phys. stat. sol. (b) 216, 599 (1999)

Subject classification: 78.20.Bh; 71.35.ÿy; S7.14

Optical Selection Rules for Hexagonal GaN

P. Tronc1) (a), Yu. E. Kitaev2) (a), G. Wang (a), M. F. Limonov2) (a),A. G. Panfilov2) (a), and G. Neu (b)

(a) Laboratoire d'Optique, Ecole Superieure de Physique et Chimie Industrielles,10 Rue Vauquelin, F-75005 Paris, France

(b) Centre de Recherche sur l'HeÂteÂroeÂpitaxie et ses Applications, Centre National de laRecherche Scientifique (CRHEA-CNRS), Rue Bernard GreÂgory, Parc Sophia-Antipolis,F-06560 Valbonne, France

(Received July 4, 1999)

We determined the possible symmetries for band (Bloch) states at various points of the Brillouinzone. The spin±orbit interaction has been taken into account. Free excitons offer larger sets ofpossible state symmetries than single carriers do. The selection rules for the optical transitions havebeen established. Experiments with polarized light are shown to be relevant to study the nature(s or p) of A, B, and C excitons.

1. Introduction

Hexagonal GaN, AlN, and InN have the wurtzite structure (space group C6v4 ) with dif-

ferent values of anion z coordinate (Table 1). In a previous paper, we performed ananalysis of state symmetries for electrons and excitons bound to impurities and defectsand derived the optical selection rules for transitions involving them [1]. Here, we ana-lyze the symmetries of band (Bloch) states at the symmetry points of the Brillouin zone(BZ) and derive the selection rules for optical transitions between band states. Excitonsare also considered. Finally, we analyze some experimental results which appeared inthe literature.

2. Band-State Symmetry

We determined the symmetry of band states in hexagonal GaN using the method ofinduced band representations of space groups [2]. It allows one to establish a symmetrycorrespondence between band states and localized (Wannier-type) states in crystals.The atoms in GaN occupy 2b Wyckoff positions with C3v site symmetry and coordi-nates shown in column 1 of Table 1. Column 2 contains those irreps of the site-symme-try group according to which the localized wave functions transform as well as thesymbols of double-valued irreps (with a bar over the irrep symbol) in the case wherethe spin±orbit interaction is taken into account. The remaining columns of Table 1 givethe labels of the induced representations in the k-basis, with the symbols of k-points,their coordinates (in units of primitive translations of the reciprocal lattice), and their

P. Tronc et al.: Optical Selection Rules for Hexagonal GaN 599

1) Corresponding author; Tel.: 331-40-79-46-05; Fax: 331-43-36-23-95;e-mail : tronc@optique.espci.fr

2) Permanent address: A. F. Ioffe Physico-Technical Institute, 194021 St. Petersburg, Russia.

point groups in rows 1 to 3, respectively, and the indices of small irreps of little groupsin subsequent rows; these determine the symmetries of band states. For example, thes-atomic-like Wannier-type orbital induces Bloch states G1, G4, A1, A4, K3, H3, M1, M4,L1, and L4. The labelling of point and space group irreps is the same as in Ref. [1]. Insuch a labelling, G5 and G6 irreps are exchanged in comparison with notations commonfor the II±VI wurtzite materials. Table 1 is also valid for free excitons (see below).Note that the a2 irrep of the site symmetry group corresponds to bound excitons only[1]. Therefore G2, G3, A2 and A3 states exist for free excitons only but not for carriers.We limit ourselves to the s and p atomic-like localized electron states since just thesefunctions form the uppermost valence-band states and the lowest conduction-bandstates and, therefore, determine the interband optical transitions. As for d orbitals, eventhough they should be taken into account to describe lower valence bands [3], they donot induce band states with symmetries different than those of states induced by s andp orbitals [2]. They could just modify the number of states with a given symmetry. Wecan see that G1 and G4 states are induced by s orbitals as well as by pz orbitals of Gaand N. G5 and G6 states are formed by px and py orbitals of Ga and N atoms. Thus, atthe G point we have four G1, four G4, two G5, and two G6 band states resulting from sand p localized states (Ga and N atoms). Finally, when the spin±orbit interaction istaken into consideration, the symmetry correspondence between the Bloch states whichtransform according to single-valued irreps (without spin±orbit coupling) and thosewhich transform according to double-valued irreps (with spin±orbit coupling) can beobtained following Ref. [4] with the spinor function D1=2 = G7. Thus, for example, at theG point of the BZ, the symmetry correspondence is G1! G7, G2 ! G7, G3! G8, G4 ! G8,G5 ! G8 + G9, G6 ! G7 + G9. The double-valued induced representations are given inTable 1. The complex-conjugated irreps forming a pair correspond to different stateswith the same energy. This degeneracy is connected with the inversion of time and canbe lifted by applying the magnetic field which does not reduce the point symmetry ofthe system (that is the one directed along the symmetry axis). The complex-conjugatedirreps are combined in so-called co-representations (co-reps) which are given in thecaption of Table 1. On applying the magnetic field, the states described by co-reps aresplit whereas the states described by doubly degenerate irreps (for example G7, G8, G9,K6, M5) are not.

600 P. Tronc et al.

Ta b l e 1Band state symmetries in wurtzite GaN (AlN, InN) crystal (space group C4

6v (P63mc))GaN : Ga (z � 0), N (z � 0.377) ; AlN : Al (z � 0), N (z = 0.382) ; InN : In (z � 0),N (z � 0.38). Co-reps : eÅ �1�1 � eÅ �2�1 ; K4 �K5;A1 �A4;A5 �A6;H1 �H2;L1 � L4;L2 � L3 :The a2 state can exist for bound excitons only

G A K H M Lq b �000� (00 1

2) (13

13 0) (1

313

12) (1

2 00) (12 0 1

2)C6v C6v C3v C3v C2v C2v

2b a1(s; pz) 1, 4 1, 4 3 3 1, 4 1, 4a2 2, 3 2, 3 3 3 2, 3 2, 3

(13

23 z) e(px, py) 5, 6 5, 6 1, 2, 3 1, 2, 3 1, 2, 3, 4 1, 2, 3, 4

(23

13 z� 1

2) eÅ �1�1 9 9 6 6 5 5C3v eÅ �2�1 9 9 6 6 5 5

eÅ 2 7, 8 7, 8 4, 5, 6 4, 5, 6 5, 5 5, 5

3. Selection Rules for Optical Transitions between Band States

The procedure used to establish the optical selection rules is given elsewhere [4]. Wepresent the k � 0 parts of Kronecker products of irreps corresponding to the variouscombinations of initial and final states at the G symmetry point of the BZ (Table 2).The transitions are allowed between those pairs of states for which Kronecker productshave irreps in common with the vector representation Dn � G1(z) � G6(x, y) (the anglebetween the x and y axes is 120�). Thus, the optical transition between the G5 and G6

band states taken as an example is allowed in the x and y polarizations and forbiddenin the z one. It can be seen that some forbidden transitions become allowed when the

Optical Selection Rules for Hexagonal GaN 601

Ta b l e 2Kronecker products of G-irreps and selection rules for direct optical transitions (the al-lowed polarizations are given in parentheses). The G2 and G3 states can exist only forexcitons

single-valued irreps (neglecting the spin±orbit interaction)

G1 G2 G3 G4 G5 G6

G1 G1 (z) G2 G3 G4 G5 G6(x, y)G2 G2 G1(z) G4 G3 G5 G6(x, y)G3 G3 G4 G1(z) G2 G6(x, y) G5

G4 G4 G3 G2 G1(z) G6(x, y) G5

G5 G5 G5 G6(x, y) G6(x, y) G1(z) � G2 � G5 G3 � G4 � G6(x, y)G6 G6(x, y) G6(x, y) G5 G5 G3 � G4 � G6(x, y) G1(z) � G2 � G5

double-valued irreps (taking into account the spin±orbit interaction)

G7 G8 G9

G7 G1(z) � G2 � G6(x, y) G3 � G4 � G5 G5 � G6(x, y)G8 G3 � G4 � G5 G1(z) � G2 � G6(x, y) G5 � G6(x, y)G9 G5 � G6(x, y) G5 � G6(x, y) G1(z) � G2 � G3 � G4

Ta b l e 3The modification of selection rules for direct optical transitions when including spin±or-bit interaction. The labels of the irreps in parentheses refer to the case when spin±orbitinteraction is not taken into account. The light polarizations in brackets (parentheses)refer to the case only without (only with) taking into account spin±orbit interaction;those in capitals are allowed in any case. The G2 and G3 states can exist only for excitons

conduction band state

G7�G1� G7�G2� G7�G6� G8�G3� G8�G4� G8�G5� G9�G5� G9�G6�valence G7�G1� (x, y) Z (x, y, z) X, Y (z) (x, y) X, Yband G7�G2� (x, y, z) (x, y) Z X, Y (z) (x, y) X, Ystates G7�G6� X, Y(z) X, Y(z) (x, y) Z X, Y X, Y (x, y) [z]

G8�G3� (x, y) Z (x, y, z) X, Y(z) X, Y (x, y)G8�G4� (x, y, z) (x, y) Z X, Y(z) X, Y (x, y)G8�G5� [x, y] X, Y(z) X, Y(z) (x, y) Z (x, y) [z] X, YG9�G5� (x, y) (x, y) X, Y X, Y X, Y (x, y) [z] Z [x, y] (z)G9�G6� X, Y X, Y (x, y) [z] (x, y) (x, y) X, Y [x, y] (z) Z

spin±orbit interaction is taken into account and, in turn, some allowed transitions be-come forbidden (Table 3). Obviously time-reversal processes obey the same selectionrules. The transitions allowed only from the spin±orbit interaction of course have aweaker oscillator strength.

4. Free-Exciton State Symmetry and Optical Selection Rules

The Hamiltonian of a free exciton has the space group symmetry since the term within theHamiltonian which describes the electron-hole coupling is invariant under any operationof the space group. The symmetry of an exciton [1] at the P-point of the BZ is described bya Pexc single-valued irrep which is included in the Ph * PePenv Kronecker product of thehole, electron, and envelope function irreps at the P-point. In exciton radiative recombina-tion, the final state is that of the void (G1 fully symmetrical irrep). Radiative recombinationis therefore allowed for excitons whose irrep Pexc belongs to the vector representation. G1

and G6 excitons can therefore radiatively recombine by emitting light polarized along the zaxis and in the (x, y) plane, respectively. Table 2 can be straightforwardly used for studyingrecombination of excitons with totally symmetric envelope function G1. It seems reason-able to assume that the ground state of an exciton corresponds to a state with a totallysymmetric envelope function. For Wannier excitons at the G-point one can thus put intocorrespondence the G1 envelope function with zero orbital momentum states (s-excitons)whereas for momentum equal to 1 (p-excitons) Genv should be G1 + G6.

5. Discussion

There is a general agreement about the symmetries at the G-point for the lowest con-duction band (G7 originating from G1) and for the uppermost valence bands (G9 origi-nating from G6, G7 from G6, and G7 from G1 for A, B, and C transitions, respectively). G7

c

and G9v give rise to G5 and G6 s-excitons (G6 when spin±orbit coupling is not taken into

account), G7c and both G7

v give rise to G1, G2, and G6 s-excitons (G6 and G1, respectively,for B and C transitions when spin±orbit coupling is not taken into account) (Table 4).For the A and B transitions, the G6 exciton optical recombination is fully allowed (bothwithout and with taking into account the spin±orbit coupling) in (x, y) polarization(light propagating along the z direction). On the contrary, the transition is allowed inthe same polarization only from spin±orbit coupling for the C exciton. These resultsare in a good agreement with measurements [5 to 7] which show that the C transition is

602 P. Tronc et al.

Ta b l e 4Kronecker products G*hGeGenv for free excitons at the G point

Transitions A B C

neglecting spin±orbit interaction

s-excitons G6(x, y) G6(x, y) G1(z)p-excitons G1(z) � G2 � G5 � G6(x, y) G1(z) � G2 � G5 � G6(x, y) G1(z) � G6(x, y)

taking into account spin±orbit interaction

s-excitons G5 � G6(x, y) G1(z) � G2 � G6(x, y) G1(z) � G2 � G6(x, y)p-excitons G1�z�+G2 � G3 � G4

� 2G5 � 2G6(x, y)2G1�z� � 2G2

� G5 � 3G6�x; y�2G1�z� � 2G2

� G5 � 3G6�x; y�

weak when compared to A and B ones. Table 4 shows that the A transition is strictlyforbidden in z polarization, whereas the B transition is weak since allowed from spin±orbit coupling only and the C one is strong since fully allowed. Reflectance measure-ments fit this result [6].

Next, Shan et al. [7] assigned in their photoreflectance spectra some structures to 2sstates of A, B, and C exciton families (light propagating along the z axis). It can readilybe seen that p-excitons with G1 or G6 symmetry are present in A, B, and C families(Table 4) both without and with taking into account the spin±orbit coupling. Thisshows that their optical recombination can be present and pretty strong in photoreflec-tivity spectra. In the case of the C transition, optical recombination of the G6 s-excitonis weakly allowed in (x,y) polarization (from spin±orbit coupling only) whereas, asshown above, the G6 p-exciton is fully allowed. Moreover, the oscillator strength ofns-excitons is known to decrease very rapidly with increasing values of n. One cantherefore wonder if the structure assigned in Ref. [7] to a C 2s-exciton does not actuallyresult from a p one.

Finally, it is worth noticing that an uniaxial stress along the six-fold (z) axis (e.g. thestress induced by a lattice mismatch with the substrate) does not modify the symmetryproperties of the crystals and hence the optical selection rules.

Acknowledgements We acknowledge the CLG 975053 NATO grant, the 99-02-18318Russian Foundation of Basic Research grant, and the support of Mairie de Paris and ofthe French Embassy in Moscow.

References

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Optical Selection Rules for Hexagonal GaN 603