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On the stability of non-stoichiometric compounds
LETTERS On the stability of non-stoichiometric
TO
THE
compounds
EDITOR
an impurity atom that may be either foreign to the compound or else a constituent atom that would not normally occupy that position. The electrons of this impurity atom will probably lie in an impurity level somewhere between the top of the valence band and the bottom of the conduction band. The presence of the foreign atom in the site may relieve, to some degree, the local strain in the structure caused by the vacancy, and thus make this impure compound the one of lower energy. An example of a non-stoichiomctric cornpou~~d whose stability can be explained by the above reasoning is the mineral stromeyerite, AgCuS. If stoichiometric, stromeyeritc would have eight valence electrons for every three atoms, or an t u ratio of 2.66. Its principal Brillouin zone, dctermined from three strong X-ray reflections(112), (130) and (004) -has an e,‘czcapacity of only 2.63.(s) The next larger zone, with a capacity of 2.69, is determined from the (1 12) and (130) reflections alone. As the (004) is the most intense X-ray reflection, it seems likely that them is :m effective energy gap between the two zones, and that the principal zone, which determines the valence band, will be filled to capacity and the excess electrons of the stoichiomctric compound must be accommodated in the conduction hatid at a higher level. Esperimcntal evidence indicates that AgCuS does not exist as a stable stoichiometric compound. SCH\VARTZ(~) and SUIIR(‘) found from polished sections and from X-ray diffraction data that either metallic silver or AgsCuSs appeared with stromeyerite when equal atomic proportions of the elements or the sulfides were sintered together. SUHR concluded that stromeyerite is either silverdeficient or else has a structure in which some copper atoms occupy silver positions. lJlore recently, DJURLE determined that the atomic proportions stable compound has _4ge.&ut.&.(5) From the structure@) it is evident that the copper positions are quite distinct from the silver positions, both in co-ordination number and in bond lengths. Therefore, the copper can definitely be considered as an impurity when it proxies for silver in the structure. It is therefore proposed that this non-stoichiometric composition is the stable one; because when the impurity
(Received 21 April 1959) IN a recent letter to this journal, F. A. KRUGER pointed out some of the electrical effects of nonstoichiometry.(r) In considering crystalline compounds where one of the components is present in excess of the concentration required by the simple several mechanisms for stoichiometric ratio, accomodating the excess atoms were described. The electrical consequences of these mechanisms were discussed, with the tacit assumption that the compounds, when stoichiometric, are ideal semiconductors with the exact electron-to-atom ratio required to fill their valence bands. This assumption is undoubtedly correct for the majority of semiconductors that have been investigated, and is easily sern in the AB compounds with the ZnS structure and with an electron-to-atom ratio of 4:l. Attention should be called to compounds that, if stoichiometric, have electrons slightly in excess of the capacity of the valence band. These excess electrons must be accommodated in the conduction band. At temperatures below the intrinsic range, a plot of resistivity versus l/T would be expected to be similar to that of a normal metal or an extrinsic semiconductor with activated donor electrons. 1Vhcn the intrinsic range is reached, the slope of the curve will reverse because of the increase in the number of carriers, and the material then behaves like a typical intrinsic semiconductor. If, however, the energy gap between the valence band and the conduction band is greater than the energy involved in forming a vacancy (Schottky defect), the stoichiometric compound may not be stable; for more energy will be involved in lifting the electron over the energy gap than in creating a vacancy. The latter can be looked upon as substituting an atom that does not contribute any electrons in a site normally occupied by one that does, thereby lowering the electron-to-atom ratio of the compound. If a sufficient number of vacancies is formed, the e/a ratio may be lowered to exactly the capacity of the valence band, and it will no longer be necessary to accommodate any electrons at a higher energy level. It is also possible that these sites are occupied by 334
LETTERS
TO
copper atoms, whose electrons lie in an impurity level below the conduction band, are substituted for the silver atoms, less energy is involved than the energy necessary to lift the electrons of the silver atoms from the valence band to the conduction band. With the purpose of confirming the above proposals, the band gap energy and electrical properties of stromeyerite are being investigated. This work is supported in part by the United States Air Force on Contract A.F. 61(052)-178. Geological Museum University of Oslo, Norway
THE
335
EDITOR
opposite faces of the cube shown in Fig. 1. The calculation of the dipole anisotropy for these sublattices was facilitated by a table for the dipole fields at the 118 grid points of a simple cubic
A. J. FRUEH, Jr.*
References 1. KRUGER F. A., J. Phys. Chem. Solids 7, 277 (1958). 2. FRUEH A. J., 2. Krist. 106, 299 (1955). 3. SCHWARTZG. M., Econ. Geol. 30, 128 (1935). 4. SUHR N., Econ. Geol. 50, 347 (1955). 5. DJUFUE S., Acta Chem, Stand. 12.1427 (1958).
Magnetic-dipole-induced normal mode gadolinium iron garnet (Received
30 March
in
1959)
THE
magnetic properties of garnets can be calculated by the NCel model in terms of intersublattice and intra-sublattice exchange fields.(l) Because of the vector relation MX M = 0, the resonant frequency solutions of the dynamic equations of motion between the sub-lattices depend only on the inter-sublattice exchange fields. To obtain by resonance methods any information about the intra-sublattice exchange fields, one must be able in some way to subdivide a sublattice into distinguishable parts. It is the purpose of this note to point out that the magnetic-dipole anisotropy allows such a distinguishable decomposition. Fig. 1 represents lj8 of the garnet unit cube with all the oxygen removed. A particular Q (or dt) sublattice is defined as all the c (or d) sites contained in parallel planes one-half a unit cell spacing apart and perpendicular to one of the three crystal axes. Two of the parallel planes will contain one pair of * Present address : Department of Geological Sciences, McGill University, Montreal, P.Q., Canada.
FIG. 1. The figure shown is l/S of a garnet unit cube with the oxygen missing.
lattice,
kindly furnished to the author by L. R. of the Bell Telephone Laboratories. The condition under which it might be possible to excite this dipole-induced mode is obviously one in which the exchange forces which act on the sublattice are not overwhelmingly larger than the dipole anisotropy fields. A favorable material for the observation of this resonance mode, then, is gadolinium iron garnet, where the c sites are relatively weakly coupled to the a+d sites and the c( and cj coupling is even weaker.(r) The resonant frequency for the c sub-lattice mode is WALKER,
HE,,pd) means the exchange field seen at the ct sites arising from all the a+d sites and similarly for H,,,. The values for these fields as a function of temperature are given graphically in reference (1). Over most of the temperature range o/y N 4x 105 G. The value of the susceptibility ~)tatet~~;a~;uency is found to be given roughly y x w (Hdd2Ma+d _ where
1o_3
H&WY
-
TO
THE
compounds
EDITOR
an impurity atom that may be either foreign to the compound or else a constituent atom that would not normally occupy that position. The electrons of this impurity atom will probably lie in an impurity level somewhere between the top of the valence band and the bottom of the conduction band. The presence of the foreign atom in the site may relieve, to some degree, the local strain in the structure caused by the vacancy, and thus make this impure compound the one of lower energy. An example of a non-stoichiomctric cornpou~~d whose stability can be explained by the above reasoning is the mineral stromeyerite, AgCuS. If stoichiometric, stromeyeritc would have eight valence electrons for every three atoms, or an t u ratio of 2.66. Its principal Brillouin zone, dctermined from three strong X-ray reflections(112), (130) and (004) -has an e,‘czcapacity of only 2.63.(s) The next larger zone, with a capacity of 2.69, is determined from the (1 12) and (130) reflections alone. As the (004) is the most intense X-ray reflection, it seems likely that them is :m effective energy gap between the two zones, and that the principal zone, which determines the valence band, will be filled to capacity and the excess electrons of the stoichiomctric compound must be accommodated in the conduction hatid at a higher level. Esperimcntal evidence indicates that AgCuS does not exist as a stable stoichiometric compound. SCH\VARTZ(~) and SUIIR(‘) found from polished sections and from X-ray diffraction data that either metallic silver or AgsCuSs appeared with stromeyerite when equal atomic proportions of the elements or the sulfides were sintered together. SUHR concluded that stromeyerite is either silverdeficient or else has a structure in which some copper atoms occupy silver positions. lJlore recently, DJURLE determined that the atomic proportions stable compound has _4ge.&ut.&.(5) From the structure@) it is evident that the copper positions are quite distinct from the silver positions, both in co-ordination number and in bond lengths. Therefore, the copper can definitely be considered as an impurity when it proxies for silver in the structure. It is therefore proposed that this non-stoichiometric composition is the stable one; because when the impurity
(Received 21 April 1959) IN a recent letter to this journal, F. A. KRUGER pointed out some of the electrical effects of nonstoichiometry.(r) In considering crystalline compounds where one of the components is present in excess of the concentration required by the simple several mechanisms for stoichiometric ratio, accomodating the excess atoms were described. The electrical consequences of these mechanisms were discussed, with the tacit assumption that the compounds, when stoichiometric, are ideal semiconductors with the exact electron-to-atom ratio required to fill their valence bands. This assumption is undoubtedly correct for the majority of semiconductors that have been investigated, and is easily sern in the AB compounds with the ZnS structure and with an electron-to-atom ratio of 4:l. Attention should be called to compounds that, if stoichiometric, have electrons slightly in excess of the capacity of the valence band. These excess electrons must be accommodated in the conduction band. At temperatures below the intrinsic range, a plot of resistivity versus l/T would be expected to be similar to that of a normal metal or an extrinsic semiconductor with activated donor electrons. 1Vhcn the intrinsic range is reached, the slope of the curve will reverse because of the increase in the number of carriers, and the material then behaves like a typical intrinsic semiconductor. If, however, the energy gap between the valence band and the conduction band is greater than the energy involved in forming a vacancy (Schottky defect), the stoichiometric compound may not be stable; for more energy will be involved in lifting the electron over the energy gap than in creating a vacancy. The latter can be looked upon as substituting an atom that does not contribute any electrons in a site normally occupied by one that does, thereby lowering the electron-to-atom ratio of the compound. If a sufficient number of vacancies is formed, the e/a ratio may be lowered to exactly the capacity of the valence band, and it will no longer be necessary to accommodate any electrons at a higher energy level. It is also possible that these sites are occupied by 334
LETTERS
TO
copper atoms, whose electrons lie in an impurity level below the conduction band, are substituted for the silver atoms, less energy is involved than the energy necessary to lift the electrons of the silver atoms from the valence band to the conduction band. With the purpose of confirming the above proposals, the band gap energy and electrical properties of stromeyerite are being investigated. This work is supported in part by the United States Air Force on Contract A.F. 61(052)-178. Geological Museum University of Oslo, Norway
THE
335
EDITOR
opposite faces of the cube shown in Fig. 1. The calculation of the dipole anisotropy for these sublattices was facilitated by a table for the dipole fields at the 118 grid points of a simple cubic
A. J. FRUEH, Jr.*
References 1. KRUGER F. A., J. Phys. Chem. Solids 7, 277 (1958). 2. FRUEH A. J., 2. Krist. 106, 299 (1955). 3. SCHWARTZG. M., Econ. Geol. 30, 128 (1935). 4. SUHR N., Econ. Geol. 50, 347 (1955). 5. DJUFUE S., Acta Chem, Stand. 12.1427 (1958).
Magnetic-dipole-induced normal mode gadolinium iron garnet (Received
30 March
in
1959)
THE
magnetic properties of garnets can be calculated by the NCel model in terms of intersublattice and intra-sublattice exchange fields.(l) Because of the vector relation MX M = 0, the resonant frequency solutions of the dynamic equations of motion between the sub-lattices depend only on the inter-sublattice exchange fields. To obtain by resonance methods any information about the intra-sublattice exchange fields, one must be able in some way to subdivide a sublattice into distinguishable parts. It is the purpose of this note to point out that the magnetic-dipole anisotropy allows such a distinguishable decomposition. Fig. 1 represents lj8 of the garnet unit cube with all the oxygen removed. A particular Q (or dt) sublattice is defined as all the c (or d) sites contained in parallel planes one-half a unit cell spacing apart and perpendicular to one of the three crystal axes. Two of the parallel planes will contain one pair of * Present address : Department of Geological Sciences, McGill University, Montreal, P.Q., Canada.
FIG. 1. The figure shown is l/S of a garnet unit cube with the oxygen missing.
lattice,
kindly furnished to the author by L. R. of the Bell Telephone Laboratories. The condition under which it might be possible to excite this dipole-induced mode is obviously one in which the exchange forces which act on the sublattice are not overwhelmingly larger than the dipole anisotropy fields. A favorable material for the observation of this resonance mode, then, is gadolinium iron garnet, where the c sites are relatively weakly coupled to the a+d sites and the c( and cj coupling is even weaker.(r) The resonant frequency for the c sub-lattice mode is WALKER,
HE,,pd) means the exchange field seen at the ct sites arising from all the a+d sites and similarly for H,,,. The values for these fields as a function of temperature are given graphically in reference (1). Over most of the temperature range o/y N 4x 105 G. The value of the susceptibility ~)tatet~~;a~;uency is found to be given roughly y x w (Hdd2Ma+d _ where
1o_3
H&WY
-