PHYSICAL REVIEW B 72, 195419 2005 Unifie kinetic moel of opant segregation uring vapor-phase growth Craig B. Arnol 1, * an Michael J. Aziz 2 1 Department of Mechanical an Aerospace Engineering an Princeton Institute for Science an Technology of Materials, Princeton University, Princeton, New Jersey 08544, USA 2 Division of Engineering an Applie Sciences, Harvar University, Cambrige, Massachusetts 02138, USA Receive 15 October 2004; revise manuscript receive 6 September 2005; publishe 18 November 2005 We evelop a unifie kinetic moel for surface segregation uring vapor phase growth that concisely an quantitatively escribes the observe behavior in silicon-base systems. A simple analytic function for the segregation length is erive by treating terrace-meiate an step-ege-meiate mechanisms in parallel. The preicte behavior of this parameter is examine through its temperature, flux, an terrace length epenence. Six istinct temperature regimes are preicte for the segregation length that epen on the relative segregation energies an activation barriers of the two mechanisms. The moel is compare to reporte behavior of Sb an PinSi001 an excellent agreement is obtaine using realistic energies an preexponential factors. The moel accounts for the experimentally observe anomalous low-temperature segregation of Sb as a consequence of the competition between step-ege-meiate segregation, ominant at low temperatures, an terrace-meiate segregation, ominant at higher temperatures. The generalize treatment of segregation mechanisms in the moel makes it applicable to other segregating systems, incluing metals an III-V semiconuctors. DOI: 10.1103/PhysRevB.72.195419 PACS numbers: 68.35.Dv, 81.15.Aa, 68.55.Ln I. INTRODUCTION Recent avances in thin-film growth technology to improve ensity, spee, or other evice properties require accurate control of impurities an opants at small length scales. Complex architectures such as elta-oping 1,2 or ban-gap engineering 3 for quantum well evices 4 6 or spintronic applications 7,8 require sharp heterostructures in semiconuctors. Similarly, innovative metal-base structures such as giant magnetoresistive 9,10 or nanomagnetic evices 11,12 rely on the fabrication of abrupt interfaces. As the nee for these sharp interfaces has become increasingly important, the problems of segregation, whereby one species of atom tens to preferentially move to the free surface uring thin-film growth, remain a hinrance. Growth of sharp interface structures is experimentally challenging but possible with well-controlle physical vapor eposition techniques such as pulse eposition, 13,14 energetic techniques, 15 low-temperature molecular beam epitaxy LT-MBE, 16 an surfactant meiate approaches. 17,18 Although one can achieve sharp jumps in composition, often a traeoff must be struck between suppresse segregation an efect accumulation 19 so as to preclue the incorporation of an arbitrary concentration-epth profile. Despite an abunance of experimental observations, there is no consensus on a physical mechanism unerlying segregation an trapping. Any successful moel for segregation must explain 1 the experimentally observe temperature epenence at low temperatures, 2 experimentally observe temperature epenence at high temperatures, an 3 the experimentally observe eposition rate epenence. Typically, in such segregating systems, the high-temperature regime is characterize by local equilibrium segregation, etermine by the thermoynamic balance between the free energies of the surface an subsurface states. 20 In this regime, the amount of segregate material ecreases as the temperature increases because more impurity is soluble in the host material. At lower temperature, the system is consiere to be in a kinetically limite segregation regime. In this case, the segregation is etermine by the kinetics of moving impurity atoms, an the relatively low mobilities cause them to become trappe in the growing film. As the temperature increases, the amount of segregate material increases because impurity atoms have enough mobility to move ahea of the growing film. In this paper, an analytic moel for segregation is evelope that successfully escribes the funamental physics of the segregation phenomenon, yet is simple enough to enable easy comparison with experimental measurements without the nee for numerical solutions or simulations. The evelopment is base on earlier moels of liqui phase growth 21,22 but with the introuction of multiple classes of exchange mechanisms for segregation at ifferent sites on the growth surface. The moel is evelope in the context of segregation for ilute opants in Si001; however, the principles are sufficiently general that it can be applicable to other systems such as metallic alloys or III-V semiconuctors. We emonstrate the effectiveness of the moel by reproucing, for the first time, the measure temperature an growth rate epenence for Sb an P in Si001. II. BACKGROUND There is a variety of moels for surface segregation uring thin film growth. For the purposes of our unifie moel, we classify them into three categories base on the mechanisms an approach to moeling. As these moels have been iscusse extensively in the literature, a etaile iscussion of each is beyon the scope of this article. A. Phenomenological In this earliest class of segregation moel, the overall process of segregation is iscusse without kinetic etails of the 1098-0121/2005/7219/19541917/$23.00 195419-1 2005 The American Physical Society
C. B. ARNOLD AND M. J. AZIZ PHYSICAL REVIEW B 72, 195419 2005 atomic processes. 23 26 Rather, these moels treat the flux of incorporate an segregate atoms at the surface in orer to erive relations between the amount of segregation an the temperature. Given certain assumptions such as incorporation proportional to surface coverage 23 or exponentially varying iffusion near the surface, 27 phenomenological formulations show a transition between kinetically limite an equilibrium segregation that agrees qualitatively with experimental observations. B. Terrace meiate The first main moification to the phenomenological moels inclue the introuction of atomic exchange between ifferent layers in the growing film. 20,28 In these works, the iniviual layers in the structures are treate as flat an terrace-like with an unspecifie kinetic pathway for exchange occurring between them. Exchange between two ajacent layers is allowe to procee provie these are at the free surface. Deposition at iscrete time intervals halts the exchange as a new surface layer is forme an another layer is burie in the bulk. Later incarnations of these moels inclue the possibility of self-limiting behavior whereby impurity atoms in one layer exchange with impurity atoms in the ajacent layer, resulting in no increase of segregation. 29,30 These terrace-meiate moels have been shown to be sufficient in escribing much of the physical behavior in the real systems. As with the phenomenological moels, these moels preict a transition between a kinetically limite lowtemperature regime an an equilibrium regime at higher temperatures. However, the explicit introuction of an exchange mechanism enable not only a better fit to experimental ata, particularly at higher temperatures, but also enable an experimental measurement of the energy barriers for this process. C. Step-ege meiate Two of the main shortcomings of the terrace-meiate moels are that it neglects the presence of steps an roughness on the surface an it is insufficient to escribe the experimentally measure behavior at low temperatures. To overcome these issues, moels were evelope to aress the roughness of the surface an exchange of atoms at steps as a low-temperature mechanism to increase segregation. 31 33 Jesson et al. further emonstrate the importance of such mechanisms through experiments an calculations of Si-Ge growth in showing that it can be energetically favorable for atoms to climb at step eges. 34,35 Significant ifferences exist among the step ege moels in how segregation an incorporation are treate. Nonetheless, the introuction of low-temperature mechanisms enable these moels to improve the agreement with experiment in certain temperature regimes. However, in many cases, these moels fell short in accurately preicting the transitions between kinetically limite an equilibrium segregation, or in preicting other experimental epenencies such as growth rate or surface miscut epenence. FIG. 1. Color online Simplifie view of surface layers an exchanges. M: aatoms above step ege, R: aatoms above terrace, S: step ege, T: terrace, E: below step ege, P: below terrace. Single arrows enote locations of iffusive exchange, ouble arrows enote regions of convective flux. Bol lines in the figure enote no exchanges allowe across bounary, an the small circle enotes the origin for the coorinate system. D. Summary of previous moels Although the iniviual moels outline in the above sections are not optimal for our evelopment objectives, each class provies important physical unerstaning to the segregation phenomenon. Phenomenological moels inicate a transition between two segregation regimes, terrace moels escribe high-temperature regimes, surface-meiate moels escribe low-temperature regimes, an simulations 36 38 enumerate kinetic pathways for segregating atoms. In eveloping a unifie moel we capitalize on the strengths of each moel class. III. THE MODEL A. Segregation kinematics The complexity of any given growth surface makes a rigorous kinetic treatment of all possible segregation mechanisms analytically untenable. Consequently we moel the surface as a perioic structure, unergoing step-flow growth, where each perio contains seven istinct structural regions, as shown in Fig. 1. The space between steps is ivie into a step-ege region M,S,E of length w, an a terrace region R,T,P of length L-w. The monolayer spacing in the z irection is a an the surface is consiere to be screw perioic with repeat vector r=lxˆ +aẑ. The surface is uniform in the y irection, but for imensional purposes, we efine a o as the lattice parameter of one conventional unit cell in the x an y irections. For an 001 surface a o =4a. The origin of our coorinate system is fixe on the top of the moving step ege an we keep track of only the motion of impurity atoms. Impurity B atoms are allowe to move between the regions via irect interchange exchange events between B atoms an host A atoms with the conition that they are immobile in the bulk region. Our moel is not concerne with the etaile kinetic pathway for a given exchange event, but rather the effective result of an impurity atom moving from one location to another. It shoul be note that this metho of simplifying the surface processes coul be extene to inclue aitional layers at the surface of the film, if necessary. There are several metrics available to quantify the segregation behavior an the most appropriate choice epens on the experimental conitions. For example, in one class of 195419-2
UNIFIED KINETIC MODEL OF DOPANT GREGATION experiments the impurity an host species are eposite concurrently over the entire course of the experiment coeposition. In this case a relevant measure for segregation is the Gibbsian surface excess. If there is negligible evaporation from the free surface, the system establishes a steay-state positive or negative surface excess of impurity with the composition of the bulk region fixe at the composition of the incient flux. If evaporation is significant, the system still establishes a steay-state surface excess, however the composition of the bulk region iffers from that of the incient flux by a term proportional to the evaporation flux. By locating the Gibbsian iviing surface at the interface between the vapor phase an the substrate surface, the Gibbsian surface excess,, is given by 1 0 a o L Cx,z xyz, 1 a o L 0 0 where is the concentration of the steay-state bulk material an Cx,z is the concentration at location x,z in the surface. We can refine this efinition for our moel by noting that the surface region, the region in which the impurity atoms are able to make exchanges, has only a finite epth,. Beyon this epth the mobility is negligibly small, the composition must equal the bulk composition, an the above integran vanishes. Therefore, 0 L = 1 Cx,zxz C L 0 bulk. 2 One important consequence of Eq. 2 is that can excee one complete monolayer of impurity atoms. Such behavior has been experimentally observe in the case of Si an Ge, 39,40 but previous moels were unable to account for this behavior. A secon class of experiments is characterize by serial eposition, in which the impurity atoms are eposite first, followe by subsequent eposition of host only. This type of experiment may be associate with elta oping or surfactant-meiate growth processes. 2,41 The surface impurity concentration ecays over time as impurity atoms become trappe in the bulk or evaporate an are not replenishe by a eposition flux. In this case a relevant measure of segregation is the segregation ratio r as efine by Jorke, 28 Areal concentration of impurity in the surface r = Volume concentration of impurity in the bulk. 3 This parameter has the units of length an is therefore sometimes referre to as the segregation length. From Eq. 2 we can fin the 2-D areal concentration of impurity in the surface region, n, by integrating the concentration over all regions of the surface an iviing by the projecte area in the x-y plane, 42 n = 1 0 a o L 0 a o 0 L Cx,zxyz. We integrate this equation over the y imension an ivie by the bulk concentration to yiel the equation for r, 4 r = PHYSICAL REVIEW B 72, 195419 2005 1 0 L L 0 Cx,zxz. The literature provies a variety of metrics to quantify the concentration profile uner serial eposition experiments. However, these measures are relate by appropriate transformations uner particular conitions. For example, the partition coefficient, 30 which is efine as the bulk concentration normalize by the surface concentration, is simply the inverse of r multiplie by the monolayer spacing a. The profile broaening,, efine as the 1/e ecay length of the 2-D surface concentration vs istance grown, reuces to the segregation ratio Eq. 3 when evaporation is negligible. 28,32 In the case where evaporation cannot be ignore, one calculates the relationship between an r by solving the equation relating surface concentration to the changing height of the surface, = n n 1 z r + kevap. 6 v z In this equation, k evap is the evaporation rate constant, given by exp E/k B T, where is an effective vibration frequency an v z is the velocity of growth in the z irection. The solution to the ifferential equation for n is recognize as an exponential with a 1/e ecay length of 1 = 1 k evap + r v z. 7 The above equation goes to the appropriate limit of r as k evap goes to zero. Although the coeposition an serial eposition experiments seem funamentally ifferent, Eqs. 2 an 5 show their measures of segregation are relate by r = bulk +. C 8 Our moel is evelope from the steay-state co-eposition case escribe above. By fining a solution for, one reaily uses Eq. 8 to etermine r for the non-steay-state case that escribes the experimental results. 43 We apply our moel to these experiments uner the quasi-stationary assumption that the surface concentration profile equilibrates rapily on the time scale of changes in itself. B. Kinetics Define C as the concentration of B atoms in region, J as the vertical iffusive flux of B atoms from region to region, D as the lateral iffusivity of B atoms within region, k,evap as the rate of evaporation from region, v as the spee of the moving step ege with respect to the lattice, F as the projecte eposition flux atoms/site s, an f as the fraction of incient B atoms in the eposition flux. Then in the moving reference frame centere on the step ege at x,z=0,0, the concentration at a particular location within each region evolves with time ue to vertical 5 195419-3
C. B. ARNOLD AND M. J. AZIZ PHYSICAL REVIEW B 72, 195419 2005 iffusion into an out of the region, lateral iffusion within the region, an a velocity-epenent convective flux ue to the moving coorinate system. Thus the equations of mass balance in regions P, E, T, S, R, an M are C P t C E t C R t C M t = 1 a JTP C T,C P J PT C T,C P v CP x + DP2 C P x 2, 9 = 1 a J C S,C E J ES C S,C E v CE x + DE2 C E x 2, C T = 1 t a JPT C T,C P J TP C T,C P + J RT C T,C R J TR C T,C R v CT x + DT2 C T x 2, C S = 1 t a JES C S,C E J C S,C E + J MS C S,C M J SM C S,C M v CS x + DS2 C S x 2, = 1 a JTR C T,C R J RT C T,C R v CR x + C R DR2 x 2 C R k R,evap + ff aa o 2, = 1 a JSM C S,C M J MS C S,C M v CM x + 2 C M DM x 2 10 11 12 13 C M k M,evap + ff 2 aa, 14 o where all C an J epen explicitly on both lateral coorinate x an time t. The above set of equations is solvable in principle with the appropriate set of bounary conitions. However, we further simplify the problem an gain physical insight by consiering only two of the four mechanisms shown in Fig. 1. For our purposes it is sufficient to inclue one step-ege mechanism an one terrace-meiate mechanism to emonstrate the relative importance of each mechanism class. In the evelopment here, we have chosen to inclue only the transitions T P an S E; we neglect lateral step-ege transitions because we conten they are less important than vertical ones. This assumption eliminates the vertical iffusive fluxes in Eqs. 13 an 14 as well as the corresponing terms in the remaining equations. Given this simplification, we make the following assertions: 1 Lateral iffusion in layers other than the aatom layer provies a negligible contribution to segregation an we therefore set D =0 for =S,T,E,P.Arigorous treatment of iffusion in the current formalism is beyon the scope of this paper. One expects that iffusion can affect the actual shape of the concentration profile within a given layer; however it shoul not significantly affect the integrate amount of impurity in that region. The assertion breaks own in the case where there is a large iffusivity in, e.g., layer T an a small vertical exchange rate T R but rapi step-ege exchange rate, leaing to a short circuit vertical pathway at the step ege. At high temperatures, such a short circuit mechanism coul cause a isproportionate change in the integrate impurity concentration for that region. The result of this assertion is that the lateral iffusion terms can be remove from Eqs. 9 12. 44 2 The extent of the step-ege region approaches the imension of a single unit cell in the plane, w a o. This assertion basically says that the availability of step-egemeiate mechanisms is limite to those atoms that are irectly on the step ege. Thus we can treat regions S an E as iscrete an convert the partial spatial erivative term in Eqs. 10 an 12 into a finite ifference. In aition, the partial time erivative becomes a full erivative. The case of extene step-ege regions is treate in Appenix A. 3 The total impurity content in aatom regions M an R oes not significantly contribute to the overall surface excess or the segregation ratio. This allows us to ignore Eqs. 13 an 14 from the set of equations 9 14. Essentially the position-epenent aatom concentration in regions M an R for xl has no consequence other than to establish a steaystate concentration at x = L through the balance of eposition, evaporation, an lateral iffusion. This value, plus the kinetics of lateral segregation at the moving step ege, etermines how much impurity gets incorporate at the step ege, thus fixing the x=0 bounary of C S. This in turn sets the scale for the concentration profiles in subsequent regions T, E, P, an bulk. The range of valiity for this assertion is iscusse in further etail in Appenix B. From these assertions, Eqs. 9 14 are rewritten with their explicit variable epenencies as C P x,t = 1 t a JTP C T x,t,c P x,t J PT C T x,t,c P x,t v CP x,t, 15 x C E t = 1 t a J C S t,c E t J ES C S t,c E t v a o C E t C T L,t, 16 C T x,t = 1 t a JPT C T x,t,c P x,t J TP C T x,t,c P x,t v CT x,t, 17 x 195419-4
UNIFIED KINETIC MODEL OF DOPANT GREGATION C S t = 1 t a JES C S t,c E t J C S t,c E t v a o C S t C inc, 18 where C inc is the concentration of B atoms that is incient on the step ege at x=l. Equations 16 an 18 make the implicit assumption that there is continuity in concentration across the bounary at x=l. The vertical iffusive fluxes, J, in the above equations are obtaine from unimolecular rate theory. 21,45 Consier the P T transition an assume the system lowers its energy by exchanging a B atom in region P with an A atom in region T, an likewise for E an S, respectively. The interchange flux from P to T is then given by J PT C T x,t,c P x,t = C P z,t 1 a 2 o ac T z,t a exp QTP k B T. 19 The reverse flux has the same form, but the barrier for this process inclues the segregation energy, or the ifference in reistribution potential between the two states TP ; 46 J TP C T x,t,c P x,t = C T z,t 1 a 2 o ac P z,t a exp QTP + TP. k B T 20 Similarly for the S E transition, J ES C S t,c E t = C E t 1 a o 2 ac S t a exp Q k B T, J C S t,c E t 21 = C S t 1 a 2 o ac E t a exp Q +. k B T 22 In these cases, the assumption that a B atom has a lower energy in the T state forces TP to be a positive value. 47 All concentrations are assume ilute an the 1 a 2 o ac terms in Eqs. 19 an 20 are set equal to unity, thereby making the iffusive fluxes linearly proportional to the concentrations. 48 In the steay state, the concentration of incient atoms equals the concentration entering the bulk, C inc =, an the ilute forms for Eqs. 19 an 20 are substitute into Eqs. 15 18, to yiel a final set of equations to be solve: 0= v TP TP k a e C T x C P x v o x CP x, 23 0= v TP a o k e TP C T x C P x v x CT x, 24 where 0=C S v k e a o PHYSICAL REVIEW B 72, 195419 2005 C E v + v a o a o + v C T L, 25 a o v a o + v, 26 a o 0=C S v k e v a o a o + C E k e TPor = exp TPor k B T, 27 v TPor = a o exp QTPor k B 28 T are the equilibrium partition coefficient an the iffusive spee, respectively, as efine for the case of soliification. 46 C. Concentration profiles The problem has now been reuce to a set of couple, linear orinary ifferential equations for C P x an C T x. We normalize these equations by the constant an the solution becomes C P x = Ae x + k TP C e B, 29 bulk C T x = Ae x + B, 30 where we have substitute the in-plane ecay length = v TP 1+k TP va e. 31 o The constants, A an B, are etermine from the bounary conitions C T a o =C S an C P a o =C E,or C S = Ae a o + B, 32 C E = Ae a o + k TP C e B. 33 bulk Finally, the solution of the simultaneous set of equations given by 25, 26, 32, an 33 provies expressions in close form for A an B: A =e a o B = e L a v o e L a o k e TP v v +1 k e v v v + k e k e TP + k e TP1+k e + v v ; 34 v v 1+e L a o +1+k e v v + k e k e TP + k e TP1+k e + v v. 35 195419-5
C. B. ARNOLD AND M. J. AZIZ PHYSICAL REVIEW B 72, 195419 2005 segregation ratio exhibits not only temperature-epenent behavior, but also velocity an miscut epenence as observe experimentally. 49,50 In the next section this behavior is explore in further etail. IV. GENERAL LIMITING BEHAVIOR AND GREGATION REGIMES FIG. 2. Possible temperature-epenent behavior for the segregation ratio. The soli line represents the preictions of this moel assuming energies of TP, Q TP =1 ev an, Q =0.5 ev, a eposition rate of 1 monolayer/secon an a terrace length of 25a o. The ashe line shows the plot of Eq. 49 representing the moel with the single S E transition. The ot-ashe line shows Eq. 44 representing the moel with the single T P transition. The roman numerals label the segregation regimes escribe in the text an summarize in Table I. D. Surface excess an segregation ratio It remains to etermine the Gibbsian surface excess,, from Eq. 2. In this formulation of the moel, which ignores the aatom regions M an R, everywhere there are two layers for the nonbulk region; hence =2a. From Eq. 2, 0 = 1 L a + 2a 0 a o C S a 0a o C E 0 xz + a xz + 2a a o L C T x xz a ao L C P x xz 2a. This equation is integrate to obtain = 1 o a La CS L C T x + x + a o a CE aa o + aa o L C P x x 2a. 36 37 After substituting Eqs. 29, 30, 32, an 33 an solving the integrals, the Gibbsian excess reuces to = ab1+k TP C e 2a. bulk Finally the segregation ratio is obtaine from Eq. 8: r = a1+k e TP B. 38 39 Equations 38 an 39 represent the major result of this moel. One of the main consequences of the moel is a transition between kinetically limite an equilibrium segregation. When temperature is varie at constant flux an step spacing, multiple transition temperatures correspon to the kinetic transition for each iniviual segregation mechanism. The Figure 2 shows some general temperature-epenent behavior of the segregation ratio as preicte by this moel for arbitrary input energy barriers. A transition occurs at a temperature T * that separates the kinetically limite segregation regime at lower temperatures an the equilibrium segregation regime at higher temperatures regimes III an IV. With the appropriate choice of energies, it is possible to introuce a secon transition temperature within the kinetically limite regime, at which the preominant segregation mechanism unergoes a change from terrace-meiate T P regime III to step-ege-meiate S E regime II. In orer to gain insight into the results of this moel, it is useful to look at a few physical limits. These limits inclue T an T 0, as well as consiering large ifferences in the energy barriers for the two mechanisms e.g., Q or Q TP. A. Limit 1: T\ In the limit that T, the atoms rapily surmount any barrier to motion an the energy ifferences among configurations become negligible; hence the system oes not maintain any concentration graients. In this limit the surface excess vanishes as all the impurity atoms are evenly istribute throughout the growing film. Mathematically, from Eqs. 27 an 28, in this limit k e an k e TP 1, an v an v TP a o. Then substituting into Eq. 35, wefinb 1. Then by Eqs. 38 an 39, 0 an r 2a. B. Limit 2: T\0 In this limit, the temperature of the system is so low that impurity atoms have no mobility uring growth an all impurity gets trappe in the growing film. Uner these conitions, we woul expect that there is no segregation an, corresponingly, no surface excess. Here, k e, k e TP, v, v TP 0 an, by Eq. 31, 0 as well. Then by Eq. 35, B 2, an 0 an r 2a as expecte. C. Limit 3: Q \ In this limit we probe the behavior of the mi-terrace T P mechanism alone by turning off the S E transition. The activation barrier for S E is allowe to iverge so that v 0 an v/v. For finite temperatures, k e an k TP e 1 an Eq. 35 becomes B 1+e L a o e L a o TP + k. 40 e This equation is combine with Eq. 39 to yiel 195419-6
UNIFIED KINETIC MODEL OF DOPANT GREGATION r a 1+e L a o e L a o TP + k. 41 e This result is plotte as the ot-ashe line in Fig. 2. For this single mechanism, there exists a transition between a kinetically limite regime an an equilibrium regime. Two turning points are observe on this plot. The first, T TP a,isthe temperature above which this mechanism is activate an segregation begins. The secon location, T TP *, is the transition temperature above which equilibrium segregation occurs. Through Eq. 41, we fin that as e L ao approaches 1; we return to limit 2. Therefore, the temperature at which this mechanism becomes activate is given by =1/L a o.we recognize that for a terrace of length L, the step velocity is given by v = RL, 42 where R is the net eposition rate monolayers/secon of all species. Then in the limit that La o we fin k B ln R 1 TP T = a Q TP. 43 For temperatures above T TP a,e L ao 1, an the segregation ratio becomes a r e L a o TP + k. 44 e The transition temperature between kinetically limite an equilibrium segregation is governe by the relationship between e L ao an k TP e. In this limit, r eq a TP for T T TP *, 45 k e r kin ae L a o for T T TP *, 46 where r eq is the segregation length in the equilibrium segregation regime an r kin is the segregation length in the kinetically limite regime. The transition temperature is etermine by equating Eqs. 45 an 46. Applying the efinitions of an k TP e we fin the transcenental equation, RL TP L a o = k BT TP * exp QTP k B T TP. 47 * It shoul be note that the preicte behavior for the single T P transition given by Eq. 44 is ientical to Eq. 7 in Jorke s treatment 28 with the aitional conition that the Jorke moel oes not inclue a step-ege region i.e., a o 0. D. Limit 4: Q TP \ In this limit, we turn off the mi-terrace T P transition an enable the step ege meiate S E transition. Here we have the similar conition as before that k e TP an k e 1, yet because v TP 0, e L a o 1. In this limit B PHYSICAL REVIEW B 72, 195419 2005 2 v v +1 1+k TP e v. v + k e 48 When we combine this equation with Eq. 39, we fin the segregation ratio r a 2 v v +1 v v + k e. 49 The result of Eq. 49 is nearly ientical to the equation 7.46 in Tsao s moel 30 with the aitional factor of 2 relate to the fact that =2a in our case. This equation is plotte in Fig. 2 as a ashe line, an again our moel preicts a transition between kinetically limite behavior an equilibrium behavior for this single mechanism. For v/v 1, we return to limit 2 where the mechanism has been turne off an there is no segregation. Thus the mechanism becomes activate as v/v 1. We use this relation to etermine that the temperature to activate this mechanism is given by k B ln 1 RL a T = a Q. 50 In the temperature regime near the transition, T *, both k e an v/v are 1. Therefore, the numerator of Eq. 49 approaches unity while the relative magnitues of two terms in the enominator etermine whether segregation is in a kinetically limite or equilibrium regime. Thus, r eq a for T T *, 51 k e r kin av for T T *, 52 v an the transition temperature is given by equating these two expressions to fin T * = + Q k B ln a. 53 v E. Temperature epenence of segregation behavior The sample plot of the segregation ratio using arbitrary energies Fig. 2 for the full two-mechanism moel shows four regimes enote I IV. If we consier the results of the previous limiting cases, the segregation behavior at these ifferent temperatures can be unerstoo. At the lowest temperatures, there is not enough thermal energy to overcome the activation barriers for exchange an therefore no segre- 195419-7
C. B. ARNOLD AND M. J. AZIZ PHYSICAL REVIEW B 72, 195419 2005 TABLE I. Approximate equations governing segregation behavior in ifferent temperature regimes. The transition conitions enote the relevant mathematical equation that istinguishes one regime from the previous one. The physical interpretation tells us which mechanism is active an whether it is kinetically limite kin or in its equilibrium segregation eq regime. Regime Segregation ratio Transition conition Physical interpretation I 2a S E off, T P off II av /v v/v =1 S E kin, T P off IIb a/k e v/v =k e S E eq, T P off III a/e L a o v/v +k e L a o =1 S E kin or eq, T P kin IIIb a/k e TP 1 e L a o v/v +k e /k e TP =1 e L a o S E kin or eq, T P kin IV a/k e TP k e TP /v/v +k e =e L a o S E kin or eq, S E eq gation occurs. In this regime, r 2a. Once the temperature increases beyon T a, a transition to regime II occurs as the step-ege-meiate mechanism becomes active in its kinetically limite regime. At somewhat higher temperatures regime III, the terrace mechanism at T TP a activates. This mechanism ominates the overall segregation behavior provie the terrace length is larger than the lattice spacing. Ultimately, we reach the transition to the equilibrium segregation regime at T * an move into regime IV. In this regime, the segregation behavior is ominate by the equilibrium segregation regime of the T P transition. The reason that the step ege region oes not play an important role in the segregation behavior in the equilibrium regime is that the terrace region is the last place an atom can make an exchange before becoming trappe in the bulk. Therefore, regarless of what segregation behavior occurs at the step ege, atoms subsequently have the entire terrace length to reequilibrate with region T. A summary of these regimes an the governing equations are given in Table I. It remains to fin the overall transition temperature, or temperature of maximum segregation, T *, in a similar fashion as before. Near T *, k e, k TP e 1, but also, from limits 3 an 4, e L ao an v/v are also 1. Therefore, the segregation ratio is rewritten in this regime as r = e L a v o a. 54 v + k TP e + k e This equation is similar to the equation for the T P transition limit 3, Eq. 44 with a slight moification to the e L a o term. This moification accounts for the fact that the step ege region is proviing aitional atomic reorganization for the overall system. Again, the regime in which we fin ourselves epens on the relative magnitues of the two terms in the enominator. As before, we calculate the transition temperature by fining the temperature, T *, that satisfies the equation TP k e =e L ao, 55 v v + k e where the enominator on the left sie of the equation will be ominate by either the v/v or k e term epening on whether the step-ege mechanism is in the kinetically limite or equilibrium segregation regime. The presence of four istinct regimes in the temperature epenence of the segregation ratio epens on the relative energies use in the calculation. The physical interpretation of the energy hierarchy is written in terms of the transition temperatures an equilibrium segregation coefficients, namely, T a T a TP, T * T a TP, an k e TP k e. If the energies are such that one or more of these conitions is no longer vali, we either get aitional regimes in the temperature epenence or fewer regimes. Figure 3 shows representative plots for the temperature epenence of the segregation ratio if one assumes alternate relative energies for the system. The energies have been chosen to catalogue the possible behavior exhibite by the moel an are liste in Table II. In Fig. 3a, one observes the situation when one mechanism ominates the other mechanism. In this case, as the temperature is raise, the terrace-meiate mechanism becomes activate well before the step-ege mechanism an clearly ominates the resulting behavior. Although lower atomic coorination at step eges might imply that the stepege mechanism shoul always exhibit a lower activation energy than the terrace mechanism, unusual circumstances, e.g., strain effects on the activation barriers, coul cause such a scenario. In aition, the epenence of segregation behavior on terrace length can cause similar effects as will be iscusse later. In Figs. 3b 3e, the system is shown for cases in which, as the temperature is raise, the step-ege mechanism becomes activate before the terrace-meiate processes. This conition is the same as for Fig. 2; however, by moifying the relative magnitue of the energies, alternative regimes 195419-8
UNIFIED KINETIC MODEL OF DOPANT GREGATION PHYSICAL REVIEW B 72, 195419 2005 kinetically limite regime as shown by the reemergence of regime III. Once the temperature reaches T * TP, the system returns to the equilibrium regime for the terrace process an returns to regime IV. In Figs. 3 an 3e, k e TP k e an a new regime appears in the segregation ratio temperature behavior. The onset of the terrace meiate mechanism for these sample energies leas to a kinetically limite segregation regime in which the segregation ratio ecreases with respect to increasing temperature as enote by regime IIIb. This occurs whether the step meiate process is in an equilibrium segregation regime Fig. 3 or a kinetically limite segregation regime Fig. 3e. Such behavior is opposite that which is expecte for kinetically limite segregation regimes ue to an inversion in the chemical potential ifference between T an P. This occurs ue to the large concentration of B atoms that convectively flow from S into T. Recall the chemical potential ifference between T an P is given by FIG. 3. Alternate possibilities for temperature-epenent segregation behavior. Dot-ashe lines represent the moel with single T P transition, ashe line represents the moel with single S E transition. Roman numerals enote the segregation regimes liste in Table I. Deposition rate of 1 monolayer/ secon, terrace length of 25a o, an energies given in Table II. are isplaye. In Figs. 3b an 3c, the terrace process becomes activate an reaches the equilibrium segregation regime of the step-ege mechanism regime IIb. Depening on the magnitue of k e TP relative to the magnitue of k e, the segregation ratio will exhibit ifferent behavior. Figure 3b shows the special case when the two values are equal. In this case, there is no aitional segregation regime an the equilibrium regime IV remains for increasing temperatures. In Fig. 3c, k e TP k e, an when the terrace-meiate mechanism becomes activate, it pushes the system back into a TABLE II. Energies use for Q TP, TP, Q, an in generating Figs. 2 an 3 along with the temperature-epenent segregation ratio regimes that are present in the plot. Figure Q TP ev TP ev Q ev ev Regimes 2 1.0 1.0 0.5 0.5 I, II, III, IV 3a 1.0 1.0 1.0 1.0 I, III, IV 3b 1.5 0.5 0.5 0.5 I, II, IV 3c 1.5 1.0 0.5 0.5 I, II, IIb, III, IV 3 1.5 0.4 0.5 0.5 I, II, IIb, IIIc, IV 3e 1.5 0.5 0.5 1.0 I, II, IIIb, IV TP = TP k B T ln CT C P. 56 When the concentration in T is sufficiently large, the sign of TP will change inicating a change in the irection of the net riving force on B atoms from T into P. We refer to this regime as the chemical potential inversion regime. Thus, when the temperature is high enough to allow the terrace meiate mechanism to exchange atoms, the net flux from T into P causes the segregation ratio to ecrease as temperature is increase. The possibility of a chemical potential inversion regime inicates that a single measurement of the sign of the slope in r vs T is insufficient to conclusively etermine whether one is in a kinetically limite or equilibrium segregation regime. A more complete ata set e.g., see rate epenence below or aitional information about the relative energies woul be neee to unambiguously make a etermination. The explicit form for the segregation ratio in this regime is given in Table I with the appropriate transition conitions given in Table III. F. Deposition rate epenence Figure 4 shows a sample plot of the eposition rate epenence at a fixe temperature for the same parameters given in Fig. 2. The labelle regimes on the plot irectly correspon to the previously iscusse segregation regimes. The connection is reaily apparent in Fig. 5 for which the temperature epenence is plotte for a variety of growth rates. One observes that changes in the eposition rate cause a shift in the transition an activation temperatures. Thus the segregation regime associate with a fixe temperature soli vertical line in Fig. 5 changes ue to the effects of growth rate. The rate at which the transitions occur are given in Table III with the relevant equation numbers. At the highest eposition rates R, regime I, there is insufficient time for the atoms to segregate before becoming burie by the moving step ege see Eq. 42. This case is equivalent to the limit T 0 an thus the segregation ratio r 2a an it is inepenent of the eposition rate. As the 195419-9
C. B. ARNOLD AND M. J. AZIZ PHYSICAL REVIEW B 72, 195419 2005 TABLE III. Equations governing the temperature an eposition rate transition between segregation regimes. For regime IV, the transition temperature an eposition rate are solutions to the transcenental functions in the given conition. The given equation numbers refer to the relevant equations in the text. In all cases, we have assume La o. In regimes IIIb an IV, T * an r * are obtaine by solving the given equation. Regime Transition temperature Transition eposition rate Equations I T 0 R II T a =Q /k B lna/rl R a =a o /L exp Q /k B T 50 an 57 IIb T * = +Q /k B lna/rl R * =a o /L exp +Q /k B T 53 III T a TP =Q TP /k B ln/r R a TP = exp Q /k B T 43 IIIb v/v +k e /k e TP =1 e L a o v/v +k e /k e TP =1 e L a o IV k e TP /v/v +k e =e L a o k e TP /v/v +k e =e L a o 55 eposition rate ecreases, eventually we reach a transition to regime II in which the S E transition occurs fast enough for impurity atoms to evae the moving step ege, but the T P transition is frozen by the motion of the step ege. This case is equivalent to temperature regime II in which the S E transition is activate, but the T P transition oes not have enough thermal energy to overcome the barrier. The segregation ratio in this regime is given in Eq. 52 showing a rate epenence of rr 1. As the eposition rate is further reuce, we enter regime III, in which the terrace meiate transitions occur fast enough for impurity atoms to evae the slowly moving growth front. In this case, the segregation ratio is given in Table I with a steeper rate epenence. Finally, at the slowest growth rates, the system has enough time to reach the equilibrium segregation regime an we return to the growth-rateinepenent regime IV with r 2a. A full summary of the segregation ratio behavior as a function of growth rate is given in Table I. The transition eposition rates between the regimes can be etermine in a similar fashion as those for the temperature epenence. The transition between regimes I an II occurs at the onset of the S E transition, R a.asineq. 50, the conition for transition between these regimes is given by v/v 1. The equation can be solve for R a at fixe temperature, R a = a o L Q exp k B T. 57 The conitions for the transitions between II-III an III-IV given in Eqs. 43 an 55 can similarly be solve for R a TP an r * an are given in Table III. Aitional flux epenence regimes corresponing to the aitional temperature regimes iscusse in the previous section are possible epening on the barrier height see Tables I an III. In regime IIb, the segregation is in the equilibrium regime an therefore inepenent of the growth rate. Interestingly, regime IIIb exhibits a growth rate epenence in which the segregation ratio ecreases as the growth ecreases inset of Fig. 4. Again, this behavior is a result of the chemical potential inversion which causes a riving force FIG. 4. General eposition rate behavior for the segregation ratio. The soli line represents the preictions of this moel assuming energies of TP, Q TP =1 ev, an, Q =0.5 ev, a temperature of 400 K, an a terrace length of 25a o. The Roman numerals label the segregation regimes escribe in the text an summarize in Table I. The inset epicts moel preictions for the energies given in Fig. 3e at a temperature of 555 K showing the inverse eposition rate epenence of regime IIIb. FIG. 5. Temperature epenence of segregation ratio for ifferent growth rates. The energies are the same as in Fig. 4. The soli line inicates the temperature at which the ata for Fig. 4 are taken. 195419-10
UNIFIED KINETIC MODEL OF DOPANT GREGATION PHYSICAL REVIEW B 72, 195419 2005 FIG. 6. Segregation ratio as a function of the terrace length for energies given in Fig. 4 at 1000/T=2.4. for atoms to bury themselves in region P rather than segregate to region T. Thus, as the rate ecreases, atoms have more time to exchange an move towar region P instea of segregating towar T. G. Terrace length epenence The general behavior of the segregation ratio as a function of terrace length can be obtaine from the expressions for r given in Table I. The terrace length enters r implicitly through the step velocity in Eq. 42. One reaily observes that the segregation ratio is inepenent of the terrace length in regimes I, IIb, an IV. In regime II, the segregation ratio is inversely proportional to the step velocity an therefore inversely proportional to the terrace length at constant R. This behavior is expecte because only the step-ege-meiate mechanism is active. Thus an increase in L effectively ecreases the number of active sites on the surface, thereby ecreasing the overall amount of segregation. Regime III shows a more complicate epenence on L Fig. 6. In this case, we see that at the largest terrace lengths, the segregation ratio varies inversely with L. As the terrace length ecreases, the segregation ratio levels off an in fact begins to ecrease. The exhibite behavior in regime III can be unerstoo by realizing that a change in the terrace length will greatly affect the step-ege contribution to the segregation behavior while having a much smaller effect on the terrace-meiate processes. This effect is shown graphically in Fig. 7 for ifferent terrace lengths. The purely T P segregation oes not change significantly with L, whereas the S E segregation shifts to higher temperatures as L increases. The result of this is that the temperature range spanne by regime III oes not change significantly with terrace length as it oes for the growth rate epenence Fig. 4. Equations 46 an 52 give the segregation ratio for the iniviual T P an S E transitions in their kinetically limite regimes. Because is proportional to 1/L, the prouct L a 0 is inepenent of L to first orer for large L. Thus, the T P transition is only slightly affecte by changes in the terrace length in the kinetically limite regime. In contrast, the S E transition clearly shows 1/L FIG. 7. Temperature epenence of the segregation ratio for terrace lengths. The energies are the same as in Fig. 4. The vertical ashe line represents the temperature plotte in Fig. 6. Subplot of terrace only T P an step-ege only S E segregation show effects on iniviual mechanisms. epenence in the kinetically limite regime. Then in regime III where both mechanisms are active, the prouct of their behavior will follow similar behavior to the S E transition for large L. Depening on T * the segregation ratio will either be inepenent of L or inversely epenent on L. To eal with the case of L small i.e., high miscut, one nees to further analyze the expression for r in regime III. Equation 54 can be rewritten with explicit L epenence, r = a exp v TP a 0 Rexp v TP RL RL 1 v + k e. 58 For small L, the secon exponential factor on the rhs no longer approaches unity an we observe a ecaying exponential behavior with respect to L that ominates the behavior. Therefore, the segregation ratio will ecrease as L ecreases in this regime. V. COMPARISON TO EXPERIMENT A. Temperature epenence Data for the segregation ratio of Sb in Si001 an P in Si001 have been obtaine from the literature 32,51 53 an are shown in Figs. 8 an 9. The ata for Sb emonstrate the rich behavior that is possible in real systems, showing multiple slopes in the kinetically limite regime. The soli line in the plot shows a goo fit of our moel to the experimental ata using parameters given in Table IV. Therefore, applying our previous analysis to the moel, we may interpret the apparent anomalous low temperature behavior as the presence of a step-ege-meiate mechanism that works in parallel with a terrace mechanism but has a lower activation barrier. This is consistent with the picture of a step ege on a surface where an ege atom has to break fewer bons in comparison to a terrace atom in orer to make an exchange. 35 Furthermore, our fit emonstrates that TP. This inicates that the Sb atoms feel less of a riving force between two states at a step ege in comparison to atoms in the mi-terrace, which is again consistent with the boning structure at a step ege. Base on the available ata, we consier TP to be a lower limit because we o not have 195419-11
C. B. ARNOLD AND M. J. AZIZ PHYSICAL REVIEW B 72, 195419 2005 TABLE IV. Parameters use for generating ata fits from moel Fitting parameters Q TP TP ev ev Q ev ev Sb in Si001 1.9 1.03 1.19 0.37 PinSi001 2.0 0.61 1.17 0.48 FIG. 8. Fit of temperature-epenent segregation ratio for Sb in Si001. The symbols represent experimental ata obtaine from Refs. 51 53. Fitting parameters for J moel Ref. 28 are E A =1.78 ev, E I =1.2 ev, an =210 12 s 1. Fitting parameters for NA moel Ref. 32, are E seg =0.75 ev, o =2 cm, an =1.6 10 14 s 1. Fitting parameters for the current moel are given in Table IV. Dashe lines labele S-E an T-P represent single S E an T P transitions. FIG. 9. Fit of temperature-epenent segregation ratio for P in Si001. The symbols represent experimental ata obtaine from Ref. 32. Fitting parameters for J moel Ref. 28 are E A =0.1 ev, E I =1.0 ev, an =26.5 s 1. Fitting parameters for NA moel Ref. 32 are E seg =0.66 ev, o =0.8 cm, an =1.610 14 s 1. Fitting parameters for the current moel are given in Table IV. Dashe lines labele S-E only an T-P only represent the single S E an T P transitions. Fixe parameters a o cm a cm L R s 1 monolayer secon 5.43210 8 1.35810 8 25a o 1 110 13 sufficient ata in the equilibrium segregation regime. An increase in TP woul have the effect of increasing the slope in regime IV see Table I an a shift in the transition temperature towar higher temperatures, neither of which can be etermine from the ata. 54 The fits obtaine through this unifie moel show significant improvement over earlier attempts that employ single segregation mechanisms. 28,32 The ot-ashe line shows a best fit to the ata using the Jorke moel terrace only. In this case, the moel can sufficiently fit the higher temperature regimes, but insufficiently escribes the ata at lower temperatures. In the case of the surface-meiate moel of Nutzel an Abstreiter otte line, the moel is able to fit only the low-temperature regime, but is unable to hanle the greater segregation at higher temperatures. In orer to generate the fits of our moel to the experimental ata, we have use only four free fitting parameters corresponing to Q TP, TP, Q, an, whereas previous unsuccessful moel fits use three 28 an two 32 fitting parameters. The values we obtain, 1.9, 1.03, 1.19, an 0.37 ev, respectively, are consistent with previously reporte barrier measurements an calculations for the ifferent segregation mechanisms. Because the unifie moel is not base on a etaile kinetic pathway, these energy barriers shoul be consiere a weighte average over all possible kinetic pathways that enable the exchange of atoms between S E or T P. First principles calculations are possible to etermine the energies inepenently, thereby permitting an inepenent metho of obtaining an interpreting these energy values. It is necessary to make an explicit assumption about the surface structure in orer to input a value for the terrace length. In all cases we have assume a fixe terrace length as a function of time, temperature, an flux. Clearly this may not be the case, particularly at low temperatures an high fluxes where layer-by-layer growth is possible. In orer to accommoate a nonconstant terrace length, we can introuce an explicit temperature an flux epenence to the terrace length in the moel for example, LD/F n Refs. 55 an 56. Such a process woul introuce another arbitrary fitting parameter as aitional information from experiments woul be require to etermine the relevant constant. Alternatively, experiments with etaile STM stuies can be use to irectly measure the terrace length. 57 The fitting behavior for phosphorous shows similar results. Previous moels are unable to fit the ata across the available temperature regimes. For instance, the terracemeiate moel is able to fit the low-temperature regime, but requires the input of nonphysical energies 0.1 ev an attempt frequencies 26.5. Similarly, the NA moel is able to fit the low-temperature ata, but oes not preict a transition to equilibrium segregation an is unable to escribe the hightemperature regimes. Base on the fit from our unifie moel, the onset of the terrace-meiate mechanism is much more slight in comparison to the Sb case an rapily procees into the equilibrium regime. It is interesting to note that the activation barrier energies Q TP an Q are almost the same as those in the Sb case whereas TP an are quite ifferent Table IV. It appears that for Si001, the barrier heights o not vary markely among the segregating species. This may inicate 195419-12