Calculation of the optical and electronic properties of the zinc-blende alloy Zn1 − xMgxSe

Superlattices and Microstructures, Vol. 30, No. 1, 2001 doi:10.1006/spmi.2001.0986 Available online at http://www.idealibrary.com on

Calculation of the optical and electronic properties of the zinc-blende alloy Zn1−x Mg x Se F. B ENKABOU , H. AOURAG† Computational Materials Science Laboratory, Physics Department, University of Sidi Bel-Abbes, 22000 Algeria

M. C ERTIER Laboratoire de Spectrométrie Optique de la Matière, Département Mesures Physiques, Technopôle Metz 2000, F57078 Metz Cedex 3, France

T EIJI KOBAYASI College of Medical Sciences, Physics Devision, Tohoku University 2-1 Seiryo-machi, Aoba-ku, Sendai 980-8575, Japan (Received 27 February 2001)

In order to clarify the electronic and optical properties of wide-energy gap zinc-blende structures ZnSe, MgSe and their alloys (ZnSe)1−x (MgSe)x , a simple pseudo-potential scheme (EPM) within an effective potential, the virtual crystal approximation (VCA) which incorporates compositional disorder as an effective potential, is presented. Various quantities, including the fundamental band gap, the energies of several optical gaps, charge densities, ionicity character, transverse effective charge, and refractive index are obtained for this alloy. c 2001 Academic Press

Key words: wide energy gap, alloy Zn1−x Mgx Se, optical properties, conicity refractive index.

1. Introduction ZnSe is one of the most attractive base materials for optoelectronic applications. This is due to its wide, direct band gap of E g = 2.7 eV at room temperature, its complete miscibility with other II–VI compounds (alloying is possible with Cd, Mg, Be, S or Te, on the respective sublattice), its excellent luminescence properties after electrical or optical excitation, and to the fact that high-quality, nearly lattice-matched, ZnSe epilayers can be grown on readily available GaAs substrates. It is well known that II–VI semiconductors have large optical gaps, but the feasibility of green–blue optoelectronic devices based on these materials has only recently been demonstrated. ZnS, ZnSe, and ZnTe are the prototype II–VI semiconductors and their cubic phase, which occurs naturally as a mineral, has been called the zinc-blende structure. † Author to whom correspondence should be addressed. E-mail: [email protected]

0749–6036/01/070009 + 11

$35.00/0

c 2001 Academic Press

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Superlattices and Microstructures, Vol. 30, No. 1, 2001

Furthermore, in recent years ZnSe has proved to be a particularly interesting dilute magnetic semiconductor when doped with Mn [1–3]. Many efforts have been made to fabricate a sustainable ZnSe blue laser. Mixed crystals of Mg chalcongenides with wide-gap II–VI compounds have recently attracted much attention due to successful fabrication of blue and blue–green injection laser diodes based on ZnSe, ZnSSe, ZnCdSe, and ZnMgSSe multilayer heterostructures grown by MBE [4–7]. Blue–violet electroluminescence in ZnSe/ZnMgSe and green in CdSe/MgZnTe/ZnTe structures has also been demonstrated [8, 9]. The structural and optical properties of other Mg-containing II–VI solid solutions have also been investigated [10–13]. Unlike ZnSe which have been extensively studied, very little is known about MgSe. The normal structure of MgSe is zinc-blende [14]. Extrapolation of experimental data on ZnMgSSe alloys to MgSe indicates a band gap of 3.6 eV for MgSe [15]. The wider band gaps of 3.6 eV for MgSe, as compared to 2.8 eV for ZnSe, occur despite increases in lattice constant from 5.67 to 5.89 in going from ZnSe to MgSe [15]. Zn1−x Mgx Se crystals were grown by the high-pressure Bridgman method without a seed and with an argon overpressure of 11 MPa following the procedure described in Ref. [16]. The alloy obtained was then crushed, put into the crucible again together with ZnSe: 20% Mg powder, and the crystallization process was performed twice with a lowering speed of 4 mm h−1 . Crystal dimensions were of diameter 10 mm and length 40–50 mm. The crystals obtained were transparent and of light yellow–green with increasing x concentration. In this paper we report a pseudopotential calculation [21] of the band structure of Zn1−x Mgx Se in an extended virtual-crystal approximation (VCA) [22, 23], which treats an alloy as a perfectly periodic crystal with an average potential at the anionic sublattice sites and does not include the effects of aperiodic fluctuations in the crystal potentials. Use of the EPM method has generally proven to produce reasonably good band structures in comparison with the self-consistent pseudopotential method in the local-density approximation (LDA), which underestimates the energy band gap [24] and the Quasi-particle method [24], which is reliable but very time-consuming. The main goal of this study is to clarify the electronic and optical properties of zinc-blende alloys Zn1−x Mgx Se which have not yet been synthesized.

2. Calculation Let us define the empirical pseudopotential (EPP) parameters of the semiconductor as a superposition of the pseudo-atomic potential of the form V (r ) = VL (r ) + VN L (r ), where VL (r ) and VN L (r ) are local and nonlocal parts, respectively. In this calculation we have omitted the nonlocal part. We regard the Fourier components of VL (r ) as the EPP local parameters. We determine the EPP parameters by the nonlinear least-squares method, in which all the parameters are simultaneously optimized under a defined criterion of minimizing the root-mean square (rms) deviation. The experimental electronic band structure is used. Our nonlinear least-squares method requires that the rms-deviation of the calculated level spacing (ls) from the experimental ones defined by 1/2  m X (1) δ =  [1E (i, j) ]2 /(M − N ) (i, j)

should be minimum: (i, j)

(i, j)

1E (i, j) = E exp − E cal , (i, j)

(i, j)

where E exp and E cal are the observed and calculated LSs between the ith state at the wave vector k = ki and the jth at k = k j , respectively, in the m chosen pairs (i, j). N is the number of EPP parameters. The calculated energies given by solving the EPP secular depend nonlinearly on the EPP parameters. The starting values of the parameters are improved by step iterations until δ is minimized. Let us denote the parameters by Pu (u = 1, 2, . . . , N ) and write as Pu (n +1) = Pu (n)+1Pu , where Pu (n) is the value at the nth iteration.

Superlattices and Microstructures, Vol. 30, No. 1, 2001

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Table 1: The adjusted local pseudopotential form factors (in Ry) and lattice constants (in at. units) used in the calculation.

ZnSe MgSe

a()

V S (3)

V S (8)

V S (11)

V a (3)

V a (8)

V a (11)

5.6699 5.8899

−0.2567 −0.2890

0.029 0.00

0.0594 0.0400

0.139 0.2402

0.062 0.090

0.016 0.040

These corrections 1Pu are determined simultaneously by solving a system of linear equations   N m m X X X (i, j) (i, j) j  (Q iu − Q uj )(Q i 0 − Q j 0 ) 1Pu = [E exp − E cal (n)](Q iu 0 − Q u 0 ), u u u=1

(i, j)

(2)

(i, j)

u 0 = (1, 2, . . . , N ),

(3)

(i, j)

where E cal (n) is the value at the nth iteration. Q u is given by   X ∂ H (ki ) j Qu = [Cqi (ki )]∗ = Cqi 0 (ki ). ∂ P 0 u q,q 0

(4)

q,q

H (ki ) is the pseudo-Hamiltonian matrix at k = ki in the plane wave representation, and the ith pseudo-wave function at k = ki is expanded as X ψki i (r ) = Cqi (ki ) exp(i(ki + kq )r ), (5) q

kq being the reciprocal lattice vector. Equation (2) shows that all of the parameters are determined automatically in an interdependent way. The adjustable parameters are then the symmetric V s (G) and antisymmetric V a (G) form factors of ZnSe and MgSe (see Table 1). The lattice constant of the alloys Ax B1−x (A: ZnSe or B: MgSe) are determined by using Vegard’s rule as a(x) = (1 − x)aA + xaB ,

(6)

where a A , a B are the lattice constants of the parent semiconductors A and B, respectively. The alloy potential is calculated using the virtual crystal approach (VCA), where we add a compositional disorder as an effective periodic potential

Valloy

Valloy = VVCA + Vdis , A B = (1 − x) VA + x VB , alloy alloy

(7) (8)

where A , B , and alloy are the volumes of the parent semiconductors A, B, and their alloy, respectively. By adding this effective disorder potential to the virtual crystal potential, we obtain the final expression for the potential [25]   p A B Vdis (r ) = −β x(1 − x) VA (r ) − VB (r ) , (9) alloy alloy where the disorder parameter β, that simulates the disorder effect, is treated in our calculations as an adjustable parameter. This parameter cannot be varied arbitrarily when a particular choice of experimental band-bowing parameter is used. For example, β can only be varied in a very narrow region to create a reasonable fit to experimental data. The calculation was performed in different positions, by varying β until agreement was achieved with the values of the bowing parameter determined by experiment and other theoretical calculations.

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Superlattices and Microstructures, Vol. 30, No. 1, 2001 Table 2: Comparison of the calculated levels spacings (eV) for experimental and other calculations.

v 015 v 015 v 015

→ → →

01c X 1C LC 1

ZnSe Cal

Exp, The

MgSe Cal

Exp, The

2.799 3.399 3.122

2.8[15], 2.5[18] 3.4[17, 20], 3.1[18] 3.8[17, 20], 3.2[18]

3.599 6.422 5.159

3.6[16], 3.6[18] — 3.3[18] — 4.1[18]

The total valence charge distribution was computed using the two-scheme of Chadi and Cohen [26] X ρn (r ) = e|ψnk (r )|2 , (10) k

where n is the band index of the energy eigenvalue associated with state k. The valence charge density distribution profiles along the bond direction h111i was used to calculate the ionicity character f i using an empirical formula [27] that is based on the fact that the total area integration under the valence charge density is divided into two parts with respect to the bond center (bc), Sa − Sc fi = , (11) Sa where Sa and Sc are the areas of the anion and cation sides, respectively, and are calculated using a parabolic formula with a higher and a pair number for the integration zone.

3. Results 3.1. Electronic properties The pseudo-potential form factors for the pure components ZnSe and MgSe are given in Table 1 and are in reasonable agreement with the experimental results for the principal energy gaps (see Table 2). All energies v ). are in reference to the top of the valence band (015 In Fig. 1A and 1B, the EPM band structures for ZnSe and MgSe in the zinc-blende phase are given. The results show that ZnSe is an indirect gap semiconductor with the minimum of the conduction band at point X, while MgSe has a direct-gap. The calculated energy gaps of ZnSe and MgSe are 2.799 eV and 3.599 eV, respectively, which are in good agreement with the experimental results [15, 17, 20] and other calculations [18] as listed in Table 2. The absence of core d electrons in Mg and the higher ionicity of the materials as compared to ZnSe are probably both responsible for the larger gaps. In ZnSe, the states at the valence-band maximum (VBM) have an appreciable d character and these are repelled towards higher energy by Zn core d electrons, thereby reducing the band gap [19]. The absence of cation core d electrons increases the band gaps of MgSe and it should also result in a lowering of the valence-band maximum (VBM) of MgSe relative to that of ZnSe by several tenths of an eV. The VCA potential for intermediate compositions is obtained from that of the pure components. Furthermore, linear variation of the lattice constant with composition has been assumed. The results of our calculation are presented in Fig. 2. A least-square fit of the curve exhibits a sublinearity, yielding: E g0 = 2.79906 − 0.90793.x + 1.71179.x 2

(12)

E gX E gL

(13)

= 3.39809 + 2.3658.x + 0.66064.x

2

= 3.12316 + 1.13748.x + 0.89917.x 2

for the linear and quadratic energy coefficient.

(14)

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A ZnSe 20

Energie (eV)

15

10

5

0

−5 Γ

B

X

W Vector d'onde (K)

L

Γ

18 MgSe

16 14 12 10

Energie (eV)

8 6 4 2 0 −2 −4 −6 −8 −10 −12 Γ

X

W Vector d'onde (K)

L

Γ

Fig. 1. A, the band structure of cubic ZnSe; B, the band structure of cubic MgSe.

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Superlattices and Microstructures, Vol. 30, No. 1, 2001 7

Γ−L

(ZnSe)1 −x Mgx

Energy gap (eV)

6

5 Γ−X 4

3 Γ−Γ 2

0.0

0.2

0.4 0.6 Concentration of Mg

0.8

1.0

Fig. 2. Calculated values of the lowest direct (0–0) and indirect (0 X, 0L) gaps of Zn1−x Mgx Se plotted as a function of alloy concentration (VCA calculation).

The quadratic term stands for the bowing parameter predicted for ZnSeMg. The complete VCA electronic band structure of ZnSeMg is shown in Fig. 3 for x = 0.50. Our VCA calculation indicates that the complete electronic band structure of Zn1−x Mgx Se can be considered as a linear function of composition (see Fig. 2). This linearity can be easily understood if we consider that the alloy potential form factors calculated within the scheme VCA turn out to be linear functions of composition. 3.2. Ionic character In order to better understand the physical mechanism behind the band-gap trends calculated, in Fig. 4 we give the results of the calculation of the total valence charge densities in the two-point scheme of Chadi and Cohen [26] with different concentration x. It is well known that ionicity character is highly dependent on total valence charge densities. We have calculated the variation of the ionicity parameter using the model which used the pseudopotential parameter [27]. Figure 5 displays the variation versus the Mg fraction x. Again, we noticed the nonlinear variation of the ionicity character for the alloys Zn1−x Mgx Se. The ionicity character increased from 0.63 for ZnSe to 0.73 for MgSe, which is more ionic. We found the dependence of the ionic character with temperature variation to be f i = 0.643 + 0.106.x − 0.015.x 2 .

(15)

This variation was confirmed by the behaviour of electronic density as a delocalization of charge to the bond center was observed with variation of concentration from x = 0 to 1, indicating an increase in the ionic character.

Superlattices and Microstructures, Vol. 30, No. 1, 2001

15

20

15

Energie (eV)

10

5

0 (ZnSe)0.5 Mg0.5 −5

− 10

Γ

X

W Vector d'onde (K)

L

Γ

Fig. 3. Energy band structure of Zn1−x Mgx Se calculated at x = 0.50 within the virtual crystal approximation.

30 (ZnSe)1 -xMgx

xc = 0.0

Charge Density (arb.units)

25 xc = 0.5 20 xc =1.0 15

10

5 − 0.5

0.0 Atomic Position (at.unit)

0.5

Fig. 4. Variation of the total valence charge densities along h111i direction for Zn1−x Mgx Se with x values.

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Superlattices and Microstructures, Vol. 30, No. 1, 2001 (ZnSe)1 -xMgx

0.74

0.72

Ionicity

0.70

0.68

0.66

0.64 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Concentration of Mg Fig. 5. The variation of ionicity character versus Mg fraction x for Zn1−x Mgx Se.

4. Optical properties It is also of interest to estimate the transverse effective charge because it gives some insight into the infrared activity of phonons in materials. Transverse dynamical effective charges are fundamental quantities in the lattice dynamics of semiconductors, determining the long-range part of the force constants in the longwavelength limit, the Froehlich electron–optical-phonon coupling, parts of the piezoelectric coefficients, etc. [28]. A number of empirical and semi-empirical models have been developed [29–32]. In this paper, the procedure of determining the transverse effective charge, was based on the model of Vogl [33]. The composition dependence of effective charge determined from these procedures is shown in Fig. 6. One feature of this curve is the strong dependence of f on volume since the effective charge increased continuously with increasing volume. We find the dependence of effective charge with x concentration as: eT∗ = 4.039 + 3.507.x − 1.526.x 2 .

(16)

This increase suggests the strong influence of anharmonicity on the effective charge. The most important optical property of the alloy Zn1−x Mgx Se is refractive index, because the use of fast nondestructive optical techniques for epitaxial layer characterization (determination of thickness or alloy composition) is limited by the accuracy with which refractive indices can be related to alloy composition. These applications require an analytical expression or known accuracy to relate the wavelength dependence of the refractive index to alloy composition, as determined from simple techniques such as photoluminescence. Therefore, we used the Hervé and Vandamme [34] model for determination of the refractive index with the concentration x. The refractive index is related to the energy band gap of the semiconductor, as

Superlattices and Microstructures, Vol. 30, No. 1, 2001

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(ZnSe)1 -xMgx 6.0

eT*

5.5

5.0

4.5

4.0

0.0

0.1

0.2

0.3

0.4 0.5

0.6

0.7

0.8

0.9

1.0

Concentration of Mg Fig. 6. The variation of the effective charge with alloy composition for Zn1−x Mgx Se.

(ZnSe)1 -xMgx

2.45

Refractive index

2.40

2.35

2.30

2.25

2.20

2.15 0.0 0.1

0.2

0.3 0.4 0.5 0.6 0.7 Concentration of Mg

0.8

0.9

1.0

Fig. 7. The variation of the refractive index with alloy composition for Zn1−x Mgx Se.

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Superlattices and Microstructures, Vol. 30, No. 1, 2001

follows: s

2 A n = 1+ (17) Eg + B where E g is the energy band gap and A = 13.6 eV and B = 3.4 eV. This expression is valuable for w  w0 : the ultraviolet resonance frequency. Using this model we calculated the variation of refractive index with alloy concentration. The results are displayed in Fig. 7 where composition dependence of the refractive index was determined by polynomial fitting. Our best fit yielded: 

n = 2.414 + 0.251.x − 0.487.x 2 .

(18)

Again, we observed strong nonlinear dependence of alloy properties with Mg concentration. The results also indicate that, for certain concentrations, the alloy exhibits a larger ionicity character than the basic compounds.

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