Ab initio investigation of phosphorus and boron diffusion in germanium

ARTICLE IN PRESS Materials Science in Semiconductor Processing 11 (2008) 324–327

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Ab initio investigation of phosphorus and boron diffusion in germanium ¨ berg c, P.R. Briddon d C. Janke a,, R. Jones a, J. Coutinho b, S. O a

School of Physics, University of Exeter, Stocker Road, Exeter EX4 4QL, UK I3N and Department of Physics, University of Aveiro, 3810-193 Aveiro, Portugal ˚ University of Technology, Lulea ˚ S-97187, Sweden Department of Mathematics, Lulea d School of Natural Sciences, University of Newcastle upon Tyne, Newcastle upon Tyne NE1 7RU, UK b c

a r t i c l e i n f o

abstract

Available online 15 August 2008

P and B diffusion has been modeled in Ge using ab initio methods along with the formation energies and electrical levels of various Px Vy defects expected to be important in the deactivation of P in heavily n-doped Ge. The calculated activation barrier for B diffusion is found to be substantially lower than the measured barrier. However, the exceptionally large pre-exponential factor in the measured diffusivity points to a Meyer–Neldel rule operating and accounting for the discrepancy. The magnitude of the theoretical diffusivity is about a factor 10 lower than observed. For P diffusion, the experimental and theoretical results are in much closer agreement. The formation energy calculations show that all Px Vy clusters are stable with respect to their component defects, and all but P4 V are predicted to introduce acceptor levels into the band gap. A simple analysis of possible formation mechanisms and Coulombic contributions suggests that as in Si, P3 V is the most important compensating center in heavily n-doped Ge. & 2008 Elsevier Ltd. All rights reserved.

PACS: 61.72.Bb 61.72.Cc 61.72.J 61.72.Uf 71.15.Mb 71.55.Cn Keywords: Germanium Phosphorus Boron Diffusion Clustering Ab initio

1. Introduction Diffusion of dopants is a process of fundamental importance in semiconductor technology. In germanium, p-type doping is often achieved by boron implantation. Diffusion of boron in germanium is seen to be very slow experimentally [1,2], and unlike in Si, transient enhanced diffusion (TED) is very weak and has only recently been observed [3]. Diffusion barriers of 4.5–4.65 eV have been measured experimentally, but these are accompanied by unusually large prefactors ranging from 6  108 down to 1:97  105 cm2 s1 [1,4,5]. In Si, B diffusion is seen to progress with a barrier of 3:5 eV [6], with prefactors

 Corresponding author.

E-mail addresses: [email protected], [email protected] (C. Janke). 1369-8001/$ - see front matter & 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.mssp.2008.07.002

close to 0:9 cm2 s1 . In both Si and Ge, B diffusion is expected to proceed via an interstitial mechanism [1,7,8], and the differences in the diffusion behavior are believed to stem from the differences in the behavior of the selfinterstitial in the two materials. Theoretical work on B diffusion in Ge has been performed previously, leading to a calculated diffusion barrier of 4.5 eV for the interstitialmediated path, in good agreement with the experimental results, though the origin of the prefactor was not discussed in this work [9]. In germanium, n-type doping has proved more problematic. Phosphorus is often used and high activation values are much harder to achieve with a maximum concentration around 3  1019 cm3 obtained by ion implantation although a large proportion 90%, of the donor atoms remain inactive for rapid thermal annealing (RTA) below 700 1C [10]. Experimentally measured diffusion barriers for P in Ge barriers range from 2.07 to 2.85 eV [4,11–13],

ARTICLE IN PRESS C. Janke et al. / Materials Science in Semiconductor Processing 11 (2008) 324–327

with diffusion mediation attributed to doubly [13] or singly [12] negatively charged phosphorus vacancy (PV) defects. While the lowest of these barriers has been attributed to TED effects [13], annealing at 700 1C would still be expected to produce significant motion of the dopants and disruption of the sharply defined shallow junctions required for modern devices. In Si, the barrier for P diffusion is a little higher, with a value of 3.46 eV derived from experiment [6], very close to the value for B diffusion. The mechanism is also slightly different, with P diffusion shown to exhibit either interstitial- [14] or vacancy-mediated [15] diffusion depending on the position of the Fermi level within the band gap. As with B diffusion, the difference in diffusion behavior can be linked to the differences between interstitial and vacancy behavior in Si and Ge. Theoretical studies of donor-vacancy (DV) complexes have been performed in the past with the intent of improving understanding of how these defects contribute to donor deactivation in Ge. Single DV, and di-antimony vacancy ðSb2 VÞ as well as antimony divacancy ðSbV2 Þ defects have been studied [16,17], revealing an acceptor behavior for all the defects and a common structure with donor atoms replacing Ge adjacent to the vacancy. A more recent study investigating Sbx V ð1pxp4Þ revealed acceptor behavior for defects up to Sb3 V while Sb4 V was seen to be electrically inactive. It also revealed a negative formation energy for Sb3 V and Sb4 V [18]. Calculations on the behavior of P and B in Ge were performed using ab initio methods described elsewhere [19,20], with the aim of studying the diffusion mechanisms of B and P as well as the P-related defects likely to account for the high inactive fraction in Ge.

2. Boron diffusion B diffusion in Ge has been the focus of recent theoretical work by the present authors [19]. In that work, boron diffusion was calculated to proceed via neutral or positively charged boron interstitial (BI) complexes. In the neutral charge state, the diffusion

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occurs via a kick-out mechanism with a diffusion barrier relative to negatively charged substitutional B Ge of 3:6 eV þ me where me is the Fermi level position measured from the top of the valence band. In the positive charge state the boron diffuses through forming a complex with an interstitial Ge atom but without entering a true interstitial configuration itself. For this process the diffusion barrier relative to B Ge is calculated to be 3:1 eV þ 2me . Taking the Fermi level to lie around midgap, where it is expected to be found during the hightemperature diffusion experiments, yields barriers around 3.9 and 3.8 eV for the two charge states. The diffusion barrier dependence on Fermi level is shown in Fig. 1. The results calculated for the structure of the BI defects are in good agreement with previous theoretical calculations [9], though the barrier calculated here does not agree with prior work. These results are also not in agreement with experimental results of 4:6 eV, and neither can the experimental prefactor of 105 cm2 s1 be explained by the methods used. The diffusivity calculated from this work is not, however, in especially poor agreement with prior work. For a temperature of 800 1C, for example, experimental studies give diffusivities of 3  1017 cm2 s1 [1], 6  1015 cm2 s1 [5], and 4  1013 cm2 s1 [4]. The calculated diffusion barrier is very similar to that found in Si both experimentally and theoretically [6,21], and so it does not seem unreasonable to assume for this estimate that the prefactor measured in Si of 0:87 cm2 s1 [6] would apply. Using this prefactor with the diffusion barrier calculated here leads to a diffusivity of 1018 cm2 s1 . This value is slightly lower than those found from experiment, but not excessively. A possible explanation of the differences between experiment and theory is found in the form of a theoretical description of the effect of high temperature excitations on observed diffusion barriers and preexponential factors for diffusion in Ge and Si [22]. In the paper, calculations are performed for temperatures approaching the melting point, suggesting an increase of 1 eV on the energy barrier and 105 cm2 s1 of the diffusion prefactor beyond that which would be expected from zero-temperature theory. These modifications

4.6

4.0

4.2

Diffusion Barrier (eV)

Diffusion Barrier (eV)

BI(−) BI(0)

3.8 BI(+) 3.4 3.0

PI(+)

3.6 3.2

PI(0)

2.8 PV(−) 2.4 PV(=)

2.0 0.0

0.1 0.2 0.3 0.4 0.5 Fermi energy (eV above Ev)

0.6

0.0

0.1 0.2 0.3 0.4 0.5 Fermi energy (eV above Ev)

0.6

Fig. 1. Dependence of the diffusion energy on Fermi level position for phosphorus complexes as calculated in Ref. [20] and boron complexes from Ref. [19].  Barriers are given relative to isolated substitutional Pþ Ge and BGe , respectively. Only the lowest barrier for each charge state is displayed, and the labels on the graphs indicate the species and charge state corresponding to each line.

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C. Janke et al. / Materials Science in Semiconductor Processing 11 (2008) 324–327

would then bring the calculated results into line with experimental measurements. The weak TED observed for B in Ge can be attributed to the interstitial mechanism for diffusion, as vacancies are the dominant defect in germanium. 3. Phosphorus diffusion P diffusion in Ge has also been studied by theoretical methods recently. In this case, the interstitial mediated diffusion path is seen to be important only in p-type material, with me p0:2 eV. As phosphorus is an n-type dopant, and most diffusion studies on P in Ge are concerned with heavily doped regions, this process is not expected to play an important role in explaining experimental data. The PV defect, or E-center is seen to be the most important mediating species for P diffusion for the remainder of the band gap, with diffusion barriers relative to the isolated Pþ Ge of 4.0 eV–3me for the most active doubly negative charge state. For a me of 0:5 eV, this yields a diffusion barrier of 2.5 eV, in good agreement with the experimental results. The barrier dependence across the band gap is shown in Fig. 1. 4. PV clusters During the diffusion of negative or doubly negative PV defects through n-type Ge, interaction between the mobile defects and positively charged Pþ Ge would be enhanced by a Coulomb attraction. The possibility then exists for the formation of Px V and even Px V2 clusters. Energy levels for these defects were calculated using the marker method [23] in cluster calculations, with the antimony-vacancy (SbV) donor level at Ev þ 0:09 eV and acceptor levels at Ev þ 0:31 and Ec  0:30 eV as markers [24,25]. Formation energies were calculated for the neutral defects relative to Pþ Ge using supercell. The results are given in Tables 1 and 2. The suffixes for the P3 V2 and P4 V2 defects indicate different structures with differing numbers of P atoms around the first and second vacancy. From these results, it can be seen that all studied Px Vy complexes aside from P4 V have at least one acceptor level within the band gap. This suggests that, if they are mobile,

P2 V and P3 V could interact with more P atoms to produce larger Px V clusters. It also suggests that the formation of Px V2 clusters would be impeded by Coulomb repulsion. Even so, the formation energies indicate that all the defects studied are stable with respect to their component defects, and that the larger defects (P3 V, P4 V, and P5 V2 ) have a negative formation energy for values of me sufficiently high in the gap. It is not believed that the defects will form spontaneously, however, as this would require several phosphorus atoms to exist close together, and more importantly the calculation does not take into account the energy of the ejected Ge self-interstitial or interstitials. Furthermore, while both the PV and P2 V defects are expected to be mobile, due to lying on one sixmembered ring, the P3 V defect does not and so it is not clear how this defect could diffuse. Therefore, the formation path for P4 V is unclear and so it is not expected to form in substantial concentrations. These energy levels are in good agreement with the previous theoretical work using similar methods for the single vacancy defects [16–18]. All together, these results suggest that the diffusion of P will proceed via the formation of PV defects, and can thus be enhanced by a supersaturation of vacancies leading to the observed TED. The deactivation of P in Ge can be explained via the formation of Px V defects, with x ¼ 3 being the highest value expected for large concentrations of P. Larger clusters are not expected to form in significant quantities due to the Coulombic repulsion between the negatively charged Px V defects. This is very similar to the situation in Si where P3 V has been detected as the dominant compensation center in highly doped regions while a larger defect tentatively identified as P5 V2 is also seen, though not with as high concentrations [26,27]. 5. Conclusion Calculations performed on P and B in Ge crystals suggest that the diffusion of B proceeds via an interstitial mechanism, though neither the experimentally measured value for the diffusion barrier or prefactor can be explained by the zero temperature theory used. Inclusion

Table 1 Calculated energy levels for a number of Px Vy clusters Level

PV

P2 V

P3 V

P4 V

P3 V2 .21

P3 V2 .30

P4 V2 .22

P3 V2 .31

P5 V2

ð0=þÞ-Ev ð=0Þ-Ev ð¼ =Þ-Ev

0.13 0.43 0.38

0.26 0.50 0.47

0.06 0.31 1.51

0.57 1.62 1.49

0.01 0.26 0.48

0.15 0.49 0.41

0.01 0.40 0.31

0.34 0.55 0.54

0.06 0.32 1.47

Energies are given in eV above the valence band top. The SbV levels were used as markers for all defects [24,25].

Table 2 Formation energies calculated for the Px Vy clusters being studied Charge

PV

P2 V

P3 V

P4 V

P3 V2 .21

P3 V2 .30

P4 V2 .22

P4 V2 .31

P 5 V2

0

2.23me

1.78–2me

1.51–3me

0.92–4me

3.01–3me

2.84–3me

2.60–4me

2.36–4me

2.13–5me

Energies are given with respect to isolated

Pþ Ge .

ARTICLE IN PRESS C. Janke et al. / Materials Science in Semiconductor Processing 11 (2008) 324–327

of electronic excitations predicted to be present at the high temperatures the experiments were performed at raises the calculated barrier and prefactor to values much closer to those found from the experimental results, and so it is believed that these effects are important in the high-temperature diffusion of B. P diffusion is calculated to occur almost exclusively via a vacancy-mediated mechanism, with interstitial diffusion being important only in p-type regions. The diffusion barrier calculated is in good agreement with experimental values, while the experimental prefactors are not as anomalously high as in the B case, likely due to the lower temperature at which the experiment was conducted. Simulations of Px Vy clusters revealed that, as in Si, the P3 V defect is expected to be the dominant compensating defect in n-type Ge. P4 V defects are not expected to form in significant quantity due to the low expected mobility of P3 V, while Px V2 cluster formation would be impeded by Coulombic repulsion between the component Px V clusters, all of which exhibit acceptor behavior. References [1] Uppal S, Willoughby AFW, Bonar JM, Cowern NEB, Grasby T, Morris RJH, et al. J Appl Phys 2004;96:1376. [2] Stolwijk NA. In: Schulz M, Landolt-Bo¨rnstein, editors. Impurities and defects in group IV elements and III–V compounds, vol. 22A. Berlin: Springer; 1989. [3] Satta A, Van Daele B, Simoen E, Vandervorst W. Unpublished. [4] Dunlap Jr. WC. Phys Rev 1954;94:1531.

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