FIVE - FOLD d- ELECTRON DEGENERACY IN THE HUBBARD MODEL OF TRANSITION METAL MAGNETISM Thesis for the Degree of Ph. D. MICHIGAN STATE UNIVERSITY EDWARD I. ‘SIEGEL 1970 J'HESIS 2552483Z IIIIIIllll‘lIlllHlIlIUlIlIIHIIIHlIIIII‘IIII‘HIIIIIIIII L 3 1293 00766 597 This is to certify that the thesis entitled FIVE-FOLD d-ELECTRON DEGENERACY IN THE HUBBARD MODEL OF TRANSITION METAL MAGNETISM presented by Edward J. Siegel has been accepted towards fulfillment of the requirements for Ph .D degree in METALLURGY / Major professor Date October 30.1970 0-7639 ABSTRACT FIVE-FOLD d-ELECTRON DEGENERACY IN THE HUBBARD MODEL OF TRANSITION METAL MAGNETISM BY Edward J. Siegel The purpose of this inquiry is to elucidate the basic mechanisms and criteria for the formation of long range order magnetic phases in the Hubbard Model of transition metals. we include intrasite d-d electron interactions exactly, rather than in a simplified picture used by Penn (Phys. Rev., Vol. 142, Number 2, February, 1966). In this work Penn utilized seband electrons in the tight binding approximation and solved the Heisenberg equation of motion for the energy of the system as a function of the direct coulomb coupling constant divided by the band energy. He plotted the total energy at T = O as a function of this ratio, and of the ratio of number of electrons to the total number of states in the system (the filling of the band). The result was a phase diagram for the various magnetic states, ferromagnetism, antiferromagnetism, and paramagnetism, in which Edward J. Siegel stability regions were chosen on the basis of minimization of the total energy as a function of the two parameters mentioned. we have repeated Penn's calculation, but with inclusion of the full five-fold degeneracy of the d-band that exists in real transition metals. This treatment leads to a similar phase diagram, but with a third axis, that of exchange coupling constant divided by band energy. The reason for this is that in a seband calculation, only an up spin and a down spin elec- tron may interact in each Wigner-Seitz cell due to the Pauli principle. With five-fold degenerate d-electrons, there is an exchange energy contribution to the total energy. We also try to extend this calculation to non-zero temperatures and compare the evolution of our phase diagram as T varies with the 3d, 4d, and 5d transition metal Curie and Neel points. Lastly, a com- parison with a tdmatrix calculation by Kemeny and Caron will be attempted, in which we hope to show Whether doing a many—body calculation on s-electrons or a one-body Hartree Fock calcula- tion of five-fold degenerate d-electrons gives a more realistic fit to the real transition metals, both of these calculations being done in the Hubbard Model. This should allow a conclusion as to how one should proceed to improve the Hubbard Model, with- out including s-d electron interactions and s-d and d-d screening. FIVE-FOLD d-ELECTRON DEGENERACY IN THE HUBBARD MODEL OF TRANSITION METAL MAGNETISM By w “d V Edward J? Siegel A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Metallurgy, Mechanics, and Materials Science 1970 6670273 ACKNOWLEDGMENTS I would like to express my sincere gratitude to Dr. Gabor Kemeny, my thesis advisor, whose patient tutelage and insight into both physics and how one should approach physics have benefited and inspired me greatly. I am also indebted to Dr. Donald Montgomery, my department chairman, for his guidance and hospitality during my stay at Michigan State University. Thanks are also in order to the other members of my committee, Dr. Frank Blatt, Dr. Michael Harrison, and Dr. Robert Little, for many stimulating discus- sions; and to two "unofficial members", Dr. William Hartmann and Dr. Laurent G. Caron, who so kindly gave of their time when requested. I must sincerely thank Mrs. Dorothy Vikstrom of the General Motors Tachnical Center, who donated her superb typing skills so generously. My parents, Mr. and Mrs. Sidney Siegel, have helped me so very much through the years, from early inspiration to continuous ii encouragement and financial support at the graduate level. My gratitude to them can never be sufficiently expressed. Last, but not least, I am sincerely indebted to my wife, Beverley, whose understanding and patience throughout the ups and downs of thesis research and writing was of great moral support, and is so very deeply appreciated. iii TABLE OF CONTENTS Page ACKNOWLEDGMENTS ii LIST OF TABLES v LIST OF FIGURES vi I. INTRODUCTION 1 II. NON-DEGENERATE MODEL AND RESULTS 23 III. THE TWO-FOLD AND FIVE-FOLD DEGENERATE PROBLEMS 73 IV. DISCUSSION AND CONCLUSIONS 181 BIBLIOGRAPHY 204 GENERAL REFERENCES 209 iv LIST OF TABLES Table Page 1. Magnetic Wave Function Coefficient Relations 38 . . . (2) 2. Number Operator Combinations in V Bloch 76 3. Decomposition of Number Operator Combinations 76 . . (2) 4. Gorkov Factorized Terms in V Bloch 78 5 V(2) in Terms of k Space Correlation ' Bloch -— Functions 85 6. Running FM-PM Program 158 7. Properties of Transition Metals 198 LIST OF FIGURES Figure l. s-Electron Tight Binding Band 2. s-Electron Density of States 3. AFM Energy Gap 4. AEM Energy Gap on Dispersion Relation 5a. Penn's Phase Diagram 5b. Comparison of Penn's and the t-matrix Phase Diagram 5c. Comparison of Penn's and the t-matrix Phase Diagram in Inverted Coordinate System 6. Itinerant Electron Contributions to FM Array 7. Itinerant Electron Contributions to PM Array 8. Itinerant Electron Contributions to AFM Array 9. One to One Mapping of k_and k_+ Q in Two- Dimensional Brillouin Zone Projection 10. One to One Mapping of k and k_+ Q in DiSpersion Relation 11. Many to One Mapping of k_and k_+ Q_in Two- Dimensional Brillouin Zone Projection 12. Many to One Mapping of k_and k.+ Q_in Dispersion Relation 13. Electron Dispersion Relation l4. Allowed Wave Vectors in k_Space 15. Fermi Factor Dependence on k, Including AFM Gap 16. Direct and Exchange Couplings in 2-, 3-, 4-, and S-Fold Degenerate Cases vi Page 24 24 69 69 7O 71 72 113 113 114 125 126 127 127 135 136 137 139 LIST OF FIGURES - continued Figure 17. FM-PM Flow Chart 18. AFM-PM Flow Chart (Any T, K/4T 4.3.5) 19. AFM-PM Flow Chart (Any T, K/4T 2. 3.5) 20. Variation in Chemical Potential as a Function of Temperature 21. Spread in Chemical Potential Versus Temperature as a Function of Band Filling 22. Variation of Chemical Potential Versus Tempera- ture for Various Band Fillings as a Function of the AFM Gap Parameter 23. Electron and Hole Parabolic Band Approximation Near k = 0 and TT/a Respectively 24. Parabolic Band Approximation Reexpressed in Terms of D = ka 25. AFM Gap Parameter Dependence on Temperature 26. AFM Gap Dispersion Relation 27. Jump in Chemical Potential at AFM Gap 28. Peak in Derivitive of Chemical Potential with Respect to Band Filling Versus Band Filling at Low TEmperature Around AFM Gap (n/2N = 0.5) 29. T = 1.1609°K. FM-PM Phase Boundary 30. Isometric View of Isothermal Phase Surface in (K/4T, J/4T, n/2N) Parameter Space 31. 'T = 300°K. FM-PM Phase Boundary 32. T = 1000°K. FM-PM Phase Boundary 33. Inverted T = 1.1609°K. FM-PM Phase Boundary vii Page 147 149 153 170 172 173 174 175 177 178 179 180 183 184 186 187 188 LIST OF FIGURES continued Figure 34. Inverted T = 300°K. FM—PM Phase Boundary 35. Inverted T = 1000°K. FM-PM Phase Boundary 36. J/4T = 0.0 FM-PM Phase Boundary 37. J/4T = 1/2 K/4T FM—PM Phase Boundary 38. J/4T = K/4T FM-PM Phase Boundary 39. Inverted J/4T = 0.0 (100) Plane. FM-PM Phase Boundary 40. Inverted J/4T = 1/2 K/4T (102) Plane. FM-PM Phase Boundary 41. Inverted J/4T = K/4T (101) Plane. FM-PM Phase Boundary 42. Slater-Pauling Curve for 3d Transition Metals viii Page 189 190 192 193 194 195 196 197 198 INTRODUCTION In this thesis, we shall investigate the relative stability of ferromagnetic, antiferromagnetic, and paramagnetic phases of a Hubbard Model of the transition metals by numerically calculating the free energies of these three phases. We take into account the full five fold degeneracy of the d electrons which con- tribute to condensed magnetic phases in the transition metals, thus including d-d exchange energy, and base our calculation technique upon the general formulation ofl, and the explicit s electron Hubbard Model calculation of2. We do not take into account static nor dynamic d-d interactions nor s-d interactions explicitly, nor local correlations within any one Wigner-Seitz cell, although the minimization of local polarity energy would tend to produce a fairly uniform d electron or d hole density. Thus, we suppress polarity fluctuations, which would reduce the energy. Following Penn, we plot a phase diagram for a few values of temperature of direct Coulomb coupling constant, exchange Coulomb coupling constant and band filling. Emphasis throughout is on the Bloch (5) space interpretation of condensed magnetic phase correlations between electrons as opposed to the Wannier (configuration) space interpretation of Hubbard. The theory of the magnetic properties of transition metals has been divided up into two parts, the ferromagnetic-paramagnetic stability question and the paramagnetic-antiferromagnetic sta— bility question. The ferromagnetic-antiferromagnetic transition is thus considered indirectly, through the intermediary of the paramagnetic state. In the ferromagnetic-paramagnetic question, two schools of thought have arisen; the localized spin Heisenberg model3 and the model of itinerant band theory of ferromagnetism4. For transition metals, experiments have shown that the d electron bands contribute to the possibility of large magnetic moments, and these d electrons are indeed in itinerant energy bands, and not localized5’6'7’8. On the theoretical sideg’lo, a detailed itinerant band model has been developed in which a partially filled narrow d band is used as a basis for calculating and correlating electronic specific heats, Curie temperatures, magnetic moments, etc. of ferromagnetic and nonferromagnetic 11, by a more exact calculation transition elements and alloys than the Hartree-Fock technique of the itinerant d electron advocates, who argue that the "free" electron gas could not be ferromagnetic at any density. He was not concerned with Bloch electrons in a periodic potential so that there was some doubt of the applicability of his "free electron gas correlation energy" correction to real transition metals. An alternative model for ferromagnetism utilizing polarity buildup on lattice 12,13 sites was latter proposed A compromise theory involving competition between bands with itinerancy energy and quasi- localized polarity energy was arrived atl4. In a coupled model for nickel, the d electrons were envisioned as distributed among atoms in states closely resembling d10 and d9 states of the free nickel atom. The holes (d9 configuration) migrate through the lattice, avoiding one another owing to their electrostatic repulsion, producing minimum polarity contributions at each lattice site, lowering the 15 energy of the system A major question in all of these theories was the effect of d band degeneracy certainly present in transition metals. This exchange contribution to the energy would be negative, enhancing the net magnetic susceptibility, 16. Whether this degener- and would even be present in an 3 band acy, or Hund's rule of intra-atomic coupling, as for example between more than one 3d hole per atom in iron cobalt and nickel, dominated the enhancement of magnetic susceptibility at tempera- tures less than the Curie temperature was open to questionl7. One necessity in answering these questions is a knowledge of the band structure of transition metals which has been reviewedls'19 20'21'22. In addition to Hund's rule, coupling versus exchange (d electron degeneracy) as a criterion, the possibilities of s-d and d-d screening to minimize polarity energy have to be con- 23’24. In addition to these realistic considerations. a sidered purely statistical mechanical model school, considering magnetic phase transitions as a symmetry breaking operation, just as other types of phase transitions, has been activezs. Clearly, all of these approaches may be partially valid and one must pursue one or the other. . . . . . . 26 . A fairly modern, semi-experimental rev1ew is available as is an 27 8 older one . Herring2 has reviewed the many and varied theories. The extent of polar fluctuations in d band metals has been 29 30,31,32 summarized , with the s-d question being considered The correlation idea has been taken up by a large number of authors33'34’35'36. The antiferromagnetic-paramagnetic question seems to be quite a bit harder to handle because there arises in the electron dis- persion relation a self consistent energy gap which must be calculated iteratively. The approach by way of spiral spin density37 has been thoroughly summarized38’39. The basic idea that an exchange induced splitting of the energy band resulting in a self-consistent gap resembling the superconductivity B.C.S. gap equation40, has been derived41 based on the earlier ideas of Slater42. For the half-filled d band, this would produce an antiferromagnetic insulator since the Fermi energy would coincide with the middle of the gap in the split d energy band43. This split d band model of antiferromagnetism is reviewed by Herring in great detail44. Herring points out that the usefulness of the Slater Hartree-Fock picture as developed by Slater45 and des Cloizeaux46 is tempered with the need for inclusion of a good deal of configuration interaction and polarity energy, which must be considered separately for different metals as their filling of the d band at a given temperature is different since their valences and ionic structures vary. Clearly, a general theory of antiferromagnetism must cut across the different atomic parameters of the various transition metals, lest we wind up with a completely separate theory for each metal. Des Cloizeaux showed that one could write Bloch functions as linear combina- tions of a kn, and a 54-9, 0 state, and that such linear combina- tion wave functions determined a gap A, the exchange field parameter, independent of 5. It is unclear what "exchange” means in this context since explicit five-fold d electron subband degeneracy was never included. His work is in agreement with 47 48 that of Matsubara and Yokota and of Kemeny and Caron in regard to the description of the antiferromagnetic state. The inclusion of exchange among electrons in narrow d bands is 49 for Bloch electrons in a periodic potential in which described semi-quantitative comparisons between the various transition metals are made for the negative exchange energy contribution. The Hubbard Modelso'51 was proposed to satisfy the need for a sbmple, itinerant band theory of magnetism and magnetic phase transitions at zero temperature. Hubbard worked in configuration space and developed the configuration space Green's function for the one electron density matrix, and hence, the expectation values of all one-electron operators, the relevant quantities in his mOdel. He did this only because he used the Gorkov factoriza- tion technique52 to renormalize the constants in the potential energy part of the system Hamiltonian and express its two-body interaction, four operator product as an equivalent kinetic energy two operator product, making it identical in operator form to the kinetic band term in his Hamiltonian. He found that the poles of the Green's function play the role of quasi-particle energies, and that the Fermi surface volume was altered. Since the electron-electron interaction strength had been proven to be unchanged by electron interactions to all orders of perturbation theory53, this lead Hubbard to conclude that as it was increased, the electrons could undergo a phase change leading to a super- lattice antiferromagnetic order or ferromagnetic order54. Herring pointed out that Hubbard's conclusion destroys the implicit assumptions of his model, that of equal electron and spin densities on each lattice site,since an antiferromagnetic or ferrOmagnetic does not possess this property. Nevertheless, the Hubbard model is one of the best general methods of attacking the magnetic phase transition problem, and yields to an analysis that could never be applied to the real transition metals with any hope of success. A shift of thinking of major importance occurred between Hubbard's work and that of des Cloizeaux, Matsubara, Yokota, Caron, Kemeny. Hubbard worked out his correlations completely in configuration space with Wannier functions for the electrons at each lattice site so that his Green’s functions were configuration space expectation values. The latter authors all paired 5 states and completely worked in a Bloch space with Bloch functions for each electron state 5. They thus made the description of the antiferromagnetic state quite transparent and easy to deal with. The treatment of the antiferromagnetic-paramagnetic question and the ferromagnetic-paramagnetic question, and thus indirectly, the ferromagnetic-antiferromagnetic question, was carried out simultaneously by Pennss. His work will be described extensively in the next chapter, but it is briefly summarized here to fit it into its historical perspective. Like des Cloizeaux, Matsubara, YOkoto, Caron, and Kemeny, he used a Bloch space description pairing states kg with k +.Q, o in a simple cubic lattice (Q = l/2§ = l/Zn) for the antiferromagnetic state. His basic Hamiltonian was that of Hubbard which he then Gorkov factorized to get an equivalent one-body form. This is thus a "Hartree- Fock" theory. Penn used correlation functions instead of Green's functions and calculated, for zero temperature, which magnetic phases would exist for various values of the interaction energy-band energy ratio and filling of the d band. This was done for the simple magnetic phases by direct energy comparison in the para- meter space of the phase diagram and by a susceptibility argu- ment56 for the more complicated ferrimagnetic and spiral spin density wave states. He arrived at a phase diagram for the three simple phases that illustrates a dominance of paramagnetism for an almost empty and an almost full d band. A dominance of antiferromagnetism for the half-filled band which disappears as the electron-electron interaction strength goes to zero, and disappears as the electron-electron interaction strength goes to infinity. The ferromagnetic state, which is sandwiched inbetween the paramagnetic and antiferromagnetic states, grows toward the half-filled band region57. A particularly illuminating general treatment of the theory of fermion phase transitions from a very general and unified point of view is the work of Mattuck and Johanssonsa. They use the spin correlation function S'(£T£') = to describe the correlation functions in configuration space of spins at sites 5 and 5' in a lattice. For the short range ordered para- magnetic state, they require S'(L£1£'I>O) ¢ 0 and lim S'(L£:£'|)=O and for the long range ordered ferromagnetic staEEEt 2; require 1im i'(1§1£'|) i 0. Their spin correlation functions in con- r-r' 4m figuration space are equivalent to our Bloch space correlation functions, to be introduced in the next chapter, by a simple Fourier transformation. An alternative way of viewing magnetic phases is by brdken symmetrysgxflz For the paramagnetic-ferro- magnetic phase transition, the broken symmetry is "rotation“ and the long range order parameter is "magnetization", as we shallsee in the next chapter describing Penn's definitions of magnetizations for the various magnetic states. For the paramagnetic-antiferromagnetic phase transition, the broken symmetries are "rotation" and "translation" and the long range order parameter is "S the amplitude of the ch Fourier com- .9 ponent of spin density." This concept is also used by Penn and will be delved into in greater length in the next chapter, since S corresponds to the exact Fourier transformation to <9 Bloch space we had mentioned before, being equivalent to our (a + swam When a source field is allowed to exist (a very weak magnetic ) type of terms we shall use throughout this work. field which breaks the 2N fold rotational degeneracy (symmetry) of a randomly initiated ferromagnetic state in the atomic limit), we may describe the long range order of the magnetic state in a simpler way than by the spin correlation function description. Mattuck and Johanssen define the average value of spin density itself <§> and write the relative magnetiza— tion (ratio of actual magnetization and magnetization) as (1.1) M.= <§>, N being the electron number. M is called the zho long range order parameter and is finite in the ferromag- netic state. A broken symmetry is needed to use M instead of 8' since if all directions of total spin were equally likely, 10 the above <§> = 0, so that M would be zero even in the ferro- magnetic phase. The spin correlation function S' is always nonzero in a ferromagnet since §(£)1§(£') is a scalar, and hence independent of total spin. Thus the spin correlation function is more generally utilized, but harder to apply. Actually one may calculate M from (1.2) M = lim g (Y '§|Y0v> where You is v40 N ovI the ground state for a given v where the external source field is vH. This is the quasi-average method of Bogoliubov. For the general spiral spin density wave antiferromagnetic state P reciprocal lattice vectors of the spiral: Zn/R, 4n/R . . ., (1.3) S = <‘YO(§)I/_\S_O I ‘I’o(_s_)> = $89!,ng where I QI is one of the and can include all harmonics of.g (whereas Overhauser's work61 included only perfect spirals, with lQl = 2n/R), and s is the direction of the spiral axis. The particular 9 dealt with in this thesis isig =-§ =1g, one half of each component of the reciprocal lattice vector, which is explicitly defined only when we consider a specific lattice structure (simple cubic). Mattuck and JOhannson further discuss the general difference between a first order and a second order phase transition. In the second order phase transition the long range order parameter (changes continuously from a nonmagnetic to a magnetic state (as in the paramagnetic-ferromagnetic phase transition) while in the first order phase transition, the long range order parameter changes discontinuously at the transition point. They emphasize 11 a very important result of Uhlenbeck, namely that in a finite system all dynamic variables (such as the long range order para- meter) Change continuously as a function of temperature, density, and interaction strength so that discontinuous changes of dynamic variables imply that only an infinitely large system can undergo a first order phase transition. A contact is made with Heisenberg's effective internal field concept in the following way: A long range effective internal field is defined (1.4) F = FSTATIC + FDYNAMIC where the static field is due to all the other particles of the system considered as stationary, and the dynamic field is due to the motion of the other particles of the system which correl- 1ates with the motion of the particular particle being considered. The long range order parameter is related to the internal field by a self-consistent equation: (1,5) 6 = 9(F(9)I where 6 is the long range order parameter and F is the internal self-consistent field. They further show that the condensed phases in a Fermi system may be described field theoretically by a matrix propagator whose off diagonal elements are directly related to the long range order parameter (our spin correlation function in Bloch space). Equation (1.5) indicates that the 12 rest of the system produces an internal field which extends throughout the system and ”produces" [or is "produced by", depending upon how one reads equation (1.5)] the condensed phase. It is of only limited range and "produces" (or is "produced by") the local, short range order. The field in a magnetic phase problem is called the spin aligning field or the spiral spin aligning field. It was introduced by Weiss in a theory of ferromagnetism, molecular field being a synonym for it. This molecular field acts somewhat like a magnetic field, but it is not a real magnetic field since the charged particle orbits are unbent by a Lorentz force. They further assume that (1.6) 6 = 9(CO, T, p) where Co is the interparticle interaction strength, 0 is the particle density (equivalent to our band filling in chapter 3), and T is the absolute temperature. Thus (1.7) 6(CO,T,p)=9(F ( 9(Co,T,p))) = e(F(co.T.p)) The self-consistency is seen clearly in (1.7). 8 must be found self-consistently, i.e. we assume a nonzero value for 9. find F from it, substitute into (1.5) to get a new value for 9. compare it with the initial 9 value, etc., until our iterative cycles start reproducing the same values of e and F on every cycle. This proceedure dominates our calculational flow charts 13 at the end of chapter 3. If the self-consistent values of 6 is nonzero, the system is indeed in a condensed (magnetic) phase. There are inherent dangers in such a technique which will concern us very deeply in this work. There is no way to determine which magnetic phases exist for a particular set of p, Co, and T values and the number of magnetic phases one can study must be limited to a finite number. At a given 0, Co, and T, the self-consistent equations for F and 9 possess more than one solution. 6 can be zero or nonzero for the same p, Co, and T values, so that a relative comparison of the free energies of the two phases must be used as a criterion for ascertaining which are the thermodynamically stable by virtue of a minimum free energy. We must make a guess, based on the experimental relevance of our analysis to the particular metals we desire to study, The transition metals. They then write a Hartree-Fock equation for wk: 2 .— (1.8) [- €711 V2 + V(_§)+A(cp1,...:CON):I Cpk (.E) " EJS—wlgg) where the wk's determine the trial wave function Yo by a Slater determinant: 1 (1.9) ‘IO = ;- cpNU.) . . . cpN(N) 14 Since a single Slater determinant best characterizes the anti- symmetry of one electron wave function, the single m's give symmetry to Yo, so that Yo has the symmetry and long range order 8 chosen. This is done by putting in the correct self- consistent A value appropriate to the particular phase studied, as A is an integral operator giving the potential felt by a particle in state wk due to its interaction with all the other particles; A is the self-consistent potential of the Weiss internal field. The long range order parameter can then be obtained from Yo as seen in (1.1) and (1.2), and hence from the m's. (1.8) and (1.9) have the same form as the ordinary Hartree- Fock theory, but here the trial wave function Yo has the (less than perfect) symmetry of the condensed (magnetic) phase built into it; the normal phase has perfect symmetry included in Y0. Thus in the condensed (magnetic) phase, operator A describes a long range internal field; in the normal phase it describes a short range field. The Hartree-Fock theory thus presented STATIC . DYNAMIC yields only the F , the F is due to correlations DYNAMIC between the particle motions. This F may possibly destroy the condensed (magnetic) phase. Since particles, propagators in the normal system, directly pro- portional to the correlation functions, are defined as (1.10) GPARAQS'O ,t'-t:)=—i<‘l’0 lII‘oP' {aISoG (t')a£'o (t)} Y0) 15 or A (1.11) G ‘£,G,t'-t)= -i where "+" denotes an up spin electron and "-" a down spin electron, in addition to the normal propagator (1.10) and (1.11). These anomalous propagators are zero in the normal (paramagnetic) phase. They arise from multiple scatterings from the internal spin aligning potential, which depopulates state |k,o> and repopulates Ik,o'>. The difference in pro- pagating spin up or spin down particles (with respect to the small, external source field) is given by: (1.15) AG = G(_]5+'k__'t'-t) - G(_k_’_lg+'t -t) This is zero in the normal (paramagnetic) phase since F = 0, but nonzero in the condensed (ferromagnetic) phase since F i O. In the condensed (antiferromagnetic) phase the characteristic anomalous propagator is: A (1.16) G = -i ANTIFERRO O 199' . 0 3+9. 0 I 0 This seemingly describes a particle picking up momentum 9:9' from scattering against the periodic structure of the spiral 17 internal field, and having its spin flipped from o to o' by the spin-aligning character of the internal field. What actually happens is neithera momentum pick up nor spin slip by the particle: YO is a linear combination of Slater determinants, so that the aa+ connect different Slater determinants, both of which occur in the linear combination with nonzero amplitude. In this paper, we will choose Q = 0,.9' redefined as.g and o = 0'; thus no spin flip takes place and only Bloch electrons with wave vector k are scattered off the spiral internal field. Mattuck and Johannson further generalize their work by defining a general anomalous propagator: (1.17) G (5',o',a'zh,o,a;t'-t) = -i or (1018) G (E'IU'IG.7£IUIa7t"t—)= -1 in configuration space, where a, a' = + or 1, so that the creation operator may or may not precede the annihilation opera- tor. Generalth is a function of 515' since exact translational invariance does not hold in many condensed phases, as in the antiferromagnetic state. The major power of this technique is that it applied to all sorts of "condensed" phases: ferromagnets, antiferromagnets, superconductors, and solids. These authors 18 then derive the long range order parameters from the propagators: For the ferromagnetic phase: 1 __ + + (1.19) M — N E (Y0I§5:+§Ef++éha‘ékt*lw0> and for the antiferromagnetic phase: + ak.-ékflQ.-IWO> _ + (1.20) SQ — 1}: alS'IO'aJSlOIYO In the ferromagnetic phase: (1.22) A = 6 6 .5'1030'15.o.a .E'Lhao',-o—5o',+ a,+ as an example. The general relation can be written directly in terms of the propagator as: (1.23) 9= )3 A I I I, G (Jilo'la';l<_'lnla7t'_t=6) .k',o',a' '5 ’0 ,a .k,o,a ANOMALOUS £1000: where D is a negative infinetesimal. Thus, in the normal phase when GANOMALOUS = 0, the long range order parameter, 6 = O; a normal phase can only have short range order at best. A detailed investigation of the mathematical structure of the condensed versus normal phase is given. As pointed out by them, the normal perturbation series for the normal propagator breaks down in the condensed phase since anomalous propagators cannot 19 be taken into account. The reason that this is so is that for the normal perturbation expansion of a propagator to be valid, the interacting groung state Y0> must have the same "structure" as the non-interacting ground state li0>; i.e. it must overlap §O> so that: (1.24) # 0 (nonorthogonality) Since at infinite volume the structure of the condensed phase differs markedly from that of the normal phase, we always have (1.25) E O (orthogonality) violating the criterion (1.24) for a normal perturbation expan- sion of the propagator to be valid. For example, in the condensed (ferromagnetic state in the Hartree-Fock approximation: (1.26) Y > = 1 , 1 ...l ; 1 _,1 _...1 _,000... I 0 1'51.+ k2’+ EMA 151, k k where M>P, i.e. there are more up spins than down spins. This is clearly orthogonal to the normal non-interacting state > 6o (for free electrons) which has M.5 P. Mattuck and Johannson construct a perturbation expansion of the anomalous propagator, valid for the condensed phase, by replacing each normal propagator by a matrix propagator. Its diagonal elements are normal propagators and its off diagonal elements are anomalous propagators. For the ferromagnetic phase: 20 ko+ k, + kp+ k, - (1.27) G(k,t'-t) = -i FERRO kl+ k: - ]_(I- If the interaction between electrons was switched off, this woul become: -[ur(€_k‘u)+15:]-l G0 02"”): '- FERRO (1.28) L o [un-(eh-'11)+i€>3§1mld the free electron propagators are on the diagonal and the anomalous propagators vanish: the system reverts to a normal (paramagnetic) state. The antiferromagnetic general spiral spin density wave state has an infinite matrix propagator since the 9 that an electron can absorb or emit to the spiral spin structure is unspecified and can be ng, n being any integer; and Q being unspecified itself: -' I.<\I OITOP’{a(t')a+ (t)}I‘I’O>1 d + + -<é5'+ ak'+: Gk: + flath. >° + + pali-I-Q,—> (8:49, 199:1“ ANTIFERRO . . . —' '—+ <"“k+251. 31;. +> In this work, we shall put Q = n/a = g/Z, as before, and n = 1, so that we reduce (1.29) to a 2 x 2 matrix propagator (for spins 21 up or spins down): + + (1.30) a”) Qc.t'-t) = -i + + ANTIFERRO (éktflaiéhoi><ébi94ia.Lt9.i> No spin flipping occurs and only one wavevector exchange of crystal momentum, Q, is allowed. Finally, Mattuck and JOhannson note that the long range order parameter is directly related to the source term in the Hamiltonian (i.e. to the source field Since F = -aHSOURCE/ aspace): variable _ 1 (1.31) e — CONSTANT x v For the ferromagnetic phase: (1.32) ERRO = vgBHxSx = external magnetic potential OURCE vgBHxi: (it -a_]_<_v +4612. +3351-) where 8 is the BOhr magneton number of an electron and g is the electron g-factor, and the v's in (1.31) and (1.32) cancel. For the spin density wave antiferromagnetic phase: S ODOW O IFERRO _ + + (1'33) HSMOIEIRCE " V95H§(ak_.+ak+g.++ak+Q.-als.-) where the constants are defined as before and the v's cancel. For our chose of.Q = n/a =.§/2 and n = l, we drop the r and get: .9 HANTIFERR0_ + + (1°34) SOURCE ’VgBHE(ak.+ak+Q.++ak~IQ.-als.-) 22 We have thoroughly emphasized the Mattuck and JOhannson approach since it seems to be a general, unified description of phase transitions, being based on Brout's work quoted earlier. In subsequent Chapters we will not utilize the propagator approach, but rather the correlation function approach, and all relations will differ from those of this Chapter by constants, i.e. factors of i. We will not delve into perturbation theoretic series calculation of propagators as we will be able to find them exactly by expressing them in terms of known coefficients in the condensed state wave func- tions. However, Mattuck and JOhannson's development is a general basis for all of the calculations that we shall under- take and should be kept in mind throughout. NON-DEGENERATE MODEL AND RESULTS Penn62 treated the Hubbard Model at zero temperature in solving for the phase stability of model transition metals by neglecting intersite interaction and treating the tight binding narrow d band. (2.1) 6k = -E'(cos kxa + cos kya + cos kza) as a tight binding 5 band in a simple cubic lattice. His treatment was not explicitly worked out, so that this chapter had to be derived by us. The effect of the non-inclusion of intersite interaction between electrons, the basic Simplicity of the Hubbard Model, is that one can treat the lattice of isolated Wigner-Seitz cells in configuration space as a geometric point lattice so that the electrons hop only from site to site. The region within the Wigner-Seitz cell surrounding the lattice point has no meaning in this problem, being suppressed by integrals extending over its volume which are associated with the lattice point, so that electrons can be found at the lattice point. The electron wave functions are thus modified by: (2.2) 6(_13j),‘§j = lattice sites Tight binding energy bands nearly accomplish this, and still allow a hopping term in the Hubbard Hamiltonian, the band being 23 24 .UcmmIOGHUCHm DSOHB :ouuooamum .H ouomflm .moumum mo.NMflmCOQ COHDOOHMIm .m Opsmfim Amy E ~.>.xx 2.3x am \uI 25 extremely narrow in energy. A simple cubic Brillouin zone is definable: (2.3) --§ <'ki <-§ , i = x,y,z with a being the nearest neighbor distance in the crystal. Figures 1 and 2 indicate the [111] direction energy band dis- persion relation in‘k space and the associated density of states. Penn correctly pointed out that these curves are not exactly equal to the band form and density of states of real transition metal d bands. There are basically three reasons for this nonequivalence. Firstly, there may be actually a different d band form owing to the non-tight binding of d electrons to the degree implied here; the d band tight binding form differs from that of the 5 band. This fault will not be corrected in this paper as it involves some use of empirical results for the dispersion relation and density of states, an approach we want to avoid. Secondly, Penn has not actually treated a d band, but only an 3 band. The crucial question of band degeneracy has been ignored as it is very difficult to include. We will correct this fault as much as possible, the extent of our being able to take into account all five degenerate states of each 5 26 state in the d band being governed by computational difficulties. We decide on this correction because it is of inherent physical interest in real transition metals. Thirdly, the crystal symmetry makes Figures 1 and 2 idealizations because it is not simple cubic. In Penn's non-degenerate treatment, the Pauli principle forces each site to have no more than two electrons on it, and those to be of antiparallel spin. As site-site interaction is neglected (except for the trivial hopping kinetic energy in the Hubbard Hamiltonian), no parallel spin interactions can exist. Thus the all important exchange energy is not included in the model and the resultant energies of the various magnetic phases considered are thus lacking in a large, negative interaction contribution. If this exchange energy were the same for all states, it would only shift the energy origin on Penn's final magnetic phase diagram uniformly for all states, making his results correct for real d bands as well as the d bands modeled as 3 bands that he actually used. All evidence would indicate that in states of different magnetic ordering there is no a priori reason why this exchange energy should be the same, and if so, it would only be fortuitous, since the number of parallel spins varies from state to state. 27 Starting with the Hubbard Hamiltonian in Wannier (configuration) space: wannier + n (2.4) HHubbard=12j30Tij Ci :7ch + J 01;” he transforms the operators into their equivalent in Bloch space (5 space) which operate on states of a definite k: (2.5) ci = l 24.15331 '0 N k —"’ + 1 ikoRi + (2.6) C. = -— 2 L-—-— a 1,n IN .5 .L,n where N is the number of sites. Thus, the Hubbard Hamiltonian becomes: 'k-Ri 1k-Rj (2.7) = Z Tij-YL1-- 2!. a H'Hubbard i'j'a ls amok 35,0 J + + 1 2E: Li(- +k- -k )_ +- . a ,a a - 1-2 -3 -4 N 01;,0.a33.a 34'" l$1""'32""N 151,132 123.15, Now .1 i(k +k -k- ) Ri _ _ _ (2.8) NEIL 1 —2 —3 —4 — 531*152153354) (2.9) ETijL1-'-BJ 2 (2(5) 3 T _, I (2.10) 1%- ): 1105 k )'R3 tag-5') 3 so that (2.11) 1°Ch =22e(_) +— H2 5L+k -k k-__) + ubbard ”km 1 —2 —3 —4 333,4. kl'TiE2 a]; -ak + ak - 3 4 4' _1' —2' 28 where + means a and - means 0'. Penn first transformed the Wannier space form of this Hamiltonian, which contains products of four creation and annihilation opera- tors (and is thus an explicit two-body Hamiltonian) into an effective quasi-particle Hamiltonian containing products of two of these operators multiplied by the expectation value of the other two . Interaction Bloch __ + + _- (2'12) HHubbard ’ C°LO §ct.a01.; + CL.O CL,O] Interaction This Govkov factorization S8 is of a Hartree form and H is now formally equivalent to a one—body kinetic energy for the quasi-particles. Co is a constant representing the particle- particle interaction strength. This model forces the Hamiltonian to be specified by two para- meters, the total particle number n and the quasi-particle- quasi-particle interaction strength Co (Measured relative to E“). Thus the phase diagram can be represented in the versus plane. Penn notes that actually Co and E“ should depend on the magnetic state of the system, but neglects it since including this dependence would be a hard computational problem. To do so, one would Choose a Y magnetic, compute Co and E” from it, use Co and E" to determine a Y magnetic at a certain line in 29 the Co/E"-n/2N plane, and then iteratively correct Co and E" to approach self-consistency. This would be an arduous task since the dependence of Co and E” upon Y magnetic could only be guessed at roughly. Thus, all Penn could do was to compute relative state energies and compare them, absolute cohesive energies of the system in various magnetic states being out of the question. In general, one can imagine an infinite number of magnetic states, and handling them all is a difficult task. Penn could only calculate the energy of a few of the magnetic states that he chose to consider, the rest being compared by a magnetic susceptibility calculation which we will avoid because the com- parison of energy (or free energy) slopes as a function of band filling uniquely defines state stability. Penn chooses six magnetic states defined by an operator Yk' a wave function Yk' and equivalently by their correlations, defined as the sum over- occupied 3 states of the expectation values of 2-operator products taken in the magnetic states of each electron with wave vector k: = + (2.13) Amn iCo § '- k k these wave functions combining to form Y magnetic by summing over all filled 3 states. 30 The other two types of definitions are generally, in operator form: (2 .14) Y]; = Blah.++BZal<_.-+B3ak+g.++B4als+flI' being composed of Bloch function operators, and in wave function form: 1 (2.15) 1k = Bl‘i’L,++Bz‘*'_]S,—+B3I’1_g+g.++B4YJs4g.- being composed of Bloch functions proper. Penn chooses g =IQ = g/Z ='fl, where Q is a reciprocal lattice vector equal to 2n/a (i, j or k) with a equal to unity being a unit lattice constant in configuration space, so that the magnetic state function component contributed by the electron in Bloch state k,+ is paired with a component from Bloch state 15,-, and k+Q,+ and E+Qw components also interfere generally. Clearly a simple group of states must be singled out for com- parison, and Penn Chooses: ferromagnetic E lFM > (the two that he chooses paramagnetic 5 12M > are equivalent under a antiferromagnetic E lAFM > coordinate axis rotation) ferrimagnetic E 1FIM > spiral spin density wave lsSDW > which are defined as: 31 (2.16) Yk = lPM > = Bl¥§J+ or BZYk _ (2.17) Y}- = 1PM > = 1311112,+ or 132115,- (2.18) YEFM= lAFM > = Biig,++33151g,+ or BZYlSI-IP‘IYEQI- (2.19) Tim: 1FIM > = 13113,+ +32Yk.-+83Yk&.++B4YL+Q:- (2.20) YSSDW= lSSDW>= BN1“ +B4Yr+gr°r Bzir.-+B3I’r+2.+ The correlation function language for describing these states involves the sums over matrix elements: (2.21) Ao+- = -Co E < a£'+ IE.“ > (2.22) Ao++ = Co E < a£’+ ab'+ > (2.23) AO__ = Co E < agfi a£'_ > (2.24) AQ+_ = -Co 12$ < flak aliri' > (2.25) AQ++ = C0 g < a£&’_ a£'_ > (2.26) 59-“ = C0“: < I£t9.+ a£’+ > where z is over the whole Brillouin zone. k When the various Y magnetic states are put into the matrix elements to measure the Correlations in those states, certain A's become equal or vanish: 32 >: = ; =; =; =; :0 (2.27) lPM Ao++-Ao,-- Ao,+- o RQJ+_ o AQI++ o A Five Relations >0 0 0 = 0 = 0 = ; = (2.28) lFM . Ao,++’Ao,--’Ao,+- o’éQ.+o O'AQ:++ o AQ.+' 0 Four Relations . >3 " ; = ; = ; = (2 29) MPH Ao'H-Ao'__ AQ'H AQ'__ AO'+_ o AQ’+_ 0 Four Relations >: : ; : = : = ; = (2.30) ISSDW Ao,++-Ao,-- §Qm+' AQI++ o Ao,+- o.RQ'__ 0 Three Relations (2 31) 1FIM A0.++ A0," AQ’++ AQ'__ Ag“? 0 Ao’+_ 0 Two Relations These are derived as follows: We assume all coefficients B are real and independent of 3. (2.32) lRM> For L=l,2 or 3,4 by translation of all indices a B Y >=B 26 . + by Q. '= O if.g ¥.E. 0 ¢ 0' aq,o|BLYk.o' (2.34) 1EM>: For L = 1,2 or 3,4 by translation of all indices by Q: : Here we cannot see by inspection which A's are non- zero, so we must work them all out, the vanishing of certain A's being a result of the properties of Fermion annihilation operator: + 2 a number +B Y 31' operator (2.35) <31? 3.3+Qo+ a£’+ah'+ Bl?§fi2.+>= 50+ Using abbreviated notation for the magnetic states and dropping commas in the subscripts. (2.36) <1,3la'_]:_ a}: l1,3>=o (2.37) (1'3lé£+9+é3i9+|1'3>=B32' number operator (2.38) =o (2.40) <1,3|s£+ éh- I1,3>=o (2.41) =B1B3, anomalous operator + (2.42) =o + (2 .43) <1,3|a_]E+ a£+9_|l,3>=0 (2 .44) (2 .45) (2.46) (2.47) (2 .48) (2 .49) (2.50) (2 .51) (2.52) (2 .53) (2 .54) (2.55) (2 .56) (2.57) (2 .58) (2 .59) (2 .60) <1,3 hi4 éEfQ+ll (1'3 (‘3‘;- aL-Ia- '1 + <1 ’ 3 labia-ab + <1 ' 3 Iakfl-ay =B l2. l2. l2. 34 '3>=B3Bl' anomalous operator ,3>=0 |1,3>=o |1,3>=o ,3>'=O ,4>=O ,4>=B22, number operator ,4>=o 2 ,4>=B4 , number operator ,4>=o 4>=O |2,4>=o 2B4, anomalous operator 4>=0 4k0 4>=B4B2, anomalous operator 4>=0 35 + (2.61) <2,4Ia18+ éEtQ‘I2'4>=O + (2.62) <2,4lak+Q_ah+ I2,4>=o There are four nonzero number operator matrix elements and four nonzero anomalous operator matrix elements. We note that the physically relevant quantities are still sums of these nonzero matrix elements multiplied by occupation number Fk between zero and one, so that the unfilled 5 states cannot contribute to these sums: ISSDW>: + 2 (2.63) =B , number 13+ 4362 lay 3+| 13+ 4126: 1 operator + (2.64) <1.4|a£+ a5+ |1,4>=o + (2.65) =0 + 2 O < - (2 66) 1'4lé5t9'éhf9‘l 1,4>-B4 , number operator + (2.67) <1,4lal<-_ a]? |1,4>.=o (2.68) <1,4|a]:+ ék‘ | 1,4>=0 (2.69) <1'4|é£+gfifih+ I l,4>=0 (2 .70). <1,4 Mia}? | 1,4>=o (2.71) (1'4|§£49?§E+ l1,4>=BlB4, anomalous operator (2.72) (2.73) (2 .74) (2.75) (2.76) (2.77) (2 .78) (2.79) (2.80) (2 .81) (2.82) (2.83) (2 .84) (2 .85) (2.86) (2.87) (2 .88) 36 <1 , 4 Piaf};- |1 , 4>=o =o <1,4|a;_ ak+grll'4>=0 + <1,4|a15+ 4>=B4 1, anomalous operator \ alsfil- '1' <1,4|a£_ a£+Q+|l,4>=0 = 2 2 , number operator <2,3|a;_ a k+ (2’3>=B 2 3 , number operator (2'3la:+g+§5+g+l2'3>=3 <2,3 laifl‘ahfl“ l2,3>=o <2,3|a£_ a12+ |2,3>=o <2,3|a£+ a}? [2,3>=o <2,3|a_]*<:+£2+a15+ [2,3>=o <2,3|a£+g_a_]s_ [2,3x0 <2,3|a£+ a£+9+|2,3>=0 <2,3la;_ ak+9_|2,3>=0 <2,3|a1]*('+9_a12+ |2,3>=o <2,3la:+Q+ak_ '2’3>=BZB3' anomalous operator 37 (2.89) <2,3Ia J 2,3>=B , anomalous operator + k-ayg 332 + (2.90) <2,3|a£_a£+QJ 2,3>=o There are again four nonzero anomalous operator matrix elements and four nonzero number operator matrix elements. FIM>: In general, all four B's in each wave function are nonzero so that every correlation exists and is nonzero. We now seek the relations between the A's and the coefficients B in a general way. Penn finds: (2.91) AO+_ = - %9 EPIS— (Ble+B3B4) (2.92) AO++ = £9 EFk (822+B42) (2.93) A0“ = {312 ?— (812+B32) (2.94) AQ+_ = - -§—° 33$ (B1B4+B2B3) (2.95) AQ++ = 1%?- 13F}- 3234 (2.96) AQ__ = €19- EFE 3133 Now we choose a wave function for a state, i.e., we find which B's in the general four term linear combination of states that is Y magnetic, are zero. 38 For a given state we wrote down the relations among the A's and which A's were zero. Utilizing the definitions of the A's in terms of the B's directly above, we see that we will obtain relations among the B's with the ZFk multiplier being E..— needed only when we discuss which states are filled and which are empty, but not their structure. Explicitly: Table 1. Magnetic wave Function Coefficient Relations. Correlation 5.233223... “:2: iii???“ Relation ’ g " PM AQ+_=O B1B4=-B2B3 EEBIB4=k £(-B2B3) AQ-H-=O BlB3=O EF—BJ'B3=O B1=B2 AQ__=0 13213 4=0 1:33.323 4:0 Ao+_=0 131132913334 EFEB1B2=Fh(-B3B4) _ 2 2_ 2 2 - 2 2 _ B =13 =0 Ao++—AO-- B2+B4—B1+B3 ZBE(B2+B4)— 3 4 H‘k(Bi+B§) 1(- _ FM AQ++=O B1B3=O E£B1B3=O r- . BZ¥B1 AQ__—0 13213 4=0 5531321350 AQ+_=0 B1B4=-B2B3 35131134??? 132133) — — B =B = = =-— = 4 3 Ao+- 9 B132 B334 2FkBle 2Fk( B334) 39 Table 1 (cont'd.) Correlation . State State Structures . State (Function ‘ Structure .and Filling) Conclu51ons Relation _ 2 2_ 2 2 2 2 _ AFM AO++5AO__ B2+B4—B1+B3 ERE(BZ+B4) _ '— B =3 2 2 l 2 . (B +B ) 13.3 1 3 BQ++=BQ__ B1B3=B2B4 EBEB1B3=EBEB2B4 B3=B4,B2=B4 AQ+_=O B1B4=-B2B3 E?EB1B4=k.K(BZB3) B1=B3 and B1,B3 - - orB2,B4 pairs Ao+-=O BlBZ=—B3B4 E§_B1B2=E?h(-BBB4) are zero when the —' '_ other pair is not SSDW BQ++=O B1B3=O §B_B1B3=O Bléo, B2#O __ or AQ__=0 B2B4=0 ETEBZB4=0 3220, 3320 AO+_=0 B1B2=-B3B4 EFEB1B2=EFh(-B3B4) FIM 4: 3233/31 } BQ+_=O B1B4=-B2B3 EBEBlB4=EBh(-B2B3) 4=-B1B2/B3 B1=B3orB2=B4 Ao+-=0 B1B2='B3B4 E553132=E53(’B3B4) thus _ _ 32:32 1 3 Thus, from the correlations, he concludes: (2.97) YPM = B V or B Y e uall o ulated 1.3+ 2.K" g Y P P PM (2.98) Y = BlYE+ or B2Y£_, unequally populated AFM 2.99 = ( ) Y 3113+ + B3YE+Q+or B2YJ£_+B4Y£+Q_ 4O SSDW _ (2 .100) Y 81Y5++B4Yk+gfir 92 Yk-+B3Y_Ig+g+ FIM _ , (2.101) Y — B1Y£++B2YE_+B3YEfQ++B2YE+Q_ or Bl‘l’k ++B2Y£_+B1Y£ +52 +13 4Y1: +0- as written earlier. We now seek to graphically illustrate the spatial spin structure of these states using a plane wave basis set for the Y's: IEM>=YPM:B £75); and B £732; and SE B 2 = 2F 1B 2: 1 + 2 - k'3 1 k-3 2 IPM) t t I t Li B¥=l or 0, B3,: 1 or 0, but when Fk=l,Fk,=l (is occupied) also. IFM>: $112,131,133; and 33423335 and ZFkBlzaéZ'Fk,B2 ‘ .32 - k - lFM) I 1 I t L53 Here FkiFk" so we have unequal occupation of spin up state and 2 spin down state on each site. 41 AFM ik-r ik-r iQ-r >. = _ — — _ _ IAFM . Y BlLl + 331,! L and 3225555 + B4255”; 629:3 the L79”; term oscillating in sign from site to site. fl Z IAFM) 3"5—9— ( i 1...}; EK'ER I We note that if our wave function is: FM _ it '2 112:1: 19 :1: (A _ Bl.+ + B3.+ . and 3223315 - B4th‘5 L‘Q‘E we get: 35413, '3r' % t__‘.- X) We get a one site phase shift of the lower AFM.wave with respect to the upper, leading to a new spin density cancellation of each lattice site, but a net charge density fluctuation from site to site, a charge density wave. ISSDW>: YSSEW = 612:5i5 + B42fkfi£ L191; 152E 15.. 19:; and B2L_ + 334+ 2 42 which is the site by site sum of the following waveforms: BI a)" 3‘ elk r eio-r 1 I T I 1 9’ eir-r Their site by site sum is a Spiral Spin Density Wave: 41 ”111110 We note the change in axis orientation. The SSDW spirals along A A the Z, not the X axis. We have graphed the SSDW for a particular choice of Coefficient magnitudes: B1>B2>B4>BB. The SSDW will look different if a different relation among the B's is chosen, A but will still spiral along the Z axis. Some of these other SSDW states are: Other ISSDW> States _L_.___J__.___l_ with varying relative amplitudes, phases with respect to the four components that were added to arrive at each of these spin structures. These are only a special case of the general SSDW, that for Ao+_=0 and Q = g/Z = n. The above SSDW spin structures A differ only in pitch angle along the Z axis. There are again, as in the SSDW case, a few possible FIM.struc- tures, depending on relative B magnitudes. These are the same as A the SSDW spin structures in form, but directed along the X axis, A making the spins point parallel or antiparallel to Z. The meaning of these states in terms of the nonzero correlation functions can best be seen by writing out the magnetization and numbers of electrons in terms of them. ‘ Penn defines these (2 .102) 15((0) (2 .103) MX (9) (2.104) (2.105) (2.106) (2.107) For all the states M2 (9) 142(0) n (0) n(Q) 45 JEEQA 7 Co 0+- SELEA ‘ Co ,Q+- u _._2 _ = C0 (§Q++ AQ—-) u =_§ _ 7 Co (Ao++vAo--) =_1_ " Co (Ao+++Ao--) 1 Co (AQ+++AQr-) he cons1dered.Ao+_s 0 so that Mx(0) = 0, states are seen to possess the following properties: IPM>= (2.108) (2.109) (2.110) (2.111) (2.112) (2.113) IFM>: wa) Mxm) M2 (0) M2 (52) n(0) n(Q) 2 2 _ 2 315(31 + 132) _ 35(261) ll 0 The 46 ll 0 (2 .114) Mx(0) (2.115) MX(Q) = O (2.116) MZ(0) =-—§ Eng (Bi - Bi) (2.117) MZ(Q) = 0 (2.118) n(0) = §s£(si + B3) (2.119) n(Q) = o £11122 (2.120) Mx(0) = 0 (2.121) ux(g) = 0 (2.122) MZ(0) = 0 (2.123) MZ(Q) = 0 (2.124) n(0)=-%5 Egk(si+6§+3§+sfi)=-%3 EBE(Bi+B§) _ 1. _ _2_ (2.125) n(Q)— ZFk (BlB3+B3B1+B2B4+B4BZ)— Cokml<_(BlB3+B2B4) SS >: _zuB (2.127) MXQ) = a;- 35081134) IJ. _ s _ 2 2_ 2 2 47 (2.129) 142(9) = 0 __1_ 2 2 2 2 (2.130) n(0) — Co ER_(B1+BZ+BB+B4) (2.131) n(Q) = 0 FIM>: (2.132) MX(O) = o (2.133) Mx(g) = 0 (2 134) (0) = :1ng (B2+B B -B2-B B )where B =B B =B °“z 015123224 13'24 (2 135) (Q) = fl3233' (B B +B2-B B ~32) ° Mz 00k; 4 2 32 1 (2 136) n(0) = i- (B2+B B +B2 + B B ) ' COR 5 1 2 3 2 2 4 (2 137) n(Q) =-;—XF (B B +BZ+B B +32) ' Coklg 24 2 23 1 Penn utilized the correlation function, A, to derive the energy of a magnetic state at some point of the phase diagram .§% --§§), and compared this with that of another state whose energy is cal- culated with the same value of those two parameters in order to see which states are stable for that particular point on the magnetic phase diagram (considering the five states previously mentioned as the only possible magnetic states of the system in the Hubbard Model). we note that his SSDW and PIN states are special cases since generally AO+_#0, so that, especially in the 48 FIM case, all four coefficients B are not related at all, while in Penn's FIM state, Ao+~ = 0 forces one B to be related to the other three. Also, the special chose of.g ='Q = Q/Z =.fl only brings out SSDW and FIM.states with a certain pitch, while in general, the pitch or phase is arbitrary for both spin structures. He computes the energy by solving the Heisenberg equation of motion for the state operator Yk' (20138) [HIV—12] " YkEk or equivalently: (2.139) [H.YJS] n .< br+ m Ix for its adjoint. We first write H out in the Bloch representation and in quasi- particle form where the Gorkov factorization has been done, as before, to the two-body (four operator) potential term to make it an effective one-body (two operator) quasi-particle term. Co Bloch J 2: (2.140) H = Zekakak+ zflzk 2 5kk" 0 0' .[Eakua ahua> I kwhole Brillouin zong;° + _ + ak"o'a}_<_"o' akuau a7_<"c] where the bkk.comes from the Fourier transformation of fiWannier. 49 We rewrite this as: Blodh _ + .99 _ _ (2.141) H — Elsa-12+ 2N E. k k k6 051+152 153 154) - -1'-2'—3'—4 - 2 ak o'ak c'-ak a'ak oJ 6.0' -3 —2 -4 —1 —3 —1° —4 —2 0#O' Summing over 5" = k and k + g Bloch _ + ‘99 .1 + (2.142) H — Zekakak+ 2N Z Z . 2 (33020) .5'-'_ .5 0,0 whole 0¢O' Brillouin zone + + + alsa'akv' + (akamaymc) a1<.+g.oar+g,a' + _ + > + __ + + “1630' aBa 'azso “114905160 ' > a1<.+90' a1+go ' + + + + + alfid'alsc , Kak-(Qoalg-(Qa) aka ,alsU , “31103349? aiaa'aro' + (aiaoazcmgc'alsflf «ago also > $4.00 "33490 ' Kaiaoahflo mic ' aha ' 7<§£o§kc' >i§£iQfi'§hfiQp - <§EiQPé§fiQQ'>ha£B'éEQ ' aims 'ahfilo - «£62659 ' > all; ”6an + + _ + + “(ahfialso' > 5149031420 “140031400” aBo'aBo where the extra 1/2 factor is put into account for double counting, i.e., the fact that, for example, the second and third terms are 50 equivalent by the translation of -Q in the Brillouin zone. Summing over 0 and 0' we find: Bloch _ + _1 £9 + + (2.143) H — 12265 _a1<_+ 2 2 E [a}£_alg_+ — - + + whole' < a > a Brillouin a_]$_ 35 83+ 25+ zone + < a+ a > a+ a + < + > + a 1<+9+ 362+ 5+9: k+9_- aha-alwa- a15+c_2+ §+g+ + + + + - < a > a - < > k+al$+ £- a’i“ ak- ak- ak+ak+ ' < 3.1263140? alga-3+2: (aim-aka? ai+9+als+9+ + < $39 > age-6‘39““ (ii-‘31- > £424,314... + < a£+g+agg+g+> 6%: - + 311.6366. "’ < a1<+2+al<+2+> aka}- + (aha-511314.23 a15+aJs+ + + - < a > a a - < + + 19a):- 192+ 3+2- a1<-“"1<+ > a1+9+a1+9- + + + _ > .. + < swam- 2162-162? 32+ _- + + _ + + ‘ < ak+a1_<+_q- > ak+._g-alk.+ “bag-10+ > ak +ak- + + + + - > - < alc+Q_+a1_<- a_k-ak+Q_+ aka-3140+ ‘ (at-as > als+2+azs+9- + + + “£03603 3312+ " (alga-arm? ale-tab] Now, we put in our correlation function definitions (2.21 - 2.26) and find: (2 '144) HBIOCh=E€l$a£alS+ % [Am-ii" _-+A0++~Ea£+ k+ +Ao+-§ i3}? + AQ+- i 3; Js+Q+ + Am- ; a£+als+9- Ms- + 26...; a1... .- + 3,; 6:..- +AQ+—§ a£+9+ak+ + A9.++ i It+a1s+0+ + An" g egg-aha- +Ao+—}§ a£+sz+"“k+.9-+ Act- g alts: 31409“ Ao++ E ai+9+a3+9+ +Ao__J5 a£&-al<_+Q-+ Ao-I-- li a):- 3+] NOW: (2.145) v; = Bla;++132a£_+ 33a£+9++ B4a£+52- So that: (2.146) yin-Hy; = yi = ' ii'fl-fifls'bw = ("31:49) .. . _+ __+ _ and a similar commutator yields ( akfiQ-) — ( a3+91_) by trans lational symmetry. These are terms (5) and (9). . + + = + + _ + + IKE—L" ,E.[a1+.a1-+a1'-J E.‘a1<.+a1'+a.t'- a1'+aB'-als+’ _ _ + = _ E.ak'+6]_ a3a4+a1a2 -ala4 and tabulate the resultant term subscripts. All such averages as and of two creation and two annihilation, which occur in superconductivity theory correlation operators, are ignored. . . (2) Table 4. Gorkov Factorized Terms in V Bloch. Sign a+ a Sign a+ a + .51+ .51+ '52- '52- + '52- k2— kl+ k1+ - 151+ 152- 15,2- 51+ - 52- 51+ 51+ 52- + 151+ 1991+ 152- 1992- + 52- 5+92- 51+ 5+91+ - 51+ 5+92- 52- 5+Q_1+ - 52- 5+Ql+ 51+ 5+92- + 51+ 51+ 5492- 5+g_2- + 52- 52- 5+_g_1+ 5+Ql+ - 51+ 399.2- 102- 121+ - 5+22- 31+ 51+ 1922- + 51— 51- 52+ 52+ + 52+ 52+ 51- 51- - 51+ 52+ 52+ 51- - 52+ 51- 51- 52+ + 51- 5+Ql- 5+QZ+ 52+ + 5+92+ 52+ 51— 5+_Q_l— - 51- 52+ 5+92+ 5+Ql- - 52+ 51- 5+91- 5+QZ+ 79 Table 4 (cont, d.) Sign a+ a Sign + a + _ - 151- 5+92+ 5+92+ + 5+92+ 5+92+ 51- 51- - 51- 5+92+ 5+92+ 51- - 5+92+ 51- 51- 5+92 + + 51+ 51+ 51- 51- + 151- 51- 51+ 51+ - 51+ 51- 51- 51+ - 51- 51+ 51+ 51- + 51+ 5+91+ 5+91+ 51 i- + 5+91- 51— 51+ 5+91+ - 151+ 121- 5+Ql- l<_+Q1+ - 1<_+Ql- 1<_+Q1+ 121+ 51- + 51+ 51+ 5491- 5+91- + 5+91- 5+91— 51+ 51+ - 51+ 5&1- 5+Ql- 51+ - 5+Ql- 121+ 51+ 5+91- + 52+ 52+ 52- 52- + 52- 52- 52+ 52+ - 52+ 52- 52- 52+ - 52— 52+ 52+ 52- + 52+ 5+02+ 52- 5+02- + 152- 5+02- 352+ 5+92+ - 52- 5+92+ 52+ 5+92- - 52+ 5+92- 52- 5+92+ + 52+ 52+ 5+92- 5+92- + 5+92- 5+92- 52+ 52+ - 52+ 5+92- 5+92- 52+ - 5+92- 52+ 52+ 5+92- + 52+ 52+ 51+ 51+ + 51+ 51+ 52+ 52+ - 52+ 51+ 51+ 52+ - 51+ 52+ 52+ 51+ + 52+ 5+92+ 5+91+ 51+ + 5+91+ 51+ 52+ 5+92+ - 52+ 51+ 5+91+ 5+92+ - 51+ 52+ 5+92+ 5+91+ + 51+ 51+ 5+92+ 5+92+ + 52+ 52+ 5+91+ 5+91+ - 51+ 5+92+ 5+92+ 51+ - 5+92+ 51+ 51+ 5+92+ + 51- 51- 52- 52— + 52- 52- 51- 51- - 51- 52- 52- 51- - 52- 51- 51- 52— + 121- Js+£21- 1992- 52- + 5&2- 52- 121- 5+91- - l'Sl" 32- 5+92: Lfll- - 122- 1d- l<_+Ql- b92- + 1&1- Jsl- 1<.+QZ- 199.2- + 1992- 5&2- 151- 121- 1 151- Js+QZ- 549.2- 51- - 19522- 51- 1:1- 19522- We exclude all correlations of form Amnklz or AmnkZl as "orbital ordering". (These are terms on lines 2, 4, 6, 8, 10, 12, 26, 28, 80 30,32,34,36), a spin arrangement in which the orbital that the spin is in is as important as what lattice site it is on and what its nearest neighbors are. We do this since no evidence for any such ordering exists in the transition metals. We arrive at Véioch in terms of four types of correlation functions. If we define our correlation functions as: Brillouin zone _ + (3.13) A - E (aabcadec> as type A(obecc) or type A(Qbecc), where c is the orbital index, b and e are the spin indices. 0 (representing 5) and Q (repre— senting 5+9) are the crystal momentum indices. This is unlike Penn's definitions (2.21) - (2.26) in that all are positive, and Co and N are not included in the definition. We prefer to carry them along in the calculation explicitly. Penn's Co will no longer be a simple s subband term, so we prefer to keep it visible throughout all steps we make. Type A(obecc) and A(Qbecc) correspond to the pair (a,d) being (5,5), 3+9, 5+9) 01‘ (1% 3+9). (hfli) respectivelyo (2) Bloch zation represented in Table 4 in terms of the nonzero correlation We can rewrite V (equation 3.11) after the Gorkov factori- functions for a general magnetic state without "12" type "orbital" correlations. The specific relations between the correlation functions for the specific magnetic states are held in abeyance until later. 81 |_ T d“ CI!!— .1_ On each site we have one up spin or one down spin in either of two orbital states in the two-fold degenerate case (or in the five orbital states in the five-fold degenerate case). In a zero frequency probe, which measures X(0) = M(O)/H(0), these would all appear the same as viewed from the top as: Top View where "." and "X" denote up or down spins on each site. Physi- cally, this is because of the zero frequency probe n(0) (where n(0) is the applied magnetic field; n(0), the resulting macro- scopic magnetization; and X(0) is the magnetic susceptibility relating the two) which does not excite orbital transitions, i.e. does not probe the orbital degeneracy. The definitions of our 82 We must examine the physical reasons for this neglect of "12", cross-orbit correlations, A(oggij), where i and j here are two orbital indices. ij can be pairs 12, 21, in the two-fold de- generate case or 12, l3, 14, 15, 21, 23, 24, 25, 31, 32, 34, 35, 41, 32, 43, 45, 51, 52, 53, 54, in the five-fold degenerate case. Consider an AFM spin structure: Nondegenerote lAFM) Spin Structure In the two-fold degenerate case (the same considerations hold in the five-fold degenerate case) this array could look like any of the following: Five fold degenerate IAFM) Spin Structure i i i i i I i I i Of : + + +—+— Of 83 three magnetic states, ARM, EM, and FM, depend on macroscopic measurements which are probes of net spin on a particular lattice site, regardless of which degenerate orbital state that spin is in. Hence, orbital indices only play a physical role in putting new correlation terms into the total energy of a particular state, which adds more direct; and exchange terms with coefficient J. The interactions term in the Hamiltonian is also (2) Bloch correlations with K and J coefficients enter as we shall see in altered as seen by V being considerable enlarged. New equation (3.14) By mathematical reasoning, we can represent a sequence of . . . 3 correlation functions as a symmetric array6 : ++ ++ A(Oi-zll) A(Oizzl) ++ ++ A(Oi:12) A(Oi:22) ++ ++ Since A(otle) E A(o+:21), diagonalization by a similar trans- formation becomes, trivially: ++ A(Oizll) 0 ++ 0 A(Ot:22) since these two correlations are assumed to be zero. Since we need only two independent one-electron wave functions, then: 1 ik- 'k° ' - . ++ ‘1']: )(5) = 3111: 5 + 1335:: -r- 1,19 5 (contains A(oitllH (2) __ i5-5 i5-5 iQ-g . ‘H’ *3 (5) — B5L2+ + B7L2+ L (contains A(oi:22)) 84 (3) _ 1.12:; i_k -_r 19.-.2 - ++_ Yk (R) — BlLl+ + B5L2+ L (contains A(oi_12)) (4) _ ik-r ik-r iQ'r . +f Yk (R) — BSL2+ + B1L1+ L (contains A(o+_21)) Only two are independent. We assume the other two are zero by assume A(o$:12) E A(o;:21) = 0. Electrons in different orbital states interact (as seen in the diagonalization where this is included) but are not correlated, i.e. our magnetic state one-electron wave functions are not constructed of mixed orbital electron states. A physical justification for neglecting spin-orbit splitting is the large overlap of the d orbital subbands so that the total d electron band width is much greater than the spin-orbit splitting in transition metals. In our neglect of cross-orbit correlations, we can further be assured that we are not probing the orbital structure so that the orbital angular momentum operator L2 is never used. It is usually quenched in transition metals in a cubic crystal field so that orbital angular momentum eigenvalues do not influence the magnetization definition of our magnetic states (2.108 - 2.125) ; they are electron spin-produced magnetization. $2, the spin angular momentum operator, is used to operate on our electron wave functions; its eigenvalues are the important ones 0 85 Table 5. V(2) in Terms of 3_Space Correlation Functions. Bloch We find: 2 K (3,14) vélgch =[1EE A(o++11)an)'("2_ak2_+A(o-——22)a.;:l_'_akl+ + -_ +A<2++11> mm +A<2 ”we...“ +A(o++ll) a+ +A(o- -22)a+ 3&2- a3+g2+ 3+g1+a 3+9_1- __ + +A (o 11) ak2+a£2++A (o++22)ak1_alsl_ +1A(o++ll)a+-HA(o--22)a+ kl- ak1- +A(Q&+ll)ak2_a k2— +A(Q- -22)a+ 3+522+a 3+g2+ k1+a k+Q 1+ +A (o++11)ak-|Q1-ak+Ql-+A (o--22)ak+91+ak+gl+ +A(o—-11)a+ +.A(o++22)a+ 3+Ql+a k+Ql+ k+Q2- alk+QZ- +A (Q--11)a+ +A (Q++22) a+ k+92+a k2+ kl- ak+Ql- +A (0"11) 33492+a3492++A (04-4-22) algal—3154.9]... +A (o++11) ail-31(1-4'A (0"11) a£l+al$1+ _ _ + _ _ + A(o+ 11)akl-§31+ A(o+ ll)ak1+ak1_ +A(Q++ll)a+ +A(_- -ll)a+ 3+Q1- a31- 31+a3+<21+ -A(O+-ll)é£le-éEfQI+-A(O+-ll)§£fgl+§bfgl- -A(Q+- ll)a A(Q+- ll)a 3+_g1-a31++31+a3+01- -A(o+- 22)a+ -A(o+-22)a+ 32- a32+ 32+3 32- * “3H2” a32-a3492-+A (9“22) a1<2+a1~z4392+ -A (Q+-22) a322+alfi£2_-A (52+-22) £23349“ -A (Q+-22) ai+92_al(2+-A (Q+-22) i2;alfi92_] 86 Table 5 (cont'dJ 2l + + [ A (o++22)ah1+alsl++A (OH11)31_<_2+332+ +A (Q++22) a+ IKWfl +.A(Q++ll)a+ k4Q1+ak1+ k2+a k+92+ +A (o++11)a;_&2+ak_&2++A (o++22) a}:+g1+ak+gl+ + +A(o-- ll)ak2_a k2- +A(o--22)ahl_ahl_ +A(_Q—-11)ak+92_ ak2_ +A(Q- -22)ak1_a k+Ql- +A(o--ll)a+ k+92- ak+Q2_ +A(o- 2-2)a+ 3+g1— a3+g1- -] Lastly, degeneracies in the physical makeup of the spin structure of our states do not have to be all of an orbital filling nature as we have just indicated. The GM state has M.z with any magni- tude from MEAX, when all the spins are polarized along 9 to M.z g50, and few spins are polarized along 2. Also Q can have any direction in space. Therefore, there is a 2N fold spatial rotation de- generacy in defining FM. The AFM state can be represented even in the single s-band calculation of Penn, as: A i e 87 or as: L 1 with a phase shift in its spatial spin structure of one lattice site. This will not matter macroscopically in a M.X or Mz de- finition. In conclusion, we must look upon our magnetic states as being defined solely by their nonzero correlation functions, and not take intuitive spatial spin structures too seriously. This approach is a necessity in any practical calculation. Macro- scopic Mfs and n's determine our magnetic states, and these depend solely upon the correlation functions as in (2.102) - (2.107). + 32+ 531+ + . to ak1+a52+ we can further group them in terms of a smallest We group the terms in (3.14). Noting that a is equivalent set of independent two-operator product: Bloch- 2 (3.15) V(2) - -%{a; _a£2_[KA(o++ll)+KA(o++22)+KA(o--ll)j Dru 88 +9£1+a31+EKA(°"11)+KA(°--22)+(K~J)A(o++22)J +a;2+ak2+[KA(o--11)+KA(o—-22)+(K~J)A(o++11)J +a:1-akl_[KA(O++ll)+KA(O++22)+(K-J)A(o--22)] +a;2_ak+02_[KA(Q++ll)+KA(Q&+22)+(K-J)A(Qf-ll)j +agfigz+abz+[KA(Qe-ll)+KA(Qt-22)+(K-J)A(Q++ll)J +a£l+ak+Ql+fKA(Q--ll)+KA(Q--22)+(KrJ)A(Q++22)] +afgI_ak +91_[KA (Q++11) +KA (Q++22) + (K-J)A (g--22) ] +a£1+ak+gl_[-KA (52+-11) -KA (Q+-11) 1 +a]:+91+ak+gl_[—KA (o+-11) -KA (o+e11)] +a£2_alg+gz+[-KA (Q+—22) +a£2+a£&2_[-KA (Q+-22) -KA (52+-22) -KA (_Q+-22)] +a£l+ah1_[-KA(o+-ll)-KA(o-+11)] +a£2+a52_[-KA(o-+22)-KA(o—+22)j} k l + where the first (3+9 2 -) denotes the subscript of an a+ and the second denotes the subscript of an a. This is still the Gorkov factorized one-body, quasi—particle Hamiltonian. We label the above terms in (3.5) in brackets as C, D, E, F, K, L, M, N, O, P, R, S, T, U, V, W, X, Y, Z, respectively. 89 (2) + _ + . We must now commute [VBloch' YE] - gkvb to find the energy "eigenvalues", Ek' The relevant commutators are such that for (23' Bloch' operators making up y: are nonzero. The rest have delta func- each term of V only two of the eight commutators with the tions, 6+ _ or 51 2, which are automatically zero. In all of I these terms, i is summed from one to eight to represent the eight terms in the linear combination of a 's that is Y;' Each term has a "-" sign, since the commutator [a:,a;ak] has been factored ++ + +_++ +__+_+ , out as (aiajak ajahai) — aj(aia5+a_ai) — aj aia3]+. Note that here i and j have nothing to do with lattice site labels; this is just an unfortunate duplication of notation (also 3 above is just a dummy index, note a wave vector). The actual anti-commutators are: (3'16) ii. 33'2-[a:,§k'2-]+= ‘36§£2+ ' B3§£+QZ+ (3°17) ii. aiHmJaLar'uh "" "Bia£1+‘33a£&l+ (3'18) 1:. a£'+92+[air,al<_'2+]+ = _Bsai2+ -B7al:+522+ (3.19) Ei' a£.+91_[a:'ak.l_]+ = -B2a£l_ -B4a£391_ (3.20) E. ag.2_[a:'al$.&2_]+ = -385132- -B63£+Q2_ (3.21) Eg. a£.+92+[a:'ah.2+]+ = -B5a£+92+ -B7a£2+ + + (3.22) 22 a . ‘33é31+ 'Blékin+ 13' -— + ._ + - + «3.23) §;.a._.-1-tai,ag.m-l+- 3233491- B4331- (3.24) (3.25) (3.26) (3.27) (3.28) (3 .29) 90 + + Z}: a I _[a' a l .1 i3} E.l 1' E.+Q} + + ii. a3f1+[ai ak'agl+]+ '_ + + V. ak'2-[a'k, k'+02+]+ 13_ —- -— + + 73. ak'2+[ai,ak'+QZ-]+ 13_ -— '— -— + + ' _ F3. ak-1+Eal,ak-1-1+ - 13. - + + 2? a , [a. a , _] = 15) 3.2+ 1, k 2 + Gathering up all of these terms and tators calculated before, yields: where (3.30) (3.31) (3.32) (3.33) (3.34) (3.35) (3.36) (3.37) > n 0 (V C) m FI co m <3 «I o H m C) 0 r4 0 (V H H :4 0 cu o H H o m o c) H m ‘i‘ 73,. We. 0 312E? Kg w lav Ill 0 m III v +g; ”fit 2 O o H o N H H 22's,. I I IF! .543. I H o N o z -F< 94 Here we have only two independent matrix element correlations, A(o—-) and A(gr—) as defined in (3.39). The 33 term is equivalent to the 11 term. For brevity we have labeled redundant matrix elements by their row-column indices in the matrix format pre- sented here. These indices bear no relation to the physical indices "1" or "2" in the A's, which are subband labels in this two-subband degenerate model. In the two-fold degenerate PM case, the correlation functions are related as: (3.41) A(o++11) = A(o—-ll) = A(o++22) = A(o--22) # 0 All other A's are identically zero. There is only one independent matrix element since the 33 and 11 terms are again equivalent. (3.42) N (ex-Eh) + (BK-J) o 0 o A (o--) E 11 0 11 0 0 N (61242"? 0 + (3K—J) 0 0 A (o--) 0 E 33 0 0 0 33 = 0 ll 0 0 O 0 ll 0 O 0 0 33 0 0 0 33 95 In the two-fold degenerate FM.case we have the following correlation function relations: (3.43) A(o--11) = A(o--22) A(o--) i 0, A(o++ll) = A(o++22) s A(o++) i 0 All other A's are identically zero. There are 339 independent matrix elements, A(o-fi_ and A(o++). Thus, the 11 and 33 matrix elements are not equivalent. (3.44) N(€k-Ek) +2KX(d:-) -+(K-J) 0 0 o A(o++) E 11 0 ll 0 O N “Mg-Bk) +2KA (o++-)- O 0 +(K-J) 0 A(o--) E 33 0 0 0 33 96 As it should be, we note that only the AFM.state had_Q depen- dent correlation functions in its secular equation. We can easily generalize to the five-fold d-subband degeneracy problem by writing down the answer modeled after the two-fold results as shown in equation (3.44). (We spread out the matrix subblocks on a few pages due to lack of room to include the 20 x 20 determinant.) There are five 4 x 4 subblocks. (Note that the symbols A, B, C, D, E, here are for the subblocks and just notational. They in no way should be mixed up with A's (correlation functions) or B's (linear combination coefficients) used before.) The subblocks of these subblocks have the correlation A; and, therefore, the coefficient B in them. (4x4) (4x4) (4x4) " (4x4) (4x4) 97 In the five-fold degenerate AFM case, the correlation functions are related as: (3.45) A(Q++ii) = A(Qr-ii) #0, A(o++ii) = A(o--ii)#0, i= l---5 There will be two independent A's, since 11 and 22 are equiva— lent. The 4 x 4 subblocks will be: (3.46) A = N(e -E ) 5 k k 'g '— k.$ A(Q--ii) +K z A(o--ii) 1=l i=1 0 5 . . o 5 +(K-J) .2 A(Q++11) +(K-J) >3 A(o++ii) l=2 i=2 513 all “‘63"? 5 .. 5 K. 2 A (Q++11) +k z: A(o-I—I-ii) 1=l 0 i=1 0 5 5 +(K-J) >3 A(Q--ii) +(K-J) z A o--ii i=2 i=2 ( ) 524 522 all 11, but with 13 o €3+Q instead of o €12 11, but with o 24 0 €k+Q instead of ‘53 98 l ng .54 w mo oOMHQ CH 0 .¢.N o nuflz ugh .~.m 0 MW mo momHm :H . d+mw . o .m H w ruH3_.H H .NeN" .flsN" Nwmfl NxH an“ said: u 9.5+ 0 Ta 0 m I. HnH AHA++OV¢ w x+ aid: w x I m m Axmuxwvz .HT—n" m ~33 . .Hm N33 Hwfl Hun.” AHH++OVG .w C...Iv¢+ o AAH++de w Abuxv+ o m H a m n. Hua “Hauuovm m x+ :TIa: w x I I m Axmuchz 99 Mm mcHUMHmmu gmw o aim 0 its 83 ..~.~ o of mGHUMHQOH 93w . :33 ”:5 ..H.H o _.m H . Rim" _..VsN“ M‘H 9.: TH HHH O AHHflIIOV < N Ann-xv + . m 0 5.7.92 w 8.5+ Hi m HuH 33.134 My? 3H++9< w x I I m 3”,?wa . ..H.Hm . I T: ..m HI . m} HuH o HuH HHH++3< w 833+ 33.2.5: my 8.x: 0 m H1 m HnH :7qu My? :Tua: w x I I m Axmuxwvz 100 d+mw xw mcHomHmmu o .=+.N o squ pan .=~.~ o xw mcHomHmmu o+xw o .=m.H nqu usn.=H.H ...N~N" .:.V~N" .V‘H «x3 HuH HuH AHHIIOV< w Annmv+ AHHILdV< w Abuxv+ o m o m HnH Hufl AHH++oV<.w M+ xflfl++ov< w x .I .m m xxmuchz ...._...~._u" _..msH" Vkfl ex“ Hug HuH xflH++oc< w Annxv+ o AflH++ovm w AnIxc+ o m m HHH HuH AHH++oV¢ w x+ :3--de w M I m m Axmuxwvz Q 3+. 3 101 xw mCHUMHmuH ng a»? 33 ....~.~ 0 3:0. o 0 go mCHUMHmwu g.mHIw o m.H 5H3 ”:5 ....H.H ___. ..:.V~N" ::N~NIII m} {H HuH HuH . m m flu.“ HHH HHH++9¢ w M AHH++0V< w M+ m .. m Axmuxwvz ==H~Hm ::M~H" mm.“ Mm“ HuH 3319 m w. Suzi :13: w 8.5+ HuH CH 3 ¢HMHVH+ 37-9 11 m x . . m xxmuxwvz 102 For the FM state in the five-fold d-subband degeneracy problem, the correlation functions are related: (3.51) A(o—-) E A(o—-ii) i 0, A(o++) s A(o++ii) # 0 All other A's are identically zero. There are two independent correlations as in the two orbit case, A(o--) and A(o++ , but there are three more of each since 11 and 22 terms are not equivalent. We find a diagonal 20 x 20 secular determinant with five subblocks. (3.52) A = N(eh-35) +(9K-4J) A(o++) E 11 N(eh-35) +5KA(o++) +4(KrJ)A(o--) E 22 11 with e£49 re- placing eh 11 with placing eh 103 (3.53) B = 11 22 11‘ with (Eh-IQ instead of e:h 22 with €3+Q instead of €12 (3.54) C = 11 22 11 with e192 in place of 6-15 22 with €539 in place of ek 104 (3.55) D = 11 22 11 Wlth 61's+52 in place of 63 22 With 6349. in place of 63 E = 11 22 ll Wlth ehflg in place of €22 22 with e Subblocks A through E are all the same here. 3+9 in place of £3 105 For the PM state in the five-fold d-subband degeneracy problem, the correlation functions are related: (3.57) A(o++ii) = A(o--ii) s A(o++) i 0 All other A's are identically zero. There is one independent correlation function, A(o++), but there are four more of them. We find a diagonal 20 x 20 secular determinant with five exactly identical subblocks: (3.58) A = N(€k-Ek) +(9K-4J) A(o++) = 11 11 11 With €192 replacing Gk 11 With €£+Q replacing Gk 106 The following subblocks are identical. (3.59) B = A (3.60) C = A (3.61) D = A (3.62) E = A We can transform the AFM 20 x 20 secular determinant for the five- fold degenerate subband case into block form as follows: Each of the five 4 x 4 subblocks of the 20 x 20 determinant can be changed from: (3.63) A 0 B O 0 C O D E O F O 0 G O H into: (3.64) A O B O E 0 F O O C O D O G O H and then into: (3 .65) A B O O E F O 0 O O C D O O G H 107 by permuting the rows and columns. The resulting minus signs have no meaning since the secular equation is homogeneous, and the minus signs factor out of the whole 20 x 20 secular deter- minant. (gggg that A, B, C, D, E, F, G, H here are just no- tations for the elements of a 4 x 4 subblock. They are in no way related to the same symbols that were used before for correlations and for terms in our development of a shorthand (2) Bloch in (3.16) - (3.29).) expression for V in (3.15) before we did the commutations We thus get, putting A(o++ii) = A(o_-ii) E A(o++) i 0 and A(Q++ii) = A(Qr-ii) = A(Q++) i 0, five identical 4 x 4 subblocks for the 20 x 20 AFM five-fold d-subband degeneracy case: (3 .66) A = N(e - ) k Ek (9K-4J)A (Q++_) + (9K-4J) A(o++) a 12 E 11 N (51923.13) 12 +(9K-4J)A(o++) E 22 (3.67) B = A (3.68) C = A (3.69) D = A (3.70) E = A 108 We now proceed to find the eigenvalues and total energies by expanding out our six determinental secular equations. Three are for the AFM, PM and EM two-fold degeneracy case. Three are for the ARM, PM and FM five-fold degeneracy case. We will then solve the resulting equations for the eight or twenty roots as the case may be (two or five-fold degeneracy respectively), and sum those roots to get the total energy for an electron in state .3, E3. Before we do this, we note that the five-fold degeneracy case for the AFM, PM, and EM states has a seemingly complex 20 x 20 secular determinant. However, this determinant is already diagonal in the PM and FM cases with only a few independent elements. It is easily diagonalizable in the AFM case when the 4 x 4 subblocks are all transformed to diagonal block form. we will not attempt Penn's SSDW or FIM states in the two- or five-fold degenerate case since the 20 x 20 secular matrix would be difficult to diagonalize. Also, the existence of these two states in real transition metals is in some doubt (except for Overhauser's work in chromium). The AFM, PM and EM are the most common transition metal magnetic states. The secular equation for the five-fold degenerate PM state re— duces to: (3.71) [N(ek-Ek)+(9K-4J)A(o++)]10-[N(ek+9fEk)+(9K-4J) A(o++)]10 = 0 109 There are ten roots of form: (3.72) + 125-1331)- A(o++) = Eku-lo) eE ._ and ten roots of form: (3.73) e (9K-4J) (11-20) .332 + N A(o++) =33: which are the same. The secular equation for the five-fold degenerate FM state re- duces to: (3 .74) [N(ek-Ek) +5K(Ao--) +4 (K-J)A (o++) JS-[N(ek-Ek)+5KA (o++) +4 (K-J)A(o—-) JS-[N(ek+Q-Ek)+SKA(o--)+4 (K-J)A (o++) 15- [N(ek_+Q.-Ek)+5KA(o++)+4(K.-J)A(o--)J5 = 0 There are five roots of form: (3.75) Eél—S) = 6k +'§5.A(o--)+-£1§:£L A(o++) five roots of form: (3.76) RSV-10) = e + 215 A(o++)+ -(——)-4 K-J five roots to form: (3.77) Eéll-ls) = €k+Q + 135 A(o--) + fi-(fi—“D- A(o++) and five roots of form: (3.78) Elfin-20) - +21S _ —€_}£+Q N A(o++)+fl1§:l)-A(o--) 110 The secular equation in the five-fold degenerate EM state is: (3.79) [(11)(22) - (12)2]1° = 0 which becomes: (3.80) {[N(ek-Ek)+(9Kr4J)A(o++)]-[N(€k+Q-Ek)+(9K-4J) A (o++) ]-[ (9K-4J)A(Q++) 12110 = o The solution expanding this out yields: (3 81) E 2N2+E [-N26 -N26 -2N(9K-4J)A(o++)]+[-(9K-4J) ' 15. 12 1: 3+9 (N€£+N€£+Q)AO++) + (9K-4J) 2 2]le2 (O++) +N2€1s€ls+9+ (9K-4J)2][A2 (g++)] = 0 Thus, the two groups of ten roots apiece are of form: (3.82) Eéi) = N2 k+N2ek+g+2u(9K.-4J)A(o++) 1' 2E? [-112 k-Nzekfl-2N(9K-4J)A(o+—I-) 12-4N2[-(9K-4J) (N€k+N<-:k+Q)A(o++)+1\126k<-:k_‘_£2+(9K-4J)ZJUX2 (o++)-I-A2 (Q++) J e e _ k kig (9Kr4J) 33. ‘[square root ”'5' + 2 + N A(o++): 2N2 above There are ten roots Eél) with the minus sign and then roots Efil) with the plus sign: Since the band is split in the AFM case, for the less-than or just half-filled band case, (1/2 n/ZN 5 1/2), we take all twenty IOOtS‘With the minus sign. 111 The secular equation for the two-fold degenerate EM state is: (3.83) [N(ek-Ek)+(3KrJ)A(o++)]2-[N(ek+Q-Ek)+(3K-J)A(o++)]2;0 There are four roots of form: (3.84) 83‘“ = ek + 1371;;“11 A(o++) and four roots of form: (3.85) E-éS-B) = 619-9 + L3-I§'£)-A(o++) The secular equation for the two-fold degenerate FM state is: (3 .86) [N (ek-Bk) +2KA (o-—) + (K-J)A (o++) 2 - [N (ek-Ek) +2KA (o++) +(K-J)A(o++)]2-[N(e 2KA(o--)+(x.-J)A(o++)]2 E+Q-Eli) + [N(€k+Q-Ek)+2KA(o++)+(K-J)A(o--)]2 = 0 There are two roots of form: (3.87) Efil'z) = ek +'%§.A(o--)+-1§§EL A(o++) two of form: (3.88) Eé3'4) - ek + fi—K A(o++)+‘K—;IJ-)- A(o--) two of form: (38% E§&6’=e €k&+-Akrfl +i——1Au»9 and two of form: (3.90) E(7'8) +-—A(o4—i-)+-u— array coninbunon from “when It.) from olocfron It) on; Figure 7. _Linerant Electron Contributions to PM Array. 114 and an AFM array might look like: contribution 'AFM) 0" '01 from electron It.) freer electron It.) etc. Figure 8. Itinerant Electron Contributions to AFM Array. As we have stated before, in the AFM state, Q occurs. Therefore, it also appears in the AFM correlation function. A(Q,..) will be seen for AFM only. This means that the AFM wave function has an L¥Qflg = LAEKE term in our simple cubic lattice, so the spin contribution is flipped in sign on alternating sites as £1335 oscillates between i 1. This Yields the AFM array seen in Fig. 8. In this regard, Penn took exactly the same approach. We must now generalize Penn's original definitions of the correlation functions, A, in terms of the linear combinations, B, to our two--and five-fold degenerate cases where the B's could be functions ofIE in general. The two-orbit general state have a wave function as a linear combination of eight terms; hence, eight B's. The five-orbit general state had a wave function as a linear combination of twenty terms; hence, twenty B's. 115 For the two-fold degenerate case, we generalize equations (2.91) - (2.96) to: F _ _ .k Fk 2 2 2 2 (3.96) A(o++) = 121T- (B2 +B4 +36 +38 ) Fk _ __=__ 2 2 2 2 (3.97) A(o--) — i N (B1 +83 +B5 +37 ) Fk (3.98) A(g+-) = JE-fi— (B1B4+ 3233 + BSBB+BGB7) Fk (3.99) A(Q++) = i-fi— (3234143638) Fk (3.100) A(Q--) = Efi- (BIB3+BSB7) We do not include the 11, 12, 21, or 22 combinations for the orbital indices since each A is either for orbit l or orbit 2. Therefore, 11 = 22 and 12 and 21 orbital correlations have been purposely neglected, as mentioned previously. Also, the l/N and "-" factors are excluded, contrary to Penn's work, so that we may carry them along explicitly. Similarly, the interaction coupling coefficients, K (Penn's Co), and J are not included in Penn'S‘work. For the five-fold degenerate case, we generalize equations (2.91) - (2.96) to: 116 _ 1 (3.101) A(o+-) - N E?— (8182+B3B4+13586+B788+89810+811812 +BlZBl3+Bl3Bl4+BlSBl6+Bl7BlB+Bl9BZ0) _,1 2 2 2 2 2 2 2 (3.102) A(o++) - N EEE(BZ +84 +B6 +88 +310 +812 +814 2 2 2 +816 +318 +320 ) _.1 2 '2 2 2 2 2 2 (3.103) A(o—-) - N 33ml +83 +B5 +87 +89 +B11 +813 2 2 2 +815 +817 +819 ) _.1 (3.104) A(Q+-) — N §F_(3134+BZB3+BSB8+BGB7+B9312+B10B11 +313316+Bl4315+Bl7320+318319) _.1 (3.105) A(g++) — N EFE(BZB4+B6B8+B10B12+B14B16+318B20) _.1 (3.106) A(Q -) — N EEK(BlB3+BSB7+BgBll+Bl3Bl5+Bl7319) We can rewrite these as: 19 (3.101) A(o+-) = z 5: F /N (B.B. ) i=1 odd 5 -5 1 1+1 20 2 (3.102) A(o++) = X ZFk/N B. . 1 i=1 even k - 20 2 (3.103) A(o--) = 3". Z Fk/N Bi i=1 odd .3 - 20 (3'104) A(Q*-) = 1:? odd E #E/N(BiBi+3+Bi+ZBi+l) 20 (3.105) A(Q++) = 2 2 F N(B.B. ) i=1 even _15 5/ 1 1+2 19 (3.106) A(g--) = 2 2 FE/N (BiBi+2) i=1 odd .5 117 where these i's have no relation to the lattice coordinate i's used before. They are just dummy indices to be summer over. There are certain simple relations between the B's which come out when we put in the relations between the correlation functions, A, using (3.95) - (3.106). In the two-fold degenerate PM case we have: (3.107) A(o—-) = A(o++) leads to-% EF£(312+B32+B52+B72) =I%.§?-(322+B42+B62+382) (3.108) A(o+-) = 0 leads to 1%- 1221115(131132+133134+135136+137138)=0 (3.109) A(Q+-) = 0 leads to 311- EFB(BlB4+B233+BsB8+B6B7)=0 (3.110) A(Q++) = 0 leads to % 35(3234J'3638) = o (3.111) A(Q--) = 0 leads to 1%.- 1:19];(131133835137) = o In the two-fold degenerate FM case we have: (3.112) A(o+-) = 0 leads to-% iFh(BlB2+B3B4+BSB6+B7B8)=0 (3.113) A(g+-) = 0 leads to 1% 1755-—(131134+132133+135138+136137)=0 (3.114) A(g++) = 0 leads to 311- EF—(B2B4+B6B8) = o (3.115) A(Q--) = 0 leads to-% mk(8183+8537) = o 1‘.- 118 In the two-fold degenerate AFM case we have: — l — (3.116) A(O+-) — 0 leads to N EBE(B1B2+B3B4+B5B6+B7B8)—0 — l — (3.117) A(Q+-) — 0 leads to N EFE(BIB4+BZB3+BSB8+BGB7)-0 — -‘ «I; .- (3.118) A(Q++) - A(Q ) leads to N EFE(BZB4+B6B8) — ‘l VF (B B +B B ) N k.h 1 3 5 7 In the five—fold degenerate PM case we have: _ .1 2 2 2 2 2 (3.119) A(o )—A(o++) leads to N EFEGl +B3 +B5 +B7 +B9 2 2 2 2 2 _,1 2 2 2 2 +B11 +813 +315 +B17 +319 )— N EEK-(B2 +34 +36 +88 2 2 2 2 2 2 +B10 +312 +814 +316 +318 +320 ) _ .1 (3.120) A(o+-)—0 leads to N k k(3132+B3B4+BSBG+B7BB+B9BlO +311’312‘L’313E’1zf'131513163171318443191320)=0 _ .1 (3.121) A(Q+-)—0 leads to N EFE(B1 B4+B2B3+BSB8+B6B7+B9B12 320+318’319)=0 +813B16+Bl4315+317 _ .1 (3.122) A(Q++)-O leads to N EFE(BZB4+BGBB+810312+B 14316 +318320)=° _ .1 (3.123) A(Q--)—0 leads to N EFE(B1B3+B5B7+BgBll+813B15 “3171319)=0 119 In the five-fold degenerate FM case we have: _ 1 (3.124) A(o+-)—0 leads to N {F12(Ble+B3B4+B5B6+B7B8+B9B10 +BllBlZ+Bl3Bl4+Bl5B16+Bl7BlB+Bl9BZO)=0 _ .1 (3.125) A(Q+-)—O leads to N EFE(BIBZ+BZB3+BSBB+B6B7+ B9B12+BlOBll+BlBBlG+Bl4Bl5+Bl7BZO+Bl8Bl9)=0 _ l . (3.126) A(g++)—0 leads to ZFk(B2B4+3688+310312+314B16 N.E'— +BlBB20)=O _ 1 (3.127) A(g -)—0 leads to N Efik(BlB3+BSB7+B9Bll+Bl3BlS “3171319)=0 In the five-fold degenerate AFM case we have: _ 1 (3.128) A(o+—)—0 leads to N EFE(B1B2+B3B4+BSB6+B7B8 +B9BlO+BllB12+BlB314+BlSB16+Bl7BlB+Bl9BZO)=0 _ .1 (3.129) A(Q+-)-0 leads to N 1(th(B1B4+BZB3+B5B8+B6B7+B9B12 +BlOBll+Bl3Bl6+Bl4Bl5+Bl7BZO+BlBBl9)=0 _ -_ A (3.130) A(Q++)-A(Q ) leads to N ill(BZB4+B6B8+B10B12+B14Bl6 _.l +318320)‘ N E33(3133+BSB7+39311+B13315+317319) In the five-fold case, we note that relations (3.120), (3.124), and (3.129) all stem from A(o+-)=0. (3.121), (3.125), and 120 (3.129) stem from A(Q+-)=0, and so on. The same correlation function relations in different magnetic states lead to the same relations among the B's. This is also true of the two-fold state. We now express our roots, Eél) in terms of these B coefficients. In the two-fold degenerate PM case we have: (1’4)- .125221 2 2 2 2 (3.131) E3 — €£+ N 1533(82 +B4 +36 +88 ) and (5'8)- .léfizfll VP 2 2 2 2 In the two-fold degenerate FM case we have: 2+3 2+8 2)+'15391 SF 357 Nk_ (1:2) = .25 k(B12+B k _— (3.132) Ek 6k + Z 2 2 2 2 (B2 +B4 +86 +38 ) (3,4) _ 2K 2 2 2 2 (B1 +B +B5 +B7 ) 3 2K (B 2 2 2+B72)+ (K-J) 2F (5.6)_ .__ .5 .3 2 2 2 2 (32 +b4 +86 +38 ) (7.8) _ g 2 2 2 2 _(ggfl EN — €E+ N ;E_(B2 +B4 +B6 +38 )+ N EEK 2 2 2 2 (B1 +B3 +35 +37 ) In the two-fold degenerate AFM case we have: (1-4) (5-8)_E_L_ 61962, (3K-J)_ 2 2 2 2 — + T ZFk(BZ+B4+B6+BB) (3.133) BE 2 2 N k-— 121 2 2 :t __1 [N2 e k+n e +2N(3K- J)2F (B2 24+B +36 28+B 22)] 2N2 ]“9 5 Fk -4N2 [N2 €k_€_+g+(3K'J) 2[{EN-151322+B42+B62+B82) }2+EN= (B B B B )121-(3K-J) (N +N )2 3&2 2+b 2+B2 +B8 2)] 2 4 6 8 - 9; Eyck 4 6 We will use the compressed summation notation of (3.101) — (3.106) for the correlation functions in terms of the coefficients, B, in the five-fold degenerate case. In the five—fold degenerate PM case we have: 20 (3.134) Eé1'10)=ek +M2Fk( 2: Biz) - - k-— i=1 even 20 E1:11 20; + (9K 4J) 2Fk( 2 B 2) e . 5+9 N k-i=leven 1 In the five-fold degenerate FM case we have: . 20 20 (3.135) Elél-S) = (33+ 335 k( 212)+ 415N912FN( 2 Biz) 3, i=1 oddB .3 i=1 even 20 20 131:6 102-e3 N521“ ( 212)+ 11:51:13“ 2 Biz) - .k'- i=1 evenB k,k i=1 odd (11-15) 5K 20 4(K- J) 20 2 Ek ‘€k+Q+N—2Fk( 212)+ N 2Fk( 2 Bi ) - - ‘k- i=1 oddB k'- i=1 even 3‘15‘201 +3523 ( $0312) 444;“ J 2:» 20 B 2) _15 “6154.9. N k + N k J; — i=1 odd 1 In the five-fold degenerate AFM case we have: 122 (l-lO) _ 6k Gk _ 20 2 2 N ._ .k 1 ._ 1-1.E even F 20 1 . _lg 2 2 2 iENZ [N2 €k+N2€ k+Q -2N(9K- 4J);— N (121Bi )] +N’e‘lgekHg even 2 20 Pk 2 2 -4N [-(9K-4J)(N€k+N€k+Q)F 21 (z-N- 8i )+(9K-4J) - k even 20 Fk 20 Pk [z (2:--B_.-L 2)+'z(z--=B B. )1] i=1 k i= 1 k N 1 1+2 even even We note that factors (9K-4J) in the five-fold case and (3KrJ0 in the two-fold case have emerged. They will do so next in the total energy. This is very significant as will be shown later. Finally, we get the total energies of the three states in both the two- and five-orbit case in terms of the coefficients B. For the two-orbit case (where the factor of 2 is again inserted in the correlation terms denominators to prevent double counting): . B.Z F B 2 F B 2 F 2 orb1t-PM_ ' k _ ' °.d§ ' ' .3 (3.137) ETOT — 2 E 'N- EE— 2 E N €E+2 E 'N— Fk '- " '— ._ 2 2 2 2 [(3K MEN—(B2 +B4 +B6 +B8 )] f‘ B.Z. Fk B. 2. F 2 orb1t-FM_ _. k (3.138) ETOT — 2 E -N—6N+2 EN ‘[(3K— J){EE-(B2 2+ Bz+Bz 2 Fl: 2 2 2 2 4 6 +B8 )+§jN(Bl +B3 +B5 +B7 )}] 123 . __ B.Z. Fk B. 2. Pk Fk (3.139) 122 Orb“: AFM=2 z --= 6 +2 2 —=[ (31<-J)2--(132 2 TOT N k 1‘. " E. .35. 2 2 2 ’ ‘ +34 +36 +38 ) . 2 2 F13 2 2 2 2 + §5+§k+g‘ [N2 €k+N €k+Q+2N (3K+J)z— N2(B +34 +B6 +382)] I l I l : -4N2{N3€k€kig-(3KrJ)[N;£+NeEHQ] : FL 2 F E- (E N +34 +36 I2 _ _ l 2 F1; : +B82)12+[z—§(B2B4+B6B8)]2}} E l 2 2 2 2 ___]: 2 (132 +138 )+(3K-J) {[1}: N (B2 2 +B +B éh 4 6 _.J where the kinetic energy term in (3.139) should more explicitly be written as: half half Brillouin Brillouin zone Fk zone Fk+g 2 -‘= 6 + 2 e kNle 5+9 NJ§+Q For the five-orbit case: (3 140) E5 orbit-PM = 513.2. Fk B.Z. F F___ 2 TOT '3 N 2 2 2 2 2 2 2 2 2 (B2 +B +B +B +310 +B12 +Bl4 +Bl6 +B18 +320 )] 4 6 8 B.Z. F B.Z. F S orbit-EM _ .d_ _45 TOT — 5 E N e£+5 E N (9K-J)[E (3.141) E F 222222222 2:]: (32 +34 +36 +B8 +310 +312 +13l4 +316 +B18 +3 20 2)+ 2N 2 2 2 2 2 2 2 2 2 (BliB3 +35 +37 +39 +311 +313 +315 +317 +319 )] 124 B.Z. Fk Fk B. E:- GEE-+5]: E“(9K-J)[ Z 5 orbit-AFM= TOT 20 ( 2 Biz) i=1 even (3.142) E S 7: .5 F 2 _ \ 25 2 2 2 2 .E+Q+2N (9K 4J,§ N(B2 +B4 +B6 +38 2 2 2 2 2 2 2_ 2 2 +1312 +1314 +1316 +318 +1320 )] 4N {N else-3+9 Fk _ 2 2 2 2 2 .£+Q](§ N (32 +34 +36 +38 +310 [N2€k+N2 +le-0 -(9K-4J)[Nek+N€ 2 2 2 2 Pk 2 2 _ 2 Z.F +13 +B +B +B +13 ))+(9K-4J) {[2—(13 fit + 12 14 16 18 20 k N 2 €E+€kfi9_ _ 2 2 2 2 2 2 2 2 2 2 +B4 +36 +38 +B10 +312 +314 +B16 +B18 +320 )] F .15 2 J +[ T'E (32B4+BeBa+BioBlz+314Ble+318B2o) 3 } -263 k where the kinetic energy term in (3.142) could be more explicitly written as: half half Brillouin Brillouin ‘é zgne ‘5; +.§ zon; Ektg 2 k N E); 2 k N €195; We note here that the restriction to one Q, Q: g/Za has nowhere been used explicitly in the antiferromagnetic state. This choice of Q would correspond to an antiferromagnetic order with wave length a in real space. One might think that in general one could write any term containing Q in terms of a generalized-Qj = IQ/ana, with n.=l, nj>l = an interger 22. (For example'g2 = ‘g/4a would correspond to an antiferromagnetic order with wave 125 length 2a in real space.) Thus, for example, any correlation, A(Q++), occurring in an energy eigenvalue or total energy, would become $A(Qj++). J The fallacy in this argument, which seems to imply that our off-diagonal terms in the 8 x 8 or 20 x 20 secular matrix would just become sums of the form indicated is seen when one con- siders the mapping between 3 and h +IQ in the Brillouin zone. For example, in two dimensions, the h and 5 +IQ pairing in the Brillouin zone in‘E space: It) k+Q Ix? Figure 9. One to One Mapping of; and 5+ Q in Two-Dimensional Brillouin Zone Projection. corresponds to: I __t_—__ \ Io ’ _.( ”r k Figure 10. One to One Mapping of k_and E.+ Q_in Dispersion Relation. in one 5 direction in the dispersion relation. For 9 = g/Za, there is a unique 1:1 pairing of states 5 and 5 +.Q, both in the Brillouin zone and the dispersion relation. However, for ‘Q. = g/2nja, a 1:1 pairing breaks down giving a many to one 3 pairing, which one does. 127 Figure 11. Many to One Mapping of k and §9+‘Q%in Two-Dimensional Brillouin Zone‘Prgjection. k Figure 12. Many to One Mapping of k_andkl+ g in Dispersion Relation. 128 Thus, an attempt to amend the antiferromagnetic secular deter- minant to include arbitrary Qj is clearly doomed to failure because of this breakdown of 1:1 mapping ofik andefig. One 5 would need to make each off diagonal term equal to K 2 A 5 i=1 ++ . (Qj++ii)+K Z.A(Qj --ii). Then, being four in each subblock, i72 each term would become 2 x 2 larger for each Q added. The diagonal term would change from ekfig to e or Tekfig, making two new - j .Efigj diagonal terms for each new Qj This consideration will not be important for the general para- magnetic and ferromagnetic states we consider since these two states have diagonal secular matrices and no A(g)'s appear; thus, the gap parameter, A50. There is no exchange splitting of the bands, and the question never arises. We now calculate the self-consistent conditions for the energy in order to decide for which values of n/2N, J, K, and T, the ETOT is minimized, and thus the FTOT is minimized. The minima, not necessarily coinciding, tell us which of the three states, paramagnetic, ferromagnetic, or antiferromagnetic, has the lowest free energy and is thus stable. A direct comparison of all three simultaneously is complicated, so we try to compare FAFM(J, K,n/2N, T) with F% (J, K, n/ZN, T) first, FM and then FTOT (J, K, n/2N, T) with FP (J, K, n/2N, T) second. TOT 129 64 65 Matsubara and Yokota and Kemeny and Caron have derived the equation for the energy gap, A, used in the first comparison. We reproduCe Kemeny and Caron's derivation. We note at the outset that Kemeny and Caron's J becomes our K, the direct coulomb coupling constant. Our J is the exchange coulomb coupling constant, absent in their work. Also, Matsubara has an (IO-211) coupling constant, where I0 is an intra-atomic "exchange integral" and I1 is an inter-atomic "exchange integral". Since we are working in the Hubbard Model, we have no inter- atomic term in our potential part of the Hubbard Hamiltonian. Therefore, I1 is absent in our work. Also, it is not clear whether Matsubara's I0 is really an "exchange term". Our total coulomb coupling coefficient in our energy eigenvalues and total energies is (9K-4J) for the five-orbit degenerate case. To derive the energy gap equation, the cornerstone for our first comparison involving a self—consistent calculation for the AFM state, we define a single particle symmetry breaking potential HOF. (3.143) 22 Aia . id 10 where Aia is a self-consistently defined Hartree-Fock potential seen by an electron in one of the S-d subbands. We generalize Kemeny and Caron's expression for A (their J becomes K in our notation): 130 H.F.) = Kn_ (3.144) (A10 a i,-o,a putting in an a as a subband index since they were considering one s-band. Summing over the S-d subbands we find: 5 5 5 H F. H F (3.145) A.‘ = 2 (A.' ') = K 2 n _ +(K-J) 2 io d=1 lo a d=l 1, O.o 5(fa)=1 ni,-o,d = (9K-4J)ni'_ a generalized total A for all the five subbands (although we may still talk of five Ad's, one for each d-subband, and consider their gaps separately). Note that since we have introduced exchange by adding d-subband ‘degeneracy on each site, our A in a true Hartree-Fock gap para- meter. In single s-band calculations, A is actually only a Hartree gap parameter, since exchange was not included. The Hubbard Hamiltonian in the Matsubara approximation becomes: _ H.F. (3°146) HM ‘ izj g Tij Cio Cjo+f g Aia nio I . in Wannier space. The potential term is in effective one-body form, equivalent to our Gorkov factorization result previously, if AfiéF' is taken as a parameter. In‘k space, this becomes: H.F. A = (9K-4J) c + (3.147) HM i§[e(3)+ 2 JNEO+E§ 2 [ghogk+r.o + 131 where 3 =IQ = g/z (as in Penn's paper) and.where (3°l48) A - 1252411 = + A for up spins 0+ 2 A = “o _ Ao— - Aggiggl = -A for down spins is related to our Bl(k), B3(k) coefficients in the magnetic phase wave function: AFM _ (3.149) w+ _ 31(3) “121+ +B3g) “3991+ by: e k l _ (3.150) Bl(k) = -§(l- -————--) V A +th e‘ k l _ (3.151) 33(3) ={§(1+ ) VA5+OE2 where we must remember that B1 is equivalent to BZ’ B5’ B6' B9, 310,813,B14,B17,B18 since any of the pairs (B1,B3), (BS'B7)' 39,311), (313,315), (317,318) yield a YiFM'while any of the pairs (32'34)' (Be'BB)' (BlO'BIZ)’ (314'B16)' (BlB'BZO) Yields a YEFM. Any of these combinations will allow description of a pairing of a state 5 with state 5 +.Q, the antiferromagnetic correlation. Continuing our analogy with Kemeny and Caron's work, the eigen- functions and eigenvalues are, for each d-subband, d (and for Bloch wave electron states): 132 in-R (3.152) Yi‘FMa (_13.)= —l eik'gual 09.33 (ye - -J Ecifls) =" (82+ek2 +12£§£~ll (3.153) 2““ (R)= lei3'3[s3(1<_)i31(k)eifl'5] E o(£) = + A2+ e 2 + (9K-4J1 2 .k 2 which can be written as above since if A(o++) = A(o--) in the state and A(Q++) = A(Q--), then: (3.144) FF B1 = FkBZ = FkB3 = BF 2 Y B4 is a solution since: 1— 1— 1— 1— xi W 2 2_ 2 2. AFM2_ AFM2 (3.145) EFE(Bl +33 )—EFE(B2 +34 ) 1.8. Iv+ I — |w_ l the number of up spins being equal to the number of down spins. A self-consistency condition is written by Kemeny and Caron66 as: (3.154) 11+§=§ z |111k_(0)l2 k 2 k 0, when the gap shrinks. F Fk T"() T'=() :~‘~ ‘\\ / "\ ‘\ 1' \\ \\ / \ ‘ GAP -~~ I \ \ "' I \ \ \ \ l \ 1 i ‘1 k = 0 kBI kBZ (u: 5- Figure 15. Fermi Factor Dependence on 5: Including AFM Gap. 138 we note the physical ideas embodied in this equation. The first term, with a coefficient, K, favors the AFM state while the second term, with a J coefficient, favors the AFM state by tending to increase the gap parameter magnitude and, therefore, the gap width, 2A. Thus, as more and more exchange between degenerate d-subband electrons takes place on any site, the tendency toward the AFM state is increased. J = 0, from an s-band {__L- V configuration on each site, gives the least tendency to the AFM order. As J increases from zero, new terms are added on each site. This is illustrated with the following progression: 139 3 row DEGENERATE 2 FOLD DEGENERATE | 2 I k k I / ‘f’ L '1‘") J ‘ FOLD DEGENERATE 5 FOLD DEGENERATE Figure 16. Direct and Exchen e Cou lin e in 2-, 3‘1 4-1 and 5- Fold Degenerate Ceeee. 140 Of course, the extent to which this happens is governed by the n/2N filling of the whole d-electron band. Not all subband states will be occupied by electrons if the filling is not per- fect. Actually, for each J term added, a direct coulomb K term is added so that this effect would cancel out if J “ K, in Which case 9K - 4J would become 5K, the direct coulomb interaction of five electrons in the five d-subbands with no exchange coupling constant. But J # K in general, although there will be a J = K plane in the three-dimensional (n/2N, K, J) parameter phase diagram we will plot. In this J E K plane, introducing five d-electron exchange as we have done, the tendency toward the AFM state would not be changed. It would raise the direct coulomb energy from Penn's Co to 5K, a five-f01d increase in the coulomb interaction energy evaluation from the Hubbard- Hamiltonian since presumably for an s-band, Penn's Co 2 our K. We will also alter this tendency for the AFM state in the band filling, n/2N, manifested by the subtractive effect in the brackets { } in our master self-consistent equation for A for filling above the half-filled d-band. For n/2N l/2) goes to l, the completely filled d-band. When 141 n/2N E l, the gap equation degenerates and the PM state should dominate. These ideas match those of Penn, and are mirrored in his three state phase diagram of Co/E" versus n/2N. We note two further details to be used in the calculation. Firstly, to find Ftot = Etot-TS, we need S. We can use the equation for the entropy of a Fermi gas: ( -U)/1<_ T _ (E -U/1 T _ (3.171) S = fikB-lO-${[l+e 35 B ] l ln{[1+e k B ] l k (E -U)/k T _ (E -U)/£ T _ +(l-[l+e 'k B ] 1 1n{l-[l+e k B J 1}} where degeneracy in S is accounted for in the self-consistent u and Ek calculation. It is now a clothed Fermi gas, rather than a free one, since in an energy band, me gets renormalized to an effective electron mass me*. We, however, cannot use the usual formulas for the chemical potential of a free Fermi gas which are only valid when n/ZN is very small or very large, but not near the middle of the n/2N axis, i.e., the half-filled band. (i) Secondly, our expressions for Bk = ZEk must be amended as _ i— Penn does since in just summing up the eigenvalues, we are over-counting the interaction. In general: 142 (3.172) ETOT = E where: Ek = ZEk In our work: Ao+~ = 52+” = 0 for every state we consider, so that: (3.173) ETOT = k.EFE-N(A°++A°“+AQ++§Q'-) For our three magnetic states this becomes: (3.174) E20§°ld-§MEFEB£¥-NA3++Since AO++;AO__and AQij=O (3.175) Egogold_§MEFkE:M-NAO++AO__since AO++fiAo__and.AQij=0 (3.176) Egoiold_A§ME?EF§EM-NA§++-N§§++ since Ao++=Ao-- and AQ++=AQ-- The general energy expression can be written in terms of the B's as: F 5 fold_ _ .3 2 2 2 2 2 2 —zF 3k N{[E N (B2 +34 +36 +38 +310 +312 + (3.177) E F 2 _5 2 2 2 2 2 +320 J E N (B1 +33 +35 +Bll +313 + +B +B 2)]°[T-EE(B B B B +B B +B B + 17 19 g N 1 3+ 5 7 9 11 13 15 2 N 2 B15 317319)]} We have not put Co=(9K-4J) in the denominator of these subtractive expressions since we did not include any coupling constants in 143 our definition of the A's, whereas Penn did. Thus, he needed to divide by Co to make E dimensionally correct; we do not. TOT Our three final total energies are now: F 5 fold-EM RM_ .3 2 2 2 2 2 2 2 2 (3.178) ETOT — E3333 N[E'fi- (B2+B4+B6+B8+B10+B12+Bl4+Bl6 2 2 +BlBBZO)] 5 fold-EM FM; 2 2 2 2 2 2 2 (3.179) ETOT k _ _ 3 +B +B +B9+B11+Bl3+B15 k 2 (B1+B 5 7 F 2 2 _jg 2 2 2 2 2 2 2 2 2 +Bl7+B19)]-[E N (B2+B4+B6+BB+BlO+BlZ+B14+Bl6+B18 2 +BZO)] 5 fold-AFM F FM_ .3 2 2 2 2 2 (3.180) ETCT — E737: N[{E—§(BZ+B4+B6+B8+B10+B12 F 2 2 2 2 2 .3 +Bl4+Bl6+BlB+BZO)}-iN (31333537393 Fk '— E'N‘(3234363831031231431631832o)3 11313315317319)° We now derive our general subtractive expression: _ _ _ 2 _ 2 (3.181) ETOT—Esksh N(AO++AO__+AQ++AQ__ Ao+_ AQ+_) from Penn's: + + + . = > > _> (3 182) ETOT E06£ ._ > (CLaCLc (CLOCLO J =ZF B would double for completeness, simply by noting that E k k_... TOT 144 count the effect of the various correlations on E Thus, the TOT' true ETOT must have them subtracted off. We now outline the self-consistent decision procedure to decide whether at a particular point (n/2N,K,J,T) we shall find FggTfif FggT’ The minimum free energy will be the criterion for phase stability. First we solve our master gap equation (3.168): K 9 kBl F3 kF F15 (3.168) 1=[-fi 2{>: ———-——- 2 __1. j=l k=0 2 2 k=_B2 2 2 J A +8E A +6E J 4 kBl F3 kF F15 3 2 { z - z 1 1 3—1 _1_<_-0 A2+€k2 k-kBl A2+€k2 for A by guessing a value of A(=A(l)): i) . ( . ., We know. (a) BE as a function of the bi s, 93'93+Q’E3 (b) Fk as a function of T, U, Eél) (c) B1 as a function of 6k, A (d) B2 as a function of 6k, A 3 ._ (e) ek/4T = - 1/2 121 cos ”1' Q = 5:3 Now (a) and (b) together form a self-consistent calculation since we need Fk(A) and Efi1)(A) to solve the master gap equation above. 145 Second, we see if A is satisfactorily computed from our master gap equation, (=A(2)), by comparing with A(l): A(2)-A(l) s an error factor If so, we have found the correct A and can proceed to calculate Eggg as a function of A, Bi(A). If not so, we either increase or decrease our A and find a A(3) from the master gap equation. We then test A(3) as we tested A(Z), etc. This is conveniently summarized in the flow-charts (Figures 18 and 19). The testing of FPM versus FFM to determine whether PM or FM TOT TOT has a minimum for a given parameter set (n/2N,K,J,T) is different from the AFM-PM calculation since a gap equation cannot be used. Here, we must write an E+ for up spins and an E- for down spins, sum them to get an E and then minimize E with respect TOT’ TOT to 6= n+-n_ , the difference between up spin and down spin electron densities. A 6:0 determination means the PM state is stable for a particular parameter choice (n/2N,K,J,T). A 6>O implies a FM state, with varying degrees of net spin polarization from n+~n_ so that 6>O, but just barely, to n so that += ntotal 6 reaches a maximum, the completely polarized FM state. (The PM-FM.flow chart is given in Figure 17). 146 Figure 17. FM—PM Flow Chart. 147 n’QfiUALx Igygg' SPIN POLARIZATION LOOP K14! J/4T SELF CONSISTENT CHEMICAL POTENTIAL 00? YES NO shifts as 'r/4'r. n/ZN ' C e CALC 0 Change) SUM31 SUM3 Alter No U/4T‘ YES CALCULAT 8T9344T lx=o.o,o.1 x=o.o,o.1' CALCULATE FSLOPE/4T PM FM 148 Figure l8. AFM—PM Flow Chart (Any T, K/4T < 3.5)- 149 ( START ) SELF CONSISTENT BK CALCULATE EKBAND/4T . ‘ SELF CONSISTENT CHEMICAL CALCULATE POTENTIAL SUM 31 or SUM NO _ Alte -n/2N U/4T YES YES r CALCULATE E" 1’. J CALCULATE 4k; CALCULATE OT/4T (X=0.0 (48*9*K L J ___. _48*4*__ 4T 4T *2 FK*n/2N KSQRT _A 2 EKBAE-ID 2 r (’TF'”) ) PM ENERGY NO ‘SELF CONSISTENT lter DA A/4T 'ND GAP CALCULATE 81K and 83K CALCULATE EK2/4T 150 Figure 18 (cont'd.) 151 -EK/4 NO 1 YES CALCULATE ETOT/4T CALCULATE ESLOPE/4T=ETOT(PM)/4T - ETOT(AFM)/4T NO NO YES ‘ YES PHASE FM BOUNDARY STRADDLED 152 Figure 19. AFM-FM Flow Chart (Any T) Q4122 3.5) . 153 START 4“) 1! SELF PARAMETER ‘ INPUT CALCULATE 7 SELF E“3AND/4T _ CONSISTENT CONSISTENT EK CHEMICAL POTENTIAL CALCULATE SUN 31 or SO. 32 NO _ Alter -n/2N U/4T YES YES CALCULATE ‘ . , . ‘ ETOT/4T for x=o.1 [, CALCLLATE F“ J 0*? ' )L * FTOT 4T for X=0.1 /T > 0 CALCULATE . -l:u;-!F-“_-‘-_-‘ {48*925.-43*4*i_ 4T 4T *2 FK*n/2N -;~«,~ A KHAN ENERGY '— :ELF Alter ONSIGTENT A/4T AND :AP CALCULATE 81K and 33K CALCULATE EK2/4T 154 Figure 19 (cont'd.) 155. NO =EK/4T YES CALCULATE ETOT/dT CALCULATE ESLCPE/4T=ETOT (PM) /4T - ETOT (AFT-Z) YES YES PIZASE PM . . BOUNDARY Am" ST PADDLED (‘1 r 156 We programmed an interactive summation over 1540 points in 1/48 of the Brillouin zone, allowing for a 540 point overcount by weighing points on the [111] axis by 1/6 and points on the l/4 zone planes by 1/2. Therefore, we effectively count 1,000 points per 1/48 of the Brillouin zone. The renormalized energies, EKl, Ek2, were calculated as functions of J/4T, K/4T, n/2N, x, U and EBT and used in the appropriate total energy (at small kBT). In the FM—PM comparison program, U was iteratively cal- culated as a sum of Fermi factors and compared with its initial value until a self-Consistent U was achieved. In the AFM-PM com- parison, this was done for A as well as U. The A and U itera- tive loops are concentric so that one self-consistency depended upon the other. In the FMrRM comparison we calculated FTOT (X=0.0) and FTOT(X=0.1), X being the net spin polarization for constant kBT, n/2N, J/4T, K/4T and US The sign of the OCOFO. slope of the difference, [F (X=0.0)-F (X=0.1)]/increment n/2N TOT TOT yielded a measure of whether at any point in the K/4T, J/4T, n/ZN three dimensional space, the FM state had lower or higher energy than the PM state. If a small slope was found, n/2N was increased by 0.01 until a large slope appeared, indicating a FM-PM second order phase transition line had been crossed. Then a smaller K/4T value was chosen and the same process redone. Finally, J/4T was manually changed and we then had derived the FM-PM phase diagram for a given kT. Indicating electron and hole symmetry in the problem, the n/2N axis only when up to 157 n/2N 0.5 since Penn's phase diagram is symmetric about n/ZN = 0.5. As we expected our FM region to be larger at low kBT,we limited n/2N to 0.2, Penn's lowest FM phase boundary intercept of a horizontal line. Our FM phase boundary line should be outside his; hence, intercept some horizontal line (some K/4T value not necessarily-his) at a lower n/ZN value. The flow charts for both programs are given with the running FM-PM program. High kBT and low kBT cases were run on the same program with appropriate instructions for skipping the entropy calculation for low kBT (a.0.0001) and summing Fermi factors = 1 or 0, rather than doing out the actual exponential summa- tion of Fermi factors as was done at higher kBT. In this program, we sum 5 over 1,000 points in 1/48 of the cubic lattice Brillouin zone so that, since the number of states equals the number of atoms in the solid, N = 48,000. For every ; we have a l/N to normalize EFk/N to a number less than 1. N is meaningless when we compare total free energies of the various states; it cancels out. 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IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII onemeoazoo eoem mo ozm IIIIIIIIIII o xxxm.e.oamoe .amaoeezmom oem e.z~>Hoz.x.o.ozmmoxm.eommHo.eoem.oem azumm oom ¥mozHazoo omm GZHmOHNB IIIIIIIIIIIIIIIIIIIIIIIIII BZWZZOU IIIIIIIIIIIII U Heoem+xxmveoamuflxuveoem mmm AouvmeHeoemuHeoem A Axu.avIz~>HoZIAx+.avIz~>Hozo*A.ooomc\.oc~o a IA22Hoz meHz zomHmmazoo w mo ozwIezmzzoonuququ Ao.mamm .omaveng*.xm.*aoamavaaozIAx+.avaz~>aozvag.ooomc\.oc~o a IgzzeNIAoaoNm+zHNexoaoamo.x.ooome\.oemvuaaoam owe fianceexxxoavNaomxm+.avuzzagz.aommao .mmm azamm mmm Aabe.Xm.«~mzbma.xm.aamzamaa .«a« .xm.eoa .xm.«z~\zI .xm.axe .xm .aoaamzoaoz.aoumao .mmm a2amm com A *D* .xm.*NmEDm* .Xm.«amZDm* .Xm H .xm.*aa .xm.«oe.xm.*2m\z* .xm.*M¥.xm.eoaamzo«z H ozHggam ozam zN>Hoz mama mom mmogm ommN mam aoame .xoa .amav amzmom amm amm aszm omm amm oa oo Ao.o .ag. mmogmmo ma aam mmm oa 00 10.0 .ao. maOAmmv ma mam a.o\AaaoamINaoamvummogmm mam A.c.ucoov o magma 168 QZM mozaazoo mmmm mozHazoo ammo Aaaazmza 243a mmozIImmooa o azaz ooa Io aagz.aommag.omo aZHmm mmo A*B*.Xm.¥h¥.Nm.%ZN\Z«.Nm.*M*.Nm.¥UHBMZU¢ZOmmmm*V demom wmo vmo aZHma mmo ammo oa oo A*¢.am.xm.a.am.xm.a.am.xm.c.am.xma«va¢zmom amo a.ozgmoxm.zN>aoz.aommHo.amo azamm omo Aaae.xm.Io*.xm.*2m\z*.xm.*M*.xm.aoaamzo«z oZHAgHm ozcm z~>aoz mama mom maogm ommN mam aoamev aczmom ace avo aZHma oco mmo ca 06 Ao.o .ag. mmogmmv ma omo mac oa oo Ao.o .ao. mmogmmv ma oao a.o\aaaoamumaoamvnmmogmm moo Aao.ouxvaoaanaaoam IIIIIIII o Amoaoamuaaoam coo Ao.onxvaoaaumaoaa IIIIIII azmzzoo IIIIIII o Lavaoaaumaoam com I IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII ozHaoom mgooamam aoam mo o2azzaomm IIIIIIII o I. IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII zoaaaaoazoo aoam mo ozm IIIIIIII o IIIIIIIIIIIIIIIIIIIIIIIIIIII mzHaoom mgoocmam mom mooa ao.o.o.ouxrazmzzoonuunnuuuo mo2aazoo mom Aec.am.xm.c.am.xm.e.am.xm.c.am.xm.v.am.xm.c.am.xm.c.amevamzmoa mom o.x.a.ozaoz.aoam.mom aszm mom x.o.ucoov o magma 169 case. We need only heuristically prove why this was so in his calculation to show that it will also be true of our phase diagram at any temperature and for any J/4T value. The Hubbard- Hamiltonian contains, in the interaction term, the two-body + + operator product a a a + + a, which becomes a a after Gorkov factorization. These may be written in number operator form, since n = a+ a, as n n and n respectively. Whichever we choose to consider, we know that the number of d-type particles, be they electrons (to the left of 1/2 n/N) or holes (to the right of 1/2 n/N = 1/2), enter as n2. Now, the number of electrons added to the number of holes is unity. Calling n the number of electrons and h the number of holes: (3.184) n + h = 1 Thus: (3.185) n2 = (1-h)2 which becomes: (3.186) n2 = 1 - 2h + h2 Now the factor l 18 a constant and can be absorbed in HHubbard by redefining our origin of energy when we calculate the eigen— I II _ u - ' ' values of HHubbard' the UK s. The 2h factor 15 linear in the number of holes, while all interaction terms are quadratic in the number operator, be it n for electrons or h for holes. Thus, "-2h" is absorbed in the definition of ek in the kinetic energy term and 2 (3.187) n = h2 170 as far as the interaction term goes. So the 1/2 n/N > 1/2 and 1/2 n/N < 1/2 halves of the phase diagram as symmetric about the half-filled band, 1/2 n/N , 1/2, ordinate, for any k/4T, J/4T, or temperature T/4T. The utilization of five-fold degenerate electrons has no bearing on this: In Penn's diagram we have: (3.188) n 2 = h 2; S S in ours, we will have: 2 _ 2 (3.189) nd — hd . A further simple result will help us eliminate the number of iterations necessary to calculate the chemical potential as a function of temperature/4T67. Kittel plots chemical potential/EF as a function of kBT/EF for a free electron gas in three dimensions. Since our Gorkov factorization reduced our d—electron system to a system of quasi-d—electrons, we can use Kittel's diagram, re- produced here, to help us select a best first guess for the chemi- cal potential in our programs. I‘VE; 1.00—* .795_ 1/2 n/N % O, 1 .99» .98, .97» .96" .95r -—————————_— 1 1 k T/E 0 0.05 0.1 0.2 B F Figure 20. Variation in Chemical Potential as a Function of Temperature. 171 From this we can read off some interesting results. At zero temperature u = 8 since u/EF = 1.00. Now 8 z 40,000°K, so F F kBT/eF = 0.2 corresponds to 8,000°K. Thus kBT/eF = 0.1 corres- ponds to 4,000°K. Our highest Currie or Neel temperature is of the order of 1,800°K, so that we are always going to be working at kBT/eF<0.005. The ordinate value of u/eF = 0.99 and is pro- bably above u/eF = 0.995. Thus, for our highest temperature, say z 1,800"K, u = 0.995 EF, is an excellent first guess for the Chemical potential. Only, at most, four iterations will be needed to achieve the best 0 to the nearest thousandth. For low temperatures, a guess of U = 6F will yield quick results and similarly. There is one difficulty with this approach. Kittel's "free electron gas" means just that; n/2N m 0 (or z 1, since a free hole gas should behave just as a free electron gas, as we have argued before), and we must correct his curve, Fig. 20, as 1/2 n/N approaches 1/2, the half-filled band case. For the half- filled band case, u = 6 Thus, we have a family of curves as F. in Figure 211 172 .I-é. 6 2 a, - I12 . half filled band can L \ gfits 0,| empty or hole bond case .995----’- -- -- -- --- ------- i ' ’- .05 kaT 6F Figure 21. Spread in Chemical Potential Versus Temperature as a Function of Band Filling. being bounded by a constant curve for the half-filled band. The curve for largest possible variation of u with temperature occurs when the band is almost empty or almost full. In the anti- ferromagnetic state the curves will uniformly shift up or down the u/EF axis as A varies. E 6F I.O - or _. 8- 173 TI: F BECOMES Ix)- \ I 59'! .0: EF Figure 22. Variation of Chemical Potential Versus Temperature for Various Band Fillings as a Function of the AFM Gap Parameter. We must derive these results rigorously so that an idea of u u(T,n/2N,A), the explicit functional dependence, is gotton. Certainly for any fixed n/ZN value and for any temperature, the n/2N &,0, 1 curve is the largest n(T) variation so that a guess of u RIO-995 eF-l.00 6F is an excellent first guess and will minimize the number of self—consistent iterations needed to find n. In the paramagnetic and ferromagnetic states, the quasi-free electron or hole energy band is parabolic for low (E) and high '3' , so that 174 electron parabolic 3 & hOkB ° parabolic m hole 2.- 0 7/20 1170 k Figure 23. Electron and Hole Parabolic Band Approximation Near k = 0 and TT/E Respectively. the only region in question is for filling of n/2N'eIl/2 for k kgn/Za. In this region, this question for the paramagnetic and ferromagnetic states does not really arise since Penn's phase diagram, and Kemeny and Caron, have shown that the anti- ferromagnetic state will be most stable for n/2N m 1/2, the half- filled band case. For low Co (in an s-electron calculation like Penn's) the half-filled band case is paramagnetic. Here we simply note that the curve of u versus T will be within the family of curves so that u = 0.995 6F - 1.000 6F will be an excellent first guess. To prove this, aside from our knowledge that n2 = h2 which we proved before, we note that: 175 (3.190) f(€k) = [1+ epoek/4T - u/4T)/kBT) 3'1 where (3.191) 65/4T - l/2(cos kxa + cos kya + cos kza) = - 1/2(cos mx + cos my + cos m2) if mi = kia for a tight binding d electron. Near the botton of the band, we expand the cosines about m = 0 and 4 _‘ ”(1'45 Eorw “’96 7T Figure 24. Parabolic Band Approximations Reexpressed in Tarms ofgfl = ka. near the top of the band we expand the cosine about na - m = 0. We see: 2 (3.192) (cos cpi)cp=0 a 1— 012(0) 8 cpi/Bz (01(0) 176 (3.193) (cos cpi)co=a~l-[fia-cpi(0)]2 82[Ua-cpi]/02[na-cpi(0)] el-(UzaZ-Zaai(0)+miz(0))(Ha-azwi/BZCUa-w(0)11 which is approximately of the same form as the first equation so that holes behave just as electrons and have the same f(ek). Now: rah (3.194) 63/4T - 1/2(cos mx+cos my+cosmz) = -1/2 (3-3w:(0) azox/aox(0)2) since wx = my = ”z in a simple cubic lattice. We know €k = h332/ 2m* so that ek/4T = hgkz/BTm*. Thus: I .1 (3.195) -1/2(3-3w:(0) asz/ao:(0)) = figLZ/BTm* k a 2 ka x Resubstituting wx (3.196) -1/2(3-3(ka)282(ka)/a[(ka)(ka)]2=n?5?/8Tm* = 3/2 (l-azazka/a[(ka)(ka)]2)k2=figkz/8Tm* So .-'I.Jl‘ 52 3 2 82(xa)_ _ 3 _——a _-3 81“" 2 at (ka) (ka) 12 (3.197) k2( For small k, say k = n/100a 2 2 (3.198) (%;) =(_3/2 + 3/2 8 (ka) 3; ) 8T k=n/100a a[(ka)(ka)]2 104 n For large k, say k = n/3a, but still in the parabolic region, 52 Ba) TT2) _8_T “‘ 2 (3.199) q%;) =(-3/2 +3/2 k=n/3a a[(ka)(ka)]2 9 h 2 a...- 4.‘ 1; ' 177 The difference in both of these effective mass values is small, as 62(ka)/a[(ka)(ka)]2 is much larger at k = n/100a than at k = n/3a so that the 104 factor in one denominator is not grossly different in its effect from the 9 factor. We know m* (k small or large) > m* (middle of band), so there is some discrepancy since curvature depends on m* and is small at the F. «.1» center of the band. Thus m* still has meaning near the middle of the band. “I It remains to see how u = n(Temperature, A (Temperature), n/2N) I I ,3 for the antiferromagnetic state. We know L‘ i: Ta i 1 A I i. i TNacI’ T Figure 25. AFM GappParamater Dependence on Temperature. 178 the A(Temperature) dependence. This is seen64 as: (3.200) A = 9K‘4J 2[ A ( 1 k _ N k 2 2 <:L+I:3E.e.<_)1 "'vrA +(e -e ) e 15 3+9 1 ) where Es(k) and FA(k) are the split lower and upper bands respectively, and a and 8 are constants, a being determined by Matsubara and Yokota's equation (10) (that the sum of Fermi factors over all k states for the upper and lower split bands equals half the number of electrons in the system), and is thus a chemical potential, and B = l/kBT as usual. The band splitting is 2A, 6 k l l _____l 1’ I ___...— | I I I, GAP = ZA ’| I I __ ___. /— ___. _ _ ___ I I Ii Typical-Z—N l Pohfl I a—qp 77720 Tl/a 5 Figure 26. AFM Gap in Dispersion Relation. . .A- ¢_-A—-n. r1 . u“, 179 so that the 6k value at k = n/Za depends on temperature. As the temperature rises, the gap narrows and the origin about N which we expanded cos ka in Ek in the hole expression, k ~ m/2a, shifts. What has this to do with band filling? For n/ZN < 1/2, n/ZN > 1/2, there is hardly any effect. But for n/2N zgl/Z, there may be a large effect. Now we know u = 1.0 6F at n/2N = 1/2 (i.e. at k = n/Za) so that u must look like Figure 27. 1028.99.92.01. _ + upper band J22°UQSP£I_ _ ’ _ _ _ _ _ ._ lower band ,’1 7- ' I’ I” ’ 'r )0 ' I"' ' tn Tusclf’ I ? 4v” ' I 4 Figure 27. Jump_in Chemical Potential at AFM Gap. as a function of n/ZN; with a jump of 2A between n/2N slightly Lil less than 1/2 and n/2N slightly greater than 1/2. If u were a single value, continuous function of n/2N, we could plot: 180 A I 1' ‘I . ‘I 911 I I d(§,-. II, ‘I‘ 11’ \1 *- .1 2N Figure 28. Peak in_perivitiv§ of Chemical Potential with Respect to Band Filling Versus Band Filling at Low Tamperature Around AFM Gap (n/2N = 0.5). Since this cannot be true at n/2N, we represent its derivitive by a maximum at n/ZN (dotted curve). Thus the n/2N = 1/2 case corresponds to the lower curve (the most deviation from u E EF) for T = 0 in Figure 22. As T > 0, the n/2N = 1/2 curve goes into the family of curves along with all the other n/2N curve, since A decreases as T increases. Therefore, an excellent range to choose our initial u from is u = 0.995 - 1.000 EF. .-.n-'.¢- “4.“ AA A‘C‘-- 4 DISCUSSION AND CONCLUSIONS The FM-PM phase boundary straddeling program was run on the C.D.C.-6500 computer at Michigan State University. Only single precision arithmetic was used to evaluate the sign change of the energy difference (at low temperature) or the free energy difference (at high temperature) between the two magnetic phases. Three sets of phase boundaries at three different temperatures were calculated with an error of about 5%. The variables used a. in the program for temperature was TT, defined as temperature/ 1/3 x d-bandwidth x kB, a form Chosen so that temperature was normalized similarly to the direct and exchange coupling con- stants, K and J. This is in the dimensionless form. At T = 1.1609°K. (TT = 10'4), we calculated the difference between ETOT for spin polarization equal to zero and for spin I polarization equal to 0.1. This was done for J/4T = 0.0, I J/4T = 1/2 K/4T, and J/4T = K/4T, the maximum value the ratio (exchange coupling/l/3'bandwidth) can attain. é , A, At T = 300°K. (TT = 2.584 x 10‘2) and T = 1000°K. (TT = 8.614 x 10-2), we calculated the difference between F for spin polar- TOT 181 182 ization equal to zero and for spin polarization equal to 0.1. This was also done for the same three ratios (exchange coupling/ 1/3 bandwidth) ratios as the low temperature case. The results of these calculations are shown in Figures 29 to 41. In Figure 29 we see the low temperature phase boundary at the three (exchange coupling/1/3 bandwidth) ratios. It is evident that there is a progressive increase of the ferromagnetic region at the expense of the paramagnetic as exchange coupling increases. Our zero exchange coupling phase boundary does not coincide with Penn's because we have an extra coefficient of 5 multiplying the potential term in the five-fold degenerate case. There is one electron in each of five d-shells as opposed to one electron in Penn's only s-shell. In Figure 30 we illustrate the results of Figure 29 in a three- dimensional phase diagram. This is done to illustrate that the ratios J/4T = 1/2 K/4T and J/4T = K/4T do not lie on planes parallel to that of the J/4T = 0.0 calculation. we will denote these planes as (100) for J/4T = 0.0, (102) for J/4T = 1/2 K/4T, and (101) for J/4T = K/4T. These should not be confused with crystal lattice planes; they are planes in the parameter space of the phase diagram. Thus, all the two-dimensional phase boundary diagrams are projections of phase boundaries from the (101) and (102) planes onto the (100) plane in which the J/4T = 183 Figure 29. T=l.l609 K. FM-PM Phase Boundary. 9 - l I K/‘W \ ————— Penn 1 x ' .__—$———m———-£U%T==0.0 afl‘ 1 ——o——a—— J/4T = 1/2 K/41? -—-O-—--O-— J/4T = K/4T 2b 6. a $- 4L 0 3'. a 0 a—x 1 . 1 0 0 1H, PM 0 o a O 1 l - I I l 0.1 l 0.2 I 0.13 014 0.!5 n/ZN Ni Co Fe Mn Cr 184 T = 1.1609° K. K/4T 8 ir- 0.1 74m; 3'6 -- \\ .'° \ end of Penn's FM .,' 5 region at FM-AFM .3 " boundary. 4 _ 1 2 J/ E/4T / Figure 30. Isometric View of Isothermal Phase Surface in (g/4TLiJ/4T, n/ZN) Parameter Space. The phase surface looks similar at higher temperatures in this projection. 185 0.0 boundary is plotted. The convergence of the two J/4T # 0.0 boundaries in Figure 29 at low values of K/4T is simply because they are on (101) and (102) planes which merge at K/4T = 0.0. In Figure 31 we see that at T = 300°K., all boundaries have been shifted so as to decrease the ferromagnetic region at the expense of the paramagnetic. The same reverse trend on increase of J/4T is seen as in the low temperature case. Figure 32 illu- strates both of these trends at 1000°K. The scatter of computed points seems to decrease as the temperature of the calculation increases. Penn's work did not include any data points, so we must assume that his smooth phase boundary curves are best fits to some data point scatter. Anyway, only trends are really im- portant, as the Hubbard model itself is only a rough approxima- tion. Figures 33, 34, and 35 are the results illustrated in Figures 29, 31, and 32 respectively, only with the K/4T ratio inverted so as to superimpose the curves on those of Kemeny and Caron, as we saw in Chapter 2. Because this inverted phase diagram exaggerates the low K/4T region of the previous phase diagrams, the trend toward increasing the ferromagnetic region at the ex- pense of the paramamagnetic region is very evident. At higher temperatures, this trend is still evident, but it is offset by 186 ‘ .u—db—é-fik—-——- J/4T = 0.0 -——o————o————- J/4T 1/210 K/4T -—-o-———<>————— J/4T MT I l 1 J I 0.1 I 0.2 | o{3 o.[4 0.|5 n/2N Ni Co Fe Mn Cr Figure 31. T=300 K. FM—PM Phase Boundary. .187 \ -——— _— .— Penn. W4T ‘ I ~ . -——ax :5 J/4T==0.0 \\ 4' .— J/4T = 1/2 W41 \ ——o—o—— J/4T.= K/4T \_ ‘ \ \ \ \ \ \ \ \ \ \ \.....___. PM 031 ' I 01.2 I 0.} 0.14 0.I5 n/ZN Ni Co Fe Mn Cr Figure 32. T=1000 K. FM-PM Phase Boundary. a? 188 hor ' MIC t m T = l.l609°K. at IC=8.0 Weak Coupling Limit ---—-— Penn 2'5 F. t-matrix —x—-x— J /4T +0—J/4T = 1/2 K/4T _-O—O— J/4T = K/4T I Kemeny + Caron = (4T/K)Kemeny + Siegel Strong Coupling Limit 0 I , I a J n/2N 0.1 I 0.2 I 013 Ni Co Fe FM-PM Phase Boundary. Figure 33. Inverted T = l.l609°K. 189 T = 30 °K. :IWeak __ __ _ penn 3 Coupling . : Limit 2.5_ t-matrlx : ——x-—-II-—— J/4T = o —-o—o—J/4T = 1/2 K/4T .‘ —o——o— J/4T = K/4T 5 2.0— E c I Kemeny + Caron : = (4T/K)Kemeny + Siegel 5 1.5L I / PM Limit FM n/2N I I 013 014 C0 Fe Mn Cr Figure 34. Inverted T = 300°K. FM-PM Phase Boundary. Strong Coupling Fat: 9.9,. < {I .1 “o u 190 T =41900°K. weak Coupling Limit -———-Penn t-matrix ’5 " —x——x_——J/4T = 0 J/4T = 1/2 /4T_ 5 r a —o——<>— J/4T = K/4 200F- C Kemeny + Caron = (4TVK)Kemeny + Siegel 1 5- ,' ll pm / / 1 0— / / / //_ \ / \ 0.5 _' l’ ' _,1 . / fl“ . O ‘ . Strong I. ‘,/’ Coupling Limit FM 1 /2 ' AL n N 0.1 I 0.2 I 0.I3 014 0 5 Ni Co Fe Mn Cr FM—PM Phase gpundagy, Inverted T = 1000°K. Figure 35. ,— r ’0' k H‘ *1 “".<.".T'A.' up ‘ 191 the effect of the higher temperature as seen in Figures 34 and 35. The important trend that emerges is that increasing J/4T yields results opposite to Kemeny and Caron's t-matrix calculation. In Figures 36, 37, and 38, we have plotted for constant J/4T, the motion of the phase boundary as temperature increases. Again, the trend of reduced ferromagnetism as temperature in- creases is evident. In Figures 39, 40, and 41 we have plotted, on the inverted phase diagram, the phase boundaries at constant J/4T as temp- erature increases. There, we can see that the trend toward a reduced ferromagnetic region as temperature increases, seeming to parallel the result of the tdmatrix calculation at T = 0°K., that of a reduced tendency toward ferromagnetism for most band fillings as compared to Penn's prediction. The AFM-PM program was never run due to a lack of funds for computer time. These self-consistent calculations require long running time programs with even longer, more expensive debugging periods. The conclusions of the FM-PM program runs can be qualitatively compared with experiment in a way to bring out the trends of he" 192 g/nT = 0.0 (100) Plane Ni Figure 36. J/4T Mn Cr 0.0 FM-PM Phase Boundary. IQQT -—¥——X-—— T = 1.1609oK. —._.— T =. 300°K. % ——X PM I I I J I 0.1 I 0.2 I 0.p 0.P 0.; n/2N Co Fe 193 .9 . MT a 112 3/4'1‘ L102) ‘Plape ——---— —- Penn (T=0), (J20) K/4T —-x——x—-— '1'- : 1.1609°K. 8 — ' -——o—-a--- T = 300°K. -——-o——-o—-- T = 1000°K. o L l I l 0.1 I 0.2 I O.'3 O.I4 0% n/1EN Ni Co . Fe Mn Cr Figure 37. J/4T = 1/2 K/4T FM-PM Phase Boundary. 194 9| ’ JA4T a K/4T 1101) P1223 —£—¥——— '1‘ = 1.1609'K. 8 _. . hI . ——O—-O-—— T = 300°K. —-—0-—O—— T = 1000°K. 7 I- 6 .— 5 _— 4 .— 3 — 2.. 1 .. g. l l I I 0.1 I - 0.2 I 0% 0. l4 0. l5 n/ZN Ni Co Fe Mn Cr Figure 38. J/4T = K/4T FM—PM Phase Boundary. __qwu_~.n.«ou- fi.‘n— [~— w..__ Mb... 1 / '4 Y r h -...,' 195 nweak Coupling (100) Plan{ g/4T = 0.0 -— -- — —- Penn t-matrix 2.5-— 3 X T = 1.1609PK. % 4 T = 300°K. —-o-—o—— T = 1000°K. C Kemeny + Caron = (4T/K)Kemeny + Siegel Limit 0.0 (100) Plane. Figure 39. Inverted J/4T FM-PM Phase gpundary. 196 horizontal bottom L4T=y2 r/4T (10am ne at- c = 3.0 ,3w66k ——-———Penn : Coupling . ' Limit t-matrix : 2.5 - °‘ —-x-—x— T = l.l609°K. I —o—o-— T = 300°KI I —o‘-—o—T = 1000°1I. Z “*- 2.0 I- : C Kemeny + Caron : = (4T/K)Kemeny + Siegel I I JJ 1.5w- I I PM I, 1.0— x / ,— — — I // 0.5- \ / / / ’ \ Strong / AFM Coupling / Limit 01 f ,, I , I FM l n/2N 0.1 1 0.2 I o.I3 o. 0. 5 Ni Co Fe Mn Cr 1 Figure 40. Inygrtgd_J/4T = 112 K/4T (102) Plane. FM-PM Phase Boundary. 197 horizontal bottom at C = 8. :Weak .Coupling -Limit t-matrix ': 205- O —x—x—- T = 1.160 °K. ' -O-—-O-— T = 300° : -—o——o——T = 1000’ . I 20C_ : C iKemeny + Caron .’ = (4T/K)Kemeny + Siegel ' I 1.5? I 1.0+- O 0.5-4 \ trong oupling 1‘ AFM z imit 0L J, , .1 - I EM l /2N 0.1 I 0.2 l O.P 014 0.~ Ni Co Fe Mn Cr Figure 41. gnverted J/4T = K/4T (101) Plane. EM-PM Phase Boundary. 198 phase transitions in the 3d series transition metals. A survey of the experimental literature indicates the following Curie and Neel points: Table 7. Properties of Transition Metals. Atomic Spin Number of Number of Transi- Transition Metal Moment (p ) d-holes d-electrons tion Temperature B 0 LR.) Cr 0.6 5.0 5.0 AFM-PM 312 7- Mn --- 4.0 6.0 AFM-PM 500 Fe 2.0 3.0 7.0 FM-PM 1043 Ni 0.5 1.5 8.5 FM-PM 627-631 Co 1.5 2.5 7.5 FM-PM 1400 There is no recorded AFM-FM transition in transition metals. The number of d-electrons is the numerator of our n/2N abscissa in the phase diagrams. As we have shown, the electron-hole symmetry in terms of what magnetic state their intrinsic spin angular momenta will produce when coupled, we can pinpoint the abscissa corresponding to a given transition metal. For the FM-PM transition, we can place Fe, Co, and Ni at 3.0 holes per déband, 2.5 holes per deband, and 1.5 holes par deband, respect- tvely. Chromium has 5.0 holes per d-band 255.0 electrons per 199 d-band. Thus Cr occurs at the n/ZNx 5.0 abscissa, (at the very center of the AFM central phase region), a value which we would not expect to be altered if a FM~AFM.program with the full five-fold d-electron orbital degeneracy had been run. Next, in decreasing hole band filling, comes Mn, with approximately n/ZN = 4.0 holes per deband, still fairly near the half-filled band. This band filling placesan in the antiferromagnetic region of Penn's diagram, and the addition of the five—fold degeneracy should not alter its tendency to an ARM state either. On our final phase diagrams, we labeled the band fillings for Cr, Mn, Fe, Co, Ni, at which the three temperatures the calcula- tion was performed. The direct and exchange interaction constants have not been separated experimentally as yet for the transition metals, so the K/4T and J/4T intercepts for any of the transition metals are unknown. we should emphasize that we have not included the variation in crystal lattice structure between the transition metals. The energy difference between the crystal structures should dominate that between the magnetic phases. As mentioned previously, the non-isomorphic mapping makes a unique choice of a finite number of glvalues impossible so that a unique'VLkMAG or I; cannot be written in the F.C.C., B.C.C., 200 and H.C.P. lattices. Nevertheless, our calculation has shown the enhanced FM phase region at the expense of the PM phase region as the ratio (exchange interaction energy/l/3 bandwidth) increases, and the increase in the PM phase region at the ex- pense of the FM phase region as temperature rises. An explicit experimental knowledge of K, J, and the bandwidth in Cr, Mn, Fe, Co, and Ni would help fix the J/4T and K/4T intercepts for each of these particular metals. There is some doubt, however, that such K/4T, J/4T, and n/2N values would be temperature indepen- dent, so that any high temperature conclusions would involve a shift in the point (K/4T, J/4T, n/2N) in the three parameter space of the phase diagram, in addition to the shift of phase boundary we have calculated. For a comparison of Cr, Mn, Fe, Co, Ni, we illustrate a Slater- Pauling curve of magnetic moment per atom versus atomic number for the first transition period in Figure 42. There, it is seen that the magnetic moment per atom rises, peaks, and falls with atomic number quite linearly. Metals to the left of the center line tend to be B.C.C.; those to the right, tend to be FOCOCO-HOCOP. we see that fi/pB is a maximum at the quarter-filled band near Fe, which corresponds with the fact (found by both Penn and us) that Fe has the largest FM region at its hand filling, n/2N 201 Empty Band 3‘" 2L Half-Filled Band L N Figure 42. Slater-Pauling Curve for 3d Transition Metals. Ni Co Fe d-band filling Cr 202 value (= 3.0), while AFM Cr and Mn have no net fiApB, and the «fiAFB per atom is smaller. The Slater-Pauling curve thus, qualitatively, upholds our result of a PM region near the empty band (with consequent low G/uB values), and a strong FM region near the quarter-filled band (with very high E/uB values). As band filling proceeds, the AFM-FM question needs to be solved by comparing results of the AFM-PM calculation and the FM-PM calculation. The utilization of this technique also has again demonstrated the power and applicability of Kemeny's and.Mattuck and Johansson's momentum space correlation function approach to ordered magnetic phases as opposed to the configuration space correlation function approach originally taken by Hubbard. In conclusion, we see that a full five-fold degenerate Hartree- Fock treatment shifts the FM-PM phase transition curve in a direction opposite to the many-body (t-matrix) nondegenerate s-electron calculation of Kemeny and Caron, with Penn's non- degenerate s-electron Hartree-Fock approach lying somewhere inbetween. This could have been expected since a Hartree-Fock treatment of the Hubbard model, even without exchange, is certain to enhance ferromagnetism, perhaps unnaturally so. As a result, without knowing K/4T and J/4T experimentally for Fe, Ni, Co, Cr, and Mn, we cannot conclusively say whether Kemeny and Caron's ire.- 203 or our own improvement over Penn's simplified model is super- ior. At present, we only know the plane defined by the n/2N (bandfilling) for each of these metals. Such a decision must await the accurate experimental measurements of d-electron band- widths, direct interaction constant, and exchange interaction constant in order to pin down the K/4T and J/4T intercepts for a particular transition metal. B IBLIOGRAPHY 10. 11. 12. 13. 14. BIBLIOGRAPHY Mattuck, R., and Johansson, B., ”Quantum Field Theory of Phase Transitions in Fermi Systems", Advances in Physics, Volume 19, Number 1, (1969). Penn, D., Physical Review, Volume 142, Number 2, (1966). Heisenberg, W., Zeit. Physik, Volume 49, Page 619, (1928). Bloch, F., Zeit. Physik, Volume 57, Page 545, (1929). 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