Volume 91A, number 2
23 August 1982
SYMMETRY OF FINITE CRYSTALS J. ZAK’ Department ofPhysics, The University ofBritish Columbia, Vancouver, BC, Canada, V6R 2TY Received 29 March 1982
It is shown that the Born—von Kaxman boundary conditions change in a substantial way the structure of the irreducible representations of space groups. As a consequence of this, they change the symmetry labelling of the elementary excitations in solids. What remains unchanged by the boundary conditions is the labelling of localized states of band representations in solids.
The Born—von Karman boundary conditions (BKBC) are known to play an important role both in the dynamics of elementary excitations in solids  and in the representation theory of space groups . There are a number of proofs [1,3] showing that in sufficiently large crystals the density of states for elementary excitations in finite solids is essentially independent of the boundary conditions. In representations of space groups the BKBC are used as a tool for converting the space groups into finite groups . The application of these boundary conditions makes the translation group finite and consequently also the whole space group. The representations of the translation group are labelled by the k-vector and when the BKBC are applied k assumes only discrete values. However, when the crystal is sufficiently large, k varies in very small steps, and the number of representations of the translation group in the interval dk per unit vol. ume of the crystal will be dk/(2ir)3 and is essentially ,
independent of the size of the crystal. In general, this argument should also be applicable to the whole space group and one is led to believe that the irreducible representations of space groups are not, in any essential way, influenced by the Born—von Karman boundary conditions. In todays literature [2,4—6]it is accepted as an established fact that the BKBC have no essential influence on the representations of space groups. On leave from the Physics Department, Technion, Haifa, Israel.
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In this letter we re-examine the irreducible representations of space groups for finite crystals. Contrary to the commonly accepted belief it is proven that the whole stmcture of irreducible representations in finite crystals is very sensitive to the boundary conditions. The reason for this is that some k-vectors in the Brillouin zone with high symmetry are completely excluded by the particular boundary conditions no matter how large the crystal is. By changing the structure of the irreducible representations the BKBC influence substantially the labelling of elementary excitations in solids. The BKBC can be introduced in the following way, via conditions on the wave function ~1i(r): ~Li(r+S~a.)~1i(r), 1
where a~are the unit vectors of the Bravais lattice, and S, is any large integer (i = 1, 2, 3). With the condition (1) imposed the translation group will have 5 1S2S3 elements and correspondingly the whole space group will become finite-dimensional. As an example showing how the boundary conditions (1) influence the representations, let us first consider a one-dimensional crystal with inversion symmetry. The space group of this crystal C1 contains the unit element E, the inversion I, S translations and the mixed elements. The translations can be expressed as ma, with a the lattice constant, where for S even m = 0, ±1, ±(S/2 1), 5/2;while for S odd m = 0, ±1, ±(S 1)12. The group C, for Seven, S = 2N, has ...,
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the following N + 3 classes  : N + 1 classes of pure translations (ma, —ma), m = 0, 1, N, and 2 classes containing the inversion (II2ma), (II(2m + l)a) with m assuming any possible value. On the other hand, for S odd, S = 2N + 1, the group C1 has N + 2 classes: ..,
N + 1 classes of pure translations (ma, —ma), m = 0, 1, ...,Nand 1 class (II2ma), (II(2m + 1)a). The irreducible representations of C, are easily found for each symmetry point k in the Brilouin zone  When the boundary conditions (1) are imposed, k assumes the following S possible values .
r = 0, 1,
For S even, S 2N, this can be rewritten as k = (7r/Na)r, r = 0, ±1, ±N, ...,
where k = 11/a and —71/a coincide because they differ by a reciprocal lattice vector 271/a. On the other hand, when S~2N + 1, k assumes a different set of S values k = [211/(2N+ l)a] r, r = 0, ±1, ±N. (4) Note that in (4) the point k = 11/a is excluded. What this means is that while in (3) there are two points with high symmetry k = 0, 11/a (invariant under I) in (4) there is only one such point k = 0. Correspondingly, the structure of the irreducible representations of C1 differs substantially for the cases of even and odd S. For S = 2N, C, has 4 one-dimensional representations (even and odd Bloch functions with respect to I at k = 0,71/a) and N— 1, two-dimensional representations at general k’s in the Briilouin zone. The situation is completely different for S odd. In this case (S = 2N + 1), the symmetry point k = IT/a is excluded by the boundary conditions and there is only one point of high symmetry, k = 0. The irreducible representations of C1 are: 2 one-dimensional representations (even and odd Bloch functions with respect toI at k = 0) and N two-dimensional representations at general k’s in the Brillouin zone. It is interesting to point out that the number of two-dimensional representations has changed from N— 1 (for S = 2N) to N(for S = 2N + 1). The relative change is 1/N which is negligible foi large N. This result is of the same nature as the change of the density of states for elementary excitations [1, 3]. However, the number of one-dimensional representations goes down by a factor of 2 when S changes from even to odd no matter how large S is. The conclusion is that the boundary conditions change corn...~
23 August 1982
pletely the structure of the irreducible representations in one-dimensional crystals. The situation is also very similar in three-dimensional crystals. When the condition (1) is imposed the wave vector k assumes the following possible S1S2S3 values [2,6] k
where K~are the unit vectors of the reciprocal lattice, and mi = 0, 1, ,S,~ 1,1 = 1, 2,3. It can be checked that the number of general k-vectors in the Brillouin —
zone is not influenced, in any essential way, by the particular choice of the integers S~.However, the appearance (or disappearance) of points of high symmetry on the surface of the Brillouin zone turns out to be very sensitive to the S,’s. We have investigated, in detail, the two important space groups O~and D~hof the diamond and the hexagonal close-packed structures correspondingly. The results of the investigation are summarized in the tables 1 and 2. As was already mentioned above one should expect that the BKBC will exclude some k-vectors of high symmetry in the Brillouin zone. It can easily be checked that vectors of high symmetry inside the Brillouin zone are not excluded by the boundary conditions and they are not listed in the tables 1 and 2. On the other hand, symmetry points on the surface of the Brillouin zone turn out to be very sensitive to the choice of the numbers S1, S2, S3. This is clearly seen from the tables 1 and 2 where the main symmetry points for the groups O~and D~hare listed  Let us first discuss table 1 for the diamond structure. If all the three integers S1, S2, S3 are odd then all the points of high symmetry on the surface of the Table 1 Integers leading to symmetry points on the surface of the Brillouin zone for the space group O~.S1, S2. S3 are integers defining the size of the crystal [see eq. (1)]. N1, N2, N3 are arbitrary integers. The symmetry points are denoted according to ref. . -
= (0, 2it/a, 0) W (IT/a, 2it/a, 0) K = (3ir/a, 3it/a, 0) L = (IT/a, It/a, it/a) = ~ IT/a, ir/2a) __________
2N1 4N, 4N1 2N1 8N1
2N2 2N2 8N2 2N2 8N2 ________________
S3 4N3 8N3 2N3 4N3
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Table 2 Integers leading to symmetry points on the surface of the Brillouin zone for the space group D~h.S~,S 2, S3 are integers defimng the size of the crystal [see eq. (1)]. N, N2 N3 are arbitrary integers. The symmetry points are denoted according to ref. .
A = (0 0 it/c) K = (4ir/3a, 0,0) H = (4ir/3a, 0, it/c) M = (4ir/3a, 2ir/3a, 0) L = (4ir/3a, 2it/3a, it/c) P (4ir/3a, 0, k~) U - (4it/3a, 2ir/3a, kz)
3N2 3N2 52
3N1 2N1 2N, 3N, 2N1
~ 2N3 S3 2N3 S3
Brillouin zone are excluded by the boundary conditions. This is an extreme case when one is left only with symmetry points inside the Brillouin zone. The other extreme case is when all the integers Si~~2’ S3 are multiples of 8. In this case as is seen from table 1 none of the symmetry points are excluded by the boundary conditions. In between these two extreme cases there are many other possibilities. Thus, when s1 ~2’ 53 are even but are not multiples of 4, the symmetry points of the kind X and L survive the boundary conditions while all the others will disappear. A similar discussion can be carried out for the hexagonal close-packed structure in table 2. For this structure, the boundary conditions exclude all the symmetry points on the surface of the Brillouin zone if all the three integers ~1 ~2’ S3 divide neither by 2 nor by 3. In the other extreme case, when ~ S~S3 are multiples of 6, none of the symmetry points is excluded by the boundary conditions. Again, in between, there are many possibilities. Thus, when S~is even, the symmetry points M and U are not excluded, while when S3 is even the point A is a symmetry point on the surface of the Brillouin zone in the hexagonal close-packed structure, This discussion shows that the BKBC influence very strongly the irreducible representations of space groups. Since the latter are used for the specification of Bloch states, the application of boundary conditions may completely change the symmetry labelling of elementary excitations in solids. Only for very particular BKBC do the symmetry points on the surface of the Brillouin zone survive. This means that for finite crystals one is, in general, not able to label the ,
23 August 1982
symmetry of Bloch functions on the surface of the Brillouin zone. This can be clearly seen for the case of a one-dimensional crystal with the space group C.. When S is even the points k = 0 (F) and IT/a (X) have the inversion symmetry and one can choose Bloch states ~‘oand ~‘it/a to be either even or odd at these symmetry points. Let us denote by I’i~~‘2 an even and odd Bloch state at k 0 and correspondingly by X1 and X2 at k = IT/a. The symmetry of a band as a whole entity can be specified by the symmetries of the Bloch functions at different symmetry points in the Brillouin zone [7,9] In the case of a one-dimensional crystal for S even there are 4 different symmetry types ofbands: (F1, X1), (F2, X2), (F1, X2), .
(F2, X1). This is, however, no longer the case when S is odd. As was shown above, the symmetry point k = IT/a is excluded by the boundary conditions when S is odd, and we have only 2 types of bands: (F1) and (~‘2)’We see therefore that the boundary conditions change completely the structure of the representations and the symmetry labelling of elementary excitations in solids. Having come to the conclusion that the BKBC influence substantially the irreducible representations of space groups, there seems to be a need for revising the symmetry labelling of elementary excitations in solids. There is no doubt that the latter is very sensitive to the boundary conditions. On the other hand, as was already pointed out in the beginning of the paper, the density of states of the elementary excitations in sufficiently large crystals is essentially independent on the boundary conditions. The latter is an indication that the physical properties of the crystal should not depend on the BKBC. We see therefore that despite the fact that the physical properties of a crystal should not be influenced in any essential way by the boundary conditions, the symmetry labeffing of its elementary excitations depends very strongly on the BKBC. This conclusion is in full agreement with the well known fact that eigenstates in crystals are, in general, very sensitive to the boundary conditions. It is therefore natural to expect that their symmetry will also be strongly influenced by the boundary conditions. An alternative way of labelling the symmetry of Bloch states in solids is by using band representations. The latter are specified by a symmetry center q in the Wigner—Seitz cell and by a representation index I of the point group of q [7,9—11]. In band representa85
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tions the indices (q, 1) label localized states of a band. It can be shown that the (q, 1)-labelling of localized functions does not depend on the boundary conditjons. This, again, is in full agreement with the fact that localized states in solids are not influenced by the boundary conditions  We conclude that the Born—von Karman boundary conditions may change completely the structure of the irreducible representations of space groups. The change appears as a consequence of the exclusion of some or all the symmetry points on the surface of the Brillouin zone. The symmetry points inside the Brillouin zone are not influenced by the boundary conditions, while those on the surface depend very strongly on the BKBC. One can instead label the localized states by means of band representations. Unlike the values of the quasimomentum kin the Brillouin zone which label the Bloch functions and which depend very crucially on the boundary conditions, the values of the quasicoordinate q in the Wigner—Seitz cell which label the localized states are not sensitive to the boundary conditions. This is in full agreement with the fact that eigenstates in solids are sensitive to the boundary conditions while localized states are not . ,
The author profitted from discussions on the subject with Professors R. Besserman, J.L. Birman and A. Peres.
23 August 1982
References  R.E. Peierls, Quantum theory of solids (Oxford, 1955).  G.F. Koster, in: Solid state physics, Vol. 5, ed. F. Seitz andBorn D. Tumbull (Academic Press, New York, 1957).  M. and K. Huang, Dynamical theory of crystal lattices (Clarendon Press, Oxford, 1954).  R.S. Knox and A. Gold, Symmetry in the solid state (Benjamin, New York, 1964). [51infrared J.L. Birman, The theory of crystal space groups and and Raman lattice processes of insulating crystals, Handbuch der Physik, Vol. XXV/26 (Springer, Berlin, 1974).  M. Lax, Symmetry principles in solid state and molecular physics (Wiley, New York, 1974).  J. des Cloiseaux, Phys. Rev. 129 (1963) 554,  J. Zak, ed., A. Casher, Gluck and (Benjamin, Y. Gur, TheNew irreducible representations of M. space groups York, 1969).
[91 J. Zak, Phys. Rev. Lett. 45
(1980) 1025; Phys. Rev. B23
(1981) 2824.  N.F.M. Henry and K. Lonsdale, eds., International tables of X-ray crystallography, Vol. 1 (Kynoch, Birmingham, 1952).  C. Tejedor and J.A. Verges, Phys. Rev. B19 (1979) 2283.  V. Heine, In: Solid state physics, Vol. 35, ed. H. Ehrenreich, F. Seitz and D. Turnbull (Academic Press, New York, 1980).