h One of the authors (V.P.S.) acknowledges the support of Ministre de la Recherche (France).

1. Introduction Silicon is used to built integrated circuits and other devices based on charge transport phenomena (GaAs and related The knowledge of selection rules is well known to be compounds can also be used for such devices when their of a great importance in the study of optical properties of high carrier mobility is needed). Some nanostructures, such crystals, electron–phonon interaction and phase transitions as type II GaAs/AlAs SL’s have also indirect gaps.

in solids. It is evident that the generation of selection In any bulk semiconductor or semiconductor structure, rules for transitions between states related to the center it is necessary to study direct transitions also at points in k = 0 of the Brillouin zone (BZ) can be expressed in terms the BZ other than and indirect transitions when one or of representations (reps) of the crystal point group. For both states correspond to k = 0 (participation of a particle direct transitions between the states with k = 0 and for with a finite wave vector). The more frequent case is indirect transitions, the generation of selection rules is more that of phonon assisted transitions. Note that the high complex because of complicated structure of space group symmetry points in the BZ are particularly important since reps (subgroup method [1–4], full group method [5]).

they generally correspond to high density of phonon states.

The optical properties at the point in the BZ of materials Selection rules’ deduction in general case is much more used for optoelectronic devices are of crucial importance complicated since the symmetry of the Bloch states and since the materials are direct gap semiconductors. Among hence the selection rules depend of the location of the BZ them, one can find stoichiometric crystals (such as GaAs, points involved in the process.

InP or CaN) and alloys (for example, ternary compounds In Chapter 4 of Ref. [3], the procedure of generation like AlGaAs or quaternary ones such as GaInAsS and of the selection rules in crystals is based on a sufficiently GaInAsSb). In any of these materials, the fundamental refined mathematical groundwork (double and triple coset optical transition takes place at the point insuring strong decompositions of space groups, use of the Mackey theorem absorption and recombination. By varying the ratios of for induced reps). This procedure was realized in the various elements in the alloys, it is possible to tailor the computer program, the results being collected in the threeband gap value at point to fit the required operating volume Tables [4].

wavelength. Many nanostructures such as quantum wells The more simple approach to the selection rules’ problem (QW’s), superlattices (SL’s), quantum wires (QWI’s) and may be developed basing on the well-known relation quantum dots (QD’s) are made of the materials mentioned between small irreducible reps (irreps) of little space groups above. Nanostructures are usually studied in the envelope function approximation based on the properties of the Bloch and projective irreps of corresponding little co-groups [6,7].

functions at the point in the BZ of the bulk materials they We demonstrate in this paper that the procedure of are built from. generation of the selection rules for a transition between any On the other hand, germanium Ge and silicon Si (the states in crystals can be formulated in terms of projective most widely used semiconductors) have indirect gaps. irreps of point groups.

1374 V.P. Smirnov, R.A. Evarestov, P. Tronc In Section 2, all the necessary notations are introduced The little groups Gk for different points of the star k are j and the connection between irreps of space groups and isomorphous to the little group Gk projective irreps of point groups is considered in detail.

Gk = g Gkg-1. (6) j j j The general procedure of generation of the selection rules is formulated in Section 3. Each step of its realization is The irreps of G (full irreps) are labeled by the irreducible illustrated by calculations of the Kronecker products for star k of the wave vector k and by the index numbering different irreps at the W point of the BZ for nonsymmorphic the inequivalent irreps within the same star k: [G]( k). The space group O7 which is one of the most complicated space h full irrep [G]( k) of G is in a one-to-one correspondence groups for the selection rules’ generation. In Section 4, with the small irrep [Gk] of Gk G and is obtained from as an example, the general procedure is applied to obtain the latter by induction procedure [7,8] the selection rules for direct and phonon–assisted electrical dipole transitions between some states in crystals with space [G]( k) =[Gk] G. (7) group O7.

h The set of all small irreps of all little groups Gk with k being in a representation domain of the BZ determines 2. Connection between small unambiguously all the irreps of the space group G. That representations of space groups is why the Tables of space group irreps, as a rule, contain and projective representations the small irreps of little groups Gk [3,9–12].

of point groups k The matrices D([G ] )(gi,n) (gi,n Gk) of the small irreps [Gk] of Gk are in one-to-one correspondence with the Let the space group G of a crystal consist of the elements k matrices d([F ] )(Ri) of so-called projective irreps [Fk] of Fk g =(R|vR + an) G where the orthogonal operation R as follows is followed by the improper translation vR and lattice k n k translation an. The vectors an form the invariant subgroup T D([G ] )(gi,n) =e-ika d([F ] )(Ri), of the space group G (T G). The point group F of gi,n =(Ri|vi + an) Gk, Ri Fk. (8) the nF orthogonal operations R describes the symmetry of directions in the crystal and is called crystalline class or k k In particular, the matrices D([G ] )(gi,0) and d([F ] )(Ri) point symmetry group of the crystal. The set of left cosets coincide (Ri|vi)T in the decomposition of G with respect to the k k D([G ] )(gi,0) =d([F ] )(Ri), translation subgroup T gi,0 =(Ri|vi ) Gk, Ri Fk. (9) nF k The multiplication law for the matrices d([F ] )(Ri) of G = (Ri|vi)T (1) projective irreps [Fk] of a co-group Fk follows from the i=multiplication law for space group elements forms a factor group G/T isomorphic to the point group F k k k (F G/T ) of order nF. d([F ] )(Ri)d([F ] )(Ri ) =d([F ] )(RiRi )(k)(Ri, Ri ), (10) The translation group T is Abelian. All its irreps are onewhere the set of dimensional and are classified by wave vectors k in the BZ i (k)(Ri, Ri ) =e-ik(v +Rivi -vi,i ), d(k)(an) =exp(-ikan). (2) |(Ri, Ri )|2 = 1, Ri, Ri Fk (11) The elements g G leaving the wave vector k invariant up to reciprocal lattice vector Bm is a factor system for the projective irreps [Fk] (gii,0 =(Ri, Ri |vii ) Gk). The characters of these prog(k)k = R(k)k = k + Bm (3) jective irreps can be taken directly from Tables [3,9,10,12].

If all the factors (11) are equal to unit, the projective irrep form the little group Gk of the wave vector k. The becomes an ordinary one. In particular, this is the case of all group G-k consists of the same elements as the group Gk.

The little co-group Fk = F-k includes the elements R(k) the little co-groups Fk of all the symmorphic space groups (since all vj = 0).

((R(k)|vR(k)) Gk). The representatives gi =(R |vj) of left j There exist projective irreps [Fk] with another choice of cosets g Gk in the decomposition of G with respect to j the factor system (p-equivalent to [Fk]) Gk G t G = g Gk g1 =(E|0)(4) k i k j D([F ] )(gi,n) =e-ik(v +an) d([F ])(Ri), j= gi,n =(Ri|vi + an) Gk, Ri Fk, determine the so-called irreducible star k of the wave -vector k consisting of t wave vectors i (k)(Ri, Ri ) =ei(k-R k)vi, k : kj = g k = R k, j = 1, 2,..., t. (5) |(Ri, Ri )|2 = 1, Ri, Ri Fk. (12) j j Физика твердого тела, 2003, том 45, вып. A point group approach to selection rules in crystals k They are used in Ref. [11]. The matrices D([G ] )(gi,0) and be matrices of the direct product of two projective reps.

k Then i d([F ] )(Ri) differ by the factor e-ikv only k3 kd([F ])(R1) · d([F ] )(R2) k i k D([F ] )(gi,0) =e-ikv d([F ])(Ri), k1 k2 k1 k= d([F ])(R1) d([F ])(R1) d([F ])(R2) d([F ] )(R2) k1 k1 k2 kgi,0 =(Ri|vi) Gk, Ri Fk. (13) = d([F ])(R1) · d([F ])(R2) d([F ])(R1) · d([F ])(R2) ]) ) k1 Let relation (8) between the reps [Gk] of Gk and [Fk] = d([F (R1R2) · (k (R1, R2) of Fk be denoted by the symbols k2 d([F ])(R1, R2) · (k )(R1, R2) [Gk] =[Fk] Gk, [Fk] =[Gk] Fk. (14) k3 1 =d([F ])(R1R2) · (k )(R1, R2) · (k )(R1, R2) k3 Then the relation between the irreps [G]( k) of space =d([F ])(R1R2) · k (R1, R2)(18) group G and the projective irreps [Fk] or [Fk] of little for both (11) and (12) factor systems.

co-groups has the form Let the coset representatives g( j) Gk in the decompos j sition of Gk with respect to the translation group T j [G]( k) =([Fk] Gk) G, or [G]( k) =([Fk] Gk) G.

(15) Gk = g( j)T, g(1) Gk (19) j s s The basis functions of irreps [G]( k) of a space group G s can be always chosen as being the basis functions of small j) j) j) s s s irreps [Gk] of little groups Gk and projective irreps [Fk] be chosen in the form g( (R( |v( ), i. e. they are among the representatives (Ri|vi) in the decomposition (1).

(or [Fk]) of little co-groups Fk.

The element g g(1)g-1 Gk with g being taken from j s j j j For the selection rules’ generation, it is necessary to decomposition (4) (see also (6)) may differ from g( j) by s consider the direct product of small reps of two of the some lattice translation. That is why the notations of little groups. The latter is possible only for the common the small irreps of the little groups Gk and Gk (and the elements of little groups, i. e. for their intersection. Let j [Gk ] and [Gk ] be small reps of two little groups Gk projective irreps of the corresponding little co-groups) may 1 2 be different.

and Gk. The direct product of their subductions on their In particular, let g k = -k. The groups Gk and G-k jintersection ([Gk ] (Gk Gk ) [Gk ] (Gk Gk )) 1 1 2 2 1 are composed of the same elements. The whole set of is a small rep of the group (Gk Gk ). Every ele1 small irreps of the little group G-k is complex conjugated ment gi,0 (Gk Gk ) leaves invariant the wave vec1 with respect to the whole set of small irreps of the little tors k1 and k2 and, therefore, their sum k3 = k1 + k2:

group Gk, but the notations of the irreps of Gk and of G-k (Gk Gk ) Gk. The little group Gk has no other 1 2 3 may differ (see example in Section 3).

common elements either with Gk or with Gk. Indeed, 1 Let Q be a group and H be its subgroup (H Q). Let let us assume the contrary that d() and D() be irreps of H and Q, respectively. Then the frequency of the irrep D() of Q in the rep (d() Q) g Gk, g Gk, g (Gk Gk ). (16) / 3 1 1 induced by the irrep d() of H is equal to the frequency of the irrep d() of H in the rep (D() H) subduced by D() Such an element g would leave invariant k3 and k1 on H (Frobenius reciprocity theorem). This theorem can and, therefore, k2 = k3 - k1, i. e. it would be contained be applied also to the projective irreps of a group and its in Gk Gk in contradiction with the initial assumption.

1 subgroup with the same factor system [6].

k1 kLet d([F ]) and d([F ]) be the matrices of subductions of the projective reps [Fk ] (Fk Fk ) and 1 1 3. Procedure of the selection rules’ [Fk ] (Fk Fk ) of two little co-groups Fk and Fk 2 1 2 1 1 2 generation using projective with factor systems (k )(Ri, R ) and (k )(Ri, R ) j j representations of point groups (Ri, R Fk Fk ), respectively. The direct product j 1 k1 k2 kd([F ]) d([F ]) is a projective rep d([F ] ) (k3 = k1 + k2) The stationary states of a system with the symmetry of the group (Fk Fk ) Fk with the factor system 1 2 of a space group G are classified according to the irreps 3 1 (k )(Ri, R ) =(k )(Ri, R )(k )(Ri, R ). Indeed, let j j j of G and their full group–theoretical notation is as follows:

|k,, m, µ where k = k1, k2,..., kt (star k), m numbers k3 k1 kd([F ])(R) d([F ])(R) d([F ])(R), the basis vectors of the small irrep of the little group Gk, and µ numbers the independent bases of equivalent reps R (Fk Fk )(17) of Gk.

1 Физика твердого тела, 2003, том 45, вып. 1376 V.P. Smirnov, R.A. Evarestov, P. Tronc Let us consider the selection rules for the transi- Table 1. Characters of single- and double-valued projective tions between the stationary states |k( f ), ( f ), m( f ), µ( f ) irreps of the little co-groups FW D(i) (i = 1-6) and single(i) 2d and double-valued small irreps of the little groups GW (i = 1-6) (i) and |k(i), (i), m(i), µ(i) caused by an operator P (for six vectors in the star W: (102), (120), (210), (012), (021) (k(p), (p), m(p)) transforming according to the irrep and (201) in the units of /a along Cartesian axes, a being the (k(p), (p)) of G. If the operator P transforms according lattice constant) in the BZ in crystal with the space group O7, h to a reducible rep of G, one can obtain the selection rules = exp(i/4) for every of its irreducible components separately.

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