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Thermal equilibrium concentrations of point defects in gallium arsenide
J. Phys. Chem. SolIds Vol. 55, No. IO,pp. 911-929, 1994 Copyright 0 1994 Elswin Science LKI Printed in Great Britain. All rights rcsmed 0022-3697/M S7.00 + 0.00
Pergamon
THERMAL
EQUILIBRIUM CONCENTRATIONS DEFECTS IN GALLIUM ARSENIDE
OF POINT
T. Y. TAN Department of Mechanical Engineering and Materials Science, Duke University, Durham, NC 27708-0300,U.S.A. Abstract-Expressions of the thermal equilibrium concentrations of point defects in GaAs, including the neutral and charged species, are derived. These expressions are explicit functions of well-defined thermodynamic quantities, which in turn yield explicit expressions for the reaction constant Kin the usual As, pressure power law representation of the point defect thermal equilibrium concentrations. Such power laws have been of little quantitative value in the past, because values for K were not known. In the present derivation, emphases are placed upon the difference between the Gibbs free energies of an As atom in the interior of a GaAs crystal and in an As vapor phase molecule, and the role of the crystal Fermi level. Numerical values of the thermal equilibrium concentrations of the neutral and three negatively charged Ga vacancies (v”, , V&, V& and V& ), the neutral As vacancies v”*, and the two neutral antisite defects Gai, and As&, have been obtained. The calculated thermal equilibrium concentration of the anion antisite defect As:= reaches a peak value of about 1 x 10” cm-3 and is practically temperature independent, in agreement with experimental findings. The thermal equilibrium concentrations of the triply negatively charged Ga vacancy V& , CT&(n), have been found to exhibit a temperature independence or a negative temperature dependence behavior under strong ndoping conditions. That is, the Cyi; (n) value is either unchanged or increases as the temperature is lowered. This C:;-(n) property provided explanations to U. a number of outstanding experimental results, either requiring the interpretation that V& has attained its thermal equilibrium concentration at the onset of each experiment, or requiring mechanisms involving point defect nonequilibrium phenomena. The calculated CT;;(n) values are in agreement with available experimental results. Keyworcis: D. defects, D. diffusion.
1. INTRODUCTION Point defects are responsible for self-diffusion and the diffusion of substitutional impurities occurring in a crystalline solid. Therefore, it is necessary to know the thermal equilibrium point defect concentrations as a function of temperature. In the elemental semiconductor Si, the neutral point defect thermal equilibrium concentrations are described by simple Arrhenius relationships. For example, that of the neutral vacancy species V” is given by
eq cVO
a
=c=exp
[
--
1
gto k,T
,
(1)
where cz and CT are the normalized and the actual Vo concentrations, respectively, Co is the lattice site density, k, is Boltzmann’s constant, T is the absolute temperature and g& is the Gibbs free energy of forming a vacancy. The quantity g& is equal to h: - Ts;, where hc is the vacancy formation enthalpy and SE is the vacancy formation entropy. An expression analogous to eqn (1) holds for the neutral self-interstitial species IO. It is known [l] that both
vacancies and self-interstitials contribute to diffusion processes in Si and, aside from V’ and IO, charged point defect species also contribute. For metals, it is well known that the vacancies V dominate the diffusion processes. The vacancy thermal equilibrium concentration in a metal is given by an expression identical to eqn (1). For metals, however, there is no need to discuss whether V is charged. This is due to the fact that, because the electron concentration in metals is at least Co and therefore cannot be easily perturbed, no experiment can be conducted to check whether V is charged. At a given temperature, the thermal equilibrium concentration of a neutral point defect species in Si (and in Ge), cl, is single valued. The same also holds for metals. For GaAs and other compound semiconductors, the situation is different in that, at a given temperature, a range of thermodynamically allowed values exists for the thermal equilibrium concentration of a point defect species (~7). This is revealed for the six neutral point defect species by the common recognition that cy is proportional to specific power of the partial pressure of an As vapor species used in an experiment [2-4]. Recently,
917
T. Y. TAN
918
expressions of the thermal equilibrium concentrations of the six neutral point defect species have been obtained ($61. In these expressions, all involved thermodynamic quantities are explicitly defined. In this paper, the derivation of these expressions will be first presented. These expressions are then used to estimate the values of the thermal equilibrium concentrations of some neutral point defect species. Furthermore, a derivation of the general expressions relating the thermal equilibrium concentrations of charged
and neutral
also be presented. used to calculate trations of three species [7].
point
defect
species
[6] will
Subsequently, the expressions are the thermal equilibrium concennegatively charged Ga vacancy
2. NEUTRAL
POINT DEFECT THERMAL EQUILIBRIUM CONCENTRATIONS
We first discuss the situation in the elemental semiconducting crystal, e.g. Si. The quantity g$, is defined as the Gibbs free energy change, not including the crystal configurational entropy change, incurred by removing an atom from a lattice site in the crystal interior and replacing it on a crystal surface kink-site. In the presence of surface kink-sites, the creation of a single Vo incurs no change in the number of crystal atoms and only a negligibly small change in the crystal surface area. Since the number of atoms is unchanged, the crystal cohesive energy per atom, or the crystal free energy of binding per atom, g, which is the difference in the Gibbs free energy for an atom existing in the crystal vs its existence as a single atom in vacuum, does not contribute to g$. Since there is practically no surface area change, the crystal surface energy does not contribute to g&. Thus, in principle g& is identical to the free energy expended in creating the “broken bonds” (with relaxation properly accounted for) associated with the nearest neighboring atoms of V” [6]. Thus, g& is in principle a well-defined quantity. In this picture, the effect of a vapor phase consisting of the same atomic species and in thermal equilibrium with the crystal is not considered. To allow this simplification, a common explanation is that the pressure of the vapor phase is small and therefore its effect may be ignored. Actually, whether the thermal equilibrium vapor phase pressure of an elemental semiconducting crystal is large or small and whether it is maintained or not in an experiment have no first-order effect on the crystal thermal equilibrium point defect concentrations. That is, irrespective of the magnitude of the thermal equilibrium vapor phase pressure, when it is not maintained, there is no influence on c’,‘~.Instead, the crystal will simply grow or shrink in size.
It is well known that a compound crystal has a thermodynamically allowed equilibrium composition range. This can be seen from the schematic Ga-As binary phase diagram, Fig. 1, wherein the allowed composition range of the compound GaAs around Ga,,,As,, is greatly exaggerated. At a composition for which the As concentration exceeded the composition range limit b’ (solidus line) on the As-rich side, the material is a multi-phase mixture of vapor phases, GaAs of the composition b’ together with either an As-rich liquid of composition b (above -8OO’C) or with As clusters (below -800°C). At a composition for which the Ga concentration exceeded the composition range limit a’ (also a solidus line) on the Ga-rich side, the material is a multi-phase mixture of vapor phases, GaAs of composition a’ and a Ga-rich liquid of composition a. For a solid Ga-As material inside the range a’b’ the material is a GaAs crystal in thermal equilibrium with the appropriate vapor phases. There are three As vapor phases: As,, As, and As,; and there is one Ga vapor phase: the monatomic Ga, gas. Figure 2 is a diagram showing the equilibrium vapor pressures of GaAs [7] for which several important points will now be noted. Lines marked by b, b’ apply to As-rich Ga-As materials with an As concentration at or exceeding the composition range limit b’ on the As-rich side shown in Fig. 1, while those marked a, a’ apply to Ga-rich Ga-As materials with a Ga concentration at or exceeding the composition range limit a’ on the Ga-rich side. The equilibrium pressure of As, is much lower than that of As,, and As,. Over almost the whole temperature range of interest, i.e. for T > 7OO”C, the overall As equilibrium
pressure is much larger than that of Ga if the crystal is not very Ga-rich. Also, for these crystals As, dominates. For a GaAs crystal that is Ga-rich,
Ga fraction 1.0
0.8
0.6
0.4
0.2
0.0
,
1000 G e t-
800
600
As fraction Fig. 1. The Ga-As binary phase diagram. The allowed GaAs composition range is schematic and greatly exaggerated.
919
Point defects in gallium arsenide
T(t)
lo2-
0.6
0.7
0.8
0.9
Id/T
1.0
1.1
1.2
(K“)
Fig. 2. Thermal equilibrium vapor phase pressures at the allowed Ga-rich and As-rich composition boundaries of GaAs [9].
however, As, is the dominant species, for example, at a composition near a’. The equilibrium pressures of a GaAs crystal with a composition within the composition range a%’ are not expe~mentally known, although it is clear that they must be between the approp~ate a, a’ and b, b’ values shown in Fig. 2. The GaAs crystal (as well as other III-V compound crystals) is of the ZnS structure which consists of two sublattices, one occupied by Ga atoms and the other by As atoms. Because of this compound nature, six single point defect species exist: vacancies in the Ga sublattice (Voa), vacancies in the As sublattice (V,,), Ga self-interstitials (Ioa), As self-interstitials (IAs), antisite defects of a Ga atom on an As site (Ga,,) and antisite defects of an As atom on a Ga site (As,,). Any two single point defect species can form a paired species, which we will not be concerned with in this paper. The proper sum of the thermal equilibrium concentrations of the point defects constitutes the allowed GaAs crystal composition variation within
of a point defect species includes those in all charge states. In thermal ~uilib~um, therefore, the following three categories of quantities form a mutual dependence for which it cannot be strictly judged which is the cause or which is the effect: (i) the vapor phase pressures; (ii) the GaAs crystal composition; and (iii) the point defect concentrations. For the sake of convenience, we shall view the vapor phase pressures as the cause, since they are adjustable parameters in experiments. From the above discussion, it is clear that, as required by phase equilibria, the GaAs crystal point defect thermal equilibrium concentrations are dependent upon the vapor phase pressures. It is convenient to express such relationships first by kinetic reactions of the type As,, * Vi, + 4As,,
(3)
where AsASis an As atom on an As site in the GaAs crystal and Vk is the neutral As vacancy. Reaction (3) holds for the GaAs crystal with a composition within the allowed composition range a% shown in Fig. 1. At an As vapor pressure lower than that corresponding to point a‘, in equilibrium with a Ga-As material whose average composition is outside the composition range a% of Fig. 1 on the Ga-rich side, the Vi, thermal equilibrium concentration in the GaAs crystal Cy:, is that at point a’, with excess Ga atoms existing in the liquid phase a. At an As vapor pressure higher than that corresponding to point b’, in equilibrium with a Ga-As material whose average composition is outside the composition range a’b’ of Fig. 1 on the As-rich side, Cy:, is that at point b’, with excess As atoms existing in the liquid phase b or as solid As. In the past, a typical way of expressing the relationship between the point defect thermal equilibrium concentrations and an As vapor phase pressure is via the use of a reaction constant K. For example,
the range a’b’ shown in Fig. 1. For instance, consider the contributions of only the single point defects,
is the excess As concentration, while SC,, = -SC,, is the excess Ga concentration, which is responsible for the GaAs crystal composition deviation from the G%.sAsO,, stoichiometry. In eqn {2), Czl, CzSa and CT&% are the IAS, Asoa and V,, thermal equilibrium concentrations, respectively, and Cz,, C& and CT_ are the Ioat Ga,, and Vo, thermal equilibrium concentrations, respectively. Here, the concentration
where PAY is the As, pressure, obtained from reaction (3). While expressions such as eqn (4) showed qualitatively the dependence of the thermal equilibrium concentration of a point defect species on PAS, to a power, they have been rather useless on a quantitative basis, since neither the vafue nor the expression of I( is known. Because of the very small allowed composition range of GaAs, K cannot be measured and there has been no calculation of K until recently [5,6]. Here we review briefly the derivation given in Ref. [S] in which quantitatively useful expressions and
T. Y. TAN
920
values for neutral GaAs point defect species have been obtained. Formation of a point defect in GaAs (or other compound semiconductors) can be via several parallel kinetic paths or kinetic reactions, because there is more than one ~uilib~~ vapor phase. All such possible reactions, however, lead to the same thermal equilibrium concentration of the concerned point defect species, because all vapor phase species are in thermal equilibrium among themselves. Therefore, among the four available vapor phases As,, As,, As, and Ga,, only one needs to be considered, for example, As,. Assuming that As, is the involved vapor phase, we present a detailed derivation for a neutral As vacancy Vis so as to illustrate the main physical aspects involved. For the cases of PAS,V$, , I&, As& and Gak the procedures are essentially the same as for Vi, and hence only the results will be given. Consistent with that commonly adopted in the literature and analogous to that of the elemental semiconductor case, we shall first define a stundard free energy of formation of Yk, g:“,, . It is the free energy expended in creating the broken bonds at the cavity associated with Vi, formation. In determining the thermal equilibrium V” concentration in an elemental semiconductor, the standard Gibbs free energy of formation of V”, g$, is the sole energetic factor. This is the case because in forming a V”, the atom removed from the crystal interior site is replaced at an appropriate crystal surface kink site, which leads to no net atom number or surface area changes and therefore no further free energy changes. The situation associated with the elemental crystal cases does not hold for GaAs or any other compounds. In accordance with reaction (3), when forming a Vi,, the As atom number in the GaAs crystal is decreased by one while the As, molecule number is increased by one-quarter. The free energy change associated with this As atom transfer between GaAs and As, must also be accounted for. For this purpose, we adopt the convention that the Gibbs free energy of one As atom existing in vacuum is zero. For an As atom in a GaAs crystal, the Gibbs free energy is g,, (GaAs) = h,, (GaAs) - TsAs(GaAs).
In eqn (7) NV%is the number of VL in the GaAs crystal, NAs is the number of As atoms in the GaAs crystal, NAY is the number of As4 molecules in the vapor phase and S~~ is the ~~gurational entropy of the crystal As sublattice which contains Vi,. The quantity ST$? is given by
(8)
1 Asatom
(5)
In eqn (5) h,,(GaAs) is the binding or cohesive enthalpy of one As atom in the GaAs crystal. The value of h,,(GaAs) is negative. The quantity s,,(GaAs) is the entropy of one As atom in the GaAs crystal which is mainly due to thermal vibrations. For an As atom in an As, molecule existing in vacuum, the Gibbs free energy is g,,s(Ase) = &,s(Ase) - %&As.,).
In eqn (6) &(As,) is the binding or cohesive enthalpy of one As atom in the AS,+molecule. The value of hAs(As4) is negative. The quantity sAs(As4) is the entropy of one As atom in the As, molecule. This entropy is due to contributions of the internal degrees of freedom the molecule possesses: electronic, rotational, vibrational, etc. The As, molecules form a gas with a nonzero pressure. Because of their translational freedom, each molecule possesses an additional chemical potential or Gibbs free energy gi,. Based on the above discussion, the free energy situation associated with the thermodynamic ensemble of the As, vapor phase, the GaAs crystal and Vis is shown in Fig. 3, wherein the various free energy quantities denote that of one As atom and one VR. In Fig. 3 the zero value of g,,(GaAs) and gAs(As.,) is taken to be that of an As atom existing in vacuum. The quantities gAs(Asq) and gi,,/4 are additive and hence the zero value of gX,/4 is at gAs(Asq). The zero value of gbi, is at g,,(GaAs). The Gibbs free energy of the ensemble may now be written as
(6)
Fig. 3. A schematic drawing showing the free energies of binding of an As atom in the various involved materials together with the free energy of formation of a VA,.
Point defects in gallium arsenide
We shall regard g!,,%as that of an k&w1gus [lo],
g&=
921
defect species is the same as for Vi,. All the results are of the form
B
-ksTlnp,
(9)
A%
[ where B,
is the ideal gas pressure constant
BA% =
t
iht, -
hi
1 “*
[kaTl”‘,
ksT
given
by
(10)
with mA, being the mass of an As, molecule and h Plan&s constant. Note that reaction (3) requires aN,,laNv, = l/4 and aNA,/aNV~, = - 1 to hold. Using eqns @j-(10) in eqn (7) and minimizing G with respect to NQ, we obtain
1
122= As exp -- cgiJK
’
(14)
where x0 stands for any of the defect species. Expressions of the factors Ati and (g$)“” for all cases are summarized in Table 1. FOTVg, we can now determine Kvn, by comparing eqns (4) and (11): f&n, = Bzi exp[ -(g$$‘/ks
T&
(15)
The reaction constant Kfi of defect x0, as in the representation of c> by
(16)
=[%]‘“exp[-F],
(11)
where (g$)‘* is the effective Gibbs free energy of formation of Vi8 given by (S&C,,)“’= g& -
&?A,,
(12)
with Sg, being the difference of the As atom free energies of binding defined as &As
= i!,,
@aAs)
-
gA, 64~4
h
(13)
In eqn (11) eye is the normalized VP, thermal equilibrium con&tration, C”O is the actual Vi, thermal equilib~um co~centr~~~ and CO is the As sublattice site density. The prO&?dtXe of determining the thermal equilibrium concentrations of the other neutral point
where t is an appropriate power, is of a form analogous to eqn (15). Expressions of Kfl for all six neutral point defect species are also listed in Table 1. Our present expressions for the thermal equilibrium point defect concentrations are functions of the point defect standard free energy of formation and vapor phase related energetic and pressure factors (which are already known). These expressions are obtained by considering explicitly, in addition to the usual standard Gibbs free energy of formation of a defect, the contribution of an As vapor phase to the Gibbs free energy of the thermodynamic ensemble consisting of the GaAs crystal, the defect species and the As, vapor phase. The As, vapor phase contributes to the determination of the point defect thermal equilibrium concentration mainly in two ways. The first amounts to essentially quite a large enthalpy wnt~bution resulting from the binding enthalpy difference of an As atom existing in GaAs and in an As vapor molecule. The second is a large entropy
Table 1. Summary of the expressions of the thermal equilibrium concentrations of the six electrically neutral point defect species (x0) in GaAs. These expressions are of the general form c$ = A, exp[- (&F/k, 7’1.The factors A, and Cp$)” are listed for each case. Also listed are the expressions of K* in the reaction constant representation cz = K*P!+, where I is the appropriate power of P& in the factor of A, of the cz = A, exp[ -(g:‘,~ff/k, 7’j representation
922
T. Y. TAN Table 2. Enthalpy values (in eV) of one As atom in the various As-containing phases
at room temperature NBS [ll] Arthur [9] Adopted
As,
As (solid)
0 0 0
-3.1350 -3.1350 -3.1350
contribution resulting from the As, vapor phase molecule translational freedom. To calculate numerically the thermal equilibrium concentrations of the point defects, the actual values of the various involved thermodynamic quantities are needed. The values of PAq shall be the experimental ones. When expressed in units of atmospheres, eqn (10) yields BAs4 = 135T5j2atm.
As, - 1.9820 -1.9192 -2.092
GaAs
As, -2.7625 -2.5931 -2.8125
-3.5032 -3.5002 -3.5002
Figure 4 shows the calculated CT:8 and Cy;, values for PA_ values at the GaAs crystal Ga-rich solidus and As-rich solidus. A GaAs crystal in equilibrium with the As, vapor phase with PAId values between these two extreme cases will have CT;* and Cx values between the two branches of the appropriate curves. Using the Phillips-van Vechten two-band dielectric model [13], Wager [14] recently obtained
(17)
h6ga = 1.62 eV, Thus the numerical values of the pre-exponential factors in the point defect thermal equilibrium concentration expressions are now calculable. For g$ and 6g,,, etc., we shall approximate them by th: appropriate enthalpies h,;, and S/r,, , etc. This kind of practice usually yields sufficiently accurate results if the concerned quantities are all related to crystals. Using this approximation for a gas phase material, however, the inaccuracy thus incurred can be large. Fortunately, this does not seem to be the case for the As vapor species [5]. In Table 2 the values of the relevant enthalpies of the various As species obtained from the literature [9, 1I] as well as the values adopted are listed. These values are for one As atom. From Table 2 and eqn (13), we obtain ah,, z -0.69 eV.
(18)
&a”,,z 3.11 eV, Here, the uncertainties in the enthalpy values are probably even greater than in the cases of I’L and Vi,. Using eqns (17), (22), (23) and the relevant quantities in the appropriate expressions listed in Table 1, we obtain eq cG&
x 1 l.6T5’4P&:‘2 exp [-z],
eq e 0 086T-5’4P~~~ ’
(19)
The use of the values given by eqns (17)-(19) in the appropriate expressions listed in Table 1 yields CT;, x ~.~T”*PL;‘~ exp[ -31?$],
c;t. x 0.293T-5’8P”4Asrexp[ -%I.
(20)
(21)
T (“Cl lo=‘
1200 : :
:
900 :
600
:
____.
c 5 Y
JO')-
s P 2"
lo'*
f "
10"
-
'"'4: 10'0 0.6
(\ ', '*.,, '\ '\\ ', '8, ', '\ \ As-rich .)& 0.7
VL
Ga-rkh
\
'\ '\ \
‘8, “\
\
0.8
0.9
1 03/T
In view of the approximate nature of the values of the involved quantities, an uncertainty factor of at least 10 should be assigned to eqns (20) and (21).
(25)
Figure 5 shows the Cya;,and CTs;avalues calculated from eqns (24) and (25) for PAa values at the GaAs
Some efforts have been made to calculate the standard enthalpies, for example, h:a etc. Using the macroscopic cavity model, van Vechteg [12] obtained x 2.59 eV.
exp [ -%I,
‘A$,
point defect formation
h:;, x h&
(24)
'\ '\
,
1.0
1.1
1.2
(K-’ )
Fig. 4. Calculated V& and Vi, thermal equilibrium concentrations for Pti values at the GaAs crystal Ga-rich solidus and As-rich solidus.
923
Point defects in gallium arsenide
1200
1o18
T 63 900
600
As-rich
0.6
0.7
0.8
0.9
103/T
1.0
1.1
1.2
(k-’ )
Fig. 5. Calculated Gai, and As”&thermal equilibrium concentrations for PA, values at the GaAs crystal Ga-rich solidus and As-rich solidus.
crystal Ga-rich solidus and As-rich solidus, respectively. It is interesting to note that the calculated CT& values at the GaAs crystal As-rich solidus are between 1 x 1OL6and 1 x 10” cms3 and fairly temperature dependent. The AS& concentration has been found to be in the range of N 1 x 1016-1x 10” cme3 in As-rich GaAs crystals and is temperature independent [13, 16181. Our calculated A& concentration and its temperature dependence are in agreement with these experimental findings. The antisite defect Ass, is the main constituent of the deep donor EL2, which is responsible for growing modern semi-insulating GaAs crystals using the liquid encapsulated Czochralski (LEC) method under As-rich conditions. Presently, values of g$ and g$, are not available. Therefore, numer&l values of cq and c? it ‘oa cannot yet be obtained.
as well as by the Fermi level position of the semiconductor. A charged point defect whose energy corresponds to having an electronic level in either the conduction or the valence band is an unstable situation since it will increase the free energy of the ensemble. Thus, the defect will loose a charge (negative and positive, respectively) to either the conduction or the valence band edge, becoming one in the next highest charged state with an electronic level either at the band edge (the neutral ones) or in the gap. The correct and complete derivation of the thermal equilibrium concentrations of charged “flaws” (point defects or impurities) was first obtained by Shockley and Last [22]. Here we shall use a different method to obtain their results [22]. This is the Gibbs free energy formulation [6] which has the advantage of being more explicit so that certain important physical features may be better pointed out. The defect levels are assumed to be deep ones, for example, that shown in Fig. 6, wherein a few deep acceptor levels are indicated. The Fermi level of the crystal Er is determined by a shallow dopant compensated by the formation of charged point defects. Consider the case shown in Fig. 6. The thermodynamic ensemble consists of the systems of points defects and electrons. This is a grand canonical ensemble for which energy and mass transfers both occur among the constituent systems. With e denoting an electron, the kinetic reactions, xO+eox-, x-+eox*-,
(26b)
x2- + e 0 x3-,
(264
EF
3. CHARGED POINT DEFECT THERMAL EQUILIBRIUM CONCENTRATIONS Charged point defect species exist in Si and in GaAs. They contribute to the diffusion phenomena. Charged point defects can form in a semiconductor, since it can lower the total Gibbs free energy of the thermodynamic ensemble consisting of the point defect and the electronic systems. The thermal equilibrium concentrations of the stable charged point defects depend upon the crystal Fermi level [19-221. A stable charged point defect must possess an electronic energy level in the semiconductor forbidden energy gap. Thus, the concentration of the defect is influenced by its electronic level position
(264
Ev Fig. 6. GaAs band diagram showing schematically the relation of the Fermi level and the native point defect (x) deep acceptor states. The left-most arrow indicates the total ensemble (defect and electron systems) Gibbs free energy decrease of forming one x- from x0. The middle arrow indicates the additional total ensemble Gibbs free energy decrease of forming one x2- from x-. The right-most arrow indicates the additional total ensemble Gibbs free energy decrease of forming one x3- from x2-.
T. Y. TAN
924
describe the successive formation of the charged species of x. in the presence of both X- and x0, the Gibbs free energy of the ensemble is G = N&&)ee + N,_ (g;_ )“e + G, - ?-SW+
(27)
where NS and N,_ are the numbers of x0 and x -, respectively, (g$,)‘” and (gf;_>” are their effective free energies of formation, respectively, G, is the Gibbs free energy of the electron system and Sconfigis the configurational entropy of the crystal, given by
scc,a
=
ln
k
3
INo+ Nxo+ Nx-Y No! N,! N,_!
’
holds. In eqn (30) ni is the intrinsic carrier concentration and Ei is the Fermi level of the intrinsic crystal. Therefore, C,-(n) -=expr$]=[!Jrk] CX-64)
(31)
holds, where n and p are the electron and hole con~ntrations of a doped crystal, respectively. In eqn (31) the last two identities follow from the calculation of n and p for doped semiconductors. For reaction (26b), one uses the same procedure to readily obtain
1 1
G- @I
(28)
EF
-
42
k,T
c,_o=exp
(32)
and therefore where No is the number of atoms of the appropriate sublattice. Reaction (26a) requires aN,/aiV,_ = - 1 to hold. To minimize G, with respect to N,.. , we use aGJaN,_ = (aG=/aN~)(a~~/aN~_), where N, is the number of electrons in the crystal. Reaction (26a) also requires aN,/aN,_ = - 1 to hold. By definition, we further have aG,/aN* = EF and (gi-)‘“(g$,)‘e = E,, . The minimization of G with respect to N,_ now yields
G @I
2Er - Ed - & k,T
--exp C,
G-02) -= C.+ (ni)
[I
1’
-n 2. ni
(33)
(34)
In general, for x2-, we have (35)
C,_
1 1
N,_
EF-&I
C=N=exp ~ k,T XQ
x4
’
(29) (36)
where the energies are measured from the valence band edge, i.e. E, = 0 has been assumed. When the Fermi level is above the acceptor level, xbecomes more abundant than x0, with C, not influenced by the Fermi level. Note that (gf-)“” is higher than f,&)‘“, but under sufficiently heavy n-doping conditions, x- becomes more abundant than x-, because the free energy decrease of the electronic system is more than enough to compensate for the free energy increase of forming x-: for each x - formed, the ensemble energy is decreased by E, - E,, (see Fig. 6). The level E,,, is a real state but EF is not. The electron actually comes from the conduction band edge of energy EC and yet the released energy does not correspond to EC - E,, . This is because the total ensemble Gibbs free energy is minimized, including that of the electrons. This corresponds to measuring the free energy change associated with eliminating one average electron, for which its chemical potential is EF. For an intrinsic crystal,
C-
(4
-=exp C,
-4 - Eat
)
-
i
k,T
1
cx:-(n) -= C,:- (ni)
[I
(37)
In eqns (35)-(37) m and z are integers. Equations (35)-(37) may be directly obtained by considering the reaction x0 + ze 0 xi-. However, formation of x2via the simultaneous reaction of x0 and ze is a physically unlikely process. Proceeding as for deep acceptor defects, one obtains for deep donor defects
cx:+ (PI
= exp[ -w],
(38)
G-~,+(P)
C*,+(P) -=
Cxl+(ni 1
(30)
-.n ni
In eqns (38~~),
-P z. [Ini
Edr is the level of x2+.
tw
Point defects in gallium arsenide In the above defect,
including
expressions,
x may stand
an impurity
atom.
for any
singly, doubly, and triply negatively charged Ga vacancies V & , V & and V & , Baraff and Schhiter [26] have given the energy levels as E,, z O.l33E,, Es2 z 0.355E, and E,, z 0.49E,, respectively. Here, E8 is the GaAs crystal band gap energy at 0 K, _ 1.5eV. For high temperature applications we assume that the same expressions also hold. To approximate the effect of band gap narrowing at high temperatures, we further assume that Es 5 1.15 eV. Now, eqns (21) and (36) yield
V&, h, the elecare 2.05, respect-
ively. The values of h, are dependent upon the Fermi level position EF. For undoped GaAs the V& thermal equilibrium concentrations are much smaller than the intrinsic carrier concentration n, and the crystal may be approximated as intrinsic. By letting EF = Et = 0.5E,, where E, is the intrinsic Fermi-level energy, we obtain for undoped GaAs
TM’/* ex p(-s),
using
while the n value is
the charge
neutrality
con-
dition. Assuming that free electron compensations are due to the presence of only the negatively charged Ga vacancies, the charge neutrality condition is
The triply negatively charged Ga vacancies V& are responsible for Ga self-diffusion in GaAs and AlAs/GaAs superlattice crystals under intrinsic and n-doping conditions [6,23] and for the diffusion of the shallow donor Si (occupying the Ga sublattice, Sib.) in GaAs [24,25]. The thermal equilibrium concentrations of the various negatively charged Ga vacancies have been recently calculated [7’j. For the
cy;; (n,) z 0.293Pii
ture and is difficult to determine easily determined
4. NEGATIVELY CHARGED Ca VACANCY THERMAL EQUILIBRIUM CONCENTRATIONS
where h,, is the total enthalpy of forming a is the configurational enthalpy and zEF is tronic energy released, h,,. The values of h, 2.46 and 3.03 eV for the z = 1, 2 and 3 cases,
925
(42)
where h,(q) is h, under intrinsic conditions. The values of h,,(q) are 1.48, I.31 and 1.3 eV for the z = 1, 2 and 3 cases, respectively. For n-type GaAs it is more convenient to use [eqn (37)]
(43)
instead of eqn (41), because for a constant dopant concentration the value of EF changes with tempera-
n=C,-C:&)-2C;&(n)-3C:qZi;(n)+p,
(44)
where p is the crystal hole concentration, C refers to the actual concentrations and C, is the concentration of shallow donors which are assumed to be fully excited. In eqn (44) the fact that in forming a Vi,. a V& and a V&, 1, 2 and 3 electrons are consumed, respectively, has been accounted for. Using np = nf, we obtain n = f[C, - C&(n) +f{[c,-
- 2Q
(n) - 3c;;.
c:__(n)-2CT$J(n)-
+ 4$}‘;*. Using the equilibrium
(n)] 3C;&;((n)]* (45)
PA, values at the thermody-
namically allowed GaAs crystal composition limits given by Arthur [9], shown in Fig. 2, the values of C~&,), CT;+,) and CT;;(n,) can be calculated from eqn (42; and the values of CT&), Cy& (PI), CT&(n) and n are then readily calculated using eqns (43) and (45) for a given C, value. The calculated CyG*(n), Cy;. (n). C,,&(n) and n values for a few C,, values are shown in Fig. 7. Figure 7 shows that, at a given temperature, the thermal equilibrium concentrations of the charged Ga vacancies (also holding for any other point defect species) increase as the doping level is increased. This well known fact [22] is now called the Fermi-level effect [23], since the CT;I(n) changes are related to the position of the Fer@ level, EF. From eqns (41) and (43) we see that C,;;(n) increases as the Fermi level moves towards the GaAs conduction band edge or as n is increased. An unusual and previously unanticipated aspect is the temperature dependence of the defect concentration for a constant dopant concentration. It is clear from Fig. 7 that the temperature dependence of CT:;(n) is in general weaker than that of CT;&) and, in particular, that of CT;,(n) has exhibited behavior from a temperature independence to a small negative temperature dependence under heavy n-doping conditions in that, as the temperature is lowered, Cy;. (n) is unchanged or even increased. This is in strong contrast to the normal positive temperature dependence of the point defect thermal equilibrium concentrations, which decreases as the temperature is lowered [CT;,; (n,) are examples]. This is also a result of the Fermi-level effect. For a
926
TAN 1020,
lOlO,.,.,,,.,.,. 0.6 0.7 0.8
lo20,
12FO;
0.9
: 9;O
;
1.0
:
1.1
1200:
: 900
10'01.,,,.,.,.,-. 0.6 0.7 0.8
:
:
0.9
1.0
600
1.1
2
6pO
b
1OlOI 0.6
0.7
0.8
0.9
1.0
1.1
0.6
0.7
0.8
0.9
1.0
1.1
1.2
Fig. 7. (aHd) The calculated thermal equilibrium concentrations of V,, V& , V& and n, respectively. Legends le18, le19, etc. refer to shallow donor concentrations of 1 x 10’*ctn3, 1 x 10’9cm-3, etc. and are the same for all four figures. In each figure, the vertical axis is the concentration values in crnm3,the lower horizontal axis is the inverse temperature, lO’/T(K-‘) and the upper horizontal axis is temperature in “C.
very high shallow donor concentration,
EF is close to
EC for all temperatures of interest. For a moderate shallow donor concentration, the Fermi level moves closer to EC as the temperature is lowered and hence its effect becomes more prominent. With EF very close to EC, the electronic enthalpy released h,, can be so large that it may compensate the V& configurational enthalpy h,r sufficiently to lead to a very small or even negative total V& activation enthalpy ha3, resulting in the C$ (n) temperature independence or negative temperature dependence behavior. For example, if it is assumed that EF - Eg - 1.15 eV
holds, we obtain h,, N (- 0.42 eV). This is in qualitative agreement with the calculated results that, under heavy n-doping, CyLi (n) increases as the temperature is lowered. Using local vibrational mode measurements, Uematsu and Maezawa [27] and McQuaid et al. [28] have recently determined the V& concentrations and the corresponding n values in molecular beam epitaxy (MBE) grown GaAs layers doped by Si. These results are plotted in Figs 8 and 9 together with the appropriate calculated values, which are in good agreement. Since the samples are low temperature GaAs
927
Point defects in gallium arsenide T
(‘Cl
600
400
200
10M J]
0.6 0.8
1.6 1.8 2.0 2.2
1.0 1.2 1.4 103/T(
K-l)
Fig. 8. Experimentally measured V&i concentrations in MBE grown GaAs layers [27,28] fitted to the caicuiated values. Legends let9 and le20 refer to Si donor concentrations of 1 x lOI and 1 x 1020cm-3, respectively.
which are supposed to have been grown under highly nonequilibrium conditions, the agreement between the ex~rimental and our calculated results should be regarded as surprisingly good. Under intrinsic conditions, the Voa species exhibit the normal positive temperature dependence (Fig. 7). Therefore, it is expected that these vacancies can be frozen-in from a high temperature and precipitate out at a lower temperature. Such an experimental result has recently been obtained by Williams et al. [29], who observed formation of microvoids in undoped, semi-insulating GaAs grown by the LEC method and
receiving a post-growth annealing. The V, concentrations, estimated by the present author based on the void size and density reported by Williams et al. [29], is in quantitative agreement with our calculated thermal equilibrium concentration of the Vo, species for intrinsic GaAs (Fig. 10). Allowing an accuracy factor of -2, it is seen from Fig. 10 that these vacancies could be those frozen-in from either the GaAs melting temperature of 1238”C, or N 1000°C provided the crystal is As-rich. Semi-insulating LEC GaAs crystals can only be grown from an As-rich melt and therefore the data should he that of an As-rich crystal case. Associated with diffusion studies in n-type GaAs and GaAs/AlAs superlattice crystals, there are several recent experimental results requiring the interpretation that the responsible point defect species V& has attained its thermal eq~librium concentration C:&;(n) at the onset of each experiment [25,30,31]. If it is assumed that Vi; is undersaturated at the onset of these experiments, then the crystal interior C”_(n) value can only be reached by the V&i indiffusio?process which has heen found to be too slow [32]. Furthermore, in these experiments the substrate materials have not been (and cannot be) pre-annealed to introduce V& to its experimental temperature C”,_ (n) values before the onset of the experiment. Tlk only leaves the possibility that, for sufficiently highly doped n-type GaAs and GaAs/AlAs superlattice, C:,_(n) is fairly temperature independent. The pres&tly calculated
T
17 T 600
(*C) 400
3
10
:
(OCt 600
900
1200
:
:
:
200
---..“---_______ 10'8 p? E .2 c
10'6
10
10'4
0.6 0.8
1m
11
0.6 1.0
0.7
Williams etal. 0.8
(1991) 1
0.9
1.0
1.1
1.2
1.2 1.4 1.6 1.8 2.0 2.2 103/T(
K-')
Fig. 9. Experimentally measured n values in MBE grown GaAs layers 127,281, wrres~nding to the measured V& concentrations shown in Fig. 8, fitted to the calculated values. Legends le19 and le20 refer to Si donor concentrations of 1 x lOi and 1 x 10”cm-3, respectively.
103/T(k-')
Fig. 10. The vacancy concentration, in microvoids measured by Williams er al. [29] in a semi-insulating LEC GaAs crystal using TEM, fitted to calculated V, thermat equilibrium concentrations for intrinsic GaAs. Placement of the experimental data at the high temperature range is due to the present author.
928
T. Y. TAN
temperature independence behavior of Cy;; (n) satisfies this requirement. In the literature, there also exists a number of room temperature experimental results involving the nonequilibrium point defect generation phenomena. These include the dark-line defects formed during GaAs laser operations [33,34], the optical pumping damage in n-type GaAs [35] and the enhanced Be diffusion in GaAs heterojunction bipolar transistors during operations [36]. In these experiments, excess Ga self-interstitials were generated to form dislocation loops, which were attributed to carrier recombination enhanced Frenkel pair generations. This explanation pointed out the energy source for generating the point defects, but did not provide a thermodynamic reason for the fact that the vacancies so generated appear to be stable. Our calculated C$ (n) negative temperature dependence behavior offers a thermodynamic explanation to these phenomena. For these results, the difference between the crystal growth and the experimental temperatures is so large that our calculated Cyj (n) values can no longer be regarded as temperatu& independent. Instead, the CT& (n) negative temperature dependencies must be considered. Thus, at room temperature, V& is undersaturated. The carrier recombination generated V,, species of the Frenkel pair is to fulfil the larger C’y;; (n) needs and the I,, species condense into interstitial dislocation loops to further lower the Gibbs free energy of the ensemble. The details associated with these experiments and the explanation of the experimental results using the C;&(n) temperature independence and negative temperature dependence behavior have been thoroughly discussed elsewhere [A. 5. SUMMARY
On the happy occasion of celebrating Bob Balluffi’s seventieth birthday, it is only fitting that some of his disciples discuss point defects. This paper summarizes the theoretical aspects of the work on point defects in GaAs carried out at Duke University during the last five years by my students, colleagues and myself. Our work has been supported by U.S. Army Research ORice under contract DAAL03-89-K-0119. Expressions of the thermal equilibrium concentrations of point defects in GaAs, including the neutral and charged species, have been derived. These expressions are explicit functions of well-defined thermodynamic quantities, which in turn yield explicit expressions for the reaction constant K in the usual As, pressure power law representation of the point defect thermal equilibrium concentrations. Such power laws have been of little quantitative value in the past, because K were not known. In the present derivation, emphases have been placed upon the
difference between the Gibbs free energies of an As atom in the interior of a GaAs crystal and in an As vapor phase molecule, and the role of the crystal Fermi level. Numerical values of the thermal equilibrium concentrations of the neutral and three negatively charged Ga vacancies (V”,, , Vi,, V& and V&i),, the neutral As vacancies Vi, and the two neutral antisite defects Gai, and As:, have been obtained. The calculated thermal equilibrium concentration of the anion antisite defect A& reaches a peak value of about 1 x 10’7cm-3 and is practically temperature independent, in agreement with experimental findings. For the various negatively charged Ga vacancy species, the thermal equilibrium concentrations of the triply negatively charged Ga vacancy VA;, C”, (n) has been emphasized, since V& dominates Ga self-diffusion and Ga-AI interdiffusion under intrinsic and n-doping conditions, as well as the diffusion of Si donor atoms occupying Ga sites. Under strong n-doping conditions, Cy:, (n) VG.
has been found to exhibit a temperature independence or a negative temperature dependence behavior, i.e. the CT& (n) value is either unchanged or increases as the temperature is lowered. This is quite contrary to the normal point defect behavior for which the point defect thermal equilibrium concentration decreases as the temperature is lowered. This CT:,(n) property has provided explanations to a number of outstanding experimental results, either requiring the interpretation that V& has attained its thermal equilibrium concentration at the onset of each experiment, or requiring mechanisms involving point defect nonequilibrium phenomena. The calculated Cy& (n) values are seemingly in good agreement with available experimental
results.
REFERENCES I.
Tan T. Y. and Gijsele U.,
Appl.
Phys.
A
37,
I
(1985).
2. Casey H. C., Jr, in Atomic Diffusion in Semiconducrar.r (Edited by D. Shaw), p. 351. Plenum, New York
(1973). B. Tuck, Atomic DQjiuion in III-V Semiconducrors. Hilger, Bristol (1988). 4. Kriiger F. A., The Chemistry of Imperfecr Crystals, Chap. 13. North-Holland, New York (1964). 5. Tan T. Y., Mater. Sci. Engng E 10, 227 (1991). 3.
6. Tan T. Y., Giisele Marer.
7.
U. and Yu S., Crir. Rm. Solid SI.
Sci. 17, 47 (1991).
Tan T. Y.. You H.-M. and GBsele U., Appl.
Phys. A 56,
249 (1993).
New York 8. Friedel J, Dislocafionr, p. 78. Pergamon, (1965). 9. Arthur J. R., J. Phys. Chem. Solids 28, 2257 (1967). Mechanics. 10. Mayer J. E. and Mayer M. G., StarMcal Wiley, New York (1966). properties. Il. The i\iBS tables of chemical thermodynamic J. Phys. Chem. Ref. Data 11, Suppl. 2, 2.77, 2.132. American Institute of Physics, New York (1982).
Point defects in gallium arsenide 12. Van Vechten J. A., J. Electrochem.
Sot.
122, 419
(1975).
13. Van Vechten J. A., Handbook on Semiconductors (Edited by S. P. Keller), Vol. 3, Chap. 1. NorthHolland, Amsterdam (1980). 14. Wager J. F., J. A&. Phys. 69, 3022 (1991). 15. Wagner R. J., Krebbs J: J., Stauss G. H..and White A. M., Solid St. Commun. 36, 15 (1980). 16. Lagowski J., Gatos H. C., Parsey J. M., Wada K.,. Kaminska M. and Walukiewiez W., Appl. Phys. L&t. 40, 342 (1982).
17. Elliott K., Chen R. T., Greenbaum S. G. and Wagner R. J.. AIJP~. Phvs. L.ett. 44. 907 (1984). 18. Lagowski J., Gates H. C., Kang C. H.; Skowronski M., Ko K. Y. and Lin D. G., Appl. Phys. L&t. 49, 892 (1986).
19. Wagner C. and Grunwald K., Z. Phys. Chem. B 40,455 (1938).
20. Reiss H., J. Chem. Phys. 21, 1209 (1953). 21. Longini R. L. and Greene R. F., Phys. Reo. 102, 992 (1956). 22. Schodkley W. and Last J. T., Phys. Rev. 107,392 (1957). 23. Tan T. Y. and Giisele U.. APP~. __ Phvs. _ L&t. 52, 1240 (1988). 24. Yu S., Giisele U. and Tan T. Y., J. Appl. Phys. 66,2952 (1989).
929
25. You H.-M., Gijsele U. M. and Tan T. Y., J. Appl. Phys. 73, 7207 (1993). G. A. and Schliiter M., Phys. Rev. Lett. 55, 1327 (1985). 27. Uematsu M. and Maezawa K. Jup. J. Appl. Phys. 29,
26. Bard
L527 (1990). 28. McQuaid S. A., Newman R. C., Missous M. and O’Hagan S., Appl. Phys. Lett. 61, 3008 (1992); J. Cryst. Growth (in press) (1994). 29. Williams G. M., Cullis A. G. and Stirland D. J., Appl. __ Phys. Left. 59, 2585 (1991). 30. Mei P.. Yoon H. W.. Venkatesan T.. Schwarz S. A. and Harbisbn J. P., Appi. Phys. Lett. So, 1823 (1987). 31. Tan T. Y., You H.-M., Yu S., Gosele U. M., JIger W., Boeringer D. W., Zypman F., Tsu R. and Lee S.-T., J. Appl. Phys. 72, 5206 (1992). 32. You H.-M., Gosele U. M. and Tan T. Y., J. Appl. Phys. 74, 2461 (1993). 33. Petroff P. and Hartman R. L., J. Appl. Phys. 45, 3899 (1974). 34. Hutchinson P. W. and Dobson P. S., Appl. Phys. Lett. 26, 250 (1975). 35. Hutchinson P. W., Dobson P. S., Wakefield B. and O’Hara S., Solid-St. Electron. 21, 1413 (1978). 36. Uematsu M. and Wada K., Appl. Phys. Lett. 58, 2015
(1991).
Pergamon
THERMAL
EQUILIBRIUM CONCENTRATIONS DEFECTS IN GALLIUM ARSENIDE
OF POINT
T. Y. TAN Department of Mechanical Engineering and Materials Science, Duke University, Durham, NC 27708-0300,U.S.A. Abstract-Expressions of the thermal equilibrium concentrations of point defects in GaAs, including the neutral and charged species, are derived. These expressions are explicit functions of well-defined thermodynamic quantities, which in turn yield explicit expressions for the reaction constant Kin the usual As, pressure power law representation of the point defect thermal equilibrium concentrations. Such power laws have been of little quantitative value in the past, because values for K were not known. In the present derivation, emphases are placed upon the difference between the Gibbs free energies of an As atom in the interior of a GaAs crystal and in an As vapor phase molecule, and the role of the crystal Fermi level. Numerical values of the thermal equilibrium concentrations of the neutral and three negatively charged Ga vacancies (v”, , V&, V& and V& ), the neutral As vacancies v”*, and the two neutral antisite defects Gai, and As&, have been obtained. The calculated thermal equilibrium concentration of the anion antisite defect As:= reaches a peak value of about 1 x 10” cm-3 and is practically temperature independent, in agreement with experimental findings. The thermal equilibrium concentrations of the triply negatively charged Ga vacancy V& , CT&(n), have been found to exhibit a temperature independence or a negative temperature dependence behavior under strong ndoping conditions. That is, the Cyi; (n) value is either unchanged or increases as the temperature is lowered. This C:;-(n) property provided explanations to U. a number of outstanding experimental results, either requiring the interpretation that V& has attained its thermal equilibrium concentration at the onset of each experiment, or requiring mechanisms involving point defect nonequilibrium phenomena. The calculated CT;;(n) values are in agreement with available experimental results. Keyworcis: D. defects, D. diffusion.
1. INTRODUCTION Point defects are responsible for self-diffusion and the diffusion of substitutional impurities occurring in a crystalline solid. Therefore, it is necessary to know the thermal equilibrium point defect concentrations as a function of temperature. In the elemental semiconductor Si, the neutral point defect thermal equilibrium concentrations are described by simple Arrhenius relationships. For example, that of the neutral vacancy species V” is given by
eq cVO
a
=c=exp
[
--
1
gto k,T
,
(1)
where cz and CT are the normalized and the actual Vo concentrations, respectively, Co is the lattice site density, k, is Boltzmann’s constant, T is the absolute temperature and g& is the Gibbs free energy of forming a vacancy. The quantity g& is equal to h: - Ts;, where hc is the vacancy formation enthalpy and SE is the vacancy formation entropy. An expression analogous to eqn (1) holds for the neutral self-interstitial species IO. It is known [l] that both
vacancies and self-interstitials contribute to diffusion processes in Si and, aside from V’ and IO, charged point defect species also contribute. For metals, it is well known that the vacancies V dominate the diffusion processes. The vacancy thermal equilibrium concentration in a metal is given by an expression identical to eqn (1). For metals, however, there is no need to discuss whether V is charged. This is due to the fact that, because the electron concentration in metals is at least Co and therefore cannot be easily perturbed, no experiment can be conducted to check whether V is charged. At a given temperature, the thermal equilibrium concentration of a neutral point defect species in Si (and in Ge), cl, is single valued. The same also holds for metals. For GaAs and other compound semiconductors, the situation is different in that, at a given temperature, a range of thermodynamically allowed values exists for the thermal equilibrium concentration of a point defect species (~7). This is revealed for the six neutral point defect species by the common recognition that cy is proportional to specific power of the partial pressure of an As vapor species used in an experiment [2-4]. Recently,
917
T. Y. TAN
918
expressions of the thermal equilibrium concentrations of the six neutral point defect species have been obtained ($61. In these expressions, all involved thermodynamic quantities are explicitly defined. In this paper, the derivation of these expressions will be first presented. These expressions are then used to estimate the values of the thermal equilibrium concentrations of some neutral point defect species. Furthermore, a derivation of the general expressions relating the thermal equilibrium concentrations of charged
and neutral
also be presented. used to calculate trations of three species [7].
point
defect
species
[6] will
Subsequently, the expressions are the thermal equilibrium concennegatively charged Ga vacancy
2. NEUTRAL
POINT DEFECT THERMAL EQUILIBRIUM CONCENTRATIONS
We first discuss the situation in the elemental semiconducting crystal, e.g. Si. The quantity g$, is defined as the Gibbs free energy change, not including the crystal configurational entropy change, incurred by removing an atom from a lattice site in the crystal interior and replacing it on a crystal surface kink-site. In the presence of surface kink-sites, the creation of a single Vo incurs no change in the number of crystal atoms and only a negligibly small change in the crystal surface area. Since the number of atoms is unchanged, the crystal cohesive energy per atom, or the crystal free energy of binding per atom, g, which is the difference in the Gibbs free energy for an atom existing in the crystal vs its existence as a single atom in vacuum, does not contribute to g$. Since there is practically no surface area change, the crystal surface energy does not contribute to g&. Thus, in principle g& is identical to the free energy expended in creating the “broken bonds” (with relaxation properly accounted for) associated with the nearest neighboring atoms of V” [6]. Thus, g& is in principle a well-defined quantity. In this picture, the effect of a vapor phase consisting of the same atomic species and in thermal equilibrium with the crystal is not considered. To allow this simplification, a common explanation is that the pressure of the vapor phase is small and therefore its effect may be ignored. Actually, whether the thermal equilibrium vapor phase pressure of an elemental semiconducting crystal is large or small and whether it is maintained or not in an experiment have no first-order effect on the crystal thermal equilibrium point defect concentrations. That is, irrespective of the magnitude of the thermal equilibrium vapor phase pressure, when it is not maintained, there is no influence on c’,‘~.Instead, the crystal will simply grow or shrink in size.
It is well known that a compound crystal has a thermodynamically allowed equilibrium composition range. This can be seen from the schematic Ga-As binary phase diagram, Fig. 1, wherein the allowed composition range of the compound GaAs around Ga,,,As,, is greatly exaggerated. At a composition for which the As concentration exceeded the composition range limit b’ (solidus line) on the As-rich side, the material is a multi-phase mixture of vapor phases, GaAs of the composition b’ together with either an As-rich liquid of composition b (above -8OO’C) or with As clusters (below -800°C). At a composition for which the Ga concentration exceeded the composition range limit a’ (also a solidus line) on the Ga-rich side, the material is a multi-phase mixture of vapor phases, GaAs of composition a’ and a Ga-rich liquid of composition a. For a solid Ga-As material inside the range a’b’ the material is a GaAs crystal in thermal equilibrium with the appropriate vapor phases. There are three As vapor phases: As,, As, and As,; and there is one Ga vapor phase: the monatomic Ga, gas. Figure 2 is a diagram showing the equilibrium vapor pressures of GaAs [7] for which several important points will now be noted. Lines marked by b, b’ apply to As-rich Ga-As materials with an As concentration at or exceeding the composition range limit b’ on the As-rich side shown in Fig. 1, while those marked a, a’ apply to Ga-rich Ga-As materials with a Ga concentration at or exceeding the composition range limit a’ on the Ga-rich side. The equilibrium pressure of As, is much lower than that of As,, and As,. Over almost the whole temperature range of interest, i.e. for T > 7OO”C, the overall As equilibrium
pressure is much larger than that of Ga if the crystal is not very Ga-rich. Also, for these crystals As, dominates. For a GaAs crystal that is Ga-rich,
Ga fraction 1.0
0.8
0.6
0.4
0.2
0.0
,
1000 G e t-
800
600
As fraction Fig. 1. The Ga-As binary phase diagram. The allowed GaAs composition range is schematic and greatly exaggerated.
919
Point defects in gallium arsenide
T(t)
lo2-
0.6
0.7
0.8
0.9
Id/T
1.0
1.1
1.2
(K“)
Fig. 2. Thermal equilibrium vapor phase pressures at the allowed Ga-rich and As-rich composition boundaries of GaAs [9].
however, As, is the dominant species, for example, at a composition near a’. The equilibrium pressures of a GaAs crystal with a composition within the composition range a%’ are not expe~mentally known, although it is clear that they must be between the approp~ate a, a’ and b, b’ values shown in Fig. 2. The GaAs crystal (as well as other III-V compound crystals) is of the ZnS structure which consists of two sublattices, one occupied by Ga atoms and the other by As atoms. Because of this compound nature, six single point defect species exist: vacancies in the Ga sublattice (Voa), vacancies in the As sublattice (V,,), Ga self-interstitials (Ioa), As self-interstitials (IAs), antisite defects of a Ga atom on an As site (Ga,,) and antisite defects of an As atom on a Ga site (As,,). Any two single point defect species can form a paired species, which we will not be concerned with in this paper. The proper sum of the thermal equilibrium concentrations of the point defects constitutes the allowed GaAs crystal composition variation within
of a point defect species includes those in all charge states. In thermal ~uilib~um, therefore, the following three categories of quantities form a mutual dependence for which it cannot be strictly judged which is the cause or which is the effect: (i) the vapor phase pressures; (ii) the GaAs crystal composition; and (iii) the point defect concentrations. For the sake of convenience, we shall view the vapor phase pressures as the cause, since they are adjustable parameters in experiments. From the above discussion, it is clear that, as required by phase equilibria, the GaAs crystal point defect thermal equilibrium concentrations are dependent upon the vapor phase pressures. It is convenient to express such relationships first by kinetic reactions of the type As,, * Vi, + 4As,,
(3)
where AsASis an As atom on an As site in the GaAs crystal and Vk is the neutral As vacancy. Reaction (3) holds for the GaAs crystal with a composition within the allowed composition range a% shown in Fig. 1. At an As vapor pressure lower than that corresponding to point a‘, in equilibrium with a Ga-As material whose average composition is outside the composition range a% of Fig. 1 on the Ga-rich side, the Vi, thermal equilibrium concentration in the GaAs crystal Cy:, is that at point a’, with excess Ga atoms existing in the liquid phase a. At an As vapor pressure higher than that corresponding to point b’, in equilibrium with a Ga-As material whose average composition is outside the composition range a’b’ of Fig. 1 on the As-rich side, Cy:, is that at point b’, with excess As atoms existing in the liquid phase b or as solid As. In the past, a typical way of expressing the relationship between the point defect thermal equilibrium concentrations and an As vapor phase pressure is via the use of a reaction constant K. For example,
the range a’b’ shown in Fig. 1. For instance, consider the contributions of only the single point defects,
is the excess As concentration, while SC,, = -SC,, is the excess Ga concentration, which is responsible for the GaAs crystal composition deviation from the G%.sAsO,, stoichiometry. In eqn {2), Czl, CzSa and CT&% are the IAS, Asoa and V,, thermal equilibrium concentrations, respectively, and Cz,, C& and CT_ are the Ioat Ga,, and Vo, thermal equilibrium concentrations, respectively. Here, the concentration
where PAY is the As, pressure, obtained from reaction (3). While expressions such as eqn (4) showed qualitatively the dependence of the thermal equilibrium concentration of a point defect species on PAS, to a power, they have been rather useless on a quantitative basis, since neither the vafue nor the expression of I( is known. Because of the very small allowed composition range of GaAs, K cannot be measured and there has been no calculation of K until recently [5,6]. Here we review briefly the derivation given in Ref. [S] in which quantitatively useful expressions and
T. Y. TAN
920
values for neutral GaAs point defect species have been obtained. Formation of a point defect in GaAs (or other compound semiconductors) can be via several parallel kinetic paths or kinetic reactions, because there is more than one ~uilib~~ vapor phase. All such possible reactions, however, lead to the same thermal equilibrium concentration of the concerned point defect species, because all vapor phase species are in thermal equilibrium among themselves. Therefore, among the four available vapor phases As,, As,, As, and Ga,, only one needs to be considered, for example, As,. Assuming that As, is the involved vapor phase, we present a detailed derivation for a neutral As vacancy Vis so as to illustrate the main physical aspects involved. For the cases of PAS,V$, , I&, As& and Gak the procedures are essentially the same as for Vi, and hence only the results will be given. Consistent with that commonly adopted in the literature and analogous to that of the elemental semiconductor case, we shall first define a stundard free energy of formation of Yk, g:“,, . It is the free energy expended in creating the broken bonds at the cavity associated with Vi, formation. In determining the thermal equilibrium V” concentration in an elemental semiconductor, the standard Gibbs free energy of formation of V”, g$, is the sole energetic factor. This is the case because in forming a V”, the atom removed from the crystal interior site is replaced at an appropriate crystal surface kink site, which leads to no net atom number or surface area changes and therefore no further free energy changes. The situation associated with the elemental crystal cases does not hold for GaAs or any other compounds. In accordance with reaction (3), when forming a Vi,, the As atom number in the GaAs crystal is decreased by one while the As, molecule number is increased by one-quarter. The free energy change associated with this As atom transfer between GaAs and As, must also be accounted for. For this purpose, we adopt the convention that the Gibbs free energy of one As atom existing in vacuum is zero. For an As atom in a GaAs crystal, the Gibbs free energy is g,, (GaAs) = h,, (GaAs) - TsAs(GaAs).
In eqn (7) NV%is the number of VL in the GaAs crystal, NAs is the number of As atoms in the GaAs crystal, NAY is the number of As4 molecules in the vapor phase and S~~ is the ~~gurational entropy of the crystal As sublattice which contains Vi,. The quantity ST$? is given by
(8)
1 Asatom
(5)
In eqn (5) h,,(GaAs) is the binding or cohesive enthalpy of one As atom in the GaAs crystal. The value of h,,(GaAs) is negative. The quantity s,,(GaAs) is the entropy of one As atom in the GaAs crystal which is mainly due to thermal vibrations. For an As atom in an As, molecule existing in vacuum, the Gibbs free energy is g,,s(Ase) = &,s(Ase) - %&As.,).
In eqn (6) &(As,) is the binding or cohesive enthalpy of one As atom in the AS,+molecule. The value of hAs(As4) is negative. The quantity sAs(As4) is the entropy of one As atom in the As, molecule. This entropy is due to contributions of the internal degrees of freedom the molecule possesses: electronic, rotational, vibrational, etc. The As, molecules form a gas with a nonzero pressure. Because of their translational freedom, each molecule possesses an additional chemical potential or Gibbs free energy gi,. Based on the above discussion, the free energy situation associated with the thermodynamic ensemble of the As, vapor phase, the GaAs crystal and Vis is shown in Fig. 3, wherein the various free energy quantities denote that of one As atom and one VR. In Fig. 3 the zero value of g,,(GaAs) and gAs(As.,) is taken to be that of an As atom existing in vacuum. The quantities gAs(Asq) and gi,,/4 are additive and hence the zero value of gX,/4 is at gAs(Asq). The zero value of gbi, is at g,,(GaAs). The Gibbs free energy of the ensemble may now be written as
(6)
Fig. 3. A schematic drawing showing the free energies of binding of an As atom in the various involved materials together with the free energy of formation of a VA,.
Point defects in gallium arsenide
We shall regard g!,,%as that of an k&w1gus [lo],
g&=
921
defect species is the same as for Vi,. All the results are of the form
B
-ksTlnp,
(9)
A%
[ where B,
is the ideal gas pressure constant
BA% =
t
iht, -
hi
1 “*
[kaTl”‘,
ksT
given
by
(10)
with mA, being the mass of an As, molecule and h Plan&s constant. Note that reaction (3) requires aN,,laNv, = l/4 and aNA,/aNV~, = - 1 to hold. Using eqns @j-(10) in eqn (7) and minimizing G with respect to NQ, we obtain
1
122= As exp -- cgiJK
’
(14)
where x0 stands for any of the defect species. Expressions of the factors Ati and (g$)“” for all cases are summarized in Table 1. FOTVg, we can now determine Kvn, by comparing eqns (4) and (11): f&n, = Bzi exp[ -(g$$‘/ks
T&
(15)
The reaction constant Kfi of defect x0, as in the representation of c> by
(16)
=[%]‘“exp[-F],
(11)
where (g$)‘* is the effective Gibbs free energy of formation of Vi8 given by (S&C,,)“’= g& -
&?A,,
(12)
with Sg, being the difference of the As atom free energies of binding defined as &As
= i!,,
@aAs)
-
gA, 64~4
h
(13)
In eqn (11) eye is the normalized VP, thermal equilibrium con&tration, C”O is the actual Vi, thermal equilib~um co~centr~~~ and CO is the As sublattice site density. The prO&?dtXe of determining the thermal equilibrium concentrations of the other neutral point
where t is an appropriate power, is of a form analogous to eqn (15). Expressions of Kfl for all six neutral point defect species are also listed in Table 1. Our present expressions for the thermal equilibrium point defect concentrations are functions of the point defect standard free energy of formation and vapor phase related energetic and pressure factors (which are already known). These expressions are obtained by considering explicitly, in addition to the usual standard Gibbs free energy of formation of a defect, the contribution of an As vapor phase to the Gibbs free energy of the thermodynamic ensemble consisting of the GaAs crystal, the defect species and the As, vapor phase. The As, vapor phase contributes to the determination of the point defect thermal equilibrium concentration mainly in two ways. The first amounts to essentially quite a large enthalpy wnt~bution resulting from the binding enthalpy difference of an As atom existing in GaAs and in an As vapor molecule. The second is a large entropy
Table 1. Summary of the expressions of the thermal equilibrium concentrations of the six electrically neutral point defect species (x0) in GaAs. These expressions are of the general form c$ = A, exp[- (&F/k, 7’1.The factors A, and Cp$)” are listed for each case. Also listed are the expressions of K* in the reaction constant representation cz = K*P!+, where I is the appropriate power of P& in the factor of A, of the cz = A, exp[ -(g:‘,~ff/k, 7’j representation
922
T. Y. TAN Table 2. Enthalpy values (in eV) of one As atom in the various As-containing phases
at room temperature NBS [ll] Arthur [9] Adopted
As,
As (solid)
0 0 0
-3.1350 -3.1350 -3.1350
contribution resulting from the As, vapor phase molecule translational freedom. To calculate numerically the thermal equilibrium concentrations of the point defects, the actual values of the various involved thermodynamic quantities are needed. The values of PAq shall be the experimental ones. When expressed in units of atmospheres, eqn (10) yields BAs4 = 135T5j2atm.
As, - 1.9820 -1.9192 -2.092
GaAs
As, -2.7625 -2.5931 -2.8125
-3.5032 -3.5002 -3.5002
Figure 4 shows the calculated CT:8 and Cy;, values for PA_ values at the GaAs crystal Ga-rich solidus and As-rich solidus. A GaAs crystal in equilibrium with the As, vapor phase with PAId values between these two extreme cases will have CT;* and Cx values between the two branches of the appropriate curves. Using the Phillips-van Vechten two-band dielectric model [13], Wager [14] recently obtained
(17)
h6ga = 1.62 eV, Thus the numerical values of the pre-exponential factors in the point defect thermal equilibrium concentration expressions are now calculable. For g$ and 6g,,, etc., we shall approximate them by th: appropriate enthalpies h,;, and S/r,, , etc. This kind of practice usually yields sufficiently accurate results if the concerned quantities are all related to crystals. Using this approximation for a gas phase material, however, the inaccuracy thus incurred can be large. Fortunately, this does not seem to be the case for the As vapor species [5]. In Table 2 the values of the relevant enthalpies of the various As species obtained from the literature [9, 1I] as well as the values adopted are listed. These values are for one As atom. From Table 2 and eqn (13), we obtain ah,, z -0.69 eV.
(18)
&a”,,z 3.11 eV, Here, the uncertainties in the enthalpy values are probably even greater than in the cases of I’L and Vi,. Using eqns (17), (22), (23) and the relevant quantities in the appropriate expressions listed in Table 1, we obtain eq cG&
x 1 l.6T5’4P&:‘2 exp [-z],
eq e 0 086T-5’4P~~~ ’
(19)
The use of the values given by eqns (17)-(19) in the appropriate expressions listed in Table 1 yields CT;, x ~.~T”*PL;‘~ exp[ -31?$],
c;t. x 0.293T-5’8P”4Asrexp[ -%I.
(20)
(21)
T (“Cl lo=‘
1200 : :
:
900 :
600
:
____.
c 5 Y
JO')-
s P 2"
lo'*
f "
10"
-
'"'4: 10'0 0.6
(\ ', '*.,, '\ '\\ ', '8, ', '\ \ As-rich .)& 0.7
VL
Ga-rkh
\
'\ '\ \
‘8, “\
\
0.8
0.9
1 03/T
In view of the approximate nature of the values of the involved quantities, an uncertainty factor of at least 10 should be assigned to eqns (20) and (21).
(25)
Figure 5 shows the Cya;,and CTs;avalues calculated from eqns (24) and (25) for PAa values at the GaAs
Some efforts have been made to calculate the standard enthalpies, for example, h:a etc. Using the macroscopic cavity model, van Vechteg [12] obtained x 2.59 eV.
exp [ -%I,
‘A$,
point defect formation
h:;, x h&
(24)
'\ '\
,
1.0
1.1
1.2
(K-’ )
Fig. 4. Calculated V& and Vi, thermal equilibrium concentrations for Pti values at the GaAs crystal Ga-rich solidus and As-rich solidus.
923
Point defects in gallium arsenide
1200
1o18
T 63 900
600
As-rich
0.6
0.7
0.8
0.9
103/T
1.0
1.1
1.2
(k-’ )
Fig. 5. Calculated Gai, and As”&thermal equilibrium concentrations for PA, values at the GaAs crystal Ga-rich solidus and As-rich solidus.
crystal Ga-rich solidus and As-rich solidus, respectively. It is interesting to note that the calculated CT& values at the GaAs crystal As-rich solidus are between 1 x 1OL6and 1 x 10” cms3 and fairly temperature dependent. The AS& concentration has been found to be in the range of N 1 x 1016-1x 10” cme3 in As-rich GaAs crystals and is temperature independent [13, 16181. Our calculated A& concentration and its temperature dependence are in agreement with these experimental findings. The antisite defect Ass, is the main constituent of the deep donor EL2, which is responsible for growing modern semi-insulating GaAs crystals using the liquid encapsulated Czochralski (LEC) method under As-rich conditions. Presently, values of g$ and g$, are not available. Therefore, numer&l values of cq and c? it ‘oa cannot yet be obtained.
as well as by the Fermi level position of the semiconductor. A charged point defect whose energy corresponds to having an electronic level in either the conduction or the valence band is an unstable situation since it will increase the free energy of the ensemble. Thus, the defect will loose a charge (negative and positive, respectively) to either the conduction or the valence band edge, becoming one in the next highest charged state with an electronic level either at the band edge (the neutral ones) or in the gap. The correct and complete derivation of the thermal equilibrium concentrations of charged “flaws” (point defects or impurities) was first obtained by Shockley and Last [22]. Here we shall use a different method to obtain their results [22]. This is the Gibbs free energy formulation [6] which has the advantage of being more explicit so that certain important physical features may be better pointed out. The defect levels are assumed to be deep ones, for example, that shown in Fig. 6, wherein a few deep acceptor levels are indicated. The Fermi level of the crystal Er is determined by a shallow dopant compensated by the formation of charged point defects. Consider the case shown in Fig. 6. The thermodynamic ensemble consists of the systems of points defects and electrons. This is a grand canonical ensemble for which energy and mass transfers both occur among the constituent systems. With e denoting an electron, the kinetic reactions, xO+eox-, x-+eox*-,
(26b)
x2- + e 0 x3-,
(264
EF
3. CHARGED POINT DEFECT THERMAL EQUILIBRIUM CONCENTRATIONS Charged point defect species exist in Si and in GaAs. They contribute to the diffusion phenomena. Charged point defects can form in a semiconductor, since it can lower the total Gibbs free energy of the thermodynamic ensemble consisting of the point defect and the electronic systems. The thermal equilibrium concentrations of the stable charged point defects depend upon the crystal Fermi level [19-221. A stable charged point defect must possess an electronic energy level in the semiconductor forbidden energy gap. Thus, the concentration of the defect is influenced by its electronic level position
(264
Ev Fig. 6. GaAs band diagram showing schematically the relation of the Fermi level and the native point defect (x) deep acceptor states. The left-most arrow indicates the total ensemble (defect and electron systems) Gibbs free energy decrease of forming one x- from x0. The middle arrow indicates the additional total ensemble Gibbs free energy decrease of forming one x2- from x-. The right-most arrow indicates the additional total ensemble Gibbs free energy decrease of forming one x3- from x2-.
T. Y. TAN
924
describe the successive formation of the charged species of x. in the presence of both X- and x0, the Gibbs free energy of the ensemble is G = N&&)ee + N,_ (g;_ )“e + G, - ?-SW+
(27)
where NS and N,_ are the numbers of x0 and x -, respectively, (g$,)‘” and (gf;_>” are their effective free energies of formation, respectively, G, is the Gibbs free energy of the electron system and Sconfigis the configurational entropy of the crystal, given by
scc,a
=
ln
k
3
INo+ Nxo+ Nx-Y No! N,! N,_!
’
holds. In eqn (30) ni is the intrinsic carrier concentration and Ei is the Fermi level of the intrinsic crystal. Therefore, C,-(n) -=expr$]=[!Jrk] CX-64)
(31)
holds, where n and p are the electron and hole con~ntrations of a doped crystal, respectively. In eqn (31) the last two identities follow from the calculation of n and p for doped semiconductors. For reaction (26b), one uses the same procedure to readily obtain
1 1
G- @I
(28)
EF
-
42
k,T
c,_o=exp
(32)
and therefore where No is the number of atoms of the appropriate sublattice. Reaction (26a) requires aN,/aiV,_ = - 1 to hold. To minimize G, with respect to N,.. , we use aGJaN,_ = (aG=/aN~)(a~~/aN~_), where N, is the number of electrons in the crystal. Reaction (26a) also requires aN,/aN,_ = - 1 to hold. By definition, we further have aG,/aN* = EF and (gi-)‘“(g$,)‘e = E,, . The minimization of G with respect to N,_ now yields
G @I
2Er - Ed - & k,T
--exp C,
G-02) -= C.+ (ni)
[I
1’
-n 2. ni
(33)
(34)
In general, for x2-, we have (35)
C,_
1 1
N,_
EF-&I
C=N=exp ~ k,T XQ
x4
’
(29) (36)
where the energies are measured from the valence band edge, i.e. E, = 0 has been assumed. When the Fermi level is above the acceptor level, xbecomes more abundant than x0, with C, not influenced by the Fermi level. Note that (gf-)“” is higher than f,&)‘“, but under sufficiently heavy n-doping conditions, x- becomes more abundant than x-, because the free energy decrease of the electronic system is more than enough to compensate for the free energy increase of forming x-: for each x - formed, the ensemble energy is decreased by E, - E,, (see Fig. 6). The level E,,, is a real state but EF is not. The electron actually comes from the conduction band edge of energy EC and yet the released energy does not correspond to EC - E,, . This is because the total ensemble Gibbs free energy is minimized, including that of the electrons. This corresponds to measuring the free energy change associated with eliminating one average electron, for which its chemical potential is EF. For an intrinsic crystal,
C-
(4
-=exp C,
-4 - Eat
)
-
i
k,T
1
cx:-(n) -= C,:- (ni)
[I
(37)
In eqns (35)-(37) m and z are integers. Equations (35)-(37) may be directly obtained by considering the reaction x0 + ze 0 xi-. However, formation of x2via the simultaneous reaction of x0 and ze is a physically unlikely process. Proceeding as for deep acceptor defects, one obtains for deep donor defects
cx:+ (PI
= exp[ -w],
(38)
G-~,+(P)
C*,+(P) -=
Cxl+(ni 1
(30)
-.n ni
In eqns (38~~),
-P z. [Ini
Edr is the level of x2+.
tw
Point defects in gallium arsenide In the above defect,
including
expressions,
x may stand
an impurity
atom.
for any
singly, doubly, and triply negatively charged Ga vacancies V & , V & and V & , Baraff and Schhiter [26] have given the energy levels as E,, z O.l33E,, Es2 z 0.355E, and E,, z 0.49E,, respectively. Here, E8 is the GaAs crystal band gap energy at 0 K, _ 1.5eV. For high temperature applications we assume that the same expressions also hold. To approximate the effect of band gap narrowing at high temperatures, we further assume that Es 5 1.15 eV. Now, eqns (21) and (36) yield
V&, h, the elecare 2.05, respect-
ively. The values of h, are dependent upon the Fermi level position EF. For undoped GaAs the V& thermal equilibrium concentrations are much smaller than the intrinsic carrier concentration n, and the crystal may be approximated as intrinsic. By letting EF = Et = 0.5E,, where E, is the intrinsic Fermi-level energy, we obtain for undoped GaAs
TM’/* ex p(-s),
using
while the n value is
the charge
neutrality
con-
dition. Assuming that free electron compensations are due to the presence of only the negatively charged Ga vacancies, the charge neutrality condition is
The triply negatively charged Ga vacancies V& are responsible for Ga self-diffusion in GaAs and AlAs/GaAs superlattice crystals under intrinsic and n-doping conditions [6,23] and for the diffusion of the shallow donor Si (occupying the Ga sublattice, Sib.) in GaAs [24,25]. The thermal equilibrium concentrations of the various negatively charged Ga vacancies have been recently calculated [7’j. For the
cy;; (n,) z 0.293Pii
ture and is difficult to determine easily determined
4. NEGATIVELY CHARGED Ca VACANCY THERMAL EQUILIBRIUM CONCENTRATIONS
where h,, is the total enthalpy of forming a is the configurational enthalpy and zEF is tronic energy released, h,,. The values of h, 2.46 and 3.03 eV for the z = 1, 2 and 3 cases,
925
(42)
where h,(q) is h, under intrinsic conditions. The values of h,,(q) are 1.48, I.31 and 1.3 eV for the z = 1, 2 and 3 cases, respectively. For n-type GaAs it is more convenient to use [eqn (37)]
(43)
instead of eqn (41), because for a constant dopant concentration the value of EF changes with tempera-
n=C,-C:&)-2C;&(n)-3C:qZi;(n)+p,
(44)
where p is the crystal hole concentration, C refers to the actual concentrations and C, is the concentration of shallow donors which are assumed to be fully excited. In eqn (44) the fact that in forming a Vi,. a V& and a V&, 1, 2 and 3 electrons are consumed, respectively, has been accounted for. Using np = nf, we obtain n = f[C, - C&(n) +f{[c,-
- 2Q
(n) - 3c;;.
c:__(n)-2CT$J(n)-
+ 4$}‘;*. Using the equilibrium
(n)] 3C;&;((n)]* (45)
PA, values at the thermody-
namically allowed GaAs crystal composition limits given by Arthur [9], shown in Fig. 2, the values of C~&,), CT;+,) and CT;;(n,) can be calculated from eqn (42; and the values of CT&), Cy& (PI), CT&(n) and n are then readily calculated using eqns (43) and (45) for a given C, value. The calculated CyG*(n), Cy;. (n). C,,&(n) and n values for a few C,, values are shown in Fig. 7. Figure 7 shows that, at a given temperature, the thermal equilibrium concentrations of the charged Ga vacancies (also holding for any other point defect species) increase as the doping level is increased. This well known fact [22] is now called the Fermi-level effect [23], since the CT;I(n) changes are related to the position of the Fer@ level, EF. From eqns (41) and (43) we see that C,;;(n) increases as the Fermi level moves towards the GaAs conduction band edge or as n is increased. An unusual and previously unanticipated aspect is the temperature dependence of the defect concentration for a constant dopant concentration. It is clear from Fig. 7 that the temperature dependence of CT:;(n) is in general weaker than that of CT;&) and, in particular, that of CT;,(n) has exhibited behavior from a temperature independence to a small negative temperature dependence under heavy n-doping conditions in that, as the temperature is lowered, Cy;. (n) is unchanged or even increased. This is in strong contrast to the normal positive temperature dependence of the point defect thermal equilibrium concentrations, which decreases as the temperature is lowered [CT;,; (n,) are examples]. This is also a result of the Fermi-level effect. For a
926
TAN 1020,
lOlO,.,.,,,.,.,. 0.6 0.7 0.8
lo20,
12FO;
0.9
: 9;O
;
1.0
:
1.1
1200:
: 900
10'01.,,,.,.,.,-. 0.6 0.7 0.8
:
:
0.9
1.0
600
1.1
2
6pO
b
1OlOI 0.6
0.7
0.8
0.9
1.0
1.1
0.6
0.7
0.8
0.9
1.0
1.1
1.2
Fig. 7. (aHd) The calculated thermal equilibrium concentrations of V,, V& , V& and n, respectively. Legends le18, le19, etc. refer to shallow donor concentrations of 1 x 10’*ctn3, 1 x 10’9cm-3, etc. and are the same for all four figures. In each figure, the vertical axis is the concentration values in crnm3,the lower horizontal axis is the inverse temperature, lO’/T(K-‘) and the upper horizontal axis is temperature in “C.
very high shallow donor concentration,
EF is close to
EC for all temperatures of interest. For a moderate shallow donor concentration, the Fermi level moves closer to EC as the temperature is lowered and hence its effect becomes more prominent. With EF very close to EC, the electronic enthalpy released h,, can be so large that it may compensate the V& configurational enthalpy h,r sufficiently to lead to a very small or even negative total V& activation enthalpy ha3, resulting in the C$ (n) temperature independence or negative temperature dependence behavior. For example, if it is assumed that EF - Eg - 1.15 eV
holds, we obtain h,, N (- 0.42 eV). This is in qualitative agreement with the calculated results that, under heavy n-doping, CyLi (n) increases as the temperature is lowered. Using local vibrational mode measurements, Uematsu and Maezawa [27] and McQuaid et al. [28] have recently determined the V& concentrations and the corresponding n values in molecular beam epitaxy (MBE) grown GaAs layers doped by Si. These results are plotted in Figs 8 and 9 together with the appropriate calculated values, which are in good agreement. Since the samples are low temperature GaAs
927
Point defects in gallium arsenide T
(‘Cl
600
400
200
10M J]
0.6 0.8
1.6 1.8 2.0 2.2
1.0 1.2 1.4 103/T(
K-l)
Fig. 8. Experimentally measured V&i concentrations in MBE grown GaAs layers [27,28] fitted to the caicuiated values. Legends let9 and le20 refer to Si donor concentrations of 1 x lOI and 1 x 1020cm-3, respectively.
which are supposed to have been grown under highly nonequilibrium conditions, the agreement between the ex~rimental and our calculated results should be regarded as surprisingly good. Under intrinsic conditions, the Voa species exhibit the normal positive temperature dependence (Fig. 7). Therefore, it is expected that these vacancies can be frozen-in from a high temperature and precipitate out at a lower temperature. Such an experimental result has recently been obtained by Williams et al. [29], who observed formation of microvoids in undoped, semi-insulating GaAs grown by the LEC method and
receiving a post-growth annealing. The V, concentrations, estimated by the present author based on the void size and density reported by Williams et al. [29], is in quantitative agreement with our calculated thermal equilibrium concentration of the Vo, species for intrinsic GaAs (Fig. 10). Allowing an accuracy factor of -2, it is seen from Fig. 10 that these vacancies could be those frozen-in from either the GaAs melting temperature of 1238”C, or N 1000°C provided the crystal is As-rich. Semi-insulating LEC GaAs crystals can only be grown from an As-rich melt and therefore the data should he that of an As-rich crystal case. Associated with diffusion studies in n-type GaAs and GaAs/AlAs superlattice crystals, there are several recent experimental results requiring the interpretation that the responsible point defect species V& has attained its thermal eq~librium concentration C:&;(n) at the onset of each experiment [25,30,31]. If it is assumed that Vi; is undersaturated at the onset of these experiments, then the crystal interior C”_(n) value can only be reached by the V&i indiffusio?process which has heen found to be too slow [32]. Furthermore, in these experiments the substrate materials have not been (and cannot be) pre-annealed to introduce V& to its experimental temperature C”,_ (n) values before the onset of the experiment. Tlk only leaves the possibility that, for sufficiently highly doped n-type GaAs and GaAs/AlAs superlattice, C:,_(n) is fairly temperature independent. The pres&tly calculated
T
17 T 600
(*C) 400
3
10
:
(OCt 600
900
1200
:
:
:
200
---..“---_______ 10'8 p? E .2 c
10'6
10
10'4
0.6 0.8
1m
11
0.6 1.0
0.7
Williams etal. 0.8
(1991) 1
0.9
1.0
1.1
1.2
1.2 1.4 1.6 1.8 2.0 2.2 103/T(
K-')
Fig. 9. Experimentally measured n values in MBE grown GaAs layers 127,281, wrres~nding to the measured V& concentrations shown in Fig. 8, fitted to the calculated values. Legends le19 and le20 refer to Si donor concentrations of 1 x lOi and 1 x 10”cm-3, respectively.
103/T(k-')
Fig. 10. The vacancy concentration, in microvoids measured by Williams er al. [29] in a semi-insulating LEC GaAs crystal using TEM, fitted to calculated V, thermat equilibrium concentrations for intrinsic GaAs. Placement of the experimental data at the high temperature range is due to the present author.
928
T. Y. TAN
temperature independence behavior of Cy;; (n) satisfies this requirement. In the literature, there also exists a number of room temperature experimental results involving the nonequilibrium point defect generation phenomena. These include the dark-line defects formed during GaAs laser operations [33,34], the optical pumping damage in n-type GaAs [35] and the enhanced Be diffusion in GaAs heterojunction bipolar transistors during operations [36]. In these experiments, excess Ga self-interstitials were generated to form dislocation loops, which were attributed to carrier recombination enhanced Frenkel pair generations. This explanation pointed out the energy source for generating the point defects, but did not provide a thermodynamic reason for the fact that the vacancies so generated appear to be stable. Our calculated C$ (n) negative temperature dependence behavior offers a thermodynamic explanation to these phenomena. For these results, the difference between the crystal growth and the experimental temperatures is so large that our calculated Cyj (n) values can no longer be regarded as temperatu& independent. Instead, the CT& (n) negative temperature dependencies must be considered. Thus, at room temperature, V& is undersaturated. The carrier recombination generated V,, species of the Frenkel pair is to fulfil the larger C’y;; (n) needs and the I,, species condense into interstitial dislocation loops to further lower the Gibbs free energy of the ensemble. The details associated with these experiments and the explanation of the experimental results using the C;&(n) temperature independence and negative temperature dependence behavior have been thoroughly discussed elsewhere [A. 5. SUMMARY
On the happy occasion of celebrating Bob Balluffi’s seventieth birthday, it is only fitting that some of his disciples discuss point defects. This paper summarizes the theoretical aspects of the work on point defects in GaAs carried out at Duke University during the last five years by my students, colleagues and myself. Our work has been supported by U.S. Army Research ORice under contract DAAL03-89-K-0119. Expressions of the thermal equilibrium concentrations of point defects in GaAs, including the neutral and charged species, have been derived. These expressions are explicit functions of well-defined thermodynamic quantities, which in turn yield explicit expressions for the reaction constant K in the usual As, pressure power law representation of the point defect thermal equilibrium concentrations. Such power laws have been of little quantitative value in the past, because K were not known. In the present derivation, emphases have been placed upon the
difference between the Gibbs free energies of an As atom in the interior of a GaAs crystal and in an As vapor phase molecule, and the role of the crystal Fermi level. Numerical values of the thermal equilibrium concentrations of the neutral and three negatively charged Ga vacancies (V”,, , Vi,, V& and V&i),, the neutral As vacancies Vi, and the two neutral antisite defects Gai, and As:, have been obtained. The calculated thermal equilibrium concentration of the anion antisite defect A& reaches a peak value of about 1 x 10’7cm-3 and is practically temperature independent, in agreement with experimental findings. For the various negatively charged Ga vacancy species, the thermal equilibrium concentrations of the triply negatively charged Ga vacancy VA;, C”, (n) has been emphasized, since V& dominates Ga self-diffusion and Ga-AI interdiffusion under intrinsic and n-doping conditions, as well as the diffusion of Si donor atoms occupying Ga sites. Under strong n-doping conditions, Cy:, (n) VG.
has been found to exhibit a temperature independence or a negative temperature dependence behavior, i.e. the CT& (n) value is either unchanged or increases as the temperature is lowered. This is quite contrary to the normal point defect behavior for which the point defect thermal equilibrium concentration decreases as the temperature is lowered. This CT:,(n) property has provided explanations to a number of outstanding experimental results, either requiring the interpretation that V& has attained its thermal equilibrium concentration at the onset of each experiment, or requiring mechanisms involving point defect nonequilibrium phenomena. The calculated Cy& (n) values are seemingly in good agreement with available experimental
results.
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Tan T. Y. and Gijsele U.,
Appl.
Phys.
A
37,
I
(1985).
2. Casey H. C., Jr, in Atomic Diffusion in Semiconducrar.r (Edited by D. Shaw), p. 351. Plenum, New York
(1973). B. Tuck, Atomic DQjiuion in III-V Semiconducrors. Hilger, Bristol (1988). 4. Kriiger F. A., The Chemistry of Imperfecr Crystals, Chap. 13. North-Holland, New York (1964). 5. Tan T. Y., Mater. Sci. Engng E 10, 227 (1991). 3.
6. Tan T. Y., Giisele Marer.
7.
U. and Yu S., Crir. Rm. Solid SI.
Sci. 17, 47 (1991).
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