Modeling of boron diffusion in silicon–germanium alloys using Kinetic Monte Carlo

Solid-State Electronics 93 (2014) 61–65

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Modeling of boron diffusion in silicon–germanium alloys using Kinetic Monte Carlo Ignacio Dopico a,⇑, Pedro Castrillo b, Ignacio Martin-Bragado a a b

Atomistic Materials Modeling group, IMDEA Materials Institute, Eric Kandel 2, Getafe, Madrid, Spain UCAM, Universidad Católica de Murcia, Campus de los Jerónimos, Guadalupe, Murcia, Spain

a r t i c l e

i n f o

Article history: Received 22 July 2013 Received in revised form 25 November 2013 Accepted 4 December 2013 Available online 21 January 2014 The review of this paper was arranged by Prof. S. Cristoloveanu

a b s t r a c t We present an accurate atomistic physically based Kinetic Monte Carlo model for binary alloys. The model takes into account the different formation and migration energies and prefactors for both point defects and dopants due to the varying alloy composition, and it also accounts for the energy barrier the defect have to surpass in order to diffuse across the different composition regions. Model, parameters and implementation validation with several experimental results are shown. Finally, discussion of some simulation divergences and coincidences between particular experiments and our simulations are reported. Ó 2013 Elsevier Ltd. All rights reserved.

Keywords: Silicon germanium Object Kinetic Monte Carlo Atomistic modeling Segregation

1. Introduction

2. Models

Boron diffusion in SiGe has become of great interest since strained thin SiGe layers heavily doped with boron serve as a very good infrastructure in silicon-based heterojunction bipolar transistors [1]. It is well known that in silicon boron diffuses solely by interstitials [2]. Although the diffusion mechanism of boron in SiGe is not as well known as in pure silicon, it has been suggested to be also driven by interstitials [3], and its effective diffusion has been found to be lower in SiGe than in pure Si, for Ge content in the range from 0% to 40% [4,5]. For a better understanding of this phenomena a physicallybased model for binary alloys that accounts for defect diffusion, dopant diffusion and segregation has been developed based on a previous work [6]. The model has been implemented in the Object Kinetic Monte Carlo simulator (OKMC) MMonCa [7] and has been validated through comparison with selected experiments [6,8]. Finally, it has been used to elucidate whether the use of thin SiGe layers can affect the behavior in the silicon bulk, as has been suggested in the literature [9].

For a homogeneous Si1x Gex alloy in equilibrium conditions, the concentration of point-defects, C A , is given by [6]:

  EfA ; C A ¼ C 0A exp  kB T

ð1Þ

where C 0A and EfA are the concentration prefactor and the formation energy, respectively, which are composition dependent, whereas, kB T is the usual thermal energy. Also in homogeneous materials, A moves randomly with a thermally-activated migration frequency mA as [6]:



mA ¼ m0A exp 

 EmA ; kB T

ð2Þ

with EmA and m0A being the migration energy and prefactor, respectively, which are also composition dependent. The diffusivity DA is linked to its migration frequency by [6]:

DA ¼

  k2 mA EmA ; ¼ D0A exp  6 kB T

ð3Þ

where k is the distance length, and D0A is the diffusivity prefactor. The transport capacity of a point defect A (DC A ) is defined as: ⇑ Corresponding author. Tel.: +34 91 549 34 22x1036; fax: +34 91 550 30 47. E-mail addresses: [email protected] (I. Dopico), [email protected] (P. Castrillo), [email protected] (I. Martin-Bragado). 0038-1101/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.sse.2013.12.007

DC A ¼ DA

CA ; C at

ð4Þ

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being DA and C A , the diffusivity and concentration of A respectively, and C at the atom density of the lattice. In our model for SiGe alloys, only one average type of point defect A is considered. For instance, different microscopic configurations for the interstitial, such as ‘‘silicon in interstitial position’’ or ‘‘germanium in interstitial position’’, are dismissed and only an effective configuration for interstitials and vacancies is taken into account by an overall transport capacity DC I (for interstitials) or DC V (for vacancies) [6]. It is worthy to note that the values reported in the literature for DI and C I (or for DV and C V ) are very scattered, whereas the disagreement for the transport capacities DC A is much lower [10]. This is believed to be because the experimental values of C A , and DA , are distorted by the effect of traps [11]. In contrast, the diffusion fluxes, and thus the transport capacities DC A , hardly depend on the defect trap densities, excepting for the initial transient time. In this way, DC A is the relevant magnitude for point defect diffusion in regular process conditions. This means that different values of DA ; C A yielding the same DC A would produce the same result in common annealing experiments [11]. In contrast, the individual values for DA or C A are hardly accessible (and hardly important) in experiments, excepting very particular transient conditions such as those involved in cryogenic anneals [12]. Thus, the model described in this work relies on the transport capacity concept and it is valid for usual process conditions. As a consequence, model parameters will be selected to fit the experimental values of DC A , whereas the splitting of the product DA  C A will be done on the base on theoretical calculations. The model considers the effect of biaxial stress for SiGe alloys pseudomorphically grown on a silicon substrate. This effect has been accounted as a linear modification of the formation energies with stress, which, in turn, depends linearly with composition. Thus, strain effect is superimposed to the chemical composition effect. In this work, we will deal with intrinsic material, in which Fermi level effects are irrelevant [13]. Therefore, we neglect charge effects and we use only neutral defects with a parameterization mimicking the join effect of all possible charge states in intrinsic conditions. At the interface of two homogeneous material regions (i.e. SiGe/ Si) the diffusing particle moving from one to another will face an extra energy barrier, being able to surpass it with a frequency given by [6]:

mA;1!2 ¼

DC A;2 mA;1 ; DC A;1

ð5Þ

where DC A;1 and DC A;2 are the transport capacities for homogeneous materials 1 and 2 respectively, and mA;1 the jump frequency in material 1. Fig. 1 sketches the barrier that atoms face when changing to different composition regions, including the different formation and migration energies for the atom. This barrier translates into a probability for the atom movement between regions to be rejected, staying the particle in region 1. If DC A;2 P DC A;1 , the jump is always accepted. The activation energy of the ratio DC A;2 =DC A;1 becomes the extra energy barrier DEA ¼ EaA;2  EaA;1 , and the prefactor becomes the ratio of transport capacity prefactors. As it has been already pointed out, dopant diffusion in silicon is assisted by point defects. Considering a dopant X, and a point defect A, the dopant defects can migrate in the XA states [10]. These dopant defects are formed by the reactions:

A þ X $ XA; 0

A þ A X $ X;

ð6aÞ ð6bÞ

being A0 an interstitial when A is a vacancy an so on. Dissociative mechanisms (Eq. (6b)) are not considered since it is not believed

Material 1

Material 2 EmA,2

ΔEA P1→2

EaA,2

EmA,1 EaA,1

Fig. 1. Energy scheme for a point defect A jumping from material 1 to material 2, with a probability P 1!2 to surmount an energy barrier, DEA ¼ EaA;2  EaA;1 due to its different transport capacities in each material.

that these mechanisms are significant in the case of boron diffusion in SiGe since the process B ! Bi þ V has activation energy higher than 7 eV and, therefore, its rate is negligible and it is not considered in our simulations. The concentration, C XA is controlled by the kick-out/break-up mechanisms (Eq. (6a)):

C XA ¼

CX

mbk;XA

C A mmA Xcap ;

ð7Þ

with mbk;XA as the break-up rate of the XA pair, mmA the migration frequency of A, and Xcap the capture volume. Similarly as in the case of the point defect, the transport capacity of a dopant in a XA state (DXA ) is defined as:

DC XA ¼ DXA

C XA ; C at

ð8Þ

being DXA the diffusivity of the defect in the XA state, that can be obtained from Eqs. (2) and (3). The effective diffusivity of the particle X (e.g. boron) driven by A (e.g. interstitials) is given by:

DX ¼ DXA

C AX : CX

ð9Þ

As in the case of the point defect moving through two homogeneous regions with different composition, the defect in the XA state needs to surpass an extra energy barrier DEXA ¼ EaXA;2  EaXA;1 , reaching the desired energy with a frequency given by:

mAX;1!2 ¼

DC AX;2 mAX;1 DC AX;1

ð10Þ

The atomistic implementation of Eqs. (5) and (10) considers that the particle is actually migrating in the interface direction and when it is randomly selected to jump from the material region 1 to the material region 2 the probability for the atom to perform the jump is:

P1!2 ¼ A

DC A;2 : DC A;1

ð11Þ

If the jump is rejected the particle stays in material region 1. 3. Validation In order to calibrate and proof the reliability of our model and implementation, a set of simulations reproducing several experiments has been carried out. The parameter values of the model for Si1x Gex alloys are listed in Table 1.

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Interstitial

Boron

Si1x Gex strained to Si (for 0 6 x [ 0.4) ¼ 920  10x cm3

C 0I ¼ 6  1026  4x cm3

C 0Bi C 0B

D0I ¼ 5  102 cm2/s EfI ¼ 4 þ 0:2x eV EmI ¼ 0:8 eV

D0Bi ¼ 5  103  10x cm2/s EfBi  EfB ¼ 2:9 þ 0:8x eV EmBi ¼ 0:77 þ 0:5x eV EfB ¼ 0:5x eV

10-11 10

Hill Folmer Goshtagore Mathiot Okamura Plummer Schnabel Fair Simulated

-12

10-13

Diffusivity (cm2/s)

Table 1 Parameters used for native point defect diffusion and for Boron diffusion for intrinsic Si1x Gex alloys strained to Silicon in the Ge composition content from x = 0 to x = 0.4. Composition dependence involves both chemical effects and strain contributions. For x = 0, parameters correspond to unstrained silicon and pure germanium.

10

-14

10-15 10-16 10

-17

10-18 10-19 10

-20

10

-21

6.5

2

D¼

hjr  r0 j i ; 6Dt

ð12Þ

where r 0 and r are the initial and final position respectively of the diffusing atoms in the time interval Dt. Fig. 3 is a comparison between simulated results and experimental measurements [21] where boron diffuses in a doped

7

7.5

8

8.5

9

9.5

10

10.5

11

11.5

104/T (K) Fig. 2. Comparison of boron diffusivity in intrinsic pure silicon for experimental (lines) [14–20] and simulated (squares) results at different temperatures.

B as-grown [SIMS] B as-grown [Simulated] B [Simulated] B [SIMS]

4*1018

18

3*10

Concentration (cm-3)

For pure silicon (x = 0), the migration energy and diffusion prefactor of interstitials (named EmI and Dm0I , respectively) were taken from theoretical calculations [22]. The formation energy, EfI , and the concentration prefactor, C 0I , were chosen to fit the transport capacity DC I in silicon under equilibrium conditions [23]. Boron diffusivity is controlled by the parameters related to boron interstitial (Bi). The migration energy of Bi (EmBi ) in pure silicon is taken from Ref. [24] whereas the other parameters (diffusion prefactor, D0Bi , relative concentration prefactor, C 0Bi =C 0B , and formation energy enhancement, EfBi  EfB ) are taken to fit the experimental B diffusivities in Si [14–20]. For Si1x Gex , chemical composition effect is included as a potential dependence in prefactors and as a linear dependence in energies, whereas strain effect is assumed to affect only energies [6]. This assumption seems to be valid for interstitials in the whole composition range but valid for boron only for x < 0.4 [25]. In particular, the 0.2  x eV dependence of rmEfI in Table 1 is the joint result of a chemical contribution of 0.3  x eV (yielding a 3.7 eV for pure unstrained Ge) [6,26] and a strain contribution of +0.5  x eV (which is consistent to an effective volume of 0.6 atoms) [6]. Prefactor dependence is fitted to reproduce experimental Si-Ge interdiffusion results in tensile SiGe heterostructures [6,27]. The Ge concentration dependence of boron diffusivity in unstrained Si1x Gex with x < 0.4 is adjusted to experiments of Ref. [25]. Data corresponding to Si1x Gex strained to silicon [5,28] are consistent to an effective volume of 0.35 atoms for boron diffusion. This gives an additional activation energy of 0.3  x eV, which is included in the 0.8  x eV variation of EfBi  EfB in Table 1. The formation energy of boron in Si1x Gex (EfB = 0.5  x eV in Table 1) accounts for the boron segregation from Si to Si1x Gex , taking EfB ¼ 0 in pure silicon as a reference. Simulated boron diffusion in pure silicon for a wide range of temperatures, from 800 K to 1400 K, is compared with experimental results [14–20] in Fig. 2. Values agree within the range of experiments for low temperatures, and both calculated and measured data converge for higher temperatures. Anneals are performed for long times (reaching pseudo-equilibrium) in 50 K intervals. Anneals performed at high temperatures were 6 h and 36 min for 1273 K, 1 h and 24 min for 1323 K, 20 min for 1373 K and 5 min for 1432 K. As temperature increases more computational time is required in order to reach the same simulation time. For this reason shorter simulation times has been chosen for higher temperature simulations. In simulations of Fig. 2 the diffusivity is evaluated using Einstein’s relation:

2*1018

18

10

0

100

200

300

400

500

Depth (nm) Fig. 3. Experimental [21] (squares) and simulated (solid line) B profiles of an 1123 K 96 h, annealed B box profile between two 30 nm thick Si0:9 Ge0:1 layers, situated at 130 nm and 420 nm from the surface. Vertical dashed lines indicate the position and thickness of the SiGe layers.

epitaxial silicon layer surrounding Si90 Ge10 layers for 96 h at 1123 K. The figure shows how the boron piles up into SiGe layers due to the migration barriers and different formation energies between pure Si and the alloy. The good agreement validates both the implementation and the parametrization used by our model. 4. Results The validated model has been used to compare and understand the results of a specific set of published experiments [9], that are shown below, finding a divergence between one of the experimental measures and the simulated results, that will be discussed in Section 5. Fig. 4 shows the simulated results and experimental measures for the experiment of three boron spikes diffusing in silicon below a 45 nm Si0:75 Ge0:25 cap layer, after annealing for 2028 min in N2 at

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I. Dopico et al. / Solid-State Electronics 93 (2014) 61–65 1020

20

10 B annealed [Simulated] I*1010 [Simulated]

1019

1018

1017

1016 0

100

200

300

400

500

B annealed [Simulated] 10 I*10 [Simulated]

B as-grown [Carroll] B annealed [Carroll] B as-grown [Simulated]

Concentration (cm-3)

Concentration (cm-3)

B as-grown [Carroll] B annealed [Carroll] B as-grown [Simulated]

19

10

1018

1017

1016

600

0

100

200

Depth (nm)

300

400

500

600

Depth (nm)

Fig. 4. Comparison of experimental B profile measured by SIMS [9] (squares) with simulated one (solid line). Boron spikes have been grown on a Si substrate with a 45 nm Si0:75 Ge0:25 cap layer on top, and annealed for 2028 min in N2 at 1088 K. The simulated concentration of interstitials is also shown (dashed line).

Fig. 6. Comparison of experimental B profile [9] (squares) with simulated one (solid line). Boron spikes have been grown on a Si substrate without cap layers, and annealed for 2028 min in N2 at 1088 K. The concentration of interstitials calculated in the simulation is also shown (dashed line).

1088 K. The agreement with the simulations is noticeable indicating a good calibration of boron diffusivity and the correct computation of interstitial concentrations in the whole simulation domain. In particular, it is interesting to notice how the presence of the cap layer locally modifies the interstitial concentration (due to a different formation energy for interstitials in SiGe), reverting back to the regular interstitial concentration in bulk silicon below the cap layer. In this way, the observed boron diffusion in the buried silicon region would not be influenced by the SiGe cap layer. In fact, the experimental boron diffused profiles in Fig. 4 (and Fig. 5) correspond to boron diffusivity values for bulk silicon (Fig. 2). Fig. 5 shows a similar comparison, also with an excellent agreement, for a Si0:55 Ge0:45 5 nm cap layer. Fig. 6 compares simulation results to SIMS [9] data of three boron spikes diffusing in pure silicon without any layer, after annealing for 2028 min in N2 at 1088 K. The important disagreement between simulated and experimental results indicates a higher diffusion of boron in silicon bulk in the experimental case than in the simulated one: in the simulations boron diffuses similarly in all cases, and it seems to be independent of the existence or not of a surface SiGe layer. This disagreement is further discussed in the following Section 5.

5. Discussion The atomistic model and parameter calibration shown previously for dopant diffusion in both pure material and binary alloys, and specifically for boron diffusion in silicon and silicon germanium alloy, has proven to reproduce the experimental results [14–21] under similar annealing conditions. The agreement of experimental results with simulated ones for the cases with SiGe cap layer shows that boron diffuses with its regular diffusivity, indirectly proving that the concentration of interstitials is the one for bulk silicon: in silicon the boron diffusion is dominated by the kick-out mechanisms with an interstitial [2], B þ I $ Bi , so the concentration of self-interstitials in the bulk, where the boron is, will determine the diffusion of boron independently of the existence of a surface layer, as seen in Figs. 4 and 5. If we notice that boron diffusion is the same, this is because of the same silicon self-interstitials concentration in bulk, despite the difference of self-interstitial in the first 45 nm below the surface where the SiGe layer lies. Notice that the unexpected high diffusion in Fig. 6 is for pure silicon, and it implies boron diffusivity at 1088 K clearly higher than the upper bond of the commonly accepted DB values in the litera-

20

10

20

B annealed [Simulated] 10 I*10 [Simulated]

10

B annealed [Simulated] I*1010 [Simulated]

B as-grown [Carroll] B annealed [Carroll] B as-grown [Simulated]

1019

Concentration (cm-3)

Concentration (cm-3)

B as-grown [Carroll] B annealed [Carroll] B as-grown [Simulated]

18

10

17

10

19

10

1018

1017

16

10

0

100

200

300

400

500

600

Depth (nm)

1016 0

100

200

300

400

500

600

Depth (nm) Fig. 5. Comparison of experimental B profile [9] (squares) with simulated one (solid line). Boron spikes have been grown on a Si substrate with a 5 nm Si0:55 Ge0:45 cap layer on top, and annealed for 2028 min in N2 at 1088 K. The simulated concentration of interstitials is also shown (dashed line).

Fig. 7. Comparison of experimental B profile [9] (squares) with simulated one (solid line) under the same conditions of Fig. 6 but including an extra interstitial supersaturation in the simulation (dashed line).

I. Dopico et al. / Solid-State Electronics 93 (2014) 61–65

ture [14–20] (see Fig. 2). If these values are accepted, the disagreement in Fig. 6 has nothing to do with SiGe cap layers but to an anomalously high boron diffusivity in silicon in this particular sample, which can be related to an interstitial supersaturation of a factor 3 as shown in Fig. 7. The origin of this supersaturation would be unknown, but could be related to an excess of native defects due to non-optimized epitaxial growth. As it can be seen in the figures, the initial boron spikes in Fig. 6 are clearly wider and with longer tails than those in Figs. 4 and 5, indicating differences in the growth process which would be consistent to the presence of an excess defect concentration.

6. Conclusion In this work he have presented a comprehensive model to simulate the evolution of point-defects and dopants in SiGe. We have explained the correction introduced by the strain, and intrinsic material conditions. The model is able to simulate diffusion, transient enhanced diffusion and segregation of dopants at Si/SiGe interfaces using Object Kinetic Monte Carlo techniques. These simulations have proven to be useful in order to understand and verify some experimental results from the literature while at the same time providing information that, contrary to an experimental observation, let us conclude that SiGe cap layers do not seem to play a significant role reducing interstitial concentration, and then boron diffusion, in bulk Si. Acknowledgment This work was partly funded by the European Community 7th Framework Marie Curie Actions Grant FP7_PEOPLE-2011-CIG under Grant Agreement 293783 (MASTIC). References [1] Patton GL, Iyer SS, Delage S, Tiwari S, Stork J. Silicon–germanium base heterojunction bipolar transistors by molecular beam epitaxy. IEEE Electr Device Lett 1988;9(4):165–7. [2] Fahey PM, Griffin P, Plummer J. Point defects and dopant diffusion in silicon. Rev Mod Phys 1989;61(2):289. [3] Uppal S, Willoughby AF, Bonar JM, Cowern NE, Grasby T, Morris RJ, et al. Diffusion of boron in germanium at 800–900 C. J Appl Phys 2004;96(3):1376–80. [4] Kuo P, Hoyt J, Gibbons J, Turner J, Lefforge D. Effects of strain on boron diffusion in Si and SiGe. Appl Phys Lett 1995;66:580. [5] Moriya N, Feldman L, Luftman H, King C, Bevk J, Freer B. Boron diffusion in strained Si1x Gex epitaxial layers. Phys Rev Lett 1993;71(6):883–6.

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