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Band structure and Jones zone of SnTe
Solid State Communications,
Vol. 11, PP. 1397—1399, 1972. Pergamon Press.
Printed in Great Britain
BAND STRUCTURE AND JONES ZONE OF SnTe Y. Onodera Department of Physics, Kyoto University, Kyoto 606, Japan (Received 8 September 1972 by Y. Toyozawcz)
The energy band structure of SnTe as obtained by the pseudopotential calculation can be simply analyzed in terms of the Jones zone scheme. The band gap occurring on the (311) Jones zone plane accounts for the behaviour of the 62 (ce)) spectrum.
THE PURPOSE of this paper is to point out that the band structure and the optical properties of the IV—VI compound semiconductors can be understood in a simple way by introducing the concept of the Jones zone. SnTe is chosen as an example. The Jones zone schemes has already been shown to be useful for the group IV semiconductors (Si and Ge),’ and the present paper is its natural generalization to the IV—VI compounds. SnTe crystallizes in the NaC1 structure. Its Bravais lattice is face-centered cubic. Figure 1(a) shows the Brillouin zone. Since a unit cell of SnTe contains 10 valence electrons, the occupied Bloch states would cover a volume 5 times as large as that of the Brillouin zone if the extendedzone scheme is adopted. Such a volume, called the Jones zone, is depicted in Fig. 1(b). SnTe becomes a semiconductor (not a metal) owing to the energy gap produced on the zone surface. The important surface plane is the (311) plane, which bisects perpendicularly the reciprocal lattice vector (2ii/a) (311). It occupies most of the surface. The energy gap on the (311) plane produced by the (311) Fourier component of the crystal potential lowers the energy of the occupied states below the gap and hence tends to stabilize the crystal binding. To confirm the presence of such an energy gap on the (311) zone plane, knowledge of the actual band structure is needed. The hitherto
(a)
W Z
E L~)
FIG. 1. (a) The Brillouin zone for the f.c.c. lattice. The shaded plane represents the (311) Jones zone plane projected into the Brillouin zone. (b) The Jones zone for IV—VI compounds. Its volume is just five times as large as that of the Brillouin zone. published results of the band calculations, being limited to the high-symmetry points in the Brillouin zone, cannot be used for this purpose,
1397
1398
BAND STRUCTURE AND JONES ZONE OF SnTe
since the (311) zone plane does not contain highsymmetry points like F and X. Therefore, the band structure on the (311) zone plane was calculated anew, with the empirical pseudopotential 2 of Lin et al. V(200) V(220) V(222) V(111) V(311)
=
=
=
=
=
—3.17eV, —0.38,
plane. The situation is in marked contrast to the energy spectrum in the high-symmetry directions (see references 2—4), where the energy gap greatly varies with the wave vector k. This clearly demonstrates the usefulness of our Jones zone scheme. As for the magnitude of the energy gap, it is determined by V(311), the (311) Fourier component
0.22,
of the pseudopotential. But its value is too small,
0.56, 0.05.
and furthermore, inspection of the eigenfunctions tells us that it has to be negative (Bonding cornbinations lie lower than anti-bonding ones). This discrepancy shows that the other pseudopotential form factors contribute to V (311) in higher order. Thus, if we take a state I k> on the (311) plane, it is coupled to the state k 311> on the back plane by the effective potential
The number of the plane waves used is about 34, which is sufficient for our semi-quantitative purpose, and the results for the high-symmetry directions (not presented here) reproduced those of Lin et al. considerably well,
—
V~(311) ______
Q
W
L ___E
Vol. 11, No. 10.
2m\’
______
WZX
—
V(311)
=
K> k2—(k—K? —
SnTe
~
5
where K stand for the reciprocal lattice vectors. If one computes the above quantity at the W point and collects important terms, then one gets ~ff(311)V(311)+_(_~V(111)[V(200)+V(220)]
-
h2 ~27r)
5,
=0.05 eV o
.
I I
-
I (+01)
I I ~
________
(-~.+o)
_____
(fol)
Cool)
2. Band structure of SnTe on the (311) Jones zone plane (left), and along the W-Z-X direction lying on the (200) plane (right), calculated by use of the empirical pseudopotential of Li~et al. The shaded region is the forbidden gap. The zero of the energy is taken arbitrarily. The symmetry labels mdicated are for rotations about the Sn nuclei.
-
0.54
=
-
0.49eV.
The above expression shows that the combination of V(111) and V(200) gives a dominant contribution to the band gap. If the crystal were stabilized by the first, V(311) term, such a mechanism of cohesion is in no way different from that of metals5 and hence the crystal should take another different, metallic, structure. What makes SnTe different from metals and keeps it in the ionic crystal structure is the second term, which is higher order in the pseudopotential.
FIG.
The E(k) curve on the zone plane is shown in Fig. 2. We observe that the band gap is, roughly speaking, constant (‘-‘- 2eV) on the (311) zone
The above value of Veff (311) gives the energy gap 21 V~(311)1 1.0eV, which is smaller than the gap 1.7eV at W, and so the agreement is only semi-quantitative. It is simply because the above simple argument based on the perturbation theory =
does not work very well. But it by no means harms our interpretation of the band structure in terms of the Jones zone.
Vol. 11, No. 10
BAND STRUCTURE AND JONES ZONE OF SnTe
The Jones zone scheme leads us to a clearcut understanding of the optical properties.’ The wave functions of the states on the zone surface behave like cosine (below the gap) and sine (above the gap) functions. Therefore, the matrix elements associated with the direct band-to-band transitions are considerably large on and near the zone surface, while they can be neglected for the other transitions since the corresponding wave functions may be regarded pure plane waves. As a result, the imaginary part of the dielectric constant is expected to have a strong peak at about 2 eV corresponding to the (311) energy gap. This explains the sharp peak at 2eV in both experimentally observed6’7 and theoretically computed2 spectra. This outstanding feature of the optical spectra of the IV—VI compounds has so far been explained as a result of a detailed numerical calculation, but the Jones zone scheme allows a simpler explanation to it, as in the group IV semiconductors.1
1399
The optical spectrum of SnTe possesses another structure at about 7eV. It can be ascribed to another Jones zone plane (200). The energy gap on this zone plane is nearly equal to 2 V (200) I; the splitting occurs almost linearly in V(200). The corresponding band-to-band transition is indicated by an arrow in Fig. 2. In conclusion, the band structure of SnTe (and probably other IV—VI semiconductors) can be well understood with the aid of the concept of the Jones zone. On the basis of these considerations, lattice dynamics of SnTe is now being investigated.
REFERENCES
1. 2.
HEINE V. and JONES R.O., J. Phys. C2, 719 (1969). UN P.J., SASLOW W. and COHEN M.L., Solid State Commun. 5, 893 (1967).
3.
TUNG Y.W. and COHEN M.L., Phys. Rev. 180, 823 (1969).
4.
HERMAN F., KORTUM R.L., ORTENBURGER I.B. and VAN DYKE J.P., J. Phys. 29, C4-62 (1968).
5.
HEINE V. and WEAIRE D., Solid State Physics (Edited by SEITZ F., TURNBULL D. and EHRENREICH H.) Vol. 24, p. 249, Academic Press, New York (1970).
6.
CARDONA M. and GREENAWAY D.L., Phys. Rev. 133, A1685 (1964).
7.
KORN D.M. and BRAUNSTEIN R., Phys. Rev.
floKa3aHo,
85, 4837 (1972).
4’ro c’rpys’rypa
nonynpoBo~HolKa SnTe
eprer~ecKl~x3001
BbitLMcneHxaR C rIOMOULbIO nceano—
no~e~u~aiia uozeT npocTo ~H5flH301PO~~TbC05 Ha 0CHOBaHMH 3OHOi~ ~lZoH3a. 3HepreTH4eCKo~ ueji~,o nosaJt~Ion~e~ Ha (311) rxnocKoc’rot 3OHo~ o~bRcHneTc~noae~eHHe cne~~pa ~ (w)
-
Vol. 11, PP. 1397—1399, 1972. Pergamon Press.
Printed in Great Britain
BAND STRUCTURE AND JONES ZONE OF SnTe Y. Onodera Department of Physics, Kyoto University, Kyoto 606, Japan (Received 8 September 1972 by Y. Toyozawcz)
The energy band structure of SnTe as obtained by the pseudopotential calculation can be simply analyzed in terms of the Jones zone scheme. The band gap occurring on the (311) Jones zone plane accounts for the behaviour of the 62 (ce)) spectrum.
THE PURPOSE of this paper is to point out that the band structure and the optical properties of the IV—VI compound semiconductors can be understood in a simple way by introducing the concept of the Jones zone. SnTe is chosen as an example. The Jones zone schemes has already been shown to be useful for the group IV semiconductors (Si and Ge),’ and the present paper is its natural generalization to the IV—VI compounds. SnTe crystallizes in the NaC1 structure. Its Bravais lattice is face-centered cubic. Figure 1(a) shows the Brillouin zone. Since a unit cell of SnTe contains 10 valence electrons, the occupied Bloch states would cover a volume 5 times as large as that of the Brillouin zone if the extendedzone scheme is adopted. Such a volume, called the Jones zone, is depicted in Fig. 1(b). SnTe becomes a semiconductor (not a metal) owing to the energy gap produced on the zone surface. The important surface plane is the (311) plane, which bisects perpendicularly the reciprocal lattice vector (2ii/a) (311). It occupies most of the surface. The energy gap on the (311) plane produced by the (311) Fourier component of the crystal potential lowers the energy of the occupied states below the gap and hence tends to stabilize the crystal binding. To confirm the presence of such an energy gap on the (311) zone plane, knowledge of the actual band structure is needed. The hitherto
(a)
W Z
E L~)
FIG. 1. (a) The Brillouin zone for the f.c.c. lattice. The shaded plane represents the (311) Jones zone plane projected into the Brillouin zone. (b) The Jones zone for IV—VI compounds. Its volume is just five times as large as that of the Brillouin zone. published results of the band calculations, being limited to the high-symmetry points in the Brillouin zone, cannot be used for this purpose,
1397
1398
BAND STRUCTURE AND JONES ZONE OF SnTe
since the (311) zone plane does not contain highsymmetry points like F and X. Therefore, the band structure on the (311) zone plane was calculated anew, with the empirical pseudopotential 2 of Lin et al. V(200) V(220) V(222) V(111) V(311)
=
=
=
=
=
—3.17eV, —0.38,
plane. The situation is in marked contrast to the energy spectrum in the high-symmetry directions (see references 2—4), where the energy gap greatly varies with the wave vector k. This clearly demonstrates the usefulness of our Jones zone scheme. As for the magnitude of the energy gap, it is determined by V(311), the (311) Fourier component
0.22,
of the pseudopotential. But its value is too small,
0.56, 0.05.
and furthermore, inspection of the eigenfunctions tells us that it has to be negative (Bonding cornbinations lie lower than anti-bonding ones). This discrepancy shows that the other pseudopotential form factors contribute to V (311) in higher order. Thus, if we take a state I k> on the (311) plane, it is coupled to the state k 311> on the back plane by the effective potential
The number of the plane waves used is about 34, which is sufficient for our semi-quantitative purpose, and the results for the high-symmetry directions (not presented here) reproduced those of Lin et al. considerably well,
—
V~(311) ______
Q
W
L ___E
Vol. 11, No. 10.
2m\’
______
WZX
—
V(311)
=
K>
SnTe
~
5
where K stand for the reciprocal lattice vectors. If one computes the above quantity at the W point and collects important terms, then one gets ~ff(311)V(311)+_(_~V(111)[V(200)+V(220)]
-
h2 ~27r)
5,
=0.05 eV o
.
I I
-
I (+01)
I I ~
________
(-~.+o)
_____
(fol)
Cool)
2. Band structure of SnTe on the (311) Jones zone plane (left), and along the W-Z-X direction lying on the (200) plane (right), calculated by use of the empirical pseudopotential of Li~et al. The shaded region is the forbidden gap. The zero of the energy is taken arbitrarily. The symmetry labels mdicated are for rotations about the Sn nuclei.
-
0.54
=
-
0.49eV.
The above expression shows that the combination of V(111) and V(200) gives a dominant contribution to the band gap. If the crystal were stabilized by the first, V(311) term, such a mechanism of cohesion is in no way different from that of metals5 and hence the crystal should take another different, metallic, structure. What makes SnTe different from metals and keeps it in the ionic crystal structure is the second term, which is higher order in the pseudopotential.
FIG.
The E(k) curve on the zone plane is shown in Fig. 2. We observe that the band gap is, roughly speaking, constant (‘-‘- 2eV) on the (311) zone
The above value of Veff (311) gives the energy gap 21 V~(311)1 1.0eV, which is smaller than the gap 1.7eV at W, and so the agreement is only semi-quantitative. It is simply because the above simple argument based on the perturbation theory =
does not work very well. But it by no means harms our interpretation of the band structure in terms of the Jones zone.
Vol. 11, No. 10
BAND STRUCTURE AND JONES ZONE OF SnTe
The Jones zone scheme leads us to a clearcut understanding of the optical properties.’ The wave functions of the states on the zone surface behave like cosine (below the gap) and sine (above the gap) functions. Therefore, the matrix elements associated with the direct band-to-band transitions are considerably large on and near the zone surface, while they can be neglected for the other transitions since the corresponding wave functions may be regarded pure plane waves. As a result, the imaginary part of the dielectric constant is expected to have a strong peak at about 2 eV corresponding to the (311) energy gap. This explains the sharp peak at 2eV in both experimentally observed6’7 and theoretically computed2 spectra. This outstanding feature of the optical spectra of the IV—VI compounds has so far been explained as a result of a detailed numerical calculation, but the Jones zone scheme allows a simpler explanation to it, as in the group IV semiconductors.1
1399
The optical spectrum of SnTe possesses another structure at about 7eV. It can be ascribed to another Jones zone plane (200). The energy gap on this zone plane is nearly equal to 2 V (200) I; the splitting occurs almost linearly in V(200). The corresponding band-to-band transition is indicated by an arrow in Fig. 2. In conclusion, the band structure of SnTe (and probably other IV—VI semiconductors) can be well understood with the aid of the concept of the Jones zone. On the basis of these considerations, lattice dynamics of SnTe is now being investigated.
REFERENCES
1. 2.
HEINE V. and JONES R.O., J. Phys. C2, 719 (1969). UN P.J., SASLOW W. and COHEN M.L., Solid State Commun. 5, 893 (1967).
3.
TUNG Y.W. and COHEN M.L., Phys. Rev. 180, 823 (1969).
4.
HERMAN F., KORTUM R.L., ORTENBURGER I.B. and VAN DYKE J.P., J. Phys. 29, C4-62 (1968).
5.
HEINE V. and WEAIRE D., Solid State Physics (Edited by SEITZ F., TURNBULL D. and EHRENREICH H.) Vol. 24, p. 249, Academic Press, New York (1970).
6.
CARDONA M. and GREENAWAY D.L., Phys. Rev. 133, A1685 (1964).
7.
KORN D.M. and BRAUNSTEIN R., Phys. Rev.
floKa3aHo,
85, 4837 (1972).
4’ro c’rpys’rypa
nonynpoBo~HolKa SnTe
eprer~ecKl~x3001
BbitLMcneHxaR C rIOMOULbIO nceano—
no~e~u~aiia uozeT npocTo ~H5flH301PO~~TbC05 Ha 0CHOBaHMH 3OHOi~ ~lZoH3a. 3HepreTH4eCKo~ ueji~,o nosaJt~Ion~e~ Ha (311) rxnocKoc’rot 3OHo~ o~bRcHneTc~noae~eHHe cne~~pa ~ (w)
-