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STATISTICAL EXCHANGE AND THE HEISENBERG EXCHANGE INTEGRAL
J. Slater
To cite this version:
J. Slater. STATISTICAL EXCHANGE AND THE HEISENBERG EXCHANGE INTEGRAL. Jour- nal de Physique Colloques, 1972, 33 (C3), pp.C3-7-C3-11. �10.1051/jphyscol:1972302�. �jpa-00215036�
JOURNAL DE PHYSIQUE Colloque C3, supplkment au no 5-6, Tome 33, Mai-Juin 1972, page C3-7
STATISTICAL EXCHANGE
AND THE HEISENBERG EXCHANGE INTEGRAL
(*) J. C. SLATERQuantum Theory Project, University of Florida, Gainesville, Florida, USA
Abstract. - A method is proposed for using the statistical Xoc exchange approximation to the self-consistent field for studying magnetic excitations in crystals. The remarks are illustrated by the example of the ferromagnetism of EuS. In the ground state this crystal is known to be ferro- magnetic, in agreement with the energy-band calculations of Cho, each E u + ~ ion having a half- filled shell of 4 f electrons, all with parallel spin. From experiment, one can find the energy diffe- rence between this state, and a state with the magnetic moment of one Eutz ion reversed. This energy difference can be expressed in terms of the Heisenberg exchange integral J. A method is proposed for using existing computer programs to find this same energy difference by fundamental a priori methods, thus determining J. This new method is based on two recent developments.
One is the study of a so-called transition state, a state in which the electrons taking part in the transition are half in the initial state, half in the final state. In the present case, where the transition is from a state with the spin of a given ion up, to that with the spin down, the transition state is non-spin-polarized. A self-consistent solution is to be found for a crystal with one atom in this transition state. It has been shown that the energy difference between the initial and final states of such a transition can be given very accurately from differences of one-electron states computed for this transition state. To study these energy levels, one can use recently-developed methods for treating a crystal with an impurity atom. Use of these methods should be capable of leading even to the very small energy differences characteristic of the low Curie temperature (16.5 OK) known to hold for EuS. The method should be adaptable to many other similar problems.
The author and associates, in a long series of papers [I]
...
[22] have discussed the use of a statis- tical approximation to exchange in the study of the energy-band problem in solids, the nature of localized states, and the relation of these questions to problems in magnetism. Though these papers deal largely with the energy-band methods and the self-consistent field, it is well known that most of the really useful discussions of magnetic properties of crystals have been based on the use of the Heisenberg exchange method, and of localized magnetic excitations. There is a tremendous literature on this subject, much too long for references t o be given here. The weakness of this approach is that the value of the exchange integral met with the Heisenberg method is almost universally taken from experiment, rather than being derived from a fundamental microscopic theory of the behavior of the crystal. The object of the present paper is to point out that the techniques of the statistical exchange and the self-consistent field have now reached a point where we can expect t o make calculations of these exchange integrals from first principles, though such calculations have not yet been made.In a short discussion such as the present one, it is more practical t o take a specific example, and show how this would be carried through, rather
(*) Assisted by the National Science Foundation.
than to give a general discussion applicable to many varied applications. Accordingly we shall take the ferromagnetic semiconductor EuS, a case in which we have a good deal of information, and which is in principle particularly simple. This crystal has sodium chloride structure, and is formed from EU + and S-' ions. The Eu" ion has a half-filled shell of seven 4 f electrons, and its magnetic moment in the crystal corresponds t o the situation where all seven electrons have parallel spin, which we shall refer to as spin up.
Since all seven m , states are occupied by a single spin-up electron, there is no net orbital magnetic moment, and the multiplet corresponding t o the ground state of the isolated ion is 'S.
The energy bands of this crystal have been computed, using the augmented plane wave method, by Cho [23].
In this paper, Cho studies the suitable exchange to use, and from consideration of the relation between the observed and calculated energy bands, he arrives at values of the statistical exchange parameter a (see ref. [16], [22]), multiplying the original exchange suggested by the author in reference [2], which are not far from the currently preferred values. From the present point of view, the important fact about this spin-polarized calculation of Cho is that the highest occupied energy band is a narrow 4 f band corresponding t o spin up (ft), while the corresponding spin down band (fJ) lies about 6 eV, o r 0.45 Ry, higher, interacting with other excited bands to form
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1972302
a complicated conduction band. This separation of about 0.45 Ry between f and fj. bands corresponds, as the author has pointed out in reference [13], to the energy separation found in the spectrum of a Eu+' ion between its ground state configuration 'S, and a sort of weighted mean of the much higher multiplets of multiplicity 6, namely the 6P, 6D, 6F, 6G, 'H, and 61. It would require a very high tempe- rature, of the order of tens of thousands of degrees, to get any appreciable thermal excitation of these states.
We know, however, that EuS is ferromagnetic, with a Curie temperature of about 16.5 OK, corres- ponding to about 1.04 x Ry. It was pointed out in reference [13] that the only difference in the energy-band structure t o be expected in going through the Curie point would be very minor changes in the bands characteristic of sulfur. The reason is that the physical phenomenon responsible for the Curie point is related to superexchange. There is no direct interaction between the 4 f electrons in one Euf ion and those in a neighboring Euf
'.
The 4 f orbitals are too concentrated, the ions too far apart. However, the sulfur 3 p orbitals, as they overlap a E u + ~ ion, will find themselves in a location with different exchange potentials for spin up and spin down, and consequently the spin-up sulfur orbitals will have a slightly different energy from the spin-down orbitals.The effect of this slight differentiation between energy levels, and orbitals, corresponding to 3 pf and 3 pj.
sulfur electrons, will be transferred from one Eu*' ion to its neighbor, and will provide the machinery necessary to produce the superexchange coupling resulting in the ferromagnetic arrangement of the Euf ions in the crystal. As was pointed out in refe- rence [13], an antiferromagnetic arrangement would also be possible, and in fact is realized in EuTe. The energy difference between the ferromagnetic and antiferromagnetic arrangements would be of the order of magnitude of the Curie temperature, or Ry, and in reference 1131 it was indicated that at the time when that paper was written, the compu- tational techniques had not reached a point where this very small energy difference could be calculated.
Since 1968 our methods have improved, and in the remainder of the present paper it will be indicated how one could make an a priori calculation of these energy differences, using methods which are now available.
The Heisenberg exchange method, operating between nearest neighboring E u f 2 ions, is known to be rather accurately applicable t o the EuS crystal. This has been studied particularly by McCollum and Callaway [24], 1251. These authors have measured the low- temperature specific heat (0-4 OK) of EuS, in a range in which the specific heat is expected to arise almost entirely from spin-wave excitation. They have shown that the interaction energy
- 2 JZ(i, j ) S,. Sj (1)
of the Heisenberg-Dirac-Van Vleck type between the spins Si and Sj on nearest-neighboring E u f 2 ions gives a good account of the specific heat, where J is about 1.48 x eV = 1.09 x Ry. The exchange integral J of course is not an exchange integral in the usual sense, but must arise from the superexchange effect, through the agency of the sulfur orbitals, which we have mentioned above. Let us now inquire whether there is any chance of comput- ing J from self-consistent calculations using a statis- tical exchange.
The thing which we are now able to do, and which had not been developed sufficiently in 1968 when reference [13] was published, is to find easily the energies and orbitals corresponding to a localized excitation in a crystal. We can apply these new tech- niques in the following way to the present problem.
Let us ask what would be the energy difference between a ferromagnetic EuS crystal in its ground state, and in a state in which the magnetic moment of a single E u f 2 ion is reversed, so as t o point opposite to the orientation of all other ions. This energy difference on the one hand can be directly computed from the techniques mentioned above, which we shall describe shortly. On the other hand, since we have just men- tioned that the Heisenberg Hamiltonian works well in this case, and since we have just quoted a value of the integral J found experimentally, we can find the energy difference between the ground state of the crystal and the state with one reversed ion, in terms of the experimental J. Thus we can intercompare these two values of J, and hence get this integral from the direct calculation. Let us first find the energy difference in terms of the experimentally determined J.
If two spins S1 and S, couple to give a resultant S, simple vector-coupling theory tells us that
Sl.S2 = S(S
+
1) - S, (S,+
1) - S,(S2+
1)2 (2)
If the two spins are equal in magnitude, as in this case, the largest value of S, .S2 which we can have is for S = Sl
+
S, = 2 S,, while the smallest value is for S = 0. Thus the difference in the quantities S,.S2 for the two cases isThe difference in the total energy given by eq. (I), when one ion reverses its spin, is
where z is the number of nearest neighbors of the reversed ion. Since we are considering only E U + ~ ions in this simple picture, we have z = 12, the number of nearest neighbors in the face-centered cubic lattice of Euf ions. The value of S, in the present case is 712. Hence we have
Next let us ask how to find this energy difference
STATISTICAL EXCHANGE AND THE HEISENBERG EXCHANGE INTEGRAL C3-9 from fundamental calculation. In the Xa method,
outlined in references [16], [22], there is an exchange potential energy in the one-electron equation of the self-consistent field equal to
acting on an electron of spin up, with a corresponding formula for spin down. Thus an electron of spin up has a lower potential energy in a region where there is an excess of spin-up charge density than a spin- down electron, and vice versa. This effect is very important inside the spin-polarized Euf ions, resul- ting in a 4 f energy level which is about 0.45 Ry lower for spin-up electrons (assuming the E u + ~ ions to have spin-up) than the corresponding spin-down level.
1f now one E u + ~ ion is reversed, and has its spin pointing down, an electron of spin up will have a much higher potential energy inside this ion than in the other ions. While the energy levels of an electron are split into energy bands in the crystal, and the wave functions extend throughout the crystal, this reversed E u f 2 ion will act like an impurity atom, and will have localized wave functions, and quite different energy levels from the rest of the crystal.
The spin-down 4 f levels inside this special atom will lie almost precisely at the same energy as the spin-up 4 f levels in the rest of the crystal, and vice versa. At the same time, the wave functions and energy levels of the sulfur 3 p orbitals will be modified as compared to the perfect crystal, since a 3 p orbital which extends from a E u * ~ ion with spin-up to a EuC2 with spin down will find quite different exchange potentials at these two neighboring sites, and as a result it will be spin-polarized, tending to extend more to one ion than to the other. But, as pointed out in reference [13], it would be very hard to make a calculation of the energy difference between the crystal with this one reversed Euf ion, and a perfect crystal.
This energy difference, fortunately, can be found comparatively easily from a device proposed in references [20] and [22], namely a transition state.
It was shown that in any electronic transition, it is useful to solve the self-consistent problem for a state half way between the initial and the final states, which was called a transition state. This transition state is one in which the electron having a transition is half in the initial state, half in the final state. Then it was shown that the energy-difference between the one-electron energies of an electron in the initial and final energy levels of this transition state, as calculated by the Xa method, gives a very accurate value of the actual energy difference between the initial and final states, taking account very statis- factorily of the modification or relaxation of the charge distribution of all orbitals of the system in going from the initial to the final state. The
computer programs for solving self-consistent field problems can find one-electron energies and their differences very easily, whereas in many cases it is very difficult t o get values of the total energy of the system. This is particularly true in the case of a crystal with a localized perturbation, where we should really find the energy of the whole crystal in the initial and final states, enormously large quantities, and take their ditlerence. When we use the transition state concept, on the contrary, it is only the one-electron energy of the localized state that must be used.
In our case, the initial state of the Eu" ion which is having a transition has seven 4 f electrons with spin up, while the final state has seven with spin down.
In the transition state, we have 712 electrons with spin up, 712 with spin down, so that the localized E u f 2 ion is in a non-spin-polarized state. As far as the 4 f energy levels of this ion are concerned, the only reason why there should be any difference between spin-up and spin-down energy levels is that these 4 f orbitals slightly overlap the 3 p orbitals of the six neighboring sulfur ions. These sulfur 3 p orbitals in turn are somewhat spin-polarized on account of their overlapping with the other E u + ~ ions, all of which are highly spin-polarized. Thus there wid be a very slight difference between the spin-up an11 spin-down energy levels of the spin-orbitals of the E u + ~ ion which is in its non-spin-polarized transition state. These energy differences would be expected to be about the same for each of the m, values of the 4 f state, from rn, = 3 to - 3.
In the use of the transition state, we are then directed to find the difference of one-electron energies between the initial and final spin-orbitals, but calculated for the self-consistent field found in the transition state.
Thus this small energy difference between spin-up and spin-down is what we are to use, only here each of the seven electrons will have a transition from the spin-up to the spin-down state. Thus the total energy difference between the crystal in its ground state, and that with the magnetic moment of one ion reversed, will be
AE = 7 x one-electron energy difference between 4 f spin-up and spin-down levels in transition
state
.
(6)By comparison of this result with eq. (4), we see that One-electron energy difference between 4 f spin-up
and spin-down levels, experimental,
By coincidence this energy difference is almost exactly equal to the Curie temperature, expressed in Rydbergs. This is reasonable. Thermal excitation at the Curie temperature would raise Euf ions in large numbers from the spin-orientation characteristic of the ferromagnetic ground state, to the reversed orientation, and cooperative effects between different ions would lead to a disappearance of the magne-
C3-10 J. C . SLATER tization, a t approximately this temperature. We
shall not go further into the statistical aspects of the problem. Our task, rather, is to study the tran- sition state, and to try to calculate from fundamental theory this very small energy difference of about 1.04 x Ry between spin-up and spin-down states. It is instructive, as stressed in reference [13], to contrast this very small energy difference with the energy difference of about 0.45 Ry between spin-up and spin-down states met in the direct application of the energy-band theory.
Great progress has been made recently in the appli- cation of the Xa statistical exchange, and scattered- wave methods, to the study of the energy levels of clusters of atoms or ions surrounding an impurity atom. This progress has been made by K. H. Johnson, F. C. Smith, Sr., and J. W. D. Connolly [26]
...
[34].This development replaces the earlier work of Koster and Slater [5], [6] and is based on the same foundations as the methods currently being used for studying perfect crystals. Unlike the method of Koster and Slater, it does not demand finding the Wannier func- tions and carrying out difficult calculations with them. The successful application of this method to the calculation of the molecular orbitals and one- electron energies is clear from unpublished calcula- tions of Johnson and Smith at MIT. It is particularly useful for studying a central ion surrounded by a shell of nearest neighbors, though extensions to second nearest neighbors as well are now being made.
Let us now sketch the way in which this method could be adapted for making the calculations required in our discussion of EuS. First one could study a central S-2 ion surrounded by its six nearest neighbor Euf2's, of which five would be spin-polarized with spin up, and the remaining one would be non-spin- polarized. The emphasis would be placed on finding accurate spin-polarized molecular orbitals for the S-2 ion. Next one could study a central non-spin- polarized E u f 2 ion, surrounded by six S-2 ions, each modified as was found from the previous calcula- tion. Presumably one would find the spin-up 3 p orbitals of the S-2 ions extending very slightly closer to the non-spin-polarized Eu" ions than for the spin-down 3 p's. This would result in a very slight spin-polarization of the Euf 4 f spin-orbitals. One would hope that the difference of one-electron energies, between these spin-up and spin-down 4 f orbitals, would be of the order of the quantity 1.04 x Ry suggested by the experimental discussion. This is a very small energy, but it is just on the edge of what is now being attained in the use of these programs.
It should not be impractical to improve the accuracy of the methods enough so as to get a useful approxi- mation to these very small energy differences.
In the absence of any calculations of this type which have yet been made, it is not worth while trying to go
further with this rather speculative discussion. The point is that these are straightforward calculations, using computer programs already in existence, and requiring only slight increase of accuracy to carry them out. If they can really be carried through, it will emphasize the value of the type of accurate numerical calculations which are being made at the University of Florida, MIT, and other institutions.
It is obvious that without such really accurate work, the calculation of these small magnetic quantities will continue to be the rather profitless occupation which it has been in the past, when very few quanti- tative results dealing with magnetism, based on first principles, have been possible.
It should be made clear that the example we have used in this discussion is only an illustration of what can be done. A good many calculations, listed in bibliographies particularly in references 1201, [21]
and [22], have shown that the energy-band method is capable of leading to correct descriptions of the ground states of ferromagnetic and antiferromagnetic crystals, leading even to the explanation of the compli- cated helimagnetic properties of many of the rare earth elements. Methods similar to those outlined in the present paper should be adaptable to finding the low-energy excitations of all of these types of materials. The cases are somewhat more complicated to analyze than the case of EuS which we have taken up here, but in some cases they would be easier to carry out, since in many cases one would be dealing with higher Curie or Nee1 temperatures than the quite low value found in EuS. Thus the required energy accuracy would not be as great. In every case the fundamental situation would be the same : one would wish to investigate an excited state in which one or more magnetic atoms or ions had a reversed magnetic moment, and would use the transition-state method to render this problem tractable. One would analyze the problem on the one hand by the funda- mental methods suggested here, on the other hand by Heisenberg exchange methods, and would fit the Heisenberg exchange integral to the result of the a priori calculation, regarding it as a parameter to be determined in this way. Even in cases where the Heisenberg method is less applicable than it is in EuS, the fundamental calculations would still be possible, and one can look forward to a new possibility of getting quantitative calculations, from first principles, for many magnetic problems which have heretofore been impossible to treat with any accuracy.
Acknowledgments. - The author wishes to ack- nowledge assistance of the National Science Founda- tion, and to thank his associates Dr J. B. Conklin, Sr., J. W. D. Connolly, K. H. Johnson, Karlheinz Schwarz, and J. H. Wood, for valuable discussions concerning the methods proposed here.
STATISTICAL EXCHANGE AND THE HEISENBERG EXCHANGE INTEGRAL C3-11
References
[I] SLATER (J. C.), Phys. Rev., 1949, 76, 1592.
[2] SLATER (J. C.), Phys. Rev., 1951, 81, 385, 1951, 82, 538.
[3] SLATER (J. C.), Rev. Mod. Phys., 1953, 25, 199.
[4] SLATER (J. C.) and KOSTER (G. F.), Phys. Rev., 1954, 94. 1498.
[5] K O S T E ~ (G. F.) and SLATER (J. C.), Phys. Rev., 1954, 95. 1167.
[6] KOSTER (G. F.) and SLATER (J. C.), Phys. Rev., 1954, 96, 1208.
[7] SLATER (J. C.), The Electronic Structure of Solids, from Encyclopedia of Physics, Vol. XIX, Sprin- ger-Verlag, Berlin, 1956.
[8] SLATER (J. C.), from Advances in Quantum Chemistry, Vol. 1, Academic Press, Inc., New York, 1964.
[9] SLATER (J. C.), Internat. J . Quantum Chem., 1967, 1, 37.
[lo] SLATER (5. C.), Internat. J. Quantum Chem., 1967, IS, 523.
[ l l ] SLATER (J. C.), Phys. Rev., 1968, 165, 655.
1121 SLATER (J. C.), Phys. Rev., 1968, 165, 658.
[13] SLATER (J. C.), J . Appl. Phys., 1968, 39, 761.
[14] SLATER (J. C.), from Methods in Computational Physics, Vol. 8, Academic Press, Tnc., New York, 1968.
[15] SLATER (J. C.), from Energy Bands in Metals and Alloys, Gordon and Breach, New York, 1968.
[16] SLATER (J. C.), WILSON (T. M.) and WOOD (J. H.), Phys. Rev., 1969, 179, 28.
[17] SLATER (J. C.), MANN (J. B.), WILSON (T. M.) and WOOD (J. H.), Phys. Rev., 1969, 184, 672.
[18] SLATER (J. C.), Internat. J. Quantum Chem., 1970, IIIS, 727.
1191 WILSON (T. M.), WOOD (J. H.) and SLATER (J. C.), Phys. Rev., 1970, A 2, 620.
[20] SLATER (J. C.), from Computational Methods in Band Theory, Plenum Press, 1971, 447.
[21] SLATER (J. C.) and WOOD (J. H.), Internat. J. Quan- tum Chem., 1971, 4S, 3.
[22] SLATER (J. C.), from Advances in Quantum Che- mistry, Vol. 6, Academic Press, Inc., New York (in press).
[23] CHO (S. J.), Phys. Rev., 1970, B 1 , 4589. See also Phys. Rev., 1967, 157, 632.
[24] MCCOLLUM, Jr (D. C.) and CALLAWAY (J.), Phys.
Rev. Letters, 1962, 9 , 376.
[25] CALLAWAY (J.) and MCCOLLUM (D. C.), Phys. Rev., 1963, 130, 1741.
[26] JOHNSON (K. H.), J. Chem. Phys., 1966, 45, 3085.
[27] JOHNSON (K. H.), Internat. J . Quantum Chem., 1967. IS. 361.
[28] JOHNSON'(K.'H.), Internat. J. Quantum Chem., 1968, 2 s . 233.
[29] JOHNSON (K. H.), Phys. Letters, 1968, 27A, 138.
[30] JOHNSON (K. H.) and CONNOLLY (J. W. D.), Phys.
Letters, 1968, 28A, 291.
[31] JOHNSON (K. H.) and SMITH, Jr (F. C.), Phys. Rev.
Letters, 1970, 24, 139.
[32] JOHNSON (K. H.), Internat. J. Quantum Chem., 1971, 4S, 153.
[33] JOHNSON (K. H.) and SMITH, Jr (F. C.), from Compu- tational Methods in Band Theory, Plenum Press, New York, 1971, 377.
[34] SMITH, Jr (I?. C.), and JOHNSON (K. H.), Phys. Rev.
Letters, 1969, 22, 1168.
