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Phonon-assisted optical transitions in GaN with impurities and defects
Physica B 302–303 (2001) 291–298
Phonon-assisted optical transitions in GaN with impurities and defects P. Tronca,*, Yu.E. Kitaeva,1, G. Wanga, M.F. Limonova,1, G. Neub b
a Laboratoire d’Optique, Ecole Superieure de Physique et Chimie Industrielles, 10 Rue Vauquelin, F-75005 Paris, France Centre de Recherche sur l’Heteroepitaxie et ses Applications, Centre National de la Recherche Scientifique (CRHEA-CNRS), Rue Bernard Gregory, Parc Sophia-Antipolis, F-06560 Valbonne, France
Abstract We have classified all the possible types of phonon-assisted optical processes involving bound or extended initial, final, and virtual (intermediate) electron states and formulated, for the first time, in terms of site symmetry, the selection rules for corresponding transitions. We apply this theory to phonon-assisted transitions in hexagonal GaN involving substitutional impurities and vacancies with C3v site-symmetry as well as interstitial impurities and molecular point defects (paired impurities, double vacancies, and vacancy-impurity complexes) occupying sites with C3v, Cs, and C1 ones. We show that phonon-assisted optical recombination is allowed in any polarization for free and bound carriers and excitons whatever is the number of involved phonons. Just, the nature of virtual state(s) and phonon(s) can depend on the polarization of the emitted light. We discuss, in particular, the case of excitons bound to neutral donors or acceptors. Our predictions are in good agreement with experimental optical spectra published in the literature which exhibit numerous lines assigned to one- and multi-phonon-assisted transitions. # 2001 Elsevier Science B.V. All rights reserved. PACS: 78.30.Fs; 71.20.Nr; 71.55.Eq Keywords: GaN; Phonon-assisted optical transitions; Group theory
1. Introduction Site symmetry approach to solid state problems (see [1]) is very effective especially when studying the properties connected with microscopic structure of crystals. It allows one to establish *Corresponding author. Tel.: +33-1407-94804; fax: 33-143362395. E-mail address: [email protected] (P. Tronc). 1 Permanent address: Ioffe Physico-Technical Institute, 194021 St. Petersburg, Russia.
symmetry relation between localized and extended electron, exciton or phonon states [2–4]. One of the problems where this method proves to be very efficient is the study of optical transitions involving simultaneously both localized and extended states. Indeed, this method allows to establish symmetry correspondence between atomic orbitals and Bloch functions. In this paper we classify all possible types of phonon-assisted optical transitions involving bound or extended states as initial, final and virtual ones. We formulate for the first time, in terms of site
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symmetry, the selection rules for corresponding transitions. Though applied below for GaN, these selection rules are of general character and can be used for any crystal with defects. Hexagonal GaN is a very promising material to achieve optoelectronic devices operating at short (blue or near UV) wavelengths as well as transistors operating at high temperatures [5]. It has the wurtzite structure (space group C46v) with two formula units per primitive cell. AlN and InN have the same structure with different values of anion z coordinate. Many II–VI compounds, as for example CdS and ZnS, also present the same structure. In GaN, phonon-assisted transitions involving impurities are of particular importance since many states are generally optically active within the band gap. We also consider free and bound excitons. Finally, we analyze some experimental results which appeared in the literature and make some predictions about results to be observed in optical experiments.
2. Optical selection rules for phonon-assisted transitions involving crystal defects We consider below crystal defects (CDs) like substitutional and interstitial impurities, vacancies, and molecular defects (paired impurities, double vacancies and vacancy-impurity complexes). Next, we study phonon-assisted transitions involving CD states as initial, final or virtual ones. The general property of all the above processes is that they involve a virtual state thus consisting of two steps: the first step is a transition from an initial to a virtual state due to a phonon (or a photon) and the second step is a transition from the virtual to a final state due to a photon (or a phonon). The seven possible types of optical transitions which can occur in crystals with defects are shown in Fig. 1. The transitions corresponding to the time-reversal processes have to obey the same selection rules and are not shown. We formulate below, in terms of site symmetry, the optical selection rules for these processes. Let the symmetry of an electron state of a CD be described by the irreducible representation
Fig. 1. The possible phonon-assisted optical transitions involving impurity (circles) and band (lines) states as initial, final (solid lines) and/or virtual (dashed lines) ones. The phonons and photons involved are shown by zigzag and straight lines, respectively.
(irrep) da of the CD site symmetry group Gq , the symmetry of an electron state in the conduction or valence band of the host crystal by the irrep Db of the space group G of the crystal, and the symmetry of a phonon by the irrep Dph of the space group G. It should be stressed that if a transition involves a localized state of a CD, all the extended states (Db or Dph ) involved in the transition should be subduced [1] on the site symmetry group Gq (Db , Dph # Gq ) of the CD. There are two types of processes depending on whether the transition from a localized or band initial state (d ðiÞ or DðiÞ ) to a localized or band virtual one (d virt or Dvirt ) is due to a phonon (Dph ) (1,3,5-types in Fig. 1) or a photon (2,4,6,7-types in Fig. 1).
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In the former case, the possible virtual states are determined from the condition ðDvirt Þ * DðiÞ \ Dph 6¼ 0;
ð1aÞ
where for 1-type process, Dvirt and DðiÞ are the irreps Dvirt and DðiÞ of the crystal space group G, which correspond to a conduction- or valenceband state, and Dph is Dph whereas for 3,5-type processes Dvirt and DðiÞ are irreps d virt and d ðiÞ of the CD site symmetry group Gq , which describes localized states, and Dph is (Dph # Gq ). Then to determine the transitions to a final state one should use the following condition: ðDðfÞ Þ * Dvirt \ dv 6¼ 0;
ð1bÞ
where DðfÞ is either d ðfÞ for 1,3-type processes, or (DðfÞ # Gq ) for 5-type process whereas Dvirt is either d virt for 3,5-type processes, or (Dvirt # Gq ) for 1type process; dv is the vector representation of the site symmetry group Gq of the CD. In the case when the phonon is involved after the photon (2,4,6,7-type processes), that is when the transition from the initial state to a virtual state is due to a photon, the possible virtual states are determined from the condition ðDvirt Þ * DðiÞ \ dv 6¼ 0; virt
ð2aÞ
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the vector representation are the IR-active ones from the BZ center (k ¼ 0). In turn, phonons which are only Raman active, or silent ones, cannot be interchanged with photons.
3. Phonon-assisted transitions involving CD states in hexagonal GaN Now we apply the above theory to phononassisted optical transitions involving localized states in hexagonal GaN. The crystal structure and the BZ of wurtzite-like GaN are shown in Fig. 2. Detailed classification of CDs in GaN, their site symmetries and the symmetry of localized states of CDs is given in [6]. The symmetry of electron band states at the symmetry points of the BZ in GaN was determined [7]. We studied also the direct interband optical transitions as well as zero-phonon transitions involving impurities and defects. The phonon symmetry in hexagonal GaN was determined in [8]. In papers [6–8] and hereafter, the site symmetry group irreps are labeled according to [9], the labeling of the space group irreps follows [10] and symmetry (Wyckoff )
ðiÞ
and D are Dvirt and DðiÞ (2-type where D process), (Dvirt # Gq ) and d ðiÞ states (4-type process), d virt and d ðiÞ states (6-type process) and, at last, d virt and (DðiÞ # Gq ) (7-type process); dv is a vector representation of the crystal space group G (2-type process) or the site symmetry group Gq of the CD. The final states are determined by the condition ðDðfÞ Þ * Dvirt \ Dph 6¼ 0; ðfÞ
ð2bÞ
ðfÞ
where D is either d (2,4-type processes) or (DðfÞ # Gq ) for 6,7-type processes; Dvirt is either d virt (6,7-type processes) or (Dvirt # Gq ) for 2,4-type processes and Dph is (Dph # Gq ). It should be noted that, in general, the photon and the phonon cannot be interchanged within a given process. They can be interchanged in the only case where the phonon involved is transformed according to the vector representation. In this case, Eqs. (1a) and (1b) coincide with Eqs. (2a) and (2b). The phonons transforming according to
Fig. 2. The crystal structure and the BZ of wurtzite-like GaN. The corresponding Wyckoff positions are shown in parentheses.
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positions (sites in direct space) are given in accordance with [11]. We briefly list some of the results of papers [6–8] which we shall use below when considering phonon-assisted transitions involving CD states. In hexagonal GaN, according to [6], the substitutional impurities and vacancies can occupy sites with C3v symmetry (the 2b Wyckoff position) whereas interstitial impurities and molecular defects can occupy sites with C3v(2a and 2b), Cs(6c) and C1(12d) symmetries. Without the account of spin–orbit interaction, the symmetry of localized states of a C3v CD is described by site-symmetry group irreps a1 (s; pz ), a2 (only excitons), and eðpx ; py Þ with the atomic orbitals transforming according to these irreps shown in parentheses. The band state symmetries at the BZ center are G1 and G4 (induced by atomic s and pz orbitals), G2 and G3 (exciton states), G5 and G6 (induced by atomic px and py orbitals) [7]. When including spin–orbit interaction, the localized states % 2 whereas transform as a1;2 ! e% 2 ; e ! e% ð1Þ eð2Þ 1 þ% 1 þe the band ones as G1;2 ! G7 ; G3;4 ! G8 ; G5 ! G8 þ G9 ; G6 ! G7 þ G9 . At last, the phonons have the following symmetries at the BZ symmetry points: 2ðG1 þ G4 þ G5 þ G6 Þ; 2ðK1 þ K2 þ 2K3 Þ, and 2(2M1+M2+M3+2M4) [8]. Among the BZ center phonons, G1 ðA1 Þ and G6 ðE1 Þ ones are both Raman and IR-active, G5 ðE2 Þ is Raman-active, G4 ðB1 Þ corresponds to a silent mode, whereas G2 ðA2 Þ and G3 ðB2 Þ phonons are absent. Using site symmetry analysis, it can be shown that any G state can be connected with any G, K or
M state in any polarization by at least one phonon-assisted optical transition both when the spin–orbit interaction is not taken into account and when it is. The phonon involved depends on the process. To not overload the paper with too long developments we just give hereafter some examples. 3.1. Phonon-assisted transitions without spin–orbit interaction As an example, let us consider the phononassisted transitions in GaN between the bound state a1 of a CD occupying the 2b Wyckoff position (C3v site symmetry) and the band state G2 , the phonon being involved between the localized and virtual states. The direct transition (i.e. that involving no phonons) between a1 and G2 states is forbidden. Note, that the vector representation describing the symmetry of photons is either G1 ðzÞ þ G6 ðx; yÞ when both states involved are band ones or a1 ðzÞ þ eðx; yÞ when at least one state is a localized one. Next, we take into account only phonons from the symmetry points of the BZ since these phonons have the strongest contributions to the processes due to the maxima of the phonon density of states function at these points [8]. The phonons from general symmetry points of the BZ can induce any transitions but their contribution is negligible due to their low density of states. Then the above transition can be achieved at least by three different processes for z-polarized photons:
a1 - - -ðM2;3 ; K3 phononÞ - - - ! a2 - - -ðz-photonÞ - - - ! G2 ð5-type processÞ; a1 - - -ðM2;3 ; K3 phononÞ - - - ! G2 - - -ðz-photonÞ - - - ! G2 ð2-type processÞ; a1 - - -ðz-photonÞ - - - ! a1 - - -ðM2;3 ; K3 phononÞ - - - ! G2 ð6-type processÞ: (M2,3, K3 phonon means that the involved phonon can be M2, M3 or K3) and by three processes for x; y-polarized photons a1 - -ðG5;6 ; M1;2;3;4 ; K1;2;3 phononÞ ! e- -ðx; y-photonÞ ! G2 ð5-type processÞ; a1 - -ðG5;6 ; M1;2;3;4 ; K1;2;3 phononÞ ! G6 - -ðx; y-photonÞ ! G2 ð2-type processÞ; a1 - -ðx; y-photonÞ ! e- -ðG5;6 ; M1;2;3;4 ; K1;2;3 phononÞ ! G2 ð6-type processÞ:
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As a proof, let us check the first of the processes given above. The transition between a1 and virtual a2 states of a CD occupying the 2b Wyckoff position can be mediated by a K3 phonon since the Kronecker product of a1 and a2 is a2 and K3 phonon includes a a2 component when subduced onto the C3v site symmetry group of the impurity [1,8]. Next, it is seen that G2 irrep includes a a2 component when subduced onto the site-symmetry group of the impurity which makes the transition between a a2 state and a G2 one mediated by a z photon to be allowed. It should be noted that for band states and phonons from the BZ center the result of subduction and, consequently, allowed transitions depend only on the point group of the site. However, when subducing the irreps of k 6¼ 0 band states or phonons, one should specify the Wyckoff position, since the groups of different sites (even with the same point group) contain different translations which modify the result of subduction. Indeed, for example, for substitutional and interstitial impurities occupying the 2b Wyckoff position, the K2 phonon cannot induce the transition between a1 and a2 localized states since K2 # Gq ðq¼ 2bÞ ¼ e. In turn, for interstitial impurities occupying the 2a Wyckoff position, the transition between a1 and a2 localized states can be mediated by the K2 phonon since K2 # Gq ðq¼ 2aÞ ¼ a1 þa2 .
states. The direct transition is forbidden in any polarization since the irrep product G1 G2 ¼ G2 does not contain the vector representation G1 ðzÞþ G6 ðx; yÞ. In turn, phonons can connect G1 state with G1 ; G4 ; G5 , or G6 virtual states. The transitions between G1 ; G4 ; G5 virtual states and the G2 final one are forbidden in all polarizations whereas the transition between G6 and G2 is allowed in xyand forbidden in z-polarization. However, the G1 2G2 transition becomes allowed in z-polarization via virtual state of impurity at the 2b site with C3v site symmetry (a 7-type process): G1 - - ðM2;3 ; K3 phononÞ ! a2 - -ðz-photonÞ ! G2 ð7-type processÞ: 3.2. Phonon-assisted transitions with spin–orbit interaction The case when spin–orbit interaction is taken into account can be treated in the same way. Now let us consider, for example, the transition between the bound state e% 2 (originated from an a1 state) of a C3v CD occupying the 2b Wyckoff position and the band state G9 (originated from the G5 state). The direct transition e% 2 ! G9 is allowed in xy and forbidden in z polarization. However, this transition becomes weakly allowed in z-polarization when assisted by phonons:
% ð2Þ e% 2 - -ðG5;6; M1;2;3;4 ; K1;2;3 phononÞ ! e% ð1Þ 1 ; e 1 - -ðz-photonÞ ! G9 ð5-type processÞ; e% 2 - -ðG5;6; M1;2;3;4 ; K1;2;3 phononÞ ! G9 - - -ðz-photonÞ- - - ! G9 ð2-type processÞ; e% 2 - -ðz-photonÞ- - - ! e% 2 - -ðG5;6 ; M1;2;3;4; K1;2;3 phononÞ ! G9 ð6-type processÞ: Involving CD states as the virtual ones, any transition between the G band states, which is forbidden when a bound state is not involved as a virtual one, becomes allowed. Indeed, either the direct or phonon-assisted transitions via a virtual extended state between G1 2G2 ; G1 2G3 ; G2 2G4 and G3 2G4 couples of states are forbidden in z polarization due to the absence of G2 and G3 phonons. (In contrast, it can be shown that an allowed phonon-assisted transition does always exist in any polarization which connects any couple made of a G state and K(M) one). Consider as an example the transition between G1 and G2
3.3. Excitons The above analysis includes also the phononassisted optical recombination of free and bound excitons via virtual bound states. Phonon-assisted optical recombination of free and bound excitons is therefore allowed in any polarization. The phonon involved depends on the process. The only difference from the analysis of optical recombination of carriers lies in the fact that excitons can have any state symmetry, which is not true for carriers. Indeed, the symmetry of free excitons at the P-point of the BZ is given by
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Ph P*e Penv Kronecker product of the hole and electron irreps and the envelope function representation at the P-point. That is the initial exciton state is one of the irreps contained in this product. The analysis [7] shows that free excitons can have also G2 and G3 symmetries which are absent for free carriers. For bound excitons, the irreps are those of the CD site symmetry group. In this case, an a2 state becomes possible for a bound exciton [6]. The final state of an exciton radiative recombination is described by the fully symmetrical irrep (i.e. that of the void), being G1 for free excitons [7] or the fully symmetrical irrep of the CD site symmetry group for bound excitons [6]. 3.4. General properties It should be noted that if one consider an allowed process involving one or several CD states as initial, final or virtual ones, the process is still allowed if the symmetry of a CD is lowered from Gqo (Gqo may be C3v or Cs) to a subgroup Gq
Gqo (Gq may be Cs or C1). Indeed the initial or final or virtual state di of a CD with Gq symmetry would be connected by a subduction relation with the corresponding former one d0j of the higher symmetry CD group Gqo : X ai di ¼ d0j # Gq : ð3Þ As a result, if a process is allowed for d0j state it is allowed for any di state at least for the same light polarizations and the same phonon involved. Last, it has to be added that a transition from the single state of C1 impurity to any band state is allowed for any photon polarization. Transitions involving two or more phonons can be studied in the same manner. For a transition between a couple of states, the number of different possible processes increases very rapidly with the number of phonons. It is easy to show from above that an optical transition between any couple of G-states, any couple of states bound to the same CD and any couple made of a G-state and a bound state is allowed in any polarization whatever is the number of involved phonons from symmetry points of the BZ. However, for each process the phonons involved are, in general, different, depending even on the polarization of the emitted or
absorbed light (see above). This results in different energies of the corresponding transitions. Thus, when studying optical properties near the band gap of GaN crystals with impurities and/or defects, one can therefore be sure that phononassisted optical recombination of free or bound carriers or excitons is allowed in any polarization whatever is the number of phonons. Just the phonon(s) involved may depend on the polarization ((x; y) or z) of the light if the nature of the process depends on the latter. 4. Discussion In GaN, phonon-assisted transitions involving impurity states were experimentally studied in several papers (see for example [12,13]). The authors [13] observed phonon replicas in the photoluminescence spectra of wurtzite GaN which were interpreted as phonon-assisted neutral donor bound-exciton (D0X) recombination. They interpret the lines at 3.4024 and 3.3990 eV resulting from the collapse of D0X accompanied by creation of TO phonons with G1 ðA1 Þ and G6 ðE1 Þ symmetries. This process can be treated in the framework of the above theory. If we assume the interaction between the bound exciton and the electron (hole) of the donor (acceptor) to be negligible, the bound-exciton recombination does obey the selection rules derived in our previous paper [6]. When an interaction term is included, the exciton plus electron (hole) bound to a neutral donor (acceptor) behave as a whole. In the case of a donor, the symmetry of the initial state is described by one of the irreps contained in the product hðexcÞ *
daeðexcÞ db
d env dgeðdonorÞ ;
ð4Þ
where daeðexcÞ and dbhðexcÞ are irreps of the electron and hole forming the exciton bound to the donor, d env is the representation of exciton envelope function, and dgeðdonorÞ is the irrep of the electron of the donor. The symmetry of the final state is the symmetry of the electron of the donor (dxeðdonorÞ ) since the final state of the exciton is that of the void (totally symmetrical irrep). The symmetries of photon and
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phonon involved are dv (vector representation of the site symmetry group Gq ) and d ph ¼ ðDðphÞ # Gq Þ, respectively. Our analysis shows that the exciton recombination observed in [13] can be achieved by different processes involving the G1 ðA1 Þ and G6 ðE1 Þ phonons. Without taking into account spin–orbit interaction, the initial state of the donor electron (D) can be a1(D) and e(D) whereas that of the bound exciton (X) can be a1(X), a2(X), and e(X) being also the initial state of the D0X complex. After recombination, the final state of D0X is that of the donor electron, namely either a1(D) or e(D). Since being involved, the G1 ðA1 Þ phonon does not change the symmetry of a state. Therefore, the process, from the symmetry point of view, reduces to one-phonon one. The performed analysis shows that any arbitrary pair of initial and final states can be coupled by x; y or z-photon and the G1 ðA1 Þ and G6 ðE1 Þ phonons, except for a2(D0X) ! a1(D) transition which is forbidden in z-polarization. When including spin–orbit interaction, the single-valued irreps transform into the doublevalued ones as follows: % ð2Þ %2: a1 ! e% 2 ; a2 ! e% 2 ; e ! e% ð1Þ 1 þe 1 þe Then, the initial states of electron, exciton and % ð2Þ % 1 as well as the D0X complex can be e% ð1Þ 1 ; e 1 , and e final state of the electron. Our analysis shows that again any arbitrary pair of initial and final states can be coupled by x; y or z-photon and the G1 ðA1 Þ 0 % ð2Þ and G6 ðE1 Þ phonons, except for e% ð1Þ 1 ; e 1 ðD XÞ ! ð1Þ ð2Þ e% 1 ; e% 1 (D) transitions which are forbidden in z-polarization. The complete analysis shows that for arbitrary combination of initial states of the exciton and the electron (hole) of the donor (acceptor) and of final state of the electron (hole) one can find, at least, one phonon-assisted process allowed for any given polarization. Thus, the analysis of phonon-assisted optical transitions between bound states via bound or extended virtual states given in Section 3 includes also the analysis of recombination of excitons bound to neutral donors or acceptors. It can therefore be concluded that the radiative recombination of an exciton bound to a neutral donor
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(acceptor) is always allowed in any polarization whatever are the symmetries of the initial and final states of the electron (hole) of the donor (acceptor) and the number of phonons involved. However, the different allowed processes correspond, in general, to transitions with different energies due to energy transfer to phonons and donor (acceptor) electron (hole) involved.
5. Conclusions The main theoretical results obtained in this paper are of general character and can be applied for any crystal. In particular, in terms of the site symmetry approach, we have derived optical selection rules for the seven possible phonon-assisted transitions involving CD localized states as initial, virtual, and/or final ones. Our results include exciton radiative recombination processes as well. We have also established that, in optical transitions involving a virtual state, photons and phonons can be interchanged only for the BZcenter infrared-active phonons. Next, we have shown that for allowed optical transitions involving CD localized states and k 6¼ 0 band states and/or k 6¼ 0 phonons one should specify not only the CD site symmetry but also the Wyckoff position it occupies. We have applied the above theory for analysis of phonon-assisted optical transitions in wurzitetype GaN. We analyzed all possible phononassisted optical transitions without and with the inclusion of spin–orbit interaction. We have shown that optical recombination of free and bound carriers and excitons is always achievable through phonon-assisted processes whatever is the number of phonons involved. Of course, when several processes are possible for connecting an initial state to a final one with the same number of phonons, each allowed process is accompanied, in general, by different phonons resulting in different energies of optical transitions. The variety of initial, virtual, and final states is particularly rich for excitons bound to neutral donor or acceptors due to the extra electron (hole) involved in the process. In any process we considered, the phonons are those from symmetry points of
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the BZ. Their density is high making the phononassisted transitions to be rather intense as can be checked from numerous published photoluminescence spectra. Acknowledgements We acknowledge the CLG 97053 NATO grant, the Russian Foundation for Basic Research (Grant No. 99-02-18318) and the support of Mairie de Paris and of French Embassy in Moscow. References [1] R.A. Evarestov, V.P. Smirnov, Site Symmetry in Crystals: Theory and Applications, Springer Series in Solid State Sciences, Vol. 108, Springer, Heidelberg, 1993. [2] Yu.E. Kitaev, A.G. Panfilov, P. Tronc, R.A. Evarestov, J. Phys.: Condens. Matter 9 (1997) 257. [3] Yu.E. Kitaev, A.G. Panfilov, P. Tronc, R.A. Evarestov, J. Phys.: Condens. Matter 9 (1997) 277.
[4] P. Tronc, Yu.E. Kitaev, A.G. Panfilov, M.F. Limonov, G. Wang, V.P. Smirnov, Phys. Rev. B 61 (3) (2000) 1999. [5] J. Pankove, T.D. Moustakas, (Ed.), Gallium Nitride (GaN) II, Semiconductors and Semimetals, Vol. 57, Academic Press, New York, 1999. [6] P. Tronc, Yu.E. Kitaev, G. Wang, M.F. Limonov, Phys. Stat. Sol. B 210 (1998) 471. [7] P. Tronc, Yu.E. Kitaev, G. Wang, M.F. Limonov, A.G. Panfilov, G. Neu, Phys. Stat. Sol. B 216 (1999) 599. [8] V.Yu. Davydov, Yu.E. Kitaev, I.N. Goncharuk, A.N. Smirnov, J. Graul, O. Semchinova, D. Uffmann, M.B. Smirnov, A.P. Mirgorodsky, R.A. Evarestov, Phys. Rev. B 58 (1998) 12899. [9] C.J. Bradley, A.P. Cracknell, The Mathematical Theory of Symmetry in Solids, Clarendon, Oxford, 1972. [10] S.C. Miller, W.F. Love, Tables of Irreducible Representations of Space Groups and Co-Representations of Magnetic Space Groups, Pruett, Boulder, 1967. [11] T. Hahn (Ed.), International Tables for Crystallography, Vol. A, Space Group Theory, Reidel, Dordrecht, 1983. [12] M. Leroux, B. Beaumont, N. Grandjean, P. Lorenzini, S. Hafflouz, P. Venn!egu"es, J. Massies, P. Gibart, Mat. Sci. Eng. B 50 (1997) 97. [13] D.C. Reynolds, D.C. Look, B. Jogai, R.J. Molnar, Solid State Commun. 108 (1998) 49.
Phonon-assisted optical transitions in GaN with impurities and defects P. Tronca,*, Yu.E. Kitaeva,1, G. Wanga, M.F. Limonova,1, G. Neub b
a Laboratoire d’Optique, Ecole Superieure de Physique et Chimie Industrielles, 10 Rue Vauquelin, F-75005 Paris, France Centre de Recherche sur l’Heteroepitaxie et ses Applications, Centre National de la Recherche Scientifique (CRHEA-CNRS), Rue Bernard Gregory, Parc Sophia-Antipolis, F-06560 Valbonne, France
Abstract We have classified all the possible types of phonon-assisted optical processes involving bound or extended initial, final, and virtual (intermediate) electron states and formulated, for the first time, in terms of site symmetry, the selection rules for corresponding transitions. We apply this theory to phonon-assisted transitions in hexagonal GaN involving substitutional impurities and vacancies with C3v site-symmetry as well as interstitial impurities and molecular point defects (paired impurities, double vacancies, and vacancy-impurity complexes) occupying sites with C3v, Cs, and C1 ones. We show that phonon-assisted optical recombination is allowed in any polarization for free and bound carriers and excitons whatever is the number of involved phonons. Just, the nature of virtual state(s) and phonon(s) can depend on the polarization of the emitted light. We discuss, in particular, the case of excitons bound to neutral donors or acceptors. Our predictions are in good agreement with experimental optical spectra published in the literature which exhibit numerous lines assigned to one- and multi-phonon-assisted transitions. # 2001 Elsevier Science B.V. All rights reserved. PACS: 78.30.Fs; 71.20.Nr; 71.55.Eq Keywords: GaN; Phonon-assisted optical transitions; Group theory
1. Introduction Site symmetry approach to solid state problems (see [1]) is very effective especially when studying the properties connected with microscopic structure of crystals. It allows one to establish *Corresponding author. Tel.: +33-1407-94804; fax: 33-143362395. E-mail address: [email protected] (P. Tronc). 1 Permanent address: Ioffe Physico-Technical Institute, 194021 St. Petersburg, Russia.
symmetry relation between localized and extended electron, exciton or phonon states [2–4]. One of the problems where this method proves to be very efficient is the study of optical transitions involving simultaneously both localized and extended states. Indeed, this method allows to establish symmetry correspondence between atomic orbitals and Bloch functions. In this paper we classify all possible types of phonon-assisted optical transitions involving bound or extended states as initial, final and virtual ones. We formulate for the first time, in terms of site
0921-4526/01/$ - see front matter # 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 0 1 ) 0 0 4 4 3 - 4
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symmetry, the selection rules for corresponding transitions. Though applied below for GaN, these selection rules are of general character and can be used for any crystal with defects. Hexagonal GaN is a very promising material to achieve optoelectronic devices operating at short (blue or near UV) wavelengths as well as transistors operating at high temperatures [5]. It has the wurtzite structure (space group C46v) with two formula units per primitive cell. AlN and InN have the same structure with different values of anion z coordinate. Many II–VI compounds, as for example CdS and ZnS, also present the same structure. In GaN, phonon-assisted transitions involving impurities are of particular importance since many states are generally optically active within the band gap. We also consider free and bound excitons. Finally, we analyze some experimental results which appeared in the literature and make some predictions about results to be observed in optical experiments.
2. Optical selection rules for phonon-assisted transitions involving crystal defects We consider below crystal defects (CDs) like substitutional and interstitial impurities, vacancies, and molecular defects (paired impurities, double vacancies and vacancy-impurity complexes). Next, we study phonon-assisted transitions involving CD states as initial, final or virtual ones. The general property of all the above processes is that they involve a virtual state thus consisting of two steps: the first step is a transition from an initial to a virtual state due to a phonon (or a photon) and the second step is a transition from the virtual to a final state due to a photon (or a phonon). The seven possible types of optical transitions which can occur in crystals with defects are shown in Fig. 1. The transitions corresponding to the time-reversal processes have to obey the same selection rules and are not shown. We formulate below, in terms of site symmetry, the optical selection rules for these processes. Let the symmetry of an electron state of a CD be described by the irreducible representation
Fig. 1. The possible phonon-assisted optical transitions involving impurity (circles) and band (lines) states as initial, final (solid lines) and/or virtual (dashed lines) ones. The phonons and photons involved are shown by zigzag and straight lines, respectively.
(irrep) da of the CD site symmetry group Gq , the symmetry of an electron state in the conduction or valence band of the host crystal by the irrep Db of the space group G of the crystal, and the symmetry of a phonon by the irrep Dph of the space group G. It should be stressed that if a transition involves a localized state of a CD, all the extended states (Db or Dph ) involved in the transition should be subduced [1] on the site symmetry group Gq (Db , Dph # Gq ) of the CD. There are two types of processes depending on whether the transition from a localized or band initial state (d ðiÞ or DðiÞ ) to a localized or band virtual one (d virt or Dvirt ) is due to a phonon (Dph ) (1,3,5-types in Fig. 1) or a photon (2,4,6,7-types in Fig. 1).
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In the former case, the possible virtual states are determined from the condition ðDvirt Þ * DðiÞ \ Dph 6¼ 0;
ð1aÞ
where for 1-type process, Dvirt and DðiÞ are the irreps Dvirt and DðiÞ of the crystal space group G, which correspond to a conduction- or valenceband state, and Dph is Dph whereas for 3,5-type processes Dvirt and DðiÞ are irreps d virt and d ðiÞ of the CD site symmetry group Gq , which describes localized states, and Dph is (Dph # Gq ). Then to determine the transitions to a final state one should use the following condition: ðDðfÞ Þ * Dvirt \ dv 6¼ 0;
ð1bÞ
where DðfÞ is either d ðfÞ for 1,3-type processes, or (DðfÞ # Gq ) for 5-type process whereas Dvirt is either d virt for 3,5-type processes, or (Dvirt # Gq ) for 1type process; dv is the vector representation of the site symmetry group Gq of the CD. In the case when the phonon is involved after the photon (2,4,6,7-type processes), that is when the transition from the initial state to a virtual state is due to a photon, the possible virtual states are determined from the condition ðDvirt Þ * DðiÞ \ dv 6¼ 0; virt
ð2aÞ
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the vector representation are the IR-active ones from the BZ center (k ¼ 0). In turn, phonons which are only Raman active, or silent ones, cannot be interchanged with photons.
3. Phonon-assisted transitions involving CD states in hexagonal GaN Now we apply the above theory to phononassisted optical transitions involving localized states in hexagonal GaN. The crystal structure and the BZ of wurtzite-like GaN are shown in Fig. 2. Detailed classification of CDs in GaN, their site symmetries and the symmetry of localized states of CDs is given in [6]. The symmetry of electron band states at the symmetry points of the BZ in GaN was determined [7]. We studied also the direct interband optical transitions as well as zero-phonon transitions involving impurities and defects. The phonon symmetry in hexagonal GaN was determined in [8]. In papers [6–8] and hereafter, the site symmetry group irreps are labeled according to [9], the labeling of the space group irreps follows [10] and symmetry (Wyckoff )
ðiÞ
and D are Dvirt and DðiÞ (2-type where D process), (Dvirt # Gq ) and d ðiÞ states (4-type process), d virt and d ðiÞ states (6-type process) and, at last, d virt and (DðiÞ # Gq ) (7-type process); dv is a vector representation of the crystal space group G (2-type process) or the site symmetry group Gq of the CD. The final states are determined by the condition ðDðfÞ Þ * Dvirt \ Dph 6¼ 0; ðfÞ
ð2bÞ
ðfÞ
where D is either d (2,4-type processes) or (DðfÞ # Gq ) for 6,7-type processes; Dvirt is either d virt (6,7-type processes) or (Dvirt # Gq ) for 2,4-type processes and Dph is (Dph # Gq ). It should be noted that, in general, the photon and the phonon cannot be interchanged within a given process. They can be interchanged in the only case where the phonon involved is transformed according to the vector representation. In this case, Eqs. (1a) and (1b) coincide with Eqs. (2a) and (2b). The phonons transforming according to
Fig. 2. The crystal structure and the BZ of wurtzite-like GaN. The corresponding Wyckoff positions are shown in parentheses.
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positions (sites in direct space) are given in accordance with [11]. We briefly list some of the results of papers [6–8] which we shall use below when considering phonon-assisted transitions involving CD states. In hexagonal GaN, according to [6], the substitutional impurities and vacancies can occupy sites with C3v symmetry (the 2b Wyckoff position) whereas interstitial impurities and molecular defects can occupy sites with C3v(2a and 2b), Cs(6c) and C1(12d) symmetries. Without the account of spin–orbit interaction, the symmetry of localized states of a C3v CD is described by site-symmetry group irreps a1 (s; pz ), a2 (only excitons), and eðpx ; py Þ with the atomic orbitals transforming according to these irreps shown in parentheses. The band state symmetries at the BZ center are G1 and G4 (induced by atomic s and pz orbitals), G2 and G3 (exciton states), G5 and G6 (induced by atomic px and py orbitals) [7]. When including spin–orbit interaction, the localized states % 2 whereas transform as a1;2 ! e% 2 ; e ! e% ð1Þ eð2Þ 1 þ% 1 þe the band ones as G1;2 ! G7 ; G3;4 ! G8 ; G5 ! G8 þ G9 ; G6 ! G7 þ G9 . At last, the phonons have the following symmetries at the BZ symmetry points: 2ðG1 þ G4 þ G5 þ G6 Þ; 2ðK1 þ K2 þ 2K3 Þ, and 2(2M1+M2+M3+2M4) [8]. Among the BZ center phonons, G1 ðA1 Þ and G6 ðE1 Þ ones are both Raman and IR-active, G5 ðE2 Þ is Raman-active, G4 ðB1 Þ corresponds to a silent mode, whereas G2 ðA2 Þ and G3 ðB2 Þ phonons are absent. Using site symmetry analysis, it can be shown that any G state can be connected with any G, K or
M state in any polarization by at least one phonon-assisted optical transition both when the spin–orbit interaction is not taken into account and when it is. The phonon involved depends on the process. To not overload the paper with too long developments we just give hereafter some examples. 3.1. Phonon-assisted transitions without spin–orbit interaction As an example, let us consider the phononassisted transitions in GaN between the bound state a1 of a CD occupying the 2b Wyckoff position (C3v site symmetry) and the band state G2 , the phonon being involved between the localized and virtual states. The direct transition (i.e. that involving no phonons) between a1 and G2 states is forbidden. Note, that the vector representation describing the symmetry of photons is either G1 ðzÞ þ G6 ðx; yÞ when both states involved are band ones or a1 ðzÞ þ eðx; yÞ when at least one state is a localized one. Next, we take into account only phonons from the symmetry points of the BZ since these phonons have the strongest contributions to the processes due to the maxima of the phonon density of states function at these points [8]. The phonons from general symmetry points of the BZ can induce any transitions but their contribution is negligible due to their low density of states. Then the above transition can be achieved at least by three different processes for z-polarized photons:
a1 - - -ðM2;3 ; K3 phononÞ - - - ! a2 - - -ðz-photonÞ - - - ! G2 ð5-type processÞ; a1 - - -ðM2;3 ; K3 phononÞ - - - ! G2 - - -ðz-photonÞ - - - ! G2 ð2-type processÞ; a1 - - -ðz-photonÞ - - - ! a1 - - -ðM2;3 ; K3 phononÞ - - - ! G2 ð6-type processÞ: (M2,3, K3 phonon means that the involved phonon can be M2, M3 or K3) and by three processes for x; y-polarized photons a1 - -ðG5;6 ; M1;2;3;4 ; K1;2;3 phononÞ ! e- -ðx; y-photonÞ ! G2 ð5-type processÞ; a1 - -ðG5;6 ; M1;2;3;4 ; K1;2;3 phononÞ ! G6 - -ðx; y-photonÞ ! G2 ð2-type processÞ; a1 - -ðx; y-photonÞ ! e- -ðG5;6 ; M1;2;3;4 ; K1;2;3 phononÞ ! G2 ð6-type processÞ:
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As a proof, let us check the first of the processes given above. The transition between a1 and virtual a2 states of a CD occupying the 2b Wyckoff position can be mediated by a K3 phonon since the Kronecker product of a1 and a2 is a2 and K3 phonon includes a a2 component when subduced onto the C3v site symmetry group of the impurity [1,8]. Next, it is seen that G2 irrep includes a a2 component when subduced onto the site-symmetry group of the impurity which makes the transition between a a2 state and a G2 one mediated by a z photon to be allowed. It should be noted that for band states and phonons from the BZ center the result of subduction and, consequently, allowed transitions depend only on the point group of the site. However, when subducing the irreps of k 6¼ 0 band states or phonons, one should specify the Wyckoff position, since the groups of different sites (even with the same point group) contain different translations which modify the result of subduction. Indeed, for example, for substitutional and interstitial impurities occupying the 2b Wyckoff position, the K2 phonon cannot induce the transition between a1 and a2 localized states since K2 # Gq ðq¼ 2bÞ ¼ e. In turn, for interstitial impurities occupying the 2a Wyckoff position, the transition between a1 and a2 localized states can be mediated by the K2 phonon since K2 # Gq ðq¼ 2aÞ ¼ a1 þa2 .
states. The direct transition is forbidden in any polarization since the irrep product G1 G2 ¼ G2 does not contain the vector representation G1 ðzÞþ G6 ðx; yÞ. In turn, phonons can connect G1 state with G1 ; G4 ; G5 , or G6 virtual states. The transitions between G1 ; G4 ; G5 virtual states and the G2 final one are forbidden in all polarizations whereas the transition between G6 and G2 is allowed in xyand forbidden in z-polarization. However, the G1 2G2 transition becomes allowed in z-polarization via virtual state of impurity at the 2b site with C3v site symmetry (a 7-type process): G1 - - ðM2;3 ; K3 phononÞ ! a2 - -ðz-photonÞ ! G2 ð7-type processÞ: 3.2. Phonon-assisted transitions with spin–orbit interaction The case when spin–orbit interaction is taken into account can be treated in the same way. Now let us consider, for example, the transition between the bound state e% 2 (originated from an a1 state) of a C3v CD occupying the 2b Wyckoff position and the band state G9 (originated from the G5 state). The direct transition e% 2 ! G9 is allowed in xy and forbidden in z polarization. However, this transition becomes weakly allowed in z-polarization when assisted by phonons:
% ð2Þ e% 2 - -ðG5;6; M1;2;3;4 ; K1;2;3 phononÞ ! e% ð1Þ 1 ; e 1 - -ðz-photonÞ ! G9 ð5-type processÞ; e% 2 - -ðG5;6; M1;2;3;4 ; K1;2;3 phononÞ ! G9 - - -ðz-photonÞ- - - ! G9 ð2-type processÞ; e% 2 - -ðz-photonÞ- - - ! e% 2 - -ðG5;6 ; M1;2;3;4; K1;2;3 phononÞ ! G9 ð6-type processÞ: Involving CD states as the virtual ones, any transition between the G band states, which is forbidden when a bound state is not involved as a virtual one, becomes allowed. Indeed, either the direct or phonon-assisted transitions via a virtual extended state between G1 2G2 ; G1 2G3 ; G2 2G4 and G3 2G4 couples of states are forbidden in z polarization due to the absence of G2 and G3 phonons. (In contrast, it can be shown that an allowed phonon-assisted transition does always exist in any polarization which connects any couple made of a G state and K(M) one). Consider as an example the transition between G1 and G2
3.3. Excitons The above analysis includes also the phononassisted optical recombination of free and bound excitons via virtual bound states. Phonon-assisted optical recombination of free and bound excitons is therefore allowed in any polarization. The phonon involved depends on the process. The only difference from the analysis of optical recombination of carriers lies in the fact that excitons can have any state symmetry, which is not true for carriers. Indeed, the symmetry of free excitons at the P-point of the BZ is given by
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Ph P*e Penv Kronecker product of the hole and electron irreps and the envelope function representation at the P-point. That is the initial exciton state is one of the irreps contained in this product. The analysis [7] shows that free excitons can have also G2 and G3 symmetries which are absent for free carriers. For bound excitons, the irreps are those of the CD site symmetry group. In this case, an a2 state becomes possible for a bound exciton [6]. The final state of an exciton radiative recombination is described by the fully symmetrical irrep (i.e. that of the void), being G1 for free excitons [7] or the fully symmetrical irrep of the CD site symmetry group for bound excitons [6]. 3.4. General properties It should be noted that if one consider an allowed process involving one or several CD states as initial, final or virtual ones, the process is still allowed if the symmetry of a CD is lowered from Gqo (Gqo may be C3v or Cs) to a subgroup Gq
Gqo (Gq may be Cs or C1). Indeed the initial or final or virtual state di of a CD with Gq symmetry would be connected by a subduction relation with the corresponding former one d0j of the higher symmetry CD group Gqo : X ai di ¼ d0j # Gq : ð3Þ As a result, if a process is allowed for d0j state it is allowed for any di state at least for the same light polarizations and the same phonon involved. Last, it has to be added that a transition from the single state of C1 impurity to any band state is allowed for any photon polarization. Transitions involving two or more phonons can be studied in the same manner. For a transition between a couple of states, the number of different possible processes increases very rapidly with the number of phonons. It is easy to show from above that an optical transition between any couple of G-states, any couple of states bound to the same CD and any couple made of a G-state and a bound state is allowed in any polarization whatever is the number of involved phonons from symmetry points of the BZ. However, for each process the phonons involved are, in general, different, depending even on the polarization of the emitted or
absorbed light (see above). This results in different energies of the corresponding transitions. Thus, when studying optical properties near the band gap of GaN crystals with impurities and/or defects, one can therefore be sure that phononassisted optical recombination of free or bound carriers or excitons is allowed in any polarization whatever is the number of phonons. Just the phonon(s) involved may depend on the polarization ((x; y) or z) of the light if the nature of the process depends on the latter. 4. Discussion In GaN, phonon-assisted transitions involving impurity states were experimentally studied in several papers (see for example [12,13]). The authors [13] observed phonon replicas in the photoluminescence spectra of wurtzite GaN which were interpreted as phonon-assisted neutral donor bound-exciton (D0X) recombination. They interpret the lines at 3.4024 and 3.3990 eV resulting from the collapse of D0X accompanied by creation of TO phonons with G1 ðA1 Þ and G6 ðE1 Þ symmetries. This process can be treated in the framework of the above theory. If we assume the interaction between the bound exciton and the electron (hole) of the donor (acceptor) to be negligible, the bound-exciton recombination does obey the selection rules derived in our previous paper [6]. When an interaction term is included, the exciton plus electron (hole) bound to a neutral donor (acceptor) behave as a whole. In the case of a donor, the symmetry of the initial state is described by one of the irreps contained in the product hðexcÞ *
daeðexcÞ db
d env dgeðdonorÞ ;
ð4Þ
where daeðexcÞ and dbhðexcÞ are irreps of the electron and hole forming the exciton bound to the donor, d env is the representation of exciton envelope function, and dgeðdonorÞ is the irrep of the electron of the donor. The symmetry of the final state is the symmetry of the electron of the donor (dxeðdonorÞ ) since the final state of the exciton is that of the void (totally symmetrical irrep). The symmetries of photon and
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phonon involved are dv (vector representation of the site symmetry group Gq ) and d ph ¼ ðDðphÞ # Gq Þ, respectively. Our analysis shows that the exciton recombination observed in [13] can be achieved by different processes involving the G1 ðA1 Þ and G6 ðE1 Þ phonons. Without taking into account spin–orbit interaction, the initial state of the donor electron (D) can be a1(D) and e(D) whereas that of the bound exciton (X) can be a1(X), a2(X), and e(X) being also the initial state of the D0X complex. After recombination, the final state of D0X is that of the donor electron, namely either a1(D) or e(D). Since being involved, the G1 ðA1 Þ phonon does not change the symmetry of a state. Therefore, the process, from the symmetry point of view, reduces to one-phonon one. The performed analysis shows that any arbitrary pair of initial and final states can be coupled by x; y or z-photon and the G1 ðA1 Þ and G6 ðE1 Þ phonons, except for a2(D0X) ! a1(D) transition which is forbidden in z-polarization. When including spin–orbit interaction, the single-valued irreps transform into the doublevalued ones as follows: % ð2Þ %2: a1 ! e% 2 ; a2 ! e% 2 ; e ! e% ð1Þ 1 þe 1 þe Then, the initial states of electron, exciton and % ð2Þ % 1 as well as the D0X complex can be e% ð1Þ 1 ; e 1 , and e final state of the electron. Our analysis shows that again any arbitrary pair of initial and final states can be coupled by x; y or z-photon and the G1 ðA1 Þ 0 % ð2Þ and G6 ðE1 Þ phonons, except for e% ð1Þ 1 ; e 1 ðD XÞ ! ð1Þ ð2Þ e% 1 ; e% 1 (D) transitions which are forbidden in z-polarization. The complete analysis shows that for arbitrary combination of initial states of the exciton and the electron (hole) of the donor (acceptor) and of final state of the electron (hole) one can find, at least, one phonon-assisted process allowed for any given polarization. Thus, the analysis of phonon-assisted optical transitions between bound states via bound or extended virtual states given in Section 3 includes also the analysis of recombination of excitons bound to neutral donors or acceptors. It can therefore be concluded that the radiative recombination of an exciton bound to a neutral donor
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(acceptor) is always allowed in any polarization whatever are the symmetries of the initial and final states of the electron (hole) of the donor (acceptor) and the number of phonons involved. However, the different allowed processes correspond, in general, to transitions with different energies due to energy transfer to phonons and donor (acceptor) electron (hole) involved.
5. Conclusions The main theoretical results obtained in this paper are of general character and can be applied for any crystal. In particular, in terms of the site symmetry approach, we have derived optical selection rules for the seven possible phonon-assisted transitions involving CD localized states as initial, virtual, and/or final ones. Our results include exciton radiative recombination processes as well. We have also established that, in optical transitions involving a virtual state, photons and phonons can be interchanged only for the BZcenter infrared-active phonons. Next, we have shown that for allowed optical transitions involving CD localized states and k 6¼ 0 band states and/or k 6¼ 0 phonons one should specify not only the CD site symmetry but also the Wyckoff position it occupies. We have applied the above theory for analysis of phonon-assisted optical transitions in wurzitetype GaN. We analyzed all possible phononassisted optical transitions without and with the inclusion of spin–orbit interaction. We have shown that optical recombination of free and bound carriers and excitons is always achievable through phonon-assisted processes whatever is the number of phonons involved. Of course, when several processes are possible for connecting an initial state to a final one with the same number of phonons, each allowed process is accompanied, in general, by different phonons resulting in different energies of optical transitions. The variety of initial, virtual, and final states is particularly rich for excitons bound to neutral donor or acceptors due to the extra electron (hole) involved in the process. In any process we considered, the phonons are those from symmetry points of
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the BZ. Their density is high making the phononassisted transitions to be rather intense as can be checked from numerous published photoluminescence spectra. Acknowledgements We acknowledge the CLG 97053 NATO grant, the Russian Foundation for Basic Research (Grant No. 99-02-18318) and the support of Mairie de Paris and of French Embassy in Moscow. References [1] R.A. Evarestov, V.P. Smirnov, Site Symmetry in Crystals: Theory and Applications, Springer Series in Solid State Sciences, Vol. 108, Springer, Heidelberg, 1993. [2] Yu.E. Kitaev, A.G. Panfilov, P. Tronc, R.A. Evarestov, J. Phys.: Condens. Matter 9 (1997) 257. [3] Yu.E. Kitaev, A.G. Panfilov, P. Tronc, R.A. Evarestov, J. Phys.: Condens. Matter 9 (1997) 277.
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