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Trends in the ionicity in the average valence V materials
Solid State Communications, Vol. 12, pp.xiii—xvi, 1973.
Pergamon Press.
Printed in Great Britain
TRENDS IN THE IONICITY IN THE AVERAGE VALENCE V MATERIALS P. J. Stiles Physics Department, Brown University, Providence, R.I. 02912 and M. H. Brodsky IBM Thomas J. Watson Research Center, Yorktown Heights, N.Y. 10598, U.S.A. (Received 27 June 1972 by J. Tauc)
An ionicity scale is given for average valence V materials. The scale is derived via the methods of the Phillips spectroscopic theory.
WE REPORT here an extension of a form of the Phillips spectroscopic theory”2 of chemical bonding to the average valence V materials. We previously3 utilized an aspect of this approach in examining the electronic dielectric constant of amorphous materials, Here we examine those crystalline materials of average valence V with no more than two components and with all atoms having 6-fold or near 6-fold coordination. O~particular interest is the relation between the ionicities of these materials and their structural distortions. Our principal fmdings are that it is possible to calculate an ionicity scale for average valence V crystals and that the iomcity gives useful qualitative structural information, The Phillips—Van Vechten ionicity of a crystal is defmed’2’4 as C2
__
-
E~+ C2
=
E~
where E~= ii (4n Ne2/m)m is the plasma energy of the N electrons per unit volume that contributes to the electronic dielectric response. The parameter D corrects for the different oscillator strengths of d-shell electrons. The homopolar gap Eh is evaluated by an empirical scaling found to hold for group IV elements and postulated to hold for average valence N compounds, i.e. = Ej~”(aw/a)25. (3) -
where a and a~are the nearest neighbor distances of the crystals with homopolar gaps E,, and Ej~.With Eg and Eh known C and thus f~ are then evaluated. It is then found that C obeys a screened charge transfer formula of the form C
=
b
(~ \rA
(1)
_~‘1 ~
(4)
rBI
where ZA (ZB) and rA (rB) are the valence number N (8 — N) and atom radius of the A(B) of the average valence iv compound ANB~N.The exponential term contains the Thomas—Fermi screening wave number k 1 and the average radius R = (rA + rB)/2. The proportionality constant b is of order unity.
where C and Eh are, respectively, the ionic and homo-. polar components of the crystalline spectroscopic energy gap E~.The spectroscopic energy gap is evaluated from the low frequency (low with respect to electronic frequencies, but high with respect to lattice frequencies) dielectric constant e0 by 2 = l+—2D (2) E
logicThe to average valence V crystals the spectroprocedure we adopt here using is to apply the same scopic parameters for Sb to do our scaling. We need to make several simplifying assumptions and therefore xiil
xiv
TRENDS IN THE ION1CITY IN THE AVERAGE VALENCE V MATERIALS
we claim only to be calculating trends in ionicities rather than finding precise values. We confine ourselves to materials of the same crystal coordinations with effective cubic lattice constants that are within 7% of the average of the group. Therefore refinements due to changes in the ratio of Fermi energy to plasma energy are ignored.4 If follows that the plasma energy scales with volume, that is, E~~ 1/a3. We assume that the analog of equation (3) holds for average valence V materials with the same power law dependence. We next present a procedure for averaging the near neighbor bond lengths which can differ slightly for each of a crystal’s six nearest neighbors. Of the materials studied here, PbS, PbSe, PbTe and SnTe crystallize in the 6-fold coordinated NaC1 structure (f.c.c.) at room temperature and all the bond lengths are the same. However, GeTe (and SnTe at low ternperatures) has a slightly distorted NaCl structure which is ferroelectric and is face-centred rhombohedral (f.c.r.), similar to the As, Sb and Bi structures. An atom in one of these four crystals has close to a 6-fold coordination. There are three neighbors at a distance (1 — a) times the average and three additional neighbors at a correspondingly greater distance. GeSe, GeS, SnSe and SnS have an orthorhombic structure which is also a distortion of the NaCl lattice and is an antiferroelectric structure. Here the coordination is effectively six also and the six neighbors nearly divide into two groups of three neighbours each as in the f.c.r. structure. The values of a are given in Table 1. We have chosen to average the contributions to the horno-polar energy gap squared such that E~ a where the effective 6 bond length is obtained from = (,~ r~)/6and where the r 1 are the six nearest neighbor separations. In the f.c.c. structure thisSuris Just £~“~/2, where ~7is the volume of the unit cell. prisingly it is found that the ratio f2~/2ais very nearly 1 for all the materials in question, as is illustrated in the Table. This implies that although small changes in bond angle and length occur to stabilize the coordination, that basically all these materials have the same dependence on volume. It is seen that the most nearly molecular crystal listed, As, has the largest variation of the above ratio. ~,
We may calculate the ionicity of the compounds in question from lattice constants and dielectric constants using Sb data to determine the constants of proportionality. We thus use Sb rather than Bi, as the
Vol. 12, No.6
theory is expected to work better for lighter elements in that there are fewer uncertainties in determining the number of valence electrons. The value for As is unknown; we do not use the values of the lighter column V elements (P and N) as they are molecular in nature. We expect the least uncertainties in the ionicities for SnTe which is formed of elements of the same row and PbSe which is of the same average row (GePo is hard to find) as Sb. The expression for j is then,
(f,)
=
1
—
(
~— 1
\
/asb\ 2
~
i) (5) and the results are given in the Table for those cornpounds where e has been measured. Also given are the logarithmic derivatives off, with respect to n, where n is the negative power of the length dependence of E7~,and is 5 here. It is seen that the results are not strongly dependent on our choice of n, particularly for PbSe and SnTe, which have approximately the same lattice constant as Sb, and hence the same homopolar energy gap. ~Sb
—
‘
We now look at the ionic component of the bonding to gain insight into the stability of the different structures. Assuming that C
=
2b/.~Z(a)1exp (— k 9a)
(6)
where b is a constant of the order I and is determined experimentally for a class of compounds, a has replaced the covalent radü of the equivalent fàrmula used for the average valence IV materials, ~.Z is the difference in charge on the two different 2atoms Withand Eh k8 is~ the (a)5t2, Thomas—Fermi we may writescreening factor. 3t2 exp (k~a) (7) bt3.Z (l (aT \ fi ~‘~~‘) i The right side of the equation is found to be independent of a for the range 2.25 A
Vol. 12, No.6
TRENDS IN THE IONICITY IN THE AVERAGE VALENCE V MATERIALS
xv
Table 1. Swnmary ofparameters used in calculations. All symbols are as defined in text, except i which is the average bond length
Structure PbS PbS PbTe SnTe GeTe
f.c.c. f.c.c. f.c.c. f.c.c. f.c.r.
GeSe
ortho
GeS
ortho
SnSe
ortho
SnS
ortho
As
f.c.r.
Sb
f.c.r.
Bi
f.c.r.
a values where(l+a)ã are bond lengths 0 0 0 0 3 with +.O5~ 3 with — .05 +.l23,.123,.134* —.124,—.124,—.133 +.l21,.121,.150* —.126,—.126,—.140 +.082, +.082, .122* —.089, —.089, —.107 +.093,.097,.140* —.103,—.103,—.l22 3with+.ll* 3 with — .11 3 with +.07* 3 with —.07 3 with +.07* 3 with —.07
1
e
2.97 3.06 3.25 3.14 2.97
1.000 1.000 1.000 1.000 1.007
17.6t 22.8t 32.8t 42~ 36”
.77 .72 .63 .49 .52
3.3 2.5 1.7 1.0 1.1
—.07 —.03 +.13 +.05 —.17
2.83
1.007
22~
.68
2.1
—.21
2.72
1.006
—
—
—
—
3.02
.995
22
.7
2.3
—.08
2.88
1.001
14**
.8
4
—.09
2.73
1.023
6Stt
0
0
0
3.09
1.008
80~
0
0
0
3.26
1.009
100~
0
0
0
WYCKOFF R.W.G., Crystal Structures, 2nd edn., Vol. 1, Interscience, New York (1963).
t
ZEMEL J.N., JENSEN J.D. and SCHOOLAR R.B., Phys. Rev. 140, A330 (1965).
~ §
RIEDL H.R., DIXON J.R. and SCHOOLAR R.B., Phys. Rev. 162, 692 (1967). GOLDAK J., BARRETT C.S., INNES D. and YOUDELIS W., J. Chem. Phys. 44,3323 (1966).
~‘
TSU R., HOWARD W.E. and ESAKI L., Phys. Rev. 172, 779 (1968).
**
KANNEWARF C.R. and CASHMAN R.J., J. Phys. Qiem. Solids. 22,293 (1970). Preliminary values, GRUTTOLA V. DE, Private communication.
tt
Predicted by the scaling of equation (3).
14
NANNEYC.,Phys. Rev. 129,109(1963). BOYLE W.S. and BRAILSFORD A.D., Phys. Rev. 120, 1943 (1960).
§§
A consequence of this is that about twice as much charge is transferred in PbS as in GeTe, GeTe (and also SnTe) with low fe’s thus have a small charge transfer component to stabilize what would be an
ndf dn
f,
£2l~~3/2a*
*
•
f
a(A)
ft
unstable structure if it were only covalently bonded. The distortion to f.c.r. from f.c.c. appears to correlate with low fonicity. Additionally, it appears that the distortion to f.c.r. from f.c.c. does not change the
xvi
TRENDS IN THE IONICITY IN THE AVERAGE VALENCE V MATERIALS
charge transfer by a significant amount. It is known that a few % GeTe in PbTe causes the alloy to transform to f.c.r.6 GeSe, which is orthorhombic, does have a larger charge transfer than GeTe, and therefore it appears that the-crystals gain more ionicity and stability by distorting to the orthohombic structure. We know of no IV—VI alloy system which undergoes a transition from f.c.c. to orthorhombic as a function of temperature. However, it appears that the PbSe and SnSe alloy system might. It also appears that
Vol. 12, No.6
lighter group VI elements increase the ionicity and hence the charge transfer but that the trends are not obvious for lighter group IV elements. Finally the values of CA$ and Cth may be estimated from the above scaling and should have values of approximately 65 and 90. The Bi value is consistent with the measured ~ of 100, the As value has not, to our knowledge, been measured.
REFERENCES 1.
PHILUPS J.C.,Phys. Rev. Lett. 22, 645 (1969).
2.
For a review, see PHILLIPS J.C., Rev. Mod. Phys. 42, 317 (1970).
3.
BRODSKY M.H. and STILES P.J.,Phys. Rev. Lert. 25, 798 (1970).
4.
VAN VECHTEN J.A.,Phys. Rev. 182, 891 (1969).
5.
MAZELSKY R., LUBELL M.S., KRAMER W.E.,J. Chem. Phys. 37,45 (1962).
Une échelle d’ionicité est donnée pour les matériaux a valence moyenne V. Cette échelle est obtenus par les méthodes de la théorie spéctroscopique de Phillips.
Pergamon Press.
Printed in Great Britain
TRENDS IN THE IONICITY IN THE AVERAGE VALENCE V MATERIALS P. J. Stiles Physics Department, Brown University, Providence, R.I. 02912 and M. H. Brodsky IBM Thomas J. Watson Research Center, Yorktown Heights, N.Y. 10598, U.S.A. (Received 27 June 1972 by J. Tauc)
An ionicity scale is given for average valence V materials. The scale is derived via the methods of the Phillips spectroscopic theory.
WE REPORT here an extension of a form of the Phillips spectroscopic theory”2 of chemical bonding to the average valence V materials. We previously3 utilized an aspect of this approach in examining the electronic dielectric constant of amorphous materials, Here we examine those crystalline materials of average valence V with no more than two components and with all atoms having 6-fold or near 6-fold coordination. O~particular interest is the relation between the ionicities of these materials and their structural distortions. Our principal fmdings are that it is possible to calculate an ionicity scale for average valence V crystals and that the iomcity gives useful qualitative structural information, The Phillips—Van Vechten ionicity of a crystal is defmed’2’4 as C2
__
-
E~+ C2
=
E~
where E~= ii (4n Ne2/m)m is the plasma energy of the N electrons per unit volume that contributes to the electronic dielectric response. The parameter D corrects for the different oscillator strengths of d-shell electrons. The homopolar gap Eh is evaluated by an empirical scaling found to hold for group IV elements and postulated to hold for average valence N compounds, i.e. = Ej~”(aw/a)25. (3) -
where a and a~are the nearest neighbor distances of the crystals with homopolar gaps E,, and Ej~.With Eg and Eh known C and thus f~ are then evaluated. It is then found that C obeys a screened charge transfer formula of the form C
=
b
(~ \rA
(1)
_~‘1 ~
(4)
rBI
where ZA (ZB) and rA (rB) are the valence number N (8 — N) and atom radius of the A(B) of the average valence iv compound ANB~N.The exponential term contains the Thomas—Fermi screening wave number k 1 and the average radius R = (rA + rB)/2. The proportionality constant b is of order unity.
where C and Eh are, respectively, the ionic and homo-. polar components of the crystalline spectroscopic energy gap E~.The spectroscopic energy gap is evaluated from the low frequency (low with respect to electronic frequencies, but high with respect to lattice frequencies) dielectric constant e0 by 2 = l+—2D (2) E
logicThe to average valence V crystals the spectroprocedure we adopt here using is to apply the same scopic parameters for Sb to do our scaling. We need to make several simplifying assumptions and therefore xiil
xiv
TRENDS IN THE ION1CITY IN THE AVERAGE VALENCE V MATERIALS
we claim only to be calculating trends in ionicities rather than finding precise values. We confine ourselves to materials of the same crystal coordinations with effective cubic lattice constants that are within 7% of the average of the group. Therefore refinements due to changes in the ratio of Fermi energy to plasma energy are ignored.4 If follows that the plasma energy scales with volume, that is, E~~ 1/a3. We assume that the analog of equation (3) holds for average valence V materials with the same power law dependence. We next present a procedure for averaging the near neighbor bond lengths which can differ slightly for each of a crystal’s six nearest neighbors. Of the materials studied here, PbS, PbSe, PbTe and SnTe crystallize in the 6-fold coordinated NaC1 structure (f.c.c.) at room temperature and all the bond lengths are the same. However, GeTe (and SnTe at low ternperatures) has a slightly distorted NaCl structure which is ferroelectric and is face-centred rhombohedral (f.c.r.), similar to the As, Sb and Bi structures. An atom in one of these four crystals has close to a 6-fold coordination. There are three neighbors at a distance (1 — a) times the average and three additional neighbors at a correspondingly greater distance. GeSe, GeS, SnSe and SnS have an orthorhombic structure which is also a distortion of the NaCl lattice and is an antiferroelectric structure. Here the coordination is effectively six also and the six neighbors nearly divide into two groups of three neighbours each as in the f.c.r. structure. The values of a are given in Table 1. We have chosen to average the contributions to the horno-polar energy gap squared such that E~ a where the effective 6 bond length is obtained from = (,~ r~)/6and where the r 1 are the six nearest neighbor separations. In the f.c.c. structure thisSuris Just £~“~/2, where ~7is the volume of the unit cell. prisingly it is found that the ratio f2~/2ais very nearly 1 for all the materials in question, as is illustrated in the Table. This implies that although small changes in bond angle and length occur to stabilize the coordination, that basically all these materials have the same dependence on volume. It is seen that the most nearly molecular crystal listed, As, has the largest variation of the above ratio. ~,
We may calculate the ionicity of the compounds in question from lattice constants and dielectric constants using Sb data to determine the constants of proportionality. We thus use Sb rather than Bi, as the
Vol. 12, No.6
theory is expected to work better for lighter elements in that there are fewer uncertainties in determining the number of valence electrons. The value for As is unknown; we do not use the values of the lighter column V elements (P and N) as they are molecular in nature. We expect the least uncertainties in the ionicities for SnTe which is formed of elements of the same row and PbSe which is of the same average row (GePo is hard to find) as Sb. The expression for j is then,
(f,)
=
1
—
(
~— 1
\
/asb\ 2
~
i) (5) and the results are given in the Table for those cornpounds where e has been measured. Also given are the logarithmic derivatives off, with respect to n, where n is the negative power of the length dependence of E7~,and is 5 here. It is seen that the results are not strongly dependent on our choice of n, particularly for PbSe and SnTe, which have approximately the same lattice constant as Sb, and hence the same homopolar energy gap. ~Sb
—
‘
We now look at the ionic component of the bonding to gain insight into the stability of the different structures. Assuming that C
=
2b/.~Z(a)1exp (— k 9a)
(6)
where b is a constant of the order I and is determined experimentally for a class of compounds, a has replaced the covalent radü of the equivalent fàrmula used for the average valence IV materials, ~.Z is the difference in charge on the two different 2atoms Withand Eh k8 is~ the (a)5t2, Thomas—Fermi we may writescreening factor. 3t2 exp (k~a) (7) bt3.Z (l (aT \ fi ~‘~~‘) i The right side of the equation is found to be independent of a for the range 2.25 A
Vol. 12, No.6
TRENDS IN THE IONICITY IN THE AVERAGE VALENCE V MATERIALS
xv
Table 1. Swnmary ofparameters used in calculations. All symbols are as defined in text, except i which is the average bond length
Structure PbS PbS PbTe SnTe GeTe
f.c.c. f.c.c. f.c.c. f.c.c. f.c.r.
GeSe
ortho
GeS
ortho
SnSe
ortho
SnS
ortho
As
f.c.r.
Sb
f.c.r.
Bi
f.c.r.
a values where(l+a)ã are bond lengths 0 0 0 0 3 with +.O5~ 3 with — .05 +.l23,.123,.134* —.124,—.124,—.133 +.l21,.121,.150* —.126,—.126,—.140 +.082, +.082, .122* —.089, —.089, —.107 +.093,.097,.140* —.103,—.103,—.l22 3with+.ll* 3 with — .11 3 with +.07* 3 with —.07 3 with +.07* 3 with —.07
1
e
2.97 3.06 3.25 3.14 2.97
1.000 1.000 1.000 1.000 1.007
17.6t 22.8t 32.8t 42~ 36”
.77 .72 .63 .49 .52
3.3 2.5 1.7 1.0 1.1
—.07 —.03 +.13 +.05 —.17
2.83
1.007
22~
.68
2.1
—.21
2.72
1.006
—
—
—
—
3.02
.995
22
.7
2.3
—.08
2.88
1.001
14**
.8
4
—.09
2.73
1.023
6Stt
0
0
0
3.09
1.008
80~
0
0
0
3.26
1.009
100~
0
0
0
WYCKOFF R.W.G., Crystal Structures, 2nd edn., Vol. 1, Interscience, New York (1963).
t
ZEMEL J.N., JENSEN J.D. and SCHOOLAR R.B., Phys. Rev. 140, A330 (1965).
~ §
RIEDL H.R., DIXON J.R. and SCHOOLAR R.B., Phys. Rev. 162, 692 (1967). GOLDAK J., BARRETT C.S., INNES D. and YOUDELIS W., J. Chem. Phys. 44,3323 (1966).
~‘
TSU R., HOWARD W.E. and ESAKI L., Phys. Rev. 172, 779 (1968).
**
KANNEWARF C.R. and CASHMAN R.J., J. Phys. Qiem. Solids. 22,293 (1970). Preliminary values, GRUTTOLA V. DE, Private communication.
tt
Predicted by the scaling of equation (3).
14
NANNEYC.,Phys. Rev. 129,109(1963). BOYLE W.S. and BRAILSFORD A.D., Phys. Rev. 120, 1943 (1960).
§§
A consequence of this is that about twice as much charge is transferred in PbS as in GeTe, GeTe (and also SnTe) with low fe’s thus have a small charge transfer component to stabilize what would be an
ndf dn
f,
£2l~~3/2a*
*
•
f
a(A)
ft
unstable structure if it were only covalently bonded. The distortion to f.c.r. from f.c.c. appears to correlate with low fonicity. Additionally, it appears that the distortion to f.c.r. from f.c.c. does not change the
xvi
TRENDS IN THE IONICITY IN THE AVERAGE VALENCE V MATERIALS
charge transfer by a significant amount. It is known that a few % GeTe in PbTe causes the alloy to transform to f.c.r.6 GeSe, which is orthorhombic, does have a larger charge transfer than GeTe, and therefore it appears that the-crystals gain more ionicity and stability by distorting to the orthohombic structure. We know of no IV—VI alloy system which undergoes a transition from f.c.c. to orthorhombic as a function of temperature. However, it appears that the PbSe and SnSe alloy system might. It also appears that
Vol. 12, No.6
lighter group VI elements increase the ionicity and hence the charge transfer but that the trends are not obvious for lighter group IV elements. Finally the values of CA$ and Cth may be estimated from the above scaling and should have values of approximately 65 and 90. The Bi value is consistent with the measured ~ of 100, the As value has not, to our knowledge, been measured.
REFERENCES 1.
PHILUPS J.C.,Phys. Rev. Lett. 22, 645 (1969).
2.
For a review, see PHILLIPS J.C., Rev. Mod. Phys. 42, 317 (1970).
3.
BRODSKY M.H. and STILES P.J.,Phys. Rev. Lert. 25, 798 (1970).
4.
VAN VECHTEN J.A.,Phys. Rev. 182, 891 (1969).
5.
MAZELSKY R., LUBELL M.S., KRAMER W.E.,J. Chem. Phys. 37,45 (1962).
Une échelle d’ionicité est donnée pour les matériaux a valence moyenne V. Cette échelle est obtenus par les méthodes de la théorie spéctroscopique de Phillips.