Dopant Diffusion and Segregation in Semiconductor Heterostructures: Part 1

Appl. Phys. A 68, 9–18 (1999) Applied Physics A Materials Science & Processing  Springer-Verlag 1999

Dopant diffusion and segregation in semiconductor heterostructures: Part 1. Zn and Be in III-V compound superlattices

C.-H. Chen, U.M. Gösele, T.Y. Tan∗

Department of Mechanical Engineering and Materials Science, Duke University, Durham, NC 27708-0300, USA

Received: 20 August 1998/Accepted: 23 September 1998

Abstract. Distribution of shallow dopants in semiconductor nomenon, which requires the description of the dopant heterostructures in general exhibits a pronounced segregation atom diffusion and segregation processes simultaneously. phenomenon, which requires the description of the dopant We treat these class of problems in a series of three pa- atom diffusion and segregation processes simultaneously. We pers. In this present paper, which is the first part, we treat this class of problems in a series of three papers. In the present the general method of formulating such problems present paper, which is the first of the three, Zn and Be dis- with an emphasis of application to Zn and Be distributions tributions in III-V superlattice (SL) structures are discussed in III-V superlattice (SL) structures. In the second paper, in detail. The analysis method developed in this paper is gen- we analyze B distribution in GeSi/Si heterostructures [1]. erally applicable to other cases. In the second paper we ana- In the third paper, we treat the problems associated with lyze B distribution in GeSi/Si heterostructures. In the third a number of n-type dopants in a variety of semiconductor paper we treat the problems associated with a number of n- heterostructures [2]. type dopants in a variety of semiconductor heterostructures. In GaAs and in AlGaAs/GaAs-type superlattices (SL), Segregation of a dopant species between two semiconductor self-diffusion and diffusion of dopant atoms on the group III heterostructure layers is explained by a model incorporating sublattice of GaAs and other III-V compound semiconductors (i) a chemical effect on the neutral species; and (ii) a Fermi- are governed by charged point defects [3–6]. In intrinsic and level effect on the ionized species, because, in addition to the n-type materials, the triply-negatively-charged group III va- 3− chemical effect, the solubility of the species also has a depen- cancies VGa dominate the diffusion processes of self- and im- dence on the semiconductor Fermi-level position. For Zn and purity atoms occupying the group III sublattice, for example, Be in GaAs and related compounds, their diffusion process is Si, whereas in highly doped p-type materials the doubly- 2+ governed by the doubly-positively-charged group III element positively-charged group III self-interstitials IIII dominate 2+ self-interstitials (IIII ), whose thermal equilibrium concentra- the group III atom self-diffusion and diffusion of the p-dopant tion, and hence also the diffusivity of Zn and Be, exhibit also Zn and Be atoms which occupy group III sublattice sites [4– a Fermi-level dependence, i.e., in proportion to p2.Ahet- 6]. Because the point defect species are charged and hence erojunction consists of a space-charge region with an electric their thermal equilibrium concentrations are dependent upon field, in which the hole concentration is different from those the semiconductor doping type and level, the diffusivities of in the bulk of either of the two layers forming the junction. atomic species utilizing the appropriate point defects as dif- This local hole concentration influences the local concentra- fusion vehicles are also dependent upon the semiconductor 2+ − − tions of IIII and of Zn or Be , which in turn influence the doping type and level, which is known as the Fermi-level distribution of these ionized acceptor atoms. The process in- effect [3–6]. To arrive at these conclusions, in previous ana- 2+ − − volves diffusion and segregation of holes, IIII , Zn ,orBe , lyses [4–6] of dopant and self-atom diffusion profiles in and an ionized interstitial acceptor species. The junction elec- AlGaAs/GaAs SL structures, the effect of the dopant solu- tric field also changes with time and position. bility difference in the SL layers has been ignored, which is justifiable for two reasons. First, experimental results showed PACS: 61.72.Vv; 61.72.Ss; 61.72.Yx that the solubility difference of the dopants in AlAs and in GaAs is small. Second, layer disordering has occurred in the experiments which minimizes the effect arisen from the Distribution of shallow dopants in semiconductor heterostruc- dopant solubility difference. tures in general exhibits a pronounced segregation phe- In fabricating some III-V SL device structures for which the involved acceptor diffusion time is short and/or the dif- ∗ E-mail: [email protected] fusion temperature is relatively low, the p-type dopants Zn 10 and Be showed a prominent segregation behavior in the SL 3 Shallow acceptor solubilities layers [7–9]. This segregation phenomenon is caused by the dopant solubility difference in the layers when there is no In this section, we discuss the solubility issue encountered for substantial layer disordering taking place to smooth out the shallow acceptor species. It is noted, however, that the general material compositional and chemical differences. In this pa- principle applies also to shallow donor species. per, we describe a detailed quantitative model to account for A shallow acceptor species in a semiconductor at a given the dopant SL layer-dependent distribution process, includ- temperature consists of a neutral and an ionized species. ing the segregation phenomenon. The model considers the The Fermi level affects the solubilities of the ionized shal- − dopant solubility dependence on the Gibbs free energy of in- low acceptor species A . Since the semiconductor Fermi corporating a neutral dopant atom onto the semiconductor level is in turn determined by shallow dopants, we see that lattice site, which is determined chemically, and the con- there is a mutual dependence of the Fermi level and the cept of the Fermi-level dependence of the ionized shallow dopant solubility. We need to consider the ionized dopant dopant solubility. The model also considers the dopant atom solubilities because we will use the diffusion segregation diffusion process via the Fermi-level dependence of the gov- equation of You et al. [11], (1), to formulate the present erning point defect concentrations, as well as the effect of problem. carrier concentrations at the heterojunctions. A preliminary It is standard textbook knowledge that, at a given tem- such study was first carried out by the present authors on III-V perature, in the presence of one and only one kind shallow compound SL layer disordering due to Si doping [10]. dopant of a given total concentration, the partition between The overall dopant distribution process involves diffusion the ionized and neutral species is according to the Fermi– 2+ − − and segregation of holes, IIII , Zn ,orBe , and an ionized Dirac distribution function. This is a closed thermodynamic interstitial acceptor species. The junction electric field also system containing the semiconductor crystal and the dopant changes with time and position. of the given total concentration. Under the closed-system constraint that the total dopant concentration is constant, this partitioning is the consequence of the fact that the free en- 1 The diffusion–segregation equation ergy of this closed system is minimized. In this sense, the concentrations of both the neutral and ionized dopant species Because of the involvement of a large number of species are those under thermal equilibrium conditions. However, the for each of which diffusion and segregation occur simultan- ratio of the so-partitioned values will change with a change eously, the diffusion–segregation equation derived by You et in the total dopant concentrations which in practice can be al. [11], largely varied. Thus, these closed-system equilibrium quantities cannot be defined as the solubilities of the appropri- ∂C ∂ ∂C C ∂Ceq ate species. = D − , (1) ∂t ∂x ∂x Ceq ∂x At a given temperature, a unique thermal equilibrium concentration or solubility value of the neutral, the ionized, and eq the total dopant species is defined for the semiconductor crys- is used to formulate the problem. In (1) C denotes the ther- tal which is open to a unique dopant source material. This mal equilibrium concentration or solubility of the considered dopant source material is the one and only one in thermal species at the given temperature. The second term on the equilibrium coexistence with the semiconductor crystal con- right-hand side (RHS) of (1) accounts for the spatial varia- eq taining dopant atoms, which is a compound phase composed tions in C due to any physical cause, but not including that of the dopant element and elements constituting the semi- of a temperature gradient. conductor crystal, but not any other elements. If, by some means, the total dopant concentration introduced into the semiconductor is below the appropriate solubility value, sub- 2 Charged point defect thermal equilibrium sequently the closed-system equilibrium conditions apply and concentrations the dopant atoms will be partitioned accordingly. If, instead, the total dopant concentration introduced into the semicon- For III-V compounds and especially for GaAs and AlAs,it ductor is above the appropriate solubility value, the subse- is well recognized that the Fermi level plays the eminent quent thermal equilibrium process will result in that, in the role in leading to the charge dependence of the dopant diffu- semiconductor matrix material, dopant atoms will reach the sivities via its effect on the concentrations of charged point solubility values of the neutral and ionized species and sim- defects which govern the dopant diffusion processes [3–6]. ultaneously precipitates of the unique source material phase, This is one effect of the Fermi level, for which the pos- which determines the dissolved species solubility values, ition of the Fermi level is determined by a shallow dopant form. Figure 1 schematically illustrates the situations dis- species. The energy level positions of the involved charged cussed above. point defect species are assumed to be deep in the semicon- The effect of the Fermi level on the solubilities of the + − ductor band gap and hence the concentrations of the defects ionized shallow donors D or acceptors A has only been are small so that they do not have a noticeable influence on scarcely addressed. In fact, we are aware of only one discus- the Fermi level position determined by the shallow dopants. sion on this issue that is sufficiently explicit and fundamental. The Fermi-level dependence of the thermal equilibrium con- Using a thermodynamic formulation, Yu et al. [12] showed − − centrations of charged point defects has been discussed else- that, for Zn or Be in GaAs, the solubility of the ionized eq where in detail [4–6], and in the present paper we will use this acceptor species C − is proportional to 1/p,wherep is the A eq knowledge. hole concentration in the crystal. The value of CA− is also 11 For the ionized acceptor species A−, the Fermi–Dirac distribution function is 1 f = , (3) E −Eeq 1 + g exp a F kB T

− where f is the fractional concentration of A , Ea is the shallow acceptor level energy position in the semiconduc- eq tor bandgap, EF is the Fermi level energy position under thermal equilibrium conditions for the open system, kB is Boltzmann’s constant, T is the absolute temperature, and g is the hole degeneracy factor. In this paper, we adopt the convention that all semiconductor-band-related energies are measured relative to the vacuum level which is set at 0eV. The quantity f specifies the fractional A− concentration according to eq C − f = A . (4) eq + eq CAo CA− Equations 3 and 4 yield exactly Fig. 1. A schematic diagram showing the solubility values of the neutral, the 1 Eeq − E ionized, and the total concentrations of a shallow dopant species in a semi- eq = eq F a . CA− CAo exp (5) conductor, which is defined in an open system consisting also of the dopant g kBT source material which is represented by the precipitated species Appt,which in this diagram is assumed to be resulting from introducing the dopant into As has already been mentioned, the solubility of the neu- the semiconductor in supersaturated values and subsequently annealed for eq tral acceptor species CAo is a unique constant at a given infinitely long times. When introduced in undersaturation (closed-system temperature, because of the open-system assumption. The cases) the dopant will partition into neutral and ionized species according to the Fermi–Dirac distribution, but at a ratio differing from that at solubility open-system assumption can hold in experiments performed case at high temperatures. In experiments with a pre-introduced total amount of acceptor atoms to begin with, for example, as obtained using the ion implantation process, the subse- − dependent upon the formation Gibbs free energy of A−.In quent steady-state partition between the A and Ao species their formulation [12], this free energy is an implicit function will also satisfy (5). However, now the eq notation, defined of a certain material’s constants related to the semiconductor for the open system, no longer holds. This is because an EF eq band structure. value differing from EF will be obtained. Noting that The most straightforward way to illustrate the Fermi-level eq Ea ≈ Ev , (6a) effect on CA− is via a derivation using the Fermi–Dirac distribution function. The mathematical procedure of treating the problem is the same as for partitioning the dopant into the E − E n = N exp v i , (6b) neutral and ionized species for the closed system case. The i v k T physical condition imposed by the open system assumption B is fairly different from that of the closed system case, and eq eq − eq Ev EF the dependence of CA− on p (which will be designated as p = N exp , (6c) eq v p ) and on the semiconductor band structure constants will kBT be explicit. Under the open-system assumption, the thermal where peq is the hole concentration under thermal equilib- equilibrium concentration of the neutral species, as given by rium conditions, ni is the intrinsic carrier concentration, Ev and E are respectively the valence band edge energy and the i f intrinsic Fermi-level energy, and Nv is the effective density of eq = − gAo , CAo Co exp (2) states of the valence band, (5) becomes kBT eq 1 eq Nv 1 eq ni Ei − Ev C − = C o = C o exp . (7) A g A peq g A peq k T f B where Co is the crystal lattice site density and gAo is the Gibbs free energy of incorporating an Ao onto a lattice The middle expression in (7) is of a more compact form, but site, is constant at a given temperature, for two reasons. the RHS expression is more useful for the practical reason First, the source material is in principle inexhaustible. Sec- that the values of ni, Ei,andEv are usually measured from ex- o f periments and are hence available for most semiconductors. ond, because A is uncharged, gAo is determined chemically, which is independent of the semiconductor doping type In turn, peq satisfies the charge-neutrality condition f and level, i.e., it is independent of E . Furthermore, g o is q F A 1 also independent of any band-structure-related semiconduc- eq = eq + eq 2 + 2 p CA− CA− 4ni (8) tor constants. 2 12 in the presence of only one shallow acceptor element. It is seen from (7) and (8) that, for one shallow dopant species existing in a semiconductor alone, the thermal equilibrium concentration of the ionized species is dependent upon this concentration itself, since peq is determined by this concentration. With the exception of peq, all quantities in (7) eq are constants, and CAo is independent of ni or any other semiconductor band-structure-related constants, for example, Ei or Ev, or the semiconductor doping type and level. These semiconductor band-structure-related constants will be differ- eq ent for different SL layers. Therefore, in the SL layers, CA− values will also be different, which leads to the A− segregation phenomenon.

4 Charge-carrier concentrations at a heterojunction

In the immediate vicinity of a heterojunction, the electric charge-carrier concentration is different from those in the bulk of either of the two material layers forming the junction. Since the carrier concentration influences the junction region concentrations of A− and I2+, it influences also the − s III As distribution in the SL. For a single semiconductor, an electric junction is formed if the dopant distribution is inho- Fig. 2. A schematic diagram showing the band structure of III-V compounds mogeneous, which can be due to an n- or p-type dopant alone forming a heterostructure, doped and undoped. Indicated are the band bend- or two dopants producing a p–n junction. For a SL structure, ing, the excess charges (fixed and carriers), and the junction electric field in addition to the effect of doping, the band gap discontinu- ity at the heterojunctions of the semiconductor layers will also contribute in producing the electric field. The band gap dis- band-structure-related energy values are shown in Fig. 3 for continuity contribution to the junction electric behavior can a number of III-V materials used in fabricating heterostruc- be particularly strong if the differences in the semiconductor ture devices. Second, for the involved AlxGa1−x As/GaAs ex- band-structure constants are large. periments, the annealing temperatures were high and/or the The electric junction originates from the electric car- annealing times were long, so that layer disordering or inter- rier thermal equilibrium property. Under thermal equilibrium mixing due to Al−Ga interdiffusion has occurred to a signifi- conditions, the chemical potential of the charge carriers (or cant extent, which smears out the junction and hence also the the Fermi level of electrons) is constant throughout the whole effect of the junction carrier concentrations. volume of the semiconductor or semiconductor structure. At the junction, this thermal equilibrium requirement produces a local depletion of carriers on one side of the junction and an accumulation of carriers on the other side of the junction. Thus, in a small region bordering the junction on one side, the carrier concentration becomes smaller than that in the bulk of the appropriate layer while in a small region on the other side of the junction the carrier concentration becomes larger than that in the bulk of the appropriate layer. The junction region is a space-charge region associated with local band bending. It consists of two neighboring sub-regions of oppo- site charges of the same total amount, producing an electric field confined primarily in the region. The charges may both be fixed charges associated with ionized dopant or host crystal atoms, or one may be of fixed charges while the other is of carriers, but never can both be carriers. The situation associated with doping of a single semiconductor is well known, and those for the heterostructure cases are schematically illus- trated in Fig. 2. In previous analyses [4–6] of dopant and self-atom diffusion profiles in AlGaAs/GaAs SL structures, the effect of the carrier concentrations at the heterojunctions has been ignored, for two reasons. First, the band-structure energy value differences between AlxGa1−x As and GaAs are relatively Fig. 3. Room-temperature III-V compound band-structure-related energies, small when compared to those involving other III-V materi- Ec, Ev,andEg.ValuesofEc and Ev are those referenced to that of the als, for example, those for the InP/InAs heterojunctions. The vacuum level at 0eV,andEg = Ec − Ev 13

2+ Under static conditions at room temperature, the junction where DI and Di are, respectively, the diffusivities of I and + III region carrier concentration, and hence also the associated Ai , which are constants at a given temperature in a given SL electric field, is unchanging with time. There is no net car- layer material. rier or current flowing, and the distribution of dopant atoms is We assume that dynamical equilibrium has been reached − + 2+ also not changing with time. For III-V SL structure at elevated among the three species As ,Ai ,andIIII , i.e., in (10) the left- temperatures, however, the junction region carrier concentra- hand side term ∂Cs/∂t is regarded as much smaller than either 2+ tions influence the local IIII concentrations as well as the term on its RHS. This allows (10) to reduce to solubilities of A−, which in turn affects the SL layer A− eq concentrations. Ci = ≡ Ci , K eq eq (13) CsCI Cs CI

5 Formulation of the acceptor distribution problem where K = kb/kf is the equilibrium constant associated with eq reaction (9). The quantity Cs is of the form of (7), i.e., The process of Zn− or Be− distribution in experiments in- − volves diffusion and segregation of the electric carrier holes, eq = 1 eq ni Ei Ev , 2+ Cs Cso exp (14) the doubly-positively-charged group III self-interstitials IIII , g p kBT the ionized shallow acceptor atoms A−, and an ionized in- eq terstitial species of the dopant atoms, the role of which is with Cso being the thermal equilibrium concentration of the o discussed shortly. Moreover, the junction-region carrier con- neutral acceptor atoms As . 2+ centration changes with the electric field variations as a func- For IIII , one can assume either thermal equilibrium con- tion of time and position. ditions hold or not. The assumption that thermal equilibrium The p-type dopants Zn and Be are substitutional–intersti- conditions hold is applicable to [4, 6]: (i) the materials in the tial (As–Ai) species in GaAs and other III-V compounds. The experiments are rich in group III elements, and/or (ii) the thermal equilibrium concentration of As is much larger than materials used in the experiments contain a sufficiently large that of Ai, and hence the measured Zn or Be concentration density of sinks/sources for point defects of both the group III is that of As. The diffusion of As is accomplished by the and the group V sublattices. The Zn indiffusion experiment of ◦ migration of Ai and its subsequent change-over to become Weber et al. [8] was conducted at 550 C without the use of As, because the process is much faster than the migration of an As over-pressure, which apparently satisfy the above con- As atoms themselves via the vacancy mechanism. The Ai–As dition (i). The experiments of Humer-Hager et al. [7] and of change-over process is governed by the kick-out mechanism Häussler et al. [9] were performed using ion-implanted Be, involving [4–6] which apparently satisfy the above condition (ii), because of the ion-implantation damage. Thus, we will use the assump- + ⇔ − + 2+ , Ai As IIII (9) tion that thermal equilibrium conditions hold. This allows (11) to be replaced by where the interstitial species of the dopant is assumed to be a donor, A+. 2 i = eq = eq( ) p , To formulate the problem, we first account for the time CI CI CI ni (15) − + ni and spatial variations in the concentrations of As ,Ai ,and 2+ eq eq I , respectively designated as Cs, Ci,andCI. In accordance ( ) III where CI ni is CI under intrinsic conditions which is inde- with the discussion of the last paragraph, the change of Cs is pendent of the crystal Fermi-level position or carrier concen- described by trations. + ≈ ∂ Using (12)–(15), and noting that Cs Ci Cs and hence Cs ∂( + )/∂ ≈ ∂ /∂ = kfCi − kbCsCI , (10) Cs Ci x Cs x, we obtain ∂t ∂C ∂ ∂C C ∂Ceq where k and k are, respectively, the forward and backward s = eff s − s s f b ∂ ∂ Ds ∂ eq ∂ reaction constants associated with reaction (8). For the mo- t x x Cs x + 2+ bile species A and I , expressions obtained in previous ∂ ∂Cs Cs ∂p Cs ∂ni i III = Deff + − formulation of Zn/Be diffusion [6] become insufficient, be- ∂ s ∂ ∂ ∂ x x p x ni x cause inhomogenities of the thermal equilibrium concentra- eq C ∂E ∂E C ∂C o tions of these species caused by the heterojunctions were not − s i − v − s s , ∂ ∂ eq ∂ (16) included. To account for both diffusion and segregation in kBT x x Cso x the SL layers, the use of (1) in accordance with reaction (9) − 2+ + where Deff is the effective A diffusivity given by yields, respectively, for IIII and Ai s s eq 2 ∂ ∂ ∂ ∂C ∂ eff eq p CI = CI − CI I + Cs , D = KC (n ) D , (17) DI eq (11) s I i i ∂ ∂ ∂ ∂ ∂ ni t x x CI x t which is the same as that describing Zn diffusion in GaAs [6]. eq The distribution of holes in the SL layers will be described ∂C ∂ ∂C C ∂C ∂C i = i − i i − s , by a generalized hole-transport equation including the hole ∂ ∂ Di ∂ eq ∂ ∂ (12) t x x Ci x t segregation property in the different layers and the junction 14 electric field on the hole concentrations in the junction re- 6.1 Fits of experimental results gions. With these effects included, in thermal equilibrium, the hole distribution in accordance with (6c) is Using the partial differential equation solver ZOMBIE [13], (15)–(17), (19), and (20) were numerically solved to fit the − eq − Φ available experimental results [7–9], Figs. 4–6. In obtaining eq Ev EF q p = Nv exp , (18) these fits, it is assumed that there is no intermixing of the SL kBT layers, because in all cases the annealing time is sufficiently short and/or the temperature is sufficiently low, allowing the where q is the magnitude of the electron charge (taken to effect of layer intermixing to be ignored. It is also assumed Φ be positive); is the electrostatic potential throughout the that: (i) the dopant diffusivity value under intrinsic condi- SL structure, which changes rapidly in the heterojunction re- eff( ) = eq tions, Ds ni , which is obtained by letting p ni in (16), is gions; and E is the thermal equilibrium EF position, which eq( ) F the same for all layers of a given SL structure; (ii) the CI ni is constant throughout the SL structure. The quantities Nv and value is the same for the different layers of each SL. Ev are constants in each SL layer, but are different in the dif- Figure 4 shows our calculated fitting curve together with ferent SL layers. To account for the distribution of holes, the the experimental results of Humer-Hager et al. [7] obtained use of (1) and (18) yields by implanting Be into a GaAs/Al0.3Ga0.7As structure on a GaAs substrate and annealed at 860 ◦C for 3s. It is seen ∂ ∂ ∂ ∂ ∂ ∂Φ p = p − p Nv − p Ev + pq , from Fig. 3 that the fit is fairly satisfactory. The used Be in- Dp eff( ) × −13 2 −1 ∂t ∂x ∂x Nv ∂x kBT ∂x kBT ∂x trinsic diffusivity value, Ds ni ,is1 10 cm s ,and (19) the used self-interstitial thermal equilibrium concentration eq( ) . × 11 −3 under intrinsic conditions, CI ni ,is3 65 10 cm .In the experiment [7], the SL layers were pre-doped. The pre- where Dp is the hole diffusivity. The band bending schematically shown in Fig. 1 result from the last three terms on the doping conditions are listed in Table 1. Also listed in Table 1 RHS of (19). The potential Φ satisfies Poisson’s equation are the used values of the materials’ band-structure-related constants, including ni, Ei, Ev,andNv of each material layer, ∂2Φ q = [n − p + C − − C + − 2C ] , (20) ∂x2 ε A D I where ε is the SL layer dielectric constant, CA− is the total ionized acceptor density for all acceptor species including − + As ,andCD is the total ionized donor density. Here the quantities CA− and CD+ are used in (20) to account also for predoping of the SL layers. In the absence of the electric field, (20) is just the charge-neutrality condition. To obtain the distribution of A− in the SL structure, (15)–(17), (19), and (20) need to be solved. In the SL struc- eq tures the quantities ni, Ei, Ev, Nv,andCso are constants inside the bulk of each SL layer. However, these quantities change from layer to layer and hence their spatial derivatives become important in the heterojunction region.

6 Results and discussions

Some available experimental results [7–9] have been fitted Fig. 4. The Be data of Humer-Hager et al. [7] in GaAs/AlGaAs SL ob- using the present Fermi-level effect and junction carrier contained by ion implantation, together with the calculated fitting curve. The centration effect model. Furthermore, some consequences of broken line is the as-implanted Be profile, the symbols are the Be data after the model are discussed. annealing, and the solid line is the fitting curve

Table 1. SL layer pre-doping condi- / eq tions of the experiment of Humer- Layer pre-doping type ni Ei Ev Nv mso concentration Hager et al. [7], and materials’ con- − − − stants used for obtaining the ﬁt. The /cm 3 /cm 3 /eV /eV /cm 3 values of ni, Ei, Ev,andNv are those at the experimental temperature of GaAs n+/2 × 1018 (0.1 m) 7.65 × 1016 −4.85 −5.5 6.02 × 1019 12 ◦ 860 C, referenced to the vacuum level n /2 × 1017 (0.1 m) eq of 0eV. The listed GaAs m o value is . . / × 17 . × 16 − − . × 19 s eq Al0 3Ga0 7As n 2 10 2 83 10 4.86 5.629 7 85 10 1 with respect to the Al . Ga . As C 0 3 0 7 so GaAs p /3 × 1018 7.65 × 1016 −4.85 −5.5 6.02 × 1019 12 layer value 17 16 19 Al0.3Ga0.7As n /2 × 10 2.83 × 10 −4.86 −5.629 7.85 × 10 1 GaAs n−/5 × 1016 7.65 × 1016 −4.85 −5.5 6.02 × 1019 12 15 Table 2. SL layer pre-doping condi- / eq tions of the experiment of Weber et Layer pre-doping type ni Ei Ev Nv Cso al. [8], and materials’ constants used concentration / −3 / −3 / / / −3 for obtaining the ﬁt. The values of ni, cm cm eV eV cm Ei, Ev and Nv are those at the experi- ◦ mental temperature of 550 C, refer- InGaAs n+/3 × 1019 2.2 ×1016 −4.8 −5.31 8.67×1019 100 enced to the vacuum level of 0eV.The InP n /3 × 1017 1.5 ×1016 −5.2 −5.65 2 ×1019 1 listed layer meq values are with respect so InGaAsP p /3× 1017 3.24×1016 −4.95 −5.43 5.17×1019 35 to the InP layer Ceq value. so InP n /3 × 1017 1.5 ×1016 −5.2 −5.65 2 ×1019 1

Fig. 5. The hole data (open circles)inanInGaAs/InP/InGaAsP/InP SL obtained by Weber et al. [8] using Zn indiffusion, together with the calculated ﬁtting curve (solid line)

eq and the relative Cso values among the layers. The values of the materials’ band-structure-related constants are those appropriate for the diffusion temperature of 860 ◦C, which are not directly available from the literature. The procedure of obtaining such high temperature values is a complex one discussed elsewhere [14]. Figure 5 shows our calculated fitting curve together with the experimental results of Weber et al. [8] obtained by meas- uring hole concentrations after diffusing Zn at 550 ◦C for 12 min into an InGaAs/InP/InGaAsP/InP SL. It is seen that the fit is excellent. For this case, our calculated Zn profile (not shown) is nearly identical to that of the holes. The used Zn in- eff( ) × −13 2 −1 trinsic diffusivity value Ds ni is 3 10 cm s ,andthe used intrinsic self-interstitial thermal equilibrium concentra- eq( ) × 9 −3 eff( ) eq( ) tion value CI ni is 5 10 cm .TheDs ni and CI ni values are in accordance with those used by Zimmermann et Fig. 6a,b. The Be data of Häussler et al. [9] with Be implanted into the ◦ al. [15]. The SL layer pre-doping conditions [8] are listed in InGaAs layer and annealed at 850 C for a 6s,andb 26 s. Dashed lines are Table 2, together with the used values of the materials’ con- the as-implanted data, open circles are those after annealing, and the solid lines are the calculated fitting curves stants for obtaining the fit. Figure 6 shows two examples of our fits to the experimental data of Häussler et al. [9], from which it is seen that eff( ) . × −13 2 −1 the fits are excellent. They obtained four sets of results by Be intrinsic diffusivity value Ds ni is 2 1 10 cm s implanting Be into InP/InGaAs structures and annealing at and the intrinsic self-interstitial thermal equilibrium concen- ◦ eq( ) . × 11 −3 850 C for 6 or 26 s, two with the implanted Be peak con- tration value CI ni is 3 7 10 cm . The used values centration at the layer interface, and two with the Be peak of the materials’ constants for obtaining the fits are listed inside the InGaAs layer. Those shown in Fig. 5 are the two in Table 3. The SL layer pre-doping conditions [9] are not latter cases. The degree of satisfaction of our fits to all four known and we have assumed that they are intrinsic to begin sets of their data are the same. For these cases [9], the used with. 16 Table 3. SL layer nature of the experiment of / eq Häussler et al. [9], and SL materials’ con- Layer pre-doping type ni Ei Ev Nv Cso stants used for obtaining the fit. The values concentration / −3 / −3 / / / −3 of ni, Ei, Ev,andNv are those at the cm cm eV eV cm experimental temperature of 850 ◦C, referenced to the vacuum level of 0eV. The listed In . Ga . As unknown 8 × 1017 −4.936 −5.35 1 ×1020 8.1 eq 0 47 0 53 In0.47Ga0.53As layer m o value is with respect × 16 − − × 19 eq s InP unknown 6 10 5.18 5.65 3.2 10 1 to the InP layer Cso value.

6.2 The segregation phenomenon obtains. For the ionized acceptor species, (α) The most outstanding feature of the shallow acceptor distribu- Cs mA− (α, β) = (25) tion data shown in Figs. 4–6 is the segregation of dopants in Cs(β) the adjacent SL layers to a sizable magnitude. Our fittings to n (α) p(β) = meq i these data, consequently, are simultaneous descriptions of the Ao n (β) p(α) acceptor diffusion and segregation processes. The main fea- i E (α) − E (α) − E (β) + E (β) tures of our treatment of the segregation process can be more × exp − i v i v clearly understood by an examination of the steady-state seg- kBT regation phenomenon. For two adjacent SL layer materials α and β, the thermal equilibrium segregation coefficient of an obtains. ionized shallow acceptor species is obtained from (7) as An outstanding aspect of (25) is that eq eq mA− (α, β) 6= m − (α, β) , (26) eq Cs (α) A m − (α, β) = (21) A eq(β) Cs because the quantity p(β)/p(α) is not equal to peq(β)/peq(α). (α) eq(β) eq ni p In (25), p(α) and p(β) satisfy (22) provided the ‘eq’ designa- = m o A (β) eq(α) − (α, β) ni p tion is removed. Another outstanding aspect is that mA is dependent upon the doping level. Knowing either the total Ei(α) − Ev(α) − Ei(β) + Ev(β) × exp − . acceptor dose or the acceptor concentration in either layer, k T B (25) can be solved. The former condition applies to the cases In (21) the thermal equilibrium hole concentrations peq(α) shown in Figs. 4 and 6 for which a given dose of Be has been and peq(β) satisfy the charge neutrality conditions introduced into the sample by ion implantation, and the lat- q ter condition applies to cases resembling that shown in Fig. 5, 1 for which Zn is diffused into the SL using an external source peq(α) = Ceq(α) + Ceq(α) 2 + 4n2(α) , (22a) 2 s s i material, if it is not the unique equilibrium source material which determines the Zn solubility values in the SL. In fact, q since the Zn source material used by Weber et al. is ZnAs2 [8] 1 eq 2 for obtaining the results shown in Fig. 5, it is not the thermal peq(β) = Ceq(β) + C (β) + 4n2(β) , (22b) 2 s s i equilibrium source material because the InP layers contain no As atoms. There are two limiting cases of (25). For suffi- 2+ when the concentrations of I and other dopant species are ciently low doping conditions for which p ≈ ni holds in both III eq assumed to be small. Moreover, mAo in (21) is the thermal materials, (25) reduces to equilibrium segregation coefficient of the neutral acceptor − (α, β) = (α, β) species given by mA mAo (27) E (α) − E (α) − E (β) + E (β) eq(α) f (α) − f (β) × exp i v i v , eq Cso gAo gAo m (α, β) = = exp − . (23) kBT Ao eq(β) Cso kBT while for sufficiently high doping conditions for which the eq (α, β) ≈ The quantity mAo is constant at a given temperature, self-doping condition p Cs holds in both materials, (25) re- f (α) because, as has been mentioned before, gAo is deter- duces to mined chemically, which is independent of the semiconductor 1 charge-carrier type and concentrations. On the other hand, ni(α) 2 − (α, β) = (α, β) it is seen from (21) that the thermal equilibrium segregation mA mAo (β) (28) − eq ni coefficient of A , m − (α, β), is further dependent upon the A E (α) − E (α) − E (β) + E (β) charges in the semiconductor structure. × exp i v i v . In practice, an acceptor species may be introduced into 2kBT a SL at concentration levels different from its solubility value, As an example, Fig. 7 shows the calculated steady-state seg- and its distribution in the SL layers will reach a steady state regation coefficient of Zn− between the first two layers of for long annealing times. For such cases, (7) still applies. For materials of the case shown in Fig. 5. The materials involved the neutral acceptor species, are InGaAs and InP, and the results are given as a function of eq Zn concentration in InP. o (α) C o (α) (α, β) = Cs = s = eq (α, β) , mAo eq mAo (24) Equation 25 is also obtained from integrating (16) by Cso (β) C o (β) s noting that in steady state the Cs flux vanishes. Thus, our 17

Fig. 7. Calculated steady-state Zn segregation coefficient in the bulk of the Fig. 8. Calculated results for fitting the experimental data of Weber et al. [8] first two layers of materials of the example shown in Fig. 5, i.e., InGaAs shown in Fig. 5, with and without the inclusion of the junction carrier and InP.TheInP layer Zn concentration is used as the independent variable. concentration effect (or junction electric field effect). The results with the The dashed line is the chemically determined part, which is strictly valid junction carrier concentration effect included fit the experimental data well, only for the neutral Zn atoms. The solid line is that of ionized Zn,whichis see Fig. 4. The results without the junction carrier concentration effect in- determined chemically as well as by the Fermi-level effect cluded do not fit the experimental data: the Zn indiffusion process is too slow formulation of the present problem is consistent with the ther- same as that in the InP layer bulk. In the presence of junction- modynamic definitions of the involved quantities. region hole concentration effect, the junction flux becomes much larger because at the junction region on the InP side 6.3 Role of the junction carrier concentrations there is a hole accumulation, resulting in a hole concentration that is much larger than that in the InP layer bulk on The heterojunction carrier concentration influences the con- an order of magnitude basis. Meanwhile, also on an order- − 2+ − centrations of As and of IIII , and hence also the As distri- of-magnitude basis, the depletion of holes on the InGaAs or bution rate throughout the SL. Since the junction and hence InGaAsP sides of the appropriate junctions is not too sig- also its electric property physically exist, in general, the effect nificant, because the bulk hole concentrations in these layers should be taken into account. However, to avoid the com- are orders of magnitude larger than those in the adjacent plexity involved, to ignore this effect seems to be a common InP layers. practice. We believe that whether this effect can be ignored is dependent upon the nature of the problem. The effect should have played a minor role in long-time high-temperature ex- 6.4 Previous attempts periments, but a significant role in low-temperature and/or short-time experiments. In general, it will not be a prudent It appears that there exist three prior attempts in modeling approach to just ignore this effect. the acceptor diffusion–segregation phenomenon [8, 16, 17], to The junction carrier concentration effect has been found various degrees of satisfaction in fitting the experimental data. to be fairly large for the Zn indiffusion case shown in Fig. 5. In the first attempt, Weber et al. [8] provided a simu- Figure 8 shows the calculated results with and without in- lation of their own experimental data, those shown in cluding this effect. The latter results are obtained by letting Fig. 5, by assuming that the solubilities in the layers of the ∂Φ/∂x = 0 in (19) and ∂2Φ/∂x2 = 0 in (20), which sim- InGaAs/InP/InGaP/InP structure are of different constant ply amounts to the use of the charge-neutrality condition values. By diffusing Zn into individual InGaAs and InP throughout the SL structure. From Fig. 8, it is seen that, materials at 550 ◦C, they found an 80 times Zn solubility dif- without including the junction carrier concentration (or junc- ference. Their analysis provided a rough approximation to the − eq tion field) effect, the As concentrations in the three buried complicated situation, in the sense that the effect of factor mso layers, InP, InGaAlP,andInP, are significantly lower than has been in principal accounted for, but the Fermi-level effect eq those in the case of including the effect. Physically, this out- contribution to ms− was ignored. come is understood by noting that the hole thermal equilib- The second attempt is that of Bracht et al. [16], who fit- rium concentrations in the InGaAs and InGaAsP layers are ted the experimental data shown in the present Figs. 4–6 larger than those in the InP layers by orders of magnitude. approximately by assuming a constant dopant solubility dif- During the A− distribution process, one rate-limiting factor ference in the SL layers due to an electronic effect given by − s is the A fluxes across a heterojunction. Without including exp(−δε/kBT),whereδε is the Ev difference of two adjacent s eq the junction-region hole concentration effect, the flux is con- SL layers. This means that mso is taken to be 1 among the trolled by a relatively small hole concentration that is the layers. Their electronic effect is arrived at via a Fermi-level 18 stabilization energy (EFS) argument, which bears a resem- 7 Conclusions blance to the presently discussed Fermi-level effect model. We believe that the Fermi-level stabilization energy concept In conclusion, we mention that the observed p-dopant segre- is inherently illusive and difficult to understand. Furthermore, gation behavior in SL layers in short annealing time and/or eq from the presently used mso values listed in Tables 1–3 all of low annealing temperature experiments has been satisfacto- which do not equal to 1, it is most probable that the attempt is rily explained using a model incorporating three effects. The physically incorrect. overall dopant segregation behavior results from the chem- The third attempt concerns our own preliminary ef- ical effect on the neutral acceptor species thermal equilibrium fort [17]. Irrespective of the apparently satisfactory fits ob- concentrations, the Fermi-level effect on ionized acceptor and tained, which are very close to the present ones, the report charged point defect concentrations, and the junction carrier- contained errors. First, the segregation effect was attributed concentration effect on the dopant distribution kinetics in the to the junction electric field effect alone, which is incorrect. SL layers. In principle, in SL layers of different chemical Second, in what should have been the present (16), the terms compositions the diffusivities of the ionized dopant atoms concerning the role of ni and of Cso were missing. The miss- should be different. The present satisfactory fits of the ex- ing of the ni term was accidental and in the actual calculation perimental data means that, quantitatively, the contribution of the effect was included. The missing of the Cso term is due this factor to the observed dopant segregation phenomenon eq to the use of the assumption m o = 1. In obtaining the fits, is fairly small. This Fermi-level effect model has also pro- eq s the effect of mso does not equal to 1 was compensated by vided satisfactory fits to available boron distribution profiles an adjustment in the Ev value differences among the adja- in GexSi1−x /Si heterostructures, see the accompanying art- cent layers. This is possible because, in (16), Ei, Ev,andCso icle [1], and its application to a number of n-type doping constitute similarity terms in the sense that they are of an effects in a variety of semiconductor heterostructures will be identical mathematical form. It is obvious that the mathemat- reported shortly [2]. ical forms of the Ei and Ev terms in (16) are identical. By eq ∂ eq/∂ / eq noting that Cso is given by (2), the term Cso x Cso in f (16) is equal to − ∂g o /∂x /(kBT), which is also of an iden- A References tical mathematical form of that of the Ei and Ev terms. Thus, f since the values of Ei, Ev, gAo and their differences are all of the order of a couple to a few eV, it is seen that the misuse of 1. C.-H. Chen, U.M. Gösele, T.Y. Tan: Appl. Phys. A, 68, 19 (1999) 2. C.-H. Chen, U.M. Gösele, T.Y. Tan: Appl. Phys. A, to be submitted the value of one quantity can be readily compensated by using 3. D.G. Deppe, N. Holonyak Jr.: J. Appl. Phys. 64, R93 (1988) also a wrong value for another quantity. This means that, to 4. T.Y. Tan, U. Gösele, S. Yu: Crit. Rev. Solid State Mater. Sci. 17,47 solve the problem, we need to know the values of Ei, Ev,and (1991) eq 5. T.Y. Tan, U. Gösele: Appl. Phys. Lett. 52, 1240 (1988) C o accurately, or else a fitting to the experimental data can s 6. S. Yu, T.Y. Tan, U.M. Gösele: J. Appl. Phys. 69, 3547 (1991) be obtained by accounting for the effects of these quantities 7. T. Humer-Hager, R. Treichler, P. Wurzinger, H. Tews, P. Zwicknagl: by an arbitrary value for any one of the three or in any combi- J. Appl. Phys. 66, 181 (1989) nation, which can result in an erroneous interpretation of the 8. R. Weber, A. Paraskevopoulos, H. Schroeter-Janssen, H.G. Bach: eq = J. Electrochem. Soc. 138, 2812 (1991) physics involved. Under the mso 1 assumption, our previous fitting parameters for the InGaAs and InP layers of Fig. 5 9. W. Häussler, J.W. Walter, J. Müller: In Ion Beam Processing of Ad- vanced Electronic Materials, ed. by N.W. Cheung, A.D. Marwick, will lead to unacceptable Ei values that are too close to the Ec J.B. Roberto, Proc. Mat. Res. Soc. Symp. Vol. 147 (Mat. Res. Soc., values of the appropriate layer materials. Thus, presently, the Pittsburgh, PA 1989) p. 333 eq = mso 1 assumption is abandoned. The presently used materi- 10. C.-H. Chen, U. Gösele, T.Y. Tan: In Semiconductor Process and eq Device Performance Modeling, ed. by J.S. Nelson, C.D. Wilson, als’ constants and mso values given in Tables 1–3 are judged to be reasonable, but it is almost certain that their accuracies S.T. Dunham (1997 Mater. Res. Soc. Fall Meeting, Symposium Q, Boston, Dec. 1-5, 1997). In press can be further improved when more band energy information 11. H.-M. You, U.M. Gösele, T.Y. Tan: J. Appl. Phys. 74, 2461 (1993) becomes available for these materials. Since the assumption 12. S. Yu, U.M. Gösele, T.Y. Tan: J. Appl. Phys. 69, 3547 (1991) eq = of Bracht et al. [16] also involves mso 1, we suppose that 13. W. Jüngling, P. Pichler, S. Selberherr, E. Guerrero, H.W. Pötzl: IEEE the same situation will hold also for their attempt. Trans. Electron Devices ED-32, 156 (1985) From the values given in Tables 1–3, it is seen that the 14. C.-H. Chen: Ph.D. thesis, Duke University (available: December 1998) 15. H. Zimmermann, U. Gösele, T.Y. Tan: J. Appl. Phys. 73, 150 (1993) segregation coefficient of the neutral acceptor species Ao, eq 16. H. Bracht, W. Walukiewicz, E.E. 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