The behaviour of electron valence and conduction charge densities in InP under pressure

Materials Chemistry and Physics 65 (2000) 107–112

The behaviour of electron valence and conduction charge densities in InP under pressure N. Bouarissa∗ International Center for Theoretical Physics, Trieste 34100, Italy Received 12 August 1999; received in revised form 27 November 1999; accepted 12 January 2000

Abstract The electronic valence and conduction charge densities at the 0 and X k-points are calculated as a function of position in the unit cell for InP in the zinc-blende structure at various hydrostatic pressures ranging from 0 up to 100 kbar using wave functions derived from empirical pseudo-potential band-structure calculations. Detailed plots of the charge density along the [1 1 1] direction and in the (1 1 0) plane at different pressures are presented for the total valence bands and the first and second conduction ones. It is found that even the studied electronic charge densities are sensitive to the effect of hydrostatic pressure; their shape remains the same as at ambient pressure. Trends in bonding and ionicity under pressure are also discussed. © 2000 Elsevier Science S.A. All rights reserved. Keywords: Electron valence density; Conduction charge density; InP; Zinc-blende structure; Pressure

1. Introduction III–V compound semiconductors are extensively studied, in particular, due to their applicability in optical and optoelectronic devices. These semiconductors crystallise in the zinc-blende structure which corresponds to the space group Td and have a tetrahedrally oriented binding between period-III and period-V elements mediated by eight electrons per basis from the upper-most s and p shells [1]. InP is one of the III–V semiconductors, which possesses physical properties that make this material potentially interesting for high frequency and optoelectronic devices. It is, for instance, the base material for the devices involved in signal generation, transmission, regeneration and recovery. The theoretical and experimental study of electronic charge densities in semiconductors is of fundamental interest to researchers in the material sciences since it yields useful information about the bonding properties, interstitial impurities and the response of specific band states to perturbations in these studied materials [2–4]. The pressure applied influences the electronic charge densities, and hence, the electronic band structure of the material of interest. In order to see how valence and conduction electron charge densities in III–V compound semiconductors behave under hydrostatic pressure, we have carried out calculations on the pressure dependence of charge densities ∗ Permanent address: Physics Department, University of M’sila, M’sila 28000, Algeria.

in InP at selected k-points of the Brillouin zone using the empirical pseudo-potential method (EPM). The organisation of this article is as follows. In Section 1, we give a brief introduction, followed by the description of the method used in the calculations in Section 2. In Section 3, we discuss our results.

2. Calculations The band structure of zinc-blende InP is obtained using the empirical pseudo-potential method (EPM). The pseudo-potential Hamiltonian  2 ~ (1) ∇ 2 + V (r) H =− 2m contains an effective potential which is expanded as Fourier series in reciprocal lattice space. For a binary compound, the expansion is written in two parts which are symmetric and antisymmetric with respect to an interchange of two atoms about their midpoint: X [S S (G)VGS + iS A (G)VGA ]exp(−iGr) (2) V (r) = |G|≤G0

The limited summation reflects the fact that the effective pseudo-potential is sufficiently weak; because of cancellation between the kinetic and potential energies in the vicinity of atomic cores, only few Fourier terms suffice.

0254-0584/00/$ – see front matter © 2000 Elsevier Science S.A. All rights reserved. PII: S 0 2 5 4 - 0 5 8 4 ( 0 0 ) 0 0 2 1 9 - 4

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For the case of N-(8-N) compounds with a zinc-blende structure, on choosing the coordinate origin as the midpoint of the line joining the nearest A and B atoms, the structure and form factors are as follows: S S (G) = cosGτ,

S A (G) = sinGτ

VGS =

1 [V1 (G) + V2 (G)] 2

VGA =

1 [V1 (G) − V2 (G)] 2

(3)

Here, τ =a/8(1, 1, 1), half the vector between the two atoms contained in the unit cell and V1 (G) and V2 (G) are the pseudo-potential form factors of the individual atoms in the unit cell. The electronic wave functions were obtained from the band-structure calculation. These wave functions are then used to compute the charge density by noting that the probability of finding an electron in a certain spatial region of volume d is given by |ψ n,k (r)|2 d, where n is the index of the energy eigenvalue associated with the state k. When many different electronic states k are considered, it becomes meaningful to speak of a charge distribution for the electrons. In particular, the charge density for each valence band may be written as X e|ψn,k (r)|2 (4) ρn (r) = k

where the summation is over all states in the Brillouin zone for a given band n. Once the wave function is expanded in a large number of plane waves and the function is evaluated at a large grid of points in the Brillouin zone, an adequate convergence in calculating |ψ n,k (r)|2 is obtained. The total charge density for a semiconductor can be obtained by adding the charge density from all the valence bands, that is X ρn (5) ρ(r) = n

where the sum is over all occupied bands. Since we are not interested in the total valence charge density in the whole Brillouin zone, but only at the high symmetry points in this zone for specific band,

ρ(r) = e|ψn,k (r)|2

(6)

with k=2π /a(0, 0, 0) and k=2π/a(1, 0, 0) stand for the 0 and X points, respectively (a is the lattice constant) and n is equal to 5 and 6 for the first and second conduction bands, respectively. For the determination of the pressure coefficients of the main band gaps, namely, dE00 /dp, dEXX /dp, dELL /dp, dE0X /dp and dE0L /dp, we have used the Murnaghan equation of state [5]. In this equation, the equilibrium bulk modulus and its first pressure derivative are taken to be B0 =71.0 GPa [6] and B00 =4.67 GPa [7], respectively.

3. Results In Table 1, we present the local adjusted symmetric and antisymmetric pseudo-potential form factors (in Ry) of InP used in the present calculations at various pressures. The lattice constant of InP at zero pressure and room temperature is taken to be 5.8688 Å. The variation of the lattice constant of the studied material versus pressure is shown in Fig. 1, where it is clear that the lattice constant decreases monotonically with increasing pressure. The linear pressure coefficients of important band gaps, namely, (0 c −0 v ), (Xc −Xv ), (Lc −Lv ), (Xc −0 v ) and (Lc −0 v ) (with respect to the top of the valence bands, where the superscripts c and v on the level notation refer to the conduction and valence bands, respectively) of InP are given in Table 2. Available experimental values as well as calculations of other authors are also listed for comparison. Our results agree favourably with the existing experimental ones. Since the experimental data are not available for the pressure coefficients of (Xc −Xv ) and (Lc −Lv ) band gaps, our results are only for reference. Fig. 2 shows the computed total valence electron charge densities along the [1 1 1] direction and in the (1 1 0) plane at the 0 point for InP at different pressures. As can be seen from the profiles and contour plots of Fig. 1, at ambient pressure, most of the electronic charge density is shifted towards the P anion. This is because InP involves a charge transfer between the nearest neighbours, from the III to the V atom. The lowest lying bands can clearly be recognised as s-like atomic orbitals. The middle group of bands forms the covalent bond with the majority of the charge lying between the atomic sites. There is almost no charge in the

Table 1 Pseudo-potential form factors for InP at different pressures up to 100 kbar Form factors (Ry)

p=0 kbar

p=20 kbar

p=50 kbar

p=80 kbar

p= 100 kbar

VS

−0.213862 0.00 0.070499 0.088818 0.06 0.03

−0.216682 0.00 0.075973 0.084137 0.06 0.03

−0.220215 0.00 0.083671 0.077317 0.06 0.03

−0.223172 0.00 0.090922 0.070589 0.06 0.03

−0.224934 0.00 0.095570 0.066084 0.06 0.03

VS VS VA VA VA

(3) (8) (11) (3) (4) (11)

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Fig. 1. Lattice constant of InP as a function of pressure.

interstitial regions. While these features agree with those of other III–V compound semiconductors [2,10], it is not the case for elemental semiconductors, where for example in Ge [11], the charge density is concentrated half way between the two atoms. When pressure is applied, the shape of the total valence electronic charge density (profiles or contours) seems to remain the same. We can state then that the hydrostatic pressure does not change completely the topology of the valence densities in InP. However, a remarkable de-enhancement of the valence charge density around the anion P is observed, accompanied by a slight increase around the cation In. The maximum value of the valence charge density is raised slightly. This rise is accompanied by a weak charge transfer into the interstitial regions. Such a pressure behaviour is similar to that of III–V binary semiconductors [10], but differs from that of Ge [11]. It is worth noting, moreover, that the hydrostatic pressure behaviour of the ionic trends in the bonding may be seen by comparing

Table 2 Calculated and experimental linear pressure coefficients of the main band gaps (in 10−3 eV kbar−1 ) for InP Pressure coefficients (10−3 eV kbar−1 )

Calculated

Experimental

dE00 /dp

8.5a 7.81 [6] 13.4 [8]

8.5 [8] 8.4±0.2 [9]

dEXX /dp

0.69a

–

dELL /dp

4.88a

–

dE0X /dp

−1a

−1 [6] −3±1[9]

dE0L /dp

4a 3.79 [6] 6.8 [8]

4 [6]

a

Present calculations.

−2.09 [6] 1.8 [8]

Fig. 2. Total valence charge density at the 0 point (a) along the [1 1 1] direction at various pressures, (b) in the (1 1 0) plane at zero pressure, and (c) in the (1 1 0) plane at 100 kbar pressure, for InP.

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the total valence charge densities for InP at various pressures, since under hydrostatic pressure, the bond charges tend to move towards the bond center sites (Fig. 1); we may thus conclude that the ionic character becomes weaker when the material under consideration is compressed. In Fig. 3, we plot the first conduction band charge densities along the [1 1 1] direction and in the (1 1 0) plane at the 0 point for InP at various pressures. At normal pressure, a more important charge density can be seen around the anion (P) than the cation (In). However, the reverse is observed in the interstitial regions, where we notice a larger amount of charge density in the interstitial region nearest to the cation compared to that nearest to the anion. The minimum of the charge density in the bonding region is approximately half way along the bond. This indicates that the charge density for the first conduction band at the 0 point for the material of interest shows an antibonding-like state, whereas, the maximum of the charge distribution occurs at the P site, indicating that the charge distribution is predominantly s-like. The effect of pressure on this studied charge density seems to be not so important either qualitatively or quantitatively, although we notice a slight decrease in the electronic charge both at the anion site and in the interstitial region nearest to it. This weak de-enhancement is accompanied by a small increase in the charge at the cation site and its nearest interstitial region. We do believe then that, under pressure up to 100 kbar, the first conduction band charge density at the 0 point for InP is still antibonding and s-like. At zero pressure, the first conduction band charge density shows a substantially different charge distribution at the X point (Fig. 4) compared to that at the 0 point (Fig. 3). Moreover, there is an appreciable enhancement of the electronic charge density in the interstitial region nearest to the anion (P) as compared to that at the 0 point. There is practically no charge at the anion site, whereas a slight amount of charge is noticed at the cation site. When we pass to the material of interest under pressure, one observes a diminution of the charge distribution around the cation site with an enhancement of the electronic charge density in the interstitial region nearest to the anion. Qualitatively, the same behaviour of the first conduction band charge density at X point at different pressures is noticed. The situation is completely different for the second conduction band charge density at the X point, as can be seen in Fig. 5. While the charge density at the X point for the first conduction band is more pronounced in the interstitial region nearest to the anion P, the situation is reversed at the X point for the second conduction band since there is a higher charge density in the interstitial region nearest to the cation In. Furthermore, there is practically no charge at the cation site, whereas, a large charge density is observed around the anion site. As pressure increases gradually from 0 to 100 kbar passing through 50 kbar, the charge density is highly increased in the interstitial region nearest to the cation and decreases around the anion. Given the fact that there are practically no charge densities in the 0 interstitial

Fig. 3. First conduction band charge density at the 0 point (a) along the [1 1 1] direction at various pressures, (b) in the (1 1 0) plane at zero pressure, and (c) in the (1 1 0) plane at 100 kbar pressure, for InP.

N. Bouarissa / Materials Chemistry and Physics 65 (2000) 107–112

Fig. 4. First conduction band charge density at the X point (a) along the [1 1 1] direction at various pressures, (b) in the (1 1 0) plane at zero pressure, and (c) in the (1 1 0) plane at 100 kbar pressure, for InP.

111

Fig. 5. Second conduction band charge density at the X point (a) along the [1 1 1] direction at various pressures, (b) in the (1 1 0) plane at zero pressure, and (c) in the (1 1 0) plane at 100 kbar pressure, for InP.

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regions for the first conduction band at various pressures, such a behaviour of the electronic charge density in the X interstitial regions for the first and second conduction bands at normal and under pressure is relevant when considering the effect of interstitial impurities on the electronic band structure of compound semiconductors [2,3]. On the other hand, it is also important to note that, in a manner similar to the above-studied charge densities, the shape of the profiles and contour maps of the second conduction band charge density at the X point seems to be not affected by pressure. Although the variation of the charge distribution under hydrostatic pressure may not yield accurate quantitative results, observation of trends occurring in crystals at various pressures can yield a physical picture for the behaviour of crystals under pressure.

Acknowledgements The author wishes to acknowledge the support of the International Center for Theoretical Physics (ICTP), Trieste,

Italy, and to express his gratitude to the ICTP staff for aiding him whenever possible. References [1] H. Kalt, in: H.-J. Queisser (Ed.), Optical Properties of III–V Semiconductors, Springer Series in Solid-State Sciences 120, Springer, Berlin, Heidelberg, 1996. [2] S.L. Richardson, M.L. Cohen, S.G. Louie, J.R. Chelikowsky, Phys. Rev. B 33 (1986) 1177. [3] N. Bouarissa, H. Aourag, Phys. Stat. Sol. (b) 190 (1995) 227. [4] N. Bouarissa, Infrared Phys. Technol. 39 (1998) 265. [5] F.D. Murnaghan, Proc. Natl. Acad. Sci. USA 30 (1944) 244. [6] P.E. Van Camp, V.E. Van Doren, J.T. Devreese, Phys. Rev. B 41 (1990) 1598. [7] R. Trommer, H. Muller, M. Cardona, P. Vogl, Phys. Rev. B 21 (1980) 4869. [8] Y.F. Tsay, S.S. Mitra, B. Bendow, Phys. Rev. B 10 (1974) 1476. [9] H. Muller, R. Trommer, M. Cardona, P. Vogl, Phys. Rev. B 21 (1980) 4879. [10] N. Bouarissa, H. Aourag, Mater. Sci. Eng. B 33 (1995) 122. [11] N. Bouarissa, A. Tanto, H. Aourag, T. Bent-Meziane, Comput. Mater. Sci. 3 (1995) 430.