- Email: [email protected]
Binding and semiconductor properties of AIIIBV compounds
14
SESSION
A:
OPENING
An excess electron in this crystal will use a wave function for the lowest unocdupied level, which must be, of course, orthogonal to wave functions for the electrons in the valence band. The valence-band electrons are those at the highest level and are those wave functions occupied by the most loosely bound electrons of the tellurium and sulphur ions. Accordingly, we wouId assume that an excess electron uses a wave function which tends to become concentrated around the bismuth ions and to have very small values near the suphur or tellurium ions, since its wave function there must be orthogonal to the electrons of the valence band. On the basis of this picture, it follows that an excess electron has a very slight interaction with the cores of the sulphur and tellurium ions. Accordingly, its scattering by disorder among these ions will be very small. Apparently it is so small as to be imperceptible in the experiments carried out at Professor JOFFE’S laboratory. On the other hand, the wave functions used by holes are large around the tellurium and sulphur ions and consequently are affected by the potential-energy distribution in the interior of these ions. These will evidently be different for these two ions, and thus the scattering of the holes will be strongly influenced by disorder among these positive ions. Some evidence to support this interpretation is found in the crystal structure of BisTes (tellurobismuthite) and BisTesS (tetradymite).(l) These two structures differ by having 33 atomic per cent of the tellurium atoms replaced by sulphur atoms. This replacement apparently
J. Phys. Chem. Solids
BINDING A.4
Pergamon
AND
changes the interatomic distances by no more than about 2 per cent. This is in keeping with the idea that the core of the Te-- and S-- ions is not very important, so that it is largely the distribution of a loosely bound outer electron which is most important. The assumption that the electrons of the Te-- and S-- ions spread out is in keeping with the observations made on the polarizability of the oxygen O-- ion in various compounds.(s) It has been found that in the sequence of alkaline-earth ions the electronic polarizability which must be assigned to O-- in order to account for the refractive index varies approximately linearly with the volume available for the O-- ion. Assigned polar&abilities for AlsOs and Liz0 appear to fit well on this same scheme.
REFERENCES 1. WYCKOFF R. W., Crystal Structures Interscience Publishers, Inc., New York (1948). WYCKOFF quotes the following references: GARR~DO J. and FEO R., Bull. Sot. Franc. Mint. 61, 196 (1938); FRONDEL C,, Amer. Min. 24, No. 12, Part II, 7 (1939); LANGE P. W., Naturwissenschaften 27, 133 (1939). 2. SHOCKLEY W., Phys. Rev. 73, 1273 (1948), TESSMAN J. R., KAHN A. H. and SHOCKLEY W., Phys. Ra. 92, 890 (1953).
Press 1959. Vol. 8. pp. 14-20.
Printed in Great Britain
SEMICONDUCTOR PROPERTIES _/l”I BV COMPOUNDS 0.
‘Siemens-Schuckert
Abstract-A
SESSION
G. FOLBERTH
OF
and H. M’EXXER
Forschungslaboratorium,
Erlangen,
Germany
model for the interpretation of the characteristic properties of the AmBv compounds is proposed which takes into account the polarization of the valence electrons towards the BP ions. It is supposed that the polarization increases with increasing ionic part of the chemical bond and __ with increasing mean atomic weight. The influence of polarization on the lattice potential is investieated. It is shown that for AmBV compounds, especially InSb, a neutral binding is to be expected, as proposed by J. C. SLATER and G.-F. Kos’mR-(fhys; Rev. 94, 1948 (1954)). The results are used to provide an explanation of the characteristic differences between the values of the energy compounds. Some pecularities congap AE and the mobility ratio p_nlp’. in the various Am@ cerning the mutual solubilities of AmBv compounds and the solubilities of elements of the Group IV of the Periodic System in the A”‘@’ compounds are discussed. It is shown that for a continuous solid solution to be formed the polarizations must not differ too greatly. This rule allows the preferential formation of continuous solid solutions in the systems InAs-InP and GaAs-GaP t@ be. explained.
SESSION
A:
OPENING
IF we consider the various groups of semiconducting materials, we find some general rules governing the variation of the energy gap. Thus it is well known that the energy gap decreases with increasing atomic number; for instance, the energy gap of the elements of Group IV of the Periodic System decreases from diamond to gray tin. A similar rule exists also for the semiconducting A IIIBV compounds. Furthermore the AIIIBv compounds all have higher energy gaps than the corresponding isoelectronic elements of Group IV. These rules were set forth by WELKER(~)in 1952, on the basis of binding theory. The essential basis of the argument was that for the AIIIBV compounds there exists a small additional ionic bond which arises from the inequality between the two kinds of atoms, namely the AI11 and the gv atoms. The covalent bond and the ionic bond come to resonance according to PAULING(2) and the result is a larger total binding strength and from this a larger energy gap. However, it is not possible to explain by this concept alone the differences between the energy gaps of AIIIBV compounds With the same mean atomic weight, the so-called isoelectronic compounds. The binding strength increases in the isoelectronic sequences AlSb-GaAs-InP and GaSb-InAs, whereas the energy gap decreases. This behaviour cannot be accounted for by the preceding considerations, because an increasing binding strength should cause an increase in the energy gap. Therefore a further factor must exist, whose effect is greater than that of the binding strength. Such an effect has been known for a long time, but, in our opinion, it has not as yet been applied correctly to A IIIBV compounds. SERAPHIN,@) AnAVI(4) and GUBANOW(5)have shown-partly in a different manner-that the energy gap increases with increasing difference between the potential valleys at the A and B atoms, with otherwise constant binding strength, In this way it is possible to understand why the energy gap increases markedly from an element of Group IV to the isoelectronic AIII.Bv, ALLBvI, and ALBvII compounds. The very large energy gaps of nearly pure ionic crystals, like the alkali halides, can be explained only in this manner. To a first approximation all AIIBV compounds have the same
SESSION
15
difference between the potential valleys at the sites of the AI11 and BV atoms. In this approximation it is assumed that every BV atom gives one valence electron to an AILI atom, SO that all atoms now share equally four valence electrons. These charged atoms could now build up a crystal having a tetrahedral structure like the elements of Group IV. The resulting structure has the ZnS lattice. In this approximation the difference in charge between the BV and the AIIf sites amounts to two electronic charges uniformly for all AIIIBV compounds. For a better approximation it is still necessary to consider the deformation of the electronic structure caused by the above-mentioned charge difference, namely the polarization of the valence and the inner-shell electrons in the direction towards the BJ’ atoms. This polarization differs for the various AIIIBV compounds; it increases with increasing difference in electronegativity between the BV and AIII atoms, and therefore it increases with increasing ionic bond. It increases also with the atomic weight.(s) How does this polarization influence the lattice potential? If the polarization is very small, the valence electrons are almost equally distributed over the valence bridges in the same manner as in the elements of Group IV. So the potential is essentially determined by the difference of charge between BV and AI11 sites. The conduction electrons then find deeper potential valleys at the BV sites. Increasing polarization causes a variation of the potential valley in such a way that the deep BV valleys become narrower and the shallow AI11 valleys become broader. If we approximate the potential by Dirac functions, we may say that the polarization causes a reduction of the potential difference between BV and AI11 sites. This difference will be called B in the following figures; E vanishes completely if the polarization is SO large that on the average three valence electrons correspond to the AILI sites and five electrons correspond to the BV sites. On further increase in the polarization, the relations are reversed, so that for the conduction electrons the potential at an AI11 site is stronger than at a BV site. By making certain assumptions it is possible to estimate the polarization in arbitrary units by the following formula:(s)
16
SESSION
A:
OPENING
cov where Elan is the PAULING, Ecov is and ZII1 and Zv elements AIlI and
ionic extra energy according to the convalent binding energy, are the atomic numbers of the Bv, respectively.
SESSION
It is easy to see now that the experimental values of the energy gaps can be explained. To show this, the potential differences 1~1and the energy gaps of the two isoelectronic groups Al Sb-GaAs-InP and GaSb-InAs are given in Fig. 2. One sees that these differences vary in the same sense as the energy gaps. This effect should then overcompenaate the effect of binding strength discussed
FIG. 1. Hypothetical relationship between the difference in the potential strength c at A”’ and BV sites and the polarization J, The principal hypothesis is taken from SLATER and KOWER,‘~) who suggest a nearly “neutral binding” for the A IIIBV compounds. This means that the polarization is such that parameter E is in the neighbourhood of zero for AI”Bv compounds. Further we assume a linear relation between E and J, with the neutral point at a J value between InSb and InAs (Fig. 1). We can see that the polarization increases from 38 units for AlSb to 118 units for InAs, whereas the potential difference E increases from a negative value for AlSb to zero and then to a positive value for InAs.
earlier. It is worth noting that, if the proposed model is true, the increase of the energy gaps of A”*BV compounds with regard to the energy gap of the corresponding isoelectronic element of Group IV is caused to a great extent by the potential difference at AlSb and Gash, whereas the effect of binding strength is predominant in InP and InAs. GaAs occupies an intermediate position. It seems that the model developed may also explain other features of the semiconducting properties of the A”‘Bv compounds. At first we shall apply the proposed ideas to the following
SESSION
A:
OPENING
question: what is the reason for the peculiar, abnormally high values of the mobility ratio b(= ~,J,L$ in the various AIIrBv compounds) For the elements of Group IV it is known that the ratio b is of the order of 2. This ratio b is
SESSION
17
distant. This means that for the first group the jump probabilities are increased, whilst for the second group these probabilities are decreased. Movement of a hole through the lattice is possible only with the participation of this second group
2 Af
A&b -_--__
GoAs __
*.._
hP _-
I
Af t
0 1 FIG.
2. The
potential
energy gap AE and the difference in the strength between Au’ and Bv sites 1~1 for AuxBv compounds.
larger (up to b w 100) for many Axl*Bv compounds. The high values of b are not only based on an increased electron mobility pm, but also on a decreased hole mobility CL,. Such a decrease of the hole mobility pP in a ZnS lattice may be explained by the polarrzation of the valence electrons in the following way.(s) In a diamond lattice the valence electrons are distributed symmetrically over the valence bridges. If anywhere an electron is missing, a hole is created. Such a hole moves if a valence electron from one of the six neighbouring valence bridges jumps into the hole. Each of these six possibilities has the same probability. In a ZnS lattice with polarization, the mechanism of hole motion is slightly different. The valence electrons are not distributed symmetrically in the valence bridges, bring closer to the Bv than to the &II atoms. Thus from the six neighbouring positions of valence electrons which surround a hole, three have come nearer, whilst the other three are more c
of jumps. In this way the smaller jump probability determines the hole mobility. This is equivalent to an increase of the effective masses of the holes. If we remove other influences, especially the influence on the electron mass, we may expect b to increase with increasing polarization, That such a relation between the mobility ratio and the polarization really exists may be seen from Fig. 3, where the experimental values of b are plotted against the estimated values of polarization J. It is more difficult to explain the very high electron mobilities, e.g. those in InSb and InAa. These two compounds are located near the “neutral point” (see Fig. 1). Therefore the lattice scattering should be of the same order as that in the elements of Group IV. The higher binding strength is also favourable to a small lattice scattering. The second reason for the high mobility in AIIrBv compounds-the small effective mass -is not yet fully understood. The polarization effect may also explain some
18
SESSION
A:
OPENING
pecularities which are connected with the formation of solid solutions.(s) From metallic alloy systems it is known that a continuous solid solution exists if the two components crystallize in the same crystal system and if the lattice constants do not differ by more than approximately
SESSION
compounds differ too greatly, a considerable distortion of the electronic structure results. Consequently the internal energy is raised and decomposition is favoured, since the system then passes into a state with lower energy. In Table 1 the relative disparity between the lattice con-
80
b=A7+Jp 6c
41
21
0
FIG.
3. Polarization J as a function of the mobility ratio b = pnpg for PTBV compounds.
10 per cent. These conditions also apply in semiconducting systems, but they are not always sufficient. A study of the quasi-binary systems between AW3v ~orn~~ds shows that in some of these systems continuous solid solutions do not form or form only to a Limited extent. This is the case even though the lattice constants satisfy the above-mentioned conditions well. These facts may be explained in the following way, If the polarization values of the two
stants and the difference in the polarization values are listed for various quasi-binary systems. Evidently a continuous solid solution exists only in those systems for which the polarization values and the lattice constants do not differ very markedly. For the same reason, continuous solid solutions do not exist between AIWv compounds and elements of Group IV. For example, GaAs and germanium do not mix together over a wide range in spite of having almost equal Iattice
SESSION
A:
OPENING
constants. By the same reasoning it follows that continuous solution between a M*%v compound and a A*rrBv compound is not possible. Near the ends of the quasi-binary systems, the formation of solid solutions is favoured by the increased intluence of the entropy in these regions.
19
SESSION
up to 2 per cent, but germanium dissolves only much smaller amounts of GaAs. It is known that in the ArrrW compounds with components of almost equal ’ atomic radii, elements of Group IV are more likely to replace BV than A**1 atoms.@) This behaviour appears
Table 1. So&l solutions in systems with various AxxlBvcompounds Continous Without tendency to decomposition System
i
InAs in InP GaAs in GaP
I
3.2% 3.7%
i
--
Aa
I
AJ
solid solution
_ .
With tendency to decomposition System InSb in GaSb XnSbinAlSb GaSb in AlSb InAs in GaAs
3.5 11-9
I
I
_-
\
Aa
’ 6.1% ] 5.5% 0.7% 6.9%
J 1
No continuous solid solution
1
AJ
System
j
35.9 54-s
InSb in InAs
18.6
GaSb in GaAs
’ ! ’
Aa 6.7%
54.9
1
A_l 41 37-s 22 63-3 10-l
=
In these cases solid solutions may exist even if the above-mentioned rules are not satisfied. The region of solid solution, however, is larger the smaller the increase in internal energy resulting from the solution process. Such an increase is small if an AW3v compound with a small polarization is dissolved in an AffIBv compound with a high polarization. The opposite case-an ArIlBv compound with high polarization dissolved in an AW3v compound with small polarization-would give a relatively high increase of internal energy. To understand this, it is necessary to remember that in a ZnS lattice the polarization of the valence electrons exerts a stress on the lattice. On dissolving a compound with a Iower polarization, this stress is reduced and a state with lower energy is reached.* From these considerations it can be understood why InAs dissolves InSb up to 2 per cent, whereas InSb dissolves only a negligible amount of InAs.@~ For the same reason GaAs dissolves germ~ium *The ZnS lattice is held together by means of the covalent bond. If an ionic bond is added, this structure remains stable, up to values of the ionic bond of the order of the covalent bond. Consider, for example, the compound CuBr, which crystallizes in the ZnS lattice. If the ionic bond predominates, the ZnS lattice becomes unstable, because the NaCl or CsCl lattice is now more favourabb.
to arise from the fact that the ~larization is reduced more by substitution of a BV site than of an A*11 site, because the valence electrons are nearer to the BV sites. For energy gaps larger than w 1 eV, it appears that the above-mentioned effect is negligible and it becomes energetic~ly favourable, if the atoms of Group IV are distributed equally over AIlI and Bv sites.(le) Finally, it is worth pointing out that it may be possible to treat AIIBVI compounds in a similar manner. REFERENCES
1. WELKERH., 2. Naturf. 7a, 744 (1952). 2. PAULINC L., The Nature of the Chemical Bond Oxford University Press (1950). 3. SERAPHINB., 2. Natutf. 9q 450 (1954). 4. ALIAVrJ., Phys. Rw. 105, 789 (1957). 5. GUBANOW A. J., Zh. tekh. fiz., Moscow 26, 2170 (1956). 6. FOLRERTH 0. G., 2. Naturf. (to be published). In this paper a detailed description of the problem presented here can be found, 7. SLATER J. C. and KOSTER G. F., Phys. Rew. 94, 1948 (19.54). 8. SHIH C. and. PERETTI E. A., J. Amer. Chem. Sec. 75, 608 (1953). 0. G. and SCHILLIMANN E., 2. N-6 9. FOLBFJRTH 12a, 943 (1957). 10. KOLM C., KULIN S. A. and AVERBACHB. S., Phyr. Rev. 108, 965 (1957).
SESSION
A:
OPENING
An excess electron in this crystal will use a wave function for the lowest unocdupied level, which must be, of course, orthogonal to wave functions for the electrons in the valence band. The valence-band electrons are those at the highest level and are those wave functions occupied by the most loosely bound electrons of the tellurium and sulphur ions. Accordingly, we wouId assume that an excess electron uses a wave function which tends to become concentrated around the bismuth ions and to have very small values near the suphur or tellurium ions, since its wave function there must be orthogonal to the electrons of the valence band. On the basis of this picture, it follows that an excess electron has a very slight interaction with the cores of the sulphur and tellurium ions. Accordingly, its scattering by disorder among these ions will be very small. Apparently it is so small as to be imperceptible in the experiments carried out at Professor JOFFE’S laboratory. On the other hand, the wave functions used by holes are large around the tellurium and sulphur ions and consequently are affected by the potential-energy distribution in the interior of these ions. These will evidently be different for these two ions, and thus the scattering of the holes will be strongly influenced by disorder among these positive ions. Some evidence to support this interpretation is found in the crystal structure of BisTes (tellurobismuthite) and BisTesS (tetradymite).(l) These two structures differ by having 33 atomic per cent of the tellurium atoms replaced by sulphur atoms. This replacement apparently
J. Phys. Chem. Solids
BINDING A.4
Pergamon
AND
changes the interatomic distances by no more than about 2 per cent. This is in keeping with the idea that the core of the Te-- and S-- ions is not very important, so that it is largely the distribution of a loosely bound outer electron which is most important. The assumption that the electrons of the Te-- and S-- ions spread out is in keeping with the observations made on the polarizability of the oxygen O-- ion in various compounds.(s) It has been found that in the sequence of alkaline-earth ions the electronic polarizability which must be assigned to O-- in order to account for the refractive index varies approximately linearly with the volume available for the O-- ion. Assigned polar&abilities for AlsOs and Liz0 appear to fit well on this same scheme.
REFERENCES 1. WYCKOFF R. W., Crystal Structures Interscience Publishers, Inc., New York (1948). WYCKOFF quotes the following references: GARR~DO J. and FEO R., Bull. Sot. Franc. Mint. 61, 196 (1938); FRONDEL C,, Amer. Min. 24, No. 12, Part II, 7 (1939); LANGE P. W., Naturwissenschaften 27, 133 (1939). 2. SHOCKLEY W., Phys. Rev. 73, 1273 (1948), TESSMAN J. R., KAHN A. H. and SHOCKLEY W., Phys. Ra. 92, 890 (1953).
Press 1959. Vol. 8. pp. 14-20.
Printed in Great Britain
SEMICONDUCTOR PROPERTIES _/l”I BV COMPOUNDS 0.
‘Siemens-Schuckert
Abstract-A
SESSION
G. FOLBERTH
OF
and H. M’EXXER
Forschungslaboratorium,
Erlangen,
Germany
model for the interpretation of the characteristic properties of the AmBv compounds is proposed which takes into account the polarization of the valence electrons towards the BP ions. It is supposed that the polarization increases with increasing ionic part of the chemical bond and __ with increasing mean atomic weight. The influence of polarization on the lattice potential is investieated. It is shown that for AmBV compounds, especially InSb, a neutral binding is to be expected, as proposed by J. C. SLATER and G.-F. Kos’mR-(fhys; Rev. 94, 1948 (1954)). The results are used to provide an explanation of the characteristic differences between the values of the energy compounds. Some pecularities congap AE and the mobility ratio p_nlp’. in the various Am@ cerning the mutual solubilities of AmBv compounds and the solubilities of elements of the Group IV of the Periodic System in the A”‘@’ compounds are discussed. It is shown that for a continuous solid solution to be formed the polarizations must not differ too greatly. This rule allows the preferential formation of continuous solid solutions in the systems InAs-InP and GaAs-GaP t@ be. explained.
SESSION
A:
OPENING
IF we consider the various groups of semiconducting materials, we find some general rules governing the variation of the energy gap. Thus it is well known that the energy gap decreases with increasing atomic number; for instance, the energy gap of the elements of Group IV of the Periodic System decreases from diamond to gray tin. A similar rule exists also for the semiconducting A IIIBV compounds. Furthermore the AIIIBv compounds all have higher energy gaps than the corresponding isoelectronic elements of Group IV. These rules were set forth by WELKER(~)in 1952, on the basis of binding theory. The essential basis of the argument was that for the AIIIBV compounds there exists a small additional ionic bond which arises from the inequality between the two kinds of atoms, namely the AI11 and the gv atoms. The covalent bond and the ionic bond come to resonance according to PAULING(2) and the result is a larger total binding strength and from this a larger energy gap. However, it is not possible to explain by this concept alone the differences between the energy gaps of AIIIBV compounds With the same mean atomic weight, the so-called isoelectronic compounds. The binding strength increases in the isoelectronic sequences AlSb-GaAs-InP and GaSb-InAs, whereas the energy gap decreases. This behaviour cannot be accounted for by the preceding considerations, because an increasing binding strength should cause an increase in the energy gap. Therefore a further factor must exist, whose effect is greater than that of the binding strength. Such an effect has been known for a long time, but, in our opinion, it has not as yet been applied correctly to A IIIBV compounds. SERAPHIN,@) AnAVI(4) and GUBANOW(5)have shown-partly in a different manner-that the energy gap increases with increasing difference between the potential valleys at the A and B atoms, with otherwise constant binding strength, In this way it is possible to understand why the energy gap increases markedly from an element of Group IV to the isoelectronic AIII.Bv, ALLBvI, and ALBvII compounds. The very large energy gaps of nearly pure ionic crystals, like the alkali halides, can be explained only in this manner. To a first approximation all AIIBV compounds have the same
SESSION
15
difference between the potential valleys at the sites of the AI11 and BV atoms. In this approximation it is assumed that every BV atom gives one valence electron to an AILI atom, SO that all atoms now share equally four valence electrons. These charged atoms could now build up a crystal having a tetrahedral structure like the elements of Group IV. The resulting structure has the ZnS lattice. In this approximation the difference in charge between the BV and the AIIf sites amounts to two electronic charges uniformly for all AIIIBV compounds. For a better approximation it is still necessary to consider the deformation of the electronic structure caused by the above-mentioned charge difference, namely the polarization of the valence and the inner-shell electrons in the direction towards the BJ’ atoms. This polarization differs for the various AIIIBV compounds; it increases with increasing difference in electronegativity between the BV and AIII atoms, and therefore it increases with increasing ionic bond. It increases also with the atomic weight.(s) How does this polarization influence the lattice potential? If the polarization is very small, the valence electrons are almost equally distributed over the valence bridges in the same manner as in the elements of Group IV. So the potential is essentially determined by the difference of charge between BV and AI11 sites. The conduction electrons then find deeper potential valleys at the BV sites. Increasing polarization causes a variation of the potential valley in such a way that the deep BV valleys become narrower and the shallow AI11 valleys become broader. If we approximate the potential by Dirac functions, we may say that the polarization causes a reduction of the potential difference between BV and AI11 sites. This difference will be called B in the following figures; E vanishes completely if the polarization is SO large that on the average three valence electrons correspond to the AILI sites and five electrons correspond to the BV sites. On further increase in the polarization, the relations are reversed, so that for the conduction electrons the potential at an AI11 site is stronger than at a BV site. By making certain assumptions it is possible to estimate the polarization in arbitrary units by the following formula:(s)
16
SESSION
A:
OPENING
cov where Elan is the PAULING, Ecov is and ZII1 and Zv elements AIlI and
ionic extra energy according to the convalent binding energy, are the atomic numbers of the Bv, respectively.
SESSION
It is easy to see now that the experimental values of the energy gaps can be explained. To show this, the potential differences 1~1and the energy gaps of the two isoelectronic groups Al Sb-GaAs-InP and GaSb-InAs are given in Fig. 2. One sees that these differences vary in the same sense as the energy gaps. This effect should then overcompenaate the effect of binding strength discussed
FIG. 1. Hypothetical relationship between the difference in the potential strength c at A”’ and BV sites and the polarization J, The principal hypothesis is taken from SLATER and KOWER,‘~) who suggest a nearly “neutral binding” for the A IIIBV compounds. This means that the polarization is such that parameter E is in the neighbourhood of zero for AI”Bv compounds. Further we assume a linear relation between E and J, with the neutral point at a J value between InSb and InAs (Fig. 1). We can see that the polarization increases from 38 units for AlSb to 118 units for InAs, whereas the potential difference E increases from a negative value for AlSb to zero and then to a positive value for InAs.
earlier. It is worth noting that, if the proposed model is true, the increase of the energy gaps of A”*BV compounds with regard to the energy gap of the corresponding isoelectronic element of Group IV is caused to a great extent by the potential difference at AlSb and Gash, whereas the effect of binding strength is predominant in InP and InAs. GaAs occupies an intermediate position. It seems that the model developed may also explain other features of the semiconducting properties of the A”‘Bv compounds. At first we shall apply the proposed ideas to the following
SESSION
A:
OPENING
question: what is the reason for the peculiar, abnormally high values of the mobility ratio b(= ~,J,L$ in the various AIIrBv compounds) For the elements of Group IV it is known that the ratio b is of the order of 2. This ratio b is
SESSION
17
distant. This means that for the first group the jump probabilities are increased, whilst for the second group these probabilities are decreased. Movement of a hole through the lattice is possible only with the participation of this second group
2 Af
A&b -_--__
GoAs __
*.._
hP _-
I
Af t
0 1 FIG.
2. The
potential
energy gap AE and the difference in the strength between Au’ and Bv sites 1~1 for AuxBv compounds.
larger (up to b w 100) for many Axl*Bv compounds. The high values of b are not only based on an increased electron mobility pm, but also on a decreased hole mobility CL,. Such a decrease of the hole mobility pP in a ZnS lattice may be explained by the polarrzation of the valence electrons in the following way.(s) In a diamond lattice the valence electrons are distributed symmetrically over the valence bridges. If anywhere an electron is missing, a hole is created. Such a hole moves if a valence electron from one of the six neighbouring valence bridges jumps into the hole. Each of these six possibilities has the same probability. In a ZnS lattice with polarization, the mechanism of hole motion is slightly different. The valence electrons are not distributed symmetrically in the valence bridges, bring closer to the Bv than to the &II atoms. Thus from the six neighbouring positions of valence electrons which surround a hole, three have come nearer, whilst the other three are more c
of jumps. In this way the smaller jump probability determines the hole mobility. This is equivalent to an increase of the effective masses of the holes. If we remove other influences, especially the influence on the electron mass, we may expect b to increase with increasing polarization, That such a relation between the mobility ratio and the polarization really exists may be seen from Fig. 3, where the experimental values of b are plotted against the estimated values of polarization J. It is more difficult to explain the very high electron mobilities, e.g. those in InSb and InAa. These two compounds are located near the “neutral point” (see Fig. 1). Therefore the lattice scattering should be of the same order as that in the elements of Group IV. The higher binding strength is also favourable to a small lattice scattering. The second reason for the high mobility in AIIrBv compounds-the small effective mass -is not yet fully understood. The polarization effect may also explain some
18
SESSION
A:
OPENING
pecularities which are connected with the formation of solid solutions.(s) From metallic alloy systems it is known that a continuous solid solution exists if the two components crystallize in the same crystal system and if the lattice constants do not differ by more than approximately
SESSION
compounds differ too greatly, a considerable distortion of the electronic structure results. Consequently the internal energy is raised and decomposition is favoured, since the system then passes into a state with lower energy. In Table 1 the relative disparity between the lattice con-
80
b=A7+Jp 6c
41
21
0
FIG.
3. Polarization J as a function of the mobility ratio b = pnpg for PTBV compounds.
10 per cent. These conditions also apply in semiconducting systems, but they are not always sufficient. A study of the quasi-binary systems between AW3v ~orn~~ds shows that in some of these systems continuous solid solutions do not form or form only to a Limited extent. This is the case even though the lattice constants satisfy the above-mentioned conditions well. These facts may be explained in the following way, If the polarization values of the two
stants and the difference in the polarization values are listed for various quasi-binary systems. Evidently a continuous solid solution exists only in those systems for which the polarization values and the lattice constants do not differ very markedly. For the same reason, continuous solid solutions do not exist between AIWv compounds and elements of Group IV. For example, GaAs and germanium do not mix together over a wide range in spite of having almost equal Iattice
SESSION
A:
OPENING
constants. By the same reasoning it follows that continuous solution between a M*%v compound and a A*rrBv compound is not possible. Near the ends of the quasi-binary systems, the formation of solid solutions is favoured by the increased intluence of the entropy in these regions.
19
SESSION
up to 2 per cent, but germanium dissolves only much smaller amounts of GaAs. It is known that in the ArrrW compounds with components of almost equal ’ atomic radii, elements of Group IV are more likely to replace BV than A**1 atoms.@) This behaviour appears
Table 1. So&l solutions in systems with various AxxlBvcompounds Continous Without tendency to decomposition System
i
InAs in InP GaAs in GaP
I
3.2% 3.7%
i
--
Aa
I
AJ
solid solution
_ .
With tendency to decomposition System InSb in GaSb XnSbinAlSb GaSb in AlSb InAs in GaAs
3.5 11-9
I
I
_-
\
Aa
’ 6.1% ] 5.5% 0.7% 6.9%
J 1
No continuous solid solution
1
AJ
System
j
35.9 54-s
InSb in InAs
18.6
GaSb in GaAs
’ ! ’
Aa 6.7%
54.9
1
A_l 41 37-s 22 63-3 10-l
=
In these cases solid solutions may exist even if the above-mentioned rules are not satisfied. The region of solid solution, however, is larger the smaller the increase in internal energy resulting from the solution process. Such an increase is small if an AW3v compound with a small polarization is dissolved in an AffIBv compound with a high polarization. The opposite case-an ArIlBv compound with high polarization dissolved in an AW3v compound with small polarization-would give a relatively high increase of internal energy. To understand this, it is necessary to remember that in a ZnS lattice the polarization of the valence electrons exerts a stress on the lattice. On dissolving a compound with a Iower polarization, this stress is reduced and a state with lower energy is reached.* From these considerations it can be understood why InAs dissolves InSb up to 2 per cent, whereas InSb dissolves only a negligible amount of InAs.@~ For the same reason GaAs dissolves germ~ium *The ZnS lattice is held together by means of the covalent bond. If an ionic bond is added, this structure remains stable, up to values of the ionic bond of the order of the covalent bond. Consider, for example, the compound CuBr, which crystallizes in the ZnS lattice. If the ionic bond predominates, the ZnS lattice becomes unstable, because the NaCl or CsCl lattice is now more favourabb.
to arise from the fact that the ~larization is reduced more by substitution of a BV site than of an A*11 site, because the valence electrons are nearer to the BV sites. For energy gaps larger than w 1 eV, it appears that the above-mentioned effect is negligible and it becomes energetic~ly favourable, if the atoms of Group IV are distributed equally over AIlI and Bv sites.(le) Finally, it is worth pointing out that it may be possible to treat AIIBVI compounds in a similar manner. REFERENCES
1. WELKERH., 2. Naturf. 7a, 744 (1952). 2. PAULINC L., The Nature of the Chemical Bond Oxford University Press (1950). 3. SERAPHINB., 2. Natutf. 9q 450 (1954). 4. ALIAVrJ., Phys. Rw. 105, 789 (1957). 5. GUBANOW A. J., Zh. tekh. fiz., Moscow 26, 2170 (1956). 6. FOLRERTH 0. G., 2. Naturf. (to be published). In this paper a detailed description of the problem presented here can be found, 7. SLATER J. C. and KOSTER G. F., Phys. Rew. 94, 1948 (19.54). 8. SHIH C. and. PERETTI E. A., J. Amer. Chem. Sec. 75, 608 (1953). 0. G. and SCHILLIMANN E., 2. N-6 9. FOLBFJRTH 12a, 943 (1957). 10. KOLM C., KULIN S. A. and AVERBACHB. S., Phyr. Rev. 108, 965 (1957).