Mat. Res. Soc. Symp. Proc. Vol. 648, P3.11.1-7 2001 Materials Research Society Moel for Dopant an Impurity Segregation During Vapor Phase Growth Craig B. Arnol an Michael J. Aziz Division of Engineering an Applie Sciences, Harvar University, Cambrige MA 02138, USA ABSTRACT We propose a new kinetic moel for surface segregation uring vapor phase growth that takes into account multiple mechanisms for segregation, incluing mechanisms for inter-layer exchange an surface iffusion. The resulting behavior of the segregation length shows temperature an velocity epenence, both of which have been observe in experiments. We compare our analytic moel to experimental measurements for segregation of Phosphorus in Si001), an we fin an excellent agreement using realistic energies an pre-exponential factors for kinetic rate constants. INTRODUCTION The growth of extremely sharp interfaces in materials has become increasingly important in the evices we buil. For example, the evice quality for elta oping in semiconuctors [1, 2] or certain multi-layere metallic systems [3, 4] is sensitive to the reistribution of atomic species on the monolayer scale. The main physical problem to overcome is the tenency for atoms of one species or another to segregate to the free surface uring film growth. Growth of such structures is experimentally challenging, an although there are some exceptions [5], high quality crystal growth with completely suppresse segregation is not generally possible. The physical, chemical, an kinetic principles unerlying segregation are not entirely unerstoo in these systems. Several moels have been presente in the literature, but none of them have been successful in escribing the segregation behavior uner a wie variety of conitions. Our objective is to evelop a more robust moel for surface segregation uring vapor phase growth. The approach starts with successful moels for liqui phase growth which enables us to inclue multiple mechanisms for segregation which are missing in earlier moels. PREVIOUS MODELS Previous moels for segregation can be ivie into three major categories base on the type of mechanism use. The first an earliest type of moels were phenomenological in nature [6, 7]. Their important conclusion was that at sufficiently low temperatures, there exists a kinetically limite regime in which the impurity atoms cannot move quickly enough to avoi becoming trappe in the bulk. Phenomenological moels were followe by a class of moels invoking an interlayer exchange pathway for segregation [8, 9]. In these moels, an atom is first burie by the incient eposition flux an subsequently may exchange positions with an ajacent atom above or below, provie both are within the first two layers of the free surface. An atom is consiere incorporate once it is burie three layers below the surface an unable to make any further exchanges. As i the earlier moels, these moels isplay a transition temperature from equilibrium segregation to a kinetically limite segregation regime. However, the main problem P3.11.1
S D Bulk T S x=l x=0 Figure 1. Simplifie version of exchange mechanisms for segregation on a surface. There are four istinct regions: S) step ege, D) aatoms, T) terrace, an Bulk. The terrace extens from step ege to step ege with an average length L. An atom is consiere incorporate once it reaches the bulk region. Exchange events are enote with ouble ene arrows. with these moels is that they are unsuccessful at reproucing the experimentally measure results at low eposition temperatures [10, 11]. A more recently evelope class of moels invoke a surface iffusion mechanism for segregation behavior [12, 13]. In these moels, an atom can avoi incorporation by remaining on the free surface either by climbing over a step ege or by riing at the step ege. An atom is consiere incorporate once it is completely surroune by other atoms in the plane. These moels better escribe the experimental behavior at low temperatures, but i not preict a transition temperature from equilibrium segregation to kinetically limite segregation. NEW MODEL Our new moel for surface segregation combines both interlayer an surface iffusion processes base on earlier moels evelope for liqui-soli segregation [14, 15]. The first step is to reuce the multitue of atomistic processes that may be occurring on the surface to a tractable few as shown in figure 1. Here, our moel is not concerne with the etaile kinetic pathway for a given exchange event, but rather the effective result of an atom moving from one location S,D,T) to another. Hence we consier irect interchange events between atoms on ifferent sites on the surface. We label solute as B an solvent as A an a solute atom is not completely incorporate until it reaches the region labele bulk in the figure. For the purposes of eveloping this moel, we consier only two types of transitions. Refinement to inclue other transitions is currently in progress. The first transition is similar to the surface iffusion mechanism in which we take a B atom from the first terrace position T 0)) an exchange it with an A atom at the step ege T 0) S)). The secon transition takes a B atom from the final terrace position T L)) an exchanges it with an A atom at the step ege enote T L) S)). This transition can be associate with the previously escribe interlayer mechanism. If a B atom is sitting at S), then once another atom attaches to the step, it will bury the B atom laterally by changing it into a T 0)). This B atom can then unergo a T 0) S) exchange event to return it to S). In the reference frame that is moving laterally with the step ege, the atomic fraction of B atoms at T 0)) evolves in time as X T 0) t = J S T 0) J T 0) S + J c S T 0) J c T 0) T x). 1) P3.11.2
µ' B in "T0)" A in "S" k T0) k S S T0) B in "S" A in "T0)" S T0) µ' A +µ' B Q ST0) S T0) µ' B +µ' A "T0)" µ' ST0) "S" Configurational Coorinate Figure 2. Energy iagram for the T 0) S) transition. The J α β terms represent the number of B atoms iffusing into state β per unit time. Jα β c is the convective flux of B atoms from state α to state β. This is the number of B atoms that move into state β from α ue to the motion of the coorinate system. The convective term is not an activate process, so it epens only on the velocity of the moving step ege v an the fraction of B atoms in the initial state: J c α β = X α v λ. 2) The iffusive flux terms epen on the chemical potential of the A an B atoms at their particular locations. Consier the energy iagram in figure 2 which represents the barrier between states S) an T 0)). We are intereste in the motion of B atoms, but because in this moel all transitions require irect interchange events, we also nee to consier the energetics of A atoms. Then we can write the iffusive fluxes in the form J S T 0) = X S 1 X T 0) )ν exp Q ) ST0) + µ ST0) J T 0) S = X T 0) 1 X S )ν exp Q ) ST0) 3) 4) Here we have accounte for the ieal entropy of mixing by multiplying our transition rates by the number fraction of occupation in the appropriate states. We bring the secon transition into the moel by consiering a similar epenence for the atomic fraction of B atoms in the T L)) state. In this case, there is an analogous set of equations P3.11.3
1-4), with the important ifference that the convective flux equation 2) epens on X T 0) because we are not allowing any exchanges along the terrace. Next, we assume the transitions are able to reach steay state, thereby allowing us to set equation 1 an the analogous equation for T L) S) to 0. We solve these equations in the ilute limit for the ratio, κ ST0) XT 0) X S = κ ST0) e + v v v ST0) v ST0) +1 5) κ STL) XT L) X S = κ STL) e + κ ST0) v v v STL) +1 v STL). 6) Here we have mae the substitutions κ ST0) e exp µ ST0) ) 7) v ST0) νλexp Q ) ST0), 8) with similar equations for κ STL) e an v STL). The final step in the evelopment of this moel is to extract a measurable quantity from these atomic fractions. There are a variety of measures use in the literature, but for convenience, we will use the segregation ratio r as efine by Jorke [9]: r Surface Areal Concentration of Impurity. 9) Bulk Volume Concentration Then by simply aing up the total number of B atoms in the surface an normalizing by the surface area, we obtain r = λ2 κ STL) [ ) ] 1 ρ + λ ρ κ ST0), 10) where ρ is the ensity of steps on the surface an λ is the atomic spacing. P3.11.4
One of the main results of our moel is the preiction of transition between kinetically limite an equilibrium segregation. For ifferent combinations of energies for the two transitions, we can observe multiple transition temperatures corresponing to the segregation regime of each iniviual transition. Furthermore, the transition temperature will have a velocity epenence as well as a miscut epenence for the case of step flow growth. The apparent activation enthalpy of the segregation length will epen on the growth temperature relative to these transition temperatures. A complete iscussion of these behaviors is beyon the scope of this letter an is iscusse elsewhere[16]. COMPARISON TO EXPERIMENTS In orer to compare our moel results to experimental ata, we make some further assumptions about the step ensity. For the experimental ata of Phosphorus in Si001), it is suggeste that films grow in a layer by layer fashion [13]. Therefore, we assume the valiity of scaling theories which relate the step ensity to the surface iffusion rate D surface iffusivity ivie by λ 2 ) to eposition flux monolayers sec ) [17], ρ = ρ 0 D F ) 1 3 = ρ 0 E ν exp iff F ) 1 3. 11) For the purpose of comparing our moel to experiment, we use the value of E iff =1.1eV for the iffusing species in Si001)[18]. Figure 3 shows the experimental measurement symbols) of the segregation ratio for phosphorus in Si001) grown at 0.1 nm s 1 [19]. The experimental ata show a transition between the equilibrium segregation regime an the kinetically limite regime at approximately 800 K. As we can see from the figure, our moel is superior at fitting the experimental ata over the entire range of temperatures. We accurately reprouce the transition between kinetically limite an equilibrium segregation using energies an prefactors that are quite typical for segregating species. In contrast, the interlayer exchange moel of Jorke [9] an the surface iffusion moel of Nutzel an Abstreiter [13] are not sufficient to escribe the experimental ata fully. In the NA moel, the moel fits the low temperature ata, but is unable to prouce a transition temperature. In the J moel, we can fin a fit to the low temperature ata only after applying clearly non-physical parameters to the moel, an even then we are unable to preict the transition temperature accurately. SUMMARY We have evelope a new kinetic moel for segregation that can reprouce the experimental observations of the segregation process. Our moel overcomes some of the shortcomings of previous moels by incorporating both surface iffusion mechanisms an interlayer exchange mechanisms. We obtain a velocity an miscut angle epenent transition P3.11.5
temperature between an equilibrium segregation regime an a kinetically limite regime. In the kinetically limite regime, the segregation length increases with increasing temperature, increasing miscut angle, or ecreasing eposition flux. The moel assumes a particular scaling behavior for the temperature epenence of the step ensity an contains a number of parameters. In principle, some of these can be etermine by inepenent experimental measurements. We fin better agreement with previous experimental measurements of the temperature epenence of the segregation ratio of P in Si001) than is possible with earlier moels. This research was supporte by NSF-DMR-9727369. Segregation Ratio cm) 10-3 Experimental ata AA moel J moel NA moel 10-4 10-5 10-6 0.8 1.0 1.2 1.4 1.6 1.8 1000/T K -1 ) Figure 3. Segregation ratio vs. inverse temperature for the segregation of P in Si001). Symbols: ata of Nutzel an Abstreiter [19]. The soli line shows the fit to our moel AA moel) using a temperature epenent step ensity equation 11). The ashe line shows the fit to the Jorke moel [9] J moel) an the otte line shows the fit to the moel of Nutzel an Abstreiter [13]NA moel) All the relevant parameters are given in table I. Table I. Table of the parameters use in our moel to fit the ata in figure 3. The variables in parentheses represent the analogous parameters in our moel. ν was not use as fitting parameter in either our moel or the NA moel. 0 is a free parameter in the NA moel with no irect analog in our moel. AA moel J moel NA moel Q ST0) 1.00 ev E A Q STL) ) 0.1 ev E s 1E 2 iff- µ ST0) ) 0.66 ev µ ST0) 1.65 ev E I µ STL) ) 1.0 ev 0 0.8 cm Q STL) 1.00 ev ν 26.5 s 1 ν 1.6x10 14 s 1 µ STL) 1.22 ev ρ 0 3.2x10 6 cm 1 ν 1x10 13 s 1 P3.11.6
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