Due to the abrupt changes in absorption and refractive index that the quantum-confined Stark effect (QCSE) can potentially induce in a low-dimensional structure, this mechanism has excellent potential for developing ultra-fast all-optical functions for telecommunications systems using strained [111]-oriented zinc-blende multiple quantum well semiconductor optical amplifiers (MQW-SOAs)^[1,2]. Indeed, these amplifiers, compared to the massive ones, exhibit higher differential gains, lower noise figures, and, notably, an internal piezoelectric field that is mainly responsible for the QCSE when they are unbiased^[3,4]. Therefore, to estimate the temperature dependence of the QCSE in MQW-SOAs and thus be able to use it as a contribution to tune the energy of the excitonic resonances where required for a specific application, it is crucial to determine the temperature dependence of the piezoelectric constant e14. Some authors estimate e14 using linear interpolation between the piezoelectric constant values of the relevant binary semiconductors of the alloy of the quantum wells (QWs)^[3,5]. Nevertheless, this procedure generates larger e14 values than those obtained experimentally. Concerning the piezoelectric constant experimental determination in QWs, typically, e14 is used as an adjustment parameter in theoretical models to fit the calculated values of the energies of determined electronic transitions affected by the QCSE, to those obtained experimentally^[6]. e14 is also estimated by extracting key parameters from the Franz–Keldysh oscillations that allow its indirect determination^[7]. However, in structures where, during the growth of the monolayers of the QW alloy, the surface segregation phenomenon can occur, the e14 experimental determination becomes complex since this phenomenon produces a blue shift of the fundamental transition energy^[8–10]. Therefore, methods used for determining e14, based on the calculations of the energy levels, should consider this effect, although it is rarely included in them and probably because it substantially increases their degree of difficulty. Furthermore, when these methods are used to calculate e14 as a function of temperature, they have the drawback that temperature also has a marked effect on the bandgap energy of the material constituting the QWs. Thus, as the temperature fluctuates, the energy of the electronic transitions or the extremes of the Franz–Keldysh oscillations is simultaneously affected by two remarkable effects: the temperature dependence that the bandgap energy presents and that exhibited by the piezoelectric constant. These simultaneous effects may generate erroneous e14 experimental results or, at best, complicate its experimental determination.

Here, we use a simple method for experimentally determining the effective piezoelectric constant e14e, as a function of temperature, in strained zinc-blende QWs grown along the [111] direction of SOAs with a p-iMQW-n diode structure. Because the surface segregation phenomenon can impose a profile of values on e14, by effective piezoelectric constant, we mean the global magnitude that is assigned to e14. The used method, based on the determination of the relative Stark shifts that the QCSE induces in the fundamental excitonic resonance when the electrodes of the MQW-SOA under test are short- and open-circuited, is insensitive to the variation of the bandgap energy with temperature. Likewise, the method is insensitive to the bandgap energy shift that may cause the surface segregation phenomenon. Even though the method was used to determine e14e in an MQW-SOA, it can be applied to any p-iMQW-n diode structure with electrodes and strained identical zinc-blende QWs grown along the [111] direction. In particular, we estimated e14e in a temperature range of interest for telecommunication applications (18–28°C).

In strained zinc-blende QWs grown along the [111] direction of unbiased SOAs with a p-iMQW-n diode structure, the excitonic transition energy from the first electronic state to the first heavy-hole state (1s e-hh) can be represented as follows when the amplifier electrodes are short-circuited (Exsc) or open-circuited (Exocn): Exsc,ocn=Eg(T,δEhy,δEsh,ϕxi)+Eq−Eb+ΔEssc,ocn(T,Fwn),(1)where sub-subscript n indicates dependence on the input power, namely, on the used nth discrete value of the optical input power (Pinn). Eg is the QW alloy bandgap energy, which is a function of temperature T, hydrostatic deformation energy δEhy, shear deformation energy δEsh, and chemical modulation ϕxi that the QW alloy could undergo due to the surface segregation phenomenon. Eq is the ground state total quantization energy of the conduction and heavy-hole (hh) valence bands when QWs are unperturbed (Fwn=Pinn=0). Eb is the 1s e-hh exciton binding energy, and ΔEssc,ocn is the total Stark shift of the 1s e-hh excitonic resonance (1Se-hhER) caused by the total electric field Fwn acting on each of the QWs when the MQW-SOA electrodes are short-(SC) and open-circuited (OC).

In the present analysis, it was assumed the MQW-SOA structure under study is composed of identical QWs, and the Fwn and piezoelectric fields are the same in all the wells. Based on these considerations, Fwn can be given by^[4]Fwn(T)≈εiεwVbtn(T)Li−e14e(T)Li−mLwLiηxy,(2)where εi and εw are the static dielectric constants of the intrinsic layer and the QW layers, respectively; Li is the intrinsic region thickness; m is the number of QWs; Lw is the width of the QWs; ε0 is the vacuum permittivity; −ηxye14e(T) is the piezoelectric field (Fwp) acting on each QW; and ηxy is the shear strain times 23(εwε0)−1 defined as ηxy=23εwε0(C11w+2C12wC11w+2C12w+4C44w)εs,(3)where εs is the lattice mismatch strain, and C11w, C12w, and C44w are the elastic stiffness coefficients of the material of the QWs, which can be estimated, as can εi and εw, by Vergard’s Law^[4]. Furthermore, in Eq. (2), Vbtn(T) is the effective built-in potential drop across the p-i-n diode, which is a function of temperature, contact potential difference, piezoelectric fields in the QWs, and electric fields created by dipoles formed by photogenerated electron-hole pairs inside and outside the QWs (in-well and long-range screening fields). It is noteworthy that, at temperature T, Vbtn(T) can be determined experimentally as a function of the optical power using a digital multimeter operating in the diode-test mode with its test leads connected between the MQW-SOA electrodes in such a way that it is forward-biased.

Considering that Fwn causes a negligible change in Eb, from Eq. (1), the energy difference (ΔExosn=Exocn−Exsc) existing between the 1Se-hhER energies under open- and short- circuit conditions (OCC and SCC) becomes practically equal to the total Stark shift difference (ΔEsocn−ΔEssc) experiencing 1Se-hhER under OCC and SCC^[11]. Indeed, since the energies Exocn and Exsc are defined by Eq. (1), their difference, for the same temperature, causes the cancellation of Eg, Eq, and Eb, and therefore of the effects of temperature and segregation phenomenon on Eg. Consequently, ΔExosn is given in eV by^[11,12]$$\begin{gathered} ΔExosn=ΔEsocn−ΔEssc \\ =(−A¯ehhLw4Fwn2+QqLiwsnFwn)−(−A¯ehhLw4Fwsc2), \end{gathered}$$(4)where A¯ehh is a function of the electron (heavy hole) out-of-plane effective mass in the QWs and the ground state energy shift enhancement factor, at low fields, due to the finite value of the barrier height for the electrons (heavy holes)^[12]. Moreover, Fwsc is the maximum value of Fwn, which is obtained when the amplifier electrodes are short-circuited and thus Vbtn=0 [see Eq. (2)], Qq is the elementary charge times 6.2415×1018eV/J and Liwsn is the spatial separation, induced by Fwn within the QWs, of the photogenerated electron and hole wave functions. For input powers that, under OCC, produce such a piezoelectric field screening that the 1Se-hhER energy undergoes a shift of less than 1.0 meV, it can be assumed Fwp is strong enough for Fwn to induce a spatial separation between the photogenerated electron and hole wave functions close to the largest possible (Lw) ^[4]. Under these conditions, Liwsn≈Lw. Thus, substituting Eq. (2) into Eq. (4) and solving it for e14e results in the following expression: e14e=εiεwVbt1(LiLwQq−A¯ehhεiεwLw4Vbt1)−ΔExos1Li2ηxy(Li−mLw)(LiLwQq−2A¯ehhεiεwLw4Vbt1).(5)

Here, ΔExos1 and Vbt1 are the values that ΔExosn and Vbtn assume, respectively, when the input power is Pin1. On the other hand, using two input powers, Pin1 and Pin2, a system of two coupled equations can be obtained from Eq. (5), whose resolution results in the following expression for A¯ehh: $$\begin{gathered} A¯ehh=[−2W−+LwQq(Vbt12−Vbt22)εi/εw]Li4Vbt1Vbt2Lw4(Vbt1−Vbt2)εi2/εw2 \\ −LiR−4Vbt1Vbt2Li2ΔExos1ΔExos22Vbt1Vbt2Lw4(Vbt1−Vbt2)εi2/εw2, \end{gathered}$$(6)being R=W+2+LwQ(Vbt1−Vbt2)2(W++M)εi/εw,(7)W±=Li(ΔExos1Vbt2±ΔExos2Vbt1),(8)M=[14(Vbt1−Vbt2)2−Vbt1Vbt2]LwQqεi/εw,(9)where ΔExos2 and Vbt2 are the values that ΔExosn and Vbtn adopt, respectively, when the input power is Pin2.

For notation simplicity, in Eqs. (5)–(9), the temperature dependence of e14e, A¯ehh, Vbt1,2, ΔExos1,2, W±, M, and R is omitted.

According to what is established above, by substituting Eq. (3) and Eqs. (6)–(9) into Eq. (5), and measuring Vbt1,2 and ΔExos1,2 for different temperatures, it is possible to determine e14e(T).

Now, we use the procedure explained above for estimating the e14e value within the [111]-oriented strained In0.687Ga0.313As/In0.807Ga0.193As0.304P0.696 QWs of an unbiased MQW-SOA. The amplifier comprises a p-iMQW-n structure with a 2.2-µm-wide and 0.1-µm-thick intrinsic active region incorporating a central section, with eight QWs separated by seven barriers clad on both sides by a 28.4-nm-thick undoped In0.807Ga0.193As0.304P0.696 separate-confinement heterostructure (SCH). The QW and barrier widths are nominally 1.2 nm and 4.8 nm, respectively, and the QWs are subjected to a compressive lattice mismatch strain of εs=−0.0142. Moreover, εi/εw=0.904 and εw=14.122.

First, we determine ΔExos1,2=Exoc1,2−Exsc when the total input power is Pin1,2 in the temperature range from 18 to 28°C. For this purpose, we use the setup shown in Fig. 1. where a probe beam and a control beam, whose photon energy (1569 nm) is located well within the continuum spectrum, are launched in co-propagation into the amplifier with horizontal linear polarization. At a specific temperature within the range from 18 to 28°C, the probe beam wavelength is swept across the 1Se-hhER spectral width, keeping its input power constant at −15.9dBm. For each examined wavelength and each total input power, defined as the sum of the input powers of the control and probe beams, the probe beam power at the amplifier output is determined, under SCC and OCC (using switch S1 in Fig. 1), via an optical spectrum analyzer (OSA) with a wavelength accuracy of ΔλOSA=±0.02nm (see Fig. 1). Figure 2 shows the obtained transmission spectra, around 1Se-hhER, for the probe beam when the total input powers are Pin1=−15.9dBm and Pin2=−4.53dBm, and the amplifier temperatures are set at 18, 23, and 28°C under OCC. For the same temperatures but under SCC, Fig. 2 shows only transmission spectra obtained with a total input power of −15.9dBm since these are identical to those acquired with a total input power of −4.53dBm. Effectively, under steady-state and SCC, Vbtn=0 and Fwn becomes input power independent because the photogenerated carriers quickly escape from the QWs and are immediately drained by the MQW-SOA electrodes. Consequently, there are no free carriers inside or outside the QWs that can establish piezoelectric field screening mechanisms modifying the QCSE^[4].

Figure 2 clearly shows how, with rising temperature, the 1Se-hhER energy (indicated by arrows) red shifts. This is because |e14e| decreases with temperature, causing |Fwn| to increase [see Eq. (2)]. Therefore, the deformation of the QWs is enhanced, resulting in a decrease in the carrier quantization energy. In particular, under SCC, the 1Se-hhER spectral position is shifted from 774.739 to 770.741 meV when the MQW-SOA temperature changes from 18 to 28°C, which corresponds to a tuning span of 4 meV (8.3 nm). From this figure, ΔExos1,2=Exoc1,2−Exsc, at 23°C, turns out to be approximately equal to ΔExos1=774.078−772.728=1.35meV for Pin1, and ΔExos2=774.660−772.728=1.932meV for Pin2. On the other hand, with the probe beam photon energy matching the 1Se-hhER central energy, the voltages Vbt1=0.570V and Vbt2=0.602V are measured with an accuracy of ΔVmul=±111µV, under OCC, using a 6.5 digit multimeter in diode test mode (denoted as V in Fig. 1) for Pin1 and Pin2, respectively.

Now, using Eqs. (3) and (5)–(9), as well as the found values for ΔExos1,2 and Vbt1,2, e14e at 23°C can be determined. Similarly, e14e can be estimated for any temperature. For clarity, Fig. 2 only shows the transmission spectra for 18, 23, and 28°C. However, the experimental determination of the 1Se-hhER energy under SCC (Exsc) and OCC (Exoc1,2) and the voltages Vbt1,2 was performed for eleven temperatures. Expressly, the MQW-SOA temperature was varied using a Peltier element (PE) and a thermoelectric temperature controller (TTC) with ±0.2°C accuracy (see Fig. 1). The results are presented in Fig. 3, where the experimental data for Exsc, Exoc1,2, and Vbt1,2 are shown with their linear interpolations accompanied by their respective formulas.

Figure 4 shows the e14e, A¯ehh, and ΔEssc values (open crosses) calculated as a function of temperature using Eqs. (3), (5)–(9) and the last term on the right-hand side (RHS) of Eq. (4) (−A¯ehhLw4Fwsc2) together with the experimentally obtained values of Exsc,oc1,2 and Vbt1,2 reported in Fig. 3 (marks).

These parameters were also calculated using the linear interpolation formulas for Exsc,oc1,2 and Vbt1,2 given in Fig. 3, resulting in the values represented with open circles in Fig. 4. Moreover, the precisions ±Δe14e, ±ΔA¯ehh, and ±ΔΔEssc with which the e14e, A¯ehh, and ΔEssc values were extracted from the experimental data are represented in the graphs of Fig. 4 with error bars and were obtained in a manner similar to that described in Ref. [11]. Due to the high absorption of the 1Se-hhER, the OSA noise floor generates inaccuracies in the determination of Exsc,oc1,2, which is the main cause of the dispersion of the e14e, A¯ehh, and ΔEssc values represented with open crosses in Fig. 4. However, all these values, calculated directly with the experimental data reported in Fig. 3 (marks), do not exceed the limits of the error bars, which validates the precision wherewith e14e, A¯ehh, and ΔEssc are estimated. In particular, the extracted e14e values at 18, 23, and 28°C are −0.0536±0.0041, −0.0534±0.0040, and −0.0531±0.0040C·m−2, respectively, i.e., in the analyzed temperature range, the result accuracy is approximately equal to ±0.004C·m−2, which is similar to that obtained by other methods extracting the e14e value^[13]. Moreover, as expected, these e14e values, and in general all those reported in the upper graph of Fig. 4, evidence that the e14e magnitude tends to decrease as temperature increases. Indeed, as temperature rises, the electric dipole randomization increases. Hence, the strain-induced polarization decreases together with e14e.

It is relevant to mention that this behavior is contrary to that observed in other previously published work where the piezoelectric constant magnitude in In1−xGaxAs QWs increases as temperature increases, for which no convincing explanation has been presented^[13–17]. To gain more insight into this contradictory aspect, we further investigate the behavior regarding the temperature of the 1Se-hhER total Stark shift under SCC (ΔEssc). As shown in the lower graph of Fig. 4, the ΔEssc magnitude increases as the temperature increases. Vis-a-vis the behavior exhibited by e14e, the ΔEssc behavior concerning temperature would seem to be opposite to that expected since the ΔEssc magnitude is directly proportional to Fwsc2, which is in turn directly proportional to e14e2 [see the last terms on the RHS of Eqs. (4) and (2) with Vbtn=0]. However, ΔEssc also depends on the parameter A¯ehh whose magnitude increases as the temperature increases, as shown in the middle graph of Fig. 4. Although A¯ehh contributes linearly to the ΔEssc magnitude, its growth rate with increasing temperature is strong enough to overcome the antagonistic rate of the quadratic contribution from e14e. For this reason, the ΔEssc magnitude increases as temperature increases. The parameter A¯ehh, which can be determined by performing indirect measurements and applying Eq. (6), is physically a function of the out-of-plane effective masses (mwe,hh⊥) and the ground state energy shift enhancement factors (Ωe,hh), due to the finite value of the barrier height, of electrons and heavy holes^[12]. Consequently, in models where mwe,hh⊥ and Ωe,hh are used explicitly instead of A¯ehh, it becomes essential that these parameters are a function of temperature. Otherwise, the change that the 1Se-hhER total Stark shift would undergo with temperature would be solely attributed to e14e, and vice versa. If the Stark shift behavior were used to determine that of e14e, then an erroneous comportment of e14e, contrary to that found in this work, would be obtained. We speculate that the discrepancy between the piezoelectric constant behavior regarding temperature estimated in this work and that reported by other authors might be due to the omission of the temperature dependences of mwe,hh⊥ and electron (heavy hole) energy in the QWs, or an imprecise description of them, possibly because of the lack of detailed reports on these issues.

Finally, Fig. 2 shows that by varying the temperature from 18 to 28°C, the 1Se-hhER operation energy can be tuned 4 meV, and then, under OCC, fluctuating the input power from −15.9 to 5 dBm, this resonance can be shifted 3.7 meV, as shown in Fig. 5. It is important to note that if the input signal were composed of pulses whose duration was less than the escape time of the carriers in the QWs plus the time it takes them to drain through the circuit formed by parasitic elements and the amplifier electrodes, then, under SCC, the power of each input pulse would also shift 1Se-hhER.

This opens the possibility of devising ultra-fast all-optical applications using unbiased MQW-SOAs since there would be no free carriers that would generate slow tails in the falling edges of the output pulses as might occur under OCC, notably with high input powers. A study of the Stark effect dynamics under SCC and OCC is currently in progress.

We presented a procedure for determining the effective piezoelectric constant value of the [111]-oriented strained In1−xGaxAs/In1−xGaxAsyP1−y QWs of an MQW-SOA. Remarkably, the proposed method is insensitive to the temperature dependence of the bandgap energy and the segregation phenomenon. Likewise, it requires neither fitting parameters nor temperature-dependent expressions for energy and out-of-plane effective masses of electrons and heavy holes. When the procedure was applied to the MQW-SOA under study, a value of e14e=−0.0534±0.0040C·m−2 at 23°C was obtained. Unlike previously published methods, where the piezoelectric constant magnitude increased as the temperature rised without convincing explanation, we extracted an e14e magnitude that decreased as temperature increased. Even though the method was specially designed for experimentally determining e14e in MQW-SOAs, it can be applied to any p-iMQW-n structure with electrodes and strained [111]-oriented zinc-blende QWs. Finally, we found that by varying temperature, the 1Se-hhER operation energy could be tuned, and then, fluctuating the input power, this resonance could be significantly shifted. This enables the devising all-optical applications based on QCSE in unbiased MQW-SOAs. Thus, for example, the method can be applied to other MQW-SOAs, intended to implement all-optical functions based on the QCSE, to predict the shift of their 1Se-hhER with temperature, or to indirectly deduce how temperature will affect the undesirable effects that QCSE causes in some MQW LEDs.