Temperature dependence of exciton-surface plasmon polariton couplingin Ag, Au, and Al films on InxGa12xN/GaN quantum wells studied withtime-resolved cathodoluminescence

Y. Estrin,1 D. H. Rich,1,a) S. Keller,2 and S. P. DenBaars21Department of Physics and The Ilse Katz Institute for Nanoscale Science and Technology, Ben-GurionUniversity of the Negev, P.O.B 653, Beer-Sheva 84105, Israel2Electrical and Computer Engineering and Materials Departments, University of California, Santa Barbara,California 93111, USA

(Received 3 November 2014; accepted 15 January 2015; published online 30 January 2015)

The optical properties and coupling of excitons to surface plasmon polaritons (SPPs) in Ag, Au,

and Al-coated InxGa1�xN/GaN multiple and single quantum wells (SQWs) were probed with time-

resolved cathodoluminescence. Excitons were generated in the metal coated SQWs by injecting a

pulsed high-energy electron beam through the thin metal films. The Purcell enhancement factor

(Fp) was obtained by direct measurement of changes in the temperature-dependent radiative life-time caused by the SQW exciton-SPP coupling. Three chosen plasmonic metals of Al, Ag, and Au

facilitate an interesting comparison of the exciton-SPP coupling for energy ranges in which the SP

energy is greater than, approximately equal to, and less than the excitonic transition energy for the

InGaN/GaN QW emitter. A modeling of the temperature dependence of the Purcell enhancement

factor, Fp, included the effects of ohmic losses of the metals and changes in the dielectric propertiesdue to the temperature dependence of (i) the intraband behavior in the Drude model and (ii) theinterband critical point transition energies which involve the d-bands of Au and Ag. We show thatan inclusion of both intraband and interband effects is essential when calculating the x vs k SPPdispersion relation, plasmon density of states (DOS), and the dependence of Fp on frequency andtemperature. Moreover, the “back bending” in the SPP dispersion relation when including ohmiclosses can cause a finite DOS above xsp and lead to a measurable Fp in a limited energy rangeabove xsp, which can potentially be exploited in plasmonic devices utilizing Ag and Au. VC 2015AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4906850]

I. INTRODUCTION

During the past 20 years, much effort has been invested

in optimizing the fabrication of InGaN light emitting diodes

(LEDs). Advancements in the growth, doping, and process-

ing of InGaN/GaN quantum wells (QWs) and heterostruc-

tures have led to ultra-bright light emitting diodes and

functional lasers.1,2 Despite the rapidly growing consumer

markets for many solid-state lighting applications with

InGaN, many issues still remain regarding improvements in

the material quality, choice of substrates and growth orienta-

tion, and optimization of the internal quantum efficiency

(IQE) for LEDs, particularly for those emitting in the longer

green wavelengths.3,4 The use of metal layers that support

surface plasmon polaritons (SPPs), which are oscillations in

the charge density that propagate along the interface between

a metal and dielectric, offers a possible approach for enhanc-

ing the IQE of LEDs.5,6 SPPs can be excited by light or a

high-energy electron beam (e-beam) and exhibit an x vs kdispersion relation within the plane of the metal film.7,8

Several studies have demonstrated enhancements in light

absorption, fluorescence intensity of molecules, and Raman

scattering intensities in close proximity to metallic films as a

result of coupling between the transition dipole and plasmon

modes.9–11 Very strong effects of the environment on the

emission rate are predicted for emitters that are positioned in

close proximity to metal nanoparticles.12,13 If a plasmon

mode in a metal nanoparticle is resonant with an optical tran-

sition, the electromagnetic coupling between transition

dipole and plasmon mode can result in an enhancement of

the radiative decay rate by several orders of magnitude for

emitter-nanoparticle separation distances less than

�20 nm.14 Most of the recent work on interactions betweenemitters and metal nanoparticles has been done with dyes or

quantum dots (QDs) that emit at wavelengths between 500

and 900 nm.9

By an appropriate choice of a metal possessing a surface

plasmon frequency, xsp, which is reasonably close to theexcitonic transition energy of the InGaN/GaN QW, exciton

coupling to SPPs can be enhanced, leading to an improved

QE for light extraction. An enhancement of the photolumi-

nescence (PL) efficiency of InGaN/GaN QW samples coated

with thin films of Ag has been demonstrated previously.15–18

Okamoto et al. have compared the exciton-SPP coupling forAu, Ag, and Al films deposited on InGaN/GaN QWs by

measuring the enhancement in PL for each metal type.17 A

coupling of QW excitons with surface plasmons, in which a

propagating SPP that possesses similar energy to that of the

initial exciton is created, can lead to markedly faster exciton

recombination rates. The results show an enhancement of the

a)Author to whom correspondence should be addressed. Electronic mail:

danrich@bgu.ac.il

0021-8979/2015/117(4)/043105/14/$30.00 VC 2015 AIP Publishing LLC117, 043105-1

JOURNAL OF APPLIED PHYSICS 117, 043105 (2015)

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http://dx.doi.org/10.1063/1.4906850http://dx.doi.org/10.1063/1.4906850http://dx.doi.org/10.1063/1.4906850http://dx.doi.org/10.1063/1.4906850http://dx.doi.org/10.1063/1.4906850http://dx.doi.org/10.1063/1.4906850http://dx.doi.org/10.1063/1.4906850mailto:danrich@bgu.ac.ilhttp://crossmark.crossref.org/dialog/?doi=10.1063/1.4906850&domain=pdf&date_stamp=2015-01-30

IQE by a transfer of energy between the excited electron-

hole (e-h) pairs and the propagating SPPs.15–18 A similarSPP-enhanced PL has been observed in the GaAs/AlGaAs

QW system in which samples were coated with Au.19

Notably, SPP-enhanced emission was also observed in CdSe

QDs that were dispersed on evaporated Au films20 and GaN/

AlN QD samples prepared using thin Ag films.21

Cathodoluminescence (CL) has emerged as a powerful

probe of SPP propagation along planar metal surfaces and

linear gratings.22,23 CL results from light emission that is

caused by the injection of a high-energy focused e-beam andis most conveniently exploited in a scanning electron micro-

scope (SEM). CL imaging and spectroscopy were used to

determine the propagation distance of SPPs near the SP reso-

nance of Ag and Au films, and the propagation lengths var-

ied from a few hundred nanometers to several lm’s.22,23

Recently, the loss mechanisms in SPP propagation on poly-

crystalline Au surfaces were examined with CL.23 Moreover,

CL has enabled, in an unprecedented manner, the direct

imaging of SPP modes and standing waves in Au nanowires

and various plasmonic nanoresonators with spatial resolution

on the order of the e-beam diameter, which is typically�5 nm.24–26 In such work, metallic plasmonic annular nano-resonators were fabricated on Ag and Au surfaces using

focused ion beam (FIB) nanofabrication.24–26 The develop-

ment of plasmonic nanoresonators is expected to lead to an

improved spatial resolution and energy confinement in

plasmon-based photonic circuitry.27,28

We recently investigated the optical properties and car-

rier relaxation dynamics of Al- and Au-coated GaAs/AlAs/

GaAs core-shell nanowires dispersed on Si substrates using

time-resolved CL.29 The nanowires were grown by self-

assisted vapor-liquid-solid (VLS) growth in a high purity

molecular beam epitaxy (MBE) system.29 The metal-coated

nanowires, having diameters of �100 nm, provide a systemin which the electron-hole pairs can be created, diffuse, and

recombine in varying proximity to the surface. The variable

distance from the surface, at which excitons recombine,

results in a range in the expected Purcell enhancement factor

(Fp) which describes the enhanced radiative recombinationrate due to coupling of the excitons to the SPP modes of the

metal film. The SPPs can be converted to free-space photons

if the propagating SPPs in the thin metal film are scattered

by the surface/interface roughness or grain boundaries of the

polycrystalline metal film, owing to the momentum change

associated with the scattering that enables a matching with

the x vs k light dispersion relation.17,18

Due to the opacity of the metal with respect to light/laser

excitation from the top surface, CL is shown to be a particu-

larly useful probe for metal-coated QWs and nanostructure

systems, in that the high-energy e-beam can easily penetratethe metal film and create excess e-h pairs in the semiconduc-tor QW or nanostructure. In this paper, we investigate the

temperature dependence of Fp for Ag, Au, and Al films de-posited on InxGa1�xN/GaN QW samples. We exploit the

penetration ability of highly focused e-beams to study phe-nomena associated with exciton-SPP coupling in metal/

InGaN/GaN QWs using time-resolved CL. In particular, the

three chosen plasmonic metals of Al, Ag, and Au facilitate

an interesting comparison of exciton-SPP coupling for

energy ranges in which the SP energy (xsp) is greater than,approximately equal to, and less than the excitonic transition

energy of the InGaN/GaN QW emitter. We show that in

order to correctly model the temperature dependence of the

Purcell enhancement factor, Fp, the inclusion of the effectsof ohmic losses of the metals and changes in the critical

point transition energies (i.e., changes in the metal band

gaps) as a function of temperature is essential when calculat-

ing the x vs k SPP dispersion relation, plasmon density ofstates (DOS), and the dependence of Fp on frequency andtemperature. Moreover, the “back bending,” which is due toohmic losses in the SPP dispersion,30–33 can cause a finite

DOS above xsp and lead to a measurable Fp in a limitedenergy range above xsp, which can potentially be exploitedin plasmonic devices utilizing Ag and Au with even poten-

tially adjusting the ohmic losses according to the various

methods of film preparation and deposition.

II. EXPERIMENT

The InGaN MQW samples were grown on c-plane sap-phire by metal organic chemical vapor deposition (MOCVD)

using trimethylgallium, trimethylindium, disilane and ammo-

nia as precursors, as has been described previously.34–36 Two

InGaN MQW samples were examined in this study. A sche-

matic diagram of the basic structure is shown in Fig. 1. First,

a 2 lm-thick GaN:Si buffer layer was grown on c-plane sap-phire substrate at 1060 �C, followed by deposition of a 6-nmlow-temperature (LT) GaN layer. The MQW structures con-

sisted of a confinement region (CR) and a single quantum

well (SQW) in which the CR contains InGaN QWs possess-

ing a lower content of In than that of the SQW so as to

enhance the capture, transport, and collection of excited elec-

trons and holes into the SQW, as previously studied in

detail.34–36 In both samples (labeled as S4 and S8), the CR

consisted of 4 and 8 periods, respectively, of 2.5 nm-thick

In0.05Ga0.095 N/6 nm-thick GaN layers grown prior to the

growth of the 3-nm-thick In0.12Ga0.88 N SQW, respectively,

as illustrated schematically in Fig. 1 for the S4 structure. The

CR, the 3 nm-thick InxGa1�xN SQW with x � 0.12, nomi-nally, and the 15-nm-thick GaN capping layer were grown at

800 �C. The Ag, Au, and Al films were deposited at near nor-mal incidence onto the InGaN/GaN QW samples, which

were maintained at room temperature in a separate metal

evaporation system with a pressure of �1� 10�6 Torr andevaporation rate of �0.1 nm/s. A nominal thickness (t) of20 nm for each metal was deposited on separate bare regions

of the S4 and S8 samples.

The CL experiments were performed with a modified

JEOL-5910 SEM using Eb¼ 15 keV and Ib in the 0.5 to1.5 nA range.37 The e-beam was injected through the thinmetal films. Light generated by e-h recombination wasreflected and scattered by the metal-covered surface and

sample back-side and was eventually emitted in a bare region

through the sample surface. An ellipsoidal mirror was used

to collect the light from above the sample surface and trans-

fer it via a coherent optical fiber bundle to a 0.25 -m

monochromator outside the SEM vacuum system. A UV

043105-2 Estrin et al. J. Appl. Phys. 117, 043105 (2015)

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multi-alkali photomultiplier tube (PMT) operating in the 185

to 850 nm spectral range enabled photon counting of the lu-

minescence. Measurements were performed at different tem-

peratures in the 50 to 300 K temperature range. In order to

increase the range of accessible excitation densities, the exci-

tation volume was varied by defocusing the e-beam in repro-ducible steps as has been described in detail

elsewhere.35,38,39 In the very low-excitation density regime,

the e-beam was defocused to a spot with a diameter of�45 lm at the sample surface. The defocusing approach per-mits measuring CL emission with a uniform excitation den-

sity that is several orders of magnitude smaller than what can

be employed using a focused e-beam.35 Time-resolved CLexperiments were performed with the method of delayed

coincidence in an inverted single photon counting mode,

with a time resolution of � 100 ps.35,37 Electron beam pulsesof 50 ns width with a 1 MHz repetition rate were used to

excite the sample. Steady-state and time-delayed CL spec-

troscopy measurements were performed with a spectral reso-

lution of 1 nm. CL spectra for both the bare (reference) and

adjacent metal-covered regions were acquired within

�0.5 mm of each other to minimize the effects of materialvariations when acquiring data and comparing CL results for

different regions of the same sample.

Atomic force microscopy (AFM) measurements were

performed in AC-mode with an MFP-3D AFM from Asylum

Research (Oxford Instruments). The AFM measurements

were performed on the bare and metal-covered surfaces to

determine the root-mean-square (RMS) roughness and exam-

ine differences in morphology between the various samples.

III. RESULTS AND DISCUSSION

CL Spectra for sample S8 which possesses 8 QWs in the

CR were acquired for sample temperatures of 300 and 55 K

and are shown in Figs. 2 and 3, respectively. The e-beam wasuniformly defocused (i.e., the diameter of the spot was

�45 lm), yielding an excitation density of P � 1.4 W/cm2.The low excitation density ensures a relatively small screening

of the polarization field and minimal state filling in both the

CR and SQW at all temperatures. The peak emission energies

are expected to exhibit energies near the lowest transition ener-

gies (i.e., largest red-shifts) due to the Quantum Confined Stark

Effect (QCSE) under such excitation conditions.35,36 In each

panel of Figs. 2 and 3, CL spectra are shown for both the bare

(reference) region and the adjacent Ag-, Al-, and Au-covered

regions. The CL spectra are normalized to have the same peak

heights in each of the panels for the different metal coatings, as

shown in Figs. 2 and 3. The CL spectra acquired at room tem-

perature in Fig. 2 show broadened peaks with full-widths at

half-maxima (FWHM) of �150 and �90 meV for the SQWand CR emissions, respectively, which reduce to �95 and60 meV at low temperature. A weak GaN near-band edge

emission at k¼ 365 nm (3.407 eV) is also visible at room tem-perature. The relative intensities of the CR and SQW emissions

change by an order of magnitude between 300 and 55 K as a

result of much more efficient carrier transfer from the CR to

the SQW at higher temperatures, as previously modeled in

detail.35 For the CL spectra acquired at 55 K, phonon sidebands

are observed on the low-energy sides of both the SQW and CR

zero-phonon lines (ZPLs), with both one-phonon (1LO) and

two-phonon (2LO) lines clearly visible for the SQW emission.

The lifetimes (s) as a function of energy and temperaturewere obtained from the time-resolved CL measurements. CL

transients of the bare and metal-covered S8 sample are shown

in Fig. 4 for various temperatures in the 55 to 300 K range.

The transients were acquired for emission wavelengths near

the CL SQW peak intensity at each temperature. The carrier

lifetimes were extracted from the slopes of the CL transients

at each temperature, assuming a single exponential decay

whose fits are illustrated by the red lines running through the

data in Fig. 4. The lifetimes of the SQW are an order of mag-

nitude larger than those for the CR, as a result of the wider

well and larger In composition, x, for the SQW.35

The carrier lifetimes depend on the temperature and

decrease monotonically as the temperature is increased for

both the bare and metal-covered samples. The wavelength

dependence of the lifetime is shown in the semi-log plots of

Figs. 2 and 3, in the same panels as the CL spectra for the

bare and metal-covered samples at high and low tempera-

ture. At T¼ 55 K, the carrier lifetime is observed to increaseas a function of wavelength through the energy ranges of

both the CR and SQW peaks, as shown in Fig. 3. Such

FIG. 1. Schematic diagram of the S4 structure for the InGaN/GaN MQW

sample illustrating the CR and SQW regions. The SQW region for both S4

and S8 samples contains a single nominal In0.12Ga0.88N QW. The thickness

for the regions shown in the diagrams is not to scale. Ag, Au, and Al layers of

20 nm thicknesses were deposited onto the GaN-capping layers of S4 and S8.

043105-3 Estrin et al. J. Appl. Phys. 117, 043105 (2015)

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FIG. 2. CL Spectra of S8 for the bare, Ag-, Au-, and Al-covered sample at

T¼ 300 K. The decay lifetimes (s) as a function of wavelength are superim-posed on the same graph. The CL spectra for the metal-covered and bare

regions were acquired within �0.5 mm of each other to minimize effects ofmaterial variations when comparing different regions of the same sample.

The SQW emission is observed to dominate the CR emission as a result of

an efficient transfer of carriers to the SQW at room temperature.

FIG. 3. CL Spectra of S8 for the bare, Ag-, Au-, and Al-covered sample at

T¼ 55 K. The decay lifetimes (s) as a function of wavelength are superim-posed on the same graph. The CR (MQW) emission is observed to dominate

the SQW emission as a result of a reduced transfer of carriers to the SQW at

low temperatures. Phonon sidebands are observed on the low-energy sides

of both the SQW and CR ZPLs, with both one-phonon (1LO) and two-

phonon (2LO) lines clearly visible for the SQW emission.

043105-4 Estrin et al. J. Appl. Phys. 117, 043105 (2015)

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behavior for s vs k has previously been observed in InGaN/GaN QWs and is attributed to the localization of excitons in

InN-rich regions.40–43 The spectral diffusion of electrons and

holes to lower energy InN-rich regions is expected to yield

longer decay times, as observed in the zero-phonon peaks in

both the CR and SQW.42,43 An oscillation in s near the posi-tions of the 1LO and 2LO sidebands is also observed in Fig.

3, as the overlapping energy ranges of the inhomogenously

broadened ZPL, 1LO, and 2LO lines lead to a superposition

of the lifetime dependence for the three emissions. The oscil-

lations become weaker as the temperature increases to

300 K, owing to the thermal broadening and delocalization

of carriers from the InN-rich centers, which lead to a smear-

ing of the peaks at higher temperatures, as shown in Fig. 2.

The most salient aspect of Figs. 2 and 3 is the reduction

of the lifetime for the SQW emission for the metal-covered

samples, which becomes more pronounced at lower tempera-

tures. At T¼ 55 K, the deposition of Al, Au, and Ag resultsin lifetimes that are reduced by factors of �1.4, 1.7, and 2.2,

respectively, near wavelengths close to the peak of the SQW

emission. In contrast, the lifetimes in the wavelength range

of the CR emission remain unchanged after the deposition of

the three metals for all temperatures in the 55 to 300 K range.

The temperature dependence of the lifetime, s, is shown inFig. 5 for the bare and metal-covered samples. Previously,

we have shown that the decay lifetime of the SQW emission

at temperatures below �80 K saturates and is dominated byradiative recombination process.35 Small differences in the

decay lifetimes of the SQW emission, as observed for differ-

ent portions of the bare samples adjacent to the metal

regions, are most likely related to material variations

between regions of the same sample. Such variations are

expected in the magnitude of the polarization field which are

caused by small variations in the In composition or in strain

relaxation.35,40,42,43

At high temperatures approaching �300 K, nonradiativeprocesses dominate and such processes may also be influ-

enced by the deposition of the various metals. The radiative

FIG. 4. CL decay transients for the

bare S8 sample and for the Ag, Au,

and Al-covered S8 sample. The transi-

ents are shown for various tempera-

tures from 55 to 300 K. The red lines

show the results of a linear fit in the

semi-log plot for obtaining the carrier

decay lifetime, s.

043105-5 Estrin et al. J. Appl. Phys. 117, 043105 (2015)

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lifetime at a given temperature can be estimated from a com-

bined measurement of the decay lifetime and integrated in-

tensity using a standard expression for the radiative

efficiency g35,44–46

g Tð Þ ¼ s Tð Þ

sR Tð Þ¼ I Tð ÞI0

; (1)

where I0 is a normalization factor, which depends on thenumber of excited carriers and is equal to the value of the

saturated CL intensity at low temperatures. If temperature-

induced variations in both the CL intensity and decay life-

time are determined by nonradiative processes only, thenthe expression s/I is proportional to the radiative lifetime,as given by Eq. (1). The reciprocal of the spatially inte-

grated CL intensity, I�1, of the SQW peak emission multi-plied by decay lifetime s vs temperature was measured andused to determine sR vs T for both the bare and metal cov-ered regions of samples S4 and S8. The results for s(T) andsR(T) are shown together in Fig. 5 for the bare, Ag-, Au-,and Al-covered S8 sample. Similar results were obtained

for the S4 sample (not shown). The lifetimes shown in Fig.

5 are for the wavelengths associated with the peak intensity

of the SQW emission at each temperature. The approxi-

mately linear increase in sR vs T until �180 K for the bareregions is indicative of behavior for a two-dimensional

excitonic system, consistent with the radiative recombina-

tion of free excitons at low temperatures.35,46,47 The

decrease in sR for T> 180 K is largely attributed to a ther-mally activated process of carrier transfer from the underly-

ing CR region to the SQW, as previously reported.35 Other

regions in which sR is approximately independent of tem-perature, such as exhibited in the results of Figs. 5(a) and

5(b) for the bare regions, may be caused by fluctuations in

the In composition. Such fluctuations can induce an

enhanced localization of the exciton that causes a reduced

variation in sR(T) for which it can remain constant over alimited low-temperature range.35,48

In order to assess the enhancement as a result of exciton

coupling to SPPs, we obtain the temperature-dependent

Purcell enhancement factors as FP¼ sR(bare)/sR(metal) whichare shown in Fig. 6 for the Ag-, Al-, and Au-covered S4 and

S8 samples. The nominal structure of the SQW, both in In

composition and width, is identical for both samples as is the

thickness of the GaN capping layer which is 15 nm for both

samples. The difference in structure is due to the number of

QWs of the CR for the S4 and S8 samples (i.e., four and

eight QWs, respectively) and has apparently caused no sig-

nificant difference in the Fp vs T results for the two samples,as can be expected since the SQW is in closest proximity to

the surface and metal layers. The results of Fig. 6 demon-

strate that the deposition of the Ag film results in the largest

enhancement in the radiative recombination rate for all tem-

peratures in the 55 to 300 K range. The maximum values of

Fp for Al and Au are about 25% less than that for Ag at thelowest temperatures, which exhibits a maximum Fp � 2.1. Amonotonic decrease is observed in Fp for the three metals asthe temperature increases towards room temperature, with Al

exhibiting the smallest change in Fp over the 55 to 300 Krange. The largest reduction in sR and thus the largestincrease in Fp over the full temperature range is for the Agfilm and is likely indicative of an enhanced exciton-SPP cou-

pling for the Ag film in comparison to that for the Au and Al

films. Such a result is supported by the observation that the

surface plasmon frequency (xsp) for films of Ag/GaN ismuch closer to the energy of the SQW emission than that for

either films of Al/GaN or Au/GaN whose xsp are at energiesthat are considerably above and below that of the SQW

FIG. 5. The carrier decay times, s(T), and radiative lifetimes, sR(T), for the bare, Ag, Al, and Au-covered regions of S8. The results for the Ag, Al, and Aufilms on sample S8 are shown, respectively, in panels (a)–(c). The lifetime measurements of the bare region that was probed within 0.5 mm of the metal cov-

ered region are shown in each panel to minimize effects of material variations when comparing different regions of the same sample.

043105-6 Estrin et al. J. Appl. Phys. 117, 043105 (2015)

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emission energy,15–18 respectively, as will be demonstrated

by the calculations presented below. In order to understand

and properly model the temperature dependence of Fp forthe three cases of Ag, Au, and Al films on the InGaN/GaN

SQW, the temperature dependence of the ohmic losses, the

real and imaginary parts of the dielectric constants, and the

x vs k SPP dispersion relation, including the “back bending”of the dispersion, and density of states have been incorpo-

rated into the modeling, as described below in Sec. IV.

Roughness with a feature size ranging from �10 to50 nm is observed in the AFM images of the Ag-, Au-, and

Al-covered surfaces shown in Fig. 7, which were each

acquired over a 1 lm� 1 lm scan area. Moreover, the AFMimages reveal larger recessed holes that exhibit diameters

ranging from �100 to 150 nm. These are V-pit defects whichare commonly observed in MOCVD-grown InGaN/GaN

MQW samples.49 In order to assess the roughness caused by

island growth and grain size/orientation variations for the

three metal films, RMS roughness values were measured in

selected 0.3 lm� 0.3 lm regions that were free of V-pitdefects and are 7.1, 2.2, 2.1, and 1.0 nm for the Ag-, Au-, Al-

covered, and bare surfaces, respectively. The Ag film clearly

exhibits the largest observed RMS by a factor of �3.2 andsuch roughness will have consequences in attempting to

understand differences between the calculated and measured

Fp, as discussed in Sec. IV. The roughness observed for thebare surface is attributed to possible variations in the oxide

layer thickness. While surface/interface roughness and grain

boundaries of the polycrystalline metal films are essential in

enabling the conversion of SPPs to free-space photons, the

magnitude of the roughness can influence the optical proper-

ties of the metal film by increasing the losses which will

directly influence Fp, as discussed below.

IV. THEORETICAL MODELLING OF THE SPP DENSITYOF STATES AND FP

In order to develop a reasonable model and calculate FPas a function of energy and temperature for the metal/

FIG. 6. Experimental Purcell enhancement factors, Fp, vs temperature forthe Ag, Al, and Au-covered S4 (a) and S8 (b) InGaN/GaN SQW samples.

The Purcell enhancement factors are given by FP¼ sR(bare)/sR(metal) at theenergies representing the peak of the emission intensity of the SQW in the

55 to 300 K temperature range.

FIG. 7. AFM images of the Ag-, Au-, Al-covered, and bare InGaN/GaN

SQW S4 samples in (a)–(d), respectively. Images were each acquired over a

1 lm� 1 lm scan area. The RMS roughness is indicated for each image andwas obtained by averaging over 0.3 lm� 0.3 lm regions not containing V-pit defects for each sample.

043105-7 Estrin et al. J. Appl. Phys. 117, 043105 (2015)

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InGaN/GaN SQW system, we observe that the separation

between the metal film and the SQW is �15 nm and is closeto (i) the typical penetration depths of the SPP field into thesemiconductor near resonance and (ii) the typical excitondipole to metal film separation, both of which are �10 to50 nm for SPP-induced field enhancements in quantum heter-

ostructures and nanostructures.15–21 We have used a standard

two-dimensional (2D) SPP energy dispersion relation model

for a flat metal film covering a dielectric material layer (i.e.,

GaN in our case), which will provide a reasonable model

from which the x vs k dispersion relations can be calculatedfor both Au-, Ag-, and Al-covered InGaN/GaN SQW struc-

tures. From the solution of Maxwell’s equations with a

matching of boundary conditions at the interfaces of the vac-

uum/metal/GaN system, we have calculated the x vs k dis-persion relations for various temperatures between 50 and

300 K, which also include the temperature-dependent losses

for Ag, Al, and Au films possessing varying thicknesses (t)on GaN. We have used the energy and temperature depend-

ence of the dielectric constants (i.e., e(x,T)¼ e0(x,T)þ ie00(x,T)) of GaN, Ag, Au, and Al that are found in the lit-erature and which we have calculated, as described

below.50–52 The formalism for the SPP dispersion relation

calculation has been previously solved and discussed in

detail.15,16,31 Briefly, the SPP modes can be described by

matching boundary conditions (i.e., requiring continuity of

the tangential E and normal D fields) for a three layer systeminvolving the vacuum, metal film of thickness t, and GaNsubstrate which are denoted by indices i¼ 1 to 3, respec-tively. The propagation of electromagnetic waves along

the metal/GaN interface yields analytical solutions for

the in-plane electric field, Exi, in the ith layer of the form Exi/ exp(ikx)exp(�kziz) with k2zi¼ k2� (x2/c2)ei, where ei isthe complex dielectric constant (as a function of frequency

and temperature) for the ith layer. The boundary conditionslead to a dispersion relation of the form

ðe1kz2 þ e2kz1Þðe2kz3 þ e3kz2Þ� ðe3kz2 � e2kz3Þðe1kz2 � e2kz1Þ expð�2kz2tÞ ¼ 0:

(2)

The k-wavevector is in general complex (i.e., k¼ k0 þ ik00)with the solutions of Eq. (2) usually being plotted as x vs k0

and representing the SPP dispersion relation, as presented

below. Numerical solutions of the SPP x vs k0 dispersion atvarious temperatures can be obtained from Eq. (2) provided

that ei(x,T) can be measured or calculated. The measure-ments of e(x,T) were performed for AlxGa1�xN for energiesless than the bandgap and for 0� x� 1 by Brunner et al.,53which we incorporated into our calculations for the SPP dis-

persion of Au and Ag using e(x,T) for GaN. For the SPP dis-persion of aluminum, ei(x,T) is required for energiesconsiderably above the GaN bandgap and so we used the ex-

perimental data of Kim et al.54

In order to account for the temperature dependence of

e(x) for the metal films, it is necessary to first consider exist-ing approaches, as e(x) is usually modelled using aDrude–Lorentz approach.50–52 The Drude term for nearly

free valence electrons describes the intraband behavior andis given by

eD x; Tð Þ ¼ e1 �xp Tð Þ

x2 þ ixc Tð Þ ; (3)

in which e1, xp(T), and c(T) represent the high-frequencylimit of the dielectric constant, plasma frequency, and a

broadening factor (or damping rate) related to the free elec-

tron lifetime, respectively, the latter two being functions of

temperature.50 Contributions to the electron scattering fre-

quency come from both electron-electron, xe�e(T), andelectron-phonon, xe�ph(T), scattering events withc(T)¼xe�e(T)þxe�ph(T).55–57 The plasma frequency scaleswith the thermal expansion of the metal and is given by

xp Tð Þ ¼xp T0ð Þ

1þ 3a T � T0ð Þ½ �1=2; (4)

where a is the thermal expansion coefficient of the metal andthe reference temperature, T0, is taken as 298 K.

55 The e-escattering rate is given by

xe�e Tð Þ ¼p3CD12�hEF

kBTð Þ2 þ�hx2p

� �2" #; (5)

where C is the average of the scattering probability over theFermi surface, and D is the fractional Umklapp scatteringcoefficient.56 The remaining constants assume their usual

meaning. The electron-phonon scattering rate is given by

xe�ph Tð Þ ¼ x02

5þ 4T

5

h5D

ðhD=T0

z4

ez � 1 dz

264

375; (6)

where hD is the Debye temperature and x0 is a constantrelated to the dc resistivity of the metal.57

The interband contribution is typically modelled using one

or more Lorentzian oscillator terms.52,58,59 However, in order

to achieve satisfactory fits to the experimental e(x) data, alarge number of Lorentzian terms are often required and can

lead to a model that lacks physical meaning and justification.59

Moreover, the combination of the Drude and Lorentzian terms,

when fitting the model function to the experimental e(x), canlead to e00(x)< 0 over a portion of the spectral range, resultingin an unphysical negative absorption for metals.60

In order to better model the temperature dependence of

the interband contributions to the complex dielectric function

and avoid unphysical artifacts such as a negative absorption,

we have utilized and built upon a recently developed model

by Rioux et al. whose approach involves a critical point anal-ysis of the band structure of Au and Ag.60 The model is able

to reliably reproduce the dielectric function for Au and Ag

thin films (and for AuxAg1�x alloys) measured by ellipsome-

try and contains 10 fitting parameters for each metal film

alloy. The model is based on an analysis of the relevant criti-

cal points (Van Hove singularities) in the joint density of

states for interband optical transitions in the Au and Ag band

structures. The relevant critical points for optical transitions in

the visible energy range are located in k-space at the X and Lsymmetry points in the fcc Brillouin zone. Briefly, the modelinvolves parameterizing two optical transitions with energies

043105-8 Estrin et al. J. Appl. Phys. 117, 043105 (2015)

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xg1 and x01 which represent direct transitions at criticalpoints near the X-point from the d-bands to the s-band abovethe Fermi-level. A third optical transition energy, x02, isincluded in the model and represents a similar direct transition

near the L-point from the s-band below the Fermi-level tohigher unoccupied bands.60 The contributions of the critical

point transitions near the X- and L-points to the complex

dielectric function are labeled as eCP1(x,T) and eCP2(x,T),respectively, and are presented in the Appendix. In order to

account for the temperature dependence of the transition ener-

gies, we have incorporated the temperature dependence of the

band gaps, bi, into the model by allowing x01, x02, and xg1to shift at the same approximate rate of �1.0� 10�3 and�0.6� 10�3 eV/K for Au and Ag, respectively, for each criti-cal point transition, as observed in the polarimetric measure-

ments of e00(x, T) by Winsemius et al.61 Thus, after a fit of thereal and imaginary parts of the model dielectric function at

room temperature (i.e., e(x, 300 K)¼ e0(x, 300 K)þ ie00(x,300 K)), the temperature dependence is incorporated by add-

ing the band gap shift rates to the critical point transition ener-

gies. The parameters for the fits of the dielectric function

model, e(x), to the experimental data for the metal films arepresented in Table I of the Appendix.

Using our model for e(x,T)¼ e0(x,T)þ ie00(x,T) for Au,Ag, and Al films and the empirical functions obtained for

e(x,T) of GaN,53,54 we then solved Eq. (2) with t¼ 20 nm andobtained the x vs k0 SPP dispersion relation for various tem-peratures from �50 to 300 K, as shown in Fig. 8. As a resultof including the imaginary part of the dielectric constants,

e00(x,T), for the metals (and hence the presence of dampingand ohmic losses), the dispersions for Au and Ag exhibit

“back bending” in which the energy vs k0 dispersion curvereaches a finite real wavevector close to xsp which is at 2.35and 2.72 eV for Au and Ag, respectively, at 300 K.30–32 As the

ohmic losses are reduced at lower temperatures, the maximum

real wavevector associated with xsp increases as the tempera-ture decreases, resulting in smaller values of the magnitude of

the group velocity, jvgj ¼ j@x/@kj, at energies just below andabove the surface plasmon frequency, xsp. Moreover, as thetemperature is lowered to 50 K, xsp blue-shifts and reachesvalues of 2.50 and 2.77 eV for Au and Ag, respectively. Also

shown in Fig. 8 is the position of the SQW emission at room

temperature, which is reasonably close to xsp of Au and Agfor all temperatures in the 55 to 300 K range. For the alumi-

num film, the calculation yields xsp of 5.14 and 5.19 eV for300 and 50 K, respectively, which are considerably above the

SQW emission energy at �2.88 eV.As the SPP x vs k0 dispersion relation curves for Au and

Ag tail into the energy region above xsp, quasi-bound (QB)or leaky modes exist in a region sometimes referred to as the

plasmon bandgap between xsp and xp, the bulk plasmon fre-quency. Were it not for the inclusion of e00(x,T)> 0 for Auand Ag, only imaginary k-values would exist in this region.The group velocity, vg¼ @x/@k, approaches infinity beforebecoming negative in the region of the QB modes in which

anomalous dispersion exists owing to the damping and

ohmic losses.62 As the dispersion curve tails into the QB

region, the finite density of states for energies greater than

xsp can be exploited to yield an enhancement in the radiative

recombination rate and resulting Purcell enhancement factor,

Fp. Consequently, ohmic losses can lead to a broadening ofthe dispersion relation that include QB modes above xsp forwhich exciton-SPP coupling can contribute to an enhance-

ment in the radiative recombination rate for such higher

energy modes. Such losses and broadening of the dispersion

are temperature-dependent. The most pronounced

FIG. 8. The calculated x vs k0 SPP dispersion relations for various tempera-tures from 50 to 300 K for the Ag-, Au-, and Al-covered InGaN/GaN SQW

(S4 and S8 samples). The horizontal dashed lines show the SQW emission

energy at T¼ 55 K, which is greater than xsp for the Ag- and Au-coveredsamples and less than xsp for the Al-covered sample. The energy positions ofxsp (T¼ 300 K) and the bandgap of GaN (Eg) at T¼ 300 K are indicated. Thetemperatures for which the curves were calculated are 50, 60, 70, 85, 100,

120, 150, 180, 210, 250, and 300 K and are indicated by arrows in the panels.

043105-9 Estrin et al. J. Appl. Phys. 117, 043105 (2015)

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temperature-dependent dispersion relation is observed for

the case of Au in Fig. 8(b), which is a consequence of the X-

and L-point critical point energies (i.e., x01, x02, and xg1)being fairly close to xsp. As a result, small positive shifts inthe gap energies with decreasing temperature cause a large

relative decrease in e00(x,T) which in turn leads to a blue-shift in xsp for decreasing temperature, as obtained from sol-utions to Eq. (2) and shown in Fig. 8(b). The relative changes

in the dispersion relation and xsp for varying T are muchsmaller for Ag and Al, owing to the absence of critical point

transitions in the vicinity of xsp. The influence of changes inxsp due to the temperature dependence of the nearly freeelectron density, electron-electron scattering, and the

electron-phonon interaction (as expressed in Eqs. (4)–(6)) is

observed to have a much smaller effect on the x vs k0 disper-sion relation in the vicinity of xsp than for changes in the dis-persion caused by shifts in the d-band critical point transitionenergies near xsp, as for the case of Au.

For a multilayer dielectric system possessing a thin

metal film, such as for the present metal/GaN/InGaN SQW

system, Fp can be obtained analytically and is given by

Fp xð Þ ¼ 1þpc3jE zð Þj2

4x2ð1�1

uE x; z0ð Þdz0

q2D xð Þ; (7)

where uE(x,z) is the electric energy density for each materialregion and E(z) is the un-normalized plasmon electric field ata distance z from the metal/GaN interface, where the SQW islocated.15,16 The 2D density of plasmon states per unit area

can be calculated from the dispersion relations in Fig. 8 and

is given by q2DðxÞ ¼ ð1=2pÞkj@x=@kj�1. For a dielectric inthe transparency region (i.e., je00(x)j je0(x)j), the energydensity of the electric field in the layer is given by uE(x,z)¼ [@(xe0)/@x][jE(z)j2/8p].63 However, for metals with dissi-pation, the expression for the energy density is uEðx; zÞ.¼ [e0 þ 2xe00/c(T)][jE(z)j2/8p], where c(T)¼xe�e(T) þxe�ph(T)is the damping constant in the Drude model as presented

before which is obtained from Eqs. (5) and (6) and the pa-

rameters of Table I.64,65 The denominator of Eq. (7) can eas-

ily be integrated to yield an analytical solution for Fp(x) thatis given by

Fp x; Tð Þ ¼ 1þ2pc3k exp �2kz3zð Þ

x2exp �2kz2tð Þ

kz1þ e02 þ

2xe002c

� �1� exp �2kz2tð Þ

kz2

� �þ 1

kz3

@ xe03� �@x

" # ���� @x@k�����1; (8)

where ei(x,T)¼ ei0(x,T)þ iei00(x,T) is the frequency andtemperature dependent dielectric constant for the appropriate

metal and GaN, for i¼ 2 and 3, respectively.Calculations of Fp(x,T) using the dispersion relations

for Ag, Au, and Al films on the GaN/InGaN SQW are shown

for the relevant energy range and temperatures in Fig. 9. The

energy of the SQW emission (labeled with vertical dashed

lines in Fig. 9) is higher than the energies calculated for the

peaks of Fp(x,T) for Ag and Au on the SQW. Also for Agand Au, the blue-shifting of the x vs k0 dispersions and xspas the temperature decreases is observed to cause an increase

in Fp at the energy of the SQW emission at lower tempera-tures. For the case of aluminum, the relatively small change

in the dispersion relation is observed to have a negligible

effect on Fp as the temperature decreases from 300 to 50 K.The calculated dependence of Fp(xSQW,T) for temperaturesin the 50 to 300 K range is shown in Fig. 10. The trends in

Fp(xSQW,T) vs T for all three metals are similar to the exper-imental results shown in Fig. 6, but show larger calculated

values for Ag and Al, by factors of �15 and �4, respec-tively. For the experimental results, Fp for the Al film is alsoobserved to exhibit the least variation with temperature.

The case of Fp vs T for the Au film is observed to yield thebest agreement between theory and experiment. For the Ag and

Al films, large discrepancies in the magnitude of Fp betweenexperiment and theory can be attributed to the large RMS and

associated grain sizes of the polycrystalline thin metal films.

Chen et al. demonstrated that optical loss in thin (10–20 nm)

Ag films is directly correlated with the roughness of the Ag

film, owing to an enhanced collision or scattering rate of elec-

trons for smaller grain sizes.66 Moreover, in a recent PL study

of thin Ag films deposited on an InGaN/GaN QW, the PL

enhancement due to exciton-SPP coupling was also found to

depend strongly on the grain size for which a reduction in aver-

age grain size from �50 to 38 nm caused a decrease in the PLenhancement by nearly an order of magnitude.67 In view of

these studies, we suggest that the relatively large roughness

(i.e., a measured RMS of �7.1 nm for the 20-nm thick Ag film,as observed in the AFM results of Fig. 7) is primarily responsi-

ble for the observed discrepancy between the measured and cal-

culated values of Fp. In order to test this hypothesis, we havefurther calculated Fp vs a normalized damping factor, ~c¼ c/c(T), where c(T) represents the calculated damping rates fromEqs. (5) and (6) for Ag, Au, and Al, as previously discussed.

The calculations were performed using Eq. (8) and the results

are shown in Fig. 11 for temperatures of 50 and 300 K. The cal-

culations for T¼ 50 K show that as ~c increases from 1 to 20, Fpdecreases by 73% and 56% for Ag and Al, respectively. Large

reductions in Fp at room temperature for Ag and Al are simi-larly observed as ~c increases by about an order of magnitudebeyond the bulk reference value of ~c¼ 1. The model predictsthat Fp for Ag and Au decreases substantially as ~c approacheszero due to the critical point transitions in the dielectric func-

tion model. However, Fp for Al remains constant at �4.8 for~c � 3 due to the absence of a critical point transition in themodel for the dielectric function of Al.

043105-10 Estrin et al. J. Appl. Phys. 117, 043105 (2015)

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In spite of the similar RMS roughness values for the Al

and Au films from the AFM measurements of Fig. 7, the dis-

crepancy between experiment and theory for the Fp of the Al

film is much greater than that for the Au film as observed in

Figs. 6 and 10. Many factors other than RMS can influence ~cfor a particular metal, such as the grain size/shape, surface

morphology of the thin film, oxidation at the surface and at

the grain boundaries (i.e., for the case of Al), the electron

mean free path, and the electron scattering rates at the grain

boundaries. These factors are expected to be different for the

Al and Au films whose surface morphologies observed in

Fig. 7 are distinct and exhibit a wavy surface for the Al film

and well-separated rounded hillocks for the Au film.

An additional potential source for a reduction in Fp isdue to the native oxide formed on the GaN surface prior to

the metal film deposition and possibly due to the oxidation

of the Ag and Al films as well. The oxide layers separating

these metals from non-oxidized metal and GaN could be asthick as 2–3 nm, as the GaN capping layer was already oxi-

dized during the metal deposition.68 In order to demonstrate

the effect of a decrease in Fp for an increasing metal film toSQW separation distance (z), we show the calculated Fp as afunction of z for T¼ 50 K in Fig. 12. Combined with thedecrease in Fp due to an increase in damping rate (~c) due toroughness, the decrease in Fp for the Ag film and less so forthe Al film for increasing z (i.e., beyond the nominalz¼ 15 nm GaN capping layer thickness) is consistent withthe observed discrepancy between the experimental and cal-

culated Fp. For example, a 3-nm thick oxide layer for the Agfilm would yield a �35% decrease in Fp in this simplemodel, not including the effect of an increase in damping

rate. A similar discrepancy in the experimental and theoreti-

cal Fp for measurements of Al and Au films on GaAs/AlAscore-shell nanowires was attributed to the presence of oxide

layers and the effective increase in the metal to exciton sepa-

ration owing to the presence of oxide layers.29

V. CONCLUSIONS

The coupling of excitons to SPPs in Ag-, Au-, and Al-

coated InxGa1�xN/GaN multiple and SQWs was probed

FIG. 9. The calculated Fp(x,T) for the Ag-, Au-, and Al-covered InGaN/GaN SQW (S4 and S8 samples) for temperatures in the 50 to 300 K range.

The position of the SQW emission energy at T¼ 55 K is shown. For the Ag-and Au-covered samples, the blue-shifting of the x vs k0 dispersion relationand xsp as the temperature decreases is observed to cause an increase in Fpat the energy of the SQW emission at lower temperatures. The temperatures

for which the curves were calculated are 50, 60, 70, 85, 100, 120, 150, 180,

210, 250, and 300 K and are indicated by arrows in the panels.

FIG. 10. The calculated Purcell factor, Fp, vs temperature at the energy ofthe SQW emission for each temperature for the Ag-, Al-, and Au-covered

InGaN/GaN (S4 and S8) samples.

043105-11 Estrin et al. J. Appl. Phys. 117, 043105 (2015)

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using time-resolved CL. Excitons were generated in the

metal coated SQWs by injecting a pulsed high-energy elec-

tron beam through the thin metal films. Such an approach

can enable excitation of the metal-covered sample in situa-

tions where excitation by laser sources would be difficult or

impossible due to the opacity of the metal film. The Purcell

enhancement factor (Fp) was obtained by direct measure-ment of changes in the temperature-dependent radiative life-

time caused by the SQW exciton-SPP coupling. Three

chosen plasmonic metals of Al, Ag, and Au facilitated an

interesting comparison of exciton-SPP coupling for energy

ranges in which the SP energy (xsp) is greater than, approxi-mately equal to, and less than the excitonic transition energy

for an InGaN/GaN QW emitter. The deposition of the Ag

film resulted in the largest enhancement in the radiative

recombination rate of the SQW (i.e., the largest Fp) for alltemperatures in the 55 to 300 K range and was attributed to a

more closely matched xsp and SQW emission energy incomparison to the cases for Au and Al.

A modeling of the temperature dependence of Purcell

enhancement factor, Fp, was performed and included theeffects of ohmic losses of the metals and changes in the

dielectric properties due to the temperature dependence of

intraband and interband effects, respectively, associated with

the Drude model and the critical point transitions involving

the d-bands of Au and Ag. Calculations of the x vs k SPPdispersion relation, plasmon DOS, and the dependence of Fpon frequency and temperature were presented for the three

metals. Despite the relatively large difference between xspfor films of Al/GaN and the SQW emission energy, a signifi-

cant enhancement in Fp was observed for the Al-coveredSQW sample owing to the finite contribution of the plasmon

DOS even for energies significantly below the resonance.

The modelling of the temperature dependence of the Au-

covered sample was found to agree well with the experimen-

tal results. “Back bending” in the SPP dispersion relation

when including ohmic losses was shown to cause a finite

DOS above xsp for Au and Ag and lead to a measurable Fpin a limited energy range above xsp, which can potentiallybe exploited in plasmonic devices.

ACKNOWLEDGMENTS

We thank J€urgen Jopp of the Ilse Katz Institute forNanoscale Science and Technology, Ben-Gurion University

of the Negev for performing the AFM measurements.

APPENDIX: TEMPERATURE DEPENDENCE OF THEDIELECTRIC CONSTANT FOR Au, Ag, AND Al

The contributions of the critical point transitions near the

X- and L-points of the fcc Brillouin zone to the complex dielec-tric function are labeled as eCP1(x,T) and eCP2(x,T), respec-tively, and are derived in the model of Rioux et al.60 Using thenotation of Rioux et al.,60 the contributions are as follows:

eCP1 x; Tð Þ ¼ A1ð1

x01 Tð Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffixg1 Tð Þ � x01 Tð Þ

qX X2 � xþ iC1ð Þ2� dX

264

�ðxg1 Tð Þ

x01 Tð Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffixg1 Tð Þ � X

qX X2 � xþ iC1ð Þ2� dX

35; (A1)

eCP2 x; Tð Þ ¼ �A2

2 xþ iC2ð Þln 1� xþ iC2

x02 Tð Þ

� �2" #: (A2)

Each of the contributions eCP1(x,T) and eCP2(x,T) containfitting parameters A1, C1, A2, C2, x01(T), x02(T), and xg1(T)to account for the amplitudes, broadening, and critical point

transition energies of each component. The integrals in Eq.

(A1) have been solved analytically and presented in Ref. 60.

The temperature dependence of optical transition energies

x01(T), x02(T), and xg1(T) is expressed as follows:

xiðTÞ ¼ xiðT0Þ þ biðT � T0Þ; (A3)

where i¼ 01, 02, g1. In the simplest model, the three bi areassumed constant and identical for each metal.

FIG. 11. The calculated Purcell factor, Fp, vs the normalized damping fac-tor, ~c¼ c/c (T), for the Ag, Al, and Au films. The calculations are shown fortemperatures (T) of 50 and 300 K.

FIG. 12. The calculated Fp at the energy of the SQW emission as a functionof the distance of the SQW from the metal layer (z) for T¼ 50 K.

043105-12 Estrin et al. J. Appl. Phys. 117, 043105 (2015)

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The Drude parameters of xp(T), c(T) are given by Eqs.(3)–(6). The total complex dielectric constant is given by the

sum of all components involving Eqs. (3)–(6), (A1), and

(A2) as follows:

e x; Tð Þ ¼ e1 �xp Tð Þ

x2 þ ixc Tð Þþ eCP1 x; Tð Þ þ eCP2 x; Tð Þ:

(A4)

In order to determine e(x,T) for Au, we have fit the recentexperimental ellipsometry data of Kuryoz et al. obtained atT¼ 300 K who studied the effects of varying thicknesses,different adhesive seed layers, and annealing of the films on

e(x) in the 300 to 1000 nm wavelength range.69 In Table I,we show results of our fits to the room temperature e(x) dataof Kuryoz et al. (Ref. 69) obtained for a gold film thicknessof 25 nm in which the Au film preparation was similar to our

Au film growth on the InGaN/GaN MQW samples. The e(x)data of aluminum (bulk sample) and silver (80 nm-thick

film) at room temperature were acquired from Refs. 51 and

60, respectively. The fits of the experimental e(x, 300 K)data for the Ag and Au films are shown along with the calcu-

lated e(x, T) curves for various temperatures between 50 and300 K in Fig. 13. Due to the absence of d-band transitions inaluminum, the contributions of eCP1(x,T) and eCP2(x,T) areobviously not applicable (N/A) and were excluded for the

case of aluminum. Moreover, the interband absorption peak,

observed in aluminum at �1.5 eV, is not included in themodel since it is not expected to appreciably affect e(x,T) inthe range of the SQW emission (i.e., from 2.8 to 3.0 eV).70

The complete set of fitting parameters used to generate

e(x,T) for the three metal films are summarized in TableI.61,71–77 Parameters not shown with a reference were

obtained from our fits and calculations.

1S. Nakamura, M. Senoh, S. Nagahama, N. Iwasa, T. Yamada, T.

Matsushita, H. Kiyoku, and Y. Sigimoto, Appl. Phys. Lett. 68, 2105 (1996).2S. Nakamura, G. Fasol, and S. Pearton, The Blue Laser Diode (Springer,Berlin, 2000).

3H. Sato, R. B. Chung, H. Hirasawa, N. Fellows, H. Masui, F. Wu, M.

Saito, K. Fujito, J. S. Speck, S. P. DenBaars, and S. Nakamura, Appl.

Phys. Lett. 92, 221110 (2008).4M. Funato, M. Ueda, Y. Kawakami, Y. Narukawa, T. Kosugi, M.

Takahashi, and T. Mukai, Jpn. J. Appl. Phys., Part 2 45, L659 (2006).5X. Gu, T. Qiu, W. Zhang, and P. K. Chu, Nanoscale Res. Lett. 6, 199(2011).

6N. Gao, K. Huang, J. Li, S. Li, X. Yang, and J. Kang, Sci. Rep. 2, 816 (2012).7R. H. Ritchie, Phys. Rev. 106(5), 874 (1957).

FIG. 13. Fits of the dielectric function model (i.e., e(x)¼ e0(x)þ ie00(x)) tothe experimental dielectric constant data for Au and Ag thin films at room

temperature. The dielectric constant data (at T¼ 300 K) for the Au and Agfilms are from Kuryoz et al. (Ref. 69) and Rioux et al. (Ref. 60), respec-tively. The thicknesses of the deposited Au and Ag films are 25 and 80 nm,

respectively. Parameters obtained from the fits are shown in Table I. The

calculated surface plasmon energies (xsp) at room temperature for Au/GaNand Ag/GaN (both with t¼ 20 nm) from solutions of Eq. (2) are also shown.The temperatures for which the e(x) curves were calculated are 50, 60, 70,85, 100, 120, 150, 180, 210, 250, and 300 K and are indicated by arrows in

the panels.

TABLE I. Fitting parameters used to calculate e(x,T) for the Au, Ag, and Alfilms.

Metal Au Ag Al

Thickness (nm) 25 80 Bulk

�1 1.9636 1.4240 1.0124

xp(T0) (eV) 8.9306 8.4937 15.789xg1 (eV) 2.6236 4.0530 N/Ax01 (eV) 2.6067 3.8586 N/AC1 (eV) 0.17880 0.01419 N/AA1 258.27 42.745 N/A

x02 (eV) 3.5369 4.1844 N/AC2 (eV) 0.35467 0.18819 N/AA2 25.721 36.019 N/A

b (10�4 eV/K) �10a �6a N/AHD (K) 165

b 215b 394b

Ef (eV) 5.53b 5.48b 11.7b

C 0.55c 0.55c 0.85d

a (10�5 1/K) 2.26e 1.89f 2.31f

D 0.77g 0.73g 0.4h

x0 (eV) 0.0384 0.0247 0.9254xe�e (eV),

i T¼ 50 K 0.0417 0.0399 0.00307xe�e (eV),

i T¼ 300 K 0.0418 0.0400 0.00308xe�ph (eV), T¼ 50 K 0.0179 0.0106 0.2316xe�ph (eV), T¼ 300 K 0.0710 0.0355 0.7722

aReference 61.bReference 71.cReference 72.dReference 73.eReference 74.fReference 75.gReference 76.hReference 77.iCalculated at the energy of the SQW (�2.884 eV).

043105-13 Estrin et al. J. Appl. Phys. 117, 043105 (2015)

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Erratum: “Temperature dependence of exciton-surface plasmon polaritoncoupling in Ag, Au, and Al films on InxGa12xN/GaN quantum wells studiedwith time-resolved cathodoluminescence” [J. Appl. Phys. 117, 043105 (2015)]

Y. Estrin,1 D. H. Rich,1,a) S. Keller,2 and S. P. DenBaars21Department of Physics and The Ilse Katz Institute for Nanoscale Science and Technology, Ben-GurionUniversity of the Negev, P.O. Box 653, Beer-Sheva 84105, Israel2Electrical and Computer Engineering and Materials Departments, University of California, Santa Barbara,California 93111, USA

(Received 24 February 2015; accepted 26 February 2015; published online 5 March 2015)

[http://dx.doi.org/10.1063/1.4914350]

Equations (3), (A2), and (A4) contained typographical errors in the original article.1 The corrected equations are as

follows:

eD x; Tð Þ ¼ e1 �x2p Tð Þ

x2 þ ixc Tð Þ; (3)

eCP2 x; Tð Þ ¼ �A2

2 xþ iC2ð Þ2ln 1� xþ iC2

x02 Tð Þ

� �2" #; (A2)

e x; Tð Þ ¼ e1 �x2p Tð Þ

x2 þ ixc Tð Þ þ eCP1 x; Tð Þ þ eCP2 x; Tð Þ: (A4)

All calculations and model fitting described in the article were performed with the correct equations, as indicated above.1

The c/n “light line” in Fig. 8(c) was plotted with the incorrect slope.1 The corrected graph for Fig. 8(c) is as follows:

1Y. Estrin, D. H. Rich, S. Keller, and S. P. DenBaars, J. Appl. Phys. 117, 043105 (2015).

FIG. 8(c).

a)Author to whom correspondence should be addressed. Electronic mail: danrich@bgu.ac.il

0021-8979/2015/117(9)/099902/1/$30.00 VC 2015 AIP Publishing LLC117, 099902-1

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