Magnetic ordering in cubic crystals with first and second neighbor exchange

J. Phys. Chem. Solids

Pergamon

MAGNETIC

Press 1966. Vol. 27, pp. 23-32.

ORDERING

Printed in Great Britain.

IN CUBIC CRYSTALS WITH

FIRST AND SECOND NEIGHBOR EXCHANGE R. A. TAHIR-KHBLI Pakistan Atomic Energy Commission,

Karachi,

Pakistan

H. B. C!ALLEN+ Department

of Physics, University

of Pennsylvania,

Philadelphia, Pennsylvania

and

H. JARREIT Central Research Laboratory,

E. I. Du Pont de Nemours

Co., Wilmington,

Delaware

(Received 22 July 1965) Abstract-The problem of stability of various ferromagnetic and antiferromagnetic spin configurations is considered. Simple cubic, f.c.c. and b.c.c. lattices are analyzed, for all possible ratios of the strengths of the nearest-neighbor and next-nearest-neighbor exchange constants (all other exchange constants are neglected), and for all temperatures. The calculations are carried out by a Green function ‘random phase’ approximation. The results are presented in terms of phase diagrams, and the nature of the instabilities at the phase boundaries is discussed. 1. INTRODUCXION

By far the most common method of analysis of spin configurations has been the molecular field method. N&.,(s) VAN VLECI&~) and hDERSON(6) in turn considered various special cases, and SMART,(~)and later TER HMR and LINES,@) then gave a more comprehensive treatment of cubic systems with first and second neighbor exchange, and with multiple sublattice structures. It is precisely to this degree of generalization that our analysis will apply, substituting however the more powerful method of temperature-dependent Green functions for the molecular field theory. The Green function method can be thought of as an extension of spin-wave theory to higher temperature, and this paper correspondingly can be considered as a re-investigation of Smart’s model by an ‘extended spin-wave analysis. The relevant portions of the Green function analysis are simply the equations for the energies of the excitations (the renormalized spin-wave spectrum) and for the Curie temperatures. These are given in equations (3.1) and (3.2); the underlying Green function theory is indicated briefly in the Appendix, but is not essential to the body of the paper.

THE SPIN configurations in magnetic materials exhibit great variety, even in crystals of simple These configurations are chemical structure. determined by the relative signs and magnitudes of the exchange interactions, by the temperature, and in some cases by various sources of magnetic anisotropy. Accordingly, considerable theoretical interest has focussed on the analysis of magnetic structures. The most reliable analytic tool in magnetic problems is spin-wave theory, and for the calculation of magnetization curves its validity extends over an appreciable temperature range. But for the comparison of the free energies of competing magnetic structures spin-wave theory is valid only at T = 0, the leading temperature corrections being of the same order as the spin-wave interaction terms. W Consequently spin-wave anaIysis, as applied for instance by TER HAAR and LINES,(~) supplies only the zero-temperature limit of the phase stability diagram. * Supported by the Office of Naval Research. 23

24

R. A. TAHIR-KHELI,

H. B. CALLEN

The Smart model is, of course, not entirely general. In certain materials the spin configuration may be radically altered by anisotropic interactions, and this problem has been treated by TER HAAR and LINES(*) in the molecular field picture. Furthermore, there are spin colorations which are not decomposable into N&EL sublattices, ‘fans’ and ‘spirals’ being exotic examples. These configurations have been treated by LYONS and KAPLAN,@) again by the molecular field method. Despite the wide use of molecular field theory, it is far from trustworthy. It excludes from consideration the effect of correlation, of collective modes, and of the contribution to the free energy of zero-point excitations. And, of course, it does not agree with spin-wave theory at zero temperature. Several attempts at more adequate theories have been made. Lo has used the Bethe-PierlsWeiss cluster theory, and more recently, H. Callen and E. Callen have investigated the phase boundaries of ferromagnetsflf) and antiferroma~e~~ls) by a two-spin cluster approximation. The europium and manganese chalcogenides are particularly interesting and topical materiafs which provided the primary physical motivation of that work, and which similarly motivate the present investigation. The europium chalcogenides (EuO, EuS, EuSe and EuTe) consist of face-centered arrays of Errs+ ions with a positive nearest-neighbor exchange and, quite probably, with a negative next-nearest neighbor exchange.(a~1s~l*)The second neighbor exchange is relatively small in the oxide and sulfide, and these materials are simple ferromagnets at low temperature. The telluride is a simple NCel antiferromagnet (of ‘type 2 order’, later to be described). The selenide was originally throught to be ferromagnetic, but S. Pickart informs us that neutron diffraction now appears to indicate a complex spiral antiferromagnetic structure which is so marginally stable that an applied field of 5 kOe. induces simple ferromagnetic alignment. The manganese chalcogenides provide examples of similar chemical structure, but with an alteration in the sign of the nearest-neighbor exchange.(lQ MnO correspondingly exhibits type 2 order, whereas MnS exists in two forms, exhibiting type 2 orderos) (cc-MnS) and type 3 orderos) @-MnS). In Section 2 we define the various types of magnetic order and we summarize the results of

and H. JARRETT

molecular field calculations, as given by SMART,(~) and of spin-wave calculations. These results are most conveniently represented in phase diagrams. Each point in the phase plane specifies the temperature T, the nearest-neighbor exchange integral 51, and the next-nearest-neighbor exchange integral Js; only ratios of these quantities are physically significant so that a two-dimensional representation is possible. The phase diagrams then divide this phase plane into regions of stability of the various types of magnetic order. The molecular field results are not reliable, and the spin wave results are not complete, but taken together they strongly indicate the qualitative shapes of the true phase diagrams and they guide the Green function calculations. In Section 3 we summarize the results of our calculations, which are given separately in the Appendix. The physical nature of the instabilities at the phase boundaries are examined in some detail. A preliminary application of the present method was made by TAHIR-KHELI and JARRETT,~~)who investigated the stability of the ferromagnetic phase in the f.c.c. lattice with 51 > 0. Additional results for the f.c.c. lattice have been given in a recent paper by LINER, using the same method as employed in this paper. The purpose of this paper is thus to extend the calculations of Lines to other structures, and to discuss the physical nature of the ins~bilities at the phase boundaries. 2. DEFINM’IONS OF TYPES OF ORDF,R AND SUMMXRY OF PREVIOUS RESWLTS Following SMART(~)and TER HAARand LINES(*) we consider antiferromagnetic spin configurations in which each spin is (on the average) either ‘up’ or ‘down’, and in which the repetition distance is not greater than twice the cube edge of the cubic chemical unit cell. All such con~gurations have been assigned numbersf7~8) and they are shown in Figs. l-3. The NCel temperature of a given spin configuration, according to molecular field theory, is

lS* is the energy of interaction of Here -2J& spins i and j. The sums over i and j count each pair

CI

0 Y

26

R.

A.

TAHIR-KHELI,

H.

B.

CALLEN

and

H.

JARRETT

Z

(a)

t

(d)

t

“Y

-Y -Y

FIG. 3. Types of order for the f.c.c. bravais lattice. a represents Type I order; b, Type II; c, Type IIIA; d, Type IIIB; e, Type IVA; f, Type IVB.

twice, so the quantity in the brackets represents the total exchange energy divided by the number of spins N. This quantity is to be calculated considering each spin as a classical vector of length [S(S+ 1)]1’2, and considering each spin to be pointing fully upward or downward according to the specified configuration.

Anderson and Smart then adopt the assumption that the stable configuration is that with the maximum NCel (or Curie) temperature. This configuration is postulated to remain stable from T = 0°K to its NCel temperature, above which the system becomes paramagnetic. The molecular field phase diagrams for simple,

MAGNETIC

ORDERING

IN CUBIC

CRYSTALS,

FIRST

AND SECOND

NEXGHBOR

EXCHANGE

27

body-centered, and face-centered cubic lattices are shown in Figs. 4-6. In these figures the ordinate is Js/lJlj, and the abscissa kBZ’fJ1S(Sfl), thereby condensing all possible sign combinations of Jr and Js into a single diagram. However it is convenient to think of these .diagrams as two disjoint diagrams; the right-hand portion of the diagram applies to crystals with positive JI, and, the abscissa is proportional to the temperature whereas the left-hand portion of the diagram applies to crystals with negative J1, and the temperature increases to the left. In Figs. 4-6 we have plotted the N&e1 or Curie temperature of each Pooramognstic

FIG. 5. Molecular

FxG. 4. Molecular

field f>ha!se diagram ._ . cubic bravats latttce.

field phase diagram bravais lattice,

for the b.c.c.

for the simple

configuration as predicted by equation (1). The maximum N&e1 or Curie temperature is assumed to correspond to the phase boundary and is indicated by the solid line. The molecular field phase diagram suggests the probable regions of stability of various phases, but in regions in which two phases compete fairly closely the molecular field criterion is not reliable. For further information we therefore turn to spin wave theory.

pammagnetic

FIG. 6. MolecuIar

field phase diagram bravais lattice.

for the f.c.c.

28

R. A. TAHIR-KHELI,

H.

Spin wave theory provides information on the permissible ranges of J1 and Jz for which a given phase can exist at T = 0”. These ranges are indicated in Figs. 7-9. Taken in conjunction with the molecular field curves they leave few ambiguities as to the qualitative features of the phase boundaries. This understanding of the general shape of the phase boundaries is important as it permits us to carry out Green function calculations only for the physically significant phase boundaries. The nature of the instability at a phase boundary at T = 0” is also indicated by spin wave theory, as

B.

CALLEN

and

H.

JARRETT

the particular spin wave which becomes unstable heralds the periodicity of the incipient stable phase. However, we postpone further discussion of this point, as it arises again in an identical fashion in our calculations of the next section.

III*

t

J,
J,>O

J,/iJ,I

FIG. 9. Spin wave stability diagram for the b.c.c. bravais lattice.

3. EXCITATION

SPECTRUM

AND

STABILITY

CRITERIA

FIG. 7. Spin wave stability diagram for the simple cubic bravais lattice.

t J,/ lJ,I

I

J,
c(k) = 2
F

I

The results of the Green function analysis, which is given in the Appendix, are summarized by the equations for the renormalized excitation (spin wave) spectrum and for the NCel temperature. With the exception of b.c.c. type II and f.c.c. type III order, which are somewhat more complicated, the excitation spectrum can be written in a simple general form. Assuming no external field the spectrum is doubly degenerate, and the two modes of wave vector k each have the energy e(k), given by

)[Juu(O)-J,,(k) - Ju@) + Ju&]1’2

x [Jut@)- J,,(k) - Ju@) - Ju&)]“2

-3 i xt II ::‘“T I

J, >O

Iv

FIG. 8. Spin wave stability diagram for the b.c.c. bravais lattice.

(3.1)

The subscript u denotes the ‘up’ sublattices, and d denotes the ‘down’ sublattices in the particular type of order being considered. The quantity Jud(k) denotes the Fourier component of the exchange interaction between ions on u and d sublattices respectively, and similarly for Juu(k) and J&k). (Si) is the average spin component of an ion on an ‘up’ sublattice, and (S$) = - (So) because we assume no applied field. In the special cases referred to above (b.c.c. type II and f.c.c. type III order) the up and down

MAGNETIC

ORDERING IN CUBIC CRYSTALS,

FIRST AND SECOND NEIGHBOR EXCHANGE

sublattices must each be further subdivided into two non-equivalent sublattices; this presents no essential difficulty but complicates the resultant equations slightly. The NCel temperature associated with a given type of order is to be computed from the equation 4 kBTN = -S(S+1) 3

x rJuu(o) -J&4 -Jt‘d(O)+ J&k)]-’

(3.2) To illustrate the physical significance of the instabilities at the phase boundaries we consider the ferromagnetic phase in the f.c.c. lattice. We assume J1 > 0, as relevant, for instance, to EuS. The excitation energies then have the form{2o1 c(k) = 2 (P)[J(O)-J(k)] = 8J1[3 - Gz- Gy - czz]

(3.3)

where cx = cos(kZa/2) ,925= cos(k&

(3.4) (3.5)

and where a is the length of the edge of the cubic unit cell. If T = 0” then = S, and equation (3.3) reduces to the well-known spin wave spectrum of the ferromagnetic phase in the f.c.c. lattice. If (Jl+Js) > 0 the minimum of c(k) occurs at the center of the Brillouin zone (k = 0). However for Js = -J; the excitation energies vanish along the entire cube-diagonal [ill] directions in k-space. In fact, along these directions, e&11) = 24(J1 +Js)
(3.6)

For Ja < - 51 these spin wave energies then become negative and the implied instability destroys the ferromagnetic phase. In fact the dominant instability occurs at the edge of the Brillouin zone (where sina(kllla/22/3) is maximum), at which point kill = [1/(3n)/u]. This instability has a wave

29

length of h = 2~43~ or twice the distance between [l 111planes. We consequently expect that the new phase which sets in at Ja = -51 is one in which alternate [ill] planes of spins are antiparallel, and this is precisely the structure of the type II order. At non-zero temperature the various modes of equation (3.6) are thermally excited, and ( Sz ) is thereby decreased, The smaller the magnitude of (J1+ Jz) the lower are the mode energies, the greater is the thermal excitation at a given temperature, and the smaller is (SZ). For a particular temperature (SE) vanishes and the system becomes paramagnetic. It follows that this Curie temperature decreases with decreasing (Jl-f-Jg), so that the Curie temperature has the form shown in Fig. 12. We note also that the instability at the phase boundary (for T = T,) is arrived at by the collapse of all spin wave frequencies simultaneously to zero. Hence the new phase which sets in is one of general chaos-the paramagnetic phase. At T - 0” and JZ = - JI both the above effects occur s~ultaneously. The factors (Jr + Jz)2in2@m&W3 ) in equation (3.6) bring about the instability to type II order, whereas the factor (SZ) brings about the instability to the paramagnetic phase; this point consequently is a triple point, as shown in Fig. 12. An interesting aspect of the phase diagram appears when we apply a similar analysis to the antiferromagnetic type II phase. According to equation (3.4) the spectrum is given by b(k) = 2[-6J2-4J1(czcg+cycz+czc2) -4Jz,fc~fc;+E;-g)]~‘2 x [-

652 -

4J1(~,+

SAG+ szsx)

+~J&;+c;-c~,-$)]~‘~

(3.7)

where s2: = sin(k$a/2),

cz m cos(kza/2)

(3.8)

The minimum values of the excitation spectrum again are found to occur along the [ll I] direction, for which equation (3.7) reduces to E(kn1) = 12 [( - Jl-

J2)2sin2(knl~/2/3)]1’2 (3.9)

We first note that the formalism fails to resobre the ambiguity of sign of the excitation frequencies,

30

R.

A.

TAHIR-KHELI,

H.

B.

so

that one must augment equation (3.9) by a direct examination of the modes. Equivalently one appeals to spin wave theory, to which equation (3.9) must reduce at T = 0. Applying this condition we find that the positive square root is appropriate, so that ~(krlr) = 12(Si)(-Jr-Js)sin(k&d3)

(3.10)

The analysis of the instabilities of equation (3.10) now parallels that of equation (3.6). The frequencies are positive only when (-Jr-&) > 0, corroborating the spin wave criterion indicated in Fig. 9. Furthermore, decreasing values of (- Jr - Js) lead to lower excitation energies and a lower NCel temperature, in agreement with the general shape of the Neel temperature (phase boundary) curve shown in Fig. 12. Consider now the case of zero temperature, and let (-.Jl-Js) -+ 0. The dominant instability in equation (3.10) occurs at sin(&la/2/3) = 1, or /\ = 4a/2/3. This corresponds to four times the separation of [ill ] planes. An instability which would restore the ferromagnetic phase (which we

CALLEN

and

H.

JARRETT

would expect at the triple point shown in Fig. 12) would occur at h = 2a/1/3. In order to interpret the instability at the triple point we must examine the type of order obtained if the spin wave with h = 4a/d3 grows exponentially. Unfortunately this order is not uniquely specified. One possibility is the complete reversal of spins, with h = 40143 ; the resultant order is characterized by alternating pairs of [ill] planes (up, up, down, down, up, up, etc.). However, one

/

/

i

i

T-“5

\

FIG. 11. Green function (RPA) stability diagram for the b.c.c. bravais lattice.

-05

\

Poram’ognefic

FIG. 10. Green function (RPA) stability diagram for the simple cubic bravais lattice.

easily finds that this order is stable only if 51 > 0. The remaining possibility is a canting of the spins, with the canting wave-length of X = 4a[1/3. Such canted spin arrangements are omitted in our treatment of the problem. Consequently we are simply alerted to the possibility that a [ill] canting order may physically dominate the ferromagnetic order just above the indicated triple point in Fig. 12. The analysis for the various other types of order is carried out analogously. The results are summarized in the phase diagrams of Figs. 10, 11, and 12.

MAGNETIC

ORDERING

IN CUBIC

CRYSTALS,

FIRST

AND SECOND

NEIGHBOR

31

EXCHANGE

15. ROTH W. L., Phys. Reu. 110,1333 (1958). 16. CORLI~S L. M., ELLIOTT N. and HASTINGS J. M., Phys. Rev. 104,924 (1956). 17. TAHIR-KHELI R. A. and JARRETT H. S., Phys. Rew. 135, A1096 (1964). 18. LINES M., Phys. Rev. 135, Al336 (1964). 19. TAHIR-KHELI R. A. and TER HAAR D., Phys. Rew.

4

Jz lJ,l

20

t

127, 88 (1962). 20. CALLEN H. B., Phys. Rev. 130,890 (1963). 21. CALLEN H. B. and SH~RIKMAN S., Solid St. Com\

CD5

mun. 3, 5 (1965).

/ J

-D

-6

-6

-4

-2

F

4

6

6

hlT

APPENDIX IO

lUA

/I/

-05

Paramognefic

Paromognelic

Green function analysis Preparatory to a general analysis we note that all but two types of antiferromagnetic order shown in Figs. l-3 have the following simple characteristic; the sum

-1.0

-i‘ II

-15

ti

\

-2.0

FIG. 12. Green function (RPA) stability diagram for the f.c.c. bravais lattice. See also LINER.

REFERENCES 1. For a more thorough discussion of this fact see, for instance, ANDERSON F. B. and CALLEN H. B., Phys. Reo. 136, A1068 (1964). 2. TW HAAR and Lmr~s M. E., Phil. Trans. R. Sot. A225.1 (19621. 3. NOEL L:, &n. p;hrs. l&64 (1932); 5,256 (1936); 3, 137 (1948). 4. VAN VLECK J. H., J. Chem. Phys. 9, 85 (1941). 5. VAN VLECK J. H., J. Phys. Radium, Paris 12, 262

(1951). ANDERSON P. W., Phys. Rev. 79, 705 (1950). 7. SMART J. S., Phys. Rev. 86, 968 (1952). 8. TER HAAR D. and LINES M. E., Phil. Truns. R. Sot.

?

Jw%

is identical for every ion i on an ‘up’ sublattice, and is equal in magnitude but opposite in sign for every ion i on a ‘down’ sublattice. The two types of order (b.c.c. type II and f.c.c. type 111~) which are exceptions must be treated in terms of four sublattices, and we shall not give the analysis explicitly here. The results are given in the appropriate phase diagrams (Figs. 11 and 12). The formalism used is that of double-time, temperature-dependent Green functions, with the Tyablikov (or random phase) decoupling approximation. This formalism is described fully in the ferromagnetic case by TAHIR-KHELI and TER HAAR.(lg) Denoting ions on an up sublattice by Roman indices and those on a down sublattice by Greek indices we define two Green functions, Gj,c (t-t’) and Gf,,(t-t’). The Green function Gj,r (t-t’) is defined by %(t---‘I

=
-ie(t-t’)

si_(t’>> ([S;(t),

$(t’)])

and similarly for the other function. motion of Gj,i(t-t’) is

The

(Al)

equation of

6.

ili-&,l(r-t’)

= fi&$(t--I’)

([s;,s,-])

A254, 521 (1962). 9. LYONS D. H. and KAPLAN T. A., Phys. Rev. 120, 1580 (1960); see also LYONS D. H., KAPLAN T. A., DWIGHT K. and MENWK N., Phys. Rev. 126, 10. 11. 12. 13. 14:

540 (1962). Lr Y. Y., Phys. Rev. 84, 721 (1951). CALLEN H. B. and CALLEN E.. Phvs. _ Rev. 136. Al675 (1964). CALLEN H. B. and CALLN E., J. Phys. Sot. Japan (in press.) MATTHIAS B. T., BOZORTH R. M. and VAN VLECK J. H., Phqs. Rev. Left. 7, 160 (1961). MCGUIRE T. R., ARGYLE B. E., SHAFER M. W. and SMRT J. S., J. Appl. Phys. 34, 1345 (1963).

+2 1 J&Y~S;-S;S;; 1 +2 c Jj,&;S:B

S,) S?S;; S,>> 642)

The random placement

phase approximation

@is;;

ST>>+


consists

S,>

of the re-

(A3)

32

R.

A.

TAHIR-KHELI,

Then we introduce the temporal Fourier 03

G(t-t’)

H.

B.

transformation

= j G(E) exp[-iE(t-t’)/h]dE

(A4)

1 Glj = N

and

H.

JARRETT

and -%(k)

= 2
)[Juu(O)- J,,(k) - Juc@)l (AlO)

Furthermore (Si) = - is the average value of Sz for a spin on the up sublattice. Equations (A6) and (A7) give G&k,,??) directly, as

-co

and the spatial Fourier

CALLEN

transformation

c

Guu(k)exp[-~k~(rz-r~)l

(A5)

k

where 2N is the number of magnetic ions in the crystal. Correspondingly the spatial transformation of Gj, is G%&(k). Then the Green function equations of motion become

P--WWL@,~)

The roots of the Green function are the excitation energies, whence we find the spectrum given in equation (3.1). The solution for is slightly more involved, but equation (All) can be solved by several methods(lg-sl) of which the simplest has been given by H. CALLEN and S. SHTRIKMAN.@I)

where

L(k)

(S;>

= Jw( -k) = 2 Jlj exp[-ik*(rl-rj)] I -b(k)

= J&k)

(A8)

-=ez-1

= c Jfa exp[-ik.(rj-rg)] B

= 2 Jja exp[-ik*(rf-rrg)] j

(A12)

where B, denotes the Brillouin function, and x is defined by 1

= &u( - k)

= SB,(xS)

1 N c

1

El(k) - c(k) k

%k)

exp[r(k)/ksT]

-

1 W)

(A9)

from equations (A12) and (A13) one then easily finds(“) the Neel temperature, as given in equation (3.2).