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7

Temperature-dependent Optoelectronic Properties of

Quasi-2D Colloidal Cadmium Selenide Nanoplateletsa)

Sumanta Bose,1, 2, b) Sushant Shendre,1, 3 Zhigang Song,1, 4 Vijay Kumar Sharma,1, 3, 5, 6 Dao Hua Zhang,1, 2, 3, c)

Cuong Dang,1, 3, d) Weijun Fan,1, 2, e) and Hilmi Volkan Demir1, 3, 5, 6, f)1)School of Electrical and Electronic Engineering, Nanyang Technological University, 50 Nanyang Avenue,

Singapore 639798, Singapore2)OPTIMUS, Centre for OptoElectronics and Biophotonics, Nanyang Technological University, 50 Nanyang Avenue,

Singapore 639798, Singapore3)LUMINOUS! Centre of Excellence for Semiconductor Lighting & Displays and TPI – The Photonics Institute,

Nanyang Technological University, Singapore 6397984)State Key Laboratory of Superlattices and Microstructures, Institute of Semiconductors,

Chinese Academy of Sciences, Beijing 100083, People’s Republic of China5)School of Physical and Mathematical Sciences, Nanyang Technological University, 50 Nanyang Avenue,

Singapore 639798, Singapore6)Department of Physics, Department of Electrical and Electronics Engineering and UNAM,

Institute of Materials Science and Nanotechnology, Bilkent University, Turkey

(Dated: October 26, 2017)

Colloidal Cadmium Selenide (CdSe) nanoplatelets (NPLs) are a recently developed class of efficient lumines-cent nanomaterial suitable for optoelectronic device applications. A change in temperature greatly affectstheir electronic bandstructure and luminescence properties. It is important to understand how-and-why thecharacteristics of NPLs are influenced, particularly at elevated temperature, where both reversible and irre-versible quenching processes come into picture. Here we present a study on the effect of elevated temperatureon the characteristics of colloidal CdSe NPLs. We used an effective-mass envelope function theory based8-band k ·p model and density-matrix theory considering exciton-phonon interaction. We observed the pho-toluminescence (PL) spectra at various temperatures for their photon emission energy, PL linewidth andintensity by considering the exciton-phonon interaction with both acoustic and optical phonons using Bose-Einstein statistical factors. With rise in temperature we observed a fall in the transition energy (emissionredshift), matrix element, Fermi factor and quasi Fermi separation, with reduction in intraband state gapsand increased interband coupling. Also, there was a fall in the PL intensity, along with spectral broadeningdue to an intraband scattering effect. The predicted transition energy values and simulated PL spectra atvarying temperatures exhibit appreciable consistency with experimental results. Our findings have importantimplications for application of NPLs in optoelectronic devices, such as NPL lasers and LEDs, operating muchabove room temperature.

Keywords: Nanoplatelets, Photoluminescence, Cadmium Selenide, Temperature dependence, k ·p method

I. INTRODUCTION

Semiconductor nanoplatelets (NPLs) are a class ofatomically flat quasi two-dimensional (2D) quantum con-fined nanocrystals, often synthesized using wide bandgapII-VI materials.1–4 Advancements in colloidal chemistryhave led to the synthesis of high quality single crystalNPL samples.1,2 It is assisted by the saturation of dan-gling bonds on the surface, as the organic ligands blockfurther growth.5 This reduces nonradiative recombina-tion paths enhancing their optical properties. They have

a)Ancillary document available: TE/TM mode TMEs, NPLs di-mension analysis, Quasi-Fermi energy levels, Fermi factor, Intra-band state gaps, Results comparison and validation, TRPL fit &response, and comparative (experiment and theory) data tables.b)Electronic mail: sumanta001@e.ntu.edu.sgc)Electronic mail: edhzhang@ntu.edu.sgd)Electronic mail: hcdang@ntu.edu.sge)Electronic mail: ewjfan@ntu.edu.sgf)Electronic mail: hvdemir@ntu.edu.sg

attracted increasing interest as they can act as cost-effective and efficient luminophores in display devices,LEDs and lasers.3 They are excellent candidates for suchapplications owing to their morphological bandgap tun-ability, fast fluorescence lifetime and unique optical char-acteristics supporting bright and tunable spectral emis-sion with narrow full-width-at-half-maxima (FWHM)spanning the entire visible to near IR spectral range.3,4

They exhibit strong 1D confinement as their thickness isvery small (typically few monolayers (MLs)) compared tothe Bohr radius.6 Also, they have smaller fluorescent life-times than the typical colloidal quantum dots (QDs) asa result of fast band-edge exciton recombination. NPLshave been routinely synthesized with a high quantum ef-ficiency of about 50%.7 Compared to QDs, they typicallyhave narrower emission spectra, reduced inhomogeneousbroadening and suppressed Auger recombination. Re-cently for the first time, continuous wave laser operationhas been demonstrated using colloidal NPLs.8

With the recent developments and potential applica-tions of NPLs in commercial optoelectronic devices,7,8 it

2

is imperative to study their characteristics and perfor-mances at elevated temperature – when both reversibleand irreversible luminescence quenching processes comeinto play.9 In this work, we study in tandem using the-oretical modeling and experimental measurements, theunderlying physical phenomena determining the opto-electronic characteristics of CdSe NPLs across varyingtemperature, above the room temperature (RT). Acht-stein et al.10 studied NPL characteristics at the cryo-genic range, while there are several works studyingQDs and other nanocrystals across varying temperatureranges.11–13 But, a work addressing the study of NPLoptoelectronic properties at elevated temperatures aboveRT has not been reported thus far. We begin by layingdown the theoretical framework, followed by a compre-hensive discussion on the obtained results to understandthe physics of the NPLs at elevated temperatures, andexperimental methods at the end.

II. THEORETICAL FRAMEWORK

A. Electronic Bandstructure

Here we study CdSe NPLs in the zincblende (ZB)phase as they are colloidally synthesized. The surround-

ing dielectric medium formed by the ligands in colloidalsolutions helps in the optical property enhancement bysurface electronic state passivation and also to stabilizethem. In our model we construct each NPL atom-by-atom in a 3D structure and use an effective mass enve-lope function theory approach based on the 8-band k·pmethod. It is used to solve the eigenvalue equation toobtain the eigenenergy values and electronic structurearound the Γ-point of the Brillouin zone. We simul-taneously take into account the nonparabolicity of thecoupled valence band (VB) and conduction band (CB)including the split-off bands. Using the Bloch functionbasis of |s〉 ↑, |11〉 ↑, |10〉 ↑, |1− 1〉 ↑, |s〉 ↓, |11〉 ↓, |10〉 ↓,|1−1〉 ↓, the 8-band Hamiltonian can be expressed by Eq.1, where all the quadratic, linear and independent termsof k are in the first Hamiltonian matrix, with Hij = H∗

ji,

denoted by c.c. standing for complex conjugate.14 Ep

is the Kane’s theory matrix element and VNPL is theNPL confining potential. Expressions for Hamiltonianelements of Eq. 1, A through E are detailed below. Themodified wavevectors, k′x, k

′y, k

′z are calculated using the

3× 3 strain tensor matrix, ε.

H8×8 =

Ai~√

Ep(k′

x+ik′

y)2√m0

i~√

Ep

2m0

k′zi~√

Ep(k′

x−ik′

y)2√m0

0 0 0 0

c.c. B C D 0 0 0 0c.c. c.c. E C 0 0 0 0c.c. c.c. c.c. B 0 0 0 0

c.c. c.c. c.c. c.c. Ai~√

Ep(k′

x+ik′

y)2√m0

i~√

Ep

2m0

k′zi~√

Ep(k′

x−ik′

y)2√m0

c.c. c.c. c.c. c.c. c.c. B C Dc.c. c.c. c.c. c.c. c.c. c.c. E Cc.c. c.c. c.c. c.c. c.c. c.c. c.c. B

+Hso + VNPL (1)

A = − ~2

2m0γc

(

k2x + k2y + k2z)

+ ac [tr (ε)] + Eg (2a)

B = − ~2

2m0

[

L′ +M ′

2

(

k2x + k2y)

+M ′k2z

]

+av [tr (ε)] +b

2[tr (ε)− 2εzz] (2b)

C = − ~2

2m0

[

N ′ (kx − iky) kz√2

]

+√6d (εxz − iεyz) (2c)

D = − ~2

2m0

[

L′ −M ′

2

(

k2x − k2y)

− iN ′kxky

]

−i√12dεxy +

3b

2(εxx − εyy) (2d)

E = − ~2

2m0

[

M ′(

k2x + k2y)

+ L′k2z]

+av [tr (ε)] + b [3εzz − tr (ε)] (2e)

γc = −Ep

3

[

1

Eg +∆so+

2

Eg

]

+m0

m∗e

(2f)

3

k′xk′yk′z

= (I3 − ε)

kxkykz

(2g)

Using the Luttinger-Kohn effective mass parametersγ1, γ2, γ3 as enlisted in Table I, we have derived themodified Luttinger parameters for our calculation.

L′ = L− Ep

Eg= − ~

2

2m0(γ1 + 4γ2 + 1) (3a)

M ′ =M = − ~2

2m0(γ1 − 2γ2 + 1) (3b)

N ′ = N − Ep

Eg= − ~

2

2m0(6γ3) (3c)

Hso, given by Eq. 4 is the VB spin-orbit coupling Hamil-tonian.

Hso =∆so

3

0 0 0 0 0 0 0 00 0 0 0 0 0 0 0

0 0 −1 0 0 −√2 0 0

0 0 0 −2 0 0√2 0

0 0 0 0 0 0 0 0

0 0 −√2 0 0 −2 0 0

0 0 0√2 0 0 −1 0

0 0 0 0 0 0 0 0

(4)

Considering the NPL periodicities to be Lx, Ly, Lz alongthe x, y and z directions, we use plane waves to expandthe eight-dimensional hole and electron envelope wavefunction as15

φm ={

φjm}

(j = 1, 2, ..., 8) (5a)

with

φjm =1√V

∑

nx,ny,nz

ajm,nx,ny,nze[i(knxx+knyy+knzz)] (5b)

where knx = 2πnx/Lx, kny = 2πny/Ly, knz = 2πnz/Lz

and nx, ny, nz are the plane wave numbers. And V =LxLyLz. The indices of the basis and energy subbandsare given by j and m respectively.Now, the interatomic interactions in the NPL cause

atomistic relaxations which can be estimated by em-ploying the microscopic theory of a valence force field(VFF) model, wherein we account for 2-body inter-action (atomic-bond-stretching) and 3-body interaction(atomic-bond-bending). The sum total of the strain en-ergy is given by16

Table I: Physical Material Parameters of CdSe at RoomTemperature

Symbol Physical parameter Valueref.

Eg (eV) Bandgap energy 1.732a

Ep (eV) Kane Matrix Element 16.5a

∆so (eV) Spin-orbit splitting energy 0.42b

γ1 3.265c

γ2 Luttinger-Kohn parameters 1.162c

γ3 1.443c

m∗

e/m0 Effective Electron Mass 0.12d

m∗

h/m0 Effective Hole Mass 0.45d

ac (eV) Hydrostatic deformation potential of CB -2.83a

av (eV) Hydrostatic deformation potential of VB 1.15a

b (eV) Sheer deformation potential -1.05a

d (eV) Sheer deformation potential -3.10a

a0 (A) Lattice constant 6.052d

d0 (A) Interatomic bond-length 2.620e

α (N/m) Bond-stretching force constant 6.05e

β (N/m) Bond-bending force constant 6.052e

nr Refractive Index 2.5a

aB (nm) Bohr radius 5.6f

aRef. 18 bRef. 19 cRef. 20 dRef. 21 eRef. 22 fRef. 23

EVFF =∑

i(j)

3αij

16d20,ij

(

|ri − rj |2 − d20,ij

)2

+∑

i(j,k)

3βjik8d0,ijd0,ik

(

|ri − rj ||ri − rk| − cos θijk · d0,ijd0,ik)2

(6)

where the atoms in the crystal lattice are identified by theindices i, j and k. d0,ij is the ideal atomic bond-length

between the ith and jth atoms, with 4d0,ij =√3a0,ij re-

lating it to a0,ij , the lattice constant.17 Factors such asexternal piezoelectric strain or temperature cause devi-ations in the bond-length from the ideal d0,ij , and thebond-distance between the ith and jth atoms in the sys-tem studied is given by |ri − rj |, where ri is the positionvector of the ith atom. The bond-stretching force con-stant of the i − j bond is given by αij (Table I). Theideal bond-angle of the bond among the ith, jth and kth

atoms (vertexed at ith atom) is given by θijk (= 13 for

ZB structures). The bond-bending force constant of thej − i− k bond-angle is given by βjik (Table I).

B. Optical Properties

For studying the optical characteristics of NPLs, weuse the density-matrix equation dependent on the quasiFermi levels of the CB (fc) and VB (fv). We can estimateit for a two-dimensional structure with exciton effects asa sum of the spontaneous radiative rate from the exci-tonic bound states, rex,bsp and continuum-states, rcsp. The

excitonic bound state contribution is given by24,25

4

rex,bsp (E) =e2nrE

πm20ε0c

3~2t

∑

c,v

|Ψcv1s (0) |2|Pcv|2 · |Icv|2 ×

fc (1− fv)L (E − Ecv − Eb) (7a)

Ψ1s (x) =4β

aB√2πe−2xβ/aB (7b)

Eb = −4β2Ry (7c)

where symbols (e, nr, E, m0, ε0, c, ~) have standardphysical meanings. t is the NPL thickness, Ψ1s (x) is the1S exciton envelope function and x is the relative distancebetween the electron and hole along the transverse direc-tion in the NPL. fc and fv are the Fermi-Dirac distribu-tions of the CB and VB, dependent on the quasi-Fermienergy levels of the CB (Efc) and VB (Efv), respec-

tively. Icv =∫

ΨE(c) (z)ΨH(v) (z) dz is the overlap inte-gral between the electron and hole wavefunctions alongthe transverse direction. Eb is the 1S exciton bindingenergy and β is a variational parameter, taken to be 1for two-dimensional structures, based on conclusive ob-servation that for two-dimensional semiconductors, thebinding energy increment by the low dielectric constantof the surface ligands and the solvent is is almost ex-actly compensated by self-interactions of electrons andholes with self-image potential.4 aB = 4πε0εr~

2/mre2

and Ry = mre4/32π2ε20ε

2r~

2 are the excitonic Bohr ra-dius and excitonic Rydberg energy respectively, wheremr is the reduced mass of the electron-hole pair: m−1

r =m−1

e +m−1h and εr is the relative permittivity. For the

continuum state contributions, we have25,26

rcsp (E) =e2nrE

πm20ε0c

3~2V

∑

c,v

|Pcv|2fc (1− fv)×

S2D (E − Ecv)L (E − Ecv) (8a)

S2D (E − Ecv) =2

1 + exp(

−2π√

Ry/(E − Ecv)) (8b)

where S2D (E) is the 2D Sommerfeld enhancementfactor.26 V is the NPL volume in real space. The PLemission is maximum when fc = 1 (fully occupied up-per level) and fv = 0 (fully empty lower level), and theemission rate is ∝ fc (1− fv), which is called the Fermifactor.27 The calculation of the theoretical PL also takesinto consideration the spectral transition energy broad-ening (dephasing) effect accounted by the Lorentzianbroadening lineshape term Lcv(E − Ecv) in Eq. 7a and8a, given by

Lcv(E − Ecv) =1

π

~/τin

(E − Ecv)2 + (~/τin)

2 (9)

whose full width at half maximum (FWHM) is 2~/τinand scattering probability per unit time is 1/τin, whereτin is the intraband relaxation time. The term |Pcv|2in Eq. 7a and 8a is the square of the optical transi-tion matrix element (TME). It quantifies the strength oftransition between the electron- and hole-subband.28 Wecalculate it using the momentum operator, p and the realelectron and hole wavefunctions, Ψc,k and Ψv,k respec-tively: Pcv,i = 〈Ψc,k| e · pi |Ψv,k〉, i = x, y, z.29 Expres-sions for Pcv,i along the x, y and z directions are given inthe ancillary document. The average of Pcv,x and Pcv,y

(TMEs along x and y directions) contributes to the trans-verse electric (TE) mode emission, while Pcv,z, the TMEalong the z direction contributes to the transverse mag-netic (TM) mode emission. The TE mode emission is x-yplane polarized, while the TM mode emission is z direc-tion polarized. Table I enlists the material parametersof CdSe used in this work, citing their sources. Now, inSec. III, using this theoretical framework, we study theeffect of temperature on the optoelectronic properties ofCdSe NPLs and compare with experimental results.

III. RESULTS AND DISCUSSIONS

A. Theoretical Results

We have used the theoretical model described in Sec. IIto study the optoelectronic characteristics of quasi two-dimensional colloidal CdSe NPLs of lateral dimensions22 nm × 8 nm, and vertical thickness of 4 monolayer(ML), across temperature varying from room tempera-ture (RT) to 90◦C. We have modeled and studied NPLsof this particular dimension, based on experimental mea-surements as we shall describe later in Sec. III B. Underthis scheme, we have studied how temperature affectstheir electronic structure and optical properties and pre-dicted some temperature-dependent photoluminescence(PL) characteristics of CdSe NPLs. Fig. 1a shows thetheoretically calculated excitonic transition energy (andphoton emission wavelength) from the bottom of CB (E1)to the top of VB (H1) i.e. E1-H1, as a function of tem-perature. With an increase in temperature there is ared shift in the transition energy (photon emission en-ergy). The reason is that the bandgap, Eg has a negativethermal coefficient (dEg/dT ) due to (i) lattice thermaldilatation and electron-phonon interaction ∝ T at hightemperatures, and (ii) electron-phonon interaction ∝ T 2

at low temperatures. In this context, we conventionallyhave the Varshni relation30 to explicate the behavior ofEg. However, detailed systematic study for II-VI semi-conductors have shown that sometimes the Varshni rela-tion may not be very accurate.31 Another semiempiricalexpression suggested by Cardona et al.32 is more accu-rate as we shall subsequently see. Also, with an increasein temperature, the phonon concentration increases andcauses increased scattering and non-radiative recombina-tions start occurring,33 which causes the E1-H1 TME to

5

300 315 330 345 3602.36

2.38

2.40

2.42

525

521

517

512(a)

2.36 eV2.37 eV2.38 eV

2.39 eV2.40 eV

2.41 eV

2.42 eV2.43 eV

Wav

elen

gth

(nm

)

Tran

sitio

n en

ergy

(eV

)

Temperature (K)

E1-H1 Theoretical Transition Energy/Wavelength (eV/nm)

-0.15-0.10-0.050.002.4

2.6

2.8

-0.15-0.10-0.050.002.4

2.6

2.8

-0.15-0.10-0.050.002.4

2.6

2.8

0.0

0.2

0.4

0.6

0.8

1.0

= 2.42 eV (30oC)(b)Transition energy

0.0

0.2

0.4

0.6

0.8

1.0

(c)Transition energy= 2.40 eV (50oC)

Ban

d en

ergy

(eV

)

Prob

abili

ty in

ban

d m

ixin

g

0.0

0.2

0.4

0.6

0.8

1.0

Band energy Band energy

VB Split off hole VB Light hole VB Heavy hole CB electron

(d)Transition energy= 2.38 eV (70oC)

Valence (hole) state indexConduction (electron) state index E10 E7 E4 E1 H1 H4 H7 H10

Band energy

Figure 1: (a: first) Theoretical E1–H1 (bottom ofconduction band to top of valence band) excitonic

transition energy/wavelength as a function oftemperature. Frames (b–d) show the electronic

bandstructure and the probability in band mixingbetween conduction electrons and valence heavy-, light-and split off holes for 4 ML CdSe NPLs at (b: second)30◦C, (c: third) 50◦C and (d: fourth) 70◦C. E1–H1transition energies are indicated. Legends show the

band mixing probabilities.

decrease.

Of the eight temperature cases studied in Fig. 1a, wepresent the electronic bandstructure and the probabilityin band mixing between conduction electrons and valenceheavy-, light- and split off-holes due to coupling for the4 ML CdSe NPLs at 30, 50 and 70◦C, in Fig. 1b-d. TheE states represent the conduction electron states, whilethe H states represent the valence hole states. Each ofthe states is two fold degenerate, considering spin-up andspin-down alternatives. For the sake of discussion, let usconsider the 4 ML CdSe NPL at 30◦C of Fig. 1a as acase. The E1-H1 transition energy (2.42 eV) determines

300 320 340 360-0.2

0.0

0.2

0.4

0.6

0.8

1.0

300 320 340 360

2.38

2.45

2.52

2.59

2.66

2.73

Photogenerated carrier densities(×1019cm-3)

(a) Fermi factor

1 2 3 4 5

Ferm

i fac

tor f

c(1-f v

)

Temperature (K)

(b) Fermi evergyseparation & E1-H1

F

F

F

F

F

E1-H1

Ener

gy (e

V)

Temperature (K)

Figure 2: (a: left) Fermi factor fc(1 − fv) for E1–H1transition, and (b: right) Fermi energy separation

∆F = Efc − Efv compared with the E1-H1 transitionenergy in 4 ML CdSe NPLs as a function of temperaturefor varying injection carrier concentration. Both, Fermifactor and ∆F have a negative temperature gradient.

300 315 330 345 3600.448

0.450

0.452

0.454

0.456

TE mode Transition matrix element forE1-H1 transitions in 4 ML CdSe NPLs

E1-H1 TME (P02)

Tran

sitio

n m

atrix

ele

men

t

Temperature (K)

Figure 3: TE mode transition matrix element (TME)for the E1–H1 transitions as a function of temperaturein 4 ML CdSe NPL. The other interband TMEs (bothTE and TM mode) are given in the ancillary document.

the PL peak emission energy.34 The CB states mainlycomprise of electrons with very low composition of hh,lh and split-off (so) hole, which comes from the couplingeffects. The effective hh mass is much larger than the ef-fective electron mass for CdSe,21 thus the hh dispersioncurve is more flat than the conduction dispersion curve.Therefore, the first ten hole levels span only ∼140 meV,while the first ten electron levels span ∼425 meV. In theVB, the first few levels are hh dominated followed byincreased influence of lh and so. Comparing the span offirst ten electron and hole states and band-mixing proba-bilities of the three cases in Fig. 1 we have two significantobservations: (i) an increase in temperature induces afaster reduction in the intra-CB state gaps compared tothe intra-VB state gaps. For instance, a temperature riseof 20◦C between adjacent frames of Fig. 1, the ∆E1-E10 isover five times larger than the ∆H1-H10. This is by virtueof the temperature-induced thermal lattice dilatation ef-fects, given than the deformation potential of the CB ishigher than that of the VB for CdSe; and (ii) an increase

6

E1 E2 E3 E4 E5 E6 E7 E8 E9 E10

H1(hh) H2(hh) H3(hh) H4(hh) H5(hh) H6(hh) H7(mix) H8(hh) H9(lh) H10(lh)

Figure 4: The spatial charge density of the firstten electron and hole states of 4 ML CdSe NPLsat 30◦C (303.15 K) cut along the vertically centralx-y plane (z = 0) of the NPL. Warmer (reddish)colors depict higher occupation probability overcooler ones (bluish). The rectangular boundary

depicts our studied NPL (22 nm × 8 nm), but thedimensions are not to scale, for the sake of

comprehensive visual representation.

in temperature promotes e-h quantum state coupling –the conduction states start to have an increased contribu-tion from holes and vice versa. Moreover, the intrabanddynamics is also affected, such that the role of the domi-nant contributor diminishes with increasing temperature.For instance, the dominant contributors for H1 throughH8 is hh while for H9 and H10 is lh, whose contributionfalls as temperature rises. The contrast is however smallto be reflected graphically. These band-mixing probabil-ities are affected by the varying degree coupling betweenthe conduction electrons and valence hh, lh and so holes,which can be empirically determined. For each level, theband-mixing probabilities are dictated by their aggregateelectron and hole wavefunctions and associated chargedensities thereof.

The Fermi factor and energy levels are important de-terminers of the temperature-dependent optoelectronicscharacteristics. In Fig. 2a we show the Fermi factorfc(1 − fv) at E1–H1 transition and in Fig. 2b the Fermienergy separation ∆F = Efc − Efv in our 4 ML CdSeNPL as a function of temperature, for varying photo-generated carrier densities from 1 to 5×1019 cm−3. The∆F for each case is compared with the E1–H1 transitionenergy, and ∆F¿E1–H1 ensures the Bernard-Duraffourginversion condition (population inversion) necessary forlasing.26 With an increase in carrier density, both Fermifactor and Fermi energy separation increases. However,as we continue to increase the density, the extent of Fermifactor increment falls, at it approaches saturation. How-ever, a rise in temperature has detrimental effects onboth, Fermi factor and Fermi energy separation. Withan increase in temperature, the fermions (electrons andholes) get thermally excited and therefore the probabil-ity of occupying higher CB and VB energy states is in-creased; so the fc falls, while fv rises. Consequently theFermi factor decreases with temperature. Also, the quasiFermi levels of the CB and VB (Efc and Efv) approachthe band edges, and ∆F falls. In the ancillary document,we have shown the Efc and Efv in contour forms alongwith the Fermi factor and Fermi energy separation.

Another critical temperature-dependent optoelectronicfactor is the transition matrix element (TME) which isshown in Fig. 3 for the E1–H1 transition in TE modeas a function of temperature. As the thermally excitedfermions begin to occupy higher energy states the E1–H1transition weakens. A physical equivalent of the TME isthe oscillator strength fcv ∝ µ2

cv, where µcv is the tran-

sition dipole moment, and it can be calculated from theTME using µcv = e~Pcv/im0Ecv. It allows us to quan-tify the transition strength from state |c〉 to |v〉, whichdecreases with a rise in temperature.Now we shall study the spatial manifestation of the

electronic states. Fig. 4 shows the spatial charge densi-ties of the first ten electron and hole states (square ofthe wavefunction |ψ2|, i.e. probability of finding them),within the vertically central x-y plane of the 4 ML CdSeNPL at 30◦C corresponding to Fig. 1a. The charge den-sity description in terms of s and p orbitals in a quitegeneral feature of III-V semiconductors. However in II-VI semiconductors the VB is influenced by chemicallyactive d orbitals also.35 The E1, H1 and H7 states are s-like. While H1 is dominated by hh, H7 has higher lh con-tribution. The E2 and H2 are py-like, while E4, H4 andH9 are px-like – H4 has greater hh influence while H9 hasmore of lh. The E3, E7 and H3 are formed by s-p-mixing.The E5 and H5 are dxy-like, while the E6 and H6 againhave some amount of mixing. It is observed that withchanges in temperature the spatial charge densities havevery inconsequential variations because it is a measure ofthe electron and hole probability density determined bythe wavefunction localization, which is primarily affectedby piezoelectric strain and external electric fields.

B. Experimental Results

We have synthesized 4 monolayer (ML) CdSeNPLs as they have been most widely investigated inliterature7,18,36–40 and found applications in LED color-conversion layers, lasers and luminescent solar concentra-tors. We have used colloidal synthesis techniques similarto those used in Refs. 39, 41–43, as described in the Ex-perimental Methods section.Fig. 5 shows a high-resolution transmission electron

microscopy (TEM) image of our synthesized NPL en-semble population. We can observe NPLs in two orienta-tions: face view (laterally flat) and edge view (verticallystacked). We have done lateral dimension measurementsover several NPL samples using ImageJ software. Theiraverage length and width were measured to be 22 nmand 8 nm respectively. The theoretical results discussedin Sec. III A correspond to the same dimensions of oursynthesized NPLs. Measurements of stacked NPLs revealdiffering thicknesses at the two ends, which is attributed

7

Figure 5: High-resolution transmission electronmicroscopy (TEM) image of 4 ML CdSe NPL ensemblepopulation. Average lateral dimension is l = 22 nm and

w = 8 nm (Scale: 20 nm)

0 10 20 30 400.1

1

10

450 500 550 600

0

5

10

15

20 2.76 2.48 2.25 2.07

509 nm

480 nm

Photoluminesce (PL) Absorbance

Photon Energy (eV)

E1-H9 (lh)

E1-H1 (hh)

Photon Wavelength (nm)

Phot

olum

ines

ence

(a.u

.)

512 nm

0.0

0.1

0.2

0.3

0.4

Abs

orba

nce

(a.u

.)1 = 0.34 ns (30%)

2 = 2.27 ns (70%)

Inte

nsity

(a.u

.)

Time (ns)

Photon count TRPL Fit

Figure 6: Experimental Photoluminescence (PL) andAbsorption spectrum of 4 ML CdSe NPLs at 30◦C

(303.15 K). The PL excitation wavelength was 350 nm.Both PL and absorption were measured in solution

form. PL peak is at 512 nm. Primary absorption peak,at 509 nm has a Stokes shift of 3 nm, while the

secondary absorption peak is at 480 nm. Inset showsTime-Resolved PL (TRPL) spectrum and fit measured

in thin-film form at 30◦C. The fit and the systemresponse is given in the ancillary document.

to NPL tilting/folding, a common phenomenon for NPLs,also observed by several other groups.4,44 Also, the lig-ands surrounding the NPLs have a contrast in the TEMimage, so the NPL boundary is not strictly discernible.However, the confirmation of 4 ML (1.2 nm) thicknesscomes from the optical spectra characterization. This isdiscussed in the ancillary document.

Spectral characterization results in Fig. 6 show the ex-perimental PL emission and absorption spectrum (bothmeasured in solution form) for our 4 ML CdSe NPLs at30◦C. The PL was measured in a standard PL spectrom-eter with an excitation wavelength of 350 nm. We canobserve the PL emission peak at 512 nm, which is invery good agreement with the E1–H1 transition energyprediction of 2.42 eV by our model, as shown in Fig. 1a–b. Similarly, with our E1–H1 predictions at other varyingtemperatures (Fig. 1a) we can determine their expectedPL emission wavelengths.

With a Stokes shift of 3 nm from the PL spectrum, theprimary absorption peak occurs at 509 nm. This is dueto the reorganization of the solvent in solution-processedNPLs plus dissipation and vibrational relaxation of theNPLs.45 The primary absorption peak corresponds to theE1–H1 transition, where the H1 state is 2D-heavy holes(hh) dominated [as theoretically determined in Fig. 1b].A study of the TMEs make this clear. The transverseelectric (TE) mode TME for the E1–H1 transition is max-imum among all possible cases accounting to 0.455. Thesecondary absorption peak occurs at 480 nm. A study ofthe transverse magnetic (TM) mode TMEs of the tran-sitions near the secondary absorption energy shows thatthis comes from E1-H9 transitions, for which the TMEvalue is 0.358, which is weaker than the E1-H1 TMEbut is stronger than TMEs of other allowed transitions.The fraction of light holes (lh) involved in this transi-tion is much higher (66 %) than their hh counterpart (22%) [Fig. 1b]. This conforms with the idea that hh con-tributes to the primary absorption peak of a typical CdSeNPL, while lh contributes to the secondary peak.10 Wehave discussed about the TE and TM mode TME valuesfor all possible transitions between E1−10 to H1−10 in theancillary document. Note that, the PL peak position at512 nm, and the hh and lh absorption peaks at 509 nmand 480 nm respectively confirms that our synthesizedNPL samples are indeed 4 ML (1.2 nm) thick.39,46

Time-Resolved PL (TRPL) spectrum and fit at 30◦C(inset of Fig. 6) reveals a dual decay path mechanismfollowing the model I(t) = a1e

−t/τ1 + a2e−t/τ2 , with

PL decay lifetimes of τ1 = 0.34 ns and τ2 = 2.27ns with 30% and 70% contribution respectively, lead-ing to an average lifetime of 2.15 ns calculated usingτavg =

(

a1τ21 + a2τ

22

)

/ (a1τ1 + a2τ2), which is similar to

previously reported τavg of CdSe NPLs at RT.4,6 TheTRPL fit is shown again with the system response inthe ancillary document. Previous radiative transition dy-namics studies on CdSe NPLs have also observed a sim-ilar dual decay path mechanism where τ1 is assigned asthe radiative recombination decay time period relating toreversible PL losses, while τ2 as the nonradiative recombi-nation decay time period relating to the trap states whichmay cause irreversible losses in the PL intensity,10,47 orig-inating from the fast hole-trapping due to incompletepassivation of Cd atoms on the NPL surface.41

Fig. 7a and Fig. 7b shows the effect of temperature onthe experimentally measured PL spectra of the NPLs in

8

460 480 500 520 540 560 5800

3

6

9

12

15

2.70 2.58 2.48 2.38 2.30 2.21 2.14

460 480 500 520 540 560 580

0.5

1.0

1.5

2.0

2.52.70 2.58 2.48 2.38 2.30 2.21 2.14

Photon Wavelength (nm)

30oC 40oC 50oC 60oC 70oC 80oC 90oC

(a) Ascendingtemperature

90oC80oC

40oC

30oC

Photon Energy (eV)Ph

otol

umin

esen

ce (a

.u.)

50oC

60oC

70oC

Phot

olum

ines

ence

(a.u

.) (b) Descendingtemperature 30oC 30oC

40oC 50oC 60oC 70oC 80oC 90oC

Photon Energy (eV)

Photon Wavelength (nm)

40oC

50oC

60oC

Figure 7: PL spectra of 4 ML CdSe NPLs measured in thin-film form while (a: left) ascending, and (b: right)descending temperature, respectively. Temperature of each PL spectrum is indicated. With an increase in

temperature there is a redshift in the emission peak, broadening in the PL linewidth and reduction in PL intensity.Upon decreasing the temperature to RT, we observe substantial retractability in the peak position and linewidth,but not in the intensity. Fig. 8 shows the PL emission energy (peak position), linewidth and integrated intensity as

a function of temperature.

thin-film form using a laser excitation wavelength of 355nm at 0.5 mW power, while ascending and descendingtemperature respectively. For every 10◦C rise in temper-ature, there is a red-shift in the emission wavelength of∼2 nm, and the PL intensity falls as the sample qual-ity degrades. Also, the spectrum broadens due to intra-band scattering effect. Fig. 8 shows the photon emissionenergy, PL linewidth and PL integrated intensity as afunction of ascending (black ⊡) and descending (red ×)temperature. For a particular temperature the emissionenergy and linewidth are almost identical in both. Whilecooling back to RT, the signal intensity increases, but isfar from being retraced. The recovery is < 14 % (seelogarithmic comparison in inset of Fig. 8c). The effectivephoton emission energy (equivalent to the effective E1-H1 transition energy) has a negative thermal coefficientas already discussed in the context of Fig. 1a. For II-VI semiconductors, a semiempirical expression (Eq. 10)suggested by Cardona et al.32 takes into considerationthe phonon emission and absorption using Bose-Einsteinstatistical factors.

Eexc (T ) = Eexc (0)− aep (2nB + 1) (10)

Here aep is the exciton-phonon coupling constant and

nB = 1/(

eΘ/T − 1)

. Θ is the average phonon tempera-ture, such that the average phonon energy is kBΘ. Both,acoustic and optical phonons contribute to the redshiftincurred by increasing temperature. Eexc (0) is the exci-ton energy at 0 K. T is the absolute temperature. Uponfitting the ascending temperature data of Fig. 8a withEq. 10, we get Eexc (0) = 2.727 eV. This value is in goodagreement with results of Achtstein et al.,10 who reported2.709 eV. This elevated Eexc (0) compared to bulk like ZBCdSe epilayers (1.739 eV)48 is due to increased quantumconfinement effects. Further, we obtain aep = 18.2± 2.8

meV and Θ = 36±0.4 K. Chia et al.48 calculated aep = 21meV for bulk like ZB CdSe epilayers. This shows thatthe increased confinement in the NPLs has reduced theaverage exciton-phonon coupling, also evident from thesignificant reduction in average phonon temperature of305 K in bulk ZB CdSe.21

Moreover, the exciton-phonon interaction induced in-traband scattering causes transition energy broadening.The temperature dependence of the linewidth of excitonicpeaks is given by26,49

Γ (T ) = Γ0 + γACT +ΓLO

e(~ωLO/kBT ) − 1(11)

where Γ0 is the inhomogeneous broadening due to in-trinsic effects (eg. alloy disorder, impurities). γAC ac-counts for the exciton-acoustical phonon interaction. Thelast term comes from exciton-longitudinal optical (LO)phonon Frohlich interaction. For practical purposes, thecontribution of γAC compared to ΓLO is negligible, hencesometimes it is considered zero.10,50 At cryogenic range,particularly < 50 K, the linewidth gradient drasticallydecreases.10,32 therefore extrapolating our data of ≥ RTto lower temperatures would underestimate Γ0. We haveused Γ0 = 32.5 meV from ref. 10 following their linearfit at a low temperature range of 10-50 K. Further, basedon the Raman spectra measurements of CdSe NPLs10

and QDs12, we have taken ~ωLO = 25 meV, to fit theother parameters of Eq. 11 using the ascending temper-ature data from Fig. 8b. We found ΓLO = 236.3± 11.8meV and γAC = 0.38 ± 0.03 meV, suggesting that theLO phonon contribution is much larger compared to thatof the acoustic phonon, consistent with the calculationsof Rudin et al.49 Comparing with bulk-like CdSe epilay-ers (γAC = 0.084 meV)48, NPLs have much higher γAC

due to an increase in acoustic phonon coupling causedby reduced dimensionality of the system, which is theo-retically consistent51. The interaction between excitons

9

300 315 330 345 3602.36

2.38

2.40

2.42

525

521

517

512

60

70

80

90

300 315 330 345 360 14

16

18

20

0.0

0.2

0.4

0.6

0.8

1.0

300 315 330 345 3600

4

8

12

32 34 36 380.1

1

10363 341 322 305

(a)

Wav

elen

gth

(nm

)

Emis

sion

/Tra

nsn.

ene

rgy

(eV

)

Temperature (K)

Experimental ascending TExperimental descenting TTheoritical calculation

(b)

Temperature (K)

Line

wid

th (m

eV)

Rel

axat

ion

Tim

e (f

s)

(c)

Rel

ativ

e in

t. in

tens

ity (a

.u.)

Temperature (K)

inte

nsity

(×10

5 a.u.

)

Inte

grat

ed Fit with experimental

ascending temperature

(d)

Temperature (K)

log

Inte

grat

ed In

t. (a

.u.)

1/kBT (eV-1)

Figure 8: (a: top left) Experimental PL emission energy/wavelength while ascending and descending temperaturecompared with theoretically simulated E1–H1 transition energy values; (b: top right) Experimental PL linewidthwhile ascending and descending temperature and extracted intraband relaxation time τin used in theoretical model[Eq. 9] from data of ascending temperature; (c: bottom left) Experimental PL intensity while ascending temperature

compared with theoretically simulated relative PL integrated intensity; and (d: bottom right) ComparativeArrhenius plot for PL integrated intensity (log scale) while ascending and descending temperature. Consistentsymbols have been used in all the frames: black ⊡ and red × for experimental data while ascending and

descending temperature respectively, while blue + for theoretically predicted data or data used in theoretical model(comparison is discussed in Sec. III C). The green · · · dotted lines in frames (a), (b) and (c) show the fitted plots

[using Eq. 10, 11 and 12] with the data of ascending temperature.

and LO phonons is dictated by the Frohlich interaction,which is Coulombic in nature caused by the longitudi-nal electric field produced by LO phonons. Therefore,~ωLO and ΓLO are related in the sense that a larger~ωLO induces a stronger Frohlich interaction and a largerΓLO. Valerini et al.

52 have shown the presence of multi-ple temperature-dependent non-radiative processes in PLrelaxation dynamics. At higher temperature, the contri-bution of LO phonons becomes predominant, eventuallydominating the excitonic linewidth.50

Furthermore, with increasing temperature, we alsohave a substantial reduction in the PL intensity as theexcitons are trapped in the nonradiative centers, likely atNPL surfaces and dissociate into the continuum states.The integrated intensity of the PL spectra with varyingtemperature is given by48,53

I (T ) = I (0) /[

1 + C · e−Ea/kBT]

(12)

where I (0) and I (T ) are the integrated PL intensitiesat 0 K and T K, respectively. The fitting constant C is

related to the radiative and nonradiative lifetimes, Ea isthe activation energy of the nonradiative channel. We fitthe ascending temperature data of Fig. 8c with Eq. 12 toobtainEa = 494±6 meV. Heating above RT induces bothreversible and irreversible losses in the intensity, and thelatter restricts it to trace back upon cooling54 (inset inFig. 8c). Irreversible losses are generally caused by chem-ical processes like ligand loss or oxidative degradationdamaging the NPL surface which disturbs the repopula-tion of mobile holes and electrons.55 The extent to whichthe PL intensity is retraced in Fig. 7b is due to reversiblelosses, caused due to dynamic and static quenching.54

Upon heating, dynamic quenching activates the ther-mally activated non-radiative recombination paths com-pared to radiative paths; while static quenching causesthe ratio of dark to bright particles to increase inducingnon-radiative decay. While actively heating up to 90◦Ctook us a few minutes, every successive 10◦C fall tookincreasingly longer time – 2, 4, 6, 11, 19 and 32 minutestill 30◦C and even longer to RT. The Newton’s cooling

10

2.2 2.4 2.60

0.2

0.4

0.6

0.8

1 30°C

PL I

nten

sity

(a.

u.)

2.2 2.4 2.6

50°C

Photon Emission Energy (eV)2.2 2.4 2.6

70°C

Expt.PLTheor.PL

Figure 9: Comparison of experimental (solid black) vs.theoretical (dotted blue) PL spectra of 4 ML CdSe

NPLs at 30, 50 and 70◦C.

constant following T (t) = Ta +∆T0e−kt yeilds k = 0.089

min-1. An attempt to fit the descending T integratedintensity with Eq. 12 yeilds a poor Chi-square fit, as ex-pected, since the irreversible losses incurred over almosta span of 1 hr disturbs the liable excitonic properties. Ifthe cooling is conducted in a vacuum environment (toavoid losses due to oxidation), it will result in an im-proved intensity recovery.

C. Comparison of Theoretical and Experimental Results

In Sec. III A and III B, we have studied thetemperature-dependent optoelectronic properties ofCdSe NPLs from a theoretical and experimental point-of-view. Here we combine and compare our findings,and simulate the PL spectra for varying temperature,as shown in Fig. 9. A PL spectrum is characterized bythree main parameters:

• PL peak position: We have theoretically calculatedthe temperature-dependent E1–H1 transition en-ergy values (Fig. 1a), using the 8-band k·p methoddescribed in Sec. II A and material parameters fromTable I. The Hamiltonian matrix was solved by di-rect diagonalization to obtain the electronic struc-ture and eigenenergy values, and E1–H1 transitionenergy (equivalent of the PL peak position). Itmatch excellently with the experimental data of PLpeak position, as shown in Fig. 8a.

• PL linewidth: The PL spectrum undergoes spectralbroadening due to several empirical factors, suchas local and extended structural defects, size dis-persion, alloy disorders and impurities, which af-fects its linewidth.31 Thus, different 4 ML CdSeNPL samples may have varying linewidths at thesame temperature. We have used the experimentallinewidth data of ascending temperature to extractthe intraband relaxation time (τin = 2~/FWHM)used in our simulation as described in the calcu-

lation of spontaneous radiative rate from the ex-citonic bound states and continuum-states in Sec.II B. This helps to model the linewidth of thesimulated PL spectra according to the Lorentzianbroadening lineshape [Eq. 9] for an accurate esti-mate. The extracted τin is shown in Fig. 8b.

• PL intensity: We have calculated the relative PLintensity, which shows an exponential fall with tem-perature, as verified experimentally. However, theexperimental PL intensity (arbitrary units) is notthe absolute total emission. It is therefore of in-terest to study and compare the relative intensitiesvariation with temperature, which is influenced byseveral parameters such as Fermi factor, TME, etc.[see Eq. 7 and 8]. Our theoretically predicted rel-ative PL intensities have a very good comparisonwith the experimental intensities of ascending tem-perature, as shown in Fig. 8c.

The database of theoretical and experimental (ascend-ing and descending temperature) results is given in theancillary document. Employing the above mentioned ap-proach, we have studied the PL spectra of the 4 ML CdSeNPLs at varying temperature. Fig. 9 shows a compar-ison between the experimental and theoretically calcu-lated PL spectra at 30◦C, 50◦C and 70◦C. For a clearcomparison of peak positions, linewidths and intensities,we have normalized both, the experimental and theoret-ical PL spectra with respect to the PL peak at RT. Theexperimental and theoretical PL spectra match appre-ciably, consistent with the results presented in Fig. 8. Inthe ancillary document, we have compared, extended andvalidated our calculations and measurements with exist-ing low temperature results from literature. We havestudied the temperature-dependent PL peak positions(E1–H1 transition energies), PL linewidth and relativePL intensity in tandem with results from seminal worksof Achtstein et al.10 and Erdem et al.46 to validate ourresults.

IV. SUMMARY AND CONCLUSION

In summary, we have comprehensively studied the elec-tronic bandstructure, probability in band mixing, chargedensities, optical transition matrix element, characteris-tic PL emission energy, linewidth and integrated inten-sity of CdSe NPLs – as a function of temperature, whichis of critical importance particularly for commercial de-vice applications at elevated temperature. Our theoreti-cal investigations are based on a framework that relies onthe effective-mass envelope function theory and density-matrix theory. From the quantum physics and optoelec-tronics point-of-view, the implications of rise in operationtemperature of NPLs are many-fold, such as (i) reductionin effective band-edge transition energy, (ii) fall in opti-cal transition matrix element strength, (iii) reduction inintraband state gaps, faster so in CB compared to VB,

11

(iv) promotion of interband e-h coupling, (v) modifica-tion in hh-lh-so band mixing probabilities, (vi) reductionin Fermi factor, (vii) shift of CB and VB quasi Fermienergy levels towards the band edges, and reduction inquasi Fermi separation, (viii) invariance in the chargedensity profile, (ix ) redshift in peak photon emission en-ergy in PL spectra, (x ) widening of PL spectral linewidthowing to intraband scattering, (xi) exponential fall in PLintegrated intensity, and (xii) phenomenological evidenceof reversible and irreversible losses induced which affectsthe sample quality and dictates the PL traceability.

In conclusion, we have theoretically and experimen-tally studied the temperature-dependent optoelectroniccharacteristics of quasi-2D colloidal CdSe NPLs and es-tablished a model to predict and study the electronicbandstructure and PL spectra of NPLs at any arbitrarytemperature, which can be effectively used by experimen-talists to optimize device design iterations.

EXPERIMENTAL METHODS

Synthesis Techniques

We have synthesized 4 ML colloidal CdSe NPLs us-ing synthesis protocol similar to those used in Refs.39, 41–43. The following chemicals were used: Cadmiumacetate dihydrate Cd(OAc)2·2H2O, oleic acid, techni-cal grade 1-octadecene, cadmium nitrate, cadmium ni-trate, n-hexane, powder form selenium and myristic acidsodium salt. These were procured from Sigma Aldrich.The first step is to synthesize Cadmium (myristate)2, bydissolving 1.23 g of Cadmium nitrate in methanol (40mL). This was followed by dissolving sodium myristate(3.13 g) in methanol (250 mL) for 60 min with continu-ous stirring. When the two solutions were mixed, a whiteprecipitate is formed after full dissolution. This precipi-tate was washed with methanol and dried under vacuumfor about 24 h. 12 mg (0.15 mmol) of selenium (Se) and170 mg (0.3 nmol) of Cd(myristate)2 were introduced ina three-neck flask along with 15 mL of octadecene (ODE)and degassed under vacuum. Under argon (Ar) flow, themixture was heated up, until 240◦C, and left there for10 mins. When the color of the solution becomes yellow-ish, at ∼ 195◦C, 80 mg (0.30 mmol) of Cd(OAc)2·2H2Owas introduced. 5 mL of n-hexane and 0.5 mL of oleicacid was added at the end of the synthesis, and the so-lution was allowed to cool down to RT. A centrifuge wasused to precipitate the NPLs before storing them in hex-ane. This synthesis also produces some 5 ML NPLs andspherical nanocrystals with distinctly different emissionwavelength. The 4 ML NPLs were separated from therest using selective precipitation.

Purification and Film preparation

The synthesized NPLs were centrifuged for 5 min at4500 rpm, followed by removal of the supernatant solu-tion. Nitrogen was used to dry the precipitate, beforedissolving it in hexane to centrifuge for 10 min at 4500rpm. The supernatant solution was made turbid by ad-dition of ethanol, followed by centrifuging again for 10min at 4500 rpm. Finally, using a 0.20 µm filter, the dis-solved precipitate in hexane was filtered. Later, piranhasolution was used for 30 min to clean quartz substratesof size 1.5 × 1.5 cm followed by DI-water. It was driedat 80◦C for 30 min in an oven and the filtered solutionswas spin-coated for 1 min at 2000 rpm to prepare solidthin-films of NPLs.

TEM and Optical Characterization

High-resolution transmission electron microscopy(TEM) images of NPL samples were taken on a JEOLJEM-2010 HR TEM operating at 200 kV, which has aresolution of ∼0.22 nm. The steady state PL spectrameasurements were done using a cooled iDus CCD sys-tem. The time resolved PL measurements were done witha Becker & Hickl DCS 120 confocal scanning FLIM sys-tem with an excitation laser of 375 nm. The system hastemporal resolution of 200 ps.

ACKNOWLEDGMENTS

We acknowledge the Ministry of Education, Singapore(RG 182/14 and RG 86/13), A*STAR (1220703063),Economic Development Board (NRF2013SAS-SRP001-019) and Asian Office of Aerospace Research and Devel-opment (FA2386-17-1-0039).

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