J. Phys. Chem. 1995,99, 14486-14493
14486
Thermodynamic Stability of pln Junctions Jean Franqois Guillemoles,*”i*Igor Lubomirsky? Ilan Riess2 and David Cahen*f* Department of Materials and Interfaces, Weizmann Institute of Science, 76100 Rehovot, Israel, and Physics Department, Technion, Israel Institute of Technology, 32000 Haifa, Israel Received: May 23, 1995; In Final Form: July 2, 1995@
In view of our recent experimental finding of self-restoration of p/n junctions in Ag-doped (Cd,Hg)Te after their electrical or thermal perturbation, we ask the question if, and if so, when can, a mixed electronic semiconductorhonic conductor support a built-in electric field. The question is of interest because common p/n junctions are merely kinetically stabilized systems. We study the problem by deriving the thermodynamically stable states of mixed conductors. This shows that (1) as long as all components of a multicomponent system behave ideally, no stable concentration gradient and built-in field may exist; (2) a thermodynamically stable concentration gradient and thus a built-in field can exist in a multicomponent system, if at least one of its components behaves nonideally (and thus, from the Gibbs-Duhem relation, at least one additional component must behave nonideally, too); and (3) the likelihood of finding a thermodynamically stable concentration gradient increases with the number of components of the system. While the first of these results is intuitively obvious, the rigorous proof given here is necessary to deduce that actual observation of self-restoration of p/n junctions implies nonideal behavior of at least two of the mobile species in the system. We show that our results can be used to derive the built-in electric field for a given variation of activity coefficients of one or more of the mobile species and vice versa.
I. Introduction (a) General Background. A p/n junction is often explained as the equilibrium state of a system containing in it both a pand an n-type semiconductor.’ In fact, equilibrium here refers only to the electronic subsystem. The atomic subsystem is clearly not at equilibrium because the concentrations of donors and acceptors in the different parts have to be sufficiently different to yield different electronic properties, necessary for making certain parts n and other parts p. Furthermore, the builtin electric field acts, in the case of the donors and acceptors, in the same direction as their chemical potential gradients. This field results from the equilibration of electronic carriers and it keeps the electrons and holes apart by balancing their chemical potential gradients. At first sight common sense tells us that the atomic subsystem is only a kinetically stabilized state of such a system. Its “stability” is a function of the effective mobilities and diffusion coefficients of the dopants that make the semiconductor n or p. The question is now if this commonsense conclusion is also rigorously correct. This question of possible true thermodynamic stability of built-in electrical fields in physical systems is of more than academic interest. With the miniaturization of electronic devices (and thus of p/n junctions), this requirement might become more stringent as kinetic stability is less and less sufficient. On the one hand, from dimensional analysis, kinetic stability should decrease as the square of dimensions and on the other hand, the stronger built-in electrical fields2and concentration gradients in smaller devices enhance the migration of doping impurities, thus limiting the lifetime of the devices. (b) Experimental Background. A p/n junction, which is able to restore its rectifying properties after electrical or thermal perturbations, was discovered recently in systems of Ag, diffused
* Authors for correspondence.
’ On leave from CNRSEcole Nationale Superieure de Chimie de Paris. 5 @
Weizmann Institute of Science. Israel Institute of Technology. Abstract published in Advance ACS Abstracts, September 1, 1995.
0022-3654/95/2099-1486$09.00/0
into (Hg,Cd)Te (MCT).3 The origin of such behavior is not yet well understood, and the question of true thermodynamic equilibrium versus kinetic stability in this system (and in related ones) arises. Kinetic stability of the created junctions can be questioned because (a) Ag is a fast diffuser in (Hg,Cd)Te, and the measured diffusion coefficient does not fit with the longterm stability (> 16 months) of the diodes made in this way; (b) upon perturbation of the junction (by heating for instance), which causes a grading of the built-in electrical field, the device restores itself close to its initial properties, within 3- 120 days, depending on how strong the perturbation was; (c) for a given sample the Ag-rich region has a defined silver concentration, which is recovered by restoration, after it has been changed by the external perturbation. (c) Statement of the Problem. Can a p/n Junction be a Thermodynamically Stable State? If So, What Will Be the Conditions? We will give here an analysis of systems with charged mobile species to answer these questions. To start, we need ta define what we mean by the system under study. We consider a system which is (1) globally neutral and ( 2 ) sufficiently large so that we can assume that its surface energy can be neglected, compared to its total energy. (d) Outline of the Paper. The structure of the remainder of this article is as follows: In parts I1 and I11 we analyze ideal systems in terms of the existence and uniqueness of equilibrium conditions. These analyses are extended to nonideal systems in part IV. On the basis of this we propose a classification of the different cases in part V. In part VI we apply the results of these analyses to real systems. Part VI1 summarizes our findings.
11. Ideal Systems
(a) Ideal System Definition. We define ideal systems as systems for which the activity coefficients of the mobile charged species equals 1. This means that their electrochemical potential takes the form 0 1995 American Chemical Society
Thermodynamic Stability of p/n Junctions
J. Phys. Chem., Vol. 99, No. 39, 1995 14487 Reservoir
where #(x) is the position-dependentelectrical potential, whose reference energy level may be chosen arbitrary, ,uko is the electrochemical potential of species sk under standard conditions and ck is the concentration of the species. This relation holds whenever the concentration of charged species is sufficiently low so that no interactions between species occur (ideal solution): Ideal behavior is often referred to, explicitly or implicitly, when dealing with mixed ionic/electronic conductors5 or with impurities in semiconductors. In the latter case this is justified because they can be considered as very dilute systems in most cases6 This is necessary to assure that charged species do not interact among themselves. Therefore we consider a host matrix (solvent) with a low impurity concentration in it. (b) Preliminary Remarks. We will show below that, no matter what the state of charge of the species, whatever the complex between solvent (or host matrix) and solutes or between solutes that may form, no built-in electrical potential difference, A#, can exist in an ideal system in thermodynamic equilibrium. Such a proof is important (a) because standard treatments of mixed conductors deal with ideal systems5 and (b) because of its logical consequence; namely, IF there is a A#, this must be due to nonideal behavior of the system. Such behavior will be considered in the later parts of this paper. Some points need to be stressed. (1) Even if the above statements on ideal systems may seem trivial, they require careful analyses, because the thermodynamics of systems with Coulombic charges is far from straightforward and is often pr~blematic.~ (2) We did not find proofs for these statements in the literature. It is well-known that for a globally neutral system a state with no built-in electric field and a homogeneous distribution of species constitutes an extremum of free energy,8 but nothing proves that it is the only state or that it is always a minimum. If not, it would be possible to have a built-in electric field as a local minimum of the free energy of the system, Le. at least as a macroscopic metastable state.9 (3) The analysis of the ideal system will be helpful when it comes to analyzing the behavior of nonideal systems.
111. Analyses (a) Definitions and Conditions. We consider the relation between components of an ideal system. Let us consider an ideal system composed of a number N of species S k (including electrons and holes), each with its charge qzk and its concentration ck. A schematic picture of such a system is shown in Figure 1. The solvent or the host matrix can be included as a neutral component (z = 0). Unless specified, we consider no external electric field: the electrical potential results from the charged components only. For each component of an ideal system, the electrochemical potential is given by eq 1. The above expression holds for very dilute, charged impurities only. Because uncharged species cannot lead to p/n junction formation, they are not important in our case. Therefore we do not make any assumptions concerning their ideality or dilution. When the system is in equilibrium, the electrochemical potential is the same in all parts of the system for each component. This electrochemical potential, P k , is a constant for each species. Following ref 10, and to keep a uniform formalism, we use the terminology of electrochemical potential rather than of the Fermi level, also for electrons. From eq 1 it follows that
I
1
I
1
System
Figure 1. (Top) Schematic of the systems considered, here exemplified as a p/n junction. Shaded areas are the space charge regions. Dotted regions represent possible external phases that can serve as reservoirs for one or more species. (Bottom) Representation of the local charge density. The extent of the system itself is shown by the double arrow.
i.e. at equilibrium each component of the system follows a Boltzmann distribution. For instance, for free electrons (k = 1, Zk = -1)
($1
C,= n = no exp
(3)
so that the concentration of any species can be expressed as n o
where ak is a positive constant. (b) Determination of Equilibrium for an Ideal System. The condition that the electrochemical potential is constant is not sufficient to define the equilibrium states. To define these, we would have to examine the consequences of the (Gibbs) free energy of the global system being at a minimum with respect to transformationsof the system at constant temperature and pressure. We would also have to check that the second variation of the relevant thermodynamic function (d2G) is strictly positive. This path actually leads to still unsolved difficulties in the theory of the kind described in refs 7b,d, concerning the definition and handling of thermodynamic functions for single ions. However, this type of derivation is not needed for exploring the consequences of nonideal behavior for systems that are of practical interest. We present now one of the simpler alternative paths that are possible. Let us consider the system described in (a) above. For simplicity we consider a one-dimensional case. The charge density everywhere is given by N
N
k= 1
k= 1
(5) The Poisson equation for this case may written as
dE d E d n qN div(E) = - ------ --zzkakn-" dx dn dx E E k = l From eq 3 we get
so that
(6)
Guillemoles et al.
14488 J. Phys. Chem., Vol. 99, No. 39, 1995 1.2 1 and, from eq 6
0.8 3 0.6 C
Y
Analysis of eq 6 shows that at the edges of the sample, the electric field is equal to zero." If we assume that there is a built-in electric field, then P ( n ) is always definitely positive wherever there is such field. Because l? = 0 at the edges, it will have at least one maximum in between. A typical plot of l?(n) for a p/n junction is given in Figure 2. At the maximum d(P)/dn = 0, which, by using eq 8, is equivalent to
'k 0.4
0.2 0
io-6 o.oooi
0.01 1 100 n (normalised)
io4
io6
N
(9) The charges of the various species k have a lower and upper bound. Let pl be the charge of the most negative particles @I < 0) and p2 the charge of the most positive particles @ 2 > 0); then dl zk are in the range p~ 4 Zk p2. Multiplying eq 9 by nPl+' leads then to N
0.6 Q
.-8
-2
5
0.5 0.4
0.3
'k 0.2 0.1
It is possible to show that eq 10 has one and only one positive solution for n (cf. Appendix A). This stems from a characteristic of the function P(n): the coefficients of the terms of order higher than a certain integer ~GJare all positive and those of lower order are all negative; moreover, since the system is globally neutral, there is at least one non-zero negative coefficient and at least one non-zero positive coefficient. Because we have at least one positive and one negative term in eq 8, we see that
0 0.0001o.ooi 0.01 0.1
1
i o 100 1000
104
n (normalised) Figure 2. Typical plot of Ez vs n (= normalized electron concentration) for an abrupt (top) and a linearly graded (bottom) p/n junction. Note the logarithmic x-axis. N A = lo6 and ND = lo6 are the normalized acceptor and donor concentrations in the p-type and n-type regions of
the abrupt junction, respectively. The normalized gradient of ND N A was taken as 1O'O cm-I for the graded junction. All the concentrations are normalized to the intrinsic carrier concentration. not, dilute or not)
which means that the only possible solution of eq 10 is a minimum, rather than a maximum of @(n). The direct implication of this contradiction is that, in an ideal system, at equilibrium, the electric field and charge density are equal to zero everywhere. The concentration of all charged species everywhere in the sample is constant and unique for a given set of conditions. This concentration is defined by eq 9. These results have an important consequence. In many systems of practical interest, like p/n junctions in semiconductors, there are at least two regions which are both electrically neutral and differ in composition (Figure 1). It follows from the preceding discussion that these systems are either kinetically stabilized (the general case) or nonideal (a case that we will examine below) because two neutral regions with different compositions cannot coexist in equilibrium in an ideal system.
,ilk = ,il: i- kT ln(A,C,)
+ zkq@
with
A, = &(Cj,T,...) (1 1)
where &, the activity coefficient, depends on the temperature (Z') and composition (CJ. In principle the coefficient 1 k can depend also on the electric field through the influence of the electric field on the interactions between the components. Because an ideal system cannot explain the experimental observations, we will try to find which kind of nonideality, i.e. what class of functions for &, gives a stable p/n junction. The analysis given in the preceding section can give some hints for this. In equilibrium the electrochemical potential is constant throughout the sample. Therefore we can write, in analogy with eq 4
IV. Application to Nonideal Systems Systems with charged species can hardly be ideal, as is wellknown. Their nonideality, due to Coulombic interactions in them, can be taken into account by introducing in eq 1 an activity coefficient, 1,computed by Debye and H i i ~ k e l .The ~~ result is a ,general expression, valid for all species (charged or
If we consider now a pJn junction (or n+Jn and p+Jp junctions), there are at least three zones where electroneutrality holds: the neutral n region, the neutral p region, and the transition region (Figure 1). In each region we have, in analogy with eq 10
Thermodynamic Stability of p/n Junctions N
J. Phys. Chem., Vol. 99, No. 39, 1995 14489 where
In the ideal case Akv = ak everywhere and only one solution exists. In the nonideal case it is possible to have two different series of concentrations compatible with eq 13 because Akv can have different values for different sets of {Ck}. For instance, we can have A," = 1 in one region and A," = constant t 1 in another region, for a single component Sj. In that case, the two polynomials P ( n )will each have a unique solution, the solution being different for the two polynomials (Le. in the two regions) and these regions with different electronic concentrations will be in equilibrium. If we are considering a pln junction, the activity coefficients in each of the two regions must be such that n > ni in the n-type region and vice versa in the p-type region. This means that at least one electrically active impurity is not distributed uniformly at equilibrium; Le. it has segregated. We stress that the term segregation is used here a bit differently from its usual use. It describes the formation of regions with different impurity concentrations. This should be distinguished from the situation where the impurity segregates out by itself or as a major component of a different phase. At the beginning of this section, we considered a possible dependence of the & on the electrical field, but the above considerations show that this is not critical, because the existence of two solutions for eq 13 in the neutral regions does not depend on it. In the following, we will therefore not consider that possibility further. We note that the built-in electrical field will be thermodynamically stable only below a critical temperature, because of the entropic term in the free energy. These considerations give a necessary condition, but this is by no means sufficient. We still have to prove that spontaneous segregation of impurities may really occur. This question is addressed now.
V. Nonideal System Classification To be able to classify the behavior of mixed conductors, we now need a criterion that distinguishes between the cases when spontaneous segregation of impurities occurs and the cases when no such phenomena can be observed. We shall consider an initially homogeneous system (which, therefore, cannot have a built-in electrical field) and check its stability for small fluctuations. In a homogeneous system, we can write (considering a unit volume)
12 E
...
...
Here the labeling of the components (1,2, ...,N) is not important, from the mathematical point of view (Le. we do not need to keep label 1 for electrons, as the labels can be permutated). The condition det(P,",j) = 0 stems from the Gibbs-Duhem equality. (The case where det(Ff2~~,) = 0 is more complicated and is described in ref 12). Let us see how condition (15) can be fulfilled. Mathematically, this means that all minors of the matrix f i 2 * M ~are >O. To illustrate the consequences of this, we consider a system with two atomic components, of which one is ionized and dilute (CN).We assume for simplicity that the nonideality for electrons and holes arises only from Coulombic interaction in the dilute systems that we consider here (i.e. the limiting Debye-Hiickel expression is validI3). In that case, the condition of stability, if we take into account the dilute nature of the system, simplifies to
where TF stands for the thermodynamic factor.I4 As is clear from condition (16) segregation of the ionized impurity, if it is the only nonideal species, requires [TF] < 0. This can also be understood from the Gibbs-Duhem relation, &.&Vi = 0. This relation implies 'that if one component is nonideal then there should be at least one more component whose behavior deviates from ideality. If we consider now two ionized species, the same condition becomes aPN aPk
aPN 'Pk
0 (cf. section IIIb above) yield the following most restrictive condition on the activity coefficients:I2
In this case the necessary condition for spontaneous segregation is that the cross-term (a,uNlaCk)2has to be larger than the product of the two TF's, (apNlaCN)(apk/aCk). Generalizing this finding, we can make the following observation: the larger the number of components, the more likely it is that condition (15) will break down in some region of the phase diagram and the smaller the strength of interaction or the concentration of species that is needed for this (due to a cumulative effect). Relations between Activity Coefficients and Built-in Electric Field. Let us now consider the case where the stable state is not a state of zero concentration gradient, but nevertheless the system remains single phase, from the point of view that all parameters remain continuous throughout the system.
Guillemoles et al.
14490 J. Phys. Chem., Vol. 99, No. 39, I995 We will assume furthermore that formation of the quasiinterface, i.e. the region where some of the parameters (potential, concentration) change, does not cost any extra energy. For the one-dimensional case, we can write eq 8 as
where we have changed a to A; i.e. we consider the nonideal case now. Because the activity coefficients are different in different parts of the system, the requirement d(,??)ldn = 0 may be fulfilled in more than one location; i.e. a thermodynamically stable built-in field may exist. The coefficients Ak that correspond to such a profile are given by eq 19, taken as a functional equation. Thus any built-in electric field can, in principle, be thermodynamically stable, if the activity coefficients of the species involved have a suitable compositional dependence. Conversely, if the compositional dependence of the activity coefficients is known, the stable builtin electrical field can be computed through integration of eq 19. This is shown, using a simple example, in Appendix B, as illustrated in Figure 5. This example shows that the Coulombic forces tend to smooth the transition region, something that can be contrasted with the abrupt transition that results when segregation occurs with uncharged species. Abrupt vs Smooth Transition. Figure 5 (Appendix B) shows a smooth transition of charged dopant concentration between the two regions. This can be understood as follows. Suppose we have an equilibrium situation with an abrupt transition of the concentration of a charged impurity between two regions, each of which is homogeneous. Then the concentrations of the impurities are constant; i.e. their chemical potential is constant in a given region. To have an abrupt transition in such a case, the electrical potential would have to change abruptly, to keep the electrochemical potential constant, i.e. to maintain the system in equilibrium. However, because the electrical potential is a continuous function of spatial coordinate, this cannot be so. Thus, if such a situation were approached, mobile charged impurities would redistribute themselves, thus smoothing out the transition region. This is the case for a pln junction, as far as the electronic charge carriers are concerned. Thus the transition occurs over a region, the size of that over which the electric field extends; i.e. the impurity concentration will vary smoothly rather than abruptly (cf. also Appendix B). This fits with our experimental SIMS data for the Ag:(Hg,Cd)Te system, which reveal clear, smooth transitions, occurring over -1 pm.20
VI. Relevance to Experimental Systems As has become clear from the previous discussion, a central issue in our search for systems that are in thermodynamic equilibrium (6G = 0 and d2G > 0) and maintain an electric field, is the nature of the interface between the two regions with different impurity concentrations. This is not a trivial issue in materials with relatively high concentrations of defects. In them a wide gamut of situations, ranging from isolated point defects in a well-defined phase to defect ordering, leading to phase transition, can be found. This can be understood by looking at such well-studied families of compounds as metal oxides. Furthermore if the defects are charged, this will lead to an electric field. We need to consider this, especially when these defects are mobile, because it will affect the nature of the interface. This will be discussed through a simple example of a Schottky barrier.
Si
*l
7 E7
I
I
Figure 3. Electron energy band diagram of the AYp-type Si Schottky barrier.
From Isolated Defects to Defect ‘Thases”. In metal oxides, increasing concentrations of defects are accommodated in a number of ways. Initially, none of those involves necessarily long-range order. The idealized situation (where isolated, noninteracting point defects serve to accommodate nonstoichiometry) breaks down at densities that can be as low as 10l8 cm-3 or even less, depending, among others, on the degree of ionicity (dielectric constant) of the material. Ionicity enters because it will determine the maximum distance across which point defects can interact. When point defects start to interact beyond the stage of simple complexes (e.g. VCd-hCd in In-doped CdTe),I5 two main types of reconstructions are found, namely defect assimilation and defect elimination.16 In both cases intermediate stages exist, without long-range order, e.g. cluster formation (as in Fe-0, “Roth” and “Koch” clusters at increasing vacancy concentrations”) and the formation of isolated planar defects such as stacking faults (which are also found in II-VI’S~~). Stable Junctions in Two-Phase Systems. Assuming that junction stability is due to some type of phase separation, let us see if we can find a “normal” two-phase system with a selfstabilizing junction. Built-in electrical fields are found easily in nature by looking at interfaces between different phases, but self-stabilization or restoration of junctions is another matter. One of the conceptually simplest systems is the AI-Si system. Because the mutual solubility is low,l9 we expect to have an abrupt interface between a solid solution of Si in A1 and a solid solution of A1 in Si. In the absence of an electrical field at the interface, the composition in each phase at equilibrium would be given by the solubility of the solute. If we form an AVp-Si:Al Schottky barrier (Figure 3), and if the Si is not saturated with Al, there will be a chemical driving force for Al to diffuse into Si. But because AI is a p-type dopant in Si, the built-in electrical field will counterbalance the chemical diffusion (A1 in Si is an acceptor; Le. it bears a negative charge) or drive back to the metal any excess A1 in the Si.*O This is in contrast to the normal p/n junction, discussed in the Introduction. We may then obtain a stable junction. Here, we have clearly a two-phase system, each phase with very different structures and properties, and it can be considered as an example of a heterojunction. As discussed in the Introduction, the behavior of such systems might become critical for devices involving small length scales. Stability in Ag:MCT. Let us now recall the initial motivation for this analysis, i.e. the unusual behavior of Ag in MCT suggesting the existence of a pln junction in thermodynamic equilibrium. The system presented in the Introduction has no observable discontinuous transition between phases such as what occurs at a macroscopic interface (Le. there is no metallurgical interface). Therefore the interfacial energy is due only due to
Thermodynamic Stability of pln Junctions electrostatic energy in the space charge region, as is the case in homojunctions. This pln junction can thus be considered as a homojunction. However our analysis does point to a two-phase system (the formalism used in section V [eqs 15-18] is usually meant to describe phase transitions). One way to look at the interface between the two regions as a phase boundary is by viewing the impurities in the matrix as a nonideal gas which is present in different states on the two sides of the interface. We are now able to compare the results of our thermodynamic analysis with experiments. From this we find that the system has to be nonideal. Furthermore from basic thermodynamics we know that the concentrations of the various species in the two adjacent regions are fixed by external parameters (temperature, pressure, total amount of each species). The simplest system that we can consider, conforming to these requirements, is a two-component one: silver and an effective medium. In that case, all devices should have similar concentrations of Ag on both sides. However, experimental SIMS result^^^^' contradict the existence of one common Ag concentration for different MCT crystals. Therefore, on the basis of our analysis, we conclude that at least one of the other components of the system participates in the process and is mobile (most likely Hg; Cd, Te, and In, as dopant, are other possibilities). Once all this is deduced, we can ask if there is any support for the idea that Ag segregates from the point of view of interatomic interactions. From percolation theory we know that collective phenomena can occur when the average distance between interacting centers, R, is smaller or their concentration, N, is larger than what is given by the relation224/3nN(2R)3M 2.7. For the experimental conditions of ref 3 (Ag concentration in p-region of -101s-1019 ~ m - R ~ will ) be 2-4.5 nm. In other compounds (for instance PbTe:Cd23), collective behavior was found for concentrations as low as 5 x lOI9 ~ m - corresponding ~, to R x 1.2 nm. For many applications a rather large impurity concentration is required (particularly for small devices) or unavoidable (self-compensation effects in compound semiconductors), which increases the likelihood of such collective behavior. In the specific case of MCT:Ag:In, the anomalous behavior could only be observed when the Hg fraction in the MCT alloy was above a certain thre~hold.~ This suggests that Hg in MCT mediates the interaction, thus yielding a deviation from ideality that is much larger than could be expected for the nominal Ag concentration.
VII. Conclusion We have used general thermodynamic arguments to clarify the issue of stability of a built-in electrical field. First we derived the expected conclusion that only a uniform distribution of mobile species can exist in equilibrium in an ideal system. The condition of ideality for any system restricts drastically its equilibrium behavior, because, as we have shown, no builtin voltage can exist in ideal systems. Thus, the reason for apparently thermodynamically stable pln junctions has to be found in nonideal behavior of one or more components of the system. We have shown how the probability that nonideal behavior will lead to a thermodynamically stable built-in electric field increases as the number of components in the system increases. It was also possible to get an analytical relationship between the nonideal behavior (activity of the electrically active species) and the equilibrium built-in electric field. The pln junctions that result can be considered as resulting from “phase” separation, but with smooth, rather than abrupt, transitions, because
J. Phys. Chem., Vol. 99, No. 39, 1995 14491
1.5 1
0.5
0 -0.5 -1
-1.5 -2 -0.5
0
0.5
X
1
1.5
Figure 4. Examples of behavior of polynomials like that given by eq 10, written as P(n) = Xakn (to group like-charged ions together): (a) a1 = -1; (b) a1 = 0; (c) a0 is the only non-zero negative coefficient. The polynomials that are represented are of degree 5 , in each case, a0 = -1.
of the condition of continuity of the electrical potential. We conclude that indeed it is possible to get a pln junction that is thermodynamically stable over a certain range of temperature. This conclusion is also of some practical importance in view of, for instance, the miniaturization of devices.2
Acknowledgment. J.F.G. thanks the Feinberg Graduate School of the WIS, for a postdoctoral fellowship, in the framework of the Arc-en-Ciel Israel-France bilateral scientific research program. At the Weizmann Institute this work is supported in part by grants from the EC-Israel scientific research program, the US.-Israel Binational Science Foundation, and the Minerva Foundation. Appendix A: Ideally Behaving System in Equilibrium (Solution to Eq 10) The polynomial in eq 10 N
k=l
belongs to a set of polynomials having the characteristic that the coefficients of n P 2 - z k are 5 0 if (p2 - Z k ) < PI and are 2 0 if (p2 - zk) > PI. This means that in the series [zkak] all zlal, z2a2, and z3a3 until z,,a,, are negative and from then on till zna,they are positive. Remember that z k denotes the absolute value of the charge of s k . As we assume positively as well as negatively charged particles to be present, there is at least one non-zero positive coefficient and one non-zero negative coefficient and there is only one change of sign in this series. Then, according to a theorem proven by Descartes [cf. Wilf, H. S . Mathematics for the Physical Sciences; Dover: New York, 1962; p 941, the number of positive solutions is either equal to the number of times that the sign changes in the series [zkak] or less than it by an even number. Therefore in our case there is one and only one solution to eq 10 (cf. Figure 4).
Appendix B: Smooth n+/n Junction in Nonideal Systems Let us consider a semiconductor with one mobile species (a donor for instance) D+ with concentration ND. The activity coefficients for electrons and holes may be taken as 1 as long as the semiconductor is not degenerate. We assume that all donors are ionized (the Fermi level is well below the donor level).
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14492 J. Phys. Chem., Vol. 99, No. 39, 1995
-
100
7
m
v
c
.4-
2
=
1
i8
10
E
c.
1 w 1 10
1
Donor concentration (a.u.)
Figure 6. Plot of the electron concentration and of the absolute activity coefficient of the donors, AD, versus the donor concentration, ND. The -10
0 5 Spatial coordinate (a.u.)
-5
concentrations are normalized to the intrinsic carrier Concentration.
10
Figure 5. Spatial distribution of potential and corresponding equilibrium distributions of electrons and donors for an n+/n homojunction. The concentrations are normalized to the intrinsic canier concentration.
Let us consider a potential of the form
(B 1) where x is length. Then the charge density is
(B2)
The charge density may be expressed also as
(B3) The electron concentration is
So that from (B3) we can get
nevertheless at thermodynamic equilibrium because of the decrease of the activity coefficient of the donors with their concentration (Figure 6). This variation of the activity coefficient of the donors can be related to an attractive interaction between the donors depending on their concentration (correlation effect). We can check from Figure 6 that indeed TF = - 1, for the donor species in the concentration range considered. The same steps [(B2) to (B7)] could be followed for any potential 4. Therefore we have shown that for any stable builtin electric field we can reconstruct the activity coefficient, which will yield the necessary distribution of electrically active species. The opposite reasoning, i.e. computing the concentration and potential distributions for a given activity coefficient, is also possible. But, because now we have a differential equation to solve (the Poisson equation), we must specify the boundary conditions. We will consider an example that is analytically tractable. For this we will impose, as an additional condition, unidimensional symmetry on the problem; Le. the quantities will depend only on the coordinate x. We consider an infinite medium and impose the condition that at infinity the charge density and the electric field must vanish. We will take again the case of a donor species in an elemental semiconductor. Again we will assume that electrons and holes behave ideally, but not the donor (this implies that then the semiconductor is also nonideal, from the Gibbs-Duhem equation). We will assume that over a certain range of concentrations of donors, ND,the activity coefficient, AD, is given as follows, using the electrical potential, 4, as a parameter (cf. eq 12 with K = a&;): ~ , = - eKx pX( ND
l
?
kT
)
i
The electrochemical potential of donors is where A, K , and 40 are constants. ND defines the range of 4 (because only donors are present, N D 1. 0, Le., 4 I0). Such a donor distribution may take place in the case where the activity coefficient decreases with increasing concentration. The Poisson equation that we have to solve is
(B7) together with (B4) and (B5)gives the dependence of the activity coefficient as a function of donor concentration. This example is illustrated by Figures 5 and 6. Figure 5 shows the correspondence between the potential distribution and the concentration of donors ( N D )and of electrons (n),normalized with respect to the intrinsic carrier concentration ni. Note that everywhere n > ni, so.that the system is an n/n+ homojunction. The donor concentration is nonuniform, but the system is
This equation has two and only two solutions, a and b, that satisfy the boundary conditions: (a)
4 ( x ) = constant (any value from q50, 240, 340) (B1 l a )
Thermodynamic Stability of p/n Junctions
J. Phys. Chem., Vol. 99, No. 39, 1995 14493
or
3 = 40(
+ be”‘“’ +
(Bllb)
where x, = (d2qA)”*(l/q50)and b is an integration constant that translates the solution in the x direction. In an infinite medium all these solutions are physically identical. If we take b = 1, q5(x) can be written as
From the analysis in section V we find the criterion that has to be satisfied for the solution to be stable, namely that
a Wfid
>
8% Taking this condition into account then yields the following stability condition: -
