Optical Properties of Nitride Cylindrical Quantum Dots as Biosensor Transducer

Nitride cylindrical quantum dots (NCQDs) are commonly used materials in optical chemical sensors and biosensors. We investigated the exciton-related optical properties of wurtzite ZnSnN 2 /In x Ga 1− x N cylindrical quantum dots (CQDs) with finite potential barriers. The energy states in a CQD system were computed by a variational scheme using the single-band effective mass approximation, which considers the effect of the built-in electric field (BEF). The results are presented in the form of a specific function of CQD structural parameters including height, radius, and In concentration in the barrier. Our major findings are as follows. First, the excitonic transition energy decreases with increasing CQD size and increases with increasing In concentration. Second, the oscillator strength rapidly decreases with increasing CQD height and slowly increases with increasing CQD radius. Furthermore, we obtained the absorption coefficient (AC) of the exciton as a function of the incident photon energy for different CQD structural parameters. We concluded that, in general, increasing any of the aforementioned structural parameters causes either a blueshift or redshift of the AC peak. A similar phenomenon occurs for the resonant peak intensity.


Introduction
Nitride cylindrical quantum dots (NCQDs) are commonly used materials in optical chemical sensors and biosensors. Maier et al. (1) reported an optical chemical sensor that adopts nitride nanowire heterostructures grown by plasma-assisted molecular beam epitaxy on Si (111) substrates. Nötzel (2) reported an enzymatic biosensor for the detection of glucose and cholesterol for medical diagnostics that uses quantum dots (QDs) grown by plasma-assisted molecular beam epitaxy on a Si (111) substrate. Such nanomaterial-based semiconductor QD biosensors use optical readout schemes based on the luminescence properties of the QDs. He also found that, compared with optical and other sensing techniques, the nitride QD enzymatic electrochemical biosensors have excellent linearity, sensitivity, response time, stability, repeatability, reproducibility, and selectivity. Further, among the various materials frequently used for NCQDs, wurtzite (WZ) ZnSnN 2 /In x Ga 1−x N cylindrical quantum dots (CQDs) with finite potential barriers are especially important as sensor materials because of their superior optical characteristics.
Research on the structural properties of Zn-IV-N 2 compounds (IV = Si, Ge, Sn) has been drawing much attention. (3)(4)(5)(6)(7) The crystal structure of a Zn-IV-N 2 compound is derived from the WZ III-nitride, where the group III ions are replaced by alternating Zn and group IV ions. By adding Si and Ge, the direct band gaps of Zn-(Si,Ge,Sn)-N 2 alloys may cover the entire visible light spectrum. (8,9) This feature gives Zn-(Si,Ge,Sn)-N 2 alloys promising performance for fullspectrum LED applications. (10)(11)(12)(13)(14) In addition, II-IV-N 2 compounds, which are closely related to the WZ-structured III-N semiconductors, have similar electronic and optical properties to InGaN, such as direct band gaps and large optical absorption coefficients. (9,15) Moreover, Zn and the IV group elements of Si and Sn are earth abundant and environment friendly. (16) Among the II-IV-N 2 materials, ZnSnN 2 is considered one of the most promising absorber materials in photovoltaic solar cell applications due to its excellent characteristics of an ideal solar-matched band gap, robust optical absorption, and low cost. (17)(18)(19)(20)(21) However, ZnSnN 2 is one of the least studied Zn-IV-N 2 compounds.
Computational research on the properties of ZnSnN 2 did not begin until 2008, (3) and the first synthesis of ZnSnN 2 was not reported until 2013. (15) Although extensive studies on this material are now under way, its basic electronic and optical properties have not yet been fully elucidated. Because the lattice constant of ZnSnN 2 is between those of GaN and InN, it can be latticematched to that of InGaN. Wang et al. (12) reported that the conduction band minimum of ZnSnN 2 is 1.44 eV lower than that of GaN and that the valence band maximum of ZnSnN 2 is 0.39 eV higher than that of GaN. (22) Therefore, a ZnSnN 2 layer may serve as a well layer in the heterostructure system of ZnSnN 2 /In x Ga 1−x N, in which x may reach up to 0.5. Several possible applications of ZnSnN 2 /In x Ga 1−x N quantum wells (QWs) in optoelectronics and electronics have been investigated, such as the use of InGaN-ZnSnN 2 QWs for high-efficiency amber LEDs, (12) an LED tunable in the IR range that is based on InGaN-ZnSnN 2 /GaN QWs, (13) and QW-based near-IR LEDs. (14) Moreover, in a WZ In x Ga 1−x N/ZnSnN 2 /In x Ga 1−x N QW structure with the c axis parallel to the growth direction, a piezoelectric (PE) field may occur in the ZnSnN 2 due to its lattice mismatch with In x Ga 1−x N, and a sizeable local surface plasmon (SP) resonance across the ZnSnN 2 layer also appears. The total built-in electric field (BEF) is estimated to have an order of magnitude of MV/cm and to decrease as the In concentration x increases. (23) Hence, the optical properties of [0001]-oriented WZ ZnSnN 2 /In x Ga 1−x N strained QWs are strongly affected by the BEF.
Although the optical properties of III-V nitride QDs have been fully studied, WZ ZnSnN 2 / In x Ga 1−x N QDs have not yet been fully analyzed. Thus, as reported in this paper, we have theoretically studied the excitonic transition energy, the oscillator strength, and the effect of the BEF in a WZ ZnSnN 2 /In x Ga 1−x N CQD with geometrical confinement. In numerical calculations, we use a variational scheme with a single-band effective mass approximation. We also fully investigated the absorption coefficient (AC) of the exciton as a function of the incident photon energy (IPE) for different CQD structural parameters (height, radius, and In concentration in the barrier).

Theoretical Foundation
In accordance with previous theoretical studies on WZ semiconductor QDs, (24)(25)(26) we now consider a ZnSnN 2 CQD with height L along the c axis and radius R embedded in an In x Ga 1−x N barrier material (Fig. 1). On the basis of the single-band effective mass approximation, (24) the Hamiltonian of an exciton confined in a ZnSnN 2 /In x Ga 1−x N CQD is given by 2 All symbols in the above equations are described in Refs. 25   In Eqs. (2) and (3), V(ρ j ) and V(z j ) are, respectively, the in-plane and z-direction confinement potentials of the electron (hole) in the WZ ZnSnN 2 /In x Ga 1−x N CQD structure. These parameters can be calculated via 1 2 , and 1 2 , where Q j is the band offset ratio, taken as 79.5:20.5 for both the conduction band and the valence band. (22) The eigenfunction of the Hamiltonians ˆj H can be written in the form (24)(25)(26) ( , , ) ( ) ( ) , 0, 1, 2, .
The time-independent Schrödinger equation for this exciton system is given by where E ex is the eigenenergy and ( ) ex e h , r r Φ   is the eigenfunction. To find the eigenenergy, in the variation we choose the trial wave function (24)(25)(26) The ground-state energy E ex is defined as The binding energy E b and interband transition energy E ph can be calculated as (24,25) where E e + E h is the summation of the free electron and hole confinement energies in the same CQD.
The oscillator strength f ex for the exciton ground state can be computed as ( In the calculation, we set P 2 /m 0 = 1 eV. (28) The AC is given by ( and where V and ħω represent the sample volume and the IPE, respectively. Γ is the line width of the exciton, and in our calculation, we set Γ = 5 meV. (29)

Results and Discussion
Through calculations using the above equations, we obtain the excitonic transition energy and the oscillator strength as functions of the QD structural parameters in the strained WZ ZnSnN 2 /In 0.2 Ga 0.8 N CQD. In addition, the AC is investigated for different QD structural parameters. The material parameters are given in Table 1. All the material parameters of In x Ga 1−x N, are computed through linear interpolation, except for E g . The bowing parameter of the band gap for In x Ga 1−x N is set as 3.2 eV. (22) Figure 2 shows the calculated variation of E ph as a function of the CQD size in the ZnSnN 2 / In 0.2 Ga 0.8 N CQD. E ph decreases as L or R increases and drops below the bulk WZ ZnSnN 2 energy gap when L is higher than 3.5 nm for all radii of the CQD. This is because the confined energies of the electron and hole decrease with increasing CQD size. In addition, L has a more pronounced influence than R on E ph because the BEF in the ZnSnN 2 layer is along the growth direction of the heterostructure, and it may affect E ph via the quantum-confined Stark effect. The variation predicted using our proposed formula, which adopts the optical transition energy as a function of the CQD size, closely matches those of the InGaN/GaN CQD reported by Shi and Gan (24) and the GaN/AlGaN QD reported by Minimala et al. (27) We now investigate the variation of the excitonic transition energy E ph as a function of the In concentration x of WZ ZnSnN 2 /In x Ga 1−x N CQDs with L = 3 nm and R = 6 nm. Figure 3 shows that E ph is significantly influenced by the BEF, also increasing with increasing x. In contrast, without the BEF, E ph decreases with increasing In concentration x. The reason for this phenomenon is as follows. When the BEF originating from SP and PE polarizations in the Hamiltonian is included, the In concentration in the barrier affects E ph by both decreasing the confined potential of the electron and hole and decreasing the BEF. Increasing the In concentration x results in (i) a decrease in the electron and hole confined energies because a decrease in the potential barrier height leads to a redshift of E ph , and (ii) a decrease in the BEF because of the occurrence of a blueshifted energy spectrum. The latter result dominates in the case of ZnSnN 2 /In x Ga 1−x N CQDs with L = 3 nm; thus, E ph exhibits a redshift with decreasing x. However, when the BEF is ignored, only the first result occurs with increasing x; therefore, E ph decreases with increasing x. Moreover, Fig. 3 also clearly indicates that a strong BEF gives rise to a large decrease in E ph , especially for QDs with low x.  Figure 4 shows the obtained variation of the oscillator strength f ex for the ground exciton state with the dot size in the ZnSnN 2 /In 0.2 Ga 0.8 N CQD. f ex decreases rapidly with increasing CQD height L, approaching zero when L is near to 5.5 nm. The reason for this is as follows. f ex is inversely proportional to E ex and directly proportional to the square of the overlap integral between the electron and hole wave functions [see Eq. (13)]. As L increases, the electron-hole spatial separation in the z direction becomes large, which leads to an apparent reduction in the overlap integral between the electron and hole wave functions, and E ex simultaneously decreases. However, the overlap integral between the electron and hole wave functions decreases by a larger margin than E ex . Therefore, f ex decreases rapidly with increasing L. Moreover, Fig. 4 illustrates that f ex increases slowly with increasing R. The main reason for this is that the overlap integral increases as R increases, directly leading to an increase in the oscillator strength f ex . Figure 4 further shows that the influence of L on f ex is more visible than that of R because the BEF in the ZnSnN 2 layer is along the growth direction of the heterostructure. The predicted variation using our proposed formula, which adopts f ex as a function of the QD size, is similar to that of the ZnO/MgZnO QD reported by Zheng et al. (26)

Variation of AC α
We next obtain the variation of the AC α as a function of the IPE ħω with and without the BEF in the ZnSnN 2 /In 0.2 Ga 0.8 N QDs with R = 6 nm and different dot heights, as shown in Fig. 5. It is observed that the position of the peak of the AC α gradually shifts to the region where a lower photon energy occurs as L increases. Eventually, the AC peak value is obtained at the specific IPE that ħω ≈ E ph [see Eqs. (14) and (15)]. The reason for this is as follows. As L increases, the excitonic transition energy decreases. As a result, the AC α peak exhibits a redshift with increasing L. Further, the AC peak intensity decreases as L increases. This is because the  AC α is directly proportional to the square of the overlap integral between the electron and hole wave functions and inversely proportional to the sample volume [see Eq. (14)]. On the one hand, the sample volume increases with increasing L. On the other hand, as L increases, the quantum confinement attenuates, resulting in a decrease in the overlap of the electron and hole. Therefore, the AC peak intensity decreases with increasing CQD height L. Furthermore, comparing Figs. 5(a) and 5(b), we find that the magnitude of the AC α decreases and its peak is obtained at a lower IPE when the BEF in the Hamiltonian is included for a given CQD height L. For example, for L = 2 nm, when considering the BEF effect, the peak value of the AC α is 2.570 × 10 4 cm −1 and appears at an IPE of 1947.4 meV, whereas ignoring the BEF effect, the peak value of the AC α is about 4.163 × 10 4 cm −1 and appears at ħω = 1994.0 meV. The difference in the peak AC values between the cases with and without the BEF effect is about 1.593 × 10 4 cm −1 , and the position of the AC peak for these two cases has a redshift difference of 46.6 meV. For L = 4 nm, the peak value of the AC α is 1.624 × 10 3 cm −1 (2.286 × 10 4 cm −1 ) and appears at ħω = 1662.2 (1873.2) meV with (without) the BEF effect. The difference in the peak AC intensity values between the cases with and without the BEF effect is about 2.124 × 10 4 cm −1 , and the position of the AC peak for these two cases has a redshift difference of 211.0 meV. These results further illustrate the phenomenon that the influence of the BEF becomes significant with increasing QD height L. This is because the spatial separation of electrons and holes induced by the huge on-axis BEF in the WZ ZnSnN 2 /In 0.2 Ga 0.8 N heterostructure reduces their overlap. Thus, stronger BEFs reduce the intensity of the absorption peak. Moreover, it is known that the existence of BEFs always reduces the excitonic transition energy. Therefore, strong BEFs cause a redshift of the AC peak. The predicted variation using our proposed formula, which adopts the AC as a function of the CQD height, is similar to that of the ZnO/MgZnO CQD reported by Zheng et al. (26) Considering the BEF effect, we now calculate the AC α as a function of the IPE ħω in ZnSnN 2 /In 0.2 Ga 0.8 N CQDs with a height L of 3 nm and different radii R. The calculation result is shown in Fig. 6. As R increases, the AC peak moves towards a lower IPE accompanied by an apparent decrease in its intensity. For example, the AC peak intensity decreases from about 2.834 × 10 4 cm −1 for R = 3 nm to 1.078×10 4 cm −1 for R = 5 nm, and the AC peak undergoes a redshift from 1852.8 meV for R = 3 nm to 1802.8 meV for R = 5 nm. In addition, comparing Fig. 5(a) with Fig. 6 indicates that L exhibits a stronger influence than R on the AC α. Thus, our investigation of the exciton-related optical properties reveals that both the BEF and the QD height L play crucial roles.
Using Eqs. (14) and (15), we calculate the AC α as a function of the IPE ħω with and without the BEF effect in ZnSnN 2 /In x Ga 1−x N CQDs with L = 3 nm and R = 6 nm for In concentrations of x = 0.1, 0.2, and 0.3, and the obtained result is shown in Fig. 7. When the BEF is included in the Hamiltonian, the AC peak gradually moves to the region where a high photon energy occurs, and the AC peak intensity increases with increasing In concentration. However, when the BEF is ignored, the AC peak shifts towards the region of low photon energy with increasing In concentration, whereas the intensity of the AC peak is insensitive to the In concentration in the barrier. The reason for this is as follows. The AC peak value is obtained where the IPE ħω ≈ E ph . From Fig. 3, we observe that the excitonic transition energy E ph increases (decreases) with increasing In concentration x when the BEF effect is included (ignored). As a result, the AC peak with (without) the BEF effect undergoes a blueshift (redshift) with increasing In concentration. Furthermore, the intensity of the BEF in the strained ZnSnN 2 layer is weakened with increasing x, which enhances the overlap integral between the electron and hole wave functions. Hence, the absorption peak intensity is enhanced with increasing In concentration.

Conclusion
Determining the key factors affecting the optical properties of WZ ZnSnN 2 /In x Ga 1−x N CQDs with finite potential barriers is important for enhancing the sensing performance of biosensors or optical chemical sensors employing this material. In this study, considering the inclusion of a strong BEF due to SP and PE polarizations, we theoretically studied some of the optical properties of the exciton confined in WZ ZnSnN 2 /In x Ga 1−x N CQDs with finite potential barriers at all surfaces using a single-band effective mass approximation with the variational method. We calculated the transition energy as functions of the CQD structural parameters, including the CQD height L and radius R, and the In concentration x in the barrier layer. We also investigated the relationship between the oscillator strength and the CQD size. In addition, the calculated exciton states were used to evaluate the related AC as functions of the IPE for different CQD structure parameters. The main findings are as follows. The transition energy increases with decreasing CQD size and decreases with decreasing In concentration. The oscillator strength rapidly decreases with increasing CQD height and slowly increases with increasing CQD radius. Furthermore, the AC resonant peak undergoes a redshift (blueshift) as the CQD size or In concentration is increased with a decrease (an increase) in the resonant peak intensity. Moreover, we computed the AC of the exciton as a function of the IPE for different CQD structure parameters in cases with and without the BEF effect. We found that a strong BEF causes a redshift of the AC peak and dramatically decreases the resonant peak intensity. Therefore, to obtain a large AC, the adverse effects of the BEF should be minimized and the dot size should be modified. Although experimental results for exciton states and interband transitions in the WZ ZnSnN 2 /In x Ga 1−x N QD are still lacking, our obtained theoretical results will be useful for related physics experiments and for the design of some LEDs covering the visible light spectrum and photovoltaic solar cells based on WZ ZnSnN 2 /In x Ga 1−x N QD structures.