1 Introduction

Semiconductor laser diodes (LDs) are important optoelectronic devices, and their characteristics have been extensively studied [1, 2]. Recent studies have shown that there are some outstanding issues regarding semiconductor laser properties, such as the definition of the threshold [3], phase change of the laser [4,5,6], and opposing experimental phenomena mentioned below. In 1976, Paoli and Barnes observed a drop of the IdV/dI–I curve of LDs at the onset of lasing [7] and attributed it to the abrupt saturation of the junction voltage (V j ), namely the abrupt stability of the separation of the quasi-Fermi levels of electrons and holes the across the active region at the threshold. Following that study, other authors invariably observed this characteristic in narrow bandgap LDs [7,8,9,10,11,12,13]. However, recent experiments have shown that the IdV/dI–I of GaN-based LDs with a 405 nm wavelength has a slight jump or instantaneous stability rather than a drop near the threshold [14,15,16]; unexpectedly, its V j is not pinned beyond the threshold [14], which is in conflict with the above concept. Clearly, further detailed research confirming this behavior is necessary.

Here, we systematically compared the electrical behaviors of MQW LDs with different wavelengths at forward bias. Aside from the previously reported 780 and 405 nm LDs [8, 14], 650, 515, 450 nm wavelength LDs were also studied. Interestingly, the apparent behavior curves, as well as the junction behavior curves, for all of the LDs showed abrupt changes at the onset of lasing, and their start and end points exactly corresponded to the two kinks of the first derivative of P–I (Il th and Iu th shown in Figures). According to Ning’s description of the threshold [3], we defined the current ranging from Il th to Iu th as a threshold region. In this region, surprisingly, the abrupt change in the trends of the electrical parameter curves in 650 and 780 nm LDs were similar, as were the trends seen in the 450 and 405 nm LDs; however, the former is unexpectedly opposite to the latter. In the 515 nm LDs, the curves’ kink curvature was not obvious. Analysis of these opposite changes of the electrical characteristics revealed that there are two different physical mechanisms between narrow bandgap and wide-bandgap LDs, namely lasing in GaAs MQW LDs is caused by the recombination of an electron and hole, whereas lasing in GaN-MQW LDs is probably caused by exciton recombination.

2 Apparent characteristics of LDs

The apparent capacitances Cp were directly measured using an Agilent 4294A LCR meter under a small AC signal modulation at 10 kHz with an oscillation level of 10 mV. The I–V curve was measured using an Agilent 4155C Parameter Analyzer, and the light output was measured using a LD2002C3 Semiconductor Laser Parameter Analyzer made by Beijing LD Opto-electronic Company, Ltd. All measurements were performed at room temperature. A range of commercial MQW LDs that had a series of wavelengths and were made by different factories were accurately measured. In this work, only the results of the LDs made by Sony Corp were introduced.

Figure 1 shows that the changing trends of the IdV/dI–I and Cp–I curves of narrow bandgap MQWs LDs at 780 and 650 nm are similar. The IdV/dI–I curves for both LDs drop abruptly in the threshold region, and then start to increase with increasing current, as shown in Fig. 1a. This result has often been attributed to the pinning of the quasi-Fermi levels of electrons and holes across active regions (or saturation of the junction voltage) [3,4,5,6,7,8,9,10,11,12,13]. The Cp of both LDs displays a negative capacitance (NC) effect at large forward voltage [17,18,19,20,21,22,23,24] and drops down in threshold.

Fig. 1
figure 1

Apparent electrical and optical parameters of MQWs 780 (red square) and 650 nm (black dot) LDs. We only focus on the slope change of the curves, so the values of some of the curves are enlarged or reduced in the figures. a IdV/dI–I curves, in which the curves for 650 nm LD are multiplied by 0.6; b C p –I, in which the curve for 650 nm LD is multiplied by 2.2; c dP/dI–I. The two kink points of the first derivatives correspond to Il th and Iu th , respectively

Full size image

For 515 nm LDs, the slope changes of the curves between two adjacent regions are even more relatively inconspicuous. For example, in the threshold region, a slight rise of the IdVdI–I and an approximate horizontal line of the Cp–I are observed, as shown in Fig. 2. Cp displays an obvious NC; it maintains a transient stability in the threshold region and then decreases with current. Similar to the above LDs, the threshold region obtained from the slope kinks of the parameter curves is consistent with the region determined by the first derivative of the optical output power, as shown in Fig. 2c. The two kink points of the first derivative clearly correspond to Il th and Iu th , respectively.

Fig. 2
figure 2

The electrical and optical parameters of a 515 nm LD. a IdV/dI–I, b C p –I, c dP/dI–I curves of the sample

Full size image

For 450 and 405 nm LDs, the slope trends of the IdV/dI–I, C p –I curves in threshold region are the same, but are opposite to those of narrow bandgap LDs with wavelengths of 650 and 780 nm, as shown in Fig. 3. The IdV/dI–I curves of both LDs experience a jump rather than a drop in the threshold region and then increase with the current. Similar phenomena have been observed in the 405 nm LDs [14,15,16]. The Cp values of the LDs at 450 and 405 nm are very similar to those of LDs, which have wavelengths at 650 and 780 nm outside the threshold region, namely Cp displays an obvious NC. However, in the threshold region, Cp experiences a jump, rather than a drop. This opposite and abrupt change confirms a distinct set of transport properties of GaN-based MQW LD that distinguishes from narrow bandgap LDs.

Fig. 3
figure 3

The apparent electrical and optical parameters of 450 (red square) and 405 nm (black round) LDs. In figures, the values of the curves for 405 are slightly enlarged or reduced. a–c IdV/dI–I, C p –I, dP/dI–I curves of the samples, respectively

Full size image

3 Junction characteristics of LDs

The junction characteristics of the above samples were extracted by an AC-IV method. Its principles, which have been discussed in detail in a previous paper [21], are described briefly here. Comparing the equivalent circuit of a real diode with the parallel equivalent circuit used in the measurements shown in Fig. 4, under a large forward bias (\(G>>\omega C\), where ω is the AC small-signal frequency), we derive

Fig. 4
figure 4

a Actual circuit of a semiconductor diode; b the equivalent circuit of a diode using parallel mode

Full size image
$${r_s}=\frac{1}{{{G_p}}} - \frac{1}{G}$$
(1)
$$C={(1+{r_s}G)^2}{C_p}$$
(2)

where r s is the series resistance and C and G are the junction capacitance and conductance, respectively. The I–V relationship of the diodes is given by \(I={I_{\text{s}}}\left( {\exp \left( {{{q{V_j}} \mathord{\left/ {\vphantom {{q{V_j}} {nkT}}} \right. \kern-0pt} {nkT}}} \right) - 1} \right)\), where Is the saturation current, q is the electron charge, n is the ideality factor, k is Boltzmann’s constant, T is the temperature, and V j is the voltage across the junction and is expressed by V j = V − rsI when \({V_j}>>{{nkT} \mathord{\left/ {\vphantom {{nkT} q}} \right. \kern-0pt} q}\). The junction conductance G consists mainly of differential conductance and is given by

$$G=\frac{{{\text{d}}I}}{{{\text{d}}{V_j}}}=\frac{{qI}}{{nkT}}\left( {1 - \frac{{{V_j}}}{n}\frac{{{\text{d}}n}}{{{\text{d}}{V_j}}}} \right)$$
(3)

Combining the above equations, we obtain a precise dependence of rs, C, n, and V j on the forward bias voltages or currents.

Figure 5 shows the junction behaviors, V j and r s , of MQWs LDs. In the threshold region, the V j of both 780 and 650 nm wavelength LDs experiences a jump and then saturates at a higher level [8, 9]. The r s values of both LDs experience a sharp decline, then increase negligibly (and approximately keeping constant) with an increasing current injection after Iu th . Below the threshold region, the rs of 650 nm LD displays nonlinear characteristics, suggesting that it cannot be obtained from the electrical derivative technique. For 405 and 450 nm wavelength LDs, their V j values experience a slight drop or pinning in the threshold region and then increase with increasing current. However, after experiencing a sharp jump in threshold region, the rs values continue decreases.

Fig. 5
figure 5

a, c and e Separation of the quasi-Fermi levels of electrons and holes across the active region, V j , versus the injected current for LDs with 780 and 650 nm, 515 nm, and 450 and 405 nm, respectively; the curves for 650 and 450 nm LDs are multiplied by 0.835 and 1.08, respectively. b, d and f Series resistance r s of the samples versus the injected current; the curves for the 650 and 450 nm LDs are multiplied by 0.6 and 2, respectively

Full size image

4 Simulation for junction properties

The simulation was based on the following equations and parameters. The well-known Poisson equation was used

$$- \nabla \cdot \left( {\frac{{{\varepsilon _0}{\varepsilon _{{\text{dc}}}}}}{q}\nabla V} \right)= - n+p+{N_{\text{D}}}\left( {1 - {f_{\text{D}}}} \right) - {N_{\text{A}}}{f_{\text{A}}}+\sum\limits_{j} {{N_{tj}}} \left( {{\delta _j} - {f_{tj}}} \right)$$
(4)

and the electron and hole current continuity equations were

$$\nabla \cdot {J_n} - \sum\limits_{j} {R_{n}^{{tj}}} - {R_{{\text{sp}}}} - {R_{{\text{st}}}} - {R_{{\text{auger}}}}+{G_{{\text{opt}}}}\left( t \right)=\frac{{\partial n}}{{\partial t}}+{N_{\text{D}}}\frac{{\partial {f_D}}}{{\partial t}}$$
(5)
$$\nabla \cdot {J_{\text{p}}}+\sum\limits_{j} {R_{n}^{{tj}}} +{R_{{\text{sp}}}}+{R_{{\text{st}}}}+{R_{{\text{auger}}}} - {G_{{\text{opt}}}}(t)= - \frac{{\partial p}}{{\partial t}}+{N_{\text{A}}}\frac{{\partial {f_{\text{A}}}}}{{\partial t}}$$
(6)

In these equations, in addition to the traditional diffusion–drift effect, the long-range interactions of carrier transport, including quantum tunneling and hot electron emission, were also considered. The optical transverse modes are described by the scalar wave equation:

$${\nabla ^2}W\left( {x,y} \right)+k_{0}^{2}\left( {\varepsilon \left( {x,y} \right) - {\beta ^2}} \right)W\left( {x,y} \right)=0$$
(7)

The optical power of the laser is related to the number of photons and is expressed by the photon rate equation:

$$\frac{c}{{{\beta _1}}}\left[ {{g_{\text{m}}} - \frac{1}{{2L}}\ln \left( {\frac{1}{{{r_{\text{m}}}}}} \right)} \right]S+{c_{\text{m}}}\int {{R_{{\text{sp}}}}} {\text{d}}v=0$$
(8)

Combining Eqs. (4–8) results in the following equation:

$${R_{{\text{st}}}}={\varepsilon _2}\omega {\left| W \right|^2}S/\left\langle {{\varepsilon _1}} \right\rangle$$
(9)

The simulation of the device is based on the one-dimensional self-consistent solution of the above equations. In the theoretical simulation, multi-body effects are mainly reflected by the laser gain spectrum correction. The definition of the universal symbols in the equations is shown in Table 1.

Table 1 Definition of the universal symbols in the equations
Full size table

Solving Eqs. (4–9) by an advanced software package, LASTIP, which combines the band structure and free carrier gain with simulations of the 1-D wave guiding and carrier transport, we calculated a narrow bandgap GaAs-LD with an 832 nm wavelength, and its typical structure was adopted as in [25]. All of the material parameters were taken from a default database provided by LASTIP software. The carrier transport model included drift and diffusion theory with consideration of long-range carrier transport. In quantum wells, self-consistent Schrödinger and Poisson equations were recomputed at every bias point to determine the states of the quantum well levels and carrier distributions. The calculated results are shown in Fig. 6a, b. V j and rs showed the same trends as the experimental results. V j jumped up but rs dropped down slightly in threshold region, and then, V j was saturated as the current increased. The two kink points of V j (or rs) matched well to the two kink points of the first derivative of the P–I curve. The detailed simulation calculation for wide-bandgap GaN-based LDs also perfectly confirmed the experiments [14], as shown in Fig. 6c, d.

Fig. 6
figure 6

Simulation results of (a) and (b) GaAs-LD, and (c) and (d) GaN-LD [14]. a and c Dependence of the separation of the quasi-Fermi levels of electrons and holes across the active region V j (black line) and series resistance rs (red line) for both LDs; b and d Corresponding first (black line) and second derivatives (red line) of the optical output power. The calculations fully agree with the experimental results

Full size image

5 Discussion

The above calculation is close to ABC or ab initio. The calculation results are in agreement with the experimental results, indicating that our calculation method and the experimental data are indeed reasonable. In the following section, we will qualitatively discuss the reversed characteristic observed between narrow and wide-bandgap MQWs LDs.

5.1 Capacitance and series resistance

Generally, NC in light-emitting devices, in which the luminescence mechanism is carriers’ (electron and hole) recombination, is always accompanied by strong luminescence and can be interpreted in terms of the carriers’ recombination and “reclaimable” charge [18]. Then, the sudden occurrence of strong lasing in LDs must cause an abrupt increase of the absolute values of Cp in the threshold region. This physical process is the same as shown in Fig. 1b. The sharp decline of the rs values of 780 and 650 nm LDs in the threshold region (see Fig. 5b) obviously supports the traditional knowledge of physical phenomena in the threshold region of LD that high-density carrier injection will certainly form a plasma in the active region [26], which will then cause a larger increase of the electrical conductivity of LDs, namely a sharp decline in the series resistance.

As for GaN-based LDs, the decrease of the absolute value of NC (see Fig. 3b) and increase of series resistance (see Fig. 5f) are clearly not able to be explained only by the emergence of lasing. These abnormal phenomena demonstrate that the carriers’ longitudinal movement was suddenly blocked; then, the conductance weakened, which means an increase of r s in the threshold region occurred and the carriers’ recombination was also affected, leading to a decrease of the absolute value of NC. Obviously, these abnormal effects cannot be explained by the simple electron–hole recombination seen in GaAs-LDs.

5.2 Junction voltage

In LDs, V j is a very important factor that reflects the change of the quasi-Fermi levels of electrons and holes across the active region. It had already been considered to play a highly important role in phase transition research involving LDs [6]. For decades, it is well known that at the threshold the V j of LDs immediately reaches saturation, namely the sudden “pinning” of the quasi-Fermi levels of electron and holes as soon as lasing begins. However, the V j for LDs with a narrow bandgap experiences a jump in the threshold region before reaching saturation. This jump, which leads to an increase of the carrier concentration in this small range of current from I th l to I th u only occurs to meet the strong increase of the recombination rates caused by the sudden occurrence of the strong stimulated emission in this region, and then to meet the dramatic increase of the photon density of several orders of magnitude in this tiny current region. For our sample with a 780 nm wavelength, the light intensity increased by approximately 1.25 orders of magnitude in the threshold region. Therefore, this jump is understandable because the stimulated emission that replaces the spontaneous emission will lead to an increase of the total number of photons caused by carrier recombination. After I th u the approximate saturation of V j indicates the steady state of the carrier population and gain in the active region.

For wide-bandgap GaN-LDs below and in the threshold regions, V j is consistent with the universal theory of LDs, namely it increases with the current before I th l and then temporarily saturates (or decreases very slightly) in the threshold region. However, it continues to increase rather than pinning with an increasing current higher than Iu th . This interesting experiment confirms the previous theory presented by Levon et al. [27]. However, the theory implies a decrease of the efficiency η, which is somewhat in conflict with our experiments.

5.3 Exploration of a new mechanism of GaN-MQW LDs

The above analysis of opposing change trend in the electrical parameters between GaAs- and GaN-LDs confirms that the lasing mechanism of GaN is not the same as that of the recombination of electron and hole in GaAs-LDs at room temperature (RT). We found that the binding energy of the exciton varies greatly with the bandgap of the materials of the QWs. In GaN-based QWs, the exciton binding energy can be as high as 46.8 meV, which is much larger than 26 meV, which corresponds to RT [28], so it is very likely that the lasing mechanism of GaN-LDs is via excitons rather than direct recombination of holes and electrons, as it is in GaAs-LDs. In GaAs-LD QWs, the bonding energy of the excitons is extremely small (a few meV) [29]; at RT, only free carriers contribute to lasing. For InP-LDs the bonding energy of the excitons is between GaN- and GaAs-LDs [30]; at RT, the contribution of excitons to light emission barely exists. Therefore, we could believe that the fraction of exciton recombination into total recombination luminescence should be the main cause of the opposite electrical characteristics of GaAs- and GaN-LDs. That is, in GaN-MQW LDs, the emission mechanism is probably the recombination of excitons [31].

6 Summary

Precise forward electrical characteristics of MQW lasers with wavelengths 780, 650, 515, 450, and 405 nm were systematically researched by experimentation and simulation. The curves of the apparent and junction electrical behaviors display abrupt changes in the threshold region. In the threshold region, the abrupt changes in the trends of these electrical parameter curves of LDs at 650 and 780 nm are similar, and at 450 and 405 nm, they are also similar; however, the former change is the opposite of the latter. The simulation fully confirms these abnormally opposite properties. The quantitative interpretation of these opposite changes between GaAs- and GaN-MQW LDs reveals that the lasing mechanism of GaN-LDs is not recombination of electrons and holes. Further analysis shows that the exciton binding energy increases with the increasing bandgap of the material in QWs. Then, in GaN-MQW LDs, exciton recombination possibly plays a role at RT, which causes different electrical characteristics from those seen in GaAs MQW LDs. In general, these experimental observations based on simple electrical measurements provide some experimental clues to Excitonic Mott Transitions.