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Nitride dot-in-nanowire light emitters with suppressed auger process
Optical Materials 99 (2020) 109610
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Nitride dot-in-nanowire light emitters with suppressed auger process Ye Wu, Zi-Chang Zhang, Shaikh Ahmed * Department of Electrical and Computer Engineering, Southern Illinois University Carbondale, 1230 Lincoln Drive, Carbondale, IL, 62901, USA
A R T I C L E I N F O
A B S T R A C T
Keywords: Auger recombination Atomistic simulation Efficiency droop Interface grading III-Nitride emitters Dot-in-nanowire structures
In wurtzite III-Nitride nano-devices, the non-radiative Auger recombination is the primary mechanism respon sible for the degradation of internal quantum efficiency (IQE), especially under high current density. In this paper, by employing an atomistic tight-binding framework, we theoretically study the effects of Auger recom bination in recently reported InGaN/GaN dot-in-nanowire light emitters. The effects of strain and polarization, which can be strong in realistically-sized structures, have been considered. We demonstrate that the use of graded interfacial confinement leads to a weaker Auger recombination as compared to the abrupt counterpart, especially for thinner nanowires. The atomistically simulated Auger recombination coefficient for the core quantum dot buried in nanowire with different diameter is then incorporated into a TCAD simulator to obtain the device terminal (efficiency vs. current) characteristics. Overall, the simulation results indicate that increasing the diameter of the host nanowire (that is, the volume of the active region) remains the most efficient way to suppress Auger recombination.
1. Introduction InGaN/GaN light-emitting diodes (LEDs) offer full-spectrum emis sion and have excellent potential for use in ultrahigh efficiency solidstate lighting [1] (SSL), single-photon sources [2] and optically and electrically pumped lasers [3]. When nonstructurally configured (such as dot-in-nanowire, disk-in-wire, etc.) these non-classical light emitters exhibit high internal quantum efficiency (IQE), low threshold currents, large differential gain, and high-temperature endurance. IQE is deter mined by the ratio of the radiative and the non-radiative recombination of charge carriers inside the active region of the device. Experimentally, the Shockley-Read-Hall (SRH) [4] and the Auger [5,6] recombination processes have been identified as the two dominant non-radiative pro cesses in nitride-based semiconductors. While SRH dominates at low injection carrier density, the Auger recombination becomes critical and significantly degrades the efficiency mainly at high injection carrier density [7]. Both the SRH and the Auger processes are strongly dependent on the size (geometry) of the device’s active region. For example, Auger recombination coefficients measured in heterostructured dot-innanowires have been found to be ~2–3 orders of magnitude lower than those in bulk materials [8]. Although there has been some progress in the study of the role of Auger processes in efficiency droop at higher current density in GaN LEDs, experimental evidence still remains scarce
[9]. In the theoretical domain, recently, several studies based on direct first-principles [10] simulations and the k�p model [11] have been re ported. However, it is well known that ab initio simulations are limited to small (mainly closed) systems, and the k�p model fails to capture the fundamental atomistic symmetry of the underlying materials [12–14]. For nano-scale devices, the structural as well as the electronic properties and their effects on Auger recombination processes are still not very well understood. In this work, by employing a 10-band sp3s*-spin atomistic tightbinding (TB) framework as available in the open source NEMO 3-D software toolkit [15–18], we theoretically calculate the Auger co efficients and study the effects of Auger recombination in recently re ported InGaN/GaN dot-in-nanowire light emitters. The effects of long-range strain and polarization fields, which can be strong in realistically-sized structures, have been accounted for via a coupled atomistic valence-force-field (VFF) model and a 3-D Poisson solver within the simulator. The microscopically determined Auger coefficients are then incorporated into a TCAD device simulator to obtain the ter minal characteristics. Specifically, we analyze and compare the effects of nanowire thickness (diameter) and quantum dot boundary grading on the IQE characteristic of the devices.
* Corresponding author. E-mail addresses: [email protected] (Y. Wu), [email protected] (Z.-C. Zhang), [email protected] (S. Ahmed). https://doi.org/10.1016/j.optmat.2019.109610 Received 1 November 2019; Received in revised form 24 November 2019; Accepted 7 December 2019 Available online 12 December 2019 0925-3467/© 2019 Elsevier B.V. All rights reserved.
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2. Theory of auger recombination for nanostructures
#1 and #2 are both at the conduction band minimum and the wave functions of these two carriers have a complete overlap. The energy that is released due to electron and hole recombination and is transferred to the second electron is set equal to the energy bandgap of the active re gion, which is larger than the quantum barrier height (band offset) in InxGa1-xN/GaN for x < 0:5. Thus the wave function of state #4 is delo calized, extended, and becomes a constant. According to Eq. (5), clearly, the Auger matrix element in the quantum dot MQD 1234 , is strongly depen dent on the overlap of conduction and valence band wavefunctions, often denoted by a factor, Fcv . Note that, in our work, ψ QD 1 ðrÞ (and QD therefore ψ QD 2 ðrÞ) in the conduction band and ψ 3 ðrÞ in the valence band are calculated atomistically within the tight-binding framework including the effects of strain-induced internal fields; whereas, as stated before, ψ QD 4 ðrÞ is assumed to be delocalized (that is, formed outside of the dot confinement). With this assumption made, and since the Auger recombination rate is proportional to the square of the Auger matrix (see Eq. (1)), the Auger recombination coefficient (C) in the quantum dot becomes proportional to the square of Fcv and can be written as C ¼
The auger recombination process in InGaN is a four-particle process. Generally, we need to take both the direct and the indirect Auger recombination processes into consideration. In the direct Auger process, the electron at the conduction band minimum (electron #1 in Fig. 1(a)) recombines with the hole at the valence band maximum (hole #3 in Fig. 1(a)). The energy released, instead of generating photons, is trans ferred to another electron in the conduction band (electron #2 in Fig. 1 (a)) and elevates it to a higher state (empty state #4 in Fig. 1(a)). The direct Auger recombination rate is given by Ref. [19]: R¼2
2π X PjM1234 j2 δðE1 þ E2 ℏ 1234
E3
E4 Þ;
(1)
where, P ¼ f1 f2 ð1 f3 Þð1 f4 Þ and f refers to the electron Fermi-Dirac distribution. The factor P ensures that there are electrons at states #1 and #2, and there are empty states at states #3 and #4 during the recombination process. The indices 1234 represent the composite band and the k-points indices [1 � ðn1 ; k1 Þ]. The M1234 is the Auger matrix element of the screened Coulomb potential, W, and is given by
C0 jFcv j2 . Here, C0 is the Auger coefficient for the bulk material (i.e. without considering the quantum confinement and atomistic effects). In other words, the effects of atomicity, quantum confinement, and long-
(2)
M1234 ¼ 〈ψ 1 ψ 2 jWjψ 3 ψ 4 〉⋅
range internal fields are all mapped to a single scaling factor, jFcv j2 . For accurate calculation, as far as carrier transport is concerned, some other parameters (such as, effective masses, density-of-states, and en ergy bandgap) are also obtained atomistically using the NEMO 3-D simulator. Fig. 1(b) shows the indirect Auger recombination process. In general, the indirect Auger recombination process is assisted by a carrierscattering mechanism, such as electron-phonon coupling, alloy disor der, or defect scattering, which provides additional momentum and enables Auger transitions to a broader range of final conduction-band states throughout the first Brillouin zone. The same relations also apply to the indirect Auger recombination process.
Here, ψ is the electron and hole wave functions, specifically ψ 1 and ψ 2 are electron wave functions at the conduction band minimum, ψ 3 is the wave function for holes at the valence band maximum, and ψ 4 is the electron wave function for electrons at a higher energy stage. Energy conservation is ensured by the factor δðE1 þ E2 E3 E4 Þ, and the momentum conservation (δðk1 þ k2 k3 k4 Þ) is also considered within the M1234 matrix. The carrier density (n) dependent Auger co efficients can be defined by Ref. [20]. CðnÞ �
RðnÞ ⋅ n3 V
(3)
Here, V is the volume of the active region. In quantum dots (QD), the size-confinement induced discrete wave functions can be written as a superposition of bulk wave function [11]: X (4) ψ QD ðrÞ ¼ aðkÞψ k ðrÞ⋅
3. Simulation model As stated earlier, the atomistic study is carried out by open source NEMO 3-D simulator combined with a 3-D Poisson solver. Detail description of the NEMO 3-D simulator and the coupling process can be found in Refs. [15,21] and references therein. In general, the device structure is created based on individual atomic coordinates. The lattice mismatch in the underlying materials induce strain fields, which is calculated via the valence force field (VFF) method using the Keating potentials [22]. Note that the effects of thermal strain have been neglected in this work. The non-overlapping of the negative and the positive charge centers in III-nitride materials generates a spontaneous polarization and the deformations due to atomistic strain induce piezoelectric polarization, which were obtained atomistically and included in the simulation. The million-atom Hamiltonian of the struc ture was then diagonalized using the block Lanczos method for the calculation of the electron and the hole wavefunctions in conduction band and the valence band, respectively. Finally, the microscopically determined quantum dot Auger co efficients are incorporated into the Silvaco TCAD device simulator [23] to obtain the terminal characteristics and compare how nanowire thickness and interfacial (confinement) grading affect the IQE. The reference bulk Auger coefficient used here is C0 ¼ 10 31 cm6 =s [24], and the quantum-corrected Auger coefficient used in the nanowire simula tion is obtained from the atomistic calculation.
k
R ikr dr is the Here, ψ k ðrÞ is the bulk wave function and aðkÞ ¼ ψ QD k ðrÞe Fourier coefficient of the wave function. The Auger matrix M in a quantum dot can be expressed as: X M QD a*1 ðk1 Þa*2 ðk2 Þa3 ðk3 Þa4 ðk4 ÞM1234 δðk1 þ k2 k3 k4 Þ 1234 ¼ k1 k2 k3 k4
¼ M1234
(5)
Z QD ψ QD* ðrÞψ QD* ðrÞψ QD 1 2 3 ðrÞψ 4 ðrÞdr⋅
As for the Auger recombination process, we assume that the electron
4. Results and discussions We simulated nanowires having diameters of 5 nm, 10 nm and 15 nm, with a buried QD with a core thickness of 3 nm. For the QD, we considered two geometries: a) abrupt interface, and b) graded
Fig. 1. Schematic diagram of (a) the direct and (b) the indirect Auger recom bination processes. 2
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(smoothed) interface (e.g. as suggested in Ref. [25]). The schematic diagrams of the quantum confined regions are shown in the top panel of Fig. 2. The simulation results of the Auger coefficient and the energy bandgap for these structures are shown in the bottom panel of Fig. 2. The suppression of the Auger process via engineering (grading) the quantum confinement of the active region is still being debated in the community. Experiments conducted in Refs. [26,27] show that a graded (softened) interface can indeed curb the Auger process; however, Refs. [28,29] argues in the opposite. Our study shows that the Auger coefficient is strongly dependent on nanowire diameter—it decreases significantly as the nanowire diameter increases from 5 nm to 15 nm. Although to a lesser extent, the Auger coefficient is reduced due to the presence of graded interfaces too. The root cause of the reduction of Auger coefficient as a function of nanowire diameter and confinement geometry can be understood considering the HOMO (highest occupied molecular orbital or valence band) and the LUMO (lowest unoccupied molecular orbital or conduc tion band) wavefunctions as shown in Fig. 3. In Fig. 3(a), the variation of nanowire diameter is considered. Clearly, with an increase in diameter, wavefunctions become more localized in both conduction and valence bands. This is in line with the observation made in Ref. [30], where an increase of device size was found to result in a suppressed Auger recombination process due to a diminished Coulomb interaction be tween electrons and holes. In our work, the diameter dependence of Auger coefficient is correlated mainly to the strain distributions. In lattice-mismatched nanostructures, such as the device considered here, the strain distribution is atomistically inhomogeneous, involving both biaxial and shear components. In addition, in III–nitride wurtzite ma terials, the iconicity of the underlying bonds produces significantly large polarization constants. This, when coupled with the strain tensors, leads to strong polarization fields and the formation of charged dipoles (electrostatic potential) within the device active region. As with strain, the internal potential fields are long-ranged and anisotropic in the lateral and vertical planes. In the subsequent calculation of the elec tronic structure (energy levels and wavefunctions), the polarization
Fig. 3. The HOMO and the LUMO wavefunctions for (a) nanowires with varying diameter (abrupt confinement with QD thickness, t ¼ 3 nm), and (b) dots with abrupt and graded interfaces (nanowire diameter, d ¼ 5 nm). Sche matic confinements are shown on the right of each panel.
fields are incorporated in the Hamiltonian as electrostatic potential in the main diagonal resulting in localization and spatial separation (mainly in the growth direction) of the electron and hole wavefunctions. It was found that atoms in the active QD region in thicker nanowires are heavily strained and longer-ranged, compared to the thinner counter parts, leading to a stronger polarization potential. Fig. 4 displays one such comparison between three devices having diameter (d) of 5 nm, 10 nm, and 15 nm. The upper panel in this figure shows the strain distri butions in the z (growth) direction. For clarity, only the diagonal
Fig. 2. (top panel) Schematic of quantum confinement in buried quantum dots (abrupt and graded interfaces) having a core thickness of 3 nm. (bottom panel) Auger coefficient and energy bandgap as a function of nanowire diameter and quantum dot geometry.
Fig. 4. Strain (top panel) and polarization-induced potential (bottom panel) distributions in the z (growth) direction in three dot-in-nanowire devices hav ing diameter (d) of 5 nm, 10 nm, and 15 nm. 3
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component of strain is considered here. The lower panel illustrates the strain induced piezoelectric potential. Within the active QD region, strain that involves the Indium (In) atoms is mainly compressive (-ve) in nature and is strongest in the thickest device (with d ¼ 15 nm). While the Gallium (Ga) atoms experience a tensile stress resulting in þve strain values. Strain directly correlates to the induction of an internal potential (bottom panel), where, again, the thickest nanowire exhibits the highest potential (with a peak ~ 90 mV). Note that the sparsity in the strain and potential distributions within the QD region, as seen in Fig. 4, arises mainly from the atomistic granularity of the InGaN alloy used therein. The dependence of piezoelectric potential on strain distributions was studied in some more details in our previous work (e.g. see Refs. [31–33]). The diameter dependence is also schematically depicted in the rightmost panel in Fig. 3(a), where the slope corresponds to the built-in polarization field. On the other hand, Fig. 3(b) displays the effect of interfacial grading within the active dot region. With grading, conduc tion band wavefunction is affected more exhibiting stronger localization as well as increased charge separation, both leading to a decreased Auger coefficient. Next, the role of Auger processes and the possibility of performance improvement via geometry engineering have been investigated using the Silvaco TCAD simulation toolkit. Here, we mainly consider the variation of internal quantum efficiency (IQE) as a function of the drive current. We have made use of the standard ABC model for IQE calcu lation given by: IQE ¼
Bn2 An þ Bn2 þ Cn3 þ kðn
n0 Þm þ qVILKQD
(6)
We have set A ¼ 2:29 � 107 s 1 and B ¼ 1:28 � 10 11 cm3 =s, for the Shockley-Read-Hall (SRH) and radiative recombination, respectively, as suggested in Ref. [25]. As described previously, the Auger coefficient, C,
Fig. 5. (top panel) The atomistic diagram of the active semiconducting region with a hexagonal geometry simulated in this work. (bottom panel) IQE vs. current density for devices with a diameter of 5 nm, 10 nm, and 15 nm. For each device, both abrupt (A) and graded (G) interfacial confinement have also been considered.
has been scaled via atomistic calculation using C ¼ C0 jFcv j2 . As stated earlier, the reference bulk Auger coefficient used here is C0 ¼ 10 31 cm6 =s. The Auger recombination rate, in Silvaco TCAD simulator, is given by: � � � �� Ebgc RAnet ¼ Cn n þ Cp p np n2i;eff ⋅ni;eff ¼ ni exp (7) 2kT
interfacial grading does little or no benefit. In this device, for a current density great than 170 A/cm2, the structure with an abrupt interface even offers higher efficiency. This may be due to the increasing domi nation of the leakage current over the Auger process. As for further improvement in the device performance, one may consider three op tions: (a) compositional grading along the nanowire growth axis, (b) incorporation of intentional localized impurities (e.g. dopants) within the quantum disk, and/or (c) the use of multiple quantum disks (perhaps with varying separation between them). These options will allow one to engineer the wavefunction distributions in the active region and hence offer the possibility of tuning the Auger coefficient. Numerical charac terization of these structures as well as a detail comparative quantifi cation of the individual contributions of Auger, internal fields, and leakage processes will be the subject of a future study. Also, note that since the Auger process has been the focus in this work, we did not scale the coefficient of the radiative recombination process. This was studied in detail in several of our previously published work (e.g. Refs. [35–38]). For low current density, where Auger process is weak, some anomalous trend is observed for the nanowire with d ¼ 10 nm. Note that the SRH recombination is dominant in this regime, which depends, among several factors, mainly on an interplay between the bandgap and the carrier density within the active region. Overall, the simulation results indicate that increasing the diameter of the host nanowire (that is, the volume of the active region) remains the most efficient way to suppress Auger recombination and boost the internal quantum efficiency.
here, ni,eff is the effective intrinsic carrier density and Ebgc denotes bandgap correction. The factor kðn n0 Þm in Eq. (6) captures other recombination mechanisms for which material specific default values have been used. ILK and VQD denote the leakage current and the volume of the active dot region, respectively. As for the simulated device, the core active region consists of undoped In0.2Ga0.8N, which sits on an n-type (doping density of 7 � 1018 cm 3) GaN buffer layer and capped by a p-type (doping density of 2 � 1019 cm 3) GaN layer. The atomistic diagram of the active semi conducting region with a hexagonal geometry is shown in the top panel of Fig. 5. The granularity (randomness) in the dot region can be clearly seen in this panel. The bottom panel of Fig. 5 displays the variation of IQE with respect to the drive current for different device geometries considered in this work. For all the devices simulated, the peak effi ciency is limited mainly by the SRH and Auger recombination processes as well as the over the barrier leakage current. The IQE never reaches the desired value of 100%, even at a smaller drive current. The droop characteristic in IQE is clearly seen for higher values of the drive current. Looking at the slope beyond the peak IQE points, as far as droop is concerned, we can infer that the thinnest nanowire suffers the most (about 30%). In thicker nanowires, the droop is somewhat weak (about 10% for the device with d ¼ 15 nm). This observation is consistent with the experimental result [34] as well as the first-principles calculation [10]. Interestingly, however, of all the structures simulated, the thinnest device with d ¼ 5 nm benefits most from the introduction of a graded interface within the active QD region provided that such grading does not introduce any defects. For the thickest nanowire with d ¼ 15 nm,
5. Conclusion Non-radiative Auger recombination is a dominant process that 4
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degrades the internal quantum efficiency of nanostructured nitridebased light emitters. Grading of the interfacial confining potential has been proposed as a potential route to suppress the Auger process, but theoretical studies along this line, especially considering the realistic size of the device, is rare. In this work, using a 10-band sp3s* atomistic tight-binding combined with a VFF-3D Poisson simulation framework, we theoretically calculate the Auger coefficients and study the effects of Auger recombination in recently reported InGaN/GaN dot-in-nanowire light emitters. Our calculation shows that the Auger coefficient, while weakly responds to the interfacial grading, is strongly dependent on nanowire diameter—it decreases significantly as the nanowire diameter increases from 5 nm to 15 nm. The microscopically determined Auger coefficients are then incorporated into a TCAD device simulator to compare and analyze the effects of thickness (nanowire diameter) and quantum dot boundary grading on the IQE characteristic of the devices. As for the efficiency droop at higher drive current, we find that the thinnest nanowire is affected the most, yet can benefit significantly from interfacial grading provided that such grading does not introduce any defects. For thicker nanowires, interfacial grading brings little or no improvement in the droop characteristic. Overall, the device with the largest volume of the active region offers the highest IQE and the use of thicker (host) nanowire remains the most effective way to suppress Auger recombination.
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Author contribution statement Ye Wu: Conceptualization, Methodology, Software, Investigation, Writing - Original Draft. Zi-Chang Zhang: Writing - Original Draft, Visualization, Formal Analysis. Shaikh Ahmed: Supervision, Software, Data Curation, Writing Review & Editing, Project administration. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgment This work was financially supported by the U.S. National Science Foundation Grant No. 1610474. Access to Arizona State University computing system is also acknowledged. References [1] M. Auf der Maur, A. Pecchia, G. Penazzi, W. Rodrigues, A. Di Carlo, Efficiency drop in green InGaN/GaN light emitting diodes: the role of random alloy fluctuations, Phys. Rev. Lett. 116 (2) (2016), 027401. [2] S. Deshpande, J. Heo, A. Das, P. Bhattacharya, Electrically driven polarized singlephoton emission from an InGaN quantum dot in a GaN nanowire, Nat. Commun. 4 (2013) 1675. [3] T. Frost, S. Jahangir, E. Stark, S. Deshpande, A. Hazari, C. Zhao, B.S. Ooi, P. Bhattacharya, Monolithic electrically injected nanowire array edge-emitting laser on (001) silicon, Nano Lett. 14 (8) (2014) 4535–4541. [4] A. Mohantaa, D.-J. Jangb, M.-S. Wang, L.W. Tu, Time-integrated photoluminescence and pump-probe reflection spectroscopy of Si doped InN thin films, J. Appl. Phys. 115 (4) (2014). [5] N. Saulius, J. Kęstutis, V. Mikas, S. Egidijus, Y. Tomohiro, N. Yasushi, Injectionactivated defect-governed recombination rate in InN, Jpn. J. Appl. Phys. 52 (8S) (2013), 08JD02. [6] Y. Cho, X. Lue, M. Wienold, M. Ramsteiner, H.T. Grahn, O. Brandt, Auger recombination as the dominant nonradiative recombination channel in InN, Phys. Rev. B 87 (15) (2013) 155203.
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[37] S.M. Al-Qahtani, A.M.A. Abdullah, M.R.K. Nishat, S.S. Ahmed, Diameter dependent polarization in ZnO/MgO disk-in-wire emitters: Multiscale modeling of optical quantum efficiency, Superlattice Microstruct. 103 (2017) 48–55. [38] M.R.K. Nishat, M.M. Taher, S.S. Ahmed, Million-atom tight-binding modeling of non-polar a-plane InGaN light emitters, J. Comput. Electron. 17 (4) (2018) 1630–1639.
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Nitride dot-in-nanowire light emitters with suppressed auger process Ye Wu, Zi-Chang Zhang, Shaikh Ahmed * Department of Electrical and Computer Engineering, Southern Illinois University Carbondale, 1230 Lincoln Drive, Carbondale, IL, 62901, USA
A R T I C L E I N F O
A B S T R A C T
Keywords: Auger recombination Atomistic simulation Efficiency droop Interface grading III-Nitride emitters Dot-in-nanowire structures
In wurtzite III-Nitride nano-devices, the non-radiative Auger recombination is the primary mechanism respon sible for the degradation of internal quantum efficiency (IQE), especially under high current density. In this paper, by employing an atomistic tight-binding framework, we theoretically study the effects of Auger recom bination in recently reported InGaN/GaN dot-in-nanowire light emitters. The effects of strain and polarization, which can be strong in realistically-sized structures, have been considered. We demonstrate that the use of graded interfacial confinement leads to a weaker Auger recombination as compared to the abrupt counterpart, especially for thinner nanowires. The atomistically simulated Auger recombination coefficient for the core quantum dot buried in nanowire with different diameter is then incorporated into a TCAD simulator to obtain the device terminal (efficiency vs. current) characteristics. Overall, the simulation results indicate that increasing the diameter of the host nanowire (that is, the volume of the active region) remains the most efficient way to suppress Auger recombination.
1. Introduction InGaN/GaN light-emitting diodes (LEDs) offer full-spectrum emis sion and have excellent potential for use in ultrahigh efficiency solidstate lighting [1] (SSL), single-photon sources [2] and optically and electrically pumped lasers [3]. When nonstructurally configured (such as dot-in-nanowire, disk-in-wire, etc.) these non-classical light emitters exhibit high internal quantum efficiency (IQE), low threshold currents, large differential gain, and high-temperature endurance. IQE is deter mined by the ratio of the radiative and the non-radiative recombination of charge carriers inside the active region of the device. Experimentally, the Shockley-Read-Hall (SRH) [4] and the Auger [5,6] recombination processes have been identified as the two dominant non-radiative pro cesses in nitride-based semiconductors. While SRH dominates at low injection carrier density, the Auger recombination becomes critical and significantly degrades the efficiency mainly at high injection carrier density [7]. Both the SRH and the Auger processes are strongly dependent on the size (geometry) of the device’s active region. For example, Auger recombination coefficients measured in heterostructured dot-innanowires have been found to be ~2–3 orders of magnitude lower than those in bulk materials [8]. Although there has been some progress in the study of the role of Auger processes in efficiency droop at higher current density in GaN LEDs, experimental evidence still remains scarce
[9]. In the theoretical domain, recently, several studies based on direct first-principles [10] simulations and the k�p model [11] have been re ported. However, it is well known that ab initio simulations are limited to small (mainly closed) systems, and the k�p model fails to capture the fundamental atomistic symmetry of the underlying materials [12–14]. For nano-scale devices, the structural as well as the electronic properties and their effects on Auger recombination processes are still not very well understood. In this work, by employing a 10-band sp3s*-spin atomistic tightbinding (TB) framework as available in the open source NEMO 3-D software toolkit [15–18], we theoretically calculate the Auger co efficients and study the effects of Auger recombination in recently re ported InGaN/GaN dot-in-nanowire light emitters. The effects of long-range strain and polarization fields, which can be strong in realistically-sized structures, have been accounted for via a coupled atomistic valence-force-field (VFF) model and a 3-D Poisson solver within the simulator. The microscopically determined Auger coefficients are then incorporated into a TCAD device simulator to obtain the ter minal characteristics. Specifically, we analyze and compare the effects of nanowire thickness (diameter) and quantum dot boundary grading on the IQE characteristic of the devices.
* Corresponding author. E-mail addresses: [email protected] (Y. Wu), [email protected] (Z.-C. Zhang), [email protected] (S. Ahmed). https://doi.org/10.1016/j.optmat.2019.109610 Received 1 November 2019; Received in revised form 24 November 2019; Accepted 7 December 2019 Available online 12 December 2019 0925-3467/© 2019 Elsevier B.V. All rights reserved.
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2. Theory of auger recombination for nanostructures
#1 and #2 are both at the conduction band minimum and the wave functions of these two carriers have a complete overlap. The energy that is released due to electron and hole recombination and is transferred to the second electron is set equal to the energy bandgap of the active re gion, which is larger than the quantum barrier height (band offset) in InxGa1-xN/GaN for x < 0:5. Thus the wave function of state #4 is delo calized, extended, and becomes a constant. According to Eq. (5), clearly, the Auger matrix element in the quantum dot MQD 1234 , is strongly depen dent on the overlap of conduction and valence band wavefunctions, often denoted by a factor, Fcv . Note that, in our work, ψ QD 1 ðrÞ (and QD therefore ψ QD 2 ðrÞ) in the conduction band and ψ 3 ðrÞ in the valence band are calculated atomistically within the tight-binding framework including the effects of strain-induced internal fields; whereas, as stated before, ψ QD 4 ðrÞ is assumed to be delocalized (that is, formed outside of the dot confinement). With this assumption made, and since the Auger recombination rate is proportional to the square of the Auger matrix (see Eq. (1)), the Auger recombination coefficient (C) in the quantum dot becomes proportional to the square of Fcv and can be written as C ¼
The auger recombination process in InGaN is a four-particle process. Generally, we need to take both the direct and the indirect Auger recombination processes into consideration. In the direct Auger process, the electron at the conduction band minimum (electron #1 in Fig. 1(a)) recombines with the hole at the valence band maximum (hole #3 in Fig. 1(a)). The energy released, instead of generating photons, is trans ferred to another electron in the conduction band (electron #2 in Fig. 1 (a)) and elevates it to a higher state (empty state #4 in Fig. 1(a)). The direct Auger recombination rate is given by Ref. [19]: R¼2
2π X PjM1234 j2 δðE1 þ E2 ℏ 1234
E3
E4 Þ;
(1)
where, P ¼ f1 f2 ð1 f3 Þð1 f4 Þ and f refers to the electron Fermi-Dirac distribution. The factor P ensures that there are electrons at states #1 and #2, and there are empty states at states #3 and #4 during the recombination process. The indices 1234 represent the composite band and the k-points indices [1 � ðn1 ; k1 Þ]. The M1234 is the Auger matrix element of the screened Coulomb potential, W, and is given by
C0 jFcv j2 . Here, C0 is the Auger coefficient for the bulk material (i.e. without considering the quantum confinement and atomistic effects). In other words, the effects of atomicity, quantum confinement, and long-
(2)
M1234 ¼ 〈ψ 1 ψ 2 jWjψ 3 ψ 4 〉⋅
range internal fields are all mapped to a single scaling factor, jFcv j2 . For accurate calculation, as far as carrier transport is concerned, some other parameters (such as, effective masses, density-of-states, and en ergy bandgap) are also obtained atomistically using the NEMO 3-D simulator. Fig. 1(b) shows the indirect Auger recombination process. In general, the indirect Auger recombination process is assisted by a carrierscattering mechanism, such as electron-phonon coupling, alloy disor der, or defect scattering, which provides additional momentum and enables Auger transitions to a broader range of final conduction-band states throughout the first Brillouin zone. The same relations also apply to the indirect Auger recombination process.
Here, ψ is the electron and hole wave functions, specifically ψ 1 and ψ 2 are electron wave functions at the conduction band minimum, ψ 3 is the wave function for holes at the valence band maximum, and ψ 4 is the electron wave function for electrons at a higher energy stage. Energy conservation is ensured by the factor δðE1 þ E2 E3 E4 Þ, and the momentum conservation (δðk1 þ k2 k3 k4 Þ) is also considered within the M1234 matrix. The carrier density (n) dependent Auger co efficients can be defined by Ref. [20]. CðnÞ �
RðnÞ ⋅ n3 V
(3)
Here, V is the volume of the active region. In quantum dots (QD), the size-confinement induced discrete wave functions can be written as a superposition of bulk wave function [11]: X (4) ψ QD ðrÞ ¼ aðkÞψ k ðrÞ⋅
3. Simulation model As stated earlier, the atomistic study is carried out by open source NEMO 3-D simulator combined with a 3-D Poisson solver. Detail description of the NEMO 3-D simulator and the coupling process can be found in Refs. [15,21] and references therein. In general, the device structure is created based on individual atomic coordinates. The lattice mismatch in the underlying materials induce strain fields, which is calculated via the valence force field (VFF) method using the Keating potentials [22]. Note that the effects of thermal strain have been neglected in this work. The non-overlapping of the negative and the positive charge centers in III-nitride materials generates a spontaneous polarization and the deformations due to atomistic strain induce piezoelectric polarization, which were obtained atomistically and included in the simulation. The million-atom Hamiltonian of the struc ture was then diagonalized using the block Lanczos method for the calculation of the electron and the hole wavefunctions in conduction band and the valence band, respectively. Finally, the microscopically determined quantum dot Auger co efficients are incorporated into the Silvaco TCAD device simulator [23] to obtain the terminal characteristics and compare how nanowire thickness and interfacial (confinement) grading affect the IQE. The reference bulk Auger coefficient used here is C0 ¼ 10 31 cm6 =s [24], and the quantum-corrected Auger coefficient used in the nanowire simula tion is obtained from the atomistic calculation.
k
R ikr dr is the Here, ψ k ðrÞ is the bulk wave function and aðkÞ ¼ ψ QD k ðrÞe Fourier coefficient of the wave function. The Auger matrix M in a quantum dot can be expressed as: X M QD a*1 ðk1 Þa*2 ðk2 Þa3 ðk3 Þa4 ðk4 ÞM1234 δðk1 þ k2 k3 k4 Þ 1234 ¼ k1 k2 k3 k4
¼ M1234
(5)
Z QD ψ QD* ðrÞψ QD* ðrÞψ QD 1 2 3 ðrÞψ 4 ðrÞdr⋅
As for the Auger recombination process, we assume that the electron
4. Results and discussions We simulated nanowires having diameters of 5 nm, 10 nm and 15 nm, with a buried QD with a core thickness of 3 nm. For the QD, we considered two geometries: a) abrupt interface, and b) graded
Fig. 1. Schematic diagram of (a) the direct and (b) the indirect Auger recom bination processes. 2
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(smoothed) interface (e.g. as suggested in Ref. [25]). The schematic diagrams of the quantum confined regions are shown in the top panel of Fig. 2. The simulation results of the Auger coefficient and the energy bandgap for these structures are shown in the bottom panel of Fig. 2. The suppression of the Auger process via engineering (grading) the quantum confinement of the active region is still being debated in the community. Experiments conducted in Refs. [26,27] show that a graded (softened) interface can indeed curb the Auger process; however, Refs. [28,29] argues in the opposite. Our study shows that the Auger coefficient is strongly dependent on nanowire diameter—it decreases significantly as the nanowire diameter increases from 5 nm to 15 nm. Although to a lesser extent, the Auger coefficient is reduced due to the presence of graded interfaces too. The root cause of the reduction of Auger coefficient as a function of nanowire diameter and confinement geometry can be understood considering the HOMO (highest occupied molecular orbital or valence band) and the LUMO (lowest unoccupied molecular orbital or conduc tion band) wavefunctions as shown in Fig. 3. In Fig. 3(a), the variation of nanowire diameter is considered. Clearly, with an increase in diameter, wavefunctions become more localized in both conduction and valence bands. This is in line with the observation made in Ref. [30], where an increase of device size was found to result in a suppressed Auger recombination process due to a diminished Coulomb interaction be tween electrons and holes. In our work, the diameter dependence of Auger coefficient is correlated mainly to the strain distributions. In lattice-mismatched nanostructures, such as the device considered here, the strain distribution is atomistically inhomogeneous, involving both biaxial and shear components. In addition, in III–nitride wurtzite ma terials, the iconicity of the underlying bonds produces significantly large polarization constants. This, when coupled with the strain tensors, leads to strong polarization fields and the formation of charged dipoles (electrostatic potential) within the device active region. As with strain, the internal potential fields are long-ranged and anisotropic in the lateral and vertical planes. In the subsequent calculation of the elec tronic structure (energy levels and wavefunctions), the polarization
Fig. 3. The HOMO and the LUMO wavefunctions for (a) nanowires with varying diameter (abrupt confinement with QD thickness, t ¼ 3 nm), and (b) dots with abrupt and graded interfaces (nanowire diameter, d ¼ 5 nm). Sche matic confinements are shown on the right of each panel.
fields are incorporated in the Hamiltonian as electrostatic potential in the main diagonal resulting in localization and spatial separation (mainly in the growth direction) of the electron and hole wavefunctions. It was found that atoms in the active QD region in thicker nanowires are heavily strained and longer-ranged, compared to the thinner counter parts, leading to a stronger polarization potential. Fig. 4 displays one such comparison between three devices having diameter (d) of 5 nm, 10 nm, and 15 nm. The upper panel in this figure shows the strain distri butions in the z (growth) direction. For clarity, only the diagonal
Fig. 2. (top panel) Schematic of quantum confinement in buried quantum dots (abrupt and graded interfaces) having a core thickness of 3 nm. (bottom panel) Auger coefficient and energy bandgap as a function of nanowire diameter and quantum dot geometry.
Fig. 4. Strain (top panel) and polarization-induced potential (bottom panel) distributions in the z (growth) direction in three dot-in-nanowire devices hav ing diameter (d) of 5 nm, 10 nm, and 15 nm. 3
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component of strain is considered here. The lower panel illustrates the strain induced piezoelectric potential. Within the active QD region, strain that involves the Indium (In) atoms is mainly compressive (-ve) in nature and is strongest in the thickest device (with d ¼ 15 nm). While the Gallium (Ga) atoms experience a tensile stress resulting in þve strain values. Strain directly correlates to the induction of an internal potential (bottom panel), where, again, the thickest nanowire exhibits the highest potential (with a peak ~ 90 mV). Note that the sparsity in the strain and potential distributions within the QD region, as seen in Fig. 4, arises mainly from the atomistic granularity of the InGaN alloy used therein. The dependence of piezoelectric potential on strain distributions was studied in some more details in our previous work (e.g. see Refs. [31–33]). The diameter dependence is also schematically depicted in the rightmost panel in Fig. 3(a), where the slope corresponds to the built-in polarization field. On the other hand, Fig. 3(b) displays the effect of interfacial grading within the active dot region. With grading, conduc tion band wavefunction is affected more exhibiting stronger localization as well as increased charge separation, both leading to a decreased Auger coefficient. Next, the role of Auger processes and the possibility of performance improvement via geometry engineering have been investigated using the Silvaco TCAD simulation toolkit. Here, we mainly consider the variation of internal quantum efficiency (IQE) as a function of the drive current. We have made use of the standard ABC model for IQE calcu lation given by: IQE ¼
Bn2 An þ Bn2 þ Cn3 þ kðn
n0 Þm þ qVILKQD
(6)
We have set A ¼ 2:29 � 107 s 1 and B ¼ 1:28 � 10 11 cm3 =s, for the Shockley-Read-Hall (SRH) and radiative recombination, respectively, as suggested in Ref. [25]. As described previously, the Auger coefficient, C,
Fig. 5. (top panel) The atomistic diagram of the active semiconducting region with a hexagonal geometry simulated in this work. (bottom panel) IQE vs. current density for devices with a diameter of 5 nm, 10 nm, and 15 nm. For each device, both abrupt (A) and graded (G) interfacial confinement have also been considered.
has been scaled via atomistic calculation using C ¼ C0 jFcv j2 . As stated earlier, the reference bulk Auger coefficient used here is C0 ¼ 10 31 cm6 =s. The Auger recombination rate, in Silvaco TCAD simulator, is given by: � � � �� Ebgc RAnet ¼ Cn n þ Cp p np n2i;eff ⋅ni;eff ¼ ni exp (7) 2kT
interfacial grading does little or no benefit. In this device, for a current density great than 170 A/cm2, the structure with an abrupt interface even offers higher efficiency. This may be due to the increasing domi nation of the leakage current over the Auger process. As for further improvement in the device performance, one may consider three op tions: (a) compositional grading along the nanowire growth axis, (b) incorporation of intentional localized impurities (e.g. dopants) within the quantum disk, and/or (c) the use of multiple quantum disks (perhaps with varying separation between them). These options will allow one to engineer the wavefunction distributions in the active region and hence offer the possibility of tuning the Auger coefficient. Numerical charac terization of these structures as well as a detail comparative quantifi cation of the individual contributions of Auger, internal fields, and leakage processes will be the subject of a future study. Also, note that since the Auger process has been the focus in this work, we did not scale the coefficient of the radiative recombination process. This was studied in detail in several of our previously published work (e.g. Refs. [35–38]). For low current density, where Auger process is weak, some anomalous trend is observed for the nanowire with d ¼ 10 nm. Note that the SRH recombination is dominant in this regime, which depends, among several factors, mainly on an interplay between the bandgap and the carrier density within the active region. Overall, the simulation results indicate that increasing the diameter of the host nanowire (that is, the volume of the active region) remains the most efficient way to suppress Auger recombination and boost the internal quantum efficiency.
here, ni,eff is the effective intrinsic carrier density and Ebgc denotes bandgap correction. The factor kðn n0 Þm in Eq. (6) captures other recombination mechanisms for which material specific default values have been used. ILK and VQD denote the leakage current and the volume of the active dot region, respectively. As for the simulated device, the core active region consists of undoped In0.2Ga0.8N, which sits on an n-type (doping density of 7 � 1018 cm 3) GaN buffer layer and capped by a p-type (doping density of 2 � 1019 cm 3) GaN layer. The atomistic diagram of the active semi conducting region with a hexagonal geometry is shown in the top panel of Fig. 5. The granularity (randomness) in the dot region can be clearly seen in this panel. The bottom panel of Fig. 5 displays the variation of IQE with respect to the drive current for different device geometries considered in this work. For all the devices simulated, the peak effi ciency is limited mainly by the SRH and Auger recombination processes as well as the over the barrier leakage current. The IQE never reaches the desired value of 100%, even at a smaller drive current. The droop characteristic in IQE is clearly seen for higher values of the drive current. Looking at the slope beyond the peak IQE points, as far as droop is concerned, we can infer that the thinnest nanowire suffers the most (about 30%). In thicker nanowires, the droop is somewhat weak (about 10% for the device with d ¼ 15 nm). This observation is consistent with the experimental result [34] as well as the first-principles calculation [10]. Interestingly, however, of all the structures simulated, the thinnest device with d ¼ 5 nm benefits most from the introduction of a graded interface within the active QD region provided that such grading does not introduce any defects. For the thickest nanowire with d ¼ 15 nm,
5. Conclusion Non-radiative Auger recombination is a dominant process that 4
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degrades the internal quantum efficiency of nanostructured nitridebased light emitters. Grading of the interfacial confining potential has been proposed as a potential route to suppress the Auger process, but theoretical studies along this line, especially considering the realistic size of the device, is rare. In this work, using a 10-band sp3s* atomistic tight-binding combined with a VFF-3D Poisson simulation framework, we theoretically calculate the Auger coefficients and study the effects of Auger recombination in recently reported InGaN/GaN dot-in-nanowire light emitters. Our calculation shows that the Auger coefficient, while weakly responds to the interfacial grading, is strongly dependent on nanowire diameter—it decreases significantly as the nanowire diameter increases from 5 nm to 15 nm. The microscopically determined Auger coefficients are then incorporated into a TCAD device simulator to compare and analyze the effects of thickness (nanowire diameter) and quantum dot boundary grading on the IQE characteristic of the devices. As for the efficiency droop at higher drive current, we find that the thinnest nanowire is affected the most, yet can benefit significantly from interfacial grading provided that such grading does not introduce any defects. For thicker nanowires, interfacial grading brings little or no improvement in the droop characteristic. Overall, the device with the largest volume of the active region offers the highest IQE and the use of thicker (host) nanowire remains the most effective way to suppress Auger recombination.
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Author contribution statement Ye Wu: Conceptualization, Methodology, Software, Investigation, Writing - Original Draft. Zi-Chang Zhang: Writing - Original Draft, Visualization, Formal Analysis. Shaikh Ahmed: Supervision, Software, Data Curation, Writing Review & Editing, Project administration. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgment This work was financially supported by the U.S. National Science Foundation Grant No. 1610474. Access to Arizona State University computing system is also acknowledged. References [1] M. Auf der Maur, A. Pecchia, G. Penazzi, W. Rodrigues, A. Di Carlo, Efficiency drop in green InGaN/GaN light emitting diodes: the role of random alloy fluctuations, Phys. Rev. Lett. 116 (2) (2016), 027401. [2] S. Deshpande, J. Heo, A. Das, P. Bhattacharya, Electrically driven polarized singlephoton emission from an InGaN quantum dot in a GaN nanowire, Nat. Commun. 4 (2013) 1675. [3] T. Frost, S. Jahangir, E. Stark, S. Deshpande, A. Hazari, C. Zhao, B.S. Ooi, P. Bhattacharya, Monolithic electrically injected nanowire array edge-emitting laser on (001) silicon, Nano Lett. 14 (8) (2014) 4535–4541. [4] A. Mohantaa, D.-J. Jangb, M.-S. Wang, L.W. Tu, Time-integrated photoluminescence and pump-probe reflection spectroscopy of Si doped InN thin films, J. Appl. Phys. 115 (4) (2014). [5] N. Saulius, J. Kęstutis, V. Mikas, S. Egidijus, Y. Tomohiro, N. Yasushi, Injectionactivated defect-governed recombination rate in InN, Jpn. J. Appl. Phys. 52 (8S) (2013), 08JD02. [6] Y. Cho, X. Lue, M. Wienold, M. Ramsteiner, H.T. Grahn, O. Brandt, Auger recombination as the dominant nonradiative recombination channel in InN, Phys. Rev. B 87 (15) (2013) 155203.
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