Angular resolved photoemission from PbS (100) for 16.85 eV and 21.22 eV excitation energy

Solid State Communications,

Vol. 23, pp.

897—900,

~977. Pergamon Press. Printed in Great Britain.

ANGULAR RESOLVED PHOTOEMISSION FROM PbS (100) FOR 16.85 eV AND 21.22 eV EXCITATION ENERGY T. Grandke, L. Ley, and M. Cardona Max—Planck—IflStitUt für FestkorperfOrschung D-7000 Stuttgart 80, Federal Republic of Germany (Received 14 July 1977 by M. Cardona)

Abstract Angular resolved photoelectron spectra are presented for the narrow—gap semiconductor PbS. Two different photon energies and two different azimuthal orientations of the (100) surface are employed. The experimental results provide a further test for the validity of the one-dimensional density of states approach in the case of the lead chalcogenides. The experimental binding energies of 10 three-dimensiOnal critical points are compared with the prediction of two independent band structure calculations.

Angular resolved photoelectron spectra (ARPES) of crystals with non—negligible energy dispersion in the direction normal to the surface are interpreted in terms of two contradictory models: (i) the direct transition model which assumes the conservation of the three dimensional fl—vector of the electron during the 1 and (ii) the one-dimensional photoexcitation density of states model (ODDS) which only assumes the conservation of the c—vector component parallel to the surface,~~,andcompletely neglects the conservation of k~2. Perhaps the most detailed application of the second model has been undertaken by Grandke et a13, who were able to explain their ARPES measurements of the narrow—gap semiconductor PbS on the basis of the ODDS approach. In this communication we present the extension of these measurements to two photon energies and to two different azimuthal orientations of the crystal. Furthermore the spinorbit interaction is now included in the calculation of the electronic band structure of PbS. These improvements enable us to accomplish a further test of the validity of the ODDS model, The experimental data were obtained in a Vacuum Generators photoemission spectrometer model ADES 400 which has been described in detail in Ref. 4. Clean and unreconstructed (100) sur— faces of natural p—type PbS were obtained by cleav~ig in vacuum, the base pressure being 8x10 Torr. The quality and the orientation of the cleavage plane were controlled by means of its LEED—pattern. The surface proved to be rather inert, as the diffraction pattern preserved its sharpness for several days. Two different orientations were used for the ARPES measurements reported here. Designating the surface normal as the (100) direction, the electron acceptance cone with an opening angle of was chosen to lie either in the (010) plane (orientation I), or in the (011) plane (orienta— tion II) . Consequently, the projection ~Cff of the electron momentum onto the surface was parallel to the (001) direction or to the (011) direction 897

respectively. The two crystal orientations differ by an azimuthal rotation of 450 about the surface normal. A list of various angles used in the ARPES measurements is given in Table I. The photoemission spectra were taken using Nel (hu = 16.85 eV) or HeI (ha = 21.22 eV) photons delivered by a differentially pumped resonance lamp. The energy resolution of the hemispherical electron analyzer was 0.3 eV. All spectra were recorded with a multichannel analyzer operating in the multichannel scaling mode. Accurate positions of peaks and shoulders in a spectrum were determined by means of its second derivative, obtained with a numerical differen— tiation routine. All experimental binding energies given in this paper are referred to the Fermi energy EF. Since in p-Pbs E 15 expected t~. lie w~thir. C. ~eV from the valence~band edge, the binding energies thus obtained should be practically identical with the binding energies referenced to the top of the valence band. The position of EF was de— termined by measuring the spectrum of the steel sample holder, previously cleaned by 1/2 h of Ar+_iOn bombardment. In view of the low cross section of s-derived valence bands for uv photoexcitation (See Ref. 3 for a set of typical spectra), we limit our discussion to the upper three valence bands, which are mainly originating from S(3p) levels. Before comparing the results of the ARPES measurements with the prediction of an EPM band structure calculation on the basis of the ODDS model, we briefly recall the basic assumptions of this model3: (a) On account of translational symmetry parallel to the unreconstructed single crystal surface, the electron momentum component parallel to the surface is conserved to a reciprocal lattice vector during the photoemission process.5 Consequently, the wavevector component ~ of the initial valence state is related to the electron energy S

898

ANGULAR RESOLVED PROTOEMISSION FRON Pbs (100)

Vol. 23, No.

12

TABLE I Angles used in the ARPES measurements, referring to the surface normal. A typical momentum resolution ~ is calculated for electrons photoexcited from a valence state with EB = 3eV and~ 1~=~.

Nel

I Photon incidence angle, 8w Range of electron acceptance direction, 8 Steps in

~,

(001)

Hel

(011)

(001)

58°

45°

_400. ..440°

-26°...+44°

L8

58°

45°

—40.5°...425.5° ~27.O°. ..428.5°

2°

Angular resolution

1.5° 3

30

Momentum re-r solution, ~k1)

0.011

and the exit angle ~ through I~iiv’ = = sin 0. (b) Due to the small escape depth of the photoelectrons, the final states inside the solid are no eigenstates of k and conservation of this quantity nay be neglected. Therefore, an angular resolved photoemission spectrum should resemble the density of states of the one-dimensional sections of the valence bands defined by k11v = = const. In particular,

(011)

0

0.014

the peaks in a spectrum are related to the critical points of the one dimensional density of valence states, calculated along lines of fixed kHv. The positions of lines in reciprocal space defined ofbythe ~ = = relative const depend on acceptance the orientation sample to the cone of the electron analyzer. In Fig. I we have sketched the irreducible parts of these lines, within the extended zone scheme of PbS, which correspond to the two different azimuthal orientations used in our experiments. The one—dimensional density of states along these lines was calculated according to ~1

(k~ D1 (E,k1~( ~

X

K

r

A

W

L

x

K

r

A 100)

x’_ (010) Fig. 1

(1) iv

jEfl=E

with the suimnation extending Over the three p-derived valence bands. The energy bands E n V. were calculated utilizing the Empirical Pseucopotential Method and the pseudopotential coeffi— cients given by Kohn et al. 6 The spin—orbit interaction was included directly in the pseudo— hamiltonian as originally suggested by Weisz?

z

r

[

,

Extended zone scheme of PbS. Our mea- , surements were confined to wavevectors k lying (a) in the F —X--X-X—plane (001)), and (b) in the F-X-K-C—Kplane (k, U (Oil)). One-dimensional den~ities of states were calculated along k~iv = const., as indicated by the vertl— cal lines,

The energy dependence of the peaks in D~ (E,k ~ is plotted as a function of ~Jv in Fig. 2a. We note that for k11~ ((011), the rectangles F—F (0 1/2 1/2)— 1(1 1/2 1/2)—X and x—Z )O 1/2 1/2)— 5 (1 1/2 1./2)—r’ are equivalent except for a change of role of the lines of k~v=O and ~ = i.~ (cf. Fig. 1). Therefore the ~orizontal axis of Fig 2a extends only from ~ = (0 1/2 i7~) over = (000) to ~~=(OO1) Full lines correspond to critical points boated at kj.v = 0 (i.e. along the lines Z-F—t~—X) dashed lines to critical points at kjv = 2~ (i.e., along the lines Z—K—X—Z—W—Z—X) , and~ct— ted lines to critical points at some interns— diate value of ~ At k~= (0 1/2 1/2 , the dotted lines merce into the 0—point, and at

900

ANGULAR RESOLVED PHOTOEMISSIOB FROM Pbs (100)

remnants of k~—conservation. The differences

Vol. 23, No. 12

out any adjustments. The only important disagreement between the EPH band structure and our experimental results occurs at k 11(0 1/2 l/2).Wbjle the calculation pre~ictsa forbidden gap along this line of fixed k,iv extending from 1.0 eV to 2.9 eV, the experimental width of this gap is at most 1.3 eV. This result does not depend on the validity of the ODDS ~Kdel: the gap must also be present if direct transitions take place. Two Either the bulk band structure of this PbS result. is consipossible consequences arise from derably different from that at the surface which is accessible by UPS, or the empirically adjus-

between Nd and Mel peak energies while small lie beyond the experimental error and may be due to such residual final state effects. An investigation of such differences with a continuously tunable source (synchrotron radiation) should elucidate this problem. Finally we compare the experimental energies of the 10 three—dimensional critical points 6 and an OPW8 calculation of the the results band strucincluded in Figs. 2 and 3 with of an ture of PbS, of. Table II. The experimental EPM values presented in this paper are clearly in favour of the EPM calculation. This result is not surprising since this calculation has been fitted to the optical properties of PbS, while

ted band structure is only correct to within ‘~‘

the other is a first—principle calculation with—

0.4 cv. Thanks are due to W. Neu and G. Krutina

for their technical assistance.

TABLE II Comparison of theenergies of the three—dimensional critical points in the calculated E vs. k 11 curves of Fig. 2.

experimental

Symmetry designation

Calculated

E(eV), ha

r8

=

Experimental

16.85eV

ha

=

E(eV),

21.22ev

1 2.75

—

F6

(?)

1

1 4.30

2.87

2.57

3.09

2.83

3.95

3.49

4.09

3.74 5.24

4.22

J

W6

OPW, Ref.8

2.95

J

—

EPM, Ref.6

and

5.45

5.65

5.44

1.68

1.68

1.66

2.73

2.51

—

+

Lj~5 +

L6

J

j

2.90

2.32

2.99

2.36

66

(0,0,0.38)

1.68

1.57

1.76

—

55

(0,0.35, 0.35)

1.03

1.00

0.93

—

REFERENCES 1.

6.

See e.g. E. DIETZ, H. BECKER, and U. GERHARDT, Phys. Rev. Lett. 36, 1397 (1976); G.J. LAPEYRE, R.J. SMITH, and 3. ANDERSON, 3. Vac. Sci. Technol. 14, 384 (1977). See e.g. B. FEUERBACHER and N.E. CHRISTENSEN, Phys. Rev. B 10, 2373 (1974); P. HEIMANN, H. NEDER~YER, and H.F. ROLOFF, Phys. Rev. Lett. 37, 775 (1976). T. GRANDKE, L. LEY, and N. CARDONA, Phys. Rev. Lett. 38, 1033 (1977). T. GRANDKE and L. LEY, Phys. Rev. B 15, (1977) , in press. E. 0. KANE, Phys. Rev. Lett. 12, 97 (1964). S.E. KOHN, P.Y.YU, Y. PETROFF, Y.R. SHEN, Y. TSANG, and M.L. COHEN, Phys. Rev. B 8, 1477

7.

(1973). G. WEISZ, Phys. Rev. 149, 504 (1966).

8.

F. HERMAN, R.L. KORTUM,

2. 3. 4. 5.

I.B. ORTENBURGER,

—

and 3.D.

van DYKE,

3. de Physique C4, 62 (1968).

ANGULAR RESOLVED PI1OTOEMISSION FROM Pbs (100)

Vol. 23, No. 12

o

PbS (~0)

k~= (001) point.

Theory

899

one of them is connected to the W-

In Figs. 2b and 2c-we show the experimenreduced tal peak wavevector positions E component versus the ~, corresponding as obtained with photon energies of 16.85 eV and 21.22 eV. The

L. g

______

circles represent weak peaks or shoulders which crosses stand for well—defined peaks while open have been taken from the 2nd derivative of an original spectrum. We first note that it is possible to con— nect most of the discrete data points by conti-

__________

6

________

____________

II1~) L

X

X W

nuous lines similar to those obtained from the band Structure calculation (Fig. 2a). Thereby we are able to assign virtually all peaks observed in the ARPES spectra to critical points in the

ZlO~)

r

X

one-dimensional density of states. The most im-

k,~u(O1b k.~:O

k

1~I(OO1)

portant discrepancies between theory and experi—

bi

o

__________________

ment occur around ~ = (001) for both photon energies. The strong energy dependence of the observed structures on the photon energy (E =1.75 eV for ?lW 16.85 eV, E8 2.50 eV for ?w = 2~.22 eV)

~

suggests that their occurrence in tH~eARPES spectra is due to final state effects neglected in the ODDS model. Another possible explanation for the string of shoulders at 1.75 eV in the Nelexcited spectra could be a violation of the ~ conservation, because intense peaks are observed

PbS(W)

~‘~“~“

\

__________

C

th \,~

•d~m.

at the same energy, but at a different -

0.5

iD

05

k’ioth

•

~I(OO11

k)1 ~ (0 0 1/3) . All other disagreements between theory and experiment are less definite, since the related structures in the ARPES spectra seem

to appear in an irregular manner. In Fig. 3 we have overlayed the continuous

____________

PbS (W)

0

energy vs. momentum curves derived from Figs. 2b

__________________

?~ej:21.22eV

PbS (IOU)

S.

hw~2122eV —

C

~

~

~—

‘

ol

~2 ~

l68SeV

~

S....

6

--~Sb~

_______

I

.~

___________

k’~ ii (011)

I~jI1k)

~II°~ (001)

~C

I61 _________________

Fig. 2

__________________________

(a) Dependence of the energies of critical points in the one—dimensional density of states on ~Iv~ The solid lines refer to critical points at 0 (lines F 5(1 (Q i,~2 1/2 —:-—x~, the 2n (lines 1 F7~-K-X-W—x) dashed lines to critical points at LV 5 and the dotted lines to critical points at an intermediate value of k~v. = =

(b) Peak positions versus reduced electron component ~ obtained from the Ne’-excited phctoemissior. spectra. The experimental points are connected by lines to give energy vs. momen— tum curves similar in shape to those of Fig. (a). (c) Same as

)b)

for the He’—excited photo-

emission spectra.

~:(OOO) Fig. 3

~~(OO1)

Superposition of the energy vs. momentum curves obtained from the photoemission1 (solid lines) radiation. spectra with Ne’ (dashed lines) and He

and 2c. Full and dashed lines correspond to Nel— and Hel—excited ARPES spectra, respectively. We emphasize that the deviations between these two are generally small (< 0.2 eV)as far as only peak positions are concerned. This lends substantial support to the assumption that the conservation of the electron momentum component normal to the surface, k~, may be neglected to a first approximation. The ODDS model, however, is not capable of predicting whether a specific critical point is observable as a peak in an ARPES spectrum or not. Even the exact energy of a peak may be influenced by the