The effect of hydrostatic pressure on the electronic and optical properties of InP

Solid-State Electronics 44 (2000) 2193±2198

The e€ect of hydrostatic pressure on the electronic and optical properties of InP N. Bouarissa *,1 International Center for Theoretical Physics, Trieste 34100, Italy Received 30 April 2000; received in revised form 14 June 2000; accepted 27 June 2000

Abstract Based on the empirical pseudo-potential method, the electronic and optical properties of the InP compound in the zinc-blende structure at ambient and under hydrostatic pressure are reported. The ®rst-order pressure coecients of the main band gaps (at C, X, and L) are given. The agreement between our calculated hydrostatic deformation potential and the available experimental data is better than 5%, whereas for the crossover pressure from direct to indirect band gap is about 10% less. The valence bandwidth increases with increasing pressure re¯ecting the decreased ionicity in the material of interest. Besides the electronic properties, the e€ect of pressure on the dielectric function is also analysed. Ó 2000 Elsevier Science Ltd. All rights reserved. Keywords: Hydrostatic pressure; Electronic structure; Optical properties; InP

1. Introduction III±V semiconductors have a great technological interest in optoelectronics. In contrast to silicon, most of the materials in the III±V family have a direct band gap that allows their use in electroluminescent devices. Among III±V compound semiconductors, InP has attracted much attention because it is one of the most promising materials for the development of optoelectronics. It is also an interesting material for the development of devices for operation at the 1.3 and 1.55 lm wavelengths which are mostly fabricated on InP substrates and often require the growth of di€erent InP epitaxial layers, since the losses in propagation in ®bres is the lowest at these wavelengths (see Ref. [1] and references therein). The electronic and optical properties of III±V semiconductors have been of high interest both experimentally and theoretically, primarily because of their

* Address: 9, Rue 'k', Fg des martyrs, Bordj-Bou-Arreridj, Algeria. Fax: +213-5-55-0404. 1 Permanent address: Department of Physics, University of MÕsila, MÕsila 28000, Algeria.

technological importance [2±5]. Pressure is, like temperature, a basic thermodynamic variable which can be used to transport matter from one state to another. As long as the applied pressure is hydrostatic, its e€ect on some selected properties, such as the gap energies, can be modi®ed systematically without having to change the sample, provided that no structural phase transformations have occurred. The diamond anvil cell is an instrument particularly suited to perform absorption, luminescence and light-scattering measurements under high hydrostatic pressure. The continuous development of this cell technology over the last thirty years has opened a wide ®eld of high pressure science. The e€ects of pressure on the electronic and optical properties of the III±V compound semiconductors have been investigated experimentally [6±12], however, at the present time primary limitations seem to arise thereby not yet from the diamond material but rather from the measuring techniques which require special developments to obtain well characterised information from the miniature samples under pressure [13]. To gain a deeper understanding of the relevant physical parameters, e€orts have been also made on the theoretical side using methods with varying degree of sophistication [14,15]. The empirical calculations, while simple in nature, have

0038-1101/00/$ - see front matter Ó 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 3 8 - 1 1 0 1 ( 0 0 ) 0 0 1 4 7 - 7

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N. Bouarissa / Solid-State Electronics 44 (2000) 2193±2198

the drawback that a large number of ®tting parameters are required to obtain acceptable agreement with experimental results [16], whereas the ®rst-principles calculations, on the other hand, are technically involved and computationally time consuming. More importantly, due to the well-known problem of using groundstate density functional theory to calculate the electronic band structures of semiconductors, these calculations underestimate the band gaps [17,18] by as much as 50% to 100%. Quasiparticle calculations [19] which at the present are probably the most fundamental and accurate way of calculating the band gaps, produce energy gaps that are in good agreement with measured values. However, the starting point for such calculations are the results of self-consistent density-functional theory calculations, which are then used to compute complex many-body corrections to ab initio band gaps. This is a complicated and CPU-intensive process. In order to avoid some of the above-mentioned dif®culties and, at the same time, produce results that are in good agreement with experimental values, we have investigated the e€ect of hydrostatic pressure on electronic and optical properties of the III±V semiconductor InP in the zinc-blende structure using the empirical pseudopotential method (EPM). The present article is organised in the following way. First, in Section 1, a brief introduction is given. Then, the computation scheme used in the simulation is described in Section 2. Section 3 is devoted to the presentation and discussion of the results. Finally, the main results and conclusions are summarised in Section 4. 2. Computational method The band-structure calculation is based on the pseudo-potential method. Brie¯y, the method involves solving a secular equation for the pseudo-potential Hamiltonian, which has the form: H ˆ ÿ… h2 =2m†r2 ‡ V …r†:

…1†

The mathematical expression of the pseudo-potential V …r† is given by XX V …r† ˆ …1=Na † Sa …G† Va …G† exp…iGr†; …2† a

G

where

Table 1 Band-gap energies of InP at zero pressure Band-gap energy (eV)

Experimental

Calculated

E0 EgX

a

1.35 2.21a

1.35b 2.21b

EgL

2.05a

2.05b

a b

Ref. [21]. Present work.

unit cell volume and sa , the basis vector of the a-atom. In these calculations, only the six pseudo-potential form factors VS …3†, VS …8†, VS …11†, VA …3†, VA …4† and VA …11† are allowed to be nonzero. The method of optimisation of the empirical pseudo-potential parameters is the nonlinear least squares method [20] which requires that the root-mean-square deviation of the calculated level spacing from the experimental ones de®ned by dˆ

" m X 

DE

…i;j† 2

#1=2 =…m ÿ N †

…5†

…i;j† …i;j†

…i;j† should be minimum. In Eq. (5), DE…i;j† ˆ Eexp ÿ Ecalc , …i;j† …i;j† where Eexp and Ecalc are the observed and calculated level spacing between the ith state at the wave vector k ˆ ki and jth at k ˆ kj , respectively, in the m chosen pairs …i; j†. N is the number of the empirical pseudopotential parameters. The experimental energy band gaps along the principal symmetry lines used in the ®tting procedure are shown in Table 1. The calculated energies given by solving the empirical pseudo-potential parameters secular equation depend nonlinearly on the empirical pseudo-potential parameters. The starting values of the parameters are improved step by step by iterations until d is minimised. We denote the parameters by Pm …m ˆ 1; 2; . . . ; N † and write them as Pm …n ‡ 1† ˆ Pm …n† ‡ DPm , where Pm …n† is the value at the nth iteration. These corrections DPm are determined simultaneously by solving a system of linear equations, ( ) N m X X j i j i …Qm ÿ Qm †…Qm0 ÿ Qm0 † DPm mˆ1

…i;j†

m X …i;j† …i;j† ˆ …Eexp ÿ Ecalc …n††…Qim0 ÿ Qjm0 † …i;j†

…m0 ˆ 1; 2; . . . ; N†;

…6†

…i;j†

Sa …G† ˆ exp…ÿiGsa †

…3†

is the structure factor, and Z Va …G† ˆ …Na =X† dr0 Va …r0 † exp…ÿiGr0 †

…4†

is the pseudo-potential form factor associated with the a-atom. Na is the number of basis atoms; X, the primitive

where Ecalc …n† is the value at the nth iteration, Qim is given by i Xh Qim ˆ Cqi …ki † …oH …ki †=oPm †qq0 Cqi 0 …ki †: …7† q;q0

H …ki † is the pseudo-Hamiltonian matrix at k ˆ ki in the plane wave representation, and the ith pseudo-wave function at k ˆ ki is expanded as

N. Bouarissa / Solid-State Electronics 44 (2000) 2193±2198

Wiki …r† ˆ

X   Cqi …ki † exp i…ki ‡ Kq †r :

…8†

q

Kq being the reciprocal lattice vector. Eq. (6) shows that all of the parameters are determined automatically in an interdependent way. The pseudo-wave functions Wik …r† are calculated at each iteration using all the plane  waves with Kq satis 2 fying … h2 =2m† k ‡ Kq ÿ jk j2 6 Emax . Good starting values of the empirical pseudo-potential parameters were carefully chosen by test-working with the smallerto full-sized Hamiltonian matrices …Emax ˆ 8±14 in units of … h2 =2m†…2p=a†2 †. In the present calculation, Emax ˆ 14 is adopted. The imaginary part e2 …x† of the complex dielectric constant can be written as [22,23] XZ e2 …x† ˆ …4p2 e2  h=3m2 x2 † 2=…2p†3 d…xn;s …k† n;s

2 ÿ x† Mn;s …k† d3 k:

BZ

…9†

The subscripts n and s refer to ®lled and un®lled bands, respectively, and 2 xn;s …k† ˆ …Es …k† ÿ En …k††= h Mn;s …k† : …10† 2 Here, Mn;s …k† is a dipole matrix element connecting the bands given by Mn;s …k† 2 ˆ …Uk;n jrjUk;s † 2 ; …11† where Uk;n and Uk;s represent the periodic parts of Bloch functions. Expression (9) neglects lifetime broadening e€ects such as those resulting from phonon and impurity scattering. If we ignore the matrix element in the integral and the prefactor in Eq. (9), then the contribution from a given pair of bands to e2 …x† is simply Z Jn;s …x† ˆ 2=…2p†3 d…xn;s …k† ÿ x† d3 k: …12† BZ

This is the joint density of states for the two bands indexed by n and s. The integrations on Eqs. (9) and (12) are done by using a Monte-Carlo method. An 12 000

2195

random points in the reduced zone (1/48) of the Brillouin zone have been used. The pressure coecients of important band gaps are calculated by using the Murnaghan equation of state [24]. Used values of the equilibrium bulk modulus …B0 † and its ®rst pressure derivative …B00 † are of 71.0 GPa [15] and 4.67 [25], respectively.

3. Results Table 2 contains the adjusted local pseudo-potential form factors (in Ry) as well as lattice constants (in atomic units) of InP used in the present calculation at various pressures ranging from 0 to 100 kbar. Table 3 gives the calculated pressure coecients of main band gaps of InP. For comparison, the available experimental data are also listed. Our calculated pressure coecients and the existing experimental data show a good agreement. We show in Fig. 1, the electronic energy-band structure of InP at normal and under 100 kbar pressure. The valence band top is assumed as energy zero. At normal pressure, the valence-band maximum and the conduction-band minimum occur at the C point. This material has, thus, a direct band gap E0 at the C point as it is well known. The band structure has a character that is common for III±V zinc blende compounds [3]. Consequently, the upper three valence bands (VB) of the electronic band structure originate from the six atomic p orbitals while the lowest conduction band stems from the two atomic s orbitals. The conduction band always has three minima: one each at the C, at the L, and close to the X point on the D line. Applied pressure a€ects the electronic band structure of InP (Fig. 1, dashed lines). The minimum of the conduction band at the C point is highly increased. Hence, the band gap E0 at the C point is considerably increased as well. We notice also that the antisymmetric gap between the ®rst and the second valence bands at the X point is a€ected by pressure. As one can see, the e€ect of pressure leads to a decrease of this gap. It has been reported in the literature [26] that this antisymmetric gap can be taken as a qualitative measure of the crystal ionicity. This suggests that the ionicity

Table 2 Form factors and lattice constants for InP at various pressures Form factors (Ry)

p ˆ 0 kbar

p ˆ 20 kbar

p ˆ 40 kbar

p ˆ 60 kbar

p ˆ 80 kbar

p ˆ 100 kbar

VS …3† VS …8† VS …11† VA …3† VA …4† VA …11† Lattice constant (a.u.)

ÿ0.213862 0.00 0.070499 0.088818 0.06 0.03 11.0906

ÿ0.216682 0.00 0.075973 0.084137 0.06 0.03 10.9932

ÿ0.219113 0.00 0.081163 0.079573 0.06 0.03 10.9072

ÿ0.221251 0.00 0.086131 0.075072 0.06 0.03 10.8304

ÿ0.223172 0.00 0.090922 0.070589 0.06 0.03 10.7609

ÿ0.224934 0.00 0.095570 0.066084 0.06 0.03 10.6976

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Table 3 Calculated and experimental linear pressure coecients of the direct and indirect band gaps (in 10ÿ3 eV/kbar) for InP Pressure coecients (10ÿ3 eV/kbar)

Calculated

Experimental

dECC =dp

8.5a 13.4c 7.81d 0.69a 4.88a ÿ1a 1.8c ÿ2.09d 4a 6.8c 3.79d

8.4  0.2b 8.5c

dEXX =dp dELL =dp dECX =dp dECL =dp

ÿ3  1b ÿ1d 4d

a

Present work. Ref. [7]. c Ref. [14]. d Ref. [15]. b

Fig. 1. Band structure for InP.

decreases when pressure is increased which leads to the enhancement of the covalent character. Such a behaviour is consistent with that of the ionicity in InP under hydrostatic pressure calculated from the total valence electronic charge densities by using the two-point scheme of Ref. [27]. The valence bandwidth is slightly increased when InP is compressed re¯ecting once again the decreased ionicity in the material of interest. The data listed in Table 3 show that dE0 =dp is about two times dECL =dp, both coecients being positive. dELL =dp has approximately the same order of magnitude as dECL =dp. The coecients dECX =dp and dEXX =dp, on the other hand, are much smaller. Moreover, dECX =dp is negative. Thus, we expect an increase of E0 ; EgL ; EXX and ELL band gaps and a decrease of the EgX

Fig. 2. Direct and indirect band gaps in InP as a function of pressure.

one with increasing pressure. This is clearly shown in Fig. 2. This behaviour is similar to that of other III±V zinc-blende type semiconductors [28]. We may then conclude that the III±V zinc-blende type semiconductors present common characteristics in the hydrostatic pressure dependence of their band gaps. The crossing point of the E0 curve with the EgX one gives the transition pressure pt at which InP goes over from a direct band gap to an indirect band gap semiconductors is found to be 90.23 kbar, while experimentally it is estimated to occur at 104  1 kbar [7]. Our calculated value is somewhat lower than the experimental one of Ref. [7] by 10%. Based on this experimental value of the transition pressure and on the experimental ®rst- and second-order pressure coecients from Ref. [7], we have estimated the crossover band gap at pt to be 2.03 eV. This value is in reasonable agreement with that of 2.12 eV determined in our calculations within the experimental uncertainty. Another way of representing the pressure variation is through the equation, E ˆ a ‡ b ‰Da=a0 Š ‡ c ‰Da=a0 Š2 ;

…13†

where Da ˆ ap ÿ a0 , and ap and a0 are the lattice constants at pressure p and at zero pressure, respectively. In agreement with Ref. [7], a linear variation of energy band gaps with …Da=a0 † is obtained. The calculated pressure coecients b and c are presented and compared with other theoretical estimates and available experimental data in Table 4. The hydrostatic deformation potential de®ned as aD ˆ B0 dE0 =dp

…14†

is found to be 6.035 eV. This value is closer to the experimental one of 6:35  0:05 [7] than that of 5.949 eV

N. Bouarissa / Solid-State Electronics 44 (2000) 2193±2198

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Table 4 Calculated and experimental pressure coecients of the direct and indirect band gaps (b and c are in eV) for InP Band Gap Cc ±Cv Xc ±Xv Lc ±Lv Xc ±Cv Lc ±Cv

b

c

Calculated

Experimental

Calculated

Experimental

ÿ23.96a ÿ19.13c ÿ1.99a ÿ13.78a 2.82a 4.87c ÿ11.28a ÿ9.30c

ÿ19.05  0.15b

0a 5.17c ± ± 0a ÿ19.83c 0a 4.11c

0b

± ± 6.6b ±

± ± ± ±

a

Present work. Ref. [7]. c Ref. [15]. b

found in Ref. [15] using ab initio calculations. Moreover, the agreement with the experimental value of Ref. [7] is better than 5%. Fig. 3. displays the imaginary part of the dielectric function for InP at ambient and under 100 kbar pressure. At zero pressure, there are many similarities in the overall shape of this imaginary part of the dielectric function and that of GaAs which seems to be common for most of the tetrahedrally bonded semiconductors in the group III±V family [29]. The main di€erence between their dielectric functions lies in the energies of the transitions. The ®rst critical point of the dielectric function is at 1.35 eV. This point is of type M0 . This Cc ÿ Cv splitting gives the threshold for direct optical transitions between the absolute fourth valence band maximum and the ®rst conduction band minimum. This is known as the fundamental absorption edge. Beyond this point, the curve increases rapidly. This is due to the

Fig. 3. Imaginary part of the dielectric function in InP.

fact that the number of points contributing towards e2 …E† increases abruptly. e2 …E† typically rises to an asymmetric peak related to transitions occurring along the (1 1 1) directions of the Brillouin zone. The critical point with an energy near of 2.78 eV can be associated to the Lc ÿ Lv transition and is of type M1 , whereas the transition Xc ÿ Xv could be associated to the critical point with an energy near of 4.02 eV. Interestingly, there is a central sharp peak in the range 4±5 eV. This latter has a predominant e€ect of volume and contains contributions from transitions occurring over a large region of the Brillouin zone close to the edges in the [1 0 0] and [1 1 0] directions. Some of these transitions are associated with M2 critical points. The experimental results reported by Welkowsky and Braunstein for InP [30] showed that the 4.7 eV peak would appear to be due to an M1 plus M2 mixture, and it is associated with R2 ! R3 without excluding the possibility of transitions at X. This peak was also seen at 5.07 eV at 300 K in the re¯ectivity derivative data [30]. Both results agree reasonably well with ours. On the other hand, assuming constant-momentum matrix elements in the density of states calculation, the transition at 4.99 eV revealed itself as a peak in the density of states function of the Kramers±Kronig analysis, and was not observed in e2 [30]. When the material of interest is compressed, the positions of all critical points cited above are shifted with an increased energy comparative to that at normal pressure. The reason lies in the enhancement of the Cc ÿ Cv , Lc ÿ Lv and Xc ÿ Xv direct gaps under pressure. Although their positions are shifted under pressure, these points still have the same type as that at zero pressure. It is worth noting, moreover, that the central peak is shifted and its maximum has been decreased under pressure. This is an expected behaviour since the lattice constant decreases with increasing pressure. Beyond the principal peak, we note a de-enhancement of the curve at both ambient and 100 kbar pressure. These involve transitions between the valence bands and higher

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N. Bouarissa / Solid-State Electronics 44 (2000) 2193±2198

conduction bands at the zone centre and along the [1 1 1] direction, respectively. From the quantitative point of view, this transition di€ers from normal and 100 kbar pressure, since the two curves of Fig. 3 in this region di€er from each other at normal and under pressure.

4. Summary In summary, the EPM is used for the investigation of the pressure dependence of the electronic and optical properties of InP in the zinc-blende structure. The ®rstorder pressure derivatives of the main band gaps, the hydrostatic deformation potential and the crossover pressure from direct-to-indirect band gap material are calculated as well as the imaginary part of the dielectric function. Our results are generally in reasonable agreement with the available experimental ones. This suggests that while the method used in these calculations is simpler than the fundamental ®rst-principle techniques, it gives satisfactory results. Moreover, it is computationally economical.

Acknowledgements I am pleased to thank the associateship scheme of the International Centre for Theoretical Physics (ICTP) for ®nancial support and to express my gratitude to Dr. A. Qteish for valuable discussions and for a critical reading of the manuscript. Helpful discussion with Prof. E. Tosatti is also acknowledged.

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