Physics Letters A 383 (2019) 1324–1329
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Physics Letters A www.elsevier.com/locate/pla
Effects of built-in electric ﬁeld on donor binding energy in InGaN/ZnSnN2 quantum well structures Hasan Yıldırım Department of Occupational Health and Safety, Faculty of Health Sciences, Karabuk University, Karabuk 78050, Turkey
a r t i c l e
i n f o
Article history: Received 15 November 2018 Received in revised form 15 January 2019 Accepted 21 January 2019 Available online 28 January 2019 Communicated by R. Wu Keywords: GaN ZnSnN2 Impurity Donor Binding energy Quantum well
a b s t r a c t Inx Ga1−x N/ZnSnN2 quantum well structures are studied in terms of a binding energy of a donor atom. 1s and 2p ± impurity states are considered. The Schrödinger’s and Poisson’s equations are solved self-consistently. A hydrogenic type wave function to represent each impurity state is assumed. The calculations include band-bending in the potential energy proﬁle introduced by the built-in electric ﬁeld existing along the structures. The binding energy and the energy of the transition between the impurity states are represented as a function of the quantum well width, the donor position, and the indium concentration. An external magnetic ﬁeld up to 10 T is included into the calculations to compute the Zeeman splitting. The maximum value of the transition energy is around 30 meV (nearly 7.3 THz) which occurs in a 15-Å In0.3 Ga0.7 N/ZnSnN2 quantum well. Being strong, the built-in electric ﬁeld makes the transition energy drop quickly with the decreasing well width. For the same reason, the energy curves are found to be highly asymmetric function of the donor position around the well center. Compared to the bulk value, the transition energy in the quantum well structures enhances nearly two-fold. © 2019 Elsevier B.V. All rights reserved.
1. Introduction The Zn–IV-nitrides, such as ZnSnN2 and ZnGeN2 , have been considered promising materials for photovoltaic and photochemical cells as well as for applications in optoelectronic and electronic together with the group-III nitrides in recent years [1–10]. ZnSnN2 is a heterovalent ternary compound  with a crystal structure derived from that of the wurtzite III-nitride. The derived crystal structure is an orthorhombic one in which the group-III ions are replaced by alternating Zn and Sn ions . The lattice constant of ZnSnN2 is between those of GaN and InN, therefore it can be lattice-matched to InGaN . As for its band alignment to these nitrides, its conduction band minimum (CBM) is reported to be 0.3 eV lower than that of GaN . On the other hand, another recent study  reports a much higher value such as 1.44 eV. Several possible applications of ZnSnN2 in optoelectronic and electronic have been investigated. Such as short ZnSnN2 /ZnO superlattice structures , InGaN–ZnSnN2 quantum wells (QWs) for high-eﬃciency amber light emitting diodes , InGaN–ZnSnN2 /GaN-based near infrared light emitting diodes . Energy of the transition between the impurity states, such as 1s and 2p ± , which are localized below the edge of the ﬁrst subband in the conduction band of a semiconductor material can be
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enhanced because of the quantum conﬁnement when the material designed to be a QW [14,15]. In GaN/AlGaN QWs , the energy of the transition between the 1s and the 2p ± states increases up to 58 meV, which is much higher than its bulk value 23.3 meV . Also, in GaAs/AlGaAs QWs, it is possible to enhance the bulk transition energy, 5.3 meV, up to 13 meV . Since these energies lie in the THz range, several possible applications of the transitions between impurity states in a QW have been suggested such as THz emitters based on donor-assisted tunneling . In addition to the quantum conﬁnement, several other factors , such as the width of the QW, the donor position along the structure, and electric and magnetic ﬁelds have an inﬂuence on the transition energy in a QW . Therefore it would be interesting to calculate the impurity state binding energies in InGaN/ZnSnN2 QWs where the conduction band offset and the built-in electric ﬁeld can be adjusted by changing the In concentration and see where the energy of the transition is located in THz spectrum for its possible applications. In this work, the binding energy of the impurity states, 1s and 2p ± , and the energy of the transition between them are reported in Inx Ga1−x N/ZnSnN2 QWs. The energies are represented as a function of the In concentration, the QW width, and the donor position along the structure. Additionally, the effects of the external magnetic ﬁeld on the transition energy are considered in view of the Zeeman splitting. The Schrödinger’s equation, written within the effective mass and envelope function approximations, and the
H. Yıldırım / Physics Letters A 383 (2019) 1324–1329
Poisson’s equation are solved self-consistently. The internal builtin electric ﬁeld, originating from the polarization discontinuities at the boundaries of the structure, is incorporated into the calculations. For each impurity state, a trial wave function, including a hydrogenic wave function, is assumed and the binding energies are calculated through a variational approach. 2. Theory A hydrogenic donor in a single QW in the presence of an external magnetic ﬁeld has the following Hamiltonian written within the effective mass and envelope function approximations [19,20]:
Table 1 Material properties of GaN, InN and ZnSnN2 used in the calculations. alc is the lattice constant. E g is the band gap. The bowing parameters of the band gap and the spontaneous polarization for the ternary compound, Inx Ga1−x N, are given by 1.4 eV and −0.037 C/m2 , respectively. Parameter
acz (eV) act (eV) alc (Å) C 13 (GPa) C 33 (GPa) e 31 (C/m2 ) e 33 (C/m2 ) E g (eV)
3.189 106 398 −0.338 0.667 3.51 10.4 0.20 −0.0339
3.545 92 224 −0.412 0.815 0.69 15.3 0.07 −0.0413
6.753 100 306 −0.59 1.09 1.8 15.088 0.12 −0.029
1 2m∗ ( z)
m∗ (m0 ) P S P (C/m2 )
(P + eA) + E c ( z) − eV bi ( z) + H
ρ 2 + (z − zd )2
The ﬁrst term is the kinetic energy term, where m∗ ( z) is the effective mass of the electron, P is the linear momentum operator and A is the vector potential. The second term, E c ( z), is the conduction band offset. The third term, V bi ( z), is the built-in electric potential induced by the discontinuities of the spontaneous ( P S P ) and the piezoelectric ( P P Z ) polarizations at the boundaries of the QW structure. The fourth term, H , is the strain-induced contribution to the total Hamiltonian. The last one is the Coulomb interaction term, where r is the dielectric constant, ρ is the in-plane distance, and zd is the position of the donor along the growth direction z. In the Hamiltonian above, if one chooses a gauge such that A = B /2 (− yi + xj) where B is the magnetic ﬁeld then the Hamiltonian changes into [19,20]
1 2m∗ ( z)
2m∗ ( z)
− eV bi ( z) + H −
1 ωc L z + m∗w ωc2 ρ 2 4
ρ 2 + (z − zd )2
+ E c ( z)
C 13 C 33
The strain-induced contribution to the total Hamiltonian, H , is computed through the following equation :
H = 2act 1 + acz 3 = 2 act − acz
C 13 C 33
where e i j , i and C i j are the piezoelectric constants, the strain components and the elastic constants of the material constituting the QW layer, respectively. The material properties of the binaries, GaN and InN, and ZnSnN2 [2,7,11,12,21–24] are given in Table 1. All the material properties of the ternary compound, Inx Ga1−x N, are computed through a linear interpolation between the respective material properties of the binaries, except for E g and P S P . Having obtained the polarizations, −∂ P ( z)/∂ z is inserted into the Poisson’s equation and it is solved self-consistently together with the Schrödinger’s equation  by applying an iterative method based on a predictor–corrector approach  to give V bi ( z).
act and acz in the equation above are the conduction deformation potentials in the growth plane and in the growth direction, respectively. A variational method  is adapted to obtain the binding energy of the 1s and 2p ± states. For each state, a trial wave function given by 
ψ(ρ , z − zd ) = χ ( z)ξ(ρ , z − zd ),
is applied. In Eq. (5), χ ( z) is the envelope function yet to be determined and ξ(ρ , z − zd ) is a hydrogenic type wave function. The following hydrogenic wave functions  are used for the 1s and 2p ± states:
ξ1s = exp (−r /λ1s )
where m∗w is the effective mass of the electron inside the well layer, ωc = e B /m∗w is the cyclotron frequency and L z is the z-component of the angular momentum. An Inx Ga1−x N/ZnSnN2 QW structure possesses a nonzero builtin electric ﬁeld F bi (x), which originates from the discontinuities of the total polarization, P = P S P + P P Z , at the material boundaries along the structure. The ﬁeld, F bi (x), leads to a band-bending in the potential energy proﬁle of the structure. The band-bending is incorporated into the Hamiltonian by means of the term −eV bi . One can calculate P P Z inside a QW layer through the equation :
P P Z = 2e 31 1 + e 33 3 = 21 e 31 − e 33
ξ2p ± = ρ exp (±i φ) exp −r /λ2p ± ,
respectively. In the equations above, r = ρ 2 + ( z − zd )2 and φ is the polar angle, while λ1s and λ2p ± are the variational parameters yet to be determined. The binding energy of the impurity state E 1s is calculated through
E 1s = E 0 + γ − E
where E 0 is the ground state energy of the electron in the absence of the donor, γ = h¯ ωc /2 is the energy of the lowest Landau level  and E is the minimum energy obtained from Eq. (2) through the variational approach. The binding energy of the 2p ± states is calculated in the same manner. The Hamiltonian in Eq. (2) is discretized by means of ﬁnite differences. All numerical calculations are performed by using the SciPy Stack . 3. Results and discussions The CBM of ZnSnN2 is 0.3 eV lower than that of GaN . Indium cooperation into GaN results in the ternary compound, InGaN, with a lower band gap. A QW structure where InGaN and ZnSnN2 play the roles of the well and the barrier layers, respectively, can be designed . In fact, the potential energy proﬁle of a 25-Å InGaN/ZnSnN2 QW is shown in Fig. 1. Panel (a) shows the
H. Yıldırım / Physics Letters A 383 (2019) 1324–1329
Fig. 1. The conduction band proﬁle of a 25-Å InGaN/ZnSnN2 QW. The panel (a) is for x = 0.2 and the panel (b) is for x = 0.5. The orange solid line in each panel stands for the squared wave function of the lowest electronic state in the absence of the donor. (For interpretation of the colors in the ﬁgure(s), the reader is referred to the web version of this article.)
energy proﬁle with x = 0.2 while panel (b) shows the one with x = 0.5. For x < 0.2, the CBM of InGaN increases above that of ZnSnN2 , therefore InGaN layer no longer serves as a well layer. On the other hand, around x = 0.53, InGaN and ZnSnN2 layers are lattice-matched. The conduction band offset of the QW changes between 252 and 932 meV for x between 0.2 and 0.5. As x decreases, the lattice mismatch between the InGaN and ZnSnN2 layers increases and therefore the InGaN layer experiences a tensile strain which makes the polarization difference between the layers larger and thus F bi (x) grows in magnitude. Therefore, the potential energy proﬁle for x = 0.2 is very close to a sharp triangle while for x = 0.5 it is almost a square in shape. Accordingly, the ground state electronic wave function, plotted in the same ﬁg-
ure, looses its symmetry around the QW center with the decreasing x. A linear ﬁt to the calculated F bi (x) values (not shown) gives F bi (x) = −11.4x + 5.92 MV/cm where 0.2 ≤ x ≤ 0.5. The magnitude of the ﬁeld is around the same value with the one found in InGaN/GaN QWs  for small In concentrations but generally larger than that calculated for GaN/ZnGeN2 QWs . The binding energy of the 1s and 2p ± states and the corresponding energy of the transition are plotted against the QW width L w in panels (a), (b), and (c) of Fig. 2, respectively. The 1s state binding energy makes a peak around L w = 15 Å and then drops fast with the increasing L w . The largest and the smallest values of the binding energy are nearly 36 meV and 23 meV occurring for x = 0.3 and x = 0.2, respectively. When the QW is narrow, the binding energy ﬁrst goes up with the increasing x but then goes down, although one expects a continuous increase in it because of the growing potential energy barrier at the same time. This is because there are two more parameters that determine the binding energy besides the potential energy barrier: the effective mass of the electron and the dielectric constant of the medium. On increasing x, the effective mass becomes smaller and the dielectric constant becomes larger and therefore the Bohr radius  increases as it is inversely proportional to the former and directly proportional to the latter. In that case, the binding energy decreases. On the other hand, the binding energy becomes smaller with the decreasing x for large L w , as a consequence of the simultaneously growing band-bending which makes the QW effectively larger. The 2p ± state binding energy goes down with the increasing x, regardless of the rising potential energy barrier, as the increasing dielectric constant and the decreasing effective mass make the Bohr radius larger at the same time. Its range is between 5 and 7 meV. The transition energy, plotted in panel (c) of the same ﬁgure, follows a trend similar to that of the 1s state binding energy. Its range is between 15.6 and 30.1 meV. The maximum value occurs around L w = 15 Å when x = 0.3. Panel (d) of the same ﬁgure shows the bulk transition energy as a function of x. The bulk value changes from nearly 13.5 to 8.5 meV within the considered range of x. It is obvious that the energy is gener-
Fig. 2. The binding energy of the impurity states in InGaN/ZnSnN2 QWs as a function of L w : a) the 1s state and b) the 2p ± states. The panel (c) shows the energy of the transition between the states 1s and 2p ± . Four different values of the In concentration are considered: 0.2, 0.3, 0.4, and 0.5. The donor is positioned at the well center, zd = 0. The panel (d) shows the bulk value of the transition energy as a function the In concentration.
H. Yıldırım / Physics Letters A 383 (2019) 1324–1329
Fig. 3. The energy of the transition between the states 1s and 2p ± in InGaN/ZnSnN2 QWs for selected values of L w as a function of the donor position. x = 0.2 in the panel (a) while x = 0.5 in the panel (b). Note that the donor position is given in terms of L w for each curve.
Fig. 5. The conduction band proﬁle of a 25-Å ZnSnN2 /InGaN QW. The In concentration is 0 in the panel (a) and 0.5 in the panel (b). The orange solid line in each panel stands for the squared wave function of the lowest electronic state in the absence of the donor.
and 0.5L w . The Zeeman splitting at B = 10 T is around 7.6 meV for x = 0.2 and 8.6 meV for x = 0.5. The Zeeman splitting in a bulk material is given by 
Fig. 4. (a) The energy of the transition between the states 1s and 2p ± in InGaN/ZnSnN2 QWs as a function of the magnetic ﬁeld, B. Three donor positions are considered: zd = −0.5L w (dashed lines), 0 (solid lines), and 0.5L w (dashed-dotted lines). The blue lines stand for x = 0.2 and the green lines stand for x = 0.5. The upper (lower) curve is for the E 1s→2p + (E 1s→2p − ) transition in each group. (b) The Zeeman splitting for B = 10 T as a function of L w .
ally increased more than two-fold in the QW structure compared to the bulk value. The transition energy for x = 0.2 and 0.5 is plotted against the donor position inside the well layer in Fig. 3. Since F bi ( z) introduces an asymmetry to the shape of the potential energy proﬁle via the band-bending (see Fig. 1), the transition energy curves become an asymmetric function of the donor position around the QW center. This is clearly in seen in panel (a) of the ﬁgure: the transition energies are generally much higher for zd values around −0.5L w because of the effective narrowing of the QW around that region. For example, the transition energy for x = 0.2 almost doubles its value at zd = 0. On the other hand, the transition energy curves for x = 0.5 shown in panel (b) of the same ﬁgure are nearly symmetric around zd = 0 as the asymmetry is much lower because of the weak built-in electric ﬁeld in this case. The transition energy is displayed as a function of magnetic ﬁeld up to 10 T in Fig. 4(a) for x = 0.2 and 0.5. L w is set to 50 Å and three different donor positions are considered: zd = −0.5, 0,
h¯ e B m∗
When computed for In0.2 Ga0.8 N and In0.5 Ga0.5 N, it gives 6.6 and 8.6 meV, respectively. Compared to the bulk value, the splitting in the In0.5 Ga0.5 N/ZnSnN2 QW is the same, but it is enhanced in the In0.2 Ga0.8 N/ZnSnN2 QW. The Zeeman splitting increases as the QW becomes thinner as shown in Fig. 4(b) because the wave function penetrates more into the barrier layer where the effective mass is smaller. For example, it increases to 8.5 and 9 meV at B = 10 T when L w is set to 10 Å for x = 0.2 and 0.5, respectively. In a recent work , it is given that the CBM of ZnSnN2 is 1.44 eV lower than that of GaN. This is indeed quite larger than the one, 0.3 eV , we have used in the calculations so far. The conduction band offset between GaN and In0.5 Ga0.5 N is 1.232 eV. This value is smaller than the conduction band offset, 1.44 eV , between ZnSnN2 and GaN. Therefore, a ZnSnN2 layer may serve as a well layer in a ZnSnN2 /Inx Ga1−x N heterostructure system for x up to 0.5, opposite to the structure considered previously in this work (see Fig. 1). In fact, the potential energy proﬁle of a 25-Å ZnSnN2 /InGaN QW is represented in Fig. 5. The potential energy proﬁle for x = 0 looks a steep triangle in shape around the well region as the built-in electric ﬁeld is very strong. Its magnitude is 7.92 MV/cm. On the other hand, F bi (x) decreases in magnitude as x goes up. For example, it is 0.33 MV/cm for x = 0.5. Accordingly, it leads to a slightly distorted square potential proﬁle in shape as shown in panel (b) of the same ﬁgure. A linear ﬁt to the calculated F bi (x) data for x up to 0.5 gives F bi (x) = −15.2x + 7.92 MV/cm. The present F bi (x) is almost 1.33 times larger in magnitude than the one previously computed for InGaN/ZnSnN2 QWs, because, being compressively-strained, the ZnSnN2 layer in ZnSnN2 /InGaN QWs with its larger e i j , has a higher total polarization. The binding energy and the energy of the transition of the impurity states in such a ZnSnN2 /InGaN QW are plotted against L w in Fig. 6. When the QW is narrow, the 1s state binding energy increases as x goes down because of the rising potential energy barrier. Both present E 1s and E 2p ± values are generally lower and drop faster with the increasing L w than the energies computed for the InGaN/ZnSnN2 QWs, as shown in Fig. 2, because the dielectric constant is greater and the magnitude of the built-in electric
H. Yıldırım / Physics Letters A 383 (2019) 1324–1329
Fig. 6. The binding energy of the impurity states in ZnSnN2 /InGaN QWs as a function of L w : a) the 1s state and b) the 2p ± states. The panel (c) shows the energy of the transition. Five different values of the In concentration are considered: 0, 0.1, 0.2, 0.3, 0.4, and 0.5. The donor is positioned at the well center, zd = 0.
Fig. 7. The energy of the transition between the states 1s and 2p ± in ZnSnN2 /InGaN QWs as a function of the donor position for selected values of L w . The indium concentration is 0 in the panel (a) while it is 0.5 in the panel (b). Note that the donor position is given in terms of L w for each curve.
ﬁeld is larger in the current ZnSnN2 /InGaN QWs. The transition energy, shown in panel (c), changes between 7.5 and 17 meV. When L w = 10 Å and x = 0, the transition energy reaches to its maximum value. The bulk transition energy is calculated to be nearly 5.4 meV. Compared to the bulk value, the QW value is enhanced more than two-fold in narrower QWs and nearly two-fold in wider QWs. The position dependency of the transition energy for several L w is plotted against the donor position around the well center in Fig. 7. In panel (a), the energy is shown for x = 0. Since the built-in electric ﬁeld is the strongest in magnitude in that case, the energy curves are quite asymmetric around zd = 0. The largest energy values are gathered at around zd = 0.5L w because of the effective narrowing of the QW around that region. The asymmetry is greatest for L w = 50 Å. The transition energy nearly doubles its value at zd = 0 in that case. On the other hand, the transition energy curves for x = 0.5 are much less asymmetric as shown in
Fig. 8. The energy of the transition between the states 1s and 2p ± in ZnSnN2 /InGaN QWs as a function of the magnetic ﬁeld, B. Three donor positions are considered: zd = −0.5L w (dashed lines), 0 (solid lines), and 0.5L w (dashed-dotted lines). The blue lines stand for x = 0 and the green lines stand for x = 0.5. The upper (lower) curve is for the E 1s→2p + (E 1s→2p − ) transition in each group. (b) The Zeeman splitting for B = 10 T as a function of L w .
panel (b) of the same ﬁgure. As compared to the energies calculated for InGaN/ZnSnN2 QWs, shown in Fig. 3, the asymmetry is greater because of the higher built-in electric ﬁeld in magnitude. In Fig. 8(a), the transition energy is displayed for x = 0 and 0.5 as a function of the magnetic ﬁeld up to 10 T. The QW width is set to 50 Å and three different donor positions, namely zd = −0.5, 0, and 0.5L w , are considered. The Zeeman splitting at B = 10 T is around 8.9 meV for x = 0 and 9.5 meV for x = 0.5. The bulk value is calculated to be around 9.6 meV. As clearly seen, the QW value for x = 0 is lower than the bulk value. For thinner quantum wells, the splitting has lower values as shown in Fig. 8(b), as the wave function increasingly spread through the barrier material which has heavier effective mass than the well material has. For example, when L w = 10 Å, the splitting reduces to 8.4 and 8.8 meV for x = 0 and x = 0.5, respectively.
H. Yıldırım / Physics Letters A 383 (2019) 1324–1329
4. Conclusions In this work, we have studied the binding energy of the impurity states, 1s and 2p ± , and the energy of the transition between them in a Inx Ga1−x N/ZnSnN2 QW with a width ranging between 10 and 50 Å as a function of the In concentration, the donor position and the magnetic ﬁeld. The 1s state binding energy goes up to 36 meV, but it drops fast with the increasing QW width as the band-bending in the potential energy proﬁle introduced by the strong built-in electric ﬁeld grows at the same time. The calculated transition energy ranges from 15.6 to 30.1 meV, that is between 3.77 and 7.28 THz. The transition energy is a strong function of the donor position which gives highly asymmetric curves around the QW center because of the band-bending. A recent study  reports that the CBM of ZnSnN2 is 1.44 eV lower than that of GaN. Therefore in an Inx Ga1−x N/ZnSnN2 system, ZnSnN2 can be considered as a well material rather than a barrier material as ﬁrst considered in this work. The calculations reveal that the 1s state binding energy goes up to 20 meV for such a ZnSnN2 /Inx Ga1−x N QW with a width changing between 10 and 50 Å. The transition energy changes between 7.5 and 17 meV, in other words between 1.8 and 4.11 THz. The strong built-in electric ﬁeld makes the transition energy decrease faster with the increasing well width and its curves as a function of the donor position more asymmetric around the well center. The binding and transition energies are signiﬁcantly smaller than those obtained in the previously considered InGaN/ZnSnN2 QW systems. References  A. Punya, W.R.L. Lambrecht, Band offsets between ZnGeN2 , GaN, ZnO, and ZnSnN2 and their potential impact for solar cells, Phys. Rev. B 88 (2013) 075302, https://doi.org/10.1103/PhysRevB.88.075302.  P.C. Quayle, K. He, J. Shan, K. Kash, Synthesis, lattice structure, and band gap of ZnSnN2 , MRS Commun. 3 (2013) 135–138, https://doi.org/10.1557/mrc.2013. 19.  P. Narang, S. Chen, N.C. Coronel, S. Gul, J. Yano, L.W. Wang, N.S. Lewis, H.A. Atwater, Bandgap tunability in Zn(Sn, Ge)N2 semiconductor alloys, Adv. Mater. 26 (2014) 1235–1241, https://doi.org/10.1002/adma.201304473.  D.Q. Fang, Y. Zhang, S.L. Zhang, Band gap engineering of ZnSnN2 /ZnO (001) short-period superlattices via built-in electric ﬁeld, J. Appl. Phys. 120 (2016) 215703, https://doi.org/10.1063/1.4971176.  L. Han, K. Kash, H. Zhao, Designs of blue and green light-emitting diodes based on type-II InGaN–ZnGeN2 quantum wells, J. Appl. Phys. 120 (2016) 103102, https://doi.org/10.1063/1.4962280.  A.D. Martinez, A.N. Fioretti, E.S. Toberer, A.C. Tamboli, Synthesis, structure, and optoelectronic properties of II–IV–V2 materials, J. Mater. Chem. A 5 (2017) 11418–11435, https://doi.org/10.1039/C7TA00406K.  T. Wang, C. Ni, A. Janotti, Band alignment and p-type doping of ZnSnN2 , Phys. Rev. B 95 (2017) 205205, https://doi.org/10.1103/PhysRevB.95.205205.  H. Yıldırım, Donor binding energies in a GaN/ZnGeN2 quantum well, Superlattices Microstruct. 111 (2017) 529–535, https://doi.org/10.1016/j.spmi.2017.07. 008.  M.R. Karim, H. Zhao, Design of InGaN–ZnSnN2 quantum wells for higheﬃciency amber light emitting diodes, J. Appl. Phys. 124 (2018) 034303, https://doi.org/10.1063/1.5036949.
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