Electronic density of states of the cluster-Bethe-lattice and the diamond lattice with continued fraction method

Solid State Coininunlcatlons,Vol. 16, pp. 1383—1385, 1975.

Pergainon Press.

Printed in Great Britain

ELECTRONIC DENSiTY OF STATES OF THE CLUSTER-BETHE-LATTICEAND THE DIAMOND LATTICE WITH CONTINUED FRACTION METHOD W. Leung Department of Physics, Imperial College, London SW7, UK. (Received 6 March 1975 by C. W. McCombie)

The electronic density of states of diamond lattice was evaluated with the continued fraction method, up to the third and the fourth level. The result was found to be superior to the cluster-Bethe-lattice approximation to the diamond lattice.

RECENTLY Yndurain et aL1’2 proposed a method to treat tetrahedrally bonded materials. The method treats a cluster of twenty nine atoms in a diamond lattice, which includes the first, the second and those third nearest neighbours which form sixfold rings, in an exact way while a Bethe lattice with coordination number equal to four is attached to each dangling bond on the surface of the cluster to simulate the medium beyond the cluster. The local Green’s function, and in tUI•I1 the electronic density of states p(E), can be evaluated with the transfer matrix method. In this paper we showed that the cluster-Bethe-lattice can also be treated exactlyby the continued fraction method (CFM), and the result is virtually the same as that given in referecne 1. Although the density of states spectrum contains some of the main features of the exact result for p(E), it also contains spurious structures near the band edge. Then the CFM was applied to• the diamond structure exactly up to the third and the fourth level with a proper continuation beyond that. The results from CFM for a diamond lattice are superior to that of the

where i and! are site indices. For simplicity we considered nearest neighbour interaction only. In other words, the hopping integral Tu assumes the form Tu

T if i andj are nearest neighbours

=

0 otherwise The local Green’s function G~,where 0 refers to a certain site in the lattice, can be written down with the continued fraction method, as developed by Haydock, Heine, and Kelly,3 _____

G~o(E)= ‘~IE_H~o) =

E

—

1 a 2 E 1T a 2 2aT 2 E 3T

(2)

—

—

where a 1 , a2, a3... are coefficients depending on the lattice structure. These coefficients as’s can be related to the number of paths leading to sites which can be reached in n hops. For the cluster-Bethe-lattice mentioned at the beginning of the paper, the coefficients an’s assume the following values a1 = 4 a2 = 3 a3 = 5 a4 = 11/5

cluster-Bethe-lattice. The main differences between these two methods were discussed at the end of the paper. The tight-binding Hamiltoman which we considered consists of one Wannier function per site, and it can be written as H = ~T~Ii>qI (1)

a~= 3forn>4.

i.j

1383

1384

ELECTRONIC DENSITY OF STATES OF THE CLUSTER-BETHE-LATTICE I

I

I

I

3:: J\J7~\.

C

02

0’ -4

-3

-2

-1

0

1

2

Reduced Energy

3

r

0.3

Vol. 16, No. 12

E/T

FIG. 1. Electronic density of states of cluster-Bethelattice as a function of reduced energy E/T. The dashed line is the exact as calculated in reference 5

-3

-~

4

-2

-1

I

I

0

I

Reduced Energy

F

1

2

3

E/T

~

. density of states of diamond lattice, FIG. 2. Electronic to the third level in CFM, as a function of reduced energy E/T. The dashed line is the exact result as calculated in reference 5.

Then G~can be written down in a closed form

0.3

I

-

G~(E)= 1/~E—4T2E—3T2[E—5T2(E—~T2F)_1]~}’~ (3)

~0.2

//I

—

U)

/

where F is given by F

=

\\

~E±~E2

—

l2T2)/6

jol

Wechoosethe+signwhenE<—~/12T,andthe— sign whenE>—~./12T.The electronic density of states is related to the local Green’s function through the following equation

I

~

/1

/

~

2

-i

0

1

Reduced Energy

2

3

4

E/T

FIG. 3. Electronic density of states of diamond lattice,

to the fourth level in CFM, as a function of reduced p(E)

=

—

—

Im G 00(E).

(5)

The results of p(E) from equations (3)—(5), 4’5 were plotted in Fig.together 1. equation with the exact result, (2) of reference 1 gives virtually the same result, including those sharp peaks around E ±3.45 T although they were not shown in Fig. 1(a) of reference 1. We can see that there are several unsatisfactory features in Fig. 1. First there are two spurious peaks near the band edge. Secondly the band width is not correct; it is 4~/3T rather than the correct value of 8T. The peaks around E ~ is±appreciably 1 .6T are displaced toward = 0, and magnitude larger than the Ecorrect one.its —

So instead of approximating the diamond lattice with a cluster-Bethe-lattice which can be solved exactly, we studied the diamond lattice and made certain assumptions in equation (2) to obtain an approximate solution of G~.For a diamond lattice, the coefficients a~‘s assume the following values

energy E/T. The dashed line is the exact .result as calculated in reference 5. a 1 a

=

4

a2

=

3

a3

=

5

a4

=

17/5

=

We made assumption that a~= a = constant for n greater than a certain value, and we chose the constant a in such a way that p(E) will have the correct band width. This condition was satisfied by be choosing = 4. 6 G~can writtena as If we=stop at the4T2 third G~ l/{E [E level, 3T2(E 5T2F’)~ }. (6) —

—

—

‘

If we stop at the fourth level,7 G~can be written as G = 00

l/~E 4T2 {E —

—

3T2 [F

—

5T2(E

—

~ T~F’)~ J~

JJ~ (7)

}~I

Vol. 16, No. 12

ELECTRONIC DENSITY OF STATES OF THE CLUSTER-BETHE-LATFICE

The function F’ is given by F’ = (E ±~/E2 — 16T2 )/8

(8)

We choose the + sign when E <— 4T, and the sign when E ~ 4T. The results of p(E) from equations (6) and (7) were plotted in Fig. 2 and 3.. We can see that our result is much better than that of the clusterBethe-lattice. It gives good approximation to the exact result near the two outer band edges, which correspond -‘-

—

to the center of the Brillouin zone. Furthermore, there is no spurious structure near the band edge. The result around E = 0, which corresponds to regions near the Brillouin zone boundary, is not very satisfactory, We were unable to reproduce the rapid variation in p(E) around E = 0, because we have not gone to a high enough level in the continued fraction formula. The results of the cluster-Bethe-lattice with an exact G® and the diamond lattice with an approximate G~give quite different results for the density of states spectra. When we examined equations (3), (6) and (7), we found that the coefficient a~‘s agreed with each other up to n = 3. The differences are in the fourth coefficient and the continuation function F and F’. The factthat equation (3) does not give the correct hand width is due to the continuation with F. This can be corrected by continuation with F’, i.e. attaching Bethe lattices with coordination number equal to five to the surface of the cluster. Then G00 can be written as = 2{E— 3T2 [EG®5T2(E ‘~T2F’)~ ~ ~ l/~E 4T (9) This will give the correct band width. However, other features of the density of states spectrum remain essentially the same. The sharp peaks at E ±3.45T —

—

r

—

1385

are slightly broadened, and the peaks atE ~ ±1 .6T have very little change. We believe that the spurious structure near E ~ ±3.45T is due to the neglect of many closed paths beyond the twenty nine atoms which are treated exactly in the cluster-Bethe-lattice, rather than the difference between F and F’, Many eightfold ring structures are neglected in the clusterBethe-lattice, and this leads to an incorrect coefficient a 4 in the continued fraction formula. Since p(E) is zero atE = 0, this necessitates many levels in the CFM to give a fair approximation around E = 0. This fact also causes the CFM coefficients an’s to oscillate around some value. The assumption that a~becomes constant after certain level is not quite right. However, we would expect that the error due to this assumption becomes less important as we go to higher level in the CFM, as evidenced in Fig. 2 and 3. In the present case for the diamond lattice a~is assumed to be four for n >4 (or n> 3 if we stop at the third levle). We noticed that four is approximately the average of the first four a~‘s. The results from this method give good approximation to the exact result, especially near the center of the Brfflouin zone. We can conclude that the continued fraction method with proper continuation is a better way to approximate the electronic density of states for a diamond lattice, than the cluster-Beth-lattice approximation. In this paper we treated the cluster-Bethe-lattice and the diamond lattice only. However, the CFM can easily be applied to other lattice structures,and and provides a useful method to study amorphous substitutionally disordered matenals. Acknowledgement The author would like to thank ~c~’I; Jac~bsdforseveral discussions on the continued —

2.

REFERENCES YNDURAIN F., JOANNOPOULOS J.D., COHEN M.L. and FALICOV L.M., Solid State Commun. 15, 617 (1974). JOANNOPOULOS J.D. and YNDURAIN F.,Phys. Rev. BlO, 5164 (1974).

3. 4.

HAYDOCK it, HEINE V. and KELLY M., J. Phys. CS, 2845 (1972). HALL G.G.,FhiL Mag. 43, 338 (1952).

5.

THORPE M.F. and WEAIRE D.,Phys. Rev. B4, 3518 (1971).

6.

This corresponds to treating41 atoms exactly.

7.

This corresponds to treating 83 atoms exactly.

1.