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The estimation of minimum growth temperature for crystals grown from the gas phase
Journal of Crystal Growth 87 (1988) 397—407 North-Holland, Amsterdam
397
THE ESTIMA11ON OF MINIMUM GROWTH TEMPERATURE FOR CRYSTALS GROWN FROM THE GAS PHASE P.M. DRYBURGH Electrical Engineering Department, The King’s Buildings, University of Edinburgh, Edinburgh EH9 3JL, Scotland, UK
Received 10 August 1987; manuscript received in final form 19 October 1987
It is proposed that the limiting temperature, 0, for single crystal growth from the gas phase is the temperature at which the kinetic rate of arrival at each surface site is equal to the rate of removal by surface diffusion. The expression derived for 0 at atmospheric 2/m1”2, where ESD is the activation energy for surface diffusion, d is interatomic distance pressure is p3/2 exp( — ESD/kO) = 0.048d in A and m is molecular weight of the diffusing species. Surface diffusion is considered for the cases of covalent, ionic and metallic solids, and simple models used to allow the estimation of ESD in each case. Results are presented for 33 crystals including group IV elements, Ill—V compounds ionic solids and metals. Agreement with experimental results is shown to be satisfactory in those cases where results are available from the literature.
1. Infroduction Processes involving the deposition of a solid from the gas phase are used to produce a growing diversity of materials, ranging from protective coatings, structural layers and abrasives to blanks for optical fibre manufacture and epitaxial layers of 111—V compounds or silicon. The extensive range of possible precursors often allows considerable freedom in the choice of deposition temperature, since usable reactions range from pyrolysis of unstable compounds to entropy-driven reactions of high temperature species, but it is necessary to analyse the kinetics of the transport process so that favourable growth rates may be ohtamed, In attempting to grow epitaxial layers, or even bulk single crystals by sublimation, it is additionally necessary to distinguish between temperatures at which the appropriate species will be deposited at a useful rate and temperatures at which the resulting material will be in the form of a single crystal. There is at present no means of making a quantitative prediction of the temperature at which single crystal growth might reasonably be ex-
pected, although many workers rely tacitly on a generalization, based upon experience, that materials of higher melting point require higher deposition temperatures. The factors which determine whether a growing layer is a single crystal or not obviously depend upon the ability of mcident atoms to find their ways to the correct sites by surface diffusion and, since we expect some sort of connection between rates of bulk diffusion and surface diffusion, it is not unreasonable to use melting point as a guide to deposition temperature. There is, however, no quantitative general correlation between melting points and minimum growth temperatures. The need for more reliable guidance in the choice of likely growth temperature arises as a result of the great range of conditions now available. An arbitrarily chosen and inappropriate ternperature range, based largely on the properties of the precursors, may result in the attempted use of a reaction which can never provide single crystal material, or it may result in much wasted experimental work being carried out on a potentially satisfactory chemical system. This paper presents a general rule for predic-
0022-0248/88/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
398
P.M. Dryburgh
/ Minimum growth temperature for crystals grown from gas phase
adsorption
adsorption
___
C
SURFACE DIFFUSION
lattice-site attachment
r
I1~
) —
surface reaction
F
— by-products
—
SURFACE DIFFUSION
—
by products
lattice-site attachment
Fig. 1. Possible reaction schemes for crystal growth from the gas phase.
ting the minimum temperature for the growth from the gas phase of single crystals of a variety of materials. The guiding principles have been that any useful rule should be valid for as wide a range of crystals as possible and that the data required for the material, or its constituent atoms, should be readily available. The price of such generality is, of course a lack of precision and specificity.
2. Model used and underlying assumptions The simple model used in this paper depends upon three preliminary assumptions: (1) growth rate is surface-kinetically controlled and not nucleation limited; (2) growth proceeds by one of the two possible routes shown in fig. 1; (3) chemisorption involves only a small number of neighbouring atoms. This assumption has been discussed and justified by Ertl [1] for metal surfaces. The essential feature of the model used is that the rate of arrival of incident atoms or molecules at surface sites must not exceed the rate at which newly arrived adsorbed atoms or molecules are moving away. This is equivalent to assuming that there is small probability of any incident atom or molecule arriving immediately at a correct site. The rate of arrival is obtained from simple kinetic theory, equated with the rate of migration oh-
tamed from simple vibrational diffusion theory, and an expression derived for the corresponding limiting value of temperature.
3. List of symbols and units b c d E F h
Madelung constant of two-dimensional sheet Coordination number Interatomic distance Energy of interaction Incident flux Planck’s constant ~ H * Heat of atomization ~ H~° Standard heat of formation k Boltzmann’s constant m Numerical “atomic weight” m0 Atomic mass unit M N0 p q Z a
~ flAB
9 ic
4s
Molecular mass (M = m0 X m) Avogadro’s number Pressure Electronic charge Ionic charge Madelung constant Permittivity of space Electronegativity difference between atoms A and B Minimum (or limiting) temperature for ordered crystal growth Relative dielectric constant Electrostatic potential
P.M. Dryburgh
/ Minimum growth temperature for crystals grown from gas phase
4. Derivation of equation giving limiting temperahire 0 In simple kinetic theory the incident flux F of molecules on a surface is given by
399
The combined expressions (1) and (3) yield 6.65
~
rn1”2
11/2
=
2kT exp I
—~—
—
hence F= (2’2~-MkT)1”2’
03/2
so, for a pressure of 1 atm, F
2
(mT)”2 m
=
.
=
=
A fraction of the incident particles will be scattered from the surface immediately but this cannot be estimated and will be ignored for the present purpose. The particles which become adsorbed will vibrate at some characteristic frequency i’,~ due to their thermal energy. Some energy is required to detach the adsorbed atom or molecule to a sufficient extent to allow it to move to another site, and this is the activation energy for surface diffusion ESD. The probability that the thermal vibration energy will be ESD is exp(—ESD/kT) and the rate ID at which the adsorbed particle will be detached from one site to move to another is given by =
0.048(d2/rn’/’2).
(4)
—1
~
2 with the darea of A, onethen lattice site as occur r rIf we ~d,take where is in collisions at each lattice site at a rate f 0 where 2 ~1 (1) f~ 6.65 X 1 ;;;i7i~ d
fD
=
Calculation of 0 from eq. (4) requires values for three quantities: the interatomic distance in the crystal surface d, the formal molecular weight m
2 66 x 1029 ____________
exp(—ESD/kO)
of the incident species and the activation energy for surface diffusion E The first two of these 5D. quantities are readily available for most systems, but there may be some doubt about the value of m in certain cases. It should be noted,tohowever, is very insensitive errors in that m.is the The dominant parameter ESDvalue and, ofas 9the choice of its value virtually dictates the value of 0, it is necessary to examine the process of surface diffusion in more detail in order to get the best practicable estimating procedure. 5. The estimation of ESD The rules by which the atomic packing arrangement is determined in a surface are different in the extreme case of covalent, ionic and metallic crystals and these are considered separately with reference to surface diffusion and the estimation of their corresponding activation energies.
v 0 exp(
—
E5D/kT).
(2)
The only source of vibrational energy available is thermal, so we can write =
kT,
vo
=
2kT/3h.
By substituting for v0 in eq. (2) we obtain an expression for ID: 2kT I ESD \ fD=—jj--exP~—-~y).
(3)
5.1. Covalent crystals The interaction energy E5 of an atom bonded to the surface of its own covalent solid may be considered as having two components, the energy of a normal single bond, E,, and the energy of interaction with a small number x of surface neighbours, each contributing an nearest energy E2/x (see fig. 2): =
If the rate of arrival of incident molecules is equal to the rate of migration, then fO~fD
and
Tm0.
E,
+
E2.
In the case of an AB compound of the zincblende structure, the position of an adsorbed atom with respect to the surface may be considered as being
400
P.M. Dryburgh
/ Minimum growth temperature for crystals grownfrom gas phase to convert one mole of AB into one mole of free A
adatom
E
2 /
atoms and one mole of free B atoms is
~ E~
E1 \~
X/
EAB
=
z~Hf+ ~ H~+ ~ H~.
surface atoms Fig. 2. The interactions of a single atom bonded to the surface of a covalent solid,
The energy relationships are summarized in fig. 3. Half the heat of formation is associated with each consistuent of an AB compound and the number of bonds per atom is c, so the energy required to break a single bond between solid AB and a single A atom is
determined completely by theIthighly 3 bond. followsdirectional that any nature of a single sp lateral movement of the atom during surface diffusion can take place only if the single bond is effectively broken, the interaction energy then being reduced to the weaker and largely non-directional nearest-neighbour bonding. This very simplified model allows us to identify E with E SD 1 and so obtain an estimated value for E5D by calculating E1 from thermochemical data. E1 is the energy required to break a single bond between an A (or a B) atom and solid AB and is easily calculated as follows. The standard heat of formation L~ H~°of AB represents the difference in energy between the compound AB in its standard state and the consistuent elements A and B in theirs. Heat of atomization L~ H * represents the energy required to convert an element in its standard state into free gaseous atoms. It is readily seen that the energy EAR required
EA
N0 atoms B
*
L~HA 1 mole A
~
Hf°+ L~H2 )/N 0c;
similarly EB
=
(~Hf + ~iH~)/N0c.
For an element, e.g. silicon, there is only one value Table 1 Estimated single bond energies E1 for some covalent crystals Crystal ~ H~° ~ H* (kcal/mol) (kcal/g-atom) (eV) A B A B C Si Ge
170.9 108.4 90
1.86 1.18 0.98
~
~
BP
2716
13216
7515
A1N AlP
76.0 39.3
78.0 78.0
AlAs
29.3 12.03
AISh InP
N0 atoms A ~
=
~
~
~
1159
0197
113.0 75.5
1.23 1.06
1.64 1.03
78.0
69
63 75.5
1.01 0.91 0.73
0.91
18.0
750 58
InAs
13.8
58
69
0.70
*
InSb
HB
GaN
7.44 26.2 29.2 19.5 10
58 69.0
63 113.0 75.5
69.0 69.0
35.7
26.8 26.8 26.8
69 63 65.7
0.67 0.89 0.91 0.86 0.80
1 mole B
1 mo I e AB Fig. 3. Energy diagram showing standard heat of formation ~ H~° and heats of atomization zlH~
GaP GaAs GaSh CdS
—
CdSe CdTe PbTe
34.6 24.33 16.4
ZnO ZnS
83.2 49.04
69.0
46.8
49.4 46 46
31.2
59.6 65.7
0.48 0.48 0.42 0.40
0.79 0.61
0.75 0.92 0.82 0.72 1.37 0.98
0.86 0.74 0.91
0.72 0.63 0.39 1.10 0.98
ZnSe 38 31.2 49.4 0.55 0.74 ZnTe 28.8 31.2 46 0.50 0.66 _______________________________________________
P.M. Dryburgh
/
Minimum growth temperature for crystals grown from gas phase
401
of z~ H * and the value of i.’i Hf is zero by defiition:
determined to very good approximation by electrostatic forces only, the terms representing exchange effects and Van der Waals forces being
E1
insignificant. In practice, however, the derivation of a straightforward estimating procedure, based upon a plausible model, is not obvious because of the difficulty in allowing for the effect of polarization. The problem of estimating polarization contributions to defect energies in serious in an infinite solid and was first discussed by Jost [4] and Mott and Littleton [5]. More precise calculations have been done by Rittner, Hutner and DuPré [6] and general accounts can be found in standard texts such as Dekker [7]. The case of an adsorbed ion, which is essentially outside the bulk of crystal, presents a simpler problem in the sense that Jost’s approximation is unnecessary, so no arbitrary value of radius has to be assigned, but it is still convenient to regard the crystal as a semi-infinite dielectric continuum to avoid the laborious summation of induced dipoles. It is also convenient to ignore the contribution made by Born repulsion, which accounts for approximately 10% of the lattice energy of an ionic crystal [8] but may well be of less importance here. The interaction or binding energy E5 of the adsorbed ion is written as the sum of two terms, one corresponding to the potential due to the constituent ions of the crystal E,~5and the other arising from the dielectric polarization of the crystal E~:
is simply given by
~ H~*/4N
Using the standard heat of formation from Barn and Knacke [2] and heat of atomization from Skinner and Pilcher [3], E, has been calculated for some group IV elements and Ill—V compounds. In the case of AB compounds there is obviously a choice of E1 values, depending upon whether an A or a B atom is considered. The results are given in table 1. 5.2. Ionic crystals The only ionic crystals for which results are reported in this paper have the NaC1 structure and only (100) surfaces have been considered. Fig. 4 shows a (100) face of an AB compound of NaC1 structure with a single A + ion occupying what would become a normal lattice site after subsequent growth. The interaction energy of the ion with the solid is E5. We make the reasonable simplification that there are only two other types of resting place for such an ion, an antisite p05i tion (y) and a “saddle” position (6). In moving from one normal surface site (A) to the next, the adsorbed ion has to pass over either a y-position or a 8-position (see fig. 2), the activation energies for the two types of movement being (E5 E~) and (E5 Eo). Surface diffusion will take place by the route offering the lower activation energy. The estimation of E5 for a binary ionic crystal is simple in principle, since cohesive energies are —
—
Fig. 4. Sketch of an ideal (100) surface of a compound AB (NaCl structure) with a single adsorbed A + ion, at a normal site A. An antisite position ~y and a saddle position 8 are shown also.
E5
E~5+ E~. For an ion sticking onto the surface of a binary ionic solid, E~5represents that part of the binding energy arising purely from the combined electrostatic potentials of the ions in the solid. To obtain an expression for this potential, we consider first the self-potential ~ at an arbitrary lattice point in an infinite ionic solid. This potential ~ depends upon the ionic charges, their separation and the Madelung constant of the lattice. 4
=
=
aZq/4ire0d.
A detailed treatment of the evaluation of Madelung constants has been given by Tosi [9]
402
/ Minimum growth temperature for crystals grownfrom gas phase
P.M. Dryburgh
I
a
_________________________________________________________________________________________________ b~
I
II
I
I
+
I
I -
-
-
-
a
+
+•
I
I
+
•
+
,
+
+ •
+
+1
+ +•
4 (i)
(ii)
Fig. 5. Notional components of the Madelung constant.
and a more general discussion by Ladd [10]. The accepted value for the Madelung constant of the NaCl structure (in terms of interionic distance d) is a 1.7476. 4) may be considered as having three components as illustrated in fig. 5(i). The component b is due to all the ions in the sheet of ions containing the given site, the sheet being one
~1-• •
+
+
/
+
e
+
+
•
+
•
+
+
Fig. 6. The derivation of the Madelung constant for a two-dimensional sheet of ions.
=
ion in thickness. Each of the two equal components a is due to half of the remainder of the crystal. In terms of contributions to the Madelung constant, it is obvious from fig. 5(i) that a 2a + b. (5) =
Fig. 5(u) has been drawn to emphasise that a is the required surface counterpart of the Madelung constant. In order to evaluate b, the individual attractions and repulsions have to be written as a series and the sum found. From fig. 6 it may be seen that the appropriate series is b=(—4+--~--~+ (4 ~ ~-
4
/ +
—
8 +
8
—
-~-
+
8
4
4 +
—
....
As with all series defining Madelung constants, this one converges extremely slowly. Fortunately, the sum to infinity has been computed by Emersleben [11] as part of a method for obtaining other constants. Tosi [9] quotes Emersleben’s value as 1.61554. From eq. (5): a
=
~(a
—
b)
=
~ X 0.1321
=
0.066.
We can now write the expression E~5 0.066q2/4~0d. =
(6)
The polarization component E~ may be approximated by considering the single ion as a point-charge at a distance from a semi-infinite dielectric and using the method of image charges. The potential energy of the ion due to polarization of the dielectric is then given by 2 ftc 1 E~=16q 1ic0dk,1c+1)~ (7) To compound the generally dubious nature of the continuum approximation as applied to the present case, two serious difficulties arise in the use of eq. (7). The first concerns the value of dielectric constant to be used, and the other the value of d. The optical and static values of dielectric constant differ by large amounts for many ionic solids and, in view of the frequency of ionic diffusion jumps, it is not clear what value of ic would be appropriate. (Recall that v0 in eq. (4) is of the order of 1013 Hz.) The use of the normal interionic distance d in eq. (7) means that the distance between the charge and the surface is comparable with the size of natural discontinuities in an ideal surface. Eq. (7) then merely implies that a quantitative estimate of
P.M. Dryburgh
Energy
/ Minimum growth temperature for crystals grownfrom gas phase
403
0 Table 2 ionic crystals Crystal
E1~
~SD -
A
I
Z
d
(A)
(eV)
ESD
MgO Activation NaC1 energies 21 for surface 2.76 diffusion estimated 2.05 0.34 1.85 for some CaO 2 2.39 1.59 SrO 2 2.57 1.48 ______________
BaO
2
2.75
1.38
S
Fig. 7. Energy diagram of A, y and 8 sites on a (100) surface of a NaC1-type crystal.
E~could be made if appropriate values of ic and d could be assigned. It turns out that there is no need to attempt any improvement of eq. (7) because of the properties of the y and 6 sites shown in fig. 4. At a surface antisite, y, the electrostatic component E,~5will be repulsive and provide an additional energy barrier. The symmetry of a 6 site is such that ~ sitely charged pairs of ions are equidistant from it and the net sum of their potentials is always zero, At a 6-point, the component E,~,5disappears and Es E~. These energy relationships are summarized in fig. 7 in which E5D is identified with E,~,5 thus obviating the need to estimate E~. Eq. (6) now appears as 2q2/4ir E5D 0.066Z 0d =
=
for the general case of an ionic crystal of the NaCl structure. Values have been calculated for NaC1 and group II oxides and are given in table 2. 5.3. Metal crystals The growth of single crystals of metals by VPE is much less common than the growth of semiconductors or insulators but, although there are few cases of immediate practical interest, metal systems are included here for the sake of completeness. The value of a notional single-bond energy E1, is calculated as for covalent compounds by the use
of heat of atomization and co-ordination number (see section 5.1). A relationship between this energy and the activation energy for surface diffusion E5D must be established before E5D can be estimated. The model of surface diffusion used here is the extremely simple one of an atom moving from on normal lattice site to another on an ideal surface. The unit step for the movement is taken to be passage over a saddle-position between two surface atoms, even though the next available resting position may not be a normal lattice site: two such steps may be necessary before the migrating atom reaches a normal lattice site. For the present purpose we are concerned only with the rate at which the atom can move away from its initial site. The number of surface nearest neighbours depends upon the type of packing and the orientation of the particular surface. In a normal site the atom is nominally bonded to its nearest surface neighbours but during a diffusion step it will be temporarily bonded to only two. The activation
Table 3 Number of single metallic bonds contributing to the activation energy for surface diffusion for various common packings Packing c Orientation Number of Number of surface bonds nearest contributmg neighbours to ESD cpc (fcc) 12 (100) 43 21 cpc(fcc) 12 (111) bcc 8 (100) 4 2 bcc cph (hex) cph (hex)
8 12 12
(111) (1000) (0001)
3 3 4
1 1 2
404
P.M. Dryburgh
/ Minimum growth temperature for crystals grown from gas phase
Table 4 Estimated values of activation energy for surface diffusion in copper, gold, and tungsten Metal
Cu Cu Au Au W W
~H*
E
1 (eV)
Orientation
(kcal/g-atom) 81.1 81.1 88.3 88.3 201.8 201.8
0.29 0.29 0.32 0.32 1.10 1.10
(100) (111) (100) (111) (100) (111)
ESD (eV)
Crystal C
ESD
0.58 0.29 0.64 0.32 2.20 1.10
Si Ge SiC BN BP A1N AlP AlAs AlSb InP InAs InSb GaN GaP
1.18 0.98 1.95 1.77 1.59 1.64 1.06 1.01 0.91 0.92 0.82 0.72 1.37 0.98
748 570 1338 1164 629 1099 656 620 553 547 453
GaAs
0.86
472
GaSb CdS CdSe CdTe PbTe
0.80
440
0.72 0.63 0.39
293 105
ZnSe ZnTe
0.74 0.66
388 312
NaC1 MgO
0.34 1.85
68 1259
BaO Cu (100) Cu (111)
1.38 0.29
951 258 14
Au (111) (100) W (100) w (111)
0.32 0.64 2.2 1.1
294 33 1448 646
energy ESD will be the difference between the energy Es and twice the single bond energy: ESD
=
E5
—
2E,,
or ESD
=
E1 ( c5
—
Table 5 Minimum temperature U for crystal growth from the gas phase for 33 materials
2),
where CS number of nearest surface neighbours The number of single bonds contributing to ESD is shown in table 3 for some typical orientations and packing types. Only 3 metals have been used as examples, gold, copper and tungsten, and values for E5D are shown in table 4.
(eV)
1.86
0.91
8 (°C) 1223
369
894 587
537 369
=
6 Results The minimum or limiting temperature has been calculated for each of a number of materials by means of eq. (4) using the values of E5D estimated in section 3. The results are presented in table 5. The higher of the two heats of atomization has been used for all covalent compounds. The calculated values of 9 are compared below with experimental results for only a limited number of the materials listed. It is difficult to assemble enough results from the literature to provide an adequate test of the proposed equation. Lower deposition temperatures are reported from time to time as new processes are developed but they are not usually claimed to represent the minimum attainable: in many cases more attention has been
0.58
paid to selecting an optimum temperature for the purpose in hand. 6.1. Silivon (0
=
748 °C)
There is a wealth of experimental data on the deposition of silicon, the general consensus being that VPE has to be carried out at temperatures above 10000 C for good quality layers to be grown [12]. As far as the lower limit is concerned, the most significant work in the present context is
P.M. Dryburgh
/ Minimum growth temperature for crystals grown from gas phase
that of Richman, Chiang and Robinson [13],which was directed specifically at achieving minimum growth temperatures. These workers found that good layers could be grown at temperatures as low as 800°C, provided that helium was used as the carrier gas and stringent precautions taken.
405
make the point that there are very few reports in the literature of growth experiments being done at temperatures below 360 °C.It seems probable that this is the lower limit of growth temperatures.
7. Discussion 6.2. Germanium (0 = 570°C) The growth of germanium on gallium arsenide by pyrolysis of GeH4 has been reported by Kräutle, Roentgen and Beneking [14], who observed that the morphology of grown layers became poorer at deposition temperatures below 600°C. They claim that layers grown at 400°C have “smooth surfaces” but their micrographs show these layers as having a very grainy texture. The rocking curves of layers grown at 500° are broader than those for layers grown at 600° but no rocking curves are included for layers grown at any lower temperature. Kräutle et al. relate the degradation in quality of the layers to the change in temperature dependence of growth rate ohserved at about 500°C. Against these observations, a predicted 0 of 570°Cshows satisfactory agreement. 6.3. Gallium arsenide (0
=
472 °C)
Epitaxial GaAs is normally grown at temperatures in the range 550—850°C and Duchemin, Bonnet, Koelsch and Huyghe [15] have reported epitaxial growth of GaAs on Ge at 500°C. Once again, the calculated 9 gives a good indication of the lower limit of growth temperature. 6.4. Indium phosphide (0 = 547°C) Normal growth temperature for MOVPE growth of InP is 570—680°C[16]. 6.5. Cadmium telluride (9 = 293 °C) Bhat, Taskar and Ghandhi [17] have recently described an MOVPE process for the growth of CdTe at 300—375°C. No comment appears in their paper on the quality of the layers. Bhat et al. discuss the results reported by other groups and
There cannot be a sharply defined minimum temperature for ordered crystal growth and, in practice, crystal quality deteriorates as deposition temperature is reduced through a range until, according to some criterion, the layer ceases to be a single crystal. The values of 0 in table 5 are given to the nearest degree as calculated but it must be stressed that no such precision is being suggested here. The temperature 0 should be taken only as a practical but non-definitive guide in the selection of growth temperature. The methods of calculation used in this paper contain many uncertainties, of which the estimation of E5D is the most serious. The problem of choosing the correct value for m is of less importance and has been referred to above. The assumption that the diffusing species is atomic is obviously not always justified but an appropriate value for m may be used whenever the true identity of the diffusing species is known. Eq. (4) is insensitive to the value of m and the identification of the actual diffusing species will matter only if its mass is very different from that of the atom concerned. Eq. (4) has been derived by making a fairly gross assumption about the value of vibrational frequency v0 which is open to two major criticisms, first, that the term ~hi’0 implies that only the ground state is involved, and, second, that frequency rather than amplitude is made to appear temperature dependent. The first of these objections has been ignored in this paper because of the highly approximate nature of the whole treatment, already alluded to in the introduction. Little would be lost by simply choosing a constant value for vo of the right order of magnitude, about 5 x 1012 Hz, but the value 2kT/3h is preferred here because it allows some temperature dependence in the pre-exponential factor of eq. (2). In doing so, it provides a counter
406
P.M. Dryburgh
/ Minimum growth temperature for crystals grown from gas phase
to the second objection. Eq. (2) is itself a simplified one implying no variation in vibrational state occupancy. In an accurate analysis, the effective average value of v0 would be temperature dependent as higher and higher states became occupied. The first of the two sequences shown in fig. 1 is an oversimplification in many cases, while the second, although probably representing a better description, offers some complications. The most important one is that the rate-determining step precedes surface diffusion and so the rate of arrival should strictly be equated to the sum of two terms, one representing removal by desorption, and the other removal by surface diffusion. If we associate the rate-determining step with the highest overall activation energy, we may assume that this step will be the most temperature dependent. The consequence will be that, as temperature is reduced, the rate of desorption will fall until the rate of reaction (and hence, growth) becomes negligible, even though the available rate of surface diffusion has not been exceeded. The effect of reducing temperature in such a case is not to produce a polycrystalline layer but to cause a negligible deposition rate, the practical outcome being the same. If, on the other hand, the activation energy for surface diffusion is larger than for desorption, it will be dominant and the argument used for the simpler case will apply equally. The relative sizes of the two energies will be different in each deposition process and must be considered as a possible modifying factor when 0 values are ohtamed from eq. (4). It has been mentioned already (section 4) that the use of eq. (1) assumes that all incident partides become adsorbed that is, they have a sticking coefficient of unity. In certain cases it may be possible to put a better estimate on the fraction which become adsorbed but no general procedure is available. Once again, the overriding importance of the exponential in eq. (4) means that the result is not too much affected by an error in sticking coefficient. If we take silicon as an example and calculate 0 for the cases of 50% and 90% sticking of incident particles, the values are respectively 702 and 740°C as compared with 748°Cfor 100% sticking.
Most binary compounds are partly ionic in character and much effort has been expended on attempting to establish a satisfactory scale of ionicity (see, for example, Phillips [18]). The rule for predicting minimum growth temperature ü is intended as a rule-of-thumb and there is no point in attempting a precise evaluation of each compound’s ionicity. For the purpose of estimating E5D, the simplest way of deciding between the ionic and the covalent approximations is on the basis of electronegativity different. Phillips has critically reviewed the validity of earlier definitions of electronegativity and has put the concept on a satisfactory theoretical footing. There are, however, more extensive compilations of data available for the older definitions and, despite their theoretical inadequacy, they have been used in the present work. We use the covalent approximation for all crystals in which the electronegativity difference (si) for the constituent atoms is less than 2.0. Data for the elements have been taken from Gordy and Thomas [19]. Using this criterion, sodium chloride (~j= 2.1) is classed as ionic while lead telluride (3J 0.5), which has the NaCl structure, is classed as covalent. As with any arbitrary criterion, anomalies are found close to the boundary, and an important boundary material here is BeO (~ = 2.0), which has the wurtzite structure. The high lattice energy of BeO [20] characterizes it as a largely ionic material but no computation of the appropriate two-dimensional Madelung constant for the wurtzite structure has been carried out so beryllia has been omitted for the present. Another obvious anomaly arises in the case of CsI (~i= 1.8). There are no experimental values available for minimum growth temperatures of NaC1, PbTe or the metals, but the peculiarities of their 0 values in table 5 require some comment. In the absence of other data, it is impossible to deduce whether the low temperatures are genuine or whether they are a consequence of errors in E5D. Quite small changes in E5D would result in more immediately believable values, but the possibility of the low 0 values representing a real physical situation cannot be ruled out. The simple model of surface diffusion used in —
P.M. Dryburgh
/ Minimum growth temperature for crystals grown from gas phase
section 5.3 for metals gives a very large dependence upon crystal orientation which should be amenable to direct observation. Most of the values of 0 given by eq. (4) so far have been reasonable but the reliability and usefulness of the rule can be tested only by more extended use.
Acknowledgements
407
[3] [4] [5] [6]
H.A. Skinner and G. Pilcher, Quart. Rev. 17, (1963) 264. W.Jost, J. Chem. Phys. 1 (1933) 466. N.F. Mott and M. Littleton, Trans. Faraday, Soc. 34 (1938) 485. ES. Rittner, R.A. Hutner and F.K. DuPré, J. Chem. Phys. 17 (1949) 198. [7] A.J. Dekker, Solid State Physics (Macmillan, London, 1960) p. 167. [8] N.N. Greenwood, Ionic Crystals, Lattice Defects and Stoichiometry (Butterworths, London, 1968) p. 16. [9] M.P. Tosi, Solid State Phys. 16 (1964) 1. tioj M.F.C. Ladd, Structure and Bonding in Solid State Chemistry (Ellis Horwood, Chichester, 1979) p. 82.
This work was done while the author was a guest in the Inorganic Chemistry Department, University of Uppsala, and he would like to thaflk Professor Stig Rundqvist for generously affording him the facilities of the Solid State Chemistry Group. Thanks are due also to the University of Edinburgh for one year’s sabbatical leave.
[11] 0. Emersleben, Z. Physik 127 (1950) 588. [12] J. Bloem, Y.S. Oei, H.H.C. de Moor, J.H.L. Hanssen and L.J. Giling, J. Crystal Growth 65 (1983) 399. [13] D. Richman, Y.S. Chiang and RH. Robinson. RCA Rev. 31 (1970) 613. [14] H. Krautle, P. Roentgen and H. Beneking, J. Crystal
References
Soc. 134 (1987) 195. [18] J.C. Phillips, Rev. Mod. Phys. 42 (1970) 317.
[1] G. Erti, in: The Nature of the Surface Chemical Bond, Eds. T.N. Rhodin and G. Ertl (North-Holland, A.msterdam, 1979) p. 319. [2] I. Barin and 0. Knacke, The Thermochemical Properties of Inorganic Substances (Springer, Berlin, 1973).
Growth 65 (1983) 439. [15] J.P. Duchemin, Bonnet181. F. Koelsch and D. Huyghe, J. Crystal Growth M. 45 (1978) [16] M.J. Ludowise, J. Appl. Phys. 58 (1985) 31. [17] lB. Bhat, N.R. Taskar and 5K. Ghandhi, J. Electrochem.
[19] W. Gordy and W.J.O. Thomas, J. Chem. Phys. 24 (1956) [20] T.C. Waddington in: Advances in Inorganic Chemistry and Radiochenustry, Vol. 1 (Academic Press, New York, 1959) p. 193.
397
THE ESTIMA11ON OF MINIMUM GROWTH TEMPERATURE FOR CRYSTALS GROWN FROM THE GAS PHASE P.M. DRYBURGH Electrical Engineering Department, The King’s Buildings, University of Edinburgh, Edinburgh EH9 3JL, Scotland, UK
Received 10 August 1987; manuscript received in final form 19 October 1987
It is proposed that the limiting temperature, 0, for single crystal growth from the gas phase is the temperature at which the kinetic rate of arrival at each surface site is equal to the rate of removal by surface diffusion. The expression derived for 0 at atmospheric 2/m1”2, where ESD is the activation energy for surface diffusion, d is interatomic distance pressure is p3/2 exp( — ESD/kO) = 0.048d in A and m is molecular weight of the diffusing species. Surface diffusion is considered for the cases of covalent, ionic and metallic solids, and simple models used to allow the estimation of ESD in each case. Results are presented for 33 crystals including group IV elements, Ill—V compounds ionic solids and metals. Agreement with experimental results is shown to be satisfactory in those cases where results are available from the literature.
1. Infroduction Processes involving the deposition of a solid from the gas phase are used to produce a growing diversity of materials, ranging from protective coatings, structural layers and abrasives to blanks for optical fibre manufacture and epitaxial layers of 111—V compounds or silicon. The extensive range of possible precursors often allows considerable freedom in the choice of deposition temperature, since usable reactions range from pyrolysis of unstable compounds to entropy-driven reactions of high temperature species, but it is necessary to analyse the kinetics of the transport process so that favourable growth rates may be ohtamed, In attempting to grow epitaxial layers, or even bulk single crystals by sublimation, it is additionally necessary to distinguish between temperatures at which the appropriate species will be deposited at a useful rate and temperatures at which the resulting material will be in the form of a single crystal. There is at present no means of making a quantitative prediction of the temperature at which single crystal growth might reasonably be ex-
pected, although many workers rely tacitly on a generalization, based upon experience, that materials of higher melting point require higher deposition temperatures. The factors which determine whether a growing layer is a single crystal or not obviously depend upon the ability of mcident atoms to find their ways to the correct sites by surface diffusion and, since we expect some sort of connection between rates of bulk diffusion and surface diffusion, it is not unreasonable to use melting point as a guide to deposition temperature. There is, however, no quantitative general correlation between melting points and minimum growth temperatures. The need for more reliable guidance in the choice of likely growth temperature arises as a result of the great range of conditions now available. An arbitrarily chosen and inappropriate ternperature range, based largely on the properties of the precursors, may result in the attempted use of a reaction which can never provide single crystal material, or it may result in much wasted experimental work being carried out on a potentially satisfactory chemical system. This paper presents a general rule for predic-
0022-0248/88/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
398
P.M. Dryburgh
/ Minimum growth temperature for crystals grown from gas phase
adsorption
adsorption
___
C
SURFACE DIFFUSION
lattice-site attachment
r
I1~
) —
surface reaction
F
— by-products
—
SURFACE DIFFUSION
—
by products
lattice-site attachment
Fig. 1. Possible reaction schemes for crystal growth from the gas phase.
ting the minimum temperature for the growth from the gas phase of single crystals of a variety of materials. The guiding principles have been that any useful rule should be valid for as wide a range of crystals as possible and that the data required for the material, or its constituent atoms, should be readily available. The price of such generality is, of course a lack of precision and specificity.
2. Model used and underlying assumptions The simple model used in this paper depends upon three preliminary assumptions: (1) growth rate is surface-kinetically controlled and not nucleation limited; (2) growth proceeds by one of the two possible routes shown in fig. 1; (3) chemisorption involves only a small number of neighbouring atoms. This assumption has been discussed and justified by Ertl [1] for metal surfaces. The essential feature of the model used is that the rate of arrival of incident atoms or molecules at surface sites must not exceed the rate at which newly arrived adsorbed atoms or molecules are moving away. This is equivalent to assuming that there is small probability of any incident atom or molecule arriving immediately at a correct site. The rate of arrival is obtained from simple kinetic theory, equated with the rate of migration oh-
tamed from simple vibrational diffusion theory, and an expression derived for the corresponding limiting value of temperature.
3. List of symbols and units b c d E F h
Madelung constant of two-dimensional sheet Coordination number Interatomic distance Energy of interaction Incident flux Planck’s constant ~ H * Heat of atomization ~ H~° Standard heat of formation k Boltzmann’s constant m Numerical “atomic weight” m0 Atomic mass unit M N0 p q Z a
~ flAB
9 ic
4s
Molecular mass (M = m0 X m) Avogadro’s number Pressure Electronic charge Ionic charge Madelung constant Permittivity of space Electronegativity difference between atoms A and B Minimum (or limiting) temperature for ordered crystal growth Relative dielectric constant Electrostatic potential
P.M. Dryburgh
/ Minimum growth temperature for crystals grown from gas phase
4. Derivation of equation giving limiting temperahire 0 In simple kinetic theory the incident flux F of molecules on a surface is given by
399
The combined expressions (1) and (3) yield 6.65
~
rn1”2
11/2
=
2kT exp I
—~—
—
hence F= (2’2~-MkT)1”2’
03/2
so, for a pressure of 1 atm, F
2
(mT)”2 m
=
.
=
=
A fraction of the incident particles will be scattered from the surface immediately but this cannot be estimated and will be ignored for the present purpose. The particles which become adsorbed will vibrate at some characteristic frequency i’,~ due to their thermal energy. Some energy is required to detach the adsorbed atom or molecule to a sufficient extent to allow it to move to another site, and this is the activation energy for surface diffusion ESD. The probability that the thermal vibration energy will be ESD is exp(—ESD/kT) and the rate ID at which the adsorbed particle will be detached from one site to move to another is given by =
0.048(d2/rn’/’2).
(4)
—1
~
2 with the darea of A, onethen lattice site as occur r rIf we ~d,take where is in collisions at each lattice site at a rate f 0 where 2 ~1 (1) f~ 6.65 X 1 ;;;i7i~ d
fD
=
Calculation of 0 from eq. (4) requires values for three quantities: the interatomic distance in the crystal surface d, the formal molecular weight m
2 66 x 1029 ____________
exp(—ESD/kO)
of the incident species and the activation energy for surface diffusion E The first two of these 5D. quantities are readily available for most systems, but there may be some doubt about the value of m in certain cases. It should be noted,tohowever, is very insensitive errors in that m.is the The dominant parameter ESDvalue and, ofas 9the choice of its value virtually dictates the value of 0, it is necessary to examine the process of surface diffusion in more detail in order to get the best practicable estimating procedure. 5. The estimation of ESD The rules by which the atomic packing arrangement is determined in a surface are different in the extreme case of covalent, ionic and metallic crystals and these are considered separately with reference to surface diffusion and the estimation of their corresponding activation energies.
v 0 exp(
—
E5D/kT).
(2)
The only source of vibrational energy available is thermal, so we can write =
kT,
vo
=
2kT/3h.
By substituting for v0 in eq. (2) we obtain an expression for ID: 2kT I ESD \ fD=—jj--exP~—-~y).
(3)
5.1. Covalent crystals The interaction energy E5 of an atom bonded to the surface of its own covalent solid may be considered as having two components, the energy of a normal single bond, E,, and the energy of interaction with a small number x of surface neighbours, each contributing an nearest energy E2/x (see fig. 2): =
If the rate of arrival of incident molecules is equal to the rate of migration, then fO~fD
and
Tm0.
E,
+
E2.
In the case of an AB compound of the zincblende structure, the position of an adsorbed atom with respect to the surface may be considered as being
400
P.M. Dryburgh
/ Minimum growth temperature for crystals grownfrom gas phase to convert one mole of AB into one mole of free A
adatom
E
2 /
atoms and one mole of free B atoms is
~ E~
E1 \~
X/
EAB
=
z~Hf+ ~ H~+ ~ H~.
surface atoms Fig. 2. The interactions of a single atom bonded to the surface of a covalent solid,
The energy relationships are summarized in fig. 3. Half the heat of formation is associated with each consistuent of an AB compound and the number of bonds per atom is c, so the energy required to break a single bond between solid AB and a single A atom is
determined completely by theIthighly 3 bond. followsdirectional that any nature of a single sp lateral movement of the atom during surface diffusion can take place only if the single bond is effectively broken, the interaction energy then being reduced to the weaker and largely non-directional nearest-neighbour bonding. This very simplified model allows us to identify E with E SD 1 and so obtain an estimated value for E5D by calculating E1 from thermochemical data. E1 is the energy required to break a single bond between an A (or a B) atom and solid AB and is easily calculated as follows. The standard heat of formation L~ H~°of AB represents the difference in energy between the compound AB in its standard state and the consistuent elements A and B in theirs. Heat of atomization L~ H * represents the energy required to convert an element in its standard state into free gaseous atoms. It is readily seen that the energy EAR required
EA
N0 atoms B
*
L~HA 1 mole A
~
Hf°+ L~H2 )/N 0c;
similarly EB
=
(~Hf + ~iH~)/N0c.
For an element, e.g. silicon, there is only one value Table 1 Estimated single bond energies E1 for some covalent crystals Crystal ~ H~° ~ H* (kcal/mol) (kcal/g-atom) (eV) A B A B C Si Ge
170.9 108.4 90
1.86 1.18 0.98
~
~
BP
2716
13216
7515
A1N AlP
76.0 39.3
78.0 78.0
AlAs
29.3 12.03
AISh InP
N0 atoms A ~
=
~
~
~
1159
0197
113.0 75.5
1.23 1.06
1.64 1.03
78.0
69
63 75.5
1.01 0.91 0.73
0.91
18.0
750 58
InAs
13.8
58
69
0.70
*
InSb
HB
GaN
7.44 26.2 29.2 19.5 10
58 69.0
63 113.0 75.5
69.0 69.0
35.7
26.8 26.8 26.8
69 63 65.7
0.67 0.89 0.91 0.86 0.80
1 mole B
1 mo I e AB Fig. 3. Energy diagram showing standard heat of formation ~ H~° and heats of atomization zlH~
GaP GaAs GaSh CdS
—
CdSe CdTe PbTe
34.6 24.33 16.4
ZnO ZnS
83.2 49.04
69.0
46.8
49.4 46 46
31.2
59.6 65.7
0.48 0.48 0.42 0.40
0.79 0.61
0.75 0.92 0.82 0.72 1.37 0.98
0.86 0.74 0.91
0.72 0.63 0.39 1.10 0.98
ZnSe 38 31.2 49.4 0.55 0.74 ZnTe 28.8 31.2 46 0.50 0.66 _______________________________________________
P.M. Dryburgh
/
Minimum growth temperature for crystals grown from gas phase
401
of z~ H * and the value of i.’i Hf is zero by defiition:
determined to very good approximation by electrostatic forces only, the terms representing exchange effects and Van der Waals forces being
E1
insignificant. In practice, however, the derivation of a straightforward estimating procedure, based upon a plausible model, is not obvious because of the difficulty in allowing for the effect of polarization. The problem of estimating polarization contributions to defect energies in serious in an infinite solid and was first discussed by Jost [4] and Mott and Littleton [5]. More precise calculations have been done by Rittner, Hutner and DuPré [6] and general accounts can be found in standard texts such as Dekker [7]. The case of an adsorbed ion, which is essentially outside the bulk of crystal, presents a simpler problem in the sense that Jost’s approximation is unnecessary, so no arbitrary value of radius has to be assigned, but it is still convenient to regard the crystal as a semi-infinite dielectric continuum to avoid the laborious summation of induced dipoles. It is also convenient to ignore the contribution made by Born repulsion, which accounts for approximately 10% of the lattice energy of an ionic crystal [8] but may well be of less importance here. The interaction or binding energy E5 of the adsorbed ion is written as the sum of two terms, one corresponding to the potential due to the constituent ions of the crystal E,~5and the other arising from the dielectric polarization of the crystal E~:
is simply given by
~ H~*/4N
Using the standard heat of formation from Barn and Knacke [2] and heat of atomization from Skinner and Pilcher [3], E, has been calculated for some group IV elements and Ill—V compounds. In the case of AB compounds there is obviously a choice of E1 values, depending upon whether an A or a B atom is considered. The results are given in table 1. 5.2. Ionic crystals The only ionic crystals for which results are reported in this paper have the NaC1 structure and only (100) surfaces have been considered. Fig. 4 shows a (100) face of an AB compound of NaC1 structure with a single A + ion occupying what would become a normal lattice site after subsequent growth. The interaction energy of the ion with the solid is E5. We make the reasonable simplification that there are only two other types of resting place for such an ion, an antisite p05i tion (y) and a “saddle” position (6). In moving from one normal surface site (A) to the next, the adsorbed ion has to pass over either a y-position or a 8-position (see fig. 2), the activation energies for the two types of movement being (E5 E~) and (E5 Eo). Surface diffusion will take place by the route offering the lower activation energy. The estimation of E5 for a binary ionic crystal is simple in principle, since cohesive energies are —
—
Fig. 4. Sketch of an ideal (100) surface of a compound AB (NaCl structure) with a single adsorbed A + ion, at a normal site A. An antisite position ~y and a saddle position 8 are shown also.
E5
E~5+ E~. For an ion sticking onto the surface of a binary ionic solid, E~5represents that part of the binding energy arising purely from the combined electrostatic potentials of the ions in the solid. To obtain an expression for this potential, we consider first the self-potential ~ at an arbitrary lattice point in an infinite ionic solid. This potential ~ depends upon the ionic charges, their separation and the Madelung constant of the lattice. 4
=
=
aZq/4ire0d.
A detailed treatment of the evaluation of Madelung constants has been given by Tosi [9]
402
/ Minimum growth temperature for crystals grownfrom gas phase
P.M. Dryburgh
I
a
_________________________________________________________________________________________________ b~
I
II
I
I
+
I
I -
-
-
-
a
+
+•
I
I
+
•
+
,
+
+ •
+
+1
+ +•
4 (i)
(ii)
Fig. 5. Notional components of the Madelung constant.
and a more general discussion by Ladd [10]. The accepted value for the Madelung constant of the NaCl structure (in terms of interionic distance d) is a 1.7476. 4) may be considered as having three components as illustrated in fig. 5(i). The component b is due to all the ions in the sheet of ions containing the given site, the sheet being one
~1-• •
+
+
/
+
e
+
+
•
+
•
+
+
Fig. 6. The derivation of the Madelung constant for a two-dimensional sheet of ions.
=
ion in thickness. Each of the two equal components a is due to half of the remainder of the crystal. In terms of contributions to the Madelung constant, it is obvious from fig. 5(i) that a 2a + b. (5) =
Fig. 5(u) has been drawn to emphasise that a is the required surface counterpart of the Madelung constant. In order to evaluate b, the individual attractions and repulsions have to be written as a series and the sum found. From fig. 6 it may be seen that the appropriate series is b=(—4+--~--~+ (4 ~ ~-
4
/ +
—
8 +
8
—
-~-
+
8
4
4 +
—
....
As with all series defining Madelung constants, this one converges extremely slowly. Fortunately, the sum to infinity has been computed by Emersleben [11] as part of a method for obtaining other constants. Tosi [9] quotes Emersleben’s value as 1.61554. From eq. (5): a
=
~(a
—
b)
=
~ X 0.1321
=
0.066.
We can now write the expression E~5 0.066q2/4~0d. =
(6)
The polarization component E~ may be approximated by considering the single ion as a point-charge at a distance from a semi-infinite dielectric and using the method of image charges. The potential energy of the ion due to polarization of the dielectric is then given by 2 ftc 1 E~=16q 1ic0dk,1c+1)~ (7) To compound the generally dubious nature of the continuum approximation as applied to the present case, two serious difficulties arise in the use of eq. (7). The first concerns the value of dielectric constant to be used, and the other the value of d. The optical and static values of dielectric constant differ by large amounts for many ionic solids and, in view of the frequency of ionic diffusion jumps, it is not clear what value of ic would be appropriate. (Recall that v0 in eq. (4) is of the order of 1013 Hz.) The use of the normal interionic distance d in eq. (7) means that the distance between the charge and the surface is comparable with the size of natural discontinuities in an ideal surface. Eq. (7) then merely implies that a quantitative estimate of
P.M. Dryburgh
Energy
/ Minimum growth temperature for crystals grownfrom gas phase
403
0 Table 2 ionic crystals Crystal
E1~
~SD -
A
I
Z
d
(A)
(eV)
ESD
MgO Activation NaC1 energies 21 for surface 2.76 diffusion estimated 2.05 0.34 1.85 for some CaO 2 2.39 1.59 SrO 2 2.57 1.48 ______________
BaO
2
2.75
1.38
S
Fig. 7. Energy diagram of A, y and 8 sites on a (100) surface of a NaC1-type crystal.
E~could be made if appropriate values of ic and d could be assigned. It turns out that there is no need to attempt any improvement of eq. (7) because of the properties of the y and 6 sites shown in fig. 4. At a surface antisite, y, the electrostatic component E,~5will be repulsive and provide an additional energy barrier. The symmetry of a 6 site is such that ~ sitely charged pairs of ions are equidistant from it and the net sum of their potentials is always zero, At a 6-point, the component E,~,5disappears and Es E~. These energy relationships are summarized in fig. 7 in which E5D is identified with E,~,5 thus obviating the need to estimate E~. Eq. (6) now appears as 2q2/4ir E5D 0.066Z 0d =
=
for the general case of an ionic crystal of the NaCl structure. Values have been calculated for NaC1 and group II oxides and are given in table 2. 5.3. Metal crystals The growth of single crystals of metals by VPE is much less common than the growth of semiconductors or insulators but, although there are few cases of immediate practical interest, metal systems are included here for the sake of completeness. The value of a notional single-bond energy E1, is calculated as for covalent compounds by the use
of heat of atomization and co-ordination number (see section 5.1). A relationship between this energy and the activation energy for surface diffusion E5D must be established before E5D can be estimated. The model of surface diffusion used here is the extremely simple one of an atom moving from on normal lattice site to another on an ideal surface. The unit step for the movement is taken to be passage over a saddle-position between two surface atoms, even though the next available resting position may not be a normal lattice site: two such steps may be necessary before the migrating atom reaches a normal lattice site. For the present purpose we are concerned only with the rate at which the atom can move away from its initial site. The number of surface nearest neighbours depends upon the type of packing and the orientation of the particular surface. In a normal site the atom is nominally bonded to its nearest surface neighbours but during a diffusion step it will be temporarily bonded to only two. The activation
Table 3 Number of single metallic bonds contributing to the activation energy for surface diffusion for various common packings Packing c Orientation Number of Number of surface bonds nearest contributmg neighbours to ESD cpc (fcc) 12 (100) 43 21 cpc(fcc) 12 (111) bcc 8 (100) 4 2 bcc cph (hex) cph (hex)
8 12 12
(111) (1000) (0001)
3 3 4
1 1 2
404
P.M. Dryburgh
/ Minimum growth temperature for crystals grown from gas phase
Table 4 Estimated values of activation energy for surface diffusion in copper, gold, and tungsten Metal
Cu Cu Au Au W W
~H*
E
1 (eV)
Orientation
(kcal/g-atom) 81.1 81.1 88.3 88.3 201.8 201.8
0.29 0.29 0.32 0.32 1.10 1.10
(100) (111) (100) (111) (100) (111)
ESD (eV)
Crystal C
ESD
0.58 0.29 0.64 0.32 2.20 1.10
Si Ge SiC BN BP A1N AlP AlAs AlSb InP InAs InSb GaN GaP
1.18 0.98 1.95 1.77 1.59 1.64 1.06 1.01 0.91 0.92 0.82 0.72 1.37 0.98
748 570 1338 1164 629 1099 656 620 553 547 453
GaAs
0.86
472
GaSb CdS CdSe CdTe PbTe
0.80
440
0.72 0.63 0.39
293 105
ZnSe ZnTe
0.74 0.66
388 312
NaC1 MgO
0.34 1.85
68 1259
BaO Cu (100) Cu (111)
1.38 0.29
951 258 14
Au (111) (100) W (100) w (111)
0.32 0.64 2.2 1.1
294 33 1448 646
energy ESD will be the difference between the energy Es and twice the single bond energy: ESD
=
E5
—
2E,,
or ESD
=
E1 ( c5
—
Table 5 Minimum temperature U for crystal growth from the gas phase for 33 materials
2),
where CS number of nearest surface neighbours The number of single bonds contributing to ESD is shown in table 3 for some typical orientations and packing types. Only 3 metals have been used as examples, gold, copper and tungsten, and values for E5D are shown in table 4.
(eV)
1.86
0.91
8 (°C) 1223
369
894 587
537 369
=
6 Results The minimum or limiting temperature has been calculated for each of a number of materials by means of eq. (4) using the values of E5D estimated in section 3. The results are presented in table 5. The higher of the two heats of atomization has been used for all covalent compounds. The calculated values of 9 are compared below with experimental results for only a limited number of the materials listed. It is difficult to assemble enough results from the literature to provide an adequate test of the proposed equation. Lower deposition temperatures are reported from time to time as new processes are developed but they are not usually claimed to represent the minimum attainable: in many cases more attention has been
0.58
paid to selecting an optimum temperature for the purpose in hand. 6.1. Silivon (0
=
748 °C)
There is a wealth of experimental data on the deposition of silicon, the general consensus being that VPE has to be carried out at temperatures above 10000 C for good quality layers to be grown [12]. As far as the lower limit is concerned, the most significant work in the present context is
P.M. Dryburgh
/ Minimum growth temperature for crystals grown from gas phase
that of Richman, Chiang and Robinson [13],which was directed specifically at achieving minimum growth temperatures. These workers found that good layers could be grown at temperatures as low as 800°C, provided that helium was used as the carrier gas and stringent precautions taken.
405
make the point that there are very few reports in the literature of growth experiments being done at temperatures below 360 °C.It seems probable that this is the lower limit of growth temperatures.
7. Discussion 6.2. Germanium (0 = 570°C) The growth of germanium on gallium arsenide by pyrolysis of GeH4 has been reported by Kräutle, Roentgen and Beneking [14], who observed that the morphology of grown layers became poorer at deposition temperatures below 600°C. They claim that layers grown at 400°C have “smooth surfaces” but their micrographs show these layers as having a very grainy texture. The rocking curves of layers grown at 500° are broader than those for layers grown at 600° but no rocking curves are included for layers grown at any lower temperature. Kräutle et al. relate the degradation in quality of the layers to the change in temperature dependence of growth rate ohserved at about 500°C. Against these observations, a predicted 0 of 570°Cshows satisfactory agreement. 6.3. Gallium arsenide (0
=
472 °C)
Epitaxial GaAs is normally grown at temperatures in the range 550—850°C and Duchemin, Bonnet, Koelsch and Huyghe [15] have reported epitaxial growth of GaAs on Ge at 500°C. Once again, the calculated 9 gives a good indication of the lower limit of growth temperature. 6.4. Indium phosphide (0 = 547°C) Normal growth temperature for MOVPE growth of InP is 570—680°C[16]. 6.5. Cadmium telluride (9 = 293 °C) Bhat, Taskar and Ghandhi [17] have recently described an MOVPE process for the growth of CdTe at 300—375°C. No comment appears in their paper on the quality of the layers. Bhat et al. discuss the results reported by other groups and
There cannot be a sharply defined minimum temperature for ordered crystal growth and, in practice, crystal quality deteriorates as deposition temperature is reduced through a range until, according to some criterion, the layer ceases to be a single crystal. The values of 0 in table 5 are given to the nearest degree as calculated but it must be stressed that no such precision is being suggested here. The temperature 0 should be taken only as a practical but non-definitive guide in the selection of growth temperature. The methods of calculation used in this paper contain many uncertainties, of which the estimation of E5D is the most serious. The problem of choosing the correct value for m is of less importance and has been referred to above. The assumption that the diffusing species is atomic is obviously not always justified but an appropriate value for m may be used whenever the true identity of the diffusing species is known. Eq. (4) is insensitive to the value of m and the identification of the actual diffusing species will matter only if its mass is very different from that of the atom concerned. Eq. (4) has been derived by making a fairly gross assumption about the value of vibrational frequency v0 which is open to two major criticisms, first, that the term ~hi’0 implies that only the ground state is involved, and, second, that frequency rather than amplitude is made to appear temperature dependent. The first of these objections has been ignored in this paper because of the highly approximate nature of the whole treatment, already alluded to in the introduction. Little would be lost by simply choosing a constant value for vo of the right order of magnitude, about 5 x 1012 Hz, but the value 2kT/3h is preferred here because it allows some temperature dependence in the pre-exponential factor of eq. (2). In doing so, it provides a counter
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/ Minimum growth temperature for crystals grown from gas phase
to the second objection. Eq. (2) is itself a simplified one implying no variation in vibrational state occupancy. In an accurate analysis, the effective average value of v0 would be temperature dependent as higher and higher states became occupied. The first of the two sequences shown in fig. 1 is an oversimplification in many cases, while the second, although probably representing a better description, offers some complications. The most important one is that the rate-determining step precedes surface diffusion and so the rate of arrival should strictly be equated to the sum of two terms, one representing removal by desorption, and the other removal by surface diffusion. If we associate the rate-determining step with the highest overall activation energy, we may assume that this step will be the most temperature dependent. The consequence will be that, as temperature is reduced, the rate of desorption will fall until the rate of reaction (and hence, growth) becomes negligible, even though the available rate of surface diffusion has not been exceeded. The effect of reducing temperature in such a case is not to produce a polycrystalline layer but to cause a negligible deposition rate, the practical outcome being the same. If, on the other hand, the activation energy for surface diffusion is larger than for desorption, it will be dominant and the argument used for the simpler case will apply equally. The relative sizes of the two energies will be different in each deposition process and must be considered as a possible modifying factor when 0 values are ohtamed from eq. (4). It has been mentioned already (section 4) that the use of eq. (1) assumes that all incident partides become adsorbed that is, they have a sticking coefficient of unity. In certain cases it may be possible to put a better estimate on the fraction which become adsorbed but no general procedure is available. Once again, the overriding importance of the exponential in eq. (4) means that the result is not too much affected by an error in sticking coefficient. If we take silicon as an example and calculate 0 for the cases of 50% and 90% sticking of incident particles, the values are respectively 702 and 740°C as compared with 748°Cfor 100% sticking.
Most binary compounds are partly ionic in character and much effort has been expended on attempting to establish a satisfactory scale of ionicity (see, for example, Phillips [18]). The rule for predicting minimum growth temperature ü is intended as a rule-of-thumb and there is no point in attempting a precise evaluation of each compound’s ionicity. For the purpose of estimating E5D, the simplest way of deciding between the ionic and the covalent approximations is on the basis of electronegativity different. Phillips has critically reviewed the validity of earlier definitions of electronegativity and has put the concept on a satisfactory theoretical footing. There are, however, more extensive compilations of data available for the older definitions and, despite their theoretical inadequacy, they have been used in the present work. We use the covalent approximation for all crystals in which the electronegativity difference (si) for the constituent atoms is less than 2.0. Data for the elements have been taken from Gordy and Thomas [19]. Using this criterion, sodium chloride (~j= 2.1) is classed as ionic while lead telluride (3J 0.5), which has the NaCl structure, is classed as covalent. As with any arbitrary criterion, anomalies are found close to the boundary, and an important boundary material here is BeO (~ = 2.0), which has the wurtzite structure. The high lattice energy of BeO [20] characterizes it as a largely ionic material but no computation of the appropriate two-dimensional Madelung constant for the wurtzite structure has been carried out so beryllia has been omitted for the present. Another obvious anomaly arises in the case of CsI (~i= 1.8). There are no experimental values available for minimum growth temperatures of NaC1, PbTe or the metals, but the peculiarities of their 0 values in table 5 require some comment. In the absence of other data, it is impossible to deduce whether the low temperatures are genuine or whether they are a consequence of errors in E5D. Quite small changes in E5D would result in more immediately believable values, but the possibility of the low 0 values representing a real physical situation cannot be ruled out. The simple model of surface diffusion used in —
P.M. Dryburgh
/ Minimum growth temperature for crystals grown from gas phase
section 5.3 for metals gives a very large dependence upon crystal orientation which should be amenable to direct observation. Most of the values of 0 given by eq. (4) so far have been reasonable but the reliability and usefulness of the rule can be tested only by more extended use.
Acknowledgements
407
[3] [4] [5] [6]
H.A. Skinner and G. Pilcher, Quart. Rev. 17, (1963) 264. W.Jost, J. Chem. Phys. 1 (1933) 466. N.F. Mott and M. Littleton, Trans. Faraday, Soc. 34 (1938) 485. ES. Rittner, R.A. Hutner and F.K. DuPré, J. Chem. Phys. 17 (1949) 198. [7] A.J. Dekker, Solid State Physics (Macmillan, London, 1960) p. 167. [8] N.N. Greenwood, Ionic Crystals, Lattice Defects and Stoichiometry (Butterworths, London, 1968) p. 16. [9] M.P. Tosi, Solid State Phys. 16 (1964) 1. tioj M.F.C. Ladd, Structure and Bonding in Solid State Chemistry (Ellis Horwood, Chichester, 1979) p. 82.
This work was done while the author was a guest in the Inorganic Chemistry Department, University of Uppsala, and he would like to thaflk Professor Stig Rundqvist for generously affording him the facilities of the Solid State Chemistry Group. Thanks are due also to the University of Edinburgh for one year’s sabbatical leave.
[11] 0. Emersleben, Z. Physik 127 (1950) 588. [12] J. Bloem, Y.S. Oei, H.H.C. de Moor, J.H.L. Hanssen and L.J. Giling, J. Crystal Growth 65 (1983) 399. [13] D. Richman, Y.S. Chiang and RH. Robinson. RCA Rev. 31 (1970) 613. [14] H. Krautle, P. Roentgen and H. Beneking, J. Crystal
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[1] G. Erti, in: The Nature of the Surface Chemical Bond, Eds. T.N. Rhodin and G. Ertl (North-Holland, A.msterdam, 1979) p. 319. [2] I. Barin and 0. Knacke, The Thermochemical Properties of Inorganic Substances (Springer, Berlin, 1973).
Growth 65 (1983) 439. [15] J.P. Duchemin, Bonnet181. F. Koelsch and D. Huyghe, J. Crystal Growth M. 45 (1978) [16] M.J. Ludowise, J. Appl. Phys. 58 (1985) 31. [17] lB. Bhat, N.R. Taskar and 5K. Ghandhi, J. Electrochem.
[19] W. Gordy and W.J.O. Thomas, J. Chem. Phys. 24 (1956) [20] T.C. Waddington in: Advances in Inorganic Chemistry and Radiochenustry, Vol. 1 (Academic Press, New York, 1959) p. 193.