A PSEUDOPOTENTIAL APPROACH TO THE ELECTRONIC STRUCTURE OF III-IV LAYER-COMPOUNDS

(1)

HAL Id: jpa-00215076

https://hal.archives-ouvertes.fr/jpa-00215076

Submitted on 1 Jan 1972

HAL

is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire

HAL, est

destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

A PSEUDOPOTENTIAL APPROACH TO THE ELECTRONIC STRUCTURE OF III-IV

LAYER-COMPOUNDS

M. Schlüter

To cite this version:

M. Schlüter. A PSEUDOPOTENTIAL APPROACH TO THE ELECTRONIC STRUCTURE OF III-IV LAYER-COMPOUNDS. Journal de Physique Colloques, 1972, 33 (C3), pp.C3-273-C3-276.

�10.1051/jphyscol:1972342�. �jpa-00215076�

(2)

JOURNAL DE PHYSIQUE Colloque C3, supplkment au no 5-6, Tome 33, Mai-Juin 1972, page C3-273

A PSEUDOPOTENTIAL APPROACH TO THE ELECTRONIC STRUCTURE OF 111-IV LAYER-COMPOUND S

M. SCHLUTER

Ecole Polytechnique FCdkrale de Lausanne, Switzerland

R6sumB.

-

Nous avons calcule la structure de bandes du j?-GaSe par la mkthode du pseudo- potentiel. Ce calcul tridimensionnel permet d'estimer la grandeur du couplage entre les couches.

Les masses effectives des electrons et des trous ainsi obtenues ne montrent pas une anisotropie prononcke. Le seuil d'absorption dans ce compose correspond aux transitions indirectes entre les

&Vats r4et M:. Le gap direct se trouvant environ 50 MeV au-dessus du gap indirect correspond aux Btats ^{r d}et I-:. La forte anisotropie de I'absorption optique que I'on observe expkrimentale- ment est ainsi bien expliquee.

Abstract. - The bandstructure of hexagonal FGaSe has been established using the semiempi- rical pseudopotential method. The three-dimensional calculations allow to estimate the strength of the interlayer coupling. Effective masses have been calculated for electrons and holes. They do not show a pronounced anisotropy. The absorption edge in this material corresponds to indirect transitions between the states

r;

and

MI.

A direct gap about 50 MeV above the indirect one is given by the states

r,-

and

r:.

The effect on optical absorption measurement due to a change in the polarisation of light is well explained by this assignment.

1. Introduction.

-

The 111-VI compounds GaSe and Gas cristallize in a highly anisotropic layer structure [I]. Each of the layers consists of four closepacked monatomic sheets in the sequence anion-Ga-Ga- anion. The Ga atoms which form the two central sheets are tetrahedrally coordinated while the chalco- genide atoms sit at the apex of triangular pyramids whose bases are formed by three Ga atoms (Fig. 1).

FIG. 1.

-

Layer of GaSe. The Ga atoms are represented by

the small shaded circles, the Se atoms by the large open circles.

Because of the strong bonding within the layers as compared to the weak bonding between them, some attempts have been made to evaluate the electronic properties of these materials by simply ignoring the coupling between the layers. Thus Bassani et al. [2]

and Kamimura et al. [3] have calculated two-dimen-

sional bandstructures using the tight-binding method.

Although these calculations give some insight into the electronic structure, they cannot satisfactorily explain the optical properties because of the bad description of the conduction bands by the tight-binding method, and of course, they do not yield any information on the interlayer coupling. In view of these shortcomings we solved the three-dimensional case using the pseudopotential scheme. To do so, a unit cell had to be considered which extends over two layers and contains 8 atoms. The corresponding 36 electrons per unit cell pose serious convergence problems for the diagonalisation of the Hamiltonmatrix. In order to keep the computational effort low, we therefore Jused local, spherical-symmetric pseudopotentials which were adjusted semi-empirically to fit the resulting bandstructure to optical data [4], [5]. This rather simple Hamilto- nian leads to an overestimation :of the strength of the van der Waals forces between thelayers. The only factor which in the present calculation accounts for the weakness of the interlayer coupling is the relatively small potential overlap resulting from the large interlayer spacing.

2. Electronic bandstructure.

-

Figure 2 shows the bandstructure of GaSe along some symmetry-lines in the hexagonal Brillouin-zone. Because of the large ratio c/a = 4.3 the zone is very flat and a large number of reciprocal lattice vectors have to be considered for the pseudohamiltonian to get sufficient convergence.

Thus 250 plane waves have been included exactly along with another 500 waves via Lowdin's pertur-

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1972342

(3)

M U L R A

FIG. 2.

-

Energy bands along the principal symmetry axes of the Brillouin-zone in hexagonal GaSe as calculated using the pseudopotential described in the text. The band-symmetries are denoted by the irreducible representations of D&.

C

-

N FIG. 4. - Sketch of the pseudowavefunctions along the FIG. 3.

-

Twodimensional bandstructure calculated by line Ga-Ga for the Ti and T: states forming the direct

Bassani et al. [2] using the tight-binding method. gap.

bation formalism. Over the considered range of 25 eV for the valence bands only. A considerable discre- the bands have thus been calculated for a given poten- pancy exists however between the conduction bands tial to within an accuracy of f 0.2 eV. Comparison found by the two calculations. The lowest four bands of the present calculations with the tight-binding in figure 2 correspond to the deep-lying nonbonding results of Bassani [2] (Fig. 3) shows good agreement Se s-states. The bonding states of the Ga-Ga bond which include the top of the valence bands overlap with

- ~ . 5

- 1 o -

d

)I

n

the highest bonding states of the Ga-Se bond. The deepest two conduction bands correspond to antibonding states of Ga-Ga. They are separated from

-

the higher conduction bands by about 1.5 eV giving

rise to a second gap. The optical gap therefore appears essentially between bonding and antibonding states of the Ga-Ga bond. This assignment is confirmed by the form of the pseudowavefunctions of the two considered states and I'3f (Fig. 4).

Between the Ga positions, these functions exhibit the behaviour which is typical of the formation of

- a covalent bond. Further confirmation for the

(4)

A PSEUDOPOTENTIAL APPROACH TO THE ELECTRONIC STRUCTURE C3-275 above assignment has been obtained from the values

of the deformation potentials for variations of the Ga-Ga distance.

The present bandstructure was first calculated with smooth pseudopotential curves interpolating between the GaAs, GaSb, Gap and ZnSe data of Cohen and Bergstresser (CB) [6]. The resulting optical gap turned out to be as high as 4 eV which has to be compared to an experimental value of 2.1 eV 141. This disagree- ment can be understood as follows : Since the optical gap is determined by the Ga-Ga bond, it is very sensi- tive to small variations in the Ga-Ga distance d.

The available x-ray data give d-values variing between 2.1

A <

d

<

2.4

4

[I], [7]. The corresponding values of the energy gap Eg lie between 38 eV < Eg

<

5 eV.

Moreover, the coordination of the metalatoms in GaSe differs markedly from that in the zinc-blende crystals for which the CB-potentials were determined : each Ga atom in zinc-blende crystals is surrounded by four anions with a total of 8 electron per anion- cation pair, whereas in GaSe it is surrounded by another Ga atom and three anions having 9 electrons per anion-cation pair. The formation of the Ga-Ga bond therefore leads to a relatively high uncompensa- ted negative charge between the Ga atomes which tends to weaken the bond and thus produces a smaller energy gap than derived from the CB-potentials. In the pseudopotential scheme this fact can be taken care of by slightly increasing the CB-Ga potential for large radii. It would thus appear that the present case shows the limitation of the transferability of atomic formfactors from one structure to another.

Looking a t the bandstructure in somewhat more detail, we note that at symmetry lines on the top and bottom faces of the Brillouin-zone a11 bands are at least doubly degenerate because of time reversal symmetry. Elsewhere this degeneracy is lifted by the interlayer coupling which results in a doubling of the bands, the separation between corresponding bands variing between 0.1 and 0.8 eV. The average width AE, = 120 meV over the occupied valence bands along the z-direction is a measure of the strength of the interlayer coupling. One can compare this value with the experimentally observed difference of 50 meV between the band gaps of the

P-

and e-modi- fications of GaSe which also measures the interlayer interaction 181. Table 1 shows the effective masses for the gap at k ⁼0. For comparison, the experimentally determined reduced exciton mass is also indica-

Calculated efective and reduced masses parallel and perpendicular to the crystal axis C . An experimen- tally determined reduced exciton mass is also indicated.

rnll c "'1 _c Pll c

PL

c

- - -

V 0,25 1,05 0,12 0,38

C 0,25 0,6 exp. 0,15 [9]

ted [9]. The masses mllc are to small as compared to m,,. The discrepancy is probably due to the overestimation of the interlayer bonding with spherical- symmetric pseudopotentials. The selectionrules for optical dipol transitions across the gap are illustrated in figure 5. Both valence- and conductionband are

without spin with spin

direct

r,+

-

Ells with Mi Phonon

'7

FIG. 5. - Selection rules for optical dipol transitions in GaSe.

non-degenerate at k = 0. Their symmetries allow transitions only for polarisation Ell,. However, because of spin-orbit coupling transitions with polarisation EL, become weakly allowed. The absence of any structure on the absorption curves which could be ascribed to split-off bands supports this assignment which does not permit the valence or conduction bands to split under spin-orbit interaction [4], [5], [lo].

Experimental evidence has been reported of an indirect gap about 50 meV below the direct gap [4].

We attribute this gap to the transition

T'T

-+ M:, M:

being also non-degenerate. An analysis of excitonic fine-structure measurements has been carried out and will be published elsewhere. The assignment of higher reflection peaks [ l l ] remains somewhat ambigous until a reflectivity curve on the basis of the present band structure has been calculated. Nevertheless, the results obtained so far afford some informed guesses to be made. Thus we compare in figure 6 and table I1

ENERGY (eV)

FIG. 6.

-

The reflectivity as a function of photon energy in GaSe, [ll] compared with a calculated joint density of states

histogram.

(5)

M. SCHLUTER

Energy diferences in eV of principal transition in GaSe.

The values used for determining the pseudopotentials are marked by the underlines Transitions

i

⁺

r;

⁺M: K6 -, K6

-

^-

-

Calculated 2,1 2,o 3,7 4,8 6 9 8,O

Observed [4], Ell] 2, 1 2,05 3,6 4,9 7,2 7, 8

the available experimental reflectivity data [l 11 with the calculated energy bands and a rough joint density of states histogram which has been obtained from 165 energy values over the Brillouin-zone. The first peak at 3.6 eV can be attributed to transitions between states in the highest valence band and the lowest conduction band along the symmetry-lines T and Z.

Recently, some electroreflectance data [I21 have been reported which suggest a decomposition of this peak into a spin-doublet. This can be understood by the same assignment since both bands are twofold degenerate at the point K. The main peak in the reflectivity at 4.9 eV is probably due to a two-dimensional singularity in the joint density of states, i. e. to transitions between bands which are flat in the z-direction such as e. g. between some high lying valence band and the second conduction band on the line A . The highest two peaks at 7.2 eV and 7.8 eV might

possibly be attributed to transitions between the highest valence states and the second and third conductionband respectiveIy.

3. Conclusion.

-

Using the pseudopotential approach with local spherical-symmetric potentials, the bandstructure of hexagonal P-GaSe has been established semiempirically. In contrast with earlier calculations, we find that both, three- and two-dimensional bands exist in GaSe. Moreover, a new ordering of the conduction bands on the energy scale is proposed which explains all optical observations. Neither valence nor conduction bands shows an orbital degeneracy.

Effective masses have been calculated for electrons and holes. They do not exhibit a pronounced anisotropy. This fact can partially be explained by the overestimation of the layer-layer attraction using spherical-symmetric pseudopotentials.

References

[I] JELLXNEK (F.) et al., 2. Naturforsch., 1961, 16b, 713. [7] SCHUBERT (K.) et al., 2. Metallkde, 1955, 46, 217.

[2] BASSANI (F.) et al., I1 Nuovo Cimento, 1967, 50, 95. 18] BREBNER (J-) et al-7 Phys. Lett-, ¹⁹⁶⁷⁷ 24 274- [9] FRITSCHE (L.), Phys. stat. sol., 1969,34, 195.

C31

^KAMrMuRA1313. ^(H-)et J . P h ~ s ' 24, 1101 [Ill BASSANI moos^^ (Be), (F.) Helve phys. acta, et al., Proc. Intern. Conf. Physics 1967, 40, 382.

[4] MOOSER (E.) et al., Phys. stat. ol., 1969,31, 129. Semiconductors, Paris, 1964, p. 51.

[51 BREBNER

(J.1,

^J-Phys- Chem- Solids, 1964, 25, 1427. [12] BALZAROTTI (A.) et al., J . Phys. Chem. Sol., 1971, 4, [6] COHEN (M.) et al., Phys. Rev., 1966,141,789. 1273.