Percolation in Semiconductor Epilayers Rohit Garg and Kailash C. Rustagi Abstract The role of percolation theory impacts many physical properties of mixed semiconductor crystals. In tetrahedrally bonded mixed crystals of the type Ax B1−x C (III-V or II-VI compounds) percolation related phenomena have been invoked to explain departures from the virtual crystal approximation. Although in most cases, the samples are epitaxial thin films, percolation theory pertains to infinite lattices appropriate for bulk crystals. In this report we study percolation effects in thin films which have one dimension of finite extent and other two infinite. Growth direction is taken as the (100) axis.

1

Introduction

Mixed semiconductor epilayers are an important ingredient in semiconductor devices such as diode lasers, self-electroptic effect devices and high efficiency solar cells. Most of these are designed by using virtual crystal approximation (VCA) for describing mixed semiconductors. In VCA for the mixed crystal Ax B1−x C the randomly distributed A and B atoms on a given face centered cubic (FCC) sub lattice are replaced by a average fictitious atom whose properties are given by a prescribed weighted average of those of the two atoms.[1] For example, to calculate electronic structure of the VC one takes a weighted average of pseudo-potentials[2]. Another important quantity the lattice constant, is experimentally known to obey a linear interpolation law called the Vegards law[1]. For lattice vibrations of mixed crystals the modified random element iso-displacement model of Chang and Mitra [3] has been widely used. The main advantage of VCA is its simplicity and it agrees qualitatively and in some cases quantitatively with experimental observations. However, a lot of evidence has accumulated over the years, that it is important to include disorder effects for an accurate description of these mixed crystals. It is useful to keep these physical effects in mind. First, using Extended X-Ray-Absorption Fine-Structure Mikkelson and Boyce [4] in InGaAs found that the InAs and GaAs bond lengths vary only 1

slightly compared to their values in the pure compounds. In general, in most cases it is now well recognized that while the average position of C atoms in the mixed crystal Ax B1−x C is reasonably well described by Vegards law, the A-B sublattice shows substantial positional disorder[5]. Detailed calculations by Bellaich etal [5] also showed that there is a small anomaly in the lattice constant at the percolation threshold. Contributions of the substitutional disorder as well as the positional disorder on band gap bowing have been estimated to be important [3],[6]. However, disorder induced indirect transitions have not received the same attention. When departures from the virtual crystal become important one should expect percolation effects to be important. In the mixed crystal Ax B1−x C, when (1−x) > x > xc , xc being the percolation threshold for the FCC lattice the crystal consists largely of connected networks of AC and BC bonds. This would considerably reduce the effectiveness of the mixed crystal as a barrier layer since lower gap material forms a continuous network through the sample. Indeed, Kim et al [7] and others [8], have proposed that such channels may be involved in explaining the thickness-dependence of tunneling in such barriers. Experimentally, however, they observe the effect in Alx Ga1−x As only when Ga atomic fraction is more than 0.5 i.e., x < 0.5 [7]. Percolation effects in the Raman spectra of mixed crystals have been investigated by Pages et al [9],[10]. First, additional optical phonon modes not expected in the MREI model of Chang and Mitra[4] were observed in Znx Be1−x Se and Znx Be1−x T e in the percolation range. Subsequently, Pages et al[9][10] have shown that the percolation phenomena play an important role in determining the Raman spectra and infra-red absorption spectra of most 3-5 mixed semiconductor crystals. Briefly, this involves recognizing that in the percolation regime the mixed crystal has meso-scale regions which are AC and BC connected networks. (Pages et al call these as A-rich and B-rich regions which is slightly confusing, since in a perfectly √random mixed crystal the concentration of A or B can have fluctuations ∼ N , where N is the sample size and this does not change whether we are in the AC percolating cluster or BC one). Obviously, the structure of such clusters is important to determine the effective size of such mesoscale regions. Each region is then treated in the MREI model with different effective force constant for each bond. The TO modes are treated as primary modes, since it has been recognized earlier that the asymmetry of the LO modes may be attributed to the coupling of different TO modes by the macroscopic electric field [11]. Other phenomena where percolation effects are believed to be involved are metallic impurity bands[12], anomalous microharness[13] and magnetic ordering in dilute magnetic semiconductors[14]. In all these discussions percolation theory refers to infinite sample limit in two, three or more dimen2

sions. In mixed semiconductors, however, the most commonly used samples are epitaxial layers consisting of a finite number of atomic layers each of which can be taken as infinite. In such systems, some properties such as tunneling and hardness would depend on height spanning paths while other properties like in-layer conductivity would involve width spanning connectivity. In this report we examine height spanning clusters in mixed crystals of zinc blend and diamond structure.

1.1

Notations and conventions used

Then we focused our attention on actual semiconductor epilayers. We simulated asymmetric, cuboidal slabs as they mimick epitaxial layers of mixed semiconductors. We first simulated Alx Ga1−x As like mixed crystals. Alx Ga1−x As is known to crystallize in a zinc-blende structure [1]. However, since As is present only on one of the two interpenetrating lattices , when percolation threshold for FCC lattice is crossed, wall to wall connected chains of AlAs will exist. Since percolation effects occur only between Al and Ga, hence for the purposes of implementation, percolation in Alx Ga1−x As may be regarded as site percolation in an ordinary fcc lattice. Our second candidate was Six Ge1−x , which crystallizes in diamond lattice in which both Si and Ge are present on both fcc lattices. Because of its extensive use in producing strained epilayers of Si , the structure of and phonons in Six Ge1−x mixed semiconductor crystals have been investigated in great detail. [15],[16] Thus, the simplification of diamond to fcc lattice is not possible in this case. For both of these crystals, the fractional concentration denoted by x was taken as the probability of substitution. For convenience, we assume the unit vectors to be aligned along the xyz axes. The dimensions of the thin film were measured in the number of conventional cubic unit cells (for both FCC and zincblende) along the edge throughout. Hereinafter, we will refer to LLxz as simply the ratio and Lz as height.

2

Number of Clusters

First of all, we looked at the variation of number of height spanning clusters (of any shape and/or size) with probability of occupation.

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Figure 1: Number of height spannig clusters in an FCC crystal, size is 45 x 45 x 5 unit cells

2.1

FCC lattice

For FCC lattice, a representative graph of number of height spanning clusters as a function of probability, or nf cc (x) is shown in Fig. 1. To account for the inherent randomness of this data, we averaged the number of clusters for a given crystal size over 30 samples. Interesting points to note from this graph are 1. There are some height spanning clusters even for x < xc . 2. The number of clusters rises to a peak at ∼ xc 3. For x > xc + 0.08 only one cluster survives, just as in percolation in the bulk regime. We found this statement to hold if the number of layers was greater than 5, irrespective of the aspect ratio. To determine the behavior of the peak at pc mentioned above with crystal size, we attempted to find functional dependence of nf cc (x) on crystal dimensions. We find that nf cc (x) is well approximated by the relation nf cc (x) = H(x − xc ) + af cc × exp(−

(x − µ)2 ) 2σ 2

(1)

where H(x − xc ) denotes step function at xc . For illustration, the fit of the data in Fig. 1 is presented in Fig. The data of Fig. 1 was fitted to the gaussian after subtracting the step function. The relations for af cc , σ, µ are straightforward. We first consider the dependence of µ. As can be seen from the histogram of µ in Fig. 3, in most of the ratio 4

Figure 2: Data fit for Fig. 1 to equation 1

Figure 3: Histogram of µ values for FCC lattice

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Figure 4: Dependence of af cc on

Figure 5: Dependence of σ on

Lz Lx

Lz Lx

and layer combinations, we find µ to be around 0.2. Considering that xc for site percolation in FCC lattice is 0.198, we take µ = xc with the small deviation being a finite size effect. As can be seen in Fig. 4,we find that af cc can be well approximated by af cc ∝ (

Lz −α ) Lx

(2)

with αf cc = 1.8 Finally, σ does not appear to have a simple dependence on ratio as shown by Fig. 5. Further, since nf cc (p) can be very large even for finite systems, and diverges in the idealization of finite layers but infinitely wide base, the utility of σ to find the conventional 3 cutoff is limited. Hence, we define an 6

Figure 6: Dependence of pσ on Lz

Figure 7: Histogram of µ values for zincblende lattice alternate cutoff pσ , defined as the probability where the number of clusters falls below a suitable theshold, say 1.1. This way, we bypass the problem of picking a suitable factor of (3 or 4 or 5)σ to decide our measure of width of the peak. We find that pσ depends only on the number of layers (Fig. 6) and does not vary with ratio or the dimensions of base.

2.2

Zincblende structure

We found that number of height spanning clusters for the diamond structure, nzb (x) behaves quite similarly to nf cc (x). Hence we again modeled nzb (x) by equation 1. However, our results for zincblende were averaged 50 times instead of 30 for fcc. As we can see from the histogram of σ for the diamond lattice in Fig. 7, σ nearly remains constant around p = 0.43. Similiar to the 7

case of FCC lattice, we find that azb is also a power law in

2.3

Lz Lx

with αzb = 1.8.

Discussion

For both zincblende and fcc lattices, we find that α is close to 2 and σ is close to xc itself. We believe that deviations for both of them are a manifestation of finite size effect. For a qualitative understanding of why α should be 2, we present the following argument. Consider a slab of size, say 200 x 200 x 10 unit cells. We mentally divide it into 4 blocks, each having dimensions of 50 x 50 x 10 unit cells. Now, we would expect thae four sub-blocks to have equal number of height spaning clusters on an average. Hence, when we double quadruple the number of sub-blocks, we should expect to see n(x) quadruple as well. Since LLxz has halved going from 4 sub-blocks to a single block, α should be 2 to ensure quadrupling of n(x).

3

Structure of height spanning clusters

The structure of percolating clusters has been well explored [17],[18]. In a percolating cluster, if we imagine that the connected sites are actually connected by electrical resistors of equal value, and we apply a potential difference across such a cluster, then there will be a few resistors which will carry all the current [17],[18]. As such resistors will become hot due to Joule heating, hence they are named red bonds. In the context of mixed crystal compound semiconductors the structure is expected to play an important role in transport of electrons through it. To understand this let us consider specific case of tunneling of electrons through Alx Ga1−x As with electron energy below the conduction band edge of the crystal in VCA. For 0.2 < x < 0.8, the crystal will consist of wall to wall percolating networks of GaAs and AlAs. The conduction band edge of GaAs being lower than that of AlAs electrons in the GaAs region will see a lower potential than that in AlAs network. However,because of quantum confinement the kinectic energy will increase inversely proportional to the area of the GaAs channel. So within the GaAs network narrow regions act as potential barriers. Specially the red and nearly red bonds act as barriers. Therefore, it is important to see if we can estimate a width of the channel provided by a height spanning cluster.

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Figure 8: Variation of average radius in fcc lattice

3.1

Radius Of Clusters

A crude estimate of the conduciveness of the cluster to tunneling (or the absence of red and nearly red bonds) can be had by defining the radius of the cluster as s

total − number − of − atoms − in − cluster length − of − backbone − of − cluster

where we define the shortest path from the bottom plane to the top most plane to be the backbone of the height spanning cluster. The radius for a crystal sample was taken to be average of radii of all the height spanning clusters. 3.1.1

Fcc lattice

For fcc lattice, we observed that the average radius increases for x > xc as can be seen in red curve in Fig. 8. The average radius is not zero for x < xc since there are a nonvanishing number of connected clusters at both below and at xc . The standard deviation in average radii is shown in green. Here again, the results were averaged over 30 samples. As shown in Fig. 8, there is a rapid rise in the radius immediately after the percolation threshold is crossed. This may be because the peak of the gaussian lies nearly at xc itself and afterwards, the total number of clusters declines rapidly. Hence the numerator in the definition of the average radius rises rapidly in this region. In view of the very large radius or alternatively, only one cluster for x > xc + 0.1, the number of red and nearly red bonds is expected to be very small. And hence for x > 0.3 or more we expect that the percolating clusters should provide a very good path for tunneling from this consideration alone. 9

Figure 9: Variation of average radius in zincblende lattice 3.1.2

Zincblende lattice

For diamond lattice, we observed that the average radius increases for x > xc as shown in Fig. 9. As in the case of fcc lattice, the average radius is not zero for x < xc as there are a non-vanishing number of connected clusters below and at pc.The standard deviation in average radii is shown in green. Here, the results were averaged out 50 times. As shown in Fig. 9, there is a rapid rise in the radius immediately after the percolation threshold is crossed. Again, as in the case of fcc lattice, this may be because the peak of the gaussian lies nearly at xc itself and afterwards, the total number of clusters declines rapidly. Hence the numerator in the definition of the average radius rises rapidly in this region. In view of the very large radius or alternatively, only one cluster at xc +0.05, the number of red and nearly red bonds is expected to be very small. And hence for x > 0.5 we expect that the percolating clusters should provide a very good path for tunneling from this consideration alone.

3.2

Cross-section In Z Plane

A crude estimate of average radius of the cluster, per se, does not convey all information as it masks over the local variations in cluster radius. Hence, for a better estimate of conduciveness of the cluster to tunneling, we counted the total number of atoms in a cluster in each cross section of cluster perpendicular to z-axis as a function of z-coordinate. Our claim of nearly no red or red- like bonds is at the probabilities mentioned in previous section is strengthened by results in this section. The results for each lattice type in the next two sub-sections are self-contained in two pictures each. There are two plots for each type of lattice. Each shows the variation in total number 10

Figure 10: Number of atoms in cross sections, perpendicular to z axis for x = 0.21

Figure 11: Number of atoms in cross sections, perpendicular to z axis for x = 0.25

11

Figure 12: Number of atoms in cross sections, perpendicular to z axis for x = 0.43

Figure 13: Number of atoms in cross sections, perpendicular to z axis for x = 0.45

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of atoms. 3.2.1

Fcc lattice

The dimensions of crystal sample are 30 x 30 x 10 unit cells. Fig. 10 shows the variation for one of the clusters at p = 0.20 and Fig. 11 shows variation for p = 0.25. 3.2.2

Zincblende lattice

The dimensions of crystal sample are 100 x 100 x 30 unit cells. Fig. 12 shows the variation for one of the clusters at x = 0.43 and Fig. 13 shows variation for x = 0.45.

4

Conclusion

Percolation theory has typically been applied to crystals infinite in all three dimensions. Although anisotropic percolation has been studied extensively, it too has been applied to crystals which are infinite in all three dimensions. To the best of our knowledge, our work represents the first application of percolation theory to epilayers. In this contribution, we make the case for the need for application of percolation theory to mixed semiconductors. We provide estimates and an empirical formula for the number of height spanning clusters in semiconductor epilayers. We also investigate the possibility of electron transport through the low potential channels provided by wall to wall percolating clusters. We find that although there are many height spanning clusters present at x = xc , they are usually inconducive to electron transport due to their limited widths. We also find that this condition is substantially relaxed as we go marginally above the percolation threshold.

References [1] P Yu and M Cardona, Fundamentals of semiconductor Physics, Springer, Berlin (1996) [2] A Baldereschi and K. Maschke, Sol St Comm, 16, 99 (1975) [3] I.F. Chang and S.S. Mitra, Phys. Rev. 172, 924 (1968) and Phys.Rev. B 2, 1215 (1970).

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[4] J. C. Mikkelsen, Jr., and J. B. Boyce, Phys. Rev. B 28, 7130 (1983) and Physical Review Letters, 49, 1412 (1982) [5] L. Bellaiche, S. H. Wei, and Alex Zunger, Phys. Rev. B 54, 17568 - 17576 (1996) [6] Alex Zunger and J. E. Jaffe, Phys. Rev. Lett. 51, 662 - 665 (1983) [7] D. S. Kim, H. S. Ko, Y. M. Kim, S. J. Rhee, S. C. Hohng, Y. H. Yee, W. S. Kim, J. C. Woo, H. J. Choi and J. Ihm Phys Rev B 54,14580 (1996) [8] L Schrottke, R Hey, H Kostial, T Ohtsuka, HT Grahn Physica E 21 852857 (2004) [9] O. Pags, M. Ajjoun, D. Bormann, C. Chauvet, E. Tourni, and J. P. Faurie, Phys. Rev. B 65, 035213 (2002). [10] O.Pages et al Phys Rev B. 77,125208 (2008) [11] O. Pags, M. Ajjoun, T. Tite, D. Bormann, E. Tourni, and K. C. Rustagi, Phys. Rev. B 70, 155319 (2004) [12] J Osorio-Guillen, S Lany, S.V.Barabash and A.Zunger Phys Rev B 75,184421 (2007) [13] E.Schubert, Doping in III-V Semiconductors ,Cambridge University Press, Cambridge, England, (1993) [14] E. Rogacheva, J. Phys. Chem. Solids 66, 2104 (2005) and refernces therein. [15] A. P. Li and J. Shen, Science J. R. Thompson and H. H. Weitering, Appl. Phys. Lett. 86, 152507 (2005). [16] A. V. G. Chizmeshya, M. R. Bauer and J. Kouvetakis Chem. Mater. 15, 2511 (2003) [17] Hans J. Herrmann and H. Eugene Stanley Phys. Rev. Lett. 53, 1121 (1984) [18] A.Coniglio, J Phys A: Math Gen 15, 3829 (1982) [19] Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein, Introduction to Algorithms (2nd ed.). MIT Press and McGraw-Hill, (2001)

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