On the calculation of photoemission and related integrals

Solid State Communications,

Vol. 12, pp. 211—214, 1973.

Pergamon Press.

Printed in Great Britain

ON THE CALCULATION OF PHOTOEMISSION AND RELATED INTEGRALS N.W. Dalton Theoretical Physics Division A.E.R.E., Harwell, Berkshire, England and G. Gilat Department of Physics Technion—Israel Institute of Technology, Haifa, Israel (Received 1 November 1972 by W. Low)

Explicit formulae are derived for the integrals which result from the applcation of the Gilat—Raubenheimer zonal integration method to the calculation of photoemission energy distributions. Furthermore it is shown how such formulae may be used to evaluate certain integrals required for the calculation of the imaginary part of the susceptibility of dielectric constant at zero temperature.

THE CALCULATION of many physical properties in solid state physics and chemistry requires the evaluation of integrals of the form 1(E) = 1ff F(k)6(E E(k))d3k, (1) —

B.Z

where kF(k)and are smooth functions the wavevector and E isE(k) an energy parameter. The of integration is usually carried out over the first Brillouin zone. E(k)

integrals associated with two energy spectra. For example, the calculation of the photoemission energy distribution12 involves integrals of the form I(E 1,E) = 3k, (2) 115 F(k)~(E1 Ei(k))t5(E2 E2(k))d B.Z where I(E 1,E2)are related to the photon and the emitted electron energies. F(k), E1(k) and E2(k) are smooth functions of k of similar meaning, but not necessarily identical with those occurring in equation (1). A similar integral occurs13also in the case of the Integrals related to de-Haas—van-Alphen orbits. equation (2) arise when calculating certain physical properties at zero temperature. For example, the calculation of e 2(w) involves integrals of the form 3k, (3) J(E1,E2) = 1ff F(k)0 [E1—E1(k)]ô [E2—E2(k)]d where 0(x)B.Z is the unit positive step function. Contrary to the case of 1(w) of equation (1)there has been little discussion in the literature relating to the calculation of integrals of the form of I(E, ,E 9’12 2) orJ(E1,E2). Janak proposed computer-oriented algorithms for calculating I(E 1,E2) based on the approach proposed by Gilat and —

is associated with the dispersion relations (e.g. phonons, electrons, magnons, etc.) characteristics of the property to be calculated. F(k) is known under such titles as ‘matrix element’, ‘transition probability’, ‘oscillator strength’, ‘coupling constant’, is particular associated with the interaction that gives etc. rise and to the spectrum studied. Examples of such integrals are the electronic or phonon density of states function1 for which F(k) function e = 1, the imaginary parts of the dielectric and the dynamical susceptibility tm (X(w)),2(w) the incoherent cross-section of inelastic -scattering of neutrons, etc. An accurate method for calculating such integrals, originally proposed by Gilat and Raubenheimer1 has been investigated in detail by several authors.2~ In addition to the higb.ly singular integrals of the type 1(E) one often encounters other 211

—

212

CALCULATION OF PHOTOEMISSION AND RELATED INTEGRALS

Raubenheimer,’ which may be suitably generalized to apply to J(E1, E2). However, an analytical expression for 1(E1,E2) analogous to those derived for 1(E) has not been worked out before. The main purpose of this letter is therefore to provide such expressions for I(E1, E2) and to show how they may be used to obtain the corresponding expressions for J(E1, E2).

simplicity, TheE2) procedure we assume proposed toby betoGilat rectangular and(I) divide prisms of Raubenheimer’ of by here the canfollowing be applied the steps: evaluation theI(Ej, volume of integration intothree small cells which, for the sides 2a1, 2a2 and 2a3, (II) within each cell replace the functions E1(k), E,.(k) and F(k) by the first two terms in their Taylor series expansion about the centre of gravity k~of the cell, and (III) evaluate analytically the resulting integrals throughout each cell. The latter integrals will be of two types: I~,(w1,w2

;

~,

132) =

3q

(4)

Vol. 12, No.3

u~

I I

G

H

1I@

c

B

FIG. 1. A scheme of the projection of an integration celland on u2 theisprojection plane. The coordinate systemis u1 shown. The coordinate u3 (not shown) along the intersection line of the two energy planes. The various zones are also shown. If the value of any u1 contribution or u2 is outside ABFHGC then there isofno to the the polygon integral zo~

11fceii~(~” —j31.q)t5(w2 —132~q)d ~1

3q

fff~~

(5)

1jq~(wi — ~31.q)~(w2 — /32~q)d

whereq ~

VE

I~(w1,w2,l31,f32)= 1(k) (1 = 1,2) and p = we x,yneed and zonly respectively. It was shown 8 that consider the calculation by Dalton of jo (w 1, w2 ; ~i’ ~2) and that I~(w~, w2 ~1, 1~2)can be obtained from jo by I~(w1,w2 ;I3~,132) —

=

~J~—J

I1(w~,w2 ; 13k, 132)dw’1.

(6)

Geometrically Io(w1, w2 ~1,~2) represents the length of the line of intersection of the planes q = 0 and w2 q = 0 which lies within the rectangular prism centred at k~.Before we proceed to give analytical expressions for the length of this line, we ifiustrate the general situation in Fig. I. Let the line A be given by its direction cosines i~ (i = 1,2,3), where 1,17= 1~ 1. We consider plane of any givenacell andpassing refer tothrough it as thethepro&ntre plane H. In Fig. I we project the cell on H jection —

—

along the line A and we defme a Cartesian coordinate system u1 and u2 in this plane, where u1 is chosen to be along the positive side of the projected axis; u3 is along A. It can be readily shown that the matrix T, which transforms the q into the u1, is given by

=

(Si0

—

— —

12 where s1

=

(1

_l~)1~~. Since

~i

I1

(7)

13/

we consider only integrals

over the integration cells it is possible to choose the directions of qj in order that all 1 should be positive. Furthermore, there is a zone (labeled I in Fig. 1) for which the length L of the line that intersects the cell is constant. If one considers the projection Uk of the areasSk = 4a1a1 (i,j,k is a cyclic triad) onto H, where = lkSk, then one can order the ik (k = 1 ,2,3) so that u3~c2~a~~ 0, which implies a corresponding rearrangement of the q,. In this case we consider the qj to be already chosen in the manner which satisfies these requirements. In The Fig. various 1 we observe fourdefined different forinwhich L * 0. zones are byzones certain equalities that are satisfied by u 1 and u2, the coordinate system in the projection plane. These conditions specify the zone boundaries. The value of L can be worked out by calculating the length of the line which is confined within the integration cell, i.e. L = I~u3I.The zones themselves are labelled by Roman numerals, and with

Vol. 12, No.3

CALCULATION OF PHOTO EMISSION AND RELATED INTEGRALS

213

Table 1. Formulae for the integral L = Ii~u31 = 10(w1 w2 ~1~2) for the different zones. Only integrals different from zero are listed. The conditions for the zone boundaries are also given (note that 03>02>01>0) Zone

Range of u1 [s1(o1—o3)

I

1112

13

IIa

Ls1(d1 +_03)

4a213

Isi(o3

—

I— L 41z213

Ui) —

1Si(01 L 411213 03)

~

{s~(u~— ~ 4a212

b

—

—

s1(o1 —o~)_~ 4a213 13

1112

s1(oi + 03)

—

1112 13 —

—

—

12

u2,

U2,

U2,

—

411213

L~:i(a~oi) 4a213

1~

F 02 03 4a1.c1

103—02 [ 4ais~

[

02 ± 03

4a1s1

i

4a~s1

13

s1(o1+o2) + 1113 4a312 12 U2 i11~

02

Vw1 I [m1u1+ (rn2!3 — rn3!2 )u2] (8)

=

Vw21 SI

[n1u1+ (n213

—

n312)u2]

where m1 and n~are the direction cosines of~1and ~2 respectively.

u1 11s1

02

—

03

4a1s1

12/3

1 j

4a1s1

]

02+ 4111s1 03

02~03 ‘~1~I

i1

13s1

s1

l~.u3I

l~ 13

a3

112 U2

+—

12

-t- — /3

u1 +13u2+ai

Jo(w1, w2

In Table 1 we list all the values of L for the various zones as well as their respective boundary inequalities, The expressions given in Table 1 contain the necessary data for evaluating photoemission integrals. These integrals must of course be given as functions of the two variables w1 and w2. In Table 1 the variables are rather u1 and u2, but it is straightforward to transform them into w1 and w2by

I I

+ 03 1~I ‘ 4a~s~

iT1ceii0~)i

=

02

1S1

, _______

0~+ 03

the exception of zone I they all consist of two subzones labelled with subscripts a and b (see Fig. I).

034a

s1(o1 02) + -b-— ui] 41131113 1113

0~

02 4a1s1

1112

‘

=

2113 13

03~ 021 ‘

L

1

o~) 1113 1 — —U2! 411213 13 j

4a1s1

Value of L

1 s1(o2 ---ar) + ..~a_ 4a31113 lil3~1

U2j

S1 (03 —

~b

U2,

1

2]

1112 1 —u2 13 I

—

—

13

U

—

I~i(~i U2, Si(0i —03) L 4(7312+ 02) + 1~l3 ‘2 411213 — — 1112

L

s1(a2 411312o~)•+ 1113 ~

~ a

{ U2~U3

ij

13

1113

02)

—

1112 U3,

13

IIIa

411213

1

1112

S1(0301)

——U2,

411213

Range of u2

—

112

=

~1,~2)

~i1q1)~(w2 ~2q)d~q —

(9)

These integrals are the contributions to J(E1, E2) of equation (3) coming from a single integration cell. The expressions for J0 are obtainable from Io(w1, w2 ~ ~ equation (4) by ,J0(w1, w2

&,~2) =

w

f~Io(a,w~; ~1,~2)da.

(10)

In conclusion, we have shown in this article how to obtain expressions for the photoemission integrals within the framework of the method of Gilat and Raubenheimer. These expressions are also derived and listed. Acknowledgements We would like to thank Dr. J. Hubbard for various discussions relating to this work. One of us (G.G.) would like to thank the Theoretical Physics Division at andTheoretical Dr. A.B. Udiard for the opportunity of UKAEA visiting this Physics —

From the computational point of view the expressions given here are algorithms pro9’3 equivalent However, to it isthe now easy to derive posed by for Janak. formulae integrals of the form Jo(w 1, w2 ; ~ defined by

Division at Harwell during which this work was carried out.

214

CALCULATION OF PHOTOEMISSION AND RELATED INTEGRALS

Vol. 12, No.3

REFERENCES I.

GILAT G. and RAUBENHEIMER L.J., Phys. Rev. 144,390(1966).

2.

RAUBENHEIMER L.J. and GILAT G.,Phys. Rev. 157, 586 (1967).

3.

KAM Z. and GILATG.,Phys. Rev. 175, 1156 (1968).

4.

GILAT G. and KAM Z., Phys. Rev. Lert. 22, 715 (1969).

5.

GILAT G. and BOHLIN L.,Solid State Commun. 7, 1727 (1969).

6.

DALTON N.W., A.E.R.E. Rep. TP—394 (1969).

7.

DALTON N.W.,J. Phys. C.,Proc.Phys. Soc. 3,1912(1970).

8. 9. 10. 11.

DAL TON N.W., Solid State Commun. 8, 2047 (1970). JANAK J.F., Computational Methods in Band Theory, p. 323 (Edited by MARCUS P.M., JANAK J.F. and WILLIAMS A.R.) Plenum Press New York (1971). GILAT G. and DALTON N.W., Solid State Commun. 9,461 (1972). DALTON N.W. and GILAT G.,Solid State Cornmun. 10, 287 (1972).

12.

JANAK J.F., EASTMAN D.E. and WILLIAMS A.R.,Solid State Commun. 8, 271 (1970).

13.

JANAKJ.F.,So!idState Commun. 10,833(1972).

On a établi les formules explicites des intégrales qul rñultent de l’application de la méthode d’int~grationzonale de Gilat—Raubenheimer au calcul des distributions en energy de la photoémission. De plus, on a montré que de telles formules peuvent ëtre utilisees pour evaluer certaines integrales necessaires au calcul de Ia partie imaginaire de la constante di~lectriquea temperature nulle.