Probing alloy formation using different excitonic species: The particular case of InGaN
PProbing alloy formation using different excitonic species:The particular case of InGaN
G. Callsen, ∗ R. Butt´e, and N. Grandjean
Institute of Physics, ´Ecole Polytechnique F´ed´erale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland (Dated: November 7, 2018)Since the early 1960s, alloys are commonly grouped into two classes, featuring bound states inthe bandgap (I) or additional, non-discrete, band states (II). As a consequence, one can observeeither a rich and informative zoo of excitons bound to isoelectronic impurities (I), or the typicalbandedge emission of a semiconductor that shifts and broadens with rising isoelectronic dopingconcentration (II). Microscopic material parameters for class I alloys can directly be extracted fromphotoluminescence (PL) spectra, whereas any conclusions drawn for class II alloys usually remainindirect and limited to macroscopic assertions. Nonetheless, here, we present a comprehensivespectroscopic study on exciton localization in a so-called mixed crystal alloy (class II) that allows usto access microscopic alloy parameters. In order to exemplify our experimental approach we studybulk In x Ga − x N epilayers at the onset of alloy formation (0 ≤ x ≤ x -dependent linewidthanalysis we extract the length scales at which excitons become increasingly localized, meaning thatthey convert from a free to a bound particle upon alloy formation. Already at x = 2 .
4% the averageexciton diffusion length is reduced to 5 . ± . ≈ µ eV). Consequently, the present spectroscopic study allows us to extract firstmicroscopic alloy properties formerly only accessible for class I alloys. I. INTRODUCTION
Upon alloying, host atoms in insulators, semiconduc-tors, or metals are replaced by atoms with an equivalentvalence electron structure, giving rise to isoelectronic im-purities. Despite the matching valency, which does notlead to n - or p -type doping in case of a semiconductorcrystal, the total number of electrons is altered, induc-ing fundamental changes in the electronic band structureand the associated optical signatures of alloying [1]. Al-ready in 1966, Thomas pointed out [2] that such isoelec-tronic impurities can be divided into two classes: Forthe first class (I), discrete electronic levels are formed inthe bandgap that can be studied by the related emissionof bound excitons in, e.g., GaP:N [3, 4], GaP:Bi [5, 6],GaAs:N [7], ZnTe:O [8], CdS:Te [9], and ZnO:Hg [10].However, such highly localized bound excitons causedby isoelectronic centers strongly differ from the widerange of bound excitons known for shallow impuritiesin semiconductors and cannot be described by an effec-tive mass approach [11–13]. The second class of isoelec- ∗ gordon.callsen@epfl.ch tronic centers (II) evokes the formation of mixed crys-tal alloys like, e.g., SiGe [14], GaAsP [15], InGaAs [16],AlGaAs [17], InGaN [18], AlGaN [19], CdSSe [20, 21],ZnSeTe [21], and MgZnO [22]. In these cases, no newelectronic levels are formed in the bandgap, but ratherin the bands themselves, a process often described ashybridization [23]. Clearly, such a simple classificationis not always straightforward [24] and even a continu-ous, concentration-dependent transition between thesetwo classes of isoelectronic centers has been reported forthe unique case of silver halides [25].The consequences regarding the existence of these twotypes of isoelectronic centers in nature for any optical ma-terial characterization are pivotal: For class I alloys theapparent rich optical signature [26] allows to directly ex-tract microscopic material parameters from macroscopicphotoluminescence (PL) spectra. Not only the numberof isoelectronic impurities per binding center can be ex-tracted [6], but even the distance in between these impu-rities or their constellation can be determined [4, 27, 28].In contrast, for class II alloys, the spectroscopic analy-sis is hampered. Commonly, only a continuous shift ofthe bandedge luminescence is observable, which is ad-ditionally plagued by pronounced linewidth broadening a r X i v : . [ c ond - m a t . m e s - h a ll ] N ov [16, 17], preventing any detailed information to be ex-tracted from PL spectra. Consequently, in contrast toclass I alloys, only a rather indirect analysis of such mixedcrystal alloys is feasible by optical techniques. What fur-ther worsens the situation is the fact that isoelectronicimpurities related to class II are more abundant in naturein comparison to their type I counterparts [1]. Further-more, most technologically relevant alloys (e.g., InGaAs,AlGaAs [29], AlInGaP [30], InGaN, InAlN [31], and Al-GaN [32]) belong to class II. Here, especially III-nitridealloys have recently proven their importance for a widerange of applications, rendering a spectroscopic analysisbeyond the limits of their classification as mixed crystalalloys an auspicious task.Over the past decade, III-nitride-based semiconductorshave reached a high level of dissemination only second tosilicon based on a wide range of applications coveringboth, electronics and optoelectronics [33]. Many aspectsof our daily life are already impacted by III-nitride-basedlight-emitting and laser diodes (LEDs and LDs) [34, 35],while high-power electronics is also on the rise [36, 37].Here, ternary alloys like InGaN and InAlN play a cru-cial role, as essential properties like, e.g., their emissionwavelength can be tuned over wide ranges [38] in orderto suit the particular application of choice. Such alloyscontaining indium are often described as ”special alloys”,because many devices like LEDs based on InGaN/GaNquantum wells (QWs) perform astonishingly well, withinternal quantum efficiencies η int well beyond 80% [39],even though the threading dislocation density is highwith values in the 10 − cm − range [40]. Generally,the material proves particularly robust against structuraland point defects in comparison to other III-V alloys (e.g.InGaAs, AlGaAs, AlInGaP, etc.) [30] as outlined in theseminal paper of Chichibu et al. [31]. However, the phys-ical origin behind this behavior is not well understood yetand several possible causes have been discussed in the lit-erature.Early explanations accounting for the particular caseof InGaN range from large structural defects like V-pits [41] and indium clusters [42], over indium-zig-zagchains [31, 43], to the particular electronic environmentof a single indium atom [23]. All explanations sharethe common idea that localization of carriers occurs inthe alloy, ultimately leading to higher η int values. Assoon as carriers are localized, their diffusion to non-radiative centers is suppressed, which in turn enhances η int . While larger structural defects can nowadays beexcluded for modern III-nitride materials based on acombination of structural [scanning transmission electronmicroscopy, nanoscale secondary ion mass spectroscopy,etc.] and optical [cathodoluminescence (CL), micro-photoluminescence ( µ -PL), etc.] techniques, the materialanalysis at the few to even sub-nanometer scale remainschallenging. Even though recent technical progress madeby atom probe tomography (APT) indicates that InGaNis a random alloy [44–46], the impact of assemblies ofindium atoms like pairs, triplets, and larger sets on the material’s optical signature remains unclear.In this work, we show that even for class II alloysit is possible to extract microscopic material parame-ters from PL and µ -PL spectra. A detailed PL anal-ysis of bulk In x Ga − x N epilayers (0 ≤ x ≤ ∼ cm − ) allows us to extract the effective exci-ton diffusion length ( r a ) and its dependence on indiumcontent ( x ). By analyzing the linewidth broadening offree and bound excitons, one obtains probes that moni-tor the alloy formation on different length scales deter-mined by r a and the exciton Bohr radius r B , hence, inour case encompassing length scales ranging from tensto a few nanometers. With rising x we measure a de-crease of r a towards r B using the free exciton (X A ) asa probe for the formation of a random alloy in whichthe excitonic center of mass (COM) movement becomesincreasingly negligible at cryogenic temperatures. In ad-dition, the exciton-phonon coupling is studied, e.g., forsilicon-bound excitons (Si X A ), representing an alterna-tive tool to monitor the onset of alloy formation. Thespectroscopy of excitons captured at individual indium-related centers proves challenging due to an apparenthigh density of emitters. Hence, we focus on an anal-ysis of µ -PL spectra of individual silicon-bound excitonsembedded in particular configurations of indium atoms intheir direct vicinity. As a result, we observe a hierarchyof energetically well-defined emission lines [full width athalf maximum (FWHM) ≈ µ eV] that originate fromSi X A − In n complexes in the InGaN alloy. Hence, ourstudy opens a pathway towards a spectroscopic analysisof the microscopic properties of a class II alloy at thenanometer scale by probing distinct configurations of in-dium atoms at the onset of alloy formation.The paper is structured as follows: In Sec. II we firstcompare the fundamental excitonic properties of a widerange of materials, before focusing the motivation to III-V semiconductors represented by indium doped GaAsand GaN as representatives of class II alloys. We presentour results in Sec. III. This section is subdivided into fourparts labeled Secs. III A-III D. In Sec. III A we show howensembles of free and bound excitons can serve as probesin the InGaN alloy using PL spectra. Temperature-dependent PL data allows us to analyze the local dis-tribution of indium atoms around particular impuritybound excitons as demonstrated in Sec. III B. Subse-quently, Sec. III C introduces the µ -PL traces of individ-ual excitonic complexes bearing on an impurity embed-ded in a distinct environment of indium atoms. The spec-tra of such individual bound excitonic complexes are fur-ther analyzed in Sec. III D, motivating the existence ofspatially direct and indirect excitonic complexes in theInGaN alloy. In a nutshell, Secs. III A-III D describe thefollowing transition: first, ensembles of excitonic com-plexes are probed in the alloy by PL. Second, the pre-sented alloy probes become increasingly localized, cul-minating in the µ -PL observation of individual impuritybound excitonic complexes embedded in a distinct con- P b T eI n S bG eS i
G a A s
I n PC d T e Z n S e
G a N
C u - I - O S n O _ 2G a S bI n A s B N B e OC d S e C u B rG a P Z n SI n N A lNZ n O M g O s ilic aP b S e C d O C ( d ia m o n d ) r X » (cid:215) c m - 3 Exciton Bohr radius rB (nm)
B a n d g a p E g ( e V ) r X » (cid:215) c m - 3 T £ > 5 0 Exciton binding energy (meV)
051 01 52 02 53 03 54 04 55 0
C u O Density r X (cm-3) S n O I n d i u m c o n c e n t r a t i o n x ( % ) n = 0 1 3 5 7 n ‡
1 n = 2 t o 1 0
G a N : I n ( c )( b )
Probability j ( a ) G a A s : I n
I n d i u m c o n c e n t r a t i o n x ( % ) n = 0 1 3 5 7 n ‡
1 n = 2 t o 1 0
FIG. 1. (color online) (a) Hydrogen-like exciton Bohr ra-dius r B (excitonic ground state) for various material sys-tems comprising semiconductors and insulators that ex-hibit a bandgap energy E g in between ≈ r B and increasing E g the exciton binding en-ergy E bind rises (color encoded) - a trend which is also validfor all related alloys. The inverse of the exciton volume V X ( r B ) motivates a density ρ X , which allows for a first es-timate of the excitons’ sensitivity to point and structuraldefects. To illustrate this matter, we compare isoelectron-ically doped (b) GaAs:In and (c) GaN:In, whose r B valuesdiffer by a factor of ∼
4. The indium concentration x re-quired to find a certain number of indium atoms n in V X with the probability ϕ is given by the Bernoulli distribu-tion and varies by one order of magnitude in this com-parison. While bound excitons average over ≈ V X , freeexcitons monitor the alloy over a larger averaging volume V a due to their eponymous movement. figuration of indium atoms. Sec. IV finally relates ourPL and µ -PL results before Sec. V summarizes all find-ings. Experimental details regarding the spectroscopicand growth techniques can be found in the Supplemen-tary Information (SI) in Sec. I. II. CONTEXT AND MOTIVATION
In order to obtain a general understanding of the sen-sitivity of excitons in alloys to crystal defects, Fig. 1(a)introduces the reduction of r B (excitonic ground state)with rising bandgap energy E g for a large variety of mate-rials (direct as well as indirect semiconductors and insula-tors). The higher the ionicity of the crystal - following thetransition from IV-IV, over III-V, to II-VI semiconduc-tors - the larger are the effective masses of the electrons( m e ) and holes ( m h ) that move in the periodic poten-tial of the lattice. In turn, the excitonic effective mass µ increases accordingly with E g like 1 /µ = 1 /m e + 1 /m h ,which is inversely proportional to r B [47]. Such ”heavy”excitons with small r B values ultimately lead to the tran-sition from Wannier-Mott type excitons found in mostsemiconductors to self-trapped excitons frequently ob-served in silica [48] and halides [49]. As soon as r B isreduced, the exciton binding energy E bind rises as theCoulomb attraction between the electron and the holeis enhanced. Hence, E bind is proportional to µ as en-coded in the data-point colors of Fig. 1(a), leading to theexperimental stability and the herewith interconnectedaccessibility of excitons beyond just cryogenic tempera-tures for large bandgap materials like, e.g., GaN [50] andAlN [51].Already these basic considerations facilitate some con-clusions that should be valid for any alloy of class I or IIformed by the materials summarized in Fig. 1(a). In al-loys with low E g values an exciton averages at least overcomparably large exciton volumes V X ∝ r B . Hence, cor-respondingly low densities of, e.g., impurities and struc-tural defects ρ X ∝ V − X suffice in order to affect or even trap an exciton if the associated perturbation is suffi-ciently large. In this sense, ρ X represents a tentativeupper limit for the concentration of such centers affect-ing all excitons. In contrast, high E g materials with lower V X values should be less sensitive to higher defect con-centrations based on this simplistic comparison. In orderto illustrate this matter, we highlight the corresponding ρ X values for GaAs and GaN in Fig. 1(a), providing afirst sensitivity estimate for excitons occurring in relatedalloys. As the r B value of both materials varies by a fac-tor of ∼
4, the corresponding approximations of ρ X differby almost two orders of magnitude. However, this com-parison is complicated by the occurrence of free, bound(isovalent), and impurity bound (non-isovalent) excitonsconstituting two- or three-particle complexes. One caneither observe free excitons that move and hence moni-tor the alloy over an averaging volume V a ≥ V X leadingto lower critical ρ X values, or localized, shallow impu-rity bound excitons sensing the material within ≈ V X (see Sec. III A for a detailed analysis). Nevertheless, thelower r B for large E g materials, the less excitons averageover a random alloy and the more they just monitor theimmediate neighborhood of a distribution of, e.g., (non-) isovalent centers in the lattice. This simple picture isfurther supported by the common neglect of any COMmovement for excitons in an alloy [52] - a simplificationthat will be discussed in detail in Sec. III A for InGaN.We further motivate the present study by focussing ourprevious considerations to indium-doped GaAs:In (e.g.Ref. [16]) and GaN:In (this work) as representatives forthe extremal cases of relatively low and high r B valuesin III-V semiconductor alloys of class II. Based on thischoice, the associated E bind values would always remainsufficiently high to ensure a straightforward PL analysis.The reasoning that further motivates this approach isthreefold:A) High quality material is available for InGaN andInGaAs alloys for studying the onset of alloy formationfor class II alloys due to the availability of high qualitysubstrates for the epitaxial growth of thick, ternary layers (eV) Si0XA
PL intensity (arb. units)
E n e r g y ( e V )
XA Si0XA XAXB L1 T = 1 2 K ,P = 1 2 . 7 W / c m l e x c = 3 2 5 n mE n e r g y ( e V ) G a N ( n i d )
Si0XA XA XBSi0XA XA XB
I n d i u m c o n t e n t x ( % ) ~ 3 9 . 6 ± 1 . 5 m e V / % ( a ) ( b )
XA energy
FIG. 2. (color online) Overview PL spectraof the In x Ga − x N sample series (0 ≤ x ≤ X A ) and the A-exciton (X A ) can befollowed up to an indium content of x = 2.4%in (a) and (b). For the X A transition a shiftrate of 39.6 ± ≤ x ≤ B ) canbe observed at the very onset of the compositionrange. At x = 2.4% a simple fit routine basedon two Voigt profiles highlights the two mainemission peaks (illustrative purpose only). Aninversion of the FWHM values related to Si X A and X A is directly visible by, e.g., comparingthe spectra for x = 0 .
01% and 0.37% in (b). ( ≥
100 nm) as required for spectroscopy.B) It is known that InGaAs- and AlGaAs-based LEDsare rather sensitive to the density of point and structuraldefects, whereas InGaN proves to be much more robust[30, 31] against them.C) It was pointed out theoretically that individual in-dium atoms in GaAs and GaN do not lead to any boundstates in the bandgap [23], a matter that would ratherrequire several indium atoms to form larger indium con-figurations [43].In other words, both materials are unambiguous repre-sentatives of class II alloys with relatively simple spectrathat facilitate a detailed tracking of the alloy formation.For class I alloys known bound states are mostly asso-ciated to single atoms or (extended) pairs and can beobserved for a wide range of materials as summarized inSec. I. The absence of such optical signatures will allow usto study the very onset of alloy formation in In x Ga − x N(0 ≤ x ≤ ϕ to find n indium atoms in V X for a certain indium concentration x is given by theBernoulli distribution in a random alloy: ϕ ( n ) = (cid:18) KV X n (cid:19) x n (1 − x ) KV X − n . (1)Here, KV X is the cation number in the excitonic vol-ume with K ≈ − for wurtzite GaN in the in-dium composition range analyzed in this paper [18, 53].An illustration for the probability ϕ to find, e.g., 0 ≤ n ≤ V X is given for GaN and GaAs ( K ≈ − ) [17] in Figs. 1(b) and 1(c), respectively. Theindium concentration x for which one can find, e.g., oneindium atom in V X with ϕ = 0 . × larger r B value of GaAs in comparison to GaN. This fact will becrucial for the following PL linewidths analysis of the In-GaN alloy, providing insight into the material’s particularrobustness against point and structural defects [31]. III. RESULTS
Figure 2 introduces the low temperature (12 K) PLspectra of the bulk In x Ga − x N sample series at handwith an indium content ranging from 0 to 2.4% (seeSec. I in the SI for experimental details). The spec-trum of the non-intentionally doped (nid) GaN referencesample shows the common optical traces of the A- andB-exciton (X A and X B ) along with a dominant neutraldonor bound exciton line (Si X A ). The energetic split-ting in between X A and Si X A is known as the localiza-tion energy E loc ≈ A , which will always be described by E loc in the following. Previous results have shown that thedominant impurity giving rise to the main bound exci-tonic emission is the neutral silicon center Si , while othertypical trace impurities in GaN like oxygen are negligiblein our samples [54]. In addition, towards lower energiesone observes the L1 emission line in Fig. 2(b), whose ori-gin is still under debate in the literature, ranging from adeep, over an ionized donor-bound exciton, to a neutral,shallow acceptor bound exciton [55–58].Upon increasing indium content, the entire set of emis-sion lines shifts continuously towards lower energies,while the level of detail in the spectra diminishes due tolinewidth broadening as commonly observed for a class IIalloy. The spectral shift of the X A transition as a functionof the change in bandgap energy δE g with rising x is lin-ear in the given indium concentration range and amountsto δE g /δx = 39 . ± . / % at a temperature of 12 Kas shown in the inset of Fig. 2(a). Clear evidence for theemission line L1 is lost at indium concentrations exceed-ing 0.05%, before even the spectral separation betweenX A and Si X A vanishes. Nevertheless, even at indiumconcentrations of 1.5% and 2.4% we can still reveal thepresence of the Si X A centers in the corresponding spec-tra by a simple lineshape fitting procedure employing twoVoigt profiles as exemplified for x = 2 .
4% in Fig. 2(a). Wefind the Voigt profiles to be dominated by their Gaussiancontribution due to the comparable small influence of ho-mogeneous broadening at a temperature of 12 K. Notethat the Voigt profiles from Fig. 2(a) only have an illus-trative purpose, highlighting the two different spectralcomponents. In the following all reported FWHM val-ues ∆ E originate from a manual data reading in orderto avoid a troublesome and overparametrized lineshapefitting. We will subsequently analyze the temperaturedependence of the linewidths in the context of Fig. 4 inSec. III B.At first glance, the present alloying series of GaN:Inseems to match the case of GaAs:In as presented by Lau-renti et al. [16] for indium concentrations down to 0.03%in GaAs. A continuous shift and a continuous broaden-ing of all optical transitions are observed in GaN:In downto x = 0 .
01% as expected for a class II alloy. Not evenat the very onset of the alloying range one can observeany additional emission lines related to excitons bound tosingle isoelectronic indium centers in agreement with theaforementioned theoretical predictions [23]. Note that anindium concentration of x = 0 .
01% yields a probabilityof ϕ ≈ .
35 for a single indium atom to be present in theexcitonic volume V X as shown in Fig. 1(c). Hence, the in-dium concentration should be sufficiently low to observethe occurrence of any additional bound states induced bysingle indium atoms.However, a more detailed inspection of PL and µ -PLspectra will result in a manifold of interesting and so farunforeseen observations for the InGaN alloy as shown inSecs. III A-III D. The latter mark the transition from ananalysis dealing with ensembles of excitons down to in-dividual ones giving access to the microscopic propertiesof a class II alloy. A. Probing alloying with free and bound excitonsbased on macro-photoluminescence
An in-depth analysis of the emission linewidth of X A and Si X A reveals an intriguing feature for GaN:In thatis fostered by its large E bind values and, in comparison toGaAs:In, its smaller r B values. At a temperature of 12 Kand x = 0 .
01% the Si X A transition exhibits a FWHMvalue of ∆ E Si X A = 1 . ± .
06 meV while the FWHMof X A is larger with ∆ E X A = 1 . ± .
19 meV due to the distribution of free excitons in momentum ( k ) space. In-terestingly, with rising indium content this FWHM ratiois reversed as seen, e.g., in the PL spectrum recordedfor x = 0 . E X A = 2 . ± .
04 meV, while ∆ E Si X A has more than tripled to 4.47 ± E values for the X A and Si X A excitonic complexes is sum-marized in Fig. 3(a) (black and red symbols). Both setsof FWHM values show a different evolution as averagingover the alloy occurs for different effective volumes. All∆ E values are commonly related to the peak width w by∆ E = 2 √ w . Generally, such w values are composedof an inhomogeneous component due to alloy broaden-ing, σ ( x ), and a homogeneous, temperature-dependentcomponent Γ( T ), caused by phonon scattering, leadingin first approximation to w = σ + Γ [22, 59]. At lowtemperatures (12 K) σ dominates any phononic effectsgiving rise to the experimental trends shown in Fig. 3(a)for the directly measured ∆ E values. Additional broad-ening by Γ( T ) will subsequently be discussed for Si X A inSec. III B in the context of Fig. 4, while the correspond-ing analysis for X A is given in SI Sec. III. Such predomi-nance of inhomogeneous broadening due to alloy disorderat cryogenic temperatures was also observed for otherternary alloys of class II such as MgZnO [22], AlGaAs[17], CdZnTe [60], and CdSSe [20].Based on Eq. 1, the standard deviation of the bandgapenergy σ E g can be expressed by the binomial distribution[17, 20]: σ E g = δE g /δx s(cid:18) x (1 − x ) KV X (cid:19) . (2)Hence, for a two-particle system like X A , σ E g shouldequal the peak width w determined via ∆ E , while δE g /δx is deduced from Fig. 2(a). As a first approach, V X can be replaced by the standard spherical exciton vol-ume ( V sX = 4 / πr B ) - a method that commonly yieldsa good agreement with experimental data [22]. How-ever, the quantum mechanical nature of the exciton iscompletely neglected by this approach, as one assumes aconstant occupation probability density for the excitonover V X . Instead, the quantum-mechanical (qm) excitonvolume V qmX = 10 πr B should rather be considered in or-der to take the wavefunction of the exciton into accountas outlined in Refs. [20] and [52].Figure 3(a) shows that both volumes ( V sX and V qmX ) ei-ther lead to a complete over- (dashed red line) or underes-timation (dotted red line) of the experimentally observedtrend for Si X A as the effect of the donor electron in thisthree-particle complex is not yet considered. The X A complex as well as the donor electron sense the alloy for-mation, leading to energy fluctuations that are normallydistributed. Hence, the square of the entire peak width ofSi X A amounts to: w X A = [ w Si ( V D )] + [ w X A ( V qm X A )] .Here, one obtains V D = 10 πr D based on Ref. [60] with T = 1 2 K e x c i t o n B o h r r a d i u s i n G a N x c a p = 8 .7 – x c a p = 5 0 % ra of the XA averaging volume Va (nm) I n d i u m c o n t e n t x ( % ) ( b )
T = 1 2 K
S i X A ( V sX - b a s e d ) S i X A ( V q mX - b a s e d ) S i X A - ( [ V q mX + V D ] - b a s e d ) X A ( V q mX - b a s e d ) X A ( V q mX a n d V a - b a s e d ) f i t ( r a = 1 0 –
1 n m ) E x p . S i X A v a l u e s E x p . X A v a l u e s FWHM values D E (meV)
I n d i u m c o n t e n t x ( % ) k B T »
1 m e V @ T = 1 2 K ( a ) t o w a r d s ar a n d o m a l l o y
FIG. 3. (color online) (a) Experimental FWHM values ∆ E of the Si X A (red symbols) and X A (black symbols ) tran-sitions versus indium content x extracted from PL spectra.The red lines introduce step-by-step the modeling of theSi X A linewidth depending on the particular excitonic volume V X . Here, V sX denotes a spherical and V qmX the quantum-mechanical exciton volume, while V D is the volume ascribedto the donor electron. An exciton averaging volume ( V a ) hasto be introduced to model the ∆ E values of the X A com-plex (solid black line) in order to take into account its move-ment. All graphs are offset by the corresponding experimentalFWHM value at x = 0. (b) The values of r a are extractedfrom V a and converge towards the exciton Bohr radius be-cause X A gets increasingly localized with rising x in contrastto Si X A as illustrated in the inset. Hence, a random alloyis formed in which the excitonic center of mass movement isincreasingly suppressed. See the main text for details. the donor Bohr radius r D that is derived from the silicondonor binding energy of 28.5 meV [54] calculated withinthe framework of the hydrogenic model [61]. This con-sideration of Si X A as a three-particle complex leads toa reasonable agreement between experiment and theory[Fig. 3(a), solid red line].In contrast, for the two-particle complex X A the con-sideration of V qmX (dotted black line) still yields anoverestimation for most of the experimentally observed ∆ E X A values as the COM movement of the exciton wouldneed to be considered at the onset of the alloy formation.However, even the sophisticated theoretical treatment foralloys of Zimmermann given in Ref. [52] deems the COMmovement of the exciton a task for future work, whichhas - to the best of our knowledge - never been accom-plished. As a direct consequence of the excitons’ COMmotion, the entire set of X A complexes averages over alarger fraction of the alloy, hence, we suggest to consideran effective, spherical averaging volume V a = a V qmX withscaling parameter a as a measure for the X A COM move-ment. A fit to the experimental data from Fig. 3(a) for∆ E X A yields a ≈ x . This indicates that V a scales with x as describedin the following.For X A , Eq. 2 can be solved for r a using V X = V qmX = V a ( r a ) /a , yielding the indium content dependent averag-ing radius r a ( x ) of the corresponding volume as shownin Fig. 3(b) (black circles). The value of r a is continu-ously decreasing with rising indium content as X A com-plexes get increasingly localized by indium-related trap-ping sites. Ultimately, r a approaches r B as the free ex-citon transforms into a bound two-particle complex asshown in Fig. 3(b) and illustrated in the correspondinginset. In contrast, a D X A complex like Si X A alwaysremains a bound complex at the onset of alloy formationand should, as a first approximation, maintain a con-stant averaging volume. At the present onset of alloyformation, we do not expect any strong influence of x on r B , which is taken in Fig. 3(b) as that of pure GaN.Singlets, doublets, and with rising x even more extendedcomplexes of indium atoms [see Fig. 1(c)] should lead tosuch increasing localization of X A with rising x . In thisrespect, the reported r a ( x ) values represent a measure ofthe X A diffusion length. According to Bellaiche et al. [23]the hole as a building block of the X A complex can getlocalized even in the case of a single indium atom. How-ever, the corresponding energy level is still hybridizedwith the valence band states of GaN, meaning that noadditional states appear in the bandgap at the onset ofalloy formation in agreement with our observations fromFig. 2 (no new emission lines appear with rising x ) andour expectations for a class II alloy. This particular caseis fundamentally different from that of conventional ex-citons bound to isoelectronic centers (cf. examples givenin Sec. II) mostly situated on anionic lattice sites [26],whose strong localization leads to the appearance of newenergy levels in the bandgap.Based on a simple, static approximation, the proba-bility to capture X A at such indium-related sites canbe approximated by ξ cap ∝ ˜ r/r a with the average dis-tance in between the indium atoms in the dilute alloy˜ r = ( Kx ) − / . This approximation relies on the com-mon assumption made for most III-V [62] and II-VI[63] semiconductors that the exciton capture time τ cap isshort compared to the radiative decay time, τ cap < τ rad .Consequently, based on the spherical averaging volume V a ( r a ) one straightforwardly finds the following rela-tion to fit the data from Fig. 3(b): r a ∝ ξ − cap ( Kx ) − / .Clearly, only for ξ cap (cid:28) V a does not seem to provide a suffi-ciently deep potential in order to permanently capturethe X A complex at the given temperature [23]. This ob-servation represents the main result of the present simplefitting model. For instance, already for ξ cap = 50% thevalues for r a would rapidly diminish with rising x . Thisevolution is shown in Fig. 3(b) (solid grey line) and con-stitutes the reason behind the common neglect of the ex-citons’ COM movement in alloys [52], which is not highlydiluted, or in other words, just doped.The fit from Fig. 3(b) yields ξ cap ≈ . ± .
2% (solidred line), suggesting that indium-related complexes com-prising ≈
10 indium atoms predominantly contribute tothe capture of X A in the InGaN alloy at 12 K. Clearly, theoccurrence of such complexes becomes more likely withrising x , explaining the reduction of r a towards the r B value of GaN in Fig. 3(b), while V a also approaches V qmX .Hence, around 2.4% of indium are required to form anInGaN alloy for which the neglect of the excitons’ COMmovement is justified at cryogenic temperatures. There-fore, the difference in between the data points for ∆ E X A and the corresponding fit in Fig. 3(a) (solid black line)starting from x ≈ .
36% is likely caused by the increas-ing localization of X A , i.e., the transition from a free toa bound two-particle complex in a class II alloy.This transitional regime of increasing excitonic local-ization at 12 K by indium-related complexes is challeng-ing from a theoretical point of view as exciton diffusionneeds to be considered. In this context, the presentsimplistic derivation of ξ cap just represents a pragmaticapproach that seems well suited, given the error andthe scatter of the underlying data points, cf. Fig. 3(b).Furthermore, we plausibly assume that our findings arerelated to the indium zig-zag chains [43] described byChichibu et al. [31], however, the precise sub-structureat the atomic scale still remains of speculative nature.In this regard, Sec. III C and III D will describe a path-way towards an analysis of the particular configurationof indium atoms by spectroscopic means at the onset ofalloy formation. In the following we will always refer tosuch dilute indium assemblies, because larger, non-diluteindium aggregates should lead to the formation of elec-tronic states in the bandgap [43] [i.e., quantum dot (QD)like states].We suggest that the apparent localization of X A com-plexes with rising x enhances the robustness of the In-GaN alloy against point and structural defects in con-trast to other III-V alloys as outlined by Chichibu et al. in Ref. [64]. Such a picture also accounts for the inten-sity increase of X A relative to Si X A as shown in Fig. 2.As exciton diffusion is inhibited with rising x , the prob-ability to reach non-radiative exciton trapping sites isreduced. In this respect, we wish also to note that the overall intensity of the bandedge luminescence shown inFig. 2 continuously increases with rising indium contentat the onset of alloy formation (not shown) as frequentlyreported in the literature [65–68]. Naturally, with rising x the trapping potential becomes deeper as larger indiumassemblies are formed that ultimately even govern the k -distribution of X A at a temperature of 300 K with local-ization energies already in excess of k B T at x ≈ x Ga − x Nepilayers with rising x . However, the data presented inFig. 3(b) is of a more general nature as the trapping ofexcitons is monitored over a large PL excitation area (ex-citation spot diameter of ≈ µ m) and does not dependon the specific strain-driven impurity distribution arounda structural defect [70]. Obviously, such increasing exci-ton localization with rising x is less beneficial for alloyslike In x Ga − x As as already much larger r B values prevailin the related binary compounds as indicated in Fig. 1(a). B. Analysis of the local indium distribution aroundimpurity bound excitons
The transition from a free to a bound excitonic two-particle complex summarized in Fig. 3(b) represents aspectroscopic probe, whose sensitivity to the local envi-ronment in the alloy increases with rising indium contentas r a converges to r B . In contrast, an impurity boundthree-particle complex always remains localized, givingrise to a permanently local alloy probe as illustrated inthe inset of Fig. 3(b).In the following, we will show that the impurity boundexciton complex Si X A also provides an extremely ver-satile tool to study the formation of dilute indium as-semblies in GaN:In. While the temperature-dependentlinewidth of X A does not show any clear indium contentdependent trend (see Fig. S3), the broadening of Si X A upon rising temperature increases with x as shown inFig. 4(a) - a quite unexpected finding in the light of pre-vious alloy studies [20]. In addition to the data shown inFig. 4, SI Secs. II and III provide an overview about theunderlying, temperature-dependent PL spectra and thecorresponding analysis of peak positions and linewidths.Generally, the phonon-induced linewidth broadeningΓ( T ) is given byΓ( T ) = γ ac T + X i =1 γ iopt exp( E iopt /k B T ) − , (3)with the acoustical ( γ ac ) and the optical ( γ iopt ) phononcoupling constants as well as the corresponding effec-tive optical phonon energies ( E iopt ) numbered by i . The FWHM value D E of Si0XA (meV)
T e m p e r a t u r e ( K ) g a c = 4 –
1 µ e V /K g ac = 17.9 – g ac = 18.3 – g ac = 20 –
01 02 03 04 05 0
Eff. phonon energy E1opt and coup. const. g I n d i u m c o n t e n t x ( % )
S i X A c o m p l e x ,1 2 . 7 W / c m @ 3 2 5 n m ( b )( a ) t o w a r d s ar a n d o m a l l o y FIG. 4. (color online) (a) Temperature dependence of thelinewidth broadening for the Si X A complex extracted fromPL spectra. Solid lines show fits to the data based on Eq. 3.The value of the fitting parameter γ ac is inscribed, whereasthe particular scaling behavior of the fitting parameters E opt and γ opt is shown in (b). Here, the values for E opt and γ opt seem to converge with rising x (dashed lines). The temper-ature dependence of the linewidth broadening for Si X A isstrongly affected by the indium content in the alloy x , incontrast to the case of X A (cf. Fig. S3). An optical phononpopulation with an effective energy E opt (weighted by thecorresponding density of states) is responsible for the partic-ular broadening of Si X A . The maximum energy of phononswithin this phonon population corresponds to the energy ofthe E low phonon in GaN (17 . ± .
05 meV). The inset of(a) sketches the E phonon dispersion relation in the firstBrillouin zone near the Γ-point, cf. Sec III B. best fit to the evolution of ∆ E ( T ) shown in Fig. 4(a)for Si X A at x = 0 (solid black line) is obtained for γ ac = 4 ± µ eV/K and one set of γ opt , E opt values assummarized in Fig. 4(b). Here, E opt = 12 . ± . E low phonon mode in GaNof 17 . ± .
05 meV [71]. This difference can be explainedby the curvature of the corresponding phonon dispersionrelation E ( k ) around k ≈ ∂ E/∂ k | k = k < X A probes an extended phononenergy interval due to the localization of the bound ex-citon in real space, providing access to a larger frac-tion of the first Brillouin zone near the Γ-point. Hence,the temperature-dependent PL data yields an effectivephonon energy E opt that is weighted by the correspond-ing one-phonon density of states [72–74].For Si X A the effective phonon energy E opt gradu-ally decreases from 12 . ± . . ± . ≤ x ≤ .
36% as depicted in Fig. 4(b). The correspond-ing γ ac fitting parameters denoted in Fig. 4(a) approachvalues equal to the ones observed for X A within the givenerror bars with rising x (see Fig. S3 and the correspond-ing discussion in SI Sec. III). The observed trend for E opt is likely originating from an alloying-induced local varia-tion of the Si environment. Callsen et al. have found forsingle excitons trapped in GaN/AlN QDs that, e.g., theexciton-LO-phonon interaction averages over a volumewith a radius on the order of r B [75]. Hence, alloyingin thick InGaN epilayers can not only directly be moni-tored by emission energy shifts (see Fig. 2) caused by theimmediate indium atom configuration in the vicinity ofSi impurities, which in total leads to the formation ofbound excitons. More indirectly , the interaction of boundexcitons with E low phonons can also be used as an alloyprobe as shown in Fig. 4.As Γ( T ) for X A does not show any pronounced in-dium content dependence (see Fig. S3), we suggest thatthe environment of Si centers is richer in indium atoms,a feature only noticeable at the very onset of alloy for-mation. Thus, here it can be expected that the phononenergies and coupling constants (e.g., E opt and γ opt ) willrapidly converge with rising x . Any local variation inthe indium content caused by the particular distribu-tion of Si atoms will get increasingly masked by ran-dom alloy fluctuations with rising x . As a signature ofthis masking, E opt ( x ) converges towards ≈ γ opt continuously rises with x , but exhibits an op-posite convergence behavior in Fig. 4(b) at the onset ofalloy formation. Indium atoms that surround the Si cen-ter within r B can act as strongly localized isoelectroniccenters, whose broad combined distribution in k -spaceseems to foster the interaction with phonons that devi-ate from the Brillouin zone center. Therefore, the afore-mentioned bowing of the associated phonon dispersionrelation ( ∂ E/∂ k | k = k <
0) leads to an effective reduc-tion of the measured E opt values that is accompanied bya rise in γ opt . Clearly, the total reduction in the effectivephonon energy E opt cannot exclusively be explained bya local rise in indium content as even pure InN still ex-hibits an E low phonon energy of around 10 . ± . V X could have been detected by APT in state-of-the-art InGaN/GaN QW samples [44, 45] commonlycomprising silicon concentrations ≈ × cm − (cf. SISec. I). Thus, as a possible reason for the local indium en-richment in the vicinity of Si centers, we suggest tensilestrain that is commonly introduced upon silicon dopingof GaN [77]. Hence, it is energetically more favorable fora rather large atom like indium to incorporate close toa silicon atom (distance on the order of r B ), in order toapproach the lattice’s strain equilibrium at the onset ofthe alloy formation. Bezyazychnaya et al. have theoreti-cally predicted the impact of point defects (vacancies) inInGaN and InGaAs on the indium distribution in thesealloys [78]. Here, we experimentally find - to a certaindegree - a similar situation for the Si impurity. Futuretheoretical work is needed to validate this picture of apoint defect embedded in a dilute assembly of indiumatoms. This image differs from the common concept ofdirect complex formation [79] that often just considersnearest- or second-nearest-neighbor sites on the cationicsub-lattice. C. Individual bound excitonic complexes analyzedby micro-photoluminescence
Fortunately, µ -PL measurements with an excitationspot diameter ≈ µ m can provide more detailed infor-mation regarding the particular configurations of indiumatoms close to a Si donor at the onset of alloy forma-tion. Even on completely unprocessed samples one canresolve individual emission lines around the Si X A tran-sition for x = 0 .
37% and partially for 1.36% as showin Fig. 5(a), while the X A emission remains a rather un-structured emission band due to the given probe volumeand excitation power density ( P = 40 W/cm ). This ob-servation is of high importance as the subsequently de-scribed processing of metal apertures on the sample couldlead to exciton localization by any damage to the sampleas discussed in SI Sec. IV.We further reduced the size of the probe volume byprocessing apertures into an aluminum film with diam-eters down to 200 nm on the sample with x = 0 . X A along with particu-lar phonon sidebands as shown in Fig. 5(b). An equallyspaced ( ≈
18 meV) double step appears on the low energyside of the set of sharp emission lines due to coupling with E low phonons. Interestingly, the optical signature of theexciton-phonon coupling does not appear as a commonlyobserved [62] isolated peak [see the LO-replica of X A inFig. 5(b)], but as a step due to the energetically broadrange of contributing phonons with E symmetry. About7 meV below the first step related to the zero phonon linesin the spectrum of Fig. 5(b) (marked by a blue arrow) oneeven observes an additional, step-like increase in intensityand hence phonon coupling strength in accordance withthe findings of Fig. 4(b) ( E opt = 6 . x = 0 . µ -PL spectrum shown in Fig. 5(b) representsa coarse probe of the phonon density of states that standsin direct relation to the homogeneous broadening Γ( T )of the Si X A complex introduced in Fig. 4(a).The sharp emission lines from Fig. 5(b) present the a p e r t u r e ˘ : » T = 4 . 5 K E lo w2 » X B PL intensity (arb. units)
E n e r g y ( e V ) X A S i X A - I n n c o m p l e x e s X A - L O ( 9 1 m e V )E lo w2 » - 2 0 - 1 5 - 1 0 - 5 0 5 1 0 1 50 . 00 . 20 . 40 . 60 . 81 . 01 . 2 X B S i X A - I n n c o m p l e x e s X A PL intensity (arb. units)
E n e r g y r e l a t i v e t o X A ( m e V ) T = 4 . 5 KP = 8 W / c m l e x c . = 3 2 5 n me x c i t a t i o n s p o t d i a m e t e r : »
1 µ m ( a )( b )
FIG. 5. (color online) (a) µ -PL spectra of two samples con-taining 0.37% and 1.36% of indium, respectively, show a struc-tured spectrum in the energy range of Si X A , while the emis-sion band related to X A remains unstructured (spectra areshown on a relative energy scale for better comparison). Bothspectra are typical for the entire, bare surface area of our sam-ples. The density of sharp emission lines noticeable aroundSi X A increases with rising x . (b) At an indium concentrationof x = 0 .
37% and a temperature of 4.5 K the X A and X B tran-sitions still appear as unstructured bands even if measuredunder a metal aperture with a diameter of 200 nm by meansof µ -PL. In contrast, the Si X A emission band consists ofsharp emission lines with FWHM values down to ≈ µ eV.At the given low excitation density ( ∼
40 W/cm ), severalphonon replicas are noticeable [ E low and longitudinal-optical(LO) phonon replica]. Here, the blue arrow indicates a partic-ular phonon energy E opt = 6 . x = 0.37%. The inset illustrates why X A exhibits an un-structured emission band, whereas the luminescence aroundSi X A [cf. Fig. 2(b)] shows several sharp emission lines relatedto individual complexes comprising the Si center and specificindium assemblies (In n ) with a certain atomic configuration c comprising n indium atoms. most direct evidence for an alloying-induced perturba-tion of the immediate Si environment at x = 0 . ± µ eVcan be resolved, opening the perspective for a spectro-scopic study of the Si environment at the nanometer0 - 1 5 - 1 0 - 5 0 50 . 0 00 . 2 50 . 5 00 . 7 51 . 0 01 . 2 5 S i X A - I n n c o m p l e x e s A 4A 4A 3A 2f ir s t o r d e r
PL intensity (arb. units)
E n e r g y r e l a t i v e t o X A ( m e V )g r o u p I : E lo c
5 - 1 0 m e V X A a p e r t u r e d i a m e t e r : 2 0 0 - 5 0 0 n m ,T = 4 . 5 K , 0 . 3 7 % i n d i u m , P » s e c o n d o r d e r A 1 - 2 0 - 1 5 - 1 0 - 5 0 5 0 . 00 . 51 . 01 . 52 . 02 . 53 . 0g r o u p I A 1 1A 1 2A 1 0A 9A 8A 7A 6A 5 a p e r t u r e d i a m e t e r : 2 0 0 - 5 0 0 n m ,T = 4 . 5 K , 0 . 3 7 % i n d i u m , P » g r o u p I I :E lo c (cid:2) PL intensity (arb. units)
E n e r g y r e l a t i v e t o X A ( m e V ) X A S i X A - I n n c o m p l e x e s ( b )( a ) FIG. 6. (color online) The sharp emission linesintroduced in Fig. 5 appear due to the forma-tion of dilute Si X A − In n assemblies and canbe divided in two major groups I and II basedon µ -PL spectra. (a) Emission lines of group Iexhibit localization energies E loc of ≈ E loc val-ues in excess of 10 meV and exhibit pronouncedspectral shifts for any variation of the excitationpower density. This difference in the peak shiftrates for group I and II emission lines is high-lighted in Fig. 7. Generally, the emission lines ofgroup I appear more frequently than their moredeeply localized group II counterparts. A1 - A12denote the particular metal apertures selectedfrom more than 50. scale based on the emission of such dilute Si X A − In n as-semblies. In contrast to Si X A , the X A emission remainsan unstructured band in our µ -PL spectra as the density ρ c X A of X A that attach to a specific indium atom con-figuration c with n indium atoms is comparably largerthan the corresponding density of Si X A centers ρ c Si X A with an identical configuration of indium atoms in theirvicinity.Generally, the number of any (transition from c → c )per time interval decaying, indium-related complex N c X A , Si X A is proportional to the µ -PL excitation vol-ume as well as pump power ( V exc and P exc ) and dependson the corresponding radiative decay time τ rad . How-ever, for the case of impurity bound excitons, as illus-trated in the inset of Fig. 5(b), the intersecting set of acertain distribution of exemplary indium configurations(e.g., D c ) and Si X A centers ( D Si X A ) defines the totalnumber of experimentally accessible complexes ( N ) via D c ∩ D Si X A → N c Si X A . However, in this simplified de-scription no varying distances are considered in betweenthese sites. Here, r B could represent a meaningful upperlimit for the distance in between a certain indium con-figuration and the Si center. Our experimental findingsfrom Sec. III D will further elucidate this point.Hence, only the Si X A emission band falls apart intoindividual emission lines in the µ -PL spectra of Fig. 5,while a similar observation for X A would require an evenlower excitation density or a smaller probe volume. Forinstance, the impact of individual indium atoms on theemission of Si X A could possibly be resolved for smallerindium concentrations (e.g., x = 0 . x = 0 , . , and 1 . E loc values . . X A and Si X A − In n centers at x = 0 .
01% cannot straightforwardly be resolved with thepresent emission linewidths in our samples ( ≈ µ eV).So far the impact of individual indium atoms can only be monitored by the localization onset of X A as demon-strated in Figs. 3(a) and 3(b) based on PL measurements. D. Statistical analysis of individual boundexcitonic complexes
The observation of sharp emission lines in Fig. 5 causedby dilute Si X A − In n assemblies directly evokes the needfor a statistical analysis of the underlying emitters. Fig-ure 6(a) shows selected µ -PL spectra recorded for fourdifferent metal apertures labeled A1-A4. Here, the twospectra for aperture A4 (measured using the first andsecond order of the optical grating) provide a detailedview of the µ -PL spectrum shown in Fig. 5(b). All spec-tra in Fig. 6(a) show a common optical signature of sharpemission lines related to dilute Si X A − In n assemblies inaddition to a rather unstructured emission band relatedto X A . For the latter case, a smaller spectroscopic probewould be required in order to resolve any spectral details.The localization energy E loc (energetic spacing betweenX A and the Si X A − In n complexes) commonly rangesin between ≈ −
10 meV. Figure 6(b) introduces eightadditional µ -PL spectra (A5-A12) that also show emis-sion lines with E loc &
10 meV that we assign to group II.Generally, the emission lines of group I occur more fre-quently than their group II counterparts. While almostall spectra in Fig. 6 show emission lines ≈ −
10 meVbelow X A (partially overlapping likely due to a high den-sity of the associated excitonic complexes), only a fewapertures show a clear signature of the emission lines ofgroup II. Here, apertures A5-A12 illustrate the rare casesof these group II emission lines that we extracted from µ -PL measurements on more than 50 apertures.The classification of the sharp emission lines into twogroups, I and II, becomes clearer based on Fig. 7, show-ing excitation power dependent µ -PL spectra for aper-ture A3. This excitation power series shows a sequenceof spectra that is typical for all apertures showing emis-sion lines of groups I and II. When varying the excita-1tion power density by a factor of ten, group I emissionlines only exhibit minor energetic shifts on the order of ≈ µ eV. In contrast, the emission line of group IIfrom Fig. 7 shifts by ≈ X A − In n complexes comprising aunique configuration of indium atoms within the Bohr ra-dius of the exciton r B . In this respect, the two electronsand the single hole of this complex occupy a well-definedregion and can therefore be referred to as spatially direct,neutral, complex bound excitons as sketched in Fig. 7 forgroup I. Here in Fig. 7 an exemplary dilute assembly of n indium atoms is sketched, along with the Si center andthe most relevant charge carriers.The pronounced redshift of the emission lines belong-ing to group II points towards a gradual change in theeffective Coulomb interaction in between the charge par-ticles that contribute to the overall Si X A − In n emis-sion. A spatially indirect transition as drawn in Fig. 7for group II could explain such a particular sensitivityto the excitation power as well as the larger E loc val-ues with respect to the emission lines of group I. Clearly,with rising x such spatially indirect transitions involv-ing In n complexes with increasing size and density be-come more probable. At first glance, the observation ofgroup II emitters resembles the common picture of exci-tons trapped in c -plane (In)GaN QDs that are embeddedin, e.g., Al x Ga − x N (0 ≤ x ≤
1) as the matrix material[81]. Charge fluctuations commonly occur in the matrixmaterial due to point defects, which in-turn leads to apronounced linewidth broadening known to be particu-larly strong in III-nitrides [82]. Upon changing the laserexcitation power, the occupation probability of chargesin the vicinity of such QDs is altered. As a result ofthis statistical process, a pronounced red-shift of singleQD emission lines can be observed with rising excitationpower [81, 83], which resembles the emission line shift ofthe group II emitter from Fig. 7. Hence, the pronouncedenergetic shift of group II emission lines upon varying ex-citation power could originate from a spectral diffusionphenomenon, which scales with the size of the excitonicdipole moment [82]. Clearly, indirect excitonic transi-tions as sketched in Fig. 7 exhibit a larger excitonic dipolemoment and are consequently more sensitive to chargefluctuations. Future work should analyze if any preferen-tial orientation exists for this excitonic dipole. However,currently, a more detailed analysis of these group II emit-ters is hindered by the comparably long integration timesfor spectra on the order of tens of minutes and the spec- - 2 0 - 1 5 - 1 0 - 5 0 50 . 0 00 . 2 50 . 5 00 . 7 51 . 0 0
A 3 a p e r t u r e d i a m e t e r : 5 0 0 n m ,T = 4 . 5 K , 0 . 3 7 % i n d i u m g r o u p I I :
PL intensity (arb. units)
E n e r g y ( m e V ) X A g r o u p I : FIG. 7. (color online) Excitation power dependent µ -PL mea-surements for the third aperture (A3) from Fig. 6(a). Emis-sion lines of group I only exhibit shifts on the order of 100 µ eVupon a tenfold increase in excitation power (see dashed grayarrow), while the distinct emission line of group II shifts by ≈ tral overlap with other emitters, preventing a detailedlinewidths and emission line jitter analysis. IV. DISCUSSION
More detailed µ -PL measurements need to be per-formed based on further optimized samples in order toreach a conclusive picture regarding the sharp emis-sion lines shown in Figs. 5 - 7. In addition, futuretemperature-dependent µ -PL measurements should pro-vide further insights. So far we only observed that theemission lines summarized in Fig. 6 rapidly broaden andthermalize (traceable up to ∼
30 K) due to the associ-ated exciton-phonon coupling involving the E low modeand acoustic phonons as described for Figs. 4 and 5 (notshown). The distinction made between emission linesof group I and II is mainly motivated by experimentsand helps to access the mechanisms of alloying. Gener-ally, a continuous transition can be expected in betweenthese two groups of emission lines, e.g., with rising x theprobability for spatially indirect transitions should firstincrease in the dilute alloy limit.The appearance of spatially direct and indirectSi X A − In n recombinations is likely interlinked with the x -dependence of the homogeneous linewidth broadeningΓ( T, x ) (see Eq. 3) shown in Fig. 4(a). Spatially moreindirect transitions exhibit larger excitonic dipole mo-ments that would enable a strong coupling to, e.g., po-lar phonons like LO-phonons via the Fr¨ohlich interaction[84]. However, the limited E loc values of Si X A − In n render the contribution of the exciton-LO-phonon cou-pling negligible in the present temperature range (due2to the large LO-phonon energies in nitrides [71, 76])and the coupling to acoustic and non-polar, e.g., E low phonons becomes most relevant as introduced in the con-text of Fig. 4. The associated exciton-phonon interac-tion is dominated by the deformation potential coupling,which does not depend on the excitonic dipole moment.In addition, a less prominent piezoelectric coupling willoccur [85]. Hence, we can expect Si X A − In n complexeswith large E loc values (group II) to be more temperature-stable [86] than their group I counterparts as long as adominant contribution of LO-phonons can be excluded.Therefore, with rising temperature the overall emissionband related to Si X A − In n should first become increas-ingly influenced by complexes with effectively larger E loc values (group II), while the more spatially direct recom-bination channels (group I) should preferentially dissoci-ate. However, already at 12 K the emitters of group IIseem to increasingly contribute to the linewidth broad-ening of Si X A with rising x . Thus, the offset in betweenthe associated experimental ∆ E values and the model(solid red line) increases with x as shown in Fig. 3(a).Finally, the evolution of the weighting in between spa-tially direct and indirect Si X A − In n centers is likely in-volved in the indium content dependence of Γ( T, x ) atthe onset of the alloy formation. Finally, we suggest atwofold reasoning for the particular evolution of Γ(
T, x ):At the onset of alloy formation one can observe a localindium enrichment in the vicinity of Si centers that af-fects the exciton-phonon coupling. In addition, spatiallydirect and indirect Si X A − In n transitions exhibit differ-ent thermalization behaviors, contributing to the partic-ular evolution of Γ( T, x ). V. CONCLUSIONS
In summary, we have demonstrated the detailedspectroscopic analysis of a III-V mixed crystal alloy byPL and µ -PL constituting a macro- and even microscopicmaterial characterization. Thus, the present work on aclass II alloy approaches the high level of spectroscopicsophistication known for class I alloys that relies onthe emission of excitons bound to isoelectronic centers.As no such strongly localized excitons appear in theinvestigated In x Ga − x N epilayers (0 ≤ x ≤ µ -PL we directly observed a hierarchy of bound excitonsrelated to dilute silicon-indium assemblies as individual,energetically sharp (FWHM ≈ µ eV) emission linesappear. Consequently, we introduced a classification ofthe underlying emitters into spatially direct and indirectbound excitonic complexes, whose balance is weightedby the indium-induced localization of charge carriers atthe very onset of alloy formation. However, not onlysuch µ -PL data, but even conventional macro PL spectraallowed us to extract crucial material parameters for the mixed crystal alloy at hand. Based on ensemblesof impurity bound excitons (three-particle complexes)we studied the indium-enriched environment of neutralsilicon donors in InGaN at the length scale of the excitonBohr radius. The analysis of the related exciton phononcoupling revealed a reduction of the average opticalphonon energy that governs the temperature-dependentemission line broadening from 12 . ± . . ± . x = 1 . X A we foundan alloying dependence for the homogeneous emissionline broadening caused by local indium enrichment andan indium-induced delocalization of the bound excitonyielding a transition from spatially direct to indirecttransitions excitonic complexes. Based on the lumines-cence traces of free excitons that become increasinglytrapped upon alloy formation, we extracted microscopicmaterial properties from conventional PL data. Wemotivated that upon increasing indium content x theradius of the excitonic averaging volume r a reducesgradually from 27 . ± . x = 0 .
01% down to5 . ± . x = 2 . η int values of the InGaN alloy even at defectdensities that are detrimental for other III-V binarysemiconductors and alloys like, e.g, (In)GaAs [31, 87],AlInGaP [30], and AlGaAs [17, 31]. Based on ourdetailed linewidth analysis we estimated that ≈ et al. [23]. Clearly, this justifiesthe categorization of InGaN as a class II alloy. Thepresent results open the perspective to utilize ensembles,but also individual excitonic complexes as a beneficialtools for any class II alloy analysis at the few nanometerscale. In this regard, our results are not limited tothe InGaN alloy. However, the constraints regarding asuitable doping interval are strict and explain - to thebest of our knowledge - the absence of correspondingdata in the literature for any other class II alloys. ACKNOWLEDGMENTS
This work is supported by the Marie Sk lodowska-Curieaction ”PhotoHeatEffect” (Grant No. 749565) within theEuropean Union’s Horizon 2020 research and innovationprogram. The authors wish to thank I. M. Rousseau forhis support with the experimental setup. Furthermore,we highly acknowledge D. Martin for growing the samplesand K. Shojiki for processing the metal apertures.3 [1] W. Czaja,
Festk¨orperprobleme XI, Advances in SolidState Physics , edited by O. Madelung (Pergamon,Vieweg, Marburg, Braunschweig, 1971) pp. 65–85.[2] D. G. Thomas, J. Phys. Soc. Japan , 265 (1966).[3] D. G. Thomas, J. J. Hopfield, and C. J. Frosch, PhysicalReview Letters , 857 (1965).[4] Y. Zhang, B. Fluegel, A. Mascarenhas, H. Xin, andC. Tu, Physical Review B , 4493 (2000).[5] R. A. Faulkner and P. J. Dean, Journal of Luminescence
1, 2 , 552 (1970).[6] S. Francoeur, S. Tixier, E. Young, T. Tiedje, and A. Mas-carenhas, Physical Review B , 085209 (2008).[7] D. Wolford, Journal of Luminescence , 863 (1979).[8] J. L. Merz, Physical Review , 961 (1968).[9] T. Fukushima and S. Shionoya, Japanese Journal of Ap-plied Physics , 813 (1976).[10] J. Cullen, K. Johnston, D. Dunker, E. McGlynn, D. R.Yakovlev, M. Bayer, and M. O. Henry, Journal of Ap-plied Physics , 193515 (2013).[11] R. A. Faulkner, Physical Review , 991 (1968).[12] J. C. Phillips, Physical Review Letters , 285 (1969).[13] J. W. Allen, Journal of Physics C: Solid State Physics ,1936 (1971).[14] R. Braunstein, A. R. Moore, and F. Herman, PhysicalReview , 695 (1958).[15] J. J. Tietjen and L. R. Weisberg, Applied Physics Letters , 261 (1965).[16] J. P. Laurenti, P. Roentgen, K. Wolter, K. Seibert,H. Kurz, and J. Camassel, Physical Review B , 4155(1988).[17] E. F. Schubert, E. O. G¨obel, Y. Horikoshi, K. Ploog, andH. J. Queisser, Physical Review B , 813 (1984).[18] R. Butt´e, L. Lahourcade, T. K. Uˇzdavinys, G. Callsen,M. Mensi, M. Glauser, G. Rossbach, D. Martin, J.-F.Carlin, S. Marcinkeviˇcius, and N. Grandjean, AppliedPhysics Letters , 032106 (2018).[19] M. C. Wagener, G. R. James, A. W. R. Leitch, andF. Omn`es, Phys. Stat. Sol. (c) , 2322 (2004).[20] O. Goede, L. John, and D. Hennig, Physica Status SolidiB , K183 (1978).[21] A. Klochikhin, A. Reznitsky, S. Permogorov, T. Bre-itkopf, M. Gr¨un, M. Hetterich, C. Klingshirn,V. Lyssenko, W. Langbein, and J. M. Hvam, PhysicalReview B , 947 (1999).[22] M. Grundmann and C. P. Dietrich, Journal of AppliedPhysics , 123521 (2009).[23] L. Bellaiche, T. Mattila, L. W. Wang, S. H. Wei, andA. Zunger, Applied Physics Letters , 1842 (1999).[24] P. J. Dean, Journal of Luminescence , 398 (1970).[25] H. Kanzaki and S. Sakuragi, Journal of the Physical So-ciety of Japan , 109 (1969).[26] J. J. Hopfield, D. G. Thomas, and R. T. Lynch, PhysicalReview , 312 (1966).[27] D. G. Thomas and J. J. Hopfield, Physical Review ,680 (1966).[28] A. Muller, P. Bianucci, C. Piermarocchi, M. Fornari, I. C.Robin, R. Andr´e, and C. K. Shih, Physical Review B ,081306(R) (2006).[29] M. J. Manfra, Annual Review of Condensed MatterPhysics , 347 (2014). [30] G. B. Stringfellow and M. G. Craford, High Bright-ness Light Emitting Diodes: Vol 48 (Semiconductors andSemimetals) (Academic Press Inc., San Diego, 1998).[31] S. F. Chichibu, A. Uedono, T. Onuma, B. A. Haskell,A. Chakraborty, T. Koyama, P. T. Fini, S. Keller, S. P.DenBaars, J. S. Speck, U. K. Mishra, S. Nakamura, S. Ya-maguchi, S. Kamiyama, H. Amano, I. Akasaki, J. Han,and T. Sota, Nature Materials , 810 (2006).[32] M. Kneissl, T. Kolbe, C. Chua, V. Kueller, N. Lobo,J. Stellmach, A. Knauer, H. Rodriguez, S. Einfeldt,Z. Yang, N. M. Johnson, and M. Weyers, SemiconductorScience and Technology , 014036 (2011).[33] B. Gil, Physics of Wurtzite Nitrides and Oxides - Pass-port to Devices (Springer, Heidelberg, 2014).[34] S. Nakamura, T. Mukai, and M. Senoh, Japanese Journalof Applied Physics , L1998 (1991).[35] S. Nakamura, S. Pearton, and G. Fasol, The Blue LaserDiode (Springer, Berlin, 1997).[36] U. K. Mishra, L. Shen, T. E. Kazior, and Y.-f. Wu,Proceedings of the IEEE , 287 (2008).[37] S. Rajan and D. Jena, Semiconductor Science and Tech-nology , 070301 (2013).[38] C. W¨achter, A. Meyer, S. Metzner, M. Jetter,F. Bertram, J. Christen, and P. Michler, Physica StatusSolidi (B) , 605 (2011).[39] F. Nippert, S. Y. Karpov, G. Callsen, B. Galler, T. Kure,C. Nenstiel, M. R. Wagner, M. Straßburg, H. J. Lugauer,and A. Hoffmann, Applied Physics Letters , 161103(2016).[40] F. A. Ponce and D. P. Bour, Nature , 351 (1997).[41] A. Hangleiter, F. Hitzel, C. Netzel, D. Fuhrmann,U. Rossow, G. Ade, and P. Hinze, Physical Review Let-ters , 127402 (2005).[42] Y. Narukawa, Y. Kawakami, M. Funato, S. Fujita, S. Fu-jita, and S. Nakamura, Applied Physics Letters , 981(1997).[43] P. R. C. Kent and A. Zunger, Applied Physics Letters , 1977 (2001).[44] M. J. Galtrey, R. A. Oliver, M. J. Kappers, C. J.Humphreys, D. J. Stokes, P. H. Clifton, and A. Cerezo,Applied Physics Letters , 061903 (2007).[45] A. Cerezo, P. H. Clifton, M. J. Galtrey, C. J. Humphreys,T. F. Kelly, D. J. Larson, S. Lozano-Perez, E. A. Marquis,R. A. Oliver, G. Sha, K. Thompson, M. Zandbergen, andR. L. Alvis, Materials Today , 36 (2007).[46] L. Rigutti, B. Bonef, J. Speck, F. Tang, and R. A. Oliver,Scripta Materialia , 75 (2018).[47] H. Mathieu, P. Lefebvre, and P. Christol, Physical Re-view B , 4092 (1992).[48] M. A. S. Kalceff and M. R. Phillips, Physical Review B , 3122 (1995).[49] K. Tanimura and N. Itoh, Journal of Physics and Chem-istry of Solids , 901 (1981).[50] A. K. Viswanath, J. I. Lee, D. Kim, C. R. Lee, and J. Y.Leem, Physical Review B , 16333 (1998).[51] T. Onuma, T. Shibata, K. Kosaka, K. Asai, S. Sumiya,M. Tanaka, T. Sota, A. Uedono, and S. F. Chichibu,Journal of Applied Physics , 023529 (2009).[52] R. Zimmermann, Journal of Crystal Growth , 346(1990). [53] I. Vurgaftman and J. R. Meyer, Journal of AppliedPhysics , 3675 (2003).[54] G. Callsen, T. Kure, M. R. Wagner, R. Butt´e, andN. Grandjean, Journal of Applied Physics , 215702(2018).[55] B. ˇSantic, C. Merz, U. Kaufmann, R. Niebuhr, H. Obloh,and K. Bachem, Applied Physics Letters , 1837 (1997).[56] B. Monemar, W. M. Chen, P. P. Paskov, T. Paskova,G. Pozina, and J. P. Bergman, Physica Status Solidi B , 489 (2001).[57] A. Wysmo lek, M. Potemski, K. Paku la, J. M. Bara-nowski, I. Grzegory, S. Porowski, G. Martinez, andP. Wyder, Physical Review Letters , 226404 (2003).[58] S. Fischer, D. Volm, D. Kovalev, B. Averboukh,A. Graber, H. C. Alt, and B. K. Meyer, Materials Scienceand Engineering B , 192 (1997).[59] Claus Klingshirn, Semiconductor Optics , 2nd ed.(Springer, Berlin Heidelberg New York, 2005).[60] G. Neu, A. A. Mbaye, and R. Triboulet, Proceedingsof the 17th International Conference on the Physics ofSemiconductors , 1029 (1984).[61] B. Gil, P. Bigenwald, M. Leroux, P. P. Paskov, andB. Monemar, Physical Review B , 085204 (2007).[62] G. Callsen, M. Wagner, T. Kure, J. Reparaz, M. B¨ugler,J. Brunnmeier, C. Nenstiel, A. Hoffmann, M. Hoffmann,J. Tweedie, Z. Bryan, S. Aygun, R. Kirste, R. Collazo,and Z. Sitar, Physical Review B , 075207 (2012).[63] M. R. Wagner, G. Callsen, J. S. Reparaz, J. H. Schulze,R. Kirste, M. Cobet, I. A. Ostapenko, S. Rodt, C. Nen-stiel, M. Kaiser, A. Hoffmann, A. V. Rodina, M. R.Phillips, S. Lautenschlaeger, S. Eisermann, and B. K.Meyer, Physical Review B , 035313 (2011).[64] S. F. Chichibu, T. Onuma, M. Kubota, A. Uedono,T. Sota, A. Tsukazaki, A. Ohtomo, and M. Kawasaki,Journal of Applied Physics , 093505 (2006).[65] C. K. Shu, J. Ou, H. C. Lin, W. K. Chen, and M. C.Lee, Applied Physics Letters , 641 (1998).[66] X. Shen, P. Ramvall, P. Riblet, and Y. Aoyagi, JapaneseJournal of Applied Physics , L411 (1999).[67] X.-Q. Shen and Y. Aoyagi, Japanese Journal of AppliedPhysics , L14 (1999).[68] H. Kumano, K.-i. Hoshi, S. Tanaka, I. Suemune, X.-Q.Shen, P. Riblet, P. Ramvall, and Y. Aoyagi, AppliedPhysics Letters , 2879 (2001).[69] W. Liu, J.-F. Carlin, N. Grandjean, B. Deveaud, andG. Jacopin, Applied Physics Letters , 042101 (2016).[70] Y. Hayama, I. Takahashi, and N. Usami, Energy Proce-dia , 734 (2017).[71] G. Callsen, J. S. Reparaz, M. R. Wagner, R. Kirste,C. Nenstiel, A. Hoffmann, and M. R. Phillips, Applied Physics Letters , 061906 (2011).[72] V. Y. Davydov, Y. E. Kitaev, I. N. Goncharuk, A. N.Smirnov, J. Graul, O. Semchinova, D. Uffmann, M. B.Smirnov, A. P. Mirgorodsky, and R. A. Evarestov, Phys-ical Review B , 12899 (1998).[73] R. P¨assler, Journal of Applied Physics , 6235 (2001).[74] D. Y. Song, M. Basavaraj, S. A. Nikishin, M. Holtz,V. Soukhoveev, A. Usikov, and V. Dmitriev, Journalof Applied Physics , 113504 (2006).[75] G. Callsen, G. Pahn, S. Kalinowski, C. Kindel, J. Set-tke, J. Brunnmeier, C. Nenstiel, T. Kure, F. Nippert,A. Schliwa, A. Hoffmann, T. Markurt, T. Schulz, M. Al-brecht, S. Kako, M. Arita, and Y. Arakawa, PhysicalReview B , 235439 (2015).[76] V. Y. Davydov, V. V. Emtsev, I. N. Goncharuk, A. N.Smirnov, V. D. Petrikov, V. V. Mamutin, V. A. Vekshin,S. V. Ivanov, M. B. Smirnov, and T. Inushima, AppliedPhysics Letters , 3297 (1999).[77] C. Nenstiel, M. B¨ugler, and G. Callsen, Physica StatusSolidi (RRL) , 716 (2015).[78] T. V. Bezyazychnaya, D. M. Kabanau, V. V. Kabanov,Y. V. Lebiadok, A. G. Ryabtsev, G. I. Ryabtsev, V. M.Zelenkovskii, and S. K. Mehta, Lithuanian Journal ofPhysics , 10 (2015).[79] Z. Liu, B. Fu, X. Yi, G. Yuan, and J. Wang, RSC Ad-vances , 5111 (2016).[80] D. C. Reynolds, D. C. Look, and B. Jogai, Journal ofApplied Physics , 5760 (2000).[81] C. H. Kindel, Study on Optical Polarization in HexagonalGallium Nitride Quantum Dots , Ph.D. thesis, Universityof Tokyo (2010).[82] C. Kindel, G. Callsen, S. Kako, T. Kawano, H. Oishi,G. H¨onig, A. Schliwa, A. Hoffmann, and Y. Arakawa,Phys Status Solidi - Rapid Research Letters , 408(2014).[83] M. Holmes, S. Kako, K. Choi, M. Arita, and Y. Arakawa,Physical Review B , 115447 (2015).[84] G. Callsen, G. Pahn, S. Kalinowski, C. Kindel, J. Set-tke, J. Brunnmeier, C. Nenstiel, T. Kure, F. Nippert,A. Schliwa, A. Hoffmann, T. Markurt, T. Schulz, M. Al-brecht, S. Kako, M. Arita, and Y. Arakawa, PhysicalReview B , 235439 (2015).[85] I. A. Ostapenko, G. H¨onig, S. Rodt, A. Schliwa, A. Hoff-mann, D. Bimberg, M. R. Dachner, M. Richter, A. Knorr,S. Kako, and Y. Arakawa, Physical Review B ,081303(R) (2012).[86] B. K. Meyer, J. Sann, S. Eisermann, S. Lautenschlaeger,M. R. Wagner, M. Kaiser, G. Callsen, J. S. Reparaz, andA. Hoffmann, Physical Review B , 115207 (2010).[87] K. Nozawa and Y. Horikoshi, Japanese Journal of Ap-plied Physics , L668 (1991). upplementary Information-Probing alloy formation using different excitonic species:The particular case of InGaN G. Callsen, ∗ R. Butt´e, and N. Grandjean
Institute of Physics, ´Ecole Polytechnique F´ed´erale deLausanne (EPFL), CH-1015 Lausanne, Switzerland (Dated: November 7, 2018) ∗ gordon.callsen@epfl.ch a r X i v : . [ c ond - m a t . m e s - h a ll ] N ov . EXPERIMENTAL DETAILS The investigated In x Ga − x N (0 ≤ x ≤ c -plane, random core, freestanding GaN substrates [1] ( n -type) obtainedfrom a hydride vapor phase epitaxy based overgrowth technique yielding a dislocation densityof ≈ cm − . First, a 360-nm-thick GaN buffer is grown at a temperature of 1000 ◦ Cwith tri-methyl-gallium and H used as carrier gas. Second, a 500-nm-thick Al . Ga . Nlayer is deposited at 1000 ◦ C with tri-methyl-aluminum, tri-methyl-gallium, and H used ascarrier gas. Subsequently, N is used as carrier gas in order to deposit the final epilayerof InGaN with a thickness of 100 nm based on tri-methyl-gallium and tri-methyl-indium ata temperature of around 770 ◦ C. The use of an Al . Ga . N interlayer allows to inhibitcarrier migration and to study the top In x Ga − x N (0 ≤ x ≤ x = 0 − .
4% we grew a sequenceof reference InGaN epilayers at a constant flow rate of the tri-methyl-indium source, whilethe growth temperature ( T g ) was varied around 770 ◦ C in sufficiently small steps derived fromthe method described in the following. We determined the resulting indium concentrationdown to x ≈
1% by high-resolution x-ray diffraction (HRXRD) measurements that wepartially confirmed by a secondary ion mass spectroscopy (SIMS) analysis. As a result, weobtained x ( T g ), which we linearly extrapolated for x → x <
1% by just varying T g . Again, for x = 0 .
05% we cross-checked thefeasibility of our approach by SIMS. From our PL spectra we can extract a linear dependencefor the peak position of X A as shown in the inset of Fig. 2(a) for x = 0 − . x ( T g ) for x → ≤ × cm − ), while the carbon concentration amounts to ≈ × cm − .All macro-PL spectra were measured with the samples situated in a helium-flow or aclosed-cycle cryostat. A cw HeCd laser emitting at 325 nm (Kimmon Koha Co., Ltd.) wasused as excitation source. For the macro-PL measurements the excitation light was guidedtowards the sample under normal incidence ( k k c ), while being focused by a conventional2ens with a focal length of 30 cm (excitation spot size diameter: d exc ≈ µ m). Thesamples’ luminescence was also collected with the k k c geometry using a lens with anumerical aperture (NA) of ≈ µ eV in the energy regime of interest. Finally, the dispersed light was monitored witha UV-enhanced, charge-coupled-device (CCD) array from Horiba (128 × µ -PL) measurements were performed with a fully cus-tomized setup optimized for the ultraviolet (UV) spectral range. All details of the setupcan be found in the supplementary material of Ref. [2]. The samples were mounted in aclosed-cycle helium cryostat (Cryostation C2 from Montana Instruments, Inc.), while beingexcited by a cw HeCd laser emitting at 325 nm. The diameter of the laser spot amounted to d exc ≈ µ m as the laser beam was expanded in order to fill around 80% of the objective backaperture (20 × , NA = 0.4). Detection of the luminescence was based on a nitrogen-cooled,UV-enhanced charge-coupled device (Symphony II from Horiba Jobin-Yvon), while the lightwas dispersed by a FHR 640 single monochromator (Horiba Jobin-Yvon) with a focal lengthof 64 cm. We used a 1800 l/mm grating (blaze wavelength of 400 nm) in second order and slitsettings of 10-25 µ m to facilitate a spectral resolution of better than 150 µ eV in the energyrange of interest. Suitable optical filters were applied in order to lower the straylight levelin the single monochromator (required to enable integration times up to 30 min for selected µ -PL spectra comprising a high dynamic range). All spectra were calibrated based on aneon spectral calibration lamp. Subsequently, wavelength was converted to energy withoutapplying a correction for the refractive index of air in order to make the present resultsmore comparable to the commonly available literature treating bound excitons in GaN. Ap-plication of such a vacuum correction would shift in the following all reported absolute peakenergies by around 900 µ eV to lower energies.Apertures were defined in an opaque aluminum film by electron-beam- (diameters downto 200 nm) and photo-lithography (diameters down to 1 µ m), while the metal film itselfwas always deposited using electron-beam deposition. During the deposition of the film the3 . 4 5 3 . 4 6 3 . 4 7 3 . 4 8 3 . 4 9 3 . 5 0 1 0 XBXAL1 Si0XA @ 3 2 5 n m , A l G a
N i n t e r l a y e r
PL intensity (arb. units)
E n e r g y ( e V ) @ 3 2 5 n m , A l G a
Ni n t e r l a y e r
E n e r g y ( e V )
Si0XA XA XB ( c ) - 1 @ 3 2 5 n m , A l G a
Ni n t e r l a y e r
XBPL intensity (arb. units)
E n e r g y ( e V )
Si0XA XA ( b )( a )
FIG. S1. (color online) Onset of the s-like shift of the X A transition in In x Ga − x N epilayers forindium contents x up to 1.36%. (c) At x = 0 .
01% all bound (L1 and Si X A ) and free excitonicemission lines (X A and X B ) continuously shift towards lower energies with rising temperaturefollowing the evolution of the bandgap. (b) At x = 0 .
37% this redshift diminishes for X A dueto the onset of exciton localization. Finally, at x = 1 .
36% the typical, s-like shift of the X A transition becomes noticeable, while such a behavior is not present for Si X A . All peak positionsare summarized in Fig. S2. distance between the sample and the metal source heated by an electron-beam was about1 m in order to exclude any electron-beam-induced sample damage [3]. Finally, the apertureswere either etched into the metal film or generated by a metal lift-off procedure (positivevs. negative resists). For all our processing attempts we observed closely matching µ -PLresults. II. TEMPERATURE-DEPENDENT PHOTOLUMINESCENCE
Figure S1 introduces exemplary temperature-dependent PL data, which we recorded forour entire In x Ga − x N sample series (0 ≤ x ≤ X A ) and ”free” excitons (X A and X B ) can be observed. Here, theterminology of free excitons needs to be treated with care as those particles get increasinglybound in the alloy at indium-related assemblies with rising x . Figure 3(b) of the main text4llustrates this continuous transition from free to bound excitons (two-particle complexes),which must be clearly distinguished from impurity-bound complexes such as Si X A (three-particle complex). At the onset of our alloy series ( x = 0 . x increases to 0.37% [Fig. S1(b)] the redshift, e.g., for X A is suppressed at the veryonset of the temperature series and turns into a subtle blueshift reaching a maximum valueof ≈ . ≈
35 K as depicted by the spectrum of Fig. S1(c) for x = 1 . X A emission only shows a continuous redshift with risingtemperature for all x values we analyzed.Figure S2 summarizes the temperature-dependent spectral positions for Si X A , X A , andX B for 0 ≤ x ≤ x = 2 .
4% it is not possible to track the position of Si X A vs.temperature anymore as already at a temperature of 12 K the spectral separation in betweenX A and Si X A is on the order of the corresponding full width at half maximum (FWHM)values, cf. Fig. 2(a) of the main text. Clearly, for x = 0 .
37% the redshift of X A withrising temperature is delayed in comparison to pure GaN (cf. Fig. S2). However, an indiumconcentration of ∼ A . Figure S2 includes some guides to the eye in order to illustrate such emission lineshifts at the beginning of the temperature series. For x = 1 .
36% we observe a maximumblueshift of just ≈ . ≈ . x = 2 .
4% and reaches its maximum at an accordingly highertemperature of ≈
60 K, which will be explained in the following. Naturally, with increasing x the average size of indium-related assemblies increases, causing more pronounced fluctuationsin the potential landscape of the alloy. It is exactly the rising number and also the size ofsuch indium assemblies that impact the associated exciton localization energy ( E loc ) andfacilitates the transition from a free to an increasingly bound X A complex as described inthe context of Fig. 3(b) in the main text. Upon rising temperature the E loc values of suchbound X A complexes can be overcome, leading to their thermal redistribution in the potentiallandscape. As a result, higher energy states (more shallowly bound X A ) become accessiblewith rising temperature, leading to the clear, s-like shift of X A at x = 2 . A in the potential landscape. However,a blueshift is observed as soon as this thermal redistribution of X A takes over the bandgap5 IG. S2. (color online) Summary of the emission lines’ shifts for X A , X B , and Si X A partiallyextracted from Fig. S1. At an indium concentration of x = 0.37% one can observe that the commonredshift of the X A transitions with rising temperature is diminished. Here, the dashed lines areguides to the eye in order to illustrate this aspect. At x = 1 .
36% a blueshift of ≈ . A due to exciton re-distribution into deeper potential traps. With rising indiumcontent (see x = 2 . A . shrinkage. This evolution is followed by a pronounced redshift at elevated temperaturesthat is accompanied by the conversion of bound excitons into free ones, a process which isoften referred to in the context of the mobility edge [4]. Figure S2 demonstrates that theblueshift for the X B complex is delayed with respect to X A for x = 1 . X A does not exhibit any blueshiftnor delayed redshift as depicted in Fig. S2 for up to x = 1 . are distributed in the potential landscape of the alloy. However, no thermal redistributionis noticeable as the binding energy E bind (i.e., the splitting between Si X A and X A measuredat 12 K) of the exciton is always larger than the thermal energy E bind > E th , otherwisedissociation of the bound exciton occurs in a multi-step process [6]. Naturally, the twocompeting statistical processes of exciton capture by Si and the subsequent thermal releaseof the exciton contribute to the particular lineshape of the Si X A emission band with rising x and temperature. E bind of Si X A varies from 6 . ± . . ± . . ± . x = 0, 0.37%, and 1.36%, respectively, pointing towards a slight reduction inthe effective donor binding energy in the In x Ga − x N alloy. The maximum total redshiftof Si X A that occurs in between 11 K and 75 K can be compared for these three indiumcontents. Here, we observe an increasing redshift of Si X A with respect to X A that variesfrom 3 . ± . x = 0, to 4 . ± . x = 0 . . ± . x = 1 . x an increasing number of deeply bound Si X A − In n complexes isformed comprising a rising number of indium atoms n in agreement with our discussiongiven for Fig. 4 of the main text. In addition, Fig. 5 therein introduced the µ -PL signatureof such complexes, which are mainly distributed over an energy range given by the FWHM ofthe Si X A emission band (see the PL measurements from Fig. 2 in the main text). Our µ -PLstudy from Sec. IIID of the main text finally suggests two different types of transitions thatcontribute to the overall Si X A emission and the associated particular linewidth evolutionover temperature that is impacted by the indium content. III. TEMPERATURE-DEPENDENT LINEWIDTH BROADENING
Basic features of the temperature-dependent linewidth broadening for both, the Si X A and the X A complexes, are shown in Fig. S3. For X A no clear alloy-dependent trend can beidentified - the temperature-dependent FWHM values (∆ E ) overlap within the experimentalerror. The best fit to the data for the X A complex based on Eq. 3 of the main text is obtainedfor γ ac = 22 ± µ eV/K and two sets of γ iopt and E iopt values as denoted in Fig. S3. At roomtemperature the broadening of X A is mainly affected by longitudinal-optical (LO) phonons7 g a c . = 4 –
1 µ e V / KE
1o p t . = 1 2 . 9 – g
1o p t . = 2 2 –
1 m e V g a c . = 2 2 –
6 µ e V / KE
2o p t . = 9 1 . 2 m e V g
2o p t . = 5 2 9 –
1o p t . = 9 . 8 –
5 m e V g
1o p t . = 2 . 5 – S i X A X A b u l k G a N ( X A ) 0 . 0 5 % ( X A ) 0 . 3 7 % ( X A ) 1 . 3 6 % ( X A ) F i t f o r b u l k G a N ( X A ) b u l k G a N ( S i X A ) F i t f o r b u l k G a N ( S i X A ) Relative change in the FWHM D E (meV)
T e m p e r a t u r e ( K )
FIG. S3. (color online) The temperature-dependent evolution of the FWHM (∆ E ) of the X A transition does not show a pronounced dependence on the alloy content for the analyzed indiumcontent interval 0 ≤ x ≤ X A isheavily influenced by x as shown in Fig. 4 of the main text. All symbols and values are explainedin the main text of the manuscript along with the fits (solid lines) that are based on Eq. 3 therein. in GaN (energy fixed at E opt = 91 . E opt = 9 . ± γ ac = 22 ± µ eV/K is close to reportedliterature values of ≈ − µ eV/K [8] for X A and the large error for the E opt phononenergy originates from the over-parametrized fitting function, despite the fact that a fixedphonon energy of 91.2 meV was used for E opt . Nevertheless, the value of E opt is lower thanthe energy of the E low phonon mode in GaN of 17.85 meV [7], as the momentum distributionof X A allows accessing a larger fraction of the first Brillouin zone as shown in the inset ofFig. 4(a) of the main text.Figure S3 also includes the temperature-induced broadening of Si X A at x = 0 as intro-duced in Fig. 4(a) of the main text. Here, the broadening is more pronounced with respectto the case of X A as the coupling constant γ opt related to E opt is larger in comparison to X A due to the strong localization of impurity bound excitons.8 V. MICRO-PHOTOLUMINESCENCE SPECTROSCOPY
Figure 5(a) of the main text shows two µ -PL spectra on a relative energy scale recordedfor x = 0 .
37% and 1.36% on the bare, unprocessed sample surface (laser spot diameter d exc ≈ µ m). In both cases the X A emission band is unstructured, while the onset of an ensembleof sharp emission lines can be observed around the relative energy of Si X A . Generally, thedetection of these sharp emission lines at x = 0 .
37% proves very sensitive to any rise inexcitation density or temperature. Naturally, one expects that an effective reduction of d exc should lead to clearly resolved emission lines. Hence, we fabricated apertures with diametersdown to 200 nm into an opaque aluminum layer deposited on the sample surface by differentprocessing techniques (see SI Sec. I). Nevertheless, the observation shown in Fig. 5(a) ofthe main text is crucial as it proves that such sharp emission lines are not induced by theprocessing steps. Generally, any such steps (e.g., 100 kV e-beam exposure, metal deposition,lift-off, etching, etc.) could damage the InGaN material leading to exciton localization andsharp emission lines. For instance, it was found that InGaN is very sensitive to high-voltageelectron irradiation causing a migration of indium atoms [9, 10]. Hence, in the early days ofInGaN research the material was treated like a quantum dot solid comprising indium-richislands [11], while more recent measurements based on atom probe tomography [12, 13]and scanning transmission electron microscopy [14] have confirmed a random distributionof indium atoms in state-of-the-art samples. In addition, it is reassuring that the emissionaround X A shows no clear substructure of sharp emission lines in Fig. 5(b) of the main text asthe density of X A − In n complexes is much higher when being compared to their Si X A − In n counterparts (cf. the discussion about the inset of Fig. 5(b) in the main text). [1] K. M. Motoki, T. O. Kahisa, N. M. Atsumoto, M. M. Atsushima, and H. K. Imura, JapaneseJournal of Applied Physics , L140 (2001).[2] I. Rousseau, G. Callsen, G. Jacopin, J.-F. Carlin, R. Butt´e, and N. Grandjean, Journal ofApplied Physics , 113103 (2018).[3] S. Kako, Optical properties of gallium nitride self-assembled quantum dots and application togeneration of non-classical light , Ph.D. thesis, University of Tokyo (2007).[4] N. Mott, Journal of Physics C , 3075 (1987).
5] A. Rodina, M. Dietrich, A. G¨oldner, L. Eckey, A. Hoffmann, A. Efros, M. Rosen, andB. Meyer, Physical Review B , 115204 (2001).[6] D. Bimberg, M. Sondergeld, and E. Grobe, Physical Review B , 3451 (1971).[7] G. Callsen, J. S. Reparaz, M. R. Wagner, R. Kirste, C. Nenstiel, A. Hoffmann, and M. R.Phillips, Applied Physics Letters , 061906 (2011).[8] H. Morko¸c, Handbook of Nitride Semiconductors and Devices , 1st ed. (Wiley, Weinheim, 2009).[9] T. M. Smeeton, C. J. Humphreys, J. S. Barnard, and M. J. Kappers, Journal of MaterialsScience , 2729 (2006).[10] A. Cerezo, P. H. Clifton, M. J. Galtrey, C. J. Humphreys, T. F. Kelly, D. J. Larson, S. Lozano-Perez, E. A. Marquis, R. A. Oliver, G. Sha, K. Thompson, M. Zandbergen, and R. L. Alvis,Materials Today , 36 (2007).[11] Y. Narukawa, Y. Kawakami, M. Funato, S. Fujita, S. Fujita, and S. Nakamura, AppliedPhysics Letters , 981 (1997).[12] M. J. Galtrey, R. A. Oliver, M. J. Kappers, C. J. Humphreys, D. J. Stokes, P. H. Clifton, andA. Cerezo, Applied Physics Letters , 061903 (2007).[13] Y.-R. Wu, R. Shivaraman, K.-C. Wang, and J. S. Speck, Applied Physics Letters , 083505(2012).[14] T. Schulz, T. Remmele, T. Markurt, M. Korytov, and M. Albrecht, Journal of Applied Physics , 033106 (2012)., 033106 (2012).