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Modelling Simul. Mater. Sci. Eng. 4 (1996) 323–333. Printed in the UK

Quantum chemical investigations of atomic hydrogen effect on Frenkel pairs annihilation in silicon A N Nazarov†, V M Pinchuk‡, V S Lysenko† and T V Yanchuk‡ † Institute of Semiconductor Physics, National Academy of Sciences of Ukraine, Prospect Nauki 45, 252028 Kiev, Ukraine ‡ Department of General and Theoretical Physics, Kiev Polytechnical Institute, Prospect Pobedy 37, 252057 Kiev, Ukraine

Received 22 October 1994, accepted for publication 4 April 1996

Abstract. The SCF MO LCAO method in the valence approach with neglect of diatomic differential overlap (NDDO) is used to study the effect of atomic hydrogen on the silicon lattice relaxation in the nearest vicinity of a vacancy. It is shown that hydrogen atoms are localized mainly as second-nearest neighbours of the vacancies on the Si–Si bond, which results in a significant extension of the vacancy region. The potential barrier height and its dependence on the vacancy charge state were calculated for Frenkel pair annihilation with a hydrogenated vacancy in the cases of hydrogen localization inside and outside the vacancy. The results substantiate a model of enhanced annihilation of Frenkel pairs in hydrogenated crystalline Si.

1. Introduction Atomic hydrogen—the simplest elementary substance—is soluble in crystals and has high diffusivity even at low temperatures [1]. Introduction of even a small amount of hydrogen into a crystal can substantially affect its structure. Therefore, when studying the changes caused by hydrogen presence in a certain material, one should take into account not only the effect of hydrogen by itself, but also the possibility of structural transformations of the crystal lattice and the resulting modifications of its properties [2]. It was the account of crystal lattice relaxation that allowed the authors of [3, 4] to calculate the equilibrium position of hydrogen atoms in silicon on the Si–Si bond, which was validated by experimental results [5]. Hydrogen-induced relaxation of the crystal lattice can significantly affect the spatial structure and charge state of defects and impurity atoms in silicon [6]. There are two main types of hydrogen interaction with defects and impurities in semiconductors. The first type includes the processes of ‘direct’ electrical neutralization of defects and impurities due to direct chemical bonding with hydrogen. In this case, heating sooner or later breaks the bonds and recovers the properties of defects and impurities [1, 7]. In processes of the second type, the role of hydrogen is indirect; it acts as a stimulant and mediator in the reactions. Examples of such processes are: enhanced activation of implanted impurities in silicon [8, 9]; low-temperature annealing of radiation defects in hydrogenated silicon [10]; lowtemperature annealing by RF plasma treatment [11]; enhanced formation of thermodonors caused by hydrogen plasma treatment [12]. In these cases, heating does not lead to recovery of the initial charge state of the defects or impurity atoms. c 1996 IOP Publishing Ltd 0965-0393/96/030323+11$19.50

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Details of the second type of hydrogen interactions with defects and impurities are yet to be elucidated. A model proposed in [9, 13] accounts for a decrease in the temperature of vacancy defect annealing and doping impurity activation in silicon in observed in the presence of hydrogen. The main point of this model is reduction of lattice deformation in the vicinity of a vacancy due to hydrogen trapping, resulting in a lower potential barrier for interactions between the interstitial atom and defect. Thus, the authors of [9, 13] consider relaxation of the crystal lattice to be the most important moment in the process. To verify this model, we calculated the potential barrier height for diffusion of the interstitial silicon atom in the vicinity of a hydrogenated vacancy and estimated the possibility of enhanced annihilation of the hydrogenated vacancy and interstitial, taking into account relaxation of the crystal lattice and possible presence of the hydrogen in different charge states. 2. Calculation technique In this work, we used the SCF MO LCAO technique in the NDDO valence approach [14]. The calculations were performed with the help of CLUSTER-Z2 software [15] using the spd-basis. Using this software and new parameters of silicon [16], we arrived at a structure ˚ which is quite close of a silicon cluster with the equilibrium distance R(Si–Si) = 2.37 A, ˚ [17]). to the experimental values of this parameter for the crystal (Rexp (Si–Si) = 2.35 A The NDDO technique takes into account the three- and four-centre integrals representing the overlap of atomic orbitals of the same atom. In comparison with the well known MINDO/3 method, the NDDO technique takes account of the spatial direction of the p-orbitals in calculations of the two-electron integrals, thus allowing a more correct description of the repulsion of unshared orbitals. An important advantage of NDDO over MINDO/3 is that the former technique involves only the parameters of individual atoms, whereas the latter requires knowledge of parameters of both individual atoms and their paired combinations. When simulating the structure of silicon, the technique of univalent pseudo-atoms was used to passivate the extraneous valence bonds at the outer boundary of clusters containing 52 silicon atoms, so that [Si52 H35 ] clusters were formed. In our paper [18], an effective technique was proposed for fitting the quantum-chemical parameters of boundary pseudoatoms to reproduce certain characteristics of the simulated solid. This technique involved an approach based on elimination of the charge-distribution asymmetry in the extended clusters by modification of the geometric position of boundary pseudo-atoms. A similar approach was used also in [19]. This approach takes into account the relation between the radius of pseudo-atoms (r) and their electronegativity (χ). Following [20], we selected as a measure of electronegativity the force of attraction by the atom of a valent electron of its partner (i.e. the relation between χ and r is: χ = C/r 2 + D, where C and D are constants). It is evident that for substitution in the cluster of a hydrogen atom for a less electronegative atom of silicon (let us name such an H atom a pseudo-atom of silicon and denote it as Si∗ ), χSi∗ reaches the value of χSi when rSi > rH . A specific feature of the procedure of searching for the equilibrium geometry of a molecular system containing pseudo-atoms by variable metric technique is the calculation of the derivatives of the total energy (E) with respect to the internal coordinates (qi ) in such a way that in the expression 3N X ∂E ∂E ∂Xj = ∂qi ∂Xj ∂qi j =1

(1)

Atomic hydrogen effect on Frenkel pairs in silicon

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∂E/∂Xj is found analytically, and ∂Xj /∂qi is calculated numerically. In expression (1), Xj is the Cartesian coordinate of the j th atom and N is the number of atoms. Such a technique improved the efficiency of minimization of the energy functional in comparison with the standard procedures based on numerical differentiation of E with respect to the internal coordinates of the atoms (and subsequent optimization of the geometry in Cartesian coordinates). To optimize the spatial structure and distribution of the electron density in a cluster of ideal crystals, vacancy-containing crystals, and crystals containing hydrogen–vacancy complexes, we used the technique of direct search for the total energy of the functional, which gave good results for calculations of spatial and electronic structure of the clusters containing atoms of Si and 3d transitional metals [21, 22]. The issue of searching for minimum energy of the system is of crucial importance for our problem, and so it is appropriate to compare the technique used by us with those reported in the literature. The technique for approximate calculation of the optimum configurations of molecules described in [23, 24] are simpler but involve rather rough assumptions. For example, it was assumed in [23] that the transition from the initial configuration to the equilibrium one does not modify the density matrix. The authors of [24] used a sequential variation of the valent angles first, and then the interatomic distances. Generally, such approximate techniques cannot ensure detection of even a local minimum, whereas the procedure of direct search used by us, even if more laborious, is always capable of finding the minimum. 3. Results and discussion 3.1. Localization of hydrogen at a vacancy When analysing the effect of hydrogen atoms on the diffusion of silicon, we used a cluster of 52 silicon atoms with a perturbed geometry, such that an atom of silicon was moved to the interstitial site of the adjacent unit cell (thus creating a vacancy), and four hydrogen atoms were subsequently placed at various sites near the vacancy. Our calculations were based on optimization of the energy functional and involved variations of all silicon and hydrogen atoms. It was found that the favoured positions for the four hydrogen atoms are on the Si–Si bonds between the silicon atoms having dangling bonds and the atoms of the next row, so that the vacancy cavity remains empty (figure 1(a)). Since this configuration of hydrogen at a vacancy is different from the previously reported one [1, 25–27], we have performed a thorough comparison of Si–H bonding energies for different localizations of hydrogen atoms; also, we calculated the parameters of symmetrical Si–H oscillations for the localization obtained and compared them with those calculated in [27] and with experimental results [28–31]. At the first stage, hydrogen atoms were introduced into the inner region of the vacancy (figure 1(b)), which resulted in consecutive bonding of hydrogen to the dangling bonds and formation of structures VH, VH2 , VH3 and VH4 , having the symmetries C1h , C2v , C3v and Td respectively. Optimization of the equilibrium Si–H distances for the four hydrogen atoms in the inner region of the vacancy shows that the structure of the vacancy relaxes, so that the silicon atoms are displaced from their equilibrium (in the absence of hydrogen) ˚ in the direction corresponding to the extension of the vacancy. The positions by 0.20 A binding energies and equilibrium distances for the Si–H bond are presented in table 1 for each structure, along with the data reported in [27]. The results of the present work on the localization of hydrogen atoms at a vacancy are in good agreement with the data of other authors [25, 27].

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Figure 1. A hydrogenated vacancy in the silicon lattice: (a) hydrogen localized outside the vacancy (b) hydrogen localized inside the vacancy.

However, this energy minimum, corresponding to the positions of all the four hydrogen ˚ from the vacancy centre and equilibrium Si–H distance of atoms at a distance of 1.03 A ˚ is a local one. Further optimization of the energy functional, including variation 1.54 A, of all silicon and hydrogen atoms, shows that the hydrogen atoms move outward from the vacancy and rest on the Si–Si bonds between the nearest and second-nearest neighbours of the vacancy. In the case of the VH4 defect, hydrogen is localized on the Si–Si bond such ˚ and the Si13 –H distance that the Si–H–Si angle is 160◦ , the Si4 –H distance equals to 1.40 A ˚ (see inset in figure 4(a) later). It is seen from table 1 that these hydrogen equals 1.56 A atoms have higher binding energies than the hydrogen atoms localized inside the vacancy.


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Table 1. Calculated equilibrium distances R(Si–H) and binding energies 1E(Si–H) for different numbers of hydrogen atoms localized inside and outside the vacancy. ˚ R(Si–H) (A) Structure VH VH2 VH3 VH4

1E(Si–H) (eV)

˚ R(Si–H) (A)

inside the vacancy 1.512 1.529 1.534 1.540

(1.499) (1.489) (1.485) (1.490)

2.37 2.25 2.19 2.13

(2.50) (2.47) (2.41) (2.34)

1E(Si–H) (eV)

outside the vacancy 1.351 1.362 1.370 1.395

2.58 2.49 2.36 2.21

Figures in parentheses denote the results of calculations reported in [27].

Let us compare our data with the results of [25]. In that work, it was found after optimization of the geometrical parameters that hydrogen atoms move outward from the ˚ and are arranged within the vacancy with Td symmetry. This result is vacancy by 0.28 A consistent with our calculations for the first case, but it does not indicate any possibility of hydrogen localization between the nearest and second-nearest silicon neighbours of the vacancy. However, the authors of [25] used the cluster [(V + H4 )Si4 Si12 H36 ], which did not include the third and fourth layers of silicon around the vacancy and thus did not allow searching for the optimum configuration for hydrogen localization between the first and second layers. The most detailed results on localization of hydrogen atoms at the vacancy centre were obtained in [27] using a model of 32 silicon atoms forming two layers around the vacancy. Equilibrium distances and binding energies were calculated for hydrogen atoms located at the vacancy centre. These data are consistent with our results for the first case, corresponding to the hydrogen atoms positioned at the vacancy centre (see table 1). In [27], the possibility of hydrogen localization outside the vacancy was not suggested; presumably, the model contained an insufficient number of silicon atoms to allow studying this option. A calculation of the electronic structure of Si containing VH, VH2 , VH3 and VH4 defects using the Green functions technique was reported in [32]. The supercell contained a defect and 2662 lattice sites. However, the total energy of the system was not minimized, and ˚ from the vacancy centre, corresponding hydrogen atoms were compelled to rest at 0.90 A ˚ to a Si–H bond length of 1.45 A. Relaxation was allowed only for silicon atoms of the first layer, and so these results cannot give conclusive evidence about the favoured position of hydrogen atoms near the vacancy. To validate our model, we have calculated the oscillation parameters of Si–H bonds for hydrogen positions inside and outside the vacancy. Oscillations were calculated in the Si–H– Si structures along Si–H bond by taking account of the structure non-linearity. These results are shown in table 2 along with calculated [27] and experimental [28–31] data obtained by other authors. The Si–H oscillation frequencies calculated for hydrogen positions inside and outside the vacancy are close to each other; however, in case of the outside localization these frequencies are closer to the experimental results. The frequencies calculated in [27] are in good agreement with the experimental data only when using a normalizing coefficient. Without normalization, discrepancy with the experimental results is more significant for the frequencies calculated in [27] than for those obtained in the present work (see table 2). Therefore, analysis of the problem of hydrogen atom arrangement inside or outside the vacancy suggests a higher probability of hydrogen localization on the Si–Si bonds between the first and second layers of silicon atoms. Our further calculations are based mainly on this option of hydrogen localization. However, in some cases, calculations for both options

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A N Nazarov et al Table 2. Longitudinal oscillation frequencies a ω(Si–H), in cm−1 , calculated for hydrogen atom positions inside and outside the vacancy. Our calculations Structure

Calculations [27]

inside the vacancy

Our calculations outside the vacancy

Experimental results 2066 2065 2084 2083 2098 2107 2111

VH

2188

2218 (2057)

2127

VH2 VH3

2210 2246

2241 (2080) 2267 (2106)

2131 2140

VH4

2257

2270 (2109)

2144

[28, 29] [30, 31] [30, 31] [28] [30, 31] [28] [30, 31]

Figures in parentheses refer to the results obtained with the help of normalization. a Calculations were carried out for longitudinal oscillations along the Si–H bond.

are presented in order to prove that our model of enhanced annihilation of Frenkel pairs is valid for any position of hydrogen atoms. 3.2. Formation of a pair ‘vacancy–interstitial silicon’ Let us consider the stages of calculation of potential curves for interaction between the diffusing silicon and atoms of the crystal lattice. The first step in the calculation is related to vacancy formation in the absence of hydrogen. The diffusing silicon atom (Sid in figure 2(a)) moves from the unit cell centre perpendicularly to its surface and along its diagonal (crystal axes (100) and (110) accordingly). The calculated potential barrier height for a silicon atom leaving the unit cell centre is 1E(100) = 30.3 eV for a path along the (100) axis and 1E(110) = 31.2 eV for the (110) axis. The potential barrier height consists of Si–Si bond breaking energy of the silicon atom with its four neighbours, which equals 24.6 eV, and the strain energy of the Si lattice. The obtained potential barrier heights are close to those calculated in [33]. As the next step, the cluster geometry was optimized for the vacancy and an interstitial silicon atom Sid localized at the face of the second cell. The calculations showed a decrease ˚ towards in the vacancy volume, with each dangling-bond silicon atom shifted by 0.15 A the previous position of the diffusing silicon atom Sid . 3.3. Annihilation of a Frenkel pair The second step of calculations is associated with the return of the interstitial silicon atom to its own site (figure 2(b)), i.e. to the centre of the vacancy distorting the adjacent area of the crystal lattice (near the vacancy, the lattice is squeezed, resulting in an increase of interatomic distances in the adjacent unit cell). To return to its site after the relaxation of the crystal lattice near the vacancy, the silicon atom has to overcome a barrier as high as 1E(100) = 11.7 eV for the (100) direction and 1E(110) = 12.4 eV for the (110) direction. During the third stage of calculation, a cluster containing a vacancy and four hydrogen atoms localized between the dangling-bond silicon atoms and the next layer of silicon atoms was simulated (figure 2(c)). Optimization of the cluster geometry has shown that dangling-


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Figure 2. Illustration for the model of enhanced annealing of the hydrogenated vacancy. (a) ideal Si crystal; (b) interstitial Si moves towards unhydrogenated vacancy; (c) interstitial Si moves towards hydrogenated vacancy; (d ) hydrogen migrates towards other vacancies.

˚ thus expanding the vacancy. To return to the bond silicon atoms move apart by 0.21 A, vacancy centre after the relaxation of the lattice, the interstitial silicon atom Sid has to overcome potential barriers of 1E(100) = 8.6 eV or 1E(110) = 9.4 eV for the directions (100) or (110), respectively. Thus, the presence of hydrogen atoms near the vacancy lowers the barrier by a factor of about 1.36 for the (100) direction and 1.32 for (110). The effect of hydrogen is still notable for localization in the layer of the second-nearest neighbours, but disappears completely for the layer of the third-nearest neighbours (figure 3(a, b)). Table 3. Potential barriers for diffusion of the interstitial silicon atom (1E(100) , 1E(110) ) and displacements of the four silicon atoms surrounding the vacancy (1R). Path of the Si atom

1E(100) (eV)

1E(110) (eV)

Leaves the site without H Enters the vacancy without H Enters the vacancy with H0 Enters the vacancy with H+ Enters the vacancy with H−

30.3 11.7 8.6 (9.1) 7.8 9.5

31.2 12.4 9.4 (9.9) 8.9 10.2

˚ 1R (A) 0 −0.15 0.21 (0.20) 0.25 0.10

The figures in parentheses are calculated for hydrogen localization inside the vacancy.

Hydrogen localized in the inner region of the vacancy also expands its volume significantly (see table 3), which is similar to the well known results of [25]. The potential barriers calculated for this option of hydrogen localization (when the interstitial silicon moves to the silicon site of the crystalline lattice) are 9.1 and 9.9 eV, respectively, for (100) and (110) directions. Higher potential barriers for Frenkel pair annihilation in case of hydrogen localization inside the vacancy in comparison with hydrogen localization outside

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Figure 3. Energy diagrams for diffusion of a silicon atom (Sid ) to the centre of the unit cell: (a) (100) movement; (b) (110) movement. 1, vacancy in the silicon lattice, no hydrogen; 2, vacancy in the silicon lattice, with H− ; 3, vacancy in the silicon lattice, with H0 ; 4, vacancy in the silicon lattice, with H+ .

the vacancy are caused by the necessity to push the hydrogen atoms out from the vacancy. However, the Frenkel pair annihilation energy in this case is also substantially lower (by a factor of about 1.29 for the (100) direction and 1.25 for (110)) than for an unhydrogenized vacancy. Let us consider the energy relations for hydrogen atoms localized in the layer of secondnearest vacancy neighbours. In the absence of a central silicon atom, the energy of hydrogen bonding to the lattice is 2.21 eV. Localization of the Sid atom at the vacancy centre changes the localization of the hydrogen atoms (the hydrogen moves into the centre of the Si–Si bond) and lowers this energy to 1.44 eV. So, it is more favourable for hydrogen to diffuse to another vacancy than to remain localized at the annihilated vacancy (figure 2(d )). Thus, the


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calculations show that hydrogen can act as a ‘stimulant’ facilitating recovery of the crystal lattice of silicon. 3.4. Effect of the hydrogen charge state on the potential barrier height for Frenkel pair annihilation We have studied the dependence of the potential barrier height for annihilation of a vacancy and an interstitial silicon atom Sid on the charge state of hydrogen. Calculations of charged systems have been carried out by a special technique, presented in [16]. When protons (H+ ) or hydride ions (H− ) are localized at the Si–H–Si bond near the vacancy, the electron density is redistributed in such a way that H+ and H− ions are in a state close to the neutral one (H0 ). For a proton localized at the Si–H+ –Si bond, the electron density (so called ‘Mulliken population’) was found to be 1.015e, while for a hydride ion this value is 0.921e. Relaxation of the vacancy structure with protons localized at the Si–H+ –Si bond ˚ thus expanding the vacancy in results in a shift of dangling-bond silicon atoms by 0.25 A, comparison with the case of neutral hydrogen (for a neutral atom of hydrogen, this shift is ˚ Likewise, the potential barrier for an interstitial silicon atom returning to equal to 0.20 A). the vacancy centre is also lower. When a hydride ion is localized at the Si–H− –Si bond, the displacement of the silicon ˚ which results in a smaller vacancy volume than for a neutral atoms is reduced to 0.10 A, hydrogen atom. The corresponding potential barrier for diffusion of a silicon atom Sid grows to the height of 1E(100) = 9.5 eV and 1E(110) = 10.2 eV (for hydrogen localization outside the vacancy). Let us consider qualitatively the effect of the charge state of hydrogen on the polarization structure of the silicon lattice, which is known to affect significantly the diffusion of interstitial atoms [4, 34]. When diffusing in the neutral charge state (H0 ), an atom of hydrogen does not induce any noticeable polarization of silicon atoms. For the positive ion (H+ ), the electron density from the nearest silicon atoms shifts towards the proton, thus forming an almost neutral hydrogen atom (q = 1.015e), surrounded by a cloud of positive charge facilitating the motion of this atom in the presence of the electric field. For diffusion of the negative ion (H− ), the electron density shifts toward the adjacent silicon atoms, also forming an almost neutral hydrogen atom (q = 0.921e) surrounded by a cloud of negative charge which under the action of the electric field can drag this atom in the opposite direction in comparison with the proton. Thus, there are three charge states possible for a hydrogen atom; the atom becomes nearly neutral, but it may ‘travel’ surrounded by a cloud of positive or negative charge induced by it in the surrounding medium (except for the case of H0 ). Each charge state results in a certain potential barrier for vacancy filling by silicon atoms and controls the direction of diffusion of the hydrogen atom. Table 3 shows the heights of potential barriers for an Sid atom to fill the vacancy in the absence or presence of hydrogen atoms in the vicinity of a vacancy. These data indicate that the presence of hydrogen substantially affects the heights of potential barriers for interactions between the defects and interstitial atoms. 4. Conclusions The calculations performed enable us to draw some conclusions on the specific features of interactions between a vacancy and an atom of hydrogen and between an interstitial atom and a hydrogenated vacancy:

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(1) In case of a completely hydrogenated vacancy, the most favourable position for hydrogen atoms is at the Si–Si bond in the layer of second-nearest atoms. (2) Trapping of atomic hydrogen by the vacancy considerably lowers (by a factor ranging from 1.3 to 1.5) the potential barrier for Frenkel pair annihilation. (3) The effect of barrier lowering due to hydrogen trapping is strongest for the positive ion of hydrogen, and weakest for the negative ion. (4) The position of hydrogen inside the vacancy cavity or at the Si–Si bond between the layers of the nearest and second-nearest silicon atoms significantly affects the potential barrier lowering for annihilation of the pair ‘hydrogenated vacancy–interstitial silicon’. In the first case, hydrogen mainly reduces the strain energy of the lattice in the vicinity of the vacancy but not the energy of the interstitial Si interaction with neighbours in the second coordination region. In the second case hydrogen decreases both the strain energy of the lattice and the energy of Si bonding in an interstitial site. (5) After the trapping of an interstitial atom by a hydrogenated vacancy, it is more favourable for the hydrogen atom to move towards the nearest dangling bond than to remain in its original position. These conclusions are in full agreement with the model of enhanced annealing of vacancytype defects during the RF plasma treatment [11]. For interstitial atoms of doping impurities (e.g. phosphorus or arsenic), it is logical to expect that hydrogenation of silicon should decrease the temperature required for activation of implanted dopants. This was actually observed for simultaneous implantation of hydrogen and arsenic ions [8], and for the RF plasma treatment of P+ -implanted Al–SiO2 –Si structures, which results in effective introduction of atomic hydrogen into the subsurface layer of silicon [35]. Acknowledgment This work was supported by the Ukrainian State Committee for Science and Technology, Project no 4.2/43-93. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]

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