Diffusion mediated by doping and radiation-induced point defects

ARTICLE IN PRESS

Physica B 376–377 (2006) 11–18 www.elsevier.com/locate/physb

Diffusion mediated by doping and radiation-induced point defects Hartmut Bracht Institute of Materials Physics, University of Muenster, D-48149 Muenster, Germany

Abstract The growth of isotopically enriched epitaxial layers enables the preparation of material heterostructures, highly appropriate for simultaneous self- and dopant-diffusion studies. The advance in solid-state diffusion is demonstrated by experiments on the impact of dopant diffusion and proton irradiation on self-diffusion in silicon. Accurate modeling provides valuable information about the type and charge states of native point defects and the mechanisms of atomic transport. The results are compared with recent theoretical calculations. Consistencies and differences between experiment and theory are highlighted. r 2005 Elsevier B.V. All rights reserved. PACS: 61.72.Ji; 61.80.Fe; 61.82.Fk; 66.30.Hs; 66.30.Jt Keywords: Semiconductors; Silicon; Self-diffusion; Dopant diffusion; Point defects; Proton irradiation

1. Introduction The diffusion of atoms in materials is a fundamental process of mass transport that is important for numerous applications. For example, doping of Si is often performed by the diffusion of the desired foreign atom into the material. The current standard of very shallow dopant profiles formed by ion implantation and subsequent annealing requires a strong control of diffusion and reaction processes in order to minimize transient diffusion effects [1]. With the down-scaling of Si-based electronic devices, the charge concentration in the source and drain regions of a metal oxide semiconductor (MOS) must increase to maintain a low device resistance. With increasing n-type (p-type) doping level, the position of the Fermi level approaches the conduction (valence) band, thereby the formation of acceptor (donor) like point defects becomes energetically more favorable. The charge states of point defects and their energy levels in the Si band gap are of fundamental significance for controlling the diffusion of dopants during the fabrication of Si-based electronic devices.

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E-mail address: [email protected]. 0921-4526/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2005.12.006

Information about point defects in semiconductors can be obtained from spectroscopic methods such as electron paramagnetic resonance (EPR) studies, infrared (IR) spectroscopy, deep level transient spectroscopy (DLTS), and perturbed angular correlation (PAC) experiments, to name just a few. All of these methods provide results for temperatures which range from cryogenic to room temperature. However, a generally applicable spectroscopic method for studying point defects at temperatures relevant for process technology is not available. An exception is the positron annihilation spectroscopy (PAS). But this method is limited to the investigation of vacancy-like defects and unfortunately fails in the case of Si because the concentration of vacancies in thermal equilibrium is below the detection limit of the method. On the other hand, theoretical calculations of the structure and formation energy of point defects in solids are very valuable for comparison with spectroscopic results. But it remains unclear how far these theoretical results, obtained for zero Kelvin, are applicable at higher temperatures. The numerous theoretical papers published on point defects and more complex defect structures in solids reflect the need to gain more information about point defects which strongly affect the functionality of technological materials. A general method to investigate point defects in materials at high temperatures is ‘‘diffusion spectroscopy’’.

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On first sight, this appears to be an oxymoron but diffusion studies can, in fact, be performed in such a way that detailed information about point defects becomes accessible. This advance in solid-state diffusion is highlighted by the present contribution which summarizes recent results on Si self-diffusion mediated by doping and radiation. For diffusion spectroscopy, appropriate isotopically controlled heterostructures were used which enable diffusion experiments over a wide temperature range. Both selfdiffusion experiments under thermal equilibrium [2–5] and radiation-enhanced conditions [6] were performed. Moreover, the impact of dopant diffusion on self-diffusion could be investigated for the first time by means of isotope multilayer structures [7–10]. In this paper, first, the results on Si self-diffusion under thermal equilibrium are summarized and discussed in Section 2. Then, the results of Si self-diffusion under proton irradiation are presented in Section 3. Finally, the impact of doping on Si self-diffusion is discussed in Section 4. The paper concludes with a brief summary and an outlook on future experiments. 2. Si self-diffusion under thermal equilibrium The Si self-diffusion coefficient DSi was accurately measured over a wide temperature range with isotopically controlled Si heterostructures [2,3]. Typical Si profiles measured by means of secondary ion mass spectrometry (SIMS) after annealing an isotope structure with an enriched 28Si epilayer sandwiched between natural Si layers are illustrated in Fig. 1. The self-diffusion coefficients obtained from fitting the experimental profiles are summarized in Table 1. The sample numbers #1–5 reflect the specific isotope structure used for the diffusion experiment. Details about the layer structure and the analysis of the diffusion profiles are given in Refs. [2,3]. The temperature dependence of DSi is accurately described over seven orders

concentration of 30Si (cm-3)

1022 1021 1020 1019 1018 1017

0

0.2

0.4 depth (10-4cm)

0.6

0.8

Fig. 1. Depth profiles of 30Si measured with SIMS before (dashed line) and after annealing of the 28Si isotope heterostructure #4 at 1019 1C (&, t ¼ 57600 s) and 911 1C (J, t ¼ 1207200 s). For clarity, only every fourth experimental data point is shown. Solid lines show best fits to experimental data [2,3].

Table 1 Self-diffusion coefficients DSi extracted from 30Si profiles after annealing Si isotope heterostructures #1–5 at temperatures between 855 and 1388 1C and times t [2,3] Sample

T (1C)

t (s)

DSi (cm2 s
1)

#1 #1 #1 #1 #1 #2 #1 #2 #1 #2 #1 #3 #4 #3 #4 #5 #3 #4 #5 #3 #4 #5 #3 #4 #3 #4

1388 1322 1268 1220 1188 1153 1120 1095 1071 1045 1020 1019 1019 983 983 961 940 940 925 911 911 895 880 880 855 855

1680 1800 1800 3600 14 400 70 200 86 400 196 200 259 200 1 036 800 691 200 57 600 57 600 176 400 176 400 345 600 604 800 604 800 864 000 1 207 200 1 207 200 2 163 600 2 649 600 2 649 600 3 888 000 3 888 000

2.21  10
12 4.42  10
13 1.51  10
13 4.91  10
14 2.03  10
14 8.73  10
15 4.92  10
15 1.26  10
15 1.09  10
15 3.66  10
16 1.99  10
16 1.31  10
16 1.39  10
16 3.78  10
17 4.25  10
17 1.74  10
17 1.24  10
17 1.17  10
17 5.76  10
18 2.42  10
18 3.13  10
18 1.47  10
18 6.68  10
19 8.46  10
19 3.81  10
19 3.81  10
19

The experimental error for DSi is about 20% and mainly stems from the accuracy of the depth measurements of the craters left from the SIMS analysis.

of magnitude by an Arrhenius equation with one diffusion activation enthalpy of 4.7670.04 eV and the pre-exponen2
1 tial factor 560þ240 [2,3]. Additional self-diffusion
170 cm s studies with isotopically enriched Si were performed by other groups [4,5]. Their results are in good agreement with the data given in Table 1. However, Aid et al. [5] report an activation enthalpy of 4.37 eV for self-diffusion which is lower than our value. This difference is probably due to an underestimation of the two data points determined by the authors for temperatures above 1100 1C. No pronounced kink was observed in the temperature dependence of Si self-diffusion within the range of temperatures investigated so far. This does not imply that only one mechanism mediates self-diffusion but rather that one mechanism predominates in the temperature range investigated. Taking into account contributions due to vacancies (V) and self-interstitials (I) to self-diffusion, the self-diffusion coefficient is given by eq DSi ¼ xV C eq V DV þ xI C I DI .

(1)

A successful jump of a tagged Si atom to a next-nearest lattice site depends on the probability that a native point defect is located next to the tagged atom. Accordingly, DSi not only depends on the diffusion coefficients DV and DI,

ARTICLE IN PRESS H. Bracht / Physica B 376–377 (2006) 11–18

1400

1200

T (˚C) 1000

800

10-12 self-diffusion of Si (cm2s-1)

but also on the concentrations C eq V;I (in atomic fractions) of vacancies and self-interstitials under thermal equilibrium [11]. After the site exchange, the native point defect is still next to the tagged self-atom. As a consequence, the reverse jump is highly probable. Such jumps, which do not contribute to the long-range migration, are taken into account by the correlation factors xV and xI . For the diamond structure, the correlation factors for self-diffusion by vacancies and self-interstitials were calculated to be xV ¼ 0:5 [12] and xI  0:73 [13], respectively. The value of xI corresponds to the correlated diffusion of a tetrahedral self-interstitial. Since self-interstitials can exist in various configurations (tetrahedral interstitial, hexagonal interstitial, split-interstitial, bond-centered interstitial) with a prevalence of a specific defect depending on the Fermi level [14], the correlation factor can be different for each configuration with its specific migration path. Ab initio calculations of Clark and Ackland [15], and Al-Mushadami and Needs [16] reveal formation energies of the neutral /1 1 0S-oriented split-interstitial which are lower compared to that of the tetrahedral and hexagonal interstitials. In agreement with these density-functional-theory calculations, molecular dynamic (MD) simulations with the Stillinger-Weber potential also yield /1 1 0S split-interstitials as the most stable self-interstitial defects [17,18]. A correlation factor of xI ¼ 0:56 for the migration of the split-interstitial was obtained by MD simulations of Posselt et al. [18]. This value is given by the ratio of the selfdiffusion coefficient per defect and the defect diffusivity (see Table II in Ref. [18]) [19]. One first may think that the difference between the diffusion correlation factor for the split-interstitial and tetrahedral interstitial is without any consequence because the accuracy of diffusion coefficients is of similar magnitude. However, the correlation factor of self-interstitials indirectly affects the contribution of vacancies to Si self-diffusion. Taking 2
1 into account C eq I DI ¼ 2980 expð
4:95 eV=k B TÞ cm s eq 2
1 (C I DI ¼ 914 expð
4:84 eV=kB TÞ cm s ) for the contribution of self-interstitials to Si self-diffusion from Zn (Au) diffusion experiments [20, 21], the best fit of the DSi data given in Table 1 on the basis of Eq. (1) with xV ¼ 0:5 and xI ¼ 0:56 is obtained with C eq V DV ¼ 49:2 expð
4:51 eV= 2
1 kB TÞ cm2 s
1 (C eq V DV ¼ 358 expð
4:70 eV=k B TÞ cm s ). The individual contributions of self-interstitials and vacancies to Si self-diffusion with activation enthalpies of 4.95 and 4.51 eV, respectively, are displayed in Fig. 2, in comparison to the total diffusivity DSi. In the case when xI ¼ 0:73 is assumed, we get C eq V DV ¼ 0:71 exp ð
4:11 eV=kB TÞ cm2 s
1 [3]. This shows that the data of DSi and C eq I DI suggest an activation enthalpy of selfdiffusion by vacancies which ranges between 4.1 and 4.7 eV depending on the value assumed for xI . A more accurate activation enthalpy is however important for our understanding of dopant diffusion in Si. Antimony, for example, mainly diffuses by a vacancy-mediated mechanism with a diffusion activation enthalpy of 4.08 eV [22]. According to Dunham and Wu [23], the difference between the activa-

13

V

10-14

RESD 10-16 thermal diffusion 10-18 10-20 10-22

I

0.6

0.7

0.8

0.9

1.0

3 -1 10 / T (K )

Fig. 2. Temperature dependence of Si self-diffusion under thermal equilibrium (+, solid line) [2,3] and under proton irradiation (  , solid line) [6]. The long- and short-dashed lines represent the self-interstitial eq ðxI C eq I DI Þ and vacancy contribution ðxV C V DV Þ to Si self-diffusion for thermal equilibrium conditions with activation enthalpies of 4.95 and eq 4.51 eV, respectively. The solid line is the sum xV C eq V DV þ xI C I DI which accurately fits the total Si diffusion coefficient DSi . The temperature dependence of RESD (upper solid line) is described by an activation enthalpy of about 0.9 eV. Since this activation enthalpy equals 0:5H m V of the vacancy migration enthalpy, we get H m V ¼ 1:8  0:5 eV [6].

tion enthalpy of self-diffusion by vacancies and the diffusion activation enthalpy of the dopant–vacancy pair is associated with the binding energies between the substitutional dopant and the vacancy in the second and third nearest neighbor positions. Assuming either 4.11 eV or 4.70 eV for self-diffusion by vacancies, a difference of 0.03 eV or 0.62 eV is obtained. This shows that the activation enthalpy of self-diffusion by vacancies is fundamental for our understanding of the interaction between dopants and native point defects. 3. Si self-diffusion under proton irradiation The study of radiation-enhanced diffusion (RED) of selfand foreign atoms in condensed matter is a powerful approach to investigate the kinetics of point defect migration and annihilation mechanisms. The effect of radiation upon diffusion in solids is not a new topic. Extensive research on this subject has been already performed 30 years ago. However, the defect kinetics during radiation is complex, in particular in the presence of clustering and interactions with foreign atoms. This complexity has limited a comprehensive analysis of RED in the earlier days. Today, the kinetic rate equations can be fully solved numerically. The most basic process of radiation-mediated diffusion is radiation-mediated self-diffusion. Irradiation of solids with high-energy particles such as protons or electrons creates native point defects in concentrations which significantly exceed their thermal equilibrium value. As a

ARTICLE IN PRESS H. Bracht / Physica B 376–377 (2006) 11–18

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consequence, radiation enhances self-diffusion. The experiments yield direct information about the migration enthalpy of the native point defect that determines the self-diffusion under radiation [24]. In the case of Si, such information was lacking for more than 30 years because radiation-enhanced self-diffusion (RESD) experiments could not be performed due to the limitations with the short half-life of the radioactive isotope 31Si (t1=2 ¼ 2:6 h). With the use of 28Si-enriched epitaxial layers grown on a natural Si substrate, this limitation was overcome and RESD could be investigated [6]. Isotope heterostructures with more than one diffusion interface yield information about the depth dependence of RESD. A depth dependence is expected because during annealing and in situ radiation, a steady state is established between the formation of native point defects and their annihilation by direct recombination and outdiffusion to the sample surface. The depth-dependent RESD reflects the distribution of the native point defects in steady state. This steadystate distribution depends on the thermal equilibrium concentrations and diffusion coefficients of the native point defects. Fig. 3 shows the 30Si SIMS profile of a natSi/28Si/natSi isotope structure after proton irradiation at 845 1C for 63 min and a proton flux of 1 mA. Details about the experiments and analysis of the RESD profiles are given in Ref. [6]. The RESD profiles reveal an intermixing of the isotope structure which is much stronger than that expected under thermal equilibrium. Accordingly, the RESD coefficients shown in Fig. 2 significantly exceed the thermal diffusion data. Fig. 3 reveals that the intermixing at the topmost natSi/28Si interface is less pronounced than that at the deeper 28Si/natSi interface. This illustrates that the concentrations of the native point defects which mediate the self-diffusion process are not 106 105 1020 104 103

I

1018

→ V

1016

eq eq CI/C I or CV/C V

concentration of 30Si (cm-3)

1022

0

100

200 300 depth (10-7cm)

400

102 101 500

Fig. 3. RESD profile of Si(+) measured with SIMS after diffusion at 845 1C for 63 min and a proton flux of 1 mA. The as-grown Si profile is illustrated by the dashed line. The upper solid line represents the best fit to the RESD profile. The lower solid lines show the corresponding calculated eq supersaturation SI ¼ C I =C eq I and S V ¼ C V =C V . For clarity, only every fourth data points are plotted.

constant across the isotope structure. However, a steady state of the native point defect concentration is established during irradiation which was confirmed by additional experiments on the time and flux dependence of RESD [6]. Accurate modeling of the RESD profiles requires that the concentration of vacancies in thermal equilibrium must be several orders of magnitude higher than that of selfinterstitials. Accordingly, the diffusion coefficient DV must be much lower than DI because the products C eq I DI and C eq D are of the same order of magnitude. This result is at V V variance with the high mobility of vacancies observed at cryogenic temperatures [25,26] whose migration energies amount to 0.18–0.45 eV depending on the charge state of the defect. In contrast, RESD measurements provide a vacancy migration enthalpy of H m V ¼ ð1:8  0:5Þ eV [6]. In order to find an explanation for the discrepancy between the low- and high-temperature properties of vacancies, we also tried to describe the RESD profiles with high mobilities for both self-interstitials and vacancies. In this case, successful fits are only obtained when self-interstitials and vacancies are trapped by carbon and oxygen, respectively, or microdefects are present that act as internal sinks for native point defects. The concentration of oxygen and carbon must be of the order of 1019 cm
3 and the microdefect density in the range of 109 cm
2. Neither such high carbon and oxygen concentrations nor high microdefect densities were verified by means of SIMS and transmission electron microscopy. In consideration of this fact that no other reasonable explanations of RESD were found, it must be concluded that the vacancy in Si diffuses much slower at high temperatures than expected from its eq mobility at low temperatures. Accordingly, C eq V bC I eq eq holds, because the quantities C I DI and C V DV are of the same order of magnitude. Fig. 4 shows data of C eq V from modeling RESD in comparison to results given in the literature. Recent results from diffusion of Ir in Si [27] are consistent with the RESD data. The temperature dependence of C eq V from RESD and metal diffusion yields a vacancy formation enthalpy and entropy of ð2:44  0:15Þ eV and ð3:3  1:6Þ kB , respectively [27]. This result is in good agreement with the formation energy of ð2:8  0:3Þ eV for the neutral vacancy in Si estimated by Ranki and Saarinen [30] from positron annihilation measurements of the vacancy formation in highly As- and P-doped Si. Most recent ab initio calculations yield 3.3 eV [31] and 3.17 eV [32] for the neutral vacancy which are in fairly good agreement with the experimental finding. On the other hand, theory predicts an energy barrier of 0.4 eV for vacancy hops to the nearest neighbors [33] in agreement with the experimental results of Watkins [25,26], but at variance with RESD [6]. The sum of the vacancy formation and migration enthalpy equals the activation energy of self-diffusion by vacancies. The above-mentioned most recent theoretical results yield 3:17 þ 0:4 eV ¼ 3:57 eV. However, the diffusion activation energy of dopant–vacancy pairs and the results associated with the new calculations for diffusion

ARTICLE IN PRESS H. Bracht / Physica B 376–377 (2006) 11–18

T(˚C) 1400

1200

1000

800

Ceq × 5.0x1022 (cm-3) V,I

1016

1014

1012

Ceq V Ceq I

1010

Ceq V 0.6

0.7

0.8 3

0.9

1.0

-1

10 /T (K ) Fig. 4. Temperature dependence of the thermal equilibrium concentrations of vacancies and self-interstitials in Si. The thick solid line, which is 24 reproduced by C eq exp ð
2:44 eV=kB TÞ cm
3 , represents the V ¼ 1:4  10 eq best fit to the C V data determined from modeling RESD (K: [6]) and the diffusion of Ir in Si (J: [27]). The thin solid lines show C eq V data from studies of Si crystal growth and wafer heat treatments (lower thin line: [28], upper thin line: [29]). The dashed line represents the definitive upper bound for C eq I [19], which was taken into account for modeling of RESD.

correlation factors support values within 4.1 and 4.7 eV (see Section 2). Assuming an activation energy of 4.5 eV and the theoretical estimate of 3.17 eV for vacancy formation, a vacancy migration enthalpy of 1.3 eV is obtained, which is in good agreement with the RESD experiments but at variance with the properties of vacancies at low temperatures. Although theory and experiment provide similar values for the formation energy of vacancies, the sum of the formation and migration enthalpy does not equal the experimental activation enthalpy of self-diffusion by vacancies. This may suggest that the calculated migration energy does not represent the migration energy at high temperatures. Assuming the low- and high-temperature properties of vacancies to be true, the entropy and enthalpy of vacancy migration must increase with increasing temperature. This interpretation satisfies the concept of a spread-out defect first proposed by Seeger and Chik already in 1968 [34]. More sophisticated theoretical calculations are still required to understand the apparent different mobilities of vacancies at low and high temperatures. On the other hand, additional experiments on RESD in extrinsically doped Si isotope structures can provide information about the dependence of vacancy migration on the charge state. Such experiments are presently in progress. 4. Impact of dopant diffusion on self-diffusion Isotopically controlled heterostructures are well suited to investigate the impact of dopant diffusion on self-diffusion.

15

For these experiments, we used isotope multilayer structures that allow the measurement of depth-dependent selfdiffusion, which results from the diffusion of a dopant into the isotope structure. The incorporation of dopants to concentrations that exceed the intrinsic carrier concentrations makes the material extrinsic. As a consequence, the position of the Fermi level shifts, leading to a change in the thermal equilibrium concentration of charged native defects [35,36]. Simultaneous dopant and self-diffusion experiments were performed in Si stable isotope heterostructures consisting of five alternating pairs of 28Si/natSi. The dopants were introduced via implantation into an amorphous Si cap layer, thereby preventing any implantation damage from altering the equilibrium native defect concentrations in the isotope structure. The dopants B, As, and P were investigated to determine the native defects and defect charge states responsible for diffusion in Si under extrinsic p- (B) and n- (As, P) type doping conditions [7–10]. Fig. 5 illustrates typical B, As, and P profiles in the isotope multilayer structure after annealing at 950 1C. The broadening of the isotope structure expected in the case when no dopant penetrates into the structure is indicated by the dashed line in Fig. 5(a) and (b). This clearly indicates that self-diffusion is affected by the incorporation and diffusion of the dopant. In our former analyses of the simultaneous self- and dopant diffusion, just one dopant and the corresponding Si profile were considered [7–10]. The new, more comprehensive analysis is based on the energy levels of vacancies and self-interstitials. These energy level positions were used as fitting parameters for accurate modeling of all dopant and Si profiles obtained after diffusion annealing at the same temperature. Moreover, the following constraints were considered for the calculations: The contributions of the various charged native point defects to self-diffusion for intrinsic conditions must add up to the total self-diffusion coefficient given in Table 1. The sum of the contributions of self-interstitials to self-diffusion for intrinsic conditions must equal the total contribution determined from Zn diffusion studies [20]. Expressions given by Thurmond [37] were taken into account for the temperature dependence of the Si band gap Eg and the Fermi level E in f under intrinsic conditions. Data of Morin and Maita [38] were considered for the temperature dependence of the intrinsic carrier concentration ni. Correlation factors of xV ¼ 0:5 and xI ¼ 0:56 (0.73) were used for self-diffusion by vacancies and splitinterstitials (tetrahedral interstitials). The equilibrium concentrations C eq X of the native point defects are not critical for modeling. Values were chosen for intrinsic conditions which are in accord with the data of C eq I and C eq obtained from Zn diffusion [20] and the RESD [6] V experiments, respectively. A flow chart of the diffusion simulation is illustrated in Fig. 6. In addition to the energy levels of the native point defects, the dopant diffusion coefficients DB;As;P for intrinsic conditions were set as fitting parameters. The kick-out and dissociative diffusion

ARTICLE IN PRESS H. Bracht / Physica B 376–377 (2006) 11–18

16

1022

1022 t=57420 sec

t=863520 sec

30

Si concentration (cm-3)

concentration (cm-3)

1021

1020

1019

30

Si

1021

1020

1019

As

B 1018

1018 0

600

1200

1800

(a)

0

600

depth (10 cm)

1200

1800

-7 depth (10 cm)

(b)

-7

1022

concentration (cm-3)

t=86400 sec

30

Si

1021

1020

1019

1018

P

0

600

1200

1800

-7 depth (10 cm)

(c)

Fig. 5. Concentration versus depth profiles measured with SIMS for B (a), As (b), and P (c) implanted Si isotope heterostructures after annealing at 950 1C for the times indicated. The thin dashed lines in (a) and (b) illustrates the broadening of the isotope structure which is expected under thermal equilibrium and electronically intrinsic conditions, i.e., when no dopant diffuses into the isotope structure. For clarity, only a reduced number of experimental data are plotted. The solid lines illustrate successful modeling of the dopant and Si profiles on the basis of a single set of model parameters.

Energy levels of V,I * DB* , DP* , DAs

Simulation B, As, P, and Si profiles

T, ni, Efin eq CI DI (ni) DSi (ni) ξV ,ξI CXeq

Agreement with exp. profiles no

yes

Result: Energy level scheme, dopant diffusion coefficients Fig. 6. Schematic illustration of modeling the diffusion of dopants in the Si isotope heterostructures.

mechanisms were considered for B diffusion. Arsenic diffusion is assumed to take place by dopant–vacancy and dopant–interstitial pairs. For P diffusion, the kick-out and dissociative mechanisms and in addition, a contribution of dopant–vacancy pairs were taken into account. Various charge states were assumed for the point defects involved in the diffusion reactions. The diffusion model assumed in this work is similar to the integrated diffusion model proposed by Uematsu [39]. The advantage of the new diffusion simulation in comparison to the former modeling [7–10] is that all possible charge states of vacancies and self-interstitials are considered and their contributions to self-diffusion have to obey certain constraints (see above). The prevalence of a particular point defect under specific doping conditions is determined by the energy level of the defect [35]. In addition, dopant diffusion can lead to a super- and

ARTICLE IN PRESS H. Bracht / Physica B 376–377 (2006) 11–18

I

V EC

Eg(850˚C)=0.82 eV o

EV

17

o (+)

--

(+) ++

0.55…0.4 eV 0.50…0.4 eV

< 0.2 eV

… Eg(1100˚C)=0.72 eV (+) ++ o (+)

~ 0.4 eV 0.2…0.1 eV 0 eV

Fig. 7. Energy level positions of vacancies V and self-interstitials I within the band gap of Si for temperatures between 850 and 1100 1C. The range of the energy level position for a specific native point defect indicates the value for temperatures between 850 and 1100 1C.

undersaturation of native defects and hence can favor a particular point defect. The solid lines in Fig. 5 show the successful modeling of all dopant and Si profiles at 950 1C on the basis of a single set of model parameters. Similar profiles at other temperatures are also accurately described with slightly different energy levels. The energy level scheme of vacancies and self-interstitials for temperatures between 850 and 1100 1C is illustrated in Fig. 7. Accurate simulations are achieved with an energy level diagram that is essentially equal to that predicted by theory [14,40]. Two donor levels for self-interstitials and two acceptor levels for vacancies were considered. The positions of the two vacancy acceptor levels are close to each other in order to reproduce the experimental profiles. The simultaneous B and Si diffusion profiles are better described with a reverse level ordering (negative-U) for the donor levels of the selfinterstitials than with the regular level ordering. The experiments are not sensitive to the vacancy donor levels, because the diffusion of the p-type dopant B establishes a supersaturation of self-interstitials, and thereby suppresses the contribution of vacancies to self-diffusion. However, an upper bound of 0.2 eV above the valence band can be deduced for the vacancy levels. This is in agreement with the vacancy donor levels E ðþÞ=þþ ¼ 0:13 eV and E 0=ðþÞ ¼ 0:05 eV measured by Watkins [25,26]. Modeling also provides the dopant diffusion coefficient DB;As;P . The results for intrinsic conditions are consistent with the B, As, and P diffusion data reported by Antoniadis [41], Masters [42], and Makris and Masters [43], respectively. These results and the individual contributions of the native point defects to self-diffusion as well as more details about the mathematical description of modeling self- and dopant diffusion in Si isotope structures will be published elsewhere. 5. Conclusion and outlook Isotopically controlled heterostructures are highly suitable test structures for studying the mechanism of atomic transport and the properties of point defects in solids. Accurate data were obtained for Si self-diffusion under thermal equilibrium which is known to be mediated both by vacancies and self-interstitials. However, the activation enthalpy of self-diffusion by vacancies is still not yet accurately known. Values determined for this quantity

range between 4.1 and 4.7 eV depending on the diffusion correlation factor of self-interstitials. Theoretical calculations indicate that the stable configuration of selfinterstitials changes with the Fermi level [14]. In particular, under p-type doping, positively charged tetrahedral selfinterstitials are favored. With the stable defect configuration also, the preferred migration path and, thereby, the corresponding diffusion correlation factor can change. Additional molecular dynamic simulations like those performed by Posselt et al. [18] are required to determine the correlation factors for the migration of the various interstitial defects. This will certainly improve our present understanding on defect migration and self-diffusion in Si. The experiments on radiation-mediated self-diffusion demonstrate that RESD in conjunction with numerical simulations is a powerful approach to determine the thermodynamic properties of native point defects. In the past, neither the epitaxial growth of isotope structures nor a detailed modeling was possible. These limitations no longer exist and hence open up advanced studies on radiation-enhanced diffusion in materials. The RESD experiments with Si isotope structures demonstrate that vacancies are less mobile at high temperatures than expected from the low-temperature results. This suggests that the structure of the vacancy changes with temperature. Additional more sophisticated theoretical calculation of the vacancy migration enthalpy must be performed to understand the different behaviors of vacancies at low and high temperatures. The simultaneous self- and dopant diffusion in Si is accurately described with the stable native point defects predicted by theoretical calculation. The derived energy level diagram is consistent with theoretical and spectroscopic results, and shows that the enhancement of Si selfdiffusion under both p- and n-type doping conditions is due to positively charged self-interstitials and negatively charged vacancies, respectively. This doping dependence of self-diffusion and the type of the point defect involved is confirmed by recent diffusion studies under inert and oxidizing ambient with homogeneously doped Si isotope samples [44–46]. The self- and dopant diffusion experiments with isotopically enriched semiconductors show that the study of atomic transport processes can provide similar information about point defects at high temperatures that

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spectroscopic methods do for low temperatures. Diffusion experiments with isotope heterostructures can be also performed with any other material system for which enriched stable isotopes are available. Future work will concern diffusion studies in SiGe to determine the impact of the Ge content and strain on self- and dopant diffusion in this alloy.

[12] [13] [14] [15] [16] [17] [18] [19]

Acknowledgments I am very grateful to my colleagues from the Lawrence Berkeley National Laboratory (J.W. Beeman, E.E. Haller, I.D. Sharp, H.H. Silvestri), from the University of Aarhus (J. Fage Pedersen, J.L. Hansen, A. Nylandsted Larsen), and from the Forschungszentrum Rossendorf (M. Posselt) for the numerous contributions to the research reviewed here. The work has been supported in part by the Alexander von Humboldt Foundation and the Deutsche Forschungsgemeinschaft.

[22] [23] [24] [25]

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