Photoemission from Random Alloys

PHOTOEMISSION FROM RANDOM ALLOYS B. Velicky and J. Kudrnovsky Institute of Physics, Czech. Acad. Sci., 180 40 Praha, Czechoslovakia The photoelectron distribution from a random alloy of d electron metals is affected by the disorder in two ways: The k selection rule is not valid,and the transition matrix element contains a vertex correction. The first effect may be anisotropic in the k space, and heavily broadened parts of the spectra will coexist with vestigial interband structures. The vertex reflects the kinenlatical correlation in the motion of the electron- hole pair and it depends on the nature of the excited electron states. 1. FORMULATION OF THE TASK In the band theory, a dipole optical transition is subjected to a severe selection rule stating that the quasimomenta of the initial and of the final states are the same: (1 )

The optical transitions obeying this selection rule are said to be vertical, or direct. The PES, which studies details of the distribution of the photogenerated electrons is particularly suited to testing the validity of the k selection rule. In fact, the modern photoemission studies have been intimately connected with this problem, as in many instances the evidence for the k conservation was not very convincing. The verticality of the optical transitions may be relaxed either by many particle effects, or by a violation of the exact periodicity of an infinite crystal. Both these factors may be important for the photoemission, as the final states often have quite short lifetimes, and the photoelectrons originate basically from the surface region of the sample. It should not be surprising then that the k selection rule was believed to be even less important in

case of so-

lid solutions, where the randomness in the atomic distributions should lead to substantial non-periodicity of the one electron potential. Such was the situation before the CPA

when mostly the non-direct

model [1] and the virtual level model [2J were in use. The CPA improved the situation in the theory of electronic states in random alloys, because it provided a rational and systematic key for their description over a wide range of the alloy parameters, and it was tested many times by simply comparing the measured EDC profiles with computed DOS curves

[3 J ' ( 4 J .

The agreement was usually not bad, and

the previous models were recovered as limiting cases. This simplified 70

approach never got a justification, and today it is not fully appropriate, because the experimental situation has improved much, and especially the angle resolved experiments are becoming more widely accessible. Also the degree of sophistication of the alloy theory is rising. Altogether, the time seems to be ripe for a more detailed theory of the photoemission from the random alloys. This report is a sequel of two preliminary communications [5J ' [6]

and, significant-

ly, it appears in this volume along with a paper on the results of a very similar research done in Daresbury

[71 .

Our approach was to choose the simplest possible situation containing all characteristic features of the problem, but susceptible at the same time to a not overtedious treatment. (i)

We shall treat random substitutional alloys of the d electron

me~als.

Their valence bands are strongly influenced by the disorder,

and have a nearly one-electron character at the same time. (ii)

We treat the UPS for comparatively low photon energies, when

the bulk band structure regime is likely to be appropriate. and the k selection rule holds well for the respective pure metals. For the alloy we then expect to obtain the effect of the disorder-induced changes in its bulk electronic structure. (iii)

We shall study only the energy distribution of the photoelec-

trons. Among other advantages, this allows us to use the three-step model (assumed for simplicity isotropic) and to deal only with the excitation stage of the photoemission process. (iv)

To minimize the effect of the energy dependent propagation and

escape factors in the differential photoemission yield, we shall plot our

~esults

as the non-normalized constant final state (CFS) spectra.

Another advantage over the conventional EDC plots is that the CFS profiles come as close as possible to the valence DOS, as compared to the interband DOS for EDC. (v)

To describe the Hamiltonian we use the orbital representation.

In particular, the d-like valence bands are taken as tight binding with diagonal disorder, while the final stutes are virtual crystal NFE waves. This is in the spirit of the well tested combined interpolation scheme. Although somewhat inferior to the real space treatments, it has substantial computational advantages, and also serves as a convenient conceptual framework. (vi)

In dealing with the disorder, we use the configuration avera-

ging. This restores the overall periodicity of the alloy, but with an effective optical one-electron potential, or self-energy. The semantic differences with the non-direct model using the true wave functions

71

should not be misunderstood. We use the CPA in actual computations, but the whole theory is given in a more general form. This is important because of the recent &uccesses of the cluster CPA

[aJ ,

[9J '

which hint that soon better approximations may be in general use. Many people helped us to understand what we really want to do, and we especially acknowledge the conversations with I.

Barto~,

A. Liebsch, A. Mookerjee, and G. Paasch. 2. FORMALISM The object of our study is the one-electron, Golden rule expression for the primary distribution of photoelectrons with the final energy

E as a function of the photon

energy~:

P(E,w)cx;~ L l12{(f-t.J-fj)J'(E-Ed

(2)

N i,l-

In this expression the sum runs over all initial (occupied) and all final

(empty) states conserving the energy, and (iIM/f) is the transi-

tion matrix element. To make this expression tractable, we must rewrite it in an invariant form and perform the configuration average:

Here H denotes the one-electron Hamiltonian of a given configuration. Eq.

(4)

(2) should be compared with the averaged density of states

t

(E-W) =
The averages in Eqs.

(3-4) can be performed in real space, or using

an orbital representation, as we intend to do. Although the final coherent potential approximation is the same in spirit in both cases, it is applied to different quantities

[3]

as we want to stress. Sup-

pose we want to evaluate the DOS in (4) using a tight binding representation. The atomic like orbitals will differ from site to site and there is no well defined single band subspace describing the averaged situation. Thus, the averaging, and the trace are not readily interchangeable. A solution is to represent the single band by the corresponding matrix problem (using the random orbital basis) and to perform the averages on the matrix quantities. This is straightforward for the DOS and everybody using the orbital representation proceeds in fact in this way. In the case of the photoemission yield, the transition matrix represents an additional problen: while in the real space ;. may be in the simplest case represented by 72

:1 c.ompone n t.

of the mo-

mentum, and is non-random, in the basis of random orbitals the transition matrix elements are inevitably random, too. The binary system considered contains components A, B with conA B = 1. The detailed description of a confi+ c

centrations c A, c B, c

Q

guration is contained in the characteristic functions ~i=O,11 Q = A, B. For the ftarithmetic state vectors ft we shall use notation with round

brackets. Thus If, k> -tlf, k). The arithmetic basis it-

self is non-random, and the terms "Wannier states" and "Bloch states ft

are applicable. In this basis, we introduce independent Hamiltonian etc. for the initial, and for the final states: Hamiltonian resolvent self-energy

H (... Hi I Hf. )

G(1) s

ref)

,

(1 - H)"1

virtual crystal

, Green function ~(l) =(l-il- !(lJf1.

The spectral densities replacing the delta functions of the band picture are introduced by

ME)" (JeE-H» (5)

o1

A(k,E)=.-u

_

= L /k) A(kJ) (k I

Im G(k,EtiO)

,,-1

[m

L

- - (E - € (I<) - 'Re L) 1

Returning to Eq.

I

(3)

1" ( /171

r.) 1.

/

k, E+ i

0

for the differential photoelectron yield, it

should be observed that the object to be averaged is a product of four random factors. They are not independently random, however, because they all relate to the same atomic configuration. This makes the averaging difficult. It is customary to single out the product of independently averaged factors and to introduce the remainder as the socalles vertex corrections:

(6)

The disorder-induced vertex was introduced in discussed many times

[11] , [12] , [13] ,

[10J '

and it has been

mostly in connection with

the transport properties of non-crystalline solids. There is a basic reason for splitting the average in the indicated way: two of the factors are in fact resolvents, and they describe the motion of an electron-hole pair correlated kinematically by the same random environment. The independent average is then the free term in the cor73

responding Bethe-Salpeter equation. In addition to this fundamental source of the vertex, there are some other, sometimes more effective. One is a hybridization of the initial and final states. Because it may change with the disorder, it cannot be eliminated beforehand [14J. Still another source is the random matrix element. This will be discussed below, but here we point out that it has been originally introduced in connection with the electrical conductivity of alloys [ 15] • Neglecting for a moment the vertex part of (6), we may give a general discussion of the k selection rule. For a crystal, the spectral densities become delta like and

P([,W)
~[ /M(k)/1 cf ( f - 4J - €i (J<» J(E-€,(ld)

becomes the standard interband expression. As the disorder sets on, the spectral densities become broadened and cover finite strips in the k space. This smearing of the k vector leads to the relaxation of the k selection rule, although the integrand in the Eq.

(6) has seem-

ingly the vertical structure. In the limit, when one of the spectral functions will be spread evenly over the Brillouin zone, the yield will be proportional to the respective alloy DOS and the classical non-direct model will result. 3. MODEL FOR THE d BAND METALLIC ALLOYS The presently used model assumes tight binding d states (represented for simplicity by a single band) and virtual crystal NFE final states. By this we mean that the effect of disorder on the spectral density of the final states is negligible, and that the factor A can f be taken out of the average in (6). It still may be somewhat broadened by the final state life-time effects. This broadening in fact further weakens the influence of disorder, as it corresponds to drifting away from the real axis in the energy argument of the final state Green function. At the same time, the true final state wave function may have a form strongly modified by the disorder. In particular, its oscillations in the core region will depend on the occupancy of a given site. This leads to the folloWing assumption concerning the form of the transition matrix: In the mixed Wannier-Bloch representation, (7)

where

(a',I1/MIf,lc)

f~(k)

Ii C

,,,

r (k)t de" ", q

are two different functions of k, in dependence on the

type of atom on the given site. Substituting (7) (8)

P(f,W)OC 74

into (6) Yields

R""

..1. L L. L

N k n,ttI'la'

f ClC>f

R' iJc(7i'~-l?ttI) ~ (/e)l(~"

'm (t;n/I(E-tu-Hj}/£,,,,,>/ At! R'

..\

(

k,f)}

with the characteristic two-site conditional averages. This is still a general expression, exact for our model. Just how important the vertex correction will be, depends on the functions 'fA, B • First, two special limits will be analyzed. If 'f A = 'fB:: the case we shall call for obvious reasons isocoric, the Eq. (8) simplifies back to

r

r'

(9)

1 [li(k)/2 PCE,w)oc_ Ai (Ic , E- (,J ) A, (Ie, E) ,

N

k

which corresponds to an intuitive model employing the complex band structure concept. Here, the vertex is absent. An opposite situation would arise for an "isoelectronic" case, when the resolvent should be essentially non-random, while the matrix elements depending on the core oscillations would be different. The average can be obtained explicitly. Introducing (10)

r' - r'

t:

I

we can give the differential yield precisely the form of Eq. (6): ( 11)

P(f)W)oc..1.[

If(k)J1,f(f-W-E-dk»

N k + c"cBa-.(f-W)

-1...1:

A,(k,E)

}f(Ic)J!

i-

A, (k,f) .

N k While the first term has a band-like character, the vertex correction P

appears to be proportional to the valence DOS and has the general appearance of an incoherently scattered wave. To analyze the general case, we shall perform an important, still exact, transformation of Eq. (8). To this end we introduce the component projection operators in the valence arithmetic space: (12)

pfl=[/itl>'l:(ln/

PA + PB =

n

Eq. (8) becomes (13)

-

P(C,4J)oc-1..LL. N

k fI,Il'

Q*

Q'

r. I 1n>
:

Pi .

QQ'

'f (0'f (Ic)A i (k,E-c.J)A,(k/E) ,

where the new quantities AQQ - will be called the partial spectral densities and their definitions and some salient properties are:

Atl'(E) = (P'lJ(f- H L) ( 14)

: t- I c, Ie) Ai

pI/.)

'1'1'

( k, E) (c; k / 75

The importance of this transformation will emerge if we observe that in (8) we had to find the two-point quantities, which were not translationally

i~variant

and therefore were more complicated for avera-

ging. At the same time, only one matrix element of the whole

<

>

I(

~~ 1l~ E - w - HiJ was really needed. After the transformation, only the significant matrix elements are involved. and they are arranged into the translationally invariant form of the partial spectral densities. To illustrate this point, we further specialize our model by assuming the disorder in the valence band to be diagonal, that is the Hamiltonian now will consist of a periodic part Wand of a random part characterized by "atomic levels" E A

, f B .•

/./ '"' L. Ik

( 15)

I<

n

€.. (Jd (k

d

For this Hamiltonian, the component projectors can be expressed back in terms of Hi and Pi: (16 )

pA:

€~

Pt ' t

(H,'- IJ)

-

3 _

f

f A

This yields explicit expressions for

: - (C E~ p" ( 17)

: -«( E~ Pi

+ f,/ of-

Hi)

I

AQQ~, as seen from the following:

(f - Hi) ((~

W - £ ) cf (£ - Hi)

A. Pi - (Hi - hi) f"'- fB

p,.

+ W - Hi)

(€ ~ ~, .,. f,/ - t:)

/ (€ 10_ €B)

>/ (f

A_ E

'1.

B) 2

::

(3_

f".,. eL(/c)

eA-f1'f.·(k)

E:J_ f A

(10_(.8

Ai (Ic,f)

)

and similarly for the other. Substituting into (13), or performing such manipulations directly there. we get

76

15([,41) oc..i. L If(k)!1 Ai (k,F:-w) Af(kJ) N k f f," (k) + E - (f - w) _.. • o ,p E=C~€~'tc·€-.

(18 )

'V

_

-

f

-

I

_

fA

_ flJ

'

This is a physically meaningful result: structurally, the yield is just like in Eq. The ratio

I ;p / ;p /2

(9), but the matrix element is now an effective one. could be termed an Elliott factor. This factor

occurs in the theory of excitonic absorption to account for the real final state electron-hole interactions

[16] . Here it is caused by

the disorder related coupling between the particles. Unfortunately, the effective matrix element has also some unphysical properties. It is strongly energy dependent, so that it is capable of transforming the Lorentzian spectral density into its Hilbert transform. Further,

leA - fB I ~

it is divergent for

0

, although we know that the

infinities shall cancel in the final result, which is Eq.

(10). This

calls for a regularization, and it can be achieved employing the identity (19)

This leads to our final result,

-

p(f,c.J)

OC -

1

N

[.

Ie.

1

f'fd.

(20)

1m!.

€A_ Ell

}

k £-c.J,.,0 J

according to which the vertex corrections in the regularized form cause three modifications of Eq.

(9): the matrix element in the

"normal" term contains an Elliott factor, the second term involving rather ReG than AOC ImG describes a redistribution of the yield due to interference effects, and the third term has the meaning of the incoherent contribution, like in Eq. to 1m

t

ra ther than to g oc ImG.

77

(10), but is now proportional

4. NUMERICAL EXAMPLE We shall illustrate the general equations on an example which is particularly easy for computation. It is a cubium with free electron final states and a cosine valence band; the parameters chosen make the bands resemble those of copper. The final states are broadened in correspondence with the jellium self-energy for r

= 2. The band strucs ture along the symmetry lines in the Brillouin zone is in Fig. 1. The

series of Figs. 2-6 corresponds to Eq.

(9), that is with no vertex

part· On top, the matrix element is constant. In Fig. 2 the crystal DOS is compared with the CFS spectra for several final state energies up to about the Hell line. Typical band structure effects are seen, resemblance between the DOS and the PES profiles is but remote. Next figure shows the same for an alloy with 15% of atomic levels shifted by

~

of the bandwidth upwards. The profiles shown are obtained in

the CPA. For comparison, also the virtual crystal spectra are included. It is seen that in the impurity part of the band, the shape of the CFS spectra is fairly stable and quite similar to the correspond~ng

part of the DOS. Over the majority part of the band, on the other

hand, the spectra behave very much like the virtual crystal, with the typical travelling band structure related details. This can be understood from Fig. 4 which shows/lm[1 and the spectral density as a function of k for various energies. wherellm[lis large, also the spreading in the BZ is large: this is the impurity part of the band. In the majority part of the band, the spectral density is typical for the complex band structure. Thus, we have here an example of the coexistence of the band structure behaviour, and of a complete relaxation of the k selection rule in different parts of the BZ. The next two pictures show the same data, but for an alloy with a much stronger disorder. The DOS is completely changed, with two almost separated sub-bands. The CFS spectra are now in their basic structure rather similar to the DOS. The spectral densities are now indeed smeared all over the BZ: this is the case when the non-direct model applies. The last figure shows the effect of the vertex corrections. In the middle part, f~= fB • This is a repetition of Fig. 3. If we now make the

fA

larger or smaller than

fJ ,

the relative intensity of

the minority sub-band goes up or down. The effects are pronounced enough to deserve a detailed study.

78

,,-----------::;1

R

M

I I

I I

r

I ,r----

/

/

/

/

f reV]

1S

5D

b)

o ......c=;;_..... X R r

Fig. 1a:

n

r

x

The Brillouin zone of the simple cubic lattice. The points of special symmetry for which the spectral densities are plotted below are marked by heavy lines.

Fig. 1b:

The band structure used in the numerical example: Initial states: a single cosine band (dots). Final states: free electron bands. The numbers indicate the degeneracy.

79

t

,

~

reV J

......

(,)

\l

...

Q...

...

ff" 1t6.75

"', =2. IJ

;2.5

2.0

38.25

Z.IJ

H.O

2.0

2'1.7S

2.0

U.S

(.S'

2f.2S

cs

1.1. 0

is

fUS

1.0

8.5

u.s

."

v

t

£,-4) -+

II) C)

I:l

f(r)

Fig. 2:

E(X)

[(1'1)

E(R)

Photoemission from an ideal crystal. The CPS spectra as a

= Ef - Ware plotted i and widths w indicated at f f each curve. Notice that the increase of the final state function of the initial state energy E

for final states with energies E

broadening has no marked effect on the sharpness of the spectra. Lower frame: the crystal DOS. The marked energies correspond to the principal critical points of the cosine band. The small rounding of the van Hove singularities is an artefact of our method of the BZ integration.

80

t

....

1$

rev]

......

~

", ct. 0

~

....

V)

u

t

E,-w -t

">

+2.5

2.0

38.25

t.O

3+.0

2.0

tf.75

2.0

U.5

1.5

u.zs

1.5

11.0

(.S'

12.1-5

1.0

8.5

O.S'

C)

I=l

E (1'.) Fig. 3:

E (lO

E(M)

HR)

Photoemission from the A alloy. Atomic level separation 1 SB8 S is 3/8 of the bandwidth. The plot is the same as in Fig. 2. In addition, by dotted lines are drawn the CFS spectra, and the DOS in the virtual crystal approximation. On the energy scale, the points E(rl, etc. still belong to the ideal crystal.

81

Fig. 4:

E(~)

<,

--.... 00-

f(l1)

.-...

....-

~

roo

_. L

.

-?-

.E"'-A'-

-

.

~

~ ~

A

\

-«:

L ....

»r-: .. , .. " /,

-

':-....

v·~

~ ~

~

r-,

,A... ./"~

~

\

/'~

~

h

r

X R

<:>

" ..----'-

~

,~

~

~

~:-.

~

V·.

./:\

\/ ,

».

A

Ur)

..fA ..

r

x

..

~

""

N

3"

M-

-

~

LL·

/:, .I..~

./.'"

M

R

The map of the spectral density of the initial states for the A15BS 5 alloy of Fi Horizontal axis: the principal directions in the BZ as in Fig. 1. Vertical axis: gies of the initial states. For a mesh of energies, the plots are of the functio Ai(k, E / (10 + Ai(k, Eil). The dotted line: the virtual crystal band. In the s i) te frame, lIm II (the damping of the initial states) is plotted vs. energy. In the it is k -

independent.

t [e v] f,aU.1-S w,-t.O CI)

l4. ~

t

U.S

2.0

38.2S

Z.O

3'1.0

2.0

flf. 'rf

2.0

U.S

t.s

1,,1.25

I.S

11.0

'.5

12.1S

1.0

8.5

0.5

II)

c

A

etr; Fig. 5:

E(X)

E(1)

HR)

Photoemission from the A alloy. Atomic level separation 5 0B5 0 is 1/2 of the bandwidth. The plot is arranged in the manner of Figs. 2, 3.

83

Fig. 6.:

E(l()

-

.~

::....-.. ..

((M)

..

-

Erx)

./". ./"'0. ..

-'"".-:

-

.:

.

.

..

-,

.. -'""~

..

.

- "",,",

~

.

..

-

.: .

..

- .. ..

~

.

.

-

<:>

~

3"

""

...

I--.

.. .

. ~

.

.

............

.s-:«:

»<:

~

...Y""..

./"':L

..v-......

Ecr)

~

.. ~

.v-~

»<:

»< v,

~

r

r

X R

x

M

M

R

The map of the spectral density of the initial states for the ASOB alloy of F SO The arrangement of the figure is like in Fig. 4.

t

fa- D.5 f

f= 0

f= O.J

f

V)

~ <..)

E,-t.I) -

f,-I,)

~

f,-4

Fig. 7: Effect of the randomness of the transition matrix on the CFS spectra for the A 15BS no vertex, repetition of Fig. 3. The lefthand fiel alloy. The middle field:

f =-

0.5

i .

f = 0,

The righthand field:

f = 0.5

i .

References [1]

Spicer W.E.: Phys. Rev. 154, 385(1967)

1,

[2]

Friedel J.: Nuovo Cim. suppl.

[3J

Velicky B., Kirkpatrick S., Ehrenreich H.: Phys. Rev. B1,

287(1958)

[4J

Stocks G.M., Williams R.W., Faulkner J.S.: Phys. Rev. B4,

[5)

Kudrnovsky J., Velicky B.: Proc. of the 9th Annual Int.

3250(1970) 4390(1971) Symposium on Electronic Structure of Metals and Alloys, Gaussig (1979), p. 166 [6J

Kudrnovsky J., Velicky B.: Proc. of the 10th Annual Int. Symposium on Electronic Structure of Metals and Alloys, Gaussig (1980), in press

[7]

Durham P.: in this volume

[8]

Kumar V., Mookerjee A.: ICTP Trieste, preprint IC/80/142(1980)

[9]

Kaplan T., Leath P.L., Diehl H.W.: Phys. Rev. B21, 4230(1980)

l,

1020(1958)

[10]

Edwards S.F.: Phil. Mag.

[11]

Velicky B.: Phys. Rev. 184, 614(1969)

[12J

Levin K., Velicky B., Ehrenreich H.: Phys. Rev. B2, 1771 (1970)

[13J

Velicky B.: in 12. Metalltagung der DDR, Elektrische und Thermische Leitfahigkeit, Dresden (1978), p. 69

[14]

Klik I.: Diploma thesis, Charles University, Praha (1979), will be published

[15]

Fukuyama H., Krakauer H., Schwartz L.: Phys. Rev. B10, 1173

[16J

Elliott R.J.: Phys. Rev. ~, 1384(1957)

(1973)

86