J. ~ L E C R T A : On SCCF Treatment of the Hubbard Model of I-D Alloys 457

phys. stat. sol. (b) 141, 457 (1987)

Subject classification: 71.20

Department of Physics, The University of Leedsl)

On SCCF Treatment of the Hubbard Model of 1-D Alloys2)

Auger Densities

BY J. BLECHTA

There is presented an application of the SCCF method to the Hubbard model of I-D binary alloys which makes possible to progress beyond the narrow band restriction. The Hubbard split is demon- strated upon realistic densities of states provided by SCCF. There are also studied the multicentre densities relevant, e.g., for theory of the Auger effect and shown that for the alloys studied here the shapes of those densites do not depend upon the concentration of the components of those alloys. Thus clusters of more than one atom may play a role of a kind of “letters” or “words” in the genetic code.

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1. Introduction

The Hubbard model has been under a continuous attention since its basic theory had been published [l t o 31.

So far the Hubbard split has been studied upon model densities of states only, mostly parabolic ones, within the CPA approximation [4 to 91. In this work there are presented results of the application of the SCCF method 110 to 131, which gives practically precise densities of states [ll], t o that problem. Eor a selected parameters of a Hubbard model of 1-D binary alloys there are presented, in addition to the shapes of the densities of states, also the multicentre SCCF Green functions of those alloys and discussed in relation to the works on the Auger effect, prior t o this work [la t o 171.

2. Theory

In [lo t o 121 there was shown that all physical problems which can be transformed into the form

C KiiGtj = 8, (1) 1

can be tackled by the SCCF method. The matrix K = (K,,j) can have any form as this figurates nowhere in the justification of the SCCF method [lo].

l) 18 Lidgett Hill, Leeds 8, LS8, lPE, Great Britain. 2, Presented a t the 22nd Annual Solid State Physics Conference of the Institute of Physics

held in Reading in December 1985.

30 physica (b) 14112

433 J. SLECHTA

It can be shown that for the original Hubbard Hamiltonian [ l , 21

where Hij is the Hamiltonian of the physical system under consideration, without the Coulomb interaction, czu and ci ,o are the creation and annihilation operators, respectively, of an electron with its spin IS, nc,u is the operator of the number of electrons with the spin cr and I is the Coulomb interaction between the electrons with spins IS and -0, a t the site i and the approximations of the higher electron correla- tion functions as in [l to 31 the matrix K has the form

, K,z =2ncHil for i + I . (3)

The assumption of the narrow bands was not needed there.

3. Numerical Results and Discussion

There are given here the numerical results of the SCCF treatment of the Hubbard model for the 1-D binary substitutional alloys studied, in various contexts, in [lo to 13, 18 to 201. For the sake of continuity and comparison with the previous works there were used the same values of the parameters of the model as in [lo to 131.

In Fig. 1 there are presented the shapes of the total density of states for several selected combinations of the parameters involved.

These densities of states of the alloys n i th the Couloumb interaction between the electrons with opposite spins a t the same site there is best t o discuss on the back- ground of the shapes of the densities of states of those alloys without the Coloumb interaction published in [13, 191.

For example on the densities of states, for E = 0.1, separative and alternative kinds, z = 0.5 (50% of both the kinds of atoms) and I = 1 given in Fig. 1 there can be seen that, within such a comparison with the case I = 0 , each peak connected with the van Hove singularities a t the energies E = 1, 3, 4, and 6 [13, 191 is splitted into two peaks separated from each other approximately by the energy difference equal to one, in agreement with the original Hubbard theory. That can be seen most clearly on the split of those peaks for ni, --o = 0.1 a t E = 1, for ni, -u = 0.5 a t E = 3 and 4, for ni, -u = 0.9 at E = 6 and 7.

For I = 2 the peaks related to the Hubbard split of the same peak of the case I = 0 are separated by the energy difference equal 2. That can be most clearly seen in Fig. 1 a for ni, -u = 0.9 a t E = 3 and 5.

Generally, the Hubbard splits do not appear a t the middle of the bands, as in the case of the parabolic densities of states used in the works of other authors so far [l t o 91, but a t the edges of the bands as there, in the 1-D case, are the most pronounced peaks in the density of states and so also the most intensive demonstration of the Hubbard split.

The alloys under study here were also investigated for the existence of the ferro- magnetism, following the general part of [7], by the evaluation of the integral

e n," = J doJe-,(co) (4)

-m

and the test for the conditions n,F = n:,. Here E" = $ = EL, where E: is the partial Fermi level related to the filling of the conduction band by n,F electrons with their spins cr. The ferromagnetism exists only when that condition holds for n: =+ n:,.

On SCCF Treatment of the Hubbard Model of 1-D Alloys 459

33-

1 1 I I I

1 L 1 1 I I I 1 I

i\ ;=lo

f'

Fig. 1. The total densities of states e-,,(E) of the Hubbard model, as defined in the text, for E = 0.1. The case a) is separative kind and the case b) is the alternative kind. In both the cases there are presented results for z = 0.5, and several values of I . The solid line is for n-,, = 0.1, the broken line for n-, = 0.5 and the dash-dotted line for n-, = 0.9

30*

460 J. QLECHTA

The difference between the filling of the conductivity band by the electrons with the opposite spins which yield the same Fermi level is proportional t,o the total magnetic moment per a unit volume, namely [ 7 ]

( 5 )

where,uB is the Bohr magneton. This formula also gives information about the strength of the external magnetic field needed for t o achieve an additional separation of the values of n: and ns,, relative to the difference of those quantities without an external magnetic field.

I n Table 1 there are presented the values of the Fermi levels for a choice of the values of both the n, and n-, a rather wide set of x and F , both the separative and alternative cases, and the electron-electron interaction I. There can be seen that the

Table 1

Values of the Fermi levels e:, and eE_, for the Hubbard model as defined in the text, for e = 0.1 and a set of values of the partial fillings of the conduction band n-, and nu = 1 - n-, are presented. There is given here also when there occur the Peierls- Frohlich transitions (P-F, [121); the mark “Y” means their existence - ‘‘W’ their absence. Next to the mark “Y” there are given the transition temperature Tb [12]. By (w) or (s) next to it there is indicated when the transition is weak or strong, respec- tively. The total P-F is the weighted combination of the two P-F related to those partial fillings

X &nu F P-F : Th en-, F P-F : Tt total P-F : Tt -

G = l separative

n-, = 0.1 0.1 0.3 0.5 0.7 0.9

n-, = 0.5 0.1 0.3 0.5 0.7 0.9

n-, = 0.9 0.1 0.3 0.5 0.7 0.9

6.3 N 6.5 Y: 301 6.5 Y: 301 6.5 Y: 421 6.5 Y: 541

5.1 N 4.7 N 3.9 Y: 249 2.9 N 2.5 N 1.1 Y: 481 0.1 Y: 421 0.3 N 0.3 N 0.1 N

0.3 N 0.3 N 0.3 N 0.3 N 0.3 N

5.1 N 4.7 N 3.9 Y: 249 2.9 N 2.5 N 6.5 N 6.3 N 6.1 N 5.9 N 5.3 N

N Y: 301 Y: 301 Y: 421 Y: 541

N N Y: 249 Is x N N N N N

~~

alternative

m - , = 0.1 0.1 8.9 N - 1.5 N N 0.3 8.9 N -1.7 Y: 661 N 0.5 8.9 Y: 481 -1.7 N Y: 481 0.7 8.9 Y: 781 -1.7 N Y: 781 0.9 8.9 Y: 901 - 1.9 N Y: 901

On SCCF Treatment of the Hubbard Model of 1-D Alloys 46 1

Tab le 1 (continued)

n-, = 0.5 0.1 5.1 0.3 4.7 0.5 2.9 0.7 2.3 0.9 1.9

12-0 = 0.9 0.1 -1.9 0.3 -1.9 0.5 -1.9 0.7 -2.1 0.9 -2.1

N N N N N

Y: 781 Y: 661 Y: 541 N N

5.1 4.7 2.9 2.3 1.9

8.5 8.3 8.3 8.3 8.1

N N N N N

Y: 541 Y: 781 Y: 781 Y: 901 Y: 901

N N N N N

Y: 601 Y: 661 Y: 661 N N

G = 2

separative

n-, = 0.1 0.1 8.3 Y: 361 1.3 Y: 541 Y: 421 0.3 8.1 N 0.1 Y: 541 Y: 421 0.5 7.3 Y: 421 -1.1 Y: 541 Y: 361 0.7 6.9 Y: 421 -1.9 Y: 541 Y: 421 (w) 0.9 6.1 Y: 541 -2.1 N Y: 421

%-u = 0.5 0.1 4.7 N 0.3 4.1 N 0.5 3.7 N 0.7 3.1 N 0.9 2.5 N

4.7 N N 4.1 N N 3.7 N N 3.1 N N 2.5 N N

It-, = 0.9 0.1 1.7 Y: 421 7.1 N N 0.3 0.5 Y: 541 6.9 N N 0.5 0.1 Y: 541 6.7 w N 0.7 -0.1 Y: 421 5.9 Y: 241 Y: 241 0.9 -0.1 N 4.5 N N

alternative __

%-, = 0.1 0.1 0.3 0.5 0.7 0.9

It-, = 0.5 0.1 0.3 0.5 0.7 0.9

a-, = 0.9 0.1 0.3 0.5 0.7 0.9

~

8.7 8.7 8.7 8.7 8.7

4.3 3.1 2.3 1.9 1.5

-1.5 - 1.9 -2.1 -2.3 -2.3

~~ ~

N N N Y: 841 Y: 1021

N N N N N

Y: 1081 Y: 781 Y: 781 Y: 721 Y: 601

-1.7 Y: 901 Y: 541 -1.9 N N -1.9 N N -1.9 N Y: 712 -1.9 N Y: 901

4.3 N N 3.1 N N 2.3 N N 1.9 N N 1.5 N N

6.7 Y: 121 (9) Y: 121 (s) 5.9 N N 5.9 N N 5.9 N N 4.3 N N

462

Table 1 (continued)

J. SLECHTA

~

X E,, P P-F : Tt E,-, F P-F : Tt total P-F : Tt

G = 10 separative

1%-c = 0.1 0.1 0.3 0.5 0.7 0.9

7t-o 0.5 0.1 0.3 0.5 0.7 0.9

n-, = 0.9 0.1 0.3 0.5 0.7 0.9

9.7 9.7 9.7 9.3 9.1

4.5 4.1 3.5 3.1 2.5

2.3 1.3 0.7 0.3 0.3

x N N N N K N N N N Y: 301 (w) Y: 421 (w) Y: 421 (w) Y: 421 (w) Y: 781 (w)

~

2.3 ?x 0.9 N 0.5 N 0.3 N 0.3 PI;

4.5 N 4.1 N 3.5 N 3.1 R 2.5 N 9.3 Y: 541 (w) 9.7 Y: 541 (w) 9.9 Y: 541 (w)

10.1 Y: 661 (w) 10.1 Y: 541 (w)

N K N N PIT

N N P; N K Y: 421 Y: 541 Y: 541 Y: 541 (w) Y: 541

- alternative

~ ~ ~~

n-, = 0.1 0.1 9.3 Y: 541 -0.3 Y: 781 Y: 541 0.3 9.3 Y: 541 -0.7 Y: 721 Y: 541 0.5 9.3 Y: 661 -0.9 Y: 541 Y: 661 0.7 9.3 Y: 781 -0.9 Y: 361 Y: 721 0.9 9.3 Y: 961 -1.1 Y: 361 Y: 841

n-a = 0.5 0.1 4.9 N 4.9 N N 0.3 4.7 N 4.7 N N 0.5 3.7 N 3.7 N N 0.7 2.5 N 2.5 N N 0.9 2.1 N 2.1 N N

n-, = 0.9 0.1 -0.5 P: 721 9.3 Y: 601 Y: 601 0.3 -1.3 Y: 781 9.5 Y: 661 Y: 661 0.5 -1.5 Y: 721 9.5 Y: 661 Y: 661 0.7 -1.7 Y: 661 9.5 Y: 781 Y: 721 0.9 -1.9 Y: 541 9.5 Y: 901 Y: 781 __

random

n-0 = 0.1 0.1 8.7 Y: 541 2.7 Y: 301 Y: 481 0.5 8.7 N 2.1 N K 0.9 6.5 Y: 661 0.1 N N

n - u = 0.5 0.1 4.9 N 4.9 N N 0.5 3.7 N 3.7 N N 0.9 1.7 N 1.7 N N

%-, = 0.9 0.1 0.1 Y: 301 6.5 Y: 181 Y: 181 0.5 0.9 Y: 541 5.9 Y: 541 Y: 441 0.9 1.1 Y: 181 5.5 Y: 541 Y: 421

~ _ _ ~

On SCCF Treatment of the Hubbard Model of 1-D Alloys 463

relation 85 = &KO holds only for n-, = 0.5 and so no ferromagnetism can be ex- pected in those alloys. This finding is not contraversial with those results presented in [18] where there had been investigated the possibility of the existence of the long range magnetic order in 1-D binary alloys for I = 0. I n Table 1 there are also given the results of an investigation of these alloys for the existence of the Peierls-Froh- lich metal-nonmetal transitions.

Fig. 2. The two centre densities Gs,s,(E) for a) the purely random alloys; b) for the alloys with statis- tical short range order with E = 0.1, separative kind. Both the cases are for I = 0 ([17]) and results are depict- ed there for several values of x. By the solid line there is depictedGAA(E), by the dotted line GAB(E), by the broken line GBB(E) and by the dash- dotted line GBA(E). (The z values in b) are the same as in a))

E --

4. Multicentre SCCF Green Functions

I n this chapter there are given the results for the multicentre SCCP Green functions needed, for example, in the case of the Auger effect [15]. Within SCCF there is cal- culated straight not the total Green function of the material in consideration>&but the Green functions G M ( E ) of the representative M-clusters [ll, 121. For M r= 2

[

Pig. 3a, b

On SCCF Treatment of the Hubbard Model of 1-D Alloys 465

Fig. 3. The two centre densities Gsp,(E) for the Hubbard model as defined above, E = 0.1, separa- tive case, and for a) f = 1, b) I = 2, and c ) I = 10. I n dl the three cases the results are presented for several values of the filling of the conduction band n-,,. By the solid line there is depicted GAA(E), by the dotted line GAB(E), by the broken line GBB(E) and by the dash-dotted line GBA(E)

[13, 191, these Green functions are three centre ones. Out of those there can be cal- culated two and one centre Green functions by a statistical contraction [13]. Namely

From them the related densities relevant for the Auger spectroscopy [17], are given by the equation

Bij(E) = Im Gi@) .

'Ds,s,(E) = Im 'Gs,s,(E) . For example the Green function 2Gs,s,( E) yields the density

Tor the values of the parameters used within the work these densities 2Ds,s,(E) are depicted in Fig. 2a for the purely random binary alloys with I = 0; in Fig. 2 b for the substitutionary binary alloys with short range statistical order and I = 0, in Fig. 3 for the latter kind of alloys and several values of I =+ 0.

One can see that those densities: (i) for the purely random alloys, Fig, 2a, depend upon the values of the relative

concentration 5, while

466 J. SLECHTA: On SCCF Treatment of the Hubbard Model of 1-D Alloys

(ii) for the alloys with statistical short range order, Big. 2 b and 3, do not depend

It is due to the type of the averaging used in SCCE’ procedure of calculation of

Namely in the equation for calculation of the effective self-energy of the &/I-cluster

upon x.

3 ( & , ~ ~ , ~ , ( E ) and 2 G ~ , ~ , ( E ) .

{ % q ) M = 0 > (9) where T ( 2 ) is the SCCF T-matrix of the problem [13], one needs only the probab- ility of the occupation of the position S relative to the occupation of the position S, . While for the purely random alloys that probability is equal to x: for the alloys with short range statistical order it is the conditional probability which does not depend upon x. Thus for the latter kind of alloys 2Gs,s,(E) depend not on the surroundings of the atoms S,, X, and are positionally invariant. For that reason they can play a role of “word” or “letters” which can have definite spectral properties. This fact is very important for understanding and decoding of the genetic code in RNA and DNA.

(iii) show new peaks which can be interesting in connection with a plasmon-like elementary excitations. For a high value of I there appear collapsed densities. For example for I = 10, Fig. 3c, there is to notice that for n-, = 0.9 all the shapes of the densities of states are, relatively to the case I = 0, Fig. 2b, or I = 1, Fig. 3a, changed considerably, predominantely flattened - the peaks which appeared for small values of I are nonexistent collapsed ,here. Instead of them there appear new sharp peaks a t E = -2.9, 0.9, -0.7, for n-, = 0.5 and E = -2.1, 7.3, and 8.1 for

= 0.9 of these new plasmon-like localized frequencies.

Acknowledgements

The results were computed on the authors’s Micro Sinclair ZX Spectrum with 48K RAM and a microdrive. He likes to thank to the Benevolent Fund of the Institute of Physics, for their financial and moral support of himself; and the Brotherton Library, especially Mr. D. Cox, its librarian, for their hospitality.

References [l] B. J. HUBBARD, Proc. Roy. Soc. A256, 283 (1963). [2] B. J. HUBBARD, Proc. Roy. SOC. A277, 237 (1963). [3] B. J. HUBBARD, Proc. Roy. SOC. A282, 1101 (1964). [4] J. L. BEEBY, Internat. School of Physics Theory of Magnetism, Transition Metals, Ed.

[5] C. F. OSBORN, J. Phys. B 2, 1100 (1972). [GI J. SCHNEIDER and V. DRCHAL, phys. stat. sol. (b) 68, 207 (1975). [7] H. AOKI and KANIMURA, J. Phys. SOC. Japan 39,1169 (1975). [8] M. SAKON and SHIMJZU, J. Phys. Soc. Japan 40, (1976). [9] M. PETRUSKA and W. BORGIEL, Z. Phys. D 58,99 (1985).

W. MASHALL, Acad. Press, Inc., New York 1967 (p. 87).

[lo] J. SLECHTA, phys. stat. sol. (b) 1%i, 403 (1985). [ll] J. SLECHTA, phys. stat. sol. (b) 120, 329 (1983). [la] J. ~ L E C H T A , J. Phys. C 12, 1819 (1979). [13] J. SLECHTA, J. Phys. C 10, 2047 (1977). [14] &I. CINI, Solid State Commun. 20, 605 (1976). [15] V. DRCHAL and J. KUDRNOVSK+, phys. stat. sol. (b) 124, 179 (1984). [I61 V. DRCHAL and 5. KUDRNOVSK+, phys. stat. sol. (b) 12i, 611 (1985). [17] G. A. SAWATSKY, Phys. Rev. Letters 39, 504 (1977). [IS] J. QLECHTA, phys. stat. sol. (b) 133, 151 (1986). [19] J. ~ L E C H T A , phys. stat. sol. (b) 71, K165 (1975). 1203 J. SLECHTd, phys. stat. 801. (b) 104, K143 (1981).

(Received April 18, 1986; in revised form June 26, 1986)