

AlGaAs 4×10¹⁷ cm⁻³ 2.3 injector

GaAs 4×10¹⁷ cm⁻³ 2.3 AlGaAs 4×10¹⁷ cm⁻³ 2.5

GaAs 20

GaAs 2×10¹⁸ cm⁻³ 1000 bottom contact SI GaAs substrate (300µm)

subband 3 and subband 2 are τ₃ = 1.5 ps and τ₂ = 0.3 ps [99]. The complete layer structure is shown in Table 5.1.

The active region is sandwiched between two heavily n-doped (nSi = 2× 10¹⁸ cm⁻³) GaAs contact layers [Fig. 5.2 (a)]. The quantum cascade sample is processed into round mesas with a mesa radius R_m = 60µm [Fig. 5.2 (c)], which is the size typically used for electroluminescence measurements. The mesa and the surrounding area are covered with a AuGe/Ni/Au alloyed contact.

This structure shows electroluminescence due to spontaneous emission from subband 3 to subband 2 in the biased case and photo-current due to absorption from subband 1 to subband 3 in the unbiased situation as shown in Fig. 5.3 (a). At a cryostat cold-finger temperature T_CF = 77 K and for a current density of 1.8 kA/cm² (dashed line) a single electroluminescence peak can be seen, which is centered at 10.4µm. The width is 13 meV

60°

220 µm

80 µm 120 µm

Laser spot

QC-mesa Au

n +

QCS

(b) (c)

(a) GaAs substrate

Figure 5.2: (a) The active region of the quantum cascade structure (QCS) is sandwiched between twon-type GaAs layers (n= 2×10¹⁸cm⁻³, thicknesses: 1µm and 0.6µm). (b) Ge-ometry of beam propagation through the prism shaped sample. (c) Partial overlap between the mid-infrared laser spot and the disc shaped quantum cascade structure (QC-mesa).

100 120 140 160

0 1

0 1 2 3 4 5

0 1 2 3 12

3

12 3 Wavelength (µm)

Photon energy (meV)

Intensity (norm.)

13 12 11 10 9 8

(a)

(b)

Voltage (V)

Current density (kA/cm²)

Figure 5.3: (a) Electroluminescence for different current densities (dashed line: 1.8 kA/cm², dotted line: 3.5 kA/cm², solid line: 7.1 kA/cm²) and photocurrent (dash-dotted line) spectra of sample A₁ measured at TCF = 77 K [134]. Left inset: Injector and active region with applied bias. Spontaneous emission from subband 3 to subband 2 leads to the shown electro-luminescence spectra. Right inset: Injector and active region without bias. 1–3 absorption leads to photocurrent. (b) Current–voltage characteristics measured at a cryostat cold-finger temperature T_CF = 77 K with 400-ns current pulses at a repetition rate of 10 kHz [134].

(FWHM). At a current density of 7.1 kA/cm² (solid line) the electroluminescence peak shifts to 10.0µm and the width is 10 meV. For increasing voltage the tilting of the bandstructure becomes stronger. As a result, the subbands in the active region move apart leading to the observed blue shift of the electroluminescence. The photo-current is peaked at a wavelength of 9.0µm. In Fig. 5.3 (b) the current-voltage characteristics is shown. For voltages up to 1.5 V there is very low current. This voltage is needed for the alignment of the injector and the active region.

To measure the transmission change due to 3–2 population inversion, we use mid-infrared light resonant to this intersubband transition. For an efficient coupling of the intersubband dipoles with the electric field of the optical beam, we use a beam propagation geometry [Fig. 5.2 (b)] with an angle of incident α = 60^◦ as discussed in the previous chapter. The substrate thickness is 300µm. According to the considerations of section 4.2 this results in a length of a= 790µm for the base plane of the prism.

The laser device mentioned above (ridge waveguide, 2.5 mm × 24 µm) with 30 periods of a similar layer structure but with an injection barrier thickness of 5.8 nm has a threshold current density of I_th = 7.3 kA/cm² at 77 K. This laser works up to a temperature of T_th = 140K. We want to drive our sample with current densities up to I_th and below T_th. Hence, a careful consideration of the heat dissipation due to the current is important.

Im Dokument Ultrafast dynamics of coherent intersubband polarizations in quantum wells and quantum cascade laser structures (Seite 73-81)