2010_65 years of zno research – old and very recent results

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65 years of ZnO research old

and very recent resultsClaus Klingshirn*, J. Fallert, H. Zhou, J. Sartor, C. Thiele, F. Maier-Flaig, D. Schneider, and H. Kalt

Institut fur Angewandte Physik, Karlsruher Institut fur Technologie (KIT), Wolfgang Gaede Str. 1, 76131 Karlsruhe, Germany

Received 17 September 2009, revised 12 January 2010, accepted 13 January 2010

Published online 28 April 2010

Keywords excitons, photoluminescence, ZnO nanostructures

* Corresponding author: e-mail [email protected], Phone: 49 (0)721 608 3417, Fax: 49(0)721 608 8480

The research on ZnO has a long history but experiences an

extremely vivid revival during the last 10 years. We critically

discuss in this didactical review old and new results concen-

trating on optical properties but presenting shortly also a few

aspects of other fields like transport or magnetic properties. We

start generally with theproperties of bulk samples, proceed then

to epitaxial layers and nanorods, which have in many respects

properties identical to bulk samples and end in several cases

with data on quantum wells or nano crystallites. Since it is a

didactical review, we present explicitly misconceptions found

frequently in submitted or published papers, with the aim to

help young scientists entering this field to improve thequality of

their submitted manuscripts. We finish with an appendix on

quasi two- and one-dimensional exciton cavity polaritons.

2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Introduction Zinc oxide (ZnO) is a wide gapsemiconductor [Eg(0K) (3.441 0.003) eV; Eg(300 K) (3.365 0.005) eV]. It belongs to the group of IIb-VIcompound semiconductors. It crystallizes almost exclu-sively in the hexagonal wurtzite type structure (point groupC6v or 6 mm, space group C

46v or P63mc). Compared to

similar IIbVI(e.g. ZnS)or IIIV (e.g. GaN) semiconductorsit has a relatively strong polar binding and a relatively largeexciton binding energy of (59.5 0.5) meV. Due to the lightmass of oxygen, the up most LO phonon has a relativelyhighenergy of 72meV. The density of ZnO is 5.6 gcm3

corresponding to 4.2 1022 ZnO molecules per cm3.These data can be found e.g. in a recent data collection

[1] and references therein.ZnO is a long known semiconductor, withresearch going

back to the first quarter of the last century. The timementioned in the titleof this paper is simply based on the factthat the oldest references which are cited here are from theyear 1944.

The research history, with peaks occurring from time totime, is documented apart from many scientific papers by alarge numberof review articles andbooks,from which we givehere a selection[210,1216]. Duringthelast5 to 10years, theresearch on ZnO experiences a very vivid renaissance (orhype) with more than 2 000 ZnO relevant papers per year.The total number of these papers exceeds26 000 as indicated

e.g. by the databases INSPEC or web of science.

This huge number of publications makes it completelyimpossible to cite or even to read all relevant ones for thiscontribution. We restrict ourselves therefore to a small,partly arbitrary selection of references, refer the reader formore e.g. to thedatabases mentioned above or themore recentones of the reviews and apologize for this shortcoming.

After this introduction, we shortly review some aspectsof presentand future applications, growth, doping, electronictransport, band structure and deep centre luminescence.Then we present in more detail linear and nonlinear opticalproperties close to the fundamental absorption edge,including lasing and stimulated emission as well asdynamics. The contribution closes with a short outlook,

followed by an appendix in which we comment recent workon quasi one- and two-dimensional cavities.

2 Applications The present world wide researchactivities on ZnO, its alloys Zn1xAxO (with ACd, Mgor Be) and on nano- or quantumstructures based on them, aredriven by various hopes. The predominant one is possiblythe hope to obtain a material for optoelectronics covering thespectral range from the green (ACd) over the blue to thenear UV (AMg or Be), especially to obtain light emitting(LED) or laser diodes (LD) in these spectral ranges. ZnOcould in case of success supplement or replace the GaN-based devices. The main problem here is still a high, stable,

reproducible p-type doping.

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There are many reports on LED involving bothhomojunctions based on ZnO and its alloys and heterojunc-tions in which generally electrons from n-type ZnO areinjected into a p-type material like GaN, SiC etc. including

even organic semiconductors. For a few examples seeRefs. [817] and references therein. There are up to now onlyvery few claims of electrically pumped LDs [18, 19].

A common featureis the fact that the luminescence yieldis in all cases in these L(E)D very low and the emissionspectrum has frequently large contributions from deep centreemission. For a LD a light output of 0.5mW has beenreported for a forward current of 60 mA at Ub% 20V [19]. Ifwe calculate the efficiency h in the favourable way as

h photon flux out=electronhole flux in (1)

we find h% 105 even neglecting in this definition the fact

that the forward bias voltage Ub is considerably higher thanthe emitted photon energy. The present situation is (possiblyunintentionally) very well characterized by the cover photoof [15] which shows presumably the optically pumped blue/UV emission from a ZnO sample in the upper half and aLED in the lower which is still dark. Hopefully, it will emitbright blue/UV light in the future.

Another aspect driving the present research is the strongtendency of ZnO to self-organized growth in the form ofnano rods [1013] and references therein. For an example ofregular growth on a prepatterned surface see Fig. 1. Suchnanostructures could help to develop nano lasers etc.

Apart from more or less well aligned ensembles of

nanorods, there are other self organized nanostructures, likenano cabbages, corals etc. which are frequently nothing butunsuccessful attempts to grow a good epitaxial layer or nanorods.

Still another driving force for the ZnO revival is the factthat diluted magnetic or even ferromagnetic samples can begrown by doping ZnO:X with magnetic ions like Mn, Fe,Co, Ni, V or even with nonmagnetic ones like Ag, Cu, N orwith intrinsic defects [22, 23]. It is stillan openquestion if theresulting hysteresis loops observed partly up to room

temperature (RT) are due to the doped or alloyed ZnO itselfor to precipitates and clusters of other phases or even of thesubstrate alone [14, 22, 25]. For a short list of pro and contrapapers see e.g. Ref. [24] and references therein. Note that the

absence of diffraction orders other than ZnO related ones inXRD is a necessary but by no means sufficient criterion toexclude other phases. Raman scattering revealed e.g. otherphases, which have not been seen in XRD [26].

The observation of ferromagnetism immediately trig-gered heavy speculations to use ZnO:X in spintronics or evenin quantum computing. The facts that the para- orferromagnetic magnetization is tiny (apart from the signfrequently of the same order of magnitude as diamagnetism[27] and that the resulting magnetic flux densitiesB can onlybe measured with high sensitive SQUID arrangements andare orders of magnitude smaller than themagnetic field of theearth, shed serious doubt on this field of applications [28].

Other future hopes of application which are partlyalready in use or close to it are:

transparent electronics,basedon thelarge band gapof ZnO,e.g. in the form of field effect transistors (FET) or(transparent) thin film transitions (TFT or TTFT, respec-tively), which do not necessarily require a pn junction[29, 30],

theuseofZnO:Y(withYAlbutalsoInorGa)asahighlyconductive transparent oxide (TCO), in addition to themore expensive and poisonous ITO [30, 31],

the use ofZnO asa gas sensor, due to the strong sensitivityof its (surface-) conductivity on the surrounding gas

atmosphere [32], the use of pointed nanorods as field emitters [33], theuseas materialfor randomlasers asdiscussedin Section

10 below with Refs. [109111], the application in solar cells. [11]

Actually ZnO is used already by 100 000 tons per year asadditive to concrete or to the rubber of tires and in smallerquantities as additive to human and animal food, as UVblocker in sun cremes, as anti-inflammatory component incremes and ointments, as white pigment in paints and glazes,as catalyst etc. [1416].

3 Growth After this short tour dhorizon on appli-cations a few words on growth.

Bulk ZnO has been grown by gas transport methods[34], hydrothermal growth [35] or from melt [36]. Highquality epitaxial layers or quantum wells (QW) can begrown by molecular beam epitaxy (MBE), metal organicvapour phase epitaxy (MOVPE) or pulsed laser deposition(PLD). For large area growth or coatings also magnetronsputtering, spray pyrolysis and other techniques are usedalso including the oxidation of a film of metallic zinc [14,15, 22].

Nano rods are frequently grown on substrates prepat-

terned with small dots of Au or other metals in a way known

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Figure 1 An ensemble of nanorods grown on a prepatterned sur-

face, according to H.Z. in Ref. [21].

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as vapourliquidsolid process (VLS): ZnO is reduced to Znvapour at elevated temperatures by graphite or hydrogen.This vapour is transported by an inert gas to the heatedsubstrate and reacts with Au, forming a low melting eutectic.

From this liquid phase the ZnO rods grow if oxygen isadmitted. If the intermediate liquid state is not used onespeaks about vapoursolid process (VS). An example hasbeen shown already with Fig. 1 above. For an early examplesee the work by Fuller in Ref. [21].

4 Doping Doping ZnO n-type is easy with group IIIelements on Zn-place [37]. Doping levels up ton 1021 cm3 can be reached. Doping with group VIIelements on the anion site, which proved very efficient inZnSe and other IIbVI compounds ([38] and referencestherein) is much less investigated in ZnO.

Doping ZnO p-type is as mentioned already notoriously

difficult.Group I elements (Li, Na, K) on Zn place form deepacceptors, resulting in high resistivity material but not inefficient p-type conductivity. Group V elements (N, P, As,Sb) on oxygen place result sometimes in p-type conductivitysometimes also in n-type. Li may form a complex, resultingin p-type conductivity. Some authors claim also theobservation of p-type after incorporation of Ag, Cu, C andother elements. See for a few older and more recent papers[39, 40].

Unfortunately, p-type conductivityis often claimed fromHall or van der Pauw measurements, however on samples,which are inhomogeneous in their plane or into the depthwhich opens the door for misinterpretations. Furthermore Zn

may act as a p-type dopant for some substrates (e.g. for GaAsor Si) and/or enhancement-, depletion- or inversion- layersmay form at the interface to the substrate. The change ofconductivity in FET-like structures in dependence of thepolarity of the gate voltage on the other hand gives supportto the claim of p-type doping [40]. Sometimes p-type isclaimed only from the observation of excitons bound toneutral acceptors or free-to-bound or donatoracceptor pairrecombination in luminescence. Another point of concern inp-type doping is that concentrations of the dopant in the ingotor in the produced samples are claimed in the range up to5 to 10%, i.e. up to 4 1021 cm3 while the resulting holeconcentration is generally only in the 1016 to 1017 cm3

range. This means that most of the doping material does notact as the desired shallow acceptor but forms something elsee.g. precipitates [26] if it is incorporated into the sample atall. A detailed analysis of this aspect and of the solubility isfrequently missing but important, especially if dopants areintroduced with different ion radii or valence states.

In Ref. [41] it has been shown that Li concentrations inthe range of 3 000 ppm (or 1020 cm3) lead already toprecipitates with pan-cake like shape and diameters in therange of 2mm, which are seen in anisotropic light scattering.Even higher concentrations lead to macroscopic cracks andprecipitates in the samples as shown in Fig. 2 or in Ref. [41].

The sample in Fig. 2b and c has been diffusion doped by

annealing a nominally undoped ZnO bulk sample

(dimensions of several mm in all directions) for 18 days inambient atmosphere at 870 8C in a sarcophagus made fromsintered ZnO containing about 1% of LiOH. The tree-shapedprecipitates grew obviously from the cracks in one direction

only.Homogenous concentrations around or even above 1%should be considered as alloys and no longer as doping withthe usual consequences of impurity bands or localized tailstates, showing up in absorption or emission or in theoccurrence of a mobility edge [42].

In contradiction to these expectations, frequently narrowluminescence features are reported at low temperaturesindicating that the densityof incorporated species is orders ofmagnitude lower in the regions from which the luminescenceoriginates than the claimed doping levels.

Despite some success, it is still fair to say again thatp-type doping is still an unsolved problem concerning

stability, reproducibility and efficiency. Neither group I norgroup V dopants nor co-doping with either two differentacceptors or with an acceptor and a donor species broughtuntil now the breakthrough to high, stable and reproduciblep-type conductivity. However, it does not need much of aprophetic gift to predict that application oriented ZnOresearch will decay if this problem can not be solved in thenear future.

5 Electronic transport The Hall mobility of elec-trons has been measured by many authors [43]. The mobilityin good bulk samples, epilayers or nanorods is about 200 to500 cm2 /Vs at RT, limited by the intrinsic process of

scattering with LO phonons. It goes with decreasing

1426 C. Klingshirn et al.: 65 years of ZnO research old and very recent results

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Figure 2 Strongly Li-doped samples (concentrations in the rangeabove 1020 cm3) which are showing platelet type defects (a) andcracks in the sample (b) and precipitates originating from thesecracks (c). The height of the tree-like precipitates in (c) is about100mm. Partly according to Ref. [41].

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temperature through a maximum of a few 1 000 cm2/Vsaround 100 K and decays at lower temperature due toscattering with (ionized) impurities.

In quasi two-dimensional QWs and heterojunctions, this

latter process can be reduced and values between 3 000 and10000cm2/Vs have been obtained in the 1 K region [46].These values are sufficient to observe the integer quantum-Hall effect (IQHE) in ZnO [46].

If scattering at grain boundaries comes additionallyinto play, e.g. in pressed and sintered samples or in(epi-)layers consisting of small grains, the mobility may besignificantly reduced, at RT to values as low as a few tencm2/Vs [43, 44] or even below.

The hole mobility is generally considerably lower thanthe electron mobility. Typical values reported at RT rangefrom a few to some ten cm2/Vs [45].

6 Band structure and band alignments The bandstructure is shown schematically in Fig. 3. As stated already,ZnO is a wide, direct gap semiconductor. The conductionband (CB) with symmetry G7 including spin originates fromthe empty 4s levels of Zn or the antibonding sp3 hybridstates, adapting the views of ionic or covalent binding,respectively. The valence band (VB) results in this sensefrom the occupied 2p levels of O or the bonding sp3 states. Itis split by the hexagonal crystal field and by spin orbitcoupling into three, two fold degenerate sub-VBs, which arecalled from top to bottom A, B and C VB in all wurtzite typesemiconductors. The usual ordering is AG9, BG7 and CG7. InZnO the inverted VB ordering has been introduced in Ref.

[47] and since then there is a discussion about normal orinverted VB ordering in ZnO [48]. The inverted VB

ordering has been (quietly) accepted by many authors sincethen [49]. The idea of the normal VB ordering has beenrevived with the renaissance of ZnO research, partly by thesame authors [50]. It has been shownthatboth the old and the

new arguments in favour of the normal VB ordering in ZnOlack stringency [51]. These arguments against the normalordering have been written up several times [9, 14, 16].Therefore it is not necessary to repeat them here. Inagreement with [15] we state that the inverted VB-orderinghas been adopted in the meantime by the majority ofscientists and use it throughout in the following.

7 Deep centre luminescence Similarly to otherwide gap semiconductors, ZnO has a lot of defect levels inthe forbidden gap. Many of them can give rise toluminescence and absorption bands covering the spectralrange from the violet over the whole visible to the IR [52]. A

colour photograph published in Refs. [11, 14] illustratessome of these emission bands. In the past many of theseemission bands have been attributed to extrinsic defects likesubstitutional Cuzn (giving a green emission band with a wellpronounced phonon structure) [52, 53] which has also beenpartly attributed to Vanadium [54]. In the red to yellowspectral range recombination via deep Li and Na acceptorshas been identified. This identification was very thoroughlymade based on optical spectroscopy, doping experiments,EPR and thermo luminescence studies. See the work of D.Z.in Ref. [39] or Refs. [52, 55].

In recent years intrinsic defects have been stressed more,with the exception of an unstructured green emission band,

which has been attributed since decades to an oxygenvacancy V0 [56]. Now many of these emission bandsincluding one in the violet are attributed to all types ofintrinsic defects like vacancies or interstitials or anti-sitedefects. However, these interpretations are still stronglycontroversial [57] and the arguments for the identificationare often less stringent, so that more detailed results arerequired, to come to clear assignments.

The external luminescence yield of deep centres (and ofthe near edge luminescence) is typically around or below10% at low temperatures and decreases with increasingtemperature [58]. Very weak deep centre luminescence canbe a sign of good sample quality or of the presence of fast non

radiative recombination centres like Co, Fe or Ni [59] butalso Cu at higher concentrations.

8 Free excitons and the Urbach tail According tothe band-structure in Fig. 3 there are three series of excitonslabelled according to the involved hole.

The selection rules for dipole allowed transitions withoutspin flip result in strong oscillator strengths for the nB 1 Aand BG5 excitons forE?c and for the CG1 excitons inEjjc asshown in Fig. 4 where we show in reflection the series of A, Band C exciton with main quantum numbers nB 1, 2, 3 intheir respective polarizations and a close-up of the nB 1states with some line-shape fits. Further examples are found

e.g. in Refs [1, 6, 7, 60].


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Figure 3 (onlinecolourat:www.pss-b.com)Schematicofthebandstructure of ZnO. A k-linear term possible for G7 states for kperpendicular to c is shown schematically for the AG7 valenceband. The polarizations of the band-to-band transitions which

are dipole allowed and do not involve a spin flip are indicated.



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Note that the longitudinal-transverse splitting DLT is inthe case of two close lying resonances not directlyproportional to the oscillator strength f. The value ofDLTof the energetically lower resonance is reduced, that of thehigher one increased. See e.g. Refs [28, 61]. This situation is

met for the A and BG5 excitons which have roughly equalvalues offbut significantly different ones for DLT. See alsoTable 1.

The longitudinal eigen energies hvL are situated slightlybelow the reflection minima which occur at the spectralposition, where the refractive index of the UPB has the value1, the transverse ones, named frequently hv0 or hvT, areslightly below the reflection maximum. Their precisepositions can only be determined by an advanced line shapefit. See Fig. 4 for an example.

Note further that the reflection spectra are stronglypolarized. The observation of nB 1 AG5, BG5 and CG1 at10 K excitons in reflection forEjjc with approximately equal

signal strength as reported e.g. in Ref. [62] is not possible.The same is true for the absence of the CG1 exciton forunpolarized light. Either a poor sample with differentlyoriented crystal domains has been used or the notation of thestates of polarization is not correct. But even if the dotted linein the cited Figures would correspond to unpolarized lightand the solid one to E ? c the shape of the dotted spectrumremains strange. At higher temperatures, the A and B G5signals may merge to one as is evidently the case in Ref. [63].See also [14].

The free exciton resonances can also be seen inabsorption as demonstrated in Fig. 5 for low temperaturesand RT and for the two polarizations.

Again the different widths of the nB 1 A and BG5features are obvious at low temperatures. At higher energiesfollow states with nB 2, 3 and structures connected withLO-phonons, which are also known as excitonphononbound states [28]. At RT, the absorption spectra broaden. Seee.g. also [4, 65]. The position of the energy gap can be easilydeduced from the absorption spectra in Fig. 5. It is situatedby the exciton binding energy above the exciton peaki.e. at3.37 eV for T 295 K.

Note that the widths of the reflection and absorptionfeatures do not reflect the homogeneous broadening of theexciton resonances at low temperatures but are rathergiven by DLT. Only at higher temperatures the damping

G dominates the width of the absorption spectra. G varies


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Table 1 Some properties of the nB 1 excitons with large oscil-lator strength in ZnO at low temperatures (T 5 K), namely thetransverse eigen energies and the longitudinaltransverse splitting.Data according to Refs [1, 7, 47, 49, 60, 104].

state E0 (eV) DLT (meV)

AG5 3.37550.001 2.00.2BG5 3.38150.001 111CG1 3.4200.001 121

Figure 4 The A, B and C exciton series (upper panels), andthe nB 1 states with some fits (lower panels), according to Ref.[7] and the work by Hummer in [49].



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from values G% 1meV DLT at RT. See Fig. 8 or Refs. [14, 66].

An exponential tail develops in the absorption spectra

with increasing temperature on the low energy side of the

first exciton resonance, the so-called Urbach tail [67]. It canbe described by [28]

ahv;T a0 expsTEoo hv=kBT

for hv < ET:

(2)

The quantities a0 and Eoo are material parameters. Eoo issituated slightly above the transverse exciton eigen energy atlow T. The function s(T) varies only weakly with T. Indisordered systemslike amorphous semiconductors or alloysthere is an additional, essentially temperature independentcontribution to this Urbach tail. In quantum structures (wells,superlattices, wires and dots) fluctuations of the geometricalsizes can additionally contribute to disorder and broadening.

This tail is frequently misinterpreted in submitted orpublished manuscripts to determine the energy gap (some-times also called the optical gap). Usually one finds in these

texts plots like Eq. (3a, b)

a2hv fhv; (3a)

or

hv2a2hv ghv: (3b)

They are fitted by a straight line and the intersection ofthis line with thex-axis is claimed to be the gap or the opticalgap. Examples are found e.g. for pure ZnO in Ref. [68] andthereferences therein. A first hint that thisapproachis wrong,is the observation, that the linear fit of the data generally

holds for a rather small interval of a

2

(hv) only. Thenext argument is the fact that the absorption coefficient at

the thus determined gap is still rather small (i.e. the sampleis still almost transparent) in contrast to values ofa(hv Eg)! 10

5 cm1. See e.g. Fig. 5. A third point is thatmany authors state in the beginning of their manuscriptcorrectly that Eg is in ZnO at RT at (3.365 0.05) eV butbased on the misinterpretation of their data came up laterwith values around 3.25 to 3.28 eV [68], frequently withoutbothering about this discrepancy.

The approach of Eq. (3a) is based on a so-called square-root absorption edge. See e.g. Ref. [28] or the discussion inRef. [15] with e.g. (3.59).

ahv consthv Eg1=2

for hv ! Egand ahv 0 for hv < Eg:

(4)

This behaviour is expected if one assumes simplevertical band-to-band transitions without any electron-holeCoulombcorrelation (i.e. without exciton states) andwithoutbroadening mechanics. This simple approach is neverobserved in semiconductors and it is in stark contradictionwith the correct statement frequently given in the introduc-tion of the same papers that the exciton binding energy is inZnO with a value of 60 meV relatively large. Actually theapproach (4) goes back to the middle of the last century [69]

when not much was known about the optical properties of


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Figure 5 The A, B and C exciton series seen in absorption in theirrespectivepolarizations andfor 4.2 and295 K, accordingto Ref. [64].



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semiconductors. It is not clear to the authors, why such anoutdated approach is now revived.

The approach of Eq. (3b) is valid in the so-called Tauc-regime in the absorption spectrum of disordered (e.g.

amorphous) semiconductors. It is connected with theconcept of a mobility edge and occurs only at photonenergies and absorption coefficients beyond the Urbach tail.See Ref. [28] and references therein. To make it explicitlyclear, a valid fit of the absorption spectrum with (3b)extrapolates to a characteristic energy then correctly namedoptical gap only if it is situated energetically around theupper end of the Urbach tail,not somewhere at its beginning.The use of this approach for disordered semiconductorsstrongly contrasts again with the statement frequently givenin the same paper that the used ZnO samples are of highcrystalline quality as deduced e.g. from XRD rockingcurves, HRTEM and other methods.

The energy which is determined with the above methodsgives in some way the onset of the absorption edge. But evento analyse this region, a fit with the Urbach tail would bemuch more adequate.

The free excitons and their LO-phonon satellites can alsobe seen in (photo-) luminescence. Figure 6 shows thedispersion relation and the low temperature luminescence ofthe nB 1AG5 exciton polariton for the orientationsE?c andk?c (Fig. 6 a, b) or kjjc (Fig. 6c, d). The influence of the k-linear term in the A VB for k?c is obvious, which mixes theAG5 spin singlet with the A(G1G2) spin triplet states andgives them a k?c-dependent oscillator strength. For Ejjc(Fig. 6e, f) one observes the spectrally much narrower

luminescence signal from the AG1 state, which has a muchlower oscillator strength because it involves a spin flip,though being dipole allowed in this orientation. It is evidentfrom Fig. 6 that the assignments in Fig. 1 of Refs. [62a] or inRefs. [62b] are open for discussion, which attribute twoemission peaks to the transverse nB 1AG5 exciton and to itslower polariton branch (LPB) separately, while the triplet(AG6 in the band structure notation of Ref. [62]) is situatedabove the LPB and is of comparable intensity and width.

At higher temperatures the zero phonon emission of Aand BG5 excitons may be fitted with Lorentzians. See Fig. 7.Their LO-phonon replica can be modelled including thesquare-root density of states of effective mass particles

(here the excitons), a Boltzmann occupation probability andthe proper dependence of the transition probability on k, thewhole result folded with the Lorentzian used already for thezero phonon line. The exciton temperature is assumed tobe identical to the lattice temperature. This is a validassumption for low excitation levels, small excess energyand the temperature range covered in Fig. 7. For more detailssee Ref. [66] and references therein. Note that there is nophysical reason for a fit with one or several Gaussians.

The LO phononenergy is found in the distanceof the freeexciton to the low energy side of the LO phonon satellites(not to the maxima which are influenced by the thermalkinetic energy of the excitons), roughly the point of

inflection. Furthermore the luminescence may be influenced

by reabsorption through the Urbach tail at higher tempera-tures as shown in Fig. 7 for 290 K. The analysis of the

luminescence spectra allows determining the homogenouswidth ofthe A and BG5 excitons as a function of temperature(see Fig. 8 for the HWHM data) and the shift of the band gap(see Fig. 9) where a fit using the theory ofRef. [70] is shown.The equations derived there are more complicated than thefrequently used Varshni formula and are therefore notreproduced here.

We already stated that the optical properties of goodepilayers andof nano rods areessentially identical to theonesof good bulk samples with possibly two differences.Absorptionand reflection spectra may be difficult to measureon nano rods and the details of their optical properties may beinfluenced by their waveguide properties. (See Section 10).

To verify this general statement we show in Fig. 10 the


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Figure 6 Thepolaritondispersionandtheluminescenceofthezerophonon exciton polaritons for various orientations at 1.8 K, accord-ing to the work of Hummer in Refs. [49] and [14].



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photoluminescence of a single nano rod for two differenttemperatures, which looks very much like the luminescenceof a good bulk sample.

In alloys the emission and absorption features broadeninhomogeneously due to alloy disorder, resulting in asubstantial Stokes shift between the two and an overall shiftdue to the dependence of the gap and of the exciton bindingenergy on the composition. Furthermore an n-shapeddependence of the luminescence maximum can be observedwith increasing Twhich results from temperature dependentrelaxation processes in the tail states [28]. This n-shapeddependence means, as in other semiconductor alloys [28],

that the emission maximum shifts with increasing

temperature at low temperatures faster to the red than theband gap does. This region is followed by a temperatureinterval in which a blue-shift dominates caused by thermalpopulation of higher states, followed again by a red-shiftfollowing then essentially the temperature dependence of thegap. If this dependence is plotted over the temperature, theresulting curve looks like a badly written small n. Strangeenough, this dependence is called in the literature frequentlyan s-shaped behaviour.

In Fig. 11 we show a set of spectra of MgxZn1xO at4.2 K as an example. For x 0 the broadened (AB) G5nB 1 exciton resonance can be seen at 3.375 eV and abovethe onset of band-to-band absorption. The full trianglesindicate the LO phonon replica in luminescence. The fullcircles in the insert give the exciton energies.


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Figure 9 The dependence of the band gap on temperature asdeducedfrom luminescence andabsorption spectroscopy,according

to Refs. [66, 71].

Figure 8 The homogeneous width (HWHM) of the free A and B

excitons as a function of temperature, according to Refs. [66, 71].

Figure 7 (online colour at: www.pss-b.com) The luminescenceofthefree exciton polaritonand of itsLO-phonon satellites forvarioustemperatures, according to Refs. [14, 66, 71].



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In quantum structures there is in addition to alloydisorder in the well and/or the barrier also a furthercontribution to inhomogeneous broadening due to wellwidth fluctuations. For some examples see Ref. [74].

9 Bound exciton complexes, free-to-bound and

donoracceptor pair transitions The low temperature(T 50 K) near edge luminescence of high quality bulksamples, epilayers and nanorods is dominated by radiativerecombination of bound exciton complexes (BEC or BE).See Fig. 10 or Refs. [7580] for an example. The width ofindividual lines can be at low temperatures (T% 1.8 K) asnarrow as 100meV [77]. In lower quality samples, they arebroader and the whole BEC emission spectrum may mergeinto one inhomogeneously broadened band. See e.g. thecomparison of the luminescence spectra in Ref. [63].

The bound exciton lines have been labelled by Reynoldsas I0 to I10 from higher to lower emission energies [75]. In themeantime many more lines have been observed, labelled e.g.

as I6a etc. [78].The simplest, straight forward assignment of these lines

would be to attribute those with the lowest binding energiesto excitons bound to ionized donors DX (e.g. I2 and I3) thenext to excitons bound to neutral donors D0X (e.g. I4, I6, I8,and I9) and the deepest ones to A

0X (e.g. I9, I10, and I12).Excitons do not form a bound state with ionized acceptors inZnO. For some of the D0X the chemical nature of the centrecould be identified (I4:H,I6:Al,I8:GaandI9:In). See the workby Meyer in Ref. [76]. The search for and interpretation ofD0X and A0X states has varied with time. First the lines I9and I10 have been attributed to A

0X based on thethermalization properties in a B-field and on doping

experiments with Li and Na. See the work by Tomzig in

Refs. [76] or [79]. Furthermore excited states have beenobserved for these BEC in photoluminescence excitationspectroscopy containing one or two holes from the B-VB[79]. The latter is only possible for A0X. Later on even amajority of the BEC lines has been attributed to A0X [80],which is surprisingto some extend, since nominally undopedZnO is always n-type. Presently there is agreement that bothD0X and A0X may occur in the low energy part of the BECspectrum, i.e. in the range from I8 down to I12 [78]. This

means that a decision between D0

Xa ndA0

X cannot be madefrom the spectral position alone.

On the low energy side of the BEC there are furthercontributions to the near edge emission, from which wemention some in the following.

There are the LO-phonon satellites of BEC and of freeexcitons. See e.g. Fig. 10. Then there are the two electronsatellites (TES) in the interval from 3.29 to 3.33 eV. Theorigin of these TES is the radiative recombination of a D 0Xwith simultaneous transfer of the electron of the remainingD0 into an excited state like 2s or 2p. The observation allowsto determine the binding energy of the electron to the donorEbD, assuming a hydrogen like series of states, and to verify

the so-called Haynes-rule [81] which relates EbD with the


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Figure 11 Luminescence and absorption spectra of a set ofMgxZn1xO films. According to Ref. [73].

Figure 10 (online colour at: www.pss-b.com) The luminescenceofa singlenanorod at10K and 50K (a) and a close uptothefreeandbound excitons at 10 K (b). The LO phonon replica of the BEC aremarked by vertical arrows. According to Ref. [72].



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binding (or localization) energy of the exciton in the BECEb

D0X

via

EbD0X

a b EbD; (5)

originally formulated with a 0. The constants a and b canbe determined from experiment resulting in a3.8 meVand b 0.365 [76].

Furthermore, there are excitons bound to deep centrese.g. iso-electronic traps or to structural defects e.g. theY-band or so-called surface excitons. These are excitonsbound to centres at or close to the surface, i.e. in a surfacelayer with a thickness comparable to the exciton radius [82](not to be confused with surface exciton-polaritons [83]).

Free-to-bound transitions describe the recombination ofa free carrier (in ZnO usually an electron) with a bound one(in ZnO frequently a hole bound to an acceptor). Thereforethe band is also known as eA0. The thermal distribution of

the free carrier is seen in the luminescence spectrum becausek-conservation is relaxed by the localized carrier. The eA0

emission may dominate in less pure samples up to RT.Frequently its zero-phonon band coincides with the firstLO-phonon replica of the free exciton.

Donoracceptor pair recombination occurs at a photonenergy given by

hvEg Eb

D0 Eb

A0 mhvLO e2=4pee0rDA: (6)

The last term in Eq. (6) describes the Coulomb energy ofthe ionized donor and acceptor after the recombinationprocess. It results in a blue shift of the emission with

increasing (excitation-) density i.e. with decreasing (aver-age) distance rDA.The assignment of various luminescence features to

these above-mentioned processes is not unique and still apoint of controversial discussion as outlined by Waag in Ref.[16] chapter 5. Therefore we give here only one morereference as an example [84].

In alloys and QWs the BEC emission merges usuallywith the recombination of excitons in localized tail states sothat an easy distinction between an exciton localized onlyby disorder or bound additionally to some defect is hardlypossible.

10 Stimulated emission The processes for stimu-lated emission are in ZnO essentially the same for highquality bulk samples, epilayers, nano rods, nano-crystals andpartly also QWs.

Stimulated emission is usually characterized by athreshold behaviour of the input/output characteristics

ILumf(Iexc) (see e.g. Ref. [28] or Fig. 19), a narrowing ofthe emission spectrum or the appearance of a mode pattern(see e.g. Figs. 19 and 20) and spatially directed emission. SeeFig. 12.

The photo of Fig. 12 hasbeen obtained under two-photonexcitation of a rather large volume of several mm3 withpulses from a Q-switched ruby laser. Under these conditions

the laser process is the recombination of excitons under

emission of one or twoLO phonons [86, 87]. A photographic

plate has been attached just outside the cryostat window( 4 cm, distance to the ZnO sample about 10 cm) withoutany further optics except a filter, which blocked the pump-light.

As a first approach, the gain processes can be groupedas limiting cases into excitonic ones and the recombina-tion in an inverted (or degenerate) electron hole plasma(EHP) [28].

The excitonic ones include the recombination of a freeexciton under emission of one or more LO-phonons(X-mLO). The X-mLO process has relatively low thresholdand gain. Therefore it is observed preferentially if largevolumes with low losses are excited e.g. by two-photon

absorption as in Fig. 12 [8587] the positions of the emissionmaxima of the X-mLO process can be approximated by

hvmax EgT EXb mhvLO amkBT; (7)

as long as reabsorption by the Urbach tail (see above) can beneglected. In Eq. (7) Eg(T) is the temperature dependentgap energy (see Fig. 9), EXb the exciton binding energy (inZnO (59.5 0.5)meV),hvLO theLO-phonon energy(in ZnO72 meV), kB Boltzmanns constant and a(1) 3/2 anda(m> 1) 1/2. See for details of a line shape analysis alsothe explanations above with Fig. 7 and [66, 71, 87].

The next process is the inelastic excitonexcitonscattering (XX) in which one of the excitons is scattered

onto the photon-like LPB and appears as a luminescencequantum while the other reaches under energy andmomentum conservation a higher state with main quantumnumber nB 2..1, resulting in the emission bands P2 to P1.The process is called inelastic, because the total energy isconserved, but not the kinetic energy, the conservation of thelatter being the definition of elastic scattering. For ahydrogen-like exciton series the emission maxima aresituated around [87]

hvmax EgT 2 1=n2B E

Xb d3kBT; (8)

where Eg(T) is again the temperature dependent gap energy,

EXb the exciton binding energy, kB Boltzmanns constant and


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Figure 12 The spatially directed stimulated emission from a ZnOsamplejustabove(left)andwellabovethreshold(right),accordingtoRef. [85].



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d has a value of 0< d< 1, (typically d% 0.5). The laseremission occurs in this, like in many other processes [88],often on the low energy side of the maximum. The XX

process has been observed in many semiconductorsincluding ZnO [28, 91]. In Fig. 13 we give an example.With increasing temperature the nB! 2 states become

thermally populated resulting in reabsorption andan increaseof the threshold with temperature. This increase is even morepronounced, when part of the excitons become ionizedthermally.Then (for ZnO at T! 100 K) a new process sets in,namely an inelastic scattering between a free exciton and afree carrier, in ZnO generally an electron [28]. From energyand momentum conservation one can deduce that themaximum of this X-e emission process shifts with increasingtemperature faster to lower energies than the band-gap doese.g. according to [87] as

hvmax EgT EXb

3kBTM=me 1 2gM=me1=2=2; (9)

where Mis the free exciton mass, me the electron mass andgagain around 1.

The temperature dependences of the thresholds of thevarious processes mentioned above have been calculated andverified experimentally for CdS and partly for ZnO [89],though the absolute densities in the calculation for CdS aremost probably too high, since they reach well into the EHP

regime. The thresholds of the various processes dependtrivially on the losses. For the X-mLO and theXX processesit increases above approximately 100 to 150 K due toreabsorption, while the one of the X-e process decreases inthis temperature interval due to thermal ionization of freecarriers

Some other excitonic processes involve the formationand decay of biexcitons (for bulk and QWs see Refs. [9, 90])or the participation of bound exciton complexes, e.g. theiracoustic phonon side band [92, 93].

All the intrinsic processes mentioned above are shownschematically in Fig. 14. A small selection on theory andexperimental results is found in Refs. [28, 87, 93] and the

references given therein.

In Fig. 15 we show the temperature dependence ofseveral (stimulated) emission maxima.

The solid lines in Fig. 15 give the temperaturedependence of the free A and B exciton energies and oftheir LO phonon replica. The BEC and their LO replica areindicated in grey (red). The dashed lines give the Pi (i 2,1)bands for Tbelow approximately 100 K and the X-e processabove. These data are confirmed by experiment [87]. The


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Figure 14 The schematicpresentation of various excitonicrecom-bination processes in the intermediate density regime, according toRefs. [28, 87, 93].

Figure 15 Thetemperaturedependenceoftheemissionmaximaofvarious processes. See text. According to Ref. [14]. The references

[48,10,22] in the inset correspond to [94100] here.

Figure 13 Theluminescence spectrum of theXX process (P2 andP1 bands) under one photon excitation at 60 K, according to Ref.[87].



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open and closed symbols are more recent experimental data.We will come back to them later.

Most of the excitonic processes can be mapped on a fourlevel laser system (except for the X-e process) with ideallyvanishing threshold at low T. The X-LO processis inverted ifthere is one exciton in the sample and no LO phonon, the

biexciton recombination is inverted if there is one biexcitonand no exciton etc. Finite losses require of course finitethreshold densities.

If the electronhole pair density np is increased, variousthings are happening: The gap is normalized due to exchangeand correlation effects. This means that E0gnp decreasesmonotonically with increasing np. This band gap renorma-lization (BGR) is essentially temperature independent [101].The experimentally observed BGR in various II-VI semi-conductors(Fig 21.2 in Ref. [28]) is somewhat larger than thecalculated data in Ref. [101]. In Fig. 17 the solid and dashedcurves correspond to theory and an average over the II-VIdata.

The next thing is that the exciton binding energydecreases with increasing np due to Coulomb screeningand Pauli blocking, i.e. the exciton state shifts closer toE0gnp. The BGR and the decrease of E

0Xb (np) almost

compensate each other with the consequence that theexciton energy remains in absolute terms almost constant,but its homogeneous width increases as shown by thehatched area in Fig. 17. At a density nMcalled Mott densityE0Xb (np) tends to zero, i.e. excitons do no longer exist asindividual quasi particles and a new collective phase isformed, the EHP. For ZnO one finds from Fig. 17 nM 4 to9 1017 cm3. Due to the finite damping, excitons will nolonger be good quasi particles at RT already at densities np

around 1017

cm3

because then their renormalized binding

energy E0Xb (np) is smaller than the HWHM value of thehomogeneous width.

The last quantity which we introduce with Figs. 16and 17 is the chemical potential of the electronhole pairsystem m(np,T). This is the energetic distance between thequasi Fermi levels of electrons and holes describing thedistribution of the various carriers in their respective bands.The quantity m(np,T) depends strongly on both variables.Population inversion and optical gain are reached, if

mnp;T > E0

gn

p: (10)

At low temperatures the condition (10) is fulfilledalready at np nM. In Fig. 16 we show the situation for T0and in Fig. 17 for RT. Optical gain occurs in the interval

E0gnp hv mnp; T: (11)

In simplest approximation the spectral shape of the gainspectrum g(hv) is given above E0g(np) by the square rootdependence of the densityof states of the free carriers and thequasi Fermi functions according to

ghv

consthv

E0

gn

p

1=2

fenp;T fhmnp;T 1;(12)

and zero below. Actually there are some modifications: atthe low energy side a tail forms due to final state dampingand Fermi sea shake up [93, 101] and [103], at the highenergy side an enhancement of the oscillator strength occursdue to the residual electronhole Coulomb correlation [93,101].

In Fig. 18 we show gain spectra obtained for bulksamples, platelet type ones and epilayers, measured with thevariation of the excitation stripe length (Fig. 18a and c) or bypump-and probe techniques (Fig. 18b). The parameter is the

pump power. The lattice temperature is around 5 K but the


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Figure 17 The density dependencies of the BGRDEg, the excitonenergyand of thechemicalpotentialm(n

p,T 300K). According to

Refs. [14, 102].

Figure 16 Schematic representation of the recombination in aninverted (or degenerate) EHP. According to Ref. [14].



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stimulated emission in the range between a few times1017 cm3 toa few times 1018 cm3. Surprisingly most of therecent authors claim the gain mechanism for RT stimulatedemission to be inelastic XX scattering. The energetic

scattering of the data at RT in Fig. 15 immediately makesclear that different processes are involved.The inspection of Fig. 17 shows that excitons do no

longer exist at these densities at RT as individual quasi-particles since the remaining (or renormalized) bindingenergy is smaller than the homogeneous width. Therefore theassignment of gain at RT to excitonic processes will be inmost cases a misconception. The data given by open fullstarsare assigned in Ref. [95] to recombination in an invertedEHP. This interpretation is most probably a wrong one. At450 K the BGR had to be at least 300 meV resulting in npvalues well beyond 1020 cm3. These values are neithercompatible with the experimental excitation conditions nor

with the damage threshold of the sample.On the other hand it is an experimental fact thatstimulated emission is observed at RT in the density regimementioned above though neither excitons exist as good quasiparticles nor is an EHP inverted. Therefore we proposed inRefs. [14, 66] and [107] two other recombination processes,which can occur already in a non-degenerate EHP,which hassome residual Coulomb correlation between the carriers. Wenote first that in this density regime the position of therenormalized gap E0g(np) coincides approximately with theexciton energy at low densities. One of the two processes isthe recombination of an electronhole pair in the EHP underemission of a photon and of one or two LO phonons (or more

accurately plasmonphonon mixed state quanta). Thissuggestion is supported by the fact that the data given bythe open stars in Fig. 15 extrapolate the X-2LO processobserved at lower temperatures and densities. The full starsin contrast extrapolate beautifully the X-e line and couldtherefore be explained by a process in which again anelectronhole pair recombines in the EHP under emission ofa photon and simultaneously transfers part of its energy andits momentum to a third carrier, most probably anotherelectron. A detailed many-particle calculation is necessary toverify or to falsify these concepts.

The explanation developed above may be valid for bulksamples, platelets, epilayers, micro- and nano-rods, for nano

crystals and even in QWs.Nano rods have a pronounced wave guide property and

mode structure. We show in Fig. 19 the stimulated emissionfrom a single rod at 11 K under band-to-band excitation with5 ns pulses from the 3rd harmonic of a Nd-YAG laser. Thelaser modes are clearly visible. They show mode competitionwith increasing excitation (see inset and Fig. 20c) and asmall blue shift. We will come back to these aspects later.If there is no mode available for a certain emission process(e.g. the XX process at low temperatures) this process maybe suppressed [108].

Random lasinghas been observed in ZnO powders[109].Recent results of this topic including the observation of the

coexistence of spatially extended and more confined modes

are presented and discussed in Refs. [110, 111] and in aninvited contribution [113] to this conference [112]. For hintstowards random lasing in disordered ensembles of nano rodssee Ref. [114].

To conclude this section we give a few words on claimsof stimulated emission under optical pumping with powerdensities as low as 10 Wcm2 [116, 117].

An excitation power of 10 Wcm2 results with reason-able values for lifetime (%1 ns, see below) and excitationdepth (1mm) determined essentially by diffusion in a densitynp 5 10

14 cm3. This value is too low to produce an EHP[116] or significant XX scattering [117]. The appearance ofmodes observed in Ref. [116] may also occur for spon-taneous emission [118] and slopes above one [117] can beexplained without stimulated emission by simple (e.g.quadratic) recombination kinetics [28] as well.

11 Dynamics The luminescence decay time of ZnObulk samples and nano structures has been measured bymany research groups. See e.g. Refs. [9, 119] and [120] andreferences therein for some examples for bulk, epilayers,rods, QWs and nano crystallites. There are also data given ondephasing and intra- or inter (sub-) band relaxation. Weconcentrate here mainly on the lifetime. The decay dynamicsis frequently not exponential, so that one cannot speak abouta luminescence decay constant, but only about an effectivedecay time tl valid for the given density interval. Frequentlythe decaying intensity is followed in the experiments onlyover one order of magnitude or even less. This interval is byfar not sufficient to claim a single or even multi-exponential

decay. Fitting with a power law or a stretched exponential isusually also possible and partly physically even morereasonable. To claim a single exponential decay, the signalshould be followed at least over three orders of magnitude[121]. The value of tl varies from sample to sample, withtemperature and with the excitation conditions. Both for freeand bound excitons and their LO-phonon replica tl is foundgenerally in the interval

0:1 ns tl 3 ns: (13)

Differences in tl may already occur between the upperand LPBs [122]. Since the (external) luminescence yieldh of

the near edge emission is typically around 0.1 in high qualitysamples at low temperatures and tends to decrease towardshigher T [58] this luminescence decay time is significantlydifferent from the radiative lifetime tr. To identify tl with trisonly correct ifh is close to one. However, this precondition isgenerally not verified.

Under conditions of stimulated emission the decay timemay become much shorter than the range indicated by Eq.(13) reaching down to 10 ps.

We show in Fig. 20 a rather illustrative example. A singlenano rod has been excited in the band-to-band transitionregion with 150 fs pulses. The luminescence was spectrallyand temporally resolved by the combination of a spec-

trometer with a streak camera. At low cw-excitation the


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sample shows a well resolved near edge emission similar toFig. 10. At the lowest fluence F in Fig. 20 the emissionappears mainly in the M-band region. For possible origins ofthis band apart from biexciton decay see Ref. [93]. With

increasing Fthe spectrum broadens and shifts to the red. AtF 0.35mJ cm2 it is at least at the beginning due torecombination in an EHP. With further increasing F sharplaser modes develop.

We qualitatively discuss the prominent features seen inFig. 20 in the following:

Overall, the spectrum broadens and shifts to the red withincreasing F. This fact is due to the increase of the carrier pairdensity np in the EHP and the resulting increase of BGR andband filling. Which of the two effects dominates depends onnp and on T. This means that there is no such thing like asimple Burstein-Moss shift of the absorption edge (and ofother spectral features) to the blue due to filling of states.

Secondly one sees that every individual mode shifts tothe blue with increasing F. See also Fig. 20c. In the transitionto an EHP first the exciton resonances are bleached anddisappear [104, 123] and then the lower end of the band-to-band transitions is blocked between E0gnp and m(np,T).This bleaching of absorption leads via KramersKronigrelations to a decrease of the real part of the refractive indexn(np). Since the geometrical length of the rod remainsunchanged, the vacuum wavelength must have becomeshorter i.e. the photon energy must increase so that thenumber of half waves fitting in the rod length remainsconstant for a given mode.

Finally, it can be seen that within every trace the modes

shift with time back towards the red. This fact results simplyfrom recombination of the carriers and the resulting recoveryof the refractive index.

12 Conclusion and outlook We mentioned at thebeginning that the main driving force for the presentrenaissance of ZnO research is the hope to obtain green/blue/near UV light emitting (LED) or LD. Though p-typedoping is not yet solved in a stable and reproducible way,several reports exist on electrically pumped, ZnO-basedLEDs and a few on LDs. For a small selection see Refs. [817, 18] and [19]. Partly the structures are made entirely ofZnO and its alloys with MgO, BeO and CdO, partly hetero

structures are grown with n-type ZnO and p-type GaN, SiCetc.

Commonfeatures of almost all examples are at present asalready partly mentioned in chapter 2:

a low luminescence yield. See e.g. data given in theintroduction

an emission spectrum which contains a lot of deep centreluminescence or is partly even dominated by it

no information exists on the lifetime of the devices.

Because of the low valence band in ZnO [124] electronsare generally injected in heterojunctions from ZnO into the

other material and not holes into ZnO. This has the


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Figure 20 (onlinecolourat:www.pss-b.com)Timeresolvedemis-sion spectra ofa singlerodat 15 K excited with 150fs pulsesfromthesecond harmonic of a TiSa laser for various excitation fluences F.Everyhorizontaltracecovers a timeintervalof 60 ps (a),the intensityofthemodesasafunctionofF(b)andtheshiftofthemodeswithF(c).According to Ref. [115].



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unfortunate consequence that the luminescence originatesnot from the ZnO, but either from a spatially indirecttransition across the interface or from the other material ofthe hetero-junction. These materials have partly an intrinsi-

cally low luminescence yield if they have an indirect gap(like SiC) or they can also be doped n-type themselves (likeGaN), thus casting some doubt on the technical relevance ofthe heterostructures.

To conclude we may state that beautiful old and newphysics can be investigated in ZnO, its alloys and nanostructures and that many applications for ZnO exist or areemerging. The question if ZnO willsee a major breakthroughin (opto-) electronics depends strongly on the availability ofhigh, efficient, stable and reproducible p-type doping and ofefficient LEDs and LDs based on it in the next (very few!!)years to come.

Acknowledgements The authors want to thank theDeutsche Forschungsgemeinschaft for financial support in anindividual grant (Kl 354/23) and through the CFN and the Landes-stiftung Baden-Wurttemberg through the KompetenznetzFunktionelle Nanostukturen Project A1.

Appendix The fact that the deadline for the papersubmission to the proceedings of the 14th InternationalConferenceon II-VI Compounds was about 3 weeks after theendof the conference has an advantage. Some thoughts aboutcontributions to the conference can be included in thisreview. It is clear to the main author of this appendix (CK)that not all of the ideas presentedbelow are yet mature. Partly

they even contradict present state theoretical modelling.They are therefore meant as a stimulus for discussion in thescientific community.

We consider here some aspects of exciton cavitypolaritons in quasi two-dimensional and quasi one-dimen-sional systems, which are both relevant for ZnO [125131]and compare them to bulk material. Quasi two-dimensionalsystems are in the present context FabryPerot resonatorswith plane, highly reflecting mirrors on both sides (generallyBragg reflectors) containing in this cavity one or a few QW,with the exciton resonance of these QW overlappingenergetically with the two-dimensional dispersion E(kjj,k?)of the cavity modes without these QW, the so-called empty

cavity. The value ofk? is quantized by the cavity modes tomp/dwhere dis the geometrical thickness of the cavity andm 1 , 2 , 3. . .. The in-plane wave vector is given bykjj kxex kyey where ex and ey are the unit vectors in theplane of the cavity. Without loss of generality we can chooseky 0 for a system isotropic in the xy plane. Examples aree.g. GaAs, ZnO or CdTe based cavities [28, 126128] and[132].A goodoverview of this field is found in Ref. [138] andthe references therein.

The quasi one-dimensional situation has been realized inZnO micro rods of some micrometers in diameter and some10mm in length by whispering gallery modes (WGM), inwhich thek vector is quantized in the two-dimensional plane

perpendicular to the growth-, z- or c-axis of the micro rod,

while only kz can be varied (quasi-) continuously in the limitof a very long rod [129131].

We start with a relatively trivial point. It has beenclaimed that the dispersion of the empty cavity modes(i.e. of

the cavity containing no QW) and being described by theconstant refractive index of air or better of the barriermaterial of the well nb is parabolic [28, 129]. This statementis wrong. The intersection of a light cone defined over thekk k? plane by

E2kk;k? hv2 k2k k

2?c

2h2=n2b; (14a)

with the plane given by

k? mp=d; (14b)

results in a hyperbola and not a parabola. See e.g. Fig. 17.1din Ref. [28]. The curvature of this hyperbola is in itsminimum given by

@hv=@k2k ch k?=nb; (15)

If one translates thiscurvature intoan effective mass meffone obtains

meff hv=c2 nb: (16)

This meansthatthe effective mass in the minimum is justgiven by a modified Einstein relation

E hv meffc2: (17)

This corresponds indeed to the very low effective mass

quoted usually in connection with claims of Bose Einsteincondensation (BEC) of cavity polaritons. However, it mustbe kept in mind that for a hyperbolic dispersion meff is notconstant, but increases rapidly for finite kjj and asymptoti-cally reaches the value infinity in the linear part of thehyperbola. Possibly one could think to introduce a concept ofmeffbased on the first derivative as used e.g. in metal physicsaround the Fermi energy [28]

meff h2k=@E=@k: (18)

The next aspect is excitonic or polaritonic Bose-Einsteincondensation (BEC). At least one of the authors of this

review (CK) is notoriously scepticabout excitonic BEC [28],because he has seen in the 35 or more years of his scientificcareer too many claims of excitonic BEC appearing anddisappearing again. See for some recent examples e.g. Refs.[133, 134] and the references therein. However, the resultspresented at this conference [126128] start to be convinc-ing, especially since both thermal quasi-equilibrium con-ditions in the BEC ground state have been reached and thedifferences to polariton lasing have been worked out and alsoto a process analogous to two-photon or hyper-Ramanscattering in bulk samples [93] under a so-called magicangle. The first difference i.e. polariton lasing is a verycommon phenomenon in semiconductors [14, 93] and

includes among other processes the inelastic polariton


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polariton (or excitonexciton) scattering which results in thePi-bands (i 2,3. . .1). See above or Ref. [28]. This positivestatement about BEC of cavity polaritons is made beingaware of the implications (or difficulties) of BEC in a quasi

two-dimensional system, which has in meffapproximationideally a Heavyside function for the density of states insteadof the square root one and in reality tail states resulting frominhomogeneous and homogeneous broadening induced e.g.by width fluctuations of both the QW and of the cavity andfrom scattering processes e.g. with phonons.

A third point addresses the concepts of strong andweak coupling betweencavitymodes and exciton resonance,since we think that there are significant differences in thethree- and quasi two- and one-dimensional systems dis-cussed here.

In a bulk sample, the coupling between light and excitonresonances is an intrinsic feature and so is the resulting

exciton polariton dispersion curve. Consequently, one is forexcitons in direct gap semiconductors generally in the strongcoupling limit. Even for damping zero one has a significantimaginary part of k or k [28] between the transverse andlongitudinal exciton eigen-energies, which makes thedirect observation of the dispersion in this spectral range anon-trivial problem. Frequently processes of nonlinearoptics are used for its experimental verification like two-photon absorption or hyper-Raman scattering [28]. In linearoptics the polariton dispersion can be deduced by therefraction at a thin prism, from resonant Brillouin scatteringor from the FabryPerot modes of thin platelets. See forsome references [28]. Care has to be taken for the

observability of the dispersion between the transverse andlongitudinal eigen energies of the bulk exciton polaritonresonances that the optical density of the platelet typesamples a(v)dremains below or around 1, despite the factthat the reflectivity for the intensity under normal incidencereaches values around 0.8 for a single surface on the LPB.This value is almost comparable to the one of real Braggmirrors and results from the fact that the real part n(v)of thecomplex index of refraction reaches values beyond 10 in theabove mentioned range. See e.g. Figs 13.17 and 20 in Ref.[28]. If a(v)d grows considerably beyond unity, theobservability of the FabryPerot modes becomes quenchedsimply by absorption, though one is still in the strong

coupling limit.For FabryPerot (i.e. micro-cavity) modes on one side

and WGM on the other, the situations are different, too, sinceseveral round trips of the light are necessary to form thepolariton. Strong losses in the cavity caused e.g. by a largeoptical density necessarily destroy the modes [28]. Seeabove.

For the quasi two-dimensional case, the cavity isessentially filled with the barrier material of the QW, whichhas in the spectral region of interest the exciton resonancesof the QW an almost constant refractive index nb andnegligible absorption. The exciton of the QW has adispersion with much higher effective mass mexmemh

than the empty cavity. The thickness of the QW in the cavity

lz is much smaller than the width of the cavity d

10 nm % lz ( d% 500 nm: (19)

This means that only a small fraction of the cavity

volume contains the exciton resonance. The Q-value orfinesse of the cavity and the dephasing time T2 of the excitoncan be varied independently. The strong coupling regimewith the pronounced non-crossing behaviour between thedispersion curves of the empty cavity mode and of theexciton is reached ifT2 is several times the round trip time trof the light in the cavity, typically requiring T2 1ps.Furthermore, the small volume fraction occupied by the QWhas the advantage that the absorption of the QW hardlyaffects the finesse of the cavity. Even witha 105 cm1 andlz 10 nm one obtains an optical density of only a lz 0.1.The Rabi splitting is then given in the case of strong couplingby the minimal energetic distance between the lower andupper polariton branches of the cavity exciton polaritonresulting from the (quantum-mechanical) non-crossing rule.It can be much higher than in bulk samples.

In the case of weak coupling, the exciton resonance is sostrongly (in-) homogeneously broadened (and so is thecontribution of this resonance to the whole system consistingof cavity mode and exciton) that it hardly influences thehyperbolic empty cavity dispersion. Also this behaviour isfavoured by the small volume fraction occupied by the QW.

The situation is now significantly different for the quasione-dimensional longitudinal cavity formed by themicro rodor for the transverse WGM and starts to resemble again thesituation of bulk material, since in this case the whole cavityis formed by and filled with the medium containing theexciton resonance and is described by its complex index ofrefraction

~nv nv ikv: (20)

This means that these cavity modes are determinedcompletely by this n(v). The absorption coefficient a(hv)reaches in ZnO throughout the whole cavity valuesa(hv)! 2 105 /cm in the spectral range between thetransverse and longitudinal eigen energies of the bulkexciton. See Fig. 5. Consequently, alzad 20 already

for d 1mm. This fact has the consequence that the visibilityof the micro rod cavity modes is completely quenched whenapproaching this spectral range. Actually, to the memory ofone of the authors (CK), no cavity modes where visible inRefs. [129131] around 3.31 eV, the position of the A and BG5 nB 1 exciton resonances at RT, but only a broadluminescence band from the free exciton and its LO-phononreplica. See Fig. 7.

Towards lower photon energies where ZnO becomestransparent, the one-dimensional exciton cavity polaritondispersion became visible in the beautiful experimentsmentioned above, with an energetic distance and curvatureincreasing with decreasing energy of the branch due to the

decrease ofn(v) below the exciton resonance towards lower


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photon energies. The increasing curvature leads for finite kjjto a crossing between the dispersion curves of adjacentbranches and in case of coupling between them to an avoidedcrossing. Since photons in vacuum do not interact, this

coupling is based on the excitonic part of the polariton wavefunction (more generally the electronic parts, which includealso the band-to-band transitions or the presence of an EHP,see below). The wave function of an exciton polariton hassubstantial exciton-like character about more than ten timesthe longitudinal-transverse DLT splitting below and abovethe exciton energy [135]. Since the sum ofDLT is for the(AB)G5 excitons more than 10 meV, the exciton likecharacter of the polariton wave function extends quite a bitbelow the resonance. This fact is confirmed by theobservation of luminescence from the cavity polaritonmodes, since an empty cavity does not show luminescence.

The fact that the dispersion and visibility of the WGM is

completely determined by the complex index of refractionn(v) n(v) ik(v) remains true, if the (in-) homogeneousbroadeningG(T) increases fromabout 1 meV at2 K tovaluesof above 20 meV HWHM at RT. This effect influences thedispersion of the exciton cavity polariton only close toresonance in the strongly absorbing regime but not below,where cavity polaritons remain clearly visible. The width ofthe cavity modes in the transparent regime is determined bythe finesse or Q-factor of the cavity (or by the photonic decaytime) and the residual absorption, both being directlydetermined by n(v).

If a phenomenological damping constantis introduced inthe Lorentz oscillator model, the damping of the polariton

i.e. the absorption coefficient or the imaginary part of n(v)decreases strongly with increasing energetic distance belowthe resonance as is also obvious for bulk exciton polaritons.See Fig. 5 above or Figs 4.5 and 13.9 in Ref. [28]. Thereforethe spectral width of the cavity polaritons decreases, too. Formore detailed considerations of the damping of cavitypolaritons see e.g. Ref. [139]. In a microscopic model onemay consider scattering processes e.g. with phonons or withother polaritons to describe damping. The steep decrease ofthe damping especially below the bulk exciton resonancemay then be attributed to the small DOS of both bulk andcavity polaritons. It should be noted that at RT the dampingof the exciton resonances is in ZnO mainly determined by

scattering with LO phonons. See the experimental data inFig. 8 or Ref. [66].

A huge Rabi splitting of more than 300 meV (i.e. morethan five times the exciton binding energy) has been claimedin Ref. [129] from a fitting procedure. It is not clear to theauthor to which extend this value corresponds to a realsplitting between the lower and upper polariton branches atthe WGM exciton polariton resonance in the senseintroduced above, because the WG is opaque in the rangeofAa ndBG5 excitons and of their higher states including theionization continuum, or if it is simply a fitting parameterdeducedfrom the low energy data in the transparent region ofthe sample. It is difficult to imagine that a micro rod is opaque

in a simple transmission measurement through two opposite

surfaces but should become transparent and show well-resolved WGM if the light circulates in these modes.

As mentioned already, in the ZnO rods it is not possibleto vary the finesse of the resonator and the damping of the

exciton resonance within wide limits independently as in thecase of a QW exciton in a micro cavity formed by two Braggmirrors, but the whole system remains determined by n(v).This remains true if the exciton resonance is broadened byincreasing temperature T or if an increasing density np ofelectronhole pairs lead to increasing screening of theexciton up to the limit of an EHP where excitons do no longerexist as individual quasi particles. Also in this situation thesystem is described by n(v ,T, np). The influence of thedensity dependent changes of the refractive index on thelongitudinal modes in a nano rod has been presented alreadyabove with Fig. 20, including the onset of recovery with timeafter pulsed excitation. The same phenomenon was obvi-

ously observed in Ref. [129] when the spectral position of theWGM shifted in the case of stimulated emission. Theirproperties are then simply determined by n(v, T, np) and thegeometry of the micro rod.

In bulk, the dispersion of light in ZnO in the presence ofan EHP would be simply given by a modified polaritonequation

c2k2=v2 ~n2v;T; np ev;k;T; np: (21)

In the context of micro or nano rods we consider theWGM in micro rods and the longitudinal modes in nano rodsas two limiting cases with a continuous transition between

them in the following sense: For the fundamental longi-tudinal modes in a nano rod the wave vector is almostcompletely determined by the quantized kjj, while k? ismarginal. This situation changes if one goes to highertransversal modes [108]. Then k? increases at the expense ofkjj for constant energy of the light quanta orthe modes shift tothe blue for fixed kjj. For WGM one has the oppositesituation. In a mode, which circulates completely in thehexagonal cross sectionof therod,k? is quantized andmakesup the whole wave vector. For observation under an obliqueangle to the rod- or c-axis kjj increasingly comes into playnow at the expense ofk?. In principle it should be possible togive a general description of modes in a hexagonal rod for all

(k?,kjj) which includes the above limiting cases.To conclude this appendix, one may ask if it makes sense

to separate the strong and weak coupling limits in ZnO rods?A situation where one sits spectrally on the bulk excitonresonance and stillhas a high finesse cavity is not possible forthe rods as mentioned above. At this resonance, where thecoupling between electromagnetic field and the exciton isstrongest, one has necessarily strong damping and lowfinesse and in the transparent region below the exciton thefinesse and the width of the modes get better, but the couplingbetween excitons and electromagnetic field decreases.

A more detailed theoretical treatment of this problem[136, 137] came to a Rabi splitting for WGM of the order of

200 meV in agreement with the value cited above as a new


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energy scale, which is larger than the exciton binding energy,but claims that possibly only the LPB may be visible but notthe UPB, which should actually be situated already in theexcitonic ionization continuum. Since already bulk ZnO is

transparent only on the LPB up to the bottleneck, theseresults may even give some support to the tentative andqualitative considerations above.

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