On the relation between electrical and electro-optical properties of tunnelling injection quantum dot lasers

2.1 Current-voltage characteristics

Current-voltage characterizations, measured at different temperatures between 4 K and 290 K, are shown in Figure 2(a). The calculated power exponent parameter, α = d ln I / d ln V [11] dependence on bias (α–V) and temperature is shown in Figure 2(b)–(f).

Figure 2:

Temperature dependent (a) I–V, and (b)–(f) α–V characteristics. The dotted lines indicate the location of the corresponding voltages of the evolving discontinuity in Figure 2(b)–(f). Roman numerals mark the maxima and inflections points on the α–V curves.

At a very low forward bias, α exhibits a large peak that is associated with the diffusive component of the carrier flow from the cladding layers to the QW, which is highly temperature dependent. Following this large peak, at low temperatures, the α–V curve reveals an oscillatory nature which reflects discretization of the QD layers; there is one bump in the curve for each QD layer. At 3.05 V and at a temperature of 4 K, α exhibits a strong peak at the end of which appears a distinct discontinuity, which indicates carrier clamping at threshold [12]. Comparing the results with a reference QD laser having no TI region reveals a similar discretization at 4 K with the first large peak missing as there is no QW layer.

The bumps in the α–V curves are related to the step-like increment of the current with applied voltage. It is due to Coulomb blockade originating from the potential of fully charged dots in each layer, which opposes the applied bias, and also from a tunnelling process between separate QW–QD layer pairs. Each section of the I–V and α–V curves can be described by the theory of Asryan [13, 14] with the inclusion of various standard tunnelling mechanisms [15–18].

At elevated temperatures, the bumps in the α–V characteristics disappear since broadening of the homogeneous linewidth eliminates the discrete nature of the QD layers. A thermally activated branch of the I–V characteristic is dominated, at the lowest bias regime, by diffusion-recombination, in accordance with the classical Shockley-Reed-Hall model [19, 20]. At 230 K and 290 K, carrier transport is dominated by tunnelling at a somewhat increased voltage corresponding to the second and third peaks in Figure 2(e)–(f). This behavior results from the strong temperature dependence of the ideality factor and slope of the I–V characteristics, and the weak dependence of the saturation current [21].

A high-resolution α–V curve (with a 1 mV voltage step) measured at 4 K is shown in the blue trace of Figure 3(a) for the voltage range of 2.3 V–3 V. It reveals well-separated, periodic, narrow peaks which are superimposed on the bumps. These peaks are a direct imprint of the actual TI process, resembling classical resonant tunnelling. Since the QD ensemble is inhomogeneous, different voltages cause hybridization of excited states belonging to different QD groups (clusters) with the confined state of the QW injector [5–7], yielding successive resonant tunnelling events [22]. The α–V curve for a reference QD laser with no TI region is shown in the red trace of Figure 3(a) and naturally shows no resonances. The two yellow circles shown in Figure 3(a) represent the two respective threshold voltages.

Figure 3:

Temperature and bias dependent α. (a) High resolution α–V curves at 4 K measured with voltage steps of 1 mV for a TI QD laser and a conventional QD laser. The two yellow circles represent the two threshold voltages and the inset shows an enlarged view of the tunnelling resonances. (b) Evolution with temperature of the fourth and fifth bumps of the α–V curves measured with a voltage step of 10 mV.

Figure 4:

Temperature and bias dependent α and γ. (a) P _opt–V and γ–V characteristics. Comparison of γ–V and α–V dependencies measured at (b) 4 K, and (c) 60 K. (d) High resolution comparison of γ–V and α–V showing an enlarged view of the third, fourth and fifth bumps. The inset in Figure 4(c) shows the γ–V and α–V characteristics at 105 K.

The periodic reiteration in Figure 3(a) does not exhibit single peaks, but rather double peaks making a W-like shape where the pairs of peaks are not symmetric relative to the central ones. This is highlighted in the inset of Figure 3(a). The distance between them is roughly 18 mV and 25 mV and their average full width at half maximum are 5.3 mV and 13.2 mV, respectively. The appearance of the W-like shaped peaks stems from the fact that they have a relatively wide energy span since the confined state of the injector QW can hybridize simultaneously with spectrally close excited states belonging to different groups of dots [6, 7] and therefore, resonant tunnelling occurs over a finite spectral range. For the same reason, the width of the peaks is wider than the common spectral width of a single dot at 4 K which is in the μeV range. As the temperature increases, the α–V characteristics move to lower voltages and the resonance sharpness diminishes since the homogeneous linewidth of the QDs widens and the overlap between dot clusters increases. This is shown in Figure 3(b) for the fourth and fifth bumps where the resonances disappear at 80 K.

2.2 Power-voltage characteristics

The most basic electro-optical property of a diode laser is usually characterized by P _opt–I measurements. P _opt–V characteristics are not commonly used but they often show more intricate details and are easily correlated with the I–V characteristics [21]. The P _opt–V characteristic measured at 4 K is described in Figure 4(a) together with the calculated γ–V characteristic, ( γ = d ln P opt / d ln V ) [21].

The QD layer discretization is also observed in the γ–V curve at 4 K which shows the same bumps as in the α–V curve except for the absence of the first peak since there is no emission of any kind at the very low bias voltage. The sixth bump is not seen and is replaced by a sharp peak (marked I) at 2.75 V which signifies the threshold voltage. The second large peak (marked II) at 3.05 V coincides with the peak in Figure 2(b). As the temperature rises, these two peaks evolve (see Figure 4(c)). Peak I shifts to the low voltage side, while peak II sharply decreases. Beyond 60 K they coincide and represent a single threshold voltage. This is shown in the inset to Figure 4(c) for 105 K. The main features of the low temperature γ–V characteristic, in particular the oscillating bumps follow the corresponding features of the α–V characteristic. This confirms the direct impact of the resonant tunnelling on the emission process. The oscillations superimposing the bumps in the γ–V characteristics are somewhat less pronounced compared to the α–V curve as seen in Figure 4(d). This is due to the fact that different dot layers contribute differently to the laser emission. Carrier transport effects and an internal field that bends the energy levels yield a non-uniform inversion [23].

The origin of the two peaks in the γ–V curve is the dot inhomogeneity which is highlighted at low temperatures due to narrowing of the dot’s homogeneous linewidth. This is seen in the bias dependent emitted spectra presented in Figure 5 for a temperature of 6 K.

Figure 5:

Applied voltage dependent optical spectra measured at 6 K.

At low bias levels (Figure 5(a)), the laser spectrum is that of a standard Fabry–Perot laser. As the bias increases, the spectrum becomes asymmetric and at about 3 V, the laser emits in two spectral regions as seen in Figure 5(b). This is a classical behavior of an inhomogeneously broadened laser [24]. The emergence of the second emitted spectral region coincides with the voltage of the second peak of the γ–V curve and the discontinuity in the α–V characteristic that is the voltage where carrier clamping takes place. As the temperature rises and the homogeneous linewidth increases, the discrete nature of the gain spectrum diminishes and the laser exhibits but one threshold. Correspondingly, the γ–V curve exhibits a single peak which coincides with the discontinuity in the α–V curve, which represents threshold.

2.3 Capacitance voltage characteristics

A full electrical analysis including its correlation to optoelectronic properties requires also a description of the C–V characteristics. Such measurements performed at 10 kHz and 1000 kHz are shown in Figure 6 for two temperatures. Under stimulated emission conditions, as the trapping and de-trapping processes in the QDs increase, the C–V characteristics exhibit an inductive like nature (known as negative capacitance (NC)) as often seen in light emitting diodes [25–28] and QD lasers [29].

Figure 6:

Temperature dependent C–V characteristics at two frequencies. (a) 10 K, (c) 150 K, (b) and (d), enlarged views at 10 K and 150 K, respectively, of the C–V curves near the hysteresis regions.

The measured capacitance exhibits, for both frequencies, discontinuities at two different voltages what forms a clockwise hysteresis loop. It originates from charging of the QDs when the voltage varies from negative to positive and discharging for the opposite polarity. Figure 6(b) and (d) show enlarged views near the hysteresis region at 10 K and 150 K, respectively. The hysteresis loop formation resembles that of nonvolatile memory type metal–insulator–semiconductor structures or field effect transistors, with embedded metal or semiconductor nanocrystals [30, 31]. The hysteresis loop shifts towards negative bias values as the temperature changes from 10 K and 150 K.

The average QD density can be estimated by the width of the hysteresis loop which is determined, in turn, by the total sum of trapped charges in each QD [32]. Charges accumulated in the QDs induce, under bias, charges with the opposite sign in the semiconductor substrate, what shifts the capacitance to higher voltages. The InAs average dot density is estimated as [32, 33]:

The constants ɛ ₀ and q are, respectively, the vacuum permittivity and the elementary charge, while ɛ _tun = 13.4 and ɛ _InP = 12.4 are the respective dielectric coefficients of the tunnelling layer and the InP substrate layer. d _cnt = 2 nm is the barrier layer width between dot layers (playing the role of a control layer [32, 33]), d _QD = 1.38 nm (4.7 mono-layers) is the QD mean diameter. There are N = 5 control layers, and N _QD = 6 QD layers. We further evaluate the effective number of electrons captured by a single QD using ν = d eff Δ E g [30] where Δ E g = q 2 C self is the Coulomb energy gap and C _self = 2πɛ ₀ ɛ _tun d _QD is the self-capacitance. d _eff = φ _sub − φ _QD is the depth of the effective potential well of the QD region relative to substrate that depends on φ _sub = 4.65 eV [34] and φ _QD = 4.83 eV [35] which are the respective work functions of the InP substrate and InAs QD layer (the latter is similar to the work function value φ _InAs = 4.9 eV [36] for monocrystalline InAs). We obtain that d _eff = 0.18 eV, the Coulomb band gap is ΔE _g = 0.19 eV and thus ν = 0.95. Considering a hysteresis width of ΔV _C – _V = 0.53 V (based on Figure 6), the average QD density per layer is found to be n _dot = 5.3 ⋅ 10¹⁰ cm⁻² which is close to the estimated density of 3 ⋅ 10¹⁰ cm⁻² obtained by atomic force microscope [10].

Under a high stimulated emission rate, carrier injection to the QDs is dominated by carriers that originate in the QW reservoir and feed the ground state via a hybrid state and Coulomb scattering. This leads to a fast recombination rate of the ground state electrons which are replenished by carriers that originate in the hybrid state and relax fast to the ground state via Coulomb scattering [5–7]. In addition to enhancing the radiative recombination, the carriers captured by the QDs, together with the injected holes, affect the capacitance and cause the hysteresis loop whose width is determined by the sum of charges trapped in each QD.

In the forward high-voltage regime, the measured capacitance is essentially the junction capacitance of a modified quasi-P-I-N structure with an effective thickness of its intrinsic layer being lower than at low bias levels. At low bias levels, the thickness comprises the six identical QW, barrier, and QD sublayers. Since the junction capacitance is C = d Q d V j [37] (Q being the charge), and since the differential junction voltages dV _j in diodes with monotonically increasing I–V characteristics is always positive, the junction capacitance can only be negative for dQ < 0.

While negative capacitance is a rather well known effect in light emitting and laser diodes [26, 27, 37], the peak accompanying the drop in capacitance, seen in Figure 6(a), was not reported previously. Negative dQ results from a combination of various recombination processes. These include mainly a high rate of radiative recombination and non-radiative processes originating from charge trapping in interfacial states at both boundaries of the intrinsic layer [26–28]. A second reason for NC accompanied by a sharp peak is penetration of charges stored in the QW reservoir layer to the QDs in accordance with the damped resonant tunnelling model [38]. The peak value of the negative capacitance and the shape of the peak strongly depend on the damping constant and therefore can vary over a wide range.

The NC sharp narrow peak is related to the discontinuity near 3 V in the α–V curve and the peaks in the γ–V characteristics. This correlates the various electrical and electro-optical properties and proves that the electro-optical characteristics have a clear imprint on the electrical properties. The correlations are determined via the temperature dependence of characteristic voltages corresponding to the measured data as described in Figure 7.

Figure 7:

Temperature dependence of the characteristic voltages.

In the temperature range of 4 K–60 K, the main threshold voltage equals the peak NC voltage at 1000 kHz. This means that the onset of stimulated emission, and the corresponding tunnelled carrier injection from the QW, as well as the carrier clamping, induce the change in charge density that causes the NC. In this temperature range, the discontinuity in α–V, which represents the second threshold occurs at a higher voltage as is the 10 kHz NC peak. Around 100 K, the two threshold voltages merge at the value that corresponds to the NC peak at 1000 kHz. At temperatures up to 150 K, the NC peak voltage at 10 kHz is higher than that at 1000 kHz. This stems from the fact that some traps induce slow non-radiative processes which cannot be sensed at the higher frequency. In the range of 150–200 K, the characteristic voltages at both frequencies are almost the same, but are lower than that threshold voltage. Non-radiative recombination increases above 200 K due to enhancement of phonon induced processes [39], which leads to a decrease in efficiency and an increase in the threshold voltage.