Journal of the Korean Physical Society, Vol. 48, No. 4, April 2006, pp. 535∼539

First-Principles Investigation of Indium Diffusion in a Silicon Substrate

Kwan-Sun Yoon, Chi-Ok Hwang, Jae-Hyun Yoo and Taeyoung Won∗

Department of Electrical Engineering, Inha University, Incheon 402-751

(Received 17 November 2005, in final form 12 January 2006)

In this paper, we report the total energy, the minimum energy path, and the migration energy ofindium in a silicon substrate by using ab-initio calculations. Stable configurations during indiumdiffusion were obtained from the calculation of the total energy, and we estimated the minimumenergy path (MEP) with the nudged elastic band (NEB) method. After finding the MEP, we foundthe energy barrier for the diffusion of indium to be 0.8 eV from an exact calculation of the totalenergies at the minimum and the transition state.

PACS numbers: 31.15.Ar, 82.20.DdKeywords: Indium, Ab-initio calculation, Nudged elastic band method, Minimum energy path, Migrationenergy

I. INTRODUCTION

As complementary metal-oxide semiconductor(CMOS) devices are scaled down to the nanometerregion, it is even more stringent to control the impu-rity profiles at the front-end process. Especially, thenecessity for ultra-shallow junctions in nano-CMOStechnology pushes the emergence of a novel alternativematerial technology with a low diffusivity coefficientand a lower activation energy for the impurity-dopingprocess. Recently, indium has been attracting a greatdeal of interest as a candidate for a p-type dopant,especially for the fabrications of retrograde p-tubs andhalo regions for n-channel FETs. In other words, indiumis considered to be an alternative to boron due to itsrelatively heavier mass [1, 2]. Nevertheless, we do notunderstand the exact diffusion mechanism for indium,including diffusion parameters when we compared tothe case of boron.

Indium is considered to diffuse mainly through aninterstitial-mediated mechanism during the thermal an-nealing process. Recently, the kinetic Monte Carlo(KMC) method has been widely employed to model thethermal annealing process in nano-CMOS devices [3,4].The KMC method is needed to simulate the diffusion ofindium atoms on an atomistic scale. However, we do nothave enough parametric values to perform the KMC cal-culation. Therefore, we performed ab-initio calculationsin an effort to obtain parameters such as the input pa-rameters of a migration event, one of the main events inthermal annealing [5].

∗E-mail: twon@hsel.inha.ac.kr;Tel: +82-32-875-7436; Fax: +82-32-862-1350

In this work, we investigated the atomic-scale diffu-sion mechanism and tried to find the minimum energypath (MEP) of indium diffusion for acquiring the migra-tion energy in silicon by using ab-initio calculations andtransition state theory. Our theoretical study allowedus to have good insight into an understanding of indiumdiffusion and to contribute to KMC modeling of indiumatoms.

II. DETERMINATION OF THE DIFFUSIONPATH

First of all, we performed a defect structure calculationin a cubic super-cell, comprised of 216 silicon atoms witha single neutral dopant (B or In) atom. The ab-initiocalculations were implemented within density functionaltheory (DFT) by using the VASP (Vienna ab-initio Sim-ulation Package) [6–8] which combines ultrasoft pseu-dopotentials [9] and a generalized gradient approxima-tion (GGA) in the Perdew and Wang formulation. Inthis work, we employed a cutoff energy of Ec = 150.62eV, a 2 × 2 × 2 grid for the k-points mesh of Monkhorstand Pack [10], and a 3 × 3 × 3 simple cubic super-cell(216 atoms).

In this work, we initiated our ab-initio study with anassumption that the energy landscape of indium in sili-con is quite similar to that of boron because both specieshave the same number of valence electrons. Therefore,we calculated the energy for a specific defect configura-tion of indium in silicon from the recognition that indiumwill have a defect configuration similar to the boron con-figuration [11,12].

Table 1(a) shows a list of defect configurations of in--535-

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Fig. 1. Atomic structures illustrating the configurations of indium in silicon. The Indium atom is dark-colored, and thesilicon atoms are light colored for each configuration. The InSi-X consists of a self-interstitial silicon with an indium atomsharing the same lattice site and with the Si-indium dimer lying in the <110> direction. If the direction is <100> with thesame configuration, the structure is referred to as InSi-S. The Ins + SiTd

i configuration is the case when the indium atom sitson a substitutional site and stabilizes a self-interstitial silicon in a nearby tetrahedral position. The InTd

i and InHxi structures

are those with the interstitial indium atom, respectively, in the tetrahedral position and in the hexagonal position. (a) InSi-X(b) InSi-S (c) Ins + SiTd

i (d) InTdi (e) InHx

i

Table 1. Calculated energies of the Si:In and the Si:Bdefect configurations. The listed energies are relative energieswith reference to the ground-state configuration.

Configuration (In) Relative energy [eV]

InSi-X 46.11

InSi-S 58.09

InTdi 0.43

Ins + SiTdi 0.00

InHxi 1.63

(a) Indium

Configuration (B) Relative energy [eV]

BSi-X 0.56

BSi-S 1.18

BTdi 0.71

Bs + SiTdi 0.00

BHxi 0.47

(b) Boron

dium, which we can guess from our knowledge of boron.The calculated energy values for each defect configura-

tion are also shown in Table 1(a). Table 1(b) shows theenergy landscape for boron in order to illustrate the com-parison to indium. The energy values are adjusted withreference to the ground state of indium. Figure 1 shows aschematic diagram illustrating the atomic configurationsof the indium atom in the silicon lattice.

Firstly, our ab-initio calculation revealed that thelowest-energy configuration of indium is (Ins + SiTd

i ),which means that the indium atom sits on a substi-tutional site and stabilizes a self-interstitial silicon ina nearby tetrahedral position, as illustrated in Figures1(c) and 3(a), while the second lowest-energy structureis found to be InTd

i . It should be noted that the secondlowest energy configuration (InTd

i ) of indium is differentfrom the case of the boron energy landscape, BHx

i . Ourab-initio calculation produced an energy gap betweenthe ground-state Ins + SiTd

i and the next higher energylevel InTd

i at 0.43 eV. In addition, we can think of acouple of dumbbell configurations, InSi-X and InSi-S, asillustrated in Figures 1(a) and 1(b). The InSi-X consistsof a self-interstitial silicon and an indium atom sharingthe same lattice site, with the Si-indium dimer lying inthe <110> direction. Furthermore, we use the notationInSi-S if the Si-indium dimer is lying in the <100> di-

First-Principles Investigation of Indium· · · – Kwan-Sun Yoon et al. -537-

Fig. 2. Plots illustrating the calculated energy landscapefor In and boron. These plots show that the lowest-energystructure (Ins + SiTd

i ) consists of indium sitting on a substitu-tional site and stabilizing a self-interstitial silicon in a nearbytetrahedral position. Further, the second lowest-energy struc-ture is found to be InTd

i with the interstitial indium being inthe tetrahedral position. While the lowest-energy configura-tion of indium is the same as the boron structure of lowest-energy (Bs + SiTd

i ), the second lowest energy configurationof boron is different (BHx

i ) from that of indium (InTdi ). (a)

Indium (b) Boron

rection.Since the energies of the above-mentioned dumbbell

configurations, consisting of a self-interstitial silicon andan indium atom sharing the same lattice site, are foundto be 46.11 and 58.09 eV, respectively, we can concludethat indium does not constitute such a dumbbell config-uration during the diffusion process. It is well reportedin the literature [13] that the diffusion path of boron is

Bs + SiTdi → BHx

i → Bs + SiTdi .

However, we can conclude that the diffusion path of in-dium is different from that of boron due to the aforemen-tioned reasons. In other words, our ab-initio calculationrevealed that the diffusion path of neutral indium is

Ins + SiTdi → InTd

i → Ins + SiTdi .

In addition, the contribution of InHxi does not seem to be

considerable in the diffusion of indium when we compareit with the boron energy landscape. We can recognizethat the energy level for the InHx

i state is 1.63 eV, whichis relatively higher than the second energy level InTd

i at0.43 eV.

Figure 2(a) and 2(b) are schematic diagrams illustrat-ing the energy landscape for indium and boron, respec-tively. Referring to Figures 2(a) and 2(b), we can observethat the energy gap between the ground-state Ins + SiTd

i

and the second level InTdi is 0.43 eV while the energy gap

between the ground-state Bs + SiTdi and the second level

BHxi is 0.47 eV.Figure 3(a) is a schematic diagram illustrating the

atomic configuration in the three-dimensional lattice sitefor the ground state Ins + SiTd

i wherein the indiumis placed at a silicon site in a substitutional manner.The substituted indium is represented by a dark-coloredsphere while the silicon atoms are denoted by light-colored spheres. Now, the indium atom at the groundstate experiences a transition to a second level state,

Fig. 3. Plots illustrating defect configurations. The Inatom (dark-colored) and the self-interstitial Si (light-colored)are shown over the underlying diamond lattice. (a) The Ins

+ SiTdi and (b) the interstitial In at the tetrahedral position,

InTdi , are also shown. (a) Ins + SiTd

i (b) InTdi

InTdi , as illustrated in Figure 3(b). We can notice that

the indium atom is now positioned at a tetrahedral sitein accordance with the diffusion route.

During the theoretical calculation of the energy levelfor a specific transition state, we found that the calcu-lated energy level when the indium atom is located atthe middle point along an edge of a cubic cell is differentfrom the one calculated when the indium atom lies atthe center of a cubic cell by an amount of 0.02 eV. Wewould like to make a comment on this numerical issue atthis point because there may exist a technical limit onthe size of the super cell in the ab-initio calculation. Aswe mentioned earlier, the size of our super cell was 216atoms, including the indium ion. Due to the limit on thenumber of atoms employed in the ab-initio calculation,a lattice point in a certain unit cell can be erroneouslyregarded as a different lattice point even if the latticepoint in question is exactly the same lattice point fromthe standpoint of the adjacent unit cell.

Figure 4 is a schematic diagram illustrating the aboveexplanation in detail. The dark-colored sphere can beconsidered to be positioned at the center of a unit cellwhile it can also be regarded as being located at themiddle of the edge of a nearby unit cell. If the indium

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Fig. 4. A plot illustrating the equivalent lattice point forthe adjacent cells. A different tetrahedral site is shown in thecubic unit cell. The In atom (dark colored) in the middle ofthe edge is located at the body center of the other unit cell.

atom experiences a transition to the body center in thesame unit cell, the indium atom should be switched toa self-interstitial silicon in the tetrahedral site (SiTd

i ofIns + SiTd

i ) because the silicon atom is located at thebody center. We believe that the indium atom diffusesto the tetrahedral site at the middle of the edge becausethe energy barrier of the diffusion path for the switchingmechanism is relatively high.

III. MINIMUM ENERGY PATH

Now, we estimate the minimum energy path and themigration energy with reference to on our diffusion pathdescribed in the previous section by using the VASP andtransition state theory (TST). In order to find the mi-gration energy for indium diffusion, we have to calculatethe minimum energy path (MEP). In this work, we em-ploy the nudged elastic band (NEB) method [14] for thecalculation of MEP, which is the optimization procedurefor a number of intermediate images along the reactionpath. Each initial image is forced to converge to a pos-sible lowest energy while keeping the spacing relative toneighboring images constant. We believe that the NEB isan efficient method to find a saddle point and the min-imum energy path between the given initial and finalstates during diffusion.

In the previous section, we found that the initial stateis Ins + SiTd

i while the final state is InTdi . By repeat-

ing the transitions between those two states, the neutralindium diffuses in silicon. Consequently, we can now ob-

Fig. 5. A plot illustrating the relative energy along theMEP of Si:In from Ins + SiTd

i to InTdi calculated in this work

by using the nudged elastic band method. The triangles in-dicate the simulation images, and the squares are the inter-polation by using the force parallel to the band.

Fig. 6. A plot illustrating the energy and force of interpola-tion between the intermediate images (squares). The relativeenergy increased when the force is negative and the relativeenergy decreased when the force is positive. The force parallelto the band is used as information for the interpolation.

tain the energy barrier for indium migration if we inves-tigate the MEP from the initial state to the final state.

Figure 5 is a diagram illustrating the MEP for in-dium, which was calculated by using the NEB methodwith four intermediate images. The initial intermediateimages, denoted with triangles, are linearly interpolated

First-Principles Investigation of Indium· · · – Kwan-Sun Yoon et al. -539-

between the initial and the final images. Along the y-axis is shown the relative energy along the MEP of Si:In from the initial state (Ins + SiTd

i ) to the final state(InTd

i ). In Figure 5, the interval between the initial in-termediate images is interpolated with reference to theforce being calculated during the simulation. In order toobtain an estimate of the saddle point and to sketch theMEP, it is important to interpolate between the imagesof the converged elastic band [14]. The energy and theforce of converged elastic band are shown in Figure 6.

In Figure 6, the relative energy tends to exhibit a pos-itive slope when the force is negative while the relativeenergy decreases when the force is positive. The mi-gration energy is now estimated as the energy differenceto experience a transition from a local energy minimumstate to another local minimum along the diffusion path.Our simulation revealed that the migration energy of In-interstitial defects is approximately 0.8 eV, which is ingood agreement with a prior report [12]. We believe themigration energy of indium to be in the range between0.5 – 1.2 eV.

IV. CONCLUSIONS

In conclusion, we report our theoretical study on theactivation energy, as well as minimum energy path forthe diffusion of indium. We were successful in finding themigration path of the interstitial-mediated mechanism.The ab-initio study in this work is comprised of twosteps: performing the electronic structure relaxationand obtaining its total energy at the local minimum.We came up with the atomistic configurations and themigration energy during indium diffusion in silicon,and we tried to find saddle points from the minimumand the reaction pathway between two stable states byusing TST. After we found the transition state, we triedto find the energy barrier for diffusing the particle bycalculating the exact total energy at the transition state.We realized that parameter extraction for In-relateddefects could be essential for exact modeling of theexperimental diffusion profiles experienced during themanufacture of the next-generation CMOS devices.

ACKNOWLEDGMENTS

This work was supported partly by the Korean Min-istry of Information & Communication (MIC) throughthe Information Technology Research Center (ITRC)Program supervised by the Institute of InformationTechnology Assessment (IITA) and partly by the KoreaInstitute of Science and Technology Information (KISTI)through the Seventh Strategic Supercomputing Applica-tions Support Program. The authors would like to ex-press special thanks to the Supercomputing Center atthe KISTI for providing us the computing resources.

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