STRUCTURE OF THE ELECTRONIC SPECTRUM OF THE FERROMAGNETIC s-d MODEL USING THE METHOD OF MOMENTS

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STRUCTURE OF THE ELECTRONIC SPECTRUM OF THE FERROMAGNETIC s-d MODEL USING

THE METHOD OF MOMENTS

I. Barvík, V. Čápek, P. Chvosta

To cite this version:

I. Barvík, V. Čápek, P. Chvosta. STRUCTURE OF THE ELECTRONIC SPECTRUM OF THE FERROMAGNETIC s-d MODEL USING THE METHOD OF MOMENTS. Journal de Physique Colloques, 1980, 41 (C5), pp.C5-293-C5-295. �10.1051/jphyscol:1980550�. �jpa-00219985�

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JOURNAL DE PHYSIQUE Colloque C5, supplément au n° 6, Tome 41, juin 1980, page C5-293

STRUCTURE OF THE ELECTRONIC SPECTRUM OF THE FERROMAGNETIC s - d MODEL USING THE METHOD OF MOMENTS.

I . BARVIK, V. CAPEK and P. CHVOSTA*

Institute'of Physics of the Charles University, 121 16 Prague, Czechoslovakia.

*Departement of Mathematical Physios, Faculty of Mathematics and Physics, Charles University 180 00 Prague, Czechoslovakia.

Résumé.- Dans le cadre du modèle s-d, les six premiers moments de la fonction spec- trale associée à la fonction Green électronique sont exprimés en utilisant les fonctions de corrélationspin-spin, â la limite des faibles concentrations. Cette information est utilisée pour rapprocher les fonctions spectrales par des fonctions physiquement raisonnables de façon à obtenir des valeurs correctes des premiers moments. Dans le cas du couplage s-d fort, nous sommes aussi capables d'obtenir exactement les premiers moments des sous-bandes électroniques. Les spectres élec- troniques obtenus ne montrent aucun changement abrupt à la température de Curie, en accord avec l'expérience. On montre alors que le déplacement vers le rouge du front d'absorption dans la famille de matériaux comme CdCr2Se. ne peut être attribué au couplage s-d dans une bande étroite d unique.

Abstract.- The lowest six momenta of the spectral function of the electronic Green function in the s-d model are expressed using spin-spin correlation functions in the low electron-concentration limit. This nontrivial information is used to fit the spectral function by physically reasonable functions yielding correct values of the lowest momenta. For the strong s-d coupling, we are also able to deduce lowest momenta of electronic subbands exactly. Resulting electronic spectrum shows no abrupt change at the Curie temperature, in accordance with the experiment. It is then argued that the red shift of the absorption edge in CdCr-Se. family cannot be explained as being due to the influence of the s-d coupling on a single narrow d-band.

The aim of the present contribution is to report results of our attempts to obtain physical conclusions for the standard single band s-d model with the hamiltonian

ft = Hs + Hd + Hs_d= (1)

= mn<5 fcmn amSan<5~2 m * nJm nSmSn_ Im S S '

( Sm6 )6 6 ' a ma^d '

in a way which preserves the total rotational symmetry of the hamiltonian(1). Consequently, we do not break the symmetry by the infinitesimal magnetic field 5» = (0,0,5^)/limited to zero after taking the thermodynamic limit o, -»-oo/ but we direct this field parallel to a general vector v jv|=l/ and average with respect to the orientation of this field in space. / According to Bogolyubov' ' , the result is the same as if we were not brea- king the symmetry by the infinitesimal field Tt at all./ The idea is that this procedure influences surely the weights but not posi- tion / i.e. the energy structure/ of quasiparticle poles.

The method we have used is that of moments ' ' of the spectral function ' '

A(k,E)=E[a(t-t')<a ( t ) a+.6 ( f ) >

exp$-ik(r.-r.)+i E(t-t')J

x J ft '

which m i g h t be calculated exactly/all the averages here and from n o w on are m e a n t after averaging with respect to v and in the limits fi-«°, Oe+G, .y-»—<°, y being the chemi- cal potential/. W e have succeeded to express explicitly the lowest six momenta.

M(F ' = JA(k,E) Er dE

K J 2irh

(2) m x

M J V ^ H ^ ]]...] >e-

ik(r

i~

r

j

)

nx m+n=r

yielding /after suppressing usually negligi- ble terms containing explicit positive powers of J /

mn

M i =t , —h e m n . ,

k k n t ' mn M( 2 ) =t£ + I2S(S+1),

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1980550

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JOURNAL DE PHYSIQUE

Endeavour needed to express yet higher momenta increases considerably. One should mention here that within our

scheme,^':)

_-_-

dep&d on the temperature just through spin correlation functions. ? I 0 single-spin average appears /<S t > =O even in the ferromagnetic region /1/ and .&, <vS>+O when &-+O/.

In /5/, we have just used M(O!.

.

^.M⁽³⁾

k k

to fit the parameters pi(k) and ~ ~ ( k ) /i=1,2 in the approximate expression

A aPPrOx (k,E) =C- 2 pi (k) 6 (k)

1-1 (4)

in such a way that

r = 0,. . . 3 . This scheme has a solution for

all values of input parameters tmn, I end T yielding a very reasonable form of energy bands ci(k). The resulting band structure shows no abrupt change of ci(k)/or its temperature derivatives / at the Curie temperature; in a contradiction with all theo- ries where the magnetization enters the electronic energy but in a correspondence with the experiment. However, the use of

(4) is not completely satisfactory in the weak scattering r6gime IS<< W where little correspondence with the perturbation theorq Hs-d being the perturbation/ has been found

though the limit T+O is reproduced correctly.

In order to improve the scheme, we have in- volved all six momenta M';)

,

^r=o,

. .

^.5 ^to

improve ( 4 ) as

\ 0 otherwise

It is clear that (6) is good for studying the broadening of quasiparticle peaks. After determining pi (k)

,

^E~( k ) and ai (k)

,

^i=1,2

from (5) / r=0,

.

.5/, we have however found no relevant improvement for IS<< W. On the other hand, this improvement has been revealed after introducing the asymmetry effects

into the model form

/k being e.g. in the vicinity of minimum

-

of the lower band; W is an adjustable para- meter of the dimension of energy/. E'igur0e 1

shows the red shift of the/ lower edge of the /lower band calculated from (7) satis- fying (5) for r=0,1.

.

^.5,due to the change of temperature from T =

-

to T=O. We have used the simple cubic lattice with tmn=-&

for nearest neighbours and zero otherwise/

W being t h ~ original band width/. Several values of W have been chosen. Notice an improved correspondence with the result of the perturbation theory/ indicated as a dashed line.

The method of moments enables us to calculate the momenta of subbands in the strong scattering rggime IS>zW j6// the red shift

-

the quantity of primary interest for us

-

increases with increasing I though strong saturation effects appear once IS passes across W/. From (3), we get that the lowest momecta of,the density of states in the lower subband

/ indicating the integration just over the

1

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lower subband/ are

For S=-Z 3 / as for CdCr2Se4/ and the simple cubic lattice /there are no indications of any relevant change for other types of lat- tices/, one may thus estimate the band narrowing /due to increase of T from 0 to m / to be about 20% of W. Interpreting this narrowing as being the cause of the shift of the absorption edge in CdCr2Se4/

-.

^{0.3 eV4}

the relevant band width results greater than lev. It seems to be too much for a narrow d-band so that this treatment serves as an argument against interpreting the red shift in CdCr2Se4 as being due to shift of a single narrow d-band due to the s-d coupling.

To our point of view, at least some other bands hybridized and overlapping with the d-band a,re needed to interprete the red shift in the CdCr2Se4 family satisfactorily.

Fig. 1 The red shift A of the absorption band cal- culated as a shift of the edge E ₁(k)

1

_k=Q ^{of the}

lower subband in (7) due to the change of ternpera- ture from T=O' to T= for W=l eV,s = 3/2 and.dif-

References

/1/ ~ogolyubov, N.N., in Statistical Physics and Quantum Field Theory,ed. Bogolyubov N.N., (Nauka, Moscow) 1973- in Russian /2/ Kalashnikov, O.K.,Fradkin, E.S., Phys.

Status Solidi b 59 (1973) 9.

/ 3 / Nolting, W., .Phys. Status Solidi b 79

(1977) 573.

/4/ Kadanoff, L.P., Baym, G., Quantum Statistical Mechanics (Benjamin, New York) 1962.

/ 5 / Barvik, I., Egpek, V., Czech. J. ~ h y s . B.29 (1979) 1136.

/6/ Barvik, I., ?&ek, V., Chvosta, P.,

2 . Magn. and flagn. Materials 14 (1979)

97.

ferent values of W % / see the text/.