1. Introduction

During the past decade much progress has been made in III-nitride based optoelectronic devices such as light emitting diodes [1,2] and laser diodes [3,4]. The intersubband transition absorption [5,6] and quantum well infrared photodetectors [7,8] have also been demonstrated in GaN/AlGaN multi-quantum wells based devices. However, nonlinear optical effects such as second harmonic generation in III-nitride materials have received little attention so far, especially in multi-quantum wells associated with intersubband transitions [9–12]. It has been predicted that in semiconductor quantum wells, the second-order nonlinear optical susceptibility can be strongly enhanced when the input waves are in close resonance with intersubband transitions [13], while it is inherently zero in a potential with inversion symmetry. Therefore, an asymmetrical potential is essential to obtain nonzero second-order susceptibilities. For the widely used GaAs/AlGaAs, InAs/InGaAs, or SiGe/Si quantum well structures, many methods have been proposed to achieve asymmetrical potential, such as external bias voltage application [14,15], step quantum well [16] or coupled double quantum wells [17], and the second-order nonlinear optical susceptibilities have been successfully measured in some of these structures. In contrast to the above mentioned materials, GaN/AlGaN quantum wells naturally exhibit an asymmetrical triangular potential due to the large polarization-induced field. Large second-order nonlinear optical susceptibilities have been predicted in GaN/AlN quantum wells by Liu et al [18]. The experimental observation of SHG has also been demonstrated in GaN/AlN quantum wells grown on AlN/c-sapphire template by Nevou et al [19], but the susceptibility is only 114 pm/V, much smaller than the value predicted in Ref. 18. These issues prompt us to study some other structures such as GaN/AlGaN step quantum well structure for double-resonance enhancement of the susceptibility, which has not been studied so far. In such quantum well structure, an AlGaN step quantum well having a lower Al mole composition is inserted between GaN quantum well and AlGaN barrier, so that the energy levels can be tailored for double-resonance condition.

In this letter, a comparative study of second-order susceptibilities for SHG in GaN/AlGaN single quantum well and step quantum well has been conducted by solving Schrödinger and Poisson equations self-consistently, taking the polarization effect and exchange correlation potential into consideration. The influences of Al mole composition of the barrier and the well width or step well width on the susceptibilities have been investigated in detail. Specifically, the structure parameters of step quantum wells have been optimized for double-resonance condition and a significant enhancement of the susceptibility has been achieved.

2. Theoretical calculation

Considering a GaN/AlGaN quantum well structure grown on c-sapphire in the presence of an optical field with angular frequency of ω with polarization along the well growth direction (z axis), the zzz component of the second-order susceptibility tensor for SHG process is given by [16,18]

The basic unit of GaN/AlGaN single quantum well structure under investigation is$\text{(}{l}_b{\text{)Al}}_{x_1}{\text{Ga}}_{\text{1-}{x}_1}\text{N/(}{l}_w\text{)GaN/(}{l}_b{\text{)Al}}_{x_1}{\text{Ga}}_{1-x_1}\text{N,}$and that of step quantum well structure is $\text{(}{l}_b{\text{)Al}}_{x_1}{\text{Ga}}_{1-x_1}\text{N/(}{l}_w\text{)GaN/(}{l}_{sw}{\text{)Al}}_{x_s}{\text{Ga}}_{1-x_s}\text{N/(}{l}_b{\text{)Al}}_{x_1}{\text{Ga}}_{1-x_1}\text{N,}$ where l_b, l_w and l_sw are the widths of space barrier, GaN well and AlGaN step well, respectively. x₁ and x_s represent the Al mole compositions of space barrier and step well, respectively. In the calculation, l_b is fixed to 3 nm, while other parameters can be changed for various purposes.

3. Results and discussion

Figures 1(a) and 1(b) show the conduction band diagrams and squared envelope wave functions of the first three electronic levels in GaN/AlGaN single quantum well and step quantum well, respectively. It is clear that, due to the large polarization-induced field, the potential profile becomes triangular so that the inversion symmetry is broken. Therefore, a high second-order susceptibility is expected in such triangular potential profile. Unlike single quantum well structure, the susceptibility in step quantum well structure benefits from not only the polarization-induced field but also the asymmetrical structure, as shown in Fig. 1(b). The spacings between energy levels decrease significantly in step quantum well compared with that in single quantum well, which can even reach terahertz frequency range as explained in our previous study in Ref. 20.

 

Fig. 1 Conduction band diagrams and squared envelope wave functions of the first three electronic levels in GaN/AlGaN (a) single quantum well with l_w = 6 nm, x₁ = 1, and (b) step quantum well with l_w = 1 nm, l_sw = 3 nm, x₁ = 2x_s = 1. The dashed lines in (a) and (b) represent the Fermi level. Insets are the schematic one-period cross-sections of both structures, which are doped with a donor density of 1 × 10¹⁹ cm⁻³.

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Figure 2(a) shows the second-order susceptibility ${\chi }_{2\omega }^{\left(2\right)}$ as a function of the fundamental photon energy $\hslash \omega$ for different Al mole compositions of the barrier, ranging from 0.5 to 1, in GaN/AlGaN single quantum well. It can be seen that, in the frequency range, there are two main resonance peaks in the second-harmonic spectra. The lower energy peak comes from the resonance at $2\hslash \omega =E_{31}$, while the higher energy peak comes from the resonance at $\hslash \omega =E_{21}$. The two separate energy peaks indicate that the double-resonance does not occur in single quantum well, which can be confirmed by the relation $E_{21}>E_{32}$ shown in Fig. 2(b). As the Al mole composition of the barrier decreases, the maximal ${\chi }_{2\omega }^{\left(2\right)}$ increases monotonically, also seen in the inset in Fig. 1(a). When x₁ is 0.5, a high ${\chi }_{2\omega }^{\left(2\right)}$ of 4 × 10⁻⁷ m/V can be obtained in GaN/AlGaN single quantum well, which is about 4 orders of magnitude larger than that in bulk GaN [21]. To clarify which factor plays a dominant role in the enhancement of the second-order susceptibility, the intersubband transition energies (E₂₁, E₃₂, E₃₁) and the product of dipole matrix elements (z₁₂, z₂₃, z₃₁) as a function of x₁ are plotted in Fig. 2(b). The product of dipole matrix elements (z₁₂, z₂₃, z₃₁) increases by more than two times while all transition energies decrease, when x₁ decreases from 1 to 0.5. Since ${\chi }_{2\omega }^{\left(2\right)}$ is enhanced by about three times in the same decrease process, it can be concluded that the product of dipole matrix elements mainly account for the enhancement of ${\chi }_{2\omega }^{\left(2\right)}$. In addition, it should be noted that the spacing between lower energy peak and higher energy peak becomes narrower as the Al mole composition decreases, indicating that the SHG process is close to the double-resonance with an another enhancement of ${\chi }_{2\omega }^{\left(2\right)}$.

 

Fig. 2 (a) Second-order susceptibility ${\chi }_{2\omega }^{\left(2\right)}$ as a function of the fundamental photon energy $\hslash \omega$ for different Al mole compositions of the barrier, (b) intersubband transition energies (E₂₁, E₃₂, E₃₁) and the product of dipole matrix elements (z₁₂, z₂₃, z₃₁) as a function of Al mole compositions of the barrier in single quantum well, with well width of 6 nm. The inset in (a) shows the maximal ${\chi }_{2\omega }^{\left(2\right)}$ as a function of x₁.

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The influence of well width on the second-order susceptibility ${\chi }_{2\omega }^{\left(2\right)}$ is investigated and results are illustrated in Fig. 3(a). As seen in Fig. 3(a), there are still two resonance peaks in the second-harmonic spectra, which is similar to that in Fig. 2(a). The maximal ${\chi }_{2\omega }^{\left(2\right)}$ decreases significantly as the well width decreases, and simultaneously the resonance peaks have an obvious blueshift, which can be tuned to the telecommunication wavelength range by controlling well width, as shown in Fig. 3(b). It is already known that the effective depth of the triangular potential decreases while the quantum confinement effect strengthens with a decrease of the well width [20]. As a result, the whole potential profile becomes less asymmetrical leading to a decrease of the product of dipole matrix elements, meanwhile the energy spacing becomes larger making the resonance peak blueshift. However this relative small increase of the product of dipole matrix elements alone cannot completely account for the big enhancement of the second-order susceptibility ${\chi }_{2\omega }^{\left(2\right)}$ (shown in the inset in Fig. 3(a)). In the calculation, we find that the free electron density increases with increasing the well width though the dopant density is constant (not shown here). Therefore, both the carrier density and the dipole matrix elements contribute to the enhancement of ${\chi }_{2\omega }^{\left(2\right)}$.

 

Fig. 3 (a) Second-order susceptibility ${\chi }_{2\omega }^{\left(2\right)}$ as a function of the fundamental photon energy $\hslash \omega$ for different well widths, (b) intersubband transition energies (E₂₁, E₃₂, E₃₁) and the product of dipole matrix elements (z₁₂, z₂₃, z₃₁) as a function of well width in single quantum well, with x₁ = 1. The inset in (a) shows the maximal ${\chi }_{2\omega }^{\left(2\right)}$ as a function of well width.

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To highlight the advantages of double-resonance, the second-order susceptibility ${\chi }_{2\omega }^{\left(2\right)}$ in step quantum well is investigated at length in this letter. Similar to Fig. 2(a), Fig. 4(a) plots the ${\chi }_{2\omega }^{\left(2\right)}$ as a function of the fundamental photon energy $\hslash \omega$ for different Al mole compositions of the barrier. As expected, the maximal${\chi }_{2\omega }^{\left(2\right)}$ in step quantum well is larger than that in single quantum well but at much smaller resonance energy. A large value of 7 × 10⁻⁷ m/V can be gained with peak energy about 0.1 eV when x₁ = 1. It should be pointed out that the step quantum well structure is not optimized here and it is believed that the ${\chi }_{2\omega }^{\left(2\right)}$ can be further increased as discussed below when the structure is optimized for the double-resonance condition. In Fig. 4(a), it is worth noting that, the highest peaks from the resonance at $2\hslash \omega =E_{31}$ in the second-harmonic spectra are no longer located at lower energy positions, but at higher energy positions, just opposite to that in single quantum well. A very small third peak located at even higher energy position can be seen in each spectrum, which stems from the resonance at $\hslash \omega =E_{31}$. As shown in the inset in Fig. 4(a), different from the monotonic increase in single quantum well, the maximal ${\chi }_{2\omega }^{\left(2\right)}$ in step quantum well first decreases and then increases when x₁ increases from 0.5 to 1, and has a minimum at x₁ = 0.6. This strange feature is not difficult to understand. The product of the matrix elements (z₁₂, z₂₃, z₃₁) is minimum and SHG process is also far away from the double-resonance condition when x₁ = 0.6, as depicted in Fig. 4(b). When x₁ further increases from 0.8 to 1, although z₁₂z₂₃z₃₁ declines, the energy spacing between E₂₁ and E₃₂ becomes narrower, meaning that the SHG process is closer to the double-resonance condition. Therefore, the second-order susceptibility ${\chi }_{2\omega }^{\left(2\right)}$ still increases sharply.

 

Fig. 4 (a) Second-order susceptibility ${\chi }_{2\omega }^{\left(2\right)}$ as a function of the fundamental photon energy $\hslash \omega$ for different Al mole compositions of the barrier, (b) intersubband transition energies (E₂₁, E₃₂, E₃₁) and the product of dipole matrix elements (z₁₂, z₂₃, z₃₁) as a function of Al mole compositions of the barrier in step quantum well, with x₁ = 2x_s, l_w = 3 nm, and l_sw = 6 nm. The inset in (a) shows the maximal ${\chi }_{2\omega }^{\left(2\right)}$ as a function of x₁.

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The second-harmonic spectra for different step well widths in step quantum well structure are depicted in Fig. 5(a). It can be observed that the step well width has a significant impact on the second-order susceptibility${\chi }_{2\omega }^{\left(2\right)}$. Very different from that in single quantum well, the${\chi }_{2\omega }^{\left(2\right)}$ first decreases slightly and then increases sharply with an increase of the step well width, as seen in the inset in Fig. 5(a). It is interesting that ${\chi }_{2\omega }^{\left(2\right)}$ is nearly zero when step well width is 4 nm. A possible reason is that the three envelop wave functions are almost symmetrical in this asymmetrical structure, which can be confirmed by the nearly zero product of the matrix elements (z₁₂, z₂₃, z₃₁) shown in Fig. 5(b). In Fig. 5(b), the intersubband transition energies (E₃₂, E₃₁) decrease monotonically with increasing the step well width except for E₂₁, whereas the product of the matrix elements shows a zigzag change, which is very different from that in single quantum well structure. This is because there are more factors affecting the distributions of the envelop wave functions and energy levels, as described in Ref. 20.

 

Fig. 5 (a) Second-order susceptibility ${\chi }_{2\omega }^{\left(2\right)}$ as a function of the fundamental photon energy $\hslash \omega$ for different step well widths, (b) intersubband transition energies (E₂₁, E₃₂, E₃₁) and the product of dipole matrix elements (z₁₂, z₂₃, z₃₁) as a function of step well width in step quantum well, with x₁ = 2x_s = 1 and l_w = 3 nm. The inset in (a) shows the maximal ${\chi }_{2\omega }^{\left(2\right)}$ as a function of step well width.

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It should be pointed out that it is difficult to qualify double-resonance condition in single quantum well structure since the energy spacing becomes narrower as energy level increases, refering to the energy levels in Fig. 1(a). However, the situation is different in step quantum well structure where double-resonance condition can be readily satisfied since there are more degrees of freedom to be explored. To search for this condition, four structure parameters (l_w, l_sw, x₁, x_s) have been optimized and results are shown in Fig. 6 and Table 1 and Table 2. Figure 6(a) shows the energy difference of the E₃₂ and E₂₁ as a function of the Al mole composition of the barrier x₁ for different Al mole compositions of the step well x_s, meanwhile keeping l_w = 3 nm and l_sw = 6 nm. For each x_s value (except for x_s = 0.5), there exists two x₁ values at which the double-resonance condition can be realized, i.e. E₃₂ = E₂₁. The corresponding maximal second-order susceptibilities ${\chi }_{2\omega }^{\left(2\right)}$ are listed in Table 1. A huge ${\chi }_{2\omega }^{\left(2\right)}$ as large as 4 × 10⁻⁶ m/V can be obtained with peak energy $\hslash \omega$ = E₂₁ = 0.099 eV when x_s = 0.3 and x₁ = 0.68, which isabout an order of magnitude greater than that obtained in single quantum well. More importantly, all the maximal second-order susceptibilities at double-resonance condition reach 10⁻⁶ m/V magnitude, about one order of magnitude larger than that without structure optimization. It must be emphasized that the double-resonance peak energies listed in Table 1 are all very small, around 0.1 eV, meaning that the SHG process with huge ${\chi }_{2\omega }^{\left(2\right)}$ works effectively in the mid-infrared wavelength range.

 

Fig. 6 (a) Energy difference of the E₃₂ and E₂₁ as a function of the Al mole composition of the barrier x₁ for different Al mole compositions of the step well x_s with l_w = 3 nm and l_sw = 6 nm, (b) energy difference of the E₃₂ and E₂₁ as a function of the step well width for different well widths with x₁ = 1 and x_s = 0.5. The dashed horizontal line indicates zero energy difference for double-resonance condition.

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Table 1. Calculated ${\chi }_{2\omega }^{\left(2\right)}$ and Peak Energy $\hslash \omega$ at Double-resonance Condition with l_w = 3 nm and l_sw = 6 nm

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Table 2. Calculated ${\chi }_{2\omega }^{\left(2\right)}$ and Peak Energy $\hslash \omega$ at Double-resonance Condition with x₁ = 1 and x_s = 0.5

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Figure 6(b) shows the energy difference of the E₃₂ and E₂₁ as a function of the step well width for different well widths, meanwhile keeping x₁ = 1 and x_s = 0.5. Similar to Fig. 6(a), for each l_w value, there exists two l_sw values at which the double-resonance condition is satisfied. Also, the corresponding maximal second-order susceptibilities ${\chi }_{2\omega }^{\left(2\right)}$ are listed in Table 2. Different from that in Table 1, for each l_w value, the two maximal ${\chi }_{2\omega }^{\left(2\right)}$ are very different, that is, one is relatively small while the other is very large. According to Eq. (3), this difference can be attributed to the product of the dipole matrix elements and free carrier density. It is also found that the larger ${\chi }_{2\omega }^{\left(2\right)}$ corresponds to the lower peak energy while the smaller ${\chi }_{2\omega }^{\left(2\right)}$ to the higher peak energy, which further confirms that the SHG process with larger double-resonance enhanced ${\chi }_{2\omega }^{\left(2\right)}$ in step quantum well structure performs effectively in the longer wavelength range.

4. Conclusion

In summary, we have presented a study of intersubband second-order nonlinear optical susceptibilities in GaN/AlGaN single quantum well and step quantum well structures. The related envelope wave functions and energy levels have been obtained by solving Schrödinger and Poisson equations self-consistently. In single quantum well structure, large second-order susceptibilities ${\chi }_{2\omega }^{\left(2\right)}$ of several 10⁻⁷ m/V for SHG can be gained because of the large polarization-induced field. In step quantum well structure, even larger ${\chi }_{2\omega }^{\left(2\right)}$ can be obtained with structure parameters optimized for double-resonance enhancement. Specifically, a huge ${\chi }_{2\omega }^{\left(2\right)}$ as large as 4 × 10⁻⁶ m/V can be obtained when x_s = 0.3, x₁ = 0.68 and l_w = 3 nm, l_sw = 6 nm, which is about an order of magnitude greater than that obtained in single quantum well structure. In addition, the impacts of the structure parameters such as Al mole composition of the barrier and well or step well widths on the second-order susceptibility have also been investigated in detail. Calculated results show that, for single quantum well, large well width and small Al mole composition of the barrier lead to a high ${\chi }_{2\omega }^{\left(2\right)}$, but it is not correct for step quantum well where the ${\chi }_{2\omega }^{\left(2\right)}$ changes non-monotonically with the step well width and the Al mole composition of the barrier. The results indicate that nonlinear optical elements based on GaN/AlGaN step quantum wells are very promising for SHG in a wide range of wavelengths from telecommunication to mid-infrared, especially effective in longer wavelength range.